\\rho _{\\mathbb {E}}$ for $\\alpha <1$ ) and which correctly gives $\\lim _{\\alpha \\rightarrow 1}\\rho _{\\alpha =1}=\\rho _{\\mathbb {E}}$ for $\\alpha =1$ , i.e.","when the fractal ring becomes non-fractal.","Thus, a fractal ring has a higher effective mass density than the homogeneous Euclidean ring of the same overall dimensions."],["Moments of inertia","The moment of inertia of the Euclidean ring ($r\\in [0,R]$ and thickness $h$ in $z$ -direction), assuming $\\rho _{\\mathbb {E}}=\\mathrm {const}$ , is $I_{\\mathbb {E}}=\\frac{1}{2}\\pi h\\rho _{\\mathbb {E}}\\left(R^{4}-R_{D}^{4}\\right) =\\frac{1}{2}M\\left( R^{2}+R_{D}^{2}\\right) ,$ while the moment of inertia of a fractal ring is $\\begin{array}{c}I_{\\alpha }=h\\int _{B}\\rho _{\\alpha }r^{2}dB_{E}=h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }r^{2}\\rho _{\\alpha }hdS_{\\alpha } \\\\\\displaystyle 2\\pi h\\rho _{\\alpha }\\int _{R_{D}}^{R}r^{2}\\alpha \\left( \\frac{r}{R}\\right) ^{\\alpha -1}rdr=2\\pi h\\rho _{\\alpha }\\frac{\\alpha }{\\alpha +3}\\left( R^{4}-\\frac{R_{D}^{\\alpha +3}}{R^{\\alpha -1}}\\right) .\\end{array}$ Now, take the limit $\\alpha \\rightarrow 1$ : $\\lim _{\\alpha \\rightarrow 1}I_{\\alpha }=\\frac{1}{2}\\pi h\\rho _{\\mathbb {E}}\\left( R^{4}-R_{D}^{4}\\right) =I_{\\mathbb {E}},$ as expected.","Note that $I_{\\alpha }$ is an increasing function of $\\alpha $ (i.e.","we must have $I_{\\alpha }0$ .","Every circle determines the vector bundle $\\xi _r=(E_{Ur},\\pi _r,C_r)$ .","Consider the vector bundle $\\eta =(E,\\pi ,R) $ , where $E$ is the union of all $E_{Ur}$ , and the restriction of the projection map $\\pi $ to $E_{Ur}$ is equal to $\\pi _r$ .","The random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ is a random section of the above bundle, that is, $\\overline{\\mathbf {\\omega }}(r,\\varphi )\\in \\pi ^{-1}(r,\\varphi )=\\mathbb {R}^3$ .","In what follow we assume that the random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ is second-order, i.e., $\\mathsf {E}[\\Vert \\overline{\\mathbf {\\omega }}(r,\\varphi )\\Vert ^2]<\\infty $ for all $(r,\\varphi )\\in R$ .","There are at least three different (but most probably equivalent) approaches to the construction of random sections of vector bundles, the first by [4], the second by [7], [8], and the third by [3].","In what follows, we will use the second named approach.","It is based on the following fact: the vector bundle $\\eta =(E,\\pi ,R)$ is homogeneous or equivariant.","In other words, the action of the group $O(2)$ on the bundle base $R$ induces the action of $O(2)$ on the total space $E$ by $(g_0,\\mathbf {x})\\mapsto (gg_0,\\mathbf {x})$ .","This action identifies the spaces $\\pi ^{-1}(r_0,\\varphi )$ for all $\\varphi \\in [0,2\\pi )$ , while the action of the multiplicative group $\\mathbb {R}^+$ on R, $\\lambda (r,\\varphi )=(\\lambda r,\\varphi )$ , $\\lambda >0$ , identifies the spaces $\\pi ^{-1}(r,\\varphi _0)$ for all $r>0$ .","We suppose that the random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ is mean-square continuous, i.e., $\\lim _{\\Vert \\mathbf {x}-\\mathbf {x}_0\\Vert \\rightarrow 0}\\mathsf {E}[\\Vert \\overline{\\mathbf {\\omega }}(\\mathbf {x})-\\overline{\\mathbf {\\omega }}(\\mathbf {x}_0))\\Vert ^2]=0$ for all $\\mathbf {x}_0\\in R$ .","Let $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle =\\mathsf {E}[\\overline{\\mathbf {\\omega }}(\\mathbf {x})]$ be the one-point correlation vector of the random field $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ .","On the one hand, under rotation and/or reflection $g\\in O(2)$ the point $\\mathbf {x}$ becomes the point $g\\mathbf {x}$ .","Evidently, the axial vector $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ transforms according to the representation (REF ) and becomes $U(g)\\overline{\\mathbf {\\omega }}(g\\mathbf {x})$ .","The one-point correlation vector of the so transformed random field remains the same, i.e., $\\langle \\overline{\\mathbf {\\omega }}(g\\mathbf {x})\\rangle =U(g)\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle .$ On the other hand, the one-point correlation vector of the random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ should be independent upon an arbitrary choice of the $x$ - and $y$ -axes of the Cartesian coordinate systems, i.e., it should not depend on $\\varphi $ .","Then we have $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle =U(g)\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle $ for all $g\\in O(2)$ , i.e., $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle $ belongs to a subspace of $\\mathbb {R}^{3}$ where a trivial component of $U$ acts.","Then we obtain $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle =\\mathbf {0}$ , because $U$ does not contain trivial components.","Similarly, let $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x}),\\overline{\\mathbf {\\omega }}(\\mathbf {y})\\rangle =\\mathsf {E}[\\overline{\\mathbf {\\omega }}(\\mathbf {x}) \\otimes \\overline{\\mathbf {\\omega }}(\\mathbf {y})]$ be the two-point correlation tensor of the random field $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ .","Under the action of $O(2)$ we should have $\\langle \\overline{\\mathbf {\\omega }}(g\\mathbf {x}),\\overline{\\mathbf {\\omega }}(g\\mathbf {y})\\rangle =(U\\otimes U)(g)\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x}),\\overline{\\mathbf {\\omega }}(\\mathbf {y})\\rangle .$ In other words, the random field $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ is wide-sense isotropic with respect to the group $O(2)$ and its representation $U$ .","Consider the restriction of the field $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ to a circle $C_r$ , $r>0$ .","The spectral expansion of the field $\\lbrace \\,\\overline{\\mathbf {\\omega }}(r,\\varphi )\\colon \\varphi \\in C_r\\,\\rbrace $ can be calculated using [7] or [8].","The representation $U$ is the direct sum of the two irreducible representations $\\lambda _-(g)=\\det g$ and $\\lambda _1(g)=g$ .","The vector bundle $\\eta $ is the direct sum of the vector bundles $\\eta _-$ and $\\eta _1$ , where the bundle $\\eta _-$ (resp.","$\\eta _1$ ) is generated by the representation $\\lambda _-$ (resp.","$\\lambda _1$ ).","Let $\\mu _0$ be the trivial representation of the group $SO(2)$ , and let $\\mu _k$ be the representation $\\mu _k(\\varphi )=\\begin{pmatrix}\\cos (k\\varphi ) & \\sin (k\\varphi ) \\\\-\\sin (k\\varphi ) & \\cos (k\\varphi )\\end{pmatrix}.$ The representations $\\lambda _-\\otimes \\mu _k$ , $k\\ge 0$ are all irreducible orthogonal representations of the group $G=O(2)\\times SO(2)$ that contain $\\lambda _-$ after restriction to $O(2)$ .","The representations $\\lambda _1\\otimes \\mu _k$ , $k\\ge 0$ are all irreducible orthogonal representations of the group $G=O(2)\\times SO(2)$ that contain $\\lambda _1$ after restriction to $O(2)$ .","The matrix entries of $\\mu _0$ and of the second column of $\\mu _k$ form an orthogonal basis in the Hilbert space $L^2(SO(2),\\mathrm {d}\\varphi )$ .","Their multiples $e_k(\\varphi )={\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\sqrt{2\\pi }}, & \\mbox{if } k=0, \\\\\\frac{1}{\\sqrt{\\pi }}\\cos (k\\varphi ), & \\mbox{if } k\\le -1 \\\\\\frac{1}{\\sqrt{\\pi }}\\sin (k\\varphi ), & \\mbox{if } k\\ge 1\\end{array}\\right.","}$ form an orthonormal basis of the above space.","Then we have $ \\overline{\\mathbf {\\omega }}(r,\\varphi )=\\sum _{k=-\\infty }^{\\infty }e_k(\\varphi )\\mathbf {Z}^k(r),$ where $\\lbrace \\,\\mathbf {Z}^k(r)\\colon k\\in \\mathbb {Z}\\,\\rbrace $ is a sequence of centred stochastic processes with $\\begin{aligned}\\mathsf {E}[\\mathbf {Z}^k(r)\\otimes \\mathbf {Z}^l(r)]&=\\delta _{kl}B^{(k)}(r),\\\\\\sum _{k\\in \\mathbb {Z}}\\operatorname{tr}(B^{(k)}(r))&<\\infty .","\\end{aligned}$ It follows that $\\mathbf {Z}^k(r)=\\int _{0}^{2\\pi }\\overline{\\mathbf {\\omega }}(r,\\varphi )e_k(\\varphi )\\,\\mathrm {d}\\varphi .$ Then we have $ \\mathsf {E}[\\mathbf {Z}^k(r)\\otimes \\mathbf {Z}^l(s)]=\\int _{0}^{2\\pi }\\int _{0}^{2\\pi } \\mathsf {E}[\\overline{\\mathbf {\\omega }}(r,\\varphi _1)\\otimes \\overline{\\mathbf {\\omega }}(s,\\varphi _2)] e_k(\\varphi _1)\\,\\mathrm {d}\\varphi _1e_l(\\varphi _2)\\,\\mathrm {d}\\varphi _2.$ The field is isotropic and mean-square continuous, therefore $\\mathsf {E}[\\overline{\\mathbf {\\omega }}(r,\\varphi _1)\\otimes \\overline{\\mathbf {\\omega }}(s,\\varphi _2)] =B(r,s,\\cos (\\varphi _1-\\varphi _2))$ is a continuous function.","Note that $e_k(\\varphi )$ are spherical harmonics of degree $|k|$ .","Denote by $\\mathbf {x\\cdot y}$ the standard inner product in the space $\\mathbb {R}^d$ , and by $\\mathrm {d}\\omega (\\mathbf {y})$ the Lebesgue measure on the unit sphere $S^{d-1}=\\lbrace \\,\\mathbf {x}\\in \\mathbb {R}^d\\colon \\Vert \\mathbf {x}\\Vert =1\\,\\rbrace $ .","Then $\\int _{S^{d-1}}\\,\\mathrm {d}\\omega (\\mathbf {x})=\\omega _d=\\frac{2\\pi ^{d/2}}{\\Gamma (d/2)},$ where $\\Gamma $ is the Gamma function.","Now we use the Funk–Hecke theorem, see [1].","For any continuous function $f$ on the interval $[-1,1]$ and for any spherical harmonic $S_k(\\mathbf {y})$ of degree $k$ we have $\\int _{S^{d-1}}f(\\mathbf {x\\cdot y})S_k(\\mathbf {x})\\,\\mathrm {d}\\omega (\\mathbf {x})=\\lambda _kS_k(\\mathbf {y}),$ where $\\lambda _k=\\omega _{d-1}\\int _{-1}^{1}f(u)\\frac{C^{(n-2)/2}_k(u)}{C^{(n-2)/2}_k(1)} (1-u^2)^{(n-3)/2}\\,\\mathrm {d}u,$ $d\\ge 3$ , and $C^{(n-2)/2}_k(u)$ are Gegenbauer polynomials.","To see how this theorem looks like when $d=2$ , we perform a limit transition as $n\\downarrow 2$ .","By [1], $\\lim _{\\lambda \\rightarrow 0}\\frac{C^{\\lambda }_k(u)}{C^{\\lambda }_k(1)}=T_k(u),$ where $T_k(u)$ are Chebyshev polynomials of the first kind.","We have $\\omega _1=2$ , $\\mathbf {x\\cdot y}$ becomes $\\cos (\\varphi _1-\\varphi _2)$ , and $\\mathrm {d}\\omega (\\mathbf {x})$ becomes $\\mathrm {d}\\varphi _1$ .","We obtain $\\int _{0}^{2\\pi }B(r,s,\\cos (\\varphi _1-\\varphi _2))e_k(\\varphi _1)\\,\\mathrm {d}\\varphi _1 =B^{(k)}(r,s)e_k(\\varphi _2),$ where $B^{(k)}(r,s)=2\\int _{-1}^{1}B(r,s,u)T_{|k|}(u)(1-u^2)^{-1/2}\\,\\mathrm {d}u,$ Equation (REF ) becomes $\\mathsf {E}[\\mathbf {Z}^k(r)\\otimes \\mathbf {Z}^l(s)]=\\int _{0}^{2\\pi }B^{(k)}(r,s) e_k(\\varphi _2)e_l(\\varphi _2)\\,\\mathrm {d}\\varphi _2=\\delta _{kl}B^{(k)}(r,s).$ In particular, if $k\\ne l$ , then the processes $\\mathbf {Z}^k(r)$ and $\\mathbf {Z}^l(r)$ are uncorrelated.","Calculate the two-point correlation tensor of the random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ .","We have $ \\begin{aligned}\\mathsf {E}[\\overline{\\mathbf {\\omega }}(r,\\varphi _1)\\otimes \\overline{\\mathbf {\\omega }}(s,\\varphi _2)]&=\\sum _{k=-\\infty }^{\\infty }e_k(\\varphi _1)e_k(\\varphi _2)B^{(k)}(r,s)\\\\&=\\frac{1}{2\\pi }B^{(0)}(r,s)+\\frac{1}{\\pi }\\sum _{k=1}^{\\infty }\\cos (k(\\varphi _1-\\varphi _2))B^{(k)}(r,s).","\\end{aligned}$ Now we add a time coordinate, $t$ , to our considerations.","A particle located at $(r,\\varphi )$ at time moment $t$ , was located at $(r,\\varphi -\\sqrt{GM}t/r^{3/2})$ at time moment 0.","It follows that $\\overline{\\mathbf {\\omega }}(t,r,\\varphi )=\\overline{\\mathbf {\\omega }}\\left(r,\\varphi -\\frac{\\sqrt{GM}t}{r^{3/2}}\\right),$ where $G$ is Newton's gravitational constant and $M$ is the mass of Saturn.","Equation (REF ) gives $ \\overline{\\mathbf {\\omega }}(t,r,\\varphi )=\\sum _{k=-\\infty }^{\\infty }e_k\\left(\\varphi -\\frac{\\sqrt{GM}t}{r^{3/2}}\\right)\\mathbf {Z}^k(r),$ while Equation (REF ) gives $\\begin{aligned}\\mathsf {E}[\\overline{\\mathbf {\\omega }}(t_1,r,\\varphi _1)\\otimes \\overline{\\mathbf {\\omega }}(t_2,s,\\varphi _2)] &=\\frac{1}{2\\pi }B^{(0)}(r,s)\\\\&\\quad +\\frac{1}{\\pi }\\sum _{k=1}^{\\infty }\\cos \\left(k\\left(\\varphi _1-\\varphi _2-\\frac{\\sqrt{GM}(t_1-t_2)}{r^{3/2}}\\right)\\right)B^{(k)}(r,s).","\\end{aligned}$ Conversely, let $\\lbrace \\,B^{(k)}(r,s)\\colon k\\ge 0\\,\\rbrace $ be a sequence of continuous positive-definite matrix-valued functions with $ \\sum _{k=0}^{\\infty }\\operatorname{tr}(B^{(k)}(r,r))<\\infty ,$ and let $\\lbrace \\,\\mathbf {Z}_k(r)\\colon k\\in \\mathbb {Z}\\,\\rbrace $ be a sequence of uncorrelated centred stochastic processes with $\\mathsf {E}[\\mathbf {Z}^k(r)\\otimes \\mathbf {Z}^l(s)]=\\delta _{kl}B^{(|k|)}(r,s).$ The random field (REF ) may describe rotating particles inside Saturn's rings, if all the functions $B^{(k)}(r,s)$ are equal to 0 outside the rectangle $[R_0,R_1]^2$ , where $R_0$ (resp.","$R_1$ ) is the inner (resp.","outer) radius of Saturn's rings.","To make our model more realistic, we assume that all the functions $B^{(k)}(r,s)$ are equal to 0 outside the Cartesian square $F^{2}$ , where $F $ is a fat fractal subset of the interval $[R_{0},R_{1}]$ , see [12].","[9] calls these sets dusts of positive measure.","Such a set has a positive Lebesgue measure, its Hausdorff dimension is equal to 1, but the Hausdorff dimension of its boundary is not an integer number.","A classical example of a fat fractal is a fat Cantor set.","In contrast to the ordinary Cantor set, where we delete the middle one-third of each interval at each step, this time we delete the middle $3^{-n}$ th part of each interval at the $n$ th step.","To construct an example, consider an arbitrary sequence of continuous positive-definite matrix-valued functions $\\lbrace \\,B^{(k)}(r,s)\\colon k\\ge 0\\,\\rbrace $ satisfying (REF ) of the following form: $B^{(k)}(r,s)=\\sum _{i\\in I_k}\\mathbf {f}_{ik}(r)\\mathbf {f}^{\\top }_{ik}(s),$ where $\\mathbf {f}_{ik}(r)\\colon [R_0,R_1]\\rightarrow \\mathbb {R}^3$ are continuous functions, satisfying the following condition: for each $r\\in [R_0,R_1]$ the set $I_{kr}=\\lbrace \\,i\\in I_k\\colon f_i(r)\\ne 0\\,\\rbrace $ is as most countable and the series $\\sum _{i\\in I_{kr}}\\Vert \\mathbf {f}_i(r)\\Vert ^2$ converges.","The so defined function is obviously positive-definite.","Put $\\tilde{B}^{(k)}(r,s)=\\sum _{i\\in I_k}\\tilde{\\mathbf {f}}_{ik}(r)\\tilde{\\mathbf {f}}^{\\top }_{ik}(s),\\qquad r,s\\in F.$ The functions $\\tilde{B}^{(k)}(r,s)$ are the restrictions of positive-definite functions $B^{(k)}(r,s)$ to $F^2$ and are positive-definite themselves.","Consider the centred stochastic process $\\lbrace \\,\\tilde{\\mathbf {Z}}^k(r)\\colon r\\in F\\,\\rbrace $ with $\\mathsf {E}[\\tilde{\\mathbf {Z}}^k(r)\\otimes \\tilde{\\mathbf {Z}}^l(s)]=\\delta _{kl}\\tilde{B}^{(|k|)}(r,s),\\qquad r,s\\in F.$ Condition (REF ) guarantees the mean-square convergence of the series $\\overline{\\mathbf {\\omega }}(t,r,\\varphi )=\\sum _{k=-\\infty }^{\\infty }e_k\\left(\\varphi -\\frac{\\sqrt{GM}t}{r^{3/2}}\\right)\\tilde{\\mathbf {Z}}^k(r)$ for all $t\\ge 0$ , $r\\in F$ , and $\\varphi \\in [0,2\\pi ]$ ."],["Closure","This paper reports an investigation of the fractal character of Saturnian rings.","First, working with the calculus in a non-integer dimensional space, by energy arguments, we infer that the fractally structured ring is more likely than a non-fractal one.","Next, we develop a kinematics model in which angular velocities of particles form a random field."]],"string":"[\n [\n \"Saturn rings: fractal structure and random field model\"\n ],\n [\n \"Abstract This study is motivated by the observation, based on photographs from the Cassini mission, that Saturn's rings have a fractal structure in radial direction.\",\n \"Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of that fractal structure is possible?\",\n \"(2) What kinematics model of such fractal rings can be formulated?\",\n \"Both challenges are based on taking Saturn's rings' spatial structure as being statistically stationarity in time and statistically isotropic in space, but statistically non-stationary in space.\",\n \"An answer to the first challenge is given through the calculus in non-integer dimensional spaces and basic mechanics arguments (Tarasov (2006) \\\\textit{Celest.\",\n \"Mech.\",\n \"Dyn.\",\n \"Astron.}\",\n \"\\\\textbf{94}).\",\n \"The second issue is approached in Section~3 by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle.\",\n \"Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square $F^{2}$ of the radial cross-section $F$ of the rings, where $F$ is a fat fractal.\"\n ],\n [\n \"Introduction\",\n \"A recent study of the photographs of Saturn's rings taken during the Cassini mission has demonstrated their fractal structure [6].\",\n \"This leads us to ask these questions: Q1: What mechanics argument in support of that fractal structure is possible?\",\n \"Q2: What kinematics model of such fractal rings can be formulated?\",\n \"These issues are approached from the standpoint of Saturn's rings' spatial structure having (i) statistical stationarity in time and (ii) statistical isotropy in space, but (iii) statistical non-stationarity in space.\",\n \"The reason for (i) is an extremely slow decay of rings relative to the time scale of orbiting around Saturn.\",\n \"The reason for (ii) is the obviously circular, albeit disordered and fractal, pattern of rings in the radial coordinate.\",\n \"The reason for (iii) is the lack of invariance with respect to arbitrary shifts in Cartesian space which, on the contrary and for example, holds true for a basic model of turbulent velocity fields.\",\n \"Hence, the model we develop is one of rotational fields of all the particles, each travelling on its circular orbit whose radius is dictated by basic orbital mechanics.\",\n \"The Q1 issue is approached in Section 2 from the standpoint of calculus in non-integer dimensional space, based on an approach going back to [10], [11].\",\n \"We compare total energies of two rings — one of non-fractal and another of fractal structure, both carrying the same mass — and infer that the fractal ring is more likely.\",\n \"We also compare their angular momenta.\",\n \"The Q2 issue is approached in Section 3 in the following way.\",\n \"Assume that the angular velocity vector of a rotating particle is a single realisation of a random field.\",\n \"Mathematically, the above field is a random section of a special vector bundle.\",\n \"Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions $\\\\lbrace \\\\,B_{k}(r,s)\\\\colon k\\\\ge 0\\\\,\\\\rbrace $ with $\\\\sum _{k=0}^{\\\\infty }\\\\operatorname{tr}(B_{k}(r,r))<\\\\infty ,$ where the real-valued parameters $r$ and $s$ run over the radial cross-section $F$ of Saturn's rings.\",\n \"To reflect the observed fractal nature of Saturn's rings, [2] and independently [9] supposed that the set $F$ is a fat fractal subset of the set $\\\\mathbb {R}$ of real numbers.\",\n \"The set $F$ itself is not a fractal, because its Hausdorff dimension is equal to 1.\",\n \"However, the topological boundary $\\\\partial F$ of the set $F$ , that is, the set of points $x_{0}$ such that an arbitrarily small interval $(x_{0}-\\\\varepsilon ,x_{0}+\\\\varepsilon )$ intersects with both $F$ and its complement, $\\\\mathbb {R}\\\\setminus F$ , is a fractal.\",\n \"The Hausdorff dimension of $\\\\partial F$ is not an integer number.\"\n ],\n [\n \"Basic considerations\",\n \"We begin with the standard gravitational parameter, $\\\\mu =GM_{\\\\mathrm {Saturn}}$ ; its value for Saturn ($\\\\mu =37,931,187$ $km^{3}/s^{2}$ ) is known but will not be needed in the derivations that follow.\",\n \"For any particle of mass $m$ located within the ring, we take $m\\\\ll M_{\\\\mathrm {Saturn}}$ with dimensions also much smaller than the distance to the center of Saturn.\",\n \"Each particle is regarded as a rigid body, with its orbit about the spherically symmetric Saturn being circular.\",\n \"We are using the cylindrical coordinate system $\\\\left( r,\\\\theta ,z\\\\right) $ , such that the $z$ -axis is aligned with the normal to the plane of rings, Fig.\",\n \"REF .\",\n \"The particle's orbital frame of reference with the origin $O$ at its center of mass is made of three axes: $a_{1}$ in the radial direction, $a_{2}$ tangent to the orbit in the direction of motion, and $a_{3}$ normal to the orbit plane.\",\n \"All the particles orbit around Saturn in the same plane.\",\n \"The attitude of any given particle is described by the vector of body axes $\\\\left\\\\lbrace \\\\mathbf {x}\\\\right\\\\rbrace ^{T}=\\\\lbrace x_{1},x_{2},x_{3}\\\\rbrace ^{T}$ , which are related to the vector $\\\\left\\\\lbrace \\\\mathbf {a}\\\\right\\\\rbrace $ in the orbital frame of reference of the particle by $\\\\left\\\\lbrace \\\\mathbf {x}\\\\right\\\\rbrace =\\\\left[ \\\\mathbf {l}\\\\right] \\\\left\\\\lbrace \\\\mathbf {a}\\\\right\\\\rbrace .$ Here $\\\\left[ \\\\mathbf {l}\\\\right] $ is the matrix of direction cosines $l_{i}$ , $i=1,2,3$ .\",\n \"Figure: The planar ring of particles, adapted from , showing the Saturnian (Cartesian and cylindrical)coordinate systems as well as the orbital frame of reference (a 1 ,a 2 ,a 3 )\\\\mathbf {(}a_{1},a_{2},a_{3}) and the body axes (x 1 ,x 2 ,x 3 )\\\\mathbf {(}x_{1},x_{2},x_{3}) of atypical particle.Henceforth, we consider two rings: Euclidean (i.e.\",\n \"non-fractal) and a fractal one; both rings are planar, Fig.\",\n \"REF .\",\n \"Hereinafter the subscript $_{\\\\mathbb {E}}$ denotes any Euclidean object.\",\n \"Next, we must consider the mass of a Euclidean ring (body $B_{\\\\mathbb {E}}$ ) versus a fractal ring (body $B_{\\\\alpha }$ ).\",\n \"From a discrete system point of view, the ring is made of $I$ particles $\\\\lbrace i=1,...,I\\\\rbrace $ , each with a respective mass $m_{i}$ , moment of inertia $\\\\mathbf {j}_{i}$ , and positions $\\\\mathbf {x}_{i}$ .\",\n \"The mass of a Euclidean ring $B_{\\\\mathbb {E}}$ , with radius $r\\\\in [R_{D},R]$ and thickness $h$ in $z$ -direction, is now taken in a continuum sense $\\\\begin{array}{c}M_{\\\\mathbb {E}}=\\\\sum _{i=1}^{I}m_{i}\\\\rightarrow \\\\int _{B_{\\\\mathbb {E}}}\\\\rho dB_{\\\\mathbb {E}}=h\\\\int _{R_{D}}^{R}\\\\int _{0}^{2\\\\pi }\\\\rho _{\\\\mathbb {E}}hdS_{2} \\\\\\\\=2\\\\pi h\\\\rho _{\\\\mathbb {E}}\\\\int _{R_{D}}^{R}rdr=\\\\rho _{\\\\mathbb {E}}h\\\\pi \\\\left(R^{2}-R_{D}^{2}\\\\right) .\\\\end{array}$ In the above we have assumed the mass to be homogeneously distributed throughout the ring with a mass density $\\\\rho _{\\\\mathbb {E}}$ .\",\n \"To get quantitative results, one may take: $R=140\\\\times 10^{6}m$ as the outer radius of Saturn's F ring, $R_{D}=74.5\\\\times 10^{6}m$ as the radius of the (inner) D ring, and the rings' thickness $h=100m$ .\"\n ],\n [\n \"Mass densities\",\n \"All the rings constituting the fractal ring $B_{\\\\alpha }$ are embedded in $\\\\mathbb {R}^{3}$ , also with radius $r\\\\in [R_{D},R]$ and thickness $h$ in $z$ -direction.\",\n \"The parameter $\\\\alpha $ ($<1$ ) denotes the fractal dimension in the radial direction, i.e.\",\n \"on any ray(any because the ring is axially symmetric about $z$ ).\",\n \"Thus, the (planar) fractal dimension, such as seen and measured on photographs, is $D=\\\\alpha +1<2$ , consistent with the fact that Saturn's rings are partially plane-filling if interpreted as a planar body.\",\n \"In order to do any analysis involving $B_{\\\\alpha }$ , in the vein of [10], [11], we employ the integration in non-integer dimensional space.\",\n \"That is, we take the infinitesimal element $dB_{\\\\alpha }$ of $B_{\\\\alpha }$ according to [5]: $dB_{\\\\alpha }=h\\\\text{ }dS_{\\\\alpha },\\\\text{ \\\\ \\\\ }dS_{\\\\alpha }=\\\\alpha \\\\left(\\\\frac{r}{R}\\\\right) ^{\\\\alpha -1}dS,\\\\text{ \\\\ \\\\ }dS=rdrd\\\\theta .$ Now, the mass of a fractal ring is $\\\\begin{array}{c}M_{\\\\alpha }=\\\\sum _{i=1}^{I}m_{i}\\\\rightarrow \\\\int _{B}\\\\rho _{\\\\alpha }dB_{\\\\alpha }=h\\\\int _{R_{D}}^{R}\\\\int _{0}^{2\\\\pi }\\\\rho _{\\\\alpha }dS_{\\\\alpha } \\\\\\\\=\\\\displaystyle 2\\\\pi h\\\\rho _{\\\\alpha }\\\\int _{R_{D}}^{R}\\\\alpha \\\\left( \\\\frac{r}{R}\\\\right) ^{\\\\alpha -1}rdr=2\\\\pi h\\\\rho _{\\\\alpha }\\\\frac{\\\\alpha }{\\\\alpha +1}\\\\left(R^{2}-\\\\frac{R_{D}^{\\\\alpha +1}}{R^{\\\\alpha -1}}\\\\right) ,\\\\end{array}$ which involves an effective mass density $\\\\rho _{\\\\alpha }$ of a fractal ring.\",\n \"Note that the above correctly reduces to (1) for $\\\\alpha \\\\rightarrow 1$ .\",\n \"Since the rings in both interpretations must have the same mass, requiring $M_{\\\\alpha }=M_{\\\\mathbb {E}}$ for any $\\\\alpha $ , gives $\\\\rho _{\\\\alpha }=\\\\frac{\\\\alpha +1}{2\\\\alpha }\\\\rho _{\\\\mathbb {E}},$ which is a decreasing function of $\\\\alpha $ (i.e.\",\n \"we must have $\\\\rho _{\\\\alpha }>\\\\rho _{\\\\mathbb {E}}$ for $\\\\alpha <1$ ) and which correctly gives $\\\\lim _{\\\\alpha \\\\rightarrow 1}\\\\rho _{\\\\alpha =1}=\\\\rho _{\\\\mathbb {E}}$ for $\\\\alpha =1$ , i.e.\",\n \"when the fractal ring becomes non-fractal.\",\n \"Thus, a fractal ring has a higher effective mass density than the homogeneous Euclidean ring of the same overall dimensions.\"\n ],\n [\n \"Moments of inertia\",\n \"The moment of inertia of the Euclidean ring ($r\\\\in [0,R]$ and thickness $h$ in $z$ -direction), assuming $\\\\rho _{\\\\mathbb {E}}=\\\\mathrm {const}$ , is $I_{\\\\mathbb {E}}=\\\\frac{1}{2}\\\\pi h\\\\rho _{\\\\mathbb {E}}\\\\left(R^{4}-R_{D}^{4}\\\\right) =\\\\frac{1}{2}M\\\\left( R^{2}+R_{D}^{2}\\\\right) ,$ while the moment of inertia of a fractal ring is $\\\\begin{array}{c}I_{\\\\alpha }=h\\\\int _{B}\\\\rho _{\\\\alpha }r^{2}dB_{E}=h\\\\int _{R_{D}}^{R}\\\\int _{0}^{2\\\\pi }r^{2}\\\\rho _{\\\\alpha }hdS_{\\\\alpha } \\\\\\\\\\\\displaystyle 2\\\\pi h\\\\rho _{\\\\alpha }\\\\int _{R_{D}}^{R}r^{2}\\\\alpha \\\\left( \\\\frac{r}{R}\\\\right) ^{\\\\alpha -1}rdr=2\\\\pi h\\\\rho _{\\\\alpha }\\\\frac{\\\\alpha }{\\\\alpha +3}\\\\left( R^{4}-\\\\frac{R_{D}^{\\\\alpha +3}}{R^{\\\\alpha -1}}\\\\right) .\\\\end{array}$ Now, take the limit $\\\\alpha \\\\rightarrow 1$ : $\\\\lim _{\\\\alpha \\\\rightarrow 1}I_{\\\\alpha }=\\\\frac{1}{2}\\\\pi h\\\\rho _{\\\\mathbb {E}}\\\\left( R^{4}-R_{D}^{4}\\\\right) =I_{\\\\mathbb {E}},$ as expected.\",\n \"Note that $I_{\\\\alpha }$ is an increasing function of $\\\\alpha $ (i.e.\",\n \"we must have $I_{\\\\alpha }0$ .\",\n \"Every circle determines the vector bundle $\\\\xi _r=(E_{Ur},\\\\pi _r,C_r)$ .\",\n \"Consider the vector bundle $\\\\eta =(E,\\\\pi ,R) $ , where $E$ is the union of all $E_{Ur}$ , and the restriction of the projection map $\\\\pi $ to $E_{Ur}$ is equal to $\\\\pi _r$ .\",\n \"The random field $\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )$ is a random section of the above bundle, that is, $\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )\\\\in \\\\pi ^{-1}(r,\\\\varphi )=\\\\mathbb {R}^3$ .\",\n \"In what follow we assume that the random field $\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )$ is second-order, i.e., $\\\\mathsf {E}[\\\\Vert \\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )\\\\Vert ^2]<\\\\infty $ for all $(r,\\\\varphi )\\\\in R$ .\",\n \"There are at least three different (but most probably equivalent) approaches to the construction of random sections of vector bundles, the first by [4], the second by [7], [8], and the third by [3].\",\n \"In what follows, we will use the second named approach.\",\n \"It is based on the following fact: the vector bundle $\\\\eta =(E,\\\\pi ,R)$ is homogeneous or equivariant.\",\n \"In other words, the action of the group $O(2)$ on the bundle base $R$ induces the action of $O(2)$ on the total space $E$ by $(g_0,\\\\mathbf {x})\\\\mapsto (gg_0,\\\\mathbf {x})$ .\",\n \"This action identifies the spaces $\\\\pi ^{-1}(r_0,\\\\varphi )$ for all $\\\\varphi \\\\in [0,2\\\\pi )$ , while the action of the multiplicative group $\\\\mathbb {R}^+$ on R, $\\\\lambda (r,\\\\varphi )=(\\\\lambda r,\\\\varphi )$ , $\\\\lambda >0$ , identifies the spaces $\\\\pi ^{-1}(r,\\\\varphi _0)$ for all $r>0$ .\",\n \"We suppose that the random field $\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )$ is mean-square continuous, i.e., $\\\\lim _{\\\\Vert \\\\mathbf {x}-\\\\mathbf {x}_0\\\\Vert \\\\rightarrow 0}\\\\mathsf {E}[\\\\Vert \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})-\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x}_0))\\\\Vert ^2]=0$ for all $\\\\mathbf {x}_0\\\\in R$ .\",\n \"Let $\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})\\\\rangle =\\\\mathsf {E}[\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})]$ be the one-point correlation vector of the random field $\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})$ .\",\n \"On the one hand, under rotation and/or reflection $g\\\\in O(2)$ the point $\\\\mathbf {x}$ becomes the point $g\\\\mathbf {x}$ .\",\n \"Evidently, the axial vector $\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})$ transforms according to the representation (REF ) and becomes $U(g)\\\\overline{\\\\mathbf {\\\\omega }}(g\\\\mathbf {x})$ .\",\n \"The one-point correlation vector of the so transformed random field remains the same, i.e., $\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(g\\\\mathbf {x})\\\\rangle =U(g)\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})\\\\rangle .$ On the other hand, the one-point correlation vector of the random field $\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )$ should be independent upon an arbitrary choice of the $x$ - and $y$ -axes of the Cartesian coordinate systems, i.e., it should not depend on $\\\\varphi $ .\",\n \"Then we have $\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})\\\\rangle =U(g)\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})\\\\rangle $ for all $g\\\\in O(2)$ , i.e., $\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})\\\\rangle $ belongs to a subspace of $\\\\mathbb {R}^{3}$ where a trivial component of $U$ acts.\",\n \"Then we obtain $\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})\\\\rangle =\\\\mathbf {0}$ , because $U$ does not contain trivial components.\",\n \"Similarly, let $\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x}),\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {y})\\\\rangle =\\\\mathsf {E}[\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x}) \\\\otimes \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {y})]$ be the two-point correlation tensor of the random field $\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})$ .\",\n \"Under the action of $O(2)$ we should have $\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(g\\\\mathbf {x}),\\\\overline{\\\\mathbf {\\\\omega }}(g\\\\mathbf {y})\\\\rangle =(U\\\\otimes U)(g)\\\\langle \\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x}),\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {y})\\\\rangle .$ In other words, the random field $\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})$ is wide-sense isotropic with respect to the group $O(2)$ and its representation $U$ .\",\n \"Consider the restriction of the field $\\\\overline{\\\\mathbf {\\\\omega }}(\\\\mathbf {x})$ to a circle $C_r$ , $r>0$ .\",\n \"The spectral expansion of the field $\\\\lbrace \\\\,\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )\\\\colon \\\\varphi \\\\in C_r\\\\,\\\\rbrace $ can be calculated using [7] or [8].\",\n \"The representation $U$ is the direct sum of the two irreducible representations $\\\\lambda _-(g)=\\\\det g$ and $\\\\lambda _1(g)=g$ .\",\n \"The vector bundle $\\\\eta $ is the direct sum of the vector bundles $\\\\eta _-$ and $\\\\eta _1$ , where the bundle $\\\\eta _-$ (resp.\",\n \"$\\\\eta _1$ ) is generated by the representation $\\\\lambda _-$ (resp.\",\n \"$\\\\lambda _1$ ).\",\n \"Let $\\\\mu _0$ be the trivial representation of the group $SO(2)$ , and let $\\\\mu _k$ be the representation $\\\\mu _k(\\\\varphi )=\\\\begin{pmatrix}\\\\cos (k\\\\varphi ) & \\\\sin (k\\\\varphi ) \\\\\\\\-\\\\sin (k\\\\varphi ) & \\\\cos (k\\\\varphi )\\\\end{pmatrix}.$ The representations $\\\\lambda _-\\\\otimes \\\\mu _k$ , $k\\\\ge 0$ are all irreducible orthogonal representations of the group $G=O(2)\\\\times SO(2)$ that contain $\\\\lambda _-$ after restriction to $O(2)$ .\",\n \"The representations $\\\\lambda _1\\\\otimes \\\\mu _k$ , $k\\\\ge 0$ are all irreducible orthogonal representations of the group $G=O(2)\\\\times SO(2)$ that contain $\\\\lambda _1$ after restriction to $O(2)$ .\",\n \"The matrix entries of $\\\\mu _0$ and of the second column of $\\\\mu _k$ form an orthogonal basis in the Hilbert space $L^2(SO(2),\\\\mathrm {d}\\\\varphi )$ .\",\n \"Their multiples $e_k(\\\\varphi )={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{1}{\\\\sqrt{2\\\\pi }}, & \\\\mbox{if } k=0, \\\\\\\\\\\\frac{1}{\\\\sqrt{\\\\pi }}\\\\cos (k\\\\varphi ), & \\\\mbox{if } k\\\\le -1 \\\\\\\\\\\\frac{1}{\\\\sqrt{\\\\pi }}\\\\sin (k\\\\varphi ), & \\\\mbox{if } k\\\\ge 1\\\\end{array}\\\\right.\",\n \"}$ form an orthonormal basis of the above space.\",\n \"Then we have $ \\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )=\\\\sum _{k=-\\\\infty }^{\\\\infty }e_k(\\\\varphi )\\\\mathbf {Z}^k(r),$ where $\\\\lbrace \\\\,\\\\mathbf {Z}^k(r)\\\\colon k\\\\in \\\\mathbb {Z}\\\\,\\\\rbrace $ is a sequence of centred stochastic processes with $\\\\begin{aligned}\\\\mathsf {E}[\\\\mathbf {Z}^k(r)\\\\otimes \\\\mathbf {Z}^l(r)]&=\\\\delta _{kl}B^{(k)}(r),\\\\\\\\\\\\sum _{k\\\\in \\\\mathbb {Z}}\\\\operatorname{tr}(B^{(k)}(r))&<\\\\infty .\",\n \"\\\\end{aligned}$ It follows that $\\\\mathbf {Z}^k(r)=\\\\int _{0}^{2\\\\pi }\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )e_k(\\\\varphi )\\\\,\\\\mathrm {d}\\\\varphi .$ Then we have $ \\\\mathsf {E}[\\\\mathbf {Z}^k(r)\\\\otimes \\\\mathbf {Z}^l(s)]=\\\\int _{0}^{2\\\\pi }\\\\int _{0}^{2\\\\pi } \\\\mathsf {E}[\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi _1)\\\\otimes \\\\overline{\\\\mathbf {\\\\omega }}(s,\\\\varphi _2)] e_k(\\\\varphi _1)\\\\,\\\\mathrm {d}\\\\varphi _1e_l(\\\\varphi _2)\\\\,\\\\mathrm {d}\\\\varphi _2.$ The field is isotropic and mean-square continuous, therefore $\\\\mathsf {E}[\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi _1)\\\\otimes \\\\overline{\\\\mathbf {\\\\omega }}(s,\\\\varphi _2)] =B(r,s,\\\\cos (\\\\varphi _1-\\\\varphi _2))$ is a continuous function.\",\n \"Note that $e_k(\\\\varphi )$ are spherical harmonics of degree $|k|$ .\",\n \"Denote by $\\\\mathbf {x\\\\cdot y}$ the standard inner product in the space $\\\\mathbb {R}^d$ , and by $\\\\mathrm {d}\\\\omega (\\\\mathbf {y})$ the Lebesgue measure on the unit sphere $S^{d-1}=\\\\lbrace \\\\,\\\\mathbf {x}\\\\in \\\\mathbb {R}^d\\\\colon \\\\Vert \\\\mathbf {x}\\\\Vert =1\\\\,\\\\rbrace $ .\",\n \"Then $\\\\int _{S^{d-1}}\\\\,\\\\mathrm {d}\\\\omega (\\\\mathbf {x})=\\\\omega _d=\\\\frac{2\\\\pi ^{d/2}}{\\\\Gamma (d/2)},$ where $\\\\Gamma $ is the Gamma function.\",\n \"Now we use the Funk–Hecke theorem, see [1].\",\n \"For any continuous function $f$ on the interval $[-1,1]$ and for any spherical harmonic $S_k(\\\\mathbf {y})$ of degree $k$ we have $\\\\int _{S^{d-1}}f(\\\\mathbf {x\\\\cdot y})S_k(\\\\mathbf {x})\\\\,\\\\mathrm {d}\\\\omega (\\\\mathbf {x})=\\\\lambda _kS_k(\\\\mathbf {y}),$ where $\\\\lambda _k=\\\\omega _{d-1}\\\\int _{-1}^{1}f(u)\\\\frac{C^{(n-2)/2}_k(u)}{C^{(n-2)/2}_k(1)} (1-u^2)^{(n-3)/2}\\\\,\\\\mathrm {d}u,$ $d\\\\ge 3$ , and $C^{(n-2)/2}_k(u)$ are Gegenbauer polynomials.\",\n \"To see how this theorem looks like when $d=2$ , we perform a limit transition as $n\\\\downarrow 2$ .\",\n \"By [1], $\\\\lim _{\\\\lambda \\\\rightarrow 0}\\\\frac{C^{\\\\lambda }_k(u)}{C^{\\\\lambda }_k(1)}=T_k(u),$ where $T_k(u)$ are Chebyshev polynomials of the first kind.\",\n \"We have $\\\\omega _1=2$ , $\\\\mathbf {x\\\\cdot y}$ becomes $\\\\cos (\\\\varphi _1-\\\\varphi _2)$ , and $\\\\mathrm {d}\\\\omega (\\\\mathbf {x})$ becomes $\\\\mathrm {d}\\\\varphi _1$ .\",\n \"We obtain $\\\\int _{0}^{2\\\\pi }B(r,s,\\\\cos (\\\\varphi _1-\\\\varphi _2))e_k(\\\\varphi _1)\\\\,\\\\mathrm {d}\\\\varphi _1 =B^{(k)}(r,s)e_k(\\\\varphi _2),$ where $B^{(k)}(r,s)=2\\\\int _{-1}^{1}B(r,s,u)T_{|k|}(u)(1-u^2)^{-1/2}\\\\,\\\\mathrm {d}u,$ Equation (REF ) becomes $\\\\mathsf {E}[\\\\mathbf {Z}^k(r)\\\\otimes \\\\mathbf {Z}^l(s)]=\\\\int _{0}^{2\\\\pi }B^{(k)}(r,s) e_k(\\\\varphi _2)e_l(\\\\varphi _2)\\\\,\\\\mathrm {d}\\\\varphi _2=\\\\delta _{kl}B^{(k)}(r,s).$ In particular, if $k\\\\ne l$ , then the processes $\\\\mathbf {Z}^k(r)$ and $\\\\mathbf {Z}^l(r)$ are uncorrelated.\",\n \"Calculate the two-point correlation tensor of the random field $\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi )$ .\",\n \"We have $ \\\\begin{aligned}\\\\mathsf {E}[\\\\overline{\\\\mathbf {\\\\omega }}(r,\\\\varphi _1)\\\\otimes \\\\overline{\\\\mathbf {\\\\omega }}(s,\\\\varphi _2)]&=\\\\sum _{k=-\\\\infty }^{\\\\infty }e_k(\\\\varphi _1)e_k(\\\\varphi _2)B^{(k)}(r,s)\\\\\\\\&=\\\\frac{1}{2\\\\pi }B^{(0)}(r,s)+\\\\frac{1}{\\\\pi }\\\\sum _{k=1}^{\\\\infty }\\\\cos (k(\\\\varphi _1-\\\\varphi _2))B^{(k)}(r,s).\",\n \"\\\\end{aligned}$ Now we add a time coordinate, $t$ , to our considerations.\",\n \"A particle located at $(r,\\\\varphi )$ at time moment $t$ , was located at $(r,\\\\varphi -\\\\sqrt{GM}t/r^{3/2})$ at time moment 0.\",\n \"It follows that $\\\\overline{\\\\mathbf {\\\\omega }}(t,r,\\\\varphi )=\\\\overline{\\\\mathbf {\\\\omega }}\\\\left(r,\\\\varphi -\\\\frac{\\\\sqrt{GM}t}{r^{3/2}}\\\\right),$ where $G$ is Newton's gravitational constant and $M$ is the mass of Saturn.\",\n \"Equation (REF ) gives $ \\\\overline{\\\\mathbf {\\\\omega }}(t,r,\\\\varphi )=\\\\sum _{k=-\\\\infty }^{\\\\infty }e_k\\\\left(\\\\varphi -\\\\frac{\\\\sqrt{GM}t}{r^{3/2}}\\\\right)\\\\mathbf {Z}^k(r),$ while Equation (REF ) gives $\\\\begin{aligned}\\\\mathsf {E}[\\\\overline{\\\\mathbf {\\\\omega }}(t_1,r,\\\\varphi _1)\\\\otimes \\\\overline{\\\\mathbf {\\\\omega }}(t_2,s,\\\\varphi _2)] &=\\\\frac{1}{2\\\\pi }B^{(0)}(r,s)\\\\\\\\&\\\\quad +\\\\frac{1}{\\\\pi }\\\\sum _{k=1}^{\\\\infty }\\\\cos \\\\left(k\\\\left(\\\\varphi _1-\\\\varphi _2-\\\\frac{\\\\sqrt{GM}(t_1-t_2)}{r^{3/2}}\\\\right)\\\\right)B^{(k)}(r,s).\",\n \"\\\\end{aligned}$ Conversely, let $\\\\lbrace \\\\,B^{(k)}(r,s)\\\\colon k\\\\ge 0\\\\,\\\\rbrace $ be a sequence of continuous positive-definite matrix-valued functions with $ \\\\sum _{k=0}^{\\\\infty }\\\\operatorname{tr}(B^{(k)}(r,r))<\\\\infty ,$ and let $\\\\lbrace \\\\,\\\\mathbf {Z}_k(r)\\\\colon k\\\\in \\\\mathbb {Z}\\\\,\\\\rbrace $ be a sequence of uncorrelated centred stochastic processes with $\\\\mathsf {E}[\\\\mathbf {Z}^k(r)\\\\otimes \\\\mathbf {Z}^l(s)]=\\\\delta _{kl}B^{(|k|)}(r,s).$ The random field (REF ) may describe rotating particles inside Saturn's rings, if all the functions $B^{(k)}(r,s)$ are equal to 0 outside the rectangle $[R_0,R_1]^2$ , where $R_0$ (resp.\",\n \"$R_1$ ) is the inner (resp.\",\n \"outer) radius of Saturn's rings.\",\n \"To make our model more realistic, we assume that all the functions $B^{(k)}(r,s)$ are equal to 0 outside the Cartesian square $F^{2}$ , where $F $ is a fat fractal subset of the interval $[R_{0},R_{1}]$ , see [12].\",\n \"[9] calls these sets dusts of positive measure.\",\n \"Such a set has a positive Lebesgue measure, its Hausdorff dimension is equal to 1, but the Hausdorff dimension of its boundary is not an integer number.\",\n \"A classical example of a fat fractal is a fat Cantor set.\",\n \"In contrast to the ordinary Cantor set, where we delete the middle one-third of each interval at each step, this time we delete the middle $3^{-n}$ th part of each interval at the $n$ th step.\",\n \"To construct an example, consider an arbitrary sequence of continuous positive-definite matrix-valued functions $\\\\lbrace \\\\,B^{(k)}(r,s)\\\\colon k\\\\ge 0\\\\,\\\\rbrace $ satisfying (REF ) of the following form: $B^{(k)}(r,s)=\\\\sum _{i\\\\in I_k}\\\\mathbf {f}_{ik}(r)\\\\mathbf {f}^{\\\\top }_{ik}(s),$ where $\\\\mathbf {f}_{ik}(r)\\\\colon [R_0,R_1]\\\\rightarrow \\\\mathbb {R}^3$ are continuous functions, satisfying the following condition: for each $r\\\\in [R_0,R_1]$ the set $I_{kr}=\\\\lbrace \\\\,i\\\\in I_k\\\\colon f_i(r)\\\\ne 0\\\\,\\\\rbrace $ is as most countable and the series $\\\\sum _{i\\\\in I_{kr}}\\\\Vert \\\\mathbf {f}_i(r)\\\\Vert ^2$ converges.\",\n \"The so defined function is obviously positive-definite.\",\n \"Put $\\\\tilde{B}^{(k)}(r,s)=\\\\sum _{i\\\\in I_k}\\\\tilde{\\\\mathbf {f}}_{ik}(r)\\\\tilde{\\\\mathbf {f}}^{\\\\top }_{ik}(s),\\\\qquad r,s\\\\in F.$ The functions $\\\\tilde{B}^{(k)}(r,s)$ are the restrictions of positive-definite functions $B^{(k)}(r,s)$ to $F^2$ and are positive-definite themselves.\",\n \"Consider the centred stochastic process $\\\\lbrace \\\\,\\\\tilde{\\\\mathbf {Z}}^k(r)\\\\colon r\\\\in F\\\\,\\\\rbrace $ with $\\\\mathsf {E}[\\\\tilde{\\\\mathbf {Z}}^k(r)\\\\otimes \\\\tilde{\\\\mathbf {Z}}^l(s)]=\\\\delta _{kl}\\\\tilde{B}^{(|k|)}(r,s),\\\\qquad r,s\\\\in F.$ Condition (REF ) guarantees the mean-square convergence of the series $\\\\overline{\\\\mathbf {\\\\omega }}(t,r,\\\\varphi )=\\\\sum _{k=-\\\\infty }^{\\\\infty }e_k\\\\left(\\\\varphi -\\\\frac{\\\\sqrt{GM}t}{r^{3/2}}\\\\right)\\\\tilde{\\\\mathbf {Z}}^k(r)$ for all $t\\\\ge 0$ , $r\\\\in F$ , and $\\\\varphi \\\\in [0,2\\\\pi ]$ .\"\n ],\n [\n \"Closure\",\n \"This paper reports an investigation of the fractal character of Saturnian rings.\",\n \"First, working with the calculus in a non-integer dimensional space, by energy arguments, we infer that the fractally structured ring is more likely than a non-fractal one.\",\n \"Next, we develop a kinematics model in which angular velocities of particles form a random field.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05499"}}},{"rowIdx":3383,"cells":{"text":{"kind":"list like","value":[["Quantifying the Heat Dissipation from a Molecular Motor's Transport\n Properties in Nonequilibrium Steady States"],["Abstract Theoretical analysis, which maps single molecule time trajectories of a molecular motor onto unicyclic Markov processes, allows us to evaluate the heat dissipated from the motor and to elucidate its dependence on the mean velocity and diffusivity.","Unlike passive Brownian particles in equilibrium, the velocity and diffusion constant of molecular motors are closely inter-related to each other.","In particular, our study makes it clear that the increase of diffusivity with the heat production is a natural outcome of active particles, which is reminiscent of the recent experimental premise that the diffusion of an exothermic enzyme is enhanced by the heat released from its own catalytic turnover.","Compared with freely diffusing exothermic enzymes, kinesin-1 whose dynamics is confined on one-dimensional tracks is highly efficient in transforming conformational fluctuations into a locally directed motion, thus displaying a significantly higher enhancement in diffusivity with its turnover rate.","Putting molecular motors and freely diffusing enzymes on an equal footing, our study offers thermodynamic basis to understand the heat enhanced self-diffusion of exothermic enzymes."],["Aknowledgement","We thank Bae-Yeun Ha, Hyunggyu Park, Dave Thirumalai, and Anatoly Kolomeisky for helpful comments and illuminating discussions.","We acknowledge the Center for Advanced Computation in KIAS for providing computing resources.","Figure: Analysis of experimental data, digitized from Ref.",", using (NN=4)-state cyclic model.The solid lines are the fits to the dataA.","VV vs [ATP] at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).B.","VV vs ff at [ATP] = 5 μ\\mu M.C.","VV vs ff at [ATP] = 2 mM.D.","Stall force as a function of [ATP], measured by `Position clamp' (red square) or `Fixed trap' (blue circle) methods.E.","DD vs [ATP] at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).DD was estimated from r=2D/Vd 0 r=2D/Vd_0.","F. DD vs ff at [ATP] = 2 mM.Table: Parameters determined from the fit using (N=4)-state model.","The unit of {u n }\\lbrace u_n\\rbrace and {w n }\\lbrace w_n\\rbrace is s -1 s^{-1} except for u 1 o u_1^o ([u 1 o ]=μM -1 s -1 [u_1^o]={\\mu M}^{-1} s^{-1}).Supplementary Information"],["1. Derivation of the third degree polynomial dependence of $D$ on {{formula:bfe6300b-f497-4028-bece-5a1753aeadd7}} .","Here, we show a polynomial dependence of $D$ on $V$ using a few specific examples of the $N$ -state periodic reaction model [19] whose reaction scheme is demonstrated in Fig.","REF .","When $N$ =1, the master equation to solve is: $\\dot{\\pi }_\\mu (t)=u_1\\pi _{\\mu -1}(t)+w_1\\pi _{\\mu +1}(t)-(u_1+w_1)\\pi _\\mu (t),$ where $\\pi _\\mu (t)$ is the probability of motor being in the $\\mu $ -th reaction cycle at time $t$ .","Using generating function $F(z,t)=\\sum _{\\mu =-\\infty }^{\\infty }z^\\mu \\pi _\\mu (t)$ with $\\pi _\\mu (0)=\\delta _{\\mu ,0}$ [49], the master equation is written in terms of $F(z,t)$ as $\\partial _tF(z,t)&=\\left(u_1z+\\frac{w_1}{z}-(u_1+w_1)\\right)F(z,t)\\nonumber \\\\F(z,t)&=e^{\\left(u_1z+\\frac{w_1}{z}-(u_1+w_1)\\right)t}.$ Now, it is straightforward to obtain the mean velocity ($V$ ) and diffusion constant ($D$ ) using $\\partial _z\\log {F(z,t)}|_{z=1}=\\langle \\mu (t)\\rangle $ and $\\partial ^2_z\\log {F(z,t)}|_{z=1}=\\langle \\mu ^2(t)\\rangle -\\langle \\mu (t)\\rangle ^2-\\langle \\mu (t)\\rangle $ , where $\\mu (t)$ is the number of steps taken by the molecular motor until time $t$ .","$V\\equiv \\lim _{t\\rightarrow \\infty }\\frac{d_0\\langle \\mu (t)\\rangle }{t}= d_0(u_1-w_1)$ and $D\\equiv \\lim _{t\\rightarrow \\infty }\\frac{d^2_0(\\langle \\mu ^2(t)\\rangle -\\langle \\mu (t)\\rangle ^2)}{2t} = \\frac{d_0^2(u_1+w_1)}{2}.$ Provided that only $u_1$ changes (for example by increasing ATP concentration) while $w_1$ remains constant.","elimination of $u_1$ from $V(u_1)$ and $D(u_1)$ relates $D$ to $V$ as $ D(V) = D_0+\\frac{d_0}{2}V$ where $D_0\\equiv d_0^2w_1,$ showing that for ($N$ =1)-state kinetic model, $D$ is linear in $V$ ."],["($N$ =2)-state kinetic model","For the ($N$ =2)-kinetic model [20], $V = d_0\\frac{u_1 u_2 - w_1 w_2}{u_1 + u_2 + w_1 + w_2}$ and $D=\\frac{d_0^2}{2}\\left[\\frac{u_1u_2}{w_1w_2}+1-2\\left(\\frac{u_1u_2}{w_1w_2}-1\\right)^2\\frac{w_1w_2}{\\sigma ^2}\\right]\\frac{w_1w_2}{\\sigma }$ where $\\sigma =u_1+u_2+w_1+w_2$ .","Then, $D=D(V)$ is obtained by eliminating $u_1$ between Eq.REF and Eq.REF : $D(v)/d_0^2&=\\left(\\frac{u_2}{\\kappa +1}\\right)+\\frac{1}{2}\\left(\\frac{\\kappa -1}{\\kappa +1}\\right)u_2v-\\left(\\frac{u_2}{\\kappa +1}\\right)\\frac{u_2^2}{w_1w_2}v^2+\\left(\\frac{u_2}{\\kappa +1}\\right)\\frac{u_2^2}{w_1w_2}v^3$ where $\\kappa \\equiv \\frac{u_2(u_2+w_1+w_2)}{w_1w_2}$ , and $v\\equiv V/V_{max}$ ($0\\le v\\le 1$ ) with $V_{max}=d_0u_2$ .","Eq.REF confirms that $D$ is a third order polynomial in $V$ .","Incidentally, the ($N$ =2)-kinetic model is reduced to the Michaelis-Menten equation by setting $u_1=u_1^o[S]$ and $w_2=0$ .","$V= d_0\\frac{u_1 u_2}{u_1 + u_2 + w_1} = \\frac{V_{max}[S]}{ K_M + [S] }$ where $K_M = (u_2 + w_1)/u_1^o$ , and $D(v)=D_{max}v\\left[1-2\\phi v+2\\phi v^2\\right]$ with $\\phi \\equiv \\frac{k_{cat}}{u_1^oK_M}$ .","Note that $D\\le D_{max}=d_0V_{max}/2$ and that for $D(v)$ to be positive for all the range of $v$ , the parameter $\\phi $ should be in a rather narrow range of $0\\le \\phi \\le 2$ ."],["General case: $N$ -state kinetic model","The above two examples of one-dimensional hopping model was extended to the $N$ -state kinetic model by Derrida [19].","He obtained exact expressions for the mean velocity ($V^{(D)}$ ) and diffusion constant ($D^{(D)}$ ), where the superscript $(D)$ refers to Derrida's, in terms of the rate constants $\\lbrace u_n\\rbrace $ and $\\lbrace w_n\\rbrace $ .","Derrida's expression for $V^{(D)}$ and $D^{(D)}$ are related to $V$ and $D$ as $V\\equiv (d_0/N)V^{(D)}$ and $D\\equiv (d_0^2/N^2)D^{(D)}$ .","$V^{(D)} = \\frac{N}{\\sum _{n=1}^{N} r_n}\\left(1- \\frac{\\prod _{n=1}^{N} w_n}{\\prod _{n=1}^{N} u_n }\\right)$ and $D^{(D)}&=\\frac{1}{\\left(\\sum _{n=1}^Nr_n\\right)^2}\\left[V^{(D)}\\sum _{n=1}^Nq_n\\sum _{i=1}^Nir_{n+i}+N\\sum _{n=1}^Nu_nq_nr_n\\right]-V^{(D)}\\frac{N+2}{2}$ where $r_n = \\frac{1}{u_n} \\left[1 + \\sum _{i=1}^{N-1} \\prod _{j=1}^{i} \\frac{w_{n+j-1}}{u_{n+j}}\\right]$ , and $q_n=\\frac{1}{u_n}\\left[1+\\sum _{i=1}^{N-1}\\prod _{j=1}^i\\frac{w_{n-j}}{u_{n-j}}\\right]$ with periodic boundary conditions $u_{n+N} = u_n$ , and $w_{n+N} = w_n$ .","$D^{(D)} = D^{(D)}(V^{(D)})$ is obtained by eliminating $u_1$ between Eq.REF and Eq.REF .","For that, we first express various terms in Eq.REF in terms of $u_1$ : $ \\sum _{n=1}^Nr_n&=\\frac{A}{u_1}+B,\\nonumber \\\\\\sum _{n=1}^Nq_n\\sum _{i=1}^Nir_{n+i}&=\\frac{\\alpha }{u_1^2}+\\frac{\\beta }{u_1}+\\gamma ,\\nonumber \\\\\\sum _{n=1}^Nu_nq_nr_n&=\\frac{\\xi }{u_1^2}+\\frac{\\eta }{u_1}+\\zeta .$ where A, $B\\left(=\\sum _{n=2}^N\\frac{1}{u_n}\\left[1+\\sum _{i=1}^{N-n}\\prod _{j=1}^i\\frac{w_{n+j-1}}{u_{n+j}}\\right]\\right)$ , $\\alpha $ , $\\beta $ , $\\gamma $ , $\\xi $ , $\\eta $ , and $\\zeta $ are all positive constants independent of $u_1$ .","Next, $\\sum _{n=1}^Nr_n$ in Eq.REF substituted to Eq.REF gives $ V^{(D)}=N\\frac{1-C/u_1}{A/u_1+B},$ where $C\\left(= \\prod _{n=1}^Nw_n/\\prod _{n=2}^Nu_n \\right)$ , and $u_1$ is expressed in terms of $V^{(D)}$ $\\frac{1}{u_1}=\\frac{1-B\\frac{V^{(D)}}{N}}{C+A\\frac{V^{(D)}}{N}}.$ Finally, with Eqs.REF and REF , we show that $D^{(D)}$ (Eq.REF ) can be expressed as a third degree polynomial in $V^{(D)}$ $D^{(D)}&=V^{(D)}\\frac{\\left[\\alpha (1-B\\frac{V^{(D)}}{N})^2+\\beta (1-B\\frac{V^{(D)}}{N})(A\\frac{V^{(D)}}{N}+C)+\\gamma (A\\frac{V^{(D)}}{N}+C)^2\\right]}{(A+BC)^2}\\nonumber \\\\&\\hspace{14.22636pt}+N\\frac{\\xi (1-B\\frac{V^{(D)}}{N})^2+\\eta (1-B\\frac{V^{(D)}}{N})(A\\frac{V^{(D)}}{N}+C)+\\zeta (A\\frac{V^{(D)}}{N}+C)^2}{(A+BC)^2}-V^{(D)}\\frac{N+2}{2}\\nonumber \\\\&=z_0+z_1V^{(D)}+z_2(V^{(D)})^2+z_3(V^{(D)})^3\\nonumber \\\\D&=\\alpha _0+\\alpha _1V+\\alpha _2V^2+\\alpha _3V^3.$ with $\\alpha _i=(d_0/N)^{2-i}z_i$ , and $z_0&=\\frac{(\\xi +\\eta C+\\zeta C^2)}{(A+BC)^2}\\nonumber \\\\z_1&=\\left[\\frac{(\\alpha +\\beta C+\\gamma C^2)+(-2\\xi B+\\eta (A-BC)+2\\zeta AC)}{(A+BC)^2}-\\frac{N+2}{2}\\right]\\nonumber \\\\z_2&=\\frac{(-2\\alpha B+\\beta (A-BC) +2\\gamma AC)+(\\xi B^2-\\eta AB +\\zeta A^2)}{(A+BC)^2}\\nonumber \\\\z_3&=\\frac{(\\alpha B^2-\\beta AB +\\gamma A^2)}{(A+BC)^2}.","\\nonumber $"],["Alternative derivation of $D(V)$","In addition to Derrida's result [19], the sign of $\\alpha _i$ can be determined by deriving the relation between $V^{ (D) }$ and $D^{ (D) }$ using the result in ref.","[50].","From Eq.","(23) in ref.","[50], $ \\begin{aligned}V^{ (D) } = - i \\frac{c^{\\prime }_0}{c_1}\\end{aligned}$ where $c^{\\prime }_0 = i N ( \\prod _{n=1}^{N} u_n - \\prod _{n=1}^{N} w_n )$ and $c_1 = c_1( \\lbrace u_n\\rbrace , \\lbrace w_n\\rbrace )$ .","By combining two expressions of $V^{ (D) }$ , Eq.REF and Eq.REF , we get $c_1& = \\prod _{n=1}^{N} u_n\\times \\sum _{m=1}^{N-1} \\left[ \\frac{1}{u_m} \\left( 1 + \\sum _{i=1}^{N-1} \\prod _{j=1}^{i} \\frac{w_{m+j}}{u_{m+j}}\\right) \\right] \\nonumber \\\\& = u_1 \\prod _{n=2}^{N} u_n \\left[\\frac{1}{u_1} \\left( 1 + \\sum _{i=1}^{N-1} \\prod _{j=1}^{i} \\frac{w_{1+j}}{u_{1+j}} \\right)+ \\sum _{m=2}^{N-1} \\frac{1}{u_m} \\left( 1 + \\sum _{i=1}^{N-1} \\prod _{j=1}^{i} \\frac{w_{m+j}}{u_{m+j}} \\right)\\right]\\nonumber \\\\& = \\mathcal {A} u_1 + \\mathcal {B}$ where $\\mathcal {A}$ and $\\mathcal {B}$ are constants depending on ($u_2,\\ldots ,u_N$ ) and ($w_1,w_2,\\ldots w_N$ ).","Eq.","(REF ) substituted to Eq.","(REF ) gives $\\begin{aligned}V^{ (D) } = N \\frac{u_1 \\prod _{n=2}^{N} u_n - \\prod _{n=1}^{N} w_n}{\\mathcal {A}u_1 + \\mathcal {B}}\\end{aligned}$ and hence $u_1$ can be written as $\\begin{aligned}u_1 = \\frac{\\mathcal {B} V^{ (D) } + N \\prod _{n=1}^N w_n}{N \\prod _{n=2}^{N} u_n - \\mathcal {A}V^{ (D) }}.\\end{aligned}$ From Eq.REF and Eq.REF , $\\mathcal {A} = B \\prod _{n=2}^N u_n$ and $\\mathcal {B} = A \\prod _{n=2}^N u_n$ where $A$ and $B$ are the same constants used in Eq.REF .","Now using Eq.","(REF ), Eq.","(REF ), and general expression of $D^{ (D) }$ from Eq.","(24) of ref.","[50], we have $\\begin{aligned}D^{ (D) } & = \\frac{c_0^{\\prime \\prime } - 2 c_2 (V^{ (D) })^2 }{2 c_1 } \\\\&=\\frac{c_0^{\\prime \\prime } - 2 c_2 (V^{ (D) })^2 }{2 (\\frac{B}{\\prod _{n=2}^N u_n} u_1 + \\frac{A}{\\prod _{n=2}^N u_n}) } \\\\& = \\frac{( c_0^{\\prime \\prime } - 2 c_2 (V^{ (D) })^2 ) (\\prod _{n=2}^{N} u_n (N- B V^{(D)}) )}{2N( A + C B)}\\end{aligned}$ where $c_0^{\\prime \\prime } = N^2 ( u_1 \\prod _{n=2}^{N} u_n + \\prod _{n=1}^{N} w_n ) = N^2 (\\prod _{n=2}^N u_n) (u_1 + C)$ , and $c_2 = \\beta _1 u_1 + \\beta _2$ where $\\beta _1$ and $\\beta _2$ are positive constants depending on ($u_2, u_3, \\cdots , u_N$ ) and ($w_1, w_2, \\cdots , w_N$ ) (see Eq.","(53, 54) of ref.","[50]).","Since $u_1 (N - B V^{ (D) }) = NC + AV^{ (D) }$ (Eq.REF ), we get $D^{ (D) } & = \\frac{1}{2} \\frac{( c_0^{^{\\prime \\prime }} - 2 c_2 (V^{ (D) })^2 ) ( (N- B V^{(D)}) )\\prod _{n=2}^{N} u_n}{N A + N C B} \\nonumber \\\\& = \\frac{1}{2} \\frac{( N^2 (\\prod _{n=2}^N u_2) (u_1 + C) - 2 (\\beta _1 u_1 + \\beta _2) (V^{ (D) })^2 ) ( (N- B V^{(D)}) )\\prod _{n=2}^{N} u_n}{N A + N C B} \\nonumber \\\\& = \\frac{\\prod _{n=2}^N u_n}{2} \\frac{ \\left( N^2 (\\prod _{n=2}^N u_2) ( NC + A V^{(D)} + C(N-B V^{ (D) } ) ) - 2 ( \\beta _1 (N C + A V^{(D)}) + \\beta _2 ( N- B V^{(D)} ) ) (V^{ (D) })^2 \\right)}{N A + N C B} \\nonumber \\\\&= z_0 + z_1 V^{ (D) } + z_2 (V^{ (D) })^2 + z_3 (V^{ (D) })^3$ where $z_0&=\\frac{N^2 (\\prod _{n=2}^N u_n )^2 C}{ A + BC }>0\\nonumber \\\\z_1&=\\frac{N (\\prod _{n=2}^N u_n )^2 (A- BC)}{2 (A + BC)}\\nonumber \\\\z_2&=- \\frac{(\\prod _{n=2}^N u_n )( \\beta _1 C + \\beta _2 )}{A + BC}<0\\nonumber \\\\z_3&= \\frac{(\\prod _{n=2}^N u_n )( - \\beta _1 A + \\beta _2 B)}{N(A + BC)}.\\nonumber $ It is obvious that $z_0 >0$ and $z_2 < 0$ since $A$ , $B$ , $C$ , $\\beta _1$ , and $\\beta _2$ are all positive constants."],["2. The 1D hopping model with a finite processivity","Because of a probability of being dissociated from microtubules, kinesin motors display a finite processivity.","However, since the mean velocity and diffusion constant are calculated from the trajectories that remain on the track, the expressions of $V$ and $D$ in terms of the rate constants are unchanged.","To make this point mathematically more explicit, we consider the master equation assuming a constant dissociation rate $k_d$ from each chemical state.","$\\frac{d P_{\\mu ,n} (t)}{d t}&=u_{n-1} P_{\\mu , n-1}(t)+w_{n}P_{\\mu , n+1}(t)\\nonumber \\\\&-(u_n+w_{n-1}+k_d)P_{\\mu , n}(t)$ where $P_{\\mu , n}(t)$ is the probability of being in the $n$ -th chemical state at the $\\mu $ -th reaction cycle.","The probability of the motor remaining on the track (survival probability of motor) is $ S (t) \\equiv \\sum _{\\mu =\\infty }^{\\infty } \\sum _{n=1}^{N} P_{\\mu , n}(t) = e^{-k_d t}.$ The expectation value of an observable, which can be used to calculate $\\langle x(t)\\rangle $ or $\\langle x^2(t)\\rangle $ , is expressed as $\\langle A(t)\\rangle = \\sum _{\\mu =-\\infty }^{\\infty } \\sum _{n=1}^{N} \\Phi _{\\mu ,n}(t) A(\\mu (t))$ with a probability density function renormalized with respect to the survival probability $ \\Phi _{\\mu , n}(t) \\equiv \\frac{P_{\\mu ,n}(t)}{S(t)}=P_{\\mu ,n}(t)e^{k_dt}.$ Incidentally, $\\Phi _{\\mu , n}(t)$ satisfies the following master equation.","$\\frac{d\\Phi _{\\mu ,n} (t)}{dt}&=u_{n-1}\\Phi _{\\mu , n-1}(t)+w_{n}\\Phi _{\\mu , n+1}(t) \\\\&\\hspace{28.45274pt}-(u_n+w_{n-1})\\Phi _{\\mu , n}(t),$ which is identical to Eq.REF , but now the probability of interest is explicitly confined to the ensemble of trajectories remaining on the track.","For an arbitrary value of $k_d$ and for any $N$ , the expressions of $V$ , $D$ , and Eq.REF remain unchanged except that the range of ensemble is specific to the motor trajectories remaining on the track.","Furthermore, the expression of $\\dot{Q}$ , which depends only on $V$ and rate constants, remains identical in the presence of detachment (finite $k_d>0$ ).","Therefore, our formalisms remain valid for motors with a finite processivity."],["3. Mapping the master equation for $N$ -state kinetic model onto Langevin and Fokker-Planck equations","The master equation (Eq.REF ) can be mapped onto a Langevin equation for position $x(t)$ as $\\dot{x}(t)=V+\\sqrt{2D}\\eta (t)$ where for (N=1)-state model $V=d_0(u_1-w_1)$ and $D=d_0^2/2\\times (u_1+w_1)$ as in Eqs.REF and REF , and $P[\\eta (t)]\\propto \\exp {\\left(-\\frac{1}{2}\\int _0^td\\tau \\eta ^2(\\tau )\\right)}$ .","Then, with the transition probability (propagator), $P(x_{t+\\epsilon }|x_t)&=\\left(\\frac{1}{4\\pi D\\epsilon }\\right)^{1/2}e^{-\\frac{\\lbrace x_{t+\\epsilon }-x_t-\\epsilon V\\rbrace ^2}{4D\\epsilon }},$ where $x_{t}\\equiv x(t)$ , and starting from an initial condition, $P[x(0)]=\\delta [x(0)-x_0]$ , it is straightforward to obtain the position of motor at time $t$ : $P[x(t)]&=\\int dx_0\\int dx_{\\epsilon }\\cdots \\int dx_{t-\\epsilon }P(x_t|x_{t-\\epsilon })\\cdots P(x_{2\\epsilon }|x_{\\epsilon })P(x_{\\epsilon }|x_0)P(x_0) \\nonumber \\\\&=\\left(\\frac{1}{4\\pi Dt}\\right)^{1/2}\\exp {\\left(-\\frac{(x(t)-x_0-Vt)^2}{4Dt}\\right)}\\nonumber \\\\&=\\left(\\frac{1}{2\\pi d_0^2(u_1+w_1)t}\\right)^{1/2} \\exp {\\left(-\\frac{[x(t)-x_0-d_0(u_1-w_1)t]^2}{2d_0^2(u_1+w_1)t}\\right)},$ where we plugged $V$ and $D$ from Eqs.REF , REF for ($N$ =1)-state kinetic model in the last line.","Unlike the normal Langevin equation, where the noise strength determined by FDT is associated with an ambient temperature ($\\sim \\sqrt{T}$ ), the noise strength in Eq.REF is solely determined by the forward and backward rate constants, which fundamentally differs from the Brownian motion of a thermally equilibrated colloidal particle in a heat bath.","Next, Fokker-Planck equation follows from Eq.REF , $\\partial _t P(x,t) &= D \\partial _x^2 P(x,t) - V \\partial _x P(x,t)\\nonumber \\\\&=-\\partial _x j(x,t)$ with the probability current being defined as $j (x,t) = - D \\partial _x P(x,t) + V P(x,t).$ Then, mean local velocity $v(x,t)\\equiv j(x,t)/P(x,t)$ is defined $v(x,t)= V - D \\partial _x \\log { P(x,t) }.$ In order to relate this definition of the mean local velocity to heat dissipated from the molecular motor moving along microtubules in a NESS, we consider $\\gamma _{\\text{eff}}$ , an effective friction coefficient, and introduce a nonequilibrium potential $\\phi (x)\\equiv -\\log {P^{ss}(x)}$ [51], [52].","By integrating the both side of Eq.REF in a NESS with respect to the displacement corresponding to a single step, we obtain $\\int _x^{x+d_0} \\gamma _{\\text{eff}} v^{ss}(x)dx &= \\gamma _{\\text{eff}} V d_0 +\\gamma _{\\text{eff}}D(\\phi (x+d_0)-\\phi (x)).$ Following the literature on NESS thermodynamics [52], [53], [51], we endowed each term of Eq.REF with its physical meaning.","(i) housekeeping heat: $Q_{hk}=\\int _x^{x+d_0} \\gamma _{\\text{eff}} v(x)dx,$ (ii) total heat: $Q=\\gamma _{\\text{eff}} V d_0$ and (iii) excess heat: $Q_{ex}=-\\gamma _{\\text{eff}}D(\\phi (x+d_0)-\\phi (x)).$ Eqs.REF , REF , and REF satisfy $Q_{hk}=Q-Q_{ex}.$ and in fact $Q_{ex}=0$ because of the periodic boundary condition implicit to our problem of molecular motor, which leads to $\\phi (x+d_0)=\\phi (x)$ .","Hence, $Q_{hk}=Q=\\gamma _{\\text{eff}}Vd_0.$ Although we introduced the effective friction coefficient $\\gamma _{\\text{eff}}$ in Eq.REF to define the heats produced at nonequilibrium, Eq.REF finally allows us to associate $\\gamma _{\\text{eff}}$ with other physically well-defined quantities.","$\\gamma _{\\text{eff}}=\\frac{Q_{hk}}{Vd_0} = \\frac{k_B T}{d^2_0(j_+-j_-)} \\log { \\left( \\frac{j_+}{j_-} \\right) }.$ Here, note that we for the first time introduced the temperature $T$ , which was discussed neither in the master equation (Eq.REF ) nor in the Langevin equation (Eq.REF ).","Of special note is that $\\gamma _{\\text{eff}}$ does not remain constant, but depends on the steady state flux $j^{ss}=j=j_+-j_-$ .","Similar to the effective diffusion constant of bacterium, $D_{\\text{eff}}$ , discussed in the main text, $\\gamma _{\\text{eff}}$ is defined operationally.","At equilibrium, when the detailed balance (DB) is established ($j_+=j_-=j_0$ ), $\\gamma _{\\text{eff}}^{\\text{DB}}$ approaches to: $\\gamma _{\\text{eff}}^{\\text{DB}} &=\\lim _{j_+\\rightarrow j_-}\\frac{k_B T}{d^2_0(j_+-j_-)} \\log { \\left( \\frac{j_+}{j_-} \\right) }\\rightarrow \\frac{k_B T}{d^2_0j_{0}}.$ In fact, $D_0=k_BT/\\gamma _{\\text{eff}}^{\\text{DB}}=d_0^2j_0$ satisfies the FDT for passive particle at thermal equilibrium, i.e., $k_BT=D_0\\gamma ^{\\text{DB}}_{\\text{eff}}$ .","For ($N$ =1)-state model, $D_0=d_0^2w_1$ , which is identical to Eq.REF ."],["4. Nonequilibrium steady state thermodynamics.","To drive a system out of equilibrium, one has to supply a proper form of energy into the system.","Molecular motors move in one direction because transduction of chemical free energy into conformational change is processed.","Relaxation from a nonequilibrium state is accompanied with heat and entropy production.","In the presence of external nonconservative force (chemical or mechanical force), the system reaches the nonequilibrium steady state.","If one considers a Markov dynamics for microscopic state $i$ , described by the master equation $\\partial _t p_i(t)=-\\sum _j (W_{ij}p_i(t)-W_{ji}p_j(t))$ , the system relaxes to nonequilibrium steady state at long time, establishing time-independent steady state probability $\\lbrace p_i^{ss}\\rbrace $ for each state satisfying the zero flux condition $\\sum _j(W_{ij}p^{ss}_i-W_{ji}p^{ss}_j)=0$ .","A removal of the nonconservative force is led to further relaxation to the equilibrium ensemble, in which the detailed balance (DB) is (locally) established in every pair of the states such that $p_i^{eq}W_{ij}=p_j^{eq}W_{ji}$ for all $i$ and $j$ .","An important feature of the equilibrium, which differentiates itself from NESS, is the condition of DB.","Over the decade, there have been a number of endeavors to better characterize the system out-of-equilibrium [51].","One of them is to define the heat and entropy production in the context of Master equation.","The heat and entropy productions in reference to either steady state or equilibrium are defined to better characterize the process of interest.","The aim is to associate the time dependent probability for state ($\\lbrace p_i(t)\\rbrace $ ) and transition rates between the states $\\lbrace W_{ij}\\rbrace $ with newly defined macroscopic thermodynamic quantities at nonequilibrium [54].","Here, we review NESS thermodynamics formalism developed by Ge and Qian [54].","For nonequilibrium relaxation processes one can consider three relaxation processes: (i) relaxation process of a system far-from-equilibrium (FFE) to a nonequilibrium steady state (NESS); (ii) relaxation process of a system far-from-equilibrium (FFE) to an equilibrium (EQ).","(iii) relaxation process of a system in NESS to an equilibrium (EQ).","To describe these relaxation processes using the probabilities for state, we introduce a phenomenological definition of an internal energy of state $i$ at a steady state by $u^{ss}_i=-k_BT\\log {p_i^{ss}}$ , and at equilibrium by $u^{eq}_i=-k_BT\\log {p_i^{eq}}$ .","Then the following thermodynamic quantities are defined either in reference to NESS or equilibrium.","First, the thermodynamic potentials are defined in reference to the NESS: the total energy $U(t)=\\sum _i^Np_i(t)u^{ss}_i$ ; the total free energy $F(t)=U(t)-TS(t)=k_BT\\sum _i^Np_i(t)\\log {(p_i(t)/p_i^{ss})}$ .","Second, the thermodynamic potentials are defined in reference to the equilibrium: the total energy $U^{eq}(t)=\\sum _i^Np_i(t)u^{eq}_i$ ; the total free energy $F^{eq}(t)=U^{eq}(t)-TS(t)=k_BT\\sum _i^Np_i(t)\\log {(p_i(t)/p_i^{eq})}$ .","In both cases, Gibbs entropy, $S(t)=-k_B\\sum _i^Np_i(t)\\log {p_i(t)}$ , is defined as usual.","Next, the above definitions of generalized thermodynamic potentials, one can define the heat and entropy productions associated with the relaxation processes (i), (ii), (iii).","The diagram in Fig.REF depicts the relaxation processes mentioned here.","Figure: A diagram illustrating the balances between various thermodynamic quantities discussed in the text.","The curvy arrows denote the heat and entropy production from relaxation processes.$-dF(t)/dt$ is the rate of entropy production in the relaxation from FFE to NESS, $\\frac{dF(t)}{dt}\\equiv -\\dot{f}_d&=-T\\sum _{i>j}\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\right]\\log {\\left[\\frac{p_i(t)p_j^{ss}}{p_j(t)p_i^{ss}}\\right]}.$ and $-dU(t)/dt$ is the rate of heat production in the relaxation from FFE to NESS.","$\\frac{dU(t)}{dt}\\equiv -\\dot{Q}_{ex}&=-\\sum _{i>j}(W_{ij}p_i(t)-W_{ji}p_j(t))(u^{ss}_i-u^{ss}_j)\\nonumber \\\\&=T\\sum _{i>j}(W_{ij}p_i-W_{ji}p_j)\\log {\\left(\\frac{p_i^{ss}}{p_j^{ss}}\\right)}.$ Similarly, $-dF^{eq}(t)/dt$ is the rate of entropy production during the relaxation to equilibrium, $\\frac{dF^{eq}(t)}{dt}\\equiv -T\\dot{e}_p&=-T\\sum _{i>j}\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\right]\\log {\\left[\\frac{p_i(t)W_{ij}}{p_j(t)W_{ji}}\\right]}$ and $-dU^{eq}(t)/dt$ is the rate of heat production.","$\\frac{dU^{eq}(t)}{dt}\\equiv -\\dot{h}_d&=-\\sum _{i>j}(W_{ij}p_i(t)-W_{ji}p_j(t))(u^{eq}_i-u^{eq}_j)\\nonumber \\\\&=T\\sum _{i>j}(W_{ij}p_i-W_{ji}p_j)\\log {\\left(\\frac{W_{ij}}{W_{ji}}\\right)}$ where the condition of DB ($p_i^{eq}/p_j^{eq}=W_{ji}/W_{ij}$ ) was used to derive the last line.","Furthermore, the heat production involved with the relaxation from NESS to equilibrium ($\\dot{Q}_{hk}$ ), namely housekeeping heat which is introduced in NESS stochastic thermodynamics from the realization that maintaining NESS requires some energy, is defined by either using $\\dot{Q}_{hk}=(-dF^{eq}(t)/dt)-(-dF(t)/dt)=T\\dot{e}_p-\\dot{f}_d$ or $\\dot{Q}_{hk}=(-dU^{eq}(t)/dt)-(-dU(t)/dt)=\\dot{h}_d-\\dot{Q}_{ex}$ .","Explicit calculations using the representation of thermodynamic potential in terms of master equation lead to $\\dot{Q}_{hk}=T\\sum _{i>j}\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\right]\\log {\\left[\\frac{p^{ss}_iW_{ij}}{p_j^{ss}W_{ji}}\\right]}.$ Lastly, from the definition of Gibbs entropy ($S(t)=-k_B\\sum _ip_i(t)\\log {p_i(t)}$ ), or from the thermodynamic relationships $TdS/dt=dF/dt-dU/dt=dF^{eq}/dt-dU^{eq}/dt$ , it is straightforward to show that $T\\frac{dS}{dt}&=\\frac{dF}{dt}-\\frac{dU}{dt}=\\frac{dF^{eq}}{dt}-\\frac{dU^{eq}}{dt}\\nonumber \\\\&=-T\\sum _{i>j}\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\right]\\log {\\left(\\frac{p_i(t)}{p_j(t)}\\right)}\\nonumber \\\\&=\\dot{h}_d-T\\dot{e}_p.$ Now, with the various heat and entropy production defined from generalized potentials $F(t)$ , $U(t)$ , and $F^{eq}(t)$ , $U^{eq}(t)$ ($\\dot{f}_d$ , $\\dot{Q}_{ex}$ , $T\\dot{e}_p$ , $\\dot{h}_d$ , and $\\dot{Q}_{hk}$ ) we acquire two important balance laws in nonequilibrium thermodynamics: $T\\dot{e}_p&=\\dot{f}_d+\\dot{Q}_{hk}\\nonumber \\\\\\dot{h}_d&=\\dot{Q}_{hk}+\\dot{Q}_{ex}$ Thus, (i) the total entropy production of a system, $T\\dot{e}_p(=-dF^{eq}/dt)$ , is contributed by the free energy dissipation due to the relaxation to NESS, $\\dot{f}_d(=-dF/dt)$ , and the housekeeping heat, $\\dot{Q}_{hk}(=dF/dt-dF^{eq}/dt)$ , that is required to maintain the NESS.","(ii) The total heat production $\\dot{h}_d(=-dU^{eq}/dt)$ of a system is decomposed into $\\dot{Q}_{hk}(=dU/dt-dU^{eq}/dt)$ and the excess heat $\\dot{Q}_{ex}(=-dU/dt)$ .","The diagram in Fig.REF recapitulates the various heat and entropy production terms and their balance.","When the system is already in NESS, then neither the production of entropy nor excess heat is anticipated ($\\dot{f}_d=0$ , $\\dot{Q}_{ex}=0$ ), and hence it follows that the amount of heat, entropy, and housekeeping heat required to sustain NESS are identical ($T\\dot{e}_p=\\dot{Q}_{hk}=\\dot{h}_d$ ).","Figure: Relaxation dynamics of various nonequilibrium thermodynamic quantities from far-from-equilibrium states calculated using (N=2)-state system.The parameters used for the plots are:[ATP] = 1 mM, ff = 1 pN;u 1 0 =1.8s -1 μM -1 u_1^0 = 1.8 ~ s^{-1} \\mu M^{-1}, u 2 =108s -1 u_2 = 108 ~ s^{-1}, w 1 =6.0s -1 w_1 = 6.0 ~ s^{-1}, and w 2 =16s -1 w_2 = 16 ~ s^{-1} at zero load;θ 1 + =0.135,θ 2 + =0.035,θ 1 - =0.080\\theta ^+_1 = 0.135, \\theta ^+_2 = 0.035, \\theta ^-_1 = 0.080, and θ 2 - =0.75\\theta ^-_2 = 0.75.Plots were made using three different initial conditions: A. p 1 (0)=1,p 2 (0)=0p_1(0) = 1, p_2(0) = 0; B. p 1 (0)=0.5,p 2 (0)=0.5p_1(0) = 0.5, p_2(0) = 0.5; and C. p 1 (0)=0,p 2 (0)=1p_1(0)=0, p_2(0) =1.In order to gain a better insight into the energy balance of molecular motor that operates in nonequilibrium steady state, we consider the dynamics of molecular motor systems by means of a cyclic Markov model and relate the essential parameters of the model with NESS thermodynamics.","The thermodynamic quantities associated with nonequilibrium process ($\\dot{f}_d$ , $\\dot{Q}_{hk}$ , $T\\dot{e}_p$ , $\\dot{h}_d$ , $\\dot{Q}_{ex}$ ) can be evaluated explicitly using $(N=2)$ -state Markov model; the time evolution of each state is given by $p_1(t)=p_1^{ss}+(p_1(0)-p_1^{ss})e^{-\\sigma t}$ and $p_2(t)=p_2^{ss}-(p_1(0)-p_1^{ss})e^{-\\sigma t}$ with $p_1^{ss}=(u_2+w_1)/\\sigma $ , $p_2^{ss}=(u_1+w_2)/\\sigma $ , and $\\sigma =u_1+u_2+w_1+w_2$ .","Using the conditions satisfied in 2-state model ($p_{1+2}(t)=p_1(t)$ ) $W_{12}=u_1$ , $W_{21}=w_1$ , $W_{23}=u_2$ , $W_{32}=w_2$ ; otherwise $W_{ij}=0$ , we obtain $\\frac{\\dot{Q}_{hk}(t)}{T}&=[w_1p_2(t)-u_1p_1(t)]\\log {\\left[\\frac{(u_2+w_1)w_1}{(u_1+w_2)u_1}\\right]}+[w_2p_1(t)-u_2p_2(t)]\\log {\\left[\\frac{(u_1+w_2)w_2}{(u_2+w_1)u_2}\\right]}\\nonumber \\\\&= \\frac{u_1u_2-w_1w_2}{u_1+u_2+w_1+w_2}\\log {\\left[\\frac{u_1u_2}{w_1w_2}\\right]}-\\lambda (p_1(0)-p_1^{ss})e^{-\\sigma t}\\nonumber \\\\&\\xrightarrow{}(j_+-j_-)\\log {\\left(\\frac{j_+}{j_-}\\right)}\\ge 0$ where $\\lambda =\\left\\lbrace (u_1+w_1)\\log {\\left[\\frac{(u_2+w_1)w_1}{(u_1+w_2)u_1}\\right]}-(u_2+w_2)\\log {\\left[\\frac{(u_1+w_2)w_2}{(u_2+w_1)u_2}\\right]}\\right\\rbrace $ .","$\\dot{e}_p(t)&=[w_1p_2(t)-u_1p_1(t)]\\log {\\left[\\frac{p_2(t)w_1}{p_1(t)u_1}\\right]}+[w_2p_1(t)-u_2p_2(t)]\\log {\\left[\\frac{p_1(t)w_2}{p_2(t)u_2}\\right]}\\nonumber \\\\&\\xrightarrow{} \\frac{u_1u_2-w_1w_2}{u_1+u_2+w_1+w_2}\\log {\\left[\\frac{u_1u_2}{w_1w_2}\\right]}=(j_+-j_-)\\log {\\left(\\frac{j_+}{j_-}\\right)}\\ge 0$ $\\frac{\\dot{h}_d(t)}{T}&=[w_1p_2(t)-u_1p_1(t)]\\log {\\left[\\frac{w_1}{u_1}\\right]}+[w_2p_1(t)-u_2p_2(t)]\\log {\\left[\\frac{w_2}{u_2}\\right]}\\nonumber \\\\&\\xrightarrow{} \\frac{u_1u_2-w_1w_2}{u_1+u_2+w_1+w_2}\\log {\\left[\\frac{u_1u_2}{w_1w_2}\\right]}=(j_+-j_-)\\log {\\left(\\frac{j_+}{j_-}\\right)}\\ge 0$ $\\frac{\\dot{f}_d(t)}{T}&=[w_1p_2(t)-u_1p_1(t)]\\log {\\left[\\frac{p_2(t)p_1^{ss}}{p_1(t)p_2^{ss}}\\right]}+[w_2p_1(t)-u_2p_2(t)]\\log {\\left[\\frac{p_1(t)p_2^{ss}}{p_2(t)p_1^{ss}}\\right]}\\nonumber \\\\&=\\sigma (p_2(t)p_1^{ss}-p_1(t)p_2^{ss})\\log {\\left[\\frac{p_2(t)p_1^{ss}}{p_1(t)p_2^{ss}}\\right]}\\ge 0\\nonumber \\\\&\\xrightarrow{} 0$ The relaxation time to a steady state (NESS) from an arbitrary state in a far-from-equilibrium is $\\sim \\sigma ^{-1}=(u_1+u_2+w_1+w_2)^{-1}$ , and it is noteworthy that the entropy production inside the system ($T\\dot{e}_p$ ), the total heat production that will be discharged to the surrounding ($\\dot{h}_d$ ), and the housekeeping heat ($\\dot{Q}_{hk}$ ) are all identical at the steady state as $T\\dot{e}_p=\\dot{h}_d=\\dot{Q}_{hk}\\rightarrow (j_+-j_-)\\log {(j_+/j_-)}\\ge 0$ , and $\\dot{f}_d=0$ .","Here, $\\dot{Q}_{ex}$ , the residual of heat (excess heat, $\\dot{Q}_{ex}=\\dot{h}_d-\\dot{Q}_{hk}$ ) for the nonequilibrium process, is zero at the steady state.","Although obtained for 2-state model, the above expression, especially the total heat production (or housekeeping heat) at the steady state, $\\dot{h}_d=T\\dot{e}_p=\\dot{Q}_{hk}=T(j_+-j_-)\\log {j_+/j_-}$ can easily be generalized for the $N$ -state model."],["5. Relationship between motor diffusivity and heat dissipation.","For $(N=2)$ -state model one can obtain an explicit expression that relates $D$ with $\\dot{Q}$ (for the case of $f=0$ ) as follows.","From the expressions of $V$ (Eq.REF ), $D$ (Eq.REF ), and $\\dot{Q}$ , $\\dot{Q}=\\frac{V}{d_0}k_BT\\log {\\frac{u_1u_2}{w_1w_2}}$ Substitution of $u_1=u_1(V)$ from Eq.REF into Eq.REF gives an expression of $\\dot{Q}$ as a function of $V$ : $\\frac{\\dot{Q}}{k_{cat}k_BT}&=v\\log {\\left[\\frac{1+\\kappa v}{1-v}\\right]}\\nonumber \\\\&=\\sum _{n=1}^{\\infty }\\frac{1}{n}\\left(1+(-1)^{n-1}\\kappa ^n\\right)v^{n+1}$ where $v=V/V_{max}$ ($0\\le v\\le 1$ ) and $\\kappa \\equiv \\frac{k_{cat}(k_{cat}+w_1+w_2)}{w_1w_2}$ .","$\\dot{Q}$ diverge as $v\\rightarrow 1$ ; but for small $v\\ll 1$ , $\\dot{Q}/(k_{cat} k_B T) \\sim (\\kappa +1)v^2$ , thus $v\\sim \\dot{Q}^{1/2}$ .","As long as $\\dot{Q}$ is small, one should expect from Eq.REF that $D$ increases with $\\dot{Q}$ as $D=D_0+\\gamma _1\\dot{q}^{1/2}+\\gamma _2\\dot{q}+\\gamma _3\\dot{q}^{3/2}$ where $\\dot{q}\\equiv \\frac{\\dot{Q}}{k_BTk_{cat}}$ , $D_0=\\frac{d_0^2k_{cat}}{\\kappa +1}$ , $\\gamma _1=d_0^2k_{cat}\\frac{\\kappa -1}{2 (\\kappa +1)^{3/2}}$ , $\\gamma _2=-d_0^2k_{cat}\\frac{k_{cat}^2}{w_1w_2}$ , and $\\gamma _3=\\frac{d_0^2k_{cat}}{(\\kappa +1)^{5/2}}\\frac{k_{cat}^2}{w_1w_2}$ .","For arbitrary number of states $N$ , by using Eq.REF and Eq.REF , $\\dot{Q}$ can be written as $\\dot{Q}/k_BT & = \\frac{V}{d_0} \\log {\\frac{ \\prod _{i=1}^N u_i }{ \\prod _{i=1}^N w_i } } \\nonumber \\\\& = \\frac{V^{(D)}}{N} \\log {\\left( 1 + V^{(D)} \\frac{ \\sum _{n=1}^N r_n }{ N }\\frac{ \\prod _{i=1}^N u_i }{ \\prod _{i=1}^N w_i }\\right)} \\nonumber \\\\& = \\frac{V^{(D)}}{N} \\log {\\left( 1 + \\frac{V^{(D)}}{N} \\left( A + B u_1 \\right) \\frac{ 1 }{C }\\right)} \\nonumber \\\\&=\\frac{V^{ (D) }}{N} \\log { \\left( 1 + \\frac{ V^{(D)} }{N} f(V^{ (D) }/N ) \\right) }$ where we used $V^{ (D) } = V N / d_0 $ , $f(V^{ (D) }/N ) =\\frac{A}{C} + B \\left( 1 - B \\frac{V^{ (D) } }{N} \\right)^{-1} \\left( 1 + \\frac{A}{C} \\frac{ V^{ (D) }}{N} \\right)$ , and Eqs.REF , REF , REF .","The definitions of $A, B$ , and $C$ are identical to those in Eq.REF .","Here $V^{ (D) } / N$ corresponds to ATP hydrolysis rate.","For $V^{ (D) } \\rightarrow 0$ , $\\dot{Q} \\rightarrow 0$ is expected.","Also for small $V^{ (D) }$ , $\\dot{Q} \\sim ( \\frac{A}{C} + B ) \\left(\\frac{V^{(D)}}{N} \\right)^2$ .","Thus, $V \\sim \\dot{Q}^{1/2}$ .","Since $D\\sim V+\\mathcal {O}(V^2)$ for small $V$ , it follows that $D$ can be written as a function of $\\dot{q}$ as in the same form as Eq.REF ."],["6. Rate constants, enhancement of diffusion, and conversion efficiency determined from the (N=2)-state kinetic model.","The values in the Table 1 were compiled based on the followings."],["Catalase","In Ref.","[5] $(\\Delta D / D_0)=0.28$ at $ V = 1.7 \\times 10^4$ $s^{-1}$ ; however, $V= 1.7 \\times 10^4$ $s^{-1}$ is not the maximum catalytic rate.","Because $\\Delta D/D_0$ is approximately linear in $V$ , the enhancement of diffusion at the maximal turnover rate $V_{\\text{max}}=u_2= 5.8 \\times 10^4$ is estimated as $(\\Delta D / D_0)_{\\text{obs}} = 0.28 \\times \\frac{5.8 \\times 10^4}{1.7 \\times 10^4} = 0.96 \\approx 1$ ."],["Alkaline phosphatase","Similar to catalase, $(\\Delta D / D_0)_{\\text{obs}} = 0.77 \\times \\frac{1.4 \\times 10^4}{5.5 \\times 10^3} = 2.5 \\approx 3$ ."],["Estimate of $(\\Delta D/D_0)_{\\text{max}}$","Freely diffusing enzymes effectively perform no work on the surrounding environment; thus $-\\Delta \\mu _{\\text{eff}}=Q$ with $W=0$ , which leads to $e^{Q/k_B T}= u_1 u_2/w_1 w_2$ .","By assuming that the substrate concentration $[S] \\sim K_M=(u_2 +w_1) / u_1^o$ , we get $\\begin{aligned}e^{Q/k_B T}&= \\frac{ u_1 u_2 }{w_1 w_2}\\ \\sim \\frac{ u_1^o K_M u_2 }{w_1 w_2}= \\frac{ (u_2 + w_1) u_2 }{w_1 w_2} \\\\&\\ge \\frac{ u_2^2 }{w_1 w_2}= \\frac{ k_{cat}^2 }{w_1 w_2}.\\end{aligned}$ This relation allows us to estimate the upper bound of $(\\Delta D/D_0)_{\\text{max}}$ as follows when $u_2 \\gg w_1, w_2$ is satisfied.","$\\left(\\frac{\\Delta D}{D_0}\\right)_{\\text{max}}\\approx \\frac{ k_{cat}^2 }{2 w_1 w_2} \\le \\frac{1}{2}e^{Q/k_BT}.$ Alternatively, $u_1$ and $w_2$ of enzymes can be estimated by assuming (i) that the reaction is diffusion limited, $u_1^o = 10^8 s^{-1} M^{-1}$ , and (ii) that the substrate concentration $[S]$ is similar to Michaelis-Menten constant $K_M$ ($[S]\\sim K_M$ ).","The two conditions $u_1 = u_1^o [S] \\sim K_M \\times 10^8$ ($s^{-1}$ ) and $K_M(=(u_2+w_1)/u_1^o)$ , and $Q$ (heat measured by the calorimeter in ref.","[5]), $u_2$ , $K_M$ which are available in ref.","[5], provide all the rate constants including $w_1=u_1^oK_M-u_2$ and $w_2=\\frac{u_1u_2}{w_1e^{Q/k_BT}}$ , allowing us to calculate $\\left(\\frac{\\Delta D}{D_0}\\right)_{\\text{max}}=\\frac{u_2^2+(w_1+w_2)u_2-w_1w_2}{2w_1w_2}$ .","Figure: Analysis of experimental data, extracted from Ref.",", but using (NN=2)-state model.The solid lines are the fits to the dataA.","VV vs ATP at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).B.","VV vs load at [ATP] = 5 μ\\mu M.C.","VV vs load at [ATP] = 2 mM.D.","Stall force as a function of [ATP], measured by `Position clamp' (red square) or `Fixed trap' (blue circle) methods.E.","DD vs ATP at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).DD was estimated from r=2D/Vd 0 r=2D/Vd_0.","F. DD vs load at [ATP] = 2 mM.G-I.","Motor diffusivity (DD) as a function of mean velocity (VV) for kinesin-1.","(VV,DD) measured at varying [ATP] (= 0 – 2 mM) and a fixed (G) f=f=1.05 pN, (H) 3.59 pN, and (I) 5.63 pN .The black dashed lines in G and H are the fits using Eq..The solid lines in magenta in G-I are plotted using the (NN=2)-kinetic model."]],"string":"[\n [\n \"Quantifying the Heat Dissipation from a Molecular Motor's Transport\\n Properties in Nonequilibrium Steady States\"\n ],\n [\n \"Abstract Theoretical analysis, which maps single molecule time trajectories of a molecular motor onto unicyclic Markov processes, allows us to evaluate the heat dissipated from the motor and to elucidate its dependence on the mean velocity and diffusivity.\",\n \"Unlike passive Brownian particles in equilibrium, the velocity and diffusion constant of molecular motors are closely inter-related to each other.\",\n \"In particular, our study makes it clear that the increase of diffusivity with the heat production is a natural outcome of active particles, which is reminiscent of the recent experimental premise that the diffusion of an exothermic enzyme is enhanced by the heat released from its own catalytic turnover.\",\n \"Compared with freely diffusing exothermic enzymes, kinesin-1 whose dynamics is confined on one-dimensional tracks is highly efficient in transforming conformational fluctuations into a locally directed motion, thus displaying a significantly higher enhancement in diffusivity with its turnover rate.\",\n \"Putting molecular motors and freely diffusing enzymes on an equal footing, our study offers thermodynamic basis to understand the heat enhanced self-diffusion of exothermic enzymes.\"\n ],\n [\n \"Aknowledgement\",\n \"We thank Bae-Yeun Ha, Hyunggyu Park, Dave Thirumalai, and Anatoly Kolomeisky for helpful comments and illuminating discussions.\",\n \"We acknowledge the Center for Advanced Computation in KIAS for providing computing resources.\",\n \"Figure: Analysis of experimental data, digitized from Ref.\",\n \", using (NN=4)-state cyclic model.The solid lines are the fits to the dataA.\",\n \"VV vs [ATP] at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).B.\",\n \"VV vs ff at [ATP] = 5 μ\\\\mu M.C.\",\n \"VV vs ff at [ATP] = 2 mM.D.\",\n \"Stall force as a function of [ATP], measured by `Position clamp' (red square) or `Fixed trap' (blue circle) methods.E.\",\n \"DD vs [ATP] at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).DD was estimated from r=2D/Vd 0 r=2D/Vd_0.\",\n \"F. DD vs ff at [ATP] = 2 mM.Table: Parameters determined from the fit using (N=4)-state model.\",\n \"The unit of {u n }\\\\lbrace u_n\\\\rbrace and {w n }\\\\lbrace w_n\\\\rbrace is s -1 s^{-1} except for u 1 o u_1^o ([u 1 o ]=μM -1 s -1 [u_1^o]={\\\\mu M}^{-1} s^{-1}).Supplementary Information\"\n ],\n [\n \"1. Derivation of the third degree polynomial dependence of $D$ on {{formula:bfe6300b-f497-4028-bece-5a1753aeadd7}} .\",\n \"Here, we show a polynomial dependence of $D$ on $V$ using a few specific examples of the $N$ -state periodic reaction model [19] whose reaction scheme is demonstrated in Fig.\",\n \"REF .\",\n \"When $N$ =1, the master equation to solve is: $\\\\dot{\\\\pi }_\\\\mu (t)=u_1\\\\pi _{\\\\mu -1}(t)+w_1\\\\pi _{\\\\mu +1}(t)-(u_1+w_1)\\\\pi _\\\\mu (t),$ where $\\\\pi _\\\\mu (t)$ is the probability of motor being in the $\\\\mu $ -th reaction cycle at time $t$ .\",\n \"Using generating function $F(z,t)=\\\\sum _{\\\\mu =-\\\\infty }^{\\\\infty }z^\\\\mu \\\\pi _\\\\mu (t)$ with $\\\\pi _\\\\mu (0)=\\\\delta _{\\\\mu ,0}$ [49], the master equation is written in terms of $F(z,t)$ as $\\\\partial _tF(z,t)&=\\\\left(u_1z+\\\\frac{w_1}{z}-(u_1+w_1)\\\\right)F(z,t)\\\\nonumber \\\\\\\\F(z,t)&=e^{\\\\left(u_1z+\\\\frac{w_1}{z}-(u_1+w_1)\\\\right)t}.$ Now, it is straightforward to obtain the mean velocity ($V$ ) and diffusion constant ($D$ ) using $\\\\partial _z\\\\log {F(z,t)}|_{z=1}=\\\\langle \\\\mu (t)\\\\rangle $ and $\\\\partial ^2_z\\\\log {F(z,t)}|_{z=1}=\\\\langle \\\\mu ^2(t)\\\\rangle -\\\\langle \\\\mu (t)\\\\rangle ^2-\\\\langle \\\\mu (t)\\\\rangle $ , where $\\\\mu (t)$ is the number of steps taken by the molecular motor until time $t$ .\",\n \"$V\\\\equiv \\\\lim _{t\\\\rightarrow \\\\infty }\\\\frac{d_0\\\\langle \\\\mu (t)\\\\rangle }{t}= d_0(u_1-w_1)$ and $D\\\\equiv \\\\lim _{t\\\\rightarrow \\\\infty }\\\\frac{d^2_0(\\\\langle \\\\mu ^2(t)\\\\rangle -\\\\langle \\\\mu (t)\\\\rangle ^2)}{2t} = \\\\frac{d_0^2(u_1+w_1)}{2}.$ Provided that only $u_1$ changes (for example by increasing ATP concentration) while $w_1$ remains constant.\",\n \"elimination of $u_1$ from $V(u_1)$ and $D(u_1)$ relates $D$ to $V$ as $ D(V) = D_0+\\\\frac{d_0}{2}V$ where $D_0\\\\equiv d_0^2w_1,$ showing that for ($N$ =1)-state kinetic model, $D$ is linear in $V$ .\"\n ],\n [\n \"($N$ =2)-state kinetic model\",\n \"For the ($N$ =2)-kinetic model [20], $V = d_0\\\\frac{u_1 u_2 - w_1 w_2}{u_1 + u_2 + w_1 + w_2}$ and $D=\\\\frac{d_0^2}{2}\\\\left[\\\\frac{u_1u_2}{w_1w_2}+1-2\\\\left(\\\\frac{u_1u_2}{w_1w_2}-1\\\\right)^2\\\\frac{w_1w_2}{\\\\sigma ^2}\\\\right]\\\\frac{w_1w_2}{\\\\sigma }$ where $\\\\sigma =u_1+u_2+w_1+w_2$ .\",\n \"Then, $D=D(V)$ is obtained by eliminating $u_1$ between Eq.REF and Eq.REF : $D(v)/d_0^2&=\\\\left(\\\\frac{u_2}{\\\\kappa +1}\\\\right)+\\\\frac{1}{2}\\\\left(\\\\frac{\\\\kappa -1}{\\\\kappa +1}\\\\right)u_2v-\\\\left(\\\\frac{u_2}{\\\\kappa +1}\\\\right)\\\\frac{u_2^2}{w_1w_2}v^2+\\\\left(\\\\frac{u_2}{\\\\kappa +1}\\\\right)\\\\frac{u_2^2}{w_1w_2}v^3$ where $\\\\kappa \\\\equiv \\\\frac{u_2(u_2+w_1+w_2)}{w_1w_2}$ , and $v\\\\equiv V/V_{max}$ ($0\\\\le v\\\\le 1$ ) with $V_{max}=d_0u_2$ .\",\n \"Eq.REF confirms that $D$ is a third order polynomial in $V$ .\",\n \"Incidentally, the ($N$ =2)-kinetic model is reduced to the Michaelis-Menten equation by setting $u_1=u_1^o[S]$ and $w_2=0$ .\",\n \"$V= d_0\\\\frac{u_1 u_2}{u_1 + u_2 + w_1} = \\\\frac{V_{max}[S]}{ K_M + [S] }$ where $K_M = (u_2 + w_1)/u_1^o$ , and $D(v)=D_{max}v\\\\left[1-2\\\\phi v+2\\\\phi v^2\\\\right]$ with $\\\\phi \\\\equiv \\\\frac{k_{cat}}{u_1^oK_M}$ .\",\n \"Note that $D\\\\le D_{max}=d_0V_{max}/2$ and that for $D(v)$ to be positive for all the range of $v$ , the parameter $\\\\phi $ should be in a rather narrow range of $0\\\\le \\\\phi \\\\le 2$ .\"\n ],\n [\n \"General case: $N$ -state kinetic model\",\n \"The above two examples of one-dimensional hopping model was extended to the $N$ -state kinetic model by Derrida [19].\",\n \"He obtained exact expressions for the mean velocity ($V^{(D)}$ ) and diffusion constant ($D^{(D)}$ ), where the superscript $(D)$ refers to Derrida's, in terms of the rate constants $\\\\lbrace u_n\\\\rbrace $ and $\\\\lbrace w_n\\\\rbrace $ .\",\n \"Derrida's expression for $V^{(D)}$ and $D^{(D)}$ are related to $V$ and $D$ as $V\\\\equiv (d_0/N)V^{(D)}$ and $D\\\\equiv (d_0^2/N^2)D^{(D)}$ .\",\n \"$V^{(D)} = \\\\frac{N}{\\\\sum _{n=1}^{N} r_n}\\\\left(1- \\\\frac{\\\\prod _{n=1}^{N} w_n}{\\\\prod _{n=1}^{N} u_n }\\\\right)$ and $D^{(D)}&=\\\\frac{1}{\\\\left(\\\\sum _{n=1}^Nr_n\\\\right)^2}\\\\left[V^{(D)}\\\\sum _{n=1}^Nq_n\\\\sum _{i=1}^Nir_{n+i}+N\\\\sum _{n=1}^Nu_nq_nr_n\\\\right]-V^{(D)}\\\\frac{N+2}{2}$ where $r_n = \\\\frac{1}{u_n} \\\\left[1 + \\\\sum _{i=1}^{N-1} \\\\prod _{j=1}^{i} \\\\frac{w_{n+j-1}}{u_{n+j}}\\\\right]$ , and $q_n=\\\\frac{1}{u_n}\\\\left[1+\\\\sum _{i=1}^{N-1}\\\\prod _{j=1}^i\\\\frac{w_{n-j}}{u_{n-j}}\\\\right]$ with periodic boundary conditions $u_{n+N} = u_n$ , and $w_{n+N} = w_n$ .\",\n \"$D^{(D)} = D^{(D)}(V^{(D)})$ is obtained by eliminating $u_1$ between Eq.REF and Eq.REF .\",\n \"For that, we first express various terms in Eq.REF in terms of $u_1$ : $ \\\\sum _{n=1}^Nr_n&=\\\\frac{A}{u_1}+B,\\\\nonumber \\\\\\\\\\\\sum _{n=1}^Nq_n\\\\sum _{i=1}^Nir_{n+i}&=\\\\frac{\\\\alpha }{u_1^2}+\\\\frac{\\\\beta }{u_1}+\\\\gamma ,\\\\nonumber \\\\\\\\\\\\sum _{n=1}^Nu_nq_nr_n&=\\\\frac{\\\\xi }{u_1^2}+\\\\frac{\\\\eta }{u_1}+\\\\zeta .$ where A, $B\\\\left(=\\\\sum _{n=2}^N\\\\frac{1}{u_n}\\\\left[1+\\\\sum _{i=1}^{N-n}\\\\prod _{j=1}^i\\\\frac{w_{n+j-1}}{u_{n+j}}\\\\right]\\\\right)$ , $\\\\alpha $ , $\\\\beta $ , $\\\\gamma $ , $\\\\xi $ , $\\\\eta $ , and $\\\\zeta $ are all positive constants independent of $u_1$ .\",\n \"Next, $\\\\sum _{n=1}^Nr_n$ in Eq.REF substituted to Eq.REF gives $ V^{(D)}=N\\\\frac{1-C/u_1}{A/u_1+B},$ where $C\\\\left(= \\\\prod _{n=1}^Nw_n/\\\\prod _{n=2}^Nu_n \\\\right)$ , and $u_1$ is expressed in terms of $V^{(D)}$ $\\\\frac{1}{u_1}=\\\\frac{1-B\\\\frac{V^{(D)}}{N}}{C+A\\\\frac{V^{(D)}}{N}}.$ Finally, with Eqs.REF and REF , we show that $D^{(D)}$ (Eq.REF ) can be expressed as a third degree polynomial in $V^{(D)}$ $D^{(D)}&=V^{(D)}\\\\frac{\\\\left[\\\\alpha (1-B\\\\frac{V^{(D)}}{N})^2+\\\\beta (1-B\\\\frac{V^{(D)}}{N})(A\\\\frac{V^{(D)}}{N}+C)+\\\\gamma (A\\\\frac{V^{(D)}}{N}+C)^2\\\\right]}{(A+BC)^2}\\\\nonumber \\\\\\\\&\\\\hspace{14.22636pt}+N\\\\frac{\\\\xi (1-B\\\\frac{V^{(D)}}{N})^2+\\\\eta (1-B\\\\frac{V^{(D)}}{N})(A\\\\frac{V^{(D)}}{N}+C)+\\\\zeta (A\\\\frac{V^{(D)}}{N}+C)^2}{(A+BC)^2}-V^{(D)}\\\\frac{N+2}{2}\\\\nonumber \\\\\\\\&=z_0+z_1V^{(D)}+z_2(V^{(D)})^2+z_3(V^{(D)})^3\\\\nonumber \\\\\\\\D&=\\\\alpha _0+\\\\alpha _1V+\\\\alpha _2V^2+\\\\alpha _3V^3.$ with $\\\\alpha _i=(d_0/N)^{2-i}z_i$ , and $z_0&=\\\\frac{(\\\\xi +\\\\eta C+\\\\zeta C^2)}{(A+BC)^2}\\\\nonumber \\\\\\\\z_1&=\\\\left[\\\\frac{(\\\\alpha +\\\\beta C+\\\\gamma C^2)+(-2\\\\xi B+\\\\eta (A-BC)+2\\\\zeta AC)}{(A+BC)^2}-\\\\frac{N+2}{2}\\\\right]\\\\nonumber \\\\\\\\z_2&=\\\\frac{(-2\\\\alpha B+\\\\beta (A-BC) +2\\\\gamma AC)+(\\\\xi B^2-\\\\eta AB +\\\\zeta A^2)}{(A+BC)^2}\\\\nonumber \\\\\\\\z_3&=\\\\frac{(\\\\alpha B^2-\\\\beta AB +\\\\gamma A^2)}{(A+BC)^2}.\",\n \"\\\\nonumber $\"\n ],\n [\n \"Alternative derivation of $D(V)$\",\n \"In addition to Derrida's result [19], the sign of $\\\\alpha _i$ can be determined by deriving the relation between $V^{ (D) }$ and $D^{ (D) }$ using the result in ref.\",\n \"[50].\",\n \"From Eq.\",\n \"(23) in ref.\",\n \"[50], $ \\\\begin{aligned}V^{ (D) } = - i \\\\frac{c^{\\\\prime }_0}{c_1}\\\\end{aligned}$ where $c^{\\\\prime }_0 = i N ( \\\\prod _{n=1}^{N} u_n - \\\\prod _{n=1}^{N} w_n )$ and $c_1 = c_1( \\\\lbrace u_n\\\\rbrace , \\\\lbrace w_n\\\\rbrace )$ .\",\n \"By combining two expressions of $V^{ (D) }$ , Eq.REF and Eq.REF , we get $c_1& = \\\\prod _{n=1}^{N} u_n\\\\times \\\\sum _{m=1}^{N-1} \\\\left[ \\\\frac{1}{u_m} \\\\left( 1 + \\\\sum _{i=1}^{N-1} \\\\prod _{j=1}^{i} \\\\frac{w_{m+j}}{u_{m+j}}\\\\right) \\\\right] \\\\nonumber \\\\\\\\& = u_1 \\\\prod _{n=2}^{N} u_n \\\\left[\\\\frac{1}{u_1} \\\\left( 1 + \\\\sum _{i=1}^{N-1} \\\\prod _{j=1}^{i} \\\\frac{w_{1+j}}{u_{1+j}} \\\\right)+ \\\\sum _{m=2}^{N-1} \\\\frac{1}{u_m} \\\\left( 1 + \\\\sum _{i=1}^{N-1} \\\\prod _{j=1}^{i} \\\\frac{w_{m+j}}{u_{m+j}} \\\\right)\\\\right]\\\\nonumber \\\\\\\\& = \\\\mathcal {A} u_1 + \\\\mathcal {B}$ where $\\\\mathcal {A}$ and $\\\\mathcal {B}$ are constants depending on ($u_2,\\\\ldots ,u_N$ ) and ($w_1,w_2,\\\\ldots w_N$ ).\",\n \"Eq.\",\n \"(REF ) substituted to Eq.\",\n \"(REF ) gives $\\\\begin{aligned}V^{ (D) } = N \\\\frac{u_1 \\\\prod _{n=2}^{N} u_n - \\\\prod _{n=1}^{N} w_n}{\\\\mathcal {A}u_1 + \\\\mathcal {B}}\\\\end{aligned}$ and hence $u_1$ can be written as $\\\\begin{aligned}u_1 = \\\\frac{\\\\mathcal {B} V^{ (D) } + N \\\\prod _{n=1}^N w_n}{N \\\\prod _{n=2}^{N} u_n - \\\\mathcal {A}V^{ (D) }}.\\\\end{aligned}$ From Eq.REF and Eq.REF , $\\\\mathcal {A} = B \\\\prod _{n=2}^N u_n$ and $\\\\mathcal {B} = A \\\\prod _{n=2}^N u_n$ where $A$ and $B$ are the same constants used in Eq.REF .\",\n \"Now using Eq.\",\n \"(REF ), Eq.\",\n \"(REF ), and general expression of $D^{ (D) }$ from Eq.\",\n \"(24) of ref.\",\n \"[50], we have $\\\\begin{aligned}D^{ (D) } & = \\\\frac{c_0^{\\\\prime \\\\prime } - 2 c_2 (V^{ (D) })^2 }{2 c_1 } \\\\\\\\&=\\\\frac{c_0^{\\\\prime \\\\prime } - 2 c_2 (V^{ (D) })^2 }{2 (\\\\frac{B}{\\\\prod _{n=2}^N u_n} u_1 + \\\\frac{A}{\\\\prod _{n=2}^N u_n}) } \\\\\\\\& = \\\\frac{( c_0^{\\\\prime \\\\prime } - 2 c_2 (V^{ (D) })^2 ) (\\\\prod _{n=2}^{N} u_n (N- B V^{(D)}) )}{2N( A + C B)}\\\\end{aligned}$ where $c_0^{\\\\prime \\\\prime } = N^2 ( u_1 \\\\prod _{n=2}^{N} u_n + \\\\prod _{n=1}^{N} w_n ) = N^2 (\\\\prod _{n=2}^N u_n) (u_1 + C)$ , and $c_2 = \\\\beta _1 u_1 + \\\\beta _2$ where $\\\\beta _1$ and $\\\\beta _2$ are positive constants depending on ($u_2, u_3, \\\\cdots , u_N$ ) and ($w_1, w_2, \\\\cdots , w_N$ ) (see Eq.\",\n \"(53, 54) of ref.\",\n \"[50]).\",\n \"Since $u_1 (N - B V^{ (D) }) = NC + AV^{ (D) }$ (Eq.REF ), we get $D^{ (D) } & = \\\\frac{1}{2} \\\\frac{( c_0^{^{\\\\prime \\\\prime }} - 2 c_2 (V^{ (D) })^2 ) ( (N- B V^{(D)}) )\\\\prod _{n=2}^{N} u_n}{N A + N C B} \\\\nonumber \\\\\\\\& = \\\\frac{1}{2} \\\\frac{( N^2 (\\\\prod _{n=2}^N u_2) (u_1 + C) - 2 (\\\\beta _1 u_1 + \\\\beta _2) (V^{ (D) })^2 ) ( (N- B V^{(D)}) )\\\\prod _{n=2}^{N} u_n}{N A + N C B} \\\\nonumber \\\\\\\\& = \\\\frac{\\\\prod _{n=2}^N u_n}{2} \\\\frac{ \\\\left( N^2 (\\\\prod _{n=2}^N u_2) ( NC + A V^{(D)} + C(N-B V^{ (D) } ) ) - 2 ( \\\\beta _1 (N C + A V^{(D)}) + \\\\beta _2 ( N- B V^{(D)} ) ) (V^{ (D) })^2 \\\\right)}{N A + N C B} \\\\nonumber \\\\\\\\&= z_0 + z_1 V^{ (D) } + z_2 (V^{ (D) })^2 + z_3 (V^{ (D) })^3$ where $z_0&=\\\\frac{N^2 (\\\\prod _{n=2}^N u_n )^2 C}{ A + BC }>0\\\\nonumber \\\\\\\\z_1&=\\\\frac{N (\\\\prod _{n=2}^N u_n )^2 (A- BC)}{2 (A + BC)}\\\\nonumber \\\\\\\\z_2&=- \\\\frac{(\\\\prod _{n=2}^N u_n )( \\\\beta _1 C + \\\\beta _2 )}{A + BC}<0\\\\nonumber \\\\\\\\z_3&= \\\\frac{(\\\\prod _{n=2}^N u_n )( - \\\\beta _1 A + \\\\beta _2 B)}{N(A + BC)}.\\\\nonumber $ It is obvious that $z_0 >0$ and $z_2 < 0$ since $A$ , $B$ , $C$ , $\\\\beta _1$ , and $\\\\beta _2$ are all positive constants.\"\n ],\n [\n \"2. The 1D hopping model with a finite processivity\",\n \"Because of a probability of being dissociated from microtubules, kinesin motors display a finite processivity.\",\n \"However, since the mean velocity and diffusion constant are calculated from the trajectories that remain on the track, the expressions of $V$ and $D$ in terms of the rate constants are unchanged.\",\n \"To make this point mathematically more explicit, we consider the master equation assuming a constant dissociation rate $k_d$ from each chemical state.\",\n \"$\\\\frac{d P_{\\\\mu ,n} (t)}{d t}&=u_{n-1} P_{\\\\mu , n-1}(t)+w_{n}P_{\\\\mu , n+1}(t)\\\\nonumber \\\\\\\\&-(u_n+w_{n-1}+k_d)P_{\\\\mu , n}(t)$ where $P_{\\\\mu , n}(t)$ is the probability of being in the $n$ -th chemical state at the $\\\\mu $ -th reaction cycle.\",\n \"The probability of the motor remaining on the track (survival probability of motor) is $ S (t) \\\\equiv \\\\sum _{\\\\mu =\\\\infty }^{\\\\infty } \\\\sum _{n=1}^{N} P_{\\\\mu , n}(t) = e^{-k_d t}.$ The expectation value of an observable, which can be used to calculate $\\\\langle x(t)\\\\rangle $ or $\\\\langle x^2(t)\\\\rangle $ , is expressed as $\\\\langle A(t)\\\\rangle = \\\\sum _{\\\\mu =-\\\\infty }^{\\\\infty } \\\\sum _{n=1}^{N} \\\\Phi _{\\\\mu ,n}(t) A(\\\\mu (t))$ with a probability density function renormalized with respect to the survival probability $ \\\\Phi _{\\\\mu , n}(t) \\\\equiv \\\\frac{P_{\\\\mu ,n}(t)}{S(t)}=P_{\\\\mu ,n}(t)e^{k_dt}.$ Incidentally, $\\\\Phi _{\\\\mu , n}(t)$ satisfies the following master equation.\",\n \"$\\\\frac{d\\\\Phi _{\\\\mu ,n} (t)}{dt}&=u_{n-1}\\\\Phi _{\\\\mu , n-1}(t)+w_{n}\\\\Phi _{\\\\mu , n+1}(t) \\\\\\\\&\\\\hspace{28.45274pt}-(u_n+w_{n-1})\\\\Phi _{\\\\mu , n}(t),$ which is identical to Eq.REF , but now the probability of interest is explicitly confined to the ensemble of trajectories remaining on the track.\",\n \"For an arbitrary value of $k_d$ and for any $N$ , the expressions of $V$ , $D$ , and Eq.REF remain unchanged except that the range of ensemble is specific to the motor trajectories remaining on the track.\",\n \"Furthermore, the expression of $\\\\dot{Q}$ , which depends only on $V$ and rate constants, remains identical in the presence of detachment (finite $k_d>0$ ).\",\n \"Therefore, our formalisms remain valid for motors with a finite processivity.\"\n ],\n [\n \"3. Mapping the master equation for $N$ -state kinetic model onto Langevin and Fokker-Planck equations\",\n \"The master equation (Eq.REF ) can be mapped onto a Langevin equation for position $x(t)$ as $\\\\dot{x}(t)=V+\\\\sqrt{2D}\\\\eta (t)$ where for (N=1)-state model $V=d_0(u_1-w_1)$ and $D=d_0^2/2\\\\times (u_1+w_1)$ as in Eqs.REF and REF , and $P[\\\\eta (t)]\\\\propto \\\\exp {\\\\left(-\\\\frac{1}{2}\\\\int _0^td\\\\tau \\\\eta ^2(\\\\tau )\\\\right)}$ .\",\n \"Then, with the transition probability (propagator), $P(x_{t+\\\\epsilon }|x_t)&=\\\\left(\\\\frac{1}{4\\\\pi D\\\\epsilon }\\\\right)^{1/2}e^{-\\\\frac{\\\\lbrace x_{t+\\\\epsilon }-x_t-\\\\epsilon V\\\\rbrace ^2}{4D\\\\epsilon }},$ where $x_{t}\\\\equiv x(t)$ , and starting from an initial condition, $P[x(0)]=\\\\delta [x(0)-x_0]$ , it is straightforward to obtain the position of motor at time $t$ : $P[x(t)]&=\\\\int dx_0\\\\int dx_{\\\\epsilon }\\\\cdots \\\\int dx_{t-\\\\epsilon }P(x_t|x_{t-\\\\epsilon })\\\\cdots P(x_{2\\\\epsilon }|x_{\\\\epsilon })P(x_{\\\\epsilon }|x_0)P(x_0) \\\\nonumber \\\\\\\\&=\\\\left(\\\\frac{1}{4\\\\pi Dt}\\\\right)^{1/2}\\\\exp {\\\\left(-\\\\frac{(x(t)-x_0-Vt)^2}{4Dt}\\\\right)}\\\\nonumber \\\\\\\\&=\\\\left(\\\\frac{1}{2\\\\pi d_0^2(u_1+w_1)t}\\\\right)^{1/2} \\\\exp {\\\\left(-\\\\frac{[x(t)-x_0-d_0(u_1-w_1)t]^2}{2d_0^2(u_1+w_1)t}\\\\right)},$ where we plugged $V$ and $D$ from Eqs.REF , REF for ($N$ =1)-state kinetic model in the last line.\",\n \"Unlike the normal Langevin equation, where the noise strength determined by FDT is associated with an ambient temperature ($\\\\sim \\\\sqrt{T}$ ), the noise strength in Eq.REF is solely determined by the forward and backward rate constants, which fundamentally differs from the Brownian motion of a thermally equilibrated colloidal particle in a heat bath.\",\n \"Next, Fokker-Planck equation follows from Eq.REF , $\\\\partial _t P(x,t) &= D \\\\partial _x^2 P(x,t) - V \\\\partial _x P(x,t)\\\\nonumber \\\\\\\\&=-\\\\partial _x j(x,t)$ with the probability current being defined as $j (x,t) = - D \\\\partial _x P(x,t) + V P(x,t).$ Then, mean local velocity $v(x,t)\\\\equiv j(x,t)/P(x,t)$ is defined $v(x,t)= V - D \\\\partial _x \\\\log { P(x,t) }.$ In order to relate this definition of the mean local velocity to heat dissipated from the molecular motor moving along microtubules in a NESS, we consider $\\\\gamma _{\\\\text{eff}}$ , an effective friction coefficient, and introduce a nonequilibrium potential $\\\\phi (x)\\\\equiv -\\\\log {P^{ss}(x)}$ [51], [52].\",\n \"By integrating the both side of Eq.REF in a NESS with respect to the displacement corresponding to a single step, we obtain $\\\\int _x^{x+d_0} \\\\gamma _{\\\\text{eff}} v^{ss}(x)dx &= \\\\gamma _{\\\\text{eff}} V d_0 +\\\\gamma _{\\\\text{eff}}D(\\\\phi (x+d_0)-\\\\phi (x)).$ Following the literature on NESS thermodynamics [52], [53], [51], we endowed each term of Eq.REF with its physical meaning.\",\n \"(i) housekeeping heat: $Q_{hk}=\\\\int _x^{x+d_0} \\\\gamma _{\\\\text{eff}} v(x)dx,$ (ii) total heat: $Q=\\\\gamma _{\\\\text{eff}} V d_0$ and (iii) excess heat: $Q_{ex}=-\\\\gamma _{\\\\text{eff}}D(\\\\phi (x+d_0)-\\\\phi (x)).$ Eqs.REF , REF , and REF satisfy $Q_{hk}=Q-Q_{ex}.$ and in fact $Q_{ex}=0$ because of the periodic boundary condition implicit to our problem of molecular motor, which leads to $\\\\phi (x+d_0)=\\\\phi (x)$ .\",\n \"Hence, $Q_{hk}=Q=\\\\gamma _{\\\\text{eff}}Vd_0.$ Although we introduced the effective friction coefficient $\\\\gamma _{\\\\text{eff}}$ in Eq.REF to define the heats produced at nonequilibrium, Eq.REF finally allows us to associate $\\\\gamma _{\\\\text{eff}}$ with other physically well-defined quantities.\",\n \"$\\\\gamma _{\\\\text{eff}}=\\\\frac{Q_{hk}}{Vd_0} = \\\\frac{k_B T}{d^2_0(j_+-j_-)} \\\\log { \\\\left( \\\\frac{j_+}{j_-} \\\\right) }.$ Here, note that we for the first time introduced the temperature $T$ , which was discussed neither in the master equation (Eq.REF ) nor in the Langevin equation (Eq.REF ).\",\n \"Of special note is that $\\\\gamma _{\\\\text{eff}}$ does not remain constant, but depends on the steady state flux $j^{ss}=j=j_+-j_-$ .\",\n \"Similar to the effective diffusion constant of bacterium, $D_{\\\\text{eff}}$ , discussed in the main text, $\\\\gamma _{\\\\text{eff}}$ is defined operationally.\",\n \"At equilibrium, when the detailed balance (DB) is established ($j_+=j_-=j_0$ ), $\\\\gamma _{\\\\text{eff}}^{\\\\text{DB}}$ approaches to: $\\\\gamma _{\\\\text{eff}}^{\\\\text{DB}} &=\\\\lim _{j_+\\\\rightarrow j_-}\\\\frac{k_B T}{d^2_0(j_+-j_-)} \\\\log { \\\\left( \\\\frac{j_+}{j_-} \\\\right) }\\\\rightarrow \\\\frac{k_B T}{d^2_0j_{0}}.$ In fact, $D_0=k_BT/\\\\gamma _{\\\\text{eff}}^{\\\\text{DB}}=d_0^2j_0$ satisfies the FDT for passive particle at thermal equilibrium, i.e., $k_BT=D_0\\\\gamma ^{\\\\text{DB}}_{\\\\text{eff}}$ .\",\n \"For ($N$ =1)-state model, $D_0=d_0^2w_1$ , which is identical to Eq.REF .\"\n ],\n [\n \"4. Nonequilibrium steady state thermodynamics.\",\n \"To drive a system out of equilibrium, one has to supply a proper form of energy into the system.\",\n \"Molecular motors move in one direction because transduction of chemical free energy into conformational change is processed.\",\n \"Relaxation from a nonequilibrium state is accompanied with heat and entropy production.\",\n \"In the presence of external nonconservative force (chemical or mechanical force), the system reaches the nonequilibrium steady state.\",\n \"If one considers a Markov dynamics for microscopic state $i$ , described by the master equation $\\\\partial _t p_i(t)=-\\\\sum _j (W_{ij}p_i(t)-W_{ji}p_j(t))$ , the system relaxes to nonequilibrium steady state at long time, establishing time-independent steady state probability $\\\\lbrace p_i^{ss}\\\\rbrace $ for each state satisfying the zero flux condition $\\\\sum _j(W_{ij}p^{ss}_i-W_{ji}p^{ss}_j)=0$ .\",\n \"A removal of the nonconservative force is led to further relaxation to the equilibrium ensemble, in which the detailed balance (DB) is (locally) established in every pair of the states such that $p_i^{eq}W_{ij}=p_j^{eq}W_{ji}$ for all $i$ and $j$ .\",\n \"An important feature of the equilibrium, which differentiates itself from NESS, is the condition of DB.\",\n \"Over the decade, there have been a number of endeavors to better characterize the system out-of-equilibrium [51].\",\n \"One of them is to define the heat and entropy production in the context of Master equation.\",\n \"The heat and entropy productions in reference to either steady state or equilibrium are defined to better characterize the process of interest.\",\n \"The aim is to associate the time dependent probability for state ($\\\\lbrace p_i(t)\\\\rbrace $ ) and transition rates between the states $\\\\lbrace W_{ij}\\\\rbrace $ with newly defined macroscopic thermodynamic quantities at nonequilibrium [54].\",\n \"Here, we review NESS thermodynamics formalism developed by Ge and Qian [54].\",\n \"For nonequilibrium relaxation processes one can consider three relaxation processes: (i) relaxation process of a system far-from-equilibrium (FFE) to a nonequilibrium steady state (NESS); (ii) relaxation process of a system far-from-equilibrium (FFE) to an equilibrium (EQ).\",\n \"(iii) relaxation process of a system in NESS to an equilibrium (EQ).\",\n \"To describe these relaxation processes using the probabilities for state, we introduce a phenomenological definition of an internal energy of state $i$ at a steady state by $u^{ss}_i=-k_BT\\\\log {p_i^{ss}}$ , and at equilibrium by $u^{eq}_i=-k_BT\\\\log {p_i^{eq}}$ .\",\n \"Then the following thermodynamic quantities are defined either in reference to NESS or equilibrium.\",\n \"First, the thermodynamic potentials are defined in reference to the NESS: the total energy $U(t)=\\\\sum _i^Np_i(t)u^{ss}_i$ ; the total free energy $F(t)=U(t)-TS(t)=k_BT\\\\sum _i^Np_i(t)\\\\log {(p_i(t)/p_i^{ss})}$ .\",\n \"Second, the thermodynamic potentials are defined in reference to the equilibrium: the total energy $U^{eq}(t)=\\\\sum _i^Np_i(t)u^{eq}_i$ ; the total free energy $F^{eq}(t)=U^{eq}(t)-TS(t)=k_BT\\\\sum _i^Np_i(t)\\\\log {(p_i(t)/p_i^{eq})}$ .\",\n \"In both cases, Gibbs entropy, $S(t)=-k_B\\\\sum _i^Np_i(t)\\\\log {p_i(t)}$ , is defined as usual.\",\n \"Next, the above definitions of generalized thermodynamic potentials, one can define the heat and entropy productions associated with the relaxation processes (i), (ii), (iii).\",\n \"The diagram in Fig.REF depicts the relaxation processes mentioned here.\",\n \"Figure: A diagram illustrating the balances between various thermodynamic quantities discussed in the text.\",\n \"The curvy arrows denote the heat and entropy production from relaxation processes.$-dF(t)/dt$ is the rate of entropy production in the relaxation from FFE to NESS, $\\\\frac{dF(t)}{dt}\\\\equiv -\\\\dot{f}_d&=-T\\\\sum _{i>j}\\\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\\\right]\\\\log {\\\\left[\\\\frac{p_i(t)p_j^{ss}}{p_j(t)p_i^{ss}}\\\\right]}.$ and $-dU(t)/dt$ is the rate of heat production in the relaxation from FFE to NESS.\",\n \"$\\\\frac{dU(t)}{dt}\\\\equiv -\\\\dot{Q}_{ex}&=-\\\\sum _{i>j}(W_{ij}p_i(t)-W_{ji}p_j(t))(u^{ss}_i-u^{ss}_j)\\\\nonumber \\\\\\\\&=T\\\\sum _{i>j}(W_{ij}p_i-W_{ji}p_j)\\\\log {\\\\left(\\\\frac{p_i^{ss}}{p_j^{ss}}\\\\right)}.$ Similarly, $-dF^{eq}(t)/dt$ is the rate of entropy production during the relaxation to equilibrium, $\\\\frac{dF^{eq}(t)}{dt}\\\\equiv -T\\\\dot{e}_p&=-T\\\\sum _{i>j}\\\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\\\right]\\\\log {\\\\left[\\\\frac{p_i(t)W_{ij}}{p_j(t)W_{ji}}\\\\right]}$ and $-dU^{eq}(t)/dt$ is the rate of heat production.\",\n \"$\\\\frac{dU^{eq}(t)}{dt}\\\\equiv -\\\\dot{h}_d&=-\\\\sum _{i>j}(W_{ij}p_i(t)-W_{ji}p_j(t))(u^{eq}_i-u^{eq}_j)\\\\nonumber \\\\\\\\&=T\\\\sum _{i>j}(W_{ij}p_i-W_{ji}p_j)\\\\log {\\\\left(\\\\frac{W_{ij}}{W_{ji}}\\\\right)}$ where the condition of DB ($p_i^{eq}/p_j^{eq}=W_{ji}/W_{ij}$ ) was used to derive the last line.\",\n \"Furthermore, the heat production involved with the relaxation from NESS to equilibrium ($\\\\dot{Q}_{hk}$ ), namely housekeeping heat which is introduced in NESS stochastic thermodynamics from the realization that maintaining NESS requires some energy, is defined by either using $\\\\dot{Q}_{hk}=(-dF^{eq}(t)/dt)-(-dF(t)/dt)=T\\\\dot{e}_p-\\\\dot{f}_d$ or $\\\\dot{Q}_{hk}=(-dU^{eq}(t)/dt)-(-dU(t)/dt)=\\\\dot{h}_d-\\\\dot{Q}_{ex}$ .\",\n \"Explicit calculations using the representation of thermodynamic potential in terms of master equation lead to $\\\\dot{Q}_{hk}=T\\\\sum _{i>j}\\\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\\\right]\\\\log {\\\\left[\\\\frac{p^{ss}_iW_{ij}}{p_j^{ss}W_{ji}}\\\\right]}.$ Lastly, from the definition of Gibbs entropy ($S(t)=-k_B\\\\sum _ip_i(t)\\\\log {p_i(t)}$ ), or from the thermodynamic relationships $TdS/dt=dF/dt-dU/dt=dF^{eq}/dt-dU^{eq}/dt$ , it is straightforward to show that $T\\\\frac{dS}{dt}&=\\\\frac{dF}{dt}-\\\\frac{dU}{dt}=\\\\frac{dF^{eq}}{dt}-\\\\frac{dU^{eq}}{dt}\\\\nonumber \\\\\\\\&=-T\\\\sum _{i>j}\\\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\\\right]\\\\log {\\\\left(\\\\frac{p_i(t)}{p_j(t)}\\\\right)}\\\\nonumber \\\\\\\\&=\\\\dot{h}_d-T\\\\dot{e}_p.$ Now, with the various heat and entropy production defined from generalized potentials $F(t)$ , $U(t)$ , and $F^{eq}(t)$ , $U^{eq}(t)$ ($\\\\dot{f}_d$ , $\\\\dot{Q}_{ex}$ , $T\\\\dot{e}_p$ , $\\\\dot{h}_d$ , and $\\\\dot{Q}_{hk}$ ) we acquire two important balance laws in nonequilibrium thermodynamics: $T\\\\dot{e}_p&=\\\\dot{f}_d+\\\\dot{Q}_{hk}\\\\nonumber \\\\\\\\\\\\dot{h}_d&=\\\\dot{Q}_{hk}+\\\\dot{Q}_{ex}$ Thus, (i) the total entropy production of a system, $T\\\\dot{e}_p(=-dF^{eq}/dt)$ , is contributed by the free energy dissipation due to the relaxation to NESS, $\\\\dot{f}_d(=-dF/dt)$ , and the housekeeping heat, $\\\\dot{Q}_{hk}(=dF/dt-dF^{eq}/dt)$ , that is required to maintain the NESS.\",\n \"(ii) The total heat production $\\\\dot{h}_d(=-dU^{eq}/dt)$ of a system is decomposed into $\\\\dot{Q}_{hk}(=dU/dt-dU^{eq}/dt)$ and the excess heat $\\\\dot{Q}_{ex}(=-dU/dt)$ .\",\n \"The diagram in Fig.REF recapitulates the various heat and entropy production terms and their balance.\",\n \"When the system is already in NESS, then neither the production of entropy nor excess heat is anticipated ($\\\\dot{f}_d=0$ , $\\\\dot{Q}_{ex}=0$ ), and hence it follows that the amount of heat, entropy, and housekeeping heat required to sustain NESS are identical ($T\\\\dot{e}_p=\\\\dot{Q}_{hk}=\\\\dot{h}_d$ ).\",\n \"Figure: Relaxation dynamics of various nonequilibrium thermodynamic quantities from far-from-equilibrium states calculated using (N=2)-state system.The parameters used for the plots are:[ATP] = 1 mM, ff = 1 pN;u 1 0 =1.8s -1 μM -1 u_1^0 = 1.8 ~ s^{-1} \\\\mu M^{-1}, u 2 =108s -1 u_2 = 108 ~ s^{-1}, w 1 =6.0s -1 w_1 = 6.0 ~ s^{-1}, and w 2 =16s -1 w_2 = 16 ~ s^{-1} at zero load;θ 1 + =0.135,θ 2 + =0.035,θ 1 - =0.080\\\\theta ^+_1 = 0.135, \\\\theta ^+_2 = 0.035, \\\\theta ^-_1 = 0.080, and θ 2 - =0.75\\\\theta ^-_2 = 0.75.Plots were made using three different initial conditions: A. p 1 (0)=1,p 2 (0)=0p_1(0) = 1, p_2(0) = 0; B. p 1 (0)=0.5,p 2 (0)=0.5p_1(0) = 0.5, p_2(0) = 0.5; and C. p 1 (0)=0,p 2 (0)=1p_1(0)=0, p_2(0) =1.In order to gain a better insight into the energy balance of molecular motor that operates in nonequilibrium steady state, we consider the dynamics of molecular motor systems by means of a cyclic Markov model and relate the essential parameters of the model with NESS thermodynamics.\",\n \"The thermodynamic quantities associated with nonequilibrium process ($\\\\dot{f}_d$ , $\\\\dot{Q}_{hk}$ , $T\\\\dot{e}_p$ , $\\\\dot{h}_d$ , $\\\\dot{Q}_{ex}$ ) can be evaluated explicitly using $(N=2)$ -state Markov model; the time evolution of each state is given by $p_1(t)=p_1^{ss}+(p_1(0)-p_1^{ss})e^{-\\\\sigma t}$ and $p_2(t)=p_2^{ss}-(p_1(0)-p_1^{ss})e^{-\\\\sigma t}$ with $p_1^{ss}=(u_2+w_1)/\\\\sigma $ , $p_2^{ss}=(u_1+w_2)/\\\\sigma $ , and $\\\\sigma =u_1+u_2+w_1+w_2$ .\",\n \"Using the conditions satisfied in 2-state model ($p_{1+2}(t)=p_1(t)$ ) $W_{12}=u_1$ , $W_{21}=w_1$ , $W_{23}=u_2$ , $W_{32}=w_2$ ; otherwise $W_{ij}=0$ , we obtain $\\\\frac{\\\\dot{Q}_{hk}(t)}{T}&=[w_1p_2(t)-u_1p_1(t)]\\\\log {\\\\left[\\\\frac{(u_2+w_1)w_1}{(u_1+w_2)u_1}\\\\right]}+[w_2p_1(t)-u_2p_2(t)]\\\\log {\\\\left[\\\\frac{(u_1+w_2)w_2}{(u_2+w_1)u_2}\\\\right]}\\\\nonumber \\\\\\\\&= \\\\frac{u_1u_2-w_1w_2}{u_1+u_2+w_1+w_2}\\\\log {\\\\left[\\\\frac{u_1u_2}{w_1w_2}\\\\right]}-\\\\lambda (p_1(0)-p_1^{ss})e^{-\\\\sigma t}\\\\nonumber \\\\\\\\&\\\\xrightarrow{}(j_+-j_-)\\\\log {\\\\left(\\\\frac{j_+}{j_-}\\\\right)}\\\\ge 0$ where $\\\\lambda =\\\\left\\\\lbrace (u_1+w_1)\\\\log {\\\\left[\\\\frac{(u_2+w_1)w_1}{(u_1+w_2)u_1}\\\\right]}-(u_2+w_2)\\\\log {\\\\left[\\\\frac{(u_1+w_2)w_2}{(u_2+w_1)u_2}\\\\right]}\\\\right\\\\rbrace $ .\",\n \"$\\\\dot{e}_p(t)&=[w_1p_2(t)-u_1p_1(t)]\\\\log {\\\\left[\\\\frac{p_2(t)w_1}{p_1(t)u_1}\\\\right]}+[w_2p_1(t)-u_2p_2(t)]\\\\log {\\\\left[\\\\frac{p_1(t)w_2}{p_2(t)u_2}\\\\right]}\\\\nonumber \\\\\\\\&\\\\xrightarrow{} \\\\frac{u_1u_2-w_1w_2}{u_1+u_2+w_1+w_2}\\\\log {\\\\left[\\\\frac{u_1u_2}{w_1w_2}\\\\right]}=(j_+-j_-)\\\\log {\\\\left(\\\\frac{j_+}{j_-}\\\\right)}\\\\ge 0$ $\\\\frac{\\\\dot{h}_d(t)}{T}&=[w_1p_2(t)-u_1p_1(t)]\\\\log {\\\\left[\\\\frac{w_1}{u_1}\\\\right]}+[w_2p_1(t)-u_2p_2(t)]\\\\log {\\\\left[\\\\frac{w_2}{u_2}\\\\right]}\\\\nonumber \\\\\\\\&\\\\xrightarrow{} \\\\frac{u_1u_2-w_1w_2}{u_1+u_2+w_1+w_2}\\\\log {\\\\left[\\\\frac{u_1u_2}{w_1w_2}\\\\right]}=(j_+-j_-)\\\\log {\\\\left(\\\\frac{j_+}{j_-}\\\\right)}\\\\ge 0$ $\\\\frac{\\\\dot{f}_d(t)}{T}&=[w_1p_2(t)-u_1p_1(t)]\\\\log {\\\\left[\\\\frac{p_2(t)p_1^{ss}}{p_1(t)p_2^{ss}}\\\\right]}+[w_2p_1(t)-u_2p_2(t)]\\\\log {\\\\left[\\\\frac{p_1(t)p_2^{ss}}{p_2(t)p_1^{ss}}\\\\right]}\\\\nonumber \\\\\\\\&=\\\\sigma (p_2(t)p_1^{ss}-p_1(t)p_2^{ss})\\\\log {\\\\left[\\\\frac{p_2(t)p_1^{ss}}{p_1(t)p_2^{ss}}\\\\right]}\\\\ge 0\\\\nonumber \\\\\\\\&\\\\xrightarrow{} 0$ The relaxation time to a steady state (NESS) from an arbitrary state in a far-from-equilibrium is $\\\\sim \\\\sigma ^{-1}=(u_1+u_2+w_1+w_2)^{-1}$ , and it is noteworthy that the entropy production inside the system ($T\\\\dot{e}_p$ ), the total heat production that will be discharged to the surrounding ($\\\\dot{h}_d$ ), and the housekeeping heat ($\\\\dot{Q}_{hk}$ ) are all identical at the steady state as $T\\\\dot{e}_p=\\\\dot{h}_d=\\\\dot{Q}_{hk}\\\\rightarrow (j_+-j_-)\\\\log {(j_+/j_-)}\\\\ge 0$ , and $\\\\dot{f}_d=0$ .\",\n \"Here, $\\\\dot{Q}_{ex}$ , the residual of heat (excess heat, $\\\\dot{Q}_{ex}=\\\\dot{h}_d-\\\\dot{Q}_{hk}$ ) for the nonequilibrium process, is zero at the steady state.\",\n \"Although obtained for 2-state model, the above expression, especially the total heat production (or housekeeping heat) at the steady state, $\\\\dot{h}_d=T\\\\dot{e}_p=\\\\dot{Q}_{hk}=T(j_+-j_-)\\\\log {j_+/j_-}$ can easily be generalized for the $N$ -state model.\"\n ],\n [\n \"5. Relationship between motor diffusivity and heat dissipation.\",\n \"For $(N=2)$ -state model one can obtain an explicit expression that relates $D$ with $\\\\dot{Q}$ (for the case of $f=0$ ) as follows.\",\n \"From the expressions of $V$ (Eq.REF ), $D$ (Eq.REF ), and $\\\\dot{Q}$ , $\\\\dot{Q}=\\\\frac{V}{d_0}k_BT\\\\log {\\\\frac{u_1u_2}{w_1w_2}}$ Substitution of $u_1=u_1(V)$ from Eq.REF into Eq.REF gives an expression of $\\\\dot{Q}$ as a function of $V$ : $\\\\frac{\\\\dot{Q}}{k_{cat}k_BT}&=v\\\\log {\\\\left[\\\\frac{1+\\\\kappa v}{1-v}\\\\right]}\\\\nonumber \\\\\\\\&=\\\\sum _{n=1}^{\\\\infty }\\\\frac{1}{n}\\\\left(1+(-1)^{n-1}\\\\kappa ^n\\\\right)v^{n+1}$ where $v=V/V_{max}$ ($0\\\\le v\\\\le 1$ ) and $\\\\kappa \\\\equiv \\\\frac{k_{cat}(k_{cat}+w_1+w_2)}{w_1w_2}$ .\",\n \"$\\\\dot{Q}$ diverge as $v\\\\rightarrow 1$ ; but for small $v\\\\ll 1$ , $\\\\dot{Q}/(k_{cat} k_B T) \\\\sim (\\\\kappa +1)v^2$ , thus $v\\\\sim \\\\dot{Q}^{1/2}$ .\",\n \"As long as $\\\\dot{Q}$ is small, one should expect from Eq.REF that $D$ increases with $\\\\dot{Q}$ as $D=D_0+\\\\gamma _1\\\\dot{q}^{1/2}+\\\\gamma _2\\\\dot{q}+\\\\gamma _3\\\\dot{q}^{3/2}$ where $\\\\dot{q}\\\\equiv \\\\frac{\\\\dot{Q}}{k_BTk_{cat}}$ , $D_0=\\\\frac{d_0^2k_{cat}}{\\\\kappa +1}$ , $\\\\gamma _1=d_0^2k_{cat}\\\\frac{\\\\kappa -1}{2 (\\\\kappa +1)^{3/2}}$ , $\\\\gamma _2=-d_0^2k_{cat}\\\\frac{k_{cat}^2}{w_1w_2}$ , and $\\\\gamma _3=\\\\frac{d_0^2k_{cat}}{(\\\\kappa +1)^{5/2}}\\\\frac{k_{cat}^2}{w_1w_2}$ .\",\n \"For arbitrary number of states $N$ , by using Eq.REF and Eq.REF , $\\\\dot{Q}$ can be written as $\\\\dot{Q}/k_BT & = \\\\frac{V}{d_0} \\\\log {\\\\frac{ \\\\prod _{i=1}^N u_i }{ \\\\prod _{i=1}^N w_i } } \\\\nonumber \\\\\\\\& = \\\\frac{V^{(D)}}{N} \\\\log {\\\\left( 1 + V^{(D)} \\\\frac{ \\\\sum _{n=1}^N r_n }{ N }\\\\frac{ \\\\prod _{i=1}^N u_i }{ \\\\prod _{i=1}^N w_i }\\\\right)} \\\\nonumber \\\\\\\\& = \\\\frac{V^{(D)}}{N} \\\\log {\\\\left( 1 + \\\\frac{V^{(D)}}{N} \\\\left( A + B u_1 \\\\right) \\\\frac{ 1 }{C }\\\\right)} \\\\nonumber \\\\\\\\&=\\\\frac{V^{ (D) }}{N} \\\\log { \\\\left( 1 + \\\\frac{ V^{(D)} }{N} f(V^{ (D) }/N ) \\\\right) }$ where we used $V^{ (D) } = V N / d_0 $ , $f(V^{ (D) }/N ) =\\\\frac{A}{C} + B \\\\left( 1 - B \\\\frac{V^{ (D) } }{N} \\\\right)^{-1} \\\\left( 1 + \\\\frac{A}{C} \\\\frac{ V^{ (D) }}{N} \\\\right)$ , and Eqs.REF , REF , REF .\",\n \"The definitions of $A, B$ , and $C$ are identical to those in Eq.REF .\",\n \"Here $V^{ (D) } / N$ corresponds to ATP hydrolysis rate.\",\n \"For $V^{ (D) } \\\\rightarrow 0$ , $\\\\dot{Q} \\\\rightarrow 0$ is expected.\",\n \"Also for small $V^{ (D) }$ , $\\\\dot{Q} \\\\sim ( \\\\frac{A}{C} + B ) \\\\left(\\\\frac{V^{(D)}}{N} \\\\right)^2$ .\",\n \"Thus, $V \\\\sim \\\\dot{Q}^{1/2}$ .\",\n \"Since $D\\\\sim V+\\\\mathcal {O}(V^2)$ for small $V$ , it follows that $D$ can be written as a function of $\\\\dot{q}$ as in the same form as Eq.REF .\"\n ],\n [\n \"6. Rate constants, enhancement of diffusion, and conversion efficiency determined from the (N=2)-state kinetic model.\",\n \"The values in the Table 1 were compiled based on the followings.\"\n ],\n [\n \"Catalase\",\n \"In Ref.\",\n \"[5] $(\\\\Delta D / D_0)=0.28$ at $ V = 1.7 \\\\times 10^4$ $s^{-1}$ ; however, $V= 1.7 \\\\times 10^4$ $s^{-1}$ is not the maximum catalytic rate.\",\n \"Because $\\\\Delta D/D_0$ is approximately linear in $V$ , the enhancement of diffusion at the maximal turnover rate $V_{\\\\text{max}}=u_2= 5.8 \\\\times 10^4$ is estimated as $(\\\\Delta D / D_0)_{\\\\text{obs}} = 0.28 \\\\times \\\\frac{5.8 \\\\times 10^4}{1.7 \\\\times 10^4} = 0.96 \\\\approx 1$ .\"\n ],\n [\n \"Alkaline phosphatase\",\n \"Similar to catalase, $(\\\\Delta D / D_0)_{\\\\text{obs}} = 0.77 \\\\times \\\\frac{1.4 \\\\times 10^4}{5.5 \\\\times 10^3} = 2.5 \\\\approx 3$ .\"\n ],\n [\n \"Estimate of $(\\\\Delta D/D_0)_{\\\\text{max}}$\",\n \"Freely diffusing enzymes effectively perform no work on the surrounding environment; thus $-\\\\Delta \\\\mu _{\\\\text{eff}}=Q$ with $W=0$ , which leads to $e^{Q/k_B T}= u_1 u_2/w_1 w_2$ .\",\n \"By assuming that the substrate concentration $[S] \\\\sim K_M=(u_2 +w_1) / u_1^o$ , we get $\\\\begin{aligned}e^{Q/k_B T}&= \\\\frac{ u_1 u_2 }{w_1 w_2}\\\\ \\\\sim \\\\frac{ u_1^o K_M u_2 }{w_1 w_2}= \\\\frac{ (u_2 + w_1) u_2 }{w_1 w_2} \\\\\\\\&\\\\ge \\\\frac{ u_2^2 }{w_1 w_2}= \\\\frac{ k_{cat}^2 }{w_1 w_2}.\\\\end{aligned}$ This relation allows us to estimate the upper bound of $(\\\\Delta D/D_0)_{\\\\text{max}}$ as follows when $u_2 \\\\gg w_1, w_2$ is satisfied.\",\n \"$\\\\left(\\\\frac{\\\\Delta D}{D_0}\\\\right)_{\\\\text{max}}\\\\approx \\\\frac{ k_{cat}^2 }{2 w_1 w_2} \\\\le \\\\frac{1}{2}e^{Q/k_BT}.$ Alternatively, $u_1$ and $w_2$ of enzymes can be estimated by assuming (i) that the reaction is diffusion limited, $u_1^o = 10^8 s^{-1} M^{-1}$ , and (ii) that the substrate concentration $[S]$ is similar to Michaelis-Menten constant $K_M$ ($[S]\\\\sim K_M$ ).\",\n \"The two conditions $u_1 = u_1^o [S] \\\\sim K_M \\\\times 10^8$ ($s^{-1}$ ) and $K_M(=(u_2+w_1)/u_1^o)$ , and $Q$ (heat measured by the calorimeter in ref.\",\n \"[5]), $u_2$ , $K_M$ which are available in ref.\",\n \"[5], provide all the rate constants including $w_1=u_1^oK_M-u_2$ and $w_2=\\\\frac{u_1u_2}{w_1e^{Q/k_BT}}$ , allowing us to calculate $\\\\left(\\\\frac{\\\\Delta D}{D_0}\\\\right)_{\\\\text{max}}=\\\\frac{u_2^2+(w_1+w_2)u_2-w_1w_2}{2w_1w_2}$ .\",\n \"Figure: Analysis of experimental data, extracted from Ref.\",\n \", but using (NN=2)-state model.The solid lines are the fits to the dataA.\",\n \"VV vs ATP at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).B.\",\n \"VV vs load at [ATP] = 5 μ\\\\mu M.C.\",\n \"VV vs load at [ATP] = 2 mM.D.\",\n \"Stall force as a function of [ATP], measured by `Position clamp' (red square) or `Fixed trap' (blue circle) methods.E.\",\n \"DD vs ATP at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).DD was estimated from r=2D/Vd 0 r=2D/Vd_0.\",\n \"F. DD vs load at [ATP] = 2 mM.G-I.\",\n \"Motor diffusivity (DD) as a function of mean velocity (VV) for kinesin-1.\",\n \"(VV,DD) measured at varying [ATP] (= 0 – 2 mM) and a fixed (G) f=f=1.05 pN, (H) 3.59 pN, and (I) 5.63 pN .The black dashed lines in G and H are the fits using Eq..The solid lines in magenta in G-I are plotted using the (NN=2)-kinetic model.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05747"}}},{"rowIdx":3384,"cells":{"text":{"kind":"list like","value":[["Dark Matter Relics and the Expansion Rate in Scalar-Tensor Theories"],["Abstract We study the impact of a modified expansion rate on the dark matter relic abundance in a class of scalar-tensor theories.","The scalar-tensor theories we consider are motivated from string theory constructions, which have conformal as well as disformally coupled matter to the scalar.","We investigate the effects of such a conformal coupling to the dark matter relic abundance for a wide range of initial conditions, masses and cross-sections.","We find that exploiting all possible initial conditions, the annihilation cross-section required to satisfy the dark matter content can differ from the thermal average cross-section in the standard case.","We also study the expansion rate in the disformal case and find that physically relevant solutions require a nontrivial relation between the conformal and disformal functions.","We study the effects of the disformal coupling in an explicit example where the disformal function is quadratic."],["Introduction","The most recent cosmological observations support the so called $\\Lambda CDM$ model of cosmology.","These observations provide overwhelming evidence for the existence of non-baryonic (cold) dark matter (DM), constituting about $27\\%$ of the Universe's energy density budget, while another $\\sim 68\\%$ is believed to be in the form of dark energy, which can simply be a cosmological constant, $\\Lambda $ .","The other $\\sim 5\\%$ being formed by baryonic matter, described by the standard model (SM) of particles.","A popular framework to understand the origin of DM is the thermal relic scenario.","In this scenario, at very early times when the universe was at a very high temperature, thermal equilibrium was obtained and the number density of DM particles $\\chi $ was roughly equal to the number density of photons.","During equilibrium the dark matter number density decayed exponentially as $n_\\chi ^{eq} \\sim e^{-m_\\chi /T}$ for a non-relativistic DM candidate, where $m_\\chi $ is the mass of the DM particle $\\chi $ .","As the universe cooled down as it expanded, DM interactions became less frequent and eventually, the DM interaction rate dropped below the expansion rate ($\\Gamma _\\chi < H$ ).","At this point the density number froze-out and the universe was left with a “relic\" of DM particles.","Therefore, the dependence of the number density at the time of freeze-out is crucial to determine the DM relic abundance.","The longer the DM particles remain in equilibrium, the lower its density will be at freeze-out and vice-versa.","In the standard $\\Lambda CDM$ scenario, particle freeze-out happens during the radiation era and DM species with weak scale interaction cross-section freeze-out with an abundance that matches the present observed value.","The weakness of the interactions is reflected in the predicted thermally-averaged annihilation cross section, $\\langle \\sigma v \\rangle $ , which is around $3.0\\times 10^{-26}cm^3s^{-1}$ .","Despite such a small value, the Fermi-LAT and Planck experiments have been exploring upper bounds on $\\langle \\sigma v \\rangle $ (see [1], [2]).","From observations, it appears that the annihilation cross-section can be smaller than the thermal average value for lower dark matter masses ($\\le 100$ GeV), whereas an annihilation cross-section larger than the thermal average value can still be allowed for larger DM mass.","If, from future measurements, $\\langle \\sigma v \\rangle \\ne 3.0\\times 10^{-26}cm^3s^{-1}$ is established, what can we say about the origin of the dark matter?","Can it still be the thermal dark matter or do we need non-thermal origin of dark matter?","In the case of non-thermal origin, the DM can arise from the decay of a heavy particle, e.g., moduli, and can satisfy the DM content with any value of $\\langle \\sigma v \\rangle $ .","The primary motivation of this paper is to find out whether the DM content can still have a thermal origin with larger or smaller $\\langle \\sigma v \\rangle $ by utilising non-standard cosmology.","The phenomenological $\\Lambda CDM$ model complemented with the inflationary paradigm to provide the seeds of large scale structure, is very successful in describing our current universe.","However, the physics describing the universe's evolution from the end of inflation (reheating) to just before big-bang nucleosynthesis (BBN) ($t\\lesssim 200$ s) remains relatively unconstrained.","During this period the universe may have gone through a “non-standard\" period of expansion, and still be compatible with BBN.","If such modification happened during DM decoupling, the DM freeze-out may be modified with measurable consequences for the relic DM abundances.","Departures from the standard cosmology between reheating and BBN will mainly be a consequence of a modified expansion rate ($\\tilde{H}$ ), due to a modification of General Relativity (GR).","Such modifications are well motivated by attempts to embed the $\\Lambda CDM$ and inflation models into a fundamental theory of gravity and particle physics, such as theories with extra-dimensions, supergravity and string theory.","Indeed, our main motivation in this paper is to develop further tools that may allow us to connect such fundamental theories with observations.","In this paper our approach will be mostly phenomenological, but we have in mind a scenario that can be derived in the context of a fundamental theory of gravity, such as string theory.","Furthermore, we will be concerned with modifications to the standard picture due to the presence and interactions with scalar fields only.","Modifications to the relic abundances were first discussed by Catena et al.","[3] in the context of conformally coupled scalar-tensor theories (ST), such as generalisations of the Brans-Dicke theory.","Further studies on conformally coupled ST models have been performed in the last years in [4], [5], [6], [7] (see also [8], [9], [10], [11], [12], [13], [14]) with the most recent work of [15] where it is shown that the boundary conditions used in [3] cannot have $\\langle \\sigma v \\rangle \\ne 3.0\\times 10^{-26}cm^3s^{-1}$ .","Indeed ST theories where dark matter and dark energy are correlated constitute an attractive way to address the dark matter and dark energy problems via an attraction mechanism towards standard general relativity (GR) [3].","In the context of ST theories, the most general physically consistent relation between two metrics in the presence of a scalar field, is given byIn the more general case, $C$ and $D$ can be functions of $X=\\frac{1}{2} (\\partial \\phi )^2$ as well.","However, we will not consider this case in the present paper.","[16]: $\\tilde{g}_{\\mu \\nu } = C(\\phi ) g_{\\mu \\nu } + D(\\phi ) \\partial _\\mu \\phi \\partial _\\nu \\phi \\,.$ The first term in (REF ) is the well-known conformal transformation which characterises the Brans-Dicke class of scalar-tensor theories explored in [3], [4], [5], [6], [7], [15].","However, in reference [15], it is shown that there is no change in the thermal cross-section arising from the conformally modified metric compared to the standard cosmology after satisfying all the constraints based on the boundary conditions chosen in [3], [4].","The second term is the disformal contribution, which is generic in extensions of general relativity.","In particular, it arises naturally in D-brane models, as discussed in [17] in a model of coupled dark matter and dark energy.","In this paper we revisit the expansion rate modification and impact on the DM relic abundances for the conformal case, providing new interesting results with new boundary conditions to show that $\\langle \\sigma v \\rangle \\ne 3.0\\times 10^{-26}cm^3s^{-1}$ , while satisfying the DM content.","We further discuss the general modifications to the expansion rate and Boltzmann equation, due to the disformal coupling and present an explicit non-trivial example for the case in which the conformal term in (REF ) is a monomial.","The paper is organised as follows.","We start in section introducing the scalar-tensor theory conformally and disformally coupled to matter.","Then, we examine the formulation of this theory in the Einstein and Jordan frames, comment on their physical interpretation and derive the equations that describe the cosmological evolution of the Universe.","Subsequently, after discussing the expansion rate modifications caused by the presence of the conformal and disformally coupled scalar field in section , we investigate in detail its impact on the dark matter relic abundance by exploring a concrete pure conformal example as well as an example where we also turn on the disformal contribution.","Finally, in section we conclude."],["The scalar-tensor theory set-up","We are interested in scalar-tensor theories coupled to matter both conformally and disformally [16].","Our motivation comes from theories with extra dimensions and in particular string theory compactifications, where several additional scalar fields appear, from closed and open string theory sectors of the theory [17].","Our approach in this paper nonetheless, will be phenomenological and therefore our equations will be simplified.","However, we present the more general set-up, which can accommodate a realisation from concrete string theory compactifications in appendix and .","The action we want to consider is given by: $S_{EH}=\\frac{1}{2\\kappa ^2} \\!\\!\\int {\\!d^4x\\sqrt{-g}\\,R}- \\!\\!\\int {\\!d^4x\\sqrt{- g} \\left[\\frac{1}{2} (\\partial \\phi )^2+V(\\phi )\\right]}- \\!\\!\\int {\\!d^4x\\sqrt{-\\tilde{g}} \\,{\\cal L}_{M}(\\tilde{g}_{\\mu \\nu }) } \\,.$ Here the disformally coupled metric is given by $\\tilde{g}_{\\mu \\nu } = C(\\phi ) g_{\\mu \\nu } + D(\\phi ) \\partial _\\mu \\phi \\partial _\\nu \\phi \\,,$ and the inverse by: $\\tilde{g}^{\\mu \\nu } = \\frac{1}{C}\\left[ g^{\\mu \\nu } - \\frac{D\\,\\partial ^\\mu \\phi \\partial ^\\nu \\phi }{C+D(\\partial \\phi )^2}\\right]\\,.$ Moreover, $\\kappa ^2=M_{P}^{-2}=8\\pi G$ , but keep in mind that that $G$ is not in general equal to Newton's constant as measured by e.g.","local experiments.","Further, $C(\\phi ), D(\\phi )$ are functions of $\\phi $ , which can be identified as a conformal and disformal couplings of the scalar to the metric, respectively (note that the conformal coupling is dimensionless, whereas the disformal one has units of $mass^{-4}$ ).","The action in (REF ) is written in the Einstein frameIn string theory, the Einstein frame refers to the frame in which the dilaton and graviton degrees of freedom are decoupled, while the string (or Jordan) frame is that in which they are not.","Further, the dilaton field as well as all other moduli (scalar) fields not relevant for the cosmological discussion are stabilised, massive, and are therefore decoupled from the low energy effective theory.","In the literature of scalar-tensor theories however, the Einstein and Jordan frames are identified with respect to the (usually single) scalar field to which gravity is coupled, but such scalar has no particular physical nor geometrical interpretation., which is identified in the literature of scalar-tensor theories (including conformal and disformal couplings) with the frame respect to which the scalar field, gravity is coupled.","We follow this use and refer to “Jordan\" or “disformal frame\" to identify the frame in which dark matter is coupled only to the metric $\\tilde{g}_{\\mu \\nu }$ , rather than to the metric $g_{\\mu \\nu } $ and a scalar field $\\phi $ .","The equations of motion obtained from (REF ) are: $R_{\\mu \\nu } -\\frac{1}{2}g_{\\mu \\nu } R = \\kappa ^2\\left(T^\\phi _{\\mu \\nu } + T_{\\mu \\nu }\\right)\\,,$ where, in the frame relative to $g_{\\mu \\nu }$ , the energy-momentum tensors are defined as $T^\\phi _{\\mu \\nu } = -\\frac{2}{\\sqrt{-g}} \\frac{\\delta S_{\\phi }}{\\delta g^{\\mu \\nu }} \\,,\\qquad \\quad T_{\\mu \\nu }= -\\frac{2}{\\sqrt{-g}} \\frac{\\delta \\left(-\\sqrt{-\\tilde{g}} \\,{\\cal L}_{M}\\right) }{\\delta g^{\\mu \\nu }}\\,,$ and we model the energy-momentum tensor for matter and both dark components as perfect fluids, that is: $T_{\\mu \\nu }^i = P_i g_{\\mu \\nu } +(\\rho _i + P_i) u_\\mu u_\\nu $ where $\\rho _i$ , $P_i$ are the energy density and pressure for each fluid $i$ with equation of state $P_i/\\rho _i = \\omega _i$ .","For the scalar field, the energy-momentum tensor takes the form: $T_{\\mu \\nu }^{\\phi } = - g_{\\mu \\nu } \\left[ \\frac{1}{2} (\\partial \\phi )^2+ V \\right]+ \\partial _\\mu \\phi \\, \\partial _\\nu \\phi \\,,$ and one can define the energy density and pressure of the scalar field as: $\\rho _\\phi =- \\frac{1}{2}(\\partial \\phi )^2 + V \\,, \\qquad P_\\phi = - \\frac{1}{2}(\\partial \\phi )^2 - V \\,.$ Finally the equation of motion for the scalar field dark energy becomes: $&&\\hspace{-28.45274pt}- \\nabla _\\mu \\nabla ^\\mu \\phi \\!","+ V^{\\prime }- \\frac{T^{\\mu \\nu }}{2}\\!\\left[\\frac{C^{\\prime }}{C} g_{\\mu \\nu } +\\frac{D^{\\prime }}{C}\\partial _\\mu \\phi \\partial _\\nu \\phi \\right]+\\nabla _\\mu \\left[\\frac{D}{C}T^{\\mu \\nu } \\partial _\\nu \\phi \\right] =0 \\,.$ Due to the nontrivial coupling, the individual conservation equations for the two fluids are modified.","However, the conservation equation for the full system is preserved, and given in the usual way by $\\nabla _\\mu \\left(T^{\\mu \\nu }_{\\phi } + T^{\\mu \\nu }\\right) =0\\,.$ Thus using (REF ) and the equation of motion for the scalar field we can write $\\nabla _\\mu T^{\\mu \\nu }_\\phi = Q \\,\\partial ^\\nu \\phi = - \\nabla _\\mu T^{\\mu \\nu }\\,,$ where $Q\\equiv \\nabla _\\mu \\left[\\frac{D}{C} \\,T^{\\mu \\lambda } \\,\\partial _\\lambda \\phi \\right] - \\frac{T^{\\mu \\nu } }{2} \\left[\\frac{C^{\\prime }}{C} g_{\\mu \\nu } +\\frac{D^{\\prime }}{C} \\,\\partial _\\mu \\phi \\,\\partial _\\nu \\phi \\right]\\,.$ In the Jordan, or disformal frame, as defined above, matter is conserved, $\\tilde{\\nabla }_\\mu \\tilde{T}^{\\mu \\nu } =0 \\,,$ where $\\tilde{\\nabla }_\\mu $ is the covariant derivative computed with respect to the disformal metric (REF ) with the Christoffel symbols given by $\\tilde{\\Gamma }^\\mu _{\\alpha \\beta } =\\Gamma ^\\mu _{\\alpha \\beta } + \\frac{C^{\\prime }}{C}\\, \\delta ^\\mu _{(\\alpha }\\partial _{\\beta )}\\phi - \\gamma ^2 \\, \\frac{C^{\\prime }}{2C} \\,\\partial ^\\mu \\phi \\, g_{\\alpha \\beta }+\\frac{D}{C}\\,\\gamma ^2\\,\\partial ^\\mu \\phi \\left[\\!\\nabla _\\alpha \\nabla _\\beta \\phi + \\!\\left(\\frac{D^{\\prime }}{2D} -\\!\\frac{C^{\\prime }}{C}\\right) \\!\\partial _\\alpha \\phi \\partial _\\beta \\phi \\right]\\,, \\nonumber \\\\$ and we have introduced the “Lorentz factor\" $\\gamma $ defined as $\\gamma = \\frac{1}{\\sqrt{1+\\frac{D}{C} (\\partial \\phi )^2}}\\,.$ In this frame, the energy-momentum tensor is defined as $\\tilde{T}^{\\mu \\nu } = \\frac{2}{\\sqrt{-\\tilde{g}}} \\frac{\\delta S_{M}}{\\delta \\tilde{g}_{\\mu \\nu } } \\,$ and the disformal energy-momentum tensor can be written as: $\\tilde{T}^{\\mu \\nu } = (\\tilde{\\rho }+\\tilde{P}) \\tilde{u}^{\\mu } \\tilde{u}^{\\mu } + \\tilde{P}\\, \\tilde{g}^{\\mu \\nu }\\,,$ where $\\tilde{u}^{\\mu } = C^{-1/2} \\gamma \\,u^{\\mu } $ .","Using (REF ), we obtain a relation between the energy momentum tensor in both frames as: $\\tilde{T}^{\\mu \\nu } = C^{-3} \\gamma \\, T^{\\mu \\nu }\\,.$ Further using (REF ) we arrive at a relation among the energy densities and pressures in both frames, given by $\\tilde{\\rho }= C^{-2} \\gamma ^{-1} \\rho \\,, \\qquad \\tilde{P} = C^{-2}\\gamma \\, P,$ and therefore the equations of state in both frames are related by $\\tilde{\\omega }= \\omega \\, \\gamma ^2$ .","Note that in the pure conformal case, $D=0$ , $\\gamma =1$ and therefore $\\tilde{\\omega }= \\omega $ ."],["Cosmological equations","Consider an homogeneous and isotropic FRW metric $g_{\\mu \\nu }$ , $ds^2 = - dt^2 + a(t)^2 dx_i dx^i \\,,$ where $a(t)$ is the scale factor.","In this background, the Einstein and Klein-Gordon equations become, respectively $&& H^2 =\\frac{\\kappa ^2}{3} \\left[\\rho _\\phi +\\rho \\right]\\,, \\\\&& \\dot{H} + H^2 = -\\frac{\\kappa ^2}{6}\\left[ \\rho _\\phi + 3P_\\phi +\\rho +3 P \\right]\\,,\\\\&& \\ddot{\\phi }+3H\\dot{\\phi }+ V_{,\\phi }+Q_0 =0 \\,.$ where, $H= \\frac{\\dot{a}}{a}$ , dots are derivatives with respect to $t$ and we have denoted $V_{,\\phi } \\equiv \\frac{dV}{d\\phi }$ .","Also the Lorentz factor becomes $\\gamma = (1-D \\,\\dot{\\phi }^2/C)^{-1/2}.$ The continuity equations for the scalar field and matter are given by $&&\\dot{\\rho }_\\phi + 3H(\\rho _{\\phi }+P_{\\phi }) = -Q_0\\dot{\\phi }\\,, \\\\&&\\dot{\\rho }+ 3H(\\rho +P) = Q_0\\,\\dot{\\phi }\\,.$ where $Q_0$ is given by $Q_0 = \\rho \\left[ \\frac{D}{C} \\,\\ddot{\\phi }+ \\frac{D}{C} \\,\\dot{\\phi }\\left(\\!3H + \\frac{\\dot{\\rho }}{\\rho } \\right) \\!+ \\!\\left(\\!\\frac{D_{,\\phi }}{2C}-\\frac{D}{C}\\frac{C_{,\\phi }}{C}\\!\\right) \\dot{\\phi }^2 +\\frac{C_{,\\phi }}{2\\,C} (1-3\\,\\omega )\\right].","\\nonumber \\\\$ Using () we can rewrite this in a more compact and useful form as $Q_0 = \\rho \\left( \\frac{\\dot{\\gamma }}{\\dot{\\phi }\\, \\gamma } + \\frac{C_{,\\phi }}{2C} (1-3\\,\\omega \\,\\gamma ^2) -3H\\omega \\,\\frac{(\\gamma ^{2}-1) }{\\dot{\\phi }}\\right)\\,.$ Plugging this into the (non-)conservation equation for dark matter (), gives: $\\dot{\\rho }+ 3H (\\rho + P\\,\\gamma ^{2}) = \\rho \\left[\\frac{\\dot{\\gamma }}{\\gamma } + \\frac{C_{,\\phi } }{2C} \\,\\dot{\\phi }\\, (1-3\\,\\omega \\gamma ^2)\\right]\\,.$ Using the relations for the physical proper time and the scale factors in the two frames, given by $\\tilde{a} = C^{1/2} a \\,, \\qquad \\quad d\\tilde{\\tau }= C^{1/2} \\gamma ^{-1} d\\tau \\,,$ we can define the disformal-frame Hubble parameter $\\tilde{H} \\equiv \\frac{d \\ln {\\tilde{a}}}{d\\tilde{\\tau }}$ , as $\\tilde{H} = \\frac{\\gamma }{C^{1/2}}\\left[ H + \\frac{C_{,\\phi }}{2C}\\dot{\\phi }\\right]\\,,$ so that (REF ) takes the standard form in terms of $\\tilde{H}$ : $\\frac{d{\\tilde{\\rho }}}{d\\tilde{\\tau }} + 3 \\tilde{H}( \\tilde{\\rho }+\\tilde{P}) =0 \\,.$ Equations (REF ) and (REF ) give the background evolution equations for the modified expansion rate and matter's density evolution."],["Master equations","In order to solve the cosmological equations, it is convenient to replace time derivatives with derivatives with respect to the number of e-folds $N$ , defined as $ N = \\ln a/a_0$ and define $\\lambda = \\frac{V}{ \\rho }(=\\frac{\\tilde{V}}{\\tilde{\\rho }})$ .","With these definitions, we can rewrite the Friedmann equation (REF ) and $Q_0$ as: $H^2 &=& \\frac{\\kappa ^2 \\rho }{3} \\frac{(1+\\lambda )}{\\left(1- \\frac{\\kappa ^2\\phi ^{\\prime 2}}{6 }\\right)} \\,, \\\\\\!\\!\\frac{Q_0}{\\rho }& =& \\frac{\\gamma ^2 H^2}{2}\\Bigg [ \\frac{2D}{C} \\phi ^{\\prime \\prime } -\\frac{2D}{C} \\phi ^{\\prime } \\!\\left(3\\,\\omega + \\frac{\\kappa ^2\\phi ^{\\prime 2}}{2} + \\frac{3(1+\\omega ) B}{2(1+\\lambda )} \\right)+ \\left(\\frac{D}{C}\\right)_{\\!\\!,\\phi } \\!\\!\\phi ^{\\prime 2} + \\frac{C_{\\!,\\phi }}{H^2 C}(\\gamma ^{-2}-3\\omega ) \\Bigg ], \\nonumber \\\\$ where here we denote $^{\\prime }= d/dN$ .","Note also that (REF ) implies that $\\kappa \\,\\phi ^{\\prime } \\le \\pm \\sqrt{6}$ .","Using these equations and further defining a dimensionless scalar field $\\varphi = \\kappa \\,\\phi $ , we can rewrite () and () as: $&& H^{\\prime } = - H \\left[ \\frac{3B}{2(1+\\lambda )} (1+ \\omega ) + \\frac{\\varphi ^{\\prime 2}}{2}\\right], \\\\ \\nonumber \\\\&& \\varphi ^{\\prime \\prime } \\left[1\\!+\\!","\\frac{3H^2\\gamma ^2 B}{\\kappa ^2(1+\\lambda )} \\frac{D}{C} \\right]+ 3\\,\\varphi ^{\\prime } \\left[1- \\omega \\frac{3H^2\\gamma ^2 B}{\\kappa ^2(1+\\lambda )} \\frac{D}{C} \\right] +\\frac{H^{\\prime }}{H} \\varphi ^{\\prime } \\left[1+ \\frac{3H^2\\gamma ^2 B}{\\kappa ^2(1+\\lambda )} \\frac{D}{C} \\right]\\nonumber \\\\&& \\hspace{42.67912pt}+ \\frac{3B}{1+\\lambda } \\alpha (\\varphi )(1-3 \\,\\omega \\gamma ^2)+ \\frac{3B\\lambda }{(1+\\lambda )} \\frac{V_{,\\varphi }}{V} + \\frac{3 H^2 \\gamma ^2 B}{\\kappa ^2(1+\\lambda )} \\frac{D}{C}\\left[ (\\delta (\\varphi ) - \\alpha (\\varphi )) \\,\\varphi ^{\\prime 2} \\right] =0\\,,\\nonumber \\\\$ where we defined: $&&B \\equiv 1-\\frac{\\varphi ^{\\prime 2}}{6}, \\\\&& \\gamma ^{-2} = 1- \\frac{H^2}{\\kappa ^2}\\frac{D}{C} \\varphi ^{\\prime 2}\\,, \\\\&& \\alpha (\\varphi ) = \\frac{d \\ln C^{1/2}}{d\\varphi }, \\\\&& \\delta (\\varphi ) = \\frac{d \\ln D^{1/2}}{d\\varphi } \\,.$ One can solve the system of coupled equations above for $H$ and $\\varphi $ as functions of $N$ .","However, in some cases it is simpler to use (REF ) into () and solve the following disformal master equation: $\\frac{2(1+\\lambda )}{3 B} \\,\\varphi ^{\\prime \\prime } &+& \\left(2\\lambda +1 -\\omega \\right)\\varphi ^{\\prime } + 2 \\lambda \\, \\frac{d\\ln V}{d\\varphi } +2 (1 -3\\,\\omega \\,\\gamma ^{2})\\,\\alpha (\\varphi )\\nonumber \\\\&+ & \\, \\frac{2\\gamma ^2(1+\\lambda )}{3B}\\frac{D \\rho }{C} \\left(\\varphi ^{\\prime \\prime } -3\\,\\varphi ^{\\prime }\\left[ \\omega + \\frac{\\varphi ^{\\prime 2}}{6} + \\frac{(1+\\omega ) B}{2(1+\\lambda )}\\right]+ \\frac{C}{2D}\\left(\\frac{D}{C}\\right)_{\\!,\\varphi } \\varphi ^{\\prime 2} \\right) \\!=0 \\,, \\nonumber \\\\$ with $\\gamma $ given by: $\\gamma ^{-2} = 1- \\frac{(1+\\lambda )}{3B}\\frac{D \\rho }{C} \\varphi ^{\\prime 2}\\,.$ From (REF ) we see that the conformal case is recovered for $D=0$ , when the second line vanishes.","Moreover, the disformal piece appears always together with derivatives of the scalar field, as expected and also nontrivially coupled to the energy density.","This complicates considerably the analysis of the disformal case, as we will see below."],["Modified expansion rate","The effect of the expansion rate during the early time evolution due to the presence of a scalar field can be extracted from the Hubble parameter evolution in the disformal frame defined as: $\\tilde{H} = d (\\log \\tilde{a} )/d\\tilde{\\tau },$ which can be written using (REF ) as: $\\tilde{H} = \\frac{H\\gamma }{C^{1/2}} \\left(1+ \\alpha (\\varphi ) \\varphi ^{\\prime } \\right)\\,,$ where remember that $\\gamma $ depends on $H$ (or $\\rho $ ) as seen from (REF ), while in the pure conformal case $D=0$ and $\\gamma = 1$ .","Note that in principle, the factor $(1+ \\alpha (\\varphi ) \\varphi ^{\\prime } )$ can be positive or negative, indicating an expansion or contraction modified rate.","We stick to positive definite values for this factor and therefore only modified expansion rates, though in principle, one could have a brief contraction period during the early universe evolution, before the onset of BBNSee [18] for a review on scenarios with a possible contraction phase in the early universe..","Moreover, notice that while $\\tilde{H}$ can grow during the cosmological evolution, the null energy condition (NEC) is not violated.","This is because the Einstein frame expansion rate $H$ is dictated by the energy density $\\rho $ and pressure $p$ , which obey the NEC and therefore $\\dot{H}<0$ during the whole evolution, as it should (see for example [19]).","We further want to relate the modified expansion rate to the expected expansion rate in general relativity (GR), that is: $H_{GR}^2 = \\frac{\\kappa _{GR}^2}{3} \\, \\tilde{\\rho }\\,.$ We can do this be using the Friedmann equation (REF ) and the relation between the energy densities (REF ) to write $\\gamma ^{-1} H^2 = \\frac{\\kappa ^2}{\\kappa _{GR}^2} \\frac{ C^2\\,(1+\\lambda )}{B} \\, H_{GR}^2 \\,.$ Using the definition of $\\gamma $ (see (REF )) into this equation, one finds a cubic equation for $H^2$ in terms of all the other parameters.","The real positive solution to that equation can then be replaced into (REF ) to find the modified expansion rate $\\tilde{H}$ , which will thus be a complicated function of $H_{GR}$ as we now see.","The cubic equation for $H$ takes the form: $d_1 H^6-H^4 + d_2^2=0\\,,$ where $d_1=\\frac{D}{C}\\frac{\\varphi ^{\\prime 2}}{\\kappa ^2} \\,, \\hspace{28.45274pt} d_2=\\frac{\\kappa ^2}{\\kappa _{GR}^2}\\frac{C^2 (1+\\lambda )H_{GR}^2}{B}\\,.$ The solutions to (REF ) can be written as $H^2=\\frac{1}{3 d_1}\\left(1+\\left(\\frac{2}{\\Delta }\\right)^{1/3}+\\left(\\frac{\\Delta }{2}\\right)^{1/3}\\right)$ with $\\Delta =2-27d_1^2d_2^2+d_1d_2\\sqrt{27(27d_1^2d_2^2-4)}$ .","The other two solutions can be obtained by replacing $ \\left(\\frac{2}{\\Delta }\\right)^{1/3} \\rightarrow e^{2\\pi i/3} \\left(\\frac{2}{\\Delta }\\right)^{1/3}\\qquad {\\rm and } \\qquad \\left(\\frac{2}{\\Delta }\\right)^{1/3} \\rightarrow e^{4\\pi i/3} \\left(\\frac{2}{\\Delta }\\right)^{1/3}\\,.$ We are interested in real positive solutions for $H^2$ .","One possibility to get this is to have the imaginary part of $(\\Delta /2)^{1/3}$ vanish by requiring that $\\Delta >0$ , which is impossible.","Therefore, the way to obtain real solutions for $H$ is to have the imaginary parts of $(\\Delta /2)^{1/3}$ and $(\\Delta /2)^{-1/3}$ cancel each other, leaving a real positive solution.","For this, we need that $27d_1^2d_2^2 \\le 4$ , which implies the following relation between the conformal and disformal functions: $\\frac{3\\sqrt{3} \\,D C\\, \\varphi ^{\\prime 2} (1+\\lambda )}{B} \\frac{H^2_{GR}}{\\kappa _{GR}^2}\\le 2\\,.$ Under this condition, we can rewrite $\\Delta $ as: $\\Delta = 2-27d_1^2d_2^2+ i d_1d_2\\sqrt{27(4- 27d_1^2d_2^2)}\\,, $ which allows us to define a complex number $Z\\equiv \\Delta /2$ and it is easy to check that $\\bar{Z} = 2/\\Delta $ and thus $|Z|^2 =1$ .","Denoting further $Z_i$ with $i=1,2,3$ denoting the three solutions to $H$ as explained above, the solutions for $H$ , (REF ) takes the simple form: $H^2_i=\\frac{1}{3 d_1}\\left[1+Z_i^{1/3} + \\bar{Z}_i^{1/3}\\right] \\,,$ and remember that we are interested only in the real positive solution.","We can now plug in (REF ), as well as the expression for $\\gamma $ in terms of $H$ into the Jordan frame expansion rate (REF ), can be written as: $\\tilde{H}^2 = \\frac{\\kappa ^2}{\\kappa ^2_{GR}} \\frac{ \\gamma ^3\\, C(1+\\lambda ) (1+\\alpha (\\varphi )\\varphi ^{\\prime })^2 }{B} \\,\\,H_{GR}^2\\,,$ where there is a non-trivial dependence of $H_{GR}$ encoded in $\\gamma _i = \\frac{1}{3d_1d_2}\\left[1+Z_i^{1/3} + \\bar{Z}_i^{1/3}\\right] \\,.$ In the conformal case, $D=0$ , $\\gamma =1$ and therefore (REF ) is simply $\\tilde{H}^2 = \\frac{\\kappa ^2}{\\kappa _{GR}^2}\\frac{C (1+\\lambda ) (1+\\alpha (\\varphi ) \\varphi ^{\\prime })^2}{B}\\, H^2_{GR} \\,.$ From this relation we define a speed-up parameter $\\xi $ , which will be useful below to measure the departures from the $GR$ expansion rate result: $\\xi \\equiv \\frac{\\tilde{H}}{H_{GR}}\\,.$"],["Modifications of the dark matter relic abundances","In this section we discuss the modifications to the DM relic abundance's predictions due to modifications of the expansion rate before the onset of nucleosynthesis caused by the presence of a scalar field conformal and disformally coupled to matter.","We start by revisiting the conformal case, discussed originally in [3]Modifications to the Boltzmann equation due to a conformal coupling in (non-critical) string theory have been discussed in [8].. We first solve (numerically) the master equation for the scalar field (REF ) in order to compute the modified expansion rate $\\tilde{H}$ and compare it with the standard expansion rate, $H_{GR}$ .","We then use this to compute the modifications to the dark matter relic abundances by solving the Boltzmann equation using the modified expansion rate.","We start revisiting by the conformal case by exploring a wide range of initial conditions, masses and cross-sections.","We then look at an explicit disformal example.","Before solving the master equation (REF ), we would like to write it in terms of Jordan frame quantities $\\tilde{\\omega }= \\omega \\gamma ^2$ , $\\tilde{\\rho }= C^{-2}\\gamma ^{-1} \\rho $ .","Moreover, the number of e-folds $N$ can be expressed in terms of Jordan frame quantities as follows.","In this frame, the entropy is conserved and is given by $\\tilde{S}=\\tilde{a}\\, \\tilde{s}$ , where $\\tilde{s}= \\frac{2\\pi }{45} g_s(\\tilde{T}) \\tilde{T}^3$ .","So, the conservation of entropy and (REF ) show that $N$ is a function of temperature and the scalar field as: $N\\equiv \\ln \\frac{a}{a_0}=\\ln \\left[\\frac{\\tilde{T}_0}{\\tilde{T}}\\left(\\frac{g_s(\\tilde{T}_0)}{g_s(\\tilde{T})}\\right)^{1/3}\\right]+\\ln \\left[\\frac{C_0}{C}\\right]^{1/2}.$ Therefore, we can introduce the parameter, $\\tilde{N}$ , defined as $\\tilde{N} \\equiv \\ln \\left[\\frac{\\tilde{T}_0}{\\tilde{T}}\\left(\\frac{g_s(\\tilde{T}_0)}{g_s(\\tilde{T})}\\right)^{1/3}\\right].$ and transform to derivatives w.r.t.","$\\tilde{N}$ (assuming well behaved functions): $\\varphi ^{\\prime } = \\frac{1}{\\left(1-\\alpha (\\varphi ) \\frac{d\\varphi }{d\\tilde{N}}\\right)} \\frac{d\\varphi }{d\\tilde{N}} \\,,\\qquad \\varphi ^{\\prime \\prime } = \\frac{1}{\\left(1-\\alpha (\\varphi ) \\frac{d\\varphi }{d\\tilde{N}}\\right)^3} \\left( \\frac{d^2\\varphi }{d\\tilde{N}^2} + \\frac{d\\alpha }{d\\varphi } \\left(\\frac{d\\varphi }{d\\tilde{N}}\\right)^3\\right)\\,.","\\\\ \\nonumber $ In a slight abuse of notation and to keep expressions neat, in what follows we denote derivatives w.r.t.","$\\tilde{N}$ with a prime $^{\\prime }$ ."],["Conformal case","We start with the pure conformal case.","That is, we take $D(\\phi ) =0$ in (REF ) and therefore $\\gamma =1$ (and $\\tilde{\\omega }= \\omega $ ).","Moreover, during the radiation and matter dominated eras, of interest for us, the potential energy of the scalar field is subdominant and therefore, we take $\\lambda \\sim 0$ .","Therefore the master equation (REF ) simplifies to: $\\frac{2}{3 (1-\\varphi ^{\\prime 2}/6)} \\,\\varphi ^{\\prime \\prime } &+& \\left(1 -\\tilde{\\omega }\\right)\\varphi ^{\\prime } + 2 (1 -3\\,\\tilde{\\omega })\\,\\alpha (\\varphi ) =0,$ which in terms of derivatives wrt $\\tilde{N}$ takes the form: $\\frac{1 }{3B\\left[1-\\alpha (\\varphi ) \\varphi ^{\\prime } \\right]^3} \\left(\\varphi ^{\\prime \\prime } + \\frac{d\\alpha }{d\\varphi } \\left(\\varphi ^{\\prime }\\right)^3\\right)+ \\frac{(1-\\tilde{\\omega }) }{\\left[1-\\alpha (\\varphi )\\varphi ^{\\prime }\\right]} \\,\\varphi ^{\\prime }+(1-3\\,\\tilde{\\omega }) \\, \\alpha (\\varphi )=0 \\,, \\nonumber \\\\$ where $B= 1-\\frac{\\left(\\varphi ^{\\prime }\\right)^2 }{6\\left(1-\\alpha (\\varphi ) \\varphi ^{\\prime }\\right)^2} $ .","Using the relation between $\\tilde{H}$ and $H_{GR}$ defined in (REF ), we can write the speed-up parameter as $\\xi = \\frac{\\tilde{H}}{H_{GR}} = \\frac{C^{1/2}(\\varphi )}{C^{1/2}(\\varphi _0)} \\frac{1}{\\left(1-\\alpha (\\varphi ) \\varphi ^{\\prime }\\right)\\sqrt{B}} \\frac{1}{\\sqrt{1+ \\alpha ^2(\\varphi _0)}}$ where we have used the relation between the bare gravitational constant and that measured by local experiments for conformally coupled theories [20]: $\\kappa _{GR}^2 = \\kappa ^2 C(\\varphi _0) [1+\\alpha ^2(\\varphi _0)]\\,,$ where $\\varphi _0$ is the value of the scalar field at present time."],["Expansion rate modification","The scalar equation in the conformal case (REF ), as function of $N$ (for $\\lambda =0$ ) contains a term which can be interpreted as an effective potential, dictated by $V_{eff} = \\ln C^{1/2}$ .","For a strictly radiation dominated era, $\\tilde{\\omega }=1/3$ , the effective potential term vanishes and we are left with an equation that can be solved analytically [21], giving $\\varphi ^{\\prime } \\propto e^{-N}$ .","That is, any initial velocity will rapidly go to zero (remember that from the Friedmann equation (REF ), $\\varphi ^{\\prime }$ is constrained to be $\\varphi ^{\\prime } \\lesssim \\pm \\sqrt{6}$ ).","Therefore we explore the effects of having a non-zero initial velocity in our analysis below (see also Appendix for further examples).","Since the scalar field is expressed in Planck units, we focus on order one or smaller field variations $\\Delta \\varphi $ .","One can check, using the analytic solution to (REF ) deep in the radiation era, that for initial velocities $\\varphi ^{\\prime }_0 \\ll \\pm \\sqrt{6}$ , the total field displacement is of order $\\Delta \\varphi \\sim \\varphi ^{\\prime }_0$ [21].","However, given that we don't know much about the theory before BBN, we explore different initial values for $(\\varphi _0, \\varphi ^{\\prime }_0)$ and study their consequences.","In particular we explore initial values $\\varphi _0$ and $\\varphi ^{\\prime }_0 \\in (-1.0, 1.0)$ .","We now concentrate on an explicit conformal factor.","We use the same conformal factor as that studied in [3], which is given by: $C(\\varphi ) = (1+ b \\,e^{-\\beta \\, \\varphi })^2$ with the values $b=0.1$ , $\\beta =8$ , which have been shown to satisfy the constraints imposed by tests of gravity, for the parameters $\\alpha , \\beta $ , $\\xi $ .","As we will see, the requirement of reaching the GR expansion rate value by the time of the onset of BBN, drives these parameters to very small values, which are thus consistent with the constraints from gravity for their values today.","As we discussed above, in the equation of motion for $\\varphi $ , with $\\tilde{\\omega }\\ne 1/3 $ , the conformal factor acts as an effective potential on which the scalar field moves, damped by the Hubble friction (see ()).","Since any initial velocity $\\varphi ^{\\prime }$ goes rapidly to zero deep in the radiation era, in the subsequent evolution of $\\varphi $ , the term $(\\varphi ^{\\prime } )^2$ in the master equation will be negligible.","In this regime, the equation is that of a particle moving in an effective potential with a damping term.","Therefore, one can understand the evolution of $\\varphi $ from the form of the effective potential ($V_{eff}=\\ln C^{1/2}$ ) and the initial conditions chosen.","For the conformal function we are considering (REF ), one sees that there is a set of initial conditions that will give rise to an interesting behaviour in $\\varphi $ , and therefore an interesting modified expansion rate in the Jordan frame $\\tilde{H}$ (REF ), as we now explain.","In general, both the initial position and velocity of the scalar field can take any value, positive and negative.","In the runaway effective potential dictated by $\\ln C^{1/2}$ for the conformal factor (REF ) we consider here, we have the following possibilities.","i) The scalar field starting somewhere up in the runaway effective potential with zero initial velocity.","In this case the scalar field will roll-down the potential, eventually stopping due to Hubble friction, at some constant value of $\\varphi $ , which depends on the its initial value.","So long as $(1+\\alpha (\\varphi ) \\,\\varphi ^{\\prime })$ stays positive (see (REF )), $C$ will evolve rapidly towards 1.","More generally, the initial velocity can be different from zero.","If the initial velocity is positive, the behaviour will be similar to the previous case.","The field will roll-down the effective potential towards its final terminal value.","ii) A more interesting possibility arises when one allows for negative initial velocities.","In this case, the field will start rolling-up the effective potential towards smaller values of the field, eventually turning back down and moving towards its terminal value.","It is easy to see that an interesting effect happens when the field starts at an initial positive value.","Given a sufficient initial negative velocity the field will move towards negative values until its velocity becomes zero and then positive again, as it rolls back down the effective potential.","This change in sing for the scalar evolution will produce a pick in the conformal function that will give rise to a non-trivial modification of the Jordan's frame expansion rate $\\tilde{H}$ , as we are looking for.","As mentioned before, we are interested in (sub-)Planckian initial values $(\\varphi _0, \\varphi ^{\\prime }_0)$ , such that $\\tilde{H}>0$ .","With these requirements, one can see that given an initial negative velocity, there is a suitable initial value of the scalar field such that the behaviour just described holds and the expansion rate $\\tilde{H}$ has an interesting evolution before the onset of BBN.","At late times the conformal function goes to one and the GR expansion is recovered.","We show this behaviour explicitly in Figures REF and REF where we plot the numerical solution for the evolution of $\\varphi $ and $C(\\varphi )$ as functions of the temperature.","In these plots we find $\\varphi =\\varphi (\\tilde{T})$ by first solving (REF ) numerically with initial conditions $ (\\varphi _0,\\varphi ^{\\prime }_0) = (0.2, -0.99)$ and then use (REF ) to express $\\varphi (\\tilde{N})$ as function of $\\tilde{T}$ In appendix we show further examples of the thermal evolution of the scalar field.. As we can see, the conformal factor starts growing towards a maximum value as $\\varphi $ moves to negative values, to rapidly drop down towards its GR value at $C\\rightarrow 1$ as $\\varphi $ moves down the effective potential towards positive values.","This non-trivial effect will give rise to the possibility of re-annihilation, as we discuss below.","Figure: Typical evolution of the scalar field as temperature decreases.","The initial values are (ϕ,dϕ/dN ˜)=(0.2,-0.994)(\\varphi ,d\\,\\varphi /d\\,\\tilde{N}) = (0.2,-0.994).Figure: Behaviour of the conformal factor, C(ϕ)C(\\varphi ) as a function of the temperature for the same initial values as in Fig.",".Based on the discussion above, we have solved the master equation (REF ), to find the the scalar field as a function of $\\tilde{N}$ for various initial conditions, where we see the interesting behaviour explained above.","The resulting modified expansion rate and its comparison with the standard case is shown in Figure REF for the same initial conditions as in Figures REF and REF .","In our numerical exploration, we choose initial conditions for which the notch in the expansion rate (see Fig.","REF ) occurs closer to the BBN timeIn appendix we show more examples of modified expansion rate using different initial conditions..","This has interesting consequences for the dark matter annihilation, as we discuss below.","Figure: Comparing the Hubble expansion rate H ˜\\tilde{H} in the Jordan Frame with the standard Hubble expansion rate H GR H_{GR}.The presence of the scalar field enhances and decreases the expansion rate during the radiation dominated era.","This plot corresponds toinitial conditions given by (ϕ 0 ,ϕ 0 ' )=(0.2,-0.994)(\\varphi _0,\\varphi ^{\\prime }_0) = (0.2,-0.994).When solving the master equation (REF ), we have taken into account an important effect that occurs during the radiation dominated era.","Deep in this epoch, the equation of state is given by $\\tilde{\\omega }=1/3$ .","When a particle species in the cosmic soup becomes non-relativistic, $\\tilde{\\omega }$ differs slightly from $1/3$ .","When the temperature of the universe drops below the rest mass of each of the particle types, there are non-zero contributions to $1-3\\tilde{\\omega }$ .","This activates the effective potential, which can be seen in the last term of (REF ), and displaces, or “kicks“ the field along $V_{eff}$ .","To examine this effect in more detail, we start by writing $1-3\\,\\tilde{\\omega }$ during the early stages of the universe as in [3] and [15] $1-3\\,\\tilde{\\omega }= \\frac{\\tilde{\\rho }- 3\\, \\tilde{p}}{\\tilde{\\rho }} =\\sum _{A} \\frac{\\tilde{\\rho }_A - 3\\tilde{p}_A}{\\tilde{\\rho }} +\\frac{\\tilde{\\rho }_m}{\\tilde{\\rho }}\\,,$ where the sum runs over all particles in thermal equilibrium during the radiation dominated era and $\\tilde{\\rho }_m$ is the contribution from the non-relativistic decoupled and pressureless matter.","The summation over all the particle is responsible for the kicking effect discussed above.","Then, a kick function is defined as $\\Sigma (\\tilde{T}) \\equiv \\sum _{A} \\frac{\\tilde{\\rho }_A - 3\\tilde{p}_A}{\\tilde{\\rho }}\\,,$ where the energy density $\\tilde{\\rho }_A$ and pressure $\\tilde{p}_A$ of each type $A$ of particle are given by $\\tilde{\\rho }_A(\\tilde{T}) =\\frac{g_A}{2\\pi ^2}\\int ^\\infty _{m_A}\\frac{\\left(E^2-m_A^2\\right)^{1/2}}{\\exp (E/\\tilde{T})\\pm 1}E^2dE$ $\\tilde{p}_A(\\tilde{T}) =\\frac{g_A}{6\\pi ^2}\\int ^\\infty _{m_A}\\frac{\\left(E^2-m_A^2\\right)^{3/2}}{\\exp (E/\\tilde{T})\\pm 1}E^2dE$ with $g_A$ being the number of internal degrees of freedom of species of type $A$ and the plus (minus) sign in the integral corresponds to fermions (bosons).","To compute (REF ), we use the Standard Model particle spectrum.","In particular, we take into account the top quark, the Higgs boson, Z boson, W bosons, bottom quark, tau lepton, charm quark, charged pions, neutral pion, moun lepton and the electronWe present more details of the kick function in appendix .. As we show in Figure , $\\Sigma $ is mostly zero, except when the kicks happen.","During the radiation dominated era $\\tilde{\\rho }\\simeq \\pi ^2 g_{eff}(\\tilde{T}) \\tilde{T}^4/30$ , where $\\tilde{T}$ is the Jordan frame temperature and $g_{eff}$ is the total number of relativistic degrees of freedom.","Also, during this stage $\\tilde{\\rho }_m$ is negligible and as we have shown $\\Sigma $ is slightly different than zero.","Therefore, we can compute the equation of state from (REF ) as $\\tilde{\\omega }=(1-\\Sigma (\\tilde{T}))/3$ .","In Figure REF we show the evolution of $\\tilde{\\omega }$ between 10 TeV and 10 eV.","This figure shows four troughs, which are the “kicks” mentioned above.","Each kick corresponds to the transition of one or more particles to the non-relativistic regime.","For example, the trough at around 0.5 MeV is due to the electron, while the one at around 100 GeV is due to the heavy particles ($t$ , $H$ , $Z$ and $W$ ).","Towards the end of the radiation era, approaching the transition to the matter dominated era, eq.","(REF ) takes the approximate form: $1-3\\,\\tilde{\\omega }\\simeq \\frac{\\tilde{\\rho }_{m}}{\\tilde{\\rho }_m + \\tilde{\\rho }_r} \\simeq \\frac{1}{1+ \\tilde{T}/\\tilde{T}_{eq}} \\,,$ where $\\tilde{T}_{eq} \\sim {\\cal O}(10^{-9})$ GeV is the temperature at matter-radiation equality, that is, $\\tilde{\\rho }_m(\\tilde{T}_{eq}) = \\tilde{\\rho }_r(\\tilde{T}_{eq})$ .","We can now combine (REF ) and (REF ) to compute the thermal evolution of (REF ) in the radiation dominated and matter dominated eras and use it in the master equation.","Figure: Evolution of ω ˜\\tilde{\\omega } in () as function of temperature during the radiation dominated era."],["Parameter Constraints","In scalar-tensor theories of gravity, there are some constraints on the parameters that need to be taken into account.","Deviations from GR can be parametrised in terms of the post-Newtonian parameters $\\gamma _{PN}$ and $\\beta _{PN}$ , which are given in terms of $\\alpha (\\varphi _0)$ defined in () and its derivative $\\alpha ^{\\prime }_0= d\\alpha / d\\varphi |_{\\varphi _0}$ as [22], [23]: $\\gamma _{PN} -1 = -\\frac{2\\alpha _0^2}{1+\\alpha _0^2} \\,, \\qquad \\beta _{PN} -1 = \\frac{1}{2}\\frac{\\alpha ^{\\prime }_0\\alpha _0^2}{(1+\\alpha _0^2)^2} \\,,$ Solar system tests of gravity, including the perihelion shift of Mercury, Lunar Laser Ranging experiments, and the measurements of the Shapiro time delay by the Cassini spacecraft [24], [25], [26] indicate that $\\alpha _0$ should be very small, with values $\\alpha _0^2 \\lesssim 10^{-5}$ , while binary pulsar observations impose that $\\alpha ^{\\prime }_0\\gtrsim -4.5$ .","The last constraint applies to the the speed-up factor $\\xi $ , which has to be of order 1 before the onset of BBN.","In our examples we have $\\alpha _0^2 \\simeq 2\\times 10^{-5}$ , $\\alpha ^{\\prime }_0 > 0$ and $\\xi \\approx 1.05$ ."],["Impact on relic abundances ","We are now ready to discuss the impact of the modified expansion rates on the relic abundance of dark matter species.","For a dark matter species $\\chi $ with mass $m_\\chi $ and annihilation cross-section $\\langle \\sigma v \\rangle $ , where $v$ is the relative velocity, the dark matter number density $n_\\chi $ evolves according to the Boltzmann equation $\\frac{d n_\\chi }{dt} = -3 \\tilde{H} n_\\chi - \\langle \\sigma v \\rangle \\left( n_\\chi ^2 - (n_\\chi ^{eq})^2 \\right)\\,,$ where, as we have discussed above, the relevant expansion rate is the Jordan frame one, which can give interesting effects due to the presence of the scalar field.","Further $n_\\chi ^{eq}$ is the equilibrium number density.","We can rewrite this equation in terms of $x=m_\\chi /\\tilde{T}$ $\\frac{d Y}{dx} = - \\frac{\\tilde{s} \\langle \\sigma v \\rangle }{x \\tilde{H}} \\left( Y^2 - Y_{eq}^2 \\right) \\,.$ where $Y = \\frac{n_\\chi }{\\tilde{s}}$ , $\\tilde{s}= \\frac{2\\pi }{45} g_s(\\tilde{T}) \\tilde{T}^3$ .","Numerical solutions to the Boltzmann equation (REF ) with the modified expansion rate $\\tilde{H}$ were found for dark matter particles with masses ranging from 5 GeV to 1000 GeV.","For instance, we show solutions in figures REF and REF for two different masses.","As we can see from (REF ), the annihilation cross-section influences the evolution of the abundance $Y$ .","The current value of $Y$ determines the present dark matter content of the universe.","This can be seen clearly by recalling the current value of the energy density parameter $\\Omega _0=\\frac{\\rho _0}{\\rho _{c,0}}=\\frac{m\\,Y_0\\,s_0}{\\rho _{c,0}}$ , where $\\rho _{c,0}$ and $s_0$ are the well-known current values of the critical energy density and the entropy density of the universe, respectively.","So, for each single mass, the thermally-averaged annihilation cross section, $\\langle \\sigma v \\rangle $ , was chosen such as the current DM content of the universe is 27 %, so $\\Omega _0=0.27$ .","In Figure REF we show the annihilation cross-section, $\\langle \\sigma v \\rangle _{Conformal}$ , found for all masses and compare it to the annihilation cross sections for the standard cosmology model, $\\langle \\sigma v \\rangle _{Standard}$ .","As it is shown, for large masses $\\langle \\sigma v \\rangle _{Conformal}$ is larger than $\\langle \\sigma v \\rangle _{Standard}$ , up to a factor of four.","As the mass decreases $\\langle \\sigma v \\rangle _{Conformal}$ decreases up to the point where is smaller than $\\langle \\sigma v \\rangle _{Standard}$ .","Then, for masses smaller than 100 GeV the figure shows that $\\langle \\sigma v \\rangle _{Conformal} \\approx \\langle \\sigma v\\rangle _{Standard}$ .","Thus, we have found that the annihilation cross-sections can be larger or smaller than the thermal average cross-section.","Just to give an example of larger and smaller cross-section, the following table compares the numerical values of $\\langle \\sigma v \\rangle _{Conformal}$ and $\\langle \\sigma v \\rangle _{Standard}$ for two dark matter masses, 1000 GeV and 130 GeV.","Table: NO_CAPTIONFigures REF and REF show the evolution of the abundance $\\tilde{Y}(x)$ for DM particles with masses 130 GeV and 1000 GeV, respectively.","These figures also include the abundance $Y_{GR}(x)$ calculated in the standard cosmology model and the equilibrium abundance $Y_{Eq}(x)$ .","Figure: Evolution of the abundance as temperature changes for a DM particle of mass 130 GeV.The temperature evolution of the abundance for a 130 GeV mass is not noticeable affected by the presence of the scalar field $\\phi $ .","In this case, $\\tilde{Y}$ and $Y_{GR}$ are almost indistinguishable from one another.","On the other hand, the scalar field $\\phi $ has a prominent effect on the temperature evolution of the abundance for a 1000 GeV DM particle.","First of all, the freeze-out happens earlier than expected due to the enhancement of the expansion rate, $\\tilde{H}$ .","Then, an unusual effect appears.","As the temperature decreases, $\\tilde{H}$ becomes smaller than the interaction rateThe interaction rate is defined as $\\tilde{\\Gamma }\\equiv \\langle \\sigma v \\rangle _{Conformal}\\,\\tilde{s}\\,\\tilde{Y}$ .","$\\tilde{\\Gamma }$ and a short period of annihilation starts again called “re-annihilation”.","The re-annihilation process reduces the abundance of dark matter until a second and final freeze-out happens.","After this final freeze-out the abundance remains constant.","Figure: Abundance for a mass of 1000 GeV.Figure: Annihilation cross section as function of mass.","The presence of the scalar field enhances the 〈σv〉\\langle \\sigma v \\rangle for large masses,and diminishes 〈σv〉\\langle \\sigma v \\rangle for masses around 130 GeV, while small mass the effect is almost negligible.Figure: Expansion rate (as in figure ) and interaction rate as function of temperature.","The interaction rate, Γ ˜\\tilde{\\Gamma }, is given by 〈σv〉 Conformal s ˜Y ˜\\langle \\sigma v \\rangle _{Conformal}\\,\\tilde{s}\\,\\tilde{Y}.","We use Y ˜\\tilde{Y} from figures and and the values of 〈σv〉 Conformal \\langle \\sigma v \\rangle _{Conformal} presented previously for 130 GeV and 1000 GeV masses.The re-annihilation phase can be described better by discussing the relation between the expansion rate $\\tilde{H}$ and the interaction rate $\\tilde{\\Gamma }$ .","The first freeze-out happens when $\\tilde{\\Gamma }$ becomes smaller than $\\tilde{H}$ which can be seen in figure REF to happen around a temperature of 50 GeV for a 1000 GeV particle.","Then, near to 7 GeV $\\tilde{H}$ drops below $\\tilde{\\Gamma }$ , and so the re-annihilation process starts and goes on until the second freeze-out occurs.","Around 2 GeV $\\tilde{H}$ becomes much larger than $\\tilde{\\Gamma }$ and so the abundance becomes almost constant.","Our analysis shows that, as found in [3], re-annihilation occurs for this particular choice of conformal factor.","However, we found that when fully integrating the master equation, the re-annihilation occurs only for very large masses of the dark matter particles (in [3] it was found for $m=50$ GeV).","On the other hand, in [15], no re-annihilation was foundAlthough [15] used a different conformal factor to [3], we expect the re-annihilation effect to be present also in that case., which was probably due to the initial conditions used and the values of the DM masses explored."],["Disformal case","We now discuss briefly the effect of the disformal factor in the metric (REF ) to the expansion rate of the universe, $\\tilde{H}$ , and compare it to the conformal modification to $\\tilde{H}$We leave a detailed exploration for a future publication..","Hence, we explore $D(\\phi )\\ne 0$ for the same conformal factor studied before, that is, $C(\\varphi ) = (1+ b \\,e^{-\\beta \\, \\varphi })^2$ for $b=0.1$ , $\\beta =8$ .","To investigate these modifications, we first need to look at the the scalar field evolution with temperature.","In the pure conformal case studied above, we found the thermal evolution of the scalar field by solving the master equation (REF ) numerically, which is (REF ) for $D(\\phi )=0$ .","However, to study the effects of the disformal factor on the scalar field, it is more convenient to solve the system of two coupled equations (REF ) and ().","Using these equations we find solutions for the dimensionless scalar field $\\varphi $ , and for the expansion rate in the Einstein frame $H$ .","Notice that solving the system of coupled equations or solving the master equation to find the thermal evolution of the scalar field are equivalent methods (as we have explicitly checked), because (REF ) it is nothing but a combination (REF ) and ().","However, while in the pure conformal case the master equation can be made independent of $H$ (or $\\rho $ ), this is not the case for the more general disformal case, as we can see in eq.","(REF ).","In the same way as for the conformal case, we are interested mainly in the radiation and matter eras and therefore we can neglect the potential energy of the scalar field.","Thus, we consider $V\\sim 0$ and $\\lambda =0$ .","Also, while solving the coupled equations we have to express $\\omega $ in the Jordan frame by using $\\tilde{\\omega }= \\omega \\gamma ^2$ and transform all derivatives w.r.t.","$N$ to derivatives w.r.t.","$\\tilde{N}$ by using (REF ).","With this information, we solve the system of coupled equations numerically to find the dimensionless scalar field $\\varphi $ and the Hubble parameter $H$ , as functions of the number of e-folds $\\tilde{N}$ (and the temperature).","We choose the same initial conditions for the scalar field and its derivative as in the conformal case and to obtain the initial condition for $H$ , we use (REF ).","Once we have the solutions for $\\varphi $ and $H$ as functions of temperature, we can go back to (REF ) and (REF ) to obtain the expansion rate for the disformal model.","As an example, in Figure REF we show the effects of a disformal factor given by $D(\\varphi ) = D_0\\,\\varphi ^2$ with $D_0 =-4.9\\times 10^{-14}$ .","In this plot, we illustrate the effect of the disformal contribution on the expansion rate ($\\tilde{H}_{Disformal}$ ) and compare it to the modified expansion rate for the conformal case ($\\tilde{H}_{Conformal}$ ) and the standard case ($H_{GR}$ ).","We use the same initial conditions as in Figures REF and REF for the scalar field and its derivative.","Also, it is important to mention that for the case shown the parameter constraints described in section REF are satisfied.","In particular we find $\\alpha _0^2 \\simeq 2\\times 10^{-5}$ , $\\alpha ^{\\prime }_0 > 0$ and $\\xi \\approx 1.02$ .","Figure: Comparing the modified expansion rate of the universe in the disformal and conformal scenarios for the same initial conditions as in Fig.",".From our example, with $C$ and $D$ as indicated above, we can clearly see the differences from the disformally modified expansion rate $\\tilde{H}_{Disformal}$ compared to the conformally modified and standard case, $H_{GR}$ .","The evolution of $\\tilde{H}_{Disformal}$ is similar to that of $\\tilde{H}_{Conformal}$ , having an (two in our example) enhancement and a decrement compared to the standard expansion rate $H_{GR}$ .","Moreover, the main differences with respect to the conformal modification are the position of the notch and its shape.","The notch is moved to higher temperatures and it becomes a little bit sharper.","These differences between the expansion rates can be understood from ().","First, in this equation we see that the factor $\\frac{3H^2\\gamma ^2BD}{\\kappa ^2(1+\\lambda )C}$ in the coefficients of $\\varphi ^{\\prime \\prime }$ , $\\varphi ^{\\prime }$ and $\\varphi ^{\\prime 2}$ vanishes when $D=0$ .","For the disformal example shown in Figure REF , this factor is a very small correction to the equation, which is reflected in the slight shape modification of $\\tilde{H}_{Disformal}$ compared to $\\tilde{H}_{Conformal}$ .","Second, the term proportional to $\\delta (\\varphi )$ plays a more important role, being responsible for the shifting of the notch.","As was discussed previously in section REF , a modification to the expansion rate of the universe prior to BBN has tremendous consequences on the abundance, $\\tilde{Y}$ , of dark matter particles.","Also, this modification implies that the thermally-averaged annihilation cross section, $\\langle \\sigma v \\rangle $ , for dark matter particles differs significantly from the one predicted by the standard cosmological model, which is approximately $3.0 \\times 10^{-26}\\rm {cm^3/s}$ (see Figure REF ).","We have seen than the enhancement of $\\tilde{H}$ allows bigger values of $\\langle \\sigma v \\rangle $ for particles with masses within certain range, and also, the lower value of $\\tilde{H}$ implies smaller $\\langle \\sigma v \\rangle $ for particles with masses within a small interval.","Thus, the location and shape of the notch determines for which masses the annihilation cross section is smaller.","In the disformal scenario, the notch has been moved to higher temperatures, which allows particles with higher masses to have smaller and larger annihilation cross sections for the observed DM content.","We leave the detailed study of the modification of $\\tilde{H}$ for more general conformal and disformal factors for a separate forthcoming publication."],["Conclusions","Scalar-tensor theories of gravity are a useful method to explore departures of the expansion rate of the universe from the standard cosmological model in the early universe.","The expansion rate of the universe had a strong influence on the evolution of the dark matter abundance during the early stages of the universe's evolution, specially prior to BBN.","Modifications to the expansion rate during that time would be reflected in the calculation of the dark matter relic abundance and so can be used as a probe to the predictions of scalar-tensor theories.","In this paper, we explored the role played by the scalar field in the modification of the expansion rate of the universe on scalar-tensor theories coupled both conformally and disformally to matter.","For the conformal case, we explored a conformal factor of the type $C=(1+b\\,e^{-\\beta \\,\\varphi })^2$ .","We made no approximations and solved numerically the master equation for the scalar (REF ) for a suitable range of initial conditions.","Using this result we then computed the expansion rate modification $\\tilde{H}$ under the presence of the scalar field during the radiation dominated era prior to BBN.","When comparing the expansion rate, $\\tilde{H}$ , to the standard expansion rate, $H_{GR}$ , we found that the speed-up factor, $\\xi =\\frac{\\tilde{H}}{H_{GR}}$ , increases up to 200 and then become of order 1 prior to BBN (see Fig.","REF ).","Previously, in reference [15], it was shown that there is no change in the expansion rate arising from the modification due to $C$ compared to the standard cosmology, after satisfying all the constraints based on the boundary conditions chosen in [3], [4].","This enhancement on the expansion rate has important consequences on the evolution of the abundance of dark matter particles.","So, we also investigated the effect on the abundance of dark matter particles.","We observed that for dark matter particles of large mass ($m\\sim 10^3$ GeV in our example, compared to the $m\\sim 50$ GeV reported in [3] for which we found no re-annihilation), the particles undergo a second annihilation process and then freeze-out once and for all in (see Fig.","REF ).","Moreover, we found that for large masses the annihilation cross-section has to be up to four times larger than that of standard cosmology models in order to satisfy the dark matter content of the universe of 27 %.","On the other hand, for small masses this re-annihilation process is not present, but we found that for masses around 130 GeV, the annihilation cross-section can be smaller than the annihilation cross-section for the standard cosmological model (see Fig.","REF ).","We also started to investigate the effects on the early evolution of the expansion rate of a disformal factor in the metric (REF ).","We noticed that in order to have a consistent solution, i.e.","a real positive $\\tilde{H}$ , the conformal and disformal factors need to satisfy a very specific relation, (REF ).","We studied the effect of a disformal factor by turning on a small disformal contribution to the conformal case, given by $D(\\varphi )=-4.9\\times 10^{-14}\\varphi ^2$ .","To find the modified expansion rate $\\tilde{H}$ during the radiation dominated era prior to BBN, we solved numerically the disformal system of coupled equations () and (REF ).","We found that when the disformal function is turned on, $\\tilde{H}$ has a very similar profile as for pure conformal case with an enhancement and a notch compare to the standard expansion rate.","However, the position of the notch changes and there may be a second enhancement of $\\tilde{H}$ compared to $H_{GR}$ , depending on the initial conditions (see also Appendix ).","The disformal factor, is moving the notch to higher temperatures which means that we can have larger and smaller annihilation cross-section for any mass of the DM candidate for the observed DM content.","We have shown the analysis of the disformal case in a concrete example.","We plan to study the disformal effects in more detail in order to asses their general features in a forthcoming publication.","Moreover, as we mentioned at the beginning of the paper, we are aiming at models that can be embedded in a more fundamental theory of gravity such as string theory.","We plan to analyse these cases in a future publication.","We would like to thank, David Cerdeño and Nicolao Fornengo for discussions and Gianmassimo Tasinato for discussions and comments on the manuscript.","BD and EJ acknowledge support from DOE Grant DE-FG02-13ER42020."],["Numerical Implementation","In this section we describe in detail our numerical method to study the master equation and expansion rate evolution and present additional examples for different initial conditions.","As we explained in section REF the result shown in Figure REF for $\\tilde{\\omega }(\\tilde{T})$ was obtained using the so-called “kick” function, defined in (REF ).","After using $\\tilde{\\rho }\\simeq \\pi ^2 g_{eff}(\\tilde{T}) \\tilde{T}^4/30$ , (REF ) and (REF ) $\\Sigma $ becomes $\\Sigma (\\tilde{T}) = \\sum _{A} \\frac{15}{\\pi ^4} \\frac{g_A}{g_{eff}(\\tilde{T})} \\,y_A^2 \\int _{y_A}^{\\infty } dx \\frac{\\sqrt{x^2-y^2_A}}{e^x\\pm 1}\\,,$ where $g_{eff}(\\tilde{T})$ is the total number of relativistic degrees of freedomTo calculate $g_{eff}$ we follow the numerical procedure described in Appendix A of [27]., $g_A$ is the number of internal degrees of freedom of particles of type $A$ , $y_A=m_A/\\tilde{T}$ and the plus (minus) sign in the integral is for fermions (bosons).","The particle spectrum used to compute $\\Sigma $ is shown in Table REF and in Figure we plot the resulting kick function during the radiation dominated era.","Then, we compute the equation of state from (REF ) as $\\tilde{\\omega }=(1-\\Sigma (\\tilde{T}))/3$ , which is shown in Figure REF .","Table: Spectrum of particles used to calculate the kick function ().","For each particle, we show its mass and number of internal degrees of freedom, g A g_A.Figure: Thermal evolution of the kick function during the radiation dominated era.","Outside the interval of temperatures shown, Σ\\Sigma vanishes."],["Conformal case solutions","Using the numerical solution for $\\tilde{\\omega }$ , we plugged it into the master equation, and solve it numerically for the interesting initial conditions as described in section REF for the conformal case.","To look for suitable solutions, we fixed the initial value of the velocity and then look for an initial value of the scalar such that $(1+\\alpha (\\varphi )\\varphi ^{\\prime })$ stayed positive, thus giving a positive modified expansion rate.","As we explained in that section, we were aiming at solutions where the scalar field passed from positive to negative to positive values again.","In Figure REF , we show the thermal evolution of the scalar field in the pure conformal scenario for different initial velocities $\\varphi ^{\\prime }$ .","We let the initial value of $\\varphi $ , fixed at $\\varphi _i=0.2$ and solve the master equation (REF ) for the different initial velocities and initial temperature to be 1000 GeV.","This plot shows the behaviour of the scalar field as described in section REF .","In Figure REF , we show the resulting modified expansion rates for the different initial conditions considered in Fig.","REF ."],["Disformal case solutions","As discussed in the main text, when both conformal and disformal functions are non-zero, we solve the system of coupled equations () and (REF ) numerically with Mathematica.","We explored different boundary conditions for the scalar field and its first derivative.","With the solutions of these equations we used (REF ) to find the modified expansion rate.","In Figure REF we show the modified expansion rates for a disformal factor given by $D=D_0\\varphi ^2$ with $D_0=-4.9\\times 10^{-14}$ and the conformal factor being the same as before, that is $C= (1+b e^{-\\beta \\varphi })^2$ with $b=0.1$ , $\\beta = 8$ .","We have added one additional initial condition with respect to the conformal case, $\\varphi ^{\\prime }_{initial}=-1.$ It is interesting to notice that for this initial condition, the pure conformal case does not give a solution satisfying the necessary constraints explained in the main text.","In this sense, we see that the disformal contribution is important in order to find solutions otherwise excluded.","Figure: Modified expansion rate as function of temperature in the disformal scenario for various initial conditions.We use ϕ i =0.2\\varphi _{i}= 0.2 and every curve shown corresponds to a different value of ϕ i ' \\varphi ^{\\prime }_i"],["General Disformal Set-Up","The general scalar-tensor action coupled to matter, which can include a realisation in string theory compactifications is given by: $S=S_{EH} + S_\\phi + S_{m}\\,,$ where: $&& S_{EH}=\\frac{1}{2\\kappa ^2} \\!\\!\\int {\\!d^4x\\sqrt{-g}\\,R}, \\\\&& S_\\phi = - \\!\\!\\int {\\!d^4x\\sqrt{- g} \\left[\\frac{b}{2} (\\partial \\phi )^2+ M^4 C_1 ^2(\\phi )\\sqrt{1+\\frac{D_1(\\phi )}{C_1(\\phi )} (\\partial \\phi )^2}+V(\\phi )\\right]} \\,,\\\\&& S_{m} = - \\!\\!\\int {\\!d^4x\\sqrt{-\\tilde{g}} \\,{\\cal L}_{DM}(\\tilde{g}_{\\mu \\nu }) } \\,,$ and the disformally coupled metric is given by $\\tilde{g}_{\\mu \\nu } = C_2(\\phi ) g_{\\mu \\nu } + D_2(\\phi ) \\partial _\\mu \\phi \\partial _\\nu \\phi \\,.$ $b$ is a constant equal to 1 or 0, depending on the model one wants to consider; $C_i(\\phi ), D_i(\\phi )$ are functions of $\\phi $ , which can be identified as conformal and disformal couplings of the scalar to the metric, respectively.","Finally, we have introduced the mass scale $M$ to keep units right (remember that the conformal coupling is dimensionless, whereas the disformal has units of $Mass^{-4}$ .)","The connection of the general action (REF ) to the different models in the literature can be obtained as follows: the case $C_1 = D_2$ , $D_1=D_2$ , $b=0$ arises when considering a D-brane moving along an extra dimension.","This case was studied in [17] as a model of a coupled dark matter dark energy sector scenario, where scaling solutions arise naturally.","Note that in this case, the kinetic term for the scalar field, identified with dark energy for example, is automatically non-canonical and dictated by the DBI action (see [17]).","On the other hand, phenomenological models considering a disformal coupling between matter and a scalar field, usually consider a canonical kinetic term, and therefore, in that case, $C_1=D_1=0$ and $b=1$ ($C_1$ can be taken to be non-zero and will be part of the scalar potential).","Furthermore, the widely studied case of a conformal coupling is obtained for $b=1, C_1=D_1=D_2=0$ or, as in the case of a D-brane for exampleFor the system corresponding to a D-brane moving in a typically warped compactification in string theory, the functions $C(\\phi )$ and $D(\\phi )$ are identified with powers of the so-called warp factor, usually denoted as $h(\\phi )$ .","In this approach, the longitudinal and transverse fluctuations of the D-brane are identified with the dark matter and dark energy fluids respectively [17]., simply considering small velocities with $b=0, C_1=C_2$ and $D_1=D_2$ , and normalising canonically the scalar field (see Appendix ).","Finally, let us clarify further our nomenclature on frames.","The action in (REF ) is written in the Einstein frame, which in string theory, is usually related to the frame in which the dilaton and the graviton degrees of freedom are decoupled.","From this point of view, the dilaton field as well as all other moduli fields not relevant for the cosmological discussion are considered as stabilised, massive, and are therefore decoupled from the low energy effective theory.","In the literature of scalar-tensor theories however (including conformal and disformal couplings), the Einstein and Jordan frames are identified with respect to the (usually single) scalar field to which gravity is coupled.","In this paper, we follow this and call “Jordan\" or “disformal frame\" the frame in which dark matter is coupled only to the metric $\\tilde{g}_{\\mu \\nu }$ , rather than to the metric $g_{\\mu \\nu } $ and a scalar field $\\phi $ .","The equations of motion obtained from (REF ) are (REF ): $R_{\\mu \\nu } -\\frac{1}{2}g_{\\mu \\nu } R = \\kappa ^2\\left(T^\\phi _{\\mu \\nu } + T^{DM}_{\\mu \\nu }\\right)\\,,$ where in the frame relative to $g_{\\mu \\nu }$ the energy momentum tensors are defined in (REF ) and (REF ).","The energy-momentum tensor for the scalar field in the general case is modified from (REF ) to: $T_{\\mu \\nu }^{\\phi } = - g_{\\mu \\nu } \\left[M^4C_1^2 \\gamma _1^{-1} + \\frac{b}{2} (\\partial \\phi )^2+ V \\right]+ \\left(M^4 C_1D_1 \\,\\gamma _1+ b\\right) \\partial _\\mu \\phi \\, \\partial _\\nu \\phi $ where now the energy density and pressure are given by: $\\rho _\\phi =- \\frac{b}{2}(\\partial \\phi )^2 + M^4C_1^2 \\gamma _1 + V \\,, \\qquad P_\\phi = - \\frac{b}{2}(\\partial \\phi )^2 - M^4C_1^2\\gamma _1^{-1} - V \\,,$ and the “Lorentz factor\" $\\gamma _1$ introduced above is defined by $\\gamma _1 \\equiv \\left(1+ \\frac{D_1}{C_1}\\, (\\partial \\phi )^2\\right)^{-1/2}\\,.$ We can rewrite (REF ) in a more succinct way, by defining ${\\mathcal {V}} \\equiv V + C_1^2 M^4$ $\\rho _\\phi =- \\left[\\frac{b}{2}+ \\frac{M^4C_1 D_1\\gamma _1}{\\gamma +1}\\right] (\\partial \\phi )^2 + {\\mathcal {V}} \\,, \\qquad P_\\phi = - \\left[\\frac{b}{2}+ \\frac{M^4C_1 D_1\\gamma _1^{-1}}{\\gamma +1}\\right] (\\partial \\phi )^2 - {\\mathcal {V}} \\,.$ The equation of motion for the scalar field becomes (compare with (REF )) $&&\\hspace{-28.45274pt}- \\nabla _\\mu \\!\\left[(M^4 D_1C_1\\gamma _1\\, +b)\\,\\partial ^\\mu \\phi \\right] \\!+ \\!\\frac{\\gamma _1^{-1}M^4 C_1^2}{2} \\!\\left[\\!\\frac{D_1^{\\prime }}{D_1} +3\\frac{C_1^{\\prime }}{C_1} \\right] \\!+ \\!\\frac{\\gamma _1 \\,M^4 C_1^2}{2} \\!\\left[\\!\\frac{C_1^{\\prime }}{C_1} -\\frac{D_1^{\\prime }}{D_1} \\right] \\!+\\!","V^{\\prime }\\nonumber \\\\&& \\hspace{85.35826pt}- \\frac{T^{\\mu \\nu }}{2}\\!\\left[\\frac{C_2^{\\prime }}{C_2} g_{\\mu \\nu } +\\frac{D_2^{\\prime }}{C_2}\\partial _\\mu \\phi \\partial _\\nu \\phi \\right]+\\nabla _\\mu \\left[\\frac{D_2}{C_2}T^{\\mu \\nu } \\partial _\\nu \\phi \\right] =0\\,.","\\nonumber \\\\$ Finally, the energy-momentum conservation equation gives rise to (REF ), where $Q$ now is given in terms of $C_2, D_2$ : $Q\\equiv \\nabla _\\mu \\left[\\frac{D_2}{C_2} \\,T^{\\mu \\lambda } \\,\\partial _\\lambda \\phi \\right] - \\frac{T^{\\mu \\nu } }{2} \\left[\\frac{C_2^{\\prime }}{C_2} g_{\\mu \\nu } +\\frac{D_2^{\\prime }}{C_2} \\,\\partial _\\mu \\phi \\,\\partial _\\nu \\phi \\right]\\,.$"],["General cosmological equations","The equations of motion for the general system in an FRW background become: $&& H^2 =\\frac{\\kappa ^2}{3} \\left[\\rho _\\phi +\\rho \\right]\\,, \\\\&& \\dot{H} + H^2 = -\\frac{\\kappa ^2}{6}\\left[ \\rho _\\phi + 3P_\\phi +\\rho +3 P \\right]\\,,\\\\&& \\ddot{\\phi }\\left[1+ \\frac{b}{M^4C_1D_1\\gamma _1^3}\\right]+3H\\dot{\\phi }\\,\\gamma _1^{-2}\\left[\\frac{b}{M^4C_1D_1\\gamma _1}+ 1\\right] \\nonumber \\\\&&\\hspace{14.22636pt}+ \\frac{C_1}{2D_1}\\left(\\gamma _1^{-2}\\left[\\frac{5 C_1^{\\prime }}{C_1} - \\frac{D_1^{\\prime }}{D_1}\\right]+ \\frac{D_1^{\\prime }}{D_1}- \\frac{C_1^{\\prime }}{C_1} -4\\gamma ^{-3}_1 \\frac{C^{\\prime }}{C} \\right)+ \\frac{1}{M^4C_1D_1\\gamma _1^{3}}\\, ({\\mathcal {V}}^{\\prime }+Q_0) =0 \\,, \\nonumber \\\\$ where, $H= \\frac{\\dot{a}}{a}$ , dots are derivatives with respect to $t$ , $^{\\prime }=d/d\\phi $ and $\\gamma _1= (1-D_1 \\,\\dot{\\phi }^2/C_1)^{-1/2}.$ We also have the continuity equations for the scalar field and matter given by $&&\\dot{\\rho }_\\phi + 3H(\\rho _{\\phi }+P_{\\phi }) = -Q_0\\dot{\\phi }\\,, \\\\&&\\dot{\\rho }+ 3H(\\rho +P) = Q_0\\,\\dot{\\phi }\\,.$ where $Q_0$ is given by $Q_0 = \\rho \\left[ \\frac{D_2}{C_2} \\,\\ddot{\\phi }+ \\frac{D_2}{C_2} \\,\\dot{\\phi }\\left(\\!3H + \\frac{\\dot{\\rho }}{\\rho } \\right) \\!+ \\!\\left(\\!\\frac{D_2^{\\prime }}{2C_2}-\\frac{D_2}{C_2}\\frac{C_2^{\\prime }}{C_2}\\!\\right) \\dot{\\phi }^2 +\\frac{C_2^{\\prime }}{2\\,C_2} (1-3\\,\\omega )\\right].","\\nonumber \\\\$ Using () we can rewrite this in a more compact and useful form as $Q_0 = \\rho \\left( \\frac{\\dot{\\gamma }_2}{\\dot{\\phi }\\, \\gamma _2} + \\frac{C_2^{\\prime }}{2C_2} (1-3\\,\\omega \\,\\gamma _2^2) -3H\\omega \\,\\frac{(\\gamma _2 -1) }{\\dot{\\phi }}\\right) \\,,$ where $\\gamma _2= (1-D_2 \\,\\dot{\\phi }^2/C_2)^{-1/2}.$ Plugging this into the (non-)conservation equation for dark matter (), gives: $\\dot{\\rho }+ 3H (\\rho + P\\,\\gamma _2^{2}) = \\rho \\left[\\frac{\\dot{\\gamma }_2}{\\gamma _2} + \\frac{C_2^{\\prime } }{2C_2} \\,\\dot{\\phi }\\, (1-3\\,\\omega \\gamma _2^2)\\right]\\,.$ The energy densities and pressures in the Einstein and Jordan frames are now related similarly to (REF ), replacing $\\gamma \\rightarrow \\gamma _2$ : $\\tilde{\\rho }= C_2^{-2} \\gamma _2^{-1} \\rho \\,, \\qquad \\tilde{P} = C_2^{-2}\\gamma _2\\, P,$ and therefore the equation of states in both frames are related by $\\tilde{\\omega }= \\omega \\gamma _2^2$ .","Similarly the physical proper time and the scale factors in the two frames are related via $\\gamma _2$ : $\\tilde{a} = C_2^{1/2} a \\,, \\qquad \\quad d\\tilde{\\tau }= C_2^{1/2} \\gamma _2^{-1} d\\tau \\,.$ Defining the disformal frame Hubble parameter $\\tilde{H} \\equiv \\frac{d \\ln {\\tilde{a}}}{d\\tilde{\\tau }}$ , gives: $\\tilde{H} = \\frac{\\gamma _2}{C_2^{1/2}}\\left[ H + \\frac{C_2^{\\prime }}{2C_2}\\dot{\\phi }\\right]\\,.$ To solve the equations of motion one now can proceed as in section REF to write the equations in terms of derivatives w.r.t.","the number of e-folds $N$ and consider different cases by choosing appropriately the parameters $b,C_i,D_i$ .","We leave the analysis of these for a future publication."],["The conformal case in D-brane scenarios","In this section we show how to recover the pure conformal case from the D-brane picture, that is, $b=0$ , $C_1=C_2$ , $D_1=D_2$ .","We start by expanding the square root in the scalar part of the action (REF ).","Doing this we get $S_\\phi & =& -\\int {d^4x \\sqrt{-g} \\left[ M^4C_1^2 \\left( 1+ \\frac{D_1}{2C_1} (\\partial \\phi )^2 + \\dots \\right) + V(\\phi )\\right]}\\nonumber \\\\&=& -\\int {d^4x \\sqrt{-g} \\left[ \\frac{M^4C_1 D_1}{2} (\\partial \\phi )^2 + M^4C_1^2(\\phi )+ V(\\phi ) + \\dots \\right]}\\nonumber \\\\&=& -\\int {d^4x \\sqrt{-g} \\left[ \\frac{M^4C_1 D_1}{2} (\\partial \\phi )^2 + {\\mathcal {V}}(\\phi ) + \\dots \\right]} \\,,$ On the other hand, the matter Lagrangean takes the form $S_{DM} &=& -\\int {d^4x \\sqrt{-\\tilde{g}} \\,{\\cal L}_{DM}(\\tilde{g}_{\\mu \\nu })}\\nonumber \\\\&=& -\\int {d^4x \\sqrt{-g} \\, C_1^2(\\phi ) \\left(1+ \\frac{D_1}{2C_1} (\\partial \\phi )^2 + \\dots \\right){\\cal L}_{DM}( \\tilde{g}_{\\mu \\nu })} \\nonumber \\\\ &=& -\\int {d^4x \\sqrt{- g} \\, C_1^2 (\\phi ) {\\cal L}_{DM}( \\tilde{g}_{\\mu \\nu }) + \\dots } =-\\int {d^4x \\sqrt{- \\tilde{g}} \\, {\\cal L}_{DM}( \\tilde{g}_{\\mu \\nu }) + \\dots } \\nonumber \\\\$ where now $\\tilde{g}_{\\mu \\nu } = C_1(\\phi ) g_{\\mu \\nu }$ (and we have used that $\\det \\tilde{g}_{\\mu \\nu } = C_1^4 (1+ D_1/C_1 (\\partial \\phi )^2)$ ).","Finally, to compare the D-brane case with the pure conformal case, we need to canonically normalise $\\phi $ .","Calling $\\varphi $ the canonically normalised field, this is obtained from $\\phi $ as $\\varphi = \\int M^2\\sqrt{D_1C_1} \\, d\\phi \\,.$ It is clear that when $D_1=1/(M^4 C_1)$ , $\\varphi =\\phi $ and therefore the action for the scalar field (REF ) is already in the required form.","We can now take the limit $\\gamma \\rightarrow 1$ into the equations of motion (REF )-() and make the identification $D_1=1/(M^4C_1)$ to recover the conformal case equations of motion.","Note that in this limit $Q_0 \\rightarrow \\rho C_1^{\\prime }/2C_1$ , and is independent of $D_1$ (see (REF ) with $C_1=C_2, D_1=D_2$ )."]],"string":"[\n [\n \"Dark Matter Relics and the Expansion Rate in Scalar-Tensor Theories\"\n ],\n [\n \"Abstract We study the impact of a modified expansion rate on the dark matter relic abundance in a class of scalar-tensor theories.\",\n \"The scalar-tensor theories we consider are motivated from string theory constructions, which have conformal as well as disformally coupled matter to the scalar.\",\n \"We investigate the effects of such a conformal coupling to the dark matter relic abundance for a wide range of initial conditions, masses and cross-sections.\",\n \"We find that exploiting all possible initial conditions, the annihilation cross-section required to satisfy the dark matter content can differ from the thermal average cross-section in the standard case.\",\n \"We also study the expansion rate in the disformal case and find that physically relevant solutions require a nontrivial relation between the conformal and disformal functions.\",\n \"We study the effects of the disformal coupling in an explicit example where the disformal function is quadratic.\"\n ],\n [\n \"Introduction\",\n \"The most recent cosmological observations support the so called $\\\\Lambda CDM$ model of cosmology.\",\n \"These observations provide overwhelming evidence for the existence of non-baryonic (cold) dark matter (DM), constituting about $27\\\\%$ of the Universe's energy density budget, while another $\\\\sim 68\\\\%$ is believed to be in the form of dark energy, which can simply be a cosmological constant, $\\\\Lambda $ .\",\n \"The other $\\\\sim 5\\\\%$ being formed by baryonic matter, described by the standard model (SM) of particles.\",\n \"A popular framework to understand the origin of DM is the thermal relic scenario.\",\n \"In this scenario, at very early times when the universe was at a very high temperature, thermal equilibrium was obtained and the number density of DM particles $\\\\chi $ was roughly equal to the number density of photons.\",\n \"During equilibrium the dark matter number density decayed exponentially as $n_\\\\chi ^{eq} \\\\sim e^{-m_\\\\chi /T}$ for a non-relativistic DM candidate, where $m_\\\\chi $ is the mass of the DM particle $\\\\chi $ .\",\n \"As the universe cooled down as it expanded, DM interactions became less frequent and eventually, the DM interaction rate dropped below the expansion rate ($\\\\Gamma _\\\\chi < H$ ).\",\n \"At this point the density number froze-out and the universe was left with a “relic\\\" of DM particles.\",\n \"Therefore, the dependence of the number density at the time of freeze-out is crucial to determine the DM relic abundance.\",\n \"The longer the DM particles remain in equilibrium, the lower its density will be at freeze-out and vice-versa.\",\n \"In the standard $\\\\Lambda CDM$ scenario, particle freeze-out happens during the radiation era and DM species with weak scale interaction cross-section freeze-out with an abundance that matches the present observed value.\",\n \"The weakness of the interactions is reflected in the predicted thermally-averaged annihilation cross section, $\\\\langle \\\\sigma v \\\\rangle $ , which is around $3.0\\\\times 10^{-26}cm^3s^{-1}$ .\",\n \"Despite such a small value, the Fermi-LAT and Planck experiments have been exploring upper bounds on $\\\\langle \\\\sigma v \\\\rangle $ (see [1], [2]).\",\n \"From observations, it appears that the annihilation cross-section can be smaller than the thermal average value for lower dark matter masses ($\\\\le 100$ GeV), whereas an annihilation cross-section larger than the thermal average value can still be allowed for larger DM mass.\",\n \"If, from future measurements, $\\\\langle \\\\sigma v \\\\rangle \\\\ne 3.0\\\\times 10^{-26}cm^3s^{-1}$ is established, what can we say about the origin of the dark matter?\",\n \"Can it still be the thermal dark matter or do we need non-thermal origin of dark matter?\",\n \"In the case of non-thermal origin, the DM can arise from the decay of a heavy particle, e.g., moduli, and can satisfy the DM content with any value of $\\\\langle \\\\sigma v \\\\rangle $ .\",\n \"The primary motivation of this paper is to find out whether the DM content can still have a thermal origin with larger or smaller $\\\\langle \\\\sigma v \\\\rangle $ by utilising non-standard cosmology.\",\n \"The phenomenological $\\\\Lambda CDM$ model complemented with the inflationary paradigm to provide the seeds of large scale structure, is very successful in describing our current universe.\",\n \"However, the physics describing the universe's evolution from the end of inflation (reheating) to just before big-bang nucleosynthesis (BBN) ($t\\\\lesssim 200$ s) remains relatively unconstrained.\",\n \"During this period the universe may have gone through a “non-standard\\\" period of expansion, and still be compatible with BBN.\",\n \"If such modification happened during DM decoupling, the DM freeze-out may be modified with measurable consequences for the relic DM abundances.\",\n \"Departures from the standard cosmology between reheating and BBN will mainly be a consequence of a modified expansion rate ($\\\\tilde{H}$ ), due to a modification of General Relativity (GR).\",\n \"Such modifications are well motivated by attempts to embed the $\\\\Lambda CDM$ and inflation models into a fundamental theory of gravity and particle physics, such as theories with extra-dimensions, supergravity and string theory.\",\n \"Indeed, our main motivation in this paper is to develop further tools that may allow us to connect such fundamental theories with observations.\",\n \"In this paper our approach will be mostly phenomenological, but we have in mind a scenario that can be derived in the context of a fundamental theory of gravity, such as string theory.\",\n \"Furthermore, we will be concerned with modifications to the standard picture due to the presence and interactions with scalar fields only.\",\n \"Modifications to the relic abundances were first discussed by Catena et al.\",\n \"[3] in the context of conformally coupled scalar-tensor theories (ST), such as generalisations of the Brans-Dicke theory.\",\n \"Further studies on conformally coupled ST models have been performed in the last years in [4], [5], [6], [7] (see also [8], [9], [10], [11], [12], [13], [14]) with the most recent work of [15] where it is shown that the boundary conditions used in [3] cannot have $\\\\langle \\\\sigma v \\\\rangle \\\\ne 3.0\\\\times 10^{-26}cm^3s^{-1}$ .\",\n \"Indeed ST theories where dark matter and dark energy are correlated constitute an attractive way to address the dark matter and dark energy problems via an attraction mechanism towards standard general relativity (GR) [3].\",\n \"In the context of ST theories, the most general physically consistent relation between two metrics in the presence of a scalar field, is given byIn the more general case, $C$ and $D$ can be functions of $X=\\\\frac{1}{2} (\\\\partial \\\\phi )^2$ as well.\",\n \"However, we will not consider this case in the present paper.\",\n \"[16]: $\\\\tilde{g}_{\\\\mu \\\\nu } = C(\\\\phi ) g_{\\\\mu \\\\nu } + D(\\\\phi ) \\\\partial _\\\\mu \\\\phi \\\\partial _\\\\nu \\\\phi \\\\,.$ The first term in (REF ) is the well-known conformal transformation which characterises the Brans-Dicke class of scalar-tensor theories explored in [3], [4], [5], [6], [7], [15].\",\n \"However, in reference [15], it is shown that there is no change in the thermal cross-section arising from the conformally modified metric compared to the standard cosmology after satisfying all the constraints based on the boundary conditions chosen in [3], [4].\",\n \"The second term is the disformal contribution, which is generic in extensions of general relativity.\",\n \"In particular, it arises naturally in D-brane models, as discussed in [17] in a model of coupled dark matter and dark energy.\",\n \"In this paper we revisit the expansion rate modification and impact on the DM relic abundances for the conformal case, providing new interesting results with new boundary conditions to show that $\\\\langle \\\\sigma v \\\\rangle \\\\ne 3.0\\\\times 10^{-26}cm^3s^{-1}$ , while satisfying the DM content.\",\n \"We further discuss the general modifications to the expansion rate and Boltzmann equation, due to the disformal coupling and present an explicit non-trivial example for the case in which the conformal term in (REF ) is a monomial.\",\n \"The paper is organised as follows.\",\n \"We start in section introducing the scalar-tensor theory conformally and disformally coupled to matter.\",\n \"Then, we examine the formulation of this theory in the Einstein and Jordan frames, comment on their physical interpretation and derive the equations that describe the cosmological evolution of the Universe.\",\n \"Subsequently, after discussing the expansion rate modifications caused by the presence of the conformal and disformally coupled scalar field in section , we investigate in detail its impact on the dark matter relic abundance by exploring a concrete pure conformal example as well as an example where we also turn on the disformal contribution.\",\n \"Finally, in section we conclude.\"\n ],\n [\n \"The scalar-tensor theory set-up\",\n \"We are interested in scalar-tensor theories coupled to matter both conformally and disformally [16].\",\n \"Our motivation comes from theories with extra dimensions and in particular string theory compactifications, where several additional scalar fields appear, from closed and open string theory sectors of the theory [17].\",\n \"Our approach in this paper nonetheless, will be phenomenological and therefore our equations will be simplified.\",\n \"However, we present the more general set-up, which can accommodate a realisation from concrete string theory compactifications in appendix and .\",\n \"The action we want to consider is given by: $S_{EH}=\\\\frac{1}{2\\\\kappa ^2} \\\\!\\\\!\\\\int {\\\\!d^4x\\\\sqrt{-g}\\\\,R}- \\\\!\\\\!\\\\int {\\\\!d^4x\\\\sqrt{- g} \\\\left[\\\\frac{1}{2} (\\\\partial \\\\phi )^2+V(\\\\phi )\\\\right]}- \\\\!\\\\!\\\\int {\\\\!d^4x\\\\sqrt{-\\\\tilde{g}} \\\\,{\\\\cal L}_{M}(\\\\tilde{g}_{\\\\mu \\\\nu }) } \\\\,.$ Here the disformally coupled metric is given by $\\\\tilde{g}_{\\\\mu \\\\nu } = C(\\\\phi ) g_{\\\\mu \\\\nu } + D(\\\\phi ) \\\\partial _\\\\mu \\\\phi \\\\partial _\\\\nu \\\\phi \\\\,,$ and the inverse by: $\\\\tilde{g}^{\\\\mu \\\\nu } = \\\\frac{1}{C}\\\\left[ g^{\\\\mu \\\\nu } - \\\\frac{D\\\\,\\\\partial ^\\\\mu \\\\phi \\\\partial ^\\\\nu \\\\phi }{C+D(\\\\partial \\\\phi )^2}\\\\right]\\\\,.$ Moreover, $\\\\kappa ^2=M_{P}^{-2}=8\\\\pi G$ , but keep in mind that that $G$ is not in general equal to Newton's constant as measured by e.g.\",\n \"local experiments.\",\n \"Further, $C(\\\\phi ), D(\\\\phi )$ are functions of $\\\\phi $ , which can be identified as a conformal and disformal couplings of the scalar to the metric, respectively (note that the conformal coupling is dimensionless, whereas the disformal one has units of $mass^{-4}$ ).\",\n \"The action in (REF ) is written in the Einstein frameIn string theory, the Einstein frame refers to the frame in which the dilaton and graviton degrees of freedom are decoupled, while the string (or Jordan) frame is that in which they are not.\",\n \"Further, the dilaton field as well as all other moduli (scalar) fields not relevant for the cosmological discussion are stabilised, massive, and are therefore decoupled from the low energy effective theory.\",\n \"In the literature of scalar-tensor theories however, the Einstein and Jordan frames are identified with respect to the (usually single) scalar field to which gravity is coupled, but such scalar has no particular physical nor geometrical interpretation., which is identified in the literature of scalar-tensor theories (including conformal and disformal couplings) with the frame respect to which the scalar field, gravity is coupled.\",\n \"We follow this use and refer to “Jordan\\\" or “disformal frame\\\" to identify the frame in which dark matter is coupled only to the metric $\\\\tilde{g}_{\\\\mu \\\\nu }$ , rather than to the metric $g_{\\\\mu \\\\nu } $ and a scalar field $\\\\phi $ .\",\n \"The equations of motion obtained from (REF ) are: $R_{\\\\mu \\\\nu } -\\\\frac{1}{2}g_{\\\\mu \\\\nu } R = \\\\kappa ^2\\\\left(T^\\\\phi _{\\\\mu \\\\nu } + T_{\\\\mu \\\\nu }\\\\right)\\\\,,$ where, in the frame relative to $g_{\\\\mu \\\\nu }$ , the energy-momentum tensors are defined as $T^\\\\phi _{\\\\mu \\\\nu } = -\\\\frac{2}{\\\\sqrt{-g}} \\\\frac{\\\\delta S_{\\\\phi }}{\\\\delta g^{\\\\mu \\\\nu }} \\\\,,\\\\qquad \\\\quad T_{\\\\mu \\\\nu }= -\\\\frac{2}{\\\\sqrt{-g}} \\\\frac{\\\\delta \\\\left(-\\\\sqrt{-\\\\tilde{g}} \\\\,{\\\\cal L}_{M}\\\\right) }{\\\\delta g^{\\\\mu \\\\nu }}\\\\,,$ and we model the energy-momentum tensor for matter and both dark components as perfect fluids, that is: $T_{\\\\mu \\\\nu }^i = P_i g_{\\\\mu \\\\nu } +(\\\\rho _i + P_i) u_\\\\mu u_\\\\nu $ where $\\\\rho _i$ , $P_i$ are the energy density and pressure for each fluid $i$ with equation of state $P_i/\\\\rho _i = \\\\omega _i$ .\",\n \"For the scalar field, the energy-momentum tensor takes the form: $T_{\\\\mu \\\\nu }^{\\\\phi } = - g_{\\\\mu \\\\nu } \\\\left[ \\\\frac{1}{2} (\\\\partial \\\\phi )^2+ V \\\\right]+ \\\\partial _\\\\mu \\\\phi \\\\, \\\\partial _\\\\nu \\\\phi \\\\,,$ and one can define the energy density and pressure of the scalar field as: $\\\\rho _\\\\phi =- \\\\frac{1}{2}(\\\\partial \\\\phi )^2 + V \\\\,, \\\\qquad P_\\\\phi = - \\\\frac{1}{2}(\\\\partial \\\\phi )^2 - V \\\\,.$ Finally the equation of motion for the scalar field dark energy becomes: $&&\\\\hspace{-28.45274pt}- \\\\nabla _\\\\mu \\\\nabla ^\\\\mu \\\\phi \\\\!\",\n \"+ V^{\\\\prime }- \\\\frac{T^{\\\\mu \\\\nu }}{2}\\\\!\\\\left[\\\\frac{C^{\\\\prime }}{C} g_{\\\\mu \\\\nu } +\\\\frac{D^{\\\\prime }}{C}\\\\partial _\\\\mu \\\\phi \\\\partial _\\\\nu \\\\phi \\\\right]+\\\\nabla _\\\\mu \\\\left[\\\\frac{D}{C}T^{\\\\mu \\\\nu } \\\\partial _\\\\nu \\\\phi \\\\right] =0 \\\\,.$ Due to the nontrivial coupling, the individual conservation equations for the two fluids are modified.\",\n \"However, the conservation equation for the full system is preserved, and given in the usual way by $\\\\nabla _\\\\mu \\\\left(T^{\\\\mu \\\\nu }_{\\\\phi } + T^{\\\\mu \\\\nu }\\\\right) =0\\\\,.$ Thus using (REF ) and the equation of motion for the scalar field we can write $\\\\nabla _\\\\mu T^{\\\\mu \\\\nu }_\\\\phi = Q \\\\,\\\\partial ^\\\\nu \\\\phi = - \\\\nabla _\\\\mu T^{\\\\mu \\\\nu }\\\\,,$ where $Q\\\\equiv \\\\nabla _\\\\mu \\\\left[\\\\frac{D}{C} \\\\,T^{\\\\mu \\\\lambda } \\\\,\\\\partial _\\\\lambda \\\\phi \\\\right] - \\\\frac{T^{\\\\mu \\\\nu } }{2} \\\\left[\\\\frac{C^{\\\\prime }}{C} g_{\\\\mu \\\\nu } +\\\\frac{D^{\\\\prime }}{C} \\\\,\\\\partial _\\\\mu \\\\phi \\\\,\\\\partial _\\\\nu \\\\phi \\\\right]\\\\,.$ In the Jordan, or disformal frame, as defined above, matter is conserved, $\\\\tilde{\\\\nabla }_\\\\mu \\\\tilde{T}^{\\\\mu \\\\nu } =0 \\\\,,$ where $\\\\tilde{\\\\nabla }_\\\\mu $ is the covariant derivative computed with respect to the disformal metric (REF ) with the Christoffel symbols given by $\\\\tilde{\\\\Gamma }^\\\\mu _{\\\\alpha \\\\beta } =\\\\Gamma ^\\\\mu _{\\\\alpha \\\\beta } + \\\\frac{C^{\\\\prime }}{C}\\\\, \\\\delta ^\\\\mu _{(\\\\alpha }\\\\partial _{\\\\beta )}\\\\phi - \\\\gamma ^2 \\\\, \\\\frac{C^{\\\\prime }}{2C} \\\\,\\\\partial ^\\\\mu \\\\phi \\\\, g_{\\\\alpha \\\\beta }+\\\\frac{D}{C}\\\\,\\\\gamma ^2\\\\,\\\\partial ^\\\\mu \\\\phi \\\\left[\\\\!\\\\nabla _\\\\alpha \\\\nabla _\\\\beta \\\\phi + \\\\!\\\\left(\\\\frac{D^{\\\\prime }}{2D} -\\\\!\\\\frac{C^{\\\\prime }}{C}\\\\right) \\\\!\\\\partial _\\\\alpha \\\\phi \\\\partial _\\\\beta \\\\phi \\\\right]\\\\,, \\\\nonumber \\\\\\\\$ and we have introduced the “Lorentz factor\\\" $\\\\gamma $ defined as $\\\\gamma = \\\\frac{1}{\\\\sqrt{1+\\\\frac{D}{C} (\\\\partial \\\\phi )^2}}\\\\,.$ In this frame, the energy-momentum tensor is defined as $\\\\tilde{T}^{\\\\mu \\\\nu } = \\\\frac{2}{\\\\sqrt{-\\\\tilde{g}}} \\\\frac{\\\\delta S_{M}}{\\\\delta \\\\tilde{g}_{\\\\mu \\\\nu } } \\\\,$ and the disformal energy-momentum tensor can be written as: $\\\\tilde{T}^{\\\\mu \\\\nu } = (\\\\tilde{\\\\rho }+\\\\tilde{P}) \\\\tilde{u}^{\\\\mu } \\\\tilde{u}^{\\\\mu } + \\\\tilde{P}\\\\, \\\\tilde{g}^{\\\\mu \\\\nu }\\\\,,$ where $\\\\tilde{u}^{\\\\mu } = C^{-1/2} \\\\gamma \\\\,u^{\\\\mu } $ .\",\n \"Using (REF ), we obtain a relation between the energy momentum tensor in both frames as: $\\\\tilde{T}^{\\\\mu \\\\nu } = C^{-3} \\\\gamma \\\\, T^{\\\\mu \\\\nu }\\\\,.$ Further using (REF ) we arrive at a relation among the energy densities and pressures in both frames, given by $\\\\tilde{\\\\rho }= C^{-2} \\\\gamma ^{-1} \\\\rho \\\\,, \\\\qquad \\\\tilde{P} = C^{-2}\\\\gamma \\\\, P,$ and therefore the equations of state in both frames are related by $\\\\tilde{\\\\omega }= \\\\omega \\\\, \\\\gamma ^2$ .\",\n \"Note that in the pure conformal case, $D=0$ , $\\\\gamma =1$ and therefore $\\\\tilde{\\\\omega }= \\\\omega $ .\"\n ],\n [\n \"Cosmological equations\",\n \"Consider an homogeneous and isotropic FRW metric $g_{\\\\mu \\\\nu }$ , $ds^2 = - dt^2 + a(t)^2 dx_i dx^i \\\\,,$ where $a(t)$ is the scale factor.\",\n \"In this background, the Einstein and Klein-Gordon equations become, respectively $&& H^2 =\\\\frac{\\\\kappa ^2}{3} \\\\left[\\\\rho _\\\\phi +\\\\rho \\\\right]\\\\,, \\\\\\\\&& \\\\dot{H} + H^2 = -\\\\frac{\\\\kappa ^2}{6}\\\\left[ \\\\rho _\\\\phi + 3P_\\\\phi +\\\\rho +3 P \\\\right]\\\\,,\\\\\\\\&& \\\\ddot{\\\\phi }+3H\\\\dot{\\\\phi }+ V_{,\\\\phi }+Q_0 =0 \\\\,.$ where, $H= \\\\frac{\\\\dot{a}}{a}$ , dots are derivatives with respect to $t$ and we have denoted $V_{,\\\\phi } \\\\equiv \\\\frac{dV}{d\\\\phi }$ .\",\n \"Also the Lorentz factor becomes $\\\\gamma = (1-D \\\\,\\\\dot{\\\\phi }^2/C)^{-1/2}.$ The continuity equations for the scalar field and matter are given by $&&\\\\dot{\\\\rho }_\\\\phi + 3H(\\\\rho _{\\\\phi }+P_{\\\\phi }) = -Q_0\\\\dot{\\\\phi }\\\\,, \\\\\\\\&&\\\\dot{\\\\rho }+ 3H(\\\\rho +P) = Q_0\\\\,\\\\dot{\\\\phi }\\\\,.$ where $Q_0$ is given by $Q_0 = \\\\rho \\\\left[ \\\\frac{D}{C} \\\\,\\\\ddot{\\\\phi }+ \\\\frac{D}{C} \\\\,\\\\dot{\\\\phi }\\\\left(\\\\!3H + \\\\frac{\\\\dot{\\\\rho }}{\\\\rho } \\\\right) \\\\!+ \\\\!\\\\left(\\\\!\\\\frac{D_{,\\\\phi }}{2C}-\\\\frac{D}{C}\\\\frac{C_{,\\\\phi }}{C}\\\\!\\\\right) \\\\dot{\\\\phi }^2 +\\\\frac{C_{,\\\\phi }}{2\\\\,C} (1-3\\\\,\\\\omega )\\\\right].\",\n \"\\\\nonumber \\\\\\\\$ Using () we can rewrite this in a more compact and useful form as $Q_0 = \\\\rho \\\\left( \\\\frac{\\\\dot{\\\\gamma }}{\\\\dot{\\\\phi }\\\\, \\\\gamma } + \\\\frac{C_{,\\\\phi }}{2C} (1-3\\\\,\\\\omega \\\\,\\\\gamma ^2) -3H\\\\omega \\\\,\\\\frac{(\\\\gamma ^{2}-1) }{\\\\dot{\\\\phi }}\\\\right)\\\\,.$ Plugging this into the (non-)conservation equation for dark matter (), gives: $\\\\dot{\\\\rho }+ 3H (\\\\rho + P\\\\,\\\\gamma ^{2}) = \\\\rho \\\\left[\\\\frac{\\\\dot{\\\\gamma }}{\\\\gamma } + \\\\frac{C_{,\\\\phi } }{2C} \\\\,\\\\dot{\\\\phi }\\\\, (1-3\\\\,\\\\omega \\\\gamma ^2)\\\\right]\\\\,.$ Using the relations for the physical proper time and the scale factors in the two frames, given by $\\\\tilde{a} = C^{1/2} a \\\\,, \\\\qquad \\\\quad d\\\\tilde{\\\\tau }= C^{1/2} \\\\gamma ^{-1} d\\\\tau \\\\,,$ we can define the disformal-frame Hubble parameter $\\\\tilde{H} \\\\equiv \\\\frac{d \\\\ln {\\\\tilde{a}}}{d\\\\tilde{\\\\tau }}$ , as $\\\\tilde{H} = \\\\frac{\\\\gamma }{C^{1/2}}\\\\left[ H + \\\\frac{C_{,\\\\phi }}{2C}\\\\dot{\\\\phi }\\\\right]\\\\,,$ so that (REF ) takes the standard form in terms of $\\\\tilde{H}$ : $\\\\frac{d{\\\\tilde{\\\\rho }}}{d\\\\tilde{\\\\tau }} + 3 \\\\tilde{H}( \\\\tilde{\\\\rho }+\\\\tilde{P}) =0 \\\\,.$ Equations (REF ) and (REF ) give the background evolution equations for the modified expansion rate and matter's density evolution.\"\n ],\n [\n \"Master equations\",\n \"In order to solve the cosmological equations, it is convenient to replace time derivatives with derivatives with respect to the number of e-folds $N$ , defined as $ N = \\\\ln a/a_0$ and define $\\\\lambda = \\\\frac{V}{ \\\\rho }(=\\\\frac{\\\\tilde{V}}{\\\\tilde{\\\\rho }})$ .\",\n \"With these definitions, we can rewrite the Friedmann equation (REF ) and $Q_0$ as: $H^2 &=& \\\\frac{\\\\kappa ^2 \\\\rho }{3} \\\\frac{(1+\\\\lambda )}{\\\\left(1- \\\\frac{\\\\kappa ^2\\\\phi ^{\\\\prime 2}}{6 }\\\\right)} \\\\,, \\\\\\\\\\\\!\\\\!\\\\frac{Q_0}{\\\\rho }& =& \\\\frac{\\\\gamma ^2 H^2}{2}\\\\Bigg [ \\\\frac{2D}{C} \\\\phi ^{\\\\prime \\\\prime } -\\\\frac{2D}{C} \\\\phi ^{\\\\prime } \\\\!\\\\left(3\\\\,\\\\omega + \\\\frac{\\\\kappa ^2\\\\phi ^{\\\\prime 2}}{2} + \\\\frac{3(1+\\\\omega ) B}{2(1+\\\\lambda )} \\\\right)+ \\\\left(\\\\frac{D}{C}\\\\right)_{\\\\!\\\\!,\\\\phi } \\\\!\\\\!\\\\phi ^{\\\\prime 2} + \\\\frac{C_{\\\\!,\\\\phi }}{H^2 C}(\\\\gamma ^{-2}-3\\\\omega ) \\\\Bigg ], \\\\nonumber \\\\\\\\$ where here we denote $^{\\\\prime }= d/dN$ .\",\n \"Note also that (REF ) implies that $\\\\kappa \\\\,\\\\phi ^{\\\\prime } \\\\le \\\\pm \\\\sqrt{6}$ .\",\n \"Using these equations and further defining a dimensionless scalar field $\\\\varphi = \\\\kappa \\\\,\\\\phi $ , we can rewrite () and () as: $&& H^{\\\\prime } = - H \\\\left[ \\\\frac{3B}{2(1+\\\\lambda )} (1+ \\\\omega ) + \\\\frac{\\\\varphi ^{\\\\prime 2}}{2}\\\\right], \\\\\\\\ \\\\nonumber \\\\\\\\&& \\\\varphi ^{\\\\prime \\\\prime } \\\\left[1\\\\!+\\\\!\",\n \"\\\\frac{3H^2\\\\gamma ^2 B}{\\\\kappa ^2(1+\\\\lambda )} \\\\frac{D}{C} \\\\right]+ 3\\\\,\\\\varphi ^{\\\\prime } \\\\left[1- \\\\omega \\\\frac{3H^2\\\\gamma ^2 B}{\\\\kappa ^2(1+\\\\lambda )} \\\\frac{D}{C} \\\\right] +\\\\frac{H^{\\\\prime }}{H} \\\\varphi ^{\\\\prime } \\\\left[1+ \\\\frac{3H^2\\\\gamma ^2 B}{\\\\kappa ^2(1+\\\\lambda )} \\\\frac{D}{C} \\\\right]\\\\nonumber \\\\\\\\&& \\\\hspace{42.67912pt}+ \\\\frac{3B}{1+\\\\lambda } \\\\alpha (\\\\varphi )(1-3 \\\\,\\\\omega \\\\gamma ^2)+ \\\\frac{3B\\\\lambda }{(1+\\\\lambda )} \\\\frac{V_{,\\\\varphi }}{V} + \\\\frac{3 H^2 \\\\gamma ^2 B}{\\\\kappa ^2(1+\\\\lambda )} \\\\frac{D}{C}\\\\left[ (\\\\delta (\\\\varphi ) - \\\\alpha (\\\\varphi )) \\\\,\\\\varphi ^{\\\\prime 2} \\\\right] =0\\\\,,\\\\nonumber \\\\\\\\$ where we defined: $&&B \\\\equiv 1-\\\\frac{\\\\varphi ^{\\\\prime 2}}{6}, \\\\\\\\&& \\\\gamma ^{-2} = 1- \\\\frac{H^2}{\\\\kappa ^2}\\\\frac{D}{C} \\\\varphi ^{\\\\prime 2}\\\\,, \\\\\\\\&& \\\\alpha (\\\\varphi ) = \\\\frac{d \\\\ln C^{1/2}}{d\\\\varphi }, \\\\\\\\&& \\\\delta (\\\\varphi ) = \\\\frac{d \\\\ln D^{1/2}}{d\\\\varphi } \\\\,.$ One can solve the system of coupled equations above for $H$ and $\\\\varphi $ as functions of $N$ .\",\n \"However, in some cases it is simpler to use (REF ) into () and solve the following disformal master equation: $\\\\frac{2(1+\\\\lambda )}{3 B} \\\\,\\\\varphi ^{\\\\prime \\\\prime } &+& \\\\left(2\\\\lambda +1 -\\\\omega \\\\right)\\\\varphi ^{\\\\prime } + 2 \\\\lambda \\\\, \\\\frac{d\\\\ln V}{d\\\\varphi } +2 (1 -3\\\\,\\\\omega \\\\,\\\\gamma ^{2})\\\\,\\\\alpha (\\\\varphi )\\\\nonumber \\\\\\\\&+ & \\\\, \\\\frac{2\\\\gamma ^2(1+\\\\lambda )}{3B}\\\\frac{D \\\\rho }{C} \\\\left(\\\\varphi ^{\\\\prime \\\\prime } -3\\\\,\\\\varphi ^{\\\\prime }\\\\left[ \\\\omega + \\\\frac{\\\\varphi ^{\\\\prime 2}}{6} + \\\\frac{(1+\\\\omega ) B}{2(1+\\\\lambda )}\\\\right]+ \\\\frac{C}{2D}\\\\left(\\\\frac{D}{C}\\\\right)_{\\\\!,\\\\varphi } \\\\varphi ^{\\\\prime 2} \\\\right) \\\\!=0 \\\\,, \\\\nonumber \\\\\\\\$ with $\\\\gamma $ given by: $\\\\gamma ^{-2} = 1- \\\\frac{(1+\\\\lambda )}{3B}\\\\frac{D \\\\rho }{C} \\\\varphi ^{\\\\prime 2}\\\\,.$ From (REF ) we see that the conformal case is recovered for $D=0$ , when the second line vanishes.\",\n \"Moreover, the disformal piece appears always together with derivatives of the scalar field, as expected and also nontrivially coupled to the energy density.\",\n \"This complicates considerably the analysis of the disformal case, as we will see below.\"\n ],\n [\n \"Modified expansion rate\",\n \"The effect of the expansion rate during the early time evolution due to the presence of a scalar field can be extracted from the Hubble parameter evolution in the disformal frame defined as: $\\\\tilde{H} = d (\\\\log \\\\tilde{a} )/d\\\\tilde{\\\\tau },$ which can be written using (REF ) as: $\\\\tilde{H} = \\\\frac{H\\\\gamma }{C^{1/2}} \\\\left(1+ \\\\alpha (\\\\varphi ) \\\\varphi ^{\\\\prime } \\\\right)\\\\,,$ where remember that $\\\\gamma $ depends on $H$ (or $\\\\rho $ ) as seen from (REF ), while in the pure conformal case $D=0$ and $\\\\gamma = 1$ .\",\n \"Note that in principle, the factor $(1+ \\\\alpha (\\\\varphi ) \\\\varphi ^{\\\\prime } )$ can be positive or negative, indicating an expansion or contraction modified rate.\",\n \"We stick to positive definite values for this factor and therefore only modified expansion rates, though in principle, one could have a brief contraction period during the early universe evolution, before the onset of BBNSee [18] for a review on scenarios with a possible contraction phase in the early universe..\",\n \"Moreover, notice that while $\\\\tilde{H}$ can grow during the cosmological evolution, the null energy condition (NEC) is not violated.\",\n \"This is because the Einstein frame expansion rate $H$ is dictated by the energy density $\\\\rho $ and pressure $p$ , which obey the NEC and therefore $\\\\dot{H}<0$ during the whole evolution, as it should (see for example [19]).\",\n \"We further want to relate the modified expansion rate to the expected expansion rate in general relativity (GR), that is: $H_{GR}^2 = \\\\frac{\\\\kappa _{GR}^2}{3} \\\\, \\\\tilde{\\\\rho }\\\\,.$ We can do this be using the Friedmann equation (REF ) and the relation between the energy densities (REF ) to write $\\\\gamma ^{-1} H^2 = \\\\frac{\\\\kappa ^2}{\\\\kappa _{GR}^2} \\\\frac{ C^2\\\\,(1+\\\\lambda )}{B} \\\\, H_{GR}^2 \\\\,.$ Using the definition of $\\\\gamma $ (see (REF )) into this equation, one finds a cubic equation for $H^2$ in terms of all the other parameters.\",\n \"The real positive solution to that equation can then be replaced into (REF ) to find the modified expansion rate $\\\\tilde{H}$ , which will thus be a complicated function of $H_{GR}$ as we now see.\",\n \"The cubic equation for $H$ takes the form: $d_1 H^6-H^4 + d_2^2=0\\\\,,$ where $d_1=\\\\frac{D}{C}\\\\frac{\\\\varphi ^{\\\\prime 2}}{\\\\kappa ^2} \\\\,, \\\\hspace{28.45274pt} d_2=\\\\frac{\\\\kappa ^2}{\\\\kappa _{GR}^2}\\\\frac{C^2 (1+\\\\lambda )H_{GR}^2}{B}\\\\,.$ The solutions to (REF ) can be written as $H^2=\\\\frac{1}{3 d_1}\\\\left(1+\\\\left(\\\\frac{2}{\\\\Delta }\\\\right)^{1/3}+\\\\left(\\\\frac{\\\\Delta }{2}\\\\right)^{1/3}\\\\right)$ with $\\\\Delta =2-27d_1^2d_2^2+d_1d_2\\\\sqrt{27(27d_1^2d_2^2-4)}$ .\",\n \"The other two solutions can be obtained by replacing $ \\\\left(\\\\frac{2}{\\\\Delta }\\\\right)^{1/3} \\\\rightarrow e^{2\\\\pi i/3} \\\\left(\\\\frac{2}{\\\\Delta }\\\\right)^{1/3}\\\\qquad {\\\\rm and } \\\\qquad \\\\left(\\\\frac{2}{\\\\Delta }\\\\right)^{1/3} \\\\rightarrow e^{4\\\\pi i/3} \\\\left(\\\\frac{2}{\\\\Delta }\\\\right)^{1/3}\\\\,.$ We are interested in real positive solutions for $H^2$ .\",\n \"One possibility to get this is to have the imaginary part of $(\\\\Delta /2)^{1/3}$ vanish by requiring that $\\\\Delta >0$ , which is impossible.\",\n \"Therefore, the way to obtain real solutions for $H$ is to have the imaginary parts of $(\\\\Delta /2)^{1/3}$ and $(\\\\Delta /2)^{-1/3}$ cancel each other, leaving a real positive solution.\",\n \"For this, we need that $27d_1^2d_2^2 \\\\le 4$ , which implies the following relation between the conformal and disformal functions: $\\\\frac{3\\\\sqrt{3} \\\\,D C\\\\, \\\\varphi ^{\\\\prime 2} (1+\\\\lambda )}{B} \\\\frac{H^2_{GR}}{\\\\kappa _{GR}^2}\\\\le 2\\\\,.$ Under this condition, we can rewrite $\\\\Delta $ as: $\\\\Delta = 2-27d_1^2d_2^2+ i d_1d_2\\\\sqrt{27(4- 27d_1^2d_2^2)}\\\\,, $ which allows us to define a complex number $Z\\\\equiv \\\\Delta /2$ and it is easy to check that $\\\\bar{Z} = 2/\\\\Delta $ and thus $|Z|^2 =1$ .\",\n \"Denoting further $Z_i$ with $i=1,2,3$ denoting the three solutions to $H$ as explained above, the solutions for $H$ , (REF ) takes the simple form: $H^2_i=\\\\frac{1}{3 d_1}\\\\left[1+Z_i^{1/3} + \\\\bar{Z}_i^{1/3}\\\\right] \\\\,,$ and remember that we are interested only in the real positive solution.\",\n \"We can now plug in (REF ), as well as the expression for $\\\\gamma $ in terms of $H$ into the Jordan frame expansion rate (REF ), can be written as: $\\\\tilde{H}^2 = \\\\frac{\\\\kappa ^2}{\\\\kappa ^2_{GR}} \\\\frac{ \\\\gamma ^3\\\\, C(1+\\\\lambda ) (1+\\\\alpha (\\\\varphi )\\\\varphi ^{\\\\prime })^2 }{B} \\\\,\\\\,H_{GR}^2\\\\,,$ where there is a non-trivial dependence of $H_{GR}$ encoded in $\\\\gamma _i = \\\\frac{1}{3d_1d_2}\\\\left[1+Z_i^{1/3} + \\\\bar{Z}_i^{1/3}\\\\right] \\\\,.$ In the conformal case, $D=0$ , $\\\\gamma =1$ and therefore (REF ) is simply $\\\\tilde{H}^2 = \\\\frac{\\\\kappa ^2}{\\\\kappa _{GR}^2}\\\\frac{C (1+\\\\lambda ) (1+\\\\alpha (\\\\varphi ) \\\\varphi ^{\\\\prime })^2}{B}\\\\, H^2_{GR} \\\\,.$ From this relation we define a speed-up parameter $\\\\xi $ , which will be useful below to measure the departures from the $GR$ expansion rate result: $\\\\xi \\\\equiv \\\\frac{\\\\tilde{H}}{H_{GR}}\\\\,.$\"\n ],\n [\n \"Modifications of the dark matter relic abundances\",\n \"In this section we discuss the modifications to the DM relic abundance's predictions due to modifications of the expansion rate before the onset of nucleosynthesis caused by the presence of a scalar field conformal and disformally coupled to matter.\",\n \"We start by revisiting the conformal case, discussed originally in [3]Modifications to the Boltzmann equation due to a conformal coupling in (non-critical) string theory have been discussed in [8].. We first solve (numerically) the master equation for the scalar field (REF ) in order to compute the modified expansion rate $\\\\tilde{H}$ and compare it with the standard expansion rate, $H_{GR}$ .\",\n \"We then use this to compute the modifications to the dark matter relic abundances by solving the Boltzmann equation using the modified expansion rate.\",\n \"We start revisiting by the conformal case by exploring a wide range of initial conditions, masses and cross-sections.\",\n \"We then look at an explicit disformal example.\",\n \"Before solving the master equation (REF ), we would like to write it in terms of Jordan frame quantities $\\\\tilde{\\\\omega }= \\\\omega \\\\gamma ^2$ , $\\\\tilde{\\\\rho }= C^{-2}\\\\gamma ^{-1} \\\\rho $ .\",\n \"Moreover, the number of e-folds $N$ can be expressed in terms of Jordan frame quantities as follows.\",\n \"In this frame, the entropy is conserved and is given by $\\\\tilde{S}=\\\\tilde{a}\\\\, \\\\tilde{s}$ , where $\\\\tilde{s}= \\\\frac{2\\\\pi }{45} g_s(\\\\tilde{T}) \\\\tilde{T}^3$ .\",\n \"So, the conservation of entropy and (REF ) show that $N$ is a function of temperature and the scalar field as: $N\\\\equiv \\\\ln \\\\frac{a}{a_0}=\\\\ln \\\\left[\\\\frac{\\\\tilde{T}_0}{\\\\tilde{T}}\\\\left(\\\\frac{g_s(\\\\tilde{T}_0)}{g_s(\\\\tilde{T})}\\\\right)^{1/3}\\\\right]+\\\\ln \\\\left[\\\\frac{C_0}{C}\\\\right]^{1/2}.$ Therefore, we can introduce the parameter, $\\\\tilde{N}$ , defined as $\\\\tilde{N} \\\\equiv \\\\ln \\\\left[\\\\frac{\\\\tilde{T}_0}{\\\\tilde{T}}\\\\left(\\\\frac{g_s(\\\\tilde{T}_0)}{g_s(\\\\tilde{T})}\\\\right)^{1/3}\\\\right].$ and transform to derivatives w.r.t.\",\n \"$\\\\tilde{N}$ (assuming well behaved functions): $\\\\varphi ^{\\\\prime } = \\\\frac{1}{\\\\left(1-\\\\alpha (\\\\varphi ) \\\\frac{d\\\\varphi }{d\\\\tilde{N}}\\\\right)} \\\\frac{d\\\\varphi }{d\\\\tilde{N}} \\\\,,\\\\qquad \\\\varphi ^{\\\\prime \\\\prime } = \\\\frac{1}{\\\\left(1-\\\\alpha (\\\\varphi ) \\\\frac{d\\\\varphi }{d\\\\tilde{N}}\\\\right)^3} \\\\left( \\\\frac{d^2\\\\varphi }{d\\\\tilde{N}^2} + \\\\frac{d\\\\alpha }{d\\\\varphi } \\\\left(\\\\frac{d\\\\varphi }{d\\\\tilde{N}}\\\\right)^3\\\\right)\\\\,.\",\n \"\\\\\\\\ \\\\nonumber $ In a slight abuse of notation and to keep expressions neat, in what follows we denote derivatives w.r.t.\",\n \"$\\\\tilde{N}$ with a prime $^{\\\\prime }$ .\"\n ],\n [\n \"Conformal case\",\n \"We start with the pure conformal case.\",\n \"That is, we take $D(\\\\phi ) =0$ in (REF ) and therefore $\\\\gamma =1$ (and $\\\\tilde{\\\\omega }= \\\\omega $ ).\",\n \"Moreover, during the radiation and matter dominated eras, of interest for us, the potential energy of the scalar field is subdominant and therefore, we take $\\\\lambda \\\\sim 0$ .\",\n \"Therefore the master equation (REF ) simplifies to: $\\\\frac{2}{3 (1-\\\\varphi ^{\\\\prime 2}/6)} \\\\,\\\\varphi ^{\\\\prime \\\\prime } &+& \\\\left(1 -\\\\tilde{\\\\omega }\\\\right)\\\\varphi ^{\\\\prime } + 2 (1 -3\\\\,\\\\tilde{\\\\omega })\\\\,\\\\alpha (\\\\varphi ) =0,$ which in terms of derivatives wrt $\\\\tilde{N}$ takes the form: $\\\\frac{1 }{3B\\\\left[1-\\\\alpha (\\\\varphi ) \\\\varphi ^{\\\\prime } \\\\right]^3} \\\\left(\\\\varphi ^{\\\\prime \\\\prime } + \\\\frac{d\\\\alpha }{d\\\\varphi } \\\\left(\\\\varphi ^{\\\\prime }\\\\right)^3\\\\right)+ \\\\frac{(1-\\\\tilde{\\\\omega }) }{\\\\left[1-\\\\alpha (\\\\varphi )\\\\varphi ^{\\\\prime }\\\\right]} \\\\,\\\\varphi ^{\\\\prime }+(1-3\\\\,\\\\tilde{\\\\omega }) \\\\, \\\\alpha (\\\\varphi )=0 \\\\,, \\\\nonumber \\\\\\\\$ where $B= 1-\\\\frac{\\\\left(\\\\varphi ^{\\\\prime }\\\\right)^2 }{6\\\\left(1-\\\\alpha (\\\\varphi ) \\\\varphi ^{\\\\prime }\\\\right)^2} $ .\",\n \"Using the relation between $\\\\tilde{H}$ and $H_{GR}$ defined in (REF ), we can write the speed-up parameter as $\\\\xi = \\\\frac{\\\\tilde{H}}{H_{GR}} = \\\\frac{C^{1/2}(\\\\varphi )}{C^{1/2}(\\\\varphi _0)} \\\\frac{1}{\\\\left(1-\\\\alpha (\\\\varphi ) \\\\varphi ^{\\\\prime }\\\\right)\\\\sqrt{B}} \\\\frac{1}{\\\\sqrt{1+ \\\\alpha ^2(\\\\varphi _0)}}$ where we have used the relation between the bare gravitational constant and that measured by local experiments for conformally coupled theories [20]: $\\\\kappa _{GR}^2 = \\\\kappa ^2 C(\\\\varphi _0) [1+\\\\alpha ^2(\\\\varphi _0)]\\\\,,$ where $\\\\varphi _0$ is the value of the scalar field at present time.\"\n ],\n [\n \"Expansion rate modification\",\n \"The scalar equation in the conformal case (REF ), as function of $N$ (for $\\\\lambda =0$ ) contains a term which can be interpreted as an effective potential, dictated by $V_{eff} = \\\\ln C^{1/2}$ .\",\n \"For a strictly radiation dominated era, $\\\\tilde{\\\\omega }=1/3$ , the effective potential term vanishes and we are left with an equation that can be solved analytically [21], giving $\\\\varphi ^{\\\\prime } \\\\propto e^{-N}$ .\",\n \"That is, any initial velocity will rapidly go to zero (remember that from the Friedmann equation (REF ), $\\\\varphi ^{\\\\prime }$ is constrained to be $\\\\varphi ^{\\\\prime } \\\\lesssim \\\\pm \\\\sqrt{6}$ ).\",\n \"Therefore we explore the effects of having a non-zero initial velocity in our analysis below (see also Appendix for further examples).\",\n \"Since the scalar field is expressed in Planck units, we focus on order one or smaller field variations $\\\\Delta \\\\varphi $ .\",\n \"One can check, using the analytic solution to (REF ) deep in the radiation era, that for initial velocities $\\\\varphi ^{\\\\prime }_0 \\\\ll \\\\pm \\\\sqrt{6}$ , the total field displacement is of order $\\\\Delta \\\\varphi \\\\sim \\\\varphi ^{\\\\prime }_0$ [21].\",\n \"However, given that we don't know much about the theory before BBN, we explore different initial values for $(\\\\varphi _0, \\\\varphi ^{\\\\prime }_0)$ and study their consequences.\",\n \"In particular we explore initial values $\\\\varphi _0$ and $\\\\varphi ^{\\\\prime }_0 \\\\in (-1.0, 1.0)$ .\",\n \"We now concentrate on an explicit conformal factor.\",\n \"We use the same conformal factor as that studied in [3], which is given by: $C(\\\\varphi ) = (1+ b \\\\,e^{-\\\\beta \\\\, \\\\varphi })^2$ with the values $b=0.1$ , $\\\\beta =8$ , which have been shown to satisfy the constraints imposed by tests of gravity, for the parameters $\\\\alpha , \\\\beta $ , $\\\\xi $ .\",\n \"As we will see, the requirement of reaching the GR expansion rate value by the time of the onset of BBN, drives these parameters to very small values, which are thus consistent with the constraints from gravity for their values today.\",\n \"As we discussed above, in the equation of motion for $\\\\varphi $ , with $\\\\tilde{\\\\omega }\\\\ne 1/3 $ , the conformal factor acts as an effective potential on which the scalar field moves, damped by the Hubble friction (see ()).\",\n \"Since any initial velocity $\\\\varphi ^{\\\\prime }$ goes rapidly to zero deep in the radiation era, in the subsequent evolution of $\\\\varphi $ , the term $(\\\\varphi ^{\\\\prime } )^2$ in the master equation will be negligible.\",\n \"In this regime, the equation is that of a particle moving in an effective potential with a damping term.\",\n \"Therefore, one can understand the evolution of $\\\\varphi $ from the form of the effective potential ($V_{eff}=\\\\ln C^{1/2}$ ) and the initial conditions chosen.\",\n \"For the conformal function we are considering (REF ), one sees that there is a set of initial conditions that will give rise to an interesting behaviour in $\\\\varphi $ , and therefore an interesting modified expansion rate in the Jordan frame $\\\\tilde{H}$ (REF ), as we now explain.\",\n \"In general, both the initial position and velocity of the scalar field can take any value, positive and negative.\",\n \"In the runaway effective potential dictated by $\\\\ln C^{1/2}$ for the conformal factor (REF ) we consider here, we have the following possibilities.\",\n \"i) The scalar field starting somewhere up in the runaway effective potential with zero initial velocity.\",\n \"In this case the scalar field will roll-down the potential, eventually stopping due to Hubble friction, at some constant value of $\\\\varphi $ , which depends on the its initial value.\",\n \"So long as $(1+\\\\alpha (\\\\varphi ) \\\\,\\\\varphi ^{\\\\prime })$ stays positive (see (REF )), $C$ will evolve rapidly towards 1.\",\n \"More generally, the initial velocity can be different from zero.\",\n \"If the initial velocity is positive, the behaviour will be similar to the previous case.\",\n \"The field will roll-down the effective potential towards its final terminal value.\",\n \"ii) A more interesting possibility arises when one allows for negative initial velocities.\",\n \"In this case, the field will start rolling-up the effective potential towards smaller values of the field, eventually turning back down and moving towards its terminal value.\",\n \"It is easy to see that an interesting effect happens when the field starts at an initial positive value.\",\n \"Given a sufficient initial negative velocity the field will move towards negative values until its velocity becomes zero and then positive again, as it rolls back down the effective potential.\",\n \"This change in sing for the scalar evolution will produce a pick in the conformal function that will give rise to a non-trivial modification of the Jordan's frame expansion rate $\\\\tilde{H}$ , as we are looking for.\",\n \"As mentioned before, we are interested in (sub-)Planckian initial values $(\\\\varphi _0, \\\\varphi ^{\\\\prime }_0)$ , such that $\\\\tilde{H}>0$ .\",\n \"With these requirements, one can see that given an initial negative velocity, there is a suitable initial value of the scalar field such that the behaviour just described holds and the expansion rate $\\\\tilde{H}$ has an interesting evolution before the onset of BBN.\",\n \"At late times the conformal function goes to one and the GR expansion is recovered.\",\n \"We show this behaviour explicitly in Figures REF and REF where we plot the numerical solution for the evolution of $\\\\varphi $ and $C(\\\\varphi )$ as functions of the temperature.\",\n \"In these plots we find $\\\\varphi =\\\\varphi (\\\\tilde{T})$ by first solving (REF ) numerically with initial conditions $ (\\\\varphi _0,\\\\varphi ^{\\\\prime }_0) = (0.2, -0.99)$ and then use (REF ) to express $\\\\varphi (\\\\tilde{N})$ as function of $\\\\tilde{T}$ In appendix we show further examples of the thermal evolution of the scalar field.. As we can see, the conformal factor starts growing towards a maximum value as $\\\\varphi $ moves to negative values, to rapidly drop down towards its GR value at $C\\\\rightarrow 1$ as $\\\\varphi $ moves down the effective potential towards positive values.\",\n \"This non-trivial effect will give rise to the possibility of re-annihilation, as we discuss below.\",\n \"Figure: Typical evolution of the scalar field as temperature decreases.\",\n \"The initial values are (ϕ,dϕ/dN ˜)=(0.2,-0.994)(\\\\varphi ,d\\\\,\\\\varphi /d\\\\,\\\\tilde{N}) = (0.2,-0.994).Figure: Behaviour of the conformal factor, C(ϕ)C(\\\\varphi ) as a function of the temperature for the same initial values as in Fig.\",\n \".Based on the discussion above, we have solved the master equation (REF ), to find the the scalar field as a function of $\\\\tilde{N}$ for various initial conditions, where we see the interesting behaviour explained above.\",\n \"The resulting modified expansion rate and its comparison with the standard case is shown in Figure REF for the same initial conditions as in Figures REF and REF .\",\n \"In our numerical exploration, we choose initial conditions for which the notch in the expansion rate (see Fig.\",\n \"REF ) occurs closer to the BBN timeIn appendix we show more examples of modified expansion rate using different initial conditions..\",\n \"This has interesting consequences for the dark matter annihilation, as we discuss below.\",\n \"Figure: Comparing the Hubble expansion rate H ˜\\\\tilde{H} in the Jordan Frame with the standard Hubble expansion rate H GR H_{GR}.The presence of the scalar field enhances and decreases the expansion rate during the radiation dominated era.\",\n \"This plot corresponds toinitial conditions given by (ϕ 0 ,ϕ 0 ' )=(0.2,-0.994)(\\\\varphi _0,\\\\varphi ^{\\\\prime }_0) = (0.2,-0.994).When solving the master equation (REF ), we have taken into account an important effect that occurs during the radiation dominated era.\",\n \"Deep in this epoch, the equation of state is given by $\\\\tilde{\\\\omega }=1/3$ .\",\n \"When a particle species in the cosmic soup becomes non-relativistic, $\\\\tilde{\\\\omega }$ differs slightly from $1/3$ .\",\n \"When the temperature of the universe drops below the rest mass of each of the particle types, there are non-zero contributions to $1-3\\\\tilde{\\\\omega }$ .\",\n \"This activates the effective potential, which can be seen in the last term of (REF ), and displaces, or “kicks“ the field along $V_{eff}$ .\",\n \"To examine this effect in more detail, we start by writing $1-3\\\\,\\\\tilde{\\\\omega }$ during the early stages of the universe as in [3] and [15] $1-3\\\\,\\\\tilde{\\\\omega }= \\\\frac{\\\\tilde{\\\\rho }- 3\\\\, \\\\tilde{p}}{\\\\tilde{\\\\rho }} =\\\\sum _{A} \\\\frac{\\\\tilde{\\\\rho }_A - 3\\\\tilde{p}_A}{\\\\tilde{\\\\rho }} +\\\\frac{\\\\tilde{\\\\rho }_m}{\\\\tilde{\\\\rho }}\\\\,,$ where the sum runs over all particles in thermal equilibrium during the radiation dominated era and $\\\\tilde{\\\\rho }_m$ is the contribution from the non-relativistic decoupled and pressureless matter.\",\n \"The summation over all the particle is responsible for the kicking effect discussed above.\",\n \"Then, a kick function is defined as $\\\\Sigma (\\\\tilde{T}) \\\\equiv \\\\sum _{A} \\\\frac{\\\\tilde{\\\\rho }_A - 3\\\\tilde{p}_A}{\\\\tilde{\\\\rho }}\\\\,,$ where the energy density $\\\\tilde{\\\\rho }_A$ and pressure $\\\\tilde{p}_A$ of each type $A$ of particle are given by $\\\\tilde{\\\\rho }_A(\\\\tilde{T}) =\\\\frac{g_A}{2\\\\pi ^2}\\\\int ^\\\\infty _{m_A}\\\\frac{\\\\left(E^2-m_A^2\\\\right)^{1/2}}{\\\\exp (E/\\\\tilde{T})\\\\pm 1}E^2dE$ $\\\\tilde{p}_A(\\\\tilde{T}) =\\\\frac{g_A}{6\\\\pi ^2}\\\\int ^\\\\infty _{m_A}\\\\frac{\\\\left(E^2-m_A^2\\\\right)^{3/2}}{\\\\exp (E/\\\\tilde{T})\\\\pm 1}E^2dE$ with $g_A$ being the number of internal degrees of freedom of species of type $A$ and the plus (minus) sign in the integral corresponds to fermions (bosons).\",\n \"To compute (REF ), we use the Standard Model particle spectrum.\",\n \"In particular, we take into account the top quark, the Higgs boson, Z boson, W bosons, bottom quark, tau lepton, charm quark, charged pions, neutral pion, moun lepton and the electronWe present more details of the kick function in appendix .. As we show in Figure , $\\\\Sigma $ is mostly zero, except when the kicks happen.\",\n \"During the radiation dominated era $\\\\tilde{\\\\rho }\\\\simeq \\\\pi ^2 g_{eff}(\\\\tilde{T}) \\\\tilde{T}^4/30$ , where $\\\\tilde{T}$ is the Jordan frame temperature and $g_{eff}$ is the total number of relativistic degrees of freedom.\",\n \"Also, during this stage $\\\\tilde{\\\\rho }_m$ is negligible and as we have shown $\\\\Sigma $ is slightly different than zero.\",\n \"Therefore, we can compute the equation of state from (REF ) as $\\\\tilde{\\\\omega }=(1-\\\\Sigma (\\\\tilde{T}))/3$ .\",\n \"In Figure REF we show the evolution of $\\\\tilde{\\\\omega }$ between 10 TeV and 10 eV.\",\n \"This figure shows four troughs, which are the “kicks” mentioned above.\",\n \"Each kick corresponds to the transition of one or more particles to the non-relativistic regime.\",\n \"For example, the trough at around 0.5 MeV is due to the electron, while the one at around 100 GeV is due to the heavy particles ($t$ , $H$ , $Z$ and $W$ ).\",\n \"Towards the end of the radiation era, approaching the transition to the matter dominated era, eq.\",\n \"(REF ) takes the approximate form: $1-3\\\\,\\\\tilde{\\\\omega }\\\\simeq \\\\frac{\\\\tilde{\\\\rho }_{m}}{\\\\tilde{\\\\rho }_m + \\\\tilde{\\\\rho }_r} \\\\simeq \\\\frac{1}{1+ \\\\tilde{T}/\\\\tilde{T}_{eq}} \\\\,,$ where $\\\\tilde{T}_{eq} \\\\sim {\\\\cal O}(10^{-9})$ GeV is the temperature at matter-radiation equality, that is, $\\\\tilde{\\\\rho }_m(\\\\tilde{T}_{eq}) = \\\\tilde{\\\\rho }_r(\\\\tilde{T}_{eq})$ .\",\n \"We can now combine (REF ) and (REF ) to compute the thermal evolution of (REF ) in the radiation dominated and matter dominated eras and use it in the master equation.\",\n \"Figure: Evolution of ω ˜\\\\tilde{\\\\omega } in () as function of temperature during the radiation dominated era.\"\n ],\n [\n \"Parameter Constraints\",\n \"In scalar-tensor theories of gravity, there are some constraints on the parameters that need to be taken into account.\",\n \"Deviations from GR can be parametrised in terms of the post-Newtonian parameters $\\\\gamma _{PN}$ and $\\\\beta _{PN}$ , which are given in terms of $\\\\alpha (\\\\varphi _0)$ defined in () and its derivative $\\\\alpha ^{\\\\prime }_0= d\\\\alpha / d\\\\varphi |_{\\\\varphi _0}$ as [22], [23]: $\\\\gamma _{PN} -1 = -\\\\frac{2\\\\alpha _0^2}{1+\\\\alpha _0^2} \\\\,, \\\\qquad \\\\beta _{PN} -1 = \\\\frac{1}{2}\\\\frac{\\\\alpha ^{\\\\prime }_0\\\\alpha _0^2}{(1+\\\\alpha _0^2)^2} \\\\,,$ Solar system tests of gravity, including the perihelion shift of Mercury, Lunar Laser Ranging experiments, and the measurements of the Shapiro time delay by the Cassini spacecraft [24], [25], [26] indicate that $\\\\alpha _0$ should be very small, with values $\\\\alpha _0^2 \\\\lesssim 10^{-5}$ , while binary pulsar observations impose that $\\\\alpha ^{\\\\prime }_0\\\\gtrsim -4.5$ .\",\n \"The last constraint applies to the the speed-up factor $\\\\xi $ , which has to be of order 1 before the onset of BBN.\",\n \"In our examples we have $\\\\alpha _0^2 \\\\simeq 2\\\\times 10^{-5}$ , $\\\\alpha ^{\\\\prime }_0 > 0$ and $\\\\xi \\\\approx 1.05$ .\"\n ],\n [\n \"Impact on relic abundances \",\n \"We are now ready to discuss the impact of the modified expansion rates on the relic abundance of dark matter species.\",\n \"For a dark matter species $\\\\chi $ with mass $m_\\\\chi $ and annihilation cross-section $\\\\langle \\\\sigma v \\\\rangle $ , where $v$ is the relative velocity, the dark matter number density $n_\\\\chi $ evolves according to the Boltzmann equation $\\\\frac{d n_\\\\chi }{dt} = -3 \\\\tilde{H} n_\\\\chi - \\\\langle \\\\sigma v \\\\rangle \\\\left( n_\\\\chi ^2 - (n_\\\\chi ^{eq})^2 \\\\right)\\\\,,$ where, as we have discussed above, the relevant expansion rate is the Jordan frame one, which can give interesting effects due to the presence of the scalar field.\",\n \"Further $n_\\\\chi ^{eq}$ is the equilibrium number density.\",\n \"We can rewrite this equation in terms of $x=m_\\\\chi /\\\\tilde{T}$ $\\\\frac{d Y}{dx} = - \\\\frac{\\\\tilde{s} \\\\langle \\\\sigma v \\\\rangle }{x \\\\tilde{H}} \\\\left( Y^2 - Y_{eq}^2 \\\\right) \\\\,.$ where $Y = \\\\frac{n_\\\\chi }{\\\\tilde{s}}$ , $\\\\tilde{s}= \\\\frac{2\\\\pi }{45} g_s(\\\\tilde{T}) \\\\tilde{T}^3$ .\",\n \"Numerical solutions to the Boltzmann equation (REF ) with the modified expansion rate $\\\\tilde{H}$ were found for dark matter particles with masses ranging from 5 GeV to 1000 GeV.\",\n \"For instance, we show solutions in figures REF and REF for two different masses.\",\n \"As we can see from (REF ), the annihilation cross-section influences the evolution of the abundance $Y$ .\",\n \"The current value of $Y$ determines the present dark matter content of the universe.\",\n \"This can be seen clearly by recalling the current value of the energy density parameter $\\\\Omega _0=\\\\frac{\\\\rho _0}{\\\\rho _{c,0}}=\\\\frac{m\\\\,Y_0\\\\,s_0}{\\\\rho _{c,0}}$ , where $\\\\rho _{c,0}$ and $s_0$ are the well-known current values of the critical energy density and the entropy density of the universe, respectively.\",\n \"So, for each single mass, the thermally-averaged annihilation cross section, $\\\\langle \\\\sigma v \\\\rangle $ , was chosen such as the current DM content of the universe is 27 %, so $\\\\Omega _0=0.27$ .\",\n \"In Figure REF we show the annihilation cross-section, $\\\\langle \\\\sigma v \\\\rangle _{Conformal}$ , found for all masses and compare it to the annihilation cross sections for the standard cosmology model, $\\\\langle \\\\sigma v \\\\rangle _{Standard}$ .\",\n \"As it is shown, for large masses $\\\\langle \\\\sigma v \\\\rangle _{Conformal}$ is larger than $\\\\langle \\\\sigma v \\\\rangle _{Standard}$ , up to a factor of four.\",\n \"As the mass decreases $\\\\langle \\\\sigma v \\\\rangle _{Conformal}$ decreases up to the point where is smaller than $\\\\langle \\\\sigma v \\\\rangle _{Standard}$ .\",\n \"Then, for masses smaller than 100 GeV the figure shows that $\\\\langle \\\\sigma v \\\\rangle _{Conformal} \\\\approx \\\\langle \\\\sigma v\\\\rangle _{Standard}$ .\",\n \"Thus, we have found that the annihilation cross-sections can be larger or smaller than the thermal average cross-section.\",\n \"Just to give an example of larger and smaller cross-section, the following table compares the numerical values of $\\\\langle \\\\sigma v \\\\rangle _{Conformal}$ and $\\\\langle \\\\sigma v \\\\rangle _{Standard}$ for two dark matter masses, 1000 GeV and 130 GeV.\",\n \"Table: NO_CAPTIONFigures REF and REF show the evolution of the abundance $\\\\tilde{Y}(x)$ for DM particles with masses 130 GeV and 1000 GeV, respectively.\",\n \"These figures also include the abundance $Y_{GR}(x)$ calculated in the standard cosmology model and the equilibrium abundance $Y_{Eq}(x)$ .\",\n \"Figure: Evolution of the abundance as temperature changes for a DM particle of mass 130 GeV.The temperature evolution of the abundance for a 130 GeV mass is not noticeable affected by the presence of the scalar field $\\\\phi $ .\",\n \"In this case, $\\\\tilde{Y}$ and $Y_{GR}$ are almost indistinguishable from one another.\",\n \"On the other hand, the scalar field $\\\\phi $ has a prominent effect on the temperature evolution of the abundance for a 1000 GeV DM particle.\",\n \"First of all, the freeze-out happens earlier than expected due to the enhancement of the expansion rate, $\\\\tilde{H}$ .\",\n \"Then, an unusual effect appears.\",\n \"As the temperature decreases, $\\\\tilde{H}$ becomes smaller than the interaction rateThe interaction rate is defined as $\\\\tilde{\\\\Gamma }\\\\equiv \\\\langle \\\\sigma v \\\\rangle _{Conformal}\\\\,\\\\tilde{s}\\\\,\\\\tilde{Y}$ .\",\n \"$\\\\tilde{\\\\Gamma }$ and a short period of annihilation starts again called “re-annihilation”.\",\n \"The re-annihilation process reduces the abundance of dark matter until a second and final freeze-out happens.\",\n \"After this final freeze-out the abundance remains constant.\",\n \"Figure: Abundance for a mass of 1000 GeV.Figure: Annihilation cross section as function of mass.\",\n \"The presence of the scalar field enhances the 〈σv〉\\\\langle \\\\sigma v \\\\rangle for large masses,and diminishes 〈σv〉\\\\langle \\\\sigma v \\\\rangle for masses around 130 GeV, while small mass the effect is almost negligible.Figure: Expansion rate (as in figure ) and interaction rate as function of temperature.\",\n \"The interaction rate, Γ ˜\\\\tilde{\\\\Gamma }, is given by 〈σv〉 Conformal s ˜Y ˜\\\\langle \\\\sigma v \\\\rangle _{Conformal}\\\\,\\\\tilde{s}\\\\,\\\\tilde{Y}.\",\n \"We use Y ˜\\\\tilde{Y} from figures and and the values of 〈σv〉 Conformal \\\\langle \\\\sigma v \\\\rangle _{Conformal} presented previously for 130 GeV and 1000 GeV masses.The re-annihilation phase can be described better by discussing the relation between the expansion rate $\\\\tilde{H}$ and the interaction rate $\\\\tilde{\\\\Gamma }$ .\",\n \"The first freeze-out happens when $\\\\tilde{\\\\Gamma }$ becomes smaller than $\\\\tilde{H}$ which can be seen in figure REF to happen around a temperature of 50 GeV for a 1000 GeV particle.\",\n \"Then, near to 7 GeV $\\\\tilde{H}$ drops below $\\\\tilde{\\\\Gamma }$ , and so the re-annihilation process starts and goes on until the second freeze-out occurs.\",\n \"Around 2 GeV $\\\\tilde{H}$ becomes much larger than $\\\\tilde{\\\\Gamma }$ and so the abundance becomes almost constant.\",\n \"Our analysis shows that, as found in [3], re-annihilation occurs for this particular choice of conformal factor.\",\n \"However, we found that when fully integrating the master equation, the re-annihilation occurs only for very large masses of the dark matter particles (in [3] it was found for $m=50$ GeV).\",\n \"On the other hand, in [15], no re-annihilation was foundAlthough [15] used a different conformal factor to [3], we expect the re-annihilation effect to be present also in that case., which was probably due to the initial conditions used and the values of the DM masses explored.\"\n ],\n [\n \"Disformal case\",\n \"We now discuss briefly the effect of the disformal factor in the metric (REF ) to the expansion rate of the universe, $\\\\tilde{H}$ , and compare it to the conformal modification to $\\\\tilde{H}$We leave a detailed exploration for a future publication..\",\n \"Hence, we explore $D(\\\\phi )\\\\ne 0$ for the same conformal factor studied before, that is, $C(\\\\varphi ) = (1+ b \\\\,e^{-\\\\beta \\\\, \\\\varphi })^2$ for $b=0.1$ , $\\\\beta =8$ .\",\n \"To investigate these modifications, we first need to look at the the scalar field evolution with temperature.\",\n \"In the pure conformal case studied above, we found the thermal evolution of the scalar field by solving the master equation (REF ) numerically, which is (REF ) for $D(\\\\phi )=0$ .\",\n \"However, to study the effects of the disformal factor on the scalar field, it is more convenient to solve the system of two coupled equations (REF ) and ().\",\n \"Using these equations we find solutions for the dimensionless scalar field $\\\\varphi $ , and for the expansion rate in the Einstein frame $H$ .\",\n \"Notice that solving the system of coupled equations or solving the master equation to find the thermal evolution of the scalar field are equivalent methods (as we have explicitly checked), because (REF ) it is nothing but a combination (REF ) and ().\",\n \"However, while in the pure conformal case the master equation can be made independent of $H$ (or $\\\\rho $ ), this is not the case for the more general disformal case, as we can see in eq.\",\n \"(REF ).\",\n \"In the same way as for the conformal case, we are interested mainly in the radiation and matter eras and therefore we can neglect the potential energy of the scalar field.\",\n \"Thus, we consider $V\\\\sim 0$ and $\\\\lambda =0$ .\",\n \"Also, while solving the coupled equations we have to express $\\\\omega $ in the Jordan frame by using $\\\\tilde{\\\\omega }= \\\\omega \\\\gamma ^2$ and transform all derivatives w.r.t.\",\n \"$N$ to derivatives w.r.t.\",\n \"$\\\\tilde{N}$ by using (REF ).\",\n \"With this information, we solve the system of coupled equations numerically to find the dimensionless scalar field $\\\\varphi $ and the Hubble parameter $H$ , as functions of the number of e-folds $\\\\tilde{N}$ (and the temperature).\",\n \"We choose the same initial conditions for the scalar field and its derivative as in the conformal case and to obtain the initial condition for $H$ , we use (REF ).\",\n \"Once we have the solutions for $\\\\varphi $ and $H$ as functions of temperature, we can go back to (REF ) and (REF ) to obtain the expansion rate for the disformal model.\",\n \"As an example, in Figure REF we show the effects of a disformal factor given by $D(\\\\varphi ) = D_0\\\\,\\\\varphi ^2$ with $D_0 =-4.9\\\\times 10^{-14}$ .\",\n \"In this plot, we illustrate the effect of the disformal contribution on the expansion rate ($\\\\tilde{H}_{Disformal}$ ) and compare it to the modified expansion rate for the conformal case ($\\\\tilde{H}_{Conformal}$ ) and the standard case ($H_{GR}$ ).\",\n \"We use the same initial conditions as in Figures REF and REF for the scalar field and its derivative.\",\n \"Also, it is important to mention that for the case shown the parameter constraints described in section REF are satisfied.\",\n \"In particular we find $\\\\alpha _0^2 \\\\simeq 2\\\\times 10^{-5}$ , $\\\\alpha ^{\\\\prime }_0 > 0$ and $\\\\xi \\\\approx 1.02$ .\",\n \"Figure: Comparing the modified expansion rate of the universe in the disformal and conformal scenarios for the same initial conditions as in Fig.\",\n \".From our example, with $C$ and $D$ as indicated above, we can clearly see the differences from the disformally modified expansion rate $\\\\tilde{H}_{Disformal}$ compared to the conformally modified and standard case, $H_{GR}$ .\",\n \"The evolution of $\\\\tilde{H}_{Disformal}$ is similar to that of $\\\\tilde{H}_{Conformal}$ , having an (two in our example) enhancement and a decrement compared to the standard expansion rate $H_{GR}$ .\",\n \"Moreover, the main differences with respect to the conformal modification are the position of the notch and its shape.\",\n \"The notch is moved to higher temperatures and it becomes a little bit sharper.\",\n \"These differences between the expansion rates can be understood from ().\",\n \"First, in this equation we see that the factor $\\\\frac{3H^2\\\\gamma ^2BD}{\\\\kappa ^2(1+\\\\lambda )C}$ in the coefficients of $\\\\varphi ^{\\\\prime \\\\prime }$ , $\\\\varphi ^{\\\\prime }$ and $\\\\varphi ^{\\\\prime 2}$ vanishes when $D=0$ .\",\n \"For the disformal example shown in Figure REF , this factor is a very small correction to the equation, which is reflected in the slight shape modification of $\\\\tilde{H}_{Disformal}$ compared to $\\\\tilde{H}_{Conformal}$ .\",\n \"Second, the term proportional to $\\\\delta (\\\\varphi )$ plays a more important role, being responsible for the shifting of the notch.\",\n \"As was discussed previously in section REF , a modification to the expansion rate of the universe prior to BBN has tremendous consequences on the abundance, $\\\\tilde{Y}$ , of dark matter particles.\",\n \"Also, this modification implies that the thermally-averaged annihilation cross section, $\\\\langle \\\\sigma v \\\\rangle $ , for dark matter particles differs significantly from the one predicted by the standard cosmological model, which is approximately $3.0 \\\\times 10^{-26}\\\\rm {cm^3/s}$ (see Figure REF ).\",\n \"We have seen than the enhancement of $\\\\tilde{H}$ allows bigger values of $\\\\langle \\\\sigma v \\\\rangle $ for particles with masses within certain range, and also, the lower value of $\\\\tilde{H}$ implies smaller $\\\\langle \\\\sigma v \\\\rangle $ for particles with masses within a small interval.\",\n \"Thus, the location and shape of the notch determines for which masses the annihilation cross section is smaller.\",\n \"In the disformal scenario, the notch has been moved to higher temperatures, which allows particles with higher masses to have smaller and larger annihilation cross sections for the observed DM content.\",\n \"We leave the detailed study of the modification of $\\\\tilde{H}$ for more general conformal and disformal factors for a separate forthcoming publication.\"\n ],\n [\n \"Conclusions\",\n \"Scalar-tensor theories of gravity are a useful method to explore departures of the expansion rate of the universe from the standard cosmological model in the early universe.\",\n \"The expansion rate of the universe had a strong influence on the evolution of the dark matter abundance during the early stages of the universe's evolution, specially prior to BBN.\",\n \"Modifications to the expansion rate during that time would be reflected in the calculation of the dark matter relic abundance and so can be used as a probe to the predictions of scalar-tensor theories.\",\n \"In this paper, we explored the role played by the scalar field in the modification of the expansion rate of the universe on scalar-tensor theories coupled both conformally and disformally to matter.\",\n \"For the conformal case, we explored a conformal factor of the type $C=(1+b\\\\,e^{-\\\\beta \\\\,\\\\varphi })^2$ .\",\n \"We made no approximations and solved numerically the master equation for the scalar (REF ) for a suitable range of initial conditions.\",\n \"Using this result we then computed the expansion rate modification $\\\\tilde{H}$ under the presence of the scalar field during the radiation dominated era prior to BBN.\",\n \"When comparing the expansion rate, $\\\\tilde{H}$ , to the standard expansion rate, $H_{GR}$ , we found that the speed-up factor, $\\\\xi =\\\\frac{\\\\tilde{H}}{H_{GR}}$ , increases up to 200 and then become of order 1 prior to BBN (see Fig.\",\n \"REF ).\",\n \"Previously, in reference [15], it was shown that there is no change in the expansion rate arising from the modification due to $C$ compared to the standard cosmology, after satisfying all the constraints based on the boundary conditions chosen in [3], [4].\",\n \"This enhancement on the expansion rate has important consequences on the evolution of the abundance of dark matter particles.\",\n \"So, we also investigated the effect on the abundance of dark matter particles.\",\n \"We observed that for dark matter particles of large mass ($m\\\\sim 10^3$ GeV in our example, compared to the $m\\\\sim 50$ GeV reported in [3] for which we found no re-annihilation), the particles undergo a second annihilation process and then freeze-out once and for all in (see Fig.\",\n \"REF ).\",\n \"Moreover, we found that for large masses the annihilation cross-section has to be up to four times larger than that of standard cosmology models in order to satisfy the dark matter content of the universe of 27 %.\",\n \"On the other hand, for small masses this re-annihilation process is not present, but we found that for masses around 130 GeV, the annihilation cross-section can be smaller than the annihilation cross-section for the standard cosmological model (see Fig.\",\n \"REF ).\",\n \"We also started to investigate the effects on the early evolution of the expansion rate of a disformal factor in the metric (REF ).\",\n \"We noticed that in order to have a consistent solution, i.e.\",\n \"a real positive $\\\\tilde{H}$ , the conformal and disformal factors need to satisfy a very specific relation, (REF ).\",\n \"We studied the effect of a disformal factor by turning on a small disformal contribution to the conformal case, given by $D(\\\\varphi )=-4.9\\\\times 10^{-14}\\\\varphi ^2$ .\",\n \"To find the modified expansion rate $\\\\tilde{H}$ during the radiation dominated era prior to BBN, we solved numerically the disformal system of coupled equations () and (REF ).\",\n \"We found that when the disformal function is turned on, $\\\\tilde{H}$ has a very similar profile as for pure conformal case with an enhancement and a notch compare to the standard expansion rate.\",\n \"However, the position of the notch changes and there may be a second enhancement of $\\\\tilde{H}$ compared to $H_{GR}$ , depending on the initial conditions (see also Appendix ).\",\n \"The disformal factor, is moving the notch to higher temperatures which means that we can have larger and smaller annihilation cross-section for any mass of the DM candidate for the observed DM content.\",\n \"We have shown the analysis of the disformal case in a concrete example.\",\n \"We plan to study the disformal effects in more detail in order to asses their general features in a forthcoming publication.\",\n \"Moreover, as we mentioned at the beginning of the paper, we are aiming at models that can be embedded in a more fundamental theory of gravity such as string theory.\",\n \"We plan to analyse these cases in a future publication.\",\n \"We would like to thank, David Cerdeño and Nicolao Fornengo for discussions and Gianmassimo Tasinato for discussions and comments on the manuscript.\",\n \"BD and EJ acknowledge support from DOE Grant DE-FG02-13ER42020.\"\n ],\n [\n \"Numerical Implementation\",\n \"In this section we describe in detail our numerical method to study the master equation and expansion rate evolution and present additional examples for different initial conditions.\",\n \"As we explained in section REF the result shown in Figure REF for $\\\\tilde{\\\\omega }(\\\\tilde{T})$ was obtained using the so-called “kick” function, defined in (REF ).\",\n \"After using $\\\\tilde{\\\\rho }\\\\simeq \\\\pi ^2 g_{eff}(\\\\tilde{T}) \\\\tilde{T}^4/30$ , (REF ) and (REF ) $\\\\Sigma $ becomes $\\\\Sigma (\\\\tilde{T}) = \\\\sum _{A} \\\\frac{15}{\\\\pi ^4} \\\\frac{g_A}{g_{eff}(\\\\tilde{T})} \\\\,y_A^2 \\\\int _{y_A}^{\\\\infty } dx \\\\frac{\\\\sqrt{x^2-y^2_A}}{e^x\\\\pm 1}\\\\,,$ where $g_{eff}(\\\\tilde{T})$ is the total number of relativistic degrees of freedomTo calculate $g_{eff}$ we follow the numerical procedure described in Appendix A of [27]., $g_A$ is the number of internal degrees of freedom of particles of type $A$ , $y_A=m_A/\\\\tilde{T}$ and the plus (minus) sign in the integral is for fermions (bosons).\",\n \"The particle spectrum used to compute $\\\\Sigma $ is shown in Table REF and in Figure we plot the resulting kick function during the radiation dominated era.\",\n \"Then, we compute the equation of state from (REF ) as $\\\\tilde{\\\\omega }=(1-\\\\Sigma (\\\\tilde{T}))/3$ , which is shown in Figure REF .\",\n \"Table: Spectrum of particles used to calculate the kick function ().\",\n \"For each particle, we show its mass and number of internal degrees of freedom, g A g_A.Figure: Thermal evolution of the kick function during the radiation dominated era.\",\n \"Outside the interval of temperatures shown, Σ\\\\Sigma vanishes.\"\n ],\n [\n \"Conformal case solutions\",\n \"Using the numerical solution for $\\\\tilde{\\\\omega }$ , we plugged it into the master equation, and solve it numerically for the interesting initial conditions as described in section REF for the conformal case.\",\n \"To look for suitable solutions, we fixed the initial value of the velocity and then look for an initial value of the scalar such that $(1+\\\\alpha (\\\\varphi )\\\\varphi ^{\\\\prime })$ stayed positive, thus giving a positive modified expansion rate.\",\n \"As we explained in that section, we were aiming at solutions where the scalar field passed from positive to negative to positive values again.\",\n \"In Figure REF , we show the thermal evolution of the scalar field in the pure conformal scenario for different initial velocities $\\\\varphi ^{\\\\prime }$ .\",\n \"We let the initial value of $\\\\varphi $ , fixed at $\\\\varphi _i=0.2$ and solve the master equation (REF ) for the different initial velocities and initial temperature to be 1000 GeV.\",\n \"This plot shows the behaviour of the scalar field as described in section REF .\",\n \"In Figure REF , we show the resulting modified expansion rates for the different initial conditions considered in Fig.\",\n \"REF .\"\n ],\n [\n \"Disformal case solutions\",\n \"As discussed in the main text, when both conformal and disformal functions are non-zero, we solve the system of coupled equations () and (REF ) numerically with Mathematica.\",\n \"We explored different boundary conditions for the scalar field and its first derivative.\",\n \"With the solutions of these equations we used (REF ) to find the modified expansion rate.\",\n \"In Figure REF we show the modified expansion rates for a disformal factor given by $D=D_0\\\\varphi ^2$ with $D_0=-4.9\\\\times 10^{-14}$ and the conformal factor being the same as before, that is $C= (1+b e^{-\\\\beta \\\\varphi })^2$ with $b=0.1$ , $\\\\beta = 8$ .\",\n \"We have added one additional initial condition with respect to the conformal case, $\\\\varphi ^{\\\\prime }_{initial}=-1.$ It is interesting to notice that for this initial condition, the pure conformal case does not give a solution satisfying the necessary constraints explained in the main text.\",\n \"In this sense, we see that the disformal contribution is important in order to find solutions otherwise excluded.\",\n \"Figure: Modified expansion rate as function of temperature in the disformal scenario for various initial conditions.We use ϕ i =0.2\\\\varphi _{i}= 0.2 and every curve shown corresponds to a different value of ϕ i ' \\\\varphi ^{\\\\prime }_i\"\n ],\n [\n \"General Disformal Set-Up\",\n \"The general scalar-tensor action coupled to matter, which can include a realisation in string theory compactifications is given by: $S=S_{EH} + S_\\\\phi + S_{m}\\\\,,$ where: $&& S_{EH}=\\\\frac{1}{2\\\\kappa ^2} \\\\!\\\\!\\\\int {\\\\!d^4x\\\\sqrt{-g}\\\\,R}, \\\\\\\\&& S_\\\\phi = - \\\\!\\\\!\\\\int {\\\\!d^4x\\\\sqrt{- g} \\\\left[\\\\frac{b}{2} (\\\\partial \\\\phi )^2+ M^4 C_1 ^2(\\\\phi )\\\\sqrt{1+\\\\frac{D_1(\\\\phi )}{C_1(\\\\phi )} (\\\\partial \\\\phi )^2}+V(\\\\phi )\\\\right]} \\\\,,\\\\\\\\&& S_{m} = - \\\\!\\\\!\\\\int {\\\\!d^4x\\\\sqrt{-\\\\tilde{g}} \\\\,{\\\\cal L}_{DM}(\\\\tilde{g}_{\\\\mu \\\\nu }) } \\\\,,$ and the disformally coupled metric is given by $\\\\tilde{g}_{\\\\mu \\\\nu } = C_2(\\\\phi ) g_{\\\\mu \\\\nu } + D_2(\\\\phi ) \\\\partial _\\\\mu \\\\phi \\\\partial _\\\\nu \\\\phi \\\\,.$ $b$ is a constant equal to 1 or 0, depending on the model one wants to consider; $C_i(\\\\phi ), D_i(\\\\phi )$ are functions of $\\\\phi $ , which can be identified as conformal and disformal couplings of the scalar to the metric, respectively.\",\n \"Finally, we have introduced the mass scale $M$ to keep units right (remember that the conformal coupling is dimensionless, whereas the disformal has units of $Mass^{-4}$ .)\",\n \"The connection of the general action (REF ) to the different models in the literature can be obtained as follows: the case $C_1 = D_2$ , $D_1=D_2$ , $b=0$ arises when considering a D-brane moving along an extra dimension.\",\n \"This case was studied in [17] as a model of a coupled dark matter dark energy sector scenario, where scaling solutions arise naturally.\",\n \"Note that in this case, the kinetic term for the scalar field, identified with dark energy for example, is automatically non-canonical and dictated by the DBI action (see [17]).\",\n \"On the other hand, phenomenological models considering a disformal coupling between matter and a scalar field, usually consider a canonical kinetic term, and therefore, in that case, $C_1=D_1=0$ and $b=1$ ($C_1$ can be taken to be non-zero and will be part of the scalar potential).\",\n \"Furthermore, the widely studied case of a conformal coupling is obtained for $b=1, C_1=D_1=D_2=0$ or, as in the case of a D-brane for exampleFor the system corresponding to a D-brane moving in a typically warped compactification in string theory, the functions $C(\\\\phi )$ and $D(\\\\phi )$ are identified with powers of the so-called warp factor, usually denoted as $h(\\\\phi )$ .\",\n \"In this approach, the longitudinal and transverse fluctuations of the D-brane are identified with the dark matter and dark energy fluids respectively [17]., simply considering small velocities with $b=0, C_1=C_2$ and $D_1=D_2$ , and normalising canonically the scalar field (see Appendix ).\",\n \"Finally, let us clarify further our nomenclature on frames.\",\n \"The action in (REF ) is written in the Einstein frame, which in string theory, is usually related to the frame in which the dilaton and the graviton degrees of freedom are decoupled.\",\n \"From this point of view, the dilaton field as well as all other moduli fields not relevant for the cosmological discussion are considered as stabilised, massive, and are therefore decoupled from the low energy effective theory.\",\n \"In the literature of scalar-tensor theories however (including conformal and disformal couplings), the Einstein and Jordan frames are identified with respect to the (usually single) scalar field to which gravity is coupled.\",\n \"In this paper, we follow this and call “Jordan\\\" or “disformal frame\\\" the frame in which dark matter is coupled only to the metric $\\\\tilde{g}_{\\\\mu \\\\nu }$ , rather than to the metric $g_{\\\\mu \\\\nu } $ and a scalar field $\\\\phi $ .\",\n \"The equations of motion obtained from (REF ) are (REF ): $R_{\\\\mu \\\\nu } -\\\\frac{1}{2}g_{\\\\mu \\\\nu } R = \\\\kappa ^2\\\\left(T^\\\\phi _{\\\\mu \\\\nu } + T^{DM}_{\\\\mu \\\\nu }\\\\right)\\\\,,$ where in the frame relative to $g_{\\\\mu \\\\nu }$ the energy momentum tensors are defined in (REF ) and (REF ).\",\n \"The energy-momentum tensor for the scalar field in the general case is modified from (REF ) to: $T_{\\\\mu \\\\nu }^{\\\\phi } = - g_{\\\\mu \\\\nu } \\\\left[M^4C_1^2 \\\\gamma _1^{-1} + \\\\frac{b}{2} (\\\\partial \\\\phi )^2+ V \\\\right]+ \\\\left(M^4 C_1D_1 \\\\,\\\\gamma _1+ b\\\\right) \\\\partial _\\\\mu \\\\phi \\\\, \\\\partial _\\\\nu \\\\phi $ where now the energy density and pressure are given by: $\\\\rho _\\\\phi =- \\\\frac{b}{2}(\\\\partial \\\\phi )^2 + M^4C_1^2 \\\\gamma _1 + V \\\\,, \\\\qquad P_\\\\phi = - \\\\frac{b}{2}(\\\\partial \\\\phi )^2 - M^4C_1^2\\\\gamma _1^{-1} - V \\\\,,$ and the “Lorentz factor\\\" $\\\\gamma _1$ introduced above is defined by $\\\\gamma _1 \\\\equiv \\\\left(1+ \\\\frac{D_1}{C_1}\\\\, (\\\\partial \\\\phi )^2\\\\right)^{-1/2}\\\\,.$ We can rewrite (REF ) in a more succinct way, by defining ${\\\\mathcal {V}} \\\\equiv V + C_1^2 M^4$ $\\\\rho _\\\\phi =- \\\\left[\\\\frac{b}{2}+ \\\\frac{M^4C_1 D_1\\\\gamma _1}{\\\\gamma +1}\\\\right] (\\\\partial \\\\phi )^2 + {\\\\mathcal {V}} \\\\,, \\\\qquad P_\\\\phi = - \\\\left[\\\\frac{b}{2}+ \\\\frac{M^4C_1 D_1\\\\gamma _1^{-1}}{\\\\gamma +1}\\\\right] (\\\\partial \\\\phi )^2 - {\\\\mathcal {V}} \\\\,.$ The equation of motion for the scalar field becomes (compare with (REF )) $&&\\\\hspace{-28.45274pt}- \\\\nabla _\\\\mu \\\\!\\\\left[(M^4 D_1C_1\\\\gamma _1\\\\, +b)\\\\,\\\\partial ^\\\\mu \\\\phi \\\\right] \\\\!+ \\\\!\\\\frac{\\\\gamma _1^{-1}M^4 C_1^2}{2} \\\\!\\\\left[\\\\!\\\\frac{D_1^{\\\\prime }}{D_1} +3\\\\frac{C_1^{\\\\prime }}{C_1} \\\\right] \\\\!+ \\\\!\\\\frac{\\\\gamma _1 \\\\,M^4 C_1^2}{2} \\\\!\\\\left[\\\\!\\\\frac{C_1^{\\\\prime }}{C_1} -\\\\frac{D_1^{\\\\prime }}{D_1} \\\\right] \\\\!+\\\\!\",\n \"V^{\\\\prime }\\\\nonumber \\\\\\\\&& \\\\hspace{85.35826pt}- \\\\frac{T^{\\\\mu \\\\nu }}{2}\\\\!\\\\left[\\\\frac{C_2^{\\\\prime }}{C_2} g_{\\\\mu \\\\nu } +\\\\frac{D_2^{\\\\prime }}{C_2}\\\\partial _\\\\mu \\\\phi \\\\partial _\\\\nu \\\\phi \\\\right]+\\\\nabla _\\\\mu \\\\left[\\\\frac{D_2}{C_2}T^{\\\\mu \\\\nu } \\\\partial _\\\\nu \\\\phi \\\\right] =0\\\\,.\",\n \"\\\\nonumber \\\\\\\\$ Finally, the energy-momentum conservation equation gives rise to (REF ), where $Q$ now is given in terms of $C_2, D_2$ : $Q\\\\equiv \\\\nabla _\\\\mu \\\\left[\\\\frac{D_2}{C_2} \\\\,T^{\\\\mu \\\\lambda } \\\\,\\\\partial _\\\\lambda \\\\phi \\\\right] - \\\\frac{T^{\\\\mu \\\\nu } }{2} \\\\left[\\\\frac{C_2^{\\\\prime }}{C_2} g_{\\\\mu \\\\nu } +\\\\frac{D_2^{\\\\prime }}{C_2} \\\\,\\\\partial _\\\\mu \\\\phi \\\\,\\\\partial _\\\\nu \\\\phi \\\\right]\\\\,.$\"\n ],\n [\n \"General cosmological equations\",\n \"The equations of motion for the general system in an FRW background become: $&& H^2 =\\\\frac{\\\\kappa ^2}{3} \\\\left[\\\\rho _\\\\phi +\\\\rho \\\\right]\\\\,, \\\\\\\\&& \\\\dot{H} + H^2 = -\\\\frac{\\\\kappa ^2}{6}\\\\left[ \\\\rho _\\\\phi + 3P_\\\\phi +\\\\rho +3 P \\\\right]\\\\,,\\\\\\\\&& \\\\ddot{\\\\phi }\\\\left[1+ \\\\frac{b}{M^4C_1D_1\\\\gamma _1^3}\\\\right]+3H\\\\dot{\\\\phi }\\\\,\\\\gamma _1^{-2}\\\\left[\\\\frac{b}{M^4C_1D_1\\\\gamma _1}+ 1\\\\right] \\\\nonumber \\\\\\\\&&\\\\hspace{14.22636pt}+ \\\\frac{C_1}{2D_1}\\\\left(\\\\gamma _1^{-2}\\\\left[\\\\frac{5 C_1^{\\\\prime }}{C_1} - \\\\frac{D_1^{\\\\prime }}{D_1}\\\\right]+ \\\\frac{D_1^{\\\\prime }}{D_1}- \\\\frac{C_1^{\\\\prime }}{C_1} -4\\\\gamma ^{-3}_1 \\\\frac{C^{\\\\prime }}{C} \\\\right)+ \\\\frac{1}{M^4C_1D_1\\\\gamma _1^{3}}\\\\, ({\\\\mathcal {V}}^{\\\\prime }+Q_0) =0 \\\\,, \\\\nonumber \\\\\\\\$ where, $H= \\\\frac{\\\\dot{a}}{a}$ , dots are derivatives with respect to $t$ , $^{\\\\prime }=d/d\\\\phi $ and $\\\\gamma _1= (1-D_1 \\\\,\\\\dot{\\\\phi }^2/C_1)^{-1/2}.$ We also have the continuity equations for the scalar field and matter given by $&&\\\\dot{\\\\rho }_\\\\phi + 3H(\\\\rho _{\\\\phi }+P_{\\\\phi }) = -Q_0\\\\dot{\\\\phi }\\\\,, \\\\\\\\&&\\\\dot{\\\\rho }+ 3H(\\\\rho +P) = Q_0\\\\,\\\\dot{\\\\phi }\\\\,.$ where $Q_0$ is given by $Q_0 = \\\\rho \\\\left[ \\\\frac{D_2}{C_2} \\\\,\\\\ddot{\\\\phi }+ \\\\frac{D_2}{C_2} \\\\,\\\\dot{\\\\phi }\\\\left(\\\\!3H + \\\\frac{\\\\dot{\\\\rho }}{\\\\rho } \\\\right) \\\\!+ \\\\!\\\\left(\\\\!\\\\frac{D_2^{\\\\prime }}{2C_2}-\\\\frac{D_2}{C_2}\\\\frac{C_2^{\\\\prime }}{C_2}\\\\!\\\\right) \\\\dot{\\\\phi }^2 +\\\\frac{C_2^{\\\\prime }}{2\\\\,C_2} (1-3\\\\,\\\\omega )\\\\right].\",\n \"\\\\nonumber \\\\\\\\$ Using () we can rewrite this in a more compact and useful form as $Q_0 = \\\\rho \\\\left( \\\\frac{\\\\dot{\\\\gamma }_2}{\\\\dot{\\\\phi }\\\\, \\\\gamma _2} + \\\\frac{C_2^{\\\\prime }}{2C_2} (1-3\\\\,\\\\omega \\\\,\\\\gamma _2^2) -3H\\\\omega \\\\,\\\\frac{(\\\\gamma _2 -1) }{\\\\dot{\\\\phi }}\\\\right) \\\\,,$ where $\\\\gamma _2= (1-D_2 \\\\,\\\\dot{\\\\phi }^2/C_2)^{-1/2}.$ Plugging this into the (non-)conservation equation for dark matter (), gives: $\\\\dot{\\\\rho }+ 3H (\\\\rho + P\\\\,\\\\gamma _2^{2}) = \\\\rho \\\\left[\\\\frac{\\\\dot{\\\\gamma }_2}{\\\\gamma _2} + \\\\frac{C_2^{\\\\prime } }{2C_2} \\\\,\\\\dot{\\\\phi }\\\\, (1-3\\\\,\\\\omega \\\\gamma _2^2)\\\\right]\\\\,.$ The energy densities and pressures in the Einstein and Jordan frames are now related similarly to (REF ), replacing $\\\\gamma \\\\rightarrow \\\\gamma _2$ : $\\\\tilde{\\\\rho }= C_2^{-2} \\\\gamma _2^{-1} \\\\rho \\\\,, \\\\qquad \\\\tilde{P} = C_2^{-2}\\\\gamma _2\\\\, P,$ and therefore the equation of states in both frames are related by $\\\\tilde{\\\\omega }= \\\\omega \\\\gamma _2^2$ .\",\n \"Similarly the physical proper time and the scale factors in the two frames are related via $\\\\gamma _2$ : $\\\\tilde{a} = C_2^{1/2} a \\\\,, \\\\qquad \\\\quad d\\\\tilde{\\\\tau }= C_2^{1/2} \\\\gamma _2^{-1} d\\\\tau \\\\,.$ Defining the disformal frame Hubble parameter $\\\\tilde{H} \\\\equiv \\\\frac{d \\\\ln {\\\\tilde{a}}}{d\\\\tilde{\\\\tau }}$ , gives: $\\\\tilde{H} = \\\\frac{\\\\gamma _2}{C_2^{1/2}}\\\\left[ H + \\\\frac{C_2^{\\\\prime }}{2C_2}\\\\dot{\\\\phi }\\\\right]\\\\,.$ To solve the equations of motion one now can proceed as in section REF to write the equations in terms of derivatives w.r.t.\",\n \"the number of e-folds $N$ and consider different cases by choosing appropriately the parameters $b,C_i,D_i$ .\",\n \"We leave the analysis of these for a future publication.\"\n ],\n [\n \"The conformal case in D-brane scenarios\",\n \"In this section we show how to recover the pure conformal case from the D-brane picture, that is, $b=0$ , $C_1=C_2$ , $D_1=D_2$ .\",\n \"We start by expanding the square root in the scalar part of the action (REF ).\",\n \"Doing this we get $S_\\\\phi & =& -\\\\int {d^4x \\\\sqrt{-g} \\\\left[ M^4C_1^2 \\\\left( 1+ \\\\frac{D_1}{2C_1} (\\\\partial \\\\phi )^2 + \\\\dots \\\\right) + V(\\\\phi )\\\\right]}\\\\nonumber \\\\\\\\&=& -\\\\int {d^4x \\\\sqrt{-g} \\\\left[ \\\\frac{M^4C_1 D_1}{2} (\\\\partial \\\\phi )^2 + M^4C_1^2(\\\\phi )+ V(\\\\phi ) + \\\\dots \\\\right]}\\\\nonumber \\\\\\\\&=& -\\\\int {d^4x \\\\sqrt{-g} \\\\left[ \\\\frac{M^4C_1 D_1}{2} (\\\\partial \\\\phi )^2 + {\\\\mathcal {V}}(\\\\phi ) + \\\\dots \\\\right]} \\\\,,$ On the other hand, the matter Lagrangean takes the form $S_{DM} &=& -\\\\int {d^4x \\\\sqrt{-\\\\tilde{g}} \\\\,{\\\\cal L}_{DM}(\\\\tilde{g}_{\\\\mu \\\\nu })}\\\\nonumber \\\\\\\\&=& -\\\\int {d^4x \\\\sqrt{-g} \\\\, C_1^2(\\\\phi ) \\\\left(1+ \\\\frac{D_1}{2C_1} (\\\\partial \\\\phi )^2 + \\\\dots \\\\right){\\\\cal L}_{DM}( \\\\tilde{g}_{\\\\mu \\\\nu })} \\\\nonumber \\\\\\\\ &=& -\\\\int {d^4x \\\\sqrt{- g} \\\\, C_1^2 (\\\\phi ) {\\\\cal L}_{DM}( \\\\tilde{g}_{\\\\mu \\\\nu }) + \\\\dots } =-\\\\int {d^4x \\\\sqrt{- \\\\tilde{g}} \\\\, {\\\\cal L}_{DM}( \\\\tilde{g}_{\\\\mu \\\\nu }) + \\\\dots } \\\\nonumber \\\\\\\\$ where now $\\\\tilde{g}_{\\\\mu \\\\nu } = C_1(\\\\phi ) g_{\\\\mu \\\\nu }$ (and we have used that $\\\\det \\\\tilde{g}_{\\\\mu \\\\nu } = C_1^4 (1+ D_1/C_1 (\\\\partial \\\\phi )^2)$ ).\",\n \"Finally, to compare the D-brane case with the pure conformal case, we need to canonically normalise $\\\\phi $ .\",\n \"Calling $\\\\varphi $ the canonically normalised field, this is obtained from $\\\\phi $ as $\\\\varphi = \\\\int M^2\\\\sqrt{D_1C_1} \\\\, d\\\\phi \\\\,.$ It is clear that when $D_1=1/(M^4 C_1)$ , $\\\\varphi =\\\\phi $ and therefore the action for the scalar field (REF ) is already in the required form.\",\n \"We can now take the limit $\\\\gamma \\\\rightarrow 1$ into the equations of motion (REF )-() and make the identification $D_1=1/(M^4C_1)$ to recover the conformal case equations of motion.\",\n \"Note that in this limit $Q_0 \\\\rightarrow \\\\rho C_1^{\\\\prime }/2C_1$ , and is independent of $D_1$ (see (REF ) with $C_1=C_2, D_1=D_2$ ).\"\n ]\n]"},"id":{"kind":"string","value":"1612.05553"}}},{"rowIdx":3385,"cells":{"text":{"kind":"list like","value":[["Scheme variations of the QCD coupling and tau decays"],["Abstract The QCD coupling, $\\alpha_s$, is not a physical observable since it depends on conventions related to the renormalization procedure.","Here we discuss a redefinition of the coupling where changes of scheme are parametrised by a single parameter $C$.","The new coupling is denoted $\\hat \\alpha_s$ and its running is scheme independent.","Moreover, scheme variations become completely analogous to renormalization scale variations.","We discuss how the coupling $\\hat \\alpha_s$ can be used in order to optimize predictions for the inclusive hadronic decays of the tau lepton.","Preliminary investigations of the $C$-scheme in the presence of higher-order terms of the perturbative series are discussed here for the first time."],["Introduction","The perturbative expansion in the strong coupling $\\alpha _s$ is the main approach to predictions in quantum chromodynamics (QCD) at sufficiently high energies.","However, the expansion parameter, $\\alpha _s$ , is not a physical observable of the theory.","Its definition carries a dependence on conventions related to the renormalization procedure, such as the renormalization scale and renormalization scheme.","Physical observables should, of course, be independent of any such conventions.","This requirement leads, in the case of the renormalization scale, to well defined Renormalization Group Equations (RGE) that must be satisfied by physical quantities.","The situation regarding the renormalization scheme is more complicated and perturbative computations are, most often, performed in conventional schemes such as ${\\overline{\\rm MS}}$ [1].","In this work we discuss a new definition of the QCD coupling, that we denote $\\hat{\\alpha }_s$ , recently introduced in Ref.","[2], and its applications to the QCD description of inclusive hadronic $\\tau $ decays.","The running of this new coupling is renormalization scheme independent, i.e.","in its $\\beta $ function only scheme independent coefficients intervene.","The scheme dependence of $\\hat{\\alpha }_s$ is parametrised by a single continuous parameter $C$ .","The evolution of $\\hat{\\alpha }_s$ with respect to this new parameter is governed by the same $\\beta $ function that governs the scale evolution.","We refer to the coupling $\\hat{\\alpha }_s$ as the $C$ -scheme coupling.","An important aspect is the fact that perturbative expansions in $\\alpha _s$ are divergent series that are assumed to be asymptotic expansions to a “true” value, which is unknown [3].F.","Dyson formulated the first form of this reasoning in 1952, in the context of Quantum Electrodynamics [4].","In this spirit, different schemes correspond to different asymptotic expansions to the same scheme invariant physical quantity, and should be interpreted as such.","One can then use the parameter $C$ to interpolate between perturbative series with larger or smaller coupling values, and exploit this dependence in order to optimize the predictions for observables of the theory.","The idea of exploiting the scheme dependence in order to optimize the series differs from the approach of other celebrated methods used for the optimisation of perturbative predictions.","In methods such as Brodsky-Lepage-Mackenzie (BLM) [5] or the Principle of Maximum Conformality [6], [7] the idea is to obtain a scheme independent result through a well defined algorithm for setting the renormalization scale, regardless of the intermediate scheme used for the perturbative calculation (which most often is ${\\overline{\\rm MS}}$ ).","The “effective charge\" method [8], on the other hand, involves a process dependent definition of the coupling.","In the procedure described here, one defines a process independent class of schemes, parametrised by the parameter $C$ .","The optimal value of $C$ must be set independently for each process considered.","We begin with the scale running of the QCD coupling that we write as $-\\,Q\\,\\frac{{\\rm d}a_Q}{{\\rm d}Q} \\,\\equiv \\, \\beta (a_Q) \\,=\\,\\beta _1\\,a_Q^2 + \\beta _2\\,a_Q^3 + \\beta _3\\,a_Q^4 + \\cdots $ We will work with $a_Q \\equiv \\alpha _s(Q)/\\pi $ , with $Q$ being a physically relevant scale.","Since the recent five-loop computation of Ref.","[9], the first five coefficients of the QCD $\\beta $ -function are known analytically.","The coefficients $\\beta _1$ and $\\beta _2$ are scheme independent.","Let us consider a scheme transformation to a new coupling $a^{\\prime }$ , which, perturbatively, takes the general form $a^{\\prime } \\,\\equiv \\, a + c_1\\,a^2 + c_2\\,a^3 + c_3\\,a^4 + \\cdots \\,$ The QCD scale $\\Lambda $ is also different in the two schemes and obeys the relation $\\Lambda ^{\\prime } \\,=\\, \\Lambda \\,{\\rm e}^{c_1/\\beta _1}.$ The shift in $\\Lambda ^{\\prime }$ depends only on a single constant [10], governed by $c_1$ of Eq.","(REF ).","This fact motivates the definition of the new coupling $\\hat{a}_Q$ , which is scheme invariant except for shifts in $\\Lambda $ parametrised by a parameter $C$ as $\\frac{1}{\\hat{a}_Q} + \\frac{\\beta _2}{\\beta _1} \\ln \\hat{a}_Q \\,&\\equiv &\\,\\beta _1 \\Big ( \\ln \\frac{Q}{\\Lambda } + \\frac{C}{2} \\Big ) \\nonumber \\\\&& \\hspace{-51.21495pt} \\,=\\, \\frac{1}{a_Q} + \\frac{\\beta _1}{2}\\,C +\\frac{\\beta _2}{\\beta _1}\\ln a_Q - \\beta _1 \\!\\int \\limits _0^{a_Q}\\,\\frac{{\\rm d}a}{\\tilde{\\beta }(a)},$ where $\\frac{1}{\\tilde{\\beta }(a)} \\,\\equiv \\, \\frac{1}{\\beta (a)} - \\frac{1}{\\beta _1 a^2}+ \\frac{\\beta _2}{\\beta _1^2 a}$ is free of singularities in the limit $a\\rightarrow 0$ and we have used the scale invariant form of $\\Lambda $ .","The coupling $\\hat{a}_Q$ is a function of the parameter $C$ but we do not make this dependence explicit to keep the notation simple.","The definition of Eq.","(REF ) should be interpreted in perturbation theory in an iterative sense, which allows one to deduce the corresponding coefficients $c_i$ of Eq.","(REF ) (their explicit expressions are given in [2] using the ${\\overline{\\rm MS}}$ as the input scheme).","One should remark that a combination similar to (REF ), but without the logarithmic term on the left-hand side, was already discussed in Refs.","[11], [12].","However, without this term, an unwelcome logarithm of $a_Q$ remains in the perturbative relation between the couplings $\\hat{a}_Q$ and $a_Q$ .","This non-analytic term is avoided by the construction of Eq.","(REF ).","From the definition of the new coupling $\\hat{a}_Q$ we can derive its $\\beta $ function that reads $-\\,Q\\,\\frac{{\\rm d}\\hat{a}_Q}{{\\rm d}Q} \\,\\equiv \\, \\hat{\\beta }(\\hat{a}_Q) \\,=\\,\\frac{\\beta _1 \\hat{a}_Q^2}{\\left(1 - \\mbox{$\\frac{\\beta _2}{\\beta _1}$}\\, \\hat{a}_Q\\right)} .$ The function $\\hat{\\beta }$ takes a simple form and is scheme independent since only the coefficients $\\beta _1$ and $\\beta _2$ intervene.","The evolution with the parameter $C$ obeys an analogous equation $-2 \\frac{{\\rm d}\\hat{a}_Q}{{\\rm d}C} =\\,\\frac{\\beta _1 \\hat{a}_Q^2}{\\left(1 - \\mbox{$\\frac{\\beta _2}{\\beta _1}$}\\, \\hat{a}_Q\\right)}.$ Therefore, there is a complete analogy between the coupling evolution with respect to the scale and with respect to the scheme parameter $C$ .","The dependence of $\\hat{a}_Q$ on $C$ is displayed in Fig.","REF using the ${\\overline{\\rm MS}}$ as input scheme and setting the scale to the $\\tau $ mass, $M_\\tau $ .","The new coupling becomes smaller for larger values of $C$ and perturbativity breaks down for values below roughly $C=-2$ .","Therefore, we restrict our analysis to $C\\ge -2$ .","Figure: The coupling a ^(M τ )\\hat{a}(M_\\tau ) according to Eq.","() as a functionof CC, and for the MS ¯{\\overline{\\rm MS}} input value α s (M τ )=0.316(10)\\alpha _s(M_\\tau )=0.316(10).","Theyellow band corresponds to the α s \\alpha _s uncertainty."],["Application to $\\tau $ decays","As a phenomenological application of the $C$ -scheme coupling, we focus here on the perturbative expansion of the total $\\tau $ hadronic width.","The chief observable is the ratio $R_\\tau $ of the total hadronic branching fraction to the electron branching fraction.","It is conventionally decomposed as $R_\\tau \\,=\\, 3\\, S_{\\rm EW} (|V_{ud}|^2 + |V_{us}|^2)\\, ( 1 + \\delta ^{(0)}+ \\cdots ),$ where $S_{\\rm EW}$ is an electroweak correction and $V_{ud}$ , as well as $V_{us}$ , CKM matrix elements.","Perturbative QCD is encoded in $\\delta ^{(0)}$ (see Refs.","[13], [14] for details) and the ellipsis indicate further small sub-leading corrections.","The calculation of $\\delta ^{(0)}$ is performed from a contour integral of the so called Adler function in the complex energy plane, exploiting analyticity properties, which allows one to avoid the low energy region where perturbative QCD is not valid.","In doing so, one must adopt a procedure in order to deal with the renormalization scale.","The scale logarithms can be summed either before or after performing the contour integration.","The first choice, where the integrals are performed over the running QCD coupling, is called Contour Improved Perturbation Theory (CIPT), while the second, where the coupling is evaluated at a fixed scale and the integrals are performed over the logarithms, is called Fixed Order Perturbation Theory (FOPT).","Analytic results for the coefficients of the Adler function are available up to five loops, or $\\alpha _s^4$ [15].","Here we consider an estimate for the yet unknown fifth order coefficient of the Adler function, namely $c_{51}=283$ [14].","In FOPT, the perturbative series of $\\delta ^{(0)}(a_Q)$ in terms of the ${\\overline{\\rm MS}}$ coupling $a_Q$ is given by [15], [14] $\\delta _{\\rm FO}^{(0)}(a_Q) =a_Q + 5.202a_Q^2 + 26.37a_Q^3 + 127.1a_Q^4 +\\cdots $ In the $C$ -scheme coupling $\\hat{a}_Q$ , the expansion for $\\delta _{\\rm FO}^{(0)}$ is $&\\delta _{\\rm FO}^{(0)}(\\hat{a}_Q) = \\hat{a}_Q + (5.202 + 2.25 C)\\,\\hat{a}_Q^2 \\nonumber \\\\& \\hspace{5.69046pt} + (27.68 + 27.41 C + 5.063 C^2)\\,\\hat{a}_Q^3 \\nonumber \\\\& \\hspace{5.69046pt} + (148.4 + 235.5 C + 101.5 C^2 + 11.39 C^3)\\,\\hat{a}_Q^4\\nonumber \\\\& \\hspace{5.69046pt} + \\cdots $ Figure: δ FO (0) (a ^ Q )\\delta _{\\rm FO}^{(0)}(\\hat{a}_Q) of Eq.","() as a function ofCC.","The yellow band arises from either removing or doubling the fifth-orderterm.","In the red dots, the 𝒪(a ^ 5 ){\\cal O}(\\hat{a}^5) vanishes, and 𝒪(a ^ 4 ){\\cal O}(\\hat{a}^4) is taken asthe uncertainty.For further explanation, see the text.In Fig.","REF , we display $\\delta _{\\rm FO}^{(0)}(\\hat{a}_Q)$ as a function of $C$ .","Assuming $c_{5,1}=283$ , the yellow band corresponds to removing or doubling the ${\\cal O}(\\hat{a}^5)$ term.","A plateau is found for $C\\approx -1$ .","Taking $c_{5,1}=566$ and then doubling the ${\\cal O}(\\hat{a}^5)$ results in the blue curve that does not show this stability.","Hence, this scenario is disfavoured.","In the red dots, which lie at $C=-0.882$ and $C=-1.629$ , the ${\\cal O}(\\hat{a}^5)$ correction vanishes, and the ${\\cal O}(\\hat{a}^4)$ term is taken as the uncertainty, in the spirit of asymptotic series.","The point to the right has a substantially smaller error, and yields $\\delta _{\\rm FO}^{(0)}(\\hat{a}_{M_\\tau },C=-0.882) \\,=\\,0.2047 \\pm 0.0034 \\pm 0.0133 \\,.$ The second error covers the uncertainty of $\\alpha _s(M_\\tau )$ .","In this case, the direct ${\\overline{\\rm MS}}$ prediction of Eq.","(REF ) is $\\delta _{\\rm FO}^{(0)}(a_{M_\\tau }) \\,=\\, 0.1991 \\pm 0.0061 \\pm 0.0119\\,\\,\\,\\,\\, ({\\overline{\\rm MS}}) \\,.$ This value is somewhat lower, but within $1\\,\\sigma $ of the higher-order uncertainty.","In CIPT, contour integrals over the running coupling have to be computed, and hence the result cannot be given in analytical form.","The general behaviour is very similar to FOPT, with the exception that now also for $c_{5,1}=566$ a zero of the ${\\cal O}(\\hat{a}^5)$ term is found.","Employing the value of $C$ which leads to the smaller uncertainty one finds $\\delta _{\\rm CI}^{(0)}(\\hat{a}_{M_\\tau },C=-1.246) \\,=\\,0.1840 \\pm 0.0062 \\pm 0.0084 \\,.$ As has been discussed many times in the past (see e.g.","[14]) the CIPT prediction lies substantially below the FOPT results.","On the other hand, the parametric $\\alpha _s$ uncertainty in CIPT turns out to be smaller."],["Higher-order terms","The behaviour of the series at higher orders is not known exactly.","However, realistic models of the Adler function can be constructed in the Borel plane, in which the singularities of the function, namely its renormalon content, is partially known [3].","In Ref.","[14] (see also Ref.","[16]), models of the Adler function were constructed using the leading renormalons, that largely dominate the higher-order behaviour of the perturbative series.","The model is matched to the exactly known coefficients in order to fully reproduce QCD for terms up to $a_Q^4$ .","This allows for a complete reconstruction of the series, to arbitrarily high orders in the coupling, and, moreover, one is able to obtain the “true” value of the asymptotic series by means of the Borel sum.","In fact, the series is not strictly Borel summable because infra-red renormalons obstruct integration on the positive real axis.","The “true” value has, therefore, an inherent ambiguity that stems from the prescription adopted to circumvent the singularities along the contour of integration.","This ambiguity is related to non-perturbative physics [3], [14].","Here we perform a preliminary investigation of the behaviour of $\\delta ^{(0)}$ at higher orders using the $C$ -scheme coupling.","The Adler function coefficients for terms higher than $a_Q^5$ are obtained in the ${\\overline{\\rm MS}}$ scheme from the central model of Ref. [14].","The series can then be translated to the $C$ -scheme by means of the perturbative relation between the couplings $a_Q$ and $\\hat{a}_Q$ [2].","Fig.","REF shows four different series that should approach the same Borel summed result, showed as a horizontal band.","The four series use as input the coefficients exactly known in QCD with the addition of the estimate $c_{5,1}=283$ .","One observes that the optimised version of $\\delta _{\\rm FO}^{(0)}$ (filled circles) approaches the Borel sum of the series faster than the ${\\overline{\\rm MS}}$ result (empty circles).","Of course, because the optimised series has a larger coupling (see Fig.","1) asymptoticity sets in earlier and the divergent character is clearly visible already around the 10th order.","The FOPT result with $C=0.7$ shows that smaller couplings do not necessarily lead to a better approximation at lower orders, requiring many more terms to give a good approximation to the Borel summed result.","Finally, the optimal CIPT series does not give a good approximation to the Borel summed result (this is also the case in the ${\\overline{\\rm MS}}$ [14]).","Unfortunately, the use of the $C$ -scheme coupling does not make the CIPT prediction closer to FOPT.","The $C$ -scheme FOPT, on the other hand, is in excellent agreement with the central Borel model which suggests that FOPT should be the favoured expansion.","Figure: Four series for δ (0) \\delta ^{(0)} with higher-order coefficients from the central model of Ref. .","In all cases α s (M τ )=0.316\\alpha _s(M_\\tau )=0.316 which corresponds to the central value of the present world average .","The optimised FOPT (filled circles) and CIPT (filled squares) series can be compared with the FOPT MS ¯{\\overline{\\rm MS}} results (empty circles) and FOPT for C=0.7C=0.7 (triangles).","The shaded band gives the Borel summed result, the “true\" value of the series, with its associated ambiguity ."],["Acknowledgements"," It is a pleasure to thank the organisers of this very fruitful meeting.","DB is supported by the São Paulo Research Foundation (FAPESP) grant 2015/20689-9, and by CNPq grant 305431/2015-3.","The work of MJ and RM has been supported in part by MINECO Grant number CICYT-FEDER-FPA2014-55613-P, by the Severo Ochoa excellence program of MINECO, Grant SO-2012-0234, and Secretaria d'Universitats i Recerca del Departament d'Economia i Coneixement de la Generalitat de Catalunya under Grant 2014 SGR 1450."]],"string":"[\n [\n \"Scheme variations of the QCD coupling and tau decays\"\n ],\n [\n \"Abstract The QCD coupling, $\\\\alpha_s$, is not a physical observable since it depends on conventions related to the renormalization procedure.\",\n \"Here we discuss a redefinition of the coupling where changes of scheme are parametrised by a single parameter $C$.\",\n \"The new coupling is denoted $\\\\hat \\\\alpha_s$ and its running is scheme independent.\",\n \"Moreover, scheme variations become completely analogous to renormalization scale variations.\",\n \"We discuss how the coupling $\\\\hat \\\\alpha_s$ can be used in order to optimize predictions for the inclusive hadronic decays of the tau lepton.\",\n \"Preliminary investigations of the $C$-scheme in the presence of higher-order terms of the perturbative series are discussed here for the first time.\"\n ],\n [\n \"Introduction\",\n \"The perturbative expansion in the strong coupling $\\\\alpha _s$ is the main approach to predictions in quantum chromodynamics (QCD) at sufficiently high energies.\",\n \"However, the expansion parameter, $\\\\alpha _s$ , is not a physical observable of the theory.\",\n \"Its definition carries a dependence on conventions related to the renormalization procedure, such as the renormalization scale and renormalization scheme.\",\n \"Physical observables should, of course, be independent of any such conventions.\",\n \"This requirement leads, in the case of the renormalization scale, to well defined Renormalization Group Equations (RGE) that must be satisfied by physical quantities.\",\n \"The situation regarding the renormalization scheme is more complicated and perturbative computations are, most often, performed in conventional schemes such as ${\\\\overline{\\\\rm MS}}$ [1].\",\n \"In this work we discuss a new definition of the QCD coupling, that we denote $\\\\hat{\\\\alpha }_s$ , recently introduced in Ref.\",\n \"[2], and its applications to the QCD description of inclusive hadronic $\\\\tau $ decays.\",\n \"The running of this new coupling is renormalization scheme independent, i.e.\",\n \"in its $\\\\beta $ function only scheme independent coefficients intervene.\",\n \"The scheme dependence of $\\\\hat{\\\\alpha }_s$ is parametrised by a single continuous parameter $C$ .\",\n \"The evolution of $\\\\hat{\\\\alpha }_s$ with respect to this new parameter is governed by the same $\\\\beta $ function that governs the scale evolution.\",\n \"We refer to the coupling $\\\\hat{\\\\alpha }_s$ as the $C$ -scheme coupling.\",\n \"An important aspect is the fact that perturbative expansions in $\\\\alpha _s$ are divergent series that are assumed to be asymptotic expansions to a “true” value, which is unknown [3].F.\",\n \"Dyson formulated the first form of this reasoning in 1952, in the context of Quantum Electrodynamics [4].\",\n \"In this spirit, different schemes correspond to different asymptotic expansions to the same scheme invariant physical quantity, and should be interpreted as such.\",\n \"One can then use the parameter $C$ to interpolate between perturbative series with larger or smaller coupling values, and exploit this dependence in order to optimize the predictions for observables of the theory.\",\n \"The idea of exploiting the scheme dependence in order to optimize the series differs from the approach of other celebrated methods used for the optimisation of perturbative predictions.\",\n \"In methods such as Brodsky-Lepage-Mackenzie (BLM) [5] or the Principle of Maximum Conformality [6], [7] the idea is to obtain a scheme independent result through a well defined algorithm for setting the renormalization scale, regardless of the intermediate scheme used for the perturbative calculation (which most often is ${\\\\overline{\\\\rm MS}}$ ).\",\n \"The “effective charge\\\" method [8], on the other hand, involves a process dependent definition of the coupling.\",\n \"In the procedure described here, one defines a process independent class of schemes, parametrised by the parameter $C$ .\",\n \"The optimal value of $C$ must be set independently for each process considered.\",\n \"We begin with the scale running of the QCD coupling that we write as $-\\\\,Q\\\\,\\\\frac{{\\\\rm d}a_Q}{{\\\\rm d}Q} \\\\,\\\\equiv \\\\, \\\\beta (a_Q) \\\\,=\\\\,\\\\beta _1\\\\,a_Q^2 + \\\\beta _2\\\\,a_Q^3 + \\\\beta _3\\\\,a_Q^4 + \\\\cdots $ We will work with $a_Q \\\\equiv \\\\alpha _s(Q)/\\\\pi $ , with $Q$ being a physically relevant scale.\",\n \"Since the recent five-loop computation of Ref.\",\n \"[9], the first five coefficients of the QCD $\\\\beta $ -function are known analytically.\",\n \"The coefficients $\\\\beta _1$ and $\\\\beta _2$ are scheme independent.\",\n \"Let us consider a scheme transformation to a new coupling $a^{\\\\prime }$ , which, perturbatively, takes the general form $a^{\\\\prime } \\\\,\\\\equiv \\\\, a + c_1\\\\,a^2 + c_2\\\\,a^3 + c_3\\\\,a^4 + \\\\cdots \\\\,$ The QCD scale $\\\\Lambda $ is also different in the two schemes and obeys the relation $\\\\Lambda ^{\\\\prime } \\\\,=\\\\, \\\\Lambda \\\\,{\\\\rm e}^{c_1/\\\\beta _1}.$ The shift in $\\\\Lambda ^{\\\\prime }$ depends only on a single constant [10], governed by $c_1$ of Eq.\",\n \"(REF ).\",\n \"This fact motivates the definition of the new coupling $\\\\hat{a}_Q$ , which is scheme invariant except for shifts in $\\\\Lambda $ parametrised by a parameter $C$ as $\\\\frac{1}{\\\\hat{a}_Q} + \\\\frac{\\\\beta _2}{\\\\beta _1} \\\\ln \\\\hat{a}_Q \\\\,&\\\\equiv &\\\\,\\\\beta _1 \\\\Big ( \\\\ln \\\\frac{Q}{\\\\Lambda } + \\\\frac{C}{2} \\\\Big ) \\\\nonumber \\\\\\\\&& \\\\hspace{-51.21495pt} \\\\,=\\\\, \\\\frac{1}{a_Q} + \\\\frac{\\\\beta _1}{2}\\\\,C +\\\\frac{\\\\beta _2}{\\\\beta _1}\\\\ln a_Q - \\\\beta _1 \\\\!\\\\int \\\\limits _0^{a_Q}\\\\,\\\\frac{{\\\\rm d}a}{\\\\tilde{\\\\beta }(a)},$ where $\\\\frac{1}{\\\\tilde{\\\\beta }(a)} \\\\,\\\\equiv \\\\, \\\\frac{1}{\\\\beta (a)} - \\\\frac{1}{\\\\beta _1 a^2}+ \\\\frac{\\\\beta _2}{\\\\beta _1^2 a}$ is free of singularities in the limit $a\\\\rightarrow 0$ and we have used the scale invariant form of $\\\\Lambda $ .\",\n \"The coupling $\\\\hat{a}_Q$ is a function of the parameter $C$ but we do not make this dependence explicit to keep the notation simple.\",\n \"The definition of Eq.\",\n \"(REF ) should be interpreted in perturbation theory in an iterative sense, which allows one to deduce the corresponding coefficients $c_i$ of Eq.\",\n \"(REF ) (their explicit expressions are given in [2] using the ${\\\\overline{\\\\rm MS}}$ as the input scheme).\",\n \"One should remark that a combination similar to (REF ), but without the logarithmic term on the left-hand side, was already discussed in Refs.\",\n \"[11], [12].\",\n \"However, without this term, an unwelcome logarithm of $a_Q$ remains in the perturbative relation between the couplings $\\\\hat{a}_Q$ and $a_Q$ .\",\n \"This non-analytic term is avoided by the construction of Eq.\",\n \"(REF ).\",\n \"From the definition of the new coupling $\\\\hat{a}_Q$ we can derive its $\\\\beta $ function that reads $-\\\\,Q\\\\,\\\\frac{{\\\\rm d}\\\\hat{a}_Q}{{\\\\rm d}Q} \\\\,\\\\equiv \\\\, \\\\hat{\\\\beta }(\\\\hat{a}_Q) \\\\,=\\\\,\\\\frac{\\\\beta _1 \\\\hat{a}_Q^2}{\\\\left(1 - \\\\mbox{$\\\\frac{\\\\beta _2}{\\\\beta _1}$}\\\\, \\\\hat{a}_Q\\\\right)} .$ The function $\\\\hat{\\\\beta }$ takes a simple form and is scheme independent since only the coefficients $\\\\beta _1$ and $\\\\beta _2$ intervene.\",\n \"The evolution with the parameter $C$ obeys an analogous equation $-2 \\\\frac{{\\\\rm d}\\\\hat{a}_Q}{{\\\\rm d}C} =\\\\,\\\\frac{\\\\beta _1 \\\\hat{a}_Q^2}{\\\\left(1 - \\\\mbox{$\\\\frac{\\\\beta _2}{\\\\beta _1}$}\\\\, \\\\hat{a}_Q\\\\right)}.$ Therefore, there is a complete analogy between the coupling evolution with respect to the scale and with respect to the scheme parameter $C$ .\",\n \"The dependence of $\\\\hat{a}_Q$ on $C$ is displayed in Fig.\",\n \"REF using the ${\\\\overline{\\\\rm MS}}$ as input scheme and setting the scale to the $\\\\tau $ mass, $M_\\\\tau $ .\",\n \"The new coupling becomes smaller for larger values of $C$ and perturbativity breaks down for values below roughly $C=-2$ .\",\n \"Therefore, we restrict our analysis to $C\\\\ge -2$ .\",\n \"Figure: The coupling a ^(M τ )\\\\hat{a}(M_\\\\tau ) according to Eq.\",\n \"() as a functionof CC, and for the MS ¯{\\\\overline{\\\\rm MS}} input value α s (M τ )=0.316(10)\\\\alpha _s(M_\\\\tau )=0.316(10).\",\n \"Theyellow band corresponds to the α s \\\\alpha _s uncertainty.\"\n ],\n [\n \"Application to $\\\\tau $ decays\",\n \"As a phenomenological application of the $C$ -scheme coupling, we focus here on the perturbative expansion of the total $\\\\tau $ hadronic width.\",\n \"The chief observable is the ratio $R_\\\\tau $ of the total hadronic branching fraction to the electron branching fraction.\",\n \"It is conventionally decomposed as $R_\\\\tau \\\\,=\\\\, 3\\\\, S_{\\\\rm EW} (|V_{ud}|^2 + |V_{us}|^2)\\\\, ( 1 + \\\\delta ^{(0)}+ \\\\cdots ),$ where $S_{\\\\rm EW}$ is an electroweak correction and $V_{ud}$ , as well as $V_{us}$ , CKM matrix elements.\",\n \"Perturbative QCD is encoded in $\\\\delta ^{(0)}$ (see Refs.\",\n \"[13], [14] for details) and the ellipsis indicate further small sub-leading corrections.\",\n \"The calculation of $\\\\delta ^{(0)}$ is performed from a contour integral of the so called Adler function in the complex energy plane, exploiting analyticity properties, which allows one to avoid the low energy region where perturbative QCD is not valid.\",\n \"In doing so, one must adopt a procedure in order to deal with the renormalization scale.\",\n \"The scale logarithms can be summed either before or after performing the contour integration.\",\n \"The first choice, where the integrals are performed over the running QCD coupling, is called Contour Improved Perturbation Theory (CIPT), while the second, where the coupling is evaluated at a fixed scale and the integrals are performed over the logarithms, is called Fixed Order Perturbation Theory (FOPT).\",\n \"Analytic results for the coefficients of the Adler function are available up to five loops, or $\\\\alpha _s^4$ [15].\",\n \"Here we consider an estimate for the yet unknown fifth order coefficient of the Adler function, namely $c_{51}=283$ [14].\",\n \"In FOPT, the perturbative series of $\\\\delta ^{(0)}(a_Q)$ in terms of the ${\\\\overline{\\\\rm MS}}$ coupling $a_Q$ is given by [15], [14] $\\\\delta _{\\\\rm FO}^{(0)}(a_Q) =a_Q + 5.202a_Q^2 + 26.37a_Q^3 + 127.1a_Q^4 +\\\\cdots $ In the $C$ -scheme coupling $\\\\hat{a}_Q$ , the expansion for $\\\\delta _{\\\\rm FO}^{(0)}$ is $&\\\\delta _{\\\\rm FO}^{(0)}(\\\\hat{a}_Q) = \\\\hat{a}_Q + (5.202 + 2.25 C)\\\\,\\\\hat{a}_Q^2 \\\\nonumber \\\\\\\\& \\\\hspace{5.69046pt} + (27.68 + 27.41 C + 5.063 C^2)\\\\,\\\\hat{a}_Q^3 \\\\nonumber \\\\\\\\& \\\\hspace{5.69046pt} + (148.4 + 235.5 C + 101.5 C^2 + 11.39 C^3)\\\\,\\\\hat{a}_Q^4\\\\nonumber \\\\\\\\& \\\\hspace{5.69046pt} + \\\\cdots $ Figure: δ FO (0) (a ^ Q )\\\\delta _{\\\\rm FO}^{(0)}(\\\\hat{a}_Q) of Eq.\",\n \"() as a function ofCC.\",\n \"The yellow band arises from either removing or doubling the fifth-orderterm.\",\n \"In the red dots, the 𝒪(a ^ 5 ){\\\\cal O}(\\\\hat{a}^5) vanishes, and 𝒪(a ^ 4 ){\\\\cal O}(\\\\hat{a}^4) is taken asthe uncertainty.For further explanation, see the text.In Fig.\",\n \"REF , we display $\\\\delta _{\\\\rm FO}^{(0)}(\\\\hat{a}_Q)$ as a function of $C$ .\",\n \"Assuming $c_{5,1}=283$ , the yellow band corresponds to removing or doubling the ${\\\\cal O}(\\\\hat{a}^5)$ term.\",\n \"A plateau is found for $C\\\\approx -1$ .\",\n \"Taking $c_{5,1}=566$ and then doubling the ${\\\\cal O}(\\\\hat{a}^5)$ results in the blue curve that does not show this stability.\",\n \"Hence, this scenario is disfavoured.\",\n \"In the red dots, which lie at $C=-0.882$ and $C=-1.629$ , the ${\\\\cal O}(\\\\hat{a}^5)$ correction vanishes, and the ${\\\\cal O}(\\\\hat{a}^4)$ term is taken as the uncertainty, in the spirit of asymptotic series.\",\n \"The point to the right has a substantially smaller error, and yields $\\\\delta _{\\\\rm FO}^{(0)}(\\\\hat{a}_{M_\\\\tau },C=-0.882) \\\\,=\\\\,0.2047 \\\\pm 0.0034 \\\\pm 0.0133 \\\\,.$ The second error covers the uncertainty of $\\\\alpha _s(M_\\\\tau )$ .\",\n \"In this case, the direct ${\\\\overline{\\\\rm MS}}$ prediction of Eq.\",\n \"(REF ) is $\\\\delta _{\\\\rm FO}^{(0)}(a_{M_\\\\tau }) \\\\,=\\\\, 0.1991 \\\\pm 0.0061 \\\\pm 0.0119\\\\,\\\\,\\\\,\\\\,\\\\, ({\\\\overline{\\\\rm MS}}) \\\\,.$ This value is somewhat lower, but within $1\\\\,\\\\sigma $ of the higher-order uncertainty.\",\n \"In CIPT, contour integrals over the running coupling have to be computed, and hence the result cannot be given in analytical form.\",\n \"The general behaviour is very similar to FOPT, with the exception that now also for $c_{5,1}=566$ a zero of the ${\\\\cal O}(\\\\hat{a}^5)$ term is found.\",\n \"Employing the value of $C$ which leads to the smaller uncertainty one finds $\\\\delta _{\\\\rm CI}^{(0)}(\\\\hat{a}_{M_\\\\tau },C=-1.246) \\\\,=\\\\,0.1840 \\\\pm 0.0062 \\\\pm 0.0084 \\\\,.$ As has been discussed many times in the past (see e.g.\",\n \"[14]) the CIPT prediction lies substantially below the FOPT results.\",\n \"On the other hand, the parametric $\\\\alpha _s$ uncertainty in CIPT turns out to be smaller.\"\n ],\n [\n \"Higher-order terms\",\n \"The behaviour of the series at higher orders is not known exactly.\",\n \"However, realistic models of the Adler function can be constructed in the Borel plane, in which the singularities of the function, namely its renormalon content, is partially known [3].\",\n \"In Ref.\",\n \"[14] (see also Ref.\",\n \"[16]), models of the Adler function were constructed using the leading renormalons, that largely dominate the higher-order behaviour of the perturbative series.\",\n \"The model is matched to the exactly known coefficients in order to fully reproduce QCD for terms up to $a_Q^4$ .\",\n \"This allows for a complete reconstruction of the series, to arbitrarily high orders in the coupling, and, moreover, one is able to obtain the “true” value of the asymptotic series by means of the Borel sum.\",\n \"In fact, the series is not strictly Borel summable because infra-red renormalons obstruct integration on the positive real axis.\",\n \"The “true” value has, therefore, an inherent ambiguity that stems from the prescription adopted to circumvent the singularities along the contour of integration.\",\n \"This ambiguity is related to non-perturbative physics [3], [14].\",\n \"Here we perform a preliminary investigation of the behaviour of $\\\\delta ^{(0)}$ at higher orders using the $C$ -scheme coupling.\",\n \"The Adler function coefficients for terms higher than $a_Q^5$ are obtained in the ${\\\\overline{\\\\rm MS}}$ scheme from the central model of Ref. [14].\",\n \"The series can then be translated to the $C$ -scheme by means of the perturbative relation between the couplings $a_Q$ and $\\\\hat{a}_Q$ [2].\",\n \"Fig.\",\n \"REF shows four different series that should approach the same Borel summed result, showed as a horizontal band.\",\n \"The four series use as input the coefficients exactly known in QCD with the addition of the estimate $c_{5,1}=283$ .\",\n \"One observes that the optimised version of $\\\\delta _{\\\\rm FO}^{(0)}$ (filled circles) approaches the Borel sum of the series faster than the ${\\\\overline{\\\\rm MS}}$ result (empty circles).\",\n \"Of course, because the optimised series has a larger coupling (see Fig.\",\n \"1) asymptoticity sets in earlier and the divergent character is clearly visible already around the 10th order.\",\n \"The FOPT result with $C=0.7$ shows that smaller couplings do not necessarily lead to a better approximation at lower orders, requiring many more terms to give a good approximation to the Borel summed result.\",\n \"Finally, the optimal CIPT series does not give a good approximation to the Borel summed result (this is also the case in the ${\\\\overline{\\\\rm MS}}$ [14]).\",\n \"Unfortunately, the use of the $C$ -scheme coupling does not make the CIPT prediction closer to FOPT.\",\n \"The $C$ -scheme FOPT, on the other hand, is in excellent agreement with the central Borel model which suggests that FOPT should be the favoured expansion.\",\n \"Figure: Four series for δ (0) \\\\delta ^{(0)} with higher-order coefficients from the central model of Ref. .\",\n \"In all cases α s (M τ )=0.316\\\\alpha _s(M_\\\\tau )=0.316 which corresponds to the central value of the present world average .\",\n \"The optimised FOPT (filled circles) and CIPT (filled squares) series can be compared with the FOPT MS ¯{\\\\overline{\\\\rm MS}} results (empty circles) and FOPT for C=0.7C=0.7 (triangles).\",\n \"The shaded band gives the Borel summed result, the “true\\\" value of the series, with its associated ambiguity .\"\n ],\n [\n \"Acknowledgements\",\n \" It is a pleasure to thank the organisers of this very fruitful meeting.\",\n \"DB is supported by the São Paulo Research Foundation (FAPESP) grant 2015/20689-9, and by CNPq grant 305431/2015-3.\",\n \"The work of MJ and RM has been supported in part by MINECO Grant number CICYT-FEDER-FPA2014-55613-P, by the Severo Ochoa excellence program of MINECO, Grant SO-2012-0234, and Secretaria d'Universitats i Recerca del Departament d'Economia i Coneixement de la Generalitat de Catalunya under Grant 2014 SGR 1450.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05558"}}},{"rowIdx":3386,"cells":{"text":{"kind":"list like","value":[["First estimation of the fission dynamics of the spectator created in\n heavy-ion collisions"],["Abstract In peripheral high-energy heavy-ion collisions only parts of colliding nuclei interact leading to the production of a fireball.","The remnants of such nuclei are called spectators.","We estimated the excitation energy as a function of impact parameter.","Their excitation energy is of the order of 100 MeV.","The dynamical evolution of hot nuclei is described by solving a set of Langevin equations in four-dimensional collective coordinate space.","The range of nuclear masses and excitation energies suits very well to ability of our model.","Thus for the first time we investigate dynamically the fission and evaporation channels in de-excitation of the spectators produced in heavy-ions collision."],["Introduction","The physics of high-energy heavy-ion collisions develops very effectively since many years.","At CERN e.g.","$^{208}$ Pb+$^{208}$ Pb collisions were considered at SPS and LHC.","Two lead nuclei flying with velocities comparable with light velocity produce the fireball in the place of collision, but depending on the centrality of the collision the Pb parts which are not participating in the collision (called ”spectators”) follow their initial path and deexcites later on.","There are many experimental and theoretical [1], [2] investigations of the processes connected to the fireball – a piece of quark-gluon plasma.","The density of the matter in the contact area is so high that the quark-gluon plasma is created and the high-energy particles such as pions, kaons, and others emerge in the hadronisation process.","This part of the reaction requires dedicated models and methods that are beyond the scope of the present discussion.","The interesting issue is the behavior of the spectators produced as the remnants of the collision [3].","There are not so many experimental observables published up to now, which can help with constraining the physics of spectator evolution.","The fast collision of two spherical ions provides exotic, for nuclear physics, shapes of the remnants which are strongly dependent on the impact parameter.","The deformation energy of the produced system turns into excitation energy.","Thus we treat the spectator as an excited nucleus, which can deexcite by shape changes or emission of light particles and/or $\\gamma $ -rays.","This allows to apply the stochastic approach based on the solving transport equations in collective coordinate space [4], [5], [6], [7]."],["Theoretical method","The stochastic approach assumes that the evolution of the nucleus in collective coordinate space is similar to the Brownian motion of particles which interact stochastically involving a large number of internal degrees of freedom, which constitutes the surrounding \"heat bath\".","The motion of the nucleus is slowed down by a friction, considered here as the hydrodynamical friction force.","The transport coefficients are derived from the random force averaged over a time larger than the collision time scale between collective and internal variables.","The Gaussian white noise is taken for the random part, producing fluctuations of the physical observables such as the mass/charge distribution of the fission fragments or evaporation residua.","The coupled Langevin equations describing the fission reaction have the form: $\\frac{d q_{i}}{d t}&=&\\mu _{ij}p_{j}, \\\\\\frac{d p_{i}}{d t}&=&- \\frac{1}{2}p_{j}p_{k}\\frac{\\partial \\mu _{jk}}{\\partial q_{i}}- \\left( \\frac{\\partial F}{\\partial q_i} \\right)_T- \\gamma _{ij}\\mu _{jk}p_{k}+\\theta _{ij}\\xi _{j}\\left(t\\right),\\nonumber $ where ${\\bf q}$ are the collective coordinates, ${\\bf p}$ are the conjugate momenta, $F({\\bf q},K)=V({\\bf q},K) - a({\\bf q}) T^2$ is the Helmholtz free energy, $V({\\bf q})$ is the potential energy, $m_{ij}({\\bf q})$ ($\\Vert \\mu _{ij}\\Vert =\\Vert m_{ij}\\Vert ^{-1}$ ) is the tensor of inertia, and $\\gamma _{ij}({\\bf q})$ is the friction tensor.","The $\\xi _j\\left(t\\right)$ is a random variable satisfying the relations $<\\xi _{i}>&=&0, \\nonumber \\\\<\\xi _{i}(t_{1})\\xi _{j}(t_{2})>&=&2\\delta _{ij}\\delta (t_{1}-t_{2}).$ Thus, a Markovian approximation is assumed to be valid.","The strength of the random force $\\theta _{ij}$ is given by the Einstein relation $\\sum \\theta _{ik}\\theta _{kj} = T\\gamma _{ij}$ .","The temperature of the \"heat bath\" $T$ is determined by the Fermi-gas model formula $T=(E_{\\rm {int}}/a)^{1/2}$ , where $E_{\\rm {int}}$ is the internal excitation energy of the nucleus and $a({\\bf q})$ is the level-density parameter.","The energy conservation law is used in the form $E^{*}=E_{\\rm {int}}+E_{\\rm {coll}}+V+ E_{\\rm {evap}}(t)$ at each time step during a random walk along the Langevin trajectory in the collective coordinate space.","The total excitation energy of the nucleus is $E^{*}$ and the collective kinetic energy is taken as: $E_{\\rm {coll}}= 0.5 \\sum \\mu _{ij} p_i p_j$ .","The energy carried away by the evaporated particles by the time $t$ is marked as $E_{\\rm {evap}}(t)$ .","The collective coordinate space is constructed with three deformation parameters: elongation of the nucleus $(q_1)$ , its neck $(q_2)$ and octupole shape $(q_3)$ based on the ”funny hills” parametrization proposed initially by Brack [8] and the forth degree of freedom is the projection of the angular momentum vector on the fission axis.","This four collective coordinates allow to describe rich ensemble of shapes and aditinally it allows to orient the nucleus in the laboratory space, because $K$ -is the projection of the spin on the fission axis.","The 4D Langevin method describes the behavior of the hot nucleus in the reference frame connected with the center-of-mass of each nucleus, thus in the case of the fast moving systems in the laboratory frame, additional transformations are necessary as the velocities of nuclei are in relativistic regime.","The initial condition to start evolution of the nucleus are usually the mass and charge of the beam and target nuclei, their excitation energy and maximal angular momentum, which can be produced in the reaction.","Our idea of investigating the spectator with 4D Langevin method is very recent.","There are no direct measurements that could identify any fission/evaporation reactions occurring in the remnant of heavy-ions collisions, but new developments in the detection techniques [9] could make possible the investigations of new effects accompanying the main reaction.","The estimation of the shape of the spectator just after the ion collision is the gateway to introduce the fission dynamics.","We propose three possible scenarios of the non-central collisions.","Let us assume that the beam and target nuclei have the spherical forms in their rest frame.","When they fly in opposite directions, collide and they graze each other.","The central part produces the quark-gluon plasma and all kinds of high-energy particle phenomena.","Remnants could have extremely exotic forms and sometimes it is assumed that they are a collection of free nucleons.","Here we discuss the following scenarios of initial shape production: ”sphere-plane” means the sphere cut by the plane at various impact parameters; ”sphere-sphere” gives the shape of the spectator which is sphere but with missing spherical cap; ”sphere-cylinder” is the globe shot by the bullet.","These three scenarios gives shapes, very strange for nuclear physics.","These exotic deformations are impossible to describe with any standard nuclear shape parameterizations.","Thus we make an assumption that the time to change nuclear deformation from initial (complicated) to spherical, is small compared to the time-scale of the rest of the dynamics.","This rough assumption will be investigated in the future in detail.","The difference in the surface energy for spherical and deformed nucleus give the preliminary excitation energy of the spectator.","The potential energy of the nucleus is obtained with the Lublin-Strasbourg Drop formula [10], [11]: $E_{lsd} (Z,N;q)&=&E_{vol}+E_{surf}+E_{curv}+E_{_{Coul}}\\\\&=&b_{vol} (1-\\kappa _{vol} I^2) \\,A+b_{surf} (1-\\kappa _{surf} I^2)\\, A^{2/3}B_{surf} (q)\\nonumber \\\\&+&b_{curv} (1-\\kappa _{curv} I^2)\\, A^{1/3}B_{curv} (q)+\\frac{3}{5}e^2 \\frac{Z^2}{r^{ch}_{0}A^{1/3}} B_{Coul} (q),\\quad $ where $ I =(N-Z)/A$ .","The geometric factors: $B_{surf}$ , $B_{curv}$ and $B_{Coul}$ are defined as: $B_{surf}=\\frac{S(def)}{S(0)},\\quad B_{curv}=\\frac{C(def)}{C(0)}\\quad \\mathrm { and }\\quad B_{Coul}=\\frac{E_{Coul}(def)}{E_{Coul}(0)}$ that are the ratios between the surface($S$ )/curvature($C$ )/Coulomb energy ($E_{Coul}$ ) of the deformed and spherical nucleus.","The details are presented e.g.","in [10].","Thus for each of the scenarios, the surface and the volume of the spectator were calculated as functions of the impact parameter.","The volume of the spectator and its mass were obtained using the geometrical properties.","Assuming $N/Z_{Pb}=N/Z_{spectator}$ also the charge was assessed.","The deformation together with the mass and charge gives the surface energy estimation."],["Results","The surface energy of the remnant just after the collision reduced by the energy of the spherical nucleus gives the deformation energy which is presented in Fig.","REF .","The deformation energy depends on the initial condition.","The highest deformation energies (almost 500 MeV) are predicted for the case of ”sphere-sphere” grazing as the shapes are more curved than in other cases.","The smallest energies are obtained for the sphere cut by the plane as the forms are less curved and the surface of the deformed nucleus is closer to a spherical shape.","Figure: (Color on line) The deformation energy for three scenarios: ”sphere-sphere”, ”sphere-plane” and ”sphere-cylinder”.","The impact distance is normalized to the radius of the spherical lead nucleus.The impact parameter is good observable to discuss the central-peripheral collision, as it is presented in Fig.","REF , but for fission dynamics calculation the mass and charge of the spectators are necessary.","Thus in Fig.","REF the deformation energy is shown as a function of the mass of the remnant.","The deformation energy has the main contribution to the excitation energy of the spectator.","The black crosses mark the nuclei taken in further calculations.","Figure: (Color on line) The deformation energy versus the spectator mass estimated from its volume.","The crosses represent nuclei taken for dynamical evaluation.The 4D Langevin method is dedicated to describe the fission dynamics of the hot nuclei at excitation energies $E^{\\star }$ =50–250 MeV and the driving forces are derived macroscopically.","The excitation energy was taken as the deformation energy from the scenario: ”sphere-cylinder” as the energy range fits exactly limitation of our model.","From phenomenological point of view this model is also the most realistic.","Figure: (Color on line) The average values of the mass (a), total kinetic energy (c), temperature (e) and fission time (f) resulting from initial systems withM=64, 100, 142, 168 and 194.","The widths of the mass (b) and TKE distribution (d) are also displayed.","The dashed line in panel (a) shows the mean mass of the fission fragments assuming symmetric scission of the nucleus.Five nuclei are chosen arbitrarily for the fission dynamics studies: $^{64}$ Mn, $^{100}$ Y, $^{142}$ Ba, $^{168}$ Dy and $^{194}$ Os.","The maximal angular momentum was taken as 2 $\\hbar $ just to avoid possible technical problems.","For each nucleus around 300 000 trajectories were calculated and the results are presented in Fig.","REF .","The panel (a) compares the mean mass coming from an assumption of the symmetric division of the fissioning nucleus (red dashed line) and centroid of the mass distribution whose width is shown in panel (b).","The medium mass system demonstrates a slight deviation from the static assumption as during the path to fission various particles (protons.","neutrons, deuteron, $\\alpha $ -particles) are emitted.","Looking at panel (c) for the Total Kinetic Energy (TKE) of the fission fragments, it is visible that almost linear decrease of the excitation energy (Fig.","REF ) provides the exponential increase of the average TKE as the system is more and more heavy.","The effects of inertia and/or the excitation energy is also visible in panel (f) for the increased fission time."],["Summary","The preliminary results discussed here show a possibility of investigating the fission of spectators produced in ultrarelativistic heavy-ions collisions.","The three scenarios of initial shape estimation for the remnants of the Pb-Pb reaction have been considered to estimate the excitation energies available for various impact parameters.","Several observables have been extracted from the state-of-art Langevin calculations for fission dynamics.","The average values and widths of mass, charge and total kinetic energy distribution of the fission fragments as well as mean values of the temperature and fission times have been shown for the first time.","Acknowledgements The work was partially supported by the Polish National Science Centre under Contract No.","2013/08/M/ST2/00257 (LEA COPIGAL) (Project No.","18) and IN2P3-COPIN (Project No.","12-145, 09-146), and by the Russian Foundation for Basic Research (Project No.","13-02-00168)."]],"string":"[\n [\n \"First estimation of the fission dynamics of the spectator created in\\n heavy-ion collisions\"\n ],\n [\n \"Abstract In peripheral high-energy heavy-ion collisions only parts of colliding nuclei interact leading to the production of a fireball.\",\n \"The remnants of such nuclei are called spectators.\",\n \"We estimated the excitation energy as a function of impact parameter.\",\n \"Their excitation energy is of the order of 100 MeV.\",\n \"The dynamical evolution of hot nuclei is described by solving a set of Langevin equations in four-dimensional collective coordinate space.\",\n \"The range of nuclear masses and excitation energies suits very well to ability of our model.\",\n \"Thus for the first time we investigate dynamically the fission and evaporation channels in de-excitation of the spectators produced in heavy-ions collision.\"\n ],\n [\n \"Introduction\",\n \"The physics of high-energy heavy-ion collisions develops very effectively since many years.\",\n \"At CERN e.g.\",\n \"$^{208}$ Pb+$^{208}$ Pb collisions were considered at SPS and LHC.\",\n \"Two lead nuclei flying with velocities comparable with light velocity produce the fireball in the place of collision, but depending on the centrality of the collision the Pb parts which are not participating in the collision (called ”spectators”) follow their initial path and deexcites later on.\",\n \"There are many experimental and theoretical [1], [2] investigations of the processes connected to the fireball – a piece of quark-gluon plasma.\",\n \"The density of the matter in the contact area is so high that the quark-gluon plasma is created and the high-energy particles such as pions, kaons, and others emerge in the hadronisation process.\",\n \"This part of the reaction requires dedicated models and methods that are beyond the scope of the present discussion.\",\n \"The interesting issue is the behavior of the spectators produced as the remnants of the collision [3].\",\n \"There are not so many experimental observables published up to now, which can help with constraining the physics of spectator evolution.\",\n \"The fast collision of two spherical ions provides exotic, for nuclear physics, shapes of the remnants which are strongly dependent on the impact parameter.\",\n \"The deformation energy of the produced system turns into excitation energy.\",\n \"Thus we treat the spectator as an excited nucleus, which can deexcite by shape changes or emission of light particles and/or $\\\\gamma $ -rays.\",\n \"This allows to apply the stochastic approach based on the solving transport equations in collective coordinate space [4], [5], [6], [7].\"\n ],\n [\n \"Theoretical method\",\n \"The stochastic approach assumes that the evolution of the nucleus in collective coordinate space is similar to the Brownian motion of particles which interact stochastically involving a large number of internal degrees of freedom, which constitutes the surrounding \\\"heat bath\\\".\",\n \"The motion of the nucleus is slowed down by a friction, considered here as the hydrodynamical friction force.\",\n \"The transport coefficients are derived from the random force averaged over a time larger than the collision time scale between collective and internal variables.\",\n \"The Gaussian white noise is taken for the random part, producing fluctuations of the physical observables such as the mass/charge distribution of the fission fragments or evaporation residua.\",\n \"The coupled Langevin equations describing the fission reaction have the form: $\\\\frac{d q_{i}}{d t}&=&\\\\mu _{ij}p_{j}, \\\\\\\\\\\\frac{d p_{i}}{d t}&=&- \\\\frac{1}{2}p_{j}p_{k}\\\\frac{\\\\partial \\\\mu _{jk}}{\\\\partial q_{i}}- \\\\left( \\\\frac{\\\\partial F}{\\\\partial q_i} \\\\right)_T- \\\\gamma _{ij}\\\\mu _{jk}p_{k}+\\\\theta _{ij}\\\\xi _{j}\\\\left(t\\\\right),\\\\nonumber $ where ${\\\\bf q}$ are the collective coordinates, ${\\\\bf p}$ are the conjugate momenta, $F({\\\\bf q},K)=V({\\\\bf q},K) - a({\\\\bf q}) T^2$ is the Helmholtz free energy, $V({\\\\bf q})$ is the potential energy, $m_{ij}({\\\\bf q})$ ($\\\\Vert \\\\mu _{ij}\\\\Vert =\\\\Vert m_{ij}\\\\Vert ^{-1}$ ) is the tensor of inertia, and $\\\\gamma _{ij}({\\\\bf q})$ is the friction tensor.\",\n \"The $\\\\xi _j\\\\left(t\\\\right)$ is a random variable satisfying the relations $<\\\\xi _{i}>&=&0, \\\\nonumber \\\\\\\\<\\\\xi _{i}(t_{1})\\\\xi _{j}(t_{2})>&=&2\\\\delta _{ij}\\\\delta (t_{1}-t_{2}).$ Thus, a Markovian approximation is assumed to be valid.\",\n \"The strength of the random force $\\\\theta _{ij}$ is given by the Einstein relation $\\\\sum \\\\theta _{ik}\\\\theta _{kj} = T\\\\gamma _{ij}$ .\",\n \"The temperature of the \\\"heat bath\\\" $T$ is determined by the Fermi-gas model formula $T=(E_{\\\\rm {int}}/a)^{1/2}$ , where $E_{\\\\rm {int}}$ is the internal excitation energy of the nucleus and $a({\\\\bf q})$ is the level-density parameter.\",\n \"The energy conservation law is used in the form $E^{*}=E_{\\\\rm {int}}+E_{\\\\rm {coll}}+V+ E_{\\\\rm {evap}}(t)$ at each time step during a random walk along the Langevin trajectory in the collective coordinate space.\",\n \"The total excitation energy of the nucleus is $E^{*}$ and the collective kinetic energy is taken as: $E_{\\\\rm {coll}}= 0.5 \\\\sum \\\\mu _{ij} p_i p_j$ .\",\n \"The energy carried away by the evaporated particles by the time $t$ is marked as $E_{\\\\rm {evap}}(t)$ .\",\n \"The collective coordinate space is constructed with three deformation parameters: elongation of the nucleus $(q_1)$ , its neck $(q_2)$ and octupole shape $(q_3)$ based on the ”funny hills” parametrization proposed initially by Brack [8] and the forth degree of freedom is the projection of the angular momentum vector on the fission axis.\",\n \"This four collective coordinates allow to describe rich ensemble of shapes and aditinally it allows to orient the nucleus in the laboratory space, because $K$ -is the projection of the spin on the fission axis.\",\n \"The 4D Langevin method describes the behavior of the hot nucleus in the reference frame connected with the center-of-mass of each nucleus, thus in the case of the fast moving systems in the laboratory frame, additional transformations are necessary as the velocities of nuclei are in relativistic regime.\",\n \"The initial condition to start evolution of the nucleus are usually the mass and charge of the beam and target nuclei, their excitation energy and maximal angular momentum, which can be produced in the reaction.\",\n \"Our idea of investigating the spectator with 4D Langevin method is very recent.\",\n \"There are no direct measurements that could identify any fission/evaporation reactions occurring in the remnant of heavy-ions collisions, but new developments in the detection techniques [9] could make possible the investigations of new effects accompanying the main reaction.\",\n \"The estimation of the shape of the spectator just after the ion collision is the gateway to introduce the fission dynamics.\",\n \"We propose three possible scenarios of the non-central collisions.\",\n \"Let us assume that the beam and target nuclei have the spherical forms in their rest frame.\",\n \"When they fly in opposite directions, collide and they graze each other.\",\n \"The central part produces the quark-gluon plasma and all kinds of high-energy particle phenomena.\",\n \"Remnants could have extremely exotic forms and sometimes it is assumed that they are a collection of free nucleons.\",\n \"Here we discuss the following scenarios of initial shape production: ”sphere-plane” means the sphere cut by the plane at various impact parameters; ”sphere-sphere” gives the shape of the spectator which is sphere but with missing spherical cap; ”sphere-cylinder” is the globe shot by the bullet.\",\n \"These three scenarios gives shapes, very strange for nuclear physics.\",\n \"These exotic deformations are impossible to describe with any standard nuclear shape parameterizations.\",\n \"Thus we make an assumption that the time to change nuclear deformation from initial (complicated) to spherical, is small compared to the time-scale of the rest of the dynamics.\",\n \"This rough assumption will be investigated in the future in detail.\",\n \"The difference in the surface energy for spherical and deformed nucleus give the preliminary excitation energy of the spectator.\",\n \"The potential energy of the nucleus is obtained with the Lublin-Strasbourg Drop formula [10], [11]: $E_{lsd} (Z,N;q)&=&E_{vol}+E_{surf}+E_{curv}+E_{_{Coul}}\\\\\\\\&=&b_{vol} (1-\\\\kappa _{vol} I^2) \\\\,A+b_{surf} (1-\\\\kappa _{surf} I^2)\\\\, A^{2/3}B_{surf} (q)\\\\nonumber \\\\\\\\&+&b_{curv} (1-\\\\kappa _{curv} I^2)\\\\, A^{1/3}B_{curv} (q)+\\\\frac{3}{5}e^2 \\\\frac{Z^2}{r^{ch}_{0}A^{1/3}} B_{Coul} (q),\\\\quad $ where $ I =(N-Z)/A$ .\",\n \"The geometric factors: $B_{surf}$ , $B_{curv}$ and $B_{Coul}$ are defined as: $B_{surf}=\\\\frac{S(def)}{S(0)},\\\\quad B_{curv}=\\\\frac{C(def)}{C(0)}\\\\quad \\\\mathrm { and }\\\\quad B_{Coul}=\\\\frac{E_{Coul}(def)}{E_{Coul}(0)}$ that are the ratios between the surface($S$ )/curvature($C$ )/Coulomb energy ($E_{Coul}$ ) of the deformed and spherical nucleus.\",\n \"The details are presented e.g.\",\n \"in [10].\",\n \"Thus for each of the scenarios, the surface and the volume of the spectator were calculated as functions of the impact parameter.\",\n \"The volume of the spectator and its mass were obtained using the geometrical properties.\",\n \"Assuming $N/Z_{Pb}=N/Z_{spectator}$ also the charge was assessed.\",\n \"The deformation together with the mass and charge gives the surface energy estimation.\"\n ],\n [\n \"Results\",\n \"The surface energy of the remnant just after the collision reduced by the energy of the spherical nucleus gives the deformation energy which is presented in Fig.\",\n \"REF .\",\n \"The deformation energy depends on the initial condition.\",\n \"The highest deformation energies (almost 500 MeV) are predicted for the case of ”sphere-sphere” grazing as the shapes are more curved than in other cases.\",\n \"The smallest energies are obtained for the sphere cut by the plane as the forms are less curved and the surface of the deformed nucleus is closer to a spherical shape.\",\n \"Figure: (Color on line) The deformation energy for three scenarios: ”sphere-sphere”, ”sphere-plane” and ”sphere-cylinder”.\",\n \"The impact distance is normalized to the radius of the spherical lead nucleus.The impact parameter is good observable to discuss the central-peripheral collision, as it is presented in Fig.\",\n \"REF , but for fission dynamics calculation the mass and charge of the spectators are necessary.\",\n \"Thus in Fig.\",\n \"REF the deformation energy is shown as a function of the mass of the remnant.\",\n \"The deformation energy has the main contribution to the excitation energy of the spectator.\",\n \"The black crosses mark the nuclei taken in further calculations.\",\n \"Figure: (Color on line) The deformation energy versus the spectator mass estimated from its volume.\",\n \"The crosses represent nuclei taken for dynamical evaluation.The 4D Langevin method is dedicated to describe the fission dynamics of the hot nuclei at excitation energies $E^{\\\\star }$ =50–250 MeV and the driving forces are derived macroscopically.\",\n \"The excitation energy was taken as the deformation energy from the scenario: ”sphere-cylinder” as the energy range fits exactly limitation of our model.\",\n \"From phenomenological point of view this model is also the most realistic.\",\n \"Figure: (Color on line) The average values of the mass (a), total kinetic energy (c), temperature (e) and fission time (f) resulting from initial systems withM=64, 100, 142, 168 and 194.\",\n \"The widths of the mass (b) and TKE distribution (d) are also displayed.\",\n \"The dashed line in panel (a) shows the mean mass of the fission fragments assuming symmetric scission of the nucleus.Five nuclei are chosen arbitrarily for the fission dynamics studies: $^{64}$ Mn, $^{100}$ Y, $^{142}$ Ba, $^{168}$ Dy and $^{194}$ Os.\",\n \"The maximal angular momentum was taken as 2 $\\\\hbar $ just to avoid possible technical problems.\",\n \"For each nucleus around 300 000 trajectories were calculated and the results are presented in Fig.\",\n \"REF .\",\n \"The panel (a) compares the mean mass coming from an assumption of the symmetric division of the fissioning nucleus (red dashed line) and centroid of the mass distribution whose width is shown in panel (b).\",\n \"The medium mass system demonstrates a slight deviation from the static assumption as during the path to fission various particles (protons.\",\n \"neutrons, deuteron, $\\\\alpha $ -particles) are emitted.\",\n \"Looking at panel (c) for the Total Kinetic Energy (TKE) of the fission fragments, it is visible that almost linear decrease of the excitation energy (Fig.\",\n \"REF ) provides the exponential increase of the average TKE as the system is more and more heavy.\",\n \"The effects of inertia and/or the excitation energy is also visible in panel (f) for the increased fission time.\"\n ],\n [\n \"Summary\",\n \"The preliminary results discussed here show a possibility of investigating the fission of spectators produced in ultrarelativistic heavy-ions collisions.\",\n \"The three scenarios of initial shape estimation for the remnants of the Pb-Pb reaction have been considered to estimate the excitation energies available for various impact parameters.\",\n \"Several observables have been extracted from the state-of-art Langevin calculations for fission dynamics.\",\n \"The average values and widths of mass, charge and total kinetic energy distribution of the fission fragments as well as mean values of the temperature and fission times have been shown for the first time.\",\n \"Acknowledgements The work was partially supported by the Polish National Science Centre under Contract No.\",\n \"2013/08/M/ST2/00257 (LEA COPIGAL) (Project No.\",\n \"18) and IN2P3-COPIN (Project No.\",\n \"12-145, 09-146), and by the Russian Foundation for Basic Research (Project No.\",\n \"13-02-00168).\"\n ]\n]"},"id":{"kind":"string","value":"1612.05397"}}},{"rowIdx":3387,"cells":{"text":{"kind":"list like","value":[["A method for obtaining nonspreading solutions of the Schr\\\"odinger\n equation"],["Abstract In this paper, we present a simple analytical method for obtaining a nonspreading solution of the time-dependent Schr\\\"odinger equation, which is given by the Airy function.","The solution is derived by imposing a restriction on the phase factor of the ansatz that is taken to solve the differential equation.","Considering at first the free particle, we show that nonspreading solutions can also be obtained for a time-dependent linear potential.","The method is shown to work in both one and two dimensions, and can be easily extended if required.","The applicability of the method is discussed in relation to the nonlinear case."],["Introduction","Berry and Balazs showed that the free-particle Schrödinger equation admits non-trivial solutions whose amplitude is given by the Airy differential equation [1].","Specifically, they showed that $-\\frac{\\hbar ^2}{2m}\\psi _{xx} = i\\hbar \\psi _{t} \\,,$ has a unique solution given by $\\psi (x,t) = \\mathrm {Ai}\\left[\\frac{B}{\\hbar ^{2/3}}\\left(x-\\frac{B^3t^2}{4m^2}\\right)\\right]e^{\\left(iB^3t/2m\\hbar \\right)\\left[x-\\left(B^3t^2/6m^2\\right)\\right]}\\,,$ where $\\displaystyle \\mathrm {Ai}(z)$ is the Airy function and $\\displaystyle B$ is a positive constant.","Unlike plane wave solution, the Airy solution given above is not separable in the variables $\\displaystyle \\lbrace x,\\, t\\rbrace $ .","The essential feature of these solutions is that the time-evolution of the wavepacket is non-dispersive.","The solution given by Eq.","(REF ) was proved to be the only unique solution of Eq.","(REF ) having this property.","This result was originally explained in terms of the behaviour of the corresponding families of the semiclassical orbits in phase space.","Subsequently, much work has been done to interpret this surprising result, with a proper quantum mechanical derivation being given in [2].","Recently, the nonspreading solution has evoked considerable interest in the community.","It has been shown that, under the paraxial approximation the wave equation allows a nonspreading solution [3].","It has been verified experimentally by generating optical beams with wave packets that are accelerated [4].","This leads further possibility of using these solutions in various optical systems.","In this paper, we show that it is possible to obtain the result from techniques based on ordinary calculus.","It is shown that the in the ansatz $\\displaystyle A(x,t)e^{i\\phi (x,t)}$ that is used to obtain wave-like solutions of Eq.","(REF ), a particular choice of the phase $\\displaystyle \\phi $ will lead to a reduction of the Schrödinger equation to the Airy differential equation.","In this paper, we first describe a method for obtaining the nonspreading solution for the Schrödinger equation for a free particle in one dimension.","We also show that we can easily extend the solution for linear and time-dependent potentials.","In section , we show that the method can be extended to higher dimension by taking the two dimensional case as an example."],["The method for obtaining solution","We are looking for non-trivial solutions of $\\psi _{xx} + \\mathrm {i}\\kappa \\psi _t - \\bar{V}(x,t)\\psi = 0\\,,$ where $\\displaystyle \\kappa = 2m/\\hbar $ and $\\displaystyle \\bar{V} = (\\kappa /\\hbar )V$ .","We start with the free particle case first with $\\displaystyle \\bar{V}(x,t)=0$ .","After demonstrating the method, we extend the solution with a potential term."],["The one-dimensional Schrödinger equation","In this section we solve for the non-disperssive solution in one dimension.","We start with $\\displaystyle V=0$ in Eq.","(REF ).","First, we propose a trial solution of the form, $\\psi = A(x,t)\\,e^{i\\phi (x,t)}\\,,$ where $\\displaystyle A(x,t)$ and $\\displaystyle \\phi (x,t)$ are assumed to be real functions.","Separating real and imaginary parts after substituting Eq.","(REF ) in Eq.","(REF ), we obtain: $A_{xx} - A(\\phi _x)^2 -\\kappa A \\phi _t & = & 0\\,, \\nonumber \\\\A\\phi _{xx} + 2 A_x\\phi _x + \\kappa A_t &= &0\\,.$ We start with the ansatz that the phase term $\\displaystyle \\phi (x,t)$ must satisfy the Laplace equation, $\\frac{\\partial ^2\\phi }{\\partial x^2} = 0\\,, $ With this we get, $\\phi = \\phi _1 \\cdot x + \\phi _0\\,,$ where $\\displaystyle \\lbrace \\phi _1,\\phi _0\\rbrace $ are functions of time only.","Substituting $\\displaystyle \\phi $ in Eq.","(REF ), we get $A_{xx} - Ay = 0\\,,$ where, $y = \\phi _1^2+\\kappa \\left(\\dot{\\phi }_1 x+\\dot{\\phi }_0\\right)\\,.$ In the above, $\\displaystyle \\dot{\\phi }=\\frac{d\\phi }{dt}$ .","We further assume that $\\displaystyle \\partial y/\\partial x$ is constant implying $\\displaystyle \\dot{\\phi }_1 = P$ (a constant).","Then we have reduced Eq.","(REF ) to an Airy differential equation with the solution $A = \\mathrm {Ai}\\left[\\frac{y}{(\\kappa P)^{2/3}}\\right]\\,.$ The condition for $\\displaystyle \\phi _0$ is given from Eq.","(REF ) by $\\kappa \\ddot{\\phi }_0 + 4P^2t = 0\\,,$ resulting in, $\\phi _0 = -\\frac{2}{3\\kappa }P^2t^3 + Qt\\,.$ Here $\\displaystyle P$ and $\\displaystyle Q$ are constants.","It can be shown that one can get Berry's solution with $\\displaystyle Q=0$ and $\\displaystyle P=B^3/2m\\hbar $ where $\\displaystyle B$ is an arbitrary positive constant.","At this point, we would like to physically justify the ansatz used in obtaining the solution.","It is closely related to the symmetry properties of the Schrödinger equation.","The Schrödinger equation admits a Galilean transformation which preserves the form of the differential equation [5].","The transformed wave-function is then just the original wave-function multiplied by a phase which is linear in the spatial coordinates.","Thus, we must have $\\displaystyle \\partial ^2 \\phi / \\partial x^2 = 0$ ."],["With linear potential","In this section, we generalise to the case with potential.","Starting with the same trial solution to obtain, $A_{xx} - A(\\phi _x)^2 -\\kappa A \\phi _t - \\bar{V}A = 0\\,, \\nonumber \\\\A\\phi _{xx} + 2 A_x \\phi _x + \\kappa A_t = 0\\,.$ Since the second equation is the same as given in Eq.","(REF ), the form of $\\displaystyle \\phi $ is modified by an additional term due to the potential.","The equation for the amplitude $\\displaystyle A$ also takes the same form in terms of the variable $\\displaystyle y = \\phi _{1}^2+\\kappa \\left(\\dot{\\phi }_1 x + \\dot{\\phi }_0\\right) + \\bar{V}$ , as before, $\\displaystyle \\partial y/\\partial x$ is assumed constant.","Like before, we take $\\displaystyle \\mathrm {d}\\phi _{1}/\\mathrm {d}t = P$ with the addition that $\\displaystyle \\partial \\bar{V}/\\partial x = \\bar{V_1}$ , both of them being constants.","With these conditions, we have $\\frac{\\partial y}{\\partial x} = \\kappa \\frac{\\mathrm {d}\\phi _{1}}{\\mathrm {d}t} + \\frac{\\partial \\bar{V}}{\\partial x} = \\kappa P + \\bar{V_1}\\,,$ from which we finally get $\\phi _1 = Pt~,\\qquad \\bar{V} = \\bar{V_1}x + \\bar{V_0}\\,.$ Here, we have freedom to choose $\\displaystyle \\bar{V_0}$ to be a function of time.","In order to get $\\displaystyle \\phi _0$ we solve an equation similar to Eq.","(REF ) to obtain $\\phi _0 = -\\frac{1}{\\kappa ^2}\\left[\\int ^t\\bar{V_0}(t^{\\prime })\\mathrm {d}t^{\\prime } + \\frac{2}{3}\\kappa P^2t^3 -\\frac{1}{3}P\\bar{V_1}t^3\\right] + Qt~,$ where $\\displaystyle Q$ is a constant.","It can be readily seen this is a validation of the results of Berry and Balazs for more general potentials of the form $\\displaystyle F(t)x$ ."],["Solution in two dimensions","In this section, we consider the Schrödinger equation in two spatial dimensions.","For the potential-free case we wish to solve, $\\psi _{xx} + \\psi _{yy} + i\\kappa \\psi _t = 0\\,,$ with the trial solution $\\displaystyle \\psi = A(x,y,t)e^{i\\psi (x,y,t)}.$ Proceeding similarly as before, the real and imaginary parts are separated as $A_{xx} + A_{yy} - A(\\phi _x)^2 - A(\\phi _y)^2 - \\kappa A \\phi _t &=& 0\\,,\\nonumber \\\\A\\phi _{xx} + A\\phi _{yy} + 2A_x\\phi _x + 2A_y\\phi _y + \\kappa A_t & = &0\\,.$ Thus, the required conditions on $\\displaystyle \\phi $ are $\\displaystyle \\phi _{xx}=\\phi _{yy}=0$ , which gives us $\\phi = \\phi _0 + \\phi _1 x + \\phi _2 y\\,.$ Here all $\\displaystyle \\phi _i$ $\\displaystyle (i=0,1,2)$ are functions of time only.","Substituting back into Eq.","(REF ) we get $A_{xx} + A_{yy} - Az = 0\\,,$ where, $z = \\phi _{1}^2 + \\phi _{2}^2 + \\kappa \\left(\\dot{\\phi }_1 x + \\dot{\\phi }_2 y +\\dot{\\phi }_0 \\right)\\,.$ Thus, the required solution for the amplitude is $A = \\mathrm {Ai}\\left[\\frac{z}{\\kappa ^{2/3}(P_{1}^2+P_{2}^2)^{1/3}}\\right]~,$ where $\\displaystyle \\dot{\\phi }_1 =P_{1}$ and $\\displaystyle \\dot{\\phi }_2=P_{2}$ are both constants.","With these conditions, we get, $ \\phi _{0} = -\\left(\\frac{2}{3\\kappa }\\right)\\left(P_{1}^2+P_{2}^2\\right)t^3-Qt\\,,$ where $\\displaystyle Q$ is a constant as before.","A similar calculation holds for the case with potential.","This solution is essentially non-separable in the variables $\\displaystyle \\lbrace x,\\,y,\\, t\\rbrace $ Figure: The plot of the amplitude in the two dimensional as function of x\\displaystyle x and y\\displaystyle y.","Here, we take P 1 =10.1\\displaystyle P_1 = 10.1, P 2 =2.3\\displaystyle P_2 =2.3 , Q=0\\displaystyle Q = 0, κ=1\\displaystyle \\kappa = 1 and t=0\\displaystyle t = 0."],["A nonlinear example","In this section, we apply this method in solving a particular example of the nonlinear type - the Gross-Pitaevskii equation [7], which is given (in atomic units $\\displaystyle m=\\hbar =1$ ) by $i\\psi _t = -\\psi _{xx} + V\\psi + |\\psi |^2\\psi ~.$ Here, $\\displaystyle V=V(x,t)$ is the potential as given in section-REF .","With the substitution of the same trial solution as before, we find that the resulting equations obtained by separating the real and imaginary parts are $A_{xx} - A(\\phi _x)^2 - A \\phi _t -V A -A^3 &= &0\\, \\nonumber \\\\A \\phi _{xx} + 2 A_x \\phi _x + A_t = 0\\,.$ The second equation is exactly the same as before given by Eq.","(REF ), while we get an additional nonlinear term $\\displaystyle A^3$ is the first.","With the condition that $\\displaystyle \\partial ^2\\phi /\\partial x^2$ be zero, we get, after simplification, the following differential equation $A_{zz} = Az + \\mu A^3\\,,$ with $\\displaystyle z$ defined similarly to the variable $\\displaystyle y$ in section 2 (multiplied by an approriate constant) and $\\displaystyle \\mu $ is a constant.","This equation is a variant of the Type-II Painlevé transcendent [8], which are known to have solutions in terms of integral equations involving Airy functions [9].","Thus we see a specific example of the utility of the present method in obtaining special solutions of nonlinear Schrödinger equations."],["Discussion","Here, we have showed that it is possible to solve for the nondispersive solution given by Berry and Balazs in a systematic way.","It is possible to obtain the solution from a purely analytical calculation, without any direct quantum-mechanical arguments.","The method we have presented, which is due to the fact that the second derivative of the phase in the spatial coordinate can be set to zero, leads to an analogy with the modification of the wavefunction under a Galilean coordinate transformation.","This requirement is shown to work in a broad sense, through the derivation of the Airy wavepacket in two dimensions and the reduction of the nonlinear Gross-Pitaevskii equation to an ordinary diffenetial equation related to the second Painlevé transcendent.","The latter has analytical solutions and numerical approximations which have a striking resemblance with the Airy functions [10]."],["Acknowledgement","KRN would like to thank Dr Bhavtosh Bansal and Dr Vivek Vyas for fruitful discussions.","KRN would wish to acknowledge the Visiting Associateship programme of Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune.","A part of this work was carried out during the visit to IUCAA under this programme."]],"string":"[\n [\n \"A method for obtaining nonspreading solutions of the Schr\\\\\\\"odinger\\n equation\"\n ],\n [\n \"Abstract In this paper, we present a simple analytical method for obtaining a nonspreading solution of the time-dependent Schr\\\\\\\"odinger equation, which is given by the Airy function.\",\n \"The solution is derived by imposing a restriction on the phase factor of the ansatz that is taken to solve the differential equation.\",\n \"Considering at first the free particle, we show that nonspreading solutions can also be obtained for a time-dependent linear potential.\",\n \"The method is shown to work in both one and two dimensions, and can be easily extended if required.\",\n \"The applicability of the method is discussed in relation to the nonlinear case.\"\n ],\n [\n \"Introduction\",\n \"Berry and Balazs showed that the free-particle Schrödinger equation admits non-trivial solutions whose amplitude is given by the Airy differential equation [1].\",\n \"Specifically, they showed that $-\\\\frac{\\\\hbar ^2}{2m}\\\\psi _{xx} = i\\\\hbar \\\\psi _{t} \\\\,,$ has a unique solution given by $\\\\psi (x,t) = \\\\mathrm {Ai}\\\\left[\\\\frac{B}{\\\\hbar ^{2/3}}\\\\left(x-\\\\frac{B^3t^2}{4m^2}\\\\right)\\\\right]e^{\\\\left(iB^3t/2m\\\\hbar \\\\right)\\\\left[x-\\\\left(B^3t^2/6m^2\\\\right)\\\\right]}\\\\,,$ where $\\\\displaystyle \\\\mathrm {Ai}(z)$ is the Airy function and $\\\\displaystyle B$ is a positive constant.\",\n \"Unlike plane wave solution, the Airy solution given above is not separable in the variables $\\\\displaystyle \\\\lbrace x,\\\\, t\\\\rbrace $ .\",\n \"The essential feature of these solutions is that the time-evolution of the wavepacket is non-dispersive.\",\n \"The solution given by Eq.\",\n \"(REF ) was proved to be the only unique solution of Eq.\",\n \"(REF ) having this property.\",\n \"This result was originally explained in terms of the behaviour of the corresponding families of the semiclassical orbits in phase space.\",\n \"Subsequently, much work has been done to interpret this surprising result, with a proper quantum mechanical derivation being given in [2].\",\n \"Recently, the nonspreading solution has evoked considerable interest in the community.\",\n \"It has been shown that, under the paraxial approximation the wave equation allows a nonspreading solution [3].\",\n \"It has been verified experimentally by generating optical beams with wave packets that are accelerated [4].\",\n \"This leads further possibility of using these solutions in various optical systems.\",\n \"In this paper, we show that it is possible to obtain the result from techniques based on ordinary calculus.\",\n \"It is shown that the in the ansatz $\\\\displaystyle A(x,t)e^{i\\\\phi (x,t)}$ that is used to obtain wave-like solutions of Eq.\",\n \"(REF ), a particular choice of the phase $\\\\displaystyle \\\\phi $ will lead to a reduction of the Schrödinger equation to the Airy differential equation.\",\n \"In this paper, we first describe a method for obtaining the nonspreading solution for the Schrödinger equation for a free particle in one dimension.\",\n \"We also show that we can easily extend the solution for linear and time-dependent potentials.\",\n \"In section , we show that the method can be extended to higher dimension by taking the two dimensional case as an example.\"\n ],\n [\n \"The method for obtaining solution\",\n \"We are looking for non-trivial solutions of $\\\\psi _{xx} + \\\\mathrm {i}\\\\kappa \\\\psi _t - \\\\bar{V}(x,t)\\\\psi = 0\\\\,,$ where $\\\\displaystyle \\\\kappa = 2m/\\\\hbar $ and $\\\\displaystyle \\\\bar{V} = (\\\\kappa /\\\\hbar )V$ .\",\n \"We start with the free particle case first with $\\\\displaystyle \\\\bar{V}(x,t)=0$ .\",\n \"After demonstrating the method, we extend the solution with a potential term.\"\n ],\n [\n \"The one-dimensional Schrödinger equation\",\n \"In this section we solve for the non-disperssive solution in one dimension.\",\n \"We start with $\\\\displaystyle V=0$ in Eq.\",\n \"(REF ).\",\n \"First, we propose a trial solution of the form, $\\\\psi = A(x,t)\\\\,e^{i\\\\phi (x,t)}\\\\,,$ where $\\\\displaystyle A(x,t)$ and $\\\\displaystyle \\\\phi (x,t)$ are assumed to be real functions.\",\n \"Separating real and imaginary parts after substituting Eq.\",\n \"(REF ) in Eq.\",\n \"(REF ), we obtain: $A_{xx} - A(\\\\phi _x)^2 -\\\\kappa A \\\\phi _t & = & 0\\\\,, \\\\nonumber \\\\\\\\A\\\\phi _{xx} + 2 A_x\\\\phi _x + \\\\kappa A_t &= &0\\\\,.$ We start with the ansatz that the phase term $\\\\displaystyle \\\\phi (x,t)$ must satisfy the Laplace equation, $\\\\frac{\\\\partial ^2\\\\phi }{\\\\partial x^2} = 0\\\\,, $ With this we get, $\\\\phi = \\\\phi _1 \\\\cdot x + \\\\phi _0\\\\,,$ where $\\\\displaystyle \\\\lbrace \\\\phi _1,\\\\phi _0\\\\rbrace $ are functions of time only.\",\n \"Substituting $\\\\displaystyle \\\\phi $ in Eq.\",\n \"(REF ), we get $A_{xx} - Ay = 0\\\\,,$ where, $y = \\\\phi _1^2+\\\\kappa \\\\left(\\\\dot{\\\\phi }_1 x+\\\\dot{\\\\phi }_0\\\\right)\\\\,.$ In the above, $\\\\displaystyle \\\\dot{\\\\phi }=\\\\frac{d\\\\phi }{dt}$ .\",\n \"We further assume that $\\\\displaystyle \\\\partial y/\\\\partial x$ is constant implying $\\\\displaystyle \\\\dot{\\\\phi }_1 = P$ (a constant).\",\n \"Then we have reduced Eq.\",\n \"(REF ) to an Airy differential equation with the solution $A = \\\\mathrm {Ai}\\\\left[\\\\frac{y}{(\\\\kappa P)^{2/3}}\\\\right]\\\\,.$ The condition for $\\\\displaystyle \\\\phi _0$ is given from Eq.\",\n \"(REF ) by $\\\\kappa \\\\ddot{\\\\phi }_0 + 4P^2t = 0\\\\,,$ resulting in, $\\\\phi _0 = -\\\\frac{2}{3\\\\kappa }P^2t^3 + Qt\\\\,.$ Here $\\\\displaystyle P$ and $\\\\displaystyle Q$ are constants.\",\n \"It can be shown that one can get Berry's solution with $\\\\displaystyle Q=0$ and $\\\\displaystyle P=B^3/2m\\\\hbar $ where $\\\\displaystyle B$ is an arbitrary positive constant.\",\n \"At this point, we would like to physically justify the ansatz used in obtaining the solution.\",\n \"It is closely related to the symmetry properties of the Schrödinger equation.\",\n \"The Schrödinger equation admits a Galilean transformation which preserves the form of the differential equation [5].\",\n \"The transformed wave-function is then just the original wave-function multiplied by a phase which is linear in the spatial coordinates.\",\n \"Thus, we must have $\\\\displaystyle \\\\partial ^2 \\\\phi / \\\\partial x^2 = 0$ .\"\n ],\n [\n \"With linear potential\",\n \"In this section, we generalise to the case with potential.\",\n \"Starting with the same trial solution to obtain, $A_{xx} - A(\\\\phi _x)^2 -\\\\kappa A \\\\phi _t - \\\\bar{V}A = 0\\\\,, \\\\nonumber \\\\\\\\A\\\\phi _{xx} + 2 A_x \\\\phi _x + \\\\kappa A_t = 0\\\\,.$ Since the second equation is the same as given in Eq.\",\n \"(REF ), the form of $\\\\displaystyle \\\\phi $ is modified by an additional term due to the potential.\",\n \"The equation for the amplitude $\\\\displaystyle A$ also takes the same form in terms of the variable $\\\\displaystyle y = \\\\phi _{1}^2+\\\\kappa \\\\left(\\\\dot{\\\\phi }_1 x + \\\\dot{\\\\phi }_0\\\\right) + \\\\bar{V}$ , as before, $\\\\displaystyle \\\\partial y/\\\\partial x$ is assumed constant.\",\n \"Like before, we take $\\\\displaystyle \\\\mathrm {d}\\\\phi _{1}/\\\\mathrm {d}t = P$ with the addition that $\\\\displaystyle \\\\partial \\\\bar{V}/\\\\partial x = \\\\bar{V_1}$ , both of them being constants.\",\n \"With these conditions, we have $\\\\frac{\\\\partial y}{\\\\partial x} = \\\\kappa \\\\frac{\\\\mathrm {d}\\\\phi _{1}}{\\\\mathrm {d}t} + \\\\frac{\\\\partial \\\\bar{V}}{\\\\partial x} = \\\\kappa P + \\\\bar{V_1}\\\\,,$ from which we finally get $\\\\phi _1 = Pt~,\\\\qquad \\\\bar{V} = \\\\bar{V_1}x + \\\\bar{V_0}\\\\,.$ Here, we have freedom to choose $\\\\displaystyle \\\\bar{V_0}$ to be a function of time.\",\n \"In order to get $\\\\displaystyle \\\\phi _0$ we solve an equation similar to Eq.\",\n \"(REF ) to obtain $\\\\phi _0 = -\\\\frac{1}{\\\\kappa ^2}\\\\left[\\\\int ^t\\\\bar{V_0}(t^{\\\\prime })\\\\mathrm {d}t^{\\\\prime } + \\\\frac{2}{3}\\\\kappa P^2t^3 -\\\\frac{1}{3}P\\\\bar{V_1}t^3\\\\right] + Qt~,$ where $\\\\displaystyle Q$ is a constant.\",\n \"It can be readily seen this is a validation of the results of Berry and Balazs for more general potentials of the form $\\\\displaystyle F(t)x$ .\"\n ],\n [\n \"Solution in two dimensions\",\n \"In this section, we consider the Schrödinger equation in two spatial dimensions.\",\n \"For the potential-free case we wish to solve, $\\\\psi _{xx} + \\\\psi _{yy} + i\\\\kappa \\\\psi _t = 0\\\\,,$ with the trial solution $\\\\displaystyle \\\\psi = A(x,y,t)e^{i\\\\psi (x,y,t)}.$ Proceeding similarly as before, the real and imaginary parts are separated as $A_{xx} + A_{yy} - A(\\\\phi _x)^2 - A(\\\\phi _y)^2 - \\\\kappa A \\\\phi _t &=& 0\\\\,,\\\\nonumber \\\\\\\\A\\\\phi _{xx} + A\\\\phi _{yy} + 2A_x\\\\phi _x + 2A_y\\\\phi _y + \\\\kappa A_t & = &0\\\\,.$ Thus, the required conditions on $\\\\displaystyle \\\\phi $ are $\\\\displaystyle \\\\phi _{xx}=\\\\phi _{yy}=0$ , which gives us $\\\\phi = \\\\phi _0 + \\\\phi _1 x + \\\\phi _2 y\\\\,.$ Here all $\\\\displaystyle \\\\phi _i$ $\\\\displaystyle (i=0,1,2)$ are functions of time only.\",\n \"Substituting back into Eq.\",\n \"(REF ) we get $A_{xx} + A_{yy} - Az = 0\\\\,,$ where, $z = \\\\phi _{1}^2 + \\\\phi _{2}^2 + \\\\kappa \\\\left(\\\\dot{\\\\phi }_1 x + \\\\dot{\\\\phi }_2 y +\\\\dot{\\\\phi }_0 \\\\right)\\\\,.$ Thus, the required solution for the amplitude is $A = \\\\mathrm {Ai}\\\\left[\\\\frac{z}{\\\\kappa ^{2/3}(P_{1}^2+P_{2}^2)^{1/3}}\\\\right]~,$ where $\\\\displaystyle \\\\dot{\\\\phi }_1 =P_{1}$ and $\\\\displaystyle \\\\dot{\\\\phi }_2=P_{2}$ are both constants.\",\n \"With these conditions, we get, $ \\\\phi _{0} = -\\\\left(\\\\frac{2}{3\\\\kappa }\\\\right)\\\\left(P_{1}^2+P_{2}^2\\\\right)t^3-Qt\\\\,,$ where $\\\\displaystyle Q$ is a constant as before.\",\n \"A similar calculation holds for the case with potential.\",\n \"This solution is essentially non-separable in the variables $\\\\displaystyle \\\\lbrace x,\\\\,y,\\\\, t\\\\rbrace $ Figure: The plot of the amplitude in the two dimensional as function of x\\\\displaystyle x and y\\\\displaystyle y.\",\n \"Here, we take P 1 =10.1\\\\displaystyle P_1 = 10.1, P 2 =2.3\\\\displaystyle P_2 =2.3 , Q=0\\\\displaystyle Q = 0, κ=1\\\\displaystyle \\\\kappa = 1 and t=0\\\\displaystyle t = 0.\"\n ],\n [\n \"A nonlinear example\",\n \"In this section, we apply this method in solving a particular example of the nonlinear type - the Gross-Pitaevskii equation [7], which is given (in atomic units $\\\\displaystyle m=\\\\hbar =1$ ) by $i\\\\psi _t = -\\\\psi _{xx} + V\\\\psi + |\\\\psi |^2\\\\psi ~.$ Here, $\\\\displaystyle V=V(x,t)$ is the potential as given in section-REF .\",\n \"With the substitution of the same trial solution as before, we find that the resulting equations obtained by separating the real and imaginary parts are $A_{xx} - A(\\\\phi _x)^2 - A \\\\phi _t -V A -A^3 &= &0\\\\, \\\\nonumber \\\\\\\\A \\\\phi _{xx} + 2 A_x \\\\phi _x + A_t = 0\\\\,.$ The second equation is exactly the same as before given by Eq.\",\n \"(REF ), while we get an additional nonlinear term $\\\\displaystyle A^3$ is the first.\",\n \"With the condition that $\\\\displaystyle \\\\partial ^2\\\\phi /\\\\partial x^2$ be zero, we get, after simplification, the following differential equation $A_{zz} = Az + \\\\mu A^3\\\\,,$ with $\\\\displaystyle z$ defined similarly to the variable $\\\\displaystyle y$ in section 2 (multiplied by an approriate constant) and $\\\\displaystyle \\\\mu $ is a constant.\",\n \"This equation is a variant of the Type-II Painlevé transcendent [8], which are known to have solutions in terms of integral equations involving Airy functions [9].\",\n \"Thus we see a specific example of the utility of the present method in obtaining special solutions of nonlinear Schrödinger equations.\"\n ],\n [\n \"Discussion\",\n \"Here, we have showed that it is possible to solve for the nondispersive solution given by Berry and Balazs in a systematic way.\",\n \"It is possible to obtain the solution from a purely analytical calculation, without any direct quantum-mechanical arguments.\",\n \"The method we have presented, which is due to the fact that the second derivative of the phase in the spatial coordinate can be set to zero, leads to an analogy with the modification of the wavefunction under a Galilean coordinate transformation.\",\n \"This requirement is shown to work in a broad sense, through the derivation of the Airy wavepacket in two dimensions and the reduction of the nonlinear Gross-Pitaevskii equation to an ordinary diffenetial equation related to the second Painlevé transcendent.\",\n \"The latter has analytical solutions and numerical approximations which have a striking resemblance with the Airy functions [10].\"\n ],\n [\n \"Acknowledgement\",\n \"KRN would like to thank Dr Bhavtosh Bansal and Dr Vivek Vyas for fruitful discussions.\",\n \"KRN would wish to acknowledge the Visiting Associateship programme of Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune.\",\n \"A part of this work was carried out during the visit to IUCAA under this programme.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05380"}}},{"rowIdx":3388,"cells":{"text":{"kind":"list like","value":[["Origin of doping-induced suppression and reemergence of magnetism in\n LaFeAsO$_{1-x}$H$_x$"],["Abstract We investigate the evolution of magnetic properties as a function of hydrogen doping in iron based superconductor LaFeAsO$_{1-x}$H$_x$ using the dynamical mean-field theory combined with the density-functional theory.","We find that two independent consequences of the doping, the increase of the electron occupation and the structural modification, have the opposite effects on the strength of electron correlation and magnetism, resulting in the minimum of the calculated magnetic moment around the intermediate doping level as a function of $x$.","Our result provides a natural explanation for the puzzling recent experimental discovery of the two separated antiferromagnetic phases at low and high doping limits.","Furthermore, the increase of orbital occupation and correlation strength with the doping results in reduced orbital polarization of $d_{xz/yz}$ orbitals and the enhanced role of $d_{xy}$ orbital in the magnetism at high doping levels, and their possible implications to the superconductivity are discussed in line with the essential role of the magnetism."],["Introduction","Iron-based high temperature ($T$ ) superconductors including the seminal material LaFeAsO$_{1-x}$ F$_x$[1] share a common feature that the impurity doping results in the suppression of magnetic and/or structural orders in undoped parent compounds and subsequent emergence of superconductivity.","The underlying mechanism is still not well understood, however, as the impurity doping has several independent effects on the system such as the change of the electron occupancy, structural modification, and occurrence of the disorder, etc.","One of the popular explanations has been based on the itinerant picture of the antiferromagnetic (AFM) ordering in undoped samples.","As the doping changes the number of carriers and the position of the Fermi level, the Fermi surface (FS) nesting condition for the spin density wave (SDW) formation becomes poorer.","[2], [3], [4], [5] However, the non-negligible role of the electron correlation in these materials has been pointed out,[6], [7], [8] and the validity of the FS nesting picture alone, which assumes the rigid band against the carrier doping, often turns out to be doubtful in explaining the emergence and suppression of the AFM order.","On the other hand, the structural modification effect is known to alter significantly the magnetic property with the strong magnetostructural coupling in this system.","[2], [9], [10] Even the isovalent impurity doping or the hydrostatic pressure alone, which introduce no extra carrier, can give rise to the similar phase diagram with the case of the carrier doping.","[11], [12], [13], [14], [15] There is a general consensus about the importance of the magnetism in understanding the superconductivity of iron-based materials,[16], [17] as the magnetism is omnipresent in this class of materials at least in the form of the short-range spin fluctuation.","[18], [19], [20] However, the basic nature of the magnetism has been controversial among the SDW of itinerant electrons,[2], [3], [4], [5], [21] the Heisenberg type interactions of localized spins,[22], [23], [24], [25] as well as their intermediate picture.","[26], [27], [28], [29], [30], [31], [32], [33], [34], [35] Nevertheless, the spin fluctuation has been widely accepted as the most probable candidate for the pairing glue for the superconductivity.","[36], [37], [38], [39], [40], [41], [42] In the meanwhile, there are also growing arguments and evidences for the presence of the orbital order and fluctuations in these materials.","[43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54] Recent experiments on FeSe suggest that the orbital degree of freedom drives the electronic nematicity and spontaneous symmetry breaking instead of the spin degree of freedom, [55], [56] drawing attention for the alternative mechanism of the pairing mediated by the orbital fluctuation.","[57], [58] Because the spin and orbital degrees of freedom are coupled each other, [43], [59] however, there can be an inherent ambiguity in determining which order forms first and drives the other.","Recently, a series of experiments has revived the attention to the seminal material of the family by the hydrogen (H) doping, namely, LaFeAsO$_{1-x}$ H$_x$ .","[60], [61], [62] Overcoming the solubility limit of the conventional fluorine dopant, hydrogen can increase the doping concentration up to $x=0.6$ .","Surprisingly, another superconducting and AFM phases adjacent to each other are found at high doping levels, analogously to their conventional counterparts at the low doping level (see Fig. 1(a)).","Posing fundamental questions on the nature of magnetism and superconductivity, this finding is expected to give a clue for still unresolved issues mentioned earlier.","Although some theoretical attempts have been made to explain the appearance of the second AFM phase mostly focusing on the FS nesting property in the itinerant electron picture,[60], [61], [63] first-principles approach simultaneously incorporating the itinerant and localized aspects of the system is desirable when we consider the 'moderately correlated' nature of these materials [6], [7], [8], [64], [65] and some unsatisfactory conclusions from the itinerant picture such as the prediction of an incorrect magnetic ordering vector.","[61] In this paper, we investigate the magnetic and electronic properties of LaFeAsO$_{1-x}$ H$_x$ as a function of $x$ using the combined method of density-functional theory plus dynamical mean-field theory (DFT+DMFT), which captures the material-specific electronic correlation.","[66], [67] Considering changes of both electron occupancy and lattice structure caused by the hydrogen doping which turn out to have the opposite effects on the electron correlation and magnetism, we find that both static magnetic moment and local magnetic susceptibility initially decrease to the minimum at around $x=0.3$ and then increase again up to $x=0.6$ , in agreement with the experimental phase diagram of the two separate AFM phases centered at $x=0$ and 0.5.","More electron occupation at $d_{xz/yz}$ orbitals with the doping enhances the importance of the $d_{xy}$ orbital in the static magnetic moment and also in spin dynamics, while reducing the orbital polarization.","Our results emphasize the importance of the electron correlation and structural modification in understanding the doping induced evolution of the electronic structure, and also the magnetism as an indispensable ingredient for the emergence of the superconductivity in these materials."],["Calculation Method","We use the modern implementation of DFT+DMFT method within all electron embedded DMFT approach,[67] where in addition to correlated Fe atoms the itinerant states of other species are included in the Dyson self-consistent equation.","The strong correlations on the Fe ion are treated by DMFT, adding self-energy $\\Sigma (\\omega )$ on a quasi atomic orbital in real space, to ensure stationarity of the DFT+DMFT approach.","The self-energy $\\Sigma (\\omega )$ contains all Feynman diagrams local to the Fe ion.","No downfolding or other approximations were used, and the calculations are all-electron as implemented in Ref.","67, which is based on Wien2k.","[68] We used the GGA exchange-correlation functional,[69] and the quantum impurity model was solved by the continuous time quantum Monte Carlo impurity solver [70] using $U=5.0$ eV and $J=0.72$ eV.","Brillouin zone integration is done on the 12$\\times $ 12$\\times $ 6 k-point mesh for the AFM unitcell of LaFeAsO containing 4 Fe atoms.","All calculations are done for $T=150$ K. We consider both paramagnetic (PM) and AFM states at this temperature.","The AFM state is considered to represent the actual stable phase observed in experiments, while the PM state is also calculated to understand the driving force with which the AFM state is stabilized from a 'bare' state."],["Electronic correlation and magnetic strengths as functions of doping","Schematic phase diagram of LaFeAsO$_{1-x}$ H$_x$ in the $x-T$ space is depicted in Fig. 1(a).","The first AFM phase with a stripe-type order is rapidly suppressed and disappears around $x=0.05$ with the emergence of the first superconducting phase followed by the adjacent second superconducting phase.","Further doping initiates the second AFM phase, of the same ordering pattern with the first AFM phase, but with a slightly different atomic displacement.","[62] We perform the DFT+DMFT calculations to check if this suppression and reappearance of the AFM phase can be reproduced.","To take the electron doping effect into account, we adopt the virtual crystal approximation.","[60] In addition, the structural change due to the H doping is incorporated by interpolating both the lattice constants and internal atomic coordinates of the available experimental values at $x=0$ [71] and $x=0.51$ [62] in the PM states with the tetragonal lattice symmetry.","This should be a reasonable approximation considering the almost linear As height dependence on $x$ observed experimentally,[60] avoiding the difficulty in the structural optimization of alloy structures within DFT which would require rather complicated statistical treatment, besides concerns about the general reliability of DFT in predicting the accurate lattice structure of iron-based superconducting materials.","Using this doping scheme, the static magnetic moment and the local spin susceptibility, $\\chi _{local}(\\omega =0)=\\int _0^\\beta \\!","\\langle S(\\tau )S(0) \\rangle \\mathrm {d}\\tau $ , are calculated as a function of $x$ for the stripe-type AFM phase and shown in Fig. 1(b).","The magnetic moment at $x=0$ is estimated to be 0.66 $\\mu _B$ with an agreement with the measured value 0.63 $\\mu _B$ ,[72] and decreases to the minimum value of 0.12 $\\mu _B$ at $x=0.4$ , and then exhibits a rapid increase to reach 0.68 $\\mu _B$ at $x=0.6$ .","The local susceptibility shows a similar behavior with a minimum at $x=0.3$ , implying that the overall magnetic strength is suppressed and then re-enhanced with the doping.","Therefore, we verify that the DFT+DMFT method captures the essential underlying physics of two separate AFM phases in this material and produces a consistent behavior of the local magnetic strength, while the complete suppression of magnetic phase and emergence of the superconducting phase in the intermediate $x$ as shown in Fig.","1(a) could not be properly described within the current calculation scheme.","For comparison, we perform the DFT calculation on the relative stability of the AFM phase and the magnetic moment as functions of $x$ .","For $x=0$ , the AFM phase is found to be 180 meV/Fe more stable than the nonmagnetic phase with the magnetic moment of 2.15 $\\mu _B$ .","Upon increasing $x$ , the stability of AFM phase and the magnetic moment show no discernible change, suggesting that the normal DFT calculation cannot properly describe the observed evolution of the magnetism.","Then we investigate the underlying mechanism of the doping-induced change of the electronic structure by considering the electron addition and structural modification separately.","Because the nominal number of valence electrons in a Fe atom for the undoped material is six, which corresponds to an electron doped system from the half-filled orbitals, further electron doping should result in the monotonic decrease of the correlation strength.","[73] On the other hand, the H doping increases the distance between Fe and surrounding As atoms as determined experimentally,[71], [62] which would lead to the localization of Fe $d$ orbitals.","[74] To confirm this speculation, we estimate the mass enhancement $1/Z = 1-\\frac{\\partial \\Sigma (\\omega )}{\\partial \\omega }\\vert _{\\omega =0}$ for the two effects separately.","First, we calculate $1/Z$ of the $d_{xy}$ in the PM phase as a function of $x$ considering only the electron addition effect by fixing the lattice structure to that of $x=0$ as in Fig. 1(c).","Indeed, $1/Z$ monotonically increases with the electron addition.","On the contrary, when only the structural effect is included without extra electron, $1/Z$ monotonically decreases with increasing $x$ .","When these two competing effects are combined, $1/Z$ increases overall with the doping, which means that the localization by the structural modification becomes more dominant at the highly doped system.","This competing effects are also reflected on the magnetic strength of the AFM phase as shown in Fig. 1(d).","When only electron addition effect is considered, the local susceptibility is found to monotonically decrease with increasing $x$ , while it increases monotonically when only the structural modification is taken into account (with the number of extra electrons fixed to 0.6), in agreement with the behavior of the mass enhancement factors in Fig. 1(c).","Therefore, we can conclude that the initial suppression and the later re-enhancement of the magnetism with the H doping originates from the two competing effects : the electron addition and increasing Fe-As distance which suppresses and enhances the local correlation and hence the local magnetism, respectively.","Our analysis naturally draws attention to the important role of the electron correlation and also the structural effect in understanding the doping induced phase diagram of this material.","Suppression of the magnetism and the existence of the quantum criticality in phosphorus-doped Ba122 systems BaFe$_2$ As$_{2-x}$ P$_x$ [75], [76], [77] are another set of examples which demonstrate the dramatic effect of the pure structure modification, where decreased Fe-anion distance was pointed out to cause the suppression of the magnetism.","[76]"],["Fermi surfaces","We also investigate the evolution of the FS with the doping which is generally considered to be relevant to the existence of the AFM phase.","[61], [63] The FSs for three different doping levels, $x=$ 0, 0.3, and 0.5, are calculated with both the DFT+DMFT and DFT methods and displayed in Fig. 2.","Starting with a relatively good nesting between the hole and electrons surfaces at $x=0$ for the DFT case, the doping degrades the nesting with shrinking the hole surfaces at the $\\Gamma $ point and enlarging the electron surfaces at the $M$ point (see Fig.","2(g)-(i)), as the electron doping raises the Fermi level.","Our result shows a good agreement with the previous DFT calculation using the experimentally determined lattice structure,[60] confirming that our assumption of the linear dependence of the lattice constants and internal atomic coordinates is reasonable.","The DFT+DMFT results shown in Figs.","2(a)-(f) are qualitatively similar, but the $d_{xy}$ hole FS expands compared with the DFT results, because of more correlated nature of the $d_{xy}$ orbital than $d_{xz/yz}$ orbitals as pointed out in the DFT+DMFT study of LiFeAs.","[78] The decrease of the overall spectral weight with the doping and relatively larger incoherence of the $d_{xy}$ surface reflect the larger correlation at high doping levels and for the $d_{xy}$ orbital (Figs. 2(a)-(c)).","Nevertheless, both levels of the theory indicate the monotonic degradation of the FS nesting with the doping, as already indicated by previous calculations,[60], [63] manifesting that the FS nesting alone cannot explain the appearance of the second AFM phase.","Again, we conclude that the electron correlation and many-body effects should be incorporated to understand the doping-induced evolution of the magnetism."],["Spin resolved spectral function","To understand the doping-induced suppression and the reemergence of the magnetism in detail, we investigate the spin-resolved spectral function of the $d_{xy}$ orbital in the AFM phase as a function of $x$ as shown in Fig. 3.","At $x=0$ , the majority and minority spin states exhibit a large exchange splitting reflecting the overall magnetic moment of 0.66 $\\mu _B$ , with a distinct pseudo-gap feature (a dip in spectral function) at the Fermi energy induced by the coupling between the electron and hole bands at the Fermi energy.","[31] With increasing doping level up to $x=0.3$ (see Fig.","3(a)), spectral weights moves from the peak just above the Fermi level to one below the Fermi energy in the minority spin channel, suggesting the doped electrons fills the minority spin states.","The electron filling in the minority spin states with the doping naturally leads to the gradual reduction of the exchange splitting and the magnetic moment, along with the size of the pseudo-gap.","On the other hand, further doping over $x=0.3$ enhances the exchange splitting as shown in Fig.","3(b), which seems to almost retrace the evolution of the spectral function from $x=0$ to $x=0.3$ in Fig. 3(a).","However, there are several noticeable differences as well.","First, the pseudo-gap position is constantly shifting deeper in the valence states with its size and the magnetic moment increasing with increasing $x$ , which indicates the rise of the Fermi energy as a result of the electron doping.","More importantly, doping over $x=0.3$ develops a shoulder growing with $x$ near -1 eV in the majority spin channel as indicated by a arrow in Fig.","3(b), contributing to build up the magnetic moment against the electron filling on the minority spin states with the doping.","The spectral weight piled up in this position results from many-body effects and hence is incoherent, rather than from the shift of coherent quasi-particle states.","The inset of Fig.","3(b) depicts the imaginary part of the $d_{xy}$ component of the electron self energy (Im$\\Sigma $ ) along with the spectral function for $x=0.6$ .","One can identify a strong peak of the Im$\\Sigma $ around -0.7 eV, close to the $J$ value 0.72 eV adopted in this study, and the shoulder structure of the spectral function at a nearby position, suggesting that the shoulder structure originates from the incoherent excitations related to the self energy.","Similar energy scales between this incoherent excitation and $J$ is also consistent with the suggestion that iron-based superconductors are Hund's metals where $J$ plays more important role than $U$ .","[7]"],["Orbital polarization","As mentioned earlier, the orbital order is of great interest for these materials regarding the electronic nematicity and also superconductivity itself.","Here we compare the orbital polarization, i.e., the imbalance between $d_{xz/yz}$ orbitals, in the AFM state for low and high doping cases.","For $x=0$ as displayed in Fig.","4(a), $d_{xz}$ and $d_{yz}$ spectral functions show noticeable difference, where the spin polarization is larger for $d_{yz}$ as well as $d_{xy}$ orbitals while $d_{xz}$ spin splitting is smaller.","On the other hand, for the high doping case of $x=0.5$ in Fig.","4(b), $d_{xz/yz}$ components of the spectral function becomes much more similar with each other and now the $d_{xy}$ orbital has the most significant spin polarization.","Indeed, our estimated orbital polarization $(n_{xz}-n_{yz})/(n_{xz}+n_{yz})$ decreases from 3.9 $\\%$ at $x=0$ to 1.5 $\\%$ at $x=0.5$ while the magnetic moments for the two cases are comparable.","Increasing $x$ enhances the crystal field splitting pushing up the $d_{xy}$ level above $d_{xz/yz}$ level, so that the doped electrons fill $d_{xz/yz}$ orbitals first rather than the $d_{xy}$ orbital, reducing the imbalance between $d_{xz/yz}$ orbitals as well as between their spin components.","Meanwhile, besides the less electron filling, the elevated As height in the high doping case further enhances the electron correlation for the $d_{xy}$ orbital via the `kinetic frustration' [79] compared with the $d_{xz/yz}$ orbitals.","So $d_{xy}$ becomes the most significant component for the local spin fluctuations in the PM phase and for static magnetic moments in the AFM phase.","Therefore, at high doping levels, strong local magnetism appears mainly from $d_{xy}$ orbital and the orbital polarization from $d_{xz/yz}$ is largely suppressed."],["Spin excitation spectrum","So far, we have considered the doping induced evolution of the magnetically ordered state, and we will conclude our discussion by investigating the dynamic spin fluctuations in the PM state, which is more relevant to the superconductivity, as a function of the doping.","Although $T=150$ K at which the calculation is done is close to the AFM transition temperature, the spin susceptibility in the PM state is expected to be a smooth varying function of $T$ (except at the AFM ordering wave vector for which the susceptibility diverges at the AFM transition temperature), so we expect qualitatively similar results for other temperatures.","We evaluate the dynamical spin structure factor $S({\\bf q},\\omega )=\\frac{\\chi ^{\\prime \\prime }({\\bf q},\\omega )}{1-e^{\\hbar \\omega /k_BT}}$ using the DFT+DMFT method as displayed in Fig.","5, where both the one-particle Green's function and the local two-particle vertex function are determined ab-initio.","[80] For $x=0$ , the spin excitation spectrum has strong peaks near the zero energy around the wave vector ${\\bf q}=(\\pi ,0)$ , which corresponds to the magnetic ordering vector of the AFM phase, and disperses over the path shown in the spectrum, reaching a maximum energy at the zone boundary ${\\bf q}=(\\pi ,\\pi )$ , all consistent with previous results.","[80], [81], [41] As the doping level $x$ increases, the overall spin excitation spectral weights tend to shift to lower energies as the spin wave dispersion decreases with the increasing correlation strength.","The excitation near ${\\bf q}=(\\pi /2,\\pi /2)$ noticeably goes down towards the zero energy with the doping, and a new possible static magnetic order for this wave vector is suggested for $x=0.5$ .","However, the intensity of excitations has always the maximum at the conventional AFM ordering vector ${\\bf q}=(\\pi ,0)$ for all the doping cases, consistent with the experimentally found second AFM phase for the high doping levels.","[62] For $S({\\bf q},\\omega )$ at ${\\bf q}=(\\pi +\\delta ,0)$ which is slightly off the magnetic ordering vector, as shown in Fig.","5(d), the peak height near the zero energy is reduced for $x=0.3$ compared with that for $x=0$ indicating the suppressed tendency towards the static magnetic order, and then it becomes pronounced again for the higher doping level $x=0.5$ suggesting the re-enhanced magnetism, which shows a qualitative agreement with the initial decrease and re-enhancement of the calculated magnetic moment in the AFM phase shown in Fig.","1 and again also with the motivating experiments.","[61], [62] When decomposed by orbitals (see Figs.","5(e)-(g)), the $d_{xz/yz}$ components show a large anisotropy with the $d_{yz}$ component peak being dominant at $x=0$ .","As the doping level increases, the $d_{xz/yz}$ anisotropy keeps decreasing while the $d_{xy}$ component becomes most prominent.","The decreasing $d_{xz/yz}$ anisotropy and the enhancement of the $d_{xy}$ component with the increasing doping is consistent with the features observed in our result for the AFM phase."],["Discussion","Our results remind us of the indispensable role of the electron correlation in the iron-based superconducting materials, as well as the impact of the structural change.","The doping-induced evolution of the electronic and magnetic properties cannot be understood by simply adopting the rigid shift of the Fermi level or even the self-consistent addition of carriers without taking the structural effect into account.","In addition, a natural view on the doping-induced evolution of spin and orbital orders can be obtained.","The `ferro-orbital order', which is coupled to the AFM spin order in the undoped materials, is the lowest-energy configuration for the nominally half-filled orbitals to maximize the kinetic energy gain.","[43] Both the orbital and spin orders of $d_{xz}$ or $d_{yz}$ are suppressed when the doping supplies more electrons to these orbitals away from the half-filling.","The $d_{xy}$ orbital, for which the spin order can form but the orbital order is no longer relevant, becomes the dominant channel for the electron hopping to reduce the kinetic energy as discussed above.","As a result, spin order/fluctuation is present near the both superconducting domes found in LaFeAsO$_{1-x}$ H$_x$ while orbital order/fluctuation is expected to be strong only near the first superconducting phase in the lower doping level.","Our results consequently suggests that the spin fluctuation is more closely related to the superconductivity, at least for the second superconducting phase in this alloy, while the orbital fluctuation, which is significant only at low doping levels, might not be a prerequisite for the superconductivity in general.","The enhanced role of the $d_{xy}$ orbital in magnetism is expected to naturally lead to its dominant role also in the superconductivity with a larger FS hole pocket of this orbital as shown in Fig. 2.","Although the enhanced electron correlation and consequent stronger spin fluctuation in the higher doping level might contribute to the strong superconductivity, too strong correlation would be harmful to the superconductivity, out of several reasons,[79] due to the lowered magnon energy scale (Fig.","5) which is directly coupled to the size of the superconducting gap.","Further theoretical study which directly attacks the superconductivity as a function of the doping level will be desirable."],["Conclusion","In summary, by adopting the DFT+DMFT method, where the local dynamic correlation effect is taken exactly, we successfully reproduce the hydrogen-doping-induced suppression and revitalization of the magnetism in LaFeAsO$_{1-x}$ H$_x$ which has been recently established experimentally.","Taking the structural modification by the doping into account along with the carrier addition is found to be essential, as the two factors induce independent and opposite effects on the electron correlation strength and the magnetism in this alloy.","Doping reduces the orbital imbalance between $d_{xz/yz}$ orbitals as well as their magnetic activity, while the $d_{xy}$ orbital becomes the dominant electron hopping channel with increased electron correlation and the magnetic strength for high doping levels.","Indispensable role of the electron correlation and detailed atomic structure is identified in understanding the electronic and magnetic properties, and the magnetism possibly as more fundamental ingredient in realizing the superconductivity is suggested over the orbital degrees of freedom.","This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2016R1C1B1014715 and 2015R1D1A1A01059621)."]],"string":"[\n [\n \"Origin of doping-induced suppression and reemergence of magnetism in\\n LaFeAsO$_{1-x}$H$_x$\"\n ],\n [\n \"Abstract We investigate the evolution of magnetic properties as a function of hydrogen doping in iron based superconductor LaFeAsO$_{1-x}$H$_x$ using the dynamical mean-field theory combined with the density-functional theory.\",\n \"We find that two independent consequences of the doping, the increase of the electron occupation and the structural modification, have the opposite effects on the strength of electron correlation and magnetism, resulting in the minimum of the calculated magnetic moment around the intermediate doping level as a function of $x$.\",\n \"Our result provides a natural explanation for the puzzling recent experimental discovery of the two separated antiferromagnetic phases at low and high doping limits.\",\n \"Furthermore, the increase of orbital occupation and correlation strength with the doping results in reduced orbital polarization of $d_{xz/yz}$ orbitals and the enhanced role of $d_{xy}$ orbital in the magnetism at high doping levels, and their possible implications to the superconductivity are discussed in line with the essential role of the magnetism.\"\n ],\n [\n \"Introduction\",\n \"Iron-based high temperature ($T$ ) superconductors including the seminal material LaFeAsO$_{1-x}$ F$_x$[1] share a common feature that the impurity doping results in the suppression of magnetic and/or structural orders in undoped parent compounds and subsequent emergence of superconductivity.\",\n \"The underlying mechanism is still not well understood, however, as the impurity doping has several independent effects on the system such as the change of the electron occupancy, structural modification, and occurrence of the disorder, etc.\",\n \"One of the popular explanations has been based on the itinerant picture of the antiferromagnetic (AFM) ordering in undoped samples.\",\n \"As the doping changes the number of carriers and the position of the Fermi level, the Fermi surface (FS) nesting condition for the spin density wave (SDW) formation becomes poorer.\",\n \"[2], [3], [4], [5] However, the non-negligible role of the electron correlation in these materials has been pointed out,[6], [7], [8] and the validity of the FS nesting picture alone, which assumes the rigid band against the carrier doping, often turns out to be doubtful in explaining the emergence and suppression of the AFM order.\",\n \"On the other hand, the structural modification effect is known to alter significantly the magnetic property with the strong magnetostructural coupling in this system.\",\n \"[2], [9], [10] Even the isovalent impurity doping or the hydrostatic pressure alone, which introduce no extra carrier, can give rise to the similar phase diagram with the case of the carrier doping.\",\n \"[11], [12], [13], [14], [15] There is a general consensus about the importance of the magnetism in understanding the superconductivity of iron-based materials,[16], [17] as the magnetism is omnipresent in this class of materials at least in the form of the short-range spin fluctuation.\",\n \"[18], [19], [20] However, the basic nature of the magnetism has been controversial among the SDW of itinerant electrons,[2], [3], [4], [5], [21] the Heisenberg type interactions of localized spins,[22], [23], [24], [25] as well as their intermediate picture.\",\n \"[26], [27], [28], [29], [30], [31], [32], [33], [34], [35] Nevertheless, the spin fluctuation has been widely accepted as the most probable candidate for the pairing glue for the superconductivity.\",\n \"[36], [37], [38], [39], [40], [41], [42] In the meanwhile, there are also growing arguments and evidences for the presence of the orbital order and fluctuations in these materials.\",\n \"[43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54] Recent experiments on FeSe suggest that the orbital degree of freedom drives the electronic nematicity and spontaneous symmetry breaking instead of the spin degree of freedom, [55], [56] drawing attention for the alternative mechanism of the pairing mediated by the orbital fluctuation.\",\n \"[57], [58] Because the spin and orbital degrees of freedom are coupled each other, [43], [59] however, there can be an inherent ambiguity in determining which order forms first and drives the other.\",\n \"Recently, a series of experiments has revived the attention to the seminal material of the family by the hydrogen (H) doping, namely, LaFeAsO$_{1-x}$ H$_x$ .\",\n \"[60], [61], [62] Overcoming the solubility limit of the conventional fluorine dopant, hydrogen can increase the doping concentration up to $x=0.6$ .\",\n \"Surprisingly, another superconducting and AFM phases adjacent to each other are found at high doping levels, analogously to their conventional counterparts at the low doping level (see Fig. 1(a)).\",\n \"Posing fundamental questions on the nature of magnetism and superconductivity, this finding is expected to give a clue for still unresolved issues mentioned earlier.\",\n \"Although some theoretical attempts have been made to explain the appearance of the second AFM phase mostly focusing on the FS nesting property in the itinerant electron picture,[60], [61], [63] first-principles approach simultaneously incorporating the itinerant and localized aspects of the system is desirable when we consider the 'moderately correlated' nature of these materials [6], [7], [8], [64], [65] and some unsatisfactory conclusions from the itinerant picture such as the prediction of an incorrect magnetic ordering vector.\",\n \"[61] In this paper, we investigate the magnetic and electronic properties of LaFeAsO$_{1-x}$ H$_x$ as a function of $x$ using the combined method of density-functional theory plus dynamical mean-field theory (DFT+DMFT), which captures the material-specific electronic correlation.\",\n \"[66], [67] Considering changes of both electron occupancy and lattice structure caused by the hydrogen doping which turn out to have the opposite effects on the electron correlation and magnetism, we find that both static magnetic moment and local magnetic susceptibility initially decrease to the minimum at around $x=0.3$ and then increase again up to $x=0.6$ , in agreement with the experimental phase diagram of the two separate AFM phases centered at $x=0$ and 0.5.\",\n \"More electron occupation at $d_{xz/yz}$ orbitals with the doping enhances the importance of the $d_{xy}$ orbital in the static magnetic moment and also in spin dynamics, while reducing the orbital polarization.\",\n \"Our results emphasize the importance of the electron correlation and structural modification in understanding the doping induced evolution of the electronic structure, and also the magnetism as an indispensable ingredient for the emergence of the superconductivity in these materials.\"\n ],\n [\n \"Calculation Method\",\n \"We use the modern implementation of DFT+DMFT method within all electron embedded DMFT approach,[67] where in addition to correlated Fe atoms the itinerant states of other species are included in the Dyson self-consistent equation.\",\n \"The strong correlations on the Fe ion are treated by DMFT, adding self-energy $\\\\Sigma (\\\\omega )$ on a quasi atomic orbital in real space, to ensure stationarity of the DFT+DMFT approach.\",\n \"The self-energy $\\\\Sigma (\\\\omega )$ contains all Feynman diagrams local to the Fe ion.\",\n \"No downfolding or other approximations were used, and the calculations are all-electron as implemented in Ref.\",\n \"67, which is based on Wien2k.\",\n \"[68] We used the GGA exchange-correlation functional,[69] and the quantum impurity model was solved by the continuous time quantum Monte Carlo impurity solver [70] using $U=5.0$ eV and $J=0.72$ eV.\",\n \"Brillouin zone integration is done on the 12$\\\\times $ 12$\\\\times $ 6 k-point mesh for the AFM unitcell of LaFeAsO containing 4 Fe atoms.\",\n \"All calculations are done for $T=150$ K. We consider both paramagnetic (PM) and AFM states at this temperature.\",\n \"The AFM state is considered to represent the actual stable phase observed in experiments, while the PM state is also calculated to understand the driving force with which the AFM state is stabilized from a 'bare' state.\"\n ],\n [\n \"Electronic correlation and magnetic strengths as functions of doping\",\n \"Schematic phase diagram of LaFeAsO$_{1-x}$ H$_x$ in the $x-T$ space is depicted in Fig. 1(a).\",\n \"The first AFM phase with a stripe-type order is rapidly suppressed and disappears around $x=0.05$ with the emergence of the first superconducting phase followed by the adjacent second superconducting phase.\",\n \"Further doping initiates the second AFM phase, of the same ordering pattern with the first AFM phase, but with a slightly different atomic displacement.\",\n \"[62] We perform the DFT+DMFT calculations to check if this suppression and reappearance of the AFM phase can be reproduced.\",\n \"To take the electron doping effect into account, we adopt the virtual crystal approximation.\",\n \"[60] In addition, the structural change due to the H doping is incorporated by interpolating both the lattice constants and internal atomic coordinates of the available experimental values at $x=0$ [71] and $x=0.51$ [62] in the PM states with the tetragonal lattice symmetry.\",\n \"This should be a reasonable approximation considering the almost linear As height dependence on $x$ observed experimentally,[60] avoiding the difficulty in the structural optimization of alloy structures within DFT which would require rather complicated statistical treatment, besides concerns about the general reliability of DFT in predicting the accurate lattice structure of iron-based superconducting materials.\",\n \"Using this doping scheme, the static magnetic moment and the local spin susceptibility, $\\\\chi _{local}(\\\\omega =0)=\\\\int _0^\\\\beta \\\\!\",\n \"\\\\langle S(\\\\tau )S(0) \\\\rangle \\\\mathrm {d}\\\\tau $ , are calculated as a function of $x$ for the stripe-type AFM phase and shown in Fig. 1(b).\",\n \"The magnetic moment at $x=0$ is estimated to be 0.66 $\\\\mu _B$ with an agreement with the measured value 0.63 $\\\\mu _B$ ,[72] and decreases to the minimum value of 0.12 $\\\\mu _B$ at $x=0.4$ , and then exhibits a rapid increase to reach 0.68 $\\\\mu _B$ at $x=0.6$ .\",\n \"The local susceptibility shows a similar behavior with a minimum at $x=0.3$ , implying that the overall magnetic strength is suppressed and then re-enhanced with the doping.\",\n \"Therefore, we verify that the DFT+DMFT method captures the essential underlying physics of two separate AFM phases in this material and produces a consistent behavior of the local magnetic strength, while the complete suppression of magnetic phase and emergence of the superconducting phase in the intermediate $x$ as shown in Fig.\",\n \"1(a) could not be properly described within the current calculation scheme.\",\n \"For comparison, we perform the DFT calculation on the relative stability of the AFM phase and the magnetic moment as functions of $x$ .\",\n \"For $x=0$ , the AFM phase is found to be 180 meV/Fe more stable than the nonmagnetic phase with the magnetic moment of 2.15 $\\\\mu _B$ .\",\n \"Upon increasing $x$ , the stability of AFM phase and the magnetic moment show no discernible change, suggesting that the normal DFT calculation cannot properly describe the observed evolution of the magnetism.\",\n \"Then we investigate the underlying mechanism of the doping-induced change of the electronic structure by considering the electron addition and structural modification separately.\",\n \"Because the nominal number of valence electrons in a Fe atom for the undoped material is six, which corresponds to an electron doped system from the half-filled orbitals, further electron doping should result in the monotonic decrease of the correlation strength.\",\n \"[73] On the other hand, the H doping increases the distance between Fe and surrounding As atoms as determined experimentally,[71], [62] which would lead to the localization of Fe $d$ orbitals.\",\n \"[74] To confirm this speculation, we estimate the mass enhancement $1/Z = 1-\\\\frac{\\\\partial \\\\Sigma (\\\\omega )}{\\\\partial \\\\omega }\\\\vert _{\\\\omega =0}$ for the two effects separately.\",\n \"First, we calculate $1/Z$ of the $d_{xy}$ in the PM phase as a function of $x$ considering only the electron addition effect by fixing the lattice structure to that of $x=0$ as in Fig. 1(c).\",\n \"Indeed, $1/Z$ monotonically increases with the electron addition.\",\n \"On the contrary, when only the structural effect is included without extra electron, $1/Z$ monotonically decreases with increasing $x$ .\",\n \"When these two competing effects are combined, $1/Z$ increases overall with the doping, which means that the localization by the structural modification becomes more dominant at the highly doped system.\",\n \"This competing effects are also reflected on the magnetic strength of the AFM phase as shown in Fig. 1(d).\",\n \"When only electron addition effect is considered, the local susceptibility is found to monotonically decrease with increasing $x$ , while it increases monotonically when only the structural modification is taken into account (with the number of extra electrons fixed to 0.6), in agreement with the behavior of the mass enhancement factors in Fig. 1(c).\",\n \"Therefore, we can conclude that the initial suppression and the later re-enhancement of the magnetism with the H doping originates from the two competing effects : the electron addition and increasing Fe-As distance which suppresses and enhances the local correlation and hence the local magnetism, respectively.\",\n \"Our analysis naturally draws attention to the important role of the electron correlation and also the structural effect in understanding the doping induced phase diagram of this material.\",\n \"Suppression of the magnetism and the existence of the quantum criticality in phosphorus-doped Ba122 systems BaFe$_2$ As$_{2-x}$ P$_x$ [75], [76], [77] are another set of examples which demonstrate the dramatic effect of the pure structure modification, where decreased Fe-anion distance was pointed out to cause the suppression of the magnetism.\",\n \"[76]\"\n ],\n [\n \"Fermi surfaces\",\n \"We also investigate the evolution of the FS with the doping which is generally considered to be relevant to the existence of the AFM phase.\",\n \"[61], [63] The FSs for three different doping levels, $x=$ 0, 0.3, and 0.5, are calculated with both the DFT+DMFT and DFT methods and displayed in Fig. 2.\",\n \"Starting with a relatively good nesting between the hole and electrons surfaces at $x=0$ for the DFT case, the doping degrades the nesting with shrinking the hole surfaces at the $\\\\Gamma $ point and enlarging the electron surfaces at the $M$ point (see Fig.\",\n \"2(g)-(i)), as the electron doping raises the Fermi level.\",\n \"Our result shows a good agreement with the previous DFT calculation using the experimentally determined lattice structure,[60] confirming that our assumption of the linear dependence of the lattice constants and internal atomic coordinates is reasonable.\",\n \"The DFT+DMFT results shown in Figs.\",\n \"2(a)-(f) are qualitatively similar, but the $d_{xy}$ hole FS expands compared with the DFT results, because of more correlated nature of the $d_{xy}$ orbital than $d_{xz/yz}$ orbitals as pointed out in the DFT+DMFT study of LiFeAs.\",\n \"[78] The decrease of the overall spectral weight with the doping and relatively larger incoherence of the $d_{xy}$ surface reflect the larger correlation at high doping levels and for the $d_{xy}$ orbital (Figs. 2(a)-(c)).\",\n \"Nevertheless, both levels of the theory indicate the monotonic degradation of the FS nesting with the doping, as already indicated by previous calculations,[60], [63] manifesting that the FS nesting alone cannot explain the appearance of the second AFM phase.\",\n \"Again, we conclude that the electron correlation and many-body effects should be incorporated to understand the doping-induced evolution of the magnetism.\"\n ],\n [\n \"Spin resolved spectral function\",\n \"To understand the doping-induced suppression and the reemergence of the magnetism in detail, we investigate the spin-resolved spectral function of the $d_{xy}$ orbital in the AFM phase as a function of $x$ as shown in Fig. 3.\",\n \"At $x=0$ , the majority and minority spin states exhibit a large exchange splitting reflecting the overall magnetic moment of 0.66 $\\\\mu _B$ , with a distinct pseudo-gap feature (a dip in spectral function) at the Fermi energy induced by the coupling between the electron and hole bands at the Fermi energy.\",\n \"[31] With increasing doping level up to $x=0.3$ (see Fig.\",\n \"3(a)), spectral weights moves from the peak just above the Fermi level to one below the Fermi energy in the minority spin channel, suggesting the doped electrons fills the minority spin states.\",\n \"The electron filling in the minority spin states with the doping naturally leads to the gradual reduction of the exchange splitting and the magnetic moment, along with the size of the pseudo-gap.\",\n \"On the other hand, further doping over $x=0.3$ enhances the exchange splitting as shown in Fig.\",\n \"3(b), which seems to almost retrace the evolution of the spectral function from $x=0$ to $x=0.3$ in Fig. 3(a).\",\n \"However, there are several noticeable differences as well.\",\n \"First, the pseudo-gap position is constantly shifting deeper in the valence states with its size and the magnetic moment increasing with increasing $x$ , which indicates the rise of the Fermi energy as a result of the electron doping.\",\n \"More importantly, doping over $x=0.3$ develops a shoulder growing with $x$ near -1 eV in the majority spin channel as indicated by a arrow in Fig.\",\n \"3(b), contributing to build up the magnetic moment against the electron filling on the minority spin states with the doping.\",\n \"The spectral weight piled up in this position results from many-body effects and hence is incoherent, rather than from the shift of coherent quasi-particle states.\",\n \"The inset of Fig.\",\n \"3(b) depicts the imaginary part of the $d_{xy}$ component of the electron self energy (Im$\\\\Sigma $ ) along with the spectral function for $x=0.6$ .\",\n \"One can identify a strong peak of the Im$\\\\Sigma $ around -0.7 eV, close to the $J$ value 0.72 eV adopted in this study, and the shoulder structure of the spectral function at a nearby position, suggesting that the shoulder structure originates from the incoherent excitations related to the self energy.\",\n \"Similar energy scales between this incoherent excitation and $J$ is also consistent with the suggestion that iron-based superconductors are Hund's metals where $J$ plays more important role than $U$ .\",\n \"[7]\"\n ],\n [\n \"Orbital polarization\",\n \"As mentioned earlier, the orbital order is of great interest for these materials regarding the electronic nematicity and also superconductivity itself.\",\n \"Here we compare the orbital polarization, i.e., the imbalance between $d_{xz/yz}$ orbitals, in the AFM state for low and high doping cases.\",\n \"For $x=0$ as displayed in Fig.\",\n \"4(a), $d_{xz}$ and $d_{yz}$ spectral functions show noticeable difference, where the spin polarization is larger for $d_{yz}$ as well as $d_{xy}$ orbitals while $d_{xz}$ spin splitting is smaller.\",\n \"On the other hand, for the high doping case of $x=0.5$ in Fig.\",\n \"4(b), $d_{xz/yz}$ components of the spectral function becomes much more similar with each other and now the $d_{xy}$ orbital has the most significant spin polarization.\",\n \"Indeed, our estimated orbital polarization $(n_{xz}-n_{yz})/(n_{xz}+n_{yz})$ decreases from 3.9 $\\\\%$ at $x=0$ to 1.5 $\\\\%$ at $x=0.5$ while the magnetic moments for the two cases are comparable.\",\n \"Increasing $x$ enhances the crystal field splitting pushing up the $d_{xy}$ level above $d_{xz/yz}$ level, so that the doped electrons fill $d_{xz/yz}$ orbitals first rather than the $d_{xy}$ orbital, reducing the imbalance between $d_{xz/yz}$ orbitals as well as between their spin components.\",\n \"Meanwhile, besides the less electron filling, the elevated As height in the high doping case further enhances the electron correlation for the $d_{xy}$ orbital via the `kinetic frustration' [79] compared with the $d_{xz/yz}$ orbitals.\",\n \"So $d_{xy}$ becomes the most significant component for the local spin fluctuations in the PM phase and for static magnetic moments in the AFM phase.\",\n \"Therefore, at high doping levels, strong local magnetism appears mainly from $d_{xy}$ orbital and the orbital polarization from $d_{xz/yz}$ is largely suppressed.\"\n ],\n [\n \"Spin excitation spectrum\",\n \"So far, we have considered the doping induced evolution of the magnetically ordered state, and we will conclude our discussion by investigating the dynamic spin fluctuations in the PM state, which is more relevant to the superconductivity, as a function of the doping.\",\n \"Although $T=150$ K at which the calculation is done is close to the AFM transition temperature, the spin susceptibility in the PM state is expected to be a smooth varying function of $T$ (except at the AFM ordering wave vector for which the susceptibility diverges at the AFM transition temperature), so we expect qualitatively similar results for other temperatures.\",\n \"We evaluate the dynamical spin structure factor $S({\\\\bf q},\\\\omega )=\\\\frac{\\\\chi ^{\\\\prime \\\\prime }({\\\\bf q},\\\\omega )}{1-e^{\\\\hbar \\\\omega /k_BT}}$ using the DFT+DMFT method as displayed in Fig.\",\n \"5, where both the one-particle Green's function and the local two-particle vertex function are determined ab-initio.\",\n \"[80] For $x=0$ , the spin excitation spectrum has strong peaks near the zero energy around the wave vector ${\\\\bf q}=(\\\\pi ,0)$ , which corresponds to the magnetic ordering vector of the AFM phase, and disperses over the path shown in the spectrum, reaching a maximum energy at the zone boundary ${\\\\bf q}=(\\\\pi ,\\\\pi )$ , all consistent with previous results.\",\n \"[80], [81], [41] As the doping level $x$ increases, the overall spin excitation spectral weights tend to shift to lower energies as the spin wave dispersion decreases with the increasing correlation strength.\",\n \"The excitation near ${\\\\bf q}=(\\\\pi /2,\\\\pi /2)$ noticeably goes down towards the zero energy with the doping, and a new possible static magnetic order for this wave vector is suggested for $x=0.5$ .\",\n \"However, the intensity of excitations has always the maximum at the conventional AFM ordering vector ${\\\\bf q}=(\\\\pi ,0)$ for all the doping cases, consistent with the experimentally found second AFM phase for the high doping levels.\",\n \"[62] For $S({\\\\bf q},\\\\omega )$ at ${\\\\bf q}=(\\\\pi +\\\\delta ,0)$ which is slightly off the magnetic ordering vector, as shown in Fig.\",\n \"5(d), the peak height near the zero energy is reduced for $x=0.3$ compared with that for $x=0$ indicating the suppressed tendency towards the static magnetic order, and then it becomes pronounced again for the higher doping level $x=0.5$ suggesting the re-enhanced magnetism, which shows a qualitative agreement with the initial decrease and re-enhancement of the calculated magnetic moment in the AFM phase shown in Fig.\",\n \"1 and again also with the motivating experiments.\",\n \"[61], [62] When decomposed by orbitals (see Figs.\",\n \"5(e)-(g)), the $d_{xz/yz}$ components show a large anisotropy with the $d_{yz}$ component peak being dominant at $x=0$ .\",\n \"As the doping level increases, the $d_{xz/yz}$ anisotropy keeps decreasing while the $d_{xy}$ component becomes most prominent.\",\n \"The decreasing $d_{xz/yz}$ anisotropy and the enhancement of the $d_{xy}$ component with the increasing doping is consistent with the features observed in our result for the AFM phase.\"\n ],\n [\n \"Discussion\",\n \"Our results remind us of the indispensable role of the electron correlation in the iron-based superconducting materials, as well as the impact of the structural change.\",\n \"The doping-induced evolution of the electronic and magnetic properties cannot be understood by simply adopting the rigid shift of the Fermi level or even the self-consistent addition of carriers without taking the structural effect into account.\",\n \"In addition, a natural view on the doping-induced evolution of spin and orbital orders can be obtained.\",\n \"The `ferro-orbital order', which is coupled to the AFM spin order in the undoped materials, is the lowest-energy configuration for the nominally half-filled orbitals to maximize the kinetic energy gain.\",\n \"[43] Both the orbital and spin orders of $d_{xz}$ or $d_{yz}$ are suppressed when the doping supplies more electrons to these orbitals away from the half-filling.\",\n \"The $d_{xy}$ orbital, for which the spin order can form but the orbital order is no longer relevant, becomes the dominant channel for the electron hopping to reduce the kinetic energy as discussed above.\",\n \"As a result, spin order/fluctuation is present near the both superconducting domes found in LaFeAsO$_{1-x}$ H$_x$ while orbital order/fluctuation is expected to be strong only near the first superconducting phase in the lower doping level.\",\n \"Our results consequently suggests that the spin fluctuation is more closely related to the superconductivity, at least for the second superconducting phase in this alloy, while the orbital fluctuation, which is significant only at low doping levels, might not be a prerequisite for the superconductivity in general.\",\n \"The enhanced role of the $d_{xy}$ orbital in magnetism is expected to naturally lead to its dominant role also in the superconductivity with a larger FS hole pocket of this orbital as shown in Fig. 2.\",\n \"Although the enhanced electron correlation and consequent stronger spin fluctuation in the higher doping level might contribute to the strong superconductivity, too strong correlation would be harmful to the superconductivity, out of several reasons,[79] due to the lowered magnon energy scale (Fig.\",\n \"5) which is directly coupled to the size of the superconducting gap.\",\n \"Further theoretical study which directly attacks the superconductivity as a function of the doping level will be desirable.\"\n ],\n [\n \"Conclusion\",\n \"In summary, by adopting the DFT+DMFT method, where the local dynamic correlation effect is taken exactly, we successfully reproduce the hydrogen-doping-induced suppression and revitalization of the magnetism in LaFeAsO$_{1-x}$ H$_x$ which has been recently established experimentally.\",\n \"Taking the structural modification by the doping into account along with the carrier addition is found to be essential, as the two factors induce independent and opposite effects on the electron correlation strength and the magnetism in this alloy.\",\n \"Doping reduces the orbital imbalance between $d_{xz/yz}$ orbitals as well as their magnetic activity, while the $d_{xy}$ orbital becomes the dominant electron hopping channel with increased electron correlation and the magnetic strength for high doping levels.\",\n \"Indispensable role of the electron correlation and detailed atomic structure is identified in understanding the electronic and magnetic properties, and the magnetism possibly as more fundamental ingredient in realizing the superconductivity is suggested over the orbital degrees of freedom.\",\n \"This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2016R1C1B1014715 and 2015R1D1A1A01059621).\"\n ]\n]"},"id":{"kind":"string","value":"1612.05520"}}},{"rowIdx":3389,"cells":{"text":{"kind":"list like","value":[["The relation between globular cluster systems and supermassive black\n holes in spiral galaxies. The case study of NGC 4258"],["Abstract We aim to explore the relationship between globular cluster total number, $N_{\\rm GC}$, and central black hole mass, $M_\\bullet$, in spiral galaxies, and compare it with that recently reported for ellipticals.","We present results for the Sbc galaxy NGC 4258, from Canada France Hawaii Telescope data.","Thanks to water masers with Keplerian rotation in a circumnuclear disk, NGC 4258 has the most precisely measured extragalactic distance and supermassive black hole mass to date.","The globular cluster (GC) candidate selection is based on the ($u^*\\ -\\ i^\\prime$) vs. ($i^\\prime\\ -\\ K_s$) diagram, which is a superb tool to distinguish GCs from foreground stars, background galaxies, and young stellar clusters, and hence can provide the best number counts of GCs from photometry alone, virtually free of contamination, even if the galaxy is not completely edge-on.","The mean optical and optical-near infrared colors of the clusters are consistent with those of the Milky Way and M 31, after extinction is taken into account.","We directly identify 39 GC candidates; after completeness correction, GC luminosity function extrapolation and correction for spatial coverage, we calculate a total $N_{\\rm GC} = 144\\pm31^{+38}_{-36}$ (random and systematic uncertainties, respectively).","We have thus increased to 6 the sample of spiral galaxies with measurements of both $M_\\bullet$ and $N_{\\rm GC}$.","NGC 4258 has a specific frequency $S_{\\rm N} = 0.4\\pm0.1$ (random uncertainty), and is consistent within 2$\\sigma$ with the $N_{\\rm GC}$ vs. $M_\\bullet$ correlation followed by elliptical galaxies.","The Milky Way continues to be the only spiral that deviates significantly from the relation."],["[ht] -Band Shape Parameters of Globular Cluster Candidates Table: NO_CAPTION Table: NO_CAPTION Note—Table REF is published in its entirety in the machine-readable format.","A portion is shown here for guidance regarding its form and content."]],"string":"[\n [\n \"The relation between globular cluster systems and supermassive black\\n holes in spiral galaxies. The case study of NGC 4258\"\n ],\n [\n \"Abstract We aim to explore the relationship between globular cluster total number, $N_{\\\\rm GC}$, and central black hole mass, $M_\\\\bullet$, in spiral galaxies, and compare it with that recently reported for ellipticals.\",\n \"We present results for the Sbc galaxy NGC 4258, from Canada France Hawaii Telescope data.\",\n \"Thanks to water masers with Keplerian rotation in a circumnuclear disk, NGC 4258 has the most precisely measured extragalactic distance and supermassive black hole mass to date.\",\n \"The globular cluster (GC) candidate selection is based on the ($u^*\\\\ -\\\\ i^\\\\prime$) vs. ($i^\\\\prime\\\\ -\\\\ K_s$) diagram, which is a superb tool to distinguish GCs from foreground stars, background galaxies, and young stellar clusters, and hence can provide the best number counts of GCs from photometry alone, virtually free of contamination, even if the galaxy is not completely edge-on.\",\n \"The mean optical and optical-near infrared colors of the clusters are consistent with those of the Milky Way and M 31, after extinction is taken into account.\",\n \"We directly identify 39 GC candidates; after completeness correction, GC luminosity function extrapolation and correction for spatial coverage, we calculate a total $N_{\\\\rm GC} = 144\\\\pm31^{+38}_{-36}$ (random and systematic uncertainties, respectively).\",\n \"We have thus increased to 6 the sample of spiral galaxies with measurements of both $M_\\\\bullet$ and $N_{\\\\rm GC}$.\",\n \"NGC 4258 has a specific frequency $S_{\\\\rm N} = 0.4\\\\pm0.1$ (random uncertainty), and is consistent within 2$\\\\sigma$ with the $N_{\\\\rm GC}$ vs. $M_\\\\bullet$ correlation followed by elliptical galaxies.\",\n \"The Milky Way continues to be the only spiral that deviates significantly from the relation.\"\n ],\n [\n \"[ht] -Band Shape Parameters of Globular Cluster Candidates Table: NO_CAPTION Table: NO_CAPTION Note—Table REF is published in its entirety in the machine-readable format.\",\n \"A portion is shown here for guidance regarding its form and content.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05655"}}},{"rowIdx":3390,"cells":{"text":{"kind":"list like","value":[["Tuning the hopping parameter in the Oktay-Kronfeld action for charm and\n bottom quarks on a MILC HISQ ensemble"],["Abstract The first step in the calculation of semi-leptonic form factors in the decay of heavy mesons is the tuning of the hopping parameter $\\kappa$ for the charm and bottom quark masses.","Results for the Oktay-Kronfeld (OK) action are presented for one $N_f=2+1+1$ HISQ ensemble generated by the MILC collaboration at $a\\approx 0.12\\,\\mathrm{fm}$ and $M_\\pi\\approx 310$ MeV.","Estimates of hyperfine splitting of heavy-light and heavy-heavy mesons are presented and the inconsistency parameter is evaluated."],["Introduction","There are two independent methods to extract $V_{cb}$ with $B$ -meson decays.","One is the heavy quark expansion method based on QCD sum rules with inclusive $B$ -meson decays: $\\bar{B}\\rightarrow X_{c}\\ell \\bar{\\nu }$ , and the other is the lattice QCD method to calculate the semileptonic form factors in the analysis of the exclusive $B$ -meson decays: $\\bar{B} \\rightarrow D^{(\\ast )} \\ell \\bar{\\nu }$ .","There exists about $3\\sigma $ tension in $|V_{cb}|$ between the inclusive and exclusive decay channels [1], [2].","The future experiment at KEK (Belle 2) will increase statistics for $B$ -meson decays dramatically (by a factor of 50).","It is time to improve the lattice results of semileptonic form factors for the exclusive $B$ -meson decays.","Since the dominant error in lattice QCD results for $|V_{cb}|$ comes from the heavy quark discretization, we simulate the Oktay-Kronfeld (OK) action [3], a highly improved version of the Fermilab formulation.","If we use the OK action instead of the clover action (the original action of the Fermilab formulation [4]), then the power counting estimate suggests that the discretization error due to charm quarks can be reduced from 1.0% (clover) down to 0.2% (OK) for the semileptonic form factor for the $\\bar{B} \\rightarrow D^* \\ell \\bar{\\nu }$ decay at zero recoil.","The OK action is improved to $\\mathcal {O}(\\lambda ^3)$ in HQET power counting, and $\\mathcal {O}(v^6)$ in NRQCD power counting, while the clover action is improved to $\\mathcal {O}(\\lambda ^2)$ in HQET and to $\\mathcal {O}(v^4)$ in NRQCD.","One drawback is that the OK action takes significantly more computing resources (by a factor of $\\approx 50$ ) to calculate its propagator.","We measured heavy-light (HL) and heavy-heavy (HH) meson spectra to probe the improvement by the OK action, and the inconsistency parameter and hyperfine splitting showed clear improvement [5].","In this paper, we tune hopping parameters using the physical $B_s$ and $D_s$ meson spectrum on the coarse MILC HISQ ensemble at $a \\approx 0.12 \\,\\mathrm {fm}$ ."],["Simulation Details","We use the coarse ($a\\approx 0.12$ fm) ensemble of the MILC HISQ lattices [6].","The lattice geometry is $24^3\\times 64$ .","The tadpole improvement coefficient is $u_0=0.86372$ from the plaquette Wilson loop.","The sea quark masses are $am_\\ell =0.0102$ for light quarks, $am_s=0.0509$ for the strange quark, and $am_c=0.635$ for the charm quark.","For the HL mesons, $B_s^{(\\ast )}$ and $D_s^{(\\ast )}$ , we use the HISQ action for the strange quark, and the OK action for charm and bottom quarks.","Heavy quark propagators are generated using an optimized BiCGStab inverter [7].","In the OK action, the tadpole improved bare quark mass $m_0$ is related to the hopping parameter $\\kappa $ as follows, $am_0 = \\frac{1}{2u_0\\kappa } -(1+3\\zeta r_s + 18c_4)$ where $c_4$ is the tree-level matching coefficient of a dimension-7 operator in the OK action [3].","Here, we set $\\zeta =1$ for isotropic lattices and $r_s=1$ as is standard for the Wilson clover action.","To tune the hopping parameter $\\kappa $ to the physical values, we simulate four $\\kappa $ values each for the bottom and the charm quarks as shown in Table REF .","The parameters of covariant Gaussian smearing used at both the source and sink of heavy quark propagators to reduce the excited state contamination [8] are given in Table REF .","For HISQ valence quarks, we use point source and sink.","Table: Parameters used for generating the valence quark propagators.","table:valenceHISQm s v m^v_s is set to the physical strange quark mass, and ϵ\\epsilon is the coefficient of the Naik term in the HISQ action .table:valenceOK κ\\kappa values for the bottom and charmquarks.","The covariant Gaussian smearing parameters σ\\sigma and N GS N_{GS} are defined in Ref.",".We calculate HL and HH meson correlators on 500 configurations using 6 sources for bottom quarks and 3 sources for charm quarks.","We use jackknife resampling to estimate the statistical error.","We fix the time separation between sources to $\\Delta t = 6$ .","We choose the initial source time slice randomly for each configuration.","We use 11 different momentum projections for the two-point meson correlation functions.","To increase the statistics, we use the time reflection symmetry of the two-point correlation functions."],["Fits to the Meson Correlators and the Dispersion Relation","The numerical data for the two-point meson correlators is fit using $f^\\text{HL}(t; \\mathbf {p}) &= Ae^{-Et} \\big ( 1-(-1)^t re^{-\\Delta Et} \\big ) + Ae^{-E(T-t)} \\big (1-(-1)^t re^{-\\Delta E(T-t)}\\big )\\\\f^\\text{HH}(t; \\mathbf {p}) &= Ae^{-Et}+Ae^{-E(T-t)}.$ The HL meson correlator, $f^\\text{HL}$ , has 4 fit parameters: the ground state energy and amplitude ($E$ , $A$ ), an amplitude ratio ($r=A^p/A$ ), and energy difference ($\\Delta E=E^p-E$ ), where the superscript $p$ stands for the opposite parity partner state that is present in staggered fermion correlation functions.","$f^\\text{HH}$ is the function used to fit the HH mesons.","The range $12 \\le t \\le 19$ is used to fit the HL mesons and $12 \\le t \\le 16$ for the HH mesons.","The ground state energy $E(\\mathbf {p})$ is then fit using the following dispersion relation: $E(\\mathbf {p})=M_1 + \\frac{\\mathbf {p}^2}{2M_2} - \\frac{(\\mathbf {p}^2)^2}{8M_4^3} - \\frac{a^3W_4}{6}\\sum _{i=1}^3 p_i^4 \\,,$ to obtain $M_1$ the rest mass, $M_2$ the kinetic mass, $M_4$ the quartic mass, and $W_4$ the Lorentz symmetry breaking term.","In both fits we use the full covariance matrix with trivial priors."],["Kappa Tuning","We determine the hopping parameters $\\kappa _b$ and $\\kappa _c$ such that the kinetic masses are equal to the physical $B_s$ and $D_s$ masses, respectively.","We tune the kinetic mass $M_2$ rather than the rest mass $M_1$ .","The form factors and decay constants which we are interested in are independent of the rest mass $M_1$ in the Fermilab interpretation of improved Wilson fermions.","We use the HQET inspired fitting function for kinetic HL meson masses, $aM_2(\\kappa )=d_0+am_2(\\kappa )+\\frac{d_1}{am_2(\\kappa )}+\\frac{d_2}{(am_2 (\\kappa ))^2} \\,,$ where $M_2$ is the kinetic mass of the HL meson, and $m_2$ is the kinetic mass of the heavy quark.","We determine $d_0$ , $d_1$ and $d_2$ using the correlated least $\\chi ^2$ fitting.","Here, $m_2(\\kappa )$ is related to the bare mass $m_0(\\kappa )$ at the tree level as follows, $\\frac{1}{ am_2} &= \\frac{2 \\zeta ^2}{ am_0 (2+ am_0)} + \\frac{ r_s\\zeta }{1+ am_0 }.$ Figure REF shows the interpolation of $m_2$ to the physical values for the bottom and charm quarks.","Results for $\\kappa _b$ and $\\kappa _c$ are summarized in Table REF .","In Table REF , we present the $\\kappa $ -tuning results using physical values of the pseudoscalar meson mass ($M_X$ ), vector meson mass ($M_{X^\\ast }$ ), and the spin-averaged mass $(M_X+3M_{X^\\ast })/4$ for $X=B_s$ or $D_s$ .","We find that all the results for $\\kappa $ determined from different spin states are consistent within statistical uncertainty.","We also perform another fit using a simpler fitting function: $aM_2=d_0+am_2+d_1/(am_2)$ , and take the difference in $\\kappa $ as the systematic error due to ambiguity in the fitting function.","Figure: Plot of the pseudoscalar meson mass D s D_s (B s B_s) versusthe kinetic quark mass m 2 c m_2^c (m 2 b m_2^b) defined inEq. ().","The physical κ c \\kappa _c andκ b \\kappa _b (shown as red triangles and given in Table) are determined by tuning the D s D_s and B s B_s pseudoscalar massesto their experimental values.Table: Results of tuning the κ\\kappa for the bottom (κ b \\kappa _b)and charm (κ c \\kappa _c) quarks.","For converting the experimental MMto aMaM, we use a=0.12520(22) fm a=0.12520(22)\\,\\mathrm {fm} .","Inκ b,c \\kappa _{b,c}, the first error is statistical, the second erroris propagation of experimental error in M X M^X, and the third erroris systematic to account for the uncertainty in the fit ansatz."],["Inconsistency Parameter","The inconsistency parameter $I$ [10], [11] is used to see $O(\\mathbf {p}^4)$ improvement in the OK action.","Let us use $Q$ for heavy quarks and $q$ for light quarks, and define $\\delta M \\equiv M_{2}-M_{1}$ as the difference between the kinetic and rest masses.","Then the inconsistency parameter $I$ is $I\\equiv \\frac{2\\delta M_{\\bar{Q} q} -(\\delta M_{\\bar{Q} Q}+\\delta M_{\\bar{q} q})}{2M_{2\\bar{Q} q}}= \\frac{2\\delta B_{\\bar{Q} q} -(\\delta B_{\\bar{Q} Q}+\\delta B_{\\bar{q} q})}{2M_{2\\bar{Q} q}}$ Here the binding energies $B_{1,2}$ are $M_{1\\bar{Q} q}= m_{1\\bar{Q} } + m_{1 q} + B_{1\\bar{Q} q }, \\qquad M_{2\\bar{Q} q}= m_{2\\bar{Q} } + m_{2 q} + B_{2\\bar{Q} q }$ for HL mesons.","Here the quark masses $m_{1,2}$ are defined by the quark dispersion relation, which is similar to Eq.","(REF ).","We neglect the light quarkonium contribution $\\delta M_{\\bar{q} q}$ (and $\\delta B_{\\bar{q} q}$ ).","In Fig.","REF we present results for $I$ for pseudoscalar mesons.","Near the $B_s$ region, $I$ is consistent with the continuum limit, $I=0$ , within the error bars, which indicates a dramatic improvement from that of the Fermilab action: $I\\approx -0.6$ [5].","Figure: (Left) Inconsistency parameter II versus the pseudoscalarmass aM 2Q ¯q a M_{2\\bar{Q}q} for charm (green squares) and bottom (bluecircles) quarks.","(Right) Dispersion relation for the HL bottommeson with κ=0.040\\kappa =0.040.","The rest mass, M 1 =2.112(2)M_1=2.112(2), is givenby the blue circle at 𝐩=0\\mathbf {p}=0.","The kinetic mass,M 2 =3.408(176)M_2=3.408(176), is extracted from the fit and shown by the redtriangle translated to E=2.112E= 2.112.","Note that the determinationof the error in M 2 M_2 is much larger than in M 1 M_1."],["Hyperfine Splittings","We define the hyperfine splittings of HL and HH pseudoscalar mesons, $\\Delta _1$ and $\\Delta _2$ as $\\Delta _1=M_1^\\ast -M_1,\\qquad \\Delta _2=M_2^\\ast -M_2 \\,,$ and plot $\\Delta _2$ versus $\\Delta _1$ in Fig.","REF .","As illustrated in Fig.","REF , $M_2$ has much larger errors than $M_1$ since it is extracted from the slope versus momentum.","Consequently, $\\Delta _2$ has larger errors than $\\Delta _1$ .","Figure: Hyperfine splitting for HH mesonsThe HQET expansion for $\\Delta _1$ in the HL meson system is given in Ref.","[12]: $\\Delta _1 = M_1^{\\ast } - M_1 &= \\frac{4\\lambda _2}{2m_B} -\\frac{4\\rho _2}{4m_E^2}+ \\frac{8T_2}{2m_2 2m_B} + \\frac{4(T_4-T_2)}{4m_B^2} +\\mathcal {O}\\left(\\frac{1}{m^3}\\right),$ where $\\lambda _2$ , $\\rho _2$ , $T_2$ , $T_4$ are HQET matrix elements defined in Ref. [12].","For the OK action, the matching conditions are $m_2=m_B=m_E$ [3].","Thus, $\\Delta _1$ defined in Eq.","(REF ) is in terms of the kinetic quark mass, which was used to tune the $\\kappa $ to the physical value.","To analyze $\\Delta _1$ , we recast Eq.","REF as $a\\Delta _{1} = h_0 + \\frac{h_1}{am_2} + \\frac{h_2}{(am_2)^2}+ \\frac{h_3}{(am_2)^3} \\,,$ where $h_1 = 2 a^2 \\lambda _2$ and $h_2 = a^3 (-\\rho _2 + T_2 +T_4)$ .","Because we have only 4 data points, we set $h_3=0$ in the fits.","Correlated fits, shown in Fig.","REF , give $h_0 = 0$ within statistical uncertainty, consistent with the theoretical prediction.","Our results, with $h_0$ set to zero in the fits are summarized in Table REF .","The corresponding $h_i$ from fits to $\\Delta _2$ were very poorly determined.","We are performing simulations at other values of the lattice spacing and quark mass in order to perform the continuum-chiral extrapolation and compare with the experimental value.","Table: Hyperfine splittings, λ 2 \\lambda _2 and A≡-ρ 2 +T 2 +T 4 A \\equiv -\\rho _2+T_2 +T_4 for the D s D_s and B s B_s mesons at the physical valuesof κ c \\kappa _c and κ b \\kappa _b.","Δ exp \\Delta _\\text{exp} is theexperimental value ."],["Summary and Plan","We tuned the bottom and charm quark masses using physical values for $B^{(*)}_s$ and $D^{(*)}_s$ mesons.","Estimates from fits to the pseudoscalar, vector, and spin-averaged mesons masses are consistent within their statistical uncertainty (see Table REF ).","We used estimates from the pseudoscalar mesons for the analysis of the hyperfine splittings.","These values of $\\kappa _b$ and $\\kappa _c$ are now being used to measure the semileptonic form factors for the exclusive decays $\\bar{B}\\rightarrow D^{(*)} \\ell \\bar{\\nu }$ .","Figure: Plots of Δ 1 (D s )\\Delta _1(D_s) (left panel) andΔ 1 (B s )\\Delta _1(B_s) (right panel) versus the kinetic masses, m 2 c m_2^cand m 2 b m_2^b, of the quarks.","Results at the physical quark masses,tuned using the pseudoscalar meson masses, are shown by the red trianglesand given in Table .We thank Jon A. Bailey for helpful comments and suggestions.","The research of W. Lee is supported by the Creative Research Initiatives Program (No.","20160004939) of the NRF grant funded by the Korean government (MEST).","W. Lee would like to acknowledge the support from the KISTI supercomputing center through the strategic support program for the supercomputing application research (No. KSC-2014-G2-002).","Computations were carried out in part on the DAVID GPU clusters at Seoul National University.","The research of T. Bhattacharya, R. Gupta and Y-C. Jang is supported by the U.S. Department of Energy, Office of Science of High Energy Physics under contract number DE-KA-1401020, the LANL LDRD program and Institutional Computing."]],"string":"[\n [\n \"Tuning the hopping parameter in the Oktay-Kronfeld action for charm and\\n bottom quarks on a MILC HISQ ensemble\"\n ],\n [\n \"Abstract The first step in the calculation of semi-leptonic form factors in the decay of heavy mesons is the tuning of the hopping parameter $\\\\kappa$ for the charm and bottom quark masses.\",\n \"Results for the Oktay-Kronfeld (OK) action are presented for one $N_f=2+1+1$ HISQ ensemble generated by the MILC collaboration at $a\\\\approx 0.12\\\\,\\\\mathrm{fm}$ and $M_\\\\pi\\\\approx 310$ MeV.\",\n \"Estimates of hyperfine splitting of heavy-light and heavy-heavy mesons are presented and the inconsistency parameter is evaluated.\"\n ],\n [\n \"Introduction\",\n \"There are two independent methods to extract $V_{cb}$ with $B$ -meson decays.\",\n \"One is the heavy quark expansion method based on QCD sum rules with inclusive $B$ -meson decays: $\\\\bar{B}\\\\rightarrow X_{c}\\\\ell \\\\bar{\\\\nu }$ , and the other is the lattice QCD method to calculate the semileptonic form factors in the analysis of the exclusive $B$ -meson decays: $\\\\bar{B} \\\\rightarrow D^{(\\\\ast )} \\\\ell \\\\bar{\\\\nu }$ .\",\n \"There exists about $3\\\\sigma $ tension in $|V_{cb}|$ between the inclusive and exclusive decay channels [1], [2].\",\n \"The future experiment at KEK (Belle 2) will increase statistics for $B$ -meson decays dramatically (by a factor of 50).\",\n \"It is time to improve the lattice results of semileptonic form factors for the exclusive $B$ -meson decays.\",\n \"Since the dominant error in lattice QCD results for $|V_{cb}|$ comes from the heavy quark discretization, we simulate the Oktay-Kronfeld (OK) action [3], a highly improved version of the Fermilab formulation.\",\n \"If we use the OK action instead of the clover action (the original action of the Fermilab formulation [4]), then the power counting estimate suggests that the discretization error due to charm quarks can be reduced from 1.0% (clover) down to 0.2% (OK) for the semileptonic form factor for the $\\\\bar{B} \\\\rightarrow D^* \\\\ell \\\\bar{\\\\nu }$ decay at zero recoil.\",\n \"The OK action is improved to $\\\\mathcal {O}(\\\\lambda ^3)$ in HQET power counting, and $\\\\mathcal {O}(v^6)$ in NRQCD power counting, while the clover action is improved to $\\\\mathcal {O}(\\\\lambda ^2)$ in HQET and to $\\\\mathcal {O}(v^4)$ in NRQCD.\",\n \"One drawback is that the OK action takes significantly more computing resources (by a factor of $\\\\approx 50$ ) to calculate its propagator.\",\n \"We measured heavy-light (HL) and heavy-heavy (HH) meson spectra to probe the improvement by the OK action, and the inconsistency parameter and hyperfine splitting showed clear improvement [5].\",\n \"In this paper, we tune hopping parameters using the physical $B_s$ and $D_s$ meson spectrum on the coarse MILC HISQ ensemble at $a \\\\approx 0.12 \\\\,\\\\mathrm {fm}$ .\"\n ],\n [\n \"Simulation Details\",\n \"We use the coarse ($a\\\\approx 0.12$ fm) ensemble of the MILC HISQ lattices [6].\",\n \"The lattice geometry is $24^3\\\\times 64$ .\",\n \"The tadpole improvement coefficient is $u_0=0.86372$ from the plaquette Wilson loop.\",\n \"The sea quark masses are $am_\\\\ell =0.0102$ for light quarks, $am_s=0.0509$ for the strange quark, and $am_c=0.635$ for the charm quark.\",\n \"For the HL mesons, $B_s^{(\\\\ast )}$ and $D_s^{(\\\\ast )}$ , we use the HISQ action for the strange quark, and the OK action for charm and bottom quarks.\",\n \"Heavy quark propagators are generated using an optimized BiCGStab inverter [7].\",\n \"In the OK action, the tadpole improved bare quark mass $m_0$ is related to the hopping parameter $\\\\kappa $ as follows, $am_0 = \\\\frac{1}{2u_0\\\\kappa } -(1+3\\\\zeta r_s + 18c_4)$ where $c_4$ is the tree-level matching coefficient of a dimension-7 operator in the OK action [3].\",\n \"Here, we set $\\\\zeta =1$ for isotropic lattices and $r_s=1$ as is standard for the Wilson clover action.\",\n \"To tune the hopping parameter $\\\\kappa $ to the physical values, we simulate four $\\\\kappa $ values each for the bottom and the charm quarks as shown in Table REF .\",\n \"The parameters of covariant Gaussian smearing used at both the source and sink of heavy quark propagators to reduce the excited state contamination [8] are given in Table REF .\",\n \"For HISQ valence quarks, we use point source and sink.\",\n \"Table: Parameters used for generating the valence quark propagators.\",\n \"table:valenceHISQm s v m^v_s is set to the physical strange quark mass, and ϵ\\\\epsilon is the coefficient of the Naik term in the HISQ action .table:valenceOK κ\\\\kappa values for the bottom and charmquarks.\",\n \"The covariant Gaussian smearing parameters σ\\\\sigma and N GS N_{GS} are defined in Ref.\",\n \".We calculate HL and HH meson correlators on 500 configurations using 6 sources for bottom quarks and 3 sources for charm quarks.\",\n \"We use jackknife resampling to estimate the statistical error.\",\n \"We fix the time separation between sources to $\\\\Delta t = 6$ .\",\n \"We choose the initial source time slice randomly for each configuration.\",\n \"We use 11 different momentum projections for the two-point meson correlation functions.\",\n \"To increase the statistics, we use the time reflection symmetry of the two-point correlation functions.\"\n ],\n [\n \"Fits to the Meson Correlators and the Dispersion Relation\",\n \"The numerical data for the two-point meson correlators is fit using $f^\\\\text{HL}(t; \\\\mathbf {p}) &= Ae^{-Et} \\\\big ( 1-(-1)^t re^{-\\\\Delta Et} \\\\big ) + Ae^{-E(T-t)} \\\\big (1-(-1)^t re^{-\\\\Delta E(T-t)}\\\\big )\\\\\\\\f^\\\\text{HH}(t; \\\\mathbf {p}) &= Ae^{-Et}+Ae^{-E(T-t)}.$ The HL meson correlator, $f^\\\\text{HL}$ , has 4 fit parameters: the ground state energy and amplitude ($E$ , $A$ ), an amplitude ratio ($r=A^p/A$ ), and energy difference ($\\\\Delta E=E^p-E$ ), where the superscript $p$ stands for the opposite parity partner state that is present in staggered fermion correlation functions.\",\n \"$f^\\\\text{HH}$ is the function used to fit the HH mesons.\",\n \"The range $12 \\\\le t \\\\le 19$ is used to fit the HL mesons and $12 \\\\le t \\\\le 16$ for the HH mesons.\",\n \"The ground state energy $E(\\\\mathbf {p})$ is then fit using the following dispersion relation: $E(\\\\mathbf {p})=M_1 + \\\\frac{\\\\mathbf {p}^2}{2M_2} - \\\\frac{(\\\\mathbf {p}^2)^2}{8M_4^3} - \\\\frac{a^3W_4}{6}\\\\sum _{i=1}^3 p_i^4 \\\\,,$ to obtain $M_1$ the rest mass, $M_2$ the kinetic mass, $M_4$ the quartic mass, and $W_4$ the Lorentz symmetry breaking term.\",\n \"In both fits we use the full covariance matrix with trivial priors.\"\n ],\n [\n \"Kappa Tuning\",\n \"We determine the hopping parameters $\\\\kappa _b$ and $\\\\kappa _c$ such that the kinetic masses are equal to the physical $B_s$ and $D_s$ masses, respectively.\",\n \"We tune the kinetic mass $M_2$ rather than the rest mass $M_1$ .\",\n \"The form factors and decay constants which we are interested in are independent of the rest mass $M_1$ in the Fermilab interpretation of improved Wilson fermions.\",\n \"We use the HQET inspired fitting function for kinetic HL meson masses, $aM_2(\\\\kappa )=d_0+am_2(\\\\kappa )+\\\\frac{d_1}{am_2(\\\\kappa )}+\\\\frac{d_2}{(am_2 (\\\\kappa ))^2} \\\\,,$ where $M_2$ is the kinetic mass of the HL meson, and $m_2$ is the kinetic mass of the heavy quark.\",\n \"We determine $d_0$ , $d_1$ and $d_2$ using the correlated least $\\\\chi ^2$ fitting.\",\n \"Here, $m_2(\\\\kappa )$ is related to the bare mass $m_0(\\\\kappa )$ at the tree level as follows, $\\\\frac{1}{ am_2} &= \\\\frac{2 \\\\zeta ^2}{ am_0 (2+ am_0)} + \\\\frac{ r_s\\\\zeta }{1+ am_0 }.$ Figure REF shows the interpolation of $m_2$ to the physical values for the bottom and charm quarks.\",\n \"Results for $\\\\kappa _b$ and $\\\\kappa _c$ are summarized in Table REF .\",\n \"In Table REF , we present the $\\\\kappa $ -tuning results using physical values of the pseudoscalar meson mass ($M_X$ ), vector meson mass ($M_{X^\\\\ast }$ ), and the spin-averaged mass $(M_X+3M_{X^\\\\ast })/4$ for $X=B_s$ or $D_s$ .\",\n \"We find that all the results for $\\\\kappa $ determined from different spin states are consistent within statistical uncertainty.\",\n \"We also perform another fit using a simpler fitting function: $aM_2=d_0+am_2+d_1/(am_2)$ , and take the difference in $\\\\kappa $ as the systematic error due to ambiguity in the fitting function.\",\n \"Figure: Plot of the pseudoscalar meson mass D s D_s (B s B_s) versusthe kinetic quark mass m 2 c m_2^c (m 2 b m_2^b) defined inEq. ().\",\n \"The physical κ c \\\\kappa _c andκ b \\\\kappa _b (shown as red triangles and given in Table) are determined by tuning the D s D_s and B s B_s pseudoscalar massesto their experimental values.Table: Results of tuning the κ\\\\kappa for the bottom (κ b \\\\kappa _b)and charm (κ c \\\\kappa _c) quarks.\",\n \"For converting the experimental MMto aMaM, we use a=0.12520(22) fm a=0.12520(22)\\\\,\\\\mathrm {fm} .\",\n \"Inκ b,c \\\\kappa _{b,c}, the first error is statistical, the second erroris propagation of experimental error in M X M^X, and the third erroris systematic to account for the uncertainty in the fit ansatz.\"\n ],\n [\n \"Inconsistency Parameter\",\n \"The inconsistency parameter $I$ [10], [11] is used to see $O(\\\\mathbf {p}^4)$ improvement in the OK action.\",\n \"Let us use $Q$ for heavy quarks and $q$ for light quarks, and define $\\\\delta M \\\\equiv M_{2}-M_{1}$ as the difference between the kinetic and rest masses.\",\n \"Then the inconsistency parameter $I$ is $I\\\\equiv \\\\frac{2\\\\delta M_{\\\\bar{Q} q} -(\\\\delta M_{\\\\bar{Q} Q}+\\\\delta M_{\\\\bar{q} q})}{2M_{2\\\\bar{Q} q}}= \\\\frac{2\\\\delta B_{\\\\bar{Q} q} -(\\\\delta B_{\\\\bar{Q} Q}+\\\\delta B_{\\\\bar{q} q})}{2M_{2\\\\bar{Q} q}}$ Here the binding energies $B_{1,2}$ are $M_{1\\\\bar{Q} q}= m_{1\\\\bar{Q} } + m_{1 q} + B_{1\\\\bar{Q} q }, \\\\qquad M_{2\\\\bar{Q} q}= m_{2\\\\bar{Q} } + m_{2 q} + B_{2\\\\bar{Q} q }$ for HL mesons.\",\n \"Here the quark masses $m_{1,2}$ are defined by the quark dispersion relation, which is similar to Eq.\",\n \"(REF ).\",\n \"We neglect the light quarkonium contribution $\\\\delta M_{\\\\bar{q} q}$ (and $\\\\delta B_{\\\\bar{q} q}$ ).\",\n \"In Fig.\",\n \"REF we present results for $I$ for pseudoscalar mesons.\",\n \"Near the $B_s$ region, $I$ is consistent with the continuum limit, $I=0$ , within the error bars, which indicates a dramatic improvement from that of the Fermilab action: $I\\\\approx -0.6$ [5].\",\n \"Figure: (Left) Inconsistency parameter II versus the pseudoscalarmass aM 2Q ¯q a M_{2\\\\bar{Q}q} for charm (green squares) and bottom (bluecircles) quarks.\",\n \"(Right) Dispersion relation for the HL bottommeson with κ=0.040\\\\kappa =0.040.\",\n \"The rest mass, M 1 =2.112(2)M_1=2.112(2), is givenby the blue circle at 𝐩=0\\\\mathbf {p}=0.\",\n \"The kinetic mass,M 2 =3.408(176)M_2=3.408(176), is extracted from the fit and shown by the redtriangle translated to E=2.112E= 2.112.\",\n \"Note that the determinationof the error in M 2 M_2 is much larger than in M 1 M_1.\"\n ],\n [\n \"Hyperfine Splittings\",\n \"We define the hyperfine splittings of HL and HH pseudoscalar mesons, $\\\\Delta _1$ and $\\\\Delta _2$ as $\\\\Delta _1=M_1^\\\\ast -M_1,\\\\qquad \\\\Delta _2=M_2^\\\\ast -M_2 \\\\,,$ and plot $\\\\Delta _2$ versus $\\\\Delta _1$ in Fig.\",\n \"REF .\",\n \"As illustrated in Fig.\",\n \"REF , $M_2$ has much larger errors than $M_1$ since it is extracted from the slope versus momentum.\",\n \"Consequently, $\\\\Delta _2$ has larger errors than $\\\\Delta _1$ .\",\n \"Figure: Hyperfine splitting for HH mesonsThe HQET expansion for $\\\\Delta _1$ in the HL meson system is given in Ref.\",\n \"[12]: $\\\\Delta _1 = M_1^{\\\\ast } - M_1 &= \\\\frac{4\\\\lambda _2}{2m_B} -\\\\frac{4\\\\rho _2}{4m_E^2}+ \\\\frac{8T_2}{2m_2 2m_B} + \\\\frac{4(T_4-T_2)}{4m_B^2} +\\\\mathcal {O}\\\\left(\\\\frac{1}{m^3}\\\\right),$ where $\\\\lambda _2$ , $\\\\rho _2$ , $T_2$ , $T_4$ are HQET matrix elements defined in Ref. [12].\",\n \"For the OK action, the matching conditions are $m_2=m_B=m_E$ [3].\",\n \"Thus, $\\\\Delta _1$ defined in Eq.\",\n \"(REF ) is in terms of the kinetic quark mass, which was used to tune the $\\\\kappa $ to the physical value.\",\n \"To analyze $\\\\Delta _1$ , we recast Eq.\",\n \"REF as $a\\\\Delta _{1} = h_0 + \\\\frac{h_1}{am_2} + \\\\frac{h_2}{(am_2)^2}+ \\\\frac{h_3}{(am_2)^3} \\\\,,$ where $h_1 = 2 a^2 \\\\lambda _2$ and $h_2 = a^3 (-\\\\rho _2 + T_2 +T_4)$ .\",\n \"Because we have only 4 data points, we set $h_3=0$ in the fits.\",\n \"Correlated fits, shown in Fig.\",\n \"REF , give $h_0 = 0$ within statistical uncertainty, consistent with the theoretical prediction.\",\n \"Our results, with $h_0$ set to zero in the fits are summarized in Table REF .\",\n \"The corresponding $h_i$ from fits to $\\\\Delta _2$ were very poorly determined.\",\n \"We are performing simulations at other values of the lattice spacing and quark mass in order to perform the continuum-chiral extrapolation and compare with the experimental value.\",\n \"Table: Hyperfine splittings, λ 2 \\\\lambda _2 and A≡-ρ 2 +T 2 +T 4 A \\\\equiv -\\\\rho _2+T_2 +T_4 for the D s D_s and B s B_s mesons at the physical valuesof κ c \\\\kappa _c and κ b \\\\kappa _b.\",\n \"Δ exp \\\\Delta _\\\\text{exp} is theexperimental value .\"\n ],\n [\n \"Summary and Plan\",\n \"We tuned the bottom and charm quark masses using physical values for $B^{(*)}_s$ and $D^{(*)}_s$ mesons.\",\n \"Estimates from fits to the pseudoscalar, vector, and spin-averaged mesons masses are consistent within their statistical uncertainty (see Table REF ).\",\n \"We used estimates from the pseudoscalar mesons for the analysis of the hyperfine splittings.\",\n \"These values of $\\\\kappa _b$ and $\\\\kappa _c$ are now being used to measure the semileptonic form factors for the exclusive decays $\\\\bar{B}\\\\rightarrow D^{(*)} \\\\ell \\\\bar{\\\\nu }$ .\",\n \"Figure: Plots of Δ 1 (D s )\\\\Delta _1(D_s) (left panel) andΔ 1 (B s )\\\\Delta _1(B_s) (right panel) versus the kinetic masses, m 2 c m_2^cand m 2 b m_2^b, of the quarks.\",\n \"Results at the physical quark masses,tuned using the pseudoscalar meson masses, are shown by the red trianglesand given in Table .We thank Jon A. Bailey for helpful comments and suggestions.\",\n \"The research of W. Lee is supported by the Creative Research Initiatives Program (No.\",\n \"20160004939) of the NRF grant funded by the Korean government (MEST).\",\n \"W. Lee would like to acknowledge the support from the KISTI supercomputing center through the strategic support program for the supercomputing application research (No. KSC-2014-G2-002).\",\n \"Computations were carried out in part on the DAVID GPU clusters at Seoul National University.\",\n \"The research of T. Bhattacharya, R. Gupta and Y-C. Jang is supported by the U.S. Department of Energy, Office of Science of High Energy Physics under contract number DE-KA-1401020, the LANL LDRD program and Institutional Computing.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05707"}}},{"rowIdx":3391,"cells":{"text":{"kind":"list like","value":[["Nonlocal Schrodinger Equations for Integro-Differential Operators with\n Measurable Kernels and Asymptotic Potentials"],["Abstract In this paper, we investigate the existence of nonnegative solutions for the problem $$ -\\mathcal{L}_{K}u+V(x)u=f(u) $$ in $\\mathbb R^n$, where $-\\mathcal{L}_{K}$ is a integro-differential operator with measurable kernel $K$ and $V$ is a continuous potential.","Under apropriate hypothesis, we prove, using variational methods, that the above equation has solution."],["Introduction","In this article we consider the class of integro-differential Schrödinger equations $\\begin{array}{lcl}- \\mathcal {L}_{K} u +V(x)u = f(u), &\\mbox{ in }& \\mathbb {R}^n,\\end{array}\\qquad \\mathrm {(P)}$ where $- \\mathcal {L}_{K}$ is a integro-differential operator given by $- \\mathcal {L}_{K}u(x)= 2 \\int _{\\mathbb {R}^{n}}(u(x)-u(y))K(x-y)dy$ and $K$ satisfy general properties.","This study leads both to nonlocal and to nonlinear difficulties.","For example, we can not benefit from the $s$ -harmonic extension of or commutator properties (see ).","The study of nonlocal operators is important because they intervene in a quantity of applications and models.","For example, we mention their use in phase transition models (see , ), image reconstruction problems (see ), obstacle problem, optimization, finance, phase transitions.","Integro-differential equations arise naturally in the study of stochastic processes with jumps, and more precisely of Lévy processes.","This paper was motivated by , where the authors study the existence of positive solutions for the problem $\\left\\lbrace \\begin{array}{lcl}- \\Delta u +V(x)u = f(u), &\\mbox{ in }& \\mathbb {R}^n, \\\\ u \\in D^{1,2}(\\mathbb {R}^{n})\\end{array}\\right.$ where $V$ and $f$ are continuous functions with $V$ being a nonnegative function and $f$ having a subcritical or critical growth.","Our purpose is to study an analogously problem, considering the operator $-\\mathcal {L}_{k}$ instead of the Laplacian operator.","Several papers have studied the problem $(P)$ when $K (x) = \\frac{C_{n,s}}{2}|x|^{n + 2s}$ , where $C_{n,s}=\\left(\\int _{\\mathbb {R}^{n}}\\frac{1-\\cos (\\xi _{1})}{|\\xi |^{n+2s}}d \\xi \\right)^{-1},$ that is, when the operator $-\\mathcal {L}_{k}$ is the fractional Laplacian operator (see ).","Next, we will mention some of these papers.","In , the author has proved the existence of positive solutions from $(P)$ when $V$ is a constant small enough.","Also, in , the problem was studied when $f$ is asymptotically linear and $V$ is constant.","In , the authors study the problem $(P)$ when $V \\in C^{n}(\\mathbb {R}^{n},\\mathbb {R})$ , $V$ is positive and $\\lim \\limits _{n \\rightarrow \\infty }V(|x|) \\in (0, \\infty ].$ In , the authors has Studied $(P)$ when $ V $ and $ f $ are asymptotically periodic.","When $V=1$ , Felmer et al.","has studied the existence, regularity and qualitative properties of ground states solutions for problem $(P)$ (see ).","In , Teng and He have shown the existence of solution for $(P)$ when $f(x,u)=P(x)|u|^{p-2}u+Q(x)|u|^{2^{\\ast }_{s}-2}u,$ where $2 < p < 2^{\\ast }_{s}$ and the potential functions $P(x)$ and $Q(x)$ satisfy certain hypothesis.","In , the authors have shown the existence of solution for $(P)$ when $V \\in C^{n}(\\mathbb {R}^{n}, \\mathbb {R})$ and there exists $r_{0} > 0$ such that, for any $M > 0$ , $\\mbox{meas}(\\left\\lbrace x \\in B_{r_{0}}(y); V (x)\\le M\\right\\rbrace )\\rightarrow 0\\mbox{ as }|y| \\rightarrow \\infty .$ In the problem $(P)$ was studied when $V \\in C^1(\\mathbb {R}^{n},\\mathbb {R})$ , $\\liminf _{|x|\\rightarrow \\infty }V(x)\\ge V_{\\infty }$ where $V_{\\infty }$ is constant, and $f \\in C^{1}(\\mathbb {R}^{n}, \\mathbb {R})$ .","By method of the Nehari manifold , Sechi has shown that the problem $(P)$ has a solution if $V\\le V_{\\infty }$ , but $V$ is not identically equal to $V_{\\infty }$ , where $V_{\\infty }$ is a constant.","Also in , Secchi have obtained the existence of ground state solutions of $(P)$ for general $s \\in (0,1)$ when $V(x)\\rightarrow \\infty $ as $|x| \\rightarrow \\infty $ .","In , the authors obtain the existence of a sequence of radial and non radial solutions for the problem $(P)$ when $V$ and $f$ are radial functions.","Some other interesting studies by variational methods of the problem $(P)$ can be found in , , , , , , , , , , , , , and .","Many of them use strong tools that we can not use here in our problem, as the $s$ -harmonic extension and commutator properties.","Here, we will admit that the potential $V$ is continuous and satisfies, $(V_{1}-)$ $\\inf _{x \\in \\mathbb {R}^{n}}V(x)>0;$ $(V_{2}-)$ $V(x)\\le V_{\\infty }$ for same constant $V_{\\infty }>0$ and for all $x \\in B_{1}(0)$ .","Note that, $(V_{1})$ implies that $(V_{3}-)$ There are $R>0$ and $\\Lambda >0$ such that $V(x)\\ge \\Lambda $ for all $|x|\\ge R$ .","Also, we will assume that $f\\in C(\\mathbb {R},\\mathbb {R})$ is a function satisfying: $(f_{1}-)$ $|f(s)|\\le c_{0}|s|^{p-1}$ , for some constant $C>0$ and $p \\in (2,2^{\\ast }_{s})$ ; $(f_{2}-)$ There is $\\theta >2$ such that $\\theta F(s)\\le sf(s)$ for all $s>0$ ; $(f_{3}-)$ $f(t)>0$ for all $t>0$ and $f(t)=0$ for all $t<0$ .","The kernel $K:\\mathbb {R}^{n}\\rightarrow \\left(0, \\infty \\right)$ is a measurable function such that $(K_{1}-)$ $K(x)=K(-x)$ for all $x \\in \\mathbb {R}^{n}$ ; $(K_{2}-)$ There is $\\lambda >0$ and $s \\in (0,1)$ such that $\\lambda \\le K(x)|x|^{n+2s}$ almost everywere in $\\mathbb {R}^{n}$ ; $(K_{3}-)$ $\\gamma K \\in L^{1}(\\mathbb {R}^{n})$ , where $\\gamma (x)=\\min \\left\\lbrace |x|^{2},1\\right\\rbrace $ .","Note that, when $K(x)=\\frac{C_{n,s}}{2}|x|^{-(n+2s)}$ we have that $-\\mathcal {L}_{k}$ is a fractional laplacian, $(-\\Delta )^{s}$ .","Our paper is organized as follows.","In section 2, we will present some properties of the space in which we will study the problem $(P)$ .","In section 3, we will define an auxiliary problem and we will show that the functional energy associated with the auxiliary problem satisfies the condition of Palais-Smale.","By difficulty nonlocal of the operator $-\\mathcal {L}_{K}$ , we will can not use the same technique used in .","Therefore, we will present an another technique to show this result.","In section 4, we will prove that a general estimative for weak solution of $-\\mathcal {L}_{K}u+b(x)u=g(x,u),$ where $b\\ge 0$ , $|g(x,t)|\\le h(x)|t|$ and $h\\in L^{q}(\\mathbb {R}^{n})$ with $q>\\frac{n}{2s}$ .","We will show that there is $M=M(q,||h||_{L^{q}})$ such that the solution $u$ satisfies $||u||_{\\infty } \\le M||u||_{2^{\\ast }_{s}}.$ In , using the $s$ -harmonic extension of , the authors has shown the same estimate when $ -\\mathcal {L}_{K}$ is the fractional Laplacian operator.","In our case, we can not use the $s$ -harmonic extension, because we do not have an analogously extension for integro-differential operators.","In section 5, we show our main result in this paper, the Theorem REF ."],["Preliminaries","Let $s \\in (0,1)$ , we denote by $H^{s}(\\mathbb {R}^{n})$ the fractional sobolev space.","It is defined as $H^{s}(\\mathbb {R}^{n}):=\\left\\lbrace u \\in L^{2}(\\mathbb {R}^{n});\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\frac{(u(x)-u(y))^{2}}{|x-y|^{n+2s}}dxdy<\\infty \\right\\rbrace .$ The space $H^{s}(\\mathbb {R}^{n})$ is a Hilbert space with the norm $||u||_{H^{s}}=\\left(\\int _{\\mathbb {R}^{n}}|u|^{2}dx+\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\frac{(u(x)-u(y))^{2}}{|x-y|^{n+2s}}dxdy\\right)^{\\frac{1}{2}}$ We define $X$ as the linear space of Lebesgue measurable functions from $\\mathbb {R}^{n}$ to $\\mathbb {R}$ such that any function $u$ in $X$ belongs to $L^{2}(\\mathbb {R}^{n})$ and the function $(x,y)\\longmapsto (u(x)-u(y))\\sqrt{K(x-y)}$ is in $L^{2}(\\mathbb {R}^{n}\\times \\mathbb {R}^{n})$ .","The function $||u||_{X}:= \\left(\\int _{\\mathbb {R}^{n}}u^{2}dx+\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}(u(x)-u(y))^{2}K(x-y)dxdy\\right)^{\\frac{1}{2}}$ defines a norm in $X$ and $(X, ||\\cdot ||_{X})$ is a Hilbert space.","By $(K_{2})$ , the space $X$ is continuously embedded in $H^{s}(\\mathbb {R}^{n})$ .","Therefore, $X$ is continuously embedded in $L^{p}(\\mathbb {R}^{n})$ for $p \\in \\left[2,2^{\\ast }_{s}\\right]$ , where $2^{\\ast }_{s}=\\frac{2n}{n-2s}$ .","If $\\Omega \\subset \\mathbb {R}^{n}$ , we define $X_{0}(\\Omega )=\\left\\lbrace u \\in X; u=0 \\mbox{ in } \\mathbb {R}^{n}\\setminus \\Omega .\\right\\rbrace .$ The space $X_{0}(\\Omega )$ is a Hilbert Space with the norm $u \\longmapsto ||u||_{X_{0}(\\Omega )}:=\\left(\\int _{\\Omega }u^{2}dx+\\int _{Q}\\left(u(x)-u(y)\\right)^{2}K(x-y)dxdy\\right)^{\\frac{1}{2}},$ where $Q=(\\mathbb {R}^{n}\\times \\mathbb {R}^{n}) \\setminus (\\Omega ^{c} \\times \\Omega ^{c})$ (see Lemma 7 in ).","It is continuously embedded in $H_{0}^{s}(\\mathbb {R}^{n})$ .","For definition and properties of $H_{0}^{s}(\\mathbb {R}^{n})$ we indicate .","In the problem $(P)$ we will consider the space $E$ defined as $E=\\left\\lbrace u \\in X; \\int _{\\mathbb {R}^{n}}V(x)u^{2}dx< \\infty \\right\\rbrace $ The space $E$ is a Hilbert space with the norm $u \\longmapsto ||u||:=\\left(\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)^{2}K(x-y)dxdy+\\int _{\\mathbb {R}^{n}}V(x)u^{2}dx\\right)^{\\frac{1}{2}}.$ If $u,v \\in C_{0}^{\\infty }(\\mathbb {R}^{n})$ then $(-\\mathcal {L}_{k}u,v)_{L^{2}}=[u,v].$ where $[u,v]=\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)\\left(v(x)-v(y)\\right)K(x-y)dxdy.$ Therefore, we say that $u\\in E$ is a solution for the problem $(P)$ if $[u,v]+\\int _{\\mathbb {R}^{n}}V(x)uvdx = \\int _{\\mathbb {R}^{n}}f(u)vdx$ for all $v \\in E$ , that is $\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)\\left(v(x)-v(y)\\right)K(x-y)dxdy+\\int _{\\mathbb {R}^{n}}V(x)uvdx = \\int _{\\mathbb {R}^{n}}f(u)vdx.$ Let $A, B \\subset \\mathbb {R}^{n}$ and $u,v \\in X$ .","We will denote $[u,v]_{A\\times B}=\\int _{A}\\int _{B}(u(x)-u(y))(v(x)-v(y))K(x-y)dxdy$ and we will denote $[u,v]_{\\mathbb {R}^{n}\\times \\mathbb {R}^{n}}$ by $[u,v]$ .","The Euler-Lagrange functional associated with $(P)$ is given by $I(u)=\\frac{1}{2}||u||^{2}-\\int _{\\mathbb {R}^{n}}F(u)dx,$ where $F(t)=\\int _{0}^{t}f(s)ds.$ From hypothesis about $f$ , the functional is $C^{1}(E,\\mathbb {R})$ and $I^{\\prime }(u)v=[u,v]+\\int _{\\mathbb {R}^{n}}V(x)uvdx-\\int _{\\mathbb {R}^{n}}f(u)vdx.$ We will denote by $B$ the unitary ball of $\\mathbb {R}^{n}$ .","Define $I_{0}:X_{0}(B)\\longrightarrow \\mathbb {R}$ by $I_{0}(u)=:\\int _{\\mathbb {R}^{n}} \\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)^{2}K(x-y)dxdy+\\int _{\\mathbb {R}^{n}}V_{\\infty }u^{2}dx-\\int _{\\mathbb {R}^{n}}F(u)dx,$ where $V_{\\infty }$ is the constant of $(V_{2})$ .","The functional $I_{0}$ has the mountain pass geometry.","We will denote by $d$ the mountain pass level associated with $I_{0}$ , that is $d= \\inf _{\\gamma \\in \\Gamma } \\max _{t \\in [0,1]}I_{0}(\\gamma (t)),$ where $\\Gamma =\\left\\lbrace \\gamma \\in C([0,1],X_{0}(\\Omega )); \\gamma (0)=0\\mbox{ and }\\gamma (1)=e\\right\\rbrace ,$ with $e$ fixed and verifying $I_{0}(e)<0$ .","Note that $d$ depends only on $V_{\\infty }$ , $\\theta $ and $f$ ."],["An Auxiliary Problem","According to , we will modified the problem defining an auxiliary problem.","But, as the operator $-\\mathcal {L}_{K}$ is nonlocal, we can not use the same ideas of to prove that the functional associated the auxiliary problem satisfies the Palais-Smale condition.","It is necessary that we use an another technics.","For $k=\\frac{2 \\theta }{\\theta -2}$ we consider $\\begin{array}{lll}\\tilde{f}(x,t)&=& \\left\\lbrace \\begin{array}{lll}f(t) & if & kf(t) \\le V(x)t \\\\\\frac{V(x)}{k}t& if & kf(t)>V(x)t\\end{array}\\right.\\end{array}$ and $\\begin{array}{lll}g(x,t)&=& \\left\\lbrace \\begin{array}{lll}f(t) & if & |x|\\le R \\\\\\tilde{f}(x,t)& if & |x|>R.\\end{array}\\right.\\end{array}$ And we define the auxiliary problem $\\left\\lbrace \\begin{array}{lllll}-\\mathcal {L}_{K}u+V(x)u&=&g(x,u)& in &\\mathbb {R}^{n} \\\\u \\in E&&&&\\end{array}\\right.$ We have that, for all $t \\in \\mathbb {R}$ and $x \\in \\mathbb {R}^{n}$ $\\tilde{f}(x,t)\\le f(t)$ ; $g(x,t) \\le \\frac{V(x)}{k}t$ ,se $|x|\\ge R$ ; $G(x,t)=F(t)$ se $|x|\\le R$ $G(x,t)\\le \\frac{V(x)}{2k}t^{2}$ se $|x|>R$ ; where $G(x,t)=\\int _{0}^{t}g(x,s)ds.$ The Euler-Lagrange functional associated with the auxiliary problem is given by $J(u)=\\frac{1}{2}||u||^{2} - \\int _{\\mathbb {R}^{n}}G(x,u)dx.$ The functional $J \\in C^{1}(X,\\mathbb {R})$ and $\\begin{array}{ll}J^{\\prime }(u)v&=\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)\\left(v(x)-v(y)\\right)K(x-y)dxdy \\\\&+\\int _{\\mathbb {R}^{n}}V(x)uvdx - \\int _{\\mathbb {R}^{n}}g(x,u)vdx.\\end{array}$ The functional $J$ has the mountain pass geometry.","Then, there is a sequence $\\left\\lbrace u_{n}\\right\\rbrace _{n \\in \\mathbb {N}}$ such that $J^{\\prime }(u_{n})\\rightarrow 0\\mbox{ and }J(u_{n})\\rightarrow c,$ where $c>0$ is the mountain pass level associated with $J$ , that is $c= \\inf _{\\gamma \\in \\Gamma } \\max _{t \\in [0,1]}J(\\gamma (t))$ where $\\Gamma =\\left\\lbrace \\gamma \\in C([0,1],E); \\gamma (0)=0\\mbox{ and }, \\gamma (1)=e\\right\\rbrace .$ and $e$ is the function fixed in $\\ref {eq1}$ .","By definition $c \\le d$ uniformly in $R>0$ .","Lemma 3.1 The sequence $\\left\\lbrace u_{n}\\right\\rbrace _{n \\in \\mathbb {N}}$ is bounded.","By $(f_{2})$ , $(3)$ and $(4)$ $\\begin{array}{ll}&J(u)-\\frac{1}{\\theta }J^{\\prime }(u)u\\\\& = \\left(\\frac{\\theta -2}{4\\theta }\\right)||u||^{2}+ \\frac{1}{2k}||u||^{2} + \\int _{\\mathbb {R}^{n}}\\frac{1}{\\theta }g(x,u)u-G(x,u)dx \\\\&\\ge \\left(\\frac{\\theta -2}{4\\theta }\\right)||u||^{2}+ \\frac{1}{2k}||u||^{2} + \\int _{|x|>R}\\frac{1}{\\theta }g(x,u)u - \\frac{1}{2k}\\int _{|x|>R}V(x)u^{2}dx \\\\& \\ge \\left(\\frac{1}{2k}\\right)||u||^{2}.\\end{array}$ Thereby $|J(u)|+|J^{\\prime }(u)u|\\ge \\left(\\frac{\\theta -2}{4\\theta }\\right)||u||^{2}$ for all $u \\in E$ .","This last inequality ensures that the sequence is bounded.","Let $r>R$ and $A=\\left\\lbrace x \\in \\mathbb {R}^{n}; r<||x||<2r\\right\\rbrace $ .","Consider $\\eta :\\mathbb {R}^{n}\\rightarrow \\mathbb {R}$ a function such that $\\eta =1$ in $B_{2r}^{c}(0)$ , $\\eta =0$ in $B_{r}(0)$ , $0\\le \\eta \\le 1$ and $|\\nabla \\eta |<\\frac{2}{r}$ .","Note that $(B_{r} \\times B_{r})^{c} = (B_{r}^{c}\\times \\mathbb {R}^{n}) \\cup ( B_{r} \\times B_{r}^{c}),$ where $B_{r}=B_{r}(0)$ and $B_{2r}=B_{2r}(0)$ .","We will decompose $B_{r}^{c}\\times \\mathbb {R}^{n} = (A \\times \\mathbb {R}^{n}) \\cup (B_{2r}^{c}\\times B_{r}) \\cup (B_{2r}^{c}\\times A) \\cup (B_{2r}^{c}\\times B_{2r}^{c})$ and $B_{r} \\times B_{r}^{c} = (B_{r} \\times A) \\cup (B_{r} \\times B_{2r}^{c})$ Lemma 3.2 We have that $\\begin{array}{ll}&\\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))(\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y))K(x-y)dxdy\\\\&+\\int _{B_{2r}^{c}}\\int _{B_{r}}(u_{n}(x)-u_{n}(y))(\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y))K(x-y)dxdy \\\\& \\ge -\\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy\\end{array}$ $\\begin{array}{ll}&\\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))(\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y))K(x-y)dxdy \\\\& +\\int _{B_{2r}^{c}}\\int _{B_{r}}(u_{n}(x)-u_{n}(y))(\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y))K(x-y)dxdy \\\\& =\\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(x)(u_{n}(x)-u_{n}(y))K(x-y)dxdy\\\\&- \\int _{B_{2r}^{c}}\\int _{B_{r}}u_{n}(y)(u_{n}(x)-u_{n}(y))K(x-y)dxdy \\\\&= \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(x)(u_{n}(x)-u_{n}(y))K(x-y)dxdy\\\\&- \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)(u_{n}(x)-u_{n}(y))K(x-y)dydx \\\\& = \\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy \\\\&+ \\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(y)+u_{n}(x))(u_{n}(x)-u_{n}(y))K(x-y)dxdy\\\\& = \\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy \\\\&+ \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(x)^{2}-u_{n}(y)^{2}K(x-y)dxdy \\\\& \\ge -\\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy\\end{array}$ Lemma 3.3 Let $\\epsilon >0$ .","There is $r_{0}>0$ such that if $r>r_{0}$ then $\\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy<\\epsilon ,$ for all $n \\in \\mathbb {N}$ .","For each $y\\in B_{r}(0)$ $B_{r}(y) \\subset B_{2r}(0).$ Then $\\begin{array}{ll}&\\int _{B_{2r}(0)^{c}}K(x-y)dx \\\\&\\le \\int _{B_{r}(y)^{c}}K(x-y)dx\\\\& = \\int _{B_{r}(0)^{c}}K(z)dz.","\\\\\\end{array}$ By Lemma REF , there is $L>0$ such that $||u_{n}||_{L^{2}}^{2}0$ such that $\\int _{B_{r}(0)^{c}}K(z)dz<\\frac{\\epsilon }{L},$ for all $r>r_{0}$ .","Then by REF , for all $n \\in \\mathbb {N}$ and $r>r_{0}$ $\\begin{array}{ll}&\\int _{B_{r}(0)}\\int _{B_{2r}(0)^{c}}u_{n}(y)^{2}K(x-y)dxdy\\\\& =\\int _{B_{r}(0)}u_{n}(y)^{2}\\int _{B_{2r}(0)^{c}}K(x-y)dxdy\\\\& \\le \\int _{B_{r}(0)}u_{n}(y)^{2}\\int _{B_{r}(0)^{c}}K(z)dzdy \\\\& = \\int _{B_{r}(0)^{c}}K(z)dz\\int _{B_{r}(0)}u_{n}(y)^{2}dy\\\\& \\le \\epsilon \\end{array}$ Lemma 3.4 There are constants $K_{1}>0$ and $K_{2}>0$ such that $\\begin{array}{ll}&\\int _{A}\\int _{\\mathbb {R}^{n}}|u_{n}(y)||(u_{n}(x)-u_{n}(y))||(\\eta (x)-\\eta (y))|K(x-y)dxdy \\\\&\\le \\frac{2K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ Note that $\\begin{array}{ll}\\int _{\\mathbb {R}^{n}}|\\eta (x)-\\eta (y)|^{2}K(x-y)dx& = \\int _{\\mathbb {R}^{n}}|\\eta (z+y)-\\eta (y)|^{2}K(z)dz \\\\& =\\int _{B_{1}(0)}|\\eta (z+y)-\\eta (y)|^{2}K(z)dz\\\\&+\\int _{B_{1}^{c}(0)}|\\eta (z+y)-\\eta (y)|^{2}K(z)dz \\\\& \\le \\frac{4}{r^{2}}\\int _{B_{1}(0)}|z|^{2}K(z)dz +4\\int _{B_{1}(0)^{c}}K(z)dz \\\\& \\le \\frac{4}{r^{2}}P_{1}+4P_{2},\\end{array}$ where $P_{1}=\\int _{B_{1}}|z|^{2}K(z)dz$ and $P_{2}= \\int _{B_{1}^{c}}K(z)dz.$ Let $K_{1}=2\\sqrt{P_{1}}$ and $K_{2}=2\\sqrt{P_{2}}$ .","Then, by Holder inequality $\\begin{array}{ll}&\\int _{A}\\int _{\\mathbb {R}^{n}}|u_{n}(y)||(u_{n}(x)-u_{n}(y))||(\\eta (x)-\\eta (y))|K(x-y)dxdy \\\\& \\le (\\frac{2\\sqrt{P_{1}}}{r}+2\\sqrt{P_{2}})\\int _{A}|u_{n}(y)|\\left(\\int _{\\mathbb {R}^{n}}|(u_{n}(x)-u_{n}(y))|^{2}K(x-y)dx\\right)^{\\frac{1}{2}}dy\\\\& \\le (\\frac{K_{1}}{r}+K_{2})||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ Lemma 3.5 For the same constants $K_{1}>0$ and $K_{2}>0$ of the Lemma REF $\\begin{array}{ll}&\\int _{B_{r}}\\int _{A}|u_{n}(x)-u_{n}(y)||\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y)|K(x-y)dxdy \\\\&\\le \\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ Indeed, by property $(K_{1})$ of $K$ $\\begin{array}{ll}&\\int _{B_{r}}\\int _{A}|u_{n}(x)-u_{n}(y)||\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y)|K(x-y)dxdy\\\\& = \\int _{B_{r}}\\int _{A}|u_{n}(x)||u_{n}(x)-u_{n}(y)||\\eta (x)|K(x-y)dxdy \\\\& = \\int _{A}\\int _{B_{r}}|u_{n}(x)||u_{n}(x)-u_{n}(y)||n(x)-n(y)|K(x-y)dydx\\\\& =\\int _{A}\\int _{B_{r}}|u_{n}(y)||u_{n}(y)-u_{n}(x)||n(y)-n(x)|K(y-x)dxdy\\end{array}$ $Using $ K1)$ and Lemma \\ref {lem34}, we prove this lemma.$ Lemma 3.6 We have that $\\begin{array}{ll}&-\\int _{B_{2r}^{c}}\\int _{A}u_{n}(y)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy \\\\&\\le \\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ $\\begin{array}{ll}&-\\int _{B_{2r}^{c}}\\int _{A}u_{n}(y)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy\\\\&=\\int _{B_{2r}^{c}}\\int _{A}(u_{n}(x)-u_{n}(y))^{2}(\\eta (x)-\\eta (y))K(x-y)dxdy \\\\&-\\int _{B_{2r}^{c}}\\int _{A}u_{n}(x)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy \\\\& = \\int _{B_{2r}^{c}}\\int _{A}(u_{n}(x)-u_{n}(y))^{2}(n(x)-1)K(x-y)dxdy \\\\&-\\int _{B_{2r}^{c}}\\int _{A}u_{n}(x)(u_{n}(x)-u_{n}(y))(n(x)-n(y))K(x-y)dxdy \\\\& \\le -\\int _{B_{2r}^{c}}\\int _{A}u_{n}(x)(u_{n}(x)-u_{n}(y))(n(x)-n(y))K(x-y)dxdy \\\\& = -\\int _{B_{2r}^{c}}\\int _{A}u_{n}(x)(u_{n}(y)-u_{n}(x))(n(y)-n(x))K(x-y)dxdy \\\\& \\le \\int _{B_{2r}^{c}}\\int _{A}|u_{n}(x)||u_{n}(y)-u_{n}(x)||n(y)-n(x)|K(x-y)dxdy \\\\& =\\int _{A}\\int _{B_{2r}^{c}}|u_{n}(x)||u_{n}(y)-u_{n}(x)||n(y)-n(x)|K(y-x)dydx \\\\& \\le \\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ In the last inequality, we have used the Lemma REF and $(K_{1})$ .","We will prove that the functional $J$ satisfies the Palais-Smale condition.","The next proposition ensures the existence of solution in the level $c$ for the auxiliary problem (see REF ).","We emphasize that by a nonlocal difficulty, we can not repeat the same arguments used in to show that the functional energy associated at the auxiliary problem satisfies the Palais-Smale condition, therefore we use another technique to show this.","Proposition 3.7 Suppose that $f$ and $V$ satisfy $(V_{1})$ , $(f_{1})-(f_{3})$ .","Then the functional $J$ satisfies the Palais-Smale condition.","By Lemma REF the Palais-Smale sequence $\\left\\lbrace u_{n}\\right\\rbrace _{n \\in \\mathbb {N}}$ is bounded.","We can suppose that $\\left\\lbrace u_{n}\\right\\rbrace _{n \\in \\mathbb {N}}$ converges weakly to $u$ .","By $K$ properties we have that $\\eta u_{n} \\in X$ and $||\\eta u_{n}|| \\le ||u_{n}||$ (see Lemma $5.1$ in ).","Then, the sequence $\\left\\lbrace \\eta u_{n}\\right\\rbrace $ is bounded in $X$ .","Therefore $I^{^{\\prime }}(u_{n})(\\eta u_{n})=o_{n}(1)$ .","That is $\\begin{array}{ll}&[u_{n},\\eta u_{n}]+\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx = \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ o_{n}(1).\\end{array}$ But, note that $[u_{n},\\eta u_{n}] = [u_{n},\\eta u_{n}]_{C(B_{r}\\times B_{r})}$ , because $\\eta =0$ in $B_{r}$ .","By REF , REF and REF we have: $\\begin{array}{ll}&[u_{n},\\eta u_{n}]_{A \\times \\mathbb {R}^{n}}+[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times A}+[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times B_{2r}^{c}}\\\\&+[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times B_{r}}+[u_{n},\\eta u_{n}]_{B_{r}\\times B_{2r}^{c}}+[u_{n},\\eta u_{n}]_{B_{r}\\times A}\\\\&+\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx = \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ o_{n}(1)\\end{array}$ By Lemma REF , $\\begin{array}{ll}&[u_{n},\\eta u_{n}]_{A \\times \\mathbb {R}^{n}}+[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times A}+\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\&\\le \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy-[u_{n},\\eta u_{n}]_{B_{r}\\times A}+o_{n}(1)\\end{array}$ Above, we have used that $[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times B_{2r}^{c}} =[u_{n}, u_{n}]_{B_{2r}^{c}\\times B_{2r}^{c}} \\ge 0$ .","Because $\\eta = 1$ in $B_{2r}^{c}$ .","If $C$ and $D$ are subsets of $\\mathbb {R}^{n}$ and $u \\in E$ , then $\\begin{array}{ll}[u, \\eta u]_{C \\times D}&=\\int _{C}\\int _{D}(u(x)-u(y))(\\eta u(x)-\\eta u(y))K(x-y)dxdy \\\\&=\\int _{C}\\int _{D}\\eta (x)(u(x)-u(y))^{2}K(x-y)dxdy\\\\& + \\int _{C}\\int _{D}u(y)(u(x)-u(y))(\\eta (x)-\\eta (y))K(x-y)dxdy.\\end{array}$ Thereby, $\\begin{array}{ll}&\\int _{A}\\int _{\\mathbb {R}^{n}}\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy+\\\\&+\\int _{B_{2r}^{c}}\\int _{A}\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy +\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\& \\le \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy-[u_{n},\\eta u_{n}]_{B_{r}\\times A} \\\\& -\\int _{A}\\int _{\\mathbb {R}^{n}}u_{n}(y)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy \\\\& -\\int _{B_{2r}^{c}}\\int _{A}u(y)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy+o_{n}(1).\\end{array}$ From Lemmas REF , REF and REF , we obtain constants $K_{1},K_{2}>0$ such that $\\begin{array}{ll}&\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\&\\le \\int _{A}\\int _{\\mathbb {R}^{n}}\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy\\\\&+\\int _{B_{2r}^{c}}\\int _{A}\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy +\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\& \\le \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy \\\\&+\\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}+o_{n}(1).\\end{array}$ By $(2)$ , $(f_{3})$ and $r>R$ we have $\\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx \\le \\frac{1}{k}\\int _{\\mathbb {R}^{n}}\\eta V(x)u_{n}^{2}dx.$ Thereby, $\\begin{array}{ll}&\\left(1-\\frac{1}{k}\\right)\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\& \\le \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy \\\\&+\\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}+o_{n}(1)\\end{array}$ By Lemma REF , there is $C_{1}>0$ such that $||u_{n}||\\le C_{1}$ .","Then, for some constant $C>0$ $\\begin{array}{ll}&\\left(1-\\frac{1}{k}\\right)\\int _{|x|>2r}V(x)u_{n}^{2}dx \\\\& \\le \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x,y)dxdy +C(\\frac{1}{r}+1)||u_{n}||_{L^{2}(A)}+o_{n}(1)\\end{array}$ Let $\\epsilon >0$ .","By Lemma REF , we can take $r$ , large enough, such that $\\begin{array}{ll}\\int _{|x|>2r}V(x)u_{n}^{2}dx \\le \\frac{\\epsilon (k-1)}{3k}+C(\\frac{1}{r}+1)||u_{n}||_{L^{2}(A)} +o_{n}(1)\\end{array}$ for all $n \\in \\mathbb {N}$ .","Also, we can assume that $||u||_{L^{2}(A)}< \\frac{\\epsilon (k-1)}{6 C(\\frac{1}{r}+1)k}$ By property $(2)$ of $g$ $g(x,u_{n})u_{n} \\le \\frac{V(x)}{k}u_{n}^{2}.$ for all $x$ , with $|x|>2r>R$ .","Therefore, by REF $\\int _{|x|>2r}g(x,u_{n})u_{n}dx \\le \\frac{\\epsilon }{3}+C\\left(\\frac{1}{r}+1\\right)\\frac{k}{k-1}||u_{n}||_{L^{2}(A)} +o_{n}(1).$ By REF , REF and compact embedding of the fractional Sobolev spaces, we can take $n_{1}\\in \\mathbb {N}$ such that if $n>n_{1}$ then $\\int _{|x|>2r}g(x,u_{n})u_{n}dx \\le \\frac{5\\epsilon }{6}.$ Note that, we can suppose that $r>0$ satisfies $\\int _{|x|>2r}g(x,u)udx \\le \\frac{\\epsilon }{12}.$ By compact embedding of fractional sobolev spaces we have that, there is, $n_{0} \\in \\mathbb {N}$ with $n_{0}>n_{1}$ and if $n>n_{0}$ then $\\left|\\int _{|x|\\le 2r}g(x,u_{n})u_{n}dx - \\int _{|x|\\le 2r}g(x,u)udx\\right|< \\frac{\\epsilon }{12}.$ Thereby, for $n>n_{0}$ we have $\\left|\\int _{\\mathbb {R}^{n}}g(x,u_{n})u_{n}dx - \\int _{\\mathbb {R}^{n}}g(x,u)udx\\right|< \\epsilon $ that is $\\lim \\limits _{n \\rightarrow \\infty }\\int _{\\mathbb {R}^{n}}g(x,u_{n})u_{n}dx = \\int _{\\mathbb {R}^{n}}g(x,u)udx$ From $I^{\\prime }(u_{n})u_{n}=o_{n}(1)$ , we conclude that $||u_{n}|| \\rightarrow ||u||$ and therefore $\\left\\lbrace u_{n}\\right\\rbrace $ converges to $u$ in $X$ .","Corollary 3.8 Suppose $(V_{1})$ , $(f_{1})-(f_{3})$ .","Then, there is $u \\in X$ such that $J(u)=c$ and $J^{\\prime }(u)=0$ .","Moreover, $u\\ge 0$ almost everywere in $\\mathbb {R}^{n}$ .","By REF and Proposition REF , there is $u \\in X$ such that $J(u)=c$ and $J^{\\prime }(u)=0$ .","Let $A=\\left\\lbrace x \\in \\mathbb {R}^{n};|x|>R;\\right\\rbrace \\cap \\left\\lbrace x \\in \\mathbb {R}^{n}; u(x)<0;\\right\\rbrace $ .","If $x \\in A$ , then $g(x,u(x))=\\frac{V(x)}{k}u(x)$ and in if $x \\in A^{c}$ , then $g(x,u)\\ge 0$ .","We have $0\\ge [u,u^{-}]+\\int _{A^{c}}V(x)uu^{-} = \\left(\\frac{1}{k}-1\\right)\\int _{A}V(x)uu^{-} + \\int _{A^{c}}g(x,u)u^{-}dx \\ge 0$ where $u^{-}(x)=\\max \\left\\lbrace -u(x),0\\right\\rbrace $ .","Then $[u,u^{-}]=0$ This implies that $u^{-}=0$ (see proof of Lemma 4.1 in ).","As a consequence of inequalities REF and REF we have the following proposition.","Proposition 3.9 If $V$ and $f$ satisfies $(V_{1}),(V_{2}), (f_{1})-(f_{3})$ , then the solution $u$ of the auxiliary problem satisfies $||u||^{2} \\le 2kd$ uniformly in $R>0$ .","$L^{\\infty }$ estimative for solution of auxiliary problem In this section, we will prove a Brezis-Kato estimative.","We will prove that, admitting some hypothesis, there is $M>0$ such that the solution of the problem $\\left\\lbrace \\begin{array}{rlll}-\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\mathbb {R}^{n} \\\\\\end{array}\\right.$ satisfies $||u||_{L^{\\infty }(\\mathbb {R}^{n})}\\le M$ and $M$ does not depend on $||u||$ (see Proposition REF ).","We emphasize that, in the best of our knowlegends, this result is being presented for the first time in the literature.","In , the authors have shown this result when the operator $\\mathcal {L}_{K}$ is the fractional Laplacian operator, that is, when $K(x)=\\frac{C_{n,s}}{2}|x|^{-n-2s}$ .","But, in our case, we can not use the same technique used in , because we do not have a version of the $s$ -harmonic extension for more general integro-differential operators.","Therefore, we present an another technique.","Remark 4.1 Let $\\beta >1$ .","Define the real functions $m(x):= (\\lambda -1)(x^{\\beta }+x^{-\\beta })-\\lambda (x^{\\beta -1}+x^{1-\\beta })+2$ and $p(x):= (\\lambda -1)(x^{\\beta }+x^{-\\beta })+\\lambda (x^{\\beta -1}+x^{1-\\beta })-2.$ where $\\lambda :=\\frac{\\beta ^{2}}{2\\beta -1}$ .","Then $m(x)\\ge 0$ and $p(x)\\ge 0$ for all $x> 0$ .","We will omit the proof of the claim that appears in the Remark REF .","Let $\\beta >1$ and define $f(x)=x|x|^{2(\\beta -1)}$ and $g(x)=x|x|^{\\beta -1}.$ The functions $f$ and $g$ are continuous and differentiable for all $x \\in \\mathbb {R}$ .","Consider $x,y\\in \\mathbb {R}$ with $x \\ne y$ .","By Mean Value Theorem, there are $\\theta _{1}(x,y)$ , $\\theta _{2}(x,y) \\in \\mathbb {R}$ such that $f^{^{\\prime }}(\\theta _{1}(x,y))=\\frac{f(x)-f(y)}{x-y}$ and $g^{^{\\prime }}(\\theta _{2}(x,y))=\\frac{g(x)-g(y)}{x-y},$ that is $(2\\beta -1)|\\theta _{1}(x,y)|^{2(\\beta -1)}=\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}$ and $\\beta |\\theta _{2}(x,y)|^{(\\beta -1)}=\\frac{x|x|^{(\\beta -1)}-y|y|^{(\\beta -1)}}{x-y}.$ Implying that $|\\theta _{1}(x,y)|=\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}\\right)^{\\frac{1}{2(\\beta -1)}}$ and $|\\theta _{2}(x,y)|=\\left(\\frac{1}{\\beta }\\frac{x|x|^{(\\beta -1)}-y|y|^{(\\beta -1)}}{x-y}\\right)^{\\frac{1}{(\\beta -1)}}.$ We consider $\\theta _{1}(x,x)=\\theta _{2}(x,x)=0$ for all $x \\in \\mathbb {R}$ .","Remark 4.2 Note that $|\\theta _{1}(x,y)|=|\\theta _{1}(y,x)|$ and $|\\theta _{2}(x,y)| = |\\theta _{2}(y,x)|$ for all $x,y \\in \\mathbb {R}$ .","Lemma 4.3 With the same notation, if $x \\ne 0$ then $|\\theta _{1}(x,0)|\\ge |\\theta _{2}(x,0)|.$ By equalities REF and REF , we have $|\\theta _{1}(x,0)|=\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}}{x}\\right)^{\\frac{1}{2(\\beta -1)}} = \\left(\\frac{1}{2\\beta -1}\\right)^{\\frac{1}{2(\\beta -1)}}|x|$ and $|\\theta _{2}(x,0)|=\\left(\\frac{1}{\\beta }\\frac{x|x|^{\\beta -1}}{x}\\right)^{\\frac{1}{(\\beta -1)}} = \\left(\\frac{1}{\\beta }\\right)^{\\frac{1}{(\\beta -1)}}|x|.$ Thereby, $|\\theta _{1}(x,0)|\\ge |\\theta _{2}(x,0)|.$ Lemma 4.4 If $x,y \\in \\mathbb {R}$ , then $|\\theta _{1}(x,y)|\\ge |\\theta _{2}(x,y)|.$ If $x = 0$ or $y = 0$ then the inequality was proved by Lemma REF and Remark REF .","The case $x=y$ is trivial.","We can suppose that $x \\ne y$ , $x \\ne 0$ and $y \\ne 0$ .","By equalities REF and REF we have that $|\\theta _{1}(x,y)|\\ge |\\theta _{2}(x,y)|$ if, and only if $\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}\\right)^{\\frac{1}{2(\\beta -1)}} \\ge \\left(\\frac{1}{\\beta }\\frac{x|x|^{\\beta -1}-y|y|^{\\beta -1}}{x-y}\\right)^{\\frac{1}{(\\beta -1)}}.$ This last inequality is true if, and only if $\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y} \\ge \\frac{1}{\\beta ^{2}}\\left(\\frac{x|x|^{\\beta -1}-y|y|^{\\beta -1}}{x-y}\\right)^{2}.$ This last inequality occurs if, and only if $\\lambda (x-y)(x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}) \\ge \\left(x|x|^{\\beta -1}-y|y|^{\\beta -1}\\right)^{2},$ that is $\\lambda (|x|^{2\\beta }-xy|y|^{2(\\beta -1)}-yx|x|^{2(\\beta -1)}+|y|^{2\\beta })\\ge |x|^{2\\beta }-2xy|x|^{\\beta -1}|y|^{\\beta -1}+|y|^{2\\beta }.$ But, we are supposing that $x\\ne 0$ and $y\\ne 0$ , then the last inequality is equivalent to $\\begin{array}{ll}&\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] \\\\&\\ge \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - 2\\frac{x}{|x|}\\frac{y}{|y|} + \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\end{array}$ We will prove that the inequality REF is true.","If $x\\cdot y>0$ , then $\\begin{array}{ll}&\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2\\frac{x}{|x|}\\frac{y}{|y|} - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& =\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(\\frac{|y|}{|x|}\\right)^{\\beta -1}-\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2 - \\left(\\frac{|y|}{|x|}\\right)^{\\beta } \\\\& = (\\lambda -1)\\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta }+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta }\\right]-\\lambda \\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta +1}\\right]+2 \\\\& = m(\\frac{|x|}{|y|}).\\end{array}$ If $x \\cdot y<0$ , then $\\begin{array}{ll}& \\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2\\frac{x}{|x|}\\frac{y}{|y|} - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& =\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + \\left(\\frac{|y|}{|x|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right]- \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - 2 - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& = (\\lambda -1)\\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta }+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta }\\right]+\\lambda \\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta +1}\\right]-2\\\\& = p(\\frac{|x|}{|y|}).\\end{array}$ By Remark REF , we have that $m(\\frac{|x|}{|y|})\\ge 0$ and $p(\\frac{|x|}{|y|}) \\ge 0$ .","This proves the inequality REF and the Lemma REF .","Our main result of this section will establishes an important estimate involving the $L^{\\infty }(\\mathbb {R}^{n})$ norm of the solution $u$ of the auxiliary problem.","It states that: Proposition 4.5 Leq $h \\in L^{q}(\\mathbb {R}^{n})$ with $q>\\frac{n}{2s}$ , and $v \\in E \\subset X$ be a weak solution of $\\left\\lbrace \\begin{array}{rlll}-\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\mathbb {R}^{n} \\\\\\end{array}\\right.$ where $g$ is a continuous functions satisfying $|g(x,s)|\\le h(x)|s|$ for $s\\ge 0$ , $b$ is a positive function in $\\mathbb {R}^{n}$ and $E$ is definded as in REF .","Then, there is a constant $M=M(q,||h||_{L^{q}})$ such that $||v||_{\\infty }\\le M||v||_{2^{\\ast }_{s}}.$ Let $\\beta >1$ .","For any $n \\in \\mathbb {N}$ define $A_{n}=\\left\\lbrace x \\in \\mathbb {R}^{n}; |v(x)|^{\\beta -1}\\le n\\right\\rbrace $ and $B_{n}:=\\mathbb {R}^{n}\\setminus A_{n}.$ Consider $\\begin{array}{ll}f_{n}(t):=&\\left\\lbrace \\begin{array}{lll}t|t|^{2(\\beta -1)}&se&|t|^{\\beta -1}\\le n\\\\n^{2}t&se&|t|^{\\beta -1}>n\\end{array}\\right.\\end{array}$ and $\\begin{array}{ll}g_{n}(t):=&\\left\\lbrace \\begin{array}{lll}t|t|^{(\\beta -1)}&se&|t|^{\\beta -1}\\le n\\\\nt&se&|t|^{\\beta -1}>n\\end{array}\\right.\\end{array}$ Note that $f_{n}$ and $g_{n}$ are continuous functions and they are differentiable with the exception of $n^{\\frac{1}{\\beta -1}}$ and $-n^{\\frac{1}{\\beta -1}}$ and its derivatives are limited.","Then $f_{n}$ and $g_{n}$ are lipschtz.","Therefore, setting $v_{n}:=f_{n}\\circ v$ and $w_{n}:=g_{n}\\circ v$ we have that $v_{n},w_{n}\\in E$ .","Note that $\\begin{array}{ll}\\left[v,v_{n}\\right]&= \\int _{A_{n}}\\int _{A_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy \\\\ & +\\int _{B_{n}}\\int _{B_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy\\\\ &+2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ By equality REF , if $x,y \\in A_{n}$ then $v_{n}(x)-v_{n}(y)=f_{n}^{\\prime }(\\theta _{1}(x,y))(v(x)-v(y)),$ where we are denoting $\\theta _{1}(v(x),v(y))$ by $\\theta _{1}(x, y)$ .","Thereby, $\\begin{array}{ll}\\left[v,v_{n}\\right] &= \\int _{A_{n}}\\int _{A_{n}}(2\\beta -1)|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ Analogously, by REF $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &= \\int _{A_{n}}\\int _{A_{n}}\\beta ^{2}|\\theta _{2}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}},\\end{array}$ where we are denoting $\\theta _{2}(v(x),v(y))$ by $\\theta _{2}(x, y)$ .","By Lemma REF , $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &\\le \\int _{A_{n}}\\int _{A_{n}}\\beta ^{2}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ This implies that $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\left[v,v_{n}\\right]- \\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\&\\le (\\beta -1)^{2}\\int _{A_{n}}\\int _{A_{n}}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\& +2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}-2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ The equation REF implies that $\\begin{array}{ll}&\\left[v,v_{n}\\right]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx - 2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}\\\\ &\\ge (2\\beta -1)\\int _{A_{n}}\\int _{A_{n}}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy,\\end{array}$ because $b(x)vv_{n}=b(x)w_{n}^{2}\\ge 0$ .","Replacing REF in the inequality REF we obtain $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\left[v,v_{n}\\right]- \\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\ &\\le \\frac{(\\beta -1)^{2}}{2\\beta -1}\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}},\\end{array}$ that is $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx\\\\ &\\le (\\frac{(\\beta -1)^{2}}{2\\beta -1}+1)\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\\\&= \\frac{\\beta ^{2}}{2\\beta -1}\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}bvv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\\\&\\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ In short, $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &+ \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx \\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ But, if $n \\in \\mathbb {N}$ and $C=2+\\frac{2(\\beta -1)^{2}}{2\\beta -1},$ then a simple calculation shows that the function $r(s,t)=2(ns-t|t|^{\\beta -1})^{2}-C(s-t)(n^{2}s-t|t|^{2(\\beta -1)}),$ satisfies $r(s,t)\\le 0$ for all $|s|>n^{\\frac{1}{\\beta -1}}$ and $|t|\\le n^{\\frac{1}{\\beta -1}}$ .","Then, taking $s=v(x)$ and $t=v(y)$ for $x\\in B_{n}$ and $y \\in A_{n}$ and replacing in REF we obtain $2(w_{n}(x)-w_{n}(y))^{2}-C(v(x)-v(y))(v_{n}(x)-v_{n}(y))\\le 0.$ Thereby $+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\le 0.$ By inequality REF , $\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx.$ Let $S>0$ be the best constant verifying $||u||_{2^{\\ast }_{s}}^{2} \\le S||u||_{X}^{2},$ for all $u \\in X$ , that is $S=\\sup _{u \\in X}\\frac{||u||_{2^{\\ast }_{s}}^{2}}{||u||_{X}^{2}}.$ By inequality REF , $\\begin{array}{ll}\\left(\\int _{A_{n}}|w_{n}|^{2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}}&\\le \\left(\\int _{\\mathbb {R}^{n}}|w_{n}|^{2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}}\\\\&\\le S||w_{n}||^{2} \\\\& \\le S\\beta \\int _{\\mathbb {R}^{n}}g(x,v(x))v_{n}dx \\\\& \\le S\\beta \\int _{\\mathbb {R}^{n}}h(x)w_{n}^{2}dx \\\\& \\le S\\beta ||h||_{q}||w_{n}||_{\\frac{2q}{q-1}}^{2}.\\end{array}$ But, we have that $|w_{n}(x)|\\le |v(x)|^{\\beta }$ for all $x \\in B_{n}$ and $|w_{n}(x)|=|v(x)|^{\\beta }$ for all $x \\in A_{n}$ .","Thereby, $\\begin{array}{ll}\\left(\\int _{A_{n}}|v|^{\\beta 2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}} \\le S\\beta ||h||_{q}\\left[\\int _{\\mathbb {R}^{n}}|v|^{\\frac{2q\\beta }{q-1}}dx\\right]^{\\frac{q-1}{q}}.\\end{array}$ By Monotone Convergence Theorem, $||v||_{2^{\\ast }_{s}\\beta } \\le (\\beta S||h||_{q})^{\\frac{1}{2\\beta }}||v||_{2\\beta q_{1}},$ where $q_{1}=\\frac{q}{q-1}$ .","Define $\\eta :=\\frac{2^{\\ast }_{s}}{2q_{1}}.$ and note that $\\eta >1$ .","When $\\beta =\\eta $ we have that $2\\beta q_{1} = 2^{\\ast }_{s}$ .","Then, by REF $||v||_{2^{\\ast }_{s}\\eta } \\le (\\eta S||h||_{q})^{\\frac{1}{2\\eta }}||v||_{2^{\\ast }_{s}}.$ Taking $\\beta =\\eta ^{2}$ in REF we obtain $||v||_{2^{\\ast }_{s}\\eta ^{2}} \\le \\eta ^{\\frac{1}{\\eta ^{2}}} (S||h||_{q})^{\\frac{1}{2\\eta ^{2}}}||v||_{2^{\\ast }_{s}\\eta }.$ By inequalities REF and REF we have $||v||_{2^{\\ast }_{s}\\eta ^{2}} \\le \\eta ^{\\frac{1}{(\\eta ^{2}}+\\frac{1}{2\\eta })} (S||h||_{q})^{(\\frac{1}{2\\eta ^{2}}+\\frac{1}{2\\eta })}||v||_{2^{\\ast }_{s}}.$ Inductively, we can prove that $||v||_{2^{\\ast }_{s}\\eta ^{m}} \\le \\eta ^{(\\frac{1}{2\\eta }+\\frac{1}{\\eta ^{2}}+...+\\frac{m}{2\\eta ^{m}})} (S||h||_{q})^{(\\frac{1}{2\\eta }+\\frac{1}{2\\eta ^{2}}+...+\\frac{1}{2\\eta ^{m}})}||v||_{2^{\\ast }_{s}}$ for all $m \\in \\mathbb {N}$ .","But, $\\sum _{i=1}^{\\infty }\\frac{m}{2\\eta ^{m}}=\\frac{1}{2(\\eta -1)^{2}}$ and $\\sum _{i=1}^{\\infty }\\frac{1}{2\\eta ^{m}}=\\frac{1}{2(\\eta -1)}.$ Thereby, for all $m>0$ $||v||_{2^{\\ast }_{s}\\eta ^{m}} \\le \\eta ^{\\frac{1}{2(\\eta -1)^{2}}} (S||h||_{q})^{\\frac{1}{2(\\eta -1)}}||v||_{2^{\\ast }_{s}}$ Recalling that $||v||_{\\infty }=\\lim \\limits _{n \\rightarrow \\infty }||v||_{p}$ and that $\\eta >1$ we have that $||v||_{\\infty }\\le M||v||_{2^{\\ast }_{s}}$ for $M=\\eta ^{\\frac{1}{2(\\eta -1)^{2}}} (S||h||_{q})^{\\frac{1}{2(\\eta -1)}}$ and $\\eta = \\frac{n(q-1)}{q(n-2s)}$ We conclude the proof of Proposition REF noting that $M$ depends only on $q$ , $||h||_{q}$ .","Solution for Problem (P) In this section, we prove the main result, the Theorem REF .","By Corollary REF , there is $u \\in X$ such that $J(u)=c$ and $J^{\\prime }(u)=0$ .","We have the following estimate for $||u||_{\\infty }$ .","Lemma 5.1 The solution $u$ of the auxiliary problem satisfies $||u||_{\\infty } \\le M(2Skd)^{\\frac{1}{2}}$ Consider the functions $\\begin{array}{lll}H(x,t)&=&\\left\\lbrace \\begin{array}{lllll}f(t)&if&|x| \\frac{V(x)}{k}t\\end{array}\\right.\\end{array}$ and $\\begin{array}{lll}b(x)&=&\\left\\lbrace \\begin{array}{lllll}V(x)&if&|x| \\frac{V(x)}{k}u.\\end{array}\\right.\\end{array}$ Note that the function $u$ is solution of $\\left\\lbrace \\begin{array}{ll}&-\\mathcal {L}_{k}u+b(x)u=H(x,u)\\\\&u \\in E.\\end{array}\\right.$ By $(f_{1})$ , $|H(x,t)|\\le c_{0}|t|^{p-1}$ for $p \\in (2,2^{\\ast }_{s})$ .","Thereby, $|H(x,u)|\\le h(x)|u|,$ where $h(x)=C_{0}|u|^{p-2}$ .","Note that $h \\in L^{\\frac{2^{\\ast }_{s}}{p-2}}$ with $||h||_{L^{q}(\\mathbb {R}^{n})}\\le C(2ksd)^{\\frac{p-2}{2^{\\ast }_{s}}}.$ The number $p$ satisfies $p<2^{\\ast }_{s} = 2+\\frac{2s}{n}2^{\\ast }_{s}.$ Then $q =\\frac{2^{\\ast }_{s}}{p-2}>\\frac{n}{2s}.$ By Proposition REF and sobolev embedding $||u||_{\\infty }\\le M||u||_{2^{\\ast }_{s}} \\le MS^{\\frac{1}{2}}||u||,$ where $M=M(q,||h||_{q})$ .","By Proposition REF , we have $||u||_{\\infty } \\le M(2kSd)^{\\frac{1}{2}}.$ Theorem 5.2 Suppose that $V$ satisfies $(V_{1})$ -$(V_{2})$ and that $f$ satisfies $(f_{1})$ -$(f_{3})$ .","There is $\\Lambda ^{\\ast }=\\Lambda ^{\\ast }(V_{\\infty },\\theta ,p,c_{0},s)>0$ such that if $\\Lambda >\\Lambda ^{\\ast }$ in $(V_{3})$ , then the 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author=Zhang, F. title= Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in $\\mathbb {R}^{N}$ ., journal=Journal of Mathematical Physics, volume=56, date=2015, pages=, review= noncitearticle author=Zhang, X., author=Zhang, B. author=Repovs, D. title=Existence and symmetry of solutions for critical fractiona Schrödinger equations with bounded potentials, journal=Nonlinear Analysis, volume=142, date=2016, pages=48-68, review="],["$L^{\\infty }$ estimative for solution of auxiliary problem","In this section, we will prove a Brezis-Kato estimative.","We will prove that, admitting some hypothesis, there is $M>0$ such that the solution of the problem $\\left\\lbrace \\begin{array}{rlll}-\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\mathbb {R}^{n} \\\\\\end{array}\\right.$ satisfies $||u||_{L^{\\infty }(\\mathbb {R}^{n})}\\le M$ and $M$ does not depend on $||u||$ (see Proposition REF ).","We emphasize that, in the best of our knowlegends, this result is being presented for the first time in the literature.","In , the authors have shown this result when the operator $\\mathcal {L}_{K}$ is the fractional Laplacian operator, that is, when $K(x)=\\frac{C_{n,s}}{2}|x|^{-n-2s}$ .","But, in our case, we can not use the same technique used in , because we do not have a version of the $s$ -harmonic extension for more general integro-differential operators.","Therefore, we present an another technique.","Remark 4.1 Let $\\beta >1$ .","Define the real functions $m(x):= (\\lambda -1)(x^{\\beta }+x^{-\\beta })-\\lambda (x^{\\beta -1}+x^{1-\\beta })+2$ and $p(x):= (\\lambda -1)(x^{\\beta }+x^{-\\beta })+\\lambda (x^{\\beta -1}+x^{1-\\beta })-2.$ where $\\lambda :=\\frac{\\beta ^{2}}{2\\beta -1}$ .","Then $m(x)\\ge 0$ and $p(x)\\ge 0$ for all $x> 0$ .","We will omit the proof of the claim that appears in the Remark REF .","Let $\\beta >1$ and define $f(x)=x|x|^{2(\\beta -1)}$ and $g(x)=x|x|^{\\beta -1}.$ The functions $f$ and $g$ are continuous and differentiable for all $x \\in \\mathbb {R}$ .","Consider $x,y\\in \\mathbb {R}$ with $x \\ne y$ .","By Mean Value Theorem, there are $\\theta _{1}(x,y)$ , $\\theta _{2}(x,y) \\in \\mathbb {R}$ such that $f^{^{\\prime }}(\\theta _{1}(x,y))=\\frac{f(x)-f(y)}{x-y}$ and $g^{^{\\prime }}(\\theta _{2}(x,y))=\\frac{g(x)-g(y)}{x-y},$ that is $(2\\beta -1)|\\theta _{1}(x,y)|^{2(\\beta -1)}=\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}$ and $\\beta |\\theta _{2}(x,y)|^{(\\beta -1)}=\\frac{x|x|^{(\\beta -1)}-y|y|^{(\\beta -1)}}{x-y}.$ Implying that $|\\theta _{1}(x,y)|=\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}\\right)^{\\frac{1}{2(\\beta -1)}}$ and $|\\theta _{2}(x,y)|=\\left(\\frac{1}{\\beta }\\frac{x|x|^{(\\beta -1)}-y|y|^{(\\beta -1)}}{x-y}\\right)^{\\frac{1}{(\\beta -1)}}.$ We consider $\\theta _{1}(x,x)=\\theta _{2}(x,x)=0$ for all $x \\in \\mathbb {R}$ .","Remark 4.2 Note that $|\\theta _{1}(x,y)|=|\\theta _{1}(y,x)|$ and $|\\theta _{2}(x,y)| = |\\theta _{2}(y,x)|$ for all $x,y \\in \\mathbb {R}$ .","Lemma 4.3 With the same notation, if $x \\ne 0$ then $|\\theta _{1}(x,0)|\\ge |\\theta _{2}(x,0)|.$ By equalities REF and REF , we have $|\\theta _{1}(x,0)|=\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}}{x}\\right)^{\\frac{1}{2(\\beta -1)}} = \\left(\\frac{1}{2\\beta -1}\\right)^{\\frac{1}{2(\\beta -1)}}|x|$ and $|\\theta _{2}(x,0)|=\\left(\\frac{1}{\\beta }\\frac{x|x|^{\\beta -1}}{x}\\right)^{\\frac{1}{(\\beta -1)}} = \\left(\\frac{1}{\\beta }\\right)^{\\frac{1}{(\\beta -1)}}|x|.$ Thereby, $|\\theta _{1}(x,0)|\\ge |\\theta _{2}(x,0)|.$ Lemma 4.4 If $x,y \\in \\mathbb {R}$ , then $|\\theta _{1}(x,y)|\\ge |\\theta _{2}(x,y)|.$ If $x = 0$ or $y = 0$ then the inequality was proved by Lemma REF and Remark REF .","The case $x=y$ is trivial.","We can suppose that $x \\ne y$ , $x \\ne 0$ and $y \\ne 0$ .","By equalities REF and REF we have that $|\\theta _{1}(x,y)|\\ge |\\theta _{2}(x,y)|$ if, and only if $\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}\\right)^{\\frac{1}{2(\\beta -1)}} \\ge \\left(\\frac{1}{\\beta }\\frac{x|x|^{\\beta -1}-y|y|^{\\beta -1}}{x-y}\\right)^{\\frac{1}{(\\beta -1)}}.$ This last inequality is true if, and only if $\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y} \\ge \\frac{1}{\\beta ^{2}}\\left(\\frac{x|x|^{\\beta -1}-y|y|^{\\beta -1}}{x-y}\\right)^{2}.$ This last inequality occurs if, and only if $\\lambda (x-y)(x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}) \\ge \\left(x|x|^{\\beta -1}-y|y|^{\\beta -1}\\right)^{2},$ that is $\\lambda (|x|^{2\\beta }-xy|y|^{2(\\beta -1)}-yx|x|^{2(\\beta -1)}+|y|^{2\\beta })\\ge |x|^{2\\beta }-2xy|x|^{\\beta -1}|y|^{\\beta -1}+|y|^{2\\beta }.$ But, we are supposing that $x\\ne 0$ and $y\\ne 0$ , then the last inequality is equivalent to $\\begin{array}{ll}&\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] \\\\&\\ge \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - 2\\frac{x}{|x|}\\frac{y}{|y|} + \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\end{array}$ We will prove that the inequality REF is true.","If $x\\cdot y>0$ , then $\\begin{array}{ll}&\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2\\frac{x}{|x|}\\frac{y}{|y|} - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& =\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(\\frac{|y|}{|x|}\\right)^{\\beta -1}-\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2 - \\left(\\frac{|y|}{|x|}\\right)^{\\beta } \\\\& = (\\lambda -1)\\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta }+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta }\\right]-\\lambda \\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta +1}\\right]+2 \\\\& = m(\\frac{|x|}{|y|}).\\end{array}$ If $x \\cdot y<0$ , then $\\begin{array}{ll}& \\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2\\frac{x}{|x|}\\frac{y}{|y|} - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& =\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + \\left(\\frac{|y|}{|x|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right]- \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - 2 - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& = (\\lambda -1)\\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta }+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta }\\right]+\\lambda \\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta +1}\\right]-2\\\\& = p(\\frac{|x|}{|y|}).\\end{array}$ By Remark REF , we have that $m(\\frac{|x|}{|y|})\\ge 0$ and $p(\\frac{|x|}{|y|}) \\ge 0$ .","This proves the inequality REF and the Lemma REF .","Our main result of this section will establishes an important estimate involving the $L^{\\infty }(\\mathbb {R}^{n})$ norm of the solution $u$ of the auxiliary problem.","It states that: Proposition 4.5 Leq $h \\in L^{q}(\\mathbb {R}^{n})$ with $q>\\frac{n}{2s}$ , and $v \\in E \\subset X$ be a weak solution of $\\left\\lbrace \\begin{array}{rlll}-\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\mathbb {R}^{n} \\\\\\end{array}\\right.$ where $g$ is a continuous functions satisfying $|g(x,s)|\\le h(x)|s|$ for $s\\ge 0$ , $b$ is a positive function in $\\mathbb {R}^{n}$ and $E$ is definded as in REF .","Then, there is a constant $M=M(q,||h||_{L^{q}})$ such that $||v||_{\\infty }\\le M||v||_{2^{\\ast }_{s}}.$ Let $\\beta >1$ .","For any $n \\in \\mathbb {N}$ define $A_{n}=\\left\\lbrace x \\in \\mathbb {R}^{n}; |v(x)|^{\\beta -1}\\le n\\right\\rbrace $ and $B_{n}:=\\mathbb {R}^{n}\\setminus A_{n}.$ Consider $\\begin{array}{ll}f_{n}(t):=&\\left\\lbrace \\begin{array}{lll}t|t|^{2(\\beta -1)}&se&|t|^{\\beta -1}\\le n\\\\n^{2}t&se&|t|^{\\beta -1}>n\\end{array}\\right.\\end{array}$ and $\\begin{array}{ll}g_{n}(t):=&\\left\\lbrace \\begin{array}{lll}t|t|^{(\\beta -1)}&se&|t|^{\\beta -1}\\le n\\\\nt&se&|t|^{\\beta -1}>n\\end{array}\\right.\\end{array}$ Note that $f_{n}$ and $g_{n}$ are continuous functions and they are differentiable with the exception of $n^{\\frac{1}{\\beta -1}}$ and $-n^{\\frac{1}{\\beta -1}}$ and its derivatives are limited.","Then $f_{n}$ and $g_{n}$ are lipschtz.","Therefore, setting $v_{n}:=f_{n}\\circ v$ and $w_{n}:=g_{n}\\circ v$ we have that $v_{n},w_{n}\\in E$ .","Note that $\\begin{array}{ll}\\left[v,v_{n}\\right]&= \\int _{A_{n}}\\int _{A_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy \\\\ & +\\int _{B_{n}}\\int _{B_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy\\\\ &+2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ By equality REF , if $x,y \\in A_{n}$ then $v_{n}(x)-v_{n}(y)=f_{n}^{\\prime }(\\theta _{1}(x,y))(v(x)-v(y)),$ where we are denoting $\\theta _{1}(v(x),v(y))$ by $\\theta _{1}(x, y)$ .","Thereby, $\\begin{array}{ll}\\left[v,v_{n}\\right] &= \\int _{A_{n}}\\int _{A_{n}}(2\\beta -1)|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ Analogously, by REF $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &= \\int _{A_{n}}\\int _{A_{n}}\\beta ^{2}|\\theta _{2}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}},\\end{array}$ where we are denoting $\\theta _{2}(v(x),v(y))$ by $\\theta _{2}(x, y)$ .","By Lemma REF , $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &\\le \\int _{A_{n}}\\int _{A_{n}}\\beta ^{2}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ This implies that $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\left[v,v_{n}\\right]- \\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\&\\le (\\beta -1)^{2}\\int _{A_{n}}\\int _{A_{n}}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\& +2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}-2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ The equation REF implies that $\\begin{array}{ll}&\\left[v,v_{n}\\right]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx - 2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}\\\\ &\\ge (2\\beta -1)\\int _{A_{n}}\\int _{A_{n}}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy,\\end{array}$ because $b(x)vv_{n}=b(x)w_{n}^{2}\\ge 0$ .","Replacing REF in the inequality REF we obtain $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\left[v,v_{n}\\right]- \\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\ &\\le \\frac{(\\beta -1)^{2}}{2\\beta -1}\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}},\\end{array}$ that is $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx\\\\ &\\le (\\frac{(\\beta -1)^{2}}{2\\beta -1}+1)\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\\\&= \\frac{\\beta ^{2}}{2\\beta -1}\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}bvv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\\\&\\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ In short, $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &+ \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx \\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ But, if $n \\in \\mathbb {N}$ and $C=2+\\frac{2(\\beta -1)^{2}}{2\\beta -1},$ then a simple calculation shows that the function $r(s,t)=2(ns-t|t|^{\\beta -1})^{2}-C(s-t)(n^{2}s-t|t|^{2(\\beta -1)}),$ satisfies $r(s,t)\\le 0$ for all $|s|>n^{\\frac{1}{\\beta -1}}$ and $|t|\\le n^{\\frac{1}{\\beta -1}}$ .","Then, taking $s=v(x)$ and $t=v(y)$ for $x\\in B_{n}$ and $y \\in A_{n}$ and replacing in REF we obtain $2(w_{n}(x)-w_{n}(y))^{2}-C(v(x)-v(y))(v_{n}(x)-v_{n}(y))\\le 0.$ Thereby $+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\le 0.$ By inequality REF , $\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx.$ Let $S>0$ be the best constant verifying $||u||_{2^{\\ast }_{s}}^{2} \\le S||u||_{X}^{2},$ for all $u \\in X$ , that is $S=\\sup _{u \\in X}\\frac{||u||_{2^{\\ast }_{s}}^{2}}{||u||_{X}^{2}}.$ By inequality REF , $\\begin{array}{ll}\\left(\\int _{A_{n}}|w_{n}|^{2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}}&\\le \\left(\\int _{\\mathbb {R}^{n}}|w_{n}|^{2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}}\\\\&\\le S||w_{n}||^{2} \\\\& \\le S\\beta \\int _{\\mathbb {R}^{n}}g(x,v(x))v_{n}dx \\\\& \\le S\\beta \\int _{\\mathbb {R}^{n}}h(x)w_{n}^{2}dx \\\\& \\le S\\beta ||h||_{q}||w_{n}||_{\\frac{2q}{q-1}}^{2}.\\end{array}$ But, we have that $|w_{n}(x)|\\le |v(x)|^{\\beta }$ for all $x \\in B_{n}$ and $|w_{n}(x)|=|v(x)|^{\\beta }$ for all $x \\in A_{n}$ .","Thereby, $\\begin{array}{ll}\\left(\\int _{A_{n}}|v|^{\\beta 2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}} \\le S\\beta ||h||_{q}\\left[\\int _{\\mathbb {R}^{n}}|v|^{\\frac{2q\\beta }{q-1}}dx\\right]^{\\frac{q-1}{q}}.\\end{array}$ By Monotone Convergence Theorem, $||v||_{2^{\\ast }_{s}\\beta } \\le (\\beta S||h||_{q})^{\\frac{1}{2\\beta }}||v||_{2\\beta q_{1}},$ where $q_{1}=\\frac{q}{q-1}$ .","Define $\\eta :=\\frac{2^{\\ast }_{s}}{2q_{1}}.$ and note that $\\eta >1$ .","When $\\beta =\\eta $ we have that $2\\beta q_{1} = 2^{\\ast }_{s}$ .","Then, by REF $||v||_{2^{\\ast }_{s}\\eta } \\le (\\eta S||h||_{q})^{\\frac{1}{2\\eta }}||v||_{2^{\\ast }_{s}}.$ Taking $\\beta =\\eta ^{2}$ in REF we obtain $||v||_{2^{\\ast }_{s}\\eta ^{2}} \\le \\eta ^{\\frac{1}{\\eta ^{2}}} (S||h||_{q})^{\\frac{1}{2\\eta ^{2}}}||v||_{2^{\\ast }_{s}\\eta }.$ By inequalities REF and REF we have $||v||_{2^{\\ast }_{s}\\eta ^{2}} \\le \\eta ^{\\frac{1}{(\\eta ^{2}}+\\frac{1}{2\\eta })} (S||h||_{q})^{(\\frac{1}{2\\eta ^{2}}+\\frac{1}{2\\eta })}||v||_{2^{\\ast }_{s}}.$ Inductively, we can prove that $||v||_{2^{\\ast }_{s}\\eta ^{m}} \\le \\eta ^{(\\frac{1}{2\\eta }+\\frac{1}{\\eta ^{2}}+...+\\frac{m}{2\\eta ^{m}})} (S||h||_{q})^{(\\frac{1}{2\\eta }+\\frac{1}{2\\eta ^{2}}+...+\\frac{1}{2\\eta ^{m}})}||v||_{2^{\\ast }_{s}}$ for all $m \\in \\mathbb {N}$ .","But, $\\sum _{i=1}^{\\infty }\\frac{m}{2\\eta ^{m}}=\\frac{1}{2(\\eta -1)^{2}}$ and $\\sum _{i=1}^{\\infty }\\frac{1}{2\\eta ^{m}}=\\frac{1}{2(\\eta -1)}.$ Thereby, for all $m>0$ $||v||_{2^{\\ast }_{s}\\eta ^{m}} \\le \\eta ^{\\frac{1}{2(\\eta -1)^{2}}} (S||h||_{q})^{\\frac{1}{2(\\eta -1)}}||v||_{2^{\\ast }_{s}}$ Recalling that $||v||_{\\infty }=\\lim \\limits _{n \\rightarrow \\infty }||v||_{p}$ and that $\\eta >1$ we have that $||v||_{\\infty }\\le M||v||_{2^{\\ast }_{s}}$ for $M=\\eta ^{\\frac{1}{2(\\eta -1)^{2}}} (S||h||_{q})^{\\frac{1}{2(\\eta -1)}}$ and $\\eta = \\frac{n(q-1)}{q(n-2s)}$ We conclude the proof of Proposition REF noting that $M$ depends only on $q$ , $||h||_{q}$ ."],["Solution for Problem (P)","In this section, we prove the main result, the Theorem REF .","By Corollary REF , there is $u \\in X$ such that $J(u)=c$ and $J^{\\prime }(u)=0$ .","We have the following estimate for $||u||_{\\infty }$ .","Lemma 5.1 The solution $u$ of the auxiliary problem satisfies $||u||_{\\infty } \\le M(2Skd)^{\\frac{1}{2}}$ Consider the functions $\\begin{array}{lll}H(x,t)&=&\\left\\lbrace \\begin{array}{lllll}f(t)&if&|x| \\frac{V(x)}{k}t\\end{array}\\right.\\end{array}$ and $\\begin{array}{lll}b(x)&=&\\left\\lbrace \\begin{array}{lllll}V(x)&if&|x| \\frac{V(x)}{k}u.\\end{array}\\right.\\end{array}$ Note that the function $u$ is solution of $\\left\\lbrace \\begin{array}{ll}&-\\mathcal {L}_{k}u+b(x)u=H(x,u)\\\\&u \\in E.\\end{array}\\right.$ By $(f_{1})$ , $|H(x,t)|\\le c_{0}|t|^{p-1}$ for $p \\in (2,2^{\\ast }_{s})$ .","Thereby, $|H(x,u)|\\le h(x)|u|,$ where $h(x)=C_{0}|u|^{p-2}$ .","Note that $h \\in L^{\\frac{2^{\\ast }_{s}}{p-2}}$ with $||h||_{L^{q}(\\mathbb {R}^{n})}\\le C(2ksd)^{\\frac{p-2}{2^{\\ast }_{s}}}.$ The number $p$ satisfies $p<2^{\\ast }_{s} = 2+\\frac{2s}{n}2^{\\ast }_{s}.$ Then $q =\\frac{2^{\\ast }_{s}}{p-2}>\\frac{n}{2s}.$ By Proposition REF and sobolev embedding $||u||_{\\infty }\\le M||u||_{2^{\\ast }_{s}} \\le MS^{\\frac{1}{2}}||u||,$ where $M=M(q,||h||_{q})$ .","By Proposition REF , we have $||u||_{\\infty } \\le M(2kSd)^{\\frac{1}{2}}.$ Theorem 5.2 Suppose that $V$ satisfies $(V_{1})$ -$(V_{2})$ and that $f$ satisfies $(f_{1})$ -$(f_{3})$ .","There is $\\Lambda ^{\\ast }=\\Lambda ^{\\ast }(V_{\\infty },\\theta ,p,c_{0},s)>0$ such that if $\\Lambda >\\Lambda ^{\\ast }$ in $(V_{3})$ , then the problem $(P)$ has a nonnegative nontrivial solution.","Indeed, let $|x|\\ge R$ .","If $u(x)=0$ then by denfition $f(u(x))=g(x,u(x))$ .","If $u(x)>0$ then $\\begin{array}{ll}\\frac{f(u(x))}{u(x)}&\\le c_{0}|u|^{p-2} \\\\&\\le c_{0}||u||_{\\infty }^{p-2} \\\\& = \\frac{ c_{0}||u||_{\\infty }^{p-2}}{\\Lambda }\\Lambda \\\\& \\le \\frac{k^{\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\frac{p-2}{2}}}{\\Lambda }\\frac{V(x)}{k}\\end{array}$ Define $\\Lambda ^{\\ast }=k^{\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\frac{p-2}{2}}.$ If $\\Lambda >\\Lambda ^{\\ast }$ then $\\frac{f(u(x))}{u(x)} \\le \\frac{V(x)}{k}.$ By definition of $g$ we have $g(x,u(x))=f(u(x))$ .","Then $g(x,u(x))=f(u(x))$ for all $x \\in \\mathbb {R}^{n}$ .","Therefore, $u$ is a solution nonnegative and nontrivial of problem $(P)$ .","alarticle author=Alberti, G., author=Bellettini G. title=A nonlocal anisotropic model for phase transitions.","I.","The optimal profile problem, journal=Math.","Ann., volume=310, date=1998, pages=527-560, 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has solution.\"\n ],\n [\n \"Introduction\",\n \"In this article we consider the class of integro-differential Schrödinger equations $\\\\begin{array}{lcl}- \\\\mathcal {L}_{K} u +V(x)u = f(u), &\\\\mbox{ in }& \\\\mathbb {R}^n,\\\\end{array}\\\\qquad \\\\mathrm {(P)}$ where $- \\\\mathcal {L}_{K}$ is a integro-differential operator given by $- \\\\mathcal {L}_{K}u(x)= 2 \\\\int _{\\\\mathbb {R}^{n}}(u(x)-u(y))K(x-y)dy$ and $K$ satisfy general properties.\",\n \"This study leads both to nonlocal and to nonlinear difficulties.\",\n \"For example, we can not benefit from the $s$ -harmonic extension of or commutator properties (see ).\",\n \"The study of nonlocal operators is important because they intervene in a quantity of applications and models.\",\n \"For example, we mention their use in phase transition models (see , ), image reconstruction problems (see ), obstacle problem, optimization, finance, phase transitions.\",\n \"Integro-differential equations arise naturally in the study of stochastic processes with jumps, and more precisely of Lévy processes.\",\n \"This paper was motivated by , where the authors study the existence of positive solutions for the problem $\\\\left\\\\lbrace \\\\begin{array}{lcl}- \\\\Delta u +V(x)u = f(u), &\\\\mbox{ in }& \\\\mathbb {R}^n, \\\\\\\\ u \\\\in D^{1,2}(\\\\mathbb {R}^{n})\\\\end{array}\\\\right.$ where $V$ and $f$ are continuous functions with $V$ being a nonnegative function and $f$ having a subcritical or critical growth.\",\n \"Our purpose is to study an analogously problem, considering the operator $-\\\\mathcal {L}_{k}$ instead of the Laplacian operator.\",\n \"Several papers have studied the problem $(P)$ when $K (x) = \\\\frac{C_{n,s}}{2}|x|^{n + 2s}$ , where $C_{n,s}=\\\\left(\\\\int _{\\\\mathbb {R}^{n}}\\\\frac{1-\\\\cos (\\\\xi _{1})}{|\\\\xi |^{n+2s}}d \\\\xi \\\\right)^{-1},$ that is, when the operator $-\\\\mathcal {L}_{k}$ is the fractional Laplacian operator (see ).\",\n \"Next, we will mention some of these papers.\",\n \"In , the author has proved the existence of positive solutions from $(P)$ when $V$ is a constant small enough.\",\n \"Also, in , the problem was studied when $f$ is asymptotically linear and $V$ is constant.\",\n \"In , the authors study the problem $(P)$ when $V \\\\in C^{n}(\\\\mathbb {R}^{n},\\\\mathbb {R})$ , $V$ is positive and $\\\\lim \\\\limits _{n \\\\rightarrow \\\\infty }V(|x|) \\\\in (0, \\\\infty ].$ In , the authors has Studied $(P)$ when $ V $ and $ f $ are asymptotically periodic.\",\n \"When $V=1$ , Felmer et al.\",\n \"has studied the existence, regularity and qualitative properties of ground states solutions for problem $(P)$ (see ).\",\n \"In , Teng and He have shown the existence of solution for $(P)$ when $f(x,u)=P(x)|u|^{p-2}u+Q(x)|u|^{2^{\\\\ast }_{s}-2}u,$ where $2 < p < 2^{\\\\ast }_{s}$ and the potential functions $P(x)$ and $Q(x)$ satisfy certain hypothesis.\",\n \"In , the authors have shown the existence of solution for $(P)$ when $V \\\\in C^{n}(\\\\mathbb {R}^{n}, \\\\mathbb {R})$ and there exists $r_{0} > 0$ such that, for any $M > 0$ , $\\\\mbox{meas}(\\\\left\\\\lbrace x \\\\in B_{r_{0}}(y); V (x)\\\\le M\\\\right\\\\rbrace )\\\\rightarrow 0\\\\mbox{ as }|y| \\\\rightarrow \\\\infty .$ In the problem $(P)$ was studied when $V \\\\in C^1(\\\\mathbb {R}^{n},\\\\mathbb {R})$ , $\\\\liminf _{|x|\\\\rightarrow \\\\infty }V(x)\\\\ge V_{\\\\infty }$ where $V_{\\\\infty }$ is constant, and $f \\\\in C^{1}(\\\\mathbb {R}^{n}, \\\\mathbb {R})$ .\",\n \"By method of the Nehari manifold , Sechi has shown that the problem $(P)$ has a solution if $V\\\\le V_{\\\\infty }$ , but $V$ is not identically equal to $V_{\\\\infty }$ , where $V_{\\\\infty }$ is a constant.\",\n \"Also in , Secchi have obtained the existence of ground state solutions of $(P)$ for general $s \\\\in (0,1)$ when $V(x)\\\\rightarrow \\\\infty $ as $|x| \\\\rightarrow \\\\infty $ .\",\n \"In , the authors obtain the existence of a sequence of radial and non radial solutions for the problem $(P)$ when $V$ and $f$ are radial functions.\",\n \"Some other interesting studies by variational methods of the problem $(P)$ can be found in , , , , , , , , , , , , , and .\",\n \"Many of them use strong tools that we can not use here in our problem, as the $s$ -harmonic extension and commutator properties.\",\n \"Here, we will admit that the potential $V$ is continuous and satisfies, $(V_{1}-)$ $\\\\inf _{x \\\\in \\\\mathbb {R}^{n}}V(x)>0;$ $(V_{2}-)$ $V(x)\\\\le V_{\\\\infty }$ for same constant $V_{\\\\infty }>0$ and for all $x \\\\in B_{1}(0)$ .\",\n \"Note that, $(V_{1})$ implies that $(V_{3}-)$ There are $R>0$ and $\\\\Lambda >0$ such that $V(x)\\\\ge \\\\Lambda $ for all $|x|\\\\ge R$ .\",\n \"Also, we will assume that $f\\\\in C(\\\\mathbb {R},\\\\mathbb {R})$ is a function satisfying: $(f_{1}-)$ $|f(s)|\\\\le c_{0}|s|^{p-1}$ , for some constant $C>0$ and $p \\\\in (2,2^{\\\\ast }_{s})$ ; $(f_{2}-)$ There is $\\\\theta >2$ such that $\\\\theta F(s)\\\\le sf(s)$ for all $s>0$ ; $(f_{3}-)$ $f(t)>0$ for all $t>0$ and $f(t)=0$ for all $t<0$ .\",\n \"The kernel $K:\\\\mathbb {R}^{n}\\\\rightarrow \\\\left(0, \\\\infty \\\\right)$ is a measurable function such that $(K_{1}-)$ $K(x)=K(-x)$ for all $x \\\\in \\\\mathbb {R}^{n}$ ; $(K_{2}-)$ There is $\\\\lambda >0$ and $s \\\\in (0,1)$ such that $\\\\lambda \\\\le K(x)|x|^{n+2s}$ almost everywere in $\\\\mathbb {R}^{n}$ ; $(K_{3}-)$ $\\\\gamma K \\\\in L^{1}(\\\\mathbb {R}^{n})$ , where $\\\\gamma (x)=\\\\min \\\\left\\\\lbrace |x|^{2},1\\\\right\\\\rbrace $ .\",\n \"Note that, when $K(x)=\\\\frac{C_{n,s}}{2}|x|^{-(n+2s)}$ we have that $-\\\\mathcal {L}_{k}$ is a fractional laplacian, $(-\\\\Delta )^{s}$ .\",\n \"Our paper is organized as follows.\",\n \"In section 2, we will present some properties of the space in which we will study the problem $(P)$ .\",\n \"In section 3, we will define an auxiliary problem and we will show that the functional energy associated with the auxiliary problem satisfies the condition of Palais-Smale.\",\n \"By difficulty nonlocal of the operator $-\\\\mathcal {L}_{K}$ , we will can not use the same technique used in .\",\n \"Therefore, we will present an another technique to show this result.\",\n \"In section 4, we will prove that a general estimative for weak solution of $-\\\\mathcal {L}_{K}u+b(x)u=g(x,u),$ where $b\\\\ge 0$ , $|g(x,t)|\\\\le h(x)|t|$ and $h\\\\in L^{q}(\\\\mathbb {R}^{n})$ with $q>\\\\frac{n}{2s}$ .\",\n \"We will show that there is $M=M(q,||h||_{L^{q}})$ such that the solution $u$ satisfies $||u||_{\\\\infty } \\\\le M||u||_{2^{\\\\ast }_{s}}.$ In , using the $s$ -harmonic extension of , the authors has shown the same estimate when $ -\\\\mathcal {L}_{K}$ is the fractional Laplacian operator.\",\n \"In our case, we can not use the $s$ -harmonic extension, because we do not have an analogously extension for integro-differential operators.\",\n \"In section 5, we show our main result in this paper, the Theorem REF .\"\n ],\n [\n \"Preliminaries\",\n \"Let $s \\\\in (0,1)$ , we denote by $H^{s}(\\\\mathbb {R}^{n})$ the fractional sobolev space.\",\n \"It is defined as $H^{s}(\\\\mathbb {R}^{n}):=\\\\left\\\\lbrace u \\\\in L^{2}(\\\\mathbb {R}^{n});\\\\int _{\\\\mathbb {R}^{n}}\\\\int _{\\\\mathbb {R}^{n}}\\\\frac{(u(x)-u(y))^{2}}{|x-y|^{n+2s}}dxdy<\\\\infty \\\\right\\\\rbrace .$ The space $H^{s}(\\\\mathbb {R}^{n})$ is a Hilbert space with the norm $||u||_{H^{s}}=\\\\left(\\\\int _{\\\\mathbb {R}^{n}}|u|^{2}dx+\\\\int _{\\\\mathbb {R}^{n}}\\\\int _{\\\\mathbb {R}^{n}}\\\\frac{(u(x)-u(y))^{2}}{|x-y|^{n+2s}}dxdy\\\\right)^{\\\\frac{1}{2}}$ We define $X$ as the linear space of Lebesgue measurable functions from $\\\\mathbb {R}^{n}$ to $\\\\mathbb {R}$ such that any function $u$ in $X$ belongs to $L^{2}(\\\\mathbb {R}^{n})$ and the function $(x,y)\\\\longmapsto (u(x)-u(y))\\\\sqrt{K(x-y)}$ is in $L^{2}(\\\\mathbb {R}^{n}\\\\times \\\\mathbb {R}^{n})$ .\",\n \"The function $||u||_{X}:= \\\\left(\\\\int _{\\\\mathbb {R}^{n}}u^{2}dx+\\\\int _{\\\\mathbb {R}^{n}}\\\\int _{\\\\mathbb {R}^{n}}(u(x)-u(y))^{2}K(x-y)dxdy\\\\right)^{\\\\frac{1}{2}}$ defines a norm in $X$ and $(X, ||\\\\cdot ||_{X})$ is a Hilbert space.\",\n \"By $(K_{2})$ , the space $X$ is continuously embedded in $H^{s}(\\\\mathbb {R}^{n})$ .\",\n \"Therefore, $X$ is continuously embedded in $L^{p}(\\\\mathbb {R}^{n})$ for $p \\\\in \\\\left[2,2^{\\\\ast }_{s}\\\\right]$ , where $2^{\\\\ast }_{s}=\\\\frac{2n}{n-2s}$ .\",\n \"If $\\\\Omega \\\\subset \\\\mathbb {R}^{n}$ , we define $X_{0}(\\\\Omega )=\\\\left\\\\lbrace u \\\\in X; u=0 \\\\mbox{ in } \\\\mathbb {R}^{n}\\\\setminus \\\\Omega .\\\\right\\\\rbrace .$ The space $X_{0}(\\\\Omega )$ is a Hilbert Space with the norm $u \\\\longmapsto ||u||_{X_{0}(\\\\Omega )}:=\\\\left(\\\\int _{\\\\Omega }u^{2}dx+\\\\int _{Q}\\\\left(u(x)-u(y)\\\\right)^{2}K(x-y)dxdy\\\\right)^{\\\\frac{1}{2}},$ where $Q=(\\\\mathbb {R}^{n}\\\\times \\\\mathbb {R}^{n}) \\\\setminus (\\\\Omega ^{c} \\\\times \\\\Omega ^{c})$ (see Lemma 7 in ).\",\n \"It is continuously embedded in $H_{0}^{s}(\\\\mathbb {R}^{n})$ .\",\n \"For definition and properties of $H_{0}^{s}(\\\\mathbb {R}^{n})$ we indicate .\",\n \"In the problem $(P)$ we will consider the space $E$ defined as $E=\\\\left\\\\lbrace u \\\\in X; \\\\int _{\\\\mathbb {R}^{n}}V(x)u^{2}dx< \\\\infty \\\\right\\\\rbrace $ The space $E$ is a Hilbert space with the norm $u \\\\longmapsto ||u||:=\\\\left(\\\\int _{\\\\mathbb {R}^{n}}\\\\int _{\\\\mathbb {R}^{n}}\\\\left(u(x)-u(y)\\\\right)^{2}K(x-y)dxdy+\\\\int _{\\\\mathbb {R}^{n}}V(x)u^{2}dx\\\\right)^{\\\\frac{1}{2}}.$ If $u,v \\\\in C_{0}^{\\\\infty }(\\\\mathbb {R}^{n})$ then $(-\\\\mathcal {L}_{k}u,v)_{L^{2}}=[u,v].$ where $[u,v]=\\\\int _{\\\\mathbb {R}^{n}}\\\\int _{\\\\mathbb {R}^{n}}\\\\left(u(x)-u(y)\\\\right)\\\\left(v(x)-v(y)\\\\right)K(x-y)dxdy.$ Therefore, we say that $u\\\\in E$ is a solution for the problem $(P)$ if $[u,v]+\\\\int _{\\\\mathbb {R}^{n}}V(x)uvdx = \\\\int _{\\\\mathbb {R}^{n}}f(u)vdx$ for all $v \\\\in E$ , that is $\\\\int _{\\\\mathbb {R}^{n}}\\\\int _{\\\\mathbb {R}^{n}}\\\\left(u(x)-u(y)\\\\right)\\\\left(v(x)-v(y)\\\\right)K(x-y)dxdy+\\\\int _{\\\\mathbb {R}^{n}}V(x)uvdx = \\\\int _{\\\\mathbb {R}^{n}}f(u)vdx.$ Let $A, B \\\\subset \\\\mathbb {R}^{n}$ and $u,v \\\\in X$ .\",\n \"We will denote $[u,v]_{A\\\\times B}=\\\\int _{A}\\\\int _{B}(u(x)-u(y))(v(x)-v(y))K(x-y)dxdy$ and we will denote $[u,v]_{\\\\mathbb {R}^{n}\\\\times \\\\mathbb {R}^{n}}$ by $[u,v]$ .\",\n \"The Euler-Lagrange functional associated with $(P)$ is given by $I(u)=\\\\frac{1}{2}||u||^{2}-\\\\int _{\\\\mathbb {R}^{n}}F(u)dx,$ where $F(t)=\\\\int _{0}^{t}f(s)ds.$ From hypothesis about $f$ , the functional is $C^{1}(E,\\\\mathbb {R})$ and $I^{\\\\prime }(u)v=[u,v]+\\\\int _{\\\\mathbb {R}^{n}}V(x)uvdx-\\\\int _{\\\\mathbb {R}^{n}}f(u)vdx.$ We will denote by $B$ the unitary ball of $\\\\mathbb {R}^{n}$ .\",\n \"Define $I_{0}:X_{0}(B)\\\\longrightarrow \\\\mathbb {R}$ by $I_{0}(u)=:\\\\int _{\\\\mathbb {R}^{n}} \\\\int _{\\\\mathbb {R}^{n}}\\\\left(u(x)-u(y)\\\\right)^{2}K(x-y)dxdy+\\\\int _{\\\\mathbb {R}^{n}}V_{\\\\infty }u^{2}dx-\\\\int _{\\\\mathbb {R}^{n}}F(u)dx,$ where $V_{\\\\infty }$ is the constant of $(V_{2})$ .\",\n \"The functional $I_{0}$ has the mountain pass geometry.\",\n \"We will denote by $d$ the mountain pass level associated with $I_{0}$ , that is $d= \\\\inf _{\\\\gamma \\\\in \\\\Gamma } \\\\max _{t \\\\in [0,1]}I_{0}(\\\\gamma (t)),$ where $\\\\Gamma =\\\\left\\\\lbrace \\\\gamma \\\\in C([0,1],X_{0}(\\\\Omega )); \\\\gamma (0)=0\\\\mbox{ and }\\\\gamma (1)=e\\\\right\\\\rbrace ,$ with $e$ fixed and verifying $I_{0}(e)<0$ .\",\n \"Note that $d$ depends only on $V_{\\\\infty }$ , $\\\\theta $ and $f$ .\"\n ],\n [\n \"An Auxiliary Problem\",\n \"According to , we will modified the problem defining an auxiliary problem.\",\n \"But, as the operator $-\\\\mathcal {L}_{K}$ is nonlocal, we can not use the same ideas of to prove that the functional associated the auxiliary problem satisfies the Palais-Smale condition.\",\n \"It is necessary that we use an another technics.\",\n \"For $k=\\\\frac{2 \\\\theta }{\\\\theta -2}$ we consider $\\\\begin{array}{lll}\\\\tilde{f}(x,t)&=& \\\\left\\\\lbrace \\\\begin{array}{lll}f(t) & if & kf(t) \\\\le V(x)t \\\\\\\\\\\\frac{V(x)}{k}t& if & kf(t)>V(x)t\\\\end{array}\\\\right.\\\\end{array}$ and $\\\\begin{array}{lll}g(x,t)&=& \\\\left\\\\lbrace \\\\begin{array}{lll}f(t) & if & |x|\\\\le R \\\\\\\\\\\\tilde{f}(x,t)& if & |x|>R.\\\\end{array}\\\\right.\\\\end{array}$ And we define the auxiliary problem $\\\\left\\\\lbrace \\\\begin{array}{lllll}-\\\\mathcal {L}_{K}u+V(x)u&=&g(x,u)& in &\\\\mathbb {R}^{n} \\\\\\\\u \\\\in E&&&&\\\\end{array}\\\\right.$ We have that, for all $t \\\\in \\\\mathbb {R}$ and $x \\\\in \\\\mathbb {R}^{n}$ $\\\\tilde{f}(x,t)\\\\le f(t)$ ; $g(x,t) \\\\le \\\\frac{V(x)}{k}t$ ,se $|x|\\\\ge R$ ; $G(x,t)=F(t)$ se $|x|\\\\le R$ $G(x,t)\\\\le \\\\frac{V(x)}{2k}t^{2}$ se $|x|>R$ ; where $G(x,t)=\\\\int _{0}^{t}g(x,s)ds.$ The Euler-Lagrange functional associated with the auxiliary problem is given by $J(u)=\\\\frac{1}{2}||u||^{2} - \\\\int _{\\\\mathbb {R}^{n}}G(x,u)dx.$ The functional $J \\\\in C^{1}(X,\\\\mathbb {R})$ and $\\\\begin{array}{ll}J^{\\\\prime }(u)v&=\\\\int _{\\\\mathbb {R}^{n}}\\\\int _{\\\\mathbb {R}^{n}}\\\\left(u(x)-u(y)\\\\right)\\\\left(v(x)-v(y)\\\\right)K(x-y)dxdy \\\\\\\\&+\\\\int _{\\\\mathbb {R}^{n}}V(x)uvdx - \\\\int _{\\\\mathbb {R}^{n}}g(x,u)vdx.\\\\end{array}$ The functional $J$ has the mountain pass geometry.\",\n \"Then, there is a sequence $\\\\left\\\\lbrace u_{n}\\\\right\\\\rbrace _{n \\\\in \\\\mathbb {N}}$ such that $J^{\\\\prime }(u_{n})\\\\rightarrow 0\\\\mbox{ and }J(u_{n})\\\\rightarrow c,$ where $c>0$ is the mountain pass level associated with $J$ , that is $c= \\\\inf _{\\\\gamma \\\\in \\\\Gamma } \\\\max _{t \\\\in [0,1]}J(\\\\gamma (t))$ where $\\\\Gamma =\\\\left\\\\lbrace \\\\gamma \\\\in C([0,1],E); \\\\gamma (0)=0\\\\mbox{ and }, \\\\gamma (1)=e\\\\right\\\\rbrace .$ and $e$ is the function fixed in $\\\\ref {eq1}$ .\",\n \"By definition $c \\\\le d$ uniformly in $R>0$ .\",\n \"Lemma 3.1 The sequence $\\\\left\\\\lbrace u_{n}\\\\right\\\\rbrace _{n \\\\in \\\\mathbb {N}}$ is bounded.\",\n \"By $(f_{2})$ , $(3)$ and $(4)$ $\\\\begin{array}{ll}&J(u)-\\\\frac{1}{\\\\theta }J^{\\\\prime }(u)u\\\\\\\\& = \\\\left(\\\\frac{\\\\theta -2}{4\\\\theta }\\\\right)||u||^{2}+ \\\\frac{1}{2k}||u||^{2} + \\\\int _{\\\\mathbb {R}^{n}}\\\\frac{1}{\\\\theta }g(x,u)u-G(x,u)dx \\\\\\\\&\\\\ge \\\\left(\\\\frac{\\\\theta -2}{4\\\\theta }\\\\right)||u||^{2}+ \\\\frac{1}{2k}||u||^{2} + \\\\int _{|x|>R}\\\\frac{1}{\\\\theta }g(x,u)u - \\\\frac{1}{2k}\\\\int _{|x|>R}V(x)u^{2}dx \\\\\\\\& \\\\ge \\\\left(\\\\frac{1}{2k}\\\\right)||u||^{2}.\\\\end{array}$ Thereby $|J(u)|+|J^{\\\\prime }(u)u|\\\\ge \\\\left(\\\\frac{\\\\theta -2}{4\\\\theta }\\\\right)||u||^{2}$ for all $u \\\\in E$ .\",\n \"This last inequality ensures that the sequence is bounded.\",\n \"Let $r>R$ and $A=\\\\left\\\\lbrace x \\\\in \\\\mathbb {R}^{n}; r<||x||<2r\\\\right\\\\rbrace $ .\",\n \"Consider $\\\\eta :\\\\mathbb {R}^{n}\\\\rightarrow \\\\mathbb {R}$ a function such that $\\\\eta =1$ in $B_{2r}^{c}(0)$ , $\\\\eta =0$ in $B_{r}(0)$ , $0\\\\le \\\\eta \\\\le 1$ and $|\\\\nabla \\\\eta |<\\\\frac{2}{r}$ .\",\n \"Note that $(B_{r} \\\\times B_{r})^{c} = (B_{r}^{c}\\\\times \\\\mathbb {R}^{n}) \\\\cup ( B_{r} \\\\times B_{r}^{c}),$ where $B_{r}=B_{r}(0)$ and $B_{2r}=B_{2r}(0)$ .\",\n \"We will decompose $B_{r}^{c}\\\\times \\\\mathbb {R}^{n} = (A \\\\times \\\\mathbb {R}^{n}) \\\\cup (B_{2r}^{c}\\\\times B_{r}) \\\\cup (B_{2r}^{c}\\\\times A) \\\\cup (B_{2r}^{c}\\\\times B_{2r}^{c})$ and $B_{r} \\\\times B_{r}^{c} = (B_{r} \\\\times A) \\\\cup (B_{r} \\\\times B_{2r}^{c})$ Lemma 3.2 We have that $\\\\begin{array}{ll}&\\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))(\\\\eta (x)u_{n}(x)-\\\\eta (y)u_{n}(y))K(x-y)dxdy\\\\\\\\&+\\\\int _{B_{2r}^{c}}\\\\int _{B_{r}}(u_{n}(x)-u_{n}(y))(\\\\eta (x)u_{n}(x)-\\\\eta (y)u_{n}(y))K(x-y)dxdy \\\\\\\\& \\\\ge -\\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy\\\\end{array}$ $\\\\begin{array}{ll}&\\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))(\\\\eta (x)u_{n}(x)-\\\\eta (y)u_{n}(y))K(x-y)dxdy \\\\\\\\& +\\\\int _{B_{2r}^{c}}\\\\int _{B_{r}}(u_{n}(x)-u_{n}(y))(\\\\eta (x)u_{n}(x)-\\\\eta (y)u_{n}(y))K(x-y)dxdy \\\\\\\\& =\\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(x)(u_{n}(x)-u_{n}(y))K(x-y)dxdy\\\\\\\\&- \\\\int _{B_{2r}^{c}}\\\\int _{B_{r}}u_{n}(y)(u_{n}(x)-u_{n}(y))K(x-y)dxdy \\\\\\\\&= \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(x)(u_{n}(x)-u_{n}(y))K(x-y)dxdy\\\\\\\\&- \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(y)(u_{n}(x)-u_{n}(y))K(x-y)dydx \\\\\\\\& = \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy \\\\\\\\&+ \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}(u_{n}(y)+u_{n}(x))(u_{n}(x)-u_{n}(y))K(x-y)dxdy\\\\\\\\& = \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy \\\\\\\\&+ \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(x)^{2}-u_{n}(y)^{2}K(x-y)dxdy \\\\\\\\& \\\\ge -\\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy\\\\end{array}$ Lemma 3.3 Let $\\\\epsilon >0$ .\",\n \"There is $r_{0}>0$ such that if $r>r_{0}$ then $\\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy<\\\\epsilon ,$ for all $n \\\\in \\\\mathbb {N}$ .\",\n \"For each $y\\\\in B_{r}(0)$ $B_{r}(y) \\\\subset B_{2r}(0).$ Then $\\\\begin{array}{ll}&\\\\int _{B_{2r}(0)^{c}}K(x-y)dx \\\\\\\\&\\\\le \\\\int _{B_{r}(y)^{c}}K(x-y)dx\\\\\\\\& = \\\\int _{B_{r}(0)^{c}}K(z)dz.\",\n \"\\\\\\\\\\\\end{array}$ By Lemma REF , there is $L>0$ such that $||u_{n}||_{L^{2}}^{2}0$ such that $\\\\int _{B_{r}(0)^{c}}K(z)dz<\\\\frac{\\\\epsilon }{L},$ for all $r>r_{0}$ .\",\n \"Then by REF , for all $n \\\\in \\\\mathbb {N}$ and $r>r_{0}$ $\\\\begin{array}{ll}&\\\\int _{B_{r}(0)}\\\\int _{B_{2r}(0)^{c}}u_{n}(y)^{2}K(x-y)dxdy\\\\\\\\& =\\\\int _{B_{r}(0)}u_{n}(y)^{2}\\\\int _{B_{2r}(0)^{c}}K(x-y)dxdy\\\\\\\\& \\\\le \\\\int _{B_{r}(0)}u_{n}(y)^{2}\\\\int _{B_{r}(0)^{c}}K(z)dzdy \\\\\\\\& = \\\\int _{B_{r}(0)^{c}}K(z)dz\\\\int _{B_{r}(0)}u_{n}(y)^{2}dy\\\\\\\\& \\\\le \\\\epsilon \\\\end{array}$ Lemma 3.4 There are constants $K_{1}>0$ and $K_{2}>0$ such that $\\\\begin{array}{ll}&\\\\int _{A}\\\\int _{\\\\mathbb {R}^{n}}|u_{n}(y)||(u_{n}(x)-u_{n}(y))||(\\\\eta (x)-\\\\eta (y))|K(x-y)dxdy \\\\\\\\&\\\\le \\\\frac{2K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}}.\\\\end{array}$ Note that $\\\\begin{array}{ll}\\\\int _{\\\\mathbb {R}^{n}}|\\\\eta (x)-\\\\eta (y)|^{2}K(x-y)dx& = \\\\int _{\\\\mathbb {R}^{n}}|\\\\eta (z+y)-\\\\eta (y)|^{2}K(z)dz \\\\\\\\& =\\\\int _{B_{1}(0)}|\\\\eta (z+y)-\\\\eta (y)|^{2}K(z)dz\\\\\\\\&+\\\\int _{B_{1}^{c}(0)}|\\\\eta (z+y)-\\\\eta (y)|^{2}K(z)dz \\\\\\\\& \\\\le \\\\frac{4}{r^{2}}\\\\int _{B_{1}(0)}|z|^{2}K(z)dz +4\\\\int _{B_{1}(0)^{c}}K(z)dz \\\\\\\\& \\\\le \\\\frac{4}{r^{2}}P_{1}+4P_{2},\\\\end{array}$ where $P_{1}=\\\\int _{B_{1}}|z|^{2}K(z)dz$ and $P_{2}= \\\\int _{B_{1}^{c}}K(z)dz.$ Let $K_{1}=2\\\\sqrt{P_{1}}$ and $K_{2}=2\\\\sqrt{P_{2}}$ .\",\n \"Then, by Holder inequality $\\\\begin{array}{ll}&\\\\int _{A}\\\\int _{\\\\mathbb {R}^{n}}|u_{n}(y)||(u_{n}(x)-u_{n}(y))||(\\\\eta (x)-\\\\eta (y))|K(x-y)dxdy \\\\\\\\& \\\\le (\\\\frac{2\\\\sqrt{P_{1}}}{r}+2\\\\sqrt{P_{2}})\\\\int _{A}|u_{n}(y)|\\\\left(\\\\int _{\\\\mathbb {R}^{n}}|(u_{n}(x)-u_{n}(y))|^{2}K(x-y)dx\\\\right)^{\\\\frac{1}{2}}dy\\\\\\\\& \\\\le (\\\\frac{K_{1}}{r}+K_{2})||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}}.\\\\end{array}$ Lemma 3.5 For the same constants $K_{1}>0$ and $K_{2}>0$ of the Lemma REF $\\\\begin{array}{ll}&\\\\int _{B_{r}}\\\\int _{A}|u_{n}(x)-u_{n}(y)||\\\\eta (x)u_{n}(x)-\\\\eta (y)u_{n}(y)|K(x-y)dxdy \\\\\\\\&\\\\le \\\\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}}.\\\\end{array}$ Indeed, by property $(K_{1})$ of $K$ $\\\\begin{array}{ll}&\\\\int _{B_{r}}\\\\int _{A}|u_{n}(x)-u_{n}(y)||\\\\eta (x)u_{n}(x)-\\\\eta (y)u_{n}(y)|K(x-y)dxdy\\\\\\\\& = \\\\int _{B_{r}}\\\\int _{A}|u_{n}(x)||u_{n}(x)-u_{n}(y)||\\\\eta (x)|K(x-y)dxdy \\\\\\\\& = \\\\int _{A}\\\\int _{B_{r}}|u_{n}(x)||u_{n}(x)-u_{n}(y)||n(x)-n(y)|K(x-y)dydx\\\\\\\\& =\\\\int _{A}\\\\int _{B_{r}}|u_{n}(y)||u_{n}(y)-u_{n}(x)||n(y)-n(x)|K(y-x)dxdy\\\\end{array}$ $Using $ K1)$ and Lemma \\\\ref {lem34}, we prove this lemma.$ Lemma 3.6 We have that $\\\\begin{array}{ll}&-\\\\int _{B_{2r}^{c}}\\\\int _{A}u_{n}(y)(u_{n}(x)-u_{n}(y))(\\\\eta (x)-\\\\eta (y))K(x-y)dxdy \\\\\\\\&\\\\le \\\\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}}.\\\\end{array}$ $\\\\begin{array}{ll}&-\\\\int _{B_{2r}^{c}}\\\\int _{A}u_{n}(y)(u_{n}(x)-u_{n}(y))(\\\\eta (x)-\\\\eta (y))K(x-y)dxdy\\\\\\\\&=\\\\int _{B_{2r}^{c}}\\\\int _{A}(u_{n}(x)-u_{n}(y))^{2}(\\\\eta (x)-\\\\eta (y))K(x-y)dxdy \\\\\\\\&-\\\\int _{B_{2r}^{c}}\\\\int _{A}u_{n}(x)(u_{n}(x)-u_{n}(y))(\\\\eta (x)-\\\\eta (y))K(x-y)dxdy \\\\\\\\& = \\\\int _{B_{2r}^{c}}\\\\int _{A}(u_{n}(x)-u_{n}(y))^{2}(n(x)-1)K(x-y)dxdy \\\\\\\\&-\\\\int _{B_{2r}^{c}}\\\\int _{A}u_{n}(x)(u_{n}(x)-u_{n}(y))(n(x)-n(y))K(x-y)dxdy \\\\\\\\& \\\\le -\\\\int _{B_{2r}^{c}}\\\\int _{A}u_{n}(x)(u_{n}(x)-u_{n}(y))(n(x)-n(y))K(x-y)dxdy \\\\\\\\& = -\\\\int _{B_{2r}^{c}}\\\\int _{A}u_{n}(x)(u_{n}(y)-u_{n}(x))(n(y)-n(x))K(x-y)dxdy \\\\\\\\& \\\\le \\\\int _{B_{2r}^{c}}\\\\int _{A}|u_{n}(x)||u_{n}(y)-u_{n}(x)||n(y)-n(x)|K(x-y)dxdy \\\\\\\\& =\\\\int _{A}\\\\int _{B_{2r}^{c}}|u_{n}(x)||u_{n}(y)-u_{n}(x)||n(y)-n(x)|K(y-x)dydx \\\\\\\\& \\\\le \\\\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}}.\\\\end{array}$ In the last inequality, we have used the Lemma REF and $(K_{1})$ .\",\n \"We will prove that the functional $J$ satisfies the Palais-Smale condition.\",\n \"The next proposition ensures the existence of solution in the level $c$ for the auxiliary problem (see REF ).\",\n \"We emphasize that by a nonlocal difficulty, we can not repeat the same arguments used in to show that the functional energy associated at the auxiliary problem satisfies the Palais-Smale condition, therefore we use another technique to show this.\",\n \"Proposition 3.7 Suppose that $f$ and $V$ satisfy $(V_{1})$ , $(f_{1})-(f_{3})$ .\",\n \"Then the functional $J$ satisfies the Palais-Smale condition.\",\n \"By Lemma REF the Palais-Smale sequence $\\\\left\\\\lbrace u_{n}\\\\right\\\\rbrace _{n \\\\in \\\\mathbb {N}}$ is bounded.\",\n \"We can suppose that $\\\\left\\\\lbrace u_{n}\\\\right\\\\rbrace _{n \\\\in \\\\mathbb {N}}$ converges weakly to $u$ .\",\n \"By $K$ properties we have that $\\\\eta u_{n} \\\\in X$ and $||\\\\eta u_{n}|| \\\\le ||u_{n}||$ (see Lemma $5.1$ in ).\",\n \"Then, the sequence $\\\\left\\\\lbrace \\\\eta u_{n}\\\\right\\\\rbrace $ is bounded in $X$ .\",\n \"Therefore $I^{^{\\\\prime }}(u_{n})(\\\\eta u_{n})=o_{n}(1)$ .\",\n \"That is $\\\\begin{array}{ll}&[u_{n},\\\\eta u_{n}]+\\\\int _{\\\\mathbb {R}^{n}}V(x)u_{n}^{2}\\\\eta dx = \\\\int _{\\\\mathbb {R}^{n}}g(x,u_{n})\\\\eta u_{n}dx+ o_{n}(1).\\\\end{array}$ But, note that $[u_{n},\\\\eta u_{n}] = [u_{n},\\\\eta u_{n}]_{C(B_{r}\\\\times B_{r})}$ , because $\\\\eta =0$ in $B_{r}$ .\",\n \"By REF , REF and REF we have: $\\\\begin{array}{ll}&[u_{n},\\\\eta u_{n}]_{A \\\\times \\\\mathbb {R}^{n}}+[u_{n},\\\\eta u_{n}]_{B_{2r}^{c}\\\\times A}+[u_{n},\\\\eta u_{n}]_{B_{2r}^{c}\\\\times B_{2r}^{c}}\\\\\\\\&+[u_{n},\\\\eta u_{n}]_{B_{2r}^{c}\\\\times B_{r}}+[u_{n},\\\\eta u_{n}]_{B_{r}\\\\times B_{2r}^{c}}+[u_{n},\\\\eta u_{n}]_{B_{r}\\\\times A}\\\\\\\\&+\\\\int _{\\\\mathbb {R}^{n}}V(x)u_{n}^{2}\\\\eta dx = \\\\int _{\\\\mathbb {R}^{n}}g(x,u_{n})\\\\eta u_{n}dx+ o_{n}(1)\\\\end{array}$ By Lemma REF , $\\\\begin{array}{ll}&[u_{n},\\\\eta u_{n}]_{A \\\\times \\\\mathbb {R}^{n}}+[u_{n},\\\\eta u_{n}]_{B_{2r}^{c}\\\\times A}+\\\\int _{\\\\mathbb {R}^{n}}V(x)u_{n}^{2}\\\\eta dx \\\\\\\\&\\\\le \\\\int _{\\\\mathbb {R}^{n}}g(x,u_{n})\\\\eta u_{n}dx+ \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy-[u_{n},\\\\eta u_{n}]_{B_{r}\\\\times A}+o_{n}(1)\\\\end{array}$ Above, we have used that $[u_{n},\\\\eta u_{n}]_{B_{2r}^{c}\\\\times B_{2r}^{c}} =[u_{n}, u_{n}]_{B_{2r}^{c}\\\\times B_{2r}^{c}} \\\\ge 0$ .\",\n \"Because $\\\\eta = 1$ in $B_{2r}^{c}$ .\",\n \"If $C$ and $D$ are subsets of $\\\\mathbb {R}^{n}$ and $u \\\\in E$ , then $\\\\begin{array}{ll}[u, \\\\eta u]_{C \\\\times D}&=\\\\int _{C}\\\\int _{D}(u(x)-u(y))(\\\\eta u(x)-\\\\eta u(y))K(x-y)dxdy \\\\\\\\&=\\\\int _{C}\\\\int _{D}\\\\eta (x)(u(x)-u(y))^{2}K(x-y)dxdy\\\\\\\\& + \\\\int _{C}\\\\int _{D}u(y)(u(x)-u(y))(\\\\eta (x)-\\\\eta (y))K(x-y)dxdy.\\\\end{array}$ Thereby, $\\\\begin{array}{ll}&\\\\int _{A}\\\\int _{\\\\mathbb {R}^{n}}\\\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy+\\\\\\\\&+\\\\int _{B_{2r}^{c}}\\\\int _{A}\\\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy +\\\\int _{\\\\mathbb {R}^{n}}V(x)u_{n}^{2}\\\\eta dx \\\\\\\\& \\\\le \\\\int _{\\\\mathbb {R}^{n}}g(x,u_{n})\\\\eta u_{n}dx+ \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy-[u_{n},\\\\eta u_{n}]_{B_{r}\\\\times A} \\\\\\\\& -\\\\int _{A}\\\\int _{\\\\mathbb {R}^{n}}u_{n}(y)(u_{n}(x)-u_{n}(y))(\\\\eta (x)-\\\\eta (y))K(x-y)dxdy \\\\\\\\& -\\\\int _{B_{2r}^{c}}\\\\int _{A}u(y)(u_{n}(x)-u_{n}(y))(\\\\eta (x)-\\\\eta (y))K(x-y)dxdy+o_{n}(1).\\\\end{array}$ From Lemmas REF , REF and REF , we obtain constants $K_{1},K_{2}>0$ such that $\\\\begin{array}{ll}&\\\\int _{\\\\mathbb {R}^{n}}V(x)u_{n}^{2}\\\\eta dx \\\\\\\\&\\\\le \\\\int _{A}\\\\int _{\\\\mathbb {R}^{n}}\\\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy\\\\\\\\&+\\\\int _{B_{2r}^{c}}\\\\int _{A}\\\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy +\\\\int _{\\\\mathbb {R}^{n}}V(x)u_{n}^{2}\\\\eta dx \\\\\\\\& \\\\le \\\\int _{\\\\mathbb {R}^{n}}g(x,u_{n})\\\\eta u_{n}dx+ \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy \\\\\\\\&+\\\\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}}+o_{n}(1).\\\\end{array}$ By $(2)$ , $(f_{3})$ and $r>R$ we have $\\\\int _{\\\\mathbb {R}^{n}}g(x,u_{n})\\\\eta u_{n}dx \\\\le \\\\frac{1}{k}\\\\int _{\\\\mathbb {R}^{n}}\\\\eta V(x)u_{n}^{2}dx.$ Thereby, $\\\\begin{array}{ll}&\\\\left(1-\\\\frac{1}{k}\\\\right)\\\\int _{\\\\mathbb {R}^{n}}V(x)u_{n}^{2}\\\\eta dx \\\\\\\\& \\\\le \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy \\\\\\\\&+\\\\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\\\frac{1}{2}}+o_{n}(1)\\\\end{array}$ By Lemma REF , there is $C_{1}>0$ such that $||u_{n}||\\\\le C_{1}$ .\",\n \"Then, for some constant $C>0$ $\\\\begin{array}{ll}&\\\\left(1-\\\\frac{1}{k}\\\\right)\\\\int _{|x|>2r}V(x)u_{n}^{2}dx \\\\\\\\& \\\\le \\\\int _{B_{r}}\\\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x,y)dxdy +C(\\\\frac{1}{r}+1)||u_{n}||_{L^{2}(A)}+o_{n}(1)\\\\end{array}$ Let $\\\\epsilon >0$ .\",\n \"By Lemma REF , we can take $r$ , large enough, such that $\\\\begin{array}{ll}\\\\int _{|x|>2r}V(x)u_{n}^{2}dx \\\\le \\\\frac{\\\\epsilon (k-1)}{3k}+C(\\\\frac{1}{r}+1)||u_{n}||_{L^{2}(A)} +o_{n}(1)\\\\end{array}$ for all $n \\\\in \\\\mathbb {N}$ .\",\n \"Also, we can assume that $||u||_{L^{2}(A)}< \\\\frac{\\\\epsilon (k-1)}{6 C(\\\\frac{1}{r}+1)k}$ By property $(2)$ of $g$ $g(x,u_{n})u_{n} \\\\le \\\\frac{V(x)}{k}u_{n}^{2}.$ for all $x$ , with $|x|>2r>R$ .\",\n \"Therefore, by REF $\\\\int _{|x|>2r}g(x,u_{n})u_{n}dx \\\\le \\\\frac{\\\\epsilon }{3}+C\\\\left(\\\\frac{1}{r}+1\\\\right)\\\\frac{k}{k-1}||u_{n}||_{L^{2}(A)} +o_{n}(1).$ By REF , REF and compact embedding of the fractional Sobolev spaces, we can take $n_{1}\\\\in \\\\mathbb {N}$ such that if $n>n_{1}$ then $\\\\int _{|x|>2r}g(x,u_{n})u_{n}dx \\\\le \\\\frac{5\\\\epsilon }{6}.$ Note that, we can suppose that $r>0$ satisfies $\\\\int _{|x|>2r}g(x,u)udx \\\\le \\\\frac{\\\\epsilon }{12}.$ By compact embedding of fractional sobolev spaces we have that, there is, $n_{0} \\\\in \\\\mathbb {N}$ with $n_{0}>n_{1}$ and if $n>n_{0}$ then $\\\\left|\\\\int _{|x|\\\\le 2r}g(x,u_{n})u_{n}dx - \\\\int _{|x|\\\\le 2r}g(x,u)udx\\\\right|< \\\\frac{\\\\epsilon }{12}.$ Thereby, for $n>n_{0}$ we have $\\\\left|\\\\int _{\\\\mathbb {R}^{n}}g(x,u_{n})u_{n}dx - \\\\int _{\\\\mathbb {R}^{n}}g(x,u)udx\\\\right|< \\\\epsilon $ that is $\\\\lim \\\\limits _{n \\\\rightarrow \\\\infty }\\\\int _{\\\\mathbb {R}^{n}}g(x,u_{n})u_{n}dx = \\\\int _{\\\\mathbb {R}^{n}}g(x,u)udx$ From $I^{\\\\prime }(u_{n})u_{n}=o_{n}(1)$ , we conclude that $||u_{n}|| \\\\rightarrow ||u||$ and therefore $\\\\left\\\\lbrace u_{n}\\\\right\\\\rbrace $ converges to $u$ in $X$ .\",\n \"Corollary 3.8 Suppose $(V_{1})$ , $(f_{1})-(f_{3})$ .\",\n \"Then, there is $u \\\\in X$ such that $J(u)=c$ and $J^{\\\\prime }(u)=0$ .\",\n \"Moreover, $u\\\\ge 0$ almost everywere in $\\\\mathbb {R}^{n}$ .\",\n \"By REF and Proposition REF , there is $u \\\\in X$ such that $J(u)=c$ and $J^{\\\\prime }(u)=0$ .\",\n \"Let $A=\\\\left\\\\lbrace x \\\\in \\\\mathbb {R}^{n};|x|>R;\\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace x \\\\in \\\\mathbb {R}^{n}; u(x)<0;\\\\right\\\\rbrace $ .\",\n \"If $x \\\\in A$ , then $g(x,u(x))=\\\\frac{V(x)}{k}u(x)$ and in if $x \\\\in A^{c}$ , then $g(x,u)\\\\ge 0$ .\",\n \"We have $0\\\\ge [u,u^{-}]+\\\\int _{A^{c}}V(x)uu^{-} = \\\\left(\\\\frac{1}{k}-1\\\\right)\\\\int _{A}V(x)uu^{-} + \\\\int _{A^{c}}g(x,u)u^{-}dx \\\\ge 0$ where $u^{-}(x)=\\\\max \\\\left\\\\lbrace -u(x),0\\\\right\\\\rbrace $ .\",\n \"Then $[u,u^{-}]=0$ This implies that $u^{-}=0$ (see proof of Lemma 4.1 in ).\",\n \"As a consequence of inequalities REF and REF we have the following proposition.\",\n \"Proposition 3.9 If $V$ and $f$ satisfies $(V_{1}),(V_{2}), (f_{1})-(f_{3})$ , then the solution $u$ of the auxiliary problem satisfies $||u||^{2} \\\\le 2kd$ uniformly in $R>0$ .\",\n \"$L^{\\\\infty }$ estimative for solution of auxiliary problem In this section, we will prove a Brezis-Kato estimative.\",\n \"We will prove that, admitting some hypothesis, there is $M>0$ such that the solution of the problem $\\\\left\\\\lbrace \\\\begin{array}{rlll}-\\\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\\\mathbb {R}^{n} \\\\\\\\\\\\end{array}\\\\right.$ satisfies $||u||_{L^{\\\\infty }(\\\\mathbb {R}^{n})}\\\\le M$ and $M$ does not depend on $||u||$ (see Proposition REF ).\",\n \"We emphasize that, in the best of our knowlegends, this result is being presented for the first time in the literature.\",\n \"In , the authors have shown this result when the operator $\\\\mathcal {L}_{K}$ is the fractional Laplacian operator, that is, when $K(x)=\\\\frac{C_{n,s}}{2}|x|^{-n-2s}$ .\",\n \"But, in our case, we can not use the same technique used in , because we do not have a version of the $s$ -harmonic extension for more general integro-differential operators.\",\n \"Therefore, we present an another technique.\",\n \"Remark 4.1 Let $\\\\beta >1$ .\",\n \"Define the real functions $m(x):= (\\\\lambda -1)(x^{\\\\beta }+x^{-\\\\beta })-\\\\lambda (x^{\\\\beta -1}+x^{1-\\\\beta })+2$ and $p(x):= (\\\\lambda -1)(x^{\\\\beta }+x^{-\\\\beta })+\\\\lambda (x^{\\\\beta -1}+x^{1-\\\\beta })-2.$ where $\\\\lambda :=\\\\frac{\\\\beta ^{2}}{2\\\\beta -1}$ .\",\n \"Then $m(x)\\\\ge 0$ and $p(x)\\\\ge 0$ for all $x> 0$ .\",\n \"We will omit the proof of the claim that appears in the Remark REF .\",\n \"Let $\\\\beta >1$ and define $f(x)=x|x|^{2(\\\\beta -1)}$ and $g(x)=x|x|^{\\\\beta -1}.$ The functions $f$ and $g$ are continuous and differentiable for all $x \\\\in \\\\mathbb {R}$ .\",\n \"Consider $x,y\\\\in \\\\mathbb {R}$ with $x \\\\ne y$ .\",\n \"By Mean Value Theorem, there are $\\\\theta _{1}(x,y)$ , $\\\\theta _{2}(x,y) \\\\in \\\\mathbb {R}$ such that $f^{^{\\\\prime }}(\\\\theta _{1}(x,y))=\\\\frac{f(x)-f(y)}{x-y}$ and $g^{^{\\\\prime }}(\\\\theta _{2}(x,y))=\\\\frac{g(x)-g(y)}{x-y},$ that is $(2\\\\beta -1)|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}=\\\\frac{x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}}{x-y}$ and $\\\\beta |\\\\theta _{2}(x,y)|^{(\\\\beta -1)}=\\\\frac{x|x|^{(\\\\beta -1)}-y|y|^{(\\\\beta -1)}}{x-y}.$ Implying that $|\\\\theta _{1}(x,y)|=\\\\left(\\\\frac{1}{2\\\\beta -1}\\\\frac{x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}}{x-y}\\\\right)^{\\\\frac{1}{2(\\\\beta -1)}}$ and $|\\\\theta _{2}(x,y)|=\\\\left(\\\\frac{1}{\\\\beta }\\\\frac{x|x|^{(\\\\beta -1)}-y|y|^{(\\\\beta -1)}}{x-y}\\\\right)^{\\\\frac{1}{(\\\\beta -1)}}.$ We consider $\\\\theta _{1}(x,x)=\\\\theta _{2}(x,x)=0$ for all $x \\\\in \\\\mathbb {R}$ .\",\n \"Remark 4.2 Note that $|\\\\theta _{1}(x,y)|=|\\\\theta _{1}(y,x)|$ and $|\\\\theta _{2}(x,y)| = |\\\\theta _{2}(y,x)|$ for all $x,y \\\\in \\\\mathbb {R}$ .\",\n \"Lemma 4.3 With the same notation, if $x \\\\ne 0$ then $|\\\\theta _{1}(x,0)|\\\\ge |\\\\theta _{2}(x,0)|.$ By equalities REF and REF , we have $|\\\\theta _{1}(x,0)|=\\\\left(\\\\frac{1}{2\\\\beta -1}\\\\frac{x|x|^{2(\\\\beta -1)}}{x}\\\\right)^{\\\\frac{1}{2(\\\\beta -1)}} = \\\\left(\\\\frac{1}{2\\\\beta -1}\\\\right)^{\\\\frac{1}{2(\\\\beta -1)}}|x|$ and $|\\\\theta _{2}(x,0)|=\\\\left(\\\\frac{1}{\\\\beta }\\\\frac{x|x|^{\\\\beta -1}}{x}\\\\right)^{\\\\frac{1}{(\\\\beta -1)}} = \\\\left(\\\\frac{1}{\\\\beta }\\\\right)^{\\\\frac{1}{(\\\\beta -1)}}|x|.$ Thereby, $|\\\\theta _{1}(x,0)|\\\\ge |\\\\theta _{2}(x,0)|.$ Lemma 4.4 If $x,y \\\\in \\\\mathbb {R}$ , then $|\\\\theta _{1}(x,y)|\\\\ge |\\\\theta _{2}(x,y)|.$ If $x = 0$ or $y = 0$ then the inequality was proved by Lemma REF and Remark REF .\",\n \"The case $x=y$ is trivial.\",\n \"We can suppose that $x \\\\ne y$ , $x \\\\ne 0$ and $y \\\\ne 0$ .\",\n \"By equalities REF and REF we have that $|\\\\theta _{1}(x,y)|\\\\ge |\\\\theta _{2}(x,y)|$ if, and only if $\\\\left(\\\\frac{1}{2\\\\beta -1}\\\\frac{x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}}{x-y}\\\\right)^{\\\\frac{1}{2(\\\\beta -1)}} \\\\ge \\\\left(\\\\frac{1}{\\\\beta }\\\\frac{x|x|^{\\\\beta -1}-y|y|^{\\\\beta -1}}{x-y}\\\\right)^{\\\\frac{1}{(\\\\beta -1)}}.$ This last inequality is true if, and only if $\\\\frac{1}{2\\\\beta -1}\\\\frac{x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}}{x-y} \\\\ge \\\\frac{1}{\\\\beta ^{2}}\\\\left(\\\\frac{x|x|^{\\\\beta -1}-y|y|^{\\\\beta -1}}{x-y}\\\\right)^{2}.$ This last inequality occurs if, and only if $\\\\lambda (x-y)(x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}) \\\\ge \\\\left(x|x|^{\\\\beta -1}-y|y|^{\\\\beta -1}\\\\right)^{2},$ that is $\\\\lambda (|x|^{2\\\\beta }-xy|y|^{2(\\\\beta -1)}-yx|x|^{2(\\\\beta -1)}+|y|^{2\\\\beta })\\\\ge |x|^{2\\\\beta }-2xy|x|^{\\\\beta -1}|y|^{\\\\beta -1}+|y|^{2\\\\beta }.$ But, we are supposing that $x\\\\ne 0$ and $y\\\\ne 0$ , then the last inequality is equivalent to $\\\\begin{array}{ll}&\\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - \\\\left(xy\\\\frac{|y|^{\\\\beta -2}}{|x|^{\\\\beta }}\\\\right)-\\\\left(\\\\frac{xy|x|^{\\\\beta -2}}{|y|^{\\\\beta }}\\\\right)+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right] \\\\\\\\&\\\\ge \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - 2\\\\frac{x}{|x|}\\\\frac{y}{|y|} + \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\end{array}$ We will prove that the inequality REF is true.\",\n \"If $x\\\\cdot y>0$ , then $\\\\begin{array}{ll}&\\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - \\\\left(xy\\\\frac{|y|^{\\\\beta -2}}{|x|^{\\\\beta }}\\\\right)-\\\\left(\\\\frac{xy|x|^{\\\\beta -2}}{|y|^{\\\\beta }}\\\\right)+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right] - \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } + 2\\\\frac{x}{|x|}\\\\frac{y}{|y|} - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\\\\\& =\\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta -1}-\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right] - \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } + 2 - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta } \\\\\\\\& = (\\\\lambda -1)\\\\left[\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta }+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{-\\\\beta }\\\\right]-\\\\lambda \\\\left[\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{-\\\\beta +1}\\\\right]+2 \\\\\\\\& = m(\\\\frac{|x|}{|y|}).\\\\end{array}$ If $x \\\\cdot y<0$ , then $\\\\begin{array}{ll}& \\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - \\\\left(xy\\\\frac{|y|^{\\\\beta -2}}{|x|^{\\\\beta }}\\\\right)-\\\\left(\\\\frac{xy|x|^{\\\\beta -2}}{|y|^{\\\\beta }}\\\\right)+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right] - \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } + 2\\\\frac{x}{|x|}\\\\frac{y}{|y|} - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\\\\\& =\\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } + \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right]- \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - 2 - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\\\\\& = (\\\\lambda -1)\\\\left[\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta }+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{-\\\\beta }\\\\right]+\\\\lambda \\\\left[\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{-\\\\beta +1}\\\\right]-2\\\\\\\\& = p(\\\\frac{|x|}{|y|}).\\\\end{array}$ By Remark REF , we have that $m(\\\\frac{|x|}{|y|})\\\\ge 0$ and $p(\\\\frac{|x|}{|y|}) \\\\ge 0$ .\",\n \"This proves the inequality REF and the Lemma REF .\",\n \"Our main result of this section will establishes an important estimate involving the $L^{\\\\infty }(\\\\mathbb {R}^{n})$ norm of the solution $u$ of the auxiliary problem.\",\n \"It states that: Proposition 4.5 Leq $h \\\\in L^{q}(\\\\mathbb {R}^{n})$ with $q>\\\\frac{n}{2s}$ , and $v \\\\in E \\\\subset X$ be a weak solution of $\\\\left\\\\lbrace \\\\begin{array}{rlll}-\\\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\\\mathbb {R}^{n} \\\\\\\\\\\\end{array}\\\\right.$ where $g$ is a continuous functions satisfying $|g(x,s)|\\\\le h(x)|s|$ for $s\\\\ge 0$ , $b$ is a positive function in $\\\\mathbb {R}^{n}$ and $E$ is definded as in REF .\",\n \"Then, there is a constant $M=M(q,||h||_{L^{q}})$ such that $||v||_{\\\\infty }\\\\le M||v||_{2^{\\\\ast }_{s}}.$ Let $\\\\beta >1$ .\",\n \"For any $n \\\\in \\\\mathbb {N}$ define $A_{n}=\\\\left\\\\lbrace x \\\\in \\\\mathbb {R}^{n}; |v(x)|^{\\\\beta -1}\\\\le n\\\\right\\\\rbrace $ and $B_{n}:=\\\\mathbb {R}^{n}\\\\setminus A_{n}.$ Consider $\\\\begin{array}{ll}f_{n}(t):=&\\\\left\\\\lbrace \\\\begin{array}{lll}t|t|^{2(\\\\beta -1)}&se&|t|^{\\\\beta -1}\\\\le n\\\\\\\\n^{2}t&se&|t|^{\\\\beta -1}>n\\\\end{array}\\\\right.\\\\end{array}$ and $\\\\begin{array}{ll}g_{n}(t):=&\\\\left\\\\lbrace \\\\begin{array}{lll}t|t|^{(\\\\beta -1)}&se&|t|^{\\\\beta -1}\\\\le n\\\\\\\\nt&se&|t|^{\\\\beta -1}>n\\\\end{array}\\\\right.\\\\end{array}$ Note that $f_{n}$ and $g_{n}$ are continuous functions and they are differentiable with the exception of $n^{\\\\frac{1}{\\\\beta -1}}$ and $-n^{\\\\frac{1}{\\\\beta -1}}$ and its derivatives are limited.\",\n \"Then $f_{n}$ and $g_{n}$ are lipschtz.\",\n \"Therefore, setting $v_{n}:=f_{n}\\\\circ v$ and $w_{n}:=g_{n}\\\\circ v$ we have that $v_{n},w_{n}\\\\in E$ .\",\n \"Note that $\\\\begin{array}{ll}\\\\left[v,v_{n}\\\\right]&= \\\\int _{A_{n}}\\\\int _{A_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy \\\\\\\\ & +\\\\int _{B_{n}}\\\\int _{B_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy\\\\\\\\ &+2\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ By equality REF , if $x,y \\\\in A_{n}$ then $v_{n}(x)-v_{n}(y)=f_{n}^{\\\\prime }(\\\\theta _{1}(x,y))(v(x)-v(y)),$ where we are denoting $\\\\theta _{1}(v(x),v(y))$ by $\\\\theta _{1}(x, y)$ .\",\n \"Thereby, $\\\\begin{array}{ll}\\\\left[v,v_{n}\\\\right] &= \\\\int _{A_{n}}\\\\int _{A_{n}}(2\\\\beta -1)|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\\\\\&+n^{2}\\\\int _{B_{n}}\\\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ Analogously, by REF $\\\\begin{array}{ll}\\\\left[w_{n},w_{n}\\\\right] &= \\\\int _{A_{n}}\\\\int _{A_{n}}\\\\beta ^{2}|\\\\theta _{2}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\\\\\&+n^{2}\\\\int _{B_{n}}\\\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}},\\\\end{array}$ where we are denoting $\\\\theta _{2}(v(x),v(y))$ by $\\\\theta _{2}(x, y)$ .\",\n \"By Lemma REF , $\\\\begin{array}{ll}\\\\left[w_{n},w_{n}\\\\right] &\\\\le \\\\int _{A_{n}}\\\\int _{A_{n}}\\\\beta ^{2}|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\\\\\&+n^{2}\\\\int _{B_{n}}\\\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ This implies that $\\\\begin{array}{ll}&\\\\left[w_{n},w_{n}\\\\right] + \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\\\left[v,v_{n}\\\\right]- \\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\\\\\&\\\\le (\\\\beta -1)^{2}\\\\int _{A_{n}}\\\\int _{A_{n}}|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\\\\\& +2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}-2\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ The equation REF implies that $\\\\begin{array}{ll}&\\\\left[v,v_{n}\\\\right]+\\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx - 2\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}\\\\\\\\ &\\\\ge (2\\\\beta -1)\\\\int _{A_{n}}\\\\int _{A_{n}}|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy,\\\\end{array}$ because $b(x)vv_{n}=b(x)w_{n}^{2}\\\\ge 0$ .\",\n \"Replacing REF in the inequality REF we obtain $\\\\begin{array}{ll}&\\\\left[w_{n},w_{n}\\\\right] + \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\\\left[v,v_{n}\\\\right]- \\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\\\\\ &\\\\le \\\\frac{(\\\\beta -1)^{2}}{2\\\\beta -1}\\\\left([v,v_{n}]+\\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\right)\\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}},\\\\end{array}$ that is $\\\\begin{array}{ll}&\\\\left[w_{n},w_{n}\\\\right] + \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx\\\\\\\\ &\\\\le (\\\\frac{(\\\\beta -1)^{2}}{2\\\\beta -1}+1)\\\\left([v,v_{n}]+\\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\right)\\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}} \\\\\\\\&= \\\\frac{\\\\beta ^{2}}{2\\\\beta -1}\\\\left([v,v_{n}]+\\\\int _{\\\\mathbb {R}^{n}}bvv_{n}dx\\\\right)\\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}} \\\\\\\\&\\\\le \\\\beta \\\\int _{\\\\mathbb {R}^{n}}g(x,v)v_{n}dx\\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ In short, $\\\\begin{array}{ll}\\\\left[w_{n},w_{n}\\\\right] &+ \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\\\le \\\\beta \\\\int _{\\\\mathbb {R}^{n}}g(x,v)v_{n}dx \\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ But, if $n \\\\in \\\\mathbb {N}$ and $C=2+\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1},$ then a simple calculation shows that the function $r(s,t)=2(ns-t|t|^{\\\\beta -1})^{2}-C(s-t)(n^{2}s-t|t|^{2(\\\\beta -1)}),$ satisfies $r(s,t)\\\\le 0$ for all $|s|>n^{\\\\frac{1}{\\\\beta -1}}$ and $|t|\\\\le n^{\\\\frac{1}{\\\\beta -1}}$ .\",\n \"Then, taking $s=v(x)$ and $t=v(y)$ for $x\\\\in B_{n}$ and $y \\\\in A_{n}$ and replacing in REF we obtain $2(w_{n}(x)-w_{n}(y))^{2}-C(v(x)-v(y))(v_{n}(x)-v_{n}(y))\\\\le 0.$ Thereby $+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}} \\\\le 0.$ By inequality REF , $\\\\left[w_{n},w_{n}\\\\right] + \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\\\le \\\\beta \\\\int _{\\\\mathbb {R}^{n}}g(x,v)v_{n}dx.$ Let $S>0$ be the best constant verifying $||u||_{2^{\\\\ast }_{s}}^{2} \\\\le S||u||_{X}^{2},$ for all $u \\\\in X$ , that is $S=\\\\sup _{u \\\\in X}\\\\frac{||u||_{2^{\\\\ast }_{s}}^{2}}{||u||_{X}^{2}}.$ By inequality REF , $\\\\begin{array}{ll}\\\\left(\\\\int _{A_{n}}|w_{n}|^{2^{\\\\ast }_{s}}dx\\\\right)^{\\\\frac{2}{2^{\\\\ast }_{s}}}&\\\\le \\\\left(\\\\int _{\\\\mathbb {R}^{n}}|w_{n}|^{2^{\\\\ast }_{s}}dx\\\\right)^{\\\\frac{2}{2^{\\\\ast }_{s}}}\\\\\\\\&\\\\le S||w_{n}||^{2} \\\\\\\\& \\\\le S\\\\beta \\\\int _{\\\\mathbb {R}^{n}}g(x,v(x))v_{n}dx \\\\\\\\& \\\\le S\\\\beta \\\\int _{\\\\mathbb {R}^{n}}h(x)w_{n}^{2}dx \\\\\\\\& \\\\le S\\\\beta ||h||_{q}||w_{n}||_{\\\\frac{2q}{q-1}}^{2}.\\\\end{array}$ But, we have that $|w_{n}(x)|\\\\le |v(x)|^{\\\\beta }$ for all $x \\\\in B_{n}$ and $|w_{n}(x)|=|v(x)|^{\\\\beta }$ for all $x \\\\in A_{n}$ .\",\n \"Thereby, $\\\\begin{array}{ll}\\\\left(\\\\int _{A_{n}}|v|^{\\\\beta 2^{\\\\ast }_{s}}dx\\\\right)^{\\\\frac{2}{2^{\\\\ast }_{s}}} \\\\le S\\\\beta ||h||_{q}\\\\left[\\\\int _{\\\\mathbb {R}^{n}}|v|^{\\\\frac{2q\\\\beta }{q-1}}dx\\\\right]^{\\\\frac{q-1}{q}}.\\\\end{array}$ By Monotone Convergence Theorem, $||v||_{2^{\\\\ast }_{s}\\\\beta } \\\\le (\\\\beta S||h||_{q})^{\\\\frac{1}{2\\\\beta }}||v||_{2\\\\beta q_{1}},$ where $q_{1}=\\\\frac{q}{q-1}$ .\",\n \"Define $\\\\eta :=\\\\frac{2^{\\\\ast }_{s}}{2q_{1}}.$ and note that $\\\\eta >1$ .\",\n \"When $\\\\beta =\\\\eta $ we have that $2\\\\beta q_{1} = 2^{\\\\ast }_{s}$ .\",\n \"Then, by REF $||v||_{2^{\\\\ast }_{s}\\\\eta } \\\\le (\\\\eta S||h||_{q})^{\\\\frac{1}{2\\\\eta }}||v||_{2^{\\\\ast }_{s}}.$ Taking $\\\\beta =\\\\eta ^{2}$ in REF we obtain $||v||_{2^{\\\\ast }_{s}\\\\eta ^{2}} \\\\le \\\\eta ^{\\\\frac{1}{\\\\eta ^{2}}} (S||h||_{q})^{\\\\frac{1}{2\\\\eta ^{2}}}||v||_{2^{\\\\ast }_{s}\\\\eta }.$ By inequalities REF and REF we have $||v||_{2^{\\\\ast }_{s}\\\\eta ^{2}} \\\\le \\\\eta ^{\\\\frac{1}{(\\\\eta ^{2}}+\\\\frac{1}{2\\\\eta })} (S||h||_{q})^{(\\\\frac{1}{2\\\\eta ^{2}}+\\\\frac{1}{2\\\\eta })}||v||_{2^{\\\\ast }_{s}}.$ Inductively, we can prove that $||v||_{2^{\\\\ast }_{s}\\\\eta ^{m}} \\\\le \\\\eta ^{(\\\\frac{1}{2\\\\eta }+\\\\frac{1}{\\\\eta ^{2}}+...+\\\\frac{m}{2\\\\eta ^{m}})} (S||h||_{q})^{(\\\\frac{1}{2\\\\eta }+\\\\frac{1}{2\\\\eta ^{2}}+...+\\\\frac{1}{2\\\\eta ^{m}})}||v||_{2^{\\\\ast }_{s}}$ for all $m \\\\in \\\\mathbb {N}$ .\",\n \"But, $\\\\sum _{i=1}^{\\\\infty }\\\\frac{m}{2\\\\eta ^{m}}=\\\\frac{1}{2(\\\\eta -1)^{2}}$ and $\\\\sum _{i=1}^{\\\\infty }\\\\frac{1}{2\\\\eta ^{m}}=\\\\frac{1}{2(\\\\eta -1)}.$ Thereby, for all $m>0$ $||v||_{2^{\\\\ast }_{s}\\\\eta ^{m}} \\\\le \\\\eta ^{\\\\frac{1}{2(\\\\eta -1)^{2}}} (S||h||_{q})^{\\\\frac{1}{2(\\\\eta -1)}}||v||_{2^{\\\\ast }_{s}}$ Recalling that $||v||_{\\\\infty }=\\\\lim \\\\limits _{n \\\\rightarrow \\\\infty }||v||_{p}$ and that $\\\\eta >1$ we have that $||v||_{\\\\infty }\\\\le M||v||_{2^{\\\\ast }_{s}}$ for $M=\\\\eta ^{\\\\frac{1}{2(\\\\eta -1)^{2}}} (S||h||_{q})^{\\\\frac{1}{2(\\\\eta -1)}}$ and $\\\\eta = \\\\frac{n(q-1)}{q(n-2s)}$ We conclude the proof of Proposition REF noting that $M$ depends only on $q$ , $||h||_{q}$ .\",\n \"Solution for Problem (P) In this section, we prove the main result, the Theorem REF .\",\n \"By Corollary REF , there is $u \\\\in X$ such that $J(u)=c$ and $J^{\\\\prime }(u)=0$ .\",\n \"We have the following estimate for $||u||_{\\\\infty }$ .\",\n \"Lemma 5.1 The solution $u$ of the auxiliary problem satisfies $||u||_{\\\\infty } \\\\le M(2Skd)^{\\\\frac{1}{2}}$ Consider the functions $\\\\begin{array}{lll}H(x,t)&=&\\\\left\\\\lbrace \\\\begin{array}{lllll}f(t)&if&|x| \\\\frac{V(x)}{k}t\\\\end{array}\\\\right.\\\\end{array}$ and $\\\\begin{array}{lll}b(x)&=&\\\\left\\\\lbrace \\\\begin{array}{lllll}V(x)&if&|x| \\\\frac{V(x)}{k}u.\\\\end{array}\\\\right.\\\\end{array}$ Note that the function $u$ is solution of $\\\\left\\\\lbrace \\\\begin{array}{ll}&-\\\\mathcal {L}_{k}u+b(x)u=H(x,u)\\\\\\\\&u \\\\in E.\\\\end{array}\\\\right.$ By $(f_{1})$ , $|H(x,t)|\\\\le c_{0}|t|^{p-1}$ for $p \\\\in (2,2^{\\\\ast }_{s})$ .\",\n \"Thereby, $|H(x,u)|\\\\le h(x)|u|,$ where $h(x)=C_{0}|u|^{p-2}$ .\",\n \"Note that $h \\\\in L^{\\\\frac{2^{\\\\ast }_{s}}{p-2}}$ with $||h||_{L^{q}(\\\\mathbb {R}^{n})}\\\\le C(2ksd)^{\\\\frac{p-2}{2^{\\\\ast }_{s}}}.$ The number $p$ satisfies $p<2^{\\\\ast }_{s} = 2+\\\\frac{2s}{n}2^{\\\\ast }_{s}.$ Then $q =\\\\frac{2^{\\\\ast }_{s}}{p-2}>\\\\frac{n}{2s}.$ By Proposition REF and sobolev embedding $||u||_{\\\\infty }\\\\le M||u||_{2^{\\\\ast }_{s}} \\\\le MS^{\\\\frac{1}{2}}||u||,$ where $M=M(q,||h||_{q})$ .\",\n \"By Proposition REF , we have $||u||_{\\\\infty } \\\\le M(2kSd)^{\\\\frac{1}{2}}.$ Theorem 5.2 Suppose that $V$ satisfies $(V_{1})$ -$(V_{2})$ and that $f$ satisfies $(f_{1})$ -$(f_{3})$ .\",\n \"There is $\\\\Lambda ^{\\\\ast }=\\\\Lambda ^{\\\\ast }(V_{\\\\infty },\\\\theta ,p,c_{0},s)>0$ such that if $\\\\Lambda >\\\\Lambda ^{\\\\ast }$ in $(V_{3})$ , then the problem $(P)$ has a nonnegative nontrivial solution.\",\n \"Indeed, let $|x|\\\\ge R$ .\",\n \"If $u(x)=0$ then by denfition $f(u(x))=g(x,u(x))$ .\",\n \"If $u(x)>0$ then $\\\\begin{array}{ll}\\\\frac{f(u(x))}{u(x)}&\\\\le c_{0}|u|^{p-2} \\\\\\\\&\\\\le c_{0}||u||_{\\\\infty }^{p-2} \\\\\\\\& = \\\\frac{ c_{0}||u||_{\\\\infty }^{p-2}}{\\\\Lambda }\\\\Lambda \\\\\\\\& \\\\le \\\\frac{k^{\\\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\\\frac{p-2}{2}}}{\\\\Lambda }\\\\frac{V(x)}{k}\\\\end{array}$ Define $\\\\Lambda ^{\\\\ast }=k^{\\\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\\\frac{p-2}{2}}.$ If $\\\\Lambda >\\\\Lambda ^{\\\\ast }$ then $\\\\frac{f(u(x))}{u(x)} \\\\le \\\\frac{V(x)}{k}.$ By definition of $g$ we have $g(x,u(x))=f(u(x))$ .\",\n \"Then $g(x,u(x))=f(u(x))$ for all $x \\\\in \\\\mathbb {R}^{n}$ .\",\n \"Therefore, $u$ is a solution nonnegative and nontrivial of 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H., author=Xu, J. author=Zhang, F. title= Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in $\\\\mathbb {R}^{N}$ ., journal=Journal of Mathematical Physics, volume=56, date=2015, pages=, review= noncitearticle author=Zhang, X., author=Zhang, B. author=Repovs, D. title=Existence and symmetry of solutions for critical fractiona Schrödinger equations with bounded potentials, journal=Nonlinear Analysis, volume=142, date=2016, pages=48-68, review=\"\n ],\n [\n \"$L^{\\\\infty }$ estimative for solution of auxiliary problem\",\n \"In this section, we will prove a Brezis-Kato estimative.\",\n \"We will prove that, admitting some hypothesis, there is $M>0$ such that the solution of the problem $\\\\left\\\\lbrace \\\\begin{array}{rlll}-\\\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\\\mathbb {R}^{n} \\\\\\\\\\\\end{array}\\\\right.$ satisfies $||u||_{L^{\\\\infty }(\\\\mathbb {R}^{n})}\\\\le M$ and $M$ does not depend on $||u||$ (see Proposition REF ).\",\n \"We emphasize that, in the best of our knowlegends, this result is being presented for the first time in the literature.\",\n \"In , the authors have shown this result when the operator $\\\\mathcal {L}_{K}$ is the fractional Laplacian operator, that is, when $K(x)=\\\\frac{C_{n,s}}{2}|x|^{-n-2s}$ .\",\n \"But, in our case, we can not use the same technique used in , because we do not have a version of the $s$ -harmonic extension for more general integro-differential operators.\",\n \"Therefore, we present an another technique.\",\n \"Remark 4.1 Let $\\\\beta >1$ .\",\n \"Define the real functions $m(x):= (\\\\lambda -1)(x^{\\\\beta }+x^{-\\\\beta })-\\\\lambda (x^{\\\\beta -1}+x^{1-\\\\beta })+2$ and $p(x):= (\\\\lambda -1)(x^{\\\\beta }+x^{-\\\\beta })+\\\\lambda (x^{\\\\beta -1}+x^{1-\\\\beta })-2.$ where $\\\\lambda :=\\\\frac{\\\\beta ^{2}}{2\\\\beta -1}$ .\",\n \"Then $m(x)\\\\ge 0$ and $p(x)\\\\ge 0$ for all $x> 0$ .\",\n \"We will omit the proof of the claim that appears in the Remark REF .\",\n \"Let $\\\\beta >1$ and define $f(x)=x|x|^{2(\\\\beta -1)}$ and $g(x)=x|x|^{\\\\beta -1}.$ The functions $f$ and $g$ are continuous and differentiable for all $x \\\\in \\\\mathbb {R}$ .\",\n \"Consider $x,y\\\\in \\\\mathbb {R}$ with $x \\\\ne y$ .\",\n \"By Mean Value Theorem, there are $\\\\theta _{1}(x,y)$ , $\\\\theta _{2}(x,y) \\\\in \\\\mathbb {R}$ such that $f^{^{\\\\prime }}(\\\\theta _{1}(x,y))=\\\\frac{f(x)-f(y)}{x-y}$ and $g^{^{\\\\prime }}(\\\\theta _{2}(x,y))=\\\\frac{g(x)-g(y)}{x-y},$ that is $(2\\\\beta -1)|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}=\\\\frac{x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}}{x-y}$ and $\\\\beta |\\\\theta _{2}(x,y)|^{(\\\\beta -1)}=\\\\frac{x|x|^{(\\\\beta -1)}-y|y|^{(\\\\beta -1)}}{x-y}.$ Implying that $|\\\\theta _{1}(x,y)|=\\\\left(\\\\frac{1}{2\\\\beta -1}\\\\frac{x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}}{x-y}\\\\right)^{\\\\frac{1}{2(\\\\beta -1)}}$ and $|\\\\theta _{2}(x,y)|=\\\\left(\\\\frac{1}{\\\\beta }\\\\frac{x|x|^{(\\\\beta -1)}-y|y|^{(\\\\beta -1)}}{x-y}\\\\right)^{\\\\frac{1}{(\\\\beta -1)}}.$ We consider $\\\\theta _{1}(x,x)=\\\\theta _{2}(x,x)=0$ for all $x \\\\in \\\\mathbb {R}$ .\",\n \"Remark 4.2 Note that $|\\\\theta _{1}(x,y)|=|\\\\theta _{1}(y,x)|$ and $|\\\\theta _{2}(x,y)| = |\\\\theta _{2}(y,x)|$ for all $x,y \\\\in \\\\mathbb {R}$ .\",\n \"Lemma 4.3 With the same notation, if $x \\\\ne 0$ then $|\\\\theta _{1}(x,0)|\\\\ge |\\\\theta _{2}(x,0)|.$ By equalities REF and REF , we have $|\\\\theta _{1}(x,0)|=\\\\left(\\\\frac{1}{2\\\\beta -1}\\\\frac{x|x|^{2(\\\\beta -1)}}{x}\\\\right)^{\\\\frac{1}{2(\\\\beta -1)}} = \\\\left(\\\\frac{1}{2\\\\beta -1}\\\\right)^{\\\\frac{1}{2(\\\\beta -1)}}|x|$ and $|\\\\theta _{2}(x,0)|=\\\\left(\\\\frac{1}{\\\\beta }\\\\frac{x|x|^{\\\\beta -1}}{x}\\\\right)^{\\\\frac{1}{(\\\\beta -1)}} = \\\\left(\\\\frac{1}{\\\\beta }\\\\right)^{\\\\frac{1}{(\\\\beta -1)}}|x|.$ Thereby, $|\\\\theta _{1}(x,0)|\\\\ge |\\\\theta _{2}(x,0)|.$ Lemma 4.4 If $x,y \\\\in \\\\mathbb {R}$ , then $|\\\\theta _{1}(x,y)|\\\\ge |\\\\theta _{2}(x,y)|.$ If $x = 0$ or $y = 0$ then the inequality was proved by Lemma REF and Remark REF .\",\n \"The case $x=y$ is trivial.\",\n \"We can suppose that $x \\\\ne y$ , $x \\\\ne 0$ and $y \\\\ne 0$ .\",\n \"By equalities REF and REF we have that $|\\\\theta _{1}(x,y)|\\\\ge |\\\\theta _{2}(x,y)|$ if, and only if $\\\\left(\\\\frac{1}{2\\\\beta -1}\\\\frac{x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}}{x-y}\\\\right)^{\\\\frac{1}{2(\\\\beta -1)}} \\\\ge \\\\left(\\\\frac{1}{\\\\beta }\\\\frac{x|x|^{\\\\beta -1}-y|y|^{\\\\beta -1}}{x-y}\\\\right)^{\\\\frac{1}{(\\\\beta -1)}}.$ This last inequality is true if, and only if $\\\\frac{1}{2\\\\beta -1}\\\\frac{x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}}{x-y} \\\\ge \\\\frac{1}{\\\\beta ^{2}}\\\\left(\\\\frac{x|x|^{\\\\beta -1}-y|y|^{\\\\beta -1}}{x-y}\\\\right)^{2}.$ This last inequality occurs if, and only if $\\\\lambda (x-y)(x|x|^{2(\\\\beta -1)}-y|y|^{2(\\\\beta -1)}) \\\\ge \\\\left(x|x|^{\\\\beta -1}-y|y|^{\\\\beta -1}\\\\right)^{2},$ that is $\\\\lambda (|x|^{2\\\\beta }-xy|y|^{2(\\\\beta -1)}-yx|x|^{2(\\\\beta -1)}+|y|^{2\\\\beta })\\\\ge |x|^{2\\\\beta }-2xy|x|^{\\\\beta -1}|y|^{\\\\beta -1}+|y|^{2\\\\beta }.$ But, we are supposing that $x\\\\ne 0$ and $y\\\\ne 0$ , then the last inequality is equivalent to $\\\\begin{array}{ll}&\\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - \\\\left(xy\\\\frac{|y|^{\\\\beta -2}}{|x|^{\\\\beta }}\\\\right)-\\\\left(\\\\frac{xy|x|^{\\\\beta -2}}{|y|^{\\\\beta }}\\\\right)+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right] \\\\\\\\&\\\\ge \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - 2\\\\frac{x}{|x|}\\\\frac{y}{|y|} + \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\end{array}$ We will prove that the inequality REF is true.\",\n \"If $x\\\\cdot y>0$ , then $\\\\begin{array}{ll}&\\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - \\\\left(xy\\\\frac{|y|^{\\\\beta -2}}{|x|^{\\\\beta }}\\\\right)-\\\\left(\\\\frac{xy|x|^{\\\\beta -2}}{|y|^{\\\\beta }}\\\\right)+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right] - \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } + 2\\\\frac{x}{|x|}\\\\frac{y}{|y|} - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\\\\\& =\\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta -1}-\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right] - \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } + 2 - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta } \\\\\\\\& = (\\\\lambda -1)\\\\left[\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta }+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{-\\\\beta }\\\\right]-\\\\lambda \\\\left[\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{-\\\\beta +1}\\\\right]+2 \\\\\\\\& = m(\\\\frac{|x|}{|y|}).\\\\end{array}$ If $x \\\\cdot y<0$ , then $\\\\begin{array}{ll}& \\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - \\\\left(xy\\\\frac{|y|^{\\\\beta -2}}{|x|^{\\\\beta }}\\\\right)-\\\\left(\\\\frac{xy|x|^{\\\\beta -2}}{|y|^{\\\\beta }}\\\\right)+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right] - \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } + 2\\\\frac{x}{|x|}\\\\frac{y}{|y|} - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\\\\\& =\\\\lambda \\\\left[ \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } + \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\right]- \\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta } - 2 - \\\\left(\\\\frac{|y|}{|x|}\\\\right)^{\\\\beta }\\\\\\\\& = (\\\\lambda -1)\\\\left[\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta }+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{-\\\\beta }\\\\right]+\\\\lambda \\\\left[\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{\\\\beta -1}+\\\\left(\\\\frac{|x|}{|y|}\\\\right)^{-\\\\beta +1}\\\\right]-2\\\\\\\\& = p(\\\\frac{|x|}{|y|}).\\\\end{array}$ By Remark REF , we have that $m(\\\\frac{|x|}{|y|})\\\\ge 0$ and $p(\\\\frac{|x|}{|y|}) \\\\ge 0$ .\",\n \"This proves the inequality REF and the Lemma REF .\",\n \"Our main result of this section will establishes an important estimate involving the $L^{\\\\infty }(\\\\mathbb {R}^{n})$ norm of the solution $u$ of the auxiliary problem.\",\n \"It states that: Proposition 4.5 Leq $h \\\\in L^{q}(\\\\mathbb {R}^{n})$ with $q>\\\\frac{n}{2s}$ , and $v \\\\in E \\\\subset X$ be a weak solution of $\\\\left\\\\lbrace \\\\begin{array}{rlll}-\\\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\\\mathbb {R}^{n} \\\\\\\\\\\\end{array}\\\\right.$ where $g$ is a continuous functions satisfying $|g(x,s)|\\\\le h(x)|s|$ for $s\\\\ge 0$ , $b$ is a positive function in $\\\\mathbb {R}^{n}$ and $E$ is definded as in REF .\",\n \"Then, there is a constant $M=M(q,||h||_{L^{q}})$ such that $||v||_{\\\\infty }\\\\le M||v||_{2^{\\\\ast }_{s}}.$ Let $\\\\beta >1$ .\",\n \"For any $n \\\\in \\\\mathbb {N}$ define $A_{n}=\\\\left\\\\lbrace x \\\\in \\\\mathbb {R}^{n}; |v(x)|^{\\\\beta -1}\\\\le n\\\\right\\\\rbrace $ and $B_{n}:=\\\\mathbb {R}^{n}\\\\setminus A_{n}.$ Consider $\\\\begin{array}{ll}f_{n}(t):=&\\\\left\\\\lbrace \\\\begin{array}{lll}t|t|^{2(\\\\beta -1)}&se&|t|^{\\\\beta -1}\\\\le n\\\\\\\\n^{2}t&se&|t|^{\\\\beta -1}>n\\\\end{array}\\\\right.\\\\end{array}$ and $\\\\begin{array}{ll}g_{n}(t):=&\\\\left\\\\lbrace \\\\begin{array}{lll}t|t|^{(\\\\beta -1)}&se&|t|^{\\\\beta -1}\\\\le n\\\\\\\\nt&se&|t|^{\\\\beta -1}>n\\\\end{array}\\\\right.\\\\end{array}$ Note that $f_{n}$ and $g_{n}$ are continuous functions and they are differentiable with the exception of $n^{\\\\frac{1}{\\\\beta -1}}$ and $-n^{\\\\frac{1}{\\\\beta -1}}$ and its derivatives are limited.\",\n \"Then $f_{n}$ and $g_{n}$ are lipschtz.\",\n \"Therefore, setting $v_{n}:=f_{n}\\\\circ v$ and $w_{n}:=g_{n}\\\\circ v$ we have that $v_{n},w_{n}\\\\in E$ .\",\n \"Note that $\\\\begin{array}{ll}\\\\left[v,v_{n}\\\\right]&= \\\\int _{A_{n}}\\\\int _{A_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy \\\\\\\\ & +\\\\int _{B_{n}}\\\\int _{B_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy\\\\\\\\ &+2\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ By equality REF , if $x,y \\\\in A_{n}$ then $v_{n}(x)-v_{n}(y)=f_{n}^{\\\\prime }(\\\\theta _{1}(x,y))(v(x)-v(y)),$ where we are denoting $\\\\theta _{1}(v(x),v(y))$ by $\\\\theta _{1}(x, y)$ .\",\n \"Thereby, $\\\\begin{array}{ll}\\\\left[v,v_{n}\\\\right] &= \\\\int _{A_{n}}\\\\int _{A_{n}}(2\\\\beta -1)|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\\\\\&+n^{2}\\\\int _{B_{n}}\\\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ Analogously, by REF $\\\\begin{array}{ll}\\\\left[w_{n},w_{n}\\\\right] &= \\\\int _{A_{n}}\\\\int _{A_{n}}\\\\beta ^{2}|\\\\theta _{2}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\\\\\&+n^{2}\\\\int _{B_{n}}\\\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}},\\\\end{array}$ where we are denoting $\\\\theta _{2}(v(x),v(y))$ by $\\\\theta _{2}(x, y)$ .\",\n \"By Lemma REF , $\\\\begin{array}{ll}\\\\left[w_{n},w_{n}\\\\right] &\\\\le \\\\int _{A_{n}}\\\\int _{A_{n}}\\\\beta ^{2}|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\\\\\&+n^{2}\\\\int _{B_{n}}\\\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ This implies that $\\\\begin{array}{ll}&\\\\left[w_{n},w_{n}\\\\right] + \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\\\left[v,v_{n}\\\\right]- \\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\\\\\&\\\\le (\\\\beta -1)^{2}\\\\int _{A_{n}}\\\\int _{A_{n}}|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\\\\\& +2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}-2\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ The equation REF implies that $\\\\begin{array}{ll}&\\\\left[v,v_{n}\\\\right]+\\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx - 2\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}\\\\\\\\ &\\\\ge (2\\\\beta -1)\\\\int _{A_{n}}\\\\int _{A_{n}}|\\\\theta _{1}(x,y)|^{2(\\\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy,\\\\end{array}$ because $b(x)vv_{n}=b(x)w_{n}^{2}\\\\ge 0$ .\",\n \"Replacing REF in the inequality REF we obtain $\\\\begin{array}{ll}&\\\\left[w_{n},w_{n}\\\\right] + \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\\\left[v,v_{n}\\\\right]- \\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\\\\\ &\\\\le \\\\frac{(\\\\beta -1)^{2}}{2\\\\beta -1}\\\\left([v,v_{n}]+\\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\right)\\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}},\\\\end{array}$ that is $\\\\begin{array}{ll}&\\\\left[w_{n},w_{n}\\\\right] + \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx\\\\\\\\ &\\\\le (\\\\frac{(\\\\beta -1)^{2}}{2\\\\beta -1}+1)\\\\left([v,v_{n}]+\\\\int _{\\\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\right)\\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}} \\\\\\\\&= \\\\frac{\\\\beta ^{2}}{2\\\\beta -1}\\\\left([v,v_{n}]+\\\\int _{\\\\mathbb {R}^{n}}bvv_{n}dx\\\\right)\\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}} \\\\\\\\&\\\\le \\\\beta \\\\int _{\\\\mathbb {R}^{n}}g(x,v)v_{n}dx\\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ In short, $\\\\begin{array}{ll}\\\\left[w_{n},w_{n}\\\\right] &+ \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\\\le \\\\beta \\\\int _{\\\\mathbb {R}^{n}}g(x,v)v_{n}dx \\\\\\\\&+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}}.\\\\end{array}$ But, if $n \\\\in \\\\mathbb {N}$ and $C=2+\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1},$ then a simple calculation shows that the function $r(s,t)=2(ns-t|t|^{\\\\beta -1})^{2}-C(s-t)(n^{2}s-t|t|^{2(\\\\beta -1)}),$ satisfies $r(s,t)\\\\le 0$ for all $|s|>n^{\\\\frac{1}{\\\\beta -1}}$ and $|t|\\\\le n^{\\\\frac{1}{\\\\beta -1}}$ .\",\n \"Then, taking $s=v(x)$ and $t=v(y)$ for $x\\\\in B_{n}$ and $y \\\\in A_{n}$ and replacing in REF we obtain $2(w_{n}(x)-w_{n}(y))^{2}-C(v(x)-v(y))(v_{n}(x)-v_{n}(y))\\\\le 0.$ Thereby $+2\\\\left[w_{n},w_{n}\\\\right]_{A_{n}\\\\times B_{n}}+(-2-\\\\frac{2(\\\\beta -1)^{2}}{2\\\\beta -1})\\\\left[v,v_{n}\\\\right]_{A_{n}\\\\times B_{n}} \\\\le 0.$ By inequality REF , $\\\\left[w_{n},w_{n}\\\\right] + \\\\int _{\\\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\\\le \\\\beta \\\\int _{\\\\mathbb {R}^{n}}g(x,v)v_{n}dx.$ Let $S>0$ be the best constant verifying $||u||_{2^{\\\\ast }_{s}}^{2} \\\\le S||u||_{X}^{2},$ for all $u \\\\in X$ , that is $S=\\\\sup _{u \\\\in X}\\\\frac{||u||_{2^{\\\\ast }_{s}}^{2}}{||u||_{X}^{2}}.$ By inequality REF , $\\\\begin{array}{ll}\\\\left(\\\\int _{A_{n}}|w_{n}|^{2^{\\\\ast }_{s}}dx\\\\right)^{\\\\frac{2}{2^{\\\\ast }_{s}}}&\\\\le \\\\left(\\\\int _{\\\\mathbb {R}^{n}}|w_{n}|^{2^{\\\\ast }_{s}}dx\\\\right)^{\\\\frac{2}{2^{\\\\ast }_{s}}}\\\\\\\\&\\\\le S||w_{n}||^{2} \\\\\\\\& \\\\le S\\\\beta \\\\int _{\\\\mathbb {R}^{n}}g(x,v(x))v_{n}dx \\\\\\\\& \\\\le S\\\\beta \\\\int _{\\\\mathbb {R}^{n}}h(x)w_{n}^{2}dx \\\\\\\\& \\\\le S\\\\beta ||h||_{q}||w_{n}||_{\\\\frac{2q}{q-1}}^{2}.\\\\end{array}$ But, we have that $|w_{n}(x)|\\\\le |v(x)|^{\\\\beta }$ for all $x \\\\in B_{n}$ and $|w_{n}(x)|=|v(x)|^{\\\\beta }$ for all $x \\\\in A_{n}$ .\",\n \"Thereby, $\\\\begin{array}{ll}\\\\left(\\\\int _{A_{n}}|v|^{\\\\beta 2^{\\\\ast }_{s}}dx\\\\right)^{\\\\frac{2}{2^{\\\\ast }_{s}}} \\\\le S\\\\beta ||h||_{q}\\\\left[\\\\int _{\\\\mathbb {R}^{n}}|v|^{\\\\frac{2q\\\\beta }{q-1}}dx\\\\right]^{\\\\frac{q-1}{q}}.\\\\end{array}$ By Monotone Convergence Theorem, $||v||_{2^{\\\\ast }_{s}\\\\beta } \\\\le (\\\\beta S||h||_{q})^{\\\\frac{1}{2\\\\beta }}||v||_{2\\\\beta q_{1}},$ where $q_{1}=\\\\frac{q}{q-1}$ .\",\n \"Define $\\\\eta :=\\\\frac{2^{\\\\ast }_{s}}{2q_{1}}.$ and note that $\\\\eta >1$ .\",\n \"When $\\\\beta =\\\\eta $ we have that $2\\\\beta q_{1} = 2^{\\\\ast }_{s}$ .\",\n \"Then, by REF $||v||_{2^{\\\\ast }_{s}\\\\eta } \\\\le (\\\\eta S||h||_{q})^{\\\\frac{1}{2\\\\eta }}||v||_{2^{\\\\ast }_{s}}.$ Taking $\\\\beta =\\\\eta ^{2}$ in REF we obtain $||v||_{2^{\\\\ast }_{s}\\\\eta ^{2}} \\\\le \\\\eta ^{\\\\frac{1}{\\\\eta ^{2}}} (S||h||_{q})^{\\\\frac{1}{2\\\\eta ^{2}}}||v||_{2^{\\\\ast }_{s}\\\\eta }.$ By inequalities REF and REF we have $||v||_{2^{\\\\ast }_{s}\\\\eta ^{2}} \\\\le \\\\eta ^{\\\\frac{1}{(\\\\eta ^{2}}+\\\\frac{1}{2\\\\eta })} (S||h||_{q})^{(\\\\frac{1}{2\\\\eta ^{2}}+\\\\frac{1}{2\\\\eta })}||v||_{2^{\\\\ast }_{s}}.$ Inductively, we can prove that $||v||_{2^{\\\\ast }_{s}\\\\eta ^{m}} \\\\le \\\\eta ^{(\\\\frac{1}{2\\\\eta }+\\\\frac{1}{\\\\eta ^{2}}+...+\\\\frac{m}{2\\\\eta ^{m}})} (S||h||_{q})^{(\\\\frac{1}{2\\\\eta }+\\\\frac{1}{2\\\\eta ^{2}}+...+\\\\frac{1}{2\\\\eta ^{m}})}||v||_{2^{\\\\ast }_{s}}$ for all $m \\\\in \\\\mathbb {N}$ .\",\n \"But, $\\\\sum _{i=1}^{\\\\infty }\\\\frac{m}{2\\\\eta ^{m}}=\\\\frac{1}{2(\\\\eta -1)^{2}}$ and $\\\\sum _{i=1}^{\\\\infty }\\\\frac{1}{2\\\\eta ^{m}}=\\\\frac{1}{2(\\\\eta -1)}.$ Thereby, for all $m>0$ $||v||_{2^{\\\\ast }_{s}\\\\eta ^{m}} \\\\le \\\\eta ^{\\\\frac{1}{2(\\\\eta -1)^{2}}} (S||h||_{q})^{\\\\frac{1}{2(\\\\eta -1)}}||v||_{2^{\\\\ast }_{s}}$ Recalling that $||v||_{\\\\infty }=\\\\lim \\\\limits _{n \\\\rightarrow \\\\infty }||v||_{p}$ and that $\\\\eta >1$ we have that $||v||_{\\\\infty }\\\\le M||v||_{2^{\\\\ast }_{s}}$ for $M=\\\\eta ^{\\\\frac{1}{2(\\\\eta -1)^{2}}} (S||h||_{q})^{\\\\frac{1}{2(\\\\eta -1)}}$ and $\\\\eta = \\\\frac{n(q-1)}{q(n-2s)}$ We conclude the proof of Proposition REF noting that $M$ depends only on $q$ , $||h||_{q}$ .\"\n ],\n [\n \"Solution for Problem (P)\",\n \"In this section, we prove the main result, the Theorem REF .\",\n \"By Corollary REF , there is $u \\\\in X$ such that $J(u)=c$ and $J^{\\\\prime }(u)=0$ .\",\n \"We have the following estimate for $||u||_{\\\\infty }$ .\",\n \"Lemma 5.1 The solution $u$ of the auxiliary problem satisfies $||u||_{\\\\infty } \\\\le M(2Skd)^{\\\\frac{1}{2}}$ Consider the functions $\\\\begin{array}{lll}H(x,t)&=&\\\\left\\\\lbrace \\\\begin{array}{lllll}f(t)&if&|x| \\\\frac{V(x)}{k}t\\\\end{array}\\\\right.\\\\end{array}$ and $\\\\begin{array}{lll}b(x)&=&\\\\left\\\\lbrace \\\\begin{array}{lllll}V(x)&if&|x| \\\\frac{V(x)}{k}u.\\\\end{array}\\\\right.\\\\end{array}$ Note that the function $u$ is solution of $\\\\left\\\\lbrace \\\\begin{array}{ll}&-\\\\mathcal {L}_{k}u+b(x)u=H(x,u)\\\\\\\\&u \\\\in E.\\\\end{array}\\\\right.$ By $(f_{1})$ , $|H(x,t)|\\\\le c_{0}|t|^{p-1}$ for $p \\\\in (2,2^{\\\\ast }_{s})$ .\",\n \"Thereby, $|H(x,u)|\\\\le h(x)|u|,$ where $h(x)=C_{0}|u|^{p-2}$ .\",\n \"Note that $h \\\\in L^{\\\\frac{2^{\\\\ast }_{s}}{p-2}}$ with $||h||_{L^{q}(\\\\mathbb {R}^{n})}\\\\le C(2ksd)^{\\\\frac{p-2}{2^{\\\\ast }_{s}}}.$ The number $p$ satisfies $p<2^{\\\\ast }_{s} = 2+\\\\frac{2s}{n}2^{\\\\ast }_{s}.$ Then $q =\\\\frac{2^{\\\\ast }_{s}}{p-2}>\\\\frac{n}{2s}.$ By Proposition REF and sobolev embedding $||u||_{\\\\infty }\\\\le M||u||_{2^{\\\\ast }_{s}} \\\\le MS^{\\\\frac{1}{2}}||u||,$ where $M=M(q,||h||_{q})$ .\",\n \"By Proposition REF , we have $||u||_{\\\\infty } \\\\le M(2kSd)^{\\\\frac{1}{2}}.$ Theorem 5.2 Suppose that $V$ satisfies $(V_{1})$ -$(V_{2})$ and that $f$ satisfies $(f_{1})$ -$(f_{3})$ .\",\n \"There is $\\\\Lambda ^{\\\\ast }=\\\\Lambda ^{\\\\ast }(V_{\\\\infty },\\\\theta ,p,c_{0},s)>0$ such that if $\\\\Lambda >\\\\Lambda ^{\\\\ast }$ in $(V_{3})$ , then the problem $(P)$ has a nonnegative nontrivial solution.\",\n \"Indeed, let $|x|\\\\ge R$ .\",\n \"If $u(x)=0$ then by denfition $f(u(x))=g(x,u(x))$ .\",\n \"If $u(x)>0$ then $\\\\begin{array}{ll}\\\\frac{f(u(x))}{u(x)}&\\\\le c_{0}|u|^{p-2} \\\\\\\\&\\\\le c_{0}||u||_{\\\\infty }^{p-2} \\\\\\\\& = \\\\frac{ c_{0}||u||_{\\\\infty }^{p-2}}{\\\\Lambda }\\\\Lambda \\\\\\\\& \\\\le \\\\frac{k^{\\\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\\\frac{p-2}{2}}}{\\\\Lambda }\\\\frac{V(x)}{k}\\\\end{array}$ Define $\\\\Lambda ^{\\\\ast }=k^{\\\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\\\frac{p-2}{2}}.$ If $\\\\Lambda >\\\\Lambda ^{\\\\ast }$ then $\\\\frac{f(u(x))}{u(x)} \\\\le \\\\frac{V(x)}{k}.$ By definition of $g$ we have $g(x,u(x))=f(u(x))$ .\",\n \"Then $g(x,u(x))=f(u(x))$ for all $x \\\\in \\\\mathbb {R}^{n}$ .\",\n \"Therefore, $u$ is a solution nonnegative and nontrivial of problem $(P)$ .\",\n \"alarticle author=Alberti, G., author=Bellettini G. title=A nonlocal anisotropic model for phase transitions.\",\n \"I.\",\n \"The optimal profile problem, journal=Math.\",\n \"Ann., volume=310, date=1998, pages=527-560, review= alv2article author=Alves, C. 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notice that $L \\cap (-\\infty , t_1]$ is a compact set with zero Lebesgue measure, and so with empty interior.","On the other hand, we have the following Corollary: $d(t)$ is not a Hölder continuous function.","Proof: Suppose by contradiction that $d(t)$ is Hölder continuous with exponent $\\alpha >0$ .","By the previous theorem, there is ${\\varepsilon }>0$ such that $00$ , the sets $\\text{Exact}(c)=\\lbrace \\alpha \\in {\\mathbb {R}}| \\left|\\alpha -\\frac{p}{q}\\right|<\\frac{c}{q^2}\\text{ for infinitely many }(p, q)\\in {\\mathbb {Z}}\\times {\\mathbb {N}}^*\\text{ but, for every } {\\varepsilon }>0,$ $\\left|\\alpha -\\frac{p}{q}\\right|<\\frac{c-{\\varepsilon }}{q^2}\\text{ has only a finite number of solutions }(p, q)\\in {\\mathbb {Z}}\\times {\\mathbb {N}}^*\\rbrace \\text{ and}$ $\\text{Exact}^{\\prime }(c)=\\lbrace \\alpha \\in {\\mathbb {R}}|\\text{ For every } {\\varepsilon }>0, \\left|\\alpha -\\frac{p}{q}\\right|<\\frac{c+{\\varepsilon }}{q^2}\\text{ for infinitely many }(p, q)\\in {\\mathbb {Z}}\\times {\\mathbb {N}}^*\\text{ but}$ $\\left|\\alpha -\\frac{p}{q}\\right|<\\frac{c}{q^2}\\text{ has only a finite number of solutions }(p, q)\\in {\\mathbb {Z}}\\times {\\mathbb {N}}^*\\rbrace .$ Clearly Exact$(c) \\,\\cup \\,$ Exact$^{\\prime }(c)=k^{-1}(c^{-1})$ .","Theorem 2: $\\lim _{c\\rightarrow 0}HD(\\text{Exact}(c))=\\lim _{c\\rightarrow 0}HD(\\text{Exact}^{\\prime }(c))=1$ .","Consequently, $\\lim _{t\\rightarrow \\infty }HD(k^{-1}(t))=1$ and $\\lim _{t\\rightarrow \\infty }HD(k^{-1}(-\\infty ,t))=\\lim _{t\\rightarrow \\infty }D(t)=1$ .","This solves affirmatively Problem 4 of [B].","We also prove a result on the topological structure of the Lagrange spectrum $L$ .","Theorem 3: $L^{\\prime }$ is a perfect set, i.e., $L^{\\prime \\prime }=L^{\\prime }$ .","The proof of this theorem uses the fact that an element of the Lagrange spectrum associated to an infinite sequence $\\underline{\\theta }$ is accumulated by infinitely many sums of the type ${\\alpha }_n+{\\beta }_n$ , which is not necessarily true for elements of the Markov spectrum.","The question of whether $M^{\\prime \\prime }=M^{\\prime }$ is still open.","There are still some important questions left on the structure of the Markov and Lagrange spectra.","For instance: 1) Consider the function $d_{loc}: L^{\\prime }\\rightarrow {\\mathbb {R}}$ given by $d_{loc}(t)=\\lim _{{\\varepsilon }\\rightarrow 0} HD(L\\cap (t-{\\varepsilon }, t+{\\varepsilon }))$ .","Is $d_{loc}$ a non-decreasing function ?","2) Describe the geometric structure of the difference set $M\\setminus L$ .","3) Let, as before, $t_1=\\min \\lbrace t \\in {\\mathbb {R}}\\mid d(t)=1\\rbrace <\\sqrt{12}$ .","Is it true that $t_1=\\inf $ int $L$ ?","This would imply that int$(L \\cap (-\\infty , \\sqrt{12}])\\ne \\emptyset $ , which in its turn implies that int$(C_2+C_2)\\ne \\emptyset $ , where $C_2=\\lbrace {\\alpha }=[0; a_1, a_2,\\ldots ]\\in [0,1]| a_n\\le 2, \\forall n\\ge 1\\rbrace $ .","We should mention that, in relation to question 2), there are some progresses in recent preprints by the author and C. Matheus: in [MM3] and [MM1], the authors describe the intersections of $M\\setminus L$ with the maximal intervals not intersecting $L$ containing the first examples $\\gamma =3.11812017815993...$ and $\\alpha _{\\infty }=3.293044265...$ by Freiman of elements in $M\\setminus L$ .","These intersections have positive Hausdorff dimensions (which coincide with the Hausdorff dimensions of certain regular Cantor sets we describe in these works).","In [MM4], the authors exhibit a regular Cantor set diffeomorphic to $C_2$ (and so with Hausdorff dimensions larger than $0.53128$ ) near $3.7096998597502$ contained in $M\\setminus L$ .","And, in [MM3], they prove that the Hausdorff dimension of $M\\setminus L$ is smaller than $0.986927$ , and indicate, using heuristic estimates, that this Hausdorff dimension is smaller than $0.888$ .","In relation to question 4), the question whether int$(C_2+C_2)\\ne \\emptyset $ was posed in page 71 of [CF].","I would like to thank Yann Bugeaud, Aline Gomes Cerqueira, Carlos Matheus, Túlio Carvalho and Yuri Lima for helpful comments and suggestions which substantially improved this work."],["A dimension formula for arithmetic sums of regular Cantor sets","We say that $K \\subset {\\mathbb {R}}$ is a regular Cantor set of class $C^k$, $k\\ge 1$ , if: i) there are disjoint compact intervals $I_1,I_2,\\dots ,I_r$ such that $K\\subset I_1 \\cup \\cdots \\cup I_r$ and the boundary of each $I_j$ is contained in $K$ ; ii) there is a $C^k$ expanding map $\\psi $ defined in a neighbourhood of $I_1\\cup I_2\\cup \\cdots \\cup I_r$ such that $\\psi (I_j)$ is the convex hull of a finite union of some intervals $I_s$ satisfying: ii.1) for each $j$ , $1\\le j\\le r$ , and $n$ sufficiently big, $\\psi ^n(K\\cap I_j)=K$ ; ii.2) $K=\\bigcap \\limits _{n\\in \\mathbb {N}} \\psi ^{-n}(I_1\\cup I_2\\cup \\cdots \\cup I_r)$ .","We say that $\\lbrace I_1,I_2,\\dots ,I_r\\rbrace $ is a Markov partition for $K$ and that $K$ is defined by $\\psi $ .","Let $K$ be regular Cantor sets of class $C^2$ defined by the expansive function $\\psi $ .","It is a general fact, due originally to Poincaré, that, given a periodic point $p$ of period $r$ of $\\psi $ , there is a $C^2$ diffeomorphism $h$ of the support interval $I$ of $K$ such that $\\tilde{\\psi }=h^{-1}\\circ \\psi ^r\\circ h$ is affine in $h^{-1}(J)$ , where $J$ is the connected component of the domain of $\\psi ^r$ which contains $p$ .","We say that $K$ is non essentially affine if ${\\tilde{\\psi }}^{^{\\prime \\prime }}(x)\\ne 0$ for some $x\\in h^{-1}(K)$ .","In [Mo], we use the Scale Recurrence Lemma of [MY] in order to prove the following Theorem.","If $K$ and $K^{\\prime }$ are regular Cantor sets of class $C^2$ and $K$ is non essentially affine, then $HD(K+K^{\\prime })=\\min \\lbrace HD(K)+HD(K^{\\prime }), 1\\rbrace $.","This result will be a central tool in the proof of Theorem 1."],["Regular Cantor sets defined by the Gauss map","The Gauss map is the map $g\\colon (0,1]\\rightarrow [0,1]$ given by $g(x)=\\lbrace \\frac{1}{x}\\rbrace =\\frac{1}{x}-\\lfloor \\frac{1}{x}\\rfloor , \\,\\forall \\, x\\in (0,1].$ It acts as a shift on continued fractions: if $a_n\\in {\\mathbb {N}}^*, \\forall n\\ge 1$ then $g([0; a_1, a_2, a_3, \\dots ]=[0; a_2, a_3, a_4,\\dots ]$ .","Regular Cantor sets defined by the Gauss map (or iterates of it) restricted to some finite union of intervals are closely related to continued fractions with bounded partial quotients.","We will often consider such regular Cantor sets associated to complete shifts.","A complete shift is associated to finite sets of finite sequences of positive integers in the following way: given a finite set $B=\\lbrace {\\beta }_1,{\\beta }_2,\\dots ,{\\beta }_m\\rbrace $ , $m\\ge 2$ , where ${\\beta }_j\\in ({\\mathbb {N}}^*)^{r_j}$ , $r_j\\in {\\mathbb {N}}^*$ , $1\\le j\\le m$ and ${\\beta }_i$ does not begin by ${\\beta }_j$ for $i \\ne j$ , the complete shift associated to $B$ is the set $\\Sigma (B)\\subset ({\\mathbb {N}}^*)^{{\\mathbb {N}}}$ of the infinite sequences obtained by concatenations of elements of $B$ : $\\Sigma (B)=\\lbrace ({\\alpha }_0,{\\alpha }_1,{\\alpha }_2,\\dots ) \\, \\mid \\, {\\alpha }_j\\in B,\\,\\, \\forall \\, j\\in {\\mathbb {N}}\\rbrace .$ Here (and in the rest of the paper), we use the following notation for concatenations of finite sequences: if ${\\alpha }_j=(a_j^{(1)},a_j^{(2)},\\dots ,{\\alpha }_j^{(m_j)})$ then $({\\alpha }_0,{\\alpha }_1,{\\alpha }_2,\\dots )$ means the sequence $(a_0^{(1)},a_0^{(2)},\\dots ,{\\alpha }_0^{(m_0)},a_1^{(1)},a_1^{(2)},\\dots ,{\\alpha }_1^{(m_1)},a_2^{(1)},a_2^{(2)},\\dots ,{\\alpha }_2^{(m_2)},\\dots )$ .","In some cases, when there is no ambiguity, we will write ${\\alpha }_0 {\\alpha }_1 {\\alpha }_2\\cdots $ and also ${\\alpha }_0^N$ to represent the concatenation of $N$ copies of ${\\alpha }_0$ .","In some cases some of the ${\\alpha }_j$ are finite sequences and some cases are single numbers, which are viewed as one-element sequences.","Associated to $\\Sigma (B)$ is the Cantor set $K(B)\\subset [0,1]$ of the real numbers whose continued fractions are of the form $[0;{\\gamma }_1,{\\gamma }_2,{\\gamma }_3,\\dots ]$ , where ${\\gamma }_j\\in B$ , $\\forall \\, j\\ge 1$ .","This is a regular Cantor set.","Indeed, if $a_j$ and $b_j$ are respectively the smallest and largest elements of $K(B)$ whose continued fractions begin by $[0; {\\beta }_j]$ , for $1\\le j\\le m$ , and $I_j=[a_j,b_j]$ , then $K(B)$ is the regular Cantor set defined by the map $\\psi $ with domain $\\bigcup \\limits _{j=1}^m I_j$ given by $\\psi |_{I_j}=g^{r_j}$ , $1\\le j\\le m$ .","We have the following Proposition 1.","The Cantor sets $K(B)$ defined by the Gauss map associated to complete shifts are non essentially affine.","Proof: Let $B=\\lbrace {\\beta }_1,{\\beta }_2,\\dots ,{\\beta }_m\\rbrace $ , ${\\beta }_j=(b_1^{(j)},b_2^{(j)},\\dots ,b_{r_j}^{(j)})\\in ({\\mathbb {N}}^*)^{r_j}$ , $1\\le j\\le m$ .","For each $j\\le m$ , let $x_j=[0; {\\beta }_j,{\\beta }_j,{\\beta }_j,\\dots ]\\in I_j$ be the fixed point of $\\psi |_{I_j}=g^{r_j}$ .","Notice that, since ${\\beta }_i$ does not begin by ${\\beta }_j$ for $i \\ne j$ , the $x_j, 1\\le j\\le m$ are all distinct.","Moreover, according to the classical theory of continued fractions, if $p_k^{(j)}/q_k^{(j)}:=[0;b_1^{(j)},b_2^{(j)},\\dots ,b_k^{(j)}]$ , for $1\\le j\\le m$ , $1\\le k\\le r_j$ , we have $I_j\\subset \\lbrace [0;{\\beta }_j,{\\alpha }],\\,\\, {\\alpha }\\ge 1\\rbrace $ and $\\psi |_{I_j}(x)$ is given by $\\psi |_{I_j}(x)=\\frac{q_{r_j}^{(j)}x-p_{r_j}^{(j)}}{-q_{r_j-1}^{(j)}x+p_{r_j-1}^{(j)}}$ (see the appendix); so $x_j$ is the positive root of $q_{r_j-1}^{(j)}x^2+(q_{r_j}^{(j)}-p_{r_j-1}^{(j)})x-p_{r_j}^{(j)}$ (since $x_j$ is the fixed point of $\\psi |_{I_j}$ ).","For each $j\\le m$ , since $\\psi |_{I_j}$ is a Möbius function with a hyperbolic fixed point $x_j$ , there is a Möbius function ${\\alpha }_j(x)=\\frac{a_jx+b_j}{c_jx+d_j}$ with ${\\alpha }_j(x_j)=x_j$ , ${\\alpha }_j^{\\prime }(x_j)=1$ such that ${\\alpha }_j\\circ (\\psi |_{I_j})\\circ {\\alpha }_j^{-1}$ is an affine map.","If we show that the Möbius functions ${\\alpha }_1\\circ (\\psi |_{I_2})\\circ {\\alpha }_1^{-1}$ is not affine then we are done, since the second derivative of a non-affine Möbius function never vanishes.","Suppose by contradiction that ${\\alpha }_1\\circ (\\psi |_{I_2})\\circ {\\alpha }_1^{-1}$ is affine.","Since ${\\alpha }_1\\circ (\\psi |_{I_1})\\circ {\\alpha }_1^{-1}$ is also affine these two functions have a common fixed point at $\\infty $ , so ${\\alpha }_1^{-1}(\\infty )=-d_1/c_1$ is a common fixed point of $\\psi |_{I_2}$ and $\\psi |_{I_1}$ , which implies that ${\\alpha }_1^{-1}(\\infty )$ is a common root of $q_{r_1-1}^{(1)}x^2+(q_{r_1}^{(1)}-p_{r_1-1}^{(1)})x-p_{r_1}^{(1)}$ and $q_{r_2-1}^{(2)}x^2+(q_{r_2}^{(2)}-p_{r_2-1}^{(2)})x-p_{r_2}^{(2)}$ .","Since these polynomials of ${\\mathbb {Q}}[x]$ are irreducible (indeed their roots $x_1$ and $x_2$ are irrational because their continued fractions expansions are infinite), they must be associates in ${\\mathbb {Q}}[x]$ , and so their remaining roots $x_1$ and $x_2$ must coincide, which is a contradiction.","$\\Box $ Corollary 1.","$HD(K(B)+K(B^{\\prime }))=\\min \\lbrace 1,HD(K(B))+HD(K(B^{\\prime }))\\rbrace $ , for every sets $B$ , $B^{\\prime }$ of finite sequences of positive integers.","Definition 2: If ${\\beta }=(b_1,b_2,\\dots ,b_{n-1},b_n)$ , then ${\\beta }^t:=(b_n,b_{n-1},\\dots ,b_2,b_1)$ .","Given a set of finite sequences $B$ , we define $B^t:=\\lbrace {\\beta }^t, {\\beta }\\in B\\rbrace $ .","Proposition 2.","$HD(K(B))=HD(K(B^t))$ , for any finite set $B$ of finite sequences.","Proof: This follows from $q_n({\\beta })=q_n({\\beta }^t)$ , $\\forall {\\beta }$ (see the appendix of [CF] on properties of continuants), and from the fact that, if $\\psi |_{I_j(x)}=\\frac{q_{n}^{(j)}x-p_{n}^{(j)}}{-q_{n-1}^{(j)}x+p_{n-1}^{(j)}}$ , then $\\psi ^{\\prime }|_{I_j(x)}=\\frac{-(p_{n}^{(j)}q_{n-1}^{(j)}-p_{n-1}^{(j)}q_{n}^{(j)})}{(-q_{n-1}^{(j)}x+p_{n-1}^{(j)})^2}=\\frac{(-1)^n}{(-q_{n-1}^{(j)}x+p_{n-1}^{(j)})^2}$ satisfies $(q_{n}^{(j)})^2\\le |\\psi ^{\\prime }|_{I_j(x)}|\\le 4(q_{n}^{(j)})^2$ , since $\\frac{1}{2q_{n}^{(j)}}\\le \\frac{1}{q_{n}^{(j)}+q_{n-1}^{(j)}} \\le |q_{n-1}^{(j)}x-p_{n-1}^{(j)}|\\le \\frac{1}{q_{n}^{(j)}}.$ $\\Box $ Corollary 2.","$HD(K(B)+K(B^t))=\\min \\lbrace 1,2 \\cdot HD(K(B))\\rbrace $ , for every set $B$ of finite sequences of positive integers."],["Fractal dimensions of the spectra","We recall that the Lagrange spectrum is given by $L=\\lbrace \\ell (\\underline{{\\theta }}),\\,\\, \\underline{{\\theta }}\\in \\Sigma \\rbrace $ , where $\\Sigma =({\\mathbb {N}}^*)^{{\\mathbb {Z}}}$ and, for $\\underline{{\\theta }}=(a_n)_{n\\in {\\mathbb {Z}}}\\in \\Sigma $ , $\\ell (\\underline{{\\theta }}):=\\limsup _{n\\rightarrow +\\infty }({\\alpha }_n+{\\beta }_n)$ , where ${\\alpha }_n$ and ${\\beta }_n$ are defined as the continued fractions ${\\alpha }_n:=[a_n; a_{n+1},a_{n+2},\\dots ]$ and ${\\beta }_n:=[0;a_{n-1},a_{n-2},\\dots ]$ , while the Markov spectrum is given by $M=\\lbrace m(\\underline{{\\theta }}),\\underline{{\\theta }}\\in \\Sigma \\rbrace $ , where $m(\\underline{{\\theta }})=\\sup \\lbrace {\\alpha }_n+{\\beta }_n,\\,\\, n\\in {\\mathbb {Z}}\\rbrace $ .","Given a finite sequence ${\\alpha }=(a_1,a_2,\\dots ,a_n)\\in ({\\mathbb {N}}^*)^n$ , we define its size by $s({\\alpha }):=|I({\\alpha })|$ , where $I({\\alpha })$ is the interval $\\lbrace x\\in [0,1] \\mid x=[0; a_1,a_2,\\dots ,a_n,{\\alpha }_{n+1}]$ , ${\\alpha }_{n+1}\\ge 1\\rbrace $ .","If we take $p_0=0$ , $q_0=1$ , $p_1=1$ , $q_1=a_1$ and, for $k\\ge 0$ , $p_{k+2}=a_{k+2}p_{k+1}+p_k$ and $q_{k+2}=a_{k+2} q_{k+1}+q_k$ , then $I({\\alpha })$ is the interval with extremities $[0;a_1,a_2,\\dots ,a_n]=p_n/q_n$ and $[0;a_1,a_2,\\dots ,a_{n-1},a_n+1]=\\frac{p_n+p_{n-1}}{q_n+q_{n-1}}$ and so $s({\\alpha })=\\left| \\frac{p_n}{q_n}-\\frac{p_n+p_{n-1}}{q_n+q_{n-1}} \\right| = \\frac{1}{q_n(q_n+q_{n-1})},$ since $p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}$ .","We define $r({\\alpha })=\\lfloor \\log s({\\alpha })^{-1} \\rfloor $ which controls the order of magnitude of the size of $I({\\alpha })$ .","We also define, for $r \\in {\\mathbb {N}}, P_r= \\lbrace {\\alpha }=(a_1,a_2,\\dots ,a_n) \\mid r({\\alpha }) \\ge r$ and $r((a_1,a_2,\\dots ,a_{n-1}))\\alpha _n\\ge a_n$ for every $n$ , and so $m(\\underline{{\\theta }})>\\sup \\lbrace a_n,n\\in {\\mathbb {Z}}\\rbrace $ .","So, if $m(\\underline{{\\theta }})\\le t$ we have $a_n\\le \\lfloor t \\rfloor $ , $\\forall \\,\\, n\\in {\\mathbb {N}}$ .","Given $t\\in [3,+\\infty )$ and $r\\in {\\mathbb {N}}$ , let $T:=\\lfloor t \\rfloor $ and $C(t,r)$ be the set $\\lbrace {\\alpha }=(a_1,a_2,\\dots ,a_n) \\in P_r \\mid K_t\\cap I({\\alpha })\\ne \\emptyset \\rbrace $ .","Here $K_t:=\\lbrace [0;{\\gamma }]| {\\gamma }\\in \\pi _+(\\Sigma _t)\\rbrace $ , where $\\pi _+\\colon \\Sigma \\rightarrow \\Sigma ^+$ is the projection associated to the decomposition $\\Sigma =\\Sigma ^-\\times \\Sigma ^+$ .","Since $\\Sigma _t$ is invariant by transposition and by $\\sigma $ , $K_t$ is invariant by the Gauss map $g$ and $M\\cap (-\\infty , t) \\subset ({\\mathbb {N}}^* \\cap [1, T])+K_t+K_t$ .","We define $N(t,r):=|C(t,r)|$ , where $|\\cdot |$ denotes cardinality.","Notice that if $r\\le s$ then $N(t,r)\\le N(t,s)$ and, if $t\\le \\tilde{t}$ , then $N(t,r)\\le N(\\tilde{t},r)$ .","For any finite sequences ${\\alpha },{\\beta }$ and any positive integers $k_1, k_2\\le T$ we have $r({\\alpha }{\\beta }k_1 k_2)\\ge r({\\alpha })+r({\\beta })$ (see the appendix), so if $C(t,r)=\\lbrace {\\alpha }_1,{\\alpha }_2,\\dots ,{\\alpha }_u\\rbrace $ and $C(t,s)=\\lbrace {\\beta }_1,{\\beta }_2,\\dots ,{\\beta }_v\\rbrace $ , we may cover $K_t$ by the $T^2uv=T^2N(t,r)N(t,s)$ intervals $I({\\alpha }_i{\\beta }_j k_1 k_2)$ , $1\\le i \\le u$ , $1\\le j\\le v$ , $1\\le k_1, k_2\\le T$ , which satisfy $r({\\alpha }_i{\\beta }_j k_1 k_2)\\ge r+s$ , $\\forall \\,\\, i,j,k$ .","Replacing, if necessary, some of these intervals by larger intervals $I({\\gamma })$ in $P_{r+s}$ , we conclude that $N(t,r+s)\\le T^2N(t,r)N(t,s)$ and so $\\log (T^2N(t,r+s))\\le \\log (T^2N(t,r))+\\log (T^2N(t,s)),\\quad \\forall \\,\\, r,s.$ This implies that $\\lim _{m\\rightarrow \\infty }\\frac{1}{m} \\log (T^2N(t,m))=\\inf _{m\\in {\\mathbb {N}}^*}\\frac{1}{m} \\log (T^2N(t,m))=\\lim _{m\\rightarrow \\infty }\\frac{1}{m}\\log (N(t,m))$ exists.","We will call this limit $D(t)$ (which coincides with the (upper) box dimension of $K_t$ , as follows easily from its definition).","Notice that $D(t)$ is a non-decreasing function.","We will see in the proof of Theorem 1 that $D(t)$ is continuous and that $HD(k^{-1}(-\\infty ,t))=D(t)$ .","Lemma 1.","$D(t)$ is right-continuous: given $t_0\\in [3,+\\infty )$ and $\\eta >0$ there is $\\delta >0$ such that for $t_0\\le t\\le t+\\delta $ we have $D(t_0)\\le D(t) \\le D(t+\\delta ) t_0$ , $r$ large, $\\frac{\\log N(t,r)}{r} \\ge D(t_0)+\\eta $ we would have $D(t_0) \\ge D(t_0) + \\eta $ , contradiction (indeed $C(t,r) \\subset C(s,r)$ for $t \\le s$ , and, by compacity, $C(t_0,r) ={\\bigcap }_{t>t_0}\\,C(t,r)$ ).","$\\Box $ Lemma 2.","Given $t\\in (3,+\\infty )$ and $\\eta \\in (0,1)$ there is $\\delta >0$ and a Cantor set $K(B)$ defined by the Gauss map associated to a complete shift $\\Sigma (B)\\subset \\Sigma $ such that $\\Sigma (B)\\subset \\Sigma _{t-\\delta }$ and $HD(K(B))>(1-\\eta )D(t)$ .","Since the proof of this Lemma is somewhat technical, we will postpone it to Section 6.","Lemma 3.","Given a complete shift $\\Sigma (X)\\subset \\Sigma $ (where X is a finite set of finite sequences of positive integers), we have $HD(\\ell (\\Sigma (X)))=HD(m(\\Sigma (X)))=\\overline{\\dim }(\\ell (\\Sigma (X)))=\\overline{\\dim }(m(\\Sigma (X)))=\\min \\lbrace 2 \\cdot HD(K(X)), 1 \\rbrace .$ Proof: Let $T$ be the largest element of a sequence in $X$ .","First of all we clearly have $\\ell (\\Sigma (X)) \\subset m(\\Sigma (X))\\subset \\bigcup \\limits _{1\\le a\\le T\\atop 1\\le i, j\\le R} (a+g^i(K(X))+g^j(K(X))),\\;\\;\\text{where $R$ is the length of}$ the largest word of $X$ , so $HD(\\ell (\\Sigma (X)) \\le HD(m(\\Sigma (X)) \\le \\min \\lbrace 2 \\cdot HD(K(X)), 1 \\rbrace $ .","Let ${\\varepsilon }>0$ be given.","We will show that there are regular Cantor sets $K, K^{\\prime }$ defined by iterates of the Gauss map with $HD(K),HD(K^{\\prime })>HD(K(X))-{\\varepsilon }$ such that $K+K^{\\prime } \\subset \\ell (\\Sigma (X)) \\subset m(\\Sigma (X))$ .","Since, by the dimension formula stated in section 2, $HD(K+K^{\\prime })=\\min \\lbrace HD(K)+HD(K^{\\prime }), 1\\rbrace >\\min \\lbrace 2 \\cdot HD(K(X)), 1 \\rbrace -2{\\varepsilon }$ and ${\\varepsilon }>0$ is arbitrary, the result will follow.","Given a positive integer $n$ , let $X^n=\\lbrace ({\\gamma }_1,{\\gamma }_2,\\dots ,{\\gamma }_n)|{\\gamma }_j \\in X, \\forall j \\le n \\rbrace $ .","We have $\\Sigma (X^n)=\\Sigma (X)$ and $K(X^n)=K(X)$ .","Replacing $X$ by $X^n$ for some $n$ large, we may assume without loss of generality that for any $A \\subset X$ (resp.","$A^t \\subset X^t$ ) with $|A|\\le 2$ (resp.","$|A^t|\\le 2$ ), we have $HD(K(X\\setminus A))>HD(K(X))-{\\varepsilon }$ (resp.","$HD(K(X^t\\setminus A^t))>HD(K(X^t))-{\\varepsilon }=HD(K(X))-{\\varepsilon }$ ).","Order $X$ and $X^t$ in the following way: given ${\\gamma }, \\tilde{{\\gamma }}\\in X \\, (\\text{resp.","}{\\gamma }, \\tilde{{\\gamma }}\\in X^t)$ , we say that ${\\gamma }< \\tilde{{\\gamma }}$ if and only if $[0;{\\gamma }]<[0;\\tilde{{\\gamma }}]$ .","Suppose that the maximum of $m(\\Sigma (X))$ is attained at $\\tilde{\\underline{\\theta }}= (\\dots ,\\tilde{{\\gamma }}_{-1},\\tilde{{\\gamma }}_0,\\tilde{{\\gamma }}_1,\\dots ),\\tilde{{\\gamma }}_i \\in X, \\forall i \\in {\\mathbb {Z}}$ , in a position belonging to the sequence $\\tilde{{\\gamma }}_0$ .","Let $X^* = X\\backslash \\lbrace \\min X, \\max X\\rbrace $ , $(X^t)^* = X^t\\backslash \\lbrace \\min X^t, \\max X^t\\rbrace $ .","Essentially, $K(X^*)$ and $K((X^t)^*)$ will be the required Cantor sets, but first we have to control the positions where the $\\limsup $ is attained (the idea is somewhat similar to the proof that Hall's theorem ([H]) on sums of continued fractions with coefficients bounded by 4 implies that the Lagrange spectrum contains $[6,+\\infty )$ ) and which words can appear in the beginning of the elements.","For each positive integer $m$ , let $C^m$ be the set of sequences $(\\dots , {\\gamma }_{-m-2}, {\\gamma }_{-m-1}, \\tilde{{\\gamma }}_{-m}, \\tilde{{\\gamma }}_{-m+1}, \\dots , \\tilde{{\\gamma }}_{-1}, \\tilde{{\\gamma }}_0, \\tilde{{\\gamma }}_1, \\dots , \\tilde{{\\gamma }}_{m-1}, \\tilde{{\\gamma }}_m, {\\gamma }_{m+1}, {\\gamma }_{m+2}, \\dots )$ where ${\\gamma }_k \\in X^*$ for $k \\ge m+1$ , ${\\gamma }_k^t \\in (X^t)^*$ for $k \\le -m-1$ .","Then, for $m$ large enough, there is $\\eta >0$ such that for each $\\underline{\\theta }\\in C^m$ , $\\text{sup}({\\alpha }_n+{\\beta }_n) =m(\\underline{\\theta })$ is attained only for values of $n$ corresponding to the piece $\\tau =(\\tilde{{\\gamma }}_{-m}, \\tilde{{\\gamma }}_{-m+1}, \\dots , \\tilde{{\\gamma }}_{-1}, \\tilde{{\\gamma }}_0, \\tilde{{\\gamma }}_1, \\dots , \\tilde{{\\gamma }}_{m-1}, \\tilde{{\\gamma }}_m)$ of $\\underline{\\theta }$ and, if $n$ does not correspond to the piece $\\tau $ , then ${\\alpha }_n+{\\beta }_nm_k$ such that ${\\alpha }_{n_k}(\\underline{\\theta }^{(k)})+{\\beta }_{n_k}(\\underline{\\theta }^{(k)})>m(\\underline{\\theta }^{(k)})-1/k$ .","Since $\\underline{\\theta }^{(k)}$ converges to $\\tilde{\\underline{\\theta }}$ , $m(\\underline{\\theta }^{(k)})$ converges to $m(\\tilde{\\underline{\\theta }})$ and, by compacity, if $N_k$ denotes the size of the sequence $\\tilde{{\\gamma }}_0, \\tilde{{\\gamma }}_1, \\dots , \\tilde{{\\gamma }}_{m_k-1}, \\tilde{{\\gamma }}_{m_k}, {\\gamma }_{m_k+1}, {\\gamma }_{m_k+2}, \\dots , {\\gamma }_{r(k)-1}$ , $(\\sigma ^{N_k}(\\underline{\\theta }^{(k)}))$ has a subsequence which converges to some $\\hat{\\underline{\\theta }}= (\\dots ,\\hat{{\\gamma }}_{-1},\\hat{{\\gamma }}_0,\\hat{{\\gamma }}_1,\\dots ) \\in \\Sigma (X)$ , with $\\hat{{\\gamma }}_i \\in X^*, \\forall i \\ge 0$ , such that $\\text{sup}({\\alpha }_n+{\\beta }_n)=m(\\hat{\\underline{\\theta }})=m(\\tilde{\\underline{\\theta }})$ is attained for some $n$ corresponding to the piece $\\hat{{\\gamma }}_0$ .","This is a contradiction, since $m(\\tilde{\\underline{\\theta }})$ is the maximum of $m(\\Sigma (X))$ and, changing $\\hat{{\\gamma }}_1$ by $\\min X$ or $\\max X$ , we strictly increase the value of $m(\\hat{\\underline{\\theta }})$ .","Notice that the same argument shows that for any $\\underline{\\theta }\\in C^m$ and $\\underline{\\theta }^* \\in \\Sigma (X^*)$ , we have $m(\\underline{\\theta }^*)HD(K(X))-{\\varepsilon }$ and $HD(K^{\\prime })=HD(K((X^t)^*))>$$HD(K(X^t))-{\\varepsilon }=HD(K(X))-{\\varepsilon }$ .","$\\Box $"],["Proofs of the main results","Proof of Theorem 1: Applying Lemma 3 to the complete shift $\\Sigma (B)$ obtained in Lemma 2, we get that, for any $\\eta >0$ , there is $\\delta >0$ such that $\\min \\lbrace 2(1-\\eta )D(t),1\\rbrace \\le \\min \\lbrace 2HD(K(B)),1\\rbrace \\le HD(L\\cap (-\\infty ,t-\\delta ]) \\le HD(L \\cap (-\\infty ,t)) \\le HD(M \\cap (-\\infty ,t)) \\le \\overline{\\dim }(M\\cap (-\\infty ,t))\\le \\min \\lbrace 2 \\cdot HD(K_t),1\\rbrace \\le \\min \\lbrace 2 \\cdot D(t),1\\rbrace $ (since $\\ell (\\Sigma (B))\\subset L\\cap (-\\infty ,t-\\delta ]$ , $L\\cap (-\\infty , t) \\subset M\\cap (-\\infty , t) \\subset ({\\mathbb {N}}^* \\cap [1, T])+K_t+K_t$ and $D(t)$ is the upper box dimension of $K_t$ ), and so, if $d(t):=HD(L\\cap (-\\infty ,t))$ , we have $d(t)= HD(M\\cap (-\\infty ,t))=\\overline{\\dim }(L\\cap (-\\infty ,t))=\\overline{\\dim }(M\\cap (-\\infty ,t))=\\min \\lbrace 2 \\cdot D(t),1\\rbrace $ .","In order to conclude the proof of the first assertion of i), it is enough to show that $HD(k^{-1}(-\\infty ,t))=D(t)$ .","In the notation of Lemma 3, let $x\\in K, y\\in K^{\\prime }$ .","For each $z=[0; {\\alpha }_1, {\\alpha }_2, \\dots ]\\in K(X^*)$ , define $\\underline{{\\lambda }}(z)={\\underline{{\\lambda }}}_{x,y}(z)=({\\alpha }_{1!","},\\tau ^{(1)},{\\alpha }_{2!","},\\tau ^{(2)},{\\alpha }_3,{\\alpha }_4,{\\alpha }_5,{\\alpha }_{3!","},\\tau ^{(3)},{\\alpha }_7,\\dots ,{\\alpha }_{4!","},\\tau ^{(4)},\\\\{\\alpha }_{25},{\\alpha }_{26},\\dots ,{\\alpha }_{5!","},\\tau ^{(5)},\\;\\dots \\;,{\\alpha }_{r!","},\\tau ^{(r)},{\\alpha }_{r!+1},\\dots ),$ and $h(z)=[0;\\underline{{\\lambda }}(z)]$ .","We have, as before, $k(h(z))=x+y$ .","On the other hand, given any $\\rho >0$ , we have $|z-z^{\\prime }|=O(|h(z)-h(z^{\\prime })|^{1-\\rho })$ for $|z-z^{\\prime }|$ small, so $HD(k^{-1}(x+y)) \\ge HD(K(B^*))>HD(K(B))-{\\varepsilon }$ .","As before, we get $HD(k^{-1}(-\\infty ,t)) \\ge HD(k^{-1}(-\\infty ,t-\\delta ])$ $\\ge HD(k^{-1}(x+y)) > HD(K(B))-{\\varepsilon }> (1-\\eta )D(t)-{\\varepsilon }.$ Since $\\eta $ and ${\\varepsilon }$ are arbitrary, $HD(k^{-1}(-\\infty ,t)) \\ge D(t)$ .","For the reverse inequality, let $w \\in k^{-1}(-\\infty ,t)$ .","We have $\\limsup _{n\\rightarrow \\infty }({\\alpha }_n(w)+{\\beta }_n(w))=k(w)0$ such that $D(t-\\delta )\\ge HD(K(B))>(1-\\eta )D(t)$ , so $\\lim \\limits _{s\\rightarrow t-} D(s) = D(t)$ .","In order to conclude, notice that, for each positive integer $m$ , $\\Sigma (\\lbrace 21^{2m}2,21^{2m+2}2\\rbrace ) \\subset \\Sigma _{3+2^{-m}}$ (notice that $[2;1, 1, 1,\\ldots ]+[0;2, 1, 1, 1,\\ldots ]=3$ ), so $D(3+{\\varepsilon })>0$ for every ${\\varepsilon }>0$ and, since $\\Sigma _{\\sqrt{12}}=\\lbrace 1,2\\rbrace ^{{\\mathbb {Z}}}$ , $D(\\sqrt{12})=HD(K_{\\sqrt{12}})=HD(K(\\lbrace 1,2\\rbrace ))=HD(C_2)=0,53128\\ldots >1/2$ , so, since $D(t)$ is continuous, there is $\\delta >0$ such that $D(\\sqrt{12}-\\delta )>1/2$ , and thus we have $d(\\sqrt{12}-\\delta )=\\min \\lbrace 2 \\cdot D(\\sqrt{12}-\\delta ),1\\rbrace =1$ .","$\\Box $ Remark: It follows from the above proof and from the general estimates of fractal dimensions of regular Cantor sets of Chapter 4 of [PT] that there is a constant $C>0$ such that, for each positive integer $m$ , $D(3+2^{-m})\\ge HD(K(\\lbrace 21^{2m}2,21^{2m+2}2\\rbrace ))>C/m$ .","This gives another proof of the fact that the functions $D(t)$ and $d(t)$ are not Hölder continuous.","Proof of Theorem 2: Given $m \\ge 2$ , let $C_m=\\lbrace {\\alpha }=[0;a_1,a_2,a_3,\\ldots ] \\in [0,1]| a_k \\le m, \\forall k \\ge 1\\rbrace $ .","M. Hall proved in [H] that $C_4+C_4=\\lbrace {\\alpha }+{\\beta }| {\\alpha },{\\beta }\\in C_4\\rbrace =[\\sqrt{2}-1,4(\\sqrt{2}-1)]$ .","On the other hand, we have $\\lim _{m\\rightarrow \\infty }HD(C_m)=1$ .","In fact, Jarník proved in [J] that $HD(C_m)>1-\\frac{1}{m \\cdot \\log 2}, \\forall m>8.$ Let now $t \\ge 7$ be given.","Let $m=\\lfloor t \\rfloor -3$ .","There are an integer $n \\in \\lbrace m+2,m+3\\rbrace $ and ${\\alpha }=[0;a_1,a_2,a_3,\\ldots ], {\\beta }=[0;b_1,b_2,b_3,\\ldots ] \\in C_4$ such that $t=n+{\\alpha }+{\\beta }$ .","For each $r\\ge 1$ , let $\\tilde{\\tau }^{(r)}$ and $\\hat{\\tau }^{(r)}$ be respectively the sequences $(m+1,b_{2r-1},b_{2r-2},\\ldots ,b_2,b_1,n,a_1,a_2,\\ldots ,a_{2r-2},a_{2r-1},m+1)$ and $(m+1,b_{2r},b_{2r-1},\\ldots ,b_2,b_1,n,a_1,a_2,\\ldots ,a_{2r-1},a_{2r},m+1)$ .","Consider now the maps $\\tilde{h}, \\hat{h}: C_m \\rightarrow [0,1]$ given by $\\tilde{h}(z) = \\tilde{h}([0;c_1,c_2,c_3,\\ldots ] )=[0;c_{1!","},\\tilde{\\tau }^{(1)},c_{2!","},\\tilde{\\tau }^{(2)}, c_3,c_4,c_5,c_{3!","},\\tilde{\\tau }^{(3)}, \\\\c_7, c_8,\\dots ,c_{4!","},\\tilde{\\tau }^{(4)}, c_{25}, \\dots ,c_{5!","}, \\tilde{\\tau }^{(5)},\\: \\dots \\:, c_{r!","},\\tilde{\\tau }^{(r)},c_{r!+1},\\dots ].$ $\\hat{h}(z) = \\hat{h}([0;c_1,c_2,c_3,\\ldots ] )=[0;c_{1!","},\\hat{\\tau }^{(1)},c_{2!","},\\hat{\\tau }^{(2)}, c_3,c_4,c_5,c_{3!","},\\hat{\\tau }^{(3)}, \\\\c_7, c_8,\\dots ,c_{4!","},\\hat{\\tau }^{(4)}, c_{25}, \\dots ,c_{5!","}, \\hat{\\tau }^{(5)},\\: \\dots \\:, c_{r!","},\\hat{\\tau }^{(r)},c_{r!+1},\\dots ].$ It is easy to see that $k(\\tilde{h}(z))=t$ for every $z \\in C_m$ .","Moreover, since $[x_0;x_1,x_2,x_3,x_4,\\dots ]$ is increasing in $x_0, x_2, x_4,\\ldots $ and decreasing in $x_1, x_3, x_5, \\ldots $ , $\\tilde{h}(z)\\in \\text{Exact}(t^{-1})$ and $\\hat{h}(z)\\in \\text{Exact}^{\\prime }(t^{-1})$ .","On the other hand, given any $\\rho >0$ , we have $|z-z^{\\prime }|=O(|\\tilde{h}(z)-\\tilde{h}(z^{\\prime })|^{1-\\rho })$ and $|z-z^{\\prime }|=O(|\\hat{h}(z)-\\hat{h}(z^{\\prime })|^{1-\\rho })$ for $|z-z^{\\prime }|$ small, so $HD(\\text{Exact}(t^{-1})) \\ge HD(C_m)$ and $HD(\\text{Exact}^{\\prime }(t^{-1})) \\ge HD(C_m)$ .","Since $\\lim _{m\\rightarrow \\infty }HD(C_m)=1$ , we are done.","$\\Box $ Proof of Theorem 3: Let $x \\in L^{\\prime }$ .","Consider a sequence $x_n$ converging to $x$ , $x_n \\in L$ , $x_n \\ne x$ .","Choose $\\underline{{\\theta }}^{(n)} \\in \\Sigma $ such that $x_n = \\ell (\\underline{{\\theta }}^{(n)})$ .","Let $\\underline{{\\theta }}^{(n)}=(b_j^{(n)})_{j\\in {\\mathbb {Z}}}$ and assume $b_j^{(n)} \\le 4$ , $\\forall \\, j$ , $\\forall \\, n$ (which is possible since me may assume that the $x_n$ are not in Hall's ray).","We have $x_n ={\\limsup }_{j\\rightarrow \\infty }({\\alpha }_j^{(n)} + {\\beta }_j^{(n)})$ .","Given $\\delta >0$ , $\\exists \\, n_0 \\in {\\mathbb {N}}$ large such that $n \\ge n_0 \\Rightarrow |\\ell (\\underline{{\\theta }}^{(n)})-x| < \\delta $ and there are infinitely many $j \\in {\\mathbb {N}}$ such that $|{\\alpha }_j^{(n)} + {\\beta }_j^{(n)}-x| < \\delta $ .","Let $N =\\lceil \\delta ^{-1} \\rceil $ .","Given such a pair $(j,n)$ consider the finite sequence with $2N+1$ terms $(b_{j-N}^{(n)},b_{j-N+1}^{(n)},\\dots , b_j^{(n)},\\dots , b_{j+N}^{(n)}) =: S(j,n)$ .","There is a sequence $S$ such that for infinitely many values of $n$ , $S$ appears infinitely many times as $S(j,n), \\, j \\in {\\mathbb {N}}$ , i.e., there are $j_1(n) < j_2(n) <\\dots $ with $\\lim _{i\\rightarrow \\infty } (j_{i+1}(n)-j_i(n))=\\infty $ and $S(j_i(n),n) = S$ , $\\forall \\,i \\ge 1$ , for all $n$ in some infinite set $A \\subset {\\mathbb {N}}$ .","Consider the sequences ${\\beta }(i,n)$ for $i\\ge 1$ , $n \\in A$ given by ${\\beta }(i,n) = (b_{j_i(n)+N+1}^{(n)}, b_{j_i(n)+N+2}^{(n)},\\dots ,b_{j_{i+1}(n)+N}^{(n)}).$ There are $(i_1,n_1)$ and $(i_2,n_2)$ for which there is no sequence ${\\gamma }$ such that ${\\beta }(i_1,n_1)$ and ${\\beta }(i_2,n_2)$ are concatenations of copies of ${\\gamma }$ , otherwise $x_n$ would be constant for $n \\in A$ .","This implies that, taking $B = \\lbrace {\\beta }(i_1,n_1) {\\beta }(i_2,n_2), {\\beta }(i_2,n_2) {\\beta }(i_1,n_1)\\rbrace $ , $K(B)$ is a regular Cantor set, so, as in Lemma 3, $\\ell (K(B))$ contains a regular Cantor set $\\hat{K}$ with $d(x,\\hat{K}) \\le 2\\delta $ .","$\\Box $"],["Proof of Lemma 2","Let $\\tau =\\eta /40$ .","Since $t>3$ , we have $D(t)>0$ , and so we may choose $r_0\\in {\\mathbb {N}}$ large such that, for $r\\ge r_0$ , $|\\frac{\\log N(t,r)}{r}-D(t)|<\\frac{\\tau }{2} D(t)$ .","Let $B_0:=C(t,r_0)$ and $N_0:=N(t,r_0)=|B_0|$ .","Let $k=8N_0^2\\lceil 2/\\tau \\rceil $ .","Take $\\tilde{B}=\\lbrace {\\beta }={\\beta }_1{\\beta }_2\\cdots {\\beta }_k \\mid {\\beta }_j\\in B_0,1\\le j\\le k $ and $K_t\\cap I({\\beta })\\ne \\emptyset \\rbrace $ .","Given ${\\beta }={\\beta }_1{\\beta }_2\\cdots {\\beta }_k\\in \\tilde{B}$ (with ${\\beta }_i\\in B_0$ , $1\\le i\\le k$ ), we say that $j$ , $1\\le j\\le k$ , is a right-good position of ${\\beta }$ if there are elements ${\\beta }^{(s)}={\\beta }_1{\\beta }_2\\cdots {\\beta }_{j-1} {\\beta }_j^{(s)} {\\beta }_{j+1}^{(s)}\\cdots {\\beta }_k^{(s)}$ , $s=1,2$ , of $\\tilde{B}$ such that we have the following inequality of continued fractions: $[0; {\\beta }_j^{(1)}]<[0;{\\beta }_j]<[0;{\\beta }_j^{(2)}]$ .","We say that $j$ is a left-good position if there are elements ${\\beta }^{(s)}={\\beta }_1^{(s)}{\\beta }_2^{(s)}\\cdots {\\beta }_{j-1}^{(s)}{\\beta }_j^{(s)}{\\beta }_{j+1}{\\beta }_{j+2}\\cdots {\\beta }_k$ , $s=3,4$ , of $\\tilde{B}$ such that $[0;({\\beta }_j^{(3)})^t]<[0;{\\beta }_j^t]<[0;({\\beta }_j^{(4)})^t]$ .","Finally, we say that $j$ is a good position if it is both right-good and left-good.","We will show that most positions of most words of $\\tilde{B}$ are good.","Let us first estimate $|\\tilde{B}|$ .","It follows from Lemma A2 of the appendix that, for ${\\beta }\\in \\tilde{B}$ , $s({\\beta })<(2e^{-r_0})^k& 2\\,e^{k(r_0-2)D(t)},\\quad \\text{since $k$ is large} \\nonumber \\\\&\\ge & 2\\,e^{(1-\\tau /2)r_0 kD(t)},\\quad \\text{since $r_0$ is large} \\nonumber \\\\&>& 2\\,e^{(1-\\tau )(1+\\tau /2)r_0kD(t)}\\nonumber \\\\&>& 2\\,N_0^{(1-\\tau )k},\\quad \\text{since}\\: N(t,r_0) 2N_0^{(1-\\tau )k}-2^{1+21k/20}\\cdot N_0^{19k/20} > N_0^{(1-\\tau )k}$ words of $\\tilde{B}$ , the number of good positions is at least $9k/10$ .","Let us call such an element of $\\tilde{B}$ an excellent word.","If ${\\beta }={\\beta }_1{\\beta }_2\\cdots {\\beta }_k\\in \\tilde{B}$ (with ${\\beta }_j\\in B_0$ , $1\\le j\\le k$ ) is an excellent word, we may find $\\lceil 2k/5\\rceil $ positions $i_1,i_2,\\dots ,i_{\\lceil 2k/5\\rceil }\\le k$ with $i_{s+1}\\ge i_s+2$ , $\\forall \\, s< \\lceil 2k/5\\rceil $ , such that the positions $i_1,i_1+1$ , $i_2, i_{2}+1,\\dots ,i_{\\lceil 2k/5\\rceil }$ , $i_{\\lceil 2k/5\\rceil }+1$ are good.","Since $k=8N_0^2\\lceil 2/\\tau \\rceil $ , we may take, for $1\\le s\\le 3N_0^2$ , $j_s:=i_{s\\lceil 2/\\tau \\rceil }$ (notice that $3N_0^2\\lceil 2/\\tau \\rceil <\\frac{16}{5}N_0^2\\lceil 2/\\tau \\rceil =2k/5$ ), so we have $j_{s+1}-j_s\\ge 2\\lceil 2/\\tau \\rceil $ , $\\forall \\, s<3N_0^2$ , and the positions $j_s,j_s+1$ are good for $1\\le s\\le 3N_0^2$ .","Now, the number of possible choices of $(j_1,j_2,\\dots ,j_{3N_0^2})$ is bounded by $\\binom{k}{3N_0^2}<2^k$ and, given $(j_1,j_2,\\dots ,j_{3N_0^2})$ the number of choices of $({\\beta }_{j_1},{\\beta }_{j_1+1},\\dots ,{\\beta }_{j_{3N_0^2}},{\\beta }_{j_{3N_0^2}+1})$ is bounded by $N_0^{6N_0^2}$ .","So, we may choose $\\hat{\\j }_1,\\hat{\\j }_2,\\dots ,\\hat{\\j }_{3N_0^2}$ with $\\hat{\\j }_{s+1}-\\hat{\\j }_s\\ge 2\\lceil 2/\\tau \\rceil $ , $\\forall \\, s<3N_0^2$ , and words $\\hat{{\\beta }}_{\\hat{\\j }_1},\\hat{{\\beta }}_{\\hat{\\j }_1+1}, \\hat{{\\beta }}_{\\hat{\\j }_2},\\hat{{\\beta }}_{\\hat{\\j }_2+1},\\dots ,\\hat{{\\beta }}_{\\hat{\\j }_{3N_0^2}},\\hat{{\\beta }}_{\\hat{\\j }_{3N_0^2}+1} \\in B_0$ such that the set $X:=\\lbrace {\\beta }={\\beta }_1{\\beta }_2\\cdots {\\beta }_k\\in \\tilde{B}$ excellent$\\,|\\,\\, \\hat{\\j }_s,\\hat{\\j }_s+1$ are good positions and ${\\beta }_{\\hat{\\j }_s}=\\hat{{\\beta }}_{\\hat{\\j }_s}, {\\beta }_{\\hat{\\j }_s+1}=\\hat{{\\beta }}_{\\hat{\\j }_s+1}, \\forall \\, s\\le 3N_0^2\\rbrace $ has at least $\\frac{N_0^{(1-\\tau )k}}{2^k \\cdot N_0^{6N_0^2}} > N_0^{(1-2\\tau )k}$ elements, as $N_0$ and $k$ are large.","Since $N_0=|B_0|$ , there are $N_0^2$ possible choices for the pairs $(\\hat{{\\beta }}_{\\hat{\\j }_s},\\hat{{\\beta }}_{\\hat{\\j }_s+1})$ .","We will consider, for $1\\le ss$ , the sum of $(\\hat{\\j }_t-\\hat{\\j }_s)$ for $(s,t) \\in \\cal P$ is at least $2N_0^2\\lceil 2/\\tau \\rceil $ .","Since for each pair $(s,t) \\in \\cal P$ we have $|\\pi _{s,t}(X)|(2(T+1)^2 e^{r_0})^{-(\\hat{\\j }_t-\\hat{\\j }_s)} > e^{-(\\hat{\\j }_t-\\hat{\\j }_s)(r_0+\\lceil \\log (2(T+1)^2) \\rceil )}.$ So, the Hausdorff dimension of $K(B)$ is at least $\\frac{(1-10\\tau )\\log N_0}{r_0+\\lceil \\log (2(T+1)^2) \\rceil } > \\frac{(1-10\\tau )r_0}{r_0+\\lceil \\log (2(T+1)^2) \\rceil }\\cdot (1-\\frac{\\tau }{2})D(t) > \\left(1-12\\tau \\right)D(t)> (1-\\eta )D(t).$ On the other hand, if $\\tilde{k}:=\\hat{\\j }_t-\\hat{\\j }_s$ , ${\\gamma }_1:=\\hat{{\\beta }}_{\\hat{\\j }_s+1}=\\hat{{\\beta }}_{\\hat{\\j }_t+1}$ and ${\\gamma }_2:=\\hat{{\\beta }}_{\\hat{\\j }_t}=\\hat{{\\beta }}_{\\hat{\\j }_s}$ , all elements of $B$ are of the form ${\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2$ , where ${\\gamma }_1,{\\beta }_2,{\\beta }_3,\\dots ,{\\beta }_{\\tilde{k}-1}$ , ${\\gamma }_2\\in B_0$ and there are ${\\gamma }_1^{\\prime },{\\gamma }_1^{\\prime \\prime }$ , ${\\gamma }_2^{\\prime },{\\gamma }_2^{\\prime \\prime }\\in B_0$ with $[0;{\\gamma }_2^{\\prime }]<[0;{\\gamma }_2]< [0;{\\gamma }_2^{\\prime \\prime }]$ and $[0;({\\gamma }_1^{\\prime })^t]<[0;{\\gamma }_1^t]<[0;({\\gamma }_1^{\\prime \\prime })^t]$ such that $I({\\gamma }_1^{\\prime }{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2{\\gamma }_1)\\cap K_t\\ne \\emptyset ,\\quad I({\\gamma }_1^{\\prime \\prime }{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2{\\gamma }_1)\\cap K_t\\ne \\emptyset , \\nonumber \\\\I({\\gamma }_2{\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2^{\\prime })\\cap K_t\\ne \\emptyset ,\\quad I({\\gamma }_2{\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2^{\\prime \\prime })\\cap K_t\\ne \\emptyset .\\nonumber $ We will show that this implies the existence of $\\delta >0$ such that $\\Sigma (B)\\subset \\Sigma _{t-\\delta }$ .","Let ${\\gamma }_1^t=(c_1,c_2,\\dots ,c_{m_1})$ , with $c_j\\in {\\mathbb {N}}^*$ , $\\forall \\, j\\le m_1$ , and ${\\gamma }_2=(d_1,d_2,\\dots ,d_{m_2})$ with $d_j\\in {\\mathbb {N}}^*$ , $\\forall \\, j\\le m_2$ .","Let ${\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2\\in B$ where ${\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}=a_1a_2\\cdots a_{\\tilde{m}}$ with $a_j\\in {\\mathbb {N}}^*$ , $\\forall \\, j\\le \\tilde{m}$ .","We want to estimate three kinds of sums of continued fractions.","The first one are sums of continued fractions beginning by $[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2,{\\gamma }_1,\\dots ]+[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\dots ].$ Let us assume, without loss of generality, that $q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2) \\le q_{m_1+j-1}(a_{j-1},\\ldots , a_1, {\\gamma }_1^t)$ (the other case, when the reverse inequality $q_{m_1+j-1}(a_{j-1},\\ldots , a_1, {\\gamma }_1^t) \\le q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2)$ holds, is symmetric).","Assume also that $[a_j; a_{j+1},\\ldots ,a_{\\tilde{m}},{\\gamma }_2]<[a_j; a_{j+1},\\ldots ,a_{\\tilde{m}},{\\gamma }_2^{\\prime }]$ (otherwise we change ${\\gamma }_2^{\\prime }$ by ${\\gamma }_2^{\\prime \\prime }$ ).","This allows us to exhibit $\\delta > 0$ such that, for any $\\underline{\\theta }^{(i)} \\in \\lbrace 1, 2, \\dots , T\\rbrace ^{{\\mathbb {N}}},\\; 1 \\le i \\le 4$ , ${[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2,\\underline{\\theta }^{(1)}]+[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(2)}] <}\\\\& <[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2^{\\prime },\\underline{\\theta }^{(3)}]+[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(4)}]-\\delta .$ Indeed, by the Lemma A1 of the appendix, $[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2^{\\prime },\\underline{\\theta }^{(3)}]-[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2,\\underline{\\theta }^{(1)}]>\\frac{1}{(T+1)(T+2)q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2)^2}$ $\\text{ and }\\,\\mid [0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(4)}]-[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(2)}]\\mid < \\\\\\frac{1}{q_{m_1+m_2+j-1}(a_{j-1},\\ldots , a_1, {\\gamma }_1^t, {\\gamma }_2^t)^2}<\\frac{1}{(F_{m_2+1}q_{m_1+j-1}(a_{j-1}, \\ldots , a_1, {\\gamma }_1^t))^2} \\le \\\\\\frac{1}{(F_{m_2+1}q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2))^2}\\le \\frac{1}{2(T+1)(T+2)q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2)^2}\\\\(\\text{here we use that}\\; m_2 \\text{ is large};\\\\ (F_n)\\text{ denotes Fibonacci's sequence, given by } F_0=0, F_1=1, F_{n+2}=F_{n+1}+F_n, \\forall n \\ge 0).\\\\$ So, the inequality holds with $\\delta :=\\frac{1}{(T+1)^{2(m_1+m_2+\\tilde{m})}}<\\frac{1}{2(T+1)(T+2)q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2)^2}.$ On the other hand, $I({\\gamma }_2{\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2^{\\prime })\\cap K_t\\ne \\emptyset $ , so there are $\\underline{\\theta }^{(3)}$ and $\\underline{\\theta }^{(4)}$ such that $(\\underline{\\theta }^{(4)})^t {\\gamma }_2{\\gamma }_1 {\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2^{\\prime } \\underline{\\theta }^{(3)} \\in \\Sigma _t$ , and thus $[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2^{\\prime },\\underline{\\theta }^{(3)}]+ [0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(4)}]\\le t$ , which implies that, for any $\\underline{\\theta }^{(i)} \\in \\lbrace 1, 2, \\dots , T\\rbrace ^{{\\mathbb {N}}}, i=1,2$ , $[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2,\\underline{\\theta }^{(1)}]+[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(2)}] < t-\\delta $ .","The other two kinds of sums of continued fractions we want to estimate are sums of continued fractions beginning by $[d_j; d_{j+1},\\ldots ,d_{m_2},{\\gamma }_1,\\ldots ]+[0; d_{j-1},\\ldots , d_1, a_{\\tilde{m}},\\ldots , a_1, {\\gamma }_1^t,\\ldots ]$ and, symmetrically, sums of continued fractions beginning by $[0; c_{j+1},\\ldots , c_{m_1}, {\\gamma }_2^t,\\ldots ]+[c_j; c_{j-1},\\ldots , c_1, a_1,\\ldots , a_{\\tilde{m}}, {\\gamma }_2,\\ldots ]$ .","We have: $q_{m_2-j+m_1}(d_{j+1},\\ldots ,d_{m_2}, {\\gamma }_1) \\le q_{j-1+\\tilde{m} +m_1}(d_{j-1},\\ldots , d_1, a_{\\tilde{m}},\\ldots , a_1, {\\gamma }_1^t)$ (indeed, $\\tilde{m}/(m_1+m_2)$ is large when $\\eta $ and $\\tau $ are small, depending on $T$ ).","Assume that $[d_j; d_{j+1},\\ldots ,d_{m_2}, {\\gamma }_1] < [d_j; d_{j+1},\\ldots ,d_{m_2}, {\\gamma }_1^{\\prime }]$ (otherwise we change ${\\gamma }_1^{\\prime }$ by ${\\gamma }_1^{\\prime \\prime }$ ).","Since $I({\\gamma }_2{\\gamma }_1 {\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2{\\gamma }_1^{\\prime })\\cap K_t\\ne \\emptyset $ , estimates analogous to the previous ones imply that, for any $\\underline{\\theta }^{(i)} \\in \\lbrace 1, 2, \\dots , T\\rbrace ^{{\\mathbb {N}}},\\: i=1,2$ , we have $[d_j; d_{j+1},\\ldots ,d_{m_2}, {\\gamma }_1,\\underline{\\theta }^{(1)}]+$$[0; d_{j-1},\\ldots , d_1, a_{\\tilde{m}},\\ldots , a_1, {\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(2)}] < t-\\delta $ .","This implies that the complete shift $\\Sigma (B)$ satisfies the conditions of the statement, which concludes the proof of the Lemma.","$\\Box $ As we said before, this proof doesn't give any estimate on the modulus of continuity of $d(t)$ .","Indeed, in the beginning of the proof of Lemma 2, we used the fact that $u(r)=\\log (T^2N(t,r))$ is subadditive in order to guarantee the existence of $r_0\\in {\\mathbb {N}}$ large such that, for $r\\ge r_0$ , $|\\frac{\\log N(t,r)}{r}-D(t)|<\\frac{\\tau }{2} D(t)$ (recall that $D(t)=\\lim _{m\\rightarrow \\infty }\\frac{1}{m} \\log (T^2N(t,m))=\\lim _{m\\rightarrow \\infty }\\frac{1}{m}\\log (N(t,m))$ ).","However, this gives no estimate on $r_0$ .","Consider, for instance, the function $v(n)$ given by $v(n)=2n$ for $n\\le M_0$ and $v(n)=n+M_0$ for $n>M_0$ , where $M_0$ is a large positive integer.","It is subadditive, increasing, $\\lim _{n\\rightarrow \\infty } v(n)/n=1$ but $v(M_0)/M_0=2$ , and $M_0$ can be taken arbitrarily large.","However it is possible to adapt the proof in order to give an estimate on the modulus of continuity of $d(t)$ , using an idea of [FMM].","Given ${\\varepsilon }>0$ (which we may assume to be smaller than $\\frac{1}{7}<\\frac{1}{10\\log 2}$ ), we want to obtain $\\delta \\in (0,1)$ as an explicit function of ${\\varepsilon }$ such that $D(t-\\delta )>D(t)-{\\varepsilon }$ .","Of course there is no loss of generality in assuming $D(t)\\ge {\\varepsilon }$ .","We may also assume that $T=\\lfloor t \\rfloor < 4+{\\varepsilon }^{-1}/\\log 2$ (and thus $t1-\\frac{1}{m\\log 2}>1-{\\varepsilon }$ (and so $D(t-1)>D(t)-{\\varepsilon }$ ).","Under these hypothesis, we will apply the conclusions of Lemma 2 for $\\eta ={\\varepsilon }$ .","In its proof, in this case, it is enough to assume $r_0\\ge 1/\\tau ^2$ and that, for $k=8N(t,r_0)^2\\lceil 2/\\tau \\rceil $ , $\\frac{\\log N(t,r_0)}{r_0}<(1+\\tau /2)\\frac{\\log N(t,k(r_0-1))}{k(r_0-1)}$ (indeed, assuming the above bounds for $t$ , it is not difficult to check that, except for this inequality relating $N(t,r_0)$ and $N(t,k(r_0-1))$ , the claims in other parts of the proof of the Lemma that use the assumptions that $r_0$ and $k$ are large are satisfied provided $r_0\\ge 1/\\tau ^2$ ).","We define a sequence $(c_n)_{n\\ge 0}$ recursively by $c_0=\\lceil \\frac{1}{\\tau ^2}\\rceil $ and, for every $n\\ge 0$ , $c_{n+1}=8N(t,c_n)^2\\lceil \\frac{2}{\\tau }\\rceil (c_n-1)$ .","We claim that, for some integer $s_0<(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })$ , we will have $\\frac{\\log N(t,c_{s_0})}{c_{s_0}}<(1+\\frac{\\tau }{2})\\frac{\\log N(t,c_{s_0+1})}{c_{s_0+1}}=(1+\\frac{\\tau }{2})\\frac{\\log N(t,k(c_{s_0}-1))}{k(c_{s_0}-1)}$ , with $k=8N(t,c_{s_0})^2\\lceil \\frac{2}{\\tau }\\rceil $ .","Indeed, if it is not the case, then $\\frac{\\log N(t,c_{n+1})}{c_{n+1}}\\le (1+\\frac{\\tau }{2})^{-1}\\frac{\\log N(t,c_n)}{c_n}$ for $0\\le n<(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })$ , and so, for $M=\\lceil (1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })\\rceil $ , we would have $\\frac{\\log N(t,c_M)}{c_M}\\le (1+\\frac{\\tau }{2})^{-M}\\cdot \\frac{\\log N(t,c_0)}{c_0}<({\\varepsilon }/4)\\cdot \\frac{\\log N(t,c_0)}{c_0}$ , since $(1+\\frac{\\tau }{2})^{-(1+\\frac{2}{\\tau })}{\\varepsilon }-\\frac{2\\log T}{c_M}\\ge {\\varepsilon }/2$ , since $c_M\\ge c_0\\ge \\frac{1600}{{\\varepsilon }^2}$ and $T<3/{\\varepsilon }$ .","Now let $r_0=c_{s_0}$ .","By the previous discussion, the proof of Lemma 2 works for this $r_0$ (and $k=8N(t,r_0)^2\\lceil 2/\\tau \\rceil $ ), so, for $\\delta =\\frac{1}{(T+1)^{2(m_1+m_2+\\tilde{m})}}\\ge \\frac{1}{(T+1)^{2k\\cdot \\text{max}\\lbrace |\\beta |, \\beta \\in C(t,r_0)\\rbrace }}\\ge \\frac{1}{(T+1)^{2k\\cdot r_0/\\log 2}},$ we have $D(t-\\delta )>(1-{\\varepsilon })D(t)>D(t)-{\\varepsilon }$ .","We will now give an explicit positive lower bound for $\\delta $ in terms of ${\\varepsilon }$ .","In order to do this we define recursively, for each integer $n\\ge 0$ and $x\\in {\\mathbb {R}}$ , the functions ${\\cal T}(x,n)$ and ${\\cal T}(n)$ by ${\\cal T}(x,0)=x$ , ${\\cal T}(x,n+1)=e^{{\\cal T}(x,n)}$ and ${\\cal T}(n)={\\cal T}(1,n)$ .","We have, for every $n\\ge 0$ , $c_{n+1}=8N(t,c_n)^2\\lceil \\frac{2}{\\tau }\\rceil (c_n-1)\\le 8e^{4c_n}\\cdot \\frac{3}{\\tau }\\cdot c_ne^{-e^{e^{r_0}}}>\\frac{1}{{\\cal T}(c_0,2s_0+3)}.$ Finally, since $2^k\\ge k^2$ for every $k\\ge 4$ , it follows by induction that, for every $n\\ge 4$ , ${\\cal T}(n)\\ge (n+1)^6$ for every $n\\ge 0$ .","Indeed, ${\\cal T}(4)>2^{16}>5^6$ and, for $n\\ge 4$ , ${\\cal T}(n+1)>2^{{\\cal T}(n)}\\ge {\\cal T}(n)^2\\ge (n+1)^{12}>(n+2)^6$ .","This implies that ${\\cal T}(\\lfloor 1/{\\varepsilon }\\rfloor )\\ge (1/{\\varepsilon })^6>1601/{\\varepsilon }^2>\\lceil \\frac{1600}{{\\varepsilon }^2}\\rceil =\\lceil \\frac{1}{\\tau ^2}\\rceil =c_0$ (recall that $0<{\\varepsilon }<1/7$ ), so ${\\cal T}(c_0,2s_0+3)<{\\cal T}({\\cal T}(\\lfloor 1/{\\varepsilon }\\rfloor ),2s_0+3)={\\cal T}(\\lfloor 1/{\\varepsilon }\\rfloor +2s_0+3)$ , and, since $s_0<(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })$ , we have $\\lfloor 1/{\\varepsilon }\\rfloor +2s_0+3<3+\\lfloor 1/{\\varepsilon }\\rfloor +2(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })\\le 3+1/{\\varepsilon }+2(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })=$ $=3+1/{\\varepsilon }+2(1+\\frac{80}{{\\varepsilon }})\\log (4/{\\varepsilon })<\\frac{161}{{\\varepsilon }}\\log (4/{\\varepsilon }),$ and therefore $\\delta >\\frac{1}{{\\cal T}(c_0,2s_0+3)}>\\frac{1}{{\\cal T}(\\lfloor 1/{\\varepsilon }\\rfloor +2s_0+3)}\\ge \\frac{1}{{\\cal T}(\\lfloor \\frac{161}{{\\varepsilon }}\\log (4/{\\varepsilon })\\rfloor )}.$"],["Appendix: Basic facts and estimates on continued fractions","We will prove here some elementary facts on continued fractions used in the previous sections.","We refer to [CF] for the facts used but not proved here.","Let $x = [a_0;a_1,a_2,a_3,\\dots ]$ be a real number, and $(\\frac{p_n}{q_n})_{n \\in \\mathbb {N}}, \\frac{p_n}{q_n}= [a_0;a_1,a_2,\\dots ,a_n]$ its sequence of convergents.","We have, for every $n\\in {\\mathbb {N}}$ , $p_{n+1}q_n-p_nq_{n+1}=(-1)^n$ , $x=[a_0; a_1, a_2, \\dots , a_n, {\\alpha }_{n+1}]=\\frac{\\alpha _{n+1}p_n+p_{n-1}}{\\alpha _{n+1}q_n+q_{n-1}},$ and so $\\alpha _{n+1}=\\frac{p_{n-1}-q_{n-1}x}{q_{n}x-p_{n}}$ and $x-\\frac{p_n}{q_n}= \\frac{(-1)^n}{(\\alpha _{n+1}+\\beta _{n+1})q_n^2},$ where $\\beta _{n+1}= \\frac{q_{n-1}}{q_n}= [0;a_n,a_{n-1},a_{n-2},\\dots ,a_1].$ In particular, $ \\left| x-\\frac{p_n}{q_n} \\right|=\\frac{1}{(\\alpha _{n+1}+\\beta _{n+1})q_n^2}.","$ Recall that, given a finite sequence ${\\alpha }=(a_1,a_2,\\dots ,a_n)\\in ({\\mathbb {N}}^*)^n$ , we define its size by $s({\\alpha }):=|I({\\alpha })|$ , where $I({\\alpha })$ is the interval $\\lbrace x\\in [0,1] \\mid x=[0; a_1,a_2,\\dots ,a_n,{\\alpha }_{n+1}], {\\alpha }_{n+1}\\ge 1\\rbrace $ , whose endpoints are $p_n/q_n$ and $\\frac{p_n+p_{n-1}}{q_n+q_{n-1}}$ , $r({\\alpha })=\\lfloor \\log s({\\alpha })^{-1} \\rfloor $ and, for $r \\in {\\mathbb {N}}, P_r= \\lbrace {\\alpha }=(a_1,a_2,\\dots ,a_n) \\mid r({\\alpha }) \\ge r$ and $r((a_1,a_2,\\dots ,a_{n-1}))\\frac{1}{(T+1)(T+2)q_n^2},$ where $q_n=q_n(a_1, a_2,\\ldots , a_n)$ .","Proof: Let $x=[a_0;a_1, a_2,\\ldots ,a_n, a_{n+1}, a_{n+2}, \\ldots ]$ and $y=[a_0;a_1, a_2,\\ldots ,a_{n-1}, b_n, b_{n+1}, b_{n+2}, \\ldots ]$ .","We have $x=\\frac{\\alpha _{n}(x)p_{n-1}+p_{n-2}}{\\alpha _{n}(x)q_{n-1}+q_{n-2}}, y=\\frac{\\alpha _{n}(y)p_{n-1}+p_{n-2}}{\\alpha _{n}(y)q_{n-1}+q_{n-2}},$ and so $|x-y|=\\left|\\frac{\\alpha _{n}(x)p_{n-1}+p_{n-2}}{\\alpha _{n}(x)q_{n-1}+q_{n-2}}-\\frac{\\alpha _{n}(y)p_{n-1}+p_{n-2}}{\\alpha _{n}(y)q_{n-1}+q_{n-2}}\\right|=\\left|\\frac{(\\alpha _{n}(x)-\\alpha _{n}(y))(p_{n-1}q_{n-2}-p_{n-2}q_{n-1})}{(\\alpha _{n}(x)q_{n-1}+q_{n-2})(\\alpha _{n}(y)q_{n-1}+q_{n-2})}\\right|$ $=\\left|\\frac{(\\alpha _{n}(x)-\\alpha _{n}(y))(-1)^{n-2}}{(\\alpha _{n}(x)q_{n-1}+q_{n-2})(\\alpha _{n}(y)q_{n-1}+q_{n-2})}\\right|=\\frac{|\\alpha _{n}(x)-\\alpha _{n}(y)|}{(\\alpha _{n}(x)q_{n-1}+q_{n-2})(\\alpha _{n}(y)q_{n-1}+q_{n-2})}.$ We have $\\lfloor \\alpha _n(x) \\rfloor =a_n$ , $\\lfloor \\alpha _n(y) \\rfloor =b_n$ and $a_n\\ne b_n$ , so $|\\alpha _{n}(x)-\\alpha _{n}(y)|>[1;T+1]-[0;1,T+1]=\\frac{1}{T+1}+\\frac{1}{T+2}>\\frac{2}{T+2}$ .","Moreover, $\\alpha _{n}(x)q_{n-1}+q_{n-2}<(1+a_n)q_{n-1}+q_{n-2}=q_n+q_{n-1}<2q_n$ and $\\alpha _{n}(y)q_{n-1}+q_{n-2}<\\alpha _{n}(y)(q_{n-1}+q_{n-2})\\le \\alpha _{n}(y)q_n<(T+1)q_n$ , and so $|x-y|>\\frac{2}{T+2} \\cdot \\frac{1}{2q_n\\cdot (T+1)q_n}=\\frac{1}{(T+1)(T+2)q_n^2}.\\Box $ Lemma A2.","If $\\alpha =a_1a_2\\cdots a_m$ and $\\beta =b_1b_2\\cdots b_n$ are finite words, then $\\dfrac{1}{2}s(\\alpha )s(\\beta )&q_m(\\alpha )q_n(\\beta )+q_m(\\alpha )q_{n-1}(\\beta )+\\\\&&q_{m-1}(\\alpha )q_n(\\beta )+q_{m-1}(\\alpha )q_{n-1}(\\beta )\\\\\\iff \\hspace{14.22636pt} q_m(\\alpha )q_n(\\beta )+q_m(\\alpha )q_{n-1}(\\beta )&>&q_{m-1}(\\alpha )q_n(\\beta )+q_{m-1}(\\alpha )q_{n-1}(\\beta ),$ which is obviously true.","For the other inequality, proceed analogously: $s(\\alpha \\beta )&=&\\dfrac{1}{q_{m+n}(\\alpha \\beta )\\left[q_{m+n}(\\alpha \\beta )+q_{m+n-1}(\\alpha \\beta )\\right]}\\\\&>&\\dfrac{1}{2}\\cdot \\dfrac{1}{q_m(\\alpha )q_n(\\beta )\\left[q_{m+n}(\\alpha \\beta )+q_{m+n-1}(\\alpha \\beta )\\right]}\\\\&>&\\dfrac{1}{2}\\cdot \\dfrac{1}{q_m(\\alpha )q_n(\\beta )\\left[q_m(\\alpha )+q_{m-1}(\\alpha )\\right]\\left[q_n(\\beta )+q_{n-1}(\\beta )\\right]}\\,,$ as $q_{m+n}(\\alpha \\beta )+q_{m+n-1}(\\alpha \\beta )&<&\\left[q_m(\\alpha )+q_{m-1}(\\alpha )\\right]\\left[q_n(\\beta )+q_{n-1}(\\beta )\\right]\\\\\\iff \\hspace{5.69046pt} q_{m-1}(\\alpha ){\\tilde{q}}_{n-1}(\\beta )+q_{m-1}(\\alpha ){\\tilde{q}}_{n-2}(\\beta )&<&q_{m-1}(\\alpha )q_n(\\beta )+q_{m-1}(\\alpha )q_{n-1}(\\beta )\\\\\\iff \\hspace{78.24507pt}{\\tilde{q}}_{n-1}(\\beta )+{\\tilde{q}}_{n-2}(\\beta )&<&q_n(\\beta )+q_{n-1}(\\beta ),$ where ${\\tilde{q}}_{n-1}(\\beta )=q_{n-1}(b_2b_3\\cdots b_n)$ and ${\\tilde{q}}_{n-2}(\\beta )=q_{n-2}(b_2b_3\\cdots b_{n-1})$ , and the last inequality is true, since we have ${\\tilde{q}}_{n-1}(\\beta )((T+1)^2e^r)^{-1}$ .","Proof: We have $r(a_1,a_2,\\dots ,a_{n-1})e^{-r}$ , and thus $s({\\alpha })^{-1}=q_n(q_n+q_{n-1})\\le (Tq_{n-1}+q_{n-2})((T+1)q_{n-1}+q_{n-2})<(T+1)q_{n-1}\\cdot (T+1)(q_{n-1}+q_{n-2})=(T+1)^2q_{n-1}(q_{n-1}+q_{n-2})<(T+1)^2e^r$ , so $s({\\alpha })>((T+1)^2e^r)^{-1}$ .","$\\Box $ Proposition A1.","Let $r, k$ be positive integers and $\\alpha _i, 1\\le i \\le k$ finite sequences which belong to $P_r$ and whose elements are bounded by $T$ .","Then, if ${\\alpha }={\\alpha }_1{\\alpha }_2\\cdots {\\alpha }_k$ , we have $s(\\alpha )>(2(T+1)^2 e^r)^{-k}$ .","Proof: For $1\\le i\\le k$ we have, by lemma A3, $s({\\alpha }_i)>((T+1)^2e^r)^{-1}$ .","So, using the lemma A2, we get $s({\\alpha })=s({\\alpha }_1{\\alpha }_2\\cdots {\\alpha }_k)>\\left(\\frac{1}{2}\\right)^{k-1}s({\\alpha }_1)s({\\alpha }_2)\\cdots s({\\alpha }_k)>$ $>\\left(\\frac{1}{2}\\right)^{k-1}(((T+1)^2e^r)^{-1})^k>(2(T+1)^2 e^r)^{-k}.\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\Box $ References [B] Bugeaud, Y.","- Sets of exact approximation order by rational numbers.","II - Unif.","Distrib.","Theory v. 3 , n. 2, p. 9-20, 2008.","[CF] Cusick, T. and Flahive, M. - The Markov and Lagrange spectra - Mathematical Surveys and Monographs, n. 30 - AMS.","[Fa] Falconer, K. -The geometry of fractal sets - Cambridge Tracts in Mathematics, 85.","Cambridge University Press, Cambridge, 1986.","[FMM] Ferenczi, S., Mauduit, C. and Moreira, C.G.","- An algorithm for the word entropy - https://arxiv.org/abs/1803.05533 [F] Freiman, G.A.","- Diofantovy priblizheniya i geometriya chisel (zadacha Markova) [Diophantine approximation and geometry of numbers (the Markov spectrum)], Kalininskii Gosudarstvennyi Universitet, Kalinin, 1975.","[H] Hall, M. - On the sum and products of continued fractions - Annals of Math.","v. 48, p. 966-993, 1947.","[J] Jarník, V. - Zur metrischen Theorie der diophantischen Approximationen - Prace Mat.-Fiz.","36, p. 91-106, 1928.","[Ma] Markov, A.","- Sur les formes quadratiques binaires indéfinies, Math.","Ann.","v. 15, p. 381-406, 1879.","[MM1] Matheus, C. and Moreira, C.G., $HD(M\\setminus L) > 0.353$ - https://arxiv.org/abs/1703.04302 [MM2] Matheus, C. and Moreira, C.G., $HD(M\\setminus L) < 0.986927$ - https://arxiv.org/abs/1708.06258 [MM3] Matheus, C. and Moreira, C.G., Markov spectrum near Freiman's isolated points in $M\\setminus L$ - https://arxiv.org/abs/1802.02454 [MM4] Matheus, C. and Moreira, C.G., New numbers in $M\\setminus L$ beyond $\\sqrt{12}$ : solution to a conjecture of Cusick.","https://arxiv.org/abs/1803.01230 [Mo] Moreira, C.G.","- Geometric properties of images of cartesian products of regular Cantor sets by differentiable real maps.","https://arxiv.org/abs/1611.00933 [MY] Moreira, C.G.","and Yoccoz, J.-C. - Stable intersections of regular Cantor sets with large Hausdorff Dimensions - Annals of Math.","v. 154, n.1, p. 45 - 96, 2001.","[P] Perron, O.","- Über die Approximation irrationaler Zahlen durch rationale II - S.-B.","Heidelberg Akad.","Wiss., Abh.","8, 1921, 12 pp.","[PT] Palis, J. and Takens, F. - Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Fractal dimensions and infinitely many attractors.","Cambridge Studies in Advanced Mathematics, 35.","Cambridge University Press, Cambridge, 1993. x+234 pp."]],"string":"[\n [\n \"Geometric properties of the Markov and Lagrange spectra\"\n ],\n [\n \"Abstract We prove several results on (fractal) geometric properties of the classical Markov and Lagrange spectra.\",\n \"In particular, we prove that the Hausdorff dimensions of intersections of both spectra with half-lines always coincide, and may assume any real value in the interval $[0, 1]$.\"\n ],\n [\n \"Introduction and statement of the results\",\n \"Let ${\\\\alpha }$ be an irrational number.\",\n \"According to Dirichlet's theorem, the inequality $|{\\\\alpha }-\\\\frac{p}{q}|<\\\\frac{1}{q^2}$ has infinitely many rational solutions $\\\\frac{p}{q}$ .\",\n \"Hurwitz improved this result by proving that $|{\\\\alpha }-\\\\frac{p}{q}|<\\\\frac{1}{\\\\sqrt{5} q^2}$ also has infinitely many rational solutions $\\\\frac{p}{q}$ for any irrational ${\\\\alpha }$ , and that $\\\\sqrt{5}$ is the largest constant that works for any irrational ${\\\\alpha }$ .\",\n \"However, for particular values of ${\\\\alpha }$ we can improve this constant.\",\n \"More precisely, if we define $k({\\\\alpha }):=\\\\sup \\\\lbrace k>0\\\\mid |{\\\\alpha }-\\\\frac{p}{q}|<\\\\frac{1}{kq^2}$ has infinitely many rational solutions $\\\\frac{p}{q}\\\\rbrace =\\\\limsup _{q\\\\rightarrow +\\\\infty }\\\\, ((q|q {\\\\alpha }-p|)^{-1})$ , we have $k({\\\\alpha })\\\\ge \\\\sqrt{5}$ , $\\\\forall {\\\\alpha }\\\\in {\\\\mathbb {R}}\\\\setminus Q$ and $k\\\\left(\\\\frac{1+\\\\sqrt{5}}{2} \\\\right)=\\\\sqrt{5}$ .\",\n \"Definition 1: The Lagrange spectrum is the set $L=\\\\lbrace k({\\\\alpha }) \\\\mid {\\\\alpha }\\\\in {\\\\mathbb {R}}\\\\setminus Q$ , $k({\\\\alpha })<+\\\\infty \\\\rbrace $ .\",\n \"Hurwitz-Markov theorem determines the first element of $L$ , which is $\\\\sqrt{5}$ .\",\n \"This set $L$ encodes many diophantine properties of real numbers.\",\n \"The study of the geometric structure of $L$ is a classical subject, which began with Markov, proving in 1879 ([Ma]) that $L\\\\cap (-\\\\infty , 3)=\\\\large \\\\lbrace k_1=\\\\sqrt{5}< k_2=2 \\\\sqrt{2}0$ such that $d(\\\\sqrt{12}-\\\\delta )=1$ .\",\n \"This theorem solves affirmatively Problem 3 of [B].\",\n \"It also gives some answers to Problem 5 of the same paper: the continuous function $d(t)=HD(L\\\\cap (-\\\\infty ,t))$ , which coincides (for $t>0$ ) with $\\\\sigma (1/t)$ , in the notation of [B], is a Cantor stair function: it is constant in the connected components of the complement of $L \\\\cap (-\\\\infty , t_1]$ , where $t_1:=\\\\min \\\\lbrace t \\\\in {\\\\mathbb {R}}\\\\mid d(t)=1\\\\rbrace \\\\le \\\\sqrt{12}-\\\\delta <\\\\sqrt{12}$ ; notice that $L \\\\cap (-\\\\infty , t_1]$ is a compact set with zero Lebesgue measure, and so with empty interior.\",\n \"On the other hand, we have the following Corollary: $d(t)$ is not a Hölder continuous function.\",\n \"Proof: Suppose by contradiction that $d(t)$ is Hölder continuous with exponent $\\\\alpha >0$ .\",\n \"By the previous theorem, there is ${\\\\varepsilon }>0$ such that $00$ , the sets $\\\\text{Exact}(c)=\\\\lbrace \\\\alpha \\\\in {\\\\mathbb {R}}| \\\\left|\\\\alpha -\\\\frac{p}{q}\\\\right|<\\\\frac{c}{q^2}\\\\text{ for infinitely many }(p, q)\\\\in {\\\\mathbb {Z}}\\\\times {\\\\mathbb {N}}^*\\\\text{ but, for every } {\\\\varepsilon }>0,$ $\\\\left|\\\\alpha -\\\\frac{p}{q}\\\\right|<\\\\frac{c-{\\\\varepsilon }}{q^2}\\\\text{ has only a finite number of solutions }(p, q)\\\\in {\\\\mathbb {Z}}\\\\times {\\\\mathbb {N}}^*\\\\rbrace \\\\text{ and}$ $\\\\text{Exact}^{\\\\prime }(c)=\\\\lbrace \\\\alpha \\\\in {\\\\mathbb {R}}|\\\\text{ For every } {\\\\varepsilon }>0, \\\\left|\\\\alpha -\\\\frac{p}{q}\\\\right|<\\\\frac{c+{\\\\varepsilon }}{q^2}\\\\text{ for infinitely many }(p, q)\\\\in {\\\\mathbb {Z}}\\\\times {\\\\mathbb {N}}^*\\\\text{ but}$ $\\\\left|\\\\alpha -\\\\frac{p}{q}\\\\right|<\\\\frac{c}{q^2}\\\\text{ has only a finite number of solutions }(p, q)\\\\in {\\\\mathbb {Z}}\\\\times {\\\\mathbb {N}}^*\\\\rbrace .$ Clearly Exact$(c) \\\\,\\\\cup \\\\,$ Exact$^{\\\\prime }(c)=k^{-1}(c^{-1})$ .\",\n \"Theorem 2: $\\\\lim _{c\\\\rightarrow 0}HD(\\\\text{Exact}(c))=\\\\lim _{c\\\\rightarrow 0}HD(\\\\text{Exact}^{\\\\prime }(c))=1$ .\",\n \"Consequently, $\\\\lim _{t\\\\rightarrow \\\\infty }HD(k^{-1}(t))=1$ and $\\\\lim _{t\\\\rightarrow \\\\infty }HD(k^{-1}(-\\\\infty ,t))=\\\\lim _{t\\\\rightarrow \\\\infty }D(t)=1$ .\",\n \"This solves affirmatively Problem 4 of [B].\",\n \"We also prove a result on the topological structure of the Lagrange spectrum $L$ .\",\n \"Theorem 3: $L^{\\\\prime }$ is a perfect set, i.e., $L^{\\\\prime \\\\prime }=L^{\\\\prime }$ .\",\n \"The proof of this theorem uses the fact that an element of the Lagrange spectrum associated to an infinite sequence $\\\\underline{\\\\theta }$ is accumulated by infinitely many sums of the type ${\\\\alpha }_n+{\\\\beta }_n$ , which is not necessarily true for elements of the Markov spectrum.\",\n \"The question of whether $M^{\\\\prime \\\\prime }=M^{\\\\prime }$ is still open.\",\n \"There are still some important questions left on the structure of the Markov and Lagrange spectra.\",\n \"For instance: 1) Consider the function $d_{loc}: L^{\\\\prime }\\\\rightarrow {\\\\mathbb {R}}$ given by $d_{loc}(t)=\\\\lim _{{\\\\varepsilon }\\\\rightarrow 0} HD(L\\\\cap (t-{\\\\varepsilon }, t+{\\\\varepsilon }))$ .\",\n \"Is $d_{loc}$ a non-decreasing function ?\",\n \"2) Describe the geometric structure of the difference set $M\\\\setminus L$ .\",\n \"3) Let, as before, $t_1=\\\\min \\\\lbrace t \\\\in {\\\\mathbb {R}}\\\\mid d(t)=1\\\\rbrace <\\\\sqrt{12}$ .\",\n \"Is it true that $t_1=\\\\inf $ int $L$ ?\",\n \"This would imply that int$(L \\\\cap (-\\\\infty , \\\\sqrt{12}])\\\\ne \\\\emptyset $ , which in its turn implies that int$(C_2+C_2)\\\\ne \\\\emptyset $ , where $C_2=\\\\lbrace {\\\\alpha }=[0; a_1, a_2,\\\\ldots ]\\\\in [0,1]| a_n\\\\le 2, \\\\forall n\\\\ge 1\\\\rbrace $ .\",\n \"We should mention that, in relation to question 2), there are some progresses in recent preprints by the author and C. Matheus: in [MM3] and [MM1], the authors describe the intersections of $M\\\\setminus L$ with the maximal intervals not intersecting $L$ containing the first examples $\\\\gamma =3.11812017815993...$ and $\\\\alpha _{\\\\infty }=3.293044265...$ by Freiman of elements in $M\\\\setminus L$ .\",\n \"These intersections have positive Hausdorff dimensions (which coincide with the Hausdorff dimensions of certain regular Cantor sets we describe in these works).\",\n \"In [MM4], the authors exhibit a regular Cantor set diffeomorphic to $C_2$ (and so with Hausdorff dimensions larger than $0.53128$ ) near $3.7096998597502$ contained in $M\\\\setminus L$ .\",\n \"And, in [MM3], they prove that the Hausdorff dimension of $M\\\\setminus L$ is smaller than $0.986927$ , and indicate, using heuristic estimates, that this Hausdorff dimension is smaller than $0.888$ .\",\n \"In relation to question 4), the question whether int$(C_2+C_2)\\\\ne \\\\emptyset $ was posed in page 71 of [CF].\",\n \"I would like to thank Yann Bugeaud, Aline Gomes Cerqueira, Carlos Matheus, Túlio Carvalho and Yuri Lima for helpful comments and suggestions which substantially improved this work.\"\n ],\n [\n \"A dimension formula for arithmetic sums of regular Cantor sets\",\n \"We say that $K \\\\subset {\\\\mathbb {R}}$ is a regular Cantor set of class $C^k$, $k\\\\ge 1$ , if: i) there are disjoint compact intervals $I_1,I_2,\\\\dots ,I_r$ such that $K\\\\subset I_1 \\\\cup \\\\cdots \\\\cup I_r$ and the boundary of each $I_j$ is contained in $K$ ; ii) there is a $C^k$ expanding map $\\\\psi $ defined in a neighbourhood of $I_1\\\\cup I_2\\\\cup \\\\cdots \\\\cup I_r$ such that $\\\\psi (I_j)$ is the convex hull of a finite union of some intervals $I_s$ satisfying: ii.1) for each $j$ , $1\\\\le j\\\\le r$ , and $n$ sufficiently big, $\\\\psi ^n(K\\\\cap I_j)=K$ ; ii.2) $K=\\\\bigcap \\\\limits _{n\\\\in \\\\mathbb {N}} \\\\psi ^{-n}(I_1\\\\cup I_2\\\\cup \\\\cdots \\\\cup I_r)$ .\",\n \"We say that $\\\\lbrace I_1,I_2,\\\\dots ,I_r\\\\rbrace $ is a Markov partition for $K$ and that $K$ is defined by $\\\\psi $ .\",\n \"Let $K$ be regular Cantor sets of class $C^2$ defined by the expansive function $\\\\psi $ .\",\n \"It is a general fact, due originally to Poincaré, that, given a periodic point $p$ of period $r$ of $\\\\psi $ , there is a $C^2$ diffeomorphism $h$ of the support interval $I$ of $K$ such that $\\\\tilde{\\\\psi }=h^{-1}\\\\circ \\\\psi ^r\\\\circ h$ is affine in $h^{-1}(J)$ , where $J$ is the connected component of the domain of $\\\\psi ^r$ which contains $p$ .\",\n \"We say that $K$ is non essentially affine if ${\\\\tilde{\\\\psi }}^{^{\\\\prime \\\\prime }}(x)\\\\ne 0$ for some $x\\\\in h^{-1}(K)$ .\",\n \"In [Mo], we use the Scale Recurrence Lemma of [MY] in order to prove the following Theorem.\",\n \"If $K$ and $K^{\\\\prime }$ are regular Cantor sets of class $C^2$ and $K$ is non essentially affine, then $HD(K+K^{\\\\prime })=\\\\min \\\\lbrace HD(K)+HD(K^{\\\\prime }), 1\\\\rbrace $.\",\n \"This result will be a central tool in the proof of Theorem 1.\"\n ],\n [\n \"Regular Cantor sets defined by the Gauss map\",\n \"The Gauss map is the map $g\\\\colon (0,1]\\\\rightarrow [0,1]$ given by $g(x)=\\\\lbrace \\\\frac{1}{x}\\\\rbrace =\\\\frac{1}{x}-\\\\lfloor \\\\frac{1}{x}\\\\rfloor , \\\\,\\\\forall \\\\, x\\\\in (0,1].$ It acts as a shift on continued fractions: if $a_n\\\\in {\\\\mathbb {N}}^*, \\\\forall n\\\\ge 1$ then $g([0; a_1, a_2, a_3, \\\\dots ]=[0; a_2, a_3, a_4,\\\\dots ]$ .\",\n \"Regular Cantor sets defined by the Gauss map (or iterates of it) restricted to some finite union of intervals are closely related to continued fractions with bounded partial quotients.\",\n \"We will often consider such regular Cantor sets associated to complete shifts.\",\n \"A complete shift is associated to finite sets of finite sequences of positive integers in the following way: given a finite set $B=\\\\lbrace {\\\\beta }_1,{\\\\beta }_2,\\\\dots ,{\\\\beta }_m\\\\rbrace $ , $m\\\\ge 2$ , where ${\\\\beta }_j\\\\in ({\\\\mathbb {N}}^*)^{r_j}$ , $r_j\\\\in {\\\\mathbb {N}}^*$ , $1\\\\le j\\\\le m$ and ${\\\\beta }_i$ does not begin by ${\\\\beta }_j$ for $i \\\\ne j$ , the complete shift associated to $B$ is the set $\\\\Sigma (B)\\\\subset ({\\\\mathbb {N}}^*)^{{\\\\mathbb {N}}}$ of the infinite sequences obtained by concatenations of elements of $B$ : $\\\\Sigma (B)=\\\\lbrace ({\\\\alpha }_0,{\\\\alpha }_1,{\\\\alpha }_2,\\\\dots ) \\\\, \\\\mid \\\\, {\\\\alpha }_j\\\\in B,\\\\,\\\\, \\\\forall \\\\, j\\\\in {\\\\mathbb {N}}\\\\rbrace .$ Here (and in the rest of the paper), we use the following notation for concatenations of finite sequences: if ${\\\\alpha }_j=(a_j^{(1)},a_j^{(2)},\\\\dots ,{\\\\alpha }_j^{(m_j)})$ then $({\\\\alpha }_0,{\\\\alpha }_1,{\\\\alpha }_2,\\\\dots )$ means the sequence $(a_0^{(1)},a_0^{(2)},\\\\dots ,{\\\\alpha }_0^{(m_0)},a_1^{(1)},a_1^{(2)},\\\\dots ,{\\\\alpha }_1^{(m_1)},a_2^{(1)},a_2^{(2)},\\\\dots ,{\\\\alpha }_2^{(m_2)},\\\\dots )$ .\",\n \"In some cases, when there is no ambiguity, we will write ${\\\\alpha }_0 {\\\\alpha }_1 {\\\\alpha }_2\\\\cdots $ and also ${\\\\alpha }_0^N$ to represent the concatenation of $N$ copies of ${\\\\alpha }_0$ .\",\n \"In some cases some of the ${\\\\alpha }_j$ are finite sequences and some cases are single numbers, which are viewed as one-element sequences.\",\n \"Associated to $\\\\Sigma (B)$ is the Cantor set $K(B)\\\\subset [0,1]$ of the real numbers whose continued fractions are of the form $[0;{\\\\gamma }_1,{\\\\gamma }_2,{\\\\gamma }_3,\\\\dots ]$ , where ${\\\\gamma }_j\\\\in B$ , $\\\\forall \\\\, j\\\\ge 1$ .\",\n \"This is a regular Cantor set.\",\n \"Indeed, if $a_j$ and $b_j$ are respectively the smallest and largest elements of $K(B)$ whose continued fractions begin by $[0; {\\\\beta }_j]$ , for $1\\\\le j\\\\le m$ , and $I_j=[a_j,b_j]$ , then $K(B)$ is the regular Cantor set defined by the map $\\\\psi $ with domain $\\\\bigcup \\\\limits _{j=1}^m I_j$ given by $\\\\psi |_{I_j}=g^{r_j}$ , $1\\\\le j\\\\le m$ .\",\n \"We have the following Proposition 1.\",\n \"The Cantor sets $K(B)$ defined by the Gauss map associated to complete shifts are non essentially affine.\",\n \"Proof: Let $B=\\\\lbrace {\\\\beta }_1,{\\\\beta }_2,\\\\dots ,{\\\\beta }_m\\\\rbrace $ , ${\\\\beta }_j=(b_1^{(j)},b_2^{(j)},\\\\dots ,b_{r_j}^{(j)})\\\\in ({\\\\mathbb {N}}^*)^{r_j}$ , $1\\\\le j\\\\le m$ .\",\n \"For each $j\\\\le m$ , let $x_j=[0; {\\\\beta }_j,{\\\\beta }_j,{\\\\beta }_j,\\\\dots ]\\\\in I_j$ be the fixed point of $\\\\psi |_{I_j}=g^{r_j}$ .\",\n \"Notice that, since ${\\\\beta }_i$ does not begin by ${\\\\beta }_j$ for $i \\\\ne j$ , the $x_j, 1\\\\le j\\\\le m$ are all distinct.\",\n \"Moreover, according to the classical theory of continued fractions, if $p_k^{(j)}/q_k^{(j)}:=[0;b_1^{(j)},b_2^{(j)},\\\\dots ,b_k^{(j)}]$ , for $1\\\\le j\\\\le m$ , $1\\\\le k\\\\le r_j$ , we have $I_j\\\\subset \\\\lbrace [0;{\\\\beta }_j,{\\\\alpha }],\\\\,\\\\, {\\\\alpha }\\\\ge 1\\\\rbrace $ and $\\\\psi |_{I_j}(x)$ is given by $\\\\psi |_{I_j}(x)=\\\\frac{q_{r_j}^{(j)}x-p_{r_j}^{(j)}}{-q_{r_j-1}^{(j)}x+p_{r_j-1}^{(j)}}$ (see the appendix); so $x_j$ is the positive root of $q_{r_j-1}^{(j)}x^2+(q_{r_j}^{(j)}-p_{r_j-1}^{(j)})x-p_{r_j}^{(j)}$ (since $x_j$ is the fixed point of $\\\\psi |_{I_j}$ ).\",\n \"For each $j\\\\le m$ , since $\\\\psi |_{I_j}$ is a Möbius function with a hyperbolic fixed point $x_j$ , there is a Möbius function ${\\\\alpha }_j(x)=\\\\frac{a_jx+b_j}{c_jx+d_j}$ with ${\\\\alpha }_j(x_j)=x_j$ , ${\\\\alpha }_j^{\\\\prime }(x_j)=1$ such that ${\\\\alpha }_j\\\\circ (\\\\psi |_{I_j})\\\\circ {\\\\alpha }_j^{-1}$ is an affine map.\",\n \"If we show that the Möbius functions ${\\\\alpha }_1\\\\circ (\\\\psi |_{I_2})\\\\circ {\\\\alpha }_1^{-1}$ is not affine then we are done, since the second derivative of a non-affine Möbius function never vanishes.\",\n \"Suppose by contradiction that ${\\\\alpha }_1\\\\circ (\\\\psi |_{I_2})\\\\circ {\\\\alpha }_1^{-1}$ is affine.\",\n \"Since ${\\\\alpha }_1\\\\circ (\\\\psi |_{I_1})\\\\circ {\\\\alpha }_1^{-1}$ is also affine these two functions have a common fixed point at $\\\\infty $ , so ${\\\\alpha }_1^{-1}(\\\\infty )=-d_1/c_1$ is a common fixed point of $\\\\psi |_{I_2}$ and $\\\\psi |_{I_1}$ , which implies that ${\\\\alpha }_1^{-1}(\\\\infty )$ is a common root of $q_{r_1-1}^{(1)}x^2+(q_{r_1}^{(1)}-p_{r_1-1}^{(1)})x-p_{r_1}^{(1)}$ and $q_{r_2-1}^{(2)}x^2+(q_{r_2}^{(2)}-p_{r_2-1}^{(2)})x-p_{r_2}^{(2)}$ .\",\n \"Since these polynomials of ${\\\\mathbb {Q}}[x]$ are irreducible (indeed their roots $x_1$ and $x_2$ are irrational because their continued fractions expansions are infinite), they must be associates in ${\\\\mathbb {Q}}[x]$ , and so their remaining roots $x_1$ and $x_2$ must coincide, which is a contradiction.\",\n \"$\\\\Box $ Corollary 1.\",\n \"$HD(K(B)+K(B^{\\\\prime }))=\\\\min \\\\lbrace 1,HD(K(B))+HD(K(B^{\\\\prime }))\\\\rbrace $ , for every sets $B$ , $B^{\\\\prime }$ of finite sequences of positive integers.\",\n \"Definition 2: If ${\\\\beta }=(b_1,b_2,\\\\dots ,b_{n-1},b_n)$ , then ${\\\\beta }^t:=(b_n,b_{n-1},\\\\dots ,b_2,b_1)$ .\",\n \"Given a set of finite sequences $B$ , we define $B^t:=\\\\lbrace {\\\\beta }^t, {\\\\beta }\\\\in B\\\\rbrace $ .\",\n \"Proposition 2.\",\n \"$HD(K(B))=HD(K(B^t))$ , for any finite set $B$ of finite sequences.\",\n \"Proof: This follows from $q_n({\\\\beta })=q_n({\\\\beta }^t)$ , $\\\\forall {\\\\beta }$ (see the appendix of [CF] on properties of continuants), and from the fact that, if $\\\\psi |_{I_j(x)}=\\\\frac{q_{n}^{(j)}x-p_{n}^{(j)}}{-q_{n-1}^{(j)}x+p_{n-1}^{(j)}}$ , then $\\\\psi ^{\\\\prime }|_{I_j(x)}=\\\\frac{-(p_{n}^{(j)}q_{n-1}^{(j)}-p_{n-1}^{(j)}q_{n}^{(j)})}{(-q_{n-1}^{(j)}x+p_{n-1}^{(j)})^2}=\\\\frac{(-1)^n}{(-q_{n-1}^{(j)}x+p_{n-1}^{(j)})^2}$ satisfies $(q_{n}^{(j)})^2\\\\le |\\\\psi ^{\\\\prime }|_{I_j(x)}|\\\\le 4(q_{n}^{(j)})^2$ , since $\\\\frac{1}{2q_{n}^{(j)}}\\\\le \\\\frac{1}{q_{n}^{(j)}+q_{n-1}^{(j)}} \\\\le |q_{n-1}^{(j)}x-p_{n-1}^{(j)}|\\\\le \\\\frac{1}{q_{n}^{(j)}}.$ $\\\\Box $ Corollary 2.\",\n \"$HD(K(B)+K(B^t))=\\\\min \\\\lbrace 1,2 \\\\cdot HD(K(B))\\\\rbrace $ , for every set $B$ of finite sequences of positive integers.\"\n ],\n [\n \"Fractal dimensions of the spectra\",\n \"We recall that the Lagrange spectrum is given by $L=\\\\lbrace \\\\ell (\\\\underline{{\\\\theta }}),\\\\,\\\\, \\\\underline{{\\\\theta }}\\\\in \\\\Sigma \\\\rbrace $ , where $\\\\Sigma =({\\\\mathbb {N}}^*)^{{\\\\mathbb {Z}}}$ and, for $\\\\underline{{\\\\theta }}=(a_n)_{n\\\\in {\\\\mathbb {Z}}}\\\\in \\\\Sigma $ , $\\\\ell (\\\\underline{{\\\\theta }}):=\\\\limsup _{n\\\\rightarrow +\\\\infty }({\\\\alpha }_n+{\\\\beta }_n)$ , where ${\\\\alpha }_n$ and ${\\\\beta }_n$ are defined as the continued fractions ${\\\\alpha }_n:=[a_n; a_{n+1},a_{n+2},\\\\dots ]$ and ${\\\\beta }_n:=[0;a_{n-1},a_{n-2},\\\\dots ]$ , while the Markov spectrum is given by $M=\\\\lbrace m(\\\\underline{{\\\\theta }}),\\\\underline{{\\\\theta }}\\\\in \\\\Sigma \\\\rbrace $ , where $m(\\\\underline{{\\\\theta }})=\\\\sup \\\\lbrace {\\\\alpha }_n+{\\\\beta }_n,\\\\,\\\\, n\\\\in {\\\\mathbb {Z}}\\\\rbrace $ .\",\n \"Given a finite sequence ${\\\\alpha }=(a_1,a_2,\\\\dots ,a_n)\\\\in ({\\\\mathbb {N}}^*)^n$ , we define its size by $s({\\\\alpha }):=|I({\\\\alpha })|$ , where $I({\\\\alpha })$ is the interval $\\\\lbrace x\\\\in [0,1] \\\\mid x=[0; a_1,a_2,\\\\dots ,a_n,{\\\\alpha }_{n+1}]$ , ${\\\\alpha }_{n+1}\\\\ge 1\\\\rbrace $ .\",\n \"If we take $p_0=0$ , $q_0=1$ , $p_1=1$ , $q_1=a_1$ and, for $k\\\\ge 0$ , $p_{k+2}=a_{k+2}p_{k+1}+p_k$ and $q_{k+2}=a_{k+2} q_{k+1}+q_k$ , then $I({\\\\alpha })$ is the interval with extremities $[0;a_1,a_2,\\\\dots ,a_n]=p_n/q_n$ and $[0;a_1,a_2,\\\\dots ,a_{n-1},a_n+1]=\\\\frac{p_n+p_{n-1}}{q_n+q_{n-1}}$ and so $s({\\\\alpha })=\\\\left| \\\\frac{p_n}{q_n}-\\\\frac{p_n+p_{n-1}}{q_n+q_{n-1}} \\\\right| = \\\\frac{1}{q_n(q_n+q_{n-1})},$ since $p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}$ .\",\n \"We define $r({\\\\alpha })=\\\\lfloor \\\\log s({\\\\alpha })^{-1} \\\\rfloor $ which controls the order of magnitude of the size of $I({\\\\alpha })$ .\",\n \"We also define, for $r \\\\in {\\\\mathbb {N}}, P_r= \\\\lbrace {\\\\alpha }=(a_1,a_2,\\\\dots ,a_n) \\\\mid r({\\\\alpha }) \\\\ge r$ and $r((a_1,a_2,\\\\dots ,a_{n-1}))\\\\alpha _n\\\\ge a_n$ for every $n$ , and so $m(\\\\underline{{\\\\theta }})>\\\\sup \\\\lbrace a_n,n\\\\in {\\\\mathbb {Z}}\\\\rbrace $ .\",\n \"So, if $m(\\\\underline{{\\\\theta }})\\\\le t$ we have $a_n\\\\le \\\\lfloor t \\\\rfloor $ , $\\\\forall \\\\,\\\\, n\\\\in {\\\\mathbb {N}}$ .\",\n \"Given $t\\\\in [3,+\\\\infty )$ and $r\\\\in {\\\\mathbb {N}}$ , let $T:=\\\\lfloor t \\\\rfloor $ and $C(t,r)$ be the set $\\\\lbrace {\\\\alpha }=(a_1,a_2,\\\\dots ,a_n) \\\\in P_r \\\\mid K_t\\\\cap I({\\\\alpha })\\\\ne \\\\emptyset \\\\rbrace $ .\",\n \"Here $K_t:=\\\\lbrace [0;{\\\\gamma }]| {\\\\gamma }\\\\in \\\\pi _+(\\\\Sigma _t)\\\\rbrace $ , where $\\\\pi _+\\\\colon \\\\Sigma \\\\rightarrow \\\\Sigma ^+$ is the projection associated to the decomposition $\\\\Sigma =\\\\Sigma ^-\\\\times \\\\Sigma ^+$ .\",\n \"Since $\\\\Sigma _t$ is invariant by transposition and by $\\\\sigma $ , $K_t$ is invariant by the Gauss map $g$ and $M\\\\cap (-\\\\infty , t) \\\\subset ({\\\\mathbb {N}}^* \\\\cap [1, T])+K_t+K_t$ .\",\n \"We define $N(t,r):=|C(t,r)|$ , where $|\\\\cdot |$ denotes cardinality.\",\n \"Notice that if $r\\\\le s$ then $N(t,r)\\\\le N(t,s)$ and, if $t\\\\le \\\\tilde{t}$ , then $N(t,r)\\\\le N(\\\\tilde{t},r)$ .\",\n \"For any finite sequences ${\\\\alpha },{\\\\beta }$ and any positive integers $k_1, k_2\\\\le T$ we have $r({\\\\alpha }{\\\\beta }k_1 k_2)\\\\ge r({\\\\alpha })+r({\\\\beta })$ (see the appendix), so if $C(t,r)=\\\\lbrace {\\\\alpha }_1,{\\\\alpha }_2,\\\\dots ,{\\\\alpha }_u\\\\rbrace $ and $C(t,s)=\\\\lbrace {\\\\beta }_1,{\\\\beta }_2,\\\\dots ,{\\\\beta }_v\\\\rbrace $ , we may cover $K_t$ by the $T^2uv=T^2N(t,r)N(t,s)$ intervals $I({\\\\alpha }_i{\\\\beta }_j k_1 k_2)$ , $1\\\\le i \\\\le u$ , $1\\\\le j\\\\le v$ , $1\\\\le k_1, k_2\\\\le T$ , which satisfy $r({\\\\alpha }_i{\\\\beta }_j k_1 k_2)\\\\ge r+s$ , $\\\\forall \\\\,\\\\, i,j,k$ .\",\n \"Replacing, if necessary, some of these intervals by larger intervals $I({\\\\gamma })$ in $P_{r+s}$ , we conclude that $N(t,r+s)\\\\le T^2N(t,r)N(t,s)$ and so $\\\\log (T^2N(t,r+s))\\\\le \\\\log (T^2N(t,r))+\\\\log (T^2N(t,s)),\\\\quad \\\\forall \\\\,\\\\, r,s.$ This implies that $\\\\lim _{m\\\\rightarrow \\\\infty }\\\\frac{1}{m} \\\\log (T^2N(t,m))=\\\\inf _{m\\\\in {\\\\mathbb {N}}^*}\\\\frac{1}{m} \\\\log (T^2N(t,m))=\\\\lim _{m\\\\rightarrow \\\\infty }\\\\frac{1}{m}\\\\log (N(t,m))$ exists.\",\n \"We will call this limit $D(t)$ (which coincides with the (upper) box dimension of $K_t$ , as follows easily from its definition).\",\n \"Notice that $D(t)$ is a non-decreasing function.\",\n \"We will see in the proof of Theorem 1 that $D(t)$ is continuous and that $HD(k^{-1}(-\\\\infty ,t))=D(t)$ .\",\n \"Lemma 1.\",\n \"$D(t)$ is right-continuous: given $t_0\\\\in [3,+\\\\infty )$ and $\\\\eta >0$ there is $\\\\delta >0$ such that for $t_0\\\\le t\\\\le t+\\\\delta $ we have $D(t_0)\\\\le D(t) \\\\le D(t+\\\\delta ) t_0$ , $r$ large, $\\\\frac{\\\\log N(t,r)}{r} \\\\ge D(t_0)+\\\\eta $ we would have $D(t_0) \\\\ge D(t_0) + \\\\eta $ , contradiction (indeed $C(t,r) \\\\subset C(s,r)$ for $t \\\\le s$ , and, by compacity, $C(t_0,r) ={\\\\bigcap }_{t>t_0}\\\\,C(t,r)$ ).\",\n \"$\\\\Box $ Lemma 2.\",\n \"Given $t\\\\in (3,+\\\\infty )$ and $\\\\eta \\\\in (0,1)$ there is $\\\\delta >0$ and a Cantor set $K(B)$ defined by the Gauss map associated to a complete shift $\\\\Sigma (B)\\\\subset \\\\Sigma $ such that $\\\\Sigma (B)\\\\subset \\\\Sigma _{t-\\\\delta }$ and $HD(K(B))>(1-\\\\eta )D(t)$ .\",\n \"Since the proof of this Lemma is somewhat technical, we will postpone it to Section 6.\",\n \"Lemma 3.\",\n \"Given a complete shift $\\\\Sigma (X)\\\\subset \\\\Sigma $ (where X is a finite set of finite sequences of positive integers), we have $HD(\\\\ell (\\\\Sigma (X)))=HD(m(\\\\Sigma (X)))=\\\\overline{\\\\dim }(\\\\ell (\\\\Sigma (X)))=\\\\overline{\\\\dim }(m(\\\\Sigma (X)))=\\\\min \\\\lbrace 2 \\\\cdot HD(K(X)), 1 \\\\rbrace .$ Proof: Let $T$ be the largest element of a sequence in $X$ .\",\n \"First of all we clearly have $\\\\ell (\\\\Sigma (X)) \\\\subset m(\\\\Sigma (X))\\\\subset \\\\bigcup \\\\limits _{1\\\\le a\\\\le T\\\\atop 1\\\\le i, j\\\\le R} (a+g^i(K(X))+g^j(K(X))),\\\\;\\\\;\\\\text{where $R$ is the length of}$ the largest word of $X$ , so $HD(\\\\ell (\\\\Sigma (X)) \\\\le HD(m(\\\\Sigma (X)) \\\\le \\\\min \\\\lbrace 2 \\\\cdot HD(K(X)), 1 \\\\rbrace $ .\",\n \"Let ${\\\\varepsilon }>0$ be given.\",\n \"We will show that there are regular Cantor sets $K, K^{\\\\prime }$ defined by iterates of the Gauss map with $HD(K),HD(K^{\\\\prime })>HD(K(X))-{\\\\varepsilon }$ such that $K+K^{\\\\prime } \\\\subset \\\\ell (\\\\Sigma (X)) \\\\subset m(\\\\Sigma (X))$ .\",\n \"Since, by the dimension formula stated in section 2, $HD(K+K^{\\\\prime })=\\\\min \\\\lbrace HD(K)+HD(K^{\\\\prime }), 1\\\\rbrace >\\\\min \\\\lbrace 2 \\\\cdot HD(K(X)), 1 \\\\rbrace -2{\\\\varepsilon }$ and ${\\\\varepsilon }>0$ is arbitrary, the result will follow.\",\n \"Given a positive integer $n$ , let $X^n=\\\\lbrace ({\\\\gamma }_1,{\\\\gamma }_2,\\\\dots ,{\\\\gamma }_n)|{\\\\gamma }_j \\\\in X, \\\\forall j \\\\le n \\\\rbrace $ .\",\n \"We have $\\\\Sigma (X^n)=\\\\Sigma (X)$ and $K(X^n)=K(X)$ .\",\n \"Replacing $X$ by $X^n$ for some $n$ large, we may assume without loss of generality that for any $A \\\\subset X$ (resp.\",\n \"$A^t \\\\subset X^t$ ) with $|A|\\\\le 2$ (resp.\",\n \"$|A^t|\\\\le 2$ ), we have $HD(K(X\\\\setminus A))>HD(K(X))-{\\\\varepsilon }$ (resp.\",\n \"$HD(K(X^t\\\\setminus A^t))>HD(K(X^t))-{\\\\varepsilon }=HD(K(X))-{\\\\varepsilon }$ ).\",\n \"Order $X$ and $X^t$ in the following way: given ${\\\\gamma }, \\\\tilde{{\\\\gamma }}\\\\in X \\\\, (\\\\text{resp.\",\n \"}{\\\\gamma }, \\\\tilde{{\\\\gamma }}\\\\in X^t)$ , we say that ${\\\\gamma }< \\\\tilde{{\\\\gamma }}$ if and only if $[0;{\\\\gamma }]<[0;\\\\tilde{{\\\\gamma }}]$ .\",\n \"Suppose that the maximum of $m(\\\\Sigma (X))$ is attained at $\\\\tilde{\\\\underline{\\\\theta }}= (\\\\dots ,\\\\tilde{{\\\\gamma }}_{-1},\\\\tilde{{\\\\gamma }}_0,\\\\tilde{{\\\\gamma }}_1,\\\\dots ),\\\\tilde{{\\\\gamma }}_i \\\\in X, \\\\forall i \\\\in {\\\\mathbb {Z}}$ , in a position belonging to the sequence $\\\\tilde{{\\\\gamma }}_0$ .\",\n \"Let $X^* = X\\\\backslash \\\\lbrace \\\\min X, \\\\max X\\\\rbrace $ , $(X^t)^* = X^t\\\\backslash \\\\lbrace \\\\min X^t, \\\\max X^t\\\\rbrace $ .\",\n \"Essentially, $K(X^*)$ and $K((X^t)^*)$ will be the required Cantor sets, but first we have to control the positions where the $\\\\limsup $ is attained (the idea is somewhat similar to the proof that Hall's theorem ([H]) on sums of continued fractions with coefficients bounded by 4 implies that the Lagrange spectrum contains $[6,+\\\\infty )$ ) and which words can appear in the beginning of the elements.\",\n \"For each positive integer $m$ , let $C^m$ be the set of sequences $(\\\\dots , {\\\\gamma }_{-m-2}, {\\\\gamma }_{-m-1}, \\\\tilde{{\\\\gamma }}_{-m}, \\\\tilde{{\\\\gamma }}_{-m+1}, \\\\dots , \\\\tilde{{\\\\gamma }}_{-1}, \\\\tilde{{\\\\gamma }}_0, \\\\tilde{{\\\\gamma }}_1, \\\\dots , \\\\tilde{{\\\\gamma }}_{m-1}, \\\\tilde{{\\\\gamma }}_m, {\\\\gamma }_{m+1}, {\\\\gamma }_{m+2}, \\\\dots )$ where ${\\\\gamma }_k \\\\in X^*$ for $k \\\\ge m+1$ , ${\\\\gamma }_k^t \\\\in (X^t)^*$ for $k \\\\le -m-1$ .\",\n \"Then, for $m$ large enough, there is $\\\\eta >0$ such that for each $\\\\underline{\\\\theta }\\\\in C^m$ , $\\\\text{sup}({\\\\alpha }_n+{\\\\beta }_n) =m(\\\\underline{\\\\theta })$ is attained only for values of $n$ corresponding to the piece $\\\\tau =(\\\\tilde{{\\\\gamma }}_{-m}, \\\\tilde{{\\\\gamma }}_{-m+1}, \\\\dots , \\\\tilde{{\\\\gamma }}_{-1}, \\\\tilde{{\\\\gamma }}_0, \\\\tilde{{\\\\gamma }}_1, \\\\dots , \\\\tilde{{\\\\gamma }}_{m-1}, \\\\tilde{{\\\\gamma }}_m)$ of $\\\\underline{\\\\theta }$ and, if $n$ does not correspond to the piece $\\\\tau $ , then ${\\\\alpha }_n+{\\\\beta }_nm_k$ such that ${\\\\alpha }_{n_k}(\\\\underline{\\\\theta }^{(k)})+{\\\\beta }_{n_k}(\\\\underline{\\\\theta }^{(k)})>m(\\\\underline{\\\\theta }^{(k)})-1/k$ .\",\n \"Since $\\\\underline{\\\\theta }^{(k)}$ converges to $\\\\tilde{\\\\underline{\\\\theta }}$ , $m(\\\\underline{\\\\theta }^{(k)})$ converges to $m(\\\\tilde{\\\\underline{\\\\theta }})$ and, by compacity, if $N_k$ denotes the size of the sequence $\\\\tilde{{\\\\gamma }}_0, \\\\tilde{{\\\\gamma }}_1, \\\\dots , \\\\tilde{{\\\\gamma }}_{m_k-1}, \\\\tilde{{\\\\gamma }}_{m_k}, {\\\\gamma }_{m_k+1}, {\\\\gamma }_{m_k+2}, \\\\dots , {\\\\gamma }_{r(k)-1}$ , $(\\\\sigma ^{N_k}(\\\\underline{\\\\theta }^{(k)}))$ has a subsequence which converges to some $\\\\hat{\\\\underline{\\\\theta }}= (\\\\dots ,\\\\hat{{\\\\gamma }}_{-1},\\\\hat{{\\\\gamma }}_0,\\\\hat{{\\\\gamma }}_1,\\\\dots ) \\\\in \\\\Sigma (X)$ , with $\\\\hat{{\\\\gamma }}_i \\\\in X^*, \\\\forall i \\\\ge 0$ , such that $\\\\text{sup}({\\\\alpha }_n+{\\\\beta }_n)=m(\\\\hat{\\\\underline{\\\\theta }})=m(\\\\tilde{\\\\underline{\\\\theta }})$ is attained for some $n$ corresponding to the piece $\\\\hat{{\\\\gamma }}_0$ .\",\n \"This is a contradiction, since $m(\\\\tilde{\\\\underline{\\\\theta }})$ is the maximum of $m(\\\\Sigma (X))$ and, changing $\\\\hat{{\\\\gamma }}_1$ by $\\\\min X$ or $\\\\max X$ , we strictly increase the value of $m(\\\\hat{\\\\underline{\\\\theta }})$ .\",\n \"Notice that the same argument shows that for any $\\\\underline{\\\\theta }\\\\in C^m$ and $\\\\underline{\\\\theta }^* \\\\in \\\\Sigma (X^*)$ , we have $m(\\\\underline{\\\\theta }^*)HD(K(X))-{\\\\varepsilon }$ and $HD(K^{\\\\prime })=HD(K((X^t)^*))>$$HD(K(X^t))-{\\\\varepsilon }=HD(K(X))-{\\\\varepsilon }$ .\",\n \"$\\\\Box $\"\n ],\n [\n \"Proofs of the main results\",\n \"Proof of Theorem 1: Applying Lemma 3 to the complete shift $\\\\Sigma (B)$ obtained in Lemma 2, we get that, for any $\\\\eta >0$ , there is $\\\\delta >0$ such that $\\\\min \\\\lbrace 2(1-\\\\eta )D(t),1\\\\rbrace \\\\le \\\\min \\\\lbrace 2HD(K(B)),1\\\\rbrace \\\\le HD(L\\\\cap (-\\\\infty ,t-\\\\delta ]) \\\\le HD(L \\\\cap (-\\\\infty ,t)) \\\\le HD(M \\\\cap (-\\\\infty ,t)) \\\\le \\\\overline{\\\\dim }(M\\\\cap (-\\\\infty ,t))\\\\le \\\\min \\\\lbrace 2 \\\\cdot HD(K_t),1\\\\rbrace \\\\le \\\\min \\\\lbrace 2 \\\\cdot D(t),1\\\\rbrace $ (since $\\\\ell (\\\\Sigma (B))\\\\subset L\\\\cap (-\\\\infty ,t-\\\\delta ]$ , $L\\\\cap (-\\\\infty , t) \\\\subset M\\\\cap (-\\\\infty , t) \\\\subset ({\\\\mathbb {N}}^* \\\\cap [1, T])+K_t+K_t$ and $D(t)$ is the upper box dimension of $K_t$ ), and so, if $d(t):=HD(L\\\\cap (-\\\\infty ,t))$ , we have $d(t)= HD(M\\\\cap (-\\\\infty ,t))=\\\\overline{\\\\dim }(L\\\\cap (-\\\\infty ,t))=\\\\overline{\\\\dim }(M\\\\cap (-\\\\infty ,t))=\\\\min \\\\lbrace 2 \\\\cdot D(t),1\\\\rbrace $ .\",\n \"In order to conclude the proof of the first assertion of i), it is enough to show that $HD(k^{-1}(-\\\\infty ,t))=D(t)$ .\",\n \"In the notation of Lemma 3, let $x\\\\in K, y\\\\in K^{\\\\prime }$ .\",\n \"For each $z=[0; {\\\\alpha }_1, {\\\\alpha }_2, \\\\dots ]\\\\in K(X^*)$ , define $\\\\underline{{\\\\lambda }}(z)={\\\\underline{{\\\\lambda }}}_{x,y}(z)=({\\\\alpha }_{1!\",\n \"},\\\\tau ^{(1)},{\\\\alpha }_{2!\",\n \"},\\\\tau ^{(2)},{\\\\alpha }_3,{\\\\alpha }_4,{\\\\alpha }_5,{\\\\alpha }_{3!\",\n \"},\\\\tau ^{(3)},{\\\\alpha }_7,\\\\dots ,{\\\\alpha }_{4!\",\n \"},\\\\tau ^{(4)},\\\\\\\\{\\\\alpha }_{25},{\\\\alpha }_{26},\\\\dots ,{\\\\alpha }_{5!\",\n \"},\\\\tau ^{(5)},\\\\;\\\\dots \\\\;,{\\\\alpha }_{r!\",\n \"},\\\\tau ^{(r)},{\\\\alpha }_{r!+1},\\\\dots ),$ and $h(z)=[0;\\\\underline{{\\\\lambda }}(z)]$ .\",\n \"We have, as before, $k(h(z))=x+y$ .\",\n \"On the other hand, given any $\\\\rho >0$ , we have $|z-z^{\\\\prime }|=O(|h(z)-h(z^{\\\\prime })|^{1-\\\\rho })$ for $|z-z^{\\\\prime }|$ small, so $HD(k^{-1}(x+y)) \\\\ge HD(K(B^*))>HD(K(B))-{\\\\varepsilon }$ .\",\n \"As before, we get $HD(k^{-1}(-\\\\infty ,t)) \\\\ge HD(k^{-1}(-\\\\infty ,t-\\\\delta ])$ $\\\\ge HD(k^{-1}(x+y)) > HD(K(B))-{\\\\varepsilon }> (1-\\\\eta )D(t)-{\\\\varepsilon }.$ Since $\\\\eta $ and ${\\\\varepsilon }$ are arbitrary, $HD(k^{-1}(-\\\\infty ,t)) \\\\ge D(t)$ .\",\n \"For the reverse inequality, let $w \\\\in k^{-1}(-\\\\infty ,t)$ .\",\n \"We have $\\\\limsup _{n\\\\rightarrow \\\\infty }({\\\\alpha }_n(w)+{\\\\beta }_n(w))=k(w)0$ such that $D(t-\\\\delta )\\\\ge HD(K(B))>(1-\\\\eta )D(t)$ , so $\\\\lim \\\\limits _{s\\\\rightarrow t-} D(s) = D(t)$ .\",\n \"In order to conclude, notice that, for each positive integer $m$ , $\\\\Sigma (\\\\lbrace 21^{2m}2,21^{2m+2}2\\\\rbrace ) \\\\subset \\\\Sigma _{3+2^{-m}}$ (notice that $[2;1, 1, 1,\\\\ldots ]+[0;2, 1, 1, 1,\\\\ldots ]=3$ ), so $D(3+{\\\\varepsilon })>0$ for every ${\\\\varepsilon }>0$ and, since $\\\\Sigma _{\\\\sqrt{12}}=\\\\lbrace 1,2\\\\rbrace ^{{\\\\mathbb {Z}}}$ , $D(\\\\sqrt{12})=HD(K_{\\\\sqrt{12}})=HD(K(\\\\lbrace 1,2\\\\rbrace ))=HD(C_2)=0,53128\\\\ldots >1/2$ , so, since $D(t)$ is continuous, there is $\\\\delta >0$ such that $D(\\\\sqrt{12}-\\\\delta )>1/2$ , and thus we have $d(\\\\sqrt{12}-\\\\delta )=\\\\min \\\\lbrace 2 \\\\cdot D(\\\\sqrt{12}-\\\\delta ),1\\\\rbrace =1$ .\",\n \"$\\\\Box $ Remark: It follows from the above proof and from the general estimates of fractal dimensions of regular Cantor sets of Chapter 4 of [PT] that there is a constant $C>0$ such that, for each positive integer $m$ , $D(3+2^{-m})\\\\ge HD(K(\\\\lbrace 21^{2m}2,21^{2m+2}2\\\\rbrace ))>C/m$ .\",\n \"This gives another proof of the fact that the functions $D(t)$ and $d(t)$ are not Hölder continuous.\",\n \"Proof of Theorem 2: Given $m \\\\ge 2$ , let $C_m=\\\\lbrace {\\\\alpha }=[0;a_1,a_2,a_3,\\\\ldots ] \\\\in [0,1]| a_k \\\\le m, \\\\forall k \\\\ge 1\\\\rbrace $ .\",\n \"M. Hall proved in [H] that $C_4+C_4=\\\\lbrace {\\\\alpha }+{\\\\beta }| {\\\\alpha },{\\\\beta }\\\\in C_4\\\\rbrace =[\\\\sqrt{2}-1,4(\\\\sqrt{2}-1)]$ .\",\n \"On the other hand, we have $\\\\lim _{m\\\\rightarrow \\\\infty }HD(C_m)=1$ .\",\n \"In fact, Jarník proved in [J] that $HD(C_m)>1-\\\\frac{1}{m \\\\cdot \\\\log 2}, \\\\forall m>8.$ Let now $t \\\\ge 7$ be given.\",\n \"Let $m=\\\\lfloor t \\\\rfloor -3$ .\",\n \"There are an integer $n \\\\in \\\\lbrace m+2,m+3\\\\rbrace $ and ${\\\\alpha }=[0;a_1,a_2,a_3,\\\\ldots ], {\\\\beta }=[0;b_1,b_2,b_3,\\\\ldots ] \\\\in C_4$ such that $t=n+{\\\\alpha }+{\\\\beta }$ .\",\n \"For each $r\\\\ge 1$ , let $\\\\tilde{\\\\tau }^{(r)}$ and $\\\\hat{\\\\tau }^{(r)}$ be respectively the sequences $(m+1,b_{2r-1},b_{2r-2},\\\\ldots ,b_2,b_1,n,a_1,a_2,\\\\ldots ,a_{2r-2},a_{2r-1},m+1)$ and $(m+1,b_{2r},b_{2r-1},\\\\ldots ,b_2,b_1,n,a_1,a_2,\\\\ldots ,a_{2r-1},a_{2r},m+1)$ .\",\n \"Consider now the maps $\\\\tilde{h}, \\\\hat{h}: C_m \\\\rightarrow [0,1]$ given by $\\\\tilde{h}(z) = \\\\tilde{h}([0;c_1,c_2,c_3,\\\\ldots ] )=[0;c_{1!\",\n \"},\\\\tilde{\\\\tau }^{(1)},c_{2!\",\n \"},\\\\tilde{\\\\tau }^{(2)}, c_3,c_4,c_5,c_{3!\",\n \"},\\\\tilde{\\\\tau }^{(3)}, \\\\\\\\c_7, c_8,\\\\dots ,c_{4!\",\n \"},\\\\tilde{\\\\tau }^{(4)}, c_{25}, \\\\dots ,c_{5!\",\n \"}, \\\\tilde{\\\\tau }^{(5)},\\\\: \\\\dots \\\\:, c_{r!\",\n \"},\\\\tilde{\\\\tau }^{(r)},c_{r!+1},\\\\dots ].$ $\\\\hat{h}(z) = \\\\hat{h}([0;c_1,c_2,c_3,\\\\ldots ] )=[0;c_{1!\",\n \"},\\\\hat{\\\\tau }^{(1)},c_{2!\",\n \"},\\\\hat{\\\\tau }^{(2)}, c_3,c_4,c_5,c_{3!\",\n \"},\\\\hat{\\\\tau }^{(3)}, \\\\\\\\c_7, c_8,\\\\dots ,c_{4!\",\n \"},\\\\hat{\\\\tau }^{(4)}, c_{25}, \\\\dots ,c_{5!\",\n \"}, \\\\hat{\\\\tau }^{(5)},\\\\: \\\\dots \\\\:, c_{r!\",\n \"},\\\\hat{\\\\tau }^{(r)},c_{r!+1},\\\\dots ].$ It is easy to see that $k(\\\\tilde{h}(z))=t$ for every $z \\\\in C_m$ .\",\n \"Moreover, since $[x_0;x_1,x_2,x_3,x_4,\\\\dots ]$ is increasing in $x_0, x_2, x_4,\\\\ldots $ and decreasing in $x_1, x_3, x_5, \\\\ldots $ , $\\\\tilde{h}(z)\\\\in \\\\text{Exact}(t^{-1})$ and $\\\\hat{h}(z)\\\\in \\\\text{Exact}^{\\\\prime }(t^{-1})$ .\",\n \"On the other hand, given any $\\\\rho >0$ , we have $|z-z^{\\\\prime }|=O(|\\\\tilde{h}(z)-\\\\tilde{h}(z^{\\\\prime })|^{1-\\\\rho })$ and $|z-z^{\\\\prime }|=O(|\\\\hat{h}(z)-\\\\hat{h}(z^{\\\\prime })|^{1-\\\\rho })$ for $|z-z^{\\\\prime }|$ small, so $HD(\\\\text{Exact}(t^{-1})) \\\\ge HD(C_m)$ and $HD(\\\\text{Exact}^{\\\\prime }(t^{-1})) \\\\ge HD(C_m)$ .\",\n \"Since $\\\\lim _{m\\\\rightarrow \\\\infty }HD(C_m)=1$ , we are done.\",\n \"$\\\\Box $ Proof of Theorem 3: Let $x \\\\in L^{\\\\prime }$ .\",\n \"Consider a sequence $x_n$ converging to $x$ , $x_n \\\\in L$ , $x_n \\\\ne x$ .\",\n \"Choose $\\\\underline{{\\\\theta }}^{(n)} \\\\in \\\\Sigma $ such that $x_n = \\\\ell (\\\\underline{{\\\\theta }}^{(n)})$ .\",\n \"Let $\\\\underline{{\\\\theta }}^{(n)}=(b_j^{(n)})_{j\\\\in {\\\\mathbb {Z}}}$ and assume $b_j^{(n)} \\\\le 4$ , $\\\\forall \\\\, j$ , $\\\\forall \\\\, n$ (which is possible since me may assume that the $x_n$ are not in Hall's ray).\",\n \"We have $x_n ={\\\\limsup }_{j\\\\rightarrow \\\\infty }({\\\\alpha }_j^{(n)} + {\\\\beta }_j^{(n)})$ .\",\n \"Given $\\\\delta >0$ , $\\\\exists \\\\, n_0 \\\\in {\\\\mathbb {N}}$ large such that $n \\\\ge n_0 \\\\Rightarrow |\\\\ell (\\\\underline{{\\\\theta }}^{(n)})-x| < \\\\delta $ and there are infinitely many $j \\\\in {\\\\mathbb {N}}$ such that $|{\\\\alpha }_j^{(n)} + {\\\\beta }_j^{(n)}-x| < \\\\delta $ .\",\n \"Let $N =\\\\lceil \\\\delta ^{-1} \\\\rceil $ .\",\n \"Given such a pair $(j,n)$ consider the finite sequence with $2N+1$ terms $(b_{j-N}^{(n)},b_{j-N+1}^{(n)},\\\\dots , b_j^{(n)},\\\\dots , b_{j+N}^{(n)}) =: S(j,n)$ .\",\n \"There is a sequence $S$ such that for infinitely many values of $n$ , $S$ appears infinitely many times as $S(j,n), \\\\, j \\\\in {\\\\mathbb {N}}$ , i.e., there are $j_1(n) < j_2(n) <\\\\dots $ with $\\\\lim _{i\\\\rightarrow \\\\infty } (j_{i+1}(n)-j_i(n))=\\\\infty $ and $S(j_i(n),n) = S$ , $\\\\forall \\\\,i \\\\ge 1$ , for all $n$ in some infinite set $A \\\\subset {\\\\mathbb {N}}$ .\",\n \"Consider the sequences ${\\\\beta }(i,n)$ for $i\\\\ge 1$ , $n \\\\in A$ given by ${\\\\beta }(i,n) = (b_{j_i(n)+N+1}^{(n)}, b_{j_i(n)+N+2}^{(n)},\\\\dots ,b_{j_{i+1}(n)+N}^{(n)}).$ There are $(i_1,n_1)$ and $(i_2,n_2)$ for which there is no sequence ${\\\\gamma }$ such that ${\\\\beta }(i_1,n_1)$ and ${\\\\beta }(i_2,n_2)$ are concatenations of copies of ${\\\\gamma }$ , otherwise $x_n$ would be constant for $n \\\\in A$ .\",\n \"This implies that, taking $B = \\\\lbrace {\\\\beta }(i_1,n_1) {\\\\beta }(i_2,n_2), {\\\\beta }(i_2,n_2) {\\\\beta }(i_1,n_1)\\\\rbrace $ , $K(B)$ is a regular Cantor set, so, as in Lemma 3, $\\\\ell (K(B))$ contains a regular Cantor set $\\\\hat{K}$ with $d(x,\\\\hat{K}) \\\\le 2\\\\delta $ .\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Lemma 2\",\n \"Let $\\\\tau =\\\\eta /40$ .\",\n \"Since $t>3$ , we have $D(t)>0$ , and so we may choose $r_0\\\\in {\\\\mathbb {N}}$ large such that, for $r\\\\ge r_0$ , $|\\\\frac{\\\\log N(t,r)}{r}-D(t)|<\\\\frac{\\\\tau }{2} D(t)$ .\",\n \"Let $B_0:=C(t,r_0)$ and $N_0:=N(t,r_0)=|B_0|$ .\",\n \"Let $k=8N_0^2\\\\lceil 2/\\\\tau \\\\rceil $ .\",\n \"Take $\\\\tilde{B}=\\\\lbrace {\\\\beta }={\\\\beta }_1{\\\\beta }_2\\\\cdots {\\\\beta }_k \\\\mid {\\\\beta }_j\\\\in B_0,1\\\\le j\\\\le k $ and $K_t\\\\cap I({\\\\beta })\\\\ne \\\\emptyset \\\\rbrace $ .\",\n \"Given ${\\\\beta }={\\\\beta }_1{\\\\beta }_2\\\\cdots {\\\\beta }_k\\\\in \\\\tilde{B}$ (with ${\\\\beta }_i\\\\in B_0$ , $1\\\\le i\\\\le k$ ), we say that $j$ , $1\\\\le j\\\\le k$ , is a right-good position of ${\\\\beta }$ if there are elements ${\\\\beta }^{(s)}={\\\\beta }_1{\\\\beta }_2\\\\cdots {\\\\beta }_{j-1} {\\\\beta }_j^{(s)} {\\\\beta }_{j+1}^{(s)}\\\\cdots {\\\\beta }_k^{(s)}$ , $s=1,2$ , of $\\\\tilde{B}$ such that we have the following inequality of continued fractions: $[0; {\\\\beta }_j^{(1)}]<[0;{\\\\beta }_j]<[0;{\\\\beta }_j^{(2)}]$ .\",\n \"We say that $j$ is a left-good position if there are elements ${\\\\beta }^{(s)}={\\\\beta }_1^{(s)}{\\\\beta }_2^{(s)}\\\\cdots {\\\\beta }_{j-1}^{(s)}{\\\\beta }_j^{(s)}{\\\\beta }_{j+1}{\\\\beta }_{j+2}\\\\cdots {\\\\beta }_k$ , $s=3,4$ , of $\\\\tilde{B}$ such that $[0;({\\\\beta }_j^{(3)})^t]<[0;{\\\\beta }_j^t]<[0;({\\\\beta }_j^{(4)})^t]$ .\",\n \"Finally, we say that $j$ is a good position if it is both right-good and left-good.\",\n \"We will show that most positions of most words of $\\\\tilde{B}$ are good.\",\n \"Let us first estimate $|\\\\tilde{B}|$ .\",\n \"It follows from Lemma A2 of the appendix that, for ${\\\\beta }\\\\in \\\\tilde{B}$ , $s({\\\\beta })<(2e^{-r_0})^k& 2\\\\,e^{k(r_0-2)D(t)},\\\\quad \\\\text{since $k$ is large} \\\\nonumber \\\\\\\\&\\\\ge & 2\\\\,e^{(1-\\\\tau /2)r_0 kD(t)},\\\\quad \\\\text{since $r_0$ is large} \\\\nonumber \\\\\\\\&>& 2\\\\,e^{(1-\\\\tau )(1+\\\\tau /2)r_0kD(t)}\\\\nonumber \\\\\\\\&>& 2\\\\,N_0^{(1-\\\\tau )k},\\\\quad \\\\text{since}\\\\: N(t,r_0) 2N_0^{(1-\\\\tau )k}-2^{1+21k/20}\\\\cdot N_0^{19k/20} > N_0^{(1-\\\\tau )k}$ words of $\\\\tilde{B}$ , the number of good positions is at least $9k/10$ .\",\n \"Let us call such an element of $\\\\tilde{B}$ an excellent word.\",\n \"If ${\\\\beta }={\\\\beta }_1{\\\\beta }_2\\\\cdots {\\\\beta }_k\\\\in \\\\tilde{B}$ (with ${\\\\beta }_j\\\\in B_0$ , $1\\\\le j\\\\le k$ ) is an excellent word, we may find $\\\\lceil 2k/5\\\\rceil $ positions $i_1,i_2,\\\\dots ,i_{\\\\lceil 2k/5\\\\rceil }\\\\le k$ with $i_{s+1}\\\\ge i_s+2$ , $\\\\forall \\\\, s< \\\\lceil 2k/5\\\\rceil $ , such that the positions $i_1,i_1+1$ , $i_2, i_{2}+1,\\\\dots ,i_{\\\\lceil 2k/5\\\\rceil }$ , $i_{\\\\lceil 2k/5\\\\rceil }+1$ are good.\",\n \"Since $k=8N_0^2\\\\lceil 2/\\\\tau \\\\rceil $ , we may take, for $1\\\\le s\\\\le 3N_0^2$ , $j_s:=i_{s\\\\lceil 2/\\\\tau \\\\rceil }$ (notice that $3N_0^2\\\\lceil 2/\\\\tau \\\\rceil <\\\\frac{16}{5}N_0^2\\\\lceil 2/\\\\tau \\\\rceil =2k/5$ ), so we have $j_{s+1}-j_s\\\\ge 2\\\\lceil 2/\\\\tau \\\\rceil $ , $\\\\forall \\\\, s<3N_0^2$ , and the positions $j_s,j_s+1$ are good for $1\\\\le s\\\\le 3N_0^2$ .\",\n \"Now, the number of possible choices of $(j_1,j_2,\\\\dots ,j_{3N_0^2})$ is bounded by $\\\\binom{k}{3N_0^2}<2^k$ and, given $(j_1,j_2,\\\\dots ,j_{3N_0^2})$ the number of choices of $({\\\\beta }_{j_1},{\\\\beta }_{j_1+1},\\\\dots ,{\\\\beta }_{j_{3N_0^2}},{\\\\beta }_{j_{3N_0^2}+1})$ is bounded by $N_0^{6N_0^2}$ .\",\n \"So, we may choose $\\\\hat{\\\\j }_1,\\\\hat{\\\\j }_2,\\\\dots ,\\\\hat{\\\\j }_{3N_0^2}$ with $\\\\hat{\\\\j }_{s+1}-\\\\hat{\\\\j }_s\\\\ge 2\\\\lceil 2/\\\\tau \\\\rceil $ , $\\\\forall \\\\, s<3N_0^2$ , and words $\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_1},\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_1+1}, \\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_2},\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_2+1},\\\\dots ,\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_{3N_0^2}},\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_{3N_0^2}+1} \\\\in B_0$ such that the set $X:=\\\\lbrace {\\\\beta }={\\\\beta }_1{\\\\beta }_2\\\\cdots {\\\\beta }_k\\\\in \\\\tilde{B}$ excellent$\\\\,|\\\\,\\\\, \\\\hat{\\\\j }_s,\\\\hat{\\\\j }_s+1$ are good positions and ${\\\\beta }_{\\\\hat{\\\\j }_s}=\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_s}, {\\\\beta }_{\\\\hat{\\\\j }_s+1}=\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_s+1}, \\\\forall \\\\, s\\\\le 3N_0^2\\\\rbrace $ has at least $\\\\frac{N_0^{(1-\\\\tau )k}}{2^k \\\\cdot N_0^{6N_0^2}} > N_0^{(1-2\\\\tau )k}$ elements, as $N_0$ and $k$ are large.\",\n \"Since $N_0=|B_0|$ , there are $N_0^2$ possible choices for the pairs $(\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_s},\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_s+1})$ .\",\n \"We will consider, for $1\\\\le ss$ , the sum of $(\\\\hat{\\\\j }_t-\\\\hat{\\\\j }_s)$ for $(s,t) \\\\in \\\\cal P$ is at least $2N_0^2\\\\lceil 2/\\\\tau \\\\rceil $ .\",\n \"Since for each pair $(s,t) \\\\in \\\\cal P$ we have $|\\\\pi _{s,t}(X)|(2(T+1)^2 e^{r_0})^{-(\\\\hat{\\\\j }_t-\\\\hat{\\\\j }_s)} > e^{-(\\\\hat{\\\\j }_t-\\\\hat{\\\\j }_s)(r_0+\\\\lceil \\\\log (2(T+1)^2) \\\\rceil )}.$ So, the Hausdorff dimension of $K(B)$ is at least $\\\\frac{(1-10\\\\tau )\\\\log N_0}{r_0+\\\\lceil \\\\log (2(T+1)^2) \\\\rceil } > \\\\frac{(1-10\\\\tau )r_0}{r_0+\\\\lceil \\\\log (2(T+1)^2) \\\\rceil }\\\\cdot (1-\\\\frac{\\\\tau }{2})D(t) > \\\\left(1-12\\\\tau \\\\right)D(t)> (1-\\\\eta )D(t).$ On the other hand, if $\\\\tilde{k}:=\\\\hat{\\\\j }_t-\\\\hat{\\\\j }_s$ , ${\\\\gamma }_1:=\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_s+1}=\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_t+1}$ and ${\\\\gamma }_2:=\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_t}=\\\\hat{{\\\\beta }}_{\\\\hat{\\\\j }_s}$ , all elements of $B$ are of the form ${\\\\gamma }_1{\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}{\\\\gamma }_2$ , where ${\\\\gamma }_1,{\\\\beta }_2,{\\\\beta }_3,\\\\dots ,{\\\\beta }_{\\\\tilde{k}-1}$ , ${\\\\gamma }_2\\\\in B_0$ and there are ${\\\\gamma }_1^{\\\\prime },{\\\\gamma }_1^{\\\\prime \\\\prime }$ , ${\\\\gamma }_2^{\\\\prime },{\\\\gamma }_2^{\\\\prime \\\\prime }\\\\in B_0$ with $[0;{\\\\gamma }_2^{\\\\prime }]<[0;{\\\\gamma }_2]< [0;{\\\\gamma }_2^{\\\\prime \\\\prime }]$ and $[0;({\\\\gamma }_1^{\\\\prime })^t]<[0;{\\\\gamma }_1^t]<[0;({\\\\gamma }_1^{\\\\prime \\\\prime })^t]$ such that $I({\\\\gamma }_1^{\\\\prime }{\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}{\\\\gamma }_2{\\\\gamma }_1)\\\\cap K_t\\\\ne \\\\emptyset ,\\\\quad I({\\\\gamma }_1^{\\\\prime \\\\prime }{\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}{\\\\gamma }_2{\\\\gamma }_1)\\\\cap K_t\\\\ne \\\\emptyset , \\\\nonumber \\\\\\\\I({\\\\gamma }_2{\\\\gamma }_1{\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}{\\\\gamma }_2^{\\\\prime })\\\\cap K_t\\\\ne \\\\emptyset ,\\\\quad I({\\\\gamma }_2{\\\\gamma }_1{\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}{\\\\gamma }_2^{\\\\prime \\\\prime })\\\\cap K_t\\\\ne \\\\emptyset .\\\\nonumber $ We will show that this implies the existence of $\\\\delta >0$ such that $\\\\Sigma (B)\\\\subset \\\\Sigma _{t-\\\\delta }$ .\",\n \"Let ${\\\\gamma }_1^t=(c_1,c_2,\\\\dots ,c_{m_1})$ , with $c_j\\\\in {\\\\mathbb {N}}^*$ , $\\\\forall \\\\, j\\\\le m_1$ , and ${\\\\gamma }_2=(d_1,d_2,\\\\dots ,d_{m_2})$ with $d_j\\\\in {\\\\mathbb {N}}^*$ , $\\\\forall \\\\, j\\\\le m_2$ .\",\n \"Let ${\\\\gamma }_1{\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}{\\\\gamma }_2\\\\in B$ where ${\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}=a_1a_2\\\\cdots a_{\\\\tilde{m}}$ with $a_j\\\\in {\\\\mathbb {N}}^*$ , $\\\\forall \\\\, j\\\\le \\\\tilde{m}$ .\",\n \"We want to estimate three kinds of sums of continued fractions.\",\n \"The first one are sums of continued fractions beginning by $[a_j;a_{j+1},\\\\dots ,a_{\\\\tilde{m}},{\\\\gamma }_2,{\\\\gamma }_1,\\\\dots ]+[0;a_{j-1},\\\\dots ,a_1,{\\\\gamma }_1^t,{\\\\gamma }_2^t,\\\\dots ].$ Let us assume, without loss of generality, that $q_{m_2+\\\\tilde{m} -j}(a_{j+1},\\\\ldots , a_{\\\\tilde{m}}, {\\\\gamma }_2) \\\\le q_{m_1+j-1}(a_{j-1},\\\\ldots , a_1, {\\\\gamma }_1^t)$ (the other case, when the reverse inequality $q_{m_1+j-1}(a_{j-1},\\\\ldots , a_1, {\\\\gamma }_1^t) \\\\le q_{m_2+\\\\tilde{m} -j}(a_{j+1},\\\\ldots , a_{\\\\tilde{m}}, {\\\\gamma }_2)$ holds, is symmetric).\",\n \"Assume also that $[a_j; a_{j+1},\\\\ldots ,a_{\\\\tilde{m}},{\\\\gamma }_2]<[a_j; a_{j+1},\\\\ldots ,a_{\\\\tilde{m}},{\\\\gamma }_2^{\\\\prime }]$ (otherwise we change ${\\\\gamma }_2^{\\\\prime }$ by ${\\\\gamma }_2^{\\\\prime \\\\prime }$ ).\",\n \"This allows us to exhibit $\\\\delta > 0$ such that, for any $\\\\underline{\\\\theta }^{(i)} \\\\in \\\\lbrace 1, 2, \\\\dots , T\\\\rbrace ^{{\\\\mathbb {N}}},\\\\; 1 \\\\le i \\\\le 4$ , ${[a_j;a_{j+1},\\\\dots ,a_{\\\\tilde{m}},{\\\\gamma }_2,\\\\underline{\\\\theta }^{(1)}]+[0;a_{j-1},\\\\dots ,a_1,{\\\\gamma }_1^t,{\\\\gamma }_2^t,\\\\underline{\\\\theta }^{(2)}] <}\\\\\\\\& <[a_j;a_{j+1},\\\\dots ,a_{\\\\tilde{m}},{\\\\gamma }_2^{\\\\prime },\\\\underline{\\\\theta }^{(3)}]+[0;a_{j-1},\\\\dots ,a_1,{\\\\gamma }_1^t,{\\\\gamma }_2^t,\\\\underline{\\\\theta }^{(4)}]-\\\\delta .$ Indeed, by the Lemma A1 of the appendix, $[a_j;a_{j+1},\\\\dots ,a_{\\\\tilde{m}},{\\\\gamma }_2^{\\\\prime },\\\\underline{\\\\theta }^{(3)}]-[a_j;a_{j+1},\\\\dots ,a_{\\\\tilde{m}},{\\\\gamma }_2,\\\\underline{\\\\theta }^{(1)}]>\\\\frac{1}{(T+1)(T+2)q_{m_2+\\\\tilde{m} -j}(a_{j+1},\\\\ldots , a_{\\\\tilde{m}}, {\\\\gamma }_2)^2}$ $\\\\text{ and }\\\\,\\\\mid [0;a_{j-1},\\\\dots ,a_1,{\\\\gamma }_1^t,{\\\\gamma }_2^t,\\\\underline{\\\\theta }^{(4)}]-[0;a_{j-1},\\\\dots ,a_1,{\\\\gamma }_1^t,{\\\\gamma }_2^t,\\\\underline{\\\\theta }^{(2)}]\\\\mid < \\\\\\\\\\\\frac{1}{q_{m_1+m_2+j-1}(a_{j-1},\\\\ldots , a_1, {\\\\gamma }_1^t, {\\\\gamma }_2^t)^2}<\\\\frac{1}{(F_{m_2+1}q_{m_1+j-1}(a_{j-1}, \\\\ldots , a_1, {\\\\gamma }_1^t))^2} \\\\le \\\\\\\\\\\\frac{1}{(F_{m_2+1}q_{m_2+\\\\tilde{m} -j}(a_{j+1},\\\\ldots , a_{\\\\tilde{m}}, {\\\\gamma }_2))^2}\\\\le \\\\frac{1}{2(T+1)(T+2)q_{m_2+\\\\tilde{m} -j}(a_{j+1},\\\\ldots , a_{\\\\tilde{m}}, {\\\\gamma }_2)^2}\\\\\\\\(\\\\text{here we use that}\\\\; m_2 \\\\text{ is large};\\\\\\\\ (F_n)\\\\text{ denotes Fibonacci's sequence, given by } F_0=0, F_1=1, F_{n+2}=F_{n+1}+F_n, \\\\forall n \\\\ge 0).\\\\\\\\$ So, the inequality holds with $\\\\delta :=\\\\frac{1}{(T+1)^{2(m_1+m_2+\\\\tilde{m})}}<\\\\frac{1}{2(T+1)(T+2)q_{m_2+\\\\tilde{m} -j}(a_{j+1},\\\\ldots , a_{\\\\tilde{m}}, {\\\\gamma }_2)^2}.$ On the other hand, $I({\\\\gamma }_2{\\\\gamma }_1{\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}{\\\\gamma }_2^{\\\\prime })\\\\cap K_t\\\\ne \\\\emptyset $ , so there are $\\\\underline{\\\\theta }^{(3)}$ and $\\\\underline{\\\\theta }^{(4)}$ such that $(\\\\underline{\\\\theta }^{(4)})^t {\\\\gamma }_2{\\\\gamma }_1 {\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}{\\\\gamma }_2^{\\\\prime } \\\\underline{\\\\theta }^{(3)} \\\\in \\\\Sigma _t$ , and thus $[a_j;a_{j+1},\\\\dots ,a_{\\\\tilde{m}},{\\\\gamma }_2^{\\\\prime },\\\\underline{\\\\theta }^{(3)}]+ [0;a_{j-1},\\\\dots ,a_1,{\\\\gamma }_1^t,{\\\\gamma }_2^t,\\\\underline{\\\\theta }^{(4)}]\\\\le t$ , which implies that, for any $\\\\underline{\\\\theta }^{(i)} \\\\in \\\\lbrace 1, 2, \\\\dots , T\\\\rbrace ^{{\\\\mathbb {N}}}, i=1,2$ , $[a_j;a_{j+1},\\\\dots ,a_{\\\\tilde{m}},{\\\\gamma }_2,\\\\underline{\\\\theta }^{(1)}]+[0;a_{j-1},\\\\dots ,a_1,{\\\\gamma }_1^t,{\\\\gamma }_2^t,\\\\underline{\\\\theta }^{(2)}] < t-\\\\delta $ .\",\n \"The other two kinds of sums of continued fractions we want to estimate are sums of continued fractions beginning by $[d_j; d_{j+1},\\\\ldots ,d_{m_2},{\\\\gamma }_1,\\\\ldots ]+[0; d_{j-1},\\\\ldots , d_1, a_{\\\\tilde{m}},\\\\ldots , a_1, {\\\\gamma }_1^t,\\\\ldots ]$ and, symmetrically, sums of continued fractions beginning by $[0; c_{j+1},\\\\ldots , c_{m_1}, {\\\\gamma }_2^t,\\\\ldots ]+[c_j; c_{j-1},\\\\ldots , c_1, a_1,\\\\ldots , a_{\\\\tilde{m}}, {\\\\gamma }_2,\\\\ldots ]$ .\",\n \"We have: $q_{m_2-j+m_1}(d_{j+1},\\\\ldots ,d_{m_2}, {\\\\gamma }_1) \\\\le q_{j-1+\\\\tilde{m} +m_1}(d_{j-1},\\\\ldots , d_1, a_{\\\\tilde{m}},\\\\ldots , a_1, {\\\\gamma }_1^t)$ (indeed, $\\\\tilde{m}/(m_1+m_2)$ is large when $\\\\eta $ and $\\\\tau $ are small, depending on $T$ ).\",\n \"Assume that $[d_j; d_{j+1},\\\\ldots ,d_{m_2}, {\\\\gamma }_1] < [d_j; d_{j+1},\\\\ldots ,d_{m_2}, {\\\\gamma }_1^{\\\\prime }]$ (otherwise we change ${\\\\gamma }_1^{\\\\prime }$ by ${\\\\gamma }_1^{\\\\prime \\\\prime }$ ).\",\n \"Since $I({\\\\gamma }_2{\\\\gamma }_1 {\\\\beta }_2{\\\\beta }_3\\\\cdots {\\\\beta }_{\\\\tilde{k}-1}{\\\\gamma }_2{\\\\gamma }_1^{\\\\prime })\\\\cap K_t\\\\ne \\\\emptyset $ , estimates analogous to the previous ones imply that, for any $\\\\underline{\\\\theta }^{(i)} \\\\in \\\\lbrace 1, 2, \\\\dots , T\\\\rbrace ^{{\\\\mathbb {N}}},\\\\: i=1,2$ , we have $[d_j; d_{j+1},\\\\ldots ,d_{m_2}, {\\\\gamma }_1,\\\\underline{\\\\theta }^{(1)}]+$$[0; d_{j-1},\\\\ldots , d_1, a_{\\\\tilde{m}},\\\\ldots , a_1, {\\\\gamma }_1^t,{\\\\gamma }_2^t,\\\\underline{\\\\theta }^{(2)}] < t-\\\\delta $ .\",\n \"This implies that the complete shift $\\\\Sigma (B)$ satisfies the conditions of the statement, which concludes the proof of the Lemma.\",\n \"$\\\\Box $ As we said before, this proof doesn't give any estimate on the modulus of continuity of $d(t)$ .\",\n \"Indeed, in the beginning of the proof of Lemma 2, we used the fact that $u(r)=\\\\log (T^2N(t,r))$ is subadditive in order to guarantee the existence of $r_0\\\\in {\\\\mathbb {N}}$ large such that, for $r\\\\ge r_0$ , $|\\\\frac{\\\\log N(t,r)}{r}-D(t)|<\\\\frac{\\\\tau }{2} D(t)$ (recall that $D(t)=\\\\lim _{m\\\\rightarrow \\\\infty }\\\\frac{1}{m} \\\\log (T^2N(t,m))=\\\\lim _{m\\\\rightarrow \\\\infty }\\\\frac{1}{m}\\\\log (N(t,m))$ ).\",\n \"However, this gives no estimate on $r_0$ .\",\n \"Consider, for instance, the function $v(n)$ given by $v(n)=2n$ for $n\\\\le M_0$ and $v(n)=n+M_0$ for $n>M_0$ , where $M_0$ is a large positive integer.\",\n \"It is subadditive, increasing, $\\\\lim _{n\\\\rightarrow \\\\infty } v(n)/n=1$ but $v(M_0)/M_0=2$ , and $M_0$ can be taken arbitrarily large.\",\n \"However it is possible to adapt the proof in order to give an estimate on the modulus of continuity of $d(t)$ , using an idea of [FMM].\",\n \"Given ${\\\\varepsilon }>0$ (which we may assume to be smaller than $\\\\frac{1}{7}<\\\\frac{1}{10\\\\log 2}$ ), we want to obtain $\\\\delta \\\\in (0,1)$ as an explicit function of ${\\\\varepsilon }$ such that $D(t-\\\\delta )>D(t)-{\\\\varepsilon }$ .\",\n \"Of course there is no loss of generality in assuming $D(t)\\\\ge {\\\\varepsilon }$ .\",\n \"We may also assume that $T=\\\\lfloor t \\\\rfloor < 4+{\\\\varepsilon }^{-1}/\\\\log 2$ (and thus $t1-\\\\frac{1}{m\\\\log 2}>1-{\\\\varepsilon }$ (and so $D(t-1)>D(t)-{\\\\varepsilon }$ ).\",\n \"Under these hypothesis, we will apply the conclusions of Lemma 2 for $\\\\eta ={\\\\varepsilon }$ .\",\n \"In its proof, in this case, it is enough to assume $r_0\\\\ge 1/\\\\tau ^2$ and that, for $k=8N(t,r_0)^2\\\\lceil 2/\\\\tau \\\\rceil $ , $\\\\frac{\\\\log N(t,r_0)}{r_0}<(1+\\\\tau /2)\\\\frac{\\\\log N(t,k(r_0-1))}{k(r_0-1)}$ (indeed, assuming the above bounds for $t$ , it is not difficult to check that, except for this inequality relating $N(t,r_0)$ and $N(t,k(r_0-1))$ , the claims in other parts of the proof of the Lemma that use the assumptions that $r_0$ and $k$ are large are satisfied provided $r_0\\\\ge 1/\\\\tau ^2$ ).\",\n \"We define a sequence $(c_n)_{n\\\\ge 0}$ recursively by $c_0=\\\\lceil \\\\frac{1}{\\\\tau ^2}\\\\rceil $ and, for every $n\\\\ge 0$ , $c_{n+1}=8N(t,c_n)^2\\\\lceil \\\\frac{2}{\\\\tau }\\\\rceil (c_n-1)$ .\",\n \"We claim that, for some integer $s_0<(1+\\\\frac{2}{\\\\tau })\\\\log (4/{\\\\varepsilon })$ , we will have $\\\\frac{\\\\log N(t,c_{s_0})}{c_{s_0}}<(1+\\\\frac{\\\\tau }{2})\\\\frac{\\\\log N(t,c_{s_0+1})}{c_{s_0+1}}=(1+\\\\frac{\\\\tau }{2})\\\\frac{\\\\log N(t,k(c_{s_0}-1))}{k(c_{s_0}-1)}$ , with $k=8N(t,c_{s_0})^2\\\\lceil \\\\frac{2}{\\\\tau }\\\\rceil $ .\",\n \"Indeed, if it is not the case, then $\\\\frac{\\\\log N(t,c_{n+1})}{c_{n+1}}\\\\le (1+\\\\frac{\\\\tau }{2})^{-1}\\\\frac{\\\\log N(t,c_n)}{c_n}$ for $0\\\\le n<(1+\\\\frac{2}{\\\\tau })\\\\log (4/{\\\\varepsilon })$ , and so, for $M=\\\\lceil (1+\\\\frac{2}{\\\\tau })\\\\log (4/{\\\\varepsilon })\\\\rceil $ , we would have $\\\\frac{\\\\log N(t,c_M)}{c_M}\\\\le (1+\\\\frac{\\\\tau }{2})^{-M}\\\\cdot \\\\frac{\\\\log N(t,c_0)}{c_0}<({\\\\varepsilon }/4)\\\\cdot \\\\frac{\\\\log N(t,c_0)}{c_0}$ , since $(1+\\\\frac{\\\\tau }{2})^{-(1+\\\\frac{2}{\\\\tau })}{\\\\varepsilon }-\\\\frac{2\\\\log T}{c_M}\\\\ge {\\\\varepsilon }/2$ , since $c_M\\\\ge c_0\\\\ge \\\\frac{1600}{{\\\\varepsilon }^2}$ and $T<3/{\\\\varepsilon }$ .\",\n \"Now let $r_0=c_{s_0}$ .\",\n \"By the previous discussion, the proof of Lemma 2 works for this $r_0$ (and $k=8N(t,r_0)^2\\\\lceil 2/\\\\tau \\\\rceil $ ), so, for $\\\\delta =\\\\frac{1}{(T+1)^{2(m_1+m_2+\\\\tilde{m})}}\\\\ge \\\\frac{1}{(T+1)^{2k\\\\cdot \\\\text{max}\\\\lbrace |\\\\beta |, \\\\beta \\\\in C(t,r_0)\\\\rbrace }}\\\\ge \\\\frac{1}{(T+1)^{2k\\\\cdot r_0/\\\\log 2}},$ we have $D(t-\\\\delta )>(1-{\\\\varepsilon })D(t)>D(t)-{\\\\varepsilon }$ .\",\n \"We will now give an explicit positive lower bound for $\\\\delta $ in terms of ${\\\\varepsilon }$ .\",\n \"In order to do this we define recursively, for each integer $n\\\\ge 0$ and $x\\\\in {\\\\mathbb {R}}$ , the functions ${\\\\cal T}(x,n)$ and ${\\\\cal T}(n)$ by ${\\\\cal T}(x,0)=x$ , ${\\\\cal T}(x,n+1)=e^{{\\\\cal T}(x,n)}$ and ${\\\\cal T}(n)={\\\\cal T}(1,n)$ .\",\n \"We have, for every $n\\\\ge 0$ , $c_{n+1}=8N(t,c_n)^2\\\\lceil \\\\frac{2}{\\\\tau }\\\\rceil (c_n-1)\\\\le 8e^{4c_n}\\\\cdot \\\\frac{3}{\\\\tau }\\\\cdot c_ne^{-e^{e^{r_0}}}>\\\\frac{1}{{\\\\cal T}(c_0,2s_0+3)}.$ Finally, since $2^k\\\\ge k^2$ for every $k\\\\ge 4$ , it follows by induction that, for every $n\\\\ge 4$ , ${\\\\cal T}(n)\\\\ge (n+1)^6$ for every $n\\\\ge 0$ .\",\n \"Indeed, ${\\\\cal T}(4)>2^{16}>5^6$ and, for $n\\\\ge 4$ , ${\\\\cal T}(n+1)>2^{{\\\\cal T}(n)}\\\\ge {\\\\cal T}(n)^2\\\\ge (n+1)^{12}>(n+2)^6$ .\",\n \"This implies that ${\\\\cal T}(\\\\lfloor 1/{\\\\varepsilon }\\\\rfloor )\\\\ge (1/{\\\\varepsilon })^6>1601/{\\\\varepsilon }^2>\\\\lceil \\\\frac{1600}{{\\\\varepsilon }^2}\\\\rceil =\\\\lceil \\\\frac{1}{\\\\tau ^2}\\\\rceil =c_0$ (recall that $0<{\\\\varepsilon }<1/7$ ), so ${\\\\cal T}(c_0,2s_0+3)<{\\\\cal T}({\\\\cal T}(\\\\lfloor 1/{\\\\varepsilon }\\\\rfloor ),2s_0+3)={\\\\cal T}(\\\\lfloor 1/{\\\\varepsilon }\\\\rfloor +2s_0+3)$ , and, since $s_0<(1+\\\\frac{2}{\\\\tau })\\\\log (4/{\\\\varepsilon })$ , we have $\\\\lfloor 1/{\\\\varepsilon }\\\\rfloor +2s_0+3<3+\\\\lfloor 1/{\\\\varepsilon }\\\\rfloor +2(1+\\\\frac{2}{\\\\tau })\\\\log (4/{\\\\varepsilon })\\\\le 3+1/{\\\\varepsilon }+2(1+\\\\frac{2}{\\\\tau })\\\\log (4/{\\\\varepsilon })=$ $=3+1/{\\\\varepsilon }+2(1+\\\\frac{80}{{\\\\varepsilon }})\\\\log (4/{\\\\varepsilon })<\\\\frac{161}{{\\\\varepsilon }}\\\\log (4/{\\\\varepsilon }),$ and therefore $\\\\delta >\\\\frac{1}{{\\\\cal T}(c_0,2s_0+3)}>\\\\frac{1}{{\\\\cal T}(\\\\lfloor 1/{\\\\varepsilon }\\\\rfloor +2s_0+3)}\\\\ge \\\\frac{1}{{\\\\cal T}(\\\\lfloor \\\\frac{161}{{\\\\varepsilon }}\\\\log (4/{\\\\varepsilon })\\\\rfloor )}.$\"\n ],\n [\n \"Appendix: Basic facts and estimates on continued fractions\",\n \"We will prove here some elementary facts on continued fractions used in the previous sections.\",\n \"We refer to [CF] for the facts used but not proved here.\",\n \"Let $x = [a_0;a_1,a_2,a_3,\\\\dots ]$ be a real number, and $(\\\\frac{p_n}{q_n})_{n \\\\in \\\\mathbb {N}}, \\\\frac{p_n}{q_n}= [a_0;a_1,a_2,\\\\dots ,a_n]$ its sequence of convergents.\",\n \"We have, for every $n\\\\in {\\\\mathbb {N}}$ , $p_{n+1}q_n-p_nq_{n+1}=(-1)^n$ , $x=[a_0; a_1, a_2, \\\\dots , a_n, {\\\\alpha }_{n+1}]=\\\\frac{\\\\alpha _{n+1}p_n+p_{n-1}}{\\\\alpha _{n+1}q_n+q_{n-1}},$ and so $\\\\alpha _{n+1}=\\\\frac{p_{n-1}-q_{n-1}x}{q_{n}x-p_{n}}$ and $x-\\\\frac{p_n}{q_n}= \\\\frac{(-1)^n}{(\\\\alpha _{n+1}+\\\\beta _{n+1})q_n^2},$ where $\\\\beta _{n+1}= \\\\frac{q_{n-1}}{q_n}= [0;a_n,a_{n-1},a_{n-2},\\\\dots ,a_1].$ In particular, $ \\\\left| x-\\\\frac{p_n}{q_n} \\\\right|=\\\\frac{1}{(\\\\alpha _{n+1}+\\\\beta _{n+1})q_n^2}.\",\n \"$ Recall that, given a finite sequence ${\\\\alpha }=(a_1,a_2,\\\\dots ,a_n)\\\\in ({\\\\mathbb {N}}^*)^n$ , we define its size by $s({\\\\alpha }):=|I({\\\\alpha })|$ , where $I({\\\\alpha })$ is the interval $\\\\lbrace x\\\\in [0,1] \\\\mid x=[0; a_1,a_2,\\\\dots ,a_n,{\\\\alpha }_{n+1}], {\\\\alpha }_{n+1}\\\\ge 1\\\\rbrace $ , whose endpoints are $p_n/q_n$ and $\\\\frac{p_n+p_{n-1}}{q_n+q_{n-1}}$ , $r({\\\\alpha })=\\\\lfloor \\\\log s({\\\\alpha })^{-1} \\\\rfloor $ and, for $r \\\\in {\\\\mathbb {N}}, P_r= \\\\lbrace {\\\\alpha }=(a_1,a_2,\\\\dots ,a_n) \\\\mid r({\\\\alpha }) \\\\ge r$ and $r((a_1,a_2,\\\\dots ,a_{n-1}))\\\\frac{1}{(T+1)(T+2)q_n^2},$ where $q_n=q_n(a_1, a_2,\\\\ldots , a_n)$ .\",\n \"Proof: Let $x=[a_0;a_1, a_2,\\\\ldots ,a_n, a_{n+1}, a_{n+2}, \\\\ldots ]$ and $y=[a_0;a_1, a_2,\\\\ldots ,a_{n-1}, b_n, b_{n+1}, b_{n+2}, \\\\ldots ]$ .\",\n \"We have $x=\\\\frac{\\\\alpha _{n}(x)p_{n-1}+p_{n-2}}{\\\\alpha _{n}(x)q_{n-1}+q_{n-2}}, y=\\\\frac{\\\\alpha _{n}(y)p_{n-1}+p_{n-2}}{\\\\alpha _{n}(y)q_{n-1}+q_{n-2}},$ and so $|x-y|=\\\\left|\\\\frac{\\\\alpha _{n}(x)p_{n-1}+p_{n-2}}{\\\\alpha _{n}(x)q_{n-1}+q_{n-2}}-\\\\frac{\\\\alpha _{n}(y)p_{n-1}+p_{n-2}}{\\\\alpha _{n}(y)q_{n-1}+q_{n-2}}\\\\right|=\\\\left|\\\\frac{(\\\\alpha _{n}(x)-\\\\alpha _{n}(y))(p_{n-1}q_{n-2}-p_{n-2}q_{n-1})}{(\\\\alpha _{n}(x)q_{n-1}+q_{n-2})(\\\\alpha _{n}(y)q_{n-1}+q_{n-2})}\\\\right|$ $=\\\\left|\\\\frac{(\\\\alpha _{n}(x)-\\\\alpha _{n}(y))(-1)^{n-2}}{(\\\\alpha _{n}(x)q_{n-1}+q_{n-2})(\\\\alpha _{n}(y)q_{n-1}+q_{n-2})}\\\\right|=\\\\frac{|\\\\alpha _{n}(x)-\\\\alpha _{n}(y)|}{(\\\\alpha _{n}(x)q_{n-1}+q_{n-2})(\\\\alpha _{n}(y)q_{n-1}+q_{n-2})}.$ We have $\\\\lfloor \\\\alpha _n(x) \\\\rfloor =a_n$ , $\\\\lfloor \\\\alpha _n(y) \\\\rfloor =b_n$ and $a_n\\\\ne b_n$ , so $|\\\\alpha _{n}(x)-\\\\alpha _{n}(y)|>[1;T+1]-[0;1,T+1]=\\\\frac{1}{T+1}+\\\\frac{1}{T+2}>\\\\frac{2}{T+2}$ .\",\n \"Moreover, $\\\\alpha _{n}(x)q_{n-1}+q_{n-2}<(1+a_n)q_{n-1}+q_{n-2}=q_n+q_{n-1}<2q_n$ and $\\\\alpha _{n}(y)q_{n-1}+q_{n-2}<\\\\alpha _{n}(y)(q_{n-1}+q_{n-2})\\\\le \\\\alpha _{n}(y)q_n<(T+1)q_n$ , and so $|x-y|>\\\\frac{2}{T+2} \\\\cdot \\\\frac{1}{2q_n\\\\cdot (T+1)q_n}=\\\\frac{1}{(T+1)(T+2)q_n^2}.\\\\Box $ Lemma A2.\",\n \"If $\\\\alpha =a_1a_2\\\\cdots a_m$ and $\\\\beta =b_1b_2\\\\cdots b_n$ are finite words, then $\\\\dfrac{1}{2}s(\\\\alpha )s(\\\\beta )&q_m(\\\\alpha )q_n(\\\\beta )+q_m(\\\\alpha )q_{n-1}(\\\\beta )+\\\\\\\\&&q_{m-1}(\\\\alpha )q_n(\\\\beta )+q_{m-1}(\\\\alpha )q_{n-1}(\\\\beta )\\\\\\\\\\\\iff \\\\hspace{14.22636pt} q_m(\\\\alpha )q_n(\\\\beta )+q_m(\\\\alpha )q_{n-1}(\\\\beta )&>&q_{m-1}(\\\\alpha )q_n(\\\\beta )+q_{m-1}(\\\\alpha )q_{n-1}(\\\\beta ),$ which is obviously true.\",\n \"For the other inequality, proceed analogously: $s(\\\\alpha \\\\beta )&=&\\\\dfrac{1}{q_{m+n}(\\\\alpha \\\\beta )\\\\left[q_{m+n}(\\\\alpha \\\\beta )+q_{m+n-1}(\\\\alpha \\\\beta )\\\\right]}\\\\\\\\&>&\\\\dfrac{1}{2}\\\\cdot \\\\dfrac{1}{q_m(\\\\alpha )q_n(\\\\beta )\\\\left[q_{m+n}(\\\\alpha \\\\beta )+q_{m+n-1}(\\\\alpha \\\\beta )\\\\right]}\\\\\\\\&>&\\\\dfrac{1}{2}\\\\cdot \\\\dfrac{1}{q_m(\\\\alpha )q_n(\\\\beta )\\\\left[q_m(\\\\alpha )+q_{m-1}(\\\\alpha )\\\\right]\\\\left[q_n(\\\\beta )+q_{n-1}(\\\\beta )\\\\right]}\\\\,,$ as $q_{m+n}(\\\\alpha \\\\beta )+q_{m+n-1}(\\\\alpha \\\\beta )&<&\\\\left[q_m(\\\\alpha )+q_{m-1}(\\\\alpha )\\\\right]\\\\left[q_n(\\\\beta )+q_{n-1}(\\\\beta )\\\\right]\\\\\\\\\\\\iff \\\\hspace{5.69046pt} q_{m-1}(\\\\alpha ){\\\\tilde{q}}_{n-1}(\\\\beta )+q_{m-1}(\\\\alpha ){\\\\tilde{q}}_{n-2}(\\\\beta )&<&q_{m-1}(\\\\alpha )q_n(\\\\beta )+q_{m-1}(\\\\alpha )q_{n-1}(\\\\beta )\\\\\\\\\\\\iff \\\\hspace{78.24507pt}{\\\\tilde{q}}_{n-1}(\\\\beta )+{\\\\tilde{q}}_{n-2}(\\\\beta )&<&q_n(\\\\beta )+q_{n-1}(\\\\beta ),$ where ${\\\\tilde{q}}_{n-1}(\\\\beta )=q_{n-1}(b_2b_3\\\\cdots b_n)$ and ${\\\\tilde{q}}_{n-2}(\\\\beta )=q_{n-2}(b_2b_3\\\\cdots b_{n-1})$ , and the last inequality is true, since we have ${\\\\tilde{q}}_{n-1}(\\\\beta )((T+1)^2e^r)^{-1}$ .\",\n \"Proof: We have $r(a_1,a_2,\\\\dots ,a_{n-1})e^{-r}$ , and thus $s({\\\\alpha })^{-1}=q_n(q_n+q_{n-1})\\\\le (Tq_{n-1}+q_{n-2})((T+1)q_{n-1}+q_{n-2})<(T+1)q_{n-1}\\\\cdot (T+1)(q_{n-1}+q_{n-2})=(T+1)^2q_{n-1}(q_{n-1}+q_{n-2})<(T+1)^2e^r$ , so $s({\\\\alpha })>((T+1)^2e^r)^{-1}$ .\",\n \"$\\\\Box $ Proposition A1.\",\n \"Let $r, k$ be positive integers and $\\\\alpha _i, 1\\\\le i \\\\le k$ finite sequences which belong to $P_r$ and whose elements are bounded by $T$ .\",\n \"Then, if ${\\\\alpha }={\\\\alpha }_1{\\\\alpha }_2\\\\cdots {\\\\alpha }_k$ , we have $s(\\\\alpha )>(2(T+1)^2 e^r)^{-k}$ .\",\n \"Proof: For $1\\\\le i\\\\le k$ we have, by lemma A3, $s({\\\\alpha }_i)>((T+1)^2e^r)^{-1}$ .\",\n \"So, using the lemma A2, we get $s({\\\\alpha })=s({\\\\alpha }_1{\\\\alpha }_2\\\\cdots {\\\\alpha }_k)>\\\\left(\\\\frac{1}{2}\\\\right)^{k-1}s({\\\\alpha }_1)s({\\\\alpha }_2)\\\\cdots s({\\\\alpha }_k)>$ $>\\\\left(\\\\frac{1}{2}\\\\right)^{k-1}(((T+1)^2e^r)^{-1})^k>(2(T+1)^2 e^r)^{-k}.\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\,\\\\, \\\\Box $ References [B] Bugeaud, Y.\",\n \"- Sets of exact approximation order by rational numbers.\",\n \"II - Unif.\",\n \"Distrib.\",\n \"Theory v. 3 , n. 2, p. 9-20, 2008.\",\n \"[CF] Cusick, T. and Flahive, M. - The Markov and Lagrange spectra - Mathematical Surveys and Monographs, n. 30 - AMS.\",\n \"[Fa] Falconer, K. -The geometry of fractal sets - Cambridge Tracts in Mathematics, 85.\",\n \"Cambridge University Press, Cambridge, 1986.\",\n \"[FMM] Ferenczi, S., Mauduit, C. and Moreira, C.G.\",\n \"- An algorithm for the word entropy - https://arxiv.org/abs/1803.05533 [F] Freiman, G.A.\",\n \"- Diofantovy priblizheniya i geometriya chisel (zadacha Markova) [Diophantine approximation and geometry of numbers (the Markov spectrum)], Kalininskii Gosudarstvennyi Universitet, Kalinin, 1975.\",\n \"[H] Hall, M. - On the sum and products of continued fractions - Annals of Math.\",\n \"v. 48, p. 966-993, 1947.\",\n \"[J] Jarník, V. - Zur metrischen Theorie der diophantischen Approximationen - Prace Mat.-Fiz.\",\n \"36, p. 91-106, 1928.\",\n \"[Ma] Markov, A.\",\n \"- Sur les formes quadratiques binaires indéfinies, Math.\",\n \"Ann.\",\n \"v. 15, p. 381-406, 1879.\",\n \"[MM1] Matheus, C. and Moreira, C.G., $HD(M\\\\setminus L) > 0.353$ - https://arxiv.org/abs/1703.04302 [MM2] Matheus, C. and Moreira, C.G., $HD(M\\\\setminus L) < 0.986927$ - https://arxiv.org/abs/1708.06258 [MM3] Matheus, C. and Moreira, C.G., Markov spectrum near Freiman's isolated points in $M\\\\setminus L$ - https://arxiv.org/abs/1802.02454 [MM4] Matheus, C. and Moreira, C.G., New numbers in $M\\\\setminus L$ beyond $\\\\sqrt{12}$ : solution to a conjecture of Cusick.\",\n \"https://arxiv.org/abs/1803.01230 [Mo] Moreira, C.G.\",\n \"- Geometric properties of images of cartesian products of regular Cantor sets by differentiable real maps.\",\n \"https://arxiv.org/abs/1611.00933 [MY] Moreira, C.G.\",\n \"and Yoccoz, J.-C. - Stable intersections of regular Cantor sets with large Hausdorff Dimensions - Annals of Math.\",\n \"v. 154, n.1, p. 45 - 96, 2001.\",\n \"[P] Perron, O.\",\n \"- Über die Approximation irrationaler Zahlen durch rationale II - S.-B.\",\n \"Heidelberg Akad.\",\n \"Wiss., Abh.\",\n \"8, 1921, 12 pp.\",\n \"[PT] Palis, J. and Takens, F. - Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Fractal dimensions and infinitely many attractors.\",\n \"Cambridge Studies in Advanced Mathematics, 35.\",\n \"Cambridge University Press, Cambridge, 1993. x+234 pp.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05782"}}},{"rowIdx":3393,"cells":{"text":{"kind":"list like","value":[["The geometric lattice of embedded subsets"],["Abstract This work proposes an alternative approach to the so-called lattice of embedded subsets, which is included in the product of the subset and partition lattices of a finite set, and whose elements are pairs consisting of a subset and a partition where the former is a block of the latter.","The lattice structure proposed in a recent contribution relies on ad-hoc definitions of both the join operator and the bottom element, while also including join-irreducible elements distinct from atoms.","Conversely, here embedded subsets obtain through a closure operator defined over the product of the subset and partition lattices, where elements are generic pairs of a subset and a partition.","Those such pairs that coincide with their closure are precisely embedded subsets, and since the Steinitz exchange axiom is also satisfied, what results is a geometric (hence atomic) lattice given by a simple matroid (or combinatorial geometry) included in the product of the subset and partition lattices (as the partition lattice itself is the polygon matroid defined on the edges of a complete graph).","By focusing on its M\\\"obius function, this geometric lattice of embedded subsets of a n-set is shown to be isomorphic to the lattice of partitions of a n+1-set."],["Introduction","For a finite set $N=\\lbrace 1,\\ldots ,n\\rbrace $ , the framework developed in the sequel obtains from both the lattice $(2^N,\\cap ,\\cup )$ of subsets of $N$ and the lattice $(\\mathcal {P}^N,\\wedge ,\\vee )$ of partitions of $N$ .","Generic subsets and partitions are denoted, respectively, by $A,B\\in 2^N$ and $P,Q\\in \\mathcal {P}^N$ .","Recall that a partition $P=\\lbrace A_1,\\ldots ,A_{|P|}\\rbrace $ is a family of (non-empty) pair-wise disjoint subsets $A_1,\\ldots ,A_{|P|}\\in 2^N$ , called blocks, whose union is $N$ .","As usual [2], [18], partitions are (partially) ordered by coarsening $\\geqslant $ , meaning that any $P,Q\\in \\mathcal {P}^N$ satisfy $P\\geqslant Q$ if every block $B\\in Q$ is included in a block $A\\in P$ , i.e.","$A\\supseteq B$ .","Hence the bottom partition is $P_{\\bot }=\\lbrace \\lbrace 1\\rbrace ,\\ldots ,\\lbrace n\\rbrace \\rbrace $ , while the top one is $P^{\\top }=\\lbrace N\\rbrace $ .","Denote by $X^N=2^N\\times \\mathcal {P}^N$ the product of the subset and partition lattices, whose elements $(A,P),(B,Q)\\in X^N$ are pairs consisting of a subset $A,B\\in 2^N$ and a partition $P,Q\\in \\mathcal {P}^N$ .","Lattice $(X^N,\\sqcap ,\\sqcup )$ is partially ordered by $\\sqsupseteq $ and naturally obtains through product of the meet, join and order relations for the subset and partition lattices.","That is to say, $(A,P)\\sqcap (B,Q)=(A\\cap B,P\\wedge Q)$ as well as $(A,P)\\sqcup (B,Q)=(A\\cup B,P\\vee Q)$ , while $(A,P)\\sqsupseteq (B,Q)\\Leftrightarrow A\\supseteq B,P\\geqslant Q$ , where $P\\wedge Q$ is the coarsest partition finer than both $P,Q$ and $P\\vee Q$ is the finest partition coarser than both $P,Q$ .","Thus the bottom element is $(\\emptyset ,P_{\\bot })$ while the top one is $(N,P^{\\top })$ .","The existing lattice $(\\mathfrak {C}(N)_{\\bot },\\sqcap ,\\sqcup _*)$ of embedded subsets has elements given by those pairs $(A,P)\\in X^N$ such that $A\\in P$ , together with “an artificial bottom element $\\perp $ , which is introduced for mathematical convenience and that may be considered as $(\\emptyset ,P_{\\bot })$” [6].","It is not hard to see that if $A,B\\in 2^N$ and $P,Q\\in \\mathcal {P}^N$ satisfy $A\\in P$ as well as $B\\in Q$ , then non-emptiness $A\\cap B\\ne \\emptyset $ entails $(A\\cap B,P\\wedge Q)\\in \\mathfrak {C}(N)_{\\bot }$ , since $A\\cap B$ is a block of $P\\wedge Q$ .","However, in general $A\\cup B$ is not a block of $P\\vee Q$ , i.e.","$(A\\cup B,P\\vee Q)\\notin \\mathfrak {C}(N)_{\\bot }$ .","For this reason, the existing lattice of embedded subsets $\\mathfrak {C}(N)_{\\bot }$ has the same meet $\\sqcap $ and order $\\sqsupseteq $ applying to product lattice $X^N$ , but the join $\\sqcup _*$ is different from $\\sqcup $ as detailed below.","The resulting structure [6] has $n$ atoms of the form $(i,P_{\\bot }),i\\in N$ together with additional $\\binom{n}{2}(n-2)$ join-irreducible elements [2], and thus the lattice is non-atomic (hence non-geometric) precisely because many elements cannot be decomposed as a join of atoms.","In fact, the only embedded subsets $(A,P)\\in \\mathfrak {C}(N)_{\\bot }$ admitting such a decomposition [6] are those where partition $P$ is the modular element [2], [15] of the partition lattice whose unique non-singleton block, if any, is precisely $A$ [2] [17].","This is denoted by $P=P^A_{\\bot }$ , with $P^A_{\\bot }=\\lbrace A,\\lbrace i_1\\rbrace ,\\ldots ,\\lbrace i_{n-|A|}\\rbrace \\rbrace $ and $\\lbrace i_1,\\ldots ,i_{n-|A|}\\rbrace =N\\backslash A=A^c$ .","Since $P^i_{\\bot }=P_{\\bot }$ for all $i\\in N$ and $P^N_{\\bot }=P^{\\top }$ , there are $2^n-n$ such modular partitions $P^A_{\\bot },\\emptyset \\ne A\\in 2^N$ .","They play a key role in the approach proposed here, which relies on a closure operator $cl:X^N\\rightarrow X^N$ over poset $(X^N,\\sqsupseteq )$ , meaning [2] that for all $(A,P),(B,Q)\\in X^N$ the following hold: (i) $cl(A,P)\\sqsupseteq (A,P)$ , (ii) $(A,P)\\sqsupseteq (B,Q)\\Rightarrow cl(A,P)\\sqsupseteq cl(B,Q)$ , (iii) $cl(cl(A,P))=cl(A,P)$ .","This closure is defined by $cl(\\emptyset ,P)=(\\emptyset ,P)$ for all $P\\in \\mathcal {P}^N$ , while for $\\emptyset \\ne A$ $cl(A,P)=(\\bar{A},P\\vee P^A_{\\bot })$ such that $A\\subseteq \\bar{A}\\in P\\vee P^A_{\\bot }$ .","In words, $\\bar{A}$ is the block of $P\\vee P^A_{\\bot }$ satisfying $\\bar{A}\\supseteq A$ , and $P^A_{\\bot }$ is the modular partition where the only non-singleton block, if any, is $A$ .","The remainder of this work is concerned with poset $(\\mathcal {E}^N,\\sqsupseteq )$ , with $\\mathcal {E}^N=\\lbrace (A,P):cl(A,P)=(A,P)\\rbrace $ consisting of those pairs that coincide with their closure.","As $\\perp $ corresponds to $(\\emptyset ,P_{\\bot })$ (see above), $\\mathfrak {C}(N)_{\\bot }\\subset \\mathcal {E}^N\\subset X^N$ , and in particular $\\mathcal {E}^N\\backslash \\mathfrak {C}(N)_{\\bot }=\\lbrace (\\emptyset ,P):P\\ne P_{\\bot }\\rbrace $ .","From the enumerative perspective, $|\\mathfrak {C}(N)_{\\bot }|=1+\\sum _{1\\le k\\le n}k\\mathcal {S}_{n,k}$ , where $\\mathcal {S}_{n,k}$ is the Stirling number of the second kind, i.e.","the number of partitions of a $n$ -set into $k$ blocks, while $|\\mathcal {E}^N|=\\sum _{1\\le k\\le n}(k+1)\\mathcal {S}_{n,k}=\\mathcal {B}_{n+1}$ , where $\\mathcal {B}_{n+1}$ is the Bell number, i.e.","the number of partitions of a $n+1$ -set [2], [7], [12].","The join obtained through closure $cl(\\cdot )$ coincides, over $\\mathfrak {C}(N)_{\\bot }$ , with the existing one $\\sqcup _*$ (defined in [6]), i.e.","for $(A,P),(B,Q)\\in \\mathfrak {C}(N)_{\\bot }$ , $cl((A,P)\\sqcup (B,Q))=(A,P)\\sqcup _*(B,Q).$ However, the novel enlarged lattice $\\mathcal {E}^N$ of embedded subsets is substantially different from the existing one $\\mathfrak {C}(N)_{\\bot }$ , as all its $n+\\binom{n}{2}=\\binom{n+1}{2}$ join-irreducible elements (apart from $(\\emptyset ,P_{\\bot })$ ) are atoms.","Hence $\\mathcal {E}^N$ is a geometric lattice with the same atoms as product lattice $X^N$ , and in fact the former reproduces within the latter a situation similar to the inclusion of partition lattice $(\\mathcal {P}^N,\\wedge ,\\vee )$ within Boolean lattice $(2^{N_2},\\cap ,\\cup )$ , where $N_2=\\lbrace \\lbrace i,j\\rbrace :1\\le i1$ .","The rank is $r(\\emptyset ,P^A_{\\bot })=n-(n-|A|+1)=|A|-1$ .","On the other hand, any $(B,Q)$ has rank $n-|Q|+1$ (since the case $B=\\emptyset $ is trivial precisely because $P^A_{\\bot }$ is a modular partition).","Thus $r(\\emptyset ,P^A_{\\bot })+r(B,Q)=n-|Q|+|A|$ .","Now consider that the meet has rank $r((\\emptyset ,P^A_{\\bot })\\sqcap (B,Q))=n-(|Q^A|+n-|A|)=|A|-|Q^A|$ , while the (closure of the) join yields $r(cl((\\emptyset ,P^A_{\\bot })\\sqcup (B,Q)))=n-(|Q|-|Q^A|+1)+1=n-|Q|+|Q^A|$ .","Therefore, $|A|-|Q^A|+n-|Q|+|Q^A|=n-|Q|+|A|$ as desired.","In the general case $(A,P^A_{\\bot })$ , with $r(A,P^A_{\\bot })=r(\\emptyset ,P^A_{\\bot })+1=|A|$ , the sum is $r(A,P^A_{\\bot })+r(B,Q)=n-|Q|+1+|A|$ .","The rank $r((A,P^A_{\\bot })\\sqcap (B,Q))$ of the meet equals $|A|-|Q^A|$ if $A\\cap B=\\emptyset $ and $|A|-|Q^A|+1$ if $A\\cap B\\ne \\emptyset $ .","Correspondingly, the rank $r(cl((A,P^A_{\\bot })\\sqcup (B,Q)))$ of the join equals $n-(|Q|-|Q^A|+1-1)+1=n-|Q|+|Q^A|+1$ if $A\\cap B=\\emptyset $ and $n-(|Q|-|Q^A|+1)+1=n-|Q|+|Q^A|$ if $A\\cap B\\ne \\emptyset $ .","Thus the sought sum is $|A|-|Q^A|+n-|Q|+|Q^A|+1=n-|Q|+1+|A|$ in the former case and $|A|-|Q^A|+1+n-|Q|+|Q^A|=n-|Q|+1+|A|$ in the latter, i.e.","the same.","Turning to the ‘only if’ part, it is not difficult to check that, just like the rank of partitions, the rank of embedded subsets is submodular, that is to say $r(A,P)+r(B,Q)\\ge r(cl((A,P)\\sqcup (B,Q)))+r((A,P)\\sqcap (B,Q))$ , with strict inequality whenever $(A,P),(B,Q)$ is not a modular pair [2] of lattice elements.","Let $N_+=\\lbrace 1,\\ldots ,n,n+1\\rbrace $ , i.e.","$N_+=N\\cup \\lbrace n+1\\rbrace $ and denote by $\\mathcal {P}^{N_+}$ the lattice of partitions of $N_+$ , with generic element denoted by $P^+\\in \\mathcal {P}^{N_+}$ .","From Section 1, $|\\mathcal {E}^N|=\\mathcal {B}_{n+1}=|\\mathcal {P}^{N_+}|$ .","In fact, there is a lattice isomorphism $f:\\mathcal {E}^N\\rightarrow \\mathcal {P}^{N_+}$ , meaning that for all $(A,P),(B,Q)\\in \\mathcal {E}^N$ the following holds: $f((A,P)\\sqcap (B,Q))&=&f(A,P)\\wedge f(B,Q)\\text{,}\\\\f(cl((A,P)\\sqcup (B,Q)))&=&f(A,P)\\vee f(B,Q)\\text{.","}$ To see this, consider that any partition $\\lbrace A_1,\\ldots ,A_{|P|}\\rbrace =P\\in \\mathcal {P}^N$ has associated $|P|+1$ partitions $P^+_1,\\ldots ,P^+_{|P|},P^+_{|P|+1}\\in \\mathcal {P}^{N_+}$ , all inducing the same partition $P$ of $N$ .","The first $|P|$ ones $P^+_1,\\ldots ,P^+_{|P|}$ obtain each by placing additional element $n+1$ into block $A_k\\in P,1\\le k\\le |P|$ , while the last one $P^+_{|P|+1}$ obtains by placing $n+1$ into a singleton block $\\lbrace n+1\\rbrace \\in P^+_{|P|+1}$ .","Of course, the former are pair-wise incomparable, i.e.","$P^+_k\\lnot \\geqslant P^+_{k^{\\prime }}\\lnot \\geqslant P^+_k$ for $1\\le kP^+_{|P|+1},1\\le k\\le |P|$ .","The same applies to embedded subsets $(A_1,P),\\ldots ,(A_{|P|},P),(\\emptyset ,P)\\in \\mathcal {E}^N$ , in that $(A_k,P)\\lnot \\sqsupseteq (A_{k^{\\prime }},P)\\lnot \\sqsupseteq (A_k,P)$ for $1\\le k1$ for at least one $A\\in P$ .","Segment (or interval) $[Q,P]=\\lbrace P^{\\prime }:Q\\leqslant P^{\\prime }\\leqslant P\\rbrace $ is isomorphic to product $\\times _{A\\in P}\\mathcal {P}^{k_A}$ , where $\\mathcal {P}^k$ denotes the lattice of partitions of a $k$ -set.","Let $c^{QP}_k=|\\lbrace A:A\\in P,k_A=k\\rbrace |,1\\le k\\le n$ be the number of blocks of $P$ obtained as the union of $k$ blocks of $Q$ , where $c^{QP}=(c^{QP}_1,\\ldots ,c^{QP}_n)\\in \\mathbb {Z}_+^n$ is the class (or type) of segment $[Q,P]$ .","Then [13], $\\mu _{\\mathcal {P}^N}(Q,P)=(-1)^{-n+\\sum _{1\\le k\\le n}c^{QP}_k}\\prod _{10$ for all $n\\ge 0$ , which is consistent with [3]."],["Möbius algebra, vector subspaces and games","The purpose of this section is to highlight that all cooperative games are in fact elements of subspaces of the free vector space $V(\\mathcal {E}^N)$ over $\\mathbb {R}$ generated by $\\mathcal {E}^N$ [1], [2].","That is, the elements of $V(\\mathcal {E}^N)\\subset \\mathbb {R}^{\\mathcal {B}_{n+1}}$ are real-valued lattice functions $g:\\mathcal {E}^N\\rightarrow \\mathbb {R}$ .","Recall that, for $N$ regarded as a player set, cooperative game theory deals with three types of lattice functions: coalitional games [14] or set functions $v:2^N\\rightarrow \\mathbb {R}_+,v(\\emptyset )=0$ , global games [4] or partition functions $h:\\mathcal {P}^N\\rightarrow \\mathbb {R}_+,h(P_{\\bot })=0$ , games in partition function form [6], [10], [19], hereafter referred to as PFF games for short, or functions $f:\\mathfrak {C}(N)_{\\bot }\\rightarrow \\mathbb {R}_+,f(\\emptyset ,P_{\\bot })=0$ .","A further fourth type of cooperative games may be defined, which incorporates all previous ones $(a)$ , $(b)$ and $(c)$ , namely extended PFF games, or those elements $g\\in V(\\mathcal {E}^N)$ of the free vector space under concern such that $g:\\mathcal {E}^N\\rightarrow \\mathbb {R}_+,g(\\emptyset ,P_{\\bot })=0$ .","Together with the Möbius function, the other element of the incidence algebra (of locally finite posets [2] [13]) playing a fundamental role in cooperative game theory is the zeta function $\\zeta $ , as it provides the so-called ‘unanimity games’ [14].","In the incidence algebra of $\\mathcal {E}^N$ over $\\mathbb {R}$ , the zeta function $\\zeta _{\\mathcal {E}^N}:\\mathcal {E}^N\\times \\mathcal {E}^N\\rightarrow \\lbrace 0,1\\rbrace $ is defined on ordered pairs of embedded subsets by $\\zeta _{\\mathcal {E}^N}((B,Q),(A,P))=\\left\\lbrace \\begin{array}{c} 1\\text{ if }(A,P)\\sqsupseteq (B,Q)\\text{,}\\\\0\\text{ otherwise.}","\\end{array}\\right.$ In these terms, unanimity (coalitional) games $u_B,\\emptyset \\ne B\\in 2^N$ are defined by $u_B(A)=\\zeta _{2^N}(B,A)$ for all subsets (or coalitions) $A\\in 2^N$ , where $\\zeta _{2^N}$ is the zeta function in the incidence algebra of $2^N$ .","They are linearly independendent elements of the vector space of coalitional games and thus form a basis.","This fact is essential since the very beginning of cooperative game theory.","In the present setting, for $(B,Q)\\in \\mathcal {E}^N$ , let $\\zeta ^{B,Q}(A,P)=\\zeta _{\\mathcal {E}^N}((B,Q),(A,P))$ for all $(A,P)\\in \\mathcal {E}^N$ .","Then, $\\lbrace \\zeta ^{B,Q}:(B,Q)\\in \\mathcal {E}^N\\rbrace $ is a basis of $V(\\mathcal {E}^N)$ .","Also denote by $V(\\mathcal {E}^N)_{\\bot }\\subset \\mathbb {R}^{\\mathcal {B}_{n+1}-1}_+$ the subspace of extended PFF games.","Furthermore, let the Möbius inversion of $g$ be $\\mu ^g:\\mathcal {E}^N\\rightarrow \\mathbb {R}$ , i.e.","$\\mu ^g(A,P)&=&\\sum _{\\underset{(B,Q)\\in \\mathcal {E}^N}{(B,Q)\\sqsubseteq (A,P)}}\\mu _{\\mathcal {E}^N}((B,Q),(A,P))g(B,Q)=\\\\&=&g(A,P)-\\sum _{\\underset{(B,Q)\\in \\mathcal {E}^N}{(B,Q)\\sqsubset (A,P)}}\\mu ^g(B,Q)\\text{,}$ with $\\mu ^g(\\emptyset ,P_{\\bot })=g(\\emptyset ,P_{\\bot })$ ($=0$ if $g\\in V(\\mathcal {E}^N)_{\\bot }$ ), and where $\\mu ^{\\zeta ^{B,Q}}(A,P)=\\left\\lbrace \\begin{array}{c} 1\\text{ if }(A,P)=(B,Q)\\\\0\\text{ otherwise} \\end{array}\\right.$ for every element $\\zeta ^{B,Q}$ of the chosen basis, thereby providing the canonical basis (or Kronecker delta) elements (see [2], while crucially noticing that the alternative definition $\\zeta ^{B,Q}(A,P)=\\zeta _{\\mathcal {E}^N}((A,P),(B,Q))$ applies there.)","Specifically, let $\\epsilon _{B,Q}\\in V(\\mathcal {E}^N)$ be the $\\mathcal {B}_{n+1}$ -vector with entries indexed by elements $(A,P)\\in \\mathcal {E}^N$ and such that its unique non-zero entry is the $(B,Q)$ one, which equals 1.","Then the Möbius algebra of $\\mathcal {E}^N$ over $\\mathbb {R}$ (denoted by Möb$(\\mathcal {E}^N)$ , see [1] [2]) is defined by the system of orthogonal idempotents $\\lbrace \\epsilon _{B,Q}:(B,Q)\\in \\mathcal {E}^N\\rbrace $ and linear extension to all of $V(\\mathcal {E}^N)$ through multiplication $\\epsilon _{B,Q}\\cdot \\epsilon _{A,P}=\\left\\lbrace \\begin{array}{c} 1\\text{ if }(A,P)=(B,Q)\\text{,}\\\\0\\text{ otherwise.}","\\end{array}\\right.$ In terms of the above basis $\\lbrace \\zeta ^{B,Q}:(B,Q)\\in \\mathcal {E}^N\\rbrace $ of $V(\\mathcal {E}^N)$ , this multiplication yields $\\zeta ^{B,Q}\\cdot \\zeta ^{A,P}=\\zeta ^{cl((A,P)\\sqcup (B,Q))}$ (see [2], crucially noticing again that since the alternative definition of the basis detailed above applies there, then $\\zeta ^{B,Q}\\cdot \\zeta ^{A,P}=\\zeta ^{(A,P)\\sqcap (B,Q)}$ ).","It is well known that Möbius inversion $\\mu ^g$ quantifies precisely the coefficients identifying functions $g\\in V(\\mathcal {E}^N)$ as linear combinations of the basis elements, that is $g(A,P)=\\sum _{(B,Q)\\in \\mathcal {E}^N}\\mu ^g(B,Q)\\zeta ^{B,Q}(A,P)\\text{ for all }(A,P)\\in \\mathcal {E}^N\\text{.","}$ If $Y\\subset \\mathcal {E}^N$ and $\\mu ^g(A,P)=0$ for all $(A,P)\\in \\mathcal {E}^N\\backslash Y$ , then Möbius inversion $\\mu ^g$ may be said to live only $Y$ .","In these terms, the following observations are immediate: (traditional) PFF games are elements of the subspace $\\hat{V}(\\mathcal {E}^N)\\subset \\mathbb {R}^{\\mathcal {B}_{n+1}-\\mathcal {B}_n}$ of $V(\\mathcal {E}^N)$ consisting of those extended PFF games $g\\in V(\\mathcal {E}^N)$ whose Möbius inversion lives only on embedded subsets $(A,P)$ such that $A\\ne \\emptyset $ , i.e.","$A=\\emptyset \\Rightarrow \\mu ^g(A,P)=0$ , entailing $g(\\emptyset ,P)=0$ for all $P\\in \\mathcal {P}^N$ ; global games are elements of the subspace $\\tilde{V}(\\mathcal {E}^N)\\subset \\mathbb {R}^{\\mathcal {B}_n-1}$ consisting of extended PFF games $g\\in V(\\mathcal {E}^N)$ whose Möbius inversion lives only on embedded subsets $(A,P)$ such that $A=\\emptyset $ and $P\\ne P_{\\bot }$ , entailing $g(A,P)=g(\\emptyset ,P)$ for all $A\\in P$ (thereby somehow strengthening the idea of a global level of satisfaction, common to all players, see [4]); coalitional games are elements of the subspace $\\bar{V}(\\mathcal {E}^N)\\subset \\mathbb {R}^{2^n-1}$ consisting of those $g\\in V(\\mathcal {E}^N)$ whose Möbius inversion lives only on the $2^n-1$ modular elements $(A,P^A_{\\bot })$ of $\\mathcal {E}^N$ defined in Section 3 such that $A\\ne \\emptyset $ , yielding $g(\\emptyset ,P)=0$ for all $P\\in \\mathcal {P}^N$ as well as $g(A,P)=g(A,P^A_{\\bot })$ for all $P\\in \\mathcal {P}^N$ such that $A\\in P$ .","Another subspace of $V(\\mathcal {E}^N)$ appearing in cooperative game theory is given by the so-called “additively separable” global games (or partition functions, see [4], [5]), namely those $h:\\mathcal {P}^N\\rightarrow \\mathbb {R}_+$ such that $h(P)=h_v(P):=\\sum _{A\\in P}v(A)$ for some coalitional game (or set function) $v:2^N\\rightarrow \\mathbb {R}_+$ .","The Möbius inversion of these partition functions lives only on the $2^n-n$ modular elements $P^A_{\\bot }$ of the partition lattice, see Section 1.","This is detailed hereafter (see also [4], [5] and [11]).","Proposition 5 If $h=h_v$ and $\\sum _{i\\in N}v(\\lbrace i\\rbrace )=h_v(P_{\\bot })>0$ , then a continuuum of set functions $w:2^N\\rightarrow \\mathbb {R}_+,w\\ne v$ satisfies $h=h_w$ .","Proof: By direct substitution, the Möbius inversion $\\mu ^{h_v}:\\mathcal {P}^N\\rightarrow \\mathbb {R}$ of partition function $h_v$ satisfies $\\mu ^{h_v}(P)=\\sum _{A\\in P}\\sum _{B\\subseteq A}v(B)\\sum _{Q\\leqslant P:B\\in Q}\\mu _{\\mathcal {P}^N}(Q,P)\\text{ for all }P\\in \\mathcal {P}^N\\text{.","}$ If $P\\ne P^A_{\\bot }$ , then the recursion for Möbius function $\\mu _{\\mathcal {P}^N}:\\mathcal {P}^N\\times \\mathcal {P}^N\\rightarrow \\mathbb {R}$ yields $\\sum _{Q\\leqslant P:A\\in Q}\\mu _{\\mathcal {P}^N}(Q,P)=\\sum _{P^A_{\\bot }\\leqslant Q\\leqslant P}\\mu _{\\mathcal {P}^N}(Q,P)=0\\text{,}$ and the same for proper subsets $B\\subset A$ .","Möbius inversion $\\mu ^{h_v}$ thus lives only on modular partitions, where it obtains recursively by $\\mu ^{h_v}(P_{\\bot })=\\sum _{i\\in N}v(\\lbrace i\\rbrace )$ and $\\mu ^{h_v}(P_{\\bot }^A)=\\mu ^v(A)$ for $1<|A|\\le n$ , with $P^N_{\\bot }=P^{\\top }$ .","Hence any $w\\ne v$ satisfying $\\sum _{i\\in N}v(\\lbrace i\\rbrace )=\\sum _{i\\in N}w(\\lbrace i\\rbrace )$ and $\\mu ^v(A)=\\mu ^w(A)$ for all $A\\in 2^N$ such that $|A|>1$ also additively separates $h$ , i.e.","$h_v=h_w$ .","If $\\sum _{i\\in N}v(\\lbrace i\\rbrace )>0$ , then all $w:2^N\\rightarrow \\mathbb {R}_+$ such that $h_v=h_w$ may be chosen from within a $n-1$ -dimensional simplex $\\Delta \\subset \\mathbb {R}^n_+$ .","However, if $\\sum _{i\\in N}v(\\lbrace i\\rbrace )=0$ , then there is no $w:2^N\\rightarrow \\mathbb {R}_+$ such that $h_v=h_w,v\\ne w$ .","Nevertheless, this technical distinction plays no role as soon as attention is placed on generic partition and set functions $h:\\mathcal {P}^N\\rightarrow \\mathbb {R},w:2^N\\rightarrow \\mathbb {R}$ , i.e.","not required to take only non-negative real values.","In any case, in view of observation $(ii)$ above, it is evident that additively separable partition functions may be regarded as elements of the subspace of $V(\\mathcal {E}^N)$ consisting of those extended PFF games $g$ with Möbius inversion $\\mu ^g$ such that $\\mu ^g(B,Q)\\ne 0$ only if $B=\\emptyset $ and $Q$ is a modular partition.","Developing from additively separable partition functions, now consider those $g\\in V(\\mathcal {E}^N)$ with Möbius inversion $\\mu ^g$ living only on the $2^{n+1}-(n+1)$ modular elements of $\\mathcal {E}^N$ (see Section 3).","Proposition 6 If $g\\in V(\\mathcal {E}^N)$ has Möbius inversion $\\mu ^g$ living only on modular elements of $\\mathcal {E}^N$ , then there are set functions $v,w:2^N\\rightarrow \\mathbb {R},v(\\emptyset )=0$ such that $g(A,P)=v(A)+\\sum _{B\\in P}w(B)\\text{ for all }(A,P)\\in \\mathcal {E}^N\\text{.","}$ Proof: Under the above conditions, and given expression (5) above, $g(A,P)=\\mu ^g(\\emptyset ,P_{\\bot })+\\sum _{B\\in P}\\sum _{\\emptyset \\ne B^{\\prime }\\subseteq B}\\mu ^g(\\emptyset ,P^{B^{\\prime }}_{\\bot })+\\sum _{\\emptyset \\ne A^{\\prime }\\subseteq A}\\mu ^g(A^{\\prime },P^{A^{\\prime }}_{\\bot })$ for all $(A,P)\\in \\mathcal {E}^N$ .","In view of Proposition 5, $\\mu ^g(\\emptyset ,P_{\\bot })+\\sum _{B\\in P}\\sum _{\\emptyset \\ne B^{\\prime }\\subseteq B}\\mu ^g(\\emptyset ,P^{B^{\\prime }}_{\\bot })=\\sum _{B\\in P}w(B)$ for any set function $w$ with Möbius inversion $\\mu ^w$ such that $\\mu ^g(\\emptyset ,P_{\\bot })=\\sum _{i\\in N}w(\\lbrace i\\rbrace )\\text{ and }\\mu ^g(\\emptyset ,P^{B^{\\prime }}_{\\bot })=\\mu ^w(B^{\\prime })\\text{ for }B^{\\prime }\\in 2^N,|B^{\\prime }|>1\\text{.","}$ On the other hand, $\\sum _{\\emptyset \\ne A^{\\prime }\\subseteq A}\\mu ^g(A^{\\prime },P^{A^{\\prime }}_{\\bot })=v(A)=\\sum _{A^{\\prime }\\subseteq A}\\mu ^v(A^{\\prime })$ for any set function $v$ with Möbius inversion $\\mu ^v$ satisfying $\\mu ^g(A^{\\prime },P^{A^{\\prime }}_{\\bot })=\\mu ^v(A^{\\prime })$ for $A^{\\prime }\\in 2^N$ , $|A^{\\prime }|>0$ and $\\mu ^v(\\emptyset )=v(\\emptyset )=0$ .","Like for additively separable partition functions, if the Möbius inversion of extended PFF games lives only on the modular elements of the lattice, then all the values taken on embedded subsets can be recovered from only two set functions, out of which one is unique while the other can be chosen from within a continuum (in view of Proposition 5)."],["Concluding remarks","This paper essentially proposes to look at the so-called lattice of embedded subsets from a wider perspective, such that the novel resulting structure is a combinatorial geometry obtained through a closure operator that satisfies the Steinitz exchange axiom.","With respect to the lattice previously proposed in [6], the geometric one presented here additionally includes all elements given by a (non-bottom) partition paired with the empty set, and this evidently yields a most natural lattice embedding, i.e.","of $\\mathcal {P}^N$ into $\\mathcal {E}^N$ .","However, from this perspective what seems mostly important is that the geometric lattice of embedded subsets of a $n$ set is isomorphic to the lattice of partitions of a $n+1$ -set.","In general, this enables to re-obtain a variety of results applying to partitions, ranging from the characteristic polynomial to supersolvability.","Since embedded subsets, or embedded coalitions, were firstly used as modeling tools in cooperative game theory, a meaningful direction for investigation focuses on the free vector space of functions taking real values on lattice elements.","In this view, a novel type of games has been defined, namely extended PFF games, which englobe all existing cooperative games as elements of suitable vector subspaces.","But perhaps most importantly, the geometric lattice of embedded subsets yields that all existing cooperative games are real-valued functions defined on atomic lattices, and this may have fundamental implications for the solution concept (i.e.","how to share the fruits of cooperation) associated with these games.","This shall be addressed in future work."]],"string":"[\n [\n \"The geometric lattice of embedded subsets\"\n ],\n [\n \"Abstract This work proposes an alternative approach to the so-called lattice of embedded subsets, which is included in the product of the subset and partition lattices of a finite set, and whose elements are pairs consisting of a subset and a partition where the former is a block of the latter.\",\n \"The lattice structure proposed in a recent contribution relies on ad-hoc definitions of both the join operator and the bottom element, while also including join-irreducible elements distinct from atoms.\",\n \"Conversely, here embedded subsets obtain through a closure operator defined over the product of the subset and partition lattices, where elements are generic pairs of a subset and a partition.\",\n \"Those such pairs that coincide with their closure are precisely embedded subsets, and since the Steinitz exchange axiom is also satisfied, what results is a geometric (hence atomic) lattice given by a simple matroid (or combinatorial geometry) included in the product of the subset and partition lattices (as the partition lattice itself is the polygon matroid defined on the edges of a complete graph).\",\n \"By focusing on its M\\\\\\\"obius function, this geometric lattice of embedded subsets of a n-set is shown to be isomorphic to the lattice of partitions of a n+1-set.\"\n ],\n [\n \"Introduction\",\n \"For a finite set $N=\\\\lbrace 1,\\\\ldots ,n\\\\rbrace $ , the framework developed in the sequel obtains from both the lattice $(2^N,\\\\cap ,\\\\cup )$ of subsets of $N$ and the lattice $(\\\\mathcal {P}^N,\\\\wedge ,\\\\vee )$ of partitions of $N$ .\",\n \"Generic subsets and partitions are denoted, respectively, by $A,B\\\\in 2^N$ and $P,Q\\\\in \\\\mathcal {P}^N$ .\",\n \"Recall that a partition $P=\\\\lbrace A_1,\\\\ldots ,A_{|P|}\\\\rbrace $ is a family of (non-empty) pair-wise disjoint subsets $A_1,\\\\ldots ,A_{|P|}\\\\in 2^N$ , called blocks, whose union is $N$ .\",\n \"As usual [2], [18], partitions are (partially) ordered by coarsening $\\\\geqslant $ , meaning that any $P,Q\\\\in \\\\mathcal {P}^N$ satisfy $P\\\\geqslant Q$ if every block $B\\\\in Q$ is included in a block $A\\\\in P$ , i.e.\",\n \"$A\\\\supseteq B$ .\",\n \"Hence the bottom partition is $P_{\\\\bot }=\\\\lbrace \\\\lbrace 1\\\\rbrace ,\\\\ldots ,\\\\lbrace n\\\\rbrace \\\\rbrace $ , while the top one is $P^{\\\\top }=\\\\lbrace N\\\\rbrace $ .\",\n \"Denote by $X^N=2^N\\\\times \\\\mathcal {P}^N$ the product of the subset and partition lattices, whose elements $(A,P),(B,Q)\\\\in X^N$ are pairs consisting of a subset $A,B\\\\in 2^N$ and a partition $P,Q\\\\in \\\\mathcal {P}^N$ .\",\n \"Lattice $(X^N,\\\\sqcap ,\\\\sqcup )$ is partially ordered by $\\\\sqsupseteq $ and naturally obtains through product of the meet, join and order relations for the subset and partition lattices.\",\n \"That is to say, $(A,P)\\\\sqcap (B,Q)=(A\\\\cap B,P\\\\wedge Q)$ as well as $(A,P)\\\\sqcup (B,Q)=(A\\\\cup B,P\\\\vee Q)$ , while $(A,P)\\\\sqsupseteq (B,Q)\\\\Leftrightarrow A\\\\supseteq B,P\\\\geqslant Q$ , where $P\\\\wedge Q$ is the coarsest partition finer than both $P,Q$ and $P\\\\vee Q$ is the finest partition coarser than both $P,Q$ .\",\n \"Thus the bottom element is $(\\\\emptyset ,P_{\\\\bot })$ while the top one is $(N,P^{\\\\top })$ .\",\n \"The existing lattice $(\\\\mathfrak {C}(N)_{\\\\bot },\\\\sqcap ,\\\\sqcup _*)$ of embedded subsets has elements given by those pairs $(A,P)\\\\in X^N$ such that $A\\\\in P$ , together with “an artificial bottom element $\\\\perp $ , which is introduced for mathematical convenience and that may be considered as $(\\\\emptyset ,P_{\\\\bot })$” [6].\",\n \"It is not hard to see that if $A,B\\\\in 2^N$ and $P,Q\\\\in \\\\mathcal {P}^N$ satisfy $A\\\\in P$ as well as $B\\\\in Q$ , then non-emptiness $A\\\\cap B\\\\ne \\\\emptyset $ entails $(A\\\\cap B,P\\\\wedge Q)\\\\in \\\\mathfrak {C}(N)_{\\\\bot }$ , since $A\\\\cap B$ is a block of $P\\\\wedge Q$ .\",\n \"However, in general $A\\\\cup B$ is not a block of $P\\\\vee Q$ , i.e.\",\n \"$(A\\\\cup B,P\\\\vee Q)\\\\notin \\\\mathfrak {C}(N)_{\\\\bot }$ .\",\n \"For this reason, the existing lattice of embedded subsets $\\\\mathfrak {C}(N)_{\\\\bot }$ has the same meet $\\\\sqcap $ and order $\\\\sqsupseteq $ applying to product lattice $X^N$ , but the join $\\\\sqcup _*$ is different from $\\\\sqcup $ as detailed below.\",\n \"The resulting structure [6] has $n$ atoms of the form $(i,P_{\\\\bot }),i\\\\in N$ together with additional $\\\\binom{n}{2}(n-2)$ join-irreducible elements [2], and thus the lattice is non-atomic (hence non-geometric) precisely because many elements cannot be decomposed as a join of atoms.\",\n \"In fact, the only embedded subsets $(A,P)\\\\in \\\\mathfrak {C}(N)_{\\\\bot }$ admitting such a decomposition [6] are those where partition $P$ is the modular element [2], [15] of the partition lattice whose unique non-singleton block, if any, is precisely $A$ [2] [17].\",\n \"This is denoted by $P=P^A_{\\\\bot }$ , with $P^A_{\\\\bot }=\\\\lbrace A,\\\\lbrace i_1\\\\rbrace ,\\\\ldots ,\\\\lbrace i_{n-|A|}\\\\rbrace \\\\rbrace $ and $\\\\lbrace i_1,\\\\ldots ,i_{n-|A|}\\\\rbrace =N\\\\backslash A=A^c$ .\",\n \"Since $P^i_{\\\\bot }=P_{\\\\bot }$ for all $i\\\\in N$ and $P^N_{\\\\bot }=P^{\\\\top }$ , there are $2^n-n$ such modular partitions $P^A_{\\\\bot },\\\\emptyset \\\\ne A\\\\in 2^N$ .\",\n \"They play a key role in the approach proposed here, which relies on a closure operator $cl:X^N\\\\rightarrow X^N$ over poset $(X^N,\\\\sqsupseteq )$ , meaning [2] that for all $(A,P),(B,Q)\\\\in X^N$ the following hold: (i) $cl(A,P)\\\\sqsupseteq (A,P)$ , (ii) $(A,P)\\\\sqsupseteq (B,Q)\\\\Rightarrow cl(A,P)\\\\sqsupseteq cl(B,Q)$ , (iii) $cl(cl(A,P))=cl(A,P)$ .\",\n \"This closure is defined by $cl(\\\\emptyset ,P)=(\\\\emptyset ,P)$ for all $P\\\\in \\\\mathcal {P}^N$ , while for $\\\\emptyset \\\\ne A$ $cl(A,P)=(\\\\bar{A},P\\\\vee P^A_{\\\\bot })$ such that $A\\\\subseteq \\\\bar{A}\\\\in P\\\\vee P^A_{\\\\bot }$ .\",\n \"In words, $\\\\bar{A}$ is the block of $P\\\\vee P^A_{\\\\bot }$ satisfying $\\\\bar{A}\\\\supseteq A$ , and $P^A_{\\\\bot }$ is the modular partition where the only non-singleton block, if any, is $A$ .\",\n \"The remainder of this work is concerned with poset $(\\\\mathcal {E}^N,\\\\sqsupseteq )$ , with $\\\\mathcal {E}^N=\\\\lbrace (A,P):cl(A,P)=(A,P)\\\\rbrace $ consisting of those pairs that coincide with their closure.\",\n \"As $\\\\perp $ corresponds to $(\\\\emptyset ,P_{\\\\bot })$ (see above), $\\\\mathfrak {C}(N)_{\\\\bot }\\\\subset \\\\mathcal {E}^N\\\\subset X^N$ , and in particular $\\\\mathcal {E}^N\\\\backslash \\\\mathfrak {C}(N)_{\\\\bot }=\\\\lbrace (\\\\emptyset ,P):P\\\\ne P_{\\\\bot }\\\\rbrace $ .\",\n \"From the enumerative perspective, $|\\\\mathfrak {C}(N)_{\\\\bot }|=1+\\\\sum _{1\\\\le k\\\\le n}k\\\\mathcal {S}_{n,k}$ , where $\\\\mathcal {S}_{n,k}$ is the Stirling number of the second kind, i.e.\",\n \"the number of partitions of a $n$ -set into $k$ blocks, while $|\\\\mathcal {E}^N|=\\\\sum _{1\\\\le k\\\\le n}(k+1)\\\\mathcal {S}_{n,k}=\\\\mathcal {B}_{n+1}$ , where $\\\\mathcal {B}_{n+1}$ is the Bell number, i.e.\",\n \"the number of partitions of a $n+1$ -set [2], [7], [12].\",\n \"The join obtained through closure $cl(\\\\cdot )$ coincides, over $\\\\mathfrak {C}(N)_{\\\\bot }$ , with the existing one $\\\\sqcup _*$ (defined in [6]), i.e.\",\n \"for $(A,P),(B,Q)\\\\in \\\\mathfrak {C}(N)_{\\\\bot }$ , $cl((A,P)\\\\sqcup (B,Q))=(A,P)\\\\sqcup _*(B,Q).$ However, the novel enlarged lattice $\\\\mathcal {E}^N$ of embedded subsets is substantially different from the existing one $\\\\mathfrak {C}(N)_{\\\\bot }$ , as all its $n+\\\\binom{n}{2}=\\\\binom{n+1}{2}$ join-irreducible elements (apart from $(\\\\emptyset ,P_{\\\\bot })$ ) are atoms.\",\n \"Hence $\\\\mathcal {E}^N$ is a geometric lattice with the same atoms as product lattice $X^N$ , and in fact the former reproduces within the latter a situation similar to the inclusion of partition lattice $(\\\\mathcal {P}^N,\\\\wedge ,\\\\vee )$ within Boolean lattice $(2^{N_2},\\\\cap ,\\\\cup )$ , where $N_2=\\\\lbrace \\\\lbrace i,j\\\\rbrace :1\\\\le i1$ .\",\n \"The rank is $r(\\\\emptyset ,P^A_{\\\\bot })=n-(n-|A|+1)=|A|-1$ .\",\n \"On the other hand, any $(B,Q)$ has rank $n-|Q|+1$ (since the case $B=\\\\emptyset $ is trivial precisely because $P^A_{\\\\bot }$ is a modular partition).\",\n \"Thus $r(\\\\emptyset ,P^A_{\\\\bot })+r(B,Q)=n-|Q|+|A|$ .\",\n \"Now consider that the meet has rank $r((\\\\emptyset ,P^A_{\\\\bot })\\\\sqcap (B,Q))=n-(|Q^A|+n-|A|)=|A|-|Q^A|$ , while the (closure of the) join yields $r(cl((\\\\emptyset ,P^A_{\\\\bot })\\\\sqcup (B,Q)))=n-(|Q|-|Q^A|+1)+1=n-|Q|+|Q^A|$ .\",\n \"Therefore, $|A|-|Q^A|+n-|Q|+|Q^A|=n-|Q|+|A|$ as desired.\",\n \"In the general case $(A,P^A_{\\\\bot })$ , with $r(A,P^A_{\\\\bot })=r(\\\\emptyset ,P^A_{\\\\bot })+1=|A|$ , the sum is $r(A,P^A_{\\\\bot })+r(B,Q)=n-|Q|+1+|A|$ .\",\n \"The rank $r((A,P^A_{\\\\bot })\\\\sqcap (B,Q))$ of the meet equals $|A|-|Q^A|$ if $A\\\\cap B=\\\\emptyset $ and $|A|-|Q^A|+1$ if $A\\\\cap B\\\\ne \\\\emptyset $ .\",\n \"Correspondingly, the rank $r(cl((A,P^A_{\\\\bot })\\\\sqcup (B,Q)))$ of the join equals $n-(|Q|-|Q^A|+1-1)+1=n-|Q|+|Q^A|+1$ if $A\\\\cap B=\\\\emptyset $ and $n-(|Q|-|Q^A|+1)+1=n-|Q|+|Q^A|$ if $A\\\\cap B\\\\ne \\\\emptyset $ .\",\n \"Thus the sought sum is $|A|-|Q^A|+n-|Q|+|Q^A|+1=n-|Q|+1+|A|$ in the former case and $|A|-|Q^A|+1+n-|Q|+|Q^A|=n-|Q|+1+|A|$ in the latter, i.e.\",\n \"the same.\",\n \"Turning to the ‘only if’ part, it is not difficult to check that, just like the rank of partitions, the rank of embedded subsets is submodular, that is to say $r(A,P)+r(B,Q)\\\\ge r(cl((A,P)\\\\sqcup (B,Q)))+r((A,P)\\\\sqcap (B,Q))$ , with strict inequality whenever $(A,P),(B,Q)$ is not a modular pair [2] of lattice elements.\",\n \"Let $N_+=\\\\lbrace 1,\\\\ldots ,n,n+1\\\\rbrace $ , i.e.\",\n \"$N_+=N\\\\cup \\\\lbrace n+1\\\\rbrace $ and denote by $\\\\mathcal {P}^{N_+}$ the lattice of partitions of $N_+$ , with generic element denoted by $P^+\\\\in \\\\mathcal {P}^{N_+}$ .\",\n \"From Section 1, $|\\\\mathcal {E}^N|=\\\\mathcal {B}_{n+1}=|\\\\mathcal {P}^{N_+}|$ .\",\n \"In fact, there is a lattice isomorphism $f:\\\\mathcal {E}^N\\\\rightarrow \\\\mathcal {P}^{N_+}$ , meaning that for all $(A,P),(B,Q)\\\\in \\\\mathcal {E}^N$ the following holds: $f((A,P)\\\\sqcap (B,Q))&=&f(A,P)\\\\wedge f(B,Q)\\\\text{,}\\\\\\\\f(cl((A,P)\\\\sqcup (B,Q)))&=&f(A,P)\\\\vee f(B,Q)\\\\text{.\",\n \"}$ To see this, consider that any partition $\\\\lbrace A_1,\\\\ldots ,A_{|P|}\\\\rbrace =P\\\\in \\\\mathcal {P}^N$ has associated $|P|+1$ partitions $P^+_1,\\\\ldots ,P^+_{|P|},P^+_{|P|+1}\\\\in \\\\mathcal {P}^{N_+}$ , all inducing the same partition $P$ of $N$ .\",\n \"The first $|P|$ ones $P^+_1,\\\\ldots ,P^+_{|P|}$ obtain each by placing additional element $n+1$ into block $A_k\\\\in P,1\\\\le k\\\\le |P|$ , while the last one $P^+_{|P|+1}$ obtains by placing $n+1$ into a singleton block $\\\\lbrace n+1\\\\rbrace \\\\in P^+_{|P|+1}$ .\",\n \"Of course, the former are pair-wise incomparable, i.e.\",\n \"$P^+_k\\\\lnot \\\\geqslant P^+_{k^{\\\\prime }}\\\\lnot \\\\geqslant P^+_k$ for $1\\\\le kP^+_{|P|+1},1\\\\le k\\\\le |P|$ .\",\n \"The same applies to embedded subsets $(A_1,P),\\\\ldots ,(A_{|P|},P),(\\\\emptyset ,P)\\\\in \\\\mathcal {E}^N$ , in that $(A_k,P)\\\\lnot \\\\sqsupseteq (A_{k^{\\\\prime }},P)\\\\lnot \\\\sqsupseteq (A_k,P)$ for $1\\\\le k1$ for at least one $A\\\\in P$ .\",\n \"Segment (or interval) $[Q,P]=\\\\lbrace P^{\\\\prime }:Q\\\\leqslant P^{\\\\prime }\\\\leqslant P\\\\rbrace $ is isomorphic to product $\\\\times _{A\\\\in P}\\\\mathcal {P}^{k_A}$ , where $\\\\mathcal {P}^k$ denotes the lattice of partitions of a $k$ -set.\",\n \"Let $c^{QP}_k=|\\\\lbrace A:A\\\\in P,k_A=k\\\\rbrace |,1\\\\le k\\\\le n$ be the number of blocks of $P$ obtained as the union of $k$ blocks of $Q$ , where $c^{QP}=(c^{QP}_1,\\\\ldots ,c^{QP}_n)\\\\in \\\\mathbb {Z}_+^n$ is the class (or type) of segment $[Q,P]$ .\",\n \"Then [13], $\\\\mu _{\\\\mathcal {P}^N}(Q,P)=(-1)^{-n+\\\\sum _{1\\\\le k\\\\le n}c^{QP}_k}\\\\prod _{10$ for all $n\\\\ge 0$ , which is consistent with [3].\"\n ],\n [\n \"Möbius algebra, vector subspaces and games\",\n \"The purpose of this section is to highlight that all cooperative games are in fact elements of subspaces of the free vector space $V(\\\\mathcal {E}^N)$ over $\\\\mathbb {R}$ generated by $\\\\mathcal {E}^N$ [1], [2].\",\n \"That is, the elements of $V(\\\\mathcal {E}^N)\\\\subset \\\\mathbb {R}^{\\\\mathcal {B}_{n+1}}$ are real-valued lattice functions $g:\\\\mathcal {E}^N\\\\rightarrow \\\\mathbb {R}$ .\",\n \"Recall that, for $N$ regarded as a player set, cooperative game theory deals with three types of lattice functions: coalitional games [14] or set functions $v:2^N\\\\rightarrow \\\\mathbb {R}_+,v(\\\\emptyset )=0$ , global games [4] or partition functions $h:\\\\mathcal {P}^N\\\\rightarrow \\\\mathbb {R}_+,h(P_{\\\\bot })=0$ , games in partition function form [6], [10], [19], hereafter referred to as PFF games for short, or functions $f:\\\\mathfrak {C}(N)_{\\\\bot }\\\\rightarrow \\\\mathbb {R}_+,f(\\\\emptyset ,P_{\\\\bot })=0$ .\",\n \"A further fourth type of cooperative games may be defined, which incorporates all previous ones $(a)$ , $(b)$ and $(c)$ , namely extended PFF games, or those elements $g\\\\in V(\\\\mathcal {E}^N)$ of the free vector space under concern such that $g:\\\\mathcal {E}^N\\\\rightarrow \\\\mathbb {R}_+,g(\\\\emptyset ,P_{\\\\bot })=0$ .\",\n \"Together with the Möbius function, the other element of the incidence algebra (of locally finite posets [2] [13]) playing a fundamental role in cooperative game theory is the zeta function $\\\\zeta $ , as it provides the so-called ‘unanimity games’ [14].\",\n \"In the incidence algebra of $\\\\mathcal {E}^N$ over $\\\\mathbb {R}$ , the zeta function $\\\\zeta _{\\\\mathcal {E}^N}:\\\\mathcal {E}^N\\\\times \\\\mathcal {E}^N\\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ is defined on ordered pairs of embedded subsets by $\\\\zeta _{\\\\mathcal {E}^N}((B,Q),(A,P))=\\\\left\\\\lbrace \\\\begin{array}{c} 1\\\\text{ if }(A,P)\\\\sqsupseteq (B,Q)\\\\text{,}\\\\\\\\0\\\\text{ otherwise.}\",\n \"\\\\end{array}\\\\right.$ In these terms, unanimity (coalitional) games $u_B,\\\\emptyset \\\\ne B\\\\in 2^N$ are defined by $u_B(A)=\\\\zeta _{2^N}(B,A)$ for all subsets (or coalitions) $A\\\\in 2^N$ , where $\\\\zeta _{2^N}$ is the zeta function in the incidence algebra of $2^N$ .\",\n \"They are linearly independendent elements of the vector space of coalitional games and thus form a basis.\",\n \"This fact is essential since the very beginning of cooperative game theory.\",\n \"In the present setting, for $(B,Q)\\\\in \\\\mathcal {E}^N$ , let $\\\\zeta ^{B,Q}(A,P)=\\\\zeta _{\\\\mathcal {E}^N}((B,Q),(A,P))$ for all $(A,P)\\\\in \\\\mathcal {E}^N$ .\",\n \"Then, $\\\\lbrace \\\\zeta ^{B,Q}:(B,Q)\\\\in \\\\mathcal {E}^N\\\\rbrace $ is a basis of $V(\\\\mathcal {E}^N)$ .\",\n \"Also denote by $V(\\\\mathcal {E}^N)_{\\\\bot }\\\\subset \\\\mathbb {R}^{\\\\mathcal {B}_{n+1}-1}_+$ the subspace of extended PFF games.\",\n \"Furthermore, let the Möbius inversion of $g$ be $\\\\mu ^g:\\\\mathcal {E}^N\\\\rightarrow \\\\mathbb {R}$ , i.e.\",\n \"$\\\\mu ^g(A,P)&=&\\\\sum _{\\\\underset{(B,Q)\\\\in \\\\mathcal {E}^N}{(B,Q)\\\\sqsubseteq (A,P)}}\\\\mu _{\\\\mathcal {E}^N}((B,Q),(A,P))g(B,Q)=\\\\\\\\&=&g(A,P)-\\\\sum _{\\\\underset{(B,Q)\\\\in \\\\mathcal {E}^N}{(B,Q)\\\\sqsubset (A,P)}}\\\\mu ^g(B,Q)\\\\text{,}$ with $\\\\mu ^g(\\\\emptyset ,P_{\\\\bot })=g(\\\\emptyset ,P_{\\\\bot })$ ($=0$ if $g\\\\in V(\\\\mathcal {E}^N)_{\\\\bot }$ ), and where $\\\\mu ^{\\\\zeta ^{B,Q}}(A,P)=\\\\left\\\\lbrace \\\\begin{array}{c} 1\\\\text{ if }(A,P)=(B,Q)\\\\\\\\0\\\\text{ otherwise} \\\\end{array}\\\\right.$ for every element $\\\\zeta ^{B,Q}$ of the chosen basis, thereby providing the canonical basis (or Kronecker delta) elements (see [2], while crucially noticing that the alternative definition $\\\\zeta ^{B,Q}(A,P)=\\\\zeta _{\\\\mathcal {E}^N}((A,P),(B,Q))$ applies there.)\",\n \"Specifically, let $\\\\epsilon _{B,Q}\\\\in V(\\\\mathcal {E}^N)$ be the $\\\\mathcal {B}_{n+1}$ -vector with entries indexed by elements $(A,P)\\\\in \\\\mathcal {E}^N$ and such that its unique non-zero entry is the $(B,Q)$ one, which equals 1.\",\n \"Then the Möbius algebra of $\\\\mathcal {E}^N$ over $\\\\mathbb {R}$ (denoted by Möb$(\\\\mathcal {E}^N)$ , see [1] [2]) is defined by the system of orthogonal idempotents $\\\\lbrace \\\\epsilon _{B,Q}:(B,Q)\\\\in \\\\mathcal {E}^N\\\\rbrace $ and linear extension to all of $V(\\\\mathcal {E}^N)$ through multiplication $\\\\epsilon _{B,Q}\\\\cdot \\\\epsilon _{A,P}=\\\\left\\\\lbrace \\\\begin{array}{c} 1\\\\text{ if }(A,P)=(B,Q)\\\\text{,}\\\\\\\\0\\\\text{ otherwise.}\",\n \"\\\\end{array}\\\\right.$ In terms of the above basis $\\\\lbrace \\\\zeta ^{B,Q}:(B,Q)\\\\in \\\\mathcal {E}^N\\\\rbrace $ of $V(\\\\mathcal {E}^N)$ , this multiplication yields $\\\\zeta ^{B,Q}\\\\cdot \\\\zeta ^{A,P}=\\\\zeta ^{cl((A,P)\\\\sqcup (B,Q))}$ (see [2], crucially noticing again that since the alternative definition of the basis detailed above applies there, then $\\\\zeta ^{B,Q}\\\\cdot \\\\zeta ^{A,P}=\\\\zeta ^{(A,P)\\\\sqcap (B,Q)}$ ).\",\n \"It is well known that Möbius inversion $\\\\mu ^g$ quantifies precisely the coefficients identifying functions $g\\\\in V(\\\\mathcal {E}^N)$ as linear combinations of the basis elements, that is $g(A,P)=\\\\sum _{(B,Q)\\\\in \\\\mathcal {E}^N}\\\\mu ^g(B,Q)\\\\zeta ^{B,Q}(A,P)\\\\text{ for all }(A,P)\\\\in \\\\mathcal {E}^N\\\\text{.\",\n \"}$ If $Y\\\\subset \\\\mathcal {E}^N$ and $\\\\mu ^g(A,P)=0$ for all $(A,P)\\\\in \\\\mathcal {E}^N\\\\backslash Y$ , then Möbius inversion $\\\\mu ^g$ may be said to live only $Y$ .\",\n \"In these terms, the following observations are immediate: (traditional) PFF games are elements of the subspace $\\\\hat{V}(\\\\mathcal {E}^N)\\\\subset \\\\mathbb {R}^{\\\\mathcal {B}_{n+1}-\\\\mathcal {B}_n}$ of $V(\\\\mathcal {E}^N)$ consisting of those extended PFF games $g\\\\in V(\\\\mathcal {E}^N)$ whose Möbius inversion lives only on embedded subsets $(A,P)$ such that $A\\\\ne \\\\emptyset $ , i.e.\",\n \"$A=\\\\emptyset \\\\Rightarrow \\\\mu ^g(A,P)=0$ , entailing $g(\\\\emptyset ,P)=0$ for all $P\\\\in \\\\mathcal {P}^N$ ; global games are elements of the subspace $\\\\tilde{V}(\\\\mathcal {E}^N)\\\\subset \\\\mathbb {R}^{\\\\mathcal {B}_n-1}$ consisting of extended PFF games $g\\\\in V(\\\\mathcal {E}^N)$ whose Möbius inversion lives only on embedded subsets $(A,P)$ such that $A=\\\\emptyset $ and $P\\\\ne P_{\\\\bot }$ , entailing $g(A,P)=g(\\\\emptyset ,P)$ for all $A\\\\in P$ (thereby somehow strengthening the idea of a global level of satisfaction, common to all players, see [4]); coalitional games are elements of the subspace $\\\\bar{V}(\\\\mathcal {E}^N)\\\\subset \\\\mathbb {R}^{2^n-1}$ consisting of those $g\\\\in V(\\\\mathcal {E}^N)$ whose Möbius inversion lives only on the $2^n-1$ modular elements $(A,P^A_{\\\\bot })$ of $\\\\mathcal {E}^N$ defined in Section 3 such that $A\\\\ne \\\\emptyset $ , yielding $g(\\\\emptyset ,P)=0$ for all $P\\\\in \\\\mathcal {P}^N$ as well as $g(A,P)=g(A,P^A_{\\\\bot })$ for all $P\\\\in \\\\mathcal {P}^N$ such that $A\\\\in P$ .\",\n \"Another subspace of $V(\\\\mathcal {E}^N)$ appearing in cooperative game theory is given by the so-called “additively separable” global games (or partition functions, see [4], [5]), namely those $h:\\\\mathcal {P}^N\\\\rightarrow \\\\mathbb {R}_+$ such that $h(P)=h_v(P):=\\\\sum _{A\\\\in P}v(A)$ for some coalitional game (or set function) $v:2^N\\\\rightarrow \\\\mathbb {R}_+$ .\",\n \"The Möbius inversion of these partition functions lives only on the $2^n-n$ modular elements $P^A_{\\\\bot }$ of the partition lattice, see Section 1.\",\n \"This is detailed hereafter (see also [4], [5] and [11]).\",\n \"Proposition 5 If $h=h_v$ and $\\\\sum _{i\\\\in N}v(\\\\lbrace i\\\\rbrace )=h_v(P_{\\\\bot })>0$ , then a continuuum of set functions $w:2^N\\\\rightarrow \\\\mathbb {R}_+,w\\\\ne v$ satisfies $h=h_w$ .\",\n \"Proof: By direct substitution, the Möbius inversion $\\\\mu ^{h_v}:\\\\mathcal {P}^N\\\\rightarrow \\\\mathbb {R}$ of partition function $h_v$ satisfies $\\\\mu ^{h_v}(P)=\\\\sum _{A\\\\in P}\\\\sum _{B\\\\subseteq A}v(B)\\\\sum _{Q\\\\leqslant P:B\\\\in Q}\\\\mu _{\\\\mathcal {P}^N}(Q,P)\\\\text{ for all }P\\\\in \\\\mathcal {P}^N\\\\text{.\",\n \"}$ If $P\\\\ne P^A_{\\\\bot }$ , then the recursion for Möbius function $\\\\mu _{\\\\mathcal {P}^N}:\\\\mathcal {P}^N\\\\times \\\\mathcal {P}^N\\\\rightarrow \\\\mathbb {R}$ yields $\\\\sum _{Q\\\\leqslant P:A\\\\in Q}\\\\mu _{\\\\mathcal {P}^N}(Q,P)=\\\\sum _{P^A_{\\\\bot }\\\\leqslant Q\\\\leqslant P}\\\\mu _{\\\\mathcal {P}^N}(Q,P)=0\\\\text{,}$ and the same for proper subsets $B\\\\subset A$ .\",\n \"Möbius inversion $\\\\mu ^{h_v}$ thus lives only on modular partitions, where it obtains recursively by $\\\\mu ^{h_v}(P_{\\\\bot })=\\\\sum _{i\\\\in N}v(\\\\lbrace i\\\\rbrace )$ and $\\\\mu ^{h_v}(P_{\\\\bot }^A)=\\\\mu ^v(A)$ for $1<|A|\\\\le n$ , with $P^N_{\\\\bot }=P^{\\\\top }$ .\",\n \"Hence any $w\\\\ne v$ satisfying $\\\\sum _{i\\\\in N}v(\\\\lbrace i\\\\rbrace )=\\\\sum _{i\\\\in N}w(\\\\lbrace i\\\\rbrace )$ and $\\\\mu ^v(A)=\\\\mu ^w(A)$ for all $A\\\\in 2^N$ such that $|A|>1$ also additively separates $h$ , i.e.\",\n \"$h_v=h_w$ .\",\n \"If $\\\\sum _{i\\\\in N}v(\\\\lbrace i\\\\rbrace )>0$ , then all $w:2^N\\\\rightarrow \\\\mathbb {R}_+$ such that $h_v=h_w$ may be chosen from within a $n-1$ -dimensional simplex $\\\\Delta \\\\subset \\\\mathbb {R}^n_+$ .\",\n \"However, if $\\\\sum _{i\\\\in N}v(\\\\lbrace i\\\\rbrace )=0$ , then there is no $w:2^N\\\\rightarrow \\\\mathbb {R}_+$ such that $h_v=h_w,v\\\\ne w$ .\",\n \"Nevertheless, this technical distinction plays no role as soon as attention is placed on generic partition and set functions $h:\\\\mathcal {P}^N\\\\rightarrow \\\\mathbb {R},w:2^N\\\\rightarrow \\\\mathbb {R}$ , i.e.\",\n \"not required to take only non-negative real values.\",\n \"In any case, in view of observation $(ii)$ above, it is evident that additively separable partition functions may be regarded as elements of the subspace of $V(\\\\mathcal {E}^N)$ consisting of those extended PFF games $g$ with Möbius inversion $\\\\mu ^g$ such that $\\\\mu ^g(B,Q)\\\\ne 0$ only if $B=\\\\emptyset $ and $Q$ is a modular partition.\",\n \"Developing from additively separable partition functions, now consider those $g\\\\in V(\\\\mathcal {E}^N)$ with Möbius inversion $\\\\mu ^g$ living only on the $2^{n+1}-(n+1)$ modular elements of $\\\\mathcal {E}^N$ (see Section 3).\",\n \"Proposition 6 If $g\\\\in V(\\\\mathcal {E}^N)$ has Möbius inversion $\\\\mu ^g$ living only on modular elements of $\\\\mathcal {E}^N$ , then there are set functions $v,w:2^N\\\\rightarrow \\\\mathbb {R},v(\\\\emptyset )=0$ such that $g(A,P)=v(A)+\\\\sum _{B\\\\in P}w(B)\\\\text{ for all }(A,P)\\\\in \\\\mathcal {E}^N\\\\text{.\",\n \"}$ Proof: Under the above conditions, and given expression (5) above, $g(A,P)=\\\\mu ^g(\\\\emptyset ,P_{\\\\bot })+\\\\sum _{B\\\\in P}\\\\sum _{\\\\emptyset \\\\ne B^{\\\\prime }\\\\subseteq B}\\\\mu ^g(\\\\emptyset ,P^{B^{\\\\prime }}_{\\\\bot })+\\\\sum _{\\\\emptyset \\\\ne A^{\\\\prime }\\\\subseteq A}\\\\mu ^g(A^{\\\\prime },P^{A^{\\\\prime }}_{\\\\bot })$ for all $(A,P)\\\\in \\\\mathcal {E}^N$ .\",\n \"In view of Proposition 5, $\\\\mu ^g(\\\\emptyset ,P_{\\\\bot })+\\\\sum _{B\\\\in P}\\\\sum _{\\\\emptyset \\\\ne B^{\\\\prime }\\\\subseteq B}\\\\mu ^g(\\\\emptyset ,P^{B^{\\\\prime }}_{\\\\bot })=\\\\sum _{B\\\\in P}w(B)$ for any set function $w$ with Möbius inversion $\\\\mu ^w$ such that $\\\\mu ^g(\\\\emptyset ,P_{\\\\bot })=\\\\sum _{i\\\\in N}w(\\\\lbrace i\\\\rbrace )\\\\text{ and }\\\\mu ^g(\\\\emptyset ,P^{B^{\\\\prime }}_{\\\\bot })=\\\\mu ^w(B^{\\\\prime })\\\\text{ for }B^{\\\\prime }\\\\in 2^N,|B^{\\\\prime }|>1\\\\text{.\",\n \"}$ On the other hand, $\\\\sum _{\\\\emptyset \\\\ne A^{\\\\prime }\\\\subseteq A}\\\\mu ^g(A^{\\\\prime },P^{A^{\\\\prime }}_{\\\\bot })=v(A)=\\\\sum _{A^{\\\\prime }\\\\subseteq A}\\\\mu ^v(A^{\\\\prime })$ for any set function $v$ with Möbius inversion $\\\\mu ^v$ satisfying $\\\\mu ^g(A^{\\\\prime },P^{A^{\\\\prime }}_{\\\\bot })=\\\\mu ^v(A^{\\\\prime })$ for $A^{\\\\prime }\\\\in 2^N$ , $|A^{\\\\prime }|>0$ and $\\\\mu ^v(\\\\emptyset )=v(\\\\emptyset )=0$ .\",\n \"Like for additively separable partition functions, if the Möbius inversion of extended PFF games lives only on the modular elements of the lattice, then all the values taken on embedded subsets can be recovered from only two set functions, out of which one is unique while the other can be chosen from within a continuum (in view of Proposition 5).\"\n ],\n [\n \"Concluding remarks\",\n \"This paper essentially proposes to look at the so-called lattice of embedded subsets from a wider perspective, such that the novel resulting structure is a combinatorial geometry obtained through a closure operator that satisfies the Steinitz exchange axiom.\",\n \"With respect to the lattice previously proposed in [6], the geometric one presented here additionally includes all elements given by a (non-bottom) partition paired with the empty set, and this evidently yields a most natural lattice embedding, i.e.\",\n \"of $\\\\mathcal {P}^N$ into $\\\\mathcal {E}^N$ .\",\n \"However, from this perspective what seems mostly important is that the geometric lattice of embedded subsets of a $n$ set is isomorphic to the lattice of partitions of a $n+1$ -set.\",\n \"In general, this enables to re-obtain a variety of results applying to partitions, ranging from the characteristic polynomial to supersolvability.\",\n \"Since embedded subsets, or embedded coalitions, were firstly used as modeling tools in cooperative game theory, a meaningful direction for investigation focuses on the free vector space of functions taking real values on lattice elements.\",\n \"In this view, a novel type of games has been defined, namely extended PFF games, which englobe all existing cooperative games as elements of suitable vector subspaces.\",\n \"But perhaps most importantly, the geometric lattice of embedded subsets yields that all existing cooperative games are real-valued functions defined on atomic lattices, and this may have fundamental implications for the solution concept (i.e.\",\n \"how to share the fruits of cooperation) associated with these games.\",\n \"This shall be addressed in future work.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05814"}}},{"rowIdx":3394,"cells":{"text":{"kind":"list like","value":[["Five-loop quark mass and field anomalous dimensions for a general gauge\n group"],["Abstract We present analytical five-loop results for the quark mass and quark field anomalous dimensions, for a general gauge group and in the MSbar scheme.","We confirm the values known for the gauge group SU(3) from an independent calculation, and find full agreement with results available from large-Nf studies."],["Introduction","High-precision determinations of Standard Model (SM) parameters are crucially important for precise predictions for the observables measured in collider physics experiments, such as currently performed at the LHC, or possibly at a future linear collider.","A thorough comparison of these predictions with experimental results allows to scrutinize the details of the hugely successful SM, and might shed light on possible physics beyond the SM.","In order to pin down the theory's fundamental parameters, such as coupling constants and masses, with sufficient precision, the knowledge of higher order perturbative corrections is required.","This encompasses the evaluation of multi-loop Feynman diagrams and Feynman integrals, for which significant progress has been made in recent years, mainly with respect to the formulation of advanced algorithms that allow to treat the complexity level met in those higher-loop integrals.","We focus here on the strong interactions, which are embedded into the SM via Quantum Chromodynamics (QCD).","The relevant parameters are then the (strong) gauge coupling and the quark masses, both of which run with the energy scale according to the renormalization group (RG) equations.","In order to evolve e.g.","the low-energy value of the coupling constant (measured with high precision from tau lepton decay) to high energies, the anomalous dimension of the gauge coupling (the so-called Beta function) is needed, as coefficient in the corresponding RG equation.","Likewise, a high-order evaluation of the quark mass anomalous dimension gives access to precise values for e.g.","charm and bottom quark masses, which are measured at low energies (typically a few GeV) but whose uncertainty at the high-energy scale of the Higgs mass $m_H=125$ GeV is important in Higgs decay rates into such quark pairs.","In particular, to match the precision of the known five-loop inclusive decay width of $H\\rightarrow q\\bar{q}$ [1], one should evolve the parameters (which are $\\alpha _s(\\mu )$ and the running quark mass $m_q(\\mu )$ with $\\mu $ being the renormalization scale) from low energies to $\\mu =m_H$ at the same perturbative order, for full consistency and to avoid large logarithms $\\ln (\\mu ^2/m_H^2)$ .","Given this clear phenomenological motivation, we will present new results for the quark mass anomalous dimension here, applying a number of the above-mentioned algorithmic advances.","While this renormalization constant has been studied previously up to five loops in perturbation theory [2], [3], [4], [5], [6], [7], we generalize it from the gauge group SU(3) to a semi-simple Lie group.","At the same time, we provide a truly independent check on the available SU(3) result, since we utilize largely independent methods, as described below.","We will also give results for a related quantity needed to renormalize the quark sector at five loops, namely the quark field anomalous dimension in Feynman gauge, again generalizing known SU(3) results to a semi-simple Lie group.","The paper is organized as follows.","We start by explaining our computational setup in section .","Using some notation defined in section , we then present and discuss results in sections and , before concluding in section .","For convenience, an appendix reproduces large-$N_{\\mathrm {f}}$ results from the literature, which we use for consistency checks."],["Setup","Let us start by explaining our computational setup, which closely follows the one employed and tested in [8].","We base our highly automated setup on the diagram generator qgraf [9], [10] and various in-house FORM [11], [12], [13] codes.","After generating the required fermionic 2-point functions, we apply projectors and perform the group algebra with color [14].","To extract the ultraviolet (UV) divergences, we then use the freedom to change the low-momentum behavior and make all propagators massive, which regulates the infrared at the cost of introducing a new counterterm for the unphysical regulator mass.","Expanding to sufficient depth in the external momentum [15], [16], [17] and keeping all potentially UV divergent structures results in nullifying the external momentum.","The coefficients of this expansion can then be mapped onto a family of fully massive vacuum integrals, which are labelled by 15 indices (corresponding to maximally 12 propagators plus 3 scalar products) at five loops [18].","As a next and fairly time-consuming step, we reduce those integrals to a small set of master integrals, powered by our own codes crusher [19] and TIDE [18], which are based on integration-by-parts (IBP) identities [20] and use a Laporta-type algorithm [21] for a systematic integral reduction.","At five loops, we end up with a set of 110 master integrals.","These have been evaluated in an $\\varepsilon $ -expansion around $d=4-2\\varepsilon $ dimensions in previous works [18], [22], using an approach based on IBP reductions and difference equations [21] that has been realized in C++ and uses Fermat [23] to perform the polynomial algebra that arises in solving systems of linear equations with large rational coefficients.","The resulting high-precision numerical results for the coefficients of the $\\varepsilon $ -expansions finally allow us to utilize the integer-relation finding algorithm PSLQ [24] to discover the analytic content of some of these numbers, and to find relations between others.","As a consequence, we are able to provide all our results given below in analytic form.","Figure: 5-loop master integrals with 12 lines that contribute to eqs. ()-().","Each line denotes a massive propagator 1/(k 2 +m 2 )1/(k^2+m^2), and each dot stands for an extra power of the corresponding propagator.As has already been mentioned elsewhere [8], our high-precision evaluation of all 5-loop master integrals has not yet produced results for the 12-line families, see figure REF .","These do not contain divergences in four dimensions, and could therefore be avoided in evaluations of anomalous dimensions.","It turns out, however, that with our integral reduction criteria and lexicographic ordering prescription, we do get contributions from these integral classes since they are multiplied by prefactors with spurious poles as $d\\rightarrow 4$ .","To fix the values of the 12-line master integrals that we need, we first note that in all our results, only three independent linear combinations appear: $\\ell _0&=& 689 \\,I_{30527.1.1, 5} - 3934 \\,I_{30527.1.2, 5} + 5464 \\,I_{30527.1.3, 5} + 1152 \\,I_{30527.1.4, 5} - 5228 \\,I_{30527.6.3, 5} \\;,\\\\\\ell _1&=& 689 \\,I_{30527.1.1, 6} - 3934 \\,I_{30527.1.2, 6} + 5464 \\,I_{30527.1.3, 6} + 1152 \\,I_{30527.1.4, 6} - 5228 \\,I_{30527.6.3, 6} \\nonumber \\\\&&+ 1968 \\,I_{30527.1.1, 5} + 3890 \\,I_{30527.1.2, 5} + 3844 \\,I_{30527.1.3, 5} - 6912 \\,I_{30527.1.4, 5} - 70 \\,I_{30527.6.3, 5} \\;,\\\\\\ell _2&=& 11 \\,I_{31740.1.1, 5} - 72 \\,I_{31740.1.3, 5} \\;.$ Here, each integral $I_{\\#,n}$ corresponds to the $\\varepsilon ^n$ -coefficient of the respective fully massive master integral of figure REF , divided by the fifth power of the 1-loop tadpole $J$ for normalization reasons$J=\\int {\\rm d}^dk/(k^2+m^2)\\sim \\Gamma (1-d/2)$ has a simple pole as $d\\rightarrow 4$ ; hence, finite 5-loop terms correspond to $\\varepsilon ^5$ ..","While it is conceivable that there exists a suitable basis transformation that eliminates these linear combinations altogether from the final results, we have not yet performed a systematic search of such transformations in our integral reduction tables, but opted for other criteria to fix the numerical values of the three linear combinations, with high precision, as we will explain now.","As a crude order-of-magnitude estimate, we have evaluated the set of 12-line integrals via Feynman parametric representations (see e.g.","[25]) and subsequent (primary) sector decomposition, using the strategy explained in [26], [27] and as implemented in FIESTA [28] as well as own code (see [29]).","Due to the large prefactors and cancellations in eqs.","(REF )-(), a 6-digit evaluation results in the estimates $\\ell _0\\approx -7.47(1)\\;,\\quad \\ell _1\\approx -50.6(1)\\;,\\quad \\ell _2\\approx -0.673(1)\\;,$ with 3-digit accuracy.","Turning now to high-precision evaluations, we first recall that the higher-order $\\varepsilon $ -poles of renormalization constants are completely determined by lower-order coefficients.","Checking these, we obtain one consistency condition that allows to fix $\\ell _0$ .","Second, we observe the occurrence of some rank-12 group invariants in another renormalization constant that we have evaluated using the same setup, namely for the ghost-gluon vertex [30].","Using eq.","(REF ), to determine their coefficients, we find that they vanish at least to this low accuracy.","Taking this zero for granted, we turn the argument around and require e.g.","the structure $d^{\\,444}_{FAA}\\,N_{\\mathrm {f}}$ (where the group invariant is in the notation of [14]) to be absent from the final result; this gives us another condition, fixing $\\ell _2$ .","Third, for fixing the remaining linear combination, we choose to compare our results in the SU(3) limit to the previously known 5-loop results.","To be concrete, out of the many possible coefficients we choose the $n_f$ term of $\\gamma _m$ as given in [7], giving us one constraint which fixes $\\ell _1$ .","Along these lines, we obtain numerical values for all three linear combinations with 260 digits, the first 50 of which read $\\ell _0&=& -7.4750787021276651819913288152084850401974826928834\\dots \\;,\\\\\\ell _1&=& -50.563714841071996428539372592222326105092965639946\\dots \\;,\\\\\\ell _2&=& -0.67332086607447050046759024439428336720209195028580\\dots \\;,$ and which can be seen to be consistent with our low-precision estimates of eq.","(REF ) that had been obtained by direct integration."],["Notation","Let us fix our notation here: we work with a semi-simple Lie algebra with hermitian generators $T^a$ , whose real and antisymmetric structure constants $f^{abc}$ are fixed by the commutation relation $[T^a,T^b]=i f^{abc}T^c$ .","As usual, the quadratic Casimir operators of the fundamental (adjoint) representation (of dimensions $N_{\\mathrm {F}}$ and $N_{\\mathrm {A}}$ , respectively) are defined as $T^aT^a=C_{\\mathrm {F}}1\\!\\!1$ ($f^{acd}f^{bcd}=C_{\\mathrm {A}}\\delta ^{ab}$ ).","Furthermore, traces are normalized as ${\\rm Tr}(T^aT^b)=T_{\\mathrm {F}}\\delta ^{ab}$ , we denote the number of quark flavors with $N_{\\mathrm {f}}$ , and find it convenient to define the following normalized combinations: $c_f=\\frac{C_{\\mathrm {F}}}{C_{\\mathrm {A}}} \\quad ,\\quad n_f=\\frac{N_{\\mathrm {f}}\\,T_{\\mathrm {F}}}{C_{\\mathrm {A}}} \\;.$ In our multi-loop diagrams, we will encounter traces of more than two group generators, giving rise to higher-order group invariants.","It is useful to define traces over combinations of symmetric tensors [14], of which we need the following (writing $[F^a]_{bc}=-if^{abc}$ for the generators of the adjoint representation): $d_1=\\frac{[{\\rm sTr}(T^aT^bT^cT^d)]^2}{N_{\\mathrm {A}}T_{\\mathrm {F}}^2C_{\\mathrm {A}}^2} \\;,\\;d_2=\\frac{{\\rm sTr}(T^aT^bT^cT^d)\\,{\\rm sTr}(F^aF^bF^cF^d)}{N_{\\mathrm {A}}T_{\\mathrm {F}}C_{\\mathrm {A}}^3} \\;,\\;d_3=\\frac{[{\\rm sTr}(F^aF^bF^cF^d)]^2}{N_{\\mathrm {A}}C_{\\mathrm {A}}^4} \\;.$ Here, ${\\rm sTr}$ stands for a fully symmetrized trace (such that ${\\rm sTr}(ABC)=\\frac{1}{2}{\\rm Tr}(ABC+ACB)$ etc.).","While dealing with the quark sector, it might seem more natural to normalize these traces with respect to the dimension of the fundamental representation; to this end, we note the relation $N_{\\mathrm {A}}T_{\\mathrm {F}}=N_{\\mathrm {F}}C_{\\mathrm {F}}$ which holds in general [14], [31].","For the gauge group SU($N$ ) (where $T_{\\mathrm {F}}=\\frac{1}{2}$ and $C_{\\mathrm {A}}=N$ ), the normalized group invariants introduced above read [14] $n_f=\\frac{N_{\\mathrm {f}}}{2N}\\;,\\quad c_f=\\frac{N^2-1}{2N^2}\\;,\\quad d_1=\\frac{N^4-6N^2+18}{24N^4}\\;,\\quad d_2=\\frac{N^2+6}{24N^2}\\;,\\quad d_3=\\frac{N^2+36}{24N^2}\\;.$ The corresponding SU(3) values, relevant for physical QCD, hence read $\\mbox{SU}(3):\\quad n_f=\\frac{N_{\\mathrm {f}}}{6}\\;,\\quad c_f=\\frac{4}{9}\\;,\\quad d_1=\\frac{5}{216}\\;,\\quad d_2=\\frac{5}{72}\\;,\\quad d_3=\\frac{5}{24}\\;.$"],["Quark mass renormalization","The renormalization constant for the quark mass $m_{\\rm bare}=Z_m\\,m_{\\rm ren}$ , or equivalently its anomalous dimension $\\gamma _m = -\\partial _{\\ln \\mu ^2}\\ln Z_m$ , has been known at two [2] and three loops [3], [4] for a long time.","At four loops, $\\gamma _m$ is known for SU(N) and QED [6] as well as for a general Lie group [5].","Presently, at five loops only the SU(3) value is publicly available [7].","We present our corresponding result for a general Lie group below.","The structure of the quark mass anomalous dimension is $\\partial _{\\ln \\mu ^2}\\ln m_{\\rm q}(\\mu ) &\\equiv & \\gamma _m(a) \\;=\\;-c_f\\,a\\,\\Big \\lbrace 3 +\\gamma _{m1}\\,a +\\gamma _{m2}\\,a^2 +\\gamma _{m3}\\,a^3 +\\gamma _{m4}\\,a^4 +\\dots \\Big \\rbrace \\;,\\\\a&\\equiv &\\frac{C_{\\mathrm {A}}\\,g^2(\\mu )}{16\\pi ^2}\\;,$ with $g(\\mu )$ being the renormalized QCD gauge coupling constant that depends on the renormalization scale $\\mu $ (we prefer to use the expansion parameter $a$ which is nothing but a rescaled version of the renormalized strong coupling constant $\\alpha _s=\\frac{g^2(\\mu )}{4\\pi }$ ).","We work in $d=4-2\\varepsilon $ dimensions and employ the ${\\overline{\\mbox{\\rm {MS}}}}$ scheme.","The coefficients $\\gamma _{mn}$ are polynomials in $n_f$ and can be written in terms of our normalized group factors.","Up to four loops, they read [5] $3^1\\,\\gamma _{m1} &=& n_f\\Big [-10\\Big ]+\\Big [(9c_f+97)/2\\Big ] \\;,\\\\3^3\\,\\gamma _{m2} &=& n_f^2\\Big [\\!-\\!140\\Big ] +n_f\\Big [54(24\\zeta _3\\!-\\!23)c_f-4(139\\!+\\!324\\zeta _3)\\Big ]+\\Big [(6966c_f^2\\!-\\!3483c_f\\!+\\!11413)/4\\Big ],\\\\3^4\\,\\gamma _{m3} &=& n_f^3\\Big [-8(83-144\\zeta _3)\\Big ] +n_f^2\\Big [48(19-270\\zeta _3+162\\zeta _4)c_f+2(671+6480\\zeta _3-3888\\zeta _4)\\Big ]\\nonumber \\\\&&+n_f\\Big [-216(35-207\\zeta _3+180\\zeta _5)c_f^2-3(8819-9936\\zeta _3+7128\\zeta _4-2160\\zeta _5)c_f\\nonumber \\\\&&-(65459/2+72468\\zeta _3-21384\\zeta _4-32400\\zeta _5)+2592(2-15\\zeta _3)d_1\\Big ]\\nonumber \\\\&&+\\tfrac{9}{8}\\Big [-9(1261+2688\\zeta _3)c_f^3+6(15349+3792\\zeta _3)c_f^2-2(34045+5472\\zeta _3-15840\\zeta _5)c_f\\nonumber \\\\&&+(70055+11344\\zeta _3-31680\\zeta _5)-1152(2-15\\zeta _3)d_2\\Big ] \\;,$ where we have denoted values of the Riemann Zeta function as $\\zeta _s=\\zeta (s)=\\sum _{n>0}n^{-s}$ .","At five loops, from app.",", we have LO and NLO large-$N_{\\mathrm {f}}$ terms to all orders, coinciding with the leading terms above, and predicting the first two terms of the 5-loop contributions as $6^5\\,\\gamma _{m4} &=& \\gamma _{m44}\\,\\Big [4n_f\\Big ]^4+\\gamma _{m43}\\,\\Big [4n_f\\Big ]^3+\\gamma _{m42}\\,\\Big [4n_f\\Big ]^2+\\gamma _{m41}\\,\\Big [4n_f\\Big ]+\\gamma _{m40}\\;,\\\\\\gamma _{m44} &=& -6(65+80\\zeta _3-144\\zeta _4)\\;,\\\\\\gamma _{m43} &=& 3(4483\\!+\\!4752\\zeta _3\\!-\\!12960\\zeta _4\\!+\\!6912\\zeta _5)c_f+(18667/2\\!+\\!32208\\zeta _3\\!+\\!29376\\zeta _4\\!-\\!55296\\zeta _5)\\;.$ We have evaluated the remaining coefficients, mapping all diagrams onto fully massive vacuum tadpoles; IBP-reducing them to master integrals; using high-precision numerical evaluations thereof plus some additional consistency conditions to fix linear combinations of 12-line master integrals as explained in section ; and finally employing PSLQ at 200 digits for discovery, and at 250 digits for confirmation, we confirm the two large-$N_{\\mathrm {f}}$ expressions in eqs.","() and (), and obtain the three missing coefficients of eq.","(REF ) in analytic form, containing only Zeta valuesTo make the group structure more visible, we resort to a vector notation here and below, where a dot between two curly brackets denotes a scalar product as e.g.","in $\\lbrace c_f,1\\rbrace .\\lbrace a,b\\rbrace =c_fa+b$ .",": $\\gamma _{m42} &=& \\Big \\lbrace c_f^2,c_f,d_1,1\\Big \\rbrace .\\Big \\lbrace 9 (45253-230496 \\zeta _{3}+48384 \\zeta _{3}^2+70416 \\zeta _{4}+144000 \\zeta _{5}-86400 \\zeta _{6}),\\nonumber \\\\&&375373+323784 \\zeta _{3}-1130112 \\zeta _{3}^2+905904 \\zeta _{4}-672192 \\zeta _{5}+129600 \\zeta _{6},\\nonumber \\\\&&-864 (431 - 1371 \\zeta _3 + 432 \\zeta _4 + 420 \\zeta _5),\\nonumber \\\\&&4 (13709+394749 \\zeta _{3}+173664 \\zeta _{3}^2-379242 \\zeta _{4}-119232 \\zeta _{5}+162000 \\zeta _{6})\\Big \\rbrace \\;,\\\\\\gamma _{m41} &=& \\Big \\lbrace c_f^3,c_f^2,c_fd_1,c_f,d_1,d_2,1\\Big \\rbrace .\\Big \\lbrace -54 (48797-247968 \\zeta _{3}+24192 \\zeta _{4}+444000 \\zeta _{5}-241920 \\zeta _{7}),\\nonumber \\\\&&-18 (406861+216156 \\zeta _{3}-190080 \\zeta _{3}^2+254880 \\zeta _{4}-606960 \\zeta _{5}-475200 \\zeta _{6}+362880 \\zeta _{7}),\\nonumber \\\\&&-62208 (11+154 \\zeta _{3}-370 \\zeta _{5}),\\nonumber \\\\&&753557+15593904 \\zeta _{3}-3535488 \\zeta _{3}^2-6271344 \\zeta _{4}-17596224 \\zeta _{5}+1425600 \\zeta _{6}+1088640 \\zeta _{7},\\nonumber \\\\&&1728 (3173-6270 \\zeta _{3}+1584 \\zeta _{3}^2+2970 \\zeta _{4}-13380 \\zeta _{5}),\\nonumber \\\\&&1728 (380 - 5595 \\zeta _3 - 1584 \\zeta _3^2 - 162 \\zeta _4 + 1320 \\zeta _5),\\\\&&-2 (4994047+11517108 \\zeta _{3}-57024 \\zeta _{3}^2-5931900 \\zeta _{4}-15037272 \\zeta _{5}+4989600 \\zeta _{6}+3810240 \\zeta _{7})\\Big \\rbrace \\;,\\nonumber \\\\\\gamma _{m40} &=& \\Big \\lbrace c_f^4,c_f^3,c_f^2,c_fd_2,c_f,d_2,d_3,1\\Big \\rbrace .\\Big \\lbrace 972 (50995+6784 \\zeta _{3}+16640 \\zeta _{5}),\\nonumber \\\\&&-54 (2565029+1880640 \\zeta _{3}-266112 \\zeta _{4}-1420800 \\zeta _{5}),\\nonumber \\\\&&108 (2625197+1740528 \\zeta _{3}-125136 \\zeta _{4}-2379360 \\zeta _{5}-665280 \\zeta _{7}),\\nonumber \\\\&&373248 (141+80 \\zeta _{3}-530 \\zeta _{5}),\\nonumber \\\\&&-8 (25256617+16408008 \\zeta _{3}+627264 \\zeta _{3}^2-812592 \\zeta _{4}-40411440 \\zeta _{5}+3920400 \\zeta _{6}-5987520 \\zeta _{7}),\\nonumber \\\\&&-6912 (9598+453 \\zeta _{3}+4356 \\zeta _{3}^2+1485 \\zeta _{4}-26100 \\zeta _{5}-1386 \\zeta _{7}),\\nonumber \\\\&&5184 (537 + 2494 \\zeta _3 + 5808 \\zeta _3^2 + 396 \\zeta _4 - 7820 \\zeta _5 - 1848 \\zeta _7),\\\\&&4 (22663417+\\!10054464 \\zeta _{3}+\\!1254528 \\zeta _{3}^2-\\!1695276 \\zeta _{4}-\\!41734440 \\zeta _{5}+\\!7840800 \\zeta _{6}+\\!5987520 \\zeta _{7})\\Big \\rbrace \\;.\\nonumber $ Let us now discuss some checks on this new result.","The authors of [7] have published the full 5-loop result for the case of SU(3).","To compare, we recall the definition of our expansion parameter in eq.","(REF ) and put all group invariants to their SU(3) values as given in eq.","(REF ); the $\\gamma _{m1..4}$ as listed above then coincide with the expressions given in [7].","Furthermore, the same group has very recently generalized their work to a general Lie group as well [32].","We have cross-checked their preliminary result with our $\\gamma _{m4}$ as given above, and found full agreement.","Since both five-loop results have been obtained with completely different methods (with the exception of also relying on qgraf for diagram generation, in [32] the 5-loop renormalization constants are mapped onto massless 4-loop two-point functions [33], [34], [35], and integral reduction is done via $1/d$ expansions [36], [37]), this agreement constitutes an extremely strong check."],["Quark field renormalization","For completeness, let us also present our new five-loop result for the quark field anomalous dimension $\\gamma _2 = -\\partial _{\\ln \\mu ^2}\\ln Z_2$ , where the renormalization constant $Z_2$ relates bare and renormalized quark fields as $\\psi _{\\rm bare}=\\sqrt{Z_2}\\psi _{\\rm ren}$ .","As opposed to the quark mass, this quantity is not physical and hence gauge dependent.","Lower-loop results can be found for SU(N) and covariant gauge in [38], and for a general Lie group with $\\xi ^0,\\xi ^1$ terms (which corresponds to a NLO expansion around Feynman gauge) in [39].","At five loops, $\\gamma _2$ is known for SU(3) in Feynman gauge [7], and we will below once more present the generalization to a general Lie group.","Up to four loops, we obtain ($\\xi $ being the covariant gauge parameter, such that the values $\\xi =0/1$ correspond to Feynman/Landau gauge) $\\gamma _2&=&-c_f\\,a\\,\\Big \\lbrace (1-\\xi )+\\gamma _{21}\\,a+\\gamma _{22}\\,a^2+\\gamma _{23}\\,a^3+\\gamma _{24}\\,a^4+\\dots \\Big \\rbrace \\;,\\\\2^2\\,\\gamma _{21} &=& n_f\\Big [-8\\Big ]+\\Big [-6c_f+34-10\\xi + \\xi ^2\\Big ]\\;,\\\\2^53^2\\,\\gamma _{22} &=& n_f^2\\Big [640\\Big ]+ n_f\\Big [8 (108 c_f- 1301 + 153 \\xi )\\Big ] \\\\&& + \\Big [432 c_f^2 -\\!","72 (143 \\!-\\!48 \\zeta _3)c_f+\\!2 (10559 -\\!1080 \\zeta _3) -\\!9 \\xi (371 +\\!48 \\zeta _3) +\\!27 \\xi ^2 (23 +\\!4 \\zeta _3) -\\!90 \\xi ^3 \\Big ]\\;,\\nonumber \\\\2^43^5\\,\\gamma _{23} &=&n_f^3 \\Big [ 13440 \\Big ]+ n_f^2 \\Big [ 6912 (19 - 18 \\zeta _3)c_f+ 16 (6835 + 9072 \\zeta _3) + 64 \\xi (269 - 324 \\zeta _3) \\Big ]\\nonumber \\\\&&+ n_f\\Big [5184 (19 - 48 \\zeta _3) c_f^2+ \\big (-108 (2407 - 1584 \\zeta _3 - 1296 \\zeta _4 - 5760 \\zeta _5)\\nonumber \\\\&&+ 324 \\xi (767 - 528 \\zeta _3 - 144 \\zeta _4) \\big )c_f+ 497664 d_1-(1365691 + 154224 \\zeta _3 + 97200 \\zeta _4 + 311040 \\zeta _5)\\nonumber \\\\&&+ \\xi (48865 + 152928 \\zeta _3 + 29160 \\zeta _4)- 54 \\xi ^2 (109 + 84 \\zeta _3 - 18 \\zeta _4)\\Big ]\\nonumber \\\\&&+\\Big [- 486 (1027 + 3200 \\zeta _3 - 5120 \\zeta _5) c_f^3+ 324 (5131 + 10176 \\zeta _3 - 17280 \\zeta _5) c_f^2\\nonumber \\\\&&+ \\big (-108 (23777 + 7704 \\zeta _3 + 2376 \\zeta _4 - 28440 \\zeta _5) - 1944 \\xi (6 - 7 \\zeta _3 + 10 \\zeta _5)\\big ) c_f\\nonumber \\\\&&+ 486 \\big (16 (-33 + 95 \\zeta _3 - 85 \\zeta _5) - 8 \\xi (1 + 48 \\zeta _3 - 70 \\zeta _5)- 8 \\xi ^2 (7\\zeta _3+5\\zeta _5)+ 20 \\xi ^3 (2\\zeta _3-\\zeta _5)\\nonumber \\\\&&- \\xi ^4 (7\\zeta _3-5\\zeta _5)\\big ) d_2+ (10059589/4 - 241218 \\zeta _3 + 168156 \\zeta _4 - 604260 \\zeta _5)\\nonumber \\\\&&- \\xi (2127929/8 + 164106 \\zeta _3 - 21141 \\zeta _4 - 107730 \\zeta _5)+ 27 \\xi ^2 (13883 + 9108 \\zeta _3 - 1548 \\zeta _4\\nonumber \\\\&&- 1920 \\zeta _5)/8- 81 \\xi ^3 (263 + 65 \\zeta _3 - 9 \\zeta _4 + 20 \\zeta _5)/2+ 81 \\xi ^4 (57 + \\zeta _3 + 10 \\zeta _5)/4\\Big ] \\;.$ We have presented the full gauge parameter dependence above.","Note that this fills a gap in the literature and constitutes new information at four loops: in [39] only the terms linear in $\\xi $ have been evaluated for a general Lie group, while with full gauge dependence only the SU(N) result is available [38].","However, due to the degeneracies $2d_1=6c_f^2-5c_f+13/12$ and $2d_2=7/12-c_f$ in the SU(N) limit, one cannot uniquely extract the Lie group structure from the latter reference.","Needless to say that our result for $\\gamma _{23}$ given in eq.","() reproduces the $\\xi ^0$ and $\\xi ^1$ terms given in [39], and in the SU(N) limit reduces to the respective expressions of [38], for all powers of $\\xi $ .","Expanding the all-order large-$N_{\\mathrm {f}}$ Landau gauge result of eq.","(REF ) in the coupling $a_f$ allows to confirm eqs.","(REF )-() to NLO in $n_f$ , and to predict the first two terms of $\\gamma _{24}$ in that gauge as $24^3\\,\\gamma _{24}|_{\\xi =1}&=&\\frac{83-144\\zeta _3}{72}\\,\\Big [16n_f\\Big ]^4+\\gamma _{243}^{\\,\\xi =1}\\,\\Big [16n_f\\Big ]^3+\\dots \\;,\\\\\\gamma _{243}^{\\,\\xi =1} &=& \\Big \\lbrace c_f, 1\\Big \\rbrace .\\Big \\lbrace -659/18 + 312 \\zeta _3 - 216 \\zeta _4, -1783/36 - 248 \\zeta _3 + 216 \\zeta _4 \\Big \\rbrace \\;.$ At five loops, along the same steps as explained in section , we have obtained the new Feynman gauge resultThe restriction to $\\xi =0$ is for practical reasons only.","To evaluate the $\\xi $ -dependent coefficients, one would need to enlarge the integral reduction tables as produced by crusher and TIDE to integrals with higher propagator powers (or dots), roughly one more dot per power of the gauge parameter.","Since the present calculation is at the limit of what the computing resources available to us are able to handle, we defer this to future work.","$24^3\\,\\gamma _{24} &=&\\frac{83-144\\zeta _3}{72}\\,\\Big [16n_f\\Big ]^4+\\gamma _{243}\\,\\Big [16n_f\\Big ]^3+\\gamma _{242}\\,\\Big [16n_f\\Big ]^2+\\gamma _{241}\\,\\Big [16n_f\\Big ]+\\gamma _{240}+{\\cal O}(\\xi )\\;,$ where the coefficients again contain only Zeta values up to weight 7, $\\gamma _{243} &=& \\Big \\lbrace c_f, 1\\Big \\rbrace .\\Big \\lbrace -659/18 + 312 \\zeta _3 - 216 \\zeta _4, -3443/48 - 255 \\zeta _3 + 252 \\zeta _4\\Big \\rbrace \\;,\\\\\\gamma _{242} &=& \\Big \\lbrace c_f^2, c_f, d_1, 1\\Big \\rbrace .\\Big \\lbrace -2 (2497 - 1200 \\zeta _3 + 3456 \\zeta _4 - 8640 \\zeta _5), \\nonumber \\\\&&477433/12 - 45636 \\zeta _3 + 4608 \\zeta _3^2 + 11448 \\zeta _4 - 65088 \\zeta _5 + 28800 \\zeta _6,-384 (115 - 33 \\zeta _3 - 90 \\zeta _5),\\nonumber \\\\&&3015955/72 + 69509 \\zeta _3 - 2304 \\zeta _3^2 - 12861 \\zeta _4 + 16662 \\zeta _5 - 14400 \\zeta _6 - 11907 \\zeta _7 \\Big \\rbrace \\;,\\\\\\gamma _{241} &=& \\Big \\lbrace c_f^3, c_f^2, c_fd_1, c_f, d_1, d_2, 1\\Big \\rbrace .\\Big \\lbrace 24 (29209 + 89984 \\zeta _3 + 12288 \\zeta _3^2 - 28800 \\zeta _4 - 187520 \\zeta _5 +76800 \\zeta _6), \\nonumber \\\\&& -4 (296177 + 517020 \\zeta _3 + 26784 \\zeta _3^2 - 469908 \\zeta _4 -4104720 \\zeta _5 + 1069200 \\zeta _6 + 3011904 \\zeta _7), \\nonumber \\\\&& -2304 (748 + 4536 \\zeta _3 -1368 \\zeta _3^2 - 6780 \\zeta _5 + 3255 \\zeta _7), \\nonumber \\\\&& 8 (115334 - 37764 \\zeta _3 -123012 \\zeta _3^2 - 49923 \\zeta _4 - 1124556 \\zeta _5 + 133650 \\zeta _6 +1519308 \\zeta _7), \\nonumber \\\\&& 192 (16732 + 39912 \\zeta _3 - 10944 \\zeta _3^2 - 72960 \\zeta _5 +36771 \\zeta _7), \\nonumber \\\\&&96 (6158 - 13952 \\zeta _3 - 372 \\zeta _3^2 + 2880 \\zeta _4 - 39475 \\zeta _5 - 3900 \\zeta _6 + 45696 \\zeta _7),\\\\&& -34919359/9 - 753797 \\zeta _3 + 548148 \\zeta _3^2 - 135063 \\zeta _4 + 1759474 \\zeta _5 +265350 \\zeta _6 - 2647806 \\zeta _7\\Big \\rbrace \\;,\\nonumber \\\\\\gamma _{240} &=& \\Big \\lbrace c_f^4, c_f^3, c_f^2, c_fd_2, c_f, d_2, d_3, 1\\Big \\rbrace .\\Big \\lbrace 1728 (4977 + 128000 \\zeta _3 + 19968 \\zeta _3^2 + 180800 \\zeta _5 -381024 \\zeta _7), \\nonumber \\\\&& -96 (835739 + 8494144 \\zeta _3 + 1182336 \\zeta _3^2 - 316800 \\zeta _4 +3983360 \\zeta _5 + 844800 \\zeta _6 - 17852688 \\zeta _7), \\nonumber \\\\&& 192 (825361 + 5472068 \\zeta _3 +651816 \\zeta _3^2 - 335808 \\zeta _4 - 1140420 \\zeta _5 + 950400 \\zeta _6 -8056377 \\zeta _7), \\nonumber \\\\&& 4608 (10 + 53226 \\zeta _3 - 15264 \\zeta _3^2 + 2145 \\zeta _5 -45885 \\zeta _7), \\; -16 (84040774/9 \\nonumber \\\\&& + 33396648 \\zeta _3 + 2804616 \\zeta _3^2 - 838782 \\zeta _4 - 18160944 \\zeta _5 +6252300 \\zeta _6 - 41015331 \\zeta _7), \\nonumber \\\\&& -384 (43066 + 628802 \\zeta _3 - 160998 \\zeta _3^2 + 36540 \\zeta _4 - 201125 \\zeta _5 -53475 \\zeta _6 - 403263 \\zeta _7), \\nonumber \\\\&&-72 (20566 - 218812 \\zeta _3 - 79080 \\zeta _3^2 - 13212 \\zeta _4 + 760220 \\zeta _5 + 20100 \\zeta _6 - 660667 \\zeta _7),\\; 804023630/9 \\nonumber \\\\&& + 101490400 \\zeta _3 + 3143352 \\zeta _3^2 + 7356024 \\zeta _4 -86186276 \\zeta _5 + 18372900 \\zeta _6 - 115799439 \\zeta _7 \\Big \\rbrace \\;.$ As a speculation, comparing the Landau gauge prediction eq.","(REF ) with the Feynman gauge result eq.","(REF ), the full result for $\\gamma _{243}$ could be simply adding $\\delta \\gamma _{243}=\\xi (3197/144+7\\zeta _3-36\\zeta _4)$ ; more generally however, consistency only requires that $\\delta \\gamma _{243}=c_f\\,u_1(\\xi )+u_2(\\xi )$ with $u_1(0)=0=u_1(1)$ as well as $u_2(0)=0$ and $u_2(1)=3197/144+7\\zeta _3-36\\zeta _4$ , the simplest choice being the one speculated above.","As a check, replacing the group invariants with the values of eq.","(REF ) in our Feynman gauge result for $\\gamma _{24}$ , we find perfect agreement with the known SU(3) result of [7] (see also eq.","(46) of [40])."],["Conclusions","We have provided new results for two fundamental renormalization coefficients, at five loops and for a semi-simple Lie group.","In particular, our expression for the gauge-invariant quark mass anomalous dimension $\\gamma _m$ coincides in various limits (large $N_{\\mathrm {f}}$ as well as SU(3)) with previously known results, and coincides exactly with recent results of another group [32].","We have also provided the Feynman gauge result for the quark field anomalous dimension $\\gamma _2$ , which again could be checked against known expressions in the abovementioned limits.","From these two quantities, one can reconstruct the two renormalization constants $Z_m$ and $Z_2$ of the quark sector, an electronic version of which is available by downloading the source of this article from http://arXiv.org/abs/1612.05512.","As had already been observed in [5], looking at e.g.","the 4-loop result for the quark mass anomalous dimension eq.","(REF ), all Zeta terms (and also the higher group invariants $d_n$ ) vanish at $\\lbrace c_f=1,n_f=\\frac{1}{2},d_1=d_2\\rbrace $ , which corresponds to ${\\cal N}=1$ supersymmetry.","The same had happened for the 4-loop Beta function (generalizing the last condition to $d_1=d_2=d_3$ ).","For these parameters values, from section we have $\\gamma _m &=& -a\\Big \\lbrace 3+\\!16\\,a+\\!\\tfrac{310}{3}\\,a^2+\\!\\tfrac{2228}{3}\\,a^3+\\!\\Big (\\tfrac{671075}{108} -\\!194\\,d_1+\\!\\big (\\tfrac{1483}{2} -\\!2028\\,d_1\\big ) \\zeta _3-\\!20\\big (55 +\\!354\\,d_1\\big ) \\zeta _5\\Big ) a^4\\Big \\rbrace \\;,\\nonumber $ where we observe that the cancellation pattern does not hold through five loops – although the structure becomes much simpler, due to cancellation of all terms containing $\\lbrace \\zeta _3^2,\\zeta _4,\\zeta _6,\\zeta _7\\rbrace $ .","To conclude the renormalization program at five loops, one needs to determine three more renormalization constants, which can be chosen to be those of the gluon and ghost fields, $Z_3$ and $Z_3^{c}$ , respectively, and of the ghost-gluon vertex $Z_1^{ccg}$ .","From these, due to gauge invariance, one can then construct the renormalization constant for the gauge coupling (aka the Beta function, whose five-loop coefficient is so far known for SU(3) only [41]) as well as those for the remaining vertices.","While the same methods that we have used here are sufficient to calculate those missing coefficients (which indeed already led to the NLO terms at large $N_{\\mathrm {f}}$ [8]), due to the complexity of the determination of $Z_3$ we leave the evaluation of the full coefficients for future work.","We are indebted to K. Chetyrkin for sending us their new five-loop results for $\\gamma _m$ prior to publication [32], to enable the important check discussed at the end of section .","The work of T.L.","has been supported in part by DFG grants GRK 881 and SCHR 993/2.","A.M. is supported by a European Union COFUND/Durham Junior Research Fellowship under EU grant agreement number 267209.","P.M. was supported in part by the EU Network HIGGSTOOLS PITN-GA-2012-316704.","Y.S.","acknowledges support from FONDECYT project 1151281 and UBB project GI-152609/VC.","All diagrams were drawn with Axodraw [42], [43]."],["Summary of large-$N_{\\mathrm {f}}$ results","Some coefficients of QCD anomalous dimensions are known to all loop orders, from a large $N_{\\mathrm {f}}$ expansion.","Taking $n_f$ and $c_f$ as above and defining $a_f\\equiv \\frac{N_{\\mathrm {f}}T_{\\mathrm {F}}g^2(\\mu )}{12\\pi ^2}\\;=\\;\\frac{4\\,n_f\\,a}{3}\\quad ,\\qquad \\eta (\\varepsilon )\\equiv \\frac{(2\\varepsilon -3)\\Gamma (4-2\\varepsilon )}{16\\Gamma ^2(2-\\varepsilon )\\Gamma (3-\\varepsilon )\\Gamma (\\varepsilon )}\\;,$ the all-order leading-$N_{\\mathrm {f}}$ [44] and next-to-leading-$N_{\\mathrm {f}}$ [45], [46] expressions that we need here read $\\gamma _m&=&\\frac{4c_f}{n_f}\\Big \\lbrace \\eta (a_f)+\\frac{\\eta _3(a_f)}{8\\,n_f}+{\\cal O}\\Big (\\frac{1}{n_f^2}\\Big )\\Big \\rbrace \\;,\\\\\\gamma _2|_{\\xi =1}&=&-\\frac{2a_fc_f}{n_f}\\Big \\lbrace \\eta (a_f)+\\frac{1}{n_f}\\frac{\\eta _4(a_f)}{4a_f}+{\\cal O}\\Big (\\frac{1}{n_f^2}\\Big )\\Big \\rbrace \\;,$ where the fact that the asymptotic expansions have been performed in Landau gauge only does not affect the physical and gauge invariant quark mass anomalous dimension $\\gamma _m$ .","To define the coefficient functions $\\eta _3$ and $\\eta _4$ , it is convenient to define the linear combinations $\\Psi (\\varepsilon ) &=& \\psi _0(1 - 2 \\varepsilon ) + \\psi _0(1 + \\varepsilon ) -\\psi _0(1 - \\varepsilon ) - \\psi _0(1) \\;,\\\\\\Phi (\\varepsilon ) &=& \\psi _1(1 - 2 \\varepsilon ) - \\psi _1(1 + \\varepsilon ) -\\psi _1(1 - \\varepsilon ) + \\psi _1(1) \\;,\\\\\\Theta (\\varepsilon ) &=& \\psi _1(1 - \\varepsilon ) - \\psi _1(1) \\;,$ where $\\psi _n(x)=\\partial _x^{n+1}\\ln \\Gamma (x)$ is the PolyGamma function.","Then [45], [46] $\\eta _3(\\varepsilon ) &\\equiv &\\Big (\\!-\\!\\frac{11}{4} +\\!","\\sum _{n>0}\\frac{f_n\\varepsilon ^n}{n}\\Big ) 8 \\varepsilon \\partial _\\varepsilon \\eta (\\varepsilon )-\\frac{16 \\eta ^2(\\varepsilon )}{(3 \\!-\\!","2 \\varepsilon )(1 \\!-\\!","\\varepsilon )}\\Big \\lbrace 3 (2 - \\varepsilon )^2 (1 - \\varepsilon )^2 \\Theta (\\varepsilon )-(5 + 5 \\varepsilon - 11 \\varepsilon ^2 + 4 \\varepsilon ^3) ,\\nonumber \\\\&&(88 \\!-\\!","372 \\varepsilon \\!+\\!","551 \\varepsilon ^2 \\!-\\!","380 \\varepsilon ^3 \\!+\\!","160 \\varepsilon ^4 \\!-\\!","64 \\varepsilon ^5 \\!+\\!","16 \\varepsilon ^6)\\!-\\!","4 \\varepsilon (3 \\!-\\!","2 \\varepsilon ) (1 \\!-\\!","2 \\varepsilon ) (2 \\!-\\!","\\varepsilon ) (1 \\!-\\!","\\varepsilon )^2 \\big (\\Psi ^2(\\varepsilon ) \\!+\\!","\\Phi (\\varepsilon )\\big )\\nonumber \\\\&&+ 2 (1 - \\varepsilon ) (24 - 144 \\varepsilon + 249 \\varepsilon ^2 - 146 \\varepsilon ^3 + 12 \\varepsilon ^4 + 8 \\varepsilon ^5) \\Psi (\\varepsilon )\\Big \\rbrace .\\Big \\lbrace \\tfrac{2}{2 - \\varepsilon }\\,c_f, \\tfrac{1}{4\\varepsilon (3 - 2 \\varepsilon )(1 - 2 \\varepsilon )}\\Big \\rbrace \\;,\\\\\\eta _4(\\varepsilon ) &=& \\frac{\\varepsilon \\eta _3(\\varepsilon )}{2}+\\Big (\\!-\\!\\frac{11}{4} +\\!","\\sum _{n>0}\\frac{f_n\\varepsilon ^n}{n}\\Big ) 4\\varepsilon \\eta (\\varepsilon )+\\frac{2\\eta ^2(\\varepsilon )}{3-2\\varepsilon } \\Big \\lbrace -8(1-4\\varepsilon +2\\varepsilon ^2),\\tfrac{(2-5\\varepsilon +2\\varepsilon ^2)^2}{1-\\varepsilon }\\Big \\rbrace .","\\Big \\lbrace c_f,1\\Big \\rbrace \\;,\\\\{\\rm with}&&\\sum _{j>0}f_j\\,\\varepsilon ^j \\;\\equiv \\; -\\eta (\\varepsilon )\\Big \\lbrace 4(1+\\varepsilon )(1-2\\varepsilon )c_f+\\tfrac{4\\varepsilon ^4-14\\varepsilon ^3+32\\varepsilon ^2-43\\varepsilon +20}{(1-\\varepsilon )(3-2\\varepsilon )}\\Big \\rbrace \\;.$ The coefficients $a_f$ , $\\eta (\\varepsilon )$ and $f_j$ are the same as we had defined in [8]."]],"string":"[\n [\n \"Five-loop quark mass and field anomalous dimensions for a general gauge\\n group\"\n ],\n [\n \"Abstract We present analytical five-loop results for the quark mass and quark field anomalous dimensions, for a general gauge group and in the MSbar scheme.\",\n \"We confirm the values known for the gauge group SU(3) from an independent calculation, and find full agreement with results available from large-Nf studies.\"\n ],\n [\n \"Introduction\",\n \"High-precision determinations of Standard Model (SM) parameters are crucially important for precise predictions for the observables measured in collider physics experiments, such as currently performed at the LHC, or possibly at a future linear collider.\",\n \"A thorough comparison of these predictions with experimental results allows to scrutinize the details of the hugely successful SM, and might shed light on possible physics beyond the SM.\",\n \"In order to pin down the theory's fundamental parameters, such as coupling constants and masses, with sufficient precision, the knowledge of higher order perturbative corrections is required.\",\n \"This encompasses the evaluation of multi-loop Feynman diagrams and Feynman integrals, for which significant progress has been made in recent years, mainly with respect to the formulation of advanced algorithms that allow to treat the complexity level met in those higher-loop integrals.\",\n \"We focus here on the strong interactions, which are embedded into the SM via Quantum Chromodynamics (QCD).\",\n \"The relevant parameters are then the (strong) gauge coupling and the quark masses, both of which run with the energy scale according to the renormalization group (RG) equations.\",\n \"In order to evolve e.g.\",\n \"the low-energy value of the coupling constant (measured with high precision from tau lepton decay) to high energies, the anomalous dimension of the gauge coupling (the so-called Beta function) is needed, as coefficient in the corresponding RG equation.\",\n \"Likewise, a high-order evaluation of the quark mass anomalous dimension gives access to precise values for e.g.\",\n \"charm and bottom quark masses, which are measured at low energies (typically a few GeV) but whose uncertainty at the high-energy scale of the Higgs mass $m_H=125$ GeV is important in Higgs decay rates into such quark pairs.\",\n \"In particular, to match the precision of the known five-loop inclusive decay width of $H\\\\rightarrow q\\\\bar{q}$ [1], one should evolve the parameters (which are $\\\\alpha _s(\\\\mu )$ and the running quark mass $m_q(\\\\mu )$ with $\\\\mu $ being the renormalization scale) from low energies to $\\\\mu =m_H$ at the same perturbative order, for full consistency and to avoid large logarithms $\\\\ln (\\\\mu ^2/m_H^2)$ .\",\n \"Given this clear phenomenological motivation, we will present new results for the quark mass anomalous dimension here, applying a number of the above-mentioned algorithmic advances.\",\n \"While this renormalization constant has been studied previously up to five loops in perturbation theory [2], [3], [4], [5], [6], [7], we generalize it from the gauge group SU(3) to a semi-simple Lie group.\",\n \"At the same time, we provide a truly independent check on the available SU(3) result, since we utilize largely independent methods, as described below.\",\n \"We will also give results for a related quantity needed to renormalize the quark sector at five loops, namely the quark field anomalous dimension in Feynman gauge, again generalizing known SU(3) results to a semi-simple Lie group.\",\n \"The paper is organized as follows.\",\n \"We start by explaining our computational setup in section .\",\n \"Using some notation defined in section , we then present and discuss results in sections and , before concluding in section .\",\n \"For convenience, an appendix reproduces large-$N_{\\\\mathrm {f}}$ results from the literature, which we use for consistency checks.\"\n ],\n [\n \"Setup\",\n \"Let us start by explaining our computational setup, which closely follows the one employed and tested in [8].\",\n \"We base our highly automated setup on the diagram generator qgraf [9], [10] and various in-house FORM [11], [12], [13] codes.\",\n \"After generating the required fermionic 2-point functions, we apply projectors and perform the group algebra with color [14].\",\n \"To extract the ultraviolet (UV) divergences, we then use the freedom to change the low-momentum behavior and make all propagators massive, which regulates the infrared at the cost of introducing a new counterterm for the unphysical regulator mass.\",\n \"Expanding to sufficient depth in the external momentum [15], [16], [17] and keeping all potentially UV divergent structures results in nullifying the external momentum.\",\n \"The coefficients of this expansion can then be mapped onto a family of fully massive vacuum integrals, which are labelled by 15 indices (corresponding to maximally 12 propagators plus 3 scalar products) at five loops [18].\",\n \"As a next and fairly time-consuming step, we reduce those integrals to a small set of master integrals, powered by our own codes crusher [19] and TIDE [18], which are based on integration-by-parts (IBP) identities [20] and use a Laporta-type algorithm [21] for a systematic integral reduction.\",\n \"At five loops, we end up with a set of 110 master integrals.\",\n \"These have been evaluated in an $\\\\varepsilon $ -expansion around $d=4-2\\\\varepsilon $ dimensions in previous works [18], [22], using an approach based on IBP reductions and difference equations [21] that has been realized in C++ and uses Fermat [23] to perform the polynomial algebra that arises in solving systems of linear equations with large rational coefficients.\",\n \"The resulting high-precision numerical results for the coefficients of the $\\\\varepsilon $ -expansions finally allow us to utilize the integer-relation finding algorithm PSLQ [24] to discover the analytic content of some of these numbers, and to find relations between others.\",\n \"As a consequence, we are able to provide all our results given below in analytic form.\",\n \"Figure: 5-loop master integrals with 12 lines that contribute to eqs. ()-().\",\n \"Each line denotes a massive propagator 1/(k 2 +m 2 )1/(k^2+m^2), and each dot stands for an extra power of the corresponding propagator.As has already been mentioned elsewhere [8], our high-precision evaluation of all 5-loop master integrals has not yet produced results for the 12-line families, see figure REF .\",\n \"These do not contain divergences in four dimensions, and could therefore be avoided in evaluations of anomalous dimensions.\",\n \"It turns out, however, that with our integral reduction criteria and lexicographic ordering prescription, we do get contributions from these integral classes since they are multiplied by prefactors with spurious poles as $d\\\\rightarrow 4$ .\",\n \"To fix the values of the 12-line master integrals that we need, we first note that in all our results, only three independent linear combinations appear: $\\\\ell _0&=& 689 \\\\,I_{30527.1.1, 5} - 3934 \\\\,I_{30527.1.2, 5} + 5464 \\\\,I_{30527.1.3, 5} + 1152 \\\\,I_{30527.1.4, 5} - 5228 \\\\,I_{30527.6.3, 5} \\\\;,\\\\\\\\\\\\ell _1&=& 689 \\\\,I_{30527.1.1, 6} - 3934 \\\\,I_{30527.1.2, 6} + 5464 \\\\,I_{30527.1.3, 6} + 1152 \\\\,I_{30527.1.4, 6} - 5228 \\\\,I_{30527.6.3, 6} \\\\nonumber \\\\\\\\&&+ 1968 \\\\,I_{30527.1.1, 5} + 3890 \\\\,I_{30527.1.2, 5} + 3844 \\\\,I_{30527.1.3, 5} - 6912 \\\\,I_{30527.1.4, 5} - 70 \\\\,I_{30527.6.3, 5} \\\\;,\\\\\\\\\\\\ell _2&=& 11 \\\\,I_{31740.1.1, 5} - 72 \\\\,I_{31740.1.3, 5} \\\\;.$ Here, each integral $I_{\\\\#,n}$ corresponds to the $\\\\varepsilon ^n$ -coefficient of the respective fully massive master integral of figure REF , divided by the fifth power of the 1-loop tadpole $J$ for normalization reasons$J=\\\\int {\\\\rm d}^dk/(k^2+m^2)\\\\sim \\\\Gamma (1-d/2)$ has a simple pole as $d\\\\rightarrow 4$ ; hence, finite 5-loop terms correspond to $\\\\varepsilon ^5$ ..\",\n \"While it is conceivable that there exists a suitable basis transformation that eliminates these linear combinations altogether from the final results, we have not yet performed a systematic search of such transformations in our integral reduction tables, but opted for other criteria to fix the numerical values of the three linear combinations, with high precision, as we will explain now.\",\n \"As a crude order-of-magnitude estimate, we have evaluated the set of 12-line integrals via Feynman parametric representations (see e.g.\",\n \"[25]) and subsequent (primary) sector decomposition, using the strategy explained in [26], [27] and as implemented in FIESTA [28] as well as own code (see [29]).\",\n \"Due to the large prefactors and cancellations in eqs.\",\n \"(REF )-(), a 6-digit evaluation results in the estimates $\\\\ell _0\\\\approx -7.47(1)\\\\;,\\\\quad \\\\ell _1\\\\approx -50.6(1)\\\\;,\\\\quad \\\\ell _2\\\\approx -0.673(1)\\\\;,$ with 3-digit accuracy.\",\n \"Turning now to high-precision evaluations, we first recall that the higher-order $\\\\varepsilon $ -poles of renormalization constants are completely determined by lower-order coefficients.\",\n \"Checking these, we obtain one consistency condition that allows to fix $\\\\ell _0$ .\",\n \"Second, we observe the occurrence of some rank-12 group invariants in another renormalization constant that we have evaluated using the same setup, namely for the ghost-gluon vertex [30].\",\n \"Using eq.\",\n \"(REF ), to determine their coefficients, we find that they vanish at least to this low accuracy.\",\n \"Taking this zero for granted, we turn the argument around and require e.g.\",\n \"the structure $d^{\\\\,444}_{FAA}\\\\,N_{\\\\mathrm {f}}$ (where the group invariant is in the notation of [14]) to be absent from the final result; this gives us another condition, fixing $\\\\ell _2$ .\",\n \"Third, for fixing the remaining linear combination, we choose to compare our results in the SU(3) limit to the previously known 5-loop results.\",\n \"To be concrete, out of the many possible coefficients we choose the $n_f$ term of $\\\\gamma _m$ as given in [7], giving us one constraint which fixes $\\\\ell _1$ .\",\n \"Along these lines, we obtain numerical values for all three linear combinations with 260 digits, the first 50 of which read $\\\\ell _0&=& -7.4750787021276651819913288152084850401974826928834\\\\dots \\\\;,\\\\\\\\\\\\ell _1&=& -50.563714841071996428539372592222326105092965639946\\\\dots \\\\;,\\\\\\\\\\\\ell _2&=& -0.67332086607447050046759024439428336720209195028580\\\\dots \\\\;,$ and which can be seen to be consistent with our low-precision estimates of eq.\",\n \"(REF ) that had been obtained by direct integration.\"\n ],\n [\n \"Notation\",\n \"Let us fix our notation here: we work with a semi-simple Lie algebra with hermitian generators $T^a$ , whose real and antisymmetric structure constants $f^{abc}$ are fixed by the commutation relation $[T^a,T^b]=i f^{abc}T^c$ .\",\n \"As usual, the quadratic Casimir operators of the fundamental (adjoint) representation (of dimensions $N_{\\\\mathrm {F}}$ and $N_{\\\\mathrm {A}}$ , respectively) are defined as $T^aT^a=C_{\\\\mathrm {F}}1\\\\!\\\\!1$ ($f^{acd}f^{bcd}=C_{\\\\mathrm {A}}\\\\delta ^{ab}$ ).\",\n \"Furthermore, traces are normalized as ${\\\\rm Tr}(T^aT^b)=T_{\\\\mathrm {F}}\\\\delta ^{ab}$ , we denote the number of quark flavors with $N_{\\\\mathrm {f}}$ , and find it convenient to define the following normalized combinations: $c_f=\\\\frac{C_{\\\\mathrm {F}}}{C_{\\\\mathrm {A}}} \\\\quad ,\\\\quad n_f=\\\\frac{N_{\\\\mathrm {f}}\\\\,T_{\\\\mathrm {F}}}{C_{\\\\mathrm {A}}} \\\\;.$ In our multi-loop diagrams, we will encounter traces of more than two group generators, giving rise to higher-order group invariants.\",\n \"It is useful to define traces over combinations of symmetric tensors [14], of which we need the following (writing $[F^a]_{bc}=-if^{abc}$ for the generators of the adjoint representation): $d_1=\\\\frac{[{\\\\rm sTr}(T^aT^bT^cT^d)]^2}{N_{\\\\mathrm {A}}T_{\\\\mathrm {F}}^2C_{\\\\mathrm {A}}^2} \\\\;,\\\\;d_2=\\\\frac{{\\\\rm sTr}(T^aT^bT^cT^d)\\\\,{\\\\rm sTr}(F^aF^bF^cF^d)}{N_{\\\\mathrm {A}}T_{\\\\mathrm {F}}C_{\\\\mathrm {A}}^3} \\\\;,\\\\;d_3=\\\\frac{[{\\\\rm sTr}(F^aF^bF^cF^d)]^2}{N_{\\\\mathrm {A}}C_{\\\\mathrm {A}}^4} \\\\;.$ Here, ${\\\\rm sTr}$ stands for a fully symmetrized trace (such that ${\\\\rm sTr}(ABC)=\\\\frac{1}{2}{\\\\rm Tr}(ABC+ACB)$ etc.).\",\n \"While dealing with the quark sector, it might seem more natural to normalize these traces with respect to the dimension of the fundamental representation; to this end, we note the relation $N_{\\\\mathrm {A}}T_{\\\\mathrm {F}}=N_{\\\\mathrm {F}}C_{\\\\mathrm {F}}$ which holds in general [14], [31].\",\n \"For the gauge group SU($N$ ) (where $T_{\\\\mathrm {F}}=\\\\frac{1}{2}$ and $C_{\\\\mathrm {A}}=N$ ), the normalized group invariants introduced above read [14] $n_f=\\\\frac{N_{\\\\mathrm {f}}}{2N}\\\\;,\\\\quad c_f=\\\\frac{N^2-1}{2N^2}\\\\;,\\\\quad d_1=\\\\frac{N^4-6N^2+18}{24N^4}\\\\;,\\\\quad d_2=\\\\frac{N^2+6}{24N^2}\\\\;,\\\\quad d_3=\\\\frac{N^2+36}{24N^2}\\\\;.$ The corresponding SU(3) values, relevant for physical QCD, hence read $\\\\mbox{SU}(3):\\\\quad n_f=\\\\frac{N_{\\\\mathrm {f}}}{6}\\\\;,\\\\quad c_f=\\\\frac{4}{9}\\\\;,\\\\quad d_1=\\\\frac{5}{216}\\\\;,\\\\quad d_2=\\\\frac{5}{72}\\\\;,\\\\quad d_3=\\\\frac{5}{24}\\\\;.$\"\n ],\n [\n \"Quark mass renormalization\",\n \"The renormalization constant for the quark mass $m_{\\\\rm bare}=Z_m\\\\,m_{\\\\rm ren}$ , or equivalently its anomalous dimension $\\\\gamma _m = -\\\\partial _{\\\\ln \\\\mu ^2}\\\\ln Z_m$ , has been known at two [2] and three loops [3], [4] for a long time.\",\n \"At four loops, $\\\\gamma _m$ is known for SU(N) and QED [6] as well as for a general Lie group [5].\",\n \"Presently, at five loops only the SU(3) value is publicly available [7].\",\n \"We present our corresponding result for a general Lie group below.\",\n \"The structure of the quark mass anomalous dimension is $\\\\partial _{\\\\ln \\\\mu ^2}\\\\ln m_{\\\\rm q}(\\\\mu ) &\\\\equiv & \\\\gamma _m(a) \\\\;=\\\\;-c_f\\\\,a\\\\,\\\\Big \\\\lbrace 3 +\\\\gamma _{m1}\\\\,a +\\\\gamma _{m2}\\\\,a^2 +\\\\gamma _{m3}\\\\,a^3 +\\\\gamma _{m4}\\\\,a^4 +\\\\dots \\\\Big \\\\rbrace \\\\;,\\\\\\\\a&\\\\equiv &\\\\frac{C_{\\\\mathrm {A}}\\\\,g^2(\\\\mu )}{16\\\\pi ^2}\\\\;,$ with $g(\\\\mu )$ being the renormalized QCD gauge coupling constant that depends on the renormalization scale $\\\\mu $ (we prefer to use the expansion parameter $a$ which is nothing but a rescaled version of the renormalized strong coupling constant $\\\\alpha _s=\\\\frac{g^2(\\\\mu )}{4\\\\pi }$ ).\",\n \"We work in $d=4-2\\\\varepsilon $ dimensions and employ the ${\\\\overline{\\\\mbox{\\\\rm {MS}}}}$ scheme.\",\n \"The coefficients $\\\\gamma _{mn}$ are polynomials in $n_f$ and can be written in terms of our normalized group factors.\",\n \"Up to four loops, they read [5] $3^1\\\\,\\\\gamma _{m1} &=& n_f\\\\Big [-10\\\\Big ]+\\\\Big [(9c_f+97)/2\\\\Big ] \\\\;,\\\\\\\\3^3\\\\,\\\\gamma _{m2} &=& n_f^2\\\\Big [\\\\!-\\\\!140\\\\Big ] +n_f\\\\Big [54(24\\\\zeta _3\\\\!-\\\\!23)c_f-4(139\\\\!+\\\\!324\\\\zeta _3)\\\\Big ]+\\\\Big [(6966c_f^2\\\\!-\\\\!3483c_f\\\\!+\\\\!11413)/4\\\\Big ],\\\\\\\\3^4\\\\,\\\\gamma _{m3} &=& n_f^3\\\\Big [-8(83-144\\\\zeta _3)\\\\Big ] +n_f^2\\\\Big [48(19-270\\\\zeta _3+162\\\\zeta _4)c_f+2(671+6480\\\\zeta _3-3888\\\\zeta _4)\\\\Big ]\\\\nonumber \\\\\\\\&&+n_f\\\\Big [-216(35-207\\\\zeta _3+180\\\\zeta _5)c_f^2-3(8819-9936\\\\zeta _3+7128\\\\zeta _4-2160\\\\zeta _5)c_f\\\\nonumber \\\\\\\\&&-(65459/2+72468\\\\zeta _3-21384\\\\zeta _4-32400\\\\zeta _5)+2592(2-15\\\\zeta _3)d_1\\\\Big ]\\\\nonumber \\\\\\\\&&+\\\\tfrac{9}{8}\\\\Big [-9(1261+2688\\\\zeta _3)c_f^3+6(15349+3792\\\\zeta _3)c_f^2-2(34045+5472\\\\zeta _3-15840\\\\zeta _5)c_f\\\\nonumber \\\\\\\\&&+(70055+11344\\\\zeta _3-31680\\\\zeta _5)-1152(2-15\\\\zeta _3)d_2\\\\Big ] \\\\;,$ where we have denoted values of the Riemann Zeta function as $\\\\zeta _s=\\\\zeta (s)=\\\\sum _{n>0}n^{-s}$ .\",\n \"At five loops, from app.\",\n \", we have LO and NLO large-$N_{\\\\mathrm {f}}$ terms to all orders, coinciding with the leading terms above, and predicting the first two terms of the 5-loop contributions as $6^5\\\\,\\\\gamma _{m4} &=& \\\\gamma _{m44}\\\\,\\\\Big [4n_f\\\\Big ]^4+\\\\gamma _{m43}\\\\,\\\\Big [4n_f\\\\Big ]^3+\\\\gamma _{m42}\\\\,\\\\Big [4n_f\\\\Big ]^2+\\\\gamma _{m41}\\\\,\\\\Big [4n_f\\\\Big ]+\\\\gamma _{m40}\\\\;,\\\\\\\\\\\\gamma _{m44} &=& -6(65+80\\\\zeta _3-144\\\\zeta _4)\\\\;,\\\\\\\\\\\\gamma _{m43} &=& 3(4483\\\\!+\\\\!4752\\\\zeta _3\\\\!-\\\\!12960\\\\zeta _4\\\\!+\\\\!6912\\\\zeta _5)c_f+(18667/2\\\\!+\\\\!32208\\\\zeta _3\\\\!+\\\\!29376\\\\zeta _4\\\\!-\\\\!55296\\\\zeta _5)\\\\;.$ We have evaluated the remaining coefficients, mapping all diagrams onto fully massive vacuum tadpoles; IBP-reducing them to master integrals; using high-precision numerical evaluations thereof plus some additional consistency conditions to fix linear combinations of 12-line master integrals as explained in section ; and finally employing PSLQ at 200 digits for discovery, and at 250 digits for confirmation, we confirm the two large-$N_{\\\\mathrm {f}}$ expressions in eqs.\",\n \"() and (), and obtain the three missing coefficients of eq.\",\n \"(REF ) in analytic form, containing only Zeta valuesTo make the group structure more visible, we resort to a vector notation here and below, where a dot between two curly brackets denotes a scalar product as e.g.\",\n \"in $\\\\lbrace c_f,1\\\\rbrace .\\\\lbrace a,b\\\\rbrace =c_fa+b$ .\",\n \": $\\\\gamma _{m42} &=& \\\\Big \\\\lbrace c_f^2,c_f,d_1,1\\\\Big \\\\rbrace .\\\\Big \\\\lbrace 9 (45253-230496 \\\\zeta _{3}+48384 \\\\zeta _{3}^2+70416 \\\\zeta _{4}+144000 \\\\zeta _{5}-86400 \\\\zeta _{6}),\\\\nonumber \\\\\\\\&&375373+323784 \\\\zeta _{3}-1130112 \\\\zeta _{3}^2+905904 \\\\zeta _{4}-672192 \\\\zeta _{5}+129600 \\\\zeta _{6},\\\\nonumber \\\\\\\\&&-864 (431 - 1371 \\\\zeta _3 + 432 \\\\zeta _4 + 420 \\\\zeta _5),\\\\nonumber \\\\\\\\&&4 (13709+394749 \\\\zeta _{3}+173664 \\\\zeta _{3}^2-379242 \\\\zeta _{4}-119232 \\\\zeta _{5}+162000 \\\\zeta _{6})\\\\Big \\\\rbrace \\\\;,\\\\\\\\\\\\gamma _{m41} &=& \\\\Big \\\\lbrace c_f^3,c_f^2,c_fd_1,c_f,d_1,d_2,1\\\\Big \\\\rbrace .\\\\Big \\\\lbrace -54 (48797-247968 \\\\zeta _{3}+24192 \\\\zeta _{4}+444000 \\\\zeta _{5}-241920 \\\\zeta _{7}),\\\\nonumber \\\\\\\\&&-18 (406861+216156 \\\\zeta _{3}-190080 \\\\zeta _{3}^2+254880 \\\\zeta _{4}-606960 \\\\zeta _{5}-475200 \\\\zeta _{6}+362880 \\\\zeta _{7}),\\\\nonumber \\\\\\\\&&-62208 (11+154 \\\\zeta _{3}-370 \\\\zeta _{5}),\\\\nonumber \\\\\\\\&&753557+15593904 \\\\zeta _{3}-3535488 \\\\zeta _{3}^2-6271344 \\\\zeta _{4}-17596224 \\\\zeta _{5}+1425600 \\\\zeta _{6}+1088640 \\\\zeta _{7},\\\\nonumber \\\\\\\\&&1728 (3173-6270 \\\\zeta _{3}+1584 \\\\zeta _{3}^2+2970 \\\\zeta _{4}-13380 \\\\zeta _{5}),\\\\nonumber \\\\\\\\&&1728 (380 - 5595 \\\\zeta _3 - 1584 \\\\zeta _3^2 - 162 \\\\zeta _4 + 1320 \\\\zeta _5),\\\\\\\\&&-2 (4994047+11517108 \\\\zeta _{3}-57024 \\\\zeta _{3}^2-5931900 \\\\zeta _{4}-15037272 \\\\zeta _{5}+4989600 \\\\zeta _{6}+3810240 \\\\zeta _{7})\\\\Big \\\\rbrace \\\\;,\\\\nonumber \\\\\\\\\\\\gamma _{m40} &=& \\\\Big \\\\lbrace c_f^4,c_f^3,c_f^2,c_fd_2,c_f,d_2,d_3,1\\\\Big \\\\rbrace .\\\\Big \\\\lbrace 972 (50995+6784 \\\\zeta _{3}+16640 \\\\zeta _{5}),\\\\nonumber \\\\\\\\&&-54 (2565029+1880640 \\\\zeta _{3}-266112 \\\\zeta _{4}-1420800 \\\\zeta _{5}),\\\\nonumber \\\\\\\\&&108 (2625197+1740528 \\\\zeta _{3}-125136 \\\\zeta _{4}-2379360 \\\\zeta _{5}-665280 \\\\zeta _{7}),\\\\nonumber \\\\\\\\&&373248 (141+80 \\\\zeta _{3}-530 \\\\zeta _{5}),\\\\nonumber \\\\\\\\&&-8 (25256617+16408008 \\\\zeta _{3}+627264 \\\\zeta _{3}^2-812592 \\\\zeta _{4}-40411440 \\\\zeta _{5}+3920400 \\\\zeta _{6}-5987520 \\\\zeta _{7}),\\\\nonumber \\\\\\\\&&-6912 (9598+453 \\\\zeta _{3}+4356 \\\\zeta _{3}^2+1485 \\\\zeta _{4}-26100 \\\\zeta _{5}-1386 \\\\zeta _{7}),\\\\nonumber \\\\\\\\&&5184 (537 + 2494 \\\\zeta _3 + 5808 \\\\zeta _3^2 + 396 \\\\zeta _4 - 7820 \\\\zeta _5 - 1848 \\\\zeta _7),\\\\\\\\&&4 (22663417+\\\\!10054464 \\\\zeta _{3}+\\\\!1254528 \\\\zeta _{3}^2-\\\\!1695276 \\\\zeta _{4}-\\\\!41734440 \\\\zeta _{5}+\\\\!7840800 \\\\zeta _{6}+\\\\!5987520 \\\\zeta _{7})\\\\Big \\\\rbrace \\\\;.\\\\nonumber $ Let us now discuss some checks on this new result.\",\n \"The authors of [7] have published the full 5-loop result for the case of SU(3).\",\n \"To compare, we recall the definition of our expansion parameter in eq.\",\n \"(REF ) and put all group invariants to their SU(3) values as given in eq.\",\n \"(REF ); the $\\\\gamma _{m1..4}$ as listed above then coincide with the expressions given in [7].\",\n \"Furthermore, the same group has very recently generalized their work to a general Lie group as well [32].\",\n \"We have cross-checked their preliminary result with our $\\\\gamma _{m4}$ as given above, and found full agreement.\",\n \"Since both five-loop results have been obtained with completely different methods (with the exception of also relying on qgraf for diagram generation, in [32] the 5-loop renormalization constants are mapped onto massless 4-loop two-point functions [33], [34], [35], and integral reduction is done via $1/d$ expansions [36], [37]), this agreement constitutes an extremely strong check.\"\n ],\n [\n \"Quark field renormalization\",\n \"For completeness, let us also present our new five-loop result for the quark field anomalous dimension $\\\\gamma _2 = -\\\\partial _{\\\\ln \\\\mu ^2}\\\\ln Z_2$ , where the renormalization constant $Z_2$ relates bare and renormalized quark fields as $\\\\psi _{\\\\rm bare}=\\\\sqrt{Z_2}\\\\psi _{\\\\rm ren}$ .\",\n \"As opposed to the quark mass, this quantity is not physical and hence gauge dependent.\",\n \"Lower-loop results can be found for SU(N) and covariant gauge in [38], and for a general Lie group with $\\\\xi ^0,\\\\xi ^1$ terms (which corresponds to a NLO expansion around Feynman gauge) in [39].\",\n \"At five loops, $\\\\gamma _2$ is known for SU(3) in Feynman gauge [7], and we will below once more present the generalization to a general Lie group.\",\n \"Up to four loops, we obtain ($\\\\xi $ being the covariant gauge parameter, such that the values $\\\\xi =0/1$ correspond to Feynman/Landau gauge) $\\\\gamma _2&=&-c_f\\\\,a\\\\,\\\\Big \\\\lbrace (1-\\\\xi )+\\\\gamma _{21}\\\\,a+\\\\gamma _{22}\\\\,a^2+\\\\gamma _{23}\\\\,a^3+\\\\gamma _{24}\\\\,a^4+\\\\dots \\\\Big \\\\rbrace \\\\;,\\\\\\\\2^2\\\\,\\\\gamma _{21} &=& n_f\\\\Big [-8\\\\Big ]+\\\\Big [-6c_f+34-10\\\\xi + \\\\xi ^2\\\\Big ]\\\\;,\\\\\\\\2^53^2\\\\,\\\\gamma _{22} &=& n_f^2\\\\Big [640\\\\Big ]+ n_f\\\\Big [8 (108 c_f- 1301 + 153 \\\\xi )\\\\Big ] \\\\\\\\&& + \\\\Big [432 c_f^2 -\\\\!\",\n \"72 (143 \\\\!-\\\\!48 \\\\zeta _3)c_f+\\\\!2 (10559 -\\\\!1080 \\\\zeta _3) -\\\\!9 \\\\xi (371 +\\\\!48 \\\\zeta _3) +\\\\!27 \\\\xi ^2 (23 +\\\\!4 \\\\zeta _3) -\\\\!90 \\\\xi ^3 \\\\Big ]\\\\;,\\\\nonumber \\\\\\\\2^43^5\\\\,\\\\gamma _{23} &=&n_f^3 \\\\Big [ 13440 \\\\Big ]+ n_f^2 \\\\Big [ 6912 (19 - 18 \\\\zeta _3)c_f+ 16 (6835 + 9072 \\\\zeta _3) + 64 \\\\xi (269 - 324 \\\\zeta _3) \\\\Big ]\\\\nonumber \\\\\\\\&&+ n_f\\\\Big [5184 (19 - 48 \\\\zeta _3) c_f^2+ \\\\big (-108 (2407 - 1584 \\\\zeta _3 - 1296 \\\\zeta _4 - 5760 \\\\zeta _5)\\\\nonumber \\\\\\\\&&+ 324 \\\\xi (767 - 528 \\\\zeta _3 - 144 \\\\zeta _4) \\\\big )c_f+ 497664 d_1-(1365691 + 154224 \\\\zeta _3 + 97200 \\\\zeta _4 + 311040 \\\\zeta _5)\\\\nonumber \\\\\\\\&&+ \\\\xi (48865 + 152928 \\\\zeta _3 + 29160 \\\\zeta _4)- 54 \\\\xi ^2 (109 + 84 \\\\zeta _3 - 18 \\\\zeta _4)\\\\Big ]\\\\nonumber \\\\\\\\&&+\\\\Big [- 486 (1027 + 3200 \\\\zeta _3 - 5120 \\\\zeta _5) c_f^3+ 324 (5131 + 10176 \\\\zeta _3 - 17280 \\\\zeta _5) c_f^2\\\\nonumber \\\\\\\\&&+ \\\\big (-108 (23777 + 7704 \\\\zeta _3 + 2376 \\\\zeta _4 - 28440 \\\\zeta _5) - 1944 \\\\xi (6 - 7 \\\\zeta _3 + 10 \\\\zeta _5)\\\\big ) c_f\\\\nonumber \\\\\\\\&&+ 486 \\\\big (16 (-33 + 95 \\\\zeta _3 - 85 \\\\zeta _5) - 8 \\\\xi (1 + 48 \\\\zeta _3 - 70 \\\\zeta _5)- 8 \\\\xi ^2 (7\\\\zeta _3+5\\\\zeta _5)+ 20 \\\\xi ^3 (2\\\\zeta _3-\\\\zeta _5)\\\\nonumber \\\\\\\\&&- \\\\xi ^4 (7\\\\zeta _3-5\\\\zeta _5)\\\\big ) d_2+ (10059589/4 - 241218 \\\\zeta _3 + 168156 \\\\zeta _4 - 604260 \\\\zeta _5)\\\\nonumber \\\\\\\\&&- \\\\xi (2127929/8 + 164106 \\\\zeta _3 - 21141 \\\\zeta _4 - 107730 \\\\zeta _5)+ 27 \\\\xi ^2 (13883 + 9108 \\\\zeta _3 - 1548 \\\\zeta _4\\\\nonumber \\\\\\\\&&- 1920 \\\\zeta _5)/8- 81 \\\\xi ^3 (263 + 65 \\\\zeta _3 - 9 \\\\zeta _4 + 20 \\\\zeta _5)/2+ 81 \\\\xi ^4 (57 + \\\\zeta _3 + 10 \\\\zeta _5)/4\\\\Big ] \\\\;.$ We have presented the full gauge parameter dependence above.\",\n \"Note that this fills a gap in the literature and constitutes new information at four loops: in [39] only the terms linear in $\\\\xi $ have been evaluated for a general Lie group, while with full gauge dependence only the SU(N) result is available [38].\",\n \"However, due to the degeneracies $2d_1=6c_f^2-5c_f+13/12$ and $2d_2=7/12-c_f$ in the SU(N) limit, one cannot uniquely extract the Lie group structure from the latter reference.\",\n \"Needless to say that our result for $\\\\gamma _{23}$ given in eq.\",\n \"() reproduces the $\\\\xi ^0$ and $\\\\xi ^1$ terms given in [39], and in the SU(N) limit reduces to the respective expressions of [38], for all powers of $\\\\xi $ .\",\n \"Expanding the all-order large-$N_{\\\\mathrm {f}}$ Landau gauge result of eq.\",\n \"(REF ) in the coupling $a_f$ allows to confirm eqs.\",\n \"(REF )-() to NLO in $n_f$ , and to predict the first two terms of $\\\\gamma _{24}$ in that gauge as $24^3\\\\,\\\\gamma _{24}|_{\\\\xi =1}&=&\\\\frac{83-144\\\\zeta _3}{72}\\\\,\\\\Big [16n_f\\\\Big ]^4+\\\\gamma _{243}^{\\\\,\\\\xi =1}\\\\,\\\\Big [16n_f\\\\Big ]^3+\\\\dots \\\\;,\\\\\\\\\\\\gamma _{243}^{\\\\,\\\\xi =1} &=& \\\\Big \\\\lbrace c_f, 1\\\\Big \\\\rbrace .\\\\Big \\\\lbrace -659/18 + 312 \\\\zeta _3 - 216 \\\\zeta _4, -1783/36 - 248 \\\\zeta _3 + 216 \\\\zeta _4 \\\\Big \\\\rbrace \\\\;.$ At five loops, along the same steps as explained in section , we have obtained the new Feynman gauge resultThe restriction to $\\\\xi =0$ is for practical reasons only.\",\n \"To evaluate the $\\\\xi $ -dependent coefficients, one would need to enlarge the integral reduction tables as produced by crusher and TIDE to integrals with higher propagator powers (or dots), roughly one more dot per power of the gauge parameter.\",\n \"Since the present calculation is at the limit of what the computing resources available to us are able to handle, we defer this to future work.\",\n \"$24^3\\\\,\\\\gamma _{24} &=&\\\\frac{83-144\\\\zeta _3}{72}\\\\,\\\\Big [16n_f\\\\Big ]^4+\\\\gamma _{243}\\\\,\\\\Big [16n_f\\\\Big ]^3+\\\\gamma _{242}\\\\,\\\\Big [16n_f\\\\Big ]^2+\\\\gamma _{241}\\\\,\\\\Big [16n_f\\\\Big ]+\\\\gamma _{240}+{\\\\cal O}(\\\\xi )\\\\;,$ where the coefficients again contain only Zeta values up to weight 7, $\\\\gamma _{243} &=& \\\\Big \\\\lbrace c_f, 1\\\\Big \\\\rbrace .\\\\Big \\\\lbrace -659/18 + 312 \\\\zeta _3 - 216 \\\\zeta _4, -3443/48 - 255 \\\\zeta _3 + 252 \\\\zeta _4\\\\Big \\\\rbrace \\\\;,\\\\\\\\\\\\gamma _{242} &=& \\\\Big \\\\lbrace c_f^2, c_f, d_1, 1\\\\Big \\\\rbrace .\\\\Big \\\\lbrace -2 (2497 - 1200 \\\\zeta _3 + 3456 \\\\zeta _4 - 8640 \\\\zeta _5), \\\\nonumber \\\\\\\\&&477433/12 - 45636 \\\\zeta _3 + 4608 \\\\zeta _3^2 + 11448 \\\\zeta _4 - 65088 \\\\zeta _5 + 28800 \\\\zeta _6,-384 (115 - 33 \\\\zeta _3 - 90 \\\\zeta _5),\\\\nonumber \\\\\\\\&&3015955/72 + 69509 \\\\zeta _3 - 2304 \\\\zeta _3^2 - 12861 \\\\zeta _4 + 16662 \\\\zeta _5 - 14400 \\\\zeta _6 - 11907 \\\\zeta _7 \\\\Big \\\\rbrace \\\\;,\\\\\\\\\\\\gamma _{241} &=& \\\\Big \\\\lbrace c_f^3, c_f^2, c_fd_1, c_f, d_1, d_2, 1\\\\Big \\\\rbrace .\\\\Big \\\\lbrace 24 (29209 + 89984 \\\\zeta _3 + 12288 \\\\zeta _3^2 - 28800 \\\\zeta _4 - 187520 \\\\zeta _5 +76800 \\\\zeta _6), \\\\nonumber \\\\\\\\&& -4 (296177 + 517020 \\\\zeta _3 + 26784 \\\\zeta _3^2 - 469908 \\\\zeta _4 -4104720 \\\\zeta _5 + 1069200 \\\\zeta _6 + 3011904 \\\\zeta _7), \\\\nonumber \\\\\\\\&& -2304 (748 + 4536 \\\\zeta _3 -1368 \\\\zeta _3^2 - 6780 \\\\zeta _5 + 3255 \\\\zeta _7), \\\\nonumber \\\\\\\\&& 8 (115334 - 37764 \\\\zeta _3 -123012 \\\\zeta _3^2 - 49923 \\\\zeta _4 - 1124556 \\\\zeta _5 + 133650 \\\\zeta _6 +1519308 \\\\zeta _7), \\\\nonumber \\\\\\\\&& 192 (16732 + 39912 \\\\zeta _3 - 10944 \\\\zeta _3^2 - 72960 \\\\zeta _5 +36771 \\\\zeta _7), \\\\nonumber \\\\\\\\&&96 (6158 - 13952 \\\\zeta _3 - 372 \\\\zeta _3^2 + 2880 \\\\zeta _4 - 39475 \\\\zeta _5 - 3900 \\\\zeta _6 + 45696 \\\\zeta _7),\\\\\\\\&& -34919359/9 - 753797 \\\\zeta _3 + 548148 \\\\zeta _3^2 - 135063 \\\\zeta _4 + 1759474 \\\\zeta _5 +265350 \\\\zeta _6 - 2647806 \\\\zeta _7\\\\Big \\\\rbrace \\\\;,\\\\nonumber \\\\\\\\\\\\gamma _{240} &=& \\\\Big \\\\lbrace c_f^4, c_f^3, c_f^2, c_fd_2, c_f, d_2, d_3, 1\\\\Big \\\\rbrace .\\\\Big \\\\lbrace 1728 (4977 + 128000 \\\\zeta _3 + 19968 \\\\zeta _3^2 + 180800 \\\\zeta _5 -381024 \\\\zeta _7), \\\\nonumber \\\\\\\\&& -96 (835739 + 8494144 \\\\zeta _3 + 1182336 \\\\zeta _3^2 - 316800 \\\\zeta _4 +3983360 \\\\zeta _5 + 844800 \\\\zeta _6 - 17852688 \\\\zeta _7), \\\\nonumber \\\\\\\\&& 192 (825361 + 5472068 \\\\zeta _3 +651816 \\\\zeta _3^2 - 335808 \\\\zeta _4 - 1140420 \\\\zeta _5 + 950400 \\\\zeta _6 -8056377 \\\\zeta _7), \\\\nonumber \\\\\\\\&& 4608 (10 + 53226 \\\\zeta _3 - 15264 \\\\zeta _3^2 + 2145 \\\\zeta _5 -45885 \\\\zeta _7), \\\\; -16 (84040774/9 \\\\nonumber \\\\\\\\&& + 33396648 \\\\zeta _3 + 2804616 \\\\zeta _3^2 - 838782 \\\\zeta _4 - 18160944 \\\\zeta _5 +6252300 \\\\zeta _6 - 41015331 \\\\zeta _7), \\\\nonumber \\\\\\\\&& -384 (43066 + 628802 \\\\zeta _3 - 160998 \\\\zeta _3^2 + 36540 \\\\zeta _4 - 201125 \\\\zeta _5 -53475 \\\\zeta _6 - 403263 \\\\zeta _7), \\\\nonumber \\\\\\\\&&-72 (20566 - 218812 \\\\zeta _3 - 79080 \\\\zeta _3^2 - 13212 \\\\zeta _4 + 760220 \\\\zeta _5 + 20100 \\\\zeta _6 - 660667 \\\\zeta _7),\\\\; 804023630/9 \\\\nonumber \\\\\\\\&& + 101490400 \\\\zeta _3 + 3143352 \\\\zeta _3^2 + 7356024 \\\\zeta _4 -86186276 \\\\zeta _5 + 18372900 \\\\zeta _6 - 115799439 \\\\zeta _7 \\\\Big \\\\rbrace \\\\;.$ As a speculation, comparing the Landau gauge prediction eq.\",\n \"(REF ) with the Feynman gauge result eq.\",\n \"(REF ), the full result for $\\\\gamma _{243}$ could be simply adding $\\\\delta \\\\gamma _{243}=\\\\xi (3197/144+7\\\\zeta _3-36\\\\zeta _4)$ ; more generally however, consistency only requires that $\\\\delta \\\\gamma _{243}=c_f\\\\,u_1(\\\\xi )+u_2(\\\\xi )$ with $u_1(0)=0=u_1(1)$ as well as $u_2(0)=0$ and $u_2(1)=3197/144+7\\\\zeta _3-36\\\\zeta _4$ , the simplest choice being the one speculated above.\",\n \"As a check, replacing the group invariants with the values of eq.\",\n \"(REF ) in our Feynman gauge result for $\\\\gamma _{24}$ , we find perfect agreement with the known SU(3) result of [7] (see also eq.\",\n \"(46) of [40]).\"\n ],\n [\n \"Conclusions\",\n \"We have provided new results for two fundamental renormalization coefficients, at five loops and for a semi-simple Lie group.\",\n \"In particular, our expression for the gauge-invariant quark mass anomalous dimension $\\\\gamma _m$ coincides in various limits (large $N_{\\\\mathrm {f}}$ as well as SU(3)) with previously known results, and coincides exactly with recent results of another group [32].\",\n \"We have also provided the Feynman gauge result for the quark field anomalous dimension $\\\\gamma _2$ , which again could be checked against known expressions in the abovementioned limits.\",\n \"From these two quantities, one can reconstruct the two renormalization constants $Z_m$ and $Z_2$ of the quark sector, an electronic version of which is available by downloading the source of this article from http://arXiv.org/abs/1612.05512.\",\n \"As had already been observed in [5], looking at e.g.\",\n \"the 4-loop result for the quark mass anomalous dimension eq.\",\n \"(REF ), all Zeta terms (and also the higher group invariants $d_n$ ) vanish at $\\\\lbrace c_f=1,n_f=\\\\frac{1}{2},d_1=d_2\\\\rbrace $ , which corresponds to ${\\\\cal N}=1$ supersymmetry.\",\n \"The same had happened for the 4-loop Beta function (generalizing the last condition to $d_1=d_2=d_3$ ).\",\n \"For these parameters values, from section we have $\\\\gamma _m &=& -a\\\\Big \\\\lbrace 3+\\\\!16\\\\,a+\\\\!\\\\tfrac{310}{3}\\\\,a^2+\\\\!\\\\tfrac{2228}{3}\\\\,a^3+\\\\!\\\\Big (\\\\tfrac{671075}{108} -\\\\!194\\\\,d_1+\\\\!\\\\big (\\\\tfrac{1483}{2} -\\\\!2028\\\\,d_1\\\\big ) \\\\zeta _3-\\\\!20\\\\big (55 +\\\\!354\\\\,d_1\\\\big ) \\\\zeta _5\\\\Big ) a^4\\\\Big \\\\rbrace \\\\;,\\\\nonumber $ where we observe that the cancellation pattern does not hold through five loops – although the structure becomes much simpler, due to cancellation of all terms containing $\\\\lbrace \\\\zeta _3^2,\\\\zeta _4,\\\\zeta _6,\\\\zeta _7\\\\rbrace $ .\",\n \"To conclude the renormalization program at five loops, one needs to determine three more renormalization constants, which can be chosen to be those of the gluon and ghost fields, $Z_3$ and $Z_3^{c}$ , respectively, and of the ghost-gluon vertex $Z_1^{ccg}$ .\",\n \"From these, due to gauge invariance, one can then construct the renormalization constant for the gauge coupling (aka the Beta function, whose five-loop coefficient is so far known for SU(3) only [41]) as well as those for the remaining vertices.\",\n \"While the same methods that we have used here are sufficient to calculate those missing coefficients (which indeed already led to the NLO terms at large $N_{\\\\mathrm {f}}$ [8]), due to the complexity of the determination of $Z_3$ we leave the evaluation of the full coefficients for future work.\",\n \"We are indebted to K. Chetyrkin for sending us their new five-loop results for $\\\\gamma _m$ prior to publication [32], to enable the important check discussed at the end of section .\",\n \"The work of T.L.\",\n \"has been supported in part by DFG grants GRK 881 and SCHR 993/2.\",\n \"A.M. is supported by a European Union COFUND/Durham Junior Research Fellowship under EU grant agreement number 267209.\",\n \"P.M. was supported in part by the EU Network HIGGSTOOLS PITN-GA-2012-316704.\",\n \"Y.S.\",\n \"acknowledges support from FONDECYT project 1151281 and UBB project GI-152609/VC.\",\n \"All diagrams were drawn with Axodraw [42], [43].\"\n ],\n [\n \"Summary of large-$N_{\\\\mathrm {f}}$ results\",\n \"Some coefficients of QCD anomalous dimensions are known to all loop orders, from a large $N_{\\\\mathrm {f}}$ expansion.\",\n \"Taking $n_f$ and $c_f$ as above and defining $a_f\\\\equiv \\\\frac{N_{\\\\mathrm {f}}T_{\\\\mathrm {F}}g^2(\\\\mu )}{12\\\\pi ^2}\\\\;=\\\\;\\\\frac{4\\\\,n_f\\\\,a}{3}\\\\quad ,\\\\qquad \\\\eta (\\\\varepsilon )\\\\equiv \\\\frac{(2\\\\varepsilon -3)\\\\Gamma (4-2\\\\varepsilon )}{16\\\\Gamma ^2(2-\\\\varepsilon )\\\\Gamma (3-\\\\varepsilon )\\\\Gamma (\\\\varepsilon )}\\\\;,$ the all-order leading-$N_{\\\\mathrm {f}}$ [44] and next-to-leading-$N_{\\\\mathrm {f}}$ [45], [46] expressions that we need here read $\\\\gamma _m&=&\\\\frac{4c_f}{n_f}\\\\Big \\\\lbrace \\\\eta (a_f)+\\\\frac{\\\\eta _3(a_f)}{8\\\\,n_f}+{\\\\cal O}\\\\Big (\\\\frac{1}{n_f^2}\\\\Big )\\\\Big \\\\rbrace \\\\;,\\\\\\\\\\\\gamma _2|_{\\\\xi =1}&=&-\\\\frac{2a_fc_f}{n_f}\\\\Big \\\\lbrace \\\\eta (a_f)+\\\\frac{1}{n_f}\\\\frac{\\\\eta _4(a_f)}{4a_f}+{\\\\cal O}\\\\Big (\\\\frac{1}{n_f^2}\\\\Big )\\\\Big \\\\rbrace \\\\;,$ where the fact that the asymptotic expansions have been performed in Landau gauge only does not affect the physical and gauge invariant quark mass anomalous dimension $\\\\gamma _m$ .\",\n \"To define the coefficient functions $\\\\eta _3$ and $\\\\eta _4$ , it is convenient to define the linear combinations $\\\\Psi (\\\\varepsilon ) &=& \\\\psi _0(1 - 2 \\\\varepsilon ) + \\\\psi _0(1 + \\\\varepsilon ) -\\\\psi _0(1 - \\\\varepsilon ) - \\\\psi _0(1) \\\\;,\\\\\\\\\\\\Phi (\\\\varepsilon ) &=& \\\\psi _1(1 - 2 \\\\varepsilon ) - \\\\psi _1(1 + \\\\varepsilon ) -\\\\psi _1(1 - \\\\varepsilon ) + \\\\psi _1(1) \\\\;,\\\\\\\\\\\\Theta (\\\\varepsilon ) &=& \\\\psi _1(1 - \\\\varepsilon ) - \\\\psi _1(1) \\\\;,$ where $\\\\psi _n(x)=\\\\partial _x^{n+1}\\\\ln \\\\Gamma (x)$ is the PolyGamma function.\",\n \"Then [45], [46] $\\\\eta _3(\\\\varepsilon ) &\\\\equiv &\\\\Big (\\\\!-\\\\!\\\\frac{11}{4} +\\\\!\",\n \"\\\\sum _{n>0}\\\\frac{f_n\\\\varepsilon ^n}{n}\\\\Big ) 8 \\\\varepsilon \\\\partial _\\\\varepsilon \\\\eta (\\\\varepsilon )-\\\\frac{16 \\\\eta ^2(\\\\varepsilon )}{(3 \\\\!-\\\\!\",\n \"2 \\\\varepsilon )(1 \\\\!-\\\\!\",\n \"\\\\varepsilon )}\\\\Big \\\\lbrace 3 (2 - \\\\varepsilon )^2 (1 - \\\\varepsilon )^2 \\\\Theta (\\\\varepsilon )-(5 + 5 \\\\varepsilon - 11 \\\\varepsilon ^2 + 4 \\\\varepsilon ^3) ,\\\\nonumber \\\\\\\\&&(88 \\\\!-\\\\!\",\n \"372 \\\\varepsilon \\\\!+\\\\!\",\n \"551 \\\\varepsilon ^2 \\\\!-\\\\!\",\n \"380 \\\\varepsilon ^3 \\\\!+\\\\!\",\n \"160 \\\\varepsilon ^4 \\\\!-\\\\!\",\n \"64 \\\\varepsilon ^5 \\\\!+\\\\!\",\n \"16 \\\\varepsilon ^6)\\\\!-\\\\!\",\n \"4 \\\\varepsilon (3 \\\\!-\\\\!\",\n \"2 \\\\varepsilon ) (1 \\\\!-\\\\!\",\n \"2 \\\\varepsilon ) (2 \\\\!-\\\\!\",\n \"\\\\varepsilon ) (1 \\\\!-\\\\!\",\n \"\\\\varepsilon )^2 \\\\big (\\\\Psi ^2(\\\\varepsilon ) \\\\!+\\\\!\",\n \"\\\\Phi (\\\\varepsilon )\\\\big )\\\\nonumber \\\\\\\\&&+ 2 (1 - \\\\varepsilon ) (24 - 144 \\\\varepsilon + 249 \\\\varepsilon ^2 - 146 \\\\varepsilon ^3 + 12 \\\\varepsilon ^4 + 8 \\\\varepsilon ^5) \\\\Psi (\\\\varepsilon )\\\\Big \\\\rbrace .\\\\Big \\\\lbrace \\\\tfrac{2}{2 - \\\\varepsilon }\\\\,c_f, \\\\tfrac{1}{4\\\\varepsilon (3 - 2 \\\\varepsilon )(1 - 2 \\\\varepsilon )}\\\\Big \\\\rbrace \\\\;,\\\\\\\\\\\\eta _4(\\\\varepsilon ) &=& \\\\frac{\\\\varepsilon \\\\eta _3(\\\\varepsilon )}{2}+\\\\Big (\\\\!-\\\\!\\\\frac{11}{4} +\\\\!\",\n \"\\\\sum _{n>0}\\\\frac{f_n\\\\varepsilon ^n}{n}\\\\Big ) 4\\\\varepsilon \\\\eta (\\\\varepsilon )+\\\\frac{2\\\\eta ^2(\\\\varepsilon )}{3-2\\\\varepsilon } \\\\Big \\\\lbrace -8(1-4\\\\varepsilon +2\\\\varepsilon ^2),\\\\tfrac{(2-5\\\\varepsilon +2\\\\varepsilon ^2)^2}{1-\\\\varepsilon }\\\\Big \\\\rbrace .\",\n \"\\\\Big \\\\lbrace c_f,1\\\\Big \\\\rbrace \\\\;,\\\\\\\\{\\\\rm with}&&\\\\sum _{j>0}f_j\\\\,\\\\varepsilon ^j \\\\;\\\\equiv \\\\; -\\\\eta (\\\\varepsilon )\\\\Big \\\\lbrace 4(1+\\\\varepsilon )(1-2\\\\varepsilon )c_f+\\\\tfrac{4\\\\varepsilon ^4-14\\\\varepsilon ^3+32\\\\varepsilon ^2-43\\\\varepsilon +20}{(1-\\\\varepsilon )(3-2\\\\varepsilon )}\\\\Big \\\\rbrace \\\\;.$ The coefficients $a_f$ , $\\\\eta (\\\\varepsilon )$ and $f_j$ are the same as we had defined in [8].\"\n ]\n]"},"id":{"kind":"string","value":"1612.05512"}}},{"rowIdx":3395,"cells":{"text":{"kind":"list like","value":[["How long does a quantum particle or wave stay in given region of space?"],["Abstract The delay time associated with a scattering process is one of the most important dynamical aspects in quantum mechanics.","A common measure of this is the Wigner delay time based on the group velocity description of a wave-packet, which my easily indicate super-luminal or even negative times of interaction that are unacceptable.","Many other measures such as dwell times have been proposed, but also suffer from serious deficiencies, particularly for evanescent waves.","One important way of realising a timescale that is causally connected to the spatial region of interest has been to utilize the dynamical evolution of extra degrees of freedom called quantum clocks, such as the spin of an electron in an applied magnetic field or coherent decay or growth of light in an absorptive or amplifying medium placed within the region of interest.","Here we provide a review of the several approaches developed to answer the basic question - how much time does a quantum particle (or wave) spend in a specified region of space?","While a unique answer still evades us, important progress has been made in understanding the timescales and obtaining positive definite times of interaction by noting that all such clocks are affected by spurious scattering concomitant with the very clock potentials, however, weak they be and by eliminating the spurious scattering."],["Introduction","One of the first things a student of Physics learns is to calculate the time at which an event, such as the location of a particle at a given position, occurs.","A related question would be to ask how long does a particle stay in a given region of space during the course of its motion.","Classically it is possible to simultaneously specify the position and the momentum of a particle, and these questions about the time instants or the time intervals for given events have unambiguous answers.","Even as we proceed to statistically understand systems with very large number of particles through probability distributions, where the classical motion of each particle is described only stochastically, concepts such as the first passage times [1] remain meaningful.","This well entrenched concept, however, becomes difficult to define for quantum mechanical systems, and indeed, for any form of waves.","Figure: Left panel: Schematic picture of a quantum particle that is initially localized in a fixed region and arrives at a detector array.","The first passage time refers to the first point in time when the particle is registered at a detector.","Right panel: The particle may take any virtual path toget to the detector, including ones that cross the boundary many times.","The final amplitude will be a sum over the amplitudes for any such path.Imagine an experiment where a quantum-mechanical particle is released from some fixed region inside a box.","On one side of the box there is a screen with detectors which click as soon as the particle \"arrives\" at the screen.","One expects that the time of arrival of the particle is a stochastic variable and it is interesting to ask for it's probability distribution.","This is similar to asking for the distribution of the time of absorption of a Brownian particle at some point.","However, the quantum problem turns out to be very subtle and there is as yet no clear answer to the question.","The point is that the particle that is initially localized and released must subsequently cross a given boundary in space when it is detected for the very first time.","Yet this very calculation includes amplitudes from paths that may very well extend all over space even beyond the given boundary.","Very recently the problem of first time arrival of a quantum mechanical particle has been considered satisfactorily utilizing a path integral approach that with a restricted path decomposition and appears to succeed in obtaining positive definite quantum first passage times for motion on a one dimensional lattice [2].","The principal difficulty arises from the fact that waves are infinitely deformable objects and many aspects of motion arise through interference effects.","Hence it becomes impossible to define well defined start and finish lines for wave packets.","In the quantum mechanical perspective, where all dynamical observables have a corresponding operator, there has been great difficulty in defining a “good” Hermitian operator that conforms to classical notions of time or a time interval [3].","For classical waves such as electromagnetic waves also, related difficulties exist.","For example, we would like to classically demand that a time of stay in a given region of space should be (i) Real, (ii) Positive definite, (iii) tend to classically calculable times in the limit of large energies.","Inspite of serious efforts over many years [4], no definition for such a quantum mechanical time has been universally accepted.","In fact, when we say that a wave moves, it becomes imperative to clearly define what is the quantity related to the motion of a wave that is being talked about.","Regardless of these difficulties, it must be emphasized that the times of traverse or dwell associated with a wave are calculable quantities that can be useful for understanding processes and process rates in a given system.","In this tutorial, we will discuss the several approaches, the related difficulties and some recent possibilities that have arisen in this context.","First, we will go through some of the basic definitions and concepts regarding wave motion and the times that can be related to the motion of a wave.","It turns out that there is a large variety of times arising from different definitions related to various different quantities associated with a wave.","Surprisingly, these definitions can result in apparent superluminal times or even negative delay times for the transit of a wave through dispersive media or potentials.","Subsequently, we will describe certain clocking mechanisms associated with physical processes such as the precession of a spin in a magnetic field [5] that have been developed to calculate theoretically the traversal or dwell times of a wave [4].","These clocking mechanisms have to be used with care as they can contribute to scattering and change the very problem being discussed [6].","These ideas lead to the possibility of using quantum dephasing or stochastic absorption as a clock."],["1. Time-scales based on the group velocity ","Here we will try to explore these questions in the general context of waves to cover both electromagnetism and quantum mechanics.","Both the time independent Schrodinger equation for the wavefunction of a quantum particle and the Maxwell equations in frequency domain for the amplitude of a time harmonic electromagnetic wave (of only one polarization) reduce to the Helmholtz wave equation: $\\nabla ^2 \\psi + k^2 \\psi = 0.$ For the electromagnetic wave, k is the wave vector given by $k^2 = n^2 \\omega ^2 /c^2$ , $\\omega $ is the angular frequency of the wave and $n$ is the refractive index of the medium with $c$ being the speed of light in vacuum.","For the quantum particle in comparison, $k$ is given by $k^2 = 2m(E-V_0)/\\hbar $ , where$E$ and $m$ are the energy and the mass of the particle and$V_0$ is the potential.","The principle difference between the the two systems is in free space where the refractive index is unity.","Consequently the wave-vector for an electromagnetic wave is linearly proportional to the frequency.","In the case of the quantum particle, even for $V_0=0$ , the wave-vector disperses with the quadratic root of the energy.","This has a non-trivial manifestation in the undistorted propagation of electromagnetic pulses in free space while a quantum mechanical wavepacket spreads out and disperses even in free space.","Thus, it is realized that the fundamental properties of propagation of the waves are governed by the potentials or the refractive index of the medium through the dispersion relation between the wave-vector and the frequency or energy.","Consider the scalar wave $\\psi (\\vec{r},t) = a(\\vec{r}) \\exp [i\\phi (\\vec{r}) - i\\omega t]$ where $\\phi (\\vec{r})$ is some scalar function.","For this wave, $\\phi (\\vec{r}) = \\mathrm {constant}$ denotes the constant phase surfaces.","To trace the motion of these surfaces, let us look at the condition at two points $(\\vec{r},t)$ , and $(\\vec{r} + \\delta \\vec{r}, t+\\delta t)$ .","The phase front is the same if, and only if, $\\phi (\\vec{r} + \\delta \\vec{r}) -\\omega ( t+\\delta t) \\simeq \\phi (\\vec{r}) + \\delta \\vec{r} \\cdot \\nabla \\phi (\\vec{r}),t) - \\omega t - \\omega \\delta t = \\phi (\\vec{r}) - \\omega t,$ where we have included the first order term only in the infinitesimal $ \\delta \\vec{r}$ .","FRom the above, we obtain $ v_p = \\left| \\frac{\\delta \\vec{r}}{\\delta t} \\right| = \\frac{\\omega }{| \\nabla \\phi |} $ as the phase velocity for a wave with an arbitrary wave front.","The ratio , where $\\phi (\\vec{r})$ is the phase of the wave [7], is known as the phase velocity of the wave and represents the rate at which the equiphase surfaces of the wave move through the medium.","Note that the phase velocity can be just about any number (positive or negative) depending on the phase structure (gradient) of the wave.","In one dimension or when there is transverse invariance along two dimensions and plane waves result, the phase velocity reduces to the familiar relation.","This gives rise to the conventional notion that the phase velocity for a wave is $c/n$ , and is hence mistaken for the speed of a wave in a medium.","As pointed out, the phase velocity can easily be superluminal for materials with refractive index $n < 1$ .","Further, in negative refractive index materials [8] and in waveguides that support backward wave propagation, this phase velocity is obviously negative.","Thus, the phase velocity does not really specify anything concrete about the rate of the motion of the wave or the time spent by a wave in a given region of space.","In a material medium, the refractive index (polarization) of a medium will, in general, be frequency dependent and a complex quantity.","This is a simple consequence of the medium having certain natural frequencies at which it will resonantly polarize corresponding to atomic or molecular levels of the constituents of the medium.","Due to the different refractive index at different frequencies, the constituent time harmonic waves present in a wave packet essentially travel at their own phase velocities resulting in the interference pattern changing completely in time and leading to a dispersion of the wave packet.","This is easily seen by writing down the field amplitude of the wave at different times.","If $E(\\vec{r},0)$ be the field amplitude at time$t =0$ , and $E(\\vec{k} = \\int _{-\\infty }^\\infty E(\\vec{r},0) e^{-i\\vec{k}\\cdot \\vec{r}} ~d^3r$ is is the spatial Fourier transform of the field amplitude, the field amplitude at any other time, $t$ , is given by $E(\\vec{r}, t) = \\left(\\frac{1}{2\\pi }\\right)^3 \\int _{-\\infty }^\\infty E(\\vec{k}) e^{i(\\vec{k}\\cdot \\vec{r}-\\omega t)} ~d^3k .$ If the dispersion was linear, this would just correspond to the same function shifted to a new position.","For an arbitrary dispersion $\\omega (\\vec{k})$ , it becomes difficult to say anything in general.","If we assume, however, that most of the amplitude is concentrated in a small frequency band about a central (carrier) frequency $\\omega _0$ , then one can carry out a Taylor’s series expansion $ \\omega (\\vec{k}) = \\omega (\\vec{k}_0) + (\\vec{k} -\\vec{k}_0) \\cdot \\nabla _k \\omega (\\vec{k}_0) + \\cdots , $ where the subscript $k$ indicates that the derivative is with respect to the wave-vector and retain only the linear term.","Substituting this in the expression for the field amplitude, one obtains, $E(\\vec{r}, t) = \\left(\\frac{1}{2\\pi }\\right)^3 e^{i\\varphi } \\int _{-\\infty }^\\infty E(\\vec{k}) e^{i\\vec{k}\\cdot [\\vec{r} - \\nabla _k\\omega (\\vec{k}_0) - \\omega t]} ~d^3k ,$ which is essentially the same waveform that is shifted by an amount $\\nabla _k \\omega (\\vec{k}_0)t$ in space, apart from a trivial extra phase of $\\varphi = \\omega _0 t + k_0 \\cdot \\nabla _k \\omega (\\vec{k}_0)t$ .","This brings up another rate at which the envelope of the wave propagates in the medium and defines the so-called group velocity $v_g = \\nabla _k \\omega (\\vec{k}_0)$ .","This tracks the rate at which fiducial (well recognisable) points on the waveform, such as the peak of the wavepacket, move and it is assumed that waveform is largely undistorted.","This is rarely satisfied for a quantum mechanical wave even in free space.","For an electromagnetic wave, however, this is satisfied for quasi-monochromatic fields and when absorption in the medium is minimal.","Again, it should be noted that the spread in the wavevectors is also equivalently small and the superposition should consist of waves moving along a common direction.","Noting that the group delay time accumulated upon traversing a distance $L$ in the medium is $ \\frac{L}{\\nabla _k \\omega (\\vec{k}_0)} = \\left.","\\frac{\\partial (kL)}{\\partial \\omega }\\right|_{\\omega _0} ,$ one can equivalently define a group delay time for scattering problems where the kinematic phase is replaced by the phase change upon scattering ($\\phi $ ).","In one dimension, this would be the phase shift upon reflection or transmittance.","This yields the Wigner’s group delay time $\\tau _w = \\left.","\\frac{\\partial \\phi }{\\partial \\omega } \\right|_{\\omega _0}.$ Note the extrapolation made here from a free propagation problem into the time delay obtained in a scattering event.","This time essentially measures the time difference between the entry and emergence of fiducial points into and out of the scattering volume.","Figure: The change in refractive index due to dispersion is plotted with a magnified scale.","The dispersion of the real part of the refractive index for a single Lorentz resonance (red dotted curve) and for a highly dispersive medium that can be produced, for example by electromagnetically induced transparency or by a Fano resonance.","Such dispersions enable very large changes of the refractive index, either normal or anomalous as marked in the figure.","The group velocity can easily become very small or very large and even negative in such dispersive regimes.Using the dispersion for light, $k^2 = n(\\omega )^2 \\omega ^2/c^2$ in a homogenous isotropic medium with a dispersive refractive index $n(\\omega )$ , the group velocity is also often conveniently written in the following manner $v_g = \\frac{c}{n(\\omega _0) + \\omega _0 (dn/ d\\omega )_{\\omega _0}}.$ The group velocity from this definition immediately shows that the group velocity can be very small when the dispersion is normal and large, i.e., $(dn/d\\omega ) \\gg 0$ (see Fig.","REF ).","This is the origin of ultra-slow light which has been demonstrated using the large dispersion possible in atomic gas vapours and Bose-Einstein condensates [9].","On the other, if the dispersion were anomalous and large , $(dn/d\\omega ) < 0~\\mathrm {and}~ |dn/d\\omega | \\gg 0$ (see Fig.","REF ), then we have that the group velocity can become greater than c, or even negative in such situations.","Indeed there have been several experiments on the apparent motion of light in a medium at a rate faster than light in vacuum [REF].","Negative group velocities and negative Wigner delay times are even more problematic – that would imply that the pulse exited region of space before it even entered, in a sense violating causality.","Landauer [4] had severely criticised the Wigner delay time to give information about the dwell times on grounds of causality and emphasized that there was no causal connection between the peaks (or fiducial points) of the incoming and the outgoing waveforms.","This difficulty is accentuated when the scattering potential strongly deforms the wave-packet.","The large dispersion responsible for the deformation is also usually accompanied by severe dissipation or gain that causes large spectral modifications with corresponding distortions of the temporal pulse shape.","Hence the analysis of the motion of a spectrally broad or distorted pulse in terms of the conventional group velocity is severely limited, as the pulse can lose its very identity after propagating to a large distance in the dispersive medium.","These arguments apply and hold true even for definitions of the times based on the center-of-mass of the wavepacket or in an alternative approaches where the motion of the forward edge of the wavepacket is followed.","These issues raise questions about the group velocity, or the Wigner delay time that is based on the group velocity, to actually provide an answer to the question that we seek about the time that the wave spends in a given region of space.","It has been emphasized by most authors that the superluminal or negative Wigner delay times do not violate causality or the Special theory of relativity.","It simply turns out that the functions that we considered to describe the fields have been smooth analytic functions.","This is a consequence of the fields being the solutions of the Helmhotz differential equation.","Analytic functions are problematic because they have an infinite support – the function extends all over the space, although it may be very small.","A good example of such functions is the Gaussian function.","In principle, one may always take the analytic function at a given point and carry out a Taylor's series expansion of the function using the values of the function and its derivatives at the given point, to obtain its value at any point in space, however, far off from that point.","Since the function is analytic, all the derivatives at the given point exist.","Thus, knowledge of the analytic signal even if localized in the manner of a Gaussian function, already exists at all the other points in principle, even if the value of the signal were to be infinitesimally small.","Thus, there is no extra information being conveyed to the other points with the wave motion as the information was already present.","Thus, there is no violation of the special theory of relativity or of causality.","It has been emphasized that in principle only meromorphic functions (with discontinuities in the function or its derivatives) can be used to encode information.","Any such discontinuity will generate very high frequency components in the power spectrum of the signal.","These high frequency components will always propagate at the speed of light in vacuum (c) as the refractive index as This is a consequence of finite inertia for the charge carriers in all material media."],["The Dwell times based on current fluxes or energy transfer","A second approach to this problem has been to define the time in terms of probability of finding the particle within the spatial volume of interest.","Smith [10] defined the dwell time for a mono-energetic quantum particle in the region $[0, L]$ (in one dimension) as $\\tau _d = \\frac{1}{J} \\int _0^L \\vert \\psi (x) \\vert ^2 ~dx$ where $\\psi (x)$ is the wavefunction and $ J = \\mathrm {Re} (\\hbar /im) (\\psi ^*\\;\\nabla \\psi )$ is the current flux associated with the incoming particle.","Note that the Smith dwell time, is independent of the scattering channel (reflection or transmission) and is hence, an unconditional time.","In case of a time-varying pulsed waveform, this dwell time can be generalized by integrating over all times as $\\tau _d = \\int _{-\\infty }^{\\infty } dt ~ \\int _0^L \\vert \\psi (x) \\vert ^2 ~dx.$ In the case of electromagnetic waves, a similar approach can be adopted, but using the energy of the fields.","Thus the dwell time in a region of volume $(V)$ could be defined as $ \\tau _d = \\frac{\\int _V U d^3r}{ \\int _A \\vec{S} \\cdot d\\vec{a}} , $ where $U$ is the energy density associated with the electromagnetic wave, $\\vec{S}$ is the Poynting vector denoting the power flow per unit area of the incoming wave and A is the surface area of volume through which the incoming wave is incident.","While there is no problem with the Poynting vector defined as $\\vec{S} = \\vec{E}\\times \\vec{H}$ in terms of the electric and magnetic fields, there are severe difficulties in defining an energy density solely associated with the wave in a dispersive and dissipative (or amplifying) medium.","The difficulties of separating the energy associated with the wave and the polarization in the medium are well known in this scenario where the two fields continuously exchange energy.","In fact, it is well known that the energy density defined by first order Taylor expansions of the dispersion [11] can easily be negative for severe dispersion of the material parameters.","In some sense, the above quantity and the Smith Dwell time are equivalent as both the quantities involve quadratic expressions of the underlying fields.","The issue of dispersion and polarization in material media complicate the issues further in the context of electromagnetic waves.","Another fruitful approach to define arrival times that is based on the centroid of power flow of the electromagnetic wave is noteworthy [12].","An arrival time at a point can be defined as a time average (first moment of time) over the component of the Poynting vector normal to a surface $\\langle t \\rangle _r = \\frac{\\hat{u} \\cdot \\int _{-\\infty }^\\infty t \\vec{S}(\\vec{r},t) dt}{ \\hat{u} \\cdot \\int _{-\\infty }^\\infty \\vec{S}(\\vec{r},t) dt},$ where $\\hat{u}$ denotes the unit normal to the given surface, which could very well be that of a detector.","The time for traverse between two points $\\vec{r}_i$ and $\\vec{r}_f$ can now be thought of as the difference between the arrival times at the two points as $\\Delta t = \\langle t\\rangle _{\\vec{r}_f} - \\langle t\\rangle _{\\vec{r}_i}$ .","A basic theorem was proven to show that the propagation delay could be decomposed in terms of a net group delay and a reshaping delay.","The net group delay consists of a spectrally wieghted average group delay at the final point $\\vec{r}_f$ given by $\\Delta t_G = = \\frac{\\hat{u} \\cdot \\int _{-\\infty }^\\infty t \\vec{S}(\\vec{r}_f,\\omega ) [(\\partial \\mathrm {Re}(k) / \\partial \\omega ) \\cdot \\Delta r] d\\omega }{ \\hat{u} \\cdot \\int _{-\\infty }^\\infty \\vec{S}(\\vec{r}_f,\\omega ) d\\omega },$ The reshaping delay, as the very name suggests, arises from the deformation due to spectral modulation by the medium and can be calculated in terms of the spectral fields at the initial point as $\\Delta t_R = \\wp \\lbrace \\exp [-\\mathrm {Im}(\\vec{k}) \\cdot \\Delta \\vec{r}] \\vec{E}(\\vec{r}_i, \\omega )\\rbrace \\wp \\lbrace \\vec{E}(\\vec{r}_i, \\omega )\\rbrace ,$ where the operator $ \\wp \\lbrace \\vec{E}(\\vec{r}, \\omega )\\rbrace =\\frac{\\hat{u} \\cdot \\int _{-\\infty }^\\infty \\mathrm {Re} [-i \\frac{\\partial \\vec{E}(\\vec{r}, \\omega )}{\\partial \\omega }\\; \\times \\; \\vec{H}^*(\\vec{r},\\omega )] d\\omega }{ \\hat{u} \\cdot \\int _{-\\infty }^\\infty \\vec{S}(\\vec{r},\\omega ) d\\omega }, $ This approach which is applicable to an arbitrary waveform or dispersive medium showed that both the components in general were always significant.","The most important aspect of this proposal is that it does not involve any perturbative expansion of the wave number around the carrier frequency.","A few salient points may be noted in this context: For a narrowband pulse, the reshaping delay tends to zero and the total delay time is dominated by the group delay time.","We note that even the Wigner delay time would describe the situation quite well in this case.","For a broadband pulse with only propagating components, the net group delay can become negative, but the corresponding reshaping delay causes the overall delay time to remain luminal as the pulse experiences a strong reshaping during propagation.","This makes this proposal a very strong candidate to represent the traversal time that is causal and non-negative.","It has been shown that the definition is equivalent to another definition based on the rate of energy absorbed by a detector $ \\langle t \\rangle _r = \\frac{ \\int _{-\\infty }^\\infty t \\frac{d A(\\vec{r},t)}{dt} dt}{ \\int _{-\\infty }^\\infty \\frac{d A(\\vec{r},t)}{dt} dt}, $ where$(dA/dt)$ is the rate of absorption of energy per unit volume inside the detector placed at $\\vec{r}$ , and is given by $ \\frac{dA}{dt} = \\int \\int \\omega \\left[ \\varepsilon _0 \\mathrm {Im}(\\varepsilon ) E^*(\\omega ^{\\prime }) E(\\omega ) + \\mu _0\\mathrm {Im}(\\mu )H^*(\\omega ) H(\\omega ) \\right]d\\omega ^{\\prime }~d\\omega .","$ This is integrated spatially over the detector volume to obtain the total rate of absorption within the detector.","Obviously the spatial extent of the detector is assumed to be small compared to the length scales of propagation or spatial pulse widths.","In a medium such as a plasma, however, it has been shown [13] that a positivity of the traversal time in not obtained, particularly for broad-band pulses, which have large amounts of evanescent frequency components when the carrier frequency is smaller than the plasma frequency.","This is related to the Hartmann effect [14], whereby the time for tunelling at far-sub-barrier energies becomes almost constant and negates the possibility of using this definition in principle for all situations.","Nevertheless, it should be noted that it is still operationally one of the most useful definitions of the traversal times and has been validated in experiments involving both temporally dispersive and angularly dispersive situations."],["Quantum clocks ","Due to issues with the various timescales which fail to conform to our intuitive understanding of the traversal timescales, attempts were made to develop“quantum clocks” that measure the time the particle spends in a given volume.","In analogy with a classical clock, the clock should tick only when the particle is within the region of interest.","This is accomplished by coupling other degrees of freedom to other fields localized within the region of interest.","Dynamical evolution of those degrees of freedom occur only when the particle is present in the region of interest.","Thus, the expectation values of quantities associated with those degrees of freedom will translate to expectation values of the times spent in the region of space.","While clocks have been proposed in principle for a long time [15], three powerful methods that can have direct experimental implementation have been proposed and are popular.","We will discuss the main ideas behind these proposals."],["Büttiker-Landauer oscillating barrier times","Büttiker and Landauer proposed [16] that the time of traverse of a charged particle through a potential barrier could be timed by super-imposing a time-harmonic electromagnetic field on top of the potential only within the region of interest.","Suppose is the region in which we seek the time of sojourn, the potential would be $V_0(\\vec{r}) + V_1(\\vec{r}) \\cos (\\omega t)$ where $V_0$ is the original static potential and the perturbing oscillating field has a magnitude ($V_1$ ) that is constant within the region of interest and is zero outside (see Fig.","REF for a schematic depiction).","Interaction of the charged particle with the electromagnetic field would cause cause the absorption or emission of quanta of radiation.","Thus, if the oscillation frequency was very small, the energy broadening of the transmitted or reflected spectrum would not be visible.","The particle effectively sees a static potential and the transmission and reflected fluxes will adiabatically vary in time with the potential barrier height that changes harmonically.","Thus, the times of interaction or traversal are very small compared to the time period of the oscillation.","On the other hand, at high frequencies of oscillation, the particle would see the potential undergoing many oscillations during the time of its stay in the region and it would exchange quanta with the field.","Then the energy spectrum of the reflected or transmitted particle would have energy sidebands separated from the incident energy by the energy of the quanta ($\\pm \\hbar \\omega $ ).","In this limit, the period of the electromagnetic field is clearly much smaller than the traversal time ($\\tau _s$ ).","There would be a crossover between the two regimes of low or high frequencies and the period of the electromagnetic field during the crossover regime ($\\omega \\tau _s \\simeq 1$ ) would a good measure of the time of traversal, regardless of whether the quantum wavefunction has a propagating nature or evanescent nature.","Figure: Schematic picture depicted the tunnelling of a particle through a potential barrier with a time dependent strength.","The particle exchanges quanta of energy (photons) with the radiation field and the transmission develops energy sidebands.","The particles with higher energy would tunnel through more efficiently than the particles with lower energy.","The period of the potential oscillations when the energy sidebands develop gives a timescale for the traversal of the particle through the barrierIn the one-dimensional case of a particle tunneling through a rectangular potential barrier, Buttiker and Landauer [16] obtain a tunneling time of $\\tau _{BL} = \\left[ \\frac{m}{2(V_0-E)} \\right]^{1/2} d,$ where $E$ is the energy of the tunneling particle and $d$ is the width of the barrier.","Using the WKB approximation at energies well below the barrier height, this was further generalized to spatially varying potential barrier in one-dimension as $\\tau _{BL} = \\int _{x_1}^{x_2} \\left[ \\frac{m}{2(V_0-E)} \\right]^{1/2} dx = \\int _{x_1}^{x_2} \\frac{m}{\\hbar \\kappa (x)} dx,$ where $\\hbar \\kappa (x) = \\sqrt{2m(V_0(x) - E)}$ behaves as the instantaneous momentum of the tunneling particle.","For more general potential shapes and energies, the calculations of the traversal times by this approach become very difficult.","This approach, however, clearly sets out a timescale for the problem, particularly for the limiting case of low energy tunneling, which any other valid approach would need to reproduce.","An approach where the flow of the particle through the barrier region was constructed in terms of two counter-propagating streams within the WKB approximation also obtained the traversal times consistent with the one obtained here [17]."],["Büttiker's spin clock (Larmor precession and spin flip)","A second clock related to using the Larmor precession of a quantum particle with an associated magnetic moment (or spin), such as an electron or a neutron, in an applied magnetic field (B).","The rate of the precession of the spin $\\omega _L = g\\mu _B B/\\hbar $ , ($\\mu _B$ is the Bohr magneton) is constant in a spatially constant magnetic field and could, thus, act as a possible clocking mechanism.","Consider a magnetic spin that is initially polarized along the x-axis and moving along the y-axis through a region with potential, $V(y)$ , wherein a uniform magnetic field along the z-axis is applied (see Fig.","REF for a schematic depiction).","If the particle takes a time ($\\tau _y$ ) to traverse through the region of the magnetic field, the spin components of the transmitted flux would be $\\langle S_y\\rangle = -(\\hbar /2) \\omega _L\\tau _y$ to the lowest order in the magnetic field.","Thus, measurement of the spin precession could yield the traversal time as $\\tau _y = \\frac{2}{g\\mu _B} \\lim _{B\\rightarrow 0} \\frac{\\partial \\langle S_y\\rangle }{\\partial B}$ Büttiker [5] recognized that the spin has a tendency to align along the magnetic field direction that he called spin rotation in addition to the Larmor precession in the place perpendicular to the magnetic field (spin precession).","Thus, there could be two time scales associated with the extents of the spin rotation ($\\tau _z$ ) and the spin precession ($\\tau _y$ ).","The Hamiltonian for the case of a rectangular barrier is $H = \\left\\lbrace \\begin{array}{l} (p^2/2m + V(y)) I - (\\hbar \\omega _L/2) \\sigma _z ~~~\\forall ~~~ 0 < y < L, \\\\ p^2/2m I~~~~\\mathrm {elsewhere} \\end{array} \\right.$ where $V(y)$ is the spatially constant potential, $I$ is the identity matrix and $\\sigma _z$ is the z-component of the Pauli spin matrix and $H$ acts on the spinor $ \\psi = \\left( \\begin{array}{c} \\psi _+(y) \\\\ \\psi _-(y) \\end{array} \\right)$ , where $\\psi _\\pm $ are are the Zeeman components that represent the anti-parallel and parallel spin amplitudes.","The input spinor for a particle polarized along the x-axis can be written as equal superpositions of these components outside the region of the magnetic field due to which the expectation value of the Sz component would be zero.","In the presence of the magnetic field , however, the kinetic energy for these components differ by the Zeeman energy of $\\pm \\hbar \\omega _L/2$ , and this gives a different value to the wave-vectors of the two components in the potential region.","This is particularly severe in the case of tunneling at energies $(E)$ below the barrier when the exponential decay for the wavefunctions becomes very different: $\\kappa _\\pm = \\left\\lbrace \\frac{2m}{\\hbar ^2} (V_0 -E) \\mp \\frac{m\\omega _L}{\\hbar } \\right\\rbrace ^{1/2} \\simeq \\kappa \\mp \\frac{m\\omega _L}{2\\hbar \\kappa },$ where $\\kappa ^2 = 2m(V_0-E)/\\hbar ^2$ .","Thus, one spin component has a greater probability to tunnel across than the other and the transmitted flux becomes spin polarized.","The transmittance amplitudes for the two spin components can be approximately calculated in the case of energies far below the barrier height as $T_\\pm \\simeq T \\exp (\\pm \\omega _L\\tau _z)$ , where $\\tau _z = ml/\\hbar \\kappa $ .","The expectation values of the z-spin component of the transmitted flux is easily obtained as $ \\langle S_z \\rangle = \\frac{\\hbar }{2} \\frac{T_+ - T_-}{T_++T_-} = \\frac{\\hbar }{2} \\tanh (\\omega _L\\tau _z) \\simeq \\frac{\\hbar }{2} \\omega _L\\tau _z $ where the last approximation is made in the limit of small magnetic fields.","Thus, the z-component of the spin scales linearly with the magnetic field and presents yet another clocking mechanism and the spin rotation time can be defined as $\\tau _z = \\frac{2}{g\\mu _B} \\lim _{B \\rightarrow 0} \\frac{\\partial \\langle S_z\\rangle }{\\partial B}$ It turns out that the spin precession ($S_y$ ) dominates for energies far above the barrier and the spin rotation ($S_z$ ) dominates for energies far below the barrier.For a rectangular barrier, the spin precession time for energies far above the barrier tend to the Wigner delay times and far below the barrier, the spin rotation times tend to the Büttiker-Landauer times for the oscillating barrier.","Büttiker proposed that a net traversal time could be defined as the Pythogorean sum of the two spin times as $\\tau ^2 = \\tau _y^2 + \\tau _z^2$ , which reflects the vectorial nature of the change of the spin.","This prescription does not, however, have any fundamental basis."],["Absorption and amplification as a clock","A third interesting manner to clock the time of sojourn in a given region of space would be absorption or amplification, which would be at a rate proportional to the amplitude of the wave.","Hence, a measurement of the wave amplitude as it enters and exits the region of interest can yield the time that it spends inside.","Since the growth is exponential, a logarithmic derivative of the transmittance / reflectance with respect to the imaginary potential is needed.","For light, both absorption and stimulated emission are coherent processes that leave the phase of a coherent mode unchanged, due to the Bosonic nature and this definition in terms of absorption or amplification is a natural definition.","Such a process would not be strictly applicable to Fermionic particles like electrons or neutrons.","However, the effects of absorption or amplification would eventually need to be discussed in the limit of infinitesimally small levels of absorption/amplification that tends to zero, so that the original problem remains unchanged.","Hence, we may assume that the formal procedure works for Fermionic particles as well.","For a scalar wave, coherent absorption / amplification may be implemented by adding an imaginary potential ($iV_I$ ) only in the region of interest (instead of the magnetic field as in Fig.","REF ) , which will eventually be made zero ($V_I \\rightarrow 0$ ).","The Schrodinger wave equation for the wave function of a particle in the presence of the imaginary potential becomes $i\\hbar \\frac{\\partial \\psi }{\\partial t} = -\\frac{\\hbar ^2}{2m} \\nabla ^2 \\psi + [ V_0(\\vec{r}) + i V_I] \\psi .$ Then the traversal times for the transmission and reflection can be defined as $\\tau ^{(T)} = \\frac{\\hbar }{2} \\lim _{V_I \\rightarrow 0} \\frac{\\partial \\ln \\vert T \\vert ^2 }{\\partial V_I} , ~~\\mathrm {and}~~\\tau ^{(R)} = \\frac{\\hbar }{2} \\lim _{V_I \\rightarrow 0} \\frac{\\partial \\ln \\vert R\\vert ^2 }{\\partial V_I}$ respectively, where T and R are the complex transmission and reflection coefficients.","The idea was originally suggested by by Pippard to Buttiker [18], who found that while it reproduced the spin precession times, it did not recover the spin rotation time for waves that are evanescent in the region of interest (sub-barrier tunneling).","It reproduces the Wigner delay times in the limit of large energy above the barrier height.","This identity can be understood in terms of the analytic properties of the complex reflection and transmission coefficients.","Let us consider the transmission coefficient in the complex energy plane $(E_r, E_i)$ .","Its logarithm $\\ln (T) - \\ln \\vert T \\vert + i \\mathrm {Arg}(T)$ would be an analytic function within the Riemann sheet and would satisfy the Cauchy Riemann conditions: $ \\frac{\\partial \\ln \\vert T \\vert }{\\partial E_r} = - \\frac{\\partial \\mathrm {Arg}(T) }{\\partial E_i }, ~~\\mathrm {and} ~ ~\\frac{\\partial \\ln \\vert T \\vert }{\\partial E_i} = \\frac{\\partial \\mathrm {Arg}(T) }{\\partial E_r }.","$ Thus, for energies far above the barrier, the left hand side of the second relation is directly proportional to the traversal times obtained by the imaginary potential ($E_i = V_I$ ), while the right hand side relates directly to the kinetic energy of the particle as potential energy is comparatively small ($E_r \\gg V_0$ ).","This is why the traversal times tend to the Wigner delay time for large energies.","A similar argument will hold for the reflection delay times as well.","Note, however, that it is assumed that the potentials are varied everywhere in space in these arguments.","Usually for the a quantum clock, the potential should only be varied locally within the region of interest (see Fig.","REF ).","Figure: Schematic diagram depicting a possible pathway in the Feynman path integral sense.","(a) shows the portion affected by a global variation of the potential, and (b) shows the portion (solid line) affected by a local variation of the potential.","The dotted portion of the path is not affected and should not be counted towards the traversal time.The issues with the traversal times obtained by the imaginary potential clocks refuse to remain positive for all potentials and further do not tend to the Büttiker-Landauer times for energies far below the barrier.","For the case of tunneling across a rectangular barrier, the ratio $\\tau ^{(T,R)} / \\tau _{BL} \\rightarrow 0$ in the low energy limit.","The Büttiker-Landauer times are very compelling in that limit and we would really want the traversal to remain positive definite."],[" Sojourn times: correcting the clocks","It turns out that most of the problems of defining a positive definite traversal time for the transmission arise from spurious scattering concomitant with the very clocking mechanism / potential and a procedure was duly outlined to separate out this extra scattering and correct the traversal times [6].","We will call these corrected traversal times as sojourn times as they literally relate to the times of journey through the region of interest.","We will primarily deal with the imaginary potential clock here, but will point out the exact analogies with the Larmor clock.","Figure: Schematic diagram partial transmission and reflection coefficients associated with the potentials (regions 1 and 3) on either side of a constant potential region (2).","These partial transmission(t jk t_{jk} and reflection (r jk r_{jk}) coefficients may be used to construct the net transmission and reflection coefficients using equations ( ).First of all, it is important to appreciate that the presence of the clock potential not only invokes a response in the relevant extra degree of freedom, but also modifies the scattering due to a change of potential.","One would expect this extra scattering to go to zero in the limit of zero clock potential.","Our procedure, however, involves taking a derivative with respect to the clock potential, which can have contributions that would not vanish as the clock potential is made zero.","Let us consider the transmission of a wave with energy above the barrier (case of propagating waves) through the constant potential region encased between two arbitrary potentials on either side as shown in Fig.","REF .","The space may be divided into three regions and the partial coefficients of transmission and reflection for a wave incident from region (j) unto region (k) are $t_{jk}$ and $r_{jk}$ as shown schematically in Fig.","REF .","The transmission and reflection coefficients can be written as a sum of the partial waves through the region [7] $T&=& t_{12} t_{23} e^{ik^{\\prime }L} + t_{12} r_{23} r_{21} t_{23} e^{3ik^{\\prime }L} + t_{12} r_{23} r_{21} r_{23} r_{21}t_{23} e^{5ik^{\\prime }L} + \\cdots \\nonumber \\\\R&=& r_{12} + t_{12} r_{23} t_{21} e^{2ik^{\\prime }L} + t_{12} r_{23} r_{21} r_{23} t_{21} e^{4ik^{\\prime }L} +\\cdots $ the coefficients $t_{jk}$ and $r_{jk}$ also depend on the clock potential and a derivative with respect to the total transmission or reflection coefficient would leave behind some terms arising from this dependence, which do not vanish as the clock potential is made to go to zero.","This is the origin of the spurious scattering.","An analysis of the structure of the partial wave superposition quickly reveals that the growth or attenuation related to the imaginary potential would only involve the paired combination of ($V_I L$ ) where $L$ is the lenght of the spatial region of interest.","Afterall the traversal time should directly relate to spatial region of interest.","The spurious scattering on the other hand would only involve unpaired $V_I$ .","A formal procedure to isolate the effects of this spurious scattering can now be given.","We will treat$\\xi = V_I L $ and $V_I$ as independent variables, keep $\\xi $ formally constant and let $V_IL \\rightarrow 0$ in the expression for the transmission coefficient.","The sojourn time for transmission for propagating waves is now obtained as $\\tau _s^{(T)} = \\frac{\\hbar L}{2} \\lim _{\\xi \\rightarrow 0} \\frac{\\partial \\ln \\vert T(\\xi , V_I=0) \\vert ^2}{\\partial \\xi } .$ For the case of wave tunneling (energy lesser than the barrier height $E < V_0$ ), the wave vector is principally imaginary in the barrier region.","The real part of the potential or refractive index essentially affects the rate of exponential decay / growth of the evanescent wave.","On the other hand, the imaginary potential or imaginary part of the refractive index causes a phase shift with distance of this evanescent wave [13].","Consider the complex wave-vector for propagating waves ($E > V_0$ ), $ k = \\sqrt{ \\frac{2m}{\\hbar ^2} [E - (V_0+iV_I)]} ~~\\simeq k_r - i \\frac{mV_I}{k_r \\hbar ^2} ~~ \\forall ~~ V_I \\ll E, $ where $\\hbar k_r = \\sqrt{2m(E-V_0)}$ .","When we have evanescent waves ($E < V_0$ ) on the other hand, we can write $ k = \\sqrt{ \\frac{2m}{\\hbar ^2} [E - (V_0+iV_I)]} ~~\\simeq i \\kappa _r - \\frac{mV_I}{\\kappa _r \\hbar ^2} ~~ \\forall ~~ V_I \\ll V_0, $ where $\\hbar \\kappa = \\sqrt{2m(V_0-E)}$ .We hence realise the principle effect of the clock with respect to the spatial region is in the phase of the wave and not the amplitude for the evanescent wave.","This is completely analogous to the spin rotation being predominant over the spin precession in the case of the Larmor clock.","Mathematically, we have a square root singularity for the wave-vector in the complex energy plane and we are unable to analytically continue the behaviour for propagating waves to the case of evanescent waves across the branch-cut in either case of the imaginary potential clock or the Larmor clock.","Hence, we define for the case of sub-barrier tunneling or evanescent wave, the sojourn time as $\\tau ^{(T)}_s = \\frac{\\hbar L}{2} ~ \\lim _{\\xi \\rightarrow 0}~\\frac{\\partial }{\\partial \\xi } \\left[ \\ln \\left(\\frac{T(\\xi , V_I=0)}{T^*(\\xi , V_I=0)} \\right) \\right] ,$ where $T*$ is the complex conjugate of $T$ and the ratio $T/T*$ essentially yields twice the phase of the transmission coefficient.","For the case of the region with constant potential of height $V_0$ enclosed between the two arbitrary potentials and energy above the barrier, the sojourn time reduces to a positive definite quantity: $ \\tau ^{(T)}_s = \\frac{1 - \\vert r_{21} r_23 \\vert ^2 }{1 + \\vert r_{21} r_{23} \\vert ^2 - 2\\mathrm {Re}(r_{21} r_{23} e^{2i k_r L})} \\tau _{BL}.","$ A similar positive definite expression is obtained in the case of evanesent waves (tunneling below the barrier) as $ \\tau ^{(T)}_s = \\frac{1 - \\vert r_{21} r_23 \\vert ^2 e^{-4\\kappa L} }{1 + \\vert r_{21} r_{23} \\vert ^2e^{-4\\kappa L} - 2\\mathrm {Re}(r_{21} r_{23} e^{-2\\kappa L})} \\tau _{BL}.","$ The sojourn time tends directly to the Büttiker-Landauer times for an opaque barrier (energies far below the barrier or long barrier lengths), which is very important.","In the high energy limit, it tends to the classical Wigner delay times.","Thus it has the correct limits.","Further, it was shown in Ref.","[6] that the sojourn times defined in this manner for two non-overlapping regions is additive and hence, the times for any region of space with any arbitrary applied potential can be concluded to be positive definite.","The case of reflection is a little bit more complex.","Applying the above definitions for the reflection coefficient in place of the transmission coefficient does not yield a positive definite sojourn time.","From the expansion in terms of the partial waves, one notes an essential difference between the reflection and the transmission.","All the partial waves for the transmitted wave sample the region of interest and pick the paired combination $\\xi = V_IL$ in the amplitude or the phase.","In the case of reflection, there is one partial wave $r_{12}$ corresponding to the prompt reflection from the edge of the region of interest that never enters the region of interest.","Yet, this partial wave interferes with the others to produce the net reflection amplitude.","In the spirit of our earlier arguments that the sojourn time should be causally related to the region of interest, it would be necessary to eliminate the weightage of this partial wave that should not be affected by the clocking potential.","Hence, we explicitly subtract this prompt reflection amplitude out of the total reflection amplitude $R^{\\prime } = R - r_{12}$ and define the sojourn times for reflection using $R^{\\prime }$ in place of $T$ as before for both cases of the propagating waves as well as the evanescent waves.","A simple and general result is obtained as $\\tau ^{(R)}_s = \\tau ^{(T)}_s + \\tau _{BL}$ which is consequently always greater than the sojourn time for transmission and positive definite.","An experimental measurement of this procedure is possible by interfering destructively the reflection from a modified potential whose reflection is $r_{12}$ with the reflectance from the given potential.","For example, one can use the same optical system that is index matched to the continuum form beyond the point where the absorption or gain is applied.","Alternatively, one may use the recently developed metamaterial perfect absorbers [19] for light, where there is the possibility of matching the impedance and preventing any reflection.","It should be noted that such a corection procedure can be analogously applied to the spin precession and spin rotation times for the Larmor clock as well, where the paired variable $\\xi = BL$ would be taken to zero after explicitly putting the magnetic field $B=0$ while keeping $\\xi $ formally constant.","We then obtain identical results to the imaginary potential clock.","The procedure outlined above was shown equivalent to stochastic absorption [20], whereby the absorption does not cause any scattering but contributes only to loss of the wave flux.","This can be viewed as the case when there is inelastic scattering out of the given mode of the mesoscopic and the coherence of the mode is not affected.","Only the coherent part of the wave is measured and the scattering into other modes manifests as a loss for this mode.","Another example could be that the scattering leads to decoherence of the wave.","In any interferometric measurement only the coherent part of the wave is measured and the decohered part would appear as a loss.","Thus, a finite rate of decoherence itself could be utilized as a clocking mechanism ."],["The problem of the quantum first passage time","Consider a free quantum particle that is released from a localized region at some instant of time and thereafter subjected to instantaneous projective measurements to detect its arrival at a particular region of space.","The measurements are made at regular time intervals , and the system is allowed to evolve until the time a detection occurs.","The main question addressed is: what is the probability that the particle is detected for the first time after time $t$ i.e.","at the $n=(t/\\tau )^{th}$ measurement [21].","Conversely, one can ask for the probability of particle not being detected (i.e., surviving) upto a given time.","It is proposed by a general perturbative approach for understanding the dynamics which maps the evolution operator, which consists of successive unitary transformations followed by projections, to one described by a non-Hermitian Hamiltonian.","For some examples of a particle moving on one- and two-dimensional lattices with one or more detection sites, use this approach to find exact expressions for the survival probability and find excellent agreement with direct numerical results.","For the one- and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle.","It is shown that an interesting and nontrivial connection between the dynamics of the particle in their model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites [22], [23].","If continuous projective measurements are done then famous Zeno comes in to effect and particle does not evolve.","Thus quantum first passage time indeed a nontrivial model.","It is very subtle and as mentioned before in quantum mechanics there is no dynamical operator for time travel between two points.","We may conclude this brief disussion on this topic by saying that the quantum first passage time is still remains a mystery"],["Conclusions and Outlook ","We have outlined here various attempts to answer a fundamental question, namely, “what is the time that a quantum particle or wave spends in a specified region of space ?” We do not have a self-adjoint operator for the arrival time in quantum mechanics, and the arrival time is not an observable.","Yet it is intuitive and important to ask about timescales of any physical problem and the time of stay or sojourn appears to be a calculable quantity and a useful one to compare timescales.","The sojourn time can provide for a meaningful alternative view-point within quantum mechanics.","Knowledge of the sojourn times obviously cannot provide answers to questions that cannot be answered within the paradigm of quantum mechanics.","Here we have exploited the fact that light as well as quantum particles are mathematically described by the same Helmholtz equation and talk about both systems in the same breath.","Starting with the description of wavepackets by the group velocity and its extension by Wigner, we have indicated a variety of manners in which this question has been approached.","The Smith dwell time is a positive definite time, but it is an unconditional time that does not depend on the output scattering channel.","The path integral approach [21] or an approach based on the WKB method [17] are closely related to this, but can define conditional traversal times.","Beyond these, we discuss the proposals to consider the dynamical evolution of an extra degree of freedom attached to the traversing particle due to the local interaction with a potential applied only in the region of interest.","One has to meaningfully identify the extent of evolution of the clocking mechanism, which is an observable, with the time spent in that region.","We have discussed three examples: (i) exchange of quanta via interaction with a radiation field; (ii) the spin precession; (iii) rotation in a magnetic field or the growth / attenuation due to an imaginary potential (amplifying or absorbing medium).","A rather subtle problem that arises with these clocks is that they give rise to an extra spurious scattering that interferes with the very process and effectively changes the potential in the region of interest.","This extra scatterings is carefully identified and is explicitly eliminated by a formal mathematical procedure.","Further, it is shown that the effect of the clocking potential manifests in different quantities for propagating and evanescent waves: In the case of the imaginary potentials, it manifests in the amplitude for propagating waves and in the phase for evanescent waves (tunneling below the barrier); and in the case of the Larmor spin clock, it mainfests in the spin precession for propagating waves and in the spin rotation for evanescent waves.","As a final caveat, it is shown that partial waves that reflect from the surface of the region should also be eliminated from reckoning as they do not spend anytime within the region of interest and this can, indeed, be done by interferometric measurements.","Including these considerations yields a sojourn time that is Real and positive definite; Additive for non-overlapping regions of space; Related causally to the region of interest; Calculable and directly related to a measurable quantity.","The main issue in defining traversal or sojourn times for a quantum system has been due to interference between partial waves (the alternative paths in quantum mechanics), which defies naive realism.","With these advances in understanding, however, the sojourn time becomes a calculable quantity that is practically useful for estimating other quantities and understanding physical phenomena, for example, the dephasing rates in a quantum system."],["Acknowledgement","Both the authors acknowledge illuminating and educative discussions with Prof. N. Kumar who introduced them to these topics as well as the constant encouragement they have received from him.","They would like to dedicate this tutorial to him.","SAR acknowledges funding from the DST India through a Swarna Jayanti Fellowship.","AMJ thanks DST, India for financial support (through J. C. Bose National Fellowship)"]],"string":"[\n [\n \"How long does a quantum particle or wave stay in given region of space?\"\n ],\n [\n \"Abstract The delay time associated with a scattering process is one of the most important dynamical aspects in quantum mechanics.\",\n \"A common measure of this is the Wigner delay time based on the group velocity description of a wave-packet, which my easily indicate super-luminal or even negative times of interaction that are unacceptable.\",\n \"Many other measures such as dwell times have been proposed, but also suffer from serious deficiencies, particularly for evanescent waves.\",\n \"One important way of realising a timescale that is causally connected to the spatial region of interest has been to utilize the dynamical evolution of extra degrees of freedom called quantum clocks, such as the spin of an electron in an applied magnetic field or coherent decay or growth of light in an absorptive or amplifying medium placed within the region of interest.\",\n \"Here we provide a review of the several approaches developed to answer the basic question - how much time does a quantum particle (or wave) spend in a specified region of space?\",\n \"While a unique answer still evades us, important progress has been made in understanding the timescales and obtaining positive definite times of interaction by noting that all such clocks are affected by spurious scattering concomitant with the very clock potentials, however, weak they be and by eliminating the spurious scattering.\"\n ],\n [\n \"Introduction\",\n \"One of the first things a student of Physics learns is to calculate the time at which an event, such as the location of a particle at a given position, occurs.\",\n \"A related question would be to ask how long does a particle stay in a given region of space during the course of its motion.\",\n \"Classically it is possible to simultaneously specify the position and the momentum of a particle, and these questions about the time instants or the time intervals for given events have unambiguous answers.\",\n \"Even as we proceed to statistically understand systems with very large number of particles through probability distributions, where the classical motion of each particle is described only stochastically, concepts such as the first passage times [1] remain meaningful.\",\n \"This well entrenched concept, however, becomes difficult to define for quantum mechanical systems, and indeed, for any form of waves.\",\n \"Figure: Left panel: Schematic picture of a quantum particle that is initially localized in a fixed region and arrives at a detector array.\",\n \"The first passage time refers to the first point in time when the particle is registered at a detector.\",\n \"Right panel: The particle may take any virtual path toget to the detector, including ones that cross the boundary many times.\",\n \"The final amplitude will be a sum over the amplitudes for any such path.Imagine an experiment where a quantum-mechanical particle is released from some fixed region inside a box.\",\n \"On one side of the box there is a screen with detectors which click as soon as the particle \\\"arrives\\\" at the screen.\",\n \"One expects that the time of arrival of the particle is a stochastic variable and it is interesting to ask for it's probability distribution.\",\n \"This is similar to asking for the distribution of the time of absorption of a Brownian particle at some point.\",\n \"However, the quantum problem turns out to be very subtle and there is as yet no clear answer to the question.\",\n \"The point is that the particle that is initially localized and released must subsequently cross a given boundary in space when it is detected for the very first time.\",\n \"Yet this very calculation includes amplitudes from paths that may very well extend all over space even beyond the given boundary.\",\n \"Very recently the problem of first time arrival of a quantum mechanical particle has been considered satisfactorily utilizing a path integral approach that with a restricted path decomposition and appears to succeed in obtaining positive definite quantum first passage times for motion on a one dimensional lattice [2].\",\n \"The principal difficulty arises from the fact that waves are infinitely deformable objects and many aspects of motion arise through interference effects.\",\n \"Hence it becomes impossible to define well defined start and finish lines for wave packets.\",\n \"In the quantum mechanical perspective, where all dynamical observables have a corresponding operator, there has been great difficulty in defining a “good” Hermitian operator that conforms to classical notions of time or a time interval [3].\",\n \"For classical waves such as electromagnetic waves also, related difficulties exist.\",\n \"For example, we would like to classically demand that a time of stay in a given region of space should be (i) Real, (ii) Positive definite, (iii) tend to classically calculable times in the limit of large energies.\",\n \"Inspite of serious efforts over many years [4], no definition for such a quantum mechanical time has been universally accepted.\",\n \"In fact, when we say that a wave moves, it becomes imperative to clearly define what is the quantity related to the motion of a wave that is being talked about.\",\n \"Regardless of these difficulties, it must be emphasized that the times of traverse or dwell associated with a wave are calculable quantities that can be useful for understanding processes and process rates in a given system.\",\n \"In this tutorial, we will discuss the several approaches, the related difficulties and some recent possibilities that have arisen in this context.\",\n \"First, we will go through some of the basic definitions and concepts regarding wave motion and the times that can be related to the motion of a wave.\",\n \"It turns out that there is a large variety of times arising from different definitions related to various different quantities associated with a wave.\",\n \"Surprisingly, these definitions can result in apparent superluminal times or even negative delay times for the transit of a wave through dispersive media or potentials.\",\n \"Subsequently, we will describe certain clocking mechanisms associated with physical processes such as the precession of a spin in a magnetic field [5] that have been developed to calculate theoretically the traversal or dwell times of a wave [4].\",\n \"These clocking mechanisms have to be used with care as they can contribute to scattering and change the very problem being discussed [6].\",\n \"These ideas lead to the possibility of using quantum dephasing or stochastic absorption as a clock.\"\n ],\n [\n \"1. Time-scales based on the group velocity \",\n \"Here we will try to explore these questions in the general context of waves to cover both electromagnetism and quantum mechanics.\",\n \"Both the time independent Schrodinger equation for the wavefunction of a quantum particle and the Maxwell equations in frequency domain for the amplitude of a time harmonic electromagnetic wave (of only one polarization) reduce to the Helmholtz wave equation: $\\\\nabla ^2 \\\\psi + k^2 \\\\psi = 0.$ For the electromagnetic wave, k is the wave vector given by $k^2 = n^2 \\\\omega ^2 /c^2$ , $\\\\omega $ is the angular frequency of the wave and $n$ is the refractive index of the medium with $c$ being the speed of light in vacuum.\",\n \"For the quantum particle in comparison, $k$ is given by $k^2 = 2m(E-V_0)/\\\\hbar $ , where$E$ and $m$ are the energy and the mass of the particle and$V_0$ is the potential.\",\n \"The principle difference between the the two systems is in free space where the refractive index is unity.\",\n \"Consequently the wave-vector for an electromagnetic wave is linearly proportional to the frequency.\",\n \"In the case of the quantum particle, even for $V_0=0$ , the wave-vector disperses with the quadratic root of the energy.\",\n \"This has a non-trivial manifestation in the undistorted propagation of electromagnetic pulses in free space while a quantum mechanical wavepacket spreads out and disperses even in free space.\",\n \"Thus, it is realized that the fundamental properties of propagation of the waves are governed by the potentials or the refractive index of the medium through the dispersion relation between the wave-vector and the frequency or energy.\",\n \"Consider the scalar wave $\\\\psi (\\\\vec{r},t) = a(\\\\vec{r}) \\\\exp [i\\\\phi (\\\\vec{r}) - i\\\\omega t]$ where $\\\\phi (\\\\vec{r})$ is some scalar function.\",\n \"For this wave, $\\\\phi (\\\\vec{r}) = \\\\mathrm {constant}$ denotes the constant phase surfaces.\",\n \"To trace the motion of these surfaces, let us look at the condition at two points $(\\\\vec{r},t)$ , and $(\\\\vec{r} + \\\\delta \\\\vec{r}, t+\\\\delta t)$ .\",\n \"The phase front is the same if, and only if, $\\\\phi (\\\\vec{r} + \\\\delta \\\\vec{r}) -\\\\omega ( t+\\\\delta t) \\\\simeq \\\\phi (\\\\vec{r}) + \\\\delta \\\\vec{r} \\\\cdot \\\\nabla \\\\phi (\\\\vec{r}),t) - \\\\omega t - \\\\omega \\\\delta t = \\\\phi (\\\\vec{r}) - \\\\omega t,$ where we have included the first order term only in the infinitesimal $ \\\\delta \\\\vec{r}$ .\",\n \"FRom the above, we obtain $ v_p = \\\\left| \\\\frac{\\\\delta \\\\vec{r}}{\\\\delta t} \\\\right| = \\\\frac{\\\\omega }{| \\\\nabla \\\\phi |} $ as the phase velocity for a wave with an arbitrary wave front.\",\n \"The ratio , where $\\\\phi (\\\\vec{r})$ is the phase of the wave [7], is known as the phase velocity of the wave and represents the rate at which the equiphase surfaces of the wave move through the medium.\",\n \"Note that the phase velocity can be just about any number (positive or negative) depending on the phase structure (gradient) of the wave.\",\n \"In one dimension or when there is transverse invariance along two dimensions and plane waves result, the phase velocity reduces to the familiar relation.\",\n \"This gives rise to the conventional notion that the phase velocity for a wave is $c/n$ , and is hence mistaken for the speed of a wave in a medium.\",\n \"As pointed out, the phase velocity can easily be superluminal for materials with refractive index $n < 1$ .\",\n \"Further, in negative refractive index materials [8] and in waveguides that support backward wave propagation, this phase velocity is obviously negative.\",\n \"Thus, the phase velocity does not really specify anything concrete about the rate of the motion of the wave or the time spent by a wave in a given region of space.\",\n \"In a material medium, the refractive index (polarization) of a medium will, in general, be frequency dependent and a complex quantity.\",\n \"This is a simple consequence of the medium having certain natural frequencies at which it will resonantly polarize corresponding to atomic or molecular levels of the constituents of the medium.\",\n \"Due to the different refractive index at different frequencies, the constituent time harmonic waves present in a wave packet essentially travel at their own phase velocities resulting in the interference pattern changing completely in time and leading to a dispersion of the wave packet.\",\n \"This is easily seen by writing down the field amplitude of the wave at different times.\",\n \"If $E(\\\\vec{r},0)$ be the field amplitude at time$t =0$ , and $E(\\\\vec{k} = \\\\int _{-\\\\infty }^\\\\infty E(\\\\vec{r},0) e^{-i\\\\vec{k}\\\\cdot \\\\vec{r}} ~d^3r$ is is the spatial Fourier transform of the field amplitude, the field amplitude at any other time, $t$ , is given by $E(\\\\vec{r}, t) = \\\\left(\\\\frac{1}{2\\\\pi }\\\\right)^3 \\\\int _{-\\\\infty }^\\\\infty E(\\\\vec{k}) e^{i(\\\\vec{k}\\\\cdot \\\\vec{r}-\\\\omega t)} ~d^3k .$ If the dispersion was linear, this would just correspond to the same function shifted to a new position.\",\n \"For an arbitrary dispersion $\\\\omega (\\\\vec{k})$ , it becomes difficult to say anything in general.\",\n \"If we assume, however, that most of the amplitude is concentrated in a small frequency band about a central (carrier) frequency $\\\\omega _0$ , then one can carry out a Taylor’s series expansion $ \\\\omega (\\\\vec{k}) = \\\\omega (\\\\vec{k}_0) + (\\\\vec{k} -\\\\vec{k}_0) \\\\cdot \\\\nabla _k \\\\omega (\\\\vec{k}_0) + \\\\cdots , $ where the subscript $k$ indicates that the derivative is with respect to the wave-vector and retain only the linear term.\",\n \"Substituting this in the expression for the field amplitude, one obtains, $E(\\\\vec{r}, t) = \\\\left(\\\\frac{1}{2\\\\pi }\\\\right)^3 e^{i\\\\varphi } \\\\int _{-\\\\infty }^\\\\infty E(\\\\vec{k}) e^{i\\\\vec{k}\\\\cdot [\\\\vec{r} - \\\\nabla _k\\\\omega (\\\\vec{k}_0) - \\\\omega t]} ~d^3k ,$ which is essentially the same waveform that is shifted by an amount $\\\\nabla _k \\\\omega (\\\\vec{k}_0)t$ in space, apart from a trivial extra phase of $\\\\varphi = \\\\omega _0 t + k_0 \\\\cdot \\\\nabla _k \\\\omega (\\\\vec{k}_0)t$ .\",\n \"This brings up another rate at which the envelope of the wave propagates in the medium and defines the so-called group velocity $v_g = \\\\nabla _k \\\\omega (\\\\vec{k}_0)$ .\",\n \"This tracks the rate at which fiducial (well recognisable) points on the waveform, such as the peak of the wavepacket, move and it is assumed that waveform is largely undistorted.\",\n \"This is rarely satisfied for a quantum mechanical wave even in free space.\",\n \"For an electromagnetic wave, however, this is satisfied for quasi-monochromatic fields and when absorption in the medium is minimal.\",\n \"Again, it should be noted that the spread in the wavevectors is also equivalently small and the superposition should consist of waves moving along a common direction.\",\n \"Noting that the group delay time accumulated upon traversing a distance $L$ in the medium is $ \\\\frac{L}{\\\\nabla _k \\\\omega (\\\\vec{k}_0)} = \\\\left.\",\n \"\\\\frac{\\\\partial (kL)}{\\\\partial \\\\omega }\\\\right|_{\\\\omega _0} ,$ one can equivalently define a group delay time for scattering problems where the kinematic phase is replaced by the phase change upon scattering ($\\\\phi $ ).\",\n \"In one dimension, this would be the phase shift upon reflection or transmittance.\",\n \"This yields the Wigner’s group delay time $\\\\tau _w = \\\\left.\",\n \"\\\\frac{\\\\partial \\\\phi }{\\\\partial \\\\omega } \\\\right|_{\\\\omega _0}.$ Note the extrapolation made here from a free propagation problem into the time delay obtained in a scattering event.\",\n \"This time essentially measures the time difference between the entry and emergence of fiducial points into and out of the scattering volume.\",\n \"Figure: The change in refractive index due to dispersion is plotted with a magnified scale.\",\n \"The dispersion of the real part of the refractive index for a single Lorentz resonance (red dotted curve) and for a highly dispersive medium that can be produced, for example by electromagnetically induced transparency or by a Fano resonance.\",\n \"Such dispersions enable very large changes of the refractive index, either normal or anomalous as marked in the figure.\",\n \"The group velocity can easily become very small or very large and even negative in such dispersive regimes.Using the dispersion for light, $k^2 = n(\\\\omega )^2 \\\\omega ^2/c^2$ in a homogenous isotropic medium with a dispersive refractive index $n(\\\\omega )$ , the group velocity is also often conveniently written in the following manner $v_g = \\\\frac{c}{n(\\\\omega _0) + \\\\omega _0 (dn/ d\\\\omega )_{\\\\omega _0}}.$ The group velocity from this definition immediately shows that the group velocity can be very small when the dispersion is normal and large, i.e., $(dn/d\\\\omega ) \\\\gg 0$ (see Fig.\",\n \"REF ).\",\n \"This is the origin of ultra-slow light which has been demonstrated using the large dispersion possible in atomic gas vapours and Bose-Einstein condensates [9].\",\n \"On the other, if the dispersion were anomalous and large , $(dn/d\\\\omega ) < 0~\\\\mathrm {and}~ |dn/d\\\\omega | \\\\gg 0$ (see Fig.\",\n \"REF ), then we have that the group velocity can become greater than c, or even negative in such situations.\",\n \"Indeed there have been several experiments on the apparent motion of light in a medium at a rate faster than light in vacuum [REF].\",\n \"Negative group velocities and negative Wigner delay times are even more problematic – that would imply that the pulse exited region of space before it even entered, in a sense violating causality.\",\n \"Landauer [4] had severely criticised the Wigner delay time to give information about the dwell times on grounds of causality and emphasized that there was no causal connection between the peaks (or fiducial points) of the incoming and the outgoing waveforms.\",\n \"This difficulty is accentuated when the scattering potential strongly deforms the wave-packet.\",\n \"The large dispersion responsible for the deformation is also usually accompanied by severe dissipation or gain that causes large spectral modifications with corresponding distortions of the temporal pulse shape.\",\n \"Hence the analysis of the motion of a spectrally broad or distorted pulse in terms of the conventional group velocity is severely limited, as the pulse can lose its very identity after propagating to a large distance in the dispersive medium.\",\n \"These arguments apply and hold true even for definitions of the times based on the center-of-mass of the wavepacket or in an alternative approaches where the motion of the forward edge of the wavepacket is followed.\",\n \"These issues raise questions about the group velocity, or the Wigner delay time that is based on the group velocity, to actually provide an answer to the question that we seek about the time that the wave spends in a given region of space.\",\n \"It has been emphasized by most authors that the superluminal or negative Wigner delay times do not violate causality or the Special theory of relativity.\",\n \"It simply turns out that the functions that we considered to describe the fields have been smooth analytic functions.\",\n \"This is a consequence of the fields being the solutions of the Helmhotz differential equation.\",\n \"Analytic functions are problematic because they have an infinite support – the function extends all over the space, although it may be very small.\",\n \"A good example of such functions is the Gaussian function.\",\n \"In principle, one may always take the analytic function at a given point and carry out a Taylor's series expansion of the function using the values of the function and its derivatives at the given point, to obtain its value at any point in space, however, far off from that point.\",\n \"Since the function is analytic, all the derivatives at the given point exist.\",\n \"Thus, knowledge of the analytic signal even if localized in the manner of a Gaussian function, already exists at all the other points in principle, even if the value of the signal were to be infinitesimally small.\",\n \"Thus, there is no extra information being conveyed to the other points with the wave motion as the information was already present.\",\n \"Thus, there is no violation of the special theory of relativity or of causality.\",\n \"It has been emphasized that in principle only meromorphic functions (with discontinuities in the function or its derivatives) can be used to encode information.\",\n \"Any such discontinuity will generate very high frequency components in the power spectrum of the signal.\",\n \"These high frequency components will always propagate at the speed of light in vacuum (c) as the refractive index as This is a consequence of finite inertia for the charge carriers in all material media.\"\n ],\n [\n \"The Dwell times based on current fluxes or energy transfer\",\n \"A second approach to this problem has been to define the time in terms of probability of finding the particle within the spatial volume of interest.\",\n \"Smith [10] defined the dwell time for a mono-energetic quantum particle in the region $[0, L]$ (in one dimension) as $\\\\tau _d = \\\\frac{1}{J} \\\\int _0^L \\\\vert \\\\psi (x) \\\\vert ^2 ~dx$ where $\\\\psi (x)$ is the wavefunction and $ J = \\\\mathrm {Re} (\\\\hbar /im) (\\\\psi ^*\\\\;\\\\nabla \\\\psi )$ is the current flux associated with the incoming particle.\",\n \"Note that the Smith dwell time, is independent of the scattering channel (reflection or transmission) and is hence, an unconditional time.\",\n \"In case of a time-varying pulsed waveform, this dwell time can be generalized by integrating over all times as $\\\\tau _d = \\\\int _{-\\\\infty }^{\\\\infty } dt ~ \\\\int _0^L \\\\vert \\\\psi (x) \\\\vert ^2 ~dx.$ In the case of electromagnetic waves, a similar approach can be adopted, but using the energy of the fields.\",\n \"Thus the dwell time in a region of volume $(V)$ could be defined as $ \\\\tau _d = \\\\frac{\\\\int _V U d^3r}{ \\\\int _A \\\\vec{S} \\\\cdot d\\\\vec{a}} , $ where $U$ is the energy density associated with the electromagnetic wave, $\\\\vec{S}$ is the Poynting vector denoting the power flow per unit area of the incoming wave and A is the surface area of volume through which the incoming wave is incident.\",\n \"While there is no problem with the Poynting vector defined as $\\\\vec{S} = \\\\vec{E}\\\\times \\\\vec{H}$ in terms of the electric and magnetic fields, there are severe difficulties in defining an energy density solely associated with the wave in a dispersive and dissipative (or amplifying) medium.\",\n \"The difficulties of separating the energy associated with the wave and the polarization in the medium are well known in this scenario where the two fields continuously exchange energy.\",\n \"In fact, it is well known that the energy density defined by first order Taylor expansions of the dispersion [11] can easily be negative for severe dispersion of the material parameters.\",\n \"In some sense, the above quantity and the Smith Dwell time are equivalent as both the quantities involve quadratic expressions of the underlying fields.\",\n \"The issue of dispersion and polarization in material media complicate the issues further in the context of electromagnetic waves.\",\n \"Another fruitful approach to define arrival times that is based on the centroid of power flow of the electromagnetic wave is noteworthy [12].\",\n \"An arrival time at a point can be defined as a time average (first moment of time) over the component of the Poynting vector normal to a surface $\\\\langle t \\\\rangle _r = \\\\frac{\\\\hat{u} \\\\cdot \\\\int _{-\\\\infty }^\\\\infty t \\\\vec{S}(\\\\vec{r},t) dt}{ \\\\hat{u} \\\\cdot \\\\int _{-\\\\infty }^\\\\infty \\\\vec{S}(\\\\vec{r},t) dt},$ where $\\\\hat{u}$ denotes the unit normal to the given surface, which could very well be that of a detector.\",\n \"The time for traverse between two points $\\\\vec{r}_i$ and $\\\\vec{r}_f$ can now be thought of as the difference between the arrival times at the two points as $\\\\Delta t = \\\\langle t\\\\rangle _{\\\\vec{r}_f} - \\\\langle t\\\\rangle _{\\\\vec{r}_i}$ .\",\n \"A basic theorem was proven to show that the propagation delay could be decomposed in terms of a net group delay and a reshaping delay.\",\n \"The net group delay consists of a spectrally wieghted average group delay at the final point $\\\\vec{r}_f$ given by $\\\\Delta t_G = = \\\\frac{\\\\hat{u} \\\\cdot \\\\int _{-\\\\infty }^\\\\infty t \\\\vec{S}(\\\\vec{r}_f,\\\\omega ) [(\\\\partial \\\\mathrm {Re}(k) / \\\\partial \\\\omega ) \\\\cdot \\\\Delta r] d\\\\omega }{ \\\\hat{u} \\\\cdot \\\\int _{-\\\\infty }^\\\\infty \\\\vec{S}(\\\\vec{r}_f,\\\\omega ) d\\\\omega },$ The reshaping delay, as the very name suggests, arises from the deformation due to spectral modulation by the medium and can be calculated in terms of the spectral fields at the initial point as $\\\\Delta t_R = \\\\wp \\\\lbrace \\\\exp [-\\\\mathrm {Im}(\\\\vec{k}) \\\\cdot \\\\Delta \\\\vec{r}] \\\\vec{E}(\\\\vec{r}_i, \\\\omega )\\\\rbrace \\\\wp \\\\lbrace \\\\vec{E}(\\\\vec{r}_i, \\\\omega )\\\\rbrace ,$ where the operator $ \\\\wp \\\\lbrace \\\\vec{E}(\\\\vec{r}, \\\\omega )\\\\rbrace =\\\\frac{\\\\hat{u} \\\\cdot \\\\int _{-\\\\infty }^\\\\infty \\\\mathrm {Re} [-i \\\\frac{\\\\partial \\\\vec{E}(\\\\vec{r}, \\\\omega )}{\\\\partial \\\\omega }\\\\; \\\\times \\\\; \\\\vec{H}^*(\\\\vec{r},\\\\omega )] d\\\\omega }{ \\\\hat{u} \\\\cdot \\\\int _{-\\\\infty }^\\\\infty \\\\vec{S}(\\\\vec{r},\\\\omega ) d\\\\omega }, $ This approach which is applicable to an arbitrary waveform or dispersive medium showed that both the components in general were always significant.\",\n \"The most important aspect of this proposal is that it does not involve any perturbative expansion of the wave number around the carrier frequency.\",\n \"A few salient points may be noted in this context: For a narrowband pulse, the reshaping delay tends to zero and the total delay time is dominated by the group delay time.\",\n \"We note that even the Wigner delay time would describe the situation quite well in this case.\",\n \"For a broadband pulse with only propagating components, the net group delay can become negative, but the corresponding reshaping delay causes the overall delay time to remain luminal as the pulse experiences a strong reshaping during propagation.\",\n \"This makes this proposal a very strong candidate to represent the traversal time that is causal and non-negative.\",\n \"It has been shown that the definition is equivalent to another definition based on the rate of energy absorbed by a detector $ \\\\langle t \\\\rangle _r = \\\\frac{ \\\\int _{-\\\\infty }^\\\\infty t \\\\frac{d A(\\\\vec{r},t)}{dt} dt}{ \\\\int _{-\\\\infty }^\\\\infty \\\\frac{d A(\\\\vec{r},t)}{dt} dt}, $ where$(dA/dt)$ is the rate of absorption of energy per unit volume inside the detector placed at $\\\\vec{r}$ , and is given by $ \\\\frac{dA}{dt} = \\\\int \\\\int \\\\omega \\\\left[ \\\\varepsilon _0 \\\\mathrm {Im}(\\\\varepsilon ) E^*(\\\\omega ^{\\\\prime }) E(\\\\omega ) + \\\\mu _0\\\\mathrm {Im}(\\\\mu )H^*(\\\\omega ) H(\\\\omega ) \\\\right]d\\\\omega ^{\\\\prime }~d\\\\omega .\",\n \"$ This is integrated spatially over the detector volume to obtain the total rate of absorption within the detector.\",\n \"Obviously the spatial extent of the detector is assumed to be small compared to the length scales of propagation or spatial pulse widths.\",\n \"In a medium such as a plasma, however, it has been shown [13] that a positivity of the traversal time in not obtained, particularly for broad-band pulses, which have large amounts of evanescent frequency components when the carrier frequency is smaller than the plasma frequency.\",\n \"This is related to the Hartmann effect [14], whereby the time for tunelling at far-sub-barrier energies becomes almost constant and negates the possibility of using this definition in principle for all situations.\",\n \"Nevertheless, it should be noted that it is still operationally one of the most useful definitions of the traversal times and has been validated in experiments involving both temporally dispersive and angularly dispersive situations.\"\n ],\n [\n \"Quantum clocks \",\n \"Due to issues with the various timescales which fail to conform to our intuitive understanding of the traversal timescales, attempts were made to develop“quantum clocks” that measure the time the particle spends in a given volume.\",\n \"In analogy with a classical clock, the clock should tick only when the particle is within the region of interest.\",\n \"This is accomplished by coupling other degrees of freedom to other fields localized within the region of interest.\",\n \"Dynamical evolution of those degrees of freedom occur only when the particle is present in the region of interest.\",\n \"Thus, the expectation values of quantities associated with those degrees of freedom will translate to expectation values of the times spent in the region of space.\",\n \"While clocks have been proposed in principle for a long time [15], three powerful methods that can have direct experimental implementation have been proposed and are popular.\",\n \"We will discuss the main ideas behind these proposals.\"\n ],\n [\n \"Büttiker-Landauer oscillating barrier times\",\n \"Büttiker and Landauer proposed [16] that the time of traverse of a charged particle through a potential barrier could be timed by super-imposing a time-harmonic electromagnetic field on top of the potential only within the region of interest.\",\n \"Suppose is the region in which we seek the time of sojourn, the potential would be $V_0(\\\\vec{r}) + V_1(\\\\vec{r}) \\\\cos (\\\\omega t)$ where $V_0$ is the original static potential and the perturbing oscillating field has a magnitude ($V_1$ ) that is constant within the region of interest and is zero outside (see Fig.\",\n \"REF for a schematic depiction).\",\n \"Interaction of the charged particle with the electromagnetic field would cause cause the absorption or emission of quanta of radiation.\",\n \"Thus, if the oscillation frequency was very small, the energy broadening of the transmitted or reflected spectrum would not be visible.\",\n \"The particle effectively sees a static potential and the transmission and reflected fluxes will adiabatically vary in time with the potential barrier height that changes harmonically.\",\n \"Thus, the times of interaction or traversal are very small compared to the time period of the oscillation.\",\n \"On the other hand, at high frequencies of oscillation, the particle would see the potential undergoing many oscillations during the time of its stay in the region and it would exchange quanta with the field.\",\n \"Then the energy spectrum of the reflected or transmitted particle would have energy sidebands separated from the incident energy by the energy of the quanta ($\\\\pm \\\\hbar \\\\omega $ ).\",\n \"In this limit, the period of the electromagnetic field is clearly much smaller than the traversal time ($\\\\tau _s$ ).\",\n \"There would be a crossover between the two regimes of low or high frequencies and the period of the electromagnetic field during the crossover regime ($\\\\omega \\\\tau _s \\\\simeq 1$ ) would a good measure of the time of traversal, regardless of whether the quantum wavefunction has a propagating nature or evanescent nature.\",\n \"Figure: Schematic picture depicted the tunnelling of a particle through a potential barrier with a time dependent strength.\",\n \"The particle exchanges quanta of energy (photons) with the radiation field and the transmission develops energy sidebands.\",\n \"The particles with higher energy would tunnel through more efficiently than the particles with lower energy.\",\n \"The period of the potential oscillations when the energy sidebands develop gives a timescale for the traversal of the particle through the barrierIn the one-dimensional case of a particle tunneling through a rectangular potential barrier, Buttiker and Landauer [16] obtain a tunneling time of $\\\\tau _{BL} = \\\\left[ \\\\frac{m}{2(V_0-E)} \\\\right]^{1/2} d,$ where $E$ is the energy of the tunneling particle and $d$ is the width of the barrier.\",\n \"Using the WKB approximation at energies well below the barrier height, this was further generalized to spatially varying potential barrier in one-dimension as $\\\\tau _{BL} = \\\\int _{x_1}^{x_2} \\\\left[ \\\\frac{m}{2(V_0-E)} \\\\right]^{1/2} dx = \\\\int _{x_1}^{x_2} \\\\frac{m}{\\\\hbar \\\\kappa (x)} dx,$ where $\\\\hbar \\\\kappa (x) = \\\\sqrt{2m(V_0(x) - E)}$ behaves as the instantaneous momentum of the tunneling particle.\",\n \"For more general potential shapes and energies, the calculations of the traversal times by this approach become very difficult.\",\n \"This approach, however, clearly sets out a timescale for the problem, particularly for the limiting case of low energy tunneling, which any other valid approach would need to reproduce.\",\n \"An approach where the flow of the particle through the barrier region was constructed in terms of two counter-propagating streams within the WKB approximation also obtained the traversal times consistent with the one obtained here [17].\"\n ],\n [\n \"Büttiker's spin clock (Larmor precession and spin flip)\",\n \"A second clock related to using the Larmor precession of a quantum particle with an associated magnetic moment (or spin), such as an electron or a neutron, in an applied magnetic field (B).\",\n \"The rate of the precession of the spin $\\\\omega _L = g\\\\mu _B B/\\\\hbar $ , ($\\\\mu _B$ is the Bohr magneton) is constant in a spatially constant magnetic field and could, thus, act as a possible clocking mechanism.\",\n \"Consider a magnetic spin that is initially polarized along the x-axis and moving along the y-axis through a region with potential, $V(y)$ , wherein a uniform magnetic field along the z-axis is applied (see Fig.\",\n \"REF for a schematic depiction).\",\n \"If the particle takes a time ($\\\\tau _y$ ) to traverse through the region of the magnetic field, the spin components of the transmitted flux would be $\\\\langle S_y\\\\rangle = -(\\\\hbar /2) \\\\omega _L\\\\tau _y$ to the lowest order in the magnetic field.\",\n \"Thus, measurement of the spin precession could yield the traversal time as $\\\\tau _y = \\\\frac{2}{g\\\\mu _B} \\\\lim _{B\\\\rightarrow 0} \\\\frac{\\\\partial \\\\langle S_y\\\\rangle }{\\\\partial B}$ Büttiker [5] recognized that the spin has a tendency to align along the magnetic field direction that he called spin rotation in addition to the Larmor precession in the place perpendicular to the magnetic field (spin precession).\",\n \"Thus, there could be two time scales associated with the extents of the spin rotation ($\\\\tau _z$ ) and the spin precession ($\\\\tau _y$ ).\",\n \"The Hamiltonian for the case of a rectangular barrier is $H = \\\\left\\\\lbrace \\\\begin{array}{l} (p^2/2m + V(y)) I - (\\\\hbar \\\\omega _L/2) \\\\sigma _z ~~~\\\\forall ~~~ 0 < y < L, \\\\\\\\ p^2/2m I~~~~\\\\mathrm {elsewhere} \\\\end{array} \\\\right.$ where $V(y)$ is the spatially constant potential, $I$ is the identity matrix and $\\\\sigma _z$ is the z-component of the Pauli spin matrix and $H$ acts on the spinor $ \\\\psi = \\\\left( \\\\begin{array}{c} \\\\psi _+(y) \\\\\\\\ \\\\psi _-(y) \\\\end{array} \\\\right)$ , where $\\\\psi _\\\\pm $ are are the Zeeman components that represent the anti-parallel and parallel spin amplitudes.\",\n \"The input spinor for a particle polarized along the x-axis can be written as equal superpositions of these components outside the region of the magnetic field due to which the expectation value of the Sz component would be zero.\",\n \"In the presence of the magnetic field , however, the kinetic energy for these components differ by the Zeeman energy of $\\\\pm \\\\hbar \\\\omega _L/2$ , and this gives a different value to the wave-vectors of the two components in the potential region.\",\n \"This is particularly severe in the case of tunneling at energies $(E)$ below the barrier when the exponential decay for the wavefunctions becomes very different: $\\\\kappa _\\\\pm = \\\\left\\\\lbrace \\\\frac{2m}{\\\\hbar ^2} (V_0 -E) \\\\mp \\\\frac{m\\\\omega _L}{\\\\hbar } \\\\right\\\\rbrace ^{1/2} \\\\simeq \\\\kappa \\\\mp \\\\frac{m\\\\omega _L}{2\\\\hbar \\\\kappa },$ where $\\\\kappa ^2 = 2m(V_0-E)/\\\\hbar ^2$ .\",\n \"Thus, one spin component has a greater probability to tunnel across than the other and the transmitted flux becomes spin polarized.\",\n \"The transmittance amplitudes for the two spin components can be approximately calculated in the case of energies far below the barrier height as $T_\\\\pm \\\\simeq T \\\\exp (\\\\pm \\\\omega _L\\\\tau _z)$ , where $\\\\tau _z = ml/\\\\hbar \\\\kappa $ .\",\n \"The expectation values of the z-spin component of the transmitted flux is easily obtained as $ \\\\langle S_z \\\\rangle = \\\\frac{\\\\hbar }{2} \\\\frac{T_+ - T_-}{T_++T_-} = \\\\frac{\\\\hbar }{2} \\\\tanh (\\\\omega _L\\\\tau _z) \\\\simeq \\\\frac{\\\\hbar }{2} \\\\omega _L\\\\tau _z $ where the last approximation is made in the limit of small magnetic fields.\",\n \"Thus, the z-component of the spin scales linearly with the magnetic field and presents yet another clocking mechanism and the spin rotation time can be defined as $\\\\tau _z = \\\\frac{2}{g\\\\mu _B} \\\\lim _{B \\\\rightarrow 0} \\\\frac{\\\\partial \\\\langle S_z\\\\rangle }{\\\\partial B}$ It turns out that the spin precession ($S_y$ ) dominates for energies far above the barrier and the spin rotation ($S_z$ ) dominates for energies far below the barrier.For a rectangular barrier, the spin precession time for energies far above the barrier tend to the Wigner delay times and far below the barrier, the spin rotation times tend to the Büttiker-Landauer times for the oscillating barrier.\",\n \"Büttiker proposed that a net traversal time could be defined as the Pythogorean sum of the two spin times as $\\\\tau ^2 = \\\\tau _y^2 + \\\\tau _z^2$ , which reflects the vectorial nature of the change of the spin.\",\n \"This prescription does not, however, have any fundamental basis.\"\n ],\n [\n \"Absorption and amplification as a clock\",\n \"A third interesting manner to clock the time of sojourn in a given region of space would be absorption or amplification, which would be at a rate proportional to the amplitude of the wave.\",\n \"Hence, a measurement of the wave amplitude as it enters and exits the region of interest can yield the time that it spends inside.\",\n \"Since the growth is exponential, a logarithmic derivative of the transmittance / reflectance with respect to the imaginary potential is needed.\",\n \"For light, both absorption and stimulated emission are coherent processes that leave the phase of a coherent mode unchanged, due to the Bosonic nature and this definition in terms of absorption or amplification is a natural definition.\",\n \"Such a process would not be strictly applicable to Fermionic particles like electrons or neutrons.\",\n \"However, the effects of absorption or amplification would eventually need to be discussed in the limit of infinitesimally small levels of absorption/amplification that tends to zero, so that the original problem remains unchanged.\",\n \"Hence, we may assume that the formal procedure works for Fermionic particles as well.\",\n \"For a scalar wave, coherent absorption / amplification may be implemented by adding an imaginary potential ($iV_I$ ) only in the region of interest (instead of the magnetic field as in Fig.\",\n \"REF ) , which will eventually be made zero ($V_I \\\\rightarrow 0$ ).\",\n \"The Schrodinger wave equation for the wave function of a particle in the presence of the imaginary potential becomes $i\\\\hbar \\\\frac{\\\\partial \\\\psi }{\\\\partial t} = -\\\\frac{\\\\hbar ^2}{2m} \\\\nabla ^2 \\\\psi + [ V_0(\\\\vec{r}) + i V_I] \\\\psi .$ Then the traversal times for the transmission and reflection can be defined as $\\\\tau ^{(T)} = \\\\frac{\\\\hbar }{2} \\\\lim _{V_I \\\\rightarrow 0} \\\\frac{\\\\partial \\\\ln \\\\vert T \\\\vert ^2 }{\\\\partial V_I} , ~~\\\\mathrm {and}~~\\\\tau ^{(R)} = \\\\frac{\\\\hbar }{2} \\\\lim _{V_I \\\\rightarrow 0} \\\\frac{\\\\partial \\\\ln \\\\vert R\\\\vert ^2 }{\\\\partial V_I}$ respectively, where T and R are the complex transmission and reflection coefficients.\",\n \"The idea was originally suggested by by Pippard to Buttiker [18], who found that while it reproduced the spin precession times, it did not recover the spin rotation time for waves that are evanescent in the region of interest (sub-barrier tunneling).\",\n \"It reproduces the Wigner delay times in the limit of large energy above the barrier height.\",\n \"This identity can be understood in terms of the analytic properties of the complex reflection and transmission coefficients.\",\n \"Let us consider the transmission coefficient in the complex energy plane $(E_r, E_i)$ .\",\n \"Its logarithm $\\\\ln (T) - \\\\ln \\\\vert T \\\\vert + i \\\\mathrm {Arg}(T)$ would be an analytic function within the Riemann sheet and would satisfy the Cauchy Riemann conditions: $ \\\\frac{\\\\partial \\\\ln \\\\vert T \\\\vert }{\\\\partial E_r} = - \\\\frac{\\\\partial \\\\mathrm {Arg}(T) }{\\\\partial E_i }, ~~\\\\mathrm {and} ~ ~\\\\frac{\\\\partial \\\\ln \\\\vert T \\\\vert }{\\\\partial E_i} = \\\\frac{\\\\partial \\\\mathrm {Arg}(T) }{\\\\partial E_r }.\",\n \"$ Thus, for energies far above the barrier, the left hand side of the second relation is directly proportional to the traversal times obtained by the imaginary potential ($E_i = V_I$ ), while the right hand side relates directly to the kinetic energy of the particle as potential energy is comparatively small ($E_r \\\\gg V_0$ ).\",\n \"This is why the traversal times tend to the Wigner delay time for large energies.\",\n \"A similar argument will hold for the reflection delay times as well.\",\n \"Note, however, that it is assumed that the potentials are varied everywhere in space in these arguments.\",\n \"Usually for the a quantum clock, the potential should only be varied locally within the region of interest (see Fig.\",\n \"REF ).\",\n \"Figure: Schematic diagram depicting a possible pathway in the Feynman path integral sense.\",\n \"(a) shows the portion affected by a global variation of the potential, and (b) shows the portion (solid line) affected by a local variation of the potential.\",\n \"The dotted portion of the path is not affected and should not be counted towards the traversal time.The issues with the traversal times obtained by the imaginary potential clocks refuse to remain positive for all potentials and further do not tend to the Büttiker-Landauer times for energies far below the barrier.\",\n \"For the case of tunneling across a rectangular barrier, the ratio $\\\\tau ^{(T,R)} / \\\\tau _{BL} \\\\rightarrow 0$ in the low energy limit.\",\n \"The Büttiker-Landauer times are very compelling in that limit and we would really want the traversal to remain positive definite.\"\n ],\n [\n \" Sojourn times: correcting the clocks\",\n \"It turns out that most of the problems of defining a positive definite traversal time for the transmission arise from spurious scattering concomitant with the very clocking mechanism / potential and a procedure was duly outlined to separate out this extra scattering and correct the traversal times [6].\",\n \"We will call these corrected traversal times as sojourn times as they literally relate to the times of journey through the region of interest.\",\n \"We will primarily deal with the imaginary potential clock here, but will point out the exact analogies with the Larmor clock.\",\n \"Figure: Schematic diagram partial transmission and reflection coefficients associated with the potentials (regions 1 and 3) on either side of a constant potential region (2).\",\n \"These partial transmission(t jk t_{jk} and reflection (r jk r_{jk}) coefficients may be used to construct the net transmission and reflection coefficients using equations ( ).First of all, it is important to appreciate that the presence of the clock potential not only invokes a response in the relevant extra degree of freedom, but also modifies the scattering due to a change of potential.\",\n \"One would expect this extra scattering to go to zero in the limit of zero clock potential.\",\n \"Our procedure, however, involves taking a derivative with respect to the clock potential, which can have contributions that would not vanish as the clock potential is made zero.\",\n \"Let us consider the transmission of a wave with energy above the barrier (case of propagating waves) through the constant potential region encased between two arbitrary potentials on either side as shown in Fig.\",\n \"REF .\",\n \"The space may be divided into three regions and the partial coefficients of transmission and reflection for a wave incident from region (j) unto region (k) are $t_{jk}$ and $r_{jk}$ as shown schematically in Fig.\",\n \"REF .\",\n \"The transmission and reflection coefficients can be written as a sum of the partial waves through the region [7] $T&=& t_{12} t_{23} e^{ik^{\\\\prime }L} + t_{12} r_{23} r_{21} t_{23} e^{3ik^{\\\\prime }L} + t_{12} r_{23} r_{21} r_{23} r_{21}t_{23} e^{5ik^{\\\\prime }L} + \\\\cdots \\\\nonumber \\\\\\\\R&=& r_{12} + t_{12} r_{23} t_{21} e^{2ik^{\\\\prime }L} + t_{12} r_{23} r_{21} r_{23} t_{21} e^{4ik^{\\\\prime }L} +\\\\cdots $ the coefficients $t_{jk}$ and $r_{jk}$ also depend on the clock potential and a derivative with respect to the total transmission or reflection coefficient would leave behind some terms arising from this dependence, which do not vanish as the clock potential is made to go to zero.\",\n \"This is the origin of the spurious scattering.\",\n \"An analysis of the structure of the partial wave superposition quickly reveals that the growth or attenuation related to the imaginary potential would only involve the paired combination of ($V_I L$ ) where $L$ is the lenght of the spatial region of interest.\",\n \"Afterall the traversal time should directly relate to spatial region of interest.\",\n \"The spurious scattering on the other hand would only involve unpaired $V_I$ .\",\n \"A formal procedure to isolate the effects of this spurious scattering can now be given.\",\n \"We will treat$\\\\xi = V_I L $ and $V_I$ as independent variables, keep $\\\\xi $ formally constant and let $V_IL \\\\rightarrow 0$ in the expression for the transmission coefficient.\",\n \"The sojourn time for transmission for propagating waves is now obtained as $\\\\tau _s^{(T)} = \\\\frac{\\\\hbar L}{2} \\\\lim _{\\\\xi \\\\rightarrow 0} \\\\frac{\\\\partial \\\\ln \\\\vert T(\\\\xi , V_I=0) \\\\vert ^2}{\\\\partial \\\\xi } .$ For the case of wave tunneling (energy lesser than the barrier height $E < V_0$ ), the wave vector is principally imaginary in the barrier region.\",\n \"The real part of the potential or refractive index essentially affects the rate of exponential decay / growth of the evanescent wave.\",\n \"On the other hand, the imaginary potential or imaginary part of the refractive index causes a phase shift with distance of this evanescent wave [13].\",\n \"Consider the complex wave-vector for propagating waves ($E > V_0$ ), $ k = \\\\sqrt{ \\\\frac{2m}{\\\\hbar ^2} [E - (V_0+iV_I)]} ~~\\\\simeq k_r - i \\\\frac{mV_I}{k_r \\\\hbar ^2} ~~ \\\\forall ~~ V_I \\\\ll E, $ where $\\\\hbar k_r = \\\\sqrt{2m(E-V_0)}$ .\",\n \"When we have evanescent waves ($E < V_0$ ) on the other hand, we can write $ k = \\\\sqrt{ \\\\frac{2m}{\\\\hbar ^2} [E - (V_0+iV_I)]} ~~\\\\simeq i \\\\kappa _r - \\\\frac{mV_I}{\\\\kappa _r \\\\hbar ^2} ~~ \\\\forall ~~ V_I \\\\ll V_0, $ where $\\\\hbar \\\\kappa = \\\\sqrt{2m(V_0-E)}$ .We hence realise the principle effect of the clock with respect to the spatial region is in the phase of the wave and not the amplitude for the evanescent wave.\",\n \"This is completely analogous to the spin rotation being predominant over the spin precession in the case of the Larmor clock.\",\n \"Mathematically, we have a square root singularity for the wave-vector in the complex energy plane and we are unable to analytically continue the behaviour for propagating waves to the case of evanescent waves across the branch-cut in either case of the imaginary potential clock or the Larmor clock.\",\n \"Hence, we define for the case of sub-barrier tunneling or evanescent wave, the sojourn time as $\\\\tau ^{(T)}_s = \\\\frac{\\\\hbar L}{2} ~ \\\\lim _{\\\\xi \\\\rightarrow 0}~\\\\frac{\\\\partial }{\\\\partial \\\\xi } \\\\left[ \\\\ln \\\\left(\\\\frac{T(\\\\xi , V_I=0)}{T^*(\\\\xi , V_I=0)} \\\\right) \\\\right] ,$ where $T*$ is the complex conjugate of $T$ and the ratio $T/T*$ essentially yields twice the phase of the transmission coefficient.\",\n \"For the case of the region with constant potential of height $V_0$ enclosed between the two arbitrary potentials and energy above the barrier, the sojourn time reduces to a positive definite quantity: $ \\\\tau ^{(T)}_s = \\\\frac{1 - \\\\vert r_{21} r_23 \\\\vert ^2 }{1 + \\\\vert r_{21} r_{23} \\\\vert ^2 - 2\\\\mathrm {Re}(r_{21} r_{23} e^{2i k_r L})} \\\\tau _{BL}.\",\n \"$ A similar positive definite expression is obtained in the case of evanesent waves (tunneling below the barrier) as $ \\\\tau ^{(T)}_s = \\\\frac{1 - \\\\vert r_{21} r_23 \\\\vert ^2 e^{-4\\\\kappa L} }{1 + \\\\vert r_{21} r_{23} \\\\vert ^2e^{-4\\\\kappa L} - 2\\\\mathrm {Re}(r_{21} r_{23} e^{-2\\\\kappa L})} \\\\tau _{BL}.\",\n \"$ The sojourn time tends directly to the Büttiker-Landauer times for an opaque barrier (energies far below the barrier or long barrier lengths), which is very important.\",\n \"In the high energy limit, it tends to the classical Wigner delay times.\",\n \"Thus it has the correct limits.\",\n \"Further, it was shown in Ref.\",\n \"[6] that the sojourn times defined in this manner for two non-overlapping regions is additive and hence, the times for any region of space with any arbitrary applied potential can be concluded to be positive definite.\",\n \"The case of reflection is a little bit more complex.\",\n \"Applying the above definitions for the reflection coefficient in place of the transmission coefficient does not yield a positive definite sojourn time.\",\n \"From the expansion in terms of the partial waves, one notes an essential difference between the reflection and the transmission.\",\n \"All the partial waves for the transmitted wave sample the region of interest and pick the paired combination $\\\\xi = V_IL$ in the amplitude or the phase.\",\n \"In the case of reflection, there is one partial wave $r_{12}$ corresponding to the prompt reflection from the edge of the region of interest that never enters the region of interest.\",\n \"Yet, this partial wave interferes with the others to produce the net reflection amplitude.\",\n \"In the spirit of our earlier arguments that the sojourn time should be causally related to the region of interest, it would be necessary to eliminate the weightage of this partial wave that should not be affected by the clocking potential.\",\n \"Hence, we explicitly subtract this prompt reflection amplitude out of the total reflection amplitude $R^{\\\\prime } = R - r_{12}$ and define the sojourn times for reflection using $R^{\\\\prime }$ in place of $T$ as before for both cases of the propagating waves as well as the evanescent waves.\",\n \"A simple and general result is obtained as $\\\\tau ^{(R)}_s = \\\\tau ^{(T)}_s + \\\\tau _{BL}$ which is consequently always greater than the sojourn time for transmission and positive definite.\",\n \"An experimental measurement of this procedure is possible by interfering destructively the reflection from a modified potential whose reflection is $r_{12}$ with the reflectance from the given potential.\",\n \"For example, one can use the same optical system that is index matched to the continuum form beyond the point where the absorption or gain is applied.\",\n \"Alternatively, one may use the recently developed metamaterial perfect absorbers [19] for light, where there is the possibility of matching the impedance and preventing any reflection.\",\n \"It should be noted that such a corection procedure can be analogously applied to the spin precession and spin rotation times for the Larmor clock as well, where the paired variable $\\\\xi = BL$ would be taken to zero after explicitly putting the magnetic field $B=0$ while keeping $\\\\xi $ formally constant.\",\n \"We then obtain identical results to the imaginary potential clock.\",\n \"The procedure outlined above was shown equivalent to stochastic absorption [20], whereby the absorption does not cause any scattering but contributes only to loss of the wave flux.\",\n \"This can be viewed as the case when there is inelastic scattering out of the given mode of the mesoscopic and the coherence of the mode is not affected.\",\n \"Only the coherent part of the wave is measured and the scattering into other modes manifests as a loss for this mode.\",\n \"Another example could be that the scattering leads to decoherence of the wave.\",\n \"In any interferometric measurement only the coherent part of the wave is measured and the decohered part would appear as a loss.\",\n \"Thus, a finite rate of decoherence itself could be utilized as a clocking mechanism .\"\n ],\n [\n \"The problem of the quantum first passage time\",\n \"Consider a free quantum particle that is released from a localized region at some instant of time and thereafter subjected to instantaneous projective measurements to detect its arrival at a particular region of space.\",\n \"The measurements are made at regular time intervals , and the system is allowed to evolve until the time a detection occurs.\",\n \"The main question addressed is: what is the probability that the particle is detected for the first time after time $t$ i.e.\",\n \"at the $n=(t/\\\\tau )^{th}$ measurement [21].\",\n \"Conversely, one can ask for the probability of particle not being detected (i.e., surviving) upto a given time.\",\n \"It is proposed by a general perturbative approach for understanding the dynamics which maps the evolution operator, which consists of successive unitary transformations followed by projections, to one described by a non-Hermitian Hamiltonian.\",\n \"For some examples of a particle moving on one- and two-dimensional lattices with one or more detection sites, use this approach to find exact expressions for the survival probability and find excellent agreement with direct numerical results.\",\n \"For the one- and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle.\",\n \"It is shown that an interesting and nontrivial connection between the dynamics of the particle in their model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites [22], [23].\",\n \"If continuous projective measurements are done then famous Zeno comes in to effect and particle does not evolve.\",\n \"Thus quantum first passage time indeed a nontrivial model.\",\n \"It is very subtle and as mentioned before in quantum mechanics there is no dynamical operator for time travel between two points.\",\n \"We may conclude this brief disussion on this topic by saying that the quantum first passage time is still remains a mystery\"\n ],\n [\n \"Conclusions and Outlook \",\n \"We have outlined here various attempts to answer a fundamental question, namely, “what is the time that a quantum particle or wave spends in a specified region of space ?” We do not have a self-adjoint operator for the arrival time in quantum mechanics, and the arrival time is not an observable.\",\n \"Yet it is intuitive and important to ask about timescales of any physical problem and the time of stay or sojourn appears to be a calculable quantity and a useful one to compare timescales.\",\n \"The sojourn time can provide for a meaningful alternative view-point within quantum mechanics.\",\n \"Knowledge of the sojourn times obviously cannot provide answers to questions that cannot be answered within the paradigm of quantum mechanics.\",\n \"Here we have exploited the fact that light as well as quantum particles are mathematically described by the same Helmholtz equation and talk about both systems in the same breath.\",\n \"Starting with the description of wavepackets by the group velocity and its extension by Wigner, we have indicated a variety of manners in which this question has been approached.\",\n \"The Smith dwell time is a positive definite time, but it is an unconditional time that does not depend on the output scattering channel.\",\n \"The path integral approach [21] or an approach based on the WKB method [17] are closely related to this, but can define conditional traversal times.\",\n \"Beyond these, we discuss the proposals to consider the dynamical evolution of an extra degree of freedom attached to the traversing particle due to the local interaction with a potential applied only in the region of interest.\",\n \"One has to meaningfully identify the extent of evolution of the clocking mechanism, which is an observable, with the time spent in that region.\",\n \"We have discussed three examples: (i) exchange of quanta via interaction with a radiation field; (ii) the spin precession; (iii) rotation in a magnetic field or the growth / attenuation due to an imaginary potential (amplifying or absorbing medium).\",\n \"A rather subtle problem that arises with these clocks is that they give rise to an extra spurious scattering that interferes with the very process and effectively changes the potential in the region of interest.\",\n \"This extra scatterings is carefully identified and is explicitly eliminated by a formal mathematical procedure.\",\n \"Further, it is shown that the effect of the clocking potential manifests in different quantities for propagating and evanescent waves: In the case of the imaginary potentials, it manifests in the amplitude for propagating waves and in the phase for evanescent waves (tunneling below the barrier); and in the case of the Larmor spin clock, it mainfests in the spin precession for propagating waves and in the spin rotation for evanescent waves.\",\n \"As a final caveat, it is shown that partial waves that reflect from the surface of the region should also be eliminated from reckoning as they do not spend anytime within the region of interest and this can, indeed, be done by interferometric measurements.\",\n \"Including these considerations yields a sojourn time that is Real and positive definite; Additive for non-overlapping regions of space; Related causally to the region of interest; Calculable and directly related to a measurable quantity.\",\n \"The main issue in defining traversal or sojourn times for a quantum system has been due to interference between partial waves (the alternative paths in quantum mechanics), which defies naive realism.\",\n \"With these advances in understanding, however, the sojourn time becomes a calculable quantity that is practically useful for estimating other quantities and understanding physical phenomena, for example, the dephasing rates in a quantum system.\"\n ],\n [\n \"Acknowledgement\",\n \"Both the authors acknowledge illuminating and educative discussions with Prof. N. Kumar who introduced them to these topics as well as the constant encouragement they have received from him.\",\n \"They would like to dedicate this tutorial to him.\",\n \"SAR acknowledges funding from the DST India through a Swarna Jayanti Fellowship.\",\n \"AMJ thanks DST, India for financial support (through J. C. Bose National Fellowship)\"\n ]\n]"},"id":{"kind":"string","value":"1612.05709"}}},{"rowIdx":3396,"cells":{"text":{"kind":"list like","value":[["A continuum of accretion burst behavior in young stars observed by K2"],["Abstract We present 29 likely members of the young $\\rho$ Oph or Upper Sco regions of recent star formation that exhibit \"accretion burst\" type light curves in $K2$ time series photometry.","The bursters were identified by visual examination of their ~80 day light curves, though all satisfy the $M < -0.25$ flux asymmetry criterion for burst behavior defined by Cody et al.","(2014).","The burst sources represent $\\approx$9% of cluster members with strong infrared excess indicative of circumstellar material.","Higher amplitude burster behavior is correlated with larger inner disk infrared excesses, as inferred from $WISE$ $W1-W2$ color.","The burst sources are also outliers in their large H$\\alpha$ emission equivalent widths.","No distinction between bursters and non-bursters is seen in stellar properties such as multiplicity or spectral type.","The frequency of bursters is similar between the younger, more compact $\\rho$ Oph region, and the older, more dispersed Upper Sco region.","The bursts exhibit a range of shapes, amplitudes (~10-700%), durations (~1-10 days), repeat time scales (~3-80 days), and duty cycles (~10-100%).","Our results provide important input to models of magnetospheric accretion, in particular by elucidating the properties of accretion-related variability in the low state between major longer duration events such as EX Lup and FU Ori type accretion outbursts.","We demonstrate the broad continuum of accretion burst behavior in young stars -- extending the phenomenon to lower amplitudes and shorter timescales than traditionally considered in the theory of pre-main sequence accretion history."],["Introduction","Variable mass flux has long been recognized as an important element of protostellar and pre-main sequence accretion.","Accretion rates are believed to be higher ($\\sim 10^{-5} M_\\odot $ yr$^{-1}$ ) during the first 10$^5$ years of protostellar evolution, with frequent outbursts of up to $10^{-4} M_\\odot $ yr$^{-1}$ [70].","The bursts are predicted as a consequence of unstable pile-up of gas in the inner disk, which then intermittently releases a cascade of material onto the star due to viscous-thermal disk instabilities [18], [168], [45].","The burst frequency and perhaps amplitude decline over time [71], [161].","While most of the stellar mass is thought to accumulate in the early phases of protostellar evolution, accretion at rates of $10^{-10}$ –$10^{-6} M_\\odot $ yr$^{-1}$ persists through the T Tauri phase (ages up to a few Myr), with less frequent bursts [69].","The currently accepted picture of T Tauri star accretion involves magnetic funnel flows channeling gas from the inner disk onto the central star.","Where this material impacts the surface, shocks arise and thermal hot spots form.","There may be one spot near each magnetic pole, or multiple spot complexes that are distributed about the stellar surface.","The number and geometry of the funnel flows is thought to depend on the accretion rate [138].","Empirically, time series monitoring of young stellar objects (YSOs; ages $<$ 1–10 Myr) has an extensive history.","It has been known since or before e.g.","[83] that T Tauri stars display flux variations at a wide variety of timescales and magnitudes.","The outbursting FU Ori stars and their lower amplitude, repeating cousins the EX Lup stars, are at the extreme end of the variabiity spectrum – and rare.","More common photometric variability is characterized by smaller amplitude and shorter timescale fluctuations.","The studies of [75], [60], [145], and [55] have illustrated and quantified much of the “typical\" young star photometric phenomena occurring from sub-hour to multi-decade time scales.","One cause of the routine brightness variations is sporadic infall of material from the surrounding disk.","Even in their predominant low-state accretion phases, young stars are understood as variable accretors.","The photometric variability is complemented by highly variable emission line profiles and veiling [34], [40] including in stars with relatively low accretion rates such as TW Hya and V2129 Oph [2], [3].","Nevertheless, the variability time scales and accretion rate changes remain poorly quantified for typical young accreting star/disk systems.","There may be a continuum of “burst\" behavior with a range of amplitudes and time scales that have not yet been appropriately sampled or appreciated in existing ground-based data sets.","While most of the stellar mass has been assumed to accumulate in the episodic and dramatic fashion of the rare large FU Ori and Ex Lup type events, the role and implications of discrete lower amplitude accretion events [155] and continuously stochastic accretion behavior [156] is not well understood in the context of stellar mass accumulation and inner disk evolution.","In probing accretion variability, space-based photometric campaigns have several advantages over ground-based work, including near-continuous sampling (versus interruptions for daytime, weather, etc), higher measurement precision, and fainter signal detection limits.","A detailed analysis of optical and infrared variability among disk-bearing stars in the $\\sim $ 3 Myr NGC 2264 was conducted by [38] based on a forty-day optical time series from the CoRoT space telescope at 10-minute cadence, along with 30 days of Spitzer Space Telescope monitoring at 100-minute cadence.","These data enabled an unprecedented view of YSO brightness changes on a variety of timescales.","Among the detected variability groups was a new class of “stochastic accretion burst” light curves – dominated by brightening events of duration 0.1-1 days and amplitude 5-50% the quiescent flux value [155].","It was speculated that these events were caused by the unsteady infall of material onto the stellar surface, as predicted by [143].","They may thus represent “normal\" discrete accretion variations and bursts, in contrast to the FU Ori and EX Lup outbursts described above.","The NASA $K2$ mission Campaign 2 observations included the young $\\rho $ Ophiuchus molecular cloud region at $<$ 1-2 Myr, and the adjacent Upper Scorpius OB association, which is debated from analysis of HR diagrams to be either $\\sim $ 3-5 Myr based on the low mass stellar population [122], [76] or $\\sim $ 11 Myr based on the solar and super-solar mass population [115]; the latter age is beginning to be favored by results on eclipsing binaries [85], [47] and asteroseismology [136].","By sampling stars with ages comparable to and extending to much older than NGC 2264 (in $\\rho $ Oph and Upper Sco, respectively), the $K2$ time series data can be used to compare accretion burst behavior as a function of age and therefore presumably disk properties which are expected to evolve with time.","We report here on a continuum of accretion burst behavior among members of $\\rho $ Oph and Upper Sco.","We observe discrete brightening events that range in amplitude from 0.1 to 2.5 magnitudes and in timescale from $<$ 1 day to $>$ 1 week.","We demonstrate that the burst phenomenon is seen only in those stars with evidence for strongly accreting disks, distinct from the typical disks in the region with weaker infrared excess and H$\\alpha $ emission.","Section 2 contains description of the $K2$ observations and pixel file processing, and of follow-up high dispersion spectroscopy.","Section 3 presents the light curve analysis and identification of burst type variables, Section 4 a discussion of the corresponding accretion and disk properties, and Section 5 the spatial and time domain characteristics of bursting sources.","We discuss the implications of these observations in Section 6 and summarize the results in Section 7.","The $K2$ mission [80] observed nearly 2000 stars in the young $\\rho $ Ophiuchus and Upper Scorpius regions during Campaign 2.","We have mainly considered objects submitted under programs GO2020, GO2047, GO2052, GO2056, GO2063, and GO2085 of the Campaign 2 solicitation, which comprise both secure cluster members and less secure candidates.","We later noted aperiodically variable stars among a number of other programs targeting cool dwarfs and therefore added objects from programs GO2104, GO2051, GO2069 GO2029, GO2106, GO2089, GO2092, GO2049, GO2045, GO2107, GO2075, and GO2114 if they also had proper motions consistent with Upper Sco.","[37], cull some of the less confident young star candidates using WISE photometry to eliminate giant stars and other contaminants.","This vetting results in a reduced set of 1443 $\\rho $ Oph and Upper Sco candidate members.","For each star in this set, we downloaded the target pixel file (TPF) from the Mikulski Archive for Space Telescopes (MAST).","Each target is stored under its Ecliptic Plane Input Catalog (EPIC) identification number, as listed in Table 1.","Data for each object includes 3811 $\\sim $ 10$\\times $ 12 pixel stamp images, obtained between 2014 23 August and 10 November.","Since the loss of a second reaction wheel during the Kepler mission in May of 2013, telescope pointing for $K2$ has suffered reduced stability and requires corrective thruster firings approximately every six hours.","We find a corresponding target centroid drift at a rate of $\\sim $ 0.1, or $\\sim $ 0.02 pixels, per hour.","While this movement is relatively small, associated detector sensitivity variations at the few percent level per pixel compromise the otherwise exquisite photometry, introducings jumps in measured flux on the same six-hour timescales.","It is helpful to track the $x$ -$y$ position drift for each star over time, as this can be used for aperture placement and later detrending of the light curves.","TPF headers provide a rough world coordinate system solution which is the same for all images, but these are not precise enough to center the target.","We therefore cut out a 5$\\times $ 5 pixel region around the specified target position, and used this to calculate a flux-weighted centroid.","We carried out photomety with moving apertures, the centers of each specified by the measured centroid locations.","This approach helps to minimize the effect of detector drift on the photometry.","Circular apertures were used with radii ranging from 1.0 to 4.0 pixels, in intervals of 0.5 pixels.","We found that photometric noise levels after detrending for position jump effects were generally minimized with the 2-pixel aperture, though for a few objects we selected the 1.5 or 3-pixel apertures.","These sizes have the additional advantage of being small enough so as to avoid flux contamination from other stars lying $\\sim $ 12 away.","To clean the data, we discarded the first 93 light curve points, for which the pixel positions were particularly errant compared to the rest of the time series.","We also removed points with detector anomaly flags.","Finally, we pruned points lying more than five standard deviations off the median light curve trend.","This was accomplished by median smoothing on $\\sim $ 2-day timescales, removing outliers, and then adding the median trend back in.","In general, our raw moving aperture photometry consists of lower levels of pointing-related systematic jitter than for the fixed aperture case.","For all of the stars discussed in this work, the amplitude of intrinsic variability dwarfs these systematics, and no further corrections are needed.","However, this is not the case for the less variable cluster members which form a control population for comparison of variability demographics.","In these light curves, a prominent sawtooth pattern appears on the $\\sim $ 6-hour timescales corresponding to thruster firings, an effect that was mitigated with the detrending procedure described in [1], as described in [37].","In one exceptional case (EPIC 203954898/2MASS J16263682-2415518), the light of a highly variable star was contaminated by a close neighbor 8 away.","Single pixel photometry shows that this object undergoes high amplitude bursts, while the neighboring star is relatively constant and slightly fainter.","Restricting the aperture to encompass only EPIC 203954898 results in degraded precision.","We therefore used a 3-pixel aperture encircling both stars, and then removed the average flux of the companion by scaling the data to match the amplitude of the bursts in the single pixel light curves (the lower precision here affects only the detailed light curve morphology and not its overall amplitude).","The result is a light curve with maximum burst amplitude of nearly eight times the quiescent flux level."],["New spectra and compiled spectroscopic data","We collected both archival and new spectroscopic data at high dispersion with the Keck/HIRES spectrograph [160].","The new observations were obtained on one of: 2016 May 17 and 20, 2015 June 1 and 2, or 2013 June 4, UT, and covered the spectral range $\\sim $ 4800 Å to 9200 Å at resolution $R\\approx 36,000$ .","The images were processed and the spectra were extracted and calibrated using the $makee$ software written by Tom Barlow.","H$\\alpha $ emission line strengths and spectral type estimates are tabulated in Table ; other emission lines that were observed in these high dispersion data are discussed in the Appendix notes on individual stars.","Table 1 as well as the notes section includes literature information in addition to our spectroscopic findings."],["Speckle imaging","We obtained high-resolution speckle imaging for six of our young stars to assess multiplicity properties.","Our observations used the Differential Speckle Survey Instrument [77] on the Gemini-South telescope in 2016 June.","Speckle observations were simultaneously made in two medium band filters with central wavelengths and bandpass FWHM values of ($\\lambda _c$ , $\\delta \\lambda $ ) = (692,47) and (883,54) nm.","Each star was observed for approximately 10 minutes during which time we obtained 3-5 image sets consisting of 1000, 60 ms simultaneous frames.","These observations were made during clear weather at airmass 1.0 to 1.3, when the native seeing was 0.4-0.6 arcsec.","Details of speckle observations using the Gemini telescope and our data reduction procedures can be found in [78] and [79].","We conducted visual examination of all 1443 young star light curves in order to identify stars in a “bursting\" state.","Such objects were selected by identifying behavior consistent with that presented in [38] and [155].","To confirm our visually identified bursters, we also computed quantitative variability metrics, specifically the $M$ and $Q$ statistics defined by us in the papers above.","$M$ describes the degree of symmetry of the light curve about its mean value.","It is calculated by determining the ratio of the mean of magnitude data in the top and bottom deciles to the median of all light curve points.","$M$ achieves negative values when there is a significant number of points brighter than the median, but not so many faint points.","Unlike what was done in [38], we calculated the $M$ statistic from flux, rather than magnitude values.","The main effect of this change is to lower $M$ values (by $\\sim $ 10% on average), since the magnitude to flux conversion makes bright peaks more pronounced.","We argue that flux units are a more natural choice here, as flux correlates with luminosity and accretion rate.","Bursting light curves are highly asymmetric with frequent flux increases over the mean, resulting in $M$ values from approximately -0.3 to -1.3.","While the boundary is somewhat subjective, all objects selected by eye meet the previously defined $M<-0.25$ criterion for burster status.","The $Q$ statistic defined in [38] describes tendency towards or away from periodicity over the time series.","It is a measurement of how much the standard deviation shrinks when the light curve is phased to its dominant periodicity and the associated pattern is repeated and subtracted out from the raw time series.","Strictly periodic behavior (i.e., complete removal of the phase pattern) returns $Q=0$ while light curves with no repeating behavior have $Q=1$ .","In a few cases for which the light curves are entirely aperiodic, this removal process actually increases the underlying standard deviation; this is why the computed $Q$ value is occasionally greater than 1.0.","In previous work, we have denoted light curves with moderate $Q$ values of 0.15–0.60 as “quasi-periodic.” We emphasize here that this range is somewhat subjective and was based on a by-eye analysis of CoRoT data on the NGC 2264 cluster.","As explained in Cody et al.","2016, the present K2 dataset contains some objects with $Q>0.6$ that nevertheless display repeating components upon visual examination.","The $M$ and $Q$ values for the selected bursters are provided in Table 2.","They are also plotted in Fig.","REF , alongside the values for other disk-bearing young stars in the $K2$ field, as selected in [37].","The $K2$ burster sample behavior ranges from periodic to quasi-periodic to aperiodic, with light curves most tending towards aperiodicity.","Objects falling in the bursting section of the diagram but not highlighted as such tend to be long-timescale variables for which the trend removal failed.","We favor our by-eye classification over the $M$ and $Q$ statistics here and thus do not consider these bursters.","Several other objects with highly negative $M$ values are dominated by periodic modulation and only display zero or one bursting event.","We also leave these out of the sample, in light of classification ambiguity.","Overall, 18 objects with $M<-0.25$ (i.e., a value that would qualify them as busters) were removed from the sample.","Most of these may be seen in Figure REF as the black points lying above the $M=-0.25$ line (apart from five that are hidden under orange burster points); the majority are only marginally above it.","In total, we have selected 29 stars as bursters in $\\rho $ Oph and Upper Sco; division into the two regions was based on a $1.2\\times 1.2$ square surrounding the position RA=246.79, Dec=-24.60 to define the extent of $\\rho $ Oph [37].","We list the basic properties of the bursters in Table 1 and show their light curves in Figure REF .","The burster class exhibits several subsets of behavior, with some light curves displaying a nearly continuous series of events, and others exhibiting more discrete brightening events.","We discuss the timescales of bursting in Section 6. llcccccc 8 0pt Young stars exhibiting bursting behavior in $K2$ Campaign 2 EPIC id 2MASS id Other ids SpT EW H$\\alpha $ H$\\alpha $ 10% Refs Region (Å) (km s$^{-1}$ ) 203382255 J16144265-2619421 M4-M5.5 -77 154 1 USco 203725791 J16012902-2509069 USco CTIO 7 M2/M3.5 -170, -129 437 1, 2 USco 203786695 J16245974-2456008 WSB 18 M3.5 -8.4, -140 - 3 $\\rho $ Oph 203789507 J15570490-2455227 - - - USco 203794605 J16302339-2454161 WSB 67 M3.5-M5 -69 485 1 $\\rho $ Oph 203822485 J16272297-2448071 WSB 49, MHO 2111, DROXO 57 M4.25 -37 - 4 $\\rho $ Oph 203856109 J16095198-2440197 M5-M5.5 -15 155 1 USco 203899786 J16252434-2429442 V852 Oph, SR 22, DoAr 19, WSB 23 M4.5/M3 -31, -170 - 4, 5, 6 $\\rho $ Oph 203905576 J16261886-2428196 VSSG 1, Elias 20, YLW 31, ISO-Oph 24, K7–mid-M/M0 -70 416 1,7 $\\rho $ Oph IRAS 16233-2421, MHO 2103 203905625 J16284527-2428190 V853 Oph, SR 13, DoAr 40, WSB 62, M3.75 -30, -48, -46 - 4 $\\rho $ Oph ISO-Oph 199, HBC 266 203913804 J16275558-2426179 V2059 Oph, DoAr 37, SR 10, ISO-Oph 187, M2 -43, -56, -108 - 4,8 $\\rho $ Oph WSB 57, YLW 56, HBC 265, SVS 1771 203928175 J16282333-2422405 SR 20W K5 -35 - 3 $\\rho $ Oph 203935537 J16255615-2420481 V2058 Oph, DoAr 20, Elias 13, K4.5 -220, -87, -67 - 4 $\\rho $ Oph SR 4, WSB 25, YLW 25, IRAS 16229-2413, MHA 365-12, ISO-Oph 6 203954898 J16263682-2415518 ISO-Oph 51 M0 -10 - 9 $\\rho $ Oph 204130613 J16145026-2332397 BV Sco M4.5 -108 - 10 USco 204226548 J15582981-2310077 USco CTIO 33, USco 42 M3 -158, -250 - 11,12 USco 204233955 J16072955-2308221 M3 -150 - 10 USco 204342099 J16153456-2242421 VV Sco, IRAS 16126-2235, PDS 82a M1/K9-M0 -20, -31 337 13, 1 USco 204347422 J16195140-2241266 - - - USco 204360807 J16215741-2238180 M6 -140 341 1 USco 204397408 J16081081-2229428 M5.75/M5 -22, -49, -31 - 10, 14, 15 USco 204440603 J16142312-2219338 M5.75 -95 - 10 USco 204830786 J16075796-2040087 IRAS 16050-2032 M1/G6-K5 -357, -165 684 16, 1 USco 204906020 J16070211-2019387 KSA 68 M5 -8, -30 - 11, 17 USco 204908189 J16111330-2019029 M1/M3 -160 324 1,18 USco 205008727 J16193570-1950426 K7-M3 -55 322 1 USco 205061092 J16145178-1935402 M5-M6 -70 173 1 USco 205088645 J16111237-1927374 M5, M6 -50, -50 - 12, 19 USco 205156547 J16121242-1907191 M5-M6 -15 127 1 USco Stars in the $K2$ Campaign 2 burster sample, in order of EPIC id.","EPIC 203786695/2MASS J16245974-2456008 has a companion at 1.1 separation, and the two H$\\alpha $ values belong to the distinct components of the system.","The H$\\alpha $ 10% widths in column 6 are derived from data presented in this paper.","We have highlighted in bold the other new values derived as part of this work.","References: 1) this work, 2) [137], 3) [27], 4) [165], 5) [121], 6) [100], 7) [6], 8) [8], 9) [52], 10) [96], 11) [43], 12) [122], 13) [124], 14) [154], 15) [44], 16) [88], 17) [123], 18) [99], 19) [101].","Figure: Light curves of selected bursters over the 80-day duration of K2K2 Campaign 2, in approximate order of amplitude.Figure: Cont.Figure: Cont.Figure: Cont.Figure: Cont.Figure: Cont."],["Disk and accretion properties of the burst sample","A hypothesis for the short-lived, often repeatable, brightening events in our 29 $K2$ light curves is that they are caused by episodic accretion from the circumstellar disk onto the young star.","A specific requirement for the accretion-driven burst hypothesis is that the objects exhibit both infrared excess indicative of circumstellar dust, serving as the reservoir for the accretion, and either ultraviolet excess or line emission from hot gas in the nearby circumstellar environment.","The dust criterion is satisfied by our burster sample, as all stars have been already selected as infrared excess sources in [37].","As shown in that work, there are 344 disk bearing stars in the entire $K2$ Campaign 2 sample, of which 299 are bright enough to obtain light curves.","This number includes all 29 burster stars identified here.","Therefore, the fraction of bursters among the total disk-bearing sample is at least 8$\\pm $ 2%.","The error comes from consideration of the Poisson uncertainties.","These values can also be considered separately for the $\\rho $ Oph (128 disked stars; 92 with light curves) and Upper Sco (216 disked stars; 207 with light curves) samples.","There are 10 bursters in $\\rho $ Oph and 19 bursters in Upper Sco; both numbers lead to roughly the same value of 8–10%, with a $\\sim $ 2% error on this fraction."],["Circumstellar Dust","Infrared color-magnitude diagrams can also shed light on what photometric aspects, if any, separate burster stars from other disk bearing sources.","We show the color-color diagrams $J-K$ versus $K-W3$ and $J-K$ versus $K-W4$ in Figure REF and the spectral energy distributions in Appendix Figure REF , to illustrate the strength of emission at farther, cooler locations in the disk.","The vast majority of the bursters have excesses in the three longest wavelength WISE bands $W2$ , $W3$ , and $W4$ ($4.6 \\mu $ m, $12 \\mu $ m, and $22 \\mu $ m, respectively).","The individual spectral energy distributions in the Appendix (Fig.","REF ) provide a finer look at the circumstellar flux patterns.","This finding is in stark contrast to the overall disk sample, in which only 50% of objects (or a total of 172) have excesses in all three bands.","Thus, the presence of a full, minimally evolved disk appears to be preferred for bursting behavior.","Figure: Near and mid-infrared color-color diagram of stars in the Upper Sco/ρ\\rho Oph regions (grey), with disk-bearing stars in black and bursters highlighted in red.","Burster point sizes are scaled by light curve amplitude.It has traditionally been thought that young stars with prominent fading events are surrounded by nearly edge-on disks; the dust clumps routinely obscure the central star [25], [24], causing decrements in the light curve.","Conversely, one might imagine that for disk systems farther from edge-on orientation, we would have a more direct view of the accretion columns and shocks near the stellar poles.","This could allow relatively unfettered observation of accretion bursts.","Looking at the selection of 172 full disks identified in [37], 90% are variable, but only 17% are bursting.","If this is a reflection of geometric selection, then one might hypothesize that burster disks are viewed at angles ranging from face-on to $\\sim $ 34.","However, this assumes that face-on is the best angle at which to view bursting; as suggested by the ALMA data described below, this may not be the case.","The idea that bursting behavior may be a function of viewing angle can be further explored by considering resolved disk imaging.","Five of the bursters discussed in this paper have been observed with ALMA at 0.88 mm [31], [15].","EPIC 204830786 (2MASS J16075796-2040087) has a broad CO $J=3-2$ line detection, with velocities from -17 to 17 km s$^{-1}$ ; the profile suggests that the disk is not face-on.","EPIC 204342099 (2MASS J16153456-2242421) is the only other disk with a CO detection, albeit a weak one.","The broad velocity distribution again suggests that this system is not oriented face-on.","The other three disk-bearing sources in our sample (EPIC 204906020/2MASS J16070211-2019387, EPIC 204226548/2MASS J15582981-2310077, EPIC 204908189/2MASS J16111330-2019029) have no CO detection.","The latter two do show continuum, so CO may be highly depleted or the disks in these cases could be physically small.","The continuum fluxes can also be used to infer disk properties.","[14] infers from modeling SEDs and size constraints that EPIC 204830786 (2MASS J16075796-2040087) has inclination 42$^{+12}_{-9}$ , while EPIC 204342099 (2MASS J16153456-2242421) is inclined at 43$^{+15}_{-16}$ .","Dust masses were inferred by [15] and range from $<$ 0.5 $M_\\oplus $ (EPIC 204906020/2MASS J16070211-2019387; non-detection) to 9.3 $M_\\oplus $ in the case of EPIC 204830786/2MASS J16075796-2040087.","All five bursters in the ALMA sample are noted as having been classified as full disks (as opposed to evolved or transitional) by [99].","These intermediate inclination values are consistent with the idea proposed above that we are not looking through disk material, which would be expected to produce “dipping\" rather than “bursting” light curves.","But they are inconclusive regarding whether we have a direct view of material accreting onto the central star."],["Gas and Accretion","In addition to infrared disk indicators, we obtained spectroscopic data from both the literature and our own high-resolution follow-up spectroscopy (§2.2).","The H$\\alpha $ emission equivalent width (EW) and 10% width values (Table 1) are indicative of significant accretion.","As a control sample, we gathered H$\\alpha $ EWs for other disk-bearing (but not necessarily bursting) stars in Upper Sco from [137], [43], and [122].","We cross-matched them against our Upper Sco/$\\rho $ Oph K2 $WISE$ excess star list and eliminated any objects not in common.","We then compared the H$\\alpha $ values of this general Upper Sco sample with those of the bursters in Figure REF .","Since some of the bursters have multiple H$\\alpha $ measurements (see Table 1), we have taken the average of all available values.","We find that the bursters occupy a large range of H$\\alpha $ EW values, from -10Å (presumably a low-state value) to several hundred angstroms in emission.","The non-burster stars, on the other hand, display weak H$\\alpha $ , with EWs primarily from 0 to -15Å and a tail out to -50Å with one value around -120Å.","Figure REF illustrates the emission line profiles for the 12 out of the 29 bursters for which we have high resolution spectra in H$\\alpha $ , Ca2 8542Å and He1 5876Å.","[163] advocated the designation of accreting stars based on H$\\alpha $ line widths at 10% of the maximum line strength that are larger than 270 km s$^{-1}$ .","Although all 12 sources have velocities larger than 100 km s$^{-1}$ (see Table 1), only slightly more than 1/2 meet the [163] requirement.","There is a tendency for the narrower velocity stars (EPIC 203856109/2MASS J16095198-2440197, EPIC 205156547/2MASS 16121242-1907191, EPIC 203382255/2MASS J16144265-2619421, EPIC 205061092/2MASS J16145178-1935402, EPIC 205008727/2MASS J16193570-1950426) to have lower duty cycles in their burst patterns (Figure REF ) (Where available, H$\\alpha $ velocity and duty cycle are correlated at a significance level of 1.2$\\times 10^{-3}$ ).","It may be that our spectra were taken at non-burst epochs.","This is speculative at best, however.","The weak H$\\alpha $ sources all exhibit only narrow component emission in their Ca2 profiles, as does EPIC 204342099 (2MASS J16153456-2242421) which has a broad H$\\alpha $ profile.","Generally, the broad H$\\alpha $ sources exhibit both broad component and narrow component Ca2.","[13] review the classic literature on Ca2 profile morphology in young stars, and discuss magnetospheric models of it.","Our profiles do not seem to exhibit the asymmetries predicted by these models (e..g blueshifted peaks, redshifted depressions), however.","Notably, our broad-lined Ca2 sources also exhibit evidence for forbidden line emission, e.g.","[O1] 6300Å.","The morphology of He1 emission lines in young stars was studied by [19] who also designated narrow line, broad line, and narrow$+$ broad profile categories.","Our stars are dominated by their narrow component emission, with widths ranging between 40-70 km s$^{-1}$ , but some may also have weak broad components that would require line decomposition to characterize.","Most of the He1 profiles appear to have slight asymmetries, however, in the sense of broader redshifted emission than blueshifted emission with the line peaks at zero-velocity.","Beyond emission line morphology, accretion rates are estimated for a handful of our burster stars by [111].","Values range from 10$^{-9.9}$ to 10$^{-6.7}$ $M_\\odot $ yr$^{-1}$ – a large range.","It should be noted that neither the H$\\alpha $ EWs presented above nor the accretion rates referenced here were necessarily measured during a time when the stars were undergoing bursting events.","Thus it is plausible that accretion is preferentially high in these sources, but only at certain times.","Under the hypothesis that burst events in light curves are due to higher than average mass flow, we can convert the flux to a quantitative increase in the mass accretion rate.","This requires several assumptions.","First, we assign to all stars in our sample a low-level baseline accretion rate, $\\dot{M}_{\\rm low}$ , which then increases to a larger rate $\\dot{M}_{\\rm high}$ during bursts.","We assume that this increase in mass flow can be equated to the ratio of the accretion flux $F_{\\rm acc}$ in and out of the burst state through the accretion luminosity, $L_{\\rm acc}$ : $\\frac{\\dot{M}_{\\rm high}}{\\dot{M}_{\\rm low}} = \\frac{L_{\\rm acc,high}}{L_{\\rm acc,low}}\\sim \\frac{F_{\\rm acc,high}}{F_{\\rm acc,low}}.$ The accretion luminosity is related to the stellar mass $M_*$ and radius $R_*$ by $L_{\\rm acc} = 1.25(GM_*\\dot{M}/R_*)$ where the pre-factor is that appropriate for an assumed magnetospheric accretion scenario.","The measured flux density $F$ (i.e., in the $Kepler$ band) contains contributions from both accretion and the underlying stellar luminosity, $L_*$ .","We label the measured flux ratio “$r$ ”: $r \\equiv \\frac{F_{\\rm high}}{F_{\\rm low}} = \\frac{F_*+F_{\\rm acc,high}}{F_*+F_{\\rm acc,low}}$ $F_{\\rm low}$ and $F_{\\rm high}$ are approximately the minimum and maximum flux values respectively attained in a given light curve.","To determine $F_{\\rm acc,high}/F_{\\rm acc,low}$ and hence $\\dot{M}_{\\rm high}/\\dot{M}_{\\rm low}$ , we must estimate the ratio of stellar to accretion luminosity.","This quantity has been studied by e.g.","[110] and we adopt the fit based on their Figure 2.","For each star in the burster sample, we estimate the stellar luminosity by considering the $J$ -band magnitudes of similar spectral type non-accreting young stars in the $K2$ Campaign 2 set, and applying bolometric and extinction corrections as outlined in [111].","Combining this with Equations 1 and 2, it can be shown that $\\frac{\\dot{M}_{\\rm high}}{\\dot{M}_{\\rm low}}\\sim \\frac{L_*}{L_{\\rm acc}}(r-1)+r.$ Given our estimates of $L_*/L_{\\rm acc}$ and $r$ as measured from the $K2$ photometry, we have calculated the increase in mass accretion rate during bursts for each of our sources.","Figure REF illustrates the resulting values as a function of spectral type, in cases where this is known to a subclass or better.","We find that they range from $\\sim $ 10 to over 400 for the most prominent burst events.","Figure: We show the cumulative distributions of Hα\\alpha equivalent width for disk-bearing stars in our K2 young star set with available spectroscopy; this set includes 28 bursters and 46 non-bursters.","Where more than one Hα\\alpha measurement is available, we adopt the mean value.","The sample was binned in 10Å wide sets to produce the distribution.","While the non-bursting stars tend to predominate between 0 and -15Å (i.e., emission), the burster Hα\\alpha values show a much wider dispersion, reaching much values up to -250Å.","Thus the burst phenomenon seems to favor stars with high accretion rates.","We note that there is a single disk-bearing star with quasi-periodic light curve that has a reported Hα\\alpha equivalent width around -120Å.","Such a value is highly unusual for a star with a non-bursting light curve.Figure: Line profiles in Hα\\alpha , Ca2 8542 Å, and He1 5876 Å measured by Keck/HIRES for 12 of the 29 bursters, labeled by EPIC identifier.","The Hα\\alpha panels include a horizontal line at 10% of the peak (un-normalized) flux in addition to the horizontal line indicating the continuum level.","Note the change in velocity scale for the helium panels.","Broad width and structured velocity profiles indicate accretion and wind phenomena.Figure: Estimated increase in accretion rate during bursts, as a function of stellar spectral type."],["Spatial distribution","The sample we have analyzed here is likely a mixture of ages from young ($<1-2$ Myr) but still optically visible stars associated with the young $\\rho $ Oph cloud, to somewhat older (5-10 Myr) stars in the off-cloud Upper Sco region which still retain accreting circumstellar disks.","Bursters as identified here make up $\\sim $ 9% of the disk-bearing young star sample in the $K2$ /C2 dataset.","We initially hypothesized that they would preferentially appear in the compact, young $\\rho $ Oph cluster, as opposed to the more dispersed and older Upper Sco region.","But the spatial distribution shown in Figure REF surprisingly reveals equal proportions of bursters in the two regions.","Furthermore, there is no correlation with global extinction measures.","This suggests that either the young population extends from $\\rho $ Oph out into the surrounding areas, or that the burster phenomenon is less dependent on age than on disk properties such as mass.","[52] found evidence for an intermediate age population ($\\sim $ 3 Myr) of YSOs outside of the main $\\rho $ Oph cloud core, but within the main L 1688 cloud.","Furthermore, [165] found a negligible age difference between sets of Upper Sco and L 1688 association members; both regions were estimated to be $\\sim $ 3 Myr old.","Figure: Top: Spatial distribution of young stars (gray), including variables (black) and specifically bursters (red), overlaid on the K2 field of view.","The concentration of stars near RA=246.8, Dec=-24.6 is the ρ\\rho Oph cluster.","Bottom: Young stars with disks (cyan) and bursters (red) overlaid on the extinction map of the ρ\\rho Oph region."],["Spectral types","To assess the mass distribution of stars displaying bursts, we have gathered spectral types from the literature.","Those found for the bursters are displayed in Table 1.","We have also selected a control sample of non-bursting stars from [99].","That work presents spectral types for hundreds of USco members; we cull the list to include only non-bursting, inner disk-bearing stars observed in K2 Campaign 2.","The resulting set of 31 objects has spectral types ranging from B8 to M8, as shown in Figure REF .","We compare this against the distribution of spectral types for the bursters identified here, finding a significant difference only for the B–F spectral types.","There are early type stars in the sample but none of their variability is of the bursting type.","K2 light curves for these objects show mainly low-level quasi-periodic modulation.","However, the spectral type distributions of bursters and non-bursters is very similar for the K–M range.","Statistically, the two sets are indistinguishable here.","Thus we conclude that the preponderance of late-M spectral types among bursters is consistent with the increased fraction of young stars with disks at low masses.","Figure: Spectral type distributions for bursting and non-bursting young disk-bearing stars observed in K2 Campaign 2.","Two distributions are very different for the B–F range, but indistinguishable for K and M spectral types."],["Multiplicity","The presence of close stellar companions to the young stars in our sample may influence their variability properties.","Indeed it has been hypothesized that, for high-amplitude outbursting stars such as FUors, events could be triggered by the presence of a perturbing companion [129], [94].","However, support for this idea is mixed [61].","It is nevertheless worthwhile to check which, if any, of our burster sample may be in binary systems.","We have vetted over half of the sample for multiplicity by examining the available high-resolution imaging as well as obtaining new speckle observations."],["Literature assessment","We mined the literature for adaptive optics and speckle imaging of these targets, and the details are tabulated as part of the individual object commentary in the Appendix.","Only three objects have evidence for a close companion published in the literature: EPIC 204906020 (2MASS J16070211-2019387), EPIC 203786695 (2MASS J16245974-2456008), EPIC 203905625 (2MASS J16284527-2428190).","The first is a close binary [89].","Likewise, the second has a companion at $\\sim $ 15 AU (100.4 mas) [84].","The last of these has an even closer companion at $\\sim $ 2 AU (13 mas) [153].","Interestingly, all three systems also have wider ($>$ 50 AU) companions.","None of their light curves shows any hint of a periodicity.","One additional object, EPIC 204342099 (2MASS J16153456-2242421), may have a 274 AU separation companion [63], but it is not clear whether the two stars are bound.","EPIC 204830786 (2MASS J16075796-2040087) is associated with another star 21.5 away ($>$ 3000 AU separation) but has not been surveyed for closer companions.","Other than these five objects, 9 other bursters in our sample have been surveyed for multiplicity, at a variety of sensitivities and separations: EPIC 204226548 (2MASS J15582981-2310077), EPIC 203899786 (2MASS J16252434-2429442), EPIC 203935537 (2MASS J16255615-2420481), EPIC 203905576 (2MASS J16261886-2428196), EPIC 203954898 (2MASS J16263682-2415518), EPIC 203822485 (2MASS J16272297-2448071), EPIC 203913804 (2MASS J16275558-2426179), EPIC 203928175 (2MASS J16282333-2422405), EPIC 203794605 (2MASS J16302339-2454161).","No companions were found in these cases.","However, the separations probed are very non-uniform and range from 10 mas (1.5 AU) in some cases to 1–30 in others ($>$ 150 AU); details are provided in the Appendix individual objects section."],["DSSI targets","In addition to data compiled from the literature, we also have speckle imaging observations of six bursters using DSSI (§2.3).","For EPIC 204342099 (2MASS J16153456-2242421), we recover the companion reported at 1.9 by [63]; however, we measure the separation to be 1.50 ($\\sim $ 218 AU), and a magnitude difference of $\\Delta $ m=3.38 at the 880 nm band.","This star previously had direct imaging and aperture masking by [87] that ruled out further objects down to 240 mas (35 AU).","EPIC 204830786 (2MASS J16075796-2040087) has a previously noted possible companion at thousands of AU separation, but the DSSI observations otherwise support the hypothesis that this is a single star.","EPIC 204440603 (2MASS J16142312-2219338) has no previous multiplicity information, but the 880 nm image suggests a possible companion at a separation of 0.1.","However, the lack of a similar detection at 692 nm and the faintness of this star makes speckle reconstruction challenging; thus the existence of such a companion remains indeterminate.","EPIC 204360807 (2MASS J16215741-2238180) has no multiplicity information in literature, but we find it to be a 0.48 separation binary (70 AU), with $\\Delta $ m=0.74 at 880 nm.","EPIC 203935537 (2MASS J16255615-2420481) has no reported evidence for multiplicity, and we do not detect companions down to 0.1 separation (15 AU) at 4–5 magnitudes of contrast in the 692 and 880 nm bands.","Finally, EPIC 203928175 (2MASS J16282333-2422405) was reported by [33] to host no companions down to 20 mas (3 AU) at 1-3 magnitude contrast; our observations support the lack of binarity, with no detections outward of 0.1 at $\\Delta m\\sim $ 4.4 magnitudes."],["Time domain behavior","In order to appreciate the diversity of the bursting behavior among our sample of objects, we must go beyond just the $M$ and $Q$ metrics discussed above.","We quantified the peak-to peak amplitudes of the bursters by first cleaning and normalizing the light curves and then computing the maximum-minus-minimum values.","There are several ways to quantify light curve timescales.","First, we measure the burst duty cycle, which is the fraction of time each object spends in a bursting state.","This is by nature somewhat subjective, as bursts display a range of amplitudes and shapes.","We identified bursting portions of each light curve by first fitting and removing a low-order median trend to the light curve.","This flattens the “continuum” level from which bursts arrive.","We then measure the typical point-to-point scatter by shifting each point by one, subtracting from the original light curve, and dividing the standard deviation of the result by $\\sqrt{2}$ .","Using this measure of scatter, we have found that burst behavior, as detected by eye, includes points that lie about 15 times the scatter above the minimum of the continuum-flattened light curve.","We thereby selected bursting and non-bursting sections for each time series.","This method only failed for three objects (EPIC 204397408/2MASS J16081081-2229428, EPIC 205156547/2MASS J16121242-1907191, and EPIC 203856109/2MASS J16095198-2440197) that displayed intermittent quasi-periodic behavior that was picked up as bursts.","We manually removed these light curve portions for the statistical analysis.","In Figure REF we display the peak-to-peak amplitudes versus duty cycle of each burster.","The duty cycles exhibit a large range of values, from almost 100% down to $\\sim $ 10%.","Typically the light curves with the highest amplitudes have higher duty cycles of 60% and above, with the exception of outlier EPIC 203954898/2MASS J16263682-2415518.","This object may represent a distinct form of bursting behavior.","We also quantify the burst timescales by applying a method similar to the one described in [38] (§6.5 of that paper).","In brief, this involves identifying peaks that rise above a particular amplitude level compared to the surrounding light curve.","Once peaks are found, the median timescale separating them is computed.","This procedure is repeated for a variety of amplitudes, from the noise level up to the maximum light curve extent.","The result is a plot of timescale versus amplitude (e.g., Fig.","32 in [38]).","Finally, we take as a “representative” timescale the value corresponding to an amplitude that is 40% of the maximum peak-to-peak value (we note that this is different from the value of 70% adopted in [38] and appears more appropriate for the burster light curves examined here).","This computation only fails for the light curve of EPIC 203382255 (2MASS J16144265-2619421), which displays only one burst event; here the timescale is indeterminate.","In Figure REF we display the peak-to-peak amplitudes versus estimated timescale for each burster.","Again, there is a large range of values, but no clear correlation with amplitude.","The burst duration is another way to quantify the observed events.","This is a challenging measurement, as there is a superposition of bursts with varying widths and heights.","We simplify as above by only considering peaks that rise to a level of at least 40% of the maximum peak-to-peak value.","For each peak, we identify the surrounding points that are more than 15 times the point to point scatter above the minimum light curve value (as was done for the burst duty cycle calculation).","We then adopt the median burst duration of all such peaks in each light curve.","The result is plotted against peak-to-peak amplitude in Figure REF .","We also compare the durations with the repeat timescales in Figure REF .","Here we find that burst duration is correlated with repeat timescale.","This is somewhat expected since, by definition, a duration is typically larger than the average timescale between bursts.","However, we observe a distinct lack of short bursts (duration $\\le $ 2 days) with long repeat timescales.","This may be rooted in the physical mechanism of the bursts.","We have also generated periodograms to identify any repeating components in the light curves.","Most bursters do not exhibit periodicity but instead adhere to stochastic behavior, with any quasi-periodicity quantified via the $Q$ statistic (see §3).","Those that do show significant periodicities (as indicated by $Q\\le 0.61$ and/or a strong, isolated periodogram peak) are EPIC 203794605/2MASS J16302339-2454161 ($P=4.5$ d), EPIC 203899786/2MASS 16252434-2429442 ($P=6.0$ d), EPIC 203928175/2MASS J16282333-2422405 ($P=4.4$ d), EPIC 203954898/2MASS J16263682-2415518 ($P=20.8$ d), and EPIC 204347422/2MASS J16195140-2241266 ($P=6.9$ d).","The measured periods are similar to the burst repeat timescales inferred above.","In addition, EPIC 203856109 (2MASS J16095198-2440197), EPIC 204233955 (2MASS J16072955-2308221), EPIC 204397408 (2MASS J16081081-2229428), and EPIC 205156547 (2MASS J16121242-1907191) display short-timescale ($P$ less than a few days) periodic behavior outside of their bursting states.","These periods are more typical of the K2 $\\rho $ Oph and Upper Sco sample as a whole [127] and likely measure stellar rotation, whereas those of the bursts are longer by factors at least several.","Figure: Maximum burst amplitude (in units of normalized flux) versus duty cycle, i.e., fraction of time spent bursting.Figure: Burst repeat timescales versus peak-to-peak light curve amplitude, in normalized flux.Figure: Burst durations versus peak-to-peak light curve amplitude, in normalized flux.Figure: Burst repeat timescale versus durations.","The dashed line indicates where these two quantities are equal.","We have left out EPIC 203382255 (2MASS J16144265-2619421) since it only has one burst and the repeat timescale is thus indeterminate.Few of the timescale metrics show any relation to [circum]stellar properties, but one potential correlation stands out in peak-to-peak amplitude versus the infrared $W1-W2$ color (Figure REF ), which is indicative of a dusty inner disk.","These two quantities are correlated at a significance level of 4$\\times 10^{-5}$ (Pearson $r$ coefficient of 0.69).","This is also borne out in Figure REF , which suggests that the dustiest objects have the highest light curve amplitudes.","Figure: Peak-to-peak light curve amplitude for bursters is shown against the W1-W2W1-W2 color.","All included objects have W1-W2W1-W2 values indicative of inner disk dust, but the highest amplitude bursters tend to have larger infrared excesses, with one prominent exception (EPIC 205008727/2MASS J16193570-1950426; W1-W2∼1.3W1-W2\\sim 1.3).llcccccccc 8 0pt Light curve metrics for bursting young stars in $K2$ Campaign 2 EPIC id 2MASS id Amplitude Q M Timescale Duration Duty cycle Period (norm.","flux) [d] [d] [d] 203382255 J16144265-2619421 1.43 1.00 -1.14 $>$ 77.74 7.95 0.08 - 203725791 J16012902-2509069 0.68 0.87 -0.29 5.18 1.90 0.66 - 203786695 J16245974-2456008 0.25 1.00 -0.59 5.52 4.54 0.43 - 203789507 J15570490-2455227 0.36 0.97 -0.41 9.12 4.45 0.06 - 203794605 J16302339-2454161 0.87 0.55 -0.25 4.64 3.67 0.93 4.46 203822485 J16272297-2448071 0.62 0.84 -0.29 9.36 7.66 0.90 - 203856109 J16095198-2440197 0.35 1.00 -1.00 3.11 5.25 0.15 - 203899786 J16252434-2429442 1.13 0.61 -0.83 6.42 3.98 0.96 5.95 203905576 J16261886-2428196 3.85 1.00 -0.66 7.48 5.97 0.88 - 203905625 J16284527-2428190 0.32 1.0 -0.31 7.80 5.17 0.72 - 203913804 J16275558-2426179 0.41 1.0 -0.37 5.47 3.56 0.70 - 203928175 J16282333-2422405 3.19 0.54 -0.66 4.68 2.53 0.73 4.39 203935537 J16255615-2420481 0.24 1.00 -0.31 4.92 3.09 0.69 - 203954898 J16263682-2415518 6.82 0.61 -1.35 31.17 6.84 0.43 20.83 204130613 J16145026-2332397 1.87 0.85 -0.35 3.43 0.90 0.62 - 204226548 J15582981-2310077 0.47 1.00 -0.53 8.54 3.44 0.73 - 204233955 J16072955-2308221 1.99 0.85 -0.82 3.42 2.08 0.98 - 204342099 J16153456-2242421 0.83 0.91 -0.72 35.53 11.26 0.88 - 204347422 J16195143-2241332 1.38 0.75 -1.11 7.23 1.65 0.10 6.94 204360807 J16215741-2238180 0.55 0.87 -0.49 5.21 2.53 0.65 - 204397408 J16081081-2229428 0.15 0.59 -0.68 5.01 5.30 0.22 1.65 204440603 J16142312-2219338 0.43 1.00 -0.93 4.55 3.15 0.28 - 204830786 J16075796-2040087 1.91 1.00 -0.67 14.70 4.31 0.91 - 204906020 J16070211-2019387 0.34 0.93 -0.47 3.90 0.91 0.63 - 204908189 J16111330-2019029 1.28 0.76 -0.59 7.99 7.97 0.90 19.23 205008727 J16193570-1950426 0.91 1.00 -0.68 10.25 7.93 0.76 - 205061092 J16145178-1935402 0.35 1.00 -0.51 4.85 1.94 0.35 - 205088645 J16111237-1927374 0.10 1.00 -0.53 11.30 3.18 0.04 - 205156547 J16121242-1907191 0.16 1.00 -1.01 4.38 3.62 0.23 - We tabulate basic statistical properties of the burster light curves.","$Q$ and $M$ are discussed in §3 as well as Cody et al.","(2014).","Amplitudes represent peak-to-peak measurements."],["Discussion and Summary","$K2$ data from Campaign 2 have probed the optical burst properties of young stars on time scales ranging from hours up to several months, with approximately 8–10% ($\\pm $ 2%) of strong disk sources exhibiting burst behavior.","This is roughly in agreement with the 13$^{+3}_{-2}$ % fraction found for NGC 2264 [38], [155].","It is possible that an even larger fraction of young stars undergo bursting, but are not detected as such if the amplitude is low (i.e., $<100$ % peak to peak).","Burst behavior could in some cases be masked by other types of variability.","Burster stars host inner circumstellar disks, as evidenced by WISE excesses and SEDs (Figure REF ).","Furthermore, there is a positive correlation between the $W1-W2$ color and light curve peak-to-peak amplitude.","Stronger inner disk excesses appear to be associated with bigger bursts.","Most of these objects have exceptionally strong H$\\alpha $ emission, as well as other Balmer lines, He1, and Ca2 in emission, as is typical for strongly accreting stars.","Viewing geometry may play a role in setting the observability of the bursting phenomenon, but thus far we only have disk inclination constraints from ALMA on two sources in the sample, both of which are tilted at $\\sim $ 42$\\pm $ 15.","Members of the bursting sample typically exhibit multiple discrete brightening events, some lasting up to or just over one week.","Time domain properties of the bursters are diverse, with some stars exhibiting a nearly continuous series of bursts and others displaying one or two isolated episodes superimposed on otherwise lower amplitude or quasi-periodic behavior.","The majority of these light curves show flux variations of less than a factor of two and are similar to the objects in the $\\sim $ 3 Myr NGC 2264 highlighted by [155].","However, seven of 29 bursters display discrete brightenings of more than 100% on day to week timescales, unlike most heretofore classified young stellar variables that we are aware of.","Of particular interest is the subset of these for which the bursts repeat quasi-periodically: EPIC 204347422 (2MASS J16195140-2241266), EPIC 203928175 (2MASS J16282333-2422405), and EPIC 203954898 (2MASS J16263682-2415518).","It is unclear as to what physical phenomenon sets the periods of 6.9 d, 4.4 d, and 20.8 d (respectively) in these cases; it appears to be relatively independent of stellar mass, as indicated by spectral type.","There also is a lack of evidence for binarity in most of the bursters, although in some cases limits from imaging are relatively shallow.","We thus speculate that burst events are not triggered by any companion, but rather by a repetitive interaction between the stellar magnetic field and inner disk.","From the theoretical perspective, magnetically channeled accretion is not predicted as steady in numerical simulations.","An inner disk is truncated at a balance point between the inward pressure from accretion and the outward pressure of the magnetosphere.","This produces variable and possibly cyclic mass flow due to instabilities and pulsational behavior.","Such variations in the mass loading of accretion columns are modeled under different physical scenarios by, e.g., [98], [59], [140], [141], [142], and [45], [46].","The disk corotation radius ($r_c$ ) and the magnetospheric radius ($r_m$ ) are critical in determining the regime of accretion under which a star-disk system falls.","When $r_m0.7r_c$ , “stable” accretion occurs [20] and gas follows two funnel streams onto the star [144], [141].","For smaller values of $r_m$ , the Rayleigh-Taylor instability sets in and accretion becomes chaotic, with many tongues of matter extending from the inner disk to the stellar surface [138], [91].","Numerous hot spots are present on the stellar surface, and the associated light curves display irregular bursting.","This regime may be responsible for the large subset of bursters that we observe with high duty cycles.","When $r_m$ is up to a factor of two larger than $r_c$ , on the other hand, gas in the inner disk rotates slower than the magnetosphere and is accelerated azimuthally.","Models [46] predict a “trapped disk\" regime in which relatively continuous accretion bursts occur on short timescales.","When $r_m>>r_c$ , material tends to be flung out azimuthally in what is known as the “strong propeller\" regime [139].","The light curves of several of our objects (EPIC 203954898/2MASS JJ16263682-2415518 and EPIC 204347422/2MASS J16195140-2241266) strongly resemble the simulated mass flow variations predicted by [167] and [92] for propeller behavior.","In this scenario, the matter accretes episodically in three phases: First, material accumulates at the inner disk boundary, unable to flow inward.","The magnetosphere is compressed toward the stellar surface.","Next, compression reaches the point at which the magnetosphere can no longer withstand the gravitational forces on the accumulated material; gas accretes rapidly onto the star in a funnel flow.","Finally, the compression pressure is relieved, accretion ceases, and the magnetosphere is able to expand outward again.","Following this sequence, the cycle repeats.","The time between bursts is longer than in other regimes, and the accretion rate can follow a flare-like pattern (rapid rise; slower decline) as a function of time.","Strong outflows may also be present.","EPIC 203382255 (2MASS 16144265-2619421) shows outflow signatures and also has the longest repeat timescale, making it an additional propeller regime candidate.","In contrast, a few other objects with outflow-related spectral lines do not display the expected alternating burst and quiescence pattern.","Further investigation of these targets is necessary to estimate magnetic and corotation radii and compare with the theoretical expectations.","The ratio $r_m/r_c$ depends on mass accretion rate, magnetic field strength, and stellar spin period– parameters that we are unable to determine independently for this sample of burster stars.","In the meantime, the burst timescales measured for our sample provide perhaps the best indication of the physical mechanisms at work.","Duty cycles range from 10% and all the way up to nearly 100%– suggesting that we are seeing different modes of accretion from continuous to episodic and less frequent.","The possible correlation between burst duration and repeat timescale (Figure REF ) may imply a relationship between mass loading timescale and accretion rate.","A major open question stemming from this work is what the origin of the day to (multi-)week repeat timescales that we observe is, and how it relates to young outbursting stars with much longer duty cycles (e.g., EXors and FUors).","Prominent examples from the literature include V899 Mon, which has had repeated bursts separated by $\\sim $ 1 year [112], and V1647 Ori, which repeats at $\\sim $ 2 years [11].","On even longer timescales, other EXor type stars tend to outburst once or twice per decade.","Unlike the $K2$ objects, the longer among these time scales are thought to be related to the viscous time scale in the disk.","Here, the material drains inward and must undergo replenishment before the next instability-driven episode can occur.","It has been speculated that the frequency and amplitude of outbursts may be set by the accretion rate, with younger and higher amplitude outbursting objects accreting more rapidly [12].","We would then expect accretion rates for our own sources to be somewhat lower.","This does indeed seem to be the case, as the median accretion rate where available for bursters is 10$^{-8.1}$ $M_\\odot $ yr$^{-1}$ (for exact values, see the notes on individual objects in the Appendix).","We have found (Section 4.2) that this increases one to two orders of magnitude during the most extreme $K2$ light curve peaks.","Further spectroscopic measurements during times of definitive bursting may confirm these estimates.","In summary, the $K2$ mission is providing an unprecedented view of the time domain properties of young stars, and showing that the optical photometric manifestations of accretion phenomena take on a wide variety of timescales and amplitudes.","Follow-up observations, including spectroscopy, should be carried out to investigate changes in spectral emission and inner disk structure in the highest amplitude objects presented here.","The work of A.M.C was supported by a NASA/NPP fellowship.","We thank the referee for useful feedback that improved this paper.","We acknowledge Luisa Rebull for calling our attention to one of the burst type objects that we had overlooked in our initial examination.","Thanks also to Nic Scott for assistance with DSSI observations on the Gemini South telescope.","This paper includes data collected by the $K2$ mission.","Funding for the $K2$ mission is provided by the NASA Science Mission directorate.","The spectroscopic data were obtained at the W.M.","Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.","The Observatory was made possible by the generous financial support of the W.M.","Keck Foundation.","These results are also based on observations obtained as part of the program GS-2016A-Q-64 at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil).","This paper has utilized the SIMBAD and Vizier services, through which data from USNO-B, APASS, 2MASS, and WISE were collected."],["Notes on individual objects","Not all of our sample stars appear in the literature, but for those that have been studied previously, we highlight the key results here.","We also incorporate information from our new speckle and spectroscopic observations."],["EPIC 203382255 / 2MASS 16144265-2619421","This source is cataloged by [95] as an astrometric and photometric member, but is not otherwise studied in the literature.","Our spectra suggest a spectral type of M4-M5.5 depending on the spectral range (earlier at bluer wavelengths).","Lithium is present with $W_{Li} = 0.30 A$ along with the H$\\alpha $ emission reported in Table , He1, weak NaD, and Ca2 are seen in emission.","Broad [O1] is also present.","The system is a clear accretion/outflow source."],["EPIC 203725791 / 2MASS J16012902-2509069","This star was first proposed as an Upper Sco member by [9], based on $RIZ$ photometry.","It was only recently confirmed by [137], based on significant lithium absorption.","No other data has been reported for this star.","Our HIRES spectrum indicates a spectral type of M2 with lithium absorption strength $W_{Li}=0.3$ Å.","Strong H$\\alpha $ emission is seen, as indicated in Table , along with lorentz-broadened He1.","NaD, weak narrow Fe2 as well as Mg1 emission, two-component Ca2, and O1 emission.","Very weak [O1] is also exhibited."],["EPIC 203786695 / 2MASS J16245974-2456008","This star, WSB 18, was first noted as part of Wilking et al.","(1987)'s H$\\alpha $ emission survey of the the $\\rho $ Ophiuchi complex.","Lithium absorption was detected by [52], confirming youth.","It is a visual binary with a separation of 1.1 (138 AU) and a 0.49 flux ratio [131].","Therefore, both components contribute to our $K2$ light curve.","According to [27], the primary has a spectral type of M2, whereas the secondary is M2.5.","Both show H$\\alpha $ in emission, but the primary does not appear to have a disk [104], whereas the secondary does.","Further, [84] found that the primary itself is a double star with 0.1 separation.","No detections of these sources at $>70$ $\\mu $ m were made with Herschel.","In the submillimeter, [5] put an upper limit of 0.003 $M_\\odot $ on the dust mass surrounding this source, although it is unclear which components were included."],["EPIC 203794605 / 2MASS J16302339-2454161","According to [166], this star displayed an H$\\alpha $ emission intensity of 3 on a scale of 1 to 5 (where 1 is weak and 5 is strong).","Simon et al.","(1995) performed a direct imaging search for companions in the 0.005-10 separation range, but did not find anything.","Neither were any companions detected in the Catalog of High Angular Resolution Measurements [134], its successor CHARM2 [135], or the [125] speckle imaging survey of $\\rho $ Oph.","For the latter, no companions were detected down to 0.04 (0.14) times the stellar brightness at 0.5 (0.15) separation.","This star was included in the Spitzer c2d legacy survey.","Our HIRES spectrum is veiled, but consistent with a spectral type of M3.5-M5.","Lithium is present with strength $W_{Li} = 0.32 Å$ .","H$\\alpha $ emission as reported in Table is strong with several components and a blueward-displaced central absorption.","Additional emission includes He1 with lorentzian wings, broad NaD, many multi-component Fe2 lines, strong Ca2 with multiple component and O1 8446.","Among the outflow lines, only multi-component [O1] is seen."],["EPIC 203822485 / 2MASS J16272297-2448071","This star, WSB 49, is in the $\\rho $ Ophiuchus cluster.","It was first detected as part of Wilking et al.","(1987)'s H$\\alpha $ emission line survey.","It is also an IRAS source [164].","It was first discovered as an x-ray source with the ROSAT High Resolution Imager [64], and also noted XMM Newton observations by [118].","[165] confirmed its youth via detection of lithium absorption.","[52] estimate a mass of 0.17 $M_\\odot $ .","The star was surveyed for multiplicity, but no companions were detected down to $>0.15$ separation at flux ratios of 0.06.","An infrared excess is detected with Spitzer [53].","[5] observed it in the submillimeter, classifying the disk as class II, with $<0.005$ $M_\\odot $ of material.","In addition, the disk is detected at 70 $\\mu $ m with Herschel/PACS [126].","It was listed as a long timescale near-infrared variable by [113], while [66] detected $\\sim $ 20% variations in the mid-infrared with Spitzer."],["EPIC 203856109 / 2MASS J16095198-2440197","There is no previous literature on this source.","Our spectrum indicates a spectral type of M5-M5.5 with lithium at $W_{Li} = 0.61 Å$ .","Moderate H$\\alpha $ emission, as indicated in Table 1, is present and has a multi-component profile.","Weak, also multi-component, He1 is also present, but no other emission lines."],["EPIC 203899786 / 2MASS J16252434-2429442","This star in the $\\rho $ Oph cluster is also known as V852 Oph and SR 22.","Its emission line spectrum was reported as early as the mid-20th century [157], with prominent hydrogen, Ca2, and Fe2 noted.","[74] and [73] included it in their emission line star catalogs.","It is a ROSAT x-ray source [32].","Irregular variability was also reported early on, by [148] and [54].","[4] label the object as class III based on its 2.2–10 $\\mu $ m slope, and despite its previous classification as a classical T Tauri star by [39].","It was relabeled as class II by [62], based on their 1.1–2.4 $\\mu $ m survey.","[5] confirmed the disk with submillimeter observations, detecting 0.002 $M_\\odot $ of material.","Similarly, [107] report 2.05 $M_{\\rm Jup}$ of material, based on SCUBA-2 850 $\\mu $ m observations.","[126] detected the disk at 70 $\\mu $ m with Herschel, and reported that there is a gap in the spectral energy distribution that suggests a hole.","The source is evidently a single star.","[125] ruled out any companions down to a flux ratio of 0.07 at a separation of 0.15, and [62] did not detect any radial velocity variations indicative of a spectroscopic binary.","[33] reported no companions down to 10 mas and contrasts of several magnitudes."],["EPIC 203905576 / 2MASS J16261886-2428196","This star– better known as VSSG 1– boasts a disk that was first detected by IRAS [36].","The mid-infrared slope, $\\alpha $ , is -0.4, making it a class II disk according to [4] (although Wilking et al.","1989 earlier classified it as class I).","[105] used SPEX to measure an $n_{2-25}$ spectral index of -1.26, confirming the class II categorization.","They also reported a small 10 $\\mu $ m silicate feature.","[6] have observed this source's disk with the Submillimeter Array at 0.87 mm and estimated a dust mass of 0.029 $M_\\odot $ , along with an accretion rate of $10^{-7}$ $M_\\odot $ yr$^{-1}$ .","H$_2$ O was detected in the Spitzer IRS spectrum obtained by [120], along with HCN, C$_2$ H$_2$ and CO$_2$ .","Salyk et al.","(2011) confirm these detections and estimate a disk mass of 0.029 $M_\\odot $ .","They infer a disk inclination of 53.","Submillimeter observations with ATCA [133] resulted in an estimated disk outer radius of 100–300 AU and a much lower dust mass of $4.5\\times 10^{-5}$ –$1.9\\times 10^{-4}$ $M_\\odot $ .","[7] looked for mid-infrared variability in this source and concluded that it is not an EXor candidate.","[111] report a fairly high accretion rate of 10$^{-7.19}$ $M_\\odot $ yr$^{-1}$ , and a Pa$\\beta $ equivalent width of 8.9Å in emission.","According to the work of [125], no companions are visible down to a flux ratio of 0.04 at a separation of 0.15.","Our spectrum exhibits strong H$\\alpha $ as indicated in Table with a double-peaked broad profile, as well as strong and broad Ca2 triplet as well as O1 8446 emission.","There is very little in the way of absorption, presumably due to heavy veiling, and spectral types from K7 to mid-M are plausible.","For the same reason, aggravated by low signal-to-noise in this region of the spectrum, there is no lithium measurement."],["EPIC 203905625 / 2MASS J16284527-2428190","EPIC 203905625, also known as V853 Oph and SR 13, is a late-type star in $\\rho $ Ophiuchus with reported spectral types from M2 to M4 [165].","Accretion signatures include H$\\alpha $ as well as calcium in emission.","[147] first reported strong veiling in the star's spectrum.","[23] found that the H$\\alpha $ emission is variable (30–48Å).","[111] estimated an accretion rate of $10^{-8.31}$ $M_\\odot $ yr$^{-1}$ , from near-infrared spectra.","They detected Pa$\\beta $ in emission, at an equivalent width of -1.7Å.","X-rays from this object were first detected with Einstein [108].","A disk around this star was also observed, with IRAS [36].","[4] used 1.3mm observations to detect the disk; they classify it as class II, based on a 2.2–10 $\\mu $ m slope of -0.8.","[120] detect H$_2$ O, OH, HCN, and C$_2$ H$_2$ in a Spitzer IRS spectrum of this target.","[126] report Herschel detections at 70 through 500 $\\mu $ m; they classify the system as transitional, based on 12–24 $\\mu $ m data.","The star is a multiple system, with a companion first detected at 0.4 separation via speckle imaging [58].","[10] and [125] confirmed the 0.4 and a 0.238 flux ratio.","[153] conducted an IR imaging survey that revealed a closer companion at separation 13 mas.","[104] found that both the primary and its 0.4 companion have class II disks.","[5] measured a total disk mass of 0.01 $M_\\odot $ using submillimeter observations, while [107] measured a similar 7.81 $M_J$ from SCUBA-2 850 $\\mu $ m observations.","Variability in this object was initially reported by [147].","[149] found optical variations from 12.6 to 14.7.","[75] observed V magnitude fluctuations from 12.83 to 13.52, while [60] reported $V$ magnitudes between 12.61 and 13.87.","Likewise, [152] conducted photometric monitoring, reporting a $V$ -band amplitude of 0.91 magnitudes.","The light curves are too sparsely sampled to identify any morphological features, although they are classified as “irregular.\"","The broadband K2 light curve exhibits 0.2 mag events, suggesting that amplitudes are higher at bluer wavelengths.","This is confirmed by Herbst et al.","(1994)'s estimate of d$U$ /d$V$ : 2.39."],["EPIC 203913804 / 2MASS J16275558-2426179","This target is also known as SR 10 and V2059 Oph.","It originally appeared in Herbig & Rao's (1972) catalog of emission line stars.","Both [148], and [90] list it among their variable star compilations; [149] labeled the variations as “irregular.\"","H$\\alpha $ emission at 40Å was reported early on by [146].","[8] also observed strong H$\\alpha $ , as well as He1 and Fe2 in emission.","[166] noted this star in their H$\\alpha $ emission survey, and [165] again measured H$\\alpha $ in emission as well as lithium in absorption.","[111] estimated the accretion rate to be 10$^{-7.95}$ $M_\\odot $ yr$^{-1}$ ; they detected Pa$\\beta $ emission at an equivalent width of -5.6Å.","[109] report a slightly lower accretion rate of 10$^{-8.3}$ $M_\\odot $ yr$^{-1}$ .","[153] conducted an imaging survey for binary companions at project separations from 0.005 to 10 down to a $K$ magnitude of 11.1.","Likewise, [125] searched for companions at separations of 0.15 and 0.50 but did not identify any down to flux ratios of 0.04 and 0.02, respectively.","[134], [135], [16], and [33] did not detect companions either, down to 10–20 mas separation at several magnitudes contrast.","Thus this object appears to be a single star.","EPIC 203913804/2MASS J16275558-2426179 is a ROSAT x-ray source [32].","The star is also an IRAS source [164].","[4] identified a class II circumstellar disk, and [67] later confirmed with Spitzer data.","[5] studied the system in the submillimeter and deduced an upper limit on the disk mass of 0.005 $M_\\odot $ , based on the 1.3 mm flux.","Similarly, [107] found an upper limit of 5.0284 $M_{\\rm Jup}$ .","[126] detected the disk at 70 $\\mu $ m with Herschel."],["EPIC 203928175 / 2MASS J16282333-2422405","EPIC 203928175 is a star in the $\\rho $ Ophiuchus region that is also known as SR 20 W [157] and ROXC J162823.4.","[165] report a spectral type of K5, along with variable H$\\alpha $ emission and an equivalent width of 35Å.","It was surveyed for binarity by [125], but no companions were detected down to 0.08 times the stellar flux at a separation of 0.15, or 0.04 times the stellar flux at 0.50.","[33] also did not find any companions down to 20 mas separation, at contrasts of 1–3 magnitudes.","With DSSI, we do not make any detections outward of 0.1 at $\\Delta m\\sim $ 4.4 magnitudes (flux contrast $\\sim $ 0.02) in the 692 or 880 nm bands.","The object is encircled by a class II disk, as reported by [53].","The disk was subsequently detected at 70, 160, 250, 350, and 500 $\\mu $ m with Herschel by [126], who tentatively classified it as transitional."],["EPIC 203935537 / 2MASS J16255615-2420481","This star is also known as SR 4 and V2058 Oph.","It has a long history of photometric and spectroscopic study, dating back to Struve & Rudkjøbing's (1949) publication of emission line stars.","[74], [166], and [73] listed it in their catalogs of emission line stars.","It has had a range of H$\\alpha $ equivalent widths measured from 84Å to 220Å as well as a low vsin$i$ of $\\sim $ 9 km s$^{-1}$ [22].","[159] acquired blue spectra of this target, which revealed H($\\beta $ , $\\gamma $ , $\\delta $ ) and Ca in emission, as well as significant veiling at 4450Å.","[51] measured the veiling factor, $r$ , to be $\\sim $ 1.5.","[165] detected lithium absorption and H$\\alpha $ emission in this source.","[130] classified the H$\\alpha $ emission line profile as type IIR, in which there are two peaks of similar height.","The accretion rate is estimated by [111] to be a fairly high 10$^{-6.74}$ $M_\\odot $ yr$^{-1}$ , and the same authors detected Pa$\\beta $ in emission at an equivalent width of -19.0Å.","As suggested by [117], the star may be the driver for a nearby Herbig Haro flow (HH 312) in the region.","The star is a ROSAT x-ray source.","It is also an IRAS point source [81], [162], [36].","[21] also observed the class II disk with ISO.","[5] detected the disk at 850 $\\mu $ m with SCUBA, and they estimated a mass of 0.004 $M_\\odot $ based on SED fitting.","[107] used SCUBA-2 to measure a larger disk mass of 9.4 $M_{\\rm Jup}$ ($\\sim $ 0.009 $M_\\odot $ ).","Spitzer/IRS observations revealed a 10 $\\mu $ m silicate feature, with a typical equivalent width of 2.29 $\\mu $ m (Furlan et al.","2009).","Interferometric data and modeling led to an inferred inner disk radius of 0.112 AU [51].","[119] estimated a very similar inner ring radius of 0.118 AU from near-infrared interferometry.","[6] observed the disk with the Submillimeter Array and found a centrally peaked morphology.","Their modeling predicts an inner disk radius of 0.07 AU and an inclination of 39; they infer an accretion rate of 10$^{-6.8}$ $M_\\odot $ yr$^{-1}$ , consistent with previous values.","Millimeter and submillimeter ATCA observations by [133] led to an outer disk radius of 100–300 AU and a dust mass of $\\sim $ 2$\\times $ 10$^{-5}$ $M_\\odot $ .","The disk is also detected with Herschel at 160 and 250 $\\mu $ m [126].","The object is a known variable, as originally reported by [148], [149] and [90].","It was followed up by [75], who found variations in the $V$ band of 12.73–12.93 during over 4000 days of monitoring.","The amplitude was larger at blue wavelengths, with a typical d$U$ /d$V$ of 2.4 magnitudes.","[152] reported a $V$ -band amplitude of 0.41 magnitudes over both short (hour–day) and long (years) timescales, with no detectable periodicity.","[60] monitored the star for over 7 years in the optical, finding a similar $V$ -magnitude range of 12.60–13.09.","EPIC 203935537/2MASS J16255615-2420481 is, to the best of our knowledge, a single star.","Ghez et al.","(1993)'s speckle imaging campaign did not reveal any companions down to 0.1 (0.2), at a flux ratio of 17 (18).","Neither did Simon et al.","(1995)'s imaging survey, which was sensitive to separations of 0.005–1.","Our own DSSI observations did not show any companions down to 0.1 separation at 4–5 magnitudes of contrast (flux ratio $\\sim $ 0.02) in the 692 and 880 nm bands.","[125] searched for companions in high resolution imaging but did not find any down to a flux ratio of 0.05 at a separation of 0.15.","Neither [106] nor [65] detected any radial velocity variations indicative of spectroscopic binary status.","Further high resolution imaging [134], [135] failed to reveal companions."],["EPIC 203954898 / 2MASS J16263682-2415518","This object was the subject of a multiplicity survey by [125], but no companions were detected down to 0.05 (0.12) times the stellar brightness at 0.5 (0.15) separation.","Likewise, [49] did not detect any companion, and reported that two faint stars observed at 6.1 and 6.3 separation [50] are likely background objects.","The star is a known X-ray emitter [64], [82], [57] and has a significant infrared excess, as indicated by ISO observations [21], Spitzer data [53], [35] and the AllWISE catalog [41].","The spectral index, $\\alpha $ , is 0.08, indicating a flat disk [53].","A spectral type range of K7-M1 was reported by [52].","Our own spectrum from the Palomar 200-inch telescope Double Spectrograph suggests K8.","We adopt M0.","[48] report an effective temperature of 3700$\\pm $ 56 K, consistent with this spectral type.","They also find a $v$ sin$i$ of 27$\\pm $ 4.7 km s$^{-1}$ .","The mass has been estimated to be 0.18 $M_\\odot $ [111], which is very low considering the temperature and youth of the object.","We suspect that several groups have confused the source with a neighboring star, 2MASS J16263713-2415599, which is $\\sim $ 9 away.","For example, [165] list the object name ROXRA22, a ROSAT X-ray source at RA=16:26:36.9, Dec=-24:15:53 [64].","However, the [165] coordinates match the companion star at RA=16:26:37.1, Dec=-24:15:59.9, and the listed spectral type is later, at M5.","The companion paper by [52] provides the same effective temperature, luminosity, and mass estimate, but an earlier spectral type range and lower extinction ($A_V$ =4.9).","Because of these discrepancies, we derive our own spectral data, apart from the $v$ sin$i$ measurement.","With the clear disk signatures, we can be confident that it is a young member of $\\rho $ Ophiuchus.","[111] list an accretion rate of $<10^{-9.71}$ $M_\\odot $ yr$^{-1}$ , based on near-infrared spectroscopy.","EPIC 203954898/2MASS J16263682-2415518 was monitored in the mid-infrared with Spitzer as part of the Young Stellar Object Variability project [128].","[66] obtained 81 datapoints spread over 34 days at 3.6 $\\mu $ m. The mean magnitude in this band was 8.15, in line with previous brightness estimates [35], [53], and the standard deviation was 0.09 magnitudes.","The star showed variability at the $\\sim $ 0.05-magnitude level for the first 15 days of monitoring, followed by a $\\sim $ 0.2-magnitude increase followed by a similar decrease over the remaining 20 days of observation.","WISE data also display a $\\sim $ 0.3-magnitude drop in the W1 band from February to August 2010."],["EPIC 204130613 / 2MASS J16145026-2332397","This star, BV Sco, was listed as an irregular variable by [149].","It has been erroneously classified as an RR Lyrae star in SIMBAD; this type of variability is inconsistent with the stochasticity seen in our $K2$ light curve.","Like EPIC 204233955/2MASS J16072955-2308221, it was labeled by [97] as a photometric non-member of Upper Scorpius (based on $ZYJHK$ data), before it was re-classified as a strongly accreting member [96].","It showed both H$\\alpha $ and He1 in emission (-108Å and -3.0Å equivalent widths, respectively).","Furthermore, they detected the calcium triplet lines and forbidden O1 emission, suggesting outflows."],["EPIC 204226548 / 2MASS J15582981-2310077","Also known as USco CTIO 33, [9] first identified this star as a candidate Upper Sco member.","[122] confirmed its youth via measurement of lithium absorption; they also detected strong H$\\alpha $ emission.","With a spectral type of M3, it is estimated to be 0.36 $M_\\odot $ by [86].","These authors also searched for spectroscopic and wide (1–30) binary companions, but did not find any.","This star was detected as a ROSAT x-ray source (RX J155829.5-231026) by [151].","[29] were the first to detect an infrared excess, at 8 and 16 $\\mu $ m. Cieza et al.","(2008) labeled it a transition disk source, with a mass of $<1.5\\times 10^{-3}$ $M_\\odot $ worth of material based on submillimeter data.","No millimeter flux was detected by [103].","[31] report a submillimeter detection with ALMA, but the disk is unresolved.","They constrain the dust mass to be $0.58\\pm 0.13$ $M_\\oplus $ .","[43] measure significant H$\\alpha $ emission from a broad, flat-topped peak; their estimated accretion rate is $10^{-9.91}$ $M_\\odot $ yr$^{-1}$ .","Similar values were derived by [42].","Molecular gas is detected with Herschel/PACS (both C$_2$ H$_2$ and HCN) by [114].","[102] estimate less than 0.9 $M_{Jup}$ worth of gas mass."],["EPIC 204233955 / 2MASS J16072955-2308221","EPIC 204233955 is a spectral type M3 low-mass star in the Upper Scorpius association [96].","It was initially classified as a photometric non-member by [97], but [96] found it to be an accreting source with strong emission lines, including H$\\alpha $ , He1, and O1 forbidden lines.","H$\\alpha $ and He1 equivalent widths are -150Å and -3.0Å, respectively.","Mid-infrared data from the AllWISE survey [41] reveal a significant infrared excess in all bands, indicative of a disk.","Other than the work of [97], [96], this star has not been studied in detail."],["EPIC 204342099 / 2MASS J16153456-2242421","EPIC 204342099, otherwise known as VV Sco, is a T Tauri star in the Upper Scorpius association.","This object is also a known X-ray emitter from ROSAT observations [68], XMM Newton observations [93].","[124] obtained low and medium resolution spectra of this star as part of an X-ray selected sample of candidate Upper Scorpius members.","They reported a spectral type of M1 and confirmed its youth via lithium absorption.","EPIC 204342099 is a known disk-bearing source, originally discovered with IRAS [81].","It was studied in detail with Spitzer/IRS by [56], who list it under the id 16126-2235.","They find a very strong 10 $\\mu $ m silicate feature.","This star was first presented as a visual binary by [63]; the separation is 1.9.","[86] measured a spectral type of M3.5 for the companion.","It is not clear whether this object is a co-moving Upper Scorpius member or a serendipitous field object; if the former, then the separation is 274 AU.","With DSSI speckle imaging, we measure the separation to be smaller at 1.50, with a magnitude difference of $\\Delta $ m=3.38 at the 880 nm band.","[87] also searched for closer companions with direct imaging and aperture masking, but did not find any within 240 mas, at a magnitude difference of 2.8.","Sparse variability data is available for EPIC 204342099/2MASS J16153456-2242421.","The star was first noted as variable by [116].","[17] monitored it for optical rotation signatures but did not detect any periodicities in the light curve.","Our HIRES spectrum suggests a spectral type of K9-M0 with lithium absorption present at strength $W_{Li}=0.45$ Å.","The H$\\alpha $ emission is consistent with accretion (Table ) and the profile exhibits a blueward asymmetry along with a redshifted absorption notch against the emission.","Very weak and narrow profiles in He1, Fe2, and Ca2 are also present in our data, along with very weak and narrow [O1]."],["EPIC 204360807 / 2MASS J16215741-2238180","There is no previous literature on this source.","As with other objects under consideration here, there is significant veiling present with the spectral type changing from M2 in the bluer orders of our HIRES to possibly as late as M6 by about 8800 Å. Lithium has strength $W_{Li} = 0.27 A$ .","H$\\alpha $ emission as reported in Table 1 is very strong and there is He1, broad NaD, and many Fe2 lines.","The Ca2 triplet has multiple components and O1 8446 is present.","Of the outflow lines only weak [O1] is seen.","With our DSSI speckle observations, we identify a companion at 0.48 separation with $\\Delta $ m=0.74 at 880 nm."],["EPIC 204397408 / 2MASS J16081081-2229428","This object was first identified as a candidate USco member based on proper motion by [97] with [26] assigning it a membership probability of 99.9% based on the USNO-B proper motion.","[154] confirmed youth via variability and spectroscopy, assigning a spectral type of M5.","They noted the star as an active accretor, based on a strong H$\\alpha $ emission line.","Likewise, [96] also spectroscopically confirmed this object as a USco member.","[44] measured a radial velocity of about -11 km/s, slightly lower than the cluster mean, and a rotation velocity of $v\\sin i=16.52\\pm 4.05$ .","[132] analyzed WISE photometry for EPIC 204397408/2MASS J16081081-2229428, concluding that it harbors a class II disk.","[99] also labeled it as a full disk."],["EPIC 204440603 / 2MASS J16142312-2219338","This very low mass star was classified by [97] as a photometric and proper motion member of Upper Scorpius based on UKIDSS data.","Following up with the Anglo-Australian Telecope AAOmega spectrograph, [96] obtained intermediate-resolution spectra of EPIC 204440603/2MASS J16142312-2219338 from 5750 to 8800Å, deriving a spectral type of M5.75 and H$\\alpha $ equivalent width of -94.5Å.","Their measured Na I and K I gravity-sensitive equivalent widths as well as detection of Li I absorption cements the classification of this object as a young low-mass star."],["EPIC 204830786 / 2MASS J16075796-2040087","This Upper Scorpius member was first identified as a strong H$\\alpha $ emission-line star by [158].","[88] obtained low-resolution spectra, which confirmed H$\\alpha $ emission at an equivalent width of -357Å and Ca2 triplet emission as well.","Detection of further emission lines (N2, S2, Fe2, Ni2, O1, and the Paschen series) suggested accretion-driven jets.","These authors also associated EPIC 204830786/2MASS J16075796-2040087 with a wide-separation companion some 21.5 (3120 AU) away.","It has a significant infrared excess, as shown with IRAS [28] and later with Spitzer and WISE by [99].","Our HIRES spectrum suggests a spectral type of late G to early K but the spectrum is clearly heavily veiled; the lithium strength is $W_{Li}=0.20$ Å.","Strong H$\\alpha $ emission is seen, as indicated in Table , with a blue-side absorption notch in the profile.","Strong and broad He1, NaD, Fe2, Ca2, O1 8446, and perhaps other emission is present.","Strong multi-component forbidden emission lines of [O1] and [S2] are seen, along with single-component [N2] and many [Fe2] lines.","Our DSSI speckle observations do not identify any companions outward of 0.1 from this star, at a magnitude difference of $\\Delta m\\sim 4$ in the 692 and 880 nm bands."],["EPIC 204906020 / 2MASS J16070211-2019387","[123] detected EPIC 204906020 as a youthful member of the Upper Sco association via spectroscopic measurement of lithium absorption.","This M5 star also has H$\\alpha $ emission, although it was weak enough in some observations to lead to a weak-lined T Tauri star classification.","[86] reported the object to be an M5/M5.5 wide binary with a 1.63 separation.","[88] found that the primary is itself a binary, with 55 mas projected separation (8 AU at the distance of Upper Sco).","[29] reported infrared excesses at 8 and 16 $\\mu $ m, indicating a disk.","[30] also detected an excess at 24 $\\mu $ m with Spitzer/MIPS.","[103] did not detect any cool dust around this system at millimeter wavelengths, at a 3-$\\sigma $ upper limit of 3.7$\\times 10^{-3}$ $M_{\\rm Jup}$ .","However, [102] used Herschel/PACS to detect 6.6$\\times 10^{-6}$ $M_\\odot $ worth of dust.","[31] detected but did not resolve the disk with ALMA.","EPIC 204906020/2MASS J16142312-2219338 is also known to have a circumstellar disk, as first reported by [132] and confirmed by [99].","The SED slope is -1.3, making it a class II disk [132].","Our speckle observations with DSSI rule out any companions beyond 0.1 at 4.6 magnitudes contrast in the 692 and 880 nm bands."],["EPIC 204908189 / 2MASS J16111330-2019029","This source appears in the literature only in the [99] $WISE$ sample and the [15] ALMA study.","Our HIRES spectrum suggests a spectral type of M1 with lithium absorption present at strength $W_{Li}=0.15$ Å.","Strong H$\\alpha $ emission is seen, as indicated in Table , along with He1, weak but broad NaD, weak and narrow Fe2, weak and narrow Ca2, but moderately broad O1 8446 emission.","Weak and narrow [O1] is also present."],["EPIC 205008727 / 2MASS J16193570-1950426","There is no previous literature on this source.","The HIRES spectrum is heavily veiled but appears to be a late K to M3 type, with lithium present at strength $W_{Li} = 0.55 A$ .","H$\\alpha $ emission as reported in Table is strong and has multiple components.","Additional emission includes He1, NaD, Fe2, Ca2 with the same profile shape as the H$\\alpha $ and O1 8446 is present.","Outflow lines of [O1], [N2], and [S2] are also present, along with [Fe2]."],["EPIC 205061092 / 2MASS J16145178-1935402","There is no previous literature on this source.","Our HIRES data indicate a spectral type of M5-M6 with lithium at $W_{Li} = 0.59 A$ .","Strong H$\\alpha $ emission, as indicated in Table 1, is present along with He1, but no other emission lines."],["EPIC 205088645 / 2MASS J16111237-1927374","[122] first identified this star as an M5 member of the USco association, based on lithium absorption and broad H$\\alpha $ emission (-50Å).","[26] assigned it a membership probability of 99.9% based on the USNO-B proper motion.","[101] found a slightly later spectral type of M6 based on low-resolution spectra, and similarly broad H$\\alpha $ emission.","The object displays an infrared excess confirmed by WISE to come from a full disk [99]."],["EPIC 205156547 / 2MASS J16121242-1907191","There is no previous literature on this source.","Our spectrum indicates a spectral type of M5-M6 with lithium at $W_{Li} = 0.56 A$ .","Moderate H$\\alpha $ emission, as indicated in Table 1, is apparent with with an asymmetric extension on the blue side of the profile.","There is also He1 but no other emission lines."]],"string":"[\n [\n \"A continuum of accretion burst behavior in young stars observed by K2\"\n ],\n [\n \"Abstract We present 29 likely members of the young $\\\\rho$ Oph or Upper Sco regions of recent star formation that exhibit \\\"accretion burst\\\" type light curves in $K2$ time series photometry.\",\n \"The bursters were identified by visual examination of their ~80 day light curves, though all satisfy the $M < -0.25$ flux asymmetry criterion for burst behavior defined by Cody et al.\",\n \"(2014).\",\n \"The burst sources represent $\\\\approx$9% of cluster members with strong infrared excess indicative of circumstellar material.\",\n \"Higher amplitude burster behavior is correlated with larger inner disk infrared excesses, as inferred from $WISE$ $W1-W2$ color.\",\n \"The burst sources are also outliers in their large H$\\\\alpha$ emission equivalent widths.\",\n \"No distinction between bursters and non-bursters is seen in stellar properties such as multiplicity or spectral type.\",\n \"The frequency of bursters is similar between the younger, more compact $\\\\rho$ Oph region, and the older, more dispersed Upper Sco region.\",\n \"The bursts exhibit a range of shapes, amplitudes (~10-700%), durations (~1-10 days), repeat time scales (~3-80 days), and duty cycles (~10-100%).\",\n \"Our results provide important input to models of magnetospheric accretion, in particular by elucidating the properties of accretion-related variability in the low state between major longer duration events such as EX Lup and FU Ori type accretion outbursts.\",\n \"We demonstrate the broad continuum of accretion burst behavior in young stars -- extending the phenomenon to lower amplitudes and shorter timescales than traditionally considered in the theory of pre-main sequence accretion history.\"\n ],\n [\n \"Introduction\",\n \"Variable mass flux has long been recognized as an important element of protostellar and pre-main sequence accretion.\",\n \"Accretion rates are believed to be higher ($\\\\sim 10^{-5} M_\\\\odot $ yr$^{-1}$ ) during the first 10$^5$ years of protostellar evolution, with frequent outbursts of up to $10^{-4} M_\\\\odot $ yr$^{-1}$ [70].\",\n \"The bursts are predicted as a consequence of unstable pile-up of gas in the inner disk, which then intermittently releases a cascade of material onto the star due to viscous-thermal disk instabilities [18], [168], [45].\",\n \"The burst frequency and perhaps amplitude decline over time [71], [161].\",\n \"While most of the stellar mass is thought to accumulate in the early phases of protostellar evolution, accretion at rates of $10^{-10}$ –$10^{-6} M_\\\\odot $ yr$^{-1}$ persists through the T Tauri phase (ages up to a few Myr), with less frequent bursts [69].\",\n \"The currently accepted picture of T Tauri star accretion involves magnetic funnel flows channeling gas from the inner disk onto the central star.\",\n \"Where this material impacts the surface, shocks arise and thermal hot spots form.\",\n \"There may be one spot near each magnetic pole, or multiple spot complexes that are distributed about the stellar surface.\",\n \"The number and geometry of the funnel flows is thought to depend on the accretion rate [138].\",\n \"Empirically, time series monitoring of young stellar objects (YSOs; ages $<$ 1–10 Myr) has an extensive history.\",\n \"It has been known since or before e.g.\",\n \"[83] that T Tauri stars display flux variations at a wide variety of timescales and magnitudes.\",\n \"The outbursting FU Ori stars and their lower amplitude, repeating cousins the EX Lup stars, are at the extreme end of the variabiity spectrum – and rare.\",\n \"More common photometric variability is characterized by smaller amplitude and shorter timescale fluctuations.\",\n \"The studies of [75], [60], [145], and [55] have illustrated and quantified much of the “typical\\\" young star photometric phenomena occurring from sub-hour to multi-decade time scales.\",\n \"One cause of the routine brightness variations is sporadic infall of material from the surrounding disk.\",\n \"Even in their predominant low-state accretion phases, young stars are understood as variable accretors.\",\n \"The photometric variability is complemented by highly variable emission line profiles and veiling [34], [40] including in stars with relatively low accretion rates such as TW Hya and V2129 Oph [2], [3].\",\n \"Nevertheless, the variability time scales and accretion rate changes remain poorly quantified for typical young accreting star/disk systems.\",\n \"There may be a continuum of “burst\\\" behavior with a range of amplitudes and time scales that have not yet been appropriately sampled or appreciated in existing ground-based data sets.\",\n \"While most of the stellar mass has been assumed to accumulate in the episodic and dramatic fashion of the rare large FU Ori and Ex Lup type events, the role and implications of discrete lower amplitude accretion events [155] and continuously stochastic accretion behavior [156] is not well understood in the context of stellar mass accumulation and inner disk evolution.\",\n \"In probing accretion variability, space-based photometric campaigns have several advantages over ground-based work, including near-continuous sampling (versus interruptions for daytime, weather, etc), higher measurement precision, and fainter signal detection limits.\",\n \"A detailed analysis of optical and infrared variability among disk-bearing stars in the $\\\\sim $ 3 Myr NGC 2264 was conducted by [38] based on a forty-day optical time series from the CoRoT space telescope at 10-minute cadence, along with 30 days of Spitzer Space Telescope monitoring at 100-minute cadence.\",\n \"These data enabled an unprecedented view of YSO brightness changes on a variety of timescales.\",\n \"Among the detected variability groups was a new class of “stochastic accretion burst” light curves – dominated by brightening events of duration 0.1-1 days and amplitude 5-50% the quiescent flux value [155].\",\n \"It was speculated that these events were caused by the unsteady infall of material onto the stellar surface, as predicted by [143].\",\n \"They may thus represent “normal\\\" discrete accretion variations and bursts, in contrast to the FU Ori and EX Lup outbursts described above.\",\n \"The NASA $K2$ mission Campaign 2 observations included the young $\\\\rho $ Ophiuchus molecular cloud region at $<$ 1-2 Myr, and the adjacent Upper Scorpius OB association, which is debated from analysis of HR diagrams to be either $\\\\sim $ 3-5 Myr based on the low mass stellar population [122], [76] or $\\\\sim $ 11 Myr based on the solar and super-solar mass population [115]; the latter age is beginning to be favored by results on eclipsing binaries [85], [47] and asteroseismology [136].\",\n \"By sampling stars with ages comparable to and extending to much older than NGC 2264 (in $\\\\rho $ Oph and Upper Sco, respectively), the $K2$ time series data can be used to compare accretion burst behavior as a function of age and therefore presumably disk properties which are expected to evolve with time.\",\n \"We report here on a continuum of accretion burst behavior among members of $\\\\rho $ Oph and Upper Sco.\",\n \"We observe discrete brightening events that range in amplitude from 0.1 to 2.5 magnitudes and in timescale from $<$ 1 day to $>$ 1 week.\",\n \"We demonstrate that the burst phenomenon is seen only in those stars with evidence for strongly accreting disks, distinct from the typical disks in the region with weaker infrared excess and H$\\\\alpha $ emission.\",\n \"Section 2 contains description of the $K2$ observations and pixel file processing, and of follow-up high dispersion spectroscopy.\",\n \"Section 3 presents the light curve analysis and identification of burst type variables, Section 4 a discussion of the corresponding accretion and disk properties, and Section 5 the spatial and time domain characteristics of bursting sources.\",\n \"We discuss the implications of these observations in Section 6 and summarize the results in Section 7.\",\n \"The $K2$ mission [80] observed nearly 2000 stars in the young $\\\\rho $ Ophiuchus and Upper Scorpius regions during Campaign 2.\",\n \"We have mainly considered objects submitted under programs GO2020, GO2047, GO2052, GO2056, GO2063, and GO2085 of the Campaign 2 solicitation, which comprise both secure cluster members and less secure candidates.\",\n \"We later noted aperiodically variable stars among a number of other programs targeting cool dwarfs and therefore added objects from programs GO2104, GO2051, GO2069 GO2029, GO2106, GO2089, GO2092, GO2049, GO2045, GO2107, GO2075, and GO2114 if they also had proper motions consistent with Upper Sco.\",\n \"[37], cull some of the less confident young star candidates using WISE photometry to eliminate giant stars and other contaminants.\",\n \"This vetting results in a reduced set of 1443 $\\\\rho $ Oph and Upper Sco candidate members.\",\n \"For each star in this set, we downloaded the target pixel file (TPF) from the Mikulski Archive for Space Telescopes (MAST).\",\n \"Each target is stored under its Ecliptic Plane Input Catalog (EPIC) identification number, as listed in Table 1.\",\n \"Data for each object includes 3811 $\\\\sim $ 10$\\\\times $ 12 pixel stamp images, obtained between 2014 23 August and 10 November.\",\n \"Since the loss of a second reaction wheel during the Kepler mission in May of 2013, telescope pointing for $K2$ has suffered reduced stability and requires corrective thruster firings approximately every six hours.\",\n \"We find a corresponding target centroid drift at a rate of $\\\\sim $ 0.1, or $\\\\sim $ 0.02 pixels, per hour.\",\n \"While this movement is relatively small, associated detector sensitivity variations at the few percent level per pixel compromise the otherwise exquisite photometry, introducings jumps in measured flux on the same six-hour timescales.\",\n \"It is helpful to track the $x$ -$y$ position drift for each star over time, as this can be used for aperture placement and later detrending of the light curves.\",\n \"TPF headers provide a rough world coordinate system solution which is the same for all images, but these are not precise enough to center the target.\",\n \"We therefore cut out a 5$\\\\times $ 5 pixel region around the specified target position, and used this to calculate a flux-weighted centroid.\",\n \"We carried out photomety with moving apertures, the centers of each specified by the measured centroid locations.\",\n \"This approach helps to minimize the effect of detector drift on the photometry.\",\n \"Circular apertures were used with radii ranging from 1.0 to 4.0 pixels, in intervals of 0.5 pixels.\",\n \"We found that photometric noise levels after detrending for position jump effects were generally minimized with the 2-pixel aperture, though for a few objects we selected the 1.5 or 3-pixel apertures.\",\n \"These sizes have the additional advantage of being small enough so as to avoid flux contamination from other stars lying $\\\\sim $ 12 away.\",\n \"To clean the data, we discarded the first 93 light curve points, for which the pixel positions were particularly errant compared to the rest of the time series.\",\n \"We also removed points with detector anomaly flags.\",\n \"Finally, we pruned points lying more than five standard deviations off the median light curve trend.\",\n \"This was accomplished by median smoothing on $\\\\sim $ 2-day timescales, removing outliers, and then adding the median trend back in.\",\n \"In general, our raw moving aperture photometry consists of lower levels of pointing-related systematic jitter than for the fixed aperture case.\",\n \"For all of the stars discussed in this work, the amplitude of intrinsic variability dwarfs these systematics, and no further corrections are needed.\",\n \"However, this is not the case for the less variable cluster members which form a control population for comparison of variability demographics.\",\n \"In these light curves, a prominent sawtooth pattern appears on the $\\\\sim $ 6-hour timescales corresponding to thruster firings, an effect that was mitigated with the detrending procedure described in [1], as described in [37].\",\n \"In one exceptional case (EPIC 203954898/2MASS J16263682-2415518), the light of a highly variable star was contaminated by a close neighbor 8 away.\",\n \"Single pixel photometry shows that this object undergoes high amplitude bursts, while the neighboring star is relatively constant and slightly fainter.\",\n \"Restricting the aperture to encompass only EPIC 203954898 results in degraded precision.\",\n \"We therefore used a 3-pixel aperture encircling both stars, and then removed the average flux of the companion by scaling the data to match the amplitude of the bursts in the single pixel light curves (the lower precision here affects only the detailed light curve morphology and not its overall amplitude).\",\n \"The result is a light curve with maximum burst amplitude of nearly eight times the quiescent flux level.\"\n ],\n [\n \"New spectra and compiled spectroscopic data\",\n \"We collected both archival and new spectroscopic data at high dispersion with the Keck/HIRES spectrograph [160].\",\n \"The new observations were obtained on one of: 2016 May 17 and 20, 2015 June 1 and 2, or 2013 June 4, UT, and covered the spectral range $\\\\sim $ 4800 Å to 9200 Å at resolution $R\\\\approx 36,000$ .\",\n \"The images were processed and the spectra were extracted and calibrated using the $makee$ software written by Tom Barlow.\",\n \"H$\\\\alpha $ emission line strengths and spectral type estimates are tabulated in Table ; other emission lines that were observed in these high dispersion data are discussed in the Appendix notes on individual stars.\",\n \"Table 1 as well as the notes section includes literature information in addition to our spectroscopic findings.\"\n ],\n [\n \"Speckle imaging\",\n \"We obtained high-resolution speckle imaging for six of our young stars to assess multiplicity properties.\",\n \"Our observations used the Differential Speckle Survey Instrument [77] on the Gemini-South telescope in 2016 June.\",\n \"Speckle observations were simultaneously made in two medium band filters with central wavelengths and bandpass FWHM values of ($\\\\lambda _c$ , $\\\\delta \\\\lambda $ ) = (692,47) and (883,54) nm.\",\n \"Each star was observed for approximately 10 minutes during which time we obtained 3-5 image sets consisting of 1000, 60 ms simultaneous frames.\",\n \"These observations were made during clear weather at airmass 1.0 to 1.3, when the native seeing was 0.4-0.6 arcsec.\",\n \"Details of speckle observations using the Gemini telescope and our data reduction procedures can be found in [78] and [79].\",\n \"We conducted visual examination of all 1443 young star light curves in order to identify stars in a “bursting\\\" state.\",\n \"Such objects were selected by identifying behavior consistent with that presented in [38] and [155].\",\n \"To confirm our visually identified bursters, we also computed quantitative variability metrics, specifically the $M$ and $Q$ statistics defined by us in the papers above.\",\n \"$M$ describes the degree of symmetry of the light curve about its mean value.\",\n \"It is calculated by determining the ratio of the mean of magnitude data in the top and bottom deciles to the median of all light curve points.\",\n \"$M$ achieves negative values when there is a significant number of points brighter than the median, but not so many faint points.\",\n \"Unlike what was done in [38], we calculated the $M$ statistic from flux, rather than magnitude values.\",\n \"The main effect of this change is to lower $M$ values (by $\\\\sim $ 10% on average), since the magnitude to flux conversion makes bright peaks more pronounced.\",\n \"We argue that flux units are a more natural choice here, as flux correlates with luminosity and accretion rate.\",\n \"Bursting light curves are highly asymmetric with frequent flux increases over the mean, resulting in $M$ values from approximately -0.3 to -1.3.\",\n \"While the boundary is somewhat subjective, all objects selected by eye meet the previously defined $M<-0.25$ criterion for burster status.\",\n \"The $Q$ statistic defined in [38] describes tendency towards or away from periodicity over the time series.\",\n \"It is a measurement of how much the standard deviation shrinks when the light curve is phased to its dominant periodicity and the associated pattern is repeated and subtracted out from the raw time series.\",\n \"Strictly periodic behavior (i.e., complete removal of the phase pattern) returns $Q=0$ while light curves with no repeating behavior have $Q=1$ .\",\n \"In a few cases for which the light curves are entirely aperiodic, this removal process actually increases the underlying standard deviation; this is why the computed $Q$ value is occasionally greater than 1.0.\",\n \"In previous work, we have denoted light curves with moderate $Q$ values of 0.15–0.60 as “quasi-periodic.” We emphasize here that this range is somewhat subjective and was based on a by-eye analysis of CoRoT data on the NGC 2264 cluster.\",\n \"As explained in Cody et al.\",\n \"2016, the present K2 dataset contains some objects with $Q>0.6$ that nevertheless display repeating components upon visual examination.\",\n \"The $M$ and $Q$ values for the selected bursters are provided in Table 2.\",\n \"They are also plotted in Fig.\",\n \"REF , alongside the values for other disk-bearing young stars in the $K2$ field, as selected in [37].\",\n \"The $K2$ burster sample behavior ranges from periodic to quasi-periodic to aperiodic, with light curves most tending towards aperiodicity.\",\n \"Objects falling in the bursting section of the diagram but not highlighted as such tend to be long-timescale variables for which the trend removal failed.\",\n \"We favor our by-eye classification over the $M$ and $Q$ statistics here and thus do not consider these bursters.\",\n \"Several other objects with highly negative $M$ values are dominated by periodic modulation and only display zero or one bursting event.\",\n \"We also leave these out of the sample, in light of classification ambiguity.\",\n \"Overall, 18 objects with $M<-0.25$ (i.e., a value that would qualify them as busters) were removed from the sample.\",\n \"Most of these may be seen in Figure REF as the black points lying above the $M=-0.25$ line (apart from five that are hidden under orange burster points); the majority are only marginally above it.\",\n \"In total, we have selected 29 stars as bursters in $\\\\rho $ Oph and Upper Sco; division into the two regions was based on a $1.2\\\\times 1.2$ square surrounding the position RA=246.79, Dec=-24.60 to define the extent of $\\\\rho $ Oph [37].\",\n \"We list the basic properties of the bursters in Table 1 and show their light curves in Figure REF .\",\n \"The burster class exhibits several subsets of behavior, with some light curves displaying a nearly continuous series of events, and others exhibiting more discrete brightening events.\",\n \"We discuss the timescales of bursting in Section 6. llcccccc 8 0pt Young stars exhibiting bursting behavior in $K2$ Campaign 2 EPIC id 2MASS id Other ids SpT EW H$\\\\alpha $ H$\\\\alpha $ 10% Refs Region (Å) (km s$^{-1}$ ) 203382255 J16144265-2619421 M4-M5.5 -77 154 1 USco 203725791 J16012902-2509069 USco CTIO 7 M2/M3.5 -170, -129 437 1, 2 USco 203786695 J16245974-2456008 WSB 18 M3.5 -8.4, -140 - 3 $\\\\rho $ Oph 203789507 J15570490-2455227 - - - USco 203794605 J16302339-2454161 WSB 67 M3.5-M5 -69 485 1 $\\\\rho $ Oph 203822485 J16272297-2448071 WSB 49, MHO 2111, DROXO 57 M4.25 -37 - 4 $\\\\rho $ Oph 203856109 J16095198-2440197 M5-M5.5 -15 155 1 USco 203899786 J16252434-2429442 V852 Oph, SR 22, DoAr 19, WSB 23 M4.5/M3 -31, -170 - 4, 5, 6 $\\\\rho $ Oph 203905576 J16261886-2428196 VSSG 1, Elias 20, YLW 31, ISO-Oph 24, K7–mid-M/M0 -70 416 1,7 $\\\\rho $ Oph IRAS 16233-2421, MHO 2103 203905625 J16284527-2428190 V853 Oph, SR 13, DoAr 40, WSB 62, M3.75 -30, -48, -46 - 4 $\\\\rho $ Oph ISO-Oph 199, HBC 266 203913804 J16275558-2426179 V2059 Oph, DoAr 37, SR 10, ISO-Oph 187, M2 -43, -56, -108 - 4,8 $\\\\rho $ Oph WSB 57, YLW 56, HBC 265, SVS 1771 203928175 J16282333-2422405 SR 20W K5 -35 - 3 $\\\\rho $ Oph 203935537 J16255615-2420481 V2058 Oph, DoAr 20, Elias 13, K4.5 -220, -87, -67 - 4 $\\\\rho $ Oph SR 4, WSB 25, YLW 25, IRAS 16229-2413, MHA 365-12, ISO-Oph 6 203954898 J16263682-2415518 ISO-Oph 51 M0 -10 - 9 $\\\\rho $ Oph 204130613 J16145026-2332397 BV Sco M4.5 -108 - 10 USco 204226548 J15582981-2310077 USco CTIO 33, USco 42 M3 -158, -250 - 11,12 USco 204233955 J16072955-2308221 M3 -150 - 10 USco 204342099 J16153456-2242421 VV Sco, IRAS 16126-2235, PDS 82a M1/K9-M0 -20, -31 337 13, 1 USco 204347422 J16195140-2241266 - - - USco 204360807 J16215741-2238180 M6 -140 341 1 USco 204397408 J16081081-2229428 M5.75/M5 -22, -49, -31 - 10, 14, 15 USco 204440603 J16142312-2219338 M5.75 -95 - 10 USco 204830786 J16075796-2040087 IRAS 16050-2032 M1/G6-K5 -357, -165 684 16, 1 USco 204906020 J16070211-2019387 KSA 68 M5 -8, -30 - 11, 17 USco 204908189 J16111330-2019029 M1/M3 -160 324 1,18 USco 205008727 J16193570-1950426 K7-M3 -55 322 1 USco 205061092 J16145178-1935402 M5-M6 -70 173 1 USco 205088645 J16111237-1927374 M5, M6 -50, -50 - 12, 19 USco 205156547 J16121242-1907191 M5-M6 -15 127 1 USco Stars in the $K2$ Campaign 2 burster sample, in order of EPIC id.\",\n \"EPIC 203786695/2MASS J16245974-2456008 has a companion at 1.1 separation, and the two H$\\\\alpha $ values belong to the distinct components of the system.\",\n \"The H$\\\\alpha $ 10% widths in column 6 are derived from data presented in this paper.\",\n \"We have highlighted in bold the other new values derived as part of this work.\",\n \"References: 1) this work, 2) [137], 3) [27], 4) [165], 5) [121], 6) [100], 7) [6], 8) [8], 9) [52], 10) [96], 11) [43], 12) [122], 13) [124], 14) [154], 15) [44], 16) [88], 17) [123], 18) [99], 19) [101].\",\n \"Figure: Light curves of selected bursters over the 80-day duration of K2K2 Campaign 2, in approximate order of amplitude.Figure: Cont.Figure: Cont.Figure: Cont.Figure: Cont.Figure: Cont.\"\n ],\n [\n \"Disk and accretion properties of the burst sample\",\n \"A hypothesis for the short-lived, often repeatable, brightening events in our 29 $K2$ light curves is that they are caused by episodic accretion from the circumstellar disk onto the young star.\",\n \"A specific requirement for the accretion-driven burst hypothesis is that the objects exhibit both infrared excess indicative of circumstellar dust, serving as the reservoir for the accretion, and either ultraviolet excess or line emission from hot gas in the nearby circumstellar environment.\",\n \"The dust criterion is satisfied by our burster sample, as all stars have been already selected as infrared excess sources in [37].\",\n \"As shown in that work, there are 344 disk bearing stars in the entire $K2$ Campaign 2 sample, of which 299 are bright enough to obtain light curves.\",\n \"This number includes all 29 burster stars identified here.\",\n \"Therefore, the fraction of bursters among the total disk-bearing sample is at least 8$\\\\pm $ 2%.\",\n \"The error comes from consideration of the Poisson uncertainties.\",\n \"These values can also be considered separately for the $\\\\rho $ Oph (128 disked stars; 92 with light curves) and Upper Sco (216 disked stars; 207 with light curves) samples.\",\n \"There are 10 bursters in $\\\\rho $ Oph and 19 bursters in Upper Sco; both numbers lead to roughly the same value of 8–10%, with a $\\\\sim $ 2% error on this fraction.\"\n ],\n [\n \"Circumstellar Dust\",\n \"Infrared color-magnitude diagrams can also shed light on what photometric aspects, if any, separate burster stars from other disk bearing sources.\",\n \"We show the color-color diagrams $J-K$ versus $K-W3$ and $J-K$ versus $K-W4$ in Figure REF and the spectral energy distributions in Appendix Figure REF , to illustrate the strength of emission at farther, cooler locations in the disk.\",\n \"The vast majority of the bursters have excesses in the three longest wavelength WISE bands $W2$ , $W3$ , and $W4$ ($4.6 \\\\mu $ m, $12 \\\\mu $ m, and $22 \\\\mu $ m, respectively).\",\n \"The individual spectral energy distributions in the Appendix (Fig.\",\n \"REF ) provide a finer look at the circumstellar flux patterns.\",\n \"This finding is in stark contrast to the overall disk sample, in which only 50% of objects (or a total of 172) have excesses in all three bands.\",\n \"Thus, the presence of a full, minimally evolved disk appears to be preferred for bursting behavior.\",\n \"Figure: Near and mid-infrared color-color diagram of stars in the Upper Sco/ρ\\\\rho Oph regions (grey), with disk-bearing stars in black and bursters highlighted in red.\",\n \"Burster point sizes are scaled by light curve amplitude.It has traditionally been thought that young stars with prominent fading events are surrounded by nearly edge-on disks; the dust clumps routinely obscure the central star [25], [24], causing decrements in the light curve.\",\n \"Conversely, one might imagine that for disk systems farther from edge-on orientation, we would have a more direct view of the accretion columns and shocks near the stellar poles.\",\n \"This could allow relatively unfettered observation of accretion bursts.\",\n \"Looking at the selection of 172 full disks identified in [37], 90% are variable, but only 17% are bursting.\",\n \"If this is a reflection of geometric selection, then one might hypothesize that burster disks are viewed at angles ranging from face-on to $\\\\sim $ 34.\",\n \"However, this assumes that face-on is the best angle at which to view bursting; as suggested by the ALMA data described below, this may not be the case.\",\n \"The idea that bursting behavior may be a function of viewing angle can be further explored by considering resolved disk imaging.\",\n \"Five of the bursters discussed in this paper have been observed with ALMA at 0.88 mm [31], [15].\",\n \"EPIC 204830786 (2MASS J16075796-2040087) has a broad CO $J=3-2$ line detection, with velocities from -17 to 17 km s$^{-1}$ ; the profile suggests that the disk is not face-on.\",\n \"EPIC 204342099 (2MASS J16153456-2242421) is the only other disk with a CO detection, albeit a weak one.\",\n \"The broad velocity distribution again suggests that this system is not oriented face-on.\",\n \"The other three disk-bearing sources in our sample (EPIC 204906020/2MASS J16070211-2019387, EPIC 204226548/2MASS J15582981-2310077, EPIC 204908189/2MASS J16111330-2019029) have no CO detection.\",\n \"The latter two do show continuum, so CO may be highly depleted or the disks in these cases could be physically small.\",\n \"The continuum fluxes can also be used to infer disk properties.\",\n \"[14] infers from modeling SEDs and size constraints that EPIC 204830786 (2MASS J16075796-2040087) has inclination 42$^{+12}_{-9}$ , while EPIC 204342099 (2MASS J16153456-2242421) is inclined at 43$^{+15}_{-16}$ .\",\n \"Dust masses were inferred by [15] and range from $<$ 0.5 $M_\\\\oplus $ (EPIC 204906020/2MASS J16070211-2019387; non-detection) to 9.3 $M_\\\\oplus $ in the case of EPIC 204830786/2MASS J16075796-2040087.\",\n \"All five bursters in the ALMA sample are noted as having been classified as full disks (as opposed to evolved or transitional) by [99].\",\n \"These intermediate inclination values are consistent with the idea proposed above that we are not looking through disk material, which would be expected to produce “dipping\\\" rather than “bursting” light curves.\",\n \"But they are inconclusive regarding whether we have a direct view of material accreting onto the central star.\"\n ],\n [\n \"Gas and Accretion\",\n \"In addition to infrared disk indicators, we obtained spectroscopic data from both the literature and our own high-resolution follow-up spectroscopy (§2.2).\",\n \"The H$\\\\alpha $ emission equivalent width (EW) and 10% width values (Table 1) are indicative of significant accretion.\",\n \"As a control sample, we gathered H$\\\\alpha $ EWs for other disk-bearing (but not necessarily bursting) stars in Upper Sco from [137], [43], and [122].\",\n \"We cross-matched them against our Upper Sco/$\\\\rho $ Oph K2 $WISE$ excess star list and eliminated any objects not in common.\",\n \"We then compared the H$\\\\alpha $ values of this general Upper Sco sample with those of the bursters in Figure REF .\",\n \"Since some of the bursters have multiple H$\\\\alpha $ measurements (see Table 1), we have taken the average of all available values.\",\n \"We find that the bursters occupy a large range of H$\\\\alpha $ EW values, from -10Å (presumably a low-state value) to several hundred angstroms in emission.\",\n \"The non-burster stars, on the other hand, display weak H$\\\\alpha $ , with EWs primarily from 0 to -15Å and a tail out to -50Å with one value around -120Å.\",\n \"Figure REF illustrates the emission line profiles for the 12 out of the 29 bursters for which we have high resolution spectra in H$\\\\alpha $ , Ca2 8542Å and He1 5876Å.\",\n \"[163] advocated the designation of accreting stars based on H$\\\\alpha $ line widths at 10% of the maximum line strength that are larger than 270 km s$^{-1}$ .\",\n \"Although all 12 sources have velocities larger than 100 km s$^{-1}$ (see Table 1), only slightly more than 1/2 meet the [163] requirement.\",\n \"There is a tendency for the narrower velocity stars (EPIC 203856109/2MASS J16095198-2440197, EPIC 205156547/2MASS 16121242-1907191, EPIC 203382255/2MASS J16144265-2619421, EPIC 205061092/2MASS J16145178-1935402, EPIC 205008727/2MASS J16193570-1950426) to have lower duty cycles in their burst patterns (Figure REF ) (Where available, H$\\\\alpha $ velocity and duty cycle are correlated at a significance level of 1.2$\\\\times 10^{-3}$ ).\",\n \"It may be that our spectra were taken at non-burst epochs.\",\n \"This is speculative at best, however.\",\n \"The weak H$\\\\alpha $ sources all exhibit only narrow component emission in their Ca2 profiles, as does EPIC 204342099 (2MASS J16153456-2242421) which has a broad H$\\\\alpha $ profile.\",\n \"Generally, the broad H$\\\\alpha $ sources exhibit both broad component and narrow component Ca2.\",\n \"[13] review the classic literature on Ca2 profile morphology in young stars, and discuss magnetospheric models of it.\",\n \"Our profiles do not seem to exhibit the asymmetries predicted by these models (e..g blueshifted peaks, redshifted depressions), however.\",\n \"Notably, our broad-lined Ca2 sources also exhibit evidence for forbidden line emission, e.g.\",\n \"[O1] 6300Å.\",\n \"The morphology of He1 emission lines in young stars was studied by [19] who also designated narrow line, broad line, and narrow$+$ broad profile categories.\",\n \"Our stars are dominated by their narrow component emission, with widths ranging between 40-70 km s$^{-1}$ , but some may also have weak broad components that would require line decomposition to characterize.\",\n \"Most of the He1 profiles appear to have slight asymmetries, however, in the sense of broader redshifted emission than blueshifted emission with the line peaks at zero-velocity.\",\n \"Beyond emission line morphology, accretion rates are estimated for a handful of our burster stars by [111].\",\n \"Values range from 10$^{-9.9}$ to 10$^{-6.7}$ $M_\\\\odot $ yr$^{-1}$ – a large range.\",\n \"It should be noted that neither the H$\\\\alpha $ EWs presented above nor the accretion rates referenced here were necessarily measured during a time when the stars were undergoing bursting events.\",\n \"Thus it is plausible that accretion is preferentially high in these sources, but only at certain times.\",\n \"Under the hypothesis that burst events in light curves are due to higher than average mass flow, we can convert the flux to a quantitative increase in the mass accretion rate.\",\n \"This requires several assumptions.\",\n \"First, we assign to all stars in our sample a low-level baseline accretion rate, $\\\\dot{M}_{\\\\rm low}$ , which then increases to a larger rate $\\\\dot{M}_{\\\\rm high}$ during bursts.\",\n \"We assume that this increase in mass flow can be equated to the ratio of the accretion flux $F_{\\\\rm acc}$ in and out of the burst state through the accretion luminosity, $L_{\\\\rm acc}$ : $\\\\frac{\\\\dot{M}_{\\\\rm high}}{\\\\dot{M}_{\\\\rm low}} = \\\\frac{L_{\\\\rm acc,high}}{L_{\\\\rm acc,low}}\\\\sim \\\\frac{F_{\\\\rm acc,high}}{F_{\\\\rm acc,low}}.$ The accretion luminosity is related to the stellar mass $M_*$ and radius $R_*$ by $L_{\\\\rm acc} = 1.25(GM_*\\\\dot{M}/R_*)$ where the pre-factor is that appropriate for an assumed magnetospheric accretion scenario.\",\n \"The measured flux density $F$ (i.e., in the $Kepler$ band) contains contributions from both accretion and the underlying stellar luminosity, $L_*$ .\",\n \"We label the measured flux ratio “$r$ ”: $r \\\\equiv \\\\frac{F_{\\\\rm high}}{F_{\\\\rm low}} = \\\\frac{F_*+F_{\\\\rm acc,high}}{F_*+F_{\\\\rm acc,low}}$ $F_{\\\\rm low}$ and $F_{\\\\rm high}$ are approximately the minimum and maximum flux values respectively attained in a given light curve.\",\n \"To determine $F_{\\\\rm acc,high}/F_{\\\\rm acc,low}$ and hence $\\\\dot{M}_{\\\\rm high}/\\\\dot{M}_{\\\\rm low}$ , we must estimate the ratio of stellar to accretion luminosity.\",\n \"This quantity has been studied by e.g.\",\n \"[110] and we adopt the fit based on their Figure 2.\",\n \"For each star in the burster sample, we estimate the stellar luminosity by considering the $J$ -band magnitudes of similar spectral type non-accreting young stars in the $K2$ Campaign 2 set, and applying bolometric and extinction corrections as outlined in [111].\",\n \"Combining this with Equations 1 and 2, it can be shown that $\\\\frac{\\\\dot{M}_{\\\\rm high}}{\\\\dot{M}_{\\\\rm low}}\\\\sim \\\\frac{L_*}{L_{\\\\rm acc}}(r-1)+r.$ Given our estimates of $L_*/L_{\\\\rm acc}$ and $r$ as measured from the $K2$ photometry, we have calculated the increase in mass accretion rate during bursts for each of our sources.\",\n \"Figure REF illustrates the resulting values as a function of spectral type, in cases where this is known to a subclass or better.\",\n \"We find that they range from $\\\\sim $ 10 to over 400 for the most prominent burst events.\",\n \"Figure: We show the cumulative distributions of Hα\\\\alpha equivalent width for disk-bearing stars in our K2 young star set with available spectroscopy; this set includes 28 bursters and 46 non-bursters.\",\n \"Where more than one Hα\\\\alpha measurement is available, we adopt the mean value.\",\n \"The sample was binned in 10Å wide sets to produce the distribution.\",\n \"While the non-bursting stars tend to predominate between 0 and -15Å (i.e., emission), the burster Hα\\\\alpha values show a much wider dispersion, reaching much values up to -250Å.\",\n \"Thus the burst phenomenon seems to favor stars with high accretion rates.\",\n \"We note that there is a single disk-bearing star with quasi-periodic light curve that has a reported Hα\\\\alpha equivalent width around -120Å.\",\n \"Such a value is highly unusual for a star with a non-bursting light curve.Figure: Line profiles in Hα\\\\alpha , Ca2 8542 Å, and He1 5876 Å measured by Keck/HIRES for 12 of the 29 bursters, labeled by EPIC identifier.\",\n \"The Hα\\\\alpha panels include a horizontal line at 10% of the peak (un-normalized) flux in addition to the horizontal line indicating the continuum level.\",\n \"Note the change in velocity scale for the helium panels.\",\n \"Broad width and structured velocity profiles indicate accretion and wind phenomena.Figure: Estimated increase in accretion rate during bursts, as a function of stellar spectral type.\"\n ],\n [\n \"Spatial distribution\",\n \"The sample we have analyzed here is likely a mixture of ages from young ($<1-2$ Myr) but still optically visible stars associated with the young $\\\\rho $ Oph cloud, to somewhat older (5-10 Myr) stars in the off-cloud Upper Sco region which still retain accreting circumstellar disks.\",\n \"Bursters as identified here make up $\\\\sim $ 9% of the disk-bearing young star sample in the $K2$ /C2 dataset.\",\n \"We initially hypothesized that they would preferentially appear in the compact, young $\\\\rho $ Oph cluster, as opposed to the more dispersed and older Upper Sco region.\",\n \"But the spatial distribution shown in Figure REF surprisingly reveals equal proportions of bursters in the two regions.\",\n \"Furthermore, there is no correlation with global extinction measures.\",\n \"This suggests that either the young population extends from $\\\\rho $ Oph out into the surrounding areas, or that the burster phenomenon is less dependent on age than on disk properties such as mass.\",\n \"[52] found evidence for an intermediate age population ($\\\\sim $ 3 Myr) of YSOs outside of the main $\\\\rho $ Oph cloud core, but within the main L 1688 cloud.\",\n \"Furthermore, [165] found a negligible age difference between sets of Upper Sco and L 1688 association members; both regions were estimated to be $\\\\sim $ 3 Myr old.\",\n \"Figure: Top: Spatial distribution of young stars (gray), including variables (black) and specifically bursters (red), overlaid on the K2 field of view.\",\n \"The concentration of stars near RA=246.8, Dec=-24.6 is the ρ\\\\rho Oph cluster.\",\n \"Bottom: Young stars with disks (cyan) and bursters (red) overlaid on the extinction map of the ρ\\\\rho Oph region.\"\n ],\n [\n \"Spectral types\",\n \"To assess the mass distribution of stars displaying bursts, we have gathered spectral types from the literature.\",\n \"Those found for the bursters are displayed in Table 1.\",\n \"We have also selected a control sample of non-bursting stars from [99].\",\n \"That work presents spectral types for hundreds of USco members; we cull the list to include only non-bursting, inner disk-bearing stars observed in K2 Campaign 2.\",\n \"The resulting set of 31 objects has spectral types ranging from B8 to M8, as shown in Figure REF .\",\n \"We compare this against the distribution of spectral types for the bursters identified here, finding a significant difference only for the B–F spectral types.\",\n \"There are early type stars in the sample but none of their variability is of the bursting type.\",\n \"K2 light curves for these objects show mainly low-level quasi-periodic modulation.\",\n \"However, the spectral type distributions of bursters and non-bursters is very similar for the K–M range.\",\n \"Statistically, the two sets are indistinguishable here.\",\n \"Thus we conclude that the preponderance of late-M spectral types among bursters is consistent with the increased fraction of young stars with disks at low masses.\",\n \"Figure: Spectral type distributions for bursting and non-bursting young disk-bearing stars observed in K2 Campaign 2.\",\n \"Two distributions are very different for the B–F range, but indistinguishable for K and M spectral types.\"\n ],\n [\n \"Multiplicity\",\n \"The presence of close stellar companions to the young stars in our sample may influence their variability properties.\",\n \"Indeed it has been hypothesized that, for high-amplitude outbursting stars such as FUors, events could be triggered by the presence of a perturbing companion [129], [94].\",\n \"However, support for this idea is mixed [61].\",\n \"It is nevertheless worthwhile to check which, if any, of our burster sample may be in binary systems.\",\n \"We have vetted over half of the sample for multiplicity by examining the available high-resolution imaging as well as obtaining new speckle observations.\"\n ],\n [\n \"Literature assessment\",\n \"We mined the literature for adaptive optics and speckle imaging of these targets, and the details are tabulated as part of the individual object commentary in the Appendix.\",\n \"Only three objects have evidence for a close companion published in the literature: EPIC 204906020 (2MASS J16070211-2019387), EPIC 203786695 (2MASS J16245974-2456008), EPIC 203905625 (2MASS J16284527-2428190).\",\n \"The first is a close binary [89].\",\n \"Likewise, the second has a companion at $\\\\sim $ 15 AU (100.4 mas) [84].\",\n \"The last of these has an even closer companion at $\\\\sim $ 2 AU (13 mas) [153].\",\n \"Interestingly, all three systems also have wider ($>$ 50 AU) companions.\",\n \"None of their light curves shows any hint of a periodicity.\",\n \"One additional object, EPIC 204342099 (2MASS J16153456-2242421), may have a 274 AU separation companion [63], but it is not clear whether the two stars are bound.\",\n \"EPIC 204830786 (2MASS J16075796-2040087) is associated with another star 21.5 away ($>$ 3000 AU separation) but has not been surveyed for closer companions.\",\n \"Other than these five objects, 9 other bursters in our sample have been surveyed for multiplicity, at a variety of sensitivities and separations: EPIC 204226548 (2MASS J15582981-2310077), EPIC 203899786 (2MASS J16252434-2429442), EPIC 203935537 (2MASS J16255615-2420481), EPIC 203905576 (2MASS J16261886-2428196), EPIC 203954898 (2MASS J16263682-2415518), EPIC 203822485 (2MASS J16272297-2448071), EPIC 203913804 (2MASS J16275558-2426179), EPIC 203928175 (2MASS J16282333-2422405), EPIC 203794605 (2MASS J16302339-2454161).\",\n \"No companions were found in these cases.\",\n \"However, the separations probed are very non-uniform and range from 10 mas (1.5 AU) in some cases to 1–30 in others ($>$ 150 AU); details are provided in the Appendix individual objects section.\"\n ],\n [\n \"DSSI targets\",\n \"In addition to data compiled from the literature, we also have speckle imaging observations of six bursters using DSSI (§2.3).\",\n \"For EPIC 204342099 (2MASS J16153456-2242421), we recover the companion reported at 1.9 by [63]; however, we measure the separation to be 1.50 ($\\\\sim $ 218 AU), and a magnitude difference of $\\\\Delta $ m=3.38 at the 880 nm band.\",\n \"This star previously had direct imaging and aperture masking by [87] that ruled out further objects down to 240 mas (35 AU).\",\n \"EPIC 204830786 (2MASS J16075796-2040087) has a previously noted possible companion at thousands of AU separation, but the DSSI observations otherwise support the hypothesis that this is a single star.\",\n \"EPIC 204440603 (2MASS J16142312-2219338) has no previous multiplicity information, but the 880 nm image suggests a possible companion at a separation of 0.1.\",\n \"However, the lack of a similar detection at 692 nm and the faintness of this star makes speckle reconstruction challenging; thus the existence of such a companion remains indeterminate.\",\n \"EPIC 204360807 (2MASS J16215741-2238180) has no multiplicity information in literature, but we find it to be a 0.48 separation binary (70 AU), with $\\\\Delta $ m=0.74 at 880 nm.\",\n \"EPIC 203935537 (2MASS J16255615-2420481) has no reported evidence for multiplicity, and we do not detect companions down to 0.1 separation (15 AU) at 4–5 magnitudes of contrast in the 692 and 880 nm bands.\",\n \"Finally, EPIC 203928175 (2MASS J16282333-2422405) was reported by [33] to host no companions down to 20 mas (3 AU) at 1-3 magnitude contrast; our observations support the lack of binarity, with no detections outward of 0.1 at $\\\\Delta m\\\\sim $ 4.4 magnitudes.\"\n ],\n [\n \"Time domain behavior\",\n \"In order to appreciate the diversity of the bursting behavior among our sample of objects, we must go beyond just the $M$ and $Q$ metrics discussed above.\",\n \"We quantified the peak-to peak amplitudes of the bursters by first cleaning and normalizing the light curves and then computing the maximum-minus-minimum values.\",\n \"There are several ways to quantify light curve timescales.\",\n \"First, we measure the burst duty cycle, which is the fraction of time each object spends in a bursting state.\",\n \"This is by nature somewhat subjective, as bursts display a range of amplitudes and shapes.\",\n \"We identified bursting portions of each light curve by first fitting and removing a low-order median trend to the light curve.\",\n \"This flattens the “continuum” level from which bursts arrive.\",\n \"We then measure the typical point-to-point scatter by shifting each point by one, subtracting from the original light curve, and dividing the standard deviation of the result by $\\\\sqrt{2}$ .\",\n \"Using this measure of scatter, we have found that burst behavior, as detected by eye, includes points that lie about 15 times the scatter above the minimum of the continuum-flattened light curve.\",\n \"We thereby selected bursting and non-bursting sections for each time series.\",\n \"This method only failed for three objects (EPIC 204397408/2MASS J16081081-2229428, EPIC 205156547/2MASS J16121242-1907191, and EPIC 203856109/2MASS J16095198-2440197) that displayed intermittent quasi-periodic behavior that was picked up as bursts.\",\n \"We manually removed these light curve portions for the statistical analysis.\",\n \"In Figure REF we display the peak-to-peak amplitudes versus duty cycle of each burster.\",\n \"The duty cycles exhibit a large range of values, from almost 100% down to $\\\\sim $ 10%.\",\n \"Typically the light curves with the highest amplitudes have higher duty cycles of 60% and above, with the exception of outlier EPIC 203954898/2MASS J16263682-2415518.\",\n \"This object may represent a distinct form of bursting behavior.\",\n \"We also quantify the burst timescales by applying a method similar to the one described in [38] (§6.5 of that paper).\",\n \"In brief, this involves identifying peaks that rise above a particular amplitude level compared to the surrounding light curve.\",\n \"Once peaks are found, the median timescale separating them is computed.\",\n \"This procedure is repeated for a variety of amplitudes, from the noise level up to the maximum light curve extent.\",\n \"The result is a plot of timescale versus amplitude (e.g., Fig.\",\n \"32 in [38]).\",\n \"Finally, we take as a “representative” timescale the value corresponding to an amplitude that is 40% of the maximum peak-to-peak value (we note that this is different from the value of 70% adopted in [38] and appears more appropriate for the burster light curves examined here).\",\n \"This computation only fails for the light curve of EPIC 203382255 (2MASS J16144265-2619421), which displays only one burst event; here the timescale is indeterminate.\",\n \"In Figure REF we display the peak-to-peak amplitudes versus estimated timescale for each burster.\",\n \"Again, there is a large range of values, but no clear correlation with amplitude.\",\n \"The burst duration is another way to quantify the observed events.\",\n \"This is a challenging measurement, as there is a superposition of bursts with varying widths and heights.\",\n \"We simplify as above by only considering peaks that rise to a level of at least 40% of the maximum peak-to-peak value.\",\n \"For each peak, we identify the surrounding points that are more than 15 times the point to point scatter above the minimum light curve value (as was done for the burst duty cycle calculation).\",\n \"We then adopt the median burst duration of all such peaks in each light curve.\",\n \"The result is plotted against peak-to-peak amplitude in Figure REF .\",\n \"We also compare the durations with the repeat timescales in Figure REF .\",\n \"Here we find that burst duration is correlated with repeat timescale.\",\n \"This is somewhat expected since, by definition, a duration is typically larger than the average timescale between bursts.\",\n \"However, we observe a distinct lack of short bursts (duration $\\\\le $ 2 days) with long repeat timescales.\",\n \"This may be rooted in the physical mechanism of the bursts.\",\n \"We have also generated periodograms to identify any repeating components in the light curves.\",\n \"Most bursters do not exhibit periodicity but instead adhere to stochastic behavior, with any quasi-periodicity quantified via the $Q$ statistic (see §3).\",\n \"Those that do show significant periodicities (as indicated by $Q\\\\le 0.61$ and/or a strong, isolated periodogram peak) are EPIC 203794605/2MASS J16302339-2454161 ($P=4.5$ d), EPIC 203899786/2MASS 16252434-2429442 ($P=6.0$ d), EPIC 203928175/2MASS J16282333-2422405 ($P=4.4$ d), EPIC 203954898/2MASS J16263682-2415518 ($P=20.8$ d), and EPIC 204347422/2MASS J16195140-2241266 ($P=6.9$ d).\",\n \"The measured periods are similar to the burst repeat timescales inferred above.\",\n \"In addition, EPIC 203856109 (2MASS J16095198-2440197), EPIC 204233955 (2MASS J16072955-2308221), EPIC 204397408 (2MASS J16081081-2229428), and EPIC 205156547 (2MASS J16121242-1907191) display short-timescale ($P$ less than a few days) periodic behavior outside of their bursting states.\",\n \"These periods are more typical of the K2 $\\\\rho $ Oph and Upper Sco sample as a whole [127] and likely measure stellar rotation, whereas those of the bursts are longer by factors at least several.\",\n \"Figure: Maximum burst amplitude (in units of normalized flux) versus duty cycle, i.e., fraction of time spent bursting.Figure: Burst repeat timescales versus peak-to-peak light curve amplitude, in normalized flux.Figure: Burst durations versus peak-to-peak light curve amplitude, in normalized flux.Figure: Burst repeat timescale versus durations.\",\n \"The dashed line indicates where these two quantities are equal.\",\n \"We have left out EPIC 203382255 (2MASS J16144265-2619421) since it only has one burst and the repeat timescale is thus indeterminate.Few of the timescale metrics show any relation to [circum]stellar properties, but one potential correlation stands out in peak-to-peak amplitude versus the infrared $W1-W2$ color (Figure REF ), which is indicative of a dusty inner disk.\",\n \"These two quantities are correlated at a significance level of 4$\\\\times 10^{-5}$ (Pearson $r$ coefficient of 0.69).\",\n \"This is also borne out in Figure REF , which suggests that the dustiest objects have the highest light curve amplitudes.\",\n \"Figure: Peak-to-peak light curve amplitude for bursters is shown against the W1-W2W1-W2 color.\",\n \"All included objects have W1-W2W1-W2 values indicative of inner disk dust, but the highest amplitude bursters tend to have larger infrared excesses, with one prominent exception (EPIC 205008727/2MASS J16193570-1950426; W1-W2∼1.3W1-W2\\\\sim 1.3).llcccccccc 8 0pt Light curve metrics for bursting young stars in $K2$ Campaign 2 EPIC id 2MASS id Amplitude Q M Timescale Duration Duty cycle Period (norm.\",\n \"flux) [d] [d] [d] 203382255 J16144265-2619421 1.43 1.00 -1.14 $>$ 77.74 7.95 0.08 - 203725791 J16012902-2509069 0.68 0.87 -0.29 5.18 1.90 0.66 - 203786695 J16245974-2456008 0.25 1.00 -0.59 5.52 4.54 0.43 - 203789507 J15570490-2455227 0.36 0.97 -0.41 9.12 4.45 0.06 - 203794605 J16302339-2454161 0.87 0.55 -0.25 4.64 3.67 0.93 4.46 203822485 J16272297-2448071 0.62 0.84 -0.29 9.36 7.66 0.90 - 203856109 J16095198-2440197 0.35 1.00 -1.00 3.11 5.25 0.15 - 203899786 J16252434-2429442 1.13 0.61 -0.83 6.42 3.98 0.96 5.95 203905576 J16261886-2428196 3.85 1.00 -0.66 7.48 5.97 0.88 - 203905625 J16284527-2428190 0.32 1.0 -0.31 7.80 5.17 0.72 - 203913804 J16275558-2426179 0.41 1.0 -0.37 5.47 3.56 0.70 - 203928175 J16282333-2422405 3.19 0.54 -0.66 4.68 2.53 0.73 4.39 203935537 J16255615-2420481 0.24 1.00 -0.31 4.92 3.09 0.69 - 203954898 J16263682-2415518 6.82 0.61 -1.35 31.17 6.84 0.43 20.83 204130613 J16145026-2332397 1.87 0.85 -0.35 3.43 0.90 0.62 - 204226548 J15582981-2310077 0.47 1.00 -0.53 8.54 3.44 0.73 - 204233955 J16072955-2308221 1.99 0.85 -0.82 3.42 2.08 0.98 - 204342099 J16153456-2242421 0.83 0.91 -0.72 35.53 11.26 0.88 - 204347422 J16195143-2241332 1.38 0.75 -1.11 7.23 1.65 0.10 6.94 204360807 J16215741-2238180 0.55 0.87 -0.49 5.21 2.53 0.65 - 204397408 J16081081-2229428 0.15 0.59 -0.68 5.01 5.30 0.22 1.65 204440603 J16142312-2219338 0.43 1.00 -0.93 4.55 3.15 0.28 - 204830786 J16075796-2040087 1.91 1.00 -0.67 14.70 4.31 0.91 - 204906020 J16070211-2019387 0.34 0.93 -0.47 3.90 0.91 0.63 - 204908189 J16111330-2019029 1.28 0.76 -0.59 7.99 7.97 0.90 19.23 205008727 J16193570-1950426 0.91 1.00 -0.68 10.25 7.93 0.76 - 205061092 J16145178-1935402 0.35 1.00 -0.51 4.85 1.94 0.35 - 205088645 J16111237-1927374 0.10 1.00 -0.53 11.30 3.18 0.04 - 205156547 J16121242-1907191 0.16 1.00 -1.01 4.38 3.62 0.23 - We tabulate basic statistical properties of the burster light curves.\",\n \"$Q$ and $M$ are discussed in §3 as well as Cody et al.\",\n \"(2014).\",\n \"Amplitudes represent peak-to-peak measurements.\"\n ],\n [\n \"Discussion and Summary\",\n \"$K2$ data from Campaign 2 have probed the optical burst properties of young stars on time scales ranging from hours up to several months, with approximately 8–10% ($\\\\pm $ 2%) of strong disk sources exhibiting burst behavior.\",\n \"This is roughly in agreement with the 13$^{+3}_{-2}$ % fraction found for NGC 2264 [38], [155].\",\n \"It is possible that an even larger fraction of young stars undergo bursting, but are not detected as such if the amplitude is low (i.e., $<100$ % peak to peak).\",\n \"Burst behavior could in some cases be masked by other types of variability.\",\n \"Burster stars host inner circumstellar disks, as evidenced by WISE excesses and SEDs (Figure REF ).\",\n \"Furthermore, there is a positive correlation between the $W1-W2$ color and light curve peak-to-peak amplitude.\",\n \"Stronger inner disk excesses appear to be associated with bigger bursts.\",\n \"Most of these objects have exceptionally strong H$\\\\alpha $ emission, as well as other Balmer lines, He1, and Ca2 in emission, as is typical for strongly accreting stars.\",\n \"Viewing geometry may play a role in setting the observability of the bursting phenomenon, but thus far we only have disk inclination constraints from ALMA on two sources in the sample, both of which are tilted at $\\\\sim $ 42$\\\\pm $ 15.\",\n \"Members of the bursting sample typically exhibit multiple discrete brightening events, some lasting up to or just over one week.\",\n \"Time domain properties of the bursters are diverse, with some stars exhibiting a nearly continuous series of bursts and others displaying one or two isolated episodes superimposed on otherwise lower amplitude or quasi-periodic behavior.\",\n \"The majority of these light curves show flux variations of less than a factor of two and are similar to the objects in the $\\\\sim $ 3 Myr NGC 2264 highlighted by [155].\",\n \"However, seven of 29 bursters display discrete brightenings of more than 100% on day to week timescales, unlike most heretofore classified young stellar variables that we are aware of.\",\n \"Of particular interest is the subset of these for which the bursts repeat quasi-periodically: EPIC 204347422 (2MASS J16195140-2241266), EPIC 203928175 (2MASS J16282333-2422405), and EPIC 203954898 (2MASS J16263682-2415518).\",\n \"It is unclear as to what physical phenomenon sets the periods of 6.9 d, 4.4 d, and 20.8 d (respectively) in these cases; it appears to be relatively independent of stellar mass, as indicated by spectral type.\",\n \"There also is a lack of evidence for binarity in most of the bursters, although in some cases limits from imaging are relatively shallow.\",\n \"We thus speculate that burst events are not triggered by any companion, but rather by a repetitive interaction between the stellar magnetic field and inner disk.\",\n \"From the theoretical perspective, magnetically channeled accretion is not predicted as steady in numerical simulations.\",\n \"An inner disk is truncated at a balance point between the inward pressure from accretion and the outward pressure of the magnetosphere.\",\n \"This produces variable and possibly cyclic mass flow due to instabilities and pulsational behavior.\",\n \"Such variations in the mass loading of accretion columns are modeled under different physical scenarios by, e.g., [98], [59], [140], [141], [142], and [45], [46].\",\n \"The disk corotation radius ($r_c$ ) and the magnetospheric radius ($r_m$ ) are critical in determining the regime of accretion under which a star-disk system falls.\",\n \"When $r_m0.7r_c$ , “stable” accretion occurs [20] and gas follows two funnel streams onto the star [144], [141].\",\n \"For smaller values of $r_m$ , the Rayleigh-Taylor instability sets in and accretion becomes chaotic, with many tongues of matter extending from the inner disk to the stellar surface [138], [91].\",\n \"Numerous hot spots are present on the stellar surface, and the associated light curves display irregular bursting.\",\n \"This regime may be responsible for the large subset of bursters that we observe with high duty cycles.\",\n \"When $r_m$ is up to a factor of two larger than $r_c$ , on the other hand, gas in the inner disk rotates slower than the magnetosphere and is accelerated azimuthally.\",\n \"Models [46] predict a “trapped disk\\\" regime in which relatively continuous accretion bursts occur on short timescales.\",\n \"When $r_m>>r_c$ , material tends to be flung out azimuthally in what is known as the “strong propeller\\\" regime [139].\",\n \"The light curves of several of our objects (EPIC 203954898/2MASS JJ16263682-2415518 and EPIC 204347422/2MASS J16195140-2241266) strongly resemble the simulated mass flow variations predicted by [167] and [92] for propeller behavior.\",\n \"In this scenario, the matter accretes episodically in three phases: First, material accumulates at the inner disk boundary, unable to flow inward.\",\n \"The magnetosphere is compressed toward the stellar surface.\",\n \"Next, compression reaches the point at which the magnetosphere can no longer withstand the gravitational forces on the accumulated material; gas accretes rapidly onto the star in a funnel flow.\",\n \"Finally, the compression pressure is relieved, accretion ceases, and the magnetosphere is able to expand outward again.\",\n \"Following this sequence, the cycle repeats.\",\n \"The time between bursts is longer than in other regimes, and the accretion rate can follow a flare-like pattern (rapid rise; slower decline) as a function of time.\",\n \"Strong outflows may also be present.\",\n \"EPIC 203382255 (2MASS 16144265-2619421) shows outflow signatures and also has the longest repeat timescale, making it an additional propeller regime candidate.\",\n \"In contrast, a few other objects with outflow-related spectral lines do not display the expected alternating burst and quiescence pattern.\",\n \"Further investigation of these targets is necessary to estimate magnetic and corotation radii and compare with the theoretical expectations.\",\n \"The ratio $r_m/r_c$ depends on mass accretion rate, magnetic field strength, and stellar spin period– parameters that we are unable to determine independently for this sample of burster stars.\",\n \"In the meantime, the burst timescales measured for our sample provide perhaps the best indication of the physical mechanisms at work.\",\n \"Duty cycles range from 10% and all the way up to nearly 100%– suggesting that we are seeing different modes of accretion from continuous to episodic and less frequent.\",\n \"The possible correlation between burst duration and repeat timescale (Figure REF ) may imply a relationship between mass loading timescale and accretion rate.\",\n \"A major open question stemming from this work is what the origin of the day to (multi-)week repeat timescales that we observe is, and how it relates to young outbursting stars with much longer duty cycles (e.g., EXors and FUors).\",\n \"Prominent examples from the literature include V899 Mon, which has had repeated bursts separated by $\\\\sim $ 1 year [112], and V1647 Ori, which repeats at $\\\\sim $ 2 years [11].\",\n \"On even longer timescales, other EXor type stars tend to outburst once or twice per decade.\",\n \"Unlike the $K2$ objects, the longer among these time scales are thought to be related to the viscous time scale in the disk.\",\n \"Here, the material drains inward and must undergo replenishment before the next instability-driven episode can occur.\",\n \"It has been speculated that the frequency and amplitude of outbursts may be set by the accretion rate, with younger and higher amplitude outbursting objects accreting more rapidly [12].\",\n \"We would then expect accretion rates for our own sources to be somewhat lower.\",\n \"This does indeed seem to be the case, as the median accretion rate where available for bursters is 10$^{-8.1}$ $M_\\\\odot $ yr$^{-1}$ (for exact values, see the notes on individual objects in the Appendix).\",\n \"We have found (Section 4.2) that this increases one to two orders of magnitude during the most extreme $K2$ light curve peaks.\",\n \"Further spectroscopic measurements during times of definitive bursting may confirm these estimates.\",\n \"In summary, the $K2$ mission is providing an unprecedented view of the time domain properties of young stars, and showing that the optical photometric manifestations of accretion phenomena take on a wide variety of timescales and amplitudes.\",\n \"Follow-up observations, including spectroscopy, should be carried out to investigate changes in spectral emission and inner disk structure in the highest amplitude objects presented here.\",\n \"The work of A.M.C was supported by a NASA/NPP fellowship.\",\n \"We thank the referee for useful feedback that improved this paper.\",\n \"We acknowledge Luisa Rebull for calling our attention to one of the burst type objects that we had overlooked in our initial examination.\",\n \"Thanks also to Nic Scott for assistance with DSSI observations on the Gemini South telescope.\",\n \"This paper includes data collected by the $K2$ mission.\",\n \"Funding for the $K2$ mission is provided by the NASA Science Mission directorate.\",\n \"The spectroscopic data were obtained at the W.M.\",\n \"Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.\",\n \"The Observatory was made possible by the generous financial support of the W.M.\",\n \"Keck Foundation.\",\n \"These results are also based on observations obtained as part of the program GS-2016A-Q-64 at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil).\",\n \"This paper has utilized the SIMBAD and Vizier services, through which data from USNO-B, APASS, 2MASS, and WISE were collected.\"\n ],\n [\n \"Notes on individual objects\",\n \"Not all of our sample stars appear in the literature, but for those that have been studied previously, we highlight the key results here.\",\n \"We also incorporate information from our new speckle and spectroscopic observations.\"\n ],\n [\n \"EPIC 203382255 / 2MASS 16144265-2619421\",\n \"This source is cataloged by [95] as an astrometric and photometric member, but is not otherwise studied in the literature.\",\n \"Our spectra suggest a spectral type of M4-M5.5 depending on the spectral range (earlier at bluer wavelengths).\",\n \"Lithium is present with $W_{Li} = 0.30 A$ along with the H$\\\\alpha $ emission reported in Table , He1, weak NaD, and Ca2 are seen in emission.\",\n \"Broad [O1] is also present.\",\n \"The system is a clear accretion/outflow source.\"\n ],\n [\n \"EPIC 203725791 / 2MASS J16012902-2509069\",\n \"This star was first proposed as an Upper Sco member by [9], based on $RIZ$ photometry.\",\n \"It was only recently confirmed by [137], based on significant lithium absorption.\",\n \"No other data has been reported for this star.\",\n \"Our HIRES spectrum indicates a spectral type of M2 with lithium absorption strength $W_{Li}=0.3$ Å.\",\n \"Strong H$\\\\alpha $ emission is seen, as indicated in Table , along with lorentz-broadened He1.\",\n \"NaD, weak narrow Fe2 as well as Mg1 emission, two-component Ca2, and O1 emission.\",\n \"Very weak [O1] is also exhibited.\"\n ],\n [\n \"EPIC 203786695 / 2MASS J16245974-2456008\",\n \"This star, WSB 18, was first noted as part of Wilking et al.\",\n \"(1987)'s H$\\\\alpha $ emission survey of the the $\\\\rho $ Ophiuchi complex.\",\n \"Lithium absorption was detected by [52], confirming youth.\",\n \"It is a visual binary with a separation of 1.1 (138 AU) and a 0.49 flux ratio [131].\",\n \"Therefore, both components contribute to our $K2$ light curve.\",\n \"According to [27], the primary has a spectral type of M2, whereas the secondary is M2.5.\",\n \"Both show H$\\\\alpha $ in emission, but the primary does not appear to have a disk [104], whereas the secondary does.\",\n \"Further, [84] found that the primary itself is a double star with 0.1 separation.\",\n \"No detections of these sources at $>70$ $\\\\mu $ m were made with Herschel.\",\n \"In the submillimeter, [5] put an upper limit of 0.003 $M_\\\\odot $ on the dust mass surrounding this source, although it is unclear which components were included.\"\n ],\n [\n \"EPIC 203794605 / 2MASS J16302339-2454161\",\n \"According to [166], this star displayed an H$\\\\alpha $ emission intensity of 3 on a scale of 1 to 5 (where 1 is weak and 5 is strong).\",\n \"Simon et al.\",\n \"(1995) performed a direct imaging search for companions in the 0.005-10 separation range, but did not find anything.\",\n \"Neither were any companions detected in the Catalog of High Angular Resolution Measurements [134], its successor CHARM2 [135], or the [125] speckle imaging survey of $\\\\rho $ Oph.\",\n \"For the latter, no companions were detected down to 0.04 (0.14) times the stellar brightness at 0.5 (0.15) separation.\",\n \"This star was included in the Spitzer c2d legacy survey.\",\n \"Our HIRES spectrum is veiled, but consistent with a spectral type of M3.5-M5.\",\n \"Lithium is present with strength $W_{Li} = 0.32 Å$ .\",\n \"H$\\\\alpha $ emission as reported in Table is strong with several components and a blueward-displaced central absorption.\",\n \"Additional emission includes He1 with lorentzian wings, broad NaD, many multi-component Fe2 lines, strong Ca2 with multiple component and O1 8446.\",\n \"Among the outflow lines, only multi-component [O1] is seen.\"\n ],\n [\n \"EPIC 203822485 / 2MASS J16272297-2448071\",\n \"This star, WSB 49, is in the $\\\\rho $ Ophiuchus cluster.\",\n \"It was first detected as part of Wilking et al.\",\n \"(1987)'s H$\\\\alpha $ emission line survey.\",\n \"It is also an IRAS source [164].\",\n \"It was first discovered as an x-ray source with the ROSAT High Resolution Imager [64], and also noted XMM Newton observations by [118].\",\n \"[165] confirmed its youth via detection of lithium absorption.\",\n \"[52] estimate a mass of 0.17 $M_\\\\odot $ .\",\n \"The star was surveyed for multiplicity, but no companions were detected down to $>0.15$ separation at flux ratios of 0.06.\",\n \"An infrared excess is detected with Spitzer [53].\",\n \"[5] observed it in the submillimeter, classifying the disk as class II, with $<0.005$ $M_\\\\odot $ of material.\",\n \"In addition, the disk is detected at 70 $\\\\mu $ m with Herschel/PACS [126].\",\n \"It was listed as a long timescale near-infrared variable by [113], while [66] detected $\\\\sim $ 20% variations in the mid-infrared with Spitzer.\"\n ],\n [\n \"EPIC 203856109 / 2MASS J16095198-2440197\",\n \"There is no previous literature on this source.\",\n \"Our spectrum indicates a spectral type of M5-M5.5 with lithium at $W_{Li} = 0.61 Å$ .\",\n \"Moderate H$\\\\alpha $ emission, as indicated in Table 1, is present and has a multi-component profile.\",\n \"Weak, also multi-component, He1 is also present, but no other emission lines.\"\n ],\n [\n \"EPIC 203899786 / 2MASS J16252434-2429442\",\n \"This star in the $\\\\rho $ Oph cluster is also known as V852 Oph and SR 22.\",\n \"Its emission line spectrum was reported as early as the mid-20th century [157], with prominent hydrogen, Ca2, and Fe2 noted.\",\n \"[74] and [73] included it in their emission line star catalogs.\",\n \"It is a ROSAT x-ray source [32].\",\n \"Irregular variability was also reported early on, by [148] and [54].\",\n \"[4] label the object as class III based on its 2.2–10 $\\\\mu $ m slope, and despite its previous classification as a classical T Tauri star by [39].\",\n \"It was relabeled as class II by [62], based on their 1.1–2.4 $\\\\mu $ m survey.\",\n \"[5] confirmed the disk with submillimeter observations, detecting 0.002 $M_\\\\odot $ of material.\",\n \"Similarly, [107] report 2.05 $M_{\\\\rm Jup}$ of material, based on SCUBA-2 850 $\\\\mu $ m observations.\",\n \"[126] detected the disk at 70 $\\\\mu $ m with Herschel, and reported that there is a gap in the spectral energy distribution that suggests a hole.\",\n \"The source is evidently a single star.\",\n \"[125] ruled out any companions down to a flux ratio of 0.07 at a separation of 0.15, and [62] did not detect any radial velocity variations indicative of a spectroscopic binary.\",\n \"[33] reported no companions down to 10 mas and contrasts of several magnitudes.\"\n ],\n [\n \"EPIC 203905576 / 2MASS J16261886-2428196\",\n \"This star– better known as VSSG 1– boasts a disk that was first detected by IRAS [36].\",\n \"The mid-infrared slope, $\\\\alpha $ , is -0.4, making it a class II disk according to [4] (although Wilking et al.\",\n \"1989 earlier classified it as class I).\",\n \"[105] used SPEX to measure an $n_{2-25}$ spectral index of -1.26, confirming the class II categorization.\",\n \"They also reported a small 10 $\\\\mu $ m silicate feature.\",\n \"[6] have observed this source's disk with the Submillimeter Array at 0.87 mm and estimated a dust mass of 0.029 $M_\\\\odot $ , along with an accretion rate of $10^{-7}$ $M_\\\\odot $ yr$^{-1}$ .\",\n \"H$_2$ O was detected in the Spitzer IRS spectrum obtained by [120], along with HCN, C$_2$ H$_2$ and CO$_2$ .\",\n \"Salyk et al.\",\n \"(2011) confirm these detections and estimate a disk mass of 0.029 $M_\\\\odot $ .\",\n \"They infer a disk inclination of 53.\",\n \"Submillimeter observations with ATCA [133] resulted in an estimated disk outer radius of 100–300 AU and a much lower dust mass of $4.5\\\\times 10^{-5}$ –$1.9\\\\times 10^{-4}$ $M_\\\\odot $ .\",\n \"[7] looked for mid-infrared variability in this source and concluded that it is not an EXor candidate.\",\n \"[111] report a fairly high accretion rate of 10$^{-7.19}$ $M_\\\\odot $ yr$^{-1}$ , and a Pa$\\\\beta $ equivalent width of 8.9Å in emission.\",\n \"According to the work of [125], no companions are visible down to a flux ratio of 0.04 at a separation of 0.15.\",\n \"Our spectrum exhibits strong H$\\\\alpha $ as indicated in Table with a double-peaked broad profile, as well as strong and broad Ca2 triplet as well as O1 8446 emission.\",\n \"There is very little in the way of absorption, presumably due to heavy veiling, and spectral types from K7 to mid-M are plausible.\",\n \"For the same reason, aggravated by low signal-to-noise in this region of the spectrum, there is no lithium measurement.\"\n ],\n [\n \"EPIC 203905625 / 2MASS J16284527-2428190\",\n \"EPIC 203905625, also known as V853 Oph and SR 13, is a late-type star in $\\\\rho $ Ophiuchus with reported spectral types from M2 to M4 [165].\",\n \"Accretion signatures include H$\\\\alpha $ as well as calcium in emission.\",\n \"[147] first reported strong veiling in the star's spectrum.\",\n \"[23] found that the H$\\\\alpha $ emission is variable (30–48Å).\",\n \"[111] estimated an accretion rate of $10^{-8.31}$ $M_\\\\odot $ yr$^{-1}$ , from near-infrared spectra.\",\n \"They detected Pa$\\\\beta $ in emission, at an equivalent width of -1.7Å.\",\n \"X-rays from this object were first detected with Einstein [108].\",\n \"A disk around this star was also observed, with IRAS [36].\",\n \"[4] used 1.3mm observations to detect the disk; they classify it as class II, based on a 2.2–10 $\\\\mu $ m slope of -0.8.\",\n \"[120] detect H$_2$ O, OH, HCN, and C$_2$ H$_2$ in a Spitzer IRS spectrum of this target.\",\n \"[126] report Herschel detections at 70 through 500 $\\\\mu $ m; they classify the system as transitional, based on 12–24 $\\\\mu $ m data.\",\n \"The star is a multiple system, with a companion first detected at 0.4 separation via speckle imaging [58].\",\n \"[10] and [125] confirmed the 0.4 and a 0.238 flux ratio.\",\n \"[153] conducted an IR imaging survey that revealed a closer companion at separation 13 mas.\",\n \"[104] found that both the primary and its 0.4 companion have class II disks.\",\n \"[5] measured a total disk mass of 0.01 $M_\\\\odot $ using submillimeter observations, while [107] measured a similar 7.81 $M_J$ from SCUBA-2 850 $\\\\mu $ m observations.\",\n \"Variability in this object was initially reported by [147].\",\n \"[149] found optical variations from 12.6 to 14.7.\",\n \"[75] observed V magnitude fluctuations from 12.83 to 13.52, while [60] reported $V$ magnitudes between 12.61 and 13.87.\",\n \"Likewise, [152] conducted photometric monitoring, reporting a $V$ -band amplitude of 0.91 magnitudes.\",\n \"The light curves are too sparsely sampled to identify any morphological features, although they are classified as “irregular.\\\"\",\n \"The broadband K2 light curve exhibits 0.2 mag events, suggesting that amplitudes are higher at bluer wavelengths.\",\n \"This is confirmed by Herbst et al.\",\n \"(1994)'s estimate of d$U$ /d$V$ : 2.39.\"\n ],\n [\n \"EPIC 203913804 / 2MASS J16275558-2426179\",\n \"This target is also known as SR 10 and V2059 Oph.\",\n \"It originally appeared in Herbig & Rao's (1972) catalog of emission line stars.\",\n \"Both [148], and [90] list it among their variable star compilations; [149] labeled the variations as “irregular.\\\"\",\n \"H$\\\\alpha $ emission at 40Å was reported early on by [146].\",\n \"[8] also observed strong H$\\\\alpha $ , as well as He1 and Fe2 in emission.\",\n \"[166] noted this star in their H$\\\\alpha $ emission survey, and [165] again measured H$\\\\alpha $ in emission as well as lithium in absorption.\",\n \"[111] estimated the accretion rate to be 10$^{-7.95}$ $M_\\\\odot $ yr$^{-1}$ ; they detected Pa$\\\\beta $ emission at an equivalent width of -5.6Å.\",\n \"[109] report a slightly lower accretion rate of 10$^{-8.3}$ $M_\\\\odot $ yr$^{-1}$ .\",\n \"[153] conducted an imaging survey for binary companions at project separations from 0.005 to 10 down to a $K$ magnitude of 11.1.\",\n \"Likewise, [125] searched for companions at separations of 0.15 and 0.50 but did not identify any down to flux ratios of 0.04 and 0.02, respectively.\",\n \"[134], [135], [16], and [33] did not detect companions either, down to 10–20 mas separation at several magnitudes contrast.\",\n \"Thus this object appears to be a single star.\",\n \"EPIC 203913804/2MASS J16275558-2426179 is a ROSAT x-ray source [32].\",\n \"The star is also an IRAS source [164].\",\n \"[4] identified a class II circumstellar disk, and [67] later confirmed with Spitzer data.\",\n \"[5] studied the system in the submillimeter and deduced an upper limit on the disk mass of 0.005 $M_\\\\odot $ , based on the 1.3 mm flux.\",\n \"Similarly, [107] found an upper limit of 5.0284 $M_{\\\\rm Jup}$ .\",\n \"[126] detected the disk at 70 $\\\\mu $ m with Herschel.\"\n ],\n [\n \"EPIC 203928175 / 2MASS J16282333-2422405\",\n \"EPIC 203928175 is a star in the $\\\\rho $ Ophiuchus region that is also known as SR 20 W [157] and ROXC J162823.4.\",\n \"[165] report a spectral type of K5, along with variable H$\\\\alpha $ emission and an equivalent width of 35Å.\",\n \"It was surveyed for binarity by [125], but no companions were detected down to 0.08 times the stellar flux at a separation of 0.15, or 0.04 times the stellar flux at 0.50.\",\n \"[33] also did not find any companions down to 20 mas separation, at contrasts of 1–3 magnitudes.\",\n \"With DSSI, we do not make any detections outward of 0.1 at $\\\\Delta m\\\\sim $ 4.4 magnitudes (flux contrast $\\\\sim $ 0.02) in the 692 or 880 nm bands.\",\n \"The object is encircled by a class II disk, as reported by [53].\",\n \"The disk was subsequently detected at 70, 160, 250, 350, and 500 $\\\\mu $ m with Herschel by [126], who tentatively classified it as transitional.\"\n ],\n [\n \"EPIC 203935537 / 2MASS J16255615-2420481\",\n \"This star is also known as SR 4 and V2058 Oph.\",\n \"It has a long history of photometric and spectroscopic study, dating back to Struve & Rudkjøbing's (1949) publication of emission line stars.\",\n \"[74], [166], and [73] listed it in their catalogs of emission line stars.\",\n \"It has had a range of H$\\\\alpha $ equivalent widths measured from 84Å to 220Å as well as a low vsin$i$ of $\\\\sim $ 9 km s$^{-1}$ [22].\",\n \"[159] acquired blue spectra of this target, which revealed H($\\\\beta $ , $\\\\gamma $ , $\\\\delta $ ) and Ca in emission, as well as significant veiling at 4450Å.\",\n \"[51] measured the veiling factor, $r$ , to be $\\\\sim $ 1.5.\",\n \"[165] detected lithium absorption and H$\\\\alpha $ emission in this source.\",\n \"[130] classified the H$\\\\alpha $ emission line profile as type IIR, in which there are two peaks of similar height.\",\n \"The accretion rate is estimated by [111] to be a fairly high 10$^{-6.74}$ $M_\\\\odot $ yr$^{-1}$ , and the same authors detected Pa$\\\\beta $ in emission at an equivalent width of -19.0Å.\",\n \"As suggested by [117], the star may be the driver for a nearby Herbig Haro flow (HH 312) in the region.\",\n \"The star is a ROSAT x-ray source.\",\n \"It is also an IRAS point source [81], [162], [36].\",\n \"[21] also observed the class II disk with ISO.\",\n \"[5] detected the disk at 850 $\\\\mu $ m with SCUBA, and they estimated a mass of 0.004 $M_\\\\odot $ based on SED fitting.\",\n \"[107] used SCUBA-2 to measure a larger disk mass of 9.4 $M_{\\\\rm Jup}$ ($\\\\sim $ 0.009 $M_\\\\odot $ ).\",\n \"Spitzer/IRS observations revealed a 10 $\\\\mu $ m silicate feature, with a typical equivalent width of 2.29 $\\\\mu $ m (Furlan et al.\",\n \"2009).\",\n \"Interferometric data and modeling led to an inferred inner disk radius of 0.112 AU [51].\",\n \"[119] estimated a very similar inner ring radius of 0.118 AU from near-infrared interferometry.\",\n \"[6] observed the disk with the Submillimeter Array and found a centrally peaked morphology.\",\n \"Their modeling predicts an inner disk radius of 0.07 AU and an inclination of 39; they infer an accretion rate of 10$^{-6.8}$ $M_\\\\odot $ yr$^{-1}$ , consistent with previous values.\",\n \"Millimeter and submillimeter ATCA observations by [133] led to an outer disk radius of 100–300 AU and a dust mass of $\\\\sim $ 2$\\\\times $ 10$^{-5}$ $M_\\\\odot $ .\",\n \"The disk is also detected with Herschel at 160 and 250 $\\\\mu $ m [126].\",\n \"The object is a known variable, as originally reported by [148], [149] and [90].\",\n \"It was followed up by [75], who found variations in the $V$ band of 12.73–12.93 during over 4000 days of monitoring.\",\n \"The amplitude was larger at blue wavelengths, with a typical d$U$ /d$V$ of 2.4 magnitudes.\",\n \"[152] reported a $V$ -band amplitude of 0.41 magnitudes over both short (hour–day) and long (years) timescales, with no detectable periodicity.\",\n \"[60] monitored the star for over 7 years in the optical, finding a similar $V$ -magnitude range of 12.60–13.09.\",\n \"EPIC 203935537/2MASS J16255615-2420481 is, to the best of our knowledge, a single star.\",\n \"Ghez et al.\",\n \"(1993)'s speckle imaging campaign did not reveal any companions down to 0.1 (0.2), at a flux ratio of 17 (18).\",\n \"Neither did Simon et al.\",\n \"(1995)'s imaging survey, which was sensitive to separations of 0.005–1.\",\n \"Our own DSSI observations did not show any companions down to 0.1 separation at 4–5 magnitudes of contrast (flux ratio $\\\\sim $ 0.02) in the 692 and 880 nm bands.\",\n \"[125] searched for companions in high resolution imaging but did not find any down to a flux ratio of 0.05 at a separation of 0.15.\",\n \"Neither [106] nor [65] detected any radial velocity variations indicative of spectroscopic binary status.\",\n \"Further high resolution imaging [134], [135] failed to reveal companions.\"\n ],\n [\n \"EPIC 203954898 / 2MASS J16263682-2415518\",\n \"This object was the subject of a multiplicity survey by [125], but no companions were detected down to 0.05 (0.12) times the stellar brightness at 0.5 (0.15) separation.\",\n \"Likewise, [49] did not detect any companion, and reported that two faint stars observed at 6.1 and 6.3 separation [50] are likely background objects.\",\n \"The star is a known X-ray emitter [64], [82], [57] and has a significant infrared excess, as indicated by ISO observations [21], Spitzer data [53], [35] and the AllWISE catalog [41].\",\n \"The spectral index, $\\\\alpha $ , is 0.08, indicating a flat disk [53].\",\n \"A spectral type range of K7-M1 was reported by [52].\",\n \"Our own spectrum from the Palomar 200-inch telescope Double Spectrograph suggests K8.\",\n \"We adopt M0.\",\n \"[48] report an effective temperature of 3700$\\\\pm $ 56 K, consistent with this spectral type.\",\n \"They also find a $v$ sin$i$ of 27$\\\\pm $ 4.7 km s$^{-1}$ .\",\n \"The mass has been estimated to be 0.18 $M_\\\\odot $ [111], which is very low considering the temperature and youth of the object.\",\n \"We suspect that several groups have confused the source with a neighboring star, 2MASS J16263713-2415599, which is $\\\\sim $ 9 away.\",\n \"For example, [165] list the object name ROXRA22, a ROSAT X-ray source at RA=16:26:36.9, Dec=-24:15:53 [64].\",\n \"However, the [165] coordinates match the companion star at RA=16:26:37.1, Dec=-24:15:59.9, and the listed spectral type is later, at M5.\",\n \"The companion paper by [52] provides the same effective temperature, luminosity, and mass estimate, but an earlier spectral type range and lower extinction ($A_V$ =4.9).\",\n \"Because of these discrepancies, we derive our own spectral data, apart from the $v$ sin$i$ measurement.\",\n \"With the clear disk signatures, we can be confident that it is a young member of $\\\\rho $ Ophiuchus.\",\n \"[111] list an accretion rate of $<10^{-9.71}$ $M_\\\\odot $ yr$^{-1}$ , based on near-infrared spectroscopy.\",\n \"EPIC 203954898/2MASS J16263682-2415518 was monitored in the mid-infrared with Spitzer as part of the Young Stellar Object Variability project [128].\",\n \"[66] obtained 81 datapoints spread over 34 days at 3.6 $\\\\mu $ m. The mean magnitude in this band was 8.15, in line with previous brightness estimates [35], [53], and the standard deviation was 0.09 magnitudes.\",\n \"The star showed variability at the $\\\\sim $ 0.05-magnitude level for the first 15 days of monitoring, followed by a $\\\\sim $ 0.2-magnitude increase followed by a similar decrease over the remaining 20 days of observation.\",\n \"WISE data also display a $\\\\sim $ 0.3-magnitude drop in the W1 band from February to August 2010.\"\n ],\n [\n \"EPIC 204130613 / 2MASS J16145026-2332397\",\n \"This star, BV Sco, was listed as an irregular variable by [149].\",\n \"It has been erroneously classified as an RR Lyrae star in SIMBAD; this type of variability is inconsistent with the stochasticity seen in our $K2$ light curve.\",\n \"Like EPIC 204233955/2MASS J16072955-2308221, it was labeled by [97] as a photometric non-member of Upper Scorpius (based on $ZYJHK$ data), before it was re-classified as a strongly accreting member [96].\",\n \"It showed both H$\\\\alpha $ and He1 in emission (-108Å and -3.0Å equivalent widths, respectively).\",\n \"Furthermore, they detected the calcium triplet lines and forbidden O1 emission, suggesting outflows.\"\n ],\n [\n \"EPIC 204226548 / 2MASS J15582981-2310077\",\n \"Also known as USco CTIO 33, [9] first identified this star as a candidate Upper Sco member.\",\n \"[122] confirmed its youth via measurement of lithium absorption; they also detected strong H$\\\\alpha $ emission.\",\n \"With a spectral type of M3, it is estimated to be 0.36 $M_\\\\odot $ by [86].\",\n \"These authors also searched for spectroscopic and wide (1–30) binary companions, but did not find any.\",\n \"This star was detected as a ROSAT x-ray source (RX J155829.5-231026) by [151].\",\n \"[29] were the first to detect an infrared excess, at 8 and 16 $\\\\mu $ m. Cieza et al.\",\n \"(2008) labeled it a transition disk source, with a mass of $<1.5\\\\times 10^{-3}$ $M_\\\\odot $ worth of material based on submillimeter data.\",\n \"No millimeter flux was detected by [103].\",\n \"[31] report a submillimeter detection with ALMA, but the disk is unresolved.\",\n \"They constrain the dust mass to be $0.58\\\\pm 0.13$ $M_\\\\oplus $ .\",\n \"[43] measure significant H$\\\\alpha $ emission from a broad, flat-topped peak; their estimated accretion rate is $10^{-9.91}$ $M_\\\\odot $ yr$^{-1}$ .\",\n \"Similar values were derived by [42].\",\n \"Molecular gas is detected with Herschel/PACS (both C$_2$ H$_2$ and HCN) by [114].\",\n \"[102] estimate less than 0.9 $M_{Jup}$ worth of gas mass.\"\n ],\n [\n \"EPIC 204233955 / 2MASS J16072955-2308221\",\n \"EPIC 204233955 is a spectral type M3 low-mass star in the Upper Scorpius association [96].\",\n \"It was initially classified as a photometric non-member by [97], but [96] found it to be an accreting source with strong emission lines, including H$\\\\alpha $ , He1, and O1 forbidden lines.\",\n \"H$\\\\alpha $ and He1 equivalent widths are -150Å and -3.0Å, respectively.\",\n \"Mid-infrared data from the AllWISE survey [41] reveal a significant infrared excess in all bands, indicative of a disk.\",\n \"Other than the work of [97], [96], this star has not been studied in detail.\"\n ],\n [\n \"EPIC 204342099 / 2MASS J16153456-2242421\",\n \"EPIC 204342099, otherwise known as VV Sco, is a T Tauri star in the Upper Scorpius association.\",\n \"This object is also a known X-ray emitter from ROSAT observations [68], XMM Newton observations [93].\",\n \"[124] obtained low and medium resolution spectra of this star as part of an X-ray selected sample of candidate Upper Scorpius members.\",\n \"They reported a spectral type of M1 and confirmed its youth via lithium absorption.\",\n \"EPIC 204342099 is a known disk-bearing source, originally discovered with IRAS [81].\",\n \"It was studied in detail with Spitzer/IRS by [56], who list it under the id 16126-2235.\",\n \"They find a very strong 10 $\\\\mu $ m silicate feature.\",\n \"This star was first presented as a visual binary by [63]; the separation is 1.9.\",\n \"[86] measured a spectral type of M3.5 for the companion.\",\n \"It is not clear whether this object is a co-moving Upper Scorpius member or a serendipitous field object; if the former, then the separation is 274 AU.\",\n \"With DSSI speckle imaging, we measure the separation to be smaller at 1.50, with a magnitude difference of $\\\\Delta $ m=3.38 at the 880 nm band.\",\n \"[87] also searched for closer companions with direct imaging and aperture masking, but did not find any within 240 mas, at a magnitude difference of 2.8.\",\n \"Sparse variability data is available for EPIC 204342099/2MASS J16153456-2242421.\",\n \"The star was first noted as variable by [116].\",\n \"[17] monitored it for optical rotation signatures but did not detect any periodicities in the light curve.\",\n \"Our HIRES spectrum suggests a spectral type of K9-M0 with lithium absorption present at strength $W_{Li}=0.45$ Å.\",\n \"The H$\\\\alpha $ emission is consistent with accretion (Table ) and the profile exhibits a blueward asymmetry along with a redshifted absorption notch against the emission.\",\n \"Very weak and narrow profiles in He1, Fe2, and Ca2 are also present in our data, along with very weak and narrow [O1].\"\n ],\n [\n \"EPIC 204360807 / 2MASS J16215741-2238180\",\n \"There is no previous literature on this source.\",\n \"As with other objects under consideration here, there is significant veiling present with the spectral type changing from M2 in the bluer orders of our HIRES to possibly as late as M6 by about 8800 Å. Lithium has strength $W_{Li} = 0.27 A$ .\",\n \"H$\\\\alpha $ emission as reported in Table 1 is very strong and there is He1, broad NaD, and many Fe2 lines.\",\n \"The Ca2 triplet has multiple components and O1 8446 is present.\",\n \"Of the outflow lines only weak [O1] is seen.\",\n \"With our DSSI speckle observations, we identify a companion at 0.48 separation with $\\\\Delta $ m=0.74 at 880 nm.\"\n ],\n [\n \"EPIC 204397408 / 2MASS J16081081-2229428\",\n \"This object was first identified as a candidate USco member based on proper motion by [97] with [26] assigning it a membership probability of 99.9% based on the USNO-B proper motion.\",\n \"[154] confirmed youth via variability and spectroscopy, assigning a spectral type of M5.\",\n \"They noted the star as an active accretor, based on a strong H$\\\\alpha $ emission line.\",\n \"Likewise, [96] also spectroscopically confirmed this object as a USco member.\",\n \"[44] measured a radial velocity of about -11 km/s, slightly lower than the cluster mean, and a rotation velocity of $v\\\\sin i=16.52\\\\pm 4.05$ .\",\n \"[132] analyzed WISE photometry for EPIC 204397408/2MASS J16081081-2229428, concluding that it harbors a class II disk.\",\n \"[99] also labeled it as a full disk.\"\n ],\n [\n \"EPIC 204440603 / 2MASS J16142312-2219338\",\n \"This very low mass star was classified by [97] as a photometric and proper motion member of Upper Scorpius based on UKIDSS data.\",\n \"Following up with the Anglo-Australian Telecope AAOmega spectrograph, [96] obtained intermediate-resolution spectra of EPIC 204440603/2MASS J16142312-2219338 from 5750 to 8800Å, deriving a spectral type of M5.75 and H$\\\\alpha $ equivalent width of -94.5Å.\",\n \"Their measured Na I and K I gravity-sensitive equivalent widths as well as detection of Li I absorption cements the classification of this object as a young low-mass star.\"\n ],\n [\n \"EPIC 204830786 / 2MASS J16075796-2040087\",\n \"This Upper Scorpius member was first identified as a strong H$\\\\alpha $ emission-line star by [158].\",\n \"[88] obtained low-resolution spectra, which confirmed H$\\\\alpha $ emission at an equivalent width of -357Å and Ca2 triplet emission as well.\",\n \"Detection of further emission lines (N2, S2, Fe2, Ni2, O1, and the Paschen series) suggested accretion-driven jets.\",\n \"These authors also associated EPIC 204830786/2MASS J16075796-2040087 with a wide-separation companion some 21.5 (3120 AU) away.\",\n \"It has a significant infrared excess, as shown with IRAS [28] and later with Spitzer and WISE by [99].\",\n \"Our HIRES spectrum suggests a spectral type of late G to early K but the spectrum is clearly heavily veiled; the lithium strength is $W_{Li}=0.20$ Å.\",\n \"Strong H$\\\\alpha $ emission is seen, as indicated in Table , with a blue-side absorption notch in the profile.\",\n \"Strong and broad He1, NaD, Fe2, Ca2, O1 8446, and perhaps other emission is present.\",\n \"Strong multi-component forbidden emission lines of [O1] and [S2] are seen, along with single-component [N2] and many [Fe2] lines.\",\n \"Our DSSI speckle observations do not identify any companions outward of 0.1 from this star, at a magnitude difference of $\\\\Delta m\\\\sim 4$ in the 692 and 880 nm bands.\"\n ],\n [\n \"EPIC 204906020 / 2MASS J16070211-2019387\",\n \"[123] detected EPIC 204906020 as a youthful member of the Upper Sco association via spectroscopic measurement of lithium absorption.\",\n \"This M5 star also has H$\\\\alpha $ emission, although it was weak enough in some observations to lead to a weak-lined T Tauri star classification.\",\n \"[86] reported the object to be an M5/M5.5 wide binary with a 1.63 separation.\",\n \"[88] found that the primary is itself a binary, with 55 mas projected separation (8 AU at the distance of Upper Sco).\",\n \"[29] reported infrared excesses at 8 and 16 $\\\\mu $ m, indicating a disk.\",\n \"[30] also detected an excess at 24 $\\\\mu $ m with Spitzer/MIPS.\",\n \"[103] did not detect any cool dust around this system at millimeter wavelengths, at a 3-$\\\\sigma $ upper limit of 3.7$\\\\times 10^{-3}$ $M_{\\\\rm Jup}$ .\",\n \"However, [102] used Herschel/PACS to detect 6.6$\\\\times 10^{-6}$ $M_\\\\odot $ worth of dust.\",\n \"[31] detected but did not resolve the disk with ALMA.\",\n \"EPIC 204906020/2MASS J16142312-2219338 is also known to have a circumstellar disk, as first reported by [132] and confirmed by [99].\",\n \"The SED slope is -1.3, making it a class II disk [132].\",\n \"Our speckle observations with DSSI rule out any companions beyond 0.1 at 4.6 magnitudes contrast in the 692 and 880 nm bands.\"\n ],\n [\n \"EPIC 204908189 / 2MASS J16111330-2019029\",\n \"This source appears in the literature only in the [99] $WISE$ sample and the [15] ALMA study.\",\n \"Our HIRES spectrum suggests a spectral type of M1 with lithium absorption present at strength $W_{Li}=0.15$ Å.\",\n \"Strong H$\\\\alpha $ emission is seen, as indicated in Table , along with He1, weak but broad NaD, weak and narrow Fe2, weak and narrow Ca2, but moderately broad O1 8446 emission.\",\n \"Weak and narrow [O1] is also present.\"\n ],\n [\n \"EPIC 205008727 / 2MASS J16193570-1950426\",\n \"There is no previous literature on this source.\",\n \"The HIRES spectrum is heavily veiled but appears to be a late K to M3 type, with lithium present at strength $W_{Li} = 0.55 A$ .\",\n \"H$\\\\alpha $ emission as reported in Table is strong and has multiple components.\",\n \"Additional emission includes He1, NaD, Fe2, Ca2 with the same profile shape as the H$\\\\alpha $ and O1 8446 is present.\",\n \"Outflow lines of [O1], [N2], and [S2] are also present, along with [Fe2].\"\n ],\n [\n \"EPIC 205061092 / 2MASS J16145178-1935402\",\n \"There is no previous literature on this source.\",\n \"Our HIRES data indicate a spectral type of M5-M6 with lithium at $W_{Li} = 0.59 A$ .\",\n \"Strong H$\\\\alpha $ emission, as indicated in Table 1, is present along with He1, but no other emission lines.\"\n ],\n [\n \"EPIC 205088645 / 2MASS J16111237-1927374\",\n \"[122] first identified this star as an M5 member of the USco association, based on lithium absorption and broad H$\\\\alpha $ emission (-50Å).\",\n \"[26] assigned it a membership probability of 99.9% based on the USNO-B proper motion.\",\n \"[101] found a slightly later spectral type of M6 based on low-resolution spectra, and similarly broad H$\\\\alpha $ emission.\",\n \"The object displays an infrared excess confirmed by WISE to come from a full disk [99].\"\n ],\n [\n \"EPIC 205156547 / 2MASS J16121242-1907191\",\n \"There is no previous literature on this source.\",\n \"Our spectrum indicates a spectral type of M5-M6 with lithium at $W_{Li} = 0.56 A$ .\",\n \"Moderate H$\\\\alpha $ emission, as indicated in Table 1, is apparent with with an asymmetric extension on the blue side of the profile.\",\n \"There is also He1 but no other emission lines.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05599"}}},{"rowIdx":3397,"cells":{"text":{"kind":"list like","value":[["A Complete Characterization of Pretty Good State Transfer on Paths"],["Abstract We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian.","If $n$ is the length of the path, and the vertices are indexed by the positive integers from 1 to $n$, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices $a$ and $b$ are that (a) $a + b = n + 1$, (b) $n + 1$ has at most one odd non-trivial divisor, and (c) if $n = 2^t r - 1$, for $r$ odd and $r \\neq 1$, then $a$ is a multiple of $2^{t - 1}$."],["Introduction and Preliminaries","Many quantum algorithms may be modelled as a quantum process occurring on a graph.","In [1], Childs shows that any quantum computation can be encoded as a quantum walk in some graph and thus quantum walks can be regarded as a quantum computation primitive.","The protocol for quantum communication through unmeasured and unmodulated spin chains was presented by Bose [2], and led to the interpretation of quantum channels implemented by spin chains as wires for transmission of states.","We model such a spin chain of $n$ interacting qubits by the graph of a path of $n$ vertices, denoted $P_n$ , with the vertices labelled from 1 to $n$ corresponding to qubits and the edges $\\lbrace i, i + 1\\rbrace $ , $1 \\le i < n$ corresponding to their interactions.","These interactions are defined by a time-independent Hamiltonian; that is, a symmetric matrix that acts on the Hilbert space of dimension $2^n$ .","We are concerned here with the $XY$ -Hamiltonian, whose action on the 1-excitation subspaces is equivalent to the action of the 01-symmetric adjacency matrix of $P_n$ on $n$ , and whose process evolves according to the transition matrix $U(t) := \\exp (itA)$ .","To be more specific, we consider the Hamiltonian $ H = \\sum _{a = 1}^{n - 1} \\sigma ^x_{a \\ a + 1}\\sigma ^y_{a \\ a + 1}, $ where $\\sigma ^x_{a \\ a + 1}$ and $\\sigma ^y_{a \\ a + 1}$ are the operators that apply the Pauli matrices $\\sigma ^x$ and $\\sigma ^y$ at the qubits located at vertices $a$ and $a + 1$ , and act as the identity at all other qubits.","The sum is over all pairs of vertices that are edges of the underlying graph.","Consequently, if $|u\\rangle $ stands for the system state in which the qubit at vertex $u$ is at $|1\\rangle $ and all others are at $|0\\rangle $ , then $ H |a\\rangle = {\\left\\lbrace \\begin{array}{ll}| 2 \\rangle , & a = 1 \\\\| n - 1 \\rangle , & a = n \\\\| a - 1 \\rangle + | a + 1 \\rangle , & a \\notin \\lbrace 1, n\\rbrace .\\end{array}\\right.}","$ Because of this, the action of $H$ on the set $\\lbrace |a\\rangle : a \\in V(G)\\rbrace $ is equivalent to the action of the adjacency matrix $A(P_n)$ on the canonical basis of $n$ .","In other words, $H$ can be block diagonalized, and one of its blocks is equal to $A(P_n)$ .","The subspace spanned by $\\lbrace |a\\rangle : a \\in V(G)\\rbrace $ is the 1-excitation subspace of the Hilbert space, and the quantum dynamics restricted to this subspace corresponds to the scenario where one qubit, say the one at $a$ , is initialized at $|1\\rangle $ and all others at $|0\\rangle $ .","Due to Schrödinger's equation $ |\\langle a |\\exp (t H) | b \\rangle |^2 $ indicates the probability that the state $|1\\rangle $ is measured at $b$ after time $t$ .","We are concerned solely with the 1-excitation subspace.","Let $_u$ denote the vector of the canonical basis of ${n}$ that is 1 at the entry corresponding to vertex $u$ at the ordering of the rows and columns of $A(G)$ .","From the remarks above, if the system is initialized with state $|1\\rangle $ at vertex $a$ and all others at $|0\\rangle $ , it follows that $ |_a^T \\exp (t A) _b|^2 $ indicates the probability that the state $|1\\rangle $ is measured at $b$ after time $t$ .","One of the major goals of quantum communication on spin chains is to transfer a state with high fidelity.","At the maximum fidelity of 1, we say we have achieved perfect state transfer.","Analogously, we say we have perfect state transfer between vertices $a$ and $b$ if there exists a time $\\tau $ such that $|| _a^T U(\\tau ) _b|| = 1$ .","The concept of perfect state transfer was first introduced by Christandl et al.","[3], who also showed perfect state transfer is only possible on spin chains of two or three qubits.","However, even without perfect state transfer, the fidelity may be quite high, and the notion of pretty good state transfer was isolated by Godsil [4] as a relaxation of perfect state transfer.","Formally, we say there is pretty good state transfer if, for any $\\epsilon > 0$ , there exists a time $\\tau $ such that the fidelity is $1 - \\epsilon $ .","Analogously, we say we have pretty good state transfer between vertices $a$ and $b$ if, for any $\\epsilon > 0$ , there exists a time $\\tau $ such that $|| U(\\tau )_{a,b} || > 1 - \\epsilon $ , or equivalently, if for any $\\epsilon > 0$ , there exists a $\\lambda \\in , $ || = 1$, such that $ || U() a - b || < $; this condition is abbreviated to $ U() a b$ for convenience.","Godsil et al.~\\cite {GKSS12} demonstrated the following result.$ [5] Pretty good state transfer occurs between the end vertices of $P_n$ if and only if $n = p - 1, 2p - 1$ , where $p$ is a prime, or $n = 2^m - 1$ .","Moreover, when pretty good state transfer occurs between the end vertices of $P_n$ , then it occurs between vertices $a$ and $n + 1 - a$ for all $a \\ne (n + 1) /2$ .","Banchi et al.","[6] showed that pretty good state transfer occurs between the $j$ th and $(n - j +1)$ -th vertices of spin chains with $XYZ$ -Hamiltonian, whose action on the 1-exication subspaces is equivalent to the action of the Laplacian adjacency matrix on $n$ , if the number of vertices is a power of 2, and that the condition is necessary for $j = 1$ , but possibly not for other values of $j$ .","Moreover, they present the open question of pretty good state transfer between inner vertices with $XY$ -Hamiltonian.","Coutinho, Guo, and van Bommel [7] investigated this question and determined the following infinite family of paths that admit pretty good state transfer between inner vertices but not between the two end vertices.","[7] Given any odd prime $p$ and positive integer $t$ , there is pretty good state transfer in $P_{2^t p -1}$ between vertices $a$ and $2^t p -a$ , whenever $a$ is a multiple of $2^{t-1}$ .","In this paper, we present necessary and sufficient conditions for pretty good state transfer between any two vertices of a path, demonstrating that the above result is the only family of paths that admit pretty good state transfer between inner vertices but not between the two end vertices.","In the next section, we present definitions and preliminary results that will be required to prove this characterization."],["Preliminaries","If $M$ is a symmetric matrix with $d$ distinct eigenvalues $\\theta _1 > \\theta _2 > \\cdots > \\theta _d$ , then the spectral decomposition of $M$ is $ M = \\sum _{j = 1}^d \\theta _j E_j, $ where $E_r$ denotes the idempotent projection onto the eigenspace corresponding to $\\theta _r$ .","If $a \\in V(G)$ , then the eigenvalue support of $a$ is the following subset of the eigenvalues: $ \\Theta _a = \\lbrace \\theta _j : E_j _a \\ne 0\\rbrace .","$ We say that vertices $a$ and $b$ are strongly cospectral if $E_r _a = \\pm E_r _b$ for all $r$ .","The following result is given by Banchi et al. [6].","[6] If pretty good state transfer occurs between $a$ and $b$ , then they are strongly cospectral vertices.","The spectrum of the adjacency matrix of $P_n$ (see [8] for example) is $ \\theta _j = 2 \\cos \\frac{\\pi j}{n + 1}, \\quad 1 \\le j \\le n, $ and the eigenvector corresponding to $\\theta _j$ is given by $ (\\beta _1, \\beta _2, \\ldots , \\beta _k), \\mbox{ where } \\beta _k = \\sin \\frac{k \\pi j}{n + 1}.","$ As stated by Coutinho, Guo, and van Bommel, the following lemma immediately follows.","[7] Vertices $a$ and $b$ of $P_n$ are strongly cospectral if and only if $a + b = n + 1$ .","Moreover, we observe that when $a + b = n + 1$ , $E_r _a = E_r _b$ when $r$ is odd, and $E_r _a = - E_r _b$ when $r$ is even.","We will derive our next result, which gives a sufficient condition to show pretty good state transfer does not occur between a given pair of vertices of $P_n$ , from Kronecker's Theorem, stated below.","[Kronecker, see [9]] Let $\\theta _0, \\ldots , \\theta _d$ and $\\zeta _0, \\ldots , \\zeta _d$ be arbitrary real numbers.","For an arbitrarily small $\\epsilon $ , the system of inequalities $ | \\theta _r y - \\zeta _r | < \\epsilon \\pmod {2 \\pi }, \\quad (r = 0, \\ldots , d), $ admits a solution for $y$ if and only if, for integers $l_0, \\ldots , l_d$ , if $ \\ell _0 \\theta _0 + \\cdots + \\ell _d \\theta _d = 0, $ then $ \\ell _0 \\zeta _0 + \\cdots + \\ell _d \\zeta _d \\equiv 0 \\pmod {2 \\pi }.","$ Let $a$ and $b$ be vertices of $P_n$ such that $a + b = n + 1$ .","If there is a set of integers $\\lbrace \\ell _j : \\theta _j \\in \\Theta _a,\\ j \\mbox{ odd}\\rbrace $ such that $ \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ odd}\\end{array}} \\ell _j \\theta _j = 0 \\quad \\mbox{and} \\quad \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ odd}\\end{array}} \\ell _j \\mbox{ is odd} $ and there is a set of integers $\\lbrace \\ell _j : \\theta _j \\in \\Theta _a,\\ j \\mbox{ even}\\rbrace $ such that $ \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ even}\\end{array}} \\ell _j \\theta _j = 0 \\quad \\mbox{and} \\quad \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ even}\\end{array}} \\ell _j \\mbox{ is odd} $ then pretty good state transfer does not occur between vertices $a$ and $b$ .","Proof: Suppose pretty good state transfer occurs between vertices $a$ and $b$ .","We see that the condition $U(\\tau ) _a \\approx \\lambda _b$ is equivalent to $ e^{i \\theta _r \\tau } E_r _a \\approx \\lambda E_r _b$ for all $r$ .","Writing $\\lambda = e^{i \\delta }$ , this condition is equivalent to $\\theta _r \\tau \\approx \\delta + q_r \\pi $ for all $r$ such that $\\theta _r \\in \\Theta _a$ , where $q_r$ is even when $r$ is odd and $q_r$ is odd when $r$ is even.","So, we wish to solve the system of inequalities $ |\\theta _r \\tau - (\\delta + \\sigma _r \\pi ) | < \\epsilon \\pmod {2 \\pi }, \\quad (r : \\theta _r \\in \\Theta _a), $ where $\\sigma _r = 0$ when $r$ is odd and $\\sigma _r = 1$ when $r$ is even.","Hence, by Kronecker's Theorem, if $ \\sum _{\\theta _r \\in \\Theta _a} \\ell _r \\theta _r = 0, $ then $ \\sum _{\\theta _r \\in \\Theta _a} \\ell _r (\\delta + \\sigma _r \\pi ) \\equiv 0 \\pmod {2 \\pi }.","$ Now, since there is a set of integers $\\lbrace \\ell _j : \\theta _j \\in \\Theta _a,\\ j \\mbox{ odd}\\rbrace $ such that $ \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ odd}\\end{array}} \\ell _j \\theta _j = 0 \\quad \\mbox{and} \\quad \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ odd}\\end{array}} \\ell _j \\mbox{ is odd} $ then we must have $ \\delta \\equiv 0 \\pmod {2 \\pi }.","$ However, since there is a set of integers $\\lbrace \\ell _j : \\theta _j \\in \\Theta _a,\\ j \\mbox{ even}\\rbrace $ such that $ \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ even}\\end{array}} \\ell _j \\theta _j = 0 \\quad \\mbox{and} \\quad \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ even}\\end{array}} \\ell _j \\mbox{ is odd} $ then we must also have $ \\delta \\equiv \\pi \\pmod {2 \\pi }, $ which is the contradiction completing the proof ."],["Necessary and Sufficient Conditions","We first state the following results about sums of cosines which we will use in the proof of the main theorem.","Their proofs are included for completeness.","The first result uses the fact that $2 \\cos x = e^{i x} + e^{-i x}$ , and then we sum the resulting geometric series.","Let $q$ be an odd integer.","Then $ 2 \\sum _{k = 1}^{\\frac{q - 1}{2}} (-1)^k \\cos \\left( \\frac{k \\pi }{q} \\right) + 1 = 0.","$ Proof: $& 2 \\sum _{k = 1}^{\\frac{q - 1}{2}} (-1)^k \\cos \\left( \\frac{k \\pi }{q} \\right) + 1 \\\\&= \\sum _{k = 1}^{\\frac{q - 1}{2}} (-1)^k (e^{\\frac{i k \\pi }{q}} + e^{\\frac{- i k \\pi }{q}}) + 1 \\\\&= \\sum _{k = 1}^{\\frac{q - 1}{2}} (-e^{\\frac{i \\pi }{q}})^k + \\sum _{k = 1}^{\\frac{q - 1}{2}} (-e^{-\\frac{i \\pi }{q}})^k + 1 \\\\&= \\frac{-e^{\\frac{i \\pi }{q}} + (-e^{\\frac{i \\pi }{q}})^{\\frac{q + 1}{2}}}{1 + e^{\\frac{i \\pi }{q}}} + \\frac{(-e^{\\frac{i \\pi }{q}})^{-1} + (-e^{\\frac{i \\pi }{q}})^{-\\frac{q + 1}{2}}}{1 + e^{-\\frac{i \\pi }{q}}} + 1 \\\\&= \\frac{-e^{\\frac{i \\pi }{q}} + (-e^{\\frac{i \\pi }{q}})^{\\frac{q + 1}{2}}}{1 + e^{\\frac{i \\pi }{q}}} - \\frac{1+ (-e^{\\frac{i \\pi }{q}})^{-\\frac{q - 1}{2}}}{1 + e^{\\frac{i \\pi }{q}}} + \\frac{1 + e^{\\frac{i \\pi }{q}}}{1 + e^{\\frac{i \\pi }{q}}} \\\\&= \\frac{(-e^{\\frac{i \\pi }{q}})^{\\frac{q + 1}{2}} - (-e^{\\frac{i \\pi }{q}})^{-\\frac{q - 1}{2}}}{1 + e^{\\frac{i \\pi }{q}}} \\\\&= 0 \\square $ The second result makes use of the following sum-product identity: $ \\cos \\alpha + \\cos \\beta = 2 \\cos \\frac{\\alpha + \\beta }{2} \\cos \\frac{\\alpha - \\beta }{2} .","$ The final step is an application of the previous lemma.","Let $n = km$ , where $m$ is an odd integer, and $0 \\le a < k$ be an integer.","Then $ \\sum _{j = 0}^{m - 1} (-1)^j \\cos \\left( \\frac{(a + jk) \\pi }{n} \\right) = 0.","$ Proof: $& \\sum _{j = 0}^{m - 1} (-1)^j \\cos \\left( \\frac{(a + jk) \\pi }{n} \\right) \\\\&= (-1)^{\\frac{m - 1}{2}} \\left[ \\cos \\left( \\frac{\\left(a + \\frac{m - 1}{2} k \\right) \\pi }{n} \\right) \\right.","\\\\ &\\quad \\left.+ \\sum _{j = 1}^{\\frac{m - 1}{2}} (-1)^j \\left(\\cos \\left( \\frac{\\left(a + (\\frac{m - 1}{2} - j) k \\right) \\pi }{n} \\right) + \\cos \\left( \\frac{\\left(a + (\\frac{m - 1}{2} + j) k \\right) \\pi }{n} \\right) \\right) \\right] \\\\&= (-1)^{\\frac{m - 1}{2}} \\left[ \\cos \\left( \\frac{\\left(a + \\frac{m - 1}{2} k \\right) \\pi }{n} \\right) \\right.","\\\\ &\\quad \\left.+ \\sum _{j = 1}^{\\frac{m - 1}{2}} (-1)^j 2 \\cos \\left( \\frac{\\left(a + \\frac{m - 1}{2} k \\right) \\pi }{n} \\right) \\cos \\left( \\frac{j k \\pi }{n} \\right) \\right] \\\\&= (-1)^{\\frac{m - 1}{2}} \\cos \\left( \\frac{\\left(a + \\frac{m - 1}{2} k \\right) \\pi }{n} \\right) \\left( 1 + 2 \\sum _{j = 1}^{\\frac{m - 1}{2}} (-1)^j \\cos \\left( \\frac{j \\pi }{m} \\right) \\right) \\\\&= 0 \\square $ We will now state and prove the main theorem.","There is pretty good state transfer on $P_n$ between vertices $a$ and $b$ if and only if $a + b = n + 1$ and either: $n = 2^t - 1$ , where $t$ is a positive integer; or, $n = 2^t p - 1$ , where $t$ is a nonnegative integer and $p$ is an odd prime, and $a$ is a multiple of $2^{t - 1}$ .","Proof: The sufficiency of the conditions is given by Theorem and Theorem .","It remains to show that the conditions are necessary.","The necessity of the first condition follows from Lemma and Lemma .","Henceforth, we need only consider the possibility of pretty good state transfer between vertices $a$ and $n + 1 - a$ .","Suppose first that there is pretty good state transfer on $P_n$ between vertices $a$ and $n + 1 - a$ when $n = 2^t r - 1$ , where $t$ is a positive integer and $r$ is an odd composite number.","Let $p$ be a prime factor of $r$ .","If $p \\mid \\frac{2^t r}{\\gcd (a, 2^t r)}$ , then if $\\theta _k \\notin \\Theta _a$ , then $k$ is a multiple of $p$ .","But then, for $c \\in \\lbrace 1, 2\\rbrace $ , we have $ \\sum _{i = 0}^{r/p - 1} (-1)^i \\theta _{c + i 2^t p} = \\sum _{i = 0}^{r/p - 1} (-1)^i \\cos \\left( \\frac{(c + i 2^t p) \\pi }{n + 1} \\right) = 0 $ by Lemma .","Hence, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , a contradiction.","Hence, we can now assume that $r \\mid a$ .","Since it follows from condition (a) that $a \\ne 2^{t-1} r$ , we have that $t \\ge 2$ and $4 \\mid \\frac{2^t r}{\\gcd (a, 2^t r)}$ , so if $\\theta _k \\notin \\Theta _a$ , then $k$ is a multiple of 4.","But then, for $c \\in \\lbrace 1, 2\\rbrace $ , we have $ \\sum _{i = 0}^{r - 1} (-1)^i \\theta _{c + i 2^t} = \\sum _{i = 0}^{r - 1} (-1)^i \\cos \\left( \\frac{(c + i 2^t) \\pi }{n + 1} \\right) = 0 $ by Lemma .","Hence, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , a contradiction.","Now suppose that there is pretty good state transfer on $P_n$ between vertices $a$ and $n + 1 - a$ when $n = r - 1$ , where $r$ is an odd composite number.","Let $p$ be a prime factor of $r$ .","If $p \\mid \\frac{2^t r}{\\gcd (a, 2^t r)}$ , then if $\\theta _k \\notin \\Theta _a$ , then $k$ is a multiple of $p$ .","But then, for $c \\in \\lbrace 1, 2\\rbrace $ , we have $&\\sum _{i = 0}^{\\frac{1}{2} (r/p - 1)} \\theta _{c + i 2 p} + \\sum _{i = 0}^{\\frac{1}{2} (r/p - 1) - 1} \\theta _{n + 1 - (c + p + i 2 p)} \\\\&= \\sum _{i = 0}^{r/p - 1} (-1)^i \\theta _{c + i p} = \\sum _{i = 0}^{r/p - 1} (-1)^i \\cos \\left( \\frac{(c + i p) \\pi }{n + 1} \\right) = 0$ by Lemma .","Hence, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , a contradiction.","Hence, we have shown the necessity of the conditions on $n$ .","It remains to show the necessity of the conditions on $a$ when $n = 2^t p - 1$ , where $t$ is a positive integer and $p$ is an odd prime.","Suppose $a$ is not a multiple of $2^{t - 1}$ .","Then again we have that if $\\theta _k \\notin \\Theta _a$ , then $k$ is a multiple of 4.","So as above, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , which is the contradiction completing the proof .","Acknowledgements The author thanks Chris Godsil for his guidance and support and Gabriel Coutinho for his introduction to this problem.","References"]],"string":"[\n [\n \"A Complete Characterization of Pretty Good State Transfer on Paths\"\n ],\n [\n \"Abstract We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian.\",\n \"If $n$ is the length of the path, and the vertices are indexed by the positive integers from 1 to $n$, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices $a$ and $b$ are that (a) $a + b = n + 1$, (b) $n + 1$ has at most one odd non-trivial divisor, and (c) if $n = 2^t r - 1$, for $r$ odd and $r \\\\neq 1$, then $a$ is a multiple of $2^{t - 1}$.\"\n ],\n [\n \"Introduction and Preliminaries\",\n \"Many quantum algorithms may be modelled as a quantum process occurring on a graph.\",\n \"In [1], Childs shows that any quantum computation can be encoded as a quantum walk in some graph and thus quantum walks can be regarded as a quantum computation primitive.\",\n \"The protocol for quantum communication through unmeasured and unmodulated spin chains was presented by Bose [2], and led to the interpretation of quantum channels implemented by spin chains as wires for transmission of states.\",\n \"We model such a spin chain of $n$ interacting qubits by the graph of a path of $n$ vertices, denoted $P_n$ , with the vertices labelled from 1 to $n$ corresponding to qubits and the edges $\\\\lbrace i, i + 1\\\\rbrace $ , $1 \\\\le i < n$ corresponding to their interactions.\",\n \"These interactions are defined by a time-independent Hamiltonian; that is, a symmetric matrix that acts on the Hilbert space of dimension $2^n$ .\",\n \"We are concerned here with the $XY$ -Hamiltonian, whose action on the 1-excitation subspaces is equivalent to the action of the 01-symmetric adjacency matrix of $P_n$ on $n$ , and whose process evolves according to the transition matrix $U(t) := \\\\exp (itA)$ .\",\n \"To be more specific, we consider the Hamiltonian $ H = \\\\sum _{a = 1}^{n - 1} \\\\sigma ^x_{a \\\\ a + 1}\\\\sigma ^y_{a \\\\ a + 1}, $ where $\\\\sigma ^x_{a \\\\ a + 1}$ and $\\\\sigma ^y_{a \\\\ a + 1}$ are the operators that apply the Pauli matrices $\\\\sigma ^x$ and $\\\\sigma ^y$ at the qubits located at vertices $a$ and $a + 1$ , and act as the identity at all other qubits.\",\n \"The sum is over all pairs of vertices that are edges of the underlying graph.\",\n \"Consequently, if $|u\\\\rangle $ stands for the system state in which the qubit at vertex $u$ is at $|1\\\\rangle $ and all others are at $|0\\\\rangle $ , then $ H |a\\\\rangle = {\\\\left\\\\lbrace \\\\begin{array}{ll}| 2 \\\\rangle , & a = 1 \\\\\\\\| n - 1 \\\\rangle , & a = n \\\\\\\\| a - 1 \\\\rangle + | a + 1 \\\\rangle , & a \\\\notin \\\\lbrace 1, n\\\\rbrace .\\\\end{array}\\\\right.}\",\n \"$ Because of this, the action of $H$ on the set $\\\\lbrace |a\\\\rangle : a \\\\in V(G)\\\\rbrace $ is equivalent to the action of the adjacency matrix $A(P_n)$ on the canonical basis of $n$ .\",\n \"In other words, $H$ can be block diagonalized, and one of its blocks is equal to $A(P_n)$ .\",\n \"The subspace spanned by $\\\\lbrace |a\\\\rangle : a \\\\in V(G)\\\\rbrace $ is the 1-excitation subspace of the Hilbert space, and the quantum dynamics restricted to this subspace corresponds to the scenario where one qubit, say the one at $a$ , is initialized at $|1\\\\rangle $ and all others at $|0\\\\rangle $ .\",\n \"Due to Schrödinger's equation $ |\\\\langle a |\\\\exp (t H) | b \\\\rangle |^2 $ indicates the probability that the state $|1\\\\rangle $ is measured at $b$ after time $t$ .\",\n \"We are concerned solely with the 1-excitation subspace.\",\n \"Let $_u$ denote the vector of the canonical basis of ${n}$ that is 1 at the entry corresponding to vertex $u$ at the ordering of the rows and columns of $A(G)$ .\",\n \"From the remarks above, if the system is initialized with state $|1\\\\rangle $ at vertex $a$ and all others at $|0\\\\rangle $ , it follows that $ |_a^T \\\\exp (t A) _b|^2 $ indicates the probability that the state $|1\\\\rangle $ is measured at $b$ after time $t$ .\",\n \"One of the major goals of quantum communication on spin chains is to transfer a state with high fidelity.\",\n \"At the maximum fidelity of 1, we say we have achieved perfect state transfer.\",\n \"Analogously, we say we have perfect state transfer between vertices $a$ and $b$ if there exists a time $\\\\tau $ such that $|| _a^T U(\\\\tau ) _b|| = 1$ .\",\n \"The concept of perfect state transfer was first introduced by Christandl et al.\",\n \"[3], who also showed perfect state transfer is only possible on spin chains of two or three qubits.\",\n \"However, even without perfect state transfer, the fidelity may be quite high, and the notion of pretty good state transfer was isolated by Godsil [4] as a relaxation of perfect state transfer.\",\n \"Formally, we say there is pretty good state transfer if, for any $\\\\epsilon > 0$ , there exists a time $\\\\tau $ such that the fidelity is $1 - \\\\epsilon $ .\",\n \"Analogously, we say we have pretty good state transfer between vertices $a$ and $b$ if, for any $\\\\epsilon > 0$ , there exists a time $\\\\tau $ such that $|| U(\\\\tau )_{a,b} || > 1 - \\\\epsilon $ , or equivalently, if for any $\\\\epsilon > 0$ , there exists a $\\\\lambda \\\\in , $ || = 1$, such that $ || U() a - b || < $; this condition is abbreviated to $ U() a b$ for convenience.\",\n \"Godsil et al.~\\\\cite {GKSS12} demonstrated the following result.$ [5] Pretty good state transfer occurs between the end vertices of $P_n$ if and only if $n = p - 1, 2p - 1$ , where $p$ is a prime, or $n = 2^m - 1$ .\",\n \"Moreover, when pretty good state transfer occurs between the end vertices of $P_n$ , then it occurs between vertices $a$ and $n + 1 - a$ for all $a \\\\ne (n + 1) /2$ .\",\n \"Banchi et al.\",\n \"[6] showed that pretty good state transfer occurs between the $j$ th and $(n - j +1)$ -th vertices of spin chains with $XYZ$ -Hamiltonian, whose action on the 1-exication subspaces is equivalent to the action of the Laplacian adjacency matrix on $n$ , if the number of vertices is a power of 2, and that the condition is necessary for $j = 1$ , but possibly not for other values of $j$ .\",\n \"Moreover, they present the open question of pretty good state transfer between inner vertices with $XY$ -Hamiltonian.\",\n \"Coutinho, Guo, and van Bommel [7] investigated this question and determined the following infinite family of paths that admit pretty good state transfer between inner vertices but not between the two end vertices.\",\n \"[7] Given any odd prime $p$ and positive integer $t$ , there is pretty good state transfer in $P_{2^t p -1}$ between vertices $a$ and $2^t p -a$ , whenever $a$ is a multiple of $2^{t-1}$ .\",\n \"In this paper, we present necessary and sufficient conditions for pretty good state transfer between any two vertices of a path, demonstrating that the above result is the only family of paths that admit pretty good state transfer between inner vertices but not between the two end vertices.\",\n \"In the next section, we present definitions and preliminary results that will be required to prove this characterization.\"\n ],\n [\n \"Preliminaries\",\n \"If $M$ is a symmetric matrix with $d$ distinct eigenvalues $\\\\theta _1 > \\\\theta _2 > \\\\cdots > \\\\theta _d$ , then the spectral decomposition of $M$ is $ M = \\\\sum _{j = 1}^d \\\\theta _j E_j, $ where $E_r$ denotes the idempotent projection onto the eigenspace corresponding to $\\\\theta _r$ .\",\n \"If $a \\\\in V(G)$ , then the eigenvalue support of $a$ is the following subset of the eigenvalues: $ \\\\Theta _a = \\\\lbrace \\\\theta _j : E_j _a \\\\ne 0\\\\rbrace .\",\n \"$ We say that vertices $a$ and $b$ are strongly cospectral if $E_r _a = \\\\pm E_r _b$ for all $r$ .\",\n \"The following result is given by Banchi et al. [6].\",\n \"[6] If pretty good state transfer occurs between $a$ and $b$ , then they are strongly cospectral vertices.\",\n \"The spectrum of the adjacency matrix of $P_n$ (see [8] for example) is $ \\\\theta _j = 2 \\\\cos \\\\frac{\\\\pi j}{n + 1}, \\\\quad 1 \\\\le j \\\\le n, $ and the eigenvector corresponding to $\\\\theta _j$ is given by $ (\\\\beta _1, \\\\beta _2, \\\\ldots , \\\\beta _k), \\\\mbox{ where } \\\\beta _k = \\\\sin \\\\frac{k \\\\pi j}{n + 1}.\",\n \"$ As stated by Coutinho, Guo, and van Bommel, the following lemma immediately follows.\",\n \"[7] Vertices $a$ and $b$ of $P_n$ are strongly cospectral if and only if $a + b = n + 1$ .\",\n \"Moreover, we observe that when $a + b = n + 1$ , $E_r _a = E_r _b$ when $r$ is odd, and $E_r _a = - E_r _b$ when $r$ is even.\",\n \"We will derive our next result, which gives a sufficient condition to show pretty good state transfer does not occur between a given pair of vertices of $P_n$ , from Kronecker's Theorem, stated below.\",\n \"[Kronecker, see [9]] Let $\\\\theta _0, \\\\ldots , \\\\theta _d$ and $\\\\zeta _0, \\\\ldots , \\\\zeta _d$ be arbitrary real numbers.\",\n \"For an arbitrarily small $\\\\epsilon $ , the system of inequalities $ | \\\\theta _r y - \\\\zeta _r | < \\\\epsilon \\\\pmod {2 \\\\pi }, \\\\quad (r = 0, \\\\ldots , d), $ admits a solution for $y$ if and only if, for integers $l_0, \\\\ldots , l_d$ , if $ \\\\ell _0 \\\\theta _0 + \\\\cdots + \\\\ell _d \\\\theta _d = 0, $ then $ \\\\ell _0 \\\\zeta _0 + \\\\cdots + \\\\ell _d \\\\zeta _d \\\\equiv 0 \\\\pmod {2 \\\\pi }.\",\n \"$ Let $a$ and $b$ be vertices of $P_n$ such that $a + b = n + 1$ .\",\n \"If there is a set of integers $\\\\lbrace \\\\ell _j : \\\\theta _j \\\\in \\\\Theta _a,\\\\ j \\\\mbox{ odd}\\\\rbrace $ such that $ \\\\sum _{\\\\begin{array}{c}\\\\theta _j \\\\in \\\\Theta _a \\\\\\\\ j \\\\mbox{ odd}\\\\end{array}} \\\\ell _j \\\\theta _j = 0 \\\\quad \\\\mbox{and} \\\\quad \\\\sum _{\\\\begin{array}{c}\\\\theta _j \\\\in \\\\Theta _a \\\\\\\\ j \\\\mbox{ odd}\\\\end{array}} \\\\ell _j \\\\mbox{ is odd} $ and there is a set of integers $\\\\lbrace \\\\ell _j : \\\\theta _j \\\\in \\\\Theta _a,\\\\ j \\\\mbox{ even}\\\\rbrace $ such that $ \\\\sum _{\\\\begin{array}{c}\\\\theta _j \\\\in \\\\Theta _a \\\\\\\\ j \\\\mbox{ even}\\\\end{array}} \\\\ell _j \\\\theta _j = 0 \\\\quad \\\\mbox{and} \\\\quad \\\\sum _{\\\\begin{array}{c}\\\\theta _j \\\\in \\\\Theta _a \\\\\\\\ j \\\\mbox{ even}\\\\end{array}} \\\\ell _j \\\\mbox{ is odd} $ then pretty good state transfer does not occur between vertices $a$ and $b$ .\",\n \"Proof: Suppose pretty good state transfer occurs between vertices $a$ and $b$ .\",\n \"We see that the condition $U(\\\\tau ) _a \\\\approx \\\\lambda _b$ is equivalent to $ e^{i \\\\theta _r \\\\tau } E_r _a \\\\approx \\\\lambda E_r _b$ for all $r$ .\",\n \"Writing $\\\\lambda = e^{i \\\\delta }$ , this condition is equivalent to $\\\\theta _r \\\\tau \\\\approx \\\\delta + q_r \\\\pi $ for all $r$ such that $\\\\theta _r \\\\in \\\\Theta _a$ , where $q_r$ is even when $r$ is odd and $q_r$ is odd when $r$ is even.\",\n \"So, we wish to solve the system of inequalities $ |\\\\theta _r \\\\tau - (\\\\delta + \\\\sigma _r \\\\pi ) | < \\\\epsilon \\\\pmod {2 \\\\pi }, \\\\quad (r : \\\\theta _r \\\\in \\\\Theta _a), $ where $\\\\sigma _r = 0$ when $r$ is odd and $\\\\sigma _r = 1$ when $r$ is even.\",\n \"Hence, by Kronecker's Theorem, if $ \\\\sum _{\\\\theta _r \\\\in \\\\Theta _a} \\\\ell _r \\\\theta _r = 0, $ then $ \\\\sum _{\\\\theta _r \\\\in \\\\Theta _a} \\\\ell _r (\\\\delta + \\\\sigma _r \\\\pi ) \\\\equiv 0 \\\\pmod {2 \\\\pi }.\",\n \"$ Now, since there is a set of integers $\\\\lbrace \\\\ell _j : \\\\theta _j \\\\in \\\\Theta _a,\\\\ j \\\\mbox{ odd}\\\\rbrace $ such that $ \\\\sum _{\\\\begin{array}{c}\\\\theta _j \\\\in \\\\Theta _a \\\\\\\\ j \\\\mbox{ odd}\\\\end{array}} \\\\ell _j \\\\theta _j = 0 \\\\quad \\\\mbox{and} \\\\quad \\\\sum _{\\\\begin{array}{c}\\\\theta _j \\\\in \\\\Theta _a \\\\\\\\ j \\\\mbox{ odd}\\\\end{array}} \\\\ell _j \\\\mbox{ is odd} $ then we must have $ \\\\delta \\\\equiv 0 \\\\pmod {2 \\\\pi }.\",\n \"$ However, since there is a set of integers $\\\\lbrace \\\\ell _j : \\\\theta _j \\\\in \\\\Theta _a,\\\\ j \\\\mbox{ even}\\\\rbrace $ such that $ \\\\sum _{\\\\begin{array}{c}\\\\theta _j \\\\in \\\\Theta _a \\\\\\\\ j \\\\mbox{ even}\\\\end{array}} \\\\ell _j \\\\theta _j = 0 \\\\quad \\\\mbox{and} \\\\quad \\\\sum _{\\\\begin{array}{c}\\\\theta _j \\\\in \\\\Theta _a \\\\\\\\ j \\\\mbox{ even}\\\\end{array}} \\\\ell _j \\\\mbox{ is odd} $ then we must also have $ \\\\delta \\\\equiv \\\\pi \\\\pmod {2 \\\\pi }, $ which is the contradiction completing the proof .\"\n ],\n [\n \"Necessary and Sufficient Conditions\",\n \"We first state the following results about sums of cosines which we will use in the proof of the main theorem.\",\n \"Their proofs are included for completeness.\",\n \"The first result uses the fact that $2 \\\\cos x = e^{i x} + e^{-i x}$ , and then we sum the resulting geometric series.\",\n \"Let $q$ be an odd integer.\",\n \"Then $ 2 \\\\sum _{k = 1}^{\\\\frac{q - 1}{2}} (-1)^k \\\\cos \\\\left( \\\\frac{k \\\\pi }{q} \\\\right) + 1 = 0.\",\n \"$ Proof: $& 2 \\\\sum _{k = 1}^{\\\\frac{q - 1}{2}} (-1)^k \\\\cos \\\\left( \\\\frac{k \\\\pi }{q} \\\\right) + 1 \\\\\\\\&= \\\\sum _{k = 1}^{\\\\frac{q - 1}{2}} (-1)^k (e^{\\\\frac{i k \\\\pi }{q}} + e^{\\\\frac{- i k \\\\pi }{q}}) + 1 \\\\\\\\&= \\\\sum _{k = 1}^{\\\\frac{q - 1}{2}} (-e^{\\\\frac{i \\\\pi }{q}})^k + \\\\sum _{k = 1}^{\\\\frac{q - 1}{2}} (-e^{-\\\\frac{i \\\\pi }{q}})^k + 1 \\\\\\\\&= \\\\frac{-e^{\\\\frac{i \\\\pi }{q}} + (-e^{\\\\frac{i \\\\pi }{q}})^{\\\\frac{q + 1}{2}}}{1 + e^{\\\\frac{i \\\\pi }{q}}} + \\\\frac{(-e^{\\\\frac{i \\\\pi }{q}})^{-1} + (-e^{\\\\frac{i \\\\pi }{q}})^{-\\\\frac{q + 1}{2}}}{1 + e^{-\\\\frac{i \\\\pi }{q}}} + 1 \\\\\\\\&= \\\\frac{-e^{\\\\frac{i \\\\pi }{q}} + (-e^{\\\\frac{i \\\\pi }{q}})^{\\\\frac{q + 1}{2}}}{1 + e^{\\\\frac{i \\\\pi }{q}}} - \\\\frac{1+ (-e^{\\\\frac{i \\\\pi }{q}})^{-\\\\frac{q - 1}{2}}}{1 + e^{\\\\frac{i \\\\pi }{q}}} + \\\\frac{1 + e^{\\\\frac{i \\\\pi }{q}}}{1 + e^{\\\\frac{i \\\\pi }{q}}} \\\\\\\\&= \\\\frac{(-e^{\\\\frac{i \\\\pi }{q}})^{\\\\frac{q + 1}{2}} - (-e^{\\\\frac{i \\\\pi }{q}})^{-\\\\frac{q - 1}{2}}}{1 + e^{\\\\frac{i \\\\pi }{q}}} \\\\\\\\&= 0 \\\\square $ The second result makes use of the following sum-product identity: $ \\\\cos \\\\alpha + \\\\cos \\\\beta = 2 \\\\cos \\\\frac{\\\\alpha + \\\\beta }{2} \\\\cos \\\\frac{\\\\alpha - \\\\beta }{2} .\",\n \"$ The final step is an application of the previous lemma.\",\n \"Let $n = km$ , where $m$ is an odd integer, and $0 \\\\le a < k$ be an integer.\",\n \"Then $ \\\\sum _{j = 0}^{m - 1} (-1)^j \\\\cos \\\\left( \\\\frac{(a + jk) \\\\pi }{n} \\\\right) = 0.\",\n \"$ Proof: $& \\\\sum _{j = 0}^{m - 1} (-1)^j \\\\cos \\\\left( \\\\frac{(a + jk) \\\\pi }{n} \\\\right) \\\\\\\\&= (-1)^{\\\\frac{m - 1}{2}} \\\\left[ \\\\cos \\\\left( \\\\frac{\\\\left(a + \\\\frac{m - 1}{2} k \\\\right) \\\\pi }{n} \\\\right) \\\\right.\",\n \"\\\\\\\\ &\\\\quad \\\\left.+ \\\\sum _{j = 1}^{\\\\frac{m - 1}{2}} (-1)^j \\\\left(\\\\cos \\\\left( \\\\frac{\\\\left(a + (\\\\frac{m - 1}{2} - j) k \\\\right) \\\\pi }{n} \\\\right) + \\\\cos \\\\left( \\\\frac{\\\\left(a + (\\\\frac{m - 1}{2} + j) k \\\\right) \\\\pi }{n} \\\\right) \\\\right) \\\\right] \\\\\\\\&= (-1)^{\\\\frac{m - 1}{2}} \\\\left[ \\\\cos \\\\left( \\\\frac{\\\\left(a + \\\\frac{m - 1}{2} k \\\\right) \\\\pi }{n} \\\\right) \\\\right.\",\n \"\\\\\\\\ &\\\\quad \\\\left.+ \\\\sum _{j = 1}^{\\\\frac{m - 1}{2}} (-1)^j 2 \\\\cos \\\\left( \\\\frac{\\\\left(a + \\\\frac{m - 1}{2} k \\\\right) \\\\pi }{n} \\\\right) \\\\cos \\\\left( \\\\frac{j k \\\\pi }{n} \\\\right) \\\\right] \\\\\\\\&= (-1)^{\\\\frac{m - 1}{2}} \\\\cos \\\\left( \\\\frac{\\\\left(a + \\\\frac{m - 1}{2} k \\\\right) \\\\pi }{n} \\\\right) \\\\left( 1 + 2 \\\\sum _{j = 1}^{\\\\frac{m - 1}{2}} (-1)^j \\\\cos \\\\left( \\\\frac{j \\\\pi }{m} \\\\right) \\\\right) \\\\\\\\&= 0 \\\\square $ We will now state and prove the main theorem.\",\n \"There is pretty good state transfer on $P_n$ between vertices $a$ and $b$ if and only if $a + b = n + 1$ and either: $n = 2^t - 1$ , where $t$ is a positive integer; or, $n = 2^t p - 1$ , where $t$ is a nonnegative integer and $p$ is an odd prime, and $a$ is a multiple of $2^{t - 1}$ .\",\n \"Proof: The sufficiency of the conditions is given by Theorem and Theorem .\",\n \"It remains to show that the conditions are necessary.\",\n \"The necessity of the first condition follows from Lemma and Lemma .\",\n \"Henceforth, we need only consider the possibility of pretty good state transfer between vertices $a$ and $n + 1 - a$ .\",\n \"Suppose first that there is pretty good state transfer on $P_n$ between vertices $a$ and $n + 1 - a$ when $n = 2^t r - 1$ , where $t$ is a positive integer and $r$ is an odd composite number.\",\n \"Let $p$ be a prime factor of $r$ .\",\n \"If $p \\\\mid \\\\frac{2^t r}{\\\\gcd (a, 2^t r)}$ , then if $\\\\theta _k \\\\notin \\\\Theta _a$ , then $k$ is a multiple of $p$ .\",\n \"But then, for $c \\\\in \\\\lbrace 1, 2\\\\rbrace $ , we have $ \\\\sum _{i = 0}^{r/p - 1} (-1)^i \\\\theta _{c + i 2^t p} = \\\\sum _{i = 0}^{r/p - 1} (-1)^i \\\\cos \\\\left( \\\\frac{(c + i 2^t p) \\\\pi }{n + 1} \\\\right) = 0 $ by Lemma .\",\n \"Hence, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , a contradiction.\",\n \"Hence, we can now assume that $r \\\\mid a$ .\",\n \"Since it follows from condition (a) that $a \\\\ne 2^{t-1} r$ , we have that $t \\\\ge 2$ and $4 \\\\mid \\\\frac{2^t r}{\\\\gcd (a, 2^t r)}$ , so if $\\\\theta _k \\\\notin \\\\Theta _a$ , then $k$ is a multiple of 4.\",\n \"But then, for $c \\\\in \\\\lbrace 1, 2\\\\rbrace $ , we have $ \\\\sum _{i = 0}^{r - 1} (-1)^i \\\\theta _{c + i 2^t} = \\\\sum _{i = 0}^{r - 1} (-1)^i \\\\cos \\\\left( \\\\frac{(c + i 2^t) \\\\pi }{n + 1} \\\\right) = 0 $ by Lemma .\",\n \"Hence, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , a contradiction.\",\n \"Now suppose that there is pretty good state transfer on $P_n$ between vertices $a$ and $n + 1 - a$ when $n = r - 1$ , where $r$ is an odd composite number.\",\n \"Let $p$ be a prime factor of $r$ .\",\n \"If $p \\\\mid \\\\frac{2^t r}{\\\\gcd (a, 2^t r)}$ , then if $\\\\theta _k \\\\notin \\\\Theta _a$ , then $k$ is a multiple of $p$ .\",\n \"But then, for $c \\\\in \\\\lbrace 1, 2\\\\rbrace $ , we have $&\\\\sum _{i = 0}^{\\\\frac{1}{2} (r/p - 1)} \\\\theta _{c + i 2 p} + \\\\sum _{i = 0}^{\\\\frac{1}{2} (r/p - 1) - 1} \\\\theta _{n + 1 - (c + p + i 2 p)} \\\\\\\\&= \\\\sum _{i = 0}^{r/p - 1} (-1)^i \\\\theta _{c + i p} = \\\\sum _{i = 0}^{r/p - 1} (-1)^i \\\\cos \\\\left( \\\\frac{(c + i p) \\\\pi }{n + 1} \\\\right) = 0$ by Lemma .\",\n \"Hence, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , a contradiction.\",\n \"Hence, we have shown the necessity of the conditions on $n$ .\",\n \"It remains to show the necessity of the conditions on $a$ when $n = 2^t p - 1$ , where $t$ is a positive integer and $p$ is an odd prime.\",\n \"Suppose $a$ is not a multiple of $2^{t - 1}$ .\",\n \"Then again we have that if $\\\\theta _k \\\\notin \\\\Theta _a$ , then $k$ is a multiple of 4.\",\n \"So as above, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , which is the contradiction completing the proof .\",\n \"Acknowledgements The author thanks Chris Godsil for his guidance and support and Gabriel Coutinho for his introduction to this problem.\",\n \"References\"\n ]\n]"},"id":{"kind":"string","value":"1612.05603"}}},{"rowIdx":3398,"cells":{"text":{"kind":"list like","value":[["Occulting Light Concentrators in Liquid Scintillator Neutrino Detectors"],["Abstract The experimental efforts characterizing the era of precision neutrino physics revolve around collecting high-statistics neutrino samples and attaining an excellent energy and position resolution.","Next generation liquid-based neutrino detectors, such as JUNO, HyperKamiokande, etc, share the use of a large target mass, and the need of pushing light collection to the edge for maximal calorimetric information.","Achieving high light collection implies considerable costs, especially when considering detector masses of several kt.","A traditional strategy to maximize the effective photo-coverage with the minimum number of PMTs relies on Light Concentrators (LC), such as Winston Cones.","In this paper, the authors introduce a novel concept called Occulting Light Concentrators (OLC), whereby a traditional LC gets tailored to a conventional PMT, by taking into account its single-photoelectron collection efficiency profile and thus occulting the worst performing portion of the photocathode.","Thus, the OLC shape optimization takes into account not only the optical interface of the PMT, but also the maximization of the PMT detection performances.","The light collection uniformity across the detector is another advantage of the OLC system.","By considering the case of JUNO, we will show OLC capabilities in terms of light collection and energy resolution."],["Introduction","The traditional goal of non-imaging light concentrators (LCs) has always been to maximize the collected light (see for instance [1] and [2]).","This is particularly true for detectors with a limited ($\\sim $ 30%) geometrical coverage, such as SNO, CTF, Borexino, etc.","Moreover, in a liquid based detector the amount of collected light depends on the vertex position, due to light absorption and scattering.","By limiting the field of view of the PMT, LCs reduce the amount of collected light for events at large radii, thus helping to increase the energy reconstruction uniformity.","In detectors as JUNO and RENO-50, planning very high geometrical coverages (around 70%), LC remains a valid tool also against another non-homogeneity effect.","Actually, in large PMTs the photoelectron (p.e.)","Detection Efficiency (DE) is uneven throughout their surface - the PMT edge being the worst-performing region.","The dispersive effects induced by such a non-homogeneous DE affect the total number of collected p.e., worsening significantly the detector-level energy resolution when compared to the intrinsic stochastic component.","Standard LCs can become Occulting Light Concentrator (OLC) with the aim of reducing PMT-related dispersive effects while maximizing light collection and making the detector response more uniform."],["OLCs in a gigantic (15 kt) liquid scintillator detector","An ideal two-dimensional Compound Tangential Concentrator (CTC) is designed in order to transmit to the photocathode all the light incident at the entrance aperture with an angle $<\\theta _i$ and to avoid the transmission of all the light with incident angle $>\\theta _i$ [1].","Thus, $\\theta _i$ is a CTC construction parameter and defines the maximum accepted incident photon angle.","In 3D, where the CTC is obtained by rotating the 2D profile around the symmetry axis, the perfect performances in light transmission are slightly degraded for angles $<\\theta _i$ .","We call $\\theta ^*$ the incident angle where this degradation starts and $\\theta _{FV}$ the maximal photon incident angle at the PMT level (see Figure REF ).","We consider two extreme configurations: OLC$_1$ , obtained by requiring $\\theta _i = \\theta _{FV}$ and OLC$_2$ , by requiring $\\theta ^* = \\theta _{FV}$ .","We simulate a detector Fiducial Volume (FV) as a finite spherical light source of 16 m radius and the PMT as a semi-sphere of 25.4 cm of radius, 4 m away from the edge of the FV.","Figure: Left: Geometry of the detector Fiducial Volume and of the PMT.","θ FV \\theta _{FV} corresponds to the maximal photon incident angle at the PMT level.","Center: 2D profile of OLC 1 _{1} (orange) and OLC 2 _2 (green), as obtained by requiring the maximum radius of the LC not to exceed the PMT radius via the Tangent Ray Method.Right: Acceptance of the optical module obtained with a 3D OLC 1 _{1} (orange) and OLC 2 _2 (green).","In our case, θ FV \\theta _{FV} is about 55 ∘ ^{\\circ }.Figure: Left: Collection Efficiency Profiles as a function of ρ PMT \\rho _{PMT}.","Center: Light Yield for different CE profiles.","Right: Detector Resolution for different CE profiles.Last panel of Figure REF shows the acceptances of the two 3D OLC designs: only OLC$_2$ has been configured in order to maintain the maximal acceptance up to $\\theta _{FV}$ .","We consider 12 possible CE profiles as a function of the distance from the PMT axis ($\\rho _{PMT}$ ), as shown in Figure REF (left).","A non-flat CE profile acts on the detector resolution by reducing the number of detected photons (increasing the stochastic resolution), as clearly shown by blue dots in central panel of Figure REF .","Moreover, it introduces a dependence on the event vertex position, resulting in an additional non-stochastic resolution term, as show in the right panel of Figure REF (cf.","empty and filled dots).","OLC focuses the light meant to hit the external crown of the photocathode (large $\\rho $ , close to the PMT equator) towards the central region of the photocathode, effectively mitigating the dispersive behaviour of the CE profiles.","This is shown again in the central and right panel of Figure REF for OLC$_1$ (orange) and OLC$_2$ (green).","Plots in Figure REF are obtained by applying a radial non-uniformity correction; we assume either perfect knowledge of CE profiles (filled dots) or flat CE (empty dots).","For CE profiles up to number 7, OLCs clearly improve the energy resolution.","Results are comparable with those obtained by full knowledge of CE profile.","Thanks to its tallest shape, OLC$_1$ not only collects more light in the case of a flat CE, but it also manages to have the best light collection across all the CE models.","However, in terms of energy resolution, OLC$_2$ has better performance than OLC$_1$ also for extreme CE profiles, due to its peculiar acceptance curve."],["OLCs in JUNO","The largest (20 kt) liquid scintillator detector currently under construction, JUNO [3], will be equipped with about 18 k 20” PMTs, with the aim to reach about 75% of geometrical coverage.","The main goal of JUNO is to determine the neutrino Mass Hierarchy by measuring the energy spectrum of $\\overline{\\nu }_e$ coming from nuclear reactors at 52 km far from the detector.","To achieve this goal, an unprecedented 3% energy resolution at 1 MeV is required.","The JUNO collaboration is presently considering to implement OLCs in the detector design, in order to: (1) occult the PMT edge, where the CE decreases; (2) improve the uniformity in the light collection across the detector; (3) recover good light in case the total number of PMTs has to be reduced.","The CE profile has been measured on a sub-sample of JUNO PMTs.","The analytical behavior as a function of $\\rho _{PMT}$ is reported in Figure REF (left), showing an almost constant CE up to $\\rho _{PMT}\\sim $ 24 cm.","The OLC profile has thus been tailored on JUNO PMT, via the CTC method, in order to occult the photocathode area at $\\rho _{PMT}>$ 24 cm, for R$_{FV}=$ 17.2 m, R$_{buffer}=$ 19.5 m and by asking $\\theta _i = \\theta _{FV}$ .","Since the complete CTC shape does not fit JUNO geometrical constraints, it must be cut in order to avoid overlapping with near-by OLCs, as shown in Figure REF (middle).","The reference clearance between two PMTs is 25 mm.","Due to mechanical constraints, larger clearances are begin considered.","To increase the clearance also means to reduce the total number of PMTs.","As shown in Figure REF (right), OLCs help to recover light when increasing the clearance.","Moreover, the light collection becomes more uniform across the detector volume.","Thanks to their occulting action and to the uniformity in the light detection, also the energy resolution improves.","In particular, when requiring a larger clearance, OLC allow to recover the energy resolution level of the standard configuration (no-OLC and 25 mm clearance), thus avoiding any degradation in the energy reconstruction."]],"string":"[\n [\n \"Occulting Light Concentrators in Liquid Scintillator Neutrino Detectors\"\n ],\n [\n \"Abstract The experimental efforts characterizing the era of precision neutrino physics revolve around collecting high-statistics neutrino samples and attaining an excellent energy and position resolution.\",\n \"Next generation liquid-based neutrino detectors, such as JUNO, HyperKamiokande, etc, share the use of a large target mass, and the need of pushing light collection to the edge for maximal calorimetric information.\",\n \"Achieving high light collection implies considerable costs, especially when considering detector masses of several kt.\",\n \"A traditional strategy to maximize the effective photo-coverage with the minimum number of PMTs relies on Light Concentrators (LC), such as Winston Cones.\",\n \"In this paper, the authors introduce a novel concept called Occulting Light Concentrators (OLC), whereby a traditional LC gets tailored to a conventional PMT, by taking into account its single-photoelectron collection efficiency profile and thus occulting the worst performing portion of the photocathode.\",\n \"Thus, the OLC shape optimization takes into account not only the optical interface of the PMT, but also the maximization of the PMT detection performances.\",\n \"The light collection uniformity across the detector is another advantage of the OLC system.\",\n \"By considering the case of JUNO, we will show OLC capabilities in terms of light collection and energy resolution.\"\n ],\n [\n \"Introduction\",\n \"The traditional goal of non-imaging light concentrators (LCs) has always been to maximize the collected light (see for instance [1] and [2]).\",\n \"This is particularly true for detectors with a limited ($\\\\sim $ 30%) geometrical coverage, such as SNO, CTF, Borexino, etc.\",\n \"Moreover, in a liquid based detector the amount of collected light depends on the vertex position, due to light absorption and scattering.\",\n \"By limiting the field of view of the PMT, LCs reduce the amount of collected light for events at large radii, thus helping to increase the energy reconstruction uniformity.\",\n \"In detectors as JUNO and RENO-50, planning very high geometrical coverages (around 70%), LC remains a valid tool also against another non-homogeneity effect.\",\n \"Actually, in large PMTs the photoelectron (p.e.)\",\n \"Detection Efficiency (DE) is uneven throughout their surface - the PMT edge being the worst-performing region.\",\n \"The dispersive effects induced by such a non-homogeneous DE affect the total number of collected p.e., worsening significantly the detector-level energy resolution when compared to the intrinsic stochastic component.\",\n \"Standard LCs can become Occulting Light Concentrator (OLC) with the aim of reducing PMT-related dispersive effects while maximizing light collection and making the detector response more uniform.\"\n ],\n [\n \"OLCs in a gigantic (15 kt) liquid scintillator detector\",\n \"An ideal two-dimensional Compound Tangential Concentrator (CTC) is designed in order to transmit to the photocathode all the light incident at the entrance aperture with an angle $<\\\\theta _i$ and to avoid the transmission of all the light with incident angle $>\\\\theta _i$ [1].\",\n \"Thus, $\\\\theta _i$ is a CTC construction parameter and defines the maximum accepted incident photon angle.\",\n \"In 3D, where the CTC is obtained by rotating the 2D profile around the symmetry axis, the perfect performances in light transmission are slightly degraded for angles $<\\\\theta _i$ .\",\n \"We call $\\\\theta ^*$ the incident angle where this degradation starts and $\\\\theta _{FV}$ the maximal photon incident angle at the PMT level (see Figure REF ).\",\n \"We consider two extreme configurations: OLC$_1$ , obtained by requiring $\\\\theta _i = \\\\theta _{FV}$ and OLC$_2$ , by requiring $\\\\theta ^* = \\\\theta _{FV}$ .\",\n \"We simulate a detector Fiducial Volume (FV) as a finite spherical light source of 16 m radius and the PMT as a semi-sphere of 25.4 cm of radius, 4 m away from the edge of the FV.\",\n \"Figure: Left: Geometry of the detector Fiducial Volume and of the PMT.\",\n \"θ FV \\\\theta _{FV} corresponds to the maximal photon incident angle at the PMT level.\",\n \"Center: 2D profile of OLC 1 _{1} (orange) and OLC 2 _2 (green), as obtained by requiring the maximum radius of the LC not to exceed the PMT radius via the Tangent Ray Method.Right: Acceptance of the optical module obtained with a 3D OLC 1 _{1} (orange) and OLC 2 _2 (green).\",\n \"In our case, θ FV \\\\theta _{FV} is about 55 ∘ ^{\\\\circ }.Figure: Left: Collection Efficiency Profiles as a function of ρ PMT \\\\rho _{PMT}.\",\n \"Center: Light Yield for different CE profiles.\",\n \"Right: Detector Resolution for different CE profiles.Last panel of Figure REF shows the acceptances of the two 3D OLC designs: only OLC$_2$ has been configured in order to maintain the maximal acceptance up to $\\\\theta _{FV}$ .\",\n \"We consider 12 possible CE profiles as a function of the distance from the PMT axis ($\\\\rho _{PMT}$ ), as shown in Figure REF (left).\",\n \"A non-flat CE profile acts on the detector resolution by reducing the number of detected photons (increasing the stochastic resolution), as clearly shown by blue dots in central panel of Figure REF .\",\n \"Moreover, it introduces a dependence on the event vertex position, resulting in an additional non-stochastic resolution term, as show in the right panel of Figure REF (cf.\",\n \"empty and filled dots).\",\n \"OLC focuses the light meant to hit the external crown of the photocathode (large $\\\\rho $ , close to the PMT equator) towards the central region of the photocathode, effectively mitigating the dispersive behaviour of the CE profiles.\",\n \"This is shown again in the central and right panel of Figure REF for OLC$_1$ (orange) and OLC$_2$ (green).\",\n \"Plots in Figure REF are obtained by applying a radial non-uniformity correction; we assume either perfect knowledge of CE profiles (filled dots) or flat CE (empty dots).\",\n \"For CE profiles up to number 7, OLCs clearly improve the energy resolution.\",\n \"Results are comparable with those obtained by full knowledge of CE profile.\",\n \"Thanks to its tallest shape, OLC$_1$ not only collects more light in the case of a flat CE, but it also manages to have the best light collection across all the CE models.\",\n \"However, in terms of energy resolution, OLC$_2$ has better performance than OLC$_1$ also for extreme CE profiles, due to its peculiar acceptance curve.\"\n ],\n [\n \"OLCs in JUNO\",\n \"The largest (20 kt) liquid scintillator detector currently under construction, JUNO [3], will be equipped with about 18 k 20” PMTs, with the aim to reach about 75% of geometrical coverage.\",\n \"The main goal of JUNO is to determine the neutrino Mass Hierarchy by measuring the energy spectrum of $\\\\overline{\\\\nu }_e$ coming from nuclear reactors at 52 km far from the detector.\",\n \"To achieve this goal, an unprecedented 3% energy resolution at 1 MeV is required.\",\n \"The JUNO collaboration is presently considering to implement OLCs in the detector design, in order to: (1) occult the PMT edge, where the CE decreases; (2) improve the uniformity in the light collection across the detector; (3) recover good light in case the total number of PMTs has to be reduced.\",\n \"The CE profile has been measured on a sub-sample of JUNO PMTs.\",\n \"The analytical behavior as a function of $\\\\rho _{PMT}$ is reported in Figure REF (left), showing an almost constant CE up to $\\\\rho _{PMT}\\\\sim $ 24 cm.\",\n \"The OLC profile has thus been tailored on JUNO PMT, via the CTC method, in order to occult the photocathode area at $\\\\rho _{PMT}>$ 24 cm, for R$_{FV}=$ 17.2 m, R$_{buffer}=$ 19.5 m and by asking $\\\\theta _i = \\\\theta _{FV}$ .\",\n \"Since the complete CTC shape does not fit JUNO geometrical constraints, it must be cut in order to avoid overlapping with near-by OLCs, as shown in Figure REF (middle).\",\n \"The reference clearance between two PMTs is 25 mm.\",\n \"Due to mechanical constraints, larger clearances are begin considered.\",\n \"To increase the clearance also means to reduce the total number of PMTs.\",\n \"As shown in Figure REF (right), OLCs help to recover light when increasing the clearance.\",\n \"Moreover, the light collection becomes more uniform across the detector volume.\",\n \"Thanks to their occulting action and to the uniformity in the light detection, also the energy resolution improves.\",\n \"In particular, when requiring a larger clearance, OLC allow to recover the energy resolution level of the standard configuration (no-OLC and 25 mm clearance), thus avoiding any degradation in the energy reconstruction.\"\n ]\n]"},"id":{"kind":"string","value":"1612.05444"}}},{"rowIdx":3399,"cells":{"text":{"kind":"list like","value":[["Semiclassical Formulation of Gottesman-Knill and Universal Quantum\n Computation"],["Abstract We give a path integral formulation of the time evolution of qudits of odd dimension.","This allows us to consider semiclassical evolution of discrete systems in terms of an expansion of the propagator in powers of $\\hbar$.","The largest power of $\\hbar$ required to describe the evolution is a traditional measure of classicality.","We show that the action of the Clifford operators on stabilizer states can be fully described by a single contribution of a path integral truncated at order $\\hbar^0$ and so are \"classical,\" just like propagation of Gaussians under harmonic Hamiltonians in the continuous case.","Such operations have no dependence on phase or quantum interference.","Conversely, we show that supplementing the Clifford group with gates necessary for universal quantum computation results in a propagator consisting of a finite number of semiclassical path integral contributions truncated at order $\\hbar^1$ , a number that nevertheless scales exponentially with the number of qudits.","The same sum in continuous systems has an infinite number of terms at order $\\hbar^1$."],["Introduction","The study of contextuality in quantum information has led to progress in our understanding of the Wigner function for discrete systems.","Using Wootters' original derivation of discrete Wigner functions [1], Eisert [2], Gross [3], and Emerson [4] have pushed forward a new perspective on the quantum analysis of states and operators in finite Hilbert spaces by considering their quasiprobability representation on discrete phase space.","Most notably, the positivity of such representations has been shown to be equivalent to non-contextuality, a notion of classicality [4], [5].","Quantum gates and states that exhibit these features are the stabilizer states and Clifford operations used in quantum error correction and stabilizer codes.","The non-contextuality of stabilizer states and Clifford operations explains why they are amenable to efficient classical simulation [6], [7].","This progress raises the question of how these discrete techniques are connected to prior established methods for simulating quantum mechanics in phase space.","A particularly relevant method is trajectory-based semiclassical propagation, which has been widely used in the continuous context.","Perhaps, when applied to the discrete case, semiclassical propagators can lend their physical intuition to outstanding problems in quantum information.","Conversely, concepts from quantum information may serve to illuminate the comparatively older field of continuous semiclassics.","Quantum information attempts to classify the “quantumness” of a system by the presence or absence of various quantum resources.","Semiclassical analysis proceeds by successive approximation using $\\hbar $ as a small parameter, where the power of $\\hbar $ required is a measure of “quantumness”.","Can these two views of quantum vs. classical be related?","In the current paper, we build a bridge from the continuous semiclassical world to the discrete world and examine the classical-quantum characteristics of discrete quantum gates found in circuit models and their stabilizer formalism.","Stabilizer states are eigenvalue one eigenvectors of a commuting set of operators making up a group which does not contain $-\\mathbb {I}$ .","The set of stabilizer states is preserved by elements of the Clifford group, which is the normalizer of the Pauli group, and can be simulated very efficiently.","More precisely, by the Gottesman-Knill Theorem, for $n$ qubits, a quantum circuit of a Clifford gate can be simulated using $\\mathcal {O}(n)$ operations on a classical computer.","Measurements require $\\mathcal {O}(n^2)$ operations with $\\mathcal {O}(n^2)$ bits of storage [6], [7].","The reason that stabilizer evolution by Clifford gates can be efficiently simulated classically has been explained in various ways.","For instance, as already mentioned, stabilizer states have been shown to be non-contextual in qudit [4] and rebit [8] systems.","The obstacle to proving this for qubits is that qubit systems possess state-independent contextuality [9].","Of course, we know how to simulate qubit stabilizer states and Clifford operations efficiently by the Gottesmann-Knill theorem [7].","For recent progress relating non-contextuality to classical simulatability for qubits we refer the reader to [10].","It has also been shown for dimensions greater than two that a state of a discrete system is a stabilizer state if and only if its appropriately defined discrete Wigner function is non-negative [3].","Therefore, when acted on by positive-definite operators, it can be considered as a proper positive-definite (classical) distribution.","Here, we instead relate the concept of efficient classical simulation to the power of $\\hbar $ that a path integral treatment must be expanded to in order to describe the quantum evolution of interest.","It is well known that Gaussian propagation in continuous systems under harmonic Hamiltonians can be described with a single contribution from the path integral truncated at order $\\hbar ^0$ [11].","We show that the corresponding case in discrete systems exists.","In the discrete case, stabilizer states take the place of Gaussians and harmonic Hamiltonians that additionally preserve the discrete phase space take the place of the general continuous harmonic Hamiltonians.","In the discrete case we will only consider $d$ -dimensional systems for odd $d$ since their center representation (or Weyl formalism) is far simpler.","As a consequence, we will show that operations with Clifford gates on stabilizer states can be treated by a path integral independent of the magnitude of $\\hbar $ and are thus fundamentally classical.","Such operations have no dependence on phase or quantum interference.","This can be viewed as a restatement of the Gottesman-Knill theorem in terms of powers of $\\hbar $ .","We also consider more general propagation for discrete quantum systems.","Quantum propagation in continuous systems can be treated by a sum consisting of an infinite number of contributions from the path integral truncated at order $\\hbar ^1$ .","In discrete systems, we show that the corresponding sum consists of a finite number of terms, albeit one that scales exponentially with the number of qudits.","This work also answers a question posed by the recent work of Penney et al.","that explored a “sum-over-paths” expression for Clifford circuits in terms of symplectomorphisms on phase-space and associated generating actions.","Penney et al.","raised the question of how to relate the dynamics of the Wigner representation of (stabilizer) states to the dynamics which are the solutions of the discrete Euler-Lagrange equations for an associated functional [12].","By relying on the well-established center-chord (or Wigner-Weyl-Moyal) formalism in continuous [13] and discrete systems [14], we show how the dynamics of Wigner representations are governed by such solutions related to a “center generating” function and that these solutions are harmonic and classical in nature.","We begin by giving an overview of the center-chord representation in continuous systems in Section .","Then, Section introduces the expansion of the path integral in powers of $\\hbar $ .","This leads us to show what “classical” simulability of states in the continuous case corresponds to and to what higher order of $\\hbar $ an expansion is necessary to treat any quantum operator.","Section then introduces the discrete variable case and defines its corresponding conjugate position and momentum operators.","The path integral in discrete systems is then introduced in Section and, in Section , we define the Clifford group and stabilizer states.","We prove that stabilizer state propagation within the Clifford group is captured fully up to order $\\hbar ^0$ and so is efficiently simulable classically.","Section shows that extending the Clifford group to a universal gate set necessitates an expansion of the semiclassical propagator to a finite sum at order $\\hbar ^1$ .","Finally, we close the paper with some discussion and directions for future work in Section ."],["Center-Chord Representation in Continuous Systems","We define position operators $\\hat{q}$ , $\\hat{q} \\left|q^{\\prime }\\right\\rangle = q^{\\prime } \\left|q^{\\prime }\\right\\rangle $ , and momentum operators $\\hat{p}$ as their Fourier transform, $\\hat{p} = \\hat{\\mathcal {F}}^\\dagger \\hat{q} \\hat{\\mathcal {F}}$ , where $\\hat{F} = h^{\\frac{n}{2}} \\int ^\\infty _{-\\infty } \\mbox{d} p \\int ^\\infty _{-\\infty } \\mbox{d} q \\exp \\left(\\frac{2 \\pi i}{\\hbar } p \\cdot q \\right) \\left|p\\right\\rangle \\left\\langle q\\right|.$ Since $[\\hat{q}, \\hat{p}] = i \\hbar $ , these operators produce a particularly simple Lie algebra and are the generators of a Lie group.","In this Lie group we can define the “boost” operator: $\\hat{Z}^{\\delta p} \\left|q^{\\prime }\\right\\rangle = e^{\\frac{i}{\\hbar } \\hat{q} \\delta p} \\left|q^{\\prime }\\right\\rangle = e^{\\frac{i}{\\hbar } q^{\\prime } \\delta p} \\left|q^{\\prime }\\right\\rangle ,$ and the “shift” operator: $\\hat{X}^{\\delta q} \\left|q^{\\prime }\\right\\rangle = e^{-\\frac{i}{\\hbar } \\hat{p} \\delta q} \\left|q^{\\prime }\\right\\rangle = \\left|q^{\\prime } + \\delta q\\right\\rangle .$ Using the canonical commutation relation and $e^{\\hat{A} + \\hat{B}} = e^{\\hat{A}} e^{\\hat{B}} e^{-\\frac{1}{2}[\\hat{A}, \\hat{B}]}$ if $[\\hat{A}, \\hat{B}]$ is a constant, it follows that $\\hat{Z} \\hat{X} = e^{\\frac{i}{\\hbar }} \\hat{X} \\hat{Z}.$ This is known as the Weyl relation and shows that the product of a shift and a boost (a generalized translation) in phase space is only unique up to a phase governed by $\\hbar $ .","We proceed to introduce the chord representation of operators and states [13].","The generalized phase space translation operator (often called the Weyl operator) is defined as a product of the shift and boost: $\\hat{T}(\\xi _p, \\xi _q) = e^{-\\frac{i}{2 \\hbar } \\xi _p \\cdot \\xi _q} \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q},$ where $\\xi \\equiv (\\xi _p, \\xi _q) \\in \\mathbb {R}^{2n}$ define the chord phase space.","$\\hat{T}(\\xi _p, \\xi _q)$ is a translation by the chord $\\xi $ in phase space.","This can be seen by examining its effect on position and momentum states: $\\hat{T}(\\xi _p, \\xi _q) \\left|q\\right\\rangle = e^{\\frac{i}{\\hbar } \\left(q + \\frac{\\xi _q}{2}\\right) \\cdot \\xi _p} \\left|q+\\xi _q\\right\\rangle ,$ and $\\hat{T}(\\xi _p, \\xi _q) \\left|p\\right\\rangle = e^{-\\frac{i}{\\hbar } \\left(p + \\frac{\\xi _p}{2}\\right) \\cdot \\xi _q} \\left|p+\\xi _p\\right\\rangle ,$ which are shown in Fig.","REF .","Changing the order of shifts $\\hat{X}$ and boosts $\\hat{Z}$ changes the phase of the translation in phase space by $\\xi $ , as given by the Weyl relation above (Eq.","REF ).","An operator $\\hat{A}$ can be expressed as a linear combination of these translations: $\\hat{A} = \\int ^\\infty _{-\\infty } \\mbox{d} \\xi _p \\int ^\\infty _{-\\infty } \\mbox{d} \\xi _q \\, A_\\xi (\\xi _p, \\xi _q) \\hat{T}(\\xi _p, \\xi _q),$ where the weights are: $A_\\xi (\\xi _p, \\xi _q) = \\operatorname{Tr}\\left( \\hat{T}(\\xi _p, \\xi _q)^\\dagger \\hat{A} \\right).$ These weights give the chord representation of $\\hat{A}$ .","Figure: Translation of a) a position state and b) a momentum state along the chord (ξ p ,ξ q )(\\xi _p, \\xi _q) in phase space.The Weyl function, or center representation, is dual to the chord representation.","It is defined in terms of reflections instead of translations.","We can define the reflection operator $\\hat{R}$ as the symplectic Fourier transform of the translation operator: $&&\\hat{R}(x_p, x_q) = \\left(2 \\pi \\hbar \\right)^{-n} \\int ^\\infty _{-\\infty } \\mbox{d} \\xi e^{\\frac{i}{\\hbar } {\\xi }^T \\mathcal {J} {x} } \\hat{T}(\\xi )$ where $x \\equiv \\left(x_p, x_q\\right) \\in \\mathbb {R}^{2n}$ are a continuous set of Weyl phase space points or centers and $\\mathcal {J}$ is the symplectic matrix $\\mathcal {J} = \\left( \\begin{array}{cc} 0 & -\\mathbb {I}_{n}\\\\ \\mathbb {I}_{n} & 0 \\end{array}\\right),$ for $\\mathbb {I}_n$ the $n$ -dimensional identity.","The association of this operator with reflection can be seen by examining its effect on position and momentum states: $\\hat{R}(x_p, x_q) \\left|q\\right\\rangle = e^{\\frac{i}{\\hbar } 2 (x_q - q) \\cdot x_p} \\left|2x_q-q\\right\\rangle ,$ and $\\hat{R}(x_p, x_q) \\left|p\\right\\rangle = e^{-\\frac{i}{\\hbar } 2 (x_p - p) \\cdot x_q} \\left|2x_p-p\\right\\rangle ,$ which are sketched in Fig.","REF .","It is thus evident that $\\hat{R}(x_p, x_q)$ reflects the phase space around $x$ .","Note that while we refer to reflections in the symplectic sense here, and in the rest of the paper; Eqs.","REF and REF show that they are in fact “an inversion around $x$ ” in every two-plane of conjugate ${x_p}_i$ and ${x_q}_i$ .","However, we will keep to the established nomenclature [13].","Figure: Reflection of a) a position state and b) a momentum state across the center (x p ,x q )(x_p, x_q) in phase space.An operator $\\hat{A}$ can now be expressed as a linear combination of reflections: $\\hat{A} = \\left(2 \\pi \\hbar \\right)^{-n} \\int ^\\infty _{-\\infty } \\mbox{d} x_p \\int ^\\infty _{-\\infty } \\mbox{d} x_q \\, A_x(x_p, x_q) \\hat{R}( x_p, x_q ),$ where $A_x(x_p, x_q) = \\operatorname{Tr}\\left( \\hat{R}(x_p, x_q)^\\dagger \\hat{A} \\right),$ and is called the center representation of $\\hat{A}$ .","This representation is of particular interest to us because we can rewrite the components $A_x$ for unitary transformations $\\hat{A}$ as: $A_x(x_p, x_q) = e^{\\frac{i}{\\hbar } S(x_p, x_q)},$ where $S(x_p, x_q)$ is equivalent to the action the transformation $A_x$ produces in Weyl phase space in terms of reflections around centers $x$ [13].","Thus, $S$ is also called the “center generating” function.","For a pure state $\\left|\\Psi \\right\\rangle $ , the Wigner function given by Eq.","REF simplifies to: $&& {\\Psi }_x(x_p, x_q) = \\left(2 \\pi \\hbar \\right)^{-n}\\\\&& \\int ^\\infty _{-\\infty } \\mbox{d} \\xi _q \\, \\Psi \\left( x_q + \\frac{\\xi _q }{2} \\right) {\\Psi ^*} \\left( x_q - \\frac{\\xi _q}{2} \\right) e^{-\\frac{i}{\\hbar } \\xi _q \\cdot x_p}.","\\nonumber $ The center representation for quantum states immediately yields the well-known Wigner function for continuous systems.","The chord representation is the symplectic Fourier transform of the Wigner function.","The center and chord representations are dual to each other, and are the Wigner and Wigner characteristic functions res[ectively.","Identifying the Wigner functions with the center representation, and the center representation as dual to the chord representation motivates the development of both center (Wigner) and chord representations for discrete systems in Section ."],["Path Integral Propagation in continuous Systems","Propagation from one quantum state to another can be expressed in terms of the path integral formalism of the quantum propagator.","For one degree of freedom, with an initial position $q$ and final position $q^{\\prime }$ , evolving under the Hamiltonian $H$ for time $t$ , the propagator is $\\left\\langle q|e^{-i H t/\\hbar }|q^{\\prime }\\right\\rangle = \\int \\mathcal {D}[q_t] \\, \\exp \\left(\\frac{i}{\\hbar } G[q_t]\\right)$ where $G[q_t]$ is the action of the trajectory $q_t$ , which starts at $q$ and ends at $q^{\\prime }$ a time $t$ later [15], [16].","Eq.","REF can be reexpressed as a variational expansion around the set of classical trajectories (a set of measure zero) that start at $q$ and end at $q^{\\prime }$ a time $t$ later.","This is an expansion in powers of $\\hbar $ : $&& \\left\\langle q^{\\prime }|e^{-\\frac{i}{\\hbar }Ht}|q\\right\\rangle =\\\\&& \\sum _j^\\text{cl.","paths} \\int \\mathcal {D}[q_{tj}] \\, e^{\\frac{i}{\\hbar } \\left( G[q_{tj}] + \\delta G[q_{tj}] + \\frac{1}{2} \\delta ^2 G[q_{tj}] + \\ldots \\right) }, \\nonumber $ where $\\delta G[q_{tj}]$ denotes a functional variation of the paths $q_{tj}$ and for classical paths $\\delta G[q_{tj}] = 0$ (For further details we refer the reader to Section 10.3 of [17]).","Terminating Eq.","REF to first order in $\\hbar $ produces the position state representation of the van Vleck-Morette-Gutzwiller propagator [18], [19], [20]: $&& \\left\\langle q^{\\prime }|e^{-\\frac{i}{\\hbar }Ht}|q\\right\\rangle = \\\\&& \\sum _j\\left( \\frac{- \\frac{\\partial ^2 G_{jt}(q,q^{\\prime })}{\\partial q \\partial q^{\\prime }}}{2 \\pi i \\hbar } \\right)^{1/2} e^{i \\frac{G_{jt}(q,q^{\\prime })}{\\hbar }} + \\mathcal {O}(\\hbar ^2).\\nonumber $ where the sum is over all classical paths that satisfy the boundary conditions.","In the center representation, for $n$ degrees of freedom, the semiclassical propagator $U_t(x_p, x_q)$ becomes [13]: $&&U_t(x_p, x_q) = \\\\&& \\sum _j \\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} e^{\\frac{i}{\\hbar } S_{tj}(x_p, x_q)} + \\mathcal {O}(\\hbar ^2),\\nonumber $ where $S_{tj}(x_p, x_q)$ is the center generating function (or action) for the center $x = (x_p, x_q)\\equiv \\frac{1}{2}\\left[(p, q)+(p^{\\prime }, q^{\\prime })\\right]$ .","In general this is an underdetermined system of equations and there are an infinite number of classical trajectories that satisfy these conditions.","The accuracy of adding them up as part of this semiclassical approximation is determined by how separated these trajectories are with respect to $\\hbar $ —the saddle-point condition for convergence of the method of steepest descents.","However, some Hamiltonians exhibit a single saddle point contribution and are thus exact at order $\\hbar ^1$ See Ehrenfest's Theorem, e.g.","in [42]: Lemma 1 There is only one classical trajectory $(p, q)\\underset{t}{\\rightarrow } (p^{\\prime }, q^{\\prime })$ that satisfies the boundary conditions $(x_p, x_q) = \\frac{1}{2}\\left[(p, q)+(p^{\\prime }, q^{\\prime })\\right]$ and $t$ under Hamiltonians that are harmonic in $p$ and $q$ .","Proof For a quadratic Hamiltonian, the diagonalized equations of motion for $n$ -dimensional $(p^{\\prime }, q^{\\prime })$ are of the form: $p^{\\prime }_i &=& \\alpha (t)_i p_i + \\beta (t)_i q_i + \\gamma (t)_i,\\\\q^{\\prime }_i &=& \\delta (t)_i p_i + \\epsilon (t)_i q_i + \\eta (t)_i,$ for $i \\in \\lbrace 1, 2, \\ldots , n\\rbrace $ .","Since $t$ is known and $(p, q)$ can be written in terms of $(p^{\\prime }, q^{\\prime })$ by using $(x_p, x_q)$ , this brings the total number of linear equations to $2n$ with $2n$ unknowns and so there exists one unique solution.$\\Box $ Since the equations of motion for a harmonic Hamiltonian are linear, we can write their solutions as: $\\left( \\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array} \\right) = \\mathcal {M}_t \\left[ \\left( \\begin{array}{c}p\\\\ q \\end{array}\\right) + \\frac{1}{2} \\alpha _t \\right] + \\frac{1}{2} \\alpha _t,$ where $\\mathcal {\\alpha }_t$ is an $n$ -vector and $\\mathcal {M}_t$ is an $n\\times n$ symplectic matrix, both with entries in $\\mathbb {R}$ .","In this case, the center generating function $S_t(x_p, x_q)$ is also quadratic, in particular $S_t(x_p, x_q) = \\alpha _t^T \\mathcal {J} \\left(\\begin{array}{c}x_p\\\\ x_q\\end{array}\\right) + (x_p, x_q) \\mathcal {B}_t \\left(\\begin{array}{c}x_p\\\\ x_q\\end{array}\\right),$ where $\\mathcal {B}_t$ is a real symmetric $n\\times n$ matrix that is related to $\\mathcal {M}_t$ by the Cayley parameterization of $\\mathcal {M}_t$ [22]: $\\mathcal {J} \\mathcal {B}_t = \\left( 1 + \\mathcal {M}_t \\right)^{-1} \\left( 1 - \\mathcal {M}_t \\right) = \\left( 1 - \\mathcal {M}_t \\right) \\left( 1 + \\mathcal {M}_t \\right)^{-1}.$ Since one classical trajectory contribution is sufficient in this case, if the overall phase of the propagated state is not important, then the expansion w.r.t.","$\\hbar $ in Eq.","REF can be truncated at order $\\hbar ^0$ .","Dropping terms that are higher order than $\\hbar ^0$ and ignoring phase is equivalent to propagating the classical density $\\rho (x)$ corresponding to the $(p, q) $ -manifold, under the harmonic Hamiltonian and determining its overlap with the $(p^{\\prime }, q^{\\prime })$ -manifold after time $t$ .","Such a treatment under a harmonic Hamiltonian results in just the absolute value of the prefactor of Eq.","REF : $\\left| \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right|^{\\frac{1}{2}}$ .","Indeed, this was van Vleck's discovery before quantum mechanics was formalized [18].","The relative phases of different classical contributions are no longer a concern and the higher order terms only weigh such contributions appropriately.","Here we are interested in propagating between Gaussian states in the center representation.","In continuous systems, a Gaussian state in $n$ dimensions can be defined as: $\\Psi _\\beta (q) = \\left[\\pi ^{-n} \\det \\left( \\operatorname{\\text{Re}}{\\Sigma }_\\beta \\right) \\right]^{\\frac{1}{4}} \\exp \\left( \\varphi \\right),$ where $\\varphi = \\frac{i}{\\hbar } p_\\beta \\cdot \\left( q - {q}_\\beta \\right) - \\frac{1}{2} \\left( q - {q}_\\beta \\right)^T {\\Sigma }_\\beta \\left( q - q_\\beta \\right).$ $q_\\beta \\in \\mathbb {R}^n$ is the central position, $p_\\beta \\in \\mathbb {R}^n$ is the central momentum, and $\\Sigma _\\beta $ is a symmetric $n\\times n$ matrix where $\\operatorname{\\text{Re}}\\Sigma _\\beta $ is proportional to the spread of the Gaussian and $\\operatorname{\\text{Im}}\\Sigma _\\beta $ captures $p$ -$q$ correlation.","This state describes momentum states ($\\delta (p - p_\\beta )$ in momentum representation) when $\\Sigma _\\beta \\rightarrow 0$ and position states $\\delta (q- q_\\beta )$ when $\\Sigma _\\beta \\rightarrow \\infty $ .","Rotations between these two cases corresponds to $\\operatorname{\\text{Re}}\\Sigma _\\beta = 0$ and $\\operatorname{\\text{Im}}\\Sigma _\\beta \\ne 0$ .","Gaussians remain Gaussians under evolution by a harmonic Hamiltonian, even if it is time-dependent.","This can be shown by simply making the ansatz that the state remains a Gaussian and then solving for its time-dependent $\\Sigma _\\beta $ , $p_\\beta $ , $q_\\beta $ and phase from the time-dependent Schrödinger equation [17], or just by applying the analytically known Feynman path integral, which is equivalent to the van Vleck path integral, to a Gaussian [23].","Moreover, with the propagator in the center representation known to only have have one saddle-point contribution for a harmonic Hamiltonian, it is fairly straight-forward to show that this is also true for its coherent state representation (that is, taking a Gaussian to another Gaussian).","Applying the propagator to an initial and final Gaussian in the center representation $&& \\left[ \\left|\\Psi _\\beta \\right\\rangle \\left\\langle \\Psi _\\beta \\right| U_t \\left|\\Psi _\\alpha \\right\\rangle \\left\\langle \\Psi _\\alpha \\right| \\right]_x (x) = \\\\&& \\left( \\pi \\hbar \\right)^{-3n} \\int ^\\infty _{-\\infty } \\mbox{d} x_1 \\int ^\\infty _{-\\infty } \\mbox{d} x_2 \\, U_t(x_1 + x_2- x) \\nonumber \\\\&& \\qquad \\qquad \\qquad \\times {\\Psi _\\beta }_x(x_2) {\\Psi _\\alpha }_x (x_1) \\nonumber \\\\&& \\qquad \\qquad \\qquad \\times e^{2\\frac{i}{\\hbar }(x_1^T \\mathcal {J} {x_2} + x_2^T \\mathcal {J} {x} + x^T \\mathcal {J} {x_1})}, \\nonumber $ we see that since $U_t$ is a Gaussian from Eq.","REF ($S_{tj}$ is quadratic for harmonic Hamiltonians) and since the Wigner representations of the Gaussians, ${\\Psi _\\alpha }_x$ and ${\\Psi _\\beta }_x$ , are also known to be Gaussians, the full integral in the above equation is a Gaussian integral and thus evaluates to produce a Gaussian with a prefactor.","This is equivalent to evaluating the integral by the method of steepest descents which finds the saddle points to be the points that satisfy $\\frac{\\partial \\phi }{\\partial x} = 0$ where $\\phi $ is the phase of the integrand's argument.","Since this argument is quadratic, its first derivative is linear and so again there is only one unique saddle point.","Indeed, such an evaluation produces the coherent state representation of the vVMG propagator [24].","Just as we found with the center representation, the absolute value of its prefactor corresponds to the order $\\hbar ^0$ term.","As a consequence of this single contribution at order $\\hbar ^0$ , it follows that the Wigner function of a state, $\\Psi _x(x)$ , evolves under the operator $\\hat{V}$ with an underlying harmonic Hamiltonian by $\\Psi _x(\\mathcal {M}_{\\hat{V}}(x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , where $\\mathcal {M}_{\\hat{V}}$ is the symplectic matric and $\\alpha _{\\hat{V}}$ is the translation vector associated with $\\hat{V}$ 's action [14].","Before proceeding to the discrete case, we note that the center representation that we have defined allows for a particularly simple way to express how far the path integral treatment must be expanded in $\\hbar $ in order to describe any unitary propagation (not necessarily harmonic) in continuous quantum mechanics.","Reflections and translations can also be described by truncating Eq.","REF at order $\\hbar ^1$ (or $\\hbar ^0$ if overall phase isn't important) since they correspond to evolution under a harmonic Hamiltonian.","In particular, translations are displacements along a chord $\\xi $ and so have Hamiltonians $H \\propto \\xi _q \\cdot p - \\xi _p \\cdot q$ .","Reflections are symplectic rotations around a center $x$ and so have Hamiltonians $H \\propto \\frac{\\pi }{4}\\left[ (p- x_p)^2 + (q- x_q)^2 \\right]$ .","From Eq.","REF we see that any operator can be expressed as an infinite Riemann sum of reflections.","Therefore, since reflections are fully described by a truncation at order $\\hbar ^1$ , it follows that an infinite Riemann sum of path integral solutions truncated at order $\\hbar ^1$ can describe any unitary evolution.","The same statement can be made by considering the chord representation in terms of translations.","Hence, quantum propagation in continuous systems can be fully treated by an infinite sum of contributions from a path integral approach truncated at order $\\hbar ^1$ .","As an aside, in general this infinite sum isn't convergent and so it is often more useful to consider reformulations that involve a sum with a finite number of contributions.","One way to do this is to apply the method of steepest descents directly on the operator of interest and use the area between saddle points as the metric to determine the order of $\\hbar $ necessary, instead of dealing with an infinity of reflections (or translations).","This results in the semiclassical propagator already presented, but associated with the full Hamiltonian instead of a sum of reflection Hamiltonians.","In summary, we have explained why propagation between Gaussian states under Hamiltonians that are harmonic is simulable classically (i.e.","up to order $\\hbar ^0$ ) in continuous systems.","We will see that the same situation holds in discrete systems for stabilizer states, with the additional restriction that the propagation takes the phase space points, which are now discrete, to themselves.","Discrete Center-Chord Representation We now proceed to the discrete case and introduce the center-chord formalism for these systems.","It will be useful for us to define a pair of conjugate degrees of freedom $p$ and $q$ for discrete systems.","Unfortunately, this isn't as straight-forward as in the continuous case, since the usual canonical commutation relations cannot hold in a finite-dimensional Hilbert space where the operators are bounded (since $\\operatorname{Tr}[\\hat{p}, \\hat{q}] = 0$ ).","We begin in one degree of freedom.","We label the computational basis for our system by $n \\in {0, 1, \\ldots , d-1}$ , for $d$ odd and we assume that $d$ is odd for the rest of this paper.","We identify the discrete position basis with the computational basis and define the “boost” operator as diagonal in this basis: $\\hat{Z}^{\\delta p} \\left|n\\right\\rangle \\equiv \\omega ^{n \\delta p} \\left|n\\right\\rangle ,$ where $\\omega $ will be defined below.","We define the normalized discrete Fourier transform operator to be equivalent to the Hadamard gate: $\\hat{F} = \\frac{1}{\\sqrt{d}}\\sum _{m,n \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{m n} \\left|m\\right\\rangle \\left\\langle n\\right|.$ This allows us to define the Fourier transform of $\\hat{Z}$ : $\\hat{X} \\equiv \\hat{F}^\\dagger \\hat{Z} \\hat{F}$ Again, as before, we call $\\hat{X}$ the “shift” operator since $\\hat{X}^{\\delta q} \\left|n\\right\\rangle \\equiv \\left|n\\oplus \\delta q\\right\\rangle ,$ where $\\oplus $ denotes mod-$d$ integer addition.","It follows that the Weyl relation holds again: $\\hat{Z} \\hat{X} = \\omega \\hat{X} \\hat{Z}.$ The group generated by $\\hat{Z}$ and $\\hat{X}$ has a $d$ -dimensional irreducible representation only if $\\omega ^d=1$ for odd $d$ .","Equivalently, there are only reflections relating any two phase space points on the Weyl phase space “grid” if $d$ is odd [25].","We take $\\omega \\equiv \\omega (d) = e^{2 \\pi i/d}$ [26].","This was introduced by Weyl [27].","Note that this means that $\\hbar = \\frac{d}{2 \\pi }$ or $h = d$ .","This means that the classical regime is most closely reached when the dimensionality of the system is reduced ($d \\rightarrow 0$ ) and thus the most “classical” system we can consider here is a qutrit (since we keep $d$ odd and greater than one).","This is the opposite limit considered by many other approaches where the classical regime is reached when $d \\rightarrow \\infty $ .","One way of interpreting the classical limit in this paper is by considering $h$ to be equal to the inverse of the density of states in phase space (i.e.","in a Wigner unit cell).","As $\\hbar \\rightarrow 0$ phase space area decreases as $\\hbar ^2$ but the number of states only decreases as $\\hbar $ leading to an overall density increase of $\\hbar $ .","This agrees with the notion that states should become point particles of fixed mass in the classical limit.","By analogy with continuous finite translation operators, we reexpress the shift $\\hat{X}$ and boost $\\hat{Z}$ operators in terms of conjugate $\\hat{ p}$ and $\\hat{ q}$ operators: $ \\hat{Z} \\left|n\\right\\rangle = e^{\\frac{2 \\pi i }{d}\\hat{q}} \\left|n\\right\\rangle = e^{\\frac{2 \\pi i}{d} n} \\left|n\\right\\rangle , $ and $ \\hat{X} \\left|n\\right\\rangle = e^{-\\frac{2 \\pi i }{d} \\hat{p}} \\left|n\\right\\rangle = \\left|n\\oplus 1\\right\\rangle .","$ Hence, in the diagonal “position” representation for $\\hat{Z}$ : $\\hat{Z} = \\left( \\begin{array}{cccccc} 1 & 0 & 0 & \\cdots & 0\\\\0 & e^{\\frac{2 \\pi i}{d}} & 0 & \\cdots & 0\\\\0 & 0 & e^{\\frac{4 \\pi i}{d}} & \\cdots & 0\\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & 0 & e^{\\frac{2 (d-1) \\pi i}{d}} \\end{array} \\right),$ and $\\hat{X} = \\left( \\begin{array}{cccccc} 0 & 0 & \\cdots & 0 & 1\\\\1 & 0 & \\cdots & 0 & 0\\\\0 & 1 & 0 & \\cdots & 0\\\\\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\0 & \\cdots & 0 & 1 & 0 \\end{array} \\right).$ Thus, $\\hat{q} = \\frac{d}{2 \\pi i} \\log \\hat{Z} = \\sum _{n \\in \\mathbb {Z}/d\\mathbb {Z}} n \\left|n\\right\\rangle \\left\\langle n\\right|,$ and $\\hat{ p} = \\hat{F}^\\dagger \\hat{ q} \\hat{F}.$ Therefore, we can interpret the operators $\\hat{ p}$ and $\\hat{ q}$ as a conjugate pair similar to conjugate momenta and position in the continuous case.","However, they differ from the latter in that they only obey the weaker group commutation relation $e^{i j \\hat{ q}/\\hbar }e^{i k \\hat{ p}/\\hbar } e^{-i j \\hat{ q}/\\hbar }e^{-i k \\hat{ p}/\\hbar } = e^{-i jk /\\hbar } \\hat{\\mathbb {I}}.$ This corresponds to the usual canonical commutation relation for $p$ and $q$ 's algebra at the origin of the Lie group ($j = k = 0$ ); expanding both sides of Eq.","REF to first order in $p$ and $q$ yields the usual canonical relation.","We proceed to introduce the Weyl representation of operators and states in discrete Hilbert spaces with odd dimension $d$ and $n$ degrees of freedom [1], [28], [29].","The generalized phase space translation operator (the Weyl operator) is defined as a product of the shift and boost with a phase appropriate to the $d$ -dimensional space: $\\hat{T}(\\xi _p, \\xi _q) = e^{-i \\frac{\\pi }{d} \\xi _p \\cdot \\xi _q} \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q},$ where $\\xi \\equiv (\\xi _p, \\xi _q) \\in (\\mathbb {Z} / d \\mathbb {Z})^{2n}$ and form a discrete “web” or “grid” of chords.","They are a discrete subset of the continous chords we considered in the infinite-dimensional context in Section and their finite number is an important consequence of the discretization of the continuous Weyl formalism.","Again, an operator $\\hat{A}$ can be expressed as a linear combination of translations: $\\hat{A} = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} A_\\xi (\\xi _p, \\xi _q) \\hat{T}(\\xi _p, \\xi _q),$ where the weights are the chord representation of the function $\\hat{A}$ : ${A}_\\xi (\\xi _p, \\xi _q) = d^{-n} \\operatorname{Tr}\\left( \\hat{T}(\\xi _p, \\xi _q)^\\dagger \\hat{A} \\right).$ When applied to a state $\\hat{\\rho }$ , this is also called the “characteristic function” of $\\hat{\\rho }$ [30].","As before, the center representation, based on reflections instead of translations, requires an appropriately defined reflection operator.","We can define the discrete reflection operator $\\hat{R}$ as the symplectic Fourier transform of the discrete translation operator we just introduced: $\\hat{R}(x_p, x_q) = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{\\frac{2 \\pi i}{d} (\\xi _p, \\xi _q) \\mathcal {J} (x_p, x_q)^T} \\hat{T}(\\xi _p, \\xi _q).$ With this in hand, we can now express a finite-dimensional operator $\\hat{A}$ as a superposition of reflections: $\\hat{A} = d^{-n} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} {A}_x(x_p, x_q) \\hat{R}( x_p, x_q ),$ where ${A}_x(x_p, x_q) = d^{-n} \\operatorname{Tr}\\left( \\hat{R}(x_p, x_q)^\\dagger \\hat{A} \\right).$ $x \\equiv (x_p, x_q) \\in (\\mathbb {Z} / d \\mathbb {Z})^{2n}$ are centers or Weyl phase space points and, like their $(\\xi _p, \\xi _q)$ brethren, form a discrete subgrid of the continuous Weyl phase space points considered in Section .","Again, the center representation is of particular interest to us because for unitary gates $\\hat{A}$ we can rewrite the components ${A}_x(x_p, x_q)$ as: ${A}_x(x_p, x_q) = \\exp \\left[\\frac{i}{\\hbar } S(x_p, x_q)\\right]$ where $S(x_p, x_q)$ is the argument to the exponential, and is equivalent to the action of the operator in center representation (the center generating function).","Aside from Eq.","REF , the center representation of a state $\\hat{\\rho }$ can also be directly defined as the symplectic Fourier transform of its chord representation, $\\rho _\\xi $ [3]: $\\rho _x(x_p, x_q) = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{\\frac{2 \\pi i}{d} (\\xi _p, \\xi _q) \\mathcal {J} (x_p, x_q)^T} \\rho _\\xi (\\xi _p,\\xi _q).$ We note again that for a pure state $\\left|\\Psi \\right\\rangle $ , the Wigner function from Eqs.","REF and REF simplifies to: ${\\Psi }_x(x_p, x_q) &=& d^{-n} \\sum _{\\begin{array}{c}\\xi _q \\in \\\\(\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{-\\frac{2 \\pi i}{d} \\xi _q \\cdot x_p} \\Psi \\left( x_q + \\frac{(d+1) \\xi _q }{2} \\right) {\\Psi ^*} \\left( x_q - \\frac{(d+1) \\xi _q}{2} \\right).$ It may be clear from this short presentation that the chord and center representations are dual to each other.","A thorough review of this subject can be found in [13].","Path Integral Propagation in Discrete Systems Rivas and Almeida [14] found that the continuous infinite-dimensional vVMG propagator can be extended to finite Hilbert space by simply projecting it onto its finite phase space tori.","This produces: $&& {U}_t(x) = \\\\&& \\left< \\sum _j \\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} e^{\\frac{i}{\\hbar } S_{tj}(x)} e^{i \\theta _k} \\right>_k + \\mathcal {O}(\\hbar ^2),\\nonumber $ where an additional average must be taken over the $k$ center points that are equivalent because of the periodic boundary conditions.","Maintaining periodicity requires that they accrue a phase $\\theta _k$ See Eq.","$5.18$ in [14] for a definition of the phase..","The derivative $\\frac{\\partial ^2 S_{tj}}{\\partial x^2}$ is performed over the continuous function $S_{tj}$ defined after Eq.","REF , but only evaluated at the discrete Weyl phase space points $x \\equiv (x_p, x_q)\\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{2n}$ .","The prefactor can also be reexpressed: $\\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} = \\left\\lbrace 2^d \\det \\left[ 1 + \\mathcal {M}_{tj} \\right] \\right\\rbrace ^{-\\frac{1}{2}},$ which is perhaps more pleasing in the discrete case as it does not involve a continuous derivative.","As in the continuous case, for a harmonic Hamiltonian $H(p, q)$ , the center generating function $S(x_p, x_q)$ is equal to $\\alpha ^T \\mathcal {J} x + x^T \\mathcal {B} x$ where Eq.","REF and Eq.","REF hold.","Moreover, if the Hamiltonian takes Weyl phase space points to themselves, then by the same equations it follows that $\\mathcal {M}$ and $\\alpha $ must have integer entries.","This implies that for $m, n \\in \\mathbb {Z}^{ n}$ , $&& \\mathcal {M} \\left(\\begin{array}{c}p+ m d + \\alpha _p/2\\\\q + n d + \\alpha _q/2\\end{array}\\right) + \\left( \\begin{array}{c} \\alpha _p/2\\\\ \\alpha _q/2 \\end{array} \\right) \\nonumber \\\\&=& \\mathcal {M} \\left(\\begin{array}{c}p + \\alpha _p/2 \\\\ q + \\alpha _q/2 \\end{array}\\right) + \\left( \\begin{array}{c} \\alpha _p/2\\\\ \\alpha _q/2 \\end{array} \\right) + d \\mathcal {M}\\left(\\begin{array}{c}m\\\\ n\\end{array}\\right) \\\\&=& \\left(\\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array}\\right) \\mod {d. \\nonumber }Therefore, phase space points (p, q) that lie on Weyl phase space points go to the equivalent Weyl phase space points (p^{\\prime }, q^{\\prime }).$ Moreover, again if $m, n \\in \\mathbb {Z}^n$ , $&& S(x_p + m d, x_q + n d) \\nonumber \\\\&=& \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right)^T A \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right) \\nonumber \\\\&& + b \\cdot \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right)\\\\&=& \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right)^T A \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right) + b \\cdot \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right) \\nonumber \\\\&& + d \\left[ 2 \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right)^T A \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right) \\right.\\nonumber \\\\&& \\qquad \\left.+ d \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right)^T A \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right) + b \\cdot \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right)\\right] \\nonumber \\\\&=& S(x_p, x_q) \\mod {d}, \\nonumber $ for some symmetric $A \\in \\mathbb {Z}^{n\\times n}$ and $b \\in \\mathbb {Z}^{n}$ .","Therefore, these equivalent trajectories also have equivalent actions (since the action is multiplied by $\\frac{2 \\pi i}{d}$ and exponentiated).","Hence, there is only one term to the sum in Eq.","REF .","Moreover, [25] showed that the sum over the phases $\\theta _k$ produces only a global phase that can be factored out.","Therefore, if we can neglect the overall phase, ${U}_t(x) = \\left| 2^d \\det \\left[ 1 + \\mathcal {M} \\right] \\right|^{\\frac{1}{2}},$ where the classical trajectories whose centers are $(x_p, x_q)$ satisfy the periodic boundary conditions.","As in the continuous case, we point out that this means that translations and reflections are fully captured by a path integral treatment that is truncated at order $\\hbar ^1$ (or order $\\hbar ^0$ if their overall phase isn't important) because their Hamiltonians are harmonic, but in the discrete case there is an additional requirement that they are evaluated at chords/centers that take Weyl phase space points to themselves.","Just as in the continuous case, the single contribution at order $\\hbar ^0$ implies that the propagator of the Wigner function of states, $\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(x)$ under gates $\\hat{V}$ with underlying harmonic Hamiltonians is captured by $\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ for $\\mathcal {M}_{\\hat{V}}$ and $\\alpha _{\\hat{V}}$ associated with $\\hat{V}$ .","Stabilizer Group Here we will show that the Hamiltonians corresponding to Clifford gates are harmonic and take Weyl phase space points to themselves.","Thus they can be captured by only the single contribution of Eq.","REF at lowest order in $\\hbar $ .","This then implies that stabilizer states can also be propagated to each other by Clifford gates with only a single contribution to the sum in Eq.","REF .","The Clifford gate set of interest can be defined by three generators: a single qudit Hadamard gate $\\hat{F}$ and phase shift gate $\\hat{P}$ , as well as the two qudit controlled-not gate $\\hat{C}$ .","We examine each of these in turn.","Hadamard Gate The Hadamard gate was defined in Eq.","REF and is a rotation by $\\frac{\\pi }{2}$ in phase space counter-clockwise.","Hence, for one qudit, it can be written as the map in Eq.","REF where $\\mathcal {M}_{\\hat{F}} = \\left( \\begin{array}{cc} 0 & 1\\\\ -1 & 0 \\end{array} \\right),$ and $\\alpha _{\\hat{F}} = (0,0)$ .","We have set $t=1$ and drop it from the subscripts from now on.","Since $\\alpha $ is vanishing and $\\mathcal {M}$ has integer entries, this is a cat map and such maps have been shown to correspond to Hamiltonians [32] $&&H(p,q) =\\\\&&f(\\operatorname{Tr}\\mathcal {M} ) \\left[ \\mathcal {M}_{12} p^2 - \\mathcal {M}_{21} q^2 + \\left(\\mathcal {M}_{11} - \\mathcal {M}_{22} \\right) pq \\right],\\nonumber $ where $f(x) = \\frac{\\sinh ^{-1}(\\frac{1}{2}\\sqrt{x^2-4})}{\\sqrt{x^2-4}}.$ For the Hadamard $\\mathcal {M}_{\\hat{F}}$ this corresponds to $H_{\\hat{F}} = \\frac{\\pi }{4} (p^2 + q^2)$ , a harmonic oscillator.","The center generating function $S(x_p, x_q)$ is thus $(x_p, x_q ) \\mathcal {B} (x_p, x_q)^T$ and solving Eq.","REF finds for the one-qudit Hadamard, $\\mathcal {B}_{\\hat{F}} = \\left(\\begin{array}{cc}1 & 0\\\\ 0 & 1 \\end{array} \\right).$ Thus, $S_{\\hat{F}}(x_p, x_q) = x_p^2 + x_q^2$ .","Indeed, applying Eq.","REF to Eq.","REF reveals that the Hadamard's center function (up to a phase) is: ${F}_x(x_p,x_q) = e^{\\frac{2 \\pi i}{d}(x_p^2+x_q^2)}.$ Eq.","REF shows how to map Weyl phase space to Weyl phase space under the Hadamard transformation.","Furthermore this map is pointwise, which implies the quadratic form of the center generating function obtained in Eq.","REF .","Phase Shift Gate The phase shift gate can be generalized to odd $d$ -dimensions [33] by setting it to: $\\hat{P} = \\sum _{j \\in \\mathbb {Z}/d \\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{2}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ Examining its effect on stabilizer states, it is clear that it is a $q$ -shear in phase space from an origin displaced by $\\frac{d-1}{2}\\equiv -\\frac{1}{2}$ to the right.","This can be expressed as the map in Eq.","REF with $\\mathcal {M}_{\\hat{P}} = \\left( \\begin{array}{cc} 1 & 1\\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{P}} = \\left( -\\frac{1}{2}, 0 \\right)$ .","This corresponds to $\\mathcal {B}_{\\hat{P}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & \\frac{1}{2} \\end{array} \\right).$ Solving Eq.","REF with this $\\mathcal {B}_{\\hat{P}}$ and $\\alpha _{\\hat{P}}$ reveals that $S_{\\hat{P}}(x_p, x_q) = -\\frac{1}{2} x_q + \\frac{1}{2} x_q^2$ .","Again, this agrees with the argument of the center representation of the phase-shift gate obtained by applying Eq.","REF to Eq.","REF : ${P}_x(x_p,x_q) = e^{\\frac{2 \\pi i}{d} \\frac{1}{2} (-x_q + x_q^2)}.$ Discretization the equations of motion for harmonic evolution for unit timesteps leads to: $\\left(\\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array}\\right) = \\left(\\begin{array}{c}p\\\\ q\\end{array}\\right) + \\mathcal {J} \\left(\\frac{\\partial H}{\\partial p}, \\frac{\\partial H}{\\partial q}\\right)^T,$ where the last derivative is on the continuous function $H$ , but only evaluated on the discrete Weyl phase space points.","It follows that $H_{\\hat{P}} = -\\frac{d+1}{2} q^2 + \\frac{d+1}{2} q.$ We have obtained the Hamiltonian for the phase-shift gate by a different procedure than that used for the Hadamard where we appealed to the result given in Eq.","REF for quantum cat maps.","However, as the phase-shift gate is a quantum cat map as well, we could have obtained Eq.","REF in this manner.","Similarly, the approach we used to find the phase-shift Hamiltonian by discretizing time in Eq.","REF would work for the Hadamard gate but it is a bit more involved since the latter contains both $p$ - and $q$ -evolution.","Nevertheless, this produces Eq.","REF as well.","We presented both techniques for illustrative purposes.","Controlled-Not Gate Lastly, the controlled-not gate can be generalized to $d$ -dimensions [33] by $\\hat{C} = \\sum _{j,k \\in \\mathbb {Z}/d \\mathbb {Z}} \\left|j,k \\oplus j\\right\\rangle \\left\\langle j,k\\right|.$ It is clear that this translates the $q$ -state of the second qudit by the $q$ -state of the first qudit.","As a result, as is evident by examining the gate's action on stabilizer states, the first qudit experiences an “equal and opposite reaction” force that kicks its momentum by the $q$ -state of the second qudit.","This is the phase space picture of the well-known fact that a CNOT examined in the $\\hat{X}$ basis has the control and target reversed with respect to the $\\hat{Z}$ basis.","This can also seen by looking at its effect in the momentum ($\\hat{X}$ ) basis: ${\\hat{F}}^\\dagger \\hat{C} \\hat{F} = \\sum _{j,k \\in \\mathbb {Z}/d \\mathbb {Z}} \\left|j \\ominus k,k\\right\\rangle \\left\\langle j,k\\right|.$ As a result, this gate is described by the map: $\\mathcal {M}_{\\hat{C}} = \\left( \\begin{array}{ccccc} 1 & -1 & 0 & 0\\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1 & 0\\\\ 0 & 0 & 1 & 1\\end{array}\\right),$ and $\\alpha _{\\hat{C}} = (0,0,0,0)$ .","This corresponds to $\\mathcal {B}_{\\hat{C}} = \\left( \\begin{array}{ccccc} 0 & 0 & 0 & 0\\\\ 0 & 0 & -\\frac{1}{2} & 0\\\\ 0 & -\\frac{1}{2} & 0 & 0\\\\ 0 & 0 & 0 & 0\\end{array}\\right).$ Hence its center generating function $S_{\\hat{C}}(x_p, x_q) = -x_{p_2} x_{q_1}$ .","Again, this corresponds with the argument of the center representation of the controlled-not gate, which can be found to be: ${C}_x(x_p, x_q) = e^{-\\frac{2 \\pi i}{d} x_{q_1} x_{p_2}}.$ Therefore, this gate can be seen to be a bilinear $p$ -$q$ coupling between two qudits and corresponds to the Hamiltonian $H_{\\hat{C}} = p_1 q_2,$ as can be found from Eq.","REF again.","As a result, it is now clear that all the Clifford group gates have Hamiltonians that are harmonic and that take Weyl phase space points to themselves.","Therefore, their propagation can be fully described by a truncation of the semiclassical propagator Eq.","REF to order $\\hbar ^0$ as in Eq.","REF and they are manifestly classical in this sense.","To summarize the results of this section, using Eq.","REF , the Hadamard, phase shift, and CNOT gates can be written as: $\\hat{F} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d \\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ -(x_p^2 + x_q^2) - d \\left(x_p \\xi _p - x_q \\xi _q \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q},$ $\\hat{P} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d \\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{2} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q},$ and $&&\\hat{C} = \\\\&& d^{-4} \\sum _{\\begin{array}{c}x_p, x_q, \\\\ \\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z}/d \\mathbb {Z})^{ 2}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ x_{q_1} x_{p_2} - d \\left(x_p \\cdot \\xi _q - x_q \\cdot \\xi _p \\right) \\right] } \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q}, \\nonumber $ (up to a phase).","This form emphasizes their quadratic nature.","As for the continuous case, there exists a particularly simple way in discrete systems to see to what order in $\\hbar $ the path integral must be kept to handle unitary propagation beyond the Clifford group.","We describe this in the next section.","We note that the center generating actions $S(x_p, x_q)$ found here are related to the $G(q^{\\prime }, q)$ found by Penney et al.","[12], which are in terms of initial and final positions, by symmetrized Legendre transform [13]: $G(q^{\\prime }, q, t) = F\\left(\\frac{q^{\\prime } + q}{2}, p(q^{\\prime } - q)\\right),$ where the canonical generating function $F\\left(\\frac{q^{\\prime } + q}{2}, p\\right) = S\\left( x_p = p, x_q = \\frac{q^{\\prime } + q}{2} \\right) + p \\cdot \\left( q^{\\prime } - q \\right),$ for $p (q^{\\prime } - q)$ given implicitly by $\\frac{\\partial F}{\\partial p} = 0$ .","Applying this to the actions we found reveals that $G_{\\hat{F}}(q^{\\prime }, q, t) = q^{\\prime } q,$ $G_{\\hat{P}}(q^{\\prime }, q, t) = \\frac{d+1}{2} (q^2 - q),$ and $G_{\\hat{C}}((q^{\\prime }_1, q^{\\prime }_2), (q_1, q_2), t) = 0,$ which is in agreement with [12].","Classicality of Stabilizer States In this subsection we show that stabilizer states evolve to stabilizer states under Clifford gates, and that it is possible to describe this evolution classically.","For odd $d \\ge 3$ the positivity of the Wigner representation implies that evolution of stabilizer states is non-contextual, and so here we are investigating in detail what this means in our semiclassical picture.","To begin, it is instructive to see the form stabilizer states take in the discrete position representation and in the center representation.","Gross proved that [3]: Theorem 1 Let $d$ be odd and $\\Psi \\in L^2((\\mathbb {Z}/d\\mathbb {Z})^n)$ be a state vector.","The Wigner function of $\\Psi $ is non-negative if and only if $\\Psi $ is a stabilizer state.","Gross also proved [3] Corollary 1 Given that $\\Psi (q) \\ne 0$ $\\forall \\, q$ , a vector $\\Psi $ is a stabilizer state if and only if it is of the form $\\Psi _{\\theta _\\beta ,\\eta _\\beta }(q) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( q^T \\theta _\\beta q + \\eta _\\beta \\cdot q \\right) \\right].$ where $\\theta _\\beta \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{n\\times n}$ and $q, \\eta _\\beta \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^n$ .","Applying Eq.","REF to Eq.","REF , the Wigner function of such maximally supported stabilizer states can be found to be: $&& {\\Psi _{\\theta _\\beta ,\\eta _\\beta }}_x(x_p, x_q) \\propto \\\\&& d^{-n} \\sum _{\\begin{array}{c}\\xi _q \\in \\\\\\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{ n}\\end{array}} \\exp \\left[\\frac{2 \\pi i}{d} \\xi _q \\cdot \\left( \\eta _\\beta - x_p + 2 \\theta _\\beta x_q \\right)\\right].\\nonumber $ Therefore, one finds that the Wigner function is the discrete Fourier sum equal to $\\delta _{\\eta _\\beta - x_p + 2\\theta _\\beta x_q}$ .","For $\\theta _\\beta = 0$ the state is a momentum state at $x_p$ .","Finite $\\theta _\\beta $ rotates that momentum state in phase space in “steps” such that it always lies along the discrete Weyl phase space points $(x_p, x_q) \\in (\\mathbb {Z}/d\\mathbb {Z})^{2n}$ .","This Gaussian expression only captures stabilizer states that are maximally supported in $q$ -space.","One may wonder what the stabilizer states that aren't maximally supported in $q$ -space look like in Weyl phase space.","Of course, it is possible that some may be maximally supported in $p$ -space and so can be captured by the following corollary: Corollary 2 If $\\Psi (p) \\ne 0$ for all $p$ 's then there exists a $\\theta _{\\beta p} \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{n\\times n}$ and an $\\eta _{\\beta p} \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^n$ such that $\\Psi _{\\theta _{\\beta p},\\eta _{\\beta p}}(p) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( p^T \\theta _{\\beta p} p + \\eta _{\\beta p} \\cdot p \\right) \\right].$ Proof This can be shown following the same methods employed by Gross [3] but in the discrete $p$ -basis.","Unfortunately, it is easy to show that Corollary REF and REF do not provide an expression for all stabilizer states (except for the odd prime $d$ case, as we shall see shortly) as there exist stabilizer states for odd non-prime $d$ that are not maximally supported in $p$ - or $q$ -space or any finite rotation between those two.","To find an expression that encompasses all stabilizer states, we must turn to the Wigner function of stabilizer states.","An equivalent definition of stabilizer states on $n$ qudits is given by states $\\hat{V} \\underbrace{\\left|0\\right\\rangle \\otimes \\cdots \\otimes \\left|0\\right\\rangle }_{n}$ where $\\hat{V}$ is a quantum circuit consisting of Clifford gates.","We know that the Clifford circuits are generated by the $\\hat{P}$ , $\\hat{F}$ and $\\hat{C}$ gates, and that the Wigner functions $\\Psi _x(x)$ of stabilizer states propagate under $\\hat{V}$ as $\\Psi _x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , it follows that the Wigner function of stabilizer states is: $\\delta _{\\Phi _0 \\cdot \\mathcal {M}_{\\hat{V}} \\cdot x, r_0},$ where $\\Phi _0 = \\left(\\begin{array}{cc} 0 & 0\\\\ 0 & \\mathbb {I}_n\\end{array} \\right)$ and $r_0 = (0, 0)$ .","We have therefore proved the next theorem: Theorem 2 The Wigner function $\\Psi _x(x)$ of a stabilizer state for any odd $d$ and $n$ qudits is $\\delta _{\\Phi \\cdot x, r}$ for $2n \\times 2n$ matrix $\\Phi $ and $2n$ vector $r$ .","As an aside, Theorem REF allows us to develop an all-encompassing Gaussian expression for stabilizer states for the restricted case that $d$ is odd prime.","In this case, the following Corollary shows that a “mixed” representation is always possible: where each degree of freedom is expressed in either the $p$ - or $q$ -basis: Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state for $n$ qudits, then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .","Proof We begin with a one-qudit case.","We examine the equation specified by $\\Phi \\cdot x = r$ : $\\alpha q_1 + \\beta p_1 = \\gamma $ for $\\alpha $ , $\\beta $ , and $\\gamma \\in \\mathbb {Z}/d\\mathbb {Z}$ .","If $\\alpha = 0$ then $\\Psi (q_1)$ is maximally supported and if $\\beta = 0$ then $\\Psi (p_1)$ is maximally supported since the equation specifies a line on $\\mathbb {Z}/d \\mathbb {Z}$ in $q_1$ and $p_1$ respectively.","If $\\alpha \\ne 0$ and $\\beta \\ne 0$ then the equation can be rewritten as $q_1 + (\\beta /\\alpha ) p_1 = \\gamma /\\alpha ,$ and it follows that $q_1$ can take any values on $\\mathbb {Z}/d \\mathbb {Z}$ and so $\\Psi (q_1)$ is maximally supported.","However, it is also possible to reexpress the equation as: $p_1 + (\\alpha /\\beta ) q_1 = \\gamma /\\beta .$ It follows that $p_1$ can also take any values on $\\mathbb {Z}/d \\mathbb {Z}$ —$\\Psi (p_1)$ is also maximally supported.","Therefore, one can always choose either a $p_1$ - or $q_1$ -basis such that the state is maximally supported and so is representable by a Gaussian function.","We now consider adding another qudit such that the state becomes ${\\Psi }_x(p_1,p_2,q_1,q_2)$ .","There are now two equations specified by $\\Phi \\cdot x = r$ and it follows that it is always possible to combine the two equations such that $p_1$ and $q_1$ are only in one equation and written in terms of each other (and generally the second degree of freedom): $\\alpha q_1 + \\beta p_1 + \\gamma q_2 + \\delta p_2 = \\epsilon ,$ for $\\alpha $ , $\\beta $ , $\\gamma $ , $\\delta $ and $\\epsilon \\in \\mathbb {Z}/d\\mathbb {Z}$ .","It will turn out that the $\\gamma q_2 + \\delta p_2$ term is irrelevant.","We can rewrite the above equation as: $q_1 + (\\beta /\\alpha ) p_1 + (\\gamma /\\alpha ) q_2 + (\\delta /\\alpha ) p_2 = \\epsilon /\\alpha ,$ if $\\alpha \\ne 0$ .","Since there is no other equation specifying $p_1$ , this is an equation for a line on $\\mathbb {Z}/ d \\mathbb {Z}$ and so $\\Psi $ is is maximally supported on $q_1$ .","Otherwise, rewriting the above equation as: $p_1 + (\\alpha /\\beta ) q_1 +(\\gamma /\\beta ) q_2 + (\\delta /\\beta ) p_2 = \\epsilon /\\beta ,$ if $\\beta \\ne 0$ shows that $\\Psi $ is maximally supported on $p_1$ .","If $\\alpha = \\beta = 0$ then both $p_1$ and $q_1$ are undetermined and so either representation produces a maximally supported state.","The same procedure can be performed to find if $q_2$ or $p_2$ produce a maximally supported state.","As can be seen, we are really just repeating the same procedure as we did when there was only one qudit because the other degrees of freedom have no impact on this determination.","Expressing $\\Psi $ in the basis that is maximally supported in every degree of freedom means that it is therefore a Gaussian.","Therefore, it follows that every degree of freedom (corresponding to a qudit) is maximally supported in either the $p$ - or $q$ - basis and so Eq.","REF always describes stabilizer states for odd prime $d$ .$\\Box $ The form of Eq.","REF is more general than Eq.","REF because it does not depend on the support of the state.","As we saw in the proof, this representation is generally not unique; for every qudit $i$ that is not a position or momentum state, $x_i$ can be either $p_i$ or $q_i$ .","However, if it is a position state then $x_i = p_i$ and if it is a momentum state then $x_i = q_i$ ; position and momentum states must be expressed in their conjugate representation in order to be captured by a Gaussian of the form in Eq.","REF instead of Kronecker deltas.","The reason this mixed representation doesn't hold for non-prime odd $d$ is that the coefficients above can be (multiples of) prime factors of $d$ and so no longer produce “lines” in $p_i$ or $q_i$ that cover all of $\\mathbb {Z}/d \\mathbb {Z}$ .","An alternative proof of this corollary that explores this case further is presented in the Appendix.","An example of the different classes of stabilizer states that are possible for odd prime $d$ , in terms of their support, is shown in Fig.","REF .","There it can be seen that a stabilizer state is either maximally supported in $p_i$ or $q_i$ , and is a Kronecker delta function in the other degree of freedom, or it is maximally supported in both.","Figure: The two classes of stabilizer states possible for odd prime d=7d=7 in terms of support: a) maximally supported in pp or qq and b) maximally supported in pp and qq.","The central grids denote the Wigner function ΨΨ x (p,q)\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(p,q) of a stabilizer state Ψ\\Psi with d=7d=7.","The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\Psi (q)|^2).On the other hand, for odd non-prime $d$ , we see in Fig.","REF that another class is possible: stabilizer states that are maximally supported in neither $p_i$ or $q_i$ .","In fact, rotating the basis in any of the discrete angles afforded by the grid still does not produce a basis that is maximally supported (as discussed in the Appendix).","Notice also, that Fig.","REF b shows that it is no longer true that a state that is maximally supported in only $q_i$ or $p_i$ is automatically a Kronecker delta when expressed in terms of the other.","Figure: The four classes of stabilizer states possible for odd non-prime d=15d=15 in terms of support: a) & b) maximally supported in pp or qq, c) maximally supported in pp and qq, and d) not maximally supported in pp or qq.","The central grids denote the Wigner function ΨΨ x (p,q)\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(p,q) for a stabilizer state Ψ\\Psi with d=15d=15.","The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\Psi (q)|^2).In summary, stabilizer states have Wigner function $\\delta _{\\Phi \\cdot x, r}$ and, for odd prime $d$ , are Gaussians in mixed representation that lie on the Weyl phase space points $(x_p, x_q)$ .","Heuristically, they correspond to Gaussians in the continuous case that spread along their major axes infinitely.","The only reason that they aren't always expressible as Gaussians in the mixed representation is that they sometimes “skip” over some of the discrete grid points due to the particular angle they lie along phase space for odd non-prime $d$ .","Wigner functions $\\Psi _x(x)$ of stabilizer states propagate under $\\hat{V}$ as $\\Psi _x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , and this preserves the form of the state.","In other words, Clifford gates take stabilizer states to other stabilizer states, as expected, just like in the continuous case Gaussians go to other Gaussians under harmonic evolution.","It is also clear that stabilizer state propagation under Clifford gates can be expressed by a path integral at order $\\hbar ^0$ .","Discrete Phase Space Representation of Universal Quantum Computing A similar statement to the one we made in Section —that any operator can be expressed as an infinite sum of path integral contribution truncated at order $\\hbar ^1$ —can be made in discrete systems.","However, there is an important difference in the number of terms making up the sum.","To see this we can follow reasoning that is similar to that employed in the continuous case.","Namely, from Eq.","REF we see that any discrete operator can also be expressed as a linear combination of reflections, but unlike the continuous case, this sum has a finite number of terms.","Since reflections can be expressed fully by the discrete path integral truncated at order $\\hbar ^1$ , as discussed previously, it follows that any unitary operator in discrete systems can be expressed as a finite sum of contributions from path integrals truncated at order $\\hbar ^1$ .","Again, the same statement can be made by considering the chord representation in terms of translations.","Hence, quantum propagation in discrete systems can be fully treated by a finite sum of contributions from a path integral approach truncated at order $\\hbar ^1$ .","To gather some understanding of this statement, we can consider what is necessary to add to our path integral formulation when we complete the Clifford gates with the T-gate, which produces a universal gate set.","The T-gate is generalized to odd $d$ -dimensions by $\\hat{T} = \\sum _{j \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{4}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ This gate can no longer be characterized by an $\\mathcal {M}$ with integer entries.","In particular, $\\mathcal {M}_{\\hat{T}} = \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{T}} = \\left( -\\frac{1}{4}, 0 \\right)$ .","This corresponds to $\\mathcal {B}_{\\hat{T}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & -\\frac{1}{4}\\end{array} \\right).$ Thus, the center function $T_x(x_p,x_q) = e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)},$ corresponding to the phase shift Hamiltonian applied for only half the unit of time.","The operator can thus be written: $\\hat{T} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d\\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{4} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q}.$ Though this operator is quadratic, it no longer takes the Weyl center points to themselves.","This means that the $\\hbar ^0$ limit of Eq.","REF is now insufficient to capture all the dynamics because the overlap with any $\\left|q\\right\\rangle $ will now involve a linear superposition of partially overlapping propagated manifolds.","It must therefore be described by a path integral formulation that is complete to order $\\hbar ^1$ .","In particular, $\\hat{T} = d^{-1} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})\\end{array}} e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)} \\hat{R}( x_p, x_q ),$ where $\\hat{R}$ should be substituted by its path integral.","Note that this does not imply efficient classical simulation of quantum computation but quite the opposite.","Indeed, for $n$ qudits, there are $d^{2n}$ terms in the sum above.","While every Weyl phase space point has only a single associated path when acted on by Clifford gates, this is no longer true in any calculation of evolution under the T-gate.","Eq.","REF expresses the T-gate as a sum over phase space operators (the reflections) evaluated on all the phase space points.","Thus, it can be interpreted as associating an exponentially large number of paths to every phase space point, instead of the single paths found for Clifford gates.","Therefore, any simulation of the T-gate naively necessitates adding up an exponential large sum over paths and so is comparably inefficient.","Conclusion The treatment presented here formalizes the relationship between stabilizer states in the discrete case and Gaussians in the continuous case, which has often been pointed out [3].","Namely, only Gaussians that lie along Weyl phase space points directly correspond to Gaussians in the continuous world in terms of preserving their form under a harmonic Hamiltonian, an evolution that is fully describable by truncating the path integral at order $\\hbar ^0$ .","Furthermore, we showed that the Clifford group gates, generated by the Hadamard, phase shift and controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order.","We found that this was because their Hamiltonians are harmonic and take Weyl phase space points to themselves.","This proves the Gottesman-Knill theorem.","The T-gate, needed to complete a universal set with the Hadamard, was shown not to satisfy these properties, and so requires a path integral treatment that is complete up to $\\hbar ^1$ .","The latter treatment includes a sum of terms for which the number of terms scales exponentially with the number of qudits.","We note that our observations pertaining to classical propagation in continuous systems have long been very well known.","In the continuous case, the Wigner function of a quantum state is non-negative if and only if the state is a Gaussian [34] and it has also long been known that quantum propagation from one Gaussian state to another only requires propagation up to order $\\hbar ^0$ [11].","Indeed, it has been shown that this is a continuous version of “stabilizer state propagation” in finite systems [35], and is therefore, in principle, useful for quantum error correction and cluster state quantum computation [36], [37].","It is also well known in the discrete case that quadratic Hamiltonians can act classically and be represented by symplectic transformations in the study of quantum cat maps [38], [14], [25] and linear transformations between propagated Wigner functions [39].","Interestingly though, this latter work appears to have predated the discovery that stabilizer states have positive-definite Wigner functions [3] and therefore, as far as we know, has not been directly related to stabilizer states and the $\\hbar ^0$ limit of their path integral formulation, which is a relatively recent topic of particular interest to the quantum information community and those familiar with Gottesman-Knill.","Otherwise, this claim has been pointed out in terms of concepts related to positivity and related concepts in past work [40], [2].","We also note that our exploration of continous systems is not meant to explore the highly related topic of continuous-variable quantum information.","Many topics therein apply to our discussion here, such as the continuous stabilizer state propagation we mentioned above.","However, our intention in introducing the continuous infinite-dimensional case was not to address these topics but to instead relate the established continuous semiclassical formalism to the discrete case, and thereby bridge the notions of phase space and dynamics between the two worlds.","There is an interesting observation to be made of the weights of the reflections that make up the complete path integral formulation of a unitary operator.","Namely, as is clear in Eqs.","REF and REF , the coefficients consist of the exponentiated center generating function multiplied by $\\frac{i}{\\hbar }$ .","This is very similar to the form of the vVMG path integral in Eqs.","REF and REF .","However, in Eqs.","REF and REF , reflections serve as the prefactors measuring the reflection spectral overlap of a propagated state with its evolute and the center generating actions provide the quantal phase.","Thus, this formulation can be interpreted as an alternative path integral formulation of the vVMG, one consisting of reflections as the underlying classical trajectory only, instead of the more tailored trajectories that result from applying the method of steepest descents directly on an operator.","The fact that any unitary operator in the discrete case can be expressed as a sum consisting of a finite number of order $\\hbar ^1$ path integral contributions, has the added interesting implication that uniformization—higher order $\\hbar $ corrections to the “primitive” semiclassical forms such as Eq.","REF —isn't really necessary in discrete systems.","Uniformization is characterized by the proper treatment of coalescing saddle points and has long been a subject of interest in continuous systems where “anharmonicity” bedevils computationally efficient implementation.","It seems that this problem isn't an issue in the discrete case since a fully complete sum with a finite number of terms, naively numbering $d^2$ for one qudit, exists.","As a last point, there is perhaps an alternative way to interpret the results presented here, one in terms of “resources”.","Much like “magic” (or contextuality) and quantum discord can be framed as a resource necessary to perform quantum operations that have more power than classical ones, it is possible to frame the order in $\\hbar $ that is necessary in the underlying path integral describing an operation as a resource necessary for quantumness.","In this vein, it can be said that Clifford gate operations on stabilizer states are operations that only require $\\hbar ^0$ resources while supplemental gates that push the operator space into universal quantum computing require $\\hbar ^1$ resources.","The dividing line between these two regimes, the classical and quantum world, is discrete, unambiguous and well-defined.","Acknowledgments The authors thank Prof. Alfredo Ozorio de Almeida for very fruitful discussions about the center-chord representation in discrete systems and Byron Drury for his help proof-reading and bringing [12] to our attention.","This work was supported by AFOSR award no.","FA9550-12-1-0046.","Appendix Gross proved that for odd prime $d$ [41]: Lemma 2 Let $\\Psi $ be a state vector with positive Wigner function for odd prime $d$ .","If $\\Psi $ is supported on two points, then it has maximal support.","With this lemma in mind, we can offer an alternative proof of Corollary REF : Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .","Proof We will show that for odd prime $d$ , every degree of freedom can only be fully supported or a Kronecker delta, for all other degrees of freedom fixed; WLOG we will consider a two-dimensional stabilizer state $\\Psi (q_1,q_2)$ and show that if $\\exists \\, q^{\\prime }_1$ such that $\\Psi (q^{\\prime }_1, q_2) \\ne 0$ $\\forall q_2$ then $\\Psi (q_1,q_2)\\ne 0$ $\\forall \\, q_1, q_2$ and vice-versa (if $\\exists \\, q^{\\prime }_1$ s.t.","$\\Psi (q^{\\prime }_1, q_2)$ is a delta function then $\\Psi (q_1,q_2)$ is a delta function in $q_2$ $\\forall q_1$ ).","Therefore, if a degree of freedom is maximally supported in one degree of freedom for all others fixed, then it is maximally supported for all values of the other degrees of freedom.","On the other hand, if it is a delta function in one degree of freedom for all others fixed, then it is a delta function for all values of the other degrees of freedom.","Assume that for $q_1, q_2 \\in \\lbrace 0, \\ldots , d-1\\rbrace $ , $\\exists \\, q^{\\prime }_1, q^{\\prime \\prime }_1$ such that $\\Psi (q^{\\prime }_1,q_2) = 0$ for some $q_2$ and $\\Psi (q^{\\prime \\prime }_1, q_2) \\ne 0$ $\\forall q_2$ .","We proceed to prove by contradiction.","Hence $\\Psi (q^{\\prime \\prime }_1, q_2) \\equiv \\Psi _{q^{\\prime \\prime }_1} \\propto \\left[ \\frac{2 \\pi i}{d} \\left( \\theta _{q^{\\prime \\prime }_1} q^2_2 + \\eta _{q^{\\prime \\prime }_1} q_2 \\right) \\right]$ by Corollary REF and $\\Psi (q^{\\prime }_1,q_2) \\equiv \\Psi _{q^{\\prime }_1}(q_2) \\propto \\delta _{q_2, q(q^{\\prime }_1)}$ for some $q(q^{\\prime }_1)\\in \\mathbb {Z}/d\\mathbb {Z}$ by [3].","We can rotate in $p_2$ -$q_2$ space to form a new basis $q^*_2$ in $d$ discrete angles (since $\\theta _{q^{\\prime \\prime }_1} \\in \\mathbb {Z}/d\\mathbb {Z}$ ) such that $\\Psi _{q^{\\prime }_1}(q^*_2) \\ne 0$ (since a delta function is not maximally supported only at the one angle perpendicular to it).","Since there exists $(d-1)$ other values of $q_1$ other than $q^{\\prime }_1$ , it follows that there exists at least one such angle such that $\\Psi _{q_1}(q^*_2) \\ne 0$ $\\forall q_1, q^*_2 \\in \\lbrace 0,\\ldots ,d-1\\rbrace $ .","We define $q^*_2$ as the basis that is rotated by this angle with respect to $q_2$ .","By Corollary REF , this means that $\\Psi ^*(q_1,q^*_2) \\propto \\exp ( \\theta ^{\\prime }_{11} q^2_1 + \\theta ^{\\prime }_{22} {q^*_2}^2 + 2 \\theta ^{\\prime }_{12} q_1 q^*_2 + \\eta ^{\\prime }_1 q_1 + \\eta ^{\\prime }_2 q_2),$ where by $\\Psi ^*$ we mean $\\Psi $ expressed in the new basis $q^*_2$ in its second degree of freedom.","Hence, $\\Psi ^*_{q_1}(q^*_2) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( \\theta ^{\\prime }_{q_1} {q^*_2}^2 + \\eta ^{\\prime }_{q_1} q^*_2 \\right) \\right] \\exp \\left[ \\frac{2 \\pi i}{d} \\theta _{12} q_1 q^*_2 \\right].$ Acting on this last equation to rotate back to $q_2$ , we must produce $\\Psi _{q^{\\prime \\prime }_1}(q_2) \\propto \\delta _{q_2,q(q^{\\prime \\prime }_1)}$ .","But then Eq.","REF implies that $\\Psi _{q^{\\prime }_1}(q^*_2)$ must also be proportional to $\\delta _{q_2,q(q^{\\prime }_1)}$ .","This is a contradiction.","Therefore, if a degree of freedom is maximally supported for all others fixed, then it is maximally supported for all values of the other degrees of freedom and vice-versa.","In the latter case, a position state in the $i$ th degree of freedom can be represented as a Gaussian by using the $p$ -basis where it becomes a plane wave ($\\theta _i = 0$ ).","In other words, one can always choose $x_i$ to be $p_i$ or $q_i$ such that Eq.","REF holds for odd prime $d$ .","$\\Box $ Finally, the reason this result does not hold for odd non-prime $d$ is that for discrete Wigner space there are $d + 1$ unique angles minus all the prime factors of $d$ .","For non-prime $d$ , there is more than one such prime factor and so there are cases when one cannot “rotate” away all non-maximally supported states."],["Discrete Center-Chord Representation","We now proceed to the discrete case and introduce the center-chord formalism for these systems.","It will be useful for us to define a pair of conjugate degrees of freedom $p$ and $q$ for discrete systems.","Unfortunately, this isn't as straight-forward as in the continuous case, since the usual canonical commutation relations cannot hold in a finite-dimensional Hilbert space where the operators are bounded (since $\\operatorname{Tr}[\\hat{p}, \\hat{q}] = 0$ ).","We begin in one degree of freedom.","We label the computational basis for our system by $n \\in {0, 1, \\ldots , d-1}$ , for $d$ odd and we assume that $d$ is odd for the rest of this paper.","We identify the discrete position basis with the computational basis and define the “boost” operator as diagonal in this basis: $\\hat{Z}^{\\delta p} \\left|n\\right\\rangle \\equiv \\omega ^{n \\delta p} \\left|n\\right\\rangle ,$ where $\\omega $ will be defined below.","We define the normalized discrete Fourier transform operator to be equivalent to the Hadamard gate: $\\hat{F} = \\frac{1}{\\sqrt{d}}\\sum _{m,n \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{m n} \\left|m\\right\\rangle \\left\\langle n\\right|.$ This allows us to define the Fourier transform of $\\hat{Z}$ : $\\hat{X} \\equiv \\hat{F}^\\dagger \\hat{Z} \\hat{F}$ Again, as before, we call $\\hat{X}$ the “shift” operator since $\\hat{X}^{\\delta q} \\left|n\\right\\rangle \\equiv \\left|n\\oplus \\delta q\\right\\rangle ,$ where $\\oplus $ denotes mod-$d$ integer addition.","It follows that the Weyl relation holds again: $\\hat{Z} \\hat{X} = \\omega \\hat{X} \\hat{Z}.$ The group generated by $\\hat{Z}$ and $\\hat{X}$ has a $d$ -dimensional irreducible representation only if $\\omega ^d=1$ for odd $d$ .","Equivalently, there are only reflections relating any two phase space points on the Weyl phase space “grid” if $d$ is odd [25].","We take $\\omega \\equiv \\omega (d) = e^{2 \\pi i/d}$ [26].","This was introduced by Weyl [27].","Note that this means that $\\hbar = \\frac{d}{2 \\pi }$ or $h = d$ .","This means that the classical regime is most closely reached when the dimensionality of the system is reduced ($d \\rightarrow 0$ ) and thus the most “classical” system we can consider here is a qutrit (since we keep $d$ odd and greater than one).","This is the opposite limit considered by many other approaches where the classical regime is reached when $d \\rightarrow \\infty $ .","One way of interpreting the classical limit in this paper is by considering $h$ to be equal to the inverse of the density of states in phase space (i.e.","in a Wigner unit cell).","As $\\hbar \\rightarrow 0$ phase space area decreases as $\\hbar ^2$ but the number of states only decreases as $\\hbar $ leading to an overall density increase of $\\hbar $ .","This agrees with the notion that states should become point particles of fixed mass in the classical limit.","By analogy with continuous finite translation operators, we reexpress the shift $\\hat{X}$ and boost $\\hat{Z}$ operators in terms of conjugate $\\hat{ p}$ and $\\hat{ q}$ operators: $ \\hat{Z} \\left|n\\right\\rangle = e^{\\frac{2 \\pi i }{d}\\hat{q}} \\left|n\\right\\rangle = e^{\\frac{2 \\pi i}{d} n} \\left|n\\right\\rangle , $ and $ \\hat{X} \\left|n\\right\\rangle = e^{-\\frac{2 \\pi i }{d} \\hat{p}} \\left|n\\right\\rangle = \\left|n\\oplus 1\\right\\rangle .","$ Hence, in the diagonal “position” representation for $\\hat{Z}$ : $\\hat{Z} = \\left( \\begin{array}{cccccc} 1 & 0 & 0 & \\cdots & 0\\\\0 & e^{\\frac{2 \\pi i}{d}} & 0 & \\cdots & 0\\\\0 & 0 & e^{\\frac{4 \\pi i}{d}} & \\cdots & 0\\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & 0 & e^{\\frac{2 (d-1) \\pi i}{d}} \\end{array} \\right),$ and $\\hat{X} = \\left( \\begin{array}{cccccc} 0 & 0 & \\cdots & 0 & 1\\\\1 & 0 & \\cdots & 0 & 0\\\\0 & 1 & 0 & \\cdots & 0\\\\\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\0 & \\cdots & 0 & 1 & 0 \\end{array} \\right).$ Thus, $\\hat{q} = \\frac{d}{2 \\pi i} \\log \\hat{Z} = \\sum _{n \\in \\mathbb {Z}/d\\mathbb {Z}} n \\left|n\\right\\rangle \\left\\langle n\\right|,$ and $\\hat{ p} = \\hat{F}^\\dagger \\hat{ q} \\hat{F}.$ Therefore, we can interpret the operators $\\hat{ p}$ and $\\hat{ q}$ as a conjugate pair similar to conjugate momenta and position in the continuous case.","However, they differ from the latter in that they only obey the weaker group commutation relation $e^{i j \\hat{ q}/\\hbar }e^{i k \\hat{ p}/\\hbar } e^{-i j \\hat{ q}/\\hbar }e^{-i k \\hat{ p}/\\hbar } = e^{-i jk /\\hbar } \\hat{\\mathbb {I}}.$ This corresponds to the usual canonical commutation relation for $p$ and $q$ 's algebra at the origin of the Lie group ($j = k = 0$ ); expanding both sides of Eq.","REF to first order in $p$ and $q$ yields the usual canonical relation.","We proceed to introduce the Weyl representation of operators and states in discrete Hilbert spaces with odd dimension $d$ and $n$ degrees of freedom [1], [28], [29].","The generalized phase space translation operator (the Weyl operator) is defined as a product of the shift and boost with a phase appropriate to the $d$ -dimensional space: $\\hat{T}(\\xi _p, \\xi _q) = e^{-i \\frac{\\pi }{d} \\xi _p \\cdot \\xi _q} \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q},$ where $\\xi \\equiv (\\xi _p, \\xi _q) \\in (\\mathbb {Z} / d \\mathbb {Z})^{2n}$ and form a discrete “web” or “grid” of chords.","They are a discrete subset of the continous chords we considered in the infinite-dimensional context in Section and their finite number is an important consequence of the discretization of the continuous Weyl formalism.","Again, an operator $\\hat{A}$ can be expressed as a linear combination of translations: $\\hat{A} = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} A_\\xi (\\xi _p, \\xi _q) \\hat{T}(\\xi _p, \\xi _q),$ where the weights are the chord representation of the function $\\hat{A}$ : ${A}_\\xi (\\xi _p, \\xi _q) = d^{-n} \\operatorname{Tr}\\left( \\hat{T}(\\xi _p, \\xi _q)^\\dagger \\hat{A} \\right).$ When applied to a state $\\hat{\\rho }$ , this is also called the “characteristic function” of $\\hat{\\rho }$ [30].","As before, the center representation, based on reflections instead of translations, requires an appropriately defined reflection operator.","We can define the discrete reflection operator $\\hat{R}$ as the symplectic Fourier transform of the discrete translation operator we just introduced: $\\hat{R}(x_p, x_q) = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{\\frac{2 \\pi i}{d} (\\xi _p, \\xi _q) \\mathcal {J} (x_p, x_q)^T} \\hat{T}(\\xi _p, \\xi _q).$ With this in hand, we can now express a finite-dimensional operator $\\hat{A}$ as a superposition of reflections: $\\hat{A} = d^{-n} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} {A}_x(x_p, x_q) \\hat{R}( x_p, x_q ),$ where ${A}_x(x_p, x_q) = d^{-n} \\operatorname{Tr}\\left( \\hat{R}(x_p, x_q)^\\dagger \\hat{A} \\right).$ $x \\equiv (x_p, x_q) \\in (\\mathbb {Z} / d \\mathbb {Z})^{2n}$ are centers or Weyl phase space points and, like their $(\\xi _p, \\xi _q)$ brethren, form a discrete subgrid of the continuous Weyl phase space points considered in Section .","Again, the center representation is of particular interest to us because for unitary gates $\\hat{A}$ we can rewrite the components ${A}_x(x_p, x_q)$ as: ${A}_x(x_p, x_q) = \\exp \\left[\\frac{i}{\\hbar } S(x_p, x_q)\\right]$ where $S(x_p, x_q)$ is the argument to the exponential, and is equivalent to the action of the operator in center representation (the center generating function).","Aside from Eq.","REF , the center representation of a state $\\hat{\\rho }$ can also be directly defined as the symplectic Fourier transform of its chord representation, $\\rho _\\xi $ [3]: $\\rho _x(x_p, x_q) = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{\\frac{2 \\pi i}{d} (\\xi _p, \\xi _q) \\mathcal {J} (x_p, x_q)^T} \\rho _\\xi (\\xi _p,\\xi _q).$ We note again that for a pure state $\\left|\\Psi \\right\\rangle $ , the Wigner function from Eqs.","REF and REF simplifies to: ${\\Psi }_x(x_p, x_q) &=& d^{-n} \\sum _{\\begin{array}{c}\\xi _q \\in \\\\(\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{-\\frac{2 \\pi i}{d} \\xi _q \\cdot x_p} \\Psi \\left( x_q + \\frac{(d+1) \\xi _q }{2} \\right) {\\Psi ^*} \\left( x_q - \\frac{(d+1) \\xi _q}{2} \\right).$ It may be clear from this short presentation that the chord and center representations are dual to each other.","A thorough review of this subject can be found in [13]."],["Path Integral Propagation in Discrete Systems","Rivas and Almeida [14] found that the continuous infinite-dimensional vVMG propagator can be extended to finite Hilbert space by simply projecting it onto its finite phase space tori.","This produces: $&& {U}_t(x) = \\\\&& \\left< \\sum _j \\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} e^{\\frac{i}{\\hbar } S_{tj}(x)} e^{i \\theta _k} \\right>_k + \\mathcal {O}(\\hbar ^2),\\nonumber $ where an additional average must be taken over the $k$ center points that are equivalent because of the periodic boundary conditions.","Maintaining periodicity requires that they accrue a phase $\\theta _k$ See Eq.","$5.18$ in [14] for a definition of the phase..","The derivative $\\frac{\\partial ^2 S_{tj}}{\\partial x^2}$ is performed over the continuous function $S_{tj}$ defined after Eq.","REF , but only evaluated at the discrete Weyl phase space points $x \\equiv (x_p, x_q)\\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{2n}$ .","The prefactor can also be reexpressed: $\\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} = \\left\\lbrace 2^d \\det \\left[ 1 + \\mathcal {M}_{tj} \\right] \\right\\rbrace ^{-\\frac{1}{2}},$ which is perhaps more pleasing in the discrete case as it does not involve a continuous derivative.","As in the continuous case, for a harmonic Hamiltonian $H(p, q)$ , the center generating function $S(x_p, x_q)$ is equal to $\\alpha ^T \\mathcal {J} x + x^T \\mathcal {B} x$ where Eq.","REF and Eq.","REF hold.","Moreover, if the Hamiltonian takes Weyl phase space points to themselves, then by the same equations it follows that $\\mathcal {M}$ and $\\alpha $ must have integer entries.","This implies that for $m, n \\in \\mathbb {Z}^{ n}$ , $&& \\mathcal {M} \\left(\\begin{array}{c}p+ m d + \\alpha _p/2\\\\q + n d + \\alpha _q/2\\end{array}\\right) + \\left( \\begin{array}{c} \\alpha _p/2\\\\ \\alpha _q/2 \\end{array} \\right) \\nonumber \\\\&=& \\mathcal {M} \\left(\\begin{array}{c}p + \\alpha _p/2 \\\\ q + \\alpha _q/2 \\end{array}\\right) + \\left( \\begin{array}{c} \\alpha _p/2\\\\ \\alpha _q/2 \\end{array} \\right) + d \\mathcal {M}\\left(\\begin{array}{c}m\\\\ n\\end{array}\\right) \\\\&=& \\left(\\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array}\\right) \\mod {d. \\nonumber }Therefore, phase space points (p, q) that lie on Weyl phase space points go to the equivalent Weyl phase space points (p^{\\prime }, q^{\\prime }).$ Moreover, again if $m, n \\in \\mathbb {Z}^n$ , $&& S(x_p + m d, x_q + n d) \\nonumber \\\\&=& \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right)^T A \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right) \\nonumber \\\\&& + b \\cdot \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right)\\\\&=& \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right)^T A \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right) + b \\cdot \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right) \\nonumber \\\\&& + d \\left[ 2 \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right)^T A \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right) \\right.\\nonumber \\\\&& \\qquad \\left.+ d \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right)^T A \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right) + b \\cdot \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right)\\right] \\nonumber \\\\&=& S(x_p, x_q) \\mod {d}, \\nonumber $ for some symmetric $A \\in \\mathbb {Z}^{n\\times n}$ and $b \\in \\mathbb {Z}^{n}$ .","Therefore, these equivalent trajectories also have equivalent actions (since the action is multiplied by $\\frac{2 \\pi i}{d}$ and exponentiated).","Hence, there is only one term to the sum in Eq.","REF .","Moreover, [25] showed that the sum over the phases $\\theta _k$ produces only a global phase that can be factored out.","Therefore, if we can neglect the overall phase, ${U}_t(x) = \\left| 2^d \\det \\left[ 1 + \\mathcal {M} \\right] \\right|^{\\frac{1}{2}},$ where the classical trajectories whose centers are $(x_p, x_q)$ satisfy the periodic boundary conditions.","As in the continuous case, we point out that this means that translations and reflections are fully captured by a path integral treatment that is truncated at order $\\hbar ^1$ (or order $\\hbar ^0$ if their overall phase isn't important) because their Hamiltonians are harmonic, but in the discrete case there is an additional requirement that they are evaluated at chords/centers that take Weyl phase space points to themselves.","Just as in the continuous case, the single contribution at order $\\hbar ^0$ implies that the propagator of the Wigner function of states, $\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(x)$ under gates $\\hat{V}$ with underlying harmonic Hamiltonians is captured by $\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ for $\\mathcal {M}_{\\hat{V}}$ and $\\alpha _{\\hat{V}}$ associated with $\\hat{V}$ ."],["Stabilizer Group","Here we will show that the Hamiltonians corresponding to Clifford gates are harmonic and take Weyl phase space points to themselves.","Thus they can be captured by only the single contribution of Eq.","REF at lowest order in $\\hbar $ .","This then implies that stabilizer states can also be propagated to each other by Clifford gates with only a single contribution to the sum in Eq.","REF .","The Clifford gate set of interest can be defined by three generators: a single qudit Hadamard gate $\\hat{F}$ and phase shift gate $\\hat{P}$ , as well as the two qudit controlled-not gate $\\hat{C}$ .","We examine each of these in turn."],["Hadamard Gate","The Hadamard gate was defined in Eq.","REF and is a rotation by $\\frac{\\pi }{2}$ in phase space counter-clockwise.","Hence, for one qudit, it can be written as the map in Eq.","REF where $\\mathcal {M}_{\\hat{F}} = \\left( \\begin{array}{cc} 0 & 1\\\\ -1 & 0 \\end{array} \\right),$ and $\\alpha _{\\hat{F}} = (0,0)$ .","We have set $t=1$ and drop it from the subscripts from now on.","Since $\\alpha $ is vanishing and $\\mathcal {M}$ has integer entries, this is a cat map and such maps have been shown to correspond to Hamiltonians [32] $&&H(p,q) =\\\\&&f(\\operatorname{Tr}\\mathcal {M} ) \\left[ \\mathcal {M}_{12} p^2 - \\mathcal {M}_{21} q^2 + \\left(\\mathcal {M}_{11} - \\mathcal {M}_{22} \\right) pq \\right],\\nonumber $ where $f(x) = \\frac{\\sinh ^{-1}(\\frac{1}{2}\\sqrt{x^2-4})}{\\sqrt{x^2-4}}.$ For the Hadamard $\\mathcal {M}_{\\hat{F}}$ this corresponds to $H_{\\hat{F}} = \\frac{\\pi }{4} (p^2 + q^2)$ , a harmonic oscillator.","The center generating function $S(x_p, x_q)$ is thus $(x_p, x_q ) \\mathcal {B} (x_p, x_q)^T$ and solving Eq.","REF finds for the one-qudit Hadamard, $\\mathcal {B}_{\\hat{F}} = \\left(\\begin{array}{cc}1 & 0\\\\ 0 & 1 \\end{array} \\right).$ Thus, $S_{\\hat{F}}(x_p, x_q) = x_p^2 + x_q^2$ .","Indeed, applying Eq.","REF to Eq.","REF reveals that the Hadamard's center function (up to a phase) is: ${F}_x(x_p,x_q) = e^{\\frac{2 \\pi i}{d}(x_p^2+x_q^2)}.$ Eq.","REF shows how to map Weyl phase space to Weyl phase space under the Hadamard transformation.","Furthermore this map is pointwise, which implies the quadratic form of the center generating function obtained in Eq.","REF ."],["Phase Shift Gate","The phase shift gate can be generalized to odd $d$ -dimensions [33] by setting it to: $\\hat{P} = \\sum _{j \\in \\mathbb {Z}/d \\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{2}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ Examining its effect on stabilizer states, it is clear that it is a $q$ -shear in phase space from an origin displaced by $\\frac{d-1}{2}\\equiv -\\frac{1}{2}$ to the right.","This can be expressed as the map in Eq.","REF with $\\mathcal {M}_{\\hat{P}} = \\left( \\begin{array}{cc} 1 & 1\\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{P}} = \\left( -\\frac{1}{2}, 0 \\right)$ .","This corresponds to $\\mathcal {B}_{\\hat{P}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & \\frac{1}{2} \\end{array} \\right).$ Solving Eq.","REF with this $\\mathcal {B}_{\\hat{P}}$ and $\\alpha _{\\hat{P}}$ reveals that $S_{\\hat{P}}(x_p, x_q) = -\\frac{1}{2} x_q + \\frac{1}{2} x_q^2$ .","Again, this agrees with the argument of the center representation of the phase-shift gate obtained by applying Eq.","REF to Eq.","REF : ${P}_x(x_p,x_q) = e^{\\frac{2 \\pi i}{d} \\frac{1}{2} (-x_q + x_q^2)}.$ Discretization the equations of motion for harmonic evolution for unit timesteps leads to: $\\left(\\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array}\\right) = \\left(\\begin{array}{c}p\\\\ q\\end{array}\\right) + \\mathcal {J} \\left(\\frac{\\partial H}{\\partial p}, \\frac{\\partial H}{\\partial q}\\right)^T,$ where the last derivative is on the continuous function $H$ , but only evaluated on the discrete Weyl phase space points.","It follows that $H_{\\hat{P}} = -\\frac{d+1}{2} q^2 + \\frac{d+1}{2} q.$ We have obtained the Hamiltonian for the phase-shift gate by a different procedure than that used for the Hadamard where we appealed to the result given in Eq.","REF for quantum cat maps.","However, as the phase-shift gate is a quantum cat map as well, we could have obtained Eq.","REF in this manner.","Similarly, the approach we used to find the phase-shift Hamiltonian by discretizing time in Eq.","REF would work for the Hadamard gate but it is a bit more involved since the latter contains both $p$ - and $q$ -evolution.","Nevertheless, this produces Eq.","REF as well.","We presented both techniques for illustrative purposes."],["Controlled-Not Gate","Lastly, the controlled-not gate can be generalized to $d$ -dimensions [33] by $\\hat{C} = \\sum _{j,k \\in \\mathbb {Z}/d \\mathbb {Z}} \\left|j,k \\oplus j\\right\\rangle \\left\\langle j,k\\right|.$ It is clear that this translates the $q$ -state of the second qudit by the $q$ -state of the first qudit.","As a result, as is evident by examining the gate's action on stabilizer states, the first qudit experiences an “equal and opposite reaction” force that kicks its momentum by the $q$ -state of the second qudit.","This is the phase space picture of the well-known fact that a CNOT examined in the $\\hat{X}$ basis has the control and target reversed with respect to the $\\hat{Z}$ basis.","This can also seen by looking at its effect in the momentum ($\\hat{X}$ ) basis: ${\\hat{F}}^\\dagger \\hat{C} \\hat{F} = \\sum _{j,k \\in \\mathbb {Z}/d \\mathbb {Z}} \\left|j \\ominus k,k\\right\\rangle \\left\\langle j,k\\right|.$ As a result, this gate is described by the map: $\\mathcal {M}_{\\hat{C}} = \\left( \\begin{array}{ccccc} 1 & -1 & 0 & 0\\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1 & 0\\\\ 0 & 0 & 1 & 1\\end{array}\\right),$ and $\\alpha _{\\hat{C}} = (0,0,0,0)$ .","This corresponds to $\\mathcal {B}_{\\hat{C}} = \\left( \\begin{array}{ccccc} 0 & 0 & 0 & 0\\\\ 0 & 0 & -\\frac{1}{2} & 0\\\\ 0 & -\\frac{1}{2} & 0 & 0\\\\ 0 & 0 & 0 & 0\\end{array}\\right).$ Hence its center generating function $S_{\\hat{C}}(x_p, x_q) = -x_{p_2} x_{q_1}$ .","Again, this corresponds with the argument of the center representation of the controlled-not gate, which can be found to be: ${C}_x(x_p, x_q) = e^{-\\frac{2 \\pi i}{d} x_{q_1} x_{p_2}}.$ Therefore, this gate can be seen to be a bilinear $p$ -$q$ coupling between two qudits and corresponds to the Hamiltonian $H_{\\hat{C}} = p_1 q_2,$ as can be found from Eq.","REF again.","As a result, it is now clear that all the Clifford group gates have Hamiltonians that are harmonic and that take Weyl phase space points to themselves.","Therefore, their propagation can be fully described by a truncation of the semiclassical propagator Eq.","REF to order $\\hbar ^0$ as in Eq.","REF and they are manifestly classical in this sense.","To summarize the results of this section, using Eq.","REF , the Hadamard, phase shift, and CNOT gates can be written as: $\\hat{F} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d \\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ -(x_p^2 + x_q^2) - d \\left(x_p \\xi _p - x_q \\xi _q \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q},$ $\\hat{P} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d \\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{2} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q},$ and $&&\\hat{C} = \\\\&& d^{-4} \\sum _{\\begin{array}{c}x_p, x_q, \\\\ \\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z}/d \\mathbb {Z})^{ 2}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ x_{q_1} x_{p_2} - d \\left(x_p \\cdot \\xi _q - x_q \\cdot \\xi _p \\right) \\right] } \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q}, \\nonumber $ (up to a phase).","This form emphasizes their quadratic nature.","As for the continuous case, there exists a particularly simple way in discrete systems to see to what order in $\\hbar $ the path integral must be kept to handle unitary propagation beyond the Clifford group.","We describe this in the next section.","We note that the center generating actions $S(x_p, x_q)$ found here are related to the $G(q^{\\prime }, q)$ found by Penney et al.","[12], which are in terms of initial and final positions, by symmetrized Legendre transform [13]: $G(q^{\\prime }, q, t) = F\\left(\\frac{q^{\\prime } + q}{2}, p(q^{\\prime } - q)\\right),$ where the canonical generating function $F\\left(\\frac{q^{\\prime } + q}{2}, p\\right) = S\\left( x_p = p, x_q = \\frac{q^{\\prime } + q}{2} \\right) + p \\cdot \\left( q^{\\prime } - q \\right),$ for $p (q^{\\prime } - q)$ given implicitly by $\\frac{\\partial F}{\\partial p} = 0$ .","Applying this to the actions we found reveals that $G_{\\hat{F}}(q^{\\prime }, q, t) = q^{\\prime } q,$ $G_{\\hat{P}}(q^{\\prime }, q, t) = \\frac{d+1}{2} (q^2 - q),$ and $G_{\\hat{C}}((q^{\\prime }_1, q^{\\prime }_2), (q_1, q_2), t) = 0,$ which is in agreement with [12]."],["Classicality of Stabilizer States","In this subsection we show that stabilizer states evolve to stabilizer states under Clifford gates, and that it is possible to describe this evolution classically.","For odd $d \\ge 3$ the positivity of the Wigner representation implies that evolution of stabilizer states is non-contextual, and so here we are investigating in detail what this means in our semiclassical picture.","To begin, it is instructive to see the form stabilizer states take in the discrete position representation and in the center representation.","Gross proved that [3]: Theorem 1 Let $d$ be odd and $\\Psi \\in L^2((\\mathbb {Z}/d\\mathbb {Z})^n)$ be a state vector.","The Wigner function of $\\Psi $ is non-negative if and only if $\\Psi $ is a stabilizer state.","Gross also proved [3] Corollary 1 Given that $\\Psi (q) \\ne 0$ $\\forall \\, q$ , a vector $\\Psi $ is a stabilizer state if and only if it is of the form $\\Psi _{\\theta _\\beta ,\\eta _\\beta }(q) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( q^T \\theta _\\beta q + \\eta _\\beta \\cdot q \\right) \\right].$ where $\\theta _\\beta \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{n\\times n}$ and $q, \\eta _\\beta \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^n$ .","Applying Eq.","REF to Eq.","REF , the Wigner function of such maximally supported stabilizer states can be found to be: $&& {\\Psi _{\\theta _\\beta ,\\eta _\\beta }}_x(x_p, x_q) \\propto \\\\&& d^{-n} \\sum _{\\begin{array}{c}\\xi _q \\in \\\\\\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{ n}\\end{array}} \\exp \\left[\\frac{2 \\pi i}{d} \\xi _q \\cdot \\left( \\eta _\\beta - x_p + 2 \\theta _\\beta x_q \\right)\\right].\\nonumber $ Therefore, one finds that the Wigner function is the discrete Fourier sum equal to $\\delta _{\\eta _\\beta - x_p + 2\\theta _\\beta x_q}$ .","For $\\theta _\\beta = 0$ the state is a momentum state at $x_p$ .","Finite $\\theta _\\beta $ rotates that momentum state in phase space in “steps” such that it always lies along the discrete Weyl phase space points $(x_p, x_q) \\in (\\mathbb {Z}/d\\mathbb {Z})^{2n}$ .","This Gaussian expression only captures stabilizer states that are maximally supported in $q$ -space.","One may wonder what the stabilizer states that aren't maximally supported in $q$ -space look like in Weyl phase space.","Of course, it is possible that some may be maximally supported in $p$ -space and so can be captured by the following corollary: Corollary 2 If $\\Psi (p) \\ne 0$ for all $p$ 's then there exists a $\\theta _{\\beta p} \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{n\\times n}$ and an $\\eta _{\\beta p} \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^n$ such that $\\Psi _{\\theta _{\\beta p},\\eta _{\\beta p}}(p) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( p^T \\theta _{\\beta p} p + \\eta _{\\beta p} \\cdot p \\right) \\right].$ Proof This can be shown following the same methods employed by Gross [3] but in the discrete $p$ -basis.","Unfortunately, it is easy to show that Corollary REF and REF do not provide an expression for all stabilizer states (except for the odd prime $d$ case, as we shall see shortly) as there exist stabilizer states for odd non-prime $d$ that are not maximally supported in $p$ - or $q$ -space or any finite rotation between those two.","To find an expression that encompasses all stabilizer states, we must turn to the Wigner function of stabilizer states.","An equivalent definition of stabilizer states on $n$ qudits is given by states $\\hat{V} \\underbrace{\\left|0\\right\\rangle \\otimes \\cdots \\otimes \\left|0\\right\\rangle }_{n}$ where $\\hat{V}$ is a quantum circuit consisting of Clifford gates.","We know that the Clifford circuits are generated by the $\\hat{P}$ , $\\hat{F}$ and $\\hat{C}$ gates, and that the Wigner functions $\\Psi _x(x)$ of stabilizer states propagate under $\\hat{V}$ as $\\Psi _x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , it follows that the Wigner function of stabilizer states is: $\\delta _{\\Phi _0 \\cdot \\mathcal {M}_{\\hat{V}} \\cdot x, r_0},$ where $\\Phi _0 = \\left(\\begin{array}{cc} 0 & 0\\\\ 0 & \\mathbb {I}_n\\end{array} \\right)$ and $r_0 = (0, 0)$ .","We have therefore proved the next theorem: Theorem 2 The Wigner function $\\Psi _x(x)$ of a stabilizer state for any odd $d$ and $n$ qudits is $\\delta _{\\Phi \\cdot x, r}$ for $2n \\times 2n$ matrix $\\Phi $ and $2n$ vector $r$ .","As an aside, Theorem REF allows us to develop an all-encompassing Gaussian expression for stabilizer states for the restricted case that $d$ is odd prime.","In this case, the following Corollary shows that a “mixed” representation is always possible: where each degree of freedom is expressed in either the $p$ - or $q$ -basis: Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state for $n$ qudits, then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .","Proof We begin with a one-qudit case.","We examine the equation specified by $\\Phi \\cdot x = r$ : $\\alpha q_1 + \\beta p_1 = \\gamma $ for $\\alpha $ , $\\beta $ , and $\\gamma \\in \\mathbb {Z}/d\\mathbb {Z}$ .","If $\\alpha = 0$ then $\\Psi (q_1)$ is maximally supported and if $\\beta = 0$ then $\\Psi (p_1)$ is maximally supported since the equation specifies a line on $\\mathbb {Z}/d \\mathbb {Z}$ in $q_1$ and $p_1$ respectively.","If $\\alpha \\ne 0$ and $\\beta \\ne 0$ then the equation can be rewritten as $q_1 + (\\beta /\\alpha ) p_1 = \\gamma /\\alpha ,$ and it follows that $q_1$ can take any values on $\\mathbb {Z}/d \\mathbb {Z}$ and so $\\Psi (q_1)$ is maximally supported.","However, it is also possible to reexpress the equation as: $p_1 + (\\alpha /\\beta ) q_1 = \\gamma /\\beta .$ It follows that $p_1$ can also take any values on $\\mathbb {Z}/d \\mathbb {Z}$ —$\\Psi (p_1)$ is also maximally supported.","Therefore, one can always choose either a $p_1$ - or $q_1$ -basis such that the state is maximally supported and so is representable by a Gaussian function.","We now consider adding another qudit such that the state becomes ${\\Psi }_x(p_1,p_2,q_1,q_2)$ .","There are now two equations specified by $\\Phi \\cdot x = r$ and it follows that it is always possible to combine the two equations such that $p_1$ and $q_1$ are only in one equation and written in terms of each other (and generally the second degree of freedom): $\\alpha q_1 + \\beta p_1 + \\gamma q_2 + \\delta p_2 = \\epsilon ,$ for $\\alpha $ , $\\beta $ , $\\gamma $ , $\\delta $ and $\\epsilon \\in \\mathbb {Z}/d\\mathbb {Z}$ .","It will turn out that the $\\gamma q_2 + \\delta p_2$ term is irrelevant.","We can rewrite the above equation as: $q_1 + (\\beta /\\alpha ) p_1 + (\\gamma /\\alpha ) q_2 + (\\delta /\\alpha ) p_2 = \\epsilon /\\alpha ,$ if $\\alpha \\ne 0$ .","Since there is no other equation specifying $p_1$ , this is an equation for a line on $\\mathbb {Z}/ d \\mathbb {Z}$ and so $\\Psi $ is is maximally supported on $q_1$ .","Otherwise, rewriting the above equation as: $p_1 + (\\alpha /\\beta ) q_1 +(\\gamma /\\beta ) q_2 + (\\delta /\\beta ) p_2 = \\epsilon /\\beta ,$ if $\\beta \\ne 0$ shows that $\\Psi $ is maximally supported on $p_1$ .","If $\\alpha = \\beta = 0$ then both $p_1$ and $q_1$ are undetermined and so either representation produces a maximally supported state.","The same procedure can be performed to find if $q_2$ or $p_2$ produce a maximally supported state.","As can be seen, we are really just repeating the same procedure as we did when there was only one qudit because the other degrees of freedom have no impact on this determination.","Expressing $\\Psi $ in the basis that is maximally supported in every degree of freedom means that it is therefore a Gaussian.","Therefore, it follows that every degree of freedom (corresponding to a qudit) is maximally supported in either the $p$ - or $q$ - basis and so Eq.","REF always describes stabilizer states for odd prime $d$ .$\\Box $ The form of Eq.","REF is more general than Eq.","REF because it does not depend on the support of the state.","As we saw in the proof, this representation is generally not unique; for every qudit $i$ that is not a position or momentum state, $x_i$ can be either $p_i$ or $q_i$ .","However, if it is a position state then $x_i = p_i$ and if it is a momentum state then $x_i = q_i$ ; position and momentum states must be expressed in their conjugate representation in order to be captured by a Gaussian of the form in Eq.","REF instead of Kronecker deltas.","The reason this mixed representation doesn't hold for non-prime odd $d$ is that the coefficients above can be (multiples of) prime factors of $d$ and so no longer produce “lines” in $p_i$ or $q_i$ that cover all of $\\mathbb {Z}/d \\mathbb {Z}$ .","An alternative proof of this corollary that explores this case further is presented in the Appendix.","An example of the different classes of stabilizer states that are possible for odd prime $d$ , in terms of their support, is shown in Fig.","REF .","There it can be seen that a stabilizer state is either maximally supported in $p_i$ or $q_i$ , and is a Kronecker delta function in the other degree of freedom, or it is maximally supported in both.","Figure: The two classes of stabilizer states possible for odd prime d=7d=7 in terms of support: a) maximally supported in pp or qq and b) maximally supported in pp and qq.","The central grids denote the Wigner function ΨΨ x (p,q)\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(p,q) of a stabilizer state Ψ\\Psi with d=7d=7.","The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\Psi (q)|^2).On the other hand, for odd non-prime $d$ , we see in Fig.","REF that another class is possible: stabilizer states that are maximally supported in neither $p_i$ or $q_i$ .","In fact, rotating the basis in any of the discrete angles afforded by the grid still does not produce a basis that is maximally supported (as discussed in the Appendix).","Notice also, that Fig.","REF b shows that it is no longer true that a state that is maximally supported in only $q_i$ or $p_i$ is automatically a Kronecker delta when expressed in terms of the other.","Figure: The four classes of stabilizer states possible for odd non-prime d=15d=15 in terms of support: a) & b) maximally supported in pp or qq, c) maximally supported in pp and qq, and d) not maximally supported in pp or qq.","The central grids denote the Wigner function ΨΨ x (p,q)\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(p,q) for a stabilizer state Ψ\\Psi with d=15d=15.","The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\Psi (q)|^2).In summary, stabilizer states have Wigner function $\\delta _{\\Phi \\cdot x, r}$ and, for odd prime $d$ , are Gaussians in mixed representation that lie on the Weyl phase space points $(x_p, x_q)$ .","Heuristically, they correspond to Gaussians in the continuous case that spread along their major axes infinitely.","The only reason that they aren't always expressible as Gaussians in the mixed representation is that they sometimes “skip” over some of the discrete grid points due to the particular angle they lie along phase space for odd non-prime $d$ .","Wigner functions $\\Psi _x(x)$ of stabilizer states propagate under $\\hat{V}$ as $\\Psi _x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , and this preserves the form of the state.","In other words, Clifford gates take stabilizer states to other stabilizer states, as expected, just like in the continuous case Gaussians go to other Gaussians under harmonic evolution.","It is also clear that stabilizer state propagation under Clifford gates can be expressed by a path integral at order $\\hbar ^0$ .","Discrete Phase Space Representation of Universal Quantum Computing A similar statement to the one we made in Section —that any operator can be expressed as an infinite sum of path integral contribution truncated at order $\\hbar ^1$ —can be made in discrete systems.","However, there is an important difference in the number of terms making up the sum.","To see this we can follow reasoning that is similar to that employed in the continuous case.","Namely, from Eq.","REF we see that any discrete operator can also be expressed as a linear combination of reflections, but unlike the continuous case, this sum has a finite number of terms.","Since reflections can be expressed fully by the discrete path integral truncated at order $\\hbar ^1$ , as discussed previously, it follows that any unitary operator in discrete systems can be expressed as a finite sum of contributions from path integrals truncated at order $\\hbar ^1$ .","Again, the same statement can be made by considering the chord representation in terms of translations.","Hence, quantum propagation in discrete systems can be fully treated by a finite sum of contributions from a path integral approach truncated at order $\\hbar ^1$ .","To gather some understanding of this statement, we can consider what is necessary to add to our path integral formulation when we complete the Clifford gates with the T-gate, which produces a universal gate set.","The T-gate is generalized to odd $d$ -dimensions by $\\hat{T} = \\sum _{j \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{4}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ This gate can no longer be characterized by an $\\mathcal {M}$ with integer entries.","In particular, $\\mathcal {M}_{\\hat{T}} = \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{T}} = \\left( -\\frac{1}{4}, 0 \\right)$ .","This corresponds to $\\mathcal {B}_{\\hat{T}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & -\\frac{1}{4}\\end{array} \\right).$ Thus, the center function $T_x(x_p,x_q) = e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)},$ corresponding to the phase shift Hamiltonian applied for only half the unit of time.","The operator can thus be written: $\\hat{T} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d\\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{4} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q}.$ Though this operator is quadratic, it no longer takes the Weyl center points to themselves.","This means that the $\\hbar ^0$ limit of Eq.","REF is now insufficient to capture all the dynamics because the overlap with any $\\left|q\\right\\rangle $ will now involve a linear superposition of partially overlapping propagated manifolds.","It must therefore be described by a path integral formulation that is complete to order $\\hbar ^1$ .","In particular, $\\hat{T} = d^{-1} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})\\end{array}} e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)} \\hat{R}( x_p, x_q ),$ where $\\hat{R}$ should be substituted by its path integral.","Note that this does not imply efficient classical simulation of quantum computation but quite the opposite.","Indeed, for $n$ qudits, there are $d^{2n}$ terms in the sum above.","While every Weyl phase space point has only a single associated path when acted on by Clifford gates, this is no longer true in any calculation of evolution under the T-gate.","Eq.","REF expresses the T-gate as a sum over phase space operators (the reflections) evaluated on all the phase space points.","Thus, it can be interpreted as associating an exponentially large number of paths to every phase space point, instead of the single paths found for Clifford gates.","Therefore, any simulation of the T-gate naively necessitates adding up an exponential large sum over paths and so is comparably inefficient.","Conclusion The treatment presented here formalizes the relationship between stabilizer states in the discrete case and Gaussians in the continuous case, which has often been pointed out [3].","Namely, only Gaussians that lie along Weyl phase space points directly correspond to Gaussians in the continuous world in terms of preserving their form under a harmonic Hamiltonian, an evolution that is fully describable by truncating the path integral at order $\\hbar ^0$ .","Furthermore, we showed that the Clifford group gates, generated by the Hadamard, phase shift and controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order.","We found that this was because their Hamiltonians are harmonic and take Weyl phase space points to themselves.","This proves the Gottesman-Knill theorem.","The T-gate, needed to complete a universal set with the Hadamard, was shown not to satisfy these properties, and so requires a path integral treatment that is complete up to $\\hbar ^1$ .","The latter treatment includes a sum of terms for which the number of terms scales exponentially with the number of qudits.","We note that our observations pertaining to classical propagation in continuous systems have long been very well known.","In the continuous case, the Wigner function of a quantum state is non-negative if and only if the state is a Gaussian [34] and it has also long been known that quantum propagation from one Gaussian state to another only requires propagation up to order $\\hbar ^0$ [11].","Indeed, it has been shown that this is a continuous version of “stabilizer state propagation” in finite systems [35], and is therefore, in principle, useful for quantum error correction and cluster state quantum computation [36], [37].","It is also well known in the discrete case that quadratic Hamiltonians can act classically and be represented by symplectic transformations in the study of quantum cat maps [38], [14], [25] and linear transformations between propagated Wigner functions [39].","Interestingly though, this latter work appears to have predated the discovery that stabilizer states have positive-definite Wigner functions [3] and therefore, as far as we know, has not been directly related to stabilizer states and the $\\hbar ^0$ limit of their path integral formulation, which is a relatively recent topic of particular interest to the quantum information community and those familiar with Gottesman-Knill.","Otherwise, this claim has been pointed out in terms of concepts related to positivity and related concepts in past work [40], [2].","We also note that our exploration of continous systems is not meant to explore the highly related topic of continuous-variable quantum information.","Many topics therein apply to our discussion here, such as the continuous stabilizer state propagation we mentioned above.","However, our intention in introducing the continuous infinite-dimensional case was not to address these topics but to instead relate the established continuous semiclassical formalism to the discrete case, and thereby bridge the notions of phase space and dynamics between the two worlds.","There is an interesting observation to be made of the weights of the reflections that make up the complete path integral formulation of a unitary operator.","Namely, as is clear in Eqs.","REF and REF , the coefficients consist of the exponentiated center generating function multiplied by $\\frac{i}{\\hbar }$ .","This is very similar to the form of the vVMG path integral in Eqs.","REF and REF .","However, in Eqs.","REF and REF , reflections serve as the prefactors measuring the reflection spectral overlap of a propagated state with its evolute and the center generating actions provide the quantal phase.","Thus, this formulation can be interpreted as an alternative path integral formulation of the vVMG, one consisting of reflections as the underlying classical trajectory only, instead of the more tailored trajectories that result from applying the method of steepest descents directly on an operator.","The fact that any unitary operator in the discrete case can be expressed as a sum consisting of a finite number of order $\\hbar ^1$ path integral contributions, has the added interesting implication that uniformization—higher order $\\hbar $ corrections to the “primitive” semiclassical forms such as Eq.","REF —isn't really necessary in discrete systems.","Uniformization is characterized by the proper treatment of coalescing saddle points and has long been a subject of interest in continuous systems where “anharmonicity” bedevils computationally efficient implementation.","It seems that this problem isn't an issue in the discrete case since a fully complete sum with a finite number of terms, naively numbering $d^2$ for one qudit, exists.","As a last point, there is perhaps an alternative way to interpret the results presented here, one in terms of “resources”.","Much like “magic” (or contextuality) and quantum discord can be framed as a resource necessary to perform quantum operations that have more power than classical ones, it is possible to frame the order in $\\hbar $ that is necessary in the underlying path integral describing an operation as a resource necessary for quantumness.","In this vein, it can be said that Clifford gate operations on stabilizer states are operations that only require $\\hbar ^0$ resources while supplemental gates that push the operator space into universal quantum computing require $\\hbar ^1$ resources.","The dividing line between these two regimes, the classical and quantum world, is discrete, unambiguous and well-defined.","Acknowledgments The authors thank Prof. Alfredo Ozorio de Almeida for very fruitful discussions about the center-chord representation in discrete systems and Byron Drury for his help proof-reading and bringing [12] to our attention.","This work was supported by AFOSR award no.","FA9550-12-1-0046.","Appendix Gross proved that for odd prime $d$ [41]: Lemma 2 Let $\\Psi $ be a state vector with positive Wigner function for odd prime $d$ .","If $\\Psi $ is supported on two points, then it has maximal support.","With this lemma in mind, we can offer an alternative proof of Corollary REF : Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .","Proof We will show that for odd prime $d$ , every degree of freedom can only be fully supported or a Kronecker delta, for all other degrees of freedom fixed; WLOG we will consider a two-dimensional stabilizer state $\\Psi (q_1,q_2)$ and show that if $\\exists \\, q^{\\prime }_1$ such that $\\Psi (q^{\\prime }_1, q_2) \\ne 0$ $\\forall q_2$ then $\\Psi (q_1,q_2)\\ne 0$ $\\forall \\, q_1, q_2$ and vice-versa (if $\\exists \\, q^{\\prime }_1$ s.t.","$\\Psi (q^{\\prime }_1, q_2)$ is a delta function then $\\Psi (q_1,q_2)$ is a delta function in $q_2$ $\\forall q_1$ ).","Therefore, if a degree of freedom is maximally supported in one degree of freedom for all others fixed, then it is maximally supported for all values of the other degrees of freedom.","On the other hand, if it is a delta function in one degree of freedom for all others fixed, then it is a delta function for all values of the other degrees of freedom.","Assume that for $q_1, q_2 \\in \\lbrace 0, \\ldots , d-1\\rbrace $ , $\\exists \\, q^{\\prime }_1, q^{\\prime \\prime }_1$ such that $\\Psi (q^{\\prime }_1,q_2) = 0$ for some $q_2$ and $\\Psi (q^{\\prime \\prime }_1, q_2) \\ne 0$ $\\forall q_2$ .","We proceed to prove by contradiction.","Hence $\\Psi (q^{\\prime \\prime }_1, q_2) \\equiv \\Psi _{q^{\\prime \\prime }_1} \\propto \\left[ \\frac{2 \\pi i}{d} \\left( \\theta _{q^{\\prime \\prime }_1} q^2_2 + \\eta _{q^{\\prime \\prime }_1} q_2 \\right) \\right]$ by Corollary REF and $\\Psi (q^{\\prime }_1,q_2) \\equiv \\Psi _{q^{\\prime }_1}(q_2) \\propto \\delta _{q_2, q(q^{\\prime }_1)}$ for some $q(q^{\\prime }_1)\\in \\mathbb {Z}/d\\mathbb {Z}$ by [3].","We can rotate in $p_2$ -$q_2$ space to form a new basis $q^*_2$ in $d$ discrete angles (since $\\theta _{q^{\\prime \\prime }_1} \\in \\mathbb {Z}/d\\mathbb {Z}$ ) such that $\\Psi _{q^{\\prime }_1}(q^*_2) \\ne 0$ (since a delta function is not maximally supported only at the one angle perpendicular to it).","Since there exists $(d-1)$ other values of $q_1$ other than $q^{\\prime }_1$ , it follows that there exists at least one such angle such that $\\Psi _{q_1}(q^*_2) \\ne 0$ $\\forall q_1, q^*_2 \\in \\lbrace 0,\\ldots ,d-1\\rbrace $ .","We define $q^*_2$ as the basis that is rotated by this angle with respect to $q_2$ .","By Corollary REF , this means that $\\Psi ^*(q_1,q^*_2) \\propto \\exp ( \\theta ^{\\prime }_{11} q^2_1 + \\theta ^{\\prime }_{22} {q^*_2}^2 + 2 \\theta ^{\\prime }_{12} q_1 q^*_2 + \\eta ^{\\prime }_1 q_1 + \\eta ^{\\prime }_2 q_2),$ where by $\\Psi ^*$ we mean $\\Psi $ expressed in the new basis $q^*_2$ in its second degree of freedom.","Hence, $\\Psi ^*_{q_1}(q^*_2) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( \\theta ^{\\prime }_{q_1} {q^*_2}^2 + \\eta ^{\\prime }_{q_1} q^*_2 \\right) \\right] \\exp \\left[ \\frac{2 \\pi i}{d} \\theta _{12} q_1 q^*_2 \\right].$ Acting on this last equation to rotate back to $q_2$ , we must produce $\\Psi _{q^{\\prime \\prime }_1}(q_2) \\propto \\delta _{q_2,q(q^{\\prime \\prime }_1)}$ .","But then Eq.","REF implies that $\\Psi _{q^{\\prime }_1}(q^*_2)$ must also be proportional to $\\delta _{q_2,q(q^{\\prime }_1)}$ .","This is a contradiction.","Therefore, if a degree of freedom is maximally supported for all others fixed, then it is maximally supported for all values of the other degrees of freedom and vice-versa.","In the latter case, a position state in the $i$ th degree of freedom can be represented as a Gaussian by using the $p$ -basis where it becomes a plane wave ($\\theta _i = 0$ ).","In other words, one can always choose $x_i$ to be $p_i$ or $q_i$ such that Eq.","REF holds for odd prime $d$ .","$\\Box $ Finally, the reason this result does not hold for odd non-prime $d$ is that for discrete Wigner space there are $d + 1$ unique angles minus all the prime factors of $d$ .","For non-prime $d$ , there is more than one such prime factor and so there are cases when one cannot “rotate” away all non-maximally supported states."],["Discrete Phase Space Representation of Universal Quantum Computing","A similar statement to the one we made in Section —that any operator can be expressed as an infinite sum of path integral contribution truncated at order $\\hbar ^1$ —can be made in discrete systems.","However, there is an important difference in the number of terms making up the sum.","To see this we can follow reasoning that is similar to that employed in the continuous case.","Namely, from Eq.","REF we see that any discrete operator can also be expressed as a linear combination of reflections, but unlike the continuous case, this sum has a finite number of terms.","Since reflections can be expressed fully by the discrete path integral truncated at order $\\hbar ^1$ , as discussed previously, it follows that any unitary operator in discrete systems can be expressed as a finite sum of contributions from path integrals truncated at order $\\hbar ^1$ .","Again, the same statement can be made by considering the chord representation in terms of translations.","Hence, quantum propagation in discrete systems can be fully treated by a finite sum of contributions from a path integral approach truncated at order $\\hbar ^1$ .","To gather some understanding of this statement, we can consider what is necessary to add to our path integral formulation when we complete the Clifford gates with the T-gate, which produces a universal gate set.","The T-gate is generalized to odd $d$ -dimensions by $\\hat{T} = \\sum _{j \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{4}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ This gate can no longer be characterized by an $\\mathcal {M}$ with integer entries.","In particular, $\\mathcal {M}_{\\hat{T}} = \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{T}} = \\left( -\\frac{1}{4}, 0 \\right)$ .","This corresponds to $\\mathcal {B}_{\\hat{T}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & -\\frac{1}{4}\\end{array} \\right).$ Thus, the center function $T_x(x_p,x_q) = e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)},$ corresponding to the phase shift Hamiltonian applied for only half the unit of time.","The operator can thus be written: $\\hat{T} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d\\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{4} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q}.$ Though this operator is quadratic, it no longer takes the Weyl center points to themselves.","This means that the $\\hbar ^0$ limit of Eq.","REF is now insufficient to capture all the dynamics because the overlap with any $\\left|q\\right\\rangle $ will now involve a linear superposition of partially overlapping propagated manifolds.","It must therefore be described by a path integral formulation that is complete to order $\\hbar ^1$ .","In particular, $\\hat{T} = d^{-1} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})\\end{array}} e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)} \\hat{R}( x_p, x_q ),$ where $\\hat{R}$ should be substituted by its path integral.","Note that this does not imply efficient classical simulation of quantum computation but quite the opposite.","Indeed, for $n$ qudits, there are $d^{2n}$ terms in the sum above.","While every Weyl phase space point has only a single associated path when acted on by Clifford gates, this is no longer true in any calculation of evolution under the T-gate.","Eq.","REF expresses the T-gate as a sum over phase space operators (the reflections) evaluated on all the phase space points.","Thus, it can be interpreted as associating an exponentially large number of paths to every phase space point, instead of the single paths found for Clifford gates.","Therefore, any simulation of the T-gate naively necessitates adding up an exponential large sum over paths and so is comparably inefficient."],["Conclusion","The treatment presented here formalizes the relationship between stabilizer states in the discrete case and Gaussians in the continuous case, which has often been pointed out [3].","Namely, only Gaussians that lie along Weyl phase space points directly correspond to Gaussians in the continuous world in terms of preserving their form under a harmonic Hamiltonian, an evolution that is fully describable by truncating the path integral at order $\\hbar ^0$ .","Furthermore, we showed that the Clifford group gates, generated by the Hadamard, phase shift and controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order.","We found that this was because their Hamiltonians are harmonic and take Weyl phase space points to themselves.","This proves the Gottesman-Knill theorem.","The T-gate, needed to complete a universal set with the Hadamard, was shown not to satisfy these properties, and so requires a path integral treatment that is complete up to $\\hbar ^1$ .","The latter treatment includes a sum of terms for which the number of terms scales exponentially with the number of qudits.","We note that our observations pertaining to classical propagation in continuous systems have long been very well known.","In the continuous case, the Wigner function of a quantum state is non-negative if and only if the state is a Gaussian [34] and it has also long been known that quantum propagation from one Gaussian state to another only requires propagation up to order $\\hbar ^0$ [11].","Indeed, it has been shown that this is a continuous version of “stabilizer state propagation” in finite systems [35], and is therefore, in principle, useful for quantum error correction and cluster state quantum computation [36], [37].","It is also well known in the discrete case that quadratic Hamiltonians can act classically and be represented by symplectic transformations in the study of quantum cat maps [38], [14], [25] and linear transformations between propagated Wigner functions [39].","Interestingly though, this latter work appears to have predated the discovery that stabilizer states have positive-definite Wigner functions [3] and therefore, as far as we know, has not been directly related to stabilizer states and the $\\hbar ^0$ limit of their path integral formulation, which is a relatively recent topic of particular interest to the quantum information community and those familiar with Gottesman-Knill.","Otherwise, this claim has been pointed out in terms of concepts related to positivity and related concepts in past work [40], [2].","We also note that our exploration of continous systems is not meant to explore the highly related topic of continuous-variable quantum information.","Many topics therein apply to our discussion here, such as the continuous stabilizer state propagation we mentioned above.","However, our intention in introducing the continuous infinite-dimensional case was not to address these topics but to instead relate the established continuous semiclassical formalism to the discrete case, and thereby bridge the notions of phase space and dynamics between the two worlds.","There is an interesting observation to be made of the weights of the reflections that make up the complete path integral formulation of a unitary operator.","Namely, as is clear in Eqs.","REF and REF , the coefficients consist of the exponentiated center generating function multiplied by $\\frac{i}{\\hbar }$ .","This is very similar to the form of the vVMG path integral in Eqs.","REF and REF .","However, in Eqs.","REF and REF , reflections serve as the prefactors measuring the reflection spectral overlap of a propagated state with its evolute and the center generating actions provide the quantal phase.","Thus, this formulation can be interpreted as an alternative path integral formulation of the vVMG, one consisting of reflections as the underlying classical trajectory only, instead of the more tailored trajectories that result from applying the method of steepest descents directly on an operator.","The fact that any unitary operator in the discrete case can be expressed as a sum consisting of a finite number of order $\\hbar ^1$ path integral contributions, has the added interesting implication that uniformization—higher order $\\hbar $ corrections to the “primitive” semiclassical forms such as Eq.","REF —isn't really necessary in discrete systems.","Uniformization is characterized by the proper treatment of coalescing saddle points and has long been a subject of interest in continuous systems where “anharmonicity” bedevils computationally efficient implementation.","It seems that this problem isn't an issue in the discrete case since a fully complete sum with a finite number of terms, naively numbering $d^2$ for one qudit, exists.","As a last point, there is perhaps an alternative way to interpret the results presented here, one in terms of “resources”.","Much like “magic” (or contextuality) and quantum discord can be framed as a resource necessary to perform quantum operations that have more power than classical ones, it is possible to frame the order in $\\hbar $ that is necessary in the underlying path integral describing an operation as a resource necessary for quantumness.","In this vein, it can be said that Clifford gate operations on stabilizer states are operations that only require $\\hbar ^0$ resources while supplemental gates that push the operator space into universal quantum computing require $\\hbar ^1$ resources.","The dividing line between these two regimes, the classical and quantum world, is discrete, unambiguous and well-defined."],["Acknowledgments","The authors thank Prof. Alfredo Ozorio de Almeida for very fruitful discussions about the center-chord representation in discrete systems and Byron Drury for his help proof-reading and bringing [12] to our attention.","This work was supported by AFOSR award no.","FA9550-12-1-0046."],["Appendix","Gross proved that for odd prime $d$ [41]: Lemma 2 Let $\\Psi $ be a state vector with positive Wigner function for odd prime $d$ .","If $\\Psi $ is supported on two points, then it has maximal support.","With this lemma in mind, we can offer an alternative proof of Corollary REF : Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .","Proof We will show that for odd prime $d$ , every degree of freedom can only be fully supported or a Kronecker delta, for all other degrees of freedom fixed; WLOG we will consider a two-dimensional stabilizer state $\\Psi (q_1,q_2)$ and show that if $\\exists \\, q^{\\prime }_1$ such that $\\Psi (q^{\\prime }_1, q_2) \\ne 0$ $\\forall q_2$ then $\\Psi (q_1,q_2)\\ne 0$ $\\forall \\, q_1, q_2$ and vice-versa (if $\\exists \\, q^{\\prime }_1$ s.t.","$\\Psi (q^{\\prime }_1, q_2)$ is a delta function then $\\Psi (q_1,q_2)$ is a delta function in $q_2$ $\\forall q_1$ ).","Therefore, if a degree of freedom is maximally supported in one degree of freedom for all others fixed, then it is maximally supported for all values of the other degrees of freedom.","On the other hand, if it is a delta function in one degree of freedom for all others fixed, then it is a delta function for all values of the other degrees of freedom.","Assume that for $q_1, q_2 \\in \\lbrace 0, \\ldots , d-1\\rbrace $ , $\\exists \\, q^{\\prime }_1, q^{\\prime \\prime }_1$ such that $\\Psi (q^{\\prime }_1,q_2) = 0$ for some $q_2$ and $\\Psi (q^{\\prime \\prime }_1, q_2) \\ne 0$ $\\forall q_2$ .","We proceed to prove by contradiction.","Hence $\\Psi (q^{\\prime \\prime }_1, q_2) \\equiv \\Psi _{q^{\\prime \\prime }_1} \\propto \\left[ \\frac{2 \\pi i}{d} \\left( \\theta _{q^{\\prime \\prime }_1} q^2_2 + \\eta _{q^{\\prime \\prime }_1} q_2 \\right) \\right]$ by Corollary REF and $\\Psi (q^{\\prime }_1,q_2) \\equiv \\Psi _{q^{\\prime }_1}(q_2) \\propto \\delta _{q_2, q(q^{\\prime }_1)}$ for some $q(q^{\\prime }_1)\\in \\mathbb {Z}/d\\mathbb {Z}$ by [3].","We can rotate in $p_2$ -$q_2$ space to form a new basis $q^*_2$ in $d$ discrete angles (since $\\theta _{q^{\\prime \\prime }_1} \\in \\mathbb {Z}/d\\mathbb {Z}$ ) such that $\\Psi _{q^{\\prime }_1}(q^*_2) \\ne 0$ (since a delta function is not maximally supported only at the one angle perpendicular to it).","Since there exists $(d-1)$ other values of $q_1$ other than $q^{\\prime }_1$ , it follows that there exists at least one such angle such that $\\Psi _{q_1}(q^*_2) \\ne 0$ $\\forall q_1, q^*_2 \\in \\lbrace 0,\\ldots ,d-1\\rbrace $ .","We define $q^*_2$ as the basis that is rotated by this angle with respect to $q_2$ .","By Corollary REF , this means that $\\Psi ^*(q_1,q^*_2) \\propto \\exp ( \\theta ^{\\prime }_{11} q^2_1 + \\theta ^{\\prime }_{22} {q^*_2}^2 + 2 \\theta ^{\\prime }_{12} q_1 q^*_2 + \\eta ^{\\prime }_1 q_1 + \\eta ^{\\prime }_2 q_2),$ where by $\\Psi ^*$ we mean $\\Psi $ expressed in the new basis $q^*_2$ in its second degree of freedom.","Hence, $\\Psi ^*_{q_1}(q^*_2) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( \\theta ^{\\prime }_{q_1} {q^*_2}^2 + \\eta ^{\\prime }_{q_1} q^*_2 \\right) \\right] \\exp \\left[ \\frac{2 \\pi i}{d} \\theta _{12} q_1 q^*_2 \\right].$ Acting on this last equation to rotate back to $q_2$ , we must produce $\\Psi _{q^{\\prime \\prime }_1}(q_2) \\propto \\delta _{q_2,q(q^{\\prime \\prime }_1)}$ .","But then Eq.","REF implies that $\\Psi _{q^{\\prime }_1}(q^*_2)$ must also be proportional to $\\delta _{q_2,q(q^{\\prime }_1)}$ .","This is a contradiction.","Therefore, if a degree of freedom is maximally supported for all others fixed, then it is maximally supported for all values of the other degrees of freedom and vice-versa.","In the latter case, a position state in the $i$ th degree of freedom can be represented as a Gaussian by using the $p$ -basis where it becomes a plane wave ($\\theta _i = 0$ ).","In other words, one can always choose $x_i$ to be $p_i$ or $q_i$ such that Eq.","REF holds for odd prime $d$ .","$\\Box $ Finally, the reason this result does not hold for odd non-prime $d$ is that for discrete Wigner space there are $d + 1$ unique angles minus all the prime factors of $d$ .","For non-prime $d$ , there is more than one such prime factor and so there are cases when one cannot “rotate” away all non-maximally supported states."]],"string":"[\n [\n \"Semiclassical Formulation of Gottesman-Knill and Universal Quantum\\n Computation\"\n ],\n [\n \"Abstract We give a path integral formulation of the time evolution of qudits of odd dimension.\",\n \"This allows us to consider semiclassical evolution of discrete systems in terms of an expansion of the propagator in powers of $\\\\hbar$.\",\n \"The largest power of $\\\\hbar$ required to describe the evolution is a traditional measure of classicality.\",\n \"We show that the action of the Clifford operators on stabilizer states can be fully described by a single contribution of a path integral truncated at order $\\\\hbar^0$ and so are \\\"classical,\\\" just like propagation of Gaussians under harmonic Hamiltonians in the continuous case.\",\n \"Such operations have no dependence on phase or quantum interference.\",\n \"Conversely, we show that supplementing the Clifford group with gates necessary for universal quantum computation results in a propagator consisting of a finite number of semiclassical path integral contributions truncated at order $\\\\hbar^1$ , a number that nevertheless scales exponentially with the number of qudits.\",\n \"The same sum in continuous systems has an infinite number of terms at order $\\\\hbar^1$.\"\n ],\n [\n \"Introduction\",\n \"The study of contextuality in quantum information has led to progress in our understanding of the Wigner function for discrete systems.\",\n \"Using Wootters' original derivation of discrete Wigner functions [1], Eisert [2], Gross [3], and Emerson [4] have pushed forward a new perspective on the quantum analysis of states and operators in finite Hilbert spaces by considering their quasiprobability representation on discrete phase space.\",\n \"Most notably, the positivity of such representations has been shown to be equivalent to non-contextuality, a notion of classicality [4], [5].\",\n \"Quantum gates and states that exhibit these features are the stabilizer states and Clifford operations used in quantum error correction and stabilizer codes.\",\n \"The non-contextuality of stabilizer states and Clifford operations explains why they are amenable to efficient classical simulation [6], [7].\",\n \"This progress raises the question of how these discrete techniques are connected to prior established methods for simulating quantum mechanics in phase space.\",\n \"A particularly relevant method is trajectory-based semiclassical propagation, which has been widely used in the continuous context.\",\n \"Perhaps, when applied to the discrete case, semiclassical propagators can lend their physical intuition to outstanding problems in quantum information.\",\n \"Conversely, concepts from quantum information may serve to illuminate the comparatively older field of continuous semiclassics.\",\n \"Quantum information attempts to classify the “quantumness” of a system by the presence or absence of various quantum resources.\",\n \"Semiclassical analysis proceeds by successive approximation using $\\\\hbar $ as a small parameter, where the power of $\\\\hbar $ required is a measure of “quantumness”.\",\n \"Can these two views of quantum vs. classical be related?\",\n \"In the current paper, we build a bridge from the continuous semiclassical world to the discrete world and examine the classical-quantum characteristics of discrete quantum gates found in circuit models and their stabilizer formalism.\",\n \"Stabilizer states are eigenvalue one eigenvectors of a commuting set of operators making up a group which does not contain $-\\\\mathbb {I}$ .\",\n \"The set of stabilizer states is preserved by elements of the Clifford group, which is the normalizer of the Pauli group, and can be simulated very efficiently.\",\n \"More precisely, by the Gottesman-Knill Theorem, for $n$ qubits, a quantum circuit of a Clifford gate can be simulated using $\\\\mathcal {O}(n)$ operations on a classical computer.\",\n \"Measurements require $\\\\mathcal {O}(n^2)$ operations with $\\\\mathcal {O}(n^2)$ bits of storage [6], [7].\",\n \"The reason that stabilizer evolution by Clifford gates can be efficiently simulated classically has been explained in various ways.\",\n \"For instance, as already mentioned, stabilizer states have been shown to be non-contextual in qudit [4] and rebit [8] systems.\",\n \"The obstacle to proving this for qubits is that qubit systems possess state-independent contextuality [9].\",\n \"Of course, we know how to simulate qubit stabilizer states and Clifford operations efficiently by the Gottesmann-Knill theorem [7].\",\n \"For recent progress relating non-contextuality to classical simulatability for qubits we refer the reader to [10].\",\n \"It has also been shown for dimensions greater than two that a state of a discrete system is a stabilizer state if and only if its appropriately defined discrete Wigner function is non-negative [3].\",\n \"Therefore, when acted on by positive-definite operators, it can be considered as a proper positive-definite (classical) distribution.\",\n \"Here, we instead relate the concept of efficient classical simulation to the power of $\\\\hbar $ that a path integral treatment must be expanded to in order to describe the quantum evolution of interest.\",\n \"It is well known that Gaussian propagation in continuous systems under harmonic Hamiltonians can be described with a single contribution from the path integral truncated at order $\\\\hbar ^0$ [11].\",\n \"We show that the corresponding case in discrete systems exists.\",\n \"In the discrete case, stabilizer states take the place of Gaussians and harmonic Hamiltonians that additionally preserve the discrete phase space take the place of the general continuous harmonic Hamiltonians.\",\n \"In the discrete case we will only consider $d$ -dimensional systems for odd $d$ since their center representation (or Weyl formalism) is far simpler.\",\n \"As a consequence, we will show that operations with Clifford gates on stabilizer states can be treated by a path integral independent of the magnitude of $\\\\hbar $ and are thus fundamentally classical.\",\n \"Such operations have no dependence on phase or quantum interference.\",\n \"This can be viewed as a restatement of the Gottesman-Knill theorem in terms of powers of $\\\\hbar $ .\",\n \"We also consider more general propagation for discrete quantum systems.\",\n \"Quantum propagation in continuous systems can be treated by a sum consisting of an infinite number of contributions from the path integral truncated at order $\\\\hbar ^1$ .\",\n \"In discrete systems, we show that the corresponding sum consists of a finite number of terms, albeit one that scales exponentially with the number of qudits.\",\n \"This work also answers a question posed by the recent work of Penney et al.\",\n \"that explored a “sum-over-paths” expression for Clifford circuits in terms of symplectomorphisms on phase-space and associated generating actions.\",\n \"Penney et al.\",\n \"raised the question of how to relate the dynamics of the Wigner representation of (stabilizer) states to the dynamics which are the solutions of the discrete Euler-Lagrange equations for an associated functional [12].\",\n \"By relying on the well-established center-chord (or Wigner-Weyl-Moyal) formalism in continuous [13] and discrete systems [14], we show how the dynamics of Wigner representations are governed by such solutions related to a “center generating” function and that these solutions are harmonic and classical in nature.\",\n \"We begin by giving an overview of the center-chord representation in continuous systems in Section .\",\n \"Then, Section introduces the expansion of the path integral in powers of $\\\\hbar $ .\",\n \"This leads us to show what “classical” simulability of states in the continuous case corresponds to and to what higher order of $\\\\hbar $ an expansion is necessary to treat any quantum operator.\",\n \"Section then introduces the discrete variable case and defines its corresponding conjugate position and momentum operators.\",\n \"The path integral in discrete systems is then introduced in Section and, in Section , we define the Clifford group and stabilizer states.\",\n \"We prove that stabilizer state propagation within the Clifford group is captured fully up to order $\\\\hbar ^0$ and so is efficiently simulable classically.\",\n \"Section shows that extending the Clifford group to a universal gate set necessitates an expansion of the semiclassical propagator to a finite sum at order $\\\\hbar ^1$ .\",\n \"Finally, we close the paper with some discussion and directions for future work in Section .\"\n ],\n [\n \"Center-Chord Representation in Continuous Systems\",\n \"We define position operators $\\\\hat{q}$ , $\\\\hat{q} \\\\left|q^{\\\\prime }\\\\right\\\\rangle = q^{\\\\prime } \\\\left|q^{\\\\prime }\\\\right\\\\rangle $ , and momentum operators $\\\\hat{p}$ as their Fourier transform, $\\\\hat{p} = \\\\hat{\\\\mathcal {F}}^\\\\dagger \\\\hat{q} \\\\hat{\\\\mathcal {F}}$ , where $\\\\hat{F} = h^{\\\\frac{n}{2}} \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} p \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} q \\\\exp \\\\left(\\\\frac{2 \\\\pi i}{\\\\hbar } p \\\\cdot q \\\\right) \\\\left|p\\\\right\\\\rangle \\\\left\\\\langle q\\\\right|.$ Since $[\\\\hat{q}, \\\\hat{p}] = i \\\\hbar $ , these operators produce a particularly simple Lie algebra and are the generators of a Lie group.\",\n \"In this Lie group we can define the “boost” operator: $\\\\hat{Z}^{\\\\delta p} \\\\left|q^{\\\\prime }\\\\right\\\\rangle = e^{\\\\frac{i}{\\\\hbar } \\\\hat{q} \\\\delta p} \\\\left|q^{\\\\prime }\\\\right\\\\rangle = e^{\\\\frac{i}{\\\\hbar } q^{\\\\prime } \\\\delta p} \\\\left|q^{\\\\prime }\\\\right\\\\rangle ,$ and the “shift” operator: $\\\\hat{X}^{\\\\delta q} \\\\left|q^{\\\\prime }\\\\right\\\\rangle = e^{-\\\\frac{i}{\\\\hbar } \\\\hat{p} \\\\delta q} \\\\left|q^{\\\\prime }\\\\right\\\\rangle = \\\\left|q^{\\\\prime } + \\\\delta q\\\\right\\\\rangle .$ Using the canonical commutation relation and $e^{\\\\hat{A} + \\\\hat{B}} = e^{\\\\hat{A}} e^{\\\\hat{B}} e^{-\\\\frac{1}{2}[\\\\hat{A}, \\\\hat{B}]}$ if $[\\\\hat{A}, \\\\hat{B}]$ is a constant, it follows that $\\\\hat{Z} \\\\hat{X} = e^{\\\\frac{i}{\\\\hbar }} \\\\hat{X} \\\\hat{Z}.$ This is known as the Weyl relation and shows that the product of a shift and a boost (a generalized translation) in phase space is only unique up to a phase governed by $\\\\hbar $ .\",\n \"We proceed to introduce the chord representation of operators and states [13].\",\n \"The generalized phase space translation operator (often called the Weyl operator) is defined as a product of the shift and boost: $\\\\hat{T}(\\\\xi _p, \\\\xi _q) = e^{-\\\\frac{i}{2 \\\\hbar } \\\\xi _p \\\\cdot \\\\xi _q} \\\\hat{Z}^{ \\\\xi _p} \\\\hat{X}^{ \\\\xi _q},$ where $\\\\xi \\\\equiv (\\\\xi _p, \\\\xi _q) \\\\in \\\\mathbb {R}^{2n}$ define the chord phase space.\",\n \"$\\\\hat{T}(\\\\xi _p, \\\\xi _q)$ is a translation by the chord $\\\\xi $ in phase space.\",\n \"This can be seen by examining its effect on position and momentum states: $\\\\hat{T}(\\\\xi _p, \\\\xi _q) \\\\left|q\\\\right\\\\rangle = e^{\\\\frac{i}{\\\\hbar } \\\\left(q + \\\\frac{\\\\xi _q}{2}\\\\right) \\\\cdot \\\\xi _p} \\\\left|q+\\\\xi _q\\\\right\\\\rangle ,$ and $\\\\hat{T}(\\\\xi _p, \\\\xi _q) \\\\left|p\\\\right\\\\rangle = e^{-\\\\frac{i}{\\\\hbar } \\\\left(p + \\\\frac{\\\\xi _p}{2}\\\\right) \\\\cdot \\\\xi _q} \\\\left|p+\\\\xi _p\\\\right\\\\rangle ,$ which are shown in Fig.\",\n \"REF .\",\n \"Changing the order of shifts $\\\\hat{X}$ and boosts $\\\\hat{Z}$ changes the phase of the translation in phase space by $\\\\xi $ , as given by the Weyl relation above (Eq.\",\n \"REF ).\",\n \"An operator $\\\\hat{A}$ can be expressed as a linear combination of these translations: $\\\\hat{A} = \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} \\\\xi _p \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} \\\\xi _q \\\\, A_\\\\xi (\\\\xi _p, \\\\xi _q) \\\\hat{T}(\\\\xi _p, \\\\xi _q),$ where the weights are: $A_\\\\xi (\\\\xi _p, \\\\xi _q) = \\\\operatorname{Tr}\\\\left( \\\\hat{T}(\\\\xi _p, \\\\xi _q)^\\\\dagger \\\\hat{A} \\\\right).$ These weights give the chord representation of $\\\\hat{A}$ .\",\n \"Figure: Translation of a) a position state and b) a momentum state along the chord (ξ p ,ξ q )(\\\\xi _p, \\\\xi _q) in phase space.The Weyl function, or center representation, is dual to the chord representation.\",\n \"It is defined in terms of reflections instead of translations.\",\n \"We can define the reflection operator $\\\\hat{R}$ as the symplectic Fourier transform of the translation operator: $&&\\\\hat{R}(x_p, x_q) = \\\\left(2 \\\\pi \\\\hbar \\\\right)^{-n} \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} \\\\xi e^{\\\\frac{i}{\\\\hbar } {\\\\xi }^T \\\\mathcal {J} {x} } \\\\hat{T}(\\\\xi )$ where $x \\\\equiv \\\\left(x_p, x_q\\\\right) \\\\in \\\\mathbb {R}^{2n}$ are a continuous set of Weyl phase space points or centers and $\\\\mathcal {J}$ is the symplectic matrix $\\\\mathcal {J} = \\\\left( \\\\begin{array}{cc} 0 & -\\\\mathbb {I}_{n}\\\\\\\\ \\\\mathbb {I}_{n} & 0 \\\\end{array}\\\\right),$ for $\\\\mathbb {I}_n$ the $n$ -dimensional identity.\",\n \"The association of this operator with reflection can be seen by examining its effect on position and momentum states: $\\\\hat{R}(x_p, x_q) \\\\left|q\\\\right\\\\rangle = e^{\\\\frac{i}{\\\\hbar } 2 (x_q - q) \\\\cdot x_p} \\\\left|2x_q-q\\\\right\\\\rangle ,$ and $\\\\hat{R}(x_p, x_q) \\\\left|p\\\\right\\\\rangle = e^{-\\\\frac{i}{\\\\hbar } 2 (x_p - p) \\\\cdot x_q} \\\\left|2x_p-p\\\\right\\\\rangle ,$ which are sketched in Fig.\",\n \"REF .\",\n \"It is thus evident that $\\\\hat{R}(x_p, x_q)$ reflects the phase space around $x$ .\",\n \"Note that while we refer to reflections in the symplectic sense here, and in the rest of the paper; Eqs.\",\n \"REF and REF show that they are in fact “an inversion around $x$ ” in every two-plane of conjugate ${x_p}_i$ and ${x_q}_i$ .\",\n \"However, we will keep to the established nomenclature [13].\",\n \"Figure: Reflection of a) a position state and b) a momentum state across the center (x p ,x q )(x_p, x_q) in phase space.An operator $\\\\hat{A}$ can now be expressed as a linear combination of reflections: $\\\\hat{A} = \\\\left(2 \\\\pi \\\\hbar \\\\right)^{-n} \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} x_p \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} x_q \\\\, A_x(x_p, x_q) \\\\hat{R}( x_p, x_q ),$ where $A_x(x_p, x_q) = \\\\operatorname{Tr}\\\\left( \\\\hat{R}(x_p, x_q)^\\\\dagger \\\\hat{A} \\\\right),$ and is called the center representation of $\\\\hat{A}$ .\",\n \"This representation is of particular interest to us because we can rewrite the components $A_x$ for unitary transformations $\\\\hat{A}$ as: $A_x(x_p, x_q) = e^{\\\\frac{i}{\\\\hbar } S(x_p, x_q)},$ where $S(x_p, x_q)$ is equivalent to the action the transformation $A_x$ produces in Weyl phase space in terms of reflections around centers $x$ [13].\",\n \"Thus, $S$ is also called the “center generating” function.\",\n \"For a pure state $\\\\left|\\\\Psi \\\\right\\\\rangle $ , the Wigner function given by Eq.\",\n \"REF simplifies to: $&& {\\\\Psi }_x(x_p, x_q) = \\\\left(2 \\\\pi \\\\hbar \\\\right)^{-n}\\\\\\\\&& \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} \\\\xi _q \\\\, \\\\Psi \\\\left( x_q + \\\\frac{\\\\xi _q }{2} \\\\right) {\\\\Psi ^*} \\\\left( x_q - \\\\frac{\\\\xi _q}{2} \\\\right) e^{-\\\\frac{i}{\\\\hbar } \\\\xi _q \\\\cdot x_p}.\",\n \"\\\\nonumber $ The center representation for quantum states immediately yields the well-known Wigner function for continuous systems.\",\n \"The chord representation is the symplectic Fourier transform of the Wigner function.\",\n \"The center and chord representations are dual to each other, and are the Wigner and Wigner characteristic functions res[ectively.\",\n \"Identifying the Wigner functions with the center representation, and the center representation as dual to the chord representation motivates the development of both center (Wigner) and chord representations for discrete systems in Section .\"\n ],\n [\n \"Path Integral Propagation in continuous Systems\",\n \"Propagation from one quantum state to another can be expressed in terms of the path integral formalism of the quantum propagator.\",\n \"For one degree of freedom, with an initial position $q$ and final position $q^{\\\\prime }$ , evolving under the Hamiltonian $H$ for time $t$ , the propagator is $\\\\left\\\\langle q|e^{-i H t/\\\\hbar }|q^{\\\\prime }\\\\right\\\\rangle = \\\\int \\\\mathcal {D}[q_t] \\\\, \\\\exp \\\\left(\\\\frac{i}{\\\\hbar } G[q_t]\\\\right)$ where $G[q_t]$ is the action of the trajectory $q_t$ , which starts at $q$ and ends at $q^{\\\\prime }$ a time $t$ later [15], [16].\",\n \"Eq.\",\n \"REF can be reexpressed as a variational expansion around the set of classical trajectories (a set of measure zero) that start at $q$ and end at $q^{\\\\prime }$ a time $t$ later.\",\n \"This is an expansion in powers of $\\\\hbar $ : $&& \\\\left\\\\langle q^{\\\\prime }|e^{-\\\\frac{i}{\\\\hbar }Ht}|q\\\\right\\\\rangle =\\\\\\\\&& \\\\sum _j^\\\\text{cl.\",\n \"paths} \\\\int \\\\mathcal {D}[q_{tj}] \\\\, e^{\\\\frac{i}{\\\\hbar } \\\\left( G[q_{tj}] + \\\\delta G[q_{tj}] + \\\\frac{1}{2} \\\\delta ^2 G[q_{tj}] + \\\\ldots \\\\right) }, \\\\nonumber $ where $\\\\delta G[q_{tj}]$ denotes a functional variation of the paths $q_{tj}$ and for classical paths $\\\\delta G[q_{tj}] = 0$ (For further details we refer the reader to Section 10.3 of [17]).\",\n \"Terminating Eq.\",\n \"REF to first order in $\\\\hbar $ produces the position state representation of the van Vleck-Morette-Gutzwiller propagator [18], [19], [20]: $&& \\\\left\\\\langle q^{\\\\prime }|e^{-\\\\frac{i}{\\\\hbar }Ht}|q\\\\right\\\\rangle = \\\\\\\\&& \\\\sum _j\\\\left( \\\\frac{- \\\\frac{\\\\partial ^2 G_{jt}(q,q^{\\\\prime })}{\\\\partial q \\\\partial q^{\\\\prime }}}{2 \\\\pi i \\\\hbar } \\\\right)^{1/2} e^{i \\\\frac{G_{jt}(q,q^{\\\\prime })}{\\\\hbar }} + \\\\mathcal {O}(\\\\hbar ^2).\\\\nonumber $ where the sum is over all classical paths that satisfy the boundary conditions.\",\n \"In the center representation, for $n$ degrees of freedom, the semiclassical propagator $U_t(x_p, x_q)$ becomes [13]: $&&U_t(x_p, x_q) = \\\\\\\\&& \\\\sum _j \\\\left\\\\lbrace \\\\det \\\\left[ 1 + \\\\frac{1}{2} \\\\mathcal {J} \\\\frac{\\\\partial ^2 S_{tj}}{\\\\partial x^2} \\\\right] \\\\right\\\\rbrace ^{\\\\frac{1}{2}} e^{\\\\frac{i}{\\\\hbar } S_{tj}(x_p, x_q)} + \\\\mathcal {O}(\\\\hbar ^2),\\\\nonumber $ where $S_{tj}(x_p, x_q)$ is the center generating function (or action) for the center $x = (x_p, x_q)\\\\equiv \\\\frac{1}{2}\\\\left[(p, q)+(p^{\\\\prime }, q^{\\\\prime })\\\\right]$ .\",\n \"In general this is an underdetermined system of equations and there are an infinite number of classical trajectories that satisfy these conditions.\",\n \"The accuracy of adding them up as part of this semiclassical approximation is determined by how separated these trajectories are with respect to $\\\\hbar $ —the saddle-point condition for convergence of the method of steepest descents.\",\n \"However, some Hamiltonians exhibit a single saddle point contribution and are thus exact at order $\\\\hbar ^1$ See Ehrenfest's Theorem, e.g.\",\n \"in [42]: Lemma 1 There is only one classical trajectory $(p, q)\\\\underset{t}{\\\\rightarrow } (p^{\\\\prime }, q^{\\\\prime })$ that satisfies the boundary conditions $(x_p, x_q) = \\\\frac{1}{2}\\\\left[(p, q)+(p^{\\\\prime }, q^{\\\\prime })\\\\right]$ and $t$ under Hamiltonians that are harmonic in $p$ and $q$ .\",\n \"Proof For a quadratic Hamiltonian, the diagonalized equations of motion for $n$ -dimensional $(p^{\\\\prime }, q^{\\\\prime })$ are of the form: $p^{\\\\prime }_i &=& \\\\alpha (t)_i p_i + \\\\beta (t)_i q_i + \\\\gamma (t)_i,\\\\\\\\q^{\\\\prime }_i &=& \\\\delta (t)_i p_i + \\\\epsilon (t)_i q_i + \\\\eta (t)_i,$ for $i \\\\in \\\\lbrace 1, 2, \\\\ldots , n\\\\rbrace $ .\",\n \"Since $t$ is known and $(p, q)$ can be written in terms of $(p^{\\\\prime }, q^{\\\\prime })$ by using $(x_p, x_q)$ , this brings the total number of linear equations to $2n$ with $2n$ unknowns and so there exists one unique solution.$\\\\Box $ Since the equations of motion for a harmonic Hamiltonian are linear, we can write their solutions as: $\\\\left( \\\\begin{array}{c}p^{\\\\prime }\\\\\\\\ q^{\\\\prime }\\\\end{array} \\\\right) = \\\\mathcal {M}_t \\\\left[ \\\\left( \\\\begin{array}{c}p\\\\\\\\ q \\\\end{array}\\\\right) + \\\\frac{1}{2} \\\\alpha _t \\\\right] + \\\\frac{1}{2} \\\\alpha _t,$ where $\\\\mathcal {\\\\alpha }_t$ is an $n$ -vector and $\\\\mathcal {M}_t$ is an $n\\\\times n$ symplectic matrix, both with entries in $\\\\mathbb {R}$ .\",\n \"In this case, the center generating function $S_t(x_p, x_q)$ is also quadratic, in particular $S_t(x_p, x_q) = \\\\alpha _t^T \\\\mathcal {J} \\\\left(\\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array}\\\\right) + (x_p, x_q) \\\\mathcal {B}_t \\\\left(\\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array}\\\\right),$ where $\\\\mathcal {B}_t$ is a real symmetric $n\\\\times n$ matrix that is related to $\\\\mathcal {M}_t$ by the Cayley parameterization of $\\\\mathcal {M}_t$ [22]: $\\\\mathcal {J} \\\\mathcal {B}_t = \\\\left( 1 + \\\\mathcal {M}_t \\\\right)^{-1} \\\\left( 1 - \\\\mathcal {M}_t \\\\right) = \\\\left( 1 - \\\\mathcal {M}_t \\\\right) \\\\left( 1 + \\\\mathcal {M}_t \\\\right)^{-1}.$ Since one classical trajectory contribution is sufficient in this case, if the overall phase of the propagated state is not important, then the expansion w.r.t.\",\n \"$\\\\hbar $ in Eq.\",\n \"REF can be truncated at order $\\\\hbar ^0$ .\",\n \"Dropping terms that are higher order than $\\\\hbar ^0$ and ignoring phase is equivalent to propagating the classical density $\\\\rho (x)$ corresponding to the $(p, q) $ -manifold, under the harmonic Hamiltonian and determining its overlap with the $(p^{\\\\prime }, q^{\\\\prime })$ -manifold after time $t$ .\",\n \"Such a treatment under a harmonic Hamiltonian results in just the absolute value of the prefactor of Eq.\",\n \"REF : $\\\\left| \\\\det \\\\left[ 1 + \\\\frac{1}{2} \\\\mathcal {J} \\\\frac{\\\\partial ^2 S_{tj}}{\\\\partial x^2} \\\\right] \\\\right|^{\\\\frac{1}{2}}$ .\",\n \"Indeed, this was van Vleck's discovery before quantum mechanics was formalized [18].\",\n \"The relative phases of different classical contributions are no longer a concern and the higher order terms only weigh such contributions appropriately.\",\n \"Here we are interested in propagating between Gaussian states in the center representation.\",\n \"In continuous systems, a Gaussian state in $n$ dimensions can be defined as: $\\\\Psi _\\\\beta (q) = \\\\left[\\\\pi ^{-n} \\\\det \\\\left( \\\\operatorname{\\\\text{Re}}{\\\\Sigma }_\\\\beta \\\\right) \\\\right]^{\\\\frac{1}{4}} \\\\exp \\\\left( \\\\varphi \\\\right),$ where $\\\\varphi = \\\\frac{i}{\\\\hbar } p_\\\\beta \\\\cdot \\\\left( q - {q}_\\\\beta \\\\right) - \\\\frac{1}{2} \\\\left( q - {q}_\\\\beta \\\\right)^T {\\\\Sigma }_\\\\beta \\\\left( q - q_\\\\beta \\\\right).$ $q_\\\\beta \\\\in \\\\mathbb {R}^n$ is the central position, $p_\\\\beta \\\\in \\\\mathbb {R}^n$ is the central momentum, and $\\\\Sigma _\\\\beta $ is a symmetric $n\\\\times n$ matrix where $\\\\operatorname{\\\\text{Re}}\\\\Sigma _\\\\beta $ is proportional to the spread of the Gaussian and $\\\\operatorname{\\\\text{Im}}\\\\Sigma _\\\\beta $ captures $p$ -$q$ correlation.\",\n \"This state describes momentum states ($\\\\delta (p - p_\\\\beta )$ in momentum representation) when $\\\\Sigma _\\\\beta \\\\rightarrow 0$ and position states $\\\\delta (q- q_\\\\beta )$ when $\\\\Sigma _\\\\beta \\\\rightarrow \\\\infty $ .\",\n \"Rotations between these two cases corresponds to $\\\\operatorname{\\\\text{Re}}\\\\Sigma _\\\\beta = 0$ and $\\\\operatorname{\\\\text{Im}}\\\\Sigma _\\\\beta \\\\ne 0$ .\",\n \"Gaussians remain Gaussians under evolution by a harmonic Hamiltonian, even if it is time-dependent.\",\n \"This can be shown by simply making the ansatz that the state remains a Gaussian and then solving for its time-dependent $\\\\Sigma _\\\\beta $ , $p_\\\\beta $ , $q_\\\\beta $ and phase from the time-dependent Schrödinger equation [17], or just by applying the analytically known Feynman path integral, which is equivalent to the van Vleck path integral, to a Gaussian [23].\",\n \"Moreover, with the propagator in the center representation known to only have have one saddle-point contribution for a harmonic Hamiltonian, it is fairly straight-forward to show that this is also true for its coherent state representation (that is, taking a Gaussian to another Gaussian).\",\n \"Applying the propagator to an initial and final Gaussian in the center representation $&& \\\\left[ \\\\left|\\\\Psi _\\\\beta \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi _\\\\beta \\\\right| U_t \\\\left|\\\\Psi _\\\\alpha \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi _\\\\alpha \\\\right| \\\\right]_x (x) = \\\\\\\\&& \\\\left( \\\\pi \\\\hbar \\\\right)^{-3n} \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} x_1 \\\\int ^\\\\infty _{-\\\\infty } \\\\mbox{d} x_2 \\\\, U_t(x_1 + x_2- x) \\\\nonumber \\\\\\\\&& \\\\qquad \\\\qquad \\\\qquad \\\\times {\\\\Psi _\\\\beta }_x(x_2) {\\\\Psi _\\\\alpha }_x (x_1) \\\\nonumber \\\\\\\\&& \\\\qquad \\\\qquad \\\\qquad \\\\times e^{2\\\\frac{i}{\\\\hbar }(x_1^T \\\\mathcal {J} {x_2} + x_2^T \\\\mathcal {J} {x} + x^T \\\\mathcal {J} {x_1})}, \\\\nonumber $ we see that since $U_t$ is a Gaussian from Eq.\",\n \"REF ($S_{tj}$ is quadratic for harmonic Hamiltonians) and since the Wigner representations of the Gaussians, ${\\\\Psi _\\\\alpha }_x$ and ${\\\\Psi _\\\\beta }_x$ , are also known to be Gaussians, the full integral in the above equation is a Gaussian integral and thus evaluates to produce a Gaussian with a prefactor.\",\n \"This is equivalent to evaluating the integral by the method of steepest descents which finds the saddle points to be the points that satisfy $\\\\frac{\\\\partial \\\\phi }{\\\\partial x} = 0$ where $\\\\phi $ is the phase of the integrand's argument.\",\n \"Since this argument is quadratic, its first derivative is linear and so again there is only one unique saddle point.\",\n \"Indeed, such an evaluation produces the coherent state representation of the vVMG propagator [24].\",\n \"Just as we found with the center representation, the absolute value of its prefactor corresponds to the order $\\\\hbar ^0$ term.\",\n \"As a consequence of this single contribution at order $\\\\hbar ^0$ , it follows that the Wigner function of a state, $\\\\Psi _x(x)$ , evolves under the operator $\\\\hat{V}$ with an underlying harmonic Hamiltonian by $\\\\Psi _x(\\\\mathcal {M}_{\\\\hat{V}}(x + \\\\alpha _{\\\\hat{V}}/2) + \\\\alpha _{\\\\hat{V}}/2)$ , where $\\\\mathcal {M}_{\\\\hat{V}}$ is the symplectic matric and $\\\\alpha _{\\\\hat{V}}$ is the translation vector associated with $\\\\hat{V}$ 's action [14].\",\n \"Before proceeding to the discrete case, we note that the center representation that we have defined allows for a particularly simple way to express how far the path integral treatment must be expanded in $\\\\hbar $ in order to describe any unitary propagation (not necessarily harmonic) in continuous quantum mechanics.\",\n \"Reflections and translations can also be described by truncating Eq.\",\n \"REF at order $\\\\hbar ^1$ (or $\\\\hbar ^0$ if overall phase isn't important) since they correspond to evolution under a harmonic Hamiltonian.\",\n \"In particular, translations are displacements along a chord $\\\\xi $ and so have Hamiltonians $H \\\\propto \\\\xi _q \\\\cdot p - \\\\xi _p \\\\cdot q$ .\",\n \"Reflections are symplectic rotations around a center $x$ and so have Hamiltonians $H \\\\propto \\\\frac{\\\\pi }{4}\\\\left[ (p- x_p)^2 + (q- x_q)^2 \\\\right]$ .\",\n \"From Eq.\",\n \"REF we see that any operator can be expressed as an infinite Riemann sum of reflections.\",\n \"Therefore, since reflections are fully described by a truncation at order $\\\\hbar ^1$ , it follows that an infinite Riemann sum of path integral solutions truncated at order $\\\\hbar ^1$ can describe any unitary evolution.\",\n \"The same statement can be made by considering the chord representation in terms of translations.\",\n \"Hence, quantum propagation in continuous systems can be fully treated by an infinite sum of contributions from a path integral approach truncated at order $\\\\hbar ^1$ .\",\n \"As an aside, in general this infinite sum isn't convergent and so it is often more useful to consider reformulations that involve a sum with a finite number of contributions.\",\n \"One way to do this is to apply the method of steepest descents directly on the operator of interest and use the area between saddle points as the metric to determine the order of $\\\\hbar $ necessary, instead of dealing with an infinity of reflections (or translations).\",\n \"This results in the semiclassical propagator already presented, but associated with the full Hamiltonian instead of a sum of reflection Hamiltonians.\",\n \"In summary, we have explained why propagation between Gaussian states under Hamiltonians that are harmonic is simulable classically (i.e.\",\n \"up to order $\\\\hbar ^0$ ) in continuous systems.\",\n \"We will see that the same situation holds in discrete systems for stabilizer states, with the additional restriction that the propagation takes the phase space points, which are now discrete, to themselves.\",\n \"Discrete Center-Chord Representation We now proceed to the discrete case and introduce the center-chord formalism for these systems.\",\n \"It will be useful for us to define a pair of conjugate degrees of freedom $p$ and $q$ for discrete systems.\",\n \"Unfortunately, this isn't as straight-forward as in the continuous case, since the usual canonical commutation relations cannot hold in a finite-dimensional Hilbert space where the operators are bounded (since $\\\\operatorname{Tr}[\\\\hat{p}, \\\\hat{q}] = 0$ ).\",\n \"We begin in one degree of freedom.\",\n \"We label the computational basis for our system by $n \\\\in {0, 1, \\\\ldots , d-1}$ , for $d$ odd and we assume that $d$ is odd for the rest of this paper.\",\n \"We identify the discrete position basis with the computational basis and define the “boost” operator as diagonal in this basis: $\\\\hat{Z}^{\\\\delta p} \\\\left|n\\\\right\\\\rangle \\\\equiv \\\\omega ^{n \\\\delta p} \\\\left|n\\\\right\\\\rangle ,$ where $\\\\omega $ will be defined below.\",\n \"We define the normalized discrete Fourier transform operator to be equivalent to the Hadamard gate: $\\\\hat{F} = \\\\frac{1}{\\\\sqrt{d}}\\\\sum _{m,n \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}} \\\\omega ^{m n} \\\\left|m\\\\right\\\\rangle \\\\left\\\\langle n\\\\right|.$ This allows us to define the Fourier transform of $\\\\hat{Z}$ : $\\\\hat{X} \\\\equiv \\\\hat{F}^\\\\dagger \\\\hat{Z} \\\\hat{F}$ Again, as before, we call $\\\\hat{X}$ the “shift” operator since $\\\\hat{X}^{\\\\delta q} \\\\left|n\\\\right\\\\rangle \\\\equiv \\\\left|n\\\\oplus \\\\delta q\\\\right\\\\rangle ,$ where $\\\\oplus $ denotes mod-$d$ integer addition.\",\n \"It follows that the Weyl relation holds again: $\\\\hat{Z} \\\\hat{X} = \\\\omega \\\\hat{X} \\\\hat{Z}.$ The group generated by $\\\\hat{Z}$ and $\\\\hat{X}$ has a $d$ -dimensional irreducible representation only if $\\\\omega ^d=1$ for odd $d$ .\",\n \"Equivalently, there are only reflections relating any two phase space points on the Weyl phase space “grid” if $d$ is odd [25].\",\n \"We take $\\\\omega \\\\equiv \\\\omega (d) = e^{2 \\\\pi i/d}$ [26].\",\n \"This was introduced by Weyl [27].\",\n \"Note that this means that $\\\\hbar = \\\\frac{d}{2 \\\\pi }$ or $h = d$ .\",\n \"This means that the classical regime is most closely reached when the dimensionality of the system is reduced ($d \\\\rightarrow 0$ ) and thus the most “classical” system we can consider here is a qutrit (since we keep $d$ odd and greater than one).\",\n \"This is the opposite limit considered by many other approaches where the classical regime is reached when $d \\\\rightarrow \\\\infty $ .\",\n \"One way of interpreting the classical limit in this paper is by considering $h$ to be equal to the inverse of the density of states in phase space (i.e.\",\n \"in a Wigner unit cell).\",\n \"As $\\\\hbar \\\\rightarrow 0$ phase space area decreases as $\\\\hbar ^2$ but the number of states only decreases as $\\\\hbar $ leading to an overall density increase of $\\\\hbar $ .\",\n \"This agrees with the notion that states should become point particles of fixed mass in the classical limit.\",\n \"By analogy with continuous finite translation operators, we reexpress the shift $\\\\hat{X}$ and boost $\\\\hat{Z}$ operators in terms of conjugate $\\\\hat{ p}$ and $\\\\hat{ q}$ operators: $ \\\\hat{Z} \\\\left|n\\\\right\\\\rangle = e^{\\\\frac{2 \\\\pi i }{d}\\\\hat{q}} \\\\left|n\\\\right\\\\rangle = e^{\\\\frac{2 \\\\pi i}{d} n} \\\\left|n\\\\right\\\\rangle , $ and $ \\\\hat{X} \\\\left|n\\\\right\\\\rangle = e^{-\\\\frac{2 \\\\pi i }{d} \\\\hat{p}} \\\\left|n\\\\right\\\\rangle = \\\\left|n\\\\oplus 1\\\\right\\\\rangle .\",\n \"$ Hence, in the diagonal “position” representation for $\\\\hat{Z}$ : $\\\\hat{Z} = \\\\left( \\\\begin{array}{cccccc} 1 & 0 & 0 & \\\\cdots & 0\\\\\\\\0 & e^{\\\\frac{2 \\\\pi i}{d}} & 0 & \\\\cdots & 0\\\\\\\\0 & 0 & e^{\\\\frac{4 \\\\pi i}{d}} & \\\\cdots & 0\\\\\\\\\\\\vdots & \\\\vdots & \\\\vdots & \\\\ddots & \\\\vdots \\\\\\\\0 & 0 & 0 & 0 & e^{\\\\frac{2 (d-1) \\\\pi i}{d}} \\\\end{array} \\\\right),$ and $\\\\hat{X} = \\\\left( \\\\begin{array}{cccccc} 0 & 0 & \\\\cdots & 0 & 1\\\\\\\\1 & 0 & \\\\cdots & 0 & 0\\\\\\\\0 & 1 & 0 & \\\\cdots & 0\\\\\\\\\\\\vdots & \\\\ddots & \\\\ddots & \\\\ddots & \\\\vdots \\\\\\\\0 & \\\\cdots & 0 & 1 & 0 \\\\end{array} \\\\right).$ Thus, $\\\\hat{q} = \\\\frac{d}{2 \\\\pi i} \\\\log \\\\hat{Z} = \\\\sum _{n \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}} n \\\\left|n\\\\right\\\\rangle \\\\left\\\\langle n\\\\right|,$ and $\\\\hat{ p} = \\\\hat{F}^\\\\dagger \\\\hat{ q} \\\\hat{F}.$ Therefore, we can interpret the operators $\\\\hat{ p}$ and $\\\\hat{ q}$ as a conjugate pair similar to conjugate momenta and position in the continuous case.\",\n \"However, they differ from the latter in that they only obey the weaker group commutation relation $e^{i j \\\\hat{ q}/\\\\hbar }e^{i k \\\\hat{ p}/\\\\hbar } e^{-i j \\\\hat{ q}/\\\\hbar }e^{-i k \\\\hat{ p}/\\\\hbar } = e^{-i jk /\\\\hbar } \\\\hat{\\\\mathbb {I}}.$ This corresponds to the usual canonical commutation relation for $p$ and $q$ 's algebra at the origin of the Lie group ($j = k = 0$ ); expanding both sides of Eq.\",\n \"REF to first order in $p$ and $q$ yields the usual canonical relation.\",\n \"We proceed to introduce the Weyl representation of operators and states in discrete Hilbert spaces with odd dimension $d$ and $n$ degrees of freedom [1], [28], [29].\",\n \"The generalized phase space translation operator (the Weyl operator) is defined as a product of the shift and boost with a phase appropriate to the $d$ -dimensional space: $\\\\hat{T}(\\\\xi _p, \\\\xi _q) = e^{-i \\\\frac{\\\\pi }{d} \\\\xi _p \\\\cdot \\\\xi _q} \\\\hat{Z}^{ \\\\xi _p} \\\\hat{X}^{ \\\\xi _q},$ where $\\\\xi \\\\equiv (\\\\xi _p, \\\\xi _q) \\\\in (\\\\mathbb {Z} / d \\\\mathbb {Z})^{2n}$ and form a discrete “web” or “grid” of chords.\",\n \"They are a discrete subset of the continous chords we considered in the infinite-dimensional context in Section and their finite number is an important consequence of the discretization of the continuous Weyl formalism.\",\n \"Again, an operator $\\\\hat{A}$ can be expressed as a linear combination of translations: $\\\\hat{A} = d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _p, \\\\xi _q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} A_\\\\xi (\\\\xi _p, \\\\xi _q) \\\\hat{T}(\\\\xi _p, \\\\xi _q),$ where the weights are the chord representation of the function $\\\\hat{A}$ : ${A}_\\\\xi (\\\\xi _p, \\\\xi _q) = d^{-n} \\\\operatorname{Tr}\\\\left( \\\\hat{T}(\\\\xi _p, \\\\xi _q)^\\\\dagger \\\\hat{A} \\\\right).$ When applied to a state $\\\\hat{\\\\rho }$ , this is also called the “characteristic function” of $\\\\hat{\\\\rho }$ [30].\",\n \"As before, the center representation, based on reflections instead of translations, requires an appropriately defined reflection operator.\",\n \"We can define the discrete reflection operator $\\\\hat{R}$ as the symplectic Fourier transform of the discrete translation operator we just introduced: $\\\\hat{R}(x_p, x_q) = d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _p, \\\\xi _q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} e^{\\\\frac{2 \\\\pi i}{d} (\\\\xi _p, \\\\xi _q) \\\\mathcal {J} (x_p, x_q)^T} \\\\hat{T}(\\\\xi _p, \\\\xi _q).$ With this in hand, we can now express a finite-dimensional operator $\\\\hat{A}$ as a superposition of reflections: $\\\\hat{A} = d^{-n} \\\\sum _{\\\\begin{array}{c}x_p, x_q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} {A}_x(x_p, x_q) \\\\hat{R}( x_p, x_q ),$ where ${A}_x(x_p, x_q) = d^{-n} \\\\operatorname{Tr}\\\\left( \\\\hat{R}(x_p, x_q)^\\\\dagger \\\\hat{A} \\\\right).$ $x \\\\equiv (x_p, x_q) \\\\in (\\\\mathbb {Z} / d \\\\mathbb {Z})^{2n}$ are centers or Weyl phase space points and, like their $(\\\\xi _p, \\\\xi _q)$ brethren, form a discrete subgrid of the continuous Weyl phase space points considered in Section .\",\n \"Again, the center representation is of particular interest to us because for unitary gates $\\\\hat{A}$ we can rewrite the components ${A}_x(x_p, x_q)$ as: ${A}_x(x_p, x_q) = \\\\exp \\\\left[\\\\frac{i}{\\\\hbar } S(x_p, x_q)\\\\right]$ where $S(x_p, x_q)$ is the argument to the exponential, and is equivalent to the action of the operator in center representation (the center generating function).\",\n \"Aside from Eq.\",\n \"REF , the center representation of a state $\\\\hat{\\\\rho }$ can also be directly defined as the symplectic Fourier transform of its chord representation, $\\\\rho _\\\\xi $ [3]: $\\\\rho _x(x_p, x_q) = d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _p, \\\\xi _q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} e^{\\\\frac{2 \\\\pi i}{d} (\\\\xi _p, \\\\xi _q) \\\\mathcal {J} (x_p, x_q)^T} \\\\rho _\\\\xi (\\\\xi _p,\\\\xi _q).$ We note again that for a pure state $\\\\left|\\\\Psi \\\\right\\\\rangle $ , the Wigner function from Eqs.\",\n \"REF and REF simplifies to: ${\\\\Psi }_x(x_p, x_q) &=& d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _q \\\\in \\\\\\\\(\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} e^{-\\\\frac{2 \\\\pi i}{d} \\\\xi _q \\\\cdot x_p} \\\\Psi \\\\left( x_q + \\\\frac{(d+1) \\\\xi _q }{2} \\\\right) {\\\\Psi ^*} \\\\left( x_q - \\\\frac{(d+1) \\\\xi _q}{2} \\\\right).$ It may be clear from this short presentation that the chord and center representations are dual to each other.\",\n \"A thorough review of this subject can be found in [13].\",\n \"Path Integral Propagation in Discrete Systems Rivas and Almeida [14] found that the continuous infinite-dimensional vVMG propagator can be extended to finite Hilbert space by simply projecting it onto its finite phase space tori.\",\n \"This produces: $&& {U}_t(x) = \\\\\\\\&& \\\\left< \\\\sum _j \\\\left\\\\lbrace \\\\det \\\\left[ 1 + \\\\frac{1}{2} \\\\mathcal {J} \\\\frac{\\\\partial ^2 S_{tj}}{\\\\partial x^2} \\\\right] \\\\right\\\\rbrace ^{\\\\frac{1}{2}} e^{\\\\frac{i}{\\\\hbar } S_{tj}(x)} e^{i \\\\theta _k} \\\\right>_k + \\\\mathcal {O}(\\\\hbar ^2),\\\\nonumber $ where an additional average must be taken over the $k$ center points that are equivalent because of the periodic boundary conditions.\",\n \"Maintaining periodicity requires that they accrue a phase $\\\\theta _k$ See Eq.\",\n \"$5.18$ in [14] for a definition of the phase..\",\n \"The derivative $\\\\frac{\\\\partial ^2 S_{tj}}{\\\\partial x^2}$ is performed over the continuous function $S_{tj}$ defined after Eq.\",\n \"REF , but only evaluated at the discrete Weyl phase space points $x \\\\equiv (x_p, x_q)\\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^{2n}$ .\",\n \"The prefactor can also be reexpressed: $\\\\left\\\\lbrace \\\\det \\\\left[ 1 + \\\\frac{1}{2} \\\\mathcal {J} \\\\frac{\\\\partial ^2 S_{tj}}{\\\\partial x^2} \\\\right] \\\\right\\\\rbrace ^{\\\\frac{1}{2}} = \\\\left\\\\lbrace 2^d \\\\det \\\\left[ 1 + \\\\mathcal {M}_{tj} \\\\right] \\\\right\\\\rbrace ^{-\\\\frac{1}{2}},$ which is perhaps more pleasing in the discrete case as it does not involve a continuous derivative.\",\n \"As in the continuous case, for a harmonic Hamiltonian $H(p, q)$ , the center generating function $S(x_p, x_q)$ is equal to $\\\\alpha ^T \\\\mathcal {J} x + x^T \\\\mathcal {B} x$ where Eq.\",\n \"REF and Eq.\",\n \"REF hold.\",\n \"Moreover, if the Hamiltonian takes Weyl phase space points to themselves, then by the same equations it follows that $\\\\mathcal {M}$ and $\\\\alpha $ must have integer entries.\",\n \"This implies that for $m, n \\\\in \\\\mathbb {Z}^{ n}$ , $&& \\\\mathcal {M} \\\\left(\\\\begin{array}{c}p+ m d + \\\\alpha _p/2\\\\\\\\q + n d + \\\\alpha _q/2\\\\end{array}\\\\right) + \\\\left( \\\\begin{array}{c} \\\\alpha _p/2\\\\\\\\ \\\\alpha _q/2 \\\\end{array} \\\\right) \\\\nonumber \\\\\\\\&=& \\\\mathcal {M} \\\\left(\\\\begin{array}{c}p + \\\\alpha _p/2 \\\\\\\\ q + \\\\alpha _q/2 \\\\end{array}\\\\right) + \\\\left( \\\\begin{array}{c} \\\\alpha _p/2\\\\\\\\ \\\\alpha _q/2 \\\\end{array} \\\\right) + d \\\\mathcal {M}\\\\left(\\\\begin{array}{c}m\\\\\\\\ n\\\\end{array}\\\\right) \\\\\\\\&=& \\\\left(\\\\begin{array}{c}p^{\\\\prime }\\\\\\\\ q^{\\\\prime }\\\\end{array}\\\\right) \\\\mod {d. \\\\nonumber }Therefore, phase space points (p, q) that lie on Weyl phase space points go to the equivalent Weyl phase space points (p^{\\\\prime }, q^{\\\\prime }).$ Moreover, again if $m, n \\\\in \\\\mathbb {Z}^n$ , $&& S(x_p + m d, x_q + n d) \\\\nonumber \\\\\\\\&=& \\\\left( \\\\begin{array}{c}x_p+ m d\\\\\\\\ x_q+ n d\\\\end{array} \\\\right)^T A \\\\left( \\\\begin{array}{c}x_p+ m d\\\\\\\\ x_q+ n d\\\\end{array} \\\\right) \\\\nonumber \\\\\\\\&& + b \\\\cdot \\\\left( \\\\begin{array}{c}x_p+ m d\\\\\\\\ x_q+ n d\\\\end{array} \\\\right)\\\\\\\\&=& \\\\left( \\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array} \\\\right)^T A \\\\left( \\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array} \\\\right) + b \\\\cdot \\\\left( \\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array} \\\\right) \\\\nonumber \\\\\\\\&& + d \\\\left[ 2 \\\\left( \\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array} \\\\right)^T A \\\\left( \\\\begin{array}{c}m\\\\\\\\ n\\\\end{array} \\\\right) \\\\right.\\\\nonumber \\\\\\\\&& \\\\qquad \\\\left.+ d \\\\left( \\\\begin{array}{c}m\\\\\\\\ n\\\\end{array} \\\\right)^T A \\\\left( \\\\begin{array}{c}m\\\\\\\\ n\\\\end{array} \\\\right) + b \\\\cdot \\\\left( \\\\begin{array}{c}m\\\\\\\\ n\\\\end{array} \\\\right)\\\\right] \\\\nonumber \\\\\\\\&=& S(x_p, x_q) \\\\mod {d}, \\\\nonumber $ for some symmetric $A \\\\in \\\\mathbb {Z}^{n\\\\times n}$ and $b \\\\in \\\\mathbb {Z}^{n}$ .\",\n \"Therefore, these equivalent trajectories also have equivalent actions (since the action is multiplied by $\\\\frac{2 \\\\pi i}{d}$ and exponentiated).\",\n \"Hence, there is only one term to the sum in Eq.\",\n \"REF .\",\n \"Moreover, [25] showed that the sum over the phases $\\\\theta _k$ produces only a global phase that can be factored out.\",\n \"Therefore, if we can neglect the overall phase, ${U}_t(x) = \\\\left| 2^d \\\\det \\\\left[ 1 + \\\\mathcal {M} \\\\right] \\\\right|^{\\\\frac{1}{2}},$ where the classical trajectories whose centers are $(x_p, x_q)$ satisfy the periodic boundary conditions.\",\n \"As in the continuous case, we point out that this means that translations and reflections are fully captured by a path integral treatment that is truncated at order $\\\\hbar ^1$ (or order $\\\\hbar ^0$ if their overall phase isn't important) because their Hamiltonians are harmonic, but in the discrete case there is an additional requirement that they are evaluated at chords/centers that take Weyl phase space points to themselves.\",\n \"Just as in the continuous case, the single contribution at order $\\\\hbar ^0$ implies that the propagator of the Wigner function of states, $\\\\left|\\\\Psi \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi \\\\right|_x(x)$ under gates $\\\\hat{V}$ with underlying harmonic Hamiltonians is captured by $\\\\left|\\\\Psi \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi \\\\right|_x(\\\\mathcal {M}_{\\\\hat{V}} (x + \\\\alpha _{\\\\hat{V}}/2) + \\\\alpha _{\\\\hat{V}}/2)$ for $\\\\mathcal {M}_{\\\\hat{V}}$ and $\\\\alpha _{\\\\hat{V}}$ associated with $\\\\hat{V}$ .\",\n \"Stabilizer Group Here we will show that the Hamiltonians corresponding to Clifford gates are harmonic and take Weyl phase space points to themselves.\",\n \"Thus they can be captured by only the single contribution of Eq.\",\n \"REF at lowest order in $\\\\hbar $ .\",\n \"This then implies that stabilizer states can also be propagated to each other by Clifford gates with only a single contribution to the sum in Eq.\",\n \"REF .\",\n \"The Clifford gate set of interest can be defined by three generators: a single qudit Hadamard gate $\\\\hat{F}$ and phase shift gate $\\\\hat{P}$ , as well as the two qudit controlled-not gate $\\\\hat{C}$ .\",\n \"We examine each of these in turn.\",\n \"Hadamard Gate The Hadamard gate was defined in Eq.\",\n \"REF and is a rotation by $\\\\frac{\\\\pi }{2}$ in phase space counter-clockwise.\",\n \"Hence, for one qudit, it can be written as the map in Eq.\",\n \"REF where $\\\\mathcal {M}_{\\\\hat{F}} = \\\\left( \\\\begin{array}{cc} 0 & 1\\\\\\\\ -1 & 0 \\\\end{array} \\\\right),$ and $\\\\alpha _{\\\\hat{F}} = (0,0)$ .\",\n \"We have set $t=1$ and drop it from the subscripts from now on.\",\n \"Since $\\\\alpha $ is vanishing and $\\\\mathcal {M}$ has integer entries, this is a cat map and such maps have been shown to correspond to Hamiltonians [32] $&&H(p,q) =\\\\\\\\&&f(\\\\operatorname{Tr}\\\\mathcal {M} ) \\\\left[ \\\\mathcal {M}_{12} p^2 - \\\\mathcal {M}_{21} q^2 + \\\\left(\\\\mathcal {M}_{11} - \\\\mathcal {M}_{22} \\\\right) pq \\\\right],\\\\nonumber $ where $f(x) = \\\\frac{\\\\sinh ^{-1}(\\\\frac{1}{2}\\\\sqrt{x^2-4})}{\\\\sqrt{x^2-4}}.$ For the Hadamard $\\\\mathcal {M}_{\\\\hat{F}}$ this corresponds to $H_{\\\\hat{F}} = \\\\frac{\\\\pi }{4} (p^2 + q^2)$ , a harmonic oscillator.\",\n \"The center generating function $S(x_p, x_q)$ is thus $(x_p, x_q ) \\\\mathcal {B} (x_p, x_q)^T$ and solving Eq.\",\n \"REF finds for the one-qudit Hadamard, $\\\\mathcal {B}_{\\\\hat{F}} = \\\\left(\\\\begin{array}{cc}1 & 0\\\\\\\\ 0 & 1 \\\\end{array} \\\\right).$ Thus, $S_{\\\\hat{F}}(x_p, x_q) = x_p^2 + x_q^2$ .\",\n \"Indeed, applying Eq.\",\n \"REF to Eq.\",\n \"REF reveals that the Hadamard's center function (up to a phase) is: ${F}_x(x_p,x_q) = e^{\\\\frac{2 \\\\pi i}{d}(x_p^2+x_q^2)}.$ Eq.\",\n \"REF shows how to map Weyl phase space to Weyl phase space under the Hadamard transformation.\",\n \"Furthermore this map is pointwise, which implies the quadratic form of the center generating function obtained in Eq.\",\n \"REF .\",\n \"Phase Shift Gate The phase shift gate can be generalized to odd $d$ -dimensions [33] by setting it to: $\\\\hat{P} = \\\\sum _{j \\\\in \\\\mathbb {Z}/d \\\\mathbb {Z}} \\\\omega ^{\\\\frac{(j-1)j}{2}} \\\\left|j\\\\right\\\\rangle \\\\left\\\\langle j\\\\right|.$ Examining its effect on stabilizer states, it is clear that it is a $q$ -shear in phase space from an origin displaced by $\\\\frac{d-1}{2}\\\\equiv -\\\\frac{1}{2}$ to the right.\",\n \"This can be expressed as the map in Eq.\",\n \"REF with $\\\\mathcal {M}_{\\\\hat{P}} = \\\\left( \\\\begin{array}{cc} 1 & 1\\\\\\\\ 0 & 1 \\\\end{array} \\\\right),$ and $\\\\alpha _{\\\\hat{P}} = \\\\left( -\\\\frac{1}{2}, 0 \\\\right)$ .\",\n \"This corresponds to $\\\\mathcal {B}_{\\\\hat{P}} = \\\\left( \\\\begin{array}{cc} 0 & 0\\\\\\\\ 0 & \\\\frac{1}{2} \\\\end{array} \\\\right).$ Solving Eq.\",\n \"REF with this $\\\\mathcal {B}_{\\\\hat{P}}$ and $\\\\alpha _{\\\\hat{P}}$ reveals that $S_{\\\\hat{P}}(x_p, x_q) = -\\\\frac{1}{2} x_q + \\\\frac{1}{2} x_q^2$ .\",\n \"Again, this agrees with the argument of the center representation of the phase-shift gate obtained by applying Eq.\",\n \"REF to Eq.\",\n \"REF : ${P}_x(x_p,x_q) = e^{\\\\frac{2 \\\\pi i}{d} \\\\frac{1}{2} (-x_q + x_q^2)}.$ Discretization the equations of motion for harmonic evolution for unit timesteps leads to: $\\\\left(\\\\begin{array}{c}p^{\\\\prime }\\\\\\\\ q^{\\\\prime }\\\\end{array}\\\\right) = \\\\left(\\\\begin{array}{c}p\\\\\\\\ q\\\\end{array}\\\\right) + \\\\mathcal {J} \\\\left(\\\\frac{\\\\partial H}{\\\\partial p}, \\\\frac{\\\\partial H}{\\\\partial q}\\\\right)^T,$ where the last derivative is on the continuous function $H$ , but only evaluated on the discrete Weyl phase space points.\",\n \"It follows that $H_{\\\\hat{P}} = -\\\\frac{d+1}{2} q^2 + \\\\frac{d+1}{2} q.$ We have obtained the Hamiltonian for the phase-shift gate by a different procedure than that used for the Hadamard where we appealed to the result given in Eq.\",\n \"REF for quantum cat maps.\",\n \"However, as the phase-shift gate is a quantum cat map as well, we could have obtained Eq.\",\n \"REF in this manner.\",\n \"Similarly, the approach we used to find the phase-shift Hamiltonian by discretizing time in Eq.\",\n \"REF would work for the Hadamard gate but it is a bit more involved since the latter contains both $p$ - and $q$ -evolution.\",\n \"Nevertheless, this produces Eq.\",\n \"REF as well.\",\n \"We presented both techniques for illustrative purposes.\",\n \"Controlled-Not Gate Lastly, the controlled-not gate can be generalized to $d$ -dimensions [33] by $\\\\hat{C} = \\\\sum _{j,k \\\\in \\\\mathbb {Z}/d \\\\mathbb {Z}} \\\\left|j,k \\\\oplus j\\\\right\\\\rangle \\\\left\\\\langle j,k\\\\right|.$ It is clear that this translates the $q$ -state of the second qudit by the $q$ -state of the first qudit.\",\n \"As a result, as is evident by examining the gate's action on stabilizer states, the first qudit experiences an “equal and opposite reaction” force that kicks its momentum by the $q$ -state of the second qudit.\",\n \"This is the phase space picture of the well-known fact that a CNOT examined in the $\\\\hat{X}$ basis has the control and target reversed with respect to the $\\\\hat{Z}$ basis.\",\n \"This can also seen by looking at its effect in the momentum ($\\\\hat{X}$ ) basis: ${\\\\hat{F}}^\\\\dagger \\\\hat{C} \\\\hat{F} = \\\\sum _{j,k \\\\in \\\\mathbb {Z}/d \\\\mathbb {Z}} \\\\left|j \\\\ominus k,k\\\\right\\\\rangle \\\\left\\\\langle j,k\\\\right|.$ As a result, this gate is described by the map: $\\\\mathcal {M}_{\\\\hat{C}} = \\\\left( \\\\begin{array}{ccccc} 1 & -1 & 0 & 0\\\\\\\\ 0 & 1 & 0 & 0\\\\\\\\ 0 & 0 & 1 & 0\\\\\\\\ 0 & 0 & 1 & 1\\\\end{array}\\\\right),$ and $\\\\alpha _{\\\\hat{C}} = (0,0,0,0)$ .\",\n \"This corresponds to $\\\\mathcal {B}_{\\\\hat{C}} = \\\\left( \\\\begin{array}{ccccc} 0 & 0 & 0 & 0\\\\\\\\ 0 & 0 & -\\\\frac{1}{2} & 0\\\\\\\\ 0 & -\\\\frac{1}{2} & 0 & 0\\\\\\\\ 0 & 0 & 0 & 0\\\\end{array}\\\\right).$ Hence its center generating function $S_{\\\\hat{C}}(x_p, x_q) = -x_{p_2} x_{q_1}$ .\",\n \"Again, this corresponds with the argument of the center representation of the controlled-not gate, which can be found to be: ${C}_x(x_p, x_q) = e^{-\\\\frac{2 \\\\pi i}{d} x_{q_1} x_{p_2}}.$ Therefore, this gate can be seen to be a bilinear $p$ -$q$ coupling between two qudits and corresponds to the Hamiltonian $H_{\\\\hat{C}} = p_1 q_2,$ as can be found from Eq.\",\n \"REF again.\",\n \"As a result, it is now clear that all the Clifford group gates have Hamiltonians that are harmonic and that take Weyl phase space points to themselves.\",\n \"Therefore, their propagation can be fully described by a truncation of the semiclassical propagator Eq.\",\n \"REF to order $\\\\hbar ^0$ as in Eq.\",\n \"REF and they are manifestly classical in this sense.\",\n \"To summarize the results of this section, using Eq.\",\n \"REF , the Hadamard, phase shift, and CNOT gates can be written as: $\\\\hat{F} = d^{-2} \\\\sum _{\\\\begin{array}{c}x_p,x_q, \\\\\\\\ \\\\xi _p,\\\\xi _q \\\\in \\\\\\\\ \\\\mathbb {Z}/d \\\\mathbb {Z}\\\\end{array}} e^{ -\\\\frac{2 \\\\pi i}{d} \\\\left[ -(x_p^2 + x_q^2) - d \\\\left(x_p \\\\xi _p - x_q \\\\xi _q \\\\right) \\\\right] } \\\\hat{Z}^{\\\\xi _p} \\\\hat{X}^{\\\\xi _q},$ $\\\\hat{P} = d^{-2} \\\\sum _{\\\\begin{array}{c}x_p,x_q, \\\\\\\\ \\\\xi _p,\\\\xi _q \\\\in \\\\\\\\ \\\\mathbb {Z}/d \\\\mathbb {Z}\\\\end{array}} e^{ -\\\\frac{2 \\\\pi i}{d} \\\\left[ \\\\frac{1}{2} (x_q - x_q^2) - d \\\\left(x_p \\\\xi _q - x_q \\\\xi _p \\\\right) \\\\right] } \\\\hat{Z}^{\\\\xi _p} \\\\hat{X}^{\\\\xi _q},$ and $&&\\\\hat{C} = \\\\\\\\&& d^{-4} \\\\sum _{\\\\begin{array}{c}x_p, x_q, \\\\\\\\ \\\\xi _p, \\\\xi _q \\\\in \\\\\\\\ (\\\\mathbb {Z}/d \\\\mathbb {Z})^{ 2}\\\\end{array}} e^{ -\\\\frac{2 \\\\pi i}{d} \\\\left[ x_{q_1} x_{p_2} - d \\\\left(x_p \\\\cdot \\\\xi _q - x_q \\\\cdot \\\\xi _p \\\\right) \\\\right] } \\\\hat{Z}^{ \\\\xi _p} \\\\hat{X}^{ \\\\xi _q}, \\\\nonumber $ (up to a phase).\",\n \"This form emphasizes their quadratic nature.\",\n \"As for the continuous case, there exists a particularly simple way in discrete systems to see to what order in $\\\\hbar $ the path integral must be kept to handle unitary propagation beyond the Clifford group.\",\n \"We describe this in the next section.\",\n \"We note that the center generating actions $S(x_p, x_q)$ found here are related to the $G(q^{\\\\prime }, q)$ found by Penney et al.\",\n \"[12], which are in terms of initial and final positions, by symmetrized Legendre transform [13]: $G(q^{\\\\prime }, q, t) = F\\\\left(\\\\frac{q^{\\\\prime } + q}{2}, p(q^{\\\\prime } - q)\\\\right),$ where the canonical generating function $F\\\\left(\\\\frac{q^{\\\\prime } + q}{2}, p\\\\right) = S\\\\left( x_p = p, x_q = \\\\frac{q^{\\\\prime } + q}{2} \\\\right) + p \\\\cdot \\\\left( q^{\\\\prime } - q \\\\right),$ for $p (q^{\\\\prime } - q)$ given implicitly by $\\\\frac{\\\\partial F}{\\\\partial p} = 0$ .\",\n \"Applying this to the actions we found reveals that $G_{\\\\hat{F}}(q^{\\\\prime }, q, t) = q^{\\\\prime } q,$ $G_{\\\\hat{P}}(q^{\\\\prime }, q, t) = \\\\frac{d+1}{2} (q^2 - q),$ and $G_{\\\\hat{C}}((q^{\\\\prime }_1, q^{\\\\prime }_2), (q_1, q_2), t) = 0,$ which is in agreement with [12].\",\n \"Classicality of Stabilizer States In this subsection we show that stabilizer states evolve to stabilizer states under Clifford gates, and that it is possible to describe this evolution classically.\",\n \"For odd $d \\\\ge 3$ the positivity of the Wigner representation implies that evolution of stabilizer states is non-contextual, and so here we are investigating in detail what this means in our semiclassical picture.\",\n \"To begin, it is instructive to see the form stabilizer states take in the discrete position representation and in the center representation.\",\n \"Gross proved that [3]: Theorem 1 Let $d$ be odd and $\\\\Psi \\\\in L^2((\\\\mathbb {Z}/d\\\\mathbb {Z})^n)$ be a state vector.\",\n \"The Wigner function of $\\\\Psi $ is non-negative if and only if $\\\\Psi $ is a stabilizer state.\",\n \"Gross also proved [3] Corollary 1 Given that $\\\\Psi (q) \\\\ne 0$ $\\\\forall \\\\, q$ , a vector $\\\\Psi $ is a stabilizer state if and only if it is of the form $\\\\Psi _{\\\\theta _\\\\beta ,\\\\eta _\\\\beta }(q) \\\\propto \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( q^T \\\\theta _\\\\beta q + \\\\eta _\\\\beta \\\\cdot q \\\\right) \\\\right].$ where $\\\\theta _\\\\beta \\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^{n\\\\times n}$ and $q, \\\\eta _\\\\beta \\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^n$ .\",\n \"Applying Eq.\",\n \"REF to Eq.\",\n \"REF , the Wigner function of such maximally supported stabilizer states can be found to be: $&& {\\\\Psi _{\\\\theta _\\\\beta ,\\\\eta _\\\\beta }}_x(x_p, x_q) \\\\propto \\\\\\\\&& d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _q \\\\in \\\\\\\\\\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^{ n}\\\\end{array}} \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\xi _q \\\\cdot \\\\left( \\\\eta _\\\\beta - x_p + 2 \\\\theta _\\\\beta x_q \\\\right)\\\\right].\\\\nonumber $ Therefore, one finds that the Wigner function is the discrete Fourier sum equal to $\\\\delta _{\\\\eta _\\\\beta - x_p + 2\\\\theta _\\\\beta x_q}$ .\",\n \"For $\\\\theta _\\\\beta = 0$ the state is a momentum state at $x_p$ .\",\n \"Finite $\\\\theta _\\\\beta $ rotates that momentum state in phase space in “steps” such that it always lies along the discrete Weyl phase space points $(x_p, x_q) \\\\in (\\\\mathbb {Z}/d\\\\mathbb {Z})^{2n}$ .\",\n \"This Gaussian expression only captures stabilizer states that are maximally supported in $q$ -space.\",\n \"One may wonder what the stabilizer states that aren't maximally supported in $q$ -space look like in Weyl phase space.\",\n \"Of course, it is possible that some may be maximally supported in $p$ -space and so can be captured by the following corollary: Corollary 2 If $\\\\Psi (p) \\\\ne 0$ for all $p$ 's then there exists a $\\\\theta _{\\\\beta p} \\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^{n\\\\times n}$ and an $\\\\eta _{\\\\beta p} \\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^n$ such that $\\\\Psi _{\\\\theta _{\\\\beta p},\\\\eta _{\\\\beta p}}(p) \\\\propto \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( p^T \\\\theta _{\\\\beta p} p + \\\\eta _{\\\\beta p} \\\\cdot p \\\\right) \\\\right].$ Proof This can be shown following the same methods employed by Gross [3] but in the discrete $p$ -basis.\",\n \"Unfortunately, it is easy to show that Corollary REF and REF do not provide an expression for all stabilizer states (except for the odd prime $d$ case, as we shall see shortly) as there exist stabilizer states for odd non-prime $d$ that are not maximally supported in $p$ - or $q$ -space or any finite rotation between those two.\",\n \"To find an expression that encompasses all stabilizer states, we must turn to the Wigner function of stabilizer states.\",\n \"An equivalent definition of stabilizer states on $n$ qudits is given by states $\\\\hat{V} \\\\underbrace{\\\\left|0\\\\right\\\\rangle \\\\otimes \\\\cdots \\\\otimes \\\\left|0\\\\right\\\\rangle }_{n}$ where $\\\\hat{V}$ is a quantum circuit consisting of Clifford gates.\",\n \"We know that the Clifford circuits are generated by the $\\\\hat{P}$ , $\\\\hat{F}$ and $\\\\hat{C}$ gates, and that the Wigner functions $\\\\Psi _x(x)$ of stabilizer states propagate under $\\\\hat{V}$ as $\\\\Psi _x(\\\\mathcal {M}_{\\\\hat{V}} (x + \\\\alpha _{\\\\hat{V}}/2) + \\\\alpha _{\\\\hat{V}}/2)$ , it follows that the Wigner function of stabilizer states is: $\\\\delta _{\\\\Phi _0 \\\\cdot \\\\mathcal {M}_{\\\\hat{V}} \\\\cdot x, r_0},$ where $\\\\Phi _0 = \\\\left(\\\\begin{array}{cc} 0 & 0\\\\\\\\ 0 & \\\\mathbb {I}_n\\\\end{array} \\\\right)$ and $r_0 = (0, 0)$ .\",\n \"We have therefore proved the next theorem: Theorem 2 The Wigner function $\\\\Psi _x(x)$ of a stabilizer state for any odd $d$ and $n$ qudits is $\\\\delta _{\\\\Phi \\\\cdot x, r}$ for $2n \\\\times 2n$ matrix $\\\\Phi $ and $2n$ vector $r$ .\",\n \"As an aside, Theorem REF allows us to develop an all-encompassing Gaussian expression for stabilizer states for the restricted case that $d$ is odd prime.\",\n \"In this case, the following Corollary shows that a “mixed” representation is always possible: where each degree of freedom is expressed in either the $p$ - or $q$ -basis: Corollary 3 For odd prime $d$ , if $\\\\Psi $ is a stabilizer state for $n$ qudits, then there always exists a mixed representation in position and momentum such that: $\\\\Psi _{\\\\theta _{\\\\beta x},\\\\eta _{\\\\beta x}}(x) = \\\\frac{1}{\\\\sqrt{d}} \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( x^T \\\\theta _{\\\\beta x} x + \\\\eta _{\\\\beta x} \\\\cdot x \\\\right) \\\\right],$ where $x_i$ can be either $p_i$ or $q_i$ .\",\n \"Proof We begin with a one-qudit case.\",\n \"We examine the equation specified by $\\\\Phi \\\\cdot x = r$ : $\\\\alpha q_1 + \\\\beta p_1 = \\\\gamma $ for $\\\\alpha $ , $\\\\beta $ , and $\\\\gamma \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ .\",\n \"If $\\\\alpha = 0$ then $\\\\Psi (q_1)$ is maximally supported and if $\\\\beta = 0$ then $\\\\Psi (p_1)$ is maximally supported since the equation specifies a line on $\\\\mathbb {Z}/d \\\\mathbb {Z}$ in $q_1$ and $p_1$ respectively.\",\n \"If $\\\\alpha \\\\ne 0$ and $\\\\beta \\\\ne 0$ then the equation can be rewritten as $q_1 + (\\\\beta /\\\\alpha ) p_1 = \\\\gamma /\\\\alpha ,$ and it follows that $q_1$ can take any values on $\\\\mathbb {Z}/d \\\\mathbb {Z}$ and so $\\\\Psi (q_1)$ is maximally supported.\",\n \"However, it is also possible to reexpress the equation as: $p_1 + (\\\\alpha /\\\\beta ) q_1 = \\\\gamma /\\\\beta .$ It follows that $p_1$ can also take any values on $\\\\mathbb {Z}/d \\\\mathbb {Z}$ —$\\\\Psi (p_1)$ is also maximally supported.\",\n \"Therefore, one can always choose either a $p_1$ - or $q_1$ -basis such that the state is maximally supported and so is representable by a Gaussian function.\",\n \"We now consider adding another qudit such that the state becomes ${\\\\Psi }_x(p_1,p_2,q_1,q_2)$ .\",\n \"There are now two equations specified by $\\\\Phi \\\\cdot x = r$ and it follows that it is always possible to combine the two equations such that $p_1$ and $q_1$ are only in one equation and written in terms of each other (and generally the second degree of freedom): $\\\\alpha q_1 + \\\\beta p_1 + \\\\gamma q_2 + \\\\delta p_2 = \\\\epsilon ,$ for $\\\\alpha $ , $\\\\beta $ , $\\\\gamma $ , $\\\\delta $ and $\\\\epsilon \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ .\",\n \"It will turn out that the $\\\\gamma q_2 + \\\\delta p_2$ term is irrelevant.\",\n \"We can rewrite the above equation as: $q_1 + (\\\\beta /\\\\alpha ) p_1 + (\\\\gamma /\\\\alpha ) q_2 + (\\\\delta /\\\\alpha ) p_2 = \\\\epsilon /\\\\alpha ,$ if $\\\\alpha \\\\ne 0$ .\",\n \"Since there is no other equation specifying $p_1$ , this is an equation for a line on $\\\\mathbb {Z}/ d \\\\mathbb {Z}$ and so $\\\\Psi $ is is maximally supported on $q_1$ .\",\n \"Otherwise, rewriting the above equation as: $p_1 + (\\\\alpha /\\\\beta ) q_1 +(\\\\gamma /\\\\beta ) q_2 + (\\\\delta /\\\\beta ) p_2 = \\\\epsilon /\\\\beta ,$ if $\\\\beta \\\\ne 0$ shows that $\\\\Psi $ is maximally supported on $p_1$ .\",\n \"If $\\\\alpha = \\\\beta = 0$ then both $p_1$ and $q_1$ are undetermined and so either representation produces a maximally supported state.\",\n \"The same procedure can be performed to find if $q_2$ or $p_2$ produce a maximally supported state.\",\n \"As can be seen, we are really just repeating the same procedure as we did when there was only one qudit because the other degrees of freedom have no impact on this determination.\",\n \"Expressing $\\\\Psi $ in the basis that is maximally supported in every degree of freedom means that it is therefore a Gaussian.\",\n \"Therefore, it follows that every degree of freedom (corresponding to a qudit) is maximally supported in either the $p$ - or $q$ - basis and so Eq.\",\n \"REF always describes stabilizer states for odd prime $d$ .$\\\\Box $ The form of Eq.\",\n \"REF is more general than Eq.\",\n \"REF because it does not depend on the support of the state.\",\n \"As we saw in the proof, this representation is generally not unique; for every qudit $i$ that is not a position or momentum state, $x_i$ can be either $p_i$ or $q_i$ .\",\n \"However, if it is a position state then $x_i = p_i$ and if it is a momentum state then $x_i = q_i$ ; position and momentum states must be expressed in their conjugate representation in order to be captured by a Gaussian of the form in Eq.\",\n \"REF instead of Kronecker deltas.\",\n \"The reason this mixed representation doesn't hold for non-prime odd $d$ is that the coefficients above can be (multiples of) prime factors of $d$ and so no longer produce “lines” in $p_i$ or $q_i$ that cover all of $\\\\mathbb {Z}/d \\\\mathbb {Z}$ .\",\n \"An alternative proof of this corollary that explores this case further is presented in the Appendix.\",\n \"An example of the different classes of stabilizer states that are possible for odd prime $d$ , in terms of their support, is shown in Fig.\",\n \"REF .\",\n \"There it can be seen that a stabilizer state is either maximally supported in $p_i$ or $q_i$ , and is a Kronecker delta function in the other degree of freedom, or it is maximally supported in both.\",\n \"Figure: The two classes of stabilizer states possible for odd prime d=7d=7 in terms of support: a) maximally supported in pp or qq and b) maximally supported in pp and qq.\",\n \"The central grids denote the Wigner function ΨΨ x (p,q)\\\\left|\\\\Psi \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi \\\\right|_x(p,q) of a stabilizer state Ψ\\\\Psi with d=7d=7.\",\n \"The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\\\Psi (q)|^2).On the other hand, for odd non-prime $d$ , we see in Fig.\",\n \"REF that another class is possible: stabilizer states that are maximally supported in neither $p_i$ or $q_i$ .\",\n \"In fact, rotating the basis in any of the discrete angles afforded by the grid still does not produce a basis that is maximally supported (as discussed in the Appendix).\",\n \"Notice also, that Fig.\",\n \"REF b shows that it is no longer true that a state that is maximally supported in only $q_i$ or $p_i$ is automatically a Kronecker delta when expressed in terms of the other.\",\n \"Figure: The four classes of stabilizer states possible for odd non-prime d=15d=15 in terms of support: a) & b) maximally supported in pp or qq, c) maximally supported in pp and qq, and d) not maximally supported in pp or qq.\",\n \"The central grids denote the Wigner function ΨΨ x (p,q)\\\\left|\\\\Psi \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi \\\\right|_x(p,q) for a stabilizer state Ψ\\\\Psi with d=15d=15.\",\n \"The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\\\Psi (q)|^2).In summary, stabilizer states have Wigner function $\\\\delta _{\\\\Phi \\\\cdot x, r}$ and, for odd prime $d$ , are Gaussians in mixed representation that lie on the Weyl phase space points $(x_p, x_q)$ .\",\n \"Heuristically, they correspond to Gaussians in the continuous case that spread along their major axes infinitely.\",\n \"The only reason that they aren't always expressible as Gaussians in the mixed representation is that they sometimes “skip” over some of the discrete grid points due to the particular angle they lie along phase space for odd non-prime $d$ .\",\n \"Wigner functions $\\\\Psi _x(x)$ of stabilizer states propagate under $\\\\hat{V}$ as $\\\\Psi _x(\\\\mathcal {M}_{\\\\hat{V}} (x + \\\\alpha _{\\\\hat{V}}/2) + \\\\alpha _{\\\\hat{V}}/2)$ , and this preserves the form of the state.\",\n \"In other words, Clifford gates take stabilizer states to other stabilizer states, as expected, just like in the continuous case Gaussians go to other Gaussians under harmonic evolution.\",\n \"It is also clear that stabilizer state propagation under Clifford gates can be expressed by a path integral at order $\\\\hbar ^0$ .\",\n \"Discrete Phase Space Representation of Universal Quantum Computing A similar statement to the one we made in Section —that any operator can be expressed as an infinite sum of path integral contribution truncated at order $\\\\hbar ^1$ —can be made in discrete systems.\",\n \"However, there is an important difference in the number of terms making up the sum.\",\n \"To see this we can follow reasoning that is similar to that employed in the continuous case.\",\n \"Namely, from Eq.\",\n \"REF we see that any discrete operator can also be expressed as a linear combination of reflections, but unlike the continuous case, this sum has a finite number of terms.\",\n \"Since reflections can be expressed fully by the discrete path integral truncated at order $\\\\hbar ^1$ , as discussed previously, it follows that any unitary operator in discrete systems can be expressed as a finite sum of contributions from path integrals truncated at order $\\\\hbar ^1$ .\",\n \"Again, the same statement can be made by considering the chord representation in terms of translations.\",\n \"Hence, quantum propagation in discrete systems can be fully treated by a finite sum of contributions from a path integral approach truncated at order $\\\\hbar ^1$ .\",\n \"To gather some understanding of this statement, we can consider what is necessary to add to our path integral formulation when we complete the Clifford gates with the T-gate, which produces a universal gate set.\",\n \"The T-gate is generalized to odd $d$ -dimensions by $\\\\hat{T} = \\\\sum _{j \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}} \\\\omega ^{\\\\frac{(j-1)j}{4}} \\\\left|j\\\\right\\\\rangle \\\\left\\\\langle j\\\\right|.$ This gate can no longer be characterized by an $\\\\mathcal {M}$ with integer entries.\",\n \"In particular, $\\\\mathcal {M}_{\\\\hat{T}} = \\\\left( \\\\begin{array}{cc} 1 & \\\\frac{1}{2} \\\\\\\\ 0 & 1 \\\\end{array} \\\\right),$ and $\\\\alpha _{\\\\hat{T}} = \\\\left( -\\\\frac{1}{4}, 0 \\\\right)$ .\",\n \"This corresponds to $\\\\mathcal {B}_{\\\\hat{T}} = \\\\left( \\\\begin{array}{cc} 0 & 0\\\\\\\\ 0 & -\\\\frac{1}{4}\\\\end{array} \\\\right).$ Thus, the center function $T_x(x_p,x_q) = e^{-\\\\frac{2 \\\\pi i}{d} \\\\frac{1}{4} (x_q-x_q^2)},$ corresponding to the phase shift Hamiltonian applied for only half the unit of time.\",\n \"The operator can thus be written: $\\\\hat{T} = d^{-2} \\\\sum _{\\\\begin{array}{c}x_p,x_q, \\\\\\\\ \\\\xi _p,\\\\xi _q \\\\in \\\\\\\\ \\\\mathbb {Z}/d\\\\mathbb {Z}\\\\end{array}} e^{ -\\\\frac{2 \\\\pi i}{d} \\\\left[ \\\\frac{1}{4} (x_q - x_q^2) - d \\\\left(x_p \\\\xi _q - x_q \\\\xi _p \\\\right) \\\\right] } \\\\hat{Z}^{\\\\xi _p} \\\\hat{X}^{\\\\xi _q}.$ Though this operator is quadratic, it no longer takes the Weyl center points to themselves.\",\n \"This means that the $\\\\hbar ^0$ limit of Eq.\",\n \"REF is now insufficient to capture all the dynamics because the overlap with any $\\\\left|q\\\\right\\\\rangle $ will now involve a linear superposition of partially overlapping propagated manifolds.\",\n \"It must therefore be described by a path integral formulation that is complete to order $\\\\hbar ^1$ .\",\n \"In particular, $\\\\hat{T} = d^{-1} \\\\sum _{\\\\begin{array}{c}x_p, x_q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})\\\\end{array}} e^{-\\\\frac{2 \\\\pi i}{d} \\\\frac{1}{4} (x_q-x_q^2)} \\\\hat{R}( x_p, x_q ),$ where $\\\\hat{R}$ should be substituted by its path integral.\",\n \"Note that this does not imply efficient classical simulation of quantum computation but quite the opposite.\",\n \"Indeed, for $n$ qudits, there are $d^{2n}$ terms in the sum above.\",\n \"While every Weyl phase space point has only a single associated path when acted on by Clifford gates, this is no longer true in any calculation of evolution under the T-gate.\",\n \"Eq.\",\n \"REF expresses the T-gate as a sum over phase space operators (the reflections) evaluated on all the phase space points.\",\n \"Thus, it can be interpreted as associating an exponentially large number of paths to every phase space point, instead of the single paths found for Clifford gates.\",\n \"Therefore, any simulation of the T-gate naively necessitates adding up an exponential large sum over paths and so is comparably inefficient.\",\n \"Conclusion The treatment presented here formalizes the relationship between stabilizer states in the discrete case and Gaussians in the continuous case, which has often been pointed out [3].\",\n \"Namely, only Gaussians that lie along Weyl phase space points directly correspond to Gaussians in the continuous world in terms of preserving their form under a harmonic Hamiltonian, an evolution that is fully describable by truncating the path integral at order $\\\\hbar ^0$ .\",\n \"Furthermore, we showed that the Clifford group gates, generated by the Hadamard, phase shift and controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order.\",\n \"We found that this was because their Hamiltonians are harmonic and take Weyl phase space points to themselves.\",\n \"This proves the Gottesman-Knill theorem.\",\n \"The T-gate, needed to complete a universal set with the Hadamard, was shown not to satisfy these properties, and so requires a path integral treatment that is complete up to $\\\\hbar ^1$ .\",\n \"The latter treatment includes a sum of terms for which the number of terms scales exponentially with the number of qudits.\",\n \"We note that our observations pertaining to classical propagation in continuous systems have long been very well known.\",\n \"In the continuous case, the Wigner function of a quantum state is non-negative if and only if the state is a Gaussian [34] and it has also long been known that quantum propagation from one Gaussian state to another only requires propagation up to order $\\\\hbar ^0$ [11].\",\n \"Indeed, it has been shown that this is a continuous version of “stabilizer state propagation” in finite systems [35], and is therefore, in principle, useful for quantum error correction and cluster state quantum computation [36], [37].\",\n \"It is also well known in the discrete case that quadratic Hamiltonians can act classically and be represented by symplectic transformations in the study of quantum cat maps [38], [14], [25] and linear transformations between propagated Wigner functions [39].\",\n \"Interestingly though, this latter work appears to have predated the discovery that stabilizer states have positive-definite Wigner functions [3] and therefore, as far as we know, has not been directly related to stabilizer states and the $\\\\hbar ^0$ limit of their path integral formulation, which is a relatively recent topic of particular interest to the quantum information community and those familiar with Gottesman-Knill.\",\n \"Otherwise, this claim has been pointed out in terms of concepts related to positivity and related concepts in past work [40], [2].\",\n \"We also note that our exploration of continous systems is not meant to explore the highly related topic of continuous-variable quantum information.\",\n \"Many topics therein apply to our discussion here, such as the continuous stabilizer state propagation we mentioned above.\",\n \"However, our intention in introducing the continuous infinite-dimensional case was not to address these topics but to instead relate the established continuous semiclassical formalism to the discrete case, and thereby bridge the notions of phase space and dynamics between the two worlds.\",\n \"There is an interesting observation to be made of the weights of the reflections that make up the complete path integral formulation of a unitary operator.\",\n \"Namely, as is clear in Eqs.\",\n \"REF and REF , the coefficients consist of the exponentiated center generating function multiplied by $\\\\frac{i}{\\\\hbar }$ .\",\n \"This is very similar to the form of the vVMG path integral in Eqs.\",\n \"REF and REF .\",\n \"However, in Eqs.\",\n \"REF and REF , reflections serve as the prefactors measuring the reflection spectral overlap of a propagated state with its evolute and the center generating actions provide the quantal phase.\",\n \"Thus, this formulation can be interpreted as an alternative path integral formulation of the vVMG, one consisting of reflections as the underlying classical trajectory only, instead of the more tailored trajectories that result from applying the method of steepest descents directly on an operator.\",\n \"The fact that any unitary operator in the discrete case can be expressed as a sum consisting of a finite number of order $\\\\hbar ^1$ path integral contributions, has the added interesting implication that uniformization—higher order $\\\\hbar $ corrections to the “primitive” semiclassical forms such as Eq.\",\n \"REF —isn't really necessary in discrete systems.\",\n \"Uniformization is characterized by the proper treatment of coalescing saddle points and has long been a subject of interest in continuous systems where “anharmonicity” bedevils computationally efficient implementation.\",\n \"It seems that this problem isn't an issue in the discrete case since a fully complete sum with a finite number of terms, naively numbering $d^2$ for one qudit, exists.\",\n \"As a last point, there is perhaps an alternative way to interpret the results presented here, one in terms of “resources”.\",\n \"Much like “magic” (or contextuality) and quantum discord can be framed as a resource necessary to perform quantum operations that have more power than classical ones, it is possible to frame the order in $\\\\hbar $ that is necessary in the underlying path integral describing an operation as a resource necessary for quantumness.\",\n \"In this vein, it can be said that Clifford gate operations on stabilizer states are operations that only require $\\\\hbar ^0$ resources while supplemental gates that push the operator space into universal quantum computing require $\\\\hbar ^1$ resources.\",\n \"The dividing line between these two regimes, the classical and quantum world, is discrete, unambiguous and well-defined.\",\n \"Acknowledgments The authors thank Prof. Alfredo Ozorio de Almeida for very fruitful discussions about the center-chord representation in discrete systems and Byron Drury for his help proof-reading and bringing [12] to our attention.\",\n \"This work was supported by AFOSR award no.\",\n \"FA9550-12-1-0046.\",\n \"Appendix Gross proved that for odd prime $d$ [41]: Lemma 2 Let $\\\\Psi $ be a state vector with positive Wigner function for odd prime $d$ .\",\n \"If $\\\\Psi $ is supported on two points, then it has maximal support.\",\n \"With this lemma in mind, we can offer an alternative proof of Corollary REF : Corollary 3 For odd prime $d$ , if $\\\\Psi $ is a stabilizer state then there always exists a mixed representation in position and momentum such that: $\\\\Psi _{\\\\theta _{\\\\beta x},\\\\eta _{\\\\beta x}}(x) = \\\\frac{1}{\\\\sqrt{d}} \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( x^T \\\\theta _{\\\\beta x} x + \\\\eta _{\\\\beta x} \\\\cdot x \\\\right) \\\\right],$ where $x_i$ can be either $p_i$ or $q_i$ .\",\n \"Proof We will show that for odd prime $d$ , every degree of freedom can only be fully supported or a Kronecker delta, for all other degrees of freedom fixed; WLOG we will consider a two-dimensional stabilizer state $\\\\Psi (q_1,q_2)$ and show that if $\\\\exists \\\\, q^{\\\\prime }_1$ such that $\\\\Psi (q^{\\\\prime }_1, q_2) \\\\ne 0$ $\\\\forall q_2$ then $\\\\Psi (q_1,q_2)\\\\ne 0$ $\\\\forall \\\\, q_1, q_2$ and vice-versa (if $\\\\exists \\\\, q^{\\\\prime }_1$ s.t.\",\n \"$\\\\Psi (q^{\\\\prime }_1, q_2)$ is a delta function then $\\\\Psi (q_1,q_2)$ is a delta function in $q_2$ $\\\\forall q_1$ ).\",\n \"Therefore, if a degree of freedom is maximally supported in one degree of freedom for all others fixed, then it is maximally supported for all values of the other degrees of freedom.\",\n \"On the other hand, if it is a delta function in one degree of freedom for all others fixed, then it is a delta function for all values of the other degrees of freedom.\",\n \"Assume that for $q_1, q_2 \\\\in \\\\lbrace 0, \\\\ldots , d-1\\\\rbrace $ , $\\\\exists \\\\, q^{\\\\prime }_1, q^{\\\\prime \\\\prime }_1$ such that $\\\\Psi (q^{\\\\prime }_1,q_2) = 0$ for some $q_2$ and $\\\\Psi (q^{\\\\prime \\\\prime }_1, q_2) \\\\ne 0$ $\\\\forall q_2$ .\",\n \"We proceed to prove by contradiction.\",\n \"Hence $\\\\Psi (q^{\\\\prime \\\\prime }_1, q_2) \\\\equiv \\\\Psi _{q^{\\\\prime \\\\prime }_1} \\\\propto \\\\left[ \\\\frac{2 \\\\pi i}{d} \\\\left( \\\\theta _{q^{\\\\prime \\\\prime }_1} q^2_2 + \\\\eta _{q^{\\\\prime \\\\prime }_1} q_2 \\\\right) \\\\right]$ by Corollary REF and $\\\\Psi (q^{\\\\prime }_1,q_2) \\\\equiv \\\\Psi _{q^{\\\\prime }_1}(q_2) \\\\propto \\\\delta _{q_2, q(q^{\\\\prime }_1)}$ for some $q(q^{\\\\prime }_1)\\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ by [3].\",\n \"We can rotate in $p_2$ -$q_2$ space to form a new basis $q^*_2$ in $d$ discrete angles (since $\\\\theta _{q^{\\\\prime \\\\prime }_1} \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ ) such that $\\\\Psi _{q^{\\\\prime }_1}(q^*_2) \\\\ne 0$ (since a delta function is not maximally supported only at the one angle perpendicular to it).\",\n \"Since there exists $(d-1)$ other values of $q_1$ other than $q^{\\\\prime }_1$ , it follows that there exists at least one such angle such that $\\\\Psi _{q_1}(q^*_2) \\\\ne 0$ $\\\\forall q_1, q^*_2 \\\\in \\\\lbrace 0,\\\\ldots ,d-1\\\\rbrace $ .\",\n \"We define $q^*_2$ as the basis that is rotated by this angle with respect to $q_2$ .\",\n \"By Corollary REF , this means that $\\\\Psi ^*(q_1,q^*_2) \\\\propto \\\\exp ( \\\\theta ^{\\\\prime }_{11} q^2_1 + \\\\theta ^{\\\\prime }_{22} {q^*_2}^2 + 2 \\\\theta ^{\\\\prime }_{12} q_1 q^*_2 + \\\\eta ^{\\\\prime }_1 q_1 + \\\\eta ^{\\\\prime }_2 q_2),$ where by $\\\\Psi ^*$ we mean $\\\\Psi $ expressed in the new basis $q^*_2$ in its second degree of freedom.\",\n \"Hence, $\\\\Psi ^*_{q_1}(q^*_2) \\\\propto \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( \\\\theta ^{\\\\prime }_{q_1} {q^*_2}^2 + \\\\eta ^{\\\\prime }_{q_1} q^*_2 \\\\right) \\\\right] \\\\exp \\\\left[ \\\\frac{2 \\\\pi i}{d} \\\\theta _{12} q_1 q^*_2 \\\\right].$ Acting on this last equation to rotate back to $q_2$ , we must produce $\\\\Psi _{q^{\\\\prime \\\\prime }_1}(q_2) \\\\propto \\\\delta _{q_2,q(q^{\\\\prime \\\\prime }_1)}$ .\",\n \"But then Eq.\",\n \"REF implies that $\\\\Psi _{q^{\\\\prime }_1}(q^*_2)$ must also be proportional to $\\\\delta _{q_2,q(q^{\\\\prime }_1)}$ .\",\n \"This is a contradiction.\",\n \"Therefore, if a degree of freedom is maximally supported for all others fixed, then it is maximally supported for all values of the other degrees of freedom and vice-versa.\",\n \"In the latter case, a position state in the $i$ th degree of freedom can be represented as a Gaussian by using the $p$ -basis where it becomes a plane wave ($\\\\theta _i = 0$ ).\",\n \"In other words, one can always choose $x_i$ to be $p_i$ or $q_i$ such that Eq.\",\n \"REF holds for odd prime $d$ .\",\n \"$\\\\Box $ Finally, the reason this result does not hold for odd non-prime $d$ is that for discrete Wigner space there are $d + 1$ unique angles minus all the prime factors of $d$ .\",\n \"For non-prime $d$ , there is more than one such prime factor and so there are cases when one cannot “rotate” away all non-maximally supported states.\"\n ],\n [\n \"Discrete Center-Chord Representation\",\n \"We now proceed to the discrete case and introduce the center-chord formalism for these systems.\",\n \"It will be useful for us to define a pair of conjugate degrees of freedom $p$ and $q$ for discrete systems.\",\n \"Unfortunately, this isn't as straight-forward as in the continuous case, since the usual canonical commutation relations cannot hold in a finite-dimensional Hilbert space where the operators are bounded (since $\\\\operatorname{Tr}[\\\\hat{p}, \\\\hat{q}] = 0$ ).\",\n \"We begin in one degree of freedom.\",\n \"We label the computational basis for our system by $n \\\\in {0, 1, \\\\ldots , d-1}$ , for $d$ odd and we assume that $d$ is odd for the rest of this paper.\",\n \"We identify the discrete position basis with the computational basis and define the “boost” operator as diagonal in this basis: $\\\\hat{Z}^{\\\\delta p} \\\\left|n\\\\right\\\\rangle \\\\equiv \\\\omega ^{n \\\\delta p} \\\\left|n\\\\right\\\\rangle ,$ where $\\\\omega $ will be defined below.\",\n \"We define the normalized discrete Fourier transform operator to be equivalent to the Hadamard gate: $\\\\hat{F} = \\\\frac{1}{\\\\sqrt{d}}\\\\sum _{m,n \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}} \\\\omega ^{m n} \\\\left|m\\\\right\\\\rangle \\\\left\\\\langle n\\\\right|.$ This allows us to define the Fourier transform of $\\\\hat{Z}$ : $\\\\hat{X} \\\\equiv \\\\hat{F}^\\\\dagger \\\\hat{Z} \\\\hat{F}$ Again, as before, we call $\\\\hat{X}$ the “shift” operator since $\\\\hat{X}^{\\\\delta q} \\\\left|n\\\\right\\\\rangle \\\\equiv \\\\left|n\\\\oplus \\\\delta q\\\\right\\\\rangle ,$ where $\\\\oplus $ denotes mod-$d$ integer addition.\",\n \"It follows that the Weyl relation holds again: $\\\\hat{Z} \\\\hat{X} = \\\\omega \\\\hat{X} \\\\hat{Z}.$ The group generated by $\\\\hat{Z}$ and $\\\\hat{X}$ has a $d$ -dimensional irreducible representation only if $\\\\omega ^d=1$ for odd $d$ .\",\n \"Equivalently, there are only reflections relating any two phase space points on the Weyl phase space “grid” if $d$ is odd [25].\",\n \"We take $\\\\omega \\\\equiv \\\\omega (d) = e^{2 \\\\pi i/d}$ [26].\",\n \"This was introduced by Weyl [27].\",\n \"Note that this means that $\\\\hbar = \\\\frac{d}{2 \\\\pi }$ or $h = d$ .\",\n \"This means that the classical regime is most closely reached when the dimensionality of the system is reduced ($d \\\\rightarrow 0$ ) and thus the most “classical” system we can consider here is a qutrit (since we keep $d$ odd and greater than one).\",\n \"This is the opposite limit considered by many other approaches where the classical regime is reached when $d \\\\rightarrow \\\\infty $ .\",\n \"One way of interpreting the classical limit in this paper is by considering $h$ to be equal to the inverse of the density of states in phase space (i.e.\",\n \"in a Wigner unit cell).\",\n \"As $\\\\hbar \\\\rightarrow 0$ phase space area decreases as $\\\\hbar ^2$ but the number of states only decreases as $\\\\hbar $ leading to an overall density increase of $\\\\hbar $ .\",\n \"This agrees with the notion that states should become point particles of fixed mass in the classical limit.\",\n \"By analogy with continuous finite translation operators, we reexpress the shift $\\\\hat{X}$ and boost $\\\\hat{Z}$ operators in terms of conjugate $\\\\hat{ p}$ and $\\\\hat{ q}$ operators: $ \\\\hat{Z} \\\\left|n\\\\right\\\\rangle = e^{\\\\frac{2 \\\\pi i }{d}\\\\hat{q}} \\\\left|n\\\\right\\\\rangle = e^{\\\\frac{2 \\\\pi i}{d} n} \\\\left|n\\\\right\\\\rangle , $ and $ \\\\hat{X} \\\\left|n\\\\right\\\\rangle = e^{-\\\\frac{2 \\\\pi i }{d} \\\\hat{p}} \\\\left|n\\\\right\\\\rangle = \\\\left|n\\\\oplus 1\\\\right\\\\rangle .\",\n \"$ Hence, in the diagonal “position” representation for $\\\\hat{Z}$ : $\\\\hat{Z} = \\\\left( \\\\begin{array}{cccccc} 1 & 0 & 0 & \\\\cdots & 0\\\\\\\\0 & e^{\\\\frac{2 \\\\pi i}{d}} & 0 & \\\\cdots & 0\\\\\\\\0 & 0 & e^{\\\\frac{4 \\\\pi i}{d}} & \\\\cdots & 0\\\\\\\\\\\\vdots & \\\\vdots & \\\\vdots & \\\\ddots & \\\\vdots \\\\\\\\0 & 0 & 0 & 0 & e^{\\\\frac{2 (d-1) \\\\pi i}{d}} \\\\end{array} \\\\right),$ and $\\\\hat{X} = \\\\left( \\\\begin{array}{cccccc} 0 & 0 & \\\\cdots & 0 & 1\\\\\\\\1 & 0 & \\\\cdots & 0 & 0\\\\\\\\0 & 1 & 0 & \\\\cdots & 0\\\\\\\\\\\\vdots & \\\\ddots & \\\\ddots & \\\\ddots & \\\\vdots \\\\\\\\0 & \\\\cdots & 0 & 1 & 0 \\\\end{array} \\\\right).$ Thus, $\\\\hat{q} = \\\\frac{d}{2 \\\\pi i} \\\\log \\\\hat{Z} = \\\\sum _{n \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}} n \\\\left|n\\\\right\\\\rangle \\\\left\\\\langle n\\\\right|,$ and $\\\\hat{ p} = \\\\hat{F}^\\\\dagger \\\\hat{ q} \\\\hat{F}.$ Therefore, we can interpret the operators $\\\\hat{ p}$ and $\\\\hat{ q}$ as a conjugate pair similar to conjugate momenta and position in the continuous case.\",\n \"However, they differ from the latter in that they only obey the weaker group commutation relation $e^{i j \\\\hat{ q}/\\\\hbar }e^{i k \\\\hat{ p}/\\\\hbar } e^{-i j \\\\hat{ q}/\\\\hbar }e^{-i k \\\\hat{ p}/\\\\hbar } = e^{-i jk /\\\\hbar } \\\\hat{\\\\mathbb {I}}.$ This corresponds to the usual canonical commutation relation for $p$ and $q$ 's algebra at the origin of the Lie group ($j = k = 0$ ); expanding both sides of Eq.\",\n \"REF to first order in $p$ and $q$ yields the usual canonical relation.\",\n \"We proceed to introduce the Weyl representation of operators and states in discrete Hilbert spaces with odd dimension $d$ and $n$ degrees of freedom [1], [28], [29].\",\n \"The generalized phase space translation operator (the Weyl operator) is defined as a product of the shift and boost with a phase appropriate to the $d$ -dimensional space: $\\\\hat{T}(\\\\xi _p, \\\\xi _q) = e^{-i \\\\frac{\\\\pi }{d} \\\\xi _p \\\\cdot \\\\xi _q} \\\\hat{Z}^{ \\\\xi _p} \\\\hat{X}^{ \\\\xi _q},$ where $\\\\xi \\\\equiv (\\\\xi _p, \\\\xi _q) \\\\in (\\\\mathbb {Z} / d \\\\mathbb {Z})^{2n}$ and form a discrete “web” or “grid” of chords.\",\n \"They are a discrete subset of the continous chords we considered in the infinite-dimensional context in Section and their finite number is an important consequence of the discretization of the continuous Weyl formalism.\",\n \"Again, an operator $\\\\hat{A}$ can be expressed as a linear combination of translations: $\\\\hat{A} = d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _p, \\\\xi _q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} A_\\\\xi (\\\\xi _p, \\\\xi _q) \\\\hat{T}(\\\\xi _p, \\\\xi _q),$ where the weights are the chord representation of the function $\\\\hat{A}$ : ${A}_\\\\xi (\\\\xi _p, \\\\xi _q) = d^{-n} \\\\operatorname{Tr}\\\\left( \\\\hat{T}(\\\\xi _p, \\\\xi _q)^\\\\dagger \\\\hat{A} \\\\right).$ When applied to a state $\\\\hat{\\\\rho }$ , this is also called the “characteristic function” of $\\\\hat{\\\\rho }$ [30].\",\n \"As before, the center representation, based on reflections instead of translations, requires an appropriately defined reflection operator.\",\n \"We can define the discrete reflection operator $\\\\hat{R}$ as the symplectic Fourier transform of the discrete translation operator we just introduced: $\\\\hat{R}(x_p, x_q) = d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _p, \\\\xi _q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} e^{\\\\frac{2 \\\\pi i}{d} (\\\\xi _p, \\\\xi _q) \\\\mathcal {J} (x_p, x_q)^T} \\\\hat{T}(\\\\xi _p, \\\\xi _q).$ With this in hand, we can now express a finite-dimensional operator $\\\\hat{A}$ as a superposition of reflections: $\\\\hat{A} = d^{-n} \\\\sum _{\\\\begin{array}{c}x_p, x_q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} {A}_x(x_p, x_q) \\\\hat{R}( x_p, x_q ),$ where ${A}_x(x_p, x_q) = d^{-n} \\\\operatorname{Tr}\\\\left( \\\\hat{R}(x_p, x_q)^\\\\dagger \\\\hat{A} \\\\right).$ $x \\\\equiv (x_p, x_q) \\\\in (\\\\mathbb {Z} / d \\\\mathbb {Z})^{2n}$ are centers or Weyl phase space points and, like their $(\\\\xi _p, \\\\xi _q)$ brethren, form a discrete subgrid of the continuous Weyl phase space points considered in Section .\",\n \"Again, the center representation is of particular interest to us because for unitary gates $\\\\hat{A}$ we can rewrite the components ${A}_x(x_p, x_q)$ as: ${A}_x(x_p, x_q) = \\\\exp \\\\left[\\\\frac{i}{\\\\hbar } S(x_p, x_q)\\\\right]$ where $S(x_p, x_q)$ is the argument to the exponential, and is equivalent to the action of the operator in center representation (the center generating function).\",\n \"Aside from Eq.\",\n \"REF , the center representation of a state $\\\\hat{\\\\rho }$ can also be directly defined as the symplectic Fourier transform of its chord representation, $\\\\rho _\\\\xi $ [3]: $\\\\rho _x(x_p, x_q) = d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _p, \\\\xi _q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} e^{\\\\frac{2 \\\\pi i}{d} (\\\\xi _p, \\\\xi _q) \\\\mathcal {J} (x_p, x_q)^T} \\\\rho _\\\\xi (\\\\xi _p,\\\\xi _q).$ We note again that for a pure state $\\\\left|\\\\Psi \\\\right\\\\rangle $ , the Wigner function from Eqs.\",\n \"REF and REF simplifies to: ${\\\\Psi }_x(x_p, x_q) &=& d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _q \\\\in \\\\\\\\(\\\\mathbb {Z} / d \\\\mathbb {Z})^{ n}\\\\end{array}} e^{-\\\\frac{2 \\\\pi i}{d} \\\\xi _q \\\\cdot x_p} \\\\Psi \\\\left( x_q + \\\\frac{(d+1) \\\\xi _q }{2} \\\\right) {\\\\Psi ^*} \\\\left( x_q - \\\\frac{(d+1) \\\\xi _q}{2} \\\\right).$ It may be clear from this short presentation that the chord and center representations are dual to each other.\",\n \"A thorough review of this subject can be found in [13].\"\n ],\n [\n \"Path Integral Propagation in Discrete Systems\",\n \"Rivas and Almeida [14] found that the continuous infinite-dimensional vVMG propagator can be extended to finite Hilbert space by simply projecting it onto its finite phase space tori.\",\n \"This produces: $&& {U}_t(x) = \\\\\\\\&& \\\\left< \\\\sum _j \\\\left\\\\lbrace \\\\det \\\\left[ 1 + \\\\frac{1}{2} \\\\mathcal {J} \\\\frac{\\\\partial ^2 S_{tj}}{\\\\partial x^2} \\\\right] \\\\right\\\\rbrace ^{\\\\frac{1}{2}} e^{\\\\frac{i}{\\\\hbar } S_{tj}(x)} e^{i \\\\theta _k} \\\\right>_k + \\\\mathcal {O}(\\\\hbar ^2),\\\\nonumber $ where an additional average must be taken over the $k$ center points that are equivalent because of the periodic boundary conditions.\",\n \"Maintaining periodicity requires that they accrue a phase $\\\\theta _k$ See Eq.\",\n \"$5.18$ in [14] for a definition of the phase..\",\n \"The derivative $\\\\frac{\\\\partial ^2 S_{tj}}{\\\\partial x^2}$ is performed over the continuous function $S_{tj}$ defined after Eq.\",\n \"REF , but only evaluated at the discrete Weyl phase space points $x \\\\equiv (x_p, x_q)\\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^{2n}$ .\",\n \"The prefactor can also be reexpressed: $\\\\left\\\\lbrace \\\\det \\\\left[ 1 + \\\\frac{1}{2} \\\\mathcal {J} \\\\frac{\\\\partial ^2 S_{tj}}{\\\\partial x^2} \\\\right] \\\\right\\\\rbrace ^{\\\\frac{1}{2}} = \\\\left\\\\lbrace 2^d \\\\det \\\\left[ 1 + \\\\mathcal {M}_{tj} \\\\right] \\\\right\\\\rbrace ^{-\\\\frac{1}{2}},$ which is perhaps more pleasing in the discrete case as it does not involve a continuous derivative.\",\n \"As in the continuous case, for a harmonic Hamiltonian $H(p, q)$ , the center generating function $S(x_p, x_q)$ is equal to $\\\\alpha ^T \\\\mathcal {J} x + x^T \\\\mathcal {B} x$ where Eq.\",\n \"REF and Eq.\",\n \"REF hold.\",\n \"Moreover, if the Hamiltonian takes Weyl phase space points to themselves, then by the same equations it follows that $\\\\mathcal {M}$ and $\\\\alpha $ must have integer entries.\",\n \"This implies that for $m, n \\\\in \\\\mathbb {Z}^{ n}$ , $&& \\\\mathcal {M} \\\\left(\\\\begin{array}{c}p+ m d + \\\\alpha _p/2\\\\\\\\q + n d + \\\\alpha _q/2\\\\end{array}\\\\right) + \\\\left( \\\\begin{array}{c} \\\\alpha _p/2\\\\\\\\ \\\\alpha _q/2 \\\\end{array} \\\\right) \\\\nonumber \\\\\\\\&=& \\\\mathcal {M} \\\\left(\\\\begin{array}{c}p + \\\\alpha _p/2 \\\\\\\\ q + \\\\alpha _q/2 \\\\end{array}\\\\right) + \\\\left( \\\\begin{array}{c} \\\\alpha _p/2\\\\\\\\ \\\\alpha _q/2 \\\\end{array} \\\\right) + d \\\\mathcal {M}\\\\left(\\\\begin{array}{c}m\\\\\\\\ n\\\\end{array}\\\\right) \\\\\\\\&=& \\\\left(\\\\begin{array}{c}p^{\\\\prime }\\\\\\\\ q^{\\\\prime }\\\\end{array}\\\\right) \\\\mod {d. \\\\nonumber }Therefore, phase space points (p, q) that lie on Weyl phase space points go to the equivalent Weyl phase space points (p^{\\\\prime }, q^{\\\\prime }).$ Moreover, again if $m, n \\\\in \\\\mathbb {Z}^n$ , $&& S(x_p + m d, x_q + n d) \\\\nonumber \\\\\\\\&=& \\\\left( \\\\begin{array}{c}x_p+ m d\\\\\\\\ x_q+ n d\\\\end{array} \\\\right)^T A \\\\left( \\\\begin{array}{c}x_p+ m d\\\\\\\\ x_q+ n d\\\\end{array} \\\\right) \\\\nonumber \\\\\\\\&& + b \\\\cdot \\\\left( \\\\begin{array}{c}x_p+ m d\\\\\\\\ x_q+ n d\\\\end{array} \\\\right)\\\\\\\\&=& \\\\left( \\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array} \\\\right)^T A \\\\left( \\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array} \\\\right) + b \\\\cdot \\\\left( \\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array} \\\\right) \\\\nonumber \\\\\\\\&& + d \\\\left[ 2 \\\\left( \\\\begin{array}{c}x_p\\\\\\\\ x_q\\\\end{array} \\\\right)^T A \\\\left( \\\\begin{array}{c}m\\\\\\\\ n\\\\end{array} \\\\right) \\\\right.\\\\nonumber \\\\\\\\&& \\\\qquad \\\\left.+ d \\\\left( \\\\begin{array}{c}m\\\\\\\\ n\\\\end{array} \\\\right)^T A \\\\left( \\\\begin{array}{c}m\\\\\\\\ n\\\\end{array} \\\\right) + b \\\\cdot \\\\left( \\\\begin{array}{c}m\\\\\\\\ n\\\\end{array} \\\\right)\\\\right] \\\\nonumber \\\\\\\\&=& S(x_p, x_q) \\\\mod {d}, \\\\nonumber $ for some symmetric $A \\\\in \\\\mathbb {Z}^{n\\\\times n}$ and $b \\\\in \\\\mathbb {Z}^{n}$ .\",\n \"Therefore, these equivalent trajectories also have equivalent actions (since the action is multiplied by $\\\\frac{2 \\\\pi i}{d}$ and exponentiated).\",\n \"Hence, there is only one term to the sum in Eq.\",\n \"REF .\",\n \"Moreover, [25] showed that the sum over the phases $\\\\theta _k$ produces only a global phase that can be factored out.\",\n \"Therefore, if we can neglect the overall phase, ${U}_t(x) = \\\\left| 2^d \\\\det \\\\left[ 1 + \\\\mathcal {M} \\\\right] \\\\right|^{\\\\frac{1}{2}},$ where the classical trajectories whose centers are $(x_p, x_q)$ satisfy the periodic boundary conditions.\",\n \"As in the continuous case, we point out that this means that translations and reflections are fully captured by a path integral treatment that is truncated at order $\\\\hbar ^1$ (or order $\\\\hbar ^0$ if their overall phase isn't important) because their Hamiltonians are harmonic, but in the discrete case there is an additional requirement that they are evaluated at chords/centers that take Weyl phase space points to themselves.\",\n \"Just as in the continuous case, the single contribution at order $\\\\hbar ^0$ implies that the propagator of the Wigner function of states, $\\\\left|\\\\Psi \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi \\\\right|_x(x)$ under gates $\\\\hat{V}$ with underlying harmonic Hamiltonians is captured by $\\\\left|\\\\Psi \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi \\\\right|_x(\\\\mathcal {M}_{\\\\hat{V}} (x + \\\\alpha _{\\\\hat{V}}/2) + \\\\alpha _{\\\\hat{V}}/2)$ for $\\\\mathcal {M}_{\\\\hat{V}}$ and $\\\\alpha _{\\\\hat{V}}$ associated with $\\\\hat{V}$ .\"\n ],\n [\n \"Stabilizer Group\",\n \"Here we will show that the Hamiltonians corresponding to Clifford gates are harmonic and take Weyl phase space points to themselves.\",\n \"Thus they can be captured by only the single contribution of Eq.\",\n \"REF at lowest order in $\\\\hbar $ .\",\n \"This then implies that stabilizer states can also be propagated to each other by Clifford gates with only a single contribution to the sum in Eq.\",\n \"REF .\",\n \"The Clifford gate set of interest can be defined by three generators: a single qudit Hadamard gate $\\\\hat{F}$ and phase shift gate $\\\\hat{P}$ , as well as the two qudit controlled-not gate $\\\\hat{C}$ .\",\n \"We examine each of these in turn.\"\n ],\n [\n \"Hadamard Gate\",\n \"The Hadamard gate was defined in Eq.\",\n \"REF and is a rotation by $\\\\frac{\\\\pi }{2}$ in phase space counter-clockwise.\",\n \"Hence, for one qudit, it can be written as the map in Eq.\",\n \"REF where $\\\\mathcal {M}_{\\\\hat{F}} = \\\\left( \\\\begin{array}{cc} 0 & 1\\\\\\\\ -1 & 0 \\\\end{array} \\\\right),$ and $\\\\alpha _{\\\\hat{F}} = (0,0)$ .\",\n \"We have set $t=1$ and drop it from the subscripts from now on.\",\n \"Since $\\\\alpha $ is vanishing and $\\\\mathcal {M}$ has integer entries, this is a cat map and such maps have been shown to correspond to Hamiltonians [32] $&&H(p,q) =\\\\\\\\&&f(\\\\operatorname{Tr}\\\\mathcal {M} ) \\\\left[ \\\\mathcal {M}_{12} p^2 - \\\\mathcal {M}_{21} q^2 + \\\\left(\\\\mathcal {M}_{11} - \\\\mathcal {M}_{22} \\\\right) pq \\\\right],\\\\nonumber $ where $f(x) = \\\\frac{\\\\sinh ^{-1}(\\\\frac{1}{2}\\\\sqrt{x^2-4})}{\\\\sqrt{x^2-4}}.$ For the Hadamard $\\\\mathcal {M}_{\\\\hat{F}}$ this corresponds to $H_{\\\\hat{F}} = \\\\frac{\\\\pi }{4} (p^2 + q^2)$ , a harmonic oscillator.\",\n \"The center generating function $S(x_p, x_q)$ is thus $(x_p, x_q ) \\\\mathcal {B} (x_p, x_q)^T$ and solving Eq.\",\n \"REF finds for the one-qudit Hadamard, $\\\\mathcal {B}_{\\\\hat{F}} = \\\\left(\\\\begin{array}{cc}1 & 0\\\\\\\\ 0 & 1 \\\\end{array} \\\\right).$ Thus, $S_{\\\\hat{F}}(x_p, x_q) = x_p^2 + x_q^2$ .\",\n \"Indeed, applying Eq.\",\n \"REF to Eq.\",\n \"REF reveals that the Hadamard's center function (up to a phase) is: ${F}_x(x_p,x_q) = e^{\\\\frac{2 \\\\pi i}{d}(x_p^2+x_q^2)}.$ Eq.\",\n \"REF shows how to map Weyl phase space to Weyl phase space under the Hadamard transformation.\",\n \"Furthermore this map is pointwise, which implies the quadratic form of the center generating function obtained in Eq.\",\n \"REF .\"\n ],\n [\n \"Phase Shift Gate\",\n \"The phase shift gate can be generalized to odd $d$ -dimensions [33] by setting it to: $\\\\hat{P} = \\\\sum _{j \\\\in \\\\mathbb {Z}/d \\\\mathbb {Z}} \\\\omega ^{\\\\frac{(j-1)j}{2}} \\\\left|j\\\\right\\\\rangle \\\\left\\\\langle j\\\\right|.$ Examining its effect on stabilizer states, it is clear that it is a $q$ -shear in phase space from an origin displaced by $\\\\frac{d-1}{2}\\\\equiv -\\\\frac{1}{2}$ to the right.\",\n \"This can be expressed as the map in Eq.\",\n \"REF with $\\\\mathcal {M}_{\\\\hat{P}} = \\\\left( \\\\begin{array}{cc} 1 & 1\\\\\\\\ 0 & 1 \\\\end{array} \\\\right),$ and $\\\\alpha _{\\\\hat{P}} = \\\\left( -\\\\frac{1}{2}, 0 \\\\right)$ .\",\n \"This corresponds to $\\\\mathcal {B}_{\\\\hat{P}} = \\\\left( \\\\begin{array}{cc} 0 & 0\\\\\\\\ 0 & \\\\frac{1}{2} \\\\end{array} \\\\right).$ Solving Eq.\",\n \"REF with this $\\\\mathcal {B}_{\\\\hat{P}}$ and $\\\\alpha _{\\\\hat{P}}$ reveals that $S_{\\\\hat{P}}(x_p, x_q) = -\\\\frac{1}{2} x_q + \\\\frac{1}{2} x_q^2$ .\",\n \"Again, this agrees with the argument of the center representation of the phase-shift gate obtained by applying Eq.\",\n \"REF to Eq.\",\n \"REF : ${P}_x(x_p,x_q) = e^{\\\\frac{2 \\\\pi i}{d} \\\\frac{1}{2} (-x_q + x_q^2)}.$ Discretization the equations of motion for harmonic evolution for unit timesteps leads to: $\\\\left(\\\\begin{array}{c}p^{\\\\prime }\\\\\\\\ q^{\\\\prime }\\\\end{array}\\\\right) = \\\\left(\\\\begin{array}{c}p\\\\\\\\ q\\\\end{array}\\\\right) + \\\\mathcal {J} \\\\left(\\\\frac{\\\\partial H}{\\\\partial p}, \\\\frac{\\\\partial H}{\\\\partial q}\\\\right)^T,$ where the last derivative is on the continuous function $H$ , but only evaluated on the discrete Weyl phase space points.\",\n \"It follows that $H_{\\\\hat{P}} = -\\\\frac{d+1}{2} q^2 + \\\\frac{d+1}{2} q.$ We have obtained the Hamiltonian for the phase-shift gate by a different procedure than that used for the Hadamard where we appealed to the result given in Eq.\",\n \"REF for quantum cat maps.\",\n \"However, as the phase-shift gate is a quantum cat map as well, we could have obtained Eq.\",\n \"REF in this manner.\",\n \"Similarly, the approach we used to find the phase-shift Hamiltonian by discretizing time in Eq.\",\n \"REF would work for the Hadamard gate but it is a bit more involved since the latter contains both $p$ - and $q$ -evolution.\",\n \"Nevertheless, this produces Eq.\",\n \"REF as well.\",\n \"We presented both techniques for illustrative purposes.\"\n ],\n [\n \"Controlled-Not Gate\",\n \"Lastly, the controlled-not gate can be generalized to $d$ -dimensions [33] by $\\\\hat{C} = \\\\sum _{j,k \\\\in \\\\mathbb {Z}/d \\\\mathbb {Z}} \\\\left|j,k \\\\oplus j\\\\right\\\\rangle \\\\left\\\\langle j,k\\\\right|.$ It is clear that this translates the $q$ -state of the second qudit by the $q$ -state of the first qudit.\",\n \"As a result, as is evident by examining the gate's action on stabilizer states, the first qudit experiences an “equal and opposite reaction” force that kicks its momentum by the $q$ -state of the second qudit.\",\n \"This is the phase space picture of the well-known fact that a CNOT examined in the $\\\\hat{X}$ basis has the control and target reversed with respect to the $\\\\hat{Z}$ basis.\",\n \"This can also seen by looking at its effect in the momentum ($\\\\hat{X}$ ) basis: ${\\\\hat{F}}^\\\\dagger \\\\hat{C} \\\\hat{F} = \\\\sum _{j,k \\\\in \\\\mathbb {Z}/d \\\\mathbb {Z}} \\\\left|j \\\\ominus k,k\\\\right\\\\rangle \\\\left\\\\langle j,k\\\\right|.$ As a result, this gate is described by the map: $\\\\mathcal {M}_{\\\\hat{C}} = \\\\left( \\\\begin{array}{ccccc} 1 & -1 & 0 & 0\\\\\\\\ 0 & 1 & 0 & 0\\\\\\\\ 0 & 0 & 1 & 0\\\\\\\\ 0 & 0 & 1 & 1\\\\end{array}\\\\right),$ and $\\\\alpha _{\\\\hat{C}} = (0,0,0,0)$ .\",\n \"This corresponds to $\\\\mathcal {B}_{\\\\hat{C}} = \\\\left( \\\\begin{array}{ccccc} 0 & 0 & 0 & 0\\\\\\\\ 0 & 0 & -\\\\frac{1}{2} & 0\\\\\\\\ 0 & -\\\\frac{1}{2} & 0 & 0\\\\\\\\ 0 & 0 & 0 & 0\\\\end{array}\\\\right).$ Hence its center generating function $S_{\\\\hat{C}}(x_p, x_q) = -x_{p_2} x_{q_1}$ .\",\n \"Again, this corresponds with the argument of the center representation of the controlled-not gate, which can be found to be: ${C}_x(x_p, x_q) = e^{-\\\\frac{2 \\\\pi i}{d} x_{q_1} x_{p_2}}.$ Therefore, this gate can be seen to be a bilinear $p$ -$q$ coupling between two qudits and corresponds to the Hamiltonian $H_{\\\\hat{C}} = p_1 q_2,$ as can be found from Eq.\",\n \"REF again.\",\n \"As a result, it is now clear that all the Clifford group gates have Hamiltonians that are harmonic and that take Weyl phase space points to themselves.\",\n \"Therefore, their propagation can be fully described by a truncation of the semiclassical propagator Eq.\",\n \"REF to order $\\\\hbar ^0$ as in Eq.\",\n \"REF and they are manifestly classical in this sense.\",\n \"To summarize the results of this section, using Eq.\",\n \"REF , the Hadamard, phase shift, and CNOT gates can be written as: $\\\\hat{F} = d^{-2} \\\\sum _{\\\\begin{array}{c}x_p,x_q, \\\\\\\\ \\\\xi _p,\\\\xi _q \\\\in \\\\\\\\ \\\\mathbb {Z}/d \\\\mathbb {Z}\\\\end{array}} e^{ -\\\\frac{2 \\\\pi i}{d} \\\\left[ -(x_p^2 + x_q^2) - d \\\\left(x_p \\\\xi _p - x_q \\\\xi _q \\\\right) \\\\right] } \\\\hat{Z}^{\\\\xi _p} \\\\hat{X}^{\\\\xi _q},$ $\\\\hat{P} = d^{-2} \\\\sum _{\\\\begin{array}{c}x_p,x_q, \\\\\\\\ \\\\xi _p,\\\\xi _q \\\\in \\\\\\\\ \\\\mathbb {Z}/d \\\\mathbb {Z}\\\\end{array}} e^{ -\\\\frac{2 \\\\pi i}{d} \\\\left[ \\\\frac{1}{2} (x_q - x_q^2) - d \\\\left(x_p \\\\xi _q - x_q \\\\xi _p \\\\right) \\\\right] } \\\\hat{Z}^{\\\\xi _p} \\\\hat{X}^{\\\\xi _q},$ and $&&\\\\hat{C} = \\\\\\\\&& d^{-4} \\\\sum _{\\\\begin{array}{c}x_p, x_q, \\\\\\\\ \\\\xi _p, \\\\xi _q \\\\in \\\\\\\\ (\\\\mathbb {Z}/d \\\\mathbb {Z})^{ 2}\\\\end{array}} e^{ -\\\\frac{2 \\\\pi i}{d} \\\\left[ x_{q_1} x_{p_2} - d \\\\left(x_p \\\\cdot \\\\xi _q - x_q \\\\cdot \\\\xi _p \\\\right) \\\\right] } \\\\hat{Z}^{ \\\\xi _p} \\\\hat{X}^{ \\\\xi _q}, \\\\nonumber $ (up to a phase).\",\n \"This form emphasizes their quadratic nature.\",\n \"As for the continuous case, there exists a particularly simple way in discrete systems to see to what order in $\\\\hbar $ the path integral must be kept to handle unitary propagation beyond the Clifford group.\",\n \"We describe this in the next section.\",\n \"We note that the center generating actions $S(x_p, x_q)$ found here are related to the $G(q^{\\\\prime }, q)$ found by Penney et al.\",\n \"[12], which are in terms of initial and final positions, by symmetrized Legendre transform [13]: $G(q^{\\\\prime }, q, t) = F\\\\left(\\\\frac{q^{\\\\prime } + q}{2}, p(q^{\\\\prime } - q)\\\\right),$ where the canonical generating function $F\\\\left(\\\\frac{q^{\\\\prime } + q}{2}, p\\\\right) = S\\\\left( x_p = p, x_q = \\\\frac{q^{\\\\prime } + q}{2} \\\\right) + p \\\\cdot \\\\left( q^{\\\\prime } - q \\\\right),$ for $p (q^{\\\\prime } - q)$ given implicitly by $\\\\frac{\\\\partial F}{\\\\partial p} = 0$ .\",\n \"Applying this to the actions we found reveals that $G_{\\\\hat{F}}(q^{\\\\prime }, q, t) = q^{\\\\prime } q,$ $G_{\\\\hat{P}}(q^{\\\\prime }, q, t) = \\\\frac{d+1}{2} (q^2 - q),$ and $G_{\\\\hat{C}}((q^{\\\\prime }_1, q^{\\\\prime }_2), (q_1, q_2), t) = 0,$ which is in agreement with [12].\"\n ],\n [\n \"Classicality of Stabilizer States\",\n \"In this subsection we show that stabilizer states evolve to stabilizer states under Clifford gates, and that it is possible to describe this evolution classically.\",\n \"For odd $d \\\\ge 3$ the positivity of the Wigner representation implies that evolution of stabilizer states is non-contextual, and so here we are investigating in detail what this means in our semiclassical picture.\",\n \"To begin, it is instructive to see the form stabilizer states take in the discrete position representation and in the center representation.\",\n \"Gross proved that [3]: Theorem 1 Let $d$ be odd and $\\\\Psi \\\\in L^2((\\\\mathbb {Z}/d\\\\mathbb {Z})^n)$ be a state vector.\",\n \"The Wigner function of $\\\\Psi $ is non-negative if and only if $\\\\Psi $ is a stabilizer state.\",\n \"Gross also proved [3] Corollary 1 Given that $\\\\Psi (q) \\\\ne 0$ $\\\\forall \\\\, q$ , a vector $\\\\Psi $ is a stabilizer state if and only if it is of the form $\\\\Psi _{\\\\theta _\\\\beta ,\\\\eta _\\\\beta }(q) \\\\propto \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( q^T \\\\theta _\\\\beta q + \\\\eta _\\\\beta \\\\cdot q \\\\right) \\\\right].$ where $\\\\theta _\\\\beta \\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^{n\\\\times n}$ and $q, \\\\eta _\\\\beta \\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^n$ .\",\n \"Applying Eq.\",\n \"REF to Eq.\",\n \"REF , the Wigner function of such maximally supported stabilizer states can be found to be: $&& {\\\\Psi _{\\\\theta _\\\\beta ,\\\\eta _\\\\beta }}_x(x_p, x_q) \\\\propto \\\\\\\\&& d^{-n} \\\\sum _{\\\\begin{array}{c}\\\\xi _q \\\\in \\\\\\\\\\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^{ n}\\\\end{array}} \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\xi _q \\\\cdot \\\\left( \\\\eta _\\\\beta - x_p + 2 \\\\theta _\\\\beta x_q \\\\right)\\\\right].\\\\nonumber $ Therefore, one finds that the Wigner function is the discrete Fourier sum equal to $\\\\delta _{\\\\eta _\\\\beta - x_p + 2\\\\theta _\\\\beta x_q}$ .\",\n \"For $\\\\theta _\\\\beta = 0$ the state is a momentum state at $x_p$ .\",\n \"Finite $\\\\theta _\\\\beta $ rotates that momentum state in phase space in “steps” such that it always lies along the discrete Weyl phase space points $(x_p, x_q) \\\\in (\\\\mathbb {Z}/d\\\\mathbb {Z})^{2n}$ .\",\n \"This Gaussian expression only captures stabilizer states that are maximally supported in $q$ -space.\",\n \"One may wonder what the stabilizer states that aren't maximally supported in $q$ -space look like in Weyl phase space.\",\n \"Of course, it is possible that some may be maximally supported in $p$ -space and so can be captured by the following corollary: Corollary 2 If $\\\\Psi (p) \\\\ne 0$ for all $p$ 's then there exists a $\\\\theta _{\\\\beta p} \\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^{n\\\\times n}$ and an $\\\\eta _{\\\\beta p} \\\\in \\\\left(\\\\mathbb {Z}/d\\\\mathbb {Z}\\\\right)^n$ such that $\\\\Psi _{\\\\theta _{\\\\beta p},\\\\eta _{\\\\beta p}}(p) \\\\propto \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( p^T \\\\theta _{\\\\beta p} p + \\\\eta _{\\\\beta p} \\\\cdot p \\\\right) \\\\right].$ Proof This can be shown following the same methods employed by Gross [3] but in the discrete $p$ -basis.\",\n \"Unfortunately, it is easy to show that Corollary REF and REF do not provide an expression for all stabilizer states (except for the odd prime $d$ case, as we shall see shortly) as there exist stabilizer states for odd non-prime $d$ that are not maximally supported in $p$ - or $q$ -space or any finite rotation between those two.\",\n \"To find an expression that encompasses all stabilizer states, we must turn to the Wigner function of stabilizer states.\",\n \"An equivalent definition of stabilizer states on $n$ qudits is given by states $\\\\hat{V} \\\\underbrace{\\\\left|0\\\\right\\\\rangle \\\\otimes \\\\cdots \\\\otimes \\\\left|0\\\\right\\\\rangle }_{n}$ where $\\\\hat{V}$ is a quantum circuit consisting of Clifford gates.\",\n \"We know that the Clifford circuits are generated by the $\\\\hat{P}$ , $\\\\hat{F}$ and $\\\\hat{C}$ gates, and that the Wigner functions $\\\\Psi _x(x)$ of stabilizer states propagate under $\\\\hat{V}$ as $\\\\Psi _x(\\\\mathcal {M}_{\\\\hat{V}} (x + \\\\alpha _{\\\\hat{V}}/2) + \\\\alpha _{\\\\hat{V}}/2)$ , it follows that the Wigner function of stabilizer states is: $\\\\delta _{\\\\Phi _0 \\\\cdot \\\\mathcal {M}_{\\\\hat{V}} \\\\cdot x, r_0},$ where $\\\\Phi _0 = \\\\left(\\\\begin{array}{cc} 0 & 0\\\\\\\\ 0 & \\\\mathbb {I}_n\\\\end{array} \\\\right)$ and $r_0 = (0, 0)$ .\",\n \"We have therefore proved the next theorem: Theorem 2 The Wigner function $\\\\Psi _x(x)$ of a stabilizer state for any odd $d$ and $n$ qudits is $\\\\delta _{\\\\Phi \\\\cdot x, r}$ for $2n \\\\times 2n$ matrix $\\\\Phi $ and $2n$ vector $r$ .\",\n \"As an aside, Theorem REF allows us to develop an all-encompassing Gaussian expression for stabilizer states for the restricted case that $d$ is odd prime.\",\n \"In this case, the following Corollary shows that a “mixed” representation is always possible: where each degree of freedom is expressed in either the $p$ - or $q$ -basis: Corollary 3 For odd prime $d$ , if $\\\\Psi $ is a stabilizer state for $n$ qudits, then there always exists a mixed representation in position and momentum such that: $\\\\Psi _{\\\\theta _{\\\\beta x},\\\\eta _{\\\\beta x}}(x) = \\\\frac{1}{\\\\sqrt{d}} \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( x^T \\\\theta _{\\\\beta x} x + \\\\eta _{\\\\beta x} \\\\cdot x \\\\right) \\\\right],$ where $x_i$ can be either $p_i$ or $q_i$ .\",\n \"Proof We begin with a one-qudit case.\",\n \"We examine the equation specified by $\\\\Phi \\\\cdot x = r$ : $\\\\alpha q_1 + \\\\beta p_1 = \\\\gamma $ for $\\\\alpha $ , $\\\\beta $ , and $\\\\gamma \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ .\",\n \"If $\\\\alpha = 0$ then $\\\\Psi (q_1)$ is maximally supported and if $\\\\beta = 0$ then $\\\\Psi (p_1)$ is maximally supported since the equation specifies a line on $\\\\mathbb {Z}/d \\\\mathbb {Z}$ in $q_1$ and $p_1$ respectively.\",\n \"If $\\\\alpha \\\\ne 0$ and $\\\\beta \\\\ne 0$ then the equation can be rewritten as $q_1 + (\\\\beta /\\\\alpha ) p_1 = \\\\gamma /\\\\alpha ,$ and it follows that $q_1$ can take any values on $\\\\mathbb {Z}/d \\\\mathbb {Z}$ and so $\\\\Psi (q_1)$ is maximally supported.\",\n \"However, it is also possible to reexpress the equation as: $p_1 + (\\\\alpha /\\\\beta ) q_1 = \\\\gamma /\\\\beta .$ It follows that $p_1$ can also take any values on $\\\\mathbb {Z}/d \\\\mathbb {Z}$ —$\\\\Psi (p_1)$ is also maximally supported.\",\n \"Therefore, one can always choose either a $p_1$ - or $q_1$ -basis such that the state is maximally supported and so is representable by a Gaussian function.\",\n \"We now consider adding another qudit such that the state becomes ${\\\\Psi }_x(p_1,p_2,q_1,q_2)$ .\",\n \"There are now two equations specified by $\\\\Phi \\\\cdot x = r$ and it follows that it is always possible to combine the two equations such that $p_1$ and $q_1$ are only in one equation and written in terms of each other (and generally the second degree of freedom): $\\\\alpha q_1 + \\\\beta p_1 + \\\\gamma q_2 + \\\\delta p_2 = \\\\epsilon ,$ for $\\\\alpha $ , $\\\\beta $ , $\\\\gamma $ , $\\\\delta $ and $\\\\epsilon \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ .\",\n \"It will turn out that the $\\\\gamma q_2 + \\\\delta p_2$ term is irrelevant.\",\n \"We can rewrite the above equation as: $q_1 + (\\\\beta /\\\\alpha ) p_1 + (\\\\gamma /\\\\alpha ) q_2 + (\\\\delta /\\\\alpha ) p_2 = \\\\epsilon /\\\\alpha ,$ if $\\\\alpha \\\\ne 0$ .\",\n \"Since there is no other equation specifying $p_1$ , this is an equation for a line on $\\\\mathbb {Z}/ d \\\\mathbb {Z}$ and so $\\\\Psi $ is is maximally supported on $q_1$ .\",\n \"Otherwise, rewriting the above equation as: $p_1 + (\\\\alpha /\\\\beta ) q_1 +(\\\\gamma /\\\\beta ) q_2 + (\\\\delta /\\\\beta ) p_2 = \\\\epsilon /\\\\beta ,$ if $\\\\beta \\\\ne 0$ shows that $\\\\Psi $ is maximally supported on $p_1$ .\",\n \"If $\\\\alpha = \\\\beta = 0$ then both $p_1$ and $q_1$ are undetermined and so either representation produces a maximally supported state.\",\n \"The same procedure can be performed to find if $q_2$ or $p_2$ produce a maximally supported state.\",\n \"As can be seen, we are really just repeating the same procedure as we did when there was only one qudit because the other degrees of freedom have no impact on this determination.\",\n \"Expressing $\\\\Psi $ in the basis that is maximally supported in every degree of freedom means that it is therefore a Gaussian.\",\n \"Therefore, it follows that every degree of freedom (corresponding to a qudit) is maximally supported in either the $p$ - or $q$ - basis and so Eq.\",\n \"REF always describes stabilizer states for odd prime $d$ .$\\\\Box $ The form of Eq.\",\n \"REF is more general than Eq.\",\n \"REF because it does not depend on the support of the state.\",\n \"As we saw in the proof, this representation is generally not unique; for every qudit $i$ that is not a position or momentum state, $x_i$ can be either $p_i$ or $q_i$ .\",\n \"However, if it is a position state then $x_i = p_i$ and if it is a momentum state then $x_i = q_i$ ; position and momentum states must be expressed in their conjugate representation in order to be captured by a Gaussian of the form in Eq.\",\n \"REF instead of Kronecker deltas.\",\n \"The reason this mixed representation doesn't hold for non-prime odd $d$ is that the coefficients above can be (multiples of) prime factors of $d$ and so no longer produce “lines” in $p_i$ or $q_i$ that cover all of $\\\\mathbb {Z}/d \\\\mathbb {Z}$ .\",\n \"An alternative proof of this corollary that explores this case further is presented in the Appendix.\",\n \"An example of the different classes of stabilizer states that are possible for odd prime $d$ , in terms of their support, is shown in Fig.\",\n \"REF .\",\n \"There it can be seen that a stabilizer state is either maximally supported in $p_i$ or $q_i$ , and is a Kronecker delta function in the other degree of freedom, or it is maximally supported in both.\",\n \"Figure: The two classes of stabilizer states possible for odd prime d=7d=7 in terms of support: a) maximally supported in pp or qq and b) maximally supported in pp and qq.\",\n \"The central grids denote the Wigner function ΨΨ x (p,q)\\\\left|\\\\Psi \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi \\\\right|_x(p,q) of a stabilizer state Ψ\\\\Psi with d=7d=7.\",\n \"The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\\\Psi (q)|^2).On the other hand, for odd non-prime $d$ , we see in Fig.\",\n \"REF that another class is possible: stabilizer states that are maximally supported in neither $p_i$ or $q_i$ .\",\n \"In fact, rotating the basis in any of the discrete angles afforded by the grid still does not produce a basis that is maximally supported (as discussed in the Appendix).\",\n \"Notice also, that Fig.\",\n \"REF b shows that it is no longer true that a state that is maximally supported in only $q_i$ or $p_i$ is automatically a Kronecker delta when expressed in terms of the other.\",\n \"Figure: The four classes of stabilizer states possible for odd non-prime d=15d=15 in terms of support: a) & b) maximally supported in pp or qq, c) maximally supported in pp and qq, and d) not maximally supported in pp or qq.\",\n \"The central grids denote the Wigner function ΨΨ x (p,q)\\\\left|\\\\Psi \\\\right\\\\rangle \\\\left\\\\langle \\\\Psi \\\\right|_x(p,q) for a stabilizer state Ψ\\\\Psi with d=15d=15.\",\n \"The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\\\Psi (q)|^2).In summary, stabilizer states have Wigner function $\\\\delta _{\\\\Phi \\\\cdot x, r}$ and, for odd prime $d$ , are Gaussians in mixed representation that lie on the Weyl phase space points $(x_p, x_q)$ .\",\n \"Heuristically, they correspond to Gaussians in the continuous case that spread along their major axes infinitely.\",\n \"The only reason that they aren't always expressible as Gaussians in the mixed representation is that they sometimes “skip” over some of the discrete grid points due to the particular angle they lie along phase space for odd non-prime $d$ .\",\n \"Wigner functions $\\\\Psi _x(x)$ of stabilizer states propagate under $\\\\hat{V}$ as $\\\\Psi _x(\\\\mathcal {M}_{\\\\hat{V}} (x + \\\\alpha _{\\\\hat{V}}/2) + \\\\alpha _{\\\\hat{V}}/2)$ , and this preserves the form of the state.\",\n \"In other words, Clifford gates take stabilizer states to other stabilizer states, as expected, just like in the continuous case Gaussians go to other Gaussians under harmonic evolution.\",\n \"It is also clear that stabilizer state propagation under Clifford gates can be expressed by a path integral at order $\\\\hbar ^0$ .\",\n \"Discrete Phase Space Representation of Universal Quantum Computing A similar statement to the one we made in Section —that any operator can be expressed as an infinite sum of path integral contribution truncated at order $\\\\hbar ^1$ —can be made in discrete systems.\",\n \"However, there is an important difference in the number of terms making up the sum.\",\n \"To see this we can follow reasoning that is similar to that employed in the continuous case.\",\n \"Namely, from Eq.\",\n \"REF we see that any discrete operator can also be expressed as a linear combination of reflections, but unlike the continuous case, this sum has a finite number of terms.\",\n \"Since reflections can be expressed fully by the discrete path integral truncated at order $\\\\hbar ^1$ , as discussed previously, it follows that any unitary operator in discrete systems can be expressed as a finite sum of contributions from path integrals truncated at order $\\\\hbar ^1$ .\",\n \"Again, the same statement can be made by considering the chord representation in terms of translations.\",\n \"Hence, quantum propagation in discrete systems can be fully treated by a finite sum of contributions from a path integral approach truncated at order $\\\\hbar ^1$ .\",\n \"To gather some understanding of this statement, we can consider what is necessary to add to our path integral formulation when we complete the Clifford gates with the T-gate, which produces a universal gate set.\",\n \"The T-gate is generalized to odd $d$ -dimensions by $\\\\hat{T} = \\\\sum _{j \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}} \\\\omega ^{\\\\frac{(j-1)j}{4}} \\\\left|j\\\\right\\\\rangle \\\\left\\\\langle j\\\\right|.$ This gate can no longer be characterized by an $\\\\mathcal {M}$ with integer entries.\",\n \"In particular, $\\\\mathcal {M}_{\\\\hat{T}} = \\\\left( \\\\begin{array}{cc} 1 & \\\\frac{1}{2} \\\\\\\\ 0 & 1 \\\\end{array} \\\\right),$ and $\\\\alpha _{\\\\hat{T}} = \\\\left( -\\\\frac{1}{4}, 0 \\\\right)$ .\",\n \"This corresponds to $\\\\mathcal {B}_{\\\\hat{T}} = \\\\left( \\\\begin{array}{cc} 0 & 0\\\\\\\\ 0 & -\\\\frac{1}{4}\\\\end{array} \\\\right).$ Thus, the center function $T_x(x_p,x_q) = e^{-\\\\frac{2 \\\\pi i}{d} \\\\frac{1}{4} (x_q-x_q^2)},$ corresponding to the phase shift Hamiltonian applied for only half the unit of time.\",\n \"The operator can thus be written: $\\\\hat{T} = d^{-2} \\\\sum _{\\\\begin{array}{c}x_p,x_q, \\\\\\\\ \\\\xi _p,\\\\xi _q \\\\in \\\\\\\\ \\\\mathbb {Z}/d\\\\mathbb {Z}\\\\end{array}} e^{ -\\\\frac{2 \\\\pi i}{d} \\\\left[ \\\\frac{1}{4} (x_q - x_q^2) - d \\\\left(x_p \\\\xi _q - x_q \\\\xi _p \\\\right) \\\\right] } \\\\hat{Z}^{\\\\xi _p} \\\\hat{X}^{\\\\xi _q}.$ Though this operator is quadratic, it no longer takes the Weyl center points to themselves.\",\n \"This means that the $\\\\hbar ^0$ limit of Eq.\",\n \"REF is now insufficient to capture all the dynamics because the overlap with any $\\\\left|q\\\\right\\\\rangle $ will now involve a linear superposition of partially overlapping propagated manifolds.\",\n \"It must therefore be described by a path integral formulation that is complete to order $\\\\hbar ^1$ .\",\n \"In particular, $\\\\hat{T} = d^{-1} \\\\sum _{\\\\begin{array}{c}x_p, x_q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})\\\\end{array}} e^{-\\\\frac{2 \\\\pi i}{d} \\\\frac{1}{4} (x_q-x_q^2)} \\\\hat{R}( x_p, x_q ),$ where $\\\\hat{R}$ should be substituted by its path integral.\",\n \"Note that this does not imply efficient classical simulation of quantum computation but quite the opposite.\",\n \"Indeed, for $n$ qudits, there are $d^{2n}$ terms in the sum above.\",\n \"While every Weyl phase space point has only a single associated path when acted on by Clifford gates, this is no longer true in any calculation of evolution under the T-gate.\",\n \"Eq.\",\n \"REF expresses the T-gate as a sum over phase space operators (the reflections) evaluated on all the phase space points.\",\n \"Thus, it can be interpreted as associating an exponentially large number of paths to every phase space point, instead of the single paths found for Clifford gates.\",\n \"Therefore, any simulation of the T-gate naively necessitates adding up an exponential large sum over paths and so is comparably inefficient.\",\n \"Conclusion The treatment presented here formalizes the relationship between stabilizer states in the discrete case and Gaussians in the continuous case, which has often been pointed out [3].\",\n \"Namely, only Gaussians that lie along Weyl phase space points directly correspond to Gaussians in the continuous world in terms of preserving their form under a harmonic Hamiltonian, an evolution that is fully describable by truncating the path integral at order $\\\\hbar ^0$ .\",\n \"Furthermore, we showed that the Clifford group gates, generated by the Hadamard, phase shift and controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order.\",\n \"We found that this was because their Hamiltonians are harmonic and take Weyl phase space points to themselves.\",\n \"This proves the Gottesman-Knill theorem.\",\n \"The T-gate, needed to complete a universal set with the Hadamard, was shown not to satisfy these properties, and so requires a path integral treatment that is complete up to $\\\\hbar ^1$ .\",\n \"The latter treatment includes a sum of terms for which the number of terms scales exponentially with the number of qudits.\",\n \"We note that our observations pertaining to classical propagation in continuous systems have long been very well known.\",\n \"In the continuous case, the Wigner function of a quantum state is non-negative if and only if the state is a Gaussian [34] and it has also long been known that quantum propagation from one Gaussian state to another only requires propagation up to order $\\\\hbar ^0$ [11].\",\n \"Indeed, it has been shown that this is a continuous version of “stabilizer state propagation” in finite systems [35], and is therefore, in principle, useful for quantum error correction and cluster state quantum computation [36], [37].\",\n \"It is also well known in the discrete case that quadratic Hamiltonians can act classically and be represented by symplectic transformations in the study of quantum cat maps [38], [14], [25] and linear transformations between propagated Wigner functions [39].\",\n \"Interestingly though, this latter work appears to have predated the discovery that stabilizer states have positive-definite Wigner functions [3] and therefore, as far as we know, has not been directly related to stabilizer states and the $\\\\hbar ^0$ limit of their path integral formulation, which is a relatively recent topic of particular interest to the quantum information community and those familiar with Gottesman-Knill.\",\n \"Otherwise, this claim has been pointed out in terms of concepts related to positivity and related concepts in past work [40], [2].\",\n \"We also note that our exploration of continous systems is not meant to explore the highly related topic of continuous-variable quantum information.\",\n \"Many topics therein apply to our discussion here, such as the continuous stabilizer state propagation we mentioned above.\",\n \"However, our intention in introducing the continuous infinite-dimensional case was not to address these topics but to instead relate the established continuous semiclassical formalism to the discrete case, and thereby bridge the notions of phase space and dynamics between the two worlds.\",\n \"There is an interesting observation to be made of the weights of the reflections that make up the complete path integral formulation of a unitary operator.\",\n \"Namely, as is clear in Eqs.\",\n \"REF and REF , the coefficients consist of the exponentiated center generating function multiplied by $\\\\frac{i}{\\\\hbar }$ .\",\n \"This is very similar to the form of the vVMG path integral in Eqs.\",\n \"REF and REF .\",\n \"However, in Eqs.\",\n \"REF and REF , reflections serve as the prefactors measuring the reflection spectral overlap of a propagated state with its evolute and the center generating actions provide the quantal phase.\",\n \"Thus, this formulation can be interpreted as an alternative path integral formulation of the vVMG, one consisting of reflections as the underlying classical trajectory only, instead of the more tailored trajectories that result from applying the method of steepest descents directly on an operator.\",\n \"The fact that any unitary operator in the discrete case can be expressed as a sum consisting of a finite number of order $\\\\hbar ^1$ path integral contributions, has the added interesting implication that uniformization—higher order $\\\\hbar $ corrections to the “primitive” semiclassical forms such as Eq.\",\n \"REF —isn't really necessary in discrete systems.\",\n \"Uniformization is characterized by the proper treatment of coalescing saddle points and has long been a subject of interest in continuous systems where “anharmonicity” bedevils computationally efficient implementation.\",\n \"It seems that this problem isn't an issue in the discrete case since a fully complete sum with a finite number of terms, naively numbering $d^2$ for one qudit, exists.\",\n \"As a last point, there is perhaps an alternative way to interpret the results presented here, one in terms of “resources”.\",\n \"Much like “magic” (or contextuality) and quantum discord can be framed as a resource necessary to perform quantum operations that have more power than classical ones, it is possible to frame the order in $\\\\hbar $ that is necessary in the underlying path integral describing an operation as a resource necessary for quantumness.\",\n \"In this vein, it can be said that Clifford gate operations on stabilizer states are operations that only require $\\\\hbar ^0$ resources while supplemental gates that push the operator space into universal quantum computing require $\\\\hbar ^1$ resources.\",\n \"The dividing line between these two regimes, the classical and quantum world, is discrete, unambiguous and well-defined.\",\n \"Acknowledgments The authors thank Prof. Alfredo Ozorio de Almeida for very fruitful discussions about the center-chord representation in discrete systems and Byron Drury for his help proof-reading and bringing [12] to our attention.\",\n \"This work was supported by AFOSR award no.\",\n \"FA9550-12-1-0046.\",\n \"Appendix Gross proved that for odd prime $d$ [41]: Lemma 2 Let $\\\\Psi $ be a state vector with positive Wigner function for odd prime $d$ .\",\n \"If $\\\\Psi $ is supported on two points, then it has maximal support.\",\n \"With this lemma in mind, we can offer an alternative proof of Corollary REF : Corollary 3 For odd prime $d$ , if $\\\\Psi $ is a stabilizer state then there always exists a mixed representation in position and momentum such that: $\\\\Psi _{\\\\theta _{\\\\beta x},\\\\eta _{\\\\beta x}}(x) = \\\\frac{1}{\\\\sqrt{d}} \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( x^T \\\\theta _{\\\\beta x} x + \\\\eta _{\\\\beta x} \\\\cdot x \\\\right) \\\\right],$ where $x_i$ can be either $p_i$ or $q_i$ .\",\n \"Proof We will show that for odd prime $d$ , every degree of freedom can only be fully supported or a Kronecker delta, for all other degrees of freedom fixed; WLOG we will consider a two-dimensional stabilizer state $\\\\Psi (q_1,q_2)$ and show that if $\\\\exists \\\\, q^{\\\\prime }_1$ such that $\\\\Psi (q^{\\\\prime }_1, q_2) \\\\ne 0$ $\\\\forall q_2$ then $\\\\Psi (q_1,q_2)\\\\ne 0$ $\\\\forall \\\\, q_1, q_2$ and vice-versa (if $\\\\exists \\\\, q^{\\\\prime }_1$ s.t.\",\n \"$\\\\Psi (q^{\\\\prime }_1, q_2)$ is a delta function then $\\\\Psi (q_1,q_2)$ is a delta function in $q_2$ $\\\\forall q_1$ ).\",\n \"Therefore, if a degree of freedom is maximally supported in one degree of freedom for all others fixed, then it is maximally supported for all values of the other degrees of freedom.\",\n \"On the other hand, if it is a delta function in one degree of freedom for all others fixed, then it is a delta function for all values of the other degrees of freedom.\",\n \"Assume that for $q_1, q_2 \\\\in \\\\lbrace 0, \\\\ldots , d-1\\\\rbrace $ , $\\\\exists \\\\, q^{\\\\prime }_1, q^{\\\\prime \\\\prime }_1$ such that $\\\\Psi (q^{\\\\prime }_1,q_2) = 0$ for some $q_2$ and $\\\\Psi (q^{\\\\prime \\\\prime }_1, q_2) \\\\ne 0$ $\\\\forall q_2$ .\",\n \"We proceed to prove by contradiction.\",\n \"Hence $\\\\Psi (q^{\\\\prime \\\\prime }_1, q_2) \\\\equiv \\\\Psi _{q^{\\\\prime \\\\prime }_1} \\\\propto \\\\left[ \\\\frac{2 \\\\pi i}{d} \\\\left( \\\\theta _{q^{\\\\prime \\\\prime }_1} q^2_2 + \\\\eta _{q^{\\\\prime \\\\prime }_1} q_2 \\\\right) \\\\right]$ by Corollary REF and $\\\\Psi (q^{\\\\prime }_1,q_2) \\\\equiv \\\\Psi _{q^{\\\\prime }_1}(q_2) \\\\propto \\\\delta _{q_2, q(q^{\\\\prime }_1)}$ for some $q(q^{\\\\prime }_1)\\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ by [3].\",\n \"We can rotate in $p_2$ -$q_2$ space to form a new basis $q^*_2$ in $d$ discrete angles (since $\\\\theta _{q^{\\\\prime \\\\prime }_1} \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ ) such that $\\\\Psi _{q^{\\\\prime }_1}(q^*_2) \\\\ne 0$ (since a delta function is not maximally supported only at the one angle perpendicular to it).\",\n \"Since there exists $(d-1)$ other values of $q_1$ other than $q^{\\\\prime }_1$ , it follows that there exists at least one such angle such that $\\\\Psi _{q_1}(q^*_2) \\\\ne 0$ $\\\\forall q_1, q^*_2 \\\\in \\\\lbrace 0,\\\\ldots ,d-1\\\\rbrace $ .\",\n \"We define $q^*_2$ as the basis that is rotated by this angle with respect to $q_2$ .\",\n \"By Corollary REF , this means that $\\\\Psi ^*(q_1,q^*_2) \\\\propto \\\\exp ( \\\\theta ^{\\\\prime }_{11} q^2_1 + \\\\theta ^{\\\\prime }_{22} {q^*_2}^2 + 2 \\\\theta ^{\\\\prime }_{12} q_1 q^*_2 + \\\\eta ^{\\\\prime }_1 q_1 + \\\\eta ^{\\\\prime }_2 q_2),$ where by $\\\\Psi ^*$ we mean $\\\\Psi $ expressed in the new basis $q^*_2$ in its second degree of freedom.\",\n \"Hence, $\\\\Psi ^*_{q_1}(q^*_2) \\\\propto \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( \\\\theta ^{\\\\prime }_{q_1} {q^*_2}^2 + \\\\eta ^{\\\\prime }_{q_1} q^*_2 \\\\right) \\\\right] \\\\exp \\\\left[ \\\\frac{2 \\\\pi i}{d} \\\\theta _{12} q_1 q^*_2 \\\\right].$ Acting on this last equation to rotate back to $q_2$ , we must produce $\\\\Psi _{q^{\\\\prime \\\\prime }_1}(q_2) \\\\propto \\\\delta _{q_2,q(q^{\\\\prime \\\\prime }_1)}$ .\",\n \"But then Eq.\",\n \"REF implies that $\\\\Psi _{q^{\\\\prime }_1}(q^*_2)$ must also be proportional to $\\\\delta _{q_2,q(q^{\\\\prime }_1)}$ .\",\n \"This is a contradiction.\",\n \"Therefore, if a degree of freedom is maximally supported for all others fixed, then it is maximally supported for all values of the other degrees of freedom and vice-versa.\",\n \"In the latter case, a position state in the $i$ th degree of freedom can be represented as a Gaussian by using the $p$ -basis where it becomes a plane wave ($\\\\theta _i = 0$ ).\",\n \"In other words, one can always choose $x_i$ to be $p_i$ or $q_i$ such that Eq.\",\n \"REF holds for odd prime $d$ .\",\n \"$\\\\Box $ Finally, the reason this result does not hold for odd non-prime $d$ is that for discrete Wigner space there are $d + 1$ unique angles minus all the prime factors of $d$ .\",\n \"For non-prime $d$ , there is more than one such prime factor and so there are cases when one cannot “rotate” away all non-maximally supported states.\"\n ],\n [\n \"Discrete Phase Space Representation of Universal Quantum Computing\",\n \"A similar statement to the one we made in Section —that any operator can be expressed as an infinite sum of path integral contribution truncated at order $\\\\hbar ^1$ —can be made in discrete systems.\",\n \"However, there is an important difference in the number of terms making up the sum.\",\n \"To see this we can follow reasoning that is similar to that employed in the continuous case.\",\n \"Namely, from Eq.\",\n \"REF we see that any discrete operator can also be expressed as a linear combination of reflections, but unlike the continuous case, this sum has a finite number of terms.\",\n \"Since reflections can be expressed fully by the discrete path integral truncated at order $\\\\hbar ^1$ , as discussed previously, it follows that any unitary operator in discrete systems can be expressed as a finite sum of contributions from path integrals truncated at order $\\\\hbar ^1$ .\",\n \"Again, the same statement can be made by considering the chord representation in terms of translations.\",\n \"Hence, quantum propagation in discrete systems can be fully treated by a finite sum of contributions from a path integral approach truncated at order $\\\\hbar ^1$ .\",\n \"To gather some understanding of this statement, we can consider what is necessary to add to our path integral formulation when we complete the Clifford gates with the T-gate, which produces a universal gate set.\",\n \"The T-gate is generalized to odd $d$ -dimensions by $\\\\hat{T} = \\\\sum _{j \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}} \\\\omega ^{\\\\frac{(j-1)j}{4}} \\\\left|j\\\\right\\\\rangle \\\\left\\\\langle j\\\\right|.$ This gate can no longer be characterized by an $\\\\mathcal {M}$ with integer entries.\",\n \"In particular, $\\\\mathcal {M}_{\\\\hat{T}} = \\\\left( \\\\begin{array}{cc} 1 & \\\\frac{1}{2} \\\\\\\\ 0 & 1 \\\\end{array} \\\\right),$ and $\\\\alpha _{\\\\hat{T}} = \\\\left( -\\\\frac{1}{4}, 0 \\\\right)$ .\",\n \"This corresponds to $\\\\mathcal {B}_{\\\\hat{T}} = \\\\left( \\\\begin{array}{cc} 0 & 0\\\\\\\\ 0 & -\\\\frac{1}{4}\\\\end{array} \\\\right).$ Thus, the center function $T_x(x_p,x_q) = e^{-\\\\frac{2 \\\\pi i}{d} \\\\frac{1}{4} (x_q-x_q^2)},$ corresponding to the phase shift Hamiltonian applied for only half the unit of time.\",\n \"The operator can thus be written: $\\\\hat{T} = d^{-2} \\\\sum _{\\\\begin{array}{c}x_p,x_q, \\\\\\\\ \\\\xi _p,\\\\xi _q \\\\in \\\\\\\\ \\\\mathbb {Z}/d\\\\mathbb {Z}\\\\end{array}} e^{ -\\\\frac{2 \\\\pi i}{d} \\\\left[ \\\\frac{1}{4} (x_q - x_q^2) - d \\\\left(x_p \\\\xi _q - x_q \\\\xi _p \\\\right) \\\\right] } \\\\hat{Z}^{\\\\xi _p} \\\\hat{X}^{\\\\xi _q}.$ Though this operator is quadratic, it no longer takes the Weyl center points to themselves.\",\n \"This means that the $\\\\hbar ^0$ limit of Eq.\",\n \"REF is now insufficient to capture all the dynamics because the overlap with any $\\\\left|q\\\\right\\\\rangle $ will now involve a linear superposition of partially overlapping propagated manifolds.\",\n \"It must therefore be described by a path integral formulation that is complete to order $\\\\hbar ^1$ .\",\n \"In particular, $\\\\hat{T} = d^{-1} \\\\sum _{\\\\begin{array}{c}x_p, x_q \\\\in \\\\\\\\ (\\\\mathbb {Z} / d \\\\mathbb {Z})\\\\end{array}} e^{-\\\\frac{2 \\\\pi i}{d} \\\\frac{1}{4} (x_q-x_q^2)} \\\\hat{R}( x_p, x_q ),$ where $\\\\hat{R}$ should be substituted by its path integral.\",\n \"Note that this does not imply efficient classical simulation of quantum computation but quite the opposite.\",\n \"Indeed, for $n$ qudits, there are $d^{2n}$ terms in the sum above.\",\n \"While every Weyl phase space point has only a single associated path when acted on by Clifford gates, this is no longer true in any calculation of evolution under the T-gate.\",\n \"Eq.\",\n \"REF expresses the T-gate as a sum over phase space operators (the reflections) evaluated on all the phase space points.\",\n \"Thus, it can be interpreted as associating an exponentially large number of paths to every phase space point, instead of the single paths found for Clifford gates.\",\n \"Therefore, any simulation of the T-gate naively necessitates adding up an exponential large sum over paths and so is comparably inefficient.\"\n ],\n [\n \"Conclusion\",\n \"The treatment presented here formalizes the relationship between stabilizer states in the discrete case and Gaussians in the continuous case, which has often been pointed out [3].\",\n \"Namely, only Gaussians that lie along Weyl phase space points directly correspond to Gaussians in the continuous world in terms of preserving their form under a harmonic Hamiltonian, an evolution that is fully describable by truncating the path integral at order $\\\\hbar ^0$ .\",\n \"Furthermore, we showed that the Clifford group gates, generated by the Hadamard, phase shift and controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order.\",\n \"We found that this was because their Hamiltonians are harmonic and take Weyl phase space points to themselves.\",\n \"This proves the Gottesman-Knill theorem.\",\n \"The T-gate, needed to complete a universal set with the Hadamard, was shown not to satisfy these properties, and so requires a path integral treatment that is complete up to $\\\\hbar ^1$ .\",\n \"The latter treatment includes a sum of terms for which the number of terms scales exponentially with the number of qudits.\",\n \"We note that our observations pertaining to classical propagation in continuous systems have long been very well known.\",\n \"In the continuous case, the Wigner function of a quantum state is non-negative if and only if the state is a Gaussian [34] and it has also long been known that quantum propagation from one Gaussian state to another only requires propagation up to order $\\\\hbar ^0$ [11].\",\n \"Indeed, it has been shown that this is a continuous version of “stabilizer state propagation” in finite systems [35], and is therefore, in principle, useful for quantum error correction and cluster state quantum computation [36], [37].\",\n \"It is also well known in the discrete case that quadratic Hamiltonians can act classically and be represented by symplectic transformations in the study of quantum cat maps [38], [14], [25] and linear transformations between propagated Wigner functions [39].\",\n \"Interestingly though, this latter work appears to have predated the discovery that stabilizer states have positive-definite Wigner functions [3] and therefore, as far as we know, has not been directly related to stabilizer states and the $\\\\hbar ^0$ limit of their path integral formulation, which is a relatively recent topic of particular interest to the quantum information community and those familiar with Gottesman-Knill.\",\n \"Otherwise, this claim has been pointed out in terms of concepts related to positivity and related concepts in past work [40], [2].\",\n \"We also note that our exploration of continous systems is not meant to explore the highly related topic of continuous-variable quantum information.\",\n \"Many topics therein apply to our discussion here, such as the continuous stabilizer state propagation we mentioned above.\",\n \"However, our intention in introducing the continuous infinite-dimensional case was not to address these topics but to instead relate the established continuous semiclassical formalism to the discrete case, and thereby bridge the notions of phase space and dynamics between the two worlds.\",\n \"There is an interesting observation to be made of the weights of the reflections that make up the complete path integral formulation of a unitary operator.\",\n \"Namely, as is clear in Eqs.\",\n \"REF and REF , the coefficients consist of the exponentiated center generating function multiplied by $\\\\frac{i}{\\\\hbar }$ .\",\n \"This is very similar to the form of the vVMG path integral in Eqs.\",\n \"REF and REF .\",\n \"However, in Eqs.\",\n \"REF and REF , reflections serve as the prefactors measuring the reflection spectral overlap of a propagated state with its evolute and the center generating actions provide the quantal phase.\",\n \"Thus, this formulation can be interpreted as an alternative path integral formulation of the vVMG, one consisting of reflections as the underlying classical trajectory only, instead of the more tailored trajectories that result from applying the method of steepest descents directly on an operator.\",\n \"The fact that any unitary operator in the discrete case can be expressed as a sum consisting of a finite number of order $\\\\hbar ^1$ path integral contributions, has the added interesting implication that uniformization—higher order $\\\\hbar $ corrections to the “primitive” semiclassical forms such as Eq.\",\n \"REF —isn't really necessary in discrete systems.\",\n \"Uniformization is characterized by the proper treatment of coalescing saddle points and has long been a subject of interest in continuous systems where “anharmonicity” bedevils computationally efficient implementation.\",\n \"It seems that this problem isn't an issue in the discrete case since a fully complete sum with a finite number of terms, naively numbering $d^2$ for one qudit, exists.\",\n \"As a last point, there is perhaps an alternative way to interpret the results presented here, one in terms of “resources”.\",\n \"Much like “magic” (or contextuality) and quantum discord can be framed as a resource necessary to perform quantum operations that have more power than classical ones, it is possible to frame the order in $\\\\hbar $ that is necessary in the underlying path integral describing an operation as a resource necessary for quantumness.\",\n \"In this vein, it can be said that Clifford gate operations on stabilizer states are operations that only require $\\\\hbar ^0$ resources while supplemental gates that push the operator space into universal quantum computing require $\\\\hbar ^1$ resources.\",\n \"The dividing line between these two regimes, the classical and quantum world, is discrete, unambiguous and well-defined.\"\n ],\n [\n \"Acknowledgments\",\n \"The authors thank Prof. Alfredo Ozorio de Almeida for very fruitful discussions about the center-chord representation in discrete systems and Byron Drury for his help proof-reading and bringing [12] to our attention.\",\n \"This work was supported by AFOSR award no.\",\n \"FA9550-12-1-0046.\"\n ],\n [\n \"Appendix\",\n \"Gross proved that for odd prime $d$ [41]: Lemma 2 Let $\\\\Psi $ be a state vector with positive Wigner function for odd prime $d$ .\",\n \"If $\\\\Psi $ is supported on two points, then it has maximal support.\",\n \"With this lemma in mind, we can offer an alternative proof of Corollary REF : Corollary 3 For odd prime $d$ , if $\\\\Psi $ is a stabilizer state then there always exists a mixed representation in position and momentum such that: $\\\\Psi _{\\\\theta _{\\\\beta x},\\\\eta _{\\\\beta x}}(x) = \\\\frac{1}{\\\\sqrt{d}} \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( x^T \\\\theta _{\\\\beta x} x + \\\\eta _{\\\\beta x} \\\\cdot x \\\\right) \\\\right],$ where $x_i$ can be either $p_i$ or $q_i$ .\",\n \"Proof We will show that for odd prime $d$ , every degree of freedom can only be fully supported or a Kronecker delta, for all other degrees of freedom fixed; WLOG we will consider a two-dimensional stabilizer state $\\\\Psi (q_1,q_2)$ and show that if $\\\\exists \\\\, q^{\\\\prime }_1$ such that $\\\\Psi (q^{\\\\prime }_1, q_2) \\\\ne 0$ $\\\\forall q_2$ then $\\\\Psi (q_1,q_2)\\\\ne 0$ $\\\\forall \\\\, q_1, q_2$ and vice-versa (if $\\\\exists \\\\, q^{\\\\prime }_1$ s.t.\",\n \"$\\\\Psi (q^{\\\\prime }_1, q_2)$ is a delta function then $\\\\Psi (q_1,q_2)$ is a delta function in $q_2$ $\\\\forall q_1$ ).\",\n \"Therefore, if a degree of freedom is maximally supported in one degree of freedom for all others fixed, then it is maximally supported for all values of the other degrees of freedom.\",\n \"On the other hand, if it is a delta function in one degree of freedom for all others fixed, then it is a delta function for all values of the other degrees of freedom.\",\n \"Assume that for $q_1, q_2 \\\\in \\\\lbrace 0, \\\\ldots , d-1\\\\rbrace $ , $\\\\exists \\\\, q^{\\\\prime }_1, q^{\\\\prime \\\\prime }_1$ such that $\\\\Psi (q^{\\\\prime }_1,q_2) = 0$ for some $q_2$ and $\\\\Psi (q^{\\\\prime \\\\prime }_1, q_2) \\\\ne 0$ $\\\\forall q_2$ .\",\n \"We proceed to prove by contradiction.\",\n \"Hence $\\\\Psi (q^{\\\\prime \\\\prime }_1, q_2) \\\\equiv \\\\Psi _{q^{\\\\prime \\\\prime }_1} \\\\propto \\\\left[ \\\\frac{2 \\\\pi i}{d} \\\\left( \\\\theta _{q^{\\\\prime \\\\prime }_1} q^2_2 + \\\\eta _{q^{\\\\prime \\\\prime }_1} q_2 \\\\right) \\\\right]$ by Corollary REF and $\\\\Psi (q^{\\\\prime }_1,q_2) \\\\equiv \\\\Psi _{q^{\\\\prime }_1}(q_2) \\\\propto \\\\delta _{q_2, q(q^{\\\\prime }_1)}$ for some $q(q^{\\\\prime }_1)\\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ by [3].\",\n \"We can rotate in $p_2$ -$q_2$ space to form a new basis $q^*_2$ in $d$ discrete angles (since $\\\\theta _{q^{\\\\prime \\\\prime }_1} \\\\in \\\\mathbb {Z}/d\\\\mathbb {Z}$ ) such that $\\\\Psi _{q^{\\\\prime }_1}(q^*_2) \\\\ne 0$ (since a delta function is not maximally supported only at the one angle perpendicular to it).\",\n \"Since there exists $(d-1)$ other values of $q_1$ other than $q^{\\\\prime }_1$ , it follows that there exists at least one such angle such that $\\\\Psi _{q_1}(q^*_2) \\\\ne 0$ $\\\\forall q_1, q^*_2 \\\\in \\\\lbrace 0,\\\\ldots ,d-1\\\\rbrace $ .\",\n \"We define $q^*_2$ as the basis that is rotated by this angle with respect to $q_2$ .\",\n \"By Corollary REF , this means that $\\\\Psi ^*(q_1,q^*_2) \\\\propto \\\\exp ( \\\\theta ^{\\\\prime }_{11} q^2_1 + \\\\theta ^{\\\\prime }_{22} {q^*_2}^2 + 2 \\\\theta ^{\\\\prime }_{12} q_1 q^*_2 + \\\\eta ^{\\\\prime }_1 q_1 + \\\\eta ^{\\\\prime }_2 q_2),$ where by $\\\\Psi ^*$ we mean $\\\\Psi $ expressed in the new basis $q^*_2$ in its second degree of freedom.\",\n \"Hence, $\\\\Psi ^*_{q_1}(q^*_2) \\\\propto \\\\exp \\\\left[\\\\frac{2 \\\\pi i}{d} \\\\left( \\\\theta ^{\\\\prime }_{q_1} {q^*_2}^2 + \\\\eta ^{\\\\prime }_{q_1} q^*_2 \\\\right) \\\\right] \\\\exp \\\\left[ \\\\frac{2 \\\\pi i}{d} \\\\theta _{12} q_1 q^*_2 \\\\right].$ Acting on this last equation to rotate back to $q_2$ , we must produce $\\\\Psi _{q^{\\\\prime \\\\prime }_1}(q_2) \\\\propto \\\\delta _{q_2,q(q^{\\\\prime \\\\prime }_1)}$ .\",\n \"But then Eq.\",\n \"REF implies that $\\\\Psi _{q^{\\\\prime }_1}(q^*_2)$ must also be proportional to $\\\\delta _{q_2,q(q^{\\\\prime }_1)}$ .\",\n \"This is a contradiction.\",\n \"Therefore, if a degree of freedom is maximally supported for all others fixed, then it is maximally supported for all values of the other degrees of freedom and vice-versa.\",\n \"In the latter case, a position state in the $i$ th degree of freedom can be represented as a Gaussian by using the $p$ -basis where it becomes a plane wave ($\\\\theta _i = 0$ ).\",\n \"In other words, one can always choose $x_i$ to be $p_i$ or $q_i$ such that Eq.\",\n \"REF holds for odd prime $d$ .\",\n \"$\\\\Box $ Finally, the reason this result does not hold for odd non-prime $d$ is that for discrete Wigner space there are $d + 1$ unique angles minus all the prime factors of $d$ .\",\n \"For non-prime $d$ , there is more than one such prime factor and so there are cases when one cannot “rotate” away all non-maximally supported states.\"\n 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"Qualitative study in Loop Quantum Cosmology"
],
[
"Abstract This work contains a detailed qualitative analysis, in General Relativity and in Loop Quantum Cosmology, of the dynamics in the associated phase space of a scalar field minimally coupled with gravity, whose potential mimics the dynamics of a perfect fluid with a linear Equation of State (EoS).",
"Dealing with the orbits (solutions) of the system, we will see that there are analytic ones, which lead to the same dynamics as the perfect fluid, and our goal is to check their stability, depending on the value of the EoS parameter, i.e., to show whether the other orbits converge or diverge to these analytic solutions at early and late times."
],
[
"Introduction",
"Soon after Einstein published the field equations of General Relativity (GR), it was realized that these equations contained singularities.",
"In particular, in a cosmological context, it was noticed that for the Friedmann-Lemaître-Robertson-Walker geometry, when the Equation of State (EoS) modeling the matter content was a linear equation with an EoS parameter greater than $-1$ , a singularity named Big Bang appeared at early times, where the energy density of the universe diverges.",
"Moreover, dealing with nonlinear EoS (see for instance, [1], where the analysis is done by expressing the EoS in the form of a power expansion), one can see that other kind of singularities such as Sudden singularity [2], [3], [4] or Big Freeze [5], [8], [9], [6], [7] appear.",
"Several attempts to remove these kinds of singularities were studied for years.",
"We can mention, for instance, the assumption of a non-linear EoS [10] with two fixed points where the pressure plus energy density vanishes, or the modification of the Friedmann equation assuming that the square of the Hubble parameter is equal to a power series of the energy density [11].",
"Another option is the introduction of higher order terms in the equations of GR via the quantum effects of a scalar field conformally coupled with gravity [12], [13], [14], [15], [16] or directly assuming a Lagrangian containing non-linear terms in the scalar curvature [17], [18], [19].",
"However, the simplest way to remove the Big Bang singularity is to adopt the viewpoint of Loop Quantum Cosmology (LQC), where the discrete nature of spacetime is assumed and Friedmann equations are modified (see for review [20], [21]).",
"Hence, all strong curvature singularities are resolved in LQC, both in isotropic [22], [23] and anisotropic models [24], [25] (see [26] for a complete review).",
"Thus, several types of singularities are avoided (for e.g.",
"Big Rip [27] and Big Crunch [28]) and the Big Bang is replaced by a non-singular bounce (see for instance [29], [30]).",
"However, a qualitative analysis of the dynamics, both in GR and LQC, is only done in few works [31], [32], [33], [34].",
"For this reason, the aim of this paper is to perform a qualitative analysis in the spatially flat FLRW, for both GR and Loop Quantum Cosmology (LQC), when the universe is filled with a scalar field whose potential leads to a solution conducing to a dynamics which mimics the well-known perfect fluid with EoS $P=\\omega \\rho $ .",
"More precisely, we will qualitatively study when we have a scalar field $\\varphi $ which mimics this fluid by solving the conservation equation with the effective potential to which this scalar field is submitted.",
"Thus, since when one deals with an scalar field the conservation equation is a second order differential equation, we obtain an infinite set of orbits which can be plotted in the phase space $(\\varphi ,\\dot{\\varphi })$ and some of which may not satisfy the equation of state, i.e., some of them do not depict a universe with a constant EoS parameter, meaning that they do not mimic the same background as the perfect fluid.",
"Then, the question is whether these orbits depicts at early or late time the same background as the perfect fluid.",
"With our analysis we can infer whether the orbits with the same background as the fluid are attractors or repellers of the dynamical system, in the sense that the other orbits starting in their neighbourhood asymptotically approach them or not.",
"This study is very important in the so-called matter bounce and matter-ekpyrotic bounce scenario [35], [36], [37], [38] (recall that singularities in ekpyrotic LQC was studied for the first time in [28], [39], [40]) because all the calculations -the power spectrum of perturbations, the spectral index, its running [41], [35], [42] and the reheating temperature [43], [44]- are done with the analytical solution, but of course these calculations are only viable if this solution is stable, in the sense that all the other solutions depict asymptotically a universe with the same properties.",
"The work is organized as follows: In section , we introduce the Friedmann equations for a flat homogeneous and isotropic space-time and we reconstruct the scalar field and the corresponding potential which leads to the same background as the fluid with EoS $P=w\\rho $ .",
"Using this potential, we perform a qualitative study of the dynamical equations, obtaining the phase portraits for the different values of the effective EoS parameter, and concluding that when the effective EoS parameter belongs to the interval $(-1,1)$ the analytical orbit is a repeller in the contracting phase and an attractor in the expanding one.",
"On the contrary, when the effective EoS parameter is greater than 1, i.e.",
"in the ekpyrotic case, the analytical orbit is an attractor in the contracting phase and a repeller in the expanding one.",
"In section , we proceed analogously as with GR, but now including in the Friedmann equations holonomic corrections that come from LQC [45].",
"Finally, we do as well a qualitative study for the dynamical equations of the scalar field, obtaining similar phase portraits as the one shown in [46] for a particular case of the ones we are going to study in the present work.",
"We will use natural units ($\\hbar =8\\pi G=c=1$ ).",
"Assuming homogeneity and isotropy of the universe, which is valid at sufficiently large scales, it can be shown that a suitable change of coordinates leads to the so-called Friedmann-Lemaître-Robertson-Walker (FLRW) metric $ds^2=dt^2-a^2(t)\\left(\\frac{dr^2}{1-\\kappa r^2}+r^2d\\theta ^2+r^2\\sin ^2\\theta d\\varphi ^2\\right),$ where $a(t)$ is a scale factor that parametrizes the relative expansion of the universe and the curvature $\\kappa $ is -1,0 or 1 if we are dealing respectively with an open, flat or closed universe.",
"To simplify, we will perform all our calculations in the flat FLRW space-time.",
"Hence, with the EoS $P=w\\rho $ , we obtain the Friedmann equations $H^2=\\frac{\\rho }{3}, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\dot{H}=-\\frac{3(1+w)}{2}H^2, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\dot{\\rho }=-3H(1+w)\\rho , $ where $H=\\frac{\\dot{a}}{a}$ is the Hubble parameter.",
"The dynamics provided by (REF ) could be mimicked by a scalar field minimally coupled with gravity.",
"Its potential, which is exponential [31], is easily obtained using the reconstruction method which goes as follows.",
"For a minimally coupled with gravity scalar field, one has $\\rho =\\frac{\\dot{\\varphi }^2}{2}+V(\\varphi )$ and $P=\\frac{\\dot{\\varphi }^2}{2}-V(\\varphi )$ .",
"Therefore, the linear EoS leads to $V(\\varphi )=\\frac{1-w}{1+w}\\frac{\\dot{\\varphi }^2}{2}.$ On the other hand, solving the Friedmann equations one can show that for the linear EoS the energy density evolves as $\\rho (t)=\\frac{4}{3(1+w)^2t^2}$ , which implies, using $\\dot{\\varphi }^2=P+\\rho =(1+w)\\rho $ , that $\\dot{\\varphi }=\\frac{2}{\\sqrt{3(1+w)}}\\frac{1}{|t|},$ where we have chosen, for simplicity, the positive sign of the square root.",
"Integrating this equation, one obtains the following two solutions $\\varphi =\\pm \\frac{1}{\\sqrt{3(1+w}}\\ln \\left( \\frac{t}{t_0} \\right)^2,$ where the sign $+$ (resp.",
"$-$ ) refers to a solution depicting a universe in the expanding (resp.",
"contracting) phase.",
"Finally, from this equation one deduces that $\\left( \\frac{t}{t_0} \\right)^2=e^{\\pm \\sqrt{3(1+w)}\\varphi }$ .",
"Then, using this expression and Equations (REF ) and (REF ), one concludes that $V_{\\pm }(\\varphi )=V_0e^{\\mp \\sqrt{3(1+\\omega )}\\varphi }, $ where $V_0=\\frac{2(1-w)}{3(1+w)^2t_0^2}$ and $V_+$ (resp.",
"$V_-$ ) denotes the potential for the expanding (resp.",
"contracting) phase."
],
[
"Qualitative analysis in GR",
"In the case of the linear EoS $P=\\omega \\rho $ with $\\omega >-1$ , we want to study the behaviour of the analytical solution in a similar way as done in [31], finding out whether it is either an attractor or a repeller and comparing its behaviour with that of other solutions of the system.",
"Using the potential found in (REF ), we will analyse the dynamics in the contracting phase from this dynamical system $\\ddot{\\varphi }+3H_-(\\varphi ,\\dot{\\varphi })+V_{\\varphi }=0,$ where $H_-(\\varphi ,\\dot{\\varphi })=-\\sqrt{\\frac{\\dot{\\varphi }^2}{2}+V(\\varphi )}$ .",
"With the change of variable $\\varphi =\\frac{-2}{\\sqrt{3(1+\\omega )}}\\ln \\psi $ , the dynamical system becomes $\\frac{d\\dot{\\psi }}{d\\varphi }=F_{-}(\\dot{\\psi }):=-\\frac{3}{2}\\sqrt{1+\\omega }\\left(\\sqrt{\\frac{2\\dot{\\psi }^2}{3(1+\\omega )}+V_0}+\\frac{\\sqrt{3}(1+\\omega )}{2\\dot{\\psi }}\\left(\\frac{2\\dot{\\psi }^2}{3(1+\\omega )}+ V_0 \\right) \\right).",
"$ The different cases to distinguish are the following ones: $\\omega =1$ : This case is known as a kination (or deflationary) phase [47], [48].",
"Equation (REF ) becomes $\\frac{d\\dot{\\psi }}{d\\varphi }=-\\sqrt{\\frac{3}{2}}(|\\dot{\\psi }|+\\dot{\\psi })$ and in the semi-plane $\\dot{\\psi }\\le 0$ ($\\dot{\\varphi }\\ge 0$ ) the solution is given by $(\\varphi (t),\\dot{\\varphi }(t))=\\left(-\\sqrt{\\frac{2}{3}}\\ln (-|C|(t-t_s)),-\\sqrt{\\frac{2}{3}}\\frac{1}{t-t_s}\\right) \\ \\quad \\mbox{with} \\quad t<t_s,$ which is an orbit that trivially fulfills the Equation of State because of the potential being null, with the sign corresponding to the contracting phase.",
"Despite not being stable because of the singularity at $t=t_s$ , all possible values of $C$ lead to solutions corresponding all the time to a universe with EoS $P=\\omega \\rho $ in the contracting phase, with $H(t)=\\frac{1}{3(t-t_s)}$ .",
"Regarding the semi-plane $\\dot{\\psi }>0$ ($\\dot{\\varphi }<0$ ), the solution becomes $(\\varphi (t),\\dot{\\varphi }(t))=\\left(\\sqrt{\\frac{2}{3}}\\ln (-|C|(t-t_s)),\\sqrt{\\frac{2}{3}}\\frac{1}{t-t_s}\\right) \\ \\quad \\mbox{with} \\quad t<t_s,$ which is analogous to the former case.",
"Finally, the case $\\dot{\\psi }=0$ , i.e.",
"$(\\varphi (t),\\dot{\\varphi }(t))=(C,0)$ , corresponds to $H=0$ .",
"$-1<\\omega <1$ : Given that $F_-(\\dot{\\psi })$ can only vanish for $\\dot{\\psi }<0$ , we have a single critical point for $\\dot{\\psi }$ $\\dot{\\psi }_-=-(1+\\omega )\\sqrt{\\frac{3V_0}{2(1-\\omega )}},$ which is a global repeller, given that $F_-(\\dot{\\psi })>0 \\ \\ \\forall \\dot{\\psi }_-<\\dot{\\psi }<0$ and $F_-(\\dot{\\psi })<0 \\ \\ \\forall \\dot{\\psi }<\\dot{\\psi }_-$ .",
"Figure: Phase portrait in the phase space (ϕ,ψ ˙)(\\varphi ,\\dot{\\psi }) for ω=1/2\\omega =1/2 and V 0 =1V_0=1.",
"The blue orbit is the analytic one which mimics the orbit given by a perfect fluid withEoS parameter ω=1/2\\omega =1/2.",
"The red orbits are obtained numerically to show that, in the contracting phase, the analytic orbit is a global repeller.We point out that it can be verified that the blue horizontal line in Figure REF verifies the Equation of State, depicting a universe with EoS $P=\\omega \\rho $ .",
"$\\omega >1$ : This case is known as an ekpyrotic phase or regime [49] and the system is only defined for $|\\dot{\\psi }|\\ge \\sqrt{\\frac{3(1+\\omega )|V_0|}{2}}$ .",
"We have three critical points $\\dot{\\psi }_-=-(1+\\omega )\\sqrt{\\frac{3|V_0|}{2|1-\\omega |}} ,\\ \\ \\ \\ \\ \\ \\ \\dot{\\psi }_0^{\\pm }=\\pm \\sqrt{\\frac{3(1+\\omega )|V_0|}{2}},$ where $\\dot{\\psi }_0^{\\pm }$ are repellers corresponding to $H=0$ .",
"On the other hand, $\\dot{\\psi }_-$ is an attractor for $\\dot{\\psi }<\\dot{\\psi }_0^-$ , solution that leads to a universe that all the time behaves as $P=\\omega \\rho $ in the contracting phase.",
"Figure: Phase portrait in the phase space (ϕ,ψ ˙)(\\varphi ,\\dot{\\psi }) for ω=10\\omega =10 and V 0 =-1V_0=-1.",
"In the left-hand side picture we have plotted in blue, in the semi-plane ψ ˙<0\\dot{\\psi }<0, the orbit ψ ˙ - \\dot{\\psi }^- which mimics the background given by a perfect fluid with EoS parameter w=10w=10 , and one corresponding to H=0H=0, i.e.",
"ψ ˙ 0 - \\dot{\\psi }_0^-.",
"The red orbits obtained numerically show thatψ ˙ - \\dot{\\psi }^- is an attractor and ψ ˙ 0 - \\dot{\\psi }_0^- is a repeller.",
"On the right-hand side, in the semi-plane ψ ˙>0\\dot{\\psi }>0, we have plotted in blue the other orbit corresponding to H=0H=0, i.e.",
"ψ ˙ 0 + \\dot{\\psi }_0^+, and in red anorbit obtained numerically which shows that ψ ˙ 0 + \\dot{\\psi }_0^+ is a repeller.We note that orbits approaching either of the fixed points $\\dot{\\psi }_0^{\\pm }$ reach such value in a finite cosmic time, which can be justified in the following way.",
"Since near $\\dot{\\psi }_0^{\\pm }$ the dynamical system behaves as $\\frac{d\\dot{\\psi }}{d\\varphi }=-\\frac{3}{2}\\sqrt{1+\\omega }\\sqrt{\\frac{2\\dot{\\psi }^2}{3(1+\\omega )}+V_0}$ , it is reached in a finite $\\varphi $ time.",
"Hence, $\\dot{\\varphi }\\left(=\\sqrt{-2V(\\varphi )}\\right)$ is finite as well and, thus, in a finite cosmic backwards time the contracting phase starts after a bounce at $\\rho =0$ enabling the system to leave the expanding phase with no singularity.",
"An analogous analysis for the expanding phase would show that for $\\omega = 1$ there are as well solutions with $H=0$ and other non-trivial ones, all satisfying the corresponding Equation of State.",
"The solution that depicts a fluid with EoS $P = \\omega \\rho $ is an attractor for $|\\omega | < 1$ and a repeller for $\\omega >1$ , also verifying the Equation of State.",
"And for $\\omega >1$ we would also find critical points corresponding to $H=0$ , as the ones found in the contracting phase, that are in this case attractors."
],
[
"Modified Friedmann equations",
"The general formula of loop quantum gravity, which takes into account the discrete nature of space-time, expresses the Hamiltonian in terms of the holonomies $h_j(\\lambda )\\equiv e^{-i\\frac{\\lambda \\beta }{2}\\sigma _j}$ , where $\\sigma _j$ are the Pauli matrices [45], [50] $\\mathcal {H}_{\\text{LQC}}=-\\frac{2V}{\\gamma ^3\\lambda ^3}\\sum \\limits _{i,j,k}\\epsilon ^{ijk}\\text{Tr}[h_i(\\lambda )h_j(\\lambda )h_i^{-1}(\\lambda )h_j^{-1}(\\lambda )\\lbrace h_k^{-1}(\\lambda ),V\\rbrace +\\rho V,$ where $\\gamma \\approx 0.2375$ is the Barbero-Immirzi parameter [51], $\\lambda =\\sqrt{\\frac{\\sqrt{3}}{4}\\gamma }$ is a parameter with the dimension of length, which is determined by invoking the quantum nature of the geometry (see for instance [52]), and $\\beta $ is the canonically conjugate variable of the volume $V=a^3$ and whose Poisson bracket satisfies $\\lbrace \\beta , V\\rbrace =\\frac{\\gamma }{2}$ , which means that at classical level, i.e.",
"when $H$ is small and holonomy corrections can be disregarded, $\\beta =\\gamma H$ .",
"The Hamiltonian expression in (REF ) leads to [53], [54] $\\mathcal {H}_{\\text{LQC}}=-3V\\frac{\\sin ^2(\\lambda \\beta )}{\\gamma ^2\\lambda ^2}+\\rho V,$ and, by imposing the Hamiltonian constraint $\\mathcal {H}_{\\text{LQC}}=0$ , one obtains the following holonomy corrected Friedmann equation $\\frac{\\sin ^2(\\lambda \\beta )}{\\gamma ^2\\lambda ^2}=\\frac{\\rho }{3}.",
"$ On the other hand, the Hamiltonian equation $\\dot{V}=\\lbrace V,\\mathcal {H}_{\\text{LQC}}\\rbrace =-\\frac{\\gamma }{2}\\frac{\\partial \\mathcal {H}_{\\text{LQC}}}{\\partial \\beta }$ leads to $H=\\frac{\\sin 2\\lambda \\beta }{2 \\lambda \\gamma } \\Longleftrightarrow \\beta =\\frac{1}{2\\lambda }\\arcsin (2\\lambda \\gamma H).$ Inserting the value of $\\beta $ in the Hamiltonian constraint (REF ), one obtains $\\frac{\\sin ^2\\left(\\frac{1}{2}\\arcsin (2\\lambda \\gamma H) \\right)}{\\gamma ^2\\lambda ^2}=\\frac{\\rho }{3}.",
"$ Note that the geometric corrections modify the geometric sector of the Friedmann equation and for small values of the Hubble parameter one recovers its usual form in General Relativity.",
"However, using the relation $\\sin ^2(\\frac{x}{2})=\\frac{1-\\cos x}{2}$ and after some algebra, this equation becomes $\\frac{1-\\sqrt{1-4\\lambda ^2\\gamma ^2 H^2}}{2\\lambda ^2\\gamma ^2}=\\frac{\\rho }{3},$ and finally isolating $H^2$ one obtains the more usual form $H^2=\\frac{\\rho }{3}\\left(1-\\frac{\\rho }{\\rho _c}\\right), $ where $\\rho _c=\\frac{3}{\\gamma ^2\\lambda ^2}$ is the so-called critical energy density (the maximum value that reaches the energy density).",
"Remark: It is important to note that the effective Friedmann equation (REF ) of LQC has been fully trusted only for some values of the parameter $\\omega $ .",
"In particular, in [55] it was verified for $\\omega =1$ .",
"Equation (REF ) corresponds to an ellipse in the plane $(\\rho , H)$ , that we can parametrize in the following form $\\left\\lbrace \\begin{array}{ll} H=\\sqrt{\\frac{\\rho _c}{12}} \\sin \\eta \\\\ \\rho =\\rho _c \\cos ^2\\frac{\\eta }{2}.",
"\\end{array} \\right.",
"$ The conservation equation does not differ from standard GR, i.e., the fluid fulfills the relation $d(\\rho V)=-PdV$ , where $V=a^3$ , and again with the EoS $P=-\\rho -f(\\rho )$ one gets $\\dot{\\rho }=3Hf(\\rho )$ .",
"Hence, we easily obtain the Raychaudhuri equation $\\dot{H}=\\frac{f(\\rho )}{2}\\left(1-\\frac{2\\rho }{\\rho _c}\\right).$ Analogously as in the case of General Relativity, one can mimic the perfect fluid that recovers the space-time with a scalar field and one can verify that the potential to which this scalar field is submitted is $V(\\varphi )=V_0\\frac{e^{\\sqrt{3(1+\\omega )}\\varphi }}{\\left(1+\\frac{V_0}{2\\rho _c(1-\\omega )}e^{\\sqrt{3(1+\\omega )}\\varphi } \\right)^2}.",
"$ Remark: This potential could be obtained using the reconstruction method as we have done in the case of GR or, as in [56], obtaining a differential equation whose solution is precisely (REF ).",
"Note that the potential depends on the critical density $\\rho _c$ and in the limit $\\rho _c\\rightarrow \\infty $ it coincides with the potential obtained in GR.",
"It seems clear that a potential which represents only matter degrees of freedom can in principle not contain the critical energy because it appears as a quantum geometrical effect, but if one wants to obtain an orbit which leads to the same dynamics in LQC as a perfect fluid with EoS $P=w\\rho $ , i.e.",
"$\\rho (t)=\\frac{\\rho _c}{\\frac{3}{4}(1+w)^2\\rho _ct^2+1}$ [57], it is mandatory to use it.",
"We do not know the reason why $\\rho _c$ appears in the potential, but what is sure is that using another potential one would obtain different orbits leading to bouncing backgrounds, none of which would mimic the one given by a perfect fluid with a linear EoS."
],
[
"Qualitative analysis in LQC",
"We want to analyse the behaviour of the solution corresponding to the EoS $P=\\omega \\rho $ with $\\omega >-1$ .",
"Using the potential found in (REF ), we will study the dynamics of the equation $\\ddot{\\varphi }+3H_{\\pm }(\\varphi ,\\dot{\\varphi })\\dot{\\varphi }+V_{\\varphi }=0, $ where $H_{\\pm }(\\varphi ,\\dot{\\varphi })=\\pm \\sqrt{\\frac{\\rho (\\varphi ,\\dot{\\varphi })}{3}\\left(1-\\frac{\\rho (\\varphi ,\\dot{\\varphi })}{\\rho _c}\\right)}$ , with $\\rho (\\varphi ,\\dot{\\varphi })=\\frac{\\dot{\\varphi }^2}{2}+V(\\varphi ).$ Firstly, we note that from the dynamical system it follows that $\\dot{\\rho }=-3H_{\\pm }(\\varphi ,\\dot{\\varphi })\\dot{\\varphi }^2.$ Therefore, the evolution in time will take place in an anti-clockwise sense throughout the ellipse, being $(0,0)$ a fixed point.",
"Before proceeding to analyze the different cases, we are going to take a glance at the geometry of the phase space $(\\varphi ,\\dot{\\varphi })$ .",
"Since we are dealing with a bi-valued dynamical system, we need a cover 2:1 (of two sheets) of the allowed region in the plane of the phase space $(\\varphi ,\\dot{\\varphi })$ , which is ramified in the curves $H(\\varphi ,\\dot{\\varphi })=0$ .",
"Hence, for the case $|\\omega |<1$ , this is a cylinder, whereas for $\\omega >1$ we have two cylinders, as we can see in Figure REF .",
"This explains why in the phase portrait that we will later obtain we can have intersecting orbits, which happens always between an orbit in the expanding phase and another one in the contracting phase.",
"Therefore, when solving the dynamical system one option would be to use local coordinates in a cylinder.",
"However, this would be cumbersome and, thus, we have opted for integrating the solution taking into account whether we are in the expanding or contracting phase, so that we change sign of $H$ when reaching the curve $H=0$ .",
"For the numerical results, we will use an RK78 method, that holds in memory the sign of $H$ and changes it when we switch from the contracting to the expanding phase.",
"Figure: Top row: the phase space (ϕ,ϕ ˙,H)(\\varphi ,\\dot{\\varphi },H) for |ω|<1|\\omega |< 1 (left) and ω>1\\omega >1 (right).",
"The green (resp.",
"red) lines recover all the region corresponding to the expanding (resp.",
"contracting) phase.",
"Bottom row: its projection in the plane (ϕ,ϕ ˙)(\\varphi ,\\dot{\\varphi }), in which the white region delimited by the curves H(ϕ,ϕ ˙)=0H(\\varphi ,\\dot{\\varphi })=0 is the allowed one.The orbits will be plotted for |ω|<1|\\omega |< 1 in Figure , and for ω>1\\omega >1 in Figure .Now, we treat separately the following cases: $\\omega =1$ : In this case, the potential is zero.",
"Thus, (REF ) becomes $\\ddot{\\varphi }=\\mp \\sqrt{\\frac{3}{2}} |\\dot{\\varphi }|\\dot{\\varphi }\\sqrt{1-\\frac{\\dot{\\varphi }^2}{2\\rho _c}}$ .",
"Since $\\rho \\le \\rho _c$ , $|\\dot{\\varphi }|\\le \\sqrt{2\\rho _c}$ .",
"Therefore, we can use the change of variables $\\dot{\\varphi }=\\sqrt{2\\rho _c}\\cos (\\xi )$ .",
"Assuming that at t=0 we are in the expanding phase of the semi-plane $\\dot{\\varphi }_0>0$ , we will have $\\tan \\xi -\\tan \\xi _0=\\sqrt{3\\rho _c}t$ .",
"Hence, $\\dot{\\varphi }(t)=\\sqrt{\\frac{2\\rho _c}{1+\\left(\\sqrt{3\\rho _c}t+C\\right)^2}} \\ , \\ \\quad \\mbox{for} \\quad t>\\frac{-C}{\\sqrt{3\\rho _c}},$ where $C=\\sqrt{\\frac{2\\rho _c}{\\dot{\\varphi }_0^2}-1}$ .",
"It is easy to see that in the contracting phase $\\left(t<\\frac{-C}{\\sqrt{3\\rho _c}}\\right)$ , the expression of $\\dot{\\varphi }(t)$ would be exactly analogous.",
"Moreover, if t=0 takes place during the contracting phase, the constant C would be defined as $C=-\\sqrt{\\frac{2\\rho _c}{\\dot{\\varphi }_0^2}-1}$ .",
"Therefore, the orbit in the phase portrait is $(\\varphi (t),\\dot{\\varphi }(t))=\\left(\\varphi _0+\\sqrt{\\frac{2}{3}}\\ln \\left(\\sqrt{3\\rho _c}t+C+\\sqrt{1+(\\sqrt{3\\rho _c}t+C)^2} \\right), \\ \\sqrt{\\frac{2\\rho _c}{1+\\left(\\sqrt{3\\rho _c}t+C\\right)^2}}\\right),$ and in the semi-plane $\\dot{\\varphi }_0<0$ it is given by $(\\varphi (t),\\dot{\\varphi }(t))=\\left(\\varphi _0-\\sqrt{\\frac{2}{3}}\\ln \\left(\\sqrt{3\\rho _c}t+C+\\sqrt{1+(\\sqrt{3\\rho _c}t+C)^2} \\right), \\ -\\sqrt{\\frac{2\\rho _c}{1+\\left(\\sqrt{3\\rho _c}t+C\\right)^2}}\\right).$ Hence, we see that these are orbits and foliate all the space $0<|\\dot{\\varphi }|\\le \\sqrt{2\\rho _c}$ , since the Equation of State is trivially fulfilled for all of them in this case, with the bounce taking place in $t_b=-\\frac{C}{\\sqrt{3\\rho _c}}$ and such that $H(t)=(t-t_b)\\rho (t)=\\frac{\\rho _c(t-t_b)}{1+3\\rho _c(t-t_b)^2}$ .",
"On the other hand, if $\\dot{\\varphi }_0=0$ , the corresponding orbit $(\\varphi ,\\dot{\\varphi })=(\\varphi _0,0)$ would lead to $\\rho (t)=H(t)=0$ .",
"Before analyzing the other cases, we will introduce the following change of variables $\\psi =\\sinh \\left(\\frac{\\varphi \\sqrt{3(1+\\omega )}}{2} + \\frac{1}{2}\\ln \\left(\\frac{V_0}{2\\rho _c(1-\\omega )}\\right)\\right).$ Therefore, using the potential found in (REF ), equation (REF ) becomes $\\ddot{\\psi }=-3H_{\\pm }(\\psi ,\\dot{\\psi })\\dot{\\psi }+\\rho (\\psi ,\\dot{\\psi })\\psi \\frac{3(1+\\omega )}{2}, $ where $H_{\\pm }(\\psi ,\\dot{\\psi })=\\pm \\sqrt{\\frac{\\rho (\\psi ,\\dot{\\psi })}{3}\\left(1-\\frac{\\rho (\\psi ,\\dot{\\psi })}{\\rho _c}\\right)}$ and $\\rho =\\frac{2}{3(1+\\omega )(1+\\psi ^2)}\\left(\\dot{\\psi }^2+\\frac{3(1-\\omega ^2)}{4}\\rho _c \\right)$ .",
"Thus, in order to deal with this dynamical system, we compute the conditions needed for $\\ddot{\\psi }$ to vanish.",
"They are $\\rho =0 \\ \\ \\text{ or } \\ \\ \\left\\lbrace \\text{sgn}(H\\dot{\\psi })=\\text{sgn}(\\psi ) \\text{ and } \\left(|\\dot{\\psi }|=\\frac{\\sqrt{3\\rho _c}}{2}(1+\\omega ) \\text{ or } \\psi ^2=\\frac{4}{3\\rho _c(1-\\omega ^2)}\\dot{\\psi }^2 \\right)\\right\\rbrace .$ Now we can proceed to analyse the rest of cases that are left: $|\\omega |<1$ : We can distinguish two types of orbits: those that cross the axis $\\psi =0$ (Type I), which physically correspond to the ones where the scalar field reaches the top of the potential (REF ) and those that cross the axis $\\dot{\\psi }=0$ (Type II), which are the ones where the scalar field does not reach the top of the potential.",
"Regarding Type I orbits, we are going to consider that at the initial point $t=0$ we are at $(\\psi ,\\dot{\\psi })=(0,\\dot{\\psi }_0)$ , where $0<\\dot{\\psi }_0\\le \\frac{\\sqrt{3\\rho _c}}{2}(1+\\omega )$ , which comes from the restriction $0< \\rho _0 \\le \\rho _c$ .",
"If $\\rho _0=\\rho _c$ , at $t=0$ we are at the bounce and, by (REF ), the value of $\\dot{\\psi }$ will be the same throughout all the contracting and expanding phase, i.e., the velocity of the scalar field does not change sign, thus reaching the top of the potential.",
"With respect to Type II orbits, the initial point $t=0$ will be at $(\\psi ,\\dot{\\psi })=(\\psi _0,0)$ where $\\psi _0\\ne 0$ , which corresponds to a change of sign in the velocity of the scalar field, meaning that the orbits do not reach the top of the potential.",
"The corresponding phase portrait is given in Figure REF , where we have represented the set $\\rho =\\rho _c$ , which is the discontinuous black line $\\dot{\\psi }=\\pm \\sqrt{\\frac{3\\rho _c}{2}(1+\\omega )\\left(\\psi ^2+\\frac{1+\\omega }{2}\\right)}$ .",
"The pointed diagonal lines refer to the set where $\\ddot{\\psi }=0$ , as seen in (REF ).",
"The blue horizontal lines are the orbits corresponding to the analytical solution.",
"Figure: Phase portrait in the phase space (ψ,ψ ˙)(\\psi ,\\dot{\\psi }) for w=-2/3w=-2/3 and ρ c =1\\rho _c=1.",
"The blue orbits correspond to the analytic solutions which mimic the background obtained when one considers a perfect fluid with constant EoS parameter.",
"Since there are two dynamics, one in the contracting and another one in the expanding phase, we use two colors to draw the orbits obtained numerically: red color in the contracting and green color in the expanding phase.",
"When the orbit reaches the bounce, i.e.",
"when ρ=ρ c \\rho =\\rho _c, the orbit changes form red to green.",
"Finally, the picture shows that the analytic orbits are repellers at early times and attractors at late ones.With respect to the rest of the curves, we have used the following colour notation: red for the contracting phase and green for the expanding phase.",
"We have plotted one possible Type I orbit and one possible Type II orbit.",
"In both we have considered that either $\\dot{\\psi }_0$ or $\\psi _0$ are in the positive axis and that $t=0$ takes place during the contracting phase.",
"We note that applying the symmetry with respect to the axis $\\dot{\\psi }=0$ and/or $\\psi =0$ we would obtain the other possibilities for these orbits, considering that the initial point is in the negative axis and/or $t=0$ takes place during the expanding phase.",
"Finally, in Fig REF we have drawn as well the invariant curves that come in and out from the saddle point (0,0), the only critical point of the dynamical system.",
"For clarity, we have only plotted the invariant curves for $\\dot{\\psi }<0$ .",
"The others could be obtained with the symmetry respect to the axis $\\dot{\\psi }=0$ .",
"The physical meaning of these orbits is easy to understand if one takes into account the shape of the potential (REF ).",
"Since it has a maximum, these orbits correspond to the ones that start and end at the top of the potential.",
"There is an orbit that ends at the top when the universe is in the contracting phase (red orbit in Figure REF ), one that ends at the top after the bounce (a piece red and the rest green in Figure REF ), another that starts at the top when the universe is in the expanding phase (green orbit in Figure REF ), and finally one that starts at the top when the universe is in the contracting phase and bounces to enter in the expanding one (a piece green and the rest red in Figure REF ).",
"So, we clearly see in each orbit the bounce at the time in which it touches the curve $\\rho =\\rho _c$ .",
"We observe that $\\rho =0$ takes place for $\\psi \\rightarrow \\infty $ .",
"The points in which the orbits intersect with the diagonal lines are where they change the sign of their slope.",
"And finally the horizontal lines corresponding to the analytical solution are attractors for the expanding phase and repellers for the contracting phase.",
"We can also characterize orbits with the following magnitude: $\\omega _{\\textit {eff}}(t):=\\frac{P(t)}{\\rho (t)}=\\frac{\\dot{\\psi }(t)^2-\\frac{3}{4}(1-\\omega ^2)\\rho _c}{\\dot{\\psi }(t)^2+\\frac{3}{4}(1-\\omega ^2)\\rho _c}.$ We observe that $-1\\le \\omega _{\\textit {eff}}(t)<1$ and that for the analytical value $|\\dot{\\psi }|=\\frac{\\sqrt{3\\rho _c}}{2}(1+\\omega )$ , $\\omega _{\\text{eff}}(t)=\\omega $ .",
"Regarding the other orbits, the bounce takes place at $\\omega _{\\textit {eff}}(t_b)>\\omega $ and, when $\\rho (t)\\rightarrow 0$ , since all the orbits converge to the analytic one, it can be easily verified that $\\omega _{\\textit {eff}}(t)\\rightarrow \\omega $ .",
"Figure: Evolution of ω 𝑒𝑓𝑓 (t)\\omega _{\\textit {eff}}(t) for the orbits represented in the phase portrait of Figure for ω=-2/3\\omega =-2/3 and ρ c =1\\rho _c=1 (color code as in Fig.",
").The picture shows that at early and late times ω 𝑒𝑓𝑓 (t)\\omega _{\\textit {eff}}(t) converges to ω=-2/3\\omega =-2/3.",
"$\\omega >1$ : In this case, since the potential is negative, we have the lower bound of $|\\dot{\\psi }|\\ge \\frac{\\sqrt{3\\rho _c}}{2}\\sqrt{\\omega ^2-1}$ .",
"Therefore, we only have Type I orbits because the velocity of the scalar field can not vanish due the Friedmann equation ($H^2$ is positive, not negative) and, thus, there are no invariant orbits as in the previous case.",
"If $\\rho _0=0$ , we are stuck in this value of $\\rho $ during all the orbit $|\\dot{\\psi }|=\\frac{\\sqrt{3\\rho _c}}{2}\\sqrt{\\omega ^2-1}$ .",
"If $\\rho _0=\\rho _c$ , analogously as in the $|\\omega |<1$ case, we stay throughout all the contracting and expanding phase in the analytical solution.",
"In the phase portrait we have represented as a discontinuous black line the curve corresponding to $\\rho =\\rho _c$ .",
"The other discontinuous horizontal lines that delimit the forbidden region refer to an orbit with $\\rho =0$ .",
"The blue horizontal line corresponds to the orbit coming from the analytical solution.",
"Figure: Phase portrait in the phase space (ψ,ψ ˙)(\\psi ,\\dot{\\psi }) for w=2w=2 and ρ c =1\\rho _c=1.",
"We observe two bounces at ρ=ρ c \\rho =\\rho _c and one bounce at ρ=0\\rho =0.",
"In the first picture we observe an orbit depicting a universe in the contracting phase (red piece) which bounces and enters in the expanding one (green piece).",
"In the second picture these orbits (green piece) enter once again tothe contracting phase (red piece) when it reaches the black discontinuous horizontal line ρ=0\\rho =0 and finally it performs a second bounce to enter in the expanding phase (green piece).Regarding the other orbit, we have used the same colour notation as before.",
"We note that near $|\\dot{\\psi }|=\\frac{\\sqrt{3\\rho _c}}{2}\\sqrt{\\omega ^2-1}$ (corresponding to $\\rho =0$ ), the dynamical system behaves as in General Relativity and, thus, as we already justified in this case, the orbit reaches this value in a finite time, observing a bounce for $\\rho =0$ .",
"As in the case $|\\omega |<1$ , there is as well a bounce for $\\rho =\\rho _c$ .",
"It is important to point out that, since $\\omega >1$ , the condition $\\psi ^2=\\frac{4}{3\\rho _c(1-\\omega ^2)}\\dot{\\psi }^2$ is never fulfilled.",
"Hence, equation (REF ) implies that the sign of the slope of the orbit can only change when reaching $\\rho =0$ .",
"This guarantees us that there will be a transition from the expanding to the contracting phase at $\\rho =0$ , thereby all orbits different from the analytical one depict in the $(H,\\rho )$ plane a cyclic universe (it runs clockwise twice the ellipse).",
"The beginning of the first contracting phase and the end of the second expanding phase will take place respectively for $t\\rightarrow -\\infty $ and $t\\rightarrow \\infty $ , since in these cases both $\\psi $ and $\\dot{\\psi }$ diverge in such a way that they approach $\\rho =0$ .",
"Thus, we can conclude that the analytical solution is a repeller in the expanding phase and an attractor (though not global) in the contracting phase.",
"In this case, the same equation (REF ) is valid.",
"We observe that $\\omega _{\\textit {eff}}>1$ and that $\\omega _{\\textit {eff}}=\\omega $ for the analytical orbit.",
"In the other orbit, the bounces at $\\rho =\\rho _c$ take place at $\\omega _{\\textit {eff}}<\\omega $ and the parameter diverges at the bounce at $\\rho = 0$ .",
"Regarding $t\\rightarrow -\\infty $ and $t\\rightarrow \\infty $ , the effective parameter asymptotically approaches 1.",
"Figure: Evolution of ω 𝑒𝑓𝑓 (t)\\omega _{\\textit {eff}}(t) for the orbits represented in the phase portrait of Figure for w=2w=2 and ρ c =1\\rho _c=1 (color code as in Fig.",
").The effective EoS parameter diverges when the universe passes from the expanding to the contracting phase and it converges asymptotically to 1 (mimicking a stiff fluid) at early and late times.In Figure REF , we clearly appreciate the two cycles of the orbit: Cycle A: During the contracting phase the orbit comes asymptotically from $|\\dot{\\psi }|\\rightarrow \\infty $ ($\\omega _{\\textit {eff}}\\rightarrow 1$ ), so that the value of $|\\dot{\\psi }|$ is above the one of the orbit of the analytic solution.",
"Then it bounces, crosses this orbit and finishes the expanding phase with $|\\dot{\\psi }|\\rightarrow \\frac{\\sqrt{3\\rho _c}}{2}\\sqrt{\\omega ^2-1}$ , i.e.",
"$\\omega _{\\textit {eff}}\\rightarrow \\infty $ , which corresponds to the transition from the expanding to the contracting phase at $\\rho =0$ .",
"Cycle B: During the beginning of the contracting phase the orbit comes from $|\\dot{\\psi }|=\\frac{\\sqrt{3\\rho _c}}{2}\\sqrt{\\omega ^2-1}$ ($\\omega _{\\textit {eff}}\\rightarrow \\infty $ ) in such a way that the value of $|\\dot{\\psi }|$ is below the one of the orbit of the analytic solution.",
"Then, it crosses this orbit, bounces and in the expanding phase it asymptotically approaches $|\\dot{\\psi }|\\rightarrow \\infty $ , i.e.",
"$\\omega _{\\textit {eff}}\\rightarrow 1$ ."
],
[
"Conclusions",
"In this manuscript, we have introduced a scalar field $\\varphi $ that accounts for the perfect fluid with a linear Equation of State $P=\\omega \\rho $ that fills the space-time and we have used its corresponding potential $V(\\varphi )$ both in GR and in LQC so as to make a qualitative study of the orbits in the phase space $(\\varphi ,\\dot{\\varphi })$ , concluding that, for a canonical scalar field (i.e., $\\omega >-1$ ), in the expanding (resp.",
"contracting) phase, the analytical solution is an attractor (resp.",
"repeller) for $|\\omega |<1$ both in GR and LQC.",
"For $\\omega >1$ , both in GR and in LQC it is a repeller (resp.",
"attractor) in the expanding (resp.",
"contracting) phase.",
"However, whereas in GR the analytical solution is a global attractor in the contracting phase and a repeller in the expanding one, in LQC the other solutions do not catch (do not converge asymptotically) the analytical one because of the bounce, and when they enter in the expanding phase they move away from the analytical orbit, depicting at late times a universe with an effective EoS parameter equal to 1."
],
[
"Acknowledgments",
"This investigation has been supported in part by MINECO (Spain), projects MTM2014-52402-C3-1-P and MTM2015-69135-P."
]
] | 1612.05480 |
[
[
"Probability Densities of the effective neutrino masses $m_{\\beta }$ and\n $m_{\\beta \\beta}$"
],
[
"Abstract We compute the probability densities of the effective neutrino masses $m_{\\beta }$ and $m_{\\beta \\beta}$ using the Kernel Density Estimate (KDE) approach applied to a distribution of points in the $(m_{\\min}, m_{\\beta\\beta })$ and $(m_{\\beta }, m_{\\beta\\beta })$ planes, obtained using the available Probability Distribution Functions (PDFs) of the neutrino mixing and mass differences, with the additional constraints coming from cosmological data on the sum of the neutrino masses.",
"We show that the reconstructed probability densities strongly depend on the assumed set of cosmological data: for $\\sum_j m_j \\leq 0.68\\ @\\ 95\\% \\ \\mathrm{CL}$ a sensitive portion of the allowed values are already excluded by null results of experiments searching for $m_{\\beta \\beta}$ and $m_{\\beta }$, whereas in the case $\\sum_j m_j \\leq 0.23\\ @\\ 95\\% \\ \\mathrm{CL}$ the bulk of the probability densities are below the current bounds."
],
[
"Introduction",
"Although the physics of neutrino oscillation is entering a precision era, with all mixing angles and absolute values of the mass differences measured at the level of some percent, there are still questions related to the nature of neutrinos that need to be answered.",
"Among these, we are interested in whether neutrinos are Majorana or Dirac particles and in the absolute value of their masses.",
"As it is well known, experiments on neutrinoless double beta decays ($0\\nu \\beta \\beta $ ) consider the possibility that the reaction $( A , Z ) \\longrightarrow (A , Z + 2) + e^-+ e^-\\, $ really occurs; in the case of positive signal, we could conclude that the total lepton number is violated by two units, although the process behind the conversion of two down quarks into two up quarks would not be uniquely determined [1], [2], [3].",
"The $0\\nu \\beta \\beta $ -decay amplitude has the form $ \\mathcal {A}(0\\nu \\beta \\beta ) = m_{\\beta \\beta }\\mathcal {M}(A, Z)$ , where $\\mathcal {M}(A, Z)$ is the nuclear matrix element of the decay in Eq.",
"(REF ) that does not depend on the neutrino masses and mixing parameters, and $m_{\\beta \\beta }$ is the effective mass which, in the case of three lepton families, is given by $m_{\\beta \\beta } \\equiv \\bigg |\\sum _j m_j U_{ej}^2 \\bigg |= \\bigg |\\cos ^2\\theta _{13} \\Big (m_1 \\cos ^2\\theta _{12} +m_2 \\sin ^2\\theta _{12}e^{i \\alpha } \\Big )+ m_3\\sin ^2\\theta _{13} e^{i \\beta }\\bigg |.$ In the previous relation, $ U_{ej}$ are the elements of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix $U_{\\mathrm {PMNS}}$ that encodes the leptonic mixing angles $\\theta _{ij}$ , whereas the phases $\\alpha $ and $\\beta $ are the so-called Majorana phases (one of which eventually absorbs the $CP$ violating phase $\\delta $ ).",
"As it is usually done, two of the three neutrino masses $m_j$ in Eq.",
"(REF ) can be expressed in terms of the lightest one $m_{\\min }$ in a way that dependents on the supposed neutrino mass hierarchy; for Normal Ordering (NO) we have: $m_1 = m_{\\min } \\qquad m_2 = \\sqrt{m_{\\min }^2 + \\Delta m^2_{21}} \\qquad m_3=\\sqrt{m_{\\min }^2 + \\Delta m^2_{31}} \\,,$ whereas for the Inverted Ordering (IO) we set: $m_1 =\\sqrt{m_{\\min }^2 - \\Delta m^2_{21}-\\Delta m^2_{32}} \\qquad m_2=\\sqrt{m_{\\min }^2 - \\Delta m^2_{32}} \\qquad m_3 = m_{\\min } \\,,$ so $m_{\\beta \\beta }$ effectively depends on the seven independent parameters $\\theta _{12}, \\theta _{13}, \\Delta m^2_{21}, \\Delta m^2_{31},\\alpha ,\\beta $ and $m_{\\min }$ .",
"The study of the electron spectrum near the end point in the nuclear reaction $^3\\mathrm {H} \\rightarrow {^3\\mathrm {He}} + e + \\overline{\\nu }_e$ allows, in presence of neutrino mixing, to get information on the other effective mass largely studied in the literature, $m_\\beta $ , defined by $m_{\\beta } \\equiv \\sqrt{\\sum _j m_j^2 |U_{ej}|^2 }= \\sqrt{\\cos ^2\\theta _{13} \\Big (m_1^2 \\cos ^2\\theta _{12} +m_2^2 \\sin ^2\\theta _{12} \\Big )+ m_3^2\\sin ^2\\theta _{13} }.$ Since absolute values of the PMNS matrix are taken, complex phases play no role and $m_\\beta $ only depends on three independent observables and it is somehow correlated, although not in a simple form, with $m_{\\beta \\beta }$ .",
"It is customary to present such a correlation varying all mixing parameters inside their 1, 2 or 3$\\sigma $ range ($[0,2\\pi ]$ for the Majorana phases in any case) and computing the maximum and minimum allowed value.",
"While this procedure certainly gives insights on the possible outcomes of an experimental search, no information whatsoever can be drawn on the probability distribution of the observable itself.",
"So, inspired by the work of [4] and [5], we computed the distributions of $m_{\\beta }$ and $m_{\\beta \\beta }$ and the Credible Regions (CR) as obtained using the available PDFs of $\\theta _{12}, \\theta _{13}, \\Delta m^2_{21}$ and $\\Delta m^2_{31}$ , with the additional constraints coming from cosmological data on the sum of the neutrino masses (see also Refs.",
"[6], [7], [8] and [9]).",
"However, unlike the procedure followed in [4] and [5], we use the KDE approach to compute PDFs of the observables in the $2D$ planes $(m_{\\min }, m_{\\beta \\beta })$ and $(m_{\\beta }, m_{\\beta \\beta })$ .",
"The use of such a procedure also allows us to save computation time which could become a critical aspect in this sort of simulations.",
"A short summary of the numerical procedure and the data set we used to get the PDFs from the available data is done in Sect.",
"whereas a short description of the KDE method and the obtained results is done in Sect.",
".",
"Finally in Sect.",
"we compare the PDFs derived from several choices of the Kernel function."
],
[
"Numerical procedure and datasets",
"The construction of the PDFs for $m_{\\beta }$ and $m_{\\beta \\beta }$ passes through the extraction of the observables $p = \\lbrace \\sin ^2\\theta _{12},\\sin ^2\\theta _{13}, \\Delta m^2_{21}, \\Delta m^2_{3\\ell } \\rbrace $ (with $\\ell = 1$ for NO and $\\ell = 2$ for IO) from which they depend; the sampling is based on the knowledge of the likelihoods $\\mathcal {L}(p)$ which in turn are functions to the single $\\Delta \\chi ^2(p)$ : $\\mathcal {L}(p) \\propto \\exp \\left( -\\frac{\\Delta \\chi ^2(p)}{2}\\right) \\,.$ For the observables $p$ (which are only midly correlated), this information is available online at the address http://www.nu-fit.org, where the $\\Delta \\chi ^2$ for the November 2016 data, based on the procedure discussed in Ref.",
"[10], are given.",
"Notice that a Bayesian analysis on the 2014 data set is available in Ref.",
"[11]; the authors found that the results generally agree (at the level of one standard deviation) with those of the frequentest method, with some differences involving the atmospheric angle $\\theta _{23}$ and the Dirac $CP$ violating phase.",
"However, $\\theta _{23}$ does not enter into the expressions of the effective masses and the information on the $CP$ phase is hidden by the presence of the Majorana phases.",
"We then decided to use the most recent data set.",
"For the sake of completeness, we report in Tab.",
"REF the central values and $3\\sigma $ errors for all the observables relevant in neutrino oscillation for both orderings; similar values are also obtained in Ref.",
"[12].",
"In addition to the oscillation data, our estimate of the PDFs also takes into account the cosmological constraints on the sum of the neutrino masses $\\sum _j m_j$ coming from the Planck experiment [13].",
"Table: Central values ±\\pm the 1σ1\\sigma errors and 3σ3\\sigma ranges for the neutrino mixing parameters as obtained in Ref.",
"(available at the website http://www.nu-fit.org).",
"Note that in the last line ℓ=1\\ell =1 for NO and ℓ=2\\ell = 2 for IO.",
"The analysis prefers a global minimum for NO with respect to the local minimum of IO, Δχ 2 =χ IO 2 -χ NO 2 =0.83\\Delta \\chi ^2= \\chi ^2_{\\mathrm {IO}} - \\chi ^2_{\\mathrm {NO}} = 0.83.The Planck Collaboration provides several likelihoods based on different assumptions among which we decide to use the following ones: a conservative estimate (set-1) based on the set of data given by PLANCK TT + lowP + Lensing, which has $\\sum _j m_j \\le 0.68$ eV at $95\\% \\ \\mathrm {CL}$ ; a more aggressive one (set-2) based on PLANCK TT + lowP + Lensing + Ext, which has $\\sum _j m_j \\le 0.23$ eV at $95\\% \\ \\mathrm {CL}$ , with a maximum of the likelihood for $\\sum _j m_j \\sim 0.05\\ \\mathrm {eV}$ .",
"The acronyms used above refer to the data on the temperature power spectrum (PLANCK TT), to the Planck polarization data in the low-$\\ell $ temperature (lowP), to the data on Cosmic Microwave Background lensing reconstruction (Lensing); with Ext the constraints from Baryon Acoustic Oscillations, Joint Light-curve Analysis of supernovae and the Hubble constant are indicated.",
"For comparison purposes, we show in Tab.",
"REF the upper limits at 95% CL on the sum of the neutrino masses for different datasets which also include the data from the temperature-polarization cross spectrum (TE) and those from the polarization power spectrum (EE).",
"Table: Upper bound at 95% confidence level on the sum of the neutrino masses (in eV) using the data of Ref.",
".With these likelihoods at our disposal, we employed the following procedure to accept or reject a given extraction of the set of observables $p$ and Majorana phases (notice that $\\delta $ is not relevant because the Majorana phase $\\beta $ hides any information on the Dirac $CP$ phase): we first extract $\\sin ^2\\theta _{12}$ , $\\sin ^2\\theta _{13}$ , $\\Delta m^2_{21}$ and $\\Delta m^2_{3\\ell }$ according to (REF ); the Majorana phases $\\alpha $ and $\\beta $ are extracted according to a flat distribution in the interval $[0, 2\\pi ]$ ; we then extract the value of $M=\\sum _j m_j$ using the Planck data obtained from Fig.",
"30 in Ref.",
"[13]; for NO, if $M \\le \\sqrt{\\Delta m^2_{21}} + \\sqrt{\\Delta m^2_{31}}$ (or $M \\le \\sqrt{-(\\Delta m^2_{21} + \\Delta m^2_{32})} + \\sqrt{-\\Delta m^2_{32}}$ for IO), we reject such an $M$ and extract a new value for the sum of the neutrino masses; once the value of $\\sum _j m_j$ is accepted, we compute the lightest neutrino mass $m_{\\min }$ using the relations $\\circ $ $m_{\\min } + \\sqrt{m_{\\min }^2 + \\Delta m^2_{21}} + \\sqrt{m_{\\min }^2 + \\Delta m^2_{31}} = \\sum _j m_j$ for NO $\\circ $ $\\sqrt{m_{\\min }^2 - \\Delta m^2_{21}-\\Delta m^2_{32}} + \\sqrt{m_{\\min }^2 - \\Delta m^2_{32}} + m_{\\min }= \\sum _j m_j$ for IO .",
"Notice that, unless the Planck distributions on $M$ are peaked around 0.06 eV assuming NO and 0.1 eV for IO (which is in fact not the case), this procedure penalizes very small values of $m_{\\min }$ .",
"Thus Eqs.",
"(REF ) and (REF ) are used to get the numerical values of $m_{\\beta }$ and $m_{\\beta \\beta }$ .",
"We generate $\\mathcal {O}(10^6)$ realizations that satisfy the constraints discussed above.",
"This order of magnitude is necessary to guarantee a $5\\sigma $ coverage for the input parameters, as discussed in Ref.",
"[4]."
],
[
"PDF analysis",
"The procedure outlined above produces two-dimensional histograms in the planes $(m_{\\min }, m_{\\beta \\beta })$ and $(m_{\\beta }, m_{\\beta \\beta })$ .",
"In order to compute from them the PDF and CRs, we used the Kernel Density Estimate (KDE) approach [14].",
"Suppose we have a $d$ -dimensional vector $\\mathbf {x}$ of observables of which we want to know the PDF, $f(\\mathbf {x})$ , and suppose also that we have $N$ different realizations of the same observables $\\lbrace \\mathbf {t}_{j}\\rbrace _{j =1}^N$ obtained according to the procedure described above; thus $f$ is estimated from $\\hat{f}(\\mathbf {x}) = \\frac{1}{N \\prod _{k = 1}^d h_k}\\sum _{j=1}^N \\left[ \\prod _{k = 1}^d \\mathcal {K}\\left(\\frac{x^k - t_{j}^k}{h_k}\\right)\\right]\\,,$ where $h_k$ is the bandwidth of the $k$ -th component of the vector $\\mathbf {x}$ , whose estimate according to the Scott's rule of thumb [15] is given by $\\hat{h}_k = \\left(\\frac{4}{d+4}\\right)^{1/(d+4)}N^{-1/(d+4)} \\sigma _k\\,,$ $\\sigma _k$ being the standard deviation of the $k$ -th observable $x^k$ .",
"The Scott's rule reduces the asymptotic expected value of the integrated square errors between the actual distribution $f$ and the estimated $\\hat{f}$ .",
"The positive function $\\mathcal {K}$ is called kernel and must satisfy the normalization condition $\\int _{\\mathbb {R}^d} \\mathrm {d}^dx\\ \\mathcal {K}(\\mathbf {x}) = 1 \\qquad \\mathcal {K}(\\mathbf {x}) \\ge 0.$ A simple but equally suited kernel is the Gaussian kernel, defined as: $\\mathcal {K}(\\mathbf {x}) = \\frac{1}{(2\\pi )^{d/2}}\\exp \\left(-\\frac{1}{2}|\\mathbf {x}|^2\\right)\\,,$ that we estimate using the same algorithm of Ref.",
"[16] The original code is available at https://people.ucsc.edu/ianc/python/kdestats.html ., based on the modified SciPy function gaussian_kde described in Ref.",
"[17].",
"The results for the PDFs as a function of $\\log _{10} m_{\\min }\\ (m_{\\beta })$ and $\\log _{10} m_{\\beta \\beta }$ at the 68%, 95% and 99% CRs obtained with the analysis performed using set-1 for the sum of the neutrino masses are shown in Fig.",
"REF in the $(m_{\\min }, m_{\\beta \\beta })$ plane and in Fig.",
"REF in the $( m_{\\beta }, m_{\\beta \\beta })$ plane.",
"The analogous results for set-2 are shown in Figs.",
"REF and REF .",
"In all planes, the excluded region for $m_{\\beta \\beta }$ is the area above the horizontal magenta dashed line, around $m_{\\beta \\beta }\\ge 0.19\\ \\mathrm {eV}$ [18] obtained using the 90% CL limit on the half-life of $^{76}$ Ge, $T^{0\\nu }_{1/2}(^{76}\\mathrm {Ge}) > 5.2 \\times 10^{25}\\ \\mathrm {years}$ , in the preliminary analysis of GERDA phase II [19].",
"A recent result using the $^{136}$ Xe, $T^{0\\nu }_{1/2}(^{136}\\mathrm {Xe}) > 1.07 \\times 10^{26}\\ \\mathrm {years}$ at $90\\% \\ \\mathrm {CL}$ obtained by the KamLAND-ZEN experiment [20], gives the lower bound indicated with green dashed lines, which excludes the region $m_{\\beta \\beta }\\ge 0.083\\ \\mathrm {eV}$ [18].",
"The bounds we quote for $m_{\\beta \\beta }$ are obtained according to Ref.",
"[21], where the Authors used the results of Ref.",
"[22] for the phase-space factor and those of Ref.",
"[23] for the nuclear matrix elements.",
"In our analysis we fixed the axial coupling constant of the nucleon $g_A=1.269$ .",
"We also outline that the large uncertainties associated to the nuclear matrix elements $ \\mathcal {M}(A, Z)$ can modify the prediction for decay amplitude $ \\mathcal {A}(0\\nu \\beta \\beta )$ ; however, the impact of such effects are beyond the scope of this paper and will not be analyzed in the following.",
"For the other observable, $m_{\\beta }$ , the red vertical dashed line indicates the expected sensitivity of the KATRIN experiment ($0.2\\ \\mathrm {eV}$ at $90\\% \\ \\mathrm {CL}$ [24], see https://www.katrin.kit.edu/128.php) and the grey vertical dashed line the expected sensitivity of the Project 8 experiment ($4 \\times 10^{-2}\\ \\mathrm {eV}$ at $90\\% \\ \\mathrm {CL}$ [25]) which has been especially designed to probe the whole IO parameter space.",
"Notice that the most stringent upper limit on $m_\\beta $ has been obtained by the Mainz and Troitzk experiments, $m_{\\beta }\\le 2.05\\ \\mathrm {eV}$ at $95\\% \\ \\mathrm {CL}$ [26], [27].",
"To better compare the different cosmological datasets we use the same scale for the PDF densities (which are normalized to one).",
"In the $(m_{\\min }, m_{\\beta \\beta })$ plane the maximum is fixed to be 10, while in the $( m_{\\beta }, m_{\\beta \\beta })$ plane it is 30.",
"Figure: Reconstructed probability density in the (m min ,m ββ )(m_{\\min }, m_{\\beta \\beta }) plane assuming NO (left panel) and IO (right panel)using the set-1 prior on ∑ j m j \\sum _j m_j.",
"The credible regions are at 68% (solid lines), 95% (dashed lines) and 99% (dotted lines) for 2 dof.",
"The horizontal pink dashed line indicates the excluded region at 90% CL assuming the 76 ^{76}Ge results , while the green dashed line refers tothe 136 ^{136}Xe results .Figure: The same as Fig.",
"but in the (m β ,m ββ )(m_{\\beta }, m_{\\beta \\beta }) plane.With vertical red dashed lines we indicate the expected sensitivity of KATRIN and with vertical grey dashed lines the one of Project 8 .Figure: The same as Fig.",
", but for set-2.Figure: The same as Fig.",
", but for set-2.A close inspection at Fig.",
"REF shows that a large portion of the 68% CR for $m_{\\beta \\beta }$ , the one corresponding to large $m_{\\min }$ , is already excluded by the KamLAND-ZEN data, for both hierarchies.",
"In practice, this is the consequence of having a non-negligible probability that $\\sum _j m_j \\gtrsim 0.5\\ \\mathrm {eV}$ , therefore the value of $m_{\\min }$ can be sufficiently large to approach ${\\cal O}(0.1)$ .",
"On the other hand, set-2 relaxes this constraint and the probability density is centered around smaller $m_{\\min }$ (and consequently smaller $m_{\\beta \\beta }$ ).",
"For both cases, the cosmological bounds on the sum of neutrino masses implies that for NO and IO the low mass region for $m_{\\beta \\beta }$ is strongly disfavoured.",
"The interesting features of Figs.",
"REF and REF is that, for both assumptions on the sum of the neutrino masses, the Project 8 experiment would be able to probe almost the whole allowed regions for $m_\\beta $ at 99% level whereas KATRIN shows only a modest ability to probe the largest possible values of $m_\\beta $ , around ${\\cal O}(10^{-2}-10^{-1})$ eV.",
"Instead of discussing the effects of the cosmological bounds on the effective masses, one can also adopt an opposite point of view, asking what would be the effect on $\\sum _j m_j$ of a possible measure of $m_{\\beta \\beta }$ at the new generation of experiments, see Refs.",
"[28], [29], [21].",
"As an example, we can explore the situation that a signal for the $0\\nu \\beta \\beta $ -decay is observed at the (near future) CUORE or (next-to-near future) nEXO experiments.",
"Following the discussion of Ref.",
"[21] we assume an optimistic scenario where a signal is in the expected 90% experimental sensitivity region, that is $m_{\\beta \\beta }= 0.073 \\pm 0.008\\ \\mathrm {eV}$ (assuming $g_A = 1.269$ ) for CUORE [30].",
"Similar values can also be achieved by GERDA Phase-II [31], MAJORANA-D [32] and NEXT [33] experiments, so our discussion applies equally well to a large number of possible future experiments.",
"In the case of nEXO experiment [34], we set $m_{\\beta \\beta }= 0.011 \\pm 0.001\\ \\mathrm {eV}$ , which is below the IO region.",
"The results of our finding are shown in Fig.",
"REF where the frequency of the sum of light neutrino masses (histograms normalized to 1), after the constraints coming from $m_{\\beta \\beta }$ , is displayed.",
"With black dashed lines we also show the Planck PDFs for set-1 (upper panels) and set-2 (lower panels).",
"In the first column of the plot, which refers to the case $m_{\\beta \\beta }= 0.073 \\pm 0.008\\ \\mathrm {eV}$ , we clearly see that there exists a cutoff in the distribution in the low mass region due to the fact that $m_{\\beta \\beta }$ cannot be arbitrarily small, with maxima around $\\sum _j m_j\\sim {\\cal O}(0.2-0.3)$ eV for both set-1 and set-2 and for both hierarchies (NO in blue and IO in red).",
"On the other hand, in the high mass region the distributions essentially follow the shape of the Planck priors since the assumed values of $m_{\\beta \\beta }$ do not impose strong constraints on $m_{\\min }$ .",
"If we assume a positive signal at the nEXO experiment $m_{\\beta \\beta }= 0.011 \\pm 0.001\\ \\mathrm {eV}$ , second column of Fig.",
"REF , we see that we cannot distinguish among different Planck datasets since the bound on the $0\\nu \\beta \\beta $ -decay effective mass constraints $\\sum _j m_j$ to be of $\\mathcal {O}(0.1)\\ \\mathrm {eV}$ .",
"Figure: Frequency of ∑ j m j \\sum _j m_j for an assumed m ββ =0.073±0.008 eV m_{\\beta \\beta }= 0.073 \\pm 0.008\\ \\mathrm {eV} (left panels) and m ββ =0.011±0.001 eV m_{\\beta \\beta }= 0.011 \\pm 0.001\\ \\mathrm {eV} (right panels), for NO (blue) and IO (red).",
"The black dashed lines are the Planck PDFs: set-1 in the upper panels and set-2 in the lower panels.",
"The darkest areas under the histograms are the 68% credible regions obtained from the cumulant distributions."
],
[
"Discussion and conclusions",
"At first sight, the results described above seem to depend on the choice of the kernel used to estimate the PDFs.",
"However, we have checked that adopting different functions $\\mathcal {K}$ the CL regions are not altered in a significant manner.",
"We test different kernels, provided by the scikit-learn package [35]: Gaussian $\\mathcal {K}(x; h) \\propto \\exp (-x^2/2h)$ ; tophat $\\mathcal {K}(x; h) \\propto 1$ for $|x| \\le h$ ; Epanechnikov $\\mathcal {K}(x; h) \\propto 1- x^2/h^2$ ; exponential $\\mathcal {K}(x; h) \\propto \\exp (-|x|/h)$ ; linear $\\mathcal {K}(x; h) \\propto 1-x/h$ for $|x| \\le h$ ; cosine $ \\mathcal {K}(x; h) \\propto \\cos (\\pi x/2 h)$ .",
"The check is performed adopting the $k$ -fold cross-validation approach, proposed in Refs.",
"[36], [37]; in few words, the sample of extracted points is split into $k$ smaller sets; of them, $k-1$ sets are used to estimate $f(\\mathbf {x})$ according to a given kernel and the resulting model is then validated on the remaining part of the dataset.",
"In Tab.",
"REF we show our result for the cross-validation analysis with $m_{\\rm min}$ and $m_{\\beta \\beta }$ as independent variables (similar results can be achieved for $m_{\\beta }$ and $m_{\\beta \\beta }$ ): we analyze ten subsets with $N = \\lbrace 1000, 5000, 10000, 50000\\rbrace $ points, then we average the results.",
"In order to investigate possible overfitting effects, each subset has been divided into two parts: a train ($N_{\\rm train} \\approx 0.6 N$ ) and a test ($N_{\\rm test} = N - N_{\\rm train}$ ) set.",
"We estimate the best bandwidth $\\hat{h}$ using twenty $k$ -folds in the train dataset.",
"The error ${\\cal E_{\\rm set}}$ between the actual distribution and the kernel estimate is defined as: ${\\cal E}_{\\rm set} = \\sqrt{ \\frac{1}{N_{\\rm set}} \\sum _j^{N_{\\rm set}^{1/2}}\\sum _k^{N_{\\rm set}^{1/2}}\\left[f(\\mathbf {x}_{j, k}) - \\hat{f}(\\mathbf {x}_{j, k})\\right]^2 } \\,,$ where set can be train or test-set.",
"The actual distribution can be obtained from the two dimensional density histogram.",
"We assume for the histogram $N_{\\rm set}^{1/2} \\times N_{\\rm set}^{1/2}$ bins.",
"Notice that the normalization factor $N^{-1/2}_{\\rm set}$ in the error (REF ) is necessary to compare datasets with different dimensions.",
"In Fig.",
"REF we show our results of $\\hat{f}(\\mathbf {x})$ for the set-1 prior on the sum of neutrino masses and for all kernels introduced above (green shaded area).",
"The PDFs are superimposed on a subset of $5 \\times 10^3$ points.",
"As we can see, the Gaussian kernel as well as the exponential one correctly reproduce the testing dataset for both orderings (for these two cases, the green areas are concentrated below the points and they do not appear in the graphs).",
"For the other kernels, the agreement does not appear to be as good as for the previous ones, since the PDFs extend over regions outside the subset of points.",
"In particular, in Tab.",
"REF we observe that the errors ${\\cal E}_{{\\rm set}}$ of the Gaussian and the exponential kernels are roughly one half those of the other kernels.",
"The cross-validation procedure is also useful to compute the best bandwidth $\\hat{h}$ that minimizes the residual error between the predictions and the actual values of the sample points.",
"Our findings are compatibles with the Scott's rule defined in (REF ), see Tab.",
"REF for a summary of the bandwidths computed using the same data of the cross-validation analysis.",
"Notice that in the cross-validation a single bandwidth is estimated for each kernel.",
"For the set-2 our conclusions remain unaltered: the Gaussian and the exponential kernels reproduce the training dataset with a good accuracy.",
"Figure: PDFs in the plane (m min ,m ββ )(m_{\\min },m_{\\beta \\beta }) obtained from the KDE analysis (green shaded areas) for NO and IO datasets; the blue (red)points are a sample of data obtained in the numerical scan.Table: Mean estimated bandwidth and mean errors for the train and test datasets assuming NO or IO.",
"The results are obtained using a sample of NN points performing twenty kk-folds cross-validation for the train subset.",
"The error is defined in ().Table: Mean values of the bandwidths evaluated for the Gaussian kernel using the Scott's rule defined in () and the same data of Tab.",
".In conclusions, we have shown that the KDE method is an efficient tool to evaluate the PDFs of interesting physical observables.",
"We have concentrated our efforts on two observables related to neutrino physics, namely the effective neutrino masses $m_{\\beta \\beta }$ and $m_\\beta $ which will help to reveal the true nature of neutrinos and the values of their absolute masses.",
"For them, we have computed the Credible Regions using the available PDFs on the mixing angles and mass differences, with the additional constraints coming from cosmological data on the sum of the neutrino masses.",
"We found that the reconstructed probability densities strongly depend on the assumed set of cosmological data and, in particular, for $\\sum _j m_j \\le 0.23$ eV at $95\\% \\ \\mathrm {CL}$ the bulk of the probability densities are below the current bounds on the analyzed observables.",
"This conclusion remains qualitatively unaffected if one uses a different choice of the kernel function."
],
[
"Acknowledgements",
"We are indebted with Carlo Giunti for useful discussion about the neutrino effective masses and Andrew Fowlie for sharing his code to compute in a different way the 1D and 2D posterior PDFs."
]
] | 1612.05453 |
[
[
"Weighted-$W^{1,p}$ estimates for weak solutions of degenerate and\n singular elliptic equations"
],
[
"Abstract Global weighted $L^{p}$-estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain.",
"The coefficient matrix is symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a weight in a Muckenhoupt class.",
"Under a smallness condition on the mean oscillation of the coefficients with the weight and a Reifenberg flatness condition on the boundary of the domain, we establish a weighted gradient estimate for weak solutions of the equation.",
"A class of degenerate coefficients satisfying the smallness condition is characterized.",
"A counter example to demonstrate the necessity of the smallness condition on the coefficients is given.",
"Our $W^{1,p}$-regularity estimates can be viewed as the Sobolev's counterpart of the H\\\"{o}lder's regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni in 1982."
],
[
"Introduction",
"The main concern of this paper is to establish a $W^{1,p}$ -regularity estimate for weak solutions of the linear boundary value problem $ \\left\\lbrace \\begin{array}{cccl}\\text{div} [\\mathbb {A}(x)\\nabla u] & = & \\text{div}[{\\bf F}] &\\quad \\text{in} \\quad \\Omega ,\\\\u & = &0 & \\quad \\text{on} \\quad \\partial \\Omega ,\\end{array}\\right.$ where $\\Omega \\subset \\mathbb {R}^n$ is an open bounded domain with boundary $\\partial \\Omega $ , ${\\bf F}:\\Omega \\rightarrow \\mathbb {R}^{n}$ is a given vector field, and the coefficient matrix $\\mathbb {A}: \\mathbb {R}^{n}~\\rightarrow ~\\mathbb {R}^{n\\times n}$ is symmetric and measurable satisfying the degenerate elliptic condition $ \\Lambda \\mu (x) |\\xi |^2 \\le \\langle \\mathbb {A}(x) \\xi , \\xi \\rangle \\le \\Lambda ^{-1} \\mu (x) |\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n, \\quad \\text{a.e.}",
"\\quad x \\in \\mathbb {R}^n,$ with fixed $\\Lambda >0$ , and a non-negative weight $\\mu $ in some Muckenhoupt class.",
"Our main result states that for a given $1< p < \\infty $ , the weak solution $u$ to (REF ) corresponding to ${\\bf F}$ with ${\\bf F}/\\mu \\in L^{p}(\\Omega ,\\mu )$ , the weighted $L^{p}$ space, satisfies the estimate $\\Vert \\nabla u\\Vert _{L^{p}(\\Omega , \\mu )} \\le C \\left\\Vert \\frac{{\\bf F}}{\\mu }\\right\\Vert _{L^{p}(\\Omega ,\\mu )},$ provided that $\\mathbb {A}$ has a small mean oscillation with weight $\\mu $ , and the boundary of $\\Omega $ is sufficiently flat.",
"We will demonstrate by an example that obtaining an estimate of type (REF ) for a solution of the degenerate elliptic equation (REF ) for large values of $p$ is not always possible even for a smooth degenerate coefficient matrix $\\mathbb {A}$ .",
"In light of the examples, this work provides the right set of conditions on the coefficients and on the boundary of $\\Omega $ so that the linear map $\\frac{{\\bf F}}{\\mu } \\mapsto \\nabla u $ is continuous on $L^{p}(\\Omega ,\\mu )$ .",
"The study of regularity of weak solutions to linear equations (REF ) when $\\mathbb {A}$ is uniformly elliptic (i.e.",
"for $\\mu =1$ in (REF )) is by now classical.",
"The celebrated De Giorgi-Nash-Möser theory [9], [30], [31], [34], for instance, shows that weak solutions to (REF ) corresponding to uniformly elliptic coefficients are Hölder's continuous, when the datum ${\\bf F}$ is sufficiently regular.",
"Regularity theory and related issues for the class of degenerate equations (REF ) with some weight $\\mu $ were also investigated in past decades.",
"In this direction, seminal contributions were made in the classical papers [14], [35].",
"In particular, B. Fabes, C. E. Kenig, and R. P. Serapioni in [14] have established, among other significant results, the existence, and uniqueness of weak solutions in the weighted Sobolev space $W^{1,2}_0(\\Omega , \\mu )$ for $\\mu $ in the Muckenhoupt class $A_2$ .",
"In addition, Harnack's inequality and Hölder's regularity of weak solutions were obtained in [14] by adapting the Möser's iteration technique to the non-uniformly elliptic equation (REF ).",
"Since then, Hölder's regularity theory of weak solutions for linear, nonlinear degenerate elliptic and parabolic equations have been extensively developed in [15], [19], [28], [29], [38], [40] by using and extending ideas and techniques in [14].",
"See also the earlier paper [41] on Gehring-type gradient estimate for solution of degenerate elliptic equations Sobolev type regularity theory for weak solutions of (REF ) have also been the focus of studied in the past but mostly for the uniformly elliptic case, i.e.",
"$\\mu =1$ .",
"In this case, and unlike the case of Hölder's regularity, the mere assumption on the uniform ellipticity of the coefficients $\\mathbb {A}$ is not sufficient for the gradient of the weak solution of (REF ) to have the same regularity as that of the data ${\\bf F}$ .",
"This can be seen from the counterexample provided by N. G. Meyers in [27].",
"In the event that $\\mathbb {A}$ is uniformly elliptic and continuous, the $L^p$ -norm of $\\nabla u$ can be controlled by the $L^p$ -norm of the datum ${\\bf F}$ and this is achieved via the Calderón-Zygmund theory of singular integrals and a perturbation technique, see [13], [22], [17] for this classical result.",
"The same approach was also used by [8], [10], [11], [24] to extend the result when the coefficient matrix $\\mathbb {A}$ is uniformly elliptic and is in Sarason's class of vanishing mean oscillation (VMO) functions [39].",
"The approach in [11], [10] is in fact based on the earlier work [8] where many fundamental results on Calderón-Zygmund operators were established.",
"A drawback of this approach is that it requires a Green's function representation of the solution to a corresponding elliptic equation used for comparison (usually a homogeneous equation with constant coefficients), which may not always be available for nonlinear equations.",
"Alternative approaches have been used in the papers [7], [21], [20] that avoid the use of singular integral theory directly but rather study the integrability of gradient of solutions, via approximation, as a function of the deviation of the coefficients from constant coefficients.",
"See also the papers [4], [5], [6], [26], [25], [37], [43], [16], to cite a few, for the implementation of these approaches for elliptic and parabolic equations.",
"Unlike the case $\\mu =1$ , estimates of type (REF ) for general $\\mu \\in A_2$ are not fully understood yet.",
"Our goal in this paper, the first of several projects, is to close this gap, by providing the right conditions on the coefficients $\\mathbb {A}$ and the boundary of the domain $\\Omega $ to obtain weighted gradient estimates for solutions of the degenerate elliptic problems (REF ) with (REF ) for $\\mu \\in A_2$ .",
"To establish (REF ), we follow the approximation method of Cafarrelli and Peral in [7] where we view (REF ) locally as a perturbation of an elliptic homogeneous equation with constant coefficients.",
"The key to the success of this approach to degenerate equations is the novel way of measuring mean oscillation of coefficients that is compatible with the degeneracy of the coefficients (see Definition REF ).",
"As far as we know, this way of measuring the mean oscillation of function relative to a given weight was first introduced in [32], [33] in connection with the study the Hilbert transform and the characterization of the dual of the weighted Hardy space.",
"The condition we give on $\\mathbb {A}$ is optimal in the sense that it coincided with the well known result in [6] when $\\mu = 1$ .",
"Via a counterexample we will also demonstrate the necessity of the smallness condition to obtain (REF ).",
"A class of coefficients satisfying our smallness conditions will be given.",
"Based on our approach and the recent developments [5], [6], [26], [25], we are also able to obtain estimates of type (REF ) near the boundary of $\\Omega $ for domain with a flatness condition on the boundary $\\partial \\Omega $ .",
"The paper is organized as follows.",
"In Section , we introduce notations, definitions, and state the main results on the interior and global $W^{1,p}$ -regularity estimates, Theorem REF and Theorem REF .",
"An example, and a counterexample are also provided.",
"Section recalls and proves several preliminary analytic results on weighted inequalities.",
"Necessary interior estimates and Theorem REF are proved in Section .",
"Section gives the boundary approximation estimates and completes the proof of Theorem REF ."
],
[
"Main results",
"To state our main results, we need some notations and definitions.",
"We first introduce the notations that we use in the paper.",
"Given a locally integrable function $\\sigma \\ge 0$ , we denote by $d\\sigma = \\sigma dx$ , a non-negative, Borel measure on $\\mathbb {R}^n$ .",
"For $U \\subset \\mathbb {R}^n$ , a non-empty open set, we write $\\sigma (U) = \\int _{U} \\sigma (x) dx.",
"$ For a locally integrable Lebesgue-measurable function $f$ on $\\mathbb {R}^n$ , we denote the average of $f$ in $U$ with respect to the measure $d\\sigma $ as $\\langle f\\rangle _{\\sigma , U} = _{U} f(x) d\\sigma = \\frac{1}{\\sigma (U)} \\int _{U}f(x) \\sigma dx.$ In particular, with Lebesgue measure $dx$ , we write $\\langle f\\rangle _{U} = \\langle f\\rangle _{dx, U} \\quad \\text{and} \\quad |U| = \\int _{U} dx.$ We now recall the definition of the class $A_p$ Muckenhoupt weights.",
"For $p \\in [1,\\infty )$ , the weight function $\\mu \\in L^1_{\\textup {loc}}(\\mathbb {R}^n)$ is said to be of class $A_p$ if $\\begin{split}[\\mu ]_{A_p} &: = \\sup _{ B\\subset \\mathbb {R}^{n}} \\left( _{B} \\mu (y) dy \\right) \\left(_{B} \\mu (y)^{-\\frac{1}{p-1}} dy \\right)^{p-1} < \\infty ,\\quad 1 < p < \\infty ,\\\\[\\mu ]_{A_1} &: = \\sup _{ B\\subset \\mathbb {R}^{n}} \\left( _{B} \\mu (y) dy \\right) \\Vert \\mu ^{-1}\\Vert _{L^{\\infty }(B)}< \\infty ,\\quad p=1,\\end{split}$ where the supremum is taken over all balls $B \\subset \\mathbb {R}^n$ .",
"Following [14], [35], for a given $\\mu \\in A_{p}$ , we can define the corresponding Lebesgue and Sobolev spaces with respect to the measure $d\\mu $ .",
"For $1 \\le p < \\infty $ , we say a locally integrable function $f$ defined on $\\Omega $ belongs to the weighted Lebesgue space $L^p(\\Omega , \\mu )$ if $\\left\\Vert f\\right\\Vert _{L^p(\\Omega ,\\mu )} =\\left( \\int _{\\Omega } |f(x)|^p \\mu (x) dx \\right)^{1/p}< \\infty .",
"$ Let $k \\in \\mathbb {N}$ .",
"A locally integrable function $f$ defined on $\\Omega $ is said to belong to the weighted Sobolev space $W^{k,p}(\\Omega , \\mu )$ if all of its distributional derivatives $D^\\alpha f$ are in $L^p(\\Omega , \\mu )$ for $\\alpha ~\\in ~(\\mathbb {N} \\cup \\lbrace 0\\rbrace )^n$ with $|\\alpha | \\le k$ .",
"The space $W^{k,p}(\\Omega ,\\mu )$ is equipped with the norm $\\left\\Vert f\\right\\Vert _{W^{k,p}(\\Omega , \\mu )} = \\left( \\sum _{|\\alpha | \\le k} \\left\\Vert D^\\alpha f\\right\\Vert _{L^p(\\Omega , \\mu )}^p \\right)^{1/p}.$ Moreover, we also denote $W^{1,p}_0(\\Omega , \\mu )$ to be the closure of $C_0^\\infty (\\Omega )$ in $W^{1,p}(\\Omega , \\mu )$ .",
"Now, we recall what we mean by weak solution of (REF ).",
"Definition 2.1 Assume that (REF ) holds and $|{\\bf F}|/\\mu \\in L^p(\\Omega , \\mu )$ with $ 1 < p < \\infty $ .",
"A function $u~\\in ~W^{1,p}_0(\\Omega , \\mu )$ is said to be a weak solution of (REF ) if $\\int _\\Omega \\langle \\mathbb {A}\\nabla u, \\nabla \\varphi \\rangle dx = \\int _\\Omega \\langle {\\bf F}, \\nabla \\varphi \\rangle dx, \\quad \\forall \\varphi \\in C_{0}^{\\infty }(\\Omega ).$ To discuss about local interior regularity, we recall the following the definition of weak solution.",
"Definition 2.2 Assume that (REF ) holds and $|{\\bf F}|/\\mu \\in L^p_{\\textup {loc}}(\\Omega , \\mu )$ with $ 1 < p < \\infty $ .",
"A function $u~\\in ~W^{1,p}_\\textup {loc}(\\Omega , \\mu )$ is said to be a weak solution of $\\textup {div}[\\mathbb {A}\\nabla u] = \\textup {div}[{\\bf F}], \\quad \\text{in} \\quad \\Omega $ if $\\int _\\Omega \\langle \\mathbb {A}\\nabla u, \\nabla \\varphi \\rangle dx = \\int _\\Omega \\langle {\\bf F}, \\nabla \\varphi \\rangle dx, \\quad \\forall \\varphi \\in C_{0}^{\\infty }(\\Omega ).$ The following definition of functions of bounded mean oscillations with weights introduced in [32], [33] will be needed in our paper.",
"Definition 2.3 Given $R_{0}>0$ , we say that a locally integrable function $f:\\mathbb {R}^{n}\\rightarrow \\mathbb {R}$ is a function of bounded mean oscillation with weight $\\mu $ in $\\Omega $ if $[f]_{\\textup {BMO}_{R_{0}}(\\Omega , \\mu )}^2 = \\sup _{\\stackrel{x\\in \\Omega }{0 < \\rho < R_{0}}} \\frac{1}{\\mu (B_{\\rho }(x))} \\int _{B_{\\rho }(x)}|f(y) - \\langle f\\rangle _{B_{\\rho }(x)}|^{2} \\mu ^{-1}(y) dy < \\infty ,$ where $\\langle f\\rangle _{B_{\\rho }(x)} = \\frac{1}{|B_{\\rho }(x)|} \\int _{B_{\\rho }(x)} f(y) dy$ is the average of $f$ in the ball $B_\\rho (x)$ .",
"Observe that this notion of bounded mean oscillation with weight is different from the weighted version of the classical John-Nirenberg BMO, see [32].",
"However, from this definition, the classical John-Nirenberg BMO space in $\\Omega $ corresponds to $\\mu =1$ and $R_{0} = \\text{diam}(\\Omega )$ .",
"Definition 2.4 Let $\\Lambda , R_0, \\delta $ be given positive numbers, and let $\\mu \\in A_{2}$ .",
"We denote $\\begin{split}\\mathcal {A}_{R_{0}} (\\delta , \\mu , \\Lambda , \\Omega ) := \\bigg \\lbrace \\mathbb {A}: & \\mathbb {R}^{n}\\rightarrow \\mathbb {R}^{n\\times n} : \\text{$\\mathbb {A}$ is measurable, symmetric such that} \\\\& (\\ref {ellip}) \\ \\text{holds, and}\\ [\\mathbb {A}]^{2}_{\\textup {BMO}_{R_{0}}(\\Omega , \\mu )} < \\delta \\bigg \\rbrace .\\end{split}$ In the above, for a given matrix function $\\mathbb {A} = (a_{ij})$ , $[\\mathbb {A}]^{2}_{\\textup {BMO}_{R_{0}}(\\Omega , \\mu )} = \\sum _{i,j=1}^{n} [a_{ij}]_{\\textup {BMO}_{R_{0}}(\\Omega , \\mu )}^{2}$ , where $[a_{ij}]_{\\textup {BMO}_{R_{0}}(\\Omega , \\mu )}^{2}$ is as given in Definition REF .",
"The first main result of this paper is about the interior higher integrability of the gradients of weak solutions for the equation (REF ) which we state now.",
"We use the notation $B_{r}$ for $B_{r}(0)$ .",
"Theorem 2.5 Let $p\\ge 2$ , $M_0 \\ge 1, \\Lambda >0$ , and let $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ .",
"There exists a sufficiently small positive number $\\delta = \\delta (\\Lambda , p, M_0,n)$ such that if $\\mathbb {A}\\in \\mathcal {A}_{4}(\\delta , \\mu , \\Lambda , B_{2})$ , ${\\bf F}/\\mu \\in L^{p}(B_{6}, \\mu )$ , and $u\\in W^{1, 2}(B_{6}, \\mu )$ is a weak solution of $\\textup {div}[\\mathbb {A} \\nabla u] = \\textup {div}({\\bf F})\\quad \\text{in\\ $B_{6}$},$ then $\\nabla u\\in L^{p}(B_{1}, \\mu )$ and $\\Vert \\nabla u\\Vert _{L^{p}(B_{1}, \\mu )} \\le C \\left( \\mu (B_{1})^{\\frac{1}{p} - \\frac{1}{2}} \\Vert \\nabla u\\Vert _{L^{2}(B_{6}, \\mu )} + \\Vert {\\bf F}/\\mu \\Vert _{L^{p}(B_{6}, \\mu )}\\right),$ for some constant $C$ depending only on $\\Lambda , p, n, M_{0}$ .",
"Next, to obtain the global integrability for the gradients of weak solutions for (REF ), we need to make precise the type of boundary the underlying domain $\\Omega $ required to have.",
"Intuitively, we require that at all boundary points and at all scale, locally, the boundary can be placed between two hyperplanes.",
"Definition 2.6 We say that $\\Omega $ is a $(\\delta , R_{0})$ -Reifenberg flat domain if, for every $x\\in \\partial \\Omega $ and every $r\\in (0, R_{0})$ , there exists a coordinate system $\\lbrace y_{1}, y_{2}, \\cdots , y_{n}\\rbrace $ which may depend on $x$ and $r$ , such that $x = 0$ in this coordinate system and that $B_{r}(0) \\cap \\lbrace y_{n} > \\delta r\\rbrace \\subset B_{r}(0) \\cap \\Omega \\subset B_{r}(0) \\cap \\lbrace y_{n} > -\\delta r\\rbrace .$ We remark that, as described in [26], if $\\Omega $ is a $(\\delta , R_{0})$ flat domain with $\\delta < 1$ , then for any point $x$ on the boundary and $ 0 < \\rho < R_{0}(1 -\\delta )$ , there exists a coordinate system ${z_1,z_2,\\cdots , z_n}$ with the origin at some point in the interior of $\\Omega $ such that in this coordinate system $x = -\\delta \\rho z_n $ and $B_{\\rho }^{+}(0) \\subset \\Omega _{\\rho }\\subset B_{\\rho }(0) \\cap \\lbrace (z_{1}, \\cdots , z_{n-1}, z_n): z_{n} > -2\\delta ^{\\prime } \\rho \\rbrace ,\\quad \\text{with $\\delta ^{\\prime } = \\frac{\\delta }{1-\\delta }.$}$ In the above and hereafter $B_{\\rho }(x)$ denotes a ball of radius $\\rho $ centered at $x$ , $B_{\\rho }^{+}(x)$ its upper-half ball, and $\\Omega _{\\rho }(x) = B_{\\rho }(x)\\cap \\Omega $ , the portion of the ball in $\\Omega $ .",
"Our global regularity estimate for the weak solution of the equation (REF ) now can stated as below.",
"Theorem 2.7 Let $1< p < \\infty $ , $M_0 \\ge 1$ and $\\Lambda >0$ .",
"There exists a sufficiently small $\\delta =\\delta (\\Lambda , n, p, M_0)>0$ such that if $\\Omega $ is $(\\delta , R_{0})$ Reifenberg flat and $\\mathbb {A}\\in \\mathcal {A}_{R_{0}} (\\delta , \\mu , \\Lambda , \\Omega )$ for some $R_0 >0$ , and some $\\mu \\in A_2 \\cap A_p$ with $[\\mu ]_{A_2 \\cap A_p} \\le M_0$ , then for each ${\\bf F}: \\Omega \\rightarrow \\mathbb {R}^n$ such that $|{\\bf F}|/\\mu \\in L^p(\\Omega , \\mu )$ , there exists unique weak solution $u \\in W^{1,p}_0(\\Omega , \\mu )$ of (REF ).",
"Moreover, there is some constants $C$ depending only on $n, \\Lambda , p, M_0, R_0$ and $\\text{diam}(\\Omega )$ such that $ \\left\\Vert \\nabla u\\right\\Vert _{L^p(\\Omega , \\mu )} \\le C\\left\\Vert \\frac{{\\bf F}}{\\mu }\\right\\Vert _{L^p(\\Omega , \\mu )}.$ Some comments regarding the results in Theorem REF and Theorem REF are in order.",
"Remark 2.8 (i) The weighted gradient regularity results in the above theorems are a natural generalization of similar results obtained in [6], [11], [24], [17] for uniformly elliptic equations to equations of type (REF ) with degenerate/singular coefficients satisfying (REF ).",
"The case $p =2$ , the existence and uniqueness of weak solution of (REF ) in $W^{1,2}_0(\\Omega , \\mu )$ is already obtained in [14], [35].",
"(ii) Equation (REF ) with (REF ) is invariant under the scaling: $\\mathbb {A}\\rightarrow \\mathbb {A}/\\lambda $ , $\\mu \\rightarrow \\mu /\\lambda $ , ${\\bf F} \\rightarrow {\\bf F}/\\lambda $ , with $\\lambda >0$ .",
"Therefore, by a simple scaling argument, we see that the usual mean oscillation smallness condition on $\\mathbb {A}$ in the classical John-Nirenberg BMO norm, i.e.",
"the smallness requirement on $[\\mathbb {A}]_{\\textup {BMO}(\\Omega , dx)}$ , as in [3], [4], [5], [6], [11], [16], [25], [26], [37] is not the right setting for equation (REF ) with condition (REF ).",
"(iii) Theorem REF will be proved first for the case $p>2$ and then use a duality argument for the case $1 < p < 2$ .",
"When $p > 2$ , it is enough to assume $\\mu \\in A_{2}$ since $A_2 \\cap A_p = A_2$ by the monotonicity of the Muckenhoupt classes.",
"In this case, we already know that a unique solution in $W^{1,2}_0(\\Omega , \\mu )$ already exists by [14].",
"The main concern is thus obtaining the estimate (REF ).",
"Once we have the estimate we may then apply [42] to conclude that $u\\in W^{1,p}_0(\\Omega , \\mu )$ .",
"When $1 < p < 2$ , the requirement on $\\mu $ reduces to being in $A_{p}$ and is needed to apply Poincaré's inequality in the weighted space $W^{1,p}_{0}(\\Omega ,\\mu )$ .",
"Finally, we conclude this subsection by indicating that our implementation of the approximation method of Caffarelli and Peral in [7] is influenced by the recent work [4], [5], [6], [25], [26], [36], [37], [43].",
"The main idea in the approach is to locally consider the equation (REF ) as the perturbation of an equation for which the regularity of its solution is well understood.",
"Key ingredients include Vitali's covering lemma, and the weak, strong $(p, p)$ estimates of the weighted Hardy-Littlewood maximal operators.",
"To be able to compare the solutions of the perturbed and un-pertured equations, we prefer to use compactness argument as in [3], [16], [25], [26], [36], [37], but on weighted spaces, since this method could be more suitable when working with nonlinear equations as in [36], [37] and non-smooth domains as in [3], [5], [6], [25], [26].",
"Essential properties of $A_2$ weights such as reverse Hölder's inequality and doubling property are properly utilized in dealing with technical issues arising from the degeneracy and singularity of the coefficient $\\mathbb {A}$ ."
],
[
"Counterexamples and examples",
"This section contains two examples.",
"The first example is a counterexample to demonstrate that solutions to degenerate homogeneous equations even with uniformly continuous coefficients $\\mathbb {A}$ do not necessarily have gradient with high $\\mu $ -integrability.",
"This example also justifies the necessity of having the smallness of the mean oscillation with $\\mu $ for $\\mathbb {A}$ .",
"The second example characterizes a class of coefficients for which our Theorem REF and Theorem REF apply.",
"This example also provides the required rates of degenerate or singular of the coefficient $\\mathbb {A}$ for the validity of the Sobolev's regularity theory of weak solutions of (REF ).",
"(i) A counterexample: Let $n \\ge 3$ , $\\alpha = \\frac{1}{n+1}$ , and $\\mu (x)= |x|^{2(\\alpha +1)}$ for $x \\in \\mathbb {R}^n$ .",
"Note that since $n \\ge 3$ , we have $ 2 (\\alpha +1)=\\frac{2(n+2)}{n+1}~<~n$ .",
"Therefore, $\\mu \\in A_2$ .",
"Also, with an $n\\times n$ identity matrix $I_n$ , we consider $ \\mathbb {A}(x) = \\mu (x) I_n, \\quad u(x)=\\frac{x_1}{|x|^{2\\alpha }}, \\quad x = (x_1, x_2, \\cdots , x_n) \\in \\Omega := B_1(0) \\subset \\mathbb {R}^n.",
"$ It is clear that $u \\in L^1(\\Omega ) \\cap L^2(\\Omega , \\mu )$ .",
"Moreover, by simple calculation, we see that the weak derivatives of $u$ are $\\begin{split}u_{x_1} & = \\frac{(1-2\\alpha ) x_1^2 +x_2^2 + \\cdots + x_n^2}{|x|^{2(\\alpha +1)}}, \\quad \\text{and} \\\\u_{x_k} & = - \\frac{2\\alpha x_1 x_k }{|x|^{2(\\alpha +1)}}, \\quad k =2,3,\\cdots , n.\\end{split}$ A simple calculation also shows that $u_{x_k} \\in L^1(\\Omega ) \\cap L^2(\\Omega , \\mu )$ for all $k =1,2,\\cdots , n$ and that $u$ is a weak solution of $\\text{div}[\\mathbb {A} \\nabla u] =0, \\quad \\text{in} \\quad \\Omega .$ Indeed, for every $\\varphi \\in C_0^\\infty (\\Omega )$ , we see that $\\begin{split}\\int _\\Omega \\langle \\mathbb {A}\\nabla u, \\nabla \\varphi \\rangle dx & =\\int _\\Omega [(1-2\\alpha ) x_1^2 +x_2^2 + \\cdots + x_n^2] \\varphi _{x_1} dx - 2\\alpha \\sum _{k=2}^n\\int _{\\Omega } x_1 x_k \\varphi _{x_k} dx \\\\& = - \\Big [1 - (n+1) \\alpha \\Big ]\\int _\\Omega \\varphi (x) x_1 dx =0.\\end{split}$ However, $\\int _\\Omega |\\nabla u|^p \\mu (x) dx \\approx C(\\alpha ) \\int _{B_1} |x|^{2\\alpha + 2 - 2\\alpha p} dx< \\infty $ if and only if $p < \\frac{2\\alpha + n + 2}{2\\alpha }.$ Remark 2.9 In this example, $\\mathbb {A}$ is uniformly continuous in $\\overline{B}_1(0)$ .",
"Therefore, it is in the Sarason $\\textup {VMO}(B_1(0))$ space.",
"In light of this and compared to [12], Theorem REF and Theorem REF give the right conditions on $\\mathbb {A}$ so that (REF ) holds.",
"(ii) Examples of coefficients with small mean oscillation with weights: In this example, we use the standard $A_2$ weight $\\mu (x) = |x|^\\alpha $ and $\\mathbb {A}(x) = \\mu (x) I_n$ .",
"We show that if $|\\alpha |$ is sufficiently small, then so is the mean oscillation of $\\mathbb {A}$ with $\\mu $ .",
"The proof of the next lemma is given in the appendix.",
"Lemma 2.10 Let $\\mu (x) = |x|^{\\alpha }$ for $x \\in \\mathbb {R}^n$ and $|\\alpha | \\le 1$ .",
"Then we have that (i) $\\mu \\in A_{2}$ and $[\\mu ]_{A_{2}} \\le M_{0} = M_{0}(n), and$ (ii) $\\int _{B_{r} (x_0)} \\Big | \\mu (x) - \\langle \\mu \\rangle _{B_{r}(x_0)}\\Big | dx \\le \\frac{2|\\alpha |4^{2n+1} }{2n-1} \\int _{B_r(x_0)} \\mu (x) dx, \\quad \\forall x_0 \\in \\mathbb {R}^n, \\quad \\forall r >0.$ Now, for a given $\\delta >0$ , the next lemma shows that there is $\\alpha _0 >0$ such that $\\mu \\in \\mathbb {A}\\in \\mathcal {A}_{R_{0}} (\\delta , 1,\\mu ,B_{1})$ for any $\\alpha \\in (-\\alpha _{0}, \\alpha _0)$ , and for every $R_0 >0$ .",
"Lemma 2.11 There exists a constant $C(n)$ such that if $|\\alpha | \\le 1$ , then $ [\\mathbb {A}]_{\\textup {BMO}(B_1(0), \\mu )}^2\\le C(n) |\\alpha |.",
"$ Observe from the proof of [32] for each $\\mu \\in A_2$ with $[\\mu ]_{A_2} \\le M_0$ , there exists a constant $C~=~C(M_0, n)$ such that $ \\begin{split}& \\sup _{B_r(x) \\subset \\mathbb {R}^n} \\frac{1}{\\mu (B_r(x))} \\int _{B_r(x)} |\\mathbb {A}(y) -\\langle \\mathbb {A} \\rangle _{B_r(x)}|^2 \\mu (y)^{-1} dy \\\\& \\le C(M_0, n) \\sup _{B_r(x) \\subset \\mathbb {R}^n} \\frac{1}{\\mu (B_r(x))} \\int _{B_r(x)} |\\mathbb {A}(y) -\\langle \\mathbb {A} \\rangle _{B_r(x)}| dy.\\end{split}$ Therefore, it follows from Lemma REF , and (REF ) that if $|\\alpha | \\le 1$ , then $ [\\mathbb {A}]^{2}_{\\textup {BMO}(B_1(0), \\mu )}\\le C(n) |\\alpha |.$ Remark 2.12 When $\\alpha <0$ , the coefficient $\\mathbb {A}= \\mu I_{n}$ is not in $L^p(B_1)$ for large $p$ , and so it does not belong to the standard John-Nirenberg BMO space.",
"Theorem REF , therefore, captures an important case that is not covered in many known work such as [3], [4], [5], [6], [8], [11], [16], [25], [26], [37], [24] in which the requirement that $\\mathbb {A}$ is sufficiently small in the John-Nirenberg $\\textup {BMO}$ is essential."
],
[
"Preliminaries on weights and weighted norm inequalities",
"This section reviews and proves some basic results related to $A_{p}$ weights which are needed later for the proofs of Theorem REF and Theorem REF .",
"We first state a result that follows from standard measure theory (see for example [26]).",
"Lemma 3.1 Assume that $g\\ge 0$ is a measurable function in a bounded subset $U\\subset \\mathbb {R}^{n}$ .",
"Let $\\theta >0$ and $\\varpi >1$ be given constants.",
"If $\\mu $ is a weight in $\\mathbb {R}^{n}$ , then for any $1\\le p < \\infty $ $g\\in L^{p}(U,\\mu ) \\Leftrightarrow S:= \\sum _{j\\ge 1} \\varpi ^{pj}\\mu (\\lbrace x\\in U: g(x)>\\theta \\varpi ^{j}\\rbrace ) < \\infty .$ Moreover, there exists a constant $C>0$ such that $C^{-1} S \\le \\Vert g\\Vert ^{p}_{L^{p}(U,\\mu )} \\le C (\\mu (U) + S),$ where $C$ depends only on $\\theta , \\varpi $ and $p$ .",
"For a given locally integrable function $f$ we define the weighted Hardy-Littlewood maximal function as $\\mathcal {M}^{\\mu }f(x) = \\sup _{\\rho > 0}_{B_{\\rho }(x)}|f| d\\mu = \\sup _{\\rho > 0} \\frac{1}{\\mu (B_{\\rho } (x)) }\\int _{B_{\\rho }(x)}|f| \\, \\mu (x) dx.$ For functions $f$ that are defined on a bounded domain, we define $\\mathcal {M}_{\\Omega }^{\\mu }f(x) = \\mathcal {M}^{\\mu }(f\\chi _{\\Omega })(x).$ Recall the Muckenhoupt class $A_{p}$ defined in the previous section.",
"For $1 < p < \\infty ,$ $A_{p}$ weights have a doubling property.",
"For any $\\mu \\in A_{p}$ , any ball $B$ and a measurable set $E\\subset B$ we have that $\\mu (B) \\le [\\mu ]_{A_{p}} \\left(\\frac{|E|}{|B|}\\right)^{p} \\mu (E)$ As a doubling measure, they also imply the boundedness of the Hardy-Littlewood maximal operator.",
"Since we mostly use $A_2$ -weights in this paper, we state the result for $A_2$ -weights in the following lemma, which is a simpler version of a classical, more general result that can be found in [18].",
"Lemma 3.2 Assume that $\\mu \\in A_2$ with $[\\mu ]_{A_2}\\le M_0$ .",
"Then, the followings hold.",
"(i) Strong $(p,p)$ : Let $1 < p < \\infty $ , then there exists a constant $C = C(M_0, n,p)$ such that $ \\Vert \\mathcal {M}^{\\mu }\\Vert _{L^{p}({\\mathbb {R}}^n, \\mu ) \\rightarrow L^{p}({\\mathbb {R}}^n, \\mu )} \\le C. $ (ii) Weak $(1,1)$ : There exists a constant $C=C(M_0, p, n)$ such that for any $\\lambda >0$ , we have $\\mu (x\\in \\mathbb {R}^n: \\mathcal {M} ^{\\mu } (f) > \\lambda ) \\le \\frac{C}{\\lambda } \\int _{\\mathbb {R}^{n}}|f|d\\mu .$ From the definition of $A_2$ -weights, it is immediate that $\\mu \\in A_2$ , then so is $\\mu ^{-1}$ with $ [\\mu ]_{A_2} = [\\mu ^{-1}]_{A_2}.",
"$ $A_{2}$ weights satisfy the so called reverse Hölder's inequality.",
"The statement of the following lemma and its proof can be found in [18].",
"Lemma 3.3 For any $M_{0} > 0$ , there exist positive constants $C = C (n, M_{0})$ and $\\gamma = \\gamma (n, M_{0})$ such that for all $\\mu \\in A_2$ satisfying $[\\mu ]_{A_{2}} \\le M_{0}$ , the following reverse Hölder conditions hold: $\\begin{split}& \\left( \\frac{1}{|B|} \\int _{B} \\mu ^{(1+\\gamma )}(x) dx \\right)^{\\frac{1}{1+\\gamma }} \\le \\frac{C}{|B|} \\int _{B} \\mu (x) dx,\\end{split}$ for every ball $B \\subset \\mathbb {R}^n$ .",
"In particular, the inequality is also valid for $\\mu ^{-1}$ .",
"Lemma REF implies the following inequalities which will be used in this paper frequently.",
"Lemma 3.4 Let $M_{0} > 0$ , let $\\gamma $ be the constant as given in Lemma REF .",
"Then for any $\\mu \\in A_{2}$ satisfying $[\\mu ]_{A_{2}} \\le M_{0}$ and any ball $B \\subset \\mathbb {R}^n$ we have the following.",
"(i) If $u \\in L^2(B, \\mu )$ , then $u \\in L^{1+\\beta }(B)$ where $\\beta = \\frac{\\gamma }{2+ \\gamma } >0$ .",
"Moreover, $\\left(_{B} |u|^{1+\\beta } dx \\right)^{\\frac{1}{1+\\beta }} \\le C(n, M_0) \\left( _{B} |u|^2 d\\mu \\right)^{1/2}.$ (ii) If $u \\in L^q(B)$ for some $q \\ge 1$ , then $u\\in L^{\\tau }(B, \\mu )$ , where $\\tau = \\frac{q \\gamma }{1+\\gamma }.",
"$ Moreover, $\\left( _{B}|u|^{\\tau }d \\mu \\right)^{1/\\tau } \\le C(n, M_0) \\Bigg ( _{B} |u|^q dx \\Bigg )^{1/q}.$ Both estimates follow from Hölder's inequality and Lemma REF .",
"We will demonstrate only (ii).",
"$\\begin{split}_{B}|u |^{\\tau }d \\mu & \\le \\frac{1}{\\mu (B)} \\Bigg ( \\int _{B} | u|^{\\tau (1 + 1/\\gamma )} dx \\Bigg )^{\\frac{\\gamma }{1 + \\gamma }} \\Bigg (\\int _{B} \\mu ^{1 + \\gamma }dx \\Bigg )^{\\frac{1}{1 + \\gamma }}= \\frac{1}{\\mu (B)} \\Bigg ( \\int _{B} |u|^{\\tau (1 + 1/\\gamma )} dx \\Bigg )^{\\frac{\\gamma }{1 + \\gamma }} \\Bigg (_{B} \\mu ^{1 + \\gamma }dx \\Bigg )^{\\frac{1}{1 + \\gamma }} |B|^{\\frac{1}{1 + \\gamma }}.\\end{split}$ We now apply Lemma REF to obtain the estimate $\\begin{split}_{B}|u |^{\\tau }d \\mu & \\le C(n, M_{0}) \\frac{1}{\\mu (B)} \\Bigg ( \\int _{B} |u|^{\\tau (1 + 1/\\gamma )} dx \\Bigg )^{\\frac{\\gamma }{1 + \\gamma }} \\frac{\\mu (B)}{|B|}|B|^{\\frac{1}{1 + \\gamma }} \\\\&= C(n, M_{0}) |B|^{\\frac{-\\gamma }{1 + \\gamma }} \\Bigg ( \\int _{B} | u|^{\\tau (1 + 1/\\gamma )} dx \\Bigg )^{\\frac{\\gamma }{1 + \\gamma }} \\\\&= C(n, M_0) \\Bigg ( _{B} | u|^q dx \\Bigg )^{\\tau / q},\\end{split}$ and the proof of (ii) is complete.",
"We remark that given $M_{0} > 0$ , there exist constants $\\varrho = \\varrho (n, M_{0}) \\in (0, 1)$ and $C = C(n,M_{0}) > 0$ such that for any ball $B\\subset \\mathbb {R}^{n}$ , a measurable subset $E\\subset B$ and any $\\mu \\in A_{2}$ with $[\\mu ]_{A_{2}} \\le M_{0}$ we have $\\mu (E) \\le C \\left(\\frac{|E|}{|B|}\\right)^{\\varrho } \\mu (B).$ This follows from Lemma REF by taking $u = \\chi _{E}$ in (ii).",
"Next, we recall the weighted Sobolev-Poincaré inequality which can be found in [14].",
"Lemma 3.5 Let $M_0 >0$ and assume that $\\mu \\in A_2$ and $[\\mu ]_{A_2} \\le M_0$ .",
"Then, there exists a constant $C = C(n, M_0)$ and $\\alpha = \\alpha (n, M_0)>0$ such that for every ball $B \\subset \\mathbb {R}^n$ of radius $r$ , and every $u \\in W^{1,2}(B, \\mu )$ , $ 1 \\le \\varsigma \\le \\frac{n}{n-1} + \\alpha $ , the following estimate holds $\\left(\\frac{1}{\\mu (B)} \\int _{B} |u - A|^{2\\varsigma } \\mu (x) dx \\right)^{\\frac{1}{2\\varsigma }} \\le C \\,r\\, \\left(\\frac{1}{\\mu (B)} \\int _{B} |\\nabla u|^2 \\mu (x) dx \\right)^{1/2},$ where either $A = \\frac{1}{\\mu (B)} \\int _{B} u(x) d\\mu (x), \\quad \\text{or} \\quad A = \\frac{1}{|B|} \\int _{B} u(x) dx.$ Finally, we state a technical lemma which is a consequence of Vitali's covering lemma.",
"The proof can be found in [25].",
"Lemma 3.6 Let $\\Omega $ be a $(\\delta , R)$ Reifenberg flat domain with $\\delta < 1/4$ and let $\\mu $ be an $A_{p}$ weight for some $p>1$ .",
"Let $r_{0}>0$ be a fixed number and $C\\subset D \\subset \\Omega $ be measurable sets for which there exists $0<\\epsilon <1$ such that (i) $\\mu (C) < \\epsilon \\mu (B_{r_{0}}(y)) $ for all $y\\in \\overline{\\Omega }$ , and (ii) for all $x\\in \\Omega $ and $\\rho \\in (0, 2r_{0}]$ , if $\\mu (C\\cap B_{\\rho }(x)) \\ge \\epsilon \\mu (B_{\\rho }(x))$ , then $B_{\\rho }(x)\\cap \\Omega \\subset D. $ Then we have the estimate $\\mu (C) \\le \\epsilon \\left(\\frac{10}{1 - 4\\delta }\\right)^{np} [\\mu ]^{2}_{p} \\,\\mu (D).$"
],
[
"Interior estimate setup",
"This section focuses on obtaining estimates for the gradient of solution to $ \\text{div}[\\mathbb {A}(x) \\nabla u] = \\text{div}[{\\bf F}] \\quad \\text{in} \\quad B_{4}.$ For a weak solution $u \\in W^{1,2}(B_4, \\mu )$ of (REF ), our aim is to obtain estimates that approximate the gradient $\\nabla u$ via a gradient of a solution to an associated homogeneous equation with constant coefficients.",
"To that end, we will find a constant elliptic, and symmetric matrix $\\mathbb {A}_{0}$ sufficiently close to $\\mathbb {A}(x)$ in an appropriate sense such that the weak solution $v$ of the equation $ \\begin{array}{ccll}\\text{div}[ \\mathbb {A}_{0} \\nabla v] &= & 0 & \\quad \\text{in} \\quad B_4,\\end{array}$ will be used in the comparison estimate.",
"Recall that $\\mathbb {A}: B_6 \\rightarrow \\mathbb {R}^{n\\times n}$ is measurable and symmetric satisfying the degenerate ellipticity condition: $ \\Lambda \\mu (x) |\\xi |^2 \\le \\langle \\mathbb {A}(x) \\xi , \\xi \\rangle \\le \\mu (x)\\Lambda ^{-1}, \\quad \\text{for a.e.\\ } x\\in B_4, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ for some fixed $\\Lambda >0$ and $\\mu \\in A_2$ .",
"For a given $ M_0> 0$ , we assume that $ [\\mu ]_{A_2} \\le M_0.$ Throughout the section, $\\gamma >0$ is the number defined in Lemma REF which depends only on $M_0$ and $n$ , and let $\\beta $ be as $ \\beta = \\frac{\\gamma }{2+\\gamma } >0.$ For now, we refer the readers to Definition REF for the definitions of weak solutions for the equations (REF ) and (REF ).",
"The following well-known result on regularity for weak solutions of linear elliptic equations with constant coefficient is also needed.",
"Lemma 4.1 Let $\\mathbb {A}_{0}$ be an elliptic and symmetric constant $n\\times n$ matrix such that there are positive numbers $\\Lambda _0, \\lambda _0$ such that $\\lambda _0 |\\xi |^2 \\le \\langle \\mathbb {A}_{0}\\xi , \\xi \\rangle \\le \\Lambda _0 |\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ Then, if for some $1 < p < \\infty $ , $v \\in W^{1,p}(B_{4})$ with is a weak solution of (REF ), then $\\left\\Vert \\nabla v\\right\\Vert _{L^\\infty (B_{7/2})} \\le C(n,p, \\Lambda _0/\\lambda _0) \\left[_{B_{4}} |\\nabla v|^p dx \\right]^{1/p}.$ Note that if $p \\ge 2$ , then $v$ is the energy solution and the lemma is the standard regularity result.",
"On the other hand, if $ 1 < p < 2$ , then it follows from [2] that the solution $v$ is in $W^{1,2}(B_r)$ for every $0 < r < 4$ .",
"From this, our lemma again follows by the classical regularity estimates, see [23] for example."
],
[
"Interior weighted Caccioppoli estimate ",
"The main result in this subsection is the following energy estimate for the difference $u - v$ .",
"Lemma 4.2 Let $\\Lambda >0$ , $M_{0}>0$ be given.",
"Let $\\mathbb {A}_{0}$ be an elliptic symmetric constant matrix.",
"Assume that (REF ) holds for some $\\mu \\in A_{2}$ with $[\\mu ]_{A_{2}} \\le M_{0}$ , $u \\in W^{1,2}(B_4, \\mu )$ is a weak solution of (REF ) and for some $q\\in (1, \\infty )$ , $v \\in W^{1,q}(B_4)$ is a weak solution of (REF ).",
"Define $w = u - \\langle u \\rangle _{\\mu , B_4}- v$ .",
"Then there exists a constant $C = C(n, \\Lambda , M_0)$ such that for any $\\varphi \\in C_{0}^{\\infty }(B_{4})$ , $\\begin{split}\\int _{B_4} |\\nabla w|^2\\varphi ^2(x) d\\mu & \\le C \\left[\\int _{B_4} \\Big | \\frac{{\\bf F}}{\\mu }\\Big |^{2} \\varphi ^2 d \\mu + (1 + \\left\\Vert \\varphi \\nabla v\\right\\Vert _{L^\\infty (B_4)}^2)\\int _{B_4} w^2 |\\nabla \\varphi |^2 d\\mu \\right.",
"\\\\& \\quad \\quad + \\left.\\left\\Vert \\varphi \\nabla v\\right\\Vert _{L^\\infty (B_4)}^2 \\int _{B_4} |\\mathbb {A}(x) -\\mathbb {A}_{0}|^2 \\mu ^{-1} dx\\right].\\end{split}$ Note that since $\\mathbb {A}_{0}$ is elliptic and constant, $v \\in C^\\infty _{\\text{loc}}(B_4)$ as in Lemma REF .",
"Hence, $w \\in W^{1,2}(\\Omega ^{\\prime }, \\mu )$ for every $\\Omega ^{\\prime } \\subset \\subset B_4$ .",
"Also, note that $w$ is a weak solution of the equation $\\text{div}[ \\mathbb {A} \\nabla w] = \\text{div}\\Big [ {\\bf F} - (\\mathbb {A} - \\mathbb {A}_{0}) \\nabla v \\Big ] \\quad \\text{in} \\quad B_4.$ By using $w\\varphi ^2$ as a test function for this equation, we obtain $\\begin{split}\\int _{B_4} \\langle \\mathbb {A}\\nabla w, \\nabla w \\rangle \\varphi ^2 dx & = -\\int _{B_4} \\langle \\mathbb {A}\\nabla w, \\nabla (\\varphi ^2) \\rangle w dx + \\int _{B_4} \\langle {\\bf F}, \\nabla (w\\varphi ^2) \\rangle dx\\\\& \\quad \\quad - \\int _{B_4} \\langle (\\mathbb {A}-\\mathbb {A}_{0}) \\nabla v, \\nabla (w\\varphi ^2) \\rangle dx.\\end{split}$ We then have the following estimate $ \\begin{split}\\left| \\int _{B_4} \\langle \\mathbb {A}\\nabla w, \\nabla w \\rangle \\varphi ^2 dx \\right| & \\le \\left|\\int _{B_4} \\langle \\mathbb {A}\\nabla w, \\nabla (\\varphi ^2) \\rangle w dx \\right| + \\int _{B_4} |{\\bf F}| \\left(|\\nabla w| |\\varphi |^2 + 2 |\\nabla \\varphi | |\\varphi ||w| \\right)dx \\\\& + \\int _{B_4} |(\\mathbb {A}-\\mathbb {A}_{0})| \\left(|\\nabla v| |\\nabla w| |\\varphi |^2+ 2|\\nabla v| |\\nabla \\varphi | |w| |\\varphi | \\right)dx.\\end{split}$ Using the ellipticity condition (REF ), we can estimate the term on the left hand side of (REF ) as $\\Lambda \\int _{B_4} |\\nabla w|^2 \\varphi ^2(x) d\\mu \\le \\int _{B_4} \\langle \\mathbb {A}\\nabla w, \\nabla w \\rangle \\varphi ^2 dx.$ For $\\epsilon >0$ , using (REF ) again, the first term on the right hand side can be estimated as $\\begin{split}\\left|\\int _{B_4} \\langle \\mathbb {A}\\nabla w, \\nabla (\\varphi ^2) \\rangle w dx \\right| &\\le 2 \\Lambda ^{-1} \\int _{B_4} |\\nabla w| |\\nabla \\varphi | |w| \\mu \\varphi dx\\\\&\\le \\epsilon \\int _{B_4} |\\nabla w|^2 |\\varphi |^2 d\\mu + C(\\Lambda , \\epsilon ) \\int _{B_4} | \\nabla \\varphi |^2 |w|^2 d\\mu \\end{split}$ where we have applied Hölder's inequality and Young's inequality.",
"The second term on the right hand side in (REF ) can be estimates as $\\begin{split}\\int _{B_4} |{\\bf F}| \\left(|\\nabla w| |\\varphi |^2 + 2 |\\nabla \\varphi | |\\varphi ||w| \\right)dx& \\le \\int _{B_4} \\frac{|{\\bf F}|}{\\mu } \\left(|\\nabla w| |\\varphi |^2 + 2 |\\nabla \\varphi | |\\varphi ||w| \\right) \\mu dx \\\\& \\le \\epsilon \\int _{B_4} |\\nabla w|^2 \\varphi ^2 d\\mu + C(\\epsilon ) \\left[ \\int _{B_4} \\Big |\\frac{{\\bf F}}{\\mu }\\Big |^2 \\varphi ^2 d\\mu + \\int _{B_4} w^2|\\nabla \\varphi |^2 d\\mu \\right].\\end{split}$ Finally, to estimate the third term in the right hand side of (REF ), we apply Hölder's inequality followed by Young's inequality as $\\begin{split}\\int _{B_4} |\\mathbb {A}-\\mathbb {A}_{0}|& | \\nabla v| \\Big [ \\varphi ^2|\\nabla w| + 2|w|\\varphi || \\nabla \\varphi | \\Big ] dx \\\\&\\le \\left\\Vert \\varphi \\nabla v \\right\\Vert _{L^\\infty (B_4)}\\int _{B_4} |\\mathbb {A}-\\mathbb {A}_{0}| \\Big [ \\varphi |\\nabla w| + 2|w| | \\nabla \\varphi | \\Big ] \\mu ^{1/2} \\frac{1}{\\mu ^{1/2}}dx\\\\& \\le \\epsilon \\int _{B_{4}}|\\nabla w|^{2}\\varphi ^{2}(x)d\\mu + C(\\epsilon ) \\left\\Vert \\varphi \\nabla v \\right\\Vert _{L^\\infty (B_4)}^{2} \\left[ \\int _{B_{4}} |\\mathbb {A} -\\mathbb {A}_{0}|^{2} \\mu ^{-1} dx + \\int _{B_{4}} w^{2}|\\nabla \\varphi |^{2}d\\mu \\right].\\end{split}$ Then, collecting all the estimates and choosing $\\epsilon $ sufficiently small to absorb the term containing $ \\int _{B_4} |\\nabla w|^2 \\varphi ^2 d\\mu $ to the left hand side, we obtain the desired result."
],
[
"Interior gradient approximation estimates",
"Our first lemma confirms that we can approximate in $L^2(B_4, \\mu )$ the weak solution $u \\in W^{1,2}(B_4, \\mu )$ of (REF ) by a weak solution $v$ of (REF ) if the coefficient has small mean oscillation with weight $\\mu $ and the data ${\\bf F}$ is sufficiently small relative to the weight.",
"Lemma 4.3 Let $\\Lambda >0, M_0 >0$ be fixed and let $ \\beta $ be as in (REF ).",
"For every $\\epsilon >0$ sufficiently small, there exists $\\delta >0$ depending on only $\\epsilon , \\Lambda , n$ , and $M_0$ such that the following statement holds true: If $\\mathbb {A}, \\mu , {\\bf F}$ such that (REF ) and (REF ) hold, and $\\frac{1}{ \\mu (B_{4})} \\int _{B_{4}} |\\mathbb {A} -\\langle \\mathbb {A}\\rangle _{B_{4}}|^{2} \\mu ^{-1} dx + \\frac{1}{ \\mu (B_{4})} \\int _{B_4} \\Big | \\frac{{\\bf F}}{\\mu } \\Big |^2 d\\mu (x) \\le \\delta ^2,$ a weak solution $u \\in W^{1,2}(B_4, \\mu )$ of (REF ) satisfies $ _{B_4} |\\nabla u|^2 d\\mu \\le 1,$ then, there exists a constant matrix $\\mathbb {A}_{0}$ and a weak solution $v \\in W^{1,1+\\beta }(B_4)$ of (REF ) such that $\\left\\Vert \\langle \\mathbb {A}\\rangle _{B_4} - \\mathbb {A}_{0}\\right\\Vert \\le \\epsilon \\ \\frac{ \\mu (B_{4})}{|B_{4}|},$ and $_{B_{7/2}} |\\hat{u} - v|^2 d\\mu \\le \\epsilon , \\quad \\text{where} \\quad \\hat{u} = u -\\langle u \\rangle _{\\mu , B_{4}}, \\quad \\langle u \\rangle _{\\mu , B_{4}} = _{B_{4}} u(x) d\\mu .$ Moreover, there is $C =C(\\Lambda , n, M_0)$ such that $ _{B_{3}}|\\nabla v|^2dx \\le C(\\Lambda , n, M_0).$ Note that for each $\\lambda >0$ , we can use the scaling $\\mathbb {A}_{\\lambda } = \\frac{1}{\\lambda }\\mathbb {A}$ , $\\mu _\\lambda = \\mu /\\lambda $ and ${\\bf F}_\\lambda = {\\bf F}/\\lambda $ , then a weak solution $u$ of (REF ) will also be a weak solution to $\\text{div}[\\mathbb {A}_\\lambda \\nabla u] = \\text{div} [{\\bf F}_\\lambda ] \\quad \\text{in} \\quad B_4.$ Moreover, $[\\mu _\\lambda ]_{A_2} = [\\mu ]_{A_2}$ $\\Lambda \\mu _{\\lambda }(x) |\\xi |^2 \\le \\langle \\mathbb {A}_\\lambda (x) \\xi , \\xi \\rangle \\le \\Lambda ^{-1}\\mu _{\\lambda }(x)|\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n, \\quad \\text{for a.e.",
"\\ } x \\in B_4,$ and Lemma REF is invariant with respect to this scaling.",
"Therefore, without loss of generality, we can prove Lemma REF with the additional assumption that $ \\langle \\mu \\rangle _{B_4} = \\frac{1}{|B_4|}\\int _{B_4} \\mu (x) dx =1.$ In this case, it follows from (REF ) and (REF ) that $ \\Lambda |\\xi |^2 \\le \\langle \\langle \\mathbb {A}\\rangle _{B_4}\\xi , \\xi \\rangle \\le \\Lambda ^{-1} |\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ To proceed, we use a contradiction argument.",
"Suppose that there exists $\\epsilon _0 >0$ such that corresponding to $k \\in \\mathbb {N}$ , there are $\\mu _k \\in A_2$ , $ \\mathbb {A}_k$ satisfying the degenerate ellipticity assumption as in (REF ) with $\\mu $ and $\\mathbb {A}$ are replaced by $\\mu _k$ and $\\mathbb {A}_k$ respectively, and ${\\bf F}_k$ and a weak solution $u_k \\in W^{1, 2}(B_{4}, \\mu _{k})$ of $ \\text{div}[\\mathbb {A}_k \\nabla u_k] = \\text{div}({\\bf F}_k) \\quad \\text{in }\\, B_4,$ satisfying $ \\left\\lbrace \\begin{split}& \\frac{1}{ \\mu _k(B_{4})} \\int _{B_4} |\\mathbb {A}_k -\\langle \\mathbb {A}_{k}\\rangle _{B_{4}}|^{2}\\mu _{k}^{-1} dx + _{B_4} \\Big |\\frac{{\\bf F}_k}{\\mu _k} \\Big |^2 d\\mu _k (x) \\le \\frac{1}{k^2}, \\\\& [\\mu _k]_{A_2} \\le M_0, \\quad \\langle \\mu \\rangle _{k, B_4} = \\frac{1}{|B_4|} \\int _{B_4} \\mu _k(x) dx = 1,\\end{split} \\right.$ with $ _{B_4} |\\nabla u_k|^2 d\\mu _k \\le 1,$ but for all constant matrix $\\mathbb {A}_{0}$ with $\\Vert \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}- \\mathbb {A}_{0} \\Vert \\le \\epsilon _{0},$ and all weak solution $v \\in W^{1, 1+\\beta }(B_4)$ of (REF ), we have $ _{B_{7/2}} |\\hat{u}_{k} - v|^2 d\\mu _k \\ge \\epsilon _0, \\quad \\text{with} \\quad \\hat{u}_k =u_k -\\langle u \\rangle _{k,\\mu _k, B_{4}}.$ The sequence of constant matrices $\\langle \\mathbb {A}_{k}\\rangle _{B_{4}} $ satisfies an estimate of the type (REF ), and therefore the sequence $\\langle \\mathbb {A}_{k}\\rangle _{B_{4}}$ is a bounded sequence in $\\mathbb {R}^{n\\times n}$ .",
"Thus, by passing through a subsequence, we can assume that there is a constant matrix $\\bar{\\mathbb {A}}$ in $ \\mathbb {R}^{n\\times n}$ such that $ \\lim _{k \\rightarrow \\infty } \\langle \\mathbb {A}_{k}\\rangle _{B_{4}} = \\bar{\\mathbb {A}}.$ From (REF ), and Poincaré-Sobolev inequality Lemma REF , we see that $_{B_4} |\\hat{u}_k|^2 d \\mu _k \\le C(n, M_{0}) _{B_{4}} |\\nabla u_k|^2 d\\mu _k \\le C(n, M_{0}),$ and therefore, for all $k\\in \\mathbb {N}$ , $\\left\\Vert \\hat{u}_k\\right\\Vert _{W^{1,2}(B_4, \\mu _k)} \\le C(n, M_0)$ .",
"As a consequence, Lemma REF implies that $\\left\\Vert \\hat{u}_k\\right\\Vert _{W^{1,1 +\\beta }(B_4)} \\le C(n, M_0) \\left\\Vert \\hat{u}_k\\right\\Vert _{W^{1,2}(B_4, \\mu _k)} \\le C(n, M_0), \\quad \\beta = \\frac{\\gamma }{2 + \\gamma } >0.$ Note that $\\gamma $ is defined in Lemma REF , which only depends on $n$ and $M_0$ .",
"Therefore, by the compact imbedding $W^{1,1+\\beta }(B_4) \\hookrightarrow L^{1+\\beta }(B_4)$ and by passing through a subsequence, we can assume that there is $u \\in W^{1,1+\\beta }(B_4)$ such that $ \\left\\lbrace \\begin{split}& \\hat{u}_k \\rightarrow u \\mbox{ strongly in } L^{1+\\beta }(B_4),\\quad \\nabla u_k \\rightharpoonup \\nabla u \\text{ weakly in } L^{1+\\beta }(B_4),\\ \\quad \\text{and} \\\\& \\hat{u}_{k} \\rightarrow u \\ \\text{a.e.",
"in} \\ \\ B_4.\\end{split} \\right.$ Moreover, $ \\left\\Vert u\\right\\Vert _{W^{1,1+\\beta }(B_4)} \\le C(n, M_0).$ We claim that $u \\in W^{1,1+\\beta }(B_4)$ is a weak solution of $ \\text{div}[\\bar{\\mathbb {A}} \\nabla u] = 0 \\quad \\text{in} \\quad B_4.",
"\\\\$ Let us fix a test function $\\varphi \\in C^\\infty _0({B}_4)$ .",
"Then, by using $\\varphi $ as a test function for the equation (REF ) of $u_k$ , we have $ \\int _{B_4} \\langle \\mathbb {A}_k \\nabla u_k, \\nabla \\varphi \\rangle dx = \\int _{B_4} \\langle {\\bf F}_k, \\nabla \\varphi \\rangle dx.$ We will take the limit $k \\rightarrow \\infty $ on both sides of the above equation.",
"First of all, observe that by Hölder's inequality and (REF ), it follows that the right hand side term of (REF ) can be estimated as $\\begin{split}\\left|_{B_4} {\\bf F}_k \\cdot \\nabla \\varphi dx \\right| & \\le \\left\\lbrace _{B_4} \\Big |\\frac{{\\bf F}_{k}}{\\mu _k}\\Big |^2 \\mu _k dx \\right\\rbrace ^{1/2} \\left\\lbrace _{B_4} |\\nabla \\varphi |^2 \\mu _k dx\\right\\rbrace ^{1/2} \\\\& \\le \\left\\Vert \\nabla \\varphi \\right\\Vert _{L^\\infty (B_4)} \\left\\lbrace \\frac{1}{\\mu _k (B_4)}\\int _{B_4}\\Big |\\frac{{\\bf F}_{k}}{\\mu _k} \\Big |^2d \\mu _k (x) \\right\\rbrace ^{1/2}\\frac{\\mu _{k}(B_{4})}{|B_{4}|}\\\\& \\le \\frac{\\left\\Vert \\nabla \\varphi \\right\\Vert _{L^\\infty (B_4)} }{k} .\\end{split}$ Therefore, taking the limit as $k \\rightarrow \\infty ,$ we have $ \\int _{B_4} \\langle {\\bf F}_k, \\nabla \\varphi \\rangle dx =0.$ On the other hand, it follows from (REF ), (REF ), and Hölder's inequality that $\\begin{split}\\left| _{B_4} \\langle (\\mathbb {A}_k - \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}) \\nabla u_k, \\nabla \\varphi \\rangle dx \\right|& \\le _{B_4} |\\mathbb {A}_k - \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}| |\\nabla u_k| \\mu _k^{1/2} |\\nabla \\varphi | \\mu _k^{-1/2} dx \\\\& \\le \\Vert \\nabla \\varphi \\Vert _{L^{\\infty }(B_{4})} \\left\\lbrace _{B_4} |\\mathbb {A}_k - \\langle \\mathbb {A}_{k}\\rangle _{B_4}|^{2} \\mu _{k}^{-1}dx\\right\\rbrace ^{1/2} \\left\\lbrace \\frac{1}{|B_4|}\\int _{B_4} |\\nabla u_k|^2 d\\mu _k \\right\\rbrace ^{1/2} \\\\&\\le C(n, M_0)\\frac{\\left\\Vert \\nabla \\varphi \\right\\Vert _{L^\\infty (B_4)}}{k} \\left\\lbrace _{B_4} |\\nabla u_k|^2 d\\mu _k \\right\\rbrace ^{1/2} \\left\\lbrace \\frac{1}{|B_4|} \\int _{B_4} \\mu _k(x) dx \\right\\rbrace ^{1/2} \\\\& \\le C(n, M_0)\\frac{\\left\\Vert \\nabla \\varphi \\right\\Vert _{L^\\infty (B_4)}}{k} \\rightarrow 0, \\quad \\text{as} \\quad k \\rightarrow \\infty .\\end{split}$ As a result we have $0 = \\lim _{k\\rightarrow \\infty } \\int _{B_4} \\langle (\\mathbb {A}_k - \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}) \\nabla u_k, \\nabla \\varphi \\rangle dx = \\lim _{k\\rightarrow \\infty } \\Big [\\int _{B_4} \\langle \\mathbb {A}_k \\nabla u_k, \\nabla \\varphi \\rangle dx - \\int _{B_4} \\langle \\langle \\mathbb {A}_{k}\\rangle _{B_{4}} \\nabla u_k, \\nabla \\varphi \\rangle dx\\Big ].$ We also observe that since $\\nabla u_k$ converges weakly in $L^{1 + \\beta }$ from (REF ) and $\\langle \\mathbb {A}_{k}\\rangle _{B_{4}} $ is a strongly converging sequence of constant symmetric matrices, we have that $\\lim _{k\\rightarrow \\infty } \\int _{B_4} \\langle \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}\\nabla u_k, \\nabla \\varphi \\rangle dx = \\int _{B_{4}}\\langle \\bar{\\mathbb {A}} \\nabla u, \\nabla \\varphi \\rangle dx.$ As a consequence we have that $ \\lim _{k \\rightarrow \\infty } \\int _{B_4} \\langle \\mathbb {A}_{k} \\nabla u_k, \\nabla \\varphi \\rangle dx =\\int _{B_4} \\langle \\bar{\\mathbb {A}} \\nabla u, \\nabla \\varphi \\rangle dx.$ Combining (REF ) and (REF ), we see that $\\int _{B_4} \\langle \\bar{\\mathbb {A}}\\nabla u, \\nabla \\varphi \\rangle dx =0, \\quad \\forall \\ \\varphi \\in C^\\infty _0({B}_4).$ Now, from (REF ), and since $\\bar{\\mathbb {A}} = \\lim _{k\\rightarrow \\infty } \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}$ , we observe that $\\Lambda |\\xi |^2 \\le \\langle \\bar{\\mathbb {A}}\\xi , \\xi \\rangle \\le \\Lambda ^{-1} |\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ Hence, Lemma REF implies that $u~\\in ~C^{\\infty }(\\overline{B}_{15/4})$ .",
"In addition, it follows from Lemma REF and (REF ) that $ _{B_{7/2}} |\\nabla u|^2 d\\mu _k \\le \\left\\Vert \\nabla u\\right\\Vert _{L^\\infty (B_{7/2})}^2 \\le C(n, \\Lambda ) \\left( _{B_4} |\\nabla u|^{1+\\beta } dx \\right)^{\\frac{2}{1+\\beta }}\\le C(n, M_0, \\Lambda ), \\quad \\forall \\ k \\in \\mathbb {N}.$ We claim that $\\lim _{k\\rightarrow \\infty } _{B_{7/2}}| \\hat{u}_k - u - c_{k}|^{2} d\\mu _{k} = 0, \\quad \\text{with} \\quad c_k = _{B_{7/2}}[\\hat{u}_k - u ] dx.$ To prove the claim, let us denote $H_{k} = \\hat{u}_k - u - c_k$ .",
"Note that since $\\mu _k \\in A_2$ and $[\\mu _k]_{A_2} \\le M_0$ , it follows from (REF ) and doubling property of $\\mu _{k}$ (REF ) that $_{B_{7/2}} |\\nabla u_k|^2 d\\mu _k \\le \\frac{\\mu _k(B_4)}{\\mu _k(B_{7/2})} _{B_4} |\\nabla u_k|^2 d\\mu _k \\le \\frac{\\mu _k(B_4)}{\\mu _k(B_{7/2})} \\le C(n, M_0), \\quad \\forall \\ k \\in \\mathbb {N}.$ This together with (REF ) yields $ _{B_{7/2}}|\\nabla H_k|^2 d\\mu _k \\le C(n, \\Lambda , M_0), \\quad \\forall \\ k \\in \\mathbb {N}.$ On the other hand, by the weighted Sobolev-Poincaré inequality [14], Lemma REF , there exists $\\varsigma >1$ such that $\\Bigg (_{B_{7/2}} |H_{k}|^{2 \\varsigma }d\\mu _{k}\\Bigg )^{\\frac{1}{2\\varsigma }} \\le C (n, M_{0}) \\Bigg (_{B_{7/2}} |\\nabla H_{k}|^{2}d\\mu _{k}\\Bigg )^{\\frac{1}{2}}.$ Let $\\tau > 0$ be a small number that will be determined later.",
"By Hölder's inequality, we have $\\Bigg (_{B_{7/2}} |H_{k}|^{2}d \\mu _{k} \\Bigg )^{1/2} \\le \\Bigg (_{B_{7/2}} |H_{k}|^{\\tau }d\\mu _{k}\\Bigg )^{\\theta /\\tau } \\Bigg (_{B_{7/2}} |H_{k}|^{2 \\varsigma }d\\mu _{k}\\Bigg )^{\\frac{1-\\theta }{2\\varsigma }},$ with $\\theta = \\frac{ \\frac{1}{2} - \\frac{1}{2\\varsigma } }{\\frac{1}{\\tau }- \\frac{1}{2\\varsigma }} \\in (0,1).$ Therefore, $ \\begin{split}\\Bigg (_{B_{7/2}} |H_{k}|^{2} d\\mu _{k} \\Bigg )^{1/2} &\\le \\Bigg (_{B_{7/2}} |H_{k} |^{\\tau }d\\mu _{k}\\Bigg )^{\\theta /\\tau } \\Bigg (_{B_{7/2}} |H_{k}|^{2 \\varsigma }d\\mu _{k}\\Bigg )^{\\frac{1-\\theta }{2\\varsigma }}\\\\&\\le (C (n, M_{0}))^{1 - \\theta } \\Bigg (_{B_{7/2}} |H_{k}|^{\\tau }d\\mu _{k}\\Bigg )^{\\theta /\\tau } \\Bigg (_{B_{7/2}} |\\nabla H_{k}|^{2}d\\mu _{k}\\Bigg )^{\\frac{1-\\theta }{2}}.\\end{split}$ On the other hand, by Lemma REF , we have $\\begin{split}_{B_{7/2}}|H_{k}|^{\\tau }d \\mu _{k} & \\le C(n, M_{0}) \\Bigg ( _{B_{7/2}} | H_{k}|^{\\tau (1 + 1/\\gamma )} dx \\Bigg )^{\\frac{\\gamma }{1 + \\gamma }}.\\end{split}$ Moreover, observe that $\\begin{split}_{B_{7/2}} |H_{k}|^{\\tau (1 + 1/\\gamma )} dx& \\le _{B_{7/2}} |\\hat{u}_{k} -u|^{\\tau (1 + 1/\\gamma )} dx + _{B_{7/2}} |c_k|^{\\tau (1 + 1/\\gamma )} dx\\\\&\\le 2 _{B_{7/2}} |\\hat{u}_{k} - u|^{\\tau (1 + 1/\\gamma )} dx.\\end{split}$ Now choose $\\tau $ small that $\\tau (1 + 1/\\gamma ) \\le 1 + \\beta $ .",
"We can then apply the strong convergence of $\\hat{u}_{k} \\rightarrow u $ in $L^{1 + \\beta }(B_{4})$ as in (REF ), to conclude that $_{B_{7/2}} |H_{k}|^{\\tau (1 + 1/\\gamma )} dx \\rightarrow 0 \\quad \\quad \\text{as $k \\rightarrow \\infty $.",
"}$ Combining inequalities (REF ), (REF ), (REF ) and (REF ) we obtain that $\\lim _{k\\rightarrow \\infty }_{B_{7/2}} |H_{k}|^{2}d \\mu _{k} = 0.$ This assertion proves our claim.",
"However, note that since $\\langle \\mathbb {A}_{k}\\rangle _{B_{4}} \\rightarrow \\bar{\\mathbb {A}}$ and $\\epsilon _0 >0,$ $\\left\\Vert \\langle \\mathbb {A}_{k}\\rangle _{B_{4}} - \\bar{\\mathbb {A}}\\right\\Vert \\le \\epsilon _0$ for sufficiently large $k$ .",
"But, this contradicts to (REF ) if we take $\\mathbb {A}_{0} = \\bar{\\mathbb {A}}$ , $ v = u - c_k$ and $k$ sufficiently large.",
"We finally prove the estimate (REF ).",
"We assume the existence of $\\beta ,$ $\\mathbb {A}_{0}$ and $v \\in W^{1,1+\\beta }(B_4)$ satisfying the first part of the lemma.",
"Then, with sufficiently small $\\epsilon $ , we can assume that $\\frac{\\Lambda \\mu (B_4) }{2|B_4|} |\\xi |^2 \\le \\langle \\mathbb {A}_{0} \\xi , \\xi \\rangle \\le 2\\frac{ \\Lambda ^{-1} \\mu (B_4)}{|B_4|} |\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ Hence, by the standard regularity theory for elliptic equations, Lemma REF , $v$ is in $C^\\infty (B_{15/4})$ .",
"Moreover, from standard regularity theory, we also have $_{B_{16/5}} |v|^2 dx \\le C(n, \\Lambda ) \\left\\lbrace _{B_{7/2}} |v|^{1 + \\beta } dx\\right\\rbrace ^{\\frac{2}{1+\\beta }}, \\quad \\text{with} \\quad \\beta = \\frac{\\gamma }{2+\\gamma }.$ Then, it follows from Lemma REF that $_{B_{16/5}} |v|^2 dx \\le C(n, \\Lambda , M_0) _{B_{7/2}} |v|^2 d\\mu .$ From this last estimate and the energy estimate for $v$ , we infer that $_{B_{3}} |\\nabla v|^2 dx \\le C(n, \\Lambda , M_0) _{B_\\frac{16}{5}} |v|^2 dx \\le C(n, \\Lambda , M_0) _{B_{7/2}} |v|^2 d\\mu .$ Therefore, $\\begin{split}_{B_3} |\\nabla v|^2 dx & \\le C(n, \\Lambda , M_0) \\left[ _{B_{7/2}} |\\hat{u} - v|^2 d\\mu + _{B_{7/2}} |u - \\langle u \\rangle _{\\mu , B_{4}} | ^2 d\\mu \\right] \\\\& \\le C(n, \\Lambda , M_0) \\left[\\epsilon + \\frac{\\mu (B_4)}{\\mu (B_{7/2})} _{B_{4}} |u - \\langle u \\rangle _{\\mu , B_{4}} | ^2 d\\mu \\right]\\\\& \\le C(n, \\Lambda , M_0) \\left[\\epsilon + \\frac{\\mu (B_4)}{\\mu (B_{7/2})} _{B_4} |\\nabla u|^2 d\\mu \\right],\\end{split}$ where we have used the Poincaré's inequality for weighted Sobolev spaces, Lemma REF .",
"Since $\\epsilon $ is small, we can assume that $\\epsilon <1$ .",
"It then follows from (REF ) and (REF ) that $_{B_3} |\\nabla v|^2 dx \\le C(n, \\Lambda , M_0) \\left[ 1 + \\frac{\\mu (B_4)}{\\mu (B_3)} _{B_4} |\\nabla u|^2 d\\mu \\right] \\le C(n, \\Lambda , M_0).$ This assertion proves the estimate (REF ) and completes the proof of the lemma.",
"The following main result of the section provides the approximation for the gradient of solution by gradient of a homogeneous equation with constant coefficients that is appropriately chosen.",
"Proposition 4.4 Let $\\Lambda >0, M_0 >0$ be fixed and let $ \\beta $ be as in (REF ).",
"For every $\\epsilon >0$ sufficiently small, there exists $\\delta >0$ depending on only $\\epsilon , \\Lambda , n, M_0$ such that the following statement holds true: If (REF ) and (REF ) hold and $\\frac{1}{ \\mu (B_{4})}\\int _{B_4} |\\mathbb {A} -\\langle \\mathbb {A}\\rangle _{B_4}|^{2} \\mu ^{-1} dx + _{B_4} \\Big |\\frac{{\\bf F}}{\\mu } \\Big |^2 d \\mu (x) \\le \\delta ^{2}, $ for every weak solution $u \\in W^{1,2}(B_4, \\mu )$ of (REF ) satisfying $ _{B_4} |\\nabla u|^2 d\\mu \\le 1,$ then, there exists a constant matrix $\\mathbb {A}_{0}$ and a weak solution $v \\in W^{1,1+\\beta }(B_4)$ of (REF ) such that $|\\langle \\mathbb {A}\\rangle _{B_4} - \\mathbb {A}_{0}| \\le \\frac{\\epsilon \\mu (B_{4})}{|B_{4}|} , \\quad \\text{and} \\quad _{B_{2}} |\\nabla u -\\nabla v|^2 d\\mu \\le \\epsilon .$ Moreover, there is $C =C(\\Lambda , n, M_0)$ such that $ _{B_3}|\\nabla v|^2dx \\le C.$ Let $\\alpha >0$ sufficiently small to be determined.",
"By Lemma REF , there exists $\\delta _1 >0$ such that if $\\frac{1}{ \\mu (B_{4})} \\int _{B_4} |\\mathbb {A} -\\langle \\mathbb {A}\\rangle _{B_4}|^{2} \\mu ^{-1} dx + _{B_4} \\Big |\\frac{{\\bf F}}{\\mu } \\Big |^2 d \\mu (x) \\le \\delta _{1}^2, $ and if $u$ is a weak solution of (REF ) satisfying $ _{B_4} |\\nabla u|^2 d\\mu \\le 1,$ there exist a constant matrix $\\mathbb {A}_{0}$ and a weak solution $v$ of (REF ) such that $ |\\mathbb {A}_{0} -\\langle \\mathbb {A}\\rangle _{B_4}| \\le \\alpha \\frac{\\mu (B_{4})}{|B_{4}|} , \\quad _{B_{7/2}}|\\hat{u} - v|^2 d\\mu \\le \\alpha , \\quad \\text{and} \\quad _{B_3} |\\nabla v|^2 dx \\le C(\\Lambda , n, M_0).$ From (REF ) and Lemma REF , we conclude that $ \\left\\Vert \\nabla v\\right\\Vert _{L^\\infty (B_{\\frac{5}{2}})} \\le C(n, \\Lambda , M_0).$ Also, without loss of generality, we can assume that $\\delta _1^2 \\le \\alpha $ .",
"Hence we have that $\\begin{split}\\frac{1}{\\mu (B_{4})}\\int _{B_4} |\\mathbb {A} -\\mathbb {A}_{0}|^{2} \\mu ^{-1} dx& \\le \\frac{2}{\\mu (B_{4})}\\int _{B_4} |\\mathbb {A} -\\langle \\mathbb {A}\\rangle _{B_4}|^{2} \\mu ^{-1} dx + \\frac{2}{\\mu (B_{4})} |\\mathbb {A}_{0} -\\langle \\mathbb {A}\\rangle _{B_4}|^{2} \\int _{B_{4}}\\mu ^{-1}dx\\\\&\\le 2\\delta _{1}^2+ 2 M_{0} \\alpha ^{2} \\le \\alpha C\\end{split}$ From this last estimate, and the estimates (REF )-(REF ), and by applying Lemma REF , we obtain $_{B_{2}} |\\nabla u - \\nabla v|^2 d\\mu \\le \\alpha \\ C.$ where $C $ depends only on $n, \\Lambda ,$ and $M_{0}$ .",
"Thus, if we choose $\\alpha $ such that $\\epsilon = \\alpha \\ C$ , the assertion of lemma follows with $\\delta = \\delta _1$ ."
],
[
"Proof of the interior $W^{1,p}$ -regularity estimates",
"We prove Theorem REF after establishing several estimates for upper-level set of the maximal function $\\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2})$ .",
"We begin with the following lemma.",
"Lemma 4.5 Suppose that $M_{0}>0$ and $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ .",
"There exists a constant $\\varpi = \\varpi (n, \\Lambda , M_0)> 1$ such that the following holds true.",
"Corresponding to any $\\epsilon > 0 $ , there exists a small constant $\\delta = \\delta (\\epsilon , \\Lambda , M_0, n)$ such that if $\\mathbb {A}\\in \\mathcal {A}_{4}(\\delta , \\mu , \\Lambda , B_{1})$ , $u\\in W^{1, 2}(B_{4}, \\mu )$ is a weak solution to $\\textup {div}[\\mathbb {A} \\nabla u] = \\textup {div}({\\bf F})\\quad \\text{in $B_{4}$}, \\quad \\text{and}$ $B_{1}\\cap \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) \\le 1 \\rbrace \\cap \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2}\\chi _{B_{6}}\\right) \\le \\delta ^{2} \\rbrace \\ne \\emptyset ,$ then $\\mu (\\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{2}\\rbrace \\cap B_{1}) < \\epsilon \\mu (B_{1}).$ With a given $\\epsilon > 0$ , let $\\eta >0$ to be chosen later, which is sufficiently small and is dependent only on $\\epsilon $ .",
"Using this $\\eta $ , Lemma REF , and Proposition REF , we can find $\\delta = \\delta (\\eta , \\Lambda , M_0, n)> 0$ such that if $u$ is a weak solution, $\\frac{1}{\\mu (B_{4})} \\int _{B_{4}} |\\mathbb {A} - \\langle \\mathbb {A}\\rangle _{B_{4}}|^{2} \\mu ^{-1}dx \\le \\delta ^2, \\quad _{B_{4}} \\Big | \\frac{{\\bf F}}{\\mu }\\Big |^2 d\\mu (x) \\le \\delta ^2, \\text{and}\\quad _{B_{4}} |\\nabla u|^{2}d\\mu \\le 1$ then there exists a constant matrix $\\mathbb {A}_{0}$ and a weak solution $v$ to $\\text{div} ( \\mathbb {A}_{0}\\nabla v )= 0$ in $B_{4}$ satisfying $|\\langle \\mathbb {A}\\rangle _{B_{4}} - \\mathbb {A}_{0}| < \\eta \\, \\frac{\\mu (B_{4})}{|B_{4}|},\\,\\, _{B_{2}}|\\nabla u-\\nabla v| ^{2} d\\mu < \\eta \\quad \\text{and\\, $\\Vert \\nabla v\\Vert _{L^{\\infty }(B_{2})} \\le C_{0}$},$ for some positive constant $C_{0}$ that depends only $n, \\Lambda $ and $M_{0}$ .",
"Next, by using this $\\delta $ in the assumption (REF ), we can find $x_{0}\\in B_{1}$ such that for any $r > 0$ $_{B_{r}(x_{0})} \\chi _{B_{6}}|\\nabla u|^{2} d\\mu (x) \\le 1,\\quad \\text{and } _{B_{r}(x_{0})} \\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{B_{6}} d\\mu (x) \\le \\delta ^{2}.$ We now make some observations.",
"First, we see that $B_{4} \\subset B_{5}(x_{0})\\subset B_{6}$ and therefore we have from (REF ) and (REF ) that $_{B_{4}} |\\nabla u|^{2} d\\mu (x)\\le \\frac{\\mu (B_{5}(x_0))}{\\mu (B_{4})}_{B_{5}(x_0)} \\chi _{B_{6}}|\\nabla u|^{2} d\\mu (x) \\le M_{0}\\left(\\frac{5}{4}\\right)^{2n},$ and similarly $_{B_{4}} \\left|\\frac{{\\bf F}}{\\mu }\\right|^{2}d\\mu (x) \\le M_{0}\\left(\\frac{5}{4}\\right)^{2n}\\delta ^{2}.$ Denote $\\kappa = M_{0}\\left(\\frac{5}{4}\\right)^{2n} $ .",
"Then since $\\mathbb {A}\\in \\mathcal {A}_{4}(\\delta , \\mu , \\Lambda , B_{1})$ by assumption, the above calculation shows that conditions in (REF ) are satisfied for $u$ replace by $ u_{\\kappa } = u/\\kappa $ and ${\\bf F}$ replaced by ${\\bf F}_{\\kappa } = {\\bf F}/\\kappa $ , where $u_{\\kappa }$ will remain a weak solution corresponding to ${\\bf F}/k$ .",
"So all in (REF ) will be true where $v$ will be replaced by $v_{\\kappa } :=v/\\kappa $ .",
"Second, with $M^{2} = \\max \\lbrace M_{0} 3^{2n}, 4C^{2}_{0} \\rbrace $ , we then claim that $\\lbrace x: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u_{\\kappa }|) ^{2} > M^{2}\\rbrace \\cap B_{1} \\subset \\lbrace x: \\mathcal {M}^{\\mu }(\\chi _{B_{2}}|\\nabla u_{\\kappa } -\\nabla v_{\\kappa }|^{2})>C_{0}^{2}\\rbrace \\cap B_{1}.",
"$ In fact, otherwise there will exist $x\\in B_{1}$ , such that $ \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u_{\\kappa }|) ^{2}(x) > M^2, \\quad \\text{and} \\quad \\mathcal {M}^{\\mu }(\\chi _{B_{2}}|\\nabla u_{\\kappa } -\\nabla v_{\\kappa }|^{2})(x)\\le C_{0}^{2}.$ We obtain a contradiction if we show that for any $r>0$ $_{B_{r}(x)}\\chi _{B_{6}}|\\nabla u_{\\kappa }|^{2}d\\mu \\le M^{2}.$ To that end, on the one hand, if $r \\le 1$ , then $B_{r}(x)\\subset B_{2}$ .",
"Using the fact that $\\Vert \\nabla v_{\\kappa }\\Vert _{L^{\\infty }(B_{r}(x))} \\le \\Vert \\nabla v_{\\kappa }\\Vert _{L^{\\infty }(B_{2})} \\le C_{0}$ , $_{B_{r}(x)}\\chi _{B_{6}}|\\nabla u_{\\kappa }|^{2}d\\mu \\le 2_{B_{r}(x)}\\chi _{B_{2}}|\\nabla u_{\\kappa }-\\nabla v_{\\kappa }|^{2} d\\mu + 2_{B_{r}(x)}|\\nabla v_{\\kappa }|^{2}d\\mu \\le 4 C_{0}^{2}.$ On the other hand, if $r > 1$ , then note first that $B_{r}(x) \\subset B_{3r}(x_0)$ and, so scaling the first inequality in (REF ) by $\\kappa > 1$ we obtain that $_{B_{r}(x)}\\chi _{B_{6}}|\\nabla u_{\\kappa }|^{2}d\\mu (x) \\le \\frac{\\mu (B_{3r}(x_{0}))}{B_{r}(x)}_{B_{3r}(x_0)} \\chi _{B_{6}}|\\nabla u_{\\kappa }|^{2}d\\mu (x) <M_{0}3^{2n}.$ Finally, set $\\varpi = \\kappa M $ .",
"Then since $\\varpi >M $ we have that $\\begin{split}\\mu ( \\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{2} \\rbrace )& \\le \\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}} |\\nabla u_{\\kappa }|^{2}) > M^{2} \\rbrace )\\\\& \\le \\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{2}}|\\nabla u_{\\kappa } -\\nabla v_{\\kappa }|^{2})>C_{0}^{2}\\rbrace )\\\\& \\le \\frac{C(n, M_{0})}{C_{0}^{2}}\\ \\mu (B_{2}) _{B_{2}} |\\nabla u_{\\kappa } - \\nabla v_{\\kappa }|^{2} d\\mu \\\\&\\le C\\ \\eta \\ \\mu (B_{2}) \\\\&\\le C\\ 2^{2n} \\ \\eta \\ \\mu (B_{1}),\\end{split}$ where $C(n, M_{0})$ comes from the weak $1-1$ estimates in the $\\mu $ measure and we have used (REF ).",
"From the last estimate, we observe that if we choose $\\eta > 0$ sufficiently small such that $ C \\,2^{2n} \\eta < \\epsilon $ , Lemma REF follows.",
"By scaling and translating, we can derive the following corollary from Lemma REF .",
"Corollary 4.6 Suppose that $M_{0}>0$ and $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ .",
"There exists a constant $\\varpi = \\varpi (n, \\Lambda , M_0)> 1$ such that the following holds true.",
"Corresponding to any $\\epsilon > 0 $ , there exists a small constant $\\delta = \\delta (\\epsilon , \\Lambda , M_0, n)$ such that for any $\\rho \\in (0, 1), $ $y\\in B_{1}$ , $\\mathbb {A}\\in \\mathcal {A}_{4\\rho }(\\delta , \\mu , \\Lambda , B_{\\rho }(y))$ , if $u\\in W^{1, 2}(B_{6}, \\mu )$ is a weak solution to $\\textup {div}[\\mathbb {A} \\nabla u] = \\textup {div}({\\bf F})\\quad \\text{in $B_{6}$}, \\quad \\text{and}$ $B_{\\rho }(y)\\cap \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) \\le 1 \\rbrace \\cap \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2}\\chi _{B_{6}}\\right) \\le \\delta ^{2} \\rbrace \\ne \\emptyset ,$ then $\\mu (\\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{2}\\rbrace \\cap B_{\\rho }(y)) < \\epsilon \\mu (B_{\\rho }(y)).$ The following statement is the contrapositive of the above corollary.",
"Proposition 4.7 Suppose that $M_{0}>0$ and $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ .",
"There exists a constant $\\varpi =\\varpi (n, \\Lambda , M_0)> 1$ such that the following holds true.",
"Corresponding to any $\\epsilon > 0$ , there exists a small constant $\\delta = \\delta (\\epsilon , \\Lambda , M_0, n)$ such that for any $\\rho \\in (0, 1), $ $y\\in B_{1}$ , $\\mathbb {A}\\in \\mathcal {A}_{4\\rho }(\\delta , \\mu , \\Lambda , B_{\\rho }(y))$ , if $u\\in W^{1, 2}(B_{6}, \\mu )$ is a weak solution to $\\textup {div}[\\mathbb {A} \\nabla u] = \\textup {div}({\\bf F})\\quad \\text{in $B_{6}$}, \\quad \\text{and}$ $\\mu ( \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{2}\\rbrace \\cap B_{\\rho }(y)) \\ge \\epsilon \\mu (B_{\\rho }(y)),$ then $B_{\\rho }(y)\\subset \\lbrace x\\in B_{\\rho }(y): \\mathcal {M}^{\\mu }(\\chi _{B_{6}} |\\nabla u|^{2}) > 1 \\rbrace \\cup \\lbrace x\\in B_{\\rho }(y): \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{B_{6}}\\right) >\\delta ^{2} \\rbrace .$ Our next statement, which is the key in obtaining the higher gradient integrability of solution, gives the level set estimate of $\\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2})$ in terms that of $\\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{B_{6}}\\right)$ .",
"Lemma 4.8 Let $M_{0} >0$ , $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ , and let $\\varpi $ be as in Proposition REF .",
"Then, for every $\\epsilon > 0 $ , there is $\\delta = \\delta (\\epsilon , \\Lambda , M_0, n) < 1/4$ such that the following holds: For $\\mathbb {A}\\in \\mathcal {A}_{4}(\\delta , \\mu , \\Lambda , B_{2})$ , ${\\bf F}/\\mu \\in L^2(B_6, \\mu )$ , if $u\\in W^{1, 2}(B_{6}, \\mu )$ is a weak solution to $\\textup {div}[\\mathbb {A} \\nabla u] = \\textup {div}({\\bf F})\\quad \\text{in} \\quad B_{6},$ and $\\mu (B_{1}\\cap \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{2}\\rbrace ) <\\epsilon \\mu (B_\\frac{1}{2}(y)), \\quad \\forall \\ y \\in B_1,$ then for any $k\\in \\mathbb {N}$ and $\\epsilon _{1} = \\left(\\frac{10}{1 - 4\\delta }\\right)^{2n} M_{0}^{2} \\epsilon $ we have that $\\begin{split}\\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{2k} \\rbrace ) &\\le \\sum _{i=1}^{k} \\epsilon _{1}^{i} \\mu \\left(\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{B_{6}}\\right) >\\delta ^{2} \\varpi ^{2(k-i)} \\rbrace \\right)\\\\&\\quad \\quad + \\epsilon _{1}^{k}\\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}} |\\nabla u|^{2}) > 1 \\rbrace ).\\end{split}$ We will use induction to prove the corollary.",
"For the case $k=1$ , we are going to apply Lemma REF , by taking $C = \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi {B_{6}}|\\nabla u|^{2}) > \\varpi ^{2} \\rbrace \\cap B_{1}$ and $D = \\left(\\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{B_{6}}\\right) >\\delta ^{2} \\rbrace \\cup \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}} |\\nabla u|^{2}) > 1 \\rbrace \\right)\\cap B_{1}.$ Clearly, $C \\subset D \\subset B_1$ .",
"Moreover, by the assumption, $\\mu (C) < \\epsilon \\mu (B_\\frac{1}{2}(y))$ , for all $y \\in B_1$ .",
"Also for any $y\\in B_{1}$ and $\\rho \\in (0, 1) $ , then $\\mathbb {A}\\in \\mathcal {A}_{4}(\\delta , \\mu , \\Lambda , B_{2})$ implies that $\\mathbb {A}\\in \\mathcal {A}_{4\\rho }(\\delta , \\mu , \\Lambda , B_{\\rho }(y))$ .",
"Moreover, if $\\mu (C \\cap B_{\\rho }(y))\\ge \\epsilon \\mu ((B_{\\rho }(y))$ , then by Proposition REF we have that $B_{\\rho }(y)\\cap B_{1}\\subset D.$ Hence, all the conditions of Lemma REF are satisfied and hence $\\mu (C) \\le \\epsilon _{1} \\mu (D).$ That proves the case when $k=1$ .",
"Assume it is true for $k$ .",
"We will show the statement for $k+1$ .",
"We normalize $u$ to $u_{\\varpi } = u/\\varpi $ and ${\\bf F}_{\\varpi } = {\\bf F}/\\varpi $ , and we see that since $\\varpi > 1$ we have $\\begin{split}\\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u_{\\varpi }|^{2}) > \\varpi ^{2} \\rbrace ) &= \\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{4} \\rbrace )\\\\& \\le \\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{2} \\rbrace ) \\le \\epsilon \\mu (B_\\frac{1}{2}(y)), \\quad \\forall y \\in B_1.\\end{split}$ By induction assumption, it follows then that $\\begin{split}\\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{2(k+1)} \\rbrace )&=\\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u_{\\varpi }|^{2}) > \\varpi ^{2k} \\rbrace )\\\\&\\le \\sum _{i=1}^{k} \\epsilon _{1}^{i} \\mu \\left(\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{F_{\\varpi }}{\\mu }\\right|^{2} \\chi _{B_{6}}\\right) >\\delta ^{2} \\varpi ^{2(k-i)} \\rbrace \\right)\\\\&\\quad \\quad + \\epsilon _{1}^{k}\\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}} |\\nabla u_{\\varpi }|^{2}) > 1 \\rbrace )\\\\& = \\sum _{i=1}^{k} \\epsilon _{1}^{i} \\mu \\left(\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{B_{6}}\\right) >\\delta ^{2} \\varpi ^{2(k+1-i)} \\rbrace \\right)\\\\&\\quad \\quad + \\epsilon _{1}^{k}\\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}} |\\nabla u|^{2}) > \\varpi ^2 \\rbrace ).\\end{split}$ Applying the case $k=1$ to the last term we obtain that $\\begin{split}\\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}}|\\nabla u|^{2}) > \\varpi ^{2(k+1)} \\rbrace )&\\le \\sum _{i=1}^{k + 1} \\epsilon _{1}^{i} \\mu \\left(\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{B_{6}}\\right) >\\delta ^{2} \\varpi ^{2(k+1-i)} \\rbrace \\right)\\\\&\\quad \\quad + \\epsilon _{1}^{k+1}\\mu (\\lbrace x\\in B_{1}: \\mathcal {M}^{\\mu }(\\chi _{B_{6}} |\\nabla u|^{2}) > 1 \\rbrace ),\\end{split}$ as desired.",
"With Lemma REF , the rest of the proof of Theorem REF is now the same as Case I in the proof of Theorem REF below.",
"We therefore skip it."
],
[
"Boundary estimate setup",
"We first introduce some notations.",
"For $r >0$ and for $x_0~=~(x^{0}_1, x^{0}_2, \\cdots , x^{0}_n)~\\in ~\\mathbb {R}^n$ , let us denote $\\begin{split}B_r^+(x_0) = \\lbrace y = (y_1, y_2, \\cdots , y_n) \\in B_r(x_0):\\ y_n >x^0_{n}\\rbrace , \\quad B_r^+ = B_r^+(0), \\\\T_r (x_0) = \\lbrace x = (x_1, x_2, \\cdots , x_n) \\in \\partial B_{r}^+(0): \\ x_n =x_{n}^{0}\\rbrace , \\quad T_r = T_r(0).\\end{split}$ For $x_0 \\in \\mathbb {R}^n$ , we also denote $\\Omega _r(x_0) = \\Omega \\cap B_r(x_0), \\quad \\partial _{w} \\Omega _r(x_0) = \\partial \\Omega \\cap B_r(x_0), \\quad \\Omega _r = \\Omega _r(0).$ In this section we localize the problem near the boundary, assume that there is a coordinate system where for some $K>0$ and $\\delta \\in (0, 1/K)$ $B_{r}^{+} \\subset \\Omega _{r}\\subset B_{r} \\cap \\lbrace (y^{\\prime }, y_{n} : y_{n} > -K\\delta r)\\rbrace ,$ and study the problems $ \\left\\lbrace \\begin{array}{cccl}\\text{div}[\\mathbb {A}(x) \\nabla u] & = & \\text{div}[{\\bf F}] & \\quad \\text{in} \\quad \\ \\Omega _r, \\\\u & =& 0 & \\quad \\text{on} \\quad \\partial _{w} \\Omega _r,\\end{array}\\right.$ and the corresponding homogeneous equation $ \\left\\lbrace \\begin{array}{cccl}\\text{div}[\\mathbb {A}_0 \\nabla v ] &= &0 & \\quad \\text{in $B_r^+$,} \\\\v &= & 0 & \\quad \\text{on $T_r$,}\\end{array}\\right.$ with a symmetric and elliptic constant matrix $\\mathbb {A}_{0}$ .",
"Definition 5.1 (i) $u\\in W^{1, 2}(\\Omega _{r}, \\mu )$ is a weak solution to (REF ) in $\\Omega _{r}$ if $\\int _{\\Omega _{r}}\\langle \\mathbb {A}\\nabla u, \\nabla \\varphi \\rangle dx = \\int _{\\Omega _{r}} \\langle {\\bf F},\\nabla \\varphi \\rangle dx,\\quad \\forall \\varphi \\in C_{0}^{\\infty }(\\Omega _{r}),$ and $u$ 's zero extension to $B_r$ is in $W^{1, 2}(B_{r}, \\mu )$ .",
"(ii) $v\\in W^{1, q}(B_{r}^{+})$ is a weak solution to (REF ) in $B_{r}^{+}$ , for some $1 < q < \\infty $ , if $\\int _{B_{r}^{+}} \\langle \\mathbb {A}_{0}\\nabla v, \\nabla \\varphi \\rangle dx = 0,\\quad \\forall \\varphi \\in C_{0}^{\\infty }(B_{r}^{+}),$ and $v$ 's zero extension to $B_r$ is also in $W^{1, q}(B_{r})$ .",
"Let us now consider the case when $r = 4$ .",
"The equation (REF ) in this case becomes $ \\left\\lbrace \\begin{array}{cccl}\\text{div}[\\mathbb {A}_{0} \\nabla v ] &= &0 & \\quad \\text{in}\\, B_4^+, \\\\v &= & 0 & \\quad \\text{in }\\,\\partial T_4^.\\end{array}\\right.$ for some constant matrix $\\mathbb {A}_{0}$ which will be chosen sufficiently close to $\\langle \\mathbb {A}\\rangle _{B_4}$ .",
"Similar to Lemma REF , the following boundary regularity result of the weak solution $v$ is also needed for our approximation estimates.",
"Lemma 5.2 Let $\\mathbb {A}_0$ be a constant symmetric matrix satisfying the ellipticity condition $\\lambda _0 |\\xi |^2 \\le \\langle \\mathbb {A}_0\\xi , \\xi \\rangle \\le \\Lambda _0 |\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n,$ for some fixed positive constants $\\Lambda _0, \\lambda _0$ .",
"Then there exists a constant $C = C(n, \\Lambda _0/\\lambda _0)$ such that if $v \\in W^{1,q}(B_4^+)$ is a weak solution of (REF ) with some $q >1$ , then $\\left\\Vert \\nabla v\\right\\Vert _{L^\\infty (B_{\\frac{7}{2}}^+)} \\le C \\left( _{B_{4}^+} |\\nabla v|^{q} dx \\right)^{\\frac{1}{q}}.$ The proof is the same as that of Lemma REF but we use the boundary version [1] of [2] instead when considering $1 < q <2$ ."
],
[
"Boundary weighted Caccioppoli estimate",
"We assume that $\\mathbb {A}$ is a measurable symmetric matrix, and there is $\\Lambda >0$ such that $ \\Lambda \\mu (x) |\\xi |^2 \\le \\langle \\mathbb {A}(x) \\xi , \\xi \\rangle \\le \\Lambda ^{-1} \\mu (x)|\\xi |^2 \\quad \\text{for a.e.\\ } x\\ \\in \\mathbb {R}^n, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ Let $u \\in W^{1,2}(\\Omega _4, \\mu )$ be a weak solution of $ \\left\\lbrace \\begin{array}{cccl}\\text{div}[\\mathbb {A}(x) \\nabla u] & = & \\text{div}[{\\bf F}] & \\quad \\text{in} \\quad \\ \\Omega _4, \\\\u & =& 0 & \\quad \\text{on} \\quad \\partial _{w} \\Omega _4.\\end{array}\\right.$ Let $v \\in W^{1,1+\\beta }(B_4^+)$ be a weak solution of (REF ).",
"Similarly to Lemma REF , the following estimate is a weighted Caccioppoli estimate up to the boundary for the difference $u-v$ .",
"Lemma 5.3 Assume that (REF ) holds and $[\\mu ]_{A_2} \\le M_0$ .",
"Let $\\mathbb {A}_0 = (a_0^{ki})_{k,i=1}^n, v$ be as in Lemma REF , and let $w = u- V$ where $V$ is the zero extension of $v$ in $B_4$ .",
"There exists a constant $C = C(\\Lambda , n,M_0)$ such that for all non-negative function $\\varphi \\in C_0^\\infty (B_{4})$ , $\\begin{split}\\frac{1}{\\mu (B_4)}\\int _{\\Omega _4} |\\nabla w|^2 \\varphi ^2 d\\mu & \\le C(\\Lambda , M_0,n) \\left[\\frac{1}{\\mu (B_4)} \\int _{\\Omega _{4}} w^2 |\\varphi |^2 d\\mu + \\frac{1}{\\mu (B_4)}\\int _{\\Omega _4} \\Big | \\frac{{\\bf F}}{\\mu } \\Big |^2 \\varphi ^2 d\\mu \\right.\\\\& \\left.",
"+\\frac{ \\left\\Vert \\varphi \\nabla v\\right\\Vert _{L^\\infty (B_{4}^+)}^2}{\\mu ({B_4})} \\int _{\\Omega _4}| \\mathbb {A} - \\mathbb {A}_0|^2 \\mu ^{-1} dx \\right.\\\\&\\left.+ |\\mathbb {A}_0^{n,\\cdot }|^2 \\left\\Vert \\varphi (x) \\nabla v(x^{\\prime },0)\\right\\Vert _{L^\\infty (\\Omega _4 \\setminus B_4^+)}^2 \\left(\\frac{|B_{4}|}{\\mu (B_{4})}\\right)^{2} \\left(\\frac{|\\Omega _4 \\setminus B_4^+|}{|B_4|}\\right)^{\\varrho }\\right],\\end{split}$ where $\\varrho $ is as in (REF ) and depends only on $n$ and $M_0$ , and ${\\mathbb {A}}_{0}^{n, \\cdot } = (a_{0}^{n1},\\dots ,a_{0}^{nn})$ is the $n^{th }$ row of $\\mathbb {A}_{0}$ .",
"Observe that by Lemma REF , $v$ is smooth in $\\overline{B}^+_r$ for $0 < r <4$ .",
"Therefore, a simple integration by parts shows that $V$ is a weak solution of $\\left\\lbrace \\begin{array}{ccll}\\text{div}[\\mathbb {A}_0 \\nabla V ] & = & \\frac{\\partial }{\\partial x_n} g & \\quad \\text{in} \\quad \\Omega _4 \\\\V & =& 0 & \\quad \\text{on} \\quad \\partial _{w} \\Omega _4,\\end{array}\\right.$ where $g (x^{\\prime }, x_n)= \\langle \\mathbb {A}_{0}^{n, \\cdot }, \\nabla v (x^{\\prime }, 0)\\rangle \\chi _{\\lbrace x_n <0\\rbrace }(x^{\\prime },x_n).",
"$ Then, it follows that $w$ is a weak solution of $\\left\\lbrace \\begin{array}{ccll}\\text{div}[\\mathbb {A} \\nabla w] & = & \\text{div}[{\\bf F} - (\\mathbb {A} - \\mathbb {A}_0) \\nabla V ] + \\frac{\\partial }{\\partial x_n} g& \\quad \\text{in} \\quad \\Omega _4, \\\\w & =& 0 & \\quad \\text{on} \\quad \\partial _{w} \\Omega _4.\\end{array} \\right.$ For any non-negative cut-off function $\\varphi \\in C_0^\\infty (B_4)$ , it follows that $w\\varphi ^{2} \\in W^{1,2}_{0}(\\Omega _{4}, \\mu )$ .",
"Therefore, by using $w\\varphi ^2$ as a test function for the above equation, we obtain $ \\begin{split}\\int _{\\Omega _4} \\langle \\mathbb {A}\\nabla w, \\nabla w \\rangle \\varphi ^2 dx &= - \\int _{\\Omega _4} \\langle \\mathbb {A}\\nabla w, \\nabla (\\varphi ^2) \\rangle w dx +\\int _{\\Omega _4} \\langle {\\bf F}, \\nabla (w\\varphi ^2) \\rangle dx \\\\& \\quad \\quad - \\int _{\\Omega _4} \\langle (\\mathbb {A}- \\mathbb {A}_0) \\nabla V, \\nabla (w\\varphi ^2) \\rangle dx +\\int _{\\Omega _{4}} g\\frac{\\partial }{\\partial x_n} [w\\varphi ^2] dx.\\end{split}$ Except the last term on the right hand side of (REF ), all terms in (REF ) can be estimated exactly as in those in Lemma REF .",
"To estimate this last term, we use Young's inequality to write $\\begin{split}\\left| \\int _{\\Omega _{4}} g \\frac{\\partial }{\\partial x_n} [w\\varphi ^2] dx \\right| &\\le \\int _{\\Omega _{4}} |g| \\Big [| \\nabla w| \\varphi ^2 + 2|\\nabla \\varphi | \\varphi |w|\\Big ] \\\\&\\le \\epsilon \\int _{\\Omega _{4}} |\\nabla w|^{2}\\varphi ^{2}\\mu (x)dx + \\epsilon \\int _{\\Omega _{4}} |w|^{2}|\\nabla \\varphi |^{2}\\mu (x)dx + C_{\\epsilon } \\int _{\\Omega _{4}}|g|^{2}\\varphi ^2 \\mu ^{-1} dx.",
"\\\\\\end{split}$ Next, we use the doubling property of weights and the fact that $g$ is supported on $\\Omega _{4}\\setminus B_{4}^{+}$ to have $\\begin{split}& C_{\\epsilon } \\int _{\\Omega _{4}}|g|^{2}\\varphi ^2 \\mu ^{-1} dx \\le \\Vert g\\varphi \\Vert ^{2}_{L^{\\infty }(\\Omega _4 \\setminus B_4^+)}\\mu ^{-1}(\\Omega _4 \\setminus B_4^+) \\\\&\\le C(M_{0}, \\epsilon , n)\\left\\Vert g\\varphi \\right\\Vert ^{2}_{L^{\\infty }(\\Omega _4 \\setminus B_4^+)} \\left(\\frac{|\\Omega _4 \\setminus B_4^+|}{|B_4|}\\right)^{\\varrho } \\frac{|B_4|^{2}}{\\mu (B_{4})},\\end{split}$ for some constant $\\varrho $ is as in (REF ) and that depends only on $n$ and $M_{0}$ .",
"Then, we can follow the proof of Lemma REF to derive the estimate in Lemma REF ."
],
[
"Boundary gradient approximation estimates",
"We begin the section with the following lemma showing the $u$ can be approximated by $v$ in $L^2(\\Omega _{7/2}, \\mu )$ .",
"Lemma 5.4 Let $K>0, \\Lambda >0, M_0 >0$ be fixed and let $\\beta $ be as in (REF ).",
"For every $\\epsilon >0$ sufficiently small, there exists $\\delta \\in (0, 1/K)$ depending on only $\\epsilon , \\Lambda , n$ , and $M_0$ such that the following statement holds true: If $\\mathbb {A}, \\mu , {\\bf F}$ such that $[\\mu ]_{A_2} \\le M_0$ , (REF ) holds, $B_4^+ \\subset \\Omega _4 \\subset B_4 \\cap \\lbrace x_n > -4K\\delta \\rbrace ,$ and $\\begin{split}& \\frac{1}{\\mu ({B_4})} \\int _{\\Omega _4}| \\mathbb {A} - \\langle \\mathbb {A}\\rangle _{B_{4}}|^2 \\mu ^{-1} dx + \\frac{1}{\\mu (B_4)}\\int _{\\Omega _4} \\Big | \\frac{{\\bf F}}{\\mu } \\Big |^2 d\\mu (x) \\le \\delta ^2,\\end{split}$ a weak solution $u \\in W^{1,2}(\\Omega _4, \\mu )$ of (REF ) that satisfies $ \\frac{1}{\\mu (B_4) }\\int _{\\Omega _4} |\\nabla u|^2 d\\mu \\le 1,$ then there exists a constant matrix $\\mathbb {A}_0$ and a weak solution $v \\in W^{1,1+\\beta }(B_4^+)$ of (REF ) such that $\\left\\Vert \\langle \\mathbb {A}\\rangle _{B_4} - \\mathbb {A}_0\\right\\Vert \\le \\frac{\\epsilon \\mu (B_{4})}{|B_{4}|},$ and $\\frac{1}{\\mu (B_{7/2})}\\int _{\\Omega _{7/2}} |u - V|^2 d\\mu \\le \\epsilon ,$ where $V$ is the zero extension of $v$ to $B_4$ .",
"Moreover, there is $C =C(\\Lambda , n, M_0)$ such that $ _{B_{3}^+}|\\nabla v|^2dx \\le C(\\Lambda , n, M_0).$ As in the proof of Lemma REF , we can use appropriate scaling to assume that $\\langle \\mu \\rangle _{B_4} = \\frac{1}{|B_4|}\\int _{B_4} \\mu (x) dx = 1.$ We again proceed the proof with a contradiction argument.",
"Suppose that there exists $\\epsilon _{0} > 0$ such that for every $k \\in \\mathbb {N}$ , there are $\\mu _k \\in A_2$ , $ \\mathbb {A}_k$ satisfying the degenerate ellipticity assumption as in (REF ) with $\\mu $ and $\\mathbb {A}$ are replaced by $\\mu _k$ and $\\mathbb {A}_k$ respectively, and domain $\\Omega _{4}^{k}$ , ${\\bf F}_k$ and a weak solution $u_k \\in W^{1, 2}(\\Omega _{4}^{k}, \\mu _{k})$ of $ \\left\\lbrace \\begin{array}{cccl}\\text{div}[\\mathbb {A}_k \\nabla u_k] & = & \\text{div}({\\bf F}_k) & \\quad \\text{in} \\quad \\Omega ^{k}_4, \\\\u & =& 0 & \\quad \\text{on} \\quad \\partial _w \\Omega ^{k}_4,\\end{array} \\right.$ with $ B_4^+ \\subset \\Omega _4^k \\subset B_4 \\cap \\Big \\lbrace x_n \\ge -\\frac{4 K}{k} \\Big \\rbrace ,$ $ \\left\\lbrace \\begin{split}& \\frac{1}{\\mu _{k}({B_4})} \\int _{\\Omega ^{k}_4}| \\mathbb {A}_{k} - \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}|^2 \\mu _{k}^{-1} dx + \\frac{1}{\\mu _{k}(B_4)}\\int _{\\Omega ^{k}_4} \\Big | \\frac{{\\bf F}_{k}}{\\mu _{k}} \\Big |^2 d\\mu _{k} (x) \\le \\frac{1}{k^2}, \\\\& [\\mu _k]_{A_2} \\le M_0, \\quad \\langle \\mu \\rangle _{k, B_4} = \\frac{1}{|B_4|} \\int _{B_4} \\mu _k(x) dx = 1\\end{split} \\right.$ and $ \\frac{1}{\\mu _{k}(B_4)} \\int _{\\Omega _4^k} |\\nabla u_k|^2 d\\mu _k \\le 1,$ but for all elliptic, symmetric and constant matrix $\\mathbb {A}_0$ with $\\left\\Vert \\langle \\mathbb {A}_{k}\\rangle _{B_{4}} - \\mathbb {A}_0\\right\\Vert \\le \\epsilon _{0},$ and all weak solution $v \\in W^{1, 1+\\beta }(B_4^+)$ of (REF ) and $V$ is its zero extension to $B_4$ , we have $ \\frac{1}{\\mu _k(B_{7/2})}\\int _{\\Omega ^k_{7/2}} |u_{k} - V|^2 d\\mu _k \\ge \\epsilon _0.$ It follows from (REF ) that $ \\Lambda |\\xi |^2 \\le \\langle \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}\\xi , \\xi \\rangle \\le \\Lambda ^{-1} |\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ Then, since the sequence $\\lbrace \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}\\rbrace _k$ is a bounded sequence in $\\mathbb {R}^{n\\times n}$ , by passing through a subsequence, we can assume that there is a constant matrix $\\bar{\\mathbb {A}}$ in $ \\mathbb {R}^{n\\times n}$ such that $ \\lim _{k \\rightarrow \\infty }\\langle \\mathbb {A}_{k}\\rangle _{B_{4}} = \\bar{\\mathbb {A}}$ From (REF ), and Poincaré-Sobolev inequality [14], we see that $\\frac{1}{\\mu _k(B_4)}\\int _{\\Omega _4^k} |u_k|^2 d \\mu _k \\le \\frac{C(n, M_{0})}{\\mu _k(B_4)} \\int _{\\Omega _{4}^k} |\\nabla u_k|^2 d\\mu _k \\le C(n, M_{0}), \\forall \\ k \\in \\mathbb {N}.",
"$ This and since $\\mu _k(B_4) = |B_4|$ for all $k$ , it follows that $\\left\\Vert u_k\\right\\Vert _{W^{1,2}(\\Omega _4^k, \\mu _k)} = \\left\\Vert u_k\\right\\Vert _{L^2(\\Omega _4^k, \\mu _k)} + \\left\\Vert \\nabla u_k\\right\\Vert _{L^2(\\Omega _4^k, \\mu _k)} \\le C(n, M_0), \\quad \\forall k \\in \\mathbb {N}.$ As a consequence, Lemma REF implies that $\\left\\Vert u_k\\right\\Vert _{W^{1,1 +\\beta }(B_4)} \\le C(n, M_0) \\left\\Vert u_k\\right\\Vert _{W^{1,2}(\\Omega _4^k, \\mu _k)} \\le C(n, M_0), \\quad \\beta = \\frac{\\gamma }{2 + \\gamma } >0.$ Here, note that in the above estimate we still denote $u_k \\in W^{1, 1+\\beta }(B_4)$ to be zero extension of $u_k$ to $B_4$ .",
"Also, recall that $\\gamma $ is defined in Lemma REF , which only depends on $n$ and $M_0$ .",
"Therefore, by the compact imbedding $W^{1,1+\\beta }(B_4) \\hookrightarrow L^{1+\\beta }(B_4)$ and by passing through a subsequence, we can assume that there is $u \\in W^{1,1+\\beta }(B_4)$ such that $ \\left\\lbrace \\begin{split}& u_k \\rightarrow u \\mbox{ strongly in } L^{1+\\beta }(B_4),\\quad \\nabla u_k \\rightharpoonup \\nabla u \\text{ weakly in } L^{1+\\beta }(B_4),\\ \\quad \\text{and} \\\\& \\hat{u}_{k} \\rightarrow u \\ \\text{a.e.",
"in} \\ \\ B_4.\\end{split} \\right.$ Moreover, $ \\left\\Vert u\\right\\Vert _{W^{1,1+\\beta }(B_4)} \\le C(n, M_0).$ We claim that $u \\in W^{1,1+\\beta }(B_4^+)$ is a weak solution of $ \\left\\lbrace \\begin{array}{cccl}\\text{div}[\\bar{\\mathbb {A}}\\nabla u] & = & 0 & \\quad \\text{in} \\quad B_4^+, \\\\u & =& 0 & \\quad \\text{on} \\quad T_4.\\end{array} \\right.$ To prove this claim, first note that from (REF ), (REF ) and (REF ), it follows that $ u = 0 \\quad \\text{for a.e.}",
"\\quad x \\in B_4 \\setminus B_4^+.$ In particular, $u = 0$ on $T_4$ in trace sense.",
"It is therefore enough to show that the weak form of the PDE in (REF ) holds.",
"To proceed, let us fix $\\varphi \\in C^\\infty _0(B_4^+)$ .",
"Then, by using $\\varphi $ as a test function for the equation (REF ) of $u_k$ , we have $ \\int _{B_4^+} \\langle \\mathbb {A}_k \\nabla u_k, \\nabla \\varphi \\rangle dx = \\int _{B^+_4} \\langle {\\bf F}_k, \\nabla \\varphi \\rangle dx.$ We will take the limit $k \\rightarrow \\infty $ on both sides of the above equation.",
"First of all, observe that by Hölder's inequality and (REF ), it follows that the right hand side term of (REF ) can be estimated as $\\begin{split}\\left|_{B^+_4} \\langle {\\bf F}_k, \\nabla \\varphi \\rangle dx \\right| & \\le \\left\\lbrace _{B^+_4} \\Big |\\frac{{\\bf F}_{k}}{\\mu _k}\\Big |^2 \\mu _k dx \\right\\rbrace ^{1/2} \\left\\lbrace _{B^+_4} |\\nabla \\varphi |^2 \\mu _k dx\\right\\rbrace ^{1/2} \\\\& \\le \\left\\Vert \\nabla \\varphi \\right\\Vert _{L^\\infty (B^+_4)} \\left\\lbrace \\frac{1}{\\mu _k (B_4)}\\int _{\\Omega ^{k}_4}\\Big |\\frac{{\\bf F}_{k}}{\\mu _k} \\Big |^2d \\mu _k (x) \\right\\rbrace ^{1/2} \\frac{\\mu _{k}(B_{4})}{|B_{4}^{+}|}\\\\&\\le 2\\left\\Vert \\nabla \\varphi \\right\\Vert _{L^\\infty (B^+_4)} \\left\\lbrace \\frac{1}{\\mu _k (B_4)}\\int _{\\Omega ^{k}_4}\\Big |\\frac{{\\bf F}_{k}}{\\mu _k} \\Big |^2d \\mu _k (x) \\right\\rbrace ^{1/2}\\\\& \\le 2 \\frac{\\left\\Vert \\nabla \\varphi \\right\\Vert _{L^\\infty (B_4^+)}}{k} .\\end{split}$ Therefore, taking the limit as $k \\rightarrow \\infty ,$ we have $ \\int _{B_4^+} \\langle {\\bf F}_k, \\nabla \\varphi \\rangle dx \\rightarrow 0.$ On the other hand, it follows from (REF ), (REF ), and Hölder's inequality that $\\begin{split}\\left| _{B_4^+} \\langle (\\mathbb {A}_{k} - \\langle \\mathbb {A}_{k})\\rangle _{B_{4}}\\nabla u_k, \\nabla \\varphi \\rangle dx \\right| & \\le _{B_4^+} |\\mathbb {A}_{k} - \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}| \\mu _{k}^{-1/2}|\\nabla u_k| \\mu _k^{1/2} |\\nabla \\varphi | dx \\\\& \\le \\Vert \\nabla \\varphi \\Vert _{L^{\\infty }(B_{4}^{+})}\\left\\lbrace _{B_4^+} |\\mathbb {A}_{k} - \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}|^{2} \\mu _{k}^{-1} dx\\right\\rbrace ^{1/2} \\left\\lbrace \\frac{1}{|B_4^+|}\\int _{B_4^+} |\\nabla u_k|^2 d\\mu _k \\right\\rbrace ^{1/2} \\\\&\\le 2\\frac{\\left\\Vert \\nabla \\varphi \\right\\Vert _{L^\\infty (B_4^+)}}{k} \\left\\lbrace \\frac{1}{\\mu _k(B_4)}\\int _{\\Omega ^{k}_4} |\\nabla u_k|^2 d\\mu _k \\right\\rbrace ^{1/2} \\\\& \\le 2\\frac{\\left\\Vert \\nabla \\varphi \\right\\Vert _{L^\\infty (B_4^+)}}{k} \\rightarrow 0, \\quad \\text{as} \\quad k \\rightarrow \\infty .\\end{split}$ As a result we have, $0 = \\lim _{k\\rightarrow \\infty } \\int _{B_4^+} \\langle (\\mathbb {A}_{k} - \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}) \\nabla u_k, \\nabla \\varphi \\rangle dx = \\lim _{k\\rightarrow \\infty } \\Big [\\int _{B_4^+} \\langle \\mathbb {A}_{k} \\nabla u_k, \\nabla \\varphi \\rangle dx - \\int _{B_4^+} \\langle \\langle \\mathbb {A}_{k}\\rangle _{B_{4}} \\nabla u_k, \\nabla \\varphi \\rangle dx\\Big ].$ We also observe that since $\\nabla u_k$ converges weakly in $L^{1 + \\beta }(B_4^+)$ from (REF ) and the constant symmetric matrix $\\langle \\mathbb {A}_{k}\\rangle _{B_{4}}$ converges to $\\bar{\\mathbb {A}}$ , we have that $\\lim _{k\\rightarrow \\infty } \\int _{B_4^+} \\langle \\langle \\mathbb {A}_{k}\\rangle _{B_{4}} \\nabla u_k, \\nabla \\varphi \\rangle dx = \\int _{B_{4}^+}\\langle \\bar{\\mathbb {A}} \\nabla u, \\nabla \\varphi \\rangle dx.$ Combining the above we conclude that, $\\int _{B_4^+} \\langle \\bar{\\mathbb {A}}\\nabla u, \\nabla \\varphi \\rangle dx =0, \\quad \\forall \\ \\varphi \\in C^\\infty _0({B}_4^+).$ Now, from (REF ), and since $\\bar{\\mathbb {A}} = \\lim _{k\\rightarrow \\infty } \\langle \\mathbb {A}_{k}\\rangle _{B_{4}}$ , we observe that $\\Lambda |\\xi |^2 \\le \\langle \\bar{\\mathbb {A}}\\xi , \\xi \\rangle \\le \\Lambda ^{-1} |\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ In other words, the symmetric constant matrix $\\bar{\\mathbb {A}}$ is uniformly elliptic.",
"Hence, Lemma REF implies that $u~\\in ~W^{1, \\infty }(\\overline{B}_{15/4}^+)$ .",
"In addition, it follows from Lemma REF , (REF ), (REF ), and (REF ) that $ _{B_{7/2}} |\\nabla u|^2 d\\mu _k \\le \\left\\Vert \\nabla u\\right\\Vert _{L^\\infty (B_{7/2}^+)}^2 \\le C(n, \\Lambda ) \\left( _{B_4^+} |\\nabla u|^{1+\\beta } dx \\right)^{\\frac{2}{1+\\delta }}\\le C(n, M_0, \\Lambda ) \\quad \\forall \\ k \\in \\mathbb {N}.$ As in the interior case, we claim that $\\lim _{k\\rightarrow \\infty } \\frac{1}{\\mu _{k}(B_{7/2})} \\int _{\\Omega ^k_{7/2}}| u_k - u|^{2} d\\mu _{k} = 0.$ The proof of this claim follows exactly as in that of Lemma REF , where use the Poincaré-Sobolev inequality, [14] instead of [14] and after noting that $\\frac{1}{\\mu _{k}(B_{7/2})} \\int _{\\Omega ^k_{7/2}}| u_k - u|^{2} d\\mu _{k} = _{B_{7/2}}| u_k - u|^{2} d\\mu _{k},$ since both $u_{k}$ and $u$ vanish in $B_{7/2}\\setminus \\Omega _{7/2}^{k}$ .",
"Now, since $\\langle \\mathbb {A}_{k}\\rangle _{B_{4}} \\rightarrow \\bar{\\mathbb {A}}$ and $\\epsilon _0 >0$ $\\left\\Vert \\langle \\mathbb {A}_{k}\\rangle _{B_{4}} - \\bar{\\mathbb {A}}\\right\\Vert \\le \\epsilon _0,$ for sufficiently large $k$ .",
"But, this and our last claim, contradict (REF ) if we take $\\mathbb {A}_0 = \\bar{\\mathbb {A}}$ , $v = u$ and $k$ sufficiently large.",
"Finally, estimate (REF ) can be established exactly the same way as (REF ) using Lemma REF .",
"The proof of Lemma REF is now complete.",
"Next, using the energy estimates in Lemma REF and Lemma REF , we can prove the following result, which is also the main result of the section.",
"Proposition 5.5 Let $K>0, \\Lambda >0, M_0 >0$ be fixed and let $\\beta $ be as in (REF ).",
"For every $\\epsilon >0$ sufficiently small, there exists $\\delta \\in (0, 1/K)$ depending on only $\\epsilon , \\Lambda , n$ , and $M_0$ such that the following statement holds true: If $\\mathbb {A}, \\mu , {\\bf F}$ such that $[\\mu ]_{A_2} \\le M_0$ , (REF ) holds, $\\begin{split}& B_4^+ \\subset \\Omega _4 \\subset B_4 \\cap \\lbrace x_n > -4K\\delta \\rbrace , \\\\& \\frac{1}{\\mu ({B_4})} \\int _{\\Omega _4}| \\mathbb {A} - \\langle \\mathbb {A}\\rangle _{B_{4}}|^2 \\mu ^{-1} dx + \\frac{1}{\\mu (B_4)}\\int _{\\Omega _4} \\Big | \\frac{{\\bf F}}{\\mu } \\Big |^2 d\\mu (x) \\le \\delta ^2,\\end{split}$ a weak solution $u \\in W^{1,2}(\\Omega _4, \\mu )$ of (REF ) that satisfies $ \\frac{1}{\\mu (B_4) }\\int _{\\Omega _4} |\\nabla u|^2 d\\mu \\le 1,$ then there exists a constant matrix $\\mathbb {A}_0$ and a weak solution $v \\in W^{1,1+\\beta }(B_4^+)$ of (REF ) such that $\\left\\Vert \\langle \\mathbb {A}\\rangle _{B_4} - \\mathbb {A}_0\\right\\Vert \\le \\frac{\\epsilon \\mu (B_{4})}{|B_{4}|},\\quad \\text{and}\\quad _{\\Omega _2} |\\nabla u -\\nabla V|^2 d\\mu \\le \\epsilon ,$ where $V$ is the zero extension of $v$ to $B_4$ .",
"Moreover, there is $C =C(\\Lambda , n, M_0)$ such that $_{B_{3}^+}|\\nabla v|^2dx \\le C(\\Lambda , n, M_0).$ It follows from ellipticity condition (REF ) of $\\mathbb {A}$ that $\\Lambda \\frac{\\mu ( B_4)}{|B_4|}|\\xi |^2 \\le \\langle \\langle \\mathbb {A} \\rangle _{B_{4}}\\xi , \\xi \\rangle \\le \\Lambda ^{-1}\\frac{\\mu (B_4)}{|B_4|}|\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ Therefore, if $\\epsilon $ is sufficiently small, and $\\mathbb {A}_{0}$ such that (REF ) holds, it follows that $\\Lambda \\frac{ \\mu (B_4)}{2 |B_4|} |\\xi |^2\\le \\langle \\mathbb {A}_{0} \\xi , \\xi \\rangle \\le \\Lambda ^{-1} \\frac{2 \\mu (B_4)}{|B_4|} |\\xi |^2, \\quad \\forall \\ \\xi \\in \\mathbb {R}^n.$ This particularly implies $ |a_0^{ni}|\\frac{|B_{4}|}{\\mu (B_4)} \\le 2\\Lambda ^{-1}, \\quad \\text{for $i = 1,\\dots ,n$}.$ Moreover, observe that by the flatness assumption $\\frac{|\\Omega _4\\setminus B_4^+|}{|B_4|} \\le C(n) K\\delta .$ The remaining part of the proof can be done similarly to that of Proposition REF using the last estimate, (REF ), and Lemmas REF -REF ."
],
[
"Level set estimates up to the boundary",
"We begin with the following result on the density of interior level sets which is a consequence of Proposition REF .",
"Proposition 5.6 Suppose that $M_{0}>0$ and $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ .",
"There exists a constant $\\varpi = \\varpi (n, \\Lambda , M_0)> 1$ such that the following holds true.",
"Corresponding to any $\\epsilon > 0 $ there exists a small constant $\\delta = \\delta (\\epsilon )$ such that for any $r > 0,$ $y\\in \\Omega $ such that $B_{6r}(y)\\subset \\Omega $ , $\\mathbb {A}\\in \\mathcal {A}_{6r}(\\delta , \\mu , \\Lambda , \\Omega )$ , if $u\\in W^{1, 2}(\\Omega , \\mu )$ is a weak solution to $\\textup {div}[\\mathbb {A} \\nabla u] = \\textup {div}({\\bf F})\\quad \\text{in $\\Omega $}, \\quad \\text{and}$ $\\mu ( \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2}\\rbrace \\cap B_{r}(y)) \\ge \\epsilon \\mu (B_{r}(y)),$ then $B_{r}(y)\\subset \\lbrace x\\in B_{r}(y): \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u|^{2}) > 1 \\rbrace \\cup \\lbrace x\\in B_{r}(y): \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{\\Omega }\\right) >\\delta ^{2} \\rbrace .$ Our next goal is to obtain similar result as Proposition REF but for balls that may intersect the boundary of the domain $\\Omega $ .",
"We begin with the following local near boundary estimate.",
"Lemma 5.7 Suppose that $M_{0}>0$ and $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ .",
"There exists a constant $\\varpi (n, \\Lambda , M_0)> 1$ such that the following holds true.",
"Corresponding to any $\\epsilon > 0 $ , there exists a small constant $\\delta ~=~\\delta (\\epsilon , \\Lambda , M_0, n)$ such that $\\mathbb {A}\\in \\mathcal {A}_{4}(\\delta , \\mu , \\Lambda , B_{1})$ , $u\\in W^{1, 2}_{0}(\\Omega , \\mu )$ is a weak solution to (REF ) where $B_{6}^{+}\\subset \\Omega _{6}\\subset B_{6}\\cap \\lbrace x_{n} > -12 \\delta \\rbrace ,$ $\\Omega _{1}\\cap \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) \\le 1 \\rbrace \\cap \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2}\\chi _{\\Omega }\\right) \\le \\delta ^{2} \\rbrace \\ne \\emptyset ,$ then $\\mu (\\lbrace x\\in \\Omega _{1}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2}\\rbrace ) < \\epsilon \\mu (B_{1}).$ Let $\\epsilon > 0$ be given.",
"The assumption (REF ) implies that $B_{4}^{+}\\subset \\Omega _{4}\\subset B_{4}\\cap \\lbrace x_{n} > -16 \\delta \\rbrace .$ By Lemma REF and Proposition REF corresponding to $K=4$ the following holds: for every $\\eta > 0$ there exits $\\delta \\in (0, 1/4)$ such that if $u$ is a weak solution to (REF ), $\\frac{1}{\\mu (B_{4})} \\int _{\\Omega _{4}} |\\mathbb {A} - \\langle \\mathbb {A}\\rangle _{B_{4}}|^{2} \\mu ^{-1}dx \\le \\delta , \\quad \\frac{1}{\\mu (B_{4})}\\int _{\\Omega _{4}} \\Big | \\frac{{\\bf F}}{\\mu }\\Big |^2 d\\mu (x) \\le \\delta ^2, \\text{and}\\quad \\frac{1}{\\mu (B_{4})}\\int _{\\Omega _{4}} |\\nabla u|^{2}d\\mu \\le 1$ then there exists a constant matrix $\\mathbb {A}_{0}$ and a weak solution $v$ to (REF ) satisfying $\\Vert \\langle \\mathbb {A}\\rangle _{B_{4}} - \\mathbb {A}_{0}\\Vert < \\eta \\, \\frac{\\mu (B_{4})}{|B_{4}|},\\,\\, \\frac{1}{\\mu (B_{2})}\\int _{\\Omega _{2}}|\\nabla u-\\nabla v| ^{2} d\\mu < \\eta ,\\quad \\text{and\\, $\\Vert \\nabla v\\Vert _{L^{\\infty }(B_{2}^+)} \\le C_{0}$},$ for some positive constant $C_{0}$ that depend only $n, \\Lambda $ and $M_{0}$ .",
"Using this $\\delta $ in the assumption (REF ) there exists $x_{0}\\in \\Omega _{1}$ such that for any $r > 0$ $\\frac{1}{\\mu (B_{r}(x_{0}))}\\int _{\\Omega _{r}(x_{0})} \\chi _{\\Omega }|\\nabla u|^{2} d\\mu (x) \\le 1,\\quad \\text{and } \\frac{1}{\\mu (B_{r}(x_{0}))}\\int _{\\Omega _{r}(x_{0})} \\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{\\Omega } d\\mu (x) \\le \\delta ^{2}.$ We now make several observations.",
"First, $\\Omega _{4}(0)\\subset \\Omega _{5}(x_{0})\\subset \\Omega $ and $B_{4}^{+}\\subset \\Omega _{4}\\subset B_{4}\\cap \\lbrace x_{n} > -16\\delta \\rbrace $ and therefore we have from (REF ) that $\\frac{1}{\\mu (B_{4})}\\int _{\\Omega _{4}} |\\nabla u|^{2} d\\mu (x)\\le \\frac{\\mu (B_{5}(x_0))}{\\mu (B_{4})} \\frac{1}{\\mu (B_{5}(x_{0}))} \\int _{\\Omega _{5}(x_0)} \\chi _{\\Omega }|\\nabla u|^{2} d\\mu (x) \\le M_{0}\\left(\\frac{5}{4}\\right)^{2n},$ and similarly $\\frac{1}{\\mu (B_{4})}\\int _{\\Omega _{4}} \\left|\\frac{{\\bf F}}{\\mu }\\right|^{2}d\\mu (x) \\le M_{0}\\left(\\frac{5}{4}\\right)^{2n}\\delta ^{2}.$ Denote $\\kappa = M_{0}\\left(\\frac{5}{4}\\right)^{2n}$ .",
"Then since $\\mathbb {A}\\in \\mathcal {A}_{6}(\\delta , \\mu , \\Lambda , B_{1})$ by assumption, the above calculation show that conditions in (REF ) are satisfied for $u$ replace by $ u_{\\kappa } = u/\\kappa $ and ${\\bf F}$ replaced by ${\\bf F} _{\\kappa } = {\\bf F}/\\kappa $ , where $u_{\\kappa }$ will remain a weak solution corresponding to ${\\bf F}/k$ .",
"So all in (REF ) will be true where $v$ will be replaced by $v_{\\kappa } := v/\\kappa $ .",
"Second, taking $M^{2} = \\max \\lbrace M_{0} 3^{2n}, 4C^{2}_{0} \\rbrace $ , we have that $\\lbrace x: \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u_{\\kappa }|) ^{2} > M^{2}\\rbrace \\cap \\Omega _{1} \\subset \\lbrace x: \\mathcal {M}^{\\mu }(\\chi _{\\Omega _{2}}|\\nabla u_{\\kappa } -\\nabla v_{\\kappa }|^{2})>C_{0}^{2}\\rbrace \\cap \\Omega _{1}.",
"$ In fact, otherwise there will exist $x\\in \\Omega _{1}$ , and $ \\mathcal {M}^{\\mu }(\\chi _{\\Omega _{2}}|\\nabla u_{\\kappa } -\\nabla v_{\\kappa }|^{2})(x)\\le C_{0}^{2}$ .",
"We show that for any $r>0$ $_{B_{r}(x)}\\chi _{\\Omega }|\\nabla u_{\\kappa }|^{2}d\\mu \\le M^{2}.$ Now, if $r \\le 1$ , then $B_{r}(x)\\subset B_{2}$ , using the fact that $\\Vert \\nabla v_{\\kappa }\\Vert _{L^{\\infty }(\\Omega \\cap B_{r}(x))} \\le \\Vert \\nabla v_{\\kappa }\\Vert _{L^{\\infty }(\\Omega _{2})} \\le C_{0}$ , $_{B_{r}(x)}\\chi _{\\Omega }|\\nabla u_{\\kappa }|^{2}d\\mu \\le 2_{B_{r}(x)}\\chi _{\\Omega _{2}}|\\nabla u_{\\kappa }-\\nabla v_{\\kappa }|^{2} d\\mu + 2_{B_{r}(x)}\\chi _{\\Omega _{2}}|\\nabla v_{\\kappa }|^{2}d\\mu \\le 4 C_{0}^{2}.$ If $r > 1$ , then note first that $B_{r}(x) \\subset B_{3r}(x_0)$ and, so scaling the first inequality in (REF ) by $\\kappa > 1$ we obtain that $\\Omega _{B_{r}(x)}\\chi _{\\Omega }|\\nabla u_{\\kappa }|^{2}d\\mu (x) \\le \\frac{\\mu (B_{3r}(x_{0}))}{\\mu (B_{r}(x))}_{B_{3r}(x_0)} \\chi _{\\Omega }|\\nabla u_{\\kappa }|^{2}d\\mu (x) <M_{0}3^{2n}.$ Finally, set $\\varpi = \\max \\lbrace \\kappa M\\rbrace $ .",
"Then since $\\varpi >M$ we have that $\\begin{split}\\mu ( \\lbrace x\\in \\Omega _{1}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2} \\rbrace )& \\le \\mu (\\lbrace x\\in \\Omega _{1}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u_{\\kappa }|^{2}) > M^{2} \\rbrace )\\\\& \\le \\mu (\\lbrace x\\in \\Omega _{1}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega _{2}}|\\nabla u_{\\kappa } -\\nabla v_{\\kappa }|^{2})>C_{0}^{2}\\rbrace )\\\\& \\le \\frac{C(n, M_{0})}{C_0^2}\\ \\mu (B_{2}) _{B_{2}}\\chi _{\\Omega _{2}} |\\nabla u_{\\kappa } - \\nabla v_{\\kappa }|^{2} d\\mu \\\\&\\le C\\ \\eta \\ \\mu (B_{2}) \\\\&\\le C \\,M_{0} 2^{2n} \\eta \\ \\mu (B_{1}),\\end{split}$ where $C(n, M_{0})$ comes from the weak $1-1$ estimates in the $\\mu $ measure.",
"Now we choose $\\eta > 0$ small, and along the way $\\delta = \\delta (\\eta )$ such that $ C M_{0} 2^{2n} \\eta < \\epsilon $ .",
"By scaling and translating, we can prove the following result by using Lemma REF .",
"Lemma 5.8 Suppose that $M_{0}>0$ and $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ .",
"There exists a constant $\\varpi = \\varpi (n, \\Lambda , M_0)> 1$ such that the following holds true.",
"Corresponding to any $\\epsilon > 0 $ , there exists a small constant $\\delta = \\delta (\\epsilon , \\Lambda , M_0,n)$ such that for any $u\\in W^{1, 2}_{0}(\\Omega , \\mu )$ a weak solution to corresponding to $\\mathbb {A}\\in \\mathcal {A}_{6r}(\\delta , \\mu , \\Lambda , \\Omega )$ , any $y = (y^{\\prime }, y_{n})\\in \\Omega $ and $r > 0$ with $B_{6r}^{+}(y)\\subset \\Omega _{6r}(y)\\subset B_{6r}(y)\\cap \\lbrace x_{n} > y_{n}-12r\\delta \\rbrace ,$ and that $\\Omega _{r}(y)\\cap \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) \\le 1 \\rbrace \\cap \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2}\\chi _{\\Omega }\\right) \\le \\delta ^{2} \\rbrace \\ne \\emptyset ,$ then $\\mu (\\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2}\\rbrace \\cap \\Omega _{r}(y)) < \\epsilon \\mu (B_{r}(y)).$ The following proposition on the density of level sets is the main result of the section.",
"Proposition 5.9 Suppose that $M_{0}>0$ and $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ .",
"Let $\\varpi = \\varpi (n, \\Lambda , M_0)> 1$ validate Proposition REF and Lemma REF .",
"For any $\\epsilon >0$ , there exists $\\delta = \\delta (\\epsilon , \\Lambda , M_0, n)> 0$ such that the following holds true.",
"Suppose that $u\\in W^{1,2}_{0}(\\Omega )$ is a weak solution, $\\Omega $ is $(\\delta , R)$ -Reifenberg flat and $\\mathbb {A}\\in \\mathcal {A}_{R}(\\delta , \\mu ,\\Lambda , \\Omega )$ for some $R>0$ .",
"Then if $y\\in \\overline{\\Omega }$ , $r > 0$ such that $0 < r < R/1000$ and $\\mu (\\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2}\\rbrace \\cap \\Omega _{r}(y)) \\ge \\epsilon \\mu (B_{r}(y))$ then $\\Omega _{r}(y)\\subset \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) >1 \\rbrace \\cup \\lbrace x\\in \\mathbb {R}^{n}: \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2}\\chi _{\\Omega }\\right) > \\delta ^{2} \\rbrace .$ Note that if $B_{8r}(y)\\subset \\Omega $ , then the result is precisely Proposition REF .",
"Therefore, we only need to prove this proposition when $B_{8r}(y)\\cap \\partial \\Omega \\ne \\emptyset $ .",
"We argue by contradiction.",
"Assume that the proposition is false.",
"Then there exists a constant $\\epsilon _{0} > 0$ such that corresponding to each $\\delta $ , we can find a $(\\delta , R)$ Reifenberg flat domain $\\Omega $ with some $R>0$ , a coefficient $\\mathbb {A}\\in \\mathcal {A}_{R}(\\delta , \\mu , \\Lambda , \\Omega )$ , a solution $u \\in W^{1,2}_0(\\Omega , \\mu )$ , and some $r \\in (0, R/1000)$ , $y \\in \\overline{\\Omega }, x_{0}\\in \\Omega _{r}(y)$ such that (REF ) holds and $\\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2})(x_{0}) \\le 1,\\quad \\text{and} \\,\\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2}\\chi _{\\Omega }\\right)(x_{0}) \\le \\delta ^{2}.$ Now, let $\\epsilon = \\frac{\\epsilon _{0}}{M_{0}144^{2n}}$ , and with this $\\epsilon $ we choose $\\delta ^{\\prime } < 1/7$ be as in Lemma REF .",
"Let $\\delta = \\frac{\\delta ^{\\prime }}{1 + \\delta ^{\\prime }}$ , and corresponding to this $\\delta $ , let $\\Omega , r >0, y, x_0$ and $u \\in W^{1,2}_0(\\Omega , \\mu )$ be as in the above statement.",
"For $y_{0}\\in \\partial \\Omega \\cap B_{8r}(y)$ , we observe that $x_{0} \\in \\Omega _r(y) = B_{r}(y)\\cap \\Omega \\subset \\Omega _{9r}(y_0) = B_{9r}(y_{0})\\cap \\Omega .$ Also, let $M=432r$ and $\\rho = \\frac{M(1-\\delta )}{6}$ .",
"We observe that $6\\rho < R(1-\\delta )$ .",
"Therefore, since $\\Omega $ is $(\\delta , R)$ Reifenberg flat domain, there exists a coordinate system $\\lbrace z_1, z_2,\\cdots , z_n\\rbrace $ in which $y_{0} = -\\delta M (1-\\delta ) z_{n}\\in \\partial \\Omega , \\quad y = \\hat{z},\\quad x_{0} = z_{0},$ and $B_{6\\rho }^{+}(0)\\subset \\Omega _{6\\rho } \\subset B_{6\\rho } \\cap \\lbrace z_{n} > -12\\rho \\delta ^{\\prime }\\rbrace .$ We claim that $z_0 \\in B_{\\rho }(0)$ .",
"Indeed, in the new coordinate system $|\\hat{z}| < 8r + \\delta M$ , and therefore $|z_{0}| \\le 9r + \\delta M \\le \\rho .$ In summary, up to a change of coordinate system, and after a simple calculation (i) $u\\in W^{1, 2}_{0}(\\Omega , \\mu )$ is a weak solution to (REF ), (ii) $B_{6\\rho }^{+}(0)\\subset \\Omega _{6\\rho }\\subset B_{6\\rho } \\cap \\lbrace z_{n}> -12\\rho \\delta ^{\\prime }\\rbrace $ , (iii) $z_{0}\\in B_{\\rho }(0)\\cap \\lbrace \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u|^{2}) \\le 1 \\rbrace \\cap \\lbrace \\mathcal {M}(\\chi _{\\Omega } \\left|\\frac{{\\bf F}}{\\mu }\\right|^{2}) \\le \\delta ^{\\prime 2}\\rbrace , and $ (iv) $B_{r}(y) \\subset B_{\\rho }(0)\\subset B_{144r}(y)$ .",
"Thus, from items $(i)$ -$(iii)$ we see that all the hypotheses of Lemma REF are satisfied with $B_{\\rho }(0)$ replacing $B_{r}(y)$ .",
"We thus conclude that $\\mu (\\Omega _{\\rho }(0)\\cap \\lbrace \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u|^{2}) > \\varpi ^{2} \\rbrace ) < \\epsilon \\mu (B_{\\rho }(0)).$ Moreover, from item $(iv)$ we have that $\\begin{split}\\mu (\\Omega _{r}(y)\\cap \\lbrace \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u|^{2}) > \\varpi ^{2} \\rbrace ) &\\le \\mu (\\Omega _{\\rho }(0)\\cap \\lbrace \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u|^{2}) > \\varpi ^{2} \\rbrace )\\\\&< \\frac{\\epsilon _{0}}{M_{0}144^{2n}}\\mu (B_{\\rho }(0)) \\le \\frac{\\epsilon _{0}}{M_{0}144^{2n}}\\mu (B_{144r}(y)) \\\\&\\le \\frac{\\epsilon _{0}}{M_{0}144^{2n}}M_{0}144^{2n} \\mu (B_{r}(y)) \\\\&= \\epsilon _{0}\\mu (B_{r}(y)),\\end{split}$ where we have used the doubling property of the $\\mu $ .",
"The last sequence of inequalities obviously contradict the hypothesis (REF ) of the theorem, and thus the proof is complete."
],
[
"Proof of the global $W^{1,p}$ -regularity estimates",
"Our first statement of the subsection, which is the key in obtaining the higher gradient integrability of solution, gives the level set estimate of $\\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2})$ in terms that of $\\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{\\Omega }\\right)$ .",
"Lemma 5.10 Let $M_{0}>0$ and $\\mu \\in A_{2}$ such that $[\\mu ]_{A_{2}} \\le M_{0}$ .",
"Let $\\varpi = \\varpi (n, \\Lambda , M_0)> 1$ be the constant defined in Proposition REF .",
"For a given $\\epsilon > 0 $ , there is $\\delta = \\delta (\\epsilon , \\Lambda , M_0) < 1/4$ such that for any $u\\in W^{1, 2}_{0}(\\Omega , \\mu )$ a weak solution of (REF ) to with $\\mathbb {A}\\in \\mathcal {A}_{R}(\\delta , \\mu , \\Lambda , \\Omega )$ , and $\\Omega $ a $(\\delta , R)$ Reifenberg flat domain, and for a fixed $0< r_{0} < R/2000$ , if $\\mu ( \\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2}\\rbrace ) <\\epsilon \\mu (B_{r_{0}}(y)), \\quad \\forall y \\in \\overline{\\Omega },$ then for any $k\\in \\mathbb {N}$ and $\\epsilon _{1} = \\left(\\frac{10}{1 - 4\\delta }\\right)^{2n} M_{0}^{2} \\epsilon $ , we have that $\\begin{split}\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2k} \\rbrace ) &\\le \\sum _{i=1}^{k} \\epsilon _{1}^{i} \\mu \\left(\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{\\Omega }\\right) >\\delta ^{2} \\varpi ^{2(k-i)} \\rbrace \\right)\\\\&\\quad \\quad + \\epsilon _{1}^{k}\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u|^{2}) > 1 \\rbrace ).\\end{split}$ We will use induction on $k$ .",
"For the case $k=1$ , we are going to apply Lemma REF , by taking $C = \\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2} \\rbrace ,$ and $D = \\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{\\Omega }\\right) >\\delta ^{2} \\rbrace \\cup \\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u|^{2}) > 1 \\rbrace .$ By assumption, $\\mu (C)< \\epsilon \\mu (B_{r_{0}} (y))$ for all $y\\in \\overline{\\Omega }$ .",
"Also for any $y\\in \\Omega $ and $\\rho \\in (0, 2r_{0}) $ , then $\\rho \\in (0, R/1000)$ and if $\\mu (C \\cap B_{\\rho }(y))\\ge \\epsilon \\mu ((B_{\\rho }(y))$ , then by Proposition REF we have that $B_{\\rho }(y)\\cap \\Omega \\subset D.$ Thus, all the conditions of Lemma REF are satisfied and therefore we have $\\mu (C) \\le \\epsilon _{1} \\mu (D).$ which is exactly the case when $k=1$ .",
"Assume it is true for $k$ .",
"We will show the statement for $k+1$ .",
"We normalize $u$ to $u_{\\varpi } = u/\\varpi $ and ${\\bf F}_{\\varpi } = {\\bf F}/\\varpi $ , and we see that since $\\varpi > 1$ we have $\\begin{split}& \\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u_{\\varpi }|^{2}) > \\varpi ^{2} \\rbrace ) \\\\&= \\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{4} \\rbrace )\\\\& \\le \\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2} \\rbrace ) \\le \\epsilon \\mu (B_{r_{0}}(y)), \\quad \\forall y \\in \\overline{\\Omega }.\\end{split}$ By induction assumption, it follows then that $\\begin{split}\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2(k+1)} \\rbrace )&=\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u_{\\varpi }|^{2}) > \\varpi ^{2k} \\rbrace )\\\\&\\le \\sum _{i=1}^{k} \\epsilon _{1}^{i} \\mu \\left(\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}_{\\varpi }}{\\mu }\\right|^{2} \\chi _{\\Omega }\\right) >\\delta ^{2} \\varpi ^{2(k-i)} \\rbrace \\right)\\\\&\\quad \\quad + \\epsilon _{1}^{k}\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u_{\\varpi }|^{2}) > 1 \\rbrace )\\\\& = \\sum _{i=1}^{k} \\epsilon _{1}^{i} \\mu \\left(\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{\\Omega }\\right) >\\delta ^{2} \\varpi ^{2(k+1-i)} \\rbrace \\right)\\\\&\\quad \\quad + \\epsilon _{1}^{k}\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u|^{2}) > \\varpi ^2 \\rbrace ).\\end{split}$ Applying the case $k=1$ to the last term we obtain that $\\begin{split}\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u|^{2}) > \\varpi ^{2(k+1)} \\rbrace )&\\le \\sum _{i=1}^{k + 1} \\epsilon _{1}^{i} \\mu \\left(\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}}{\\mu }\\right|^{2} \\chi _{\\Omega }\\right) >\\delta ^{2} \\varpi ^{2(k+1-i)} \\rbrace \\right)\\\\&\\quad \\quad + \\epsilon _{1}^{k+1}\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u|^{2}) > 1 \\rbrace ),\\end{split}$ as desired.",
"We divide the proof in two parts based on whether $p\\ge 2$ or $1 < p < 2$ .",
"Case 1: $p\\ge 2$ .",
"For this case, $A_2 \\subset A_p$ and therefore $ \\mu \\in A_2\\cap A_p = A_2$ .",
"Moreover, because ${\\bf F}/\\mu \\in L^{p}(\\Omega ,\\mu ) $ , then clearly ${\\bf F}/\\mu \\in L^{2}(\\Omega ,\\mu )$ .",
"Applying [14], a unique solution $u\\in W^{1, 2}_{0}(\\Omega ,\\mu )$ of (REF ) exists.",
"Moreover, it follows by the energy estimate that $\\Vert \\nabla u\\Vert _{L^{2}(\\Omega ,\\mu )} \\le C(\\Lambda ) \\left\\Vert \\frac{{\\bf F}}{\\mu }\\right\\Vert _{L^{2}(\\Omega ,\\mu )}.$ Our goal is to show that $\\nabla u\\in L^{p}(\\Omega ,\\mu )$ .",
"Let $\\epsilon >0$ be given, then $\\delta >0$ is chosen according to Lemma REF .",
"Also, take $r_{0} = R/2000$ and a ball $B = B_{s}(0)$ with sufficiently large $s$ depending only on $\\text{diam}(\\Omega ), r_0$ so that $ B_{r_0}(y) \\subset B, \\quad \\forall \\ y \\in \\overline{\\Omega }.$ Then by doubling property of $\\mu $ (REF ) we have $\\mu (\\Omega )\\le \\mu (B) \\le M_{0} \\left(\\frac{|B|}{|B_{r_{0}}(y)|}\\right)^{2} \\mu (B_{r_{0}}(y)) =M_0 \\left( \\frac{s}{r_0}\\right)^{2n} \\mu (B_{r_{0}}(y)) ,\\quad \\forall y\\in \\overline{\\Omega }.$ We claim we can choose $N$ large such that for $u_{N} = u/N$ , $\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u_{N}|^{2}) > \\varpi ^{2} \\rbrace )\\le \\epsilon \\mu ( B_{r_{0}} (y)),\\quad \\forall y\\in \\overline{\\Omega }.$ To see this we first assume that $\\Vert \\nabla u\\Vert _{L^{2}(\\Omega ,\\mu )} >0$ .",
"Then by weak (1,1) estimate for maximal functions there exists a constant $C(n, M_{0})>0$ such that $\\mu (\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega } |\\nabla u_{N}|^{2}) > \\varpi ^{2} \\rbrace ) \\le \\frac{C (n, M_{0})}{N^{2} \\varpi ^{2}} \\int _{\\Omega }|\\nabla u|^{2}d\\mu .$ Then, the claim follows if we select $N$ such that $\\frac{C(n, M_{0})}{N^{2} \\varpi ^{2}} \\int _{\\Omega }|\\nabla u|^{2}d\\mu =\\epsilon \\frac{\\mu (B)}{M_{0} \\left(\\frac{s}{r_0} \\right)^{2n}}.$ We observe that by the doubling property of $\\mu $ , it follows from the above estimate that $ N^2 \\mu (\\Omega ) \\le C(n, M_0, \\text{diam}(\\Omega ))\\int _{\\Omega } |\\nabla u|^2 \\mu (x) dx.$ Now consider the sum $S = \\sum _{k=1}^{\\infty } \\varpi ^{pk}\\mu (\\lbrace \\Omega : \\mathcal {M}(\\chi _{\\Omega }|\\nabla u_{N}|^{2} ) (x) > \\varpi ^{2k}\\rbrace ).$ Applying the previous corollary we have that $\\begin{split}S &\\le \\sum _{k=1}^{\\infty } \\varpi ^{pk} \\left[\\sum _{i=1}^{k } \\epsilon _{1}^{i} \\mu \\left(\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}_{N}}{\\mu }\\right|^{2} \\chi _{\\Omega }\\right) >\\delta ^{2} \\varpi ^{2(k-i)} \\rbrace \\right)\\right]\\\\&\\quad \\quad +\\sum _{k=1}^{\\infty } \\varpi ^{pk} \\epsilon _{1}^{k}\\mu (\\lbrace \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u_{N}|^{2} ) (x) > 1\\rbrace ).\\end{split}$ Applying summation by parts we have that $\\begin{split}S &\\le \\sum _{j=1}^{\\infty }( \\varpi ^{p} \\epsilon _{1})^{j} \\left[\\sum _{k=j}^{\\infty } \\varpi ^{p(k-j)} \\mu \\left(\\lbrace x\\in \\Omega : \\mathcal {M}^{\\mu }\\left(\\left|\\frac{{\\bf F}_{N}}{\\mu }\\right|^{2} \\chi _{\\Omega }\\right) >\\delta ^{2} \\varpi ^{2(k-j)} \\rbrace \\right)\\right]\\\\&\\quad \\quad +\\sum _{k=1}^{\\infty } (\\varpi ^{p} \\epsilon _{1})^{k}\\mu (\\lbrace \\Omega : \\mathcal {M}^{\\mu }(\\chi _{\\Omega }|\\nabla u_{N}|^{2} ) (x) > 1\\rbrace )\\\\&\\le C\\left(\\left\\Vert \\mathcal {M}^{\\mu }\\left(\\chi _{\\Omega }\\left|\\frac{{\\bf F}_{N}}{\\mu }\\right|^{2}\\right)\\right\\Vert ^{p/2}_{L^{p/2}(\\Omega , \\mu )} + \\Vert \\nabla u_{N}\\Vert ^{2}_{L^{2}(\\Omega , \\mu )}\\right) \\sum _{k=1}^{\\infty } (\\varpi ^{p} \\epsilon _{1})^{k}\\\\\\end{split}$ where we have applied the weak $(1, 1)$ estimate of the maximal function $\\mathcal {M}^{\\mu }$ .",
"Now chose $\\epsilon $ small so that $\\varpi ^{p} \\epsilon _{1} <1$ to obtain that $S \\le C\\left(\\left\\Vert \\mathcal {M}^{\\mu }\\left(\\chi _{\\Omega }\\left|\\frac{{\\bf F}_{N}}{\\mu }\\right|^{2}\\right)\\right\\Vert ^{p/2}_{L^{p/2}(\\Omega , \\mu )} + \\Vert \\nabla u_{N}\\Vert ^{2}_{L^{2}(\\Omega , \\mu )}\\right) \\le C\\left(\\left\\Vert \\frac{{\\bf F}_{N}}{\\mu }\\right\\Vert ^{p}_{L^{p}(\\Omega , \\mu )} + \\Vert \\nabla u_{N}\\Vert ^{2}_{L^{2}(\\Omega , \\mu )}\\right),$ where we have applied the strong $(p,p)$ estimate for the maximal function operator $\\mathcal {M}^{\\mu }$ .",
"Applying Lemma REF , we have $\\Vert \\nabla u_{N} \\Vert _{L^{p}(\\Omega , \\mu )}^{p} \\le C \\Vert \\mathcal {M}(\\chi _{\\Omega }|\\nabla u_{N}|^{2})\\Vert ^{p/2}_{L^{p/2}(\\Omega , \\mu )} \\le C (S + \\mu (\\Omega )),$ and therefore multiplying by $N^{p}$ and using formula (REF ) we have $ \\begin{split}\\Vert \\nabla u \\Vert _{L^{p}(\\Omega , \\mu )}^{p} &\\le C\\left(\\left\\Vert \\frac{{\\bf F}}{\\mu }\\right\\Vert ^{p}_{L^{p}(\\Omega , \\mu )} +N^{p-2} \\Vert \\nabla u\\Vert ^{2}_{L^{2}(\\Omega , \\mu )}+ N^{p}\\mu (\\Omega ) \\right)\\\\& \\le C\\left(\\left\\Vert \\frac{{\\bf F}}{\\mu }\\right\\Vert ^{p}_{L^{p}(\\Omega , \\mu )} +N^{p}\\mu (\\Omega ) \\right).\\end{split}$ Finally we estimate $N^{p}\\mu (\\Omega )$ using formula (REF ) and Hölder's inequality together with the energy estimate as $N^{p}\\mu (\\Omega ) \\le C(n, M_0, \\text{diam}(\\Omega )) N^{p-2} \\Vert \\nabla u\\Vert _{L^{2}(\\Omega ,\\mu )}^{2} \\le C N^{p-2}\\left\\Vert \\frac{{\\bf F}}{\\mu }\\right\\Vert ^{2}_{L^{2}(\\Omega , \\mu )} \\le C \\left\\Vert \\frac{{\\bf F}}{\\mu }\\right\\Vert ^{2}_{L^{p}(\\Omega , \\mu )} [N^{p}\\mu (\\Omega )]^{\\frac{p-2}{p}}.$ This estimate implies $N^{p}\\mu (\\Omega ) \\le C (n, M_0, \\text{diam}(\\Omega )) \\left\\Vert \\frac{{\\bf F}}{\\mu }\\right\\Vert ^{p}_{L^{p}(\\Omega , \\mu )}.$ By pluging the last estimate into (REF ), we obtain the desired estimate (REF ).",
"Case 2: $1 < p < 2$ .",
"In this case, $\\mu \\in A_2 \\cap A_p = A_p$ .",
"We use the standard duality argument.",
"Suppose that ${\\bf F}/\\mu \\in L^{p}(\\Omega ,\\mu )$ .",
"By density of $C_{c}^{\\infty }(\\Omega )$ in $L^{p}(\\Omega ,\\mu )$ , there exists a sequence of functions ${\\bf f}_{n}\\in C_{c}^{\\infty }(\\Omega ) $ such that ${\\bf f}_{n} \\rightarrow {\\bf F}/\\mu $ in $L^{p}(\\Omega ,\\mu )$ .",
"Corresponding to each ${\\bf f}_{n}$ , there exists $u_{n}\\in W^{1, 2}_{0}(\\Omega ,\\mu )$ solving the equation $\\text{div} (\\mathbb {A}(x)\\nabla u_{n}) = \\text{div}(\\mu {\\bf f}_{n})$ with the estimate $\\Vert u_{n}\\Vert _{W^{1, 2}_{0}(\\Omega ,\\mu )}\\le C \\Vert {\\bf f}_{n}\\Vert _{L^{2}(\\Omega ,\\mu )}$ where $C>0$ is independent of $u_{n}$ and ${\\bf f}_{n}$ .",
"Since $p\\in (1, 2)$ , $u_{n}\\in W^{1, p}_{0}(\\Omega ,\\mu )$ for all n. We claim that $u_{n}$ is in fact bounded in $W^{1, p}_{0}(\\Omega ,\\mu )$ .",
"To that end, let ${\\bf g}\\in L^{p^{\\prime }}(\\Omega ,\\mu )$ be given with $\\Vert {\\bf g}\\Vert _{L^{p^{\\prime }}(\\Omega ,\\mu )} \\le 1$ , where $p^{\\prime }$ is the Hölder conjugate of $p$ .",
"Since $p^{\\prime } > 2$ , by case 1, there exists a small constant $\\delta > 0$ such that whenever $\\mathbb {A}\\in \\mathcal {A}_{R_{0}}(\\delta , \\mu , \\Lambda , \\Omega )$ , there exists a function $w\\in W^{1, p^{\\prime }}_{0}(\\Omega ,\\mu )$ that solves $\\text{div} (\\mathbb {A}(x)\\nabla w) = \\text{div}(\\mu {\\bf g})\\quad \\text{in $\\Omega $} $ weakly accompanied by the estimate $\\Vert \\nabla w\\Vert _{W^{1, p^{\\prime }}_{0}(\\Omega ,\\mu )}\\le C \\Vert {\\bf g}\\Vert _{L^{p^{\\prime }}(\\Omega ,\\mu )},$ where $C$ is independent of $w$ and ${\\bf g}$ .",
"Now, we have that $\\begin{split}\\int _{\\Omega }\\langle \\nabla u_{n}(x), {\\bf g}(x) \\rangle d\\mu (x) &= \\int _{\\Omega }\\langle \\nabla u_{n}(x),\\mu {\\bf g}(x) \\rangle dx\\\\& = \\int _{\\Omega }\\langle \\nabla u_{n}(x), \\mathbb {A}(x)\\nabla w \\rangle dx = \\int _{\\Omega }\\langle \\mathbb {A}(x)\\nabla u_{n}(x), \\nabla w \\rangle dx = \\int _{\\Omega }\\langle {\\bf f}_{n}, \\nabla w \\rangle d\\mu (x),\\end{split}$ where we have used the fact for each $n$ , the function $u_{n}$ , is an allowable test function for the equation involving $w$ and vice versa.",
"And therefore by the definition of the dual norm, $\\begin{split}\\Vert \\nabla u_{n}\\Vert _{L^{p}(\\Omega ,\\mu )} = \\sup _{\\Vert {\\bf g}\\Vert _{L^{p^{\\prime }}(\\Omega ,\\mu )} \\le 1 }\\left|\\int _{\\Omega }\\langle \\nabla u_{n}(x), {\\bf g}(x)\\rangle d\\mu (x)\\right| &\\le \\sup _{\\Vert {\\bf g}\\Vert _{L^{p^{\\prime }}(\\Omega ,\\mu )} \\le 1 }\\Vert {\\bf f}_{n}\\Vert _{L^{p}(\\Omega ,\\mu )} \\Vert \\nabla w\\Vert _{L^{p^{\\prime }}(\\Omega ,\\mu )} \\\\&\\le C \\Vert {\\bf f}_{n}\\Vert _{L^{p}(\\Omega ,\\mu )} \\le C \\left\\Vert {\\bf F}/\\mu \\right\\Vert _{L^{p}(\\Omega ,\\mu )}.\\end{split}$ Therefore by Poincare's inequality, which we can apply because $\\mu \\in A_{p}$ , $u_{n}$ is bounded in $W^{1, p}_{0}(\\Omega ,\\mu )$ , and thus has a weak limit $u$ in $W^{1, p}_{0}(\\Omega ,\\mu )$ .",
"Clearly $u$ solves the equation $\\text{div} (\\mathbb {A}(x)\\nabla u) = \\text{div}({\\bf F})$ weakly.",
"Moreover, we also have the estimate $\\Vert \\nabla u\\Vert _{L^{p}(\\Omega ,\\mu )} \\le \\liminf _{n\\rightarrow \\infty } \\Vert \\nabla u_{n}\\Vert _{L^{p}(\\Omega ,\\mu )} \\le C \\left\\Vert {\\bf F}/\\mu \\right\\Vert _{L^{p}(\\Omega ,\\mu )}$ as desired.",
"What is left is to show the above solution $u$ is unique.",
"To that end it suffices to show that if $u\\in W^{1,p}_{0}(\\Omega ,\\mu )$ and $\\text{div} (\\mathbb {A}(x)\\nabla u) =0,$ then $u=0.$ To show this, we begin by noting that $|\\nabla u|^{p-2}\\nabla u \\in L^{p^{\\prime }}(\\Omega ,\\mu )$ , $p^{\\prime }>2$ and that there is a weak solution $w\\in W^{1, p^{\\prime }}_{0}(\\Omega ,\\mu )$ to $\\text{div} (\\mathbb {A}(x)\\nabla w) = \\text{div}(\\mu |\\nabla u|^{p-2}\\nabla u)$ .",
"Using $u$ as a test function for the equation of $w$ and vice versa, we obtain that $\\int _{\\Omega } |\\nabla u|^{p}d\\mu = \\int _{\\Omega }\\langle \\mu |\\nabla u|^{p-2}\\nabla u, \\nabla u\\rangle dx = \\int _{\\Omega } \\langle \\mathbb {A}(x)\\nabla w, \\nabla u \\rangle dx = \\int _{\\Omega } \\langle \\mathbb {A}(x)\\nabla u, \\nabla w \\rangle dx = 0.$ This implies that $\\nabla u = 0$ a.e.",
"and therefore $u\\equiv 0$ .",
"That concludes the proof."
],
[
"Proof of Lemma ",
"The proof of Lemma REF follows from the following two lemmas.",
"The first lemma implies that $[\\mu ]_{A_2}$ can be bounded by a uniform constant depending only on $n$ if $|\\alpha | \\le 1$ .",
"Lemma 1.1 Let $\\mu (x) = |x|^{\\alpha }$ for $x \\in \\mathbb {R}^n$ and $|\\alpha | < n$ .",
"Then we have the estimate $[\\mu ]_{A_{2}} \\le \\max \\left\\lbrace 2^{n}5^{|\\alpha |}, \\frac{2^{4n}}{(n+ \\alpha )(n-\\alpha )}\\right\\rbrace .$ In particular, if $|\\alpha | \\le n_{0} < n$ , then we can bound $[\\mu ]_{A_{2}}$ from above independent of $\\alpha $ as $[\\mu ]_{A_{2}} \\le \\max \\left\\lbrace 2^{n+3} , \\frac{2^{4n}}{n^2 - n_{0}^2}\\right\\rbrace .$ Following the calculations in [18] we classify balls $B_{r}(x_{0})$ as type I if $|x_{0}| \\ge 3r$ and type II if $|x_{0}|\\le 3r$ .",
"For type I balls, we have that $||x_{0}| + 2r| \\le 4 ||x_{0}| -r|$ and $||x_{0}| -2r| \\ge \\frac{1}{4}||x_{0}| + r|$ .",
"Then it follows from [18] that $\\begin{split}_{B_{r}(x_{0})}\\mu (x)dx _{B_{r}(x_{0})} \\mu (x)^{-1} dx &= \\frac{1}{|B_{r}(x_{0})|^{2}} \\int _{B_{r}(x_{0})}|x|^{\\alpha }dx \\int _{B_{r}(x_{0})} |x|^{-\\alpha } dx \\\\&\\le \\frac{1}{|B_{r}(x_{0})|^{2}} \\int _{B_{2r}(x_{0})}|x|^{\\alpha }dx \\int _{B_{2r}(x_{0})} |x|^{-\\alpha } dx\\\\& \\le 2^{n} \\left\\lbrace \\begin{split} \\left(\\frac{x_{0} + 2r}{x_{0} - 2r}\\right)^{\\alpha }, & \\quad \\text{if $\\alpha \\in [0, n)$}\\\\\\left(\\frac{x_{0} - 2r}{x_{0} + 2r}\\right)^{\\alpha },&\\quad \\text{if $\\alpha \\in (-n, 0)$}\\end{split}\\right.\\\\&= 2^{n} \\left( 1 + \\frac{ 4r}{x_{0} - 2r}\\right)^{|\\alpha |} \\le 2^{n}5^{|\\alpha |}.\\end{split}$ For type II balls, $B_{r}(x_{0}) \\subset B_{4r}(0)$ and therefore we have $\\begin{split}_{B_{r}(x_{0})}\\mu (x)dx _{B_{r}(x_{0})} \\mu (x)^{-1} dx &= \\frac{1}{|B_{r}(x_{0})|^{2}} \\int _{B_{4r}(0)}|x|^{\\alpha }dx \\int _{B_{4r}(0)} |x|^{-\\alpha } dx \\\\&\\le \\frac{\\nu _{n}^{2}}{|B_{r}(x_{0})|^{2}}\\frac{(4r)^{n+ \\alpha }}{n + \\alpha } \\frac{(4r)^{n-\\alpha }}{n-\\alpha } \\\\&= \\frac{\\nu _{n}^{2}}{|B_{r}(x_{0})|^{2}}\\frac{(4r)^{2n}}{(n + \\alpha )(n-\\alpha )} = \\frac{2^{4n}}{(n + \\alpha )(n-\\alpha )}.\\end{split}$ Lemma 1.2 Let $\\mu (x) = |x|^\\alpha $ for $x \\in \\mathbb {R}^n$ and $|\\alpha | < 1$ .",
"Then, $\\int _{B_{r} (x_0)} \\Big | \\mu (x) - \\bar{\\mu }_{B_{r}(x_0)}\\Big | dx \\le \\frac{2|\\alpha |4^{2n+1} }{2n-1} \\int _{B_r(x_0)} \\mu (x) dx, \\quad \\forall x_0 \\in \\mathbb {R}^n, \\quad \\forall r >0.$ We first need to perform some elementary calculations.",
"Note that $ \\mu (B_r(0)) = \\int _{B_r(0)} |x|^\\alpha dx = \\omega _n \\int _{0}^r s^{n+\\alpha -1} ds =\\frac{\\omega _n r^{n+\\alpha }}{n + \\alpha },$ where $\\omega _n$ is the Lebesgue measure of the unit sphere in $\\mathbb {R}^n$ .",
"On the other hand, for every $r >0$ , we have $\\begin{split}\\frac{1}{|B_r(0)|}\\int _{B_r(0)}\\int _{B_r (0)} \\Big | |x|^\\alpha - |y|^\\alpha \\Big | dx & = \\frac{n}{\\omega _n r^n} \\int _{B_r(0)} \\int _{B_r(0)} \\Big | |x|^\\alpha - |y|^\\alpha \\Big | dx dy \\\\& = \\frac{n \\omega _n}{ r^n} \\int _0^r \\int _0^r |s^\\alpha - t^\\alpha | s^{n-1} t^{n-1} ds dt \\\\& = \\frac{\\text{sgn}(\\alpha )2n \\omega _n}{ r^n} \\int _0^r s^{n-1} \\int _0^s (s^\\alpha - t^\\alpha ) t^{n-1} dt ds \\\\& = \\frac{\\text{sgn}(\\alpha )2n \\omega _n}{ r^n} \\int _0^r s^{n-1} \\left( \\frac{s^{n+\\alpha }}{n} - \\frac{s^{n+\\alpha }}{n+\\alpha } \\right) ds \\\\& = \\frac{\\text{sgn}(\\alpha )\\alpha 2\\omega _n}{ (n+\\alpha ) r^n} \\int _0^r s^{2n + \\alpha -1}ds .\\end{split}$ Noting that $|\\alpha | = \\text{sgn}(\\alpha ) \\alpha $ , we conclude that $ \\frac{1}{|B_r(0)|}\\int _{B_r(0)}\\int _{B_r (0)} \\Big | |x|^\\alpha - |y|^\\alpha \\Big | dx =\\frac{2 |\\alpha | \\omega _n r^{n+ \\alpha }}{ (n+\\alpha ) (2n +\\alpha ) }, \\quad \\forall \\ r >0.$ The proof now is divided in two cases depending on the locations and sizes of the balls.",
"Case I: We consider balls $B_r(x_0)$ with $ r > |x_0|/3$ .",
"In this case, note that $B_r(x_0) \\subset B_{4r}(0)$ .",
"Therefore, $ \\begin{split}\\int _{B_r(x_0)} \\Big | \\mu (x) - \\overline{\\mu }_{B_r(x_0)} \\Big | dx & \\le \\frac{1}{|B_r(x_0)|} \\int _{B_r(x_0)} \\int _{B_r(x_0)} \\Big ||x|^\\alpha - |y|^\\alpha \\Big | dx dy \\\\& \\le \\frac{4^n}{|B_{4r}(x_0)|} \\int _{B_{4r}(0)} \\int _{B_{4r}(0)} \\Big | |x|^\\alpha - |y|^\\alpha \\Big | dx dy \\\\& = \\frac{2 |\\alpha | \\omega _n 4^{2n + \\alpha } r^{n+ \\alpha }}{(n+\\alpha ) (2n +\\alpha ) },\\end{split}$ where in the last equality, we used (REF ).",
"On the other hand, we claim that $\\mu (B_r(x_0)) = \\int _{B_r(x_0)} |x|^\\alpha dx \\ge \\left\\lbrace \\begin{split} \\frac{\\omega _{n} }{n + \\alpha }r^{n + \\alpha }, &\\quad \\text{if $0 \\le \\alpha < 1$}, \\\\\\omega _{n}4^{\\alpha } r^{n + \\alpha }, &\\quad \\text{if $-1 < \\alpha \\le 0$}.\\end{split}\\right.$ Then, by combining (REF ), (REF ), and (REF ) we infer that, since $|\\alpha | < 1$ $\\int _{B_r(x_0)} \\Big | \\mu (x) - \\overline{\\mu }_{B_r(x_0)} \\Big | dx \\le \\frac{|\\alpha |2 \\cdot 4^{2n + 1} }{2n +\\alpha } \\mu (B_r(x_0)).$ Therefore, for this case, it remains to show (REF ).",
"The case when $-1 < \\alpha \\le 0$ is easy since $x\\in B_{r}(x_{0})$ and $|x_{0}| < 3r$ implies that $|x| < 4r$ .",
"To prove the inequality for the case $0 \\le \\alpha < n$ , we proceed as follow.",
"Because $\\mu $ is radial, by rotation, we can assume that $x_0 = (0^{\\prime }, a)$ , where $a = |x_0| \\ge 0$ and $0^{\\prime } $ is the origin of $\\mathbb {R}^{n-1}$ .",
"Then, we write $f(a): = \\int _{B_r(x_0)} \\mu (x) dx = \\int _{B_r(0^{\\prime })} \\int _{a - \\sqrt{r^2 -|x^{\\prime }|^2}}^{a + \\sqrt{r^2 -|x^{\\prime }|^2}} [|x^{\\prime }|^2 + (x_n)^2]^{\\alpha /2} dx_n dx^{\\prime },$ where $B_r(0^{\\prime })$ is the ball in $\\mathbb {R}^{n-1}$ centered at $0^{\\prime }$ .",
"Note that since $\\alpha \\ge 0$ , the fundamental theorem of calculus gives $f^{\\prime }(a) = \\int _{B_r(0^{\\prime })} \\left\\lbrace \\Big [|x^{\\prime }|^2 + (a + \\sqrt{r^2 -|x^{\\prime }|^2})^2\\Big ]^{\\alpha /2} -\\Big [|x^{\\prime }|^2 + (a - \\sqrt{r^2 -|x^{\\prime }|^2})^2\\Big ]^{\\alpha /2} \\right\\rbrace dx^{\\prime }\\ge 0.$ Hence, $f(a) \\ge f(0)$ which is (REF ).",
"Case II: We consider balls $B_r(x_0)$ with $0< r \\le |x_0|/3$ .",
"In this case, note that since $ 0 \\notin B_r(x_0)$ , $\\mu $ is smooth in $B_r(x_0)$ .",
"Let us first consider the case that $0\\le \\alpha < 1$ .",
"Applying mean value theorem for every $x, y \\in B_r(x_0)$ , there is $x^*$ in between $x$ and $y$ such that $\\Big | |x|^\\alpha - |y|^\\alpha \\Big | = |\\alpha | |x-y| |x^*|^{\\alpha -1} \\le 2 |\\alpha | r |x^*|^{\\alpha -1}.$ Also, since $x^* \\in B_r(x_0)$ , we have $|x^*| \\ge |x_0| - |x_0 - x^*| \\ge 3r -r = 2r.$ This together with the observation that $0\\le \\alpha <1$ , we obtain $ \\Big | |x|^\\alpha - |y|^\\alpha \\Big | \\le 2^{\\alpha } |\\alpha | r^{\\alpha } \\le 2 |\\alpha | r^{\\alpha }.$ Hence, $\\int _{B_r(x_0)} \\Big |\\mu (x) - \\overline{\\mu }_{B_r(x_0)} \\Big | dx \\le \\frac{2 \\omega _n |\\alpha | r^{n+\\alpha }}{n}.$ On the other hand, since $\\alpha \\ge 0$ , we have $\\begin{split}\\mu (B_r(x_0)) & = \\int _{B_r(x_0)} |x|^\\alpha dx = \\int _{B_r(0)} |y + x_0|^\\alpha dy\\ge \\int _{B_r(0)} (|x_0| - |y|)^\\alpha dy \\ge \\int _{B_r(0)} (2r)^\\alpha dy=\\frac{ \\omega _n 2^\\alpha r^{n+\\alpha }}{n}.\\end{split}$ Combining we obtain, $\\int _{B_r(x_0)} \\Big |\\mu (x) - \\overline{\\mu }_{B_r(x_0)} \\Big | dx \\le 2^{1-\\alpha } \\alpha \\mu (B_r(x_0)) \\le 2 \\alpha \\mu (B_r(x_0)).$ Let us do now the case $-1 < \\alpha \\le 0$ .",
"As before, for every $x, y \\in B_r(x_0)$ , and $|\\alpha | <1$ , we have $ \\Big | |x|^{|\\alpha |} - |y|^{|\\alpha |} \\Big | \\le 2^{|\\alpha |} |\\alpha | r^{|\\alpha |} $ Notice also that $|x| \\ge |x_0| -|x-x_0| \\ge 3r -r =2r$ and $|y| \\ge |x_0| -|y-x_0| \\ge |x_0| -r>0$ , thus $ \\Big | {|x|^\\alpha } - {|y|^\\alpha } \\Big |= \\Big | \\frac{|x|^{|\\alpha |}-|y|^{|\\alpha |}}{|x|^{|\\alpha |} |y|^{|\\alpha |}} \\Big | \\le \\frac{ 2^{|\\alpha |} |\\alpha | r^{\\alpha }}{(2r)^{|\\alpha |} (|x_0| -r )^{|\\alpha |}}= \\frac{ |\\alpha | }{(|x_0| -r )^{|\\alpha | }}.$ Therefore, noting that $|\\alpha | = -\\alpha $ $\\begin{split}\\int _{B_{r}(x_0)} \\Big | \\mu (x) - \\overline{\\mu }_{B_{r}(x_0)}\\Big | dx&\\le \\frac{1}{|B_{r}(x_0)|}\\int _{B_{r}(x_0)}\\int _{B_{r}(x_0)} \\Big | {|x|^{\\alpha }} - {|y|^{\\alpha }}\\Big | dydx \\\\&\\le \\frac{1}{\\omega _n r^n}\\int _{B_{r}(x_0)}\\int _{B_{r}(x_0)} \\ |\\alpha | {(|x_0| -r )^\\alpha } dydx = |\\alpha | \\omega _n r^n{(|x_0| -r )^\\alpha }.\\end{split}$ On the other hand, $\\mu (B_{r}(x_0)) = \\int _{B_{r}(x_0)} |x|^\\alpha dx \\ge \\int _{B_{r}(x_0)} {(|x_0|+r)^\\alpha }dx = \\omega _n r^n{(|x_0|+r)^\\alpha }.$ It follows that $\\begin{split}& \\int _{B_{r}(x_0)} \\Big | \\mu (x) - \\overline{\\mu }_{B_{r}(x_0)}\\Big | dx \\le \\frac{|\\alpha | (|x_0| - r )^\\alpha }{(|x_0| +r )^\\alpha } \\mu (B_{r}(x_0)) \\\\& = |\\alpha | \\left(\\frac{|x_0| + r }{|x_0| -r }\\right)^{|\\alpha |} \\mu (B_{r}(x_0)) \\le 2|\\alpha | \\mu (B_{r}(x_0)) .\\end{split}$ Lemma REF follows directly from the estimates in Case I and Case II.",
"Acknowledgement.",
"T. Mengehsa's research is partly supported by NSF grant DMS-312809.",
"T. Phan's research is supported by the Simons Foundation, grant # 354889."
]
] | 1612.05583 |
[
[
"On the history of the Ulam's Conjugacy"
],
[
"Abstract We show the results on the history of the invention of the conjugacy $h(x)=\\frac{2}{\\pi}\\arcsin\\sqrt{x}$ of one-dimensional $[0,\\, 1]\\rightarrow [0,\\, 1]$ maps $f(x)=4x(1-x)$ and $g(x)=1-|1-2x|$."
],
[
"Introduction",
"Not every classical mathematical problem has its final form in the moment of the creation.",
"Moreover, this is true about “classical methods” and “classical tools” of solving mathematical problems.",
"Almost all branches of modern mathematics appeared or obtained the form, which is closed to modern, in 18-19 century.",
"During the reading of mathematical works of that time, it is easy to regard such mistakes of bright mathematicians, which should not be made by scholars and students of firs cors of university nowadays.",
"From another hand, it is necessary to remember, that it is due to these (and only these) mistakes the modern mathematics has the level of strictness, that it has today.",
"We will pay attention to the appearance and the first attempts to study the following construction.",
"Let $f:\\, I\\rightarrow I$ be a function, where $I$ is a set of real numbers.",
"For any $n\\in \\mathbb {N}$ denote $g(x) = \\underbrace{f(f(\\ldots f}_{n \\text{times}}(x)\\ldots ))$ , write $g = f^n$ for being short and call $g$ the $n$ th iteration of $f$ .",
"At the beginning of the 19th century the problems of finding the formula for $f^n$ by given $f$ or, conversely, finding $f$ by given $g=f^n$ was formulated and studied as deeply as it was possible.",
"Non-succeed attempts to solve these problems, lead to the appearance and active development of Dynamical Systems Theory in the middle and second part of the 20th century.",
"This article is motivated by the following question.",
"It is known that equality $g(x)=h(f(h^{-1}(x)))$ holds for every $x\\in [0,\\,1]$ , where $f(x)=4x(1-x)$ , $g(x)=1-|1-2x|$ and $h(x)=\\frac{2}{\\pi }\\arcsin \\sqrt{x}$ are defined on $[0,\\, 1]$ .",
"The mentioned equality appears in all textbooks on one-dimensional dynamics in the chapter about topological conjugation (topological equivalence).",
"A lot of books on the Theory of Dynamical Systems (for instance [4], [6], [7] and [11]) refer this formula to [18], but this is just a note on Summer Meeting of the AMS in 1947, which is about one fifth of a page and contains only the remark that $g$ can be used as a generator a random numbers.",
"During the private talks with Kyiv specialists on one-dimensional dynamics the author failed to get the clearness about the history of the investigation of this formula.",
"All the answers of professors looked like: \"This is the mathematical folklore: the fact is known to everybody, but not the authorship.",
"Certainly, one cay spend a lot of time and found some history, but nobody cam be sure, wether or not this history will appear to be interesting enough\".",
"During the attempts to understand the history of the formula, which is mentioned above, the author entered the magic world of mathematical texts of 19-20 century, where the solutions of mathematical problems are strongly mixed with ambitions of great mathematicians and appear also at the pages of scientific articles.",
"But, let us look at everything in turn."
],
[
"Functional equations of J. Herschel",
"John Herschel is a talented scientist of the end of 19th century.",
"For instance, he was presented with the Gold Medal of the Royal Astronomical Society in 1826 and with the Lalande Medal of the French Academy of Sciences in 1825, was a Honorary Member of the St Petersburg Academy of Sciences.",
"Also he was one of the founders of the Royal Astronomical Society in 1820.",
"Nevertheless, John Herschel was an astronomer, but not a “professional mathematician”.",
"In the same time, J. Herschel stated in [8] series of mathematical problems and make attempts to solve them.",
"His solutions contain some disadvantages, which do not decrease the value of the problems themselves.",
"Problem 1.",
"Find $f^n$ for the function $f(x)=2x^2-1$.",
"The following solution is presented.",
"For every $x$ denote $u_0=x$ and consider the sequence $u_{n+1}=2u_n^2-1.$ To find the formula for $f^n$ is the same as to find the general formula for $u_n$ , dependent on $n$ and $u_0$ .",
"Without any explanations Herschel decides to find the solution in the form $ u_n =\\frac{1}{2}\\left(C^{2^n}+C^{-2^n}\\right),$ where $C$ is a constant, dependent on $u_0$ .",
"We may assume that the motivation for such form of the solution is the formula $u_n=\\lambda ^n$ for the solution of a linear second order difference equation $u_{n+2}=au_{n+1}+bu_n$ .",
"In any way, the direct substitution lets to check that (REF ) satisfies (REF ).",
"After the plug $u_0=x$ into (REF ), Herschel has found $ C=x+\\sqrt{x^2-1},$ whence rewrites (REF ) as $f^n(x)=\\frac{1}{2}\\left(\\left(x+\\sqrt{x^2-1}\\right)^{2^n}+\\left(x-\\sqrt{x^2-1}\\right)^{2^n}\\right).$ He does not explain the possibility of appearance of negative number under the square root.",
"Problem 2.",
"Find functions $\\varphi $ , which satisfy the equality $\\varphi ^2(x)=x.$ Like above, J. Herschel denotes $x=u_n$ and $\\varphi (u_n)=u_{n+1}$ .",
"Moreover, he considers the sequence $\\varphi (u_n)$ , which denotes $(\\varphi (u))_n$ .",
"After the “cross multiplication” of equalities $ \\left\\lbrace \\begin{array}{l}(\\varphi (u))_n=u_{n+1}\\\\(\\varphi (u))_{n+1}=u_n,\\end{array}\\right.$ he obtains $u_{n+1}(\\varphi (u))_{n+1}=u_n(\\varphi (u))_n$ , i.e.",
"$u_n(\\varphi (u))_n=c$ for all $n$ , where $c$ is a constant.",
"Also it follows from equalities (REF ) that $u_n+(\\varphi (u))_n= u_{n+1}+(\\varphi (u))_{n+1}$ , i.e.",
"$u_n+(\\varphi (u))_n + C=0$ for all $n$ and some constant $C$ .",
"Now Herschel claims that one may state $f(u_n(\\varphi (u))_n)$ instead of $C$ , where $f$ is an arbitrary function.",
"Thus the equation $ x + \\varphi (x) +f(x\\varphi (x))=0.$ appears.",
"Taking as example the function $f(x)=a+bx$ Herschel finds $\\varphi $ from equality $ x + \\varphi (x) +bx\\varphi (x) +a=0$ as $\\varphi (x) = -\\frac{a+x}{1+bx}$ and claims, that it is a solution of (REF ).",
"He does not notice, that if one plug (REF ) into (REF ), then would not get the identity.",
"Problem 3. for given function $f$ find a function $\\varphi $ such that $\\varphi ^n = f$ .",
"For this problem Herschel suggests “very simple” solution: find the general formula for $f^n$ and plug there $1/n$ instead of $n$ .",
"For example for the function $f(x)=2x^2-1$ Herschel uses (REF ) to find $\\varphi (x)=\\frac{1}{2}\\left(\\left(x+\\sqrt{x^2-1}\\right)^{\\@root n \\of {2}}+\\left(x-\\sqrt{x^2-1}\\right)^{\\@root n \\of {2}}\\right)$ and remarks that $\\@root n \\of {2}$ means the set of $n$ th complex roots of $n$ th degree.",
"Problem 4.",
"Let an hyperbola $AM$ with axis $CP$ (which coincides with $x$ -axis) and center $C$ (which is origin) be given.",
"It is necessary to find the curve $am$ with the following properties.",
"For any point $P$ on the $x$ -axis find the point on $am$ with the same $x$ -coordinate and take on the $x$ -axis the point $d$ such that $Cd = Pm$ .",
"Again find on $am$ the point $l$ with $x$ -coordinate as of $d$ and take the third point $l_3$ such that $Cl_3 = dl$ .",
"Repeat these actions $n$ times, where $n$ is chosen at the very beginning.",
"The question: find $am$ such that after $n$ steps we would get the segment $fk$ , whose length equals $PM$ (see fig.",
"REF a.)?",
"Write the equation of hyperbola $AM$ as $ y^2 = (1-e^2)(a^2-x^2),$ where $e$ is the eccentricity of the hyperbola.",
"For the function $f(x)=\\sqrt{(1-e^2)(a^2-x^2)}$ we have that $f^n(x)=\\sqrt{(e^2-1)^nx^2-\\frac{e^2-1}{e^2-2}\\left((e^2-1)^n-1\\right)a^2}$ Moreover, finding a function $\\varphi $ such that $\\varphi ^n=f$ , Herschel, according to his rule above, plugs $1/n$ instead of $n$ into (REF ) and obtains the equations of curves, which satisfy the condition of the problem.",
"He does not consider the existence of another curves, which satisfy the former condition.",
"It is interesting, that Herschel does not notice here, that square root has two values, one of which is complex.",
"Figure: Graphical interpretations"
],
[
"Conjugacy of C. Babbage and J. Ritt",
"Charles Babbage pays attention in [2] to the study of iterations of functions, precisely to the finding of the function $\\psi $ such that $ \\psi ^n(x)=x$ for all $x$ .",
"First, he suggests the graphical interpretation of this problem, which is similar to Herschel's one, which is described above.",
"At the picture REF b. it is necessary to find the cure $APQT$ (where $A$ is the origin ) with the following properties.",
"For arbitrary point $B$ find the length of perpendicular $BP$ and take on the $x$ -axis the point $C$ such that $AC = BP$ .",
"Then find the length of the perpendicular $CQ$ and take its length on the $x$ -axis.",
"After $n$ steps we have to get the perpendicular $FT$ , whose length should be equal to the length of the former segment $AB$ .",
"This geometrical construction is exactly the verbal description of the equation (REF ).",
"Babbage makes one more important remark.",
"Let $f$ be some solution of (REF ) and let $\\varphi $ be an arbitrary invertible function.",
"Then $g(x) =\\varphi (f(\\varphi ^{-1}(x)))$ will also be a solution of (REF ).",
"In fact, these reasonings about (REF ) is exactly the invention of the notion of topological conjugacy of maps.",
"Topological conjugacy of function $f$ and $g$ can be imagined as the rectangle, $ {^{f}[rr] _{\\varphi }[d] &&^{\\varphi }[d]\\\\^{g}[rr] && }$ where we know, that one can come from the left top vertex to the right bottom one either by “top root”, or by “bottom root” with the same result.",
"Coming by the root means here the sequent applying the functions near the arrows to an arbitrary former argument.",
"These diagrams are called “commutative diagrams” and are widely used in different branches of mathematics for the illustration of reasonings.",
"The important corollary of the commutative diagram is the possibility “to continue it to the right” and thus obtain $ {^{f}[rr] _{\\varphi }[d] &&^{\\varphi }[d] ^{f}[rr] && \\ldots ^{f}[rr] &&^{\\varphi }[d]\\\\^{g}[rr] && ^{g}[rr] && \\ldots ^{g}[rr] && }$ which implies that $g^n(x) = \\varphi (f^n(\\varphi ^{-1}(x)))$ .",
"From another hand, the diagram just illustrates the reasonings and the same conclusion could be obtained from the equality (REF ) without the commutative diagram too.",
"Joseph Ritt (injustively) says in [16] that C. Babbage earlier claimed that for any fixed solution $f$ of the equation (REF ) and any other solution $g$ there exists an invertible $\\varphi $ such that (REF ) holds.",
"Suppose (writes Ritt) that $\\varphi (a)=b$ for some $a$ and $b$ .",
"Then $\\varphi (f^n(a))=g^n(b)$ for every $n\\in \\mathbb {N}$ .",
"If for some other $a^{\\prime },\\, b^{\\prime }$ the equality $\\varphi (a^{\\prime })=b^{\\prime }$ holds, then for every $n$ we will have $\\varphi (f^n(a^{\\prime }))=g^n(b^{\\prime })$ (see.",
"pict.",
"REF a).",
"If $f^n(a)=f^n(a^{\\prime })$ for some $n$ , but $g^n(b)\\ne g^n(b^{\\prime })$ , the contradiction with that $\\varphi $ is a function would appear (see pict.",
"REF b).",
"Figure: Reasonings of RittRitt continues these reasonings.",
"Suppose that $\\varphi (x)=\\Psi (x)$ for all $x\\in (a,\\, a^{\\prime })$ and some $a,\\, a^{\\prime }$ , where $\\Psi $ is some function.",
"In other words, suppose that $\\varphi $ is already defined on some interval $(a,\\, a^{\\prime })$ .",
"Then for any $n$ the equality $\\varphi (f^n(x))=g^n(\\varphi ((x)))$ determines $\\varphi $ on $f^n([a,\\, a^{\\prime }])$ (see pict.",
"REF c).",
"If for some $n_1,\\,n_2$ intervals $f^{n_1}([a,\\, a^{\\prime }])$ and $f^{n_2}([a,\\, a^{\\prime }])$ intersect, the the contradiction may appear.",
"Notice, that talking about the contradiction, Ritt does not pay attention to the fact, which gives the contradiction.",
"Indeed, he assumes that two points $(a,\\, b)$ and $(a^{\\prime },\\, b^{\\prime })$ such that $\\varphi (a)=b$ and $\\varphi (a^{\\prime })=b^{\\prime }$ are given and then contradiction appears.",
"But this only means that the graph of $\\varphi $ does not pass through both of these points (but may be pass through one of them).",
"It is not clear, what can generate the conclusion that for some exact solutions $f$ and $g$ of (REF ) there in no any invertible $\\varphi $ , which transforms (REF ) to the identity."
],
[
"Lamerey's Diagrams",
"The Cartesian method as a way on plotting the graphs of functions is known from the de Cartes time, i.e.",
"from the first part of the 17th century.",
"In the same time, the problems, which were stated by J. Herschel (and were discussed above), need a bit specifical techniques, and we will pay attention to it now.",
"Figure: Lamerey's DiagramsWhen one use the Cartesian method, the arguments of a functions are understood as points of the $x$ -axis and its values – as points of $y$ -axis, whence the function appears to be a map from $x$ -axis to $y$ -axis.",
"In the same time, we can consider any function $f$ as a map from the line $y=x$ to itself, which maps a point $(x,\\, x)$ to $(f(x),\\, f(x))$ .",
"The graphical interpretation of this action can be imagined as vertical line from point $(x,\\,x)$ to the graph and then horizontal line to $(f(x),\\, f(x))$ , returning to $y=x$ .",
"For usefulness of such interpretation, let us come back to the problem of finding a continuous function $\\varphi :\\,\\mathbb {R}\\rightarrow \\mathbb {R}$ such that $\\varphi ^2(x)=\\varphi (x)$ for all $x$ from the domain.",
"Clearly, for every $x_0$ the equality $\\varphi (\\varphi (x_0))=\\varphi (x_0)$ holds.",
"Thus, denote $y_0=\\varphi (x_0)$ and obtain that $\\varphi (y_0)=y_0$ , i.e.",
"the graph of $\\varphi $ passes through the point $(y,\\, y)$ for every $y$ such that $y=\\varphi (x)$ for some $x$ .",
"This means that there exist $a,\\, b$ such that $\\varphi (\\mathbb {R}) = [a,\\, b]$ and $\\varphi (x)=x$ for every $x\\in [a,\\, b]$ (see pict.",
"REF a).",
"Also the following reasonings can be illustrated.",
"Notice that the solutions of the equation $x=\\varphi (x)$ are precisely the points of the intersection of the graph of $\\varphi $ and the line $y=x$ .",
"Suppose that for the function $\\varphi $ there exists (unknown) solution $a$ of the equation $x=\\varphi (x)$ and, moreover, $x<\\varphi (x)<a$ for every $x<a$ and $a<\\varphi (x)<x$ for each $x>a$ .",
"For arbitrary $x_0$ consider the sequence $x_k =\\varphi ^k(x_0)$ .",
"If one plot the graph, then it becomes evident that the sequence $\\lbrace x_k\\rbrace $ tends to unknown $a$ (see pict.",
"REF b).",
"This reasonings were invented in the beginning of the 19th century almost simultaneously by Evarist Galois [5] and Adrien Legendre [9] (see also [1]).",
"Figure: Pictures from the S. Pincherle's workThe work [14] contains the picture (see pict.",
"REF ), where this method is illustrated.",
"The left graph contains a line $y=x$ (for the technical calculations, mentioned earlier) and the graph $y=f(x)$ , which is used to construct the function $y=f^2(x)$ .",
"The right sketch contains the construction of the graph of the function $y(x)=\\varphi (f(x))$ by given graphs of $f$ and $\\varphi $ .",
"Nowadays the sketches like those on figure REF , are called Lamerey's Diagrams due to the work [10]."
],
[
"Trigonometry of J. Boole",
"Gorge Boole is known in the history of mathematics as one on the foundators of the mathematical logics.",
"In [3] G. Boole suggests the following way of finding the general formula for iterations of the function $f(x)=2x^2-1$ .",
"Keeping in mind the double angle formula $\\cos 2x = 2\\cos ^2x-1$ , denote $g(x)=2x$ and $h(x)=\\cos x$ , whence write the commutative diagram $ { x ^{g}[rr] _{h}[d]&& 2x^{h}[d]\\\\\\cos x ^{f}[rr] && f(h)=h(g) } $ Continue the diagram to the right and obtain ${ x ^{g}[rr] _{h}[d] && 2x^{h}[d] ^{g}[rr] &&\\ldots ^{g}[rr] && 2^nx^{h}[d]\\\\\\cos (x) ^{f}[rr] && \\cos (2x) ^{f}[rr] && \\ldots ^{f}[rr]&& \\cos (2^nx),}$ whence $f^n(\\cos x) = \\cos (2^nx).$ The substitution $t=\\cos x$ leads to $f^n(t) = \\cos (2^n\\arccos t)$ for all $t\\in [-1,\\, 1]$ ."
],
[
"The J.\nvon Neumann's and S. Ulam's generator",
"John von Neumann studied deeply quantum physics,functional analysis, sets theory and informatics.",
"His name is connected with the architecture of the most modern computers.",
"von Neumann, together with polish mathematician Stanislaw Ulam participated in Manhattan Project.",
"In the short note [18], which is the thesis of the mating of American Mathematical Society, J. von Neumann and S. Ulam suggested to use iterations of the function $ f(x)=4x(1-x)$ for obtaining “numbers with different distributions”.",
"Nevertheless, they have not explained there, what the “distribution of a number” is.",
"The complicatedness of calculations, which are dealing with the function (REF ), is known from the Pier Verhulst's [19], written in the first part of the 19th century, where he used the formula $p_{k+1} = p_k(m-np_k).$ for the expectation of the number of individuals $p_k$ in the biological population in the $k$ th generation, where $m$ and $n$ are constants.",
"Verhulst remarked the strong dependence of $p_k$ on $p_1,\\, m$ and $n$ .",
"In other words, for the huge $k$ the small changes of $p_1,\\, m$ and $n$ can lead to march more change of $p_k$ .",
"The sequence (REF ) was called “logistic” due to “logists”, who made calculations in the Ancient Greece.",
"The map (REF ) is also called logistic map, because it is of the form (REF ).",
"In the article [13] von Neumann mentioned the disadvantage of the function (REF ) as a generator, introduced in [18] and, in the same time, explained, what is meant under the distribution of a number.",
"Suppose that non-decreasing function $F:\\, [0,\\, 1]\\rightarrow [0,\\, 1]$ such that $F(0)=0$ and $F(1)=1$ is given.",
"It is necessary “in some way” to get the sequence of numbers $\\lbrace x_n\\rbrace \\subset [0,\\, 1]$ such that the probability of the event $x_i\\in [a,\\ b]$ equals $F(b)-F(a)$ .",
"The result of [18], says von Neumann, is that for almost every $x_0\\in [0,\\, 1]$ (up to Lebesgue measure), the probability $x_i\\in [a,\\, b]$ equals $b-a$ .",
"He mentions that obtaining the function, which “generates” the sequences with given distribution, is an important problem, for instance, for some calculating methods.",
"In the same time, it is mentioned in [13] that the function, suggested in [18] can not be used with the mentioned goal.",
"The arguments for such impossibility were the following.",
"For a given sequence $\\lbrace x_i\\rbrace \\subset [0,\\, 1]$ such that $x_{i+1}=4x_i(1-x_i)$ define $\\alpha _i$ by $x_i=\\sin ^2\\pi \\alpha _i$ .",
"Now notice that $\\alpha _{i+1}=2\\alpha _i\\,(mod\\ 1)$ , i.e.",
"$\\alpha _{i+1}$ is the fractional part of $2\\alpha _i$ .",
"Denote the binary decomposition of $\\alpha _1$ as $\\alpha _1=\\beta _1\\beta _2\\beta _3\\ldots $ , whence the binary decomposition of $\\alpha _k$ would be $\\alpha _k=\\beta _k\\beta _{k+1}\\beta _{k+2}\\ldots $ .",
"Now von Neumann makes conclusion that in real computer we can not take a number with infinitely many random binary digits, whence the sequence $\\lbrace \\alpha _i\\rbrace $ (and, correspondingly, $\\lbrace x_i\\rbrace $ ), being considered “on the real computer” will become 0 after finitely many iterations.",
"Moreover, von Neumann made in [13] a very non-ethical think: he wrote that is was S. Ulam, who suggested (REF ) as a random generator and, thus, “forget” that Ulam was co-author of [18] too.",
"The continuation of this history a much more interesting.",
"Is it easy no see that the formula $x_i=\\sin ^2\\pi \\alpha _i$ does not define the sequence $\\alpha _i$ with properties, which are mentioned in [13].",
"If one would find $\\alpha \\in [0,\\, 1]$ then there is no one-to-one correspondence, because $\\alpha _i = k+\\frac{(-1)^k}{\\pi }\\arcsin \\sqrt{x_i},\\, k\\in \\lbrace 0,\\, 1\\rbrace $ and it is not clear, it is necessary to take $k$ being 0 or 1.",
"If suppose that $\\alpha _i\\in [0,\\, 0.5]$ , then the formula $\\alpha _{i+1}=2\\alpha _i\\, (mod\\ 1)$ would give a number from the interval $(0.5,\\, 1]$ after finitely many steps for every $\\alpha _1\\ne 0$ .",
"Remark, that the function $\\alpha (x) =\\frac{1}{\\pi }\\arcsin {\\sqrt{x}}$ is the bijection between the intervals $[0,\\, 1]$ and $[0,\\, 0.5]$ .",
"Thus, commutative diagram $ { [0,\\, 1] ^{f}[rr] _{\\alpha }[d] && [0,\\, 1]^{\\alpha }[d]\\\\[0,\\, 0.5] ^{g}[rr] && [0,\\, 0.5] }$ defines a function $g:\\, [0,\\, 0.5]\\rightarrow [0,\\, 0.5]$ by $g(x)=\\alpha (f(\\alpha ^{-1}(x)))$ .",
"Remind that $\\alpha ^{-1}(x) =\\sin ^2\\pi x$ .",
"The evident technical calculations give the following: $ g(x)=\\frac{1}{\\pi }\\arcsin \\sqrt{4\\sin ^2(\\pi x)(1-\\sin ^2\\pi x)}=\\frac{\\arcsin (|\\sin (2\\pi x)|)}{\\pi }.$ Simplification of the absolute value function in the argument of $\\arcsin $ leats to the dichotomy either $x\\leqslant 0.25$ or $x>0.25$ (this is naturally, since we have already seen some “problems” with the middle of the interval for $x$ in the von Neumann's work).",
"In fact, $ g(x) = \\left\\lbrace \\begin{array}{ll}2x& \\text{for }x\\in [0,\\, 0.25]\\\\1-2x& \\text{for }x\\in (0.25,\\, 0.5]\\end{array}\\right.$ We have mentioned already, that commutative diagrams are connected with the conjugacy.",
"Let us give the formal definition.",
"Functions $f:\\, A\\rightarrow A$ and $g:\\, B\\rightarrow B$ , where $A$ and $B$ are sets of real numbers, are called topologically conjugated, if there exists an invertible function $h:\\, A\\rightarrow B$ such that $h(f(x))=g(h(x))$ for all $x\\in A$ .",
"The chapter of mathematics “topology” studies geometrical “figures” up to some transformations, which roughly can be imagined like if figures be made from a material, which admits stretches and compressing.",
"The word combination “topological conjugation” can be explained as the change of the scale while the construction of the graph of a function.",
"We shall explain more carefully what “the change of the scale” is.",
"Let us construct the graph of the function $f(x)=4x(1-x)$ for $x\\in [0,\\, 1]$ (it is the parabola inside the square $[0,\\, 1]\\times [0,\\, 1]$ , with branches going down).",
"Let $h:\\, [0,\\, 1]\\rightarrow [a,\\, b]$ be a continuous invertible function, which (we will say defines the change of coordinates).",
"Let us take each point on the segment $[0,\\, 1]$ of $x$ -axis and $y$ -axis and write $h(x)$ near it.",
"After this, we may say, that each point $(x,\\, y)$ of the square $[0,\\, 1]\\times [0,\\, 1]$ obtains the new coordinates $(\\widetilde{x},\\, \\widetilde{y})\\in [a,\\, b]\\times [a,\\, b]$ , but the line $y=x$ remains to be $y=x$ .",
"Nevertheless, the parabola, which was (!)",
"the graph of the function $f(x)=4x(1-x)$ appears to the the graph of some another function $\\widetilde{f}:\\, [a,\\,b]\\rightarrow [a,\\, b]$ .",
"If $\\alpha :\\, [0,\\, 1]\\rightarrow [0,\\,0.5]$ would be instead of $h$ above, then it follows from our calculation, that the obtained “new function” is $g$ of the form (REF ).",
"After $g$ has been found, it is necessary to “want” to obtain one more function, which will be “similar” to $g$ , but will be $[0,\\, 1]\\rightarrow [0,\\, 1]$ instead of $[0,\\, 0.5]\\rightarrow [0,\\, 0.5]$ .",
"If apply the topological conjugacy $\\beta (x)=2x$ to $g$ , then obtain $g_1(x)=\\beta (g(\\beta ^{-1}(x)))$ , which is $ g_1(x) = \\left\\lbrace \\begin{array}{ll}2x& \\text{for }x\\in [0,\\, 0.5],\\\\2-2x& \\text{for }x\\in (0.5,\\, 1].\\end{array}\\right.$ Notice that graphs of functions (REF ) and (REF ) looks the same if the numbers are marked uniformly on coordinate axis.",
"The conjugation of $f$ and $g_1$ can be easily illustrated by the following commutative diagram $ { [0,\\, 1]^{f}[rr] _{\\alpha }[d] @/_3pc/@{-->}_{h}[dd] && [0,\\, 1]@/^3pc/@{-->}^{h}[dd]^{\\alpha }[d]\\\\[0,\\, 0.5] ^{g}[rr] _{\\beta }[d] && [0,\\, 0.5]^{\\beta }[d]\\\\[0,\\, 1] ^{g_1}[rr] && [0,\\, 1] }$ and the conjugacy (the function, which defines the conjugation) is $ h(x)= \\beta (\\alpha (x)) =\\frac{2}{\\pi }\\, \\arcsin \\sqrt{x}.",
"$ Let us come bach to the George Boole's work [3].",
"Commutative diagram (REF ) does not define a topological conjugation, because the map $g$ there is not $B\\rightarrow B$ such that $h(x)=\\cos x$ is the bijection between $B$ and $h(B)$ .",
"Figure: Topologically conjugated mapsGraphs of (REF ) and (REF ) are given at picture REF a.",
"Picture REF b contains graphs of $y=2x^2-1$ and $y=2x$ , which appeared in the Boole's [3].",
"It is clear, that these picture are “geometrically” equal, just one of them is turned upside-down from another.",
"The conjugacy $h(x)=1-x$ turns the parabola from the Picture REF a to the form of Picture REF b.",
"The fact that coordinate axis are “at another position” mean nothing, because the conjugacy $h(x)=x+a$ , where $a$ is a constant, moves the graph along the line $y=x$ (the same line, whose importance we have already mentioned, talking about the iterations).",
"Thus, both George Boole in [3] and John von Neumann in [13], studied the iterations of a parabola (it is not important, which parabola) and almost invented the topological conjugation of a parabola and a piecewise linear function, but stopped with the consideration of just one linear part of it.",
"Also Stanislaw Ulam returns in [17] to the question, which we have discussed at the end of the chapter, dedicated to Joseph Ritt's work.",
"Let $g_1:\\, [0,\\, 1]\\rightarrow [0,\\, 1]$ be of the form (REF ) and $g_2:\\, [0,\\, 1]\\rightarrow [0,\\,1]$ be defined as follows: $g_2(x) = \\left\\lbrace \\begin{array}{ll}l(x)& \\text{for }x\\in [0,\\, v],\\\\r(x)& \\text{for }x\\in (v,\\, 1],\\end{array}\\right.$ where $v\\in (0,\\, 1)$ is a parameter, function $l$ increase, $r$ decrease such that $l(0)=r(1)=0$ .",
"In other words, the graph of $g_2$ is “similar” to the graph of $g_1$ , but $g_2$ is not necessary piecewise linear and is not necessary symmetrical in $x=0.5$ .",
"Ulam asks: which conditions should satisfy $g_2$ for being topologically equivalent to $g_1$ (or, which is the same, to $y=4x(1-x)$ )?",
"Suppose that $\\widetilde{h}:\\, [0,\\, 1]\\rightarrow [0,\\, 1]$ is such that $g_2(x)=\\widetilde{h}(g_1(\\widetilde{h}^{-1}(x)))$ for all $x\\in [0,\\, 1]$ .",
"Since $\\widetilde{h}$ is invertible, then $\\widetilde{h}(0)\\in \\lbrace 0,\\, 1\\rbrace $ .",
"Now $g_2(0)=0$ and $\\widetilde{h}(g_1(0))=g_2(\\widetilde{h}(0))$ imply $\\widetilde{h}(0)=0$ , whence $\\widetilde{h}$ increase.",
"Moreover, since $g_1^n(0)=0$ for all $n\\geqslant 1$ , then it follows from $\\widetilde{h}(g_1^n(x))=g_2^n(\\widetilde{h}(x))$ that $g_2^n(\\widetilde{h}(x))=0$ , whenever $g_1^n(x)=0$ .",
"Ulam makes a conclusion from these reasonings that $g_2$ is topologically conjugated to $g_1$ is and only if that set $M = \\lbrace x\\in [0,\\, 1]:\\, g_2^k(x)=0\\ \\text{ for some }k\\rbrace $ is dense in $[0,\\, 1]$ .",
"As about the map $\\psi $ , which satisfies the functional equation (REF ), Melvyn Nathanson has proved the following theorem in [12]: if $\\psi :\\, [0,\\, 1]\\rightarrow [0,\\,1]$ is a continuous function such that $\\psi ^p(x)=x$ for all $x\\in [0,\\, 1]$ , then $\\psi ^2(x)=x$ for all $x\\in [0,\\, 1]$ .",
"Moreover, if $p$ is even, then $\\psi (x)=x$ for all $x\\in [0,\\, 1]$ .",
"In other words, the problem, which Babbage treated, has, in sone cense, the only trivial solution.",
"Nevertheless, his does not decrease the importance or the reasonings, which were suggested by Babbage during the attempts of the solution.",
"The article [12] is written in the second part of the 20th century and is dedicated to the Li and Yorke's notion of chaos.",
"The formula (REF ) or the topological conjugacy of maps (REF ) and (REF ) was published at first by Ottis Richard in [15].",
"Nevertheless, Richard thanks there S. Ulam “for many helpful and stimulating conversations on the subject of this paper” and writes especially about the formula (REF ) (which is used for the calculating of the invariant measure) that it is S. Ulam, which noticed this formula at first.",
"Stanislaw Ulam published (REF ) the first time 8 years later in [17].",
"The article [17] does not contain any references to John von Neumann as the author of idea of getting (REF ).",
"Some modern books on Dynamical Systems theory refer (REF ) as Ulam's map."
],
[
"Final remarks",
"Sometimes mathematicians make mistakes.",
"Mathematicians of the worldwide level sometimes make mistakes too.",
"Sometimes mathematicians quarrel one with each other and tracks of this can be seen in their articles.",
"Nevertheless, the mistake of a scholar and the mistake of a mathematician of the worldwide level are “different sings”.",
"John Herschel's problems with solutions, which were published in year 1814, contain mistakes.",
"But they also contain the foundations of the theory of one-dimensional dynamical systems - a theory, which was developed in the second part of 20th century.",
"The idea, which lead to the notion of topological equivalency, is clearly formulated in year 1815 during the attempts of the description of a class of functions, which, in some thence, is empty (but it is the result of 1970th).",
"Lamerey's Diagrams, which were described in 1897, were used, in fact, by Adrien Legendre in 1808 and Evariste Galois in 1830.",
"John von Neumann refuse his participation in the preparation of the work, which contained a mistake, but this lead to that he loosed the authority for the more important result, which is mentioned now without the name of von Neumann.",
"Notice, that pictures REF (about Herschel's and Babbage's works) and REF (about Pincherle's work) are prepared by the graphical environment of , i.e.",
"they are not copies of the original articles, but saves the meaning and the notations of the original ones."
]
] | 1612.05592 |
[
[
"New perturbation bounds for the spectrum of a normal matrix"
],
[
"Abstract Let $A\\in\\mathbb{C}^{n\\times n}$ and $\\widetilde{A}\\in\\mathbb{C}^{n\\times n}$ be two normal matrices with spectra $\\{\\lambda_{i}\\}_{i=1}^{n}$ and $\\{\\widetilde{\\lambda}_{i}\\}_{i=1}^{n}$, respectively.",
"The celebrated Hoffman--Wielandt theorem states that there exists a permutation $\\pi$ of $\\{1,\\ldots,n\\}$ such that $\\left(\\sum_{i=1}^{n}\\big|\\widetilde{\\lambda}_{\\pi(i)}-\\lambda_{i}\\big|^{2}\\right)^{1\\over 2}$ is no larger than the Frobenius norm of $\\widetilde{A}-A$.",
"However, if either $A$ or $\\widetilde{A}$ is non-normal, this result does not hold in general.",
"In this paper, we present several novel upper bounds for $\\left(\\sum_{i=1}^{n}\\big|\\widetilde{\\lambda}_{\\pi(i)}-\\lambda_{i}\\big|^{2}\\right)^{1\\over 2}$, provided that $A$ is normal and $\\widetilde{A}$ is arbitrary.",
"Some of these estimates involving the \"departure from normality\" of $\\widetilde{A}$ have generalized the Hoffman--Wielandt theorem.",
"Furthermore, we give new perturbation bounds for the spectrum of a Hermitian matrix."
],
[
"1.2 New perturbation bounds for the spectrum of a normal matrix Xuefeng Xu$^{{\\rm a},{\\rm b},\\ast }$ , Chen-Song Zhang$^{{\\rm a},{\\rm b}}$ $^{\\rm a}$ Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China $^{\\rm b}$ School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China $^{\\ast }$ Corresponding author.",
"E-mail addresses: [email protected] (X. Xu), [email protected] (C.-S. Zhang).",
"Abstract Let $A\\in \\mathbb {C}^{n\\times n}$ and $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ be two normal matrices with spectra $\\lbrace \\lambda _{i}\\rbrace _{i=1}^{n}$ and $\\lbrace \\widetilde{\\lambda }_{i}\\rbrace _{i=1}^{n}$ , respectively.",
"The celebrated Hoffman–Wielandt theorem states that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\left(\\sum _{i=1}^{n}\\big |\\widetilde{\\lambda }_{\\pi (i)}-\\lambda _{i}\\big |^{2}\\right)^{1\\over 2}$ is no larger than the Frobenius norm of $\\widetilde{A}-A$ .",
"However, if either $A$ or $\\widetilde{A}$ is non-normal, this result does not hold in general.",
"In this paper, we present several novel upper bounds for $\\left(\\sum _{i=1}^{n}\\big |\\widetilde{\\lambda }_{\\pi (i)}-\\lambda _{i}\\big |^{2}\\right)^{1\\over 2}$ , provided that $A$ is normal and $\\widetilde{A}$ is arbitrary.",
"Some of these estimates involving the “departure from normality” of $\\widetilde{A}$ have generalized the Hoffman–Wielandt theorem.",
"Furthermore, we give new perturbation bounds for the spectrum of a Hermitian matrix.",
"Keywords: spectrum, perturbation, Hermitian matrix, normal matrix, departure from normality AMS subject classifications: 15A18, 65F15, 47A55, 15A42, 15B57 1.",
"Introduction Let $\\mathbb {C}^{m\\times n}$ and ${U}_{n}$ be the set of all $m\\times n$ complex matrices and the set of all $n\\times n$ unitary matrices, respectively.",
"For any $X\\in \\mathbb {C}^{m\\times n}$ , $X^{\\ast }$ , ${\\rm rank}(X)$ , $\\Vert X\\Vert _{2}$ , and $\\Vert X\\Vert _{F}$ denote the conjugate transpose, the rank, the spectral norm, and the Frobenius norm of $X$ , respectively.",
"For any $M\\in \\mathbb {C}^{n\\times n}$ , its diagonal part, strictly lower triangular part, and strictly upper triangular part are denoted by $\\mathcal {D}(M)$ , $\\mathcal {L}(M)$ , and $\\mathcal {U}(M)$ , respectively.",
"The trace of $M$ is denoted by ${\\rm tr}(M)$ .",
"The set ${U}_{n}(M)$ is defined as ${U}_{n}(M):=\\big \\lbrace U\\in {U}_{n}: U^{\\ast }MU \\ \\text{is upper triangular}\\big \\rbrace .$ The symbol $\\kappa _{2}(M)$ stands for the spectral condition number of a nonsingular matrix $M$ , namely, $\\kappa _{2}(M)=\\Vert M^{-1}\\Vert _{2}\\Vert M\\Vert _{2}$ .",
"For $M=(m_{ij})\\in \\mathbb {C}^{n\\times n}$ , we define $W_{L}(M)$ and $W_{U}(M)$ as follows: $W_{L}(M):&=\\max \\big \\lbrace i-j: m_{ij}\\ne 0, i>j\\big \\rbrace ,\\\\W_{U}(M):&=\\max \\big \\lbrace j-i: m_{ij}\\ne 0, i<j\\big \\rbrace .$ In particular, if $\\mathcal {L}(M)=0$ (resp., $\\mathcal {U}(M)=0$ ), we set $W_{L}(M)=0$ (resp., $W_{U}(M)=0$ ).",
"Another quantity $\\delta (M)$ is defined as $\\delta (M):=\\bigg (\\Vert M\\Vert _{F}^{2}-\\frac{1}{n}|{\\rm tr}(M)|^{2}\\bigg )^{1\\over 2},$ which is well-defined because $\\Vert M\\Vert _{F}^{2}\\ge \\frac{1}{n}|{\\rm tr}(M)|^{2}$ for all $M\\in \\mathbb {C}^{n\\times n}$ .",
"Obviously, $\\delta (M)\\le \\Vert M\\Vert _{F}$ , and $\\delta (M)=\\Vert M\\Vert _{F}$ if and only if ${\\rm tr}(M)=0$ .",
"For simplicity, for a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ , we define $\\mathbb {D}_{2}:=\\bigg (\\sum _{i=1}^{n}\\big |\\widetilde{\\lambda }_{\\pi (i)}-\\lambda _{i}\\big |^{2}\\bigg )^{1\\over 2},$ which characterizes the “distance” between $\\lbrace \\lambda _{i}\\rbrace _{i=1}^{n}$ and $\\lbrace \\widetilde{\\lambda }_{i}\\rbrace _{i=1}^{n}$ with respect to $\\ell ^{2}$ -norm.",
"Recall that $A\\in \\mathbb {C}^{n\\times n}$ is normal if $A$ commutes with its conjugate transpose, i.e., $AA^{\\ast }=A^{\\ast }A$ .",
"In particular, if $A=A^{\\ast }$ , then $A\\in \\mathbb {C}^{n\\times n}$ is called a Hermitian matrix.",
"Assume that $A\\in \\mathbb {C}^{n\\times n}$ and $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ are normal matrices with spectra $\\lbrace \\lambda _{i}\\rbrace _{i=1}^{n}$ and $\\lbrace \\widetilde{\\lambda }_{i}\\rbrace _{i=1}^{n}$ , respectively.",
"Let $E=\\widetilde{A}-A$ .",
"In 1953, Hoffman and Wielandt [1] proved that there is a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\Vert E\\Vert _{F},$ which is the well-known Hoffman–Wielandt theorem.",
"This theorem reveals that there is a strong global stability to the set of eigenvalues of a normal matrix.",
"Unfortunately, the inequality may fail when $\\widetilde{A}$ is non-normal.",
"For example, if $A=\\begin{pmatrix}0 & 0 \\\\0 & 3\\end{pmatrix} \\quad \\text{and} \\quad \\widetilde{A}=\\begin{pmatrix}-1 & -1 \\\\1 & 1\\end{pmatrix},$ then $A$ is a normal matrix with spectrum $\\lbrace 3,0\\rbrace $ and $\\widetilde{A}$ is a non-normal matrix with spectrum $\\lbrace 0,0\\rbrace $ .",
"For any permutation $\\pi $ of $\\lbrace 1,2\\rbrace $ , we have that $\\mathbb {D}_{2}=3>\\sqrt{7}=\\Vert E\\Vert _{F}$ .",
"Due to the limitation of the Hoffman–Wielandt theorem, many authors have developed analogous results.",
"If $A\\in \\mathbb {C}^{n\\times n}$ is normal and $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ is non-normal, Sun [2] demonstrated that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\sqrt{n}\\Vert E\\Vert _{F},$ which can be applied to characterize the variation of the spectrum of an arbitrary matrix [3].",
"If $A\\in \\mathbb {C}^{n\\times n}$ is normal and $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ can be diagonalized by $\\widetilde{X}$ , i.e., $\\widetilde{A}=\\widetilde{X}\\widetilde{\\Lambda }\\widetilde{X}^{-1}$ ($\\widetilde{\\Lambda }$ is diagonal), Sun [4] proved that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\kappa _{2}(\\widetilde{X})\\Vert E\\Vert _{F}.$ If both $A\\in \\mathbb {C}^{n\\times n}$ and $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ are diagonalizable, i.e., there are nonsingular matrices $S\\in \\mathbb {C}^{n\\times n}$ and $T\\in \\mathbb {C}^{n\\times n}$ such that $S^{-1}AS={\\rm diag}\\big (\\lambda _{1},\\ldots ,\\lambda _{n}\\big )$ and $T^{-1}\\widetilde{A}T={\\rm diag}\\big (\\widetilde{\\lambda }_{1},\\ldots ,\\widetilde{\\lambda }_{n}\\big )$ , Zhang [5] showed that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\kappa _{2}(S)\\kappa _{2}(T)\\Vert E\\Vert _{F}.$ Assume that $A\\in \\mathbb {C}^{n\\times n}$ is normal, $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ is arbitrary, and $\\widetilde{U}_{1}\\in {U}_{n}$ such that $\\widetilde{U}_{1}^{\\ast }\\widetilde{A}\\widetilde{U}_{1}={\\rm diag}\\big (\\widetilde{A}_{1},\\ldots ,\\widetilde{A}_{s}\\big )$ for some positive integer $s$ , where each $\\widetilde{A}_{i}\\in \\mathbb {C}^{n_{i}\\times n_{i}}$ is upper triangular and $\\sum _{i=1}^{s}n_{i}=n$ .",
"Li and Sun [6] proved that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\sqrt{n-s+1}\\Vert E\\Vert _{F},$ which has improved (1.4).",
"In particular, if $\\widetilde{A}$ is normal (hence $s=n$ ), then (1.5) reduces to the Hoffman–Wielandt theorem.",
"As a special case of normal matrices, if $A\\in \\mathbb {C}^{n\\times n}$ is Hermitian and $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ is non-normal, then by Kahan's result [7], there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\sqrt{2}\\Vert E\\Vert _{F}.$ It is easy to see that the upper bounds in (1.4) and (1.6) only depend on the distance $\\Vert E\\Vert _{F}$ .",
"In other words, the bounds do not change no matter how close $\\widetilde{A}\\widetilde{A}^{\\ast }$ is to $\\widetilde{A}^{\\ast }\\widetilde{A}$ .",
"By the Schur’s theorem, there exists a $\\widetilde{U}\\in {U}_{n}$ such that $ \\widetilde{A}=\\widetilde{U}\\big (\\widetilde{\\Lambda }+\\Delta \\big )\\widetilde{U}^{\\ast },$ where $\\widetilde{\\Lambda }={\\rm diag}\\big (\\widetilde{\\lambda }_{1},\\ldots ,\\widetilde{\\lambda }_{n}\\big )$ and $\\Delta $ is strictly upper triangular.",
"Hence, $\\Vert \\Delta \\Vert _{F}=\\bigg (\\Vert \\widetilde{A}\\Vert _{F}^{2}-\\sum _{i=1}^{n}\\big |\\widetilde{\\lambda }_{i}\\big |^{2}\\bigg )^{1\\over 2},$ which can be considered as a quantitative measure of the non-normality of $\\widetilde{A}$ .",
"Indeed, $\\Vert \\Delta \\Vert _{F}$ is referred to as the departure from normality (with respect to $\\Vert \\cdot \\Vert _{F}$ ) of $\\widetilde{A}$ [8].",
"Taking the quantity into account, Sun [9] established that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\min \\big \\lbrace \\Vert A\\Vert _{F},\\sqrt{n-1}\\Vert A\\Vert _{2}\\big \\rbrace \\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},$ provided that $A\\in \\mathbb {C}^{n\\times n}$ is normal and $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ is arbitrary.",
"Recently, Li and Vong [10] studied the variation of the spectrum of a Hermitian matrix and obtained that there is a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{2}\\Vert E\\Vert _{F}\\Vert \\Delta \\Vert _{F}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\Vert E\\Vert _{F}\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},$ provided that $A\\in \\mathbb {C}^{n\\times n}$ is Hermitian and $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ is arbitrary.",
"In this paper, we establish some novel estimates for $\\mathbb {D}_{2}$ (see Theorems 3.6 and 3.10 below), provided that $A\\in \\mathbb {C}^{n\\times n}$ is normal and $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ is arbitrary.",
"The main results include: $\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+(n-1)\\delta (E)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+(n-s)\\delta (E)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\sqrt{\\frac{n-1}{n}}\\delta (A)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},$ where $s$ is defined as in (1.5).",
"Note that, if ${\\rm tr}(E)\\ne 0$ (hence $\\delta (E)<\\Vert E\\Vert _{F}$ ), our estimates (1.10) and (1.11) are sharper than (1.4) and (1.5), respectively.",
"Since $\\Vert A\\Vert _{F}^{2}\\le n\\Vert A\\Vert _{2}^{2}$ , we have $\\sqrt{\\frac{n-1}{n}}\\delta (A)\\le \\min \\big \\lbrace \\Vert A\\Vert _{F},\\sqrt{n-1}\\Vert A\\Vert _{2}\\big \\rbrace ,$ which implies that the upper bound in (1.12) is smaller than that in (1.7).",
"Moreover, we establish some new perturbation bounds for the spectrum of a Hermitian matrix (see Theorem 4.2 below), which contain: $\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\delta (E)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{2}\\delta (E)\\Vert \\Delta \\Vert _{F}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\delta (E)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}}.$ Similarly, if ${\\rm tr}(E)=0$ , then our results (1.13)–(1.15) coincide with (1.6), (1.8), and (1.9), respectively; otherwise, our upper bounds are smaller.",
"The rest of this paper is organized as follows.",
"In Section 2, we introduce some relations between $\\Vert \\mathcal {U}(A)\\Vert _{F}$ and $\\Vert \\mathcal {L}(A)\\Vert _{F}$ , provided that $A$ is a normal matrix.",
"In Section 3, we establish some novel perturbation bounds for the spectrum of a normal matrix.",
"In Section 4, we present new perturbation bounds for the spectrum of a Hermitian matrix.",
"Finally, some conclusions are given in Section 5.",
"2.",
"Preliminaries In this section, we introduce several useful properties of normal matrices.",
"The following lemma gives an identity on the entries of a normal matrix (see, e.g., [2, Lemma 2.1]).",
"Lemma 2.1.",
"Let $A=(a_{ij})\\in \\mathbb {C}^{n\\times n}$ be normal.",
"Then $\\sum _{i=1}^{n-1}\\sum _{j=i+1}^{n}(j-i)|a_{ij}|^{2}=\\sum _{j=1}^{n-1}\\sum _{i=j+1}^{n}(i-j)|a_{ij}|^{2}.$ Using Lemma 2.1, we can obtain the following relations between $\\Vert \\mathcal {U}(A)\\Vert _{F}$ and $\\Vert \\mathcal {L}(A)\\Vert _{F}$ .",
"Lemma 2.2.",
"Let $A=(a_{ij})\\in \\mathbb {C}^{n\\times n}$ be normal.",
"Then $\\Vert \\mathcal {U}(A)\\Vert _{F}&\\le \\sqrt{W_{L}(A)}\\Vert \\mathcal {L}(A)\\Vert _{F},\\\\ \\Vert \\mathcal {L}(A)\\Vert _{F}&\\le \\sqrt{W_{U}(A)}\\Vert \\mathcal {U}(A)\\Vert _{F}.$ Proof.",
"According to the definition of $\\mathcal {U}(\\cdot )$ , it follows that $\\Vert \\mathcal {U}(A)\\Vert _{F}^{2}=\\sum _{i=1}^{n-1}\\sum _{j=i+1}^{n}|a_{ij}|^{2}\\le \\sum _{i=1}^{n-1}\\sum _{j=i+1}^{n}(j-i)|a_{ij}|^{2}.$ By Lemma 2.1, we have $\\Vert \\mathcal {U}(A)\\Vert _{F}^{2}\\le \\sum _{j=1}^{n-1}\\sum _{i=j+1}^{n}(i-j)|a_{ij}|^{2}\\le W_{L}(A)\\sum _{j=1}^{n-1}\\sum _{i=j+1}^{n}|a_{ij}|^{2}=W_{L}(A)\\Vert \\mathcal {L}(A)\\Vert _{F}^{2},$ which leads to (2.1a).",
"Analogously, we can prove the second inequality.",
"$\\Box $ Remark 2.3.",
"For any $A\\in \\mathbb {C}^{n\\times n}$ , it is clear that $W_{L}(A)\\le n-1 \\quad \\text{and} \\quad W_{U}(A)\\le n-1.$ Hence, from Lemma 2.2, we obtain $\\Vert \\mathcal {U}(A)\\Vert _{F}&\\le \\sqrt{n-1}\\Vert \\mathcal {L}(A)\\Vert _{F},\\\\\\Vert \\mathcal {L}(A)\\Vert _{F}&\\le \\sqrt{n-1}\\Vert \\mathcal {U}(A)\\Vert _{F},$ which are the inequalities stated in [2, Lemma 3.1].",
"The following lemma presents a modified version of (2.2a) and (2.2b).",
"For a detailed proof, we refer the interested reader to [6, Lemma 2.2].",
"Lemma 2.4.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be normal.",
"Then for any $i\\in \\lbrace 1,\\dots ,n\\rbrace $ , $\\Vert (\\mathcal {U}(A))_{(i)}\\Vert _{F}\\le \\Vert \\mathcal {L}(A)\\Vert _{F} \\quad \\emph {and} \\quad \\Vert (\\mathcal {L}(A))_{(i)}\\Vert _{F}\\le \\Vert \\mathcal {U}(A)\\Vert _{F},$ where $(\\cdot )_{(i)}$ denotes the $i$ -th row of a matrix.",
"3.",
"Perturbation bounds for the spectrum of a normal matrix In this section, we present several novel perturbation bounds for the spectrum of a normal matrix.",
"Some of our estimates have improved the existing results in [2, 6, 9].",
"3.1.",
"Two useful lemmas.",
"We first prove a simple but important lemma, which plays a key role in our further analysis.",
"Lemma 3.1.",
"Let $M=(m_{ij})\\in \\mathbb {C}^{n\\times n}$ .",
"Then $\\Vert \\mathcal {L}(M)\\Vert _{F}^{2}+\\Vert \\mathcal {U}(M)\\Vert _{F}^{2}\\le \\delta (M)^{2}.$ Proof.",
"According to the fact that $\\Vert M\\Vert _{F}^{2}=\\Vert \\mathcal {D}(M)\\Vert _{F}^{2}+\\Vert \\mathcal {L}(M)\\Vert _{F}^{2}+\\Vert \\mathcal {U}(M)\\Vert _{F}^{2},$ using the Cauchy–Schwarz’s inequality, we immediately obtain $\\Vert \\mathcal {L}(M)\\Vert _{F}^{2}+\\Vert \\mathcal {U}(M)\\Vert _{F}^{2}=\\Vert M\\Vert _{F}^{2}-\\sum _{i=1}^{n}|m_{ii}|^{2}\\le \\Vert M\\Vert _{F}^{2}-\\frac{1}{n}\\bigg (\\sum _{i=1}^{n}|m_{ii}|\\bigg )^{2}.$ Note that $ \\sum _{i=1}^{n}|m_{ii}|\\ge \\bigg |\\sum _{i=1}^{n}m_{ii}\\bigg |=|{\\rm tr}(M)|.$ We then get the desired inequality.",
"$\\Box $ For any $M\\in \\mathbb {C}^{n\\times n}$ , one may give other upper bounds for $\\Vert \\mathcal {L}(M)\\Vert _{F}^{2}+\\Vert \\mathcal {U}(M)\\Vert _{F}^{2}$ .",
"It is well-known that the Hadamard product of $A=(a_{ij})\\in \\mathbb {C}^{m\\times n}$ and $B=(b_{ij})\\in \\mathbb {C}^{m\\times n}$ is defined as $A\\circ B=(a_{ij}b_{ij})\\in \\mathbb {C}^{m\\times n}$ (see, e.g., [11, Definition 5.0.1]).",
"According to the proof of Lemma 3.1, we have $\\Vert \\mathcal {L}(M)\\Vert _{F}^{2}+\\Vert \\mathcal {U}(M)\\Vert _{F}^{2}=\\Vert M\\Vert _{F}^{2}-\\sum _{i=1}^{n}|m_{ii}^{2}|\\le \\Vert M\\Vert _{F}^{2}-\\bigg |\\sum _{i=1}^{n}m_{ii}^{2}\\bigg |,$ which yields $\\Vert \\mathcal {L}(M)\\Vert _{F}^{2}+\\Vert \\mathcal {U}(M)\\Vert _{F}^{2}\\le \\Vert M\\Vert _{F}^{2}-|{\\rm tr}(M\\circ M)|.$ For any $M=(m_{ij})\\in \\mathbb {C}^{n\\times n}$ , the entry-wise absolute value of $M$ is defined as $|M|=(|m_{ij}|)\\in \\mathbb {R}_{+}^{n\\times n}$ (see, e.g., [11, p. 124]), where $\\mathbb {R}_{+}^{n\\times n}$ denotes the set of all $n\\times n$ non-negative matrices.",
"We can also show that $\\Vert \\mathcal {L}(M)\\Vert _{F}^{2}+\\Vert \\mathcal {U}(M)\\Vert _{F}^{2}&=\\Vert M\\Vert _{F}^{2}-{\\rm tr}(|M|\\circ |M|),\\\\\\Vert \\mathcal {L}(M)\\Vert _{F}^{2}+\\Vert \\mathcal {U}(M)\\Vert _{F}^{2}&\\le \\Vert M\\Vert _{F}^{2}-\\frac{1}{n}\\left({\\rm tr}(|M|)\\right)^{2}.$ In view of the above relations, we now define $\\phi _{1}(M):&=\\Vert M\\Vert _{F}^{2}-|{\\rm tr}(M\\circ M)|,\\\\ \\phi _{2}(M):&=\\Vert M\\Vert _{F}^{2}-{\\rm tr}(|M|\\circ |M|),\\\\ \\phi _{3}(M):&=\\Vert M\\Vert _{F}^{2}-\\frac{1}{n}\\left({\\rm tr}(|M|)\\right)^{2}.$ It is clear that $\\delta (M)$ is invariant under a unitary similarity transformation, i.e., $ \\delta (U^{\\ast }MU)=\\delta (M)$ for all $U\\in {U}_{n}$ .",
"However, $\\phi _{j}(\\cdot ) \\ (j=1,2,3)$ may change under a unitary similarity transformation.",
"For instance, $M=\\begin{pmatrix}1+i & 0\\\\0 & 2\\end{pmatrix} \\quad \\text{and} \\quad U=\\frac{1}{\\sqrt{2}}\\begin{pmatrix}1 & i\\\\i & 1\\end{pmatrix},$ where $i=\\sqrt{-1}$ .",
"Then we have $U^{\\ast }MU=\\frac{1}{2}\\begin{pmatrix}3+i & -1-i\\\\1+i & 3+i\\end{pmatrix} \\quad \\text{and} \\quad |U^{\\ast }MU|=\\frac{1}{2}\\begin{pmatrix}\\sqrt{10} & \\sqrt{2}\\\\\\sqrt{2} & \\sqrt{10}\\end{pmatrix}.$ Straightforward calculations yield $\\phi _{1}(M)&=6-2\\sqrt{5}>1=\\phi _{1}(U^{\\ast }MU),\\\\\\phi _{2}(M)&=0<1=\\phi _{2}(U^{\\ast }MU),\\\\\\phi _{3}(M)&=3-2\\sqrt{2}<1=\\phi _{3}(U^{\\ast }MU).$ This example also illustrates that $\\phi _{j}(M) \\ (j=1,2,3)$ may alter under a unitary similarity transformation, even if $M$ is a normal matrix.",
"Using Lemmas 2.2 and 3.1, we can get the following estimates for $\\Vert \\mathcal {U}(A)\\Vert _{F}$ and $\\Vert \\mathcal {L}(A)\\Vert _{F}$ .",
"Lemma 3.2.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be normal.",
"Then $\\Vert \\mathcal {U}(A)\\Vert _{F}\\le \\sqrt{\\frac{W_{L}(A)}{1+W_{L}(A)}}\\delta (A) \\quad \\emph {and} \\quad \\Vert \\mathcal {L}(A)\\Vert _{F}\\le \\sqrt{\\frac{W_{U}(A)}{1+W_{U}(A)}}\\delta (A).$ Proof.",
"By (2.1a), we have $\\left(1+W_{L}(A)\\right)\\Vert \\mathcal {U}(A)\\Vert _{F}^{2}\\le W_{L}(A)\\left(\\Vert \\mathcal {L}(A)\\Vert _{F}^{2}+\\Vert \\mathcal {U}(A)\\Vert _{F}^{2}\\right).$ Using Lemma 3.1, we obtain $\\left(1+W_{L}(A)\\right)\\Vert \\mathcal {U}(A)\\Vert _{F}^{2}\\le W_{L}(A)\\delta (A)^{2},$ which yields $\\Vert \\mathcal {U}(A)\\Vert _{F}\\le \\sqrt{ \\frac{W_{L}(A)}{1+W_{L}(A)}}\\delta (A).$ Similarly, by (2.1b) and Lemma 3.1, we have $\\Vert \\mathcal {L}(A)\\Vert _{F}\\le \\sqrt{ \\frac{W_{U}(A)}{1+W_{U}(A)}}\\delta (A).$ This completes the proof.",
"$\\Box $ From Remark 2.3, we can see that the inequalities in Lemma 3.2 still hold when $W_{L}(A)$ and $W_{U}(A)$ are replaced by $n-1$ .",
"This observation yields the following corollary.",
"Corollary 3.3.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be normal.",
"Then $\\max \\bigg \\lbrace \\sup _{U\\in {U}_{n}}\\Vert \\mathcal {U}(U^{\\ast }AU)\\Vert _{F},\\sup _{U\\in {U}_{n}}\\Vert \\mathcal {L}(U^{\\ast }AU)\\Vert _{F}\\bigg \\rbrace \\le \\sqrt{\\frac{n-1}{n}}\\delta (A).$ In particular, we have $\\max \\big \\lbrace \\Vert \\mathcal {U}(A)\\Vert _{F},\\Vert \\mathcal {L}(A)\\Vert _{F}\\big \\rbrace \\le \\sqrt{\\frac{n-1}{n}}\\delta (A).$ 3.2.",
"The estimates based on Schur’s decomposition.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be a normal matrix with spectrum $\\lbrace \\lambda _{i}\\rbrace _{i=1}^{n}$ .",
"Assume that $\\widetilde{A}=A+E$ has the spectrum $\\lbrace \\widetilde{\\lambda }_{i}\\rbrace _{i=1}^{n}$ , where $E\\in \\mathbb {C}^{n\\times n}$ is an arbitrary perturbation.",
"By the Schur's theorem, there exists a $\\widetilde{U}\\in {U}_{n}$ such that $\\widetilde{A}=\\widetilde{U}\\big (\\widetilde{\\Lambda }+\\Delta \\big )\\widetilde{U}^{\\ast },$ where $\\widetilde{\\Lambda }$ is diagonal and $\\Delta $ is strictly upper triangular.",
"From $\\widetilde{A}=A+E$ , we have $\\widetilde{U}^{\\ast }A\\widetilde{U}+\\widetilde{U}^{\\ast }E\\widetilde{U}=\\widetilde{\\Lambda }+\\Delta ,$ which means $\\mathcal {L}(\\widetilde{U}^{\\ast }A\\widetilde{U})+\\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})&=0,\\\\\\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})+\\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})&=\\Delta .$ Since both $\\widetilde{\\Lambda }$ and $\\widetilde{U}^{\\ast }A\\widetilde{U}$ are normal, by the Hoffman–Wielandt theorem, we obtain that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\Vert \\widetilde{\\Lambda }-\\widetilde{U}^{\\ast }A\\widetilde{U}\\Vert _{F}=\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}.$ It follows from (3.1b) that $\\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta $ can be written as $\\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta =\\mathcal {D}(\\widetilde{U}^{\\ast }E\\widetilde{U})+\\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})-\\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U}).$ Then, $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}^{2}&=\\Vert \\mathcal {D}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}\\\\&=\\Vert E\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}.$ Hence, we obtain that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}}.$ In order to derive the upper bounds for $\\mathbb {D}_{2}$ , we need to estimate $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}$ .",
"Based on the different estimates for $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}$ , we can obtain the following lemma.",
"Lemma 3.4.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be a normal matrix with spectrum $\\lbrace \\lambda _{i}\\rbrace _{i=1}^{n}$ , and let $\\widetilde{A}=A+E$ with spectrum $\\lbrace \\widetilde{\\lambda }_{i}\\rbrace _{i=1}^{n}$ , where $E\\in \\mathbb {C}^{n\\times n}$ is an arbitrary perturbation.",
"Then there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\delta (E)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{1+W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}\\delta (E)\\Vert \\Delta \\Vert _{F}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\delta (E)\\Vert \\Delta \\Vert _{F}+\\Vert \\Delta \\Vert _{F}^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\sqrt{W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}\\delta (E)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},$ where $\\Delta =\\mathcal {U}(\\widetilde{U}^{\\ast }\\widetilde{A}\\widetilde{U})$ with $\\widetilde{U}\\in {U}_{n}(\\widetilde{A})$ .",
"Proof.",
"(a) Since $\\widetilde{U}^{\\ast }A\\widetilde{U}$ is normal, by (2.1a), we have $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}\\le W_{L}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}.$ According to (3.1a) and the definition of $W_{L}(\\cdot )$ , it follows that $W_{L}(\\widetilde{U}^{\\ast }A\\widetilde{U})=W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})$ .",
"Hence, $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}&\\le W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}\\\\&\\le W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Big (\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}\\Big )\\\\&\\le W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\delta (E)^{2},$ where we have used Lemma 3.1.",
"Then the estimate (3.3a) follows immediately from (3.2).",
"(b) Based on (2.1a), (3.1a), the Cauchy–Schwarz’s inequality, and Lemma 3.1, we obtain $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}&\\le \\sqrt{W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}\\\\&\\le \\sqrt{1+W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}\\sqrt{\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}}\\\\&\\le \\sqrt{1+W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}\\delta (E).$ From (3.1b), we have $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}\\le \\Vert \\Delta \\Vert _{F}.$ Hence, the estimate (3.3b) holds because of (3.2).",
"(c) By (3.1b), the triangle inequality, and Lemma 3.1, we have $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}&=\\Vert \\Delta -\\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}\\\\&\\le \\Big (\\Vert \\Delta \\Vert _{F}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}\\Big )^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}\\\\&=2\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}\\Vert \\Delta \\Vert _{F}+\\Vert \\Delta \\Vert _{F}^{2}\\\\&\\le 2\\sqrt{\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}}\\Vert \\Delta \\Vert _{F}+\\Vert \\Delta \\Vert _{F}^{2}\\\\&\\le 2\\delta (E)\\Vert \\Delta \\Vert _{F}+\\Vert \\Delta \\Vert _{F}^{2}.$ An application of (3.2) yields the estimate (3.3c).",
"(d) Using (3.1b) and the triangle inequality, we obtain $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}&=\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\Delta -\\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}\\\\&\\le \\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Big (\\Vert \\Delta \\Vert _{F}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}\\Big )^{2}\\\\&=2\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}.$ By (2.1a) and (3.1a), we have $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}\\le 2\\sqrt{W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}.$ Because $\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}\\le \\sqrt{\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}}\\le \\delta (E)$ , we arrive at $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}\\le 2\\sqrt{W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}\\delta (E)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}.$ It follows from (3.2) that the estimate (3.3d) holds.",
"$\\Box $ We next give another two upper bounds for $\\mathbb {D}_{2}$ , which are related to the original matrix $A$ .",
"Lemma 3.5.",
"Under the conditions of Lemma 3.4.",
"The permutation $\\pi $ in Lemma 3.4 also satisfies $\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\frac{W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}{1+W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}\\delta (A)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\sqrt{\\frac{W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}{1+W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})}}\\delta (A)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}}.$ Proof.",
"(a) Using Lemma 3.2, we have $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}\\le \\frac{W_{L}(\\widetilde{U}^{\\ast }A\\widetilde{U})}{1+W_{L}(\\widetilde{U}^{\\ast }A\\widetilde{U})}\\delta (A)^{2},$ which leads to $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}\\le \\frac{W_{L}(\\widetilde{U}^{\\ast }A\\widetilde{U})}{1+W_{L}(\\widetilde{U}^{\\ast }A\\widetilde{U})}\\delta (A)^{2}.$ Note that $W_{L}(\\widetilde{U}^{\\ast }A\\widetilde{U})=W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})$ .",
"An application of (3.2) yields (3.4a).",
"(b) In view of (3.1b), we have $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}&=\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\Delta -\\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}\\\\&\\le \\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Big (\\Vert \\Delta \\Vert _{F}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}\\Big )^{2}\\\\&=2\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}.$ Using Lemma 3.2 and (3.2), we immediately obtain the estimate (3.4b).",
"$\\Box $ We now give an explanation on the quantity $\\delta (A)$ involved in (3.4a) and (3.4b).",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be a normal matrix with spectrum $\\lbrace \\lambda _{i}\\rbrace _{i=1}^{n}$ .",
"We define $\\mathbf {1}_{n}:&=(1,\\ldots ,1),\\\\\\mathbf {\\lambda }_{R}:&=\\big ({\\rm Re}(\\lambda _{1}),\\ldots ,{\\rm Re}(\\lambda _{n})\\big ),\\\\ \\mathbf {\\lambda }_{I}:&=\\big ({\\rm Im}(\\lambda _{1}),\\ldots ,{\\rm Im}(\\lambda _{n})\\big ),$ where ${\\rm Re}(\\cdot )$ and ${\\rm Im}(\\cdot )$ denote the real part and the imaginary part of a complex number, respectively.",
"Then we have $\\frac{1}{n}|{\\rm tr}(A)|^{2}=\\frac{1}{n}\\left[(\\mathbf {\\lambda }_{R}\\cdot \\mathbf {1}_{n})^{2}+(\\mathbf {\\lambda }_{I}\\cdot \\mathbf {1}_{n})^{2}\\right]=|\\mathbf {\\lambda }_{R}|^{2}\\cos ^{2}\\theta _{1}+|\\mathbf {\\lambda }_{I}|^{2}\\cos ^{2}\\theta _{2},$ where $\\theta _{1}=\\arccos \\frac{\\mathbf {\\lambda }_{R}\\cdot \\mathbf {1}_{n}}{|\\mathbf {\\lambda }_{R}||\\mathbf {1}_{n}|} \\quad \\text{and} \\quad \\theta _{2}=\\arccos \\frac{\\mathbf {\\lambda }_{I}\\cdot \\mathbf {1}_{n}}{|\\mathbf {\\lambda }_{I}||\\mathbf {1}_{n}|}.$ The normality of $A$ implies $\\Vert A\\Vert _{F}^{2}=\\sum _{i=1}^{n}|\\lambda _{i}|^{2}=|\\mathbf {\\lambda }_{R}|^{2}+|\\mathbf {\\lambda }_{I}|^{2}.$ Thus, $\\delta (A)$ can be explicitly expressed as $\\delta (A)=\\sqrt{|\\mathbf {\\lambda }_{R}|^{2}\\sin ^{2}\\theta _{1}+|\\mathbf {\\lambda }_{I}|^{2}\\sin ^{2}\\theta _{2}}.$ A key observation is that the estimates in Lemmas 3.4 and 3.5 are still valid if $W_{L}(\\widetilde{U}^{\\ast }E\\widetilde{U})$ is replaced by $n-1$ .",
"Hence, we have the following results.",
"Theorem 3.6.",
"Under the conditions of Lemma 3.4.",
"Then there is a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+(n-1)\\delta (E)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{n}\\ \\delta (E)\\Vert \\Delta \\Vert _{F}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\delta (E)\\Vert \\Delta \\Vert _{F}+\\Vert \\Delta \\Vert _{F}^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\sqrt{n-1}\\ \\delta (E)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\frac{n-1}{n}\\delta (A)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\sqrt{\\frac{n-1}{n}}\\delta (A)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},$ where $\\Delta =\\mathcal {U}(\\widetilde{U}^{\\ast }\\widetilde{A}\\widetilde{U})$ with $\\widetilde{U}\\in {U}_{n}(\\widetilde{A})$ .",
"Remark 3.7.",
"For a given normal matrix $A$ , the upper bounds in (3.5a) and (3.5e) only depend on the perturbation $E$ .",
"They may not reduce to the Hoffman–Wielandt’s bound $\\Vert E\\Vert _{F}$ , even if the perturbed matrix $\\widetilde{A}$ is normal.",
"Nevertheless, the upper bounds in (3.5b)–(3.5d) and (3.5f) reduce to $\\Vert E\\Vert _{F}$ when $\\widetilde{A}$ is normal.",
"In other words, the estimates (3.5b)–(3.5d) and (3.5f) have extended the Hoffman–Wielandt theorem.",
"Remark 3.8.",
"Although $\\Vert \\Delta \\Vert _{F}$ can be explicitly expressed as $ \\Vert \\Delta \\Vert _{F}=\\bigg (\\Vert \\widetilde{A}\\Vert _{F}^{2}-\\sum _{i=1}^{n}\\big |\\widetilde{\\lambda }_{i}\\big |^{2}\\bigg )^{1\\over 2},$ the quantity $\\Vert \\Delta \\Vert _{F}$ is uncomputable in general because the spectrum of $\\widetilde{A}$ is unknown.",
"Hence, a natural question is how to effectively estimate $\\Vert \\Delta \\Vert _{F}$ .",
"Here we mention an applicable upper bound for $\\Vert \\Delta \\Vert _{F}$ derived by Henrici [8], that is, $\\Vert \\Delta \\Vert _{F}\\le \\left(\\frac{n^{3}-n}{12}\\right)^{1\\over 4}\\sqrt{\\Vert \\widetilde{A}\\widetilde{A}^{\\ast }-\\widetilde{A}^{\\ast }\\widetilde{A}\\Vert _{F}}.$ There is also a lower bound for $\\Vert \\Delta \\Vert _{F}$ established by Sun [9], that is, $\\Vert \\Delta \\Vert _{F}\\ge \\left(\\Vert \\widetilde{A}\\Vert _{F}^{2}-\\sqrt{\\Vert \\widetilde{A}\\Vert _{F}^{4}-\\frac{1}{2}\\Vert \\widetilde{A}\\widetilde{A}^{\\ast }-\\widetilde{A}^{\\ast }\\widetilde{A}\\Vert _{F}^{2}}\\right)^{1\\over 2}.$ The inequalities (3.6) and (3.7) justify that $\\Vert \\Delta \\Vert _{F}$ can be viewed as a measure of the non-normality of $\\widetilde{A}$ .",
"From (3.6) and (3.7), we see that $\\widetilde{A}$ is normal if and only if $\\Vert \\Delta \\Vert _{F}=0$ .",
"Remark 3.9.",
"Define $\\mathbb {D}_{\\infty }:=\\max _{1\\le i\\le n}\\big |\\widetilde{\\lambda }_{\\pi (i)}-\\lambda _{i}\\big |.$ Using $\\mathbb {D}_{\\infty }\\le \\mathbb {D}_{2}$ and Theorem 3.6, we can readily get the corresponding estimates for $\\mathbb {D}_{\\infty }$ .",
"3.3.",
"The estimates based on block decomposition.",
"Assume that $\\widetilde{U}_{1}\\in {U}_{n}$ satisfies $\\widetilde{A}=\\widetilde{U}_{1}{\\rm diag}\\big (\\widetilde{A}_{1},\\ldots ,\\widetilde{A}_{s}\\big )\\widetilde{U}_{1}^{\\ast }$ for some positive integer $s$ , where each $\\widetilde{A}_{i}\\in \\mathbb {C}^{n_{i}\\times n_{i}}$ is upper triangular and $\\sum _{i=1}^{s}n_{i}=n$ .",
"In particular, if $s=1$ , (3.8) is the Schur’s decomposition of $\\widetilde{A}$ ; if $s=n$ , (3.8) implies that $\\widetilde{A}$ is normal.",
"We then have $\\widetilde{U}_{1}^{\\ast }\\widetilde{A}\\widetilde{U}_{1}={\\rm diag}\\big (\\widetilde{A}_{1},\\ldots ,\\widetilde{A}_{s}\\big )=\\widetilde{\\Lambda }+\\Delta _{1},$ where $\\widetilde{\\Lambda }={\\rm diag}\\big (\\mathcal {D}(\\widetilde{A}_{1}),\\ldots ,\\mathcal {D}(\\widetilde{A}_{s})\\big )$ and $\\Delta _{1}={\\rm diag}\\big (\\mathcal {U}(\\widetilde{A}_{1}),\\ldots ,\\mathcal {U}(\\widetilde{A}_{s})\\big )$ .",
"Let $\\widetilde{U}_{1}^{\\ast }A\\widetilde{U}_{1}$ and $\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1}$ be partitioned as the block forms that coincide with $\\widetilde{U}_{1}^{\\ast }\\widetilde{A}\\widetilde{U}_{1}$ .",
"Set $\\widetilde{U}_{1}^{\\ast }A\\widetilde{U}_{1}=(\\widehat{A}_{ij})_{s\\times s}$ and $\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1}=(\\widehat{E}_{ij})_{s\\times s}$ .",
"Due to $\\widetilde{U}_{1}^{\\ast }A\\widetilde{U}_{1}+\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1}={\\rm diag}\\big (\\widetilde{A}_{1},\\ldots ,\\widetilde{A}_{s}\\big )$ , it follows that $\\widehat{A}_{ii}+\\widehat{E}_{ii}=\\widetilde{A}_{i}, \\quad \\forall i=1,\\ldots ,s.$ By the Hoffman–Wielandt theorem, we have that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\Vert \\widetilde{\\Lambda }-\\widetilde{U}_{1}^{\\ast }A\\widetilde{U}_{1}\\Vert _{F}=\\Vert \\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1}-\\Delta _{1}\\Vert _{F}.$ Hence, we have $\\mathbb {D}_{2}\\le \\sqrt{\\Vert \\mathcal {D}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}+\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})-\\Delta _{1}\\Vert _{F}^{2}}.$ The following theorem presents another three estimates for $\\mathbb {D}_{2}$ , which involve the quantity $s$ .",
"Theorem 3.10.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be normal, and let $\\widetilde{A}=A+E$ with decomposition (3.8), where $E\\in \\mathbb {C}^{n\\times n}$ is an arbitrary perturbation.",
"Assume that the spectra of $A$ and $\\widetilde{A}$ are $\\lbrace \\lambda _{i}\\rbrace _{i=1}^{n}$ and $\\lbrace \\widetilde{\\lambda }_{i}\\rbrace _{i=1}^{n}$ , respectively.",
"Then there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+(n-s)\\delta (E)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{n-s+1}\\ \\delta (E)\\Vert \\Delta _{1}\\Vert _{F}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\sqrt{n-s}\\ \\delta (E)\\Vert \\Delta _{1}\\Vert _{F}-\\Vert \\Delta _{1}\\Vert _{F}^{2}},$ where $\\Delta _{1}=\\mathcal {U}(\\widetilde{U}_{1}^{\\ast }\\widetilde{A}\\widetilde{U}_{1})$ .",
"Proof.",
"(a) By (3.9), we have $\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})-\\Delta _{1}\\Vert _{F}^{2}&=\\sum _{1\\le i<j\\le s}\\Vert \\widehat{E}_{ij}\\Vert _{F}^{2}+\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widehat{E}_{ii})-\\mathcal {U}(\\widetilde{A}_{i})\\Vert _{F}^{2}\\\\&\\le \\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}+\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}.$ Let $(\\mathcal {U}(\\widehat{A}_{ii}))_{(j)}$ be the $j$ -th row of $\\mathcal {U}(\\widehat{A}_{ii})$ , and let $\\Vert (\\mathcal {U}(\\widehat{A}_{pp}))_{(q)}\\Vert _{F}=\\max \\limits _{i,j}\\Vert (\\mathcal {U}(\\widehat{A}_{ii}))_{(j)}\\Vert _{F}$ for some $1\\le p\\le s$ and $1\\le q\\le n_{p}$ .",
"Then we have $\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}\\le \\sum _{i=1}^{s}(n_{i}-1)\\Vert (\\mathcal {U}(\\widehat{A}_{pp}))_{(q)}\\Vert _{F}^{2}\\le (n-s)\\Vert (\\mathcal {U}(\\widetilde{U}_{1}^{\\ast }A\\widetilde{U}_{1}))_{(k)}\\Vert _{F}^{2},$ where $k=\\sum _{i=1}^{p-1}n_{i}+q$ (if $p=1$ , we set $k=q$ ).",
"Using Lemma 2.4, we have $\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}\\le (n-s)\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }A\\widetilde{U}_{1})\\Vert _{F}^{2}=(n-s)\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}.$ Combining (3.12b) and (3.13), we obtain $\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})-\\Delta _{1}\\Vert _{F}^{2}\\le \\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}+(n-s)\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}.$ Due to (3.10) and $\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}\\le \\delta (E)$ , it follows that (3.11a) holds.",
"(b) From (3.9) and (3.12a), we have $\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})-\\Delta _{1}\\Vert _{F}^{2}=\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}+\\sum _{i=1}^{s}\\left(\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widehat{E}_{ii})\\Vert _{F}^{2}\\right).$ In view of (3.9) and the triangle inequality, we immediately obtain $\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}-\\Vert \\mathcal {U}(\\widehat{E}_{ii})\\Vert _{F}\\le \\Vert \\mathcal {U}(\\widetilde{A}_{i})\\Vert _{F}, \\quad \\forall i=1,\\ldots ,s.$ Hence, $\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})-\\Delta _{1}\\Vert _{F}^{2}\\le \\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}+\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widetilde{A}_{i})\\Vert _{F}\\Big (\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}+\\Vert \\mathcal {U}(\\widehat{E}_{ii})\\Vert _{F}\\Big ).$ By the Cauchy–Schwarz’s inequality, we have $\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widetilde{A}_{i})\\Vert _{F}\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}&\\le \\Vert \\Delta _{1}\\Vert _{F}\\bigg (\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}\\bigg )^{1\\over 2},\\\\\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widetilde{A}_{i})\\Vert _{F}\\Vert \\mathcal {U}(\\widehat{E}_{ii})\\Vert _{F}&\\le \\Vert \\Delta _{1}\\Vert _{F}\\bigg (\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widehat{E}_{ii})\\Vert _{F}^{2}\\bigg )^{1\\over 2}.$ Since $\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}\\le (n-s)\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2} \\quad \\text{and} \\quad \\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widehat{E}_{ii})\\Vert _{F}^{2}\\le \\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2},$ it follows that $\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})-\\Delta _{1}\\Vert _{F}^{2}\\le \\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}+\\Big (\\sqrt{n-s}\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}+\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}\\Big )\\Vert \\Delta _{1}\\Vert _{F}.$ Using the Cauchy–Schwarz’s inequality and Lemma 3.1, we obtain $\\sqrt{n-s}\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}+\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}\\le \\sqrt{n-s+1}\\ \\delta (E).$ Thus, $\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})-\\Delta _{1}\\Vert _{F}^{2}\\le \\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}+\\sqrt{n-s+1}\\ \\delta (E)\\Vert \\Delta _{1}\\Vert _{F}.$ Then, the estimate (3.11b) follows immediately from (3.10).",
"(c) By (3.9) and the triangle inequality, we have $\\sum _{i=1}^{s}\\Big (\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widehat{E}_{ii})\\Vert _{F}^{2}\\Big )&=\\sum _{i=1}^{s}\\Big (\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{A}_{i})-\\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}\\Big )\\\\&\\le \\sum _{i=1}^{s}\\bigg [\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}-\\Big (\\Vert \\mathcal {U}(\\widetilde{A}_{i})\\Vert _{F}-\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}\\Big )^{2}\\bigg ]\\\\&=2\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widetilde{A}_{i})\\Vert _{F}\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}-\\Vert \\Delta _{1}\\Vert _{F}^{2}.$ Using the Cauchy–Schwarz’s inequality, we obtain $\\sum _{i=1}^{s}\\left(\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widehat{E}_{ii})\\Vert _{F}^{2}\\right)\\le 2\\bigg (\\sum _{i=1}^{s}\\Vert \\mathcal {U}(\\widehat{A}_{ii})\\Vert _{F}^{2}\\bigg )^{1\\over 2}\\Vert \\Delta _{1}\\Vert _{F}-\\Vert \\Delta _{1}\\Vert _{F}^{2}.$ In view of (3.13)–(3.15), we arrive at $\\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})-\\Delta _{1}\\Vert _{F}^{2}\\le \\Vert \\mathcal {U}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}^{2}+2\\sqrt{n-s}\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}\\Vert \\Delta _{1}\\Vert _{F}-\\Vert \\Delta _{1}\\Vert _{F}^{2}.$ From (3.10) and $\\Vert \\mathcal {L}(\\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1})\\Vert _{F}\\le \\delta (E)$ , we deduce that (3.11c) holds.",
"$\\Box $ Remark 3.11.",
"For any $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ , the decomposition (3.8) is always valid for $s=1$ .",
"If $s=1$ , (3.11a)–(3.11c) reduce to (3.5a), (3.5b), and (3.5d), respectively; otherwise, the upper bounds in (3.11a)–(3.11c) are smaller.",
"Remark 3.12.",
"We remark that the upper bounds for $\\mathbb {D}_{2}$ derived in this section are all absolute type perturbation bounds.",
"As discussed in [6], we can apply our estimates for $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}$ and $\\Vert \\widetilde{U}_{1}^{\\ast }E\\widetilde{U}_{1}-\\Delta _{1}\\Vert _{F}$ to derive corresponding relative type perturbation bounds.",
"For more theories about relative perturbation bounds of spectrum, we refer to [12–15] and the references therein.",
"4.",
"Perturbation bounds for the spectrum of a Hermitian matrix Clearly, the estimates established in Section 3 are applicable for Hermitian matrices as well.",
"Nevertheless, Hermitian matrices possess some special properties, which can yield more accurate estimates.",
"In this section, we present several new perturbation bounds for the spectrum of a Hermitian matrix.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be Hermitian, and let its perturbed matrix $\\widetilde{A}\\in \\mathbb {C}^{n\\times n}$ be decomposed as $\\widetilde{A}=\\widetilde{U}\\big (\\widetilde{\\Lambda }+\\Delta \\big )\\widetilde{U}^{\\ast },$ where $\\widetilde{U}\\in {U}_{n}$ , $\\widetilde{\\Lambda }={\\rm diag}\\big (\\widetilde{\\lambda }_{1},\\ldots ,\\widetilde{\\lambda }_{n}\\big )$ , and $\\Delta $ is strictly upper triangular.",
"Using the same argument as in subsection 3.2, we can obtain $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}^{2}=\\Vert E\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}.$ Since $\\widetilde{U}^{\\ast }A\\widetilde{U}$ is Hermitian, by (3.1a), we have $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}=\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}=\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}.$ Hence, $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}^{2}=\\Vert E\\Vert _{F}^{2}+\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}.$ In order to derive perturbation bounds for the spectrum of a Hermitian matrix, we need the following estimates for $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}$ .",
"Lemma 4.1.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be Hermitian, and let $\\widetilde{A}=A+E$ , where $E\\in \\mathbb {C}^{n\\times n}$ is an arbitrary perturbation.",
"Then $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\delta (E)^{2}},\\\\\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{2}\\delta (E)\\Vert \\Delta \\Vert _{F}},\\\\\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\delta (E)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},\\\\\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\frac{1}{2}\\delta (A)^{2}},\\\\\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{2}\\delta (A)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},$ where $\\Delta =\\mathcal {U}(\\widetilde{U}^{\\ast }\\widetilde{A}\\widetilde{U})$ with $\\widetilde{U}\\in {U}_{n}(\\widetilde{A})$ .",
"Proof.",
"(a) Using (4.3) and Lemma 3.1, we obtain $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}}\\\\&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\delta (E)^{2}}.$ (b) From (3.1b), we have that $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}\\le \\Vert \\Delta \\Vert _{F}$ .",
"In addition, $\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}&=\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}\\\\&\\le \\sqrt{2}\\sqrt{\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}}\\\\&\\le \\sqrt{2}\\delta (E).$ An application of (4.2) yields the inequality (4.4b).",
"(c) By (4.2), (3.1b), and the triangle inequality, we have $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}^{2}&\\le \\Vert E\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Big (\\Vert \\Delta \\Vert _{F}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}\\Big )^{2}\\\\&=\\Vert E\\Vert _{F}^{2}+2\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}\\\\&=\\Vert E\\Vert _{F}^{2}+2\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}\\\\&\\le \\Vert E\\Vert _{F}^{2}+2\\sqrt{\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}}\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}\\\\&\\le \\Vert E\\Vert _{F}^{2}+2\\delta (E)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2},$ which gives the inequality (4.4c).",
"(d) Because $\\widetilde{U}^{\\ast }A\\widetilde{U}$ is Hermitian, by (4.2) and Lemma 3.1, we have $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}^{2}&=\\Vert E\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}-\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }E\\widetilde{U})\\Vert _{F}^{2}\\\\&\\le \\Vert E\\Vert _{F}^{2}+\\frac{1}{2}\\big (\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}\\big )\\\\&\\le \\Vert E\\Vert _{F}^{2}+\\frac{1}{2}\\delta (A)^{2},$ which means the inequality (4.4d).",
"(e) Based on the derivation in (c), we have $\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}^{2}&\\le \\Vert E\\Vert _{F}^{2}+2\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}\\\\&=\\Vert E\\Vert _{F}^{2}+\\big (\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}+\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}\\big )\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}\\\\&\\le \\Vert E\\Vert _{F}^{2}+\\sqrt{2}\\sqrt{\\Vert \\mathcal {U}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}+\\Vert \\mathcal {L}(\\widetilde{U}^{\\ast }A\\widetilde{U})\\Vert _{F}^{2}}\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}\\\\&\\le \\Vert E\\Vert _{F}^{2}+\\sqrt{2}\\delta (A)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2},$ which leads to the inequality (4.4e).",
"$\\Box $ Since both $\\widetilde{\\Lambda }$ and $\\widetilde{U}^{\\ast }A\\widetilde{U}$ are normal, by the Hoffman–Wielandt theorem, we get that there exists a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}\\le \\Vert \\widetilde{\\Lambda }-\\widetilde{U}^{\\ast }A\\widetilde{U}\\Vert _{F}=\\Vert \\widetilde{U}^{\\ast }E\\widetilde{U}-\\Delta \\Vert _{F}.$ An application of Lemma 4.1 yields the following theorem.",
"Theorem 4.2.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be a Hermitian matrix with spectrum $\\lbrace \\lambda _{i}\\rbrace _{i=1}^{n}$ , and let $\\widetilde{A}=A+E$ with spectrum $\\lbrace \\widetilde{\\lambda }_{i}\\rbrace _{i=1}^{n}$ , where $E\\in \\mathbb {C}^{n\\times n}$ is an arbitrary perturbation.",
"Then there is a permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ such that $\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\delta (E)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{2}\\delta (E)\\Vert \\Delta \\Vert _{F}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+2\\delta (E)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\frac{1}{2}\\delta (A)^{2}},\\\\\\mathbb {D}_{2}&\\le \\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{2}\\delta (A)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}},$ where $\\Delta =\\mathcal {U}(\\widetilde{U}^{\\ast }\\widetilde{A}\\widetilde{U})$ with $\\widetilde{U}\\in {U}_{n}(\\widetilde{A})$ .",
"We next give two useful estimates for $\\Vert \\Delta \\Vert _{F}$ , which are sharper than the existing results in [10, Lemma 2.1 and Corollary 2.5].",
"Theorem 4.3.",
"Let $A\\in \\mathbb {C}^{n\\times n}$ be Hermitian, and let $\\widetilde{A}=A+E \\ (\\widetilde{A}\\ne 0)$ , where $E\\in \\mathbb {C}^{n\\times n}$ is a perturbation.",
"Assume that the Schur's decomposition (4.1) satisfies that $\\big |\\widetilde{\\lambda }_{1}\\big |\\ge \\cdots \\ge \\big |\\widetilde{\\lambda }_{n}\\big |$ .",
"Then $\\Vert \\Delta \\Vert _{F}&\\le \\frac{1}{\\sqrt{2}}\\bigg (\\Vert \\widetilde{A}-\\widetilde{A}^{\\ast }\\Vert _{F}^{2}-\\frac{1}{{\\rm rank}(\\widetilde{A})}\\big |{\\rm tr}(\\widetilde{A}-\\widetilde{A}^{\\ast })\\big |^{2}\\bigg )^{1\\over 2}\\le \\frac{1}{\\sqrt{2}}\\delta \\big (\\widetilde{A}-\\widetilde{A}^{\\ast }\\big ),\\\\\\Vert \\Delta \\Vert _{F}&\\le \\frac{1}{\\sqrt{2}}\\bigg (\\Vert E-E^{\\ast }\\Vert _{F}^{2}-\\frac{1}{{\\rm rank}(\\widetilde{A})}\\big |{\\rm tr}(E-E^{\\ast })\\big |^{2}\\bigg )^{1\\over 2}\\le \\frac{1}{\\sqrt{2}}\\delta (E-E^{\\ast }).$ Proof.",
"By decomposition (4.1), we have $\\widetilde{A}-\\widetilde{A}^{\\ast }=\\widetilde{U}(\\widetilde{\\Lambda }+\\Delta -\\widetilde{\\Lambda }^{\\ast }-\\Delta ^{\\ast })\\widetilde{U}^{\\ast },$ which gives $\\Vert \\widetilde{A}-\\widetilde{A}^{\\ast }\\Vert _{F}^{2}=\\Vert \\widetilde{\\Lambda }-\\widetilde{\\Lambda }^{\\ast }\\Vert _{F}^{2}+\\Vert \\Delta -\\Delta ^{\\ast }\\Vert _{F}^{2}=\\sum _{i=1}^{\\rm rank{(\\widetilde{A})}}\\Big |\\widetilde{\\lambda }_{i}-\\overline{\\widetilde{\\lambda }}_{i}\\Big |^{2}+2\\Vert \\Delta \\Vert _{F}^{2}.$ Using the Cauchy–Schwarz’s inequality, we obtain $\\Vert \\widetilde{A}-\\widetilde{A}^{\\ast }\\Vert _{F}^{2}&\\ge \\frac{1}{{\\rm rank}(\\widetilde{A})}\\Bigg (\\sum _{i=1}^{\\rm rank{(\\widetilde{A})}}\\Big |\\widetilde{\\lambda }_{i}-\\overline{\\widetilde{\\lambda }}_{i}\\Big |\\Bigg )^{2}+2\\Vert \\Delta \\Vert _{F}^{2}\\\\&\\ge \\frac{1}{{\\rm rank}(\\widetilde{A})}\\big |{\\rm tr}(\\widetilde{A}-\\widetilde{A}^{\\ast })\\big |^{2}+2\\Vert \\Delta \\Vert _{F}^{2}.$ Consequently, we arrive at $\\Vert \\Delta \\Vert _{F}\\le \\frac{1}{\\sqrt{2}}\\bigg (\\Vert \\widetilde{A}-\\widetilde{A}^{\\ast }\\Vert _{F}^{2}-\\frac{1}{{\\rm rank}(\\widetilde{A})}\\big |{\\rm tr}(\\widetilde{A}-\\widetilde{A}^{\\ast })\\big |^{2}\\bigg )^{1\\over 2}.$ The second inequality follows immediately from $\\widetilde{A}-\\widetilde{A}^{\\ast }=E-E^{\\ast }$ .",
"$\\Box $ Example 4.4.",
"In order to illustrate the estimates in Theorem 4.2, we give an example as follows: $A=\\begin{pmatrix}0 & 1 & 0 & \\cdots & 0\\\\1 & 0 & 0 & \\cdots & 0\\\\0 & 0 & 1 & \\cdots & 0\\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 1\\\\\\end{pmatrix}\\in \\mathbb {R}^{n\\times n} \\quad \\text{and} \\quad E=\\begin{pmatrix}0 & 0 & 0 & \\cdots & 0\\\\-1 & 0 & 0 & \\cdots & 0\\\\0 & 0 & -1 & \\cdots & 0\\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & -1\\\\\\end{pmatrix}\\in \\mathbb {R}^{n\\times n} \\ (n\\ge 3).$ We can easily get that $\\mathbb {D}_{2}=\\sqrt{n}$ for any permutation $\\pi $ of $\\lbrace 1,\\ldots ,n\\rbrace $ .",
"Direct calculations yield $\\Vert A\\Vert _{F}^{2}=n, \\quad |{\\rm tr}(A)|^{2}=(n-2)^{2},\\quad \\Vert E\\Vert _{F}^{2}=n-1, \\quad |{\\rm tr}(E)|^{2}=(n-2)^{2}, \\quad \\Vert \\Delta \\Vert _{F}=1.$ We then have $ \\delta (E)=\\sqrt{3-\\frac{4}{n}}, \\quad \\delta (A)=2\\sqrt{1-\\frac{1}{n}}.$ Hence, the upper bounds in (4.6a)–(4.6e) are $\\sqrt{n-\\frac{4}{n}+2}, \\ \\ \\sqrt{n+\\sqrt{6-\\frac{8}{n}}-1},\\ \\ \\sqrt{n+2\\sqrt{3-\\frac{4}{n}}-2},\\ \\ \\sqrt{n-\\frac{2}{n}+1}, \\ \\ \\sqrt{n+2\\sqrt{2-\\frac{2}{n}}-2},$ respectively.",
"Remark 4.5.",
"From (4.5), we can readily obtain an upper bound $\\Vert E\\Vert _{F}+\\Vert \\Delta \\Vert _{F}$ via the triangle inequality.",
"Due to the fact that $\\delta (E)\\le \\Vert E\\Vert _{F}$ , it follows that $\\sqrt{\\Vert E\\Vert _{F}^{2}+\\sqrt{2}\\delta (E)\\Vert \\Delta \\Vert _{F}}&\\le \\Vert E\\Vert _{F}+\\Vert \\Delta \\Vert _{F},\\\\\\sqrt{\\Vert E\\Vert _{F}^{2}+2\\delta (E)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}}&\\le \\Vert E\\Vert _{F}+\\Vert \\Delta \\Vert _{F}.$ Hence, the upper bounds in (4.6b) and (4.6c) are non-trivial.",
"It is worth mentioning that the upper bound in (4.6c) also satisfies $\\sqrt{\\Vert E\\Vert _{F}^{2}+2\\delta (E)\\Vert \\Delta \\Vert _{F}-\\Vert \\Delta \\Vert _{F}^{2}}\\le \\sqrt{2}\\Vert E\\Vert _{F},$ which reveals that (4.6c) is sharper than (1.6).",
"Remark 4.6.",
"For a given Hermitian matrix $A\\in \\mathbb {C}^{n\\times n}$ , the upper bounds in (4.6a) and (4.6d) only depend on the perturbation $E$ , while other upper bounds in Theorem 4.2 are related to both $E$ and $\\Vert \\Delta \\Vert _{F}$ (Remark 3.8 and Theorem 4.3 have provided several applicable estimates for $\\Vert \\Delta \\Vert _{F}$ ).",
"Clearly, if $\\widetilde{A}$ is normal (i.e., $\\Vert \\Delta \\Vert _{F}=0$ ), the upper bounds in (4.6b), (4.6c), and (4.6e) all reduce to the Hoffman–Wielandt’s bound $\\Vert E\\Vert _{F}$ .",
"5.",
"Conclusions In this paper, we have established novel perturbation bounds for the spectrum of a normal matrix (including the case of Hermitian matrices).",
"Some of our estimates improve the existing results in [2, 6, 7, 9, 10].",
"Moreover, if the perturbed matrix is still normal, the upper bounds involving the “departure from normality” of the perturbed matrix reduce to the Hoffman–Wielandt’s bound.",
"Therefore, these estimates have generalized the classical Hoffman–Wielandt theorem.",
"Acknowledgements This work was supported by the National Key Research and Development Program of China (Grant No.",
"2016YFB0201304) and the Major Research Plan of National Natural Science Foundation of China (Grant Nos.",
"91430215, 91530323).",
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] | 1612.05759 |
[
[
"Representation of the German transmission grid for Renewable Energy\n Sources impact analysis"
],
[
"Abstract The increasing impact of fossil energy generation on the Earth ecological balance is pointing to the need of a transition in power generation technology towards the more clean and sustainable Renewable Energy Sources (RES).",
"This transition is leading to new paradigms and technologies useful for the effective energy transmission and distribution, which take into account the RES stochastic power output.",
"In this scenario, the availability of up to date and reliable datasets regarding topological and operative parameters of power systems in presence of RES are needed, for both proposing and testing new solutions.",
"In this spirit, I present here a dataset regarding the German 380 KV grid which contains fully DC Power Flow operative states of the grid in the presence of various amounts of RES share, ranging from realistic up to 60\\%, which can be used as reference dataset for both steady state and dynamical analysis."
],
[
"Background & Summary",
"The recent growth in the installed Renewable Energy Sources (RES) generation is completely changing the paradigms and rules associated with electrical systems management and control.",
"These systems were installed with a top-down hierarchical approach[10], where the power produced at high voltage levels by big conventional generators was distributed towards the users, located at the low voltage nodes.",
"With the renewable energy revolution, the power systems are now facing the presence of a large number of small and medium size generators distributed over all the voltage levels, which implies a total redesigning of the network structure.",
"Moreover, it is difficult to drive the system in the correct operative state, due to the intermittent nature of RES power production.",
"In order to ensure the correct functioning of power systems, new methodologies and concepts have to be found in the fields of electric system dynamics, control and economics[14], [20], [11], [6], [22], very few public power grid datasets exist, limiting the possibility to perform studies on real systems and limiting the reproducibility of the studies which relies on private data.",
"The lack of datasets is due to various factors, including difficulty in gathering data, secrecy of data for national security issues and difficulty in managing the huge amount of data that characterizes the physical and economic state of these systems.",
"The known power grid datasets, which can be used for performing dynamical studies on nation-wide systems, can be found on different repositories.",
"The Koblenz network collection project contains a dataset of the US transmission grid at http://konect.uni-koblenz.de/networks/opsahl-powergrid, which contains the full system topology.",
"MATPOWER[28] provides various datasets, including different versions of the Polish HV grid and an estimation of the European transmission grid, which description can be found in [8].",
"The most complete dataset of the European grid, including buses, generators and lines description, and a full DC power flow solution is given in [25], [12]; this dataset, which has been used as a starting point for our further improvements, comes in a fully geolocalized way.",
"These datasets have been used in various studies regarding power grid stability and resilience, mainly based on the topological aspects[26], [14], [21].",
"However, the newly proposed methodologies [17], [4], [16] are based on the description of RES as stochastic elements which feed intermittent power into the network, and the realistic positioning and variability of these elements can be very important in describing their impact.",
"Despite the quality and accuracy of most of the existing datasets, they have been produced mainly in the previous decade, when RES generation was still not an issue and include very little (if not none) information about RES generation.",
"Taking Germany as example, the RES installed capacity increased from 6.3% in 2000 to 30% in 2014[3], and is bound to further increase in the next years.",
"For this reason, it is clear how the previously described datasets cannot be used to properly describe the effects of RES generation in power systems.",
"In view of this situation, I have developed a dataset describing the German transmission grid state in presence of real RES and conventional generation.",
"The dataset is based on the network topology proposed in [25], [12], enriched with the RES generation present in Germany at the end of 2014, and updated with the 2014 load and RES production data given in [1].",
"The dataset, containing all 380 KV buses, lines and connected generators in Germany, contains the information about the RES generation, aggregated per bus and tech type, the load of each bus and the parameters and dispatch of each conventional generator, validated by means of DC Optimal Power Flow (DC OPF).",
"Moreover, it contains the estimation of the network status in presence of very high shares of RES up to 60% of the total load, under certain assumptions described in the methods section.",
"This dataset is connected with the paper [15], which uses this dataset for analyzing the islanding capabilities of the German transmission grid.",
"We expect that this dataset can be used for performing dynamics, steady state and economic studies on power grid in presence of an high share of RES generation, due to the accurate description of the operative parameters of the system.",
"In order to make the dataset of practical use for simulation purposes, the German power system obtained from the original UCTE grid data[27], [12] has been processed and enriched.",
"To do so, the data was processed in three main steps, described in detail in the next sections: The data about the German transmission system was isolated from the original dataset.",
"The obtained German system data was merged with the information about RES installed generation in Germany, coming from [1].",
"In this way, it was possible to estimate the amount of RES installed on each transmission node for values of RES penetration $P_\\% = \\lbrace 20, 30, 40, 50 , 60\\rbrace \\%$ .",
"The dispatch of conventional generation was calculated by means of a DC OPF[23], [24], for each value of renewable penetration $P_\\%$ .",
"Moreover, in order to increase the dataset completeness, the nodes' rotating inertia was estimated.",
"This information is not useful for performing steady state analysis, but can be precious for dynamical studies, mainly based on swing equation formalism.",
"The original dataset consists of the full European grid.",
"However, the data regarding the installed RES generation is public available for few European nations.",
"Here we use the data about the German system.",
"Since this data is restricted only to the German territory, it was necessary to isolate it from the bigger UCTE dataset, in order to perform the further proposed improvements.",
"The isolation of the German grid is not straightforward.",
"In order to properly isolate the system, it is vital to consider the power flowing in and out the neighboring countries.",
"In general, the European system is interconnected at the voltage levels of interest, and it is not possible to model the power flow of interconnected nations independently.",
"This is due to the fact that a change in the state of the network around the borders can cause a redirection of power flows through the border nodes, which can impact on the internal flows.",
"In order to overcome this issue, it is common practice in this type of datasets to model the foreign exchanges by creating fictitious nodes at the borders which work as power generators or sinks representing the power exchanges with the neighboring nations.",
"In order to isolate the German network correctly, I have extracted all the nodes belonging to the German power system and all the lines among them, by using the nation identifiers used in the original UCTE dataset.",
"Also, it is necessary to model the power flow in and from the neighboring countries.",
"In order to do so, the foreign border nodes with bigest flows have been kept in the system, with a load (positive for outflows, negative for inflows) equal to the expected foreign flow, extracted from the original dataset.",
"The used German and foreign nodes are shown in figure REF .",
"These nodes model the exchanges with Denmark, Netherlands, France, Switzerland, Austria, Czech Republic and Poland.",
"The values of the border flows have been taken from the original dataset and considered constant during all the further analysis.",
"Moreover, the RES generation facilities present in the original dataset have been not considered, in order to perform the proposed more precise analysis of the RES spatial distribution described in the next sections.",
"Performing precise and reliable studies regarding the impact of RES power supply on power systems requires a realistic spatial and temporal distribution of the existing renewable generation.",
"Although, the currently public available datasets about RES generation are given on aggregate, by measuring the total power production during time in the whole system or in fractions of it.",
"Therefore, the effective RES production of each node is difficult to estimate, especially on transmission level.",
"This is due to its strong dependence on the local meteorological situation and non-linear energy conversion factors of the single nodes.",
"So, it is necessary to identify methodologies that allows to estimate the RES power production on each transmission node, by making use of the available information about the system.",
"Fortunately, some European states (i.e.",
"Italy (atlasole.gse.it/atlasole/) and Germany (http://www.energymap.info/) have made the information about the geographical and technical characteristics of all their installed RES generators publicly available.",
"Starting from these datasets, it is possible to estimate, by making some assumptions that will be described later, the amount of installed RES $P^{Inst}_{i,t}$ per transmission node $i$ , aggregated per technology $t$ .",
"The complete information regarding the installed RES generators in Germany can be found on the web site http://www.energymap.info/.",
"From this web site, it is possible to download a database containing the following information for each RES generator $g_{RES}$ : the date of installation; the postcode of its geographical location; a registration code; the technology type (Solar, Wind, Biomass, Hydro); the nominal power (in KW); the amount of KWh produced in 2013; the amount of KWh/year produced on average during the time after installation; its geographic position, given in terms of lon/lat with a precision of 0.01 degrees; the responsible Transmission System Operators (TSOs) and Distribution System Operators (DSOs).",
"The RES power is mainly installed on low and medium voltage nodes, but its variability can easily change the system power balance, causing the loads of the very high voltage buses $i$ to fluctuate.",
"In order to model the impact of such fluctuations in the system, it is necessary to estimate the amount of RES generation supplied by each node $i$ .",
"The correct association of RES generators $g_{RES}$ with high voltage nodes $i$ is not an easy task, because it depends on the full system status and power flows, and it can in principle vary from time to time.",
"Moreover, a complete knowledge of the entire power system regarding all voltage levels is not available, making a fully correct association process impossible.",
"However, power grids are dynamical systems which always change their state during time, and therefore a correct, static description of the flows of power generated by RES cannot exist.",
"Ideally, a study of transmission grids which includes a correct description of RES should include the simulation of all the lowest voltage levels, including a correct description of the installed RES intermittent output, making the analysis almost impossible to perform, even by having access to all the data.",
"Consequently, an approximation describing a static association of RES generators with high voltage nodes is necessary for providing a good representation of the system.",
"Given the available data, it is possible to estimate this association on based on the assumption that each RES generator $g_{RES}$ contributes to the power production of the geographically nearest node $i$ .",
"I made such an assignment by using GIS (Geographical Information Systems) such as PostGIS (http://postgis.net/) and QGIS [18].",
"In particular, the area of Germany has been at first divided in 230 zones, each of them corresponding to an internal node, as shown in figure REF .",
"Figure: The Voronoi tassellation of the German territory, with respect to the German 380KV nodes.",
"Each red dot represents a network's node ii, and its surrounding area is its Voronoi area A i A_i.",
"The RES generators g RES g_{RES}, not shown here, were associated with each node ii by means of the Voronoi area A i A_i to which they geographically belong to.",
"The map has been created by using the software QGIS .Such zones have been obtained by means of the Voronoi analysis [2].",
"This geographical analysis takes as inputs a set of points ${i}$ distributed in a territory, and produces as output a set of areas ${A_i}$ , each of them corresponding to the area closer to the node $i$ than to any other node $j\\ne i$ .",
"Considering the dataset nodes as the Voronoi points, it was possible to obtain their Voronoi areas $A_i$ .",
"Once this analysis was performed, the German RES generators $g_{RES}$ , which positions are known, were associated with each node by checking which Voronoi area $A_i$ they belong to.",
"In this way, a 1 on 1 association between RES generators $g_{RES}$ and transmission nodes $i$ was obtained.",
"After this step, it was possible to sum up all the installed power $P_{i,t}^{Inst}$ in each node $i$ , for each type of RES technology $t$ .",
"In order to do so, each generator $g_{RES}$ was filtered on the base of its technology and its installed capacity, and all the obtained classes have been summed up for each area $A_i$ , representing the node $i$ .",
"The results of this analysis are shown in figures REF and REF .",
"Figure: The map shows the amount of installed PV generation P*inst i,PV P*{inst}_{i,PV} (in KW) for each node ii of the German transmission network.",
"The association has been made by the nearest node assumption described in section ”Installed RES estimation”.",
"The map has been created by using the software QGIS .Figure: This maps shows the amount of Wind generation P*inst i,Wind P*{inst}_{i,Wind} which has been assigned to each node ii, on the base of the nearest node assumption.",
"All powers are expressed in KW.",
"The map has been created by using the software .This distribution $P_{i,t}^{Inst}$ , representing the amount of installed RES sources for each node $i$ and technology $t$ , can be used to realistically describe the wanted amount of RES production through the nodes.",
"In the following section, this procedure will be described in detail.",
"The original dataset provides a load winter peak solution of the system.",
"However, the dataset includes very limited information regarding RES generation, in the form of few big generators representing RES.",
"Studies regarding dynamics and steady state stability in the presence of RES generation often require as inputs a more detailed description of the system.",
"Therefore, the datasets described in the previous sections were processed in order to obtain a description of the system which includes a realistic representation of RES generation.",
"This has been done in two steps: estimation of the RES production and redispatch of the conventional generation.",
"The first step makes use of the information regarding the load profile of the daily winter peak for the German system's load $L^{data}$ present in the original dataset.",
"At first, define the share of power $P_\\% = {20, 30, 40, 50, 60}$ % produced by renewable as $P_\\% = \\frac{P_{\\%}^{tot}}{L^{data}}$ , where $P_{\\%}^{tot}$ is the power provided in the whole German territory by RES generators of technology $t$ .",
"Then, knowing these amounts and the $P_{i,t}^{Inst}$ obtained in the previous section, it is possible to distribute the $P_{\\%}^{tot}$ among the German territory.",
"Assuming that all generators of the same RES technology have the same operative parameters (efficiency, wind/sun exposure, and in general capacity factors [5]) in the entire German territory, it is possible to estimate the RES production per each node $i$ , per each technology $t=\\lbrace Wind, PV\\rbrace $ , as defined in Eq.",
"REF .",
"$ P_{i,t}^{\\%} = \\frac{P_{i,t}^{Inst} * P_{\\%}^{tot}}{\\sum _i P_{i,t}^{Inst}}$ Again, this assumption is based on the lack of information regarding the capacity factors of all the installed generators and on the extreme complexity associated with a full study of the German weather data during an extended time period.",
"Once the load consumption and RES generation profiles $L^{norm}$ and $P_{i,t}^{\\%}$ are known, it is possible to estimate the power production of conventional generation by means of a DC Optimal Power Flow (OPF) analysis of the system.",
"OPF [23], [24], is described in detail in the research paper associated with this dataset[15].",
"OPF is a standard electrical engineering methodology used for power dispatch (i.e.",
"the decision regarding the patterns of power production in the system).",
"The inputs for the OPF concerning the conventional generation, such as generators' production prices, min and max power, ramps and voltages, lines' impedances and capacities was taken from the original dataset parameters.",
"The loads' consumption and the RES generation was taken from $L_i$ and $P_i^{RES}$ , and it was considered non dispatchable, thus fixed, during the entire OPF process.",
"The outcome of each OPF calculation represents the final dataset, with the full distribution of loads $L_i$ , the distribution of RES generation $P_{i,t}^{\\%}$ , and the conventional generation dispatches $P_g^{CONV}$ for each node $i$ and conventional generator $g$ .",
"Moreover, the further results of the OPF, such as nodes voltages $V_i$ and phases $\\varphi _i$ were inserted in the dataset, since they could be used for further analysis.",
"The rotating inertia (moment of inertia) of the nodes is difficult to estimate, especially for load nodes.",
"If a conventional generator (CG) is present in the node, the CG turbine rotating inertia could be used as reference value.",
"If this value is not known, as it often happens, it is possible to estimate it from the generator technology and nominal power [13].",
"In particular, as stated in [13], the estimated, average inertia for a conventional generator is expressed in Eq.",
"REF , where $S_{max}$ is the nominal apparent power, in W of the generator and $k_I^{gen} = 0.5 \\frac{s^3}{m}$ .",
"Note that if the generator is offline, its inertia contribution must count as zero.",
"$ I = k_I^{gen} \\cdot S_{max},$ Moreover, if a load is present on the node, the evaluation of inertia is difficult.",
"The 380KV network is the highest level of the power system hierarchy, and their nodes' load will likely be given by a combination of both user consumption and generation at lower levels, which in turn could come from RE and conventional sources.",
"For such a reason, a rotating inertia equal to an average motor load has been chosen to represent this nodes.",
"In particular, using the values described in [13], their rotating inertia has been estimated by equation REF , where $S_{max}$ is the load load and $k_I^{load} = 0.2 \\frac{s^3}{m}$ .",
"$ I = k_I^{load} \\cdot S_{max}.$ In principle, if a node has both a load and a conventional generator, their inertia contribution must be summed or, depending on the needs, it should be split in two different nodes.",
"In general, some datasets tend to represent load nodes with attached generators as mixed nodes.",
"In the particular case of the German grid, however, the original dataset consists on nodes that are either generator or load nodes, and no mixed nodes are present.",
"In general, other effects must be taken into account in the estimation of the inertia.",
"In particular, secondary effects due to electronically controlled devices and control systems and wind turbines should be taken into account.",
"However, a correct estimation of these effects was beyond the scope of this work."
],
[
"Estimation of lines capacity",
"The capacity $C_{ij}$ is defined here as the maximum amount of power which can flow through a line before without causing any arm.",
"This quantity, often different from the maximum transmissible power $P_{MAX} = \\frac{V^2}{Z}$ , is linked to heat production associated to resistive effects in the conductor.",
"In particular, it is considered safe a situation in which the produced heat $H_{prod}$ is less then the dispersed heat $H_{disp}$ .",
"Since $H_{disp}$ is dependant from the weather conditions, it is common to define a nominal value of $C_{ij}$ , which serves as reference for the daily use of the lines, especially in absence of real time monitoring systems.",
"The original dataset provided only the information about the $C_{ij}$ of the inter-tie lines.",
"Due to the lack of information regarding the other lines capacity $C_{ij}$ in the available datasets, these has been estimated by assuming fixed values for the existing lines, ranging from 800 MW to 4GW, with steps of 400 MW, obtaining the values $C_{allowed} = \\lbrace 800, 1200, 1600, 2000, 2400, 2800, 3200, 3600, 4000 \\rbrace $ MW.",
"These values have been considered a combination of one or more circuits representing the connection.",
"To estimate these values, all lines flows $P_{ij}^{orig}$ calculated in the original dataset have been multiplied by a common factor $c = 1.65$ .",
"After this step, the obtained capacities $C_{ij} = c\\cdot P_{ij}^{orig}$ have been rounded to the next value in $C_{allowed}$ ."
],
[
"Code availability",
"The proposed dataset is provided in two different formats: excel spreadsheet and matpower files.",
"Excel format is one of the most common data formats, highly portable and easily readable by the majority of programs of interest for the field, such as R[19], Python (Python Software Foundation, https://www.python.org/) and Matlab [28].",
"Geographical coordinates are saved in lat/lon format.",
"Map format is the most common data format for geographical data.",
"It is easily accessible from any GIS software and contains all operative and geographical parameters of the network.",
"The dataset is particularly useful for geographical processing and visualization of the data.",
"A free, open source program able to open and process this format is QGIS [18].",
"The proposed analysis represents an enrichment and extension of the dataset provided in [25] and updated in [12].",
"This dataset contains a full DC working state of the European network (ex UCTE), in the presence of the real geographical distribution of RES in the system.",
"The German Transmission Grid (GTG) dataset, whose geographical representation is shown in figure REF , is composed by 231 nodes and 304 edges.",
"Also, information about 82 Conventional Generators (CGs) is given, together with the information regarding the amount of RES installed generation over each node.",
"In the following, all the available information regarding all the network components is listed.",
"Figure: A geographical representation of the German 380 KV grid.",
"The German nodes are shown in red circles, the nodes connecting to foreign countries are represented by diamonds, coloured on the base of the country, which they connect to.Each network node (also called bus) corresponds to a transmission substation, working at a nominal voltage of 380KV.",
"For each node, it is available all the information needed for fully describing the system, extracted from the original dataset or by making use of the techniques described in the methods section.",
"In particular: The geographical position was obtained by georeferentiation, by means of a geographical reprojection of the original dataset.",
"This methodology is able to produce a quite accurate positioning of the substations into the territory.",
"In order to improve the precision of the positioning, if needed, a cross-matching with openstreetmap data is suggested.",
"The resulting position is given in lat/lon values.",
"The nodes load (in MW), was extracted from the original dataset in [27], [12].",
"The foreign connection nodes' loads were changed in order to take into account foreign power exchanges.",
"Each node code and id are available and are the same as in the original dataset.",
"This is useful for cross referencing between the different tables of the dataset.",
"Each node conventional power output (in MW) is given, defined as the sum of all conventional generation on the node, referred to the actual dispatch.",
"Each node installed wind and solar generation (in MW) are given.",
"This information has been obtained by means of the analysis described in methods section.",
"The information regarding the nodes rotating inertia (in $Kg\\cdot m^2$ ), was estimated by means of the methodology described in the methods section.",
"For each of the 314 existing connections between the network nodes, here called edges, the following information is given: The edge starting and ending nodes $i$ and $j$ , given in terms of nodes ids and codes.",
"The edge estimated length, obtained by calculating the geodetic distance between the edge terminal nodes.",
"The lines nominal powers $C_{ij}$ .",
"The edge reactance $X$ , obtained from the original UCTE dataset.",
"No resistance is given in this dataset, assuming negligible resistive processes on 380KV lines, but it can be estimated by assuming an average resistance per length and by multiplying this value with the given edge length.",
"Each of the 89 conventional generators present in the dataset is described by: id and code, same as the original UCTE dataset; id and code of the node at which the generator is attached; technology, obtained from the UCTE dataset; power output, obtained from the UCTE dataset; minimum and maximum nominal power, obtained from the UCTE dataset.",
"The published dataset includes 5 directories, one for each value of $P_\\%$ .",
"Each directory contains 4 files, one containing the nodes data, called ”nodes.csv”; one containing the edges data, called ”branches.csv”, one containing the generation data, called ”generators.csv” and the last one is the ”matpower.m” file, which contains all the data needed for running a power flow analysis.",
"It can be easily modified and loaded by matpower [28].",
"The presented data is obtained from two main databases: The German power grid data from [25], [12]; The data about the distribution of RES generators, published in [1]; The first dataset is part of a peer reviewed journal article[25], further enriched in [12], presented as a conference proceeding.",
"It is a common dataset used for power grid analysis, and contains the full representation of the UCTE (the old association of european TSOs, actually ENTSO-E [7]) network.",
"The second dataset is part of a project of a census of installed RES generation in Germany, supervised by the Bundesnetzagentur [9], the German federal agency for networks.",
"The datasets have been processed by means of fully maintained programs, which are standard leading choices of the community.",
"This has been done to ensure the maximum reliability of the performed analysis.",
"In particular: For saving and processing the geographical data, I used PostGIS, the GIS extension for PostgreSQL databases; Geographical analysis and visualization was performed with the suite QGIS [18]; The power flow analysis was performed with MATPOWER [28]; The data treatment, connection between different data process methods and I/O was done with python (Python Software Foundation, https://www.python.org/).",
"The described dataset can be used for studying the RES impact on transmission systems from various points of view.",
"In particular, it can be used for: Steady state analysis: The dataset contains all information needed for performing DC power flow analysis, DCOPF included.",
"This makes this dataset very valuable to monitor the impact of RES on markets and congestions.",
"The data contains the files ”grid.m”, which can be read directly from MATPOWER [28].",
"It allows various types of studies, mainly based on the solution of power flow equations.",
"Further information on how to read matpower files are given in the matpower manual [28].",
"Dynamic analysis: the dataset contains all information needed for performing dynamical analysis based on the swing equation, Kuramoto models, and their stochastic counterparts.",
"From this point of view, it can be a useful reference system for these analyses, useful for testing general results on a realistic network of large size.",
"The author gratefully acknowledge the support from the Federal Ministry of Education and Research (BMBF grant no.",
"03SF0472D NET-538-167)).",
"The author acknowledge the precious help and supervision of Prof. Meyer-Ortmanns.",
"Any opinion, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the funding parties.",
"The author declares no competing financial interests.",
"Mureddu, M. (2016).",
"Representation of the German transmission grid for Renewable Energy Sources impact analysis.figshare.",
"http://doi.org/10.6084/m9.figshare.4233782.v2"
]
] | 1612.05532 |
[
[
"Two-weight codes from trace codes over $R_k$"
],
[
"Abstract We construct a family of two-Lee-weight codes over the ring $R_k,$ which is defined as trace codes with algebraic structure of abelian codes.",
"The Lee weight distribution of the two-weight codes is given.",
"Taking the Gray map, we obtain optimal abelian binary two-weight codes by using the Griesmer bound.",
"An application to secret sharing schemes is also given."
],
[
"Introduction",
"For the reason that the access structures of the secret sharing schemes derived from few weights linear codes can be completely determined [5], [13], linear codes with few weights are of great interest in secret sharing schemes.",
"Two-Lee-weight codes over fields have been studied since 1970s due to their connections to strongly regular graphs, difference sets and finite geomertry [2], [3].",
"Recently, the notion of trace codes are extended from fields to rings and some trace codes over rings are introduced.",
"In [10], [11], two infinite families of two-Lee-weight codes over the rings $\\mathbb {F}_2+u\\mathbb {F}_2$ and $\\mathbb {F}_2+u\\mathbb {F}_2+v\\mathbb {F}_2+uv\\mathbb {F}_2$ are constructed.",
"On the basis of the study of the ring $R_k$ in [6], we shall construct a family of two-Lee-weight codes over $R_k$ , which is a generalization of the rings $\\mathbb {F}_2+u\\mathbb {F}_2$ and $\\mathbb {F}_2+u\\mathbb {F}_2+v\\mathbb {F}_2+uv\\mathbb {F}_2.$ In this paper, we construct a family of two-weight codes which are provably abelian but perhaps not cyclic.",
"These codes are constructed as trace codes.",
"We determine the weight distributions of these codes by exponential character sums.",
"Taking the Gray map, we obtain a family of binary abelian codes and they are shown to be optimal by the application of the Griesmer bound.",
"Moreover, an application to secret sharing schemes is also given.",
"The paper is organized as follows.",
"In the next section, we give some definitions and propositions about the ring $R_k$ , the Gray map and the Lee weight that we need in the following parts of the paper.",
"Section 3 shows some information of the codes over the ring $R_k.$ Section 4 shows that the trace codes are abelian.",
"Section 5 computes the weight distribution of our codes and proves the optimality of them by the application of the Griesmer bound.",
"Section 6 determines the minimum distance of the dual codes, proves that all these codes are minimal and then describes an application to secret sharing schemes.",
"In section 7, we make a summary of this paper and give some interesting problems."
],
[
"Rings",
"Consider the ring $R_k=\\mathbb {F}_2[u_1,u_2,...,u_k]/\\langle u_i^2=0,u_iu_j=u_ju_i\\rangle $ with $k \\ge 1.$ The ring can also be defined recursively as $R_k=R_{k-1}[u_k]/\\langle u_k^2=0,u_ku_j=u_ju_k\\rangle , j=1,2,...,k-1.$ In order to represent the elements of $R_k$ conveniently, we define $u_A:=\\prod \\limits _{i\\in A}u_i$ for any subset $A\\subseteq \\lbrace 1,2,...,k\\rbrace .$ What's more, $u_A=1$ when $A=\\emptyset $ by the convention.",
"Then any element of $R_k$ can be represented as $\\sum \\limits _{A\\subseteq \\lbrace 1,2,...,k\\rbrace }c_Au_A, c_A\\in \\mathbb {F}_2.$ The ring $R_k$ can be extended to $\\mathcal {R}=\\mathbb {F}_{2^m}[u_1,u_2,...,u_k]/\\langle u_i^2=0,u_iu_j=u_ju_i\\rangle $ with a given positive integer $m.$ The elements of $\\mathcal {R}$ are in the form of $\\sum \\limits _{A\\subseteq \\lbrace 1,2,...,k\\rbrace }c_Au_A, c_A\\in \\mathbb {F}_{2^m}.$ An element of $\\mathcal {R}$ is a unit if the coefficient of $c_\\emptyset \\ne 0$ and it is a maximal ideal if $c_\\emptyset =0.$ Let $\\mathcal {R}^*$ be a set that contains all the units of $\\mathcal {R}$ and $M$ denote maximal ideal.",
"Hence, $\\mathcal {R}=\\mathcal {R}^*\\bigcup M.$ There is a Frobenius operator $F$ which maps $\\sum \\limits _{A\\subseteq \\lbrace 1,2,...,k\\rbrace }c_Au_A$ to $\\sum \\limits _{A\\subseteq \\lbrace 1,2,...,k\\rbrace }c_A^2u_A.$ Let $Tr$ denote the Trace function, which is defined as $Tr=\\sum _{j=0}^{m-1}F^j.$ It is obvious that $Tr\\big (\\sum \\limits _{A\\subseteq \\lbrace 1,2,...,k\\rbrace }c_Au_A\\big )=\\sum \\limits _{A\\subseteq \\lbrace 1,2,...,k\\rbrace }u_Atr(c_A)$ and $tr()$ denotes the standard trace from $\\mathbb {F}_{2^m}$ to $\\mathbb {F}_2$ ."
],
[
"The Lee weight and the Gray map",
"The Lee weight was defined in [12] as the Hamming weight of the image of the codeword under the Gray map $\\phi _k$ and [6] has given a recursive definition of $\\phi _k.$ For $\\bar{c}\\in R_k^n,$ it can be represented as $\\bar{c}_1+u_k\\bar{c}_2$ with $\\bar{c}_1,\\bar{c}_2\\in R_{k-1}^n.$ Then, the Gray map is defined as follows: $\\phi _k(\\bar{c})=(\\phi _{k-1}(\\bar{c}_2),\\phi _{k-1}(\\bar{c}_1)+\\phi _{k-1}(\\bar{c}_2)).$ With the definition of the Lee weight, we obtain a distance preserving map from $R_k^n$ to $\\mathbb {F}_2^{2^kn}.$"
],
[
"Codes",
"A linear code over $R$ of length $n$ is a $R$ -submodule of $R^n.$ Assume that $\\sum _A c_Au_A$ and $\\sum _B d_Bu_B$ with $A,B\\subseteq \\lbrace 1,2,...,k\\rbrace $ are two elements of the ring $R$ , then their standard inner product $(\\sum _A c_Au_A)(\\sum _B d_Bu_B)=\\sum \\limits _{A,B\\subseteq \\lbrace 1,2,...,k\\rbrace , A\\cap B=\\emptyset }c_Ad_Bu_{A\\cup B}$ which is defined over $R.$ Denote the dual code of $C$ by $C^\\perp $ and define it as $C^\\perp =\\big \\lbrace \\sum _B d_Bu_B\\big |\\big (\\sum _A c_Au_A\\big )\\big (\\sum _B d_Bu_B\\big )=0, \\forall \\sum _A c_Au_A\\in C\\big \\rbrace .$"
],
[
"Symmetry",
"In this paper, we define $L=\\mathcal {R}^*$ , $n=|L|=(2^m-1)2^{m(2^k-1)}$ and $N=2^kn.$ For $a\\in \\mathcal {R},$ we define the vector $ev(a)$ by the evaluation map $ev(a)=(Tr(ax))_{x\\in L}.$ The code $C$ is defined by the formula $C=\\lbrace ev(a)|a\\in \\mathcal {R}\\rbrace .$ Thus, the length of the code $C$ is $n$ while the length of the code $\\phi _k (C)$ is $N.$ Proposition 4.1 The group $L$ acts regularly on the coordinates of $C.$ For any $u,v\\in L,$ the change of variables $x\\mapsto (u/v)x$ maps $v$ to $u.$ With given $u,v,$ a permutation with this property is unique.",
"The code $C$ is thus an abelian code based on the group $L.$ In other words, it is an ideal of the group ring $R[\\mathcal {R}^*]$ .",
"$L$ is not a cyclic group, thus $C$ is not likely to be cyclic."
],
[
"Weight distribution",
"Before calculating the Lee weight of the codewords of the code $C,$ we recall some classic lemmas first.",
"Lemma 5.1 ([9], $(6)$ , p. 412) If $y=(y_1,y_2,...,y_n)\\in \\mathbb {F}_2^n,$ then $2w_H(y)=n-\\sum \\limits _{i=1}^n(-1)^{y_i}.$ Lemma 5.2 ([9], Lemma 9, p. 143) For any $z\\in \\mathbb {F}_{2^m}^*,$ $\\sum \\limits _{x\\in \\mathbb {F}_{2^m}}(-1)^{tr(zx)}=0$ is set up.",
"Apparently, $\\sum \\limits _{x\\in \\mathbb {F}_{2^m}^*}(-1)^{tr(zx)}=-1.$ Theorem 5.3 For $a\\in \\mathcal {R},$ the Lee weight of the codewords of $C$ is as follows: If $a=0,$ then $w_L(ev(a))=0;$ If $a\\in M\\setminus \\lbrace 0\\rbrace $ , then if $a=c_{\\lbrace 1,...,k\\rbrace }u_{\\lbrace 1,...,k\\rbrace }$ and $c_{\\lbrace 1,...,k\\rbrace }\\in \\mathbb {F}_{2^m}^*$ , then $w_L(ev(a))=2^{2^km+k-1},$ if $a\\in M\\setminus \\lbrace 0,c_{\\lbrace 1,...,k\\rbrace }u_{\\lbrace 1,...,k\\rbrace }\\rbrace ,$ then $w_L(ev(a))=2^{k-1}(2^m-1)2^{m(2^k-1)};$ If $a\\in \\mathcal {R}^*$ , then $w_L(ev(a))=2^{k-1}(2^m-1)2^{m(2^k-1)}.$ Divide the value of $a$ into three categories to discuss the Lee weight of the codewords.",
"When $a=0,$ it is obvious that $w_L(ev(a))=0.$ If $a\\in M\\setminus \\lbrace 0\\rbrace $ and $a=c_{\\lbrace 1,...,k\\rbrace }u_{\\lbrace 1,...,k\\rbrace }$ with $c_{\\lbrace 1,...,k\\rbrace }\\in \\mathbb {F}_{2^m}^*$ , let $x=\\sum \\limits _{B\\subseteq \\lbrace 1,...,k\\rbrace } d_Bu_B\\in L.$ On the basis of the definition of the inner product, we compute $ax$ and here are the process and result: $\\begin{aligned}ax&=\\sum \\limits _{B\\subseteq \\lbrace 1,...,k\\rbrace } c_{\\lbrace 1,...,k\\rbrace }u_{\\lbrace 1,...,k\\rbrace }d_Bu_B\\\\&=c_{\\lbrace 1,...,k\\rbrace }d_\\emptyset u_{\\lbrace 1,...k\\rbrace }, \\ {\\rm where} \\ d_\\emptyset \\in \\mathbb {F}_{2^m}^*.\\end{aligned}$ Then $Tr(ax)=tr(c_{\\lbrace 1,...,k\\rbrace }d_\\emptyset )u_{\\lbrace 1,...,k\\rbrace }.$ Taking the Gray map, we get $\\phi _k(ev(a))=(\\underbrace{tr(c_{\\lbrace 1,...,k\\rbrace }d_\\emptyset ),...,tr(c_{\\lbrace 1,...,k\\rbrace }d_\\emptyset )}_{2^k}).$ According to Lemma 5.1, we have $2^k|L|-2w_L(ev(a))=2^k(\\sum \\limits _{d_\\emptyset \\in \\mathbb {F}_{2^m}^*}\\sum \\limits _{d_B\\in \\mathbb {F}_{2^m}}(-1)^{tr(c_{\\lbrace 1,...,k\\rbrace }d_\\emptyset )}),$ where $B\\ne \\emptyset \\subseteq \\lbrace 1,...,k\\rbrace $ and the right side of the formula can be written as $-2^k2^{m(2^k-1)}.$ Then, $w_L(ev(a))=2^{2^km+k-1}.$ In addition, if $a\\in M\\setminus \\lbrace 0,c_{\\lbrace 1,...,k\\rbrace }u_{\\lbrace 1,...,k\\rbrace }\\rbrace ,$ suppose $a=c_{\\lbrace 1\\rbrace }u_{\\lbrace 1\\rbrace }, c_{\\lbrace 1\\rbrace }\\in \\mathbb {F}_{2^m}^*,$ and $x=\\sum \\limits _{B\\subseteq \\lbrace 1,...,k\\rbrace }d_Bu_B\\in L.$ Next, determine the value of $ax$ in the same way as the previous case.",
"So $ax=\\sum \\limits _{B\\subseteq \\lbrace 2,...,k\\rbrace }c_{\\lbrace 1\\rbrace }d_Bu_{\\lbrace 1\\rbrace \\cup B}$ and $Tr(ax)=\\sum \\limits _{B\\subseteq \\lbrace 2,...,k\\rbrace }tr(c_{\\lbrace 1\\rbrace }d_B)u_{\\lbrace 1\\rbrace \\cup B}.$ Taking the Gray map, every element of the vector $\\phi _k(ev(a))$ contains $tr(c_{\\lbrace 1\\rbrace }d_{\\lbrace 2,...,k\\rbrace })$ which implies that $2^k|L|-2w_L(ev(a))=0$ since $\\sum \\limits _{d_{\\lbrace 2,...,k\\rbrace }\\in \\mathbb {F}_{2^m}}(-1)^{tr(c_{\\lbrace 1\\rbrace }d_{\\lbrace 2,...,k\\rbrace })}=0.$ For other $a\\in M\\setminus \\lbrace 0,u_{\\lbrace 1\\rbrace },u_{\\lbrace 1,...,k\\rbrace }\\rbrace ,$ the Lee weight equals $2^{k-1}|L|$ as well.",
"If $a\\in \\mathcal {R}^*$ , let $a=\\sum \\limits _{A\\subseteq \\lbrace 1,2,...,k\\rbrace }c_Au_A$ , where $c_{A\\setminus \\lbrace 0\\rbrace }\\in \\mathbb {F}_{2^m}$ and $c_\\emptyset \\in \\mathbb {F}_{2^m}^*.$ Moreover, $x=\\sum \\limits _{B\\subseteq \\lbrace 1,2,...,k\\rbrace }d_Bu_B$ , where $d_{B\\setminus \\lbrace 0\\rbrace }\\in \\mathbb {F}_{2^m}$ and $d_\\emptyset \\in \\mathbb {F}_{2^m}^*.$ Computing $ax$ and $Tr(ax)$ , we discover that every element of the vector $\\phi _k(ev(a))$ contains $tr(c_{\\emptyset }d_{\\lbrace 1,...,k\\rbrace })$ which implies $2^k|L|-2w_L(ev(a))=0$ since $\\sum \\limits _{d_{\\lbrace 1,...,k\\rbrace }\\in \\mathbb {F}_{2^m}}(-1)^{tr(c_{\\emptyset }d_{\\lbrace 1,...,k\\rbrace })}=0.$ Hence, $w_L(ev(a))=2^{k-1}|L|.$ Thus, we have constructed a family of two-weight codes of length $2^k(2^m-1)2^{m(2^k-1)}$ and dimension $2^km.$ Their weight distributions and respective frequencies are shown in Table I below.",
"$\\mathrm {Table~ I.",
"}~~~\\mathrm {weight~ distribution~ of}~ \\phi _k(C) $ Table: NO_CAPTIONNext, we study the optimality of $\\phi _k(C)$ by the application of the Griesmer bound.",
"Recall the Griesmer bound applied to an $[n,K,d]$ binary code first.",
"For an $[n,K,d]$ code, we have $\\sum _{i=0}^{K-1}\\lceil \\frac{d}{2^i}\\rceil \\le n.$ If the code with parameters $[n,K,d+1]$ does not exist, then the $[n,K,d]$ code is optimal.",
"Theorem 5.4 For arbitrary integers $m\\ge 2$ and $k\\ge 1,$ the code $\\phi _k (C)$ is optimal.",
"The parameters of $\\phi _k (C)$ are $[N,K,d]=[2^k(2^m-1)2^{m(2^k-1)},2^km,2^{k-1}(2^m-1)2^{m(2^k-1)}].$ We claim that $\\sum _{j=0}^{K-1}\\lceil \\frac{d+1}{2^j}\\rceil >N.$ There are two different values of $\\sum _{j=0}^{k-1}\\lceil \\frac{d+1}{2^j}\\rceil $ depending on the value of $j$ and we discuss these two cases next.",
"$0\\le j\\le m(2^k-1)+k-1 \\Rightarrow \\lceil \\frac{d+1}{2^j} \\rceil =2^{m(2^k-1)+k-1-j}(2^m-1)+1$ ; $m(2^k-1)+k-1< j\\le 2^km-1 \\Rightarrow \\lceil \\frac{d+1}{2^j} \\rceil =2^{m2^k+k-1-j}$ .",
"Then, $\\begin{aligned}\\sum _{j=0}^{K-1}\\big \\lceil \\frac{d+1}{2^j}\\big \\rceil -N &=\\sum _{j=0}^{m(2^k-1)+k-1}\\big \\lceil \\frac{d+1}{2^j}\\big \\rceil +\\sum _{j=m(2^k-1)+k}^{m2^k-1}\\big \\lceil \\frac{d+1}{2^j}\\big \\rceil -N\\\\&=(2^k-1)(m-1)+k\\end{aligned}$ In order to prove $\\sum _{j=0}^{K-1}\\lceil \\frac{d+1}{2^j}\\rceil -N>0,$ we can prove $(2^k-1)(m-1)+k>0$ equivalently.",
"Discussing the range of $m$ and $k,$ we obtain that the inequation is true when $k\\ge 1$ and $m\\ge 2.$"
],
[
"Application to secret sharing scheme",
"The access structure of secret sharing scheme constructed from linear code is hard to determine.",
"However, if all the codewords of the linear code are minimal, we can use its dual code to construct an interesting secret sharing scheme.",
"So next, we check that whether the codewords of $\\phi _k(C)$ are minimal or not.",
"The basic property minimal vector of a given binary linear code is described by the following lemma [1]."
],
[
"Minimal codewords",
"Lemma 6.1 (Ashikhmin-Barg) Denote by $w_0$ and $w_{\\infty }$ the smallest and largest nonzero weights of a binary code $C$ , respectively.",
"If $\\frac{w_0}{w_{\\infty }}>\\frac{1}{2},$ then every nonzero codeword of $C$ is minimal.",
"Theorem 6.2 All the nonzero codewords of $\\phi _k(C),$ for $m\\ge 2$ are minimal.",
"By the preceding lemma with $w_0=w_1,$ and $w_{\\infty }=w_2.$ Rewriting the inequality of Lemma 6.1 as $2w_1-w_2>0,$ we end up with the condition $2^{k-1}2^{m(2^k-1)}(2^m-2)>0,$ which is satisfied for $m\\ge 2$ .",
"Hence, the theorem is proved."
],
[
"The dual code",
"Lemma 6.3 For all $a\\in \\mathcal {R},$ if $Tr(ax)=0,$ then $x=0.$ Assume that $a=\\sum _Ac_Au_A$ and $x=\\sum _Bd_Bu_B$ where $A, B\\subseteq \\lbrace 1,...,k\\rbrace $ and $c_A, d_B\\in \\mathbb {F}_{2^m}.$ Thus $ax=\\sum \\limits _{A,B\\subseteq \\lbrace 1,2,...,k\\rbrace , A\\cap B=\\emptyset }c_Ad_Bu_{A\\cup B}.$ Then $Tr(ax)=0$ means $tr(c_Ad_B)=0$ where $A, B\\subseteq \\lbrace 1,...,k\\rbrace $ and $A\\cap B=\\emptyset .$ Using the nondegenerate character of $tr()$ [9], we obtain that $d_B=0$ for all $B\\subseteq \\lbrace 1,...,k\\rbrace $ which is equivalent to $x=0.$ Next, we determine the dual Lee distance of the two-Lee-weight code $C.$ Theorem 6.4 For all positive integers $k$ and $m\\ge 2,$ the dual Lee distance $d^{\\prime }$ of $C$ is 2.",
"Here we use the proof by contradiction.",
"Assume that $d^{\\prime }\\ge 3$ to prove that $d^{\\prime }<3.$ With the sphere-packing bound, we have $2^{2^km}\\ge 1+N.$ Rewriting the inequation, we have $2^k<2^{k-m}+1$ which is impossible for any positive integers $k$ or $m\\ge 2.$ Hence, $d^{\\prime }<3.$ Then, we prove that $d^{\\prime }=2.$ If not, $C^\\perp $ has at least one codeword with Lee weight one and assume that it has value $\\gamma $ at some $x\\in \\mathcal {R}^*.$ Since $\\mathcal {R}^*$ is the set that contains all the units of $\\mathcal {R}$ , for all $a\\in \\mathcal {R},$ $\\gamma Tr(ax)=0$ implies $Tr(ax\\gamma )=0.$ By Lemma 6.3, we know that $x=0$ , which is contradict with $x\\in \\mathcal {R}^*.$ In conclusion, the dual Lee distance $d^{\\prime }$ is equal to $2.$"
],
[
"Massey's scheme",
"Massey's scheme [13], is a secret sharing scheme which based on Coding theory.",
"When all nonzero codewords are minimal, it was shown in [4] that there is the following alternative, depending on $d^{\\prime }.$ If $d^{\\prime }\\ge 3,$ then the SSS is “democratic”: every user belongs to the same number of coalitions.",
"If $d^{\\prime }=2,$ then there are users who belong to every coalition: the “dictators”.",
"By Theorems 6.2 and 6.4, we see that the Secret Sharing Scheme built on $\\phi _k(C)$ is dictatorial."
],
[
"Conclusion",
"In the present paper, we have studied an infinite family of trace codes.",
"Because the localizing set $L$ has the structure of an abelian multiplicative group, the trace code is an abelian code, that is an ideal in the group ring of $L.$ It is not clear whether the code is cyclic or not.",
"In fact, we also study the trace code over the localizing set $\\lbrace \\sum _Ca_Cu_C:C\\subseteq \\lbrace 1,2,...,k\\rbrace ,a_{C\\setminus \\lbrace \\emptyset \\rbrace }\\in \\mathbb {F}_{2^m},a_\\emptyset \\in \\mathbf {D}\\rbrace $ [7], [8] and here $D$ is a subset of $\\mathbb {F}_{2^m}^*.$ Two families of codes are obtained.",
"One family of codes is the same as that constructed with the localizing set $L$ .",
"Another family of codes is a three-weight codes and it can be used to construct secret sharing schemes.",
"More importantly, it is worthwhile to study other trace codes over the extensions of $R_k.$"
]
] | 1612.05523 |
[
[
"Noise representation in residuals of LSQR, LSMR, and CRAIG\n regularization"
],
[
"Abstract Golub-Kahan iterative bidiagonalization represents the core algorithm in several regularization methods for solving large linear noise-polluted ill-posed problems.",
"We consider a general noise setting and derive explicit relations between (noise contaminated) bidiagonalization vectors and the residuals of bidiagonalization-based regularization methods LSQR, LSMR, and CRAIG.",
"For LSQR and LSMR residuals we prove that the coefficients of the linear combination of the computed bidiagonalization vectors reflect the amount of propagated noise in each of these vectors.",
"For CRAIG the residual is only a multiple of a particular bidiagonalization vector.",
"We show how its size indicates the regularization effect in each iteration by expressing the CRAIG solution as the exact solution to a modified compatible problem.",
"Validity of the results for larger two-dimensional problems and influence of the loss of orthogonality is also discussed."
],
[
"Introduction",
"In this paper we consider ill-posed linear algebraic problems of the form $b =Ax + \\eta ,\\qquad A\\in \\mathbb {R}^{m\\times n},\\qquad b\\in \\mathbb {R}^{m},\\qquad {\\Vert \\eta \\Vert \\ll \\Vert Ax\\Vert ,}$ where the matrix $A$ represents a discretized smoothing operator with the singular values decaying gradually to zero without a noticeable gap.",
"We assume that multiplication of a vector $v$ by $A$ or $A^{T}$ results in smoothing which reduces the relative size of the high-frequency components of $v$ .",
"The operator $A$ and the vector $b$ are supposed to be known.",
"The vector $\\eta $ represents errors, such as noise, that affect the exact data.",
"Problems of this kind are commonly referred to as linear discrete ill-posed problems or linear inverse problems and arise in a variety of applications , .",
"Since $A$ is ill-conditioned, the presence of noise makes the naive solution $x^{\\text{naive}}\\equiv A^{\\dagger }b,$ where $A^\\dagger $ denotes the Moore-Penrose pseudoinverse, meaningless.",
"Therefore, to find an acceptable numerical approximation to $x$ , it is necessary to use regularization methods.",
"Various techniques to regularize the linear inverse problem (REF ) have been developed.",
"For large-scale problems, iterative regularization is a good alternative to direct regularization methods.",
"When an iterative method is used, regularization is achieved by early termination of the process, before noise $\\eta $ starts to dominate the approximate solution .",
"Many iterative regularization methods such as LSQR , , , , CRAIG , , LSMR , and CRAIG-MR/MRNE , involve the Golub-Kahan iterative bidiagonalization .",
"Combination with an additional inner regularization (typically with a spectral filtering method) gives so-called hybrid regularization; see, for example, , , , .",
"Various approaches for choosing the stopping criterion, playing here the role of the regularization parameter, are based on comparing the properties of the actual residual to an a priori known property of noise, such as the noise level in the Morozov's discrepancy principle , or the noise distribution in the cumulative residual periodogram method , , .",
"Thus understanding how noise translates to the residuals during the iterative process is of great interest.",
"The aim of this paper is, using the analysis of the propagation of noise in the left bidiagonalization vectors provided in , to study the relation between residuals of bidiagonalization-based methods and the noise vector $\\eta $ .",
"Whereas in , white noise was assumed, here we have no particular assumptions on the distribution of noise.",
"We only assume the amount of noise is large enough to make the noise propagation visible through the smoothing by $A$ in construction of the bidiagonalization vectors.",
"This is often the case in ill-posed problems, as we illustrate on one-dimensional (1D) as well as significantly noise contaminated two-dimensional (2D) benchmarks.",
"We prove that LSQR and LSMR residuals are given by a linear combination of the bidiagonalization vectors with the coefficients related to the amount of propagated noise in the corresponding vector.",
"For CRAIG, the residual is only a multiple of a particular bidiagonalization vector.",
"This allows us to prove that an approximate solution obtained in a given iteration by CRAIG applied to (REF ) coincides with an exact solution of the (compatible) modified problem $Ax = b-\\tilde{\\eta },$ where $\\tilde{\\eta }$ is a noise vector estimate constructed from the currently computed bidiagonalization vectors.",
"These results contribute to understanding of regularization properties of the considered methods and should be considered when devising reliable stopping criteria.",
"Note that since LSQR is mathematically equivalent to CGLS and CGNR, CRAIG is mathematically equivalent to CGNE and CGME , and LSMR is mathematically equivalent to CRLS , then in exact arithmetic, the analysis applies also to these methods.",
"The paper is organized as follows.",
"In Section , after a recollection of the previous results, we study the propagation of various types of noise and the influence of the loss of orthogonality on this phenomenon.",
"Section investigates the residuals of selected methods with respect to the noise contamination in the left bidiagonalization vectors and compares their properties.",
"Section discusses validity of obtained results for larger 2D problems.",
"Section concludes the paper.",
"Unless specified otherwise, we assume exact arithmetic and the presented experiments are performed with full double reorthogonalization in the bidiagonalization process.",
"Throughout the paper, $\\Vert v\\Vert $ denotes the standard Euclidean norm of the vector $v$ , vector $e_k$ denotes the $k$ -th column of the identity matrix.",
"By $\\mathcal {P}_k$ , we denote the set of polynomials of degree less or equal to $k$ .",
"The noise level is denoted by $\\delta _\\text{noise}\\equiv \\Vert \\eta \\Vert /\\Vert Ax\\Vert $ .",
"By Poisson noise, we understand $b_i\\sim \\text{Pois}([Ax]_i)$ , i.e., the right-hand side $b$ is a Poisson random with the Poisson parameter $Ax$ .",
"The test problems were adopted from the Regularization tools .",
"For simplicity of exposition, we assume the initial approximation $x_0\\equiv 0$ throughout the paper.",
"Generalization to $x_0\\ne 0$ is straightforward."
],
[
"Basic relations",
"Given the initial vectors $w_{0}\\equiv 0$ , $s_{1}\\equiv b/\\beta _{1}$ , where $\\beta _{1}\\equiv \\Vert b\\Vert $ , the Golub-Kahan iterative bidiagonalization computes, for $k = 1,2,\\ldots $ , $\\alpha _{k}w_{k} & =A^{T}s_{k}-\\beta _{k}w_{k-1}\\,, & \\Vert w_{k}\\Vert & =1,\\\\\\beta _{k+1}s_{k+1} & =Aw_{k}-\\alpha _{k}s_{k}\\,, & \\Vert s_{k+1}\\Vert & =1,$ until $\\alpha _{k}=0$ or $\\beta _{k+1}=0$ , or until $k=\\min (m,n)$ .",
"Vectors $s_1,\\ldots ,s_k$ , and $w_1,\\ldots ,w_k$ , form orthonormal bases of the Krylov subspaces $\\mathcal {K}_k(AA^{T},b)$ and $\\mathcal {K}_k(A^{T}A,A^{T}b)$ , respectively.",
"In the rest of the paper, we assume that the bidiagonalization process does not terminate before the iteration $k + 1$ , i.e., $\\alpha _l,\\beta _{l+1} >0$ , $l = 1,\\ldots k.$ Denoting $S_k\\equiv [s_{1},\\ldots ,s_k]\\in \\mathbb {R}^{m\\times k}, \\, W_k\\equiv [w_{1},\\ldots ,w_k]\\in \\mathbb {\\mathbb {R}}^{n\\times k}$ and $L_k\\equiv \\left[\\begin{array}{cccc}\\alpha _1\\\\\\beta _2&\\alpha _2\\\\&\\ddots &\\ddots \\\\&&\\beta _k&\\alpha _k\\end{array}\\right]\\in \\mathbb {R}^{k\\times k},\\qquad L_{k+}\\equiv \\left[\\begin{array}{c}L_k\\\\e_k^T\\beta _{k+1}\\end{array}\\right]\\in \\mathbb {R}^{(k+1)\\times k},$ we can write the matrix version of the bidiagonalization as $A^TS_k = W_kL_k^T,\\qquad AW_k = S_{k+1}L_{k+}.",
"$ The two corresponding Lanczos three-term recurrences $(AA^T)S_k = S_{k+1}(L_{k+}L_k^T),\\qquad (A^TA)W_k = W_{k+1}(L_{k+1}^TL_{k+}),$ allow us to describe the bidiagonalization vectors $s_{k+1}$ and $w_{k+1}$ in terms of the Lanczos polynomials as $s_{k+1} = \\varphi _k(AA^T)b, \\qquad w_{k+1} = \\psi _k(A^TA)A^Tb \\qquad \\varphi _k,\\psi _k \\in \\mathcal {P}_k; $ see , , , , .",
"From (REF ) we have that $s_{k+1} = \\varphi _k(AA^T)b = \\varphi _k(AA^T)(Ax+\\eta ), $ giving $s_{k+1} = \\left[\\varphi _k(AA^T)Ax + (\\varphi _k(AA^T) - \\varphi _k(0))\\eta \\right] + \\varphi _k(0)\\eta .",
"$ The first component on the right-hand side of (REF ) can be rewritten as $s_{k+1}^\\text{LF} \\equiv \\left[\\varphi _k(AA^T)Ax + (\\varphi _k(AA^T) - \\varphi _k(0)) \\eta \\right]= A q_{k-1}(AA^T)\\left[x+A^T \\eta \\right] ,$ for some $q_{k-1} \\in \\mathcal {P}_k$ .",
"Since $A$ has the smoothing property, then $s_{k+1}^\\text{LF}$ is smooth for $k\\ll \\min (m,n)$ .",
"Thus $s_{k+1}$ is a sum of a low-frequency vector and the scaled noise vector $\\eta $ , $s_{k+1} = s_{k+1}^\\text{LF} + \\varphi _k(0)\\eta .",
"$ Note that this splitting corresponds to the low-frequency part and propagated (non-smoothed) noise part only when $\\Vert s_{k+1}^\\text{LF}\\Vert ^{2}+\\Vert \\varphi _k(0)\\eta \\Vert ^{2}\\approx 1$ .",
"For large $k$ s, there is a considerable cancellation between $s_{k+1}^\\text{LF}$ and $\\varphi _k(0)\\eta $ , the splitting (REF ) still holds but it does not correspond to our intuition of an underlying smooth vector and some added scaled noise.",
"Thus we restrict ourselves to smaller values of $k$ .",
"It has been shown in that whereas for $s_1$ (the scaled right-hand side) the noise part in (REF ) is small compared to the true data, for larger $k$ , due to the smoothing property of the matrix $A$ and the orthogonality between the vectors $s_k$ , the noise part becomes more significant.",
"The noise scaling factor determining the relative amplification of the non-smoothed part of noise corresponds to the constant term of the Lanczos polynomial $\\varphi _k(0) = (-1)^{k}\\frac{1}{\\beta _{k+1}}\\prod _{j=1}^{k}\\frac{\\alpha _j}{\\beta _j}$ called the amplification factor.Note that in a different notation was used.",
"The Lanczos polynomial $\\varphi _k$ was scaled by $\\Vert b\\Vert $ so that $s_{k+1} = \\tilde{\\varphi }_k(AA^T)s_{1}$ .",
"The vector $s_{k+1}$ was split into $s_{k+1} = s_{k+1}^\\text{exact}+s_{k+1}^\\text{noise}$ .",
"In our notation, $s_{k+1}^\\text{exact}=s_{k+1}^\\text{LF}$ , and $s_{k+1}^\\text{noise}=\\varphi _k(0)\\eta $ .",
"Its behavior for problems with white noise was studied in and the analysis concludes that its size increases with $k$ until the noise revealing iteration $k_{\\text{rev}}$ , where the vector $s_{k+1}$ is dominated by the non-smoothed part of noise.",
"Then the amplification factor increases at least for one iteration.",
"The noise revealing iteration $k_{\\text{rev}}$ can be defined as Note that there is no analogy for the right bidiagonalization vectors, since all vectors $w_k$ are smoothed and the factor $\\psi _k(0)$ on average grows till late iterations.",
"A recursive relation for $\\psi _k(0)$ , obtained directly from (REF ) has the form $\\psi _0(0) &= \\frac{1}{\\alpha _1\\beta _1},\\\\\\psi _k(0) &= \\frac{1}{\\alpha _{k+1}}(\\varphi _k(0) - \\beta _{k+1}\\psi _{k-1}(0)), \\quad k = 1,2,\\ldots .",
"$"
],
[
"Influence of the noise frequency characteristics",
"The phenomenon of noise amplification is demonstrated on the problems from , .",
"Figures REF and REF show the absolute terms of the Lanczos polynomials $\\varphi _k$ and $\\psi _k$ for the problem shaw polluted with white noise of various noise levels.",
"For example, for the noise level $10^{-3}$ , the maximum of $\\varphi _k(0)$ is achieved for $k = 6$ , which corresponds to the observation that the vector $s_{7}$ in Figure REF is the most dominated by propagated noise.",
"Obviously, the noise revealing iteration increases with decreasing noise level.",
"The amplification factors exhibit similar behavior before the first decrease.",
"However, the behavior of $\\varphi _k(0)$ can be more complicated.",
"In Figure REF for phillips, the sizes of the amplification factors oscillate as a consequence of the oscillations in the sizes of the spectral components of $b$ in the left singular subspaces of $A$ .",
"Thus there is a partial reduction of the noise component, which influences the subsequent iterations, even before the noise revealing iteration.",
"Figure: The problem shaw(400) polluted by white noise: (a) the left bidiagonalization vectors s k+1 s_{k+1} for the noise level 10 -3 10^{-3};(b) the size of the absolute term of the Lanczos polynomial ϕ k \\varphi _k for various noise levels;(c) the size of the absolute term of the Lanczos polynomial ψ k \\psi _k for various noise levels.Figure: Influence of the amount of noise and its frequency characteristics on the amplification factor (): (a) the problem phillips with various noise levels of white noise; (b) the problem shaw with noise of different frequency characteristics; (c) the problem gravity with Poisson noise with different noise levels achieved by scaling.Even though assumed white noise, noise amplification can be observed also for other noise settings and the formulas (REF )-(REF ) still hold.",
"However, for high-frequency noise, there is smaller cancellation between the low-frequency component $s_{k+1}^\\text{LF}$ and the noise part $\\varphi _k(0)\\eta $ in (REF ).",
"Therefore, in the orthogonalization steps succeeding the noise revealing iteration $k_\\text{rev}$ , the noise part is projected out more significantly.",
"For low-frequency noise, on the other side, this smoothing is less significant, which results in smaller drop of (REF ) after $k_\\text{rev}$ .",
"This is illustrated in Figure REF on the problem shaw polluted by red (low-frequency), white, and violet (high-frequency) noise of the same noise level.",
"For spectral characteristics of these types of noise see Figure REF .",
"Figure: Power spectral densities (or simply power spectra) for red (low-frequency dominated),white (or Gaussian), and violet (high-frequency dominated) noise η\\eta , ∥η∥=1\\Vert \\eta \\Vert =1.",
"Power spectrum is given by squaredmagnitudes of Fourier coefficients F(η)F(\\eta ) of η\\eta (see, e.g., chap.",
"2.7),here computed by the discrete Fourier transform.",
"Power spectra are normalized by the lengthof the vector.Figure REF shows the amplification factor for various levels of Poisson noise."
],
[
"Influence of the loss of orthogonality",
"First note that the splitting (REF ) remains valid even if $\\varphi _k$ are not exactly orthonormal Lanczos polynomials, since the propagated noise can be still tracked using the absolute term of the corresponding (computed) polynomial.",
"Nevertheless, it is clear that the loss of orthogonality among the left bidiagonalization vectors in finite precision arithmetic influences the behavior of the amplification factor $\\varphi _k$ , i.e.",
"the propagation of noise.",
"In the following, we denote all quantities computed without reorthogonalization by hat.",
"Loss of orthogonality can be detected, e.g., by tracking the size of the smallest singular value $\\bar{\\sigma }_\\text{min}$ of the matrix $\\hat{S}_k$ of the computed left bidiagonalization vectors.",
"In Figure REF (left) for the problem shaw and gravity we see that when $\\bar{\\sigma }_\\text{min}$ drops below one detecting the loss of orthogonality among its columns, the size of the amplification factor $\\hat{\\varphi }_k(0)$ starts to oscillate.",
"However, except of the delay, the larger values of $|\\hat{\\varphi }_k(0)|$ still match those of $|{\\varphi }_k(0)|$ .",
"If we plot $|\\hat{\\varphi }_k(0)|$ against the rank of $\\hat{S}_k$ instead of $k$ , the sizes of the two amplification factors become very similar.",
"In our experiments, the rank of $\\hat{S}_k$ was computed as rank(S(:,1:k),1e-1) in MATLAB, i.e., singular values of $\\hat{S}_k$ at least ten times smaller than they would be for orthonormal columns were considered zero.",
"A similar shifting strategy was proposed in for the convergence curves of the conjugate gradient method.",
"Note that the choice of the tolerance can be problem dependent.",
"Further study of this phenomenon is beyond the scope of this paper, but we can conclude that except of the delay the noise revealing phenomenon is in finite precision computations present.",
"Figure: Illustration of the noise amplification for the problem shaw and gravity in finite precision computations.",
"Left: The sizes of the amplification factor () computed with full double reorthogonalization (ϕ k (0)\\varphi _k(0)) and without reorthogonalization (ϕ ^ k (0)\\hat{\\varphi }_k(0)).",
"Right: |ϕ ^ k (0)||\\hat{\\varphi }_k(0)| plotted against rank(S ^ k )\\text{rank}(\\hat{S}_k) computed as rank(S(:,1:k),1e-1) in MATLAB, together with |ϕ k (0)||\\varphi _k(0)| plotted against rank(S k )=k\\text{rank}({S}_k) = k."
],
[
"Noise in the residuals of iterative methods",
"CRAIG , LSQR , and LSMR represent three methods based on the Golub-Kahan iterative bidiagonalization.",
"At the $k$ -th step, they search for the approximation of the solution in the subspace generated by vectors $w_1,\\ldots ,w_k$ , i.e., $x_k = W_ky_k, \\qquad y_k \\in \\mathbb {R}^k.",
"$ The corresponding residual has the form $r_k \\equiv b-Ax_k = b - AW_ky_k &= S_{k+1}(\\beta _1e_1 - L_{k+}y_k).",
"$ CRAIG minimizes the distance of $x_k$ from the naive solution yielding $L_ky_k^\\text{CRAIG} = \\beta _1e_1.$ LSQR minimizes the norm of the residual $r_k$ yielding $y_k^\\text{LSQR} = \\underset{y\\in \\mathbb {R}^k}{\\operatorname{argmin}}\\Vert \\beta _1e_1 - L_{k+}y\\Vert .",
"$ LSMR minimizes the norm of $A^Tr_k$ giving $y_k^\\text{LSMR} = \\underset{y\\in \\mathbb {R}^k}{\\operatorname{argmin}}\\Vert \\beta _1\\alpha _1e_1 - L_{k+1}^TL_{k+}y\\Vert .",
"$ These methods are mathematically equivalent to Krylov subspace methods based on the Lanczos tridiagonalization (particularly Lanczos for linear systems and MINRES) applied to particular normal equations.",
"The relations useful in the following derivations are summarized in Table REF .",
"Table: Interpretation of bidiagonalization-based methods (CRAIG, LSQR, LSMR) as tridiagonalization-based methods(Lanczos for linear systems, MINRES) applied to the corresponding normal equations.In last two columns, the solution xx of the bidiagonalization-based methods is obtained from their tridiagonalization counterparts as x=A T yx = A^Ty and x=A T Azx = A^TAz,respectively.",
"See also .Since Lanczos method is a Galerkin (residual orthogonalization) method, we immediately see that $r_k^\\text{CRAIG} &= (-1)^k\\Vert r_k^\\text{CRAIG}\\Vert s_{k+1},\\\\ A^Tr_k^\\text{LSQR} &= (-1)^k\\Vert A^Tr_k^\\text{LSQR}\\Vert w_{k+1}.$ Using the relation between the Galerkin an the residual minimization method, see, we obtain, $\\Vert r_k^\\text{LSQR}\\Vert & = \\frac{1}{\\sqrt{\\sum _{l=0}^k{1/\\Vert r_l^\\text{CRAIG}\\Vert ^2}}},\\\\\\Vert A^Tr_k^\\text{LSMR}\\Vert & = \\frac{1}{\\sqrt{\\sum _{l=0}^k{1/\\Vert A^Tr_l^\\text{LSQR}\\Vert ^2}}}.",
"$ Note that these equations hold, up to a small perturbation, also in finite precision computations.",
"See for more details.",
"In the rest of this section, we investigate the residuals of each particular method.",
"We focus on in which sense the residuals approximate the noise vector.",
"We discuss particularly the case when noise contaminates the bidiagonalization vectors fast and thus the noise revealing iteration is well defined.",
"More general discussion follows in Section ."
],
[
"CRAIG residuals",
"The following result relates approximate solution obtained by CRAIG for (REF ) to the solution of the problem with the same matrix and a modified right-hand side.",
"Proposition 1 Consider the first $k$ steps of the Golub-Kahan iterative bidiagonalization.",
"Then the approximation $x_{k}^\\text{CRAIG}$ defined in (REF ) and (REF ), is an exact solution to a consistent problem $Ax = b - \\varphi _k(0)^{-1}s_{k+1}.",
"$ Consequently, $\\Vert r_k^\\text{CRAIG}\\Vert = |\\varphi _k(0)|^{-1}.$ First note that we only need to show that $r_k^\\text{CRAIG} = \\varphi _k(0)^{-1}s_{k+1}$ , $k=1,2,\\ldots $ .",
"From (REF ) and (REF ) it follows that there exist $c\\in \\mathbb {R}$ , such that $r_k^\\text{CRAIG} = c \\cdot s_{k+1} = c \\cdot \\varphi _k(AA^T)b.$ Let us now determine the constant $c$ .",
"From (REF ) and (REF ), we have that $r_k^\\text{CRAIG} = \\Pi _k (AA^T)b, \\quad \\text{where} \\quad \\Pi _k \\in \\mathcal {P}_k \\ \\ \\text{and} \\ \\ \\Pi _k(0) = 1.",
"$ Combining these two equations, we obtain $r_k^\\text{CRAIG} = \\varphi _k(0)^{-1}\\varphi _k(AA^T)b.$ Substituting to (REF ) back from (REF ), we immediately have (REF ).",
"Since $\\Vert s_{k+1}\\Vert = 1$ , (REF ) is a direct consequence of (REF ).",
"$\\Box $ Although the relation (REF ) is valid for any problem of the form (REF ), it has a particularly interesting interpretation for inverse problems with a smoothing operator $A$ .",
"Suppose we neglect the low-frequency part $s_{k+1}^\\text{LF}$ in (REF ) and estimate the unknown noise $\\eta $ from the left bidiagonalization vector $s_{k+1}$ as $\\eta \\approx \\tilde{\\eta } \\equiv \\varphi _k(0)^{-1} s_{k+1}.$ Subtracting $\\tilde{\\eta }$ from $b$ in (REF ), we obtain exactly the modified problem (REF ).",
"Thus Proposition REF in fact states that in each iteration $k$ , $x_k^\\text{CRAIG}$ represents the exact solution of the problem (REF ) with noise being approximated by a particular re-scaled left bidiagonalization vector.",
"The norm of the CRAIG residual $r_k^\\text{CRAIG}$ is inversely proportional to the amount of noise propagated to the currently computed left bidiagonalization vector.",
"It reaches its minimum exactly in the noise revealing iteration $k = k_\\text{rev}$ , which corresponds to the iteration with (REF ) being the best approximation of the unknown noise vector.",
"The actual noise vector $\\eta $ and the difference $\\eta - \\tilde{\\eta }$ for $\\tilde{\\eta }$ obtained from $s_{k_\\text{rev}+1}$ are compared in Figure REF ; see also .",
"We see that in iteration $k_\\text{rev}$ , the troublesome high-frequency part of noise is perfectly removed.",
"The remaining perturbation only contains smoothed, i.e., low-frequency part of the original noise vector.",
"The match in (REF ) remains valid, up to a small perturbation, also in finite precision computations, since the noise propagation is preserved, see Section REF Figure: Illustration of the quality of the noise vector approximation η ˜\\tilde{\\eta } obtained by () for k=k rev +1k={k_\\text{rev}+1} on various test problems and various characteristics of noise.",
"Upper: The original noise vector η\\eta .",
"Lower: The difference η-η ˜\\eta - \\tilde{\\eta }.Note that due to different frequency characteristic of $\\eta $ and $s_{k+1}^\\text{LF}$ for small $k$ , there is a relatively small cancellation between them and $\\Vert s_{k+1}^\\text{LF}\\Vert ^{2}+\\Vert \\varphi _k(0)\\eta \\Vert ^{2}\\approx 1.$ This gives $\\Vert (b - \\tilde{\\eta }) - Ax\\Vert = \\Vert \\varphi _k(0)^{-1}s_{k+1}^\\text{LF}\\Vert \\approx |\\varphi _k(0)|^{-1}\\sqrt{1-\\Vert \\varphi _k(0)\\eta \\Vert ^{2}} =\\sqrt{|\\varphi _k(0)|^{-2} -\\Vert \\eta \\Vert ^{2}}$ supporting our expectation that the size of the remaining perturbation depends on how closely the inverse amplification factor $|\\varphi _k(0)|^{-1}$ approaches $\\Vert \\eta \\Vert $ .",
"We may also conclude that for ill-posed problems with a smoothing operator $A$ , the minimal error $\\Vert x_k^\\text{CRAIG}-x\\Vert $ is reached approximately at the iteration with the maximal noise revealing, i.e., with the minimal residual.",
"This is confirmed by numerical experiments in Figure REF comparing $\\Vert x_k^\\text{CRAIG}-x\\Vert $ with $\\Vert r_k^\\text{CRAIG}\\Vert $ for various test problems and noise characteristics, both with and without reorthogonalization.",
"Figure: Comparison of the size of the residual and the size of the error in CRAIG for various test problems with various noise characteristics.",
"The minimal error is achieved approximately when the residual is minimized (vertical line).",
"In Figures (g)-(i) without reorthogonalization."
],
[
"LSQR residuals",
"Whereas for CRAIG, the residual is just a scaled left bidiagonalization vector, for LSQR it is a linear combination of all previously computed left bidiagonalization vectors.",
"Indeed, $r_k^\\text{LSQR} = b - AW_ky_k^\\text{LSQR} = S_{k+1}\\left( \\beta _1e_1 - L_{k+}y_k^\\text{LSQR}\\right),$ see (REF ).",
"The entries of the residual of the projected problem $p_k^\\text{LSQR}\\equiv \\beta _1e_1 - L_{k+}y_k^\\text{LSQR},$ see (REF ), represent the coefficients of the linear combination in (REF ).",
"The following proposition shows the relation between the coefficients and the amplification factor $\\varphi _k(0)$ .",
"Proposition 2 Consider the first $k$ steps of the Golub-Kahan iterative bidiagonalization.",
"Let $r_k^\\text{LSQR} = b - Ax_k^\\text{LSQR}$ , where $x_k^\\text{LSQR}$ is the approximation defined in (REF ) and (REF ).",
"Then $r_k^\\text{LSQR} = \\frac{1}{\\sum _{l=0}^k\\varphi _l(0)^{2}}\\sum _{l=0}^k\\varphi _l(0)s_{l+1}.$ Consequently, $\\Vert r_k^\\text{LSQR}\\Vert = \\frac{1}{\\sqrt{\\sum _{l=0}^k\\varphi _l(0)^{2}}}.$ Since $y_k^\\text{LSQR} = \\underset{y}{\\operatorname{argmin}}\\Vert \\beta _1e_1 - L_{k+}y\\Vert ,$ we get $L_{k+}^Tp_k^\\text{LSQR} = 0.$ It follows from the structure of the matrix $L_{k+}$ that the entries of $p_k^\\text{LSQR}$ satisfy $\\alpha _le_l^Tp_k^\\text{LSQR} + \\beta _{l+1}e_{l+1}^Tp_k^\\text{LSQR} = 0, \\qquad \\text{for} \\qquad l = 1,\\ldots ,k.$ Thus $p_k^\\text{LSQR} = c_k\\left[\\begin{array}{c}\\varphi _0(0)\\\\\\varphi _1(0)\\\\\\vdots \\\\\\varphi _k(0)\\end{array}\\right],$ where $c_k$ is a factor that changes with $k$ .",
"From (REF ) and (REF ) it follows that $\\Vert p_k^\\text{LSQR}\\Vert = \\Vert r_k^\\text{LSQR}\\Vert = \\frac{1}{\\sqrt{\\sum _{l=0}^k\\varphi _l(0)^{2}}}.$ By comparing (REF ) and (REF ), we get $c_k = \\frac{1}{\\sum _{l=0}^k\\varphi _l(0)^{2}},$ which together with (REF ) and (REF ) yields (REF ).$\\Box $ In other words, Proposition REF says that the coefficients of the linear combination (REF ) follow the behavior of the amplification factor in the sense that representation of a particular left bidiagonalization vector $s_{l+1}$ in the residual $r_k^\\text{LSQR}$ , $k\\ge l$ , is proportional to the amount of propagated non-smoothed part of noise $\\eta $ in this vector.",
"Relation (REF ) also suggests that the norm-minimizing process (LSQR) and the corresponding Galerkin process (CRAIG) provide similar solutions whenever $\\frac{\\varphi _k(0)^{2}}{\\sum _{l=0}^{k}\\varphi _l(0)^{2}}\\approx 1,$ i.e., whenever the noise revealing in the last left bidiagonalization vector $s_{k+1}$ is much more significant than in all previous left bidiagonalization vectors $s_1,\\ldots s_k$ , i.e., typically before we reach the noise revealing iteration.",
"This is confirmed numerically in Figure REF , comparing the semiconvergence curves of CRAIG and LSQR.",
"Figure: The size of the error of LSQR and CRAIG in comparison with the inverse of the amplification factor for various test problems with various noise characteristics.",
"The semiconvergence curves exhibit similar behavior until the noise revealing iteration.",
"In Figure (f) without reorthogonalization."
],
[
"LSMR residuals",
"Before we investigate the residual of LSMR with respect to the basis $S_k$ , we should understand how it is related to the residual of LSQR.",
"It follows from Table REF that the relation between $A^Tr_k^\\text{LSMR}$ and $A^Tr_k^\\text{LSQR}$ is analogous to the relation between $r_k^\\text{CRAIG}$ and $r_k^\\text{LSQR}$ .",
"Using Proposition REF and REF , with $\\varphi _k$ substituted by $\\psi _k$ and $s_k$ substituted by $w_k$ , we obtain $A^Tr_k^\\text{LSQR} = \\psi _k(0)^{-1}w_{k+1},$ and $A^Tr_k^\\text{LSMR} = \\frac{1}{\\sum _{l=0}^k\\psi _l(0)^{2}}\\sum _{l=0}^k\\psi _l(0)w_{l+1}.$ Since $A^Tr_k^\\text{LSMR} = W_{k+1}L_{k+1}^Tp_k^\\text{LSMR},$ we obtain that $\\\\L_{k+1}^Tp_k^\\text{LSMR} &= \\frac{1}{\\sum _{l=0}^k\\psi _l(0)^{2}}\\left[\\begin{array}{c}\\psi _0(0)\\\\\\psi _1(0)\\\\\\vdots \\\\\\psi _k(0)\\end{array}\\right].$ This equality however does not provide the desired relationship between the residuals $r_k^\\text{LSMR}$ themselves and the left bidiagonalization vectors $s_1,\\ldots ,s_{k+1}$ .",
"This is given in the following proposition.",
"Proposition 3 Consider the first $k$ steps of the Golub-Kahan iterative bidiagonalization.",
"Let $r_k^\\text{LSMR} = b - Ax_k^\\text{LSMR}$ , where $x_k^\\text{LSMR}$ is the approximation defined in (REF ) and (REF ).",
"Then $r_k^\\text{LSMR} = \\frac{1}{\\sum _{l=0}^k\\psi _l(0)^2}\\sum _{l = 0}^k \\left(\\varphi _l(0)\\sum _{j=l}^k\\alpha _{j+1}^{-1}\\varphi _j(0)^{-1}\\psi _j(0)\\right) s_{l+1}.$ From (REF ) it follows that $p_k^{\\text{LSMR}}=\\frac{1}{\\sum _{l=0}^k\\psi _l(0)^2}\\,L_{k+1}^{-T}\\left[\\begin{array}{c}\\psi _0(0)\\\\\\psi _1(0)\\\\\\vdots \\\\\\psi _k(0)\\end{array}\\right],$ where $L_{k+1}^{-T}$ is an upper triangular matrix with entries $e_i^TL_{k+1}^{-T}e_j= \\left\\lbrace \\begin{array}{cc}\\frac{\\displaystyle 1}{\\displaystyle \\alpha _j} & \\text{(if $i=j$)} \\\\[2mm](-1)^{i-j}\\,\\frac{\\displaystyle \\beta _{i+1}\\cdots \\beta _j}{\\displaystyle \\alpha _i\\cdots \\alpha _j} & \\text{(if $i<j$)}\\end{array}\\right\\rbrace = \\frac{\\varphi _{i-1}(0)}{\\alpha _j\\,\\varphi _{j-1}(0)}\\,.$ Thus $\\begin{split}p_k^{\\text{LSMR}}= \\frac{1}{\\sum _{l=0}^k\\psi _l(0)^2}\\,\\mathrm {triu}\\left(\\left[\\begin{array}{c}\\varphi _0(0)\\\\\\varphi _1(0)\\\\\\vdots \\\\\\varphi _k(0)\\end{array}\\right]\\left[\\begin{array}{c}\\varphi _0(0)^{-1}\\\\\\varphi _1(0)^{-1}\\\\\\vdots \\\\\\varphi _k(0)^{-1}\\end{array}\\right]^T\\right)\\left[\\begin{array}{c}\\alpha _1^{-1}\\psi _0(0)\\\\\\alpha _2^{-1}\\psi _1(0)\\\\\\vdots \\\\\\alpha _{k+1}^{-1}\\psi _k(0)\\end{array}\\right], \\\\\\end{split}$ where $\\mathrm {triu}(\\cdot )$ extracts the upper triangular part of the matrix.",
"Multiplying out, we obtain $p_k^{\\text{LSMR}} = \\frac{1}{\\sum _{l=0}^k\\psi _l(0)^2}\\left[\\begin{array}{c}\\varphi _0(0)\\sum _{l=0}^k\\alpha _{l+1}^{-1}\\varphi _l(0)^{-1}\\psi _l(0)\\\\\\varphi _1(0)\\sum _{l=1}^k\\alpha _{l+1}^{-1}\\varphi _l(0)^{-1}\\psi _l(0)\\\\\\vdots \\\\\\varphi _{k-1}(0)\\sum _{l=k-1}^k\\alpha _{l+1}^{-1}\\varphi _l(0)^{-1}\\psi _l(0)\\\\\\varphi _k(0)\\qquad \\;\\alpha _{k+1}^{-1}\\varphi _k(0)^{-1}\\psi _k(0)\\end{array}\\right].$ $\\Box $ Here the sizes of coefficients in $p_k^{\\text{LSMR}}$ need careful discussion.",
"From (REF ) and (REF ) it follows that the absolute terms of the Lanczos polynomials $\\varphi _l(0)$ and $\\psi _l(0)$ have the same sign.",
"Thus we have $\\alpha _{l+1}^{-1}\\varphi _l(0)^{-1}\\psi _l(0) >0, \\quad \\forall l=0, 1, \\dots $ and therefore the sum $\\sum _{l=j}^k\\alpha _{l+1}^{-1}\\varphi _l(0)^{-1}\\psi _l(0) $ decreases when $j$ increases.",
"Furthermore, it was shown in that for $j<k_\\text{rev}$ $\\alpha _l \\approx \\beta _l.$ Thus (REF ) yields $\\sum _{l=j}^k\\alpha _{l+1}^{-1}\\varphi _l(0)^{-1}\\psi _l(0) \\approx \\sum _{l=j}^k \\psi _l(0).$ However, since $|\\varphi _j(0)|$ on average increases rapidly with $j$ (see Section REF ), the sizes of the entries of $p_k^\\text{LSMR}$ in (REF ) generally increase with $l$ before $k_\\text{rev}$ .",
"After $j$ reaches the noise revealing iteration $k_\\text{rev}$ , $|\\varphi _j(0)|$ decreases at least for one but typically for more subsequent iterations; see Section REF .",
"Multiplication by the decreasing (REF ) causes that the size of the entries in (REF ) can be expected to decrease after $k_\\text{rev}$ .",
"From the previous we conclude that the behavior of the entries of $p_k^\\text{LSMR}$ resembles the behavior of $\\varphi _l(0)$ , i.e., the size of a particular entry is proportional to the amount of propagated noise in the corresponding bidiagonalization vector, similarly as in the LSQR method.",
"Figure REF compares the entries of $p_k^\\text{LSMR}$ with appropriately re-scaled amplification factor $\\varphi _k(0)$ on the problem shaw with white noise.",
"We see that the difference is negligible an therefore the residuals for LSQR and LSMR resemble.",
"In early iterations, the resemblance of the residuals indicates resemblance of the solutions since the remaining perturbation only contains low frequencies, which are not amplified by $A^{\\dagger }$ .",
"Figure: The componets of p k LSQR p_k^\\text{LSQR} vs. the size of the amplification factor ϕ k (0)\\varphi _k(0) (after scaling) for several values of kk for the problem shaw with white noise, δ noise =10 -3 \\delta _\\text{noise}=10^{-3}.",
"The differences are negligible.Note also that since $\\psi _k(0)$ grows rapidly on average, see Figure REF in Section REF , we may expect $\\frac{\\psi _k(0)^{2}}{\\sum _{l=0}^{k}\\psi _l(0)^{2}}\\approx 1.$ Therefore $A^Tr_k^\\text{LSMR}$ resembles $A^Tr_k^\\text{LSQR}$ giving another explanation why LSMR and LSQR behave similarly for inverse problems with a smoothing operator $A$ , see Figure REF for a comparison on several test problems.",
"Figure: The size of the error of LSMR and LSQR in comparison with the inverse of the size of ψ k (0)\\psi _k(0) for various test problems with various noise characteristics.",
"Since |ψ k (0)||\\psi _k(0)| often grow on average till very late iterations, the semiconvergence curves exhibit similar behavior.",
"In Figure (f) without reorthogonalization.Figure REF illustrates the match between the noise vector and residual of CRAIG, LSQR and LSMR method.",
"We see that while CRAIG residual resembles noise only in the noise revealing iteration, LSQR and LSMR are less sensitive to the particular number of iterations $k$ as the residuals are combinations of bidiagonalization vectors with appropriate coefficients.",
"Moreover, the best match in LSQR and LSMR method overcomes the best match in CRAIG.",
"This is caused by the fact that the remaining low-frequency part is efficiently suppressed by the linear combination.",
"Figure: Difference between the noise vector and the residual of considered iterative methods for the problem shaw with white and red noise noise, δ noise =10 -3 \\delta _\\text{noise} = 10^{-3}.Residuals of LSQR and LSMR have similar approximation properties with respect to the noise vector."
],
[
"Numerical experiments for 2D problems",
"In this section we discuss validity of the conclusions made above for larger 2D inverse problems, where the smoothing property of $A$ (revealing itself in the decay of singular values) is typically less significant.",
"Consequently, noise propagation in the bidiagonalization process may be more complicated; see also .",
"However, we illustrate that essential aspects of the behavior described in previous sections are still present.",
"Note that all experiments in this section are computed without reorthogonalization.",
"We consider the following 2D benchmarks: Medical tomography problem — a simplified 2D model of X-ray medical tomography adopted from , function paralleltomo(256,0:179,362).",
"The data is represented the 256-by-256 discretization of the Shepp–Logan phantom projected in angles $\\theta = 0,1,\\ldots ,179$ by 362 parallel rays, resulting in a linear algebraic problem with $A\\in \\mathbb {R}^{65160\\times 65536}$ .",
"We use Poisson-type additive noise $\\eta $ generated as follows (see and ) to simulate physically realistic noise: A = paralleltomo(N,theta)/N; t = exp(-A*x); c = poissrnd(t*N0); eta = -log(c/N0); \\end{verbatim} where $N_0=10^5$ denotes the mean number of photons, resulting in the noise level $\\delta_\\text{noise} \\approx 0.028$. We refer to this test problem as \\texttt{paralleltomo}.",
"\\item{\\bf Seismic tomography problem} --- a simplified 2D model of seismic tomography adopted from \\cite{Hansen2012AIR}, function \\texttt{seismictomo(100,100,200)}.",
"The data is represented by a $100$-by-$100$ discretization of a vertical domain intersecting two tectonic plates with $100$ sources located on its right boundary and $200$ receivers (seismographs), resulting in a linear algebraic problem with $A\\in\\mathbb{R}^{20000\\times 10000}$. The right-hand side is polluted with additive white noise with $\\delta_\\text{noise} = 0.01$. We refer to this test problem as \\texttt{seismictomo}.",
"\\item{\\bf Image deblurring problem} --- an image deblurring problem with spatially variant blur adopted from \\cite{Berisha2013Iterative,Nagy2004Iterative}, data \\texttt{VariantGaussianBlur1}. The data is represented by a monochrome microscopic $316$-by-$316$ image of a grain blurred by spatially variant Guassian blur (with $49$ different point-spread functions), resulting in a linear algebraic problem with $A\\in\\mathbb{R}^{99856\\times 99856}$. The right-hand side is polluted with additive white noise with $\\delta_\\text{noise} = 0.01$.",
"We refer to this test problem as \\texttt{vargaussianblur}.",
"\\end{description} Figure~\\ref{fig:2D_factors} shows the absolute terms of the Lanczos polynomials $\\varphi_k$ and $\\psi_k$. We can identify the two phases of the behavior of $\\varphi_k(0)$ -- average growth and average decay.",
"However, the transition does not take place in one particular (noise revealing) iteration, but rather in a few subsequent steps, which we refer to as the {\\em noise revealing phase} of the bidiagonalization process.",
"The size of $\\psi_k(0)$ grows on average till late iterations, however, we often observe here that the speed of this growth slows down after the noise revealing phase. In conclusion, both curves $|\\varphi_k(0)|$ and $|\\psi_k(0)|$ can be flatter than for 1D problem considered in previous sections.",
"This can be further pronounced for problems with low noise levels.",
"\\begin{figure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth]{phi0_vargaussianblur} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth]{phi0_seismic} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth]{phi0_paralleltomo_new} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth]{psi0_vargaussianblur} \\caption{\\texttt{vargaussianblur}} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth]{psi0_seismic} \\caption{\\texttt{seismictomo}}\\label{fig:factor_seismic} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth]{psi0_paralleltomo_new} \\caption{\\texttt{paralleltomo} } \\end{subfigure} \\caption{The size of the absolute term of the Lanczos polynomials $\\varphi_k$ and $\\psi_k$ for selected 2D problems contaminated by noise as \t\t\t\t described in the text. For all problems $\\delta_\\text{noise} \\approx 10^{-2}$. Computed without reorthogonalization.",
"}\\label{fig:2D_factors} \\end{figure} Figure~\\ref{fig:2D_vectors} shows several (appropriately reshaped) left bidiagonalization vectors $s_k$ and their cumulative periodograms for the problem \\texttt{seismictomo}.",
"Even though it is hard to make clear conclusions based on the vectors $s_k$ themselves, we see that the periodogram for $k=10$ is flatter than the periodograms for smaller or larger values of $k$, meaning that $s_{10}$ resembles most white noise.",
"This corresponds to Figure~\\ref{fig:factor_seismic} showing that $s_{10}$ belongs to the noise revealing phase of the bidiagonalization process.",
"Note that similar flatter periodograms can be obtained for other few vectors belonging to this phase.",
"\\begin{figure} \\centering \\begin{subfigure}[b]{\\textwidth} \\includegraphics[width = .95\\textwidth]{S_seismic} \\caption{left bidiagonalization vectors $s_k$ (reshaped)}\\label{fig:2D_bidvect} \\end{subfigure} \\begin{subfigure}[b]{\\textwidth} \\includegraphics[width = .95\\textwidth]{S_cummulativeperiodogram_seismic} \\caption{cumulative periodograms of $s_k$}\\label{fig:2D_periodograms} \\end{subfigure} \\caption{Left bidiagonalization vectors $s_k$ for the problem \\texttt{seismictomo} and their cumulative periodograms.",
"The periodogram of the vector $s_{10}$ belonging to the noise revealing phase of the bidiagonalization process is flatter. Computed without reorthogonalization.",
"}\\label{fig:2D_vectors} \\end{figure} The absence of one particular noise revealing vector makes the direct comparison between $s_k$ and the exact noise vector $\\eta$ irrelevant here.",
"However, Propositions~\\ref{th:1}--\\ref{th:3} remain valid and the overall behavior of the terms $|\\varphi_k(0)|$ and $|\\psi_k(0)|$ is as expected, allowing comparing the bidiagonalization-based methods.",
"Figure~\\ref{fig:2D_errors} gives comparisons of CRAIG, LSQR and LSMR for all considered 2D test problems, analogous to Figure~\\ref{fig:error_all}, \\ref{fig:error_craig_lsqr}, and \\ref{fig:error_lsqr_lsmr}.",
"The first row of Figure~\\ref{fig:2D_errors} shows that the CRAIG error is minimized approximately in the noise revealing phase, i.e., when the residual is minimal, see Section~\\ref{sec:relation_CRAIG}.",
"The minimum is emphasized by the vertical line.",
"The second row of Figure~\\ref{fig:2D_errors} compares the errors of CRAIG and LSQR.",
"According to the derivations in Section~\\ref{sec:residuals}, the curves are similar before the noise revealing phase, after which they separate with CRAIG diverging more quickly.",
"Note that the size of the inverted amplification factor $\\varphi_k(0)$ is included to illustrate the noise revealing phase and has different scaling (specified on the right).",
"The third row of Figure~\\ref{fig:2D_errors} shows the errors of LSQR and LSMR with the underlying size of the inverted factor $\\psi_k(0)$ (scaling specified on the right).",
"The errors behave similarly as long as $|\\psi_k(0)|^{-1}$ decays rapidly, see Section~\\ref{ssec:LSMRres}.",
"The LSMR solution is slightly less sensitive to the particular choice of the number of bidiagonalization iterations $k$, which is a well know property \\cite{Fong2011LSMR}.",
"\\begin{figure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth]{error_Craig_vargaussianblur} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth]{error_Craig_seismic} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth]{error_Craig_paralleltomo_new} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth,trim={1.4cm 5cm .4cm 7cm},clip]{error_comparison_craig_lsqr_vargaussianblur} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth,trim={1.4cm 5cm .4cm 7cm},clip]{error_comparison_craig_lsqr_seismic} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth,trim={1.4cm 5cm .4cm 7cm},clip]{error_comparison_craig_lsqr_paralleltomo_new} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth,trim={1.4cm 5cm .4cm 7cm},clip] {error_comparison_lsqr_lsmr_vargaussianblur} \\caption{\\texttt{vargaussianblur}} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth,trim={1.4cm 5cm .4cm 7cm},clip]{error_comparison_lsqr_lsmr_seismic} \\caption{\\texttt{seismictomo}} \\end{subfigure} \\begin{subfigure}[b]{0.32\\textwidth} \\includegraphics[width = .95\\textwidth,trim={1.4cm 5cm .4cm 7cm},clip]{error_comparison_lsqr_lsmr_paralleltomo_new} \\caption{\\texttt{paralleltomo}} \\end{subfigure} \\caption{First row: The size of the residual and the size of the error in CRAIG. Vertical line illustrates the minimum.",
"Second row: The size of the error of CRAIG and LSQR, together with the rescaled inverse of the amplification factor $\\varphi_k(0)$ (vertical scale on the right). Third row: The size of the error of LSQR and CRAIG, together with the rescaled inverse of the factor $\\psi_k(0)$ (vertical scale on the right). Computed without reorthogonalization.",
"}\\label{fig:2D_errors} \\end{figure} \\section{Conclusion}\\label{sec:conclusions} We proved that approximating the solution of an inverse problem by the $k$th iterate of CRAIG is mathematically equivalent to solving consistent linear algebraic problem with the same matrix and a right-hand side, where a particular (typically high-frequency) part of noise is removed. Using the analysis of noise propagation, we showed that the size of the CRAIG residual is given by the inverted noise amplification factor, which explains why optimal regularization properties are often obtained when the minimal residual is reached. For LSQR and LSMR, the residual is a linear combination of the left bidiagonalization vectors. The representation of these vectors in the residuals is determined by the amplification factor, in particular, left bidiagonalization vectors with larger amount of propagated noise are on average represented with a larger coefficient in both methods. These results were used in 1D problems to compare the methods in terms of matching between the residuals and the unknown noise vector.",
"For large 2D (or 3D) problems the direct comparison of the vectors may not be possible, since noise reveals itself in a few subsequent bidiagonalization vectors (noise revealing phase of bidiagonalization) instead of in one particular iteration. However, the conclusions on the methods themselves remain generally valid.",
"Presented results contribute to understanding of the behavior of the methods when solving noise-contaminated inverse problems.",
"\\section*{Acknowledgment} Research supported in part by the Grant Agency of the Czech Republic under the grant 17-04150J.",
"Work of the first and the second author supported in part by Charles University, project GAUK~196216.",
"The authors are grateful to the anonymous referee for useful suggestions and comments that improved the presentation of the paper.",
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] | 1612.05551 |
[
[
"Physics of hollow Bose-Einstein condensates"
],
[
"Abstract Bose-Einstein condensate shells, while occurring in ultracold systems of coexisting phases and potentially within neutron stars, have yet to be realized in isolation on Earth due to the experimental challenge of overcoming gravitational sag.",
"Motivated by the expected realization of hollow condensates by the space-based Cold Atomic Laboratory in microgravity conditions, we study a spherical condensate undergoing a topological change from a filled sphere to a hollow shell.",
"We argue that the collective modes of the system show marked and robust signatures of this hollowing transition accompanied by the appearance of a new boundary.",
"In particular, we demonstrate that the frequency spectrum of the breathing modes shows a pronounced depression as it evolves from the filled sphere limit to the hollowing transition.",
"Furthermore, when the center of the system becomes hollow surface modes show a global restructuring of their spectrum due to the availability of a new, inner, surface for supporting density distortions.",
"We pinpoint universal features of this topological transition as well as analyse the spectral evolution of collective modes in the experimentally relevant case of a bubble-trap."
],
[
"Evolution of surface modes",
"The effect of hollowing on surface modes of a spherical condensate can be gleaned from the eigenvalue equation for the collective mode frequencies, Eq.",
"(REF ), in the limit of large angular momentum, $\\ell \\gg 1$ .",
"The centrifugal term $\\ell (\\ell +1) n_{\\rm {eq}}$ dominates in this limit and is minimized at the condensate's outer surface (due to $n_{\\rm {eq}}\\sim 0$ ), and also the inner one in the hollow case, thus causing localization of large-$\\ell $ density deviations to these surfaces.",
"At the transition between a filled and a hollow condensate, the doubling of the surfaces available for excitations creates a redistribution of radial nodes—some of the nodes can localize in the vicinity of the new, inner surface.",
"Consequently, since fewer nodes are compressed in each surface region, the frequency of any single mode (indexed by the total number of radial nodes $\\nu $ ) is reduced across the hollowing transition.",
"To provide an intuitive physical picture, we note that a similar situation is found in the energy states of a quantum double-well potential as the minima are brought into alignment.",
"At the point at which the minimum potential values are equal, the energy eigenvalue corresponding to $\\nu $ nodes jumps to corresponding to two degenerate states having $2\\nu $ and $2\\nu +1$ nodes (for instance, states having $\\nu =2$ and $\\nu =3$ radial nodes become nearly degenerate with the $\\nu =1$ state.)",
"Here, the centrifugal term in Eq.",
"(REF ) plays the role of the potential and, as we show below, the experimentally relevant bubble trap even supports the degeneracy exhibited by the simple double-well analogy.",
"More concretely, focusing on the two surfaces, the Thomas-Fermi approximation for the bubble trap identifies the inner and outer boundaries at the radii $r=R_{\\rm {in}}=\\sqrt{2\\Delta -R^2}$ and $r=R_{\\rm {out}}=R$ , respectively.",
"Linearizing the trapping potential at these boundaries gives the Thomas-Fermi equilibrium densities $n_{{\\rm {eq}}}^{{\\rm {TF}}}({x_{{\\rm {in,out}}}}) = -\\frac{{{F_{{\\rm {in,out}}}}}}{U}{x_{{\\rm {in,out}}}},$ with $F_{\\rm {in,out}}=-\\nabla V_{\\rm {bubble}}(R_{\\rm {in,out}})$ and $x_{\\rm {in,out}}$ the local variable pointing along the direction of $F_{\\rm {in,out}}$ .",
"This description of the condensate is appropriate when the trapping potential $V_{\\rm {bubble}}$ varies slowly over a distance $\\delta _{\\rm {sm}}=[\\hbar ^2/(2m|F|)]^{1/3}$ from a condensate boundary.",
"Consequently, such linearization is not appropriate in the very thin shell limit thus restricting Eq.",
"(REF ) to hollow BECs of nontrivial thickness.",
"In thin hollow condensates with thickness $\\delta _{\\rm {t}}<2\\delta _{\\rm {sm}}$ surface modes confined to the inner and the outer boundary overlap and hence cannot be treated as strictly localized at either of these surfaces.",
"Surface mode frequencies for thick BEC shells can be derived from the master equation, Eq.",
"(REF ), using these density profiles and the ansatz that the density distortions exponentially decay into the bulk [52].The resultant frequencies are given by $mS_l^2\\omega ^2_{\\rm {in,out}}=(1+2\\nu _{\\rm {in,out}})F_{\\rm {in,out}}q_{\\rm {in,out}}$ where $\\nu _{\\rm {in,out}}$ indicates the number of nodes of the collective mode confined to a particular condensate boundary in the direction transverse to its surface.",
"The wave-number associated with each surface mode is given by $q_{\\rm {in,out}}=\\ell /R_{\\rm {in,out}}$ .",
"To be more precise, these surface mode frequencies read $\\omega ^2_{\\mathrm {in,out}}=\\frac{\\omega _0^2\\ell (R^2-\\Delta )}{\\sqrt{(R^2-\\Delta )^2/4+\\Omega ^2}}(2\\nu _{\\rm {in,out}}+1).$ We note that Eq.",
"(REF ) is rather similar to the well-established result for the surface modes of a fully filled spherical condensate where $\\omega ^2_{\\rm {sp}}=\\ell \\omega _0^2(2\\nu _{\\rm {out}}+1)$ — the functional form and the $\\ell $ dependence are the same up to numerical factors accounting for the finite thickness of the hollow shell BEC [52].",
"For the bubble trap, $F_{\\rm {in}}q_{\\rm {in}}=F_{\\rm {out}}q_{\\rm {out}}$ which leads to a degeneracy in the frequency of surface modes at the inner and outer surfaces, as noted above.",
"A simple explanation for this degeneracy is that even though the inner surface is smaller in area, its lower stiffness can support more oscillations per unit distance (since $q_{\\mathrm {in}}>q_{\\mathrm {out}}$ ), bringing the frequency of oscillations with $\\nu _{\\rm {in}}=\\nu _{\\rm {out}}$ nodes on the two surfaces into alignment.",
"We note that increasing the bubble trap parameter $\\Delta $ decreases the thickness of the condensate shell and leads to coupling between the inner and outer boundary surface modes which can lift this degeneracy (as we will show in Fig.",
"REF ).",
"Figure: (Color online) Oscillation frequencies of the ℓ=20\\ell =20collective modes for the lowest four radial index valuesν=0,1,2,3\\nu =0,1,2,3 (curves from bottom to top, respectively) as thesystem is tuned from a filled sphere Δ=0\\Delta =0 to a thin shellΔ∼R 2 \\Delta \\sim R^2 at a constant chemical potential.",
"The linethrough Δ=0.5R 2 \\Delta =0.5 R^2 (or the shaded region in the inset)demarcates the filled and hollow regions and the schematics alongthe bottom show the associated density deviations.",
"Main: using theThomas-Fermi approximation for the equilibrium density, whichcorresponds to sharp boundaries, and assuming the spatial extentof the condensate does not change, i.e., RR is constant.",
"Inset:using equilibrium profiles given by numerical solution of the GPequation (axes units are defined by the interaction energyUN=10 4 UN=10^{4}), which gives realistic soft boundaries, and assumingthat the number of atoms making up the condensate, NN, isconstant.The restructuring of the surface mode spectrum following the hollowing out transition thus presents a marked contrast to the behavior of the same modes of a fully filled spherical BEC discussed in the literature [52].",
"We remark that inducing true surface modes corresponding to large $\\ell $ values presents an experimental challenge, and is yet to be achieved even in the filled sphere case (although such modes have been experimentally observed in classical fluids[55]).",
"But a comparison of the two cases highlights the difference between the collective mode spectra of filled and hollow BECs in any angular-momentum $\\ell $ regime, including the two extreme limits: $\\ell =0$ and $\\ell \\gg 1$ .",
"We proceed to discuss the $\\ell =0$ limit."
],
[
"Evolution of breathing modes",
"Collective modes involving density distortions transverse to the boundaries in a filled-sphere condensate correspond to spherically symmetric breathing modes.",
"As the system evolves to a hollow shell, the density at the center of the system decreases, thus creating a region of low stiffness where density oscillations tend to localize and their frequency decreases.",
"When the density in the center of the system vanishes, the inner boundary is created and the density deviations return to the bulk of the system.",
"Thus, the hollowing transition and the appearance of an inner boundary are signaled by a prominent decrease in the transverse mode frequencies.",
"More rigorously, in Eq.",
"(REF ), the eigen-frequency receives two contributions: one proportional to the equilibrium density, $n_{\\rm {eq}}$ , and the other to its gradient, $\\nabla n_{\\rm {eq}}$ .",
"In general, $n_{\\rm {eq}}$ is smallest at the surface(s) of the BEC while $\\nabla n_{\\rm {eq}}$ is smallest at the location of extremum density.",
"The only situation in which both of these contributions can be small is at the hollowing-out transition: at the center of the system, where the inner boundary is emerging, both the condensate density and its gradient simultaneously vanish.",
"The hydrodynamic equation therefore implies that the lowest frequency of oscillation for breathing modes can be realized close to the center of the condensate as it is starting to hollow.",
"Consequently, the density deviations for the collective modes concentrate near the nascent inner surface and reach unprecedentedly low frequencies.",
"The spectrum of the breathing modes, therefore, displays a minimum in frequency at the hollowing transition.",
"Breathing modes exhibiting this non-monotonic spectral property are among the most experimentally accessible excitations of a three-dimensional spherically symmetric BEC and thus a good candidate for an observation of the effects of the hollowing-out transition.",
"Additionally, collective modes with low angular momentum values, such as $\\ell =1$ or $\\ell =2$ exhibit similar frequency dip features.",
"In fact, in a realistic experimental system that may lack perfect spherical symmetry, these low-$\\ell $ collective modes would be the most likely candidate of study."
],
[
"Numerical analyses for bubble-trap geometries",
"Corroborating our heuristic arguments and simple derivations, we now perform an in-depth numerical analysis for collective modes in the bubble trap of Eq.",
"(REF ) by directly solving the hydrodynamic equation of motion, Eq.",
"(REF ) via a finite-difference method [56].",
"We initially use the Thomas-Fermi approximation for the equilibrium density ($n_{\\rm {eq}}$ ); this approach focuses on salient features of the topological hollowing transition by modeling a sharp inner (Thomas-Fermi) boundary.",
"In these calculations we hold the outer radius of the condensate, $R$ , fixed as $\\Delta $ is changed.",
"We then go beyond the Thomas-Fermi approximation by solving the time-independent Gross-Pitaevskii (GP) equation [54] for the equilibrium density, using an imaginary-time algorithm [57].",
"Here, calculations are performed with a fixed number of atoms $N$ in the condensate.",
"This method addresses the fate of the mode spectra in the physically realistic case of soft condensate boundaries.",
"First addressing the surface modes, Fig.",
"REF shows the evolution of the collective mode spectrum obtained by direct numerical solution of Eq.",
"(REF ) with $\\ell =20$ , for a range of values of $\\Delta $ in the bubble trap potential of Eq.",
"(REF ), tuning from the fully filled sphere to thin shell limits.",
"The transition point where an inner boundary first appears occurs at $R_{\\rm {in}}=0$ or $\\Delta /R^2=0.5$ .",
"As argued on general grounds above, the transition from filled sphere to hollow shell is indeed signaled by a reduction in frequency for a given radial index and a sudden degeneracy between modes having radial indices $2\\nu $ and $2\\nu +1$ (here, $\\nu $ denotes the total number of radial zeros for a given collective mode).",
"Density fluctuations on either side of the transition are schematically represented along the bottom of Fig.",
"REF .",
"The results described above model a sudden change from the filled sphere to the shell topology through the appearance of the sharp inner boundary in the Thomas-Fermi approximation.",
"In contrast, the inset to Fig.",
"REF shows the result of solving the hydrodynamic equation for the $\\ell =20$ collective modes using the numerically exact equilibrium density, $n_{\\rm {eq}}$ , obtained by solving the time-independent GP equation (with an interaction strength of $NU=10^4$ ).",
"This more realistic modeling of the gradually emerging inner surface yields a smearing of the sharp frequency discontinuity found in the Thomas-Fermi approximation as marked by the shaded region in the inset of the figure.",
"Nevertheless, the frequency spectrum of the collective modes still displays a notable jump and near-degeneracy of modes as the inner boundary emerges.",
"Figure: (Color online) Normalized density deviation profiles|δn(r)||\\delta n (r)| along the radial direction of surface collectivemodes (ℓ=20\\ell = 20) in filled (a,b) and hollow (c,d) condensateswith Δ/R 2 =0.45\\Delta /R^2 = 0.45 and 0.55, respectively, for varyingnumber of radial nodes ν\\nu , as denoted.",
"The density deviationprofile and radial node (circled) structure of ν=1\\nu =1 in thefilled case resembles that of ν=2\\nu =2 and ν=3\\nu =3 in the hollowcase close to the outer and inner boundaries, respectively.",
"In thebarely hollow situation shown, the centrifugal barrier given byℓ(ℓ+1)n eq \\ell (\\ell +1) n_{\\rm {eq}} in Eq.",
"() – andrepresented by the dotted line (in arbitrary units) – is loweredso as to distribute the remaining nodes of the ν=2\\nu =2 andν=3\\nu =3 to the other surface as small oscillations [insets in(c,d)].To corroborate our description of the frequency restructuring across the hollowing transition, in Fig.",
"REF we present the density deviations and radial node redistribution in the illustrative example of surface modes having $\\ell =20$ and $\\nu =1,2$ and 3 on either side of the topological transition.",
"Comparing Figs.",
"REF (a), (c) and (d), it is clear that after the emergence of the condensate's inner boundary, both modes with total number of radial nodes $\\nu =2$ and $\\nu =3$ bear similarity, in density deviations, to the mode with $\\nu =1$ radial nodes before this boundary surface was available.",
"Additional nodes for $\\nu =2$ and $\\nu =3$ collective modes are present but appear as radial oscillations of small amplitude on a boundary surface opposite to the one with the most prominent nodal structure.",
"After the hollowing change, the overall oscillation frequencies of the $\\nu =2$ and $\\nu =3$ collective modes are accordingly dominated by a single node associated with high amplitude radial motion.",
"As this motion has the highest energetic cost, these frequencies are nearly equivalent to the oscillation frequency of the collective mode with a single radial node in the fully filled spherical condensate.",
"We emphasize that this is a feature that only occurs once the system is hollow, as demonstrated by a comparison of Figs.",
"REF (a) and (b)—collective modes with $\\nu =1$ and $\\nu =2$ radial nodes have rather different radial profiles for a fully filled spherical BEC.",
"Extrapolating to more general surface modes, this redistribution of nodes explains the jump in the frequency spectrum shown in Fig.",
"REF where collective modes with total $2\\nu $ and $2\\nu +1$ radial nodes after the hollowing transition smoothly continue the spectrum of the $\\nu $ surface mode of the BEC prior to hollowing.",
"Figure: (Color online) Oscillation frequencies of the lowestthree breathing (ℓ=0\\ell =0) collective modes ν=1,2,3\\nu =1,2,3 (curvesfrom bottom to top) as Δ\\Delta is varied from the filled-sphereto the thin-shell limit at a constant chemical potential, in theThomas-Fermi approximation with constant RR (main) and using thenumerically exact equilibrium densities with constant NN (inset).The line through Δ=0.5R 2 \\Delta =0.5 R^2 (or the shaded region in theinset) demarcates the filled and hollow regions and the schematicsalong the bottom show the associated density deviations, for (fromleft to right) Δ/R 2 =0\\Delta /R^2 =0, 0.450.45, 0.550.55, and 0.80.8.In Fig.",
"REF , to exhibit the behavior of the breathing modes, we show the results of the direct numerical solution of Eq.",
"(REF ) for the case $\\ell =0$ as the parameter $\\Delta $ is varied.",
"We note the sharp depression of the frequencies at the transition point and their otherwise smooth evolution to the solid-sphere and thin-shell limits at $\\Delta /R^2=0$ and 1, respectively.",
"Additionally, we note that the density deviation for these spherically symmetric modes, as shown along the bottom of Fig.",
"REF , displays confinement to the inner boundary at the transition from filled to hollow.",
"As with the surface modes, we expect this sudden transition to be replaced by a smoother crossover in a realistic system in which the condensate's boundaries are not sharp.",
"In the inset to Fig.",
"REF , we show the result of solving the hydrodynamic equation for the $\\ell =0$ collective modes using the numerical solution to the time-independent GP equation for the equilibrium density, $n_{\\rm {eq}}$ .",
"While in this more realistic situation the transition is gradual, as indicated by the shaded region in the inset of Fig.",
"REF , the characteristic decrease in frequency from the filled sphere through the transition point clearly persists, indicating that the hollowing at the center of the condensate can be observed through the spectrum of the system's breathing modes.",
"Additionally, since the decrease in central density here occurs less sharply than when the Thomas-Fermi approximation is used, the frequency curves in the inset of Fig.",
"REF decrease to distinct values at the hollowing transition, in contrast to their near-degeneracy in the Thomas-Fermi approximation.",
"To summarize, we have explored the physics of a BEC undergoing a topological change from a filled sphere to a hollow shell and shown that the collective modes are powerful indicators of such a transition.",
"In particular, spherically symmetric breathing modes show a crossover between the fully filled and thin shell limits at the hollowing-out point where the associated frequency spectrum exhibits a tell-tale dip.",
"High-angular-momentum surface modes show marked sensitivity to the appearance of a new boundary as well.",
"We now turn to the analogous lower-dimensional BEC geometry of a disk hollowing out into an annulus, effectively represented by a toroidal geometry.",
"Both structures have been well-studied theoretically [50], [51] and experimentally [6], [7].",
"The transition regime between the two topologies has yet to be studied and is potentially experimentally more tracatable than the three-dimensional spherical case.",
"We have performed preliminary collective mode spectral analyses for the disk-annulus system using the same techniques as those presented here.",
"Our main results for the spherical case hold equally well for the two dimensional system, given the common physics (vanishing of central density at the hollowing out transition and so on).",
"Specifically, the breathing modes show a dip and surface modes a reconfiguration of radial nodes in their frequency spectra around the topological transition from a disk to an annulus.",
"While the shell and the annulus topologies host similar collective modes when their thickness is small [50], the collective mode frequencies in the overall hollow regimes differ.",
"Moreover, the collective mode spectra cannot distinguish the different topological measures of the BEC's shape such as the fundamental group for the disk-annulus transition (determined by whether or not a loop can be shrunk to a point) or the second homotopy group for the sphere-shell transition (determined by whether a spherical surface can be shrunk to a point).",
"However, we note that a fundamental difference exists between the two cases, making the study of collective mode spectra more pressing in the three-dimensional case: in two dimensions, the presence of a hollow inner region can be imaged (and visualized) directly by absorption imaging techniques.",
"However, for the spherical geometry, direct imaging of the hollow region is obstructed by the presence of a surrounding condensate in all three dimensions, rendering collective mode spectra a powerful, non-destructive probe of the hollowing transition.",
"Finally, the CAL trap experiment is expected to pioneer the realization of condensate shells thus adding to the growing interest in BECs in microgravity conditions in space [58] and Earth-based environments [23].",
"The challenge has been that under typical terrestrial experimental conditions, gravitational sag causes heavy depletion at the shell's apex and a pooling of atoms at its bottom.",
"Excluding gravitational effects, we estimate that for a BEC of $\\sim 10^5$ $^{87}$ Rb atoms forming a cloud of $10~\\mu $ m in a harmonic trap of bare frequency 500 Hz, in the transition from the filled sphere to the thin shell limit the lowest breathing (surface) mode would evolve from about 1 (4) kHz to 0.5 kHz, which are in an accessible regime and in a large enough range to probe our predictions.",
"We predict the decrease in the collective mode frequency at the hollowing point, compared to the oscillation frequency of the same mode in the fully filled spherical BEC to be rather prominent, on the order of $50\\%$ or more, for all low-lying breathing (and low-$\\ell $ ) modes.",
"Somewhat higher modes, such as $\\nu =3$ , are suitable for for experimental detection of the diminishing of collective mode frequency at the hollowing point compared to the thin-shell limit as well.",
"For instance, the collective mode with $\\nu =3$ shows a $20\\%$ change between these two regimes, which makes it a good candidate for a full observation of the non-monotonicity of the collective mode frequency spectrum at the hollowing point.",
"Further work on BEC shells from collective mode behavior, expansion and time-of-flight, and the nature of vortices in this new geometry is in order.",
"These studies are relevant to situations ranging from the microscopic scale of the CAL trap experiments to the proposed existence of BEC shells in stellar objects and would deepen our insight into topological changes in quantum systems.",
"We thank Nathan Lundblad and Michael Stone for illuminating discussions.",
"KS acknowledges support by ARO (W911NF-12-1-0334), AFOSR (FA9550-13-1-0045), NSF (PHY-1505496), and Texas Advanced Computing Center (TACC).",
"CL acknowledges support by the National Science Foundation under award DMR-1243574.",
"KP, SV, and CL acknowledge support by NASA (SUB JPL 1553869 and 1553885).",
"CL and SV thank the KITP for hospitality."
]
] | 1612.05809 |
[
[
"Interpretation of the BRITE oscillation data of the hybrid pulsator\n $\\nu$ Eridani: a call for the modification of stellar opacities"
],
[
"Abstract The analysis of the BRITE oscillation spectrum of the main sequence early B-type star $\\nu$ Eridani is presented.",
"Only models with the modified mean opacity profile can account for the observed frequency ranges as well as for the values of some individual frequencies.",
"The number of the $\\kappa$ modified seismic models is constrained by the nonadiabatic parameter $f$, which is very sensitive to the opacity changes in the subphotospheric layers where the pulsations are driven.",
"We present an example of the model that satisfies all the above conditions.",
"It seems that the OPLIB opacities are preferred over those from the OPAL and OP projects.",
"Moreover, we discuss additional consequences of the opacity modification, namely, an enhancement of the efficiency of convection in the Z-bump as well as an occurrence of close radial modes which is a kind of avoided-crossing phenomenon common for nonradial modes in standard main sequence models."
],
[
"Introduction",
"One of the main ingredients in stellar modelling are opacity data which affect energy transport and the structure of a star.",
"In addition, the values of opacities determine conditions for excitation of heat-driven pulsations as observed, e.g., in main-sequence or classical pulsators.",
"The profound consequences that can result from the use of incorrect opacities was demonstrated some quarter of a century ago when the new data were computed by the two independent teams: OPAL (Iglesias, Rogers & Wilson 1992, Rogers & Iglesias 1992) and OP (Seaton 1993, Seaton et al.",
"1994).",
"Since then, these opacity data were updated several times (Iglesias & Rogers 1996, Seaton 2005), but the main feature, i.e., the new local maximum of the Rosseland mean opacity, $\\kappa $ , at the temperature $T\\approx 200~000$ K, was identified in their first release.",
"This new opacity bump, called the metal or Z-bump, made possible, e.g., to explain pulsations of B-type main sequence stars, decrease the mass discrepancy in classical Cepheids and to improve the standard solar model (Rogers & Iglesias 1994).",
"Recently, the Los Alamos opacity database (OPLIB) was also updated (Colgan et al.",
"2015, 2016) and made publicly available (http://aphysics2.lanl.gov/cgi-bin/opacrun/astro.pl).",
"Calculations with the OPLIB data yielded wider instability strips for pulsations of $\\beta $ Cep and SPB type (Walczak et al.",
"2015).",
"The most recent laboratory measurements at solar interior temperatures (Bailey et al.",
"2015) indicate that the predicted Rosseland mean opacities for iron are underestimated by about 75%.",
"Despite various improvements in the calculation of stellar opacities, there are many indications that something is still missing in these data and/or has not been correctly included.",
"One example is the B-type main sequence pulsators, which exhibit both pressure and high-order gravity modes, e.g., $\\nu $ Eridani (Handler et al.",
"2004, Aerts et al.",
"2004), 12 Lacertae (Handler et al.",
"2006), $\\gamma $ Pegasi (Handler et al.",
"2009).",
"There are also bags of such pulsators identified from the space data from Kepler and CoRoT missions (e.g., Degroote et al.",
"2009, Balona et al.",
"2011, 2015a).",
"However, so far none of the pulsational models can account for their oscillation spectra.",
"The higher iron opacity has been investigated very recently by theoretical computations of Nahar & Pradhan (2016), but their results regarding a potential increase in total opacity have been criticized (Blancard et al.",
"2016).",
"Larger opacities could be a breakthrough in solar modelling and push in the right direction the seismic studies of massive stars.",
"The increase of the Z-bump opacity, mostly dominated by iron, was a first guess to explain the low frequencies detected in $\\nu $ Eri (Pamyatnykh, Handler & Dziembowski 2004).",
"Recently, the instability strips for B-type main sequence models with enhancement by 75 % iron and nickel opacity were published by Moravveji (2016).",
"In this paper, we make an attempt to analyse once again this well-known hybrid pulsator, $\\nu $ Eri, which was observed by the BRITE-Constellation (Weiss, Ruciński, Moffat et al.",
"2014).",
"Here, we base our studies on the pulsational frequencies extracted by Handler et al.",
"(2016) from the BRITE light curves.",
"Our goal is to modify the profile of the mean opacity to reproduce both the observed range of frequencies and the values of the frequencies themselves.",
"Moreover, in order to control the opacity changes and make our results more plausible, we have imposed a requirement for the relative amplitude of the bolometric flux variations, the so-called parameter $f$ (Cugier, Dziembowski, Pamyatnykh 1994, Daszyńska-Daszkiewicz et al.",
"2002).",
"Namely, we demand, in addition, that the empirical value of $f$ for the dominant radial mode is reproduced by a model satisfying the above conditions.",
"The use of the parameter $f$ significantly limits the modification of opacities because this parameter is very sensitive to the structure of subphotospheric layers where the pulsation driving occurs.",
"In Sect.",
"2, we recall the observed properties of pulsations of $\\nu $ Eri and compare them with the standard pulsational models.",
"By the standard models we mean the models computed with the available opacity data: OPLIB, OPAL and OP.",
"Sect.",
"3 contains results of our modelling extended by the modification of the mean opacity profile and in Sect.",
"4 we reduce the number of such seismic models by taking into account the parameter $f$ .",
"The effect of convection in the vicinity of the metal (or $Z-$ )bump on pulsational properties of models with masses typical for $\\nu $ Eri is discussed in Sect. 5.",
"A kind of avoided-crossing phenomenon for radial modes identified in models with the modified $\\kappa $ profile is presented in Sect. 6.",
"We end with Conclusions and perspectives for future works."
],
[
"Oscillations of $\\nu $ Eri and the standard pulsational models",
"The early B-type star $\\nu $ Eri (B2III, HD 29248) is one of the most studied hybrid pulsators in the last fifteen years.",
"These intensive studies began with dedicated photometric and spectroscopic multisite campaigns carried out in 2003-2005 (Handler et al.",
"2004, Jerzykiewicz et al.",
"2005, Aerts et al.",
"2004, de Ridder et al.",
"2004).",
"The analysis of these data enriched the known oscillation spectrum by new frequency peaks, that complemented the two known $\\ell =1$ triplets, as well as revealed the existence of high order g modes associated with low frequencies.",
"Subsequently, many theoretical studies have been devoted to finding the best seismic models that yield constraints on overshooting from the convective core, internal rotation and opacities (Pamyatnykh, Handler & Dziembowski 2004, Ausseloos et al.",
"2004, Daszyńska-Daszkiewicz, Dziembowski, Pamyatnykh 2005, Dziembowski & Pamyatnykh 2008, Suarez et al.",
"2009, Daszyńska-Daszkiewicz & Walczak 2010).",
"This star is a slow rotator with $V_{\\rm rot}=6$ km/s (Pamyatnykh, Handler & Dziembowski 2004) but some suggestions about non-rigid rotation have been made (Dziembowski & Pamyatnykh 2008, Suarez et al.",
"2009).",
"The BRITE project provided an opportunity, for the first time, to register the light variations of this bright star ($V=3.92$ mag) from space.",
"Handler et al.",
"(2016) extracted from these data 17 frequencies: 10 in the high frequency range (p modes) and 7 in the low frequency range (g modes).",
"Most of these peaks agree with the values derived from the previous photometric and spectroscopic observations.",
"However, there are a few differences.",
"Firstly, five more g-mode frequencies have been detected.",
"Interestingly, the \"old\" frequency $\\nu =0.61440$ d$^{-1}$ is missing in this low frequency \"forest\".",
"Moreover, two frequencies, one component of the $\\ell =1,~p_1$ triplet around 6.22 d$^{-1}$ and the peak around $\\nu =6.73$ d$^{-1}$ , have not been found.",
"These two frequencies had the lowest photometric amplitudes in the last photometric campaigns (cf.",
"Table 6 in Jerzykiewicz et al.",
"2005).",
"From a theoretical point of view, such a situation is quite plausible because the e-folding time for the amplitude growth of some pulsational modes in main sequence models may be quite short.",
"For the quadrupole g mode with $\\nu \\approx 0.61$ d$^{-1}$ , we obtained an e-folding time of about 15 years and for the dipole and quadrupole p modes with $\\nu \\approx 6.22$ and $6.73$ d$^{-1}$ , this time is of the order of several decades.",
"These estimates, besides the detection threshold, can explain the appearance and disappearance of some frequency peaks.",
"It is worth mentioning that the Strömgren $u$ amplitude of the high order g mode $\\nu \\approx 0.433$ d$^{-1}$ has dropped from about 5.5 mmag (Handler et al.",
"2004) to 3.0 mmag (Handler et al.",
"2016), ie., by a factor of almost two within 12 years.",
"In Fig.",
"1, we show the oscillation spectrum of $\\nu $ Eri as obtained by Handler et al.",
"(2016) from the BRITE light curves (the bottom panel) as well as, for comparison, the oscillation spectrum from the 2002-2004 photometric campaigns (the top panel) derived by Jerzykiewicz et al.",
"(2005).",
"Figure: The oscillation spectra of ν\\nu Eri obtained from the 2002-2004 photometric campaigns by Jerzykiewicz et al.",
"(2005) (the top panel)and from the BRITE space photometry by Handler et al.",
"(2016) (the bottom panel).",
"To highlight the peaks,different scales are used for the low- and high-frequency peaks, and the X-axis is broken in the rangeof 1.2-5.2 [d -1 ^{-1}].As a first step, we confront the observed frequency range with the theoretical results obtained from models computed with the standard opacity data commonly used in pulsational computations: OPAL (Iglesias & Rogers 1996), OP (Seaton 2005) and the recently released OPLIB library (Colgan et al.",
"2015, 2016).",
"In all cases, the solar chemical mixture was adopted (Asplund et al.",
"2009).",
"There are some differences in the run of the mean opacity profiles, $\\kappa (T)$ , and its temperature derivative, $\\kappa _T(T)$ , between these three sources of the opacity data.",
"A comparison of them for a typical $\\beta $ Cep model was recently presented by Walczak et al.",
"(2015) (see their Fig.",
"1 and 2).",
"In Fig.",
"2, we plot the normalized instability parameter, $\\eta $ , as a function of the mode frequency for the three models suitable for $\\nu $ Eri computed with the three opacity data sets: OPLIB, OPAL, OP.",
"The unstable modes have positive values of $\\eta $ .",
"There are shown modes with the spherical harmonic degree up to $\\ell =2$ .",
"The effective temperature and luminosity ($\\log T_{\\rm eff},~\\log L/L_\\odot $ ) of these models are consistent with the observed error box determined with the new parallax $\\pi =4.83\\pm 0.19$ mas (van Leeuwen 2007): $\\log T_{\\rm eff}=4.346\\pm 0.014$ , $\\log L/L_\\odot =3.886\\pm 0.044$ .",
"Here, we adopted a mass $M=9.5M_{\\odot }$ and metallicity $Z=0.015$ .",
"The values of ($\\log T_{\\rm eff},~\\log L/L_\\odot $ ) differ only slightly between the models shown in Fig.",
"2 and their approximate values are: $\\log T_{\\rm eff}\\approx 4.343$ , $\\log L/L_\\odot \\approx 3.92$ .",
"All models reproduce the radial fundamental mode $\\ell =0,~p_1$ and the centroid of the dipole mode $\\ell =1,~g_1$ corresponding to the frequencies $\\nu =5.76326$ d$^{-1}$ and $\\nu =5.63725$ d$^{-1}$ , respectively.",
"A small amount of core overshooting was needed to adjust the dipole mode frequency, which was chosen from the range $\\alpha _{\\rm ov}\\in (0.07-0.09)$ .",
"Fig.",
"2 recalls the old problem of mode excitation in the low frequency range, i.e., high order g modes, as well as in the highest frequency range for $\\nu \\gtrsim 7.5$ d$^{-1}$ (e.g.",
"Pamyatnykh, Handler & Dziembowski, 2004).",
"The widest instability in the p mode range is in the OPLIB models, reaching $\\nu \\approx 8$ d$^{-1}$ , whereas for the OP models we get the highest values of $\\eta $ in the range of high order g modes.",
"As we have checked, the changes of various parameters (e.g., $M,~Z,~X$ ) within a range allowed by the observational error box do not eliminate these shortcomings.",
"Increasing the total metallicity, $Z$ , enhanced an overall instability in both local maxima of $\\eta $ .",
"The effect of changing the mass is as follows: for higher masses we get the higher instability for the highest frequency p modes, e.g., changing the mass from 9.0 to 10.0 $M_\\odot $ extends the instability by about 1 d$^{-1}$ towards higher frequencies.",
"On the other hand, for the higher mass models we get lower instability for low frequency g modes.",
"Thus, we can say that in the allowed range of parameters there are no models that can account for the whole observed range of frequencies detected in the $\\nu $ Eri light variations.",
"In order to overcome this problem in the next section we try to modify the opacity profile, $\\kappa (T)$ , and to find models that reproduce both the observed range of frequencies and the values of some individual frequencies.",
"Figure: The normalized instability parameter, η\\eta , for representative models of ν\\nu Eri, computed withthe three sources of opacity data: OPLIB, OPAL and OP.",
"All models have Z=0.015Z=0.015, M=9.5M ⊙ M=9.5~M_\\odot ,logT eff ≈4.343\\log T_{\\rm eff}\\approx 4.343, logL/L ⊙ ≈3.92\\log L/L_\\odot \\approx 3.92 and the overshooting parameter from the rangeα ov ∈(0.07-0.09)\\alpha _{\\rm ov}\\in (0.07-0.09)."
],
[
"Models with modified opacities",
"In this section we change some amount of opacity at the depths (expressed in $\\log T$ ) that can affect the driving of pulsations in the stellar models of $\\nu $ Eri.",
"Of course, these $\\kappa $ modifications will affect also the values of eigenfrequencies.",
"The opacity profile is modified according to the formula $\\kappa (T)=\\kappa _0(T) \\left[1+\\sum _{i=1}^N b_i \\cdot \\exp \\left( -\\frac{(\\log T-\\log T_{0,i})^2}{a_i^2}\\right) \\right],$ where $\\kappa _0(T)$ is the unchanged opacity profile and $(a,~b,~T_0)$ are parameters of a Gaussian describing the width, height and position of the maximum, respectively.",
"Thus, for a fixed depth, $T_0$ , we enhance or reduce the opacity by changing $(a,~b)$ .",
"We adopted the Gaussian because the known local maxima of the mean opacity profile are well represented by this function.",
"Examples of modifications of the OPLIB values of $\\kappa (\\log T)$ are depicted in the top panels of Figs.",
"3 and 4 for a model with the parameters $M = 9.5~M_{}$ , $\\log T_{\\rm eff} = 4.3399$ , $\\log L/L_{} = 3.917$ , metallicity $Z = 0.015$ and core overshooting $\\alpha _{\\rm ov} = 0.07$ .",
"The $\\kappa $ derivative with respect to temperature, $\\kappa _T=\\partial \\log \\kappa /\\partial \\log T$ , is also shown, with its values given on the right-hand Y-axis.",
"In the first case, we increased opacities around $\\log T_0=5.3$ and 5.46 by 100% (i.e.",
"b=1), whereas in the second case (the top panel of Fig.",
"4), we added also 100% opacity around $\\log T_0=5.06$ .",
"Within the Z-bump, we have at $\\log T_0=5.3$ the maximum contribution of iron to the opacity whereas at $\\log T_0=5.46$ the maximum contribution of nickel occurs (e.g., Salmon et al.",
"2012).",
"Adding 100% of the opacity only at $\\log T_0=5.3$ increases the instability of both p- and high order g-mode frequencies whereas an additional increase of $\\kappa $ at the deeper temperature $\\log T_0=5.46$ raises the parameter $\\eta $ mostly for the low frequency modes.",
"The opacity changes at $\\log T_0=5.06$ are motivated by works of Cugier (2012, 2014), who identified the new opacity bump around this temperature in the Kurucz atmosphere models.",
"This bump was suggested, for example, as a possible cause of excitation of low frequency g modes in $\\delta $ Scuti stars as detected in the Kepler data (Balona, Daszyńska-Daszkiewicz & Pamyatnykh 2015).",
"The consequences of such $\\kappa $ modifications on pulsational properties are demonstrated in the middle panels of Figs.",
"3 and 4.",
"As an example we show the normalized differential work integral for the radial second overtone mode ($\\ell =0$ , p$_3$ ).",
"In the model computed with the standard OPLIB data this mode is stable.",
"Increasing the opacity at $\\log T=5.3$ and 5.46 makes this radial mode unstable whereas adding the third bump at $\\log T=5.06$ stabilizes it again.",
"In Fig.",
"3, we compare also the work integral for the high order quadrupole mode $\\ell =2$ , g$_{16}$ for which the parameter $\\eta $ reaches the maximum.",
"In the model computed with the standard $\\kappa $ profile this mode is stable whereas in the model computed with the modified $\\kappa $ profile at $\\log T=5.3$ and 5.46 it becomes unstable.",
"The bottom panels of Figs.",
"3 and 4 show the run of the normalized instability parameter, $\\eta $ , for the two considered cases.",
"As one can see, an enhancement of the opacity at the depths $\\log T_0=5.3$ and 5.46 causes an increase of the pulsational instability both for low and high frequency modes.",
"In the low frequency range, the quadrupole modes become unstable and dipole modes are not far from the instability.",
"For the high frequencies, the instability range becomes much wider than the observed frequency range.",
"If we add the third bump at the depth $\\log T_0=5.06$ (Fig.",
"4), the increase of $\\eta $ for the highest frequency modes is smaller whereas for low frequency modes the parameter $\\eta $ is almost unchanged.",
"Figure: The upper panel: the run of the standard (dotted line) and modified (solid line) Rosseland mean OPLIB opacity(the left hand Y-axis) and its temperature derivative (the right hand Y-axis) for the model: M=9.5M ⊙ M=9.5M_{\\odot }, Z=0.015Z=0.015,α ov =0.07\\alpha _{\\rm {ov}}=0.07, logT eff =4.3399\\log {T_{\\rm eff}}=4.3399, logL/L ⊙ =3.917\\log {L/L_{\\odot }}=3.917.",
"The κ\\kappa profile was modified by addingopacity at logT 0 =5.3\\log T_0=5.3 and 5.46 and the parameters of this modification are given in the legend.",
"The middle panels:a comparison of the differential work integral for the radial second overtone mode, ℓ=0,p 3 \\ell =0,~p_3, and high overtonequadrupole g mode, ℓ=2,g 16 \\ell =2,~g_{16}, in models computed with the standard and modified opacity profile.",
"The bottom panel:a comparison of the corresponding normalized instability parameter, η\\eta .Figure: A similar figure as Fig.",
"3 but the κ\\kappa profile was additionally modified at logT 0 =5.065\\log T_0=5.065.",
"The differentialwork integral for the mode ℓ=2,g 16 \\ell =2,~g_{16} is not shown because it looks very similar to that shown in Fig.",
"3.Then, to reproduce the observed frequency ranges as well as the values of some frequency peaks, we determined the corrections to $\\kappa (T)$ with the following steps $\\Delta a=0.001$ , $\\Delta b=0.05$ and $\\Delta T_0=0.005$ in the range $\\log T\\in (5.0 - 5.5)$ .",
"For each opacity database, we found many models that reproduce these observed features.",
"In Fig.",
"5, we give examples of such models for the three used opacity data sets: OPLIB (the top panel), OPAL (the middle panel) and OP (the bottom panel).",
"The parameters of the models are given in Table 1 and modifications of the opacity profile are as follows: OPLIB: $\\log T_{0,1}=5.06,~a_1=0.071,~b_1=0.3;\\\\~~~~~~~~~~~~~~~~~~~~~~~\\log T_{0,2}=5.46,~a_2=0.224,~b_2=1.5,$ OPAL: $\\log T_{0,1}=5.30,~a_1=0.082,~b_1=0.5;\\\\~~~~~~~~~~~~~~~~~~~~~~\\log T_{0,2}=5.46,~a_2=0.082,~b_2=1.5,$ OP: $\\log T_{0,1}=5.20,~a_1=0.071,~b_1=0.5;\\\\~~~~~~~~~~~~~~~~\\log T_{0,2}=5.46,~a_2=0.071,~b_2=1.0.$ Figure: The normalized instability parameter, η\\eta , as a function of mode frequency for models that roughlyreproduce some observed frequencies and the range of instability for both p and g modes of ν\\nu Eri.",
"For each type of opacity data(OPLIB, OPAL, OP), there is shown one model of this kind.",
"The OPLIB model:M=9.5M ⊙ M=9.5M_\\odot , logT eff =4.3367,logL/L ⊙ =3.916,Z=0.015,α ov =0.07\\log T_{\\rm eff}=4.3367,~\\log L/L_\\odot =3.916,~Z=0.015, ~\\alpha _{\\rm ov}=0.07;the OPAL model: M=9.2M ⊙ ,logT eff =4.3348,logL/L ⊙ =3.886,Z=0.015,α ov =0.15M=9.2~M_\\odot , ~\\log T_{\\rm eff}=4.3348,~\\log L/L_\\odot =3.886,~Z=0.015,~\\alpha _{\\rm ov}=0.15,the OP model: M=9.6M ⊙ ,logT eff =4.3358,logL/L ⊙ =3.890,Z=0.0185,α ov =0.0M=9.6M_\\odot ,~\\log T_{\\rm eff}=4.3358,~\\log L/L_\\odot =3.890,~Z=0.0185,~\\alpha _{\\rm ov}=0.0.The initial value of hydrogen was X 0 =0.7X_0=0.7 in each case.",
"The parameters (T 0 ,a,bT_0,~a,~b) of the modified opacity profile aregiven in the legend.As one can see, all of these models have unstable modes in the high frequency range (5-8) d$^{-1}$ which perfectly agrees with observations.",
"Moreover, the theoretical frequencies of the highest amplitude p modes agree with the observed values to the second decimal place, in line with mode identification (e.g., Daszyńska-Daszkiewicz et al.",
"2005, Daszyńska-Daszkiewicz & Walczak 2010.)",
"The instability range of high order g modes is worse reproduced and all modes with frequencies below 0.35 d$^{-1}$ do not have their theoretical counterparts.",
"There are only a few unstable dipole modes.",
"Quadrupole modes are more unstable but they are shifted to higher frequencies.",
"This deficiency can be partially explained by rotational splitting, in particular, if the core was really spinning faster as suggested by Pamyatnykh, Handler & Dziembowski (2004)."
],
[
"How to constrain the modifications of stellar opacities?",
"As we have mentioned at the end of the previous section, it is possible to find many models that can account simultaneously for the observed frequency ranges and roughly fit the values of some frequencies.",
"Thus, the question is: which modification of the opacity profile is more likely?",
"First, one can think about drawing some conclusions from chemical compositions determined from high-resolution spectra.",
"Such analysis by Morel et al (2006) and, more recently, by Nieva & Przybilla (2012) points to normal (solar-like) abundances for $\\nu $ Eri.",
"However, these photospheric abundances do not necessarily reflect the subphotospheric values.",
"Moreover, the values of opacities do not depend only on the abundance of individual elements but also on how they are computed (e.g., on the number of fine structure energy levels per ion taken into account).",
"Thus, we need an observable that would be sensitive to the opacity of the subphotospheric layers near the Z-bump.",
"Such an indicator is the relative amplitude of radiative flux perturbation at the level of the photosphere, which is called the parameter $f$ .",
"The value of $f$ is complex and results from linear non-adiabatic computations of stellar pulsations.",
"This asteroseismic probe was introduced by Daszyńska-Daszkiewicz, Dziembowski & Pamyatnykh (2003) and in the case of B-type pulsators, it is strongly sensitive to the metallicity and opacity data (Daszyńska-Daszkiewicz, Dziembowski & Pamyatnykh 2005).",
"The empirical counterparts of $f$ are derived from multicolour photometry and radial velocity data by means of the method proposed by Daszyńska-Daszkiewicz, Dziembowski & Pamyatnykh (2003).",
"An interesting result from the last studies of the parameter $f$ for $\\nu $ Eri is a preference of the OPAL tables by p modes and a better agreement with the OP opacity models for high order g modes (Daszyńska-Daszkiewicz & Walczak 2010).",
"Here, we will make use of the empirical values of $f$ for the radial fundamental mode $\\nu =5.76324$ d$^{-1}$ derived from the Strömgren amplitudes and phases and radial velocity variations from the last multi-site campaigns.",
"It should be mentioned that the empirical values of $f$ weakly depend on the stellar parameters in the allowed ranges whereas they do depend on model atmospheres.",
"In Table 1, we give the parameters of the seismic models found in the previous section and depicted in Fig. 5.",
"There are also provided the theoretical and empirical values of the absolute value of the $f$ parameter, $|f|$ , and the phase lag, $\\Psi ={\\rm arg}(f)-180^\\circ $ , for the radial fundamental mode of $\\nu $ Eri.",
"The value of $\\Psi $ gives the phase shift between the maximum temperature and the minimum radius.",
"The corresponding empirical values of $|f|$ and $\\Psi $ were derived by adopting both LTE models (Kurucz 2004) and NLTE models (Lanz & Hubeny 2007) atmospheres.",
"Table: Parameters of the seismic models of ν\\nu Eri with the modified opacity profile, shown in Fig.",
"5 and Fig.",
"6 (bestOPLIB).Columns from left to right are: opacity data, mass, M/M ⊙ M/M_\\odot , effective temperature, logT eff \\log T_{\\rm {eff}}, luminosity,logL/L ⊙ \\log {L/L_{\\odot }}, metallicity, ZZ, overshooting parameter, α ov \\alpha _{\\rm {ov}}, the theoretical values of (|f|,Ψ)(|f|,~\\Psi ) forthe radial fundamental mode and their empirical counterparts.",
"The latter were computed with both the LTE and NLTE atmosphere models.As one can see, there are significant differences in the values of $(|f|,~\\Psi )$ between models computed with different opacity data as well as a large disagreement with their empirical counterparts.",
"Thus now the aim is to find the $\\kappa -$ modified models that will reproduce additionally the empirical value of $f$ of the dominant mode.",
"It appeared that fitting the parameter $f$ significantly reduces the number of seismic models.",
"In fact, it is quite hard to find the models that simultaneously reproduce the observed frequencies, the range of instability for both p and g modes and the empirical value of the nonadiabatic parameter $f$ .",
"Figure: The OPLIB seismic model, which reproduces the observed frequencies of p modes, the range of instability forboth p and g modes, and the empirical value of the nonadiabatic parameter ff for the dominant frequency correspondingto the radial fundamental mode.",
"The parameters of the model are: M=9.0M ⊙ M=9.0~M_\\odot , logT eff =4.3314\\log T_{\\rm eff}=4.3314, Z=0.015Z=0.015, X 0 =0.7X_0=0.7,α ov =0.163\\alpha _{\\rm ov}=0.163.",
"The values of temperatures and parameters (a,b)(a,~b) at which the κ\\kappa profile was modified are givenin the legend.Figure: The run of the modified Rosseland mean opacity (the left-hand Y-axis) and its temperature derivative(the right-hand Y-axis) for the model presented in Fig.",
"6.Nevertheless, we succeeded in finding such a model with the OPLIB data and in Fig.",
"6 we depict its instability parameter $\\eta $ as a function of the frequency.",
"The model has the following parameters: $M= 9.0~M_\\odot $ , $\\log T_{\\rm eff}=4.3314$ , $\\log L/L_\\odot = 3.859$ , $Z=0.015$ , $\\alpha _{\\rm ov}=0.163$ , and the modification of the opacity profile is: bestOPLIB: $\\log T_{0,1}=5.065,~a_1=0.447,~b_1=-0.60;\\\\~~~~~~~~~~~~~~~~~~~~~~~\\log T_{0,2}=5.22,~a_1=0.088,~b_1=0.35;\\\\~~~~~~~~~~~~~~~~~~~~~~~\\log T_{0,3}=5.47,~a_2=0.061,~b_1=2.20$ .",
"Our best model has a greatly reduced value of the mean opacity near the temperature $\\log {T}=5.065$ .",
"This was necessary to fit the empirical values of the parameter $f$ .",
"On the other hand, the mean opacity has to be slightly increased near $\\log {T}=5.22$ and significantly increased near $\\log {T}=5.47$ in order to fulfill the instability requirement for p and g modes, respectively.",
"The modified $\\kappa $ profile together with its derivative with respect to temperature, $\\kappa _T$ , is shown in Fig. 7.",
"The consequences of this $\\kappa $ modification on the flux eigenfunction $f(\\log T)$ are large as can be judged from Fig.",
"8, where the absolute value, $|f|$ , and the phase lag, $\\Psi ={\\rm arg}(f)-180^\\circ $ , are plotted.",
"The photospheric values of the empirical $(|f|,~\\Psi )$ are marked with the short horizontal lines.",
"Both, the absolute value, $|f|$ , and the phase, $\\Psi $ , are significantly modified around the minimum values of the the opacity derivative, $\\kappa _T$ , with the proviso that $|f|$ reaches the maximum values at $\\log T\\approx 5.53$ where the derivative $\\kappa _T$ reaches the minimum, whereas the value of $\\Psi $ is the maximum at $\\log T\\approx 5.41$ where $\\kappa _T$ obtains the maximum.",
"It is worth mentioning that many models that reproduce the empirical parameter $f$ for the dominant frequency corresponding to the radial fundamental mode could not account for the instability of high order g modes.",
"This is because to excite the g modes one has to increase significantly the opacity which, in turn, results in increasing the theoretical value of the phase lag $\\Psi $ .",
"Despite such large modification of the opacity profile, the value coming out of the photosphere is not much changed and the parameter $f$ is mostly adjusted to the empirical value by reducing the opacity at $\\log T_{0,1}=5.065$ which does not affect the instability of g modes.",
"The theoretical values $|f|$ and $\\Psi $ of the model from Fig.",
"6 and the corresponding empirical values are given in the last line of Table 1.",
"As one can see, the theoretical and empirical values of $|f|$ are in excellent agreement if the NLTE model atmospheres are used.",
"With the LTE models, the agreement is achieved if the $2\\sigma $ error is allowed.",
"For the argument, $\\Psi $ , the compatibility is within the $3\\sigma $ error regardless of which model atmospheres are used.",
"Figure: The absolute value (the upper panel) and the phase lag (the bottom panel) of the flux eigenfunction f(logT)f(\\log T)for the model presented in Fig. 6.",
"The dashed blue horizontal line correspondsto the empirical values derived with the NLTE atmospheres."
],
[
"The effect of convection in the Z-bump",
"Despite the general considerations that convective transport is negligible in stellar models with masses corresponding to $\\beta $ Cep variable ($M\\approx 7-16~M_\\odot $ ), it can be important at the depth where the local maxima of the opacity occur.",
"As was shown by Cantiello et al.",
"(2009), the efficiency of the Z-bump convection increases with increasing metallicity, decreasing effective temperature and increasing total luminosity.",
"These three facts were confirmed by the observational data.",
"Moreover, fast rotation can enhance this subsurface convection (Maeder et al.",
"2008).",
"In the previous section we showed that the nonadiabatic parameter $f$ is a powerful probe of modification of the stellar opacities in the depth range $\\log T=5.0 - 5.5$ .",
"Because of this important diagnostic property, here we will check the sensitivity of the flux eigenfunction $f(\\log T)$ to the efficiency of convection in the Z-bump layer.",
"The convective efficiency is measured by the value of the mixing length parameter, $\\alpha _{\\rm MLT}$ .",
"We consider the modified OP opacity profile with the parameters: $\\log T_{0,1}=5.15,~a_1=0.071,~b_1=1.0;~~\\log T_{0,2}=5.46,~a_1=0.071,~b_1=1.0$ , and the two values of the MLT parameter: $\\alpha _{\\rm MLT}=0.5$ and 5.0.",
"This second large value of $\\alpha _{\\rm MLT}$ is chosen, firstly, to show the effect and, secondly, because such values of $\\alpha _{\\rm MLT}$ were considered at some depth of the solar convective zone.",
"The depth-dependence of $\\alpha _{\\rm MLT}$ was studied by, e.g., Schlattl et al.",
"(1997) and more recently by Magic, Weiss & Asplund (2015).",
"Figure: The run of the standard (dotted line) and modified (solid line) profiles of the Rosseland mean OP opacity(the left hand Y-axis) for the two values of the MLT parameter, α MLT =0.5\\alpha _{\\rm MLT}=0.5 and α MLT =5.0\\alpha _{\\rm MLT}=5.0.The corresponding values of the temperature derivatives of κ\\kappa are given on the right hand Y-axis.The parameter of the model are: Z=0.0185Z=0.0185, M=9.63M ⊙ M=9.63~M_\\odot , logT eff ≈4.334\\log T_{\\rm eff}\\approx 4.334, logL/L ⊙ ≈3.89\\log L/L_\\odot \\approx 3.89.In Fig.",
"9, we plot the standard and modified profiles of the Rosseland mean OP opacity (the left-hand Y-axis) for the two values of the MLT parameter, $\\alpha _{\\rm MLT}=0.5$ and $\\alpha _{\\rm MLT}=5.0$ .",
"The corresponding $\\kappa $ derivatives over temperature are depicted as well (the right-hand Y-axis).",
"The model parameters are: $Z=0.0185$ , $M=9.63~M_\\odot $ , $\\log T_{\\rm eff}\\approx 4.334$ , $\\log L/L_\\odot \\approx 3.89$ .",
"As one can see, the change of $\\alpha _{\\rm MLT}$ is only revealed near the local maxima of $\\kappa $ .",
"For the model with the modified opacity and $\\alpha _{\\rm MLT}=5.0$ , the efficiency of convection reaches about 45% in the vicinity of the Z-bump layer.",
"The effect of $\\alpha _{\\rm MLT}$ on the differential work integral is shown in Fig.",
"10 where its value is depicted for the radial fundamental mode, corresponding to $\\nu =5.7625$ d$^{-1}$ , and the high-overtone g mode, $\\ell =1, g_{14}$ , corresponding to $\\nu =0.5003$ d$^{-1}$ .",
"Although the effect of $\\alpha _{\\rm MLT}$ seems to be negligible, the result is that in the model with $\\alpha _{\\rm MLT}=0.5$ the radial fundamental mode is stable whereas in the model with $\\alpha _{\\rm MLT}=5.0$ it is unstable because of smaller damping at $\\log T\\approx 5.35$ and $\\log T\\approx 5.55$ .",
"In the case of the g mode, the damping around $\\log T=5.55$ is slightly reduced in the model with $\\alpha _{\\rm MLT}=5.0$ and the instability parameter, $\\eta $ , is a little larger.",
"Figure: The differential work integral for the radial fundamental mode (the top panel) and dipole gravity mode (the bottom panel)for the models with the modified κ\\kappa profile (cf.",
"Fig. 9).",
"The effect of the MLT parameter is illustrated.For α MLT =0.5\\alpha _{\\rm MLT}=0.5 the ℓ=0,p 1 \\ell =0,p_1 mode is stable whereas for α MLT =5.0\\alpha _{\\rm MLT}=5.0 it is excited.Figure: The effect of the MLT parameter on the flux eigenfunction f(logT)f(\\log T).",
"The upper and bottom panels show the absolutevalue and the phase of f(logT)f(\\log T), respectively.",
"The dotted line corresponds to f(logT)f(\\log T) in the model withthe standard OP opacities.",
"The short horizontal lines correspond to the empirical values derived with the NLTE atmospheres.How the value of $\\alpha _{\\rm MLT}$ affects the flux eigenfunction, $f(\\log T)$ , is presented in Fig. 11.",
"The dashed and solid lines correspond to the opacity-modified models with $\\alpha _{\\rm MLT}=0.5$ and 5.0, respectively.",
"The model with the standard opacity profile is plotted with the dotted line.",
"As one can see, the dependence of $f(\\log T)$ on the MLT parameter is important and the difference is larger than the observational error of the parameter $f$ .",
"The adjustment of the value of $\\alpha _{\\rm MLT}$ is beyond the scope of this paper but one has to keep in mind this fact when studying the properties of the Z-bump in the $\\beta $ Cep star models."
],
[
"Avoided crossing of radial modes",
"When studying models with the modified $\\kappa $ profile, we encountered the phenomenon occurring so far only for nonradial modes.",
"It appears that in models with a significant increase of opacity at certain temperatures, the radial modes can experience the avoided-crossing phenomenon, which in standard massive main sequence models occurs only for nonradial oscillations.",
"Till now the avoided crossing for radial modes was noticed in models of neutron stars (Gondek, Haensel & Zdunik 1997, Gondek & Zdunik 1999) and Cepheids (Buchler, Yecko & Kollath 1997).",
"Figure: Evolution of eigenfrequencies of radial modes in a model with M=11.3M M=11.3 M_{} from ZAMS to TAMS, computed with the standard OP opacities(the upper panel) and modified ones (the bottom panel) for the metallicity Z=0.0185Z=0.0185.",
"In the later case, the opacity was increased by a factor of 4at the depth logT=5.35\\log T=5.35.",
"Dotted vertical lines mark positions of the selected models before, during and after the avoidedcrossing of the fundamental mode and first overtone.In certain cases, an increase of the local opacity produces the two resonance cavities for radial modes and low order overtones can exchange their dynamic behaviour in stellar interiors.",
"An example is shown in Fig.",
"12, where the evolution of eigenfrequencies of the six radial overtones for models with $M=11.3 M_{}$ is shown.",
"The top and bottom panels correspond to the standard and modified models, respectively.",
"To emphasize the effect, in the modified model the OP opacities were increased by a factor of 4 around the depth $\\log T=5.35$ .",
"As one can see, at certain evolutionary stages the first and second overtone radial modes $(\\log T_{\\rm eff}\\approx 4.38)$ or the fundamental and first overtone radial modes $(\\log T_{\\rm eff}\\approx 4.34)$ can get close to each other.",
"The standard and modified $\\kappa $ profiles are depicted in the top panel of Fig. 13.",
"Note that maximum difference in the opacity between two models is a factor of about 2.3, not 4, and the position of this maximum is at $\\log T\\approx 5.25$ , not 5.35, as in the standard and modified opacity tables.",
"This is due to the fact that two evolutionary models differ one from another by distribution of both temperature and density in the interiors.",
"Therefore, the opacity coefficient is interpolated along different $\\log T$ - $\\log \\rho $ lines in the tables.",
"During the avoided crossing of radial modes, overtones exchange their oscillatory properties, in particular the distribution of the kinetic energy inside a model.",
"This can be seen from the lower panels of Fig.",
"13 where we show the run of the kinetic energy density as a function of the depth, $\\log T$ , for the three lowest radial modes, $\\ell =0, ~p_1,~p_2,~p_3$ , at three close evolutionary stages: before, during and after the avoided crossing of the fundamental mode and first overtone.",
"These three selected models are marked in Fig.",
"12 by vertical lines.",
"In this case after the avoided crossing the fundamental and first overtone modes exchange their dynamical behaviour in the interior: the distribution of the kinetic energy of fundamental mode is now very similar to that of the first overtone before the avoided crossing, and the distribution of the kinetic energy of the first overtone is very similar to that of the fundamental mode before the avoided crossing.",
"It should be mentioned that during the avoided crossing both modes have the same normalized kinetic energy.",
"At the closest approach the kinetic energy is nearly equally divided between the regions above and below $\\log T\\approx 5.7$ , which is the lower border of the potential barrier formed by the opacity increase around $\\log T=5.35$ .",
"Figure: The top panel: the run of the opacity and its temperature derivative in the models computed with the standardand modified κ\\kappa as described in the text.",
"The vertical line corresponds to logT=5.35\\log T=5.35 at which the opacity was increasedby 4.",
"The panels below show the kinetic energy density of the three radial modes in the κ\\kappa modified models before, atand after the avoided crossing.",
"These models are marked by vertical lines in Fig.",
"12."
],
[
"Conclusions and future works",
"Seismic modelling of pulsating stars that exhibit both p modes and higher order g modes is still a challenging task.",
"In this paper, we aimed at finding complex seismic models of the well-known pulsator $\\nu $ Eri which reproduce simultaneously the observed frequency range, the values of some individual frequencies and the empirical value of the nonadiabatic parameter $f$ for the dominant radial mode.",
"We based our computations on the three sources of the most widely used opacity data, OPLIB, OPAL and OP, and found that in each case only models with the modified profile of the Rosseland mean opacity can account for the instability of the observed modes.",
"A very important result is that adding the requirement of fitting the nonadiabatic parameter $f$ greatly reduces the number of $\\kappa $ modifications and, in fact, the good news is that it is very difficult to pick out such a model.",
"We found that the model computed with the OPLIB data modified at the three depths, $\\log T=5.06,~5.22,~5.47$ , best meets all the above conditions.",
"Thus, there is some indication that these opacities are preferred.",
"A large increase (more than three times) of the opacity at $\\log T=5.47$ was indispensable to get the instability of g modes whereas a reduction of the opacity by 65% at $\\log T=5.06$ was imposed by the need of fitting the theoretical values of $f$ to the empirical ones.",
"It is also worth mentioning that with the NLTE model atmospheres we got much better agreement.",
"It should also be highlighted that the opacity modifications can have serious consequences on the stellar structure.",
"In this paper, we discussed an enhancement of the efficiency of convection in the Z-bump if the opacity is increased.",
"In turn, more efficient convection affects the mode instability and the flux eigenfunction.",
"In addition, we found that in some models with the much increased opacity around the Z-bump an additional potential barrier can be formed.",
"Consequently, the frequencies of the two consecutive radial modes can come close to each other and undergo the avoided-crossing phenomenon considered up to now only for nonradial modes in main sequence models.",
"Naturally a number of questions arise: To what extent are these opacity modifications realistic or, more precisely, are they in the right direction?",
"To what extent is the solution general?",
"Can we apply it to other hybrid B-type pulsators?",
"A somewhat perplexing result is the need to lower the opacity at $\\log T=5.06$ whereas the new local maximum of $\\kappa (T)$ has been identified in Kurucz models at this depth.",
"This bump helped to solve the excitation problem with the low frequency modes in $\\delta $ Scuti star models which have much lower masses.",
"Can we have such differences in the opacity profile between stars?",
"Or maybe our result is only a consequence of the adopted parametrisation to satisfy the data?",
"The large factor of increase that was imposed on the opacity at $\\log T = 5.47$ is also difficult to justify since our investigations have been unable to identify any missing transitions that would support such an enhancement.",
"We will be able to try to answer all of these questions if more pulsating stars are analysed in the same way.",
"We plan such complex seismic studies for the three other main sequence B-type pulsator which show oscillation spectra of a dual character: $\\gamma $ Pegasi, 12 Lacertae, $\\alpha $ Lupi.",
"A need for such significant modifications of stellar opacity can result either from non-homogenous distribution of chemical elements or from the present-day methods of the opacity computations.",
"Our main goal was to present a new approach to the analysis of the hybrid pulsators, which, besides complex seismic modelling, involves the fitting of the mean opacity profile.",
"Such analysis can lead to important constraints on opacities under the conditions of stellar interiors."
],
[
"Acknowledgments",
"This work was financially supported by the Polish NCN grants 2015/17/B/ST9/02082.",
"PW's work was supported by the European Community's Seventh Framework Program (FP7/2007-2013) under grant agreement no.",
"269194.",
"Calculations have been partly carried out using resources provided by Wroclaw Centre for Networking and Supercomputing (http://www.wcss.pl), grant No.",
"265.",
"The Los Alamos National Laboratory is operated by Los Alamos National Security, LLC, for the NNSA of the US DOE under contract number DE-AC5206NA25396."
]
] | 1612.05820 |
[
[
"A defect in holographic interpretations of tensor networks"
],
[
"Abstract We initiate the study of how tensor networks reproduce properties of static holographic space-times, which are not locally pure anti-de Sitter.",
"We consider geometries that are holographically dual to ground states of defect, interface and boundary CFTs and compare them to the structure of the requisite MERA networks predicted by the theory of minimal updates.",
"When the CFT is deformed, certain tensors require updating.",
"On the other hand, even identical tensors can contribute differently to estimates of entanglement entropies.",
"We interpret these facts holographically by associating tensor updates to turning on non-normalizable modes in the bulk.",
"In passing, we also clarify and complement existing arguments in support of the theory of minimal updates, propose a novel ansatz called rayed MERA that applies to a class of generalized interface CFTs, and analyze the kinematic spaces of the thin wall and AdS3-Janus geometries."
],
[
"Introduction",
"In the last decade, two of the most successful approaches to studying conformal field theories—holographic duality and tensor networks—have turned out to be intimately tied to entanglement.",
"In the AdS/CFT correspondence [1], [2], the Ryu-Takayanagi proposal [3], [4] revealed that holographic spacetimes function as maps of CFT entanglement; meanwhile, the Multi-scale Entanglement Renormalization Ansatz (MERA) [5], [6] arose largely from considering the scale dependence of entanglement entropies in conformal field theories.",
"The fact that quantum entanglement plays a clarifying role in both approaches suggests that holographic spacetimes and MERA networks may be linked by a more direct relationship.",
"Such a relationship was first proposed by Swingle [7], [8] (see also [9], [10], [11], [12], [13]) who pointed out that the MERA network for a CFT ground state bears a striking resemblance to the geometry of anti-de Sitter (AdS) space.",
"An alternative proposal [14], [15] argued that the translation between MERA and holography is mediated by an auxiliary construct termed kinematic space.",
"But both proposals are largely qualitative and would benefit from a broader class of examples, other than the case of the CFT vacuum / pure AdS geometry.",
"Some steps in that direction were taken in refs.",
"[15], [16] (see also refs.",
"[7], [8], [17]) who compared MERA representations of CFT$_2$ thermal states to the BTZ geometries.",
"The analysis in ref.",
"[15] also included Virasoro descendants of the CFT vacuum, related to other locally AdS$_3$ space-times.",
"One commonality of all these examples is that they rely on the extended conformal symmetry in two dimensions.",
"To further explore how MERA and the holographic duality may come together, we need to consider holographic duals and MERA representations of CFT states, which are not related to the vacuum by the application of an anomalous symmetry.",
"This is the subject of the present paper.Other tensor network realizations of broader classes of geometries, mostly set in the context of the ER=EPR [18] and the complexity=action [19] conjectures, include [20], [21], [22], [23], [24], [25], [26], [27].",
"Those works concentrate on the dynamics of space-times while our interest here is on bulk duals of ground states of more general classes of CFTs.",
"We consider the ground states of two-dimensional conformal field theories whose global symmetry has been broken from $SO(2,2)$ down to $SO(2,1)$ by the presence of a defect, an interface or a boundary.In order not to clutter the text, we will refer to all these setups as `defects' unless the context requires distinguishing defects, interfaces and boundaries.",
"On the tensor network side, the `theory of minimal updates' [28] governs the structure of the MERA representations of such states.",
"In holography, there have been many discussions and several explicit examples of holographic defect/interface [29], [30], [31] and boundary CFTs [32], [33], [34] in two dimensions.",
"Our goal is to compare these MERA networks and holographic geometries and analyze in what way, if at all, they relate to one another.",
"Our principal findings are the following: In Sec.",
", we complement existing arguments [35], [28] which support the validity of the minimally updated MERA and clarify the circumstances under which it is expected to hold.",
"It applies to actual defect and interface CFTs, but not to generic two-dimensional theories with $SO(2,1)$ symmetry.",
"In Sec.",
", we propose rayed MERA—a simple generalization of MERA, which should capture ground states of generic two-dimensional theories with $SO(2,1)$ symmetry.",
"In holography, the cases where the minimally updated MERA suffices versus those requiring rayed MERA are distinguished by the boundary region where non-normalizable modes are supported.",
"In Sec.",
", we discuss two examples of holographic defect CFTs.",
"We conclude that a naïvely local relation between MERA networks and AdS$_3$ geometries, in which a specific region of the MERA network corresponds to a specific region of (the spatial slice of) AdS$_3$ , does not hold.",
"This applies both to the direct AdS-MERA correspondence of refs.",
"[7], [8] and to the kinematic proposal of refs.",
"[14], [15].",
"Instead, a key ingredient in relating tensor networks to holographic geometries is that every bond should be associated with the amount of entanglement across it and not with more naïve measures such as the bond dimension.",
"This point was already made in ref.",
"[15]; here we exemplify it.",
"We expect this conclusion to apply to all tensor network models of holography, not just to MERA.",
"Combining these observations leads to the following holographic interpretation of the prescription of [28]: the theory of minimal updates specifies which tensors do / do not register the effect of turning on non-normalizable modes in the bulk.",
"We expand on this statement and put our work in a broader context in the Discussion section."
],
[
"Remarks:",
"Our paper assumes a familiarity with MERA, the AdS/CFT correspondence and their mutual connections.",
"A good review of MERA is ref.",
"[36], reviews of AdS/CFT include ref.",
"[37], [38], [39] while relevant discussions of parallels between MERA and the holographic duality include refs.",
"[7], [40], [15] (see also [41]).",
"MERA is a variational ansatz and gives the description of a desired state only after optimization; throughout the text we will be referring to optimized MERA networks.",
"We also assume that all the gauge freedom in the network was used to exhibit it in a maximally symmetric form."
],
[
"Minimal Updates",
"Consider a 1+1-dimensional CFT deformed by a localized defect.",
"The defect traces a 0+1-dimensional world-line and introduces a preferred location in space.",
"Thus, it breaks the global symmetry from $SO(2,2)$ down to $SO(2,1)$ or a subgroup thereof.",
"We are interested in theories, where the full $SO(2,1)$ consistent with a defect is preserved.",
"We shall refer to such theories as dCFTs, though it should be remembered that this class of theories includes interface and boundary CFTs.",
"We emphasize that the symmetries of dCFTs do not include translations (broken by the defect), but do include scale transformations centered at points on the defect world-line.",
"The minimal updates proposal (MUP) [28] is a simple tensor network ansatz for the ground state wavefunction of a dCFT.",
"As an input, it starts with an optimized MERA network representing the ground state of the undeformed (parent) CFT$_2$ .",
"The MUP asserts that a dCFT ground state can be captured by `updating' in the input MERA only those tensors, which live in the causal coneThe `causal structure' in MERA was introduced in ref.",
"[6]; see ref.",
"[15] for a discussion relevant to holographic duality.",
"of the defect location (see Fig.",
"REF ).",
"Figure: The minimal updates prescription (MUP): When we deform a CFT by a defect, only the tensors in the `causal cone' of the defect (shaded blue) need to be replaced in order to account for the defect.The MUP is a remarkably powerful ansatz.",
"The computational simplifications owed to reusing the undeformed CFT ground state MERA are enormous.",
"Empirically, the MUP achieves a remarkable accuracy on benchmark examples [35], [28], including the case of topological defects [42]."
],
[
"Rationales for Minimal Updates",
"Two rationales have been offered by its authors in support of minimal updates.",
"First, minimal updates guarantee that a local defect remains local after coarse-graining [28].",
"As explained in footnote 36 of that reference, initially allowing the update to extend away from the causal cone will, after optimization, lead to a generally location-dependent set of tensors: a defect not confined to a causal cone can `spill out' under renormalization.",
"The motivation behind MUP is to forestall this undesirable scenario.",
"This rationale, however, is not a proof of validity.",
"The symmetry of the problem does not guarantee that tensors in the description of a dCFT ground state are location-independent (see Sec.",
"below.)",
"Second, there is an algorithmic procedure which takes a discretized (Trotter-Suzuki) version of the Euclidean path integral and transforms it into a MERA representation of the ground state [43].",
"In effect, Tensor Network Renormalization (TNR) is a derivation of MERA.",
"Applied to Euclidean path integrals of dCFTs, TNR can return a MERA network with a structure predicted by MUP [44].",
"This seems to provide a derivation of minimal updates, but here too there is a caveat.",
"A key step in TNR is a local substitution of tensors in the discretized path integral, which is justified by bounding the resulting error (cost function) to a desired tolerance.",
"When the cost function takes into account only the local environment of the tensor to be replaced, the TNR algorithm yields the minimally updated MERA.",
"However, as discussed in Sec.",
"VIII (B) of ref.",
"[42], the TNR algorithm with a global cost function may not produce a MERA with the MUP-dictated structure.",
"Since the conditions under which it suffices to work with a local environment are not known, the status of this second rationale for MUP is also unclear.",
"In summary, refs.",
"[28] and [44] give two independent rationales for the validity of the minimal updates proposal, neither of which is foolproof.",
"Here we offer a third argument, which relies on symmetry and known properties of dCFTs:"
],
[
"Minimal Updates and the Boundary Operator Expansion",
"A key new ingredient in a dCFT is the appearance of the Boundary Operator Expansion (BOE) [46], [45]: $\\mathcal {O}_\\eta {(x)} = \\sum _{i} \\frac{B_{\\mathcal {O}_\\eta }^{\\mathcal {\\hat{O}}_{\\hat{\\eta }_i}}}{{(2y)}^{\\eta -\\hat{\\eta }_{i}}} \\hat{O}_{\\hat{\\eta }_i}{(\\textbf {x})}$ Here we set up coordinates $x = (y, \\textbf {x})$ where $y$ is the direction perpendicular to the defect and $\\textbf {x}$ are the directions along the defect world-volume.",
"Hats mark operators living on the codimension-1 world-volume of the defect.",
"In the formula above we also assumed that $\\hat{O}_{\\hat{\\eta }_i}$ are scaling operators, i.e.",
"they have well-defined scaling dimensions $\\hat{\\eta }_{i}$ under dilations centered at the defect location.",
"The BOE allows us to decompose the action of any local operator according to irreducible representations of the residual $SO(1,2)$ symmetry.",
"In an ordinary CFT all correlation functions can in principle be reduced to kinematic invariants multiplied by products of OPE coefficients, which are the only dynamical data in the theory.",
"In a dCFT, there is an analogous statement: the complete set of dynamical data consists of the BOE coefficients $B_{\\mathcal {O}_\\eta }^{\\mathcal {\\hat{O}}_{\\hat{\\eta }_i}}$ together with the familiar OPE coefficients used for fusing operators away from the defect.",
"For example, one-point functions of local operators in a dCFT are generically non-vanishing and can be read off from fusing the local operators with the defect using the BOE: $\\langle \\mathcal {O}_\\eta (x) \\rangle = \\frac{B_{\\mathcal {O}_\\eta }^{\\hat{1}}}{(2y)^\\eta }$ Similarly, a correlation function of two local away-from-defect operators $\\mathcal {O}_{\\eta _1}$ and $\\mathcal {O}_{\\eta _2}$ can be obtained by first fusing them using the OPE into $\\mathcal {O}_\\eta $ and then applying eq.",
"(REF ) or, in a different channel, by sequentially fusing $\\mathcal {O}_{\\eta _1}$ and $\\mathcal {O}_{\\eta _2}$ with the defect via a double application of the BOE.",
"To verify the validity of the minimal updates proposal, we only need to confirm that the ansatz is powerful enough to correctly encode the away-from-defect OPE and the BOE coefficients.",
"It is well known that the optimized tensors of the ordinary MERA essentially compute the OPE coefficients of a CFT.",
"This is manifest in the way in which OPE coefficients are extracted from MERA; see e.g.",
"[47].",
"By reusing the undeformed CFT ground state MERA, the minimal updates proposal effectively borrows the undeformed theory's OPE coefficients for fusing away-from-defect local operators.",
"Indeed, a ground state ansatz that departs from the minimally updated MERA would contaminate the fusion rules for operators applied away from the defect.",
"The above logic implies that the role of the updated region is to encode the remaining dynamical data—the BOE coefficients.",
"Is the ansatz powerful enough to do so?",
"As a computational problem, finding the correct update has the same structure (the same set of inputs and outputs) as the problem of finding the OPE coefficients in the familiar applications of MERA to ordinary CFTs.",
"In both cases, we are looking for tensors that represent a super-operator, which fuses two given sets of operators into one.",
"This argument reduces the question of the validity of the MUP for describing dCFT ground states to the long-settled question of whether the ordinary MERA captures ground states of ordinary CFTs.",
"This justification for the MUP was not spelled out in [28] or subsequent papers, though similar arguments appeared in [48].",
"We believe it is important to emphasize the relation between minimal updates and the dCFT technology, especially with a view to the following generalization."
],
[
"Rayed MERA",
"Thus far we have considered dCFTs—theories obtained from ordinary CFTs by introducing codimension-1 defects.",
"In general, however, the class of two-dimensional theories with $SO(2,1)$ invariance is much larger.",
"One way to obtain such a theory is by a deformation and (if the deformation is not exactly marginal) an RG flow to a new fixed point.",
"To preserve the symmetry, the sources entering the deformation should have a power-law dependence with $y$ , the distance from the world-line fixed by the $SO(2,1)$ symmetry.",
"Still more generally, we can consider a more abstract CFT-like theory in which `OPE coefficients' for fusing $\\mathcal {O}_i(x)$ and $\\mathcal {O}_j(x^{\\prime })$ have an explicit dependence on $\\xi = \\frac{(x-x^{\\prime })^2}{4yy^{\\prime }}\\,,$ which is the $SO(2,1)$ invariant built from $x$ and $x^{\\prime }$ discussed e.g.",
"in [46].",
"Representing the ground state of a generic, two-dimensional, $SO(2,1)$ -invariant theory is outside the scope of the minimal updates proposal.",
"For an arbitrary such theory, there may not exist a CFT whose ground state MERA could be appropriately minimally updated.",
"This is most easily recognized when we consider `OPE coefficients' that depend on $\\xi $ from eq.",
"(REF ).",
"We observed previously that in the minimally updated MERA, the region that is directly imported from the parent MERA is responsible for correctly merging away-from-defect operators according to the fusion rules of the parent theory.",
"A theory with $\\xi $ -dependent `OPE coefficients' does not emulate the fusion rules of any parent theory.",
"Despite the huge freedom in constructing two-dimensional $SO(2,1)$ -invariant theories, it is possible to write down a simple MERA-like ansatz, which ought to capture the ground states of such theories.",
"To do so, note that the tensor network is supposed to represent the wavefunction of the theory at an equal time slice.",
"The only generator of $SO(2,1)$ that acts within a time slice builds dilations about the origin—where the `defect' (the world-line fixed by $SO(2,1)$ ) and the time slice intersect.",
"The action of the conformal group on the MERA network was studied in [16] (see also [49]).",
"It was found that the orbits of dilations about the origin are tensors, which live on rays emanating from the origin.",
"Thus, the invariance under $SO(2,1)$ dictates that all tensors inhabiting the same ray must be identical, though tensors living on different rays may be distinct.",
"Such an ansatz, which we call rayed MERA, is displayed in Fig. 2.",
"Figure: Rayed MERA: Tensors on each `ray' (color coded) are the same because they are related by a scaling symmetry about the origin (defect location).",
"Tensors inhabiting different rays are in general distinct.Several remarks are in order.",
"First, the minimally updated MERA is a special case of the rayed MERA in which only the vertical ray is distinct from the others.",
"Second, distinct rays are labeled by different values of: $\\xi = \\frac{(x-x^{\\prime })^2}{4yy^{\\prime }}\\quad \\xrightarrow[\\textrm {(equal time)}]{\\textbf {x} = \\textbf {x}^{\\prime }} \\quad \\frac{(y-y^{\\prime })^2}{4yy^{\\prime }}\\,.$ Here $y$ and $y^{\\prime }$ denote a pair of locations such that if two local operators are inserted there, their causal cones will merge on the ray labeled by $\\xi $ .",
"If we think of local groups of tensors as encoding OPE coefficients, making the tensors explicitly dependent on $\\xi $ amounts to choosing $\\xi $ -dependent `OPE coefficients.'",
"In the minimally updated MERA, the only $\\xi $ -dependence distinguishes the parent OPE coefficients from the BOE coefficients, which are encoded on the vertical ray."
],
[
"Holographic Interpretations",
"We will now look at two holographic realizations of interface CFTs and discuss how, if at all, they relate to either the minimally updated MERA of [28] or the rayed MERA of Sec. .",
"To set the context for our discussion, let us briefly recap how prior proposals related the ordinary MERA to pure anti-de Sitter space."
],
[
"MERA and holography without defects",
"Ref.",
"[7] observed a resemblance between the MERA network and a static slice of AdS$_3$ , i.e.",
"the hyperbolic disk.",
"Both have a self-similar structure near the cut-off surface and both contain closely related notions of a minimal cut.",
"Geodesics in AdS$_3$ , which by the Ryu-Takayanagi proposal compute entanglement entropies of CFT$_2$ regions, resemble minimal cuts through the MERA network.",
"This correspondence is consistent insofar as every bond in a minimal cut through MERA contributes an equal amount to the entanglement entropy of the subtended CFT region.",
"Based on the conclusions of [16], we recognize this fact (first observed in [5]) as a consequence of the $SO(2,2)$ symmetry of the CFT.",
"The kinematic proposal of [14], [15] instead views individual tensors in MERA as discrete counterparts of geodesics.",
"This does not run into obvious contradictions with [7] because every minimal cut in MERA selects a unique tensor, which lives in its top corner.",
"In the kinematic proposal, a key to understanding geodesic lengths and entanglement entropies is the Crofton formula, which schematically reads [14]: $\\textrm {length of a curve} = \\int _{\\rm intersecting} \\mathcal {D}{\\rm (geodesics)}.$ Here $\\mathcal {D}{\\rm (geodesics)}$ is the unique measure over the set of geodesics in $\\mathbb {H}_2$ invariant under its isometries.",
"The correspondence between MERA tensors and geodesics advocated in [15] translates eq.",
"(REF ) into simply counting tensors in certain regions of the MERA network."
],
[
"Thin Wall Models: A Naïve Realization of Minimal Updates",
"Note that under both holographic interpretations, the directly imported (i.e., not updated) regions of the MUP MERA account for two halves of (the spatial slice of) pure anti-de Sitter space.",
"This is most obvious in the kinematic interpretation: the unaltered regions consist of geodesics with both endpoints on the same side of the defect and both sets (left and right of defect) of such geodesics span one half of the hyperbolic disk.",
"In the original proposal of [7], the minimally updated region should be viewed as a discrete counterpart of a radial geodesic, with one half of $\\mathbb {H}_2$ on each side of it.",
"This is because MERA does not accommodate a notion of locality narrower than the width of one causal cone [47].",
"Whichever proposal we adopt, the regions that remain unaltered by the minimal updates should be viewed as two halves of the hyperbolic disk, each ending on a geodesic diameter.",
"From this observation, one could venture the following, naïve holographic interpretation of the minimal updates proposal: that the holographic dual of a dCFT should contain two undeformed halves of pure anti-de Sitter space separated by some `wall.'",
"Whatever the wall is, on either side of it should be (at least) one half of pure anti-de Sitter space.",
"We shall see later that this holographic reading of the minimal updates proposal is too naïve because it is too stringent.",
"But before that, let us inspect a class of models that realize this naïvely stringent interpretation of minimal updates:"
],
[
"Thin wall models",
"Consider a simple toy model for the holographic dual of a dCFT, which consists of two AdS$_{3}$ patches glued together with a tensionful brane.",
"Such models were discussed for example in [50], [51], [52], [53] (for early geometric analyses see [54], [55]), building up on an embedding in string theory [29], [30].",
"As we clarify below, the holographic duals of boundary CFTs discussed in [32], [33], [34] also fall into this class.",
"The setup is illustrated in Fig.",
"REF .",
"The two AdS patches can have different curvatures, which would correspond to coupling along an interface two CFTs with central charges $c_L$ and $c_R$ .",
"(The special case $c_L = c_R \\equiv c$ are actual defect CFTs, as opposed to the more general variety of interface CFTs.)",
"The famous Brown-Henneaux formula [56] relates the central charges to the radii of curvature: $\\frac{L}{G} = \\frac{2}{3} c_L \\qquad {\\rm and} \\qquad \\frac{R}{G} = \\frac{2}{3} c_R.$ Here $G$ is the bulk Newton's constant and $L , R$ are the AdS radii on the two sides.",
"Figure: A thin wall geometry consists of two wedges of pure AdS 3 _3 (pink and green regions) glued along a tensionful wall.The wall occupies a `straight line' in the Poincaré coordinates, which delimits each AdS 3 _3 chunk.",
"The two straight lines are identified.We will adopt the familiar Poincaré patch coordinates $(x,z)$ on both sides of the brane: $ds^2 = L^2 \\frac{-dt^2 + dx^2 + dz^2}{z^2}\\qquad {\\rm and} \\qquad ds^2 = R^2 \\frac{-dt^2 + dx^2 + dz^2}{z^2}$ The wall occupies a surface of constant extrinsic curvature, which in this coordinate system turns out to be a `straight line' in the $z$ -$x$ plane.",
"Each patch of AdS$_3$ on one side of the wall is characterized by the slope of that line, which we express in terms of $\\alpha $ and $\\beta $ : $z = -x \\tan \\beta ~{\\rm (left)}\\qquad {\\rm and} \\qquad z = x \\tan \\alpha ~{\\rm (right)}$ Fig.",
"REF depicts one example geometry, in which $\\alpha $ and $\\beta $ are both less than $\\pi /2$ .",
"Note that $\\alpha = \\pi /2$ denotes one half of the hyperbolic disk delimited by a radial geodesic.",
"Thus, the naïve holographic interpretation of minimal updates predicts that $\\alpha , \\beta \\ge \\pi /2$ .",
"We now verify that the thin wall models conform to this prediction.",
"In the thin wall geometry, Einstein's equations reduce to the Israel junction conditions [57], which we re-derive in Appendix .",
"For a brane of tension $\\lambda $ , these take the form: $\\frac{L}{\\sin \\beta } = \\frac{R}{\\sin \\alpha } = -\\frac{\\cot \\alpha + \\cot \\beta }{8\\pi G\\lambda }\\,.$ These three quantities are equal to the radius of intrinsic curvature on the brane.",
"Observe that eqs.",
"(REF ) accommodate the duals of boundary CFTs discussed in [32], [33], [34] simply by setting $\\beta = \\pi /2$ .",
"This introduces a fictitious left chunk of AdS$_3$ with curvature $L = R/\\sin \\alpha $ which decouples, because it exerts no force on the bulk wall.",
"Although eqs.",
"(REF ) have formal solutions with arbitrary $\\alpha $ and $\\beta $ , in fact only $\\alpha , \\beta \\ge \\pi /2$ are physical.",
"When $\\alpha , \\beta < \\pi /2$ , the tension $\\lambda $ is forced to be negative, which violates the weak energy condition in the bulk.Ref.",
"[53] contains a thorough discussion of energy conditions in the context of holographic dCFTs.",
"Such a situation gives rise to rather exotic features associated with strong subadditivity, which we detail in Appendix .",
"The remaining case, $\\alpha \\ge \\pi /2 > \\beta $ , is also unphysical.",
"As we show in Appendix , in this regime the wall is necessarily unstable so it cannot be the dual of the ground state of a dCFT.",
"Studying geodesics in the thin wall space-time built by a wall with positive tension turns out to involve an interesting application of Snell's law.",
"Because we have not found a solution of this problem anywhere in the literature, in Appendix we explain how to find such geodesics and compute the kinematic space of the thin wall geometry."
],
[
"Summary",
"The thin wall geometry is consistent with the naïve holographic interpretation of the minimally updated MERA.",
"This is true regardless of whether we adopt the direct [7] or the kinematic [15] proposal for relating MERA to holographic geometries.",
"However, the direct proposal is arguably subject to some awkward caveats.",
"This is because $\\alpha , \\beta > \\pi /2$ means that the thin wall geometry is strictly larger than it would have been in the absence of a defect.",
"Thus, the causal cone of the defect must be simultaneously interpretable as the radial geodesic (in the dual of the undeformed CFT) and as the extra thickness of space-time grown by the thin wall (quantified by $\\alpha + \\beta -\\pi $ .)",
"This caveat does not arise in the kinematic proposal where, with or without the wall, we are always dealing with the same set of geodesics.",
"We will not dwell on this issue further because more general models will anyway force us to revise our assumptions."
],
[
"Thick Walls: Not All Bonds Are Created Equal",
"The exercise of studying thin wall models is useful because it immediately illustrates why the `naïve holographic interpretation' of the minimally updated MERA is naïve.",
"As soon as our wall is no longer thin, it will involve non-trivial profiles of various bulk fields whose tails extend all the way to the asymptotic boundary.",
"Indeed, the non-vanishing one-point functions (REF ) of holographic dCFTs are read off precisely from such tails of normalizable modes of bulk fields.",
"Looking for two greater-than-half chunks of pure AdS$_3$ on both sides of the wall can only work in a thin wall model.",
"There is another reason why the naïve interpretation is too naïve.",
"When we discussed the direct [7] and the kinematic [15] readings of MERA, the full $SO(2,2)$ symmetry of the theory appeared to be a key ingredient.",
"In the direct proposal, the connection between minimal cuts in MERA and geodesics in AdS$_3$ was only sensible because every MERA bond contributed an equal amount to the entropy count [5].",
"This feature relies on the global $SO(2,2)$ symmetry.",
"To see this, recall that changing the UV cut in MERA corresponds to applying a conformal transformation [16].",
"Any bond in MERA can become a part of the UV cut under the action of $SO(2,2)$ and therefore all bonds are related to one another by this symmetry.",
"In the kinematic proposal, on the other hand, the $SO(2,2)$ entered via the choice of measure $\\mathcal {D}{\\rm (geodesics)}$ , which translated into uniformly counting different MERA tensors.",
"In the case at hand, the symmetry is broken to $SO(2,1)$ .",
"On the spatial slice modeled by the tensor network, the only symmetry we have are dilations about the origin.",
"In order to relate thick wall models to MUP, we must assign different weights to different tensors and bonds in the minimally updated MERA."
],
[
"Assigning relative weights to bonds and tensors",
"Ref.",
"[15] explained how to weigh different regions of MERA in the kinematic interpretation.",
"To explain this prescription, we need a few basic facts.",
"In the present context, the kinematic space is the space of intervals on a spatial slice of a CFT$_2$ .",
"When a holographic dual is available, it is also the space of bulk geodesics.",
"The kinematic space has a Lorentzian metric of the form: $ds^2_{\\rm K.S.}",
"= \\frac{\\partial ^2 S_{\\rm ent}(u,v)}{\\partial u \\partial v}\\, dudv\\,,$ where $u$ and $v$ are the two endpoints of a CFT interval / bulk geodesic and $S_{\\rm ent}$ is the entanglement entropy of the interval / length of the geodesic.",
"This metric turns out to be de Sitter space in the case of a locally AdS geometry, and has many attractive properties which were discussed in [14], [15] and elsewhere [58], [59], [60], [61].",
"For example, the volume form derived from this metric defines a measure on the space of bulk geodesics $\\mathcal {D}{\\rm (geodesics)}$ such that eq.",
"(REF ) holds.",
"The claim of [15] is that we can think of MERA as a discrete version of kinematic space.",
"To do so, consider two pairs of nearby points, $(u, u-\\Delta u)$ and $(v, v+\\Delta v)$ , on the UV cut of MERA.",
"We can impose on MERA a discretized version of metric (REF ): $ds^2_{\\rm MERA} =S_{\\rm ent}(u - \\Delta u, v) + S_{\\rm ent}(u, v+\\Delta v) - S(u-\\Delta u, v + \\Delta v) - S(u,v)\\,.$ In this `metric', the light-like directions $u$ and $v$ agree with the causal structure of MERA, which we mentioned in Sec. .",
"The quantity (REF ) coincides with a discretized `volume form' on the tensors of MERA, which can be compared with $\\mathcal {D}{\\rm (geodesics)}$ .",
"In the ground state of an $SO(2,2)$ -invariant theory, eq.",
"(REF ) defines a discrete version of two-dimensional de Sitter space.For other observations relating MERA to de Sitter space, see [62], [63].",
"But in a theory with only $SO(2,1)$ invariance, the `volumes' assigned to different regions of MERA will differ.",
"The only fact guaranteed by the symmetry is that identical regions living on the same ray (as discussed in Sec. )",
"carry equal volumes."
],
[
"A case study in thick walls: the AdS$_3$ -Janus solution",
"One holographic pair which illustrates this non-uniformity is the Janus deformation of AdS$_3$ and its dual interface CFT.",
"Following earlier developments in AdS$_5$ [64], Refs.",
"[31], [65], [66], [67], [68], [69] studied a scalar field $\\phi $ (the `dilaton') coupled to Einstein gravity with a negative cosmological constant in three dimensions and found the following solution: ds2 = L2 ( du2 + (u)2 dsAdS22 ) dsAdS22 = -2rdt2 + dr2 (u)2 = 12(1+1-222u) (u) = 0 + 12(1+1-22 + 2u 1+1-22 - 2u) They also explained how this solution is holographically dual to the ground state of a marginal deformation of the D1-D5 CFT whose strength is proportional to $\\gamma $ .",
"The deformation has a different sign on the two halves of the boundary, so the resulting theory is an interface CFT.",
"In the bulk, the AdS$_3$ -Janus solution contains a thick wall.",
"We do not have an optimized tensor network which prepares the ground state of this theory, so we cannot make quantitative comparisons with MERA.",
"But we can compute its kinematic space (eq.",
"REF ) and observe qualitative features.",
"We carried out this computation for small $\\gamma $ in Appendix .",
"Up to an overall factor of $L/2G$ , the result to first non-trivial order in $\\gamma $ reads: $ds^2_{\\rm K.S.-Janus} =\\frac{du\\, dv}{(u-v)^2} \\left[ 1 -\\frac{\\gamma ^{2}}{2}\\bigg (\\eta ^2 + 3 - \\frac{1}{2}\\left(\\eta ^3 + 3\\eta ^{-1} \\right)\\log {\\left|\\frac{1+\\eta }{1-\\eta }\\right|} \\bigg )\\right]$ Here $\\eta = (v-u)/(v+u)$ is a kinematic $SO(2,1)$ invariant, related to $\\xi $ from eq.",
"(REF ) via: $\\eta = (\\xi ^{-1} + 1)^{-1}\\,.$ The inside of the causal cone of the interface has $\\eta > 1$ while the regions in MERA that are imported from the parent without updates have $\\eta < 1$ .",
"Indeed, the effect of the interface spills out beyond the causal cone of the interface, and increases the kinematic volume there.",
"It is UV-finite and in fact vanishes in the UV limit $\\eta \\rightarrow 0$ , where the effect of the interface is the smallest.",
"Within the causal cone, on the other hand, the interface causes the overall kinematic volume to decrease.",
"This is to be expected because according to eq.",
"(REF ) the volume of this region computes the entanglement entropy of the two sides of the interface."
],
[
"Summary:",
"The bulk duals of holographic dCFTs generically involve thick walls.",
"In relating such theories to tensor networks, we cannot count all tensors or bonds with equal weight.",
"Instead, we must account for different weights that occur at different values of the $SO(2,1)$ invariant $\\xi $ (see eq.",
"REF ).",
"In a minimally updated MERA, even though all tensors outside the causal cone are identical, their weights differ depending on the location relative to the defect."
],
[
"Non-normalizable Modes: From the Minimally Updated MERA to Rayed MERA",
"The above conclusion poses one residual question.",
"On the one hand, the MUP mandates that some tensors do not register the presence of a defect; on the other hand, those tensors count with different weights when we calculate entropies.",
"What then distinguishes states constructible using the minimally updated MERA versus the rayed MERA?",
"We would like to answer this question in a way that makes contact with the AdS/CFT correspondence.",
"Recall that the minimally updated MERA is designed for theories constructed by coupling two $SO(2,2)$ invariant parent theories along a common interface.",
"The rayed MERA is for a generic $SO(2,1)$ -invariant theory, which could be constructed in multiple ways.",
"One such way is to deform a parent theory by an appropriately selected source, which is either $SO(2,1)$ -invariant or designed to recover the $SO(2,1)$ after an RG flow.",
"In holography, deforming theories by the introduction of sources is effected by turning on non-normalizable modes in the bulk [70].",
"Thus, a ground state of a holographic theory whose bulk dual involves a thick wall can be prepared by either one of the two types of networks—the minimally updated MERA or the rayed MERA—depending on whether the thick wall contains condensates of non-normalizable modes away from the `interface.'",
"Here by `interface' we mean the fixed world-line of the residual $SO(2,1)$ symmetry.",
"As an example, the holographic dual of the AdS$_3$ -Janus solution is a marginal deformation of the D1-D5 CFT [31]: $S = S_{\\,{\\rm D1D5}}+ \\tilde{\\gamma } \\int _{x>0}\\!\\!dx\\,dt\\,\\mathcal {O}_\\phi (x,t)- \\tilde{\\gamma } \\int _{x<0}\\!\\!dx\\,dt\\,\\mathcal {O}_\\phi (x,t)$ Here $\\tilde{\\gamma }$ is a deformation parameter, which agrees with the $\\gamma $ from eqs.",
"(REF ) and (REF ) to leading order, $\\tilde{\\gamma } = \\gamma + O(\\gamma ^2)$ .",
"The bulk solution involves a non-normalizable mode for the dilaton, which asymptotes to different constant values on the boundary $\\phi \\rightarrow \\phi _\\pm = \\phi _0 \\pm \\frac{1}{\\sqrt{2}}\\tanh ^{-1} \\sqrt{2}\\gamma $ and accounts for the deformation (REF ).",
"Eq.",
"(REF ) is a marginal deformation of the parent CFT with a piece-wise constant source that jumps at the interface.",
"If, in principle, we had at our disposal MERA representations of the ground states of the theories $S = S_{\\,{\\rm D1D5}}\\pm \\tilde{\\gamma } \\int _{{\\rm all}~x}\\!\\!dx\\,dt\\,\\mathcal {O}_\\phi (x,t)\\,,$ we could use them as input in the minimal updates prescription.",
"Thus, the ground state of the theory dual to the AdS$_3$ -Janus solution belongs to the class of states, which can in principle be represented in the form of a minimally updated MERA.",
"Of course, the tensors comprising that network would be different from those which prepare the ground state of the undeformed theory.",
"However, if we turn on more general deformations while preserving $SO(2,1)$ , the resulting ground states can only be prepared using the rayed MERA.",
"For example, we could deform a holographic CFT with irrelevant operators coupled to sources with a power-law dependence on the distance from a select line.",
"If the interior of the resulting bulk geometry were then compared to a MERA-type tensor network, it would have to be a rayed MERA."
],
[
"Summary:",
"The distinction between the minimally updated MERA and the rayed MERA is whether we simply couple two parent CFTs along an interface or do something more generic.",
"One option in the latter category is to deform the theory globally by an irrelevant operator to find a new fixed point in the IR.",
"Such a theory is generally outside the scope of the minimal updates prescription, but if it preserves $SO(2,1)$ , it can in principle be captured by a rayed MERA."
],
[
"Discussion",
"There is now a considerable literature which seeks ways to relate spacetimes that arise in holographic duality to tensor networks.",
"In this paper, we initiate a new chapter of this endeavor: studying space-times which are neither pure anti-de Sitter nor its quotients nor Virasoro descendants.",
"For this initial study we chose to consider holographic defect, interface and boundary CFTs (dCFTs) and tensor networks in the class of the Multi-scale Entanglement Renormalization Ansatz (MERA).",
"We concentrated on MERA for 1+1-dimensional CFTs because this class of networks is best understood.",
"In particular, in MERA we know (a) how to realize conformal transformations (by changing the UV cut [16]), (b) how the spectrum of conformal dimensions and OPE coefficients are encoded (for details, see [47]), and (c) how to represent ground states of dCFTs (the minimal updates proposal [28]).",
"Concerning the class of theories, we focused on dCFTs because they obey a residual $SO(2,1)$ global symmetry, which has a clarifying power.",
"It organizes data in both MERA (on rays emanating from the origin) and in the holographic geometry (which is foliated by AdS$_2$ slices.)",
"Some of our conclusions concern specifically the MERA class of tensor networks.",
"We clarified and complemented arguments supporting the validity of the minimal updates proposal (Sec. )",
"and proposed an extension for generic, $SO(2,1)$ -invariant theories (rayed MERA, Sec. ).",
"Our other conclusions should hold more generically.",
"In particular, we expect that in every meaningful instance of a holographic bulk geometry-tensor network correspondence, the following rule should hold: Changing tensors in the ground state network represents turning on non-normalizable modes in the bulk.",
"In the case of MERA, because of its causal structure, the effect of locally turning on a non-normalizable mode is contained in the causal cone of the deformation.",
"We propose this as the holographic interpretation of the theory of minimal updates [28].",
"But in other types of networks such as [71], [72], [73], [74], [75], the effect of a deformation should also be cleanly identifiable and likely localized in a subregion of the network.",
"At the same time, we should remember that local properties of a tensor network state in general depend non-locally on the tensors.",
"One example considered in this paper (see Sec.",
"REF ) is the set of entanglement entropies, which underlie both the direct [7] and the kinematic [15] holographic interpretation of MERA.",
"We can think of such local but non-locally determined properties of tensor network states as akin to the normalizable bulk modes.",
"In AdS/CFT, these encode responses to boundary conditions set elsewhere.",
"Other familiar examples of such quantities are CFT one-point functions, which in MERA depend on the entire causal future of the given point."
],
[
"Next steps",
"It would be interesting to realize some of these ideas in other types of tensor networks, which were specifically designed for the AdS/CFT correspondence [71], [72], [73], [74], [75], and also consider the Kondo problem as an example [76].",
"Many questions await answers: How do these networks encode OPE coefficients of the CFT?",
"Can we see how deforming the CFT changes the ground state tensors and thus observe the effect of a non-normalizable mode?",
"How to represent ground states of defect CFTs?",
"More specifically, how to deform those networks to construct an analogue of a thin wall geometry?",
"This last problem is further pertinent for understanding how those classes of tensor networks can accommodate the backreaction of bulk matter fields.",
"Departing from tensor networks, our paper is the first study of the kinematic space of dCFTs.",
"For ordinary CFTs, studying fields local in kinematic space led to enlarging the holographic dictionary by the addition of OPE blocks, which at leading order in $1/N$ are dual to bulk fields integrated along geodesics [58], [59].",
"It would be interesting to generalize these findings to holographic dCFTs, perhaps starting with thin wall bulk duals.",
"Interesting work in this direction is forthcoming [77]."
],
[
"Acknowledgments",
"We thank Xi Dong, Glen Evenbly, Markus Hauru, Michał Heller, Lampros Lamprou, Samuel McCandlish, Ashley Milsted, Rob Myers, James Sully and Guifré Vidal for helpful discussions, comments and insights.",
"BC is supported by the Peter Svennilson Membership in the Institute for Advanced Study while PN and SS are supported by the National Science Foundation under Grant Number PHY-1620610.",
"SS would like to thank the Perimeter Institute for Theoretical Physics (PI) for hospitality during part of this work.",
"Research at PI is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.",
"BC dedicates this paper to Bayu Miłosz Czech and his patient mother, Stella Christie."
],
[
"Israel Junction Conditions and Wall Stability",
"We consider three-dimensional geometries, which preserve $SO(1, 2)$ symmetry: $ds^2 = du^2 + \\rho (u)^2 (-\\cosh ^2 r dt^2 + dr^2)$ For a dual of a general holographic dCFT, we should also include other fields and their backreactions; one such example is discussed in Sec.",
"REF and Appendix .",
"Here we assume that the geometry contains a thin wall of tension $\\lambda $ .",
"To have a locally AdS$_3$ geometry to the left of the wall, we must have $\\rho (u) = L \\cosh (u/L),$ where $L$ is the left AdS$_3$ curvature radius.",
"To the right of the wall, we will have a similar expression with $L \\rightarrow R$ , the curvature radius on the right.",
"On the static slice $t = 0$ , the change of coordinates from (REF ) to (REF ) is: $z = e^{r}{\\rm sech}\\,{u/L} \\qquad {\\rm and} \\qquad x = -e^{r}\\tanh {u/L}\\,.$ Away from a spatial slice the formulas are more involved, but we do not need them in this paper.",
"In eq.",
"(REF ), the asymptotic boundary of space-time is approached as $u \\rightarrow -\\infty $ while the wall sits at some specific value $u_*$ .",
"The $u=0$ slice of metric (REF ) is a minimal surface in AdS$_3$ , so depending on the sign of $u$ the constant-$u$ slices are contracting (for $u<0$ ) or expanding (for $u>0$ ) in the direction of increasing $u$ , that is toward the wall.",
"This distinction will be important for our considerations.",
"To find a static configuration of the AdS$_{3}$ chunks and the wall, we consider the Einstein-Hilbert action with a Gibbons-Hawking-York (GHY) term and an explicit wall contribution: $S = \\frac{1}{16\\pi G} \\int _{\\rm left} d^3x \\sqrt{-g} (\\mathbf {R} - 2 \\Lambda ) + \\frac{1}{8\\pi G} \\int _{\\rm wall} d^2y \\sqrt{-h} K_L + (L \\rightarrow R) -\\lambda \\int _{\\rm wall} d^2y \\sqrt{-h}$ Additional GHY terms arise at the asymptotic boundary of space-time, but these will play no role in our analysis.",
"The Ricci scalar in metric (REF ) takes the form: $\\mathbf {R} = -2 \\, \\frac{1 + \\rho ^{\\prime 2} + 2 \\rho \\rho ^{\\prime \\prime }}{L^2 \\rho ^2}$ We can confirm the correctness of this expression by substituting (REF ), which gives $\\mathbf {R} = -6/L^2$ .",
"Plugging eq.",
"(REF ) and $\\Lambda = -L^{-2}$ into (REF ), the action takes the form: $S \\propto -\\frac{L}{8\\pi G} \\int ^{u_*} du \\left(1 + \\rho ^{\\prime 2} + 2 \\rho \\rho ^{\\prime \\prime }- \\rho ^2\\right) + \\frac{L^2 {\\rho (u_*)}^2 K_L}{8\\pi G} + (L \\rightarrow R) - L^2 {\\rho (u_*)}^2 \\lambda $ Here we have dropped an overall infinite factor, which stands for the volume of AdS$_{2}$ with unit curvature.",
"Expression (REF ) contains two terms, which can be combined and simplified.",
"To get a standard variational problem, we need to eliminate $\\rho ^{\\prime \\prime }$ via integration by parts.",
"This introduces a boundary term, which the GHY term is designed to cancel: $- L \\int ^{u_*} du\\, 2\\rho \\rho ^{\\prime \\prime } + L^2 {\\rho (u_*)}^2 K_L =- L \\int ^{u_*} du\\, 2\\rho \\rho ^{\\prime \\prime } + L \\frac{d}{du} \\rho ^2\\Big |_{u_*} =L \\int ^{u_*} du\\, 2\\rho ^{\\prime 2}$ After this substitution, action (REF ) becomes: $S =\\frac{L}{8\\pi G} \\int ^{u_*} du \\left(\\rho ^{\\prime 2} -1 + \\rho ^2\\right) + (L \\rightarrow R) - L^2 {\\rho (u_*)}^2 \\lambda $ We may now plug in the known solution (REF ) for $\\rho (u)$ and its right counterpart to obtain: $S = \\frac{L}{4\\pi G} \\int ^{u_*} du\\, \\sinh ^2 u+ \\frac{R}{4\\pi G} \\int ^{v_*} dv\\, \\sinh ^2 v- \\lambda \\, L^2 \\cosh ^2 u_*.$ For continuity of the metric, the intrinsic geometry of the wall must be the same in both the $u$ and $v$ metrics.",
"This leads to the first Israel junction condition [57], which is the first equality in eq.",
"(REF ): $L \\cosh u_* = R \\cosh v_*.$ Note that we have two distinct branches of $v_*$ , which correspond to having a `smaller-than-half' or `bigger-than-half' chunks of AdS$_{3}$ to the right of the wall: $\\sinh v_* = \\pm \\sqrt{{(L/R)}^2 \\cosh ^2 u_* - 1}$ To the left of the wall, the analogous distinction is controlled by the sign of $u_*$ .",
"It is now trivial to find the equilibrium configuration of the AdS$_{3}$ patches and the wall.",
"Setting $dS/du_* = 0$ gives: $\\frac{L}{4\\pi G} \\sinh ^2u_* + \\frac{R}{4\\pi G} \\sinh ^2 v_* \\cdot \\frac{dv_*}{du_*} - 2 \\lambda L^2 \\cosh u_* \\sinh u_* = 0$ Substituting $\\frac{dv_*}{du_*} =\\frac{L \\sinh u_*}{R \\sinh v_*}$ which follows from (REF ), we get: $\\sinh u_* \\big ( \\sinh u_* + \\sinh v_* - 8 \\pi G \\lambda L \\cosh u_* \\big ) = 0.$ Setting $u_* = 0$ is not a solution of the equations of motion; rather, it signals a breakdown of $u_*$ as a collective coordinate.",
"Equating the other factor of (REF ) to zero gives the second Israel junction condition, which is the second equality in (REF ).",
"To check the stability of the solution, we compute: $\\frac{d^2S}{du_*^2}\\,\\Bigg |_{\\rm EOM} = 2 \\lambda L^2\\, \\frac{\\sinh u_*}{\\sinh v_*}\\,.$ Thus, stability requires that the product of $\\lambda $ , $u_*$ and $v_*$ must be positive.",
"Excluding negative tensions leaves out $u_*, v_* < 0$ ($\\lambda > 0$ forbids this by the equation of motion) and $u_*, v_* > 0$ , i.e.",
"$\\alpha , \\beta > \\pi /2$ .",
"This is the only consistent, stable configuration."
],
[
"Geodesics in the Thin Wall Geometry",
"It is interesting to find the geodesics of the thin wall geometry explicitly.",
"We denote the endpoints of the geodesic with $a, b$ and assume $a > b$ ."
],
[
"Geodesics in the presence of a stable thin wall",
"The stable configuration has $\\alpha , \\beta > \\pi /2$ .",
"Geodesics that begin and end on the same side of the wall are same as in pure AdS$_3$ .",
"Their lengths are $S(a,b) = 2L \\log \\frac{a-b}{\\mu }\\,,$ where $\\mu $ is a large scale cutoff in the geometry.",
"In the following we will drop the cutoffs, which in three bulk dimensions are simple additive constants.",
"To find the geodesics crossing the wall ($b < 0 < a$ ), observe that the geodesic motion in the hyperbolic plane is analogous to the propagation of a light ray in a medium whose index of refraction is $n(z) = L/z$ .",
"Due to the first Israel junction condition, the index of refraction at the brane is continuous.",
"Thus, by Snell's law, a geodesic crossing the brane consists of two circular arcs, which meet at the location on the brane where no refraction occurs.",
"The angles can be read off directly from the $x$ -$z$ plane, which is conformal to the geometry.",
"Thus, we are looking for two arcs which meet the wall at the same location and the same angle in the $x$ -$z$ plane.",
"One such a geodesic is plotted in Fig.",
"REF .",
"Figure: A wall-crossing geodesic in a thin wall geometry consists of two arcs, which meet the wall at the same angle and location.Finding this location is a simple minimization exercise.",
"Consider a family of piecewise geodesic curves, each of which consists of two circular arcs meeting at an arbitary junction on the brane.",
"Let $y=\\sqrt{x^2 + z^2}$ be the coordinate distance of the junction from the defect; note that $y$ -values on the two sides of the wall agree.",
"One can easily write down the length of such a curve as a function of $y$ : $S{(\\alpha ,\\beta ,a,b,y)} =L\\log {\\left(\\frac{b^{2}+y^{2}-2|b|y\\cos {\\beta }}{y\\sin {\\beta }}\\right)}+R\\log {\\left(\\frac{a^{2}+y^{2}-2ay\\cos {\\alpha }}{y\\sin {\\alpha }}\\right)}$ To find the actual geodesic among this family of curves, we minimize the length formula above with respect to $y$ .",
"The critical value of $y$ , which we denote $y_{*}$ , is given by: $y_{*} = \\frac{1}{2} \\csc \\left(\\frac{\\alpha +\\beta }{2}\\right) \\left[ (a-|b|)\\sin {\\left(\\frac{\\beta -\\alpha }{2}\\right)} + \\sqrt{(a+|b|)^{2}\\sin ^{2}{\\left( \\frac{\\beta -\\alpha }{2}\\right)} + 4a|b|\\sin {\\alpha }\\sin {\\beta } } \\right].$ Substituting this expression in (REF ) gives the desired geodesic length.",
"For the kinematic space metric component, we would then take the second partial with respect to $a$ and $b$ as in eq.",
"(REF ).",
"We do not give the full expression here because it is not illuminating."
],
[
"Negative wall tension and strong subadditivity",
"The pathological case when both $\\alpha , \\beta < \\pi /2$ has some further exotic properties.",
"Geodesics corresponding to regions with $\\xi $ greater than a certain critical value are `squashed' by the wall: they consist of two semi-circular arcs that are tangent to the wall plus a finite segment along the wall.",
"A family of such geodesics spanning three adjacent boundary intervals are depicted in Fig.",
"REF .",
"Figure: Illustration of SSA saturation for squashed geodesics.",
"A family of squashed geodesics spanning three adjacent boundary intervals.If we assume that this geometry obeys the Ryu-Takayanagi proposal for some dual CFT state, we immediately see that intervals depicted in Fig.",
"REF saturate the strong subadditivity (SSA) of entanglement entropy.",
"In kinematic space, SSA saturation results in a degenerate metric in certain wedge-shaped regions near the edges of the defect's causal cone.",
"Saturation of SSA places a strong constraint on the entanglement structure of a quantum state [78].",
"Saturating it over a continuous family of intervals in a field theory is a powerful constraint, even if it is subject to $\\mathcal {O}(1/N)$ corrections.",
"It would be interesting to prove that such a set-up cannot be realized in a real CFT."
],
[
"Kinematic Space of the Janus Solution",
"In this appendix, we compute the entanglement entropy and kinematic space of the Janus solution perturbatively for small $\\gamma ^{2}$ .",
"On a constant time slice of the Janus solution, expanding the metric (REF -REF ) to first non-trivial order in $\\gamma $ gives: $ds^{2} = L^2 \\left(\\cosh ^{2}{u} - \\frac{\\gamma ^2}{2}\\cosh {2u} \\right) dr^{2} + L^2 du^{2}$ Applying the coordinate change (REF ) brings this metric to the form: $ds^{2} = L^2 \\frac{dx^{2}+dz^{2}}{z^{2}} - \\gamma ^{2}L^2\\frac{(z^{2}+2x^{2})}{2z^{2}(x^{2}+z^{2})^{2}}(xdx+zdz)^{2}$ Perturbations of geodesic lengths generally arise from two effects: the shift in the metric and the shift in the coordinate trajectory of the geodesic.",
"To lowest order, however, we can ignore the latter and only consider the former.",
"Thus, we will take the geodesics to be semi-circles in the $x$ -$z$ plane.",
"The perturbed induced metric on the semi-circle, which connects $u = x_0 - R$ and $v = x_0 + R$ takes the form: $ds^{2} = L^2 \\left[\\frac{R^{2}}{(R^{2}-(x-x_{0})^{2})^{2}} - \\gamma ^{2} \\frac{x_{0}^{2}(R^{2}+x^{2}+2xx_{0}-x_{0}^{2})}{2(R^{2}+2xx_{0}-x_{0}^{2})^{2}(R^{2}-x^{2}+2xx_{0}-x_{0}^{2})}\\right] dx^{2}$ The perturbation of the length is: $\\delta S = \\frac{1}{2} \\int \\sqrt{g_{xx}} g^{xx} \\delta g_{xx} dx$ Evaluating the integral gives: $\\delta S{(R,x_{0})} = -\\gamma ^{2}\\frac{L x_{0}^{2}}{2R} \\int _{x_{0}-R}^{x_{0}+R} \\!\\!\\frac{R^{2}+x^{2}+2xx_{0}-x_{0}^{2}}{(R^{2}+2xx_{0}-x_{0}^{2})^{2}}dx =-\\frac{\\gamma ^{2}L}{8Rx_{0}} \\left(4Rx_{0} + 2(R^{2}-3x_{0}^{2})\\log {\\bigg |\\frac{R-x_{0}}{R+x_{0}}\\bigg |} \\right)$ The correction to the kinematic space metric due to $\\gamma $ is then found by differentiation of $\\delta S$ : $\\frac{\\partial ^{2}}{\\partial u\\partial v}\\delta S &=& \\frac{1}{4}\\left(\\frac{\\partial ^{2}}{\\partial x_{0}^{2}} - \\frac{\\partial ^{2}}{\\partial R^{2}}\\right)\\delta S \\nonumber \\\\&=& -\\frac{\\gamma ^{2}L}{16 R^{3}x_{0}^{3}} \\left(4Rx_{0}(R^{2}+3x_{0}^{2}) + 2(R^{4}+3x_{0}^{4})\\log {\\bigg |\\frac{R-x_{0}}{R+x_{0}}\\bigg |} \\right)$ This is eq.",
"(REF ) from the main text."
]
] | 1612.05698 |
[
[
"Leggett-Garg tests of macro-realism for multi-particle systems including\n double-well Bose-Einstein condensates"
],
[
"Abstract We construct quantifiable generalisations of Leggett-Garg tests for macro/ mesoscopic realism and noninvasive measurability that apply when not all outcomes of measurement can be identified as arising from one of two macroscopically distinguishable states.",
"We show how quantum mechanics predicts a negation of the LG premises for proposals involving ideal-negative-result, weak and quantum non-demolition measurements on dynamical entangled systems, as might be realised with two-well Bose-Einstein condensates, path-entangled NOON states and atom interferometers."
],
[
"3 Leggett-Garg tests of macro-realism for multi-particle systems including two-well Bose-Einstein condensates L. Rosales-Zárate$^{1}$ , B. Opanchuk$^{1}$ , Q. Y. He$^{2}$ and M. D. Reid$^{1}$ $^{1}$ Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne 3122 Australia $^{\\text{2}}$ State Key Laboratory of Mesoscopic Physics, School of Physics, Peking University, Beijing 100871 China We construct quantifiable generalisations of Leggett-Garg tests for macro/ mesoscopic realism and noninvasive measurability that apply when not all outcomes of measurement can be identified as arising from one of two macroscopically distinguishable states.",
"We show how quantum mechanics predicts a negation of the LG premises for proposals involving ideal-negative-result, weak and quantum non-demolition measurements on dynamical entangled systems, as might be realised with two-well Bose-Einstein condensates, path-entangled NOON states and atom interferometers.",
"Schrodinger raised the apparent inconsistency between macroscopic realism and quantum macroscopic superposition states [1].",
"Leggett and Garg (LG) suggested to test macroscopic realism against quantum mechanics in an objective sense by comparing the predictions of quantum mechanics with those based on two very general classical premises [2].",
"The first premise is macroscopic realism (MR), that a system which has two macroscopically distinguishable states available to it is at any time in one or other of the states.",
"The second premise is noninvasive measurability (NIM), that for such a system it is possible to determine which state the system is in, without interfering with the subsequent evolution of that system.",
"Leggett and Garg showed how the two premises constrain the dynamics of a two-state system.",
"Considering three successive times $t_{3}>t_{2}>t_{1}$ , the variable $S_{i}$ denotes which of the two states the system is in at time $t_{i}$ , the respective states being denoted by $S_{i}=+1$ or $-1$ .",
"The LG premises implblacky the LG inequality [2], [3] $LG=\\langle S_{1}S_{2}\\rangle +\\langle S_{2}S_{3}\\rangle -\\langle S_{1}S_{3}\\rangle & \\le & 1.$ and also the “disturbance” or “no signalling in time (NST)” inequality $d_{\\sigma }=\\langle S_{3}|\\hat{M}_{2},\\sigma \\rangle -\\langle S_{3}|\\sigma \\rangle =0$ [4], [5].",
"Here $\\langle S_{3}|\\hat{M}_{2},\\sigma \\rangle $ (and $\\langle S_{3}|\\sigma \\rangle $ ) is the expectation value of $S_{3}$ given that a measurement $\\hat{M}_{2}$ is performed (or not performed) at time $t_{2}$ , conditional on the system being prepared in a state denoted $\\sigma $ at time $t_{1}$ .",
"These inequalities can be violated for quantum systems [6], [2], [9], [8], [7], [10], [11], [5], [4], [3], [13], [12].",
"blueThe work of LG represented an advance, since it extended beyond the quantum framework to show how the macroscopic superposition defies classical macroscopic reality.",
"The LG approach raised new ideas about how to test quantum mechanics even at the microscopic level [8], [9], [6], [7], [10].",
"Failure of the inequalities implies no classical trajectory exists between successive measurements: either the system cannot be viewed as being in a definite state independent of observation, or there cannot be a way to determine that state, without interference by the measurement.",
"Noninvasive measurability would seem “vexing” to justify, however, because of the plausibility of the measurement process disturbing the system.",
"LG countered this problem by proposing an ideal negative result measurement (INR): the argument is conditional on the first postulate being true e.g.",
"if a photon does travel through one slit or the other, a null detection beyond one slit is justified to be noninvasive [2], [8].",
"A second approach is to perform weak measurements [14], [15] that enable calculation of the moment $\\langle S_{2}S_{3}\\rangle $ in a limit where there is a vanishing disturbance to the system [15], [6], [9], [10].",
"To date, experimental investigations involving INR or weak measurements have focused mostly on microscopic systems e.g.",
"a single photon.",
"An exception is a recent experiment which gives evidence for violation of MR using a simpler form of LG inequality that quantifies the invasiveness of “clumsy” measurements, and is applied to superconducting flux qubits [4].",
"There have also been recent proposals for LG tests involving macroscopic mechanical oscillators [12] and for macroscopic states of cold atoms, using quantum non-demolition (QND) measurements [13]blue.blackred An illuminating LG test would be for a mesoscopic massive system in a quantum superposition of being at two different locations [16].",
"An example of such a superposition is the path-entangled NOON state, written as $|\\psi \\rangle =\\frac{1}{\\sqrt{2}}\\lbrace |N\\rangle _{a}|0\\rangle _{b}+|0\\rangle _{a}|N\\rangle _{b}\\rbrace $ where $|N\\rangle _{a/b}$ is the $N$ -particle state for a mode $a$ ($b$ ) [17].",
"In this case the ideal negative result measurement can be applied, and justified as noninvasive by the assumption of Bell's locality [18].",
"A method is then given to (potentially) negate that the system must be located either “here” or “there”, or else to conclude there is a significant disturbance to a massive system due to a measurement performed at a different location.",
"In this paper, we show how such tests may be possible on a mesoscopic scale.",
"As one example, we show that LG violations are predicted for Bose-Einstein condensates (BEC) trapped in two separated potential wells of an optical latttice.",
"Here dynamical oscillation of large groups of atoms to form NOON macroscopic superposition states is predicted at high nonlinearities [20], [19], [22], [23], [21].",
"A key problem however for an actual experimental realisation is the fragility of the macroscopic superpositions.",
"To address this problem, we derive modified LG inequalities that can be used to test LG premises for superpositions that deviate from the ideal NOON superposition by allowing mode population differences not equal to $-N$ or $N$ .",
"The ideal negative result measurement is difficult to apply where there are residual atoms in both modes, and we thus develop weak and QND measurement strategies for testing LG premises and demonstrating mesoscopic quantum coherence in this case.",
"Tests of LG realism are also possible for NOON states incident on an interferometer.",
"Finally, we propose a simple LG test for matter waves passing through an atom interferometer could demonstrate the no-classical trajectories result for atoms.",
"Idealised dynamical two-state oscillation: The Hamiltonian $H_{I}$ for an $N$ -atom condensate constrained to a double well potential reveals a regime of macroscopic two-state dynamics.",
"The two-well system has been reliably modelled by the Josephson two-mode Hamiltonian [24], [20], [21], [25]: $H_{I}=2\\kappa \\hat{J}_{z}+g\\hat{J}_{z}^{2}$ Here $\\hat{J}{}_{z}=(a^{\\dagger }a-b^{\\dagger }b)/2$ , $\\hat{J}{}_{x}=(a^{\\dagger }b+b^{\\dagger }a)/2$ , $\\hat{J}{}_{y}=(a^{\\dagger }b-b^{\\dagger }a)/2i$ are the Schwinger spin operators defined in terms of blackthe boson operators $\\hat{a}^{\\dagger },\\hat{a}$ and $\\hat{b}^{\\dagger },\\hat{b}$ , for the modes describing particles in each of the wells, labelled $a$ and $b$ respectively.red blackThe $\\kappa $ models interwell hopping and $g$ the nonlinear self-interaction due to the medium.",
"In a regime of high interaction strength ($Ng/\\kappa \\gg 1$ black), a regime exists where if the system is initially prepared with all $N$ atoms in one well, a two-state oscillation can take place with period $T_{N}$ (Fig.",
"1) [19], [20].",
"In one state, $|N\\rangle _{a}|0\\rangle _{b}$, all $N$ atoms are in the well $a$ ($S_{i}=1$ ), and in the second state, $|0\\rangle _{a}|N\\rangle _{b}$, all atoms are in the well $b$ ($S_{i}=-1$ ) [20].",
"If the system is prepared in $|N\\rangle _{a}|0\\rangle _{b}$, then at a later time $t^{\\prime }$ , the state vector is red black(apart from phase factors)red $|\\psi (t)\\rangle & = & \\cos (\\tau )|N\\rangle |0\\rangle +\\sin (\\tau )|0\\rangle |N\\rangle $ where $\\tau =E_{\\Delta }t^{\\prime }/\\hbar $ and $E_{\\Delta }$ is the energy splitting of the energy eigenstates $|N\\rangle |0\\rangle \\pm |0\\rangle |N\\rangle $ under $H_{I}$ .",
"The quantum solution (REF ) predicts a violation of the LG inequality.",
"The two-time correlation is $\\langle S_{i}S_{j}\\rangle =\\cos \\left[2(t_{j}-t_{i})\\right]$ and is independent of the initial state, whether $|N\\rangle |0\\rangle $ or $|0\\rangle |N\\rangle $ .",
"Choosing $t_{1}=0$ , $t_{2}=\\pi /6$ , $t_{3}=\\pi /3$ (or $t_{3}=5\\pi /12$ ), it is well-known that for this two-time correlation the quantum prediction is $LG=1.5$ ($1.37$ ) which gives a violation of (REF ) [2].",
"blackThe tunnelling times in the highly nonlinear regime however are impractically high for proposals based on Rb atoms [26], [27].",
"The fragility of the macroscopic superposition state will make any such experiment unfeasible [28].",
"Noting however that the modes $a$ , $b$ of $H_{I}$ may also describe occupation of two atomic hyperfine levels, $\\kappa $ being the Rabi frequency as in the experiments of [25], the NOON oscillation may well be achievable for other physical realisations of $H_{I}$ .",
"Alternatively, for more practical oscillation times one can use a different initial state $|N-n_{L}\\rangle |n_{L}\\rangle $ , $0<n_{L}<N$ , where there are atoms in both wells, or else a tilted well [20].",
"Here, we denote the sign of the spin $J_{z}$ at time $t_{i}$ by $S_{i}$ ($S_{i}=1$ if $J_{z}\\ge 0$ ; $S_{i}=-1$ if $J_{z}<0$ ).",
"The dynamical solutions presented in Fig.",
"2 reveal a mesoscopic two-state oscillations over reduced time scales, mimicking the experimentally observations of Albiez et al [27] for $N=1000$ atoms where oscillations were observed over milliseconds.",
"Figure: blackOscillating NOON two-state dynamicsblack:(a) Probabilities of nn atoms in mode aa at times 0,T N /6,T N /30,\\ T_{N}/6,\\ T_{N}/3and (b)red the two-state oscillation for N=100N=100, g=1g=1.",
"(c) Plot gives an upper bound on the backaction δ\\delta dueto the INR measurement that can be tolerated for an LG violation.redredblueredOur objective is to provide practical strategies for testing the LG inequality in such multiparticle experiments.",
"Two questions to be addressed are how to perform (or access the results of) the NIM (assuming it exists), and how to handle the case where the values of $S_{i}$ may not always correspond to macroscopically distinct outcomes.",
"To address the first question: As explained in the literature [2], black$\\langle S_{1}S_{2}\\rangle $ and $\\langle S_{1}S_{3}\\rangle $ can be inferred using deterministic state preparation and projective measurements at $t_{2}$ and $t_{3}$ .",
"To measure $\\langle S_{1}S_{3}\\rangle $ no intervening measurement is made at $t_{2}$ based on the assumption that the NIM at $t_{2}$ will not affect the subsequent statistics.",
"blackFor $\\langle S_{2}S_{3}\\rangle $ , the evaluation of $S_{2}$ is difficult, since with any practical measurement it could be argued that a measurement $M$ made at $t_{2}$ is not the NIM, and does indeed influence the subsequent dynamics.",
"Three methods have been used to counter this objection: INR measurements; weak measurements; and quantifiable QND measurements.",
"We next propose LG tests for each case.",
"(1) Ideal negative result measurement (INR): A particularly strong test is possible for experiments involving a NOON superposition (3) where the two modes correspond at time $t_{2}$ to spatially separated locations.",
"In this case, the INR strategy similar to that outlined by LG can be applied.",
"A measurement apparatus at time $t_{2}$ couples locally to only one mode $a$ , enabling measurement of the particle number $n_{a}$ .",
"Either $n_{a}=0$ or $n_{a}=N$ .",
"Based on the first LG premise, if one obtains the negative result $n_{a}=0$ , it is assumed that there were prior to the measurement no atoms in the mode $a$ .",
"Hence the measurement that gives a negative result is justified to be noninvasive (since $\\langle S_{2}S_{3}\\rangle $ can be evaluated using only negative result outcomes [2]).",
"For such an experiment, to assume noninvasive measurability there is implicit the assumption of locality: that there is no change to mode $b$ because of the measurement at $a$ (otherwise a change to the subsequent dynamics could be expected).",
"Figure: Mesoscopic two-state oscillations:red Probabilityof nn atoms in mode aa at t 2 t_{2} and t 3 t_{3}.",
"N=100N=100.",
"Theinitial state is n L =10n_{L}=10 particles in the right well.Here (a) g=3g=3 and (b) g=1g=1.",
"(a) LG=1.49LG=1.49 using the weak measurementof J ^ z \\hat{J}_{z} at t 2 t_{2} redredand(b)purple LG=1.43LG=1.43 using the minimal QND measurementof S 2 S_{2} at t 2 t_{2}.redblack blackredQuantification of the NIM premise: We can introduce a quantification of the second LG premise: We suppose that the measurement at mode $a$ ($b$ ) can induce a back-action effect on the macroscopic state of the other mode, so that there may be a change of the state of mode $b$ ($a$ ) of up to $\\delta $ particles, where $\\delta \\le N$ .",
"The change $\\delta $ may be microscopic, not great enough to switch the system between states $|0\\rangle |N\\rangle $ and $|N\\rangle |0\\rangle $ , but can alter the subsequent dynamics.",
"The change to the dynamics is finite however, and can be established within quantum mechanics, to give a range of prediction for $\\langle S_{2}S_{3}\\rangle $ .",
"We have carried out this calculation, and plot the effect of $\\delta $ for various $N$ in Fig.",
"1c, noting that a relatively small backaction $\\delta $ to the quantum state of one mode due to measurement on the other will destroy violations of the LG inequality even for large $N$ .",
"(2) Weak and minimally invasive QND measurements: A second strategy is to construct a measurement that can be shown to give a negligible disturbance to the system being measured.",
"We consider a QND measurement described by the Hamiltonian $H_{Q}=\\hbar G\\hat{J}_{z}\\hat{n}_{c}$ .",
"The $H_{Q}$ models QND measurements of the atomic spin $\\hat{J}_{z}$ based on an ac Stark shift [30].",
"An optical “meter” field is prepared in a coherent state $|\\gamma \\rangle $ and coupled to the system for a time $\\tau _{0}$ .",
"The meter field is a single mode with boson operator $\\hat{c}$ and number operator $\\hat{n}_{c}=\\hat{c}^{\\dagger }\\hat{c}$ .",
"Writing the state of the system at time $t_{2}$ as $\\sum _{m=0}^{N}d_{m}|m\\rangle _{a}|N-m\\rangle _{b}$ ($d_{m}$ are probability amplitudes), the output state immediately after measurement is (we set $\\tau _{0}=\\pi /2NG$ )blue $|\\psi \\rangle & = & \\sum _{m=0}^{N}d_{m}|m\\rangle _{a}|N-m\\rangle _{b}|\\gamma e^{i\\pi (N-2m)/2N}\\rangle _{c}$ Homodyne detection enables measurement of the meter quadrature phase amplitude $\\hat{p}=(\\hat{c}-\\hat{c}^{\\dagger })/i$ .",
"For $\\gamma $ large, the measurement is “strong”, or projective, and the different values of $\\hat{J}_{z}$ (and hence $S_{2}$ ) are precisely measurable as distinct regions of outcomes for $\\hat{p}$ .",
"Ideally, a minimal QND measurement of $S_{2}$ is devised, that does not discriminate the different values of $\\hat{p}$ apart from the sign, and leaves all states (REF ) with definite sign $S_{2}$ unchanged.",
"First, we discuss the weak measurement limit, attained as $\\gamma \\rightarrow 0$ .",
"In this limit, we see from (REF ) that the state of the system is minimally disturbed by the measurement.",
"The cost is no clear resolution of the value $S_{2}$ for any single measurement.",
"Yet, using (REF ), we show in the Supplemental Materials that if the system at time $t_{2}$ is in a NOON state $d_{0}|N-n_{L}\\rangle _{a}|n_{L}\\rangle _{b}+d_{N}|n_{L}\\rangle |N-n_{L}\\rangle $ , thenpurple red $\\langle S_{2}S_{3}\\rangle & = & -\\frac{1}{2\\gamma }\\langle pS_{3}\\rangle $ blackwhere $\\langle S_{2}S_{3}\\rangle $ blueblackis the value obtained by the projective measurement.",
"blackThus, for arbitrarily small $\\gamma $ , the value $\\langle S_{2}S_{3}\\rangle $ can be obtained by averaging over many trials.",
"The weak measurement strategy enables a convincing test of the LG premises, since one can experimentally demonstrate the noninvasiveness of the weak measurement, by showing the invariance of $\\langle S_{1}S_{3}\\rangle $ as $\\gamma \\rightarrow 0$ when the measurement is performed at $t_{2}$ .",
"The weak measurement relation (REF ) does not hold for all input states.",
"However, the minimal (“non-clumsy”) QND measurement of $S$ gives a strategy for LG tests, based on extra assumptions.",
"For systems such as in Fig.",
"2, the state at time $t_{2}$ is a superposition of states $|\\psi _{+}\\rangle $ and $|\\psi _{-}\\rangle $ that give, respectively, outcomes $S=\\pm 1$ .",
"The first LG premise is that the system is either in a state of positive $S$ or in a state with negative $S$ .",
"The minimal QND strategy requires a second set of measurements, in order to experimentally establish that states with definite value of $S$ are unchanged by the QND measurement black[13], [31], [4].",
"The noninvasiveness of the measurement is then justified by the first LG premise.",
"blackWe note that in cases where the measurement is not ideal (“clumsy”), the amount of disturbance can be measured and accounted for in a modified inequality as discussed in the Refs.",
"[13], [31], [4].",
"Strictly speaking, the QND approach is limited to testing a modified LG assumption that the system is always in a quantum state with definite $S$ at the time $t_{2}$ .",
"This is because it is difficult to prove that all hidden variable states with definite outcome of $S$ are not changed by the QND measurement.",
"Regardless, the approach rigorously demonstrates the quantum coherence between the states $|\\psi _{+}\\rangle $ and $|\\psi _{-}\\rangle $ .",
"Fig.",
"2 shows LG violations using weak and QND measurements.",
"Figure: blackViolation of ss-scopic LG inequalitiesusing nonlinear interferometers:black redblackredblackredblack(a)The NOON state (3) is created at time t 2 t_{2} (N=5N=5, τ=π/6\\tau =\\pi /6)and evolves to time t 3 t_{3} according to H I H_{I} with nonlinearitygg.purple blackredblackContoursshow regimes for violation of the ss-scopic inequalitypurpleblack() where (dark to light) s=4,2,0s=4,2,0.redredblack(b)Schematic of the probability distribution for results 2J z 2J_{z}blackat t 3 t_{3} depicting three regions 1, 0,and 2.The $s$ -scopic LG inequalities: We now address how to test macroscopic realism where the system deviates from the ideal of two macroscopically distinguishable states.",
"This occurs when there is a nonzero probability for $J_{z}$ different to $\\pm N/2$ as in Figure 3b.",
"Adapting the approach put forward by LG and Refs.",
"[32], we define three regions of $J_{z}$ : region “1”, $J_{z}<-s/2$ ; region “0”, $-s/2\\le J_{z}\\le s/2$ ; and region “2”, $J_{z}>s/2$ .",
"blue For arbitrary $s$ , the MR assumption is accordingly renamed, to s-scopic realism (sR).",
"In the generalised case, the meaning of sR is that the system is in a probabilistic mixture of two overlapping states: the first that gives outcomes in regions “1” or “0” (denoted by hidden variable $\\tilde{S}=-1$ ); the second that gives outcomes in regions “0” or “2” (denoted by $\\tilde{S}=1$ ).",
"The second LG premise is generalised to $s$ -scopic noninvasive measurability which asserts that such a measurement can be made at time $t_{2}$ without changing the result $J_{z}$ at time $t_{3}$ by an amount $s$ or more.",
"blackThe $s$ -scopic LG premises imply a quantifiable inequality.",
"This is because any effects due to the overlapping region are limited by the finite probability of observing a result there.",
"Defining the measurable marginal probabilities of obtaining a result in region $j\\in \\left\\lbrace 0,1,2\\right\\rbrace $ at the time $t_{k}$ by $P_{j}^{(k)}$ , the $s$ -scopic premises are violated if [29]red $LG_{s}=P_{2}^{(2)}-P_{1}^{(2)}+\\langle S_{2}S_{3}\\rangle -(P_{2}^{(3)}-P_{1}^{(3)})-2P_{0|M}^{(3)}-P_{0}^{(3)}>1$ blackwhere we have used that the system is prepared initially in region 2 and here we restrict to scenarios satisfying $P_{0}^{(2)}=0$ .black The $\\langle S_{2}S_{3}\\rangle $ is to be measured using a noninvasive measurement at $t_{2}$ .",
"redA similar modification is given for the disturbance inequality: The $s$ R premises are violated if $d_{\\sigma ,s}=|P_{2|M}^{(3)}-P_{1|M}^{(3)}-(P_{2}^{(3)}-P_{1}^{(3)})|-(P_{0}^{(3)}+P_{0|M}^{(3)})>0$ where $P_{j|M}^{(3)}$ ($P_{j}^{(3)}$ ) is the probability with (without) the measurement $M$ performed at $t_{2}$ .",
"Figure: Multiparticle linear interferometers:black (a)redblackNN bosons pass through an interferometer.",
"AQND measurement M ^\\hat{M} (purple shading) is made on the state createdat t 2 t_{2}.",
"(b) (blue solid curve) Violation of LG inequalityfor optimal rotation angles θ\\theta , φ\\phi where M ^\\hat{M}blackmeasures the number of particles in arm cc, showing “no classicaltrajectories”red blackfor individualatoms.red black(b) (dashed curves)LG violations when M ^\\hat{M} is reda minimal (non-clumsy)measurement of S 2 S_{2}.",
"blackGreen dotted-dashed curveshows the disturbance d σ =2d_{\\sigma }=2 (for all Δ<N\\Delta <N) wheremesoscopic superposition states |ψ Δ 〉|\\psi _{\\Delta }\\rangle are createdat t 2 t_{2}black.",
"Red dashed curve shows LG value forodd NN where Δ=0\\Delta =0.",
"For Δ=N-1\\Delta =N-1, the NOON state (3)with τ=θ\\tau =\\theta is created at t 2 t_{2}.",
"M ^\\hat{M} can thenbe realised as an INR or weak measurement.redredblackFig (c) shows violationsin that case using a phase shift φ\\phi and a 50/50 BS2BS2 (withoptimal θ\\theta ).",
"redblackredblackredblackredredNonlinear and linear interferometers: Figure 3a shows predictions for $s$ -scopic violations.",
"A NOON state (3) is created at $t_{2}$ and a weak measurement or INR performed.",
"The NOON state might be created via the nonlinear $H_{I}$ or by the conditional methods that have been blackapplied to photonic states [33].",
"Subsequently, the system evolves according to the nonlinear interaction $H_{I}$ and a measurement is made of $J_{z}$ at $t_{3}$ .",
"For realistic timescales, there is a spread of $J_{z}$ at the times $t_{3}$ (Fig.",
"3b).",
"These regimes are realisable for finite $g$ and $N\\sim 100$ in BEC nonlinear interferometers [25].red LG tests with mesoscopic superposition/ NOON states are also possible without nonlinearity at $t>t_{2}$ , if, after the weak/ INR measurement at $t_{2}$ , the two modes are combined across a variable-angle beam splitter (or beam splitter with phase shift $\\phi $ ) and $J_{z}$ of the outputs measured (Fig.",
"4).black redblackThe macroscopic Hong-Ou-Mandel technique conditions on $|J_{z}|>\\Delta /2$ ($\\Delta <N$ ) to create at $t_{2}$ an $N$ -atom mesoscopic superposition state $|\\psi _{\\Delta }\\rangle =|\\psi _{+}\\rangle +|\\psi _{-}\\rangle $ where states $|\\psi _{\\pm }\\rangle $ are distinct by more than $\\Delta $ particles in each arm of the interferometer [33].",
"Violations of the disturbance and LG inequalities are plotted in Figs.",
"4b and c. Results indicate small violations for $s\\sim 2$ over a range of $N$ and $\\Delta $ [29].",
"No-classical trajectories for atom interferometers: Finally, we propose a simple test to falsify classical trajectories in the multi-particle case for simple interferometers.",
"At $t_{1}$ , $N$ particles pass through a polariser beam splitter (or equivalent) (BS1) rotated at angle $\\theta $ (Fig.",
"4).",
"The number difference of the outputs if measured indicate the value of $J_{\\theta }$ (and $S_{2}$ ) at $t_{2}$ .",
"The particles are then incident on a second beam splitter $BS2$ at angle $\\phi $ whose output number difference gives $S_{3}$ at $t_{3}$ .",
"blackWe invoke the premise, that the system is always in a state of definite $J_{\\theta }$ prior to measurement at $t_{2}$ .",
"This is based on the hypothesis that blackeachblack atom goes one way blackorblack the other, through the paths of the interferometer.",
"blackA second premise is also invoked, that a measurement could be performed of $J_{\\theta }$ at $t_{2}$ that does not disturb the subsequent evolution.",
"blackThe second premise is justified by the first, and can be supported by experiments that create a spin eigenstate, and then demonstrate the complete invariance of the state after the QND number measurement.black If the premises are valid, the LG inequalities (REF ) will hold, but by contrast are predicted violated by quantum mechanics (Fig.",
"4b (blue solid curve)).",
"blackWhile not the macroscopic test LG envisaged, this gives an avenue for workable tests of the “classical trajectories” hypothesis that could be applied to atoms [34]."
]
] | 1612.05726 |
[
[
"Symmetry of the Definition of Degeneration in Triangulated Categories"
],
[
"Abstract Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety.",
"Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit.",
"Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences.",
"Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories.",
"The definition has an intrinsic non-symmetry.",
"Suppose that we have a triangulated category in which idempotents split and either for which the endomorphism rings of all objects are artinian, or which is the category of compact objects in an algebraic compactly generated triangulated K-category.",
"Then we show that the non-symmetry in the algebraic definition of the degeneration is inessential in the sense that the two possible choices which can be made in the definition lead to the same concept."
],
[
"Introduction",
"For a finite dimensional $K$ -algebra over an algebraically closed field $K$ the set of $d$ -dimensional $A$ -modules is just the space of $K$ -algebra homomorphisms from $A$ to the algebra of $d$ by $d$ matrices over $K$ .",
"It carries therefore the structure of an algebraic variety $mod(A,d)$ , and allows a $GL_d(K)$ -action given by conjugation of matrices.",
"$GL_d(K)$ -orbits correspond to isomorphism classes of modules, and we say that a $d$ -dimensional module $M$ corresponding to the point $m\\in mod(A,d)$ degenerates to the module $N$ with corresponding point $n\\in mod(A,d)$ if $n$ belongs to the Zariski-closure of the orbit $GL_d(K)\\cdot m$ .",
"We write in this case $M\\le _{\\text{deg}}N$ .",
"It is clear that $\\le _{\\text{deg}}$ is a partial order on the set of isomorphism classes of finite dimensional $A$ -modules.",
"Zwara and Riedtmann defined another relation between $A$ -modules, namely $M\\le _{\\text{Zwara}} N$ if and only if there is a finite dimensional $A$ -module $Z$ and a short exact sequence $0\\rightarrow N\\rightarrow M\\oplus Z\\rightarrow Z\\rightarrow 0$ .",
"Moreover, they showed in [9], [20] $M\\le _{\\text{deg}}N\\Leftrightarrow M\\le _{\\text{Zwara}} N.$ Zwara showed in [19] by a purely algebraic arguments that in the category of finite dimensional modules over an algebra there is $Z$ and a short exact sequence $0\\rightarrow N\\rightarrow M\\oplus Z\\rightarrow Z\\rightarrow 0$ if and only if there is $Z^{\\prime }$ and a short exact sequence $0\\rightarrow Z^{\\prime }\\rightarrow M\\oplus Z^{\\prime }\\rightarrow N\\rightarrow 0$ .",
"In joint work with Jensen and Su [4], and independently by Yoshino in [15] for the (triangulated) stable category of maximal Cohen Macaulay modules over local Gorenstein $k$ -algebras, the concept $\\le _{\\text{Zwara}}$ was generalised in the obvious way to general triangulated categories.",
"More precisely, for a triangulated category $\\mathcal {T}$ we define for two objects $M$ and $N$ that $M\\le _\\Delta N$ if and only if there is an object $Z$ and a distinguished triangle $Z\\stackrel{u\\atopwithdelims ()v}{\\longrightarrow } M\\oplus Z\\stackrel{}{\\longrightarrow } N\\rightarrow Z[1]$ .",
"Yoshino insisted in the point that one should ask that the induced endomorphism $v$ of $Z$ is nilpotent.",
"Using Fitting's lemma and possibly replacing $Z$ by a suitable direct summand, this is automatic if one assumes Krull-Schmidt properties and artinian endomorphism rings for all objects.",
"We denote $M\\le _{\\Delta +\\text{nil}}N$ if $M\\le _\\Delta N$ and the induced endomorphism $v$ on $Z$ is nilpotent.",
"The concept $\\le _\\Delta $ was used in an essential way in work of Keller and Scherotzke on Nakajima quiver varieties.",
"[4] concentrated on partial order properties of $\\le _\\Delta $ .",
"Further conditions guaranteeing partial order properties of $\\le _\\Delta $ can be found for various situations in [17], [16] and [11].",
"In this latter reference a geometric setting was developed replacing the module variety $mod(A,d)$ for general triangulated categories, mimicking for this purpose Yoshino's scheme theoretic approach [15].",
"Various results were given that ensure that $\\le _\\Delta $ or $\\le _{\\Delta +\\text{nil}}$ define partial orders on the isomorphism classes of objects of $\\mathcal {T}$ .",
"Some authors define $\\le _\\Delta $ (resp.",
"$\\le _{\\Delta +\\text{nil}}$ ) by the existence of a distinguished triangle $N\\rightarrow M\\oplus Z\\stackrel{(u_r\\;v_r)}{\\longrightarrow } Z\\rightarrow N[1],$ and some define it as the existence of a distinguished triangle $Z\\stackrel{{u_\\ell }\\atopwithdelims (){v_\\ell }}{\\longrightarrow } M\\oplus Z\\rightarrow N\\rightarrow Z[1].$ Passing to the opposite category the two definitions relate to each other.",
"Note that the opposite category of a triangulated category is triangulated as well.",
"However, we show in this paper that actually the situation is even better.",
"The two possible definitions lead to the same relation on the isomorphism classes of objects in two important cases.",
"Our main result is the following.",
"In its statement and in the rest of the paper `artinian ring' means `left and right artinian'.",
"Theorem 1 Let $K$ be a commutative ring and let $\\mathcal {T}$ be a $K$ -linear triangulated category satisfies one of the following two hypotheses Idempotents split in $\\mathcal {T}$ and all endomorphism algebras of objects are artinian, $\\mathcal {T}$ is the category of compact objects in an algebraic compactly generated triangulated $K$ -category.",
"Then for any objects $M,N$ of $\\mathcal {T}$ , the following assertions are equivalent: There is an object $Z_\\ell $ of $\\mathcal {T}$ and a distinguished triangle in $\\mathcal {T}$ $Z_\\ell \\stackrel{\\begin{pmatrix} v_\\ell \\\\ u_\\ell \\end{pmatrix}}{\\longrightarrow }Z_\\ell \\oplus M\\stackrel{}{\\longrightarrow } N\\longrightarrow Z_\\ell [1],$ where $v_\\ell $ is a nilpotent endomorphism of $Z_\\ell $ .",
"There is an object $Z_r$ of $\\mathcal {T}$ and a distinguished triangle in $\\mathcal {T}$ $N\\stackrel{}{\\longrightarrow }M\\oplus Z_r\\stackrel{\\begin{pmatrix} u_r & v_r\\end{pmatrix}}{\\longrightarrow } Z_r\\longrightarrow N[1],$ where $v_r$ is a nilpotent endomorphism of $Z_r$ .",
"It should be noted that since any (left or right) artinian ring is semiperfect (see [12])), under the first situation of the theorem, the category $\\mathcal {T}$ is Krull-Schmidt (see [2]).",
"Moreover, under this hypothesis the assumption that $v_\\ell $ (resp.",
"$v_r$ ) is nilpotent is inessential.",
"Indeed, a Fitting lemma type argument can then by applied and this shows that we can split off a trivial distinguished triangle as direct factor such that the remaining direct factor distinguished triangle satisfies the nilpotency hypothesis (cf Remark REF below for more details).",
"We cannot avoid the artinian hypothesis in the first and the nilpotency hypothesis in both cases.",
"The proof in the first case follows Zwara's arguments in [19] in the classical case, but there are quite a few subtleties arising by the non-uniqueness in the TR3-axiom of triangulated categories.",
"Zwara frequently uses pushouts and pullbacks and in particular universal properties which come along with these concepts.",
"We replace these constructions by homotopy cartesian squares, and have to cope with the lack of uniqueness of the related construction.",
"The proof in the second case is much more involved and heavily uses the concepts developed in [10].",
"The main idea in this approach is to use a dualisation functor like the $K$ -duals for ordinary $K$ -algebras $A$ .",
"However the situation is more involved here.",
"The hypothesis that the triangulated category is the category of compact objects in an algebraic and compactly generated triangulated category gives that it is actually equivalent to the category of compact objects in the derived category of some small dg-category.",
"Then, the new approach is to see this derived category as the derived category $\\mathcal {D}(A)$ of some dg algebra without unit $A$ , but with sufficiently many idempotents in a certain sense.",
"Then, it can be shown that one may dualise with respect to $A$ , using the derived functor of the suitable contravariant $Hom$ functor to $A$ .",
"Further we use in particular the main result of [11] in full generality.",
"The theory of dg algebras with enough idempotents parallels in a certain sense the development of dg categories as given by Keller but the situation is new.",
"The approach is presented in [10], and we believe that such a theory is highly useful and should provide many further applications.",
"The paper is organised as follows.",
"In Section we give a summary of the contents of reference [10], in order to provide the vocabulary needed to understand the proof of the main result in the main body of the paper, without being obliged to go into the full details of that reference.",
"In Section we give the relevant background, facts and definitions of degenerations of objects in module categories, as well as in triangulated categories as it was shown in our earlier papers [4], [11].",
"Section then proves the main result Theorem REF under the hypothesis (a), i.e.",
"in case all objects in the triangulated category have artinian endomorphism ring.",
"The final Section then gives the proof of Theorem REF under hypothesis (b), i.e.",
"in the case of a triangulated category which is the category of compact objects in an algebraic compactly generated triangulated category."
],
[
"Review on triangulated categories, dg-categories and dg-algebras with enough idempotents",
"For the proof of Theorem REF under hypothesis (b), which will cover Section , we shall need some concepts and statements from the theory of dg-algebras, dg-categories and triangulated categories in general which are not standard.",
"In particular in case of categories which do not satisfy Krull-Schmidt theorem, we proceed by considering dg algebras without units, but having enough idempotents.",
"The complete theory can be found in [10].",
"In order to facilitate the reading we summarize the results of this latter reference and introduce this way also the notations used in Section .",
"All throughout the rest of the paper, let $K$ be a commutative ring with unit and all categories which appear all assumed to be $K$ -categories.",
"The unadorned symbol $\\otimes $ will stand for the tensor product over $K$ ."
],
[
"dg categories and dg functors",
"Recall that a differential graded (dg) $K$ -module is a ${\\mathbb {Z}}$ -graded $K$ -module $V$ with a graded endomorphism $d:V\\longrightarrow V$ of degree 1 and square 0, called the differential (here and all throughout the paper, when the term `differential' is used to denote a graded map $d$ , it will be assumed, without further remark, that $d\\circ d=0$ and that $d$ is graded and of degree $+1$ ).",
"We denote by $Dg-K$ or $\\mathcal {C}_{dg}K$ the category of dg $K$ -modules.",
"The morphism space $\\text{HOM}_K(V,W)$ in this category is again a dg $K$ -module, where the homogeneous component of degree $n$ , denoted $\\text{HOM}_K^n(V,W)$ , consists of the homogeneous morphisms of degree $n$ .",
"The differential is given by $d_{Hom}(\\alpha )=d_W\\circ \\alpha -(-1)^{|\\alpha |}\\alpha \\circ d_V$ , for any homogeneous morphism $\\alpha \\in \\text{HOM}_K(V,W)$ , where $|\\;\\cdot \\;|$ denotes the degree.",
"A dg category $\\mathcal {A}$ (see [5] or [6]) is a category such that the morphism spaces are dg $K$ -modules and the composition map $\\text{Hom}_\\mathcal {A}(B,C)\\otimes \\text{Hom}_\\mathcal {A}(A,B)\\longrightarrow \\text{Hom}_\\mathcal {A}(A,C)$ satisfies Leibniz rule $d(g\\circ f)=d(g)\\circ f+(-1)^{|g|}g\\circ d(f)$ , for all homogeneous morphisms $f\\in \\text{Hom}_\\mathcal {A}(A,B)$ and $g\\in \\text{Hom}_\\mathcal {A}(B,C)$ , where, abusing notation, we have denoted by $d$ the differential on any of the appearing $Hom$ spaces.",
"The category $Dg-K$ (denoted by $\\mathcal {C}_{dg}K$ in [6]) is the prototype of a dg category.",
"With any such category, one canonically associates its 0-cycle category $Z^0\\mathcal {A}$ and its 0-homology category $\\mathcal {H}^0\\mathcal {A}$ .",
"Both of them have the same objects as $\\mathcal {A}$ , and as morphisms one puts $\\text{Hom}_{Z^0\\mathcal {A}}(A,A^{\\prime })=Z^0(\\mathcal {A}(A,A^{\\prime }))$ and $\\text{Hom}_{H^0\\mathcal {A}}(A,A^{\\prime })=H^0(\\mathcal {A}(A,A^{\\prime }))$ , for all $A,A^{\\prime }\\in \\text{Ob}(\\mathcal {A})$ , the composition of morphisms in both cases being induced by the composition in $\\mathcal {A}$ .",
"A dg functor $F:\\mathcal {A}\\longrightarrow \\mathcal {B}$ between dg categories is just a functor which preserves the grading and the differential of Hom spaces.",
"Any dg functor $F:\\mathcal {A}\\longrightarrow \\mathcal {B}$ induces corresponding functors $F=Z^0F:Z^0\\mathcal {A}\\longrightarrow Z^0\\mathcal {B}$ and $F:H^0F:H^0\\mathcal {A}\\longrightarrow H^0\\mathcal {B}$ .",
"Associated to $\\mathcal {A}$ , there is also the opposite dg category $\\mathcal {A}^{op}$ and, given another dg category $\\mathcal {B}$ , there is a definition of tensor product of dg categories $\\mathcal {A}\\otimes \\mathcal {B}$ .",
"A homological natural transformation of dg functors $\\tau :F\\longrightarrow G$ is a natural transformation such that $\\tau _A\\in Z^0(\\text{Hom}_\\mathcal {B}(F(A),G(A)))$ , for any object $A\\in \\mathcal {A}$ .",
"If we have dg functors $F:\\mathcal {A}\\longrightarrow \\mathcal {B}$ and $G:\\mathcal {B}\\longrightarrow \\mathcal {A}$ , then we have induced dg functors $\\mathcal {A}^{op}\\otimes \\mathcal {B}\\longrightarrow Dg-K$ , given by $\\text{Hom}_\\mathcal {B}(F(?),?",
")$ and $\\text{Hom}_\\mathcal {A}(?,G(?",
"))$ .",
"A dg adjunction is just an adjunction $(F,G)$ of dg functors such that the natural isomorphism $\\text{Hom}_\\mathcal {B}(F(?),?",
")\\stackrel{\\cong }{\\longrightarrow }\\text{Hom}_\\mathcal {A}(?,G(?",
"))$ is a homological natural transformation.",
"See [5] and [10] for the details concerning dg categories and dg functors."
],
[
"dg categories and dg algebras with enough idempotents",
"Any small $K$ -category can be viewed as an algebra with enough idempotents.",
"The latter is a $K$ -algebra $A$ with a distinguished family $(e_i)_{i\\in I}$ of orthogonal idempotents such that $\\bigoplus _{i\\in I}e_iA=A=\\bigoplus Ae_i$ .",
"When such an algebra comes with a grading (as an algebra) such that the $e_i$ are homogeneous of zero degree, and with a differential $d:A\\longrightarrow A$ such that $d(e_i)=0$ , for all $i\\in I$ , and $d$ satisfies Leibniz rule, then $A$ or the pair $(A,d)$ is called a differential graded (dg) algebra with enough idempotents.",
"It is also shown in [10] that such an algebra may be viewed as a small dg category with $I$ as set of objects.",
"To any such algebra $A$ one canonically associates a (non-small) dg category $Dg-A$ , whose objects are right dg $A$ -modules.",
"A right dg $A$ module is just a graded right $A$ -module $M$ together with a differential $d_M:M\\longrightarrow M$ such that $d_M(xa)=d_M(x)a+(-1)^{|x|}xd_A(a)$ , for all homogeneous elements $x\\in M$ and $a\\in A$ .",
"Here and in the rest of the paper, unless otherwise specified, all modules are assumed to be unitary.",
"That is, we assume that $M=MA$ in our case.",
"We denote by $Gr-A$ the category with objects the graded right $A$ -modules and morphisms the graded $A$ -homomorphisms of degree zero.",
"This category comes with a canonical equivalence $?",
"[1]:Gr-A\\longrightarrow Gr-A$ , and we put by $?[n]:=(?",
"[1])^n$ for each $n\\in \\mathbb {Z}$ .",
"Then, for each pair $(M,N)$ of right dg $A$ -modules, the corresponding space of morphisms in $Dg-A$ is given by $\\text{HOM}_A(M,N)=\\bigoplus _{n\\in \\mathbb {Z}}\\text{HOM}_A^n(M,N)$ , where $\\text{HOM}_A^n(M,N):=\\text{Hom}_{Gr-A}(M,N[n])$ .",
"The differential of $\\text{HOM}_A(M,N)$ is the restriction of the differential of $\\text{HOM}_K(M,N)$ (see the first paragraph of Section REF .)",
"One similarly defines the opposite dg algebra with enough idempotents $A^{op}$ and the tensor product $A\\otimes B$ of dg algebras with enough idempotents.",
"One then defines the dg category $A-Dg$ of left dg modules and that of dg $A-B-$ bimodules, which are equivalent to $Dg-A^{op}$ and $Dg-(B\\otimes A^{op})$ , respectively.",
"This allows to treat the theories of left dg modules or dg bimodules over dg algebras with enough idempotents just as right dg modules."
],
[
"Stable and derived category of a dg algebra with enough idempotents",
"The 0-cycle (resp.",
"0-homology) category of $Dg-A$ is denoted by $\\mathcal {C}(A)$ (resp.",
"$\\mathcal {H}(A)$ ).",
"The category $\\mathcal {C}(A)$ is a bicomplete abelian category, with exact sequences as in $Gr-A$ , and, apart from this abelian structure, it also has a Quillen exact structure, called the semi-split exact structure, where the conflations (=admissible short exact sequences) are those exact sequences which split in $Gr-A$ (see [1] for the terminology and main properties of exact categories).",
"With this latter structure $\\mathcal {C}(A)$ is Frobenius, that is, $\\mathcal {C}(A)$ has enough projectives and injectives and the injective objects coincide with the projective ones.",
"The stable category of $\\mathcal {C}(A)$ , which is then triangulated (see [3]), is precisely $\\mathcal {H}(A)$ .",
"This latter (triangulated) category is called the homotopy category of $A$ .",
"The class of quasi-isomorphisms in $\\mathcal {H}(A)$ (i.e.",
"those morphisms which induce isomorphisms on homology) is a multiplicative system compatible with the triangulation in the terminology of Verdier (see [13]).",
"The localization of $\\mathcal {H}(A)$ with respect to the class of quasi-isomorphism, denoted $\\mathcal {D}(A)$ , is the derived category of $A$ .",
"It then has a unique structure of triangulated category such that the canonical functor $q:\\mathcal {H}(A)\\longrightarrow \\mathcal {D}(A)$ is triangulated.",
"In [10] (see, Theorem 3.1 in that reference) it is proved that the theory of dg modules over dg algebras with enough idempotents and their homotopy and derived categories is equivalent to the corresponding theory over small dg categories (see [5] and [6] for the details of this latter theory).",
"As a consequence of Keller's famous theorem (see [5]), one gets that any algebraic compactly generated triangulated category is equivalent to $\\mathcal {D}(A)$ , for some dg algebra with enough idempotents $A$ (see [10]).",
"Recall that a triangulated category $\\mathcal {T}$ is algebraic when it is equivalent to the stable category of some Frobenius exact category, and that it is called compactly generated when $\\mathcal {T}$ has coproducts and there is a set of compact objects $\\mathcal {C}$ in $\\mathcal {T}$ such that $\\bigcap _{C\\in \\mathcal {C},n\\in \\mathbb {Z}}\\text{Ker}(\\text{Hom}_\\mathcal {T}(C[n],?",
"))=0$ .",
"Recall that an object $C$ is compact when the functor $\\text{Hom}_\\mathcal {T}(C,?",
"):\\mathcal {T}\\longrightarrow Ab$ preserves arbitrary coproducts."
],
[
"Derived functors",
"A (right) dg $A$ -module $P$ (resp.",
"$I$ ) is homotopically projective (resp.",
"homotopically injective) when the functor $\\text{HOM}_A(P,?",
"):Dg-A\\longrightarrow Dg-K$ (resp.",
"$\\text{HOM}_A(?,I):(Dg-A)^{op}\\longrightarrow Dg-K$ ) preserves acyclic dg modules, something which is equivalent to saying that the induced functor $\\text{Hom}_{\\mathcal {H}(A)}(P,?",
"):\\mathcal {H}(A)\\longrightarrow \\text{Mod}-K$ (resp.",
"$\\text{Hom}_{\\mathcal {H}(A)}(?,I):\\mathcal {H}(A)^{op}\\longrightarrow \\text{Mod}-K$ ) vanishes on acyclic dg $A$ -modules.",
"As in the case of small dg categories, the canonical functor $q_A:\\mathcal {H}(A)\\longrightarrow \\mathcal {D}(A)$ has a left adjoint functor $\\Pi _A :\\mathcal {D}(A)\\longrightarrow \\mathcal {H}(A)$ , called the homotopically projective resolution functor, and a right adjoint $\\Upsilon _A:\\mathcal {D}(A)\\longrightarrow \\mathcal {H}(A)$ , called the homotopically injective resolution functor, both of which are fully faithful and triangulated.",
"They are so named because $\\text{Im}(\\Pi _A)$ (resp.",
"$\\text{Im}(\\Upsilon _A)$ ) consists of homotopically projective (resp.",
"homotopically injective) dg $A$ -modules.",
"The counit $\\pi :\\Pi _A\\circ q_A\\longrightarrow 1_{\\mathcal {H}(A)}$ (resp.",
"unit $\\iota :1_{\\mathcal {H}(A)}\\longrightarrow q_A\\circ \\Upsilon _A$ ) of the adjunction $(\\Pi _A,q_A)$ (resp.",
"$(q_A,\\Upsilon _A)$ ) has the property that $\\pi _M$ (resp.",
"$\\iota _M$ ) is a quasi-isomorphism, for each dg module $M$ , and it is even an isomorphism when $M$ is homotopically projective (resp.",
"homotopically injective).",
"Given a dg functor $F:Dg-A\\longrightarrow Dg-B$ which preserves contractible dg modules, one defines its left derived functor (resp.",
"right derived functor) $\\mathbb {L}F:\\mathcal {D}(A)\\longrightarrow \\mathcal {D}(B)$ (resp.",
"$\\mathbb {R}F:\\mathcal {D}(A)\\longrightarrow \\mathcal {D}(B)$ ), as the composition $\\mathcal {D}(A)\\stackrel{\\Pi _A}{\\longrightarrow }\\mathcal {H}(A)\\stackrel{F}{\\longrightarrow }\\mathcal {H}(B)\\stackrel{q_B}{\\longrightarrow }\\mathcal {D}(B)$ (resp.",
"$\\mathcal {D}(A)\\stackrel{\\Upsilon _A}{\\longrightarrow }\\mathcal {H}(A)\\stackrel{F}{\\longrightarrow }\\mathcal {H}(B)\\stackrel{q_B}{\\longrightarrow }\\mathcal {D}(B)$ ).",
"When the dg functor is contravariant, meaning that $F: (Dg-A)^{op}\\longrightarrow Dg-B$ a dg functor, which preserves contractibility, then we define its right derived functor $\\mathbb {R}F$ as the composition $\\mathcal {D}(A)^{op}\\stackrel{\\Pi _A^o}{\\longrightarrow }\\mathcal {H}(A)^{op}\\stackrel{F}{\\longrightarrow }\\mathcal {H}(B)\\stackrel{q_B}{\\longrightarrow }\\mathcal {D}(B).$ All these derived functors are triangulated since they are composition of triangulated functors.",
"If moreover $G:Dg-A\\longrightarrow Dg-B$ is another dg functor as above and $\\tau :F\\longrightarrow G$ is a homological natural transformation of dg functors, then one obtains corresponding natural transformations of triangulated functors, still denoted the same, $\\tau :\\mathbb {L}F\\longrightarrow \\mathbb {L}G$ and $\\tau :\\mathbb {R}F\\longrightarrow \\mathbb {R}G$ in the covariant case, and just $\\tau :\\mathbb {R}F\\longrightarrow \\mathbb {R}G$ in the contravariant case.",
"Not only that, but any dg adjunction $(F,G)$ of dg functors gives rise to a corresponding triangulated adjunction $(\\mathbb {L}F,\\mathbb {R}G)$ in the covariant case, and $((\\mathbb {R}F)^o,\\mathbb {R}G)$ in the contravariant case (see [10]).",
"This somehow classical picture is extended in [10] to dg bifunctors.",
"Concretely, if $A$ , $B$ and $C$ are dg algebras with enough idempotents and $F:(Dg-A)\\otimes (Dg-C)\\longrightarrow Dg-B$ is a dg functor which preserves contractibility on both variables, then one defines $\\mathbb {L}F:\\mathcal {D}(A)\\otimes \\mathcal {D}(C)\\stackrel{\\Pi _A\\otimes \\Pi _C}{\\longrightarrow }\\mathcal {H}(A)\\otimes \\mathcal {H}(C)\\stackrel{H^0F}{\\longrightarrow }\\mathcal {H}(B)\\stackrel{q_B}{\\longrightarrow }\\mathcal {D}(B),$ and $\\mathbb {R}F:\\mathcal {D}(A)\\otimes \\mathcal {D}(C)\\stackrel{\\Upsilon _A\\otimes \\Upsilon _C}{\\longrightarrow }\\mathcal {H}(A)\\otimes \\mathcal {H}(C)\\stackrel{H^0F}{\\longrightarrow }\\mathcal {H}(B)\\stackrel{q_B}{\\longrightarrow }\\mathcal {D}(B).$ When $F$ is contravariant on the first variable, i.e.",
"when $F:(Dg-A)^{op}\\otimes (Dg-C)\\longrightarrow Dg-B$ is a dg functor, one also defines $\\mathbb {R}F:\\mathcal {D}(A)^{op}\\otimes \\mathcal {D}(C)\\stackrel{\\Pi _A^o\\otimes \\Upsilon _C}{\\longrightarrow }\\mathcal {H}(A)\\otimes \\mathcal {H}(C)\\stackrel{H^0F}{\\longrightarrow }\\mathcal {H}(B)\\stackrel{q_B}{\\longrightarrow }\\mathcal {D}(B).$ The point is that, under suitable conditions (see [10] for details), these later bifunctors are triangulated on each variable."
],
[
"Derived $\\text{Hom}$ and {{formula:070f780c-dde8-4331-b58f-b135b585653d}} functors",
"Given dg algebras with enough idempotents $A$ , $B$ and $C$ and dg bimodules ${}_CM_A$ , ${}_BX_A$ and ${}_CU_B$ , the dg $K$ -modules $\\text{HOM}_A(M,X)$ and $U\\otimes _BX$ have canonical structures of dg $B-C-$ bimodule and dg $C-A-$ bimodule, respectively, but the first one is non-unitary.",
"This forces to define the `unitarization' $\\overline{\\text{HOM}}_A(M,X):=B\\text{HOM}_A(M,X)C,$ which is then a (now unitary!)",
"dg $B-C-$ bimodule.",
"It is proved in [10] that the assignments $(M,X)\\rightsquigarrow \\overline{\\text{HOM}}_A(M,X)$ and $(U,X)\\rightsquigarrow U\\otimes _BX$ are the definition on objects of dg functors $\\overline{\\text{HOM}}_A(?,?",
"):(C-Dg-A)^{op}\\otimes (B-Dg-A)&\\longrightarrow & B-Dg-C\\\\?\\otimes _B?",
":(C-Dg-B)\\otimes (B-Dg-A)&\\longrightarrow & C-Dg-A.$ One then puts $?\\otimes _B^\\mathbb {L}X:=\\mathbb {L}(?\\otimes _BX):\\mathcal {D}(B\\otimes C^{op})&\\longrightarrow &\\mathcal {D}(A\\otimes C^{op}),\\\\U\\otimes _B^\\mathbb {L}?",
":=\\mathbb {L}(U\\otimes _B?",
"):\\mathcal {D}(A\\otimes B^{op})&\\longrightarrow &\\mathcal {D}(A\\otimes C^{op}),\\\\\\mathbb {R}\\text{Hom}_A(M,?",
"):=\\mathbb {R}(\\overline{\\text{HOM}}_A(M,?",
")):\\mathcal {D}(A\\otimes B^{op})&\\longrightarrow &\\mathcal {D}(C\\otimes B^{op})\\\\\\mathbb {R}\\text{Hom}_A(?,X):=\\mathbb {R}(\\overline{\\text{HOM}}_A(?,X)):\\mathcal {D}(A\\otimes C^{op})^{op}&\\longrightarrow &\\mathcal {D}(C\\otimes B^{op}).$ By [10], the pairs $(?\\otimes _BX:C-Dg-A\\longrightarrow C-Dg-A,\\overline{\\text{HOM}}_A(X,?",
"):C-Dg-A\\longrightarrow C-Dg-B)$ and $(\\overline{\\text{HOM}}_{B^{op}}(?,X)^o:B-Dg-C\\rightarrow (C-Dg-A)^{op},\\overline{\\text{HOM}}_A(?,X):(C-Dg-A)^{op}\\rightarrow B-Dg-C)$ are dg adjunctions and, hence, we get adjunctions of triangulated functors $(?\\otimes _B^\\mathbb {L}X:\\mathcal {D}(B\\otimes C^{op})\\longrightarrow \\mathcal {D}(A\\otimes C^{op}),\\mathbb {R}\\text{Hom}_A(X,?",
"):\\mathcal {D}(A\\otimes C^{op})\\longrightarrow \\mathcal {D}(B\\otimes C^{op}))$ and $(\\mathbb {R}\\text{Hom}_{B^{op}}(?,X)^o:\\mathcal {D}(C\\otimes B^{op})\\longrightarrow \\mathcal {D}(A\\otimes C^{op}))^{op},\\mathbb {R}\\text{Hom}_A(?,X):\\mathcal {D}(A\\otimes C^{op}))^{op}\\longrightarrow \\mathcal {D}(C\\otimes B^{op})).$ By the previous paragraph, one also defines $\\mathbb {R}\\text{HOM}_A(?,?",
"):=\\mathbb {R}(\\overline{\\text{HOM}}_A(?,?",
")):\\mathcal {D}(A\\otimes C^{op})^{op}\\otimes \\mathcal {D}(A\\otimes B^{op})\\longrightarrow \\mathcal {D}(C\\otimes B^{op}),$ which is then a functor which is triangulated in each variable.",
"Moreover, precise conditions are given in [10] to have a natural isomorphisms triangulated functors $\\mathbb {R}\\text{HOM}_A(M,?",
")\\cong \\mathbb {R}\\text{Hom}_A(M,?",
")$ and $\\mathbb {R}\\text{HOM}_A(?,X)\\cong \\mathbb {R}\\text{Hom}_A(?,X)$ .",
"In particular, by taking $C=K$ in [10] and its proof, one gets the following consequence, which will frequently be used in Section : Proposition 2 Let $A$ and $B$ be dg algebras with enough idempotents and let $\\mathbb {R}\\text{HOM}_A(?,?",
"):=\\mathbb {R}(\\overline{\\text{HOM}}_A(?,?",
")):\\mathcal {D}(A^{op})\\otimes \\mathcal {D}(A\\otimes B^{op})\\longrightarrow \\mathcal {D}(B^{op})$ be the associated bi-triangulated functor.",
"There are natural isomorphisms of triangulated functors, for all dg $B-A-$ bimodules $X$ and all right dg $A$ -modules $M$ : $\\mathbb {R}\\text{HOM}_A(?,X)\\cong \\mathbb {R}\\text{Hom}_A(?,X):\\mathcal {D}(A)^{op}\\longrightarrow \\mathcal {D}(B^{op})$ .",
"$\\mathbb {R}\\text{HOM}_A(M,?",
")\\cong \\mathbb {R}\\text{Hom}_A(\\Pi _A(M),?",
"):\\mathcal {D}(A\\otimes B^{op})\\longrightarrow \\mathcal {D}(B^{op})$ .",
"On the other hand, when $X={}_AA_A$ is the regular dg bimodule associated to the dg algebra with enough idempotents $A$ , one has that the adjunction $(\\mathbb {R}\\text{Hom}_{A^{op}}(?,A)^o:\\mathcal {D}(A^{op})\\longrightarrow \\mathcal {D}(A)^{op},\\mathbb {R}\\text{Hom}_A(?,A):\\mathcal {D}(A)^{op}\\longrightarrow \\mathcal {D}(A^{op}))$ gives by restriction quasi-inverse dualities $\\text{per}(A^{op})\\stackrel{\\cong ^o}{\\longrightarrow }\\text{per}(A)$ , where $\\text{per}(A)=\\mathcal {D}^c(A)$ (resp.",
"$\\text{per}(A^{op})=\\mathcal {D}^c(A^{op})$ ) is the right (resp.",
"left) perfect derived category of $A$ , i.e.",
"the full subcategory of $\\mathcal {D}(A)$ (resp.",
"$\\mathcal {D}(A^{op})$ ) consisting of the compact objects.",
"It is this duality what will allow us to pass from the left version of degeneration to the right version, and vice versa, in the proof of Theorem REF under hypothesis (b)."
],
[
"Review of degeneration in triangulated categories",
"We start to recall from [4], [11] a few facts on the concept of degeneration of objects in triangulated categories."
],
[
"The module case",
"We first recall a classical result due to Zwara and Riedtmann [9], [20].",
"Let $A$ be a $k$ -algebra over an algebraically closed field $k$ and two finite dimensional $A$ -modules $M$ and $N$ .",
"Then $N$ belongs to the closure of the orbit of $M$ if and only if there is a finite dimensional $A$ -module $Z_r$ and a short exact sequence $0\\longrightarrow N\\stackrel{}{\\longrightarrow }M\\oplus Z_r\\stackrel{\\begin{pmatrix} u_r & v_r\\end{pmatrix}}{\\longrightarrow } Z_r\\longrightarrow 0,$ where $v_r$ is a nilpotent endomorphism of $Z_r$ .",
"We say in this case that $M$ degenerates to $N$ .",
"Zwara shows in [19] that $M$ degenerates to $N$ if and only if there is a finite dimensional $A$ -module $Z_\\ell $ of $\\mathcal {T}$ and an exact sequence triangle $0\\longrightarrow Z_\\ell \\stackrel{\\begin{pmatrix} v_\\ell \\\\ u_\\ell \\end{pmatrix}}{\\longrightarrow }Z_\\ell \\oplus M\\stackrel{}{\\longrightarrow } N\\longrightarrow 0,$ where $v_\\ell $ is a nilpotent endomorphism of $Z_\\ell $ ."
],
[
"Generalising degeneration to triangulated categories",
"In our previous work [4], [11], and independently by work of Yoshino [15] in the case of stable categories of maximal Cohen-Macaulay modules of local Gorenstein algebras, the concept of degeneration for modules was generalised to triangulated categories.",
"Yoshino discovered in particular the importance of the hypothesis that the induced endomorphism of $Z_\\ell $ (resp.",
"$Z_r$ ) is nilpotent.",
"Definition 3 For two objects $M$ and $N$ of a triangulated category $\\mathcal {T}$ we say that $M$ degenerates to $N$ in the triangle sense and write $M\\le _{\\Delta +nil} N$ if and only if there is an object $Z_r$ of $\\mathcal {T}$ and a distinguished triangle in $\\mathcal {T}$ $Z_\\ell \\stackrel{\\begin{pmatrix} u_\\ell \\\\ v_\\ell \\end{pmatrix}}{\\longrightarrow }M\\oplus Z_\\ell {\\longrightarrow } N\\longrightarrow Z_\\ell [1],$ where $v_\\ell $ is a nilpotent endomorphism of $Z_\\ell $ .",
"The main purpose of [11] was to define a geometric notion of degeneration along the lines of [15], and to prove that this notion is equivalent with the notion of degeneration in the triangle sense.",
"More precisely we gave the following definition.",
"Definition 4 Let $K$ be a commutative ring and let ${\\mathcal {C}}_K^\\circ $ be a $K$ -linear triangulated category with split idempotents.",
"A degeneration data for ${\\mathcal {C}}_K^\\circ $ is given by a triangulated category ${\\mathcal {C}}_K$ with split idempotents and a fully faithful embedding ${\\mathcal {C}}_K^\\circ \\longrightarrow {\\mathcal {C}}_K$ , a triangulated category ${\\mathcal {C}}_V$ with split idempotents and a full triangulated subcategory ${\\mathcal {C}}_V^\\circ $ , triangulated functors $\\uparrow _K^V:{\\mathcal {C}}_K\\longrightarrow {\\mathcal {C}}_V$ and $\\Phi :{\\mathcal {C}}_V^\\circ \\rightarrow {\\mathcal {C}}_K$ , so that $({\\mathcal {C}}_K^\\circ )\\uparrow _K^V\\subseteq {\\mathcal {C}}_V^\\circ $ , when we view ${\\mathcal {C}}_K^\\circ $ as a full subcategory of ${\\mathcal {C}}_K$ , a natural transformation $\\text{id}_{{\\mathcal {C}}_V}\\stackrel{t}{\\longrightarrow } \\text{id}_{{\\mathcal {C}}_V}$ of triangulated functors These triangulated categories and functors should satisfy the following axioms: For each object $M$ of ${\\mathcal {C}}_K^\\circ $ the morphism $\\Phi (M\\uparrow _K^V)\\stackrel{\\Phi (t_{M\\uparrow _K^V})}{\\longrightarrow }\\Phi (M\\uparrow _K^V)$ is a split monomorphism in ${\\mathcal {C}}_K$ .",
"For all objects $M$ of ${\\mathcal {C}}_K^\\circ $ we get $\\Phi (\\text{cone}(t_{M\\uparrow _K^V}))\\simeq M$ .",
"Degeneration is then given by the following concept.",
"Definition 5 Given two objects $M$ and $N$ of ${\\mathcal {C}}_K^\\circ $ we say that $M$ degenerates to $N$ in the categorical sense if there is a degeneration data for ${\\mathcal {C}}_K^\\circ $ and an object $Q$ of ${\\mathcal {C}}_V^\\circ $ such that $p(Q)\\simeq p(M\\uparrow _k^V)\\mbox{ in ${\\mathcal {C}}_V^\\circ [t^{-1}]$and }\\Phi (\\text{cone}(t_Q))\\simeq N,$ where $p:{\\mathcal {C}}_V^\\circ \\longrightarrow {\\mathcal {C}}_V^\\circ [t^{-1}]$ is the canonical functor.",
"In this case we write $M\\le _{cdeg}N$ .",
"Example 6 Yoshino observed that in a triangulated category $\\mathcal {T}$ for all objects $X$ we get $0\\le _{\\Delta +nil} X\\oplus X[1]$ .",
"Indeed, $X \\rightarrow 0\\rightarrow X[1]\\stackrel{id}{\\rightarrow } X[1]$ and $0\\rightarrow X\\stackrel{id}{\\rightarrow } X\\rightarrow 0$ are distinguished triangles for each object $X$ .",
"Hence their direct sum $X\\stackrel{0}{\\rightarrow } X\\oplus 0\\rightarrow X\\oplus X[1]\\rightarrow X[1]$ is a distinguished triangle as well.",
"Taking $Z=X$ we get the result (in the left version of $\\le _{\\Delta +nil}$ ).",
"Now, what about the degeneration data interpretation, which is equivalent to the triangle version in important cases?",
"Then there is an object $Q$ in some triangulated category ${\\mathcal {C}}_V^\\circ $ and an element $t$ in its centre such that $\\Phi (\\text{cone}(t_Q))\\simeq X\\oplus X[1]$ and $p(Q)\\simeq 0$ in ${\\mathcal {C}}_V^\\circ [t^{-1}]$ , where $p$ is the localisation functor.",
"The latter isomorphism is equivalent to the fact that $t_Q$ is nilpotent on $Q$ .",
"Hence we cannot assume, and actually do not assume, that $Q$ is $t$ -flat, as Yoshino does in the case of modules [14]."
],
[
"When triangle degeneration is the same as categorical degeneration",
"The main result of [11] is the following.",
"Theorem 7 Let $K$ be a commutative ring and let ${\\mathcal {C}}_K^\\circ $ be a triangulated $K$ -category with split idempotents.",
"If $M$ and $N$ are objects of $\\mathcal {C}_K^\\circ $ , then $M\\le _{cdeg}N\\Rightarrow M\\le _{\\Delta +nil} N$ .",
"When $\\mathcal {C}_K^\\circ $ is equivalent to the category of compact objects of a compactly generated algebraic triangulated $K$ -category, the converse is also true.",
"In order to prove that $\\le _{\\Delta +nil}$ implies $\\le _{cdeg}$ for the category of compact objects of a compactly generated algebraic triangulated $K$ -category, we need to construct a degeneration data.",
"By a result of Keller [5], we know that $\\mathcal {T}$ is equivalent to the category $\\mathcal {D}^c(\\mathcal {A})$ of compact objects of the derived category $\\mathcal {D}(\\mathcal {A})$ of some small dg-category $\\mathcal {A}$ .",
"We construct then the degeneration data for $\\mathcal {C}_K^o=\\mathcal {D}^c(\\mathcal {A})$ very explicitly, constructing a dg category $\\mathcal {A}[[T]]$ from $\\mathcal {A}$ and our proof of Theorem REF under hypothesis (b), which is given in Section , uses this construction.",
"More precisely, recall from the proof of [11] that if $\\mathcal {A}$ is a small dg category, then, considering a variable $T$ , one can form a new dg category $\\mathcal {A}[[T]]$ with the same set of objects as $\\mathcal {A}$ and where one defines $\\text{Hom}_{\\mathcal {A}[[T]]}^n(A,A^{\\prime })=\\lbrace \\sum _{k\\in \\mathbb {N}}\\alpha _kT^k\\text{: }\\alpha _k\\in \\text{Hom}_\\mathcal {A}^n(A,A^{\\prime })\\text{, for all }k\\in \\mathbb {N}\\rbrace .$ Moreover, one gets a canonical functor $?\\hat{\\otimes }V:\\mathcal {C}(\\mathcal {A})\\longrightarrow \\mathcal {C}(\\mathcal {A}[[T]])$ which takes a right dg $\\mathcal {A}$ -module $M$ to the right dg $\\mathcal {A}[[T]]$ -module $M[[T]]:\\mathcal {A}^{op}\\longrightarrow \\mathcal {C}_{dg}K$ acting on objects as $M[[T]]^n(A)=M^n(A)[[T]]$ , for all $n\\in \\mathbb {Z}$ and all $A\\in \\mathcal {A}$ .",
"The degeneration data for $\\mathcal {C}_K^o$ is then given by taking $\\mathcal {C}_K=\\mathcal {D}(\\mathcal {A})$ , with the corresponding inclusion functor as $\\mathcal {C}_K^o\\longrightarrow \\mathcal {C}_K$ , $\\mathcal {C}_V=\\mathcal {D}(\\mathcal {A}[[T]])$ , $\\mathcal {C}_V^o=\\mathcal {D}^c(\\mathcal {A}[[T]])$ , the triangulated version of $?\\hat{\\otimes }V$ as functor $\\uparrow _K^V:\\mathcal {C}_K=\\mathcal {D}(\\mathcal {A})\\longrightarrow \\mathcal {D}(\\mathcal {A}[[T]])=\\mathcal {C}_V$ , the restriction of scalars functor as $\\phi :\\mathcal {C}_V^o=\\mathcal {D}^c(\\mathcal {A}[[T]])\\longrightarrow \\mathcal {D}(\\mathcal {A})$ and the natural transformation $t:id_{\\mathcal {C}_V}\\longrightarrow id_{\\mathcal {C}_V}$ is defined by the maps $t_Q:Q\\longrightarrow Q$ , for each dg $\\mathcal {A}[[T]]$ -module $Q$ , given by multiplication by $T$ .",
"We take the opportunity to mention that in [11] we forgot to mention the grading, although it was implicit in all of the proofs because, when dealing with dg categories and dg modules, normally one only uses homogeneous elements.",
"However, a potential reader of [11] might think that there is an error in the definition of $\\mathcal {A}[[T]]$ and of $M[[T]]$ for, as it is written, they are not a graded category or a graded module.",
"The proof of [11] does need not to be modified."
],
[
"The case of triangulated categories whose objects have artinian endomorphism algebras",
"In this section we shall prove the first part of the theorem.",
"We will mimic Zwara's proof to give the analogous statement for triangle degeneration."
],
[
"Generalities on homotopy cartesian squares in triangulated categories",
"As in [19] we first need some preparation.",
"Throughout this section let $\\mathcal {T}$ be a triangulated $K$ -category for a commutative base ring $K$ .",
"The crucial concept is that of a homotopy cartesian square.",
"Recall from [8] that a commutative diagram ${A[r]^a[d]_b&C[d]^c\\\\B[r]_d&D}$ is homotopy cartesian if there is a map $e:D\\rightarrow A[1]$ such that ${A[r]^-{b\\atopwithdelims ()a}&B\\oplus C[r]^-{(-c,d)}&D[r]^e&A[1]}$ is a distinguished triangle.",
"In the rest of this section, denote by $C_f$ the cone of any morphism $f$ in $\\mathcal {T}$ .",
"By [8], if a commutative square ${A[r]^a[d]_b&C[d]^c\\\\B[r]_d&D}$ is homotopy cartesian, then there is an isomorphism $\\sigma :C_b\\stackrel{\\cong }{\\longrightarrow }C_d$ such that ${A[r]^a[d]_b&C[d]^c[r]&C_a[d]^\\sigma [r]&A[1][d]^{b[1]}\\\\B[r]_d&D[r]&C_d[r]&B[1]}$ is commutative with rows being distinguished triangles.",
"[8] gives a partial converse.",
"If ${A[r]^a[d]_b&C[d]^c[r]&C_a[d]^\\sigma [r]&A[1][d]^{b[1]}\\\\B[r]_d&D[r]&C_d[r]&B[1]}$ is commutative, where the rows are distinguished triangles and $\\sigma $ is an isomorphism, then there is a possible different $b^{\\prime }:A\\rightarrow B$ such that still ${A[r]^a[d]_{b^{\\prime }}&C[d]^c[r]&C_a[d]^\\sigma [r]&A[1][d]^{b^{\\prime }[1]}\\\\B[r]_d&D[r]&C_d[r]&B[1]}$ is commutative and furthermore ${A[r]^a[d]_{b^{\\prime }}&C[d]^c\\\\B[r]_d&D}$ is homotopy cartesian.",
"We may alternatively modify $c$ instead of $b$ .",
"This problem has the annoying consequence that if we have two homotopy cartesian squares ${A[r]^a[d]_b&[d]^cC&\\mbox{ and }&C[r]^e[d]_c&[d]^fE\\\\B[r]^d&D&&D[r]^g&F}$ then there is $b^{\\prime }$ and $f^{\\prime }$ fitting in certain morphisms of triangles, such that ${A[r]^{ea}[d]_{b^{\\prime }}&[d]^{f^{\\prime }}E\\\\B[r]^{gd}&F}$ is a homotopy cartesian square.",
"We would like to be able to assume that $b^{\\prime }=b$ and $f^{\\prime }=f$ .",
"However, we do not know if this is true.",
"Nevertheless, we prove a weaker statement which satisfy our needs.",
"Lemma 8 Let ${X_1[r]^{u_1}[d]^{}&X_2[d]^{f_2}&\\text{and}&X_2[r]^{\\nu _1}[d]^{f_2}&X_3[d]^{f_3}\\\\0[r]^{}&Y_2&&Y_2[r]^{\\nu _2}&Y_3}$ be homotopy cartesian squares.",
"Then ${X_1[r]^{\\nu _1u_1}[d]^{}&X_3[d]^{f_3}\\\\0[r]^{}&Y_3}$ is a homotopy cartesian square.",
"We first apply the octahedron axiom to the composition $\\nu _1u_1$ .",
"We hence obtain a morphism $v:X_3\\rightarrow C_{\\nu _1u_1}$ , a morphism $\\omega :Y_2\\rightarrow C_{\\nu _1u_1}$ , a morphism $C_{\\nu _1u_1}\\rightarrow C_{\\nu _1}$ , and $C_{\\nu _1}\\rightarrow Y_2[1]$ as indicated below, ${X_1[r]^{u_1}[drr]_{\\nu _1u_1}&X_2[rd]^{\\nu _1}[rrr]^{f_2}&&&Y_2[r]^+@{-->}^{\\omega }[dd]&X_1[1]\\\\&&X_3[ddrr]^v[drr]&&\\\\&&&&C_{\\nu _1u_1}[drr]^+@{-->}[d]\\\\&&&&C_{\\nu _1}[dr]^+@{-->}[d]^+&&X_1[1]\\\\&&&&Y_2[1]&X_2[1]}$ such that all straight sequences represent distinguished triangles, and such that the diagram is commutative.",
"In particular, $v\\nu _1=\\omega f_2$ .",
"Neeman's interpretation of the octahedral axiom [8] implies that we may choose $v$ and $\\omega $ such that ${X_2[d]_{f_2}[r]^{\\nu _1}&X_3[d]^{v}\\\\ Y_2[r]_\\omega &C_{\\nu _1u_1} }$ is a homotopy cartesian square.",
"Since ${X_2[r]^{\\nu _1}[d]_{f_2}&X_3[d]^{f_3}\\\\Y_2[r]^{\\nu _2}&Y_3}$ is homotopy cartesian by hypothesis, there is an isomorphism $\\varphi :C_{\\nu _1u_1}\\stackrel{\\cong }{\\longrightarrow } Y_3$ such that ${X_2[r]^{\\nu _1}[d]_{f_2}&X_3[d]_{v}@/^/[ddr]^{f_3}\\\\Y_2[r]^{\\omega }@/_/[drr]_{\\nu _2}&C_{\\nu _1u_1}[dr]^{\\varphi }\\\\&&Y_3}$ is commutative.",
"This then shows that ${X_1[r]^{\\nu _1u_1}&X_3[r]^{f_3}&Y_3[r]&X_1[1]}$ is a distinguished triangle, and hence ${X_1[r]^{\\nu _1u_1}[d]^{}&X_3[d]^{f_3}\\\\0[r]^{}&Y_3}$ is a homotopy cartesian square as claimed.",
"A partial converse is true in general however.",
"Lemma 9 Let ${X_1[r]^{u_1}[d]^{f_1}&X_2[d]^{f_2}&&\\text{and}&&X_2[r]^{\\nu _1}[d]^{f_2}&X_3[d]^{f_3}\\\\Y_1[r]^{u_2}&Y_2&&&&Y_2[r]^{\\nu _2}&Y_3}$ be commutative diagrams.",
"If the first square and ${X_1[r]^{\\nu _1u_1}[d]^{f_1}&X_3[d]^{f_3}\\\\Y_1[r]^{\\nu _2u_2}&Y_3}$ are homotopy cartesian squares, then there is a morphism $\\hat{\\nu }_2$ with $\\nu _2u_2=\\hat{\\nu }_2u_2$ such that ${X_2[r]^{\\nu _1}[d]^{f_2}&X_3[d]^{f_3}\\\\Y_2[r]^{\\hat{\\nu }_2}&Y_3}$ is a homotopy cartesian square.",
"We supposed that ${X_1[r]^{u_1}[d]^{f_1}&X_2[d]^{f_2}&\\text{and}&X_1[r]^{\\nu _1u_1}[d]^{f_1}&X_3[d]^{f_3}\\\\Y_1[r]^{u_2}&Y_2&&Y_1[r]^{\\nu _2u_2}&Y_3}$ are homotopy cartesian squares.",
"Hence we get distinguished triangles ${&&X_2[rd]^{-f_2}\\\\Y_2[-1][r]^{\\sigma _1[-1]}&X_1[ur]^{u_1}[rd]^{f_1}&\\oplus &Y_2[r]^{\\sigma _1}&X_1[1]\\\\&&Y_1[ur]_{u_2}}$ and ${&&X_3[rd]^{-f_3}\\\\Y_3[-1][r]^{\\sigma _2[-1]}&X_1[ur]^{\\nu _1u_1}[rd]^{f_1}&\\oplus &Y_3[r]^{\\sigma _2}&X_1[1]\\\\&&Y_1[ur]_{\\nu _2u_2}}$ We further get a morphism of distinguished triangles by the fact that the second comes from a square which factors through the first one.",
"${&&X_2[rd]^{-f_2}@/^1pc/[ddddd]^{\\nu _1}\\\\Y_2[-1][r]^{\\sigma _1[-1]}[ddddd]_{\\nu _2[-1]}&X_1[ur]_{u_1}[rd]^{f_1}@{=}[ddddd]&\\oplus &Y_2[r]^{\\sigma _1}[ddddd]^{\\nu _2}&X_1[1]@{=}[ddddd]\\\\&&Y_1[ur]_{u_2}@/_1pc/@{=}[ddddd]\\\\\\\\\\\\&&X_3[rd]^{-f_3}\\\\Y_3[-1][r]^{\\sigma _2[-1]}&X_1[ur][rd]_{f_1}&\\oplus &Y_3[r]^{\\sigma _2}&X_1[1]\\\\&&Y_1[ur]_{\\nu _2u_2}}$ By Neeman's axiom TR4' we maybe need to modify $\\nu _2$ to another map $\\hat{\\nu }_2$ , forming still a map of distinguished triangles, so that the cone of this commutative diagram is a distinguished triangle.",
"${&&X_2[rd]^{-f_2}@/^1pc/[ddddd]^{\\nu _1}\\\\Y_2[-1][r]^{\\sigma _1[-1]}[ddddd]_{\\hat{\\nu }_2[-1]}&X_1[ur]_{u_1}[rd]^{f_1}@{=}[ddddd]&\\oplus &Y_2[r]^{\\sigma _1}[ddddd]^{\\hat{\\nu }_2}&X_1[1]@{=}[ddddd]\\\\&&Y_1[ur]_{u_2}@/_1pc/@{=}[ddddd]\\\\\\\\\\\\&&X_3[rd]_{-f_3}\\\\Y_3[-1][r]^{\\sigma _2[-1]}&X_1[ur][rd]_{f_1}&\\oplus &Y_3[r]^{\\sigma _2}&X_1[1]\\\\&&Y_1[ur]_{\\nu _2u_2}}$ This implies in particular that $\\nu _2u_2=\\hat{\\nu }_2u_2$ .",
"The cone of this has a direct factor isomorphic to ${&&X_3[rd]^{-f_3}\\\\Y_2[-1][r]^{\\sigma _3[-1]}&X_2[ur]^{\\nu _1}[rd]^{f_2}&\\oplus &Y_3[r]^{\\sigma _3}&X_2[1]\\\\&&Y_2[ur]_{\\hat{\\nu }_2}}$ which is therefore a distinguished triangle.",
"This proves the statement."
],
[
"Triangle degeneration is symmetric with respect to left or right existence of the object $Z$",
"Recall the following result of Xiao-Wu Chen, Yu Ye and Phu Zhang.",
"Proposition 10 [2] An additive category $\\mathcal {C}$ is Krull-Schmidt if and only if any idempotent splits, and the endomorphism ring of any object of $\\mathcal {C}$ is semiperfect.",
"Our main result of this section now is the following.",
"Theorem 11 Let $K$ be a commutative ring and let ${\\mathcal {T}}$ be a $K$ -linear triangulated category with split idempotents and such that the endomorphism ring of each object is artinian, and let $M$ and $N$ be two objects.",
"Then there is an object $Z_r$ and a distinguished triangle $N\\rightarrow M\\oplus Z_r\\stackrel{(u_r, v_r)}{\\longrightarrow }Z_r\\longrightarrow N[1]$ (with nilpotent $v_r$ ) if and only if there is an object $Z_\\ell $ and a distinguished triangle $Z_\\ell \\stackrel{{u_\\ell }\\atopwithdelims (){v_\\ell }}{\\longrightarrow }M\\oplus Z_\\ell \\longrightarrow N\\longrightarrow Z_\\ell [1]$ (with nilpotent $v_\\ell $ ).",
"Remark 12 The hypothesis that each object in $\\mathcal {T}$ has artinian endomorphism ring implies that $\\mathcal {T}$ is Krull-Schmidt, and moreover that we get Fitting's lemma for $\\mathcal {T}$ .",
"In particular, and endomorphism $\\nu :Z\\rightarrow Z$ can be decomposed into $\\nu =\\left(\\begin{array}{cc}\\nu ^{\\prime }&0\\\\0&\\nu ^{\\prime \\prime }\\end{array}\\right):Z=Z^{\\prime }\\oplus Z^{\\prime \\prime }\\rightarrow Z^{\\prime }\\oplus Z^{\\prime \\prime }=Z$ and such that $\\nu ^{\\prime }$ is an automorphism and $\\nu ^{\\prime \\prime }$ is nilpotent.",
"Splitting off the trivial triangle $Z_\\ell ^{\\prime }\\stackrel{\\nu ^{\\prime }}{\\longrightarrow }Z^{\\prime }_\\ell \\rightarrow 0\\rightarrow Z^{\\prime }_\\ell [1]$ , respectively $0\\rightarrow Z_r^{\\prime }\\stackrel{\\nu ^{\\prime }}{\\longrightarrow }Z^{\\prime }_r\\rightarrow 0[1]$ we may hence assume that $v_\\ell $ and $v_r$ are nilpotent.",
"(of Theorem REF ) Set $N_1:=N$ .",
"Let $Z_\\ell $ be an object and let $Z_\\ell \\stackrel{{u_\\ell }\\atopwithdelims (){v_\\ell }}{\\longrightarrow }M\\oplus Z_\\ell \\stackrel{(\\nu _1,h_1)}{\\longrightarrow }N_1\\stackrel{\\rho _0}{\\longrightarrow }Z_\\ell [1]$ be a distinguished triangle with nilpotent endomorphism $v_\\ell $ .",
"Then we form the homotopy pushout (cf [8]) ${Z_\\ell [r]^-{{{u_\\ell }\\atopwithdelims (){v_\\ell }}}[d]^{h_1} &M\\oplus Z_\\ell [r]^-{(\\nu _1,h_1)}[d]^{(\\nu _2,h_2)}&N_1[r]^{\\rho _0}[d]^{\\text{id}}& Z_\\ell [1][d]^{h_1[1]}\\\\N_1[r]^{s_1}&N_2[r]^{t_1}&N_1[r]^{\\rho _1}&N_1[1]}$ We get $h_1=t_1h_2$ from the commutativity of the middle square and $h_1[1]\\rho _0=\\rho _1$ from the right most square.",
"We form the homotopy pushout ${Z_\\ell [r]^{{{u_\\ell }\\atopwithdelims (){v_\\ell }}}[d]^{h_2} &M\\oplus Z_\\ell [d]^{(\\nu _3,h_3)}\\\\N_2[r]^{s_2}&N_3}$ and obtain a commutative diagram with homotopy pushouts on the front face and on the back face.",
"Denote for short $\\hat{v}:={v_\\ell \\atopwithdelims ()u_\\ell }$ .",
"${&Z_\\ell [rr]^{\\hat{v}}[dd]^{h_2}[dddl]_{h_1}&&Z_\\ell \\oplus M[dd]^{(h_3,\\nu _3)}[dddl]_{(h_2,\\nu _2)}\\\\\\\\&N_2[rr]^{s_2}[ld]^{t_1}&&N_3\\\\N_1[rr]^{s_1}&&N_2}$ Since the back face is a homotopy pushout, and since the diagram ${&Z_\\ell [rr]^{\\hat{v}}[dd]^{h_2}&&Z_\\ell \\oplus M[dd]^{(h_3,\\nu _3)}[dddl]_{(h_2,\\nu _2)}\\\\\\\\&N_2[rr]^{s_2}[ld]^{t_1}&&N_3\\\\N_1[rr]^{s_1}&&N_2}$ is commutative, there is a non-unique map $t_2:N_3\\rightarrow N_2$ making the diagram ${&Z_\\ell [rr]^{\\hat{v}}[dd]^{h_2}[dddl]_{h_1}&&Z_\\ell \\oplus M[dd]^{(h_3,\\nu _3)}[dddl]_{(h_2,\\nu _2)}\\\\\\\\&N_2[rr]^{s_2}[ld]^{t_1}&&N_3[ld]^{t_2}\\\\N_1[rr]^{s_1}&&N_2}$ commutative.",
"Lemma REF then allows to modify $t_2$ such that the bottom face of the diagram is homotopy cartesian, such that all already shown commutativity properties still hold, and such that the above diagram with modified $t_2$ is still commutative.",
"Here, in order to simplify the notation, we denote the modified $t_2$ again by $t_2$ .",
"We proceed now by induction on the degree $n$ .",
"Suppose we have a commutative diagram with rows being distinguished triangles and whose square faces are homotopy cartesian squares ${&Z_\\ell [rr]^{\\hat{v}}[dd]^{h_{n-1}}[dddl]_{h_{n-2}}&&Z_\\ell \\oplus M[dd]^{(h_n,\\nu _n)}[dddl]_{(h_{n-1},\\nu _{n-1})}\\\\\\\\&N_{n-1}[rr]^{s_{n-1}}[ld]^{t_{n-2}}&&N_n[ld]^{t_{n-1}}\\\\N_{n-2}[rr]^{s_{n-2}}&&N_{n-1}}$ We form the homotopy pushout (on the back face of the diagram defining $N_{n+1}$ , $h_{n+1}$ , $s_n$ and $\\nu _{n+1}$ ) ${&Z_\\ell [rr]^{\\hat{v}}[dd]^{h_{n}}[dddl]_{h_{n-1}}&&Z_\\ell \\oplus M[dd]^{(h_{n+1},\\nu _{n+1})}[dddl]_{(h_n,\\nu _n)}\\\\\\\\&N_{n}[rr]^{s_{n}}[ld]^{t_{n-1}}&&N_{n+1}\\\\N_{n-1}[rr]^{s_{n-1}}&&N_{n}}$ Since the diagram ${&Z_\\ell [rr]^{\\hat{v}}[dd]^{h_{n}}&&Z_\\ell \\oplus M[dd]^{(h_{n+1},\\nu _{n+1})}[dddl]_{(h_n,\\nu _n)}\\\\\\\\&N_{n}[rr]^{s_{n}}[ld]^{t_{n-1}}&&N_{n+1}\\\\N_{n-1}[rr]^{s_{n-1}}&&N_{n}}$ is commutative, there is a morphism $t_n:N_{n+1}\\rightarrow N_n$ making the diagram ${&Z_\\ell [rr]^{\\hat{v}}[dd]^{h_{n}}[dddl]_{h_{n-1}}&&Z_\\ell \\oplus M[dd]^{(h_{n+1},\\nu _{n+1})}[dddl]_{(h_n,\\nu _n)}\\\\\\\\&N_{n}[rr]^{s_{n}}[ld]^{t_{n-1}}&&N_{n+1}[dl]^{t_n}\\\\N_{n-1}[rr]^{s_{n-1}}&&N_{n}}$ commutative.",
"By Lemma REF we may modify $t_n$ without changing commutativity of what is already shown (we denote the modified $t_n$ again by $t_n$ ), such that the bottom face of the diagram is a homotopy pushout and such the above diagram is still commutative.",
"We now continue as in the proof of [19].",
"We define $\\omega _1:=\\nu _1$ and $\\omega _{j+1}:=(\\nu _{j+1},s_j\\omega _j)\\in Hom_{\\mathcal {T}}(M\\oplus M^{j},N_{j+1})=Hom_{\\mathcal {T}}(M^{j+1},N_{j+1})$ for all $j\\ge 1$ .",
"In the diagram ${Z_\\ell [r]^{{{u_\\ell }\\atopwithdelims (){v_\\ell }}}[d]^{h_j}&M\\oplus Z_\\ell [rr]^{(\\nu _1,h_1)}[d]^{(\\nu _{j+1},h_{j+1})}&&N[r]^{\\rho }[d]^{\\text{id}}& Z_\\ell [1][d]^{h_2[1]}\\\\N_j[r]^{s_j}&N_{j+1}[rr]^{v_j}&&N[r]^{\\rho _j}&N_j[1]}$ the left square is a homotopy cartesian square.",
"Therefore we get a distinguished triangle ${Z_\\ell [rr]^-{\\left(\\begin{array}{c}u_\\ell \\\\v_\\ell \\\\h_j\\end{array}\\right)}&&M\\oplus Z_\\ell \\oplus N_j[rrr]^-{(\\nu _{j+1},h_{j+1},-s_j)}&&&N_{j+1}[r]^{\\sigma _j}&Z_\\ell [1]}$ Since ${M^j[r]^{\\text{id}}&M^j[r]&0[r]&M^{j}[1]}$ is a distinguished triangle, also the direct sum of these two distinguished triangles ${M^j\\oplus Z_\\ell [rr]^-{\\left(\\begin{array}{cc}1_{M^j}&0\\\\0&u_\\ell \\\\0&v_\\ell \\\\0&h_j\\end{array}\\right)}&&M^j\\oplus M\\oplus Z_\\ell \\oplus N_j[rrr]^-{(0,\\nu _{j+1},h_{j+1},-s_j)}&&&N_{j+1}[r]^-{(0,\\sigma _j)}&(M^j\\oplus Z_\\ell )[1]}$ is distinguished.",
"For the series of morphisms $\\omega _j:M^j\\rightarrow N_j$ satisfying $\\omega _1=\\nu _1$ and $\\omega _{j+1}=(\\nu _{j+1},s_j\\omega _j)$ for all $j$ , this distinguished triangle is isomorphic to the triangle ${M^j\\oplus Z_\\ell [rr]^-{\\left(\\begin{array}{cc}1_{M^j}&0\\\\0&u_\\ell \\\\0&v_\\ell \\\\\\omega _j&h_j\\end{array}\\right)}&&M^j\\oplus M\\oplus Z_\\ell \\oplus N_j[rrr]^-{(\\omega _{j+1},\\nu _{j+1},h_{j+1},-s_j)}&&&N_{j+1}[r]^-{(0,\\sigma _j)}&(M^j\\oplus Z_\\ell )[1]}$ Indeed, we get morphisms of triangles ${M^j[rr]^{\\text{id}}&&M^j[rrr]&&&0[r]&M^j[1]\\\\ \\\\M^j\\oplus Z_\\ell [uu]^{(1,0)}[rr]^-{\\left(\\begin{array}{cc}1_{M^j}&0\\\\0&u_\\ell \\\\0&v_\\ell \\\\\\omega _j&h_j\\end{array}\\right)}&&M^j\\oplus M\\oplus Z_\\ell \\oplus N_j[rrr]^-{(\\omega _{j+1},\\nu _{j+1},h_{j+1},-s_j)}[uu]_{(1,0,0,0)}&&&N_{j+1}[r][uu]&(M^j\\oplus Z_\\ell )[1][uu]\\\\ \\\\ \\\\M^j[rr]^{\\text{id}}[uuu]^{\\left(\\begin{array}c 1\\\\0\\end{array}\\right)}&&M^j[rrr][uuu]^{\\left(\\begin{array}c 1\\\\0\\\\0\\\\0\\\\\\omega _j\\end{array}\\right)}&&&0[r][uuu]&M^j[1][uuu]}$ and therefore the middle triangle has a direct factor ${M^j[r]^{\\text{id}}&M^j[r]&0[r]&M^{j}[1]}$ and the remaining direct factor is the original distinguished triangle.",
"Hence, ${M^j\\oplus Z_\\ell [rr]^{\\scriptsize \\left(\\begin{array}{cc}1_{M^j}&0\\\\0&u_\\ell \\\\0&v_\\ell \\end{array}\\right)}[d]_{(\\omega _j,h_j)}&&M^{j+1}\\oplus Z_\\ell [d]^{(\\omega _{j+1},h_{j+1})}\\\\N_j[rr]^{s_j}&&N_{j+1}}$ is a homotopy cartesian square.",
"Moreover, ${Z_\\ell [r]^{u_\\ell \\atopwithdelims ()v_\\ell }[d]&M\\oplus Z_\\ell [d]^{(\\omega _1,h_1)}\\\\ 0[r]&N}$ is a homotopy cartesian square by hypothesis.",
"Now, since ${Z_\\ell [r]^{u_\\ell \\atopwithdelims ()v_\\ell }[d]&M\\oplus Z_\\ell [d]^{(\\omega _1,h_1)}&\\mbox{ and }&M^j\\oplus Z_\\ell [rr]^{\\scriptsize \\left(\\begin{array}{cc}1_{M^j}&0\\\\0&u_\\ell \\\\0&v_\\ell \\end{array}\\right)}[d]_{(\\omega _j,h_j)}&&M^{j+1}\\oplus Z_\\ell [d]^{(\\omega _{j+1},h_{j+1})}\\\\0[r]&N&&N_j[rr]^{s_j}&&N_{j+1}}$ are homotopy cartesian squares, applying Lemma REF , we get a homotopy cartesian square ${Z_\\ell [r]^{\\psi _2\\atopwithdelims ()v_\\ell ^2}[d]&M^2\\oplus Z_\\ell [d]^{(\\omega _2,h_2)}\\\\ 0[r]&N_2}$ and by induction on $j$ we get that there is a morphism $\\psi _j:Z_\\ell \\rightarrow M^j$ and a homotopy cartesian square ${Z_\\ell [r]^{\\psi _j\\atopwithdelims ()v_\\ell ^j}[d]&M^j\\oplus Z_\\ell [d]^{(\\omega _j,h_j)}\\\\ 0[r]&N_j}$ for all $j\\in {\\mathbb {N}}$ .",
"Hence for all $j>0$ there is a morphism $\\psi _j:Z_\\ell \\rightarrow M^j$ such that $Z_\\ell \\stackrel{\\psi _j\\atopwithdelims ()v_\\ell ^j}{\\longrightarrow }M^j\\oplus Z_\\ell \\stackrel{(\\omega _j,h_j)}{\\longrightarrow }N_j\\longrightarrow Z_\\ell [1]$ is a distinguished triangle.",
"Now $v_\\ell $ is nilpotent of degree $k_0$ , say.",
"For any $k\\ge k_0$ consider the commutative diagram with distinguished triangles in the horizontal rows ${Z_\\ell [r]^{\\psi _k\\atopwithdelims (){v_\\ell ^k}}[d]&M^k\\oplus Z_\\ell [rr]^{(\\omega _k,h_k)}[d]^{(0,\\text{id})}&&N_k[r]&Z_\\ell [1][d]\\\\0[r]&Z_\\ell [rr]^{\\text{id}}&&Z_\\ell [r]&0}$ which can be completed by a map $z_k$ to a morphism of distinguished triangles, using TR3 (see e.g.",
"[8] or [18]), ${Z_\\ell [r]^{\\psi _k\\atopwithdelims (){v_\\ell ^k}}[d]&M^k\\oplus Z_\\ell [rr]^{(\\omega _k,h_k)}[d]^{(0,\\text{id})}&&N_k[r][d]^{z_k}&Z_\\ell [1][d]\\\\0[r]&Z_\\ell [rr]^{\\text{id}}&&Z_\\ell [r]&0}$ Therefore $z_kh_k=\\text{id}_{Z_\\ell }$ , and $h_k$ is a split monomorphism.",
"We recall that we constructed the sequence $h_j$ as iterated homotopy pushouts, and hence, by definition we have a homotopy cartesian square ${Z_\\ell [r]^-{\\hat{v}}[d]_{h_k}&Z_\\ell \\oplus M[d]^{(h_{k+1},\\nu _{k+1})}\\\\N_k[r]_-{s_n}&N_{k+1}}$ where $h_k$ (and $h_{k+1}$ ) are split monomorphisms.",
"This shows first (cf e.g.",
"[18]) that $N_k\\simeq Z_\\ell \\oplus C_{h_k}$ , for $C_{h_k}$ being the mapping cone of $h_k$ .",
"Moreover, we get a distinguished triangle ${Z_\\ell [r]^-{\\hat{v}\\atopwithdelims ()h_k}&Z_\\ell \\oplus M\\oplus N_k[rrr]^-{(h_{k+1},\\nu _{k+1},-s_k)}&&&N_{k+1}[r]&Z_\\ell [1]}.$ Since $h_k$ is split monomorphism, this distinguished triangle is isomorphic to the direct sum of the trivial distinguished triangle ${Z_\\ell [r]^{\\text{id}}&Z_\\ell [r]&0[r]&Z_\\ell [1]}$ and ${0[r]&0\\oplus C_{h_k}\\oplus Z_\\ell \\oplus M[r]&N_{k+1}[r]&0[1].",
"}$ Hence (cf e.g.",
"[18]), $(\\dagger )\\;\\;N_{k+1}\\simeq C_{h_k}\\oplus Z_\\ell \\oplus M\\simeq N_k\\oplus M.$ Note that we could have argued also that since $z_k$ is left inverse to $h_k$ , $(0,z_k)$ is left inverse to $\\hat{v}\\atopwithdelims ()h_k$ , and so the above triangle splits.",
"This then gives the desired isomorphism $(\\dagger )$ via Remark REF .",
"Recall that, posing $N_0:=0$ , for all $j\\ge 1$ we have by construction homotopy cartesian squares ${N_j[r]^{s_j}[d]_{t_j}&N_{j+1}[d]^{t_{j+1}}\\\\N_{j-1}[r]^{s_j}&N_{j}}$ for all $j\\ge 1$ .",
"Using Lemma REF and an obvious induction as above this implies that we get an cartesian square ${N[r][d]&N_{k+1}[d]\\\\ 0[r]&N_k}$ which gives a distinguished triangle ${N[r]&N_{k+1}[r]&N_k[r]&N[1]}$ where we have chosen $k$ such that $v_\\ell ^k=0$ .",
"By equation $(\\dagger )$ we get $N_{k+1}\\simeq N_k\\oplus M$ , which shows that there is a distinguished triangle $N\\longrightarrow N_k\\oplus M\\longrightarrow N_k\\longrightarrow N[1].$ Posing $N_k=:Z_r$ this gives the statement, except that we do not get yet that the induced endomorphism of $Z_r$ is nilpotent.",
"Since the endomorphism ring of all objects in $\\mathcal {T}$ are artinian and idempotents split, $\\mathcal {T}$ is Krull-Schmidt (cf Proposition REF ), and then we may split off $f^{\\prime }$ in a nilpotent endomorphism of a direct factor and an automorphism of a direct factor, using Fitting's lemma.",
"The automorphism part splits in the distinguished triangle, and we obtain the statement.",
"The other direction is done applying the statement proved above to the opposite category ${\\mathcal {T}}^{op}$ of $\\mathcal {T}$ .",
"Remark 13 A triangulated category with split idempotents for which each object has artinian endomorphism rings is Krull-Schmidt and a Fitting-like theorem holds (cf Remark REF ).",
"However, if $\\mathcal {T}$ is a general triangulated category, and in particular if we do not assume that $\\mathcal {T}$ is Krull-Schmidt, then we get a weaker statement in Theorem REF .",
"The hypothesis that $\\mathcal {T}$ has artinian endomorphism rings is only used at the very end of the proof of the theorem in order to be able to split off a direct factor in order to get a nilpotent endomorphism on the remaining factor.",
"If $\\mathcal {T}$ is a general triangulated category we proved that the existence of a distinguished triangle $Z_\\ell \\stackrel{{u_\\ell }\\atopwithdelims (){v_\\ell }}{\\longrightarrow }M\\oplus Z_\\ell \\longrightarrow N\\longrightarrow Z_\\ell [1],$ with nilpotent $v_\\ell $ implies the existence of a distinguished triangle $N\\longrightarrow M\\oplus Z_r\\stackrel{(u_r\\;v_r)}{\\longrightarrow }Z_r[1]\\longrightarrow N[1],$ but we are unable to deduce that $v_r$ is nilpotent."
],
[
"The case of a category of compact objects in an algebraic compactly generated category",
"Recall from Section REF the construction of the category $\\mathcal {A}[[T]]$ .",
"When we have a dg algebra with enough idempotents $A$ and view it as a dg category (see [10]), we can undertake the same construction, but there is a classical notion of power series algebra $A[[T]]$ which does not correspond to the `power series dg category' mentioned in Section REF .",
"The corresponding dg algebra with enough idempotents is the subalgebra of $A[[T]]$ given as $\\tilde{A}[[T]]=\\bigoplus _{i,j\\in I}\\bigoplus _{n\\in \\mathbb {Z}}e_iA^ne_j[[T]].$ That is, $\\tilde{A}[[T]]$ consists of the power series $\\sum _{k\\in \\mathbb {N}}a_kT^k$ for which there are finite subsets $J\\subset I$ and $F\\subset \\mathbb {Z}$ such that $a_k\\in \\bigoplus _{i,j\\in J}\\bigoplus _{n\\in F}e_iA^ne_j$ , for all $k\\in \\mathbb {N}$ .",
"The subalgebra $\\tilde{A}[[T]]$ is made into a dg algebra by defining $\\tilde{A}[[T]]^n=\\bigoplus _{i,j\\in I}e_iA^ne_j[[T]]$ , for each $n\\in \\mathbb {Z}$ , and by defining its differential by the rule $d(\\sum _{k\\in \\mathbb {N}}a_kT^k)=\\sum _{i\\in \\mathbb {N}}d(a_k)T^k$ .",
"Note that then $(e_i)_{i\\in I}$ is also a distinguished family of orthogonal idempotents of $\\tilde{A}[[T]]$ .",
"Moreover, we have a canonical inclusion $\\iota : A\\hookrightarrow \\tilde{A}[[T]]$ ($a\\rightsquigarrow a=aT^o$ ) and a canonical augmentation map $\\rho :\\tilde{A}[[T]]\\longrightarrow A$ ($\\sum _{k\\in \\mathbb {N}}a_kT^k\\rightsquigarrow a_0$ ) such that $\\rho \\circ \\iota =1_A$ .",
"Both $\\iota $ and $\\rho $ are homomorphisms of dg algebras making the codomain into a unitary bimodule over the domain.",
"We can apply to $\\iota $ and $\\rho $ the results of [10] (see Corollary 9.4 and Section 10 in that reference) concerning homomorphisms of dg algebras with enough idempotents.",
"Likewise for right $A$ -modules, there is a classical notion of `module of power series' $M[[T]]$ which does not correspond the the `dg $\\mathcal {A}[[T]]$ -module of power series' mentioned in Section REF .",
"The corresponding right dg $\\tilde{A}[[T]]$ -module is $\\tilde{M}[[T]]=\\bigoplus _{i\\in I}\\bigoplus _{n\\in \\mathbb {Z}}M^ne_i[[T]],$ which is the $K$ -submodule of $M[[T]]$ consisting of those power series $\\sum _{k\\in \\mathbb {N}}m_kT^k$ for which there exist finite subsets $J\\subset I$ and $F\\subset \\mathbb {Z}$ , both depending on the power series, such that $m_k\\in \\oplus _{i\\in J,n\\in F}M^ne_i$ , for all $k\\in \\mathbb {N}$ .",
"Defining the grading by the rule $\\tilde{M}[[T]]^n=\\bigoplus _{i\\in I}M^ne_i[[T]]$ and the differential $d:\\tilde{M}[[T]]\\longrightarrow \\tilde{M}[[T]]$ by the rule $d(\\sum _{k\\in \\mathbb {N}}m_kT^k)=\\sum _{k\\in \\mathbb {N}}d(m_k)T^k$ , we clearly endow $\\tilde{M}[[T]]$ with a structure of right dg $\\tilde{A}[[T]]$ -module.",
"We warn the reader of the possible confusion of this construction when applied to the regular right dg $A$ -module $A_A$ for the resulting right dg $\\tilde{A}[[T]]$ -module is not equal to the regular right dg module $\\tilde{A}[[T]]_{\\tilde{A}[[T]]}$ ."
],
[
"Dualising degeneration data",
"We now give, and actually extend, the functor corresponding to $?\\hat{\\otimes }V$ to the context of dg algebras with enough idempotents and dg modules over them.",
"Proposition 14 The assignment $M\\rightsquigarrow \\tilde{M}[[T]]$ is the definition on objects of a dg functor $?\\hat{\\otimes }V:Dg-A\\longrightarrow Dg-\\tilde{A}[[T]]$ which satisfies the following properties: $?\\hat{\\otimes }V$ takes contractible dg modules to contractible dg modules.",
"The associated functor on 0-cycle categories $?\\hat{\\otimes }V:Z^0(Dg-A)=\\mathcal {C}(A)\\longrightarrow \\mathcal {C}(\\tilde{A}[[T]])=Z^0(Dg-\\tilde{A}[[T]])$ is exact with respect to the respective abelian structures.",
"$?\\hat{\\otimes }V$ preserves acyclic dg modules.",
"In particular, it induces a triangulated functor $?\\hat{\\otimes }V:\\mathcal {D}(A)\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]])$ which is both the left and right derived functor of `itself'.",
"We will denote this functor by $\\uparrow _K^V$ .",
"The multiplication map $M\\otimes _A\\tilde{A}[[T]]\\longrightarrow \\tilde{M}[[T]]$ defines a homological natural transformation of dg functors $\\mu :\\iota ^*\\longrightarrow ?\\hat{\\otimes }V$ The induced natural transformation $\\mu :\\mathbb {L}\\iota ^*\\longrightarrow \\uparrow _K^V$ of triangulated functors $\\mathcal {D}(A)\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]])$ is a natural isomorphism when evaluated at objects of $\\text{per}(A)$ .",
"If $f:M\\longrightarrow N$ is a homogeneous morphism in $Dg-A$ , we define $\\tilde{f}:=(?\\hat{\\otimes }V)(f)$ by the rule $\\tilde{f}(\\sum _{k\\in \\mathbb {N}}m_kT^k)=\\sum _{k\\in \\mathbb {N}}f(m_k)T^k$ .",
"It is routine to verify that the assignments $M\\rightsquigarrow \\tilde{M}[[T]]$ and $f\\rightsquigarrow \\tilde{f}$ give a graded functor $GR-A\\longrightarrow GR-\\tilde{A}[[T]]$ .",
"In order to check that they define a dg functor $?\\hat{\\otimes }V:Dg-A\\longrightarrow Dg-\\tilde{A}[[T]]$ , we need to check that if $M,N$ are right dg $A$ -modules and $d:\\text{HOM}_{A}(M,N)\\longrightarrow \\text{HOM}_{A}(M,N)$ and $\\delta :\\text{HOM}_{\\tilde{A}[[T]]}(\\tilde{M}[[T]],\\tilde{N}[[T]])\\longrightarrow \\text{HOM}_{\\tilde{A}[[T]]}(\\tilde{M}[[T]],\\tilde{N}[[T]])$ are the respective differentials on Hom spaces, then one has $\\delta (\\tilde{f})=\\widetilde{d(f)}$ , for any homogeneous element $f\\in \\text{HOM}_A(M,N)$ .",
"On one hand, we have that $\\delta (\\tilde{f})=d_{\\tilde{N}[[T]]}\\circ \\tilde{f}-(-1)^{|f|}\\tilde{f}\\circ d_{\\tilde{M}[[T]]}.",
"$ On the other hand, if we let act $\\widetilde{d(f)}$ on a homogeneous element $\\sum _{k\\in \\mathbb {N}}m_kT^k\\in \\tilde{M}[[T]]$ (whence the degree $deg(m_k)$ is independent of $k$ ), then we get: $\\widetilde{d(f)}\\left(\\sum _{k\\in \\mathbb {N}}m_kT^k\\right)&=&\\sum _{k\\in \\mathbb {N}}d(f)(m_k)T^k\\\\&=&\\sum _{k\\in \\mathbb {N}}\\left[(d_N\\circ f-(-1)^{|f|}f\\circ d_M)(m_k)\\right]T^k\\\\&=&\\sum _{k\\in \\mathbb {N}}d_N(f(m_k))T^k-(-1)^{|f|}\\sum _{k\\in \\mathbb {N}}f(d_M(m_k))T^k\\\\&=&d_{\\tilde{N}[[T]]}\\left(\\sum _{k\\in \\mathbb {N}}f(m_k)T^k\\right)-(-1)^{|f|}\\tilde{f}\\left(\\sum _{k\\in \\mathbb {N}}d_M(m_k)T^k\\right)\\\\&=&\\left[d_{\\tilde{N}[[T]]}\\circ \\tilde{f}-(-1)^{|f|}\\tilde{f}\\circ d_{\\tilde{M}[[T]]}\\right]\\left(\\sum _{k\\in \\mathbb {N}}m_kT^k\\right).$ This shows that $\\delta (\\tilde{f})=\\widetilde{d(f)}$ , as desired.",
"Finally, it is also routine to see that $(?\\hat{\\otimes }V)(\\text{cone}(1_M))\\cong \\text{cone}(1_{(?\\hat{\\otimes }V)(M)})$ , which ends the proof of assertion (REF ).",
"(REF ) Let $L\\stackrel{f}{\\longrightarrow } M\\stackrel{g}{\\longrightarrow } N$ be an exact sequence in $\\mathcal {C}(A)$ , when this category is considered with its natural abelian structure, and consider the corresponding sequence $\\tilde{L}[[T]]\\stackrel{\\tilde{f}}{\\longrightarrow }\\tilde{M}[[T]]\\stackrel{\\tilde{g}}{\\longrightarrow }\\tilde{N}[[T]]$ .",
"Since $\\tilde{f}$ and $\\tilde{g}$ are both morphisms in $Gr-\\tilde{A}[[T]]$ we just need to check that the induced sequence $\\tilde{L}[[T]]^ne_i=L^ne_i[[T]]\\stackrel{\\tilde{f}}{\\longrightarrow }\\tilde{M}[[T]]^ne_i=M^ne_i[[T]]\\stackrel{\\tilde{g}}{\\longrightarrow }\\tilde{N}[[T]]^ne_i=N^ne_i[[T]]$ is exact, for all $i\\in I$ and $n\\in \\mathbb {Z}$ .",
"But, given $\\sum _{k\\in \\mathbb {N}}m_kT^k\\in M^ne_i[[T]]$ , we have that $\\tilde{g}(\\sum _{k\\in \\mathbb {N}}m_kT^k)=0$ if and only if $g(m_k)=0$ for all $k\\in \\mathbb {N}$ .",
"This in turn is equivalent to saying that, for each $k\\in \\mathbb {N}$ , there exists a $l_k\\in L^ne_i$ such that $f(l_k)=m_k$ .",
"That is, we have that $\\sum _{k\\in \\mathbb {N}}m_kT^k\\in \\text{Ker}(\\tilde{g})$ if and only if $\\sum _{k\\in \\mathbb {N}}m_kT^k=\\tilde{f}(\\sum _{k\\in \\mathbb {N}}l_kT^k)$ , for some $\\sum _{k\\in \\mathbb {N}}l_kT^k\\in L^ne_i[[T]]$ .",
"(REF ) Let $M$ be an acyclic right dg $A$ -module, let $\\sum _{k\\in \\mathbb {N}}m_kT^k$ be an element of $\\text{Ker}(d^n:\\tilde{M}[[T]]^n\\longrightarrow \\tilde{M}[[T]]^{n+1})$ and let $J\\subset I$ be a finite subset such that $m_k\\in \\bigoplus _{i\\in J}M^ne_i$ , for all $k\\in \\mathbb {N}$ .",
"By the acyclicity condition of $M$ , for each $k\\in \\mathbb {N}$ , we have an $m^{\\prime }_k\\in \\bigoplus _{i\\in J}M^{n-1}e_i$ such that $d(m^{\\prime }_k)=m_k$ .",
"It follows $\\sum _{k\\in \\mathbb {N}}m^{\\prime }_kT^k$ is an element of $\\tilde{M}[[T]]^n$ such that $d(\\sum _{k\\in \\mathbb {N}}m^{\\prime }_kT^k)=\\sum _{k\\in \\mathbb {N}}m_kT^k$ .",
"Therefore $\\tilde{M}[[T]]$ is an acyclic right dg $\\tilde{A}[[T]]$ -module.",
"The last comment of the assertion follows from [10].",
"(REF ) We clearly have that $\\mu _M:M\\otimes _A\\tilde{A}[[T]]\\longrightarrow \\tilde{M}[[T]]$ is a morphism (of zero degree) in $GR-\\tilde{A}[[T]]$ .",
"In addition, if we denote by $d:M\\otimes _A\\tilde{A}[[T]]\\longrightarrow M\\otimes _A\\tilde{A}[[T]]$ and $\\tilde{d}:\\tilde{M}[[T]]\\longrightarrow \\tilde{M}[[T]]$ the respective differentials, then, for all homogeneous elements $m\\in M$ and $\\sum _{k\\in \\mathbb {N}}a_kT^k$ , we have $(\\mu _M\\circ d)\\left[m\\otimes (\\sum _{k\\in \\mathbb {N}}a_kT^k)\\right]&=&\\mu _M \\left(d_M(m)\\otimes (\\sum _{k\\in \\mathbb {N}}a_kT^k)+(-1)^{|m|}m\\otimes (\\sum _{k\\in \\mathbb {N}}d(a_k)T^k)\\right)\\\\&=&\\sum _{k\\in \\mathbb {N}}d_M(m)a_kT^k +(-1)^{|m|}md(a_k)T^k\\\\&=&\\sum _{k\\in \\mathbb {N}}(d_M(m)a_k+(-1)^{|m|}md(a_k))T^k\\\\&=&\\sum _{k\\in \\mathbb {N}}d_M(ma_k)T^k\\\\&=&\\tilde{d}(\\sum _{k\\in \\mathbb {N}}ma_kT^k)\\\\&=&(\\tilde{d}\\circ \\mu _M)\\left[m\\otimes (\\sum _{k\\in \\mathbb {N}}a_kT^k)\\right].$ Then, once the naturality $\\mu $ is proved, we will have that it is actually a homological natural transformation of dg functors (see [10]).",
"But that naturality is clear since we have $(\\tilde{f}\\circ \\mu _M)\\left[m\\otimes (\\sum _{k\\in \\mathbb {N}}a_kT^k)\\right]&=&\\tilde{f}(\\sum _{k\\in \\mathbb {N}}ma_kT^k)=\\sum _{k\\in \\mathbb {N}}f(ma_k)T^k=\\sum _{k\\in \\mathbb {N}}f(m)a_kT^k\\\\&=&\\mu _N\\left[f(m)\\otimes (\\sum _{k\\in \\mathbb {N}}a_kT^k)\\right]\\\\&=&(\\mu _N\\circ (f\\otimes 1_{\\tilde{A}[[T]]}))\\left[m\\otimes (\\sum _{k\\in \\mathbb {N}}a_kT^k)\\right],$ for any homogeneous morphism $f:M\\longrightarrow N$ in $GR-A$ and all homogeneous elements $m\\in M$ and $\\sum _{k\\in \\mathbb {N}}a_kT^k\\in \\tilde{A}[[T]]$ .",
"(REF ) Let $\\Pi _A:\\mathcal {D}(A)\\longrightarrow \\mathcal {H}(A)$ denote the homotopically projective resolution functor.",
"Since each $e_iA$ is homotopically projective (see [10]), we have that $\\Pi _A(e_iA)\\cong e_iA$ in $\\mathcal {H}(A)$ .",
"Moreover, we have an isomorphism $\\mu _{e_iA}:e_iA\\otimes _A\\tilde{A}[[T]]\\stackrel{\\cong }{\\longrightarrow }e_i\\tilde{A}[[T]]=e_iA\\hat{\\otimes }V$ in $Dg-\\tilde{A}[[T]]$ (see the proof of Proposition 10.5 in [10]).",
"One then gets from [10] that $\\mu _{e_iA}:\\mathbb {L}\\iota ^*(e_iA)\\longrightarrow (e_iA)\\uparrow _K^V$ is an isomorphism in $\\mathcal {D}(\\tilde{A}[[T]])$ , from which it follows that $\\mu _M:\\mathbb {L}\\iota ^*(M)\\longrightarrow M\\uparrow _K^V$ is an isomorphism, for all $M\\in \\text{per}(A)=\\text{thick}_{\\mathcal {D}(A)}(e_iA\\text{: }i\\in I)$ .",
"Note that we have a canonical isomorphism of dg algebras with enough idempotents $\\tilde{A}^{op}[[T]]\\cong \\tilde{A}[[T]]^{op}$ .",
"We will still denote by $\\iota ^*$ the dg functor $\\tilde{A}[[T]]\\otimes _A?",
":A-Dg\\longrightarrow \\tilde{A}[[T]]$ and by $\\mathbb {L}\\iota ^*$ its left derived functor $\\mathcal {D}(A^{op})\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]]^{op})$ .",
"We will denote by $V\\hat{\\otimes }?",
":A-Dg\\longrightarrow \\tilde{A}[[T]]-Dg$ and $\\uparrow _k^V:\\mathcal {D}(A^{op})\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]]^{op})$ the corresponding dg functor and triangulated functor, respectively.",
"We now get: Corollary 15 Consider the compositions of triangulated functors $\\mathcal {D}(A)^{op}\\stackrel{\\mathbb {R}\\text{Hom}_A(?,A)}{\\longrightarrow }\\mathcal {D}(A^{op})\\stackrel{\\mathbb {L}\\iota ^*}{\\longrightarrow }\\mathcal {D}(\\tilde{A}[[T]]^{op})$ and $\\mathcal {D}(A)^{op}\\stackrel{(\\mathbb {L}\\iota ^*)^o}{\\longrightarrow }\\mathcal {D}(\\tilde{A}[[T]])^{op}\\stackrel{\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,\\tilde{A}[[T]])}{\\longrightarrow }\\mathcal {D}(\\tilde{A}[[T]]^{op}).",
"$ There are natural isomorphisms of triangulated functors $\\eta :(\\mathbb {L}\\iota ^*\\circ \\mathbb {R}\\text{Hom}_A(?,A))_{| per(A)^{op}}\\stackrel{\\cong }{\\longrightarrow }[\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,\\tilde{A}[[T]])\\circ \\mathbb {L}\\iota ^*]_{| per(A)^{op}}$ and $\\eta :\\uparrow _K^V\\circ \\mathbb {R}\\text{Hom}_A(?,A))_{| per(A)^{op}}\\stackrel{\\cong }{\\longrightarrow }[\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,\\tilde{A}[[T]])\\circ \\uparrow _K^V]_{| per(A)^{op}}.$ The first natural isomorphism is a direct consequence of [10].",
"On the other hand, Proposition REF and its left-right symmetric version gives natural isomorphisms $(\\mathbb {L}\\iota ^*)_{| per(A)}\\cong (\\uparrow _K^V)_{| per(A)}$ and $(\\mathbb {L}\\iota ^*)_{| per(A^{op})}\\cong (\\uparrow _K^V)_{| per(A^{op})}$ .",
"Using now the duality $\\mathbb {R}\\text{Hom}_A(?,A):\\text{per}(A)\\stackrel{\\cong ^o}{\\longrightarrow }\\text{per}(A^{op}),$ the result follows.",
"We now revisit and generalize some point of [11].",
"Note that the variable $T$ is not an element of $\\tilde{A}[[T]]$ , unless $A$ has a unit.",
"However, if $Q$ is a right dg $\\tilde{A}[[T]]$ -module and $x\\in Q$ is a homogeneous element, then the product $xT$ makes sense.",
"Indeed since $x=\\sum _{i\\in I}xe_i$ , with $xe_i=0$ for almost all $i\\in I$ , the element $xT:=\\sum _{i\\in I}x(e_iT)$ is a well-defined element of $Q$ with $deg(xT)=deg(x)$ .",
"Furthermore, if $f:Q\\longrightarrow Q^{\\prime }$ is a morphism in $Dg-\\tilde{A}[[T]]$ , then we have $f(xT)=f(\\sum _{i\\in I}x(e_iT))=\\sum _{i\\in I}f(x)e_iT=f(x)T$ .",
"We can now prove: Lemma 16 For each right dg $\\tilde{A}[[T]]$ -module $Q$ , the map $t_Q:Q\\longrightarrow Q$ ($x\\rightsquigarrow xT$ ) is a morphism of zero degree in $Dg-\\tilde{A}[[T]]$ and, when $Q$ varies, the $t_Q$ give a homological natural transformation of dg functors $t:1_{Dg-\\tilde{A}[[T]]}\\longrightarrow 1_{Dg-\\tilde{A}[[T]]}$ .",
"Let $x\\in Q$ and $\\sum _{k\\in \\mathbb {N}}a_kT^k\\in \\tilde{A}[[T]]$ be homogeneous elements.",
"By definition of $\\tilde{A}[[T]]$ and by the fact that $Q=\\bigoplus _{i\\in I}Qe_i$ , we have a finite subset $F\\subset I$ such that $xe_i=0$ and $a_ke_i=0$ , for all $i\\in I\\setminus F$ and all $k\\in \\mathbb {N}$ .",
"It follows that $(x\\sum _{k\\in \\mathbb {N}}a_kT^k)e_i=0$ , for all $i\\in I\\setminus F$ .",
"We then have $t_Q(x\\sum _{k\\in \\mathbb {N}}a_kT^k)&=&(x\\sum _{k\\in \\mathbb {N}}a_kT^k)T=\\sum _{i\\in F}(x\\sum _{k\\in \\mathbb {N}}a_kT^k)e_iT=\\sum _{i\\in F}x(\\sum _{k\\in \\mathbb {N}}a_ke_iT^{k+1})\\\\&=&x\\sum _{k\\in \\mathbb {N}}a_kT^{k+1}=(xT)\\sum _{k\\in \\mathbb {N}}a_kT^k=t_Q(x)\\sum _{k\\in \\mathbb {N}}a_kT^k,$ which shows that $t_Q$ is a morphism of zero degree in $Dg-\\tilde{A}[[T]]$ .",
"If $f:Q\\longrightarrow Q^{\\prime }$ is a homogeneous morphism in $Dg-\\tilde{A}[[T]]$ then we have $(t_{Q^{\\prime }}\\circ f)(x)=f(x)T=f(xT)=(f\\circ t_Q)(x)$ , for each $x\\in Q$ .",
"This proves that, when $Q$ varies, the $t_Q$ give a natural transformation of dg functors $t:1_{Dg-\\tilde{A}[[T]]}\\longrightarrow 1_{Dg-\\tilde{A}[[T]]}$ .",
"This natural transformation is homological since we have $(d_Q\\circ t_Q)(x)&=&d_Q(xT)=d_Q(\\sum _{i\\in I}x(e_iT))=\\sum _{i\\in I}d_Q(x(e_iT))\\\\&=&\\sum _{i\\in I}(d_Q(x)e_i)T=d_Q(x)T=(t_Q\\circ d_Q)(x),$ for each homogeneous element $x\\in Q$ , due to the fact that $d_{\\tilde{A}[[T]]}(e_iT)=d(e_i)T=0$ (see [10]).",
"Note that the associated natural transformation of triangulated functors $t:1_{\\mathcal {D}(\\tilde{A}[[T]])}\\longrightarrow 1_{\\mathcal {D}(\\tilde{A}[[T]])}$ is the one given in [11], after translation to the language of dg algebras with enough idempotents.",
"Replacing $A$ by $A^{op}$ in Lemma REF and interpreting right dg modules over $\\tilde{A}[[T]]^{op}\\cong \\tilde{A}^{op}[[T]]$ as left dg $\\tilde{A}[[T]]$ -modules, we get a natural transformation of dg functors $t:1_{\\tilde{A}[[T]]-Dg}\\longrightarrow 1_{\\tilde{A}[[T]]-Dg}$ which in turn gives a natural transformation of triangulated functors $t:1_{\\mathcal {D}(\\tilde{A}[[T]]^{op})}\\longrightarrow 1_{\\mathcal {D}(\\tilde{A}[[T]]^{op})}$ .",
"These natural transformations do not have correspondents for dg $\\tilde{A}[[T]]-\\tilde{A}[[T]]-$ bimodules, because the action of $T$ by multiplication on a dg $\\tilde{A}[[T]]-\\tilde{A}[[T]]-$ bimodule need not be the same on the left and on the right.",
"We say that a dg $\\tilde{A}[[T]]-\\tilde{A}[[T]]-$ bimodule $X$ is T-symmetric when $Tx=xT$ , for each $x\\in X$ .",
"Note that, with a suitable modification of the argument used in the proof of Lemma REF , one easily sees that if $X$ is a $T$ -symmetric dg $\\tilde{A}[[T]]$ -bimodule, then the assignment $x\\rightsquigarrow xT=Tx$ is a morphism of $\\tilde{A}[[T]]-\\tilde{A}[[T]]-$ bimodules, which we also denote by $t_X$ .",
"Proposition 17 Let us consider the bi-triangulated functor $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(?,?",
"):\\mathcal {D}(\\tilde{A}[[T]])^{op}\\otimes \\mathcal {D}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]]^{op})$ (see Proposition REF ) and let $Q$ be a right dg $\\tilde{A}[[T]]$ -module and $X$ be a $T$ -symmetric dg $\\tilde{A}[[T]]-\\tilde{A}[[T]]-$ bimodule.",
"Then $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(t_Q^o,1_X)$ and $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(1_Q^o,t_X)$ , considered as maps $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(Q,X)\\longrightarrow \\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(Q,X)$ are equal.",
"Moreover they are equal to the evaluation of the natural transformation $t:1_{\\mathcal {D}(\\tilde{A}[[T]]^{op})}\\longrightarrow 1_{\\mathcal {D}(\\tilde{A}[[T]]^{op})}$ at $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(Q,X)$ .",
"By Proposition REF , we have a natural isomorphism of triangulated functor $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(?,X)\\cong \\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,X).$ So in order to see that $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(t_Q^o,1_X)$ is the evaluation of $t$ at $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(Q,X)$ it is enough to check that $t_Q^*=\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,X)(t_Q)$ is precisely $t_{\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(Q,X)}$ , where $\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(Q,X):=\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,X)(Q)$ in the rest of the proof.",
"Note that $t_Q^*$ is the map $\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(Q,X)=\\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (Q),X)\\stackrel{\\Pi (t_Q)^*}{\\longrightarrow }\\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (Q),X)=\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(Q,X).$ Here and in the rest of the proof $\\Pi :=\\Pi _{\\tilde{A}[[T]]}:\\mathcal {D}(\\tilde{A}[[T]])\\longrightarrow \\mathcal {H}(\\tilde{A}[[T]])$ and $\\Upsilon :=\\Upsilon _{\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op}}:\\mathcal {D}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})\\longrightarrow \\mathcal {H}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})$ are the homotopically projective and the homotopically injective resolution functors, respectively.",
"It is convenient to have a careful look at a special case of the action of $\\Pi $ and $\\Upsilon $ on morphisms.",
"Let $Q$ and $X$ be as in the statement and let $f:Q\\longrightarrow Q$ and $\\alpha :X\\longrightarrow X$ be morphisms in $\\mathcal {H}(\\tilde{A}[[T]])$ and $\\mathcal {H}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})$ , respectively.",
"Abusing notation, we put $q(f)=f$ and $q(\\alpha )=\\alpha $ , where $q$ is the functor from the homotopy to the derived category in each case.",
"Viewing $Q$ and $X$ as objects of the respective derived categories, we have a counit map $\\pi _Q:(\\Pi \\circ q)(Q)=\\Pi (Q)\\longrightarrow Q$ and a unit map $\\iota _X:X\\longrightarrow (\\Upsilon \\circ q)(X)=\\Upsilon (X),$ which are quasi-isomorphism.",
"Then $\\Pi (f):=(\\Pi \\circ q)(f)$ is a morphism $\\Pi (Q)\\longrightarrow \\Pi (Q)$ in $\\mathcal {H}(\\tilde{A}[[T]])$ such that $\\pi _Q\\circ \\Pi (f)=f\\circ \\pi _Q\\;\\; (*),$ due to the naturality of the counit $\\pi $ .",
"But since we have an isomorphism $\\text{Hom}_{\\mathcal {H}(\\tilde{A}[[T]])}(\\Pi (Q),\\Pi (Q))\\stackrel{\\cong }{\\longrightarrow }\\text{Hom}_{\\mathcal {D}(\\tilde{A}[[T]])}(Q,Q)$ (which maps $\\varphi \\rightsquigarrow q(\\pi )\\circ \\varphi \\circ q(\\pi )^{-1}$ ), we see that $\\Pi (f)$ is the unique morphism in $\\mathcal {H}(\\tilde{A}[[T]])$ satisfying the equality (*).",
"Similarly, $\\Upsilon (\\alpha )$ is the unique morphism $\\Upsilon (X)\\longrightarrow \\Upsilon (X)$ in $\\mathcal {H}(\\tilde{A}[[T]])\\otimes \\tilde{A}[[T]]^{op})$ such that $\\Upsilon (\\alpha )\\circ \\iota _X=\\iota _X\\circ \\alpha .$ By taking $f=t_Q$ in this argument, we readily see that $\\Pi (t_Q)=t_{\\Pi (Q)}$ since, due to the naturality of $t:1_{Dg-\\tilde{A}[[T]]}\\longrightarrow 1_{Dg-\\tilde{A}[[T]]}$ , we have that $\\pi \\circ t_{\\Pi (Q)}=t_Q\\circ \\pi $ in $\\mathcal {H}(\\tilde{A}[[T]])$ .",
"The analogous fact does not work for $\\alpha =t_X$ since we do not have a correspondent of the natural transformation $t$ for $\\tilde{A}[[T]]-\\tilde{A}[[T]]$ -bimodules.",
"In any case, these comments together with the previous paragraph show that $\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,X)(t_Q)=t_{\\Pi (Q)}^*$ .",
"Using the naturality of $t$ , we see that $t_{\\Pi (Q)}^*$ is a morphism $\\overline{\\text{HOM}}_A(\\Pi (Q),X)\\longrightarrow \\overline{\\text{HOM}}_A(\\Pi (Q),X)$ of left dg $\\tilde{A}[[T]]$ -modules which maps $f\\rightsquigarrow (-1)^{|t_{\\Pi (Q)}| |f|}f\\circ t_{\\Pi (Q)}=f\\circ t_{\\Pi (Q)}=t_X\\circ f.$ We then have that $[t_{\\Pi (Q)}^*(f)](z)=(f\\circ t_X)(z)=f(zT)=f(z)T=Tf(z)=(Tf)(z)=[t_{\\overline{\\text{HOM}}_A(\\Pi (Q),X)}(f)](z), $ for all homogeneous elements $z\\in \\Pi (Q)$ , using the definition of the left $\\tilde{A}[[T]]$ -module structure on $\\overline{\\text{HOM}}_A(\\Pi (Q),X)$ (see [10]) and the $T$ -symmetry of $X$ .",
"Therefore we have $\\mathbb {R}\\text{Hom}_A(?,X)(t_Q)=t_{\\mathbb {R}\\text{Hom}_A(Q,X)}$ , as desired.",
"On the other hand, by Proposition REF , there is a natural isomorphism $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(Q,?",
")\\cong \\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(\\Pi (Q),?",
")=q\\circ \\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (Q),?",
")\\circ \\Upsilon $ of triangulated functors $\\mathcal {D}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]]^{op}).$ Then $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(1_Q^o,t_X)$ is the morphism $\\overline{HOM}_{\\tilde{A}[[X]]}(\\Pi (Q),\\Upsilon (X))&\\longrightarrow &\\overline{HOM}_{\\tilde{A}[[X]]}(\\Pi (Q),\\Upsilon (X))\\\\f&\\rightsquigarrow & (-1)^{|1_Q| |q(\\Upsilon (t_X)|}\\Upsilon (t_X)\\circ f\\circ 1_Q)=\\Upsilon (t_X)_*(f).$ In other words, we have that $\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(1_Q^o,t_X)=q(\\Upsilon (t_X)_*),$ where $q:\\mathcal {H}(\\tilde{A}[[T]]^{op})\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]]^{op})$ is the canonical functor and $\\Upsilon (t_X)_*=\\overline{HOM}_A(\\Pi (M),?",
")(\\Upsilon (t_X)):\\overline{HOM}_A(\\Pi (M),\\Upsilon (X))\\longrightarrow \\overline{HOM}_A(\\Pi (M),\\Upsilon (X)).$ But the induced functor $\\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (M),?",
"):\\mathcal {H}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})\\longrightarrow \\mathcal {H}(\\tilde{A}[[T]]^{op})$ preserves quasi-isomorphisms since $\\Pi (M)$ is homotopically projective in $\\mathcal {H}(\\tilde{A}[[T]])$ .",
"If now $\\iota :=\\iota _X:X\\longrightarrow \\Upsilon (X)$ is as above, then $\\iota _*:=\\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (M),\\iota ):\\overline{HOM}_{\\tilde{A}[[X]]}(\\Pi (Q),X)\\longrightarrow \\overline{HOM}_{\\tilde{A}[[X]]}(\\Pi (Q),\\Upsilon (X)))$ is a quasi-isomorphism of left dg $\\tilde{A}[[T]]$ -modules.",
"Applying to the equality $\\Upsilon (t_X)\\circ \\iota =\\iota \\circ t_X$ the functor $\\overline{HOM}_A(\\Pi (M),?",
"):\\mathcal {H}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})\\longrightarrow \\mathcal {H}(\\tilde{A}[[T]]^{op}),$ we get the following commutative diagram in $\\mathcal {H}(\\tilde{A}[[T]]^{op})$ , where the horizontal arrows are quasi-isomorphisms.",
"${\\overline{HOM}_{{\\widetilde{A}}[[T]]}(\\Pi (M),X)[r]^{\\iota _*}[d]_{(t_X)_*}&\\overline{HOM}_{{\\widetilde{A}}[[T]]}(\\Pi (M),\\Upsilon (X))[d]_{\\Upsilon (t_X)_*}\\\\\\overline{HOM}_{{\\widetilde{A}}[[T]]}(\\Pi (M),X)[r]^{\\iota _*}&\\overline{HOM}_{{\\widetilde{A}}[[T]]}(\\Pi (M),\\Upsilon (X))}$ Moreover, the left vertical arrow takes $f\\rightsquigarrow t_X\\circ f$ , for each homogeneous element $f\\in \\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (M),X)$ .",
"But, in turn, we have that $(t_X\\circ f)(v)=Tf(v)=(Tf)(v)=t_{\\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (M),X)}(f)(v),$ for each homogeneous element $v\\in \\Pi (M)$ .",
"Therefore the left vertical arrow of last diagram is the evaluation of the natural transformation of dg functors $t:1_{\\tilde{A}[[T]]-Dg}\\longrightarrow 1_{\\tilde{A}[[T]]-Dg}$ at $\\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (Q),X)$ .",
"The fact that $t$ is a natural transformation of dg functors implies that we also have an equality $t_{\\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (Q),\\Upsilon (X))}\\circ \\iota _*=\\iota _*\\circ t_{\\overline{HOM}_{\\tilde{A}[[T]]} (\\Pi (Q),X)}$ in $\\tilde{A}[[T]]-Dg$ and, hence, also in $\\mathcal {H}(\\tilde{A}[[T]]^{op})$ .",
"We then have that $t_{\\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (Q),\\Upsilon (X))}\\circ \\iota _*=\\Upsilon (t_X)_*\\circ \\iota _*$ in $\\mathcal {H}(\\tilde{A}[[T]]^{op})$ .",
"Applying the functor $q:\\mathcal {H}(\\tilde{A}[[T]]^{op})\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]]^{op})$ to this last equality and bearing in mind that $q(\\iota _*)$ is an isomorphism, we conclude that $q((\\Upsilon (t_X))_*)=q(t_{\\overline{HOM}_{\\tilde{A}[[T]]}(\\Pi (Q),\\Upsilon (X))})=t_{\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(Q,X)}.$ For our next result we adopt the terminology of [11] and, for the given dg algebra with enough idempotents $A$ , we put $\\mathcal {C}_V^o=\\text{per}(\\tilde{A}[[T]])$ , we denote by $\\mathcal {C}_V^o[t^{-1}]$ the localization of $\\mathcal {C}_V^o$ with respect to natural transformation $t$ given above (see [11] for the definition) and we let $p:\\mathcal {C}_V^o\\longrightarrow \\mathcal {C}_V^o[t^{-1}]$ be the canonical functor.",
"We also put ${}_V\\mathcal {C}^o=\\text{per}(\\tilde{A}[[T]]^{op})$ , ${}_V\\mathcal {C}^o[t^{-1}]$ and $p:{}_V\\mathcal {C}^o\\longrightarrow {}_V\\mathcal {C}^o[t^{-1}]$ for the corresponding concepts on the left.",
"Lemma 18 Let $p:\\mathcal {C}_V^o\\longrightarrow \\mathcal {C}_V^o[t^{-1}]$ and $p^{\\prime }:{}_V\\mathcal {C}^o\\longrightarrow {}_V\\mathcal {C}^o[t^{-1}]$ be the canonical triangulated functors given by localization, and let $Q_1$ and $Q_2$ be objects of $\\mathcal {C}_V^o=\\text{per}(\\tilde{A}[[T]])$ .",
"There is an isomorphism $p(Q_1)\\cong p(Q_2)$ if, and only if, there is an isomorphism $p^{\\prime }(Q_1^\\star )\\cong p^{\\prime }(Q_2^\\star )$ , where $(?",
")^\\star :=\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,\\tilde{A}[[T]]):\\mathcal {D}(\\tilde{A}[[T]])^{op}=\\mathcal {C}_V^{op}\\longrightarrow {}_V\\mathcal {C}=\\mathcal {D}(\\tilde{A}^{op}[[T]])$ is the usual triangulated functor.",
"The fact that $p(Q_1)$ and $p(Q_2)$ are isomorphic in $\\mathcal {C}_V^o[t^{-1}]$ means that we have morphisms $f:Q_1\\longrightarrow Q_2$ and $g:Q_2\\longrightarrow Q_1$ in $\\mathcal {C}_V^o=\\text{per}(\\tilde{A}[[T]])$ such that $g\\circ f\\circ t_{Q_1}^r=t_{Q_1}^s$ and $f\\circ g\\circ t_{Q_2}^m=t_{Q_2}^n$ , for some $r,s,m,n\\in \\mathbb {N}$ .",
"If we now apply the duality $(?",
")^\\star =\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,\\tilde{A}[[T]]):\\text{per}(\\tilde{A}[[T]])=\\mathcal {C}_V^o\\stackrel{\\cong ^o}{\\longrightarrow }{}_V\\mathcal {C}^o=\\text{per}(\\tilde{A}[[T]]^{op}),$ then we get that $(t_{Q_2}^\\star )^m\\circ g^\\star \\circ f^\\star =(t_{Q_2}^\\star )^n$ .",
"But Propositions REF and REF tell us that we have $(t_{Q_k})^\\star =t_{Q_k^\\star }$ for $k=1,2$ , which implies that $p^{\\prime }(g^\\star )\\circ p^{\\prime }(f^\\star )$ and $p^{\\prime }(f^\\star )\\circ p^{\\prime }(g^\\star )$ are isomorphisms in ${}_V\\mathcal {C}^o[t^{-1}]$ , and hence that $p^{\\prime }(Q_1^\\star )\\cong p^{\\prime }(Q_2^\\star )$ in ${}_V\\mathcal {C}^o[t^{-1}]$ .",
"The reverse implication follows by exchanging the roles of $A$ and $A^{op}$ and of $Q_k$ and $Q_k^\\star $ , bearing in mind that $Q_k$ is isomorphic to $Q_k^{\\star \\star }$ , for $k=1,2$ (see [10]).",
"The first assertion of the following Lemma REF seems to be folklore, but we include a short proof.",
"A right dg $\\tilde{A}[[T]]$ -module $Q$ will be called T-torsion-free when $t_Q:Q\\longrightarrow Q$ is monomorphism in $Gr-\\tilde{A}[[T]]$ .",
"Note that this is equivalent to saying that $t_Q$ is a monomorphism for the abelian structure of $\\mathcal {C}(\\tilde{A}[[T]])$ .",
"Lemma 19 Let $A$ be a dg algebra with enough idempotents and let $q_A:\\mathcal {H}(A)\\longrightarrow \\mathcal {D}(A)$ be the canonical functor.",
"The following assertions hold: The induced functor $q:\\text{thick}_{\\mathcal {H}(A)}(e_iA\\text{: }i\\in I)\\longrightarrow \\text{per}(A)=\\mathcal {D}^c(A)$ is an equivalence of triangulated categories.",
"Each $Q\\in \\text{thick}_{\\mathcal {H}(\\tilde{A}[[T]])}(e_i\\tilde{A}[[T]]\\text{: }i\\in I)$ is isomorphic in $\\mathcal {H}(\\tilde{A}[[T]])$ to a T-torsion-free right dg $\\tilde{A}[[T]]$ -module.",
"(1) The subcategory $\\text{thick}_{\\mathcal {H}(A)}(e_iA\\text{: }i\\in I)$ of $\\mathcal {H}(A)$ consists of homotopically projective objects and the restriction of $q$ to the subcategory of homotopically projective objects is fully faithful.",
"In order to prove the density, recall that $\\text{per}(A)=\\text{thick}_{\\mathcal {D}(A)}(e_iA\\text{: }i\\in I)$ (see [5]).",
"This implies in particular that each $X\\in \\text{per}(A)$ is a direct summand in $\\mathcal {D}(A)$ of a right dg $A$ -module $P$ for which there is a sequence of morphisms $0=P_0\\stackrel{f_1}{\\longrightarrow }P_1\\stackrel{f_2}{\\longrightarrow }\\cdots \\stackrel{f_{n-1}}{\\longrightarrow }P_{n-1}\\stackrel{f_n}{\\longrightarrow }P_n$ in $\\mathcal {D}(A)$ such that $P_n=P$ and $\\text{cone}(f_k)\\cong e_{i_k}A[m_k]$ , for some $i_k\\in I$ and some $m_k\\in \\mathbb {Z}$ , for $k=1,...,n$ .",
"We will prove by induction on $n$ that $P\\cong q(Q)$ , for some $Q\\in \\text{thick}_{\\mathcal {H}(A)}(e_iA\\text{: }i\\in I)$ .",
"For $n=0$ there is nothing to prove, so we assume that $n>1$ .",
"By the induction hypothesis, we can choose $Q_{n-1}\\in \\text{thick}_{\\mathcal {H}(A)}(e_iA\\text{: }i\\in I)$ such that $q(Q_{n-1})\\cong P_{n-1}$ .",
"We then get a distinguished triangle $Q_{n-1}\\longrightarrow P\\longrightarrow e_iA[m]\\stackrel{f[1]}{\\longrightarrow }Q_{n-1}[1]$ , for some $i\\in I$ , some $m\\in \\mathbb {Z}$ and some morphism $f:e_iA[m]\\longrightarrow Q_{n-1}$ in $\\mathcal {D}(A)$ .",
"But the functor $q$ gives an isomorphism $\\text{Hom}_{\\mathcal {H}(A)}(e_iA,Q_{n-1})\\stackrel{\\cong }{\\longrightarrow }\\text{Hom}_{\\mathcal {D}(A)}(e_iA,Q_{n-1})$ .",
"This means that we may view $f$ as a morphism in $\\mathcal {H}(A)$ , and then the triangulated cone $Q=\\text{cone}_{\\mathcal {H}(A)}(f)$ is in $\\text{thick}_{\\mathcal {H}(A)}(e_iA\\text{: }i\\in I)$ and satisfies that $q(Q)\\cong P$ .",
"Let now $X$ , $P$ and $Q$ be as above and let $e\\in \\text{End}_{\\mathcal {D}(A)}(P)$ be the idempotent endomorphism corresponding to the direct summand $X$ of $P$ .",
"Since $q$ gives an algebra isomorphism $\\text{End}_{\\mathcal {H}(A)}(Q)\\stackrel{\\cong }{\\longrightarrow }\\text{End}_{\\mathcal {D}(A)}(P)$ , we have a unique $\\epsilon =\\epsilon ^2\\in \\text{End}_{\\mathcal {H}(A)}(Q)$ such that $q(\\epsilon )=e$ .",
"Since $\\mathcal {H}(A)$ has arbitrary (set-indexed) coproducts, we know that idempotents split in $\\mathcal {H}(A)$ (see [8]).",
"We then get a direct summand $Y$ of $Q$ in $\\mathcal {H}(A)$ corresponding to $\\epsilon $ , and we clearly have that $q(Y)\\cong X$ .",
"(2) Let $Q\\in \\text{thick}_{\\mathcal {H}(\\tilde{A}[[T]])}(e_i\\tilde{A}[[T]]\\text{: }i\\in I)$ be any object.",
"By the obvious adaptation of [5] to the language of dg algebras with enough idempotents, we know that there is a chain of inflations in $\\mathcal {C}(\\tilde{A}[[T]])$ $0=P_0\\hookrightarrow P_1\\hookrightarrow ...\\hookrightarrow P_n\\hookrightarrow ... $ such that $\\text{Coker}(P_{n-1}\\hookrightarrow P_n)$ is a direct summand in $\\mathcal {C}(\\tilde{A}[[T]])$ of a (possibly infinite) coproduct of dg right $\\tilde{A}[[T]]$ -module of the form $e_i\\tilde{A}[[T]] [m]$ , with $i\\in I$ and $m\\in \\mathbb {Z}$ , and $P=\\bigcup _{n\\in \\mathbb {N}}P_n$ is isomorphic to $Q$ in $\\mathcal {D}(\\tilde{A}[[T]])$ .",
"Since all the exact sequences $0\\rightarrow P_{n-1}\\hookrightarrow P_n\\longrightarrow P_n/P_{n-1}\\rightarrow 0$ split in $Gr-\\tilde{A}[[T]]$ , we readily see that $P$ is projective in this category.",
"In particular $P$ is $T$ -torsion-free.",
"But $P$ and $Q$ are homotopically projective objects of $\\mathcal {H}(\\tilde{A}[[T]])$ , which implies that the canonical functor $q:\\mathcal {H}(\\tilde{A}[[T]])\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]])$ induces bijections $\\text{Hom}_{\\mathcal {H}(\\tilde{A}[[T]])}(X,Y)\\stackrel{\\cong }{\\longrightarrow }\\text{Hom}_{\\mathcal {D}(\\tilde{A}[[T]])}(X,Y)$ , for $X,Y\\in \\lbrace P,Q\\rbrace $ .",
"We deduce that any isomorphism $P\\stackrel{\\cong }{\\longrightarrow }Q$ in $\\mathcal {D}(\\tilde{A}[[T]])$ can be lifted to a corresponding isomorphism in $\\mathcal {H}(\\tilde{A}[[T]])$ ."
],
[
"The main theorem under hypothesis (b)",
"We can now complete the proof of Theorem REF .",
"Theorem 20 Let $\\mathcal {C}_k^0$ be the category of compact objects of an algebraic compactly generated triangulated category.",
"For any objects $M,N\\in \\text{Ob}(\\mathcal {C}_k^0)$ , the following assertions are equivalent: There is a distinguished triangle $Z_\\ell \\stackrel{\\begin{pmatrix} v\\\\ u\\end{pmatrix}}{\\longrightarrow }Z_\\ell \\oplus M\\stackrel{\\begin{pmatrix} h & j \\end{pmatrix}}{\\longrightarrow } N\\longrightarrow Z_\\ell [1]$ , where $v$ is a nilpotent endomorphism of $Z_\\ell $ .",
"There is a distinguished triangle $N\\stackrel{\\begin{pmatrix} j\\\\ h \\end{pmatrix}}{\\longrightarrow } M\\oplus Z_r\\stackrel{\\begin{pmatrix} u & v\\end{pmatrix}}{\\longrightarrow } Z_r\\longrightarrow N[1]$ , where $v$ is a nilpotent endomorphism of $Z_r$ .",
"Using the version of Keller's theorem for dg algebras with enough idempotents (see [10]), we can and shall assume that $\\mathcal {C}_k^o=\\mathcal {D}(A)^c=\\text{per}(A)$ , where $A$ is a dg algebra with enough idempotents.",
"$(1)\\Longrightarrow (2):$ In [11] we showed that if there is a distinguished triangle as in assertion 1, then the quintuple $(\\mathcal {C}_k,\\mathcal {C}_V,\\mathcal {C}_V^o,\\uparrow _k^V,t)$ give degeneration data for $\\mathcal {C}_k^o$ , where $\\mathcal {C}_k=\\mathcal {D}(A)$ , $\\mathcal {C}_V=\\mathcal {D}(\\tilde{A}[[T]])$ , $\\mathcal {C}_V^o=\\mathcal {D(\\tilde{A}[[T]])}^c=\\text{per}(\\tilde{A}[[T]])$ and $\\uparrow _k^V:\\mathcal {D}(A)\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]])$ and $t:1_{\\mathcal {D}(\\tilde{A}[[T]])}\\longrightarrow 1_{\\mathcal {D}(\\tilde{A}[[T]])}$ are as in the previous results of this section.",
"Moreover, in the above mentioned result [11] it was also proved that there exists an object $Q\\in \\mathcal {C}_V^o=\\text{per}(\\tilde{A}[[T]])$ so that both required conditions for categorical degeneration are satisfied, namely: If $p:\\mathcal {C}_V^o\\longrightarrow \\mathcal {C}_V^o[t^{-1}]$ is the canonical functor, then $p(Q)\\cong p(M\\uparrow _k^V)$ ; $\\phi (\\text{cone}(t_Q))\\cong N$ , where $\\phi :\\mathcal {C}_V^o=\\text{per}(\\tilde{A}[[T]])\\longrightarrow \\mathcal {D}(A)=\\mathcal {C}_k$ is the restriction of $\\iota _*:\\mathcal {D}(\\tilde{A}[[T]])\\longrightarrow \\mathcal {D}(A)$ to $\\text{per}(\\tilde{A}[[T]])$ .",
"Here and in the rest of the proof $\\text{cone}(f)$ denotes the triangulated cone.",
"With this information in mind, we give the proof of the theorem, which is divided in two steps: Step 1: If $Q_1$ is a T-torsion-free right dg $\\tilde{A}[[T]]$ -module in $\\text{thick}_{\\mathcal {H}(\\tilde{A}[[T]])}(e_i\\tilde{A}[[T]]\\text{: }i\\in I)$ and if we put $Q_1^\\star =\\overline{HOM}_{\\tilde{A}[[T]]}(Q_1,\\tilde{A}[[T]])$ , then $\\phi (\\text{cone}(t_{Q_1^\\star }))\\cong \\mathbb {R}\\text{Hom}_A(?,A)(\\phi (\\text{cone}(t_{Q_1})))$ Note that $Q_1$ is homotopically projective, so that we also have $Q_1^\\star =\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,\\tilde{A}[[T]])(Q_1)\\cong \\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(Q_1,\\tilde{A}[[T]])$ (see Proposition REF ).",
"On the other hand, the homomorphism of dg algebras $\\rho \\otimes \\rho ^o :\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op}\\longrightarrow A\\otimes A^{op}$ gives a restriction of scalars functor $(\\rho \\otimes \\rho ^o)_*:A-Dg-A\\longrightarrow \\tilde{A}[[T]]-Dg-\\tilde{A}[[T]].$ In particular $A$ is a dg $\\tilde{A}[[T]]-\\tilde{A}[[T]]-$ bimodule by defining $(\\sum _{k\\in \\mathbb {N}}a_kT^k)a=a_0a$ and $a (\\sum _{k\\in \\mathbb {N}}a_kT^k))=aa_0$ , for all homogeneous elements $\\sum _{k\\in \\mathbb {N}}a_kT^k\\in \\tilde{A}[[T]]$ and $a\\in A$ .",
"Note that we then have an exact sequence of T-symmetric $\\tilde{A}[[T]]-\\tilde{A}[[T]]-$ bimodules $0\\rightarrow \\tilde{A}[[T]]\\stackrel{t_{\\tilde{A}[[T]]}}{\\longrightarrow }\\tilde{A}[[T]]\\stackrel{\\rho }{\\longrightarrow }A\\rightarrow 0$ in $Gr-(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})$ and in (the abelian structure of) $\\mathcal {C}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})$ .",
"The last sequence gives a distinguished triangle $\\tilde{A}[[T]]\\stackrel{t_{\\tilde{A}[[T]]}}{\\longrightarrow }\\tilde{A}[[T]]\\stackrel{\\rho }{\\longrightarrow }A\\longrightarrow \\tilde{A}[[T]] [1]$ in $\\mathcal {D}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})$ .",
"By Propositions REF and REF , application of the functor $\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(Q_1,?",
"):\\mathcal {D}(\\tilde{A}[[T]]\\otimes \\tilde{A}[[T]]^{op})\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]]^{op})$ to the last distinguished triangle gives a distinguished triangle $Q_1^\\star \\stackrel{t_{Q_1^\\star }}{\\longrightarrow }Q_1^\\star \\longrightarrow \\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(Q_1,A)\\longrightarrow Q_1^\\star [1]$ in $\\mathcal {D}(\\tilde{A}[[T]]^{op})$ , so that $\\text{cone}(t_{Q_1^\\star })\\cong \\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(Q_1,A):=\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(Q_1,?",
")(A)$ .",
"It is important to notice that, by Proposition REF again, we have isomorphisms in $\\mathcal {D}(\\tilde{A}[[T]]^{op})$ : $\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(Q_1,?",
")(A)\\cong \\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(Q_1,A)\\cong \\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,A)(Q_1).",
"$ When we apply the contravariant triangulated functor $\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,A):\\mathcal {D}(\\tilde{A}[[T]])\\longrightarrow \\mathcal {D}(\\tilde{A}[[T]]^{op})$ to the morphisms $t_{Q_1}:Q_1\\longrightarrow Q_1$ we obtain the zero map.",
"Indeed, due to the homotopically projective condition of $Q_1$ , we have that $\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,A)(Q_1)=\\overline{HOM}_{\\tilde{A}[[T]]}(Q_1,A)$ .",
"But since multiplication by $T$ kills the elements of $A$ , for each homogeneous element $f\\in \\overline{HOM}_{\\tilde{A}[[T]]}(Q_1,A)$ we have $[\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,A)(t_{Q_1})](f)=(-1)^{|f| |t_{Q_1}|}f\\circ t_{Q_1}=f\\circ t_{Q_1},$ and this is a morphism of right dg $\\tilde{A}[[T]]$ -modules $Q_1\\longrightarrow A$ such that $(f\\circ t_{Q_1})(x)= f(xT)=f(x)T=0$ for all $x\\in Q_1.$ On the other hand, the contravariant dg functor $\\overline{HOM}_{\\tilde{A}[[T]]}(?,A):Gr-\\tilde{A}[[T]]\\longrightarrow \\tilde{A}[[T]]-Gr$ is left exact and we have an exact sequence $0\\rightarrow Q_1\\stackrel{t_{Q_1}}{\\longrightarrow }Q_1\\stackrel{p}{\\longrightarrow } Q_1/TQ_1\\rightarrow 0$ in $Gr-\\tilde{A}[[T]]$ (which is actually an exact sequence in $\\mathcal {C}(\\tilde{A}[[T]])$ ) due to the T-torsion-free condition of $Q_1$ .",
"It follows that we have an exact sequence $0\\rightarrow \\overline{HOM}_{\\tilde{A}[[T]]}(Q_1/TQ_1,A)\\stackrel{p^*}{\\longrightarrow }\\overline{HOM}_{\\tilde{A}[[T]]}(Q_1,A)\\stackrel{0}{\\longrightarrow }\\overline{HOM}_{\\tilde{A}[[T]]}(Q_1,A)$ in $\\tilde{A}[[T]]-Gr$ .",
"Therefore we have an isomorphism $p^*:\\overline{HOM}_{\\tilde{A}[[T]]}(Q_1/TQ_1,A)\\stackrel{\\cong }{\\longrightarrow }\\overline{HOM}_{\\tilde{A}[[T]]}(Q_1,A).$ We will see that $p^*$ commutes with the differentials, which will show that we have an isomorphism $\\text{cone}(t_{Q_1^\\star })\\cong \\overline{HOM}_{\\tilde{A}[[T]]}(Q_1/TQ_1,A)$ in $\\mathcal {D}(\\tilde{A}[[T]]^{op})$ .",
"Indeed if $d:\\overline{HOM}_{\\mathcal {A}[[T]]}(Q_1/Q_1T,A)\\longrightarrow \\overline{HOM}_{\\mathcal {A}[[T]]}(Q_1/Q_1T,A)$ and $\\delta :\\overline{HOM}_{\\mathcal {A}[[T]]}(Q_1,A)\\longrightarrow \\overline{HOM}_{\\mathcal {A}[[T]]}(Q_1,A)$ denote the respective differentials, then, for each homogeneous element $g\\in \\overline{HOM}_{\\mathcal {A}[[T]]}(Q_1/Q_1T,A)$ , we have $(\\delta \\circ p^*)(g)=\\delta (g\\circ p)=d_A\\circ g\\circ p-(-1)^{|g\\circ p|}g\\circ p\\circ d_{Q_1}=d_A\\circ g\\circ p-(-1)^{|g|}g\\circ p\\circ d_{Q_1}.",
"$ But $p$ is a morphism in $Z^0(\\tilde{A}[[T]]-Dg)=\\mathcal {C}(\\tilde{A}[[T]]^{op})$ , so that $d_{Q_1/Q_1T}\\circ p-p\\circ d_{Q_1}=0$ .",
"We then get $(\\delta \\circ p^*)(g)=d_A\\circ g\\circ p-(-1)^{|g|}g\\circ d_{Q_1/Q_1T}\\circ p=d(g)\\circ p=(p^*\\circ d)(g), $ and hence $\\delta \\circ p^*=p^*\\circ d$ as desired.",
"If we apply the restriction of scalars functor $\\phi =\\iota _*:\\tilde{A}[[T]]-Dg\\longrightarrow A-Dg$ , then $\\overline{HOM}_{\\tilde{A}[[T]]}(Q_1/TQ_1,A)$ is taken to $\\overline{HOM}_A(\\phi (Q_1/Q_1T),A)$ .",
"But, as right dg $A$ -modules, we have an isomorphism $\\phi (Q_1/Q_1T)=\\rho ^*(Q_1)$ where $\\rho ^*=?\\otimes _{\\tilde{A}[[T]]}A:Dg-\\tilde{A}[[T]]\\longrightarrow Dg-A$ is the extension of scalars along the morphism of dg algebras with enough idempotents $\\rho :\\tilde{A}[[T]]\\longrightarrow A$ .",
"Then the induced triangulated functor $\\rho ^*=H^0(\\rho ^*):H^0(Dg-\\tilde{A}[[T]])=\\mathcal {H}(\\tilde{A}[[T]])\\longrightarrow \\mathcal {H}(A)=H^0(Dg-A)$ has the property that $\\rho ^*(Q_1)=\\phi (Q_1/Q_1T)\\in \\text{thick}_{\\mathcal {H}(A)}(\\rho ^*(e_i\\tilde{A}[[T]])\\text{: }i\\in I)=\\text{thick}_{\\mathcal {H}(A)}(e_iA\\text{: }i\\in I).$ In particular, we get that $\\phi (Q_1/Q_1T)$ is homotopically projective in $\\mathcal {H}(A)$ and, hence, that $\\overline{HOM}_A(\\phi (Q_1/Q_1T),A)\\cong \\mathbb {R}\\text{Hom}_A(?,A)(\\phi (Q_1/Q_1T))\\cong \\mathbb {R}\\text{HOM}_A(\\phi (Q_1/Q_1T),A).$ (see Proposition REF ).",
"Bearing in mind that the exact sequence $0\\rightarrow Q_1\\stackrel{t_{Q_1}}{\\longrightarrow }Q_1\\longrightarrow Q_1/TQ_1\\rightarrow 0$ in $\\mathcal {C}(\\tilde{A}[[T]])$ (with respect to the abelian exact structure of $\\mathcal {C}(\\tilde{A}[[T]])$ ) gives a distinguished triangle $Q_1\\stackrel{t_{Q_1}}{\\longrightarrow }Q_1\\longrightarrow Q_1/TQ_1\\longrightarrow Q_1[1]$ in $\\mathcal {D}(\\tilde{A}[[T]])$ , we get that $Q_1/Q_1T\\cong \\text{cone}(t_{Q_1})$ in $\\mathcal {D}(\\tilde{A}[[T]])$ .",
"We then get isomorphisms $\\phi (\\text{cone}(t_{Q_1^\\star }))&\\cong &\\phi (\\overline{HOM}_{\\tilde{A}[[T]]}(Q_1/TQ_1,A))\\\\&\\cong &\\overline{HOM}_A(\\phi (Q_1/Q_1T),A)\\\\&\\cong & \\mathbb {R}\\text{Hom}_A(?,A)(\\phi (\\text{cone}(t_{Q_1}))).$ Step 2: End of the proof: Let now $M$ and $N$ be as in assertion (1) and let $Q\\in \\mathcal {C}_V^o=\\text{per}(\\tilde{A}[[T]])$ be the right dg $\\tilde{A}[[T]]$ -module considered in the second paragraph of this proof.",
"Using Lemma REF , without loss of generality, we may assume that $M,N\\in \\text{thick}_{\\mathcal {H}(A)}(e_iA\\text{: }i\\in I)$ and that $Q$ is a T-torsion-free right $\\tilde{A}[[T]]$ -module in $\\text{thick}_{\\mathcal {H}(\\tilde{A}[[T]])}(e_i\\tilde{A}[[T]]\\text{: }i\\in I)$ .",
"Note that then $\\tilde{M}[[T]]\\in \\text{thick}_{\\mathcal {H}(\\tilde{A}[[T]])}(e_i\\tilde{A}[[T]]\\text{: }i\\in I)$ and $\\tilde{M}[[T]]$ is T-torsion-free.",
"According to the Step 1, we have that $\\phi (\\text{cone}(t_{Q^\\star }))\\cong \\mathbb {R}\\text{Hom}_A(?,A)(\\phi (\\text{cone}(t_{Q})))=\\mathbb {R}\\text{Hom}_A(?,A)(N)=:N^*.$ On the other hand, by Corollary REF and Proposition REF , we know that $(M\\uparrow _k^V)^\\star :=\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,\\tilde{A}[[T]])(M\\uparrow _k^V)\\cong M^*\\uparrow _k^V,$ where $M^*:=\\mathbb {R}\\text{Hom}_A(?,A)(M)$ .",
"Moreover, by Lemma REF , we get that $p^{\\prime } ((M\\uparrow _k^V)^\\star )$ and $p^{\\prime }(Q^\\star )$ are isomorphic in ${}_V\\mathcal {C}^o[t^{-1}]$ , where $p^{\\prime }:{}_V\\mathcal {C}^o\\longrightarrow {}_V\\mathcal {C}^o[t^{-1}]$ is the canonical functor.",
"It follows from this that the left dg $\\tilde{A}[[T]]$ -module $Q^\\star :=\\mathbb {R}\\text{Hom}_{\\tilde{A}[[T]]}(?,\\tilde{A}[[T]])(Q)=\\mathbb {R}\\text{HOM}_{\\tilde{A}[[T]]}(Q,\\tilde{A}[[T]])$ defines a categorical degeneration $M^*\\le _{cdeg}N^*$ , where $(?",
")^*=\\mathbb {R}\\text{Hom}_A(?,A):\\text{per}(A)\\stackrel{\\cong ^o}{\\longrightarrow }\\text{per}(A^{op})$ is the duality defined by the regular dg bimodule $X=A$ (see [10]).",
"But categorical degeneration implies triangle degeneration by Theorem REF (see [11]), so that we have $M^*\\le _{\\Delta +\\text{nil}}N^*$ .",
"That is, we have a distinguished triangle $U\\stackrel{\\begin{pmatrix} v^{\\prime }\\\\ u^{\\prime } \\end{pmatrix}}{\\longrightarrow }U\\oplus M^*\\stackrel{\\begin{pmatrix} h^{\\prime } & j^{\\prime } \\end{pmatrix}}{\\longrightarrow }N^*\\longrightarrow U[1]$ in ${}_k\\mathcal {C}^o=\\text{per}(A^{op})$ , where $v^{\\prime }$ is a nilpotent endomorphism of $U$ .",
"Applying the duality $\\mathbb {R}\\text{Hom}_{A^{op}}(?,A):\\text{per}(A^{op})\\stackrel{\\cong ^o}{\\longrightarrow }\\text{per}(A)$ to this last distinguished triangle, we obtain a distinguished triangle $N\\stackrel{\\begin{pmatrix} j^{\\prime }* \\\\ h^{\\prime *}\\end{pmatrix}}{\\longrightarrow }M\\oplus U^*\\stackrel{\\begin{pmatrix} u^{\\prime *} & v^{\\prime *}\\end{pmatrix}}{\\longrightarrow }U^*\\longrightarrow N[1],$ where $v^{\\prime *}$ is clearly a nilpotent endomorphism of $U^*$ .",
"The proof of the implication ends by taking $Z_r:=U^*$ .",
"$(2)\\Longrightarrow (1):$ By applying the duality $\\mathbb {R}\\text{Hom}_{A}(?,A):\\mathcal {C}_k^o=\\text{per}(A)\\stackrel{\\cong ^o}{\\longrightarrow }\\text{per}(A^{op})$ to the distinguished triangle in assertion (2) we get that $M^*\\le _{\\Delta +\\text{nil}}N^*$ in ${}_k\\mathcal {C}^o:=\\text{per}(A^{op})$ .",
"The proof of the implication $1)\\Longrightarrow 2)$ , when applied with $A^{op}$ instead of $A$ , shows that we have $M\\cong M^{**}\\le _{\\Delta +\\text{nil}}N^{**}\\cong N$ , so that the distinguished triangle of assertion 1) exists.",
"Observe that the proof actually gives the following additional statement.",
"Corollary 21 Let $\\mathcal {C}_k^0={\\mathcal {D}}(A)^c$ be the category of compact objects in the derived category ${\\mathcal {D}}(A)$ of a dg algebra $A$ with enough idempotents.",
"For any objects $M,N\\in \\text{Ob}(\\mathcal {C}_k^0)$ , we get $M\\le _{\\Delta +\\text{nil}}N\\Leftrightarrow \\mathbb {R}\\text{Hom}_{A}(?,A)(M)\\le _{\\Delta +\\text{nil}}\\mathbb {R}\\text{Hom}_{A}(?,A)(N).$"
]
] | 1612.05771 |
[
[
"On D.Y. Gao and X. Lu paper \"On the extrema of a nonconvex functional\n with double-well potential in 1D\""
],
[
"Abstract The aim of this paper is to discuss the main result in the paper by D.Y.",
"Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew.",
"Math.",
"Phys.",
"(2016) 67:62].",
"More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\\in [1,\\infty]$ endowed with its usual norm.",
"We show that the objective function has not local extrema with the mentioned constraints for $p\\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer.",
"We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y.",
"Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]."
],
[
"Introduction",
"For a faithful presentation of the the problem and result discussed in [1] we quote from this paper: “The fourth-order polynomial defined by $H(x):=\\nu /2(1/2x^{2}-\\lambda )^{2}$ , where $x\\in \\mathbb {R},$ $\\nu ,\\lambda $ are positive constants (1) is the well-known Landau's second-order free energy, each of its local minimizers represents a possible phase state of the material, while each local maximizer characterizes the critical conditions that lead to the phase transitions.",
"...",
"The purpose of this paper is to find the extrema of the following nonconvex total potential energy functional in 1D, $I[u]:=\\left( \\int _{a}^{b}H\\left( \\frac{du}{dx}\\right) -fu\\right) dx$ .",
"(2) The function $f\\in C[a,b]$ satisfies the normalized balance condition $\\int _{a}^{b}f(x)dx=0$ , (3) and there exists a unique zero root for $f$ in $[a,b]$ .",
"(4) Moreover, its $L^{1}$ -norm is sufficiently small such that $\\left\\Vert f\\right\\Vert _{L^{1}(a,b)}<2\\lambda \\nu \\sqrt{2\\lambda }/(3\\sqrt{3})$ .",
"(5) The above assumption is reasonable since large $\\left\\Vert f\\right\\Vert _{L^{1}(a,b)}$ may possibly lead to instant fracture, which is represented by nonsmooth solutions.",
"The deformation $u$ is subject to the following two constraints, $u\\in C^{1}[a,b]$ , (6) $\\frac{du}{dx}(a)=\\frac{du}{dx}(b)=0$ .",
"(7) ... Before introducing the main result, we denote $F(x):=-\\int _{a}^{x}f(\\rho )d\\rho ,~~x\\in [a,b]$ .",
"Next, we define a polynomial of third order as follows, $E(y):=2y^{2}(\\lambda +y/\\nu ),~~y\\in [-\\nu \\lambda ,+\\infty )$ .",
"Furthermore, for any $A\\in [0,8\\lambda ^{3}\\nu ^{2}/27)$ , $E_{3}^{-1}(A)\\le E_{2}^{-1}(A)\\le E_{1}^{-1}(A)$ stand for the three real-valued roots for the equation $E(y)=A$ .",
"At the moment, we would like to introduce the main theorem.",
"Theorem 1.1.",
"For any function $f\\in C[a,b]$ satisfying (3)–(5), one can find the local extrema for the nonconvex functional (2).",
"• For any $x\\in [a,b]$ , $\\overline{u}_{1}$ defined below is a local minimizer for the nonconvex functional (2), $\\overline{u}_{1}(x)=\\int _{a}^{x}F(\\rho )/E_{1}^{-1}(F^{2}(\\rho ))d\\rho +C_{1},~~\\forall C_{1}\\in \\mathbb {R}$ .",
"(9) • For any $x\\in [a,b]$ , $\\overline{u}_{2}$ defined below is a local minimizer for the nonconvex functional (2), $\\overline{u}_{2}(x)=\\int _{a}^{x}F(\\rho )/E_{2}^{-1}(F^{2}(\\rho ))d\\rho +C_{2},~~\\forall C_{2}\\in \\mathbb {R}$ .",
"(10) • For any $x\\in [a,b]$ , $\\overline{u}_{3}$ defined below is a local maximizer for the nonconvex functional (2), $\\overline{u}_{3}(x)=\\int _{a}^{x}F(\\rho )/E_{3}^{-1}(F^{2}(\\rho ))d\\rho +C_{3},~~\\forall C_{3}\\in \\mathbb {R}$ .",
"(11)” As mentioned in [1], in getting the above result the authors use “the canonical duality method\".",
"Let us observe from the beginning that nothing is said about the norm (and the corresponding topology) on $C^{1}[a,b]$ when speaking about local extrema (minimizers or maximizers).",
"In the following we discuss a slightly more general problem and compare our conclusions with those of Theorem 1.1 in [1].",
"We don't analyze the method by which the conclusions in Theorem 1.1 of [1] are obtained even if this is worth being done.",
"Similar problems are considered by Gao and Ogden in [2] and [3] which are discussed by Voisei and Zălinescu in [5] and [6], respectively.",
"More precisely consider $\\theta \\in C[a,b]$ such that $\\theta (x)>0$ for $x\\in [a,b]$ , the polynomial $H$ defined by $H(y):=\\tfrac{1}{2}(\\tfrac{1}{2}y^{2}-\\lambda )^{2}$ with $\\lambda >0$ , and the function $J:=J_{f}:C^{1}[a,b]\\rightarrow \\mathbb {R},\\quad J_{f}(u):=\\int _{a}^{b}\\theta \\cdot \\left( H\\circ u^{\\prime }-fu\\right) ,$ where, $\\int _{a}^{b}h$ denotes the Riemann integral $\\int _{a}^{b}h(x)dx$ of the function $h:[a,b]\\rightarrow \\mathbb {R}$ (when it exists).",
"Of course, taking $\\theta $ the constant function $\\nu $ $(>0)$ and replacing $f$ by $\\nu ^{-1}f$ we get the functional $I$ considered in [1].",
"Let us set $X & :=C_{0}[a,b]:=\\lbrace v\\in C[a,b]\\mid v(a)=v(b)=0\\rbrace ,\\\\Y & :=C_{1,0}[a,b]:=\\lbrace u\\in C^{1}[a,b]\\mid u^{\\prime }:=du/dx\\in C_{0}[a,b]\\rbrace .$ Of course $X$ is a linear subspace of $C[a,b]$ ; it is even a closed subspace (and so a Banach space) if $C[a,b]$ is endowed with the supremum norm $\\left\\Vert \\cdot \\right\\Vert _{\\infty }$ .",
"Clearly, other norms could be considered on $X$ .",
"Observe that the function $F$ defined in [1] (and quoted above) is in $C^{1}[a,b]\\cap X$ with $F^{\\prime }:=dF/dx=-f$ .",
"Moreover, condition (5) implies that $\\left\\Vert F\\right\\Vert _{\\infty }<2\\lambda \\sqrt{2\\lambda }/(3\\sqrt{3})=(2\\lambda /3)^{3/2}$ because $\\left|F(x)\\right|=\\left|\\int _{a}^{x}f(\\xi )d\\xi \\right|\\le \\int _{a}^{x}\\left|f(\\xi )\\right|d\\xi \\le \\int _{a}^{b}\\left|f(\\xi )\\right|d\\xi =\\left\\Vert f\\right\\Vert _{L^{1}(a,b)}.$ Furthermore, condition (4) implies that $F(x)>0$ for $x\\in (a,b)$ , or $F(x)<0$ for $x\\in (a,b)$ .",
"For $u\\in Y$ and $v:=u^{\\prime }$ we have that $\\int _{a}^{b}uf=-\\int _{a}^{b}uF^{\\prime }=-\\left.",
"u(x)F(x)\\right|_{a}^{b}+\\int _{a}^{b}u^{\\prime }F=\\int _{a}^{b}vF.",
"$ Using this fact, for $u$ satisfying the constraints (6) and (7), and $v:=u^{\\prime }$ , one has $J(u)=\\int _{a}^{b}\\theta \\left( H\\circ v-Fv\\right) =:K(v).$"
],
[
"Study of local extrema of the function $K$",
"As mentioned above, in the sequel $H:\\mathbb {R}\\rightarrow \\mathbb {R}$ is defined by $H(y):=\\tfrac{1}{2}\\left( \\tfrac{1}{2}y^{2}-\\lambda \\right) ^{2}$ with $\\lambda >0$ , $\\theta \\in C[a,b]$ is such that $\\mu :=\\min _{x\\in [a,b]}\\theta (x)>0$ ; moreover $F\\in C^{1}[a,b]\\cap X$ is such that $F(x)\\ne 0$ for $x\\in (a,b)$ and $\\left\\Vert F\\right\\Vert _{\\infty }<(2\\lambda /3)^{3/2}$ .",
"Our first purpose is to find the local extrema of $K:=K_{F}:X\\rightarrow \\mathbb {R},\\quad K_{F}(v):=\\int _{a}^{b}\\theta \\cdot \\left(H\\circ v-Fv\\right) $ on $X=C_{0}[a,b]$ endowed with the norm $\\left\\Vert \\cdot \\right\\Vert _{p}$ , where $p\\in [1,\\infty ]$ .",
"First we study the Fréchet and Gâteaux differentiability of $K.$ Lemma 1 Let $g\\in C[a,b]\\setminus \\lbrace 0\\rbrace $ , $s\\in \\mathbb {N}^{\\ast }\\setminus \\lbrace 1\\rbrace $ and $p\\in [1,\\infty ]$ .",
"Then, with $h\\in X,$ $\\lim _{\\left\\Vert h\\right\\Vert _{p}\\rightarrow 0}\\frac{1}{\\left\\Vert h\\right\\Vert _{p}}\\int _{a}^{b}gh^{s}=0\\iff p\\ge s.$ Proof.",
"Set $\\gamma :=\\left\\Vert g\\right\\Vert _{\\infty }$ $(>0)$ .",
"For $s<p<\\infty $ and $h\\in X$ we have that $\\left|\\int _{a}^{b}gh^{s}\\right|\\le \\gamma \\int _{a}^{b}\\left|h\\right|^{s}\\cdot 1\\le \\gamma \\left( \\int _{a}^{b}\\left( \\left|h\\right|^{s}\\right) ^{p/s}\\right) ^{s/p}\\left( \\int _{a}^{b}1^{p/(p-s)}\\right) ^{(p-s)/p},$ and so $\\left|\\int _{a}^{b}gh^{s}\\right|\\le \\gamma (b-a)^{^{(p-s)/p}}\\left\\Vert h\\right\\Vert _{p}^{s}\\quad \\forall h\\in X.$ The above inequality is true also, as easily seen, for $p=s$ and $p=\\infty $ (setting $(p-s)/p:=1$ in the former case); from it we get $\\lim _{\\left\\Vert h\\right\\Vert _{p}\\rightarrow 0}\\frac{1}{\\left\\Vert h\\right\\Vert _{p}}\\int _{a}^{b}gh^{s}=0$ because $s>1.$ Assume now that $p<s$ .",
"Since $g\\in C[a,b]\\setminus \\lbrace 0\\rbrace $ , there exist $\\delta >0$ , and $a^{\\prime },b^{\\prime }\\in [a,b]$ with $a^{\\prime }<b^{\\prime }$ such that $g(x)\\ge \\delta $ for $x\\in [a^{\\prime },b^{\\prime }]$ or $g(x)\\le -\\delta $ for $x\\in [a^{\\prime },b^{\\prime }]$ .",
"Doing a translation, we suppose that $a^{\\prime }=0$ .",
"For $n\\in \\mathbb {N}^{\\ast }$ with $n\\ge n_{0}$ $(\\ge 2/b^{\\prime })$ consider $h_{n}(x):=\\left\\lbrace \\begin{array}[c]{ll}\\alpha _{n}x & \\text{if }x\\in [0,1/n],\\\\\\alpha _{n}(2/n-x) & \\text{if }x\\in (1/n,2/n),\\\\0 & \\text{if }x\\in [a,0)\\cup [2/n,b],\\end{array}\\right.",
"$ with $\\alpha _{n}:=n^{1+\\gamma /p}>0$ , where $\\frac{p-1}{s-1}<\\gamma <1$ .",
"Clearly, $h_{n}\\in X=C_{0}[a,b]$ .",
"In this situation $\\left|\\int _{a}^{b}gh_{n}^{s}\\right|=\\int _{0}^{2/n}\\left|g\\right|h_{n}^{s}\\ge 2\\delta \\int _{0}^{1/n}(\\alpha _{n}x)^{s}dx=2\\delta \\alpha _{n}^{s}\\frac{1}{s+1}\\frac{1}{n^{s+1}}=\\frac{2\\delta }{s+1}n^{\\frac{s\\gamma -p}{p}},$ while a similar argument gives $\\left\\Vert h_{n}\\right\\Vert _{p}=\\left( 2\\alpha _{n}^{p}\\frac{1}{p+1}\\frac{1}{n^{p+1}}\\right) ^{1/p}=\\left( \\frac{2}{p+1}\\right) ^{1/p}n^{\\frac{\\gamma -1}{p}}\\rightarrow 0.$ On the other hand, $\\frac{1}{\\left\\Vert h_{n}\\right\\Vert _{p}}\\left|\\int _{a}^{b}gh_{n}^{s}\\right|\\ge \\frac{2\\delta }{s+1}\\left( \\frac{p+1}{2}\\right)^{1/p}n^{\\frac{\\gamma (s-1)-(p-1)}{p}}\\rightarrow \\infty ,$ which proves our assertion.",
"The proof is complete.",
"$\\square $ Proposition 2 Let $X=C_{0}[a,b]$ be endowed with the norm $\\left\\Vert \\cdot \\right\\Vert _{p}$ , where $p\\in [1,\\infty ]$ .",
"Then $K$ is Gâteaux differentiable; moreover, for $v\\in X$ , $K$ is Fréchet differentiable at $v$ if and only if $p\\ge 4$ .",
"Proof.",
"Let us set $g_{2}:=\\tfrac{1}{2}\\theta \\left( \\tfrac{3}{2}v^{2}-\\lambda \\right) $ , $g_{3}:=\\tfrac{1}{2}\\theta v$ and $g_{4}:=\\tfrac{1}{8}\\theta $ ; of course, $g_{2},g_{3},g_{4}\\in C[a,b]$ .",
"Set also $\\beta :=\\max \\lbrace \\left\\Vert g_{2}\\right\\Vert _{\\infty },\\left\\Vert g_{3}\\right\\Vert _{\\infty }\\rbrace $ .",
"Observe that for all $v,h\\in X$ we have that $K(v+h)=K(v)+\\int _{a}^{b}\\theta \\left[ v(\\tfrac{1}{2}v^{2}-\\lambda )-F\\right]h+\\int _{a}^{b}\\tfrac{1}{2}\\theta \\left( \\tfrac{3}{2}v^{2}-\\lambda \\right)h^{2}+\\int _{a}^{b}\\tfrac{1}{2}\\theta vh^{3}+\\int _{a}^{b}\\tfrac{1}{8}\\theta h^{4}.",
"$ For $v\\in X$ consider $T_{v}:X\\rightarrow \\mathbb {R},\\quad T_{v}(h):=\\int _{a}^{b}\\theta \\left[v(\\tfrac{1}{2}v^{2}-\\lambda )-F\\right] h\\quad (h\\in X).",
"$ Clearly, $T_{v}$ is a linear operator; $T_{v}$ is also continuous for every $p\\in [1,\\infty ]$ .",
"Indeed, setting $\\gamma _{v}:=\\left\\Vert \\theta \\left[v(\\tfrac{1}{2}v^{2}-\\lambda )-F\\right] \\right\\Vert _{\\infty }\\in \\mathbb {R}_{+}$ we have that $\\left|T_{v}(h)\\right|\\le \\gamma _{v}\\int _{a}^{b}\\left|h\\right|\\le \\gamma _{v}\\left\\Vert h\\right\\Vert _{p}\\cdot \\left\\Vert 1\\right\\Vert _{p^{\\prime }}=\\gamma _{v}(b-a)^{1/p^{\\prime }}\\left\\Vert h\\right\\Vert _{p}\\quad \\forall h\\in X$ for $p,p^{\\prime }\\in [1,\\infty ]$ with $p^{\\prime }$ the conjugate of $p,$ that is $p^{\\prime }:=p/(p-1)$ for $p\\in (1,\\infty )$ , $p^{\\prime }:=\\infty $ for $p=1$ and $p^{\\prime }:=1$ for $p=\\infty $ .",
"Hence $T_{v}$ is continuous.",
"Let $p\\in [1,\\infty ]$ and $v\\in X$ be fixed.",
"Using (REF ) we have that $\\left|\\frac{K(v+h)-K(v)-T_{v}(h)}{\\left\\Vert h\\right\\Vert _{p}}\\right|\\le \\frac{1}{\\left\\Vert h\\right\\Vert _{p}}\\left( \\left|\\int _{a}^{b}g_{2}h^{2}\\right|+\\left|\\int _{a}^{b}g_{3}h^{3}\\right|+\\left|\\int _{a}^{b}g_{4}h^{4}\\right|\\right)$ for $h\\ne 0$ .",
"Using Lemma REF for $p\\ge 4$ , we obtain that $\\lim _{\\left\\Vert h\\right\\Vert _{p}\\rightarrow 0}\\frac{K(v+h)-K(v)-T_{v}(h)}{\\left\\Vert h\\right\\Vert _{p}}=0$ .",
"Hence $K$ is Fréchet differentiable at $v.$ Assume now that $p<4$ .",
"Using again (REF ) we have that $K(v+h)-K(v)-T_{v}(h)\\ge \\tfrac{\\mu }{8}\\int _{a}^{b}h^{4}-\\beta \\int _{a}^{b}\\left|h\\right|^{3}-\\beta \\int _{a}^{b}h^{2}\\quad \\forall h\\in X.$ Take $a=a^{\\prime }=0<b^{\\prime }=b$ (possible after a translation), $\\alpha _{n}:=n^{1+\\gamma /p}$ with $\\frac{3}{s-1}<\\gamma <1$ and $h:=h_{n}$ defined by (REF ).",
"Using the computations from the proof of Lemma REF , we get $\\int _{a}^{b}\\left|h_{n}\\right|^{s}=2\\int _{0}^{1/n}(\\alpha _{n}x)^{s}dx=\\frac{2}{s+1}n^{\\frac{s\\gamma -p}{p}},\\quad \\left\\Vert h_{n}\\right\\Vert _{p}=\\left( \\frac{2}{p+1}\\right) ^{1/p}\\frac{1}{n^{(1-\\gamma )/p}}\\rightarrow 0, $ whence $\\frac{K(v+h_{n})-K(v)-T_{v}(h_{n})}{\\left\\Vert h_{n}\\right\\Vert _{p}} &\\textstyle \\ge \\left( \\frac{p+1}{2}\\right) ^{1/p}n^{\\frac{1-\\gamma }{p}}\\left( \\tfrac{\\mu }{8}\\cdot \\tfrac{2}{5}n^{\\frac{4\\gamma -p}{p}}-\\tfrac{2}{4}\\beta _{3}n^{\\frac{3\\gamma -p}{p}}-\\tfrac{2}{3}\\beta _{2}n^{\\frac{2\\gamma -p}{p}}\\right) \\\\& \\textstyle =\\left( \\frac{p+1}{2}\\right) ^{1/p}n^{\\frac{1-p+3\\gamma }{p}}\\left( \\tfrac{\\mu }{20}-\\tfrac{1}{2}\\beta _{3}n^{-\\frac{\\gamma }{p}}-\\tfrac{2}{3}\\beta _{2}n^{-\\frac{2\\gamma }{p}}\\right) \\rightarrow \\infty .$ This shows that $K\\ $ is not Fréchet differentiable at $v.$ Because $K:(X,\\left\\Vert \\cdot \\right\\Vert _{\\infty })\\rightarrow \\mathbb {R}$ is Fréchet differentiable at $v\\in X$ , it follows that $\\lim _{t\\rightarrow 0}\\frac{K(v+th)-K(v)}{t}=T_{v}(h)\\in \\mathbb {R}\\quad \\forall h\\in X.",
"$ Because $T_{v}:(X,\\left\\Vert \\cdot \\right\\Vert _{p})\\rightarrow \\mathbb {R}$ is linear and continuous, it follows that $K$ is Gâteaux differentiable at $v$ for every $p\\in [1,\\infty ]$ with $\\nabla K(v)=T_{v}$ .",
"$\\square $ We consider now the problem of finding the stationary points of $K$ , that is those points $v\\in X$ with $T_{v}=0.$ Proposition 3 The functional $K$ has only one stationary point $\\overline{v}$ .",
"More precisely, for each $x\\in [a,b]$ , $\\overline{v}(x)$ is the unique solution from $(-\\sqrt{2\\lambda /3},\\sqrt{2\\lambda /3})$ of the equation $z(\\tfrac{1}{2}z^{2}-\\lambda )=F(x)$ .",
"Proof.",
"Assume that $v\\in X$ is stationary; hence $T_{v}h=\\int _{a}^{b}Vh=0$ for every $h\\in X$ , where $V:=\\theta v(\\tfrac{1}{2}v^{2}-\\lambda )-F$ $(\\in X\\subset C[a,b])$ .",
"We claim that $V=0$ .",
"In the contrary case, since $V$ is continuous, there exists $x_{0}\\in (a,b)$ with $V(x_{0})\\ne 0$ .",
"Suppose that $V(x_{0})>0$ .",
"By the continuity of $V$ there exist $a^{\\prime },b^{\\prime }\\in \\mathbb {R}$ such that $a<a^{\\prime }<x_{0}<b^{\\prime }<b$ and $V(x)>0$ for every $x\\in [a^{\\prime },b^{\\prime }]$ .",
"Take $\\overline{h}:[a,b]\\rightarrow \\mathbb {R},\\quad \\overline{h}(x):=\\left\\lbrace \\begin{array}[c]{ll}\\frac{x-a^{\\prime }}{b^{\\prime }-a^{\\prime }} & \\text{if }x\\in (a^{\\prime },\\tfrac{1}{2}(a^{\\prime }+b^{\\prime })],\\\\\\frac{b^{\\prime }-x}{b^{\\prime }-a^{\\prime }} & \\text{if }x\\in (\\tfrac{1}{2}(a^{\\prime }+b^{\\prime }),b^{\\prime }],\\\\0 & \\text{if }x\\in [a,a^{\\prime }]\\cup (b^{\\prime },b].\\end{array}\\right.$ Then $\\overline{h}\\in X$ and $\\overline{h}(x)>0$ for $x\\in (a^{\\prime },b^{\\prime })$ .",
"Since $0=\\int _{a}^{b}V\\overline{h}=\\int _{a^{\\prime }}^{b^{\\prime }}V\\overline{h}$ and $V\\overline{h}$ is continuous and nonnegative on $[a^{\\prime },b^{\\prime }]$ we obtain that $V(x)\\overline{h}(x)=0$ for $x\\in [a^{\\prime },b^{\\prime }]$ , and so $0=V(x_{0})\\overline{h}(x_{0})>0$ .",
"This contradiction shows that $V=0$ .",
"The proof in the case $V(x_{0})<0$ reduces to the preceding one replacing $V$ by $-V$ .",
"Hence $\\theta v(\\tfrac{1}{2}v^{2}-\\lambda )=F~~\\text{on~~}[a,b].",
"$ Consider the polynomial function $G:\\mathbb {R}\\rightarrow \\mathbb {R}$ defined by $G(z):=z\\left( \\tfrac{1}{2}z^{2}-\\lambda \\right) $ .",
"Then $G^{\\prime }(z)=\\tfrac{3}{2}z^{2}-\\lambda $ having the zeros $\\pm \\kappa $ , where $\\kappa :=\\sqrt{2\\lambda /3}.",
"$ The behavior of $G$ is given in the table below.",
"Table: NO_CAPTIONThis table shows that the equation $G(z)=A$ with $A\\in (-\\sqrt{8\\lambda ^{3}/27},\\sqrt{8\\lambda ^{3}/27})$ has three real solutions, more precisely, $z_{1}(A)\\in (-2\\kappa ,-\\kappa ),\\quad z_{2}(A)\\in (-\\kappa ,\\kappa ),\\quad z_{3}(A)\\in (\\kappa ,2\\kappa ).",
"$ Moreover, the mappings $z_{i}:(-\\kappa ^{3},\\kappa ^{3})\\rightarrow \\mathbb {R}$ are continuous with $z_{1}(0)=-\\sqrt{3}\\kappa $ , $z_{2}(0)=0$ , $z_{3}(0)=\\sqrt{3}\\kappa $ .",
"This shows that $z_{i}\\circ F\\in X$ if and only $i=2,$ and so the only solution in $X$ of the equation $v(\\tfrac{1}{2}v^{2}-\\lambda )=F$ is $\\overline{v}:=z_{2}\\circ F$ .",
"$\\square $ Let us analyze if $\\overline{v}:=z_{2}\\circ F$ is a local extremum of $K.$ Proposition 4 Let $\\overline{v}\\in X$ be the stationary point of $K$ .",
"Then $\\overline{v}:=z_{2}\\circ F$ [with $z_{2}$ defined in (REF )] is a local maximizer for $K$ with respect to $\\left\\Vert \\cdot \\right\\Vert _{\\infty }$ , and $\\overline{v}$ is not a local extremum point of $K$ with respect to $\\left\\Vert \\cdot \\right\\Vert _{p}$ for $p\\in [1,4).$ Proof.",
"Let us consider first the case $p=\\infty $ .",
"From (REF ) we get $K(\\overline{v}+h)-K(\\overline{v})=\\int _{a}^{b}\\theta \\left[ \\tfrac{1}{2}\\left( \\tfrac{3}{2}\\overline{v}^{2}-\\lambda \\right) +\\tfrac{1}{2}\\overline{v}h+\\tfrac{1}{8}h^{2}\\right] h^{2}\\quad \\forall h\\in X.$ Since $F\\in C[a,b]$ , there exists some $x_{0}\\in [a,b]$ such that $\\left\\Vert F\\right\\Vert _{\\infty }=\\left|F(x_{0})\\right|<(2\\lambda /3)^{3/2}$ , and so $\\left|\\overline{v}(x)\\right|\\le \\left|\\overline{v}(x_{0})\\right|=:\\gamma <\\sqrt{2\\lambda /3}$ for $x\\in [a,b]$ .",
"It follows that $\\tfrac{1}{2}\\left( \\tfrac{3}{2}\\overline{v}^{2}-\\lambda \\right) \\le \\tfrac{1}{2}\\left( \\tfrac{3}{2}\\gamma ^{2}-\\lambda \\right) =:-\\eta <\\tfrac{1}{2}\\left( \\tfrac{3}{2}\\frac{2\\lambda }{3}-\\lambda \\right) =0$ .",
"Hence $\\tfrac{1}{2}\\left( \\tfrac{3}{2}\\overline{v}^{2}-\\lambda \\right) +\\tfrac{1}{2}\\overline{v}h+\\tfrac{1}{8}h^{2}\\le -\\eta +\\tfrac{1}{2}\\gamma \\left\\Vert h\\right\\Vert _{\\infty }+\\tfrac{1}{8}\\left\\Vert h\\right\\Vert _{\\infty }^{2}<0\\quad \\forall h\\in X,~\\left\\Vert h\\right\\Vert _{\\infty }<\\varepsilon ,$ where $\\varepsilon :=2\\big (\\sqrt{\\gamma ^{2}+2\\eta }-\\gamma \\big )$ .",
"It follows that $\\overline{v}$ is a (strict) local maximizer of $K.$ Assume now that $p\\in [1,4)$ .",
"Of course, there exists a sequence $(h_{n})_{n\\ge 1}\\subset X\\setminus \\lbrace 0\\rbrace $ such that $\\left\\Vert h_{n}\\right\\Vert _{\\infty }\\rightarrow 0$ .",
"Taking into account (REF ), we have that $K(\\overline{v}+h_{n})<K(\\overline{v})$ for large $n$ .",
"Since $\\left\\Vert h_{n}\\right\\Vert _{p}\\rightarrow 0$ , $\\overline{v}$ is not a local minimizer of $K$ with respect to $\\left\\Vert \\cdot \\right\\Vert _{p}$ .",
"In the proof of Proposition REF we found a sequence $(h_{n})_{n\\ge 1}\\subset X\\setminus \\lbrace 0\\rbrace $ such that $\\left\\Vert h_{n}\\right\\Vert _{p}\\rightarrow 0$ and $\\left\\Vert h_{n}\\right\\Vert _{p}^{-1}\\left( K(\\overline{v}+h_{n})-K(\\overline{v})-T_{\\overline{v}}h_{n}\\right) \\rightarrow \\infty $ .",
"Since $T_{\\overline{v}}=0$ , we obtain that $K(\\overline{v}+h_{n})-K(\\overline{v})>0$ for large $n$ , proving that $\\overline{v}$ is not a local maximizer of $K$ .",
"Hence $\\overline{v}$ is not a local extremum point of $K$ .",
"$\\square $ We don't know if $\\overline{v}$ is a local maximizer of $K$ for $p\\in [4,\\infty )$ ; having in view (REF ), surely, $\\overline{v}$ is not a local minimizer of $K.$ Proposition REF shows the importance of the norm (and more generally, of the topology) on a space when speaking about local extrema.",
"Let us establish now the relations between the local extrema of $J$ with the constraints (6) and (7) in [1], that is local extrema of $J$ restricted to $C_{1,0}[a,b]$ , and the local extrema of $K$ in the case in which $C^{1}[a,b]$ is endowed with the (usual) norm defined by $\\left\\Vert u\\right\\Vert :=\\left\\Vert u\\right\\Vert _{\\infty }+\\left\\Vert u^{\\prime }\\right\\Vert _{\\infty }\\quad (u\\in C^{1}[a,b]), $ and $C_{0}[a,b]$ is endowed with the norm $\\left\\Vert \\cdot \\right\\Vert _{\\infty }.$ Proposition 5 Consider the norm $\\left\\Vert \\cdot \\right\\Vert $ (defined in (REF )) on $C^{1}[a,b]$ and the norm $\\left\\Vert \\cdot \\right\\Vert _{\\infty }$ on $C_{0}[a,b]$ .",
"If $\\overline{u}$ is a local minimizer (maximizer) of $J$ on $C_{1,0}[a,b]$ , then $\\overline{u}^{\\prime }$ is a local minimizer (maximizer) of $K$ .",
"Conversely, if $\\overline{v}$ is a local minimizer (maximizer) of $K$ , then $\\overline{u}\\in C^{1}[a,b]$ defined by $\\overline{u}(x):=u_{0}+\\int _{a}^{x}\\overline{v}(\\xi )d\\xi $ for $x\\in [a,b]$ and a fixed $u_{0}\\in \\mathbb {R}$ is a local minimizer (maximizer) of $J$ on $C_{1,0}[a,b].$ Proof.",
"Assume that $\\overline{u}$ is a local minimizer of $J$ on $C_{1,0}[a,b]$ ; hence $\\overline{u}\\in C_{1,0}[a,b]$ .",
"It follows that there exists $r>0$ such that $J(\\overline{u})\\le J(u)$ for every $u\\in C_{1,0}[a,b]$ with $\\left\\Vert u-\\overline{u}\\right\\Vert <r$ .",
"Set $\\overline{v}:=\\overline{u}^{\\prime }$ and take $v\\in X=C_{0}[a,b]$ with $\\left\\Vert v-\\overline{v}\\right\\Vert _{\\infty }<r^{\\prime }:=r/(1+b-a)$ .",
"Define $u:[a,b]\\rightarrow \\mathbb {R}$ by $u(x):=\\overline{u}(a)+\\int _{a}^{x}v(\\xi )d\\xi $ for $x\\in [a,b]$ .",
"Then $u\\in C_{1,0}[a,b]$ and $u^{\\prime }=v$ .",
"Since $\\overline{u}(x)=\\overline{u}(a)+\\int _{a}^{x}\\overline{v}(\\xi )d\\xi $ , we get $\\left\\Vert u-\\overline{u}\\right\\Vert =\\left\\Vert u-\\overline{u}\\right\\Vert _{\\infty }+\\left\\Vert u^{\\prime }-\\overline{u}^{\\prime }\\right\\Vert _{\\infty }\\le (b-a)\\left\\Vert v-\\overline{v}\\right\\Vert _{\\infty }+\\left\\Vert v-\\overline{v}\\right\\Vert _{\\infty }<r^{\\prime }(1+b-a)=r.$ Hence $K(\\overline{v})=J(\\overline{u})\\le J(u)=K(v)$ .",
"This shows that $\\overline{v}$ is a local minimizer for $K$ .",
"Conversely, assume that $\\overline{v}$ is a local minimizer for $K$ .",
"Then there exists $r>0$ such that $K(\\overline{v})\\le K(v)$ for $v\\in C_{0}[a,b]$ with $\\left\\Vert v-\\overline{v}\\right\\Vert <r$ , and take $u_{0}\\in \\mathbb {R}$ and $\\overline{u}:[a,b]\\rightarrow \\mathbb {R}$ defined by $\\overline{u}(x):=u_{0}+\\int _{a}^{x}\\overline{v}(\\xi )d\\xi $ for $x\\in [a,b]$ .",
"Then $\\overline{u}\\in C_{1,0}[a,b]$ .",
"Consider $u\\in C_{1,0}[a,b]$ with $\\left\\Vert u-\\overline{u}\\right\\Vert <r$ , that is $\\left\\Vert u-\\overline{u}\\right\\Vert _{\\infty }+\\left\\Vert u^{\\prime }-\\overline{u}^{\\prime }\\right\\Vert _{\\infty }=\\left\\Vert u-\\overline{u}\\right\\Vert _{\\infty }+\\left\\Vert u^{\\prime }-\\overline{v}\\right\\Vert _{\\infty }<r;$ then $\\left\\Vert u^{\\prime }-\\overline{v}\\right\\Vert _{\\infty }<r$ .",
"Since $u^{\\prime }\\in C_{0}[a,b]$ , it follows that $J(u)=K(u^{\\prime })\\ge K(\\overline{v})=J(\\overline{u})$ , and so $\\overline{u}$ is a local minimizer of $J$ on $C_{1,0}[a,b]$ .",
"The case of local maximizers for $J$ and $K$ is treated similarly.",
"$\\square $ Putting together Propositions REF , REF and REF we get the next result.",
"Theorem 6 Consider the norm $\\left\\Vert \\cdot \\right\\Vert $ (defined in (REF )) on $C^{1}[a,b]$ and the norm $\\left\\Vert \\cdot \\right\\Vert _{\\infty }$ on $C_{0}[a,b]$ .",
"Let $\\overline{u}\\in C_{1,0}[a,b]$ and set $\\overline{v}:=\\overline{u}^{\\prime }$ .",
"Then the following assertions are equivalent: (i) $\\overline{u}$ is a local maximum point of $J$ restricted to $C_{1,0}[a,b].$ (ii) $\\overline{u}$ is a local extremum point of $J$ restricted to $C_{1,0}[a,b].$ (iii) $\\overline{v}$ is a stationary point of $K.$ (iv) $\\overline{v}$ is a local extremum point of $K.$ (v) $\\overline{v}$ is a local maximum point of $K.$ (vi) $\\overline{v}=z_{2}\\circ F$ , where $z_{2}(A)$ is the unique solution of the equation $z\\left(\\tfrac{1}{2}z^{2}-\\lambda \\right) =A$ in the interval $(-\\sqrt{2\\lambda /3},\\sqrt{2\\lambda /3}]$ for $A\\in (-(2\\lambda /3)^{3/2},(2\\lambda /3)^{3/2}).$ (vii) there exists $u_{0}\\in \\mathbb {R}$ such that $\\overline{u}(x)=u_{0}+\\int _{a}^{x}z_{2}(F(\\rho ))d\\rho $ for every $x\\in [a,b].$"
],
[
"Discussion of Theorem 1.1 from Gao and Lu's paper {{cite:7f1456e0f6766d0975e5e493d02a27097f74ff8b}}",
"First of all, we think that in the formulation of [1], “local extrema for the nonconvex functional (2)” must be replaced by “local extrema for the nonconvex functional (2) with the constraints (6) and (7)”, “local minimizer for the nonconvex functional (2)” must be replaced by “local minimizer for the nonconvex functional (2) with the constraints (6) and (7)” (2 times), and “local maximizer for the nonconvex functional (2)” must be replaced by “local maximizer for the nonconvex functional (2) with the constraints (6) and (7)”.",
"Below, we interpret [1] with these modifications.",
"As pointed in Introduction, no norms are considered on the spaces mentioned in [1].",
"For this reason in Theorem REF we considered the usual norms on $C^{1}[a,b]$ and $C_{0}[a,b]$ ; these norms are used in this discussion.",
"Moreover, let $\\theta (x):=1$ for $x\\in [a,b]$ in Theorem REF and $\\nu =1$ in [1].",
"In the conditions of [1] $F(x)>0$ for $x\\in (a,b)$ or $F(x)<0$ for $x\\in (a,b).$ For the present discussion we take the case $F>0$ on $(a,b).$ Assume that the mappings $\\rho \\mapsto F(\\rho )/E_{j}^{-1}\\left( F^{2}(\\rho \\right) )=:v_{j}(\\rho )$ [where “$E_{3}^{-1}(A)\\le E_{2}^{-1}(A)\\le E_{1}^{-1}(A)$ stand for the three real-valued roots for the equation $E(y)=A$ ” with $E(y)=2y^{2}(y+\\lambda )$ and $A\\in [0,8\\lambda ^{3}/27)$ ] are well defined for $\\rho \\in \\lbrace a,b\\rbrace $ [there are no problems for $\\rho \\in (a,b)$ ].",
"If [1] is true, then $v_{1},v_{2},v_{3}\\in C_{0}[a,b]$ ; moreover, $v_{1}$ and $v_{2}$ are local minimizers of $K$ , and $v_{3}$ is a local maximizer of $K$ .",
"This is of course false taking into account Theorem REF because $K$ has not local minimizers.",
"Because $z_{2}\\circ F$ is the unique local maximizer of $K$ , we must have that $v_{3}=z_{2}\\circ F$ .",
"Let us see if this is true.",
"Because $z_{i}(A)$ are solutions of the equation $G(z)=A$ and $E_{j}^{-1}(A)$ are solutions of the equation $E(y)=A$ , we must study the relationships among these numbers.",
"First, the behavior of $E$ is given in the next table.",
"Table: NO_CAPTIONSecondly, for $y,z,A\\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace $ such that $yz=A$ we have that $G(z)=A\\Leftrightarrow \\frac{A}{y}\\left( \\frac{1}{2}\\frac{A^{2}}{y^{2}}-\\lambda \\right) =A\\Leftrightarrow 2y^{2}(y+\\lambda )=A^{2}\\Leftrightarrow E(y)=A^{2}.",
"$ Analyzing the behavior of $G$ and $E$ (recall that $\\kappa =\\sqrt{2\\lambda /3}$ ), and the relation $yz=A$ for $A\\ne 0$ (mentioned above), the correspondence among the solutions of the equations $G(z)=A$ and $E(y)=A^{2}$ for $A\\in (0,(2\\lambda /3)^{3/2})$ is: $z_{1}(A)=A/E_{2}^{-1}(A^{2}),\\quad z_{2}(A)=A/E_{3}^{-1}(A^{2}),\\quad z_{3}(A)=A/E_{1}^{-1}(A^{2}) $ for all $A\\in (0,\\left( 2\\lambda /3\\right) ^{3/2})$ .",
"This shows that only the third assertion of [1] is true (of course, considering the norm defined in (REF ) on $C^{1}[a,b]$ )."
],
[
"Discussion of Theorem 1.1 from Lu and Gao's paper {{cite:799d20b3dab8d68f90127eaddc66a21bdb38c28d}}",
"A similar problem to that in [1], discussed above, is considered in [4].",
"In the abstract of this paper one finds: “In comparison with the 1D case discussed by D. Gao and R. Ogden, there exists huge difference in higher dimensions, which will be explained in the theorem\".",
"More precisely, in [4] it is said: “In this paper, we consider the fourth-order polynomial defined by $H(|\\vec{\\gamma }|):=\\nu /2\\left(1/2|\\vec{\\gamma }|^{2}-\\lambda \\right) ^{2},$ $\\vec{\\gamma }\\in \\mathbb {R}^{n}$ , $\\nu ,\\lambda >0$ are constants, $|\\vec{\\gamma }|^{2}=\\vec{\\gamma }\\cdot \\vec{\\gamma }$ .",
"...",
"The purpose of this paper is to find the extrema of the following nonconvex total potential energy functional in higher dimensions, (1) $I[u]:=\\int _{\\Omega }\\left( H(|\\nabla u|)-fu\\right) dx,$ where $\\Omega =$ Int$\\left\\lbrace \\mathbb {B}(O,R_{1})\\setminus \\mathbb {B}(O,R_{2})\\right\\rbrace $ , $R_{1}>R_{2}>0,$ $\\mathbb {B}(O,R_{1})$ and $\\mathbb {B}(O,R_{2})$ denote two open balls with center $O$ and radii $R_{1}$ and $R_{2}$ in the Euclidean space $\\mathbb {R}^{n}$ , respectively.",
"“Int” denotes the interior points.",
"In addition, let $\\Sigma _{1}:=\\lbrace x:|x|=R_{1}\\rbrace $ , and $\\Sigma _{2}:=\\lbrace x:|x|=R_{2}\\rbrace $ , then the boundary $\\partial \\Omega =\\Sigma _{1}\\cup \\Sigma _{2}$ .",
"The radially symmetric function $f\\in C(\\overline{\\Omega })$ satisfies the normalized balance condition (2) $\\int _{\\Omega }f(|x|)dx=0$ , and (3) $f(|x|)=0$ if and only if $|x|=R_{3}\\in (R_{2},R_{1})$ .",
"Moreover, its $L^{1}$ -norm is sufficiently small such that (4) $\\left\\Vert f\\right\\Vert _{L^{1}(\\Omega )}<4\\lambda \\nu R_{2}^{n-1}\\sqrt{2\\lambda \\pi ^{n}}/(3\\sqrt{3}\\Gamma (n/2)),$ where $\\Gamma $ stands for the Gamma function.",
"This assumption is reasonable since large $\\left\\Vert f\\right\\Vert _{L^{1}(\\Omega )}$ may possibly lead to instant fracture.",
"The deformation $u$ is subject to the following three constraints, (5) $u$ is radially symmetric on $\\overline{\\Omega }$ , (6) $u\\in W^{1,\\infty }(\\Omega )\\cap C(\\overline{\\Omega })$ , (7) $\\nabla u\\cdot \\vec{n}=0$ on both $\\Sigma _{1}$ and $\\Sigma _{2}$ , where $\\vec{n}$ denotes the unit outward normal on $\\partial \\Omega $ .",
"By variational calculus, one derives a correspondingly nonlinear Euler–Lagrange equation for the primal nonconvex functional, namely, (8) $\\operatornamewithlimits{div}\\left( \\nabla H(|\\nabla u|)\\right) +f=0$ in $\\Omega $ , equipped with the Neumann boundary condition (7).",
"Clearly, (8) is a highly nonlinear partial differential equation which is difficult to solve by the direct approach or numerical method [2, 15].",
"However, by the canonical duality method, one is able to demonstrate the existence of solutions for this type of equations.",
"... Before introducing the main result, we denote $F(r):=-1/r^{n}\\int _{R_{2}}^{r}f(\\rho )\\rho ^{n-1}d\\rho .~~r\\in [R_{2},R_{1}]$ .",
"Next, we define a polynomial of third order as follows, $E(y):=2y^{2}(\\lambda +y/\\nu ),~~y\\in [-\\nu \\lambda ,+\\infty )$ .",
"Furthermore, for any $A\\in [0,8\\lambda ^{3}\\nu ^{2}/27)$ , $E_{3}^{-1}(A)\\le E_{2}^{-1}(A)\\le E_{1}^{-1}(A)$ stand for the three real-valued roots for the equation $E(y)=A$ .",
"At the moment, we would like to introduce the theorem of multiple extrema for the nonconvex functional (2).",
"Theorem 1.1.",
"For any radially symmetric function $f\\in C(\\overline{\\Omega })$ satisfying (2)–(4), we have three solutions for the nonlinear Euler–Lagrange equation (8) equipped with the Neumann boundary condition, namely • For any $r\\in [R_{2},R_{1}]$ , $\\overline{u}_{1}$ defined below is a local minimizer for the nonconvex functional (2), $(9)\\quad \\overline{u}_{1}(\\left|x\\right|)=\\overline{u}_{1}(r):=\\int _{R_{2}}^{r}F(\\rho )\\rho /E_{1}^{-1}(F^{2}(\\rho )\\rho ^{2})d\\rho +C_{1},~~\\forall C_{1}\\in \\mathbb {R}$ .",
"• For any $r\\in [R_{2},R_{1}]$ , $\\overline{u}_{2}$ defined below is a local minimizer for the nonconvex functional (2) in 1D.",
"While for the higher dimensions $n\\ge 2$ , $\\overline{u}_{2}$ is not necessarily a local minimizer for (2) in comparison with the 1D case.",
"$(10)\\quad \\overline{u}_{2}(\\left|x\\right|)=\\overline{u}_{2}(r):=\\int _{R_{2}}^{r}F(\\rho )\\rho /E_{2}^{-1}(F^{2}(\\rho )\\rho ^{2})d\\rho +C_{2},~~\\forall C_{2}\\in \\mathbb {R}$ .",
"• For any $r\\in [R_{2},R_{1}]$ , $\\overline{u}_{3}$ defined below is a local maximizer for the nonconvex functional (2), $(11)\\quad \\overline{u}_{3}(\\left|x\\right|)=\\overline{u}_{3}(r):=\\int _{R_{2}}^{r}F(\\rho )\\rho /E_{3}^{-1}(F^{2}(\\rho )\\rho ^{2})d\\rho +C_{3},~~\\forall C_{3}\\in \\mathbb {R}$ .",
"...",
"In the final analysis, we apply the canonical duality theory to prove Theorem 1.1.” First, observe that one must have (1) instead of (2) just before the statement of [4], as well as in its statement, excepting for (2)–(4).",
"Secondly, (even from the quoted texts) one must observe that the wording in [1] and [4] is almost the same; the mathematical part is very, very similar, too.",
"To avoid any confusion, in the sequel the Euclidian norm on $\\mathbb {R}^{n}$ will be denoted by $\\left|\\cdot \\right|_{n}$ instead of $\\left|\\cdot \\right|.$ Remark that it is said $f\\in C(\\overline{\\Omega })$ , which implies $f$ is applied to elements $x\\in \\overline{\\Omega }$ , while a line below one considers $f(\\left|x\\right|)$ (that is $f(\\left|x\\right|_{n})$ with our notation); because the (Euclidean) norm $\\left|x\\right|_{n}$ of $x\\in \\overline{\\Omega }$ belongs to $[R_{2},R_{1}]$ , writing $f(\\left|x\\right|)$ shows that $f:[R_{2},R_{1}]\\rightarrow \\mathbb {R}$ .",
"Of course, these create ambiguities.",
"Probably the authors wished to say that a function $g:\\overline{\\Omega }\\rightarrow \\mathbb {R}$ is radially symmetric if there exists $\\psi :[R_{2},R_{1}]\\rightarrow \\mathbb {R}$ such that $g(x)=\\psi (\\left|x\\right|_{n})$ for every $x\\in \\overline{\\Omega }$ , that is $g=\\psi \\circ \\left|\\cdot \\right|_{n}$ on $\\overline{\\Omega }$ ; observe that $\\psi $ is continuous if and only if $\\psi \\circ \\left|\\cdot \\right|_{n}$ is continuous.",
"Because also the functions $u$ in the definition of $I$ are asked to be radially symmetric on $\\overline{\\Omega }$ (see [4]), it is useful to observe that for a Riemann integrable function $\\psi :[R_{2},R_{1}]\\rightarrow \\mathbb {R}$ , using the usual spherical change of variables, we have that $\\int _{\\Omega }\\left( \\psi \\circ \\left|\\cdot \\right|_{n}\\right)(x)dx=\\frac{2\\pi ^{n/2}}{\\Gamma (n/2)}\\cdot \\int _{R_{2}}^{R_{1}}r^{n-1}\\psi (r)dr=\\gamma _{n}\\int _{R_{2}}^{R_{1}}\\theta \\psi , $ where $\\gamma _{n}:=\\frac{2\\pi ^{n/2}}{\\Gamma (n/2)},\\text{~~and~~}\\theta :[R_{2},R_{1}]\\rightarrow \\mathbb {R},~~\\theta (r):=r^{n-1}.",
"$ So, in the sequel we consider that $f:[R_{2},R_{1}]\\rightarrow \\mathbb {R}$ is continuous.",
"Condition [4] becomes $\\int _{R_{2}}^{R_{1}}\\theta f=0$ [for the definition of $\\theta $ see (REF )], condition [4] is equivalent to the existence of a unique $R_{3}\\in (R_{2},R_{1})$ such that $f(R_{3})=0$ (that is $(\\theta f)(R_{3})=0$ ), while condition [4] is equivalent to $\\big \\Vert \\theta f\\big \\Vert _{L^{1}[R_{2},R_{1}]}<\\nu R_{2}^{n-1}(2\\lambda /3)^{3/2}.$ Moreover, condition [4] is equivalent to the existence of $\\upsilon :[R_{2},R_{1}]\\rightarrow \\mathbb {R}$ such that $u=\\upsilon \\circ \\left|\\cdot \\right|_{n}$ , while the condition $u\\in C(\\overline{\\Omega })$ is equivalent to $\\upsilon \\in C[R_{2},R_{1}].$ Which is the meaning of $\\nabla u(x)$ in condition [4] for $u\\in W^{1,\\infty }(\\Omega )$ and $x\\in \\Sigma _{1}$ (or $x\\in \\Sigma _{2}$ )?",
"For example, let us consider $\\upsilon :[1,3]\\rightarrow \\mathbb {R}$ defined by $\\upsilon (t):=(t-1)^{2}\\sin \\frac{1}{t-1}$ for $t\\in (1,2)$ .",
"Is $u:=\\upsilon \\circ \\left|\\cdot \\right|_{n}$ in $W^{1,\\infty }(\\Omega )$ for $R_{2}:=1$ and $R_{1}:=2?$ If YES, which is $\\nabla u(x)$ for $x\\in \\mathbb {R}^{n}$ with $\\left|x\\right|_{n}=1?$ Let us assume that $\\upsilon \\in C^{1}(R_{2}-\\varepsilon ,R_{1}+\\varepsilon )$ for some $\\varepsilon \\in (0,R_{2})$ and take $u:=\\upsilon \\circ \\left|\\cdot \\right|_{n}$ .",
"Then clearly $u\\in C^{1}(\\Delta )$ , where $\\Delta :=\\lbrace x\\in \\mathbb {R}^{n}\\mid \\left|x\\right|_{n}\\in (R_{2}-\\varepsilon ,R_{1}+\\varepsilon )\\rbrace $ , and $\\nabla u(x)=\\upsilon ^{\\prime }(\\left|x\\right|_{n})\\left|x\\right|_{n}^{-1}x,\\quad \\left|\\nabla u(x)\\right|_{n}=\\left|\\upsilon ^{\\prime }(\\left|x\\right|_{n})\\right|$ for all $x\\in \\Delta $ .",
"Without any doubt, $u|_{\\Omega }\\in W^{1,\\infty }(\\Omega )$ ; moreover, $\\nabla u(x)\\vec{n}=\\upsilon ^{\\prime }(\\left|x\\right|_{n})\\left|x\\right|_{n}^{-1}x\\cdot (\\left|x\\right|_{n}^{-1}x)=\\upsilon ^{\\prime }(R_{1})$ for every $x\\in \\Sigma _{1}$ and $\\nabla u(x)\\vec{n}=-\\upsilon ^{\\prime }(R_{2})$ for $x\\in \\Sigma _{2}$ .",
"Hence such a $u|_{\\Omega }$ satisfies condition [4] if and only if $\\upsilon ^{\\prime }(R_{1})=\\upsilon ^{\\prime }(R_{2})=0.$ Having in view the remark above, we discuss the result in [4] for $W^{1,\\infty }(\\Omega )$ replaced by $C^{1}(\\overline{\\Omega }),$ more precisely the result in [4] concerning the local extrema of $I$ defined in [4] (quoted above) on the space $U:= & \\lbrace u:=\\upsilon \\circ \\left|\\cdot \\right|_{n}\\mid \\upsilon \\in C^{1}[R_{2},R_{1}],~\\upsilon ^{\\prime }(R_{1})=\\upsilon ^{\\prime }(R_{2})=0\\rbrace \\\\= & \\lbrace \\upsilon \\circ \\left|\\cdot \\right|_{n}\\mid \\upsilon \\in C_{1,0}[R_{2},R_{1}]\\rbrace \\subset C^{1}\\left( \\overline{\\Omega }\\right)$ when $C^{1}\\left( \\overline{\\Omega }\\right) $ (and $U$ ) is endowed with the norm $\\left\\Vert u\\right\\Vert :=\\left\\Vert u\\right\\Vert _{\\infty }+\\left\\Vert \\nabla u\\right\\Vert _{\\infty }; $ moreover, in the sequel, $V:=C_{0}[R_{2},R_{1}]$ is endowed with the norm $\\left\\Vert \\cdot \\right\\Vert _{\\infty }.$ Unlike [4], let us set $F(r):=-\\frac{1}{r^{n-1}}\\int _{R_{2}}^{r}f(\\rho )\\rho ^{n-1}d\\rho =-\\frac{1}{r^{n-1}}\\int _{R_{2}}^{r}\\theta f,\\quad r\\in [R_{2},R_{1}],$ where $\\theta $ is defined in (REF ).",
"Remark 7 Notice that our $F(r)$ is $r$ times the one introduced in [4].",
"From (REF ) and the hypotheses on $f$ , we have that $F(R_{1})=F(R_{2})=0$ and $(\\theta F)^{\\prime }=-\\theta f$ on $[R_{2},R_{1}]$ .",
"Since $(\\theta F)^{\\prime }(r)=0\\iff (\\theta f)(r)=0\\iff f(r)=0\\iff r=R_{3}$ and $(\\theta F)(R_{1})=(\\theta F)(R_{2})=0$ , it follows that $\\theta F>0$ or $\\theta F<0$ on $(R_{2},R_{1})$ , that is $F>0$ or $F<0$ on $(R_{2},R_{1}).$ Moreover, from the definition of $F$ we get $R_{2}^{n-1}\\left|F(r)\\right|\\le \\left|r^{n-1}F(r)\\right|=\\left|\\int _{R_{2}}^{r}\\theta f\\right|\\le \\int _{R_{2}}^{R_{1}}\\left|\\theta f\\right|=\\big \\Vert \\theta f\\big \\Vert _{L^{1}[R_{2},R_{1}]}<R_{2}^{n-1}(2\\lambda /3)^{3/2}$ for every $r\\in [R_{2},R_{1}]$ , whence $\\left|F(r)\\right|<(2\\lambda /3)^{3/2}$ for $r\\in [R_{2},R_{1}].$ Let $u\\in U$ , that is $u:=\\upsilon \\circ \\left|\\cdot \\right|_{n}$ with $\\upsilon \\in C_{1,0}[R_{2},R_{1}]$ , and set $v:=\\upsilon ^{\\prime }$ $(\\in C_{0}[R_{2},R_{1}])$ .",
"We have that $\\int _{R_{2}}^{R_{1}}\\theta f\\upsilon =-\\int _{R_{2}}^{R_{1}}(\\theta F)^{\\prime }\\upsilon =-(\\theta F\\upsilon )|_{R_{2}}^{R_{1}}+\\int _{R_{2}}^{R_{1}}\\theta F\\upsilon ^{\\prime }=\\int _{R_{2}}^{R_{1}}\\theta Fv.",
"$ Using (REF ) and (REF ) we get $I[u] & =\\int _{\\Omega }\\left[ H(\\left|\\nabla u(x)\\right|)-f(\\left|x\\right|)u(x)\\right] dx=\\int _{\\Omega }\\left[ H(\\left|\\upsilon ^{\\prime }(\\left|x\\right|_{n})\\right|)-f(\\left|x\\right|)\\upsilon (\\left|x\\right|)\\right] dx\\\\& =\\gamma _{n}\\int _{R_{2}}^{R_{1}}\\theta (H\\circ \\left|v\\right|-Fv)=\\gamma _{n}\\int _{R_{2}}^{R_{1}}\\theta (H\\circ v-Fv),$ that is $I[u]=\\gamma _{n}K(v),$ where $K$ is defined in (REF ) and $[a,b]:=[R_{2},R_{1}]$ .",
"Therefore, Theorem REF applies also in this situation.",
"Applying it we get that $I$ defined in [4] has no local minimizers and $\\overline{u}\\in C^{1}(\\overline{\\Omega })$ is a local maximizer of $I|_{U}$ if and only if there exists $u_{0}\\in \\mathbb {R}$ such that $\\overline{u}(x)=u_{0}+\\int _{R_{2}}^{\\left|x\\right|_{n}}z_{2}(F(\\rho ))d\\rho $ for every $x\\in \\overline{\\Omega }$ , where $z_{2}(A)$ is the unique solution of the equation $z\\left( \\tfrac{1}{2}z^{2}-\\lambda \\right)=A$ in the interval $(-\\sqrt{2\\lambda /3},\\sqrt{2\\lambda /3})$ for $A\\in (-(2\\lambda /3)^{3/2},(2\\lambda /3)^{3/2}).$ For the present discussion we take the case in which $F>0$ on $(a,b)$ .",
"In this case observe that $z_{2}(A)=A/E_{3}^{-1}(A^{2})$ for $A\\in (0,(2\\lambda /3)^{3/2})$ .",
"This proves that the first and second assertions of [4] are false; in particular, $\\overline{u}_{2}$ is not a local minimizer of $I|_{U}$ (exactly as in the 1D case).",
"Moreover, from the discussion above, we can conclude that also the assertion “In comparison with the 1D case discussed by D. Gao and R. Ogden, there exists huge difference in higher dimensions\" from the abstract of [4] is false."
]
] | 1612.05620 |
[
[
"Secure Estimation and Zero-Error Secrecy Capacity"
],
[
"Abstract We study the problem of securely estimating the states of an unstable dynamical system subject to nonstochastic disturbances.",
"The estimator obtains all its information through an uncertain channel which is subject to nonstochastic disturbances as well, and an eavesdropper obtains a disturbed version of the channel inputs through a second uncertain channel.",
"An encoder observes and block-encodes the states in such a way that, upon sending the generated codeword, the estimator's error is bounded and such that a security criterion is satisfied ensuring that the eavesdropper obtains as little state information as possible.",
"Two security criteria are considered and discussed with the help of a numerical example.",
"A sufficient condition on the uncertain wiretap channel, i.e., the pair formed by the uncertain channel from encoder to estimator and the uncertain channel from encoder to eavesdropper, is derived which ensures that a bounded estimation error and security are achieved.",
"This condition is also shown to be necessary for a subclass of uncertain wiretap channels.",
"To formulate the condition, the zero-error secrecy capacity of uncertain wiretap channels is introduced, i.e., the maximal rate at which data can be transmitted from the encoder to the estimator in such a way that the eavesdropper is unable to reconstruct the transmitted data.",
"Lastly, the zero-error secrecy capacity of uncertain wiretap channels is studied."
],
[
"Introduction",
"With the increasing deployment and growing importance of cyber-physical systems, the question of their security has recently become a focus of research activity in control theory [1].",
"One central vulnerability of networked control or estimation is the communication channel from the system which is to be controlled/estimated to the controller/estimator and possibly the feedback channel.",
"A possible attack on the channels is to actively interfere with transmitted information with the goal of degrading the control or estimation performance.",
"However, if the state of a system is estimated remotely, e.g., in order to decide on the next control action at a remote controller, another possible attack is eavesdropping.",
"An adversary might have the chance to overhear the transmitted information, to make its own state estimate and thus obtain sensitive information.",
"For example, if the system processes health information, leakage of its state might breach privacy.",
"If the system is a production line, knowledge of its state could be valuable information for competitors or for criminals.",
"This paper addresses the question how to protect the transmitted information from such attackers.",
"We consider an unstable scalar, discrete-time, time-invariant linear system subject to nonstochastic disturbances, where both the initial state and the disturbances are arbitrary elements of a bounded interval.",
"An estimator has the goal of estimating the system states in such a way that the supremum over time of the absolute differences between the true state and its estimate is bounded uniformly over all possible system state trajectories.",
"We call this reliability.",
"The estimator does not have direct access to the system states.",
"Instead, an encoder observes the system state and is linked to the estimator through an uncertain channel, where every input is disturbed in a nonstochastic manner and the input and output alphabets are possibly finite.",
"The encoder transforms blocks of state observations into codewords using an encoding function, while the estimator applies a decoding function for estimating the system states from the channel outputs.",
"Together, the encoding and decoding functions form a transmission scheme.",
"Figure: An unstable system has to be estimated remotely.",
"It obtains state information through an uncertain wiretap channel.",
"The outputs obtained by an eavesdropper at the other channel output need to satisfy an operational security criterion.Through another, different, uncertain channel, an adversary called the eavesdropper obtains a disturbed version of the encoder's channel input and hence information about the system state.",
"In addition to reliability, our goal is to make the information transmission from the encoder to the estimator secure in such a way that the eavesdropper obtains as little information as possible about the system state, in a sense to be defined.",
"The main question of this paper is under which conditions there exists a transmission scheme such that reliability and security are achieved simultaneously.",
"See Fig.",
"REF for a sketch of the problem setting.",
"Contributions: We introduce the uncertain wiretap channel, defined as the pair consisting of the uncertain channel from the encoder to the estimator and the uncertain channel from the encoder to the eavesdropper.",
"We also define the zero-error secrecy capacity of the uncertain wiretap channel, which describes the maximal block encoding data rate such that not only the estimator can decode the transmitted message, but at the same time the eavesdropper always has at least two messages among which it cannot distinguish which one was actually transmitted.",
"We show that it either equals zero or the zero-error capacity of the uncertain channel between encoder and estimator.",
"The latter capacity was introduced by Shannon [2].",
"By definition, it is the maximal rate at which, using block encoding, data can be transmitted from the encoder to the estimator through the uncertain channel in such a way that every possible channel output is generated by a unique message.",
"A criterion to distinguish the cases of zero and positive zero-error secrecy capacity can be given in a special case.",
"For the study of the zero-error secrecy capacity of uncertain wiretap channels, we introduce a hypergraph structure on the input alphabet in addition to the graph structure which is applied in the study of the zero-error capacity of uncertain channels and which also goes back to Shannon's original paper [2].",
"With these information-theoretic tools, we address the main question formulated above.",
"We define two security criteria for secure estimation.",
"The first, called d-security, is that there is no possibility for the eavesdropper to process the data it receives in order to obtain a bounded estimation error.",
"The other security criterion is v-security, which requires that the volume of the set of system states at a given time which are possible according to the eavesdropper's information should tend to infinity.",
"We identify a sufficient condition which says that reliability and both d- and v-security are achievable if the zero-error secrecy capacity of the uncertain wiretap channel is strictly larger than the logarithm of the coefficient of the unstable system.",
"In the construction of reliable and d- or v-secure transmission schemes, we separate quantization/estimation from channel coding.",
"We also give bounds on the speed of growth of the eavesdropper's estimation error and of the volume of the set of states at a given time which are possible according to the eavesdropper's information.",
"A necessary condition for the simultaneous achievability of reliability, d- and v-security can be given for a subclass of uncertain wiretap channels.",
"Related work: Good overviews over the area of estimation and control under information constraints can be found in the introduction of [3] and in [4].",
"Matveev and Savkin [3] proved that if the system and channel disturbances are stochastic and the estimator's goal is to obtain an almost surely bounded estimation error, the crucial property of the channel is its Shannon zero-error capacity.",
"This led Nair [5] to introduce a nonstochastic information theory for studying the zero-error capacity of uncertain channels and to consider the problem of estimation and control of linear unstable systems, where the information between sensor and estimator has to be transmitted over an uncertain channel.",
"There exists a large body of work on information-theoretically secure communication, see [6] and [7].",
"Stochastic wiretap channels were introduced by [8].",
"Security in the context of estimation and control has so far mostly meant security against active adversaries, e.g., in [9], [10], [11], [12], [13].",
"To our knowledge, only [14] and [15] have combined estimation and security against a passive adversary for an unstable system so far.",
"[14] considers general stochastic disturbances in the system and a stochastic wiretap channel with Gaussian noise and uses a non-operational security criterion based on entropy whose implications are not immediately clear.",
"[15] considers a linear system with Gaussian disturbances and Gaussian observation noise, whereas the stochastic wiretap channel randomly and independently deletes input symbols.",
"As a security criterion, [15] requires that the eavesdropper's estimation error tends to infinity.",
"Uncertain channels were introduced by Nair [5], but were previously considered implicitly in the study of the zero-error capacity of channels with stochastic disturbances as introduced by Shannon [2].",
"The calculation of the zero-error capacity is known as a difficult problem which nowadays is mainly treated in graph theory [16].",
"Notation: The cardinality of a finite set $\\mathcal {A}$ is denoted by $\\operatorname{\\sharp }\\mathcal {A}$ .",
"If $\\operatorname{\\sharp }\\mathcal {A}=1$ , we call $\\mathcal {A}$ a singleton.",
"An interval $\\mathcal {I}$ will also be written $\\mathcal {I}=[\\mathcal {I}_{\\min },\\mathcal {I}_{\\max }]$ .",
"We define the length of $\\mathcal {I}$ by $\\vert \\mathcal {I}\\vert $ .",
"For two subsets $\\mathcal {A},\\mathcal {B}$ of the real numbers and a scalar $\\lambda $ , we set $\\lambda \\mathcal {A}+\\mathcal {B}:=\\lbrace \\lambda a+b:a\\in \\mathcal {A},b\\in \\mathcal {B}\\rbrace $ .",
"A sequence $(a(t))_{t=t_0}^{t_1}$ is denoted by $a(t_0\\!",
":\\!t_1)$ , where $t_1$ is allowed to equal $\\infty $ .",
"Outline: In Section , uncertain wiretap channels are introduced and the main results concerning their zero-error secrecy capacity are stated.",
"The problem of secure estimation is formulated and the corresponding results are presented in Section .",
"In Section , the quantizers applied in this work are introduced and analyzed.",
"This analysis is used in Section for the proof of the results on secure estimation.",
"Section discusses d- and v-security, including a numerical example.",
"After the conclusion in Section , Appendix contains the proofs of the results concerning uncertain wiretap channels and some additional discussion, and Appendix provides the proofs from Section ."
],
[
"Uncertain Channels and Uncertain Wiretap Channels",
"Before we can present the model for secure estimation, we need to introduce the model for data communication between the encoder and the receiving parties.",
"This model is the uncertain wiretap channel.",
"Since it is new and since some results concerning uncertain wiretap channels are relevant for secure estimation, we devote the complete section to this topic.",
"Our model for secure estimation will be defined in Section ."
],
[
"Uncertain Channels",
"Let $\\mathcal {U},\\mathcal {V}$ be arbitrary nonempty sets.",
"An uncertain channel from $\\mathcal {U}$ to $\\mathcal {V}$ is a mapping $\\mathbf {U}:\\mathcal {U}\\rightarrow 2^{\\mathcal {V}}_*:=2^{\\mathcal {V}}\\setminus \\lbrace \\varnothing \\rbrace $ .",
"For any $u\\in \\mathcal {U}$ , the set $\\mathbf {U}(u)$ is the family of all possible output values of the channel given the input $u$ .",
"When transmitting $u$ , the output of $\\mathbf {U}$ will be exactly one element of $\\mathbf {U}(u)$ .",
"That $\\mathbf {U}(u)\\ne \\varnothing $ for all $u$ means that every input generates an output.",
"Note that every mapping $\\varphi :\\mathcal {U}\\rightarrow \\mathcal {V}$ can be regarded as an uncertain channel $\\Phi :\\mathcal {U}\\rightarrow 2^{\\mathcal {V}}_*$ with singletons as outputs, i.e., $\\Phi (u)=\\lbrace \\varphi (u)\\rbrace $ .",
"Henceforth, we will not make any notational difference between a mapping and the corresponding uncertain channel.",
"Remark 1 Note that there are no probabilistic weights on the elements of $\\mathbf {U}(u)$ .",
"Thus $\\mathbf {U}$ models a channel with nonstochastic noise, where $\\mathbf {U}(u)$ describes the effect of the noise if the input is $u$ .",
"We call the set $\\operatorname{\\mathrm {ran}}(\\mathbf {U}):=\\cup _{u\\in \\mathcal {U}}\\mathbf {U}(u)$ the range of $\\mathbf {U}$.",
"Given two uncertain channels $\\mathbf {U}_1:\\mathcal {U}\\rightarrow 2^{\\mathcal {V}}_*$ and $\\mathbf {U}_2:\\mathcal {V}\\rightarrow 2^{\\mathcal {W}}_*$ , then first applying $\\mathbf {U}_1$ and then $\\mathbf {U}_2$ leads to a new uncertain channel $\\mathbf {U}_2\\circ \\mathbf {U}_1:\\mathcal {U}\\rightarrow 2^{\\mathcal {W}}_*$ called the composition of $\\mathbf {U}_1$ and $\\mathbf {U}_2$.",
"Formally, we have for any $u\\in \\mathcal {U}$ $(\\mathbf {U}_2\\circ \\mathbf {U}_1)(u):=\\mathbf {U}_2(\\mathbf {U}_1(u)):=\\bigcup _{v\\in \\mathbf {U}_1(u)}\\mathbf {U}_2(v).$ Every uncertain channel $\\mathbf {U}$ defines a reverse channel $\\mathbf {U}^{-1}:\\operatorname{\\mathrm {ran}}(\\mathbf {U})\\rightarrow 2^{\\mathcal {U}}_*$ by $\\mathbf {U}^{-1}(v)=\\lbrace u\\in \\mathcal {U}:v\\in \\mathbf {U}(u)\\rbrace .$ Obviously, $\\mathbf {U}^{-1}$ again is an uncertain channel.",
"Remark 2 We call $\\mathbf {U}^{-1}$ the reverse instead of the inverse because usually, $\\operatorname{\\sharp }\\mathbf {U}^{-1}(\\mathbf {U}(u))> 1$ .",
"We have $\\mathbf {U}^{-1}(\\mathbf {U}(u))=\\lbrace u\\rbrace $ for all $u\\in \\mathcal {U}$ if and only if every output $v\\in \\operatorname{\\mathrm {ran}}(\\mathbf {U})$ is generated by exactly one input $u$ .",
"If this is the case, we call $\\mathbf {U}$ injective.",
"If the uncertain channel $\\mathbf {U}$ is injective, then $\\mathbf {U}^{-1}$ is an ordinary mapping, in the sense that $\\mathbf {U}^{-1}(v)$ is a singleton.",
"Given uncertain channels $\\mathbf {U}_i:\\mathcal {U}_i\\rightarrow 2^{\\mathcal {V}_i}_*\\;(1\\le i\\le n)$ , their product is the channel $\\mathbf {U}_1\\times \\cdots \\times \\mathbf {U}_n:\\mathcal {U}_1\\times \\cdots \\times \\mathcal {U}_n\\longrightarrow 2^{\\mathcal {V}_1}_*\\times \\cdots \\times 2^{\\mathcal {V}_n}_*,\\\\(\\mathbf {U}_1\\times \\cdots \\times \\mathbf {U}_n)(u(1\\!",
":\\!n))=\\mathbf {U}_1(u_1)\\times \\cdots \\times \\mathbf {U}_n(u_n).$ If $\\mathbf {U}_1=\\ldots =\\mathbf {U}_n=:\\mathbf {U}$ , we write $\\mathbf {U}_1\\times \\cdots \\times \\mathbf {U}_n=:\\mathbf {U}^n$ .",
"The reverse of $\\mathbf {U}_1\\times \\cdots \\times \\mathbf {U}_n$ is given by $\\mathbf {U}_1^{-1}\\times \\cdots \\times \\mathbf {U}_n^{-1}$ .",
"We write $\\mathbf {U}^{-n}$ for the reverse of $\\mathbf {U}^n$ .",
"Figure: (a) An uncertain channel 𝐓\\mathbf {T}.",
"If one sets 𝐅(0)={a 1 },𝐅(1)={a 3 }\\mathbf {F}(0)=\\lbrace a_1\\rbrace ,\\mathbf {F}(1)=\\lbrace a_3\\rbrace , then 𝐅\\mathbf {F} is a zero-error 2-code for 𝐓\\mathbf {T}.",
"(b) An uncertain wiretap channel (𝐓 B ,𝐓 C )(\\mathbf {T}_B,\\mathbf {T}_C).",
"The uncertain channel 𝐅:{0,1,2}→2 * 𝒜 \\mathbf {F}:\\lbrace 0,1,2\\rbrace \\rightarrow 2^{\\mathcal {A}}_* defined by 𝐅(0)={a 1 },𝐅(1)={a 2 ,a 3 },𝐅(2)={a 4 }\\mathbf {F}(0)=\\lbrace a_1\\rbrace ,\\mathbf {F}(1)=\\lbrace a_2,a_3\\rbrace ,\\mathbf {F}(2)=\\lbrace a_4\\rbrace is a zero-error wiretap 3-code for (𝐓 B ,𝐓 C )(\\mathbf {T}_B,\\mathbf {T}_C)."
],
[
"Zero-Error Codes",
"An $M$ -code on an alphabet $\\mathcal {A}$ is a collection $\\lbrace \\mathbf {F}(m):0\\le m\\le M-1\\rbrace $ of nonempty and mutually disjoint subsets of $\\mathcal {A}$ .",
"This is equivalent to an uncertain channel $\\mathbf {F}:\\lbrace 0,\\ldots ,M-1\\rbrace \\rightarrow 2^{\\mathcal {A}}_*$ with disjoint output sets, so we will often denote such a code just by $\\mathbf {F}$ .",
"The elements of $\\operatorname{\\mathrm {ran}}(\\mathbf {F})$ are called codewords.",
"If $\\operatorname{\\sharp }\\mathbf {F}(m)=1$ for all $0\\le m\\le M-1$ , then we call $\\mathbf {F}$ a singleton code.",
"Zero-error codes which are not singleton codes are introduced here for the first time.",
"Let $\\mathbf {T}:\\mathcal {A}\\rightarrow 2^{\\mathcal {B}}_*$ be an uncertain channel over which data are to be transmitted.",
"A nonstochastic $M$ -code $\\mathbf {F}$ on $\\mathcal {A}$ is called a zero-error $M$ -code for $\\mathbf {T}$ if for any $m,m^{\\prime }\\in \\lbrace 0,\\ldots ,M-1\\rbrace $ with $m\\ne m^{\\prime }$ $\\mathbf {T}(\\mathbf {F}(m))\\cap \\mathbf {T}(\\mathbf {F}(m^{\\prime }))=\\varnothing .$ Thus every possible channel output $y\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}\\circ \\mathbf {F})$ can be associated to a unique message $m$ .",
"In other words, the channel $\\mathbf {T}\\circ \\mathbf {F}$ is injective, or equivalently, $\\mathbf {F}^{-1}\\circ \\mathbf {T}^{-1}$ is an ordinary mapping associating to each output $y$ the message $\\mathbf {F}^{-1}(\\mathbf {T}^{-1}(y))$ by which it was generated (cf.",
"Remark REF ).",
"See Fig.",
"REF (a) for an illustration."
],
[
"Uncertain Wiretap Channels and Zero-Error Wiretap Codes",
"Given an additional finite alphabet $\\mathcal {C}$ , an uncertain wiretap channel is a pair of uncertain channels $(\\mathbf {T}_B:\\mathcal {A}\\rightarrow 2^{\\mathcal {B}}_*,\\mathbf {T}_C:\\mathcal {A}\\rightarrow 2^{\\mathcal {C}}_*)$ .",
"The interpretation is that the outputs of channel $\\mathbf {T}_B$ are received by an intended receiver, whereas the outputs of $\\mathbf {T}_C$ are obtained by an eavesdropper who should not be able to learn the data transmitted over $\\mathbf {T}_B$ .",
"An $M$ -code $\\mathbf {F}$ is called a zero-error wiretap $M$ -code for $(\\mathbf {T}_B,\\mathbf {T}_C)$ if it is a zero-error $M$ -code for $\\mathbf {T}_B$ and additionally $\\operatorname{\\sharp }\\mathbf {F}^{-1}(\\mathbf {T}_C^{-1}(c))\\ge 2$ for every $c\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C\\circ \\mathbf {F})$ .",
"Thus every output $c\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C\\circ \\mathbf {F})$ can be generated by at least two messages.",
"Due to the lack of further information like stochastic weights on the messages conditional on the output, the eavesdropper is unable to distinguish these messages.",
"See Fig.",
"REF (b) for an example."
],
[
"Zero-Error Capacity and Zero-Error Secrecy Capacity",
"Given an uncertain channel $\\mathbf {T}:\\mathcal {A}\\rightarrow 2^{\\mathcal {B}}_*$ , an $M$ -code $\\mathbf {F}$ on $\\mathcal {A}^n$ is called a zero-error $(n,M)$ -code for $\\mathbf {T}$ if it is a zero-error $M$ -code for $\\mathbf {T}^n$ .",
"We call $n$ the blocklength of $\\mathbf {F}$ .",
"We set $N_{\\mathbf {T}}(n)$ to be the maximal $M$ such that there exists a zero-error $(M,n)$ -code for $\\mathbf {T}$ and define the zero-error capacity of $\\mathbf {T}$ by $C_0(\\mathbf {T}):=\\sup _n\\frac{\\log N_{\\mathbf {T}}(n)}{n}.$ Due to the superadditivity of the sequence $\\log N_{\\mathbf {T}}(0\\!",
":\\!\\infty )$ and Fekete's lemma [17], see also [18], the supremum on the right-hand side of (REF ) can be replaced by a $\\lim _{n\\rightarrow \\infty }$ .",
"Thus $C_0(\\mathbf {T})$ is the asymptotically largest exponential rate at which the number of messages which can be transmitted through $\\mathbf {T}$ free of error grows in the blocklength.",
"Given an uncertain wiretap channel $(\\mathbf {T}_B,\\mathbf {T}_C)$ , a zero-error $(n,M)$ -code $\\mathbf {F}$ for $\\mathbf {T}_B$ is called a zero-error wiretap $(n,M)$ -code for $(\\mathbf {T}_B,\\mathbf {T}_C)$ if it is a zero-error wiretap $M$ -code for $(\\mathbf {T}_B^n,\\mathbf {T}_C^n)$ .",
"We define $N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n)$ to be the maximal $M$ such that there exists a zero-error wiretap $(M,n)$ -code for $(\\mathbf {T}_B,\\mathbf {T}_C)$ .",
"If no zero-error wiretap $(n,M)$ -code exists, we set $N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n)=1$ .",
"The zero-error secrecy capacity of $(\\mathbf {T}_B,\\mathbf {T}_C)$ is defined as $C_0(\\mathbf {T}_B,\\mathbf {T}_C):=\\sup _n\\frac{\\log N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n)}{n}.$ Again by superadditivity and Fekete's lemma [17], [18], the supremum in (REF ) can be replaced by a limit.",
"Obviously, $C_0(\\mathbf {T}_B,\\mathbf {T}_C)\\le C_0(\\mathbf {T}_B)$ ."
],
[
"Capacity Results",
"The zero-error capacity of general uncertain channels is unknown, only a few special cases have been solved so far [16].",
"However, it is possible to relate the zero-error secrecy capacity of an uncertain wiretap channel $(\\mathbf {T}_B,\\mathbf {T}_C)$ to the zero-error capacity of $\\mathbf {T}_B$ in a surprisingly simple way.",
"Theorem 1 The zero-error secrecy capacity of an uncertain wiretap channel $(\\mathbf {T}_B,\\mathbf {T}_C)$ either equals 0 or $C_0(\\mathbf {T}_B)$ .",
"The proof of this result can be found in Appendix .",
"The simple observation behind the proof is that the possibility of sending one bit securely over $(\\mathbf {T}_B,\\mathbf {T}_C)$ as a prefix to an arbitrary zero-error code $\\mathbf {F}$ for $\\mathbf {T}_B$ generates a zero-error wiretap code for $(\\mathbf {T}_B,\\mathbf {T}_C)$ whose rate is approximately the same as that of $\\mathbf {F}$ .",
"What is missing in Theorem REF is a necessary and sufficient criterion for the zero-error secrey capacity to be positive.",
"We can give one in the case that $\\mathbf {T}_B$ is injective and the input alphabet is finite.",
"Theorem 2 Let $(\\mathbf {T}_B,\\mathbf {T}_C)$ be an uncertain wiretap channel with finite input alphabet $\\mathcal {A}$ such that $\\mathbf {T}_B$ is injective.",
"Then $C_0(\\mathbf {T}_B,\\mathbf {T}_C)=0$ if and only if $N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(1)=1$ .",
"If $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ , then $C_0(\\mathbf {T}_B,\\mathbf {T}_C)=\\log (\\operatorname{\\sharp }\\mathcal {A})$ .",
"The proof can be found in Appendix .",
"Theorem REF gives a characterization of the positivity of the zero-error secrecy capacity if $\\mathbf {T}_B$ is injective which only involves $(\\mathbf {T}_B,\\mathbf {T}_C)$ at blocklength 1.",
"Its proof also contains a simple procedure for finding $N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(1)$ .",
"If $\\mathbf {T}_B$ is not injective, finding $N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(1)$ is harder, but can be done by brute-force search for reasonably sized alphabets.",
"More importantly, if $\\mathbf {T}_B$ is not injective, it is possible that $N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(1)=1$ and $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ , see Example REF in Appendix .",
"For general uncertain wiretap channels, one can use the procedure from the proof of Theorem REF to reduce a zero-error code for $\\mathbf {T}_B$ to a zero-error wiretap code.",
"However, the code thus generated might have rate 0 although $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ .",
"The question when $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ for general uncertain wiretap channels seems to be a hard problem and has to be left open for now.",
"Further discussion of zero-error secrecy capacity is included in Appendix ."
],
[
"Degree of Eavesdropper Ignorance",
"In order to measure the achieved degree of security in greater detail, we introduce the number of messages that can generate a given eavesdropper output as an additional parameter.",
"We call a zero-error wiretap $(n,M)$ -code a zero-error wiretap $(n,M,\\gamma )$ -code if for every $c(1\\!",
":\\!n)\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^n\\circ \\mathbf {F})$ , $\\operatorname{\\sharp }\\mathbf {F}^{-1}(\\mathbf {T}_C^{-n}(c(1\\!",
":\\!n)))\\ge \\gamma .$ Clearly, $M\\ge \\gamma \\ge 2$ .",
"This parameter can be interpreted as a measure of the minimal eavesdropper's confusion about the transmitted message guaranteed by the $(n,M,\\gamma )$ -code.",
"It will be important in the analysis of one of the security criteria we apply for secure estimation."
],
[
"The Model",
"Let $\\mathcal {I}_0$ be a closed real interval and let $\\Omega \\ge 0$ and $\\lambda >1$ be real numbers such that $\\vert \\mathcal {I}_0\\vert +\\Omega >0$ .",
"We then consider the real-valued time-invariant unstable linear system $x(t+1)&=\\lambda x(t)+w(t),\\\\x(0)&\\in \\mathcal {I}_0.$ The initial state $x(0)$ can assume any value in $\\mathcal {I}_0$ and is not known before its observation.",
"The noise sequence $w(0:\\infty )$ can be any sequence in $[-\\Omega /2,\\Omega /2]^\\infty $ .",
"We call $x(t)$ the system state at time $t$.",
"The system states are directly observable.",
"Due to $\\vert \\mathcal {I}_0\\vert + \\Omega >0$ , the system suffers from nontrivial disturbances in the initial state or in the evolution.",
"The set of possible system trajectories $x(0:t)$ until time $t$ is denoted by $\\mathcal {X}_{0:t}$ .",
"Assume that an entity called the encoder is located at the system output and at time $t$ records the corresponding system state $x(t)$ .",
"At every system time step, it has the possibility of using an uncertain wiretap channel $(\\mathbf {T}_B:\\mathcal {A}\\rightarrow 2^{\\mathcal {B}}_*,\\mathbf {T}_C:\\mathcal {A}\\rightarrow 2^{\\mathcal {C}}_*)$ exactly once, i.e., the system (REF ) and the channel are synchronous.",
"At the output of $\\mathbf {T}_B$ , an estimator has the task of obtaining reliable estimates of the system states.",
"An eavesdropper has access to the outputs of $\\mathbf {T}_C$ which should satisfy a security criterion.",
"At time $t$ , the encoder only knows $x(0\\!",
":\\!t)$ and the system dynamics (REF ), i.e., it has no acausal knowledge of future states.",
"The estimator and the eavesdropper know the system dynamics (REF ), but the only information about the actual system states they have is what they receive from the encoder through $\\mathbf {T}_B$ and $\\mathbf {T}_C$ , respectively.",
"The eavesdropper also knows the transmission protocol applied by encoder and estimator.",
"The encoder also has knowledge of the complete uncertain wiretap channel, in particular the characteristics of the uncertain channel to the eavesdropper.",
"This knowledge can be justified by assuming that the eavesdropper is part of the communication network without access rights for the system state, e.g., an “honest but curious” node in the home network.",
"Uncertain wiretap channels can also be regarded as models of stochastic wiretap channels where the transition probabilities are unknown.",
"In the other direction, there exist information-theoretic techniques for wiretap channels which do not require precise knowledge about the channel to the eavesdropper, but the case with eavesdropper channel knowledge serves as a building block and as a benchmark [19], [20].",
"The allowed protocols are defined next.",
"Definition 1 A transmission scheme consists of a positive integer $n$ called the blocklength of the transmission scheme together with a sequence of pairs $(f_k,\\varphi _k)_{k=0}^\\infty $ .",
"Setting $\\tau _k:=kn+1$ and $t_k:=(k+1)n$ , for every $k\\ge 0$ the $k$ -th encoding function $f_k:\\mathcal {X}_{0:\\tau _k-1}\\rightarrow 2^{\\mathcal {A}^{n}}_*$ is an uncertain channel, the first decoding function $\\varphi _1:\\mathcal {B}^n\\rightarrow \\mathbb {R}$ is an ordinary mapping, for $k\\ge 2$ , the $k$ -th decoding function $\\varphi _k:\\mathcal {B}^{t_k}\\rightarrow \\mathbb {R}^{n}$ is an ordinary mapping.",
"Figure: The kk-th step of a transmission scheme (f k ,ϕ k ) k=0 ∞ (f_k,\\varphi _k)_{k=0}^\\infty with blocklength nn.The concept is illustrated in Fig.",
"REF .",
"The encoding function $f_k$ takes the system path $x(0\\!",
":\\!\\tau _k-1)$ until time $\\tau _k-1$ as input and maps this into a codeword of length $n$ .",
"The blocks of new observations also have length $n$ , except for the first one of length 1.",
"Thus the initial state gets a special treatment, but this is a technical detail the reason of which will become clear in the proof of Theorem REF below.",
"We allow $f_k$ to be an uncertain channel for two reasons.",
"One is that we do not have to distinguish between open and closed quantizing sets—if a path or state is on the boundary, we make an uncertain decision.",
"The more important reason is that uncertain encoding has to be allowed in order for uncertain wiretap channels to achieve capacity, see Example REF in Appendix .",
"The decoder $\\varphi _k$ takes the first $t_k$ outputs of $\\mathbf {T}_B$ and calculates an estimate of the states $x(\\tau _{k-1}),\\ldots ,x(\\tau _k-1)$ (where we set $\\tau _{-1}=0$ ), which have not been estimated before.",
"When we define the performance criterion for a transmission scheme, it will be seen that by not allowing $\\varphi _k$ to be an uncertain channel we do not lose generality.",
"Next we come to the definition of reliability and security of a transmission scheme $(f_k,\\varphi _k)_{k=0}^\\infty $ .",
"Every such transmission scheme induces the (uncertain) channels $f_{0:k}:=f_0\\times \\cdots \\times f_k,\\qquad \\varphi _{0:k}:=\\varphi _0\\times \\cdots \\times \\varphi _k.$ Observe that, given a sequence $\\hat{x}(0\\!",
":\\!\\tau _k-1)$ of system estimates, i.e., of outputs of $\\varphi _{0:k}$ , we can write the set of system states which can generate this output sequence as $(f_{0:k}^{-1}\\circ \\mathbf {T}_B^{-t_k}\\circ \\varphi _{0:k}^{-1})(\\hat{x}(0\\!",
":\\!\\tau _k-1))$ .",
"Let $T$ be a positive integer or $\\infty $ .",
"The $\\infty $ -norm of a real sequence $y(0\\!",
":\\!T)$ is given by $&\\Vert y(0\\!",
":\\!T)\\Vert _\\infty :={\\left\\lbrace \\begin{array}{ll}\\max _{0\\le t\\le T}\\vert y(t)\\vert &\\text{if }T<\\infty ,\\\\\\sup _{0\\le t<\\infty }\\vert y(t)\\vert &\\text{if }T=\\infty .\\end{array}\\right.",
"}$ For a set $\\mathcal {E}\\subset \\mathbb {R}^{T+1}$ , where $T$ is a positive integer or infinity, we define its diameter by $&\\operatorname{\\mathrm {diam}}_{T+1}(\\mathcal {E})\\\\&:=\\sup \\lbrace \\Vert y(0\\!",
":\\!T)-y^{\\prime }(0\\!",
":\\!T)\\Vert _\\infty :y(0\\!",
":\\!T),y^{\\prime }(0\\!",
":\\!T)\\in \\mathcal {E}\\rbrace .$ Definition 2 The transmission scheme $(f_k,\\varphi _k)_{k=0}^\\infty $ is called reliable if the estimation error is bounded uniformly in the estimates, i.e., if there exists a constant $\\kappa >0$ such that for every possibleDue to the application of the $\\infty $ -norm, the reliability criterion is a “pointwise\" criterion.",
"Using $p$ -norms of the form $\\Vert y(0\\!",
":\\!T)\\Vert _p:=(\\sum _{t=0}^T\\vert y(t)\\vert ^p)^{1/p}$ for some $1\\le p<\\infty $ would always lead to an infinite estimation error if $\\Omega >0$ and $\\mathbf {T}_B$ can transmit at most a finite number of messages in finite time, since the sequence $\\vert x(t)-\\hat{x}(t)\\vert :t\\ge 0$ would not tend to zero for all state sequences $x(0\\!",
":\\!\\infty )$ .",
"$\\hat{x}(0\\!",
":\\!\\infty )\\in \\operatorname{\\mathrm {ran}}(\\varphi _{0:\\infty }\\circ \\mathbf {T}_B^\\infty \\circ f_{0:\\infty })$ , $\\sup _{k}\\operatorname{\\mathrm {diam}}_{\\tau _k}\\bigl ((f_{0:k}^{-1}\\circ \\mathbf {T}_B^{-t_k}\\circ \\varphi _{0:k}^{-1})(\\hat{x}(0\\!",
":\\!\\tau _k-1))\\bigr )\\le \\kappa .$ Remark 3 One would not gain anything by allowing the decoding functions $\\varphi _k$ to be uncertain channels since this generalization could only increase the left-hand side of (REF ).",
"A transmission scheme only defines a decoder at the output of the estimator's channel $\\mathbf {T}_B$ .",
"But every system path $x(0\\!",
":\\!\\infty )$ also generates a sequence $c(0\\!",
":\\!\\infty )\\in \\mathbf {T}_C^\\infty (f_{0:\\infty }(x(0\\!",
":\\!\\infty )))$ of outputs obtained by the eavesdropper.",
"The two security criteria we define next require state information to be secure no matter how the eavesdropper further processes its channel output sequence.",
"The first criterion just ensures that the eavesdropper's estimation error grows unbounded with time.",
"Definition 3 The transmission scheme $(f_k,\\varphi _k)_{k=0}^\\infty $ is called d-secure if there exists a function $\\delta (k)$ with $\\operatorname{\\mathrm {diam}}_{\\tau _k}\\bigl ((f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_k-1))\\bigr )\\ge \\delta (k)$ for all $c(0\\!",
":\\!\\infty )\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^{\\infty }\\circ f_{0:\\infty })$ and $\\delta (k)\\rightarrow \\infty $ as $k\\rightarrow \\infty $ .",
"Upon receiving any sequence $c(0\\!",
":\\!\\infty )$ of channel outputs generated by a d-secure transmission scheme, the eavesdropper's estimate of the system path $x(0\\!",
":\\!\\infty )$ that generated $c(0\\!",
":\\!\\infty )$ grows to infinityNote that d-security as defined via the $\\infty $ -norm is stronger than the analogous criteria with the $p$ -norm instead of the $\\infty $ -norm for all $1\\le p<\\infty $ because $\\Vert x(0\\!",
":\\!\\infty )\\Vert _\\infty \\le \\Vert x(0\\!",
":\\!\\infty )\\Vert _p$ .",
"uniformly in $c(0:\\infty )$ .",
"Note that since $\\mathcal {X}_{0:t}$ is bounded for every $t\\ge 0$ , the diameter of $(f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_k-1))$ cannot be infinite for any $k$ .",
"Thus the eavesdropper's estimation error will always be finite, though increasingly large, in finite time.",
"Next one can ask the question how many system paths could be the possible generators of an eavesdropper sequence $c(0\\!",
":\\!\\infty )$ .",
"This is considered in the following secrecy criterion.",
"For a set $\\mathcal {E}$ of real sequences of finite length $T+1$ and $0\\le t\\le T$ , we write $\\mathcal {E}\\vert _t:=\\lbrace x\\in \\mathbb {R}:x=x(t)\\text{ for some }x(0\\!",
":\\!T)\\in \\mathcal {E}\\rbrace $ .",
"The volume $\\operatorname{\\mathrm {vol}}(\\mathcal {E}^{\\prime })$ of a subset $\\mathcal {E}^{\\prime }$ of the real numbers is measured in terms of the Lebesgue measure.",
"Definition 4 A transmission scheme $(f_k,\\varphi _k)_{k=0}^\\infty $ is called v-secure if there exists a function $\\nu (k)$ such that $\\operatorname{\\mathrm {vol}}((f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_{k}-1))\\vert _{\\tau _k-1})\\ge \\nu (k)$ for all $c(0\\!",
":\\!\\infty )\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^{\\infty }\\circ f_{0:\\infty })$ and $\\nu (k)\\rightarrow \\infty $ as $k\\rightarrow \\infty $ .",
"Like in the definition of d-security, we require uniform divergence to infinity.",
"Since $\\mathcal {X}_{0:t}$ is bounded for all $t\\ge 0$ , the volume in Definition REF cannot be infinite in finite time.",
"Remark 4 Clearly, v-security implies d-security.",
"The volume is measured at time $\\tau _k-1$ because it would trivially tend to infinity if the $\\tau _k$ -dimensional volume of the set $(f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_{k}-1))$ were measured.",
"If the volume of the set of states tends to infinity along $\\tau _k-1$ as $k\\rightarrow \\infty $ , then the same holds for the volume measured at all other infinite, increasing sequences of time instances."
],
[
"Results for Secure Estimation",
"We first state a sufficient condition the uncertain wiretap channel has to satisfy in order for reliability as well as d- or v-security to be possible.",
"Theorem 3 There exists a transmission scheme which is reliable, d-secure and v-secure if $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>\\log \\lambda $ .",
"The proof of Theorem REF can be found in Section REF .",
"The transmission schemes applied there separate quantization/estimation from coding for uncertain wiretap channels by concatenating a quantizer defined below with a wiretap zero-error code.",
"Note that the condition $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ is weak: Nair [5] proved that $C_0(\\mathbf {T}_B)>\\log \\lambda $ is sufficient and $C_0(\\mathbf {T}_B)\\ge \\log \\lambda $ is necessary to achieve reliability.",
"Thus by Theorem REF , the additional requirement in Theorem REF is nothing but $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ .",
"This is the minimal condition one would expect to be necessary to also achieve security.",
"For general $(\\mathbf {T}_B,\\mathbf {T}_C)$ we do not know that $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ really has to be satisfied for secure estimation to be possible.",
"For injective channels, however, the condition from Theorem REF is “almost” necessary to achieve reliability and d-security, hence also for v-security.",
"Theorem 4 If $\\mathbf {T}_B$ is injective and $\\mathcal {C}$ finite, then the existence of a reliable and d-secure transmission scheme implies $\\operatorname{\\sharp }\\mathcal {A}\\ge \\lambda $ and $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ .",
"The proof of this theorem can be found in Section REF .",
"Since $\\mathbf {T}_B$ is injective, the condition $\\operatorname{\\sharp }\\mathcal {A}\\ge \\lambda $ means nothing but $C_0(\\mathbf {T}_B)\\ge \\log \\lambda $ .",
"As noted above, $C_0(\\mathbf {T}_B)\\ge \\log \\lambda $ was shown by Nair [5] to follow from reliability for general uncertain channels.",
"The additional condition $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ , which follows from d-security, implies $C_0(\\mathbf {T}_B,\\mathbf {T}_C)\\ge \\log \\lambda $ by Theorem REF .",
"The problem of finding a tight necessary condition for secure estimation over general uncertain wiretap channels $(\\mathbf {T}_B,\\mathbf {T}_C)$ remains open.",
"We conjecture that it depends on a criterion for $C_0(\\mathbf {T}_B,\\mathbf {T}_C)$ to be positive.",
"We only have such a criterion in the case that $\\mathbf {T}_B$ is injective from Theorem REF .",
"As a refinement of Theorem REF , we have a closer look at the exponential rate at which the estimation error or the volume of the set of states at a given time which are possible according to the eavesdropper's information tend to infinity.",
"The higher the speed of divergence, the higher is the degree of security.",
"Lemma 1 There exists a reliable transmission scheme $(f_k,\\varphi _k)_{k=0}^\\infty $ such that for every $c(0\\!",
":\\!\\infty )\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^\\infty \\circ f_{0:\\infty })$ there exist system paths $x(0\\!",
":\\!\\infty ),x^{\\prime }(0\\!",
":\\!\\infty )\\in (f_{0:\\infty }^{-1}\\circ \\mathbf {T}_C^{-\\infty })(c(0\\!",
":\\!\\infty ))$ satisfying $\\lim _{t\\rightarrow \\infty }\\frac{\\log \\Vert x(0\\!",
":\\!t)-x^{\\prime }(0\\!",
":\\!t)\\Vert _\\infty }{t}=\\log \\lambda .$ This lemma is proved in Section REF .",
"Clearly, $\\log \\lambda $ is the largest exponential rate at which two trajectories can diverge.",
"For v-security, the speed of increase of the volume of the set of possible states according to the eavesdropper's information will in general increase at an exponential rate smaller than $\\log \\lambda $ .",
"Lemma 2 For every zero-error wiretap $(n,M,\\gamma )$ -code $\\mathbf {F}$ , upon setting $\\frac{\\log M}{n}=:R,\\quad \\frac{\\log \\gamma }{n}=:\\Gamma ,$ there exists a reliable transmission scheme $(f_k,\\varphi _k)_{k=0}^\\infty $ with blocklength $n$ such that for all $c(0\\!",
":\\!\\infty )\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^{\\infty }\\circ f_{0:\\infty })$ , $&\\lim _{k\\rightarrow \\infty }\\frac{\\log \\operatorname{\\mathrm {vol}}((f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_k-1))\\vert _{\\tau _k-1})}{\\tau _k}\\\\&\\quad \\ge {\\left\\lbrace \\begin{array}{ll}\\Gamma +\\log \\lambda -R &\\text{if }\\Omega =0,\\\\\\frac{\\Gamma \\log \\lambda }{R+2\\log \\lambda +\\varepsilon _n} &\\text{if }\\Omega >0,\\end{array}\\right.",
"}$ where $\\varepsilon _n=\\varepsilon _n(R,\\lambda )$ is positive and $\\varepsilon _n\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"This lemma is proved in Section REF .",
"For $\\Omega =0$ , a positive rate is achievable by choosing $R<\\Gamma +\\log \\lambda $ .",
"Lemmas REF and REF are discussed in detail in Section ."
],
[
"Quantizer Analysis",
"Both Lemmas REF and REF follow from analyzing the transmission scheme we apply in the proof of Theorem REF .",
"For proving Theorem REF , we separate quantization/estimation from channel coding.",
"Next, we will therefore describe the quantizer used in the proof of Theorem REF .",
"More precisely, we analyze the behavior of the system (REF ) with an appropriate quantization of every single state $x(t)$ .",
"Later, when concatenating the quantizer with a channel code of blocklength $n>1$ , we will use an analogous quantizer for the $n$ -sampled version of (REF ).",
"Definition 5 Consider the system (REF ) and let $M\\ge 2$ be an integer, called the number of quantizer levels.",
"Let $\\hat{x}(m(0\\!",
":\\!-1))$ be the mid point of $\\mathcal {I}(m(0\\!",
":\\!-1)):=\\mathcal {I}_0$ .",
"For every integer $t\\ge 0$ and every sequence $m(0\\!",
":\\!t)\\in \\lbrace 0,\\ldots ,M-1\\rbrace ^{t+1}$ , we then recursively set $&\\mathcal {P}(m(0\\!",
":\\!t)):=\\mathcal {I}(m(0\\!",
":\\!t-1))_{\\min }\\\\&\\qquad \\qquad +\\frac{\\vert \\mathcal {I}(m(0\\!",
":\\!t-1))\\vert }{M}\\left[m(t),m(t)+1\\right],\\\\&\\hat{x}(m(0\\!",
":\\!t)):=\\text{mid point of }\\mathcal {P}(m(0\\!",
":\\!t)),\\\\&\\mathcal {I}(m(0\\!",
":\\!t)):=\\lambda \\mathcal {P}(m(0\\!",
":\\!t))+\\left[-\\frac{\\Omega }{2},\\frac{\\Omega }{2}\\right].$ (in (REF ), recall our notation for intervals).",
"Finally we define for every $t\\ge 0$ the $t$ -th quantizer channel, an uncertain channel $\\mathbf {Q}_{t}$ which maps any message sequence $m(0\\!",
":\\!t-1)$ and any $x(t)\\in \\mathcal {I}(m(0\\!",
":\\!t-1))$ to an element of $\\mathbf {Q}_{t}(x(t),m(0\\!",
":\\!t\\!-\\!1))\\!=\\!\\lbrace m\\!",
":\\!x(t)\\!\\in \\!\\mathcal {P}(m(0\\!",
":\\!t\\!-\\!1),m)\\rbrace .$ The sets $\\mathcal {P}(\\cdot )$ will be referred to as quantizer intervals.",
"The numbers $0,\\ldots ,M-1$ are messages.",
"Equations (REF )-(REF ) define the quantizer of the system (REF ) with $M$ quantizer levels.",
"Every state sequence $x(0\\!",
":\\!\\infty )$ generates a message sequence $m(0\\!",
":\\!\\infty )$ via the uncertain channels $\\mathbf {Q}_{t}$ .",
"Assume that the state sequence $x(0\\!",
":\\!t-1)$ has generated message sequence $m(0\\!",
":\\!t-1)$ until time $t-1$ .",
"The interval $\\mathcal {I}(m(0\\!",
":\\!t-1))$ consists of all states $x(t)$ which are possible in the next time step.",
"Upon observation of $x(t)$ , the message $m(t)$ is generated as an element$m(t)$ is not determined deterministically from $x(t)$ and $m(0\\!",
":\\!t-1)$ because in this way we can have all intervals $\\mathcal {P}(m(0\\!",
":\\!t))$ closed.",
"Note that $\\operatorname{\\sharp }\\mathbf {Q}_{t}(x(t),m(0\\!",
":\\!t-1))\\ge 2$ only if $x(t)$ is on the boundary of two neighboring quantizer intervals.",
"of $\\mathbf {Q}_{t}(x(t),m(0\\!",
":\\!t-1))$ .",
"From the sequence $m(0\\!",
":\\!t)$ one can then infer that $x(t)\\in \\mathcal {P}(m(0\\!",
":\\!t))$ .",
"Accordingly, the estimate of $x(t)$ is $\\hat{x}(m(0\\!",
":\\!t))$ .",
"Note that for every message sequence $m(0\\!",
":\\!\\infty )$ there exists a system path $x(0\\!",
":\\!\\infty )$ which generates $m(0\\!",
":\\!\\infty )$ .",
"Most of the quantizer analysis we do in the following serves the proof of Lemma REF .",
"We are interested in the disjointness of quantizer intervals at a given time in order to find a lower bound on the volume of the set of states which are possible according to the eavesdropper's information: If a set of quantizer intervals at a common time instant is disjoint, the volume covered by their union equals the sum over their individual volumes.",
"Thus two questions need to be answered: 1) What is the volume of a quantizer interval?",
"2) How many disjoint quantizer intervals are there (from the eavesdropper's view)?",
"An answer to the first question is the following lemma.",
"Lemma 3 If $\\lambda \\ne M$ , then for every $t\\in \\mathbb {N}$ and $m(0\\!",
":\\!t)\\in \\lbrace 0,\\ldots ,M-1\\rbrace ^{t+1}$ we have $\\vert \\mathcal {P}(m(0\\!",
":\\!t))\\vert =\\frac{\\lambda ^t}{M^t}\\left(\\frac{\\vert \\mathcal {I}_0\\vert }{M}-\\frac{\\Omega }{M-\\lambda }\\right)+\\frac{\\Omega }{M-\\lambda }.$ In particular, we have $\\sup _t\\vert \\mathcal {P}(m(0\\!",
":\\!t))\\vert <\\infty $ for every infinite message sequence $m(0\\!",
":\\!\\infty )$ if $\\lambda <M$ .",
"In that case $\\sup _{t\\ge 0}\\vert \\mathcal {P}(m(0\\!",
":\\!t))\\vert =\\max \\left\\lbrace \\frac{\\vert \\mathcal {I}_0\\vert }{M},\\frac{\\Omega }{M-\\lambda }\\right\\rbrace .$ Further, the length of $\\mathcal {P}(m(0\\!",
":\\!t))$ only depends on $t$ , not on $m(0\\!",
":\\!t)$ .",
"Thus we can define $\\ell _{t}:=\\vert \\mathcal {P}(m(0\\!",
":\\!t))\\vert .$ The proof can be found in Appendix .",
"Lemma REF not only is useful in the security analysis, but it also essentially establishes reliability for $M>\\lambda $ , a result which of course is not surprising in view of the existing literature.",
"Concerning question 2), life is simple in the case $\\Omega =0$ because of the following lemma.",
"Lemma 4 If $\\Omega =0$ , then at each time $t\\ge 0$ , the interiors of the intervals $\\mathcal {P}(m(0\\!",
":\\!t))$ are disjoint, where $m(0\\!",
":\\!t)$ ranges over $\\lbrace 0,\\ldots ,M-1\\rbrace ^{t+1}$ .",
"For the proof, see Appendix .",
"Thus at time $t$ , we have $M^{t+1}$ disjoint quantizer intervals of the same length.",
"If $\\Omega >0$ , then the situation is more complicated: Quantizer intervals belonging to different message sequences of the same length can overlap.",
"This is the reason for the two different lower bounds on the rate of volume increase in (REF ).",
"Example 1 Consider the system (REF ) with $\\lambda =1.2$ , $\\Omega =.1$ , $\\mathcal {I}_0=[-1,1]$ and its quantizer with $M=3$ .",
"Then $\\mathcal {P}(0)=[-1,-1/3]$ and $\\mathcal {P}(1)=[-1/3, +1/3]$ .",
"In the next time step, one has $\\mathcal {P}(0, 1)=\\left[-.6,-.35\\right],\\quad \\mathcal {P}(1, 0)=\\left[-.45, -.15\\right],$ so $\\mathcal {P}(0,1)$ and $\\mathcal {P}(1,0)$ are not disjoint.",
"The closer a state $x(t)$ is to the origin (and the larger $t$ ), the more paths there are which can be in this particular state at time $t$ .",
"Example REF shows that one can only hope to obtain disjoint quantizer sets for a strict subset of all message sequences.",
"To find such a subset, we derive an important formula for the sequence $\\hat{x}(m(0\\!",
":\\!\\infty ))$ given a message sequence $m(0\\!",
":\\!\\infty )$ .",
"Lemma 5 Consider the system (REF ) and consider the quantizer for (REF ) with $M$ quantizer levels.",
"Let $m(0\\!",
":\\!\\infty )$ be a message sequence.",
"Then for every $t=0,1,2,\\ldots $ $&\\!\\!\\hat{x}(m(0\\!",
":\\!t))\\\\&\\!\\!=\\lambda ^t\\biggl \\lbrace \\hat{x}(m(0\\!",
":\\!-1))\\\\&\\!\\!+\\!\\frac{1}{2}\\!\\sum _{i=0}^{t}\\!\\left(\\!\\frac{\\Omega M}{M\\!-\\!\\lambda }\\!\\left(\\frac{1}{\\lambda ^i}\\!-\\!\\frac{1}{M^i}\\!\\right)\\!+\\!\\frac{\\vert \\mathcal {I}_0\\vert }{M^i}\\!\\right)\\!\\!\\left(\\frac{2m(i)\\!+\\!1}{M}\\!-\\!1\\right)\\!\\biggr \\rbrace .$ See Appendix for the proof.",
"In order to find disjoint quantizer intervals, the idea is to look at the distance between points $\\hat{x}(m(0:t))$ and $\\hat{x}(m^{\\prime }(0:t))$ and ask how the distances between the estimate sequences will evolve in the future.",
"Lemma 6 Assume that $M>\\lambda $ .",
"Let $m(0\\!",
":\\!\\infty ),m^{\\prime }(0\\!",
":\\!\\infty )$ be two message sequences and let $T\\ge 0$ .",
"If $\\vert \\hat{x}(m(0\\!",
":\\!T))-\\hat{x}(m^{\\prime }(0\\!",
":\\!T))\\vert \\ge \\frac{\\Omega }{M-\\lambda }\\frac{M-1}{\\lambda -1}+\\ell _T,$ then for every $t\\ge 0$ , the interiors of the intervals $\\mathcal {P}(m(0\\!",
":\\!T+t))$ and $\\mathcal {P}(m^{\\prime }(0\\!",
":\\!T+t))$ are disjoint.",
"The proof can be found in Appendix .",
"Finally, assume that at each time instant at least $\\gamma $ different messages are possible according to the eavesdropper's view.",
"For every $t\\ge 0$ let $\\mathcal {M}_t:=\\lbrace m_{t,1}<m_{t,2}<\\ldots <m_{t,\\gamma }\\rbrace \\subseteq \\lbrace 0,\\ldots ,M-1\\rbrace $ be a subset of the possible messages at time $t$ which has exactly $\\gamma $ elements.",
"In particular, $\\mathcal {M}_t$ may differ from $\\mathcal {M}_{t^{\\prime }}$ for $t\\ne t^{\\prime }$ .",
"Now fix a $T\\ge 1$ .",
"For $j\\ge 1$ and $\\xi (1\\!",
":\\!j)\\in \\lbrace 1,\\ldots ,\\gamma \\rbrace ^j$ , we define the message sequence $m_{\\xi (1:j)}(0\\!",
":\\!jT-1)$ by $m_{\\xi (1:j)}(s)=m_{s,\\xi (i)}\\in \\mathcal {M}_s$ if $1\\le i\\le j$ and $(i-1)T\\le s\\le iT-1$ .",
"On the $j$ -th block of times $(j-1)T,\\ldots ,jT-1$ , the sequences $m_{\\xi (1:j)}(0\\!",
":\\!jT-1)$ , where $\\xi (1\\!",
":\\!j-1)$ is kept fixed and $\\xi (j)$ ranges over $\\lbrace 1,\\ldots ,\\gamma \\rbrace $ , are an ordered set of $\\gamma $ message sequences with the order induced by componentwise ordering.",
"The corresponding quantizer intervals $\\mathcal {P}(m_{\\xi (1:j)}(0\\!",
":\\!jT-1))$ , where $1\\le \\xi (j)\\le \\gamma $ , will therefore diverge due to the instability of the system (REF ).",
"The following lemma is proved in Appendix .",
"Lemma 7 Let $\\Omega >0$ and $M>\\lambda $ and choose a $T\\in \\mathbb {N}$ satisfying $T\\ge 1+\\frac{\\log (M-1)+\\log (M+\\lambda -1)-\\log (M-\\lambda )}{\\log \\lambda }$ Then for every $j\\ge 1$ , the interiors of the sets $\\mathcal {P}(m_{\\xi (1:j)}(0\\!",
":\\!jT-1))$ , where $\\xi (1\\!",
":\\!j)$ ranges over $\\lbrace 1,\\ldots \\gamma \\rbrace ^j$ , are disjoint.",
"Thus we have obtained a lower bound on the number of disjoint quantizer intervals at times $t=jT-1$ , for positive $j$ .",
"This will be sufficient when we put everything together in the next section to prove v-security and obtain the lower bound of Lemma REF for the case $\\Omega >0$ ."
],
[
"Definition of the Transmission Scheme",
"We start by defining a transmission scheme $(f_k,\\varphi _k)_{k=0}^\\infty $ .",
"We choose its blocklength $n$ such that $M:=N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n)>\\lambda ^n$ , which is possible because $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>\\log \\lambda $ .",
"Let $\\gamma \\ge 2$ be chosen such that there exists a zero-error wiretap $(n,M,\\gamma )$ -code $\\mathbf {F}$ .",
"Since we use the channel in blocks of length $n$ , we also observe the system only at intervals of length $n$ .",
"If we look at the outputs of (REF ) at times $0,n,2n,\\ldots $ , we obtain a new dynamical system which satisfies $x^{(n)}(k+1)&=\\lambda ^nx^{(n)}(k)+w^{(n)}(k),\\\\x^{(n)}(0)&\\in \\mathcal {I}_0,$ where $w^{(n)}(k)=\\sum _{j=0}^{n-1}\\lambda ^{n-j-1}w(kn+j).$ Note that $w^{(n)}(k)$ is a nonstochastic disturbance in the range $[-\\Omega ^{(n)}/2,\\Omega ^{(n)}/2]$ for $\\Omega ^{(n)}=\\frac{\\Omega }{\\lambda -1}(\\lambda ^{n}-1).$ Therefore the quantizer for (REF ) with $M$ quantization levels is well-defined as in Definition REF and all results derived in the previous section for (REF ) and its quantizer carry over to (REF ) with the obvious modifications of the parameters.",
"We define the encoding and decoding functions of our transmission scheme by separating quantization/estimation from channel coding like it has been done frequently in settings without security, e.g., [21].",
"For every $k\\ge 0$ , let $\\mathbf {Q}_k^{(n)}$ be the $k$ -th quantizer channel of the quantizer of (REF ) (see (REF )).",
"The transmission scheme is defined by recursively concatenating the $\\mathbf {Q}_k^{(n)}$ with $\\mathbf {F}$ .",
"We set $f_0(x(0))=\\mathbf {F}(\\mathbf {Q}_{0}^{(n)}(x(0)))$ and for $k\\ge 1$ , assuming that the quantizer channels have produced the message sequence $m(0\\!",
":\\!k-1)$ so far, we set $f_k(x(0\\!",
":\\!\\tau _k-1))=\\mathbf {F}(\\mathbf {Q}_{k}^{(n)}(x^{(n)}(k),m(0\\!",
":\\!k-1))).$ For the definition of the decoding functions, recall that $\\mathbf {F}$ is a zero-error code.",
"Thus for every $k\\ge 0$ and $b(0\\!",
":\\!t_k-1)\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_B^{t_k}\\circ \\mathbf {F}^{k+1})$ , the set $(\\mathbf {F}^{-(k+1)}\\circ \\mathbf {T}_B^{-t_k})(b(0\\!",
":\\!t_k-1))$ contains precisely one element, namely the message sequence $m(0\\!",
":\\!k)$ sent by the encoder.",
"The 0-th decoding function has a 1-dimensional output which is defined by $\\varphi _0(b(0\\!",
":\\!t_0-1))=\\hat{x}^{(n)}((\\mathbf {F}^{-1}\\circ \\mathbf {T}_B^{-t_0})(b(0\\!",
":\\!t_0-1)))$ .",
"Here $\\hat{x}^{(n)}(m(0:k))$ for any $m(0:k)$ is the mid point of the quantizer interval $\\mathcal {P}^{(n)}(m(0:k))$ belonging to the quantizer of (REF ).",
"For $k\\ge 1$ , the output of the $k$ -th decoding function $\\varphi _k$ is $n$ -dimensional.",
"If, with a little abuse of notation, we write $\\varphi _k(b(0\\!",
":\\!t_k-1))=:(\\hat{x}_{\\tau _{k-1}}(b(0\\!",
":\\!t_k-1)),\\ldots ,\\hat{x}_{\\tau _k-1}(b(0\\!",
":\\!t_k-1)))$ , then we set $\\hat{x}_{\\tau _k-1}(b(0\\!",
":\\!t_k-1))=\\hat{x}^{(n)}((\\mathbf {F}^{-(k+1)}\\circ \\mathbf {T}_B^{-t_k})(b(0\\!",
":\\!t_k-1))).$ Since (REF ) does not grow to infinity in finite time, the values $\\hat{x}_{\\tau _{k-1}}(b(0\\!",
":\\!t_k-1)),\\ldots ,\\hat{x}_{\\tau _k-2}(b(0\\!",
":\\!t_k-1))$ can be defined in an arbitrary way as long as their distance from $\\hat{x}_{\\tau _k-1}(b(0\\!",
":\\!t_k-1))$ is uniformly bounded in $k$ and $b(0\\!",
":\\!\\infty )$ ."
],
[
"Reliability",
"Although it is not surprising and well-known in the literature, for completeness we show the reliability of the transmission scheme.",
"Since the states of (REF ) cannot diverge to infinity in finite time, we only need to make sure that the estimation errors at the observation times $\\tau _0-1,\\tau _1-1,\\ldots $ are bounded.",
"To see this, let $k\\ge 0$ and $m(0\\!",
":\\!k)$ any message sequence and observe that $(f_{0:k}^{-1}\\circ \\mathbf {T}_B^{-t_k}\\circ \\varphi _{0:k}^{-1})(\\hat{x}^{(n)}(m(0\\!",
":\\!k)))\\vert _{\\tau _k-1}=\\mathcal {P}^{(n)}(m(0\\!",
":\\!k)).$ Since $M>\\lambda ^n$ , the length of $\\mathcal {P}^{(n)}(m(0\\!",
":\\!k))$ is bounded by Lemma REF .",
"This shows that the transmission scheme is reliable."
],
[
"d-Security and Lemma ",
"Let $c(0\\!",
":\\!\\infty )\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^\\infty \\circ f_{0:\\infty })$ .",
"Let $m(0)\\ne m^{\\prime }(0)\\in \\mathbf {F}^{-1}(\\mathbf {T}_C^{-n}(c(0\\!",
":\\!t_0-1)))$ and $m(k)\\in \\mathbf {F}(\\mathbf {T}_C^{-n}(c(t_{k-1}\\!",
":\\!t_k-1)))$ .",
"Then there are two system trajectories $x(0\\!",
":\\!\\infty ),x^{\\prime }(0\\!",
":\\!\\infty )$ such that $x(\\tau _k-1)=\\hat{x}^{(n)}(m(0\\!",
":\\!k))$ and $x^{\\prime }(\\tau _k-1)=\\hat{x}^{(n)}(m^{\\prime }(0)m(1\\!",
":\\!k))$ for all $k\\ge 0$ .",
"With Lemma REF one immediately sees that $x(0\\!",
":\\!\\infty )$ and $x^{\\prime }(0\\!",
":\\!\\infty )$ diverge at exponential rate $\\log \\lambda $ .",
"Thus $x(0\\!",
":\\!\\infty ),x^{\\prime }(0\\!",
":\\!\\infty )$ satisfy (REF ).",
"This proves Lemma REF and the achievability of d-security."
],
[
"v-Security and Lemma ",
"For the proof of v-security of the transmission scheme, we consider the two subcases $\\Omega =0$ and $\\Omega >0$ .",
"We first assume $\\Omega =0$ , hence $\\vert \\mathcal {I}_0\\vert >0$ .",
"In this case, hardly anything remains to be proved.",
"By Lemma REF , for given $k\\ge 0$ , the interiors of all $\\mathcal {P}^{(n)}(m(0\\!",
":\\!k))$ are disjoint.",
"Now assume that the eavesdropper receives the sequence $c(0\\!",
":\\!t_k-1)$ .",
"Since $\\mathbf {F}$ is a $(n,M,\\gamma )$ -code, $ \\operatorname{\\sharp }(\\mathbf {F}^{-(k+1)}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_k-1))\\ge \\gamma ^{k+1}$ .",
"Hence $&\\operatorname{\\mathrm {vol}}((f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_k-1))\\vert _{\\tau _k-1})\\\\&\\quad =\\sum _{m(0:k)\\in (\\mathbf {F}^{-(k+1)}\\circ \\mathbf {T}_C^{-t_k})(c(0:t_k-1))}\\ell _k^{(n)}\\\\&\\quad \\quad \\ge \\gamma ^{k+1}\\ell _k^{(n)}=\\left(\\frac{\\gamma \\lambda ^n}{M}\\right)^k\\frac{\\gamma \\vert \\mathcal {I}_0\\vert }{M},$ where $\\ell _k^{(n)}$ is the length of the quantizer intervals at time $k$ of the quantizer of (REF ).",
"This gives the possibly negative growth rate $(\\log \\gamma )/n+\\log \\lambda -(\\log M)/n$ , as claimed in Lemma REF for the case $\\Omega =0$ .",
"Since $(\\log M)/n$ can be chosen strictly smaller than $(\\log \\gamma )/n+\\log \\lambda $ , this also proves that v-security is achievable for $\\Omega =0$ , and thus completes the proof of Theorem REF for the case $\\Omega =0$ .",
"Next we assume that $\\Omega >0$ .",
"Define $T^{(n)}:=\\left\\lceil 1+\\frac{\\log M}{n\\log \\lambda }+\\frac{\\log (M+\\lambda ^n)-\\log (M-\\lambda ^n)}{n\\log \\lambda }\\right\\rceil .$ Choose a $j\\ge 1$ and set $k(j):=jT^{(n)}-1$ .",
"Let $c(0\\!",
":\\!t_{k(j)}-1)$ be an eavesdropper output sequence.",
"Then by choice of $\\mathbf {F}$ $\\operatorname{\\sharp }(\\mathbf {F}^{-(k(j)+1)}\\circ \\mathbf {T}_C^{-t_{k(j)}})(c(0\\!",
":\\!t_{k(j)}-1))\\ge \\gamma ^{k(j)+1}.$ $T^{(n)}$ satisfies (REF ) for (REF ).",
"By Lemma REF applied to (REF ), within the set on the left-hand side of (REF ), the $\\gamma ^j$ message sequences of the form $m_{\\xi (1:j)}(0\\!",
":\\!k(j))$ produce sets $\\mathcal {P}^{(n)}(m_{\\xi (1:j)}(0\\!",
":\\!k(j)))$ with disjoint interiors.",
"Therefore $&\\operatorname{\\mathrm {vol}}((f_{0:k(j)}^{-1}\\circ \\mathbf {T}_C^{-t_{k(j)}})(c(0\\!",
":\\!t_{k(j)}-1))\\vert _{\\tau _{k(j)}-1})\\\\&\\ge \\sum _{\\xi (1:j)\\in \\lbrace 1,\\ldots ,\\gamma \\rbrace ^j}\\ell _{k(j)}^{(n)}=\\gamma ^j\\ell _{k(j)}^{(n)}.$ Since $\\ell _{k(j)}^{(n)}$ tends to a constant as $j$ tends to infinity, the asymptotic rate of volume growth is lower bounded by $\\lim _{k\\rightarrow \\infty }\\frac{\\log (\\gamma ^j\\ell _{k(j)}^{(n)})}{\\tau _k}=\\frac{\\log \\gamma }{nT^{(n)}}.$ With the notation (REF ) and setting $\\varepsilon _n:=\\frac{\\log (M+\\lambda ^n)-\\log (M-\\lambda ^n)}{n},$ we obtain $\\frac{\\log \\gamma }{nT^{(n)}}\\ge \\frac{\\Gamma \\log \\lambda }{R+2\\log \\lambda +\\varepsilon _n}.$ Clearly, $\\varepsilon _n$ is positive and tends to 0 as $n$ tends to infinity.",
"This proves that v-security can be achieved in the case $\\Omega >0$ as well, and at the rate claimed in Lemma REF .",
"Altogether, this completes the proof of Theorem REF and Lemmas REF and REF ."
],
[
"Proof of Theorem ",
"Assume that $\\mathbf {T}_B$ is injective and $\\mathcal {C}$ is finite.",
"Let $(f_k,\\varphi _k)_{k=0}^\\infty $ be a reliable and d-secure transmission scheme with blocklength $n$ .",
"In particular, choose $\\kappa >0$ in such a way that (REF ) is satisfied for every possible sequence of estimates $\\hat{x}(0\\!",
":\\!\\infty )$ .",
"The necessity of $C_0(\\mathbf {T}_B)\\ge \\log \\lambda $ was shown in [5].",
"Due to the injectivity of $\\mathbf {T}_B$ , this condition can be reformulated as $\\operatorname{\\sharp }\\mathcal {A}\\ge \\lambda $ .",
"It remains to show that $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ .",
"By the uniform divergence requirement in the definition of d-security, it is possible to choose a $k$ such that $\\operatorname{\\mathrm {diam}}_{\\tau _k}((f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_k-1)))>\\kappa $ for every $c(0\\!",
":\\!t_k-1)\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^{t_k}\\circ f_{0:k})$ .",
"Let $\\tilde{c}(0\\!",
":\\!t_k-1)\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^{t_k}\\circ f_{0:k})$ .",
"Recursively, we define the sets $\\mathcal {T}_0(\\tilde{c}(0\\!",
":\\!t_k-1)):=\\operatorname{\\mathrm {ran}}(f_{0:k})\\cap \\mathbf {T}_C^{-t_k}(\\tilde{c}(0\\!",
":\\!t_k-1))$ and $\\mathcal {T}_j(\\tilde{c}(0\\!",
":\\!t_k-1)):=\\operatorname{\\mathrm {ran}}(f_{0:k})\\cap (\\mathbf {T}_C^{-t_k}\\circ \\mathbf {T}_C^{t_k})(\\mathcal {T}_{j-1}(\\tilde{c}(0\\!",
":\\!t_k-1)))$ for $j\\ge 1$ .",
"Let $j_*$ be the maximal $j$ which satisfiesWithout going into the details, we would like to mention here that $\\mathcal {T}_{j_*}(c(0\\!",
":\\!t_k-1))$ is an equivalence class in the taxicab partition of the joint range of $f_{0:k}$ and the corresponding outputs of $\\mathbf {T}_C$ , see [5].",
"$\\mathcal {T}_j(\\tilde{c}(0\\!",
":\\!t_k-1))\\supsetneq \\mathcal {T}_{j-1}(\\tilde{c}(0\\!",
":\\!t_k-1))$ .",
"If $a_0(0\\!",
":\\!t_k-1),\\ldots ,a_{M-1}(0\\!",
":\\!t_k-1)$ is an enumeration of the elements of $\\mathcal {T}_{j_*}(\\tilde{c}(0\\!",
":\\!t_k-1))$ , then the $(M,t_k)$ -code $\\mathbf {G}_k$ defined by $\\mathbf {G}_k(m)=\\lbrace a_m(0\\!",
":\\!t_k-1)\\rbrace $ is a zero-error code.",
"This is due to the injectivity of $\\mathbf {T}_B$ .",
"But $\\mathbf {G}_k$ even is a wiretap zero-error code.",
"To show this, let $c(0\\!",
":\\!t_k-1)\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^{t_k}\\circ \\mathbf {G}_k)$ .",
"The definition of $j_*$ implies that $\\mathbf {T}_C^{-t_k}(c(0\\!",
":\\!t_k-1))\\subseteq \\mathcal {T}_{j_*}(\\tilde{c}(0\\!",
":\\!t_k-1))=\\operatorname{\\mathrm {ran}}(\\mathbf {G}_k)$ .",
"Due to (REF ) and since $(f_k,\\varphi _k)_{k=0}^\\infty $ satisfies (REF ), we have $\\operatorname{\\sharp }(\\mathbf {G}_k^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_k-1))=\\operatorname{\\sharp }\\mathbf {T}_C^{-t_k}(c(0\\!",
":\\!t_k-1))\\ge 2$ .",
"Hence $c(0\\!",
":\\!t_k-1)$ can be generated by at least two different messages.",
"This implies that $\\mathbf {G}_k$ also is a wiretap zero-error code, hence $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ ."
],
[
"Discussion: d- and v-Security",
"We have a closer look at d- and v-security, in particular the rates derived in Lemmas REF and REF .",
"First consider the system (REF ) with $\\Omega =0$ .",
"Let $(\\mathbf {T}_B,\\mathbf {T}_C)$ be any uncertain wiretap channel and $\\mathbf {F}$ an $(n,M,\\gamma )$ -code for $(\\mathbf {T}_B,\\mathbf {T}_C)$ .",
"Then the proof of Lemma REF shows that the lower bound on the right-hand side of (REF ) is tight.",
"On the other hand, the growth rate $\\log \\lambda $ of the eavesdropper's estimation error derived in Lemma REF will in general be strictly larger.",
"This means that the set $(f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0:t_k-1))$ is not connected, i.e., it has holes.",
"If $\\Omega >0$ , we have seen in Example REF and the proof of Lemma REF that the situation is more complicated than for $\\Omega =0$ .",
"For an illustration, let $(\\mathbf {T}_B,\\mathbf {T}_C)$ and $\\mathbf {F}$ be the channel and code from Fig.",
"REF (b).",
"Assume the system (REF ) with $\\lambda = 1.2, \\mathcal {I}_0=[-1,1]$ and $\\Omega =1.2$ .",
"As in the proof of Theorem REF , we construct a blocklength-1 transmission scheme $(f_k,\\varphi _k)_{k=0}^\\infty $ by concatenating the quantizer for (REF ) with $\\mathbf {F}$ by mapping the quantizer message $m$ to $\\mathbf {F}(m)$ .",
"For example, if $x(0)\\in [1/3,1]$ , the quantizer outputs message 2, which $\\mathbf {F}$ maps to the set $\\mathbf {F}(2)= \\lbrace a_4\\rbrace $ .",
"Sending $a_4$ through $\\mathbf {T}_C$ generates the output $c_2$ , from which the eavesdropper concludes that message 1 or 2 has been sent.",
"By choice of parameters, the length of the quantizer intervals remains constant over time.",
"Fig.",
"REF illustrates this situation under the assumption that the eavesdropper receives the symbols $c(0\\!",
":\\!7)=c_2c_1c_2c_1c_1c_2c_2c_1$ .",
"There are $2^8$ possible message sequences from the eavesdropper's point of view, one of which corresponds to the actual sequence generated by the quantizer.",
"Notice the growth of $\\operatorname{\\mathrm {vol}}((f_{0:7}^{-1}\\circ \\mathbf {T}_C^{-8})(c(0\\!",
":\\!7)))$ , which also implies the growth of the eavesdropper's estimation error in the sense of d-security.",
"Further observe how quantizer intervals overlap and even “cross paths”.",
"Figure: The state space of () with parameters as in the text.",
"The thick grey lines mark the outer bounds of the state space.",
"For the received eavesdropper sequence c(0:7)c(0\\!",
":\\!7) as in the text, the vertical black lines show the set of states which are possible according to the eavesdropper's view.",
"Further, for four possible message sequences m(0:7)m(0:7), the evolution of the corresponding 𝒫(m(0:7))\\mathcal {P}(m(0:7)) is shown for illustration purposes.Generally, if $\\Omega >0$ and $\\Gamma =R$ , then the eavesdropper has no information about the transmitted message, and $\\operatorname{\\mathrm {vol}}((f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_k-1)))$ grows at rate $\\log \\lambda $ .",
"The ratio of the left- and the right-hand side of (REF ) tends to 1 as $\\lambda \\searrow 1$ .",
"Thus the lower bound of Lemma REF is asymptotically tight for $\\lambda $ tending to the boundary of the instability region.",
"Moreover, the lower bound (REF ) for $\\Omega >0$ is independent of $\\Omega $ and of $\\mathcal {I}_0$ .",
"This behavior can be expected by the asymptotic dominance of $\\lambda $ in the system dynamics.",
"Fig.",
"REF shows numerical evidence for the correctness of this independence.",
"For the system parameters, we fix $\\lambda =1.2$ and consider four variations of $\\Omega $ and $\\mathcal {I}_0$ as shown in Fig.",
"REF .",
"We assume the same uncertain wiretap channel as in Fig.",
"REF and apply the same blocklength-1 transmission scheme.",
"Because of the symmetry of the channel and the transmission scheme, $\\operatorname{\\mathrm {vol}}((f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-(k+1)})(c(0\\!",
":\\!k)))$ is independent of the eavesdropper's received sequence and can be calculated in closed form.",
"For each of the four combinations of $\\Omega $ and $\\mathcal {I}_0$ we plot the ratio of the left-hand side of (REF ) (“empirical rate”) and the right-hand side of (REF ) (“rate”) versus time.",
"After different initial values mainly due to the differing lengths of the initial interval, the ratios converge.",
"At time 100, the maximal absolute value of all differences between them equals 0.417, at time 1000 it reduces to 0.042.",
"The maximum ratio of empirical rate and rate in the previous example at time 1000 equals 3.36, quite a bit away from 1.",
"This is due to the fact that $\\operatorname{\\mathrm {vol}}((f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-(k+1)})(c(0\\!",
":\\!k)))$ grows at rate $\\log \\lambda $ .",
"The reason for this is that the symmetry of the situation allows without loss of generality to assume that the eavesdropper always receives the symbol $c_1$ .",
"The volume of states compatible with this sequence is essentially given by the difference of the largest and smallest paths which are possible according to this information, which by Lemma REF grows at rate $\\log \\lambda $ .",
"Since the extreme paths compatible with a given eavesdropper information always diverge at rate $\\log \\lambda $ by Lemma REF , a smaller volume growth rate is only possible if there are gaps in the set of possible states, as occur in the case $\\Omega =0$ (see above).",
"We expect these gaps to increase if the difference between $\\Gamma $ and $R$ increases.",
"A major problem for the general analysis of $\\operatorname{\\mathrm {vol}}((f_{0:k}^{-1}\\circ \\mathbf {T}_C^{-t_k})(c(0\\!",
":\\!t_k-1)))$ is that a brute-force approach quickly becomes infeasible because with every secure transmission scheme, at least $2^{t+1}$ different message sequences are possible at time $t$ from the eavesdropper's point of view.",
"A general analysis without relying on symmetry might require techniques from fractal set theory.",
"Symmetry as in the example above is simpler to analyze.",
"To achieve this symmetry, the association of quantizer messages to the code sets is crucial, an issue we have neglected here.",
"We also expect the gap between the left- and the right-hand side of (REF ) to decrease at higher blocklengths, not least because the $\\varepsilon _n$ term in the lower bound at blocklength $n=1$ and with $M=3,\\lambda =1.2$ as in the example equals $1.22$ and is not negligible.",
"Figure: The ratio of the left- and right-hand side of () for different combinations of Ω\\Omega and ℐ 0 \\mathcal {I}_0, with other parameters as in the text."
],
[
"Conclusion",
"In this paper we introduced uncertain wiretap channels and their zero-error secrecy capacity.",
"We introduced methods from hypergraph theory which together with the already established graph theoretic methods for the zero-error capacity of uncertain channels facilitate the analysis of zero-error secrecy capacity.",
"We showed how the zero-error secrecy capacity of an uncertain wiretap channel relates to the zero-error capacity of the uncertain channel to the intended receiver of the wiretap channel.",
"In the case that the uncertain channel to the intended receiver is injective, we gave a full characterization of the zero-error secrecy capacity of the corresponding uncertain wiretap channel.",
"We also analyzed how unstable linear systems can be estimated if the system state information has to be transmitted to the estimator through an uncertain wiretap channel, such that the eavesdropper should obtain as little information about the system states as possible.",
"We introduced two security criteria, called d-security and v-security.",
"We gave a sufficient criterion which uncertain channels have to satisfy in order for the estimator to obtain a bounded estimation error as well as both d- and v-security to hold.",
"In the case of an injective uncertain channel from encoder to estimator, we showed that this sufficient criterion essentially is necessary as well.",
"We gave lower bounds on the exponential rates at which the eavesdropper's state information diverges under the two security criteria.",
"Some problems have been left open in the paper, like a complete characterization of the zero-error secrecy capacity of uncertain wiretap channels, a characterization of when secure estimation of unstable systems is possible over uncertain wiretap channels and a complete answer to the question of optimality of the lower bounds from Lemma REF .",
"Apart from that, there are several points where the paper could be extended in the future.",
"One would be that the encoder has less knowledge about the uncertain wiretap channel.",
"Another one would be an extension to multi-dimensional secure estimation, possibly with distributed observations.",
"Finally, it would be interesting to link the zero-error secrecy capacity of uncertain wiretap channels to Nair's nonstochastic information theory [5] (cf.",
"Footnote REF )."
],
[
"Uncertain Wiretap Channels: Proofs and Further Discussion",
"This appendix contains the proofs of Theorems REF and REF and sone additional discussion.",
"First we prove Theorem REF .",
"For the proof of Theorem REF we then introduce a graph and a hypergraph structure on the input alphabet induced by the uncertain wiretap channel.",
"Using these structures, we prove Theorem REF ."
],
[
"Proof of Theorem ",
"Assume that $C_0(\\mathbf {T}_B,\\mathbf {T}_C)>0$ , which implies $C_0(\\mathbf {T}_B)>0$ .",
"Let $\\mathbf {F}$ be a zero-error wiretap $(n_1,M_1)$ -code and let $\\mathbf {G}$ be a zero-error $(n_2,M_2)$ -code, where $M_1=N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n_1)\\ge 2$ and $M_2=N_{\\mathbf {T}_B}(n_2)$ .",
"Consider the concatenated $(n_1+n_2,M_1M_2)$ -code $\\mathbf {F}\\times \\mathbf {G}$ .",
"Clearly, it is a zero-error code.",
"But it also is a zero-error wiretap code: Choose $(m_1,m_2)\\in \\lbrace 0,\\ldots ,M_1-1\\rbrace \\times \\lbrace 0,\\ldots ,M_2-1\\rbrace $ and choose $c(1\\!",
":\\!n_1)\\in \\mathbf {T}_C^{n_1}(\\mathbf {F}(m_1))$ and $c(n_1+1\\!",
":\\!n_2)\\in \\mathbf {T}_C^{n_2}(\\mathbf {G}(m_2))$ .",
"Since $\\mathbf {F}$ is a zero-error wiretap code, there exists an $m_1^{\\prime }\\in (\\mathbf {F}^{-1}\\circ \\mathbf {T}_C^{-n_1})(c(1\\!",
":\\!n_1))$ with $m_1^{\\prime }\\ne m_1$ .",
"Therefore the two different message pairs $(m_1,m_2),(m_1^{\\prime },m_2)$ both can generate the output $c(1\\!",
":\\!n_1+n_2)$ .",
"Thus $\\mathbf {F}\\times \\mathbf {G}$ is a zero-error wiretap code.",
"This construction implies $\\frac{\\log N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n_1\\!+\\!n_2)}{n_1+n_2}\\!\\ge \\!\\frac{\\log N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n_1)\\!+\\!\\log N_{\\mathbf {T}_B}(n_2)}{n_1+n_2},$ and the term on the right-hand side tends to $C_0(\\mathbf {T}_B)$ as $n_2$ tends to infinity.",
"This proves Theorem REF ."
],
[
"Zero-Error Capacity and Graphs",
"It was observed by Shannon [2] that the zero-error capacity of an uncertain channel $\\mathbf {T}:\\mathcal {A}\\rightarrow 2^{\\mathcal {B}}_*$ can be determined from a graph structure induced on the input alphabet $\\mathcal {A}$ by $\\mathbf {T}$ .",
"To see this, let $n$ be a blocklength.",
"Two words $a(1\\!",
":\\!n),a^{\\prime }(1\\!",
":\\!n)\\in \\mathcal {A}^n$ cannot be used as codewords for the same message if they have a common output word $b(1\\!",
":\\!n)\\in \\mathcal {B}^n$ .",
"If we draw a line between every two elements of $\\mathcal {A}^n$ which generate a common output message $b(1\\!",
":\\!n)$ , we obtain a graph on $\\mathcal {A}^n$ which we denote by $G(\\mathbf {T}^n)$ .",
"Thus $G(\\mathbf {T}^n)$ is nothing but a binary relation $\\sim $ on $\\mathcal {A}^n$ , where $a(1\\!",
":\\!n)\\sim a^{\\prime }(1\\!",
":\\!n)$ if and only if $\\mathbf {T}^n(a(1\\!",
":\\!n))\\cap \\mathbf {T}^n(a(1\\!",
":\\!n))\\ne \\varnothing $ .",
"Since the blocklength should always be clear from the context, we omit it in the $\\sim $ -notation.",
"We call a family $\\lbrace \\mathbf {F}(0),\\ldots \\mathbf {F}(M-1)\\rbrace $ of disjoint subsets of $\\mathcal {A}^n$ an independent system in $G(\\mathbf {T}^n)$ if for all $m,m^{\\prime }\\in \\lbrace 0,\\ldots ,M-1\\rbrace $ with $m\\ne m^{\\prime }$ , we have $a(1\\!",
":\\!n)\\lnot \\sim a^{\\prime }(1\\!",
":\\!n)$ for all $a(1\\!",
":\\!n)\\in \\mathbf {F}(m),a^{\\prime }(1\\!",
":\\!n)\\in \\mathbf {F}(m^{\\prime })$ .",
"Clearly, every independent system consisting of $M$ disjoint subsets of $\\mathcal {A}$ is a zero-error $(n,M)$ -code for $\\mathbf {T}$ and vice versa.",
"Finding the zero-error capacity of $\\mathbf {T}$ therefore amounts to finding the asymptotic behavior as $n\\rightarrow \\infty $ of the sizes of maximum independent systems of the graphs $G(\\mathbf {T}^n)$ .",
"Given two blocklengths $n_1,n_2$ and elements $a(1\\!",
":\\!n_1+n_2),a^{\\prime }(1\\!",
":\\!n_1+n_2)$ of $\\mathcal {A}^{n_1+n_2}$ , note that $a(1\\!",
":\\!n_1+n_2)\\sim a^{\\prime }(1\\!",
":\\!n_1+n_2)$ if and only if one of the following holds: $a(1\\!",
":\\!n_1)\\!=\\!a^{\\prime }(1\\!",
":\\!n_1)$ and $a(n_1\\!+\\!1\\!",
":\\!n_2)\\!\\sim \\!",
"a^{\\prime }(n_1\\!+\\!1\\!",
":\\!n_2)$ , $a(1\\!",
":\\!n_1)\\!\\sim \\!",
"a^{\\prime }(1\\!",
":\\!n_1)$ and $a(n_1\\!+\\!1\\!",
":\\!n_2)\\!=\\!a^{\\prime }(n_1\\!+\\!1\\!",
":\\!n_2)$ , $a(1\\!",
":\\!n_1)\\!\\sim \\!",
"a^{\\prime }(1\\!",
":\\!n_1)$ and $a(n_1\\!+\\!1\\!",
":\\!n_2)\\!\\sim \\!",
"a^{\\prime }(n_1\\!+\\!1\\!",
":\\!n_2)$ .",
"We can therefore say that $G(\\mathbf {T}^{n_1+n_2})$ is the strong graph product of $G(\\mathbf {T}^{n_1})$ and $G(\\mathbf {T}^{n_2})$ , see [22].",
"In particular, $G(\\mathbf {T}^n)$ is the $n$ -fold product of $G(\\mathbf {T})$ with itself."
],
[
"Zero-Error Secrecy Capacity and Hypergraphs",
"Let $(\\mathbf {T}_B,\\mathbf {T}_C)$ be an uncertain wiretap channel and $n$ a blocklength.",
"In order to use the above graph-theoretic framework for zero-error capacity also in the treatment of the zero-error secrecy capacity of $(\\mathbf {T}_B,\\mathbf {T}_C)$ , we introduce an additional structure on $\\mathcal {A}^n$ , which is induced by $\\mathbf {T}_C$ .",
"Every output $c(1\\!",
":\\!n)$ of $\\mathbf {T}_C^n$ generates the set $e^{(n)}(c(1\\!",
":\\!n)):=\\mathbf {T}_C^{-n}(c(1\\!",
":\\!n))\\subseteq \\mathcal {A}^n$ .",
"We set $\\mathcal {E}(\\mathbf {T}_C^n):=\\lbrace e^{(n)}(c(1\\!:\\!n)):c(1\\!",
":\\!n)\\in \\operatorname{\\mathrm {ran}}(\\mathbf {T}_C^n)\\rbrace $ .",
"Every element $e^{(n)}$ of $\\mathcal {E}(\\mathbf {T}_C^n)$ is called a hyperedge and the pair $(\\mathcal {A}^n,\\mathcal {E}(\\mathbf {T}_C^n))$ a hypergraph denoted by $H(\\mathbf {T}_C^n)$ .",
"Now let $\\mathbf {F}$ be a zero-error $(n,M)$ -code for $\\mathbf {T}_B$ .",
"Then by definition, it is a zero-error wiretap $(n,M)$ -code for $(\\mathbf {T}_B,\\mathbf {T}_C)$ if and only if $\\operatorname{\\sharp }\\lbrace m:\\mathbf {F}(m)\\cap e^{(n)}\\rbrace \\ge 2$ for every $e^{(n)}\\in \\mathcal {E}(\\mathbf {T}_C^n)$ .",
"In other words, together with the above observation about zero-error codes and graphs we obtain the following lemma.",
"Lemma 8 A family $\\lbrace \\mathbf {F}(0),\\ldots ,\\mathbf {F}(M-1)\\rbrace $ of disjoint subsets of $\\mathcal {A}^n$ is a zero-error wiretap $(n,M)$ -code for $(\\mathbf {T}_B,\\mathbf {T}_C)$ if and only if it is an independent system in $G(\\mathbf {T}_B^n)$ and if $\\operatorname{\\sharp }\\lbrace m:\\mathbf {F}(m)\\cap e^{(n)}\\rbrace \\ge 2$ for every $e^{(n)}\\in \\mathcal {E}(\\mathbf {T}_C^n)$ .",
"Observe that every $e^{(n)}\\in \\mathcal {E}(\\mathbf {T}_C^n)$ has the form $e_1\\times \\cdots \\times e_n$ for some $e_1,\\ldots ,e_n\\in \\mathcal {E}(\\mathbf {T}_C)$ , and that every Cartesian product $e_1\\times \\cdots \\times e_n$ of elements of $\\mathcal {E}(\\mathbf {T}_C)$ is an element of $\\mathcal {E}(\\mathbf {T}_C^n)$ .",
"This means that $H(\\mathbf {T}_C^n)$ is the square product of $H(\\mathbf {T}_C)$ (see [23]).",
"For the uncertain wiretap channel from Fig.",
"REF (b), the corresponding graph/hypergraph pair at blocklength 1 and a zero-error wiretap code are illustrated in Fig.",
"REF .",
"Figure: (a): The pair (G(𝐓 B ),H(𝐓 C ))(G(\\mathbf {T}_B),H(\\mathbf {T}_C)) corresponding to the uncertain wiretap channel (𝐓 B ,𝐓 C )(\\mathbf {T}_B,\\mathbf {T}_C) from Fig. (b).",
"The black, solid line means that a 2 a_2 and a 3 a_3 are adjacent to each other in G(𝐓 B )G(\\mathbf {T}_B).",
"The blue, dotted lines are the boundaries of the hyperdeges of H(𝐓 C )H(\\mathbf {T}_C).",
"(b): The number inscribed in each node indicates to which set 𝐅(m)\\mathbf {F}(m) the node belongs, where 𝐅\\mathbf {F} is the zero-error wiretap code defined in Fig.",
"(b)."
],
[
"Proving Theorem ",
"Theorem REF will follow from a slightly more general lemma which holds for general wiretap channels.",
"This lemma analyzes a procedure, to be presented next, which eliminates elements $a(1\\!",
":\\!n)$ from $\\mathcal {A}^n$ which do not satisfy a necessary condition for being a codeword of a zero-error wiretap code.",
"The idea behind the procedure is that by Lemma REF no $a(1\\!",
":\\!n)\\in \\mathcal {A}^n$ can be a codeword which is contained in an $e^{(n)}\\in \\mathcal {E}(\\mathbf {T}_C^n)$ which is a singleton or where all elements of $e^{(n)}$ are connected in $G(\\mathbf {T}_B^n)$ .",
"Thus these elements can be neglected when looking for a zero-error wiretap code.",
"This amounts to deleting those elements from the input alphabet and to restricting the wiretap channel to the reduced alphabet.",
"But not using a certain subset of the input alphabet may generate yet another set of unusable input words.",
"Thus a further reduction of the input alphabet may be necessary, and so on, see Fig.",
"REF .",
"We now formalize this procedure and analyze the result.",
"We apply the graph/hypergraph language developed above and start with introducing some related terminology.",
"Let $(\\mathbf {T}_B,\\mathbf {T}_C)$ be an uncertain wiretap channel with input alphabet $\\mathcal {A}$ .",
"For any subset $\\mathcal {A}^{\\prime }$ of $\\mathcal {A}$ , one can consider the uncertain wiretap channel restricted to inputs from $\\mathcal {A}^{\\prime }$ , thus creating an uncertain wiretap channel $(\\mathbf {T}_B\\vert _{\\mathcal {A}^{\\prime }}:\\mathcal {A}^{\\prime }\\rightarrow 2^{\\mathcal {B}}_*,\\mathbf {T}_C\\vert _{\\mathcal {A}^{\\prime }}:\\mathcal {A}^{\\prime }\\rightarrow 2^{\\mathcal {C}}_*)$ satisfying $\\mathbf {T}_B\\vert _{\\mathcal {A}^{\\prime }}(a)=\\mathbf {T}_B(a)$ and $\\mathbf {T}_C\\vert _{\\mathcal {A}^{\\prime }}(a)=\\mathbf {T}_C(a)$ for all $a\\in \\mathcal {A}^{\\prime }$ .",
"Thus, $\\mathbf {T}_B\\vert _{\\mathcal {A}^{\\prime }}$ generates a graph $G(\\mathbf {T}_B\\vert _{\\mathcal {A}^{\\prime }})$ on $\\mathcal {A}^{\\prime }$ and $\\mathbf {T}_C\\vert _{\\mathcal {A}^{\\prime }}$ generates a hypergraph $H(\\mathbf {T}_C\\vert _{\\mathcal {A}^{\\prime }})$ on $\\mathcal {A}^{\\prime }$ .",
"If we say that we eliminate a set $\\mathcal {V}$ from $G(\\mathbf {T}_B)$ or $H(\\mathbf {T}_C)$ , we mean that we pass from $G(\\mathbf {T}_B)$ to $G(\\mathbf {T}_B\\vert _{\\mathcal {A}\\setminus \\mathcal {V}})$ or from $H(\\mathbf {T}_C)$ to $H(\\mathbf {T}_C\\vert _{\\mathcal {A}\\setminus \\mathcal {V}})$ , respectively.",
"Further, a clique in $G(\\mathbf {T}_B)$ is a subset $\\mathcal {V}\\subseteq \\mathcal {A}$ such that $a\\sim a^{\\prime }$ for all $a,a^{\\prime }\\in \\mathcal {V}$ .",
"We write $&\\mathcal {E}(G(\\mathbf {T}_B),H(\\mathbf {T}_C))_{s,c}\\\\&:=\\lbrace e\\in \\mathcal {E}(\\mathbf {T}_C):\\operatorname{\\sharp }e=1\\text{ or }e\\text{ is clique in }G(\\mathbf {T}_B)\\rbrace .$ Figure: (a): The original graph/hypergraph pair (G(𝐓 B ),H(𝐓 C ))(G(\\mathbf {T}_B),H(\\mathbf {T}_C)) of some uncertain wiretap channel (𝐓 B ,𝐓 C )(\\mathbf {T}_B,\\mathbf {T}_C).",
"a 1 a_1 cannot be used in any zero-error wiretap code.",
"(b): If a 1 a_1 is not used in any zero-error wiretap code, then a 2 a_2 is unusable as well.",
"(c): Having eliminated a 1 a_1 and a 2 a_2, there are no singletons or cliques left among the hyperedges.Finally, we can formalize the procedure of deleting some of the unusable input words from the input alphabet of an uncertain wiretap channel.",
"Let $(\\mathbf {T}_B,\\mathbf {T}_C)$ be an uncertain wiretap channel with input alphabet $\\mathcal {A}$ and fix a blocklength $n\\ge 1$ .",
"For the sake of shorter notation, we use the notation $a^n$ for elements of $\\mathcal {A}^n$ in the rest of the section.",
"Put $\\mathcal {A}_{s,c}^{(n)}(-1)=\\varnothing $ and for $i\\ge 0$ set $&\\!\\!G^{(n)}(i):=G(\\mathbf {T}_B^n\\vert _{\\mathcal {A}^n\\setminus \\mathcal {A}_{s,c}^{(n)}(i-1)}),\\\\&\\!\\!H^{(n)}(i):=H(\\mathbf {T}_C^n\\vert _{\\mathcal {A}^n\\setminus \\mathcal {A}_{s,c}^{(n)}(i-1)}),\\\\&\\!\\!\\mathcal {A}_{s,c}^{(n)}(i)\\!",
":=\\!\\lbrace a^n\\!\\!",
":\\exists \\, e^{(n)}\\!\\!\\in \\!\\mathcal {E}(G^{(n)}\\!(i),\\!H^{(n)}\\!",
"(i))_{s,c}:a^n\\in e^{(n)}\\rbrace \\\\&\\qquad \\qquad \\cup \\mathcal {A}_{s,c}^{(n)}(i-1).$ Note that $\\mathcal {A}_{s,c}^{(n)}(-1)\\subseteq \\mathcal {A}_{s,c}^{(n)}(0)\\subseteq \\mathcal {A}_{s,c}^{(n)}(1)\\subseteq \\cdots $ .",
"Define $I^{(n)}&:=[\\min \\lbrace i\\ge -1:\\mathcal {A}_{s,c}^{(n)}(i+1)=\\mathcal {A}_{s,c}^{(n)}(i)\\rbrace ]_+,\\\\\\mathcal {A}_{s,c}^{(n)}&:=\\mathcal {A}_{s,c}^{(n)}(I^{(n)})$ where we set $[x]_+=\\max \\lbrace x,0\\rbrace $ for any real number $x$ .",
"Thus $I^{(n)}+1$ is the number of steps of the procedure (REF )-() where the input alphabet is strictly reduced.",
"The reason for defining $I^{(n)}$ in the way we have done will become clear in the proof of Lemma REF below.",
"Since $\\mathcal {A}$ is finite, clearly $I^{(n)}<\\infty $ .",
"The next lemma says that not being an element of $\\mathcal {A}_{s,c}^{(n)}$ is a necessary condition for any $a^n\\in \\mathcal {A}^n$ to be the codeword of a zero-error wiretap code.",
"Lemma 9 If $\\mathbf {F}$ is a zero-error wiretap $(n,M)$ -code for $(\\mathbf {T}_B,\\mathbf {T}_C)$ and any $M\\ge 2$ , then $\\operatorname{\\mathrm {ran}}(\\mathbf {F})\\cap \\mathcal {A}_{s,c}^{(n)}=\\varnothing $ .",
"We use induction over the reduction steps $i$ .",
"Let $M\\ge 2$ and assume that $\\mathbf {F}$ is a zero-error wiretap $(n,M)$ -code for $(\\mathbf {T}_B,\\mathbf {T}_C)$ .",
"By Lemma REF it is clear that $\\operatorname{\\mathrm {ran}}(\\mathbf {F})\\cap \\mathcal {A}_{s,c}^{(n)}(0)=\\varnothing $ .",
"Thus $\\mathbf {F}$ also is a zero-error wiretap $M$ -code for the reduced uncertain wiretap channel $(\\mathbf {T}_B^n\\vert _{\\mathcal {A}^n\\setminus \\mathcal {A}_{s,c}^{(n)}(0)},\\mathbf {T}_C^n\\vert _{\\mathcal {A}^n\\setminus \\mathcal {A}_{s,c}^{(n)}(0)})$ .",
"In particular, if $e^{(n)}\\in \\mathcal {E}(G^{(n)}(1),H^{(n)}(1))_{s,c}$ , then $e^{(n)}\\cap \\operatorname{\\mathrm {ran}}(\\mathbf {F})=\\varnothing $ .",
"Now note that the union of all $e^{(n)}\\in \\mathcal {E}(G^{(n)}(1),H^{(n)}(1))_{s,c}$ equals $\\mathcal {A}_{s,c}^{(n)}(1)\\setminus \\mathcal {A}_{s,c}^{(n)}(0)$ .",
"Therefore $\\operatorname{\\mathrm {ran}}(\\mathbf {F})\\cap \\mathcal {A}_{s,c}^{(n)}(1)=\\varnothing $ .",
"Repeating this argument $I^{(n)}$ times, one obtains the statement of the lemma.",
"The crucial point about the above elimination procedure is that one can relate $\\mathcal {A}_{s,c}^{(n)}$ to $\\mathcal {A}_{s,c}^{(1)}$ , which in turn will give us Theorem REF .",
"Lemma 10 For any uncertain wiretap channel $(\\mathbf {T}_B,\\mathbf {T}_C)$ and every blocklength $n\\ge 1$ , the corresponding set $\\mathcal {A}_{s,c}^{(n)}$ satisfies $\\mathcal {A}_{s,c}^{(n)}=(\\mathcal {A}_{s,c}^{(1)})^n$ .",
"Before proving Lemma REF , we show how Theorem REF follows from it.",
"[Proof of Theorem REF ] Observe that one can restrict attention to singleton zero-error wiretap codes because the injectivity of $\\mathbf {T}_B$ implies that no vertices are connected in $G(\\mathbf {T}_B^n)$ for any $n$ .",
"Further, since $H(\\mathbf {T}_C\\vert _{\\mathcal {A}^n\\setminus \\mathcal {A}_{s,c}^{(n)}})$ has no singletons as hyperedges by construction of $\\mathcal {A}_{s,c}^{(n)}$ , we conclude that $N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n)=(\\operatorname{\\sharp }\\mathcal {A})^n-\\operatorname{\\sharp }\\mathcal {A}_{s,c}^{(n)}$ .",
"By Lemma REF , we have $\\operatorname{\\sharp }\\mathcal {A}_{s,c}^{(n)}=(\\operatorname{\\sharp }\\mathcal {A}_{s,c}^{(1)})^n$ .",
"Thus if $\\mathcal {A}_{s,c}^{(1)}$ is a strict subset of $\\mathcal {A}$ , then $C_0(\\mathbf {T}_B,\\mathbf {T}_C)=\\lim _{n\\rightarrow \\infty }\\frac{\\log N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n)}{n}=\\log \\operatorname{\\sharp }\\mathcal {A}.$ Otherwise, $C_0(\\mathbf {T}_B,\\mathbf {T}_C)$ obviously equals 0.",
"This proves Theorem REF .",
"[Proof of Lemma REF ] Fix $n\\ge 2$ .",
"We set $\\sigma :=I^{(1)}$ and define a mapping $\\iota :\\mathcal {A}\\rightarrow \\lbrace 0,\\ldots ,\\sigma \\rbrace \\cup \\lbrace \\infty \\rbrace $ , $\\iota (a)={\\left\\lbrace \\begin{array}{ll}\\text{the }i\\text{ with }a\\in \\mathcal {A}_{s,c}^{(1)}(i)\\setminus \\mathcal {A}_{s,c}^{(1)}(i-1)&\\text{if }a\\in \\mathcal {A}_{s,c}^{(n)},\\\\\\infty & \\text{otherwise}.\\end{array}\\right.",
"}$ We also define $\\iota ^{(n)}(a^n)=(\\iota (a_1),\\ldots ,\\iota (a_n)).$ Similarly, for $e\\in \\mathcal {E}(\\mathbf {T}_C)$ with $e\\subset \\mathcal {A}_{s,c}^{(1)}$ we set $\\iota (e):=\\max \\lbrace \\iota (a):a\\in e\\rbrace $ , and for any $e^{(n)}\\in \\mathcal {E}(\\mathbf {T}_C^n)$ we define $\\iota ^{(n)}(e^{(n)})=(\\iota (e_1),\\ldots ,\\iota (e_n))$ .",
"For any $i^n\\in (\\lbrace 0,\\ldots ,\\sigma \\rbrace \\cup \\lbrace \\infty \\rbrace )^n$ we set $f(i^n)=\\lbrace a^n\\in (\\mathcal {A}_{s,c}^{(1)})^n:\\iota ^{(n)}(a^n)=i^n\\rbrace ,\\quad w(i^n)=\\sum _{t=1}^ni_t$ and for $0\\le \\mu \\le n\\sigma $ $F(\\mu ):=\\bigcup _{i^n\\in \\lbrace 0,\\ldots ,\\sigma \\rbrace ^n:w(i^n)\\le \\mu }f(i^n).$ Note that $F(n\\sigma )=(\\mathcal {A}_{s,c}^{(1)})^n$ .",
"We will now prove $F(\\mu )&=\\mathcal {A}_{s,c}^{(n)}(\\mu ) \\quad \\text{for }0\\le \\mu \\le n\\sigma ,\\\\I^{(n)}&=n\\sigma =nI^{(1)}.$ Together, (REF ) and () imply $(\\mathcal {A}_{s,c}^{(1)})^n=F(nI^{(1)})=\\mathcal {A}_{s,c}^{(n)}$ , which is what we want to prove.",
"We first prove (REF ) by induction over $\\mu $ .",
"Let $\\mu =0$ .",
"Then $F(0)=(\\mathcal {A}_{s,c}^{(1)}(0))^n$ .",
"This is easily seen to equal $\\mathcal {A}_{s,c}^{(n)}(0)$ .",
"Next let $0\\le \\mu \\le n\\sigma -1$ and assume (REF ) has been proven for all $0\\le \\mu ^{\\prime }\\le \\mu $ .",
"We need to show that (REF ) holds for $\\mu +1$ .",
"First we show that $F(\\mu +1)\\subseteq \\mathcal {A}_{s,c}^{(n)}(\\mu +1)$ .",
"Let $i^n\\in \\lbrace 0,\\ldots ,\\sigma \\rbrace ^n$ with $w(i^n)=\\mu +1$ .",
"We have to show that $f(i^n)\\subseteq \\mathcal {A}_{s,c}^{(n)}(\\mu +1)$ .",
"Choose an $a^n$ with $\\iota (a^n)=i^n$ .",
"Then by (), for every $1\\le t\\le n$ , there exists an $e_t\\in \\mathcal {E}(\\mathbf {T}_C)$ such that $a^n\\in e^{(n)}=e_1\\times \\cdots \\times e_n$ and $\\iota ^{(n)}(e^{(n)})=i^n$ .",
"Therefore $\\!\\!\\!\\!\\!\\!\\!&e^{(n)}\\setminus \\mathcal {A}_{s,c}^{(n)}(\\mu )\\stackrel{(a)}{=}e^{(n)}\\setminus F(\\mu )\\\\&\\stackrel{(b)}{=}(e_1\\!\\setminus \\!\\mathcal {A}_{s,c}^{(1)}(\\iota (e_1)\\!-\\!1))\\!\\times \\!\\cdots \\!\\times \\!",
"(e_n\\!\\setminus \\!\\mathcal {A}_{s,c}^{(1)}(\\iota (e_n)\\!-\\!1)),$ where $(a)$ is due to the induction hypothesis and $(b)$ holds because $e_t\\setminus \\mathcal {A}_{s,c}^{(1)}(\\iota (e_t))=\\varnothing $ .",
"By definition of the mapping $\\iota $ , every set $e_t\\setminus \\mathcal {A}_{s,c}^{(1)}(\\iota (e_t)-1)$ is a singleton or a clique, hence so is the right-hand side of (REF ).",
"Thus $e^{(n)}\\setminus \\mathcal {A}_{s,c}^{(n)}(\\mu )\\in \\mathcal {E}(G^{(n)}(\\mu +1),H^{(n)}(\\mu +1))_{s,c}$ , hence $a^n\\in \\mathcal {A}_{s,c}^{(n)}(\\mu +1)$ .",
"This proves $F(\\mu +1)\\subseteq \\mathcal {A}_{s,c}^{(n)}(\\mu +1)$ .",
"Now we prove that $\\mathcal {A}_{s,c}^{(n)}(\\mu +1)\\subseteq F(\\mu +1)$ , which is equivalent to showing that $\\mathcal {A}^n\\setminus F(\\mu +1)\\subseteq \\mathcal {A}^n\\setminus \\mathcal {A}_{s,c}^{(n)}(\\mu +1)$ .",
"Let $a^n\\in \\mathcal {A}^n\\setminus F(\\mu +1)$ .",
"Thus $a^n\\in \\mathcal {A}^n\\setminus F(\\mu )=\\mathcal {A}^n\\setminus \\mathcal {A}_{s,c}^{(n)}(\\mu )$ , where the equality is due to the induction hypothesis.",
"We need to show that $e^{(n)}\\setminus \\mathcal {A}_{s,c}^{(n)}(\\mu )\\lnot \\subseteq \\mathcal {A}_{s,c}^{(n)}(\\mu +1)$ for every $e^{(n)}\\in \\mathcal {E}(\\mathbf {T}_C^n)$ containing $a^n$ , since then $a^n\\in \\mathcal {A}^n\\setminus \\mathcal {A}_{s,c}^{(n)}(\\mu +1)$ .",
"Choose any $e^{(n)}=e_1\\times \\cdots \\times e_n\\in \\mathcal {E}(\\mathbf {T}_C^n)$ containing $a^n$ .",
"Let $i^n=\\iota ^{(n)}(a^n)$ .",
"Thus $\\iota (e_t)\\ge i_t$ for every $t\\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"Choose any $t_*\\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"If $0\\le i_{t_*}\\le \\sigma $ , there exists an $a_{t_*}^{\\prime }\\in e_{t_*}\\setminus \\mathcal {A}_{s,c}^{(1)}(i_{t_*}-2)$ with $a_{t_*}\\lnot \\sim a_{t_*}^{\\prime }$ because otherwise, $e_t\\setminus \\mathcal {A}_{s,c}^{(1)}(i_{t_*}-2)$ would be a singleton or a clique in $G^{(1)}(i_{t_*}-1)$ , hence a subset of $\\mathcal {A}_{s,c}^{(1)}(i_{t_*}-1)$ , which we know not to be true because $\\iota (e_{t_*})\\ge i_{t_*}$ .",
"A similar argument shows that there exists an $a^{\\prime }_{t_*}\\in \\mathcal {A}\\setminus \\mathcal {A}_{s,c}^{(1)}$ with $a_{t_*}\\lnot \\sim a_{t_*}$ if $i_{t_*}=\\infty $ .",
"Consequenctly, the sequence $\\tilde{a}^n=(a_1,\\ldots ,a_{t_*-1},a_{t_*}^{\\prime },a_{t_*+1},\\ldots ,a_n)$ is an element of $e^{(n)}$ satisfying $a^n\\lnot \\sim \\tilde{a}^n$ because $G(\\mathbf {T}_B^n)$ is the $n$ -fold strong graph product of $G(\\mathbf {T}_B)$ .",
"Notice that $w(\\tilde{a}^n)\\ge w(a^n)-1\\ge \\mu +1$ because $\\iota (a_{t_*}^{\\prime })\\ge i_{t_{*}}-1$ .",
"In particular, $\\tilde{a}^n\\notin F(\\mu )=\\mathcal {A}_{s,c}^{(n)}(\\mu )$ .",
"Thus we have found two different $a^n,\\tilde{a}^n\\in e^{(n)}\\setminus \\mathcal {A}_{s,c}^{(n)}(\\mu )$ which are not adjacent to each other, which implies $e^{(n)}\\setminus \\mathcal {A}_{s,c}^{(n)}(\\mu )\\lnot \\subseteq \\mathcal {A}_{s,c}^{(n)}(\\mu +1)$ .",
"This is what we had to prove to show that $\\mathcal {A}^n\\setminus F(\\mu +1)\\subseteq \\mathcal {A}^n\\setminus \\mathcal {A}_{s,c}^{(n)}(\\mu +1)$ , and this completes the proof of (REF ).",
"To show (), observe that (REF ) implies $nI^{(1)}\\le I^{(n)}$ .",
"If $I^{(n)}>nI^{(1)}=n\\sigma $ , then$\\mathcal {E}(G^{(n)}(n\\sigma ),H^{(n)}(n\\sigma ))_{s,c}\\ne \\varnothing $ , i.e., there exists an $e^{(n)}=e_1\\times \\cdots \\times e_n\\in \\mathcal {E}(\\mathbf {T}_C^n)$ such that $e^{(n)}\\setminus \\mathcal {A}_{s,c}^{(n)}(n\\sigma )=e^{(n)}\\setminus F(n\\sigma )$ is a clique or a singleton.",
"But if $e^{(n)}\\setminus F(n\\sigma )\\ne \\varnothing $ , there exists a $t$ such that $e_t\\setminus \\mathcal {A}_{s,c}^{(1)}\\ne \\varnothing $ , which implies the existence of $a_t,a_t^{\\prime }\\in e_t\\setminus \\mathcal {A}_{s,c}^{(1)}$ with $a_t\\ne a_t^{\\prime }$ and $a_t\\lnot \\sim a_t^{\\prime }$ .",
"If the $t$ -th component of $a^n\\in e^{(n)}$ equals $a_t$ , then $a^n\\notin F(n\\sigma )=\\mathcal {A}_{s,c}^{(n)}(n\\sigma )$ .",
"But then $\\tilde{a}^n:=(a_1,\\ldots ,a_{t-1},a_t^{\\prime },a_{t+1},\\ldots ,a_n)\\in e^{(n)}\\setminus \\mathcal {A}_{s,c}^{(n)}(n\\sigma )$ as well.",
"Since $a^n\\lnot \\sim \\tilde{a}^n$ , this implies that $e^{(n)}\\setminus \\mathcal {A}_{s,c}^{(n)}(n\\sigma )$ is neither a clique nor a singleton.",
"Thus $I^{(n)}=nI^{(1)}$ , which proves ().",
"This completes the proof of Theorem REF ."
],
[
"Examples and Discussion",
"Example 2 The uncertain wiretap channel $(\\mathbf {T}_B,\\mathbf {T}_C)$ shown in Fig.",
"REF (b) is an example of the fact that at finite blocklengths $n$ , non-singleton zero-error wiretap codes may be necessary to achieve $N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(n)$ .",
"If one applies the zero-error wiretap code $\\mathbf {F}=\\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2,a_3\\rbrace ,\\lbrace a_4\\rbrace \\rbrace $ , then three messages can be distinguished at the intended receiver's output and every eavesdropper output can be generated by two different messages.",
"Hence $\\mathbf {F}$ is a zero-error wiretap $(1,3)$ -code.",
"On the other hand, the maximal $M$ for which a singleton zero-error wiretap $(1,M)$ -code exists is $M=2$ , for example $\\mathbf {F}=\\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_4\\rbrace \\rbrace $ .",
"$M=4$ is not possible because $N_{\\mathbf {T}_B}(1)=3$ .",
"For $M=3$ , either $c_1$ or $c_2$ would be generated by only one message.",
"We conjecture that non-singleton zero-error wiretap codes are also necessary to achieve $C_0(\\mathbf {T}_B,\\mathbf {T}_C)$ .",
"One can also construct examples which show the following: If there exists a zero-error wiretap $(n,M)$ -code, then it is necessary to have non-singleton codes to also find a zero-error wiretap $(M^{\\prime },n)$ -code for every $2\\le M^{\\prime }\\le M$ .",
"Another open question is when the zero-error wiretap capacity of general uncertain wiretap channels is positive.",
"Example 3 Figure: (a): An uncertain wiretap channel (𝐓 B ,𝐓 C )(\\mathbf {T}_B,\\mathbf {T}_C).",
"(b): 𝒜\\mathcal {A} with G(𝐓 B )G(\\mathbf {T}_B) and H(𝐓 C )H(\\mathbf {T}_C).",
"(c): 𝒜 2 \\mathcal {A}^2 with G(𝐓 B 2 )G(\\mathbf {T}_B^2) and H(𝐓 C 2 )H(\\mathbf {T}_C^2).",
"Vertices connected by a solid black line are connected in G(𝐓 B )G(\\mathbf {T}_B) or G(𝐓 B 2 )G(\\mathbf {T}_B^2), respectively.",
"Vertices within the boundary of a blue dotted line belong to the same hyperedge of H(𝐓 C )H(\\mathbf {T}_C) or H(𝐓 C 2 )H(\\mathbf {T}_C^2), respectively.",
"A zero-error wiretap (2,4)(2,4)-code is indicated on the right-hand figure.Consider the wiretap channel $(\\mathbf {T}_B,\\mathbf {T}_C)$ from Fig.",
"REF (a).",
"Fig.",
"REF (b) shows $\\mathcal {A}$ with $G(\\mathbf {T}_B)$ and $H(\\mathbf {T}_C)$ and Fig.",
"REF (c) shows $\\mathcal {A}^2$ with $G(\\mathbf {T}_B^2)$ and $H(\\mathbf {T}_C^2)$ .",
"The code shown in Fig.",
"REF (c) shows that $C_0(\\mathbf {T}_B,\\mathbf {T}_C)\\ge 1$ .",
"Since $C_0(\\mathbf {T}_B)=1$ by [2], we can even conclude $C_0(\\mathbf {T}_B,\\mathbf {T}_C)=1$ .",
"Note that $N_{(\\mathbf {T}_B,\\mathbf {T}_C)}(1)=1$ .",
"Thus the number of messages which can be transmitted securely jumps from none at blocklength 1 to 4 at blocklength 2.",
"This behavior is remarkable when compared to the behavior of zero-error codes for uncertain channels: An uncertain channel $\\mathbf {T}$ has $C_0(\\mathbf {T})>0$ if and only if $N_{\\mathbf {T}}(1)\\ge 2$ .",
"This is a simple criterion to decide at blocklength 1 whether or not the zero-error capacity of an uncertain channel is positive.",
"We do not yet have a general simple criterion for deciding whether the zero-error secrecy capacity of an uncertain wiretap channel is positive.",
"Of course, if $\\mathbf {T}_B$ is injective, then Theorem REF provides such a criterion."
],
[
"Proofs from Quantizer Analysis",
"For reference, we note the following simple lemma which is easily proved by induction.",
"Lemma 11 Let $\\mu $ be a real number and let $y(0\\!",
":\\!\\infty ),v(0\\!",
":\\!\\infty )$ be two sequences of real numbers satisfying $y(t+1)=\\mu y(t)+v(t)$ for every $t\\ge 0$ .",
"Then for every $t\\ge 0$ $y(t)=\\mu ^ty(0)+\\sum _{i=0}^{t-1}\\mu ^{t-i-1}v(i).$ [Proof of Lemma REF ] Note that the quantizer set $\\mathcal {P}(m(0\\!",
":\\!t))$ is an interval.",
"Thus () implies $\\vert \\mathcal {I}(m(0\\!",
":\\!t+1))\\vert =\\lambda \\vert \\mathcal {P}(m(0\\!",
":\\!t))\\vert +\\Omega $ .",
"Hence by (REF ) $\\vert \\mathcal {P}(m(0\\!",
":\\!t\\!+\\!1))\\vert \\!=\\!\\frac{\\vert \\mathcal {I}(m(0\\!",
":\\!t\\!+\\!1))\\vert }{M}\\!=\\!\\frac{\\lambda }{M}\\vert \\mathcal {P}(m(0\\!",
":\\!t))\\vert \\!+\\!\\frac{\\Omega }{M}.$ Therefore by Lemma REF , $\\vert \\mathcal {P}(m(0\\!",
":\\!t))\\vert &=\\left(\\frac{\\lambda }{M}\\right)^{t}\\vert \\mathcal {P}(m(0\\!",
":\\!0))\\vert +\\frac{\\Omega }{M}\\sum _{i=0}^{t-1}\\left(\\frac{\\lambda }{M}\\right)^{t-i-1}\\\\&=\\left(\\frac{\\lambda }{M}\\right)^t\\left(\\frac{\\vert I_0\\vert }{M}-\\frac{\\Omega }{M-\\lambda }\\right)+\\frac{\\Omega }{M-\\lambda },$ which proves (REF ).",
"The other statements of the lemma are immediate from (REF ).",
"[Proof of Lemma REF ] Let $m(0\\!",
":\\!t)\\ne m^{\\prime }(0\\!",
":\\!t)$ .",
"It is sufficient to show that the minimal distance between $\\hat{x}(m(0\\!",
":\\!t))$ and $\\hat{x}(m^{\\prime }(0\\!",
":\\!t))$ is lower-bounded by $\\ell _t$ .",
"By Lemma REF , $\\hat{x}(m(0\\!",
":\\!t))-\\hat{x}(m^{\\prime }(0\\!",
":\\!t))=\\lambda ^t\\frac{\\vert \\mathcal {I}_0\\vert }{M}\\underbrace{\\sum _{i=0}^t\\frac{m(i)-m^{\\prime }(i)}{M^{i}}}_{=:n(m,m^{\\prime },t)}.$ Since $m(i)-m^{\\prime }(i)\\ne 0$ for at least one $i\\in \\lbrace 0,\\ldots ,t\\rbrace $ , the absolute value of $n(m,m^{\\prime },t)$ is at least $1/M^t$ .",
"Thus by (REF ), $\\vert \\hat{x}(m(0\\!",
":\\!t))-\\hat{x}(m^{\\prime }(0\\!",
":\\!t))\\vert \\ge \\frac{\\vert \\mathcal {I}_0\\vert }{M}\\left(\\frac{\\lambda }{M}\\right)^t.$ By Lemma REF , the right-hand side of (REF ) equals $\\ell _t$ .",
"Hence the lemma is proven.",
"[Proof of Lemma REF ] Recall the notation $\\mathcal {I}=[\\mathcal {I}_{\\min },\\mathcal {I}_{\\max }]$ for real intervals $\\mathcal {I}$ .",
"For $t\\ge 0$ , $&\\hat{x}(m(0\\!",
":\\!t+1))\\\\&\\stackrel{(a)}{=}\\mathcal {I}(m(0\\!",
":\\!t))_{\\min }+\\left(m(t+1)+\\frac{1}{2}\\right)\\ell _{t+1}\\\\&\\stackrel{(b)}{=}\\lambda \\mathcal {P}(m(0\\!",
":\\!t))_{\\min }-\\frac{\\Omega }{2}+\\left(m(t+1)+\\frac{1}{2}\\right)\\ell _{t+1}\\\\&\\stackrel{(c)}{=}\\lambda \\hat{x}(m(0\\!",
":\\!t))-\\frac{\\lambda \\ell _t}{2}-\\frac{\\Omega }{2}+\\left(m(t+1)+\\frac{1}{2}\\right)\\ell _{t+1}\\\\&\\stackrel{(d)}{=}\\lambda \\hat{x}(m(0\\!",
":\\!t))\\!-\\!\\frac{\\lambda \\ell _t}{2}\\!-\\!\\frac{\\Omega }{2}\\!+\\!\\left(\\!m(t+1)+\\frac{1}{2}\\right)\\!\\!\\left(\\frac{\\lambda }{M}\\ell _t+\\frac{\\Omega }{M}\\right)\\\\&=\\lambda \\hat{x}(m(0\\!",
":\\!t))+\\frac{\\lambda \\ell _t+\\Omega }{2}\\left(\\frac{2m(t+1)+1}{M}-1\\right),$ where $(a)$ is due to (REF ) and (), $(b)$ is due to (), $(c)$ is again due to () and $(d)$ is due to (REF ).",
"Therefore, $&\\hat{x}(m(0\\!",
":\\!t+1))\\\\&\\!\\stackrel{(e)}{=}\\!\\lambda \\hat{x}(m(0\\!",
":\\!t))\\!+\\!\\left(\\!\\frac{\\lambda ^{t+1}\\vert \\mathcal {I}_0\\vert }{2M^{t+1}}\\!-\\!\\frac{\\lambda ^{t+1}}{2M^t}\\frac{\\Omega }{M\\!-\\!\\lambda }\\!+\\!\\frac{\\lambda }{2}\\frac{\\Omega }{M\\!-\\!\\lambda }\\!+\\!\\frac{\\Omega }{2}\\right)\\!\\times \\\\&\\qquad \\qquad \\qquad \\qquad \\times \\left(\\frac{2m(t\\!+\\!1)\\!+\\!1}{M}-1\\right)\\\\&=\\lambda \\hat{x}(m(0\\!",
":\\!t))+\\frac{1}{2}\\!\\left(\\frac{\\lambda ^{t+1}}{M^{t+1}}\\vert \\mathcal {I}_0\\vert +\\frac{\\Omega M}{M-\\lambda }\\!\\left(1-\\frac{\\lambda ^{t+1}}{M^{t+1}}\\right)\\right)\\times \\\\&\\qquad \\qquad \\qquad \\qquad \\times \\left(\\frac{2m(t+1)+1}{M}-1\\right),$ where $(e)$ is due to (REF ) and (REF ).",
"Consequently, $&\\hat{x}(m(0\\!",
":\\!t))\\\\&\\stackrel{(f)}{=}\\lambda ^t\\biggl \\lbrace \\hat{x}(m(0\\!",
":\\!0))\\\\&\\quad +\\frac{1}{2}\\sum _{i=0}^{t-1}\\frac{1}{\\lambda ^{i+1}}\\left(\\frac{\\lambda ^{i+1}}{M^{i+1}}\\vert \\mathcal {I}_0\\vert +\\frac{\\Omega M}{M-\\lambda }\\left(1-\\frac{\\lambda ^{i+1}}{M^{i+1}}\\right)\\right)\\times \\\\&\\qquad \\qquad \\qquad \\times \\left(\\frac{2m(i+1)+1}{M}-1\\right)\\biggr \\rbrace \\\\&\\stackrel{(g)}{=}\\lambda ^t\\biggl \\lbrace \\hat{x}(m(0\\!",
":\\!-1))+\\frac{\\vert \\mathcal {I}_0\\vert }{2}\\left(\\frac{2m(0)+1}{M}-1\\right)\\\\&\\quad +\\frac{1}{2}\\sum _{i=1}^{t}\\!\\left(\\!\\frac{\\vert \\mathcal {I}_0\\vert }{M^i}\\!+\\!\\frac{\\Omega M}{M-\\lambda }\\!\\left(\\!\\frac{1}{\\lambda ^i}\\!-\\!\\frac{1}{M^i}\\!\\right)\\!\\right)\\!\\left(\\!\\frac{2m(i)+1}{M}\\!-\\!1\\!\\right)\\!\\biggr \\rbrace \\\\&=\\lambda ^t\\biggl \\lbrace \\hat{x}(m(0\\!",
":\\!-1))\\\\&\\quad +\\frac{1}{2}\\sum _{i=0}^{t}\\!\\left(\\!\\frac{\\Omega M}{M-\\lambda }\\!\\left(\\!\\frac{1}{\\lambda ^i}\\!-\\!\\frac{1}{M^i}\\!\\right)\\!+\\!\\frac{\\vert \\mathcal {I}_0\\vert }{M^i}\\!\\right)\\!\\left(\\!\\frac{2m(i)+1}{M}\\!-\\!1\\!\\right)\\!\\biggr \\rbrace .$ where $(f)$ is due to Lemma REF and the recursion formula for $\\hat{x}(m(0\\!",
":\\!t))$ derived in (REF ) and in $(g)$ we applied (REF ) to find the relation between $\\hat{x}(m(0\\!",
":\\!0))$ and $\\hat{x}(m(0\\!",
":\\!-1))$ .",
"This completes the proof.",
"[Proof of Lemma REF ] Without loss of generality, we may assume that $\\hat{x}(m(0\\!",
":\\!T))>\\hat{x}(m^{\\prime }(0\\!",
":\\!T))$ .",
"Then it is sufficient to show that if (REF ) is satisfied, then $\\hat{x}(m(0\\!",
":\\!T+t))-\\hat{x}(m^{\\prime }(0\\!",
":\\!T+t))\\ge \\ell _{T+t}$ for all $t\\ge 0$ .",
"We have $&\\hat{x}(m(0\\!",
":\\!T+t))-\\hat{x}(m^{\\prime }(0\\!",
":\\!T+t))\\\\&\\stackrel{(a)}{=}\\lambda ^t\\biggl \\lbrace \\hat{x}(m(0\\!",
":\\!T))-\\hat{x}(m^{\\prime }(0\\!",
":\\!T))\\\\&\\quad +\\!\\lambda ^T\\!\\sum _{i=T+1}^{T+t}\\!\\left(\\!\\frac{\\Omega }{M\\!-\\!\\lambda }\\!\\left(\\!\\frac{1}{\\lambda ^i}\\!-\\!\\frac{1}{M^i}\\!\\right)\\!+\\!\\frac{\\vert \\mathcal {I}_0\\vert }{M^{i+1}}\\!\\right)\\!",
"(m(i)\\!-\\!m^{\\prime }(i))\\!\\biggr \\rbrace \\\\&\\stackrel{(b)}{\\ge }\\lambda ^t\\biggl \\lbrace \\hat{x}(m(0\\!",
":\\!T))-\\hat{x}(m^{\\prime }(0\\!",
":\\!T))\\\\&\\quad -\\!\\lambda ^T(M\\!-\\!1)\\!\\sum _{i=T+1}^{T+t}\\left(\\frac{\\Omega }{M-\\lambda }\\left(\\frac{1}{\\lambda ^i}-\\frac{1}{M^i}\\right)+\\frac{\\vert \\mathcal {I}_0\\vert }{M^{i+1}}\\right)\\biggr \\rbrace \\\\&=\\lambda ^t\\biggl \\lbrace \\hat{x}(m(0\\!",
":\\!T))\\!-\\!\\hat{x}(m^{\\prime }(0\\!",
":\\!T))\\!-\\frac{\\Omega (M\\!-\\!1)}{(M\\!-\\!\\lambda )(\\lambda \\!-\\!1)}(1\\!-\\!\\lambda ^{-t})\\\\&\\quad -\\frac{\\lambda ^T}{M^T}\\left(\\frac{\\vert \\mathcal {I}_0\\vert }{M}-\\frac{\\Omega }{M-\\lambda }\\right)(1-M^{-t})\\biggr \\rbrace $ where $(a)$ is due to Lemma REF and $(b)$ holds because $m(i)-m^{\\prime }(i)\\ge -(M-1)$ for all $i$ .",
"Thus one obtains $&\\frac{\\hat{x}(m(0\\!",
":\\!T+t))-\\hat{x}(m^{\\prime }(0\\!",
":\\!T+t))-\\ell _{T+t}}{\\lambda ^t}\\\\&\\stackrel{(c)}{\\ge }\\hat{x}(m(0\\!",
":\\!T))-\\hat{x}(m^{\\prime }(0\\!",
":\\!T))\\\\&\\quad -\\!\\frac{\\Omega }{M\\!-\\!\\lambda }\\!\\left(\\!\\!\\frac{M\\!-\\!1}{\\lambda \\!-\\!1}(1\\!-\\!\\lambda ^{-t})\\!-\\!\\frac{\\lambda ^T}{M^T}(1\\!-\\!M^{-t})\\!+\\!\\frac{1}{\\lambda ^t}\\!-\\!\\frac{\\lambda ^T}{M^{T+t}}\\!\\right)\\\\&\\quad -\\frac{\\vert \\mathcal {I}_0\\vert }{M}\\left(\\frac{\\lambda ^T}{M^T}(1-M^{-t})+\\frac{\\lambda ^T}{M^{T+t}}\\right)\\\\&=\\hat{x}(m(0\\!",
":\\!T))-\\hat{x}(m^{\\prime }(0\\!",
":\\!T))\\\\&\\quad -\\frac{\\Omega }{M-\\lambda }\\left(\\frac{M-1}{\\lambda -1}(1-\\lambda ^{-t})-\\frac{\\lambda ^T}{M^T}+\\frac{1}{\\lambda ^t}\\right)-\\frac{\\vert \\mathcal {I}_0\\vert }{M}\\frac{\\lambda ^T}{M^T}.$ where (REF ) and Lemma REF were used in $(c)$ .",
"Since we want (REF ) to be positive for every $t\\ge 0$ , it is sufficient by () to have $&\\hat{x}(m(0\\!",
":\\!T))-\\hat{x}(m^{\\prime }(0\\!",
":\\!T))\\\\&\\ge \\max _{t\\ge 0}\\left\\lbrace \\!\\frac{\\Omega }{M\\!-\\!\\lambda }\\!\\left(\\!\\frac{M\\!-\\!1}{\\lambda \\!-\\!1}(1\\!-\\!\\lambda ^{-t})\\!-\\!\\frac{\\lambda ^T}{M^T}\\!+\\!\\frac{1}{\\lambda ^t}\\!\\right)\\!+\\!\\frac{\\vert \\mathcal {I}_0\\vert }{M}\\frac{\\lambda ^T}{M^T}\\!\\right\\rbrace \\\\&=\\frac{\\Omega }{M-\\lambda }\\left(\\frac{M-1}{\\lambda -1}-\\frac{\\lambda ^T}{M^T}+1\\right)+\\frac{\\vert \\mathcal {I}_0\\vert }{M}\\frac{\\lambda ^T}{M^T}\\\\&\\stackrel{(d)}{=}\\frac{\\Omega }{M-\\lambda }\\frac{M-1}{\\lambda -1}+\\ell _T,$ where $(d)$ is due to Lemma REF .",
"Thus the inequality holds if (REF ) is satisfied, which proves the lemma.",
"[Proof of Lemma REF ] If we can show $&\\hat{x}(m_{\\xi (1:j-1)\\xi (j)}(0\\!",
":\\!jT-1))-\\hat{x}(m_{\\xi (1:j-1)\\xi ^{\\prime }(j)}(0\\!",
":\\!jT-1))\\\\&>\\frac{\\Omega }{M-\\lambda }\\frac{M-1}{\\lambda -1}+\\ell _{jT-1},$ for every $j\\ge 1$ , every $\\xi (1\\!",
":\\!j-1)\\in \\lbrace 1,\\ldots ,\\gamma \\rbrace ^{j-1}$ and every $\\xi (j),\\xi ^{\\prime }(j)\\in \\lbrace 1,\\ldots ,\\gamma \\rbrace $ with $\\xi (j)>\\xi ^{\\prime }(j)$ , then the claim of the lemma follows from Lemma REF .",
"We have $&\\hat{x}(m_{\\xi (1:j-1)\\xi (j)}(0\\!",
":\\!jT-1))-\\hat{x}(m_{\\xi (1:j-1)\\xi ^{\\prime }(j)}(0\\!",
":\\!jT-1))\\\\&\\!\\stackrel{(a)}{=}\\!\\!\\lambda ^{jT-1}\\!\\!\\!\\!\\!\\sum _{i=(j-1)T}^{jT-1}\\!\\!\\!\\left(\\!\\frac{\\Omega M}{M\\!-\\!\\lambda }\\!\\!\\left(\\!\\frac{1}{\\lambda ^i}\\!-\\!\\frac{1}{M^i}\\!\\!\\right)\\!\\!+\\!\\frac{\\vert \\mathcal {I}_0\\vert }{M^i}\\!\\right)\\!\\frac{m_{\\xi (j)}\\!",
"(i)\\!-\\!m_{\\xi ^{\\prime }(j)}\\!",
"(i)}{M}\\\\&\\!\\stackrel{(b)}{\\ge }\\frac{\\Omega }{M-\\lambda }\\frac{\\lambda ^T-1}{\\lambda -1}\\!+\\!\\left(\\frac{\\vert \\mathcal {I}_0\\vert }{M}\\!-\\!\\frac{\\Omega }{M-\\lambda }\\right)\\!\\frac{\\lambda ^{jT-1}}{M^{(j-1)T-1}}\\frac{1-M^{-T}}{M-1}\\\\&\\!\\stackrel{(c)}{=}\\frac{\\Omega }{M-\\lambda }\\frac{M-1}{\\lambda -1}+\\ell _{jT-1}+\\frac{\\Omega }{M-\\lambda }\\frac{\\lambda ^T-M-\\lambda +1}{\\lambda -1}\\\\&\\qquad +\\left(\\frac{\\vert \\mathcal {I}_0\\vert }{M}-\\frac{\\Omega }{M-\\lambda }\\right)\\left(\\frac{\\lambda }{M}\\right)^{jT-1}\\frac{M^T-M}{M-1}\\\\&\\!=:\\frac{\\Omega }{M-\\lambda }\\frac{M-1}{\\lambda -1}+\\ell _{jT-1}+A_{jT},$ where $(a)$ is due to Lemma REF , $(b)$ uses $m_{\\xi (j)}(i)-m_{\\xi ^{\\prime }(j)}(i)\\ge 1$ which holds due to the choice of $\\xi (j),\\xi ^{\\prime }(j)$ , and Lemma REF was used in $(c)$ .",
"It remains to show that $A_{jT}\\ge 0$ .",
"Since $\\lambda ^T\\ge M+\\lambda -1$ for $T$ satisfying (REF ), this is clear in the case that $\\vert \\mathcal {I}_0\\vert /M\\ge \\Omega /(M-\\lambda )$ .",
"Otherwise, we lower-bound $A_{jT}$ by $A_T$ , for which we have $&A_T+\\frac{\\Omega (M+\\lambda -1)}{(M-\\lambda )(\\lambda -1)}\\\\&\\ge \\frac{\\Omega }{M-\\lambda }\\lambda ^T\\left(\\frac{1}{\\lambda -1}-\\frac{M}{\\lambda (M-1)}\\right)\\\\&\\stackrel{(d)}{\\ge }\\frac{\\Omega }{M-\\lambda }\\frac{\\lambda (M-1)(M+\\lambda -1)}{M-\\lambda }\\frac{M-\\lambda }{\\lambda (\\lambda -1)(M-1)}\\\\&=\\frac{\\Omega (M+\\lambda -1)}{(M-\\lambda )(\\lambda -1)}$ where $(d)$ is due to (REF ).",
"This implies $A_T\\ge 0$ , hence $A_{jT}\\ge 0$ for all $j\\ge 1$ .",
"With (REF ), this implies (REF ) for all choices of $j$ , of $\\xi (1\\!",
":\\!j-1)$ and of $\\xi (j)>\\xi ^{\\prime }(j)$ and hence completes the proof of the lemma.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION Dr. Skoglund has worked on problems in source-channel coding, coding and transmission for wireless communications, communication and control, Shannon theory and statistical signal processing.",
"He has authored and co-authored more than 130 journal and 300 conference papers, and he holds six patents.",
"Dr. Skoglund has served on numerous technical program committees for IEEE sponsored conferences.",
"During 2003–08 he was an associate editor with the IEEE Transactions on Communications and during 2008–12 he was on the editorial board for the IEEE Transactions on Information Theory."
]
] | 1612.05552 |
[
[
"Fundamental polytopes of metric trees via parallel connections of\n matroids"
],
[
"Abstract We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik in 2010.",
"In this paper we consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid.",
"We give explicit formulas for the face numbers of fundamental polytopes and Lipschitz polytopes of all tree-like metrics, and we characterize the metric trees for which the fundamental polytope is simplicial."
],
[
"Polytopes associated to metric spaces",
"The study of fundamental polytopes of finite metric spaces was proposed by Vershik [21] as an approach to a combinatorial classification of metric spaces, motivated by its connections to the transportation problem.",
"Indeed, the Kantorovich-Rubinstein norm associated to the finite case of the transportation problem is an extension (uniquely determined by some conditions) of the Minkowski-Banach norm associated to the fundamental polytope (see [15] for details).",
"The polar dual of the fundamental polytope affords a more direct description: it consists of all real-valued functions with Lipschitz constant 1, and it is called Lipschitz polytope.",
"As polar duality preserves all combinatorial data, the combinatorial classification of Lipschitz polytopes is equivalent to that of fundamental polytopes.",
"Very little is known to date about the combinatorics of these polytopes, aside from the aforementioned work of Vershik.",
"For instance, their $f$ -vectorsThe $f$ -vector of a polytope (or of any polyhedral complex) is the list of integers encoding the number of faces of each dimension.",
"are unknown in general.",
"Gordon and Petrov [8] obtained bounds for the number of possible different $f$ -vectors given the size of the metric space.",
"The same authors also examined “generic metric spaces”,A finite metric space is called generic in [8] if the triangle inequality is always strict and its fundamental polytope is simplicial.",
"computing their $f$ -vectors (which, in this class, only depend on the number of elements in the space).",
"In this paper we compute the $f$ -vectors of Lipschitz polytopes for all tree-like pseudometric spaces.",
"More generally, to every split-decomposable pseudometric space we associate a combinatorial object (a matroid) which, in the case of tree-like pseudometrics, allows us to give formulas for the face numbers of the corresponding polytopes."
],
[
"Metric spaces from phylogenetics",
"Metric subspaces of metric trees (see Definition REF ) represent a class of spaces that is important and well-studied in pure mathematics as well as in applications, for instance in phylogenetics [18].",
"Indeed, phylogenetic analysis is often based on distances between taxa which are calculated through comparison of DNA sequences.",
"The finite metric space thus obtained is, by its very nature, a so-called “split-decomposable metric space” (see Definition REF ).",
"The main biological assumption is that genes develop according to trees and only split locally, which leads to tree-like metrics (Definition REF ).",
"On the other hand there is also horizontal gene transfer, which is relevant in the context of evolution of bacteria and leads to non-tree-like split metrics, which are in general less understood than tree-like metrics despite having been in the focus of a substantial amount of research — both from a theoretical point of view (e.g., by Buneman [2] and Bandelt-Dress [1]) as well as in view of their applications in phylogenetics (see [7], [18] for an overview and [11] for a computational approach).",
"One of the points of interest in studying combinatorial invariants of general split-decomposable metrics derives from the quest for a natural “stratification” of the space of such metrics, e.g., in order to set up statistical tests."
],
[
"Arrangements of hyperplanes and matroids",
"We call \"arrangement of hyperplanes\" a finite set of hyperplanes (i.e., linear codimension 1 subspaces) of a real vectorspace and refer to Section REF for some basics about these well-studied objects.",
"Here we only point out that the enumerative combinatorics of such an arrangement is governed by the associated matroid, an abstract combinatorial object encoding the intersection pattern of the hyperplanes (see Remark REF ).",
"In particular, such an arrangement subdivides the unit sphere into a polyhedral complex $K_{{A}}$ which is “combinatorially dual” (see Remark REF ) to the zonotopes arising as Minkowski sum of any choice of normal vectors for the hyperplanes.",
"The enumeration of the faces of these polyhedral complexes in terms of the arrangement's matroid, due to Thomas Zaslavsky [23], has been one of the earliest successful applications of matroid theory."
],
[
"Structure of this paper and main results",
"We start Section by stating the main definitions and some results about polytopes associated to metric spaces, tree-like metrics, arrangements of hyperplanes and systems of splits.",
"(1) We define, for every system of splits $\\mathcal {S}$ , an arrangement of hyperplanes ${A}(\\mathcal {S})$ with one hyperplane for each split of $\\mathcal {S}$ — and, thus, a matroid with one element for every split (Definition REF ).",
"By the decomposition theorem of Bandelt and Dress [1], every “split–decomposable metric space” gives rise to a unique system of splits.",
"(2) We show that the fundamental polytope of any tree-like finite pseudometric space is combinatorially isomorphic to the complex $K_{{A}(\\mathcal {S})}$ , where $\\mathcal {S}$ is the unique system of splits in the Bandelt-Dress decomposition of the given space.",
"We do this by showing that the Lipschitz polytope of such spaces is the zonotope defined as the Minkowski sum of a certain choice of normal vectors for the hyperplanes of ${A}(\\mathcal {S})$ (Theorem REF ).",
"(3) Using a theorem of Zaslavsky, then, we give precise formulas for the face numbers of the fundamental polytope and the Lipschitz polytope of any tree-like pseudometric space (Theorem REF ).",
"Our formulas are exact, but they require knowledge of the arrangement's matroid.",
"We describe how this matroid can be computed, and for specific examples one can easily implement the computation using some widely available software (e.g., SAGE's matroid package [17]).",
"However, one would like to give general explicit formulas, at least for some class of tree metrics.",
"(4) We give a tool allowing to compute the intersection poset of ${A}(\\mathcal {S})$ (and more precisely the closure operator of the associated matroid) from the combinatorics of the split system (Theorem REF ).",
"(5) As an illustration, we close by applying our technique in order to derive explicit formulas for $f$ -numbers of some classes of tree-like metrics."
],
[
"Beyond tree metrics",
"The arrangement ${A}(\\mathcal {S})$ can be defined for all split-decomposable metrics: in this sense, our construction provides a combinatorial stratification of the space of split-decomposable metrics.",
"The stratification of the subset of tree metrics coincides with that given by the combinatorial type of the fundamental polytope, but we believe that also outside the space of tree metrics (e.g., for circular split systems [18]) the matroid- and polytope-stratifications are related.",
"For instance, from the matroid one might be able to recover at least some bounds for the face numbers of the fundamental polytope.",
"We leave these questions for future study."
],
[
"Related work",
"The study of metric spaces by means of associated polyhedral complexes, and its application to phylogenetic trees, is a classical topic, going back at least to work of Buneman [2] and Bandelt-Dress [1].",
"After a first version of this paper circulated, we learned about further recent literature that helped us to contextualize our work.",
"Koichi [14] recently gave a uniform description of the above two seminal approaches building on Hirai's [10] polyhedral split decomposition method, where a metric is viewed as a polyhedral \"height function\" defined on a point configuration.",
"(We remark that the hyperplanes associated to splits in [10] in Hirai's work do not coincide with ours.)",
"Motivated by the connections to tropical convexity [20], [4], Herrmann and Joswig [9] studied split complexes of general polytopes and, in the process, consider an arrangement of “split hyperplanes” associated to every split metric.",
"In this respect we notice that, even if each of our hyperplanes can be expressed in the form [9], our arrangement ${A}(\\mathcal {S})$ is not one of the arrangements considered in [9] (see Remark REF ).",
"Moreover, the matroid we consider is different from the matroid whose basis polytope is cut from the hypersimplex by a set of compatible split hyperplanes, which is studied by Joswig and Schröter in [13].",
"Lipschitz polytopes of finite metric spaces are weighted digraph polyhedra in the sense of Joswig and Loho [12], who give some general results about dimension, face structure and projections [12] but mostly focus on the case of “braid cones” which does not apply to our context.",
"We close by mentioning that the polyhedra considered, e.g., in the above-mentioned work of Hirai [10] are different from the Lipschitz polytopes we consider here: in fact, such polyhedra are (translated) zonotopes for all split-decomposable metrics [10], while – for instance – the Lipschitz polytope of any split-decomposable metric on 4 points is only a zonotope if the associated split system is compatible."
],
[
"Acknowledgements",
"Both authors are supported by the first author's Swiss National Science Foundation Professorship grant PP00P2_150552/1.",
"We thank Andreas Dress for a friendly e-mail exchange and Yaokun Wu for discussing an announcement of his joint work with Zeying Xu on metric spaces with zonotopal Lipschitz polytopes, as well as for pointing out [22].",
"After a first version of this paper was put on ArXiv, we received many valuable pointers to relevant extant literature: we thank an anonymous referee for his comments as well as Michael Joswig and Benjamin Schröter for very informative discussions that took place during the program on tropical geometry at the institute Mittag-Leffler, whose excellent hospitality we also acknowledge."
],
[
"Metric spaces and their polytopes",
"Definition 2.1 Let $X$ be a set.",
"A metric on $X$ is a symmetric function $d: X\\times X \\rightarrow \\mathbb {R}_{\\ge 0} $ with the following properties.",
"(1) For all $x,y\\in X$ , $d(x,y)=0$ implies $x=y$ .",
"(2) For all $x,y,z\\in X$ , $d(x,y) + d(y,z) \\ge d ( x,z)$ (“triangle inequality\").",
"If requirement (1) is dropped, then $d$ is called a pseudometric.",
"The pair $(X,d)$ is then called a metric space (resp.",
"pseudometric space).",
"Remark 2.2 In this paper we will focus on finite metric spaces, i.e., metric spaces $(X,d)$ where $\\vert X \\vert <\\infty $ .",
"We will tacitly assume so throughout.",
"Definition 2.3 Let $(X,d)$ be a (finite) metric space.",
"Consider the vectorspace $\\mathbb {R}^X$ with its standard basis $\\lbrace \\mathbb {1}_k\\rbrace _{k\\in X}$ , i.e., $(\\mathbb {1}_{k})_i:={\\left\\lbrace \\begin{array}{ll} 1 & \\mbox{if }i = k \\\\0 & \\mbox{otherwise. }",
"\\end{array}\\right.}",
"$ Following [21] we define the fundamental polytope of $(X,d)$ as $P_d(X):=\\operatorname{conv}\\lbrace e_{i,j} \\mid i,j\\in X,\\, i\\ne j\\rbrace ,$ where $e_{i,j}:=\\frac{\\mathbb {1}_i-\\mathbb {1}_j }{ d(i,j)}.$ This polytope is contained (and full-dimensional) in the subspace $V_0(X) = \\lbrace x\\in \\mathbb {R}^X \\mid \\textstyle {\\sum }_i{x_i}=0\\rbrace .$ Definition 2.4 Let $(X,d)$ be a (finite) pseudometric space.",
"The Lipschitz polytope of $(X,d)$ is given as an intersection of halfspaces by $\\operatorname{LIP}(X,d):=\\left\\lbrace x\\in \\mathbb {R}^X \\mid \\textstyle {\\sum }_i{x_i}=0, \\, x_i-x_j\\le d(i,j) \\,\\,\\forall i,j \\in X\\right\\rbrace .$ This polytope is contained (and full-dimensional) in the subspace $V(X,d):=\\lbrace x\\in \\mathbb {R}^X \\mid \\textstyle {\\sum }_i{x_i}=0,\\,\\,x_i=x_j\\textrm { whenever }d(i,j)=0\\rbrace .$ Remark 2.5 (On Lipschitz polytopes) For metric spaces our definition specializes to the standard definition of the Lipschitz polytope, e.g., as given in [21].",
"We remark that, although related, this is not the set of Lipschitz functions considered in the work of Wu, Xu and Zhu on graph indexed random walks [22].",
"Remark 2.6 (On polytopes) We point the reader to the book by Ziegler [24] for terminology and basic facts about polytopes and fans.",
"Here let us only mention that the combinatorics of a given polytope $P$ is encoded in its poset of faces ${F}(P)$ which, here, we take to be the set of all faces of $P$ including the empty face ordered by inclusion.",
"A rougher, but very important enumerative invariant of a polytope are its face numbers $f_0^P,\\ldots ,f_{\\operatorname{dim}(P)}^P$ , where $f_i^P = \\vert \\lbrace i-\\textrm {dimensional faces of }P\\rbrace \\vert .$ It is customary to consider the empty face as a face of “dimension $-1$ ”, thus to write $f_{-1}^P = 1$ and to fit these numbers into the $f$ -polynomial of $P$ , defined as $f^P(t):= f_{-1}^P t^{m+1} + f_{0}^P t^{m} + \\ldots + f_{m}^P $ where we write $m:=\\operatorname{dim}(P)$ .",
"The problem posed by Vershik [21] is to study the face numbers and face structure of the fundamental polytope of a metric space.",
"We will do so by focussing on the associated Lipschitz polytope, whose combinatorics is \"dual\" to that of the fundamental polytope in the following precise sense.",
"Remark 2.7 A look at Theorem 2.11.",
"(vi) of [24] shows that indeed, for every metric space $(X,d)$ the polytopes $P_d(X)$ and $\\operatorname{LIP}(X,d)$ are polar dual to each other (with respect to the ambient space $V_0(X)$ , cf.",
"[24]).",
"Polar duality induces an isomorphism of posets ${F}(P_d(X)) \\cong {F}(\\operatorname{LIP}(X,d))^{op}$ and in particular $f_i^{P_d(X)} = f_{m-1-i}^{\\operatorname{LIP}(X,d)}$ , i.e., $f^{\\operatorname{LIP}(X,d)}(t) = t^{m+1} f^{P_d(X)}\\left(\\frac{1}{t}\\right)$ where again we write $m:=\\operatorname{dim}(P_d(X))$ ."
],
[
"Metric spaces associated to graphs",
"In this paper we will consider only finite, connected and simple graphs (i.e., without parallel edges or loops).",
"Given a graph $G$ , write $V(G)$ and $E(G)$ for the set of vertices, resp.",
"edges of $G$ .",
"We assume familiarity with the basics of graph theory, and point to any textbook (e.g., [6]) as a reference.",
"Example 2.8 (Metric spaces from weighted graphs) A weighting of a graph $G$ is any function $w: E(G) \\rightarrow \\mathbb {R}_{>0}$ , and the pair $(G,w)$ is called a weighted graph.",
"Then, setting $d_w(v,v^{\\prime }):=\\min \\left\\lbrace w(e_1)+\\ldots + w(e_k) \\mid e_1,\\ldots ,e_k \\textrm { an edge-path joining }v\\textrm { with } v^{\\prime }\\right\\rbrace $ the pair $(V(G),d_w)$ is a metric space.",
"We now introduce a class of metric spaces that are related to graphs in a subtler manner.",
"Recall that a tree is a graph in which every pair of vertices is connected by a unique path.",
"Definition 2.9 (Tree-like metrics) Let $X$ be a finite set.",
"An $X$ -tree is a pair $(T,\\phi )$ , where $T$ is a tree and $\\phi :X\\rightarrow V(T)$ is a map whose image contains every vertex of $V$ that is incident to at most two edges, i.e., $\\lbrace v\\in V(T) \\mid \\operatorname{deg}(v) \\le 2 \\rbrace \\subseteq \\phi (X) $ .",
"A (pseudo)metric $d$ on a set $X$ is called a tree-like (pseudo)metric if there exists an $X$ -tree $(T,\\phi )$ and a weighting $w$ of $T$ such that for all $x,y\\in X$ $d(x,y)=d_w(\\phi (x),\\phi (y)).$ (“$d$ is induced by a weighted $X$ -tree”).",
"The pseudometric $d$ is a metric if and only if $\\phi $ is injective.",
"When $\\phi $ is bijective, we call $(X,d)$ a full tree."
],
[
"Arrangements of hyperplanes",
"Let $V$ denote a finite-dimensional real vectorspace, say of dimension $m$ .",
"A hyperplane in $V$ is any linear subspace of codimension 1.",
"An arrangement of hyperplanes (or, for short, arrangement) in $V$ , is a finite set ${A}:=\\lbrace H_1,\\ldots ,H_n \\rbrace $ of hyperplanes.",
"Such an arrangement defines a polyhedral fan in $V$ , and we let ${F}({A})$ denote the poset of all faces of this fan, partially ordered by inclusion.",
"As is customary, for every $i=0,\\ldots , m$ we let $f^{{A}}_i$ be the number of faces of dimension $i$ , $f^{{A}}_i := \\vert \\left\\lbrace F\\in {F}({A}) \\mid \\operatorname{dim}(F)=i \\right\\rbrace \\vert $ and we arrange these numbers into the $f$ -polynomial of ${A}$ , $f^{{A}}(t) := f^{{A}}_0t^m + f^{{A}}_1t^{m-1}+\\ldots + f^{{A}}_m.$ Our next tool is a theorem of Zaslavsky which expresses the polynomial $f^{{A}}(t)$ in terms of the intersection pattern of ${A}$ .",
"To this end, we introduce the poset of intersections of ${A}$ , i.e., the set $L({A}) :=\\lbrace \\cap B\\mid B\\subseteq {A}\\rbrace ,\\quad x\\le y \\Leftrightarrow x\\supseteq y$ of all subspaces that arise as intersections of hyperplanes in ${A}$ , partially ordered by reverse inclusion.",
"We note that, in particular, $L({A})$ has a unique maximal element (corresponding to $\\cap {A}$ ) and a unique minimal element (corresponding to $V$ , the intersection over the empty set).",
"The poset $L({A})$ is ranked by the function $\\operatorname{rk}(x):= m - \\operatorname{dim}(x).$ and we define the rank of ${A}$ to be $r:=\\operatorname{rk}(\\cap {A})$ We define the Möbius polynomial of ${A}$ as $M_{{A}}(u,v):=\\sum _{x, y \\in L({A})} \\mu (x,y)u^{\\operatorname{rk}(x)}v^{r-\\operatorname{rk}(y)}$ where $\\mu $ denotes the Möbius function of $L({A})$ (see e.g.",
"[19]).",
"Theorem 2.10 (Zaslavsky [23]) $f^{{A}}(x)=(-1)^{r}\\, M_{{A}}(-x,-1).$ The strength of this result comes at the price of having to compute the Möbius function of $L({A})$ .",
"As we will see in Section REF , this can be a laborious and difficult task.",
"One can still recover some of the information with less effort, through the characteristic polynomial of ${A}$ , defined as follows.",
"$\\chi _{A}(t) := \\sum _{\\mathcal {K} \\subseteq {A}} (-1)^{\\vert \\mathcal {K} \\vert } t^{r - \\operatorname{rk}(\\cap \\mathcal {K})} $ Corollary 2.11 (Zaslavsky [23]) $f_m^{{A}} = (-1)^r\\chi _{A}(-1)$ Elementary manipulations show that indeed $\\chi _{A}(t) = \\sum _{x\\in L({A})} \\mu (V,x) t^{r-\\operatorname{rk}(x)} = M_{{A}}(0,t)$ hence the result follows immediately from Theorem REF , because $f_m^{{A}}=f^{{A}}(0)$ .",
"Remark 2.12 The combinatorial structure lurking in the back of these considerations is the matroid of the arrangement ${A}$ .",
"This is an abstract combinatorial object that encodes the structure of the poset $\\mathcal {L}({A})$ or, equivalently, the linear dependency relations among (any choice of) normal vectors of the hyperplanes.",
"We will not formally introduce these objects because they are not strictly necessary fin the context of this paper (except for a better understanding of the example in Section REF ), and refer the interested reader e.g.",
"to Oxley's textbook [16]."
],
[
"Zonotopes",
"Associated to every set of nonzero real vectors $v_1,\\ldots , v_k \\in \\mathbb {R}^m \\setminus \\lbrace 0\\rbrace $ there is a polytope obtained as the Minkowski sum $Z(v_1,\\ldots ,v_k):=\\sum _{i=1}^k [-1,1]v_i$ where $[-1,1] \\subseteq \\mathbb {R}$ denotes the 1-dimensional unit cube (the Minkowski sum of subsets of a vectorspace is their “pointwise sum”, see [24] for a precise definition).",
"Polytopes of this form are called zonotopes.",
"Strongly related to a zonotope is the arrangement of normal hyperplanes to the $v_i$ , i.e., ${A}:=\\lbrace v_i^\\perp \\mid i=1,\\ldots ,k\\rbrace $ .",
"For instance, one readily sees that $r=\\operatorname{rk}(\\cap {A}) = \\operatorname{dim}Z(v_1,\\ldots ,v_k).$ Moreover, we will make use of the well-known (see, e.g., [24]) isomorphism of posets ${F}({A})^{\\textrm {op}} \\cong {F}(Z(v_1,\\ldots , v_k))\\setminus \\lbrace \\emptyset \\rbrace $ which in particular implies that the $f$ -polynomials are related by the following equality.",
"$f^{Z(v_1,\\ldots ,v_k)}(t) - t^{r+1}= t^{2m-r} f^{{A}}(\\frac{1}{t})$ Remark 2.13 The hyperplane arrangement is invariant up to non-zero rescaling of the defining vectors.",
"Thus, the poset of faces of the zonotope also does not change upon rescaling of the $v_i$ ."
],
[
"Split systems",
"We now introduce a class of pseudometric spaces that has arisen from research in biology, and especially genomics.",
"As a comprehensive reference and for more context on the applied side we point to [18], whose terminology we keep in order to avoid confusion.",
"Definition 2.14 Let $X$ be a finite set.",
"A split of $X$ is a bipartition of $X$ , i.e., a pair of nonempty and disjoint subsets $A,B\\subseteq X $ (the sides of the split) such that $A\\cup B = X$ .",
"Such a pair $A,B$ will be written $A|B$ .",
"Clearly, $A\\vert B$ and $B\\vert A$ describe the same split.",
"In fact, every split $\\sigma =A\\vert B$ corresponds to a nontrivial equivalence relation $\\sim _\\sigma $ on $X$ , whose equivalence classes are $A$ and $B$ .",
"Given a split $\\sigma $ and any element $i\\in X$ we write $[i]_\\sigma $ for the equivalence class of $i$ with respect to the equivalence relation $\\sim _\\sigma $ .",
"Thus, to any split $\\sigma $ we can associate the indicator function $\\delta _{\\sigma }$ defined as $\\delta _{\\sigma }(i,j)={\\left\\lbrace \\begin{array}{ll} 0 &i \\sim _\\sigma j \\\\1 & \\mbox{otherwise.}",
"\\end{array}\\right.",
"}$ A split $\\sigma $ is customarily called trivial if one of its sides is a singleton.",
"We will use the shorthand $\\sigma =k\\vert k^c$ in order to denote a trivial split whose singleton side contains only the element $k$ .",
"Two splits $A\\vert B$ and $C\\vert D$ are compatible if at least one of the sets $A\\cap C,\\quad A\\cap D,\\quad B\\cap C,\\quad B\\cap D$ is empty.",
"A system of splits on $X$ is just a set of splits of $X$ ; the system is called compatible if its elements are pairwise compatible.",
"Definition 2.15 A weighted split system is a pair $(\\mathcal {S},\\alpha )$ where $\\mathcal {S}$ is a system of splits on $X$ and $\\alpha \\in (\\mathbb {R}_{\\ge 0})^{\\mathcal {S}}$ is any weighting.",
"Any such weighted split system defines a symmetric nonnegative function $d_{\\alpha }: X\\times X \\rightarrow \\mathbb {R}$ via $d_\\alpha (x,y)=\\sum _{\\sigma \\in \\mathcal {S}} \\alpha _\\sigma \\delta _{\\sigma }(x,y)$ where $\\delta _\\sigma $ is as in Equation (REF ).",
"The functions of the form $d_\\alpha $ are called split-decomposable pseudometrics associated to $\\mathcal {S}$ .",
"In fact, the pair $(X,d_\\alpha )$ is a pseudometric space.",
"We will write $V(\\mathcal {S}) := V(X,d_\\alpha )$ as this subspace clearly does not depend on $\\alpha $ .",
"A positively weighted split system is one where $\\alpha _\\sigma > 0 $ for all $\\sigma \\in \\mathcal {S}$ .",
"Remark 2.16 Such metric spaces are also known as cut (pseudo)metrics [5].",
"Theorem 2.17 (See [18]) Let $(X,d)$ be a pseudometric space.",
"The following are equivalent: $d$ satisfies the “four point condition”: for all $x,y,z,w\\in X$ , $d(x,y)+d(z,w)\\le \\max \\vert \\left\\lbrace d(x,z)+d(y,w),d(x,w)+d(z,y)\\right\\rbrace \\vert $ $d$ is a tree-like pseudo-metric on $X$ (in the sense of Definition REF ).",
"$d$ is a split-decomposable pseudometric associated to a positively weighted system of compatible splits.",
"Moreover, this system is unique.",
"Remark 2.18 Under the equivalence of (ii) with (iii), splits in the decomposition of the metric correspond bijectively to edges in the tree."
],
[
"Arrangements associated to split systems",
"Definition 2.19 Let $X$ be a finite set and consider a split $\\sigma =A\\vert B$ of $X$ , where $|X|=n$ .",
"To $\\sigma $ we associate the line segment (one-dimensional polytope) $S_\\sigma := \\operatorname{conv}\\left\\lbrace \\frac{|B|}{n}\\cdot \\mathbb {1}_A-\\frac{|A|}{n}\\cdot \\mathbb {1}_B,\\frac{|A|}{n}\\cdot \\mathbb {1}_B-\\frac{|B|}{n}\\cdot \\mathbb {1}_A \\right\\rbrace \\subseteq V(\\mathcal {S}) \\subseteq \\mathbb {R}^X$ where $\\mathbb {1}_A:=\\sum _{x\\in A}\\mathbb {1}_x$ , as well as a hyperplane $H_\\sigma := (S_{\\sigma })^{\\perp }.$ Accordingly, we define the hyperplane arrangement associated to $\\mathcal {S}$ ${A}(\\mathcal {S}):=\\lbrace H_\\sigma \\mid \\sigma \\in \\mathcal {S}\\rbrace .$ and the corresponding zonotope, defined by the Minkowski sum $Z(\\mathcal {S}):= \\sum _{\\sigma \\in \\mathcal {S}} S_\\sigma .$ Remark 2.20 Both the arrangement ${A}(\\mathcal {S})$ and the zonotope $Z(\\mathcal {S})$ are full-rank, resp.",
"full-dimensional, inside the natural “ambient space” $V(\\mathcal {S})$ .",
"This means that $r:=\\operatorname{rk}({A}(\\mathcal {S}))=\\operatorname{dim}Z(\\mathcal {S}) = \\operatorname{dim}V(\\mathcal {S}) $ .",
"We can then rewrite Equation (REF ) as follows.",
"$f^{Z(\\mathcal {S})}(t) - t^{r+1}= t^{r} f^{{A}(\\mathcal {S})}(\\frac{1}{t})$ Remark 2.21 As was already mentioned in Remark REF , the abstract combinatorial object on which our enumerative considerations rest is $M(\\mathcal {S}): \\textrm { the matroid of }{A}(\\mathcal {S}).$ We would like to point out that this matroid is defined for every system of splits of a finite set.",
"Although at the moment we do not have any enumerative results beyond compatible split systems, this “matroid stratification” of the space of more general split-decomposable metrics opens many interesting theoretical questions (aside from its potential applications to phylogenetics).",
"Remark 2.22 Each of our hyperplanes $H_{\\sigma }$ has the form of an $(A,B,\\mu )$ -hyperplane as described in [9], for $\\mu = p \\vert B\\vert $ and $k=pn$ where $p$ is any positive integer.",
"However, such values of $\\mu ,k$ are excluded in [9]."
],
[
"Lipschitz polytopes of compatible systems of splits",
"Lemma 3.1 Let $(X,d_1)$ and $(X,d_2)$ be pseudometric spaces.",
"Then we have $\\operatorname{LIP}(X,d_1+d_2)\\supseteq \\operatorname{LIP}(X,d_1)+\\operatorname{LIP}(X,d_2)$ Recall the definition of the Lipschitz-polytope (Equation (REF )) and consider $x\\in \\operatorname{LIP}(X,d_1),y\\in \\operatorname{LIP}(X,d_2)$ By verifying the conditions of the definition of the Lipschitz polytope we see that $x+y\\in \\operatorname{LIP}(X,d_1+d_2)$ Corollary 3.2 Consider a finite set $X$ , a system $\\mathcal {S}$ of splits of $X$ and a nonnegative weighting $\\alpha \\in \\mathbb {R}^{X}_{\\ge 0}$ .",
"Let $(X,d_\\alpha )$ be the associated pseudometric space (see Definition REF ).",
"Then $\\operatorname{LIP}(X,d_\\alpha )\\supseteq \\sum _{\\sigma \\in \\mathcal {S}}\\alpha _{\\sigma }S_\\sigma .$ The claim follows by repeated application of Lemma REF .",
"Lemma 3.3 Let $(X,d)$ be a finite pseudometric space and let $A\\subseteq X$ be such that $d(i,j)=0$ for all $i,j\\in A$ .",
"Write $\\sigma :=A\\mid A^c$ .",
"Then, for every $\\alpha _\\sigma \\ge 0$ : $\\operatorname{LIP}(X,d+\\alpha _{\\sigma } \\delta _{\\sigma } )= \\operatorname{LIP}(X,d)+\\alpha _{\\sigma }S_\\sigma .$ The right-to-left containment is a special case of Lemma REF .",
"In order to check the left-to-right containment we consider a point $x\\in \\operatorname{LIP}(X,d+\\alpha _{\\sigma } \\delta _{\\sigma })$ and prove that it is contained in the right-hand side.",
"The definition of the Lipschitz polytope implies immediately that, for all $i,j\\in X$ $x_i-x_j \\le d(i,j)+\\alpha _\\sigma $ Define $\\alpha :=max_{i\\in A, j\\in A^c}\\lbrace 0, x_i-x_j -d(i,j), x_j-x_i - d(i,j)\\rbrace .$ If $\\alpha =0$ , then all defining equations for $\\operatorname{LIP}(X,d)$ are satisfied by $x$ , thus $x\\in \\operatorname{LIP}(X,d)$ and there is nothing to show.",
"Otherwise, choose $i_0,j_0$ such that $\\alpha = x_{i_0}-x_{j_0} - d(i_0,j_0)$ .",
"Assume w.l.o.g.",
"$i_0\\in A$ and $j_0\\in A^c$ (otherwise switch $A$ and $A^c$ in the following).",
"Since now $0\\le \\alpha \\le \\alpha _\\sigma $ (the first inequality by definition, the second by (REF ) above), it is enough to show that $y:=x - \\alpha v_{\\sigma }\\in \\operatorname{LIP}(X,d)$ where $v_\\sigma :=\\frac{|A^c|}{n}\\cdot \\mathbb {1}_A-\\frac{|A|}{n}\\cdot \\mathbb {1}_{A^c}$ .",
"We will do so by verifying that $y$ satisfies the equations of $\\operatorname{LIP}(X,d)$ (Equation REF ), noticing immediately that $\\sum y_i = 0$ .",
"Another immediate observation is that $y_i-y_j = x_i-x_j \\le d(i,j) + \\underbrace{\\alpha _\\sigma \\delta _\\sigma (i,j)}_{=0} \\textrm { when } i,j\\in A \\textrm { or } i,j\\in A^c.$ Thus we only have to consider the case $i\\in A$ , $j\\in A^c$ .",
"Then, $y_j-y_i = x_j - \\alpha \\frac{\\vert A\\vert }{n} -x_i- \\alpha \\frac{\\vert A^c \\vert }{n} = x_j - x_i - \\alpha \\le d(j,i)$ while $y_i-y_j = x_i + \\alpha \\frac{\\vert A\\vert }{n} -x_j+ \\alpha \\frac{\\vert A^c \\vert }{n} = x_j - x_i + \\alpha $ $= x_j \\underbrace{-x_i + x_{i_0}}_{\\begin{array}{c}=0 \\textrm { by (\\ref {eq_xx})}\\end{array}}- x_{j_0} - d(i_0,j_0) =\\underbrace{ x_j - x_{j_0}}_{\\begin{array}{c}\\le d(j,j_0)\\\\ \\textrm { by (\\ref {eq_xx})}\\end{array}}- d(i_0,j_0) \\le d(j,i_0)=d(j,i)$ where we used the triangle inequality for the pseudometric $d$ .",
"Remark 3.4 The decomposition theorem of Bandelt and Dress, [1] says that any metric $(X,d)$ can be uniquely decomposed into $d=d_0+\\sum _{\\sigma \\in \\mathcal {S}} \\alpha _\\sigma \\delta _\\sigma $ , where $d_0$ is split prime and $\\mathcal {S}$ is a (unique) system of splits.A metric is called “split prime” in [1] if it is not further decomposable with respect to split metrics.",
"In this sense, Lemma REF can be applied to general metrics.",
"Theorem 3.5 Let $(X,d)$ be a tree-like pseudometric space.",
"Then, $LIP(X,d)= \\sum _{\\sigma \\in \\mathcal {S}}\\alpha _{\\sigma }S_\\sigma $ where $(\\mathcal {S},\\alpha )$ is the unique weighted system of compatible splits of $X$ such that $d=d_\\alpha $ (cf.",
"Theorem REF ).",
"The proof is by induction on the cardinality of $\\mathcal {S}$ .",
"If $\\vert \\mathcal {S} \\vert =0$ there is nothing to prove.",
"Let then $\\vert \\mathcal {S} \\vert >0$ and suppose that the theorem holds for all weighted systems of compatible splits of smaller cardinality.",
"By Theorem REF to the space $(X,d)$ is associated a weighted $X$ -tree $(T,\\phi )$ in the sense of Definition REF .",
"The corresponding tree metric can be expressed as a split metric with a split for every edge in the tree.",
"The uniqueness part of Bandelt and Dress' decomposition theorem (see Remark REF ) says that the associated split system must be $\\mathcal {S}$ .",
"In particular, the tree $T$ has at least one edge and then a basic theorem of graph theory says that this tree must have at least one leaf vertex (i.e., a vertex incident to exactly one edge).",
"Choose then such a leaf vertex, say $v$ , and let $\\sigma \\in \\mathcal {S}$ be the split corresponding to the unique edge incident to $v$ .",
"Then, $\\sigma = A \\mid A^c \\textrm { with } A:=\\phi ^{-1}(v).$ Let $\\mathcal {S}^{\\prime }:=\\mathcal {S}\\setminus \\lbrace \\sigma \\rbrace $ and let $(X,d^{\\prime })$ be the pseudometric space defined by $\\mathcal {S}^{\\prime }$ and the appropriate restriction of $\\alpha $ .",
"Now notice that $d=d^{\\prime }+\\alpha _\\sigma \\delta _{\\sigma }$ and that, for all $i,j\\in A$ , we have $d^{\\prime }(i,j)=0$ .",
"By Lemma REF , then, $\\operatorname{LIP}(X,d) = \\operatorname{LIP}(X,d^{\\prime }) + \\alpha _\\sigma S_\\sigma $ and with the inductive hypothesis applied to $(X,d^{\\prime })$ we obtain $\\operatorname{LIP}(X,d) = \\left(\\sum _{\\sigma ^{\\prime } \\in \\mathcal {S}^{\\prime }} \\alpha _{\\sigma ^{\\prime }} S_{\\sigma ^{\\prime }}\\right) + \\alpha _\\sigma S_\\sigma = \\sum _{\\sigma \\in \\mathcal {S}} \\alpha _\\sigma S_\\sigma $ as required.",
"Theorem 3.6 Let $(X,d)$ be a tree-like pseudometric space with associated system of compatible splits $\\mathcal {S}$ .",
"Then the $f$ -vector of the associated Lipschitz polytope can be computed as follows.",
"$f^{\\operatorname{LIP}(X,d)}(x)=(-x)^{\\operatorname{rk}(\\cap {A}(\\mathcal {S}))} M_{{A}(\\mathcal {S})}\\left(-\\frac{1}{x},-1\\right) + x^{\\operatorname{rk}(\\cap {A}(\\mathcal {S}))+1}$ If additionally $(X,d)$ is a metric space, then the $f$ -vector of the associated fundamental polytope is $f^{P_d(X)}(x)=(-1)^{\\operatorname{rk}(\\cap {A}(\\mathcal {S}))} M_{{A}(\\mathcal {S})}(-x,-1)x+1$ Theorem REF implies that ${F}(\\operatorname{LIP}(X,d)) \\simeq {F}(Z(\\mathcal {S}))$ with which we can write $f^{\\operatorname{LIP}(X,d)}(x)=f^{Z(\\mathcal {S})}(x)=x^{\\operatorname{rk}(\\cap {A}(\\mathcal {S}))}f^{{A}(\\mathcal {S})}(\\frac{1}{x}) + x^{\\operatorname{rk}(\\cap {A}(\\mathcal {S}))+1}$ $=(-x)^{\\operatorname{rk}(\\cap {A}(\\mathcal {S}))} M_{{A}(\\mathcal {S})}(-\\frac{1}{x},-1)+ x^{\\operatorname{rk}(\\cap {A}(\\mathcal {S}))+1}$ where the second equality follows from Remark REF and the last equality is Theorem REF .",
"This proves the first of the claimed equalities.",
"The second follows by the duality relation of Remark REF .",
"Corollary 3.7 For any tree-like metric space $(X,d)$ $f_i^{P_d(X)} = f_{i+1}^{{A}(\\mathcal {S})} = f_{|X|-1-i}^{\\operatorname{LIP}(X,d)}$ where $\\mathcal {S}$ denotes the associated system of (compatible) splits and the index $i$ runs from 0 to $\\operatorname{dim}(P_d(X))=|X|-1$ ."
],
[
"Computational aspects and examples",
"Our previous theoretical considerations have led us to explicit formulas for the $f$ -vectors of the polytopes associated to compatible split systems.",
"The question is now, how practical it is to compute these numbers for specific examples.",
"We will start with two easy cases and then offer a general tool allowing to compute the intersection lattice of the associated hyperplane arrangement, which we'll test on a class of examples.",
"Example 4.1 (Points in $\\mathbb {R}^1$ ) We can represent the metric space defined by any set of $n$ points in $\\mathbb {R}^1$ by just taking its metric graph in a line, considering the associated set of splits and choosing the coefficients in the split-metric accordingly.",
"The arrangement corresponds to $(n-1)$ independent vectors in $n-1$ -dimensional space, i.e.",
"it is isomorphic to the coordinate arrangement.",
"The corresponding matroid is the uniform matroid $\\mathcal {U}_{n-1}^{n-1}$ and, in particular, $f^{{A}}_i=2^{i}\\binom{n-1}{i}$ .",
"Example 4.2 (The root polytope of type $A_{n-1}$ ) Let us consider a star graph, i.e., a tree with $n>2$ leaves and a unique internal vertex .",
"If we assign each edge the length $\\frac{1}{2}$ , we define the structure of a metric space on the set $X$ of leaves of our star graph.",
"The corresponding split system consists exactly of all the elementary splits, and any two points are at distance 1.",
"Then, by definition, the fundamental polytope of this space is the convex hull of the vectors $e_{i,j}=\\mathbb {1}_i-\\mathbb {1}_j$ , where $i\\ne j \\in [n]$ .",
"This is also called the root polytope of type $A_{n-1}$, and its face numbers have been computed via algebraic-combinatorial considerations by Cellini and Marietti [3].",
"Of course, one could compute these numbers by computing the Möbius function of the corresponding matroid, i.e., the uniform matroid $\\mathcal {U}_{n}^{n-1}$ ."
],
[
"The intersection lattice of ${A}(\\mathcal {S})$",
"Definition 4.3 Let $(X,d)$ be a pseudometric space.",
"The function $d$ induces a partition $\\pi (d)$ of the set $X$ given as the set of equivalence classes of the equivalence relation defined by: $i\\sim _d j$ if and only if $d(i,j)=0$ .",
"If the space $(X,d)$ arises from a positively weighted system of splits $(\\mathcal {S} , \\alpha )$ , the partition $\\pi (d)$ does not depend on $\\alpha $ and we only write $\\pi (\\mathcal {S})$ .",
"We have an order preserving map of posets $\\pi : 2^{\\mathcal {S}}\\rightarrow \\Pi _X; \\quad \\quad \\mathcal {S}^{\\prime }\\mapsto \\pi (\\mathcal {S}^{\\prime })$ where $2^{\\mathcal {S}}$ denotes the poset of all subsets of $\\mathcal {S}$ ordered by inclusion, and $\\Pi _X$ is the poset of all partitions of $X$ ordered by refinement.",
"Theorem 4.4 Let $(\\mathcal {S},\\alpha )$ be an arbitrary weighted system of compatible splits of a finite set $X$ and write $\\pi :=\\pi (\\mathcal {S})$ .",
"Then, $\\cap _{\\sigma \\in \\mathcal {S}} H_{\\sigma }= \\big \\langle e_{i,j} : i\\sim _{\\pi } j \\big \\rangle .",
"$ The right-to-left inclusion holds by definition.",
"We will prove the left-to-right inclusion by induction on the cardinality of the system of splits.",
"If $\\vert \\mathcal {S} \\vert = 1$ the claim is evident.",
"Let then $m>0$ , assume that the statement holds for any weighted system of up to $m$ compatible splits and consider a weighted system of splits $(\\mathcal {S},\\alpha )$ with $\\vert \\mathcal {S} \\vert = m+1$ .",
"By Theorem REF , $(\\mathcal {S},\\alpha )$ can be represented by an $X$ -tree $(T,\\phi )$ with at least one edge, hence with at least one leaf vertex $v$ .",
"In particular, with $A:=\\phi ^{-1}(v)$ , we know that $\\sigma :=A\\vert A^c \\in \\mathcal {S}$ and we can consider $\\mathcal {S}^{\\prime }:= \\mathcal {S} \\setminus \\lbrace \\sigma \\rbrace ,\\quad \\quad \\alpha ^{\\prime }:= \\alpha _{\\vert \\mathcal {S}^{\\prime }}, \\quad \\quad d^{\\prime }:=d_{\\alpha ^{\\prime }}, \\quad \\quad \\pi ^{\\prime }:=\\pi (\\mathcal {S}^{\\prime }).$ The $X$ -tree $(T^{\\prime },\\phi ^{\\prime })$ associated to $(\\mathcal {S}^{\\prime }, \\alpha ^{\\prime })$ must have a vertex $v^{\\prime }$ with $\\phi ^{\\prime }(A)=v^{\\prime }$ (otherwise there would be $i,j\\in A$ with $d_\\alpha (i,j)\\ge d_\\alpha ^{\\prime }(i,j)>0$ ).",
"In a neighborhood of $v^{\\prime }$ , the $X$ -trees associated to $(\\mathcal {S}^{\\prime },\\alpha ^{\\prime })$ , resp.",
"$(\\mathcal {S},\\alpha )$ , differ as in Figure REF .",
"In particular, ($\\ast $ ) $\\quad $ $\\pi = \\lbrace A,C,\\pi _1,\\ldots ,\\pi _k\\rbrace $ ; $\\quad \\quad $ $\\pi ^{\\prime }=\\lbrace A\\sqcup C,\\pi _1,\\ldots ,\\pi _k \\rbrace $ (here and in the following we think of a partition as a set of blocks) Figure: The neighborhood of the vertex v ' v^{\\prime } in the XX-tree T ' T^{\\prime } (left-hand side) and TT (right-hand side).Let $v_{\\sigma }:=\\frac{|A|}{n}\\cdot \\mathbb {1}_{A^c}-\\frac{|A^c|}{n}\\cdot \\mathbb {1}_{A}$ so that $(v_\\sigma )^\\perp = H_\\sigma $ .",
"By induction hypothesis $\\bigcap _{\\tau \\in \\mathcal {S}} H_{\\tau } = \\bigcap _{\\tau \\in \\mathcal {S}^{\\prime }} H_{\\tau } \\cap H_\\sigma =\\langle e_{i,j}: i\\sim _{\\pi ^{\\prime }}j \\rangle \\cap (v_\\sigma )^\\perp $ In view of $(\\ast )$ , the subspace $\\langle e_{i,j}: i\\sim _{\\pi ^{\\prime }}j \\rangle $ decomposes as $\\bigoplus _{b\\in \\pi ^{\\prime }}\\left\\langle e_{i,j}\\mid i,j\\in b \\right\\rangle =\\left\\langle e_{i,j}\\mid i,j\\in A\\sqcup C \\right\\rangle \\oplus \\underbrace{\\bigoplus _{b\\in \\pi ^{\\prime }\\setminus \\lbrace A\\sqcup C\\rbrace }\\left\\langle e_{i,j}\\mid i,j\\in b \\right\\rangle }_{=:W}$ A straightforward check shows that $W\\subseteq (v_\\sigma )^{\\perp }$ .",
"Therefore, the intersection of this direct sum with $(v_\\sigma )^\\perp $ equals $\\left(\\left\\langle e_{i,j}\\mid i,j\\in A\\sqcup C \\right\\rangle \\cap (v_\\sigma )^\\perp \\right) \\oplus W $ .",
"On the other hand, $\\left\\langle e_{i,j}\\mid i,j\\in A\\sqcup C \\right\\rangle \\cap (v_\\sigma )^\\perp =\\left\\langle e_{i,j}\\mid i,j\\in A \\right\\rangle \\oplus \\left\\langle e_{i,j}\\mid i,j\\in C \\right\\rangle $ Thus, we can rewrite the right-hand side of Equation (REF ) as $\\left\\langle e_{i,j}\\mid i,j\\in A \\right\\rangle \\oplus \\left\\langle e_{i,j}\\mid i,j\\in C \\right\\rangle \\oplus W$ and in particular, recalling the block structure of $\\pi $ from ($\\ast $ ), $\\langle e_{i,j}\\mid i\\sim _{\\pi ^{\\prime }} j \\rangle \\cap (v_\\sigma )^\\perp = \\langle e_{i,j}\\mid i\\sim _{\\pi } j \\rangle $ which, together with Equation (REF ), concludes the proof.",
"Corollary 4.5 There is a poset isomorphism ${L} ({A}(\\mathcal {S})) \\simeq \\operatorname{im}\\pi $ where the right-hand is considered as an induced sub-poset of $\\Pi _X$ .",
"More precisely, if we identify the set of elements of the matroid of ${A}(\\mathcal {S})$ with $\\mathcal {S}$ itself, we can write the closure operator of the matroid as $\\operatorname{cl (\\mathcal {S}^{\\prime })} = \\lbrace \\mathcal {S}^{\\prime \\prime } \\subseteq \\mathcal {S} \\mid \\pi (\\mathcal {S}^{\\prime \\prime })=\\pi (\\mathcal {S}^{\\prime })\\rbrace .$ Example 4.6 (Full trees) If $(X,d)$ is a “full tree” (in the sense of Definition REF ), then it can be represented by an $X$ -tree where each vertex is labeled by exactly one point of $X$ .",
"Therefore it is apparent that $\\pi $ is injective, and thus the poset $\\mathcal {L} ({A}(\\mathcal {S}))$ is boolean.",
"Immediately we recover $f_i^{P_d(X)} = 2^i{n-1 \\atopwithdelims ()i},$ generalizing, as expected, Example REF ."
],
[
"Caterpillar graphs",
"As a further motivation for the usefulness of the matroidal point of view, let us consider the tree metrics whose underlying $X$ -tree is a caterpillar graph (see Figure REF ) with every leaf labelled by exactly one point of $X$ , and no internal vertices labelled.",
"Figure: The nn-caterpillar graphIf we consider some $\\mathcal {S}^{\\prime }\\subseteq \\mathcal {S}$ , we see that the intersection $\\bigcap _{\\sigma \\in \\mathcal {S}^{\\prime }} H_\\sigma $ is contained in $H_{\\tau }$ if and only if there are $\\sigma _1,\\sigma _2\\in \\mathcal {S}^{\\prime }$ such that $\\sigma _1,\\sigma _2,\\tau $ correspond to three edges incident to the same (inner) vertex.",
"The same rule describes the closure of the set $\\mathcal {S}^{\\prime }$ in the matroid ${M}(\\mathcal {S})$ which, thus, is easily seen as being isomorphic to the graphic (“cycle-”) matroid of the graph $\\mathcal {G}_n$ depicted in Figure REF .",
"Figure: The graph 𝒢 n \\mathcal {G}_nTable REF shows the $f$ -polynomials of the fundamental polytopes of these metric spaces for the first few values of $n$ , as computed with SAGE.",
"It took around 10 seconds to compute the $f$ -polynomial of the biggest example, the 6-caterpillar graph, on the sage cloud (run on a free server).",
"We can, however, give a formula for the number of vertices of $\\operatorname{LIP}(X,d)$ (which equals the number of facets of $P_d(X)$ ).",
"In fact, by Corollary REF , both these numbers equal $(-1)^{\\operatorname{rk}({M})}\\chi _{{M}(\\mathcal {S})}(-1)$ and we know that the rank of a graphic matroid is the cardinality of any spanning forest in the graph.",
"In our case, every spanning tree of $\\mathcal {G}_n$ has cardinality $n-1$ .",
"Moreover, the characteristic polynomial of ${M}(\\mathcal {S})$ can be computed as $\\chi _{{M}(\\mathcal {S})}(t) = \\frac{1}{t}\\operatorname{chr}_{\\mathcal {G}_n}(t)$ where the polynomial on the right-hand side is the chromatic polynomial of $\\mathcal {G}_n$ .",
"Now, the latter polynomial can be computed easily by looking at the graph and remembering that for every integer $k$ the value $\\operatorname{chr}_{\\mathcal {G}_n}(k)$ is the number of $k$ -colorings of $\\mathcal {G}_n$ .",
"This gives immediately $\\operatorname{chr}_{\\mathcal {G}_n}(t) = t (t-1)(t-2)^{n-2}\\quad \\quad \\textrm { for }n\\ge 3$ (for $n< 3$ we are in the case of Example REF ).",
"Thus we obtain that, if $(X,d)$ is any metric space on $n\\ge 3$ elements representable as leafs of a metric $n$ -caterpillar tree, $\\textrm { number of vertices of }\\operatorname{LIP}(X,d) = (-1)^{n-1}(-2)(-3)^{n-2}=2\\cdot 3^{n-2}.$ Table: NO_CAPTIONFigure: NO_CAPTION"
]
] | 1612.05534 |
[
[
"Continuous fuzzy measurement on two-level systems revisited"
],
[
"Abstract Imposing restrictions on the Feynman paths of the monitored system has in the past been proposed as a universal model-free approach to continuous quantum measurements.",
"Here we revisit this proposition, and demonstrate that a Gaussian restriction, resulting in a sequence of many highly inaccurate (weak) von Neumann measurements, is not sufficiently strong to ensure proximity between a readout and the Feynman paths along which the monitored system evolves.",
"Rather, in the continuous limit, the variations of a typical readout become much larger than the separation between the eigenvalues of the measured quantity.",
"Thus, a typical readout is not represented by a nearly constant curve, correlating with one of the eigenvalues of the measured quantity $\\a$, even when decoherence, or Zeno effect are achieved for the observed two-level system, and does not point directly to the system's final state.",
"We show that the decoherence in a \"free\" system can be seen as induced by a Gaussian random walk with a drift, eventually directing the system towards one of the eigenstates of $\\a$.",
"A similar mechanism appears to be responsible for the Zeno effect in a driven system, when its Rabi oscillations are quenched by monitoring.",
"Alongside the Gaussian case, which can only be studied numerically, we also consider a fully tractable model with a \"hard wall\" restriction, and show the results to be similar."
],
[
"Introduction ",
"Almost twenty years ago Audretsch and Mensky [1] considered continuous measurements performed on a two level system by means of a device which restricts virtual (Feynman) paths of the system according to the observed readout $f(t)$ .",
"The closeness of a path to the readout is measured by the time average of the square of the deviation of the path from $f(t)$ .",
"The allowed deviation is determined by the resolution of the device.",
"The authors suggested that the proximity of the Feynman paths to a registered readout would allow one to read off the state vector of the system directly from $f(t)$ .",
"They also predicted a rapid decoherence of a pure initial state if the measured quantity $\\hat{A}$ commutes with the Hamiltonian $\\hat{H}$ of the system, and formulated the conditions for the Zeno effect in case the two do not commute.",
"The analysis of [1], and its continuation in [2], is based on a more general approach [3]-[5], which advocated the restricted path integrals of the described type as a universal model for the decoherence typically caused by a wide class of environments and measuring devices.",
"More recent work on the formalism can be found in [6].",
"The general subject of continuos quantum measurements is reviewed, for example, in [7]-[11], with a recent pedagogical version given in [10].",
"The purpose of this paper is to re-examine both propositions of [1].",
"In particular, we will show that the Gaussian restriction imposed on Feynman paths in [1] cannot guarantee their closeness to readouts which, in the continuous limit, tend to become infinite, rather than lie close to one of the eigenvalues of $\\hat{A}$ .",
"With this, the estimates of the decoherence rates and Zeno times, based on the properties of constant readouts which align with one of the eigenvalues of $\\hat{A}$ [1] become inconclusive.",
"We will, therefore, look for a different decoherence mechanism in a \"free\" system, and a different reason for a Zeno effect in a driven one.",
"The rest of the paper is organised as follows.",
"In Sect.",
"II we will briefly re-derive the basic equations for a \"measurement medium\" consisting of a large number of highly inaccurate von Neumann meters.",
"In Sect.",
"III we will show on a simple example that as the continuous limit is approached, a typical readout would alternate on an ever larger scale, which will eventually become infinite.",
"In section IV we briefly revisit the formulation of the problem based on a Schroedinger equation with a non-Hermitian Hamiltonian.",
"In Section V we consider decoherence in the simplest case of a \"free\" system.",
"Section VI analyses the Zeno effect in a \"driven\" system, where continuous monitoring quenches the Rabi oscillations.",
"Our conclusions are in Sect.",
"VII."
],
[
"Restricted path integrals ",
"Perhaps the simplest way to arrive at the Mensky's equations [1] is to consider a set of $K$ identical von Neumann meters, with positions $f_k$ , acting on the system after equal intervals at $t_k=k\\tau $ , $k=1,..,K$ , where $\\tau =T/K$ , and $T$ is the duration of the monitoring.",
"Each meter is coupled to the system via (we use $\\hbar =1$ ) $\\hat{H}_{int}=-i\\partial _{f_k}\\hat{A}\\delta (t-t_k)$ , where an operator $\\hat{A}$ represents the measured quantity, and $\\delta (x)$ is the Dirac delta.",
"The system starts in an initial state $|\\psi _0{\\rangle }=\\alpha _0 |a_1{\\rangle }+\\beta _0 |a_2{\\rangle }, \\quad \\hat{A}|a_i{\\rangle }=a_i|a_i{\\rangle },\\quad i=1,2,\\quad $ with the meters prepared in the same states $|M_k{\\rangle }$ , such that $G(f_k)= {\\langle }f_k|M_k{\\rangle }$ is a real function, which peaks around $f_k=0$ , has a width $\\Delta f$ , and vanishes rapidly as $|f_k|\\rightarrow \\infty $ .",
"We note that $G(f)$ determines the initial (quantum) uncertainty of the pointer's position.",
"Since the position is determined accurately after the measurement, it determines also the measurement's accuracy, which is high for small, and low for large values of $\\Delta f$ , respectively.",
"The meters are read immediately, so that just before $t=t_k$ , the results $f_i$ , $i=1,2,..,k-1$ are known, and the state of the system is $|\\psi _{k-1}{\\rangle }=\\alpha _{k-1}(f_1, ..,f_{k-1}) |a_1{\\rangle }+\\beta _{k-1}(f_1, ..,f_{k-1}) |a_2{\\rangle }$ .",
"The $k$ -th meter interacts with the system, turning a product state into an entangled one, $|\\psi _{k-1}{\\rangle }G(f_k) \\rightarrow \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\\\ \\nonumber \\alpha _{k-1} G(f_k-a_1) |a_1{\\rangle }+\\beta _{k-1}G(f_k-a_2) |a_2{\\rangle }.\\quad \\quad $ Thus, if a complete observed readout is $\\underline{f}=(f_1,f_2,...,f_K)$ , the system undergoes an evolution with a non-unitary operator $\\hat{U}(T,\\underline{f})=\\prod _{k=1}^KG(f_k-\\hat{A})\\exp (-i\\hat{H}\\tau )$ , $|\\psi _{K}(\\underline{f}){\\rangle }=\\prod _{k=1}^KG(f_k-\\hat{A})\\exp (-i\\hat{H}\\tau )|\\psi _0{\\rangle },\\quad \\quad $ where $\\hat{H}$ is the system's own Hamiltonian.",
"Suppose we have a set of Gaussian meters, $G(f)=C^{-1/2}\\exp (-f^2/2\\Delta f^2), \\quad C= (\\pi \\Delta f^2)^{1/2},$ and send $\\Delta f \\rightarrow \\infty $ , so that each measurement becomes highly inaccurate or \"weak\", does not perturb the system's evolution, and cannot give us much information about the value of $\\hat{A}$ [12].",
"However, if the number of such measurements is also increased, the combined effect on the system may be considerable.",
"In particular, we can choose [13] $\\tau \\rightarrow 0, \\quad \\Delta f\\rightarrow \\infty , \\quad 2\\tau \\Delta f ^2 = \\kappa ^{-1} = const.$ Now, by the Lie-Trotter's formula, [14], we also have $G(f_k-\\hat{A})\\exp (-i\\hat{H}\\tau ) \\rightarrow \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\\\ \\nonumber C^{-1/2}exp\\lbrace -i[\\hat{H}-i\\kappa (f_k-\\hat{A})^2]\\tau \\rbrace ,\\quad \\quad $ even when $\\hat{A}$ and $\\hat{H}$ do not commute.",
"If a discrete readout, $\\underline{f}$ , is replaced by a continuos one, $f(t)$ , the product over $k$ in Eq.",
"(REF ) becomes proportional to the evolution operator $\\hat{U}(T, f(t))$ for a time dependent non-Hermitian Hamiltonian $\\hat{H}^{\\prime }=\\hat{H}-i\\kappa (f(t)-\\hat{A})^2$ .",
"The probability to obtain a readout $f$ is now given by a functional $W[f]={\\langle }\\psi (T,[f])|\\psi (T,[f]){\\rangle }, \\quad \\int Df W[f(t)]=1,$ where $Df = lim_{\\tau \\rightarrow 0} \\frac{df_1}{C}\\frac{df_2}{C}...\\frac{df_K}{C}$ also determines the normalisation of $|\\psi (T,[f]){\\rangle }\\equiv \\hat{U}(T, f(t))|\\psi _0{\\rangle }.$ At the end of monitoring, the observed system ends up in a mixed state, $\\hat{\\rho }(T)= \\int Df |\\psi (T,[f]){\\rangle }{\\langle }\\psi (T,[f])|, \\quad \\rm {Tr}[\\hat{\\rho }]=1,$ where $\\rm {Tr}[\\hat{A}]$ denotes the trace of $\\hat{A}$ .",
"We are interested in the Feynman path analysis.",
"Multiplying each term in the product (REF ) by a unity $\\hat{I}=\\sum _{i_k=1}^2|a_{i_k}{\\rangle }{\\langle }a_{i_k}|$ , and sending $\\tau \\rightarrow 0$ , we can write the amplitude ${\\langle }a_j|\\psi (T,[f]){\\rangle }$ , $j=1,2$ , as a path sum (integral), ${\\langle }a_j|\\psi (T,[f]){\\rangle }=\\sum _{all\\quad a(t)}F[a(t)]\\times \\\\ \\nonumber \\exp \\lbrace -\\frac{1}{\\Delta a_T^2} \\frac{\\int _0^T [f(t)-a(t))]^2dt}{T}\\rbrace .$ The new notations are: a path $a(t)$ is a function taking only the values $a_1$ or $a_2$ at any time $0\\le t \\le T$ .",
"The factor $F[a(t)]=lim_{K\\rightarrow \\infty }{\\langle }a_j|\\exp (-i\\hat{H}\\tau )|a_{i_{K-1}}{\\rangle }...{\\langle }a_{i_1}|\\exp (-i\\hat{H}\\tau )|\\psi _0{\\rangle }$ is the probability amplitude to reach $|a_j{\\rangle }$ from $|\\psi _0{\\rangle }$ via $a(t)$ with no meters present.",
"Finally, $\\Delta a_T\\equiv 1/\\sqrt{\\kappa T}=\\Delta f\\sqrt{2/K}$ (we maintain the notations of [1]), and the factor multiplying $1/\\Delta a_T^2$ is the time averaged square of the deviation of the path $a(t)$ from the observed readout $f(t)$ .",
"Equation (REF ) has the form of a restricted path integral (RPI).The role of the meters is to modify the amplitudes of the system's Feynman paths, suppressing them for the paths deviating from a readout $f(t)$ , and leaving them untouched for $a(t)$ close to $f(t)$ .",
"Given the form of the integral in (REF ) it is tempting to assume, as was done in [1], that for $\\Delta a_T<< |a_1-a_2|$ , $f(t)$ and $a(t)$ must be point-wise close, with $a(t)$ rarely differing from $f(t)$ by more than $\\Delta a_T$ .",
"By the same token, one may expect, the observed readouts to be not too different from one of the Feynman paths, $a(t)$ , i.e., to alternate between the values $a_1$ and $a_2$ .",
"In particular, in the simple case of $\\hat{A}$ and $\\hat{H}$ commuting, we have ${\\langle }a_j|\\hat{H}|a_i{\\rangle }=E_i \\delta _{ij}$ , $E_i$ being the energies of $|a_i{\\rangle }$ , and there are only two Feynman paths present: one connecting the state $|a_1{\\rangle }$ with $|a_1{\\rangle }$ , $a(t)=a_1$ , $F[a(t)=a_1]=\\exp (-iE_1T)$ , and a similar constant one, connecting $|a_2{\\rangle }$ with $|a_2{\\rangle }$ , $a(t)=a_2$ , $F[a(t)=a_2]=\\exp (-iE_2T)$ .",
"Let a two-level system be prepared in a state (REF ), and subject it to a continuous monitoring by Gaussian meters.",
"Based on the above, the authors of [1] predicted that (i) for a small $\\Delta a_T<< |a_1-a_2|$ , e.g., in the case of $T\\rightarrow \\infty $ , one would observe only the readouts lying in very narrow bands close to the constant curves $f(t)\\equiv a_i$ , such that for most of the monitoring one has $|f(t)-a_i| \\lesssim \\Delta a_T<<|a_1-a_2|$ .",
"(ii) The initial superposition (REF ) would undergo complete decoherence if the duration of the monitoring exceeds $1/\\kappa |a_1-a_2|^2$ , i.e., a pure state $|\\psi _0{\\rangle }$ will be turned into a mixture $\\hat{\\rho }(T)=|1{\\rangle }|\\alpha _0|^2{\\langle }1|+|2{\\rangle }|\\beta _0|^2{\\langle }2|$ for $T \\gtrsim 1/\\kappa |a_1-a_2|^2$ .",
"Our purpose here is to show that the assumption (i) is incorrect, and to explain how (ii) is possible without (i)."
],
[
"The single-path case",
"To make things as simple as possible, we assume that $|\\psi _0{\\rangle }=|a_1{\\rangle }$ , and leave the problem of decoherence aside at first.",
"We might as well put $\\hat{H}\\equiv 0$ , and subject the system permanently residing in the state $|a_1{\\rangle }$ to monitoring by a set of identical Gaussian meters, as discussed above.",
"If the assumption (i) of the previous Section is correct, by choosing a $T$ sufficiently large we should observe only the readouts clinging to the constant curve $f(t)=a_1$ .",
"This appears to be unlikely, since now we have $K>>1$ independent measurements of a normally distributed variable $f$ .",
"The meter firing at a time $t_k$ has no knowledge of what has happened in the past, at $t_i$ , $1\\le i <k$ .",
"Thus, there is no reason to expect its output to fit into a narrow band around $a_1$ .",
"Rather, the mean value of $[f(t)-a_1]^2$ should be determined only by $\\Delta f =1/\\sqrt{\\kappa \\tau }=\\sqrt{K}\\Delta a_T$ , which is very large if $\\tau $ is small.",
"Returning to the discrete form of Eq.",
"(REF ), we have $W(\\underline{f})=C^{-K}\\exp \\left[-\\frac{1}{\\Delta a_T^{2}}\\frac{\\sum _{k=1}^K(f_k-a_1)^2}{K}\\right].$ Now the statement (i) is equivalent to the assumption that the most probable readouts are those for which $\\Theta (\\underline{f}) \\equiv \\sum _{k=1}^N(f_k-a_1)^2/{K}\\lesssim \\Delta a_T^2$ , but this is incorrect.",
"To determine the most probable value of $\\Theta $ we also need to take into account the corresponding density of states.",
"An output $\\underline{f}$ is represented by a point in a $K$ -dimensional space, and $R^2= \\sum _{k=1}^N(f_k-a_1)^2$ is just the square of its distance from $a_1$ .",
"Other readouts sharing the same value of the $R^2$ lie on an $K$ -dimensional sphere centred at $a_1$ , and the probability to find a value of $R$ between $r$ and $r+dr$ is, therefore, given by $C^{-K}dV_K(r)/dr\\exp (-r^2/\\Delta a_T^2K)dr$ , where $V_K$ is the volume a $K$ -dimensional ball.",
"The derivative is just the surface area of a $K-1$ -dimensional sphere, and is well known to be $dV_K(r)/dr=2\\pi ^{K/2}r^{K-1}/\\Gamma (K/2)$ , where $\\Gamma (z)=\\int _0^\\infty y^{z-1}\\exp (-y)dy$ is the Gamma function [15].",
"Thus, for the probability $dP(x)$ to have the value of $\\Theta $ between $x$ and $x+dx$ , we find (for a more detailed derivation see Appendix A) $\\frac{dP}{dx}=\\Delta a_T^{-2}\\left(\\frac{x}{\\Delta a_T^2}\\right)^{K/2-1}\\exp (-\\frac{x}{\\Delta a_T^2})/\\Gamma (K/2).\\quad \\quad $ The r.h.s.",
"of Eq.",
"(REF ) peaks at $x_{0}=\\Delta a_T^2 (K/2-1)\\approx \\Delta f^2/2$ , which means that we are most likely to see the readouts wildly fluctuating around $a_1$ on the scale of $\\pm \\Delta a_T\\sqrt{K/2}\\sim \\Delta f$ , rather than those lying in a narrow band of a width $\\sim \\Delta a_T$ (see Fig.1).",
"Moreover, for such paths, the exponent in Eq.",
"(REF ) will, in the limit (REF ), tend to infinity as $\\Delta f^2\\sim \\tau ^{-1}$ .",
"Figure: (Color online)A randomly chosen readout f k /Δaf_k/\\Delta a, Δa≡a 2 -a 1 =2\\Delta a\\equiv a_2-a_1=2, for K=10 9 K=10^9 Gaussian meters defined by Eq.",
"() (only 10 5 10^5 values are shown), for the system in the first state |a 1 〉|a_1{\\rangle }, β 0 =0\\beta _0=0, H ^=0\\hat{H}=0, andΔa T =0.03\\Delta a_T=0.03.",
"Also shown by a horizontal white line is a 1 /Δa=-1/2a_1/\\Delta a=-1/2."
],
[
"The complex Hamiltonian approach",
"Next we briefly revisit the approach used, for example, in [1] in order to predict the behaviour of the measurement readouts.",
"From Eq.",
"(REF ) it follows that, for a given readout $f(t)$ , the system's evolution is described by a Schroedinger equation (SE) with a non-hermitian Hamiltonian [1]-[4], $i\\partial _t |\\psi (t,[f]){\\rangle }= \\left\\lbrace \\hat{H}-i\\kappa [\\hat{A}-f(t)]^2\\right\\rbrace |\\psi (t){\\rangle },\\\\ \\nonumber |\\psi (t=0,[f]){\\rangle }=|\\psi _0{\\rangle }.$ If the Hamiltonian, $\\hat{H}$ , commutes with the measured operator, $\\hat{A}$ , $[\\hat{H},\\hat{A}]=0, \\quad \\hat{H}|a_i{\\rangle }=E_i|a_i{\\rangle }, \\quad i=1,2,$ Eq.",
"(REF ) is easily solved to yield $|\\psi (t,[f]){\\rangle }=\\alpha (t,[f])|a_1{\\rangle }+\\beta (t,[f])|a_2{\\rangle }$ with [ ${\\langle }(\\hat{A}-f)^2{\\rangle }_T\\equiv T^{-1}\\int _0^T(\\hat{A}-f)^2dt$ ] $\\alpha (T,[f]) =\\exp \\left[-iE_1T-\\frac{{\\langle }(f-a_1)^2{\\rangle }_T}{{\\Delta a_T}^2}\\right]\\alpha _0,$ and similarly for $\\beta (T)$ , with $a_1$ and $E_1$ replaced by $a_2$ and $E_2$ .",
"In the single-path case of the previous Section we can put $\\alpha _0=1$ and $\\beta _0=0$ , to obtain from Eq.",
"(REF ) $W(f)=\\exp \\left[\\frac{-2{\\langle }(f-a_1)^2{\\rangle }_T}{{\\Delta a_T}^2}\\right].\\quad $ For a small $\\Delta a_T$ , Eq.",
"(REF ) seems to suggest that the only possible readouts are the constant one, $f(t)=a_1$ and, perhaps, some others in its immediate vicinity [1].",
"However, in the previous Section we have demonstrated this assumption to be incorrect.",
"The reason is the factor $C^{-K}=1/(\\pi ^{K/2}\\Delta f^K)$ , which multiplies the contribution from each readout in the path integral (REF ).",
"While it is true that the contribution of the constant readout $f(t)=a_1$ is far greater that the one from a readout for which ${\\langle }(f-a_1)^2{\\rangle }_T >> \\Delta a_T^2$ , the contribution itself vanishes as the number of meters increases.",
"At the same time, the readouts with smaller individual probabilities are by far more numerous, and therefore more likely, as discussed in the previous Section.",
"The same argument applies in the two paths case outlined at the end of Sect.",
"II, where $|\\psi _0{\\rangle }$ is chosen to be a superposition (REF ).",
"Also in this case, by choosing $\\Delta a_T<< |a_1-a_2|$ , one would not obtain readouts clinging to the constant curves $a(t)=a_i$ .",
"Rather, the spread of the readings would greatly exceed the separation between the eigenvalues $a_1$ and $a_2$ , making it impossible to decide immediately which of the two states the system is in.",
"This poses a further question.",
"If the readouts were an eigenvalue curve $f(t)=a_1$ or $f(t)=a_2$ , it would be easy to conclude that, as a result of the decoherence, the system has indeed settled into one of the eigenstates of $\\hat{A}$ .",
"But since this is not the case, how sure can we be that decoherence has taken place?",
"In other words, is the statement (ii) of the Section II correct, and if it is, what is the precise mechanism of the decoherence?"
],
[
"Decoherence of a \"free\" system",
"First we check whether the statement (ii) of Sect.II is correct.",
"If $\\hat{H}$ commutes with $\\hat{A}$ , ${\\langle }a_i|\\hat{H}|a_j{\\rangle }=E_i\\delta _{ij}$ , for $|\\psi _K(\\underline{f}){\\rangle }$ in (REF ) we have $|\\psi _K(\\underline{f}){\\rangle }=\\alpha _0\\exp (-iE_1K\\tau ) \\prod _{k=1}^K G(f_k-a_1)|a_1{\\rangle }\\\\ \\nonumber +\\beta _0\\exp (-iE_2K\\tau ) \\prod _{k=1}^K G(f_k-a_2)|a_2{\\rangle }.\\quad $ We may as well choose $E_1=E_2=0$ , in which case Eq.",
"(REF ) yields $ \\nonumber {\\langle }a_1| \\hat{\\rho }(T)|a_2{\\rangle }= \\alpha _0\\beta _0^*\\left[\\int df G(f-a_1)G(f-a_2)\\right] ^K=\\\\\\alpha _0\\beta _0^*\\exp [-\\kappa T|a_1-a_2|^2/2],\\quad \\quad \\quad \\quad $ where we have evaluated the Gaussian integral, and used Eqs.",
"(REF ).",
"Coherence (REF ) vanishes if $\\kappa T = \\Delta a_T^2>> 1/(a_1-a_2)^2$ , leaving the system in a mixed state $\\hat{\\rho }(T)= |a_1{\\rangle }|\\alpha _0|^2{\\langle }a_1|+|a_2{\\rangle }|\\beta _0|^2{\\langle }a_2|.$ Thus, assumption (ii) of Sect.II is indeed correct.",
"We still need to see how this is possible.",
"Instead of aligning with one of the eigenvalues of $\\hat{A}$ , a typical readout would alternate wildly, and give no apparent indication as to the state the system has ended up in.",
"Yet such information must be available since, according to Eq.",
"(REF ), a given readout uniquely determines the system's final destination."
],
[
"Decoherence by \"sudden reduction\"",
"To see how this happens, we first resort to a simpler model similar to the one used in [5].",
"The new \"measuring medium\" consists of a set of non-Gaussian meters, with $G(f)$ having the shape of a \"rectangular window\" of a width $\\Delta f> |a_1-a_2|$ , $G(f)=1/\\sqrt{\\Delta f}, \\quad for \\quad |f|\\le \\Delta f/2,$ and zero otherwise.",
"[This can be seen as imposing a \"hard wall\" restriction on the system's Feynman paths: If $G(f)$ is written as $1/\\sqrt{\\Delta f} \\exp [-g(f)]$ , $g(f)$ would need to be 0 for $|f|< \\Delta f/2$ and infinite for $|f|\\ge \\Delta f/2$ .]",
"Now in Eq.",
"(REF ) the state of the system after the $k$ -th meter has fired is (assuming $a_2 > a_1$ ).",
"$\\nonumber \\alpha _{k-1}|a_1{\\rangle }/\\sqrt{\\Delta f},\\quad if \\quad f \\in [a_1-\\Delta f/2,a_2-\\Delta f/2]\\equiv A,\\quad \\quad \\\\ \\nonumber |\\psi _{k-1}{\\rangle }/\\sqrt{\\Delta f},\\quad if \\quad f \\in [a_2-\\Delta f/2,a_1+\\Delta f/2]\\equiv C,\\quad \\quad \\\\ \\nonumber \\beta _{k-1}|a_2{\\rangle }/\\sqrt{\\Delta f},\\quad if \\quad f \\in [a_1+\\Delta f/2,a_2+\\Delta f/2]\\equiv B.\\quad \\quad $ $$ Here $C$ is the region where $G(f-a_1)$ and $G(f-a_2)$ overlap, and if $f_k$ happens to lie there, the state before the meter has fired, $|\\psi _{k-1}{\\rangle }$ , remains unaltered.",
"If $f_k$ falls into the regions $A$ or $B$ , $|\\psi _{k-1}{\\rangle }$ is reduced to $|a_1{\\rangle }$ , or $|a_2{\\rangle }$ , respectively.",
"With no Hamiltonian to rotate the state between the measurements, it will remain the same for the rest of the monitoring.",
"An elementary calculation shows that the probabilities $P(J)$ to have $f_k$ in a region $J=A,B,C$ are $P(A)= |a_1-a_2||\\alpha _{k-1}|^2/\\Delta f,\\\\ \\nonumber P(B)= |a_1-a_2||\\beta _{k-1}|^2/\\Delta f,\\\\ \\nonumber P(C)=1- |a_1-a_2|/\\Delta f.$ As before, we wish to lower the resolution of each measurement $\\Delta f$ , and increase their number, albeit in a slightly different manner, $\\tau \\rightarrow 0, \\quad \\Delta f\\rightarrow \\infty , \\quad {\\tau }\\Delta f \\rightarrow \\kappa ^{\\prime -1} = const.$ With $P(A)$ and $P(B)$ extremely small, each meter is now likely to leave the state of the system unchanged.",
"It will, therefore, propagate unaltered until an unlikely fluctuation will put $f_k$ in, say, the region $A$ .",
"After that the system will continue in the state $|a_1{\\rangle }$ , and subsequent meters will produce the reading in a very broad interval $[a_1-\\Delta f/2, a_1+\\Delta f/2]$ , as illustrated in Fig.2.",
"Thus, the reduction of $|\\psi _0{\\rangle }$ to $|a_1{\\rangle }$ is achieved instantaneously, but the precise moment at which it occurs is hidden from the viewer by the noise of the readout and, thus, remains unknown without further analysis.",
"It is easy to evaluate the number of measurements and, therefore, the time after which the system will have collapsed into one of the two states almost certainly.",
"From Eqs.",
"(REF ), the probability to survive in the initial state $|\\psi _0{\\rangle }$ after $K$ measurements is $P_{surv}(T)=P(C)^K=(1- \\kappa ^{\\prime } T{|a_1-a_2|}/K)^K\\\\ \\nonumber \\rightarrow _{K\\rightarrow \\infty }\\exp (-\\kappa ^{\\prime } T |a_1-a_2|),$ and after waiting for $T >> \\kappa ^{\\prime }|a_1-a_2|$ one can be sure that either region $A$ or $B$ has been hit, the initial state has been reduced, and system's density matrix is given by Eq.",
"(REF ).",
"Figure: (Color online)a) A randomly chosen readout f k /Δaf_k/\\Delta a, Δa≡a 2 -a 1 \\Delta a\\equiv a_2-a_1, for K=10 9 K=10^9 non-Gaussian meters defined by Eq.",
"() (only 10 5 10^5 values are shown).",
"The system is prepared in an initial state, |ψ 0 〉=(|a 1 〉+|a 2 〉)/2|\\psi _0{\\rangle }=(|a_1{\\rangle }+|a_2{\\rangle })/\\sqrt{2}, a 2 =-a 1 =1a_2=-a_1=1, H ^=0\\hat{H}=0,Δf/Δa=4*10 8 \\Delta f/\\Delta a=4*10^8; b) the probability to find the system in the state |a 1 〉|a_1{\\rangle } after kk meters have fired."
],
[
"Decoherence by \"random walk\"",
"A somewhat similar mechanism must be responsible for the decoherence of a system monitored by a set of Gaussian meters (REF ).",
"In this case it is unrealistic to expect a single fluctuation capable of eliminating one of the states from the superposition (REF ).",
"Indeed, for $\\Delta f >>|a_1-a_2|$ to have, for example, $G(f-a_1) << G(f-a_2)$ requires an $f >>f_0\\equiv \\Delta f^2/|a_1-a_2|$ .",
"The probability to have any $f > f_0$ is then expressed in terms of the complimentary error function [15], $Prob(f>f_0)\\sim erfc(f_0/\\Delta f) \\approx (\\Delta f/f_0)\\exp (-f_0^2/\\Delta f^2) \\sim \\exp (-\\Delta f^2/|a_1-a_2|^2)$ and is extremely small.",
"With decoherence \"by sudden death\" unlikely, we should find another mechanism.",
"Consider the ratio $\\xi _k\\equiv |\\alpha _k/\\beta _k|^2$ , such that $\\xi _k=0$ if the particle is in the state $|a_2{\\rangle }$ and $\\xi _k=\\infty $ , if it is in the state $|a_1{\\rangle }$ .",
"With the help of Eqs.",
"(REF ) and (REF ) it can be written as $\\xi _K=\\exp (-X_K)|\\alpha _0/\\beta _0|^2,\\quad \\quad $ where $X_K\\equiv \\frac{2(a_2-a_1)}{\\Delta f^{2}}\\sum _{k=1}^K\\left(f_{k}-\\frac{a_1+a_2}{2}\\right),$ so that the ratio is determined by the value of the sum $X_k$ .",
"For the system to be ultimately driven into one of the eigenstates of $\\hat{A}$ , $X_k$ must be a large positive or a large negative number.",
"To show that this is always the case, we look at the distribution of the random variable $X_k$ .",
"First, using Eqs.",
"(REF ) and (REF ), we note that the probability distribution of a sum $Y_K=\\sum _{k=1}^Kf_{k}$ is given by (see Appendix B) $W(Y_K)=|\\alpha _0|^2\\mathcal {N}(Y_K|Ka_1, K\\Delta a_T/{2}) + \\\\ \\nonumber |\\beta _0|^2\\mathcal {N}(Y_K|Ka_2, K\\Delta a_T/{2}),\\quad $ where $\\mathcal {N}(x|\\mu ,\\sigma )$ denotes a normal distribution [16] with a mean $\\mu $ and a standard deviation $\\sigma $ , $\\mathcal {N}(x|\\mu ,\\sigma )\\equiv (2\\pi \\sigma ^2)^{-1/2}\\exp [-(x-\\mu )^2/2\\sigma ^2].$ For the re-scaled and shifted variable $X_K$ , in the limit (REF ), we then find [$X(T) \\equiv X_{T/\\tau }$ ] $\\nonumber W(X(T))=|\\alpha _0|^2\\mathcal {N}(X(T)|2\\kappa T (a_1-a_2)^2,2\\sqrt{\\kappa T}|a_1-a_2|)\\\\+|\\beta _0|^2\\mathcal {N}(X(T)|-2\\kappa T (a_1-a_2)^2,2\\sqrt{\\kappa T}|a_1-a_2|),\\quad \\quad \\quad $ where $T=K\\tau $ .",
"A brief inspection shows that we have a case of two Gaussian random walks with opposite drifts.",
"A walk can be visualised as a process, in which the displacement of a walker at the $k$ -th step consists of a constant \"drift\" $\\pm 2\\kappa \\tau (a_1-a_2)^2$ and a random shift $y$ , drawn from a normal distribution ${N}(y|0,2\\sqrt{\\kappa \\tau }|a_1-a_2|)$ .",
"The sum $X(T)$ is then the displacement of the walker at a time $T$ .",
"It is readily seen that the distribution of $X(T)$ consists of two Gaussians moving, as time progresses, in opposite directions, and becoming broader at the same time.",
"The broadening, however, is much slower then the separation, and for $T>> 1/\\kappa (a_1-a_2)^2$ , i.e., for $\\Delta a_T<< |a_1-a_2|$ , the Gaussians are separated completely (see Fig.3).",
"Thus, there are just two possibilities.",
"Either a walk ends far to the right, $X(T)>>1$ , and leaves the system in the state $|a_2{\\rangle }$ since $\\xi (T)\\equiv \\xi _{T/\\tau } \\rightarrow 0$ , or it ends far to the left, $X(T) << -1$ , and leaves the system in the state $|a_1{\\rangle }$ .",
"The relative frequency, with which both types of the walks occur, is given by the ratio $|\\alpha _0|^2/|\\beta _0|^2$ , in accordance with Eq.",
"(REF ) Figure: (Color online)The distribution () of the sum X(T)X(T) in Eq.",
"() for different values of the parameter γ=2κT(a 1 -a 2 ) 2 \\gamma =2\\kappa T (a_1-a_2)^2.",
"The system is prepared in an initial state, |ψ 0 〉=(|a 1 〉+|a 2 〉)/2|\\psi _0{\\rangle }=(|a_1{\\rangle }+|a_2{\\rangle })/\\sqrt{2}, a 2 =-a 1 =1a_2=-a_1=1, H ^=0\\hat{H}=0,Δf/(a 2 -a 1 )=250\\Delta f/(a_2-a_1)=250.",
"The histograms show the corresponding results of numerical simulations involving 2*10 4 2*10^4 random realisations, obtained with the help of the algorithm described in Appendix C.In summary, for a free system, complete decoherence of an arbitrary pure state (REF ) is indeed achieved for $T>> 1/\\kappa (a_1-a_2)^2$ , but by a mechanism different from the one assumed in [1].",
"A typical readout does not align with one of the eigenvalues of the measured operator, and remains irregular at all times as shown in Fig.",
"4a.",
"To find out into which of the two states the system is driven as a result, we must use all the readings to evaluate the exponent in Eq.",
"(REF ), and then see whether the result is a large positive, or a large negative number (see Fig.",
"4b).",
"This analysis is easily generalised to systems with any number of states $N>2$ , in which case the large-time distribution of $X(T)$ will be a multi-modal sum of Gaussians, to one of which a random walk can always be traced.",
"A randomly chosen graph $|\\alpha _k|^2=\\xi _0\\exp (-X_k)/[1+\\xi _0\\exp (-X_k)]$ vs. $k$ , is shown in Fig.",
"3c.",
"The irregular patterns, with clearly visible ups and downs, reflect, albeit indirectly, the behaviour of the underlying random walk $X_k$ in Fig.",
"3b.",
"As $X_k$ increases, its fluctuations are damped be the factor $\\exp (X_k)$ , and the curve $|\\alpha _k|^2$ becomes smoother.",
"Figure: (Color online)a) A randomly chosen readout f k /Δaf_k/\\Delta a, Δa≡a 2 -a 1 \\Delta a\\equiv a_2-a_1, for K=10 9 K=10^9 Gaussian meters (only 10 5 10^5 values are shown).",
"The system is prepared in an initial state, |ψ 0 〉=(|a 1 〉+|a 2 〉)/2|\\psi _0{\\rangle }=(|a_1{\\rangle }+|a_2{\\rangle })/\\sqrt{2}, a 2 =-a 1 =1a_2=-a_1=1, H ^=0\\hat{H}=0,Δf/Δa=10 4 \\Delta f/\\Delta a=10^4; b) displacement of the random walker, X k X_k, defined in Eq.",
"() and c) the probability to find the system in the state |a 1 〉|a_1{\\rangle } after kk meters have fired."
],
[
"Zeno effect in a \"driven\" system",
"In [1] the authors considered also monitoring of a system, capable of making transitions between the state $|a_1{\\rangle }$ and $|a_2{\\rangle }$ , and described by a Hamiltonian ${\\langle }a_i|\\hat{H}|a_i{\\rangle }=0 , \\quad {\\langle }a_1 |\\hat{H}|a_2 {\\rangle }={\\langle }a_2 |\\hat{H}|a_1 {\\rangle }\\equiv \\omega .\\quad $ In the absence of the meters, such a system performs Rabi oscillations with a period $T_R=2\\pi /\\omega $ .",
"Following [1], we choose to measure an operator $\\hat{A}$ , ${\\langle }a_j|\\hat{A}|a_i{\\rangle }=a_i \\delta _{ij}$ .",
"In the Zeno regime, i.e., for $1/\\kappa |a_1-a_2|^2 << T_R << T$ , the authors of [1] made the following suggestions: (I) Only those measurement outputs $f(t)$ that are close to one of the constant curves $f(t)=a_1$ and $f(t)=a_2$ have high probability.",
"(II) The probability of the output to be close to $a_1$ or $a_2$ is given by the initial values of the decomposition coefficients $|\\alpha _0|^2$ or $|\\beta _0|^2$ correspondingly.",
"(III) In the case of the output being close to $a_1$ or $a_2$ the final state is correspondingly the eigenstate $|a_1{\\rangle }$ or $|a_2{\\rangle }$ .",
"Having found (I) incorrect in Sect.",
"III, we need to re-examine the other two points as well."
],
[
"Zeno effect by \"sudden reduction\"",
"We start with the simple model (REF )-(REF ) of the previous Section.",
"As before, reduction of the state to either $|a_1{\\rangle }$ or $|a_2{\\rangle }$ is achieved whenever a rare fluctuation puts an $f_k$ into the regions $A$ or $B$ .",
"A typical time between two fluctuations is of order of $T^{\\prime }_{LR}$ (we use the notations of [1], and \"LR\" stand for \"level resolution\"), where $T^{\\prime }_{LR}$ is the average time after which the first fluctuation occurs, $T^{\\prime }_{LR}=-\\int _0^\\infty t \\frac{d}{dt} P_{surv}(t)dt = \\frac{1}{\\kappa ^{\\prime }|a_1-a_2|}.$ What happens to the system between two subsequent reductions depends of the relation between $T^{\\prime }_{LR}$ and the Rabi period $T_R$ .",
"For $T_R \\lesssim T^{\\prime }_{LR}$ , the system may have a chance to perform a number of Rabi oscillations, and a typical curve $|\\alpha (t,[f])|^2$ will consist of several pieces of regular oscillation $\\sim \\cos ^2(\\omega T)$ , with arbitrary relative phases where the curve becomes discontinuous (see Fig.",
"5a).",
"For $T_R \\gtrsim T^{\\prime }_{LR}$ , the system would, on average, have no time to complete a single oscillation before it is interrupted by the next reduction, and the curve will typically have an irregular shape shown in Fig.",
"5b.",
"Finally, for $T^{\\prime }_{LR}<< T_R$ , and $\\exp (-i\\hat{H}T^{\\prime }_{LR}) = 1 + O( T^{\\prime }_{LR}/T_R)$ , we return to the situation of the previous Section.",
"The initial state (REF ) is reduced for the first time after approximately $T^{\\prime }_{LR}$ , after which it continues almost unchanged until $t=T$ .",
"Close to this Zeno regime, $|\\alpha (T,[f])|^2$ takes a form characteristic of a \"telegraph noise\" (see, for example, [17]), with the system spending, on average, a duration $T^{stay}$ in $|a_1{\\rangle }$ , then making a sudden transition, and spending a similar amount of time in $|a_2{\\rangle }$ , and so on (see Fig.",
"5c).",
"The time $T^{stay}$ can be evaluated by noting that after free evolution during $T^{\\prime }_{LR}$ , the probability for the system to have changed its state is approximately $|{\\langle }a_i|\\hat{H}T^{\\prime }_{LR}|a_j{\\rangle }|^2 \\approx \\omega ^2{T^{\\prime }_{LR}}^2$ , $j\\ne i$ .",
"The system succeeds in changing its state after approximately $n_{att} \\approx 1/\\omega ^2{T^{\\prime }_{LR}}^2$ attempts, and $T^{stay} \\approx n_{att}T^{\\prime }_{LR} = \\frac{T_R^2}{4\\pi ^2 T^{\\prime }_{LR}}.$ Thus, the Zeno regime is reached as $T_R/T^{\\prime }_{LR}\\rightarrow \\infty $ , and the system remains in one state for any finite $T$ .",
"Figure: (Color online) Probabilities |α(t,[f])| 2 |\\alpha (t,[f])|^2 vs. tt for a randomly chosen readout ff.A \"driven\" system, with H ^\\hat{H} given by Eqs.",
"(), is monitored for 0≤t≤T0\\le t \\le T, ωT=25\\omega T=25, by K=10 9 K=10^9 non-Gaussian meters.The system's initial state is |ψ 0 〉=|a 1 〉|\\psi _0{\\rangle }=|a_1{\\rangle }, and T LR ' /T R =T^{\\prime }_{LR}/T_R= a) 0.5; b) 0.08, and c) 0.008.The dashed lines show the Rabi oscillations of the system with no meters present.",
"The vertical dashed linesin (a) indicate the moments the system's state is suddenly reduced to |a 1 〉|a_1{\\rangle } or |a 2 〉|a_2{\\rangle }."
],
[
"Zeno effect by \"random walk\"",
"The case of Gaussian meters is similar, and Section VB suggests a possible mechanism.",
"However, now we need to take into account all, and not just two, of the system's Feynman's paths in Eq.",
"(REF ).",
"Considering for simplicity the case where the system starts in the state $|a_1{\\rangle }$ , we can write the state (REF ) after $K$ measurements in a matrix form, $\\nonumber \\begin{pmatrix}\\alpha _K \\\\\\beta _K\\end{pmatrix}=\\tilde{C}_K\\prod _{k=1}^K\\begin{pmatrix}\\exp [-\\frac{(f_k-a_1)^2}{2\\Delta f^2}] & 0 \\\\0& \\exp [-\\frac{(f_k-a_2)^2}{2\\Delta f^2}]\\end{pmatrix}\\\\\\times \\begin{pmatrix}U_{11}(\\tau )& U_{12}(\\tau ) \\\\U_{21}(\\tau )& U_{22}(\\tau )\\end{pmatrix}\\begin{pmatrix}1 \\\\0\\end{pmatrix}, \\quad \\quad \\quad \\quad \\quad \\quad \\quad $ where $\\tilde{C}_K=(\\pi \\Delta f^2)^{-K/4}$ , $U_{ij}(\\tau )\\equiv {\\langle }a_i|\\exp (-i\\hat{H}\\tau )|a_j{\\rangle }$ , and $U_{11}(\\tau )=U_{22}(\\tau )=\\cos (\\omega \\tau )\\approx 1-\\omega ^2\\tau ^2/2,\\quad \\quad \\quad \\\\ \\nonumber U_{21}(\\tau )=-U_{12}(\\tau )=-i\\sin (\\omega \\tau )\\approx -i\\omega \\tau ,\\quad \\quad \\quad \\quad $ .",
"We can uncouple the system from the meters by choosing $a_1=a_2=a$ , so that in Eq.",
"(REF ) the diagonal matrices would commute with the evolution operator $\\hat{U}(\\tau )$ .",
"With the Rabi oscillations unhampered, we have $\\alpha ^{unc}(T,[f]) =\\cos (\\omega T)\\prod _{k=1}^KG(f_k-a),\\\\ \\nonumber \\beta ^{unc}(T,[f])=-i \\sin (\\omega T)\\prod _{k=1}^KG(f_k-a).$ Next we ask whether the Rabi oscillations will be quenched by the monitoring in the continuous limit (REF ), for times $T=K\\tau $ large enough to ensure $\\Delta a_T=1/\\sqrt{\\kappa T} << |a_1-a_2|$ ?",
"Thus, a Zeno effect will be found if we could prove that for $\\kappa T(a_1-a_2)^2 >>1$ one would almost certainly find $\\alpha (T,[f]) \\approx 1, \\quad \\text{or}\\quad \\alpha (T,[f]) \\approx 0.$ We will provide a demonstration in the weak coupling limit, $\\omega T <<1$ , choosing, for simplicity, $a_1=0$ and $a_2=a$ .",
"Now the system can reach $|a_2{\\rangle }$ by Feynman paths which remain in $|a_1{\\rangle }$ until some $0\\le t^{\\prime }\\le T$ , and then change once to $|a_2{\\rangle }$ , in which they continue until $T$ .",
"Let $F^{}_{K^{\\prime },K^{\\prime }+1}(\\underline{f})$ , be the sum of the probability amplitudes for the paths which change from $|a_1{\\rangle }$ to $|a_2{\\rangle }$ within an interval $\\tau $ between $T_{K^{\\prime }}=\\tau K^{\\prime }$ and $T_{K^{\\prime }+1}=\\tau (K^{\\prime }+1)$ .",
"To the first order in $\\omega $ , the amplitude $\\beta ^{}(T,\\underline{f})$ is the sum over all $K^{\\prime }$ of the amplitudes $F^{}_{K^{\\prime },K^{\\prime }+1}(\\underline{f})$ , $\\beta ^{}(T,\\underline{f}) \\approx \\sum _{K^{\\prime }=1}^{K-1}F^{}_{K^{\\prime },K^{\\prime }+1}(\\underline{f}).$ For an uncoupled system we have $F^{unc}_{K^{\\prime },K^{\\prime }+1}(\\underline{f})= -i\\omega \\tau \\prod _{k=1}^KG(f_k),$ while for a monitored system, with the help of Eq.",
"(REF ), we find $F_{K^{\\prime },K^{\\prime }+1}(\\underline{f})=- i\\omega \\tau \\prod _{k=1}^{K^{\\prime }}G(f_k) \\prod _{k=K^{\\prime }+1}^KG(f_k-a)\\\\ \\nonumber \\equiv \\exp (Z_{K^{\\prime }})F^{unc}_{K^{\\prime },K^{\\prime }+1}(\\underline{f}).$ Thus, the presence of the meters modifies each amplitude $F^{unc}_{K^{\\prime },K^{\\prime }+1}(\\underline{f})$ by a factor $\\exp (Z_{K^{\\prime }})$ , with $Z_{K^{\\prime }}\\equiv - \\frac{2a}{\\Delta {f}^2}\\sum _{k=K^{\\prime }+1}^K(f_k-a/2).$ To see what effect this factor would have we need the probability distribution of the readouts.",
"Using Eq.",
"(REF ), we obtain $W(\\underline{f})=|\\alpha (T,\\underline{f})|^2+|\\beta (T,\\underline{f})|^2\\approx \\prod _{k=1}^{K}G^2(f_k),$ and acting as in Sect.",
"IV, we find $Z_{K^{\\prime }}$ normally distributed, $W(Z_{K^{\\prime }})=\\\\ \\nonumber \\mathcal {N}\\left(Z_{K^{\\prime }}|-\\frac{a^2(K-K^{\\prime })}{\\Delta f^{2}},\\frac{a\\sqrt{2(K-K^{\\prime })}}{\\Delta f}\\right).$ Thus, the factor $\\exp (Z_{K^{\\prime }})$ will reduce the contribution of a Feynman path, provided it spends a sufficient amount of time in $|a_2{\\rangle }$ , i.e., for $a^2(K-K^{\\prime })/ \\Delta f^{2}\\gtrsim 1$ .",
"In the limit (REF ) this condition reads $2\\kappa a^2 (T-T^{\\prime })$ , where $T^{\\prime }=K^{\\prime }\\tau $ is the time at which a Feynman path changes from $|a_1{\\rangle }$ to $|a_2{\\rangle }$ .",
"With the contribution from most of the paths reduced, and all terms in (REF ) having the same phase, we can expect also a reduction in the probability $|\\beta (T)|^2$ .",
"This reduction can be evaluated directly since, for a given readout, the probability to find the system in $|a_2{\\rangle }$ is given by $|\\beta (T,\\underline{f})|^2 = |\\sum _{K^{\\prime }=1}^KF_{K^{\\prime },K^{\\prime }+1}(\\underline{f})|^2=\\omega ^2\\tau ^2\\sum _{K^{\\prime },K^{\\prime \\prime }=1}^K \\prod _{k=K_>+1}^KG^2(f_k-a) \\prod _{k=K_{<}+1}^{K_>}G(f_k-a)G(f_k)\\prod _{k=1}^{K_<}G^2(f_k)$ , where $K_{^>_<}=\\text{max(min)}\\lbrace K^{\\prime },K^{\\prime \\prime }\\rbrace $ .",
"The net probability for the system to make the transition by $t=T$ is found by summing over all possible readouts, $|\\beta (T)|^2=\\int d\\underline{f}|\\beta (T,\\underline{f})|^2$ .",
"Evaluating Gaussian integrals, we then have $|\\beta (T)|^2=\\omega ^2 \\tau ^2\\sum _{K^{\\prime },K^{\\prime \\prime }=1}^K\\exp (-|K^{\\prime }-K^{\\prime \\prime }|a^2/4\\Delta f^2)\\quad \\quad \\\\ \\nonumber \\approx \\omega ^2 \\int _0^TdT^{\\prime } \\int _0^T dT^{\\prime \\prime }\\exp (-\\kappa |T^{\\prime }-T^{\\prime \\prime }|a^2/4).$ For $\\kappa T a^2 >>1$ the last integral is approximately $8T/\\kappa a^2$ and we find $|\\beta (T)|^2$ significantly reduced by the monitoring, $\\frac{|\\beta (T)|^2}{|\\beta ^{unc}(T)|^2}\\approx \\frac{8}{\\kappa a^2 T}\\rightarrow 0.$ Figure: (Color online)Probabilities |α(t,[f])| 2 |\\alpha (t,[f])|^2 vs. tt for a randomly chosen readout ff.A \"driven\" system, with H ^\\hat{H} given by Eqs.",
"(), is monitored for 0≤t≤T0\\le t \\le T, ωT=10π\\omega T=10\\pi , by K=10 9 K=10^9 Gaussian meters.The system's initial state is |ψ 0 〉=|a 1 〉|\\psi _0{\\rangle }=|a_1{\\rangle }, and T LR /T R =T_{LR}/T_R= a) 0.4; and b) 0.03.The dashed lines show the Rabi oscillations of the system with no meters present.While our discussion suggests a way in which monitoring can suppress Rabi oscillations in a system, it provides no proof that this will occur beyond the weak coupling limit (REF ) for the simple Hamiltonian (REF ).",
"In general, it is impossible to consider separately the evolution of the system and the pointers, as was done in Eq.",
"(REF ) and in Sect.",
"V, and the rest of the analysis will have to be performed numerically.",
"The results, shown in Figs.",
"6 and 7, are broadly similar to those presented in Fig.",
"5.",
"Following [1], we can introduce a time $T_{LR}$ , similar to $T^{\\prime }_{LR}$ in Eq.",
"(REF ) $T_{LR}=1/{\\kappa (a_1-a_2)^2},$ and study the evolution of the system's state as function of $T_{LR}/T_R$ .",
"For $T_R \\sim T_{LR} << T$ , the system performs regular oscillations which gradually get out of phase with the uncoupled Rabi oscillations (Fig.",
"6a).",
"For $T_R \\gtrsim T^{\\prime }_{LR}$ , the curve $|\\alpha (T)|^2$ is highly irregular (Fig.",
"6b).",
"For $T^{\\prime }_{LR}<< T_R$ , the system is near a Zeno regime and $|\\alpha (T)|^2$ curve has a \"telegraph noise\" shape (Fig.7b), although we cannot easily evaluate the typical duration of $T^{stay}$ , as was done in the previous sub-Section.",
"Figure 7c shows that each time the system changes the state, the corresponding random walk changes direction.",
"With evolutions of the system and the pointers intertwined, we are unable to say whether the change of the system state affects the direction of the walk, or if the change of direction causes the system to alter its state.",
"As in the previous sub-Section, the Zeno regime is reached when $T^{stay}\\rightarrow \\infty $ , and the system remains in one state for any finite $T$ .",
"In summary, for $T_{LR}<< T_R << T$ we do have a Zeno effect, although the conclusions of [1] must be modified as follows: (I') The measurement outputs $f(t)$ that are close to one of the constant curves $f(t)=a_1$ and $f(t)=a_2$ are by far not the most probable ones.",
"A typical readout will look like the ones shown in Figs.",
"1 and 3a.",
"(II') The probability of a readout being close to $a_1$ or $a_2$ is proportional to the initial values of the decomposition coefficients $|\\alpha _0|^2$ or $|\\beta _0|^2$ , respectively.",
"However, an analysis of the evolutions induced by these constant readouts, does not explain the mechanism of the Zeno effect, since such scenarios will never occur in practice.",
"(III') Even with most readouts not close to $a_1$ or $a_2$ the Rabi oscillations are quenched, and final state is the eigenstate $|a_1{\\rangle }$ or $|a_2{\\rangle }$ .",
"Figure: (Color online) a) A randomly chosen readout f k /Δaf_k/\\Delta a, Δa≡a 2 -a 1 \\Delta a\\equiv a_2-a_1, a 2 =-a 1 =1a_2=-a_1=1, for K=10 9 K=10^9 Gaussian meters (only 10 5 10^5 values are shown);b) Corresponding probability |α(t,[f])| 2 |\\alpha (t,[f])|^2 vs. tt for a driven system with H ^\\hat{H} given by Eqs.",
"().The system's initial state is |ψ 0 〉=|a 1 〉|\\psi _0{\\rangle }=|a_1{\\rangle }, and T LR /T R =T_{LR}/T_R=0.002.The dashed lines show the Rabi oscillations of the system with no meters present;c) displacement of the random walker defined in Eq.",
"()."
],
[
"Ensemble averages",
"Although our interest has been in individual realisations of a continuous measurements, we conclude by briefly discussing the averages obtained if a measurement is repeated several times.",
"Let us assume that the system starts in a state $|\\psi _0{\\rangle }$ at $t=0$ , and is post selected at $t=T$ in some final state $|\\psi _F{\\rangle }=\\alpha _F|a_1{\\rangle }+\\beta _F|a_2{\\rangle }$ .",
"What is the average value, ${\\langle }f(t|\\psi _F){\\rangle }$ , of a readout $f(t)$ , evaluated over many runs of the experiment?",
"The general expression is ${\\langle }f(t|\\psi _F){\\rangle }=\\frac{\\int Df f(t)|{\\langle }\\psi (T,[f])|\\psi _T{\\rangle }|^2}{\\int Df|{\\langle }\\psi (T,[f])|\\psi _T{\\rangle }|^2},$ and we illustrate the main points on the simplest example of decoherence of a free system, for the \"sudden reduction\" model of Sect.VA.",
"We choose $|\\psi _0{\\rangle }=(|a_1{\\rangle }+|a_2{\\rangle })/\\sqrt{2}$ , $a_1=-a_2$ , and consider first $|\\psi _F{\\rangle }=|a_2{\\rangle }$ .",
"If $K$ is chosen big enough to ensure full decoherence, by symmetry, post selection in $|\\psi _F{\\rangle }$ will be successful in one half of all trials.",
"Let the system's state be reduced at some $t_{k_0}$ , and consider the subset of readouts consistent with this condition.",
"For $t_k < t_{k_0}$ , all such readouts are bound to lie within the region $C$ defined in Eq.",
"(REF ), and their average is zero.",
"For $t_k > t_{k_0}$ this average is $a_2$ .",
"Finally, at $t_k=t_{k_0}$ the readouts must lie in the region $B$ , and their mean is $\\Delta f/2$ .",
"Summing over all $k_0$ , while taking into account (REF ), yields ${\\langle }f(t|a_2){\\rangle }\\equiv a_2$ for all $t$ .",
"Repeating the calculation for $|\\psi _F{\\rangle }=|a_1{\\rangle }$ then yields ${\\langle }f(t|a_1){\\rangle }= a_1, \\quad {\\langle }f(t|a_2){\\rangle }= a_2, \\quad \\\\ \\nonumber {\\langle }f(t|\\text{all}){\\rangle }\\equiv [{\\langle }f(t|a_1){\\rangle }+{\\langle }f(t|a_2){\\rangle }]/2=0,$ for any $0\\le t \\le T$ .",
"The result (REF ) also follows directly from Eq.",
"(REF ), and remains valid for any choice of $G(f)$ , provided $\\int f G^2(f-a)df=a$ .",
"It holds, therefore, also for the Gaussian meters of Sect.VB.",
"In practice, to evaluate these averages, we will need $M$ realisations of the same experiment.",
"To estimate how many, we note from Eq.",
"(REF ) that the standard deviation of $f(t)$ , $\\sigma $ , is of order of $\\Delta f$ .",
"According to the Central Limit Theorem, for a sample of a size $M>>1$ , the mean of $f(t)$ is normally distributed with a standard deviation $\\sigma _M= \\Delta f/\\sqrt{M}$ .",
"If the $M$ is finite, the measured values of ${\\langle }f(t|a_{1,2}){\\rangle }$ remain noisy.",
"To reduce the noise below the level $\\sim |a_2-a_1|$ , we need $\\sigma _M << a_2$ or, equivalently, $M >> \\Delta f^2/a_2^2$ .",
"Results of a simulation are shown in Fig.8a for $M=5\\times 10^5$ trials.",
"Thus, while most probable readouts remain noisy, their conditional averages do align with the eigenvalues of the measured operator.",
"Note, however, that, as the continuous limit is approached, the number of trials required to free the curves in Fig.8a from the noise, tends to infinity.",
"We note also, that if no post-selection is performed, the average readout $ {\\langle }f(t|\\text{all}){\\rangle }$ aligns with the mean $(a_1+a_2)/2$ and contains no information as to the final state of the system.",
"A similar argument applies in the case of a \"driven\" system, and for the Gaussian meters of Sect.VB [20], as illustrated in Fig.8b.",
"Hence the main conclusion of this Section: close to the continuous limit, the number of realisations needed to recover the average readouts from the noise of individual readouts becomes prohibitively large.",
"Figure: (Color online)Readouts averaged over 5×10 5 5\\times 10^5 trials.",
"a) for the \"sudden reduction\" model of Sect.",
"VA,with κ ' =2.5\\kappa ^{\\prime }=2.5, K=100K=100, and Δf/Δa=20\\Delta f/\\Delta a =20;b) for the Gaussian model of Sect.",
"VB,with κ=5\\kappa =5, K=2000K=2000, and Δf/Δa=10\\Delta f/\\Delta a =10The upper and lower curves are for the system post selected in the states |a 1 〉|a_1{\\rangle } and |a 2 〉|a_2{\\rangle },respectively.",
"The central curve shows the results without post selection.In both cases |ψ 0 〉=(|a 1 〉+|a 2 〉)/2|\\psi _0{\\rangle }=(|a_1{\\rangle }+|a_2{\\rangle })/\\sqrt{2}, and a 2 =-a 1 =1a_2=-a_1=1."
],
[
"Conclusions and discussion",
"In summary, we have considered a \"measuring medium\" consisting of a large number of individual meters of accuracy $\\Delta f$ , arranged in such a way, that their combined action amounts to a Gaussian restriction (REF ) imposed on the Feynman paths of a two-level system.",
"We have shown that, for a fixed period of monitoring, $T$ , as the number of meters, $K$ , increases, typical readouts $f_k$ become highly irregular, as shown in Figs.1, 2a, 3a and 7a, and do not align with one of the eigenvalues of the measured quantity, as suggested in [1] even when decoherence of an initial state is achieved, or Zeno effect is imposed on the system.",
"Thus, a different description of the decoherence process and the Zeno effect was required, and we presented it in Sections V and VI, using a fully tractable non-Gaussian \"hard wall\" model as a guide.",
"In particular, for a system prepared in a pure state (REF ), in the case its Hamiltonian $\\hat{H}$ does not facilitate transitions between the eigenstates of the measured quantity $\\hat{A}$ , decoherence can be linked to a fictitious \"random walk\", which is bound to lead to one of two outcomes, which, in turn, determine the final state of the system, More precisely, we have shown that for $\\Delta a_T= \\Delta f/\\sqrt{K}<< |a_1-a_2|$ , the restriction imposed on the paths in the RPI (REF ) does not limit the readouts $f(t)$ , to the classes (i=1,2) $f(t) \\in \\mathcal {F}_i, \\quad \\mathcal {F}_i=\\lbrace [f] |T^{-1}\\int _0^T (f-a_i)^2dt \\lesssim \\Delta a_T^2\\rbrace \\quad $ as proposed in Eq.",
"(22) of [1].",
"Rather, Eq.",
"(REF ) shows that in this limit a readout $f$ would belong to one of the two classes $f(t) \\in \\mathcal {F}_i^{\\prime }, \\quad \\mathcal {F}_i^{\\prime }=\\lbrace [f] |[T^{-1}\\int _0^T(f-a_i)dt]^2 \\lesssim \\Delta a_T^2\\rbrace , \\quad \\quad $ where, as in (REF ) the integral is understood as the limit of a discrete sum, $ T^{-1}\\int _0^T(f-a_i)dt =lim_{K\\rightarrow \\infty }K^{-1}\\sum _{k=1}^K(f_k-a_i)$ .",
"Condition (REF ) is weaker than (REF ), and allows the measurement readouts to be nowhere differentiable in the continuous limit $K\\rightarrow \\infty $ .",
"It is, however, sufficient to ensure decoherence of a superposition (REF ) into a mixture (REF ) provided $\\Delta a_T<< |a_1-a_2|$ .",
"In practice, this means that a typical readout obtained in an experiment with $K$ meters would look like the one shown in Fig.",
"3a, rather than align with an eigenvalue $a_i$ , as it would do if (REF ) where true.",
"To find out in which of the two eigenstates our monitoring has left the system, we would need to evaluate the (finite) sum $\\sum _{k=1}^K f_k/K$ , in order to see whether its value is closer to $a_1$ and $a_2$ .",
"The \"random walk\" analogy remains useful also in a case of a driven system, subject to Rabi oscillations.",
"For such a system, a typical readout is highly irregular (see Fig.",
"7a) even in a near-Zeno regime, where Rabi oscillations of the system's state are replaced by a telegraph noise (Fig.",
"7b).",
"In this case, as seen in Fig.",
"7c, the corresponding random walk changes direction every time the system jumps from one state to the other.",
"The two evolutions should be considered together, and it is difficult to say whether it is the walker, which causes the system to change its state, or the system, which causes the walker to change direction."
],
[
"Acknowledgements",
"Support of MINECO and the European Regional Development Fund FEDER, through the grant FIS2015-67161-P (MINECO/FEDER) (DS), through MINECO grant SVP-2014-068451 (SR), and through MINECO MTM2013-46553-C3-1-P (EA).",
"are gratefully acknowledged.",
"The SGI/IZOSGIker UPV/EHU and the i2BASQUE academic network are acknowledged for computational resources.",
"This research is also supported by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323."
],
[
"Appendix A. The chi-squared distribution",
"The distribution of the values $x$ of the functional $\\Theta (\\underline{f})=\\sum _{k=1}^K f_k^2/K$ (with $a_1=0$ ) is given by $\\frac{dP(x)}{dx}=\\int df_1..df_K W(\\underline{f})\\delta (X(\\underline{f})-x),$ where $\\delta (x)=(2\\pi )^{-1}\\int \\exp (i\\lambda x) d\\lambda $ is the Dirac delta.",
"Using (REF ), we find $\\frac{dP(x)}{dx}=2^{-1}\\pi ^{-K/2-1} \\Delta {f}^{-K}\\int d\\lambda \\exp (i\\lambda x) I(\\lambda )^K,\\quad $ with $I(\\lambda )=\\int df \\exp [-(\\kappa T +i\\lambda )f^2/K]=[K\\pi /(\\kappa T+i \\lambda )]^{1/2}$ .",
"We then have $\\frac{dP(x)}{dx}=(2\\pi )^{-1}(\\kappa T) ^{K/2}\\int d\\lambda \\frac{ \\exp (i\\lambda x)}{(\\kappa T+i\\lambda )^{K/2}} .\\quad $ The last integral must be evaluated for $K$ both even and odd.",
"For $K=2M$ the contour of integration can be closed in the upper half-plane, and application of the Cauchy integral formula [15] yields (REF ) immediately.",
"For $K=2M+1$ we cut the complex $\\lambda $ -plane from $\\lambda = i\\kappa T$ to $+i\\infty $ , and deform the contour integration to run up and down along the opposite sides of the cut.",
"Integration along the cut then gives (up to a constant factor) $x^{M-1/2}\\exp (-\\kappa Tx) \\Gamma (1/2-M)$ .",
"Using the relation $\\Gamma (1-z)\\Gamma (z)=\\pi /\\sin (\\pi z)$ [15] then yields (REF ).",
"In statistics, this result is also known a the \"chi-squared distribution\" [18]."
],
[
"Appendix B. Normal distributions and the central limit theorem",
"Consider $K$ independent normally distributed variables $f_k$ , $1\\le k\\le K$ , $W(f_k)=\\mathcal {N}(f_k|\\mu ,\\sigma ).$ By the Central Limit Theorem [19], the sum $Y=\\sum _{k=1}^K f_k$ is also normally distributed, $W(Y)=\\mathcal {N}(Y|K\\mu ,\\sqrt{K}\\sigma ).$ A rescaled, and shifted variable $X=A(Y-B)$ is then distributed according to $W(X)=\\mathcal {N}(X|A(K\\mu -B),A\\sqrt{K}\\sigma ).$ Using (REF ), we obtain Eqs.",
"(REF ) and (REF ) from (REF ) and (REF )."
],
[
"Appendix C. Stochastic simulation algorithm",
"We consder an $N \\ge 1$ -level system, with a Hamiltonian $\\hat{H}$ and operator $\\hat{A}$ , representing the measured quantity, $[\\hat{A},\\hat{H}] \\ne 0$ , $ \\hat{A} |a_j \\rangle = a_j |a_j \\rangle $ , $\\langle a_i | a_j \\rangle = \\delta _{i,j}$ , with $i,j=1,..,N$ .",
"Below we describe a Monte Carlo (MC) procedure to draw a single realisation of the system's dynamics, during the simulated time interval $[0,T]$ .",
"By repeating it $S$ times, it is possible to get a MC sample of the random process, and evaluate the required statistics.",
"Given the number of measurements $K$ and the monitoring time $T$ , the procedure is as follows: —————————————————————————————————– 1.",
"Assign the measured operator $\\hat{A}=\\sum _{j=1}^N|a_j{\\rangle }a_j{\\langle }a_j|$ 2.",
"Assign the number $K \\ge 1$ of measurements and the time step $\\tau ={T}{K}$ 3.",
"Assign the evolution operator $\\hat{U} = \\mbox{exp}(-i\\hat{H}\\tau )$ 4.",
"Assign the measure $G(f)$ , $\\int \\!",
"G^2(f) \\, df =1$ , with $ \\int \\!",
"f \\, G^2(f) \\, df = 0 $ 5.",
"Assign initial state of the system $ | \\psi _0 \\rangle = \\sum _{j=1}^N \\alpha ^j_0 |a_j\\rangle $ , with $\\sum _{j=1}^N |\\alpha ^j_0|^2=1$ for $k=0,..,K-1$ do 6.",
"Assign time $t_k =k \\tau $ 7.",
"Evolve the state of the system: $ | \\phi _{k} \\rangle = \\hat{U} |\\hat{\\psi }_k \\rangle $ 8.",
"Compute the probabilities $ \\underline{p}_{k} = \\left\\lbrace {\\langle }a_1|\\phi _k{\\rangle }|^2,{\\langle }a_2|\\phi _k{\\rangle }|^2,..,{\\langle }a_N|\\phi _k{\\rangle }|^2 \\right\\rbrace $ 9.",
"Select the state index $i_{k} \\in \\left\\lbrace 1,..,N\\right\\rbrace $ with probabilities $\\underline{p}_{k}$ 10.",
"Draw the observed value of $f_k \\sim G^2(f-a_{i_k})$ 11.",
"Compute the normalisation $M_{f_k} = \\sqrt{ \\sum _{j=1}^N G^2(f_k-a_j) |{\\langle }a_j|\\phi _k{\\rangle }|^2 }$ 12.",
"Use $f_k$ to construct $|\\psi _{k+1}\\rangle $ : $|{\\psi }_{k+1}\\rangle = \\sum _{j=1}^N \\frac{G(f_k-a_j)}{M_{f_k}} {\\langle }a_1|\\phi _k{\\rangle }|a_j\\rangle $ end ——————————————————————————————————-"
]
] | 1612.05796 |
[
[
"Wormholes from cosmological reconstruction based on Gaussian processes"
],
[
"Abstract We study the model-independent traversable wormholes from cosmological reconstruction based on Gaussian processes (GP).",
"Using a combination of Union 2.1 SNe Ia data, the latest observational Hubble parameter data and recent Planck's shift parameter, we find that our GP method can give a tighter constraint on the normalized comoving distance, its derivatives and the dark energy equation of state than the previous work \\cite{1}.",
"Subsequently, two specific traversable wormhole solutions are obtained, i.e., the cases of a constant redshift function and a linear shape function.",
"We find that, with decreasing cosmic acceleration, the traversal velocity $v$ of the former case increases and the amounts of exotic matter $I_V$ of the latter case decreases."
],
[
"Introduction",
"Modern cosmology has already entered a precise, data-driven era.",
"In 1998, the elegant discovery that the universe is undergoing the phase of accelerated expansion [2], [3], has motivated a great deal of studies concentrated on how best to parameterize the dark energy and measure its properties.",
"An accompanying task comes naturally into being, i.e., explore the corresponding behaviors at astrophysical scales when investigating the expansion history of the universe for different cosmological theories at cosmological scales.",
"Recently, because of the discovery of the accelerating universe, theorists have gradually paid more and more attention to the exotic spacetime configurations, especially, the renewed field—wormholes.",
"To be more concrete, in both cases (accelerating universe and wormholes) the null energy condition (NEC) is violated and consequently all of other energy conditions.",
"Thus, an appealing overlap between two seemingly separated subjects occurs.",
"So far, there is no doubt that together with black holes, white dwarfs, pulsars and quasars, etc., wormholes constituting the most intriguing celestial bodies may provide a new window for physical discovery.",
"In the literature, all the authors almost study the wormhole solutions only for a given dark energy model.",
"However, with booming astronomical data, one can investigate better the wormhole spacetime configurations supported by cosmological observations through implementing highly precise constraints on a given dark energy model, in order to avoid choosing the values of the cosmological parameters arbitrarily.",
"In our previous works [4], [5], we have studied geometrical and holographical dark energy wormholes constrained by astrophysical observations for details, and verified that the exotic spacetime configurations wormholes can actually exist in our universe.",
"It is worth noting that the obtained wormhole configurations are static and spherically symmetrical solutions existing at some certain low redshift.",
"Nonetheless, the previous results depend apparently on given dark energy models.",
"Hence, one shall ask whether the model-independent wormholes exist in the universe.",
"In other words, whether can one investigate the existence of wormholes directly starting from cosmological observations ?",
"To answer this question, one should determine the dark energy equation of state (EoS) $\\omega $ by using the model-independent methods, in order to satisfy the requirement $\\omega <-1$ which ensures the wormhole spacetime structure open.",
"During the past few years, there were many model-independent methods to study the evolutional behaviors of dark energy EoS $\\omega $ in the literature, for instance, principal component analysis (PCA) [6] and Gaussian processes (GP) [1], etc.",
"In this work, we would like to use the GP method to reconstruct the dark energy EoS $\\omega $ and investigate the correspondingly model-independent wormhole solutions.",
"The GP algorithm is a fully Bayesian approach for smoothing data, and can preform a reconstruction of a function directly from data without assuming a parameterization of the function.",
"As a result, one can determine any cosmological quantity from cosmic data, and the key requirement of the GP algorithm is just the covariance function which entirely depends on the cosmic data.",
"In light of the special advantage, the GP method has been widely applied in exploring the expansion dynamics of the universe [1], [7], [8], the test of the base cosmological model [9], the cosmography [10], the distance duality relation [11], the determination of the interaction between dark energy and dark matter and cosmic curvature [12], [13], dodging the cosmic curvature to probe the constancy of the speed of light [14], dodging the matter degeneracy to determine the dynamics of dark energy [15], the slowing down of cosmic acceleration [16], etc.",
"It is noteworthy that the analysis in paper [17] has indicated that the Matérn ($\\nu =9/2$ ) covariance function is a better choice to carry out the reconstruction and it has been used in papers [9], [12].",
"The outline of the rest paper is as follows: In the next section, we make a brief review on GP method.",
"In Section III, we reconstruct the dark energy EoS $\\omega $ in order to study the corresponding wormhole spacetime configurations.",
"In Section IV, we obtain two specific wormhole solutions including the cases of a constant redshift function and a linear shape function, and investigate the related physical properties.",
"The discussions and conclusions are presented in the final section (we use the units $8\\pi G=c=1$ )."
],
[
"GP method",
"In a spatially flat Friedmann-Robertson-Walker (FRW) universe, the luminosity distance $d_L(z)$ can be expressed as $d_L(z)=\\frac{1+z}{H_0}\\int ^{z}_{0}\\frac{dz^{\\prime }}{E(z^{\\prime })}, $ where $z$ denotes the redshift, $H_0$ the present-day value of the Hubble parameter and $E(z)$ the dimensionless Hubble parameter, respectively.",
"Subsequently, using the normalized comoving distance $D(z)=H_0(1+z)^{-1}d_L(z)$ , the dark energy EoS is written as $\\omega (z)=\\frac{2(1+z)D^{\\prime \\prime }-3D^{\\prime }}{3D^{\\prime }[(1+z)^3\\Omega _{m0}D^{\\prime 2}-1]}, $ where the prime denotes the derivative with respect to (w.r.t) the redshift $z$ and $\\Omega _{m0}$ is present-day value of the matter density ratio parameter.",
"To study the model-independent wormhole configurations, firstly, one needs to reconstruct the dynamic dark energy EoS $\\omega (z)$ .",
"Then, as described in our previous works [4], [5], [18], [19], one shall concentrate on the parts in which the condition $\\omega (z)<-1$ is satisfied and study the static and spherically symmetrical wormhole solutions at some certain redshift.",
"The GP method can reconstruct a function directly from observational data without assuming a concrete parameterization for the function.",
"Here we utilize the package GaPP (Gaussian Processes in Python) [1] to implement the reconstruction, which is firstly invented by Seikel et al.",
"for a pedagogical introduction to GP.",
"Usually, the GP is a generalization of a Gaussian distribution which is the distribution of a random variable.",
"In addition, the GP exhibits a distribution over functions.",
"At each point $x$ , the reconstructed function $f(x)$ is a Gaussian distribution with a mean value and Gaussian error.",
"The key of the GP method is a covariance function $k(x,\\tilde{x})$ which correlates the function $f(x)$ at different reconstruction points.",
"The covariance function $k(x,\\tilde{x})$ depends entirely on two hyperparameters $l$ and $\\sigma _f$ , which characterize the coherent scale of the correlation in $x$ -direction and typical change in the $y$ -direction, respectively.",
"In general, the choice is the squared exponential covariance function $k(x,\\tilde{x})=\\sigma _f^2 \\mathrm {exp}[-|x-\\tilde{x}|^2/(2l^2)]$ .",
"However, the analysis in [17] has verified that the Matérn ($\\nu =9/2$ ) covariance function is a better choice to carry out the reconstruction.",
"Hence, we will adopt the Matérn ($\\nu =9/2$ ) covariance function in the following analysis: $k(x,\\tilde{x})=\\sigma _f^2 \\mathrm {exp}(-\\frac{3|x-\\tilde{x}|}{l})\\times [1+\\frac{3|x-\\tilde{x}|}{l}+\\frac{27(x-\\tilde{x})^2}{7l^2}+\\frac{18|x-\\tilde{x}|^3}{7l^3}+\\frac{27(x-\\tilde{x})^4}{35l^2}].",
".",
"$ In our reconstruction, we use the Union 2.1 data sets [20] which consist of 580 SNe Ia data and cover the redshift range [0.015,1.4].",
"As noted in paper [1], we transform the distance modulus $m-M$ to $D$ in the following manner $m-M-25+5\\lg (\\frac{H_0}{c})=5\\lg [(1+z)D] $ with $H_0=70kms^{-1}Mpc^{-1}$ .",
"Furthermore, we set the initial conditions $D(z=0)$ and $D^{\\prime }(z=0)=1$ in the reconstruction process.",
"Notice that the values of $D$ just depend on a combination of the absolute magnitude $M$ and $H_0$ .",
"Different from the previous literature [1], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], we adopt the latest 36 observational Hubble parameter data (OHD) from the paper [21] and the shift parameter $\\mathcal {R}=1.7488\\pm 0.0074$ from the recent Planck's result [22] as important supplements of SNe Ia data, since the GP algorithm in GaPP does need the data of $D$ and $D^{\\prime }$ at the same time.",
"To be more precise, we conclude the relations between the above-mentioned data sets and the normalized comoving distance $D$ as follows $m-M\\Longrightarrow D,$ $\\frac{H_0}{H(z)}\\Longrightarrow D^{\\prime },$ $\\mathcal {R}=\\sqrt{\\Omega _{m0}}\\int ^{z_c}_0\\frac{dz^{\\prime }}{E(z^{\\prime })}\\Longrightarrow D,$ where $z_c=1089.0$ denotes the redshift of recombination and we use the values $H_0=70kms^{-1}Mpc^{-1}$ and $\\Omega _{m0}=0.27\\pm 0.015$ [20].",
"Subsequently, we verify the correctness of our method by using the Union 2.1 data sets to run the modified code.",
"Obviously, the result in Fig.",
"REF is consistent with that in Fig.",
"6 of the paper [16].",
"Figure: The GP reconstruction of f(x),f ' (x)f(x), f^{\\prime }(x) and f '' (x)f^{\\prime \\prime }(x) using the Union 2.1 data sets.",
"The shaded regions are reconstructions with 68%68\\% and 95%95\\% confidence level.",
"The blue lines represents the underlying true model (the mean value of the reconstructions).",
"Since this plot is aimed at verifying the correctness of our GP method, we take the labels x,f(x),f ' (x)x, f(x), f^{\\prime }(x) and f '' (x)f^{\\prime \\prime }(x) as distinctions with those z,D(z),D ' (z)z, D(z), D^{\\prime }(z) and D '' (z)D^{\\prime \\prime }(z) used in Fig.",
".Figure: The GP reconstruction of D(z),D ' (z)D(z), D^{\\prime }(z) and D '' (z)D^{\\prime \\prime }(z) using a combination of the Union 2.1 data sets, 36 OHD data points and Planck's shift parameter.",
"The 36 OHD data points are shown in the lower right panel.",
"The blue and magenta lines represents the underlying true model (the mean value of the reconstructions) and the base cosmological model, respectively.Figure: The GP reconstruction of dark energy EoS ω(z)\\omega (z) using a combination of the Union 2.1 data sets, 36 OHD data points and Planck's shift parameter.",
"The blue and magenta lines represents the underlying true model (the mean value of the reconstructions) and the base cosmological model, respectively.",
"The left and right panels represent the longitudinal coordinate ranges [-10,5] and [-1.5,0], respectively."
],
[
"Reconstruction results",
"In this section, we implement the reconstruction by using a combination of Union 2.1 data sets, 36 OHD data points and Planck's shift parameter, and the result is shown in Fig.",
"REF .",
"It is easy to find that the combined reconstructions $D(z), D^{\\prime }(z)$ and $D^{\\prime \\prime }(z)$ in Fig.",
"REF give apparently a tighter constraint than those in Fig.",
"REF , and that the base cosmological model is consistent with the GP reconstructions at $2\\sigma $ level.",
"More precisely, the OHD data provide a tighter constraint in the low redshift range, and the Planck's shift parameter gives a substantially strict high-redshift constraint avoiding the divergence in the high-redshift range of Fig.",
"REF .",
"Therefore, we can reconstruct the dark energy EoS $\\omega (z)$ better in light of the stricter constraints on $D(z), D^{\\prime }(z)$ and $D^{\\prime \\prime }(z)$ .",
"Then, according to Eq.",
"(REF ), we carry out the reconstruction in Fig.",
"REF .",
"To exhibit better the evolving behaviors of $\\omega (z)$ , we make two plots in the longitudinal coordinate ranges [-10,5] and [-1.5,0], respectively.",
"We find that we give a tighter constraint on $\\omega (z)$ by comparing Fig.",
"REF with Fig.",
"9 in paper [1].",
"Furthermore, we find that the base cosmological model is well compatible with our reconstruction at $2\\sigma $ level, and that the dark energy EoS $\\omega (z)$ may exhibit an oscillating behavior at $1\\sigma $ level in the low redshift range [0, 2].",
"Concentrating on the parts $\\omega <-1$ in Fig.",
"REF , this result implies that the evolving wormhole spacetime configurations supported by cosmological observations may actually exist in the universe.",
"However, in the present paper, we just investigate the static and spherically symmetrical wormhole configurations at some certain redshift."
],
[
"Wormhole models",
"Consider a general metric for the wormhole models as follows $ds^2=-e^{2\\Phi (r)}dt^2+\\frac{dr^2}{1-\\frac{b(r)}{r}}+r^2(d\\theta ^2+\\sin ^2\\theta d\\phi ^2), $ where $r$ , $\\theta $ , $\\phi $ , $\\phi (r)$ and $b(r)$ denote the radial coordinate, angular coordinates, the redshift function and shape function, respectively.",
"Usually speaking, there are four necessary ingredients to form a wormhole configuration [23]: $\\diamond $ The NEC is violated.",
"$\\diamond $ Satisfy the so-called flaring-out conditions, i.e., $b(r_0)=r_0$ , $b^{\\prime }(r_0)<1$ and $b(r)<r$ when $r>r_0$ .",
"$\\diamond $ $\\Phi (r)$ shall be finite anywhere so that an horizon can be avoided.",
"$\\diamond $ The asymptotically flatness condition shall be satisfied, i.e., $b/r\\rightarrow 0$ and $\\Phi \\rightarrow 0$ when $r\\rightarrow \\infty $ .",
"In an orthonormal reference frame, using the Einstein field equation, $G_{\\mu \\nu }=T_{\\mu \\nu }$ , we express the stress-energy scenario as: $\\rho =\\frac{b^{\\prime }}{r^2}, $ $p_r=\\frac{b}{r^3}-2\\frac{\\Phi ^{\\prime }}{r}(1-\\frac{b}{r}), $ $p_t=(1-\\frac{b}{r})[\\Phi ^{\\prime \\prime }+(\\Phi ^{\\prime })^2-\\Phi ^{\\prime }\\frac{b^{\\prime }r-b}{2r^2(1-b/r)}+\\frac{\\Phi ^{\\prime }}{r}-\\frac{b^{\\prime }r-b}{2r^3(1-b/r)}], $ where $\\rho (r)$ , $p_r(r)$ , $p_t(r)$ and the prime denote the matter energy density, the radial pressure, the tangential pressure and the derivative w.r.t.",
"$r$ , respectively.",
"Using the stress-energy conservation equation, $T^{\\mu \\nu }_{\\hspace{8.53581pt};\\nu } = 0$ , one can have $p^{\\prime }_r=\\frac{2}{r}(p_t-p_r)-\\Phi ^{\\prime }(\\rho +p_r), $ which can be regarded as the relativistic Euler equation or the hydrostatic equation of equilibrium for the matter threading a wormhole spacetime structure.",
"Subsequently, we would like to calculate the wormhole solutions by adopting the mean value $\\omega _{m}=-1.075$ , which lies in the $1\\sigma $ range $\\omega (0.5)\\in [-1.188, -0.982]$ at redshift $z=0.5$ (see Fig.",
"REF )."
],
[
"A Constant Redshift Function",
"Take into account a constant redshift function $\\Phi =C$ , substituting the dark energy EoS $p=\\omega _{m}\\rho $ and Eq.",
"(REF ) into Eq.",
"(REF ), we obtain $b(r)=r_0(\\frac{r_0}{r})^{\\frac{1}{\\omega _{m}}}.",
"$ One can easily check that this shape function $b(r)<r$ when $r>r_0$ , which satisfies the flaring-out conditions.",
"Subsequently, evaluating at the wormhole throat $r_0$ , we get $b^{\\prime }(r_0)=-\\frac{1}{\\omega _{m}}$ .",
"Furthermore, taking the mean value $\\omega _{m}=-1.075$ from GP reconstruction, one can also get $b^{\\prime }(r_0)=-0.93<1$ .",
"Because the constant redshift function $\\Phi $ is finite everywhere and $b/r\\rightarrow 0$ when $r\\rightarrow \\infty $ , this wormhole spacetime structure is both asymptotically flat and traversable.",
"Thus, the dimension of this wormhole configuration can be substantially large in principle.",
"In what follows, we would like to analyze the the traversability of the wormhole configuration as our previous works [4], [5], [18], [19].",
"Generally speaking, for a traveler in the spaceship who plans to journey successfully through the wormhole throat, there are three necessary ingredients [23]: $\\diamond $ The tidal acceleration shall not exceed 1 Earth's gravitational acceleration $g_\\star $ .",
"$\\diamond $ The acceleration felt by the travelers shall not exceed 1 Earth's gravitational acceleration $g_\\star $ .",
"$\\diamond $ The traverse time measured by the travelers and the observers who keep static at the space station shall satisfy the quantitative relations.",
"More detailed descriptions can be found in paper [23].",
"For simplicity, we only exhibit the key formula through some derivations in the following manner $v\\leqslant r_0\\sqrt{\\frac{\\omega _{m} g_\\star }{\\omega _{m}+1}}, $ Figure: In the left panel, we exhibit the relation between the traversal velocity vv and the throat radius of the wormhole r 0 r_0 by assuming ω m =-1.075\\omega _{m}=-1.075 and g ☆ =9.8g_\\star =9.8 m//s 2 ^{2}.",
"In the medium panel, we exhibit the relation between the traversal velocity vv and the dark energy EoS ω m \\omega _{m} by assuming r 0 =100r_0=100 m and g ☆ =9.8g_\\star =9.8 m//s 2 ^{2}.",
"In the right panel, we exhibit the relation between the traversal velocity vv and the Earth's gravitational acceleration g ☆ g_\\star by assuming r 0 =100r_0=100 m and ω m =-1.075\\omega _{m}=-1.075.",
"The shaded regions and the red (dashed) lines correspond to the allowable regions for the traversal velocity vv and the extreme cases the equality signs in inequality ().where $v$ is the traversal velocity.",
"Subsequently, assuming the height of the traverser $h=2$ m, $r_0=100$ m, $\\omega _{m}=-1.075$ and $g_\\star =9.8$ m$/$ s$^{2}$ , we can have the traversal velocity $v\\approx 1185.19$ m/s, which is approximately equivalent to 3 times the speed of sound.",
"Furthermore, we can also get the traverse time $\\Delta \\tau \\approx \\Delta t\\approx 2r_\\star /v\\approx 16.875s$ by setting the junction radius $r_\\star =10000$ m. Moreover, one can find that the value of the traversal velocity $v$ in inequality (REF ) depends on the throat radius of the wormhole $r_0$ , the dark energy EoS $\\omega _{m}$ from the GP reconstruction and the Earth's gravitational acceleration $g_\\star $ .",
"In addition, we are also of interest to investigate the dependence of the traversal velocity $v$ on the above-mentioned physical quantities.",
"In the left panel of Fig.",
"REF , it is easy to be seen that the traversal velocity $v$ is a linear function of the throat radius of the wormhole $r_0$ , meaning that with increasing $r_0$ , $v$ will also gradually increase.",
"In the medium panel of Fig.",
"REF , we find that the traversal velocity $v$ increase monotonically with gradually dark energy EoS $\\omega _{m}$ , which means that $v$ will increase with decreasing cosmic acceleration.",
"Notice that the dark energy EoS $\\omega _{m}$ will not approach $-1$ since the right hand side of inequality (REF ) diverges.",
"In the right panel of Fig.",
"REF , assuming that the largest gravitational acceleration a human being can bear is 6.5 times Earth's gravitational acceleration, we find that the traversal velocity $v$ still increase monotonically with increasing gravitational acceleration $g_\\star $ ."
],
[
"A Linear Shape Function",
"Take into account a specific shape function $b(r)=r_0-\\frac{1}{\\omega _{m}}(r-r_0)$ and utilize the dark energy EoS $p=\\omega _{m}\\rho $ , we have $\\Phi ^{\\prime }(r)=-\\frac{1}{2r} \\qquad and \\qquad \\Phi (r)=-\\frac{1}{2}\\ln r+C, $ where $C$ denotes an arbitrary integration constant.",
"We find that this solution is non-asymptotically flat since it diverges directly when $r\\rightarrow \\infty $ .",
"Thus, this solution is non-traversable for an interstellar traveler.",
"However, in theory, one can construct a traversable wormhole geometry through gluing an exterior flat geometry into the interior geometry at a junction radius $r_\\diamond $ .",
"So the constant $C$ can be written as $C=\\Phi (r_\\diamond )+\\frac{1}{2}\\ln (\\frac{r_\\diamond }{r})$ .",
"Subsequently, we will utilize the the method of “ volume integral quantifier ” (VIQ) [24] to quantify the exotic matter constructing the traversable wormhole in the finite range $r_0\\leqslant r\\leqslant r_\\diamond $ .",
"For simplicity, one can obtain the amounts of the exotic matter by calculating the definite integral $\\int T_{\\mu \\nu }k^\\mu k^\\nu dV$ , where $T_{\\mu \\nu }$ is still the stress-energy tensor and $k^{\\mu }$ any future directed null vector.",
"In what follows, computing the quantity $I_V=\\int [p_r(r)+\\rho ]dV$ , we obtain $I_V=\\int ^{r_\\diamond }_{r_0}(r-b)[\\ln (\\frac{e^{2\\Phi }}{1-\\frac{b}{r}})]^{\\prime }dr. $ Through some easy derivations, we get $I_V=(1+\\frac{1}{\\omega _{m}})(r_0-r_\\diamond ).",
"$ Since the amounts of exotic matter $I_V$ is associated with the throat radius of the wormhole $r_0$ , the dark energy EoS $\\omega _{m}$ from the GP reconstruction and the junction radius $r_\\diamond $ , it is necessary to study the the dependence of the amounts of exotic matter $I_V$ on the above-mentioned physical quantities.",
"In the left and right panels of Fig.",
"REF , we find that $I_V$ increases with increasing $r_0$ and $r_\\diamond $ and approaches zero finally, since it is a linear function of $r_0$ and $r_\\diamond $ .",
"In the medium panel of Fig.",
"REF , however, we find that $I_V$ increases very slowly with increasing $\\omega _{m}$ .",
"This can be ascribed to the value of $r_0-r_\\diamond $ is very large.",
"It is not difficult to verify that the amounts of exotic matter $I_V\\longrightarrow 0$ when $r_j\\rightarrow r_0$ and $\\omega _{m}$ is fixed.",
"For instance, if we still use the dark energy EoS $\\omega _{m}=-1.075$ from GP reconstruction, Eq.",
"(REF ) can be rewritten as $I_V=0.0698(r_0-r_\\diamond ).",
"$ This result indicates that, in theory, one can construct a traversable wormhole with infinitesimal amounts of averaged-NEC (ANEC) violating dark energy fluid in the present situation.",
"It is worth noting that we have obtained the same result by using the model-independent GP method, which does not depend on any concretely cosmological model.",
"Figure: In the left panel, we exhibit the relation between the amounts of exotic matter I V I_V and the throat radius of the wormhole r 0 r_0 by assuming ω m =-1.075\\omega _{m}=-1.075 and r ⋄ =500r_\\diamond =500 m. In the medium panel, we exhibit the relation between the amounts of exotic matter I V I_V and the dark energy EoS ω m \\omega _{m} by assuming r 0 =100r_0=100 m and r ⋄ =500r_\\diamond =500 m. In the right panel, we exhibit the relation between the amounts of exotic matter I V I_V and the junction radius r ⋄ r_\\diamond by assuming r 0 =100r_0=100 m and ω m =-1.075\\omega _{m}=-1.075."
],
[
"Discussions and conclusions",
"The elegant discovery that the universe is in accelerating expansion has motivated theorists to pay afresh attention to the old field—wormholes, since in both fields an surprising overlap occurs, i.e., the violation of the NEC.",
"However, in the literature, all the authors almost study the wormhole spacetime configurations for a given cosmological model.",
"Therefore, in light of rapidly increasing cosmic data, we are aiming at exploring the model-independent dark energy wormholes by using the GP method.",
"In this work, first of all, we make a brief review on the GP method and demonstrate the correctness of our reconstruction method.",
"Subsequently, using a combination of Union 2.1 580 SNe Ia data, the latest 36 OHD data points and recent Planck's shift parameter, we find that our GP method can give a tighter constraint on the normalized comoving distance $D(z)$ , its derivatives $D^{\\prime }(z), D^{\\prime \\prime }(z)$ and the dark energy EoS $\\omega (z)$ than the previous literature [1].",
"In the mean while, we find that the base cosmological model is well compatible with our reconstruction at $2\\sigma $ level.",
"In what follows, two specific traversable wormhole solutions are obtained in the framework of a perfect fluid, i.e., the cases of a constant redshift function and a linear shape function.",
"By choosing the mean value $\\omega _{m}=-1.075$ , which lies in the $1\\sigma $ range $\\omega (0.5)\\in [-1.188, -0.982]$ at redshift $z=0.5$ (see Fig.",
"REF ), we analyze the traversabilities of the former case and investigate the dependence of the traversal velocity $v$ on the throat radius of the wormhole $r_0$ , the dark energy EoS $\\omega _{m}$ from the GP reconstruction and the gravitational acceleration $g_\\star $ .",
"Furthermore, we also calculate the amounts of exotic matter of the latter case and study the the dependence of the amounts of exotic matter $I_V$ on the throat radius of the wormhole $r_0$ , the dark energy EoS $\\omega _{m}$ from the GP reconstruction and the junction radius $r_\\diamond $ .",
"We find that, with decreasing cosmic acceleration, the traversal velocity $v$ of the former case increases and the amounts of exotic matter $I_V$ of the latter case decreases (see Figs.",
"REF -REF ).",
"The dark energy EoS $\\omega (z)$ from our GP reconstruction may exhibit an oscillating behavior at $1\\sigma $ level in the low redshift range [0,2], which indicates that the evolving wormhole spacetime configurations supported by cosmological observations may actually exist in the universe.",
"Furthermore, the evolving wormhole spacetime structure originated from cosmic acceleration may open at a relatively high redshift and close at a low redshift based on some unknown mechanism (e.g., quantum fluctuations).",
"According to the papers [25], [26], the obtained wormhole configurations from our GP construction can be stable by choosing appropriately the values of the throat radius and junction radius.",
"However, the interesting stability analysis is beyond the scopic of the present work.",
"We expect more and more high quality cosmic data can help the human beings to explore the nature of dark energy and dark matter."
],
[
"acknowledgements",
"The author Xin-He Meng warmly thanks Professors Bharat Ratra and S. D. Odintsov for helpful feedbacks on astrophysics and cosmology.",
"The author Deng Wang thanks Qi-Xiang Zou for beneficial discussions and programming.",
"The authors acknowledge partial support from the National Science Foundation of China."
]
] | 1612.05725 |
[
[
"Inferential framework for two-fluid model of cryogenic chilldown"
],
[
"Abstract We report a development of probabilistic framework for parameter inference of cryogenic two-phase flow based on fast two-fluid solver.",
"We introduce a concise set of cryogenic correlations and discuss its parameterization.",
"We present results of application of proposed approach to the analysis of cryogenic chilldoown in horizontal transfer line.",
"We demonstrate simultaneous optimization of large number of model parameters obtained using global optimization algorithms.",
"It is shown that the proposed approach allows accurate predictions of experimental data obtained both with saturated and sub-cooled liquid nitrogen flow.",
"We discuss extension of predictive capabilities of the model to practical full scale systems."
],
[
"Introduction",
"Autonomous management of two-phase cryogenic flows is a subject of great interest to many spacefarers including effective human exploration of the Solar System [1], [2], [3].",
"It requires development of models that can recognize and predict cryogenic fluid dynamics on-line regime in nominal and faulty flow regimes without human interaction.",
"However, predicting the behavior of two-phase flows is a long standing problem of great complexity [4], [5].",
"It becomes especially challenging when flowing fluids are far away from thermal equilibrium (e.g.",
"during chilldown) and the analysis has to include heat and mass transfer correlations [6], [7], [8], [9].",
"During past decades a number of efficient algorithms [6], [10], [11], [12] and advanced correlation relations for heat and mass transfer [6], [13], [14] have been developed for analysis of multi-phase flows [5], [9], [15], [16], [17].",
"Despite this progress the state of the art in two-phase modeling lacks a general agreement regarding the fundamental physical models that describe the complex phenomena [12].",
"As a consequence, uncertainties in modeling source terms may ultimately have a bigger impact on the results than the particular numerical method adopted [4].",
"Analysis of cryogenic fluids introduces further complications due to relatively poor knowledge of heat and mass transfer correlations in boiling cryogenic flows [18], [19], [20], [21], [22], [23].",
"Even less is known about flow boiling correlations of cryogenic fluids in microgravity [3], [24].",
"To address these and other mission critical issues NASA has developed and implemented an impressive program of research, see e.g.",
"[1], [2], [25], [26], that resulted in emergence of space based fluid management technologies.",
"Under this program a number of important experimental and modeling results have been obtained related to cryogenic two-phase flows (see e.g.",
"[3], [22], [23], [27], [28], [29], [30], [31], [32] and references therein).",
"Specifically, two-phase separated flow models were developed for some the flow regimes [27], [33], [34].",
"A number of optimization techniques have become commercially available for analysis of the model parameters and data correlations [35].",
"However, small time steps and instabilities [27], [34] or implicitness of numerical scheme [36], [37] impose substantial limitations on the speed of the solution, efficiency of multi-parametric optimization, and possibility of on-line application.",
"As a result accurate predictions of transient cryogenic flows remain a challenge [23], [35] and extensive research is currently under way [3], [22].",
"Some of the grand challenges of this analysis include inference of parameters of cryogenic correlations from experimental time-series data and extension of the results obtained from small experimental subsystems to full scale practical systems.",
"In this paper we report on the development of separated two-fluid model suitable for fast on-line analysis of cryogenic flows and introduce model-based inferential framework capable of efficient multi-parametric optimization of the model parameters.",
"We demonstrate an application of this inferential framework to the problem of modeling chilldown in horizontal cryogenic line.",
"This problem has been shown to be a difficult one to solve in the earlier research [35].",
"Using proposed approach we obtain accurate predictions for transient liquid nitrogen flow both under sub-cooled and saturated conditions.",
"The paper is organized as follows.",
"In the next Section we briefly describe the model and algorithm of its integration.",
"In the Section we introduce probabilistic framework for inference of the model parameters, discuss the uncertainties in the source terms and their parameterization.",
"In the Section we introduced constitutive relations used to model source teams.",
"The approach to the inference of model parameters is discussed in Section .",
"In the Section we describe an application of the proposed technique to an analysis of cryogenic chilldown in horizontal pipe.",
"Finally, in the Conclusions we summarize the obtained results and discuss directions of future work."
],
[
"Model",
"We limit our analysis to one-dimensional flow networks having in mind fast on-line applications of the solver.",
"To this end we have developed and tested a number of algorithms [38], [39], [40], [41], [42], [43], [44], [45]) to see if their speed and accuracy can satisfy requirements of real-time application.",
"It was shown that the nearly-implicit algorithm, similar to one developed in [10], can be applied successfully for on-line predictions of non-homogeneous ($u_g \\ne u_l$ ) and non-equilibrium ($T_g\\ne T_l$ ) flows.",
"In this section we will describe briefly the corresponding model equations and the algorithm of their integration.",
"Extensive details can be found in [38], [39], [46], [40], see also [44], [45]."
],
[
"Model Equations",
"In nearly implicit algorithm a closed system of equations is obtained assuming equal local pressure values for the both phases [11], [47], [15].",
"The corresponding six-equation model consists of a set of conservation laws for the mass, momentum, and energy of the gas (see e.g.",
"[10], [48], [6], [38], [39] ) $\\hspace{-8.5359pt}\\begin{array}{l}{\\left( {A{\\alpha }{\\rho _g}} \\right)_{,t}} + {\\left( {A{\\alpha }{\\rho _g}{u_g}} \\right)_{,x}} = A{{\\rm {\\Gamma }}_g}\\\\{\\left( {A{\\alpha }{\\rho _g}{u_g}} \\right)_{,t}} + {\\left( {A{\\alpha }{\\rho _g}u_g^2} \\right)_{,x}} + A{\\alpha }{p_{,x}} = - A{\\alpha }{\\rho _g}{z_{,x}}\\\\~~~~~~~~ - {\\tau _{gw}}{l_{wg}} - {\\tau _{gi}}{l_i} + A{{\\rm {\\Gamma }}_g}{u_{ig}}\\\\{\\left( {A{\\alpha }{\\rho _g}{E_g}} \\right)_{,t}} + {\\left( {A{\\alpha }{\\rho _g}{E_g}{u_g}} \\right)_{,x}} = -Ap {\\alpha _{,t}} - {\\left( {pA\\alpha {u_g}} \\right)_{,x}} \\\\~~~~~~~~ + {{\\dot{q}}_{gw}}{l_{wg}} + {{\\dot{q}}_{gi}}{l_i} + A{\\rm \\Gamma }_{g}H_{g}\\end{array}$ and liquid phases $\\hspace{-8.5359pt}\\begin{array}{l}{\\left( {A\\beta {\\rho _l}} \\right)_{,t}} + {\\left( {A\\beta {\\rho _l}{u_l}} \\right)_{,x}} = - A{{\\rm {\\Gamma }}_g}\\\\{\\left( {A\\beta {\\rho _l}{u_l}} \\right)_{,t}} + {\\left( {A\\beta {\\rho _l}u_l^2} \\right)_{,x}} + A\\beta {p_{,x}} = - A\\beta {\\rho _l}{z_{,x}} -\\\\~~~~~~~~ {\\tau _{lw}}{l_{wl}} - {\\tau _{li}}{l_i} - A{{\\rm {\\Gamma }}_g}{u_{il}}\\\\{\\left( {A\\beta {E_l}{\\rho _l}} \\right)_{,t}} + {\\left( {A\\beta {E_l}{\\rho _l}{u_l}} \\right)_{,x}} = -Ap {\\beta _{,t}} - {\\left( {pA\\beta {u_l}} \\right)_{,x}} +\\\\~~~~~~~~ {{\\dot{q}}_{lw}}{l_{wl}} + {{\\dot{q}}_{li}}{l_i} - A{\\rm \\Gamma }_{g}H_{l}.\\\\\\end{array}$ Here $p$ , $\\alpha $ , $T$ , and $ \\rho $ are pressure, temperature, and density of the fluid.",
"$ E $ is the total specific energy, $ H_{g(l)}$ is the specific enthalpy of the gas generated (liquid evaporated) at the interface and near the wall.",
"$u$ is the fluid velocity, $\\tau $ is the wall shear stress, and $\\dot{q}$ is the heat flux at the wall and at the interface.",
"The total mass flux $\\Gamma _{g}=\\Gamma _{wg}+\\Gamma _{ig}$ has two components corresponding to the mass transfer at the wall $\\Gamma _{wg}$ and at the interface $\\Gamma _{ig}$ .",
"The fluid dynamics equations are coupled to the equation for the wall temperature $T_w$ $\\begin{array}{l}{\\rho _w}{c_w}{d_w}\\frac{{\\partial {T_w}}}{{\\partial t}} = {h_{wg}}\\left( {{T_g} - {T_w}} \\right)\\\\ \\qquad + {h_{wl}}\\left( {{T_l} - {T_w}} \\right) + {h_{amb}}\\left( {{T_{amb}} - {T_w}} \\right).\\end{array}$ Here $\\rho $ , $c$ , and $d$ are density, specific heat, and thickness of the pipe wall, $h$ is the heat transfer coefficient corresponding to the ambient ($h_{amb}$ ) and internal heat flowing to the wall from the gas ($h_{wg}$ ) and liquid ($h_{wl}$ ) phases.",
"The characteristic feature of the model (REF ), (REF ) is its non-hyperbolicity [15], [49] related to the assumption of exclusively algebraic terms describing the interfacial drag and incomplete formulation for the interfacial momentum coupling.",
"It can be shown that this system does not have a complete set of real eigenvalues and does not represent a well-posed system of equations [38], [48], [50].",
"It is also known that this system displays lack of positivity and instabilities due to phase appearance/disappearance process [11], [51].",
"In addition, the effect of algebraic source terms represents a system of “stiff” differential equations [15] and roundoff errors may significantly contribute to numerical instabilities.",
"Despite these difficulties a number of algorithms were developed [6], [10], [11] and successfully employed to predict two-phase flows of boiling water in large scale system.",
"In our development of the algorithm we were following the guidelines of earlier research."
],
[
"Algorithm",
"The choice of the algorithm was guided by the fact [11] that all current reactor thermal-hydraulics codes [6], [10], [52] originate from Liles and Reed [53] extension of Harlow and Asden [54], [55] all-speed implicit continuous-fluid Eulerian algorithm.",
"These codes enhance the stability of the method and eliminate material CFL restrictions using a couple of extensions: stability-enhancing two-step [6] and nearly implicit [10] algorithms.",
"In this work the discretization and integration of the model equations (REF ) - (REF ) follow closely the nearly-implicit method described in RELAP5-3D [10] (see [39], [44] for the details).",
"The integration was performed in two steps.",
"The first step of the algorithm can be briefly summarized as follows: (i) Solve expanded equation with respect to pressure expressed in terms of new velocities; (ii) Solve momenta equations written in the form of block tri-diagonal matrix for the new velocities; (iii) Find new pressure; (iv) Find provisional values for energies and void fractions using expanded equations of states; (v) Find provisional values of mass fluxes and heat transfer coefficients using provisional values of temperatures obtained.",
"At the second step new values of the densities, void fractions, and energies are found by solving the unexpanded conservation equations for the phasic masses and energies using provisional values for the heat and mass fluxes in source terms.",
"The solution is reduced to independent solution of four tri-diagonal matrices.",
"The values of pressure and velocities in these matrices are taken at the new time step.",
"The resulting scheme is efficient and fast and can integrate 1000 sec of real time chilldown in a few seconds of computational time.",
"For a model consisting of $N$ control volumes it involves inversion of N $4\\times 4$ matrices, solution of $2\\times (N-1)$ tree-block-diagonal matrix equation, solution of four $N\\times N$ tridiagonal matrix equations, and $N\\times m$ explicit computations.",
"A special attention was paid to the stability of the code.",
"Various methods are available for regularization of the solution including standard upwinding and staggered grid methods as well as ad hoc smoothing and multiple time step controls techniques, see [39] for the details.",
"Specifically, multiple techniques can be used to suppress [10], [49] the non-hyperbolicity.",
"In this work to suppress the non-hyperbolicty we are using so-called virtual mass term [10] ${M_V} = C\\alpha \\beta {\\rho _m}\\left[ {\\frac{{\\partial \\left( {{u_g} - {u_l}} \\right)}}{{\\partial t}} + {u_l}\\frac{{\\partial {u_g}}}{{\\partial x}} - {u_g}\\frac{{\\partial {u_l}}}{{\\partial x}}} \\right]$ in the right hand sides of the momentum equations.",
"In practical computations the terms corresponding to spatial derivatives were neglected.",
"The stability of the algorithm was further enhanced by using the time step control to insure that all the thermodynamic variables remain within the predetermined limits and that the change of these variables at any given time step does not exceed 25% of their values obtained at the previous time step.",
"If these conditions are not satisfied the time step is halved and integration is repeated.",
"If time step goes beyond limiting value the integration is terminated.",
"Similar control is applied to enforce mass conservation in each control volume and in the system as whole.",
"In addition, smoothing mentioned above was found to be a very important tool to ensure stability of the scheme.",
"In this work we followed recommendations provided by Liou [56] and adjust temperature, velocity, and density according to the following expression ${\\phi _{adj}} = g(x){\\phi _d} + \\left( {1 - g(x)} \\right){\\phi _c},$ where $ g(x) = {x^2}\\left( {2x - 3} \\right);\\quad {\\rm and}\\quad x = \\frac{{{\\alpha _d} - {x_{\\min }}}}{{{x_{\\max }} - {x_{\\min }}}}.$ Here “d” stands for disappearing phase and “c” for conducting phase.",
"The exact values of the minimum and maximum void fraction $x_{\\min }$ and $x_{\\max }$ were established using extensive numerical experimentation as$\\sim 10^{-7}$ and $\\sim 10^{-2}$ respectively.",
"The set of equations (REF ), (REF ), and (REF ) is incomplete and a number of closure relations is required to close it.",
"For cryogenic flows, however, the number of available experimental results is limited and further research is required to establish flow boiling correlations (see e.g.",
"[18], [20]).",
"It is, therefore, important that the cryogenic modeling is embedded within optimization framework that allows efficient inference of the correlation parameters and systematic comparison between various functional forms of the constitutive relations.",
"We note that corresponding optimization framework is also one of the key tools required for autonomous control of cryogenic flows.",
"Accordingly in the current work we were focused on development of an efficient optimization framework.",
"Below we briefly outline this approach."
],
[
"Probabilistic framework",
"The most time consuming step in development of the cryogenic flow models is accurate correlation of the model predictions against experimental data (see e.g.",
"[9] and references therein).",
"This step is crucial for practical applications of the model including e.g.",
"reliable design of cryogenic hardware [35], analysis of nominal regimes of operation, fault detection and isolation, and efficient recovery from off-nominal regimes.",
"It becomes even more important when one attempts to extend model predictions to untestable conditions [35], [57] or to practical full-scale systems [58].",
"At present the main approach to correlation of experimental data is based on fitting (mainly by hand) extensive databases obtained in various flow regimes under carefully controlled experimental conditions [6], [59], [23].",
"The rationale behind this approach is an attempt to reduced a very large number of uncertainties inherent to the model and to obtain solution of the fitting problem using traditional techniques.",
"However, such an approach becomes prohibitively expensive in development two-phase flow correlations and autonomous fluid management in microgravity.",
"An efficient solution of the problem in this case has to rely on more intelligent and less expensive methods of inferring correlation parameters.",
"This work is an attempt to establish feasibility of such methods.",
"Below we briefly review the uncertainties of the model and outline probabilistic approach that can be applied to their analysis."
],
[
"Uncertainties ",
"Fundamentally, the probabilistic nature of the model predictions is related to the fact that the interface between two phases is continuously fluctuating and neither location nor the shape of the interface can be resolved by the model.",
"The spatial and time scales of these fluctuations are continuously changing depending on the flow regime.",
"The intensity of these fluctuations is especially significant during chilldown, when liquid and vapor phases coexist under strongly non-equilibrium conditions, see e.g. [31].",
"Another major source of uncertainty is related to the choice of the functional form of the correlations.",
"There have been literally hundreds of various modifications proposed for the flow boiling correlations [60], [61] and the corresponding functional space is continuously expanding [22], [62].",
"To illustrate this point let us consider as an example one of the key correlation parameters so-called critical heat flux, $\\dot{q}_{chf,0} $ , corresponding to the maximum heat transfer from boiling fluid to the wall.",
"One of the best known correlations for the pool boiling value of $\\dot{q}_{chf,0} $ was proposed by Zuber [63] in the form $\\dot{q}_{chf,0} = \\frac{\\pi }{24} h_{lg} \\rho _g \\left(\\frac{\\sigma g (\\rho _l-\\rho _g)}{\\rho _g ^{2}}\\right)^{1/4} \\left(\\frac{\\rho _l }{\\rho _l+\\rho _g}\\right)^{1/2}.$ []$h_{lg}$ latent heat of evaporationZuber's model assumes several approximations, including e.g.",
"the following: (i) rising jets with radius $R_j$ form a square grid with a pitch equal to the fastest growing wavelength due to Taylor instability, (ii) the rising jets are assumed to have critical velocity dictated by Helmholtz instability, (iii) the neutral wavelength of the rising jet is assumed to be $ 2\\pi R_j $ .",
"It is clear from the list of assumptions that numerical constants in Zuber's correlation have to be viewed only as an approximation.",
"Furthermore, this approximation does not take into account surface wettability, pipe curvature, sub-cooling, and surface orientation.",
"Accordingly, several corrections are known [9] that modify functional form of this correlation.",
"In boiling flows further corrections have to be introduced to take into account the dependence of the heat flux on the void fraction, velocity, and sub-cooling of the flow.",
"For example, Griffith et all use the following functional form of the corresponding corrections for cryogenic flows [64], [65] $ \\dot{q}_{chf}& =& \\dot{q}_{chf,0} (\\alpha _{cr} - \\alpha )\\left( 1 + a_1\\left(\\frac{\\rho _l c_l \\Delta T_{sub}}{\\rho _g h_{lg}}\\right)\\right.",
"\\\\&& +\\left.",
"a_2 Re_l + a_3\\left(\\frac{Re_l \\rho _l c_l \\Delta T_{sub}}{\\rho _g h_{lg}}\\right)^{1/2}\\right), \\nonumber $ where $\\alpha _{cr}$ is the critical value of the void fraction and $a_i$ are constants, e.g.",
"$a_1 = 0.0144$ , $a_2 = 10^{-6}$ , $a_3 = 0.5\\times 10^{-3}$ [66], and $\\alpha _{cr} = 0.96$ [64] for water.",
"Different functional forms of similar corrections are also known and will be considered below.",
"In practice, we often used a simpler expression, cf [64], [67] $\\dot{q}_{chf} =\\dot{q}_{chf,0}\\cdot a_1 \\cdot (\\alpha _{cr}-\\alpha )^{a_2} (1+a_3 G^{a_5}),$ where typical values of parameters used in simulations are $a_1$ =1.0, $\\alpha _{cr}$ =0.96, $a_2$ =2.0, $a_3$ =0.16, and $a_4$ =0.2.",
"Another source of uncertainty is added to the problem by the fact that models are often correlated against multiple datasets obtained for different flow conditions.",
"Some of these conditions (e.g.",
"wettability) are not well known.",
"As a result of multiple approximations the number of parameters that have to be established in different flow regimes for practical full-scale systems is of the order of thousand.",
"It becomes clear that computer base intelligent methods are required to handle complexity of this scale in realistic time frame."
],
[
"Probabilistic approach ",
"Here we consider briefly one of the approaches to the solution of this problem based on probabilistic Bayesian method [68].",
"Using this technique one can [68] estimate the probability of unknown model parameters $P(\\theta |d,m) = \\frac{P(d|\\theta ,m)P(\\theta |m) }{P(d|m)},$ compare different models $P(d|m) = \\frac{P(m|d)P(m) }{P(d)}$ and forecast system response $d_{n}$ to untested experimental conditions $P(d_{n}|d,m) =\\int P(d_{n}|\\theta ,d,m)P(\\theta |d,m) d\\theta .$ []$M$ model []$d$ time-series data [G]$\\theta $ set of model parameters [U]$w$ wall Here, $d$ is the experimental time-series data, $m$ is the model, and $\\theta $ is the set of model parameters.",
"There are two important advantages of this approach to bear in mind.",
"The first one is its ability to select simpler models over more complex models, thus resolving so-called “Ockham's razor problem” of optimization.",
"The second advantage is its flexibility in the choice model parameters.",
"It is known that the best predictive performance is often obtained [68] using the most flexible system that can better adapt to the complexity of the data.",
"Accordingly, this approach allows for development of a flexible model with a set of parameters large enough to capture all the required properties of the data.",
"The main outcome of the method is distribution of the model parameters that contains maximum statistical information available in a given experimental data set.",
"Importantly, this information can be updated when new time-series data or new database become available.",
"As a result, the approach tends to provide the best fit to all available data."
],
[
"Equivalent State Space Model",
"One of the key steps in developing probabilistic inferential framework is the solution of the optimization problem.",
"In general terms this problem is formulated by presenting results of integration of equations (REF ) - (REF ) on one time step $t$ in the form of discrete-time state-space model (SSM) $\\begin{array}{ll}x_{t+1} = f(x_t,c) + \\varepsilon _t,\\\\y_{t} = g(x_t,b) + \\chi _t.\\end{array}$ Here $c$ is the set of the model parameters and $x_t$ is a set of dynamical variables $ \\left\\lbrace \\rho _g, \\rho _l, T_g, T_l, u_g, u_l, p, \\alpha \\right\\rbrace ^L_t $ at time $t$ discretized in space on a set of control volumes $V_L$ .",
"The observations $y_t$ in the SSM are related to the unobserved states $x_t$ via nonlinear function $g(x_t,b)$ .",
"$\\varepsilon _t$ and $\\chi _t$ in equations (REF ) are independent identically distributed sources of Gaussian noise.",
"The latter assumption is standard within this approach when sources of noise have multiple origin and are not well established (cf [68], [69], [70]).",
"Although it is possible to determine simultaneously parameter and state of a systems within proposed framework (see e.g.",
"[68], [71], [70]), here for simplicity we neglect measurement noise and assume that the key dynamical variables such as pressure $\\hat{p}$ , wall temperature $\\hat{T}_w$ , and fluid temperature $\\hat{T}_f$ can be measured directly in the experiment.",
"This is indeed the case for the time-series data obtained during chilldown experiment at National Bureau of Standards [72] that will be considered below, see Sec. .",
"In the simplest case of general importance the problem can be reduced to the curve fitting problem (cf [35]).",
"The model $m$ in this case is the set of equations (REF ) - (REF ) completed with constitutive relations and equations of state for the liquid and gas [10], [39].",
"Data $d$ correspond to the time-series data $\\lbrace \\hat{p}, \\hat{T}_f, \\hat{T}_w\\rbrace $ of pressure, fluid and wall temperature obtained in the experiment.",
"$\\theta $ correspond to the model parameters that will be discussed in more details in the following section.",
"The goal of the probabilistic approach is to use time-series data $d$ to update initial guess for the distribution of the model parameters $\\theta $ .",
"Below we provide an example of development and application of this approach to the analysis of chilldown in cryogenic horizontal line."
],
[
"Constitutive relations",
"As was discussed above, the model (REF ) - (REF ) has to be completed with the equations of state and the constitutive relations.",
"The equations of state can be included into the model in the form of NIST tables [39], [73].",
"Functional and parametric form of constitutive relations, on the other hand, represent one of the main source of uncertainties in the model.",
"The corresponding constitutive relations [46] define boundaries between flow and boiling regimes, interphase friction, the coefficient of virtual mass, wall friction, wall heat transfer, interphase heat and mass transfer.",
"In practical calculations the boundaries between flow and boiling regimes have to be determined first.",
"Frictional losses and coefficients for the heat and mass transfer are defined at the next step for each flow regime.",
"In this work the boundaries between flow regimes are estimated using simplified Wojtan et al [74] map.",
"The map was simplified in two ways.",
"First we used only a few transition boundaries as shown below.",
"Next, we estimated the location of these boundaries in the coordinates of mass flow rate ($\\dot{m}$ ) and vapor quality ($\\chi $ ) using original expressions.",
"Finally, we approximated the location of these boundaries using low-dimensional polynomials and used polynomial coefficients as fitting parameters.",
"The rationale behind approximate description of the boundaries of flow regimes is twofold.",
"It is known [75] that Wojtan maps can only be considered as an approximation to the flow regimes for cryogenic fluids.",
"In the experiments performed at NBS [72] the flow regimes were not established experimentally and therefore cannot be validated.",
"Stratified-Wavy-to-stratified transition.",
"For the stratified to wavy stratified transition flow on the plain $(\\dot{m}, \\chi )$ we have ${\\dot{m}_{strat}} = {\\left\\lbrace {\\frac{{{{\\left( {226.3} \\right)}^2}{A_{ld}}A_{gd}^2{\\rho _g}\\left( {{\\rho _l} - {\\rho _g}} \\right){\\mu _l} g}}{{{\\chi ^2}\\left( {1 - \\chi } \\right){\\pi ^3}}}} \\right\\rbrace ^{1/3}} + 20\\chi ,$ []$A$ cross-sectional area []$D$ internal pipe diameter []$We$ Weber number []$Fr$ Froude number []$\\dot{m}$ mass flow rate [G]$\\chi $ vapor quality [G]$\\mu $ dynamic viscocity [U]$ld$ dimensionless quantity for liquid cross-section [U]$gd$ dimensionless quantity for gas cross-section where $A_{gd}$ and $A_{ld} $ are dimensionless cross-sectional area of the gas and liquid fractions.",
"Stratified-Wavy-to-annular-intermittent transition.",
"The transition boundary from wavy-stratified to annular or intermittent flow is given by the following relation $\\begin{array}{*{20}{l}}\\dot{m}_{wavy} = \\left\\lbrace \\frac{16 A_{gd}^3 g D \\rho _l \\rho _g}{\\chi ^2 \\pi ^2}\\left[ 1 - \\left( 2h_{ld} - 1\\right)^2 \\right]^{.5}\\right.\\\\\\qquad \\left.\\times \\left[ {\\frac{{{\\pi ^2}}}{{25 h_{ld}^2}}{{\\left( {1 - \\chi } \\right)}^{ - {F_1}\\left( q \\right)}}\\left( {\\frac{{We}}{{Fr}}} \\right)_l^{ - {F_2}\\left( q \\right)} + 1} \\right] \\right\\rbrace ^{.5} + 50.\\end{array}$ Here $We_l$ and $Fr_l$ are Weber and Froude numbers ($We_l = \\frac{{\\dot{m}_l^2D}}{{{\\rho _l}\\sigma }}$ and $Fr_l = \\frac{{\\dot{m}_l^2}}{{\\rho _l^2 g D}}$ ), while $ h_{ld} $ is dimensionless height of the liquid level.",
"[]$h_{ld}$ dimensionless height of the liquid level Dryout transition that takes into account heat flux from the wall has the form $\\begin{array}{*{20}{l}}\\dot{m}_{dry} = \\left[ 4.25\\left( ln\\left( {\\frac{{0.58}}{\\chi }} \\right) + 0.52 \\right)\\left( {\\frac{{{\\rho _g}\\sigma }}{D}} \\right)^{.17}\\right.\\\\\\qquad \\times \\left.",
"{{\\left( g D{\\rho _g}\\left( {{\\rho _l} - {\\rho _g}} \\right) \\right)}^{.37}}{{\\left( {\\frac{{{\\rho _l}}}{{{\\rho _g}}}} \\right)}^{.25}}{\\left( \\frac{1}{\\tilde{q}} \\right)}^{.7} \\right]^{.93}.\\end{array}$ Here $\\tilde{q}$ is the heat flux at the wall normalized by the characteristic heat flux corresponding to the departure from nucleate boiling in the form [76] $ {q_{DNB}} = K\\cdot \\rho _g^{1/2}{h_{lg}}{\\left[ {\\sigma g\\left( {{\\rho _l} - {\\rho _g}} \\right)} \\right]^{1/4}}.$ The location of these boundaries determines transitions between various regimes of heat transfer and pressure losses as will be discussed briefly below."
],
[
"Heat and mass transfer",
"The total mass transfer $\\Gamma _g$ in equations (REF ), (REF ) is the sum of the mass transfer at the wall and at the interface ${\\Gamma _g} = {\\Gamma _{wg}} + {\\Gamma _{ig}},$ where ${\\Gamma _{wg}} = \\frac{{{\\dot{q}}_{wl}}}{{H_g^* - H_l^*}};\\quad \\quad {\\Gamma _{ig}} = \\frac{{{\\dot{q}}_{li} + {{\\dot{q}}_{gi}}}}{{H_g^* - H_l^*}};\\\\$ and $ H_g^* - H_l^* = \\left\\lbrace {\\begin{array}{*{20}{c}}{{H_{g,s}} - {H_l},\\quad \\Gamma > 0}\\\\{{H_g} - {H_{l,s}},\\quad \\Gamma < 0}\\end{array}} \\right..$ The heat transfer correlations are subject of extensive research [7], [8].",
"Here we briefly outline a subset of these correlations selected in the present work.",
"The heat fluxes at the wall and at the interface are defined as follows $\\begin{array}{*{20}{l}}{\\dot{q}_{wg} = h_{wg}\\left( {T_w - T_g} \\right);\\quad \\dot{q}_{ig} = h_{ig}\\left( {T_{l,s} - T_g} \\right);}\\\\{\\dot{q}_{wl} = h_{wl}\\left( {T_w - T_l} \\right);\\quad \\dot{q}_{il} = h_{il}\\left( {T_{l,s} - T_l} \\right).",
"}\\end{array}$ In the current work we are interested in relatively low mass fluxes, $G < 600 kg/m^2/s$ .",
"In this limit correlations for the heat flux are often based on the multiplicative or additive corrections to the values obtained for pool boiling [64], [65], [66], [63], [77].",
"The following heat transfer mechanisms are included in the analysis: (i) convection, (ii) nucleate boiling, (iii) transition boiling, (iv) film boiling, and (v) transition to dryout regime.",
"Convective heat transfer in horizontal pipes distinguishes four flow regimes: (F-L) forced laminar, (F-T) forced turbulent, (N-L) natural laminar, and (N-T) natural turbulent convection.",
"The corresponding correlations for the convective heat transfer can be taken in the form e.g.",
"[6], [14] $h_{cb} = \\frac{\\kappa }{D_h}\\left\\lbrace {\\begin{array}{*{20}{l}}{4.36,\\qquad \\qquad \\qquad ~ \\hbox{\\rm {F-L~~\\cite {Nellis:09}}};}\\\\{0.023\\cdot R{e^{0.8}}P{r^{0.4}},\\quad ~ \\hbox{\\rm {F-T~~\\cite {Nellis:09}}};}\\\\{0.1\\cdot (Gr\\cdot Pr)^{1/3}},\\quad ~~ \\hbox{\\rm {N-L~~\\cite {Holman:89}}};\\\\{0.59\\cdot (Gr\\cdot Pr)^{1/4}},\\quad ~ \\hbox{\\rm {N-T~~\\cite {Holman:89}}}.\\end{array}} \\right.$ Here $Pr=\\frac{\\mu C_p}{\\kappa }$ and $Gr=\\frac{\\rho ^2 g \\beta _T (T_w - T_{l(g)})D^3}{\\mu ^2}$ are Prandtl and Grashof numbers respectively, $\\beta _T$ is the coefficient of thermal expansion, and $D_h$ is the hydraulic diameter.",
"To guarantee a smooth transition between various regimes the maximum value of $h_{cb}$ is taken as the value for the convective heat transfer.",
"[U]$cb$ convective boiling [U]$s$ satturation [U]$sub$ subcooled []$D_h$ hydraulic pipe diameter [G]$\\kappa $ thermal conduciviy We note that convective heat transfer in the stratified flow does not significantly affect the chilldown process, because the fluid temperature in this regime is close to (or lower than) saturation temperature $T_s$ .",
"The first critical temperature that defines the shape of the boiling curve and influences the chilldown corresponds to the onset of nucleation boiling $T_{onb}$ ."
],
[
"Onset of nucleate boiling",
"The correlations for onset of nucleate boiling are based on the analysis of the balance between mechanical and thermodynamical equilibrium [9].",
"Using this analysis the $T_{onb}$ and the corresponding heat flux $\\dot{q}_{onb}$ can be written in the form [79], [80], [81] $T_{onb} &=& T_s+F\\left( {1 + \\sqrt{1 + \\frac{2 \\Delta T_{sub}}{F}} } \\right),\\\\\\dot{q}_{onb} &=& \\frac{B}{Pr^2}~\\Delta T^2_{sat} = h_{cb}(T_{onb} - T_{l})$ where $ B = \\frac{{{\\rho _g}{h_{lg}}{\\kappa _l}}}{{8\\sigma {T_s}}}$ , $F = \\frac{h_{cb} Pr_l^2}{2B}$ , $\\Delta T_{sat}=T_{onb}-T_s$ is the wall superheat, and $\\Delta T_{sub}=T_s-T_l$ is liquid subcooling temperature.",
"The convective heat transfer coefficient is given by (REF ).",
"[U]$onb$ onset of nucleate boiling When the wall superheat exceeds $\\Delta T_{sat} = T_{onb} - T_s$ the nucleation boiling begins and the heat flux to the wall may increase by more than an order of magnitude significantly affecting the chilldown process.",
"This increase continues until the heat flux approaches its critical value $\\dot{q}_{chf}$ ."
],
[
"Critical heat flux",
"The values of critical heat flux $\\dot{q}_{chf}$ and the corresponding critical wall superheat $T_{chf}$ are crucial for predicting chilldown and dryout phenomena in non-equilibriums flows.",
"In nuclear reactor codes (see e.g.",
"[14], [6]) these values are determined using look-up tables based on extensive experimental measurements obtained under various flow conditions.",
"For cryogenic fluids experimental data remain sparse and values of $\\dot{q}_{chf}$ and $T_{chf}$ are often estimated using mechanistic models, see e.g.",
"[7], [9], [65], [82], see also [57] for a recent review.",
"The correlations for the critical heat flux where introduced in Sec.",
"REF .",
"The temperature $T_{chf}$ for the critical heat flux was estimated in this work using approach proposed by Theler [83] $T_{chf} = \\frac{T_s}{1-\\frac{T_s R_g}{h_{lg}}log(2 k_g +1)},$ where $k_g$ is the isoentropic expansion factor that for ideal diatomic gases is 7/2 and $R_g$ is the specific gas constant.",
"When wall superheat exceeds $\\Delta T_{chf} = T_{chf} - T_s$ , the transition boiling begins and the heat flux to the wall decreases sharply as a function of the wall temperature until the latter reaches minimum film boiling temperature $T_{mfb}$ ."
],
[
"Minimum film boiling",
"In the film foiling regime the fluid flow is completely separated from the wall by the vapor film.",
"The minimum value of the wall superheat $\\Delta T_{mfb} = T_{mfb} - T_s$ corresponding to this regime was estimated by Berenson as [84], [85] $\\begin{array}{l} \\Delta {T_{mfb,0}} = 0.127\\frac{\\rho _g h_{lg}}{\\kappa _g}\\times \\\\\\qquad \\left[ {\\frac{{g\\left( {{\\rho _l} - {\\rho _g}} \\right)}}{{{\\rho _l} + {\\rho _g}}}} \\right]^{2/3} {\\left[ {\\frac{\\sigma }{{g\\left( {{\\rho _l} - {\\rho _g}} \\right)}}} \\right]^{1/2}}{\\left[ {\\frac{{{\\mu _g}}}{{\\left( {{\\rho _l} - {\\rho _g}} \\right)}}} \\right]^{1/3}}\\end{array}$ []$c_p$ specific heat for constant pressure [U]$fb$ film boiling Iloeje [64], [67] has corrected Berenson equation to take into account the dependence of the $\\Delta T_{mfb}$ on the quality and mass flux of the boiling flows in the form $\\Delta {T_{mfb}} = c_1 \\Delta {T_{mfb,0}} (1-c_2 X_e^{c_3}) (1+c_4 G^{c_5}),$ where $X_e$ is the equilibrium quality, $G$ is liquid mass flux and $a_i$ are constants, e.g.",
"$a_1 = 0.0144$ , $a_2 = 10^{-6}$ , $a_3 = 0.5\\times 10^{-3}$ [66], and $\\alpha _{cr} = 0.96$ [64] for water.",
"[]$G$ mass flux [U]$mfb$ minimum film boiling The heat flux in the film boiling flow can be chosen following e.g.",
"recommendations of Groeneveld and Rousseau [86].",
"In this work the heat flux to the wall in the film boiling regime was taken in the form of Bromley correlations $h_{br} = C \\cdot {\\left[ {\\frac{{g{\\rho _g}\\kappa _g^2\\left( {{\\rho _l} - {\\rho _g}} \\right){{\\tilde{h}}_{lg}}{c_{pg}}}}{{D\\left( {{T_w} - {T_{spt}}} \\right)P{r_g}}}} \\right]^{0.25}},$ corrected using Iloeje-type correlations [64], [67] $h_{fb} = c_1 h_{br} (1-c_2 X_e^{c_3}) (1+c_4 G^{c_5})$ Typical values of the parameters used in simulations are the following: (i) $c_1 = 2.0$ ; (ii) $c_2 = 1.04$ ; (iii) $c_3 = 2.0$ ; (iv) $c_4 = 0.2$ ; (v) $c_5 = 0.1$ .",
"The minimum film boiling heat flux can now be defined as $\\dot{q}_{mfb} = h_{fb} \\Delta T_{mfb}.$ [U]$wg$ wall to gas []$X$ mass quality [U]$e$ equilibrium To complete the discussion of the boiling heat transfer we notice that in the region of single phase gas flow the heat transfer is given by equations (REF ) with appropriately modified parameters.",
"Transition to the single phase heat transfer is initiated when dryout transition is detected."
],
[
"Parametrization",
"It follows from the discussion above that boundaries between various flow boiling regimes are characterized by a number of critical points including onset of nucleate boiling, critical heat flux, minimum film boiling, and onset of dry-out.",
"The heat flux to the wall at these points may differ by an order of magnitude.",
"To simplify the analysis of correlations in two-phase flow-boiling regimes the corresponding values of the heat flux can be anchored to the values at critical points as follows.",
"In the regime of nucleate flow boiling when the wall superheat increases from $\\Delta T_{onb}$ to $\\Delta T_{chf}$ .",
"the heat flux can be defined using simple interpolation $\\dot{q}_{nb} = y^n \\dot{q}_{onb} + (1-y^n) \\dot{q}_{chf},$ where $n$ is constant, $y$ is defined as $(T_w-T_{onb})/(T_{chf}-T_{onb})$ , while $\\dot{q}_{chf}$ and $\\dot{q}_{onb}$ are given by the equations (REF ) and () respectively.",
"Similar correlations were applied to interpolate transition boiling in the form [6] $\\dot{q}_{tb} = f_{tb}\\cdot \\dot{q}_{chf}+(1-f_{tb})\\dot{q}_{mfb},$ where $f_{tb} = \\left(\\frac{T_w-T_{mfb}}{T_{chf}-T_{mfb}}\\right)^2$ , where $T_{chf}$ , $T_{mfb}$ , and $\\dot{q}_{mfb}$ are given by equations (REF ), (REF ), and (REF ) respectively.",
"We note that $\\dot{q}_{chf}$ is the same as in eq.",
"(REF ) and $T_{chf}$ was corrected using Iloeje-type correlations [67] similar to the one applied in eq.",
"(REF ).",
"Within this approach the flow boiling correlations are essentially controlled by parameterization of the set of characteristic points on the boiling curve $ c_{cr} = \\lbrace (T_{onb}, \\dot{q}_{onb})$ , $(T_{chf}, \\dot{q}_{chf})$ , $(T_{mfb}, \\dot{q}_{mfb})$ , $(T_{dry}, \\dot{q}_{dry}) \\rbrace $ .",
"By introducing corrections to temperature and heat transfer coefficient for critical heat flux and minimum film boiling in the form (REF ) and (REF ) we were able to obtain smooth transformation of the boiling surface between the pool and flow boiling regimes.",
"An example of such transformation is shown in Fig.",
"REF .",
"In this figure the wall heat flux was calculated as a function of the wall temperature and Reynolds number of the liquid nitrogen for three different values of pressure.",
"Figure: Heat flux from the liquid to the wetted wall as a function of theReynolds number of the liquid flow and wall temperature T w T_w calculatedfor three different pressures: 1, 3, and 7 atm."
],
[
"Pressure drop",
"To complete the discussion of the constitutive relations, we briefly consider pressure drop correlations used in this research.",
"For the single phase flow the wall drag was calculated using following relations $\\tau _{wl} = f_{wl}\\frac{\\rho _l u_l^2}{2}, \\quad \\quad \\tau _{wg} = f_{wg}\\frac{\\rho _g u_g^2}{2},$ Here the friction factors for turbulent and laminar flow are given by Churchill approximation $f_{wg(l)}=2\\left[\\left(\\frac{8}{Re}\\right)^{12}+\\frac{1}{\\left(a+b\\right)^{3/2}}\\right]^{1/12},$ with Reynolds numbers $Re_{m,L} = \\frac{\\rho _{m,L}u_{m,L}D_{m,L}}{\\mu _{g(l)}}$ based on volume centered velocities $u_{m,L}$ and hydraulic diameter $D_m = \\frac{4~A_L}{l_{m,L}}$ for each control volume.",
"Index $m$ takes values $m=\\lbrace g,~l,~i\\rbrace $ for gas, liquid, and interface in a given control volume.",
"The coefficients $a$ and $b$ have the following form $\\begin{array}{l}a=\\left\\lbrace 2.475\\cdot log\\left[ \\frac{1}{\\left(\\frac{7}{Re}\\right)^{0.9}+0.27\\left(\\frac{\\epsilon }{D_h}\\right)}\\right]\\right\\rbrace ^{16}, \\\\b=\\left( \\frac{3.753\\times 10^4}{Re}\\right)^{16}.\\end{array}$ The two-phase friction pressure drop $\\left( {\\frac{{dp}}{{dz}}} \\right)_{2\\phi }$ is defined using Lockhart-Martinelli correlations [87].",
"The pressure losses are partitioned between the phases as follows [10] $\\begin{array}{l}{\\tau _{wg}}{l_{wg}} = {\\alpha _g}{\\left( {\\frac{{dp}}{{dz}}} \\right)_{2\\phi }}\\left( {\\frac{1}{{{\\alpha _g} + {\\alpha _l}{Z^2}}}} \\right),\\\\{\\tau _{wl}}{l_{wl}} = {\\alpha _l}{\\left( {\\frac{{dp}}{{dz}}} \\right)_{2\\phi }}\\left( {\\frac{{{Z^2}}}{{{\\alpha _g} + {\\alpha _l}{Z^2}}}} \\right).\\end{array}$ Here $Z^2$ is given by ${Z^2} = {{\\left( f_{wl}Re_l{\\rho _l}u_l^2\\frac{\\alpha _{wl}}{\\alpha _l} \\right)} \\mathord {\\left\\bad.",
"{\\vphantom{{\\left( {{f_l}R{e_l}{\\rho _l}u_l^2\\frac{{{\\alpha _{wl}}}}{{{\\alpha _l}}}} \\right)} {\\left( {{f_{wg}}R{e_g}{\\rho _g}u_g^2\\frac{\\alpha _{wg}}{\\alpha _g}} \\right)}}} \\right.\\hspace{0.0pt}} \\left( f_{wg}R{e_g}{\\rho _g}u_g^2\\frac{\\alpha _{wg}}{\\alpha _g} \\right)},$ friction factor $f_{wg(l)}$ is in eq.",
"(REF ), while coefficients $\\alpha _{wl}$ and $\\alpha _{wg}$ depend on the flow pattern [10].",
"The interface drag is given by ${\\tau _{ig}} = - {\\tau _{il}} = \\frac{1}{2}{C_D}{\\rho _g}\\left| {{u_g} - {u_l}} \\right|\\left( {{u_g} - {u_l}} \\right),$ where interfacial drag coefficient $C_D$ depends on the flow pattern [6].",
"We note that the functional form of the correlations adopted in this work is not unique and a number of alternative presentations can be used, see e.g.",
"[14], [6], [88], [46] for further details.",
"The main goal of the present analysis is to develop an efficient approach to the parameter inference and systematic comparison between alternative functional forms of these correlation."
],
[
"Inference of the model parameters",
"The discussion in previous sections has emphasized the fact that modeling of cryogenic flows involves a large number of unknown parameters.",
"We will now show that proposed probabilistic framework allows for their efficient simultaneous estimation.",
"The following steps are included into the process: (i) choice of the model parameters; (ii) definition of the objective (cost) function; (iii) estimation of the initial distribution of the model parameters via sensitivity study; (iv) simplified direct search for approximate globally optimized parameter values; (v) refined estimation of the optimal parameter values using global optimization; and (vi) estimation of the variance of the model parameters."
],
[
"Model parameters",
"Analysis of the correlations of the two-phase boiling flows in full scale industrial systems may involve hundreds of model parameters [14].",
"In the present simplified model of the chilldown in horizontal straight line we limited studies to a set of 47 parameters divided into several groups, including e.g.",
"parameters for: (i) onset of nucleate boiling; (ii) critical heat flux; (iii) film boiling; (iii) convective heat transfer; (iv) flow regime boundaries; and (v) frictional losses.",
"For example, parameters related to the Iloeje's corrections (REF ) to the minimum film boiling temperature $ T_{mfb} $ are combined in a group shown in Table REF .",
"Similar subsets of parameters were formed for other groups, see [46] for further details.",
"Table: Example of parameters for the temperature T mfb T_{mfb} .Not all the parameters are equally important/sensitive for the system dynamics.",
"Relative significance of the model parameters depends strongly on the objective of optimization, the stage of the chilldown process, and the location of the sensors in the system.",
"Accordingly, the first step in the analysis of the sensitivity is an appropriate choice of the objective function."
],
[
"Cost function",
"The primary goal of modeling large scale cryogenic systems is the ability to reproduce and predict system response in a variably of nominal and off-nominal regimes.",
"The natural choice of the objective in this case is to minimize the sum of square difference between model predictions (${x}^k_{n}$ ) and data ($\\hat{x}^k_n$ ) measured by different types of sensors at various locations.",
"Typically, the fluid and wall temperatures and the fluid pressure are available for the measurements during chilldown.",
"Taking into account time discretization of measured data, the cost function can be written in the form, cf.",
"[35] $&&S({\\bf c}) = \\sum _{n=0}^{N}\\sum _{k=1}^{K}\\left[ \\eta _{T_w}\\left({T}^k_{w,n}({\\bf c})-\\hat{T}^k_{w,n}\\right)^2 +\\right.",
"\\\\\\nonumber &&~ \\left.",
"\\eta _{T_f}\\left({T}^k_{f,n}({\\bf c})-\\hat{T}^k_{f,n}\\right)^2+\\eta _{p} \\left({p}^k_{n}({\\bf c})-\\hat{p}^k_{n}\\right)^2\\right],$ where $\\eta _i$ are weighting coefficients for different types of measurements, index $k$ runs through different locations of the sensors, and the index $n$ corresponds to discrete time instants $t_0,...,t_N$ ."
],
[
"Sensitivity analysis",
"Once the objective function of optimization is chosen we proceed with the analysis of input-output relations for the model to determine the most sensitive model parameters.",
"At this step we evaluate how much each model parameter is contributing into the model uncertainty.",
"Figure: Results of the sensitivity analysis for GwscGwsc experimental time-series data obtained at NIST.",
"Data recorded at different locations are shown by black solid lines for fluid temperature at three locations.",
"Colored dashed lines show model predictions at: (red) 0.6m from the entrance; (blue) 24 m; (green) 43 m; (pink) 60 mWe perform this test for each motel parameter.",
"An example of the test outcome for overall scaling coefficient for mass transfer at the wall $Gwsc$ is shown in Fig.",
"REF .",
"The sensitivity can be estimated as the relative change of the cost function normalized by the relative change of the parameter.",
"In this particular example 12 % change in the parameter value results in 88 % change in the cost function, i.e.",
"the sensitivity is considered to be very high, except for the data obtained at station 1.",
"The results of the sensitivity test were used primarily to simplify the model by fixing parameters that have no effect on the output and to rank the most sensitive parameters and to learn their effect on the output of the model at various sensor locations.",
"Typically it was found that only 20 parameters can be retained for subsequent model calibration."
],
[
"Direct search",
"At the first step of the model calibration we used a simplified direct search to determine roughly the values of globally optimal model parameters.",
"Simplified direct search algorithm developed in this work has proven to be highly efficient at this stage.",
"The algorithm is searching for a minimum of the cost function on a regular grid in multi-dimensional parameter space by scanning one parameter at a time.",
"The search is repeated several times with randomly changing order of the scanning directions.",
"The convergence of the algorithm is illustrated in the Fig.",
"REF .",
"Figure: (a) Estimated values of the model parameters.",
"(b) Convergence ofthe simplified direct search algorithm for simultaneous optimization of6 model parameters.In this example the following 6 parameters were analyzed: scaling coefficients of the mass transfer at the interface (Gisc) and at the welted wall (Gwsc), characteristic time of the heat transfer to the wall (tauw), scaling coefficient for the film boiling heat transfer (qmfbsc), coefficients $c_2$ (Gmfbsc) and $c_3$ (Emfbsc) in eq.",
"(REF ) for correction of the minimum heat flux.",
"The main advantage of this algorithm is that allows to determine quickly an approximate location of the global minimum in a given subspace of parameter space for poorly defined initial guess.",
"tie: example, the algorithm can scan within one hour uo to 30 parameters of the NIST model using 10 different scanning orders.",
"An approximation to the values of the model parameters found at this step can be further refined using one of the global optimization algorithms."
],
[
"Global optimization",
"We note that casting the problem of fitting model predictions for two-phase flow in the standard form (REF ), (REF ) allows one to use any available standard library for the solution of the optimization problem.",
"In this work we performed global optimization using a set of optimization algorithms available in MATLAB.",
"We have verified the convergence of the model predictions towards experimental time series using pattern search, genetic algorithm, simulated annealing, and particle swarm algorithms.",
"Figure: (a) Convergence of the simulated annealing algorithm for simultaneous optimization of 14 model parameters.",
"(b) Best values of the model parameters.The convergence of the model predictions using simulated annealing algorithm is illustrated in Fig.",
"REF .",
"We note that convergence is achieved for simultaneous optimization of 14 parameters of the model.",
"Besides 6 parameters listed above the following parameters were added to simultaneous optimization: scaling for the the Ditus-Boetler exponents in the heat transfer correlations on both sides of liquid vapor interface (hgi0esc and hli0esc) and at the dry wall (hg0esc), overall scaling for the heat transfer to the dry wall (hg0sc) and to the interface on the gas side (hgisc), scaling for the temperatures of the critical heat flux (Tchfsc) and minimum film boiling (Tmfbsc), and parameters of the transition boundary to the dispersed flow regime (xmin).",
"Once the estimation of the optimal values of the model parameters are refined we can formally complete inference procedure by estimating the variance of the model parameters."
],
[
"Variance of the model parameters",
"To estimate variance we repeat optimization using local search with multiple restarts in the vicinity of the quasi–optimal parameter value.",
"Essentially, at this stage we enhance original sensitivity analysis using simplex algorithm.",
"Figure: Example of estimation of the variances of the parameter values obtained using local search with multiple restarts.An example of estimation of the variance of parameter value is shown in the Fig REF .",
"In this example the distribution function for the parameter values obtained by direct calculations of the cost function for various values of the model parameter close to its optimal value are shown in figure by open symbols.",
"The results of the direct numerical estimation of the distribution of the model parameters were fitted by Gaussian function $F = A(c_0)\\cdot \\exp \\left(-\\frac{1}{2}(c-c_0)\\frac{\\partial ^2 S(c_0)}{\\partial c^2}(c-c_0)\\right)$ The results of the fitting are shown in the figure by thin solid lines.",
"We note that the fit by Gaussian function is quite satisfactory close to the maximum of the distribution.",
"However, numerical simulations also reveal strong deviations from Gaussian fit for some values of the parameters.",
"Specifically, analysis shows the range of parameter values were simulations diverge.",
"These results also provide enhanced sensitivity analysis.",
"For example, for parameter corresponding to the scale of the mass transfer coefficient at the wall ($Gwsc$ ) the dispersion $\\sigma ^2\\approx 0.025$ indicated the fact that the value of this parameter can be determined quite accurately using optimization procedure.",
"On the other hand, the dispersion of the scaling coefficient of the heat transfer from the gas to the wall in the regime of forced convection ($hg0sc$ ) is very large $\\sigma ^2\\approx 12$ , indicating that this parameter value can not be estimated accurately during optimization.",
"The described optimization procedure is robust and sufficiently fast.",
"Simultaneous optimization of 14 model parameters for the NIST (see next section) model with 30 control volumes, including sensitivity analysis, direct search, and global optimization can be computed in several hours on the laptop.",
"Importantly, the proposed approach allows one to cast the fitting problem within a general inferential framework.",
"Indeed, we begin with initial guess followed by rough estimation of the distribution of the model parameters and then we use available experimental time-series data to update these distributions by estimating globally optimal values of the model parameters and their variance.",
"This procedure can be systematically continued as soon as new experimental data become available.",
"Furthermore, the approach can encompass comparison between various alternative functional forms for two-phase flow correlations using time-series data available in multiple databases.",
"Using this approach we were able to demonstrate convergence of the model predictions towards experimental time-series obtained for chilldown of the cryogenic transfer lines under various experimental conditions [45], [89], [90], [91], [92].",
"An example of such convergence is provided in the next section.",
"To validate this approach we used a set of experimental data obtained for chilldown in horizontal transfer line at National Bureau of Standards (currently NIST) [72] and chilldown large scale experimental transfer line at KSC [26].",
"Here we describe the result of the application of our approach to the analysis of chilldown in NIST experiment.",
"In the chilldown experiment [72] the vacuum jacketed line was 61 m long.",
"The internal diameter of the copper pipe was 3/4 inches.",
"Four measurement stations were located at the distance 6, 24, 42, and 60 m from the input valve.",
"Three particular experimental data sets were considered in this work: (i) subcooled liquid nitrogen and pressure in the storage tank was 4.2 atm; (ii) saturated liquid nitrogen flow driven by 3.4 atm pressure in the tank; and (iii) saturated liquid nitrogen flow driven by 2.5 atm pressure in the tank.",
"This set of experiments was selected for our analysis because it possesses a well-known difficulty for modeling, see e.g.",
"[35]."
],
[
"Sub-cooled flow",
"The results of modeling chilldown of cryogenic transfer line with sub-cooled liquid nitrogen flow under tank pressure 4.2 atm are shown in the next four figures.",
"The corresponding time-series data include fluid and wall temperature, the heat flux coefficient, and fluid pressure.",
"Figure: Comparison of the model predictions (dashed colored lines) with the experimental time-series data (solid lines) for the fluid temperature measured at four locations along the pipe.",
"Dashed colored lines and lines with colored open symbols correspond to the model predictions with two different sets of parameters.The results of comparison of the model predictions with the experimental data for the fluid temperature are shown in the Fig.",
"REF .",
"The corresponding comparison for the wall temperature is shown in the Fig.",
"REF Three different regions can be noticed in the figure.",
"A fast cooling region in the beginning of the pipe.",
"A region near the second station with long characteristic cooling time (order of 100 sec).",
"And a region in the second half of the pipe that the remains hot for an extended period of time.",
"Figure: Comparison of the model predictions (dashed lines) with the experimental time-series data (solid lines) for the wall temperature measured at four locations along the pipe.",
"Color codding is the same as in previous figure.It can be seen from the figure that all three regions are reproduced by the model quite accurately both for the fluid and wall temperature.",
"In general the solution of the optimization problem is not unique.",
"Given different initial conditions the algorithm may converge to a slightly different values of parameters.",
"Example of such convergence to two different sets of parameter values is illustrated in Figs.",
"REF to REF by different color codding.",
"Both sets of parameters converged to the experimental time-series data within accepted tolerance and correspond to sub-optimal values of the cost function (REF ).",
"Figure: Comparison of the model predictions (dashed lines with open symbols) with the experimental time-series data (solid black lines) for the heat transfer coefficient measured at four locations along the pipe.",
"Color codding is the same as in previous figures.The non-uniqueness of the solution is a generic feature of the two-phase flow models that stems from the complex landscape of the cost function with multiple local minima.",
"Regularization of the solution can be achieved e.g.",
"by measurements of the additional flow variables or by testing the flow under different flow conditions.",
"For example, the comparison of the model predictions with experimental time-series for the heat transfer coefficient and for the pressure are shown in Fig.",
"REF and Fig.",
"REF respectively.",
"Figure: Comparison of the model predictions obtained for two different sets of the model parameters (blue dashed-dotted and red solid lines) with the experimental time-series data (dashed black lines) for the pressure measured at three locations along the pipe.",
"Solid colored lines correspond to the model predictions with a different set of parameters.It can be seen from the figures that experimentally estimated values of the total heat transfer coefficient to the wall are nearly constant at all locations and times except for a few narrow peaks.",
"Therefore, the analysis of the heat transfer coefficient can provide in this case only semi-quantitative validation of the model predictions.",
"The comparison of the model predictions with experimental time-series data for the pressure shown in Fig.",
"REF (note that the pressure time-series data are available only at three locations) are more informative.",
"The model can capture semi-quantitatively the frequency and the mean values of the pressure oscillations.",
"However, large amplitude oscillations of pressure signal cannot be reproduce by the model.",
"The most likely reason for this discrepancy is the dynamics of the input valve, which parameters are unknown.",
"Therefore, during numerical experiments we usually limited contribution of the pressure signal to the cost function by setting values of $\\eta _p$ to $\\sim 0.1$ in eq.",
"(REF )."
],
[
"Saturated flow",
"As was mentioned above the convergence of the may be further improved by extending analysis to encompass time-series data obtained under different flow conditions.",
"Following this idea we have included into our analysis the time-series data obtained in NIST experiment [85] for saturated flows for two different driving pressures in the storage tank.",
"Here we consider chilldown in the horizontal line observed for saturated nitrogen flow driven by the tank pressure 3.4 atm, see Fig.",
"REF .",
"Figure: Comparison of the model predictions (dashed colored lines) with the experimental time-series data (solid lines) for the fluid temperature measured at four locations along the pipe.",
"The nitrogen was under saturated conditions in the tank with pressure 3.4 atm.It can be seen from the figure that the main effect of the reduced tank pressure (and corresponding reduction of nitrogen mass flow rate through the inlet valve) is an increase of the chilldown time by approximately 70 sec.",
"Note, that the shape of the temperature signals remains essentially the same, cf.",
"Fig.",
"REF .",
"A good agreement between model predictions and experimental time-series data can be obtained using the same sets of the model parameters discussed above with small ( within 10% ) adjustment of parameter $ tauw $ .",
"Similar results are obtained for saturated nitrogen flow under tank pressure 2.5 atm.",
"We note, however, that the uncertainty in the inference of model parameters could not be resolved.",
"We believe that the main reason for this is threefold: (i) the complexity of the temperature dynamics at the location of the 2-nd measurement station; (ii) the limited set of correlations adopted in this work for modeling cryogenic flow boiling during chilldown; and (iii) the limited information about system dynamics available in NIST time-series data.",
"All these issues will be addressed in the future work in more details.",
"[]$u$ fluid velocity []$p$ pressure []$T$ temperature[]$E$ total specific energy []$H$ total enthalpy[]$\\dot{q}$ heat flux[]$h$ heat transfer coefficient[]$Re$ Reynolds number []$Gr$ Grashoff number[]$Pr$ Prandtl number[]$Pr$ Prandtl number[G]$\\alpha $ gas void fraction[G]$\\beta $ liquid void fraction[G]$\\Gamma $ mass flow rate[G]$\\rho $ density[G]$\\sigma $ surface tension[G]$\\tau $ shear stress[U]$n$ index for the time step"
],
[
"Conclusion",
"To summarize, we developed fast and reliable solver for separated two-fluid cryogenic flow based on nearly-implicit algorithm and proposed a concise set of cryogenic two-phase flow boiling correlations capable of reproducing a wide range of experimental time-series data.",
"The main emphasis in this work were placed on development of an efficient algorithm for simultaneous learning of a large number of parameters of cryogenic correlations that could ensure convergence of the model predictions towards experimental time-series data.",
"Such an algorithm was proposed within inferential probabilistic framework.",
"It involves the following steps: (i) sensitivity analysis of the model parameters, (ii) simplified direct search for approximate globally optimal values of these parameters, (iii) global stochastic optimization that refines the estimate for parameter values obtained at the previous step, and (iv) estimation of variance of the model parameters using local non-linear optimization.",
"The proposed approach was used to analyze chilldown in the horizontal transfer line with liquid nitrogen flow.",
"It was shown that the algorithm can reliably converge towards experimental time-series data in the space of $\\sim $ 20 model parameters both for sub-cooled and saturated flows.",
"At the same time the analysis revealed the non-uniqueness of inferred set of model parameters.",
"The latter results indicates that to obtain more accurate and reliable predictions the set of correlations will have to be extended and validated on a larger database of experimental data.",
"These issues will be addressed in the future work.",
"Another direction of future research will involve development an automation of the proposed approach using machine learning framework.",
"It is important to note that the machine learning approach will most likely underly autonomous control and fault management of two-phase flows in the future space missions.",
"Therefore, its development may accelerate and improve both learning required correlation parameters and reliable design of future exploration missions relying on two-phase flow management in space."
],
[
"Acknowledgments",
"This work was supported by the Advanced Exploration Systems and Game Changing Development programs at NASA HQ."
]
] | 1612.05379 |
[
[
"Complete spectral sets and numerical range"
],
[
"Abstract We define the complete numerical radius norm for homomorphisms from any operator algebra into ${\\mathcal B}({\\mathcal H})$, and show that this norm can be computed explicitly in terms of the completely bounded norm.",
"This is used to show that if $K$ is a complete $C$-spectral set for an operator $T$, then it is a complete $M$-numerical radius set, where $M=\\frac12(C+C^{-1})$.",
"In particular, in view of Crouzeix's theorem, there is a universal constant $M$ (less than 5.6) so that if $P$ is a matrix polynomial and $T \\in {\\mathcal B}({\\mathcal H})$, then $w(P(T)) \\le M \\|P\\|_{W(T)}$.",
"When $W(T) = \\overline{\\mathbb D}$, we have $M = \\frac54$."
],
[
"The main theorem",
"We begin with a key observation which yields one direction of our theorem.",
"Lemma 2.1 If $\\Vert T\\Vert \\le 1$ and $\\Vert S\\Vert \\, \\Vert S^{-1}\\Vert \\le C$ , then $ w(S^{-1}TS) \\le \\frac{1}{2}(C+C^{-1}) .$ Using polar decomposition, $S=U|S|$ , we may replace $T$ by the unitarily equivalent $U^*TU$ and suppose that $S>0$ .",
"After scaling, we may suppose that $C^{-1/2}I \\le S \\le C^{1/2}I$ .",
"Since $\\Vert T\\Vert \\le 1$ we have that $\\begin{bmatrix}0 &0\\\\0&0 \\end{bmatrix} \\le \\begin{bmatrix}S^{-1} &0\\\\0& S \\end{bmatrix}\\begin{bmatrix}I & T\\\\ T^* & I \\end{bmatrix}\\begin{bmatrix}S^{-1} &0\\\\0& S \\end{bmatrix} =\\begin{bmatrix}S^{-2} & S^{-1}TS\\\\ ST^*S^{-1}& S^2 \\end{bmatrix}$ By Ando's numerical radius formula, we obtain that $w(S^{-1}TS) &\\le \\frac{1}{2} \\Vert S^{-2} + S^2 \\Vert \\\\&\\le \\sup \\lbrace \\frac{1}{2}(t+t^{-1}) : C^{-1} \\le t \\le C\\rbrace \\\\&= \\frac{1}{2}(C+C^{-1}) .",
"$ To establish the converse, we first need a simple computational lemma.",
"Lemma 2.2 Let $B \\in \\mathcal {B}(\\mathcal {H})$ and let $T = \\begin{bmatrix}\\alpha I & B\\\\0& \\alpha I \\end{bmatrix} $ .",
"Then $\\Vert T\\Vert = \\frac{ \\Vert B\\Vert + \\sqrt{\\Vert B\\Vert ^2 + 4 |\\alpha |^2}}{2} \\quad \\text{and}\\quad w(T) = |\\alpha | +\\tfrac{1}{2} \\Vert B\\Vert .$ In particular, $\\Vert T\\Vert =1$ if and only if $|\\alpha |^2+\\Vert B\\Vert =1$ .",
"It is straightforward to show that $\\Vert T\\Vert = \\left\\Vert {\\left[{\\begin{matrix} \\Vert B\\Vert &|\\alpha | \\\\ |\\alpha | &0 \\end{matrix}}\\right]} \\right\\Vert $ , and computation of the eigenvalues of this self-adjoint matrix yields the desired formula.",
"Routine manipulation now shows that $\\Vert T\\Vert =1$ if and only if $|\\alpha |^2+\\Vert B\\Vert =1$ .",
"It is also easy to see that $W\\left( {\\left[{\\begin{matrix} 0&B\\\\ 0 &0 \\end{matrix}}\\right]} \\right) = W\\left( {\\left[{\\begin{matrix} 0&\\Vert B\\Vert \\\\ 0 &0 \\end{matrix}}\\right]} \\right)$ is a disc centred at 0 of radius $\\Vert B\\Vert /2$ .",
"Hence $W(T) = \\alpha + \\frac{\\Vert B\\Vert }{2} \\overline{\\mathbb {D}}$ , and therefore $w(T) = |\\alpha | + \\frac{1}{2} \\Vert B\\Vert $ .",
"Theorem 2.3 Let $\\mathcal {A}$ be a unital operator algebra, and let $\\Phi $ be a unital completely bounded homomorphism.",
"Then $ \\Vert \\Phi \\Vert _{wcb} = \\frac{1}{2} \\big ( \\Vert \\Phi \\Vert _{cb} + \\Vert \\Phi \\Vert _{cb}^{-1} \\big ) .$ Let $C = \\Vert \\Phi \\Vert _{cb}$ .",
"By Paulsen's similarity theorem [11], there is an invertible operator $S$ so that $\\operatorname{Ad}S \\circ \\Phi $ is completely contractive and $\\Vert S\\Vert \\,\\Vert S^{-1}\\Vert = C$ .",
"(Here $\\operatorname{Ad}S(T) = STS^{-1}$ .)",
"Let $A \\in \\mathcal {M}_n(\\mathcal {A})$ with $\\Vert A\\Vert =1$ .",
"Then $T := (\\operatorname{Ad}S\\circ \\Phi )^{(n)}(A)$ satisfies $\\Vert T\\Vert \\le 1$ and $\\Phi (A) = \\operatorname{Ad}S^{-1(n)}(T)$ .",
"Hence by Lemma REF , $w(\\Phi ^{(n)}(A)) \\le \\frac{1}{2}(C+C^{-1})$ .",
"Thus $ \\Vert \\Phi \\Vert _{wcb} \\le \\frac{1}{2}(C+\\frac{1}{C}) .$ Conversely, suppose that $A \\in \\mathcal {M}_n(\\mathcal {A})$ with $\\Vert A\\Vert =1$ such that $\\Vert \\Phi ^{(n)}(A)\\Vert > C-\\varepsilon $ for some $\\varepsilon >0$ .",
"Define $B \\in \\mathcal {M}_{2n}(\\mathcal {A})$ by $ B = \\begin{bmatrix}C^{-1} I_n&(1-C^{-2})A\\\\0&C^{-1} I_n \\end{bmatrix} .$ Then by Lemma REF , $\\Vert B\\Vert =1$ .",
"Moreover by the second part of that lemma, $\\Vert \\Phi \\Vert _{wcb} &\\ge w(\\Phi ^{(2n)}(B)) \\\\&= w\\Big ( \\begin{bmatrix}C^{-1} I_n&(1-C^{-2})\\Phi ^{(n)}(A)\\\\0&C^{-1} I_n \\end{bmatrix} \\Big ) \\\\&> C^{-1} + \\frac{1}{2} (1-C^{-2}) (C-\\varepsilon ) \\\\&> \\frac{1}{2}(C + \\frac{1}{C}) - \\frac{\\varepsilon }{2}.$ As $\\varepsilon >0$ was arbitrary, we obtain $ \\Vert \\Phi \\Vert _{wcb} = \\frac{1}{2}(C+\\frac{1}{C}) = \\frac{1}{2} \\big ( \\Vert \\Phi \\Vert _{cb} + \\Vert \\Phi \\Vert _{cb}^{-1} \\big ) .",
"$ Remark 2.4 Inverting the above function shows that for a unital homomorphism $\\Phi $ , $ \\Vert \\Phi \\Vert _{cb} = \\Vert \\Phi \\Vert _{wcb} + \\sqrt{\\Vert \\Phi \\Vert _{wcb}^2 - 1}.$"
],
[
"Consequences",
"As an immediate application, we obtain the second theorem stated in the introduction.",
"Note that convexity of $K$ is not required.",
"Theorem 3.1 Let $C\\ge 1$ and set $C^{\\prime }=\\frac{1}{2}(C+C^{-1})$ .",
"A compact subset $K\\subset \\mathbb {C}$ is a complete $C$ -spectral set for $T\\in \\mathcal {B}(\\mathcal {H})$ if and only if it is a complete $C^{\\prime }$ -numerical radius set for $T$ .",
"If $K$ is a complete $C$ -spectral set for $T$ , then the map $\\Phi _T(f) = f(T)$ for $f \\in R(K)$ has $\\Vert \\Phi _T\\Vert _{cb} \\le C$ .",
"Hence by Theorem REF , $ \\Vert \\Phi _T\\Vert _{wcb} = \\frac{1}{2} \\big ( \\Vert \\Phi _T\\Vert _{cb} + \\Vert \\Phi _T\\Vert _{cb}^{-1} \\big ) \\le \\frac{1}{2}(C+C^{-1}) = C^{\\prime } .",
"$ Thus $K$ is a complete $C^{\\prime }$ -numerical radius set for $T$ .",
"Conversely, since $\\Vert A\\Vert \\le 2 w(A)$ , if $K$ is a complete $C^{\\prime }$ -spectral set for $T$ , it follows that $\\Phi $ is completely bounded.",
"Then $ \\frac{1}{2}(C+C^{-1}) = C^{\\prime } \\ge \\Vert \\Phi \\Vert _{wcb} = \\frac{1}{2} \\big ( \\Vert \\Phi \\Vert _{cb} + \\Vert \\Phi \\Vert _{cb}^{-1} \\big ) $ implies that $\\Vert \\Phi _T\\Vert _{cb} \\le C$ .",
"So $K$ is a complete $C$ -spectral set for $T$ .",
"We apply this to the family of $C_\\rho $ -contractions.",
"For these operators, the set $K$ is the unit disc.",
"Corollary 3.2 Suppose that $T$ is a $C_\\rho $ -contraction for $\\rho \\ge 1$ .",
"If $F:\\mathbb {D} \\rightarrow M_n$ is a matrix polynomial $($ or has coefficients in $A(\\mathbb {D}))$ , then $ w(F(T)) \\le \\frac{1}{2}(\\rho +\\rho ^{-1}) \\Vert F\\Vert _\\infty .$ By [10], there is an invertible operator $S$ such that $\\Vert S^{-1}TS\\Vert \\le 1$ and $\\Vert S\\Vert \\,\\Vert S^{-1}\\Vert \\le \\rho $ .",
"After scaling, we may suppose that $\\Vert F\\Vert _\\infty = 1$ .",
"Then by the generalized von Neumann inequality, we have $1 \\ge \\Vert F(S^{-1}TS) \\Vert = \\Vert (S^{-1}\\otimes I_n) F(T) (S \\otimes I_n) \\Vert .$ Now an application of Lemma REF yields the conclusion.",
"The case $\\rho =2$ includes all operators $T$ with $w(T) \\le 1$ .",
"This provides a matrix polynomial version of Drury's scalar inequality [7].",
"Corollary 3.3 Suppose that $T$ has $w(T) \\le 1$ .",
"If $F:\\mathbb {D} \\rightarrow M_n$ is a matrix polynomial $($ or has coefficients in $A(\\mathbb {D}))$ , then $ w(F(C)) \\le \\frac{5}{4} \\Vert F\\Vert _\\infty .$ Remark 3.4 Note that the class of operators which have the disc an a complete 2-spectral set contains many operators which do not have numerical radius 1.",
"For example, let $ T = \\begin{bmatrix}1/2 &3/2\\\\0&1/2 \\end{bmatrix} \\quad \\text{and}\\quad S = \\begin{bmatrix}2 &0\\\\0& 1 \\end{bmatrix} .$ Then $\\Vert S^{-1}TS\\Vert = 1$ and $\\Vert S\\Vert \\,\\Vert S^{-1}\\Vert = 2$ but $w(T) = 5/4$ .",
"As we mentioned in the introduction, Crouzeix showed [6] that the numerical range $W(T)$ is a complete $C$ -spectral set for $T$ for a universal constant $C < 11.08$ .",
"Crouzeix conjectures [5] that the optimal constant is 2, which is the case for a disc by [10].",
"The following are immediate from Theorem REF .",
"Corollary 3.5 Let $T$ be a bounded operator on $\\mathcal {H}$ .",
"Suppose that $W(T)$ has a complete Crouzeix constant of $C$ , and let $C^{\\prime }= \\frac{1}{2}(C+C^{-1})$ .",
"If $F:W(T) \\rightarrow M_n$ is a matrix polynomial $($ or has coefficients in $A(W(T)))$ , then $ w(F(T)) \\le C^{\\prime } \\Vert F\\Vert _{W(T)} .$ In particular, the constant $C^{\\prime }=5.6$ is valid.",
"Corollary 3.6 Let $T$ be a bounded operator on $\\mathcal {H}$ .",
"Then $W(T)$ is a complete 2-spectral set for $T$ if and only if $w(F(T)) \\le \\frac{5}{4} \\Vert F\\Vert _{W(T)}$ for every matrix polynomial $F$ .",
"Thus Crouzeix's conjecture is true for the norm case if and only if the above $5/4$ 's inequality holds for every operator $T$ .",
"Also, we know that 2 and $\\frac{5}{4}$ are the best possible constants in each case.",
"Acknowledgment.",
"The research was conducted while the third author was visiting the Institute for Quantum Computing at the University of Waterloo.",
"He gratefully acknowledges the hospitality of many at the University of Waterloo, including his gracious host Vern Paulsen."
]
] | 1612.05683 |
[
[
"Approximating Approximate Distance Oracles"
],
[
"Abstract Given a finite metric space $(V,d)$, an approximate distance oracle is a data structure which, when queried on two points $u,v \\in V$, returns an approximation to the the actual distance between $u$ and $v$ which is within some bounded stretch factor of the true distance.",
"There has been significant work on the tradeoff between the important parameters of approximate distance oracles (and in particular between the size, stretch, and query time), but in this paper we take a different point of view, that of per-instance optimization.",
"If we are given an particular input metric space and stretch bound, can we find the smallest possible approximate distance oracle for that particular input?",
"Since this question is not even well-defined, we restrict our attention to well-known classes of approximate distance oracles, and study whether we can optimize over those classes.",
"In particular, we give an $O(\\log n)$-approximation to the problem of finding the smallest stretch $3$ Thorup-Zwick distance oracle, as well as the problem of finding the smallest P\\v{a}tra\\c{s}cu-Roditty distance oracle.",
"We also prove a matching $\\Omega(\\log n)$ lower bound for both problems, and an $\\Omega(n^{\\frac{1}{k}-\\frac{1}{2^{k-1}}})$ integrality gap for the more general stretch $(2k-1)$ Thorup-Zwick distance oracle.",
"We also consider the problem of approximating the best TZ or PR approximate distance oracle \\emph{with outliers}, and show that more advanced techniques (SDP relaxations in particular) allow us to optimize even in the presence of outliers."
],
[
"Introduction",
"Given a finite metric space $(V,d)$ , an approximate distance oracle is a data structure which can approximately answer distance queries.",
"It is usually a combination of a preprocessing algorithm to compute a data structure, and a query algorithm which returns a distance $d^{\\prime }(u,v)$ whenever queried on a pair of vertices $u,v\\in V$ .",
"An approximate distance oracle is said the have stretch $t$ if $d(u,v)\\le d^{\\prime }(u,v)\\le t\\cdot d(u,v)$ .",
"Note that there is a trivial stretch 1 distance oracle that uses $\\Theta (n^2)$ space: we could just store the entire metric space.",
"So the goal is to reduce the space, i.e., to build a small data structure that also has small stretch and small query time.",
"The seminal work on approximate distance oracles is due to Thorup and Zwick [21].",
"They showed that for every integer $k \\ge 1$ , every finite metric space has an approximate distance oracle with stretch $(2k-1)$ and query time $O(k)$ which uses only $O(kn^{1+\\frac{1}{k}})$ space.",
"A significant fraction of more recent results have built off of the ideas developed in [21], and much of this follow-up work has stored the exact same (or very similar) data structure, just with improved query algorithms or slightly different information in the storage (see, e.g., [18], [23], [7], [8]).",
"Most notably, Pǎtraşcu and Roditty [17] gave a different distance oracle (still using some of the basic ideas from [21]) that has multiplicative stretch of 2 and additive stretch of 1, with size $O(n^\\frac{5}{3})$ .",
"This broke through the stretch 3 barrier from [21].",
"Later this result was improved to more general multiplicative/additive stretches [1].",
"In this paper we ask a natural but very different type of question about approximate distance oracles: can we find (or approximate) the best approximate distance oracle?",
"If we are given an input metric space and a stretch bound, is it possible to find the smallest approximate distance oracle for that particular input?",
"This is an unusual question in two ways.",
"First, most data structures are by design forced to store all of the input data; the question is how to store it and what extra information should be stored.",
"This is the case in other settings where instance-optimality of data structures has been considered, e.g., static or dynamic optimality of splay trees.",
"Second, it is not clear whether this question is even well-defined: lower bounds on data structures are commonly arrived at through information or communication complexity (see, e.g., [16]) but when we ask for the optimal data structure on one particular instance this approach becomes meaningless.",
"However, approximate distance oracles are different in ways which allow us to make meaningful progress towards these optimization questions.",
"First, since we are allowed to return only approximate distances (up to some stretch factor), we are allowed to store only part of the input (and indeed this is the entire point of such an oracle).",
"The second problem is a bit more tricky: given an input, how can we optimize over “the space of all approximate distance oracles\"?",
"What does this mean, and what does this space look like?",
"To get around this issue, we make an observation: many modern distance oracles (and in particular Thorup-Zwick, Pǎtraşcu-Roditty, and almost all of their variants) have a similar structure.",
"The preprocessing algorithm chooses a subset of the original distances to store which has some particular structure, and the query algorithm can return a valid distance estimate efficiently as long as the stored distances satisfy the required structure.",
"Thus we can optimize for these particular distance oracles by choosing the best possible set of distances to remember subject to the required structure.",
"By characterizing this structure for different types of distance oracles, we can optimize over those types.",
"For example, the stretch-3 Thorup-Zwick distance oracle uses a subtle but simple method to choose the set of distances to store.",
"It randomly samples a subset of approximately $\\sqrt{n}$ vertices, without using any information about the original metric space, and then creates a data structure which is related (in a well-defined, important way) to these vertices.",
"The correctness of the query algorithm does not depend on the choice of the vertices.",
"Thus instead of simply choosing the subset of vertices uniformly at random, we can instead try to optimize the set of chosen vertices with respect to the actual input metric space.",
"In this paper, we give matching $\\Theta (\\log n)$ upper and lower bounds for optimizing stretch-3 Thorup-Zwick distance oracles, and matching $\\Theta (\\log n)$ upper and lower bounds for optimizing the Pǎtraşcu-Roditty distance oracle.",
"These upper bounds both use a similar LP relaxation, but by giving an $\\Omega (n^{\\frac{1}{k}-\\frac{1}{2^{k-1}}})$ integrality gap for optimizing stretch-$(2k-1)$ Thorup-Zwick distance oracles, we show that this relaxation is not enough to give nontrivial approximations when extended to larger stretch values.",
"As an extension, we also study the problem of optimizing distance oracles with outliers: if we are allowed to not answer queries for some of the vertices (of our choosing), can we have much smaller storage space?",
"We give an $(O(\\log n),1+\\varepsilon )$ -bicriteria approximation to both stretch-3 Thorup-Zwick and Pǎtraşcu-Roditty distance oracles with outliers.",
"We also give a true approximation to stretch-3 Thorup-Zwick distance oracle with outliers when the number of outliers is small."
],
[
"Relationship to Spanners.",
"It is worth noting that this paper is motivated by a similar line of research on graph spanners (subgraphs which approximately preserve distances).",
"Spanners and distance oracles tend to be related (although there is no known formal connection between them), and the traditional questions asked of spanners (what is the tradeoff between the stretch and the size?)",
"are similar to the traditional questions asked of distance oracles.",
"Recently, there has been significant progress in looking at spanners from an optimization point of view: given an input graph and an allowed stretch bound, can we find the sparsest possible spanner meeting that stretch bound?",
"In the last few years, upper and lower bounds have been developed for these problems in the basic case, the directed case, with a degree objective, with fault-tolerance, etc.",
"See, e.g., [11], [4], [12], [10], [9].",
"It is natural to ask these kinds of optimization questions for distance oracles as well, but the definitions become much more difficult.",
"For spanners, the space we are optimizing over (all subgraphs) is very clear and well-defined.",
"But for distance oracles, as discussed, it is much harder to define the space of all data structures.",
"Thus in this paper we optimize over restricted classes, where this space is more well-defined.",
"We view our definitions of these restricted optimization questions as one of the major contributions of this work."
],
[
"Definitions and Preliminaries",
"We begin with some basic definitions, including formal definitions of the problems that we will be working on.",
"Definition 2.1 An approximate distance oracle with $(m,a)$ -stretch, size $s$ , preprocessing time $g$ , and query time $h$ is a pair of algorithms, $preprocess$ and $query$ , with the following properties.",
"$preprocess$ is a randomized preprocessing algorithm $preprocess(V,d,m,a,r)$ which takes as input a metric space $(V,d)$ , stretch bound $(m,a)$ , and random string $r$ and outputs a data structure $S$ where the expected output size is at most $\\mathbb {E}_r[|S|]\\le s(|V|,m,a)$ and the expected preprocessing time is at most $g(|V|,m,a)$ .",
"$query$ takes as input a data structure $S=preprocess(V,d,m,a,r)$ (the output of the preprocess algorithm) with two vertices $u,v \\in V$ , and outputs a value $d^{\\prime }(u,v) \\in \\mathbb {R}$ such that $d(u,v)\\le d^{\\prime }(u,v)\\le m\\cdot d(u,v)+a$ .",
"The running time of $query$ is at most $h(|V|,m,a)$ .",
"We will frequently refer to these just as “distance oracles\" rather than “approximate distance oracles\" when the stretch bound is clear from context.",
"The query algorithm guarantees here are deterministic: the randomness only affects the size of the data structure.",
"Note that one could easily define distance oracles so that either the correctness (with respect to the stretch bound) or the query running time (or both) hold only in expectation or with high probability, but as discussed in Section , essentially all existing distance oracles (and in particular the Thorup-Zwick distance oracle) have deterministic guarantees on the queries.",
"This naturally leads us to the following question: If we fix a particular distance oracle and metric space, can we find the best possible data structure?",
"Here we will focus on the output size, not the preprocessing time (as long as the preprocessing time is polynomial).",
"In other words, since the query algorithm work on any of the possible data structures which the preprocessing algorithm might output, can we actually find the smallest such data structure?",
"This gives the following natural optimization problem.",
"Definition 2.2 Given an approximate distance oracle $\\mathcal {A}=(preprocess,query)$ , the $\\mathcal {A}$ -optimization problem takes as input a metric space $(V,d)$ and a stretch bound $(m,a)$ , and the goal is to find a string $r$ which minimizes $|preprocess(V,d,m,a,r)|$ .",
"In this paper we will focus on two distance oracles (Thorup-Zwick [21] and Pǎtraşcu-Roditty [17]), so we now introduce these oracles."
],
[
"Thorup-Zwick Distance Oracle",
"For every integer $k \\ge 1$ , Thorup and Zwick [21] provided an approximate distance oracle with $(2k-1,0)$ -stretch, size $O(n^{1+\\frac{1}{k}})$ , preprocessing time $O(kn^{2+\\frac{1}{k}})$ , and query time $O(k)$ .",
"We call this distance oracle $TZ_k$ .",
"Their preprocessing algorithm first constructs a chain of subsets $\\varnothing =A_k\\subseteq A_{k-1}\\subseteq \\ldots \\subseteq A_0=V$ by repeated sampling.",
"Each set $A_i$ , where $i\\in [k-1]$ , is obtained by including each element of $A_{i-1}$ independently with probability $n^{-\\frac{1}{k}}$ .",
"Let $R_{iu}=\\lbrace v\\in A_{i-1}\\mid d(u,v)<\\min _{w\\in A_i}d(u,w)\\rbrace $ for all $u\\in V$ and $i\\in [k]$ (where by convention $\\min _{w\\in \\varnothing }d(u,w)=\\infty $ for all $u\\in V$ to handle the $i=k$ case).",
"The output data structure is obtained by storing (in a 2-level hash table) the distance from each node $u$ to each node in $\\bigcup _{i=1}^kR_{iu}$ .",
"The data structure also stores a little more information.",
"Each vertex $u$ remembers $k-1$ pivots: $\\arg \\min _{w\\in A_i}d(u,w)$ for all $i\\in [k-1]$ , and the distance from $u$ to these pivots.",
"However, this is a fixed space cost, and also negligible, so when analyzing the size of the oracle we will ignore the cost of storing the pivots Clearly the output data structure is determined once $A_1,\\ldots ,A_{k-1}$ are fixed.",
"The size of the data structure is: $cost(A_1,\\ldots ,A_{k-1},V,d)=\\sum _{u\\in V}\\sum _{i=1}^k|R_{iu}|=\\sum _{u\\in V}\\sum _{i=1}^k\\left|\\lbrace v\\in A_{i-1}\\mid d(u,v)<\\min _{w\\in A_i}d(u,w)\\rbrace \\right|.$ We will refer to $\\sum _{u\\in V}|R_{iu}|$ as the cost in level $i$ .",
"Let us look back on the definition of approximate distance oracle.",
"The random string $r$ is only used to generate $A_i$ 's, and the query algorithm will return a correct distance estimate no matter what the sets $A_i$ are, but the size is determined by the sets.",
"Therefore, the $TZ_k$ -optimization problem is to find the subsets $\\varnothing =A_k\\subseteq A_{k-1}\\subseteq \\ldots \\subseteq A_0=V$ in order to minimize the total cost."
],
[
"Pǎtraşcu-Roditty Distance Oracle",
"Pǎtraşcu and Roditty [17] provided an approximate distance oracle with $(2,1)$ -stretch, size $O(n^\\frac{5}{3})$ , preprocessing time $O(n^2)$ , and query time $O(1)$ .",
"We call this distance oracle $PR$ .",
"Note that $PR$ works only for metric spaces with integer distances.",
"Their preprocessing algorithm first construct a set $A\\subseteq V$ via a complicated correlated sampling (informally, they sample a large set and a small set, and then define $A$ to be everything in the large set and everything contained in a ball around the small set delimited by the large set).",
"The data structure consists of a 2-level hash table for the distance from each node in $A$ to each node in $V$ , as well as a 2-level hash table storing the distance between each pair $\\lbrace u,v\\rbrace \\subseteq V$ such that $d(u,v)<\\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1$ As with Thorup-Zwick, the output data structure is completely determined once $A$ is fixed.",
"Let $R=\\left\\lbrace \\lbrace u,v\\rbrace \\subseteq V\\mid d(u,v)<\\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1\\right\\rbrace $ .",
"Then the size of the data structure is $cost(A,V,d)=n\\cdot |A|+|R|=n\\cdot |A|+\\left|\\left\\lbrace \\lbrace u,v\\rbrace \\subseteq V\\mid d(u,v)<\\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1\\right\\rbrace \\right|.$ As before, the random string $r$ is only used to generate the set $A$ , and any $A\\subseteq V$ gives a data structure on which the query algorithm works.",
"Therefore, the $PR$ -optimization problem is to find the subset $A\\subseteq V$ in order to minimize the total cost."
],
[
"Distance Oracles With Outliers",
"In some cases, a small set of outlier vertices may make the size of the data structure blow up.",
"Yet in some applications it is acceptable to ignore these outliers.",
"This was the motivation behind a line of work on distance oracles with slack ([5], [6]), in which the data structure could ignore the stretch bound on a small fraction of the distances.",
"In this paper, we consider the case that we can refuse to answer distance queries for some outlier vertices.",
"In other words, we can essentially remove an outlier set $F$ out of $V$ when computing the distance oracle.",
"This gives us the problem of optimizing distance oracle with outliers, in which we not only need to find the random string to determine the output data structure, we also need to find the set of outliers to minimize the final cost.",
"More formally, we have the following type of problem.",
"Definition 2.3 Given an approximate distance oracle $\\mathcal {A}=(preprocess,query)$ , the $\\mathcal {A}$ -optimization problem with outliers takes as input a metric space $(V,d)$ , a stretch bound $(m,a)$ , and a bound on the number of outliers $f\\in \\mathbb {N}$ .",
"The goal is to find a string $r$ as well as a set $F\\subseteq V$ where $|F|\\le f$ , in order to minimize $|preprocess(V\\backslash F,d,m,a,r)|$ .",
"We will provide both true approximation results and ($\\alpha ,\\beta $ )-bicriteria results, in which we slightly violate the bound on the number of outliers.",
"Formally, an $(\\alpha , \\beta )$ -approximation algorithm for the $\\mathcal {A}$ -optimization problem with outliers is on algorithm which on any input $((V,d), (m,a), f)$ returns a solution with cost at most $\\alpha \\cdot OPT$ that has at most $\\beta \\cdot f$ outliers (where $OPT$ is the minimum cost of any solution with at most $f$ outliers)."
],
[
"Our Results and Techniques",
"With these definitions in hand, we can now formally state our results.",
"In Section we discuss the problem of optimizing the 3-stretch Thorup-Zwick distance oracle, i.e., the $TZ_2$ -optimization problem.",
"It is straightforward to obtain an $O(\\log n)$ -approximation by reducing to the non-metric facility location problem.",
"Theorem 2.4 There is an $O(\\log n)$ -approximation algorithm for the $TZ_2$ -optimization problem.",
"To prove a matching lower bound, we use a reduction from Label Cover to the $TZ_2$ -optimization problem.",
"We use a proof which is similar to the proof of the hardness of Set Cover in [22] (based on [14]).",
"However, we cannot use a reduction directly from Set Cover since we will need some extra properties of the starting instances, and thus are forced to start from Label Cover.",
"We introduce a new notion of $(m, l, \\delta )$ -set families and show that these can still be plugged into existing hardness results to get the extra structural properties that we need.",
"This lets us prove the following theorem: Theorem 2.5 Unless $\\mathbf {NP}\\subseteq \\mathbf {DTIME}(n^{O(\\log \\log n)})$ , the $TZ_2$ -optimization problem does not admit a polynomial-time $o(\\log n)$ -approximation.",
"For larger stretch values, a natural approach is to realize that a simple LP relaxation suffices to give Theorem REF in the stretch 3 case, and try to extend this basic LP to larger stretches.",
"In Section , we show that this does not work for the more general $TZ_k$ -optimization problem: the integrality gap jumps up to become a polynomial.",
"The instance is very simple: it is just the metric space formed by shortest paths on the $n$ -cycle.",
"It turns out to be straightforward to calculate the optimal fractional LP cost, but proving that the optimal integral solution is large is surprisingly involved.",
"Theorem 2.6 The basic LP relaxation for the $TZ_k$ -optimization problem has an $\\Omega (n^{\\frac{1}{k}-\\frac{1}{2^{k-1}}})$ integrality gap when $k>2$ .",
"In Section we discuss the problem of optimizing the Pǎtraşcu-Roditty distance oracle.",
"The basic LP and a simple rounding algorithm gives us an $O(\\log n)$ -approximation algorithm.",
"Theorem 2.7 There is an $O(\\log n)$ -approximation algorithm for $PR$ -optimization problem.",
"A reduction from set cover problem also gives us a matching lower bound.",
"Theorem 2.8 Unless $\\mathbf {P}=\\mathbf {NP}$ , the $PR$ -optimization problem does not admit a polynomial-time $o(\\log n)$ -approximation.",
"In Section we move to the outliers setting.",
"For both $TZ_2$ - and $PR$ -optimization problems, a semidefinite programming relaxation and a simple rounding algorithm gives us an $(O(\\frac{\\log n}{\\varepsilon }),1+\\varepsilon )$ -approximation algorithm.",
"Using an SDP relaxation seems to be necessary – the corresponding LP relaxation requires violating the number of outliers by a factor of 2 rather than a factor of $1+\\varepsilon $ .",
"We can also get a true approximation on $TZ_2$ -optimization problem with outliers if the number of outliers is low.",
"These results form the following theorems.",
"Theorem 2.9 There is an $(O(\\frac{\\log n}{\\varepsilon }),1+\\varepsilon )$ -approximation algorithm for the $TZ_2$ -optimization problem with outliers.",
"Theorem 2.10 There is an $O(\\log n)$ -approximation algorithm for $TZ_2$ -optimization problem with outliers if the number of outliers is at most $\\sqrt{n}$ .",
"Theorem 2.11 There is an $(O(\\frac{\\log n}{\\varepsilon }),1+\\varepsilon )$ -approximation algorithm for the $PR$ -optimization problem with outliers."
],
[
"$TZ_2$ -Optimization Problem",
"We first give an $O(\\log n)$ -approximation for $TZ_2$ -optimization (Theorem REF ), and follow this with a matching lower bound."
],
[
"Upper Bound",
"We will prove our upper bound by a reduction to the non-metric facility location problem.",
"Definition 3.1 In the non-metric facility location problem we are given a set $F$ of facilities, a set $D$ of clients, an opening cost function $f:F\\rightarrow \\mathbb {R}^+$ , and a connection cost function $c:D\\times F\\rightarrow \\mathbb {R}^+$ .",
"The goal is to find the set $S \\subseteq F$ which minimizes $\\sum _{i\\in S}f(i)+\\sum _{i\\in D}\\min _{j\\in S}c(i,j)$ (i.e.",
"the sum of the opening and connection costs).",
"Non-metric facility location is a classic problem, and much is known about it, including the following upper bound due to Hochbaum.",
"Theorem 3.2 ([15]) There is a polynomial time algorithm which gives an $O(\\log n)$ -approximation to the non-metric facility location problem.",
"Hochbaum's algorithm is a greedy algorithm, but it is also straightforward to design an algorithm with similar bounds using an LP relaxation.",
"Since it is not necessary we do not present the relaxation here, but generalizations of the relaxation will prove important in the more general $TZ_k$ setting (see Section ).",
"We now show that the $TZ_2$ -optimization problem is essentially a special case of non-metric facility location problem.",
"First, simple arithmetic manipulation of the cost function of the $TZ_2$ -optimization problem gives the following: $cost(A_1,V,d)=&\\sum _{u\\in V}|R_{1u}|+\\sum _{u\\in V}|R_{2u}| \\\\=&\\sum _{u\\in V}\\left|\\lbrace v\\in V\\mid d(u,v)<\\min _{w\\in A_1}d(u,w)\\rbrace \\right| +\\sum _{u\\in V}\\left|\\lbrace v\\in A_1\\mid d(u,v)<\\infty \\right|\\\\=&\\sum _{u\\in V}\\left|\\lbrace v\\in V\\mid d(u,v)<\\min _{w\\in A_1}d(u,w)\\rbrace \\right|+n|A_1| \\\\=&\\sum _{w\\in A_1}n+\\sum _{u\\in V}\\min _{w\\in A_1}\\left|\\lbrace v\\in V\\mid d(u,v)<d(u,w)\\rbrace \\right|.$ Given an instance $(V,d)$ of the $TZ_2$ -optimization problem, we create an instance of non-metric facility location by setting $F=D=V$ , opening costs $f(v)=n$ for all $v\\in V$ , and connection costs $c(u,w)=|\\lbrace v\\in V\\mid d(u,v)<d(u,w)\\rbrace |$ for all $u,w\\in V$ .",
"Then the cost function of the $TZ_2$ -optimization problem is exactly the same as the cost function of non-metric facility location problem.",
"Therefore $TZ_2$ is a special case of non-metric facility location, which together with Theorem REF implies Theorem REF ."
],
[
"Lower Bound",
"Proving an $\\Omega (\\log n)$ hardness of approximation (Theorem REF ) turns out to be surprisingly difficult.",
"Details appear in Appendix ; here we provide an informal overview.",
"Technically we reduce directly to $TZ_2$ -optimization from a version of the Label Cover problem that corresponds to applying parallel repetition [19] to 3SAT-5, which is a standard starting point for hardness reductions.",
"Informally, though, we are “really\" reducing from Set Cover: given an instance of Set Cover, we show how to create an instance of $TZ_2$ -optimization where the cost of the optimal solution is the same (up to a constant and a polynomial scaling factor).",
"But in order for our reduction to work, we actually need more than just an arbitrary Set Cover instance: we need a version of Set Cover in which it is hard even to cover most of the elements, not just all of them.",
"So we have to also give a new reduction from Label Cover to Set Cover, showing that even this version of Set Cover is hard.",
"It turns out that Feige's reduction [14], reinterpreted by Vazirani [22], essentially already gives us what we need.",
"We just need to analyze it a bit more carefully.",
"In particular, a key component of this reduction is what Vazirani called $(m, l)$ -set systems, which can be thought of as nearly-unbiased sample spaces.",
"We generalize this notion to $(m, l, \\delta )$ -set systems, given in the following definition.",
"Definition 3.3 A set $B$ (the universe) and a collection of subsets $C_1,\\ldots ,C_m$ of $B$ form an ($m,l,\\delta $ )-set system if any collection of $l$ sets in $\\lbrace C_1,\\ldots ,C_m,\\overline{C_1},\\ldots ,\\overline{C_m}\\rbrace $ whose union contains at least $(1-\\delta )|B|$ elements must include both $C_i$ and $\\overline{C_i}$ for some $i$ .",
"An $(m, l)$ -set system is just a $(m, l, 0)$ -set system.",
"While not all $(m, l)$ -set systems are $(m, l \\delta )$ -set systems for larger $\\delta $ , the construction of $(m, l)$ -set systems in [22] actually does generalize directly to larger values of $\\delta $ .",
"With this tool in hand, we follow through the rest of the reduction and it gives us the type of Set Cover instances which we need.",
"Technically our reduction skips this step by going directly from Label Cover to $TZ_2$ -optimization, but generating these kinds of Set Cover instances is intuitively what the first part of the reduction is doing."
],
[
"$TZ_k$ -Optimization Problem",
"We now move to the more general $TZ_k$ -optimization problem.",
"While we are not able to give nontrivial upper bounds for this problem, we can at least show that the basic LP relaxation (as discussed in Section REF ) does not give polylogarithmic bounds in this more general setting."
],
[
"The LP",
"Let $B_u(v)=\\lbrace w\\in V\\mid d(u,w)\\le d(u,v)\\rbrace $ .",
"For every $v\\in V$ and $i\\in [k]$ , let $x_v^{(i)}$ be a variable which is supposed to be an indicator for whether $v\\in A_i$ .",
"Similarly, for all $u,v\\in V$ and $i\\in [k]$ , let $y_{uv}^{(i)}$ be a variable which is supposed to be an indicator for whether $v\\in R_{iu}$ .",
"(Recall that $R_{iu}=\\lbrace v\\in A_{i-1}\\mid d(u,v)<\\min _{w\\in A_i}d(u,w)\\rbrace $ ) We can easily write an LP relaxation for this problem: $\\begin{array}{rll}(LP_{TZ_k}):\\min &\\sum _{i=1}^k\\sum _{u,v\\in V}y_{uv}^{(i)}\\\\s.t.&0=x_v^{(k)}\\le x_v^{(k-1)}\\le \\ldots \\le x_v^{(1)}\\le x_v^{(0)}=1&\\forall v\\in V\\\\&y_{uv}^{(i)}\\ge x_v^{(i-1)}-\\sum _{w\\in B_u(v)}x_w^{(i)}&\\forall u,v\\in V,i\\in [k]\\\\&y_{uv}^{(i)}\\ge 0&\\forall u,v\\in V,i\\in [k]\\\\\\end{array}$ It can easily be shown that this is a valid relaxation (the proof can be found in Appendix ).",
"When restricted to the special case of $k=2$ , it is not hard to see that this LP is essentially a special case of the basic LP relaxation for non-metric facility location, which can be used to prove the $O(\\log n)$ bound of Theorem REF .",
"But for larger values of $k$ the behavior is different, and does not result in a polylogarithmic integrality gap."
],
[
"Integrality Gap",
"The integrality gap instance is quite simple: the metric $(V,d)$ induced by shortest-path distances in a cycle.",
"Slightly more formally, we let $V=[n]$ , and use the cycle distance $d(u,v)=\\min \\lbrace |u-v|,n+\\min \\lbrace u,v\\rbrace -\\max \\lbrace u,v\\rbrace \\rbrace $ .",
"Details can be found in Appendix .",
"It turns out to be relatively easy to find a fractional solution to $LP_{TZ_k}$ with cost $O(n^{1+\\frac{1}{2^{k-1}}})$ on this instance.",
"The tricky part is lower bounding the optimal solution, i.e., showing that the optimal integral solution has cost at least $\\Omega (n^{1+\\frac{1}{k}})$ .",
"Combining these two results gives us an $\\Omega (n^{\\frac{1}{k}-\\frac{1}{2^k-1}})$ integrality gap, proving Theorem REF ."
],
[
"$PR$ -Optimization Problem",
"We now move from Thorup-Zwick distance oracles to Pǎtraşcu-Roditty distance oracles.",
"We show that from an optimization perspective, they are similar to $TZ_2$ in that we can find matching bounds: an $O(\\log n)$ -approximation, and $\\Omega (\\log n)$ -hardness."
],
[
"Upper Bound",
"In this section we prove Theorem REF by using an LP and randomized rounding to give an $O(\\log n)$ -approximation to the $PR$ -optimization problem.",
"Let $B_u(v)=\\lbrace w\\in V\\mid d(u,w)\\le d(u,v)\\rbrace $ , and $B(u,r)=\\lbrace w\\in V\\mid d(u,w)\\le r\\rbrace $ .",
"We can see $B_u(v)=B(u,d(u,v))$ .",
"Now, let $x_v$ be a variable which is supposed to be an indicator for whether $v\\in A$ , and let $y_{uv}$ be a variable which is supposed to be an indicator for whether $\\lbrace u,v\\rbrace \\in R$ .",
"(Recall that $R=\\left\\lbrace \\lbrace u,v\\rbrace \\subseteq V\\mid d(u,v)<\\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1\\right\\rbrace $ ).",
"We can write the following LP relaxation: $\\begin{array}{rll}(LP_{PR}):\\min &\\sum _{v\\in V}n\\cdot x_v+\\sum _{\\lbrace u,v\\rbrace \\subseteq V}y_{uv}\\\\s.t.&y_{uv}\\ge 1-\\sum _{w\\in B_u(r)\\cup B_v(d(u,v)-r)}x_w&\\forall u,v\\in V,\\forall r\\in [0,d(u,v)]\\\\&x_v\\in [0,1]&\\forall v\\in V\\\\&y_{uv}\\ge 0&\\forall u,v\\in V\\\\\\end{array}$ At first blush it may not be obvious that the first type of constraint in this LP really captures the characterization of paris in $R$ .",
"But it is actually not that hard to see that this is a valid relaxation (a formal proof can be found in Appendix ).",
"Note that while the number of constraints appears to be exponential (recall that we assume integer weights, but not necessarily unit weights, and hence $d(u,v)$ is not necessarily polynomial in the input size), it is in fact possible to solve this LP in polynomial time.",
"We can do this by noting that for each $u, v \\in V$ , only at most $n$ different value of $r$ actually yield different constraints, so we can simply write the constraints for those values.",
"Our algorithm is relatively straightforward.",
"We first solve $LP_{PR}$ and get an optimal fractional solution $(x_v^*,y_{uv}^*)$ .",
"We then use independent randomized rounding, adding each $v \\in V$ to $A$ independently with probability $\\min \\lbrace 4\\ln n\\cdot x_v^*,1\\rbrace $ .",
"Lemma 5.1 If $y_{uv}^*\\le \\frac{1}{2}$ , then the probability that $\\lbrace u,v\\rbrace \\in R$ is at most $\\frac{1}{n}$ .",
"If $y_{uv}^*\\le \\frac{1}{2}$ , then the first constraint implies that $\\sum _{w\\in B_u(r)\\cup B_v(d(u,v)-r)}x_w^*\\ge \\frac{1}{2}$ for all $r\\in [0,d(u,v)]$ .",
"Therefore, the probability that $A\\cap (B_u(r)\\cup B_v(d(u,v)-r))=\\varnothing $ for a specific $r\\in [0,d(u,v)]$ is at most $\\prod _{w\\in B_u(r)\\cup B_v(d(u,v)-r)}(1-\\min \\lbrace 4\\ln n\\cdot x_w^*,1\\rbrace )\\le e^{-\\sum _{w\\in B_u(r)\\cup B_v(d(u,v)-r)}4\\ln n\\cdot x_w^*}\\le \\frac{1}{n^2}$ A union bound over all the different values of $r$ we used in our LP implies that the probability that there exists an $r\\in [0,d(u,v)]$ where $A\\cap (B_u(r)\\cup B_v(d(u,v)-r))=\\varnothing $ is at most $\\frac{1}{n^2}\\cdot n=\\frac{1}{n}$ .",
"We claim that the existence of such an $r$ is implied by $\\lbrace u,v\\rbrace \\in R$ , and hence the probability that $\\lbrace u,v\\rbrace \\in R$ is at most $\\frac{1}{n}$ .",
"To see this, suppose that $\\lbrace u,v\\rbrace \\in R$ , i.e.",
"suppose that $d(u,v)<\\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1$ .",
"Then if we set $r = \\min _{w \\in A} d(u,w) - 1$ , this implies that $\\min _{w \\in A} d(v,w) > d(u,v) - r$ .",
"But then this would imply that no element of $A$ is in $B_u(r)\\cup B_v(d(u,v)-r)$ .",
"Let $OPT_{LP_{PR}}$ denote the optimal cost of $LP_{PR}$ .",
"Then the above lemma implies that the expected cost of the rounding algorithm is at most $\\textbf {E}[n|A| + |R|] &\\le \\sum _{v\\in V}n\\cdot x_v^*\\cdot 4\\ln n+2\\cdot \\sum _{u,v\\in V}y_{uv}^*+n^2\\cdot \\frac{1}{n}\\le O(\\log n)\\cdot OPT_{LP_{PR}}+n\\\\&\\le O(\\log n)\\cdot OPT$ (where we use the fact that $OPT\\ge \\Omega (n)$ ).",
"This completes the proof of Theorem REF ."
],
[
"$\\Omega (\\log n)$ -hardness",
"We now show a matching hardness bound for the $PR$ -optimization problem by reducing from the Set Cover problem.",
"Consider a set cover instance $(\\mathcal {U},\\mathcal {S})$ where $|\\mathcal {U}|+|\\mathcal {S}|=n$ .",
"For each $e\\in \\mathcal {U}$ , we create a group of vertices $G_e$ where $|G_e|=3n$ .",
"For each $S\\in \\mathcal {S}$ , we also create a group of vertices $G_S$ where $|G_S|=3n$ .",
"Now we construct the following metric space: $V=(\\bigcup _{e\\in \\mathcal {U}}G_e)\\cup (\\bigcup _{S\\in \\mathcal {S}}G_S)$ and $d(u,v)={\\left\\lbrace \\begin{array}{ll}1,&\\mbox{if }u\\in G_e,v\\in G_e\\\\1,&\\mbox{if }u\\in G_S,v\\in G_S\\\\1,&\\mbox{if }u\\in G_e,v\\in G_S,e\\in S\\\\2,&\\mbox{otherwise.}\\end{array}\\right.",
"}$ In Appendix we show that if there is a solution $\\mathcal {S}^*$ to the set cover instance $(\\mathcal {U},\\mathcal {S})$ where $|\\mathcal {S}^*|=t$ , then there is a set $A$ where $cost(A,V,d)\\le t|V|$ .",
"We also show that if there is a set $A \\subseteq V$ where $cost(A,V,d)\\le t|V|$ , then there exists a solution $\\mathcal {S^*}$ to the set cover instance $(\\mathcal {U},\\mathcal {S})$ where $|\\mathcal {S}^*|=t$ .",
"These two claims, together with an appropriate hardness theorem for Set Cover [13], imply Theorem REF ."
],
[
"Distance Oracles With Outliers",
"We now move to the more difficult outliers setting, where we can also optimize over a set of vertices to ignore.",
"Recall that for an approximate distance oracle $\\mathcal {A}$ , our goal is now to find a set of vertices $F$ (the outliers) where $|F| \\le f$ as well as a string $r$ in order to minimize $|preprocess(V \\setminus F,d,m,a,r)|$ .",
"In other words, we are going to try to solve the same problems as before, but where we can choose a set $F$ to remove.",
"We begin with $TZ_2$ , and then move to $PR$ ."
],
[
"$TZ_2$ -Optimization Problem With Outliers",
"For this problem, it is easy to see that the cost function becomes: $cost(A,F,V,d)=(n-f)|A|+\\sum _{u\\in V\\backslash F}|R_{1u}|=(n-f)|A|+\\sum _{u\\in V\\backslash F}\\left|\\lbrace v\\in V\\backslash F\\mid d(u,v)<\\min _{w\\in A}d(u,w)\\rbrace \\right|$ A natural approach is to use an LP which is similar to $LP_{TZ_k}$ to solve this problem (but for $TZ_2$ ), suitably adapted to handle outliers.",
"Let $x_v$ be a variable which is supposed to be an indicator for whether $v\\in A$ , let $y_{uv}$ be a variable which is supposed to be an indicator for whether $v\\in R_{1u}$ , and let $z_v$ be a variable which is supposed to be an indicator for whether $v\\in F$ .",
"Then we can write the following natural LP relaxation: $\\begin{array}{rll}(LP_{TZ_2O}):\\min &\\sum _{v\\in V}(n-f)\\cdot x_v+\\sum _{u,v\\in V}y_{uv}\\\\s.t.&y_{uv}\\ge 1-z_u-z_v-\\sum _{w\\in B_u(v)}x_w&\\forall u,v\\in V\\\\&\\sum _{v\\in V}z_v\\le f\\\\&x_v\\in [0,1]&\\forall v\\in V\\\\&y_{uv}\\ge 0&\\forall u,v\\in V\\\\&z_v\\in [0,1]&\\forall v\\in V\\\\\\end{array}$ Unfortunately, this LP can not give an $(\\alpha ,\\beta )$ -approximation with $\\beta =2-\\epsilon $ .",
"To see this, consider the case that $f=\\frac{n}{2}$ .",
"Then the optimal solution to $LP_{TZ_2O}$ is 0, by setting all $z_v$ to $\\frac{1}{2}$ , all $x_v$ to 0, and all $y_{uv}=0$ .",
"Thus any integral solution, to be competitive with this fractional solution, must treat all nodes as outliers, requiring $\\beta $ to be at least 2.",
"Fortunately we can give a stronger semidefinite programming relaxation, allowing for a better approximation.",
"As in $LP_{TZ_2O}$ , let $\\vec{x}_v$ be a variable which is supposed to be an indicator for whether $v\\in A$ , let $\\vec{y}_{uv}$ be a variable which is supposed to be an indicator for whether $v\\in R_{1u}$ , and let $\\vec{z}_v$ be a variable which is supposed to be an indicator for whether $v\\in F$ .",
"We can then write this SDP: $\\begin{array}{rll}(SDP_{TZ_2O}):\\min &\\sum _{v\\in V}(n-f)\\cdot \\Vert \\vec{x}_v\\Vert ^2+\\sum _{u,v\\in V}\\Vert \\vec{y}_{uv}\\Vert ^2\\\\s.t.&\\Vert \\vec{y}_{uv}\\Vert ^2\\ge 1-\\vec{z}_u\\cdot \\vec{z}_v-\\sum _{w\\in B_u(v)}\\Vert \\vec{x}_w\\Vert ^2&\\forall u,v\\in V\\\\&\\sum _{v\\in V}\\Vert \\vec{z_v}\\Vert ^2\\le f\\\\&\\Vert \\vec{x}_v\\Vert ^2\\le 1&\\forall v\\in V\\\\&\\Vert \\vec{y}_{uv}\\Vert ^2\\le 1&\\forall u,v\\in V\\\\&\\Vert \\vec{z}_v\\Vert ^2\\le 1&\\forall v\\in V\\\\\\end{array}$ Our approximation algorithm first solves $SDP_{TZ_2O}$ to get an optimal solution $(\\vec{x}_v^*,\\vec{y}_{uv}^*,\\vec{z}_v^*)$ .",
"We then use independent randomized rounding to construct $A$ , adding each $v \\in V$ to $A$ independently with probability $\\min \\lbrace \\frac{3\\ln n}{\\varepsilon }\\cdot \\Vert \\vec{x}_v^*\\Vert ^2,1\\rbrace $ where $\\varepsilon $ is a small constant.",
"Finally, we use threshold rounding to construct $F$ by adding each $v\\in V$ to $F$ if $\\Vert \\vec{z}_v^*\\Vert ^2\\ge \\frac{1}{1+\\varepsilon }$ .",
"We want to show that this is an ($O(\\log n),1+\\varepsilon $ )-approximation.",
"It is easy to see that $|F|\\le (1+\\varepsilon )f$ because $\\sum _{v\\in V}\\Vert \\vec{z}_v^*\\Vert ^2\\le f$ .",
"In order to prove Theorem REF it only remains to prove that the expected cost is at most $O(\\log n)\\cdot OPT$ .",
"This proof can be found in Appendix .",
"When $f \\le \\sqrt{n}$ we can actually give a true $O(\\log n)$ -approximation (Theorem REF ).",
"The algorithm is almost the same; we just need to change the threshold rounding for outliers to instead pick the $f$ vertices with largest $\\Vert \\vec{z}_v\\Vert ^2$ value.",
"Details appear in Appendix ."
],
[
"$PR$ -Optimization Problem With Outliers",
"For this problem, the cost function becomes: $cost(A,F,V,d)=&(n-f)\\cdot |A|+|R|\\\\=&(n-f)\\cdot |A|+\\left|\\lbrace \\lbrace u,v\\rbrace \\subseteq V\\backslash F\\mid d(u,v)<\\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1\\rbrace \\right|.$ We will again use an SDP relaxation.",
"Let $\\vec{x}_v$ be a variable which is supposed to be an indicator for whether $v\\in A$ , let $\\vec{y}_{uv}$ be a variable which is supposed to be an indicator for whether $\\lbrace u,v\\rbrace \\in R$ , and let $\\vec{z}_v$ be a variable which is supposed to be an indicator for whether $v\\in F$ .",
"We have the following relaxation which is similar to both $LP_{PR}$ and $SDP_{TZ_2O}$ : $\\begin{array}{rll}(SDP_{PR}):\\min &\\sum _{v\\in V}(n-f)\\cdot \\Vert \\vec{x}_v\\Vert ^2+\\sum _{\\lbrace u,v\\rbrace \\subseteq V}\\Vert \\vec{y}_{uv}\\Vert ^2\\\\s.t.&\\Vert \\vec{y}_{uv}\\Vert ^2\\ge 1-\\vec{z}_u\\cdot \\vec{z}_v-\\sum _{w\\in B_u(r)\\cup B_v(d(u,v)-r)}\\Vert \\vec{x}_w\\Vert ^2&\\forall u,v\\in V,r\\in [0,d(u,v)]\\\\&\\sum _{v\\in V}\\Vert \\vec{z_v}\\Vert ^2\\le f\\\\&\\Vert \\vec{x}_v\\Vert ^2\\le 1&\\forall v\\in V\\\\&\\Vert \\vec{y}_{uv}\\Vert ^2\\le 1&\\forall u,v\\in V\\\\&\\Vert \\vec{z}_v\\Vert ^2\\le 1&\\forall v\\in V\\\\\\end{array}$ Note that this $SDP_{PR}$ is solvable in polynimial time for the same reason that $LP_{PR}$ is solvable: for each pair of $(u,v)$ , we can find $n$ different values of $r$ that give all of the distinct constraints.",
"The rounding algorithm is basically the same as the $TZ_2$ -optimization problem with outliers.",
"We first solve the $SDP_{PR}$ and get an optimal solution $(\\vec{x}_v^*,\\vec{y}_{uv}^*,\\vec{z}_v^*)$ .",
"We then use independent randomized rounding to get $A$ , adding each $v\\in V$ to $A$ independently with probability $\\min \\lbrace \\frac{6\\ln n}{\\varepsilon }\\cdot \\Vert \\vec{x}_v^*\\Vert ^2,1\\rbrace $ where $\\varepsilon $ is a small constant.",
"Then we use threshold rounding to get $F$ , adding each $v\\in V$ to $F$ if $\\Vert \\vec{z}_v^*\\Vert ^2\\ge \\frac{1}{1+\\varepsilon }$ .",
"This is an ($O(\\log n),1+\\varepsilon $ )-approximation.",
"It is easy to see that $|F|\\le (1+\\varepsilon )f$ because $\\sum _{v\\in V}\\Vert \\vec{z}_v^*\\Vert ^2\\le f$ .",
"The proof that the expected cost is at most $O(\\log n)\\cdot OPT$ is in Appendix , which completes the proof of Theorem REF ."
],
[
"Conclusion and Future Work",
"In this paper we initiate the study of approximating approximate distance oracles.",
"This is a different take on the question of optimizing data structures, where we attempt to find the best data structure for a particular input, rather than for a class of inputs.",
"In order to make this tractable (or even well-defined), we restrict our attention to known classes of distance oracles, and show that it is sometimes possible to find the best of these restricted oracles.",
"We also extended our approaches to optimize in the presence of outliers.",
"For future work, the major question is clearly whether we can approximately optimize higher level (i.e., higher stretch) Thorup-Zwick distance oracles.",
"Although we show an integrality gap for the basic LP, it is quite conceivable that a stronger LP or SDP could be used to give a logarithmic approximation ratio.",
"Beyond this, there are other distance oracles which could be optimized – we chose Thorup-Zwick and Pǎtraşcu-Roditty since they are well-known and in some ways canonical, but it would be interesting to extend these ideas to other oracles.",
"At a higher level, we believe that the definitions and ideas we have introduced here could lead to many interesting questions about optimizing data structures for given inputs: can we find near-optimal distance labels?",
"Or compact routing schemes?",
"Or connectivity oracles?",
"Or fault-tolerant oracles?",
"Essentially any data structure question in which there is a choice of which data to store, rather than how to store it, can be put into our optimization framework.",
"Exploring this space is an exciting future direction."
],
[
"Label Cover Problem",
"For the lower bound, we start from a hard Label Cover instance, and following the steps in proving the hardness of approximating Set Cover problem.",
"Since the definition of the Label Cover problem is somewhat complex, we break it into parts: first defining an instance, a labelling, and then defining the problem.",
"Note that we are using a specific setting where the parameters in the graph are strongly related, so it is slightly different from the definition of classic/general Label Cover problem.",
"Definition A.1 A label cover instance consists of ($G=(V_1,V_2,E),\\Sigma ,\\Pi $ ) where $G$ is a bipartite graph between vertex sets $V_1$ and $V_2$ and an edge set $E$ .",
"Let $V^{\\prime }=V_1\\cup V_2$ $G$ is left and right regular.",
"Denote by $\\Delta _1$ and $\\Delta _2$ the degrees of vertices in $V_1$ and $V_2$ respectively.",
"For each edge $e$ , there is a constraint $\\Pi _e$ which is a bijection function from $\\Sigma $ to itself.",
"The set of all constraints in $G$ are $\\Pi = \\lbrace \\Pi _e:\\Sigma \\rightarrow \\Sigma \\mid e\\in E\\rbrace $ Definition A.2 A labelling of the graph, is a mapping $\\sigma :V^{\\prime }\\rightarrow \\Sigma $ which assigns a label for each vertex of $G$ .",
"A labelling $\\sigma $ is said to satisfy an edge $e=(v_1,v_2)$ if and only if $\\Pi _e(\\sigma (v_1))=\\sigma (v_2)$ .",
"The following definition is the problem which will be the starting point of our reduction.",
"Definition A.3 In the $\\text{LabelCover}_{n,r,\\varepsilon }$ problem, we are given an instance $(G, \\Sigma , \\Pi )$ of Label Cover where $|V_1|=(5n)^r,|V_2|=(5n)^r,|\\Sigma |=7^r,\\Delta _1=15^r,\\Delta _2=15^r$ , and one of the following is true: There exists a labelling $\\sigma $ such that it satisfies all the edges $e$ in $G$ (in which case we say that the input is a YES instance), or For any labelling $\\sigma $ of the vertices, no more than $\\varepsilon ^r|E|$ edges are satisfied by $\\sigma $ (in which case we say that the input is a NO instance).",
"The goal is to determine whether the input is a YES or a NO instance.",
"Label Cover forms the starting point of many hardness of approximation results.",
"Its hardness is a now-classical application of the PCP theorem [3] and Raz's parallel repetition lemma [19], which give the following theorem.",
"Theorem A.4 ([19]) There exists a constant $\\varepsilon \\ge 0$ such that $\\text{LabelCover}_{n,r,\\varepsilon }$ is not in $\\mathbf {P}$ unless $\\mathbf {NP}\\subseteq \\mathbf {DTIME}(n^{O(r)})$ .",
"For example, there exists a constant $\\varepsilon \\ge 0$ such that $\\text{LabelCover}_{n,3\\log \\log n,\\varepsilon }$ is not in $\\mathbf {P}$ unless $\\mathbf {NP}\\subseteq \\mathbf {DTIME}(n^{O(\\log \\log n)})$ .",
"Starting from here, we will fix $r=3\\log \\log n$ , and $\\varepsilon $ be the constant which makes $\\text{LabelCover}_{n,3\\log \\log n,\\varepsilon }$ hard."
],
[
"$(m,l,\\delta )$ -Set System",
"We also need a $(m,l,\\delta )$ -set system (see Definition REF ) to do the reduction.",
"We can construct a ($m,l,\\delta $ )-set system by using a ($l,\\gamma $ )-independent collection of length $m$ strings.",
"Definition A.5 Let $B$ be a collection (may contains repetitions) of binary strings of length $m$ .",
"$B$ is $(l,\\gamma )$ -independent if the following inequality holds for every $i_1,i_2,\\ldots ,i_l$ and $a\\in \\lbrace 0,1\\rbrace ^l$ : $\\left|\\Pr _{x\\in B}[x_{i_1}=a_1\\wedge \\ldots \\wedge x_{i_l}=a_l]-2^{-l}\\right|\\le \\gamma .$ A corollary of lemma 1 and construction 3 in [2] provides a explict construction of ($l,\\gamma $ )-independent collection.",
"Corollary A.6 For any $l\\le m$ , there is an explict construction of a ($l,(1-2^{-l})\\cdot 2^{-l-1}$ )-independent collection of length $m$ strings with $|B|=4^{l+1}m^2$ in $|B|^{O(1)}$ time.",
"With the corollary in hand, we can construct a ($m,l,\\delta $ )-set system with the parameters we want.",
"Lemma A.7 For any $l\\le m$ , there is an explicit construction of a ($m,l,2^{-l-1}$ )-set system with $|B|=4^{l+1}m^2$ in $|B|^{O(1)}$ time.",
"Let $B$ be the collection of length $m$ strings in Corollary REF .",
"Define $C_i=\\lbrace x\\in B\\mid x_i=1\\rbrace $ for all $i\\in [m]$ , we will show that ($B;C_1,\\ldots ,C_m$ ) is a ($m,l,2^{-l-1}$ )-set system.",
"Assume that there exist $D_{i_1},D_{i_2},\\ldots ,D_{i_l}$ such that $\\left|\\bigcup _{j=1}^lD_{i_j}\\right|\\ge (1-2^{-l-1})|B|$ , where each $D_{i_j}$ is either $C_{i_j}$ or $\\overline{C_{i_j}}$ (note that this implies that there are no $j$ and $k$ such that $D_{i_j}=\\overline{D_{i_k}}$ ).",
"Define $a_j={\\left\\lbrace \\begin{array}{ll}0,&\\mbox{if }D_{i_j}=C_{i_j}\\\\1,&\\mbox{if }D_{i_j}=\\overline{C_{i_j}}\\end{array}\\right.",
"}.$ Let $S=\\lbrace x\\mid x\\in B,x_{i_1}=a_1,x_{i_2}=a_2,\\ldots ,x_{i_l}=a_l\\rbrace $ .",
"Because $B$ is a ($l,(1-2^{-l})\\cdot 2^{-l-1}$ )-independent collection, we have $|S|=|\\lbrace x\\mid x\\in B,x_{i_1}=a_1,x_{i_2}=a_2,\\ldots ,x_{i_l}=a_l\\rbrace |>(2^{-l}-(1-2^{-l})\\cdot 2^{-l-1})|B|>2^{-l-1}|B|.$ On the other hand, note by construction, for all $x\\in S$ and $j\\in [l]$ , we have $x\\notin D_{i_j}$ , which implies that $\\left|\\bigcup _{j=1}^lD_{i_j}\\right|\\le |B|-|S|<(1-2^{-l-1})|B|$ : a contradiction."
],
[
"Reduction",
"We now show how to use the set systems from the previous section to give a reduction from Label Cover to $TZ_2$ -optimization problem.",
"Let $(G=(V_1,V_2,E),\\Sigma ,\\Pi )$ be a $\\text{LabelCover}_{n,r,\\varepsilon }$ instance with $r=3\\log \\log n$ , and let $(B;C_1,\\ldots ,C_m)$ be a ($m,l,2^{-l-1}$ )-set system with $m=|\\Sigma |=7^r,l=r\\log n$ .",
"We first create a universe $\\mathcal {U}=E\\times B$ , and a set of sets $\\mathcal {S}=\\lbrace S_{v,x}\\mid v\\in V^{\\prime },x\\in [m]\\rbrace $ (recall that $V^{\\prime } = V_1 \\cup V_2$ ).",
"Here $S_{v,x}=\\bigcup _{e:v\\in e,e\\in E}\\lbrace e\\rbrace \\times C_{\\Pi _e(x)},\\mbox{ if }v\\in V_1,$ $S_{v,x}=\\bigcup _{e:v\\in e,e\\in E}\\lbrace e\\rbrace \\times \\overline{C_x},\\mbox{ if }v\\in V_2.$ We know that $|E|=(15n)^r$ , $|B|=4^{r\\log n+1}\\cdot 7^{2r}=n^{\\Theta (1)\\cdot r}$ and $|\\mathcal {S}|=m\\cdot |V^{\\prime }|=7^r\\cdot 2\\cdot (5n)^r$ , so $|\\mathcal {U}|\\gg |\\mathcal {S}|$ .",
"Without lose of generality and for simplicity of our proof, we can assume $|\\mathcal {U}|$ is dividable by $|\\mathcal {S}|$ , so that we can replicate $\\mathcal {S}$ for $\\frac{|\\mathcal {U}|}{|\\mathcal {S}|}$ times, and get a set of sets $\\mathcal {S}^{\\prime }=\\mathcal {S}^{(1)} \\cup \\ldots \\cup \\mathcal {S}^{\\left(\\frac{|\\mathcal {U}|}{|\\mathcal {S}|}\\right)}$ which has the same size as $\\mathcal {U}$ .",
"It is also easy to see that each $u=((v_1,v_2),b)\\in \\mathcal {U}$ appears in exactly $m$ sets in $\\mathcal {S}$ because for each $x\\in [m]$ , either $u\\in S_{v_1,x}$ or $u\\in S_{v_2,\\Pi _{(v_1,v_2)}(x)}$ .",
"Therefore each $u\\in \\mathcal {U}$ appears in $\\frac{m|\\mathcal {U}|}{|\\mathcal {S}|}=\\frac{|\\mathcal {U}|}{|V^{\\prime }|}$ sets in $\\mathcal {S}^{\\prime }$ .",
"The metric space is defined as $V=\\mathcal {U}\\cup \\mathcal {S}^{\\prime }$ and the distance is defined as following: $d(u,v)={\\left\\lbrace \\begin{array}{ll}1.2,&\\mbox{if }u\\in \\mathcal {S}^{\\prime },v\\in \\mathcal {S}^{\\prime }\\\\1.4,&\\mbox{if }u\\in v\\mbox{ or }v\\in u\\\\1.6,&\\mbox{if }u\\in \\mathcal {U},v\\in \\mathcal {U}\\\\1.8,&\\mbox{otherwise}\\end{array}\\right.",
"}$ This metric space $(V, d)$ will form the instance of $TZ_2$ -optimization which we analyze.",
"It is easy to see that the reduction is polynomial because $|V|$ is polynomial of $|E|$ ."
],
[
"Analysis",
"Lemma A.8 If ($G,\\Sigma ,\\Pi $ ) is a YES instance in the $\\text{LabelCover}_{n,r,\\varepsilon }$ problem.",
"Then the reduction $(V,d)$ to the $TZ_2$ -optimization problem has a solution with cost $\\le (|V^{\\prime }|+1)\\cdot |V|$ .",
"Let $\\sigma :V^{\\prime }\\rightarrow [m]$ denote a labelling of G which satisfies all the edges in $E$ .",
"Let $A_1=\\lbrace S_{v,\\sigma (v)}\\mid v\\in V^{\\prime }\\rbrace $ .",
"We claim that $A_1$ is a solution with cost at most $(|V^{\\prime }|+1)\\cdot |V|$ .",
"Note that in Section REF we showed that the level 2 cost $\\sum _{u\\in V}|R_{2u}|=|V|\\cdot |A_1|=|V^{\\prime }|\\cdot |V|$ , the only thing left is to show that the level 1 cost is $\\sum _{u\\in V}|R_{1u}|\\le |V|$ .",
"We will prove this by showing $R_{1u}\\subseteq \\lbrace u\\rbrace $ for all $u\\in V$ .",
"For any $u\\in \\mathcal {S}^{\\prime }$ , we have that $d(u,v)\\ge 1.2$ for all $v\\in V$ because the definition of $d$ , and $\\min _{w\\in A_1}d(u,w)=1.2$ because $A_1\\cap \\mathcal {S}^{\\prime }\\ne \\varnothing $ .",
"Thus $R_{1u}=\\lbrace v\\in V\\mid d(u,v)<\\min _{w\\in A_1}d(u,w)\\rbrace \\subseteq \\lbrace u\\rbrace $ .",
"For any $u=((v_1,v_2),b)\\in \\mathcal {U}$ , we have that $d(u,v)\\ge 1.4$ for all $v\\in V$ because the definition of $d$ .",
"We also know that either $u\\in S_{v_1,\\sigma (v_1)}$ or $u\\in S_{v_1,\\Pi _{(v_1,v_2)}(\\sigma (v_1))}$ by the definition of $\\mathcal {S}$ , and $\\Pi _{(v_1,v_2)}(\\sigma (v_1))=\\sigma (v_2)$ because edge $(v_1,v_2)$ is satisfied by labelling $\\sigma $ .",
"Therefore $u\\in S_{v_1,\\sigma (v_1)}\\cup S_{v_2,\\sigma (v_2)}$ .",
"From the fact that both $S_{v_1,\\sigma (v_1)}$ and $S_{v_2,\\sigma (v_2)}$ are in $A_1$ , we have $\\min _{w\\in A_1}d(u,w)=1.4$ .",
"Thus $R_{1u}=\\lbrace v\\in V\\mid d(u,v)<\\min _{w\\in A_1}d(u,w)\\rbrace \\subseteq \\lbrace u\\rbrace $ .",
"Therefore $R_{1u}\\subseteq \\lbrace u\\rbrace $ for all $u\\in V$ , so that $A_1$ is a solution with cost at most $\\le (|V^{\\prime }|+1)\\cdot |V|$ .",
"Lemma A.9 If ($G,\\Sigma ,\\Pi $ ) is a No instance in the $\\text{LabelCover}_{n,r,\\varepsilon }$ problem.",
"Then the reduction $(V,d)$ to the $TZ_2$ -optimization problem has no solution with cost $<\\frac{l}{8}|V^{\\prime }|\\cdot |V|$ .",
"We prove the lemma by showing that if the optimal solution of the reduction $(V,d)$ to the $TZ_2$ -optimization problem has cost $<\\frac{l}{8}|V^{\\prime }|\\cdot |V|$ , then there exists a labelling $\\sigma $ such that it satisfies more than $\\varepsilon ^r|E|$ edges.",
"Assume the optimal solution $A_1\\subseteq V$ has $cost(A_1,V,d)<\\frac{l}{8}|V^{\\prime }|\\cdot |V|$ , then $|A_1|<\\frac{l}{8}|V^{\\prime }|$ because the level 2 cost is $\\sum _{u\\in V}|R_{2u}|=|V||A_1|$ .",
"Let $L_v=\\lbrace x\\in [m]\\mid \\exists j\\in \\left[\\frac{|\\mathcal {U}|}{|\\mathcal {S}|}\\right]\\text{ s.t.",
"}S_{v,x}^{(j)}\\in A_1\\cap \\mathcal {S}^{\\prime }\\rbrace $ for all $v\\in V$ , then $\\sum _vL_v\\le |A_1\\cap \\mathcal {S}^{\\prime }|\\le |A_1|<\\frac{l}{8}|V^{\\prime }|$ .",
"Therefore at least $\\frac{3}{4}|V^{\\prime }|$ vertices has $|L_v|<\\frac{l}{2}$ , because otherwise $\\sum _vL_v\\ge (1-\\frac{3}{4})\\cdot |V^{\\prime }|\\cdot \\frac{l}{2}\\ge \\frac{l}{8}|V^{\\prime }|$ .",
"Let $E_1=\\lbrace e=(v_1,v_2)\\in E\\mid |L_{v_1}|<\\frac{l}{2},|L_{v_2}|<\\frac{l}{2}\\rbrace $ .",
"Then $|E_1|\\ge \\frac{|E|}{2}$ because $|V_1|=|V_2|=\\frac{|V^{\\prime }|}{2}$ and $G$ is regular.",
"On the other hand, we define a $u\\in \\mathcal {U}$ is “uncovered” if $\\lbrace v\\in A_1\\cap \\mathcal {S}^{\\prime }\\mid u\\in v\\rbrace =\\varnothing $ .",
"Then for any uncovered $u\\in \\mathcal {U}$ , we know that $\\min _{w\\in A_1}d(u,w)=1.6$ .",
"Thus $R_{1u}=&\\lbrace v\\in V\\mid d(u,v)<\\min _{w\\in A_1}d(u,w)\\rbrace \\\\\\ge &\\lbrace v\\in \\mathcal {S}^{\\prime }\\mid d(u,v)<1.6\\rbrace \\\\=&\\lbrace v\\in \\mathcal {S}^{\\prime }\\mid u\\in v\\rbrace =\\frac{|\\mathcal {U}|}{|V^{\\prime }|}.$ Therefore $|\\lbrace u\\in \\mathcal {U}\\mid u\\text{ is uncovered}\\rbrace |<\\frac{\\frac{l}{8}|V^{\\prime }|\\cdot |V|}{\\frac{|\\mathcal {U}|}{|V^{\\prime }|}}<\\frac{l}{4}|V^{\\prime }|^2$ .",
"Let $E_2=\\lbrace e\\in E\\mid |\\lbrace u=(e,b)\\in \\mathcal {U}\\mid u\\text{ is uncovered}\\rbrace |<\\frac{l|V^{\\prime }|^2}{|E|}\\rbrace $ .",
"Then $|E_2|\\ge \\frac{3}{4}|E|$ because otherwise $|\\lbrace u\\in \\mathcal {U}\\mid u\\text{ is uncovered}\\rbrace |\\ge (|E|-\\frac{3}{4}|E|)\\cdot \\frac{l|V^{\\prime }|^2}{|E|}\\rbrace \\ge \\frac{l}{4}|V^{\\prime }|^2$ .",
"Let $E^{\\prime }=E_1\\cap E_2$ , we know that $|E^{\\prime }|\\ge \\frac{|E|}{4}$ .",
"Now, we will show that if we uniformly random sample labels from $L_v$ for each $v\\in V^{\\prime }$ , the expected number of the edges satisfied in $E^{\\prime }$ is at least $\\frac{|E|}{l^2}$ .",
"For each edge $e=(v_1,v_2)\\in E^{\\prime }$ where $v_1\\in V_1$ and $v_2\\in V_2$ .",
"Assume $L_{v_1}=\\lbrace a_1,\\ldots ,a_p\\rbrace $ , $L_{v_2}=\\lbrace b_1,\\ldots ,b_q\\rbrace $ .",
"Note that for every $e\\in E_2$ we have $\\Big |\\lbrace u=(e,b)\\in \\mathcal {U}\\mid \\exists v\\in A_1\\cap \\mathcal {S}^{\\prime },u\\in v\\rbrace \\Big |\\ge |B|-\\frac{l|V^{\\prime }|^2}{|E|},$ and for all $u=((v_1,v_2),b)\\in \\mathcal {U}$ , there exists $v\\in A_1\\cap \\mathcal {S}^{\\prime }$ where $u\\in v$ iff $u\\in S_{v_1,a_i}$ or $u\\in S_{v_2,b_i}$ .",
"Thus we have $\\left|(\\lbrace e\\rbrace \\times B)\\cap \\left((\\bigcup _{i=1}^pS_{v_1,a_i})\\cup (\\bigcup _{j=1}^qS_{v_2,b_j})\\right)\\right|\\ge |B|-\\frac{l|V^{\\prime }|^2}{|E|},$ which means $\\left|(\\bigcup _{i=1}^pC_{\\Pi _e(a_i)})\\cup (\\bigcup _{j=1}^q\\overline{C_{b_j}})\\right|\\ge |B|-\\frac{l|V^{\\prime }|^2}{|E|}=(1-\\frac{l|V^{\\prime }|^2}{|E||B|})|B|=(1-(\\frac{5}{147n})^rl)|B|>(1-2^{-l-1})|B|.$ Thus by the definition of ($m,l,2^{-l-1})$ )-set system, we know that there exists $i,j$ such that $\\Pi _e(a_i)=b_j$ .",
"Therefore, $e$ is satisfied with probability $\\frac{1}{|L_{v_1}|\\cdot |L_{v_2}|}\\ge \\frac{4}{l^2}$ because the labels are uniformly sampled.",
"Thus the expected number of the edges satisfied in $E^{\\prime }$ is at least $\\frac{4}{l^2}\\cdot \\frac{|E|}{4}=\\frac{|E|}{l^2}$ , which means, there is a way to label all the vertices in $V^{\\prime }$ and satisfies at least $\\frac{|E|}{l^2}$ edges.",
"Finally, because $r=3\\log \\log n$ and $l=r\\log n$ , we know that at most $\\varepsilon ^r\\cdot |E|<\\frac{|E|}{l^2}$ edges can be satisfied by any labelling, which is a contradiction.",
"With these lemmas, we can prove our lower bound on the $TZ_2$ -optimization problem.",
"Proof of Theorem REF : By Lemma REF and Lemma REF , we have a polynomial reduction from $\\text{LabelCover}_{n,r,\\varepsilon }$ problem to $TZ_2$ -optimization problem, which maps a YES instance of $\\text{LabelCover}_{n,r,\\varepsilon }$ to a $TZ_2$ -optimization instance with optimal cost at most $(|V^{\\prime }|+1)\\cdot |V|$ , and maps a NO instance of $\\text{LabelCover}_{n,r,\\varepsilon }$ to a $TZ_2$ -optimization instance with optimal cost at least $\\frac{l}{8}|V^{\\prime }|\\cdot |V|$ .",
"The gap is $\\frac{\\frac{l}{8}|V^{\\prime }|\\cdot |V|}{(|V^{\\prime }|+1)\\cdot |V|}=\\Theta (\\frac{l}{8})=\\Theta (\\log |V|)$ .",
"Combined with the hardness Theorem REF , we know that unless $\\mathbf {NP}\\subseteq \\mathbf {DTIME}(n^{O(\\log \\log n)})$ , the $TZ_2$ -optimization problem does not admit a polynomial-time $o(\\log n)$ -approximation.",
"$\\Box $"
],
[
"Relaxation validity",
"We first prove that our LP relaxation is indeed valid, i.e., we prove the following claim.",
"Claim B.1 $LP_{TZ_k}$ is a valid relaxation to the $TZ_k$ -optimization problem.",
"Let $A_1,\\ldots ,A_{k-1}$ be a valid solution to the $TZ_k$ -optimization problem.",
"Let $x_v^{(i)}={1}_{v\\in A_i}$ and $y_{uv}^{(i)}={1}_{v\\in R_{iu}}$ for all $i\\in [k]$ and $u,v\\in V$ .",
"We can see that the objective value $\\sum _{i=1}^k\\sum _{u,v\\in V}y_{uv}^{(i)}=\\sum _{i=1}^k\\sum _{u\\in V}|R_{iu}|=cost(A_1,\\ldots ,A_{k-1},V,d)$ , which is the cost function.",
"We can also see that the first constraint is satisfied by $x_v^{(i)}$ and $y_{uv}^{(i)}$ because $\\varnothing =A_k\\subseteq A_{k-1}\\subseteq \\ldots \\subseteq A_0=V$ .",
"The second constraint is satisfied because if $v\\in A_{i-1}$ and there is no vertex in $A_i\\cap B_u(v)$ , then $v\\in R_{iu}$ .",
"The third constraint is trivially satisfied.",
"Therefore $x_v^{(i)}$ and $y_{uv}^{(i)}$ is a valid solution to $LP_{TZ_k}$ which makes the LP objective value equal to the actuall cost function.",
"Thus the claim is proved."
],
[
"Integrality gap",
"Let's consider an instance $(V,d)$ with $V=[n]$ .",
"All $n$ vertices lie on a circle and they evenly split the cycle.",
"The cycle distance $d(u,v)=\\min \\lbrace |u-v|,n+\\min \\lbrace u,v\\rbrace -\\max \\lbrace u,v\\rbrace \\rbrace $ .",
"We first show that on this instance, $LP_{TZ_k}$ has a solution with low cost.",
"Lemma B.2 $LP_{TZ_k}$ has a solution with cost $O(n^{1+\\frac{1}{2^{k-1}}})$ on instance $(V,d)$ .",
"Consider the following setting of the LP variables: let $x_{v}^{(i)}=n^{-\\frac{2^i-1}{2^{k-1}}}$ for all $v\\in V$ and $i\\in [k-1]$ , and let $y_{uv}^{(i)}=\\max \\lbrace 0,x_v^{(i-1)}-\\sum _{w\\in B_u(v)}x_w^{(i)}\\rbrace $ for all $u,v\\in V$ and $i\\in [k]$ .",
"We can see that $x_{v}^{(i)}=n^{-\\frac{2^i-1}{2^{k-1}}}\\ge n^{-\\frac{2^{i+1}-1}{2^{k-1}}}=x_{v}^{(i+1)}$ which satisfies the first constraint of $LP_{TZ_k}$ , $y_{uv}^{(i)}\\ge x_v^{(i-1)}-\\sum _{w\\in B_u(v)}x_w^{(i)}$ which satisfies the second constraint of $LP_{TZ_k}$ , and $y_{uv}^{(i)}\\ge 0$ which satisfies the third constraint of $LP_{TZ_k}$ .",
"Therefore $x_{v}^{(i)},y_{uv}^{(i)}$ is a valid solution to $LP_{TZ_k}$ .",
"The objective value of this solution is $\\sum _{i=1}^k\\sum _{u,v\\in V}y_{uv}^{(i)}=&\\sum _{i=1}^k\\sum _{u,v\\in V}\\max \\lbrace 0,x_v^{(i-1)}-\\sum _{w\\in B_u(v)}x_w^{(i)}\\rbrace \\\\=&\\sum _{i=1}^{k-1}\\sum _{u,v\\in V}\\max \\lbrace 0,x_v^{(i-1)}-|B_u(v)|x_v^{(i)}\\rbrace +\\sum _{u,v\\in V}(x_v^{(k-1)}-0)\\\\=&\\sum _{i=1}^{k-1}\\sum _{u,v\\in V}\\max \\lbrace 0,n^{-\\frac{2^{i-1}-1}{2^{k-1}}}-(2\\cdot d(u,v)+1)\\cdot n^{-\\frac{2^i-1}{2^{k-1}}}\\rbrace +\\sum _{u,v\\in V}n^{-\\frac{2^{k-1}-1}{2^{k-1}}}\\\\=&\\sum _{i=1}^{k-1}\\sum _{u\\in V}\\Big (n^{-\\frac{2^{i-1}-1}{2^{k-1}}}-n^{-\\frac{2^i-1}{2^{k-1}}}+n^{-\\frac{2^{i-1}-1}{2^{k-1}}}-3\\cdot n^{-\\frac{2^i-1}{2^{k-1}}}+\\ldots \\Big ) +n^{1+\\frac{1}{2^{k-1}}}\\\\=&\\sum _{i=1}^{k-1}n\\cdot \\Big (O(n^{-\\frac{2^{i-1}-1}{2^{k-1}}})\\cdot O(n^{\\frac{2^i-1}{2^{k-1}}-\\frac{2^{i-1}-1}{2^{k-1}}})\\Big )+n^{1+\\frac{1}{2^{k-1}}}\\\\=&\\sum _{i=1}^{k-1}O(n^{1+\\frac{1}{2^{k-1}}})+n^{1+\\frac{1}{2^{k-1}}}\\\\=&O(n^{1+\\frac{1}{2^{k-1}}})$ Here equation (REF ) holds because of all $x_v^{(i)}$ are equal and all $x_v^{(k)}=0$ .",
"Equation () holds because of the definition of circle distance.",
"Equation () is a unrolling, and equation () is a summation over arithmetic progression.",
"The last equation holds because of $k$ is a constant.",
"Next we will show that the optimal solution of this instance is large.",
"Lemma B.3 The optimal solution to the instance $(V,d)$ has cost at least $\\Omega (n^{1+\\frac{1}{k}})$ .",
"We will prove this lemma using a stronger claim.",
"The lemma holds by setting $a=1,b=\\lfloor \\frac{n}{2}\\rfloor $ , and $l=k$ in this claim: Claim B.4 For a segment $[a,b]$ of the cycle where $a,b\\in [n]$ , $b-a<\\frac{n}{2}$ , and all the vertices in $[a,b]$ are NOT in $A_l$ , we have $\\sum _{i=1}^l\\sum _{u\\in [a,b]\\cap [n]}|R_{iu}|\\ge \\left(\\frac{b-a+1}{4^l}\\right)^{1+\\frac{1}{l}}$ for each $l\\in [k]$ .",
"We prove this by doing induction on $l$ .",
"The base case is $l=1$ .",
"For each vertex $u\\in [a,b]$ , We know that $R_{1u}=\\lbrace v\\in V\\mid d(u,v)<\\min _{w\\in A_1}d(u,w)\\rbrace \\subseteq \\lbrace v\\in [a,b]\\mid |u-v|\\le \\min \\lbrace u-a,b-u\\rbrace \\rbrace ,$ so $|R_{1u}|\\ge 2\\cdot \\min \\lbrace u-a,b-u\\rbrace $ because all the vertices in $[a,b]$ are NOT in $A_1$ .",
"So $\\sum _{u\\in [a,b]}|R_{1u}|\\ge 2\\cdot (1+2+\\ldots +\\left\\lfloor \\frac{b-a+2}{2}\\right\\rfloor +\\ldots +2+1)\\ge \\left(\\frac{b-a+1}{4}\\right)^2.$ Now we consider general case $l\\ge 2$ , and assume the claim is established on $l-1$ .",
"Assume there are $m$ vertices $t_1,\\ldots ,t_m\\in [a,\\lceil \\frac{a+b}{2}\\rceil ]\\cap A_{l-1}$ , and $[a,\\lceil \\frac{a+b}{2}\\rceil ]$ are splitted to small segments $[a_0,b_0],\\ldots ,[a_m,b_m]$ where all the vertices in $[a_i,b_i]$ are not in $A_{l-1}$ (if a segment has no vertex inside, we let $b_i=a_i-1$ without lose of generality).",
"Then for each $i\\in [m]$ and $u\\in [a_i,b_i+1]$ , we have $t_1,\\ldots ,t_i\\in R_{lu}$ because $R_{lu}=\\lbrace v\\in A_{l-1}\\mid d(u,v)<\\min _{w\\in A_l}d(u,w)\\rbrace $ .",
"Thus $\\sum _{i=1}^l\\sum _{u\\in [a,b]\\cap [n]}|R_{iu}|\\ge &\\sum _{i=0}^m\\left(\\sum _{j=1}^{l-1}\\sum _{u\\in [a_i,b_i]\\cap [n]}|R_{ju}|+\\sum _{u\\in [a_i,b_i]\\cap [n]}|R_{lu}|\\right)\\\\\\ge &\\sum _{i=0}^m\\left(\\left(\\frac{b_i-a_i+1}{4^{l-1}}\\right)^{1+\\frac{1}{l-1}}+i\\cdot (b_i-a_i+2)\\right).$ If $m>\\frac{b-a+1}{4}$ , $\\sum _{i=0}^mi\\cdot 1$ is already at least $\\left(\\frac{b-a+1}{4^l}\\right)^{1+\\frac{1}{l}}$ .",
"If $m\\le \\frac{b-a+1}{4}$ , we have a stronger inequality which we will prove later: Lemma B.5 If $\\alpha \\in [1,2]$ and $x_i\\ge 0$ for all $i\\in [m]$ , then $\\sum _{i=0}^m(x_i^\\alpha +4i\\cdot x_i)\\ge \\left(\\sum _{i=0}^mx_i\\right)^{2-\\frac{1}{\\alpha }}$ Using this inequality, by setting $x_i=\\frac{b_i-a_i+1}{4^{l-1}}$ and $\\alpha =1+\\frac{1}{l-1}$ we have $\\sum _{i=1}^l\\sum _{u\\in [a,b]\\cap [n]}|R_{iu}|\\ge &\\left(\\sum _{i=0}^m\\frac{b_i-a_i+1}{4^{l-1}}\\right)^{2-\\frac{1}{1+\\frac{1}{l-1}}}\\\\\\ge &\\left(\\frac{\\lceil \\frac{a+b}{2}\\rceil -a+1-m}{4^{l-1}}\\right)^{1+\\frac{1}{l}}\\\\\\ge &\\left(\\frac{\\frac{b-a}{2}+1-\\frac{b-a}{4}}{4^{l-1}}\\right)^{1+\\frac{1}{l}}\\ge \\left(\\frac{b-a+1}{4^l}\\right)^{1+\\frac{1}{l}}.$ With these lemma in hand, we can now prove Theorem REF .",
"Proof of Theorem REF : Combine Lemma REF and Lemma REF , there is an $\\Omega (\\frac{n^{1+\\frac{1}{k}}}{n^{1+\\frac{1}{2^k-1}}})=\\Omega (n^{\\frac{1}{k}-\\frac{1}{2^k-1}})$ integrality gap for the basic LP relaxation $LP_{TZ_k}$ .",
"$\\Box $ Proof of Lemma REF : Let $M=(\\sum _{i=1}^mx_i)^\\frac{\\alpha -1}{\\alpha }$ .",
"We first split the problem to 2 cases, depending on whether $m\\le M$ .",
"Case 1: $m\\le M$ .",
"In this case, by Hölder's inequality, we have $(\\sum _{i=0}^mx_i^\\alpha )^\\frac{1}{\\alpha }\\cdot (\\sum _{i=0}^m1^\\frac{\\alpha }{\\alpha -1})^\\frac{\\alpha -1}{\\alpha }\\ge (\\sum _{i=0}^mx_i\\cdot 1)$ thus $\\sum _{i=0}^mx_i^\\alpha \\ge \\frac{(\\sum _{i=0}^mx_i)^\\alpha }{m^{\\alpha -1}}\\ge \\frac{(\\sum _{i=0}^mx_i)^\\alpha }{M^{\\alpha -1}}\\ge (\\sum _{i=0}^mx_i)^{\\alpha -\\frac{\\alpha -1}{\\alpha }\\cdot (\\alpha -1)}=(\\sum _{i=0}^mx_i)^{2-\\frac{1}{\\alpha }}$ Case 2: $m>M$ .",
"Let's fix $T=\\sum _{i=0}^mx_i$ and consider the $\\mathbf {x}^*$ which minimizes the left side: $l(\\mathbf {x})=\\sum _{i=0}^m(x_i^\\alpha +4i\\cdot x_i)$ .",
"Consider any two consecutive variables $x_j$ and $x_{j+1}$ , we claim that, in $\\mathbf {x}^*$ , for each $0\\le j<m$ , either $x_{j+1}^*=0$ , or $(x_j^*)^{\\alpha -1}-(x_{j+1}^*)^{\\alpha -1}=\\frac{4}{\\alpha }$ .",
"This is because, if we replace the $x_{j+1}$ in $l(\\mathbf {x})$ by $T-\\sum _{i\\ne (j+1)}x_i$ and do partial derivative with respect of $x_j$ , we have $&\\frac{\\partial }{\\partial x_j}\\left(\\sum _{i\\ne (j+1)}(x_i^\\alpha +4i\\cdot x_i)+\\left(T-\\sum _{i\\ne (j+1)}x_i\\right)^\\alpha +4(j+1)\\cdot \\left(T-\\sum _{i\\ne (j+1)}x_i \\right)\\right)\\\\=&\\frac{\\partial }{\\partial x_j}\\left(x_j^\\alpha +4j\\times x_j+\\left(T-\\sum \\nolimits _{i\\ne (j+1)}x_i\\right)^\\alpha +4(j+1)\\cdot \\left(T-\\sum \\nolimits _{i\\ne (j+1)}x_i \\right)\\right)\\\\=&\\alpha \\cdot x_j^{\\alpha -1}+4j-\\alpha \\cdot \\left(T-\\sum \\nolimits _{i\\ne (j+1)}x_i \\right)^{\\alpha -1}-4(j+1)\\\\=&\\alpha \\cdot \\left(x_j^{\\alpha -1}-\\left(T-\\sum \\nolimits _{i\\ne (j+1)}x_i \\right)^{\\alpha -1}\\right)-4\\\\=&\\alpha (x_j^{\\alpha -1}-x_{j+1}^{\\alpha -1})-4.$ If we fix $x_i$ for all $i\\in [0,m]\\cap \\mathbb {N}\\backslash \\lbrace j,j+1\\rbrace $ , this partial derivative monotonically increases as $x_j$ increases.",
"Thus when $l(\\mathbf {x})$ is minimized, either the partial derivative equals 0, which means $(x_j^*)^{\\alpha -1}-(x_{j+1}^*)^{\\alpha -1}=\\frac{4}{\\alpha }$ , or $x_j$ hits the ceiling, which means $x_j^*=T-\\sum _{i\\ne j,(j+1)}x_i^*$ , so $x_{j+1}^*=0$ .",
"This result shows that, the number series $(x_0^*)^{\\alpha -1},(x_1^*)^{\\alpha -1},\\ldots ,(x_m^*)^{\\alpha -1}$ is in decreasing order, where $(x_i^*)^{\\alpha -1}={\\left\\lbrace \\begin{array}{ll}(x_{i-1}^*)^{\\alpha -1}-\\frac{4}{\\alpha },&\\mbox{if }(x_i^*)^{\\alpha -1}>\\frac{4}{\\alpha }\\\\0,&\\mbox{otherwise}\\end{array}\\right.",
"}$ If the number of non-zero entries in $\\mathbf {x}^*$ is at most $M$ , then this comes back to the Case 1.",
"Otherwise, there are more than $M$ non-zero entries in $\\mathbf {x}^*$ , thus $x_0,x_1,\\ldots ,x_M$ are all non-zero, and $(x_i^*)^{\\alpha -1}\\ge \\frac{4}{\\alpha }\\cdot (M-i)$ for $i\\le M$ .",
"Therefore $\\sum _{i=0}^m(x_i^*)^\\alpha &\\ge \\sum _{i=0}^M\\left(\\frac{4}{\\alpha }\\cdot (M-i)\\right)^\\frac{\\alpha }{\\alpha -1}\\\\&\\ge \\sum _{i=1}^M\\left(\\frac{4i}{\\alpha }\\right)^\\frac{\\alpha }{\\alpha -1}\\\\&\\ge \\frac{(\\sum _{i=1}^M\\frac{4i}{\\alpha })^\\frac{\\alpha }{\\alpha -1}}{M^\\frac{1}{\\alpha -1}}\\\\&\\ge \\frac{(\\frac{2M^2}{\\alpha })^\\frac{\\alpha }{\\alpha -1}}{M^\\frac{1}{\\alpha -1}}\\\\&\\ge (\\frac{2}{\\alpha })^\\frac{\\alpha }{\\alpha -1}\\cdot (\\sum _{i=0}^mx_i)^{\\frac{\\alpha -1}{\\alpha }\\cdot (\\frac{2\\alpha }{\\alpha -1}-\\frac{1}{\\alpha -1})}\\\\&\\ge (\\sum _{i=0}^mx_i)^{2-\\frac{1}{\\alpha }}$ Here inequality () holds because of $\\alpha \\in [1,2]$ .",
"Inequality (REF ) holds because of Hölder's inequality $(\\sum _{i=1}^m1^\\alpha )^\\frac{1}{\\alpha }\\cdot (\\sum _{i=1}^my_i^\\frac{\\alpha }{\\alpha -1})^\\frac{\\alpha -1}{\\alpha }\\ge \\sum _{i=1}^my_i\\cdot 1$ $\\Box $"
],
[
"Proof of Valid Relaxation",
"We prove the following claim: Claim C.1 $LP_{PR}$ is a valid relaxation to the $PR$ -optimization problem.",
"Let $A$ be a valid solution to the $PR$ -optimization problem.",
"Let $x_v={1}_{v\\in A}$ and $y_{uv}={1}_{\\lbrace u,v\\rbrace \\in R}$ for all $u,v\\in V$ .",
"We can see that the objective value $\\sum _{v\\in V}n\\cdot x_v+\\sum _{\\lbrace u,v\\rbrace \\subseteq V}y_{uv}=n\\cdot |A|+|R|=cost(A,V,d)$ , which is the cost function.",
"We can also see that the first constraint is satisfied by $x_v$ and $y_{uv}$ because if $y_{uv}=0$ , we have $d(u,v)\\ge \\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1$ , then for all $r\\in [0,d(u,v)]$ , there must be a vertex in $A\\cap (B_u(r)\\cup B_v(d(u,v)-r))$ , which makes $0\\ge 1-\\sum _{w\\in B_u(r)\\cup B_v(d(u,v)-r)}x_w$ satisfied.",
"The second and the third constraints are trivially satisfied.",
"Therefore $x_v$ and $y_{uv}$ is a valid solution to $LP_{PR}$ which makes the LP objective value equal to the actuall cost function.",
"Thus the claim is proved.",
"$\\Box $"
],
[
"Lower Bound Proofs",
"We start from the following theorem: Theorem C.2 ([20]) Unless $\\mathbf {P}=\\mathbf {NP}$ , there is no $o(\\log n)$ -approximation to the set cover problem.",
"We now prove two lemmas about the reduction (completeness and soundness).",
"Lemma C.3 If there is a solution $\\mathcal {S}^*$ to the set cover instance $(\\mathcal {U},\\mathcal {S})$ where $|\\mathcal {S}^*|=t$ , then there is a set $A$ where $cost(A,V,d)\\le t|V|$ .",
"For each $S\\in S^*$ , we add an arbitrary element from $G_S$ to $A$ .",
"Then for every vertex in $V$ , the closest vertex in $A$ has distance at most 1 to it.",
"Therefore $R=&\\left\\lbrace \\lbrace u,v\\rbrace \\subseteq V\\mid d(u,v)<\\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1\\right\\rbrace \\\\=&\\left\\lbrace \\lbrace u,v\\rbrace \\subseteq V\\mid d(u,v)<1+1-1\\right\\rbrace =\\varnothing $ Thus the total cost is at most $|V|\\cdot |A|+|R|=t|V|$ .",
"Lemma C.4 If there is a set $A \\subseteq V$ where $cost(A,V,d)\\le t|V|$ , then there exists a solution $\\mathcal {S^*}$ to the set cover instance $(\\mathcal {U},\\mathcal {S})$ where $|\\mathcal {S}^*|=t$ .",
"First, we say that a group $G=G_e$ or $G=G_S$ is “covered” if there exists a vertex $u\\in G$ , which $\\min _{w\\in A}d(u,w)=1$ .",
"Then by the definition of $d$ , it's easy to see that if a group $G$ is covered, then for all vertices $u\\in G$ , we have $\\min _{w\\in A}d(u,w)=1$ .",
"In addition, if a group $G_e$ is covered, then either $G_e\\cap A\\ne \\varnothing $ , or there is a $S\\in \\mathcal {S}$ , where $e\\in S$ and $G_S\\cap A\\ne \\varnothing $ .",
"We can also see that, if a group $G$ is not covered, then let $R_G=&\\left\\lbrace \\lbrace u,v\\rbrace \\subseteq G\\mid d(u,v)<\\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1\\right\\rbrace \\\\=&\\left\\lbrace \\lbrace u,v\\rbrace \\subseteq G\\mid d(u,v)<2+2-1\\right\\rbrace \\\\=&\\lbrace \\lbrace u,v\\rbrace \\subseteq G\\rbrace $ Thus $|R_G|\\ge \\frac{3n(3n-1)}{2}>3n^2>|V|$ .",
"Therefore if we add an arbitrary element from $G$ to $A$ , then $|R|$ decreases by at least $|R_G|\\ge |V|$ , and $|V|\\cdot |A|$ increases by $|V|$ , which makes $cost(A,V,d)$ only decrease.",
"If we keep doing this operation, there will be a set $A$ which makes sure that all the groups are covered, and $cost(A,V,d)\\le t|V|$ .",
"Now, for every vertex $u\\in V$ , we have $\\min _{w\\in A}d(u,w)=1$ , so $R=\\varnothing $ .",
"We can keep modifying $A$ to the form we want.",
"If there is a vertex $v\\in G_e\\cap A$ , removing $v$ and simultaneously adding a vertex in any $S\\ni e$ to $A$ does not increase the cost.",
"This is because this operation keeps the fact that all the groups are covered.",
"Finally, we have a set $A$ where only contains vertices in $\\bigcup _{S\\in \\mathcal {S}}G_S$ and $cost(A,V,d)\\le t|V|$ .",
"Let $\\mathcal {S}^*=\\lbrace S\\in \\mathcal {S}\\mid G_S\\cap A\\ne \\varnothing \\rbrace $ .",
"Then $|\\mathcal {S}^*|\\le t$ because $cost(A,V,d)=|A|\\cdot |V|+|R|\\ge |\\mathcal {S}^*|\\cdot |V|$ , and $\\mathcal {S}^*$ covers $\\mathcal {U}$ because all the group $G_e$ are covered.",
"These lemmas, combined with Theorem REF , imply Theorem REF"
],
[
"Proof of expected cost",
"Lemma D.1 If $\\Vert \\vec{y}_{uv}^*\\Vert ^2\\le \\frac{\\varepsilon }{2}$ , then the probability that $uv\\in R_{1u}$ is at most $\\frac{1}{n}$ .",
"If $\\Vert \\vec{z}_u^*\\Vert ^2\\ge \\frac{1}{1+\\varepsilon }$ or $\\Vert \\vec{z}_v^*\\Vert ^2\\ge \\frac{1}{1+\\varepsilon }$ , then $u$ or $v$ is in $F$ , so $v\\notin R_{1u}$ .",
"Thus we only consider the case that $\\Vert \\vec{z}_u^*\\Vert ^2<\\frac{1}{1+\\varepsilon }$ and $\\Vert \\vec{z}_v^*\\Vert ^2<\\frac{1}{1+\\varepsilon }$ , which means $\\vec{z}_u^*\\cdot \\vec{z}_v^*<\\frac{1}{1+\\varepsilon }$ .",
"Since $\\Vert \\vec{y}_{uv}^*\\Vert ^2\\le \\frac{\\varepsilon }{2}$ , we have $\\sum _{w\\in B_u(v)}\\Vert \\vec{x}_w^*\\Vert ^2\\ge 1-\\frac{\\varepsilon }{2}-\\frac{1}{1+\\varepsilon }\\ge \\frac{\\varepsilon }{3}.$ Therefore, the probability that $d(u,v)<\\min _{w\\in A}d(u,w)$ is at most $\\prod _{w\\in B_{u}(v)}(1-\\min \\lbrace \\frac{3\\ln n}{\\varepsilon }\\cdot \\Vert \\vec{x}_v^*\\Vert ^2,1\\rbrace )\\le e^{-\\sum _{w\\in B_{u}(v)}\\frac{3\\ln n}{\\varepsilon }\\cdot \\Vert \\vec{x}_v^*\\Vert ^2}\\le \\frac{1}{n}$ Therefore, let $OPT_{SDP_{TZ_2O}}$ denotes the optimal cost of $SDP_{TZ_2O}$ , then the expected cost of the rounding algorithm is at most $\\sum _{v\\in V}(n-f)\\cdot \\frac{3\\ln n}{\\varepsilon }\\cdot \\Vert \\vec{x}_v^*\\Vert ^2+\\frac{2}{\\varepsilon }\\cdot \\sum _{u,v\\in V}\\Vert \\vec{y}_{uv}^*\\Vert ^2+n^2\\cdot \\frac{1}{n}\\le O(\\log n)\\cdot SDP_{TZ_2O}+n\\le O(\\log n)\\cdot OPT$ because $OPT\\ge \\Omega (n)$ , which proves Theorem REF ."
],
[
"True approximation",
"When the number of outliers is low, in particular when $f \\le \\sqrt{n}$ , we can find an actual $O(\\log n)$ -approximation.",
"The SDP and rounding algorithm are the same, except we will choose $f$ vertices with the highest $\\Vert \\vec{z}_v^*\\Vert ^2$ values as $F$ , rather than a threshold rounding of $\\frac{1}{1+\\varepsilon }$ .",
"Now there are two cases when $\\Vert \\vec{y}_{uv}^*\\Vert ^2\\le \\frac{\\varepsilon }{2}$ .",
"One case is the same as before, where $\\sum _{w\\in B_u(v)}\\Vert \\vec{x}_w^*\\Vert ^2\\ge \\frac{\\varepsilon }{3}$ .",
"In this case, the probability that $v\\in R_{1u}$ is at most $\\frac{1}{n}$ .",
"The other case is $\\sum _{w\\in B_u(v)}\\Vert \\vec{x}_w^*\\Vert ^2<\\frac{\\varepsilon }{3}$ , which means $\\vec{z}_u^*\\cdot \\vec{z}_v^*\\ge 1-\\frac{\\varepsilon }{2}-\\frac{\\varepsilon }{3}=1-\\frac{5}{6}\\varepsilon $ .",
"However, this case will not appear a lot.",
"Whenever $\\vec{z}_u^*\\cdot \\vec{z}_v^*\\ge 1-\\frac{5}{6}\\varepsilon $ , both $\\Vert \\vec{z}_u^*\\Vert $ and $\\Vert \\vec{z}_v^*\\Vert $ should be at least $1-\\frac{5}{6}\\varepsilon $ , which means $\\Vert \\vec{z}_u^*\\Vert ^2$ and $\\Vert \\vec{z}_v^*\\Vert ^2$ is at least $\\frac{1}{2}$ .",
"Because $\\sum _{v\\in V}\\Vert \\vec{z}_v^*\\Vert ^2\\le f$ , we know that there are at most $2f$ of $\\Vert \\vec{z}_v^*\\Vert ^2$ are at least $\\frac{1}{2}$ .",
"Therefore the number of $u,v$ pairs that $\\Vert \\vec{y}_{uv}^*\\Vert ^2\\le \\frac{\\varepsilon }{2}$ and $\\sum _{w\\in B_u(v)}\\Vert \\vec{x}_w^*\\Vert ^2<\\frac{\\varepsilon }{3}$ is at most $2f\\cdot 2f=4n$ .",
"Therefore, let $OPT_{SDP_{TZ_2O}}$ denotes the optimal cost of $SDP_{TZ_2O}$ , then the expected cost of the rounding algorithm is at most $\\sum _{v\\in V}(n-f)\\cdot \\frac{3\\ln n}{\\varepsilon }\\cdot \\Vert \\vec{x}_v^*\\Vert ^2+\\frac{2}{\\varepsilon }\\cdot \\sum _{u,v\\in V}\\Vert \\vec{y}_{uv}^*\\Vert ^2+n^2\\cdot \\frac{1}{n}+4n\\le O(\\log n)\\cdot OPT_{SDP_{TZ_2O}}+5n\\le O(\\log n)\\cdot OPT$ because $OPT\\ge \\Omega (n)$ , which proves Theorem REF ."
],
[
"$PR$ -Optimization Problem With Outliers",
"Lemma D.2 If $\\Vert \\vec{y}_{uv}^*\\Vert ^2\\le \\frac{\\varepsilon }{2}$ , then the probability that $\\lbrace u,v\\rbrace \\in R$ is at most $\\frac{1}{n}$ .",
"If $\\Vert \\vec{z}_u^*\\Vert ^2\\ge \\frac{1}{1+\\varepsilon }$ or $\\Vert \\vec{z}_v^*\\Vert ^2\\ge \\frac{1}{1+\\varepsilon }$ , then $u$ or $v$ is in $F$ , so $\\lbrace u,v\\rbrace \\notin R$ .",
"Thus we only consider the case that $\\Vert \\vec{z}_u^*\\Vert ^2<\\frac{1}{1+\\varepsilon }$ and $\\Vert \\vec{z}_v^*\\Vert ^2<\\frac{1}{1+\\varepsilon }$ , which means $\\vec{z}_u^*\\cdot \\vec{z}_v^*<\\frac{1}{1+\\varepsilon }$ .",
"Since $\\Vert \\vec{y}_{uv}^*\\Vert ^2\\le \\frac{\\varepsilon }{2}$ , we have $\\sum _{w\\in B_u(r)\\cup B_v(d(u,v)-r)}\\Vert \\vec{x}_w^*\\Vert ^2\\ge 1-\\frac{\\varepsilon }{2}-\\frac{1}{1+\\varepsilon }\\ge \\frac{\\varepsilon }{3}.$ Therefore, the probability that $A\\cap (B_u(r)\\cup B_v(d(u,v)-r))=\\varnothing $ for a specifiic $r\\in [0,d(u,v)]$ is at most $\\prod _{w\\in B_u(r)\\cup B_v(d(u,v)-r)}(1-\\min \\lbrace \\frac{6\\ln n}{\\varepsilon }\\cdot \\Vert \\vec{x}_w^*\\Vert ^2,1\\rbrace )\\le e^{-\\sum _{w\\in B_{u}(v)}\\frac{6\\ln n}{\\varepsilon }\\cdot \\Vert \\vec{x}_w^*\\Vert ^2}\\le \\frac{1}{n^2}$ By using union bound over all the different $r$ we used in our SDP, the probability that there exists an $r\\in [0,d(u,v)]$ where $A\\cap (B_u(r)\\cup B_v(d(u,v)-r))=\\varnothing $ is at most $\\frac{1}{n^2}\\cdot n=\\frac{1}{n}$ , which means $d(u,v)<\\min _{w\\in A}d(u,w)+\\min _{w\\in A}d(v,w)-1$ with probability at most $\\frac{1}{n}$ , so the probability that $\\lbrace u,v\\rbrace \\in R$ is at most $\\frac{1}{n}$ .",
"Therefore, let $OPT_{SDP_{PR}}$ denotes the optimal cost of $SDP_{PR}$ , then the expected cost of the rounding algorithm is at most $\\sum _{v\\in V}(n-f)\\cdot \\frac{3\\ln n}{\\varepsilon }\\cdot \\Vert \\vec{x}_v^*\\Vert ^2+\\frac{2}{\\varepsilon }\\cdot \\sum _{u,v\\in V}\\Vert \\vec{y}_{uv}^*\\Vert ^2+n^2\\cdot \\frac{1}{n}\\le O(\\log n)\\cdot OPT_{SDP_{PR}}+n\\le O(\\log n)\\cdot OPT$ because $OPT\\ge \\Omega (n)$ , which proves Theorem REF ."
]
] | 1612.05623 |
[
[
"Structure of chambers cut out by Veronese arrangements of hyperplanes in\n the real projective spaces"
],
[
"Abstract We study arrangements of $m$ hyperplanes in the $n$-dimensional real projective space, with a special focus on $m=n+3$ and $n=3$ or $n=4$."
],
[
"Introduction",
"Consider several lines in the real projective plane.",
"The lines divide the plane in various chambers.",
"We are interested in the arrangement of the chambers.",
"This problem is quite naive and simple.",
"But if the cardinality $m$ of the lines are large, there are no way to control in general.",
"If we assume that no three lines meet at a point (line arrangements in general position), the situation does not improve much.",
"In this paper we always assume this.",
"Let us observe when $m$ is small.",
"Since we are mathematicians, we start from $m=0$ : The projective plane itself.",
"When $m=1$ , what remains is the euclidean plane, of course.",
"When $m=2$ , the plane is divided in two di-angular chambers.",
"When $m=3$ , the lines cut out four triangles; if we think one of the lines is the line at infinity, then the remaining two can be considered as two axes dividing the euclidean plane in four parts.",
"Another explanation: if we think that the three lines bound a triangle, then around the central triangle there is a triangle along each edge/vertex.",
"When $m=4$ , four triangles and three quadrilaterals are arranged as follows; if we think one of the lines is the line at infinity, then the remaining three lines bound a triangle, and around this triangle there are a quadrilateral along each edge, and a triangle kissing at each vertex.",
"Another explanation: put the four lines as the symbol $\\#$ , then around the central quadrilateral are a triangle along each edge, and a quadrilateral kissing at each pair of opposite vertices.",
"The situation is already not so trivial.",
"When $m=5$ , there is a unique pentagon, and if we see the arrangement centered at this pentagon, the situation can be described simply: there are five triangles adjacent to five edges, and five quadrilaterals kissing at five vertices.",
"If we think one of the lines is the line at infinity, the remaining four lines can not be like $\\#$ since the five lines should be in general position; the picture does not look simple, but one can find a pentagon.",
"Thanks to the central pentagon, we can understand well the arrangement.",
"When $m=6$ , there are four types of arrangements.",
"The simplest one has a hexagon surrounded by six triangles along the six edges; there are six quadrilaterals kissing at the six vertices, and three quadrilaterals away from the hexagon.",
"Another one with icosahedral symmetry: If we identify the antipodal points of the dodeca-icosahedron projected from the center onto a sphere we get six lines in the projective plane: there are $20/2=10$ triangles and $12/2=6$ pentagons.",
"We do not describe here the two other types.",
"For general $m$ there are so many types, and we can not find a way to control them, unless the $m$ lines bound an $m$ -gon, from which we can see the arrangement.",
"We can think this $m$ -gon as the center of this arrangement, and we would like to study higher dimensional versions of this kind of arrangements with center, if such a chamber exists.",
"Before going further, since we are mathematicians, we start from the very beginning: the $(-1)$ -dimensional projective space is an empty set; well it is a bit difficult to find something interesting.",
"The 0-dimensional projective space is a point; this point would be the center, OK. Now we proceed to the projective line.",
"$m$ points on the line divide the line into $m$ intervals.",
"(In this case `in general position' means `distinct'.)",
"Very easy, but there is no center in the arrangement!",
"How about 3-dimensional case?",
"We have $m$ planes in general position (no four planes meet at a point).",
"$m=0$ : The projective space itself.",
"When $m=1$ , what remains is the euclidean space, of course.",
"When $m=2$ , the space is divided into two di-hedral chambers.",
"When $m=3$ , the space is divided into four tri-hedral chambers.",
"When $m=4$ , the planes cut out eight tetrahedra; if we think one of the planes is the plane at infinity, then the remaining three can be considered as the coordinate planes dividing the euclidean space in eight parts.",
"Another explanation: if we think that the four planes bound a tetrahedron, then around the central tetrahedron there is a tetrahedron adjacent to each face (kisses at the opposite vertex), and a tetrahedron touching along a pair of opposite edges: $1+4+6/2=8$ .",
"When $m=5$ , if we think one of the planes is the plane at infinity, then the remaining four bounds a tetrahedron.",
"Around this central tetrahedron, there is a (triangular) prism adjacent to the face, a prism touching along an edge, and a tetrahedron kissing at a vertex.",
"($1T+4P+6P+4T.$ ) Since the central tetrahedron is bounded only by four planes, this can not be considered as a center of the arrangement.",
"Though a prism is bounded by five planes, there are several prisms, and none can be considered as the center.",
"When $m=6$ , we can not describe the arrangement in a few lines; we will see that there is no center of the arrangement.",
"The last author made a study of this arrangement in [2], but he himself admits the insufficiency of the description.",
"To make the scenery of this arrangement visible, we spend more than thirty pages.",
"The chamber bounded by six faces which is not a cube plays an important role: this is bounded by two pentagons, two triangles and two quadrilaterals.",
"This fundamental chamber seems to have no name yet.",
"So we name it as a dumpling (gyoza in [1], [2]).",
"We study arrangements of $m$ hyperplanes in the real projective $n$ -space in general position.",
"As we wrote already, there would be no interesting things in general.",
"So we restrict ourselves to consider the Veronese arrangements (this will be defined in the text); when $n=2$ , it means that $m$ lines bound an $m$ -gon.",
"In general such an arrangement can be characterized by the existence of the action of the cyclic group $\\mathbb {Z}_m=\\mathbb {Z}/m\\mathbb {Z}$ of order $m$ .",
"Note the fact: if $m\\le n+3$ then every arrangement is Veronese.",
"When $n$ is even, there is a unique chamber which is stable under the group; this chamber will be called the central one.",
"When $n=0$ , it is the unique point, and when $n=2$ , it is the $m$ -gon.",
"We are interested in the next case $n=4$ , which we study in detail.",
"In particular, when $(n,m)=(4,7)$ , the central chamber is bounded by seven dumplings.",
"This study gives a light to general even dimensional central chamber.",
"Remember that central chambers will be higher dimensional versions of a point and a pentagon (and an $m$ -gon).",
"When $n$ is odd, other than the case $(n,m)=(3,6)$ , we study higher dimensional versions of a dumpling.",
"Note that the 1-dimensional dumpling should be an interval."
],
[
"Preliminaries",
"We consider arrangements of $m$ hyperplanes in the $n$ -dimensional real projective space $\\mathbb {P}^n:=\\mathbb {R}^{n+1}-\\lbrace 0\\rbrace \\ {\\rm modulo}\\ \\mathbb {R}^\\times .$ $\\mathbb {P}^{-1}$ is empty, $\\mathbb {P}^0$ is a singleton, $\\mathbb {P}^1$ is called the projective line, $\\mathbb {P}^2$ the projective plane, and $\\mathbb {P}^n$ the projective $n$ -space.",
"We always suppose the hyperplanes are in general position (i.e.",
"no $n+1$ hyperplanes meet at a point).",
"We often work in the $n$ -dimensional sphere $\\mathbb {S}^n:=\\mathbb {R}^{n+1}-\\lbrace 0\\rbrace \\ {\\rm modulo}\\ \\mathbb {R}_{>0},$ especially when we treat inequalities.",
"This is just the double cover of $\\mathbb {P}^n$ , which is obtained by identifying the anti-podal points of $\\mathbb {S}^n$ .",
"So, a hyperplane in $\\mathbb {S}^n$ is nothing but a punctured vector hyperplane modulo $\\mathbb {R}_{>0}$ ."
],
[
"Projective spaces",
"Let us give some geometric idea of the projective spaces.",
"Two distinct hyperplanes meet along a projective space two dimensional lower.",
"Two distinct lines in $\\mathbb {P}^2$ meet at a point.",
"Two distinct points in $\\mathbb {P}^1$ do not meet.",
"If you think a hyperplane as the hyperplane at infinity, what remains is the usual euclidean space.",
"A tubular neighborhood of a line – the complement of a disc – in $\\mathbb {P}^2$ is a Möbius strip.",
"If you keep walking along a straight line then you eventually come back along the same line from behind to the point you started.",
"Two parallel lines meet at a point at infinity.",
"$\\mathbb {P}^1$ as well as $\\mathbb {S}^1$ is just a circle.",
"$\\mathbb {P}^2$ can not be embedded into $\\mathbb {R}^3$ unless you permit a self-intersection.",
"Odd dimensional ones are orientable, and even dimensional ones are non-orientable."
],
[
"Hyperplane arrangements",
"We consider arrangements of $m$ hyperplanes in $\\mathbb {P}^n$ .",
"We always assume that an arrangement is in general position, which means no $n+1$ hyperplanes meet at a point.",
"Let $x_0:x_1:\\cdots :x_n$ be a system of homogeneous coordinates on $\\mathbb {P}^n$ .",
"A hyperplane $H$ is defined by a linear equation $a_0x_0+a_1x_1+\\cdots +a_nx_n=0,\\quad (a_0,\\dots ,a_n)\\ne (0,\\dots ,0).$ By corresponding a hyperplane $H$ its coefficients $a_0:\\cdots :a_n$ of the defining equation, we have an isomorphism between the set of hyperplanes and the set of points in the dual projective space (i.e.",
"$a$ -space).",
"A hyperplane arrangement is in general position if and only if the corresponding point arrangement is in general position, which means no $n+1$ points are on a hyperplane.",
"By definition an arrangement of $m$ hyperplanes $H_1,\\dots ,H_m$ , is defined up to the action of the symmetry group on the indices $1,\\dots ,m$ ."
],
[
"Grassmann isomorphism",
"An arrangement of $m(\\ge n+2)$ hyperplanes $H_j: a_{0j}x_0+a_{1j}x_1+\\cdots +a_{nj}x_n=0,\\quad j=1,\\dots ,m$ defines an $(n+1)\\times m$ -matrix $A=(a_{ij})$ .",
"We can regard $A$ as a matrix representing a linear map from a linear space of dimension $m$ to that of dimension $n+1$ .",
"Note that the arrangement is in general position if and only if no $(n+1)$ -minor vanish.",
"Choose a basis of the kernel of this map and arrange them vertically, and we get an $m\\times (m-n-1)$ -matrix $B$ such that $AB=0$ .",
"The matrix ${}^tB$ defines an arrangement of $m$ hyperplanes in $\\mathbb {P}^{m-n-2}$ .",
"The choice of the bases is not unique; the ambiguity gives projective transformations of $\\mathbb {P}^{m-n-2}$ .",
"No $(m-n-1)$ -minor of $B$ vanish (linear (in)dependence of the first $n+1$ columns of $A$ implies that of the last $m-n-1$ lines of $B$ ).",
"Summing up, we get an isomorphism between the arrangements of $m$ hyperplanes in $\\mathbb {P}^n$ and those in $\\mathbb {P}^{m-n-2}$ ."
],
[
"Arrangements when $m=n+2$",
"It is well-known that any three distinct points on the projective line $\\mathbb {P}^1=\\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ can be transformed projectively into $\\lbrace 0,1,\\infty \\rbrace $ .",
"In general, we want to prove that any two systems of $n+2$ points in general position in $\\mathbb {P}^n$ can be transformed projectively to each other.",
"Since Grassmann isomorphism doesn't make sense when $m-n-2=0$ , we prove this fact using linear algebra.",
"Proposition 1 Any $n+2$ points in general position in $\\mathbb {P}^n$ can be transformed projectively into the $n+2$ points: $1:0:\\cdots :0,\\quad 0:1:0:\\cdots :0,\\quad \\dots ,\\quad 0:\\cdots :0:1,\\quad 1:\\cdots :1.$ proof: Let $A$ be the corresponding $(n+1)\\times (n+2)$ -matrix.",
"Multiplying a suitable matrix from the left, we can assume that $A$ is of the form $(I_{n+1}\\ a)$ , where $a=^t(a_0,\\dots ,a_n)$ .",
"Since the arrangement is in general position, $a_j\\ne 0\\ (j=0,\\dots ,n)$ .",
"We then multiplying diag$(1/a_0, \\dots , 1/a_n)$ from the left.",
"Projective transformations still operate on these points as permutations of $n+2$ points."
],
[
"Arrangements when $m=n+3$",
"Proposition 2 The set of arrangements of $n+3$ hyperplanes in general position in $\\mathbb {P}^n$ is connected.",
"We can use the Grassmann isomorphism (note that $m-n-2=1$ ).",
"The arrangements of $m$ points in $\\mathbb {P}^1$ is topologically unique, but not projectively of course.",
"We give direct proof without using Grassmann isomorphism.",
"We prove the dual statement: The set of arrangements of $n+3$ points in $\\mathbb {P}^n$ in general position is connected.",
"proof: Put $n+2$ points as in Proposition REF .",
"Where can we put the ($n+3$ )-th point so that the $n+3$ points are in general position, that is, no $n+1$ points are on a hyperplane?",
"$n$ points out of these $n+2$ points span hyperplanes defined by: $x_i=0,\\quad x_j=x_k\\quad (i,j,k=1,\\dots ,n+1,\\ j\\ne k).$ These hyperplanes are places which are not allowed to put the ($n+3$ )-th point.",
"These hyperplanes divide the space $\\mathbb {P}^n$ into simplices (if non-empty) defined by $x_{i_1}<x_{i_2}<\\cdots <x_{i_{n+1}},\\quad \\lbrace i_1,\\dots ,i_{n+1}\\rbrace \\subset \\lbrace 0,1,\\dots ,n+1\\rbrace ,$ where $x_0=0,x_{n+1}=1$ .",
"The symmetric group on $n+2$ letters acts transitively on these simplices.",
"This completes the proof.",
"We can rephrase these propositions as: Corollary 1 If $m\\le n+2$ there is only one arrangement up to linear transformations.",
"If $m=n+3$ there is only one arrangement up to continuous move keeping the intersection pattern."
],
[
"Curves of degree $n$",
"Consider a curve $x(t)$ in $\\mathbb {P}^n$ given by $x_0=x_0(t),\\quad x_1=x_1(t),\\quad \\dots ,\\quad x_n=x_n(t),\\qquad t\\in \\mathbb {R}.$ At $t=\\tau $ , if there is a hyperplane $y_0(\\tau )x_0+y_1(\\tau )x_1+\\cdots +y_n(\\tau )x_n=0$ passing through $x(\\tau )$ such that the derived vectors $x^{\\prime }(\\tau ), x^{\\prime \\prime }(\\tau ),\\dots ,x^{(n-1)}$ lie on it, that is, $X(\\tau )\\left(\\begin{array}{c}y_0(\\tau )\\\\y_1(\\tau )\\\\\\vdots \\\\ y_n(\\tau )\\end{array}\\right)=0,\\quad X(\\tau )=\\left(\\begin{array}{cccc}x_0(\\tau )&x_1(\\tau )&\\cdots &x_n(\\tau )\\\\x^{\\prime }_0(\\tau )&x^{\\prime }_1(\\tau )&\\cdots &x^{\\prime }_n(\\tau )\\\\\\cdots &&&\\\\x^{(n-1)}_0(\\tau )&x^{(n-1)}_1(\\tau )&\\cdots &x^{(n-1)}_n(\\tau )\\end{array}\\right),$ this hyperplane is called an osculating hyperplane of the curve at $x(\\tau )$ .",
"If the rank of $X(\\tau )$ is $n$ , there is a unique osculating hyperplane at $x(\\tau )$ .",
"The correspondence $\\tau \\longmapsto y_0(\\tau ):y_1(\\tau ):\\cdots :y_n(\\tau )$ defines a curve in the dual projective space (i.e.",
"$y$ -space).",
"Notice that (since $(\\sum x_ky_k)^{\\prime }=\\sum x^{\\prime }_ky_k+\\sum x^{\\prime }_ky_k, (\\sum x^{\\prime }_ky_k)^{\\prime }=\\sum x^{\\prime \\prime }_ky_k+\\sum x^{\\prime }_ky^{\\prime }_k,$ etc) we have $Y(\\tau )\\left(\\begin{array}{c}x_0(\\tau )\\\\\\vdots \\\\ x_n(\\tau )\\end{array}\\right)=0,\\quad Y(\\tau )=\\left(\\begin{array}{ccc}y^{\\prime }_0(\\tau )&\\cdots &y_n(\\tau )\\\\y_0(\\tau )&\\cdots &y^{\\prime }_n(\\tau )\\\\\\cdots &&\\\\y^{(n-1)}_0(\\tau )&\\cdots &y^{(n-1)}_n(\\tau )\\end{array}\\right),$ that is, the hyperplane $x_0(\\tau )y_0+\\cdots +x_n(\\tau )y_n=0$ in $y$ -space is an osculating hyperplane of the curve $y(t)$ at $y(\\tau )$ .",
"This curve $y(t)$ is called the dual curve.",
"Notice also that if $x(t)$ is of degree $n$ , so is $y(\\tau )$ .",
"Thanks to this dual correspondence, in place of proving Proposition 3 For any $n+3$ hyperplanes in general position in $\\mathbb {P}^n$ , there is a unique rational curve of degree $n$ osculating these hyperplanes.",
"we prove Proposition 4 For any $n+3$ points in general position in $\\mathbb {P}^n$ , there is a unique rational curve of degree $n$ passing through these points.",
"This is a generalization of a well-known fact that there is a unique conic passing through given five points in general position.",
"Without loss of generality, we put $n+3$ points as: $\\begin{array}{cccccc}&x_1:&x_2&:\\cdots :&x_n:&x_{n+1}\\\\&&&&&\\\\p_0=&1:&1&:\\cdots :&1:&1,\\\\p_1=&1:&0&:\\cdots :&0:&0,\\\\p_2=&0:&1&:\\cdots :&0:&0,\\\\\\vdots &&&&&\\\\p_n=&0:&0&:\\cdots :&1:&0,\\\\p_{n+1}=&0:&0&:\\cdots :&0:&1,\\\\p_{n+2}=&a_1:&a_2&:\\cdots :&a_n:&a_{n+1},\\end{array}$ where $0<a_1<\\cdots <a_n<a_{n+1}$ .",
"Sorry, in this proof we use coordinates $x_1:\\cdots :x_{n+1}$ instead of $x_0:\\cdots :x_n$ .",
"We will find a curve $C:t\\longmapsto x_1(t):\\cdots :x_{n+1}(t),$ such that $x_j(t)$ is a polynomial in $t$ of degree $n$ , and that $ C(q_0)=p_0,\\quad C(q_1)=p_1,\\ \\dots ,\\ C(q_{n+1})=p_{n+1},\\quad C(r)=p_{n+2}.$ If we normalize as $q_0=\\infty , \\quad q_1=0, \\quad q_2=1,$ then the above condition is equivalent to the system of equations $\\begin{array}{lll}(x_1(r)=)&c(r-q_2)(r-q_3)(r-q_4)\\cdots (r-q_{n+1})&=a_1,\\\\(x_2(r)=)&c(r-q_1)(r-q_3)(r-q_4)\\cdots (r-q_{n+1})&=a_2,\\\\(x_3(r)=)&c(r-q_1)(r-q_2)(r-q_4)\\cdots (r-q_{n+1})&=a_3,\\\\\\vdots &&\\\\(x_{n+1}(r)=)&c(r-q_1)(r-q_2)(r-q_4)\\cdots (r-q_{n})&=a_{n+1},\\end{array}$ with $n+1$ unknowns $q_3,\\dots ,q_{n+1}, r$ and $c$ .",
"From the first and the second equations, $r$ is solved, from the second and the third equation, $q_3$ is solved,..., and we obtain a unique set of solutions: $r=\\frac{a_2}{a_2-a_1},\\quad q_j=\\frac{(a_j-a_1)a_2}{(a_2-a_1)a_j}\\quad (j=3,\\dots n+1)$ We do not care the value of $c$ .",
"Since $q_3-1=\\frac{(a_3-a_2)a_1}{(a_2-a_1)a_3},\\quad q_j-q_i=\\frac{a_1a_2}{a_2-a_1}\\cdot \\frac{a_j-a_i}{a_ja_i},\\quad r-q_j=\\frac{a_1a_2}{a_2-a_1}\\cdot \\frac{1}{a_j}$ we have $q_1=0<q_2=1<q_3<\\cdots <q_{n+1}<r.$"
],
[
"Number of chambers",
"Let $\\#(n,m)$ be the number of chambers cut out from the projective $n$ -space $\\mathbb {P}^n$ by $m$ hyperplanes in general position.",
"We have $\\#(1,m)=m,\\quad \\#(n,1)=1.$ Consider $m-1$ hyperplanes in $\\mathbb {P}^n$ .",
"The $m$ -th hyperplane intersects $m-1$ hyperplanes, which cut outs $\\#(n-1,m-1)$ chambers of dimension $n-1$ .",
"Each $(n-1)$ -dimensional chamber cut an $n$ -dimensional chamber into two.",
"Thus we have $ \\#(n,m)=\\#(n,m-1)+\\#(n-1,m-1),$ and so $\\#(n,m)=1+\\sum _{\\ell =1}^{m-1}\\#(n-1,\\ell ).$ This can be readily solved: $\\begin{array}{lll}\\#(n,m)&=\\displaystyle {{m\\atopwithdelims (){n}}+{m\\atopwithdelims (){n-2}}+\\cdots +{m\\atopwithdelims ()2}+{m\\atopwithdelims ()0}},&n:{\\rm even},\\\\[6mm]\\#(n,m)&=\\displaystyle {{m\\atopwithdelims (){n}}+{m\\atopwithdelims (){n-2}}+\\cdots +{m\\atopwithdelims ()3}+{m\\atopwithdelims ()1}},&n:{\\rm odd},\\\\\\end{array}$ where ${m\\atopwithdelims ()n}=0$ if $m<n$ .",
"In particular, $\\#(1,m)=m$ , $\\#(2,m)={m\\atopwithdelims ()2}+1,\\quad \\#(3,m)={m\\atopwithdelims ()3}+m,\\quad \\#(4,m)={m\\atopwithdelims ()4}+{m\\atopwithdelims ()2}+1.$"
],
[
"Rotating a polytope around a face",
"Let $P$ be an $n$ -polytope in $\\mathbb {R}^{n}$ canonically embedded in $\\mathbb {R}^{n+1}$ , and $\\sigma $ one of its facets ($(n-1)$ -dimensional faces).",
"The convex hull of $P$ and its image by a rotation in $\\mathbb {R}^{n+1}$ of an angle less than $\\pi $ and centered at $\\sigma $ is an $(n+1)$ -polytope, which will be denoted by $P^\\sigma $ , and will be called a polytope obtained from $P$ by rotating it around $\\sigma $ .",
"It can be abstractly defined as $P^\\sigma =P\\times [0,1]/{\\sigma \\over \\sim }, \\quad (x,t)\\ {\\sigma \\over \\sim }\\ (x,0), \\ x\\in \\sigma ,\\quad t\\in [0,1],$ in other words it is obtained by crushing the facet $\\sigma \\times [0,1]$ of the product $P\\times [0,1]$ by the projection (see Figure REF ) $\\sigma \\times [0,1]\\ni (x,t)\\longmapsto x\\in \\sigma .$ Note that the boundary of $P^\\sigma $ consists of $P\\times \\lbrace 0\\rbrace ,\\quad P\\times \\lbrace 1\\rbrace ,\\quad (\\tau \\times [0,1])/{\\sigma \\over \\sim }\\quad (\\tau :\\ {\\rm face\\ of\\ }P,\\ \\tau \\ne \\sigma ).$ For example, rotating a pentagon around a side, we get a polytope bounded by two pentagons, two triangles, and two quadrilaterals.",
"In other words, it is obtained by crushing a rectangular face of the pentagonal prism.",
"As is shown in Figure REF (right), it looks like a dumpling, so it will be called a dumpling.",
"This is called a gyoza in [1], [2].",
"In general, rotating an $m$ -gon around a side, we get a polytope bounded by two $m$ -gon's, two triangles, and $m-3$ quadrilaterals; this polytope will also be called a (3-dimensional $m$ -) dumpling.",
"A 3-dumpling is a tetrahedron, a 4-dumpling is a triangular prism.",
"High dimensional ones will be also called dumplings.",
"Figure: Dumplings n=3,m=3,4,5n=3, m=3,4,5"
],
[
"Veronese arrangements",
"A hyperplane arrangement $A=\\lbrace H_j\\rbrace _{j=1,\\dots ,m}$ in $\\mathbb {S}^n$ (or $\\mathbb {P}^n$ ) is said to be Veronese if under a suitable linear change of coordinates, $H_j$ is given by $f_j=0$ , where $f_j=f(t_j,x)=x_0+t_jx_1+t_j^2x_2+\\cdots +t_j^nx_n,\\quad j=1,\\dots ,m.$ Here $t_1<t_2<\\cdots <t_m$ are real numbers, and $(x_0,\\dots ,x_n)$ are coordinates on $\\mathbb {R}^{n+1}$ .",
"Note that by proposition REF , if $m\\le n+3$ , every arrangement of hyperplanes in general position is Veronese.",
"Proposition 5 For the curve $V_n$ of degree $n$ defined by $\\mathbb {R}\\ni t\\longmapsto x(t)=(-t)^n:n(-t)^{n-1}:\\cdots :{n\\atopwithdelims ()k}(-t)^{n-k}:\\cdots :1,$ the osculating hyperplane at the point $x(t_j)$ is given by $f_j=0$ .",
"The curve $V_n$ is often called a Veronese embedding of $\\mathbb {P}^1$ into $\\mathbb {S}^n$ (or $\\mathbb {P}^n$ ).",
"This is the reason for coining Veronese arrangements.",
"From here to the end of next section, we work in $\\mathbb {S}^n$ .",
"The binomial theorem tells $f_j(x(t))=(t_j-t)^n;$ so that if $n$ is even, $f_j(x(t))> 0\\quad {\\rm if}\\quad t\\ne t_j,$ if $n$ is odd, $f_j(x(t))< 0\\quad {\\rm if}\\quad t<t_j,\\qquad f_j(x(t))>0\\quad {\\rm if}\\quad t>t_j.$"
],
[
"Chambers cut out by Veronese arrangements",
"The closure of each connected component of the complement of a hyperplane arrangement is called a chamber.",
"Let a Veronese arrangement be given as in the previous section.",
"Each chamber is given by a system of inequalities $\\varepsilon _jf_j\\ge 0,\\quad j=1,\\dots ,m, \\quad (\\varepsilon _j=\\pm 1)$ which is often denoted by the sequence $\\varepsilon _1,\\ \\dots ,\\ \\varepsilon _m.$ Group action: Since the Veronese arrangement is determined by a sequence $ t_1, t_2,\\dots ,t_m$ of points in $\\mathbb {P}^1$ arranged in this order, the shift $\\iota : j\\mapsto j+1 \\quad {\\rm mod}\\ m$ acts on the set of chambers by $\\varepsilon _1,\\ \\dots ,\\ \\varepsilon _m\\longmapsto \\left\\lbrace \\begin{array}{ll}-\\varepsilon _m,\\ \\varepsilon _1,\\ \\dots ,\\ \\varepsilon _{m-1}\\quad &m: {\\rm \\ odd,}\\\\\\varepsilon _m,\\ \\varepsilon _1,\\ \\dots ,\\ \\varepsilon _{m-1}\\quad &m: {\\rm \\ even.",
"}\\end{array}\\right.$ The cyclic group generated by $\\iota $ will be denoted by $\\mathbb {Z}_m$ .",
"Note that if the points $t_1,t_2,\\dots , t_m$ are so arranged that the transformation $t_j\\rightarrow t_{j+1}$ is given by a projective one, then the corresponding action on $\\mathbb {P}^n$ is also projective.",
"Thus we have the well-defined $\\mathbb {Z}_m$ action on the set of chambers as above for any sequence $ t_1, t_2, \\ldots , t_m \\in \\mathbb {P}^1 $ ."
],
[
"A specific feature of Veronese arrangements",
"Let a Veronese arrangement $A=\\lbrace H_j\\rbrace $ of $m$ hyperplanes $H_j: f_j=f(t_j,x)=0,\\quad \\cdots <t_j<t_{j+1}<\\cdots $ in $\\mathbb {P}^n$ be given as in the previous section.",
"If we let $t_j$ tends near to $t_{j+1}$ , the intersection pattern does not change, that is, there is no vertex (intersection point of $n$ hyperplanes in $A$ ) in the slit between the hyperplanes $H_j$ and $H_{j+1}$ .",
"More precisely, the hyperplanes $H_j$ and $H_{j+1}$ divide the space into two parts, and one of them does not contain any vertex.",
"This fact gives the following information on the chambers cut out by $A$ .",
"If a chamber in the slit between the hyperplanes $H_j$ and $H_{j+1}$ does not touch the intersection $H_j\\cap H_{j+1}$ , then this chamber is the direct product of a chamber of the restricted arrangement $A_j$ and the unit interval.",
"If a chamber in the slit touches the intersection of the two hyperplanes, then (since the hyperplanes are in general position) there is a chamber $P$ of the restricted arrangement $A_j$ with a facet $\\sigma $ included in $H_j\\cap H_{j+1}$ and the chamber in the slit is the polyhedron $P^\\sigma $ obtained from $P$ by rotating around $\\sigma $ .",
"A polyhedron (of dimension greater than 1) is said to be irreducible if it is not a direct product of two polyhedron nor is obtained from a lower one by rotation described in §1.4.",
"For example, triangles, rectangles, tetrahedra, prisms, cubes and dumplings are reducible, while pentagon is irreducible.",
"A combinatorial study on Veronese arrangements is made in [1].",
"We do not use in this paper the result obtained there.",
"To give a general idea, we just quote one of the results: Consider a Veronese arrangement $A(m,n)$ of $m\\ (\\ge n+3)$ hyperplanes in $\\mathbb {P}^n$ $(n\\ge 2)$ .",
"If $n$ is odd, then every chamber is a direct product of the unit interval and a chamber of $A(m-1,n-1)$ , or is obtained from a chamber of $A(m-1,n-1)$ by rotating it around a facet.",
"If $n$ is even, then there is a unique irreducible chamber.",
"Other chamber is obtained from $A(m-1,n-1)$ like the above.",
"In this paper, for a Veronese arrangement of even dimension, the unique irreducible chamber will be called the central chamber."
],
[
"Six planes in the 3-space",
"In this case there is no central chamber.",
"The arrangement is Veronese.",
"We are going to choose a particular arrangement so that the action of a group $G\\cong \\mathfrak {S}_3\\times \\mathbb {Z}_2$ of orientation preserving projective transformations will be geometrically visible.",
"The elements of the group are given by matrices (up to scalar multiplication) with coefficients $0,1,-1$ .",
"The group $G$ is a subgroup of $PGL_+(4,\\mathbb {Z})$ .",
"Notice that $GL_+(4,\\mathbb {Z})$ is not a group since inverse operation is not always defined.",
"However it is defined up to scalar matrix multiplication, so that the quotient $PGL_+(4,\\mathbb {Z})$ has a group structure.",
"The group $G$ acts on the 26 chambers with four orbits: 12 prisms, 6 tetrahedra, 6 dumplings, 2 cubes.",
"The orbit of a prism together with a cube is a solid torus.",
"The orbit of a dumpling is a second solid torus linked with the first one.",
"The orbit of a tetrahedron is the complementary of the two solid tori.",
"Actually, the union of one of these solid tori and the orbit of a tetrahedron is a solid torus as well."
],
[
"Cutting a tetrahedron and a prism",
"Chambers in the plane bounded by five or less lines are diangle, triangle, quadrilateral, and pentagon.",
"Chambers in the space bounded by four or less planes are dihedron, trihedron, tetrahedron; to see further we cut a tetrahedron by a plane.",
"There are two ways to cut (see Figure REF ).",
"Figure: Cutting a tetrahedron (tetracut)If we denote a tetrahedron by $T$ , and a (triangular) prism by $P$ , then the two cuttings can be presented as $T\\rightarrow T+P\\quad ({\\rm triangle}),\\quad P+P\\quad ({\\rm quadrilateral}),$ where $T+P$ (triangle) means $T$ and $P$ share a triangle.",
"We cut next a prism.",
"There are five ways to cut (see Figure REF ).",
"Figure: Cutting a prism (prismcut)If we denote a cube by $C$ , and a (pentagonal) dumpling by $D$ , then the five cuttings can be presented as $T+D\\quad ({\\rm triangle}),\\quad P+D\\quad ({\\rm quadrilateral}),\\quad P+C\\quad ({\\rm quadrilateral}),$ $ P+P\\quad ({\\rm triangle}),\\quad D+D\\quad ({\\rm pentagon}).$ The last one will appear again in §REF ."
],
[
"A few facts seen from the above cuttings",
"Since five planes cut out tetrahedra and prisms, if two chambers cut out by an arrangement of six planes are adjacent along a face, the union is a tetrahedron or a prism.",
"Thus though $C$ and $D$ have quadrilateral faces, they are not face to face.",
"When two $D$ 's are face to face along a pentagon, the remaining two pentagons do not share an edge.",
"Since there is one pentagon on each plane (see Introduction), and since the group $\\mathbb {Z}_6$ acts on the arrangement, we conclude that there are six $D$ 's, which are glued to form a solid torus (see §3.4)."
],
[
"Setting",
"If we can choose coordinates on $\\mathbb {P}^n$ and the equations of $n+3$ hyperplanes so nicely that the $\\mathbb {Z}_{n+3}$ -action is clearly seen, it would be nice.",
"Though we can not expect this in general, when $n=3$ , there is a very good choice ([3] and remark after proposition REF ).",
"We work in the real projective space coordinatized by $x:y:z:t$ .",
"Our six planes are $\\begin{array}{lll}H_x: x=0,& H_y: y=0,& H_z: z=0,\\\\[3mm]H^x:h^x=0,& H^y: h^y=0,& H^z: h^z=0,\\end{array}$ where $h^x:=y-z-t,\\quad h^y:= z-x-t,\\quad h^z:= x-y-t.$ Note that if we put $h^t=x+y+z$ , then we have $h^y-h^z-h^t=-3x,\\quad h^z-h^x-h^t=-3y,\\quad h^x-h^y-h^t=-3z.$ Note also that if we change from $t$ to $-t$ , the new arrangement is the mirror image of the original one.",
"We often work in the Euclidean 3-space coordinatized by $(x,y,z)=x:y:z:1,$ especially when we speak about distance and/or angle, without saying so explicitly."
],
[
"A group action and a cubic curve $K$",
"The six planes admit the transformations $\\rho :H_x\\rightarrow H_y\\rightarrow H_z\\rightarrow H_x,\\ H^x\\rightarrow H^y\\rightarrow H^z\\rightarrow H^x,$ $\\sigma : H_x\\rightarrow H_y, H_y\\rightarrow H_x, H_z\\rightarrow H_z,\\ H^x\\rightarrow H^y, H^y\\rightarrow H^x, H^z\\rightarrow H^z,\\quad \\mbox{and}$ $\\tau :H_a\\rightarrow H^a\\rightarrow H_a,\\quad (a=x,y,z)$ which are of order 3, 2 and 2, respectively.",
"These are given by the projective transformations $\\rho =\\left(\\begin{array}{cccc}0&1&0&0\\\\0&0&1&0\\\\1&0&0&0\\\\0&0&0&1\\end{array}\\right),\\quad \\sigma =\\left(\\begin{array}{cccc}0&\\bar{1}&0&0\\\\\\bar{1}&0&0&0\\\\0&0&\\bar{1}&0\\\\0&0&0&1\\end{array}\\right)\\quad {\\rm and}\\quad \\tau =\\left(\\begin{array}{cccc}0&1&\\bar{1}&\\bar{1}\\\\\\bar{1}&0&1&\\bar{1}\\\\1&\\bar{1}&0&\\bar{1}\\\\1&1&1&0\\end{array}\\right),$ where $\\bar{1}=-1$ .",
"Note that $\\tau ^2=-3I_4$ .",
"In the euclidean space, $\\rho $ acts as the $2\\pi /3$ -rotation around the axis generated by the vector $(1,1,1)$ , and $\\sigma $ the $\\pi $ -rotation about the axis generated by the vector $(1,-1,0)$ .",
"The transformation $\\tau $ exchanges the plane $H^{\\infty }:x+y+z=0$ and the plane $H_\\infty :t=0$ at infinity.",
"They generate the orientation preserving projective transformation group $G \\cong \\langle \\rho ,\\sigma \\rangle \\times \\langle \\tau \\rangle \\cong \\mathfrak {S}_3\\times \\mathbb {Z}_2,$ which is a subgroup of $PGL_+(4,\\mathbb {Z})$ of order 12 with center $\\langle \\tau \\rangle =\\lbrace 1,\\tau \\rbrace $ .",
"The relations $\\sigma ^2=1$ , $(\\rho \\tau )^6=1$ and $(\\rho \\tau \\sigma )^2=1$ show that $G$ is also isomorphic to the dihedral group $D_6\\cong \\langle \\rho \\tau \\rangle \\rtimes \\langle \\sigma \\rangle \\cong \\mathbb {Z}_6\\rtimes \\mathbb {Z}_2.$ Proposition 6 There is no invariant plane by $G$ .",
"There are two orbits of order two, $\\lbrace H^\\infty ,H_\\infty \\rbrace \\quad \\text{and}\\quad \\lbrace H^{\\sqrt{3}},H_{\\sqrt{3}}\\rbrace ,$ where $H^{\\sqrt{3}}: x+y+z-t\\sqrt{3}=0,\\quad H_{\\sqrt{3}}: x+y+z+t\\sqrt{3}=0.",
"$ There is no orbit of order three.",
"proof.",
"If $H$ is invariant by $\\tau $ , it is different from $H_\\infty $ .",
"If, in addition, it is invariant by $\\rho $ it is orthogonal (in the euclidean space) to the vector $(1,1,1)$ .",
"Such a plane is not invariant by $\\sigma $ , so that there is no plane invariant by $G$ .",
"Suppose now that $H$ belongs to an orbit of order 2, say $\\lbrace H,H^{\\prime }\\rbrace $ , different from $\\lbrace H^\\infty ,H_\\infty \\rbrace $ .",
"The group $\\langle \\rho \\rangle \\cong \\mathbb {Z}_3$ acts necessarily trivially on a set of order 2 so that $\\rho (H)=H$ .",
"Then, as above, $H$ is orthogonal to the vector $(1,1,1)$ and is not invariant by $\\sigma $ so that $\\sigma (H)=H^{\\prime }$ .",
"The equation of $H$ writes $x+y+z-\\alpha t=0$ , and the one of $H^{\\prime }$ writes $x+y+z+\\alpha t=0$ .",
"Writing that $\\tau (H)=H$ or $\\tau (H)=H^{\\prime }$ , we find that $\\alpha =\\pm \\sqrt{3}$ .",
"Now, suppose that $H$ belongs to an orbit of order 3, then the order two elements $\\sigma $ and $\\tau $ act trivially on this orbit, so that $\\sigma (H)=\\tau (H)=H$ , and similarly for $\\rho (H)$ and $\\rho ^2(H)$ .",
"In particular $H\\ne H_\\infty $ and, since $\\sigma $ is a half-turn rotation about the vector $(1,-1,0)$ , either $H$ is orthogonal to the vector $(1,-1,0)$ or $H$ contains the line generated by the vector $(1,-1,0)$ .",
"In the first case, $\\rho (H)$ is no longer orthogonal to the vector $(1,-1,0)$ so that $\\rho (H)$ , and $\\rho ^2(H)$ as well, must contain the line generated by the vector $(1,-1,0)$ , so that $\\rho (H)$ , and then $H$ itself, is orthogonal to the vector $(1,1,1)$ .",
"Therefore $H$ would be invariant, which is impossible.",
"In the second case, since $H$ is not invariant and then not orthogonal to the vector $(1,1,1)$ , the plane $\\rho (H)$ doesn't contain the line generated by the vector $(1,-1,0)$ , and therefore is orthogonal to the vector $(1,-1,0)$ .",
"We are brought back to the first case.",
"$\\square $ Each plane intersects the remaining five; they cut out a pentagon, five triangles and five rectangles.",
"See Figures REF , and REF for $H_\\bullet $ and $H^\\circ $ , respectively.",
"The six planes form a Veronese arrangement in the order: $H_x,\\quad H^y,\\quad H_z,\\quad H^x,\\quad H_y,\\quad H^z,\\quad H_x,$ by which we mean, there is a unique rational cubic curve $K$ osculating these six planes in this order.",
"Note that $\\rho $ and $\\tau $ respect the order, and $\\sigma $ just reverses the order.",
"Proof goes as follows.",
"If the regular hexagon (the projective line coordinatized by $s$ ) is given by $s=0,\\quad 1,\\quad 3/2,\\quad 2,\\quad 3,\\quad \\infty ,$ (here, regular means there is a projective transformation $s\\rightarrow 3/(3-s)$ sending 0 to 1, 1 to $3/2$ , ..., $\\infty $ to 0) then the osculating curve is given by $K: x=s^3,\\quad y=(s-3)^3,\\quad z=-8(s-3/2)^3,\\quad t=9(s-1)(s-2),$ which osculates the planes $H_x,\\dots ,H^z$ at $\\begin{array}{lll}K(0)=0:3:\\bar{3}:2, &K(1)=1:\\bar{8}:1:0, &K(\\frac{3}{2})=\\bar{3}:3:0:2,\\\\[3mm] K(2)=\\bar{8}:1:1:0,&K(3)=3:0:\\bar{3}:2, &K(\\infty )=1:1:\\bar{8}:0.\\qquad \\square \\end{array}$ Remark.",
"The chosen arrangement can be called selfadjoint in the following sense.",
"The dual coordinates of the planes are $H_x=1000,\\quad H^y=\\bar{1}01\\bar{1},\\quad H_z=0010,\\quad H^x=01\\bar{1}\\bar{1},\\quad H_y=0100,\\quad H^z=1\\bar{1}0\\bar{1}.$ For instance the equation of the plane $H^y=\\bar{1}01\\bar{1}$ writes $-x+z-t=0$ .",
"The six planes turn out to be the points $P_x=1000,\\quad P^y=\\bar{1}01\\bar{1},\\quad P_z=0010,\\quad P^x=01\\bar{1}\\bar{1},\\quad P_y=0100,\\quad P^z=1\\bar{1}0\\bar{1}$ in the dual space coordinatized by $\\check{x}:\\check{y}:\\check{z}:\\check{t}$ .",
"They are on the cubic curve $\\check{K}$ given by $\\check{x}=2(s-1)(s-3/2)(s-2),\\quad \\check{y}=-s(s-1),\\quad \\check{z}=(s-2)(s-3),\\quad \\check{t}=-3(s-1)(s-2),$ in this order.",
"Indeed we have $\\check{K}(\\infty )=P_x,\\quad \\check{K}(0)=P^y,\\quad \\check{K}(1)=P_z,\\quad \\check{K}(3/2)=P^x,\\quad \\check{K}(2)=P_y,\\quad \\check{K}(3)=P^z.$ The planes supported by three consecutive points $\\langle P_x,P^y,P_z\\rangle \\quad \\langle P^y,P_z,P^x\\rangle \\quad \\langle P_z,P^x,P_y\\rangle \\quad \\langle P^x,P_y,P^z\\rangle \\quad \\langle P_y,P^z,P_x\\rangle \\quad \\langle P^z,P_x,P^y\\rangle $ happen to be the original ones: $H_x,\\quad H^y,\\quad H_z,\\quad H^x,\\quad H_y,\\quad H^z.$ We sometimes code these six planes $H_x,H^y,\\dots $ as $H_1,\\dots ,H_6$ .",
"Then the transformation of $\\mathbb {P}^1$ above induces a projective transformation of $\\mathbb {P}^3$ sending $H_j \\rightarrow H_{j+1}$ mod 6.",
"This generates the cyclic group $\\langle \\rho ,\\tau \\rangle \\cong \\mathbb {Z}_6$ ."
],
[
"The twenty points",
"Convention: $H_{a\\cdots }^{b\\cdots }:=H_a\\cap \\cdots \\cap H^b\\cap \\cdots $ .",
"Three planes meet at a point; there are ${6\\atopwithdelims ()3}=20$ of them.",
"They are divided into three $G$ -orbits: $ \\text{\\ding {108}}\\quad P_0=H_{xyz}=0:0:0:1,\\quad \\bigcirc \\quad P^0=H^{xyz}=1:1:1:0;$ $$ $\\begin{array}{lll}\\blacksquare \\quad H^{yz}_x=0:\\bar{1}:1:1,& H^{zx}_y=1:0:\\bar{1}:1,& H^{xy}_z=\\bar{1}:1:0:1,\\\\[3mm]\\square \\quad H_{yz}^x=1:0:0:0,& H_{zx}^y=0:1:0:0,& H_{xy}^z=0:0:1:0;\\\\[7mm]\\bullet \\quad H_{xy}^x=0:0:\\bar{1}:1,& H_{yz}^y=\\bar{1}:0:0:1,& H_{zx}^z=0:\\bar{1}:0:1,\\\\[3mm]\\circ \\quad H^{xy}_x=0:2:1:1,& H^{yz}_y=1:0:2:1,& H^{zx}_z=2:1:0:1;\\\\[3mm]\\bullet \\quad H_{yx}^y=0:0:1:1,& H_{zy}^z=1:0:0:1,& H_{xz}^x=0:1:0:1,\\\\[3mm]\\circ \\quad H^{yx}_y=\\bar{2}:0:\\bar{1}:1,& H^{zy}_z=\\bar{1}:\\bar{2}:0:1,& H^{xz}_x=0:\\bar{1}:\\bar{2}:1,\\end{array}$ $$ where $\\bar{1}=-1, \\bar{2}=-2$ .",
"Note that the distances from the vertices to the origin $P_0$ are either $\\text{\\ding {108}}:0,\\qquad \\bullet :1,\\quad \\blacksquare :\\sqrt{2},\\quad \\circ :\\sqrt{5},\\qquad \\square ,\\ \\bigcirc :\\infty .$ In the $1\\cdots 6$ -coding, the three orbits above are represented by $135,\\quad 123, \\quad 124,$ respectively, where 135 stands for $H_1\\cap H_3\\cap H_5$ ."
],
[
"An invariant quadratic form",
"While there are many $G$ -invariant quadratic forms in $(x,y,z,t)$ , if we ask the zero set passes through the six vertices marked $\\square $ and $\\blacksquare $ , then it is (up to scalar multiplication) given by $Q=xy+yz+zx+t^2.$ The surface $Q=0$ and the curve $K$ do not meet: in fact, substituting the expression of the curve in $s$ into $Q$ , we have $-15(s^2-3s+3)^3$ .",
"By identifying the space $\\mathbb {P}^3$ and its dual, we regard the cubic curve $\\check{K}$ live in our space, that is, the curve is defined by $x=2(s-1)(s-3/2)(s-2),\\quad y=-s(s-1),\\cdots $ Then this is the unique rational cubic curve passing through the six points marked $\\square $ and $\\blacksquare $ : $\\check{K}(\\infty )=H^x_{yz},\\quad \\check{K}(0)=H^{zx}_y,\\quad \\check{K}(1)=H^z_{xy},\\quad \\check{K}(3/2)=H^{yz}_x,\\quad \\check{K}(2)=H^y_{zx},\\quad \\check{K}(3)=H^{xy}_z.$ This curve $\\check{K}$ is on the surface $Q=0$ ."
],
[
"The twenty six chambers",
"The six planes in $\\mathbb {P}^3$ cut out two cubes $C$ , twelve prisms $P$ , six tetrahedra $T$ , six dumplings $D$ .",
"The cubic curve $K$ stays in the dumplings, and osculates each pentagon.",
"The quadratic surface $Q=0$ lies in the union of the tetrahedra and the dumplings (see Figures REF , REF )."
],
[
" The six planes",
"For the planes $H^x,H^y,H^z$ , the intersection with other planes are shown in Figure REF .",
"Figure: The planes H ∘ H^\\circ (planeH-)Here $TP, DP,\\dots $ stand for $T\\cap P, D\\cap P,\\dots .$ If the plane $H^x$ , defined by $z=y-1$ , is coordinatized by $(y,x)$ , then the dotted curve is the conic (a hyperbola) $Q=xy+y(y-1)+(y-1)x+1=\\left(y-\\frac{1}{2}\\right)\\left(y+2x-\\frac{1}{2}\\right)+\\frac{3}{4}=0.$ For the planes $H_x,H_y,H_z$ , the intersection with other planes are shown in Figure REF .",
"Figure: The planes H • H_\\bullet (planeH)If the plane $H_x$ , defined by $x=0$ , is coordinatized by $(y,z)$ , then the dotted curve is the conic (hyperbola) $Q=yz+1=0$ , and the point $\\heartsuit $ is the point $K(0)$ where the curve $K$ kisses the plane $H_x$ ."
],
[
"Bottom and top",
"It is convenient to consider the triangle with the three vertices $\\blacksquare $ in the plane $H^\\infty : x+y+z=0$ as the ground floor, and the triangle with the three vertices $\\square $ in the plane $H_\\infty $ at infinity as the ceiling.",
"The planes $H^x,H^y,H^z$ are orthogonal to the ground.",
"The intersection (lines) of the six planes with the floor and the ceiling are shown in Figure REF .",
"The dotted curves are the intersections with the surface $Q=0$ .",
"Figure: Bottom and top (bottom-top)"
],
[
"Octant chambers",
"The finite space coordinatized by $(x,y,z)=x:y:z:1$ is divided by the three planes $H_x,H_y$ and $H_z$ into eight chambers.",
"We denote them as $(+++)=\\lbrace x\\ge 0,y\\ge 0,z\\ge 0\\rbrace ,\\quad (++-)=\\lbrace x\\ge 0,y\\ge 0,z\\le 0\\rbrace ,\\dots .$ In this section, we see how these chambers are cut by the three planes $H^x,H^y$ and $H^z$ .",
"In the chambers $(+++)$ and $(---)$ , the happenings are similar, and in the remaining six chambers, similar things happen.",
"Set $(+++;---)=\\lbrace (x,y,z)\\in (+++)\\mid h^x\\le 0,h^y\\le 0,h^z\\le 0\\rbrace ,...$ If we write these as $({\\varepsilon }_1{\\varepsilon }_2{\\varepsilon }_3;\\eta _1\\eta _2\\eta _3)$ , they permit the $\\mathbb {Z}_3$ -action $({\\varepsilon }_1{\\varepsilon }_2{\\varepsilon }_3;\\eta _1\\eta _2\\eta _3)\\rightarrow ({\\varepsilon }_2{\\varepsilon }_3{\\varepsilon }_1;\\eta _2\\eta _3\\eta _1).$ In the chamber $(+++)$ : There are four $\\mathbb {Z}_3$ -orbits represented by $\\begin{array}{ll}(+++;+++)=\\emptyset ,\\quad &(\\dot{+}\\ \\dot{+}\\ \\dot{+};\\dot{-}\\ \\dot{-}\\ \\dot{-}):{\\ \\rm cube}\\\\[3mm](\\dot{+}+\\dot{+};\\dot{+}\\ \\dot{-}\\ \\dot{-}): {\\ \\rm pri\\ }P^{\\prime },\\quad &(+\\ \\dot{+}\\ +;-\\ \\dot{+}\\ \\dot{+}): {\\ \\rm sm\\ }P^{\\prime \\prime }.\\end{array}$ Here effective ones are marked by dots.",
"For example, the last one is defined by $y\\ge 0,h^y\\ge 0$ and $h^z\\ge 0$ ; these three inequalities imply the other ones $x\\ge 0,z\\ge 0$ and $h^x\\le 0$ .",
"The number of dots corresponds to the number of walls.",
"As a whole, there are a cube and three $P^{\\prime }$ and three $P^{\\prime \\prime }$ (see Figure REF ).",
"One $P^{\\prime }$ and another $P^{\\prime \\prime }$ in the opposite chamber $(---)$ are glued along the plane at infinity forming a (full) prism (see Figure REF ).",
"Figure: Unbounded prism P=P ' ∪P '' P=P^{\\prime }\\cup P^{\\prime \\prime } (inftyprism80)If we cut this chamber by a big sphere centered at the origin, the intersection with the chamber is surrounded by three arcs ($x=0, y=0, z=0$ ), and the arc-triangle is cut by the three lines ($h^x=0,h^y=0,h^z=0$ ).",
"Around a triangle (cube), there are a triangle ($P^{\\prime \\prime }$ ), a pentagon ($P^{\\prime }$ ), a triangle ($P^{\\prime \\prime }$ ),... (see Figure REF and Figure REF left).",
"In the chamber $(---)$ : There are four $\\mathbb {Z}_3$ -orbits represented by $\\begin{array}{ll}(---;+++)=\\emptyset ,\\quad &(\\dot{-}\\dot{-}\\dot{-};\\dot{-}\\dot{-}\\dot{-}):{\\ \\rm cube}\\\\[3mm](\\dot{-}\\dot{-}-;\\dot{+}\\dot{-}\\dot{-}): {\\ \\rm pri\\ }P^{\\prime },\\quad &(--\\dot{-};-\\dot{+}\\dot{+}): {\\ \\rm sm\\ }P^{\\prime \\prime }.\\end{array}$ Figure: Chambers (+++)(+++) and (++-)(++-) (octants)In the chamber $(++-)$ : This chamber does not admit the group action.",
"Since $x\\ge 0$ and $z\\le 0$ imply $z-x\\le 0$ , and so $h^y=z-x-1<0$ , we have $(++-;*+*)=\\emptyset .$ Among the remaining four, there is a unique compact one: $(\\dot{+}\\dot{+}\\dot{-};\\dot{-}-\\dot{-}): \\ {\\rm prism}.$ (See Figures REF (right) and REF (left).)",
"The others are $(+\\dot{+}\\dot{-};\\dot{-}-\\dot{+}): \\ {\\rm tetrahedron},$ and $(\\dot{+}\\dot{+}\\dot{-};\\dot{+}-\\dot{-}): \\ {\\rm dump}\\ D^{\\prime },\\quad (+\\dot{+}\\dot{-};\\dot{+}-\\dot{+}): \\ {\\rm ling}\\ D^{\\prime \\prime }.$ One $D^{\\prime }$ (resp.",
"$D^{\\prime \\prime }$ ) and another $D^{\\prime \\prime }$ (resp.",
"$D^{\\prime }$ ) in the opposite chamber $(--+)$ are glued along the plane at infinity forming a (full) dumpling (see Figure REF ).",
"If we cut this chamber by a big sphere centered at the origin, the chamber is surrounded by three arcs ($x=0, y=0, z=0$ ), and the arc-triangle is cut by the two lines ($h^x=0,h^z=0$ ).",
"There are a triangle (tetrahedron) and two quadrilaterals, which are sections of $D^{\\prime }$ and $D^{\\prime \\prime }$ (see Figure REF and Figure REF right).",
"In the chamber $(--+)$ : Since $y\\le 0$ and $z\\ge 0$ imply $y-z\\le 0$ , and so $h^x=y-z-1<0$ , we have $(--+;+**)=\\emptyset $ .",
"Among the remaining four, there is a unique compact one: $(\\dot{-}\\dot{-}\\dot{+};-\\dot{-}\\dot{-}): \\ {\\rm prism}.$ The others are $(\\dot{-}-\\dot{+};-\\dot{-}\\dot{+}): \\ {\\rm tetrahedron},$ and $(\\dot{-}\\dot{-}\\dot{+};-\\dot{+}\\dot{-}): \\ {\\rm dump}\\ D^{\\prime },\\quad (\\dot{-}-\\dot{+};-\\dot{+}\\dot{+}): \\ {\\rm ling}\\ D^{\\prime \\prime }.$ Figure: (right) A dumpling DD is cut by the plane at infinity into two parts dump D ' D^{\\prime } and ling D '' D^{\\prime \\prime }.",
"(left) They are in octants of type (++-)(++-).",
"The faces of DD have names1 ' +1 '' ,2,3 ' +3 '' ,4,5,6 ' +6 '' 1^{\\prime } + 1^{\\prime \\prime }, 2, 3^{\\prime }+3^{\\prime \\prime }, 4, 5, 6^{\\prime }+6^{\\prime \\prime }.In the octant, a wall is 6 ' 6^{\\prime } seen from D ' D^{\\prime }, and is 6 '' 6^{\\prime \\prime } seen from D '' D^{\\prime \\prime }.",
"(octant70)"
],
[
"The solid torus made of the two cubes and the twelve prisms",
"The two cubes kiss at the two vertices $P_0$ and $P^0$ , which are opposite vertices of each cube.",
"Around the two cubes are twelve prisms, forming with the two cubes a solid torus.",
"We explain how they are situated."
],
[
"The two cubes and the six prisms around $P_0$",
"The two cubes and the six prisms around $P_0$ , form in the finite (Euclidean) space an infinitely long triangular cylinder called the big-cylinder and bounded by $H^x,H^y,H^z$ .",
"It is defined by the inequations (figure REF ): $h^x\\leqslant 0,\\quad h^y\\leqslant 0,\\quad h^z\\leqslant 0.$ Figure: (left) The big-cylinder.",
"(middle) The two truncated cubes slightly moved away from each other.",
"(right)Exploded view of the two cubes and the six prisms.",
"(amy-fig13.eps)Figure: The big-cylinder made by two truncated cubes and six compact prisms (right).",
"A truncated cube (left) and a prism (center)are shown with their vertices.Part of the pictures in Figure REF is enlarged in Figure REF ."
],
[
"The small cylinder",
"Considering the planes $H^{\\sqrt{3}}$ and $H_{\\sqrt{3}}$ defined in proposition REF , we have $\\tau (H_{\\sqrt{3}})=H^{\\sqrt{3}}$ where $\\tau $ is the involution defined in REF .",
"Let us denote by $C_+$ the heptahedron shown in figure REF and defined by $\\frac{x}{t}\\geqslant 0,\\quad \\frac{y}{t}\\geqslant 0,\\quad \\frac{z}{t}\\geqslant 0,\\quad \\frac{x}{t}-\\frac{y}{t}\\leqslant 1,\\quad \\frac{y}{t}-\\frac{z}{t}\\leqslant 1,\\quad \\frac{z}{t}-\\frac{x}{t}\\leqslant 1,\\quad \\frac{x+y+z}{t}\\leqslant \\sqrt{3}.$ Figure: The heptahedron C + C_+ (left) and its 1-skeleton (center).",
"Cutting a cube into two heptahedra (right)We set $C_-:=\\sigma (C_+)$ .",
"The faces of the heptahedron $C_+$ are three triangles, three pentagons and one hexagon.",
"We consider the solid $SC$ (called the small cylinder) given by the union of the two heptahedrons and the six prisms.",
"It is homeomorphic to a triangular prism by a homeomorphism preserving the generating lines (see figure REF ).",
"Figure: Heptahedrons C + C_+ and C - C_- kissing at P 0 P_0 (left).The small cylinder SCSC and two quadrilaterals DPDP and two triangles TPTP on its boundary (middle).The triangular prism homeomorphic to SCSC (right)."
],
[
"The prism torus",
"The $G$ -orbit of one of the six compact prisms around $P_0$ is given by twelve prisms, the six extra ones being located around $P^0$ .",
"Now the image of the small cylinder $SC$ by $\\sigma $ , or equivalently, the $G$ -orbit of one prism and the heptahedron $C_+$ , form the union of the big cylinder and the six prisms around $P^0$ .",
"These two solid cylinders $SC$ and $\\sigma (SC)$ are attached by $\\sigma $ along their bases and therefore form a solid torus.",
"Let us call it the prism torus.",
"Thus, this prism torus is decomposed into sixteenteen cells, four heptahedrons and twelve prisms, on which acts the group $G$ ."
],
[
"The boundary of the prism torus",
"The prism torus consists also of the two cubes and the twelve prisms.",
"Note that this torus is a tubular neighborhood of the projective line joining $P_0$ and $P^0$ .",
"Therefore, as mentioned in the introduction of section , the complementary of this torus is a solid torus as well.",
"The above construction shows that the boundary of the torus consists of the faces of the prisms on the lateral boundary on the small cylinder $SC$ and their images by $\\sigma $ .",
"In particular, no face of the cubes belongs to the boundary of the prism torus.",
"The boundary of the prism torus on each plane consists of two quadrilaterals $DP$ and two triangles $TP$ (see figure REF ).",
"They are shown more explicitly the pattern below: $\\begin{array}{ccc}\\begin{array}{ccccccc}\\bullet & - &\\square &-&\\bullet &&\\\\&\\backslash & | & & | &&\\\\& &\\circ &-&\\circ &&\\\\& & | & & | &\\backslash &\\\\& &\\bullet &-&\\square & - &\\bullet \\end{array}&\\qquad &\\begin{array}{ccccccc}\\circ & - &\\blacksquare &-&\\circ &&\\\\&\\backslash & | & & | &&\\\\& &\\bullet &-&\\bullet &&\\\\& & | & & | &\\backslash &\\\\& &\\circ &-&\\blacksquare & - &\\circ \\end{array}\\end{array}$ Figure: The boundary of the prism torus consists of the faces of the prisms (planeH2-70)The twelve quadrilaterals and the twelve triangles tessellate the boundary torus as below.",
"To indicate the identification, four copies of the plane number 1 are shown in addition.",
"After identification, there are $3\\ \\square ,\\quad 3\\ \\blacksquare ,\\quad 6\\ \\bullet ,\\quad 6\\ \\circ ;$ note that among the twenty vertices, only $P^0$ and $P_0$ are missing.",
"$\\begin{array}{cccccccccccccccccc}& & & & & & & & & & &{\\bullet }& - &{\\square }& - &{\\bullet }& &\\\\& & & & & & & & & & & &{\\backslash }& | & & | & &\\\\& & & & & & & & &{\\circ }& - &{\\blacksquare }& - &{\\circ }& 1 &{\\circ }& &\\\\& & & & & & & & & &{\\backslash }& | & & | & & | &{\\backslash }&\\\\&{\\circ }& - &{\\square }& - &{\\bullet }& &{\\bullet }& - &{\\square }& - &{\\bullet }& 2 &{\\bullet }& - &{\\square }& - &{\\bullet }\\\\& &{\\backslash }& | & & | & & &{\\backslash }& | & & | & & | &{\\backslash }& & &\\\\& & &{\\circ }& 1 &{\\circ }& - &{\\blacksquare }& - &{\\circ }& 3 &{\\circ }& - &{\\blacksquare }& - &{\\circ }& &\\\\& & & | & & | &{\\backslash }& | & & | & & | &{\\backslash }& & & & &\\\\& & &{\\bullet }& - &{\\square }& - &{\\bullet }& 4 &{\\bullet }& - &{\\square }& - &{\\bullet }& & & &\\\\& & & &{\\backslash }& | & & | & & | &{\\backslash }& | & & | & & & &\\\\&{\\circ }& - &{\\blacksquare }& - &{\\circ }& 5 &{\\circ }& - &{\\blacksquare }& - &{\\circ }& 1 &{\\circ }& & & &\\\\& &{\\backslash }& | & & | & & | &{\\backslash }& & & | & & | &{\\backslash }& & &\\\\- &{\\square }& - &{\\bullet }& 6 &{\\bullet }& - &{\\square }& - &{\\bullet }& &{\\bullet }& - &{\\square }& - &{\\bullet }& &\\\\{\\backslash }& | & & | & & | &{\\backslash }& & & & & & & & & & &\\\\&{\\circ }& 1 &{\\circ }& - &{\\blacksquare }& - &{\\circ }& & & & & & & & & &\\\\& | & & | &{\\backslash }& & & & & & & & & & & & &\\\\&{\\bullet }& - &{\\square }& - &{\\bullet }& & & & & & & & & & & &\\\\\\end{array}$"
],
[
"The solid torus made of the six dumplings",
"Each of the six dumplings is glued along the two pentagonal faces with two others.",
"The six dumplings form a solid torus; let us call it the dumpling torus.",
"This section is devoted to understand it."
],
[
"A dumpling",
"A dumpling has two rectangular faces, two triangular faces, and two pentagonal faces (see Figure REF ), which share an edge with two vertices marked white and black squares.",
"This edge will be called a special edge (see Figure REF ).",
"Figure: Two kinds of pentagons bounding the dumplings (planeH3-70)Figure: A dumpling, and two dumplings glued along a pentagon (gyoza170)Each dumpling $D$ is cut into two by the plane at infinity: the part with two vertices $\\bullet $ is denoted by $D^{\\prime }$ , and the other by $D^{\\prime \\prime }$ .",
"They are shown in Figure REF ."
],
[
"The dumpling torus",
"The six dumplings glued along their pentagonal faces form the dumpling torus.",
"Figure REF shows the six pentagons glued along the special edges forming a circle.",
"In the figure, special edges are shown by thick segments.",
"Figure: Theix pentagons glued along the special edges (pentagons70)Remember that the planes $H_x, H^y, H_z, H^x, H_y, H^z, H_x\\quad \\mbox{\\rm are\\ numbered}\\quad 1,\\dots ,6;$ and the curve $K$ lives inside the dumpling torus, and touches six pentagons in this order.",
"In Figure REF , the six dumplings are glued along their pentagonal faces.",
"The smallest pentagon on the top and the biggest one at the bottom are on the plane number 6; they should be identified after $2\\pi /5$ turn.",
"The pentagon on the plane 2 is next to the top, and so on.",
"The edges are labeled as 12, 23, ...; for example, 12 indicates the intersection of two pentagons on the planes 1 and 2.",
"The edges labeled by two consecutive numbers are the special edges.",
"The special edges form a circle: $\\underset{612}{\\blacksquare }{12\\over --}\\underset{123}{\\square }{23\\over --}\\underset{234}{\\blacksquare }{34\\over --}\\underset{345}{\\square }{45\\over --}\\underset{456}{\\blacksquare }{56\\over --}\\underset{561}{\\square }{61\\over --}\\underset{612}{\\blacksquare }$ Figure: Piles of six dumplings glued along their pentagonal faces (pileofG1)One can see in Figure REF , in front, the quadrilateral consisting of two triangles and two rectangles with edges labeled 12 at the top and 61 in the bottom, and segments 31, 41 and 51 are inside; this rectangle together with the pentagon $12-13-14-15-61$ is on the plane 1."
],
[
"Some curves on the dumpling torus",
"Trace the edge 51 on this rectangle along non-special edges, and we have the close curve: $\\underset{561}{\\square }{51\\over --}\\bullet {52\\over --}\\bullet {53\\over --}\\underset{345}{\\square }{35\\over --}\\bullet {36\\over --}\\bullet {31\\over --}\\underset{123}{\\square }{13\\over --}\\bullet {14\\over --}\\bullet {15\\over --}\\underset{561}{\\square }.$ Trace the edge 41 on this rectangle along non-special edges, and we have the close curve: $\\underset{612}{\\blacksquare }{62\\over --}\\circ {63\\over --}\\circ {64\\over --}\\underset{456}{\\blacksquare }{46\\over --}\\circ {41\\over --}\\circ {42\\over --}\\underset{234}{\\blacksquare }{24\\over --}\\circ {25\\over --}\\circ {26\\over --}\\underset{612}{\\blacksquare }.$ The other edges form a single closed curve: $\\underset{561}{\\square }{61\\over --}\\underset{4}{\\circ }{61\\over --}\\underset{3}{\\bullet }{61\\over --}\\underset{612}{\\blacksquare }{12\\over --}\\underset{5}{\\bullet }{12\\over --}\\underset{4}{\\circ }{12\\over --}\\underset{123}{\\square }{23\\over --}\\underset{6}{\\circ }{23\\over --}\\underset{5}{\\bullet }{23\\over --}\\underset{234}{\\blacksquare }$ $\\underset{234}{\\blacksquare }{34\\over --}\\underset{1}{\\bullet }{34\\over --}\\underset{6}{\\circ }{34\\over --}\\underset{345}{\\square }{45\\over --}\\underset{2}{\\circ }{45\\over --}\\underset{3}{\\bullet }{45\\over --}\\underset{456}{\\blacksquare }{56\\over --}\\underset{2}{\\bullet }{56\\over --}\\underset{3}{\\circ }{56\\over --}\\underset{561}{\\square }.$ Boundary of a(ny) pentagon; they will be called $m_D$ later: $\\blacksquare --\\circ --\\circ --\\blacksquare --\\square --\\blacksquare ,\\quad \\square --\\bullet --\\bullet --\\square --\\blacksquare --\\square .$ The diagonal of the front quadrilateral in Figure REF connecting $\\underset{612}{\\blacksquare }$ crossing the edges $31,41$ and 51; this will be called $L$ later."
],
[
"Another description of the dumpling torus",
"The dumpling torus presented as the pile of dumplings above can be understood also as follows, which may help the understanding: Consider the cylinder made by six copies of a pentagon times the interval $[0,1]$ , glued along pentagonal faces; the top and the bottom with $2\\pi /5$ twist.",
"The side of the cylinder is tessellated by $5\\times 6$ rectangular tiles.",
"Now choose six squares diagonally arranged (thanks to the twist, one can make it consistently) and compress these squares vertically, then you end up with the pile of dumplings above.",
"In Figure REF , it is shown in particular, how the dumpling with pentagonal faces 5 and 6 is made by a pentagon times the interval.",
"Figure: Compressing the pile of six pentagonal prisms to the dumpling torus (chochin)"
],
[
"The boundary of the dumpling torus and the line $L$",
"The boundary of the dumpling torus on each plane is shown in Figures REF and REF , and more explicitly again in Figure REF .",
"It consists of two quadrilaterals $DP$ and two triangles $DT$ .",
"This is the front rectangle shown in Figure REF , and a column of Figure REF left.",
"We show it again below by a diagram which will fit the diagram of the boundary of the prism torus shown in §REF .",
"$\\begin{array}{ccc}\\begin{array}{cccc}& & &{\\blacksquare }\\\\& & / & | \\\\&{\\square }& - &{\\bullet }\\\\& | & & | \\\\&{\\circ }& - &{\\circ }\\\\& | & & | \\\\&{\\bullet }& - &{\\square }\\\\& | & / & \\\\&{\\blacksquare }& &\\end{array}&\\qquad \\qquad \\qquad \\qquad &\\begin{array}{cccc}& & &{\\square }\\\\& & / & | \\\\&{\\blacksquare }& - &{\\circ }\\\\& | & & | \\\\&{\\bullet }& - &{\\bullet }\\\\& | & & | \\\\&{\\circ }& - &{\\blacksquare }\\\\& | & / & \\\\&{\\square }& &\\end{array}\\end{array}$ Note that in the left diagram, the two opposite ${\\blacksquare }$ stands for the same vertex; in the right diagram, two opposite ${\\square }$ stands for the same vertex.",
"The diagonal joining two opposite ${\\blacksquare }$ in the left (in Figure REF left, the line $x-y+2=0$ ) and the diagonal joining two opposite ${\\square }$ in the right (in Figure REF right, a vertical line $y=1/2$ ) are closed curves, which are homotopically equivalent on the boundary of the dumpling torus; Let us call one of them $L$ .",
"Figure: Boundary of the dumpling torus on each plane (planeH4-70)We now show the six such forming the boundary of the dumpling torus.",
"The twelve quadrilaterals and the twelve triangles tessellate the boundary torus as below.",
"To indicate the identification, four copies of the plane number 1 are shown.",
"After identification, there are $3\\ \\square ,\\quad 3\\ \\blacksquare ,\\quad 6\\ \\bullet ,\\quad 6\\ \\circ ;$ note that among the twenty vertices, only $P^0$ and $P_0$ are missing.",
"$\\begin{array}{cccccccccccccccccc}& & & & & & & & & & & & & & &{\\blacksquare }& &\\\\& & & & & & & & & & & & & & / & | & &\\\\& & & & & & & & & & & & &{\\square }& - &{\\bullet }& &\\\\& & & & & & & & & & & & / & | & & | & &\\\\& & & & &{\\blacksquare }& & & & & &{\\blacksquare }& - &{\\circ }& 1 &{\\circ }& &\\\\& & & & / & | & & & & & / & | & & | & & | & &\\\\& & &{\\square }& - &{\\bullet }& & & &{\\square }& - &{\\bullet }& 2 &{\\bullet }& - &{\\square }& &\\\\& & & | & & | & & & / & | & & | & & | & / & & &\\\\& & &{\\circ }& 1 &{\\circ }& &{\\blacksquare }& - &{\\circ }& 3 &{\\circ }& - &{\\blacksquare }& - &{\\circ }& &\\\\& & & | & & | & / & | & & | & & | & / & | & & & &\\\\& & &{\\bullet }& - &{\\square }& - &{\\bullet }& 4 &{\\bullet }& - &{\\square }& - &{\\bullet }& & & &\\\\& & & | & / & | & & | & & | & / & | & & | & & & &\\\\& & &{\\blacksquare }& - &{\\circ }& 5 &{\\circ }& - &{\\blacksquare }& - &{\\circ }& 1 &{\\circ }& & & &\\\\& & / & | & & | & & | & / & & & | & & | & & & &\\\\&{\\square }& - &{\\bullet }& 6 &{\\bullet }& - &{\\square }& - &{\\bullet }& &{\\bullet }& - &{\\square }& & & &\\\\& | & & | & & | & / & & & & & | & / & & & & &\\\\&{\\circ }& 1 &{\\circ }& - &{\\blacksquare }& - &{\\circ }& & & &{\\blacksquare }& & & & & &\\\\& | & & | & / & & & & & & & & & & & & &\\\\&{\\bullet }& - &{\\square }& & & & & & & & & & & & & &\\\\& | & / & & & & & & & & & & & & & & &\\\\&{\\blacksquare }& & & & & & & & & & & & & & & &\\end{array}$"
],
[
"Tetrahedra",
"Let us compare the (tessellated) boundary of the prism torus and that of the dumpling torus.",
"They share the quadrilateral faces.",
"The two solid tori almost fill the space; six tetrahedra are the notches.",
"Each tetrahedron has two triangular faces adjacent to prisms and other two triangular faces adjacent to dumplings: $\\begin{array}{ccccccccccccccccc}&{\\circ }& - &{\\blacksquare }& & \\qquad \\qquad &{\\circ }& - &{\\blacksquare }& & \\qquad \\qquad & &{\\circ }& - &{\\blacksquare }&\\\\\\partial & | &\\times & | & & = & | &{\\backslash }& | & & \\cup & & | & {\\bf /} & | &\\\\&{\\square }& - &{\\bullet }& & &{\\square }& - &{\\bullet }& & & &{\\square }& - &{\\bullet }&\\end{array}$ In Figures REF (right) and REF (left), we see that each tetrahedron has bounded base (triangle $(\\bullet ,\\circ ,\\blacksquare )\\subset H^\\circ $ ) on the big-cylinder, an infinitely long face on the prism torus, and two infinitely long faces on the dumpling torus; intersection of the last two faces is a special edge.",
"The six tetrahedra form a rosary (see Figure REF (left) for a combinatorial idea).",
"The curve $\\check{K}$ , introduced in §3.2.3 lives in the union of the six tetrahedra.",
"Figure: The rosary of tetrahedra and two tetrahedra extracted from Figure (right)"
],
[
"Intersection with a big sphere",
"The aim of this section is to visualize what happens near the plane at infinity $H_{\\infty }$ by looking at the intersection with a big sphere.",
"The sphere at infinity is shown in Figure REF ; the plane at infinity (see Figure REF ) is obtained by identifying antipodal points.",
"The three circles represent (the intersections with the planes) $H_x,H_y,H_z,$ showing an octahedron, and the three lines represent (the intersections with the planes) $H^x,H^y,H^z$ .",
"They cut out 24 triangles.",
"Each triangle is a section of a prism or a dumpling.",
"Dotted circles represent (the intersection with) the surface $Q=0$ .",
"Figure: Intersection of the chambers with the sphere at infinity (sphere0)A very big sphere is shown in Figure REF .",
"The three circles (showing an octahedron) and the three lines represent as in the previous figure.",
"The triangle appeared at the center is the section of a cube; the other cube situates out of the picture.",
"The six small triangles are the sections of the tetrahedra.",
"Figure: Intersection of the chambers with a big sphere (sphere1)Each prism $P$ is the union of two parts $P^{\\prime }$ and $P^{\\prime \\prime }$ as is shown in Figure REF , glued along the shaded triangle.",
"Now the shaded triangle is thickened; the sections of $P^{\\prime }$ and $P^{\\prime \\prime }$ with the big sphere are a triangle and a pentagon, respectively.",
"(See Figure REF left.)",
"Each dumpling $D$ is the union of two parts $D^{\\prime }$ and $D^{\\prime \\prime }$ as is shown in Figure REF , glued along the shaded triangle.",
"Now the shaded triangle is thickened; the sections of $D^{\\prime }$ and $D^{\\prime \\prime }$ with the big sphere are both quadrangle.",
"(See Figure REF right.)",
"Figure: Enlargement of two faces of the octahedron (sphere3)Intersection with the big sphere with the surface $Q=0$ is shown in Figure REF ; which shows that the surface does not path through any prisms nor cubes, and that the surface can be considered as an approximation of (the boundary of) the prism torus.",
"Figure: Enlargement around the intersection with a tetrahedron together with that of the surface Q=0Q=0 (sphere2)"
],
[
"Seven hyperplanes in the 4-space",
"Seven hyperplanes in general position in $\\mathbb {P}^4$ cut out a unique chamber, bounded by the seven hyperplanes, stable under the action of the cyclic group $\\mathbb {Z}_7$ ; let us call it the central chamber CC.",
"This action can be projective if the hyperplanes are well-arranged.",
"The intersection of CC and a hyperplane is a dumpling.",
"So the boundary of CC is a 3-sphere $\\mathbb {S}^3$ tessellated by seven dumplings.",
"We would like to know how they are arranged, especially the $\\mathbb {Z}_7$ -action on the tessellation.",
"In [1], the seven dumplings in $\\mathbb {S}^3$ is shown (see Figure ), in which the $\\mathbb {Z}_7$ -action can be hardly seen.",
"We first observe the tessellation.",
"We label the dumplings as $D_1,D_2,\\dots ,D_7$ so that the dumplings $D_k$ and $D_{k+1}$ , modulo 7, share a pentagon, and the group action induces a transformation $D_k\\rightarrow D_{k+1}$ modulo 7.",
"The special edges form a circle $C$ .",
"(Recall that a dumpling is bounded by two pentagons, two triangles and two quadrilaterals, and that the intersection of the two pentagons is called the special edge.)",
"If we remove seven pentagons $D_k\\cap D_{k+1}$ from the 2-skeleton of the tessellation, it remains a Möbius strip $M$ with the curve $C$ as the boundary.",
"We find, in the 2-skeleton, a subskeleton homeomorphic to a disc which bound the curve $C$ .",
"This implies that $C$ is unknotted, and that $M$ is twisted only by $\\pm \\pi .$ We give another description of the tessellation so that the action of $\\mathbb {Z}_7$ can be seen; we start from a vertical pentagonal prism, horizontally sliced into seven thinner prisms.",
"We collapse vertically the seven rectangular faces diagonally situated on the boundary of the prism; this process changes each thinner prism into a dumpling.",
"Identifying the top and the bottom after a suitable rotation, we get a solid torus $ST$ made of seven dumplings, so that the solid torus $ST$ admits a $Z_7$ -action.",
"Consider a usual solid torus $UT$ in our space $\\mathbb {S}^3$ and put $ST$ outside of $UT$ ; the boundary of $ST$ , which is also the boundary of $UT$ , is a torus $T$ .",
"We collapse $UT$ by folding the torus $T$ into a Möbius strip, which can be identified with $M$ , whose boundary can be identified with $C$ .",
"We will see that the result, say $X_7$ , of this collapsing is homeomorphic to $\\mathbb {S}^3$ .",
"In this way, we recover the tessellation of $\\mathbb {S}^3$ by seven dumplings.",
"To make our idea and a possible inductive process clear, we start this chapter by studying a chamber bounded by five (as well as six) hyperplanes stable under the action of the cyclic group $\\mathbb {Z}_5$ (resp.",
"$\\mathbb {Z}_6$ ); though such a chamber is not unique, the above statement with obvious modification is still true, if we understand the special edge of $D_i$ as $D_{i-1}\\cap D_i\\cap D_{i+1}$ .",
"A 3-dumpling is a tetrahedron, and a 4-dumpling is a prism.",
"After the collapsing process as above, we get the manifolds $X_5$ (resp.",
"$X_6$ ).",
"We prove in detail that $X_5$ is homeomorphic to $\\mathbb {S}^3$ ; for other cases proof is similar."
],
[
"Five hyperplanes in the 4-space",
"Five hyperplanes in the projective 4-space cut out sixteen 4-simplices.",
"If a hyperplane is at infinity, then the remaining four can be considered as the four coordinate hyperplanes, dividing the space into $2^4$ chambers.",
"The boundary of each 4-simplex is of course made by five tetrahedra.",
"In other words, five tetrahedra are glued together to make a (topological) 3-sphere: around a tetrahedron, glue four tetrahedra along their faces, and further use the valley of each pair of triangles in order to close the pair like a book."
],
[
"Labeling and a study of 2-skeleton",
"We study the above tessellation more precisely.",
"Label the five hyperplanes as $H_1,\\dots ,H_5$ .",
"Choose one 4-simplex, out of sixteen, and call it CC.",
"We use the convention $H_{ij}=H_i\\cap H_j,\\quad H_{ijk}=H_i\\cap H_j\\cap H_k, \\ \\dots $ The five vertices of CC are: $H_{2345},\\quad H_{1345},\\ \\cdots ,\\ H_{1234}$ which are often denoted simply by $ 1,\\ 2,\\ \\dots ,\\ 5,$ respectively.",
"The boundary of CC is made by five tetrahedra: $D_1=2345,\\quad D_2=1345,\\ \\dots .$ (Tetrahedron $D_1$ has vertices $2,3,4,5$ .)",
"Any two of them share a triangle: $D_i\\cap D_j=\\Delta (\\lbrace 1,\\dots ,5\\rbrace -\\lbrace i,j\\rbrace ),\\quad i\\ne j;$ there are ten of them.",
"These tetrahedra tessellate the 3-sphere $\\mathbb {S}^3$ .",
"We consider the sequence $D_1,D_2,\\dots $ , in this order, and name the intersections of the two consecutive ones as: $D_1\\cap D_2=\\Delta (345),\\quad D_2\\cap D_3=\\Delta (451),\\ \\cdots D_5\\cap D_1=\\Delta (234).$ (Note that the union of these five triangles form a Möbius strip, but for a while just forget it.)",
"Remove these five triangles from the 2-skeleton of the tessellation.",
"Then five triangles remain, forming another Möbius strip $M:\\quad \\scriptsize \\begin{array}{ccccccc }3&-&4&-&5&-&1\\\\\\quad {\\backslash }& & /\\quad {\\backslash }&&/\\quad {\\backslash }&& /\\quad \\\\&1&-&2&-&3&\\end{array}.$ Its boundary $C=\\partial M:\\quad 1 \\ -\\ 2 \\ -\\ 3 \\ -\\ 4 \\ -\\ 5 \\ -\\ 1$ is an unknotted circle because $C$ bounds also the disc $\\Delta (234)\\cup \\Delta (124)\\cup \\Delta (451): \\qquad \\scriptsize \\begin{array}{ccccc }3&-&4&-&5\\\\\\quad {\\backslash }& & /\\quad {\\backslash }&&/\\quad \\\\&2&-&1&\\end{array}.$"
],
[
"Visualization",
"Since the situation is quite simple, we visualize the happenings.",
"Consider a tetrahedron with vertices $\\lbrace 1,2,3,4\\rbrace $ in our space, and put vertex 5 at the barycenter.",
"The barycentric subdivision of this tetrahedron (which yields $D_1,\\dots ,D_4$ ) together with its complementary tetrahedron $D_5$ fills the 3-sphere.",
"Figure REF shows the 2-skeleton consisting of ten triangles as the union of the Möbius strips $M$ and the complementary one.",
"You can find the disc, given in the previous subsection, bounded by the curve $C$ .",
"When you make a paper model of the 2-skelton out of Figure REF , be ware that the two Möbius strips have different orientations.",
"Figure: Möbius strips MM (left) and the complementary one (right) before identification along appropriate edges.Figure: (left) Möbius strip MM.",
"(right) Boundary ∂M\\partial M."
],
[
"From a solid torus",
"We would like to give another description, which will prepare the next step where the number $m$ of hyperplanes is $m=6,7,\\dots $ , in such cases, naive description above would hardly work.",
"We start with a vertical triangular prism, horizontally sliced into five thinner triangular prisms.",
"Its top and the bottom triangles are identified after a $4\\pi /3$ rotation; this makes the vertical prism a solid torus.",
"The boundary torus is tessellated by fifteen rectangles.",
"Now collapse vertically five diagonal rectangles.",
"This changes every remaining rectangles into triangles, and the thinner triangular prisms into tetrahedra.",
"Further, use each collapsed edge as a valley of each pair of triangles face-to-face in order to close the pair like a book.",
"This identification makes the solid torus a 3-sphere tessellated by five tetrahedra, recovering the tessellation described in §REF .",
"A vertical triangular prism horizontally sliced.",
"We show/repeat the above process by using the labels $1,2,3,4,5$ (they correspond the labels of the five hyperplanes $H_1,\\ H_2,\\dots $ ).",
"Consider five triangles labeled by two consecutive numbers: $51,\\ 12, \\dots $ (they correspond the triangle $H_{51}\\cap {\\rm CC},\\dots $ ).",
"The triangle 51, for example, has three vertices with labels $51:\\quad \\lbrace 5124,\\ 5123, \\ 5134\\rbrace $ (they correspond the points $\\lbrace H_{4512},H_{5123},H_{3451}\\rbrace $ ).",
"Now operate the group $\\mathbb {Z}_5$ on the labels: $j\\rightarrow j+1$ mod 5, and we get four other triangles, for example, the triangle 12 has three vertices: $12:\\quad \\lbrace 1235, \\ 1234, \\ 1245\\rbrace .$ We make a triangular prism by putting the triangle 51 above the triangle 12; the two vertices of 51 having the same labels of the ones of 12 should be put just above each other.",
"In a word this prism is the sandwich made by the two triangles 51 and 12.",
"In this way, we make five sandwiches and pile them to form a vertical triangular prism.",
"The table in Figure REF repeats what we described.",
"Each off-diagonal rectangle is labeled by the two numbers common to the four vertices.",
"Figure: (left) Table of five piles.",
"(right) Triangular prism split up into five slices.The label 14 denotes the upper back rectangle.Collapsing rectangles.",
"If we identify two points with the same indices as sets, for example, $4512=5124=1245$ , you find five rectangles with pairwise coincide vertices (marked by $\\times $ ), situated diagonally.",
"We collapse such rectangles vertically to a segment, each called a special edge.",
"Note that a special edge has two ends which are labeled by consecutive numbers: $\\lbrace 1234,2345\\rbrace ,\\lbrace 2345,3451\\rbrace ,\\dots $ Accordingly all the remaining rectangles become triangles and, the five thin prisms become tetrahedra.",
"But as a whole it still remains to be a cylinder.",
"This collapsing process is described in Figure REF .",
"Here the prisms are hollow so that we can peep inside.",
"Figure: (left) Collapsing rectangles.",
"(right) Tiling the boundary of the solid cylinder by ten triangles.The dashed line is the special loop CC.Identifying the top and the bottom to get a solid torus $ST$ .",
"We identify the top triangle and the bottom triangle of the collapsed vertical prism according to the labels; note that we must twist the triangle by $4\\pi /3$ .",
"By the identification we get a solid torus, say $ST$ ; its meridean ${\\bf mer}$ is represented by the boundary of a(ny) triangle, in Figure REF , say 51 (or 12, 23,...).",
"Note that it is still an abstract object.",
"The five special edges now form a circle: $C:\\quad 1234-2345-3451-4512-5123-1234.$ We take a vertical line, in Figure REF (left), with a $4\\pi /3$ twist, say, ${\\bf par}: 5123 - 3451 - 5123=4-2-4$ as a parallel.",
"The torus $T$ and the solid torus $UT$ .",
"We consider in our space ($\\sim \\mathbb {S}^3$ ) a usual (unknotted) solid torus, say $UT$ , and put our solid torus $ST$ fills outside of $UT$ , so that the tessellated boundary torus $T:=\\partial ST\\ (=\\partial UT)$ has par as the meridean, and mer as the parallel.",
"From the description of the vertical prism, we see ten triangles tessellating the torus $T$ arranged in a hexagonal way (Figure REF -left).",
"Figure: (left) Ten triangles tesselating the hexagon.",
"(right) Identifying the left and right vertical sides we get a cylinder topologicallyequivalent to that of Figure -right.The special curve CC is dotted while STST-parallel is dashed.Note that the triangulated torus above is not a simplicial complex; indeed there are two edges with the same vertices, for example, $5123-3451$ .",
"Anyway, you can see the curve $C$ lies on the torus $T=\\partial ST$ as a $(2,1)$ -curve (see Figure REF ).",
"Figure: (left) Tesselation of the torus ∂ST\\partial ST. Special curve CC is dotted while STST-parallel is dashed.",
"(right) The special curve CC.Collapsing the solid torus $UT$ by folding $T$ .",
"On the torus $T$ , there are pairs of triangles with the same vertices.",
"We identify these triangles by folding the torus $T$ along the curve $C$ .",
"This collapses the solid torus $UT$ ; we prove below that the resulting space, say $X_5$ , is still $\\mathbb {S}^3$ .",
"Figure: (left) Tesselation of the torus ∂UT\\partial UT.",
"Special curve CC is dotted while UTUT-meridian is dashed.",
"(right) Möbius strip MM.The torus $T$ is folded to be a Möbius strip $M$ with boundary $C$ : $M:\\quad \\begin{array}{ccccccc }4512&-&5123&-&1234&-&2345\\\\\\quad {\\backslash }&25 & /\\ 35\\ {\\backslash }&31&/\\ 41\\ {\\backslash }&42& /\\quad \\\\&2345&-&3451&-&4512&\\end{array}$ Figure: (from left to right) Folding the torus TT along the special curve CC to be the Möbius strip MM.",
"A sector is cut out to show the folding.Note that $M$ is a simplicial complex.",
"If we use complementary labeling, for example, 5 for 1234, this is exactly the same as we got several pages before.",
"On the hexagonal expression of the torus $T$ , this folding is done as closing a book along the diagonal.",
"The process of collapsing the solid torus $UT$ onto the Möbius strip $M$ can be explicited by the following homotopy parametrized by $t\\in [0,1]$ : $F(\\vartheta ,\\eta ,r,t)=O^{\\prime }+(1-t)r\\overrightarrow{w}(\\vartheta ,\\eta +\\frac{\\vartheta }{2})+tr\\cos \\eta \\cdot \\overrightarrow{w}(\\vartheta ,\\frac{\\vartheta }{2}),$ where $O^{\\prime }=\\sqrt{2}(\\cos \\vartheta ,\\sin \\vartheta ,0),\\quad \\overrightarrow{w}(\\vartheta ,\\eta )=(\\cos \\eta \\cos \\vartheta ,\\cos \\eta \\sin \\vartheta ,\\sin \\eta )$ and $(\\exp (i\\vartheta ),\\exp (i\\eta ),r)$ parametrizes the solid torus $\\mathbb {S}^1\\times \\mathbb {D}^2$ (Figure REF ).",
"Figure: Meridian section of UTUT.",
"The homotopy collapsing UTUT onto MM follows the dashed line.Puffing the Möbius strip to make a torus.",
"The inverse operation of folding the (unknotted) torus $T$ to the Möbius strip $M$ (with $\\pm \\pi $ twist) can be described as follows: Consider a Möbius strip $M$ being double sheeted like a flat tire folded along $C$ .",
"Then puff the tire to be a air-filled tire $T$ .",
"It is illustrated in Figure REF seen from right to left.",
"A disc bounding $C$ .",
"Let us repeat the disc in §REF bounding $C$ , using complementary labeling: $\\begin{array}{ccccc }4512&-&5123&-&1234\\\\[2mm]\\quad {\\backslash }&51 & /\\quad 35\\quad {\\backslash }&23&/\\quad \\\\[2mm]&3451&-&2345&\\end{array}.$ Proof that $X_5$ is homeomorphic to $\\mathbb {S}^3$ .",
"The collapsing of $\\mathbb {S}^3=UT\\cup _T ST$ by folding $T$ along $C$ and collapsing $UT$ on $M$ is nothing but the gluing of the set of the five tetrahedra tiling $ST$ in such a way that the tetrahedra facets are identified by pairs.",
"In doing so we must get $ST$ back in one hand, and, on the other hand, the triangles of the boundary $\\partial ST$ are identified by pairs to get the Möbius strip $M$ .",
"The result $X_5$ , is a closed 3-dimensional manifold since the link of each vertex is homeomorphic to a 2-sphere, namely the 2-skeleton of a tetrahedron.",
"For instance the link of the vertex 1234 is the union of the four triangles $\\lbrace 15,25,35,45\\rbrace $ , where 15 for example, is the triangle with vertices $1523,1534$ and 1542.",
"The 3-manifold $X_5$ , as a continuous image of $\\mathbb {S}^3$ , is clearly connected.",
"Now, computing the fundamental group $\\pi (X_5,\\ast )$ will prove that $X_5$ is simply connected and therefore, by Poincaré-Perelman theorem, homeomorphic to $\\mathbb {S}^3$ .",
"Indeed, since $X_5$ is endowed with a structure of simplicial complex, $\\pi (X_5,\\ast )$ is isomorphic to $\\pi (X_5^2,\\ast )$ , where $X_5^2$ denotes the 2-skeleton.",
"The group $\\pi (X_5^2,\\ast )$ can be described by generators and relators as follows.",
"Firstly, we observe that the 1-skeleton $X_5^1$ is the complete graph over the five vertices $1,2,3,4,5$ (using complementary labeling) (Figure REF ).",
"Figure: (left) The complete graph X 5 1 X_5^1 over five vertices and a maximal tree base at 5.",
"(right) One of the six generators of π(X 5 1 ,5)\\pi (X_5^1,5):α=(5,1,2)\\alpha =(5,1,2).Recall that we can use a maximal tree (that is a tree containing all vertices) to compute the fundamental group $\\pi (X_5^1,5)$ .",
"Indeed, to each egde not contained in the tree, we associate in a obvious way a loop based at 5.",
"The set of these loops is a basis of the free group $\\pi (X_5^1,5)$ .",
"Therefore we see that $\\pi (X_5^1,\\ast )$ is the free group over the six following generators (still using complementary labeling for the sequence of vertices and chosing 5 as a base point): $\\alpha =(5,1,2),\\quad \\beta =(5,1,3),\\quad \\gamma =(5,1,4),\\quad \\delta =(5,2,3),\\quad \\varepsilon =(5,2,4),\\quad \\zeta =(5,3,4)$ Secondly, the relators of $\\pi (X_5^2,\\ast )$ are associated with the ten facets coming from the five tetrahedra tiling $X_5$ : $12,\\quad 13,\\quad 14,\\quad 15,\\quad 23,\\quad 24,\\quad 25,\\quad 34,\\quad 35,\\quad 45.$ Figure: (left) The generators δ=(5,2,3)\\delta =(5,2,3) and ζ=(5,3,4)\\zeta =(5,3,4).",
"(right) The generator ε=(5,2,4)\\varepsilon =(5,2,4)and the relator δζε -1 =(2,3,4)\\delta \\zeta \\varepsilon ^{-1}=(2,3,4).It yields to the following relators (Figure REF ): $\\zeta ,\\quad \\varepsilon ,\\quad \\delta ,\\quad \\delta \\zeta \\varepsilon ^{-1},\\quad \\gamma ,\\quad \\beta ,\\quad \\beta \\zeta \\gamma ^{-1},\\quad \\alpha ,\\quad \\alpha \\varepsilon \\gamma ^{-1},\\quad \\alpha \\delta \\beta ^{-1},$ something which proves that $\\begin{array}{l}\\pi (X_5,\\ast )=\\pi (X_5^2,\\ast )=\\\\<\\alpha ,\\beta ,\\gamma ,\\delta ,\\varepsilon ,\\zeta :\\zeta =\\varepsilon =\\delta =\\delta \\zeta \\varepsilon ^{-1}=\\gamma =\\beta =\\beta \\zeta \\gamma ^{-1}=\\alpha =\\alpha \\varepsilon \\gamma ^{-1}=\\alpha \\delta \\beta ^{-1}=1>\\\\=1.\\end{array}$ Conclusion.",
"In the sphere $\\mathbb {S}^3=UT\\cup _T ST$ , the solid torus $UT$ is collapsed to an unknotted Möbius strip $M$ , which is twisted only by $\\pm \\pi $ (see Figure REF ), and the resulting 3-manifold $X_5$ is a 3-sphere.",
"In this way we recover the tessellation of $\\partial \\Delta _4=\\partial {\\rm CC}$ described in §REF ."
],
[
"Six hyperplanes in the 4-space",
"Six hyperplanes in the projective 4-space cut out six 4-simplices, fifteen prisms of type $\\Delta _3\\times \\Delta _1$ and ten prisms of type $\\Delta _2\\times \\Delta _2$ .",
"If a hyperplane is at infinity, then the remaining five bound a simplex.",
"Other chambers touch this simplex along 3-simplices (five $\\Delta _3\\times \\Delta _1$ ), along 2-simplices (ten $\\Delta _2\\times \\Delta _2$ ), along 1-simplices (ten $\\Delta _3\\times \\Delta _1$ ) and along 0-simplex (five $\\Delta _4$ ).",
"Though a prism of type $\\Delta _3\\times \\Delta _1$ is bounded by six hyperplanes, it does not admit an action of the group $\\mathbb {Z}_6$ .",
"So we consider one of the prisms of type $\\Delta _2\\times \\Delta _2$ , which is bounded by six prisms (of type $\\Delta _2\\times \\Delta _1$ ).",
"We study how these six prisms tessellate a 3-sphere."
],
[
"Labeling and a study of 2-skeleton",
"Let the six hyperplanes $H_1,\\dots ,H_6$ bound a chamber CC of type $\\Delta _2\\times \\Delta _2$ .",
"The chamber CC has six faces $D_1,\\dots ,D_6$ , which form the $\\mathbb {Z}_6$ -orbit of the prism $D_1={\\rm CC}\\cap H_1:\\scriptsize \\begin{array}{ccccc}1346& --&--&-- &1456\\\\|& {\\backslash }\\quad &4&\\quad / &|\\\\|& 3\\quad 1234&--& 1245\\quad 5 &|\\\\|& /\\quad & 2 &\\quad {\\backslash }&|\\\\1236& --&--&-- &1256\\end{array}\\normalsize ,$ where $1234=H_{1234}=H_1\\cap \\cdots \\cap H_4$ .",
"Note that the rectangular face of $D_1$ behind is $D_1\\cap D_6$ , and the rectangular face in front below is $D_1\\cap D_2$ .",
"Using the notation introduced in Chapter 2, CC is represented by $++++++$ , and its boundary consistes of $D_1=0+++++,\\quad D_2=-0++++,\\ \\dots ,\\ D6=-----0.$ The boundary of $D_1$ consists of (cf.",
"§) $\\begin{array}{lll}00++++:{\\rm 4-gon}&0-0+++:{\\rm triangle}&0--0++:{\\rm rectangle}\\\\0---0+:{\\rm triangle}&0----0:{\\rm 4-gon}\\end{array}\\normalsize $ The 2-skeleton of CC consists of six triangles and nine rectangles.",
"Remove the intersections of the two consecutive ones: $D_1\\cap D_2,\\quad D_2\\cap D_3,\\quad \\dots ,\\quad D_6\\cap D_1:$ $\\scriptsize \\begin{array}{cccccccccc}5612& -&6123& -&2356& & & &&\\\\| &12& | &23& | & & & &&\\\\1245& -&1234& -&2345& -&4512& &&\\\\& & | &34& | &45& | & &&\\\\& &3461& -&3456& -&4561& -&6134&\\\\& & & & | &56& | &61& | &\\\\& & & &5623& -&5612& -&6123&.\\end{array}\\normalsize $ (Note that the union of these six rectangles form a cylinder, not a Möbius strip.)",
"Then the remaining six triangles and three rectangles form a Möbius strip $M:\\quad \\scriptsize \\begin{array}{ccccccccc }5612&-&6123&-&1234&-&2345&-&3456\\\\\\quad {\\backslash }&26 & /\\quad {\\backslash }&31&/\\quad {\\backslash }&42&/\\quad {\\backslash }&53&/\\\\&2356& &3461& &1245& &2356\\\\&\\quad {\\backslash }&36 & /\\ 46\\ {\\backslash }&41&/\\ 51\\ {\\backslash }&52&/\\quad & \\\\& &5634&-&6145&- &5612&&\\\\\\end{array}\\normalsize $ Möbius 6 Its boundary $C=\\partial M:\\quad 1234-2345-3456-4561-5612-6123-1234$ is an unknotted circle because $C$ bounds also the disc $\\scriptsize \\begin{array}{ccccccc}&6123\\quad &-&\\quad 1234&\\\\[2mm]\\quad /&&{\\backslash }\\quad 13\\quad /&&{\\backslash }\\quad \\\\[2mm]5612&61\\quad &6134&34&2345\\\\[2mm]\\quad {\\backslash }&&/\\quad 46\\quad {\\backslash }&&/\\quad \\\\[2mm]&4561\\quad &-&\\quad 3456&\\end{array}\\normalsize .$"
],
[
"From a solid torus",
"We start with a vertical rectangular prism, horizontally sliced into six thinner rectangular prisms.",
"Its top and the bottom rectangles are identified after a $4\\pi /4$ rotation; this makes the vertical prism a solid torus.",
"The boundary torus is tessellated by 24 rectangles.",
"Now collapse vertically six diagonal rectangles.",
"This changes eighteen rectangles (out of 24) into six segments (forming a circle) and twelve triangles.",
"Accordingly, the six thinner rectangular prisms (actually cubes) into prisms.",
"Next use each collapsed edge as a valley of each pair of triangles face-to-face in order to close the pair like a book, and further identify three pairs of rectangles.",
"This identification changes the solid torus into a 3-sphere tessellated by six prisms.",
"A vertical rectangular prism horizontally sliced.",
"We show/repeat the above process by using the labels $1,2,3,4,5,6$ (they correspond the labels of the six hyperplanes $H_1,\\ H_2,\\dots $ ).",
"Consider six rectangles labeled by two consecutive numbers: $61,\\ 12, \\dots $ (they correspond the rectangle $H_{61}\\cap {\\rm CC},\\dots $ ).",
"The rectangle 61, for example, has four vertices with labels $61:\\quad \\lbrace 6125,\\ 6123, \\ 6134,\\ 6145\\rbrace $ (they correspond the points $\\lbrace H_{5612},H_{6123},H_{6134},H_{6145}\\rbrace $ ).",
"Now operate the group $\\mathbb {Z}_6$ on the labels: $j\\rightarrow j+1$ mod 6, and we get five other rectangles, for example, the rectangle 12 has three vertices: $12:\\quad \\lbrace 1236, \\ 1234, \\ 1245,\\ 1256\\rbrace .$ We make a rectangular prism by putting the rectangle 61 above the rectangle 12; the two vertices of 61 having the same labels of the ones of 12 should be put just above each other.",
"In a word this prism is the sandwich made by the two rectangles 61 and 12.",
"In this way, we make six sandwiches and pile them to form a vertical rectangular prism.",
"The following table repeats what we described.",
"Each off-diagonal rectangle is labeled by the two numbers common to the four vertices.",
"$\\scriptsize \\begin{array}{rccccccccc}61:&6125&-&6123&-&6134&-&6145&-&6125\\\\&\\Vert &\\times &\\Vert &13&| &14&| &15&\\Vert \\\\12:&1256&-&1236&-&1234&-&1245&-&1256\\\\&| &26 &\\Vert &\\times &\\Vert &24&| &25&| \\\\23:&2356&-&2361&-&2341&-&2345&-&2356\\\\&| &36 &| &31&\\Vert &\\times &\\Vert &35&| \\\\34:&3456&-&3461&-&3412&-&3452&-&3456\\\\&\\Vert &46 &| &41&| &42&\\Vert &\\times &\\Vert \\\\45:&4563&-&4561&-&4512&-&4523&-&4563\\\\&\\Vert &\\times &\\Vert &51&| &52&| &53&\\Vert \\\\56:&5634&-&5614&-&5612&-&5623&-&5634\\\\&| &64&\\Vert &\\times &\\Vert &62 &| &63&| \\\\61:&6134&-&6145&-&6125&-&6123&-&6134\\\\&| &14&| &15&\\Vert & \\times &\\Vert &13&| \\\\\\end{array}\\normalsize $ Table of piles 6 Collapsing rectangles.",
"If we identify two points with the same consecutive indices as sets, for exmaple, $6123=1236=2361$ but $2356\\ne 5623$ , you find six rectangles with pairwise coincide vertices (marked by $\\times $ ), situated diagonally.",
"We collapse such rectangles vertically to a segment, each called a special edge.",
"Note that a special edge has two ends which are labeled by consecutive numbers: $\\lbrace 1234,2345\\rbrace ,\\lbrace 2345,3451\\rbrace ,\\dots $ Accordingly the remaining rectangles next to the special edges become triangles, and the six thin rectangular prisms become triangular prisms ($\\sim \\Delta _2\\times \\Delta _1$ ).",
"But as a whole it still remains to be a cylinder.",
"Identifying the top and the bottom to get a solid torus $ST$ .",
"We identify the top rectangle and the bottom rectangle of the collapsed vertical prism according to the labels; note that we must twist the triangle by $4\\pi /4$ .",
"By the identification we get a solid torus, say $ST$ ; its meridean ${\\bf mer}$ is represented by the boundary of a(ny) rectangle, in Table of piles 6, say 61 (or 12, 23,...).",
"Note that it is still an abstract object.",
"The six special edges now form a circle: $C: 1234-2345-3456-4561-5612-6123-1234.$ We take a vertical line, in Table of piles 6, with a $4\\pi /4$ twist, say, ${\\bf par}: 6123 - 3461 - 4561 - 6123$ as a parallel.",
"The torus $T$ and the solid torus $UT$ .",
"We consider in our space ($\\sim \\mathbb {S}^3$ ) a usual (unknotted) solid torus, say $UT$ , and think our solid torus $ST$ fills outside of $UT$ , so that the tessellated boundary torus $T:=\\partial \\ ST\\ (=\\partial \\ UT)$ has par as the meridean, and mer as the parallel.",
"From the description of the vertical prism, we see twelve triangles and six rectangles tessellating the torus $T$ arranged in a hexagonal way (identify the opposite sides of the hexagon): $\\scriptsize \\begin{array}{ccccccccccc}5612&-&5623&-&3456&&& & &\\\\| &26{\\backslash }62&| &63&| &64{\\backslash }& && &\\\\2356&-&6123&-&6134&-&4561&&&\\\\| &36 &| &31{\\backslash }13&| &14 &| &15{\\backslash }& &\\\\3456&-&3461&-&1234&-&1245&-&5612&\\\\& {\\backslash }46 &| &41 &| &42{\\backslash }24&| &25 &| &\\\\& &4561&-&4512&-&2345&-&2356&\\\\& & &{\\backslash }51 &| &52 &| &53{\\backslash }35&| & \\\\&&&&5612&-&5623&-&3456&&\\\\\\end{array}\\normalsize $ Torus (hexagon) 6 Collapsing the solid torus $UT$ by folding $T$ .",
"We identify two vertices with the same indices as sets.",
"Along each collapsed edge (special edge), there are two triangles with the same vertices; these triangles are folded along the special edge into one triangle.",
"Further there are three pairs of rectangles with the same vertices; they are also identified.",
"Consequently the torus $T$ is folded along the curve $C$ to be a Möbius strip, which turns out to be the same as the one in the previous subsection (shown as Möbius 6).",
"In this way, we get a space $X_6$ from the solid torus $ST$ made by six prisms by folding the torus $T$ (collapsing $UT$ ).",
"Since the curve $C$ is unknotted, this folding is done exactly the same as in the previous section.",
"Then $X_6$ is homeomorphic to $\\mathbb {S}^3$ .",
"A model.",
"Put three vertices with labels without numeral 6: $1245, 2345, 1234$ inside the prism $(6123,5612,6235)*(6134,6145,6345)$ .",
"The Möbius strip above is shown in Figure REF ([2]).",
"From this picture, one can hardly see the $\\mathbb {Z}_6$ action.",
"Figure: Five prisms pack a prism.",
"Möbius strip made by six triangles and three rectangles is shown(Moe6)"
],
[
"$6-1=5$",
"If we remove the sixth hyperplane, then the remaining five hyperplanes bounds a 4-simplex.",
"This process can be described as follows: the prism on the sixth plane reduces to a segment.",
"More precisely, the two triangles of the boundary of the sixth prism reduce to two points, and the prism reduces to a segment connecting these two points.",
"Combinatorial explanation: The prism on the sixth plane with vertices $\\scriptsize \\begin{array}{ccccc}1256&-&-&-&1456\\\\|&{\\backslash }\\qquad &&\\qquad /&|\\\\|&2356&-&3456&|\\\\|&/\\qquad &&\\qquad {\\backslash }&|\\\\1236&-&-&-&1346\\end{array}\\normalsize \\quad {\\rm on\\ 6}$ reduces ($1256,2356,1236\\rightarrow 1235,\\ 1456,3456,1346\\rightarrow 1345$ ) to the segment ${\\bf 1235\\ ---\\ 1345}.$ Note that $\\begin{array}{ll}\\lbrace 1,2,3,5\\rbrace &=\\lbrace 1,2,5,6\\rbrace \\cup \\lbrace 2,3,5,6\\rbrace \\cup \\lbrace 1,2,3,6\\rbrace -\\lbrace 6\\rbrace ,\\\\[2mm]\\lbrace 1,3,4,5\\rbrace &=\\lbrace 1,4,5,6\\rbrace \\cup \\lbrace 3,4,5,6\\rbrace \\cup \\lbrace 1,3,4,6\\rbrace -\\lbrace 6\\rbrace .\\end{array}$ Accordingly the other prisms reduce to tetrahedra, for example: $ \\begin{array}{cccl}\\scriptsize \\begin{array}{ccc}3412&---&3612\\\\|&{\\backslash }\\qquad \\qquad /&|\\\\|&4512-5612&|\\\\|&/\\qquad \\qquad {\\backslash }&|\\\\3452&---&3562\\end{array}\\normalsize &\\longrightarrow &\\scriptsize \\begin{array}{cl}3412&\\\\|&{\\backslash }\\qquad \\ {\\backslash }\\\\|&4512-{\\bf 1235}\\\\|&/\\qquad \\ /\\\\3452&\\end{array}\\normalsize &\\normalsize {\\rm on\\ 2} \\ ({\\rm similar\\ on}\\ 4),\\\\\\\\\\scriptsize \\begin{array}{ccc}6145&---&6345\\\\|&{\\backslash }\\qquad \\qquad /&|\\\\|&1245-2345&|\\\\|&/\\qquad \\qquad {\\backslash }&|\\\\6125&---&6235\\end{array}\\normalsize &\\longrightarrow &\\scriptsize \\begin{array}{c}{\\bf 1345}\\\\/\\quad |\\quad {\\backslash }\\\\1245-2345\\\\{\\backslash }\\quad |\\quad /\\\\{\\bf 1235}\\end{array}\\normalsize &{\\rm on\\ 5}\\ ({\\rm similar\\ on}\\ 1\\ {\\rm and}\\ 3).\\end{array}\\normalsize $ Remark.",
"Consider an $n$ -gon in the plane bounded by $n$ lines.",
"If you remove the $n$ -th line, then you get an $(n-1)$ -gon.",
"During this process, the $n-3$ sides away from the $n$ -th sides do not change.",
"Consider an $n$ -hedron in the 3-space bounded by $n$ planes.",
"If you remove the $n$ -th plane, you get an $(n-1)$ -hedron.",
"During this process, faces away from the $n$ -th face remain unchanged.",
"In the present case, all the prism-faces change into tetrahedra."
],
[
"$5+1=6$",
"If we add the sixth hyperplane to the five hyperplanes, then among the sixteen 4-simplices 4, some are untouched, some are divided into 4 and $3\\times 1$ (with section 3), and some are divided into $2\\times 2$ and $3\\times 1$ (with section $2\\times 1$ .)",
"We will describe the change of 4 into $2\\times 2$ via that of these boundaries: the 3-sphere tessellated by five tetrahedra into that tessellated by six prisms.",
"As a preparation, we would like to explain this cutting process by using a 1-dimension less model: we truncate a vertex of a polyhedron in 3-space.",
"Let us consider a polyhedron in the 3-space, say for example a tetrahedron bounded by four faces $\\lbrace 0,1,2,3\\rbrace $ , and cut the vertex $123=1\\cap 2\\cap 3$ by the new plane named 4.",
"The tetrahedron is truncated and is divided into a prism and a small tetrahedron; they share a triangle which should be called 4.",
"Instead of describing this process by 3-dimensional pictures, we express it by 2-dimensional pictures describing the change near the vertex 123: the plane is divided into three chambers $1,2,3$ (imagine the letter Y) sharing a unique point representing the vertex 123, which we call the center.",
"The cutting is expressed by a blowing-up at the center, which means inserting a triangle in place of the center point (see Figure REF , first line).",
"Though this is not so honest as 3-dimensional pictures, this is still a fairly honest way to describe the cutting process.",
"There is a way to make these pictures simpler without loosing any information: We watch only one chamber, say 3, and show the cutting/blowing-up of the center only in the chamber 3 (see Figure REF , second line, right).",
"You see that the terminal figure of marked triangle tells everything.",
"Figure: Blowing up a point to be a triangle (blowup2)Now we are ready.",
"Label the tetrahedra as $1,2,\\dots ,5$ as in the previous subsection.",
"We choose the edge $E$ joining two vertices 1235 and 1345.",
"Along this edge $E$ , there are three tetrahedra $1, 3$ and 5.",
"Other than these three, tetrahedron 2 touches the vertex 1235, and tetrahedron 4 touches the vertex 1345.",
"We cut the edge by a new hyperplane 6.",
"The tetrahedron 2 (resp.",
"4) has vertex 1235 (resp.",
"1345) as the intersection with $E$ ; this versex is cut, and the tetrahedron 2 becomes a prism (see Figure REF ).",
"Figure: Truncating a vertex of a tetrahedron (tetratoprism1)The tetrahedron 5 (as well as 1 and 3) has the edge $E$ ; this edge is cut, and the tetrahedron 5 becomes a prism (see Figures REF and REF ).",
"Figure: Truncating an edge of a tetrahedron (tetratoprism2)The new face 6 is a prism bounded by two triangles labeled 2 and 4, and three rectangles labeled 1, 3 and 5.",
"Recall the above 1-dimension lower process, then the blow-up of the edge $E$ to become the prism 6 can be summerized by Figure REF .",
"Figure: Blowing up a segment to be a prism (blowup56)In the case of seven or more hyperplanes, we describe the cutting/blow-up process only by this kind of figure."
],
[
"Seven hyperplanes in the 4-space",
"Seven hyperplanes in the projective 4-space cut out a unique chamber CC stable under the action of $\\mathbb {Z}_7$ .",
"It is bounded by the seven dumplings.",
"We study how the seven dumplings tessellate a 3-sphere."
],
[
"Labeling and a study of 2-skeleton",
"The central chamber CC has seven faces $D_1,\\dots ,D_7$ , which form the $\\mathbb {Z}_7$ -orbit of the dumpling $D_1={\\rm CC}\\cap H_1:\\quad \\scriptsize \\begin{array}{ccccccccc}& & & &1457& & & &\\\\& &/\\quad &4 &| & 5 &\\quad {\\backslash }& &\\\\1347& & & &1245& & & &1567\\\\&\\quad {\\backslash }& &/& &{\\backslash }& &/\\quad &\\\\\\quad {\\backslash }& 3 &1234& & & &1256& 6 &/\\quad \\\\&{\\backslash }\\quad &| & & 2 & & | & \\quad / &\\\\& &1237&-&-- & -&1267& &\\end{array}.$ Note that the pentagonal face of $D_1$ behind is $D_1\\cap D_7$ and the pentagon in front is $D_1\\cap D_2$ .",
"Using the notation introduced in [1], CC is represented by $+++++++$ , and its boundary consistes of $D_1=0++++++,\\quad D_2=-0++++,\\ \\dots ,\\ D7=------0.$ The boundary of $D_1$ consists of $\\begin{array}{lll}00+++++:{\\rm 5-gon}&0-0++++:{\\rm triangle}&0--0+++:{\\rm rectangle}\\\\0---0++:{\\rm rectangle}&0----0+:{\\rm triangle}&0-----0:{\\rm 5-gon}\\end{array}$ The 2-skeleton of CC consists of seven pentagons, seven triangles and seven rectangles.",
"Remove the intersections of the two consecutive ones: $D_1\\cap D_2,\\quad D_2\\cap D_3,\\quad \\dots ,\\quad D_7\\cap D_1:$ $\\scriptsize \\begin{array}{cccccccccc}& &2367-2356 & &\\cdots & &\\\\&\\qquad \\qquad / & &{\\backslash }\\qquad \\qquad /& &{\\backslash }\\qquad \\qquad & \\\\&7123 & 23 &2345 & \\cdots &6712 &\\\\&/\\qquad \\qquad {\\backslash }& &/\\qquad \\qquad {\\backslash }& &/\\qquad \\qquad {\\backslash }&\\\\6712 & 12 &1234 & 34 &\\cdots &71 &7123 \\\\\\qquad {\\backslash }& &/\\qquad \\qquad {\\backslash }& &/\\qquad &{\\backslash }\\qquad \\qquad \\qquad /& \\\\&1256-1245 & &3471-3467 & &1457-1347&\\end{array}\\normalsize $ (Note that the union of these seven pentagons form a Möbius strip, but for a while just forget it.)",
"Then the remaining seven triangles and seven rectangles form a Möbius strip $M:\\quad \\scriptsize \\begin{array}{ccccccccccc }6712&-&7123&-&1234&-&2345&-&3456&-&4567\\\\\\quad {\\backslash }&27 & /\\quad {\\backslash }&31&/\\quad {\\backslash }&42&/\\quad {\\backslash }&53&/\\quad {\\backslash }&64&/\\quad \\\\&2367&&3471& &1245& &2356& &3467&\\\\&\\quad {\\backslash }&37 & /\\quad {\\backslash }&41&/\\quad {\\backslash }&52&/\\quad {\\backslash }&63&/\\quad & \\\\& &6734&&7145& &5612& &2367&&\\\\&&\\quad {\\backslash }&47 & /\\ 57\\ {\\backslash }&51&/\\ 61\\ {\\backslash }&62&/\\quad && \\\\&&&4567&-&5671&-&6712&&&\\end{array}\\normalsize $ Möbius 7 Its boundary $C=\\partial M:\\quad 1234-2345-3456-4567-5671-6712-7123-1234$ is an unknotted circle because $C$ bounds also the disc $\\scriptsize \\begin{array}{cccccccc}7123&-&-&1234&-&-&2345\\\\[2mm]|&{\\backslash }&13&/\\qquad {\\backslash }&24&/&|\\\\[2mm]|&&3471&14&1245&&|\\\\[2mm]|&&&{\\backslash }\\qquad /&&&|\\\\[2mm]6712&&71&7145&45&&3456\\\\[2mm]&{\\backslash }&&/\\quad 57\\quad {\\backslash }&&/&\\\\&&5671&-&4567&&&\\end{array}\\normalsize .$"
],
[
"From a solid torus",
"We start with a vertical pentagonal prism, horizontally sliced into seven thinner pentagonal prisms.",
"Its top and the bottom pentagons are identified after a $4\\pi /5$ rotation; this makes the vertical prism a solid torus with a tessellation of the lateral boundary into 35 rectangles.",
"Now collapse vertically seven diagonal rectangles.",
"This changes 21 rectangles (out of 35) into seven segments (forming a circle) and fourteen triangles.",
"Accordingly, the seven thinner pentagonal prisms into dumplings.",
"Next use each collapsed edge as a valley of each pair of triangles face-to-face in order to close the pair like a book, and further identify seven pairs of rectangles.",
"This identification changes the solid torus into a sphere tessellated by seven dumplings.",
"A vertical pentagonal prism horizontally sliced.",
"We show the above process by using the labels $1,2,3,4,5,6,7$ of the seven hyperplanes.",
"The boundary of the vertical pentagonal prism; each line, say 71 is a pentagon on the plane $7\\cap 1$ .",
"Two pentagons 71 and 12 sandwich a thin pentagonal prism.",
"6712 is the point $6\\cap 7\\cap 1\\cap 2$ .",
"$\\scriptsize \\begin{array}{rccccccccccc}71:&7126&-&7123&-&7134&-&7145&-&7156&-&7126\\\\&\\Vert &\\times &\\Vert &13&| &14&| &15&| &16&\\Vert \\\\12:&1267&-&1237&-&1234&-&1245&-&1256&-&1267\\\\&| &27 &\\Vert &\\times &\\Vert &24&| &25&| &26&|\\\\23:&2367&-&2371&-&3412&-&2345&-&2356&-&2367\\\\&| &37 &| &31&\\Vert &\\times &\\Vert &35&| &36&|\\\\34:&3467&-&3471&-&3412&-&3452&-&3456&-&3467\\\\&| &47 &| &41&| &42&\\Vert &\\times &\\Vert &46&|\\\\45:&4567&-&4571&-&4512&-&4523&-&4563&-&4567\\\\&\\Vert &57 &| &51&| &52&| &53&\\Vert &\\times &\\Vert \\\\56:&5674&-&5671&-&5612&-&5623&-&5634&-&5674\\\\&\\Vert &\\times &\\Vert &61&| &62 &| &63&| &64&\\Vert \\\\67:&6745&-&6715&-&6712&-&6723&-&6734&-&6745\\\\&| &75&\\Vert &\\times &\\Vert &72 &| &73&| &74&|\\\\71:&7145&-&7156&-&7126&-&7123&-&7134&-&7145\\\\&| &15 &| &16&\\Vert &\\times &\\Vert &13&| &14&|\\\\\\end{array}\\normalsize $ Table of piles 7 Collapsing rectangles.",
"If we identify two points with the same consecutive indices as sets, for exmaple, $7123=1237=2371$ but $2356\\ne 5623$ , you find seven rectangles with pairwise coincide vertices (marked by $\\times $ ), situated diagonally.",
"We collapse such rectangles vertically to a segment, each called a special edge.",
"Note that a special edge has two ends which are labeled by consecutive numbers: $\\lbrace 1234,2345\\rbrace ,\\lbrace 2345,3451\\rbrace ,\\dots $ Accordingly the remaining rectangles next to the special edges become triangles, and the seven thin rectangular prisms become dumplings.",
"But as a whole it still remains to be a cylinder.",
"Identifying the top and the bottom to get a solid torus $ST$ .",
"We identify the top pentagon and the bottom pentagon of the collapsed vertical prism according to the labels; note that we must twist the triangle by $4\\pi /5$ .",
"By the identification we get a solid torus, say $ST$ ; its meridean ${\\bf mer}$ is represented by the boundary of a(ny) pentagon, in Table of piles 7, say 71 (or 12, 23,...).",
"Note that it is still an abstract object.",
"The seven special edges now form a circle: $C: 1234-2345-3456-4561-5612-6712-7123-1234.$ We take a vertical line, in Table of piles 7, with a $4\\pi /5$ twist, say, ${\\bf par}: 7123 - 3471 - 4571 - 5671 - 7123$ as a parallel.",
"The torus $T$ and the solid torus $UT$ .",
"We consider in our space ($\\sim \\mathbb {S}^3$ ) a usual (unknotted) solid torus, say $UT$ , and think our solid torus $ST$ fills outside of $UT$ .",
"Let us see the boundary torus $T:=\\partial \\ ST\\ (=\\partial \\ UT)$ has par as the meridean, and mer as the parallel.",
"From the description of the vertical prism, we see fourteen triangles and fourteen rectangles tessellating the torus $T$ arranged in a hexagonal way (identify the opposite sides of the hexagon): $\\scriptsize \\begin{array}{ccccccccccc}6712&-&6723&-&6734&-&4567& & &&\\\\| &27{\\backslash }72&| &73&| &74&| &75{\\backslash }& &&\\\\2367&-&7123&-&7134&-&7145&-&5671&&\\\\| &37 &| &31{\\backslash }13&| & 14&| &15 &| &16{\\backslash }&\\\\3467&-&3471&-&1234&-&1245&-&1256&-&6712\\\\| &47 &| &41 &| &42{\\backslash }24&| &25 &| & 26&|\\\\4567&-&4571&-&4512&-&2345&-&2356&-&2367\\\\&{\\backslash }57 &| &51 &| &52 &| &53{\\backslash }35&| &36 &|\\\\& &5671&-&5612&-&5623&-&3456&-&3467\\\\& & &{\\backslash }61&| &62 &| &63 &| &64{\\backslash }46&|\\\\& & & &6712&-&6723&-&6734&-&4567\\\\\\end{array}\\normalsize $ Torus (hexagon) 7 Collapsing the solid torus $UT$ by folding $T$ .",
"We identify two vertices with the same indices as sets.",
"Along each collapsed edge (special edge), there are two triangles with the same vertices; these triangles are folded along the special edge into one triangle.",
"Further there are seven pairs of rectangles with the same vertices; they are also identified.",
"Consequently the torus $T$ is folded along the curve $C$ to be a Möbius strip, which turns out to be the same as the one in the previous subsection (shown as Möbius 7).",
"In this way, we get a space $X_7$ from the solid torus $ST$ made by six prisms by folding the torus $T$ (collapsing $UT$ ).",
"Since the curve $C$ is unknotted, this folding is done exactly the same as in the previous section.",
"Then $X_7$ is homeomorphic to $\\mathbb {S}^3$ .",
"A model.",
"In [1], the tessellation is shown, as we quote in Figure REF .",
"In the figure, 0 stands for 7, and the edge connecting 0346 and 0134 is not drawn to avoid complication.",
"Six dumplings are shown; they are packed in a dumpling.",
"The other one is infinitely large.",
"From this diagram, one can hardly see the $\\mathbb {Z}_7$ action.",
"To understand this group action is the motivation of our prism-collapse construction.",
"Figure: Six dumplings pack a dumpling"
],
[
"$7-1=6$",
"If we remove the seventh hyperplane, then the remaining six hyperplanes bounds a hyperhexahedron.",
"This process can be described as follows: the dumpling on the seventh plane reduces to the union of two adjacent segments.",
"More precisely, the two triangles of the boundary of the dumpling reduce to two points, the segment joining the remaining two vertices to a point, and the dumpling to the union of two segments joining the last one to the former ones.",
"Combinatorial explanation: The dumpling on the seventh plane with vertices $\\scriptsize \\begin{array}{ccccc}&&3467&&\\\\&/&|&{\\backslash }&\\\\2367&&|&&4567\\\\|&{\\backslash }\\qquad &|&\\qquad /&|\\\\|&1267&---&1567&|\\\\|&/\\qquad &|&\\qquad {\\backslash }&|\\\\1237&&|&&1457\\\\&{\\backslash }&|&/&\\\\&&1347\\end{array}\\normalsize \\quad {\\rm on\\ 7}$ reduces to the union of two segments $\\bf 1236\\ --\\ 1346\\ --\\ 1456.$ Note that $\\begin{array}{ll}\\lbrace 1,2,3,6\\rbrace &=\\lbrace 2,3,6,7\\rbrace \\cup \\lbrace 1,2,6,7\\rbrace \\cup \\lbrace 1,2,3,7\\rbrace -\\lbrace 7\\rbrace ,\\\\[2mm]\\lbrace 1,3,4,6\\rbrace &=\\lbrace 1,3,4,7\\rbrace \\cup \\lbrace 3,4,6,7\\rbrace -\\lbrace 7\\rbrace ,\\\\[2mm]\\lbrace 1,4,5,6\\rbrace &=\\lbrace 4,5,6,7\\rbrace \\cup \\lbrace 1,5,6,7\\rbrace \\cup \\lbrace 1,4,5,7\\rbrace -\\lbrace 7\\rbrace .\\end{array}$ Accordingly the other dumplings reduce to prisms (cf.",
"Figure REF ), for example, $\\scriptsize \\begin{array}{ccc}&1245&\\\\&/\\qquad |\\qquad {\\backslash }&\\\\7145&|&2345\\\\|&{\\backslash }\\qquad |\\qquad /&|\\\\|&6745--6345&|\\\\|&/\\qquad |\\qquad {\\backslash }&|\\\\6715&|&6235\\\\&{\\backslash }\\qquad |\\qquad /&\\\\&6125&\\end{array}\\normalsize \\longrightarrow \\scriptsize \\begin{array}{ccc}&1245&\\\\&/\\qquad |\\qquad {\\backslash }&\\\\\\quad \\qquad /&|&2345\\\\\\qquad /&\\qquad |\\qquad /&|\\\\\\bf 1465&----6345&|\\\\\\qquad {\\backslash }&\\qquad |\\qquad {\\backslash }&|\\\\\\quad \\qquad {\\backslash }&|&6235\\\\&{\\backslash }\\qquad |\\qquad /&\\\\&6125&\\end{array}\\normalsize \\quad \\normalsize {\\rm on\\ 5} \\ ({\\rm similar\\ on}\\ 2),$ $\\scriptsize \\begin{array}{ccc}&6723&\\\\&/\\qquad |\\qquad {\\backslash }&\\\\5623&|&7123\\\\|&{\\backslash }\\qquad |\\qquad /&|\\\\|&4523--4123&|\\\\|&/\\qquad |\\qquad {\\backslash }&|\\\\4563&|&4713\\\\&{\\backslash }\\qquad |\\qquad /&\\\\&4673&\\end{array}\\normalsize \\longrightarrow \\scriptsize \\begin{array}{ccc}5623&----&\\bf 1236\\\\|&{\\backslash }\\qquad \\qquad /&|\\\\|&4523--4123&|\\\\|&/\\qquad \\qquad {\\backslash }&|\\\\4563&----&\\bf 1346\\\\\\end{array}\\normalsize \\quad {\\rm on\\ 3}\\ ({\\rm similar\\ on}\\ 4),$ $\\scriptsize \\begin{array}{ccc}&2356&\\\\&/\\qquad |\\qquad {\\backslash }&\\\\1256&|&3456\\\\|&{\\backslash }\\qquad |\\qquad /&|\\\\|&7156--7456&|\\\\|&/\\qquad |\\qquad {\\backslash }&|\\\\7126&|&7346\\\\&{\\backslash }\\qquad |\\qquad /&\\\\&7236&\\end{array}\\normalsize \\longrightarrow \\scriptsize \\begin{array}{ccccc}&&2356&&\\\\&/&/&{\\backslash }&\\\\1256&&&&3456\\\\|&{\\backslash }&&/&\\\\|&/\\quad &\\bf 1456&\\quad /&\\\\\\bf 1236&&|&&\\\\&{\\backslash }&|&/&\\\\&&\\bf 1346&&\\end{array}\\normalsize \\quad {\\rm on\\ 6}\\ ({\\rm similar\\ on}\\ 1).$"
],
[
"$6+1=7$",
"Let us describe the change of 3-sphere tessellated by six prisms into that by seven dumplings.",
"As we explained in §REF , we express this change – blowing-up a connected union of two edges (of a prism) – by cutting a prism 6 by a hyperplane 7 and dividing the prism into two dumplings, which share the pentagonal face that should be called 6.",
"(See Figure REF ).",
"Figure: Blowing up the union of two segments to be a dumpling (blowup67)"
],
[
"$m(\\ge 8)$ hyperplanes in the 4-space",
"A Veronese arrangement of $m(\\ge 8)$ hyperplanes in the projective 4-space cut out a unique chamber CC, stable under the action of $\\mathbb {Z}_m$ .",
"It is bounded by $m$ $(m-2)$ -dumplings.",
"(An $(m-2)$ -dumpling is bounded by two $(m-2)$ -gons, two triangles and $m-5$ rectangles.)",
"The reader is expected to imagine how these $(m-2)$ -dumplings tessellate a 3-sphere.",
"Here we show a 6-dumpling ($m=8$ ): $D_1={\\rm CC}\\cap H_1:\\quad \\scriptsize \\begin{array}{ccccccc}1458 &- & - &-& - & - &1568\\\\| & {\\backslash }& & 5 & & / &|\\\\|& & 1245 &--& 1256 & &|\\\\|&4& | & & |&6&|\\\\1348 & - &1234 & 2 & 1267& - &1678\\\\&{\\backslash }\\quad 3&| & & | &7\\quad / &\\\\& &1238&-- & 1278& &\\end{array}\\normalsize .$ The central chamber CC has $m$ faces $D_1,\\dots ,D_m$ , on which $\\mathbb {Z}_m$ acts as $D_k\\rightarrow D_{k+1}$ mod $m$ .",
"The 2-skeleton of CC consists of $m$ $(m-2)$ -gons, $m$ triangles and $m(m-5)/2$ rectangles.",
"Remove the intersections of the two consecutive ones: $D_1\\cap D_2,\\quad D_2\\cap D_3,\\quad \\dots ,\\quad D_m\\cap D_1.$ Then the remaining $m$ triangles and $m(m-5)/2$ rectangles form a Möbius strip $M$ , which the reader can make following the previous sections.",
"Its boundary $C=\\partial M:\\quad 1234-2345-\\cdots -m123-1234$ is an unknotted circle because $C$ bounds also a disc.",
"To find a disc is not so obvious so we give a hint when $m=8$ (the reader is expected to guess the codes for $A,B,C,D,E$ , and then to guess what happens when $m\\ge 9$ ) $\\scriptsize \\begin{array}{cccccccc}8123&-&1234&-&2345&-&3456\\\\[2mm]|&{\\backslash }&/\\quad \\quad {\\backslash }&&/\\quad \\quad {\\backslash }&/&|\\\\[2mm]|&\\qquad A&&B&&C\\qquad &|\\\\[2mm]|&&{\\backslash }\\quad \\quad /&&{\\backslash }\\quad \\quad /&&|\\\\[2mm]|&&D&15&E&&|\\\\[2mm]|&&&{\\backslash }\\qquad /&&&|\\\\[2mm]7812&&81&8156&56&&4567\\\\[2mm]&{\\backslash }&&/\\quad 68\\quad {\\backslash }&&/&\\\\&&6781&-&5678&&&\\end{array}\\normalsize .$ (Answer: $A=3481,B=1245,C=2356,D=8145,E=1256$ .)"
],
[
"From a solid torus",
"You start with a vertical $(m-2)$ -gonal prism, horizontally sliced into $m$ thinner $(m-2)$ -gonal prisms.",
"Its top and the bottom triangles are identified after a $4\\pi /(m-2)$ rotation.",
"This makes the vertical prism a solid torus; the boundary is tessellated by $m(m-2)$ rectangles.",
"Now collapse vertically $m$ diagonal rectangles.",
"This changes $3m$ rectangles into $m$ segments (special edges) and $2m$ triangles.",
"Accordingly, the $m$ thinner pentagonal prisms deform into $(m-2)$ -dumplings, forming a solid torus $ST$ .",
"The special edges make a circle $C$ .",
"We consider in our space ($\\sim \\mathbb {S}^3$ ) a usual solid torus, say $UT$ , and think our solid torus $ST$ fills outside of $UT$ .",
"In the tessellation on the boundary torus $T:=\\partial \\ ST= \\partial \\ UT,$ each special edge is shared by the two triangles with the same code.",
"The special edge is used as a valley to close the pair of triangles like a book, and further identify pairs of the rectangles.",
"This identification changes the torus $T$ into the Möbius strip $M$ with boundary $C$ , and accordingly collapse the solid torus $UT$ , and changes the solid torus $ST$ into a 3-sphere tessellated by $m$ $(m-2)$ -dumplings, recovering the tessellation on $\\partial \\ {\\rm CC}$ ."
],
[
"From $m$ hyperplanes to {{formula:5d2a49d0-05a3-4246-ae59-b65cc08dbf8b}} hyperplanes",
"Consider the central chamber bounded by $m$ hyperplanes.",
"If we remove the $m$ -th hyperplane, then $m-1$ hyperplanes bound the central chamber bounded by $m-1$ hyperplanes.",
"This process can be described as follows: the $(m-2)$ -dumpling on the $m$ -th plane reduces to $m-4$ connected segments.",
"Combinatorial explanation: The $(m-2)$ -dumpling on the $m$ -th plane with vertices $\\scriptsize \\begin{array}{ccccccc}&&\\bullet &-\\ \\bullet \\ \\cdots \\ -\\cdots \\ \\bullet \\ -&\\bullet &&\\\\\\bullet &&|&&|&&\\bullet \\\\|&\\bullet &-&--\\cdots \\ -\\cdots \\ --&-&\\bullet &|\\\\\\bullet &&|&&|&&\\bullet \\\\&&\\bullet &-\\ \\bullet \\ \\cdots \\ -\\cdots \\ \\bullet \\ -&\\bullet &&\\end{array}\\normalsize $ reduces to the union of $m-4$ segments $\\bullet \\ -\\ \\ \\bullet \\ \\ -\\ \\ \\bullet \\ \\cdots \\ -\\cdots \\ \\ \\bullet \\ -\\ \\ \\bullet \\ -\\ \\ \\bullet .$ Accordingly, the other $(m-2)$ -dumplings reduce to $(m-3)$ -dumplings."
],
[
"From $m$ hyperplanes to {{formula:253764bf-9ad7-494f-b641-9a4a981cc47b}} hyperplanes",
"Let us describe the change of 3-sphere tessellated by $m$ $(m-2)$ -dumplings into that by $m+1$ $(m-1)$ -dumplings.",
"As we explained in §REF , we express this change – blowing-up a connected union of $m-4$ edges (of a $(m-2)$ -dumpling) – by cutting the $(m-2)$ -dumpling (labeled by $m$ ) by the hyperplane (labeled by $m+1$ ) and dividing the $(m-2)$ -dumpling into two $(m-1)$ -dumplings, which share the $(m-1)$ -gonal face which should be labeled by $m$ .",
"(See Figure REF ).",
"Figure: Blowing up the union of m-4m-4 segments to be a (m-1)(m-1)-dumpling (blowup89)"
],
[
"Appendix: Higher dimensional cases",
"We describe higher dimensional dumplings $D_n$ for odd $n$ , and central chambers CC$_n$ for even $n$ in arrangements of $n+3$ hyperplanes in $\\mathbb {P}^n$ .",
"Description will be just combinatorial.",
"We start from dimension 0: 0-dimensional central chamber CC$_0$ is a point."
],
[
"Review",
"- D$_1$ : 1-dimensional dumpling is a segment, whose boundary consists of two CC$_0$ 's.",
"- CC$_2$ : The 2-dimensional central chamber CC$_2$ (usually called a pentagon) is bounded by five $D_1$ 's.",
"Each $D_1$ share CC$_0$ -faces with two $D_1$ 's.",
"- D$_3$ : Consider the direct product of CC$_2$ and an interval $I$ .",
"Choose a boundary component of CC$_2$ , and call it ${\\bf D_1}$ .",
"Push down ${\\rm \\bf D_1}\\times I $ to ${\\rm \\bf D_1}$ , to get a dumpling $D_3$ .",
"The crashed ${\\bf D_1}$ is called the special edge of $D_3$ .",
"Accordingly, for a boundary component $D_1$ adjacent to ${\\bf D_1}$ (there are two of them), the rectangle ${ D_1}\\times I $ is pushed to become a triangle.",
"For a boundary component $D_1$ non-adjacent to ${\\bf D_1}$ (there are two of them), the rectangle ${ D_1}\\times I $ leaves as it is.",
"Thus the boundary of a dumpling consists of two CC$_2$ 's and two triangles and two quadrilaterals (The following notation is explained in [1]): $\\begin{array}{lll}\\partial (++++++)&=0+++++ &\\cdots \\ \\rm CC_2\\\\&\\cup \\ -0++++ &\\cdots \\ \\rm triangle \\\\&\\cup \\ --0+++ &\\cdots \\ \\rm quadrilateral\\\\& \\cup \\ ---0++ &\\cdots \\ \\rm quadrilateral\\\\&\\cup \\ ----0+ &\\cdots \\ \\rm triangle\\\\& \\cup \\ -----0 &\\cdots \\ \\rm CC_2\\end{array}\\normalsize $ - CC$_4$ : The 4-dimensional central chamber CC$_4$ is bounded by seven $D_3$ 's.",
"Each $D_3$ shares CC$_2$ -faces with two $D_3$ 's.",
"The 2-skeleton of the $D_3$ -tessellation of $\\mathbb {S}^3$ minus the seven CC$_2$ 's form a Möbius strip $M_2$ with boundary $C_1$ consisting of seven special edges $D_1$ 's of seven $D_3$ 's.",
"On the other hand, we start with a vertical prism with base CC$_2$ , horizontally sliced into seven thinner prisms with base CC$_2$ .",
"Collapse the seven diagonal $D_1\\times I$ 's on the boundary into seven $\\bf D_1$ 's; this makes each thinner prism a $D_3$ with the said $\\bf D_1$ as special one.",
"Identify the top and the bottom CC$_2$ after $4\\pi /5$ rotation so that the seven special edges $\\bf D_1$ form a circle $C_1$ .",
"In this way, we get a solid torus $ST_3$ made of seven $D_3$ 's.",
"The (tessellated) boundary torus $T_2$ of $ST_3$ is just the double of the (tessellated) Möbius strip $M_2$ branching along $C_1$ .",
"Fold the boundary torus $T_2$ along the curve $C_1$ to $M_2$ , Then the solid torus $ST_3$ changes into the tessellated boundary of CC$_4$ ."
],
[
"$D_{2k-1},D_{2k}$ {{formula:b19d1be3-9a0d-4c68-b418-f4cddc002b60}}",
"- D$_{2k-1}$ : Consider the direct product of CC$_{2k-2}$ and an interval $I$ .",
"Choose a boundary component $D_{2k-3}$ of CC$_{2k-2}$ , and call it ${\\bf D_{2k-3}}$ .",
"Push down ${\\bf D_{2k-3}}\\times I $ to ${\\bf D_{2k-3}}$ , to get a dumpling $D_{2k-1}$ .",
"The resulting $\\bf D_{2k-3}$ is called the special edge of $\\bf D_{2k-1}$ .",
"- CC$_{2k}$ : The $2k$ -dimensional central chamber CC$_{2k}$ is bounded by $2k+3$ dumplings $D_{2k-1}$ .",
"Each $D_{2k-1}$ shares CC$_{2k-2}$ -faces with two $D_{2k-1}$ 's.",
"The ($2k-2$ )-skeleton of the $D_{2k-1}$ -tessellation of $\\mathbb {S}^{2k-1}$ minus the $2k+3$ chambers CC$_{2k-2}$ form a CW-complex $M_{2k-2}$ with boundary $C_{2k-3}$ $(\\sim \\Delta _{2k-4}\\times \\mathbb {S}^1$ ) consisting of $2k+3$ special edges $\\bf D_{2k-3}$ of $2k+3$ dumplings $D_{2k-1}$ .",
"On the other hand, we start with a vertical prism with base CC$_{2k-2}$ , horizontally sliced into $2k+3$ thinner prisms with base CC$_{2k-2}$ .",
"Collapse the $2k+3$ diagonal $D_{2k-3}\\times I$ on the boundary into $2k+3$ dumplings $\\bf D_{2k-3}$ ; this makes each thinner prism a dumpling $D_{2k-1}$ with the said $\\bf D_{2k-3}$ as special edge.",
"Identify the top and the bottom CC$_{2k-2}$ after $4\\pi /(2k+1)$ rotation so that the $2k+3$ special edges $\\bf D_{2k-3}$ form $C_{2k-3}$ .",
"In this way, we get a solid torus $ST_{2k-1}\\ (\\sim \\Delta _{2k-2}\\times \\mathbb {S}^1)$ made of $2k+3$ dumplings $D_{2k-1}$ .",
"The (tessellated) boundary torus $T_{2k-2}\\ (\\sim \\mathbb {S}^{2k-3}\\times \\mathbb {S}^1)$ of $ST_{2k-1}$ can be folded along the solid ($2k-3$ )-torus $C_{2k-3}$ (as a hinge) to $M_{2k-2}$ .",
"Then the solid torus $ST_{2k-1}$ changes into the tessellated boundary of CC$_{2k}$ .",
"Acknowledgement: The last author is grateful to K. Cho for his help.",
"This collaboration started on 11 March 2011 , the very day of the Fukushima disaster in Japan.",
"Francois Apéry Université de Haute-Alsace 2, rue des Frères Lumière 68093 Mulhouse Cedex France [email protected] Bernard Morin villa Beausoleil 32, avenue de la Résistance 92370 Chaville France Masaaki Yoshida Kyushu University Nishi-ku, Fukuoka 819-0395 Japan [email protected]"
]
] | 1612.05434 |
[
[
"A point particle model of lightly bound skyrmions"
],
[
"Abstract A simple model of the dynamics of lightly bound skyrmions is developed in which skyrmions are replaced by point particles, each carrying an internal orientation.",
"The model accounts well for the static energy minimizers of baryon number $1\\leq B\\leq 8$ obtained by numerical simulation of the full field theory.",
"For $9\\leq B\\leq 23$, a large number of static solutions of the point particle model are found, all closely resembling size $B$ subsets of a face centred cubic lattice, with the particle orientations dictated by a simple colouring rule.",
"Rigid body quantization of these solutions is performed, and the spin and isospin of the corresponding ground states extracted.",
"As part of the quantization scheme, an algorithm to compute the symmetry group of an oriented point cloud, and to determine its corresponding Finkelstein-Rubinstein constraints, is devised."
],
[
"Introduction",
"The Skyrme model is an effective theory of nuclear physics in which nucleons emerge as topological solitons in a field whose small amplitude travelling waves represent pions.",
"It thus provides a unified treatment of both nucleons and the mesons which, in the Yukawa picture, are responsible for the strong nuclear forces between them.",
"While the Skyrme model has been superceded as a fundamental model of strong interactions by QCD, interest in the model revived once it was recognized to be a possible low energy reduction of QCD in the limit of large $N_c$ (number of colours) [17], [16], and much work has been conducted to extract phenomenological predictions about nuclei from standard versions of the model [11], [12], [10], [3].",
"Many of these predictions are in good qualitative agreement with experiment, and recent improvements in skyrmion quantization schemes offer hope of significant further improvement to come [6], [7].",
"One area in which standard versions of the model perform poorly, however, is that of nuclear binding energies: typically, classical skyrmions are much more tightly bound than the nuclei they are meant to represent (by a factor of 15 or so).",
"In recent years, no fewer than three variants of the model have been proposed which seek to remedy this problem.",
"In each case, the model is, by design, a small perturbation of a Skyrme model in which the binding energies vanish exactly.",
"Perhaps the most radical proposal, due to Sutcliffe and motivated by holography, couples the Skyrme field to an infinite tower of vector mesons [15].",
"Small but nonvanishing binding energies are (conjecturally) introduced by truncating this infinite tower at some high but finite level.",
"This proposal, while elegant, has so far not been amenable to detailed analysis.",
"A second proposal, due to Adam, Sanchez-Guillen and Wereszczynzki, starts with a model which is invariant under volume preserving diffeomorphisms of space, then perturbs it by mixing with a small fraction of the conventional Skyrme energy [1].",
"Skyrmions in this model have the attractive feature of being somewhat akin to liquid drops.",
"However, the large (in fact, infinite dimensional) symmetry group of the unperturbed model is extremely problematic for numerical simulations, and the shapes and symmetries of classical skyrmions, even for rather low baryon number ($B\\ge 3$ ) are, so far, not known in this model in the regime of realistically small binding energy [5].",
"In this paper we will study the third (and arguably least radical) proposal, originally due to one of us [8].",
"This amounts to making a nonstandard choice of potential term in the standard Skyrme lagrangian and, more importantly, radically shifting the weighting of the derivative terms from the quadratic to the quartic.",
"The resulting model is still amenable to numerical simulation, but its classical solutions are quite different from conventional skyrmions: the lowest energy Skyrme field of baryon number $B$ now resembles a loosely bound collection of $B$ spherically symmetric unit skyrmions, rather than a tightly bound object in which the skyrmions have merged and lost their individual identities.",
"In the terminology of [14], which studied a $(2+1)$ dimensional analogue of the model, skyrmions in this lightly bound Skyrme model prefer to hold themselves aloof from one another.",
"Numerical analysis reveals [5] that they also prefer to arrange themselves on the vertices of a face centred cubic spatial lattice, with internal orientations dictated by their lattice position.",
"This suggests that, unlike conventional skyrmions, lightly bound skyrmions can be modelled as point particles, each carrying an internal orientation, interacting with one another through some pairwise interaction potential whose minimum encourages them to sit at a fixed separation with their internal orientations correlated.",
"The aim of this paper is to derive such a simple point particle model, compare its predictions with numerical simulations of the full field theory, and use it to extract, via rigid body quantization, phenomenological predictions about nuclei with baryon number $2\\le B\\le 23$ .",
"A similar programme (minus quantization) for the $(2+1)$ dimensional analogue model was completed in [14].",
"As we shall see, the point particle model accounts almost flawlessly for static skyrmions with $1\\le B\\le 8$ , where comparison with simulations of the full field theory is available.",
"For $B\\ge 9$ , it predicts a rapid proliferation of nearly degenerate skyrmions as $B$ grows, all rather close to size $B$ subsets of the face centred cubic lattice.",
"In comparison with conventional skyrmions, these typically have rather little symmetry, and anisotropic mass distribution.",
"Determining the symmetries of these configurations is an interesting and important task, nonetheless, as they determine the Finkelstein-Rubinstein constraints on quantization.",
"Usually, symmetries of skyrmions are determined by ad hoc means: one looks at suitable pictures of the skyrmion, predicts a symmetry by eye, then checks it by operating on the numerical data.",
"By contrast, we will develop an algorithm which automatically computes the symmetry group of any point particle configuration.",
"This allows us to completely automate the rigid body quantization scheme.",
"The result is, as a phenomenological model of nuclei, moderately successful: rigid body ground states plausibly account for the lightest nucleus of baryon number $B$ for 12 of the 23 values considered.",
"Presumably this can be improved by replacing rigid body quantization by something more sophisticated.",
"The rest of the paper is structured as follows.",
"In section we review the lightly bound Skyrme model, focussing on its spin-isospin symmetry and associated inertia tensors.",
"In section we introduce the point particle model, then in section we describe a numerical scheme to find its energy minimizers, and present the results of this scheme.",
"In section we formulate the rigid body quantization of our classical energy minimizers, focussing particularly on the Finkelstein-Rubinstein constraints.",
"Some concluding remarks and possible future directions of development are presented in section ."
],
[
"The lightly bound Skyrme model",
"The field theory of interest is defined as follows.",
"There is a single Skyrme field $U:{\\mathbb {R}}^{3,1}\\rightarrow \\mathrm {SU}(2)$ , required to satisfy the boundary condition $U(t,\\mathbf {x})=1$ as $|\\mathbf {x}|\\rightarrow \\infty $ for all $t$ .",
"Such a field, if smooth, has at each $t$ , a well-defined integer valued topological charge $B = -\\frac{1}{24\\pi ^2}\\int _{{\\mathbb {R}}^3}\\epsilon _{ijk}\\mathrm {Tr}(R_iR_jR_k) {\\rm d}^3 x,$ the topological degree of the map $U(t,\\cdot ):{\\mathbb {R}}^3\\cup \\lbrace \\infty \\rbrace \\rightarrow \\mathrm {SU}(2)\\cong S^3$ .",
"Since the field is smooth, $B(t)$ is smooth and integer valued, hence automatically conserved.",
"Physically it is interpreted as the baryon number of the field $U$ .",
"The right invariant current associated with $U$ is $R_\\mu =(\\partial _\\mu U)U^\\dagger $ , in terms of which the lagrangian density is $\\mathcal {L} = \\frac{F_\\pi ^2}{16\\hbar } \\mathrm {Tr}(R_\\mu R^\\mu ) + \\frac{\\hbar }{32e^2} \\mathrm {Tr}([R_\\mu ,R_\\nu ],[R^\\mu ,R^\\nu ]) \\\\- \\frac{F_\\pi ^2 m_\\pi ^2}{8\\hbar ^3}\\mathrm {Tr}(1-U) - \\frac{F_\\pi ^4e^2\\alpha }{32(1-\\alpha )^2} ({\\textstyle \\frac{1}{2}}\\mathrm {Tr}(1-U))^4 .$ Here $F_\\pi $ is the pion decay constant, $m_\\pi $ the pion mass, and $e>0$ , $0\\le \\alpha <1$ are dimensionless parameters.",
"In [5] the following values were chosen for these parameters so that classical binding energies in the model are comparable with experimentally-measured nuclear binding energies:The value for $F_\\pi $ recorded here corrects a typographical error in [5] $F_\\pi = 36.1\\,\\mathrm {MeV},\\quad m_\\pi = 303\\,\\mathrm {MeV},\\quad e= 3.76,\\quad \\alpha =0.95.$ There is certainly room for improvement in this calibration: for example, obtaining the correct pion mass was not a priority in [5], and we expect that a more thorough analysis could result in a parameter set for which $m_\\pi $ is closer to its experimental value of 137MeV.",
"However, the aim in the present paper is not to fine-tune the parameters, but rather to study qualitative properties of static solutions, which we expect to be insensitive to details of the calibration.",
"It will be convenient to use $F_\\pi /4e\\sqrt{1-\\alpha }$ as a unit of energy and $2\\sqrt{1-\\alpha }/F_\\pi e$ as a unit of length; in these units the lagrangian takes the form $L=T-V$ , where $T &= \\int _{{\\mathbb {R}}^3}\\Big [ -\\frac{1}{2}(1-\\alpha )\\mathrm {Tr}(R_0R_0) - \\frac{1}{8}\\mathrm {Tr}([R_0,R_i][R_0,R_i]) \\Big ]{\\rm d}^3 x, \\\\\\nonumber V &= \\int _{{\\mathbb {R}}^3} \\Big [ (1-\\alpha )\\left( -\\frac{1}{2}\\mathrm {Tr}(R_iR_i) + m^2\\mathrm {Tr}(1-U) \\right) \\\\& \\qquad \\qquad - \\frac{1}{16}\\mathrm {Tr}([R_i,R_j][R_i,R_j]) + \\alpha ({\\textstyle \\frac{1}{2}}\\mathrm {Tr}(1-U))^4 \\Big ] {\\rm d}^3 x,$ and $m:=(2m_\\pi \\sqrt{1-\\alpha }/F_\\pi e)$ .",
"In the parameter set given above, $m=1.00$ .",
"Note that when $\\alpha =0$ , $L$ is the lagrangian of the conventional Skyrme model with pion mass, while for $\\alpha =1$ this is a completely unbound model [8]: there is a topological energy bound of the form $V\\ge \\mathrm {const}|B|$ , but this is attained only when $|B|\\le 1$ .",
"The first approximation to a nucleus containing $B$ nucleons is a static Skyrme field $U:{\\mathbb {R}}^3\\rightarrow \\mathrm {SU}(2)$ of degree $B$ which minimizes the potential energy $V$ .",
"Thus it is important to identify static classical energy minimizers.",
"These are referred to as skyrmions.",
"A better approximation to a nucleus is obtained by allowing solitons to carry spin and isospin.",
"The lagrangian is invariant under a left action of the group $G:=\\mathrm {SU}(2)_I\\times \\mathrm {SU}(2)_J$ , defined by $[(g,h)\\cdot U](t,\\mathbf {x}):= gU(t,h^{-1}\\mathbf {x}h) g^{-1}$ where we have identified physical space ${\\mathbb {R}}^3$ with the Lie algebra $\\mathfrak {su}(2)$ via $\\mathbf {x}\\cong ix^j\\sigma _j$ , $\\sigma _1,\\sigma _2,\\sigma _3$ being the Pauli matrices, to define the action of $h$ on $\\mathbf {x}$ .",
"Equivalently, $[(g,h)\\cdot U](t,\\mathbf {x}):= gU(t,R(h)^{-1}\\mathbf {x}) g^{-1}$ where $R(h)$ is the $SO(3)$ matrix with entries $R(h)_{ij}=\\frac{1}{2}\\mathrm {Tr}(h\\sigma _ih^{-1}\\sigma _j).$ The conserved quantities associated with these symmetries are isospin and spin.",
"We refer to transformations $g\\in \\mathrm {SU}(2)_I$ as isorotations, in analogy with rotations $h\\in \\mathrm {SU}(2)_J$ .",
"Every $\\omega \\in \\mathfrak {g}:=\\mathfrak {su}(2)_I\\oplus \\mathfrak {su}(2)_J$ defines a one-parameter subgroup $\\lbrace \\exp (t\\omega )\\: :\\: t\\in {\\mathbb {R}}\\rbrace $ of $G$ isomorphic to $S^1$ , whose action on a static skyrmion $U$ generates a rigidly isorotating and rotating skyrmion, $U_\\omega =\\exp (t\\omega )\\cdot U$ , of constant kinetic energy $T[U_\\omega ]$ .",
"The mapping $\\omega \\mapsto T[U_\\omega ]$ is a quadratic form on $\\mathfrak {g}$ , and hence defines a unique symmetric bilinear form $\\Lambda :\\mathfrak {g}\\times \\mathfrak {g}\\rightarrow {\\mathbb {R}}$ called the inertia tensor of the skyrmion $U$ .",
"By its definition, $\\Lambda $ vanishes on the subspace of $\\mathfrak {g}$ tangent to the isotropy group $G^U$ of $U$ (that is, the subgroup $G^U:=\\lbrace (g,h)\\in G\\: :\\: (g,h)\\cdot U=U\\rbrace <G$ which leaves $U$ unchanged).",
"If $G^U$ is discrete, as is the case for all the skyrmions studied in this paper except when $B=1$ , then $\\Lambda $ is a positive bilinear form, and thus defines a left invariant Riemannian metric on $G$ .",
"In order to identify spin and isospin quantum numbers of skyrmions corresponding to those of nuclei, isorotations and rotations needed to be treated quantum mechanically rather classically.",
"The inertia tensor plays an important role in the simplest quantization scheme, known as rigid body quantization, which will be reviewed in section , and amounts to quantizing geodesic motion on $(G,\\Lambda )$ , subject to certain symmetry constraints required to give skyrmions fermionic exchange statistics.",
"Clearly, by choosing a basis for $\\mathfrak {su}(2)$ , we obtain a basis for $\\mathfrak {g}$ which can be used to represent $\\Lambda $ as a real symmetric $6\\times 6$ matrix.",
"We shall consistently represent inertia tensors in this way, having chosen the basis $[-{\\textstyle \\frac{{\\rm i}}{2}}\\sigma _1,-{\\textstyle \\frac{{\\rm i}}{2}}\\sigma _2,-{\\textstyle \\frac{{\\rm i}}{2}}\\sigma _3]$ for $\\mathfrak {su}(2)$ ."
],
[
"The point particle model",
"Extensive numerical simulations reported in [5] showed that skyrmions in the lightly bound Skyrme model with $B>0$ invariably resemble collections of $B$ particles.",
"Encouraged by this observation, we have developed a point particle model in which a Skyrme field $U$ with baryon number $B$ is replaced by $B$ oriented point particles in ${\\mathbb {R}}^3$ .",
"To explain how the model is derived, we begin by recalling the structure of the simplest skyrmion, which has $B=1$ , and is of “hedgehog” form $U_H(\\mathbf {x}) = \\exp (f(r){\\rm i}\\sigma _j x_j/r),$ with $f(r)$ a real function satisfying $f(0)=\\pi $ , $f(r)\\rightarrow 0$ as $r\\rightarrow \\infty $ , and $r=|\\mathbf {x}|$ .",
"The profile function is determined by solving (numerically) the Euler-Lagrange equation for $V$ restricted to fields of hedgehog form, a certain nonlinear second order ODE for $f$ .",
"One finds that $U_H$ has total energy $M_H:=V[U_H]\\approx 87.49$ , and its energy density is monotonically decreasing with $r$ and concentrated around the origin.",
"The 1-skyrmion has a high degree of symmetry: if $g\\in \\mathrm {SU}(2)$ then $gU_H(R(g)^{-1}\\mathbf {x})g^{-1} = U_H(\\mathbf {x}).$ In other words, $G^{U_H}$ is the diagonal subgroup of $\\mathrm {SU}(2)_I\\times \\mathrm {SU}(2)_J$ .",
"This basic skyrmion can be moved and rotated using symmetries of the model.",
"A 1-skyrmion with position $\\mathbf {x}_0\\in {\\mathbb {R}}^3$ and orientation $q_0\\in \\mathrm {SU}(2)$ is given by $U(\\mathbf {x};\\mathbf {x}_0,q_0) = U_H(R(q_0)(\\mathbf {x}-\\mathbf {x}_0)).$ The energy-minimizers with $2\\le B\\le 8$ resemble superpositions of fields of this type [5].",
"More precisely, their energy densities are concentrated at $B$ well-separated points $\\mathbf {x}_1,\\ldots ,\\mathbf {x}_B$ , and near each such point $\\mathbf {x}_a$ the field $U$ is approximately of the above form for some $q_a$ .",
"These positions and orientations are the basic degrees of freedom in our point particle model, and will be allowed to depend on time $t$ .",
"The lagrangian for this point particle model takes the form $L_{pp}=\\sum _{a=1}^B \\left(\\frac{1}{2}M|\\dot{\\mathbf {x}}_a|^2 + \\frac{1}{2}L |\\dot{q}_a|^2\\right) - BM - V(\\mathbf {x}_1,\\ldots ,\\mathbf {x}_B,q_1,\\ldots ,q_B),$ where $|\\dot{q}|^2:=\\frac{1}{2}\\mathrm {Tr}(\\dot{q}\\dot{q}^\\dagger )$ and $V(\\mathbf {x}_1,\\ldots ,\\mathbf {x}_B,q_1,\\ldots ,q_B) = \\sum _{1\\le a<b\\le B} V_{int}(\\mathbf {x}_a,q_a,\\mathbf {x}_b,q_b),$ is an interaction potential.",
"The terms involving time derivatives of $\\mathbf {x}_a$ and $q_a$ represent the kinetic energy of a moving skyrmion.",
"Their coefficients could be deduced from the Skyrme model.",
"It is known that the 1-skyrmion has inertia tensor $ \\Lambda _H=L_H \\begin{pmatrix} \\mathrm {Id}_3 & -\\mathrm {Id}_3 \\\\ -\\mathrm {Id}_3 & \\mathrm {Id}_3 \\end{pmatrix}, $ where $ L_H = \\frac{16\\pi }{3} \\int _0^\\infty \\sin ^2f\\big ((1-\\alpha )r^2+r^2(f^{\\prime })^2+\\sin ^2f\\big ){\\rm d}r\\approx 53.49.",
"$ From this it follows that the kinetic energy of a rigidly rotating skyrmion should take the form $\\frac{1}{2}L_H |\\dot{q}_0|^2$ , suggesting that $L=L_H$ in the lagrangian (REF ).",
"Similarly, the kinetic energy of a 1-skyrmion moving with velocity $\\dot{\\mathbf {x}}_0$ is $\\frac{1}{2}M_H|\\dot{\\mathbf {x}}_0|^2$ , where $M_H\\approx 87.49$ is the potential energy of a static 1-skyrmion.",
"This suggests choosing $M=M_H$ in the lagrangian.",
"However, we have chosen to fix the coefficients by an alternative phenomenological method that will be explained in the next section."
],
[
"Symmetries of the interaction potential",
"The point particle model inherits an action of $G=\\mathrm {SU}(2)_I\\times \\mathrm {SU}(2)_J$ from the Skyrme model.",
"The action of $(g,h)\\in G$ on the field $U(\\mathbf {x};\\mathbf {x}_0,q_0)$ defined in equation (REF ) is $U(\\mathbf {x};\\mathbf {x}_0,q_0) &\\mapsto gU(R(h)^{-1}\\mathbf {x};\\mathbf {x}_0,q_0)g^{-1}\\\\& = gU_H(R(q_0)(R(h)^{-1}\\mathbf {x}-\\mathbf {x}_0))g^{-1} \\\\& = U_H(R(g)R(q_0)R(h)^{-1}(\\mathbf {x}-R(h)\\mathbf {x}_0)) \\\\& = U(\\mathbf {x};R(h)\\mathbf {x}_0,gq_0h^{-1}).$ Therefore the action of $(g,h)$ on a point particle configuration is $ (\\mathbf {x}_a,q_a)\\mapsto (R(h)\\mathbf {x}_a,gq_ah^{-1}),\\qquad a=1,\\ldots ,B.",
"$ The point particle lagrangian should be invariant under these transformations, and under translations $\\mathbf {x}_a\\mapsto \\mathbf {x}_a+\\mathbf {c}$ for $\\mathbf {c}\\in {\\mathbb {R}}^3$ .",
"It should be invariant under changes of the signs of any of the $q_a$ , because $U(\\mathbf {x};\\mathbf {x}_0,-q_0)=U(\\mathbf {x};\\mathbf {x}_0,q_0)$ .",
"It should also be invariant under permutations of the particles, because configurations of particles that are the same up to a re-ordering describe the same Skyrme field.",
"Finally, the Skyrme model is invariant under the inversion $U(\\mathbf {x})\\mapsto U(-\\mathbf {x})^{\\dagger },$ which is equivalent, for a field of the form (REF ), to $(\\mathbf {x}_0,q_0)\\mapsto (-\\mathbf {x},q_0)$ .",
"Hence, our point particle lagrangian should be invariant under $(\\mathbf {x}_a,q_a)\\mapsto (-\\mathbf {x}_a,q_a).$ The kinetic terms in (REF ) obviously have these symmetries.",
"Demanding that the potential (REF ) is also invariant imposes constraints on the function $V_{int}(\\mathbf {x}_1,q_1,\\mathbf {x}_2,q_2)$ which we now describe.",
"Translation symmetry implies that $V_{int}(\\mathbf {x}_1,q_1,\\mathbf {x}_2,q_2)$ depends on the positions of the skyrmions only through their relative position $\\mathbf {X}:=\\mathbf {x}_1-\\mathbf {x}_2$ .",
"Isorotation symmetry implies that it depends on $q_1,q_2$ only through the isorotation-invariant combination $Q=q_1^{-1}q_2$ .",
"Thus $ V_{int}(\\mathbf {x}_1,q_1,\\mathbf {x}_2,q_2) = V_{red}(\\mathbf {X},Q), $ for some function $V_{red}$ on ${\\mathbb {R}}^3\\backslash \\lbrace 0\\rbrace \\times \\mathrm {SU}(2)$ .",
"Invariance under $q_1\\mapsto -q_1$ implies $V_{red}(\\mathbf {X},-Q)=V_{red}(\\mathbf {X},Q),$ while rotational symmetry demands that $V_{red}(R(h)\\mathbf {X},hQh^{-1}) = V_{red}(\\mathbf {X},Q)\\quad \\forall h\\in \\mathrm {SU}(2)_J.$ A permutation $(\\mathbf {x}_1,q_1,\\mathbf {x}_2,q_2)\\mapsto (\\mathbf {x}_2,q_2,\\mathbf {x}_1,q_1)$ changes the sign of $\\mathbf {X}$ and inverts $Q$ , so permutation invariance implies that $V_{red}(-\\mathbf {X},Q^{-1}) = V(\\mathbf {X},Q).$ Finally, symmetry under inversion (REF ), implies $V_{red}(-\\mathbf {X},Q) = V_{red}(\\mathbf {X},Q).$ To proceed further, it is helpful to think of $V_{red}$ as a one-parameter family of real functions $V_\\rho $ on $S^2\\times {\\mathrm {SU}}(2)$ , parametrized by $\\rho :=|\\mathbf {X}|\\in (0,\\infty )$ .",
"We may expand each such function in a convenient basis for $L^2(S^2\\times {\\mathrm {SU}}(2))$ , for example, the basis of eigenfunctions of the Laplacian.",
"A natural truncation to finite dimensions is obtained by keeping only eigenfunctions up to a fixed finite eigenvalue.",
"The effect of this truncation is to exclude from $V_{red}$ terms with fast orientation dependence.",
"This motivates the following definition: for each $\\lambda $ in the spectrum of $\\Delta _{S^2\\times S^3}$ , let $E_\\lambda $ denote the corresponding eigenspace, and for any $\\mu \\ge 0$ , $F_\\mu =\\bigoplus _{\\lambda \\le \\mu }E_\\lambda .$ Let $C^\\infty _\\mu $ denote the space of smooth functions on $V:{\\mathbb {R}}^3\\backslash \\lbrace 0\\rbrace \\times {\\mathrm {SU}}(2)\\rightarrow {\\mathbb {R}}$ such that $V_\\rho \\in F_\\mu $ for all $\\rho $ .",
"Proposition 1 Let $\\mu \\in [0,20)$ and $V$ be a function in $C^\\infty _\\mu $ invariant under the symmetries (REF )-(REF ).",
"Then there exist functions $V_i:(0,\\infty )\\rightarrow {\\mathbb {R}}$ , $i=0,1,2$ , such that $V(\\mathbf {X},Q) = V_0(|\\mathbf {X}|) + V_1(|\\mathbf {X}|)\\mathrm {Tr}(R(Q)) + V_2(|\\mathbf {X}|) \\frac{\\mathbf {X}\\cdot R(Q)\\mathbf {X}}{|\\mathbf {X}|^2}.$ Recall that the eigenvalues of the Laplacian on $S^n$ are $\\lambda _d^{(n)}=d(d+n-1)$ , $d=0,1,2,\\ldots $ , and the corresponding eigenspaces, ${\\mathbb {E}}_d^{(n)}$ , are spanned by (the restrictions to $S^n\\subset {\\mathbb {R}}^{n+1}$ of) harmonic homogeneous polynomials in ${\\mathbb {R}}^{n+1}$ of degree $d$ [2].",
"It follows that the eigenvalues of $\\Delta _{S^2\\times S^3}$ are $\\lambda _d^{(2)}+\\lambda _{d^{\\prime }}^{(3)}$ with eigenspaces ${\\mathbb {E}}_{d}^{(2)}\\otimes {\\mathbb {E}}_{d^{\\prime }}^{(3)}$ .",
"By (REF ), (REF ), $V$ is invariant under both $\\mathbf {X}\\mapsto -\\mathbf {X}$ and $Q\\mapsto -Q$ , so we may restrict $d$ and $d^{\\prime }$ to only even values (homogeneous polynomials of odd degree are parity odd).",
"Further, since $V\\in C^\\infty _\\mu $ with $\\mu <20$ , each restriction $V_\\rho $ lies in $E_0\\oplus E_6\\oplus E_8\\oplus E_{14}=({\\mathbb {E}}_0^{(2)}\\otimes {\\mathbb {E}}_0^{(3)})\\oplus ({\\mathbb {E}}_2^{(2)}\\otimes {\\mathbb {E}}_0^{(3)})\\oplus ({\\mathbb {E}}_0^{(2)}\\otimes {\\mathbb {E}}_2^{(3)})\\oplus ({\\mathbb {E}}_2^{(2)}\\otimes {\\mathbb {E}}_2^{(3)}).$ Now ${\\mathrm {SU}}(2)$ acts on both ${\\mathbb {E}}^{(2)}_d$ (by rotations of $S^2$ ) and ${\\mathbb {E}}^{(3)}_{d^{\\prime }}$ (by conjugation on ${\\mathrm {SU}}(2)$ ), and, by(REF ), each $V_\\rho $ is invariant under the combined action.",
"In fact ${\\mathbb {E}}^{(2)}_d\\cong {\\mathbb {R}}^{2d+1}$ and carries the irreducible spin $d$ representation of ${\\mathrm {SU}}(2)$ , while ${\\mathbb {E}}^{(3)}_{d^{\\prime }}\\cong {\\mathbb {R}}^{d^{\\prime }+1}\\otimes {\\mathbb {R}}^{d^{\\prime }+1}$ where, for $d^{\\prime }=2\\ell $ , ${\\mathbb {R}}^{d^{\\prime }+1}$ carries the irreducible spin $\\ell $ representation of ${\\mathrm {SU}}(2)$ .",
"In particular, ${\\mathbb {E}}_0^{(3)}={\\mathbb {R}}$ , on which ${\\mathrm {SU}}(2)$ acts trivially, and ${\\mathbb {E}}_0^{(3)}$ decomposes into irreducible representations as ${\\mathbb {E}}_0^{(3)}={\\mathbb {R}}\\oplus {\\mathbb {R}}^3\\oplus {\\mathbb {R}}^5.$ Now the tensor product ${\\mathbb {R}}^{2d+1}\\otimes {\\mathbb {R}}^{2\\ell +1}$ contains no trivial subrepresentation if $d\\ne \\ell $ , and exactly one if $d=\\ell $ .",
"Hence, of the summands in (REF ), $E_0$ , $E_8$ and $E_{14}$ each contain a one-dimensional subspace on which ${\\mathrm {SU}}(2)$ acts trivially (while $E_6$ does not) and, by (REF ), $V_\\rho $ lies in the three-dimensional space spanned by these.",
"Clearly $E_0^{triv}=E_0$ which is spanned by the constant function $(\\mathbf {X},Q)\\mapsto 1$ .",
"Consider the functions $(\\mathbf {X},Q)\\mapsto \\mathrm {Tr}(Q),\\qquad (\\mathbf {X},Q)\\mapsto \\mathbf {X}\\cdot R(Q)\\mathbf {X}-\\frac{1}{2}\\mathrm {Tr}R(Q)|\\mathbf {X}|^2.$ These are manifestly ${\\mathrm {SU}}(2)$ invariant and extend to homogeneous polynomials on ${\\mathbb {R}}^3\\times {\\mathbb {R}}^4$ of bidegree $(0,2)$ and $(2,2)$ respectively.",
"Furthermore, one may readily check that these polynomials are harmonic (separately with respect to $\\mathbf {X}$ and $Q$ ).",
"Hence, they span $E_8^{triv}$ and $E_{14}^{triv}$ respectively.",
"Noting that $|\\mathbf {X}|^2\\equiv 1$ on $S^2$ , the claim follows.",
"From now on, we assume that $V_{red}$ lies in the truncated function space $C^\\infty _{14}$ , so that it has the structure prescribed by Proposition REF .",
"Recall that, in the standard Skyrme model, the interaction potential for well separated skyrmion pairs can be modelled using the dipole formalism [13]: far from its centre, a unit skyrmion looks like the field induced in the linearization of the Skyrme model about the vacuum, $U=1$ , by an orthogonal triplet of scalar dipoles placed at the skyrmion's centre.",
"The interaction potential for a skyrmion pair with relative position $\\mathbf {X}$ and orientation $Q$ can then be approximated by the interaction energy of a pair of triplets of dipoles held at relative displacement $\\mathbf {X}$ and orientation $Q$ , interacting via the linear theory.",
"This approximation introduces another useful constant associated with the unit skyrmion, namely the strength of the (necessarily equal) dipoles.",
"In practice this is determined numerically by reading off a coefficient $C$ in the large $r$ asymptotics of the skyrmion profile function.",
"This formalism is readily adapted to the lightly bound Skyrme model, producing an interaction potential of the form (REF ) with $V_0(r) &=& 0 \\nonumber \\\\V_1(r) &=& -8\\pi C^2(1-\\alpha )\\left(\\frac{m}{r^2}+\\frac{1}{r^3}\\right)e^{-mr} \\nonumber \\\\V_2(r) &=& 8\\pi C^2(1-\\alpha )\\left(\\frac{m^2}{r}+\\frac{3m}{r^2}+\\frac{3}{r^3}\\right)e^{-mr}.$ The dipole strength (for $\\alpha =0.95$ and $m=1$ ) is found numerically to be $C\\approx 14.58$ .",
"These formulae reproduce the usual prediction of attractive and repulsive channels for well-separated skyrmions.",
"That is, $V_{red}$ is maximally attractive (increases fastest with $|\\mathbf {X}|$ ) if the orientations of the skyrmions differ by a rotation by $\\pi $ about any direction orthogonal to $\\mathbf {X}$ , is maximally repulsive if the orientations differ by a rotation by $\\pi $ about $\\mathbf {X}$ , and is nonmaximally repulsive if their orientations are equal.",
"We refer to these three situations as the attractive, repulsive and product channels respectively.",
"The existence of these three channels allows us to fix the functions $V_0,V_1,V_2$ numerically by conducting scattering simulations of skyrmion pairs in the full field theory, in similar fashion to Salmi and Sutcliffe's work on the $(2+1)$ dimensional model [14].",
"We begin with a Skyrme field of the form $U_a(x_1,x_2,x_3)=U_H(x_1+\\frac{s}{2},x_2,x_3)U_H(-(x_1-\\frac{s}{2}),-x_2,x_3)$ where $s>0$ is large and $U_H$ is a unit hedgehog skyrmion defined (numerically) in a ball of radius less than $s/2$ (so $U_H(\\mathbf {x})=1$ for all $|\\mathbf {x}|\\ge s/2$ , and the product above commutes).",
"Such a field represents a pair of skyrmions located at $\\mathbf {x}=(\\pm s/2,0,0)$ , that is, with separation $s$ , in the attractive channel.",
"Here, and henceforth, we define the skyrmion positions of a Skyrme field $U:{\\mathbb {R}}^3\\rightarrow {\\mathrm {SU}}(2)$ to be those points where $U=-1$ .",
"We now allow $U$ to evolve with time according to the dynamics defined by the lagrangian (REF ), using the fourth order spatial discretization employed by the energy minimization scheme of [5], and a fourth order Runge-Kutta scheme with fixed time step for the time evolution.",
"This numerical scheme conserved total energy $E=T+V$ to extremely high accuracy, $\\max _t \\frac{|E(t)-E(0)|}{E(0)}<2.4\\times 10^{-5},$ for all the dynamical processes presented here.",
"As the dipole model predicts, the skyrmions with these initial data slowly move towards one another, attain a minimum separation, then recede again.",
"By recording their separation $s(t)$ and potential energy $V(t)$ at each time step, we recover a numerical approximation to the attractive channel interaction potential which, according to (REF ) is related to $V_0,V_1,V_2$ by $V_a(s)=V_0(s)-V_1(s)-V_2(s).$ We then repeat the process with intial data $U_r(x_1,x_2,x_3)&=&U_H(x_1+\\frac{s}{2},x_2,x_3)U_H(-(x_1-\\frac{s}{2}),x_2,-x_3)\\\\U_p(x_1,x_2,x_3)&=&U_H(x_1+\\frac{s}{2},x_2,x_3)U_H(x_1-\\frac{s}{2},x_2,x_3)$ which are in the repulsive and product channels respectively.",
"To make the skyrmions approach one another and interact, we now Galilean boost them towards one another at low speed (v=0.1).",
"Note that the reflexion symmetries of the initial data trap these fields in their respective channels for all time.",
"From these numerical solutions we obtain numerical approximations to the repulsive and product channel interaction potentials, which are related to $V_0,V_1,V_2$ by $V_r(s) &=& V_0(s) -V_1(s)+V_2(s), \\\\V_p(s) &=& V_0(s) +3V_1(s) + V_2(s).$ It is clear that $V_a,V_r,V_p$ uniquely determine $V_0,V_1,V_2$ and hence, within the ansatz (REF ), $V_{int}$ .",
"Figure: Interaction energies of skyrmions pairs with separation ss in the attractive (blue), product (red) and repulsive (green) channels.",
"In each case the thick curve represents numerical data extracted from a scattering process, the thin curve is a fit to this, and the dashed curve is the interaction energy predicted by the dipole model.Graphs of $V_a,V_r,V_p$ , determined numerically as described above, are presented in figure REF .",
"These curves also show the potentials predicted by the dipole model (with dipole strength $C=14.58$ ).",
"Clearly, the dipole formulae (REF ) do not provide an accurate quantitative picture of skyrmion interactions in the lightly bound model at any separation where the interactions are not negligible.",
"This is, perhaps, not surprising, since the dipole formalism replaces the full field theory by terms originating only in the quadratic and pion mass potential terms of the lagrangian, and these are precisely the terms which are given very low weighting, $1-\\alpha $ , in the lightly bound regime.",
"The qualitative predictions of the dipole picture are reliable however: the interaction potentials appear to decay exponentially fast, and the three channels identified have the behaviour predicted (attractive, repulsive, more weakly repulsive).",
"For later use, it is convenient to have explicit functions which approximate the numerical data for $V_a,V_r,V_p$ .",
"For our purposes, it is important that these functions decay exponentially with $s$ and accurately fit the numerical data for $s\\ge s_0$ , where $s_0$ is somewhat smaller then the equilibrium separation defined by $V_a$ (that is, the separation at which $V_a$ is minimal).",
"The behaviour for $s<s_0$ is not so important, provided the formulae introduce a repulsive core interaction, and is, in any case, inaccessible to our numerical scheme (since close approach of lightly bound skyrmions is forbidden in low energy scattering processes).",
"Figure REF also depicts the following fit functions $\\begin{aligned}V_a(s) &= {\\left\\lbrace \\begin{array}{ll} \\frac{7.7479-4.5997s+0.8297s^2-0.0473s^3}{1-0.4751s+0.0843s^2+0.0331s^3-0.0049s^4} & 0\\le s<7.096 \\\\ -94.6178\\frac{e^{-s}}{s} & s\\ge 7.096, \\end{array}\\right.}",
"\\\\V_r(s) &= \\left(\\frac{2476}{s}-\\frac{20322}{s^2}+\\frac{50254}{s^3}\\right)e^{-s},\\\\V_p(s) &= \\left(\\frac{2126}{s}-\\frac{18325}{s^2}+\\frac{47298}{s^3}\\right)e^{-s}.\\\\\\end{aligned}$ Of these, the most elaborate is $V_a$ , a Padé approximant on $[0,7.096]$ spliced to an exponentially decaying tail, the splice being chosen so that $V_a$ is continuously differentiable.",
"Unlike $V_r$ and $V_p$ , $V_a$ is well defined at $s=0$ , where it is chosen to equal the static energy of the axially symmetric $B=2$ solution (a saddle point of the Skyrme energy), obtained numerically by a different scheme, a choice made mainly for aesthetic reasons.",
"From now on, we choose $V_{red}$ to be the function defined by (REF ), where $V_0(s) &= \\frac{1}{2} V_a(s)+\\frac{1}{4}V_p(s)+\\frac{1}{4}V_r(s) \\\\V_1(s) &= \\frac{1}{4} V_p(s)-\\frac{1}{4} V_r(s) \\\\V_2(s) &= -\\frac{1}{2} V_a(s) + \\frac{1}{2} V_r(s)$ and $V_a,V_p,V_r$ are the functions defined in (REF ).",
"It is straightforward to show that this function $V_{red}$ is bounded below as, on physical grounds, it should be."
],
[
"The FCC lattice",
"We have seen that the interaction potential $V_{int}$ prefers particles to be in the attractive channel, i.e.",
"such that their relative orientation corresponds to a rotation about an axis perpendicular to their line of separation through angle $\\pi $ .",
"It is therefore desirable to find a way to pack them together such that all neighbouring pairs of particles are in the attractive channel.",
"The face-centred-cubic (FCC) lattice provides a solution to this problem.",
"The face-centred cubic lattice may be defined to be $ \\lbrace (n_1\\lambda ,n_2\\lambda ,n_3\\lambda )\\::\\:\\mathbf {n}\\in {\\mathbb {Z}}^3,\\,n_1+n_2+n_3 = 0\\mod {2} \\rbrace , $ with $\\lambda >0$ defining a lattice scale.",
"The underlying cubic lattice is given by points $(n_1\\lambda ,n_2\\lambda ,n_3\\lambda )$ for which $n_1,n_2,n_3$ are all even.",
"Those points for which some of the coordinates $n_i$ are odd lie on faces of the underlying cubic cells.",
"We assign orientations to these points as follows: those points on the vertices have orientation $1\\in \\mathrm {SU}(2)$ , those on faces perpendicular to the $x$ -axis have orientation $\\mathbf {i}$ , those on faces perpendicular to the $y$ -axis have orientation $\\mathbf {j}$ , and those perpendicular to the $z$ -axis have orientation $\\mathbf {k}$ .",
"Here we have implicitly identified elements $q\\in \\mathrm {SU}(2)$ with unit quaternions $q\\in \\mathbb {H}$ , such that $\\mathbf {i}=-{\\rm i}\\sigma _1$ , $\\mathbf {j}=-{\\rm i}\\sigma _2$ , $\\mathbf {k}=-{\\rm i}\\sigma _3$ and 1 is the identity matrix.",
"Put differently, the orientation $q$ of a particle at lattice site $(n_1,n_2,n_3)\\lambda $ is such that $R(q) = \\begin{pmatrix} (-1)^{n_1}&0&0\\\\0&(-1)^{n_2}&0\\\\0&0&(-1)^{n_3}\\end{pmatrix}.$ The reader may verify that any pair of nearest neighbours, separated by a distance $\\lambda \\sqrt{2}$ , is in the attractive channel.",
"One might expect that minimizers of the potential energy derived from (REF ) resemble subsets of the FCC lattice.",
"This was certainly true of all global minima of the Skyrme energy identified in [5], and all but one of the local minima."
],
[
"Inertia tensors",
"The point particle model (REF ) makes simple predictions for the inertia tensors of lightly bound skyrmions.",
"These are obtained by calculating the kinetic energy of a rotating and isorotating oriented point cloud.",
"Let $\\lbrace (\\mathbf {x}_a,q_a)\\rbrace $ be a minimizer of the potential energy derived from (REF ).",
"Choose any pair of angular velocities $(\\omega _I,\\omega _J)\\in \\mathfrak {su}(2)\\oplus \\mathfrak {su}(2)$ .",
"It is useful to identify each $\\omega \\in \\mathfrak {su}(2)$ with a vector $\\mbox{$\\omega $}\\in {\\mathbb {R}}^3$ by choosing $-{\\textstyle \\frac{{\\rm i}}{2}}\\sigma _j$ , $j=1,2,3$ , as a basis for $\\mathfrak {su}(2)$ (so $\\omega =-{\\textstyle \\frac{{\\rm i}}{2}} \\mbox{$\\omega $}\\cdot \\mbox{$\\sigma $}$ ).",
"Consider the following configuration, which is isorotating and rotating at constant angular velocity $\\omega =(\\omega _I,\\omega _J)$ : $(\\mathbf {x}_a(t),q_a(t)) &= \\exp (\\omega t)\\cdot (\\mathbf {x}_a,q_a) \\\\&= \\left( R(\\exp (\\omega _J t))\\mathbf {x}_a,\\, \\exp (\\omega _I t)q_a\\exp (-\\omega _J t) \\right).$ We find $\\dot{\\mathbf {x}}_a(0) &= \\mbox{$\\omega $}_J\\times \\mathbf {x}_a,\\\\\\dot{q}_a(0) &= \\omega _Iq_a-q_a\\omega _J=(\\omega _I-q_a\\omega _Jq_a^{-1})q_a,$ whence $|\\dot{\\mathbf {x}}_a(0)|^2 &= |\\mbox{$\\omega $}_J|^2|\\mathbf {x}_a|^2-(\\mbox{$\\omega $}_J\\cdot \\mathbf {x}_a)^2,\\\\|\\dot{q}_a(0)|^2 &= \\frac{1}{2}\\mathrm {Tr}[(\\omega _I-q_a\\omega _Jq_a^{-1})q_a\\, q_a^\\dagger (\\omega _I^\\dagger -q_a\\omega _J^\\dagger q_a^{-1})]=|\\mbox{$\\omega $}_I|^2-2\\mbox{$\\omega $}_I\\cdot R(q_a)\\mbox{$\\omega $}_J+|\\mbox{$\\omega $}_J|^2.$ Therefore the kinetic energy is $\\frac{1}{2}\\sum _{a=1}^B \\left( M|\\dot{\\mathbf {x}}_a|^2+L|\\dot{q}_a|^2 \\right) = \\left(\\begin{array}{cc} \\mbox{$\\omega $}_I & \\mbox{$\\omega $}_J \\end{array}\\right)\\Lambda \\left(\\begin{array}{c} \\mbox{$\\omega $}_I \\\\ \\mbox{$\\omega $}_J \\end{array}\\right),$ where the inertia tensor is $\\Lambda = \\sum _{a=1}^B \\left( M \\left(\\begin{array}{c|c} 0_3 & 0_3 \\\\ \\hline 0_3 & |\\mathbf {x}_a|^2\\mathrm {Id}_3-\\mathbf {x}_a\\mathbf {x}_a^T \\end{array}\\right)+ L \\left(\\begin{array}{c|c} \\mathrm {Id}_3 & -R(q_a) \\\\ \\hline -R(q_a)^T & \\mathrm {Id}_3 \\end{array}\\right) \\right).$ The point particle model predicts that this is a good approximation to the inertia tensor of a lightly bound degree $B$ skyrmion.",
"We will test this prediction in the next section."
],
[
"Light nuclei",
"Having introduced the point particle model for lightly bound skyrmions, in this section we present our results for energy-minimizing configurations of point particles.",
"We begin by discussing our results for eight particles or fewer, where comparison can be made with energy minima in the lightly bound Skyrme model found in [5].",
"We have developed an iterative zero-temperature annealing algorithm to minimize the energy of a configuration of particles.",
"We applied this algorithm both to randomly-chosen initial ensembles of particles and to initial ensembles that are subsets of the FCC lattice.",
"We ran a large number of simulations for each value of $B$ , typically obtaining several local energy minima, and record here only the lowest local minimum and up to two closest competitors.",
"Energies of these local minima with $2\\le B\\le 8$ are presented in table REF .",
"The particle ensembles themselves are depicted in figure REF .",
"The corresponding binding energies in the lightly bound Skyrme model are also recorded in the table.",
"These are defined to be the energy of the $B$ -skyrmion minus $B$ times the energy of the 1-skyrmion.",
"Table: Energies and numbers of bonds of the lowest-energy local minima in the point particle model, and energies of the corresponding lightly bound skyrmions (taken from , except those marked * {}^*, which result from new simulations conducted with the same numerical scheme).Figure: Local energies minimizers in the point particle model.",
"Each ball is centred on a point skyrmion position 𝐱 a \\mathbf {x}_a and its colour represents theinternal orientation q a q_a.",
"Each picture also depicts the FCC lattice configuration of size BB to which the minimizer best fits.Thick grey line segments indicate interskyrmion bonds no more than 10% longer than the FCC bond length, while thin magenta line segments show nearest neighbour bonds in thebest fit lattice configuration.",
"In most cases the fit is so good that the thin bonds are not visible.",
"They show quite clearly on 5(a), 5(b), 6(b), 6(c), 7(b) and 8(c) however.Our results are almost entirely consistent with the results obtained for the lightly bound Skyrme model in [5].",
"For $1\\le B\\le 5$ we obtained the same global minima as in the lightly bound Skyrme model.",
"For $B=6,7,8$ multiple local minima were previously obtained in the lightly bound Skyrme model.",
"All of these occured as local minima in the point particle model.",
"For $B=7,8$ the ordering of energies in the point particle also agreed with the ordering of energies in the lightly bound Skyrme model.",
"The only failure of the point particle model is for 6 particles: here the energies of the two lowest-energy local minima appear in the wrong order.",
"In addition to reproducing previously-known minimizers from the lightly bound Skyrme model our point particle model also predicted some new local minima.",
"Most interestingly, the global energy-minimizer in the point particle model for $B=7$ , labelled $7a$ in figure REF , did not correspond to any solution of the lightly bound Skyrme model found in [5].",
"Based on this discovery, we constructed an approximate Skyrme field with a similar shape to the point particle energy-minimizer, and minimized its energy using the same numerical scheme that was used in [5].",
"After relaxation this Skyrme field had a lower energy than any of the configurations discovered in [5], as predicted by the point particle model.",
"Thus we have a new candidate global energy minimizer at charge seven.",
"Similarly, new simulations find local energy minimizers in the lightly bound model of similar shape to $5b$ , $6c$ and $8c$ , and these have energies ordered exactly as the point particle model predicts (so $E_{8c}>E_{8b}>E_{8a}$ , for example).",
"In every case, the minimizers found look, to the naked eye, like subsets of the FCC lattice.All minimizers in this paper can be found at http://www1.maths.leeds.ac.uk/$\\sim $ pmtdgh/lightlybound It is an interesting problem to measure this property quantitatively.",
"Given an oriented point cloud $(X,Q)=(\\mathbf {x}_1,\\ldots ,\\mathbf {x}_B,q_1,\\ldots ,q_B)$ , we wish to identify the FCC subset of size $B$ which best approximates it.",
"To do this, we consider the orbit of $(X,Q)$ under the group $S$ of similitudes of ${\\mathbb {R}}^3$ , ${\\mathbb {R}}^3\\times (0,\\infty )\\times {\\mathrm {SU}}(2)\\ni (\\mathbf {c},\\lambda ,h):\\mathbf {x}\\mapsto R(h)\\frac{(\\mathbf {x}-\\mathbf {c})}{\\lambda }.$ For each $s\\in S$ , we define $d^2$ to be the squared distance from $s\\cdot X$ to the FCC lattice, i.e.",
"$d(s)^2=\\sum _{a=1}^B \\min \\lbrace |\\mathbf {x}_a-\\mathbf {n}|^2\\: :\\: \\mathbf {n}\\in {\\mathbb {Z}}^3,\\: n_1+n_2+n_3=0\\, \\mod {2}\\rbrace .$ Now, given a neighbouring triple of particles in $X$ (a particle $\\mathbf {x}$ , its nearest neighbour $\\mathbf {x}^{\\prime }$ and next-nearest neighbour $\\mathbf {x}^{\\prime \\prime }$ ), we construct a similitude $s_0$ which maps $\\mathbf {x}$ to 0, $\\mathbf {x}^{\\prime }$ to $(1,1,0)$ and $\\mathbf {x}^{\\prime \\prime }$ to the plane spanned by $(1,1,0)$ , $(0,1,1)$ .",
"We then solve the gradient flow equation of $d^2:S\\rightarrow {\\mathbb {R}}$ , with $s(0)=s_0$ , to find a local minimum of $d^2$ close to $s_0$ .",
"Repeating over all neighbouring triples, we keep the lowest local minimum $s_{min}$ of $d^2$ found (note that $d^2$ never has a global minimum since $s\\cdot X$ can be made arbitrarily close to $(0,0,0)$ by taking $\\lambda $ sufficiently large).",
"In this way we identify the closest FCC subset to $X$ and its root mean square distance from $X$ , namely $d_{RMS}=\\sqrt{d(s_{min})^2/B}$ .",
"Having found $s_{min}\\cdot X$ , the FCC colouring rule predicts the internal orientations $(q_1^{\\prime },\\ldots ,q_B^{\\prime })$ the particles should have.",
"These should be compared with $(q_1h_{min}^{-1},\\ldots ,q_B h_{min}^{-1})$ , bearing in mind that orientations are defined only up to sign, and that the system is isospin invariant.",
"Thus we minimize $d_{iso}^2:{\\mathrm {SU}}(2)\\rightarrow {\\mathbb {R}},\\qquad g\\mapsto \\sum _{a=1}^B \\min \\lbrace |gq_ah_{min}^{-1}- q_a^{\\prime }|^2 , |gq_ah_{min}^{-1}+ q_a^{\\prime }|^2\\rbrace $ over $g\\in {\\mathrm {SU}}(2)_I$ , again by gradient flow.",
"This gives us a measure of the root mean squared distance of the internal orientations of the configuration $(X,Q)$ from those imposed by the colouring rule applied to its closest FCC approximant, namely $d_{RMS}^{iso}=\\sqrt{d_{iso}^2(g_{min})/B}$ .",
"It also allows us to “coarse grain” the internal orientations, that is, map each $q_a$ to the element of $\\lbrace \\pm 1,\\pm \\mathbf {i},\\pm \\mathbf {j},\\pm \\mathbf {k}\\rbrace $ to which $gq_ah_{min}^{-1}$ is closest.",
"We used this method to determine the particle colours and FCC bonds in figure REF .",
"We will present graphs of $d_{RMS}$ and $d_{RMS}^{iso}$ in the next section.",
"In addition to comparing energies we have also compared inertia tensors in the point particle and lightly bound Skyrme models.",
"Under isorotations and rotations $(g,h)\\in \\mathrm {SU}(2)_I\\times \\mathrm {SU}(2)_J$ inertia tensors transform as $\\Lambda \\mapsto \\left(\\begin{array}{c|c} R(g) & 0 \\\\ \\hline 0 & R(h) \\end{array}\\right) \\Lambda \\left(\\begin{array}{c|c} R(g)^{-1} & 0 \\\\ \\hline 0 & R(h)^{-1} \\end{array}\\right).$ In comparing the inertia tensors of a charge $B$ skyrmion, obtained by solving the field theory, and a charge $B$ point particle energy minimizer, we must account for the fact that the orientations of these two objects are completely unrelated.",
"We do this by introducing a standard form for inertia tensors which fixes these symmetries.",
"We say that an inertia tensor $\\Lambda $ is in standard form if $\\Lambda = \\left(\\begin{array}{ccc|ccc}\\ast & \\ast & \\ast & \\mu _1 & \\nu _3 & \\nu _2 \\\\\\ast & \\ast & \\ast & 0 & \\mu _2 & \\nu _1 \\\\\\ast & \\ast & \\ast & 0 & 0 & \\mu _3 \\\\\\hline \\mu _1 & 0 & 0 & \\lambda _1 & 0 & 0 \\\\\\nu _3 & \\mu _2 & 0 & 0 & \\lambda _2 & 0 \\\\\\nu _2 & \\nu _1 & \\mu _3 & 0 & 0 & \\lambda _3\\end{array}\\right),$ where $\\lambda _1,\\lambda _2,\\lambda _3$ satisfy $|\\lambda _1-\\lambda _2|\\le |\\lambda _2-\\lambda _3|\\le |\\lambda _1-\\lambda _3|$ ; if $\\lambda _1\\ne \\lambda _2$ then $\\mu _1,\\mu _2,\\mu _3$ are either all non-negative or all non-positive and $\\nu _1,\\nu _2,\\nu _3$ are either all non-negative or all non-positive; and if $\\lambda _1=\\lambda _2$ then $\\nu _3=0$ , $|\\mu _1|>|\\mu _2|$ , and $\\mu _1,\\mu _2,\\mu _3,\\nu _1,\\nu _2$ are either all non-negative or all non-positive.",
"Any inertia tensor has a matrix of standard form in its ${\\mathrm {SU}}(2)_I\\times {\\mathrm {SU}}(2)_J$ orbit, and, in generic cases this matrix is unique.",
"Note that we have chosen not to define standard form as being a form in which both the upper-left and lower-right blocks of $\\Lambda $ are diagonal, even though such a form is arguably simpler than the one described above.",
"The reason is that the upper-left block of any inertia tensor obtained in the point particle model is proportional to the identity, so diagonalising the upper left block does not fix the isorotation symmetry.",
"We shall measure the distance between inertia tensors by the distance between their standard forms, using the usual Euclidean norm on the space of real matrices, that is $\\Vert \\Lambda \\Vert ^2:=\\mathrm {Tr}(\\Lambda ^T\\Lambda ).$ In table REF the distances between inertia tensors obtained in the point particle and lightly bound Skyrme models are recorded.",
"The errors recorded in the table are normalised by dividing through by $\\Vert \\Lambda \\Vert $ , where $\\Lambda $ is the lightly bound Skyrme model inertia tensor.",
"The configurations chosen in this comparison correspond to global energy minima in the lightly bound Skyrme model.",
"The values of $L$ and $M$ have been chosen to optimise the agreement between the two models, in other words, to minimize the sum over all chosen configurations of the distance between the lightly bound Skyrme and point particle inertia tensors.",
"The precise values are $ M = 93.09,\\quad L=54.30.",
"$ These are quite close to the values $M_H\\approx 87.49$ and $L_H\\approx 53.49$ obtained directly from the 1-skyrmion (REF ).",
"As with energies, agreement of inertia tensors is generally good (within 6%), with one exception at baryon number 6.",
"Table: Percentage error in inertia tensors calculated in the point particle model, as compared with the lightly bound Skyrme model"
],
[
"Heavier nuclei",
"When searching for local energy minima with large numbers of particles, one faces the problem that the number of connected subsets of the FCC lattice grows rapidly with the number of particles, and hence the number of candidate local minima of the energy grows rapidly.",
"We addressed this problem by seeking only local minima corresponding to FCC lattice subsets with a large number of bonds.",
"More precisely, we used as initial conditions in our relaxation algorithm only lattice subsets whose number of bonds is at most two less than the maximum possible for the given number of particles.",
"In the end we found that global energy minima always had at most one less than the maximum number of bonds, so this restriction seems reasonable.",
"Even with this simplification, the number of initial conditions to consider is large and it is difficult to be sure that enough simulations have been run to find the global energy minimizer.",
"To solve this problem we separated our minimization algorithm into two stages: in the first stage, a list of distinct lattice subsets is generated, and in the second stage these subsets are relaxed as before.",
"Our method for telling whether two lattice subsets are distinct is to compute their energy: if two lattice subsets have the same energy to high precision we assume that they are identical and discard one.",
"In doing so we run the risk that a lattice subset whose energy happens to coincide with another is wrongfully discarded.",
"For example, the initial FCC subsets used to generate solutions 6b and 6c have exactly the same spectrum of bonds, and hence exactly the same energy.",
"Only after relaxation away from the FCC lattice do their energies separate.",
"To mitigate against this danger we ran extensive simulations up to 16 particles starting from randomly chosen lattice subsets satisfying the bond number constraint; in all cases we obtained the same minimum energy as when we started with a list of distinct lattice subsets.",
"One distinct advantage of our method is that it makes it easy to identify not just the global energy-minimizer but also local energy minima.",
"Another is that it allows one to tell with reasonable confidence when sufficiently many lattice subsets have been sampled.",
"Throughout the procedure the number of occurences of each subset is recorded, and when all of these numbers are above a fixed minimum one may assume that all distinct lattice subsets have been found and terminate the algorithm.",
"In order to generate lattice subsets to use as initial conditions we developed a crystal-growing algorithm.",
"Again, this algorithm proceeds in two stages.",
"In the first stage a connected subset of the FCC lattice is generated iteratively.",
"This scheme starts with a lattice subset consisting of a single point.",
"At each step of the iteration a member of the lattice subset is chosen at random and one of its twelve nearest neighbours is chosen at random.",
"If the neighbour is not already a member of the lattice subset, it is appended, otherwise it is discarded.",
"This continues until the lattice subset has the required number of particles.",
"In the second stage of the algorithm the subset is modified so as to increase the number of bonds while maintaining a fixed number of particles.",
"At each step the algorithm chooses at random one member of the subset and a neighbour of another member.",
"If the neighbour is not already a member of the subset, and replacing the original member with this neighbour increases the number of bonds, the algorithm makes this replacement; otherwise, nothing happens.",
"This continues for a fixed number of steps.",
"At each step of the second stage the lattice subset is recorded, so running the algorithm once generates a large number of lattice subsets.",
"The crystal-growing algorithm was run repeatedly and distinct subsets with sufficiently many bonds saved until it was deemed that enough lattice subsets had been sampled, according to the above-defined criteria.",
"The maximum number of bonds and the number of crystals identified satisfying our criteria are recorded in table REF .",
"Table: Maximum number of bonds in an FCC lattice subset of given size, and the number of lattice subsets identified by our algorithm with at most two fewer bonds than the maximum.The output of our algorithm is recorded in table REF .",
"The total number of local energy minima found was huge; in this table we list all local minima whose energy is within 0.1 of the lowest energy found, together with configurations 5b, 6b and 6c.",
"For the most part, energy minimizers have the maximum number of bonds possible (exceptions in the table are marked by asterisks), and have the most even distribution of particle “colours\" (after coarse graining) possible.",
"A notable exception to both these rules is 23a, which has one less bond than maximal, and a rather uneven colour distribution (8,5,5,5) but is, nonetheless, the lowest energy $B=23$ configuration found.",
"This minimizer also has unusually high symmetry, as can be seen from figure REF , which also depicts the highly symmetric minimizers 10b and 19b.",
"One should note, however, that the point particle model does not always favour highly symmetric configurations.",
"The $B=13$ configuration, let us call it 13sym, obtained by augmenting a single point by all its nearest neighbours, for example, has the maximal number of bonds, but has energy $-8.556$ , which is much higher than the 13a.",
"It also has a very uneven colour distribution: 4,4,4,1.",
"So for $B=13$ , unlike $B=23$ , the model prefers to sacrifice symmetry in favour of uniform colour distribution.",
"These two charge 13 configurations are also depicted in figure REF .",
"Note that all particles in 13a are contained in just two planes of the FCC lattice, a feature it has in common with all global minimizers for $4\\le B\\le 15$ .",
"Table: The lowest energy local energy minima in the point particle model.",
"Asterisks in column 2 indicate that the configuration has one or two fewer bonds than the maximum bond number found for that particle number.",
"Column 3 indicates the number of particles of each internal orientation, after coarse-graining.",
"The classical energy is the potential VV of eq.",
"(), and the quantum energy is V+ΔV+\\Delta , where Δ\\Delta is defined in section .",
"The experiment columnidentifies the lightest nucleus for given baryon number BB if this nucleus has the spin and isospin predicted.Figure: Selected energy minimizers for 10≤B≤2310\\le B\\le 23.23a is a large octohedron with a tetrahedron glued to one face.",
"This unusually symmetric configuration is the global minimizer for B=23B=23, despite having less than maximal bond number and rather uneven colour distribution.",
"By contrast, the exceptionally symmetric configuration 13sym has much higher energy than 13a.",
"Also depicted are the local minimizers 10b and 19b, a large tetrahedron and octohedron respectively.The corresponding predictions for nuclear binding energies per nucleon, defined to be $-V_{int}/B$ , are plotted in figure REF .",
"Here, as in [5], energies in table REF have been converted to MeV by multiplying with 10.72.",
"The curve shows that ensembles of 4 and 16 particles have unusually high binding energies, in agreement with nuclear experiment, although these effects are less pronounced in the point particle model than in experiment.",
"The energy minimizers corresponding to these two peaks are particularly special: they both have tetrahedral symmetry.",
"Note that our binding energy curve lacks the peak seen at baryon number 12 in the nuclear binding energy curve.",
"Figure: Predictions for nuclear binding energies from the point particle model.In order to analyze the overall shape of the energy minimizers we have calculated for each its second moment matrix $M_{ij}$ , defined by $ M_{ij} := \\sum _{a=1}^N (x^i_a-x^i_0)(x^j_a-x^j_0),\\quad \\mathbf {x}_0:=\\frac{1}{N}\\sum _{a=1}^N \\mathbf {x}_a.",
"$ This matrix can be decomposed $M=M_1+M_0$ , where $M_1=\\frac{1}{3}\\mathrm {Tr}(M)\\mathrm {Id}_3$ and $M_0$ is traceless.",
"The trace part $M_1$ provides a measure of the size of the point cloud $\\lbrace \\mathbf {x}_a\\rbrace $ .",
"If the cloud is approximately round, then $M$ has nearly equal eigenvalues and the traceless part $M_0$ is close to zero.",
"Therefore $\\Vert M_0\\Vert $ provides a measure of anisotropy of the point cloud (recall that the norm of a matrix is defined in (REF )).",
"For a symmetric matrix such as $M$ , $\\Vert M\\Vert ^2=\\lambda _1^2+\\lambda _2^2+\\lambda _3^2$ , where $\\lambda _i$ are its eigenvalues.",
"Clearly, $\\Vert M_1\\Vert ^2=3\\overline{\\lambda }^2$ where $\\overline{\\lambda }=(\\lambda _1+\\lambda _2+\\lambda _3)/3$ , and the eigenvalues of $M_0$ are $\\lambda _i-\\overline{\\lambda }$ .",
"Hence $\\Vert M_0\\Vert ^2&=&(\\lambda _1-\\overline{\\lambda })^2+(\\lambda _2-\\overline{\\lambda })^2+(\\lambda _3-\\overline{\\lambda })^2\\nonumber \\\\&=&(\\lambda _1+\\lambda _2+\\lambda _3)^2-2(\\lambda _1\\lambda _2+\\lambda _2\\lambda _3+\\lambda _3\\lambda _1)-3\\overline{\\lambda }^2\\nonumber \\\\&=&6\\overline{\\lambda }^2-2(\\lambda _1\\lambda _2+\\lambda _2\\lambda _3+\\lambda _3\\lambda _1)\\le 6\\overline{\\lambda }^2=2\\Vert M_1\\Vert ^2$ since $M$ is positive definite.",
"In figure REF we have plotted $\\Vert M_0\\Vert $ against $\\Vert M_1\\Vert $ for the minimizers listed in table REF .",
"Overall there seems to be a downward trend in $\\Vert M_0\\Vert /\\Vert M_1\\Vert $ as $\\Vert M_1\\Vert $ increases, indicating that larger minimizers are closer to being round than small minimizers.",
"However, even for large nuclei the level of anisotropy is substantial.",
"The determinant $\\det (M_0)$ measures the qualitative nature of the anisotropy.",
"Let us order the eigenvalues of $M$ so that $\\lambda _1\\le \\lambda _2\\le \\lambda _3$ .",
"If the point cloud is long and thin then $\\lambda _1\\le \\lambda _2<\\overline{\\lambda }<\\lambda _3$ , so $M_0$ has two negative eigenvalues and one positive, whence $\\det (M_0)>0$ .",
"By contrast, if the point cloud is flat and round then $\\lambda _1<\\overline{\\lambda }<\\lambda _2\\le \\lambda _3$ , so $\\det (M_0)<0$ .",
"It is useful to define $\\mu _i=\\lambda _i-\\overline{\\lambda }$ , the eigenvalues of $M_0$ .",
"By extremizing the function $\\det M_0=\\mu _1\\mu _2\\mu _3$ on the circle obtained by intersecting the sphere of radius $\\Vert M_0\\Vert $ with the plane $\\mu _1+\\mu _2+\\mu _3=0$ , one finds that $-\\frac{1}{3\\sqrt{6}}\\Vert M_0\\Vert ^3 \\le \\det (M_0) \\le \\frac{1}{3\\sqrt{6}}\\Vert M_0\\Vert ^3,$ with equality precisely when two of the eigenvalues (of $M_0$ or, equivalently, $M$ ) coincide.",
"Figure REF also displays a plot of $\\@root 3 \\of {\\det (M_0)}$ against $\\Vert M_0\\Vert $ for the minimizers listed in table REF .",
"Interestingly, $\\det (M_0)$ is close to either its maximum or its minimum value in the majority of cases, indicating that both extremes of anisotropy are well-represented.",
"Figure: Graphs showing (a) the anisotropy of energy minimizers listed in table as a function of their size, and(b) the type of anisotropy of these energy minimizers.",
"The dashed lines represent the bounds on ∥M 0 ∥\\Vert M_0\\Vert and detM 0 \\det M_0 given by(), ().Just as for $2\\le B\\le 8$ , we can measure the distance of each local minimum found from its closest FCC lattice approximant, both in space and in internal (orientation) space, as defined in (REF ) and (REF ).",
"The results of this analysis are presented in figure REF .",
"With very few exceptions the minimizers match up very closely with FCC subsets, and their internal orientations are very close to the FCC prediction.",
"Note, however, that the optimal lattice scale varies quite significantly with $B$ (rightmost graph), so it is not a good approximation to fix this at the start (to match the FCC bond length to the optimal separation of a single skyrmion pair, for example) and minimize energy only over FCC subsets of that fixed scale.",
"Any attempt to proceed in this way always gets the relative energy ordering of local minima wrong for several values of $B$ .",
"Figure: Comparison of energy minimizers of the point particle model with subsets of the FCC lattice for baryon number 3≤B≤233\\le B\\le 23.",
"In each case we shift, scale and rotate the configuration until it matches, as closely as possible, a connected subset of the standard FCC lattice (with integer coordinates).",
"Plot (a) shows the root mean square distance of the particles in each transformed configuration from the FCC lattice, whileplot (b) shows the root mean square distance of their internal orientations from those predicted by the FCC colouring rule.",
"Plot (c) shows the scale factor used.",
"In each case, data corresponding to global energy minima (i.e.",
"configurations labelled “a\")are circled in red.",
"The dashed line in (c) marks the optimal lattice scale for B=2B=2."
],
[
"Rigid body quantization",
"Nuclei are inherently quantum-mechanical, so to make a direct comparison between skyrmions and nuclei it is necessary to include quantum effects in the Skyrme model.",
"Traditionally this is done semiclassically, treating the classical skyrmions as rigidly rotating and isorotating bodies.",
"The quantized wavefunction is required to satisfy the Finkelstein-Rubinstein constraints.",
"In practice these constraints restrict the spin and isospin quantum numbers of a quantized skyrmion.",
"For example, they guarantee that quantized skyrmions have either half-integer or integer spin and isospin according to whether the baryon number is odd or even.",
"In cases where solitons have symmetry they yield more nontrivial information.",
"In the present section we describe how to apply rigid body quantization and the Finkelstein-Rubinstein constraints in the point particle model.",
"The procedure reduces to a numerical algorithm which we have implemented and applied to the minima presented in the previous section."
],
[
"Finkelstein-Rubinstein constraints",
"We begin by recalling the definition of the Finkelstein-Rubinstein constraints (readers not interested in topological details could skip this subsection and continue reading at the start of the next subsection).",
"The classical configuration space of solitons with baryon number $B$ is the space $\\mathcal {S}_B$ of continuous maps $U:{\\mathbb {R}}^3\\rightarrow \\mathrm {SU}(2)$ of topological degree $B$ , satisfying the boundary condition $U(\\mathbf {x})\\rightarrow 1$ as $|\\mathbf {x}|\\rightarrow \\infty $ .",
"This space is topologically nontrivial: it contains non-contractible loops, so has a nontrivial fundamental group.",
"In fact, $\\pi _1(\\mathcal {S}_B)=\\mathbb {Z}_2$ for all $B\\in {\\mathbb {Z}}$ .",
"$\\mathcal {S}_B$ has a universal covering space $\\tilde{\\mathcal {S}}_B$ together with a two-to-one map $\\pi _{\\mathcal {S}}:\\tilde{\\mathcal {S}}_B\\rightarrow \\mathcal {S}_B$ , such that all loops in $\\tilde{\\mathcal {S}}_B$ are contractible and a loop in $\\mathcal {S}_B$ is contractible if and only if it can be lifted to a closed loop in $\\tilde{\\mathcal {S}}_B$ .",
"The soliton wavefunction is a function $\\Psi :\\tilde{\\mathcal {S}}_B\\rightarrow {\\mathbb {C}}$ .",
"The Finkelstein-Rubinstein constraint on $\\Psi $ states that for every pair $y$ and $y^{\\prime }$ of distinct points in $\\tilde{\\mathcal {S}}_B$ such that $\\pi _\\mathcal {S}(y)=\\pi _\\mathcal {S}(y^{\\prime })$ , $ \\Psi (y) = - \\Psi (y^{\\prime }).",
"$ A configuration of $B$ point particles consists of $B$ vectors $\\mathbf {x}_1,\\ldots \\mathbf {x}_B$ in ${\\mathbb {R}}^3$ and $B$ elements $q_1,\\ldots ,q_B\\in \\mathrm {SU}(2)$ .",
"The energy function disfavours vectors $\\mathbf {x}_a$ from being too close, so for practical purposes we may demand that their separations are greater than some fixed minimum $\\delta >0$ .",
"Therefore the naive configuration space for the point particle model is $\\tilde{\\mathcal {C}}_B &:= \\mathrm {SU}(2)^B \\times {\\mathbb {R}}^{3B\\ast } \\\\{\\mathbb {R}}^{3B\\ast }&:= \\big \\lbrace (\\mathbf {x}_1,\\ldots \\mathbf {x}_B)\\in ({\\mathbb {R}}^3)^B\\::\\:|\\mathbf {x}^a-\\mathbf {x}^b|>\\delta \\mbox{ whenever } a\\ne b \\big \\rbrace .$ The map to Skyrme configuration space is given by a so-called “relativised product ansatz” $\\hat{F}:(\\mathbf {x}_1,\\ldots ,\\mathbf {x}_B,q_1,\\ldots ,q_B)\\mapsto P\\left( \\frac{1}{N!",
"}\\sum _{\\sigma \\in \\Sigma _B} \\prod _{a=1}^B U(\\mathbf {x};\\mathbf {x}_{\\sigma (a)},q_{\\sigma (a)}) \\right).$ Here $U(\\mathbf {x};\\mathbf {x}_a,q_a)$ is defined in in (REF ), $\\Sigma _B$ is the group of permutations of the set $\\lbrace 1,\\ldots ,B\\rbrace $ , and $ P(U) = \\mathrm {Tr}(U^{\\dagger }U)^{-1/2} U.",
"$ Applying $P$ to a sum of products of matrices in $\\mathrm {SU}(2)$ yields an $\\mathrm {SU}(2)$ matrix, as long as the sum of products is everywhere nonvanishing.",
"The argument of $P$ in eq.",
"(REF ) is nonvanishing if the positions $\\mathbf {x}_a$ are sufficiently well-separated, so the right hand side is a map ${\\mathbb {R}}^3\\rightarrow \\mathrm {SU}(2)$ .",
"The map $\\hat{F}:\\tilde{\\mathcal {C}}_B\\rightarrow \\mathcal {S}_B$ in eq.",
"(REF ) is not injective since flipping the sign of any orientation $q_a$ , and permuting the particle labels, leave the associated Skyrme field unchanged.",
"To be precise, let $\\Sigma _B$ be the group of permutations of $\\lbrace 1,2,\\ldots ,B\\rbrace $ and ${\\mathbb {Z}}_2=\\lbrace 1,-1\\rbrace $ .",
"To each pair $(\\sigma ,s)\\in \\Sigma _B\\times ({\\mathbb {Z}}_2)^B$ , associate the map $\\Phi _{(\\sigma ,s)}:\\tilde{\\mathcal {C}}_B\\rightarrow \\tilde{\\mathcal {C}}_B,\\qquad (\\mathbf {x}_1,\\ldots ,\\mathbf {x}_B,q_1,\\ldots ,q_B)\\mapsto (x_{\\sigma (1)},\\ldots ,x_{\\sigma (B)},s_{\\sigma (1)}q_{\\sigma (1)},\\ldots ,s_{\\sigma (B)}q_{\\sigma (B)}).$ This defines a right action of $\\Sigma _B \\ltimes ({\\mathbb {Z}}_2)^B$ on $\\tilde{\\mathcal {C}}_B$ by homeomorphisms, where the semi-direct product carries group operation $(\\sigma ,{s})\\cdot (\\mu ,{t})=(\\sigma \\circ \\mu ,(s_1t_{\\sigma ^{-1}(1)},\\ldots ,s_Bt_{\\sigma ^{-1}(B)})).$ This action leaves $\\hat{F}$ invariant, that is, $\\hat{F}\\circ \\Phi _{(\\sigma ,s)}=\\hat{F}$ , so $\\hat{F}$ descends to a continuous map $F:\\mathcal {C}_B\\rightarrow \\mathcal {S}_B$ where $ \\mathcal {C}_B := \\tilde{\\mathcal {C}}_B/\\Sigma _B \\ltimes ({\\mathbb {Z}}_2)^B $ is the true point particle configuration space.",
"Since $\\tilde{\\mathcal {C}}_B$ is simply connected and the action is free, $\\tilde{\\mathcal {C}}_B$ is the universal cover of $\\mathcal {C}_B$ and $\\pi _1(\\mathcal {C}_B)\\cong \\Sigma _B\\ltimes ({\\mathbb {Z}}_2)^B$ .",
"Clearly, $\\hat{F}=F\\circ \\pi _\\mathcal {C}$ where $\\pi _\\mathcal {C}:\\tilde{\\mathcal {C}}_B\\rightarrow \\mathcal {C}_B$ is the canonical projection.",
"Choose any pair of points $x_0\\in \\tilde{\\mathcal {C}}_B$ , $y_0\\in \\tilde{\\mathcal {S}}_B$ such that $\\hat{F}(x_0)=\\pi _\\mathcal {S}(y_0)$ .",
"By a standard theorem of topology (see [9] for example), $\\hat{F}$ has a unique continuous lift $\\tilde{F}:\\tilde{\\mathcal {C}}_B\\rightarrow \\tilde{\\mathcal {S}}_B$ with $\\tilde{F}(x_0)=y_0$ .",
"The situation is summarized in the following commutative diagram $\\begin{xy}(0,20)*+{\\tilde{\\mathcal {C}}_B}=\"a\"; (20,20)*+{\\tilde{\\mathcal {S}}_B}=\"b\";(0,0)*+{\\mathcal {C}_B}=\"c\"; (20,0)*+{\\mathcal {S}_B}=\"d\";{\"a\";\"b\"}?",
"*!/_3mm/{\\tilde{F}};{@{->}_{\\pi _{\\mathcal {C}}} \"a\";\"c\"};{@{->}^{\\pi _{\\mathcal {S}}} \"b\";\"d\"};{\"c\";\"d\"}?",
"*!/_3mm/{{F}};{@{->} \"a\";\"d\"};?",
"*!/_3mm/{\\hat{F}};\\end{xy}.$ Note that $\\widetilde{F}$ is a lift of $F$ .",
"Any wavefunction $\\Psi :\\tilde{\\mathcal {S}}_B\\rightarrow {\\mathbb {C}}$ defines a wavefunction $\\psi =\\Psi \\circ \\tilde{F}$ on $\\tilde{\\mathcal {C}}_B$ , which must satisfy some nontrivial constraints derived from the Finkelstein-Rubinstein constraints: Proposition 2 If $x,x^{\\prime }$ are two points in $\\tilde{\\mathcal {C}}_B$ such that $\\pi _\\mathcal {C}(x)=\\pi _\\mathcal {C}(x^{\\prime })$ , then $ \\psi (x^{\\prime }) = \\mathrm {sgn}(\\sigma )\\prod _{a=1}^Bs_a\\, \\psi (x), $ where $(\\sigma ,s)\\in \\Sigma _B\\ltimes ({\\mathbb {Z}}_2)^B$ is the unique group element that maps $x$ to $x^{\\prime }$ .",
"This result follows almost directly from two important results of Finkelstein and Rubinstein [4].",
"First, if $x\\in \\tilde{\\mathcal {C}}_B$ , and $t^a\\in ({\\mathbb {Z}}_2)^B$ is the transformation that changes the sign of $q_a$ (only) and $\\alpha $ is a path in $\\tilde{\\mathcal {C}}_B$ from $x$ to $\\Phi _{({\\mathrm {Id}},t^a)}(x)$ then $F\\circ \\pi _C\\circ \\alpha $ is non-contractible.",
"Second, if $\\sigma \\in \\Sigma _B$ is a transposition and $\\beta $ is a path in $\\tilde{\\mathcal {C}}_B$ from $x$ to $\\Phi _{(\\sigma ,1)}(x)$ then $F\\circ \\pi _C\\circ \\alpha $ is also non-contractible.",
"Thus the constraint (REF ) implies that $ \\psi (\\Phi _{({\\mathrm {Id}},t^a)}(x)) = -\\psi (x)\\qquad \\mbox{and} \\qquad \\psi (\\Phi _{(\\sigma ,1)}(x))=-\\psi (x).",
"$ Now any element of $\\Sigma _B\\ltimes ({\\mathbb {Z}}_2)^B$ can be written as a product of sign flips and transpositions, so the claim follows."
],
[
"Rigid body quantization",
"In rigid body quantization, motion is restricted to the rotation-isorotation orbit of a fixed minimum $x=(\\mathbf {x}_1,\\ldots ,\\mathbf {x}_B,q_1,\\ldots ,q_B)$ of the classical energy.",
"Thus the classical configuration space is taken to be $G=\\mathrm {SU}(2)_I\\times \\mathrm {SU}(2)_J$ with each $(g,h)\\in G$ identified with $ (g,h)\\cdot (\\mathbf {x}_a,q_a) = (R(h)\\mathbf {x}_a,hq_ag^{-1})\\in \\tilde{\\mathcal {C}}_B.",
"$ The wavefunction $\\psi :G\\rightarrow {\\mathbb {C}}$ is required to solve a Schrödinger equation $\\hat{H}\\psi =E\\psi $ , where $\\hat{H}$ is (up to a constant factor) the Laplacian operator on $G$ associated with the left invariant metric $\\Lambda $ , the inertia tensor of $x$ .",
"In order to model a nucleus of definite spin and isospin one assumes that $\\psi $ is an eigenstate of the total isospin and spin operators with isospin $I$ and spin $J$ .",
"This is consistent with the Schrödinger equation because the hamiltonian commutes with these operators.",
"By the Peter-Weyl theorem, any such $\\psi $ is a finite sum of functions of the form $ \\psi (g,h) = \\langle w, \\rho _I(g)\\otimes \\rho _J(h) v\\rangle ,\\quad v,w\\in V_{I,J}:={\\mathbb {C}}^{2I+1}\\otimes {\\mathbb {C}}^{2J+1}, $ where for $\\ell \\in \\frac{1}{2}{\\mathbb {N}}$ , $\\rho _\\ell :\\mathrm {SU}(2)\\rightarrow \\mathrm {SU}(2\\ell +1)$ denotes the spin-$\\ell $ representation of $\\mathrm {SU}(2)$ .",
"For each $w\\in V_{I,J}$ denote by $V^{(w)}$ the subspace of functions with $w$ fixed.",
"Clearly $V^{(w)}\\cong V_{I,J}$ for all $w\\ne 0$ .",
"Furthermore, $\\hat{H}$ preserves $V^{(w)}$ , and its action on every $V^{(w\\ne 0)}$ is unitarily equivalent.",
"Hence we may, without loss of generality, fix $w\\ne 0$ , and represent $\\hat{H}$ by a linear operator $H_{I,J}$ on $V^{(w)}\\cong V_{I,J}$ .",
"To write this operator down explicitly, it is useful to introduce the usual basis for $\\mathfrak {g}=\\mathfrak {su}(2)_I\\oplus \\mathfrak {su}(2)_J$ , namely $K_i=-\\frac{{\\rm i}}{2}\\sigma _i\\oplus 0,\\qquad K_{i+3}=0\\oplus \\left(-\\frac{{\\rm i}}{2}\\sigma _i\\right),\\qquad i=1,2,3,$ with respect to which $\\Lambda $ is a symmetric $6\\times 6$ real matrix.",
"Denote its entries $\\Lambda _{ab}$ and those of its inverse $\\Lambda ^{ab}$ .",
"Then $\\hat{H}$ acts on $V^{(w)}\\cong V_{I,J}$ as $H_{I,J}:v\\mapsto -\\frac{\\hbar ^2}{2}\\Lambda ^{ab}\\rho ^\\ast _{I,J}(K_a)\\rho ^\\ast _{I,J}(K_b)v,$ where $\\rho ^{\\ast }_{I,J}:\\mathfrak {su}(2)_I\\oplus \\mathfrak {su}(2)_J\\rightarrow \\mathfrak {su}((2I+1)(2J+1))$ is the Lie algebra representation associated to $\\rho _I\\otimes \\rho _J$ : $ \\rho ^\\ast _{I,J}(K_a) = {\\left\\lbrace \\begin{array}{ll} \\rho ^\\ast _I(K_a)\\otimes \\mathrm {Id}_{2J+1} & a=1,2,3 \\\\ \\mathrm {Id}_{2I+1}\\otimes \\rho ^\\ast _J(K_a) & a=4,5,6.",
"\\end{array}\\right.}",
"$ Suppose that $(g_0,h_0)\\in \\mathrm {SU}(2)_I\\times \\mathrm {SU}(2)_S$ is a symmetry of $x$ , i.e.",
"that there exists a $(\\sigma ,s)\\in \\Sigma _B\\ltimes ({\\mathbb {Z}}_2)^B$ such that $ (R(h_0)\\mathbf {x}_a,h_0q_ag_0^{-1}) = (\\mathbf {x}_{\\sigma (a)},s_{\\sigma (a)}q_{\\sigma (a)}).",
"$ Then the Finkelstein-Rubinstein constraints described in the previous section imply that $ \\psi (gg_0,hh_0) = \\mathrm {sgn}(\\sigma )\\left(\\prod _{a=1}^Bs_a\\right)\\, \\psi (g,h)\\quad \\forall (g,h)\\in \\mathrm {SU}(2)_I\\times \\mathrm {SU}(2)_J.",
"$ This in turn implies that $\\rho _I(g_0)\\otimes \\rho _J(h_0) v &= \\chi (g_0,h_0)v, \\\\\\chi (g_0,h_0) &:= \\mathrm {sgn}(\\sigma )\\prod _{a=1}^Bs_a.$ Thus each element of the symmetry group of $x$ determines a linear constraint on $v$ , and $v$ therefore must belong to the subspace $V^{(x)}_{I,J}\\subseteq {\\mathbb {C}}^{2I+1}\\otimes {\\mathbb {C}}^{2J+1}$ on which all of these constraints are satisfied simultaneously.",
"Therefore, to find the lowest energy quantized state of a configuration with isospin $I$ and spin $J$ one needs to find the smallest eigenvalue $\\Delta $ of the restriction of $H_{I,J}$ to $V^{(x)}_{I,J}$ .",
"One important consequence of equation (REF ) is that nucleons have half-integer spin and isospin.",
"This can be deduced using the trivial symmetries $(g_0,h_0)=(1,-1)$ and $(-1,1)$ , which are symmetries of any configuration.",
"Both of these transformations negate all of the orientations $q_a$ , so the sign appearing on the right of eq.",
"(REF ) is $(-1)^B$ .",
"Now $\\rho _I(-1)$ equals $-\\mathrm {Id}_{2I+1}$ if $I$ is half-integer and $\\mathrm {Id}_{2I+1}$ if $I$ is integer, so $V^{(x)}_{I,J}=\\lbrace 0\\rbrace $ if $I$ is half integer and $B$ is even, or if $I$ is integer and $B$ is odd.",
"Hence, there are no half integer isospin energy eigenstates when $B$ is even, and no integer isospin energy eigenstates when $B$ is odd.",
"Similar comments apply to spin."
],
[
"Two particles",
"We now illustrate the quantization procedure for the simple example of two particles.",
"After rotation and centreing, the energy minimizer 2a is $x_{2a}=(\\mathbf {x}_1,\\mathbf {x}_2,q_1,q_2) = \\left( -\\frac{\\lambda }{\\sqrt{2}}\\mathbf {e}_1,\\,\\frac{\\lambda }{\\sqrt{2}}\\mathbf {e}_1,\\,1,\\,\\mathbf {k} \\right),$ where the lattice scale parameter $\\lambda $ takes the value $2.9$ to minimize energy and $\\mathbf {e}_1=(1,0,0)$ .",
"This configuration has $D_2$ dihedral symmetry, and the nontrivial elements of the symmetry group are $(\\mathbf {i},\\mathbf {j})$ , $(\\mathbf {i},\\mathbf {i})$ , and $(1,\\mathbf {k})$ .",
"The actions of these transformations, and the corresponding signs $\\chi (g,h)$ , are as follows: $(\\mathbf {i},\\mathbf {j}):(\\mathbf {x}_1,\\mathbf {x}_2,q_1,q_2)&\\mapsto \\left( \\frac{\\lambda }{\\sqrt{2}}\\mathbf {e}_1,\\,-\\frac{\\lambda }{\\sqrt{2}}\\mathbf {e}_1,\\,\\mathbf {k},\\,1\\right) & \\chi (\\mathbf {i},\\mathbf {j}) &= -1 \\\\(\\mathbf {i},\\mathbf {i}):(\\mathbf {x}_1,\\mathbf {x}_2,q_1,q_2)&\\mapsto \\left( -\\frac{\\lambda }{\\sqrt{2}}\\mathbf {e}_1,\\,\\frac{\\lambda }{\\sqrt{2}}\\mathbf {e}_1,1,\\,\\,-\\mathbf {k} \\right) & \\chi (\\mathbf {i},\\mathbf {i}) &= -1 \\\\(1,\\mathbf {k}):(\\mathbf {x}_1,\\mathbf {x}_2,q_1,q_2)&\\mapsto \\left( \\frac{\\lambda }{\\sqrt{2}}\\mathbf {e}_1,\\,-\\frac{\\lambda }{\\sqrt{2}}\\mathbf {e}_1,\\,\\mathbf {k},\\,-1 \\right) & \\chi (1,\\mathbf {k}) &= 1.",
"\\\\$ We will briefly explain how these signs $\\chi (g_0,h_0)$ have been determined.",
"The first transformation $(\\mathbf {i},\\mathbf {j})$ permutes the two particles but does not change any signs.",
"It therefore has $\\mathrm {sgn}(\\sigma )=-1$ , $s_1=s_2=1$ , and consequently $\\chi (\\mathbf {i},\\mathbf {j})=-1$ .",
"The second does not permute the particles but does change the sign of exactly one orientation, so has $\\chi (\\mathbf {i},\\mathbf {i}) = -1$ .",
"The third permutes the particles and changes one sign, so has $\\chi (1,\\mathbf {k})=(-1)^2=1$ .",
"Now we determine the subspaces $V^{(2a)}_{I,J}$ allowed by the constraint (REF ).",
"As explained above, the isospin and spin quantum numbers $I,J$ are necessarily integers.",
"Since some of the symmetries have negative signs in (REF ) states with $(I,J)=(0,0)$ are forbidden.",
"Thus we consider the possibilities $(I,J)=(1,0)$ or $(0,1)$ .",
"If $V^{(2a)}_{1,0}$ or $V^{(2a)}_{0,1}$ is nontrivial, states with higher spin and isospin certainly have higher energy.",
"We need to use the spin 1 representation $\\rho _1$ of $\\mathrm {SU}(2)$ , for which $ \\rho _1(\\mathbf {i}) = \\begin{pmatrix}1&0&0\\\\0&-1&0\\\\0&0&-1\\end{pmatrix},\\,\\rho _1(\\mathbf {j}) = \\begin{pmatrix}-1&0&0\\\\0&1&0\\\\0&0&-1\\end{pmatrix},\\,\\rho _1(\\mathbf {k}) = \\begin{pmatrix}-1&0&0\\\\0&-1&0\\\\0&0&1\\end{pmatrix}.",
"$ In the case $(I,J)=(1,0)$ the constraints (REF ) reduce to $\\rho _1(\\mathbf {i})v=-v$ , so their solution space is $ V^{(2a)}_{1,0} = \\lbrace (0,v_2,v_3)^T\\::\\: v_2,v_3\\in {\\mathbb {C}}\\rbrace .",
"$ In the case $(I,J)=(0,1)$ the constraints (REF ) say that $\\rho _1(\\mathbf {k})v=v$ and $\\rho _1(\\mathbf {i})v=\\rho _1(\\mathbf {j})v=-v$ , and their solution space is $ V^{(2a)}_{0,1} = \\lbrace (0,0,v_3)^T\\::\\: v_3\\in {\\mathbb {C}}\\rbrace .",
"$ The inertia tensor of the configuration (REF ) is $\\Lambda = \\left( \\begin{array}{ccc|ccc}2L & 0 & 0 & 0 & 0 & 0 \\\\0 & 2L & 0 & 0 & 0 & 0 \\\\0 & 0 & 2L & 0 & 0 & -2L \\\\\\hline 0 & 0 & 0 & 2L & 0 & 0 \\\\0 & 0 & 0 & 0 & 2L+M\\lambda ^2 & 0 \\\\0 & 0 & -2L & 0 & 0 & 2L+M\\lambda ^2\\end{array}\\right),$ whose inverse is $\\Lambda ^{-1} = \\left( \\begin{array}{ccc|ccc}\\frac{1}{2L} & 0 & 0 & 0 & 0 & 0 \\\\0 & \\frac{1}{2L} & 0 & 0 & 0 & 0 \\\\0 & 0 & \\frac{2L+M\\lambda ^2}{2LM\\lambda ^2} & 0 & 0 & \\frac{1}{M\\lambda ^2} \\\\\\hline 0 & 0 & 0 & \\frac{1}{2L} & 0 & 0 \\\\0 & 0 & 0 & 0 & \\frac{1}{2L+M\\lambda ^2} & 0 \\\\0 & 0 & \\frac{1}{M\\lambda ^2} & 0 & 0 & \\frac{1}{M\\lambda ^2}\\end{array}\\right).$ The representation $\\rho _1^\\ast $ of the Lie algebra $\\mathfrak {su}(2)$ is $\\rho _1^\\ast \\left({\\textstyle \\frac{\\mathbf {i}}{2}}\\right) = \\begin{pmatrix} 0&0&0\\\\0&0&-1\\\\0&1&0\\end{pmatrix},\\,\\rho _1^\\ast \\left({\\textstyle \\frac{\\mathbf {j}}{2}}\\right) = \\begin{pmatrix} 0&0&1\\\\0&0&0\\\\-1&0&0\\end{pmatrix},\\,\\rho _1^\\ast \\left({\\textstyle \\frac{\\mathbf {k}}{2}}\\right) = \\begin{pmatrix} 0&-1&0\\\\1&0&0\\\\0&0&0\\end{pmatrix}.$ In the case $(I,J)=(1,0)$ the hamiltonian defined in (REF ) is $H_{1,0} &= -\\frac{\\hbar ^2}{4L}\\left(\\rho _1^\\ast \\left({\\textstyle \\frac{\\mathbf {i}}{2}}\\right)^2+\\rho _1^\\ast \\left({\\textstyle \\frac{\\mathbf {j}}{2}}\\right)^2\\right)-\\frac{\\hbar ^2(2L+M\\lambda ^2)}{4LM\\lambda ^2}\\rho _1^\\ast \\left({\\textstyle \\frac{\\mathbf {k}}{2}}\\right)^2 \\\\&=\\begin{pmatrix}\\frac{\\hbar ^2}{2L}+\\frac{\\hbar ^2}{2M\\lambda ^2}&0&0\\\\0&\\frac{\\hbar ^2}{2L}+\\frac{\\hbar ^2}{2M\\lambda ^2}&0\\\\0&0&\\frac{\\hbar ^2}{2L}\\end{pmatrix}.$ In the case $(I,J)=(0,1)$ it is $H_{0,1} &= -\\frac{\\hbar ^2}{4L}\\rho _1^\\ast \\left({\\textstyle \\frac{\\mathbf {i}}{2}}\\right) - \\frac{\\hbar ^2}{4L+2M\\lambda ^2}\\rho _1^\\ast \\left({\\textstyle \\frac{\\mathbf {j}}{2}}\\right)^2 -\\frac{\\hbar ^2}{2M\\lambda ^2}\\rho _1^\\ast \\left({\\textstyle \\frac{\\mathbf {k}}{2}}\\right)^2 \\\\&= \\begin{pmatrix}\\frac{\\hbar ^2}{4L+2M\\lambda ^2}+\\frac{\\hbar ^2}{2M\\lambda ^2}&0&0\\\\0&\\frac{\\hbar ^2}{4L}+\\frac{\\hbar ^2}{2M\\lambda ^2}&0\\\\0&0&\\frac{\\hbar ^2}{4L}+\\frac{\\hbar ^2}{4L+2M\\lambda ^2}\\end{pmatrix}.$ Therefore, the lowest eigenvalue of the restriction of $H$ to $V^{(2a)}_{I,J}$ is $\\hbar ^2/2L$ in the case $(I,J)=(1,0)$ and $\\hbar ^2/4L+\\hbar ^2/(4L+2M\\lambda ^2)$ in the case $(I,J)=(0,1)$ .",
"Since $ \\frac{\\hbar ^2}{4L}+\\frac{\\hbar ^2}{4L+2M\\lambda ^2}<\\frac{\\hbar ^2}{2L}, $ the groundstate has isospin 0 and spin 1.",
"The quantum mechanical correction to the energy is $ \\Delta = \\frac{\\hbar ^2}{4L}+\\frac{\\hbar ^2}{4L+2M\\lambda ^2} \\approx 4.123, $ using the value $\\lambda =2.922$ of the lattice corresponding the minimum of $V$ .",
"Hence, rigid body quantization of the $B=2$ energy minimizer correctly reproduces the spin and isospin of the deuteron."
],
[
"Automation of the quantization procedure",
"Most of the quantization procedure described above is linear algebraic and so easily automated, but determining the symmetries of a configuration and the associated signs arising in the Finkelstein-Rubinstein constraints can be tricky.",
"We have developed an algorithm that finds symmetries of configurations of particles, and hence have been able to automate the entire quantization procedure.",
"Our symmetry-finding algorithm first identifies rotational symmetries of the set of particle positions, and then determines which of these can be lifted to symmetries in $\\mathrm {SU}(2)_I\\times \\mathrm {SU}(2)_S$ .",
"To find spatial symmetries it first translates the configuration so that its centre of mass is at the origin and rotates it so that its second moment matrix (REF ) is diagonal.",
"It then treats separately three cases corresponding to different degeneracies of the diagonal entries, i.e.",
"eigenvalues, of the second moment matrix.",
"If the eigenvalues are all different then the symmetry group is a subgroup of the group $D_2\\in \\mathrm {SO}(3)$ consisting of rotations about the three coordinate axes through $\\pi $ .",
"Each element of this group is applied to the particle positions and a distance between the resulting configuration and the original configuration is measured (taking account of permutations).",
"If the distance is sufficiently small then the group element is accepted as a symmetry.",
"If two eigenvalues are equal and the third distinct then the spatial symmetry group is a subgroup of $O(2)$ .",
"Since, by observation, none of the minimizers with three or more particles have continuous symmetry, the symmetry group is assumed to be dihedral or cyclic.",
"To identify cyclic symmetries, rotations through angle $\\theta $ are applied to the configuration and the distance from the resulting configuration to the original configuration measured as a function of $\\theta $ .",
"Minima of this function close to zero are interpreted as symmetries.",
"Dihedral symmetries are identified in a similar way.",
"If all three eigenvalues are equal then the symmetry group is likely to be a discrete subgroup of $\\mathrm {SO}(3)$ which is neither dihedral nor cyclic.",
"Since icosahedral symmetry is not compatible with the FCC lattice our algorithm works on the assumption that the symmetry group is either the octahedral group $O$ or the tetrahedral group $T< O$ .",
"It computes the fourth moment tensor $ M^{(4)}_{ijkl} = \\sum _{a=1}^B x_i^a x_j^a x_k^a x_l^a $ and finds minima or maxima of the function $S^2\\rightarrow {\\mathbb {R}}$ defined by $ \\mathbf {n}\\mapsto M^{(4)}_{ijkl}n_in_jn_kn_l,\\quad \\mathbf {n}\\cdot \\mathbf {n}=1.",
"$ This is a polynomial function on the sphere of degree less than or equal to 4 containing terms of even degree only.",
"It is known that there are only two linearly independent functions of this type with tetrahedral symmetry, namely the constant function and the function $ \\mathbf {n}\\mapsto n_1^4+n_2^4+n_3^4.",
"$ The latter furthermore has octahedral symmetry and its maxima are at the points where the coordinate axes intersect the sphere.",
"Therefore, if the configuration has either octahedral or tetrahedral symmetry and the function constructed from the fourth moment tensor is non-constant then either its maxima are at mutually-orthogonal points on the sphere or its minima are.",
"The algorithm seeks either a pair of orthogonal maxima or a pair of orthogonal minima and rotates these to lie on two of the coordinate axes.",
"It then tests whether each element of the octahedral group is a symmetry of the configuration.",
"Once the configuration's rotational symmetries are known, the algorithm determines whether each lifts to a full rotation-isorotation symmetry of $(\\mathbf {x}_1,\\ldots ,\\mathbf {x}_B,q_1,\\ldots ,q_B)$ .",
"Given a rotational symmetry $R$ , we choose $h\\in {\\mathrm {SU}}(2)$ such that $R=R(h)$ .",
"Being a rotational symmetry means precisely that $R(h)\\mathbf {x}_a=\\mathbf {x}_{\\sigma (a)}$ for some permutation $\\sigma $ .",
"Note that spatial rotations change the orientations also, $q_a\\mapsto q_a h^{-1}$ .",
"The spatial symmetry $h$ lifts to a full symmetry if there exists $g\\in {\\mathrm {SU}}(2)$ such that $gq_ah^{-1}=\\pm q_{\\sigma (a)}$ for all $a$ .",
"If such an isorotation $g$ exists, it is unique up to sign.",
"In fact it must be $g=q_{\\sigma (1)}hq_1^{-1}$ (or minus this).",
"Hence, our algorithm computes $q_a^{\\prime }:=q_{\\sigma (1)}hq_1^{-1}q_ah^{-1}$ for each $a=2,\\ldots ,B$ and tests whether $q_a^{\\prime }=\\pm q_{\\sigma (a)}$ (to some numerical tolerance) for all $a$ .",
"If so, $(g,h)$ is accepted as a full symmetry, and its FR factor $\\chi (g,h)$ is readily computed from the sign of the permutation $\\sigma $ and the signs occuring in $q_a^{\\prime }=\\pm q_a$ .",
"If not, the spatial symmetry $R(h)$ is discarded.",
"The output of our algorithm is recorded in table REF .",
"For each classical energy minimizer we record the spin and isospin quantum numbers corresponding to the ground state, and the quantum mechanical energy (defined to be the sum of the classical energy $V$ and the $O(\\hbar ^2)$ correction $\\Delta $ calculated by our algorithm).",
"The numerical value of $\\hbar $ is fixed by the calibration proposed in [5].",
"We have been using energy and length units $F_\\pi /4g\\sqrt{1-\\alpha }$ and $2\\sqrt{1-\\alpha }/F_\\pi g$ .",
"In natural units Planck's constant is 1, so in our units it equals $ \\frac{4g\\sqrt{1-\\alpha }}{F_\\pi }\\frac{F_\\pi g}{2\\sqrt{1-\\alpha }}=2g^2.",
"$ In [5] the dimensionless parameter $g$ was determined to be 3.96 by comparing the charge radii of the one-skyrmion and the proton, so the numerical value for $\\hbar ^2$ is $4\\times 3.96^4\\approx 799.5$ .",
"Figure: Binding energies calculated using rigid body quantization.In most cases the configuration with the lowest quantum energy is the same as the configuration with the lowest classical energy.",
"There are two exceptions to this trend.",
"The quantum corrections to the 6-particle minimizers are relatively large and their order is reversed, so that 6c is the lightest and 6a the heaviest.",
"The two configurations 10a and 10b have almost identical classical energies, and after quantization the order of their energies is reversed.",
"The table also lists spin and isospin quantum numbers of the lightest nucleus for each mass number.",
"In 12 out of 22 cases there is a quantized point particle configuration with the same quantum numbers.",
"Sometimes the configuration with the correct quantum numbers is not that with lowest energy: for example, 5b has the same spin and isospin as ${}^5\\mathrm {He}_2$ , but its energy exceeds that of 5a.",
"Including the quantum corrections gives new predictions for nuclear masses and binding energies, which are plotted in figure REF .",
"The mass of a quantized configuration of $B$ particles is $BM+V+\\Delta $ , where $V+\\Delta $ is the quantum energy recorded in REF .",
"The binding energy is the difference between this quantity and $B$ times the quantized mass of 1 particle.",
"The calculation of the quantum mechanical correction to the mass of one particle is a standard calculation similar to those described above; the end result is $ M_1 = M + \\frac{3\\hbar ^2}{8L}.",
"$ The binding energy per nucleon is therefore $ \\frac{1}{B}\\left( B\\left(M + \\frac{3\\hbar ^2}{8L}\\right) - (BM+V+\\Delta )\\right) = \\frac{3\\hbar ^2}{8L} - \\frac{V+\\Delta }{B}.",
"$ Note that $3\\hbar ^2/8L \\approx 5.521$ .",
"Rigid body quantization has the effect of increasing the binding energy of nuclei, so that they are roughly 5% of the 1-skyrmion mass rather than 1%.",
"It is easy to see why: the quantum correction to the 1-skyrmion mass represents about 5% of its total mass, whereas the quantum corrections to the masses of larger nuclei represent a much smaller percentage of their total mass.",
"This means that the quantum corrections to binding energies are also around 5% of the 1-skyrmion mass, and much larger than the classical binding energies.",
"The fact that binding energies calculated by rigid body quantization are too large does not represent a failure of the lightly bound Skyrme model, but rather illustrates the pitfalls of rigid body quantization itself.",
"A collection of $B$ point particles has $6B-3$ degrees of freedom, but in rigid body quantization at most 6 of these are quantized.",
"Only in the case $B=1$ are all degrees of freedom quantized, so rigid body quantization systematically underestimates the mass of configurations with a large number of particles.",
"From the point of view of the lightly bound Skyrme model, the degrees of freedom corresponding to moving particles are almost massless, because 1-skyrmions interact only weakly, so arguably these are of comparable importance to the massless degrees of freedom studied in rigid body quantization."
],
[
"Concluding remarks",
"We have constructed a simple point particle model of lightly bound skyrmions which almost flawlessly reproduces the results of numerical field theoretic energy minimization for charges 1 to 8.",
"The only exception is charge 6.",
"Here, the point particle model predicts minimizers with shapes, in order of ascending energy, octahedron, bowtie and pyramid-plus-one, whereas full field simulations find that the correct order is bowtie, octohedron, pyramid-plus-one, albeit with the first two of these very close to degenerate.",
"Alongside this minor blemish one should set some unexpected successes: the point particle model predicted previously unknown energy minimizers at charges 5, 7 and 8, all of which corresponded to local energy minimizers of the field theory with correct energy ordering.",
"This includes the (so far) lowest energy skyrmion at charge 7.",
"The point particle model makes a simple prediction for the inertia tensors of lightly bound skyrmions which, with only two free parameters, fits the field theoretic data for the global minimizers with $1\\le B\\le 8$ to within 10%.",
"In judging this, one should bear in mind that an inertia tensor is not a single number, but rather (after accounting for symmetries) 15 independent numbers, so we are actually fitting 120 independent quantities here.",
"Having checked consistency with field theory simulations for $1\\le B\\le 8$ , we then proceeded to generate local energy minimizers of the point particle model for $9\\le B\\le 23$ , where full field simulations are, so far, unavailable.",
"We found that the number of nearly degenerate local energy minimizers grows rapidly with $B$ , that minimizers consistently resemble subsets of a face centred cubic lattice, with internal orientations correlated with lattice position, and that minimizers often have one fewer than the maximum possible number of nearest-neighbour bonds.",
"We have, furthermore, implemented a simple rigid body quantization scheme for all the local minima we found ($1\\le B\\le 23$ ).",
"As part of this, we devised an automated algorithm to compute the spin-isospin symmetry group of an oriented point cloud, which simultaneously computes the Finkelstein-Rubinstein constraint associated with each symmetry.",
"This allowed us to compute the spin and isospin of the quantum ground states associated (in rigid body quantization) with each local energy minimizer.",
"Since classical binding energies are so small in the lightly bound model, quantization occasionally altered the energy ordering of local minima with a given charge $B$ .",
"For 12 baryon numbers (out of 23), this simple quantization procedure produced states corresponding to the spin-isospin data of the lightest nucleus of that baryon number.",
"Our numerical scheme to find energy minimizers of the point particle model had two steps: first a crystal-building algorithm was run to generate a subset of the FCC lattice with sufficiently many (nearest neighbour) bonds.",
"Given such a subset, an initial point particle configuration was constructed with particles at the occupied vertices, their internal orientations being fixed by a simple colouring rule, the lattice length scale being chosen to minimize total interaction energy.",
"The second step was to relax this initial FCC subset using a simulated annealing algorithm which allowed the particle positions, and internal orientations, to vary continuously.",
"The results suggest that, in retrospect, this second step is actually superfluous, since the relaxed configuration always stays very close to some FCC lattice (see figure REF ).",
"If one merely wishes to find good approximations to classical energy minimizers of the lightly bound Skyrme model, it would seem that considering only FCC lattice subsets, with the lattice scale left as a free parameter, is a fast and effective strategy.",
"It is possible that low energy FCC subsets may also provide useful sets of initial data for energy minimization in more standard variants of the Skyrme model.",
"Certainly this is a quick and convenient means to generate rather uniform initial data of a qualitatively new kind, not obtainable from rational map or alpha particle clustering methods.",
"A more interesting problem is to find a better quantization scheme than rigid body quantization.",
"In principle, one could attempt to solve the full Schödinger equation on $\\tilde{\\mathcal {C}}_B$ , subject to the FR constraints.",
"For $B=2$ , there is sufficient symmetry that this may well be tractable.",
"For larger $B$ , however, it is clearly hopeless.",
"Instead, one should attempt to implement some form of “vibrational” quantization scheme, as used for the conventional Skyrme model in [6], [7].",
"This requires one to find a low-dimensional moduli space of configurations, including all relevant local energy minima, but also configurations interpolating between them, which captures the most important vibrational processes of the classical skyrmion.",
"In this regard, the full point particle model, with positions and orientations allowed to leave the set of FCC configurations, will be essential.",
"Indeed the model of point skyrmions introduced here may well prove to be an ideal testing ground for vibrational quantization techniques.",
"Acknowledgements We are grateful to Paul Sutcliffe for helpful conversations.",
"The field theory simulations were performed using code originally developed in collaboration with Juha Jäykkä."
]
] | 1612.05481 |
[
[
"On Natural Deduction for Herbrand Constructive Logics II: Curry-Howard\n Correspondence for Markov's Principle in First-Order Logic and Arithmetic"
],
[
"Abstract Intuitionistic first-order logic extended with a restricted form of Markov's principle is constructive and admits a Curry-Howard correspondence, as shown by Herbelin.",
"We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov's principle.",
"Starting from classical natural deduction, we restrict the excluded middle and we obtain a natural deduction system and a parallel Curry-Howard isomorphism for the logic.",
"We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possible witnesses.",
"As corollary, we derive that the logic is Herbrand constructive: whenever it proves any existential formula, it proves also an Herbrand disjunction for the formula.",
"Finally, using the techniques just introduced, we also provide a new computational interpretation of Arithmetic with Markov's principle."
],
[
"Introduction",
"Markov's Principle was introduced by Markov in the context of his theory of Constructive Recursive Mathematics (see [15]).",
"Its original formulation is tied to Arithmetic: it states that given a recursive function $f: \\mathbb {N} \\rightarrow \\mathbb {N}$ , if it is impossible that for every natural number $n$ , $f(n)\\ne 0$ , then there exists a $n$ such that $f(n)=0$ .",
"Markov's original argument for justifying it was simply the following: if it is not possible that for all $n$ , $f(n)\\ne 0$ , then by computing in sequence $f(0), f(1), f(2), \\ldots $ , one will eventually hit a number $n$ such that $f(n)=0$ and will effectively recognize it as a witness.",
"Markov's principle is readily formalized in Heyting Arithmetic as the axiom scheme $\\lnot \\lnot \\exists \\alpha ^{ {\\tt N} } {P}\\rightarrow \\exists \\alpha ^{ {\\tt N} } {P}$ where ${P}$ is a primitive recursive predicate [14].",
"When added to Heyting Arithmetic, Markov's principle gives rise to a constructive system, that is, one enjoying the disjunction and the existential witness property [14] (if a disjunction is derivable, one of the disjoints is derivable too, and if an existential statement is derivable, so it is one instance of it).",
"Furthermore, witnesses for any provable existential formula can be effectively computed using either Markov's unbounded search and Kleene's realizability [9] or much more efficient functional interpretations [7], [3].",
"The very shape of Markov's principle makes it also a purely logical principle, namely an instance of the double negation elimination axiom.",
"But in pure logic, what exactly should Markov's principle correspond to?",
"In particular, what class of formulas should $P$ be restricted to?",
"Since Markov's principle was originally understood as a constructive principle, it is natural to restrict $P$ as little as possible, while maintaining the logical system as constructive as possible.",
"As proven by Herbelin [8], it turns out that asking that $P$ is propositional and with no implication $\\rightarrow $ symbols guarantees that intuitionistic logic extended with such a version of Markov's principle is constructive.",
"The proof of this result employs a Curry-Howard isomorphism based on a mechanism for raising and catching exceptions.",
"As opposed to the aforementioned functional interpretations of Markov's principle, Herbelin's calculus is fully isomorphic to an intuitionistic logic: there is a perfect match between reduction steps at the level of programs and detour eliminations at the level of proofs.",
"Moreover, witnesses for provable existential statements are computed by the associated proof terms.",
"Nevertheless, as we shall later show, the mechanism of throwing exceptions plays no role during these computations: intuitionistic reductions are entirely enough for computing witnesses.",
"A question is now naturally raised: as no special mechanism is required for witness computation using Herbelin's restriction of Markov's principle, can the first be further relaxed so that the second becomes stronger as well as computationally and constructively meaningful?",
"Allowing the propositional matrix $P$ to contain implication destroys the constructivity of the logic.",
"It turns out, however, that Herbrand constructivity is preserved.",
"An intermediate logic is called Herbrand constructive if it enjoys a strong form of Herbrand's Theorem [5], [4]: for every provable formula $\\exists \\alpha \\, A$ , the logic proves as well an Herbrand disjunction $A[m_{1}/\\alpha ]\\vee \\ldots \\vee A[m_{k}/\\alpha ]$ So the Markov principle we shall interpret in this paper is $\\mathsf {MP}: \\lnot \\lnot \\exists \\alpha \\, {P}\\rightarrow \\exists \\alpha \\, {P} \\mbox{\\qquad ($P$ propositional formula)}$ and show that when added to intuitionistic first-order logic, the resulting system is Herbrand constructive.",
"This is the most general form of Markov's principle that allows a significant constructive interpretation: we shall show how to non-trivially compute lists of witnesses for provable existential formulas thanks to an exception raising construct and a parallel computation operator.",
"$\\mathsf {MP}$ can also be used in conjunction with negative translations to compute Herbrand disjunctions in classical logic, something which is not possible with Herbelin's form of Markov's principle."
],
[
"Restricted Excluded Middle",
"The Curry-Howard correspondence we present here is by no means an ad hoc construction, only tailored for Markov's principle.",
"It is a simple restriction of the Curry-Howard correspondence for classical first-order logic introduced in [4], where classical reasoning is formalized by the excluded middle inference rule: $\\Gamma , a: \\forall x \\, {\\mathsf {Q}} \\vdash u: C$ $\\Gamma , a: \\exists x\\, \\lnot {\\mathsf {Q}} \\vdash v: C$ $\\mathsf {EM}$ $\\Gamma \\vdash { u \\parallel _{a} v} : C$ It is enough to restrict the conclusion $C$ of this rule to be a simply existential statement and the ${\\mathsf {Q}}$ in the premises $\\forall x \\, {\\mathsf {Q}}, \\exists x\\, \\lnot {\\mathsf {Q}}$ to be propositional.",
"We shall show that the rule is intuitionistically equivalent to $\\mathsf {MP}$ .",
"With our approach, strong normalization is just inherited and the transition from classical logic to intuitionistic logic with $\\mathsf {MP}$ is smooth and natural."
],
[
"Markov's Principle in Arithmetic",
"We shall also provide a computational interpretation of Heyting Arithmetic with $\\mathsf {MP}$ .",
"The system is constructive and witnesses for provable existential statements can be computed.",
"This time, we shall restrict the excluded middle as formalized in $\\cite {ABB}$ and we shall directly obtain the desired Curry-Howard correspondence.",
"As a matter of fact, the interpretation of $\\mathsf {MP}$ in Arithmetic ends up to be a simplification of the methods we use in first-order logic, because the decidability of atomic formulas greatly reduces parallelism and eliminates case distinction on the truth of atomic formulas."
],
[
"Plan of the Paper",
"In sec:herb, we provide a simple computational interpretation of first-order intuitionistic logic extended with Herbelin's restriction of Markov's principle.",
"We also show that the full Markov principle $\\mathsf {MP}$ cannot be proved in that system.",
"In sec:ilemeno, we provide a Curry-Howard correspondence for intuitionistic logic with $\\mathsf {MP}$ , by restricting the excluded middle, and show that the system is Herbrand constructive.",
"In sec:arithmetic, we extend the Curry-Howard to Arithmetic with $\\mathsf {MP}$ and show that the system becomes again constructive."
],
[
"Herbelin's Restriction of Markov's Principle",
"In [8] Herbelin introduced a Curry-Howard isomorphism for an extended intuitionistic logic.",
"By employing exception raising operators and new reduction rules, he proved that the logic is constructive and can derive the axiom scheme $\\mathsf {HMP}: \\lnot \\lnot \\exists \\alpha \\, P \\rightarrow \\exists \\alpha \\, P \\text{\\qquad ($P$ propositional and $\\rightarrow $ not occurring in $P$)}$ Actually, Herbelin allowed $P$ also to contain existential quantifiers, but in that case the axiom scheme is intuitionistically equivalent to $ \\lnot \\lnot \\exists \\alpha _{1}\\ldots \\exists \\alpha _{n} \\, P \\rightarrow \\exists \\alpha _{1}\\ldots \\exists \\alpha _{n} \\, P$ , again with $P$ propositional and $\\rightarrow $ not occurring in $P$ .",
"All of the methods of our paper apply to this case as well, but for avoiding trivial details, we keep the present $\\mathsf {HMP}$ .",
"Our first goal is to show that $\\mathsf {HMP}$ has a simpler computational interpretation and to provide a straightforward proof that, when added on top of first-order intuitionistic logic, $\\mathsf {HMP}$ gives rise to a constructive system.",
"In particular, we show that the ordinary Prawitz reduction rules for intuitionistic logic and thus the standard Curry-Howard isomorphism [13] are enough for extracting witnesses for provable existential formulas.",
"The crucial insight, as we shall see, is that $\\mathsf {HMP}$ can never actually appear in the head of a closed proof term having existential type.",
"It thus plays no computational role in computing witnesses; it plays rather a logical role, in that it may be used to prove the correctness of the witnesses.",
"To achieve our goal, we consider the usual natural deduction system for intuitionistic first-order logic [12], [13], to which we add $\\mathsf {HMP}$ .",
"Accordingly, we add to the associated lambda calculus the constants $\\mathcal {M}_{P}: \\lnot \\lnot \\exists \\alpha \\, P \\rightarrow \\exists \\alpha \\, P$ .",
"The resulting Curry-Howard system is called $\\mathsf {IL}+\\mathsf {HMP}$ and is presented in fig:system.",
"The reduction rules for $\\mathsf {IL}+\\mathsf {HMP}$ presented in fig:red-ilmp are just the ordinary ones of lambda calculus.",
"On the other hand, $\\mathcal {M}_{P}$ has no computational content and thus no associated reduction rule.",
"Of course, the strong normalization of $\\mathsf {IL}+\\mathsf {HMP}$ holds by virtue of the result for standard intuitionistic Curry-Howard.",
"The system $\\mathsf {IL}+\\mathsf {HMP}$ is strongly normalizing Figure: Reduction Rules for 𝖨𝖫\\mathsf {IL} + 𝖧𝖬𝖯\\mathsf {HMP}As we shall see in thm:construct-il-mp, the reason why $\\mathsf {HMP}$ cannot be appear in the head of a closed proof term having existential type is that its premise $\\lnot \\lnot \\exists \\alpha \\, P$ is never classically valid, let alone provable in intuitionistic logic.",
"Proposition Assume that the symbol $\\rightarrow $ does not occur in the propositional formula $P$ .",
"Then $\\lnot \\lnot \\exists \\alpha \\, P$ is not classically provable.",
"We provide a semantical argument.",
"$\\lnot \\lnot \\exists \\alpha \\, P$ is classically provable if and only if it is classically valid and thus if and only if $\\exists \\alpha \\, P$ is classically valid.",
"For every such a formula, we shall exhibit a model falsifying it.",
"Consider the model $\\mathfrak {M}$ where every $n$ -ary predicate is interpreted as the empty $n$ -ary relation.",
"We show by induction on the complexity of the formula $P$ that $P^\\mathfrak {M} = \\bot $ for every assignment of individuals to the free variables of $P$ , and therefore $(\\exists \\alpha \\, P)^\\mathfrak {M} = \\bot $ .",
"If $P$ is atomic, then by definition of $\\mathfrak {M}$ , we have $P^\\mathfrak {M} = \\bot $ for every assignment of the variables.",
"If $P=P_1 \\wedge P_2$ , then since by induction $P_1^\\mathfrak {M} = \\bot $ , $(P_1\\wedge P_2)^\\mathfrak {M} = \\bot $ If $P=P_1 \\vee P_2$ , then since by induction $P_1^\\mathfrak {M} = \\bot $ and $P_2^\\mathfrak {M} = \\bot $ , $(P_1\\vee P_2)^\\mathfrak {M} = \\bot $ In order to derive constructivity of $\\mathsf {IL}+\\mathsf {HMP}$ , we shall just have to inspect the normal forms of proof terms.",
"Our main argument, in particular, will use the following well-known syntactic characterization of the shape of proof terms.",
"Proposition (Head of a Proof Term) Every proof-term of $\\mathsf {IL}+\\mathsf {HMP}$ is of the form $\\lambda z_1 \\dots \\lambda z_n.\\, r u_1 \\dots u_k $ where $r$ is either a variable or a constant or a term corresponding to an introduction rule: $\\lambda x .",
"t$ , $\\lambda \\alpha .",
"t$ , $\\langle t_1, t_2 \\rangle $ , $_{i}(t)$ , $(m, t)$ $u_1, \\dots u_k$ are either proof terms, first order terms, or one of the following expressions corresponding to elimination rules: $ \\pi _{i}$ , $[x.w_1, y.w_2]$ , $[(\\alpha ,x).t]$ .",
"Standard.",
"We are now able to prove that $\\mathsf {IL}+\\mathsf {HMP}$ is constructive.",
"[Constructivity of $\\mathsf {IL}+\\mathsf {HMP}$ ] If $\\mathsf {IL}+\\mathsf {HMP}\\vdash t: \\exists \\alpha \\, A$ , and $t$ is in normal form, then $t=(m, u)$ and $\\mathsf {IL}+\\mathsf {HMP}\\vdash u: A[m/\\alpha ]$ .",
"If $\\mathsf {IL}+\\mathsf {HMP}\\vdash t: A \\vee B$ and $t$ is in normal form, then either $t=_{0}(u)$ and $\\mathsf {IL}+\\mathsf {HMP}\\vdash u: A$ or $t=_{1}(u)$ and $\\mathsf {IL}+\\mathsf {HMP}\\vdash u: B$ .",
"By theorem:head-form, $t$ must be of the form $r u_1\\dots u_k$ .",
"Let us consider the possible forms of $r$ .",
"Since $t$ is closed, $r$ cannot be a variable.",
"We show that $r$ cannot be $\\mathcal {M}_{P}$ .",
"If $r$ were $\\mathcal {M}_{P} : \\lnot \\lnot \\exists x \\, P \\rightarrow \\exists \\alpha \\, P$ for some $P$ , then $\\mathsf {IL}+\\mathsf {MP} \\vdash u_1 : \\lnot \\lnot \\exists \\alpha \\, P$ .",
"Since $\\mathsf {IL}+\\mathsf {HMP}$ is contained in classical logic, we have that $\\lnot \\lnot \\exists \\alpha \\, P$ is classically provable.",
"However we know from theorem:no-sigma-taut that this cannot be the case, which is a contradiction.",
"We also show that $r$ cannot be ${\\mathtt {H}^{{\\mathsf {\\bot \\rightarrow P}}}}$ .",
"Indeed, if $r$ were ${\\mathtt {H}^{{\\mathsf {\\bot \\rightarrow P}}}}$ for some $P$ , then $\\mathsf {IL}+\\mathsf {MP} \\vdash u_1 : \\bot $ , which is a contradiction.",
"The only possibility is thus that $r$ is one among $\\lambda x .",
"t$ , $\\lambda \\alpha .",
"t$ , $\\langle t_1, t_2 \\rangle $ , $_{i}(t)$ , $(m, t)$ .",
"In this case, $k$ must be 0 as otherwise we would have a redex.",
"This means that $t=r$ and thus $t=(m, u)$ with $\\mathsf {IL}+\\mathsf {HMP}\\vdash u : A(m)$ .",
"The proof goes along the same lines of case 1.",
"Finally, we prove that $\\mathsf {IL}+\\mathsf {HMP}$ is not powerful enough to express full Markov's principle $\\mathsf {MP}$ .",
"Intuitively, the reason is that $\\mathsf {IL}+\\mathsf {HMP}$ is a constructive system and thus cannot be strong enough to interpret classical reasoning.",
"This would indeed be the case if $\\mathsf {IL}+\\mathsf {HMP}$ proved $\\mathsf {MP}$ , an axiom which complements very well negative translations.",
"Proposition $\\mathsf {IL}+\\mathsf {HMP}\\nvdash \\mathsf {MP}$ .",
"Suppose for the sake of contradiction that $\\mathsf {IL}+\\mathsf {HMP}\\vdash \\mathsf {MP}$ .",
"Consider any proof in classical first-order logic of a simply existential statement $\\exists \\alpha \\,{\\mathsf {P}}$ .",
"By the Gödel-Gentzen negative translation (see [14]), we can then obtain an intuitionistic proof of $\\lnot \\lnot \\exists \\alpha \\, {\\mathsf {P}}^{N}$ , where ${\\mathsf {P}}^{N}$ is the negative translation of ${\\mathsf {P}}$ , and thus $\\mathsf {IL}+\\mathsf {HMP}\\vdash \\exists \\alpha \\, {\\mathsf {P}}^{N}$ .",
"By thm:construct-il-mp, there is a first-order term $m$ such that $\\mathsf {IL}+\\mathsf {HMP}\\vdash {\\mathsf {P}}^{N}[m/\\alpha ]$ .",
"Since ${\\mathsf {P}}^{N}[m/\\alpha ]$ is classically equivalent to ${\\mathsf {P}}[m/\\alpha ]$ , we would have a single witness for every classically valid simply existential statement.",
"But this is not possible: consider for example the first-order language $\\mathcal {L}=\\lbrace P,a,b\\rbrace $ and the formula $F = (P(a) \\vee P(b)) \\rightarrow P(\\alpha )$ where $P$ is an atomic predicate.",
"Then the formula $\\exists \\alpha \\, F$ is classically provable, but there is no term $m$ such that $F[m/\\alpha ]$ is valid, let alone provable: it cannot be $m=a$ , as it is shown by picking a model where $P$ is interpreted as the set $\\lbrace a\\rbrace $ it cannot be $m=b$ , because we can interpret $P$ as the set $\\lbrace b\\rbrace $ ."
],
[
"Full Markov Principle and Restricted Excluded Middle in First-Order Logic",
"In this section we describe the natural deduction system and Curry-Howard correspondence $\\mathsf {IL}+\\mathsf {EM}_{1}^{-}$ , which arise by restricting the excluded-middle in classical natural deduction [4].",
"This computational system is based on delimited exceptions and a parallel operator.",
"We will show that on one hand full Markov principle $\\mathsf {MP}$ is provable in $\\mathsf {IL}+\\mathsf {EM}_{1}^{-}$ and, on the other hand, that $\\mathsf {IL}+\\mathsf {MP}$ derives all of the restricted classical reasoning that can be expressed in $\\mathsf {IL}+\\mathsf {EM}_{1}^{-}$ , so that the two systems are actually equivalent.",
"Finally, we show that the system $\\mathsf {IL}+\\mathsf {EM}_{1}^{-}$ is Herbrand constructive and that witnesses can effectively be computed.",
"All of the classical reasoning in $\\mathsf {IL}+\\mathsf {EM}_{1}^{-}$ is formally restricted to negative formulas.",
"[Negative, Simply Universal Formulas] We denote propositional formulas as ${\\mathsf {P_1}},\\dots {\\mathsf {P_n}},{\\mathsf {Q}},{\\mathsf {R}},\\dots $ .",
"We say that a propositional formula is negative whenever $\\vee $ does not occur in it.",
"Formulas of the form $\\forall \\alpha _{1}\\ldots \\forall \\alpha _{n}\\, {\\mathsf {P}}$ , with ${\\mathsf {P}}$ negative, will be called simply universal.",
"In order to computationally interpret Markov's principle, we consider the rule $\\mathsf {EM}_{1}^{-}$ , which is obtained by restricting the conclusion of the excluded middle $\\mathsf {EM}_{1}$ [4], [2] to be a simply existential formula, $\\Gamma , a: \\forall \\alpha \\, {\\mathsf {P}} \\vdash u: \\exists \\beta \\, {\\mathsf {Q}}$ $\\Gamma , a: \\exists \\alpha \\, \\lnot {\\mathsf {P}} \\vdash v: \\exists \\beta \\, {\\mathsf {Q}}$ $\\mathsf {EM}_{1}^-$ $\\Gamma \\vdash { u \\parallel _{a} v} : \\exists \\beta \\, {\\mathsf {Q}}$ where both ${\\mathsf {P}}$ and ${\\mathsf {Q}}$ are negative formulas.",
"This inference rule is complemented by the axioms: $\\Gamma , a:{\\forall \\alpha {\\mathsf {P}}}\\vdash {\\mathtt {H}_{a}^{\\forall {\\alpha } {{\\mathsf {P}}}}}: \\forall \\alpha {\\mathsf {P}}$ $\\Gamma , a:{\\exists \\alpha \\lnot {\\mathsf {P}}}\\vdash \\mathtt {W}_{a}^{\\exists {\\alpha } {\\lnot {\\mathsf {P}}} }: \\exists \\alpha \\lnot {\\mathsf {P}}$ These last two rules correspond respectively to a term making an Hypothesis and a term waiting for a Witness and these terms are put in communication via $\\mathsf {EM}_{1}^-$ .",
"A term of the form ${\\mathtt {H}_{a}^{\\forall {\\alpha } {{\\mathsf {P}}}}} m$ , with $m$ first-order term, is said to be active, if its only free variable is $a$ : it represents a raise operator which has been turned on.",
"The term ${ u \\parallel _{a} v}$ supports an exception mechanism: $u$ is the ordinary computation, $v$ is the exceptional one and $a$ is the communication channel.",
"Raising exceptions is the task of the term ${\\mathtt {H}_{a}^{\\forall {\\alpha } {{\\mathsf {P}}}}}$ , when it encounters a counterexample $m$ to $\\forall \\alpha \\, {\\mathsf {P}}$ ; catching exceptions is performed by the term $\\mathtt {W}_{a}^{\\exists {\\alpha } {\\lnot {\\mathsf {P}}} }$ .",
"In first-order logic, however, there is an issue: when should an exception be thrown?",
"Since the truth of atomic predicates depends on models, one cannot know.",
"Therefore, each time ${\\mathtt {H}_{a}^{\\forall {\\alpha } {{\\mathsf {P}}}}}$ is applied to a term $m$ , a new pair of parallel independent computational paths is created, according as to whether ${\\mathsf {P}}[m/\\alpha ]$ is false or true.",
"In one path the exception is thrown, in the other not, and the two computations will never join again.",
"To render this computational behaviour, we add the rule $\\mathsf {EM}_{0}$ of propositional excluded middle over negative formulas $\\Gamma , a: \\lnot {\\mathsf {P}} \\vdash u: A$ $\\Gamma , a: {\\mathsf {P}} \\vdash v: A$ $\\mathsf {EM}_{0}$ $\\Gamma \\vdash { u\\, |_{}\\, v} : A$ even if in principle it is derivable from $\\mathsf {EM}_{1}^{-}$ ; we also add the axiom $\\Gamma , a:{\\mathsf {P}} \\vdash {\\mathtt {H}^{{\\mathsf {P}}}}: {\\mathsf {P}}$ We call the resulting system $\\mathsf {IL}+\\mathsf {EM}_1^-$ (fig:system-ilem) and present its reduction rules in fig:red; they just form a restriction of the system $\\mathsf {IL}+\\mathsf {EM}$ described in [4].",
"The reduction rules are in fig:red and are based on the following definition, which formalizes the raise and catch mechanism.",
"[Exception Substitution] Suppose $v$ is any proof term and $m$ is a term of $\\mathcal {L}$ .",
"Then: If every free occurrence of $a$ in $v$ is of the form $\\mathtt {W}_{a}^{\\exists {\\alpha } { {\\mathsf {P}}} }$ , we define $v[a:=m]$ as the term obtained from $v$ by replacing each subterm $\\mathtt {W}_{a}^{\\exists {\\alpha } {{\\mathsf {P}}} }$ corresponding to a free occurrence of $a$ in $v$ by $(m, {\\mathtt {H}^{{\\mathsf {P}}[m/\\alpha ]}})$ .",
"If every free occurrence of $a$ in $v$ is of the form ${\\mathtt {H}_{a}^{\\forall {\\alpha } {{\\mathsf {P}}}}}$ , we define $v[a:=m]$ as the term obtained from $v$ by replacing each subterm ${\\mathtt {H}_{a}^{\\forall {\\alpha } {{\\mathsf {P}}}}}m$ corresponding to a free occurrence of $a$ in $v$ by ${\\mathtt {H}^{\\mathsf {P}[m/\\alpha ]}}$ .",
"Figure: Term Assignment Rules for 𝖨𝖫+𝖤𝖬 1 - \\mathsf {IL}+\\mathsf {EM}_{1}^-Figure: Reduction Rules for 𝖨𝖫\\mathsf {IL} + 𝖤𝖬 1 - \\mathsf {EM}_{1}^-As we anticipated, our system is capable of proving the full Markov Principle $\\mathsf {MP}$ and thus its particular case $\\mathsf {HMP}$ .",
"Proposition (Derivability of $\\mathsf {MP}$ ) $\\mathsf {IL}+ \\mathsf {EM}_1^- \\vdash \\mathsf {MP}$ First note that with the use of $\\mathsf {EM}_{0}$ , we obtain that $\\mathsf {IL}+\\mathsf {EM}_{1}^{-}\\vdash P \\vee \\lnot P$ for any atomic formula $P$ .",
"Therefore $\\mathsf {IL}+\\mathsf {EM}_{1}^{-}$ can prove any propositional tautology, and in particular $\\mathsf {IL}+\\mathsf {EM}_{1}^{-}\\vdash {\\mathsf {P}} \\vee {\\mathsf {Q}} \\leftrightarrow \\lnot (\\lnot {\\mathsf {P}} \\wedge \\lnot {\\mathsf {Q}})$ for any propositional formulas ${\\mathsf {P}}, {\\mathsf {Q}}$ , thus proving that each propositional formula is equivalent to a negative one.",
"Consider now any instance $\\lnot \\lnot \\exists \\alpha \\, {\\mathsf {Q}} \\rightarrow \\exists \\alpha \\, {\\mathsf {Q}}$ of $\\mathsf {MP}$ .",
"Thanks to the previous observation, we obtain $\\mathsf {IL}+\\mathsf {EM}_{1}^{-}\\vdash \\big (\\lnot \\lnot \\exists \\alpha \\, {\\mathsf {Q}} \\rightarrow \\exists \\alpha \\, {\\mathsf {Q}}\\big ) \\leftrightarrow \\big (\\lnot \\lnot \\exists \\alpha \\, {\\mathsf {P}} \\rightarrow \\exists \\alpha \\, {\\mathsf {P}}\\big )$ for some negative formula ${\\mathsf {P}}$ logically equivalent to ${\\mathsf {Q}}$ .",
"The following formal proof shows that $\\mathsf {IL}+\\mathsf {EM}_{1}^- \\vdash \\lnot \\lnot \\exists \\alpha \\, {\\mathsf {P}} \\rightarrow \\exists \\alpha \\, {\\mathsf {P}}$ .",
"$[\\lnot \\lnot \\exists \\alpha \\, {\\mathsf {P}}]_{(2)}$ $[\\exists \\alpha \\, {\\mathsf {P}}]_{(1)}$ $[\\forall \\alpha \\, \\lnot {\\mathsf {P}}]_{\\mathsf {EM}_{1}^-}$ $\\lnot {\\mathsf {P}}$ $[P]_{\\exists }$ $\\bot $ $\\exists $ $\\bot $ $_{(1)}$ $\\lnot \\exists \\alpha \\, {\\mathsf {P}}$ $\\bot $ $\\exists \\alpha \\, {\\mathsf {P}}$ $[\\exists \\alpha \\, \\lnot \\lnot {\\mathsf {P}}]_{\\mathsf {EM}_{1}^-}$ $[{\\mathsf {P}}]_{\\mathsf {EM}_{0}}$ $[\\lnot \\lnot \\, {\\mathsf {P}}]_{\\exists }$ $[\\lnot {\\mathsf {P}}]_{\\mathsf {EM}_{0}}$ $\\bot $ ${\\mathsf {P}}$ $\\mathsf {EM}_{0}$ ${\\mathsf {P}}$ $\\exists $ ${\\mathsf {P}}$ $\\exists \\alpha \\, {\\mathsf {P}}$ $\\mathsf {EM}_{1}^-$ $\\exists \\alpha \\, {\\mathsf {P}}$ $_{(2)}$ $\\lnot \\lnot \\exists \\alpha \\, {\\mathsf {P}} \\rightarrow \\exists \\alpha \\,{\\mathsf {P}}$ Finally, this implies $\\mathsf {IL}+\\mathsf {EM}_{1}^- \\vdash \\lnot \\lnot \\exists \\alpha \\, {\\mathsf {Q}} \\rightarrow \\exists \\alpha \\, {\\mathsf {Q}}$ .",
"Conversely, everything which is provable within our system can be proven by means of first-order logic with full Markov principle.",
"If $\\mathsf {IL}+\\mathsf {EM}_{1}^- \\vdash F$ , then $\\mathsf {IL}+ \\mathsf {MP}\\vdash F$ .",
"We just need to show that $\\mathsf {IL}+ \\mathsf {MP}$ can prove the rules $\\mathsf {EM}_{1}^-$ and $\\mathsf {EM}_{0}$ .",
"For the case of $\\mathsf {EM}_{0}$ , note that $\\mathsf {IL}+\\mathsf {MP}\\vdash \\lnot \\lnot {\\mathsf {P}} \\rightarrow {\\mathsf {P}}$ for all propositional formulas ${\\mathsf {P}}$ , thanks to $\\mathsf {MP}$ .",
"Since for every propositional ${\\mathsf {Q}}$ we have $\\mathsf {IL}+\\mathsf {MP}\\vdash \\lnot \\lnot ({\\mathsf {Q}}\\vee \\lnot {\\mathsf {Q}})$ , we obtain $\\mathsf {IL}+\\mathsf {MP}\\vdash {\\mathsf {Q}} \\vee \\lnot {\\mathsf {Q}}$ , and therefore $\\mathsf {IL}+\\mathsf {MP}$ can prove $\\mathsf {EM}_{0}$ by mean of an ordinary disjunction elimination.",
"In the case of $\\mathsf {EM}_{1}^-$ , if we are given the proofs of $\\forall \\alpha \\, {\\mathsf {P}}$ $\\vdots $ $\\exists \\alpha \\, {\\mathsf {C}}$ and $\\exists \\alpha \\lnot {\\mathsf {P}}$ $\\vdots $ $\\exists \\alpha {\\mathsf {C}}$ in $\\mathsf {IL}+\\mathsf {MP}$ , the following derivation shows a proof of $\\exists \\alpha \\, {\\mathsf {C}}$ in $\\mathsf {IL}+\\mathsf {MP}$ .",
"$[\\forall \\alpha {\\mathsf {P}}]_{(1)}$ $\\vdots $ $\\exists \\alpha {\\mathsf {C}}$ $[\\lnot \\exists \\alpha {\\mathsf {C}}]_{(4)}$ $\\bot $ (1) $\\lnot \\forall \\alpha {\\mathsf {P}}$ $[\\exists \\alpha \\lnot {\\mathsf {P}}]_{(2)}$ $\\vdots $ $\\exists \\alpha {\\mathsf {C}}$ $[\\lnot \\exists \\alpha {\\mathsf {C}}]_{(4)}$ $\\bot $ $_{(2)}$ $\\lnot \\exists \\alpha \\lnot {\\mathsf {P}}$ $[\\lnot {\\mathsf {P}}]_{(3)}$ $\\exists \\alpha \\, \\lnot {\\mathsf {P}}$ $\\bot $ $_{(3)}$ $\\lnot \\lnot {\\mathsf {P}}$ $\\lnot \\lnot \\, {\\mathsf {P}} \\rightarrow \\, {\\mathsf {P}}$ ${\\mathsf {P}}$ $\\forall \\alpha \\, {\\mathsf {P}}$ $\\bot $ (4) $\\lnot \\lnot \\exists \\alpha {\\mathsf {C}}$ $\\lnot \\lnot \\exists \\alpha \\, {\\mathsf {C}} \\rightarrow \\exists \\alpha \\, {\\mathsf {C}}$ $\\exists \\alpha \\, {\\mathsf {C}}$ As in [4], all of our main results about witness extraction are valid not only for closed terms, but also for quasi-closed ones, which are those containing only pure universal assumptions.",
"[Quasi-Closed terms] An untyped proof term $t$ is said to be quasi-closed, if it contains as free variables only hypothesis variables $a_{1}, \\ldots , a_{n}$ , such that each occurrence of them is of the form ${\\mathtt {H}_{a_{i}}^{\\forall {\\vec{\\alpha }} {{\\mathsf {P}}_i}}}$ , where $\\forall \\vec{\\alpha }\\, {\\mathsf {P}}_{i}$ is simply universal.",
"$\\mathsf {IL}+\\mathsf {EM}_{1}^-$ with the reduction rules in figure fig:red enjoys the Subject Reduction Theorem, as a particular case of the Subject Reduction for $\\mathsf {IL}+\\mathsf {EM}$ presented in [4].",
"[Subject Reduction] If $\\Gamma \\vdash t : C$ and $t \\mapsto u$ , then $\\Gamma \\vdash u : C$ .",
"No term of $\\mathsf {IL}+\\mathsf {EM}_{1}^-$ gives rise to an infinite reduction sequence [4].",
"[Strong Normalization] Every term typable in $\\mathsf {IL}+\\mathsf {EM}_{1}^-$ is strongly normalizing.",
"We now update the characterization of proof-terms heads given in theorem:head-form to the case of $\\mathsf {IL}+\\mathsf {EM}_{1}^-$ .",
"[Head of a Proof Term] Every proof term of $\\mathsf {IL}+\\mathsf {EM}_{1}^-$ is of the form: $\\lambda z_1 \\dots \\lambda z_n .",
"r u_1 \\dots u_k $ where $r$ is either a variable $x$ , a constant ${\\mathtt {H}^{P}}$ or ${\\mathtt {H}_{a}^{\\forall {\\alpha } {A}}}$ or $\\mathtt {W}_{a}^{\\exists {\\alpha } {{\\mathsf {P}}} }$ or an excluded middle term ${ u \\parallel _{a} v}$ or ${ u\\, |_{}\\, v}$ , or a term corresponding to an introduction rule $\\lambda x .",
"t$ , $\\lambda \\alpha .",
"t$ , $\\langle t_1, t_2 \\rangle $ , $_{i}(t)$ , $(m, t)$ $u_1, \\dots u_k$ are either lambda terms, first order terms, or one of the following expressions corresponding to elimination rules: $ \\pi _{i}$ , $[x.w_1, y.w_2]$ , $[(\\alpha ,x).t]$ Standard.",
"We now study the shape of the normal terms with the most simple types.",
"Proposition (Normal Form Property) Let ${\\mathsf {P}},{\\mathsf {P}}_1,\\dots {\\mathsf {P}}_n$ be negative propositional formulas, $A_1, \\dots , A_m$ simply universal formulas.",
"Suppose that $ \\Gamma = z_1: {\\mathsf {P}}_1, \\dots z_n: {\\mathsf {P}}_n, a_1 : \\forall \\alpha _1 A_1, \\dots a_m : \\forall \\alpha _m A_m $ and $\\Gamma \\vdash t:\\exists {\\alpha }\\, {\\mathsf {P}}$ or $\\Gamma \\vdash t: {\\mathsf {P}}$ , with $t$ in normal form and having all its free variables among $z_1, \\dots z_n, a_1, \\dots a_m $ .",
"Then: Every occurrence in $t$ of every term ${\\mathtt {H}_{a_{i}}^{\\forall {\\alpha _{i}} {A_i}}}$ is of the active form ${\\mathtt {H}_{a_{i}}^{\\forall {\\alpha _{i}} {A_i}}}m$ , where $m$ is a term of $\\mathcal {L}$ .",
"$t$ cannot be of the form $u\\parallel _{a} v$ .",
"We prove 1. and 2. simultaneously and by induction on $t$ .",
"There are several cases, according to the shape of $t$ : $t=(m, u)$ , $\\Gamma \\vdash t:\\exists {\\alpha }\\, {\\mathsf {P}}$ and $\\Gamma \\vdash u: {\\mathsf {P}}[m/\\alpha ]$ .",
"We immediately get 1. by induction hypothesis applied to $u$ , while 2. is obviously verified.",
"$t=\\lambda x\\, u$ , $\\Gamma \\vdash t: {\\mathsf {P}}={\\mathsf {Q}}\\rightarrow {\\mathsf {R}}$ and $\\Gamma , x: {\\mathsf {Q}} \\vdash u: {\\mathsf {R}}$ .",
"We immediately get 1. by induction hypothesis applied to $u$ , while 2. is obviously verified.",
"$t=\\langle u, v\\rangle $ , $\\Gamma \\vdash t: {\\mathsf {P}}={\\mathsf {Q}}\\wedge {\\mathsf {R}}$ , $\\Gamma \\vdash u: {\\mathsf {Q}}$ and $\\Gamma \\vdash v: {\\mathsf {R}}$ .",
"We immediately get 1. by induction hypothesis applied to $u$ , while 2. is obviously verified.",
"$t={ u\\, |_{}\\, v}$ , $\\Gamma , a: \\lnot {\\mathsf {Q}} \\vdash u: \\exists {\\alpha }\\, {\\mathsf {P}}$ (resp.",
"$u: {\\mathsf {P}}$ ) and $\\Gamma , a: {\\mathsf {Q}} \\vdash v: \\exists {\\alpha }\\,{\\mathsf {P}}$ (resp.",
"$v: {\\mathsf {P}}$ ).",
"We immediately get the thesis by induction hypothesis applied to $u$ and $v$ , while 2. is obviously verified.",
"$t={ u \\parallel _{a} v}$ .",
"We show that this is not possible.",
"Note that $a$ must occur free in $u$ , otherwise $t$ is not in normal form.",
"Since $\\Gamma , a: \\forall \\beta \\, A \\vdash u: \\exists {\\alpha }\\, {\\mathsf {P}}$ , we can apply the induction hypothesis to $u$ , and obtain that all occurrences of hypothetical terms must be active; in particular, this must be the case for the occurrences of ${\\mathtt {H}_{a}^{\\forall {\\beta } {A}}}$ , but this is not possible since $t$ is in normal form.",
"$t={\\mathtt {H}_{a_{i}}^{\\forall {\\alpha } {A_{i}}}}$ .",
"This case is not possible, for $\\Gamma \\vdash t:\\exists {\\alpha }\\, {\\mathsf {P}}$ or $\\Gamma \\vdash t: {\\mathsf {P}}$ .",
"$t={\\mathtt {H}^{{\\mathsf {P}}}}$ .",
"In this case, 1. and 2. are trivially true.",
"$t$ is obtained by an elimination rule and by theorem:head-form-em we can write it as $r\\, t_{1}\\,t_{2}\\ldots t_n$ .",
"Notice that in this case $r$ cannot correspond to an introduction rule neither be a term of the form ${ u \\parallel _{a} v}$ , because of the induction hypothesis, nor ${ u\\, |_{}\\, v}$ , because of the permutation rules and $t$ being in normal form; moreover, $r$ cannot be $\\mathtt {W}_{b}^{\\exists {\\alpha } {P} }$ , otherwise $b$ would be free in $t$ and $b\\ne a_{1}, \\ldots , a_{n}$ .",
"We have now two remaining cases: $r=x_{i}$ (resp.",
"$r={\\mathtt {H}^{{\\mathsf {P}}}}$ ).",
"Then, since $\\Gamma \\vdash x_{i}: {\\mathsf {P}}_{i}$ (resp.",
"$\\Gamma \\vdash {\\mathtt {H}^{{\\mathsf {P}}}}: {\\mathsf {P}}$ ), we have that for each $i$ , either $t_{i}$ is $\\pi _{j}$ or $\\Gamma \\vdash t_{i}: {\\mathsf {Q}}$ , where ${\\mathsf {Q}}$ is a negative propositional formula.",
"By induction hypothesis, each $t_{i}$ satisfies 1. and also $t$ .",
"2. is obviously verified.",
"$r={\\mathtt {H}_{a_{i}}^{\\forall {\\alpha _i} {{A}_{i}}}}$ .",
"Then, $t_{1}$ is $m$ , for some closed term of $\\mathcal {L}$ .",
"Let $A_{i}=\\forall \\gamma _1\\ldots \\forall \\gamma _l \\, {\\mathsf {Q}}$ , with ${\\mathsf {Q}}$ propositional, we have that for each $i$ , either $t_i$ is a closed term $m_i$ of $\\mathcal {L}$ or $t_{i}$ is $\\pi _{j}$ or $\\Gamma \\vdash t_{i}: {\\mathsf {R}}$ , where ${\\mathsf {R}}$ is a negative propositional formula.",
"By induction hypothesis, each $t_{i}$ satisfies 1. and thus also $t$ , while 2. is obviously verified.",
"If we omit the parentheses, we will show that every normal proof-term having as type an existential formula can be written as ${ { { v_0\\, |_{}\\, v_{1}}\\, |_{}\\, }\\ldots \\, |_{}\\, v_{n}}$ , where each $v_{i}$ is not of the form ${ u\\, |_{}\\, v}$ ; if for every $i$ , $v_i$ is of the form $(m_i,u_i)$ , then we call the whole term an Herbrand normal form, because it is essentially a list of the witnesses appearing in an Herbrand disjunction.",
"Formally: [Herbrand Normal Forms] We define by induction a set of proof terms, called Herbrand normal forms, as follows: Every normal proof-term $(m,u)$ is an Herbrand normal form; if $u$ and $v$ are Herbrand normal forms, ${ u\\, |_{}\\, v}$ is an Herbrand normal form.",
"Our last task is to prove that all quasi-closed proofs of any existential statement $\\exists \\alpha \\, A$ include an exhaustive sequence $m_{1}, m_{2}, \\ldots , m_{k}$ of possible witnesses.",
"This theorem is stronger than the usual Herbrand theorem for classical logic [4], since we are stating it for any existential formula and not just for formulas with a single and existential quantifier.",
"[Herbrand Disjunction Extraction] Let $\\exists \\alpha \\,A$ be any closed formula.",
"Suppose $\\Gamma \\vdash t: \\exists \\alpha \\, A$ in $\\mathsf {IL}+\\mathsf {EM}_{1}^-$ for a quasi closed term $t$ , and $t\\mapsto ^{*} t^{\\prime }$ with $t^\\prime $ in normal form.",
"Then $\\Gamma \\vdash t^{\\prime }: \\exists \\alpha \\, A$ and $t^{\\prime }$ is an Herbrand normal form ${ { { (m_{0}, u_{0})\\, |_{}\\, (m_{1}, u_{1})}\\, |_{}\\, }\\ldots \\, |_{}\\, (m_{k}, u_{k})}$ Moreover, $\\Gamma \\vdash A[m_{1}/\\alpha ]\\vee \\dots \\vee A[m_{k}/\\alpha ]$ .",
"By the Subject Reduction Theorem , $\\Gamma \\vdash t^{\\prime }: \\exists \\alpha \\, A$ .",
"We proceed by induction on the structure of $t^\\prime $ .",
"According to theorem:head-form-em, we can write $t^\\prime $ as $r u_1\\dots u_n$ .",
"Note that since $t^{\\prime }$ is quasi closed, $r$ cannot be a variable $x$ ; moreover, $r$ cannot be a term ${\\mathtt {H}^{{\\mathsf {P}}}}$ or ${\\mathtt {H}_{b}^{\\forall {\\alpha } {B}}}$ , otherwise $t^{\\prime }$ would not have type $\\exists \\alpha \\, A$ , nor a term $\\mathtt {W}_{b}^{\\exists {\\alpha } {{\\mathsf {P}}} }$ , otherwise $t^{\\prime }$ would not be quasi closed.",
"$r$ also cannot be of the shape ${ u \\parallel _{a} v}$ , otherwise $\\Gamma \\vdash { u \\parallel _{a} v} : \\exists \\alpha \\, {\\mathsf {Q}}$ , for some negative propositional ${\\mathsf {Q}}$ , but from prop:pnf we know that this is not possible.",
"By theorem:head-form-em, we are now left with only two possibilities.",
"$r$ is obtained by an introduction rule.",
"Then $n=0$ , otherwise there is a redex, and thus the only possibility is $t^\\prime = r = (n, u)$ which is an Herbrand Normal Form.",
"$r = { u\\, |_{}\\, v}$ .",
"Again $n=0$ , otherwise we could apply a permutation rule; then $t^\\prime = r = { u\\, |_{}\\, v}$ , and the thesis follows by applying the induction hypothesis on $u$ and $v$ .",
"We have thus shown that $t^\\prime $ is an Herbrand normal form ${ { { (m_{0}, u_{0})\\, |_{}\\, (m_{1}, u_{1})}\\, |_{}\\, }\\ldots \\, |_{}\\, (m_{k}, u_{k})}$ Finally, we have that for each $i$ , $\\Gamma _{i}\\vdash u_{i}: A[m_{i}/\\alpha ]$ , for the very same $\\Gamma _{i}$ that types $(m_{i}, u_{i})$ of type $\\exists \\alpha \\, A$ in $t^{\\prime }$ .",
"Therefore, for each $i$ , $\\Gamma _{i}\\vdash u_{i}^{+}: A[m_{1}/\\alpha ]\\vee \\dots \\vee A[m_{k}/\\alpha ]$ , where $u_{i}^{+}$ is of the form $_{i_{1}}(\\ldots _{i_{k}}(u_{i})\\ldots )$ .",
"We conclude that $\\Gamma \\vdash { { { u_{0}^{+}\\, |_{}\\, u_{1}^{+}}\\, |_{}\\, }\\ldots \\, |_{}\\, u_{k}^{+}}: A[m_{1}/\\alpha ]\\vee \\dots \\vee A[m_{k}/\\alpha ]$"
],
[
"Markov's Principle in Arithmetic",
"The original statement of Markov's principle refers to Arithmetic and can be formulated in the system of Heyting Arithmetic $\\mathsf {HA}$ as $ \\lnot \\lnot \\exists \\alpha \\, {\\mathsf {P}} \\rightarrow \\exists \\alpha \\, {\\mathsf {P}} \\text{, for } {\\mathsf {P}} \\text{ atomic}$ By adapting $\\mathsf {IL}+\\mathsf {EM}_{1}^-$ to Arithmetic, following [2], we will now provide a new computational interpretation of Markov's principle.",
"Note first of all that propositional formulas are decidable in intuitionistic Arithmetic $\\mathsf {HA}$ : therefore we will not need the rule $\\mathsf {EM}_{0}^-$ and the parallelism operator.",
"For the very same reason, we can expect the system $\\mathsf {HA}+\\mathsf {EM}_{1}^-$ to be constructive and the proof to be similar to the one of Herbrand constructivity for $\\mathsf {IL}+\\mathsf {EM}_{1}^-$ .",
"In this section indeed we will give such a syntactic proof.",
"We could also have used the realizability interpretation for $\\mathsf {HA}+\\mathsf {EM}_{1}$ introduced in [2] (see [11])."
],
[
"The system $\\mathsf {HA}+\\mathsf {EM}_{1}^-$",
"We will now introduce the system $\\mathsf {HA}+\\mathsf {EM}_{1}^-$ .",
"We start by defining the language: [Language of $\\mathsf {HA}+ \\mathsf {EM}_{1}^-$ ] The language $\\mathcal {L}$ of $\\mathsf {HA}+ \\mathsf {EM}_1$ is defined as follows.",
"The terms of $\\mathcal {L}$ are inductively defined as either variables $\\alpha , \\beta ,\\ldots $ or 0 or $\\mathsf {S}(t)$ with $t\\in \\mathcal {L}$ .",
"A numeral is a term of the form $\\mathsf {S}\\ldots \\mathsf {S}0$ .",
"There is one symbol $\\mathcal {P}$ for every primitive recursive relation over $\\mathbb {N}$ ; with $\\mathcal {P}^{\\bot }$ we denote the symbol for the complement of the relation denoted by $\\mathcal {P}$ .",
"The atomic formulas of $\\mathcal {L}$ are all the expressions of the form $\\mathcal {P}(t_{1}, \\ldots , t_{n})$ such that $t_{1}, \\ldots , t_{n}$ are terms of $\\mathcal {L}$ and $n$ is the arity of $\\mathcal {P}$ .",
"Atomic formulas will also be denoted as ${\\mathsf {P}}, \\mathsf {Q}, {\\mathsf {P_i}}, \\ldots $ and $\\mathcal {P}(t_{1}, \\ldots , t_{n})^{\\bot }:=\\mathcal {P}^{\\bot }(t_{1}, \\ldots , t_{n})$ .",
"The formulas of $\\mathcal {L}$ are built from atomic formulas of $\\mathcal {L}$ by the connectives $\\vee ,\\wedge ,\\rightarrow , \\forall ,\\exists $ as usual, with quantifiers ranging over numeric variables $\\alpha ^{ {\\tt N} }, \\beta ^{ {\\tt N} }, \\ldots $ .",
"The system $\\mathsf {HA}+\\mathsf {EM}_{1}^-$ in fig:haemeno extends the usual Curry-Howard correspondence for $\\mathsf {HA}$ with our rule $\\mathsf {EM}_{1}^-$ and is a restriction of the system introduced in [2].",
"The purely universal arithmetical axioms are introduced by means of Post rules, as in Prawitz [12].",
"As we anticipated, there is no need for a parallelism operator.",
"Therefore $\\mathsf {EM}_{1}^-$ introduces a pure delimited exception mechanism, explained by the reduction rules in fig:F: whenever we have a term ${ u \\parallel _{a} v}$ and ${\\mathtt {H}_{a}^{\\forall {\\alpha } {{\\mathsf {P}}}}}m$ appears inside $u$ , we can recursively check whether ${\\mathsf {P}}[m/\\alpha ]$ holds, and switch to the exceptional path if it doesn't; alternatively, if it does hold we can remove the instance of the assumption.",
"When there are no free assumptions relative to $a$ left in $u$ , we can forget about the exceptional path.",
"Figure: Term Assignment Rules for 𝖧𝖠+𝖤𝖬 1 \\mathsf {HA}+\\mathsf {EM}_{1}Figure: Reduction Rules for 𝖧𝖠\\mathsf {HA} + 𝖤𝖬 1 \\mathsf {EM}_{1}Similarly to the previous sections, we extend the characterization of the proof-term heads to take into account the new constructs.",
"[Head of a Proof Term] Every proof term of $\\mathsf {HA}+\\mathsf {EM}_{1}^-$ is of the form: $\\lambda z_1 \\dots \\lambda z_n .",
"r u_1 \\dots u_k $ where $r$ is either a variable $x$ , a constant ${\\mathtt {H}_{a}^{\\forall {\\alpha } {P}}}$ , $\\mathtt {W}_{a}^{\\exists {\\alpha } {P} }$ , $\\mathsf {r}$ or $\\mathsf {R}$ , an excluded middle term ${ u \\parallel _{a} v}$ , or a term corresponding to an introduction rule $\\lambda x .",
"t$ , $\\lambda \\alpha .",
"t$ , $\\langle t_1, t_2 \\rangle $ , $_{i}(t)$ , $(m, t)$ $u_1, \\dots u_k$ are either lambda terms, first order terms, or one of the following expressions corresponding to elimination rules: $ \\pi _{i}$ , $[x.w_1, y.w_2]$ , $[(\\alpha ,x).t]$ The new system proves exactly the same formulas that can be proven by making use of Markov's principle in Heyting Arithmetic.",
"For any formula $F$ in the language $\\mathcal {L}$ , $\\mathsf {HA}+\\mathsf {MP}\\vdash F$ if and only if $\\mathsf {HA}+ \\mathsf {EM}_1^- \\vdash F$ The proof is identical as the one in the previous section.",
"$\\mathsf {HA}+\\mathsf {EM}_{1}^-$ with the reduction rules in figure fig:red enjoys the Subject Reduction Theorem [2], [11].",
"[Subject Reduction] If $\\Gamma \\vdash t : C$ and $t \\mapsto u$ , then $\\Gamma \\vdash u : C$ .",
"No term of $\\mathsf {HA}+\\mathsf {EM}_{1}^-$ gives rise to an infinite reduction sequence [1].",
"[Strong Normalization] Every term typable in $\\mathsf {HA}+\\mathsf {EM}_{1}^-$ is strongly normalizing."
],
[
"$\\mathsf {HA}+\\mathsf {EM}_{1}^-$ is Constructive",
"We can now proceed to prove the constructivity of the system, that is the disjunction and existential properties.",
"We will do this again by inspecting the normal forms of the proof terms; the first thing to do is adapting prop:pnf to $\\mathsf {HA}+\\mathsf {EM}_{1}^-$ .",
"Proposition (Normal Form Property) Let ${\\mathsf {P}},{\\mathsf {P}}_1,\\dots {\\mathsf {P}}_n$ be negative propositional formulas, $A_1, \\dots A_m$ simply universal formulas.",
"Suppose that $ \\Gamma = z_1: {\\mathsf {P}}_1, \\dots z_n: {\\mathsf {P}}_n, a_1 : \\forall \\alpha _1 A_1, \\dots a_m : \\forall \\alpha _m A_m $ and $\\Gamma \\vdash t:\\exists {\\alpha }\\, {\\mathsf {P}}$ or $\\Gamma \\vdash t: {\\mathsf {P}}$ , with $t$ in normal form and having all its free variables among $z_1, \\dots z_n, a_1, \\dots a_m $ .",
"Then: Every occurrence in $t$ of every term ${\\mathtt {H}_{a_{i}}^{\\forall {\\alpha _{i}} {A_i}}}$ is of the active form ${\\mathtt {H}_{a_{i}}^{\\forall {\\alpha _{i}} {A_i}}}m$ , where $m$ is a term of $\\mathcal {L}$ $t$ cannot be of the form $u\\parallel _{a} v$ .",
"The proof is identical to the proof of prop:pnf.",
"We just need to consider the following additional cases: $t=\\mathsf {r} t_1 t_2 \\dots t_n$ .",
"Then $\\Gamma \\vdash t_i : {\\mathsf {Q}}_i$ for some atomic ${\\mathsf {Q}}_i$ and for $i=1 \\dots n$ ; 1. holds by applying the inductive hypothesis to the $t_i$ , while 2. is obviously verified.",
"$t = \\mathsf {R}t_1 \\dots t_n$ .",
"This case is not possible, otherwise since $t_{3}$ is a numeral and thus $t$ would not be in normal form.",
"Thanks to this, we can now state the main theorem.",
"The proof of the existential property is the same as the one for theorem-extraction: we just need to observe that since we don't have a parallelism operator in $\\mathsf {HA}+\\mathsf {EM}_{1}^-$ , every Herbrand disjunction will consist of a single term.",
"The disjunction property will follow similarly.",
"[Constructivity of $\\mathsf {HA}+\\mathsf {EM}_{1}^-$ ] If $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash t : \\exists \\alpha A$ , then there exists a term $t^{\\prime } = (n,u)$ such that $t\\mapsto ^{*}t^{\\prime }$ and $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash u : A[n/\\alpha ]$ If $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash t : A \\vee B$ , then there exists a term $t^{\\prime }$ such that $t\\mapsto ^{*}t^{\\prime }$ and either $t^{\\prime }=_{0}(u)$ and $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash u : A$ , or $t^{\\prime }=_{1}(u)$ and $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash u: B$ For both cases, we start by considering a term $t^{\\prime }$ such that $t \\mapsto ^* t^{\\prime }$ and $t^{\\prime }$ is in normal form.",
"By the Subject Reduction subjectred we have that $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash t^{\\prime } : \\exists \\alpha A$ (resp.",
"$\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash t^{\\prime }: A \\vee B$ ).",
"By theorem:head-form-em we can write $t^{\\prime }$ as $r t_1 \\dots t_n$ .",
"Since $t^{\\prime }$ is closed, $r$ cannot be a variable $x$ or a term ${\\mathtt {H}_{a}^{\\forall {\\alpha } {{\\mathsf {P}}}}}$ or $\\mathtt {W}_{a}^{\\exists {\\alpha } {{\\mathsf {P}}} }$ ; moreover it cannot be $\\mathsf {r}$ , otherwise the type of $t^{\\prime }$ would have to be atomic, and it cannot be $\\mathsf {R}$ , otherwise the term would not be in normal form.",
"$r$ also cannot have been obtained by $\\mathsf {EM}_{1}^-$ , otherwise $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash r : \\exists \\alpha {\\mathsf {P}}$ , for ${\\mathsf {P}}$ atomic and $r={ t_1 \\parallel _{a} t_2}$ ; but this is not possible due to prop:pnf2.",
"Therefore, $r$ must be obtained by an introduction rule.",
"We distinguish now the two cases: $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash t^{\\prime } : \\exists \\alpha B$ .",
"Since the term is in normal form, $n$ has to be 0, that is $t^{\\prime }=r$ and $r=(n, u)$ ; hence also $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash u : A(n)$ .",
"$\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash t^{\\prime } : A \\vee B$ .",
"Then either $t^{\\prime }=_{0}(u)$ , and so $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash u : A$ , or $t^{\\prime }=_{1}(u)$ , and so $\\mathsf {HA}+\\mathsf {EM}_{1}^- \\vdash u: B$ ."
]
] | 1612.05457 |
[
[
"Pressure-Induced Metallization in Iron-Based Ladder Compounds\n Ba$_{1-x}$Cs$_x$Fe$_2$Se$_3$"
],
[
"Abstract Electrical resistivity measurements have been performed on the iron-based ladder compounds Ba$_{1-x}$Cs$_x$Fe$_2$Se$_3$ ($x$ = 0, 0.25, 0.65, and 1) under high pressure.",
"A cubic anvil press was used up to 8.0 GPa, whereas further higher pressure was applied using a diamond anvil cell up to 30.0 GPa.",
"Metallic behavior of the electrical conductivity was confirmed in the $x$ = 0.25 and 0.65 samples for pressures greater than 11.3 and 14.4 GPa, respectively, with the low-temperature $\\log T$ upturn being consistent with weak localization of 2D electrons due to random potential.",
"At pressures higher than 23.8 GPa, three-dimensional Fermi-liquid-like behavior was observed in the latter sample.",
"No metallic conductivity was observed in the parent compounds BaFe$_2$Se$_3$ ($x $ = 0) up to 30.0 GPa and CsFe$_2$Se$_3$ ($x$ = 1) up to 17.0 GPa.",
"The present results indicate that the origins of the insulating ground states in the parent and intermediate compounds are intrinsically different; the former is a Mott insulator, whereas the latter is an Anderson insulator owing to the random substitution of Cs for Ba."
],
[
"Introduction",
"Iron-based ladder compounds $A$ Fe$_2X_3$ ($A$ = K, Rb, Cs, and Ba; $X$ = S, Se, and Te) [1], [2], [3] attract considerable attention as low-dimensional analogues of iron-based superconductors.",
"Figure REF shows crystal structure of iron-based ladder compounds [4].",
"The compounds have quasi-one-dimensional two-leg Fe ladders separated by the $A$ cations.",
"The two-leg ladder is formed by edge-sharing [Fe$X_4$ ] tetrahedral structures, of which the connectivity is the same as those found for the two-dimensional square lattice of Fe in iron-based superconductors.",
"Earlier neutron diffraction experiments show that at low temperatures, several different magnetic structures are stabilized depending on $A$ and $X$ .",
"For instance, block magnetism, in which four magnetic moments on Fe atoms form a ferromagnetic block and each block is aligned antiferromagnetically along the leg direction, was observed in BaFe$_2$ Se$_3$ [5], [6], [7], [8] and stripe magnetism was observed in $A$ Fe$_2$$X_3$ ($A$ = K, Cs, Ba, $X$ = S, Se) [9], [10], [11].",
"These magnetic structures are also analogues to the ones found in the iron-based superconductors, such as the block magnetic structure in the 245 system and the single stripe structure in 1111 systems.",
"Despite the similarity in the [Fe$X_4$ ] edge-sharing connectivity and also in the magnetic structures, the compounds are insulators, in contrast to the metallic nature of iron-based superconductors.",
"Therefore, the metallization of iron-based ladder compounds has been highly desired.",
"Carrier doping is one plausible way of realizing metallization.",
"However, previous studies on hole-doped [11], [12], [13], [14] and electron-doped [15], [14] iron-based ladder compounds concluded that the compounds are insulating for all compositions.",
"Several intriguing findings were, nevertheless, reported.",
"The study of Ba$_{1-x}$ Cs$_x$ Fe$_2$ Se$_3$ revealed a large decrease in resistivity at intermediate compositions [13].",
"The block magnetic order is suppressed completely at $x$ = 0.25, where no magnetic signal was observed in powder neutron diffraction profiles down to 7 K. On the other hand, the low-temperature divergence of the resistivity is most suppressed at $x$ = 0.65, indicating that the system is the closest to the metallic state among the intermediate compounds.",
"Another way of realizing metallization may be to apply pressure.",
"Indeed, Takahashi et al.",
"reported metallization and the appearance of superconductivity in BaFe$_2$ S$_3$ under a pressure of 11 GPa [9].",
"Combining the two observations, i.e.",
"the decrease in the resistivity of career-doped Ba$_{1-x}$ Cs$_x$ Fe$_2$ Se$_3$ and the pressure-induced metallization of BaFe$_2$ S$_3$ , we can expect the metallization of Ba$_{1-x}$ Cs$_x$ Fe$_2$ Se$_3$ under high pressure.",
"In the present study, we have performed resistivity measurements under high pressure for the parent compounds BaFe$_2$ Se$_3$ and CsFe$_2$ Se$_3$ , and for the intermediate compounds with $x$ = 0.25 (Ba$_{0.75}$ Cs$_{0.25}$ Fe$_2$ Se$_3$ ) and $x $ = 0.65 (Ba$_{0.35}$ Cs$_{0.65}$ Fe$_2$ Se$_3$ ).",
"No superconductivity is observed in all the samples, nevertheless, our results show metallic behavior of the $x$ = 0.25 and $x$ = 0.65 samples.",
"It may be noted that in this work, we use ”metallic” when the sample shows a negative temperature coefficient of resistance.",
"We discuss the mechanism of the insulating behavior and metal-insulator transition in iron-based ladder compounds, Ba$_{1-x}$ Cs$_x$ Fe$_2$ Se$_3$ , based on the different responses to pressure of the parent and intermediate compounds."
],
[
"Experimental Procedure",
"Single crystals with $x$ = 0, 0.25, 0.65, and 1 were synthesized by the slow-cooling method[8], [10].",
"In this report, $x$ denotes the nominal composition.",
"The samples were characterized using powder X-ray diffraction with Cu $K\\alpha $ radiation (not shown), and the composition dependence of the lattice parameters almost obeys Vegard's law.",
"Electrical resistivity was measured by the four-probe dc technique with current flow along the leg direction in the $T$ range between 4.2 and 300 K and in the pressure range between 2.0 and 30.0 GPa.",
"Resistivity at ambient pressure was measured by Physical Property Measurement System (PPMS, Quantum Design) [13].",
"A cubic anvil press consisting of tungsten carbide anvils was used to measure resistivity at pressures of 2-8 GPa.",
"The pressure-transmitting medium for the cubic anvil press experiments was a mixture of Fluorinert FC70 and FC77 (3M Company) with a 1:1 ratio.",
"Four gold wires were attached to the sample using gold paste (Tokuriki 8560).",
"A diamond anvil cell (DAC) was used to measure resistance at pressures of 3.8-30.0 GPa.",
"The pressure-transmitting medium was NaCl powder.",
"Thin platinum ribbons were pressed and attached to samples as electrodes.",
"A rhenium gasket was used, and a thin BN layer was installed as electric insulation between the platinum electrodes and the gasket.",
"The cubic anvil press and DAC were placed in a cryostat and cooled with liquid $^{4}$ He."
],
[
"Results",
"Figures REF (a)-REF (d) show the temperature and pressure dependences of electric resistivity for $x = 0, 0.25, 0.65$ , and 1, respectively, measured using a cubic anvil press.",
"Overall, the resistivity decreases with increasing pressure.",
"Upon careful inspection, we found that the parent compounds with $x$ = 0 and 1 show stronger suppression of resistivity than the mixed compounds.",
"The resistivity of the $x = 0, 0.25,$ and 0.65 samples reaches less than $1 \\times 10^{-2} $ $\\Omega $ cm at 8.0 GPa, but there is no metallization or superconductivity.",
"Interestingly, at 8.0 GPa, the $x$ = 0, 0.25, and 0.65 samples show similar resistivities at room temperature, although their crystal and magnetic structures at ambient pressure are different.",
"On the other hand, the $x$ = 1 sample still shows much higher resistivity than the other compounds at room temperature.",
"A structural transition was reported in a previous study [16] for $x$ = 0 at 6 GPa.",
"However, there is no corresponding anomaly in our results.",
"For further higher pressures, we performed resistance measurements using a DAC.",
"Figures REF (e)-REF (h) show the results of electrical resistance for $x$ = 0, 0.25, 0.65, and 1 samples under higher pressures.",
"As seen in Figs.",
"REF (e) and REF (h), the resistance of the parent compounds with $x$ = 0 and 1 increases with decreasing temperature, indicating that the parent compounds are still insulators under pressures of 30.0 GPa and 17.0 GPa, respectively.",
"In contrast to the insulating behavior in the parent compounds, metallic behavior was observed in the $x$ = 0.25 and 0.65 samples, as detailed below.",
"Figure REF (f) shows the pressure dependence of the resistance in the $x$ = 0.25 sample.",
"The resistance is greatly suppressed as the pressure is increased, and indeed, the $R$ -$T$ curves become almost flat on a log scale for pressures greater than 11 GPa.",
"Figure REF (a) shows the same $R$ -$T$ curves of the $x$ = 0.25 sample on a linear scale.",
"As clearly seen in this figure, the resistance between 100 and 200 K decreases with decreasing temperature for pressures greater than 11 GPa, indicating that the metallic state is realized.",
"On further reducing the temperature, resistivity shows an upturn for all the pressures, indicating that the lowest-temperature state is again insulating for $x$ = 0.25.",
"Such an upturn can also be seen in the cuprate ladder compound Sr$_{2.5}$ Ca$_{11.5}$ Cu$_{24}$ O$_{41}$ [17] and the two-dimensional organic conductor (DOET)$_2$ BF$_4$ [18], indicating the existence of a metal-insulator transition at lower temperatures.",
"The arrows in the figure denote the metal-insulator transition temperature, which was determined using d$R$ /d$T$ = 0.",
"Applying higher pressure generally tends to suppress the metal-insulator transition; however, the insulating state for $x = 0.25$ at the lowest temperature is even robust at the highest pressure of 18.2 GPa applied in this work.",
"The temperature dependence of the conductivity for $x$ = 0.25 is shown in a semilogarithmic plot [Fig.",
"REF (c)].",
"One can see that the conductivity is proportional to $\\log T$ below 50 K in the pressure range between 11.3 and 18.2 GPa, where a low-temperature insulating phase can be seen, and slightly deviates from the relation below 7 K. In a two-dimensional system with perturbative random potential, namely, a weakly localized system, it is known that conductivity shows the relation $\\sigma = \\sigma _0 + A\\log T$ , $A > 0$ [19], [20], [21].",
"Hence, the $\\log T$ dependence of the conductivity suggests that this iron-based ladder compound with $x$ = 0.25 can be regarded as a disordered two-dimensional system at these pressures.",
"For $x =$ 0.65, metallization was achieved at 14.4 GPa [Fig.",
"REF (b)].",
"At this pressure, a metal-insulator transition similar to that of the $x$ = 0.25 sample was observed at $\\sim $ 30 K. Figure REF (d) shows the $\\log T$ dependence of the conductivity for $x$ = 0.65.",
"In the low-temperature insulating phase found for 14.4 $\\le $ pressure ($P$ ) $\\le $ 22.0 GPa, the conductivity obeys $\\sigma = \\sigma _0 + A\\log T$ , indicating a two-dimensional feature similar to that of the $x$ = 0.25 sample.",
"The metal-insulator transition is completely suppressed at 23.8 GPa, where we found the system is metallic down to the base temperature.",
"The low-temperature part of the resistance in the range of 23.8 to 27.9 GPa is shown in Fig.",
"REF (e).",
"As seen in the figure, the resistance of the $x$ = 0.65 sample shows fully metallic behavior in the measured temperature range and is well expressed by the power law $\\rho = \\rho _0 + AT^\\alpha $ .",
"The results of the power-law fitting are also shown in Fig.",
"REF (e).",
"The resistivity follows the power law below 50 K. The $\\alpha $ values are almost 2, indicating that the system is in a nearly three-dimensional Fermi liquid state.",
"Figure: (Color online) (a), (b) Metallic behavior of resistance for (a) x=x = 0.25 and (b) x=x = 0.65.",
"Note that the vertical axis is a linear scale.",
"Arrows indicate the metal-insulator transition.",
"(c), (d) Temperature dependence on semilogarithmic scale for (c) x=x = 0.25 at 9.5-18.2 GPa and (d) x=x = 0.65 at 9.7-23.8 GPa.",
"(e) Metallic behavior in the x=x = 0.65 sample at low temperatures at 23.8-27.9 GPa.",
"Note that all points in (c)-(e) are shifted for clarity.",
"Solid lines show the results of fitting with (c), (d) σ=σ 0 +AlogT\\sigma = \\sigma _0 + A \\log {T} and (e) the power law ρ=ρ 0 +AT α \\rho = \\rho _0 + AT^\\alpha ."
],
[
"Discussion",
"As described above, BaFe$_2$ Se$_3$ ($x = 0$ ) and CsFe$_2$ Se$_3$ ($x = 1$ ) show no metallization or superconductivity up to 30.0 and 17.0 GPa, respectively.",
"In contrast, BaFe$_2$ S$_3$ shows metallization and superconductivity at 11 GPa[9].",
"This tendency is consistent with the report that photoemission spectroscopy at ambient pressure revealed a sizable energy gap for all these compounds, with the largest (smallest) energy gap in CsFe$_2$ Se$_3$ (BaFe$_2$ S$_3$ ) [22].",
"The Mott gap of BaFe$_2$ S$_3$ is small enough to be suppressed by a pressure of 11 GPa, hence metallization is achieved.",
"On the other hand, the gaps of BaFe$_2$ Se$_3$ and CsFe$_2$ Se$_3$ are too large to be suppressed by the pressure of 30.0 and 17.0 GPa, respectively.",
"Next, we discuss the resistivity of the intermediate compounds.",
"The intermediate compounds, Ba$_{1-x}$ Cs$_x$ Fe$_2$ Se$_3$ , correspond to career-doped Mott insulator, so they could be metallic at ambient pressure.",
"However, our results show that they are insulators even at ambient pressure and low pressures below 10 GPa.",
"Additionally, the $x$ = 0.25 and 0.65 samples show metallic behavior at higher pressures, in contrast to the robust insulating behavior in the parent compounds.",
"These quantitatively different behaviors of resistivity at high pressure for the intermediate compounds from the parent compounds suggest intrinsically different mechanisms behind the insulating state.",
"To consider the origin of the insulating behavior in the intermediate compounds, their unique temperature dependence of resistivity can be helpful.",
"The resistivity can be well fitted by a one-dimensional variable-range-hopping (VRH) [23] type temperature dependence at ambient pressure [13] and a two-dimensional weakly localized system-type dependence at high pressures.",
"Both temperature dependences are based on the same idea: Anderson localization [24].",
"Therefore, based on the Anderson localization mechanism, we here offer the most plausible scenario for the behavior of resistivity in the intermediate compounds, Ba$_{1-x}$ Cs$_x$ Fe$_2$ Se$_3$ , as follows.",
"At ambient pressure, electrons are strongly localized because of the randomness of the potential, and the sample shows one-dimensional VRH-type resistivity at low temperatures.",
"The Arrhenius-type resistivity, instead of the VRH-type, is observed at room temperature (not shown), but this behavior does not contradict the Anderson localization scenario; electrons can hop to much higher energy states in the nearest sites at high temperature [25].",
"At high pressures, owing to the decrease in atom distances, hopping integral and dimensionality increase, and the random potential is effectively weakened and samples show metallic behavior.",
"However, the effective random potential is still valid as a perturbation, hence the conductivity of the $x$ = 0.25 and 0.65 samples shows a $\\log T$ dependence at low temperatures, reflecting the weakly localized two-dimensional nature.",
"The complete suppression of the effect of the random potential upon the application of further higher pressure realizes the three-dimensional Fermi liquid behavior of the $x$ = 0.65 sample.",
"In addition, the finite $\\gamma $ term of the specific heat for the $x$ = 0.25 and 0.65 samples [13] supports the idea of Anderson localization; the $\\gamma $ term could result from the density of states of electrons at the Fermi level.",
"The random potential could originate from a large difference of iconic radii between Ba (1.49 Å) and Cs (1.81 Å) for Ba$_{1-x}$ Cs$_x$ Fe$_2$ Se$_3$ .",
"The low dimensionality originating from the two-leg ladder structure of Fe lattice enhances effective random potential.",
"The above Anderson localization scenario seems likely, but is a speculation.",
"To confirm this scenario, electrical resistivity and Hall effect measurements under a magnetic field should be important [26].",
"Photoemission spectroscopy is also meaningful for observing the density of states of electrons at the Fermi level.",
"Such tasks are out of the scope of the present study, and a future study along these lines is highly desired."
],
[
"Conclusion",
"We have performed electrical resistivity measurements in the iron-based ladder compounds Ba$_{1-x}$ Cs$_x$ Fe$_2$ Se$_3$ under high pressure using a cubic anvil press and a diamond anvil cell.",
"We achieved metallization in the $x$ = 0.25 and 0.65 samples at 11.3 and 14.4 GPa, respectively.",
"The samples show insulating behavior at low temperatures, indicating that the samples are weakly localized two-dimensional systems.",
"Fully metallic behavior was observed for the $x$ = 0.65 samples in the measured temperature range at 23.8 GPa, and the metallic state shows a three-dimensional Fermi liquid-like temperature dependence below 50 K. We speculate that iron-based ladder compounds have two origins of the insulating state: the first is Mott gap for the parent compounds; the second is the random potential for intermediate compounds."
],
[
"Acknowledgments",
"This research was partly supported by a Grants-in-Aid for Scientific Research (Nos.",
"15F15023, 15H03681, 16H04019, 16H04007, and 23244068) from MEXT of Japan, the Nano-Macro Materials, Devices and System Research Alliance, and the Mitsubishi Foundation."
]
] | 1612.05394 |
[
[
"A Dual Ascent Framework for Lagrangean Decomposition of Combinatorial\n Problems"
],
[
"Abstract We propose a general dual ascent framework for Lagrangean decomposition of combinatorial problems.",
"Although methods of this type have shown their efficiency for a number of problems, so far there was no general algorithm applicable to multiple problem types.",
"In his work, we propose such a general algorithm.",
"It depends on several parameters, which can be used to optimize its performance in each particular setting.",
"We demonstrate efficacy of our method on graph matching and multicut problems, where it outperforms state-of-the-art solvers including those based on subgradient optimization and off-the-shelf linear programming solvers."
],
[
"Introduction",
"Computer vision and machine learning give rise to a number of powerful computational models.",
"It is typical that inference in these models reduces to non-trivial combinatorial optimization problems.",
"For some of the models, such as conditional random fields (CRF), powerful specialized solvers like [46], [47], [11], [50] were developed.",
"In general, however, one has to resort to off-the-shelf integer linear program (ILP) solvers like CPLEX [2] or Gurobi [35].",
"Although these solvers have made a tremendous progress in the past decade, the size of the problems they can tackle still remains a limiting factor for many potential applications, as the running time scales super-linearly in the problem size.",
"The goal of this work is to partially fill this gap between practical requirements and existing computational methods.",
"It is an old observation that many important optimization ILPs can be efficiently decomposed into easily solvable combinatorial sub-problems [31].",
"The convex relaxation, which consists of these sub-problems coupled by linear constraints is known as Lagrangean or dual decomposition [30], [48].",
"Although this technique can be efficiently used in various scenarios to find approximate solutions of combinatorial problems, it has a major drawback: In the most general setting only slow (sub)gradient-based techniques [49], [55], [48], [40], [59] can be used for optimization of the corresponding convex relaxation.",
"In the area of conditional random fields, however, it is well-known [39] that message passing or dual (block-coordinate) ascent algorithms (like e.g.",
"TRW-S [46]) significantly outperform (sub)gradient-based methods.",
"Similar observations were made much earlier in [57] for a constrained shortest path problem.",
"Although dual ascent algorithms were proposed for a number of combinatorial problems (see the related work overview below), there is no general framework, which would (i) give a generalized view on the properties of such algorithms and more importantly (ii) provide tools to easily construct such algorithms for new problems.",
"Our work provides such a framework."
],
[
"Related Work",
"Dual ascent algorithms optimize a dual problem and guarantee monotonous improvement (non-deterioration) of the dual objective.",
"The most famous examples in computer vision are block-coordinate ascent (known also as message passing) algorithms like TRW-S [46] or MPLP [27] for maximum a posteriori inference in conditional random fields [39].",
"To the best of our knowledge the first dual ascent algorithm addressing integer linear programs belongs to Bilde and Krarup [10] (the corresponding technical report in Danish appeared 1967).",
"In that work an uncapacitated facility location problem was addressed.",
"A similar problem (simple plant location) was addressed with an algorithm of the same class in [29].",
"In 1980 Fisher and Hochbaum [21] constructed a dual ascent-based algorithm for a problem of database location in computer networks, which was used to optimize the topology of Arpanet [1], predecessor of Internet.",
"The generalized linear assignment problem was addressed by the same type of algorithms in [22].",
"The Authors considered a Lagrangean decomposition of this problem into multiple knapsack problems, which were solved in each iteration of the method.",
"An improved version of this algorithm was proposed in [33].",
"Efficient dual ascent based solvers were also proposed for the min-cost flow in [24], for the set covering and the set partitioning problems in [23] and the resource-constrained minimum weighted arborescence problem in [34].",
"The work [32] describes basic principles for constructing dual ascent algorithms.",
"Although the authors provide several examples, they do not go beyond that and stick to the claim that these methods are structure dependent and problem specific.",
"The work [17] suggests to use the max-product belief propagation [71] to decomposable optimization problems.",
"However, their algorithm is neither monotone nor even convergent in general.",
"In computer vision, dual block coordinate ascent algorithms for Lagrangean decomposition of combinatorial problems were proposed for multiple targets tracking [7], graph matching (quadratic assignment) problem [76] and inference in conditional random fields [46], [47], [27], [72], [73], [60], [36], [54], [70].",
"From the latter, the TRW-S algorithm [46] is among the most efficient ones for pairwise conditional random fields according to [39].",
"The SRMP algorithm [47] generalizes TRW-S to conditional random fields of arbitrary order.",
"In a certain sense, our framework can be seen as a generalization of SRMP to a broad class of combinatorial problems."
],
[
"Contribution.",
"We propose a new dual ascent based computational framework for combinatorial optimization.",
"To this end we: (i) Define the class of problems, called integer-relaxed pairwise-separable linear programs (IRPS-LP), our framework can be used for.",
"Our definition captures Lagrangean decompositions of many known discrete optimization problems (Section ).",
"(ii) Give a general monotonically convergent message-passing algorithm for solving IRPS-LP, which in particular subsumes several known solvers for conditional random fields (Section ).",
"(iii) Give a characterization of the fixed points of our algorithm, which subsumes such well-known fixed point characterizations as weak tree agreement [46] and arc-consistency [72] (Section ).",
"We demonstrate efficiency of our method by outperforming state-of-the-art solvers for two famous special cases of IRPS-LP, which are widely used in computer vision: the multicut and the graph matching problems.",
"(Section ).",
"A C++-framework containing the above mentioned solvers and the datasets used in experiments are available under http://github.com/pawelswoboda/LP_MP.",
"We give all proofs in the supplementary material."
],
[
"Notation.",
"Undirected graphs will be denoted by $G=(V,E)$ , where $V$ is a finite node set and $E\\subseteq {{V}\\atopwithdelims (){2}}$ is the edge set.",
"The set of neighboring nodes of $v \\in V$ w.r.t.",
"graph $G$ is denoted by $\\mathcal {N}_G(v) := \\lbrace u: uv \\in E\\rbrace $ .",
"The convex hull of a set $X\\subset \\mathbb {R}^n$ is denoted by $\\operatornamewithlimits{conv}(X)$ .",
"Disjoint union is denoted by $\\dot{\\cup }$ ."
],
[
"Integer-Relaxed Pairwise-Separable Linear Programs (IRPS-LP)",
"Combinatorial problems having an objective to minimize some cost $\\theta (x)$ over a set $X\\subseteq \\lbrace 0,1\\rbrace ^n$ of binary vectors often have a decomposable representation as $\\min _{x_i\\in X_i\\atop i=1,\\dots ,k} \\sum _{i=1}^k \\langle \\theta _i,x_i \\rangle $ for $X_i\\subseteq \\lbrace 0,1\\rbrace ^{d_i}$ being sets of binary vectors, typically corresponding to subsets of the coordinates of $X$ .",
"This decomposed problem is equivalent to the original one under a set of linear constraints $A_{(i,j)}x_i = A_{(j,i)} x_j$ , which guarantee the mutual consistency of the considered components.",
"Replacing $X_i$ by its convex hull $\\operatornamewithlimits{conv}(X_i)$ and therefore switching to real-valued vectors from binary ones one obtains a convex relaxationMore precisely, this is a linear programming relaxation, since a convex hull of a finite set can be represented in terms of linear inequalities of the problem, which reads: $\\min _{\\mu \\in \\Lambda _{\\mathbb {G}}} \\sum _{i=1}^k \\langle \\theta _i,\\mu _i \\rangle \\,,\\ \\text{where}\\ \\Lambda _{\\mathbb {G}}\\ \\text{is defined as}$ $\\Lambda _{\\mathbb {G}} :=\\left\\lbrace (\\mu _1\\dots \\mu _k)\\left|\\begin{array}{ll}\\mu _i \\in \\operatornamewithlimits{conv}(X_i) & i\\in \\mathbb {F}\\\\A_{(i,j)}\\mu _i = A_{(j,i)} \\mu _j & \\forall ij \\in \\mathbb {E}\\end{array}\\right\\rbrace \\right..$ Here $\\mathbb {F}:= \\lbrace 1,\\ldots ,k\\rbrace $ are called factors of the decomposition and $\\mathbb {E}\\subseteq \\begin{pmatrix} \\mathbb {F}\\\\ 2 \\end{pmatrix}$ are called coupling constraints.",
"The undirected graph $\\mathbb {G}= (\\mathbb {F},\\mathbb {E})$ is called factor graph.",
"We will use variable names $\\mu $ whenever we want to emphasize $\\mu _i \\in \\operatornamewithlimits{conv}(X_i)$ and $x$ whenever $x_i \\in X_i$ , $i\\in \\mathbb {F}$ .",
"Definition 1 (IRPS-LP) Assume that for each edge $ij \\in \\mathbb {E}$ the matrices of the coupling constraints $A_{(i,j)}, A_{(j,i)}$ are such that $A_{(i,j)} \\in \\lbrace 0,1\\rbrace ^{K \\times d_i}$ and $A_{(i,j)} x_i \\in \\lbrace 0,1\\rbrace ^K$ $\\forall x_i \\in X_i$ for some $K \\in \\mathbb {N}$ , analogously for $A_{(j,i)}$ .",
"The problem $\\min _{\\mu \\in \\Lambda _{\\mathbb {G}}} \\sum _{i\\in \\mathbb {F}} \\langle \\theta _i, \\mu _i \\rangle $ is called an Integer-Relaxed Pairwise-Separable Linear Program, abbreviated by IRPS-LP.",
"In the following, we give several examples of IRPS-LP.",
"To distinguish between notation for the factor graph of IRPS-LP, where we stick to bold letters (such as $\\mathbb {G}$ , $\\mathbb {F}$ , $\\mathbb {E}$ ) we will use the straight font (such as $\\mathsf {G}$ , $\\mathsf {V}$ , $\\mathsf {E}$ ) for the graphs occurring in the examples.",
"[MAP-inference for CRF] A conditional random field is given by a graph $\\mathsf {G}= (\\mathsf {V},\\mathsf {E})$ , a discrete label space $X= \\prod _{u \\in \\mathsf {V}} X_u$ , unary $\\theta _u : X_u \\rightarrow \\mathbb {R}$ and pairwise costs $\\theta _{uv} : X_u \\times X_v \\rightarrow \\mathbb {R}$ for $u \\in \\mathsf {V}$ , $uv \\in \\mathsf {E}$ .",
"We also denote $X_{uv}:=X_u\\times X_v$ .",
"The associated maximum a posteriori (MAP)-inference problem reads $\\min _{x \\in X} \\sum \\nolimits _{u \\in \\mathsf {V}} \\theta _u(x_u) + \\sum \\nolimits _{uv \\in \\mathsf {E}} \\theta _{uv}(x_{uv})\\,,$ where $x_u$ and $x_{uv}$ denote the components corresponding to node $u \\in \\mathsf {V}$ and edge $uv\\in \\mathsf {E}$ respectively.",
"The well-known local polytope relaxation [72] can be seen as an IRPS-LP by setting $\\mathbb {F}= \\mathsf {V}\\cup \\mathsf {E}$ , that is associating to each node $v \\in \\mathsf {V}$ and each edge $uv \\in \\mathsf {E}$ a factor, and introducing two coupling constraints for each edge of the graphical model, i.e.",
"$\\mathbb {E}= \\lbrace \\lbrace u,uv\\rbrace , \\lbrace v,uv\\rbrace : uv \\in \\mathsf {E}\\rbrace $ .",
"For the sake of notation we will assume that each label $s\\in \\ X_u$ is associated a unit vector $(0,\\dots ,0,\\mathop {\\vtop {\\m@th {#\\crcr \\hfil \\displaystyle {1}\\hfil \\crcr {\\hspace{3.0pt}}\\tiny \\crcr {\\hspace{3.0pt}}}}}\\limits _{{s}},0\\dots ,0)$ with dimensionality equal to the total number of labels $|X_u|$ and 1 on the $s$ -th position.",
"Therefore, the notation $\\operatornamewithlimits{conv}(X_u)$ makes sense as a convex hull of all such vectors.",
"After denoting an $N$ -dimensional simplex as $\\Delta _N:=\\lbrace \\mu \\in \\mathbb {R}_+^N\\colon \\sum _{i=1}^N \\mu _i=1 \\rbrace $ the resulting relaxation reads $\\min _{\\mu \\in \\mathsf {L}_{\\mathsf {G}}} \\quad \\langle \\theta ,\\mu \\rangle :=\\sum _{u\\in \\mathsf {V}}\\langle \\theta _u,\\mu _u\\rangle + \\sum _{uv\\in \\mathsf {E}}\\langle \\theta _{uv},\\mu _{uv}\\rangle $ in the overcomplete representation [69] and $\\mathsf {L}_{\\mathsf {G}}$ is defined as $\\hspace{-10.0pt} \\begin{array}{ll}\\underline{\\mu _u\\in \\operatornamewithlimits{conv}(X_{u}):} & \\mu _u\\in \\Delta _{|X_u|}, u\\in \\mathsf {V}\\\\\\underline{\\mu _{uv}\\in \\operatornamewithlimits{conv}(X_{uv}):} & \\mu _{uv}\\in \\Delta _{|X_{uv}|}, uv\\in \\mathsf {E}\\\\\\underline{A_{(uv,u)}\\mu _{uv}=A_{(u,uv)}\\mu _u:} & \\hspace{-7.0pt}\\sum \\limits _{x_v\\in X_v}\\hspace{-3.0pt}\\mu _{uv}(x_u,x_v) = \\mu _{u}(x_u),\\\\& uv\\in \\mathsf {E}, (x_u,x_v)\\in X_{uv},\\\\ & u\\in uv, x_u\\in X_u\\,.\\end{array}$ Here $\\mu _u(x_u)$ and $\\mu _{uv}(x_u,x_v)$ denote those coordinates of vectors $\\mu _u$ and $\\mu _{uv}$ , which correspond to the label $x_u$ and the pair of labels $(x_u,x_v)$ respectively.",
"[Graph Matching] The graph matching problem, also known as quadratic assignment [12] or feature matching, can be seen as a MAP-inference problem for CRFs (as in Example ) equipped with additional constraints: The label set of $\\mathsf {G}$ belongs to a universe $\\mathcal {L}$ , i.e.",
"$X_u \\subseteq \\mathcal {L}$ $\\forall u \\in \\mathsf {V}$ and each label $s \\in \\mathcal {L}$ can be assigned at most once.",
"The overall problem reads $\\min _{x} \\sum _{u \\in \\mathsf {V}} \\theta _u(x_u) + \\sum _{uv \\in \\mathsf {E}} \\theta _{uv}(x_u,x_v) \\ \\text{s.t.}",
"\\ x_u \\ne x_v \\forall u\\ne v\\,.$ Graph matching is a key step in many computer vision applications, among them tracking and image registration, whose aim is to find a one-to-one correspondence between image points.",
"For this reason, a large number of solvers have been proposed in the computer vision community [17], [74], [76], [68], [51], [67], [61], [28], [77], [37], [52], [15].",
"Among them two recent methods [68], [76] based on Lagrangean decomposition show superior performance and provide lower bounds for their solutions.",
"The decomposition we describe below, however, differs from those proposed in [68], [76].",
"Our IRPS-LP representation for graph matching consists of two blocks: (i) the CRF itself (which further decomposes into node- and edge-subproblems with variables $(\\mu _u)_{u \\in \\mathsf {V}}$ and (ii) additional label-factors keeping track of nodes assigned the label $s$ .",
"We introduce these label-factors for each label ${s \\in \\mathcal {L}}$ .",
"The set of possible configurations of this factor $X_s := \\lbrace u \\in \\mathsf {V}: s \\in X_u\\rbrace \\cup \\lbrace \\#\\rbrace $ consists of those nodes $u \\in \\mathsf {V}$ which can be assigned the label $s$ and an additional dummy node $\\#$ .",
"The dummy node $\\#$ denotes non-assignment of the label $s$ and is necessary, as not every label needs to be taken.",
"As in Example , we associate a unit binary vector with each element of the set $X_s$ , and $\\operatornamewithlimits{conv}(X_s)$ denotes the convex hull of such vectors.",
"The set of factors becomes $\\mathbb {F}= \\mathsf {V}\\dot{\\cup }\\mathsf {E}\\dot{\\cup }\\mathcal {L}$ , with the set $\\mathbb {E}= \\lbrace \\lbrace u,uv\\rbrace , \\lbrace v, uv\\rbrace : uv \\in \\mathsf {E}\\rbrace \\cup \\lbrace \\lbrace u,l\\rbrace : u \\in \\mathsf {V}, l \\in X_u \\rbrace $ of the factor-graph edges.",
"The resulting IRPS-LP formulation reads $ & \\min _{\\mu ,\\tilde{\\mu }}\\sum _{u\\in \\mathsf {V}}\\langle \\theta _u,\\mu _u\\rangle + \\sum _{uv\\in \\mathsf {E}}\\langle \\theta _{uv},\\mu _{uv}\\rangle + \\sum _{s\\in \\mathcal {L}}\\langle \\tilde{\\theta }_s,\\tilde{\\mu }_s\\rangle \\\\& \\begin{array}{ll}\\mu \\in \\mathsf {L}_{\\mathsf {G}} \\\\\\tilde{\\mu }_{s} \\in \\operatornamewithlimits{conv}(X_s), & s \\in \\mathcal {L}\\\\\\mu _{u}(s) = \\tilde{\\mu }_{s}(u), & s \\in X_u\\,.\\end{array} \\nonumber $ Here we introduced (i) auxiliary variables $\\tilde{\\mu }_{s}(u)$ for all variables $\\mu _u(s)$ and (ii) auxiliary node costs $\\tilde{\\theta }_{s} \\equiv 0$ $\\forall s \\in \\mathcal {L}$ , which may take other values in course of optimization.",
"Factors associated with the vectors $\\mu _u$ and $\\mu _{uv}$ correspond to the nodes and edges of the graph $\\mathsf {G}$ (node- and edge-factors), as in Example and are coupled in the same way.",
"Additionally, factors associated with the vectors $\\tilde{\\mu }_s$ ensure that the label $s$ can be taken at most once.",
"These label-factors are coupled with node-factors (last line in (REF )).",
"[Multicut] The multicut problem (also known as correlation clustering) for an undirected weighted graph $\\mathsf {G}=(\\mathsf {V},\\mathsf {E})$ is to find a partition $(\\Pi _1,\\ldots ,\\Pi _k)$ , $\\Pi _i\\subseteq \\mathsf {V}$ , $\\mathsf {V}=\\dot{\\cup }_{i=1}^{k}\\Pi _i$ of the graph vertexes, such that the total cost of edges connecting different components is minimized.",
"The number $k$ of components is not fixed but is determined by the algorithm.",
"See Fig.",
"for an illustration.",
"Although the problem has numerous applications in computer vision [3], [4], [5], [75] and beyond [6], [58], [13], [14], there is no scalable solver, which could provide optimality bounds.",
"Existing methods are either efficient primal heuristics [64], [56], [26], [18], [19], [8], [9] or combinatorial branch-and-bound/branch-and-cut/column generation algorithms, based on off-the-shelf LP solvers [41], [42], [45], [75].",
"Move-making algorithms do not provide lower bounds, hence, one cannot judge their solution quality or employ them in branch-and-bound procedures.",
"Off-the-shelf LP solvers on the other hand scale super-linearly, limiting their application in large-scale problems.",
"Instead of directly optimizing over partitions (which has many symmetries making optimization difficult in a linear programming setting), we follow [16] and formulate the problem in the edge domain.",
"Let $\\theta _e$ , $e\\in \\mathsf {E}$ denote the cost of graph edges and let $\\mathsf {C}$ be the set of all cycles of the graph $\\mathsf {G}$ .",
"Each edge that belongs to different components is called a cut edge.",
"The multicut problem reads $\\hspace{-10.0pt}\\min _{x_{\\mathsf {E}}\\in \\lbrace 0,1\\rbrace ^{|\\mathsf {E}|}}\\sum _{e\\in \\mathsf {E}}\\theta _{e}x_{e}\\,,\\ \\ \\text{s.t.",
"}\\ \\forall \\mathsf {C}\\ \\forall e^{\\prime }\\in \\mathsf {C}\\colon \\sum _{e\\in \\mathsf {C}\\backslash \\lbrace e^{\\prime }\\rbrace }\\hspace{-5.0pt}x_{e} \\ge x_{e^{\\prime }} \\,.$ Here $x_e=1$ signifies a cut edge and the inequalities force each cycle to have none or at least two cut edges.",
"The formulation (REF ) has exponentially many constraints.",
"However, it is well-known that it is sufficient to consider only chordless cycles [16] in place of the set $\\mathsf {C}$ in (REF ).",
"Moreover, the graph can be triangulated by adding additional edges with zero weights and therefore the set of chordless cycles reduces to edge triples.",
"Such triangulation is refered to as chordal completion in the literature [25].",
"The number of triples is cubic, which is still too large for practical efficiency and therefore violated constraints are typically added to the problem iteratively in a cutting plane manner [41], [42].",
"To simplify the description, we will ignore this fact below and consider all these cycles at once.",
"Assuming a triangulated graph and redefining $\\mathsf {C}$ as the set of all chordless cycles (triples) we consider the following IRPS-LP relaxation of the multicut problem One can show that this relaxation coincides with the standard LP relaxation for the multicut problem [16]: $& \\min _{\\mu ,\\tilde{\\mu }}\\sum _{e\\in \\mathsf {E}}\\theta _{e}\\mu _{e} + \\sum _{c\\in \\mathsf {C}}\\sum _{e\\in c}\\tilde{\\theta }_{e,c}\\tilde{\\mu }_{e,c}\\,,\\quad \\text{s.t.",
"}\\\\&{\\small \\left\\lbrace \\begin{array}{l}\\mu _{e}\\in \\operatornamewithlimits{conv}\\lbrace \\lbrace 0,1\\rbrace \\rbrace = [0,1],\\ e\\in \\mathsf {E}\\\\\\forall c\\in \\mathsf {C},\\ e\\in c\\colon \\\\\\tilde{\\mu }_{c}:=(\\tilde{\\mu }_{e,c})_{e\\in c}\\in \\operatornamewithlimits{conv}\\lbrace \\lbrace 0,1\\rbrace ^3|\\ \\forall e^{\\prime }\\in \\ c\\colon \\hspace{-5.0pt} \\sum \\limits _{e\\in c\\backslash \\lbrace e^{\\prime }\\rbrace }\\hspace{-9.0pt}\\tilde{\\mu }_{e,c} \\ge \\tilde{\\mu }_{e^{\\prime },c}\\rbrace \\\\\\hspace{10.0pt}\\equiv \\operatornamewithlimits{conv}\\lbrace \\lbrace 0,0,0\\rbrace ,\\lbrace 0,1,1\\rbrace ,\\lbrace 1,0,1\\rbrace ,\\lbrace 1,1,0\\rbrace ,\\lbrace 1,1,1\\rbrace \\rbrace \\\\\\mu _e = \\tilde{\\mu }_{e,c}\\end{array}\\right.",
"}$ For the sake of notation we shortened a feasible set definition $\\tilde{\\mu }\\in \\operatornamewithlimits{conv}\\lbrace \\mu ^{\\prime }\\in \\lbrace 0,1\\rbrace ^n\\colon \\text{constraints on}\\ \\mu ^{\\prime }\\rbrace $ to $\\tilde{\\mu }\\in \\operatornamewithlimits{conv}\\lbrace \\lbrace 0,1\\rbrace ^n\\colon \\text{constraints on}\\ \\tilde{\\mu }\\rbrace $ .",
"Here $\\mu _e$ is the relaxed (potentially non-integer) variable corresponding to $x_e$ .",
"Variable $\\tilde{\\mu }_{e,c}$ is a copy of $\\mu _e$ , which corresponds to the cycle $c$ .",
"Therefore, each $\\mu _e$ gets as many copies $\\tilde{\\mu }_{e,c}$ , as many chordless cycles $c$ contain the edge $e$ .",
"For each cycle the set of binary vectors satisfying the cycle inequality is considered.",
"For a cycle with 3 edges this set can be written explicitly as in ().",
"Along with copies of $\\mu _e$ , $e\\in \\mathsf {E}$ we copy the corresponding cost $\\theta _e$ and create auxiliary costs $\\tilde{\\theta }_{e,c}\\equiv 0$ for each cycle $c$ containing the edge $e$ .",
"During optimization, the cost $\\theta _e$ will be redistributed between $\\theta _e$ itself and its copies $\\tilde{\\theta }_{e,c}$ , $c\\in \\mathsf {C}$ .",
"The factors of the IRPS-LP are associated with each edge (variable $\\mu _e$ ) and each chordless cycle (variable $\\tilde{\\mu }_{c}$ ).",
"Coupling constraints connect edge-factors with those cycle-factors, which contain the corresponding edge (see the last constraint in ()).",
"An in-depth discussion of message passing for the multicut problem with tighter relaxations can be found in [66].",
"decorations.pathmorphing cut-edge=[densely dotted,thick,reddy] vertex=[circle, draw, fill=white, inner sep=0pt, minimum width=1ex] every picture/.append style=baseline,scale=1.1 [scale=0.7] [draw=mycolor1!30, fill=mycolor1!30] plot[smooth cycle, tension=0.5] coordinates (-0.3, 2.3) (2.3, 2.3) (2.3, 0.7) (0.7, 0.7) (0.7, 1.7) (-0.3, 1.7); [draw=mycolor1!30, fill=mycolor1!30] plot[smooth cycle, tension=0.5] coordinates (-0.3, -0.3) (-0.3, 1.3) (0.3, 1.3) (0.3, 0.3) (1.3, 0.3) (1.3, -0.3); [draw=mycolor1!30, fill=mycolor1!30] plot[smooth cycle, tension=0.5] coordinates (1.7, -0.3) (1.7, 0.3) (2.7, 0.3) (2.7, 2.3) (3.3, 2.3) (3.3, -0.3); (0, 0) – (0, 1); (0, 0) – (1, 0); (0, 2) – (1, 2); (1, 1) – (1, 2); (1, 1) – (2, 1); (1, 2) – (2, 2); (2, 1) – (2, 2); (2, 0) – (3, 0); (3, 0) – (3, 1); (3, 1) – (3, 2); [style=cut-edge] (0, 1) – (0, 2); [style=cut-edge] (0, 1) – (1, 1); [style=cut-edge] (1, 0) – (1, 1); [style=cut-edge] (1, 0) – (2, 0); [style=cut-edge] (2, 0) – (2, 1); [style=cut-edge] (2, 1) – (3, 1); [style=cut-edge] (2, 2) – (3, 2); style=vertex] at (0, 0) ; style=vertex] at (1, 0) ; style=vertex] at (0, 1) ; style=vertex] at (0, 2) ; style=vertex] at (1, 1) ; style=vertex] at (1, 2) ; style=vertex] at (2, 1) ; style=vertex] at (2, 2) ; style=vertex] at (2, 0) ; style=vertex] at (3, 0) ; style=vertex] at (3, 1) ; style=vertex] at (3, 2) ; Illustration of Example .",
"A multicut of a graph induced by three connected components $\\Pi _1,\\Pi _2,\\Pi _3$ (green).",
"Red dotted edges indicate cut edges $x_e = 1$ ."
],
[
"Dual Problem and Admissible Messages",
"Since our technique can be seen as a dual ascent, we will not optimize the primal problem (REF ) directly, but instead maximize its dual lower bound."
],
[
"Dual IPS-LP",
"The Lagrangean dual to (REF ) w.r.t.",
"the coupling constraints reads $\\begin{array}{ll}\\max _{\\phi } & D(\\phi ) := \\sum _{i \\in \\mathbb {F}} \\min _{x_i \\in X_i} \\langle \\theta ^{\\phi }_i, x_i \\rangle \\\\\\text{s.t.}",
"& \\theta ^{\\phi }_i := \\theta _i + \\sum _{j : ij \\in \\mathbb {E}} A_{(i,j)}^_{(i,j)} \\quad \\forall i \\in \\mathbb {F}\\\\& \\phi _{(i,j)} = -\\phi _{(j,i)} \\quad \\forall ij \\in \\mathbb {E}\\end{array}$ Here $\\phi _{(i,j)}\\in \\mathbb {R}^{K}$ for $A_{(i,j)}\\in \\lbrace 0,1\\rbrace ^{K\\times d_i}$ for some $K \\in \\mathbb {N}$ .",
"The function $D(\\phi )$ is called lower bound and is concave in $\\phi $ .",
"The modified primal costs $\\theta ^{\\phi }$ are called reparametrizations of the potentials $\\theta $ .",
"We have duplicated the dual variables by introducing $\\phi _{(i,j)}: = - \\phi _{(j,i)}$ to symmetrize notation.",
"In practice, only one copy is stored and the other is computed on the fly.",
"Note that in this doubled notation the reparametrized node and edge potentials of the CRF from Example read $\\begin{array}{l}\\theta _u^{\\phi }(x_u) = \\theta _u(x_u) +\\sum _{v\\colon uv\\in \\mathsf {E}}\\phi _{u,uv}(x_u)\\\\\\theta _{uv}^{\\phi }(x_u,x_v) = \\theta _{uv}(x_u,x_v) + \\phi _{uv,v}(x_v) + \\phi _{uv,u}(x_u)\\\\\\phi _{u,uv} = - \\phi _{uv,u}\\end{array}$ It is well-known for CRFs that cost of feasible solutions are invariant under reparametrization.",
"We generalize this to the IRPS-LP-case.",
"Proposition 1 $\\sum _{i\\in \\mathbb {F}} \\langle \\theta _i,\\mu _i\\rangle = \\sum _{i\\in \\mathbb {F}} \\langle \\theta ^{\\phi }_i,\\mu _i\\rangle $ , whenever $\\mu _1,\\ldots ,\\mu _k$ obey the coupling constraints."
],
[
"Admissible Messages",
"While Proposition REF guarantees that the primal problem is invariant under reparametrizations, the dual lower bound $D(\\phi )$ is not.",
"Our goal is to find $\\phi $ such that $D(\\phi )$ is maximal.",
"By linear programming duality, $D(\\phi )$ will then be equal to the optimal value of the primal (REF ).",
"First we will consider an elementary step of our future algorithm and show that it is non-decreasing in the dual objective.",
"This property will ensure the monotonicity of the whole algorithm.",
"Let $\\theta ^{\\phi }$ be any reparametrization of the problem and $D(\\phi )$ be the corresponding dual value.",
"Let us consider changing the reparametrization of a factor $i$ by a vector $\\Delta $ with the only non-zero components $\\Delta _{(i,j)}$ and $\\Delta _{(j,i)}$ .",
"This will change reparametrization of the coupled factors $j$ (such that $ij\\in \\mathbb {E}$ ) due to $\\Delta _{(i,j)}=-\\Delta _{(j,i)}$ .",
"The lemma below states properties of $\\Delta _{(i,j)}$ which are sufficient to guarantee improvement of the corresponding dual value $D(\\phi +\\Delta )$ : Lemma 1 (Monotonicity Condition) Let $ij\\in \\mathbb {E}$ be a pair of factors related by the coupling constraints and $\\phi _{(i,j)}$ be a corresponding dual vector.",
"Let $x_i^* \\in \\operatornamewithlimits{arg min}\\limits _{x_i\\in X_i} \\langle \\theta ^{\\phi }_i, x_i \\rangle $ and $\\Delta _{(i,j)}$ satisfy $\\Delta _{(i,j)}(s){\\left\\lbrace \\begin{array}{ll}\\ge 0, & \\nu (s) =1\\\\\\le 0, & \\nu (s) =0\\end{array}\\right.",
"},\\ \\text{where}\\ \\nu :=A_{(i,j)} x^*_i\\,.$ Then $x_i^* \\in \\operatornamewithlimits{arg min}\\limits _{x_i\\in X_i} \\langle \\theta ^{\\phi +\\Delta }_i, x_i \\rangle $ implies ${D(\\phi ) \\le D(\\phi +\\Delta )}$ .",
"Let us apply Lemma REF to Example .",
"Let $ij$ correspond to $\\lbrace u,uv\\rbrace $ , where $u\\in \\mathsf {V}$ is some node and $uv\\in \\mathsf {E}$ is any of its incident edges.",
"Then $x_i^*$ corresponds to a locally optimal label $x^*_u\\in \\arg \\min _{s\\in X_u}\\theta _u(s)$ and $\\nu (s)=\\llbracket s= x^*_u\\rrbracket $ .",
"Therefore we may assign $\\Delta _{u,uv}(s)$ to any value from $[0,\\theta _u(x^*_u) - \\theta _u(s)]$ .",
"This assures that (REF ) is fulfilled and $x^*_u$ remains a locally optimal label after reparametrization even if there are multiple optima in $X_u$ .",
"Lemma REF can be straightforwardly generalized to the case, when more than two factors must be reparametrized simultaneously.",
"In terms of Example this may correspond to the situation when a graph node sends messages to several incident edges at once: Definition 2 Let ${i\\in \\mathbb {F}}$ be a factor and ${J = \\lbrace j_1,\\ldots ,j_l\\rbrace \\subseteq \\mathcal {N}_{\\mathbb {G}}(i)}$ be a subset of its neighbors.",
"Let $\\theta _i^{\\Delta }:=\\theta _i+\\sum _{j \\in J} A^{(i,j)} \\Delta _{(i,j)}$ , $\\Delta _{(i,j)}(=-\\Delta _{(j,i)})$ satisfies (REF ) for all $j\\in J$ and all other coordinates of $\\Delta $ are zero.",
"If there exists $x_i^* \\in \\operatornamewithlimits{arg min}\\nolimits _{x_i\\in X_i} \\langle \\theta _i, x_i \\rangle $ such that ${x_i^* \\in \\operatornamewithlimits{arg min}\\nolimits _{x_i\\in X_i} \\langle \\theta ^{\\Delta }_i, x_i \\rangle }$ , the dual vector $\\Delta $ is called admissible.",
"The set of admissible vectors is denoted by $AD(\\theta _i,x^*_i,J)$ .",
"Lemma 2 Let $\\Delta \\in AD(\\theta ^{\\phi }_i,x^*_i,J)$ then $D(\\phi )\\le D(\\phi +\\Delta )$ .",
"Procedure Input:Factor $i\\in \\mathbb {F}$ , neighboring factors $J = \\lbrace j_1,\\ldots ,j_l\\rbrace \\subseteq \\mathcal {N}_{\\mathbb {G}}(i)$ , dual variables $\\phi $ $\\hspace{-70.0pt}\\text{Compute}\\ x^*_i\\in \\arg \\min \\nolimits _{x_i\\in X_i}\\langle \\theta ^{\\phi },x_i\\rangle \\hfill $ $\\hspace{-13.0pt} \\text{Choose}\\ \\delta \\in \\mathbb {R}^{d_i}\\ \\text{s.t.",
"}\\ \\delta (s) \\left\\lbrace \\begin{array}{ll} > 0, & x^*_i(s) = 1 \\\\ < 0,& x^*_i(s) = 0 \\end{array} \\right.$ Maximize admissible messages to $J$ : $\\Delta ^*_{(i,J)} \\in \\operatornamewithlimits{arg max}\\limits _{\\Delta \\in AD(\\theta ^{\\phi }_i,x^*_i,J)} \\langle \\delta ,\\theta _i^{\\phi +\\Delta } \\rangle $ Output: $\\Delta ^*_{(i,J)}$ .",
"Message-Passing Update Step."
],
[
"Message-Passing Update Step",
"To maximize $D(\\phi )$ , we will iteratively visit all factors and adjust messages $\\phi $ connected to it, monotonically increasing the lower bound (REF ).",
"Such an elementary step is defined by Procedure REF .",
"Procedure REF is defined up to the vector $\\delta $ , which satisfies (REF ) (see Proc.",
"REF ).",
"Usually, $\\delta (s)=\\left\\lbrace \\begin{array}{rl} 1, & x^*_i(s) = 1 \\\\ -1 ,& x^*_i(s) = 0 \\end{array}\\right.$ is a good choice.",
"Although different $\\delta $ may result in different efficiency of our framework, fulfillment of (REF ) is sufficient to prove its convergence properties.",
"The reparametrization adjustment problem (REF ) serves an intuitive goal to move as much slack as possible from the factor $i$ to its neighbors $J$ .",
"For example, for the setting of Example REF its solution reads $\\Delta _{u,uv}(s) = \\theta ^{\\phi }_u(x^*_u) - \\theta ^{\\phi }_u(s)$ .",
"Depending on the selected $\\delta $ it might correspond to maximization of the dual objective in the direction defined by admissible reparametrizations.",
"Although maximization (REF ) is not necessary to prove convergence of our method (as we show below, only a feasible solution of (REF ) is required for the proof), (i) it leads to faster convergence; (ii) for the case of CRFs (as in Example ) it makes our method equivalent to well established techniques like TRW-S [46] and SRMP [47], as shown in Section REF .",
"The following proposition states that the elementary update step defined by Procedure REF can be performed efficiently.",
"That is, the size of the reparametrization adjustment problem (REF ) grows linearly with the size of the factor $i$ and its attached messages: Proposition 2 Let $\\operatornamewithlimits{conv}(X_i) = \\lbrace \\mu _i : A_i \\mu _i \\le b_i\\rbrace $ with $A \\in \\lbrace 0,1\\rbrace ^{n \\times m}$ .",
"Let the messages in problem (REF ) have size $n_1,\\ldots ,n_{\\vert J\\vert }$ .",
"Then (REF ) is a linear program with $O(n + n_1 + \\ldots + n_{\\vert J\\vert })$ variables and $O(m + n_1 + \\ldots + n_{\\vert J\\vert })$ constraints."
],
[
"Message Passing Algorithm",
"Now we combine message passing updates into Algorithm .",
"It visits every node of the factor graph and performs the following two operations: (i) Receive Messages, when messages are received from a subset of neighboring factors, and (ii) Send Messages, when messages to some neighboring factors are computed and reweighted by $\\omega $ .",
"Distribution of weights $\\omega $ may influence the efficiency of Algorithm just like it influences the efficiency of message passing for CRFs (see [47]).",
"We provide typical settings in Section REF .",
"Usually, factors are traversed in some given a-priori order alternately in forward and backward direction, as done in TRW-S [46] and SRMP [47].",
"We refer to [47] for a motivation for such a schedule of computations.",
"$i \\in \\mathbb {F}$ in some order Receive Messages: Choose a subset of connected factors $J_{receive} \\subseteq \\mathcal {N}_{\\mathbb {G}}(i)$ $j \\in J_{receive}$ Compute $\\Delta ^*_{(j,\\lbrace i\\rbrace )}$ with Procedure REF .",
"Set $\\phi = \\phi + \\Delta ^*_{(j,\\lbrace i\\rbrace )}$ .",
"Send Messages: Choose partition $J_1 \\dot{\\cup } \\ldots \\dot{\\cup } J_l \\subseteq \\mathcal {N}_{\\mathbb {G}}(i)$ .",
"$J \\in \\lbrace J_1,\\ldots ,J_l\\rbrace $ Compute $\\Delta ^*_{(i,J)}$ with Procedure REF .",
"Choose weights $\\omega _{J_1},\\ldots ,\\omega _{J_l} \\ge 0$ such that $ \\omega _{J_1} + \\ldots +\\omega _{J_l} \\le 1$ .",
"$J \\in \\lbrace J_1,\\ldots ,J_l\\rbrace $ Set $\\phi = \\phi + \\omega _{J} \\Delta ^*_{(i,J)}$ .",
"One Iteration of Message-Passing We will discuss parameters of Algorithm (factor partitioning $\\lbrace J_i\\rbrace $ , weights $w_{J_i}$ ) right after the theorem stating monotonicity for any choice of parameters.",
"Theorem 1 Algorithm monotonically increases the dual lower bound (REF )."
],
[
"Parameter Selection for Algorithm ",
"There are the following free parameters in Algorithm : (i) The order of traversing factors of $\\mathbb {F}$ ; (ii) for each factor the neighboring factors from which to receive messages $J_{receive} \\subseteq \\mathcal {N}_{\\mathbb {G}}(i)$ ; (iii) the partition $J_1 \\dot{\\cup } \\ldots \\dot{\\cup } J_l \\subseteq \\mathcal {N}_{\\mathbb {G}}(i)$ of factors to send messages to and (iv) the associated weights $\\omega _{J_1},\\ldots , \\omega _{J_l}$ for messages.",
"Although for any choice of these parameters Algorithm monotonically increases the dual lower bound (as stated by Theorem REF ), its efficiency may significantly depend on their values.",
"Below, we will describe the parameters for Examples -, which we found the most efficient empirically.",
"Additionally, in the supplement we discuss parameters, which turn our algorithm into existing message passing solvers for CRFs (as in Example ).",
"Sending a message by some factor automatically implies receiving this message by another, coupled factor.",
"Therefore, usually there is no need to go over all factors in Algorithm .",
"It is usually sufficient to guarantee that all coupling constraints are updated by Procedure REF .",
"Formally, we can always exclude processing some factors by setting $J_{receive}$ and $J_i$ , $i=1,\\dots ,l$ to the empty set.",
"Instead, we will explicitly specify, which factors are processed in the loop of Algorithm in the examples below."
],
[
"Parameters for Example ",
"Pairwise CRFs have the specific feature that node factors are coupled with edge factors only.",
"This implies that processing only node factors in Algorithm is sufficient.",
"Below, we describe parameters, which turn Algorithm into SRMP [47] (which is up to details of implementation equivalent to TRW-S [46] for pairwise CRFs).",
"Other settings, given in the supplement, may turn it to other popular message passing techniques like MPLP [27] or min-sum diffusion [62].",
"We order node factors and process them according to this ordering.",
"The ordering naturally defines the sets of incoming $\\mathsf {E}_u^{+}$ and outgoing $\\mathsf {E}_u^{-}$ edges for each node $u\\in \\mathsf {V}$ .",
"Here $uv\\in \\mathsf {E}$ is incoming for $u$ if $v < u$ and outgoing if $v > u$ .",
"Each node $u\\in \\mathsf {V}$ receives messages from all incoming edges, which is $J_{receive} = \\mathcal {N}_{\\mathbb {G}}(u)=\\mathsf {E}_u^{+}$ .",
"The messages are send to all outgoing edges, each edge $uv\\in \\mathsf {E}$ in the partition in line of Algorithm is represented by a separate set.",
"That is, the partition reads $\\dot{\\cup }_{e\\in \\mathsf {E}_u^{-}}\\lbrace e\\rbrace $ .",
"Weights are distributed uniformly and equal to $w_e=\\lbrace \\frac{1}{\\max \\lbrace |\\mathsf {E}_u^{-}|,|\\mathsf {E}_u^{+}|\\rbrace }\\rbrace $ , $e\\in \\mathsf {E}_u^{-}$ .",
"After each outer iteration, when all nodes were processed, the ordering is reversed and the process repeats.",
"We refer to [47] for substantiation of these parameters."
],
[
"Parameters for Example ",
"Additionally to the node and edge factors, the corresponding IRPS-LP has also label factors (REF ).",
"To this end all node factors are ordered, as in Example .",
"Each node factor $u\\in \\mathsf {V}$ receives messages from all incoming edge factors and label factors $J_{receive}(u) = \\mathsf {E}_u^{+} \\cup X_u$ and sends them to all outgoing edges and label factors.",
"The corresponding partition reads $\\dot{\\cup }_{f\\in \\mathcal {N}_{\\mathbb {G}}(u)\\backslash \\mathsf {E}_u^{+}}\\lbrace f\\rbrace \\ \\dot{\\cup }X_u$ .",
"The weights are distributed uniformly with $w_f=\\lbrace \\frac{1}{1+\\max \\lbrace |\\mathsf {E}_u^{-}|,|\\mathsf {E}_u^{+}|\\rbrace }\\rbrace $ .",
"The label factors are processed after all node factors were visited.",
"Each label factor receives messages from all connected node factors and send messages back as well: $J_{receive}(s) = \\lbrace u \\in \\mathsf {V}: s \\in X_u\\rbrace $ .",
"We use the same single set for sending messages, i.e.",
"$J_1 = J_{receive}$ .",
"After each iteration we reverse the factor order."
],
[
"Parameters for Example ",
"Similarly to Example , it is sufficient to go only over all edge factors in the loop of Algorithm , since each coupling constraint contains exactly one cycle and one edge factor.",
"Each edge factor $e$ receives messages from all coupled cycle factors $J_{receive}=\\mathcal {N}_{\\mathbb {G}}(\\lbrace c \\in C : e \\in c\\rbrace )$ and sends them to the same factors.",
"As in Example , each cycle factor forms a trivial set in the partition in line of Algorithm , the partition reads $\\dot{\\cup }_{c \\in C : e \\in c}\\lbrace c\\rbrace $ .",
"Weights are distributed uniformly with $w_e=\\frac{1}{\\vert c \\in C : e \\in c\\vert }$ .",
"After each iteration the processing order of factors is reversed."
],
[
"Obtaining Integer Solution",
"Eventually we want to obtain a primal solution $x \\in X$ of (REF ), not a reparametrization $\\theta ^{\\phi }$ .",
"We are not aware of any rounding technique which would work equally well for all possible instances of IRPS-LP problem.",
"According to our experience, the most efficient rounding is problem specific.",
"Below, we describe our choices for the Examples – ."
],
[
"Rounding for Example ",
"coincides with the one suggested in [46]: Assume we have already computed a primal integer solution $x^*_v$ for all $v<u$ and we want to compute $x^*_u$ .",
"To this end, right before the message receiving step of Algorithm for $i=u$ we assign $x_u^* \\in \\operatornamewithlimits{arg min}_{x_u} \\theta _u(x_u) + \\sum _{v < u: uv \\in \\mathsf {E}} \\theta _{uv}(x_u, x_v^*)\\,.$"
],
[
"Rounding for Example ",
"is the same except that we select the best label $x_u$ among those, which have not been assigned yet, to satisfy uniqueness constraints: $x_u^* \\in \\operatornamewithlimits{arg min}_{x_u : x_v^* \\ne x_u \\forall v < u} \\theta ^{\\phi }_u(x_u) + \\sum _{v < u: uv \\in \\mathsf {E}} \\theta ^{\\phi }_{uv}(x_u, x_v^*)\\,.$"
],
[
"Rounding for Example ",
"We use the efficient Kernighan&Lin heuristic [43] as implemented in [44].",
"Costs for the rounding are the reparametrized edge potentials."
],
[
"Fixed Points and Comparison to Subgradient Method",
"Algorithm does not necessarily converge to the optimum of (REF ).",
"Instead, it may get stuck in suboptimal points, similar to those correspoding to the \"weak tree agreement\" [46] or \"arc consistency\" [72] in CRFs from Example .",
"Below we characterise these fixpoints precisely.",
"Definition 3 (Marginal Consistency) Given a reparametrization $\\theta ^{\\phi }$ , let for each factor $i \\in \\mathbb {F}$ a non-empty set $\\mathbb {S}_i \\subseteq \\operatornamewithlimits{arg min}_{x_i \\in X_i} \\langle \\theta ^{\\phi }, x_i \\rangle $ , $i\\in \\mathbb {F}$ be given.",
"Define $\\mathbb {S}= \\prod _{i \\in \\mathbb {F}} \\mathbb {S}_i$ .",
"We call reparametrization $\\theta ^{\\phi }$ marginally consistent for $\\mathbb {S}$ on $ij \\in \\mathbb {E}$ if $A_{(i,j)} \\left( \\mathbb {S}_i \\right) = A_{(j,i)} \\left(\\mathbb {S}_j\\right)\\,.$ If $\\theta ^{\\phi }$ is marginally consistent for $\\mathbb {S}$ on all $ij \\in \\mathbb {E}$ , we call $\\theta ^{\\phi }$ marginally consistent for $\\mathbb {S}$.",
"Note that marginal consistency is necessary, but not sufficient for optimality of the relaxation (REF ).",
"This can be seen in the case of CRFs (Example ), where it exactly corresponds to arc-consistency.",
"The latter is only necessary, but not sufficient for optimality [72].",
"Theorem 2 If $\\theta ^{\\phi }$ is marginally consistent, the dual lower bound $D(\\phi )$ cannot be improved by Algorithm ."
],
[
"Comparison to Subgradient Method.",
"Decomposition IRPS-LP and more general ones can be solved via the subgradient method [48].",
"Similar to Algorithm , it operates on dual variables $\\phi $ and manipulates them by visiting each factor sequentially.",
"Contrary to Algorithm , subgradient algorithms converge to the optimum.",
"Moreover, on a per-iterations basis, computing subgradients is cheaper than using Algorithm , as only (REF ) needs to be computed, while Algorithm needs to solve (REF ) additionally.",
"However, for MAP-inference, the study [39] has shown that subgradient-based algorithms converge much slower than message passing algorithms like TRWS [46].",
"In Section we confirm this for the graph matching problem as well.",
"The reason for this large empirical difference is that one iteration of the subgradient algorithm only updates those coordinates of dual variables $\\phi $ that are affected by the current minimal labeling $x_i^* \\in \\operatornamewithlimits{arg min}_{x_i \\in X_i} \\langle \\theta ^{\\phi }_i, x_i \\rangle $ (i.e.",
"coordinates $k: (A^{(i,j)} x_i^*)_k = 1$ ), while in Algorithm all coordinates of $\\phi $ are taken into account.",
"Also message passing implicitly chooses the stepsize so as to achieve monotonical convergence in Algorithm REF , while subgradient based algorithms must rely on some stepsize rule that may either make too large or too small changes to the dual variables $\\phi $ ."
],
[
"Experimental Evaluation",
"Our experiments' goal is to illustrate applicability of the proposed technique, they are not an exhaustive evaluation.",
"The presented algorithms are only basic variants, which can be further improved and tuned to the considered problems.",
"Both issues are addressed in the specialized studies [65], [66] which are appended.",
"Still, we show that the presented basic variants are already able to surpass state-of-the-art specialized solvers on challenging datasets.",
"All experiments were run on a computer with a 2.2 GHz i5-5200U CPU and 8 GB RAM.",
"Figure: Runtime plots comparing averaged log(primalenergy-duallowerbound)\\log (\\text{primal energy} - \\text{dual lower bound}) values on car, motor and worms graph matching datasets.Both axes are logarithmic.Figure: Runtime plots comparing averaged primal/dual values on the three knott-3d-{150|300|450} multicut datasets.",
"Values are averaged over all instances in the dataset.",
"The x-axis are logarithmic.",
"Continuous lines are dual lower bounds while corresponding dashed lines show primal solutions obtained by rounding."
],
[
"Solvers.",
"We compare against two state-of-the-art algorithms: (i) the subgradient based dual decomposition solver [68] abbreviated by DD and (ii) the recent “hungarian belief propagation” message passing algorithm [76], abbreviated as HBP.",
"While the authors of [76] have embedded their solver in a branch-and-bound routine to produce exact solutions, we have reimplemented their message passing component but did not use branch and bound to make the comparison fair.",
"Both algorithms DD and HBP outperformed alternative solvers at the time of their publication, hence we have not tested against [17], [74], [51], [67], [61], [28], [77], [37], [52], [15].",
"We call our solver AMP."
],
[
"Datasets.",
"We selected three challenging datasets.",
"The first two are the standard benchmark datasets car and motor, both used in [53], containing 30 pairs of cars and 20 pairs of motorbikes with keypoints to be matched 1:1.",
"The images are taken from the VOC PASCAL 2007 challenge [20].",
"Costs are computed from features as in [53].",
"Instances are densely connected graphs with 20 – 60 nodes.",
"The third one is the novel worms datasets [38], containing 30 problem instances coming from bioimaging.",
"The problems are made of sparsely connected graphs with up to 600 nodes and up to 1500 labels.",
"To our knowledge, the worms dataset contains the largest graph matching instances ever considered in the literature.",
"For runtime plots showing averaged logarithmic primal/dual gap over all instances of each dataset see Fig.",
"REF ."
],
[
"Results.",
"Our solver AMP consistently outperforms HBP and DD w.r.t.",
"primal/dual gap and anytime performance Most markedly on the largest worms dataset, the subgradient based algorithm DD struggles hard to decrease the primal/dual gap, while AMP gives reasonable results."
],
[
"Solvers.",
"We compare against state-of-the-art multicut algorithms implemented in the OpenGM [39] library, namely (i) the branch-and-cut based solver MC-ILP [42] utilizing the ILP solver CPLEX [2], (ii) the heuristic primal “fusion move” algorithm CC-Fusion [8] with random hierarchical clustering and random watershed proposal generator, denoted by the suffixes -RHC and -RWS and (iii) the heuristic primal “Cut, Glue & Cut” solver CGC [9].",
"Those solvers were shown to outperform other multicut algorithms [8].",
"Algorithm MC-ILP provides both upper and lower bounds, while CC-Fusion and CGC are purely primal algorithms.",
"We call our message passing solver with cycle constraints added in a cutting plane fashion MP-C."
],
[
"Datasets.",
"A source of large scale problems comes from electron microscopy of brain tissue, for which we wish to obtain neuron segmentation.",
"We have selected three datasets knott-3d-{150|300|450} of increasingly large size [39], each consisting of 8 instances.",
"Instances have $\\le 972$ , 5896 and 17074 nodes and $\\le 5656$ , 36221, and 107060 edges respectively."
],
[
"Results.",
"For plots showing dual bounds and primal solution objectives over time see Figure REF .",
"Our algorithm MP-C combines advantages of LP-based techniques awith those of primal heuristics: It delivers high dual lower bounds faster than MC-ILP.",
"Its has fast primal convergence speed and delivers primal solutions comparable/superior to CGC's and CC-Fusion's."
],
[
"Acknowledgments",
"The authors would like to thank Vladimir Kolmogorov for helpful discussions.",
"This work is partially funded by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 616160."
],
[
"Proof of Proposition ",
"Proposition $\\sum _{i\\in \\mathbb {F}} \\langle \\theta _i,\\mu _i\\rangle = \\sum _{i\\in \\mathbb {F}} \\langle \\theta ^{\\phi }_i,\\mu _i\\rangle $ , whenever $\\mu _1,\\ldots ,\\mu _k$ obey the coupling constraints.",
"$\\sum _{i \\in \\mathbb {F}} \\langle \\theta ^{\\phi }, \\mu _i \\rangle =\\sum _{i \\in \\mathbb {F}} \\langle \\theta , \\mu _i \\rangle + \\underbrace{\\sum _{ij \\in \\mathbb {E}} \\langle \\phi _{(i,j)}, A_{(i,j)} \\mu _i \\rangle + \\langle \\phi _{(j,i)}, A_{(j,i)} \\mu _j \\rangle }_{(*)}=\\sum _{i \\in \\mathbb {F}} \\langle \\theta , \\mu _i \\rangle \\,,$ where $(*) = 0$ due to $\\phi _{(i,j)} = -\\phi _{j,i)}$ and $A_{(i,j)} \\mu _i = A_{(j,i)} \\mu _j$ ."
],
[
"Proof of Proposition ",
"Proposition Let $\\operatornamewithlimits{conv}(X_i) = \\lbrace \\mu _i : A_i \\mu _i \\le b_i\\rbrace $ with $A \\in \\lbrace 0,1\\rbrace ^{n \\times m}$ .",
"Let the messages in problem (REF ) have size $n_1,\\ldots ,n_{\\vert J\\vert }$ .",
"Then (REF ) is a linear program with $O(n + n_1 + \\ldots + n_{\\vert J\\vert })$ variables and $O(m + n_1 + \\ldots + n_{\\vert J\\vert })$ constraints.",
"From LP-duality we know that $\\mu _i^* \\in \\operatornamewithlimits{arg min}_{\\mu _i : A \\mu _i \\le b_i} \\langle c, \\mu _i \\rangle $ iff $\\exists y\\ge 0 : A_i^y̰ = c_i$ and $\\langle b_i - A_i \\mu _i^* , y \\rangle = 0$ .",
"Hence, (REF ) can be rewritten as $\\begin{array}{cl}\\max \\limits _{y \\ge 0, \\Delta _{(i,j_1)},\\ldots ,\\Delta _{(i,j_l)}}& \\langle \\delta ,\\theta ^{\\phi +\\Delta } \\rangle \\\\\\text{s.t.",
"}& \\langle b_i - A_i \\mu _i^*, y \\rangle = 0 \\\\& A_i^y̰ = \\theta ^{\\phi +\\Delta } \\\\& \\Delta _{(i,j)}(s){\\left\\lbrace \\begin{array}{ll}\\le 0, & \\nu _i(s) =0\\\\\\ge 0, & \\nu _i(s) =1\\end{array}\\right.",
"}\\\\&\\hfill \\text{where}\\ \\nu _i:=A_{(i,j)} \\mu ^*_i\\end{array}$ $\\theta ^{\\phi + \\Delta }$ is a linear expression and $\\mu ^*$ is constant during the computation, hence (REF ) is a LP."
],
[
"Proof of Lemma ",
"Lemma Let $ij\\in \\mathbb {E}$ be a pair of factors related by the coupling constraints and $\\phi _{(i,j)}$ be a corresponding dual vector.",
"Let $x_i^* \\in \\operatornamewithlimits{arg min}\\limits _{x_i\\in X_i} \\langle \\theta ^{\\phi }_i, x_i \\rangle $ and $\\Delta _{(i,j)}$ satisfy $\\Delta _{(i,j)}(s){\\left\\lbrace \\begin{array}{ll}\\ge 0, & \\nu (s) =1\\\\\\le 0, & \\nu (s) =0\\end{array}\\right.",
"},\\ \\text{where}\\ \\nu :=A_{(i,j)} x^*_i\\,.$ Then $x_i^* \\in \\operatornamewithlimits{arg min}\\limits _{x_i\\in X_i} \\langle \\theta ^{\\phi +\\Delta }_i, x_i \\rangle $ implies ${D(\\phi ) \\le D(\\phi +\\Delta )}$ .",
"Let $x_j^* \\in \\operatornamewithlimits{arg min}_{x_j \\in X_j} \\langle \\theta ^{\\phi }_j,x_j \\rangle $ be a solution of (REF ) at which the dual lower bound (REF ) is attained before the update and $x^{**}_j \\in \\operatornamewithlimits{arg min}_{x_j \\in X_j} \\langle \\theta ^{\\phi } - A_{(j,i)}^^*_{(i,j)}, x_j \\rangle $ be an integral solution at which the dual lower bound is attained after $\\phi $ has been updated.",
"Variable $x_i^*$ as chosen in (REF ) is optimal for $\\theta ^{\\phi }$ and for $\\theta ^{\\phi + \\Delta }$ by construction.",
"We need to prove $\\langle \\theta ^{\\phi }_i,x_i^*\\rangle + \\sum _{j \\in J} \\langle \\theta ^{\\phi }_j, x_j^*\\rangle \\\\\\le \\langle \\theta ^{\\phi }_i + \\sum _{j \\in J} A^{(i,j)} \\Delta ^*_{(i,j)} ,x^*_i\\rangle + \\sum _{j \\in J} \\langle \\theta ^{\\phi }_j - A^{(j,i)} \\Delta ^*_{(i,j)} , x^{**}_j\\rangle \\,.$ We shuffle all terms with variables $\\Delta ^*_{(i,j)}$ , $j \\in J$ to the right side and all other terms to the left side.",
"$\\langle \\theta _i^{\\phi }, x^*_i - x^*_i \\rangle + \\sum _{j \\in J} \\langle \\theta _j^{\\phi }, x^*_j - x^{**}_j \\rangle \\\\\\le \\langle \\sum _{j \\in } A^{(i,j)} \\Delta ^*_{(i,j)}, x^*_i \\rangle - \\sum _{j \\in J} \\langle A^{(j,i)} \\Delta ^*_{(i,j)}, x^{**}_j \\rangle $ All terms on the left side are smaller than zero due to the choice of $x^*_j$ being minimizers w.r.t.",
"$\\theta ^{\\phi }_j$ .",
"Hence, it will be enough to prove the above inequality when assuming the left side to be zero.",
"We rewrite the scalar products by transposing $A^{(i,j)}$ and $A^{(j,i)}$ .",
"$0 \\le \\sum \\nolimits _{j \\in J} \\left\\lbrace \\langle \\Delta ^*_{(i,j)}, A_{(i,j)} x_i^* - A_{(j,i)} x^{**}_j \\rangle \\right\\rbrace $ Due to $A_{(j,i)} x^{**}_j \\in \\lbrace 0,1\\rbrace ^{\\dim (\\phi _{(i,j)})}$ and $A_{(i,j)} x_i^* \\in \\lbrace 0,1\\rbrace ^{\\dim (\\phi _{(i,j)})}$ by Definition REF and $\\Delta ^*_{(i,j)} \\lessgtr 0$ whenever $A_{(i,j)} x^*_i \\lessgtr 0$ , the result follows.",
"Lemma Let $\\Delta \\in AD(\\theta ^{\\phi }_i,x^*_i,J)$ then $D(\\phi )\\le D(\\phi +\\Delta )$ .",
"Analoguous to the proof of Lemma REF ."
],
[
"Proof of Theorem ",
"Theorem Algorithm monotonically increasis the dual lower bound (REF ).",
"We prove that (i) the receiving messages and (ii) the sending messages step improve (REF ).",
"(i) Directly apply Lemma REF .",
"(ii) The difficulty here is that we compute descent directions from the current dual variables $\\phi $ in parallel and then apply all of them simultaneously.",
"By Lemma REF , the send message step is non-decreasing when called for each set $J_1,\\ldots ,J_l$ in Algorithm .",
"The dual lower bound $L(\\phi )$ is concave, hence we apply Jensen's inequality and note that $\\omega _1 + \\ldots + \\omega _l \\le 1$ to obtain the result."
],
[
"Proof of Theorem ",
"Theorem If $\\theta ^{\\phi }$ is marginally consistent, the dual lower bound $D(\\phi )$ cannot be improved by Algorithm .",
"First, we need two technical lemmata.",
"Lemma 3 Let $X \\subset \\lbrace 0,1\\rbrace ^n$ , $A \\in \\lbrace 0,1\\rbrace ^{K \\times n}$ and $Ax \\in \\lbrace 0,1\\rbrace ^K$ $\\forall x \\in X$ .",
"Let $x^* \\in X$ be given and define $\\nu ^* := Ax^*$ .",
"Let $\\Delta \\in \\mathbb {R}^K$ be given such that $\\Delta (s) {\\left\\lbrace \\begin{array}{ll} \\ge 0,& \\nu ^*(s) =1 \\\\ \\le 0,& \\nu ^*(s) = 0 \\end{array}\\right.",
"}$ .",
"Then (i) $x^* \\in \\operatornamewithlimits{arg min}_{x \\in X} \\langle -\\Delta , A x \\rangle $ and (ii) for $x^{**} \\in \\operatornamewithlimits{arg min}_{x \\in X} \\langle -\\Delta , A x \\rangle $ , $\\nu ^{**} = A x^{**}$ it holds that $\\Delta (s) = 0$ whenever $\\nu ^*(s) \\ne \\nu ^{**}(s)$ .",
"Let $x \\in X$ and define $\\nu = A x$ .",
"Then $\\langle -\\Delta , A x \\rangle \\\\=\\underbrace{\\sum _{s: \\nu ^*(s) = 1 = \\nu (s) } \\hspace*{-11.38092pt} - \\Delta (s)}_{(*)}+\\underbrace{\\sum _{s: \\nu (s) = 1 > 0 = \\nu ^*(s) } \\hspace*{-17.07182pt} - \\Delta (s)}_{(**)} \\\\\\ge \\underbrace{\\sum _{s : \\nu ^*(s) = 1}-\\Delta (s) }_{(***)} \\\\= \\langle -\\Delta , A x^* \\rangle $ because $(*) \\ge (***)$ due to $\\Delta (s) \\ge 0$ for $\\nu ^*(s) = 1$ and $(**) \\ge 0$ due to $\\Delta (s) \\le 0$ for $\\nu ^*(s) = 0$ .",
"This proves (i) and (ii) is proven by observing that $(**) = 0$ and $(*) = (***)$ must also hold.",
"Lemma 4 Let $x_i^*,x_i^{**} \\in \\operatornamewithlimits{arg min}_{x_i \\in X_i} \\langle \\theta ^{\\phi }, x_i \\rangle $ be two solutions to the $i$ -th factor for the current reparametrization $\\theta ^{\\phi }$ .",
"If $\\Delta $ is admissible w.r.t.",
"$x_i^*$ then $\\Delta $ is also admissible w.r.t.",
"$x_i^{**}$ .",
"As both $x_i^*$ and $x_i^{**}$ are optimal to $\\theta ^{\\phi }$ and $x_i^*$ is also optimal to $\\theta ^{\\phi + \\Delta }$ , we have $\\langle \\Delta _{(i,j)}, A_{(j,i)} x_i^* \\rangle \\le \\langle \\Delta _{(i,j)}, A_{(j,i)} x_i^{**} \\rangle $ .",
"By Lemma REF , (i) also $\\langle -\\Delta _{(i,j)}, A_{(j,i)} x_i^* \\rangle \\le \\langle -\\Delta _{(i,j)}, A_{(j,i)} x_i^{**} \\rangle $ holds, hence equality must hold.",
"This shows $x^{**}_i \\in \\operatornamewithlimits{arg min}_{x_i \\in X_i} \\langle \\theta ^{\\phi + \\Delta },x_i \\rangle $ .",
"Second, Lemma REF , (ii) implies that $\\Delta (s) = 0$ whenever $\\nu ^*(s) \\ne \\nu ^{**}(s)$ .",
"This proves that $ \\Delta _{(i,j)}(s){\\left\\lbrace \\begin{array}{ll}\\ge 0, & \\nu ^{**}(s) =1\\\\\\le 0, & \\nu ^{**}(s) =0\\\\\\end{array}\\right.",
"}, \\nu ^{**}:=A_{(i,j)} x^{**}_i$ .",
"[Proof of Theorem REF ] It is sufficient to show that for marginally consistent $\\theta ^{\\phi }$ for $\\mathbb {S}$ , the update $\\Delta $ computed by Algorithm REF on an arbitrary factor $i \\in \\mathbb {F}$ and some set $J \\subset \\mathcal {N}_{\\mathbb {G}}(i)$ has the following properties: (i) $L(\\phi ) = L(\\phi + \\Delta )$ , (ii) $\\theta ^{\\phi + \\Delta }$ is marginally consistent for $\\mathbb {S}$ .",
"For an easier proof, we only consider the case $J = \\lbrace j\\rbrace $ .",
"The general case can be proven analoguously.",
"(i) Let $x_i^* \\in \\mathbb {S}_i$ , $x_j^* \\in \\mathbb {S}_j$ with $A_{(i,j)} x_i^* = A_{(j,i)} x_j^*$ .",
"We have to show that $\\min _{x_i \\in X_i} \\langle \\theta _i^{\\phi }, x_i \\rangle + \\min _{x_j \\in X_j} \\langle \\theta _j^{\\phi }, x_j \\rangle =\\min _{x_i \\in X_i} \\langle \\theta _i^{\\phi + \\Delta }, x_i \\rangle + \\min _{x_j \\in X_j} \\langle \\theta _j^{\\phi + \\Delta }, x_j \\rangle $ Due to $x^*_i$ optimal to $\\theta _i^{\\phi + \\Delta }$ , since by Lemma REF the update $\\Delta $ is admissible for $x_i^*$ , it remains to show that $x_j^* \\in \\operatornamewithlimits{arg min}_{x_j \\in X_j} \\langle \\theta ^{\\phi + \\Delta }, x_j \\rangle $ .",
"As $x_j^* \\in \\operatornamewithlimits{arg min}_{x_j \\in X_j} \\langle \\theta ^{\\phi }, x_j \\rangle $ , it is sufficient to prove that $x_j^* \\in \\operatornamewithlimits{arg min}_{x_j \\in X_j} \\langle -\\Delta _{(i,j)}, A_{(j,i)} x_j \\rangle $ .",
"This follows from Lemma REF (i).",
"We conclude by noting $\\langle \\theta ^{\\phi }_i, x_i^* \\rangle + \\langle \\theta ^{\\phi }_j x_j \\rangle = \\langle \\theta ^{\\phi +\\Delta }_i, x_i^* \\rangle + \\langle \\theta ^{\\phi +\\Delta }_j x_j \\rangle $ .",
"(ii) The computations in (i) show that $\\mathbb {S}_i \\subseteq \\operatornamewithlimits{arg min}_{x_i \\in X_i} \\langle \\theta ^{\\phi + \\Delta }_i, x_i \\rangle $ and $\\mathbb {S}_j \\subseteq \\operatornamewithlimits{arg min}_{x_j \\in X_j} \\langle \\theta ^{\\phi + \\Delta }_j, x_j \\rangle $ .",
"The reparametrizations of all other factors stay the same: $\\theta ^{\\phi + \\Delta }_k = \\theta ^{\\phi }_k$ for $k \\in \\mathbb {F}\\backslash \\lbrace i,j\\rbrace $ .",
"Hence, $\\theta ^{\\phi + \\Delta }$ is marginally consistent for $\\mathbb {S}$ after the update."
],
[
"Special Cases: Graphical Model Solvers",
"We will show how Algorithm subsumes known message-passing algorithms MSD [72], TRWS [46], SRMP [47] and MPLP [27] for MAP-inference with common graphical models, considered in Example ."
],
[
"Solver Primitives (",
"As it can be seen, all factors in (REF ) are of the form ${X_i = \\lbrace (1,0,\\ldots ,0),(0,1,0,\\ldots ,0),\\ldots ,(0,\\ldots ,0,1)\\rbrace }$ and $\\operatornamewithlimits{conv}(X_i) = \\lbrace \\mu \\ge 0 : \\langle {1},\\mu \\rangle = 1\\rbrace $ is a $\\dim _(X_i)$ -dimensional simplex.",
"In all message passing algorithms [46], [47], [72], [27], there are two types of invokations of Algorithm REF together with solutions of the accompanying optimization problem (REF ) and (REF ): Table: NO_CAPTION"
],
[
"MAP-inference Solvers.",
"In Table REF we state solvers MSD [72], TRWS [46], SRMP [47] and MPLP [27] as special cases of our framework.",
"Factors are visited in the order they are read in.",
"Table: , , , as special cases of Algorithm .Remark 1 We have only treated the case of unary $\\theta _u, u \\in \\mathsf {V}$ and pairwise potentials $\\theta _{uv}, uv \\in \\mathsf {E}$ here.",
"MPLP [27] and SRMP [47] can be applied to higher order potentials as well, which we do not treat here.SRMP [47] is a generalisation of TRWS [46] to the higher-order case.",
"Remark 2 There are convergent message-passing algorithms such that factors comprise trees [70], [63].",
"Their analysis is more difficult, hence we omit it here.",
"Note that our framework generalizes upon [46], [47], [27], [72], [63], [70] in several ways: (i) Our factors need not be simplices or trees.",
"(ii) Our messages need not be marginalization between unary/pairwise/triplet/$\\ldots $ factors.",
"(iii) We can compute message updates on more than one coupling constraint simultaneously, i.e.",
"we may choose $J_1\\dot{\\cup }\\ldots \\dot{\\cup }J_l$ in Algorithm to be different than singleton sets.",
"(i) and (ii) affect LP-modeling, (iii) affects computational efficiency: By considering multiple messages at once in Procedure REF , we may be able to make larger updates $\\Delta ^*$ , resulting in faster convergence."
]
] | 1612.05460 |
[
[
"Strong Federations: An Interoperable Blockchain Solution to Centralized\n Third-Party Risks"
],
[
"Abstract Bitcoin, the first peer-to-peer electronic cash system, opened the door to permissionless, private, and trustless transactions.",
"Attempts to repurpose Bitcoin's underlying blockchain technology have run up against fundamental limitations to privacy, faithful execution, and transaction finality.",
"We introduce \\emph{Strong Federations}: publicly verifiable, Byzantine-robust transaction networks that facilitate movement of any asset between disparate markets, without requiring third-party trust.",
"\\emph{Strong Federations} enable commercial privacy, with support for transactions where asset types and amounts are opaque, while remaining publicly verifiable.",
"As in Bitcoin, execution fidelity is cryptographically enforced; however, \\emph{Strong Federations} significantly lower capital requirements for market participants by reducing transaction latency and improving interoperability.",
"To show how this innovative solution can be applied today, we describe \\emph{\\liquid}: the first implementation of \\emph{Strong Federations} deployed in a Financial Market."
],
[
"Introduction",
"Bitcoin, proposed by Satoshi Nakamoto in 2008, is based on the idea of a blockchain [1].",
"A blockchain consists of a series of blocks, each of which is composed of time-stamped sets of transactions and a hash of the previous block, which connects the two together, as presented in Figure REF .",
"The underlying principle of Bitcoin's design is that all participants in its network are on equal footing.",
"They jointly trust proof-of-work [2] to validate and enforce the network's rules, which obviates the need for central authorities such as clearinghouses.",
"As a result, Bitcoin empowers a wide range of participants to be their own banks – storing, transacting, and clearing for themselves without the need for a third-party intermediary.",
"Bitcoin's network automatically enforces settlements between participants using publicly verifiable algorithms that avoid security compromises, expensive (or unavailable) legal infrastructure, third-party trust requirements, or the physical transportation of money.",
"For the first time, users of a system have the ability to cryptographically verify other participants' behaviors, enforcing rules based on mathematics that anyone can check and no one can subvert.",
"Due to its design, Bitcoin has characteristics that make it a vehicle of value unlike anything that previously existed.",
"First, it eliminates most counterparty risk from transactions [3].",
"Second, it offers cryptographic proof of ownership of assets, as the knowledge of a cryptographic key defines ownership [4].",
"Third, it is a programmable asset, offering the ability to pay to a program, or a “smart contract\", rather than a passive account or a singular public key [5].",
"Fourth, and finally, it is a disruptive market mechanism for use cases such as point-to-point real-time transfers, accelerated cross-border payment, B2B remittance, asset transfers, and micropayments [6].",
"Because it is a global consensus system, Bitcoin's decentralized network and public verifiability come with costs.",
"Speed of execution and insufficient guarantees of privacy are two of Bitcoin's limitations.",
"Bitcoin's proof-of-work methodology was designed to process transactions on average only once every ten minutes, with large variance.",
"As a result, Bitcoin is slow from a real-time transaction processing perspective.",
"This creates spontaneous illiquidity for parties using bitcoinThe capitalized “Bitcoin\" is used to talk about the technology and the engine, while the lowercase “bitcoin\" is used to refer to the currency.",
"as an intermediary, volatility exposure for those holding bitcoin for any length of time, and obstacles for the use of Bitcoin's contracting features for fast settlements.",
"Even after a transaction is processed, counterparties must generally wait until several additional blocks have been created before considering their transaction settled.",
"This is because Bitcoin's global ledger is at constant risk of reorganization, wherein very recent history can be modified or rewritten.",
"This latency undermines many commercial applications, which require real-time, or nearly instant, executionIn most traditional systems, the speed of transaction is achieved by instant execution and delayed settlement.. Today, solving this requires a centralized counterparty, which introduces a third-party risk.",
"Despite issues of short-term validation, Bitcoin excels on settlement finality, providing strong assurance against transaction reversals after adequate block confirmations.",
"In contrast, legacy payment networks leave absolute final settlement in limbo for up to 120 days typically, though chargebacks have been allowed up to 8 years late [7], depending on policies imposed by the centralized network owner [8] [9].",
"While a popular prevailing belief is that Bitcoin is anonymous [10], its privacy properties are insufficient for many commercial use cases.",
"Every transaction is published in a global ledger, which allows small amounts of information about users' financial activity (e.g., the identities of the participants in a single transaction [11]) to be amplified by statistical analysis [12].",
"This limits the commercial usefulness of the network and also harms individual privacy [13], as user behavior frequently reflects the pervasive assumption that Bitcoin is an anonymous system.",
"Further, it can damage the fungibility of the system, as coins that have differing histories can be identified and valued accordingly.",
"Overcoming these two problems would be of significance and have positive impact on the Bitcoin industry and the broader global economy [12].",
"Unfortunately, previous attempts to solve similar tasks with electronic money have encountered a variety of issues: they fail to scale (e.g., BitGold [14]); they are centralized (e.g., the Liberty Reserve or Wei Dai's B-money [15]); or they raise other security concerns [16].",
"Moreover, higher trust requirements are often imposed through reliance on a centrally controlled system or on a single organization.",
"This effectively replicates the problems of pre-Bitcoin systems by establishing highly permissioned arenas that have substantial regulatory disadvantages and user costs, that create onboarding and offboarding friction, and that introduce restrictions on both users and operators of the system [17].",
"Needless to say, if a solution is run by a central party, it is inevitably subject to systemic exposure, creating a single point of failure (SPOF) risk [18].",
"The recent Ripple attack is an example of such a situation.",
"It has been shown, that although interesting and successful otherwise, both Ripple and Stellar face the SPOF risk [19].",
"Similarly, introduction of stronger trust requirements can lead to dangerous risks of consensus failure, as the consensus methods of Tendermint and Ethereum have proven [4].",
"Finally, there are exchanges and brokerages that require explicit trust in a third-party [20].",
"Such systems leak their intrinsic insecurity into any solutions built on top of them, creating a “house of cards\" arrangement where any instability in the underlying system may result in a collapse of the dependent arrangements.",
"Figure: A merkle tree connecting transactions to a block header merkle root."
],
[
"Contributions",
"This paper describes a new blockchain-based system that addresses these problems and contributes to the field in the following ways: Public Verifiability – While not fully decentralized, the system is distributed and publicly verifiable, leaving users with the ultimate spending authority over their assets.",
"Liquidity – Users can move their assets into and out of the system, giving them access to its unique characteristics while also allowing them to exit at any time.",
"No Single Point of Failure – The system maintains Bitcoin's permissionless innovation and avoids introducing SPOFs, all while providing novel features.",
"Multiple Asset–Type Transfers – The system supports multiple asset–type transfer on the same blockchain, even within the same atomic transaction.",
"Privacy – By extending earlier work on Confidential Transactions [21] through Confidential Assets, the system supports nearly instant, trustless, atomic exchange of arbitrary goods, in a publicly verifiable but completely private way.",
"Implementation – Liquid, an implementation of a Strong Federation, is presented with lessons learned from the use case of high speed inter-exchange transfers of bitcoin.",
"The rest of the paper is organized as follows: the next section discusses the general design of the solution to the problems identified above – Strong Federations.",
"Next, in Section more in-depth, technical details are provided.",
"Section is devoted to the applications of Strong Federations in different areas.",
"Here, Liquid is presented, the first market implementation of the system.",
"Strong Federations are very novel in many aspects, thus some time is spent discussing various innovations in Section , to then move to a thorough evaluation of the security and comparison of the system in Section .",
"Finally, Section discusses methodologies to further improve them and Section presents conclusions.",
"As mentioned in Section REF , a consensus mechanism based on proof-of-work introduces the problem of latency.",
"However, moving to a centralized system would create significant risks of its own.",
"To combat these problems, this paper builds on a design introduced by Back et.",
"al.",
"called “Federated Pegs\" [22], a methodology for two-way movement of assets between Bitcoin and sidechains.",
"Sidechains are parallel networks that allow parties to transfer assets between blockchains by providing explicit proofs of possession within transactions, as shown in Figure REF ."
],
[
"Sidechains",
"Sidechains are blockchains that allow users to transfer assets to and from other blockchains.",
"At a high level, these transfers work by locking the assets in a transaction on one chain, making them unusable there, and then creating a transaction on the sidechain that describes the locked asset.",
"Effectively, this moves assets from a parent chain to a sidechain.",
"This works as follows: The user sends their asset to a special address that is designed to freeze the asset until the sidechain signals that asset is returned.",
"Using the “in\" channel of a federated peg, the user embeds information on the sidechain stating that the asset was frozen on the main chain and requests to use it on the sidechain.",
"Equivalent assets are unlocked or created on the sidechain, so that the user can participate in an alternative exchange under the sidechain rules, which can differ from the parent chain.",
"When the user wishes to move her asset, or a portion thereof, back via the “out channel\", she embeds information in the sidechain describing an output on the main blockchain.",
"The Strong Federation reaches consensus that the transaction occurred.",
"After consensus is reached, the federated peg creates such an output, unfreezing the asset on the main blockchain and assigning it as indicated on the sidechain."
],
[
"Improving Sidechains with Strong Federations",
"Bitcoin demonstrates one method of signing blocks: the use of a Dynamic Membership Multiparty Signature (DMMS) [22] using a dynamic set of signers called miners.",
"A dynamic set introduces the latency issues inherent to Bitcoin.",
"A federated model offers another solution, with a fixed signer set, in which the DMMS is replaced with a traditional multisignature scheme.",
"Reducing the number of participants needed to extend the blockchain increases the speed and scalability of the system, while validation by all parties ensures integrity of the transactions.",
"A Strong Federation is a federated sidechain where the members of the federation serve as a protocol adapter between the main chain and the sidechain.",
"One could say, essentially, that together they form a Byzantine-robust smart contract.",
"In a Strong Federation the knowledge of private keys is sufficient for the “right to spend\" without the permission of any third-party, and the system has a mechanism that allows settlement back to a parent chain in the case of a complete failure of the federation.",
"Not only are the code updates open and auditable and rejectable in case of coercive behavior, but the state of the system also provides a consistently reliable log that maintains immutability of state.",
"Most importantly: the members of the federation cannot directly control any users' money inside the system other than their own.",
"The network operators of a Strong Federation consist of two types of functionaries.",
"Functionaries are entities that mechanically execute defined operations if specific conditions are met [23].",
"To enhance security, certain operations are split between entities to limit the damage an attacker can cause.",
"In a Strong Federation, functionaries have the power to control the transfer of assets between blockchains and to enforce the consensus rules of the sidechain.",
"In the next section further details will be provided on why dividing those responsibilities is critical.",
"The two types of functionaries are: Blocksigners, who sign blocks of transactions on the sidechain, defining its consensus history.",
"Watchmen, who are responsible for moving the assets out of the sidechain by signing transactions on the main chain.",
"The two components can be independent.",
"Blocksigners are required to produce the blockchain consensus and to advance the sidechain ledger, which they do by following the protocol described in the next section.",
"Watchmen are only required to be online when assets are to be transferred between blockchains.",
"As an extreme example, one could imagine a scheme where watchmen were only brought online once daily to settle a pre-approved batch of inbound and outbound transactions.",
"These two functions are performed by separate dedicated hardened boxes, configured by their owners with the secret key material required for their operation.",
"The interaction between the elements of the network is presented in Figure REF .",
"Between blocksigners and watchmen, only the former are required to produce consensus, which they do by following the protocol described in the next section.",
"Figure: Schematic overview of how a Strong Federation interoperates with another blockchain."
],
[
"Technical Details",
"Supporting Strong Federations on a technical level requires the development of two types of federation: the Federated Peg and Federated Blocksigning."
],
[
"Federated Peg",
"The authors of “Enabling Blockchain Innovations with Pegged Sidechains\" [22] suggested a way to deploy federated sidechains without requiring any alterations to the consensus rules of Bitcoin's blockchain.",
"In their methodology, a sidechain used a $k$ -of-$n$ federation of mutually distrusting participants, called the functionaries, who validate and sign the blocks of the chain (blocksigners) and the pegs (watchmen) respectively.",
"A Federated Peg is a mechanism that uses functionaries to move assets between two chains.",
"The functionaries observe at least the two chains – the Bitcoin blockchain and the sidechain – to validate asset transfers between them.",
"To meet the criteria of a Strong Federation, a set of geographically and jurisdictionally distributed servers is used, creating a compromise-resistant network of functionaries.",
"This network retains a number of the beneficial properties of a fully decentralized security model.",
"Members of the Federated Peg each operate a secure server that runs Bitcoin and sidechain nodes along with software for creating and managing cross-chain transactions.",
"Each server contains a hardware security module that manages cryptographic keys and signs with them.",
"The module's job is primarily to guard the security of the network, and if a compromise is detected, to delete all of its keys, causing the network to freeze by design.",
"If one or a few functionaries are attacked – even if their tamper-resistant hardware is totally compromised – the system is unaffected, as long as enough other functionaries are still intact.",
"Successfully tampering with the federated peg system requires a compromise of at least the majority of functionaries, both blocksigners and watchmen.",
"Even then, tampering is always detectable and usually immediately observable because the blockchain is replicated and validated on machines other than the functionaries.",
"A compromise of a majority of blocksigners would be observable as soon as a non-conforming block was published.",
"If the majority of watchmen remain secure, the value held by the sidechain can be redeemed on the parent blockchain."
],
[
"Byzantine Robustness",
"One of the most important aspects of Bitcoin's mining scheme is that it is Byzantine robust, meaning that anything short of a majority of bad actors cannot rewrite history or censor transactions [24].",
"The design was created to be robust against even long-term attacks of sub-majority hash rate.",
"Bitcoin achieves this by allowing all miners to participate on equal footing and by simply declaring that the valid chain history with the majority of hashpowerThe hashpower, or hash rate, is the measuring unit of the processing power used to secure the Bitcoin network behind it is the true one.",
"Would-be attackers who cannot achieve a majority are unable to rewrite history (except perhaps a few recent blocks and only with low probability) and will ultimately waste resources trying to do so.",
"This incentivizes miners to join the honest majority, which increases the burden on other would-be attackers.",
"However, as discussed in Section REF , this setup leads to latency due to a network heartbeat on the order of tens of minutes and introduces a risk of reorganization even when all parties are behaving honestly."
],
[
"Achieving Consensus in Strong Federations",
"It is critical that functionaries have their economic interests aligned with the correct functioning of the Federation.",
"It would obviously be a mistake to rely on a random assortment of volunteers to support a commercial sidechain holding significant value.",
"Beyond the incentive to attempt to extract any value contained on the sidechain, they would also have little incentive to ensure the reliability of the network.",
"Federations are most secure when each participant has a similar amount of value held by the federation.",
"This kind of arrangement is a common pattern in business [25].",
"Incentives can be aligned through the use of escrow, functionary allocation, or external legal constructs such as insurance policies and surety bonds."
],
[
"Blocksigning in Strong Federations",
"In order for a Strong Federation to be low latency and eliminate the risk of reorganization from a given hostile minority, it replaces the dynamic miner set with a fixed signer set.",
"As in Private Chains [26], the validation of a script (which can change subject to fixed rules or be static) replaces the proof-of-work consensus rules.",
"In a Strong Federation, the script implements a $k$ -of-$n$ multisignature scheme.",
"This mechanism requires blocks be signed by a certain threshold of signers; that is, by $k$ -of-$n$ signers.",
"As such, it can emulate the Byzantine robustness of Bitcoin: a minority of compromised signers will be unable to affect the system.",
"Figure REF presents how the consensus is achieved in a Strong Federation.",
"It is referred to as federated blocksigning and consists of several phases: Step 1: Blocksigners propose candidate blocks in a round-robin fashion to all other signing participants.",
"Step 2: Each blocksigner signals their intent by pre-committing to sign the given candidate block.",
"Step 3: If threshold X is met, each blocksigner signs the block.",
"Step 4: If threshold Y (which may be different from X) is met, the block is accepted and sent to the network.",
"Step 5: The next block is then proposed by the next blocksigner in the round-robin.",
"Due to the probabilistic generation of blocks in Bitcoin, there is a propensity for chain reorganizations in recent blocks [27].",
"Because a Strong Federation's block generation is not probabilistic and is based on a fixed set of signers, it can be made to never reorganize.",
"This allows for a significant reduction in the wait time associated with confirming transactions.",
"Of course, as with any blockchain-based protocol, one could imagine other ways of coordinating functionary signing.",
"However, the proposed scheme improves the latency and liquidity of the existing Bitcoin consensus mechanism, while not introducing SPOF or higher trust requirements as discussed in Section REF .",
"Figure: Federated Blocksigning in a Strong Federation."
],
[
"Security Improvements",
"Byzantine robustness provides protection against two general classes of attack vectors.",
"In the first case,a majority of nodes could be compromised and manipulated by the attacker, breaking integrity of the system.",
"In the second case, a critical portion of nodes could be isolated from the network, breaking availability.",
"Blocksigning in a Strong Federation is robust against up to $2k - n - 1$ attackers.",
"That is, only $2k - n$ Byzantine attackers will be able to cause conflicting blocks to be signed at the same height, forking the network.",
"For instance, a 5-of-8 threshold would be 1-Byzantine robustIf “Byzantine failures\" in a network are caused by nodes that operate incorrectly by corrupting, forging, delaying, or sending conflicting messages to other nodes, then Byzantine robustness is defined as a network exhibiting correct behavior while a threshold of arbitrarily malfunctioning nodes (nodes with Byzantine failures) participate in the network.",
"[28], while 6-of-8 would be 3-Byzantine robust.",
"On the other hand, if at least $n - k + 1$ signers fail to sign, blocks will not be produced.",
"Thus, increasing the threshold $k$ provides stronger protection against forks, but reduces the resilience of the network against signers being unavailable.",
"Section REF explains how the same strategy can be used for applying functionary updates, which is planned as future work."
],
[
"Use Cases",
"Strong Federations were developed as a technical solution to problems blockchain users face daily: transaction latency, commercial privacy, fungibility, and reliability.",
"Many applications for blockchains require Strong Federations to avoid these issues, two of which are highlighted here."
],
[
"International Exchange and Liquid",
"Bitcoin currently facilitates remittance and cross-border payment, but its performance is hampered by technical and market dynamics [29].",
"The high latency of the public Bitcoin network requires bitcoin to be tied up in multiple exchange and brokerage environments, while its limited privacy adds to the costs of operation.",
"Due to market fragmentation, local currency trade in bitcoin can be subject to illiquidity.",
"As a result, many commercial entities choose to operate distinct, higher-frequency methods of exchange [20].",
"These attempts to work around Bitcoin's inherent limitations introduce weaknesses due to centralization or other failings [29].",
"We have developed a specific solution called Liquid designed to make international exchanges more efficient by utilizing bitcoin.",
"The solution is presented in Figure REF , and it is the first implementation of a Strong Federation.",
"As a Strong Federation, Liquid has novel security and trust assumptions, affording it much lower latency than Bitcoin's blockchain, with a trust model stronger than that of other, more centralized, systems (though nonetheless weaker than Bitcoin) [30].",
"Today, the implementation allows for one-minute blocks.",
"It will be possible to reduce the time to the amount required for the pre-commit and agreement threshold time of the network traversal, as discussed in Section REF .",
"This trade-off is worthwhile in order to enable new behaviors, serving commercial needs that neither the Bitcoin blockchain nor centralized third parties can provide.",
"Liquid is a Strong Federation where functionaries are exchanges participating in the network, and an asset is some currency that is transferred from Alice to Bob.",
"As shown in Figure REF , when Alice wants to send money to Bob, she contacts her preferred exchange.",
"The local node of that exchange takes care of finding an appropriate local node of the exchange willing to trade within the Liquid Strong Federation to move assets to Bob.",
"They negotiate the terms, meaning the exchange rate and execution time, and notify Alice about the result.",
"If she agrees, the assets are transferred to Bob.",
"Normally, a very similar scheme would happen on the Blockchain, only the transaction would have to be approved by the whole network, thus causing horrible delay in settlement.",
"Because Liquid operates on a sidechain, we use multisig and if 8 out of 11 participants of the Strong Federation agree on the settlement, Bob receives his money.",
"Figure: International Trade Flow with Liquid.A decrease in latency in Liquid results in an increase in the speed of transaction finality.",
"This in turn reduces the risk of bitcoin valuation changes during transaction settlement time – a key component of successful arbitrage and remittance operations [31].",
"The remitter will eventually receive the initial sender's bitcoin, but will have mitigated a substantial portion of the downside volatility risk by executing closer to the time of sale.",
"Thanks to the decrease in transfer times reducing the cost of arbitrage, Liquid participant markets will function as if they were a unified market.",
"In addition, because Liquid assets are available at multiple fiat on- and off-ramps with relatively little delay, a remitter can settle for fiat in two or more locations in different currencies at price parity.",
"Essentially, Liquid lowers capital constraints relative to money held at varied end-points in the exchange cycle as a result of the network structure.",
"By moving the bitcoin-holding risk, intrinsic to the operation of exchange and brokerage businesses, from a SPOF introduced by a single institution to a federation of institutions, Liquid improves the underlying security of the funds held within the network.",
"By increasing the security of funds normally subject to explicit counterparty risk, Liquid improves the underlying reliability of the entire Bitcoin market.",
"Improvements in privacy come thanks to the adoption of Confidential Transactions, a specific addition to Strong Federations that are discussed further in Section .",
"This provides users of the system stronger commercial privacy guarantees.",
"Strong Federations such as Liquid improve privacy, latency, and reliability without exposing users to the weaknesses introduced by third-party trust.",
"By moving business processes to Liquid, users may improve their efficiency and capital-reserve requirements."
],
[
"Other Financial Technology",
"Significant portions of current financial service offerings are dependent on trusted intermediaries (and shared legal infrastructure when this trust breaks down) or centralized systems for operation [32].",
"They have the potential to be supplanted by new, publicly verifiable consensus systems such as Bitcoin, which offer improvements to security and reliability [33].",
"As an example, liquidity provisioning is the primary business model of Prime Brokerages and Investment Banks [34].",
"Fund managers commit their funds to a single location's custodianship under the premise of reducing costs associated with investment management and improving access to both investment opportunities and liquidity.",
"Third-party broker-dealers then grant each participant access to the liquidity of their respective counterparties – a function of aggregation of capital under a single trusted third-party custodian [35].",
"This system offers investors a means of preferential access to liquidity by enabling customers to buy, sell, and hedge trades with their respective counterparties in a single location.",
"These centralized systems provide convenience to market participants, but are not without risks.",
"One realized example of these risks is that of the Eurosystem following the global financial crisis, in the wake of the financial default of the Lehman Brothers.",
"The effort of the Eurosystem to liquidate assets collateralized by 33 complex securities took more than four years, and resulted in over EUR 1 billion in losses [34].",
"The implicit centralization and dependent trust that arise from systems like these can be resolved via a Strong Federation.",
"It can remove the element of trust when claiming ownership and prevent transactions of uncollateralized assets, while also allowing auditing by existing and new members of the system.",
"Furthermore, ownership of assets can be proven and verified publicly."
],
[
"Innovations",
"In this section, major highlights of the presented design are discussed including: improvements to Determinism, Latency, and Reliability; expansions of Privacy and Confidentiality; improvements to system integrity with Hardware Security; modifications to Native Assets; and Bitcoin wallet protections via Peg-out Authorization."
],
[
"Determinism, Latency, and Reliability",
"While Bitcoin's proof-of-work is a stochastic process, the Strong Federation scheme is deterministic, where each block is expected to be produced by a single party.",
"Therefore reorganizations cannot happen, unlike in Bitcoin where they are an ordinary fact of life.",
"In a Strong Federation, blocksigners need only to obtain consensus amongst themselves before extending history; since they are a small, well-defined set, the network heartbeat can be significantly faster than in Bitcoin.",
"This means that users of Strong Federations can consider a single confirmation to indicate irreversibility; that confirmation can occur as quickly as information can be broadcast between the federation members and processed into a block.",
"It also means that blocks will be produced reliably and on schedule, rather than as a stochastic process where the heartbeat is actually a mean time."
],
[
"Privacy and Confidentiality",
"Though many users presume that a blockchain inherently provides strong privacy, this has repeatedly been shown to be false [36] [37] [38] [39] [40].",
"The Liquid implementation of Strong Federations uses Confidential Transactions (CT) [21] to cryptographically verify users' behavior without providing full transparency of transaction details.",
"As a result, the transfer of assets within a Strong Federation is guaranteed to be private between counterparties, while verifiably fair to network participants.",
"In order to protect confidentiality, CT blinds the amounts of all outputs to avoid leaks of information about the transaction size to third parties.",
"It is also possible to combine inputs and outputs using traditional Bitcoin privacy techniques, such as CoinJoin [41].",
"In typical application, such mechanisms are substantially weakened by the presence of public amounts [12], which can be used to determine mappings between owners of inputs and outputs, but in Liquid, the transaction graph no longer exposes these correlations [42].",
"The use of CT in Liquid is important for two main reasons: commercial usability and fungibility.",
"When it comes to the former, most companies would not be able to operate if their internal ledgers and financial actions were entirely public, since private business relationships and trade secrets can be inferred from transaction records.",
"When CT is introduced this is no longer a problem as the detailed information about the trades is hidden.",
"It is also important to improve the fungibility, because otherwise the history of an asset can be traced through the public record.",
"This can be problematic in the case of “tainted money\", which the authorities in a given jurisdiction define as illegal or suspicious [43].",
"If an asset's history can be backtraced, then users of the network may find themselves obligated to ensure they are not receiving those assets.",
"Such forensic work puts a large technical burden on users and operators of a network and may not even be possible across multiple jurisdictions whose definitions of taint are conflicting or ill-defined [43].",
"This is a potential danger for any type of system that enables the passing back and forth of value with a history, but one that can be corrected with improved fungibility.",
"Unfortunately, CT comes at a technical cost: transactions are much largerThe range of values CT can support include proofs that are often order-of-magnitude larger size than ordinary Bitcoin outputs and can be made larger depending on user requirements.",
"and take correspondingly longer to verify.",
"All transactions in Liquid use CT by default, making operation of the network computationally intensive.",
"Mimblewimble [44] introduces a scheme by which full security may be achieved without full historical chain data, and by which transactions within blocks can no longer be distinguished from each other.",
"This gives stronger privacy than CT alone with better scaling properties than even Bitcoin without CT.",
"The benefits to fungibility and privacy of such a system are readily apparent.",
"Further research will be allocated towards investigating Mimblewimble as a means of confidentially transacting."
],
[
"Hardware Security",
"In Strong Federations, the $k$ -of-$n$ signing requirement requires full security of the hardware, which will be distributed across multiple unknown locations and conditions.",
"The signing keys need to be stored on the devices and not on the server for a simple reason: otherwise, even if the application code was flawless and the userspace code was minimized, a networking stack vulnerability could be exploited in order to gain access to the host and then any keys.",
"While efforts have been made over the years to segment memory and create boundaries through virtualization, memory protection, and other means, the industry has not yet been completely successful [45].",
"The best solution today is to use simplified interfaces and physical isolation; Liquid specifically creates a separate hardened device for key storage and signing in order to significantly reduce the number of avenues of attack.",
"While it is true that public review of cryptographic algorithms and protocols improves the security of a system, the same cannot be said for public review of hardware designs.",
"Indeed, any measure will eventually be defeated by an attacker with an infinite supply of sample hardware.",
"However, if a piece of hardware requires expensive, highly specialized equipment and skills to examine, it reduces the set of people who might be interested in (and capable of) attacking it.",
"This is even more true when a technique used to break a system is destructive, requiring multiple copies of any given hardware [46].",
"Unfortunately, the value of hardware obfuscation for security purposes holds only until the system is broken.",
"After an attack is published, the only way to protect the hardware is to change its design.",
"Thus, Strong Federation hardware includes a reactive system that, when under attack, either sends an alert or simply deletes the information that it determines is likely to be targeted.",
"Traditionally, hardware security modules do this when they register a significant environmental change such as sudden heating or cooling, a temperature out of expected operating ranges, persistent loss of access to the internet, or other environmental fluctuations [47]."
],
[
"Native Assets",
"Strong Federations support accounting of other digital assets, in addition to bitcoin.",
"These native assets can be issued by any user and are accounted for separately from the base bitcoin currency.",
"A participant issues such assets by means of an asset-generating transaction, optionally setting the conditions by which additional issuance can take place in the future: The asset issuer decides on policy for the asset it is generating, including out-of-band conditions for redemption.",
"The issuer creates a transaction with one or more special asset-generating inputs, whose value is the full issuance of the asset.",
"This transaction, and an asset's position in it, uniquely identifies the asset.",
"Note that the initial funds can be sent to multiple different outputs.",
"The asset-generating transaction is confirmed by a Strong Federation participant and the asset can now be transacted.",
"The issuer distributes the asset as necessary to its customer base, using standard Strong Federation transactions.",
"Customers wishing to redeem their asset tokens transfer their asset holdings back to the issuer in return for the out-of-band good or service represented.",
"The issuer can then destroy the tokens (i.e.",
"by sending to an unspendable script like OP_RETURN).",
"Today, users can only trade with one asset type, however the design allows for multiple assets to be involved in a single transaction.",
"In such cases, consensus rules ensure that the accounting equation holds true for each individual asset grouping.",
"This allows the exchange of assets to be trustless and conducted in a single transaction without any intermediary.",
"For that to happen, two participants who wish to trade asset A and asset B would jointly come to an agreement on an exchange rate out-of-band and produce a transaction with an A input owned by the first party and an A output owned by the second.",
"Then the participants create another transaction with a B input owned by the second party and a B output owned by the first.",
"This would result in a transaction that has equal input and output amounts, and will therefore be valid if and only if both parties sign it.",
"To finalize it, both parties would need to sign the transaction, thus executing the trade.",
"What is amazing is that this innovation allows not only for exchanging currencies but any other digital assets: data, goods, information.",
"The protocol could be further improved with more advanced sighash mechanisms."
],
[
"Peg-out Authorization",
"When moving assets from any private sidechain with a fixed membership set but stronger privacy properties than Bitcoin, it is desirable that the destination Bitcoin addresses be provably in control of some user of the sidechain.",
"This prevents malicious or erroneous behavior on the sidechain (which can likely be resolved by the participants) from translating to theft on the wider Bitcoin network (which is irreversible).",
"Since moving assets back to Bitcoin is mediated by a set of watchmen, who create the transactions on the Bitcoin side, they need a dynamic private whitelist of authorized keys.",
"That is, the members of the sidechain, who have fixed signing keys, need to be able to prove control of some Bitcoin address without associating their own identity to it, only the fact that they belong to the group.",
"We call such proofs peg-out authorization proofs and have accomplished it with the following design: Setup.",
"Each participant $i$ chooses two public-private keypairs: $(P_i, p_i)$ and $(Q_i, q_i)$ .",
"Here $p_i$ is an “online key\" and $q_i$ is the “offline key\".",
"The participant gives $P_i$ and $Q_i$ to the watchmen.",
"Authorization.",
"To authorize a key $W$ (which will correspond to a individually controlled Bitcoin address), a participant acts as follows.",
"She computes $L_j = P_j + H(W + Q_j)(W + Q_j)$ for every participant index $j$ .",
"Here $H$ is a random-oracle hash that maps group elements to scalars.",
"She knows the discrete logarithm of $L_i$ (since she knows the discrete logarithm of $P_i$ and chooses $W$ so she knows that of $W + Q_i$ ), and can therefore produce a ring signature over every $L_i$ .",
"She does so, signing the full list of online and offline keys as well as $W$ .",
"She sends the resulting ring signature to the watchmen, or embeds it in the sidechain.",
"Transfer.",
"When the watchmen produce a transaction to execute transfers from the sidechain to Bitcoin, they ensure that every output of the transaction either (a) is owned by them or (b) has an authorization proof associated to its address.",
"The security of this scheme can be demonstrated with an intuitive argument.",
"First, since the authorization proofs are ring signatures over a set of keys computed identically for every participant, they are zero-knowledge for which participant produced them [48].",
"Second, the equation of the keys $L_j = P_j + H(W + Q_j)(W + Q_j)$ is structured such that anyone signing with $L_i$ knows either: The discrete logarithms of $W$ , along with $p_i$ and $q_i$ ; or $p_i$ , but neither $q_i$ nor the discrete logarithm of $W$ .",
"In other words, compromise of the online key $p_i$ allows an attacker to authorize “garbage keys\" for which nobody knows the discrete logarithm.",
"Only compromising both $p_i$ and $q_i$ (for the same $i$ ) will allow an attacker to authorize an arbitrary key.",
"However, compromise of $q_i$ is difficult because the scheme is designed such that $q_i$ need not be online to authorize $W$ , as only the sum $W + Q_i$ is used when signing.",
"Later, when $i$ wants to actually use $W$ , she uses $q_i$ to compute its discrete logarithm.",
"This can be done offline and with more expensive security requirements."
],
[
"Evaluation",
"The information moved through Strong Federations will be very sensitive.",
"As a result, a thorough understanding of the potential security threats is crucial.",
"This is particularly important when dealing with Bitcoin, where transactions are irrevocable.",
"In other words, continued operation of the network is a secondary priority; few would choose to have their money move rapidly into the hands of a thief over a delayed return to their own pockets.",
"As the aggregate value of assets inside a Strong Federation increases, the incentives for attackers grow, and it becomes crucial they cannot succeed when targeting any functionary nor the maintainer of the codebase.",
"Thankfully, as participants in a Strong Federation scale up the value of assets flowing through the system, they will be naturally incentivized to take greater care of access to the federated signers under their control.",
"Thus, the federated security model neatly aligns with the interests of its participants."
],
[
"Comparison to Existing Solutions",
"Existing proposals to form consensus for Bitcoin-like systems generally fall into two categories: those that attempt to preserve Bitcoin's decentralization while improving efficiency or throughput and those that adopt a different trust model altogether.",
"In the first category are GHOST [49], block DAGs [50] [51], and Jute [52].",
"These schemes retain Bitcoin's model of blocks produced by a dynamic set of anonymous miners, and depend on complex and subtle game-theoretic assumptions to ensure consensus is maintained in a decentralized way.",
"The second category includes schemes such as Stellar [53] and Tendermint [54], which require new participants to choose existing ones to trust.",
"These examples have the failure risks associated with trusted parties, which when spread across complex network topologies lead to serious but difficult-to-analyze failure modes [55].",
"Our proposal works in the context of a fixed set of mutually distrusting but identifiable parties, and therefore supports a simple trust model: as long as a quorum of participants act honestly, the system continues to work.",
"Parallel to consensus systems are systems that seek to leverage existing consensus systems to obtain faster and cheaper transaction execution.",
"The primary example of this is the Lightning network [56], which allows parties to transact by interacting solely with each other, only falling back to the underlying blockchain during a setup phase or when one party fails to follow the protocol.",
"We observe that since these systems work on top of existing blockchains, they complement new consensus systems, including the one described in this paper.",
"A novel proposal has been presented recently by Eyal et al. [57].",
"Although not yet available on the market, their Bitcoin-NG scheme is a new blockchain protocol designed to scale.",
"Based on the experiments they conduct it seems like their solution scales optimally, with bandwidth limited only by the capacity of the individual nodes and latency limited only by the propagation time of the network.",
"However, there may be game-theoretic failings or denial-of-service vectors inherent to their design that have yet to be explicated."
],
[
"Protection Mechanisms",
"An attacker must first communicate with a system to attack it, so the communication policies for a Strong Federation have been designed to isolate it from common attack vectors.",
"Several different measures are taken to prevent untrustworthy parties from communicating with functionaries: Functionary communications are restricted to hard-coded Tor Hidden Service addresses known to correspond to known-peer functionaries.",
"Inter-functionary traffic is authenticated using hard-coded public keys and per-functionary signing keys.",
"The use of Remote Procedure Calls (RPC) is restricted on functionary hardware and on Liquid wallet deployments to callers on the local system only.",
"Above and beyond, the key policy works to protect the network.",
"While the blocksigners are designed with secret keys that are unrecoverable in any situation, the watchmen keys must be created with key recovery processes in mind.",
"Loss of the blocksigner key would require a hard-fork of the Strong Federation's consensus protocol.",
"This, while difficult, is possible and does not risk loss of funds.",
"However, loss of sufficient watchmen keys would result in the loss of bitcoin and is unacceptable.",
"Although the Strong Federation design is Byzantine robust, it is still very important that functionaries avoid compromise.",
"Given tamper-evident sensors designed to detect attacks on functionaries, if an attack is determined to be in progress, it is important to inform other functionaries in the network that its integrity can no longer be guaranteed.",
"In this case, the fallback is to shutdown the individual system, and in a worst case scenario, where the Byzantine robustness of the network is potentially jeopardized, the network itself should shutdown.",
"This ensures a large safety margin against system degradation – assuring both the direct security of users' funds and the users' confidence in the system's continued correct operation."
],
[
"Backup Withdrawal",
"The blockchain for the Liquid implementation of Strong Federations is publicly verifiable, and it should be possible, in principle, for holders of bitcoin in Liquid to move their coins back to Bitcoin even under conditions where the Liquid network has stalled (due to DoS or otherwise).",
"The most straightforward way of doing this would be for watchmen to provide time-locked Bitcoin transactions, returning the coins to their original owners.",
"However, this updates the recipient-in-case-of-all-stall only at the rate that time-locked transactions are invalidated, which may be on the order of hours or days.",
"The actual owners of coins on the Liquid chain will change many times in this interval, so this solution does not work.",
"Bitcoin does not provide a way to prove ownership with higher resolution than this.",
"However, it is possible to set a “backup withdrawal address\" that is controlled by a majority of network participants, functionaries, and external auditors.",
"This way, if Liquid stalls, it is possible for affected parties to collectively decide on appropriate action."
],
[
"Availability and Denial of Service",
"There are two independent thresholds involved when signing blocks in a Strong Federation: the signing and precommit thresholds.",
"The former is an unchangeable property of the network and may be set with resilience in mind.",
"It may also be adjusted to a more advanced policy that supports backup blocksigners that are not normally online.",
"The precommit threshold, on the other hand, is determined only by the signers themselves and may be set to a high level (even requiring unanimity of signers) and changed as network conditions require.",
"This means that even if the network block signature rules in principle allow Byzantine attackers to cause forks, in practice malicious users are (at worst) limited to causing a denial of service to the network, provided that the blocksigners set a high enough precommit threshold.",
"A software bug or hardware failure could lead to a breakdown in a single functionary such that it temporarily no longer functions.",
"Such a participant would no longer be able to take part in the consensus protocol or be able to approve withdrawals to Bitcoin.",
"Unless enough functionaries fail so that the signing threshold is not achievable, the network will be unaffectedDowntime for a functionary does cause degradation in throughput performance as that signer's turn will have to be “missed\" each round..",
"In such a case, funds will be unable to move (either within the sidechain or back to Bitcoin) for the duration of the outage.",
"Once the functionaries are restored to full operation, the network will continue operating, with no risk to funds."
],
[
"Hardware Failure",
"If a blocksigner suffers hardware failure, and cryptographic keys are not recoverable, the entire network must agree to change signing rules to allow for a replacement blocksigner.",
"A much more serious scenario involves the failure of a watchman, as its keys are used by the Bitcoin network and cannot be hard-forked out of the current Bitcoin signature set.",
"If a single watchman fails, it can be replaced and the other watchmen will be able to move locked coins to ones protected by the new watchman's keys.",
"However, if too many watchmen fail at once, and their keys are lost, bitcoin could become irretrievable.",
"As mentioned in Section REF , this risk can be mitigated by means of a backup withdrawal mechanism.",
"Prevention mechanisms include extraction and backup withdrawal of watchman key material, so that bitcoin can be recovered in the event of such a failure.",
"The encryption of extracted keys ensures that they can only be seen by the original owner or an independent auditor.",
"This prevents individual watchmen operators (or anyone with physical access to the watchmen) from extracting key material that could be used to operate outside of the sidechain."
],
[
"Rewriting History",
"It is possible that blocksigners could attempt to rewrite history by forking a Strong Federation blockchain.",
"Compared to Bitcoin, it is quite cheap to sign conflicting histories if one is in possession of a signing key.",
"However, rewriting the chain would require compromising the keys held in secure storage on a majority of blocksigners.",
"Such an attack is an unlikely scenario, as it would require determining the locations of several signers, which are spread across multiple countries in multiple continents, and either bypassing the tamper-resistant devices or else logically accessing keys through an exploit of the underlying software.",
"Further, such an attack is detectable, and a proof (consisting simply of the headers of the conflicting blocks) can be published by anybody and used to automatically stop network operation until the problem is fixed and compromised signers replaced.",
"If the network was forked in this way, it might be possible for active attackers to reverse their own spending transactions by submitting conflicting transactions to both sides of the fork.",
"Therefore, any “valid” blocks that are not unique in height should be considered invalid."
],
[
"Transaction Censorship",
"By compromising a threshold number of blocksigners, an attacker can potentially enforce selective transaction signing by not agreeing to sign any blocks that have offending transactions and not including them in their own proposed blocks.",
"Such situations might occur due to a conflict between legitimate signers or the application of legal or physical force against them.",
"This type of censorship is not machine-detectable, although it may become apparent that specific blocksigners are being censored if they have many unsuccessful proposal rounds.",
"It will be obvious to the affected network participants that something is happening, and in this case the Strong Federation may use the same mechanism to replace or remove the attacking signers that is used to resolve other attacks."
],
[
"Confiscation of Locked Bitcoins",
"If enough watchmen collude, they can overcome the multisig threshold and confiscate all the bitcoin currently in the sidechain.",
"The resilience against such attacks can be improved by setting a high signing threshold on the locked bitcoins.",
"This can exclude all but the most extreme collusion scenarios.",
"However, this weakens resilience against failures of the watchmen whose key material is lost.",
"The cost-benefit analysis will have to be done as federated signing technology matures."
],
[
"Future Research",
"While Strong Federations introduce new technology to solve a variety of longstanding problems, these innovations are far from the end of the road.",
"The ultimate design goal is to have a widely distributed network in which the operators are physically unable to interfere or interact with application-layer processes in any way, except possibly by entirely ceasing operation, with backup plans to retrieve the funds to the parent chain."
],
[
"Further Hardening of Functionaries",
"More research should be done to ensure that functionaries cannot be physically tampered with, and that network interactions are legitimate and auditable.",
"Methods for future hardening could include specific design improvements or further cryptographic arrangements.",
"In a Strong Federation, compromised functionaries are unable to steal funds, reverse transactions, or influence other users of the system in any way.",
"However, enough malicious functionaries can always stall the network by refusing to cooperate with other functionaries or by shutting down completely.",
"This could freeze funds until an automatic withdrawal mechanism starts.",
"As such, it would be beneficial to research possibilities for creating incentive structures and methods to encourage functionary nodes to remain online under attack.",
"This could be done, for example, by requiring that they periodically sign time-locked transactions.",
"These incentives could prevent certain denial-of-service attacks."
],
[
"Enlightening Liquid",
"The privacy and speed of a Strong Federation could be further improved by combining it with Lightning [56].",
"Just as with Bitcoin's network, the throughput of initial systems built with this architecture is limited on purpose, as the transactions are published in blocks that must be made visible to all participants in the network.",
"This threshold is set by the need for everyone to see and validate each operation.",
"Even with a private network that mandates powerful hardware, this is a serious drawback.",
"With Lightning, individual transactions only need to be validated by the participating parties [56].",
"This dramatically reduces the verification load for all participants.",
"Because end-to-end network speed is the only limiting factor [58], it also greatly reduces the effects of network latency.",
"Furthermore, nodes in a Strong Federation could route payments via Lightning, a network of bidirectional payment channel smart contracts.",
"This may allow for even more efficient entry and exit from the Liquid network.",
"Finally, Lightning can replace inter-chain atomic swap smart contracts and probably hybrid multi-chain transitive trades [26] without having the limitation of a single DMMS chain."
],
[
"Confidential Assets",
"Confidential Transactions (CT) hide the amounts but not the types of transacted assets, so its privacy is not as strong as it could be.",
"However, CT could be extended to also hide the asset type.",
"For any transaction it would be impossible to determine which assets were transacted or in what amounts, except by the parties to the transaction.",
"Called Confidential Assets (CA), this technology improves user privacy and allows transactions of unrelated asset classes to be privately spent in a single transaction.",
"However, the privacy given to assets is qualitatively different than that of CT.",
"Consider a transaction with inputs of asset types A and B.",
"All observers know that the outputs have types A and B, but they are unable to determine which outputs have what types (or how they are split up or indeed anything about their amounts).",
"Therefore all outputs of this transaction will have type “maybe A, maybe B\" from the perspective of an observer.",
"Suppose then that an output of type A, which is a “maybe A, maybe B\" to those not party to the transaction, is later spent in a transaction with an asset type C. The outputs of this transaction would then be “maybe A, maybe B, maybe C\" to outside observers, “maybe A, maybe C\" to those party to the first transaction, and known to those party to the second.",
"As transactions occur, outputs become increasingly ambiguous as to their type, except to individual transactors, who know the true types of the outputs they own.",
"If issuing transactions always have multiple asset–types, then non-participant observers never learn the true types of any outputs."
],
[
"Byzantine-Robust Upgrade Paths",
"Most hardening approaches rely on a central, trusted third-party who can provide upgrades: operating systems and other critical software wait for signed software packages, generated in locked down build labs, then hosts retrieve these packages, verify signatures, and apply them, often automatically.",
"This would undermine the threat model of a Strong Federation, as any SPOF can be compromised or coerced to comply.",
"All aspects of the change control system must instead be defensibly Byzantine secure.",
"In any large system, one must assume some part of it may be in a state of failure or attack at any point in time.",
"This means that what can be a simple process for a central authority becomes somewhat more complex.",
"Unfortunately, creating an agile network, or a system that is upgradable, requires a security tradeoff.",
"An ideal balance is hard to strike: as a network's independence grows, the cost and difficulty in upgrading also increases.",
"As such, it is important that all changes in the code should be opt-in for all parties and the process should be consensus ($k$ -of-$n$ ) driven across the functionary set.",
"These changes should also be fully auditable and transparent prior to application.",
"Ultimately, the processes of maintenance, of new member additions, or of strict improvements to the network must also be Byzantine secure for the whole system to be Byzantine secure.",
"For Bitcoin, this is achieved by a long-tail upstream path which is an audited and open-source procedure, and ultimately the consensus rules each user decides to validate are self-determined (i.e., there may be permanent chain splits in case of controversial changes).",
"Figure: Byzantine-Robust Upgrade schema planned for Strong Federations.For Strong Federations, this will be achieved through the design and implementation of an upgrade procedure that enables iterative improvement to the system without enabling attack surfaces by emulating Bitcoin's soft-fork upgrade path.",
"This is presented in Figure REF , and follows the steps: An upstream software provider (USP) writes software updates for the functionary network and provides those updates to the functionaries for implementation.",
"An external security auditor may be used to review the software update and documentation for correctness, verifying the accuracy of the documentation and/or the codebase itself.",
"Each functionary verifies the signatures from the USP and possibly the third-party auditor, and may also review or audit the updates if it wishes.",
"Each functionary signs the update on the server and returns the resulting signature to the USP.",
"Once a supermajority of functionaries have signed the update, the USP combines their signatures and the update image into a single package.",
"This file, consisting of the update image, documentation, and a supermajority of functionary signatures, is then distributed to each functionary.",
"Each functionary receives the USP and supermajority signatures on their server.",
"Each functionary verifies the package contents and applies the update.",
"Note that this situation assumes honest participants.",
"There are scenarios in which, for instance, a single group of collaborating malicious functionaries can collectively reject any given upgrade path.",
"Methods of combating this scenario will be further investigated."
],
[
"Conclusion",
"The popularity of Bitcoin shows that permissionless proof-of-work is an effective mechanism for developing an infrastructure, with hundreds of millions of dollars [59] across dozens of companies being invested in new innovations spanning chip and network design, datacenter management, and cooling systems [60].",
"The value of the security offered by this conglomerate of resources is immense.",
"There is, however, a drawback of the proof-of-work underlying Bitcoin [2]: the addition of latency (the block time) to establish widely distributed checkpoints for the shared, current state of the ledger.",
"This paper introduces the Strong Federation: a federated consensus mechanism which significantly mitigates a number of real-world systemic risks when used in conjunction with proof-of-work.",
"The solution is resilient against broad categories of attacks via specific implementation decisions and minimization of attack surfaces.",
"Strong Federations improve blockchain technology by leveraging sidechain technology.",
"Furthermore, market enhancements utilizing Confidential Transactions and Native Assets are proposed.",
"This paper proposes a methodology that utilizes hardware security modules (HSMs) for enforcing consensus.",
"Currently HSMs have limited ability to verify that their block signatures are only used on valid histories that do not conflict with past signatures.",
"This arises both because of the performance limitations in secure hardware and because anything built into an HSM becomes unchangeable, making complex rule sets difficult and risky to deploy.",
"Improved verification requires HSMs to support an upgrade path that is sufficiently capable while being hardened against non-authorized attempts at upgrading.",
"Alternatively, every software deployment may imply a new hardware HSM deployment, but that's not cost efficient.",
"The first working implementation of a Strong Federation is Liquid – a Bitcoin exchange and brokerage multi-signature sidechain that bypasses Bitcoin's inherent limitations while leveraging its security properties.",
"In Liquid, Bitcoin's proof-of-work is replaced with a $k$ -of-$n$ multisignature scheme.",
"In this model, consensus history is a blockchain where every block is signed by the majority of a deterministic, globally distributed set of functionaries running on hardened platforms, a methodology that directly aligns incentives for the participants.",
"Strong Federations will be useful in many general-purpose industries – especially those that seek to represent and exchange their assets digitally and must do so securely and privately without a single party that controls the custodianship, execution, and settling of transactions."
],
[
"Acknowledgements",
"We thank Matt Corallo and Patrick Strateman for their substantial commentary and contribution in the formation of the ideas and process behind this paper.",
"We'd also like to thank Eric Martindale, Jonas Nick, Greg Sanders, and Kat Walsh for their extensive review.",
"Finally, we thank Kiara Robles for her excellent figures and diagrams."
]
] | 1612.05491 |
[
[
"Bulk viscous corrections to photon production in the quark-gluon plasma"
],
[
"Abstract Photons radiated in heavy-ion collisions are a penetrating probe, and as such can play an important role in the determination of the quark-gluon plasma (QGP) transport coefficients.",
"In this work we calculate the bulk viscous correction to photon production in two-to-two scattering reactions.",
"Phase-space integrals describing the bulk viscous correction are evaluated explicitly in order to avoid the forward scattering approximation which is shown to be poor for photons at lower energies.",
"We furthermore present hydrodynamical simulations of AA collisions focusing on the effect of this calculation on photonic observables.",
"Bulk corrections are shown to reduce the elliptic flow of photons at higher $p_T$."
],
[
"Introduction",
"Photons are an important probe in relativistic heavy-ion collisions: they are created at all stages of the collisions and leave the medium undistorted by the strong interaction.",
"At leading order in perturbation theory, ${\\cal O}(\\alpha _s)$ , photon production in the QGP phase receives contributions from two-to-two scattering channels [1], [2] which comprise Compton scattering and quark-antiquark annihilation, and from channels which consider quark bremsstrahlung, off-shell pair annihilation, and coherence between different scattering sites (Landau-Pomeranchuk-Migdal effect) [3].",
"The early works assumed a medium in thermal equilibrium.",
"However, hydrodynamical simulations show that there are sizeable viscous effects in the evolution of the created medium due to shear and bulk viscosity [4].",
"For theoretical consistency, one thus needs to include the effect of viscosity on photon production.",
"In return, this allows for the extraction of QGP transport coefficients from electromagnetic observables: a worthy reward.",
"Table II in Ref.",
"[5] presents a summary of the viscous corrections (which include both shear and bulk) to photon production included so far in photon calculations.",
"The contribution presented here addresses the correction owing to bulk viscosity in a specific channel: two-to-two scattering channels, for which estimates of the bulk viscous correction existed using the forward scattering approximation (see below).",
"We present the results of a field-theoretical calculation, and then explore its phenomenological effects using relativistic hydrodynamics."
],
[
"Bulk viscous correction to two-to-two channels",
"The rate of photon emission, $R$ , in a unit volume can be written in a way which is valid out of equilibrium, as long as the medium is static and spatially uniform [6] $k \\frac{dR}{d^3k} = \\frac{i}{2 (2\\pi )^3} \\Pi _{12}(K)^{\\,\\mu }_{\\hspace{5.0pt}\\mu }$ Here, $k$ is the photon energy and $\\Pi _{12}$ is one component of the photon polarization tensor in the real-time formalism [7].",
"Figure: Top row: Diagrams contributing at leading order for hard quark loops.",
"Bottom row: Diagrams contributing at leading order for hard quark loops.The diagrams in the top row of Fig.",
"REF contribute at leading order to the photon polarization tensor when the loop momentum is hard.",
"Using finite-temperature cutting rules these diagrams can be transformed into a kinetic theory equation for the rates [2] $k \\frac{dR}{d^3 k} = \\sum _{\\mathrm {channels}} \\int _P \\, \\int _{P^{^{\\prime }}} \\, \\int _{K^{^{\\prime }}} \\frac{1}{2(2\\pi )^3}\\\\ (2\\pi )^4 \\delta ^{(4)}(P+P^{\\prime }-K-K^{\\prime }) \\,\\left| \\mathcal {M}\\right|^2 f(P) \\, f(P^{\\prime })\\, (1 \\pm f(K^{\\prime })).$ Here $\\mathcal {M}$ is the amplitude for two-to-two scattering with a photon in the final state.",
"Additionally, diagrams with soft loop momenta contribute at the same order, see the bottom row of Fig.",
"REF .",
"In order to avoid infrared divergences and include all leading order effects one must use resummed quark propagators for soft momenta.",
"This is done using the method of hard thermal loops (HTL) [8].",
"We included the viscous correction to the resummed propagator.",
"The different structure of hard and soft loop diagrams necessitates a cutoff scale $q_{\\mathrm {cut}}$ between hard and soft loop momenta, such that $gT \\ll q_{\\mathrm {cut}} \\ll T$ .",
"At leading order the results should be independent of $q_{\\mathrm {cut}}$ .",
"Figure: Dependence of Γ bulk \\Gamma _{\\mathrm {bulk}} on q cut q_{\\mathrm {cut}}, the cutoff between hard and soft loops, for two values of the strong coupling constant gg.The crucial ingredient in the calculation of viscous corrections is the momentum distribution.",
"The equilibrium distribution, $f_0$ , is replaced by $f = f_0 + \\delta f_{\\mathrm {bulk}}.$ where $\\delta f_{\\mathrm {bulk}}$ is the bulk viscous correction derived using kinetic theory: $\\delta f_{\\mathrm {bulk}} = f_{0} (1 \\pm f_{0}) \\; (E-\\frac{m_{\\mathrm {th}}^2}{E}) \\;\\frac{\\Pi }{15 (\\epsilon + P) (\\frac{1}{3} - c_s^2)}$ where $E = \\sqrt{p^2 +m_{\\mathrm {th}}^2}$ and $\\Pi $ is the bulk viscous pressure.",
"This equation was obtained in the relaxation time approximation for particles with thermal masses $m_{\\mathrm {th}}$ [5].",
"We calculated rates up to first order in $\\delta f$ evaluating the loop and phase space integrals numerically.",
"For further details see the discussion of shear viscous corrections to photon production using a diagrammatic approach in Ref.",
"[9].",
"The photon production rate can be written as $k \\frac{dR}{d^3 k} = T^2 \\left(\\Gamma _{\\mathrm {eq}} + \\frac{\\Pi }{15(\\epsilon + P)(\\frac{1}{3}-c_s^2)} \\Gamma _{\\mathrm {bulk}}\\right).$ The quantities $\\Gamma _{\\mathrm {eq}}$ and $\\Gamma _{\\mathrm {bulk}}$ contain the contribution of $f_0$ and $\\delta f_{\\mathrm {bulk}}$ respectively to photon production in a static, homogenous brick of QGP.",
"Fig.",
"REF shows the dependence of $\\Gamma _{\\mathrm {bulk}}$ on the cut between soft and hard loops $q_{\\mathrm {cut}}$ .",
"At very low values of $g$ there is a wide range of cuts that give the same value of $\\Gamma _{\\mathrm {bulk}}$ .",
"At more realistic values for heavy-ion collisions, $g = 2$ , $\\Gamma _{\\mathrm {bulk}}$ depends more on the cut which only cancels at leading order.",
"We chose the minimal value of $\\Gamma _{\\mathrm {bulk}}$ which occurs at roughly $q_{\\mathrm {cut}}/T = \\sqrt{g}$ .",
"Figure: Comparison of a full calculation of bulk viscous corrections to photon production and a calculation using the forward scattering approximation.",
"Top panel: Single particle spectrum elements as defined in Eq.",
"().",
"Bottom panel: Ratio of forward-scattering result to that of the full calculation.The full calculation of $\\Gamma _{\\mathrm {bulk}}$ can be compared with its value obtained in the forward scattering approximation [10], see Fig.",
"REF .",
"This approximation assumes that the exchanged momentum in the two-to-two scattering diagrams is soft which is valid when the external particles have high momenta.",
"It is furthermore only correct at leading logarithm in $g_s$ and ignores the viscous correction to the HTL resummation.",
"As expected, the two calculations are similar at high photon momenta but different at lower momenta."
],
[
"Hydrodynamic modelling",
"Up until now we have only discussed viscous corrections to the photon production rate in a homogeneous brick of QGP.",
"To make contact with experiments we need to integrate these calculations with a hydrodynamic simulation of heavy-ion collisions.",
"The spectrum of thermal photons produced in the medium is then described by $k \\frac{dN_{\\mathrm {thermal}}}{d^3 k} = \\int d^4 x \\left[ \\left.",
"k \\frac{d R}{d^3 k} \\left(T(x),E_k \\right)\\,\\right|_{E_k = k \\cdot u(x)} \\right].$ $k d R/d^3 k$ only depends on position through hydrodynamic variables such as $u^\\mu $ , $T$ , and $\\Pi $ .",
"For our bulk viscous correction the integral is over all cells in the QGP phase.",
"The hydrodynamical events were generated with MUSIC [11] and used IP-Glasma initial conditions.",
"We looked at Au-Au collisions at $200\\,\\mathrm {GeV}$ in a $0-40\\%$ centrality class.",
"For further details see [5].",
"We model all photon production channels as in [5], with the exception of the bulk viscous correction to two-to-two scattering in the QGP phase where the full calculation was used instead of the forward scattering approximation.",
"Figure: Elliptic flow of thermal photons coming from two-to-two channels in QGP.",
"The curves refer to the treatment of the bulk viscous correction.Figure: Elliptic flow of direct photons.",
"The curves refer to the treatment of the bulk viscous correction to two-to-two scattering in QGP.The bulk viscous correction effect is moderate for the photon spectrum, and becomes more considerable for the elliptic flow.",
"Correcting for bulk viscosity creates a $\\sim 20\\%$ decrease in the yield for photons from $ 2 \\rightarrow 2$ processes with $p_T\\sim 2$ GeV.",
"Fig.",
"REF shows the elliptic flow of photons coming from two-to-two scattering in the QGP phase, and compares results obtained using different bulk viscous corrections to that channel.",
"The underlying hydrodynamical events are the same in all cases and include bulk viscosity.",
"At lower $p_T$ the bulk viscous correction has little effect but at higher $p_T$ it reduces the elliptic flow considerably.",
"Fig.",
"REF shows the elliptic flow of direct photons, i.e.",
"prompt photons, thermal photons from the hadronic and QGP phase and viscous corrections where they have been calculated.",
"The curves only differ in the treatment of the bulk viscous correction to two-to-two scattering in the QGP phase.",
"The difference between the curves is smaller than that in Fig.",
"REF , owing to the contribution from other channels, but a complete calculation is clearly required.",
"Figure: Bulk viscosity as a function of temperature .Figure: Distribution of photon yield in different p t p_t bins with the temperature of the emitting fluid cells, for Au Au collisions in the 0-20%0-20\\% centrality class.",
"This figure is from Ref.",
".",
"The events used to make this figure are slightly different from the ones used in this work but the general trend is the same.The effect of bulk viscous corrections on the photon elliptic flow can be understood in a simple way.",
"Bulk viscosity peaks sharply at a temperature of $T_c = 180$ MeV [4], see Fig.",
"REF .",
"Thus the bulk correction mostly affects photon production from cells with temperature around $T_c$ .",
"Furthermore $\\delta f_{\\mathrm {bulk}}$ is rotationally invariant so elliptic flow of the bulk viscous correction is solely due to the flow of the emitting fluid cell.",
"Fig.",
"REF shows the yield of photons as a function of temperature and for different windows of photon momentum [12].",
"Concentrating, for the sake of illustration, on photons with $p_T \\approx 3$ GeV: according to Fig.",
"REF , those are either blueshifted photons created in low temperature cells with high flow, or photons from high temperature cells with lower flow.",
"The bulk correction will suppress emission from the former, which in turn leads to a net reduced elliptic flow [13]."
],
[
"Conclusions, outlook, and acknowledgments",
"We have presented a calculation of the bulk viscous correction to photon production in QGP through two-to-two channels.",
"This calculation was integrated with hydrodynamic simulations of heavy-ion collisions.",
"The bulk correction appreciably reduces the elliptic flow at higher photon $p_T$ , while having little effect at lower $p_T$ .",
"The viscous corrections to all leading-order QGP channels have not been computed so far, as the corresponding Feynman diagrams can have a complicated structure.",
"They can be evaluated for a medium in equilibrium by using the Kubo-Martin-Schwinger relation.",
"In a future publication we will explain how these diagrams can be evaluated out of equilibrium, thereby allowing the computation of viscous corrections to all leading order channels of photon production in the QGP in a theoretically consistent fashion [13].",
"This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, and in part by the DOE under Contract No.",
"DE-SC0012704.",
"C.S.",
"gratefully acknowledges a Goldhaber Distinguished Fellowship from Brookhaven Science Associates, and C. G. gratefully acknowledges support from the Canada Council for the Arts through its Killam Research Fellowship program."
]
] | 1612.05517 |
[
[
"Understanding quantum work in a quantum many-body system"
],
[
"Abstract Based on previous studies in a single particle system in both the integrable [Jarzynski, Quan, and Rahav, Phys.~Rev.~X {\\bf 5}, 031038 (2015)] and the chaotic systems [Zhu, Gong, Wu, and Quan, Phys.~Rev.~E {\\bf 93}, 062108 (2016)], we study the the correspondence principle between quantum and classical work distributions in a quantum many-body system.",
"Even though the interaction and the indistinguishability of identical particles increase the complexity of the system, we find that for a quantum many-body system the cumulative quantum work distribution still converges to its classical counterpart in the semiclassical limit.",
"Our results imply that there exists a correspondence principle between quantum and classical work distributions in an interacting quantum many-body system, especially in the large particle number limit, and further justify the definition of quantum work via two point energy measurements in quantum many-body systems."
],
[
"Introduction",
"In recent years, the field of nonequilibrium statistical mechanics in small systems [1], [2], [3], [4] has attracted lots of attention.",
"A major breakthrough in this field in the past two decades is the discovery of exact fluctuation relations, which hold true for systems driven arbitrarily far from equilibrium.",
"Their validity has been confirmed in various experimental and numerical studies [5], [6], [7], [8], [9], [10], [11], [12], [13].",
"Now, these relations are collectively known as fluctuation theorems (FTs).",
"The FTs have provided insights into the physics of nonequilibrium processes in small systems where fluctuations are important [3].",
"Despite these great developments, there are still some aspects of these FTs that have not been fully understood.",
"The definition of the quantum work is one example.",
"There have been many definitions of quantum work for an isolated system [14].",
"However, only the work defined through two projective measurements of the system's instantaneous energy, i.e., at the start ($t=0$ ) and at the end ($t=\\tau $ ) of the driving process [11], [15], [16], [17], [18], [19], [20], satisfies the FTs.",
"Although this definition of quantum work satisfies quantum nonequilibrium work relations, it might seem ad hoc.",
"This is because the collapse of the wave function [21], when measuring the final energy, brings profound interpretational difficulty to the definition of quantum work [22].",
"Therefore, it is necessary to find other independent evidence (besides the validity of the FTs) to justify the definition of quantum work via two-point energy measurements.",
"Recently, the quantum work defined via two-point energy measurements has been justified in both a one-dimensional integrable system [22] and a chaotic system [23], [24] through the correspondence between quantum and classical work distributions.",
"By using the semiclassical method [25], [26] and the numerical simulation, it is shown that in the semiclassical limit, i.e., $\\hbar \\rightarrow 0$ , the quantum work distribution converges to the classical work distribution after ignoring the effect due to interference of classical trajectories [22].",
"Therefore, there is a quantum-classical correspondence principle of work distributions.",
"Thus, these studies provide some justification to the definition of the quantum work, because the classical work is well defined without any ambiguity.",
"Nevertheless, for quantum many-body systems, the correspondence between quantum and classical work distributions has not been studied so far.",
"The indistinguishability of identical particles [27], [28] and the interaction makes the properties of quantum work even more elusive.",
"Also, the nonequilibrium dynamic evolution of a quantum many-body system is extremely difficult to solve.",
"Following a similar argument to that in Refs.",
"[19], [20], it can be checked that quantum work defined via two-point energy measurements in a quantum many-body system satisfies FT. For example, the work fluctuations in bosonic Josephson junctions has been studied in Ref. [29].",
"But a deeper understanding about quantum work in a quantum many-body system is still lacking.",
"And the quantum work mentioned above has not been justified in these systems.",
"In this article we aim to explore the properties of quantum work in a quantum many-body system, i.e., a one-dimensional (1D) Bose-Hubbard (BH) model, and study the correspondence principle of work distributions when both indistinguishability and interaction play an important role.",
"The BH model which describes an interacting boson in a lattice potential£¬ constitutes one of the most extensively studied and most fundamental Hamiltonians in the field of condensed matter theory and quantum simulation.",
"It undergoes a transition from a superfluid phase to an insulator phase as the strength of the potential is increased [30], [31], [32], [33], [34], [35], [36], [37], [38].",
"Meanwhile, this quantum many-body system has a classical limit.",
"The classical limit of this model is described by the celebrated discrete nonlinear Schödinger equation [39], which possesses rich properties in both static and dynamic aspects.",
"We study the work distribution of this system in both quantum and classical regimes.",
"The results show that there indeed exists a quantum-classical correspondence between work distributions in this quantum many-body system.",
"Furthermore, we investigate when the correspondence principle between work distributions will break down with the decrease of the number of particles.",
"Our study justifies the definition of the quantum work via two-point energy measurements in a quantum many-body system.",
"The remainder of this article is organized as follows.",
"In Sec.",
", we introduce the 1D BH model, briefly review its properties, and discuss the classical limit of it.",
"The quantum and classical work distributions are compared in Sec.",
"where we prove that the correspondence principle between quantum and classical work distributions can be reduced to the correspondence principle between the quantum and classical transition probabilities.",
"Then we give definitions and discussions of the quantum and classical transition probabilities between different energy eigenstates.",
"Our numerical results and analysis are provided in Sec.",
"where we show that the quantum and classical transition probabilities in the 1D two-site and three-site BH models converge in the semiclassical limit.",
"Finally, conclusions and discussions are given in Sec. .",
"Figure: (Color online) Energy spectrum of the 1D two-site BH Hamiltonian () as a function of the work parameter JJ for N=100N=100.Inset: The details of the energy spectrum in the red rectangle."
],
[
"1D Bose-Hubbard model",
"The Hamiltonian of the standard 1D BH model is written as $\\hat{H}=\\sum _j^L\\left[-J(\\hat{a}_j^\\dag \\hat{a}_{j+1}+\\hat{a}_{j+1}^\\dag \\hat{a}_{j})+\\frac{U}{2}\\hat{a}^\\dag _j\\hat{a}_j(\\hat{a}^\\dag _j\\hat{a}_j-1)\\right],$ where $\\hat{a}_j, \\hat{a}_j^\\dag $ are bosonic annihilation and creation operators for the $j$ th site and satisfy the usual bosonic commutation rules $[\\hat{a}_i, \\hat{a}_j^\\dag ]=\\delta _{ij}$ and $L$ denotes the number of sites.",
"$U$ is a measure for the on-site two-body interaction strength depending on the $s$ -wave scattering length, and $J$ denotes the tunneling amplitude, which depends on the barrier height [38], [40].",
"Here the periodic boundary condition, i.e., $a_{L+1}=a_1$ , has been assumed.",
"Obviously, it is straightforward to check that $[\\hat{H}, \\hat{N}]=0$ with $\\hat{N}=\\sum _j\\hat{a}_j^\\dag \\hat{a}_j$ .",
"The total number of particles $N=\\sum _j n_j$ is a conserved quantity, and the dimension of the Hilbert space is $\\mathrm {dim}[H]=C_{N+L-1}^N$ .",
"Such a model can be experimentally realized by using cold atoms in an optical lattice [37], [38], [41], [42], [40].",
"The interactions between the bosons can be characterized by a dimensionless coupling parameter [36], [43], [44], [45], [46], [47], [48] $\\lambda =\\frac{UN}{J}.$ For the two-site case, depending on the values of $\\lambda $ , one can identify three qualitatively different regimes [46], [48], [47], [43], [44], [45].",
"The Rabi regime ($\\lambda <1$ ), the Josephson regime ($1<\\lambda \\ll N^2$ ), and the Fock regime ($\\lambda \\gg N^2$ ).",
"Due to the interplay between the tunneling and the on-site interaction among the bosons, the BH model exhibits rich and interesting dynamical properties.",
"Figure: (Color online) Quantum [Eq.",
"()] and classical [Eq.",
"()] transition probabilities forthe 1D two-site BH model with different number of particles (a) N=100N=100, (b) N=200N=200.",
"The solid blue curve represents the quantum case P Q (n B |m A )P^Q(n^B|m^A),while the dashed red curve represents the classical case P C (n B |m A )P^C(n^B|m^A).",
"For the quantum case, the initial state is the ground stateof H(t=0)H(t=0) with |m A 〉=|𝐧 A 〉=|N/2,N/2〉|m^A\\rangle =|\\mathbf {n}^A\\rangle =|N/2,N/2\\rangle .For the classical case, the initial state is a collection of microscopic states Ψ A ={ψ 1 A ,ψ 2 A }={(N/2,φ 1 A ),(N/2,φ 2 A )}\\Psi ^A=\\lbrace \\psi _1^A,\\psi _2^A\\rbrace =\\lbrace (N/2,\\phi _1^A),(N/2,\\phi _2^A)\\rbrace ,with φ 1 A \\phi _1^A and φ 2 A \\phi _2^A the independent random numbersevenly sampled in the range [0,2π)[0,2\\pi ).Here, due to J(t=τ)=0J(t=\\tau )=0, the finial energy eigenstates are given by the Fock states: |n B 〉=|n 1 B ,N-n 1 B 〉|n^B\\rangle =|n_1^B,N-n_1^B\\rangle with n 1 B =0,1,...,Nn_1^B=0,1,\\ldots ,N.Inset: Time dependence of the work parameter J(t)J(t).The semiclassical limit of this model can be achieved when $N\\rightarrow \\infty $ ; in other words, the effective Planck constant is given by $\\hbar _{\\mathrm {eff}}=1/N$ [49].",
"With $N\\rightarrow \\infty $ , one can replace the annihilation and creation operators by complex numbers [48], [36], [53], [50], [54], [52], [55], [56], [57], [51], [58], [59], [49]: $\\hat{a}_j\\rightarrow \\psi _j,\\quad \\hat{a}^\\dag _j\\rightarrow \\psi _j^\\ast ,$ with $\\psi _j=\\sqrt{n_j+\\frac{1}{2}}\\exp \\lbrace i\\phi _j\\rbrace .$ Then, one finds that the classical counterpart of the Hamiltonian (REF ) is given by $\\mathcal {H}_c=\\sum _j^L\\left[-J(\\psi _j^\\ast \\psi _{j+1}+\\psi _{j+1}^\\ast \\psi _j)+\\frac{U}{2}|\\psi _j|^4\\right],$ with Poisson brackets $\\lbrace \\psi _i,\\psi _j^\\ast \\rbrace =\\delta _{ij},$ and $\\lbrace \\mathcal {H}_c,\\mathcal {N}\\rbrace =0,$ where $\\mathcal {N}=\\sum _j|\\psi _j|^2=N+L/2$ [50], [51], [59], [49].",
"The time evolution of the complex valued mean-field amplitudes $\\psi _j$ are given by the following equation [52]: $i\\hbar \\frac{\\partial \\psi _j}{\\partial t}=\\frac{\\partial \\mathcal {H}_c}{\\partial \\psi _j^\\ast }=-J(\\psi _{j+1}+\\psi _{j-1})+U|\\psi _j|^2\\psi _j.",
"$ This equation can be regarded as the Hamilton equation of the mean field $\\psi _j$ .",
"In the following sections, we will study the quantum-classical correspondence of work distributions in the 1D BH model based on the quantum and classical pictures given above.",
"Figure: (Color online) Cumulative quantum and classical transition probabilities for the 1D two-site BH model with different number of particles:(a) N=10N=10, (b) N=50N=50, (c) N=100N=100, (d) N=200N=200.The jagged blue solid curve shows the quantum case, ∑ n B P Q (n B |m A )\\sum _{n^B}P^Q(n^B|m^A),while the smooth dashed red curve shows the classical case, ∑ n B P C (n B |m A )\\sum _{n^B}P^C(n^B|m^A).",
"For the quantum case, the initial state is chosento be the ground state of H(t=0)H(t=0) with |m A 〉=|𝐧 A 〉=|N/2,N/2〉|m^A\\rangle =|\\mathbf {n}^A\\rangle =|N/2,N/2\\rangle .",
"The corresponding classical initial state is a collection of microscopic statesΨ A ={ψ 1 A ,ψ 2 A }={(N/2,φ 1 A )(N/2,φ 2 A )}\\Psi ^A=\\lbrace \\psi _1^A,\\psi _2^A\\rbrace =\\lbrace (N/2,\\phi _1^A)(N/2,\\phi _2^A)\\rbrace , where φ 1 A \\phi _1^A and φ 2 A \\phi _2^A are the uniformly distributedrandom numbers in the range [0,2π)[0,2\\pi ).Due to the fact that J(t=τ)=0J(t=\\tau )=0, the final energy eigenstates are Fock states: |n B 〉=|n 1 B ,N-n 1 B 〉|n^B\\rangle =|n_1^B,N-n_1^B\\rangle with n 1 B =0,...,Nn_1^B=0,\\ldots ,N."
],
[
"Quantum, semiclassical and classical transition probabilities",
"Consider a quantum system, described by a Hamiltonian $\\hat{H}(J)$ , where $J$ is an externally controlled parameter, usually called the work parameter of the system in the field of nonequilibrium statistical mechanics [3].",
"We study the time evolution of the system when the work parameter $J$ is varied with time from initial value $J(t=0)=A$ to the finial value $J(t=\\tau )=B$ .",
"We assume that the the system at $t=0$ is in a thermal equilibrium state at an inverse temperature $\\beta $ .",
"The system is then detached from the heat bath and work is applied when the work parameter $J$ is varied.",
"Then following the definition of quantum work [15], the work distribution of this nonequilibrium process is given by [15], [22] $P^Q(W)=\\sum _{n^B,m^A}P^Q(n^B|m^A)P^Q(m^A)\\delta (W-E_n^B+E_m^A),$ where $E_n^B$ and $E_m^A$ are the $n$ th and the $m$ th eigenvalues of the final and initial Hamiltonian $\\hat{H}(t=\\tau )$ , $\\hat{H}(t=0)$ , respectively.",
"And the corresponding eigenstates are given by $|n^B\\rangle $ and $|m^A\\rangle $ , respectively.",
"$P^Q(m^A)$ is the probability of sampling the $m$ th eigenstate of $\\hat{H}(t=0)$ from the initial thermal equilibrium state when making the initial energy measurement: $P^Q(m^A)=\\frac{1}{Z_A^Q}\\exp \\left[-\\beta E_m^A\\right],$ with $Z_A^Q=\\sum _m\\exp [-\\beta E_m^A]$ .",
"Given the initial $m$ th eigenstate of $\\hat{H}(t=0)$ , the conditional probability of obtaining the $n$ th eigenstate of $\\hat{H}(t=\\tau )$ is given by the quantum transition probability $P^Q(n^B|m^A)\\equiv |\\langle n^B|\\hat{U}(\\tau )|m^A\\rangle |^2,$ with $\\hat{U}(\\tau )=\\hat{\\mathcal {T}}\\exp \\left[-\\frac{i}{\\hbar }\\int _0^\\tau dt\\hat{H}(t)\\right],$ where $\\hat{\\mathcal {T}}$ is the time ordering operator.",
"For the classical case, we can follow the same lines as we do in the quantum case, except that we are now in the phase space instead of the Hilbert space.",
"The classical work distribution can be expressed in the following form [22]: $P^C(W)\\approx \\sum _{n^B,m^A}P^C(n^B|m^A)P^C(m^A)\\delta (W-E_n^B+E_m^A),$ where $P^C(n^B|m^A)$ and $P^C(m^A)$ are the classical counterparts of $P^Q(n^B|m^A)$ and $P^Q(m^A)$ , respectively.",
"With Eqs.",
"(REF ) and (REF ), the classical and quantum work distributions can be compared directly.",
"We begin with comparing the classical and quantum initial probabilities $P^C(m^A)$ and $P^Q(m^A)$ .",
"Following Ref.",
"[22], we know that the initial distribution for a $d$ -dimensional classical system reads $P^C(m^A)=\\int _{E_m^A}^{E_{m+1}^A}\\frac{1}{Z_A^C}\\bar{\\rho }(E)e^{-\\beta E}dE,$ where $Z_A^C=\\int \\frac{d^dp d^dq}{(2\\pi \\hbar )^d}\\exp [-\\beta \\mathcal {H}_c(p,q)],$ is the classical partition function and $\\bar{\\rho }(E)$ is the density of states (DOS) of the classical system.",
"For the BH model which we study here, $\\bar{\\rho }(E)$ has the following expression [60]: $\\bar{\\rho }(E)=&\\left(\\frac{4}{\\pi }\\right)^L\\int d^Lp d^Lq\\delta [E-\\mathcal {H}_c(\\mathbf {p},\\mathbf {q})] \\\\&\\times \\delta \\left(\\mathbf {p}^2+\\mathbf {q}^2-N-\\frac{L}{2}\\right),$ with $\\psi _j=q_j+ip_j$ and $\\mathbf {q}=(q_1,\\ldots ,q_L)$ , $\\mathbf {p}=(p_1,\\ldots ,p_L)$ .",
"For the quantum case, $P^Q(m^A)$ has the same form as the classical case except that the partition function is given by quantum expression and the DOS now reads $\\rho (E)=\\sum _n\\delta (E-E_n),$ where $E_n$ are the eigenvalues of the Hamiltonian in Eq.",
"(REF ).",
"According to Gutzwiller [61], in the semiclassical limit, i.e., $\\hbar \\rightarrow 0$ , $\\rho (E)$ has the generic form [60] $\\rho (E)=\\bar{\\rho }(E)+\\tilde{\\rho }(E).$ Here the smooth part $\\bar{\\rho }(E)$ is purely classical, known as the Weyl term, while the oscillatory part $\\tilde{\\rho }(E)$ comes from the quantum fluctuations and can be expressed in terms of classical quantities, which are encoded in the classical periodic orbits.",
"In our study, the energy scale that we consider is much larger than the periodicity of $\\tilde{\\rho }(E)$ , therefore, we can ignore the oscillatory part and approximately take the DOS of the quantum system to be $\\bar{\\rho }(E)$ .",
"Finally, we find that the initial distributions of the quantum and classical cases are approximately equal [22], [23]: $P^Q(m^A)&=\\int _{E^A_m}^{E^A_{m+1}} dE\\frac{1}{Z_A^Q}\\rho (E)e^{-\\beta E} \\\\&\\approx \\int _{E_m^A}^{E_{m+1}^A} dE\\frac{1}{Z_A^C}\\bar{\\rho }(E)e^{-\\beta E}=P^C(m^A).$ Thus, in order to compare the quantum and classical work distributions, the only thing one needs to clarify is the relationship between the classical and quantum transition probabilities $P^C(n^B|m^A)$ and $P^Q(n^B|m^A)$ .",
"In the following, we study these transition probabilities in the 1D BH model explicitly.",
"In our study, we change $J$ from $J(t=0)=0$ to $J(t=\\tau )=0$ .",
"Therefore, $A=0$ , $B=0$ , and both the initial and the final energy eigenstates are given by the Fock states.",
"The transition probability between different energy eigenstates is given by the transition probability between different Fock states.",
"The classical counterpart of the Fock state is a collection of microscopic states $\\Psi ^A\\equiv \\lbrace \\psi _1^A,\\ldots ,\\psi _L^A\\rbrace =\\lbrace (n^A_1,\\phi ^A_1),\\ldots ,(n_L^A,\\phi _L^A)\\rbrace $ , with $n^A_j$ 's equal to the number of particles on the $j$ th site and $\\phi ^A_j$ 's are the independent random numbers which have a uniform distribution in the range $[0,2\\pi )$ ."
],
[
"Quantum transition probability",
"In order to calculate the quantum transition probability, we expand the wave function evolving under $\\hat{H}[J(t)]$ as follows: $|\\Phi (t)\\rangle =\\sum _{n_2,\\ldots ,n_L}^Nc_{n_1,n_2,\\ldots ,n_L}(t)|n_1,\\ldots ,n_L\\rangle ,$ where $|n_1,\\ldots ,n_L\\rangle $ are the Fock basis, and the sum is constrained by $N=\\sum _{j=1}^L n_j$ .",
"Therefore, the particle number on the first lattice site is given by $n_1=N-\\sum _{j=2}^Ln_j$ .",
"$c_{n_1,n_2,\\ldots ,n_L}$ 's are expansion coefficients and satisfy the normalization condition $\\sum _{n_2,\\ldots ,n_L}^N|c_{n_1,n_2,\\ldots ,n_L}(t)|^2=1.$ Inserting Eq.",
"(REF ) into Schrödinger equation $i\\hbar \\frac{\\partial }{\\partial t}|\\Phi (t)\\rangle =\\hat{H}[J(t)]|\\Phi (t)\\rangle ,$ after some algebra, we finally get the equations of these coefficients $c_{n_1,n_2,\\ldots ,n_L}(t)$ : $ &i\\hbar \\dot{c}_{n_1,\\ldots ,n_j,\\ldots ,n_L}=\\frac{U}{2}\\sum _{j=1}^L n_j(n_j-1){c}_{n_1,\\ldots ,n_j,\\ldots ,n_L} \\\\&-J(t)\\sum _{j=1}^L\\left(c_{n_1,\\ldots ,n_j-1,n_{j+1}+1,\\ldots ,n_L}\\sqrt{n_j(n_{j+1}+1)}\\right.",
"\\\\&\\left.+c_{n_1,\\ldots ,n_j+1,n_{j+1}-1,\\ldots ,n_L}\\sqrt{(n_j+1)n_{j+1}}\\right),$ where the dot denotes the time derivative.",
"The quantum transition probability between different Fock states, which we denote by $P^Q(\\mathbf {n}^B|\\mathbf {n}^A)$ with $|\\mathbf {n}^{A/B}\\rangle =|n_1^{A/B},\\ldots ,n_L^{A/B}\\rangle $ , reads $P^Q(\\mathbf {n}^B|\\mathbf {n}^A)=|c_{n_1,n_2,\\ldots ,n_L}(\\tau )|^2,$ where $c_{n_1,n_2,\\ldots ,n_L}(\\tau )$ solves Eq.",
"(REF ) with the initial condition given by $c_{n_1,n_2,\\ldots ,n_L}(0)$ .",
"These results will be used in Sec.",
"."
],
[
"Semiclassical and classical transition probabilities",
"According to Refs.",
"[59], [49], one can write down the semiclassical transition probability between different Fock states of the BH model as follows: $P^{\\mathrm {SC}}(\\mathbf {n}^B|\\mathbf {n}^A)=|K^{\\mathrm {SC}}(\\mathbf {n}^{B},\\tau ;\\mathbf {n}^{A},0)|^2,$ where $K^{\\mathrm {SC}}(\\mathbf {n}^{B},\\tau ;\\mathbf {n}^{A},0)$ is the semiclassical propagator, and given by [59], [49] $K^{\\mathrm {SC}}&(\\mathbf {n}^{B},\\tau ;\\mathbf {n}^{A},0) \\nonumber \\\\&=\\sum _\\gamma \\sqrt{\\mathrm {det}^{\\prime }\\frac{1}{(-2\\pi i\\hbar )}\\frac{\\partial ^2R^\\gamma (\\mathbf {n}^B,\\tau ;\\mathbf {n}^A,0)}{\\partial \\mathbf {n}^B\\partial \\mathbf {n}^A}} \\nonumber \\\\&\\times \\exp \\left[\\frac{i}{\\hbar }R^\\gamma (\\mathbf {n}^{B},\\tau ;\\mathbf {n}^{A},0)+i\\mu ^\\gamma \\frac{\\pi }{2}\\right].$ Here, $\\gamma $ indexes all classical trajectories satisfying Eq.",
"(REF ) and the boundary conditions $&&|\\psi _j(t=0)|^2=n^A_j+\\frac{1}{2}, \\\\&&|\\psi _j(t=\\tau )|^2=n_j^B+\\frac{1}{2},$ with $j=1,\\ldots ,L$ and $\\arg \\psi _1(t=0)=0$ and $\\mu ^\\gamma $ denotes the Maslov index of the $\\gamma $ th trajectory, while $R^\\gamma (\\mathbf {n}^B,\\tau ;\\mathbf {n}^A,0)$ is the classical action of the $\\gamma $ th trajectory $R^\\gamma (\\mathbf {n}^B,\\tau ;\\mathbf {n}^A,0)={\\int }_0^\\tau \\Big [\\sum _j\\phi _j^\\gamma (t)\\dot{n}_j^\\gamma (t)-\\mathcal {H}^\\gamma _c(t)/\\hbar \\Big ]dt.$ The derivatives of the action $R^\\gamma $ with respect to $n_j^A$ and $n_j^B$ are $\\frac{\\partial R^\\gamma (\\mathbf {n}^B,\\tau ;\\mathbf {n}^A,0)}{\\partial n_j^A}&=-\\hbar \\phi _j^\\gamma (0), \\\\\\frac{\\partial R^\\gamma (\\mathbf {n}^B,\\tau ;\\mathbf {n}^A,0)}{\\partial n_j^B}&=\\hbar \\phi _j^\\gamma (\\tau ).$ The prime in the determinant $\\mathrm {det}^{\\prime }\\left(\\frac{\\partial ^2 R^\\gamma }{\\partial \\mathbf {n}^A\\partial \\mathbf {n}^B}\\right)\\equiv \\mathrm {det}\\left(\\frac{\\partial ^2 R^\\gamma }{\\partial n_j^A\\partial n_k^B}\\right)_{j,k=2,\\ldots ,L}$ indicates that the derivatives skip the first component.",
"This is a consequence of the conservation of the total number of particles [59], [49].",
"Following the same procedure as in Ref.",
"[22], we can further simplify the expression of the transition probability by ignoring the interference terms between different classical trajectories [49] $P^{\\mathrm {SC}}(\\mathbf {n}^B|\\mathbf {n}^A)\\overset{\\mathrm {diag}}{\\approx }\\left(\\frac{1}{2\\pi \\hbar }\\right)^{L-1}\\sum _\\gamma \\left|\\mathrm {det}^{\\prime }\\left[\\frac{\\partial \\mathbf {\\phi }(0)}{\\partial \\mathbf {n}^B}\\right]\\right|,$ where $\\mathbf {\\phi }(0)$ represents the vector of the initial phases for the $\\gamma $ th trajectory, and has been obtained in Eq.",
"(REF ).",
"For the classical case, the transition probability is given by [49] $P^C(\\mathbf {n}^B|\\mathbf {n}^A)={\\int }_0^{2\\pi }d^{L-1}\\phi ^A\\prod _{j=2}^L\\delta \\left[|\\psi _j(\\mathbf {n}^A,\\mathbf {\\phi }^A;\\tau )|^2\\right.",
"\\\\\\left.-(n_j^B+1/2)\\right].",
"$ Using the property of $\\delta $ function, Eq.",
"(REF ) can be rewritten as [59] $P^C(\\mathbf {n}^B|\\mathbf {n}^A)=\\left(\\frac{1}{2\\pi \\hbar }\\right)^{L-1}\\sum _\\gamma \\left|\\mathrm {det}^{\\prime }\\left[\\frac{\\partial \\mathbf {\\phi }(0)}{\\partial \\mathbf {n}^B}\\right]\\right|.$ By comparing Eqs.",
"(REF ) and (REF ), we find that the semiclassical transition probability (REF ) converges to the classical transition probability (REF ) after taking the diagonal approximation [22], [62], [63], [64].",
"Thus, similar to the single-particle system [22], [23], we have analytically proved that the quantum work distribution will converge to the classical work distribution in a quantum many-body system when ignoring the interference effect of different classical trajectories.",
"In the following we will provide some numerical results of both quantum and classical transition probabilities to demonstrate our central result.",
"Figure: (Color online) RMSE R(N)R(N) (blue pentagrams) as a function of the number of particles NN.The other parameters are U=5/NU=5/N, τ=10\\tau =10, ℏ=1\\hbar =1."
],
[
"Numerical results",
"In this section, we give our numerical results of the 1D two-site and three-site BH models.",
"We set $\\hbar =1$ , $U=5/N$ , $\\tau =10$ and vary the work parameter $J$ according to the following protocol: $J(t)=J_0\\left(t-\\frac{t^2}{\\tau }\\right),$ with $J_0=5$ .",
"In our study we also set the particle number $N$ to be an even number.",
"Here, we stress that qualitatively similar results can be obtained for any $L$ -site BH model with $L\\ge 2$ .",
"To calculate the quantum transition probability between different Fock states, we first use a Runge-Kutta method to solve the set of coupled ordinary differential equations given by Eq.",
"(REF ), then use Eq.",
"(REF ) to obtain the quantum transition probability.",
"For the classical case, the shooting method [65] has been employed to find all classical trajectories from $|\\mathbf {n}^A\\rangle $ to $|\\mathbf {n}^B\\rangle $ at the fixed transit time $\\tau $ .",
"Then we calculate the classical transition probability via Eq.",
"(REF )."
],
[
"1D Two-site Bose-Hubbard model",
"In this section we study the transition probability in the 1D two-site BH model without periodic boundary condition $\\hat{H}=-J(\\hat{a}_1^\\dag \\hat{a}_2+\\hat{a}_2^\\dag \\hat{a}_1)+\\frac{U}{2}(\\hat{a}_1^\\dag \\hat{a}_1^\\dag \\hat{a}_1\\hat{a}_1+\\hat{a}_2^\\dag \\hat{a}_2^\\dag \\hat{a}_2\\hat{a}_2).$ This is an extensively studied [67], [43], [68], [44], [45], [69], [70], [71], [66], [72], [73], [74], [57], [56], [55], [50], [54], [48], [47], [46], [75], [76], [77], [78], [79], [80] paradigmatic model and can be realized in various systems, for example, particles in a harmonic well [72].",
"Under the well-known two-mode approximation, the 1D two-site BH Hamiltonian in Eq.",
"(REF ) can also be used to describe the dynamics of an atomic Bose-Einstein condensate in a double-well potential [68], [73], [74].",
"Figure: (Color online) Energy spectrum of the 1D three-site BH model () with the work parameter JJ for N=20N=20.Inset: The details of the red rectangle.We choose the ground state of $\\hat{H}(t=0)$ as our initial state.",
"The corresponding Fock state is the twin-Fock state, therefore, we have $|\\mathbf {n}^A\\rangle =|N/2,N/2\\rangle $ .",
"Its classical counterpart is a collection of microscopic states $\\Psi ^A=\\lbrace \\psi _1^A,\\psi _2^A\\rbrace =\\lbrace (N/2,\\phi _1^A),(N/2,\\phi _2^A)\\rbrace $ , with $\\phi _1^A$ and $\\phi _2^A$ the uniformly distributed random numbers in the range $[0,2\\pi )$ .",
"Here we should point out that for small $J$ all excited energy levels are doubly degenerate and it splits with the increase of $J$ (see Fig.",
"REF ).",
"However, the quantity that we studied is the transition probability between different energy eigenstates, therefore we do not need to consider the effect of the degeneracy.",
"Due to the fact that both the initial and the final values of $J$ are equal to zero, the Fock states $|\\mathbf {n}^{A/B}\\rangle $ are also the energy eigenstates at the initial and the final moments.",
"Hence, the quantum and classical transition probabilities between different energy eigenstates can be expressed as the transition probabilities between different Fock states: $&&P^Q(n^B|m^A)=P^Q(\\mathbf {n}^B|\\mathbf {n}^A), \\\\&&P^C(n^B|m^A)=P^C(\\mathbf {n}^B|\\mathbf {n}^A).",
"$ Here, the relation between the energy eigenstates $|m^{A}\\rangle $ , $|n^B\\rangle $ and the Fock states $|\\mathbf {n}^{A}\\rangle $ , $|\\mathbf {n}^B\\rangle $ are defined in the captions of Figs.",
"REF , REF , and REF .",
"In Fig.",
"REF , we plot the quantum transition probability for different number of particles as a function of the final energy eigenstates $|n^B\\rangle $ (solid line).",
"Comparing with the classical case (dashed line), we find that the quantum probability oscillates rapidly with $n^B$ .",
"This feature has an origin in the wave nature of the quantum system.",
"Obviously, the correspondence between $P^Q(n^B|m^A)$ and $P^C(n^B|m^A)$ is visually evident.",
"Figure: (Color online) Quantum (solid blue curve) and classical (dashed red curve) transition probabilitiesof the 1D three-site BH model:(a) Transition probabilities between different energy eigenstates, (b) Cumulative transition probabilities.For the quantum case, the number of bosons is N=20N=20 and the initial state is one of the three degenerate eigenstatesof the 19th energy level of H(t=0)H(t=0) with |m A 〉=|𝐧 A 〉=|5,5,10〉|m^A\\rangle =|\\mathbf {n}^A\\rangle =|5,5,10\\rangle .The classical counterpart of |m A 〉|m^A\\rangle is a collection of microscopic states Ψ A ={ψ 1 A ,ψ 2 A ,ψ 3 A }={(5,φ 1 A ),(5,φ 2 A ),(10,φ 3 A )}\\Psi ^A=\\lbrace \\psi _1^A,\\psi _2^A,\\psi _3^A\\rbrace =\\lbrace (5,\\phi _1^A),(5,\\phi _2^A),(10,\\phi _3^A)\\rbrace ,with φ j \\phi _j's (j=1,2,3)(j=1,2,3) the uniformly distributed random numbers in the range [0,2π)[0,2\\pi ).Due to the fact that the final value of JJ is zero, the energy eigenstates of H(t=τ)H(t=\\tau ) are the Fock states: |n B 〉=|n 1 B ,n 2 B ,N-n 1 B -n 2 B 〉|n^B\\rangle =|n_1^B,n_2^B,N-n_1^B-n_2^B\\rangle with n 1 B =0,n 2 B =0,...,N;n 1 B =1,n 2 B =0,...,N-1;...;n 1 B =N,n 2 B =0n_1^B=0,n_2^B=0,\\ldots ,N; n_1^B=1,n_2^B=0,\\ldots ,N-1;\\ldots ;n_1^B=N,n_2^B=0.In order to smooth out the rapid oscillations and to compare these two probabilities in a better way, we plot the cumulative transition probabilities $\\sum _{n^B}P^Q(n^B|m^A)$ and $\\sum _{n^B}P^C(n^B|m^A)$ in Fig.",
"REF for different number of particles $N$ .",
"Obviously, the agreements between these two probabilities are not very good for small $N$ , but the convergence is improved when $N$ increases.",
"The deviation observed in small $N$ can be explained as follows: when the number of particles $N$ is small, the characteristic actions of the system are not much larger than the effective Planck's constant $\\hbar _{\\mathrm {eff}}$ .",
"Therefore, the classical approximations adapted in Sec.",
"[cf.",
"Eqs.",
"(REF )-(REF )] are expected to be a poor approximation.",
"We can also see that the jagged quantum cumulative transition probability oscillates around the classical cumulative transition probability.",
"This phenomenon stems from the interference between different classical trajectories [22].",
"The convergence displayed in Fig.",
"REF suggests that there indeed exists a correspondence principle between quantum and classical work distributions, despite the nonclassical feature visible in Fig.",
"REF .",
"The convergence of the quantum and classical transition probabilities depends on the number of particles $N$ (see Fig.",
"REF ).",
"In order to understand the correspondence of work distribution in a better way, we use the root-mean-square error (RMSE) [81] to quantify the difference between the quantum and classical cumulative probabilities.",
"For certain $N$ , the RMSE, which we denote by $R(N)$ , between these two cumulative probabilities is given by $R(N)\\equiv \\sqrt{\\frac{1}{M}\\sum _{l=0}^{N}\\left[S_l^Q(N)-S^C_l(N)\\right]^2},$ where $M=N+1$ represents the total number of eigenstates and $S_l^{Q/C}(N)=\\sum _{n^B=0}^{l}P^{Q/C}(n^B|m^A),$ with $l=0,\\ldots ,N$ .",
"The RMSE $R(N)$ quantifies the average deviations between two different probability distributions.",
"If two probability distributions are identical, we have $R(N)=0$ .",
"The closer the two cumulative probability distributions $S_l^Q$ and $S^C_l$ are, the smaller $R(N)$ is.",
"The vanishing of $R(N)$ implies the correspondence principle [23].",
"Hence, the validity of the correspondence principle can be quantitatively characterized by the vanishing of the RMSE.",
"RMSE $R(N)$ as a function of particle numbers $N$ is shown in Fig.",
"REF .",
"It is seen that the value of $R(N)$ decreases with the increase of particle numbers $N$ .",
"In order to satisfy the classical limit ($N\\rightarrow \\infty $ ), large $N$ is necessary.",
"The behavior of $R(N)$ implies that its value will approach zero when the particle numbers go to infinity, i.e., $\\lim _{N\\rightarrow \\infty }R(N)\\rightarrow 0.$ This is in accordance with the well-known correspondence principle that quantum mechanics and classical mechanics give the same result in the classical limit."
],
[
"1D three-site Bose-Hubbard model",
"The 1D two-site BH model is simple and a special case of BH model, in order to study a general case we extend our study to the 1D three-site case.",
"The Hamiltonian of the three-site BH reads $\\hat{H}=-J\\sum _{j=1}^3\\left(\\hat{a}_j^\\dag \\hat{a}_{j+1}+\\hat{a}_{j+1}^\\dag \\hat{a}_j\\right)+\\frac{U}{2}\\sum _{j=1}^3n_j(n_j-1),$ where the periodic boundary condition (i.e., a ring geometry) $\\hat{a}_{L+1}=\\hat{a}_1$ has been assumed.",
"The three-site system is a non-integrable system and its energy spectrum (Fig.",
"REF ) is less regular than that of the two-site system (Fig.",
"REF ).",
"The dynamics of its classical counterpart is chaotic due to the nonlinear dynamics in a four-dimensional phase space, and its behavior is much richer than the two-site setup [51], [58], [82], [83], [84], [85], [86], [87], [88].",
"In our study, we choose the initial state to be one of three degenerate eigenstates of the 19th energy level of the initial Hamiltonian.",
"Its corresponding Fock state is $|\\mathbf {n}^A\\rangle =|5,5,10\\rangle $ .",
"The classical counterpart of $|\\mathbf {n}^A\\rangle $ is a collection of microscopic states $\\Psi ^A=\\lbrace \\psi _1^A,\\psi _2^A,\\psi _3^A\\rbrace =\\lbrace (5,\\phi _1^A),(5,\\phi _2^A),(10,\\phi _3^A)\\rbrace $ , where $\\phi _1^A$ , $\\phi _2^A$ , and $\\phi _3^A$ are the uniformly distributed random numbers in the range $[0,2\\pi )$ .",
"The classical counterpart of the Hamiltonian (REF ) can be found in Sec. .",
"And the classical dynamics of the system satisfies three coupled differential equations of $\\psi _j$ $(j=1,2,3)$ [cf.",
"Eq.",
"(REF )].",
"Figure REF (a) shows the quantum and classical transition probabilities of the three-site BH model with $N=20$ .",
"It can be seen that unlike the 1D two-site case where the behavior of the classical transition probability is regular, in the three-site system the classical transition probability is irregular.",
"This phenomenon stems from the fact that the dynamics of the 1D three-site BH model is nonintegrable and becomes more and more chaotic as $\\lambda $ increases.",
"Surprisingly, for the three-site BH model, the agreement between the quantum and classical cumulative transition probabilities is very good even for small $N$ [see Fig.",
"REF (b)]."
],
[
"Conclusions and discussions",
"The quantum-classical correspondence principle for work distribution in a quantum many-body system, i.e., 1D BH model, has been studied in this article.",
"Since the initial quantum and classical probability distribution functions are approximately equal, the correspondence principle between quantum and classical work distributions is equivalent to the correspondence between the quantum and classical transition probabilities between different energy eigenstates.",
"We first analytically demonstrate the convergence of the quantum and the classical transition probabilities by utilizing the analytical expression of the semiclassical propagator between Fock states [59], [49], and then we numerically calculate the quantum and classical transition probabilities in the two-site and three-site 1D BH models.",
"We find that the numerical results agree with the analytic result.",
"A direct comparison of the quantum and classical transition probabilities shows that the quantum transition probability oscillates rapidly along the classical transition probabilities due to the interference of different classical trajectories, while the classical transition probability is smooth and continuous for the integrable case and irregular for the nonintegrable case.",
"Therefore, the classical and quantum probabilities are manifestly different.",
"However, for the cumulative probabilities, we have observed good agreement between them.",
"Our results also demonstrate that the convergence, which is characterized by the vanishing of the statistical quantity RMSE, between cumulative quantum and classical probabilities becomes better with the increase of the particle numbers of the system, and vanishes as $N\\rightarrow \\infty $ .",
"This behavior of RMSE implies that in the classical limit the quantum work distribution converge to the classical work distribution.",
"Therefore, there indeed exists a quantum-classical correspondence principle of work distributions in a quantum many-body system, even though the indistinguishability and interaction make the properties of quantum work elusive.",
"Finally, we stress that the quantum-classical correspondence of the BH models studied in this article is a dynamic one [22], [23], namely, for a system governed by a time dependent Hamiltonian, the quantum and classical transition probabilities converge to each other in the classical limit.",
"Whereas, the usual studies of the quantum-classical correspondence in the BH models [43], [44], [45], [50], [51], [82], [89], [90] are the static case, where the Hamiltonian of the system is time independent.",
"Our work, therefore, complements the previous static correspondence principle in the BH model, which has been studied extensively.",
"Furthermore, this work also complements the recent progress established in Refs.",
"[22] and [23], and justifies the definition of quantum work via two-point energy measurements in a quantum many-body system.",
"H.T.Q.",
"gratefully acknowledges support from the National Science Foundation of China under Grants No.",
"11375012 and No.",
"11534002, and The Recruitment Program of Global Youth Experts of China."
]
] | 1612.05692 |
[
[
"The role of homophily in the emergence of opinion controversies"
],
[
"Abstract Understanding the emergence of strong controversial issues in modern societies is a key issue in opinion studies.",
"A commonly diffused idea is the fact that the increasing of homophily in social networks, due to the modern ICT, can be a driving force for opinion polariation.",
"In this paper we address the problem with a modelling approach following three basic steps.",
"We first introduce a network morphogenesis model to reconstruct network structures where homophily can be tuned with a parameter.",
"We show that as homophily increases the emergence of marked topological community structures in the networks raises.",
"Secondly, we perform an opinion dynamics process on homophily dependent networks and we show that, contrary to the common idea, homophily helps consensus formation.",
"Finally, we introduce a tunable external media pressure and we show that, actually, the combination of homophily and media makes the media effect less effective and leads to strongly polarized opinion clusters."
],
[
"Introduction",
"In modern society we observe the emergence of several controversial issues that can challenge the organization of the society.",
"We have less severe issues, like the diffusion of conspiracy theories and more severe issues, like religious fundamentalisms, that can lead to violent attacks and terrorism.",
"In our society, where the communication patterns are so rapidly changing, understanding how these opinion niches are created and reinforced is a key issue: only a full comprehension of these phenomena can suggest the most suitable communication strategies to control or diffuse some ideas.",
"Several authors pointed out that a possible responsible of the strong opinion polarization in the society is the particular organization of online social networks ([1],[2]), enormously enlarging the social pool where the social actors can look for peers (geographical and demographic barriers are broken down), allowing people to preferentially enter in contact with people sharing very similar ideas and socio-cultural traits.",
"Moreover, the same filtering algorithms used by the social networks, in order to provide personalized information, amplify this effect favoring the membership in coherent groups and the connection between similar opinions.",
"Online social networks amplify the homophily principle, a well known tendency supported by several studies in social psychology ([3]), defined as the individual tendency to interact preferentially with people perceived as similar.",
"The literature on homophily principle has rapidly developed in the last years, both with theoretical papers ([4],[5]) and with experimental approaches ([6],[7],[8]).",
"Several authors propose, therefore, that this mechanism of network formation based on homophily directly generates isolated echo chambers where the information flows remain trapped ([9],[10]).",
"On the other side, all these papers mostly deal with the observation of the opinion flows at a quite initial stage of the opinion formation process: using microblogging data we can easily observe how an information spreads among the users, but it is difficult to track, on long time scales, if and how single users change their opinions.",
"In this paper we investigate the process of opinion formation in an homophilous environment on the long term, using an ad-hoc simulation framework.",
"Agent-based modeling approach is being broadly used in order to capture the emergent phenomena in opinion dynamics, when relevant individual mechanisms, postulated by social psychology theories, are applied to several agents ([11],[12]).",
"In the context of agent based simulations, several studies on rumor spreading ([13]) and controversies formation mechanisms ([14]) have been published in the last decade, but none of these is focused on the social network structure.",
"The basic research question we want to answer here is instead \"is homophily in social network a possible origin of the strong opinion polarization observed in our society?\".",
"Opinion polarization, is defined,in our case, as the emergence of two cohesive opinion groups with radical opinions on a certain subject.",
"Clearly this is not the only possible definition of a so articulated concept.",
"Different definitions of polarization can be found in ([15]).",
"Here we develop our analysis on three levels, gradually extending the complexity of the model.",
"First, we focus on the network morphogenesis, where we describe a network generation model, simplifying the implementation proposed in ([16]), where the local rules for peer selection are based on homophily preferences.",
"We show which are the basic topological properties of these network structures and, in particular, that these local rules, based on opinion, automatically generate the emergence of topological communities in the society.",
"This first result is significant by itself since only few existing models are able to reproduce the emergence of the structural partitions that characterize real social networks, using few and essential ingredients.",
"Second, we investigate the opinion formation processes on networks displaying homophily.",
"In particular, our interest is oriented to the bounded confidence models ([17] and [18]), agent based models, allowing to consider at the same time two central mechanisms of the social influence: the tendency toward conformity - explained by social comparison theory ([19]) and social balance theory ([20],[21])- and confirmation bias - the tendency to filter out informations that are too far from our points of view.",
"In particular, we will consider the Deffuant BC model (DW), where the opinion evolution is based on peer interactions.",
"It has been shown, in ([22]), that the outcomes of the bounded confidence models are topology independent on static networks.",
"On the contrary, in ([23],[24], [13], [25]) the authors showed that the results can strongly change on (co)-evolving networks.",
"Here we address the key question about the connection between homophily and opinion polarization ([26]): is homophily promoting opinion controversies in the society as evoked, for example, in ([9],[10])?",
"We show that, on the stable final configurations, the contrary is true: networking based on homophily promotes consensus, due to the interplay of the dynamics inside and between the community structures.",
"Third, we add in the simulation framework a further ingredient represented by the traditional media, spreading with a more or less marked pressure, the opinion of the society's empowerment.",
"To model the media we extend the bounded confidence framework to an asymmetrical interaction between human agents and media, considering that the confirmation bias in the media exposure is a well documented fact, usually called selective exposure ([27]).",
"In every society, although the dominant tendencies usually follow the message promoted by dominant institutions (i.e.",
"corporations for mass media, religion and educational institutions) some other less dominant tendencies/opinions always appear.",
"A lot of effort has been devoted in trying to understand the mechanism leading to this evident opinion diversification.",
"A counterintuitive effect, regarding the effect of mass media, has been observed in BC models.",
"It has been shown in several papers ([28],[29], and [30]) that, if the media pressure is low, media are able to attract all the opinions but, on the contrary, oppressive propaganda gives rise to the emergence of opposite extreme opinions both in the Deffuant model (DW) and in the Hegselmann and Krause (HK) model.",
"A similar effect has been also observed in the Axelrod's model for the dissemination of culture, in ([31], [32]) and in particular in this recent study ([33]) where, coherently to our case, the interplay between word-of-mouth and media is considered.",
"This over-exposure phenomenon is well known in marketing studies ([34]).",
"Although all these studies address the research question of the formation of counter-message competing with the dominant mass media, in this work we explicitly focus on the structure of the non-aligned states.",
"In ([28],[29], and [30]) it has been shown that, when the interaction is constrained by non-homophilous connectivity, the final outcome of the model, for high media pressure, is a strong cluster aligned with the media and a large number of unclustered opponent opinions.",
"A second cohesive counter opinion cluster cannot be formed in BC models over regular complex networks.",
"Here we show that, on the contrary, the presence of homophily in social networks over a media dominated system, plays a central role in the recomposition of a strong opponent cluster, leading to the final polarization of the opinions.",
"At the same time we show that homophilous system are much more robust to media propaganda, allowing the formation of counter-clusters also for lower values of the media pressure.",
"Differently by the co-evolution frameworks ([23],[24], [13], [25]) where opinions are updated together withe the network structure, in this paper we consider that network morphogenesis and opinion dynamics take place at different time scales.",
"The formation of social network is a slow process and, in this case, the opinion on which we base the homophily choices, is an abstract representation of a global vision of the world.",
"Opinion dynamics processes represent the formation of a global opinion on a concrete subject (a new law, a referendum, a piece of news, etc).",
"This processes have a fast dynamics that do not allow to the networks to coherently reshape.",
"In this case, what we define opinion is the particular judgement that an individual, with a certain vision of the world, has on this subject.",
"In this sense, the representation of the opinion in these processes is a sort of local characterization of the opinion on which network morphogenesis is based.",
"The paper is organized as follows: In section 2 we present the network morphogenesis model and the topological properties of the obtained networks.",
"In section 3 we present the opinion evolution on homophily-based networks.",
"In section 4 we show the combined effect of media propaganda and homophily.",
"Conclusions are presented in section 5."
],
[
"The Model",
"We first propose a growing network model allowing to fine tune the homophily level.",
"The growing network approach for network morphogenesis is a dynamical process where at each time step new nodes and new links enter the network and/or old links are canceled or rewired.",
"We will consider the simplest case where at each time step a single node is added to the network with a set of associated links to the pre-existing nodes.",
"The probability that the new node $N$ gets connected to the pre-existing node $i$ , $\\Pi _{N\\rightarrow i}\\sim \\varphi _N(i)$ contains the selected mechanism for network growth.",
"The fitness function $\\varphi _N(i)$ , associated to each pre-existing node, represents how attractive is a pre-existing node $i$ , for the new node $N$ , to establish a link.",
"It is well known that a fitness function based on the degree ($k_i$ ) of the pre-existing nodes $\\varphi _N(i)= k_i$ , namely a situation where a node with a large connectivity (measured as its degree) has a larger chance to attract new links, gives the preferential attachment mechanism generating scale free networks with a power law degree distribution, hereby named BA-networks, ([35]).",
"Implementing a fitness function based on the degree is motivated by the larger visibility that highly connected nodes have: more friends I have, more probable is that I am present in different social circles and more probable is to meet new friends.",
"Moreover this is one of the mechanism on which friendship recommendations in online social networks are based.",
"To include homopily in the morphogenesis without forgetting the connectivity issue, in our setup, we construct the fitness function $\\varphi _N(i)$ so that the new node has a preference to get connected both to high degree nodes and with nodes with similar opinion: $\\varphi _N(i)= k_i \\exp (- \\beta | \\theta _N - \\theta _i| ),$ where $k_i$ is the degree of the ancient node $i$ , $\\theta _i$ its opinion, $\\theta _N$ the opinion of the new node and $\\beta $ a coefficient tuning the homophily effect.",
"Let us note that when $\\beta =0$ the growing mechanism follows the classical preferential attachment, leading to the usual Barabasi-Albert network, with power-law degree distribution $P(k)\\sim k^{-3}$ .",
"When $\\beta \\ne 0$ the homophily comes into play and a competition between the preferential degree attachment and the opinion similarity takes place.",
"In the limit $\\beta \\rightarrow \\infty $ only homoplily matters.",
"The model follows the following steps: Each node is initialized with an opinion randomly selected in a continuos interval between $\\theta _i\\in [-1,1]$ .",
"The network generation process starts from an initial fully connected structure with $N_{ini}=5$ nodes.",
"At each step a new agent, $N$ , enters in the network The new agent $N$ gets connected to $m$ pre-existing agents ($m$ new links) using a roulette-wheel selection process (or fitness proportional selection), based on the probabilities: $\\Pi _{N\\rightarrow i}=\\frac{\\varphi _N(i)}{\\sum _{i=0}^N\\varphi _N(i)}$ Notice that the parameter $m$ , namely the number of new links added to each new node has no influence on the global properties of the network (like degree distribution, clustering, mixing, etc.).",
"This parameter defines the minimum degree of the network and the total number of edges.",
"In Fig.REF we report the the Python implementation of the algorithm for the network morphogenesis.",
"Figure: Python function to generate networks."
],
[
"Results",
"Due to the presence of the connectivity in the fitness function $\\varphi _N(i)$ (Eq.",
"REF ), the final degrees of the network are distributed for a large range, as in the original BA model ($\\beta =0$ ), on a power law distribution.",
"Increasing $\\beta $ , the homophily mechanism leads to a cutoff in the distribution, decreasing the maximum value of degree (figure REF -A).",
"In figure REF -B we show the average opinion distance between connected nodes.",
"The opinion similarity between connected nodes increases extremely fast as the homopily parameter $\\beta $ is switched on.",
"Figure: A) Power-law degree distribution for several values of β\\beta over Scale Free (SF) networks with N=1000N=1000,m=3m=3 and N ini N_{ini} (initial core) =5=5.",
"The homophily mechanism (β≠0\\beta \\ne 0) leads to structures following the same power-lawdegree distribution but with a cutoff, imposing a maximum value of degree by increasing β\\beta .",
"B) Average opinion distance betweenconnected nodes.",
"Connection between more similar nodes is clearly increasing with the homophily parameterThe results obtained in Fig.",
"REF are the direct consequence of the preference function structure.",
"A more relevant emergent property can be observed in figure REF : networks with high homophily exhibit meaningful community structure.",
"Topological communities are groups of nodes that are more strongly connected among them than with the rest of the network.",
"Notice, however, that there is a strong difference between community structures and network disconnected components: communities are largely connected among them but links exist also between the communities, connected components are totally disconnected among them.",
"Several real network structure present signature of this properties - citation networks, mobility networks, social networks, semantic networks.. ([36]).",
"At the same time few network morphogenesis models are able to reproduce these patterns ([37]).",
"Several algorithm exist to identify community structures For a review see for example ([38]).",
"In the following we used the Louvain algorithm ([39], http://perso.crans.org/aynaud/communities/.",
"The goodness of a partition is measured by the modularity ($Q$ ), a function comparing the concentration of edges within communities, in the network, with a rewired network with a random distribution of links (obtained with the configuration model).",
"Large values of the modularity ($Q\\rightarrow 1$ ) signify that the community structure is highly significant, namely the fraction of links within the community largely exceed the fraction of links between the same nodes in a random configuration.",
"For a mathematical definition of the modularity see ([40]) The best partition is the configuration that maximize modularity.",
"At the same time, if the community structure is not a significant marker of a network, modularity remains low also for the best partition.",
"In figure REF we display the modularity measure for the best partition of the network, as a function of the homophily parameter $\\beta $ .",
"Modularity monotonically increases with $\\beta $ , implying that the presence of communities is a natural emergent effect of the homophily preference in network morphogenesis.",
"In the lower plots A1-4 of figure REF we present a network visualization for different values of the homophily parameter.",
"The used visualization layout (based on a force algorithm) has a repulsive force to push away disconnected nodes and a spring-like force attracting connected node.",
"The result of this visualization algorithm is the spatial separation of the network communities: when the community structure is significant few connections (represented by the large black lines in the plot) exist between the communities that will be therefore pushed away among them, while the large number of links inside the communities will spatially group the nodes of the same community.",
"We can observe that for the BA network ($\\beta =0$ ) communities are not visible, while a more and more structured shape appear as homophily is switched on.",
"Notice that the communities are largely uniform in term of opinion.",
"At the same time links exist between all the communities.",
"This factor is central for understanding the opinion propagation dynamics that will be described in the next section.",
"In the upper plots A1-4 of figure REF we show the agents opinions inside each community.",
"Each line in the plots represents a community, ranked from the largest (on the bottom) to the smaller (on the top).",
"We observe that for the BA network the opinions are randomly distributed in all the communities and the average opinions of the communities coincide with the center of the opinion interval.",
"When the homophily is present the communities specialize: their average opinion moves from the center.",
"Largest it the value of $\\beta $ smaller is the dispersion of the opinions around the average opinion of the community.",
"Figure: A1-4) Upper plot: each line represent a community (ordered from the larger on the bottom to the smaller on the top); the red points represent the agents opinions inside each community, the blue square is the average opinion of the community.",
"Increasing β\\beta the opinion range inside the communities is smaller.",
"Lower plot: network visualization using a force layout (allowing to visualize the partitions).",
"The color of the nodes depends on their opinion.B) Community modularity as a function of β\\beta .",
"The results are the average values of 100 replicas of the morphogenesis process, for a networkwith N=1000N=1000 and m=3m=3.",
"Modulatity increases with β\\beta , meaning that more significant community structures are formed.In all our experiments the initial opinion has been initialized according to a uniform distribution.",
"We tested that the aggregated results we presented in Fig.",
"REF and Fig.",
"REF B are robust to a changing in the simulation paradigm: before fixing the same opinion vector and after building different networks on this opinions.",
"Notice that using different distributions for the initialization (like i.e.",
"a Gaussian) could change the final outcome of the process.",
"Exploring this issue is out from the scopes of this paper (addressed mostly to understand the relationship between homophily and opinion propagation) and we leave this direction open for subsequent studies."
],
[
" Central result:",
"Homoplily as ingredient for network morphogenesis leads to networks where the community structure is a fundamental marker."
],
[
"The model",
"In the previous section we described a morphogenesis algorithm to build network structures based on homophily.",
"In this section we will show how this network structure influences opinion dynamics processes.",
"Several different mechanisms can drive opinion formation.",
"In peer interactions two main factors have been observed as fundamental forces for mutual influence.",
"The first on is conformity, a mechanism due to the psychological need to reduce conflict among peers that consists in the reduction of opinion distances after an opinion exchange.",
"The second one is the confirmation bias that is the selective filtering of opinions too far from ours.",
"A well known model taking into account both these mechanisms is the bounded confidence (BC) model ([17]).",
"This model depends on a tolerance parameter $\\varepsilon $ tuning the importance of confirmation bias.",
"According to this model, once two agents $(i,j)$ are selected as peer for an interactions, they will update their opinion using the following threshold rule: $if |\\theta _i-\\theta _j|<\\varepsilon \\Rightarrow {\\left\\lbrace \\begin{array}{ll}\\theta _i=\\theta _i+\\mu (\\theta _j-\\theta _i)\\\\\\theta _j=\\theta _j+\\mu (\\theta _i-\\theta _j)\\end{array}\\right.",
"}$ If the velocity parameter is $\\mu =1/2$ , as it is usually fixed, the two opinions will converge to their average.",
"The Python implementation of the function for a single interaction with the BC model is presented in Fig.",
"REF .",
"Figure: Python function for a single update of the BC model.It is well known that, independently from the choice of the parameter $\\mu $ the repeated application of this rule, on random pairs of peers in a fully connected population, drives to two possible scenarios: total consensus, where all the opinions converge to the initial opinion average, for $\\varepsilon \\ge 0.5$ and opinion clustering, where two or more opinions coexist, for $\\varepsilon < 0.5$ ([42]) In the original description of the BC model ([17]), all the agents can interact with all the others.",
"This complete mixing assumption can be quite unrealistic when we consider a large number of agents that cannot be in contact with anyone else in the society.",
"People can interact and exchange opinions only with the peers that they meet in their everyday life (online or offline), namely with the neighbors in their social network.",
"A second important step, after ([17]), is therefore to constrain the peer selection only to couples of nodes connected by a link in a network structure.",
"It has been shown in ([22]) that the same scenarios and the same transition threshold to consensus $\\varepsilon _c=0.5$ is observed if the peers are selected only between the edges of a more static complex network structure (random graphs, small world networks, scale free networks).",
"On the other side, it has been shown in ([25], [23],[24]) that the consensus threshold can change on dynamical networks, once some rewiring based on nodes' opinion can take place.",
"In the following we will study how the parameter $\\beta $ , tuning the homophily level in networks, can influence the outcome of the BC model.",
"Namely we will address the question: Does a larger homophily in the network structure implies the formation of opinion bubbles?",
"The model evolves according to the following steps (see Fig.",
"REF for a Python implementation): A random uniform opinion distribution and a network structure (with $N$ nodes and with a $\\beta $ parameter) are created.",
"At each time step $N$ couples of neighboring nodes (connected by a link of the network) are selected and a pairwise interaction with BC model is performed (asynchronous update).",
"The loop is halted when, on all the edges no more successful interactions are possible ($\\Vert \\theta _i-\\theta _j|>\\varepsilon $ or $\\Vert \\theta _i-\\theta _j|=0$ ) Figure: Python implementation of the opinion dynamics model."
],
[
"Results",
"In figure REF we show that the answer to the question \"Does a larger homophily in the network structure implies the formation of opinion bubbles?",
"\", is clearly negative and that, on the opposite, a large homophily reduce the existence of radical issues.",
"In figure REF A we plot the fraction of replicas ending up to consensus for different values of the homophily parameter.",
"For $\\beta =0$ (Barabasi-Albert network), we observe the transition at $\\varepsilon _c=0.5$ predicted in [22].",
"But for higher values of $\\beta $ we observe that the transition threshold becomes smaller and smaller ($\\lim _{\\beta \\rightarrow \\infty }\\varepsilon _c(\\beta )=0$ ), meaning that the system will always end up to consensus.",
"Figure REF C shows how it happens.",
"Remember that the system can evolve until on some links of the social network the agents connected by the link have two different opinions at a distance smaller than the tolerance.",
"At the moment when, on all the links the agents have the same opinion or their opinion difference is larger than $2\\varepsilon $ , whatever pair is selected for the opinion dynamics, opinions will not change anymore.",
"For low values of $\\beta $ the system get frozen at a very initial phase.",
"After few iterations the agents cannot find in their neighborhood any peers with whom having \"positive exchanges\" (for all the couples $|\\theta _i-\\theta _j|\\ge 2\\varepsilon $ ).",
"The agents with a moderate opinion rapidly converge forming a major cluster (located around the average opinion of the system $\\langle \\theta \\rangle =0$ ).",
"Larger is $\\varepsilon $ , larger is the central cluster size.",
"Since the link construction is independent by the opinion, there is therefore a large probability that the radical agents are connected with agents in the majoritarian cluster (positioned at an opinion distance larger than $2\\varepsilon $ from their actual opinion) and not among them.",
"The radical agents, remain therefore isolated, keeping their initial opinion.",
"For larger values of the homophily $\\beta $ the dynamics is slower, but much more \"inclusive\".",
"Since radicals agents are now mostly connected with similar, they do not remain isolated.",
"Agents always find peers with an opinion sufficiently near to interact, and therefore their opinions change gradually at each interaction.",
"As we can observe in Figure REF B (where each color represents the opinion span in each community) the dynamics happens at two levels: a rapid convergence inside the communities and a slower one between the communities.",
"The large number of links inside the communities allow the fast convergence to the average opinion of the communities, at the same time the links between the communities (in their turn connecting communities with similar average opinions) allow a slow dynamics of the average opinions of the communities, toward consensus.",
"To use a visual conceptualization, the homophily structure, provides a sort of continuous path allowing the radical opinions to join the central ones."
],
[
" Central result:",
"In a situation where opinion evolve in time, homophily in social networks favors consensus formation.",
"Therefore, contrarily to the common idea that online social networks are directly responsible for the installation of sever opinion wars in the society, we can argue that, on larger time scales, homophily helps the resilience of the society to the presence of radical issues.",
"Figure: A) Fraction of replicas leading the system to consensus as a function of ε\\varepsilon .",
"For each value of β=0,5,20,100\\beta =0,5,20,100, the simulation has been performed N replicas =100N_{replicas}=100 times.",
"Consensus is reached for lower values of β\\beta for homophilous networks.",
"B) Each color represent the opinion span and the average opinion of a community for a single replica of a system with ε=0.3\\varepsilon =0.3 and β=100\\beta =100.",
"There is an interplay between the fast dynamics inside each community and the slower dynamics between the communities.",
"C1,2) Single replica representation of the evolution of the individual opinions (each line represents the opinion of an agent), for C1→ε=0.2C1\\rightarrow \\varepsilon =0.2 and C2→ε=0.3C2\\rightarrow \\varepsilon =0.3, and for different values of β=0,5,20,100\\beta =0,5,20,100"
],
[
"Opinion dynamics in the presence of dominant media",
"In this section we add a further ingredient in the system: the presence of an external media, diffusing with a certain pressure $p_m$ a constant opinion $\\theta _M$ .",
"In several previous papers it has been observed, in the context of BC models, that the dominant media lose their capacity to attract people opinion, (i.e., the effectiveness of propaganda) after a certain pressure threshold.",
"In [30] the external media pressure has been modeled as a heterogeneous open mindedness distribution and some interesting particularities are reported, due to the specific conditions considered.",
"In Carletti et al.",
"([28]) the mass media has been modeled as a periodic perturbation.",
"In this paper the authors divided the systems response into four regimes, where the efficiency of the message is explained in terms of the people open-mindedness threshold, ($\\varepsilon $ ), and the period of the message.",
"In this work the authors stress the importance of the collapse into clusters before the exposure to propaganda, given the influential role that community structures can develops to profile the opinions around a message.",
"This phenomenon is a natural connection with our work, where the effect of the dominant message faces a strong community interaction.",
"Opinion dynamics with media is an asymmetrical opinion update, meaning that, after an agent interacts with media, she can change her opinion, while the opinion of the media ($\\theta _M$ ) will remain identical.",
"We extend to this asymmetrical framework the structure of the BC model: $if |\\theta _i-\\theta _M|<2\\varepsilon \\Rightarrow {\\left\\lbrace \\begin{array}{ll}\\theta _i=\\theta _i+\\mu (\\theta _M-\\theta _i)\\\\\\theta _M=\\theta _M\\end{array}\\right.",
"}$ In the following we will fix $\\mu =0.5$ , as in the previous case, and the opinion of the media, $\\theta _M=1$ .",
"The Python function defining this asymmetrical opinion update is described in Fig.",
"REF Figure: Python implementation of the BC interaction between an agent and mediaTo analyze the effect we introduce a parameter $p_m$ representing the exposure to dominant media messages.",
"If $p_m=0$ the agents have no probability to interact with the media, while if $p_m=1$ the agents will interact only with the media.",
"The model evolves according to the following steps (for a Python implementation see Fig.",
"REF ): A random uniform opinion distribution and a network structure (with $N$ nodes and with a $\\beta $ parameter) are created.",
"At each time step, for $N$ times, an agent $i$ and a real number in the interval $r\\in [0,1]$ are randomly extracted.",
"If $r<p_m$ : agent $i$ makes opinion dynamics with the media according to Eq.",
"REF If $r\\ge p_m$ : a second agent $j$ is selected and an opinion dynamics update according to Eq.",
"REF is performd on the pair ($i,j$ ).",
"The loop is halted when, on all the edges no more successful interactions are possible ($|\\theta _i-\\theta _j|\\ge 2\\varepsilon $ or $|\\theta _i-\\theta _j|=0$ ) Figure: Python implementation of the opinion dynamics model with media.In order to compare the same scenarios according to the unforced opinion dynamics, first we will consider the case $\\varepsilon =0.5$ where, without forcing, consensus is obtained both for homophilous ($\\beta =30$ ) and non-homophilous ($\\beta =0$ ) situations.",
"In a second step we will extend these results to a larger spectrum of values for the tolerance threshold."
],
[
"Results",
"Our results are reported in figure REF .",
"In figure REF A we show the result of the evolution of a single replica in the case where the media pressure is fixed to $p_m=0.5$ .",
"When agents are highly affected by mass-media, like in this case ($p_m=0.5$ ), several opinions remains in opposition to the dominant message.",
"This effect has been observed over BC models in all the previously cited papers ([30],[28],[29]) as well as in the Axelrod model ([32]).",
"Notice, however, (Fig.",
"REF A) that when homophily is not present in the social network, these counter-messages remain a non-homogeneous set of separate opinions (the parallel red lines in the figure) because the agents carrying these opinions are not connected by the social network and therefore cannot interact among them.",
"The presence of homophily in the social network, makes more probable that the individuals with radical opinions opposing to the media cluster, are connected among them by the social network.",
"The interactions between these radical agents are therefore possible allowing the recomposition of a secondary composite opinion counter-cluster (the final blue state in the figure), describing a real situation of opinion polarization.",
"In the following we analyze this effect more deeply as the result of aggregate replicas of the model and for various values of the media pressure parameter $p_m$ .",
"In figure REF B is shown the probability density function for the final states in terms of mass-media intensity.",
"The red shapes represent the case for $\\beta = 0$ , the blue shapes represent the case for $\\beta = 30$ , where homophily strongly influence the network structure.",
"In figure REF C and D we respectively display the number of opinion clusters and the size of the two largest clusters as a function of the media pressure, $p_m$ .",
"In general we observe that the alignment of all the agents to the mass media state (an unique opinion cluster at $\\theta =\\theta _M$ ) occurs, counterintuitively, only for low values of the media intensity, as previously found in several works, and explained in the introduction.",
"This happens because the fast drift toward media opinion leaves several isolated agents that cannot find a peer to interact with.",
"For networks without homophily, the threshold value for the mass-media to have this self-defeating effect is around $p_m = 0.4$ , when some very small non-aligned states start to appear (Fig.",
"REF C), and become macroscopic (Fig.",
"REF D).",
"A first effect of the presence of homophily in network morphogenesis is the lowering of the threshold for the media to be effective.",
"In the case where homophily is present we observe the formation of counter messages (and the decrease of the size of the cluster aligned with the media) already for $p_m=0.2$ (Fig.",
"REF B).",
"At the same time, if we look at the blue plots in figure REF B, for high values of $p_m$ , it is clear how the homophily-based networking structures the non-aligned states around one powerful cluster (as we observed for the single replica plot in Fig REF A) .",
"The fact that the cluster has a larger amplitude than the single peak observed in (Fig.",
"REF A) is due to the statistical fluctuations of the position of the second cluster among the different replicas of the system (for each replica the single cluster observed in Fig.",
"REF A has a different position), but we can observe, as well, that the final opinions are much less sparse than in the case without homophily.",
"This effect can be better observed in Fig.",
"REF C and D. The community structure resulting from the homophily effect during the network morphogenesis, re-organizes all the small non-aligned states into a macroscopic one, competing with the dominant message: for the homophilous case, for $p_m \\ge 0.4$ the size of the second cluster is much larger while the number of cluster is much smaller.",
"At the same time we can observe a relevant reduction of the size of the dominant cluster (aligned to the media).",
"Figure: A) Single replica evolution of the system for ε=0.5,p m =0.5,N=1000\\varepsilon =0.5, p_m=0.5, N=1000, for two values of the homophily parameter: β=(0,30)\\beta =(0,30).",
"B) Final state representation for 20 replicas of the system and for different values of the media pressure.",
"As p m p_m increases the cluster aligned with media loses mass.",
"C) Average number of clusters as a function of the media pressure p m p_m for two values of the homophily parameter: β=0,30\\beta =0,30 and for 20 replicas of the system.",
"When homophily is present less clusters are formed.",
"D) Average size of the first (filled markers) and second (empty markers) clusters as a function of the media pressure p m p_m for two values of the homophily parameter: β=(0,30)\\beta =(0,30) and for 20 replicas of the system.",
"When homophily is present the secondary cluster becomes larger.In Fig.",
"REF we extend the analysis to other values of the tolerance threshold $\\varepsilon $ .",
"Notice however that in these cases the output of the model is a priori different already at the level of the opinion dynamics without media.",
"The number of clusters (in upper panel) and the fraction of agents aligned with the external mass-media (in the bottom panel) are shown, varying the media pressure $p_m$ but also for several values of tolerance.",
"This global picture first shows that, as in the case where media are not present, when the tolerance is low more clusters are formed (REF A).",
"This effect is more pronounced for the case where community structures is not present ($\\beta =0$ ).",
"In this case the effect of the media pressure on the number of clusters is remarkable only for high values of the tolerance ($\\varepsilon >0.4$ ), when high values of the media pressure produce the increase of the number of opinion clusters.",
"When community structures is present ($\\beta =20$ ), the basic scenario of the loss of control by the strong mass media exposure, takes a different shape: First we can observe that the tuning of media pressure has the general role of creating, when a media pressure threshold is reached, a general bi-polarized configuration as previously described for the $\\varepsilon =0.5$ case.",
"In Fig.",
"REF B-C we display the fraction of agents aligned with the media opinion $\\theta _M$ .",
"For both the network topologies (homophilous, REF B, and non homophilous, REF C) lower values of tolerance create a stronger resistance to the dominant media independently from its pressure.",
"At same time we observe that the maximum power of attraction of the mass media is obtained around $pm=0.2$ and it monotonically decreases as the media pressure increases.",
"However, Fig.",
"REF C shows that as a consequence of the community structures, when homophily is present, this maximum value no longer reaches all the population.",
"Therefore in this case, the large mass of the counter-cluster is able to subtract support to the media.",
"Figure: Upper Panel (A) : Number of clusters as a function of the media pressure, pmpm, for two values of the homophily parameter: β=0,20\\beta = 0, 20 andsix values of ε\\varepsilon .",
"Each point is the average over 100 replicas.",
"Bottom panels: Averaged proportion of agents aligned with the imposed mass media, for β=0\\beta = 0 (in B) and 20 (in C).",
"Results in B and C show how the strength of community structures undermines the dominantrole of mass media reported in several ABM's.",
"On the other hand, results in A (upper panel) show how the community structurere-organizes all the possible different opinions in one counter-message position, independent of the general tolerance of thesociety.",
"However, the lower the tolerance, the faster the counter message appears."
],
[
" Central result:",
"In general, independently of the network topology, the media messages are less effective if the media pressure is too high.",
"In the case where homophily is present in the network structure, the threshold where the media become ineffective is lower, showing that these structures are more resistant to propaganda.",
"Furthermore, the community structures resulting from the homophily-based networking, when facing a dominant message, aggregate the non-aligned states into just one second strong opinion counter-cluster, decreasing the size of dominant one."
],
[
"Conclusions",
"In this work three related subjects have been addressed.",
"First, we showed how a network morphogenesis model, including at the same time the preferential attachment mechanism and an homophily effect, can structure networks with the same power-law degree distribution as the BA networks, but with marked communities of nodes sharing similar opinions.",
"Second, the bounded confidence model has been used on such topology showing that, contrary to established ideas, homophily in social networks favors consensus formation: we show that the critical value of tolerance ( $\\varepsilon _c= 0.5$ ), previously reported as the threshold for BC models to shift between total consensus and different opinions, loses its \"universal\" character when considered in more realistic networks, as the ones formed with community structures.",
"In the case where homophily is present consensus is reached also for lower values of the tolerance parameter (less open-minded societies).",
"Finally, the effect of mass media over the BC models with homophily scale-free networks has been reported.",
"We showed that homophily has a double effect: first, it decreases the effectiveness of the media pressure, facilitating the emergence of counter opinions also for lower values of the media pressure.",
"On the other hand, we showed that, when the community structures (typical of homophily-based networks) face dominant messages, disaggregated non-aligned states converge into just one (or few) strong counter-opinion cluster, representing a strong polarization of the opinions in the societies.",
"Moreover, the strong polarization against the dominant message is promoted by low values of tolerances.",
"Social networks and social media are nowadays the backbone of the diffusion of controversial subjects.",
"Using data analytics tools (to personalize the advertisements) and new tools of the digital economy like the click-farms (to increase visibility to a content), new debates, often deviant from the dominant vision of the state authorities, spread everyday on the web.",
"In several cases, when these debates, can trigger risky behaviors, the authorities answer with strong media campaigns (think for example to the case of vaccines).",
"According to our findings, these risky opinions would be naturally controlled, on long time scale.",
"If the subject is too risky and an immediate response is needed, media campaigns are probably the worst method.",
"Probably the best solution would be to use the same social networks to propagate the counter-messages.",
"A first further direction of analysis, that could be pursued in the future, concerns this last point: if traditional media apparently have lost their centrality in the communication, how to better veicolate counter-messages to prevent risky behaviors?",
"A second central direction to be addressed is the role of the click-market on the opinion formation.",
"How the new instruments to capture users' attention are different from tradition media?",
"Is the click economy responsible for the fact that opinions that once were considered deviant are now dominating?"
]
] | 1612.05483 |
[
[
"Outskirts of Nearby Disk Galaxies: Star Formation and Stellar\n Populations"
],
[
"Abstract The properties and star formation processes in the far-outer disks of nearby spiral and dwarf irregular galaxies are reviewed.",
"The origin and structure of the generally exponential profiles in stellar disks is considered to result from cosmological infall combined with a non-linear star formation law and a history of stellar migration and scattering from spirals, bars, and random collisions with interstellar clouds.",
"In both spirals and dwarfs, the far-outer disks tend to be older, redder and thicker than the inner disks, with the overall radial profiles suggesting inside-out star formation plus stellar scattering in spirals, and outside-in star formation with a possible contribution from scattering in dwarfs.",
"Dwarf irregulars and the far-outer parts of spirals both tend to be gas dominated, and the gas radial profile is often non-exponential although still decreasing with radius.",
"The ratio of H-alpha to far-UV flux tends to decrease with lower surface brightness in these regions, suggesting either a change in the initial stellar mass function or the sampling of that function, or a possible loss of H-alpha photons."
],
[
"Introduction",
"The outer parts of galaxies represent a new frontier in observational astronomy at the limits of faint surface brightness.",
"We know little about these regions except that galaxies viewed deeply enough can usually be traced out to 10 stellar scale lengths or more, without any evident edge.",
"We do not know in detail how the stars and gas got there and whether stars actually formed there or just scattered from the inner parts.",
"Neither do we know as much as we'd like about the properties, elemental abundances, scale heights and kinematics of outer disk stars except for a limited view in the Milky Way (e.g., [15]) and the Andromeda galaxy ([21]).",
"Yet the outer parts of disks are expected to be where galaxy growth is occurring today, and where the left-over and recycled cosmological gas accretes or gets stored for later conversion into stars in the inner disk ([75], [93]).",
"The outer parts should also show the history of a galaxy's interactions with other galaxies, as the orbital time is relatively long.",
"A high fraction of outer disks are lopsided too, correlating with the stellar mass fraction in the outer parts (i.e., the ratio of the stellar mass to the total from the rotation curve; [139]), perhaps because of uneven accretion, interactions, or halo sloshing ([45]).",
"This chapter reviews disk structure, star formation and stellar populations in the outer parts of nearby galaxies.",
"General properties of these outer disks are in Sect.",
"to , and a focus on dwarf irregular galaxies (dIrrs) is in Sect. .",
"The observational difficulties in observing the faint outer parts of disks are discussed in other chapters in this volume."
],
[
"Outer disk Structure from Collapse Models of Galaxy Formation",
"A fundamental property of galaxy disks is their exponential or piece-wise exponential radial light profile ([24]).",
"[42] noted that this profile gives a distribution of cumulative angular momentum versus radius that matches that of a flattened uniformly rotating sphere ([86]), but this match is only good for about four disk scale lengths.",
"The problem is that an exponential disk has very little mass and a lot of angular momentum in the far-outer parts, unlike a power-law halo which has both mass and angular momentum increasing with radius in proportion ([26]).",
"Nevertheless, observations show some disks with 8 to 10 scale lengths ([133], [13], [48], [56], [104], [128], [6] , [58], [89], [127]).",
"These large extents compared to the predicted four scale lengths from pure collapse models need to be explained ([39]).",
"Thus we have a problem: if the halo collapses to about four scale lengths in a disk, then how can we get the observed eight or more scale lengths in the stars that eventually form?",
"The answer may lie with the conversion of incoming gas into stars.",
"In a purely gaseous medium, interstellar collapse proceeds at a rate per unit area that is proportional to the square of the mass column density, $\\Sigma _{\\rm gas}$ ([29]).",
"One factor of $\\Sigma _{\\rm gas}$ accounts for the amount of fuel available for star formation and the other factor accounts for the rate of conversion of this fuel into stars.",
"This squared Kennicutt-Schmidt law converts four scale lengths of primordial gas into eight scale lengths of stars after they form (Sect.",
").",
"Stellar scattering from clouds and other irregularities could extend or smooth out this exponential further ([33], [34]).",
"There is an additional observation in [131] that in local gas-rich galaxies, the outer gas radial profiles are all about the same when scaled to the radius where $\\Sigma _{\\rm HI} = 1\\,M_\\odot $ pc$^{-2}$ .",
"[10] found a similar universality to the gas profile when normalized to $R_{25}$ , the radius at 25 magnitudes per square arcsec in the $V$ band.",
"[131] found that the ratio of the radius at 1 $M_\\odot $ pc$^{-2}$ to the gaseous scale length in the outer disk is about four, the same as the maximum number of scale lengths in a pure halo collapse.",
"This similarity may not be a coincidence (Sect.",
").",
"Cosmological simulations now have a high enough resolution to form individual galaxies with reasonable properties ([129], [118]).",
"Zoom-in models in a cosmological environment show stellar exponential radial profiles in these galaxies ([106]) even though specific angular momentum is not preserved during the collapse and feedback moves substantial amounts of gas around, especially for low-mass galaxies ([27]).",
"For example, [2] ran models with rotating halo gas aligned in various ways with respect to the dark matter symmetry axis.",
"They found broken exponential disks with a break radius related to the maximum angular momentum of the gas in the halo, increasing with time as the outer disk cooled and formed stars.",
"Star formation is from the inside-out.",
"Angular momentum was redistributed through halo torques, but still the disks were approximately exponential.",
"[3] further studied 16 simulated galaxies with various masses.",
"All of them produced near-exponential disks.",
"In a systematic study of angular momentum, [51] found a transition from exponentials with up-bending outer profiles (Type III—Sect. )",
"at low specific angular momentum ($\\lambda $ ) to Type I (single exponential) and Type II (down-bending outer parts) at higher $\\lambda $ .",
"An intermediate value of $\\lambda =0.035$ , similar to what has been expected theoretically ([92]), corresponded to the pure exponential Type I.",
"The reason for this change of structure with $\\lambda $ was that collapse at low spin parameter produces a high disk density in a small initial radius, and this leads to significant stellar scattering and a large redistribution of mass to the outer disk, making the up-bending Type III.",
"Conversely, large $\\lambda $ produces a large and low-density initial disk, which does not scatter much and nearly preserves the initial down-bending profile of Type II."
],
[
"Outer Disk Structure: Three Exponential Types",
"Galaxy radial profiles are often classified as exponential Types I, II or III according to whether the outer parts continue with the same scale length as the inner parts, continue with a shorter scale length (i.e., bend down a little) or continue with a larger scale length (bend up a little), respectively (Fig.",
"REF ).",
"The Sersic profile with $n=1$ corresponds to Type I; the other types do not have a constant Sersic index.",
"For a review, see [125], and for early surveys, see [101] and [37].",
"Figure: Three types of exponential or piece-wise exponential profiles[49] determined the proportion of the three exponential profile types for barred and non-barred galaxies of various Hubble types, including 183 large local face-on galaxies from three separate studies.",
"For S0 and earlier, the three profile types are nearly evenly divided.",
"For Sab to Sbc, Types II and III are about equal and Type I becomes relatively rare (10%).",
"For Scd to Sdm, Type II dominates with $\\sim 80$ % of the total.",
"[52] continued this study to dIrrs and blue compact dwarfs (BCDs, see also Sect.",
"REF ).",
"dIrr galaxies are dominated (80%) by Type IIs, while BCDs have steep inner parts from a starburst and are usually Type III.",
"A general caution should be mentioned about possible contamination at faint light levels from scattered light.",
"[115] showed $R$ and $I$ band radial profiles for NGC 4102 that were fit to a single exponential model with a broad point spread function from the instrument.",
"The usual Type III profile for this galaxy turned into a Type I when the outer excess was corrected for the instrumental profile."
],
[
"Outer Disk Stellar Populations: Colour and Age Gradients",
"Radial profiles become more complex when changes in stellar colours and ages are considered.",
"[4] noted that Type II light profiles tend to correspond to U-shaped $B-V$ colour profiles, which means that the inner part of the disk gets bluer with radius at first, and then the outer part of the disk gets red again.",
"This colour change presumably corresponds to a change in the mass-to-light ratio, with large ratios in the outer parts.",
"Then the down-bending Type II in a light profile tends to straighten out and become Type I in a mass profile.",
"That is, the outer red trend gives an increasing mass-to-light ratio, causing an increasing conversion factor from surface brightness to mass surface density.",
"The red outer parts could be from old stars that scattered there from the inner regions ([109]), as distinct from the common model of inside-out growth for spiral galaxies.",
"A larger survey recently confirmed this result.",
"[142] included 700 galaxies using deep images from the Pan-STARRS survey.",
"The average $g$ -band (peak at 5150 Å) light profile was the down-bending Type II for low-mass galaxies ($<10^{10}\\,M_\\odot $ ) and slightly less bent for high mass galaxies ($<10^{10.5}\\,M_\\odot $ ), as usual, and the average $g-i$ colour profiles ($i$ band peaks at $7490Å$ ) were U-shaped to various degrees, so the average mass profile became Type I for all galaxy masses.",
"[96] made average radial profiles separated into eight mass bins for $\\sim 2400$ galaxies using 3.6 $\\mu $ m emission from the Spitzer Survey of Stellar Structure in Galaxies.",
"Such long wavelength emission is a nearly direct probe of galaxy mass, although there is some PAH emission from dust in it too.",
"All masses showed Type II profiles on average, with a straighter trend like Type I from a central bulge in the more massive galaxies.",
"This suggests that the mass profile for most galaxies is not exactly Type I, but still tapers off more steeply in the outer parts, beyond 1 kpc for low mass galaxies ($<10^9\\,M_\\odot $ ) and beyond 10 kpc for high mass galaxies ($>10^{10.5}\\,M_\\odot $ ).",
"The most telling observations are of stellar age gradients because colour gradients can be from a mixture of age gradients and metallicity gradients.",
"[107] determined radial age profiles from photometry and stellar population models of 64 Virgo cluster disk galaxies.",
"They found U-shaped age profiles in 15% of Type I's, and also in 36% of both Types II and III.",
"In one-third of all exponential types, the age increased steadily with radius.",
"[137] found about the same mixture of age profiles, measuring ages from spectra in 12 galaxies.",
"[22] determined star formation histories for 15 nearby galaxies with masses in the range $10^8\\,M_\\odot $ to $10^{11}\\,M_\\odot $ using ultraviolet and infrared data; they also found U-shaped age profiles.",
"These results imply that outer disks generally have old stars, although most also still have some star formation.",
"A recent integral field unit survey of 44 nearby spiral galaxies (CALIFA) by [110] also found U-shaped age profiles in Types I and II when the stars were weighted by brightness, as would be the case from integrated spectra or photometry.",
"This is in agreement with the previous surveys mentioned above.",
"The galaxies were observed beyond their break radii or for at least three scale lengths.",
"In contrast, [110] found constant age profiles when the stars were weighted by mass.",
"They suggested that the entire disk formed early with star formation stopping in the inner parts first, and then quenching from inside-out.",
"This is unlike cosmological simulations that have the outer disk form more slowly than the inner disk, and also unlike models where the outer disk stars migrate there from the inner disk.",
"[132] viewed three nearby spirals with very deep images, covering a range of about 10 magnitudes in surface brightness for $B$ band.",
"They found smooth and red stellar distributions with no spiral arms in the far-outer disks.",
"For the typical colour of $B-V \\sim 0.8$ mag in the outer parts, and from a lack of FUV light, they concluded that the star formation rate (SFR) had to be less than $3-5\\times 10^{-5}\\,M_\\odot $ pc$^{-2}$ Myr$^{-1}$ .",
"This seemed to be too low for continuous star formation and disk building, suggesting some radial migration.",
"However, the lack of spiral arms makes the usually invoked churning mechanism ([119], [109], [9]) inoperable.",
"Churning is a process of stellar migration back and forth around corotation.",
"Perhaps stellar scattering off local gas irregularities makes the outer exponential structure ([33])."
],
[
"Mono-age Structure of Stellar Populations",
"Age profiles in galaxy disks can be viewed in another way too.",
"A series of galaxy simulations have looked at the distributions of stars of various ages in the final model.",
"For example, [12] did a simulation of the Milky Way and found that older stars in the present-day disk have shorter radial scale lengths and thicker perpendicular scale heights than younger stars.",
"Other mono-age population studies of simulated disks are in [114], [121], [81], [91] and [1], giving the same result.",
"[15] found structure related to this in the Milky Way using 14700 red clump stars.",
"Higher metallicity populations are more centrally concentrated than lower metallicity populations (not considering the $\\alpha $ -enhanced “thick disk” component).",
"Each narrow metallicity range tends to have a maximum surface density of stars at a particular radius where the disk has that average metallicity.",
"Plus, each mono-metallicity population has a perpendicular scale height that increases with radius, producing a flare.",
"The correspondence between metallicity and peak surface density for a population of stars suggests that star formation, feedback, halo recycling, and other processes establish an equilibrium metallicity in a region that depends primarily on local conditions, such as the local mass surface density ([15]).",
"Stellar migration then broadens this distribution to produce the observed total profiles.",
"This local equilibrium concept is consistent with the results of [108], who found for 2000 Hii regions in nearby galaxies that metallicity depends mostly on stellar mass surface density, as determined from photometry.",
"[16] present a similar result: that the metallicity gradients in galaxies are all the same when expressed in units of the disk scale length."
],
[
"Outer Disk Structure: Environmental Effects and the Role of Bulges and Bars",
"Environment may also affect outer disk structure.",
"[138] showed that prograde minor mergers can drive mass inward and outward, creating a Type III profile.",
"[14] also suggested that Type III S0 galaxies can result from a merger.",
"This is consistent with observations in [38] that S0 galaxies in Virgo have proportionally more Types I and III, suggesting that interactions or mergers have been important.",
"Erwin et al.",
"also found that bars have little effect on the proportion of exponential types.",
"[1] simulated a gas-rich major merger and showed that it formed an exponential disk in the final system.",
"On the other hand, [79] measured the $V$ -band radial profiles of 330 galaxies observed with Hubble Space Telescope over a half-degree field surrounding a galaxy supercluster at redshift 0.165.",
"They found no dependence on environment, cluster versus field, for the ratio of the outer to the inner disk scale length or the outer scale length itself.",
"[50] got a similar result looking at S0 galaxies in the Coma cluster; using a profile decomposition algorithm to remove the bulge, they found that bars are important for disk structure, correlating with Types II and III (contrast this with the [38] result above), but that location in the cluster is not important.",
"According to [50], the relative proportion of Types I, II, and III is the same in the core, at intermediate radii, and in the outskirts of Coma (in fact, most of the S0 galaxies were Type I).",
"Some of the appearance of Type III could be from a bright halo or extended bulge and not from stars in the disk ([36]).",
"[80] suggested that half of the S0 Type III structures in various environments come from extended bulge light, although this fraction is only 15% in later Hubble type spirals.",
"This implies that disk fading can make an S0 from a spiral, preserving the scale length.",
"Simulations by [19] also found that the outer stellar structure can be in a halo and not a disk, as a result of mergers.",
"Bars and spirals seem to be important in determining the break radius for down-bending (Type II) exponentials.",
"[95] suggested that the break radius for Type II's is either at the outer Lindblad resonance (OLR) of a bar or the OLR of a spiral that is outside of a bar.",
"The spiral and bar are assumed to have their pattern speeds in a resonance with the inner 4:1 resonance of the spiral at the corotation radius of the bar (see [101] and [37]).",
"[70] also found that bars and spirals are important: 94% of Type II breaks are associated with some type of feature; 48% are in early type galaxies with an outer ring or pseudoring; 8% are with a lens, assumed to be the OLR of a bar, and if there is no outer ring, then the breaks are at 2 times the radius of an inner ring (this being the ratio of radii for outer and inner ring resonances); 14% are in late type galaxies associated with an end to strong star formation, and 24% are at the radius where the spiral arms end.",
"For Type III breaks studied by [70], 30% are associated with inner or outer lenses or outer rings."
],
[
"Outer Disk Structure: Star Formation Models",
"The outer disks of spiral galaxies and most dIrrs are dominated by gas in an atomic form, and not stars.",
"Because stars form in molecular gas, it is difficult to observe directly how stars form in these regions.",
"Moreover, outer disks and dIrrs tend to be stable by the Toomre $Q$ condition ([31]).",
"Nevertheless, star formation usually looks normal there, forming clusters and associations at low density ([84], [59]), although it may stop short of the full extent of the gas disk (see, for example, Fig.",
"REF .)",
"Figure: Stellar mass surface density Σ * \\Sigma _*, H\\,i++He surface density Σ HI + He \\Sigma _{\\rm HI+He}, and SFR density Σ SFR ,Hα \\Sigma _{\\rm SFR, H\\alpha } plotted as a function ofradius for two very luminous (M V =-22M_V=-22 to -23-23) Sc-type spiral galaxies, NGC 801and UGC 2885.",
"The radius is normalized to the optical VV-band disk scale lengthR D R_{\\rm D}.",
"The gas and stellar mass surface densities have been corrected to face-on.The logarithmic interval is the same for all three quantities, but the SFR zeropoint is different.",
"Adapted from [68] formulated a model for star formation in these conditions that considers the existence of a two-phase atomic medium (i.e., a warm neutral medium in pressure equilibrium with a cool neutral medium) and the molecular fraction in such a medium.",
"He then assumed that star formation occurs in the molecular medium at a rate given by an efficiency per unit free fall time, $\\epsilon _{\\rm ff}\\sim 0.01$ , times the molecular mass divided by the free fall time in the molecular gas: $\\Sigma _{\\rm SFR}=\\epsilon _{\\rm ff}\\Sigma _{\\rm mol}/t_{\\rm ff,mol}.$ The free fall time depends on the molecular cloud density, which for outer disks in their model, depends on the molecular cloud mass and a fiducial value of the molecular cloud surface density, $\\Sigma _{\\rm GMC}=85\\,M_\\odot $ pc$^{-2}$ .",
"The molecular cloud mass was taken to be the turbulent Jeans mass in the interstellar medium (ISM), $M_{\\rm GMC}=\\sigma ^4/(G\\Sigma _{\\rm gas})$ for turbulent speed $\\sigma $ and average ISM gas surface density $\\Sigma _{\\rm gas}$ .",
"For inner disks, the free fall time was taken to be the value for an average disk density where the Toomre $Q$ parameter equals unity.",
"To span the inner and outer regions, the minimum of these two free fall times was used.",
"One uncertainty in the [68] model is the assumption that a two-phase medium is present, because it need not be present everywhere in the outer disk.",
"But this assumption seems reasonable for star formation because cool gas greatly facilitates cloud formation ([32], [117], [41]).",
"A second assumption is that the SFR is given only by the molecular gas mass and density, and that this is related to the total density by the molecular fraction, which depends on the ratio of the radiation field to the cool cloud density.",
"In this model, molecule formation is calculated separately as a precursor to star formation, and then whatever is calculated for molecules is used to determine the SFR.",
"Another model considers that the average SFR is determined by the average ISM dynamics and that molecule formation is incidental, i.e., molecule formation happens along the way but it is not a limiting factor.",
"Star formation in predominantly atomic gas has been predicted by [47] and [67] and suggested by observations in [88] and [35].",
"In this model, the gas mass available for star formation is the total gas mass in all forms, even atomic gas, and the free fall rate of this gas is given by the average midplane density, regardless of molecular content ([29]).",
"Cool clouds are still required so the ISM cannot be purely warm phase.",
"Also, because molecular hydrogen is slow to form at low density ([77]), there could be a substantial fraction of H$_2$ in stagnant, diffuse clouds without significant CO emission and with little connection to star formation ([31]).",
"Such a diffuse H$_2$ medium was found in simulations by [54] and [112] but has not been observed yet.",
"These diffuse H$_2$ clouds, along with more atom-rich clouds, would presumably come together during localized ISM collapse as a precursor to star formation.",
"In this model the SFR per unit area is given by $\\Sigma _{\\rm SFR} = \\epsilon _{\\rm ff} \\Sigma _{\\rm gas} / t_{\\rm ff,gas}$ for midplane free-fall time $t_{\\rm ff}$ at the density $\\rho = \\Sigma _{\\rm gas}/(2H)$ and scale height $H = \\sigma ^2 / ( \\pi G \\Sigma _{\\rm gas})$ .",
"The result is ([29]) $\\Sigma _{\\rm SFR} = \\epsilon _{\\rm ff} ( 4/3^{1/2} ) ( G / \\sigma ) \\Sigma _{\\rm gas}^2= 1.7 \\times 10^{-5} (\\Sigma _{\\rm gas}/[1\\,M\\odot /{\\rm pc}^2])^2 (\\sigma /6\\,{\\rm km\\, s}^{-1})^{-1}$ where $\\sigma $ is the gas velocity dispersion and $\\epsilon _{\\rm ff}\\sim 1$ % is the efficiency of star formation per unit free fall time.",
"Both of the above models compare well with observations ([68], [29]).",
"[54] simulated dwarf galaxies with a chemical model to form H$_2$ , CO and other molecules, cloud self-shielding from radiation, and a SFR given by Eq.",
"(REF ) at a threshold density of 100 cm$^{-3}$ and a temperature less than 100 K. They found that $\\Sigma _{\\rm SFR}$ decreases faster than $\\Sigma _{\\rm gas}$ but not because of a flare (the extra $\\Sigma _{\\rm gas}$ factor in Eq.",
"REF ).",
"Rather, $\\Sigma _{\\rm SFR}$ follows the cold gas with a rate that scales directly with the cold gas fraction, i.e., a linear law, and this cold gas fraction decreases with radius.",
"The linear law in their model is because of the assumed constant threshold density.",
"Most star-forming gas is close to this fixed density, so the characteristic dynamical time is the fixed value at this density.",
"This point about a fixed density is similar to the explanation for the linear star formation law in [29], where it was pointed out that if CO, HCN, and other star formation tracers emit mostly at their fixed excitation density, as determined by the Einstein $A$ coefficient, then the effective free fall time is the fixed value at this density.",
"The SFR then scales only with the amount of gas at or above this observationally selected density.",
"The fixed density in this case is not because of an assumption about a star formation threshold, as there is no threshold in the [29] model.",
"There is just a continuous collapse of ISM gas at a rate given by the midplane density, and a feedback return of the dense gas to a low density form.",
"The existence of a fixed threshold density for star formation is something to be tested observationally.",
"[29] suggest there is no threshold density because clouds are strongly self gravitating when $\\pi G\\Sigma _{\\rm cloud}^2>P_{\\rm ISM}$ for cloud surface density $\\Sigma _{\\rm cloud}$ and ambient pressure $P$ .",
"Because the interstellar pressure varies with the square of the total surface density of gas and stars inside the gas layer, there is a large range in pressure over several exponential scale lengths in a galaxy disk—a range that may exceed a factor of 100 for dIrrs, and 1000 for spirals.",
"Thus if there is a threshold for star formation, the above equation suggests that it might be $\\Sigma _{\\rm cloud}>\\left(P/\\pi G \\right)^{1/2},$ in which case it should vary with radius.",
"We return to a point made in Sect.",
"about the gas surface density profile in the far-outer regions of gas-rich galaxies.",
"[131] noted that the gas exponential scale length beyond the radius $R_1$ , where $\\Sigma _{\\rm gas}=1\\,M_\\odot $ pc$^{-2}$ , is always about 0.25 times this radius.",
"We can see this here also from Eq.",
"(REF ), which states that $\\Sigma _{\\rm SFR} = 2\\times 10^{-5}\\,M_\\odot \\,{\\rm pc}^{-2}\\,{\\rm Myr}^{-1}\\,{\\rm at}\\,\\Sigma _{\\rm gas}=1\\,M_\\odot \\,{\\rm pc}^{-2}.$ After a Hubble Time of $10^4$ Myr, $\\Sigma _{\\rm stars}$ is approximately $0.2\\,M_\\odot $ pc$^{-2}$ .",
"According to the average disk mass profiles in [142], this outer stellar surface density is lower than that at the disk centre by $10^{-3.5}$ on average, which represents eight scale lengths in stars.",
"But eight scale lengths in stars is four scale lengths in gas for Eq.",
"(REF ).",
"Thus the radius at 1 $M_\\odot $ pc$^{-1}$ is about four times the scale length in the gas, as observed further out by [131]."
],
[
"The Disks of Dwarf Irregular Galaxies",
"Dwarf irregular galaxies are like the outer parts of spiral galaxies in terms of gas surface density, SFR, and gas consumption time.",
"Tiny dIrrs have extended exponential disks as well.",
"For example, [113] traced the Large Magellanic Cloud (LMC) to 12 $R_{\\rm D}$ , an effective surface brightness of 34 mag arcsec$^{-2}$ in $I$ , and [116] found stars in IC 10 to $\\sim 10$ $R_{\\rm D}$ .",
"[8] detected stars associated with Sextans A and Sextans B to 6 $R_{\\rm D}$ , and Hunter et al.",
"(2011) measured surface brightness profiles in four nearby dIrrs and one BCD to 29.5 mag arcsec$^{-2}$ in $V$ , corresponding to $3-8$ $R_{\\rm D}$ .",
"These extended stellar disks represent extreme galactic environments for star formation and are potentially sensitive probes of galaxy evolutionary processes, and yet they are relatively unexplored.",
"In this Section we examine what is known about outer disks of dIrr galaxies."
],
[
"The Gas Disk",
"The H$\\,$i gas often dominates the stellar component of dIrr galaxies, both in extent and mass.",
"How much further the gas extends compared to the stars was demonstrated by [69] for DDO 154 where the H$\\,$i was traced to 8 $R_{25}$ at a column density of $2\\times 10^{19}$ atoms cm$^{-2}$ ($0.22\\,M_\\odot $ pc$^{-2}$ ).",
"In the LITTLE THINGS sample of 41 nearby ($<10.3$ Mpc), relatively isolated dIrrs ([57]), most systems have gas extending to $2-4$ $R_{25}$ or $3-7$ $R_{\\rm D}$ at that same (face-on) column density.",
"Some spiral galaxies also have extended H$\\,$i; [103] found that the Sbc galaxy NGC 765, for example, has gas extending to 4 $R_{25}$ .",
"Large holes (up to 2.3 kpc diameter) are also sometimes found in the gas beyond 2 $R_{\\rm D}$ ([25], R.N.",
"Pokhrel, in preparation).",
"In most dIrrs, the galaxy is gas-dominated and becomes increasingly gas-rich with radius (Fig.",
"REF ).",
"This implies a decreasing large-scale star formation efficiency ([76], [11]).",
"The lack of sharp transitions in the star-to-gas ratio, including at breaks in the optical exponential surface brightness profiles, suggests that the factors dominating the drop in star formation with radius are changing relatively steadily.",
"Figure: Azimuthally averaged stellar mass to gas mass ratios as a function of radiusnormalized to the disk scale length (top) and radius at which the VV-band surfacebrightness profile changes slope R Br R_{\\rm Br} (bottom).",
"These galaxies are from the LITTLETHINGS sample with stellar mass profiles determined by and gas massprofiles from Hunter et al.",
"(2012)The gas surface density drops off with radius usually in a non-exponential fashion.",
"In Sect.",
"$2-7$ , we have approximated the radial gas profiles of spiral galaxies as exponentials.",
"However, especially in dIrrs, the gas profiles are rarely pure exponentials, and since dIrrs are gas-dominated, the shape of the gas profile is crucial.",
"Note, however, that in dIrrs, the radial stellar profiles are usually exponential in shape.",
"In a sub-sample of the THINGS spirals ([130]), [103] found that the gas is approximately constant at $5-10\\times 10^{20}$ atoms cm$^{-2}$ and then drops off rapidly.",
"A Sersic function fits the profiles with indices of $n=0.14-0.22$ .",
"For comparison, an exponential disk has an $n$ of 1.0.",
"In five THINGS dIrrs the gas density dropped more shallowly with radius, and the distribution of $n$ peaked around 0.3.",
"For the LITTLE THINGS sample of dwarfs, the shape of the H$\\,$i radial profiles varied from galaxy to galaxy, and $n$ varied from 0.2 to 1.65 with most having values $0.2-0.8$ .",
"The lack of correlations between the H$\\,$i profile index $n$ and characteristics of the stellar disk suggest that the role of the gas distribution in determining the stellar disk properties is complex."
],
[
"The Stellar Disk",
"[141] performed spectral energy distribution fitting to azimuthally-averaged surface photometry of the LITTLE THINGS galaxies.",
"The fitting included up to 11 passbands from the FUV to the NIR.",
"From these fits they constructed SFRs as a function of radius over three broad timescales: 100 Myr, 1 Gyr, and galaxy lifetime.",
"Zhang et al found that the bulk star formation activity has been shrinking with radius over the lifetime of dwarf galaxies, and they adopted the term “outside-in” disk growth.",
"Although Zhang et al found that “outside-in” disk growth applied primarily to dIrrs with baryonic masses $<10^8\\,M_\\odot $ , [44] and [85] found the same phenomenon in the LMC, a more massive irregular galaxy.",
"Similarly, [98] suggested from colour profiles that the same process is occurring in a large sample of Sloan Digital Sky Survey galaxies with stellar masses up to $10^{10}\\,M_\\odot $ .",
"This outside-in disk growth is in contrast to the inside-out disk growth identified in spirals ([135], [92], [94], [136], but see [110]).",
"Figure: Left: VV-band image of DDO 133 from .",
"The white ellipse marks theextent of the galaxy measured to 29.5 mag arcsec -2 ^{-2}.",
"The white contours tracecolumn densities of 5, 30, 100, 300, 500, 1000, and 3000×10 18 3000\\times 10^{18} atoms cm -2 ^{-2}in H\\,i (Hunter et al.",
"2012).",
"Top right: Surface photometry in VV, FUV,and B-VB-V as a function of radius normalized to the disk scale length R D R_{\\rm D}.",
"Bottomright: VV-band surface photometry of all five galaxies from .",
"IZw115 is a BCD; the rest are dIrrs[56] carried out an ultra-deep imaging program on four nearby dIrrs and one BCD.",
"They measured surface photometry in this sample to 29.5 mag arcsec$^{-2}$ in $V$ , and also obtained deep $B$ images of three of the galaxies and deep FUV and NUV images with the Galaxy Evolution Explorer (GALEX, [82]).",
"Fig.",
"REF shows the $V$ -band image and photometry of DDO 133, illustrating what they found.",
"What does a surface brightness of 29.5 mag arcsec$^{-2}$ mean?",
"In DDO 133, that is a factor of $\\sim $ 160 down in brightness from the centre.",
"A 1 kpc-wide annulus at 29.5 mag arcsec$^{-2}$ corresponds to a SFR of $0.0004\\,M_\\odot $ yr$^{-1}$ , assuming a mass-to-light ratio from the $B-V$ colour and a constant SFR for 12 Gyr.",
"This is roughly seven Orion nebulae every 10 Myr.",
"In their five dwarfs, [56] found that the stellar surface brightnesses in $V$ and FUV continue exponentially as far as could be measured.",
"Furthermore, the stellar disk profiles are exponential and extraordinarily regular in spite of the fact that dIrr galaxies are clumpy in gas and SFR and star formation is sporadic.",
"[113] found the same thing for the LMC, and [8], for Sextans B.",
"However, Bellazzini et al.",
"also found that, by contrast, Sextans A has a very complex surface brightness profile and suggested that that is the consequence of past outside perturbations, assuming that a regular profile is “normal” for an isolated galaxy.",
"The deep FUV surface photometry of [56] also shows that there is a continuity of star formation with radius.",
"The [123] model, in which star formation is driven by two-dimensional gravitational instabilities in the gas, predicts a precipitous end to star formation where the gas surface density drops below a critical level.",
"Nevertheless, these data show that young stars extend into the realm where the gas is a few percent of the critical gas density and should be stable against spontaneous gravitational collapse ([63]).",
"Models suggest that dIrrs need to be treated as three-dimensional systems, in which case the $Q$ parameter is not a good measure of total stability.",
"Also, the dynamical time at the mid-plane density is more important than the growth time of a two-dimensional instability, which is more closely related to spiral arms than star formation ([29], [31]).",
"The presence of FUV emission in outer disks poses a stringent test of star formation models by extending measures of star formation activity to the regime of low gas densities.",
"How low can the gas density get and still have star formation?",
"H$\\,$ii regions have been found in the far-outer disks of spirals ([40]), and GALEX has found FUV-bright regions out to $2-3$ times the optical radius of the spiral ([46], [122]).",
"[18] proposed that these FUV regions could be due to spiral density waves from the inner disk propagating into the outer disk and raising local gas regions above a threshold for star formation.",
"In fact, [7] found evidence for greater instability in outer disk spirals compared to inner disk spirals in eight nearby spiral galaxies.",
"Dwarf irregular galaxies, however, do not have spiral density waves, and neither do the far-outer parts of the galaxies observed by [132], so the problem still remains of how stars form in or get scattered to extreme outer disks.",
"Recently, GALEX images have been used to identify FUV-bright knots in the outer disks of dIrrs in order to determine how far-out young star clusters are formed in situ and the nature of the star clusters found there.",
"[59] identified the furthest-out FUV knot of emission in the LITTLE THINGS galaxies, and found knots at radii of $1-8\\,R_{\\rm D}$ (see Fig.",
"REF ).",
"Most of these outermost regions are found intermittently where the H$\\,$i surface density is $\\sim 2\\, M_\\odot $ pc$^{-2}$ , although both the H$\\,$i and dispersed old stars go out much further (also true of some spiral galaxies; e.g., [48]).",
"In a sample of 11 of the LITTLE THINGS dwarfs within 3.6 Mpc, [84] identified all of the FUV knots and modelled their UV, optical, and NIR colours to determine masses and ages.",
"They found no radial gradients in region masses and ages (see Fig.",
"REF for an example), even beyond the realm of H$\\alpha $ emission, although there is an exponential decrease in the luminosity density and number density of the regions with radius.",
"In other words, young objects in outer disks cover the same range of masses and ages as inner disk star clusters.",
"Figure: Left: Histogram of the distance from the centre to the furthest knot of FUVemission in the LITTLE THINGS dIrrs relative to the disk scale length R D R_{\\rm D} ().",
"Right: Distance from the galaxy centre to the FUV knot vs average H\\,i surface density at the radius of the knot.The Σ HI \\Sigma _{\\rm HI} have been corrected for a scaling error as described in Figure: Plots of mass vs. age, mass vs. galactocentric radius, and age vs. radius for WLM,one of the galaxies from Fig.",
"3 of .",
"The radius corresponding tothe extent of Hα\\alpha is marked with a vertical dashed red line, and the regionsoutside the Hα\\alpha extent are in red.",
"The horizontal dashed line in the leftpanel is the mass limit for completeness to an age of 500 Myr.",
"The slanted dashedline is a fit by eye to the upper envelope of the cluster distribution.",
"The slantedsolid line shows the slope for a fading relationship in which the minimumobservable mass scales as log\\log mass ∝\\propto 0.69 log\\log age"
],
[
"The H$\\alpha $ /FUV Ratio",
"H$\\alpha $ and FUV emission are often used to trace star formation in galaxies, including dwarfs.",
"However, commonly the H$\\alpha $ emission drops off faster than, and ends before, the FUV emission as one traces star formation into the outer disk (for example, [55], and Fig.",
"REF here).",
"In addition global ratios of H$\\alpha $ /FUV have been found to be a function of galactic surface brightness (for example, [87], [124]).",
"[71], for example, find that the H$\\alpha $ /FUV ratio is lower than expected by a factor of $2-10$ in the nearby 11HUGS galaxies with the lowest SFRs ($<0.003\\,M_\\odot $ yr$^{-1}$ ).",
"The decline of the ratio H$\\alpha $ /FUV with radius in galaxies and variations between galaxies have been the subject of much debate.",
"Causes that have been considered include variations in ([87], [17]) or sampling issues with ([43], [20]) the stellar initial mass function.",
"Other explanations include variations in star formation history, loss of ionizing photons from the galaxy, and undetectability of diffuse H$\\alpha $ emission in outer disks (for example, [55], [56], [72], [28], [105], [134]).",
"Since escape of ionizing photons from galaxies, and preferentially from small galaxies, is believed to have been responsible for the epoch of re-ionization in the early Universe, measuring the amount of leakage has been an important motivation for observations of Lyman continuum emission around galaxies in the nearby and more distant Universe.",
"These observations give us the opportunity to see if leakage of ionizing photons from galaxies or outer disks could explain low H$\\alpha $ /FUV ratios.",
"Lyman continuum escape fractions have been measured of order $6\\%-13\\%$ in compact star-forming galaxies at $z\\sim 0.3$ and Ly$\\alpha $ escape fractions of order $20\\%-40\\%$ ([62]).",
"[111] have placed a limit of $\\le 2.1$ % on the Lyman continuum escape fraction of a sample of most star-forming dwarf galaxies at $z\\sim 1$ , and [61] measured an escape fraction of order 8% in a relatively low-mass star-forming galaxy at $z\\sim 0.3$ .",
"In a sample of four nearby galaxies, [74] detected Lyman continuum in one, yielding an escape fraction of 2.4%, but [140] mapped [Siii]/[Sii] in six nearby dwarf starburst galaxies and found that the fraction of emission that escapes may depend on the orientation of the galaxy to the observer, the morphology of the ISM, and the age and concentration of the starburst producing the emission.",
"Nevertheless, we see that escape fractions are not high enough to explain the lowest ratios of H$\\alpha $ /FUV.",
"On the other hand, [58], in a study of two luminous spirals, suggest that the drop in H$\\alpha $ emission with radius is due to low gas densities in outer disks and the resulting loss of Lyman continuum photons from the vicinity of star forming regions, making them undetectable in H$\\alpha $ , and not from a loss of photons out of the galaxy altogether.",
"Figure: Hα\\alpha images of UGC 5456 displayed on a linear scale to emphasize emission fromdiffuse ionized gas.",
"Contours are at 10σ10\\sigma , 30σ30\\sigma , and 100σ100\\sigma abovethe background.",
"On the left is the standard continuum subtracted narrow band imagefrom the 11HUGS/LVL survey () and on the right is the deeperimage from .",
"Reproduced from with permission from theAASCould we instead be under-estimating the amount of H$\\alpha $ emission that is actually there?",
"To test the idea that significant amounts of H$\\alpha $ emission have been missed in outer disks, [73] performed very deep imaging in H$\\alpha $ of three nearby dwarf galaxies, reaching flux limits of order 10 times lower than that of the 11HUGS/LVL survey ([64]).",
"Their new images (Fig.",
"REF ) do show emission extending up to 2.5 times further than the previous survey data, but this additional emission only contributes $\\sim $ 5% more H$\\alpha $ flux.",
"Therefore, the additional emission found in these deep images does not account for the radial trend in H$\\alpha $ /FUV.",
"The emission measure of individual Hii regions in outer disks can be very low, however, because of the extremely low average density.",
"Following [56], one can consider the possible values for emission measure if the far-outer disks in Fig.",
"REF are completely ionized.",
"The limits of the stellar disks in these galaxies correspond to radii of 60 kpc in NGC 801 ($R/R_{\\rm D}=4.2$ in the figure) and 71 kpc in UGC 2885 ($R/R_{\\rm D}=5.9$ ).",
"The total surface densities at these radii can be used to determine the gas disk thicknesses assuming a velocity dispersion of 10 km s$^{-1}$ .",
"These thicknesses are $T=26.2$ kpc and 11.4 kpc, respectively, if we consider thickness to be two isothermal scale heights.",
"When combined with the Hi surface densities, the corresponding average densities are only $n=0.00052$ cm$^{-3}$ and 0.0031 cm$^{-3}$ .",
"If the entire disk thicknesses were ionized at these densities, then the emission measures would be $n^2T=0.0069$ cm$^{-6}$ pc and 0.11 cm$^{-6}$ pc.",
"We convert emission measure to surface brightness as $\\Sigma _{{\\rm H}\\alpha }({\\rm erg\\,s}^{-1}\\,{\\rm pc}^{-2})=7.7\\times 10^{30}{\\rm EM}({\\rm cm}^{-6}\\,{\\rm pc})$ .",
"Converting this to intensity units, we get $I_{\\rm H\\alpha }({\\rm erg\\,s}^{-1}\\,{\\rm cm}^{-2}\\,{\\rm arcsec}^{-2})=1.5\\times 10^{-18}{\\rm EM}({\\rm cm}^{-6}\\,{\\rm pc})$ .",
"The limit of detection in the very deep survey by [73] was $8\\times 10^{-18}{\\rm erg\\,s}^{-1}\\,{\\rm cm}^{-2}\\,{\\rm arcsec}^{-2}$ , which is still too high to see the fully ionized far-outer disks in Figure REF by a factor of $\\sim 50$ or more."
],
[
"Breaks in Radial Profiles in dIrr Galaxies",
"Figure REF (right) illustrates another common feature of outer dIrr disks: abrupt breaks in azimuthally-averaged surface brightness profiles ([52]).",
"Most often the profile drops in brightness into the outer disk more steeply than in the inner disk (Type II profiles; Sect. )",
"but occasionally it drops less steeply (Type III).",
"Surface brightness profiles without breaks (Type I) are relatively rare.",
"Radial profile breaks are common in spirals as well and were first discovered there ([126], [120], [23], [66], [102], [78], [65], [36], [101], [37], [49]).",
"They are also found in high redshift disks ([99]).",
"[4] and [110] found that the Type II downturn at mid-radius decreases significantly in spirals when stellar mass profiles are considered instead of surface brightness.",
"However, this is not the case for most dIrrs, as found by [53].",
"Thus, $R_{\\rm Br}$ appears to represent a change in stellar population in spirals but a change in stellar surface mass density, at least in part, in dwarfs.",
"Figure: Left: Representative VV and FUV Type II and III surface brightness profiles withparameters for M B =-16M_B = -16 from .VV highlights older stars, and FUV reveals younger stars.The steep FUV slope of the Type III profile interior to R Br R_{\\rm Br} implies an inner accretion trend.The steeper FUV slope in the Type II outskirts isevidence of outside-in shrinking of star formation activity.Right: Number of galaxies with the given fraction of the stellar mass beyond R Br R_{\\rm Br}.Type II profiles with bluing colour trends with radius (IIB)and with reddening colour trends with radius (IIR)are shown separately.",
"Type IIR profiles have a larger fractionof their stellar mass beyond R Br R_{\\rm Br} thanType IIB or Type III[52] examined the surface brightness profiles of 141 dwarfs in up to 11 passbands, and typical Type II and Type III profiles are sketched in Fig.",
"REF (left).",
"[53] further examined the colour and mass surface density trends.",
"They found that, although brighter galaxies tend to have larger $R_{\\rm Br}$ , the surface brightness in $V$ , $\\mu _V$ , at $R_{\\rm Br}$ is about 24$\\pm $ 1 mag arcsec$^{-2}$ , independent of $M_B$ and independent of galaxy type.",
"The $B-V$ colour at $R_{\\rm Br}$ is also nearly constant.",
"However, when surface photometry is converted to stellar mass surface density for Type II profiles, values for dwarfs are a factor of $\\sim $ 6 lower than those for spirals ([53], [4]).",
"When separated by radially averaged colour trends, Type II profiles with reddening colour trends (IIR) have a larger fraction of their stellar mass beyond $R_{\\rm Br}$ than Type IIs with a bluing colour trend (IIB) or Type IIIs ([53]; Fig.",
"REF , right).",
"What is happening at $R_{\\rm Br}$ ?",
"Simulations of spirals by [109], [83], [5], and [90] suggest that the break radius $R_{\\rm Br}$ grows with time and that for Type II profiles stars formed inside $R_{\\rm Br}$ migrate outward beyond $R_{\\rm Br}$ as a result of secular processes involving bar potentials or spiral arms (see observations by [104]).",
"However, scattering of stars from spiral arms is not applicable to dIrrs and observations of some spirals are inconsistent with this scenario as well ([137]).",
"Another possibility is that there is a change in the dominate star formation process or efficiency at $R_{\\rm Br}$ (e.g., [117], [100], [30], [7]; but for models of star formation without a sharp change with radius see [97] and [68]).",
"[109] suggest that, for spirals, it is a combination of a radial star formation cutoff and stellar mass redistribution (see also [142]).",
"The different radial surface brightness and colour profiles in dwarfs can be understood empirically as the result of different evolutionary histories (Fig.",
"REF , left): Type III galaxies are building their centres, perhaps through accretion of gas, while in Type IIR galaxies star formation is retreating to the inner regions of the galaxy (outside-in disk growth as suggested by [141]) and Type IIB galaxies may be systems in which star formation in the inner regions is winding down.",
"Regardless, the near-constant surface brightness and colour at $R_{\\rm Br}$ in dwarfs and spirals argue that whatever is happening at $R_{\\rm Br}$ is common to both types of disk galaxies."
],
[
"Summary",
"The outer parts of spiral and dwarf irregular galaxies usually have a regular structure with an exponentially declining surface brightness in FUV, optical, and near-infrared passbands, somewhat flatter or irregular radial profiles in atomic gas, and frequent evidence for azimuthal asymmetries.",
"Models suggest that these outer parts form by a combination of gas accretion from the halo or beyond, in situ star formation, and stellar scattering from the inner disk.",
"The exponential shape is not well understood, but cosmological simulations get close to exponential shapes by approximate angular momentum conservation.",
"Wet mergers also get exponentials after the combined stellar systems relax, and stellar scattering from gas irregularities and spiral arm corotation resonances get exponentials too, all probably for different reasons.",
"Star formation persists in far outer disks without any qualitative change in the properties of individual star-forming regions.",
"This happens even though the gas density is very low, H$\\alpha $ is often too weak to see, and the dynamical time is long.",
"Gas also tends to dominate stars by mass in the outer parts, but the gas appears to be mostly atomic, making star formation difficult to understand in comparison to inner disks, where it is confined to molecular clouds.",
"Colour and age gradients suggest that most spiral galaxies have their earliest star formation in the inner disk, with a scale length that increases in time and an outward progress of gas depletion or quenching too.",
"The result is a tendency for spirals to get bluer with increasing radius.",
"Eventually the blue trend stops and spirals get redder after that.",
"This gradient change occurs in all types of spiral galaxies, regardless of the exponential shapes of their radial profiles, and suggests a different process for the formation of inner and outer stellar disks.",
"Most likely stellar scattering from the inner disk to the outer disk is part of the explanation, including stellar scattering from bars and spirals, but there could be other processes at work too, including minor mergers and interactions with other galaxies.",
"Colour gradients in dIrrs are usually the opposite of those in spiral galaxies.",
"Dwarfs tend to get systematically redder with radius in what looks like outside-in star formation.",
"This could reflect an enhanced role for stellar scattering with the first star formation still near the centre, as for spirals, or it could result from radial gas accretion or other truly outside-in processes.",
"The advent of new surveys that probe galaxies to very faint stellar surface brightnesses, combined with new maps of atomic and molecular emission from the far-outer regions of galaxies, should help us to better understand the origin and evolution of galaxy disks.",
"DAH appreciates the support of the Lowell Observatory Research Fund in writing this chapter."
]
] | 1612.05615 |
[
[
"A correlation coefficient of belief functions"
],
[
"Abstract How to manage conflict is still an open issue in Dempster-Shafer evidence theory.",
"The correlation coefficient can be used to measure the similarity of evidence in Dempster-Shafer evidence theory.",
"However, existing correlation coefficients of belief functions have some shortcomings.",
"In this paper, a new correlation coefficient is proposed with many desirable properties.",
"One of its applications is to measure the conflict degree among belief functions.",
"Some numerical examples and comparisons demonstrate the effectiveness of the correlation coefficient."
],
[
"Introduction",
"Dempster-Shafer evidence theory (D-S theory) [1], [2] is widely used in many real applications[3], [4], [5], [6], [7], [8] due to its advantages in handling uncertain information, since decision-relevant information is often uncertain in real systems[9], [10], [11].",
"However, in D-S theory, the results with Dempster's combination rule are counterintuitive[12] when the given evidence highly conflict with each other.",
"Until now, how to manage conflict is an open issue in D-S theory.",
"In recent years, hundreds of methods have been proposed to address this issue[13], [14], [15], [16], [17], [18], [6], [19], [20], [21], [22].",
"These solutions are generally divided into two categories: one is to modify the combination rule and redistribute the conflict; the other is to modify the data before combination and keep the combination rule unchanged.",
"Obviously, how to measure the degree of conflict between the evidences is the first step, since we need to know whether the evidence to be combined is in conflict before doing anything in conflict management[23], [24], [25].",
"So far there are no general mechanisms to measure the degree of conflict other than the classical conflict coefficient $k$ .",
"But since the classical conflict coefficient $k$ is the mass of the combined belief assigned to the emptyset and ignores the difference between the focal elements, using $k$ to indicate a conflict between the evidences may be incorrect.",
"Several conflict measures, such as Jousselme's evidence distance[26], Liu's two-dimensional conflict model[27], Song et al.",
"'s conflict measurement based on correlation coefficient[28], have been proposed to measure the conflict in D-S theory.",
"Although some improvements have been made, there are still some shortcomings in the existing conflict measure methods.",
"How to measure the the degree of conflict between the evidences is not yet solved.",
"In D-S theory, the conflict simultaneously contains the non-intersection and the difference among the focal elements[27].",
"Only when these two factors are considered simultaneously, the effective measure of conflict can be realized.",
"In this paper, we try to measure the conflict from the perspective of the relevance of the evidence, based on simultaneously considering the non-intersection and the difference among the focal elements.",
"A correlation coefficient can quantify some types of correlation relationship between two or more random variables or observed data values.",
"In D-S theory, a correlation coefficient is usually used to measure the similarity or relevance of evidence, which can be applied in conflict management, evidence reliability analysis, classification, etc[29], [28].",
"Recently, various types of correlation coefficient are presented.",
"For example, in [30], a correlation coefficient was introduced to calculate the similarity of template data and detected data, then the basic probability assignments (BPAs) were obtained based on classification results for fault diagnosis.",
"In [29], a correlation coefficient is proposed based on the fuzzy nearness to characterize the divergence degree between two basic probability assignments (BPAs).",
"In [28], Song et al.",
"defined a correlation coefficient to measure the conflict degree of evidences.",
"Moreover, some different correlation coefficients are proposed respectively according to specific applications [31], [32], [30].",
"In this paper, a new correlation coefficient, which takes into consider both the non-intersection and the difference among the focal elements, is proposed.",
"One of its applications is to measure the conflict degree among belief functions.",
"Based on this, a new conflict coefficient is defined.",
"Some numerical examples illustrate that the proposed correlation coefficient could effectively measure the conflict degree among belief functions.",
"The paper is organised as follows.",
"In section 2, the preliminaries D-S theory and the existing conflicting measurement are briefly introduced.",
"Section 3 presents the new correlation coefficient and proofs many desirable properties.",
"In section 4, some numerical examples are illustrated to show the efficiency of the proposed coefficient.",
"Finally, a brief conclusion is made in Section 5."
],
[
"Preliminaries",
"In this section, some preliminaries are briefly introduced."
],
[
"Dempster-Shafer evidence theory",
"D-S theory was introduced by Dempster [1], then developed by Shafer [2].",
"Owing to its outstanding performance in uncertainty model and process, this theory is widely used in many fields [33], [34], [35].",
"Definition 2.1 Let $\\Theta = \\lbrace {\\theta _1},{\\theta _2}, \\cdots {\\theta _i}, \\cdots ,{\\theta _N}\\rbrace $ be a finite nonempty set of mutually exclusive hypothesises, called a discernment frame.",
"The power set of $\\Theta $ , ${2^{\\Theta }}$ , is indicated as: ${2^{\\Theta }} = \\lbrace \\emptyset ,\\lbrace {\\theta _1}\\rbrace , \\cdots \\lbrace {\\theta _N}\\rbrace ,\\lbrace {\\theta _1},{\\theta _2}\\rbrace , \\cdots ,\\lbrace {\\theta _1},{\\theta _2}, \\cdots {\\theta _i}\\rbrace , \\cdots ,\\Theta \\rbrace $ Definition 2.2 A mass function is a mapping $\\mathbf {m}$ from ${2^{\\Theta }}$ to [0,1], formally noted by: ${\\bf {m}}:{2^{\\Theta }} \\rightarrow [0,1]$ which satisfies the following condition: $\\begin{array}{*{20}c}{m(\\emptyset ) = 0} & {and} & {\\sum \\limits _{A \\in {2^{\\Theta }} } {m(A)} = 1} \\\\\\end{array}$ When m(A)$>$ 0, A is called a focal element of the mass function.",
"In D-S theory, a mass function is also called a basic probability assignment (BPA).",
"Given a piece of evidence with a belief between [0,1], noted by $m(\\cdot )$ , is assigned to the subset of $\\Theta $ .",
"The value of 0 means no belief in a hypothesis, while the value of 1 means a total belief.",
"And a value between [0,1] indicates partial belief.",
"Definition 2.3 Evidence combination in D-S theory is noted as $\\oplus $ .",
"Assume that there are two BPAs indicated by $m_1$ and $m_2$ , the evidence combination of the two BPAs with Dempster's combination rule [1] is formulated as follows: $ m(A) = \\left\\lbrace {\\begin{array}{*{20}{c}}{0\\begin{array}{*{20}{c}},&{}\\end{array}\\begin{array}{*{20}{c}}{}&{}&{\\begin{array}{*{20}{c}}{}&{}&{}\\end{array}}\\end{array}}\\\\{\\frac{1}{{1 - k}}\\sum \\limits _{B \\cap C = A} {{m_1}(B){m_2}(C)} }\\end{array}\\begin{array}{*{20}{c}}{}\\\\{}\\end{array}} \\right.\\begin{array}{*{20}{c}}{}\\\\{}\\end{array}\\begin{array}{*{20}{c}}{A = {\\emptyset } }\\\\{A \\ne {\\emptyset } }\\end{array}$ with $k = \\sum \\limits _{B \\cap C = \\emptyset } {{m_1}(B){m_2}(C)}$ Where $k$ is a normalization constant, called conflict coefficient because it measures the degree of conflict between $m_1$ and $m_2$ .",
"$k=0$ corresponds to the absence of conflict between $m_1$ and $m_2$ , whereas $k=1$ implies complete contradiction between $m_1$ and $m_2$ .",
"Note that the Dempster's rule of combination is only applicable to such two BPAs which satisfy the condition $k<1$ ."
],
[
"Evidence distance",
"Jousselme et al.",
"[26] proposed a distance measure for evidence.",
"Definition 2.4 Let $m_1$ and $m_2$ be two BPAs on the same frame of discernment $\\Theta $ , containing N mutually exclusive and exhaustive hypotheses.",
"The distance between $m_1$ and $m_2$ is represented by: ${d_{BBA}}({m_1},{m_2}) = \\sqrt{\\frac{1}{2}{{({{\\overrightarrow{m} }_1} - {{\\overrightarrow{m} }_2})}^T}\\mathop D\\limits _ = ({{\\overrightarrow{m} }_1} - {{\\overrightarrow{m} }_2})}$ Where ${{{\\overrightarrow{m} }_1}}$ and ${{{\\overrightarrow{m} }_2}}$ are the respective BPAs in vector notation, and ${\\mathop D\\limits _ = }$ is an ${2^N} \\times {2^N}$ matrix whose elements are $D(A,B) = \\frac{{\\left| {A \\cap B} \\right|}}{{\\left| {A \\cup B} \\right|}}$ , where $A,B \\in 2^\\Theta $ are derived from $m_1$ and $m_2$ , respectively."
],
[
"Pignistic probability distance",
"In the transferable belief model(TBM) [36], pignistic probabilities are typically used to make decisions and pignistic probability distance can be used to measure the difference between two bodies of evidence.",
"Definition 2.5 Let $m$ be a BPA on the frame of discernment $\\Theta $ .",
"Its associated pignistic probability transformation (PPT) $BetP_m$ is defined as $BetP_m (\\omega ) = \\sum \\limits _{A \\in {2^{\\Theta }} ,\\omega \\in A} {\\frac{1}{{\\left| A \\right|}}} \\frac{{m(A)}}{{1 - m(\\emptyset )}}$ where $\\left| A \\right|$ is the cardinality of subset $A$ .",
"The PPT process transforms basic probability assignments to probability distributions.",
"Therefore, the pignistic betting distance can be easily obtained using the PPT.",
"Definition 2.6 Let $m_1$ and $m_2$ be two BPAs on the same frame of discernment $\\Theta $ and let $BetP_{m_1}$ and $BetP_{m_2}$ be the results of two pignistic transformations from them respectively.",
"Then the pignistic probability distance between $BetP_{m_1}$ and $BetP_{m_2}$ is defined as $difBetP = \\mathop {\\max }\\limits _{A \\in {2^{\\Theta }} } (\\left| {BetP_{m_1 } (A) - BetP_{m_2 } (A)} \\right|)$"
],
[
"Liu's conflict model",
"In [27], Liu noted that the classical conflict coefficient $k$ cannot effectively measure the degree of conflict between two bodies of evidence.",
"A two-dimensional conflict model is proposed by Liu [27], in which the pignistic betting distance and the conflict coefficient $k$ are united to represent the degree of conflict.",
"Definition 2.7 Let $m_1$ and $m_2$ be two BPAs on the same frame of discernment $\\Theta $ .",
"The two-dimensional conflict model is represented by: $cf(m_1 ,m_2 ) = \\langle k,difBetP\\rangle $ Where $k$ is the classical conflict coefficient of Dempster combination rule in Eq.",
"(REF ), and $difBetP$ is the pignistic betting distance in Eq.",
"(REF ).",
"Iff both $k>\\varepsilon $ and $difBetP>\\varepsilon $ , $m_1$ and $m_2$ are defined as in conflict, where $\\varepsilon $ is the threshold of conflict tolerance.",
"Liu's conflict model simultaneously considers two parameters to realize conflict management.",
"To some extent, the two-dimensional conflict model could effectively discriminate the degree of conflict.",
"But in most cases, an accurate value, which represents the degree of conflict, is needed for the following process, such as evaluate the reliability of the evidence with assigning different weights."
],
[
"Correlation coefficient of evidence",
"In [28], Song et al.",
"proposed a correlation coefficient for the relativity between two BPAs, which can be used to measure the conflict between two BPAs.",
"Definition 2.8 Let $m_1$ and $m_2$ be two BPAs on the same frame of discernment $\\Theta $ , containing N mutually exclusive and exhaustive hypotheses.",
"Use the Jaccard matrix $D$ , defined in Eq.",
"(REF ), to modify the BPA: $\\left\\lbrace {\\begin{array}{*{20}c}{m^{\\prime }_1 = m_1 D} \\\\{m^{\\prime }_2 = m_2 D} \\\\\\end{array}} \\right.$ Then the correlation coefficient between two bodies of evidence is defined as: $cor(m_1 ,m_2 ) = \\frac{{\\langle m^{\\prime }_1 ,m^{\\prime }_2 \\rangle }}{{\\left\\Vert {m^{\\prime }_1 } \\right\\Vert \\cdot \\left\\Vert {m^{\\prime }_2 } \\right\\Vert }}$ where ${\\langle m^{\\prime }_1 ,m^{\\prime }_2 \\rangle }$ is the inner product of vectors, ${\\left\\Vert {m^{\\prime }_1 } \\right\\Vert }$ is the norm of vector.",
"Song et al.",
"'s correlation coefficient measures the degree of relevance between two bodies of evidence: the higher the conflict is, the lower the value of the correlation coefficient is.",
"In Song et al.",
"'s correlation coefficient, the Jaccard matrix $D$ is used to modify BPA in order to process BPA including the multi-element subsets.",
"But this modification will repeatedly allocate the belief value of BPA , so the modified BPA does not satisfy the condition $\\sum \\limits _{A \\in {2^{\\Theta }}} {m(A)} = 1$ .",
"Thus, Song et al.",
"'s correlation coefficient could not satisfy the property $cor(m_1 ,m_2 ) = 1 \\Leftrightarrow m_1 {\\rm { = }}m_2$ and sometimes will yield incorrect results."
],
[
"A new correlation coefficient",
"The nature of conflict between two BPAs is there exists the difference between the beliefs of two bodies of evidence on the same focal elements, so the conflict could be quantified by the relevance between two bodies of evidence.",
"If the value of the relevance between two bodies of evidence is higher, the degree of the similarity between two bodies of evidence is higher and the degree of conflict between two bodies of evidence is lower; Conversely, if the value of the relevance between two bodies of evidence is lower, the degree of the similarity between two bodies of evidence is lower and the degree of conflict between two bodies of evidence is higher.",
"In order to measure the degree of relevance between two bodies of evidence, a new correlation coefficient, which considers both the non-intersection and the difference among the focal elements, is proposed.",
"Firstly, some desirable properties for correlation coefficient are shown as follows.",
"Definition 3.1 Assume $m_1,\\ m_2$ are two BPAs on the same discernment frame $\\Theta $ , $r_{BPA}(m_1,m_2)$ is denoted as a correlation coefficient for two BPAs, then $r_{BPA}(m_1,m_2)=r_{BPA}(m_2,m_1)$ ; $0 \\le r_{BPA}(m_1,m_2)\\le 1$ ; if $m_1=m_2$ , $r_{BPA}(m_1,m_2)=1$ ; $r_{BPA}(m_1,m_2)=0\\Leftrightarrow (\\bigcup A_i)\\bigcap (\\bigcup A_j)=\\emptyset $ , $A_i,\\ A_j$ is the focal element of $m_1,\\ m_2$ , respectively.",
"In D-S theory, a new correlation coefficient is defined as follows.",
"Definition 3.2 For a discernment frame $\\Theta $ with $N$ elements, suppose the mass of two pieces of evidence denoted by $m_1$ , $m_2$ .",
"A correlation coefficient is defined as: $r_{BPA}\\left( {{m_1},{m_2}} \\right) = \\frac{{c\\left( {{m_1},{m_2}} \\right)}}{{\\sqrt{c\\left( {{m_1},{m_1}} \\right) \\cdot c\\left( {{m_2},{m_2}} \\right)} }}$ Where $c(m_1,m_2)$ is the degree of correlation denoted as: $c({m_1},{m_2}) = \\sum \\limits _{i = 1}^{{2^N}} {\\sum \\limits _{j = 1}^{{2^N}} {{m_1}({A_i}){m_2}({A_j})\\frac{{\\left| {{A_i} \\cap {A_j}} \\right|}}{{\\left| {{A_i} \\cup {A_j}} \\right|}}} }$ Where $i,j=1,\\dots , 2^N$ ;$A_i$ , $A_j$ is the focal elements of mass, respectively; and $\\left| \\cdot \\right|$ is the cardinality of a subset.",
"The correlation coefficient $r_{BPA}\\left( {{m_1},{m_2}} \\right)$ measures the relevance between $m_1$ and $m_2$ .",
"The larger the correlation coefficient, the high the relevance between $m_1$ and $m_2$ .",
"$r_{BPA}=0$ corresponds to the absence of relevance between $m_1$ and $m_2$ , whereas $r_{BPA}=1$ implies $m_1$ and $m_2$ complete relevant, that is, $m_1$ and $m_2$ are identical.",
"In the following, the mathematical proofs are given to illustrate that the proposed correlation coefficient satisfies all desirable properties defined in Definition REF .",
"Before the proofs, a lemma is introduced as follows: Lemma 3.1 For the vector 2-norm, $\\forall $ non-zero vector $\\xi = (\\xi _1,\\xi _2,\\ldots ,\\xi _n)^T,$ $\\eta =(\\eta _1,\\eta _2,\\ldots ,\\eta _n)^T$ , the condition for the equality in triangle inequality $||\\xi +\\eta ||_2\\le ||\\xi ||_2+||\\eta ||_2$ is if and only if $\\xi =k\\eta $ .",
"Proof $||\\xi +\\eta ||_2^2 &\\le (||\\xi ||_2+||\\eta ||_2)^2 \\\\(\\xi _1+\\eta _1)^2+(\\xi _2+\\eta _2)^2+\\ldots +(\\xi _n+\\eta _n)^2 &\\le (\\sqrt{\\xi _1^2+\\xi _2^2+\\ldots +\\xi _n^2}+\\sqrt{\\eta _1^2+\\eta _2^2+\\ldots +\\eta _n^2})^2\\\\\\xi _1\\eta _1+\\xi _2\\eta _2+\\ldots +\\xi _n\\eta _n &\\le \\sqrt{(\\xi _1^2+\\xi _2^2+\\ldots +\\xi _n^2)(\\eta _1^2+\\eta _2^2+\\ldots +\\eta _n^2)}$ From the Cauchy-Buniakowsky-Schwarz Inequality, we can see that the condition of the equality is if and only if $\\frac{\\xi _1}{\\eta _1}=\\frac{\\xi _2}{\\eta _2}=\\ldots =\\frac{\\xi _n}{\\eta _n}=k$ , namely $\\xi =k\\eta $ .$\\Box $ The proofs are details as follows: Proof Sort the $2^N$ subsets in $\\Theta $ as $\\lbrace \\emptyset ,a,b,\\ldots ,N,ab,ac,\\ldots ,abc,\\ldots \\rbrace $ , then $m_1,m_2$ are arranged in column vectors in this order, ${{\\bf {m}}_1}:x=(m_1(\\emptyset ),m_1(a),m_1(b),\\ldots ,m_1(N),m_1(ab),m_1(ac),\\ldots )^T,$ ${{\\bf {m}}_2}:y=(m_2(\\emptyset ),m_2(a),m_2(b),\\ldots ,m_2(N),m_2(ab),m_2(ac),\\ldots )^T.$ Let $D=\\frac{|A_i\\cap A_j|}{|A_i\\cup A_j|}$ , where $A_i,\\ A_j\\in 2^\\Theta $ and the subsets are arranged in the same order as described above.",
"$D$ is positive definite so $\\exists C\\in R^{2^N\\times 2^N}_{2^N}$ and satisfies $D=C^TC$ .",
"Then it is obvious that $c(m_1,m_1)=x^TDx=x^TC^TCx$ , and similarly $c(m_1,m_2)=x^TC^TCy$ , $c(m_2,m_2)=y^TC^TCy$ .",
"$r_{BPA}(m_1,m_2)=\\frac{x^TC^TCy}{\\sqrt{x^TC^TCx}\\sqrt{y^TC^TCy}}$ .",
"$r_{BPA}(m_1,m_2)=\\frac{x^TC^TCy}{\\sqrt{x^TC^TCx}\\sqrt{y^TC^TCy}}$ , $r_{BPA}(m_2,m_1)=\\frac{y^TC^TCx}{\\sqrt{x^TC^TCx}\\sqrt{y^TC^TCy}}$ .",
"Because $x^TC^TCy$ is a real number, $x^TC^TCy=(x^TC^TCy)^T=y^TC^TCx$ .",
"Thereby, $r_{BPA}(m_1,m_2)=r_{BPA}(m_2,m_1)$ .",
"All the elements in $x,\\ y,\\ D$ are non-negative real numbers, so it is clear that $r_ {BPA}(m_1,m_2)=\\frac{x^TDy}{\\sqrt{x^TDx}\\sqrt{y^TDy}}\\ge 0$ .",
"Note that $x^TC^TCx=(Cx)^T(Cx)=||Cx||_2^2$ , then using the trigonometric inequality on the vector 2-norm for vector $Cx,\\ Cy$ , the following inequalities are obtained.",
"$||C(x+y)||_2^2 &\\le (||Cx||_2+||Cy||_2)^2 \\\\(x+y)^TC^TC(x+y) &\\le (\\sqrt{x^TC^TCx}+\\sqrt{y^TC^TCy})^2 \\\\x^TC^TCx+x^TC^TCy+y^TC^TCx+y^TC^TCy &\\le x^TC^TCx+y^TC^TCy+2\\sqrt{x^TC^TCxy^TC^TCy} \\\\x^TC^TCy &\\le \\sqrt{x^TC^TCxy^TC^TCy}\\ \\ (x^TC^TCy=y^TC^TCx)$ Accordingly, $r_ {BPA}(m_1,m_2)=\\frac{x^TC^TCy}{\\sqrt{x^TC^TCx}\\sqrt{y^TC^TCy}} \\le 1.$ $r_{BPA}(m_1,m_2)=1$ , and now $x^TC^TCy=\\sqrt{x^TC^TCxy^TC^TCy}$ , that is $||C(x+y)|| _2=||Cx||_2+||Cy||_2$ .",
"We can get $Cx=kCy$ from Lemma REF .",
"$C$ is an invertible matrix, so $x=ky$ .",
"The vectors $x$ and $y$ each represent a BPA, their length are both 1, thus $x=y$ , $m_1=m_2$ .",
"If $r_{BPA}(m_1,m_2)=\\frac{c(m_1,m_2)}{\\sqrt{c(m_1,m_1)\\times c(m_2,m_2)}}=0$ , then $c(m_1,m_2)=\\sum _{i=1}^{2^N}\\sum _{j=1}^{2^N}m_1(A_i)m_2(A_j)\\frac{|A_i\\cap A_j|}{|A_i\\cup A_j|}=0.$ The above formula shows if $m_1(A_i)m_2(A_j)\\ne 0$ , namely $A_i,\\ A_j$ are the focal elements of $m_1,\\ m_2$ , respectively, $|A_i\\cap A_j|=0$ must occur.",
"In other words, $\\forall A_i\\cap \\forall A_j=\\emptyset $ when $A_i,\\ A_j$ are the focal elements of $m_1,\\ m_2$ , respectively.",
"Thereby, $(\\bigcup A_i)\\bigcap (\\bigcup A_j)=\\emptyset $ and vice versa.",
"$\\Box $ In summary, the new correlation coefficient satisfies all the desirable properties and could measure the relevance between two bodies of evidence.",
"Based on this, a new conflict coefficient is proposed.",
"Definition 3.3 For a discernment frame $\\Theta $ with $N$ elements, suppose the mass of two pieces of evidence denoted by $m_1$ , $m_2$ .",
"A new conflict coefficient between two bodies of evidence ${k_r}$ is defined with the proposed correlation coefficient as: $\\begin{array}{*{20}{l}}{{k_r}({m_1},{m_2})}& = &{1 - {r_{BPA}}({m_1},{m_2})}\\\\{}& = &{1 - \\frac{{c({m_1},{m_2})}}{{\\sqrt{c({m_1},{m_1}) \\cdot c({m_2},{m_2})} }}}\\end{array}$ The conflict coefficient ${{k_r}({m_1},{m_2})}$ measures the degree of conflict between $m_1$ and $m_2$ .",
"The larger the conflict coefficient, the high the degree of conflict between $m_1$ and $m_2$ .",
"$k_r=0$ corresponds to the absence of conflict between $m_1$ and $m_2$ , that is, $m_1$ and $m_2$ are identical, whereas $k_r=1$ implies complete contradiction between $m_1$ and $m_2$ ."
],
[
"Numerical examples",
"In this section, we use some numerical examples to demonstrate the effectiveness of the proposed conflict coefficient.",
"Example 1.",
"Suppose the discernment frame is $\\Theta =\\lbrace A_1,A_2,A_3,A_4\\rbrace $ , two bodies of evidence are defined as following: $\\begin{array}{l}{\\bf {m}}_{\\bf {1}} :m_1 (A_1 ,A_2 ) = 0.9,m_1 (A_3 ) = 0.1,m_1 (A_4 ) = 0.0 \\\\{\\bf {m}}_2 :m_2 (A_1 ,A_2 ) = 0.0,m_2 (A_3 ) = 0.1,m_2 (A_4 ) = 0.9 \\\\\\end{array}$ The various conflict measure value are calculated as follows: The classical conflict coefficient [1] $k=0.99$ .",
"Jousselme's evidence distance [26] $d_{BPA}=0.9$ .",
"Song et al.",
"'s correlation coefficient [28] $cor=0.3668$ .",
"The proposed correlation coefficient $r_{BPA}=0.0122$ .",
"The proposed conflict coefficient $k_r=1-r_{BPA}=0.9878$ .",
"In this example, the first BPA is almost certain that the true hypothesis is either $A_1$ or $A_2$ , whilst the second BPA is almost certain that the true hypothesis is $A_4$ .",
"Hence these two BPAs largely contradict with each other, that is, the two BPAs are in high conflict.",
"The results of the classical conflict coefficient $k$ , Jousselme's evidence distance $d_{BPA}$ and the proposed conflict coefficient $k_r$ are all consistent with the fact, but Song et al.",
"'s correlation coefficient $cor=0.3668$ , which means there are a relatively great correlation between these two BPAs.",
"The value of Song et al.",
"'s correlation coefficient is unreasonable.",
"Let us discuss the condition when two BPAs totally contradict.",
"In Example 1, if we revise the two BPAs as: $\\begin{array}{l}{\\bf {m^{\\prime }}}_{\\bf {1}} :m^{\\prime }_1 (A_1 ,A_2 ) = 1.0,m^{\\prime }_1 (A_3 ) = 0.0,m^{\\prime }_1 (A_4 ) = 0.0 \\\\{\\bf {m^{\\prime }}}_2 :m^{\\prime }_2 (A_1 ,A_2 ) = 0.0,m^{\\prime }_2 (A_3 ) = 0.0,m^{\\prime }_2 (A_4 ) = 1.0 \\\\\\end{array}$ then these two BPAs totally contradict with each other.",
"This is the situation when the maximal conflict occurs.",
"The results are shown as follows: The classical conflict coefficient [1] $k=1.0$ .",
"Jousselme's evidence distance [26] $d_{BPA}=1.0$ .",
"Song et al.",
"'s correlation coefficient [28] $cor=0.3229$ .",
"The proposed correlation coefficient $r_{BPA}=0.0$ .",
"The proposed conflict coefficient $k_r=1-r_{BPA}=1.0$ .",
"In this case, the values of $k$ , $d_{BPA}$ , and $K_r$ are all equal to 1.0, which indicates that these two BPAs are in total conflict, whilst $cor=0.3229$ , which means there are some correlation between the two BPAs.",
"The value of Song et al.",
"'s correlation coefficient is unreasonable.",
"Let us continue to discuss the following two pairs of BPAs.",
"Example 2.",
"Suppose the discernment frame is $\\Theta =\\lbrace A_1,A_2,A_3,A_4,A_5,A_6\\rbrace $ , two pairs of BPAs are defined as following: $\\begin{array}{l}{\\bf {m}}_{\\bf {1}} :m_1 (A_1 ) = 0.5,m_1 (A_2 ) = 0.5,m_1 (A_3 ) = 0.0,m_1 (A_4 ) = 0.0 \\\\{\\bf {m}}_{\\bf {2}} :m_2 (A_1 ) = 0.0,m_2 (A_2 ) = 0.0,m_2 (A_3 ) = 0.5,m_2 (A_4 ) = 0.5 \\\\\\end{array}$ $\\begin{array}{l}{\\bf {m}}_3 :m_3 (A_1 ) = 1/3,m_3 (A_2 ) = 1/3,m_3 (A_3 ) = 1/3,m_3 (A_4 ) = 0.0,m_3 (A_5 ) = 0.0,m_3 (A_6 ) = 0.0 \\\\{\\bf {m}}_4 :m_4 (A_1 ) = 0.0,m_4 (A_2 ) = 0.0,m_4 (A_3 ) = 0.0,m_4 (A_4 ) = 1/3,m_4 (A_5 ) = 1/3,m_4 (A_6 ) = 1/3 \\\\\\end{array}$ The summary of $k$ , $d_{BPA}$ , $cor$ , and $k_r$ values of the two pairs of BPAs is shown in Table REF .",
"Obviously, these two pairs of BPAs both totally contradict with each other.",
"$k$ , and $k_r$ values are equal to 1.0, which is consistent with the fact, whilst $d_{BPA}$ , and $cor$ values show that there is some similarity or relevance between the two BPAs, that is, these two BPAs are not in total conflict, especially $cor=0.5606$ for $m_3,m_4$ shows that there is high relevance between $m_3$ and $m_4$ .",
"$d_{BPA}$ , and $cor$ values are unreasonable.",
"In summary, Jousselme's evidence distance and Song et al.",
"'s correlation coefficient could not always give us the correct conflict measurement.",
"Table: Comparison of kk, d BPA d_{BPA}, corcor, and k r k_r values of the two pairs of BPAs in Example 2On the other hand, the total absence of conflict occurs when two BPAs are identical.",
"In this situation, whatever supported by one BPA is equally supported by the other BPA and there is no slightest difference in their beliefs.",
"The following example is two identical BPAs.",
"Example 3.",
"Suppose the discernment frame is $\\Theta =\\lbrace A_1,A_2,A_3,A_4,A_5\\rbrace $ , two bodies of evidence are defined as following: $\\begin{array}{l}{\\bf {m}}_{\\bf {1}} :m_1 (A_1 ) = 0.2,m_1 (A_2 ) = 0.2,m_1 (A_3 ) = 0.2,m_1 (A_4 ) = 0.2,m_1 (A_5 ) = 0.2 \\\\{\\bf {m}}_2 :m_2 (A_1 ) = 0.2,m_2 (A_2 ) = 0.2,m_2 (A_3 ) = 0.2,m_2 (A_4 ) = 0.2,m_2 (A_5 ) = 0.2 \\\\\\end{array}$ In this case, $k_r=0$ , $d_{BPA}=0$ , and $cor=1$ all indicates that the two BPAs are identical, which are consistent with the fact.",
"But the classical conflict coefficient $k=0.8$ , which indicates that these two BPAs are in high conflict.",
"Obviously, the value of $k=0.8$ is incorrect.",
"With regard to this example, using $k$ as a quantitative measure of conflict is not always suitable.",
"According the above examples, it can be concluded that the proposed correlation coefficient and the proposed conflict coefficient always give the correct quantitative measure of relevance and conflict between two bodies of evidence.",
"In order to further verify the effectiveness of the proposed method, consider the following example: Example 4.",
"Suppose the number of elements in the discernment frame is 20, such as $\\Theta =\\lbrace 1,2,3,4,...,20\\rbrace $ , two bodies of evidence are defined as following: ${{\\bf {m}}_1}:{m_1}(2,3,4) = 0.05 ,{m_1}(7) = 0.05 ,{m_1}(\\Theta ) = 0.1, {m_1}({\\rm A}) = 0.8$ ${{\\bf {m}}_2}:{m_2}(1,2,3,4,5) = 1$ where the A is a variable set taking values as follow: {1},{1,2} {1,2,3},{1,2,3,4},...,{1,2,3,4,...,19,20} .",
"In terms of conflict analysis, the comparative behavior of the two BPAs are shown in Table REF and Fig.",
"REF .",
"From which we can find the proposed conflict coefficient $k_r$ adopts the similar behavior as Jousselme's evidence distance, that is, when set A tends to the set {1,2,3,4,5}, both the values of $k_r$ and $d_{BPA}$ tend to their minimum.",
"On the contrary, the two values will increase when the set A departs from the set {1,2,3,4,5}.",
"Fig.",
"REF shows that the trends of $k_r$ , and $d_{BPA}$ value are consistent with intuitive analysis, when the size of set A changes, whilst the classical conflict coefficient $k$ fails to differentiate the changes of evidence.",
"So both the proposed conflict coefficient and Jousselme's evidence distance are appropriate to measure the conflict degree of evidence in this example.",
"However, the major drawback of $d_{BPA}$ is its inability to full consider the non-intersection among the focal elements of BPAs.",
"In Example 2, Jousselme's evidence distance can not give us the correct conflict measurement.",
"As to the classical conflict coefficient $k$ , it only takes into consider the non-intersection of the focal elements, but not the difference among the focal elements.",
"Because of the lack of information, $k$ is not sufficient as the quantitative measure of conflict between two BPAs.",
"Compared with Jousselme's evidence distance and the classical conflict coefficient, the proposed correlation coefficient takes into consider both the non-intersection and the difference among the focal elements of BPAs.",
"Therefore, to some extent, the proposed correlation coefficient combine the classical conflict coefficient and Jousselme's evidence distance, and overcome their respective demerits.",
"Thus, the proposed method can measure the degree of relevance and conflict between belief functions correctly and effectively.",
"Table: Comparison of k r k_r, d BPA d_{BPA}, and kk values in Example 4Figure: Comparison of correlation degree"
],
[
"Conclusions",
"In D-S theory, it is necessary to measure the conflicts of belief functions.",
"A correlation coefficient provides a promising way to address the issue.",
"In this paper, a new correlation coefficient of belief functions is presented.",
"It can overcome the drawbacks of the existing methods.",
"Numerical examples in conflicting management are illustrated to show the efficiency of the proposed correlation coefficient of belief functions."
],
[
"Acknowledgment",
"The work is partially supported by National Natural Science Foundation of China (Grant No.",
"61671384), Natural Science Basic Research Plan in Shaanxi Province of China (Program No.",
"2016JM6018), the Fund of Shanghai Aerospace Science and Technology (Program No.",
"SAST2016083)."
]
] | 1612.05497 |
[
[
"Edge-exchangeable graphs and sparsity (NIPS 2016)"
],
[
"Abstract Many popular network models rely on the assumption of (vertex) exchangeability, in which the distribution of the graph is invariant to relabelings of the vertices.",
"However, the Aldous-Hoover theorem guarantees that these graphs are dense or empty with probability one, whereas many real-world graphs are sparse.",
"We present an alternative notion of exchangeability for random graphs, which we call edge exchangeability, in which the distribution of a graph sequence is invariant to the order of the edges.",
"We demonstrate that edge-exchangeable models, unlike models that are traditionally vertex exchangeable, can exhibit sparsity.",
"To do so, we outline a general framework for graph generative models; by contrast to the pioneering work of Caron and Fox (2015), models within our framework are stationary across steps of the graph sequence.",
"In particular, our model grows the graph by instantiating more latent atoms of a single random measure as the dataset size increases, rather than adding new atoms to the measure."
],
[
"Introduction",
"In recent years, network data have appeared in a growing number of applications, such as online social networks, biological networks, and networks representing communication patterns.",
"As a result, there is growing interest in developing models for such data and studying their properties.",
"Crucially, individual network data sets also continue to increase in size; we typically assume that the number of vertices is unbounded as time progresses.",
"We say a graph sequence is dense if the number of edges grows quadratically in the number of vertices, and a graph sequence is sparse if the number of edges grows sub-quadratically as a function of the number of vertices.",
"Sparse graph sequences are more representative of real-world graph behavior.",
"However, many popular network models (see, e.g., [37] for an extensive list) share the undesirable scaling property that they yield dense sequences of graphs with probability one.",
"The poor scaling properties of these models can be traced back to a seemingly innocent assumption: that the vertices in the model are exchangeable, that is, any finite permutation of the rows and columns of the graph adjacency matrix does not change the distribution of the graph.",
"Under this assumption, the Aldous-Hoover theorem [2], [31] implies that such models generate dense or empty graphs with probability one [39].",
"This fundamental model misspecification motivates the development of new models that can achieve sparsity.",
"One recent focus has been on models in which an additional parameter is employed to uniformly decrease the probabilities of edges as the network grows (e.g., [5], [6], [47], [7]).",
"While these models allow sparse graph sequences, the sequences are no longer projective.",
"In projective sequences, vertices and edges are added to a graph as a graph sequence progresses—whereas in the models above, there is not generally any strict subgraph relationship between earlier graphs and later graphs in the sequence.",
"Projectivity is natural in streaming modeling.",
"For instance, we may wish to capture new users joining a social network and new connections being made among existing users—or new employees joining a company and new communications between existing employees.",
"[19] have pioneered initial work on sparse, projective graph sequences.",
"Instead of the vertex exchangeability that yields the Aldous-Hoover theorem, they consider a notion of graph exchangeability based on the idea of independent increments of subordinators [33], explored in depth by [45].",
"However, since this Kallenberg-style exchangeability introduces a new countable infinity of latent vertices at every step in the graph sequence, its generative mechanism seems particularly suited to the non-stationary domain.",
"By contrast, we are here interested in exploring stationary models that grow in complexity with the size of the data set.",
"Consider classic Bayesian nonparametric models as the Chinese restaurant process (CRP) and Indian buffet process (IBP); these engender growth by using a single infinite latent collection of parameters to generate a finite but growing set of instantiated parameters.",
"Similarly, we propose a framework that uses a single infinite latent collection of vertices to generate a finite but growing set of vertices that participate in edges and thereby in the network.",
"We believe our framework will be a useful component in more complex, non-stationary graphical models—just as the CRP and IBP are often combined with hidden Markov models or other explicit non-stationary mechanisms.",
"Additionally, Kallenberg exchangeability is intimately tied to continuous-valued labels of the vertices, and here we are interested in providing a characterization of the graph sequence based solely on its topology.",
"In this work, we introduce a new form of exchangeability, distinct from both vertex exchangeability and Kallenberg exchangeability.",
"In particular, we say that a graph sequence is edge exchangeable if the distribution of any graph in the sequence is invariant to the order in which edges arrive—rather than the order of the vertices.",
"We will demonstrate that edge exchangeability admits a large family of sparse, projective graph sequences.",
"In the remainder of the paper, we start by defining dense and sparse graph sequences rigorously.",
"We review vertex exchangeability before introducing our new notion of edge exchangeability in sec:exchangeability, which we also contrast with Kallenberg exchangeability in more detail in sec:relwork.",
"We define a family of models, which we call graph frequency models, based on random measures in sec:generative.",
"We use these models to show that edge-exchangeable models can yield sparse, projective graph sequences via theoretical analysis in sec:poissp and via simulations in sec:simulations.",
"Along the way, we highlight other benefits of the edge exchangeability and graph frequency model frameworks."
],
[
"Exchangeability in graphs: old and new",
"Let $(G_n)_n := G_1, G_2, \\ldots $ be a sequence of graphs, where each graph $G_n = (V_n, E_n)$ consists of a (finite) set of vertices $V_n$ and a (finite) multiset of edges $E_n$ .",
"Each edge $e \\in E_n$ is a set of two vertices in $V_n$ .",
"We assume the sequence is projective—or growing—so that $V_{n} \\subseteq V_{n+1}$ and $E_{n} \\subseteq E_{n+1}$ .",
"Consider, e.g., a social network with more users joining the network and making new connections with existing users.",
"We say that a graph sequence is dense if $|E_n| = \\Omega (|V_n|^2)$ , i.e., the number of edges is asymptotically lower bounded by $c \\cdot |V_n|^2$ for some constant $c$ .",
"Conversely, a sequence is sparse if $|E_n| = o(|V_n|^2)$ , i.e., the number of edges is asymptotically upper bounded by $c \\cdot |V_n|^2$ for all constants $c$ .",
"In what follows, we consider random graph sequences, and we focus on the case where $|V_n| \\rightarrow \\infty $ almost surely."
],
[
"Vertex-exchangeable graph sequences",
"If the number of vertices in the graph sequence grows to infinity, the graphs in the sequence can be thought of as subgraphs of an “infinite” graph with infinitely many vertices and a correspondingly infinite adjacency matrix.",
"Traditionally, exchangeability in random graphs is defined as the invariance of the distribution of any finite submatrix of this adjacency matrix—corresponding to any finite collection of vertices—under finite permutation.",
"Equivalently, we can express this form of exchangeability, which we henceforth call vertex exchangeability, by considering a random sequence of graphs $(G_n)_n$ with $V_n = [n]$ , where $[n] := \\lbrace 1,\\ldots ,n\\rbrace $ .",
"In this case, only the edge sequence is random.",
"Let $\\pi $ be any permutation of the integers $[n]$ .",
"If $e = \\lbrace v,w\\rbrace $ , let $\\pi (e) := \\lbrace \\pi (v),\\pi (w)\\rbrace $ .",
"If $E_n = \\lbrace e_1,\\ldots ,e_m\\rbrace $ , let $\\pi (E_n) := \\lbrace \\pi (e_1),\\ldots ,\\pi (e_m)\\rbrace $ .",
"Consider the random graph sequence $(G_n)_n$ , where $G_n$ has vertices $V_n = [n]$ and edges $E_n$ .",
"$(G_n)_n$ is (infinitely) vertex exchangeable if for every $n \\in $ and for every permutation $\\pi $ of the vertices $[n]$ , $G_n \\tilde{G}_n$ , where $\\tilde{G}_n$ has vertices $[n]$ and edges $\\pi (E_n)$ .",
"A great many popular models for graphs are vertex exchangeable; see app:vertex and [37] for a list.",
"However, it follows from the Aldous-Hoover theorem [2], [31] that any vertex-exchangeable graph is a mixture of sampling procedures from graphons.",
"Further, any graph sampled from a graphon is almost surely dense or empty [39].",
"Thus, vertex-exchangeable random graph models are misspecified models for sparse network datasets, as they generate dense graphs."
],
[
"Edge-exchangeable graph sequences",
"Vertex-exchangeable sequences have distributions invariant to the order of vertex arrival.",
"We introduce edge-exchangeable graph sequences, which will instead be invariant to the order of edge arrival.",
"As before, we let $G_n = (V_n, E_n)$ be the $n$ th graph in the sequence.",
"Here, though, we consider only active vertices—that is, vertices that are connected via some edge.",
"That lets us define $V_n$ as a function of $E_n$ ; namely, $V_n$ is the union of the vertices in $E_n$ .",
"Note that a graph that has sub-quadratic growth in the number of edges as a function of the number of active vertices will necessarily have sub-quadratic growth in the number of edges as a function of the number of all vertices, so we obtain strictly stronger results by considering active vertices.",
"In this case, the graph $G_n$ is completely defined by its edge set $E_n$ .",
"As above, we suppose that $E_{n} \\subseteq E_{n+1}$ .",
"We can emphasize this projectivity property by augmenting each edge with the step on which it is added to the sequence.",
"Let $E^{\\prime }_n$ be a collection of tuples, in which the first element is the edge and the second element is the step (i.e., index) on which the edge is added: $E^{\\prime }_n = \\lbrace (e_1,s_1),\\ldots ,(e_m,s_m) \\rbrace $ .",
"We can then define a step-augmented graph sequence $(E^{\\prime }_n)_n = (E^{\\prime }_1,E^{\\prime }_2,\\ldots )$ as a sequence of step-augmented edge sets.",
"Note that there is a bijection between the step-augmented graph sequence and the original graph sequence.",
"Figure: Upper, left four: Step-augmented graph sequence from Ex. .",
"At each step nn,the step value is always at least the maximum vertex index.",
"Upper, right two: Two graphswith the same probability under vertex exchangeability.",
"Lower, left four: Step-augmented graph sequencefrom Ex. .",
"Lower, right two: Two graphs with the same probability under edge exchangeability.In the setup for vertex exchangeability, we assumed $V_n = [n]$ and every edge is introduced as soon as both of its vertices are introduced.",
"In this case, the step of any edge in the step-augmented graph is the maximum vertex value.",
"For example, in fig:ex-graphs, we have E'1 = , E'2 = E'3 = {({1,2},2)}, E'4 = {({1,2},2), ({1,4},4),({2,4},4),({3,4},4)}.",
"In general step-augmented graphs, though, the step need not equal the max vertex, as we see next.",
"Suppose we have a graph given by the edge sequence (see fig:ex-graphs): E1 = E2 = {{2,5}, {5,5}}, E3 = E2 {{2,5} }, E4 = E3 { {1,6} }.",
"The step-augmented graph $E^{\\prime }_4$ is $\\lbrace (\\lbrace 2,5\\rbrace ,1),(\\lbrace 5,5\\rbrace ,1), (\\lbrace 2,5\\rbrace ,3), (\\lbrace 1,6\\rbrace ,4)\\rbrace .$ Roughly, a random graph sequence is edge exchangeable if its distribution is invariant to finite permutations of the steps.",
"Let $\\pi $ be a permutation of the integers $[n]$ .",
"For a step-augmented edge set $E^{\\prime }_n = \\lbrace (e_1,s_1),\\ldots ,(e_m,s_m) \\rbrace $ , let $\\pi (E^{\\prime }_n) =\\lbrace (e_1,\\pi (s_1)),\\ldots ,(e_m,\\pi (s_m)) \\rbrace $ .",
"Consider the random graph sequence $(G_n)_n$ , where $G_n$ has step-augmented edges $E^{\\prime }_n$ and $V_n$ are the active vertices of $E_n$ .",
"$(G_n)_n$ is (infinitely) edge exchangeable if for every $n \\in $ and for every permutation $\\pi $ of the steps $[n]$ , $G_n \\tilde{G}_n$ , where $\\tilde{G}_n$ has step-augmented edges $\\pi (E^{\\prime }_n)$ and associated active vertices.",
"See fig:ex-graphs for visualizations of both vertex exchangeability and edge exchangeability.",
"It remains to show that there are non-trivial models that are edge exchangeable (sec:generative) and that edge-exchangeable models admit sparse graphs (sec:poissp)."
],
[
"Edge-exchangeable graph frequency models",
"We next demonstrate that a wide class of models, which we call graph frequency models, exhibit edge exchangeability.",
"Consider a latent infinity of vertices indexed by the positive integers $= \\lbrace 1,2,\\ldots \\rbrace $ , along with an infinity of edge labels $(\\theta _{\\lbrace i,j\\rbrace })$ , each in a set $\\Theta $ , and positive edge rates (or frequencies) $(w_{\\lbrace i,j\\rbrace })$ in $\\mathbb {R}_+$ .",
"We allow both the $(\\theta _{\\lbrace i,j\\rbrace })$ and $(w_{\\lbrace i,j\\rbrace })$ to be random, though this is not mandatory.",
"For instance, we might choose $\\theta _{\\lbrace i,j\\rbrace } = (i,j)$ for $i \\le j$ , and $\\Theta = \\mathbb {R}^2$ .",
"Alternatively, the $\\theta _{\\lbrace i,j\\rbrace }$ could be drawn iid from a continuous distribution such as $\\textrm {Unif}[0,1]$ .",
"For any choice of $(\\theta _{\\lbrace i,j\\rbrace })$ and $(w_{\\lbrace i,j\\rbrace })$ , W := {i,j}: i,j w{i,j} {i,j} is a measure on $\\Theta $ .",
"Moreover, it is a discrete measure since it is always atomic.",
"If either $(\\theta _{\\lbrace i,j\\rbrace })$ or $(w_{\\lbrace i,j\\rbrace })$ (or both) are random, $W$ is a discrete random measure on $\\Theta $ since it is a random, discrete-measure-valued element.",
"Given the edge rates (or frequencies) $(w_{\\lbrace i,j\\rbrace })$ in $W$ , we next show some natural ways to construct edge-exchangeable graphs."
],
[
"Single edge per step",
"If the rates $(w_{\\lbrace i,j\\rbrace })$ are normalized such that $\\sum _{\\lbrace i,j\\rbrace : i, j \\in } w_{\\lbrace i,j\\rbrace } = 1$ , then $(w_{\\lbrace i,j\\rbrace })$ is a distribution over all possible vertex pairs.",
"In other words, $W$ is a probability measure.",
"We can form an edge-exchangeable graph sequence by first drawing values for $(w_{\\lbrace i,j\\rbrace })$ and $(\\theta _{\\lbrace i,j\\rbrace })$ —and setting $E_0 = \\emptyset $ .",
"We recursively set $E_{n+1} = E_n \\cup \\lbrace e \\rbrace $ , where $e$ is an edge $\\lbrace i,j\\rbrace $ chosen from the distribution $(w_{\\lbrace i,j\\rbrace })$ .",
"This construction introduces a single edge in the graph each step, although it may be a duplicate of an edge that already exists.",
"Therefore, this technique generates multigraphs one edge at a time.",
"Since the edge every step is drawn conditionally iid given $W$ , we have an edge-exchangeable graph."
],
[
"Multiple edges per step",
"Alternatively, the rates $(w_{\\lbrace i,j\\rbrace })$ may not be normalized.",
"Then $W$ may not be a probability measure.",
"Let $f(m | w)$ be a distribution over non-negative integers $m$ given some rate $w \\in \\mathbb {R}_{+}$ .",
"We again initialize our sequence by drawing $(w_{\\lbrace i,j\\rbrace })$ and $(\\theta _{\\lbrace i,j\\rbrace })$ and setting $E_0 = \\emptyset $ .",
"In this case, recursively, on the $n$ th step, start by setting $F = \\emptyset $ .",
"For every possible edge $e = \\lbrace i,j\\rbrace $ , we draw the multiplicity of the edge $e$ in this step as $m_{e} f(\\cdot | w_{e})$ and add $m_{e}$ copies of edge $e$ to $F$ .",
"Finally, $E_{n+1} = E_n \\cup F$ .",
"This technique potentially introduces multiple edges in each step, in which edges themselves may have multiplicity greater than one and may be duplicates of edges that already exist in the graph.",
"Therefore, this technique generates multigraphs, multiple edges at a time.",
"If we restrict $f$ and $W$ such that finitely many edges are added on every step almost surely, we have an edge-exchangeable graph, as the edges in each step are drawn conditionally iid given $W$ .",
"Given a sequence of edge sets $E_0, E_1, \\dots $ constructed via either of the above methods, we can form a binary graph sequence $\\bar{E}_0, \\bar{E}_1, \\dots $ by setting $\\bar{E}_i$ to have the same edges as $E_i$ except with multiplicity 1.",
"Although this binary graph is not itself edge exchangeable, it inherits many of the properties (such as sparsity, as shown in sec:poissp) of the underlying edge-exchangeable multigraph.",
"The choice of the distribution on the measure $W$ has a strong influence on the properties of the resulting edge-exchangeable graph sampled via one of the above methods.",
"For example, one choice is to set $w_{\\lbrace i,j\\rbrace } = w_i w_j$ , where the $(w_i)_i$ are a countable infinity of random values generated according to a Poisson point process (PPP).",
"We say that $(w_i)_i$ is distributed according to a Poisson point process parameterized by rate measure $\\nu $ , $(w_i)_i \\sim \\textrm {PPP}(\\nu )$ , if (a) $\\#\\lbrace i: w_i \\in A\\rbrace \\sim \\textrm {Poisson}(\\nu (A))$ for any set $A$ with finite measure $\\nu (A)$ and (b) $\\#\\lbrace i: w_i \\in A_j\\rbrace $ are independent random variables across any finite collection of disjoint sets $(A_j)_{j=1}^{J}$ .",
"In sec:poissp we examine a particular example of this graph frequency model, and demonstrate that sparsity is possible in edge-exchangeable graphs."
],
[
"Related work and connection to nonparametric Bayes",
"Given a unique label $\\theta _i$ for each vertex $i\\in \\mathbb {N}$ , and denoting $g_{ij} = g_{ji}$ to be the number of undirected edges between vertices $i$ and $j$ , the graph itself can be represented as the discrete random measure $G = \\sum _{i,j}g_{ij}\\delta _{(\\theta _i, \\theta _j)}$ on $_+^2$ .",
"A different notion of exchangeability for graphs than the ones in sec:exchangeability can be phrased for such atomic random measures: a point process $G$ on $^2$ is (jointly) exchangeable if, for all finite permutations $\\pi $ of $$ and all $h > 0$ , G(Ai Aj) G(A(i) A(j)), for (i,j) 2, where Ai :=[h(i-1), hi].",
"This form of exchangeability, which we refer to as Kallenberg exchangeability, can intuitively be viewed as invariance of the graph distribution to relabeling of the vertices, which are now embedded in $^2$ .",
"As such it is analogous to vertex exchangeability, but for discrete random measures [19].",
"Exchangeability for random measures was introduced by Aldous [3], and a representation theorem was given by Kallenberg [33], [32].",
"The use of Kallenberg exchangeability for modeling graphs was first proposed by [19], and then characterized in greater generality by [45] and [8].",
"Edge exchangeability is distinct from Kallenberg exchangeability, as shown by the following example.",
"[Edge exchangeable but not Kallenberg exchangeable] Consider the graph frequency model developed in sec:generative, with $w_{\\lbrace i,j\\rbrace } = (ij)^{-2}$ and $\\theta _{\\lbrace i,j\\rbrace } = \\lbrace i,j\\rbrace $ .",
"Since the edges at each step are drawn iid given $W$ , the graph sequence is edge exchangeable.",
"However, the corresponding graph measure $G = \\sum _{i,j} n_{ij}\\delta _{(i, j)}$ (where $n_{ij} = n_{ji} \\sim \\mathrm {Binom}(N, (ij)^{-2})$ ) is not Kallenberg exchangeable, since the probability of generating edge $\\lbrace i,j\\rbrace $ is directly related to the positions $(i,j)$ and $(j, i)$ in $\\mathbb {R}_+^2$ of the corresponding atoms in $G$ (in particular, the probability is decreasing in $ij$ ).",
"Figure: A comparison of a graph frequency model (sec:generative and eq:graphmodel)and the generative model of .Any interval [0,y][0,y] contains a countablyinfinite number of atoms with a nonzero weight in the random measure; a drawfrom the random measure is plotted at the top (and repeated on the right side).Each atom correspondsto a latent vertex.Each point (θ i ,θ j )(\\theta _i,\\theta _j) corresponds to a latent edge.",
"Darker point colorson the left occur for greater edge multiplicities.",
"On the left, more latent edges are instantiatedas more steps nn are taken.",
"On the right, the edges within [0,y] 2 [0,y]^2 are fixed, but more edgesare instantiated as yy grows.Our graph frequency model is reminiscent of the [19] generative model, but has a number of key differences.",
"At a high level, this earlier model generates a weight measure $W = \\sum _{i,j} w_{ij}\\delta _{(\\theta _i,\\theta _j)}$ ([19] used, in particular, the outer product of a completely random measure), and the graph measure $G$ is constructed by sampling $g_{ij}$ once given $w_{ij}$ for each pair $i, j$ .",
"To create a finite graph, the graph measure $G$ is restricted to the subset $[0, y]\\times [0, y]\\subset \\mathbb {R}_+^2$ for $0<y<\\infty $ ; to create a projective growing graph sequence, the value of $y$ is increased.",
"By contrast, in the analogous graph frequency model of the present work, $y$ is fixed, and we grow the network by repeatedly sampling the number of edges $g_{ij}$ between vertices $i$ and $j$ and summing the result.",
"Thus, in the [19] model, a latent infinity of vertices (only finitely many of which are active) are added to the network each time $y$ increases.",
"In our graph frequency model, there is a single collection of latent vertices, which are all gradually activated by increasing the number of samples that generate edges between the vertices.",
"See fig:cffreq for an illustration.",
"Increasing $n$ in the graph frequency model has the interpretation of both (a) time passing and (b) new individuals joining a network because they have formed a connection that was not previously there.",
"In particular, only latent individuals that will eventually join the network are considered.",
"This behavior is analogous to the well-known behavior of other nonparametric Bayesian models such as, e.g., a Chinese restaurant process (CRP).",
"In this analogy, the Dirichlet process (DP) corresponds to our graph frequency model, and the clusters instantiated by the CRP correspond to the vertices that are active after $n$ steps.",
"In the DP, only latent clusters that will eventually appear in the data are modeled.",
"Since the graph frequency setting is stationary like the DP/CRP, it may be more straightforward to develop approximate Bayesian inference algorithms, e.g., via truncation [17].",
"Edge exchangeability first appeared in work by [20], [21], [46], and [11], [12], [15].",
"[11], [12] established the notion of edge exchangeability used here and provided characterizations via exchangeable partitions and feature allocations, as in app:edge.",
"[11], [15] developed a frequency model based on weights $(w_i)_{i}$ generated from a Poisson process and studied several types of power laws in the model.",
"[20] established a similar notion of edge exchangeability in the context of a larger statistical modeling framework.",
"[21], [20] provided sparsity and power law results for the case where the weights $(w_i)_{i}$ are generated from a Pitman-Yor process and power law degree distribution simulations.",
"[46] described a similar notion of edge exchangeability and developed an edge-exchangeable model where the weights $(w_i)_i$ are generated from a Dirichlet process, a mixture model extension, and an efficient Bayesian inference procedure.",
"In work concurrent to the present paper, [22] re-examined edge exchangeability, provided a representation theorem, and studied sparsity and power laws for the same model based on Pitman-Yor weights.",
"By contrast, we here obtain sparsity results across all Poisson point process-based graph frequency models of the form in eq:graphmodel below, and use a specific three-parameter beta process rate measure only for simulations in sec:simulations."
],
[
"Sparsity in Poisson process graph frequency models",
"We now demonstrate that, unlike vertex exchangeability, edge exchangeability allows for sparsity in random graph sequences.",
"We develop a class of sparse, edge-exchangeable multigraph sequences via the Poisson point process construction introduced in sec:generative, along with their binary restrictions."
],
[
"Model",
"Let $\\mathcal {W}$ be a Poisson process on $[0, 1]$ with a nonatomic, $\\sigma $ -finite rate measure $\\nu $ satisfying $\\nu ([0, 1]) = \\infty $ and $\\int _0^1 w \\nu (\\mathrm {d}w) < \\infty $ .",
"These two conditions on $\\nu $ guarantee that $\\mathcal {W}$ is a countably infinite collection of rates in $[0,1]$ and that $\\sum _{w\\in \\mathcal {W}} w< \\infty $ almost surely.",
"We can use $\\mathcal {W}$ to construct the set of rates: $w_{\\lbrace i,j\\rbrace } = w_iw_j$ if $i\\ne j$ , and $w_{\\lbrace i, i\\rbrace }=0$ .",
"The edge labels $\\theta _{\\lbrace i,j\\rbrace }$ are unimportant in characterizing sparsity, and so can be ignored.",
"To use the multiple-edges-per-step graph frequency model from sec:generative, we let $f(\\cdot | w)$ be Bernoulli with probability $w$ .",
"Since edge $\\lbrace i, j\\rbrace $ is added in each step with probability $w_iw_j$ , its multiplicity $M_{\\lbrace i,j\\rbrace }$ after $n$ steps has a binomial distribution with parameters $n, w_iw_j$ .",
"Note that self-loops are avoided by setting $w_{\\lbrace i, i\\rbrace } =0$ .",
"Therefore, the graph after $n$ steps is described by: W PPP() M{i,j} Binom(n, wiwj) for i < j N. As mentioned earlier, this generative model yields an edge-exchangeable graph, with edge multiset $E_n$ containing $\\lbrace i, j\\rbrace $ with multiplicity $M_{\\lbrace i, j\\rbrace }$ , and active vertices $V_n = \\lbrace i: \\sum _j M_{\\lbrace i, j\\rbrace }>0\\rbrace $ .",
"Although this model generates multigraphs, it can be modified to sample a binary graph $(\\bar{V}_n, \\bar{E}_n)$ by setting $\\bar{V}_n = V_n$ and $\\bar{E}_n$ to the set of edges $\\lbrace i, j\\rbrace $ such that $\\lbrace i, j\\rbrace $ has multiplicity $\\ge 1$ in $E_n$ .",
"We can express the number of vertices and edges, in the multi- and binary graphs respectively, as |Vn| = |Vn| = i 1(ji M{i,j} > 0) , |En| = 12ijM{i, j}, |En| = 12ij 1(M{i, j}>0)."
],
[
"Moments",
"Recall that a sequence of graphs is considered sparse if $|E_n| = o(|V_n|^2)$ .",
"Thus, sparsity in the present setting is an asymptotic property of a random graph sequence.",
"Rather than consider the asymptotics of the (dependent) random sequences $|E_n|$ and $|V_n|$ in concert, lem:xvsex allows us to consider the asymptotics of their first moments, which are deterministic sequences and can be analyzed separately.",
"We use $\\sim $ to denote asymptotic equivalence, i.e., $a_n\\sim b_n \\iff \\lim _{n\\rightarrow \\infty } \\frac{a_n}{b_n} = 1$ .",
"For details on our asymptotic notation and proofs for this section, see app:proofs.",
"Lemma The number of vertices and edges for both the multi- and binary graphs satisfy |Vn| = |Vn| a.s. (|Vn|), |En| a.s. (|En|), |En| a.s. (|En|), n. Thus, we can examine the asymptotic behavior of the random numbers of edges and vertices by examining the asymptotic behavior of their expectations, which are provided by prop:moments.",
"Lemma The expected numbers of vertices and edges for the multi- and binary graphs are (|Vn|) = (|Vn|) = [1 - (-(1- (1-wv)n) (dv)) ] (dw), (|En|) = n2wv (dw)(dv), (|En|) = 12(1 - (1-wv)n) (dw)(dv)."
],
[
"Sparsity",
"We are now equipped to characterize the sparsity of this random graph sequence: Suppose $\\nu $ has a regularly varying tail, i.e., there exist $\\alpha \\in (0, 1)$ and $\\ell : \\mathbb {R}_+ \\rightarrow \\mathbb {R}_+$ s.t.",
"x1(d w) x-(x-1), x0 and c > 0, x (cx)(x) = 1.",
"Then as $n\\rightarrow \\infty $ , |Vn| (n(n)), |En| (n), |En| O((n1/2)(n1+2,(n)n32)).",
"thm:asymp implies that the multigraph is sparse when $\\alpha \\in ({1}{2}, 1)$ , and that the restriction to the binary graph is sparse for any $\\alpha \\in (0, 1)$ .",
"See Remark REF for a discussion.",
"Thus, edge-exchangeable random graph sequences allow for a wide range of sparse and dense behavior."
],
[
"Simulations",
"In this section, we explore the behavior of graphs generated by the model from sec:poissp via simulation, with the primary goal of empirically demonstrating that the model produces sparse graphs.",
"We consider the case when the Poisson process generating the weights in eq:graphmodel has the rate measure of a three-parameter beta process (3-BP) on $(0,1)$ [43], [13]: (dw) = (1+)(1-)(+) w-1- (1 - w)+-1 dw, with mass $\\gamma > 0$ , concentration $\\beta > 0$ , and discount $\\alpha \\in (0, 1)$ .",
"In order for the 3-BP to have finite total mass $\\sum _j w_j < \\infty $ , we require that $\\beta > -\\alpha $ .",
"We draw realizations of the weights from a $\\text{3-BP}(\\gamma ,\\beta ,\\alpha )$ according to the stick-breaking representation given by [13].",
"That is, the $w_i$ are the atom weights of the measure ${W}$ for W = i=1j=1Ci Vi,j(i) l=1i-1 (1 - Vi,j())i,j, Ci Pois(), Vi,j() Beta(1-, + ), i,j B0 and any continuous (i.e., non-atomic) choice of distribution $B_0$ .",
"Since simulating an infinite number of atoms is not possible, we truncate the outer summation in $i$ to 2000 rounds, resulting in $\\sum _{i=1}^{2000} C_i$ weights.",
"The parameters of the beta process were fixed to $\\gamma =3$ and $\\theta =1$ , as they do not influence the sparsity of the resulting graph frequency model, and we varied the discount parameter $\\alpha $ .",
"Given a single draw ${W}$ (at some specific discount $\\alpha $ ), we then simulated the edges of the graph, where the number of Bernoulli draws $N$ varied between 50 and 2000. fig:type1multi shows how the number of edges varies versus the total number of active vertices for the multigraph, with different colors representing different random seeds.",
"To check whether the generated graph was sparse, we determined the exponent by examining the slope of the data points (on a log-scale).",
"In all plots, the black dashed line is a line with slope 2.",
"In the multigraph, we found that for the discount parameter settings $\\alpha = 0.6, 0.7$ , the slopes were below 2; for $\\alpha = 0, 0.3$ , the slopes were greater than 2.",
"This corresponds to our theoretical results; for $\\alpha < 0.5$ the multigraph is dense with slope greater than 2, and for $\\alpha > 0.5$ the multigraph is sparse with slope less than 2.",
"Furthermore, the sparse graphs exhibit power law relationships between the number of edges and vertices, i.e., $|{E}_N| c\\, |{V}_N|^b, \\, N \\rightarrow \\infty $ , where $b \\in (1,2)$ , as suggested by the linear relationship in the plots between the quantities on a log-scale.",
"Note that there are necessarily fewer edges in the binary graph than in the multigraph, and thus this plot implies that the binary graph frequency model can also capture sparsity.",
"fig:type1bin confirms this observation; it shows how the number of edges varies with the number of active vertices for the binary graph.",
"In this case, across $\\alpha \\in (0,1)$ , we observe slopes that are less than 2.",
"This agrees with our theory from sec:poissp, which states that the binary graph is sparse for any $\\alpha \\in (0, 1)$ .",
"Figure: Data simulated from agraph frequency model with weights generatedaccording to a 3-BP.Colors represent different random draws.The dashed line has a slope of 2."
],
[
"Conclusions",
"We have proposed an alternative form of exchangeability for random graphs, which we call edge exchangeability, in which the distribution of a graph sequence is invariant to the order of the edges.",
"We have demonstrated that edge-exchangeable graph sequences, unlike traditional vertex-exchangeable sequences, can be sparse by developing a class of edge-exchangeable graph frequency models that provably exhibit sparsity.",
"Simulations using edge frequencies drawn according to a three-parameter beta process confirm our theoretical results regarding sparsity.",
"Our results suggest that a variety of future directions would be fruitful—including theoretically characterizing different types of power laws within graph frequency models, characterizing the use of truncation within graph frequency models as a means for approximate Bayesian inference in graphs, and understanding the full range of distributions over sparse, edge-exchangeable graph sequences."
],
[
"Acknowledgments",
"We would like to thank Bailey Fosdick and Tyler McCormick for helpful conversations."
],
[
"Overview",
"In app:vertex, we provide more examples of graph models that are either vertex exchangeable or Kallenberg exchangeable.",
"In app:edge, we establish characterizations of edge exchangeability in graphs via existing notions of exchangeability for combinatorial structures such as random partitions and feature allocations.",
"In app:proofs, we provide full proof details for the theoretical results in the main text."
],
[
"More exchangeable graph models",
"Many popular graph models are vertex exchangeable.",
"These models include the classic Erdős–Rényi model [24], as well as Bayesian generative models for network data, such as the stochastic block model [30], the mixed membership stochastic block model [1], the infinite relational model [34], [48], the latent space model [29], the latent feature relational model [38], the infinite latent attribute model [40], and the random function model [37].",
"See [39] and [37] for more examples and discussion.",
"Recently, a number of extensions to the Kallenberg-exchangeable model of [19], which builds on early work on bipartite graphs by [18], have also been developed.",
"These models include extensions to stochastic block models [28], mixed membership stochastic block models [44], and dynamic network models [41]."
],
[
"Characterizations of edge-exchangeable graph sequences",
"We introduced edge exchangeability, a new notion of exchangeability for graphs.",
"Just as the Aldous-Hoover theorem provides a characterization of the distribution of vertex-exchangeable graphs, it is desirable to provide a characterization of edge exchangeability in graphs.",
"Below we show how characterization theorems that already exist for other combinatorial structures can be readily applied to provide characterizations for edge exchangeability in graphs.",
"We first develop mappings from edge-exchangeable graph sequences to familiar combinatorial structures—such as partitions [42], feature allocations [10], and trait allocations [14], [16]—showing that edge exchangeability in the graph corresponds to exchangeability in those structures.",
"In this manner, we provide characterizations of the case where one edge is added to the graph per step in sec:part, where multiple unique edges may be added per step in sec:feat, and where multiple (non)unique edges may be added in sec:trait.",
"A limitation of these connections is that it is not immediately clear how to recover the connectivity in the graph from the mapped combinatorial object; for instance, given a particular feature allocation, the graph to which it corresponds is not identifiable.",
"This issue has been addressed in a purely combinatorial context via vertex allocations and the graph paintbox [16] using the general theory of trait allocations.",
"In sec:labeledcomb, we provide an alternative connection to ordered combinatorial structures [10], [16] under the assumption that vertex labels are provided.",
"This assumption is often reasonable in the setting of network data where the vertices and edges are observed directly.",
"By contrast, it is unusual to assume that labels are provided for blocks in the case of partitions, feature allocations, and trait allocations since, in these cases, the combinatorial structure is typically entirely latent in real data analysis problems.",
"For instance, in clustering applications, finding parameters that describe each cluster is usually part of the inference problem.",
"In the graph case, though, the use of an ordered structure identifies the particular pair of vertices corresponding to each edge in the graph, allowing recovery of the graph itself."
],
[
"The step collection sequence and connections to other forms of\ncombinatorial exchangeability",
"In order to analyze edge-exchangeable graphs using the existing combinatorial machinery of random partitions, feature allocations, and trait allocations, we introduce a new combinatorial structure, the step collection sequence, which can take the form of a sequence of partitions, feature allocations, or trait allocations.",
"As we will now see, the step collection sequence can be constructed from the step-augmented graph sequence in the following way.",
"Suppose we assign a unique label $\\phi $ to each pair of vertices.",
"Then if a pair of vertices is labeled $\\phi $ , we may imagine that any particular edge between this pair of vertices is assigned label $\\phi $ when it appears.",
"Let $\\phi _j$ be the $j$ th such unique edge label.",
"Recall that we consider a sequence of graphs defined by its step-augmented edge sequence $E^{\\prime }_n$ .",
"Let $S_j$ be the set of steps up to the current step $n$ in which any edge labeled $\\phi _j$ was added.",
"If $m$ edges labeled $\\phi _j$ were added in a single step $s$ , $s$ appears in $S_j$ with multiplicity $m$ .",
"So each element $s \\in S_j$ is an element of $[n]$ .",
"Let $K_n$ be the number of unique vertex pairs seen among edges introduced up until the current step $n$ .",
"Then we may define $C_n$ to be the collection of step sets across edges that have appeared by step $n$ : Cn = { S1, ..., SKn }.",
"Finally, we can define the step collection sequence $C = (C_1, C_2,\\ldots )$ as the sequence of $C_n$ for $n=1,2,\\ldots $ .",
"Note that it is not clear how to recover the original edge connectivity of the graph from the step collection sequence, or whether it is possible to modify the sequence (or the labels $\\phi _j$ ) such that it is easy to recover connectivity while maintaining the (non-trivial) connections to combinatorial exchangeability provided in sec:part,sec:feat,sec:trait below.",
"Suppose we have the edge sequence E1 = {{2,3}, {3,6}}, E2 = {{2,3}, {3,6}}, E3 = {{2,3}, {3,6}, {6,6}, {3,6}}, E4 = {{2,3},{1,4},{3,6}, {6,6}, {3,6} }, with step-augmentation $E^{\\prime }_4 = \\lbrace (\\lbrace 2,3\\rbrace ,1),(\\lbrace 1,4\\rbrace ,4),(\\lbrace 3,6\\rbrace ,1), (\\lbrace 6,6\\rbrace ,3), (\\lbrace 3,6\\rbrace ,3)\\rbrace $ for $E_4$ .",
"Now we label the unique edges in $E^{\\prime }_n$ .",
"Using an order of appearance scheme [10] to index the labels, $E^{\\prime }_4$ becomes {(1,1), (2,1), (3,3), (1,3), (4,4)}, where the labels $\\phi _j$ correspond to the four unique vertex pairs: $\\phi _1 = \\lbrace 3,6\\rbrace ,\\phi _2 = \\lbrace 2,3\\rbrace ,\\phi _3 = \\lbrace 6,6\\rbrace ,\\phi _3 = \\lbrace 1,4\\rbrace $ .",
"The step collection sequence for $C_1,\\ldots ,C_4$ is C1 = { {1}1 }, C2 = { {1}1 }, C3 = { {1, 3}1, {3}3 }, C4 = { {1, 3}1, {3}3, {4}4 }.",
"Here each element of $C_n$ is a set corresponding to one of the four unique labels $\\phi _j$ and contains all step indices up to step $n$ in which an edge with that label was added to the graph sequence.",
"To see that the step collection sequence can be interpreted as a familiar combinatorial object, we recall the following definitions.",
"A partition $C_n$ of $[n]$ is a set $\\lbrace S_1, \\ldots , S_{K_n}\\rbrace $ whose blocks, or clusters, are mutually exclusive, i.e., $S_i \\cap S_j = \\emptyset , i\\ne j$ , and exhaustive, i.e., $\\bigcup _j S_j = [n]$ .",
"Feature allocations relax the definition of partitions by no longer requiring the blocks to be mutually exclusive and exhaustive.",
"A feature allocation $C_n$ of $[n]$ is a multiset $\\lbrace S_1, \\ldots , S_{K_n}\\rbrace $ of subsets of $[n]$ , such that any datapoint in $[n]$ occurs in finitely many features $S_j$ [10].",
"A trait allocation generalizes the feature allocation where now each $S_j$ , called a trait, may itself be a multiset [14], [16].",
"We see that the step collection $C_n$ can be interpreted as follows.",
"If a single edge is added to the graph at each round, $C_n$ is a partition of $[n]$ , and the step collection sequence is a projective partition sequence.",
"If at most one edge is added between any pair of vertices at each step, $C_n$ is a feature allocation of $[n]$ , and the step collection sequence is a projective sequence of feature allocations.",
"In the most general case, when multiple edges may be added between any pair of vertices at each step, $C_n$ is a trait allocation of $[n]$ , and the step collection sequence is a projective sequence of trait allocations.",
"In the following examples, corresponding to Figure REF , we show different step collection sequences that correspond to a partition, a feature allocation, and a trait allocation.",
"[Partition] Consider the step collection $C_5 = \\lbrace \\lbrace 1,3\\rbrace ,\\lbrace 2\\rbrace ,\\lbrace 4\\rbrace ,\\lbrace 5\\rbrace \\rbrace $ .",
"The edges form a partition of the steps.",
"Here exactly one edge arrives in each step.",
"[Feature allocation] Consider the step collection $C_5 = \\lbrace \\lbrace 1,3\\rbrace ,\\lbrace 1\\rbrace ,\\lbrace 1,5\\rbrace ,\\lbrace 3,4\\rbrace \\rbrace $ .",
"This step collection forms a feature allocation of the steps.",
"Thus in this case, there may be multiple unique edges arriving in each step.",
"[Trait allocation] In a trait allocation, there may be multiple edges (not necessarily unique) at each step.",
"Consider the step collection $C_5 = \\lbrace \\lbrace 1,3,3,3\\rbrace ,\\lbrace 1\\rbrace ,\\lbrace 1,5\\rbrace ,\\lbrace 3\\rbrace ,\\lbrace 4,4\\rbrace \\rbrace $ .",
"This collection forms a trait allocation of the steps, where elements of $C_5$ are now multisets.",
"Figure: Connection of edge-exchangeable graphs with partitions, featureallocations, and trait allocations.Light blocks represent 0, dark blocks either represent 1 or thespecified count.In a partition, exactly one edge arrives in each step.In a feature allocation, multiple edges may arrive at each step, but at most one edge arrives between any two vertices at each step.In a trait allocation, there may be multiple edges of any type.In this section, we have connected certain types of edge-exchangeable graphs to partitions and feature allocations.",
"In the next two sections, we make use of known characterizations of these combinatorial objects to characterize edge exchangeability in graphs.",
"First consider the connection to partitions.",
"In this case, suppose that each index in $[n]$ appears exactly once across all of the subsets of $C_n$ .",
"This assumption on $C_n$ is equivalent to assuming that in the original graph sequence $E_1, E_2, \\ldots $ , we have that $E_{n+1}$ always has exactly one more edge than $E_{n}$ .",
"In this case, $C_n$ is exactly a partition of $[n]$ ; that is, $C_n$ is a set of mutually exclusive and exhaustive subsets of $[n]$ .",
"If the edge sequence $(E_n)$ is random, then $(C_n)$ is random as well.",
"We say that a partition sequence $C_1, C_2, \\ldots $ , where $C_n$ is a (random) partition of $[n]$ and $C_m \\subseteq C_n$ for all $m \\le n$ , is infinitely exchangeable if, for all $n$ , permuting the indices in $[n]$ does not change the distribution of the (random) partitions [42].",
"Permuting the indices $[n]$ in the partition sequence $(C_m)$ corresponds to permuting the order in which edges are added in our graph sequence $(E_m)$ .",
"As an example of a model that generates a step collection sequence corresponding to a partition sequence, consider the frequency model we introduced in sec:generative where the weights are normalized.",
"At each step, we choose a single edge according the resulting probability distribution over pairs of vertices.",
"Given this connection to exchangeable partitions, the Kingman paintbox theorem [36] provides a characterization of edge exchangeability in graph sequences that introduce one edge per step: in particular, it guarantees that a graph sequence that adds exactly one edge per step is edge exchangeable if and only if the associated step collection sequence $(C_n)$ has a Kingman paintbox representation.",
"An alternate characterization of edge exchangeability in graph sequences that introduce one edge per step is provided by exchangeable partition probability functions (EPPFs) [42].",
"In particular, a graph sequence that introduces one edge per step is edge-exchangeable if and only if the marginal distribution of $C_n$ (the step collection at step $n$ ) is given by an EPPF for all $n$ ."
],
[
"Feature allocation connection",
"Next we notice that it need not be the case that exactly one edge is added at each step of the graph sequence, e.g.",
"between $E_{n}$ and $E_{n+1}$ .",
"If we allow multiple unique edges at any step, then the step collection $C_n$ is just a set of subsets of $[n]$ , where each subset has at most one of each index in $[n]$ .",
"Suppose that any $m$ belongs to only finitely many subsets in $C_n$ for any $n$ .",
"That is, we suppose that only finitely many edges are added to the graph at any step.",
"Then $C_n$ is an example of a feature allocation [10].",
"Again, if $(E_n)$ is random, then $(C_n)$ is random as well.",
"We say that a (random) feature allocation sequence $(C_m)$ is infinitely exchangeable if, for any $n$ , permuting the indices of $[n]$ does not change the distribution of the (random) feature allocations [9], [10].",
"Permuting the indices $[n]$ in the sequence $(C_m)$ corresponds to permuting the steps when edges are added in the edge sequence $(E_m)$ .",
"Consider the following example of a graph frequency model that produces a step collection sequence corresponding to an exchangeable feature allocation.",
"For $n=1,2,\\ldots $ , we draw whether the graph has an edge $\\lbrace i,j\\rbrace $ at time step $n$ as Bernoulli with probability $w_{\\lbrace i,j\\rbrace } = w_i w_j$ .",
"Thus, in each step, we draw at most one edge per unique vertex pair.",
"But we may draw multiple edges in the same step.",
"Similarly to the partition case in part, we can apply known results from feature allocations to characterize edge exchangeability in graph models of this form.",
"For instance, we know that the feature paintbox [10], [16] characterizes distributions over exchangeable feature allocations (and therefore the step collection sequence for graphs of this form) just as the Kingman paintbox characterizes distributions over exchangeable partitions (and therefore the step collection sequence for edge-exchangeable graphs with exactly one new edge per step).",
"We may also consider feature paintbox distributions with extra structure.",
"For instance, the step collection sequence is said to have an exchangeable feature probability function (EFPF) [10] if the probability of each step collection $C_n$ in the sequence can be expressed as a function only of the total number of steps $n$ and the subset sizes within $C_n$ (i.e.",
"the edge multiplicities in the graph), and is symmetric in the subset sizes.",
"As another example, the step collection sequence is said to have a feature frequency model if there exists a (random) sequence of probabilities $(w_j)_{j=1}^\\infty $ associated with edges $j=1, 2, \\dots $ and a number $\\lambda > 0$ , conditioned on which the step collection sequence arises from the graph built by adding edge $j$ at each step independentlyThis is conditional independence since the $(w_j)$ may be random.",
"with probability $w_j$ for all values of $j\\in \\mathbb {N}$ , along with an additional $\\mathrm {Poiss}(\\lambda )$ number of edges that never share a vertex with any other edge in the sequence.",
"In other words, the graph is constructed with a graph frequency model as in the main text of the present work (modulo the aforementioned additional Poisson number of edges).",
"Theorem 17 (“Equivalence of EFPFs and feature frequency models”) from [10] shows that these two examples are actually equivalent: if the step collection sequence has an EFPF, it has a feature frequency model, and vice versa."
],
[
"Further extensions",
"Finally, we may consider the case where at every step, any non-negative (finite) number of edges may be added and those edges may have non-trivial (finite) multiplicity; that is, the multiplicity of any edge at any step can be any non-negative integer.",
"By contrast, in feat, each unique edge occurred at most once at each step.",
"In this case, the step collection $C_n$ is a set of subsets of $[n]$ .",
"The subsets need not be unique or exclusive since we assume any number of edges may be added at any step.",
"And the subsets themselves are multisets since an edge may be added with some multiplicity at step $n$ .",
"We say that $C_n$ is a trait allocation, which we define as a generalization of a feature allocation where the subsets of $C_n$ are multisets.",
"As above, if $(E_n)$ is random, $(C_n)$ is as well.",
"We say that a (random) trait allocation sequence $(C_m)$ is infinitely exchangeable if, for any $n$ , permuting the indices of $[n]$ does not change the distribution of the (random) trait allocation.",
"Here, permuting the indices of $[n]$ corresponds to permuting the steps when edges are added in the edge sequence $(E_m)$ .",
"A graph frequency model that generates a step collection sequence as a trait allocation sequence is the multiple-edge-per-step frequency model sampling procedure described in sec:generative.",
"Here, at each step, multiple edges can appear each with multiplicity potentially greater than 1, requiring the full generality of a trait allocation sequence.",
"[16] characterize exchangeable trait allocations via, e.g., probability functions and paintboxes and thereby provide a characterization over the corresponding step collection sequences of such edge-exchangeable graphs."
],
[
"Connections to exchangeability in ordered combinatorial structures",
"As noted earlier, it is not immediately clear how to recover the connectivity in an edge-exchangeable graph from the step collection sequence, nor how to do so in a way that preserves non-trivial connections to other exchangeable combinatorial structures.",
"[16] considers an alternative to the step collection sequence in which the (multi)subsets in the combinatorial structure correspond to vertices rather than edges, known as a vertex allocation.",
"This allows for the characterization of edge-exchangeable graphs via the graph paintbox using the general theory of trait allocations, while maintaining an explicit representation of the structure of the graph, i.e., the connection between edges that share a vertex.",
"If we are willing to eschew the unordered nature of the step collection sequence, and assume that we have an a priori labeling on the vertices, there is yet another alternative using the ordered step collection sequence.",
"The availability of labeled vertices is often a reasonable assumption in the setting of network data, where the vertices and edges are typically observed directly.",
"Suppose the vertices are labeled using the natural numbers $1, 2, \\dots $ .",
"Then we can use the ordering of the vertex labels to order the vertex pairs in a diagonal manner, i.e.",
"$\\lbrace 1, 1\\rbrace , \\, \\lbrace 1, 2\\rbrace , \\, \\lbrace 2, 2\\rbrace , \\, \\lbrace 1, 3\\rbrace , \\, \\lbrace 2, 3\\rbrace , \\dots $ .",
"Note that, for the purpose of building this diagonal ordering, we consider the lowest-valued index in each vertex pair first.",
"We build the step collection sequence $(C_n)$ in the same manner as before, except that each step collection $C_n$ is no longer an unordered collection of subsets; the subsets derive their order from the vertex pairs they represent.",
"For example, if we observe edges at vertex pairs $\\lbrace 1, 1\\rbrace $ and $\\lbrace 1, 2\\rbrace $ at step 1, and edges at vertex pairs $\\lbrace 1, 1\\rbrace $ and $\\lbrace 2, 3\\rbrace $ at step 2, then C1 = ( {1}, {1}, , , ...) and C2 = ({1, 2}, {1}, , , {2}, , ...).",
"Since we know the order of the subsets in each $C_n$ as they relate to the vertex pairs in the graph and their connectivity, we can recover the graph sequence from the ordered step collection sequence $(C_n)$ .",
"Exchangeability in an ordered step collection sequence means that the distribution is invariant to permutations of the indices within the subsets (although the ordering of the subsets themselves cannot be changed).",
"Given this notion of exchangeability, the earlier connections to exchangeable partitions, feature allocations, and trait allocations remain true, modulo the fact that they must themselves be ordered.",
"[10] provides a paintbox characterization of ordered exchangeable feature allocations, thereby providing characterizations (via the earlier connections to partitions and feature allocations) of edge-exchangeable graphs that add either one or multiple unique edges per step.",
"Note that, in these cases, this is a full characterization of edge-exchangeable graphs, by contrast to app:stepconnections, where we provided a characterization only of edge exchangeability in graphs.",
"We suspect that a similar characterization of edge-exchangeable graphs with multiple (non)unique edges per step is available by examining characterizations of exchangeable ordered trait allocations."
],
[
"Proofs",
"The proof of the main theorem in the paper (Theorem REF ) follows from a collection of lemmas below.",
"Lemma REF characterizes the expected number of vertices and edges; Lemma REF establishes a useful transformation of those expectations; and Lemma REF shows that the two sets of expectations are asymptotically equivalent, so it is enough to consider the transformed expectation.",
"Lemma REF provides the asymptotics of the transformed expectations.",
"Finally, Lemma REF shows that the random sequences converge almost surely to their expectations, yielding the final result."
],
[
"Notation",
"We first define the asymptotic notation used in the main paper and appendix.",
"We use the notation “a.s.” to mean almost surely, or with probability 1.",
"Let $(X_n)_{n \\in },(Y_n)_{n \\in }$ be two random sequences.",
"We say that $X_n O(Y_n)$ if $\\limsup _{n \\rightarrow \\infty } \\frac{X_n}{Y_n} <\\infty $ a.s., and that $X_n \\Omega (Y_n)$ if $Y_n O(X_n)$ a.s. We say that $X_n o(Y_n)$ if $\\lim _{n \\rightarrow \\infty } \\frac{X_n}{Y_n} = 0$ a.s. Lastly, we say that $X_n \\Theta (Y_n)$ if $X_n O(Y_n)$ and $Y_n O(X_n)$ .",
"Let $V_n, E_n$ be the respective sets of active vertices and edges at step $n$ in the multigraph, and $|V_n|, |E_n|$ be their respective cardinalities, as defined in the main text.",
"We use the notation $\\bar{V}_n$ and $\\bar{E}_n$ to represent these analogous vertex and edge sets for the binary graph.",
"Note that $\\bar{V}_n$ is the same as $V_n$ ."
],
[
"Useful results",
"We present two useful theorems for analyzing expectations involving random sums of functions of points from Poisson point processes.",
"Below, we will apply these theorems repeatedly to get expectations of graph quantities.",
"The first theorem is Campbell's theorem, which is used to compute the moments of functionals of a Poisson process.",
"We state it below for completeness, and refer to [35] for details.",
"[Campbell's theorem] Let $\\Pi $ be a Poisson point process on $S$ with rate measure $\\nu $ , and let $f:S \\rightarrow $ be measurable.",
"If $\\int _S \\min (|f(x)|,1) \\,\\nu (dx) <\\infty $ , then ( (c x f(x)) ) = ( S (( c f(x)) - 1) (dx) ) for any $c \\in \\mathbb {C}$ , and furthermore, (x f(x) ) = S f(x) (dx).",
"The second theorem is a specific form of the Slivnyak-Mecke theorem, which is useful for computing the expected sum of a function of each point $x \\in \\Pi $ and $\\Pi \\setminus \\lbrace x\\rbrace $ over all points in a Poisson point process $\\Pi $ .",
"If each point in $\\Pi $ is thought of as relating to a particular vertex in a graph, the Slivnyak-Mecke theorem allows us to take expectations of the sum (over all possible vertices in the graph) of a function of each vertex and all its possible edges.",
"For example, it is used below to compute the expected number of active vertices by taking the expected sum of vertices that have nonzero degree.",
"We state it below for completeness, and refer to [23] and [4] for details.",
"[Slivnyak-Mecke theorem] Let $\\Pi $ be a Poisson point process on $S$ with rate measure $\\nu $ , and let $f:S \\times \\Omega \\rightarrow _+$ be measurable.",
"Then (x f(x, {x})) = S ( f(x, ) ) (dx).",
"In this section, we give the expected number of vertices and expected number of edges for the multi- and binary graph cases.",
"We begin by defining the degree $D_i$ of vertex $i$ in the multigraph and the degree $\\bar{D}_i$ of vertex $i$ in the binary graph, respectively, as Di = j M{i, j} Di = j 1(M{i, j} > 0).",
"Now we present the expected number of edges and vertices.",
"We note that both the multi- and binary graphs have the same number of (active) vertices, and so their expectations are the same.",
"[REF , main text] The expected number of vertices and edges for the multi- and binary graphs are (|Vn|) = (|Vn|) = [ 1 - (-1- (1-wv)n (dv)) ] (dw), (|En|) = n2wv (dw) (dv), (|En|) = 12(1 - (1-wv)n) (dw) (dv).",
"Using the tower property of conditional expectation and Fubini's theorem, we have that the expected number of vertices is ( |Vn| ) = (( i 1(Di > 0) | W )) = ( i P(Di > 0 | W )), followed by the definition of degree in eq:degree and the binomial density, ( |Vn| ) = ( i [1 - j P(M{i,j} = 0 | W ) ] ) = ( w W [1 - v W {w} (1 - wv)n ] ).",
"Using the Slivnyak-Mecke theorem (thm:slivnyakmecke), $\\left(|V_n|\\right) & = \\int \\left(1 - \\prod _{v \\in \\mathcal {W}} (1 - w v)^n\\right) \\nu (\\mathrm {d}w)\\\\&= \\int \\left[ 1 - \\left(\\exp \\left(n\\sum _{v \\in \\mathcal {W}} \\log (1 - w v) \\right)\\right) \\right]\\nu (\\mathrm {d}w),$ and finally by Campbell's theorem (thm:campbell) on the inner expectation, (|Vn|) = [ 1 - (-(1- (1-wv)n) (dv)) ] (dw).",
"For the expected number of edges, we can again apply the tower property and Fubini's theorem followed by repeated applications of Slivnyak-Mecke to the expectations to get: (|En|) = (( 12 ij M{i,j} | W )) = 12 ( v W nwv ) (dw) = n2 w v (dw) (dv).",
"The expected number of edges for the binary case is obtained similarly via Fubini and Slivnyak-Mecke: ( |En| ) = (12 ij P(M{i,j} > 0 | W ) ) = 12 ( w W, v W {w} (1 - (1 - w v)n )) = 12 (1 - (1 - w v)n ) (dw) (dv).",
"The asymptotic behavior of these quantities is difficult to derive directly due to the discreteness of the indices $n$ .",
"Therefore, we rely on a technique called Poissonization, which allows us to bypass this difficulty by instead considering a continuous analog of the quantities in order to get asymptotic behaviors.",
"Below, we introduce primed notation $V^{\\prime }_t, E^{\\prime }_t, \\bar{E}^{\\prime }_t, D^{\\prime }_{t,i}$ to represent the Poissonized quantities for the vertices, multigraph edges, binary edges, and the degree of a vertex, where the index $t$ now represents a continuous quantity.",
"These will be defined such that $V^{\\prime }_N$ has the same asymptotic behavior as $V_N$ , $E^{\\prime }_N$ has the same asymptotic behavior as $E_N$ , and so on.",
"Given $\\mathcal {W}$ , let $\\Pi _{ij}$ be the Poisson process generated with rate $w_i w_j$ if $i < j$ and rate 0 if $i = j$ , and let $\\Pi _{ji} = \\Pi _{ij}$ .",
"Let $\\Pi _i := \\bigcup _{j = 1}^\\infty \\Pi _{ij}$ , which is a Poisson process with rate $u_i := \\sum _{j: j \\ne i} w_i w_j$ via Poisson process superposition [35].",
"If we think of $t$ as continuous time passing, the process $\\Pi _{ij}$ represents the times at which new edges are added between vertices $i$ and $j$ , and $\\Pi _i$ represents the times at which any new edges involving vertex $i$ are added.",
"Thus, we define the Poissonized degree of vertex $i$ in the multi- and binary graph cases, respectively, to be a function of the continuous parameter $t > 0$ , Dt,i' = |i [0,t]|, Dt,i' = j 1(|ij [0,t]|>0).",
"We can define the Poissonized graph quantities of interest using these two quantities: |V't| = |V't| = i 1(D't,i > 0), |E't| = 12 i=1D't,i, |E't| = 12iD't,i.",
"The expected number of Poissonized vertices and edges for the multi- and binary graphs is (|V't|) = [ 1 - ( -(1-e-t wv) (dv)) ] (dw) (|E't|) = t2 w v (dw) (dv) (|E't|) = 12(1-(-twv)) (dw) (dv).",
"For the expected number of Poissonized vertices, we apply the tower property and Fubini's theorem to get ( |V't| ) = (( i 1(D't,i > 0) | W ) ) = ( i 1-P ( Dt,i = 0 | W) ).",
"Using the fact that $D^{\\prime }_{t,i} | \\mathcal {W}$ is Poisson-distributed, ( |V't| ) = ( i 1 - (-tui) ) = ( w W 1 - (-t w v W{w} v) ).",
"Finally, by the Slivnyak-Mecke theorem and Campbell's theorem, ( |V't| ) = (1 - (-t w v W v)) (dw) = [ 1 - ( (e-t wv-1) (dv)) ] (dw).",
"For the expected number of Poissonized edges, after applying Fubini's theorem and Slivnyak-Mecke we have ( |E't| ) = ( 12 i D't,i ) = ( 12 i (D't,i | W) ) = ( 12 i ui ) = ( 12 w W, v W {w} wv ) = 12 wv (dw) (dv).",
"For the expected number of Poissonized edges in the binary case, noting that $|\\Pi _{ij} \\cap [0,t]|$ is Poisson-distributed with rate $t w_i w_j$ , and applying Fubini's theorem and Slivnyak-Mecke, we have: (|E't|) = ( ( i D't,i | W )) = ( w W, v W {w} (1 - (-twv))) = (1 - (-twv)) (dw) (dv)."
],
[
"Asymptotics",
"We have defined the expected number of vertices and edges for the multigraph and binary graph cases (Lemma REF ) and presented the Poissonized version of these expectations (Lemma REF ).",
"We now show in Lemma REF that the expected graph quantities and their Poissonized expectations are asymptotically equivalent.",
"The Poissonized expectations for the number of vertices and the number of edges in the multi- and binary graphs are asymptotically equivalent to the original expectations; i.e., as $n \\rightarrow \\infty $ , (|V'n|) (|Vn|), (|E'n|) (|En|), (|E'n|) (|En|).",
"For the vertices, we have (|Vn|-|V'n|) = [ (-(1-e-nwv) (dv)) - ( -(1-(1-wv)n) (dv)) ] (dw) .",
"Using the elementary inequalities 0 e-nx - (1-x)n nx2e-nx, x[0, 1], n>0 0 e-a - e-b b-a, 0a b, we have 0(|Vn|-|V'n|) n(wv)2 e-nwv (dv) (dw).",
"Finally, note that n> 0, w, v [0, 1], n w v e-n wv e-1 and e-1 wv (dw) (dv) = e-1(w (dw))2 < .",
"Therefore by Lebesgue dominated convergence, 0 n (|Vn|-|V'n|) n n(wv)2 e-nwv (dv) (dw) = 0, so we have that $\\lim _{n\\rightarrow \\infty } \\left(|V_n|-|V^{\\prime }_n|\\right) = 0$ .",
"Since $(|V_n|)$ , $(|V^{\\prime }_n|)$ are monotonically increasing by inspection, $(|V_n|)\\sim (|V^{\\prime }_n|)$ , $n\\rightarrow \\infty $ , as required.",
"For the binary graph edges, (|En|-|E'n|) = 12 ((-nwv) - (1-wv)n) (dv) (dw).",
"Using the earlier elementary inequalities, 0 (|En|-|E'n|) = 12 n(wv)2e-nwv (dv) (dw).",
"This is (modulo the constant factor of 12) the exact expression in eq:vertexpdiff.",
"Therefore, the same analysis can be performed, and the result holds.",
"For multigraph edges, (|En|-|E'n|) = n2 (wv - wv) (dv) (dw) = 0, so $\\left(|E_n|\\right) \\sim \\left(|E^{\\prime }_n|\\right)$ , $n\\rightarrow \\infty $ .",
"Lemma REF allows us to analyze the asymptotics of the Poissonized expectations and apply the result directly to the asymptotics of the original graph quantities.",
"To achieve the desired asymptotics for the Poissonized expectations, we will make a further assumption on the rate measure $\\nu $ generating the vertex weights in eq:graphmodel.",
"Namely, we assume that the tails of $\\nu $ decay at a rate that will yield the appropriate weight decay in the weights $(w_j)$ —and thereby the appropriate decay in vertex creation to finally yield sparsity in the graph itself.",
"In particular, the tail of a measure $\\nu $ is said to be regularly varying if there exists a function $\\ell : \\mathbb {R}_+ \\rightarrow \\mathbb {R}_+$ and $\\alpha \\in (0, 1)$ such that x1(d w) x-(x-1), x0, c >0, x (cx)(x) = 1.",
"The condition on the function $\\ell $ is equivalent to saying that $\\ell $ is slowly varying.",
"For additional details on regular and slow variation, see [25].",
"An important equivalent formulation of eq:regvar that we will use in our following proof of the asymptotics of Poissonized expectations is provided by Lemma REF (see [27] and [13] for the proof).",
"[[27], [13]] The tail of $\\nu $ is regularly varying iff there exists a function $\\ell : \\mathbb {R}_+ \\rightarrow \\mathbb {R}_+$ and $\\alpha \\in (0, 1)$ such that 0x u (du) x1- (x-1), x 0, c >0, x (cx)(x) = 1.",
"Lemma REF is often easier to use than eq:regvar when checking whether a particular measure $\\nu $ has a regularly varying tail.",
"For example, for the three-parameter beta process, we have 0x u (du) = (1+)(1-)(+) 0x u- (1-u)+ - 1 du (1+)(1-)(+) 0x u- du, x 0 = (1+)(1-)(+) 11- x1-, so the tail of $\\nu $ is regularly varying when the discount parameter $\\alpha $ satisfies $\\alpha \\in (0,1)$ with $\\ell (x^{-1})$ equal to the constant function (x-1) = 1- (1+)(1-)(+).",
"Note that the two-parameter beta process does not exhibit this behavior (since in this case, $\\alpha = 0$ ).",
"Given the two formulations of a measure $\\nu $ with a regularly varying tail above, we are ready to characterize the asymptotics of the earlier Poissonized expectations.",
"If the tail of $\\nu $ is regularly varying as per eq:regvar, then as $n\\rightarrow \\infty $ , (|V'n|) =(n(n)), (|E'n|) = (n), (|E'n|) = O((n)(n1+2,(n)n32)).",
"Throughout this proof we use $c$ to denote a constant; the precise value of $c$ changes but is immaterial.",
"We also define the tail of $\\nu $ as $\\bar{\\nu }(x) := \\int _x^1\\nu (\\mathrm {d}w)$ , for notational brevity.",
"Furthermore, recall that we assume the rate measure $\\nu $ satisfies $\\int w \\nu (dw) <\\infty $ .",
"We first examine the expected number of Poissonized vertices, (|V'n|) = [1 - ( -(1-e-n wv) (dv))] (dw), by splitting the integral into two parts.",
"First, by the assumption that the tail of $\\nu $ is regularly varying, n-11 [1 - ( -(1-e-n wv) (dv))] (dw) n-11 (dw) c n (n).",
"Next, we upper bound the integral term 0n-1 [1 - ( -(1-e-n wv) (dv))] (dw) 0n-1(1-e-nwv) (dv)(dw) 0n-1nwv(dv)(dw) (v(dv)) n 0n-1w(dw) c n(n), where the asymptotic behavior in the last line follows from Lemma REF .",
"Thus, combining the upper bounds on pois-verts-upper1 and pois-verts-upper2 gives the bound for the entire integral: $\\left(|V^{\\prime }_n|\\right) = O(n^\\alpha \\ell (n))$ .",
"Now we bound the integral below: n-11 [ 1 - ( -(1-e-n wv) (dv))] (dw) n-11 [ 1 - (-(1-e- v ) (dv)) ] (dw) =(n-11(dw))(1 - ( -(1-e- v)(dv))) c n(n), where the last line follows from the assumption that the tail of $\\nu $ is regularly varying.",
"The second piece of the integral on $[0, n^{-1}]$ is bounded below by 0, and in combination, we have that $n^\\alpha \\ell (n) = O\\left(\\left(|V^{\\prime }_n|\\right)\\right)$ .",
"Now combining this with the previous upper bound result, we have $\\left(|V^{\\prime }_n|\\right) = \\Theta (n^\\alpha \\ell (n))$ .",
"The expected number of Poissonized multigraph edges satisfies $\\left(E^{\\prime }_n\\right) =\\Theta (n)$ , since (|En'|) = n2 wv (dw)(dv) = n2 w(dw) v(dv) = c2 2 n. For the Poissonized binary graph edges, we split the integral into two pieces.",
"We first upper bound the integral on the interval $[0, n^{-{1}{2}}]$ and apply gnedin13 to get the following asymptotic behavior: 120n-1/2 (1-(-nwv)) (dw) (dv) 120n-1/2 nwv (dw) (dv) = n2(w(dw))0n-1/2v (dv) c n (n-1/2)1-(n1/2) = cn1+2(n1/2).",
"We then bound the second portion on the interval $[n^{-{1}{2}}, 1]$ by linearizing at $v = n^{-1/2}$ .",
"Using integration by parts and an Abelian theorem [25] for the Laplace transform, for some constants $b, d > 0$ , we have 12 n-1/21 (1-(-nwv)) (dw) (dv) 12n-1/21 (1-(-n1/2 w) + nw(-n1/2 w) (v-n-1/2)) (dw) (dv) = 12 (n-1/21 (dv)) n1/2 (-n1/2 w) (w) dw +12n-1/21 (nv-n1/2) (dv) w(-n1/2w) (dw) b n2(n1/2) +12 01v (dv) n1/2 n1/2 ((-n1/2 w) - n1/2 w(-n1/2 w)) (w) dw b n2(n1/2) +1201v (dv) n1/2 n1/2 (-n1/2 w) (w) dw b n2(n1/2) +d n1/2 n/2 (n1/2) = O(n1+2(n1/2)).",
"Therefore we have that $\\left(|\\bar{E}^{\\prime }_n|\\right) = O(n^{\\frac{1+\\alpha }{2}}\\ell ({n}^{1/2}))$ .",
"To get the other bound, we split the integral into three pieces.",
"First, 120n-1 (1-(-nwv)) (dw) (dv) 120n-1 nwv (dw) (dv) = n2(w (dw)) 0n-1v (dv) c n (n-1)1-(n) = cn(n).",
"Next, integration by parts yields 12 n-1/21 (1-(-nwv)) (dw) (dv) 12n-1/21(1-(-nw)) (dw) (dv) = 12(n-1/21 (dv))n(-nw) (w) dw c (n-1/2)- (n1/2) n(n) = cn32(n)(n1/2).",
"Finally, integration by parts yields the final upper bound 12 n-1n-1/2 (1-(-nwv)) (dw) (dv) 12n-1n-1/2 (1-(-n1/2 w)) (dw) (dv) = 12(n-1n-1/2 (dv)) n1/2 (-n1/2 w) (w) dw (c1 n(n) - c2n2(n1/2)) (c3n/2 (n1/2)) cn32(n)(n1/2).",
"Therefore $\\left(|\\bar{E}^{\\prime }_n|\\right) =O(\\ell (n)\\ell ({n}^{1/2}) \\,n^{\\frac{3\\alpha }{2}})$ .",
"Finally, we show that $|E_n|$ , $|\\bar{E}_n|$ , and $|V_n|$ are asymptotically equivalent to their expectations almost surely; thus, the asymptotic results for the expectation sequences applies to the random sequences.",
"[REF , main text] The number of edges and vertices for both the multi- and binary graphs satisfy |En| a.s. (|En|), |En| a.s. (|En|) |Vn| = |Vn| a.s. (|Vn|), n. We use $X_n$ to refer to $|E_n|$ , $|\\bar{E}_n|$ , or $|V_n|$ , since the proof technique is the same for all.",
"Since we need to show $X_n/\\left(X_n\\right) \\overset{\\text{a.s.}}{\\rightarrow } 1$ , by the Borel-Cantelli lemma it is sufficient to show that for any $\\epsilon > 0$ , n P(|Xn-(Xn)| > (Xn)) < .",
"By the union bound, and the fact that $X_n$ can be expressed as a countable sum of indicators combined with the note after Theorem 4 in [26], P(|Xn-(Xn)| > (Xn)) P(Xn > (1+) (Xn)) + P(Xn < (1-)(Xn)) 2 (-2 (Xn)2 ).",
"Since $(X_n) \\ge n^\\beta $ for some $\\beta > 0$ , the expression is summable and the result holds.",
"Combining the results of Lemmas REF , REF , and REF gives us the main theorem, which we state here for completeness.",
"[REF , main text] If the tail of $\\nu $ is regularly varying as per eq:regvar, then as $n\\rightarrow \\infty $ , |Vn| (n(n)), |En| (n), |En| O((n1/2)(n1+2,(n)n32)).",
"Finally, to conclude that there exists a class of sparse, edge-exchangeable graphs, we examine the asymptotics from this result in more detail.",
"In the multigraph case, we see that the number of vertices increases at the same rate as $n^{\\alpha }\\ell (n)$ , and the number of edges increases linearly in $n$ .",
"So $|E_n|$ grows at the same rate as $|V_n|^{1/\\alpha }\\ell (n)^{-1/\\alpha }$ .",
"When $\\alpha \\in ({1}{2}, 1)$ , the exponent $1/\\alpha $ lies in the range $(1,2)$ , and thus this parameter range for $\\alpha $ results in sparse graph sequences.",
"For binary graphs, the number of edges $|\\bar{E}_n|$ grows at a rate that is bounded by $\\ell (\\sqrt{n})\\min \\left\\lbrace |V_n|^{\\frac{1+\\alpha }{2\\alpha }} \\ell (n)^{-\\frac{1+\\alpha }{2\\alpha }}, |V_n|^{\\frac{3}{2}} \\ell (n)^{-\\frac{1}{2}}\\right\\rbrace $ .",
"Since $\\min \\left\\lbrace \\frac{1+\\alpha }{2\\alpha }, \\frac{3}{2}\\right\\rbrace \\le {3}{2} < 2$ , binary graphs are sparse for any $\\alpha \\in (0, 1)$ .",
"Note that $\\ell (n)$ does not affect the growth rate throughout since it is a slowly-varying function; i.e., for all $c>0$ , $\\ell (cn) \\sim \\ell (n)$ .",
"For the three-parameter beta process, which we examined in our simulations, the function $\\ell $ is a constant function, as in eq-3bp-const.",
"We have shown that edge exchangeability admits sparse graphs by proving the existence of sparse graph sequences in a wide subclass of graph frequency models: those frequency models with weights generated from Poisson point processes whose rate measures have power law tails.",
"Notably, we have shown the existence of a range of sparse and dense behavior in this wide class of graph frequency models, as desired."
]
] | 1612.05519 |
[
[
"Fluid-solid-electric energy transport along piezoelectric flags"
],
[
"Abstract The fluid-solid-electric dynamics of a flexible plate covered by interconnected piezoelectric patches in an axial steady flow are investigated using numerical simulations based on a reduced-order model of the fluid loading for slender structures.",
"Beyond a critical flow velocity, the fluid-solid instability results in large amplitude flapping of the structure.",
"Short piezoelectric patches positioned continuously along the plate convert its local deformation into electrical currents that are used within a single internal electrical network acting as an electric generator for the external output circuit.",
"The relative role of the internal and external impedance on the energy harvesting of the system is presented and analyzed in the light of a full modeling of the electric and mechanical energy exchanges and transport along the structure."
],
[
"Introduction",
"Flow-induced vibrations have been extensively studied for the last 50 years: stemming from fundamental instabilities in the coupled dynamics of a moving solid body and a surrounding flow, they generate spontaneous, self-sustained and often large amplitude vibrations, that effectively convert some of an incoming flow's kinetic energy into solid kinetic or elastic energy [1], [2].",
"Because of their critical and often damaging impact in industrial applications, most existing research has focused on the control of their linear dynamics in order to prevent the development of large amplitude vibrations [3], [4].",
"The last decade has seen a renewed interest for these classical instabilities as energy harvesting systems, converting with an electric generator the vibration energy resulting from transverse galloping [5], airfoil flutter [6], vortex-induced vibrations [7] and axial flutter of flexible structures [8], [9].",
"The latter, also known as “flapping flag” instability, is the result of the coupling of solid inertia and rigidity, to the destabilizing fluid forces resulting from the unsteady deflection of the flow by the moving structure [4], [10]: beyond a critical flow velocity, large amplitude flapping develops, characterized by bending waves propagating along the plate [11], [12], [13].",
"Two main approaches have been proposed to harvest the associated energy: (i) the mechanical coupling of the flapping motion to a generator through its rotating mast [14], and (ii) the use of electro-active materials (e.g.",
"piezoelectric materials) to directly convert the plate's deformation into an electric current [15], [16], [17].",
"The present work focuses on the modeling of a piezoelectric flapping plate, for which an explicit description of the two-way electro-mechanical coupling and a more relevant definition of the harvesting efficiency have been obtained [17], [9], in contrast with empirical damping models for the harvesting process [18], [8].",
"Modeling of such piezoelectric flags have so far followed two distinct routes: (i) a continuous approach, where the energy associated with the local bending is used locally into independent circuits [17], [9], [19] and (ii) a discrete approach, where the structure is covered by a single element (or a small number) powering a single circuit [20], [21], [22], [23].",
"Beyond its formal simplicity, the main advantage of the former is its ability to exploit the entire structure's deformation, regardless of the deformation mode excited by the fluid-solid coupling.",
"The latter is however the most relevant for applications as it corresponds to a single output circuit, but the use of a single piezoelectric element effectively performs an average of the deformation, reducing the efficiency of the system [21].",
"The present work investigates an alternative approach that fully exploits the complex deformation of the structure, by using many short interconnected piezoelectric elements to create a single internal electrical network that can be connected to an external load.",
"This electrical structure allows for the coupling of propagating bending and electrical waves, and richer electromechanical energy exchanges between the flapping flag and the output circuit.",
"The paper is organized as follows.",
"In Section , the model and equations governing the dynamics of the piezoelectric flag are presented, in particular focusing on the original nonlocal circuit design and the resulting electromechanical exchanges within this fluid-solid-electric system.",
"The resulting efficiency is then discussed in Section , focusing in particular on the role of the circuit's properties.",
"Building upon those results, Section analyzes in detail the electrical energy fluxes along the flag, and potential routes of optimization of the harvester's design.",
"Finally, conclusions and perspectives are presented in Section .",
"The energy harvester considered in this work is a thin inextensible flexible plate (or “flag”) placed in an incoming uniform flow of velocity $U_\\infty $ and density $\\rho $ , and covered by piezoelectric patches on each side.",
"The plate is rectangular with dimensions $L$ and $H$ in the stream-wise and cross-flow directions, and its thickness is $h\\ll H,\\, L$ .",
"The piezoelectric plate assembly is supposed to have homogeneous structural properties and the effective mass per unit length and flexural rigidity are noted $\\rho _s$ and $B$ , respectively.",
"The plate is clamped parallel to the flow at its leading edge, and is free to deform under the effect of its internal dynamics and of the flow forces.",
"For simplicity, we condider here only purely planar deformations of the structure (i.e.",
"twisting and cross-flow displacement are neglected).",
"The deformation of the plate periodically stretches and compresses the piezoelectric layers positioned on each side of the flag's surface, leading to a reorganization of their internal electrical structure and to an electric charge transfer between the electrodes of each patch.",
"These patches are all identical and positioned by pairs (i.e.",
"one patch on each side), shunted through the flag's surface; the polarities of the patches within each pair are reversed so that the effect of stretching of one patch and compression of the other during the flag's bending motion are additive [24], [17].",
"The remaining two electrodes of each pair are connected to the electrical network (Figure REF ).",
"The electric state of the piezoelectric pair is characterized by the electric current and voltage between its free electrodes, noted respectively $\\dot{Q}_i$ and $V_i$ for the $i$ -th pair.",
"The electro-mechanical coupling is two-fold a direct coupling: the deformation of the flag induces a charge transfer so that $Q_i=CV_i+\\chi [\\theta (s_i^+)-\\theta (s_i^-)],$ where $C$ is the internal capacitance of the patch pair, $s_i^\\pm $ the Lagrangian coordinate of the leading and trailing edge of the patch along the flag's centerline, and $\\chi $ the electro-mechanical coupling that includes material and geometrical properties of the assemply [24], [17] a reverse or feedback coupling: the voltage within the pair induces an electric field inside the patch, resulting in a mechanical stress and an additional torque on the structure, $-\\chi V_i$ applied between $s_i^-$ and $s_i^+$ ."
],
[
"Piezoelectric coverage",
"Our previous work on piezoelectric flags exclusively focused on local circuits: the energy extracted from the mechanical deformation is dissipated in an electric loop connected solely to that region, and there is no electrical energy exchange between different piezoelectric pairs.",
"Such local circuits can take two forms: (i) one or a few patches cover the flag and energy is transferred to a small number of output circuits [20], [21], [22] or (ii) a large number of piezoelectric patches is considered so that a continuous limit can be used [17], [9], [19].",
"The advantage of the former is its simplicity and relevance to experiments (single output circuit).",
"However, from a modeling point of view, this introduces discontinuities in the piezoelectric forcing on the flag; more importantly, the finite length of the piezoelectric patch effectively acts as an averaging filter in space: the forcing on the electric circuit is only a function of the change in orientation between $s_i^-$ and $s_i^+$ , and not of the detailed bending.",
"As a result, more energy can be harvested in the continuous limit consisting of many short piezoelectric patches and associated circuits, although a careful design of a finite number of a few piezoelectric patches allows to approach almost the same efficiency as that of the continuous limit [21].",
"We consider here the alternative approach of interconnecting the different piezoelectric patch pairs electrically, so that energy can be transferred along the flag both mechanically and electrically.",
"Adjacent pairs $i$ and $i+1$ are connected by two impedances (one on each side) $Z_i^A$ and $Z_i^B$ (Figure REF ).",
"The advantage of this approach is twofold: (i) focusing on the limit of many small patches, providing a continuous coverage of the flag (i.e.",
"$s_i^+=s_{i+1}^-$ and $s_i^+-s_i^-=\\mathrm {d}s\\rightarrow 0$ ) allows for a maximum forcing of the circuit by removing any spatial average introduced by a finite patch length $l$ ; (ii) the integrated form of this connection provides the possibility to power a single output circuit with the entire apparatus by connecting the output load to the free electrodes located at the leading or trailing edge.",
"Figure: Piezoelectric flag in a uniform flow.",
"The surface of the flag is covered on both sides by piezoelectric patches (in grey) that are connected to their immediate neighbors.Applying Kirchhoff's circuit laws (Figure REF ) leads to $\\dot{Q}_i=- I_i^A + I_{i-1}^A= - I_i^B + I_{i-1}^B, \\\\V_{i+1}-V_{i}=-Z_i^AI_i^A-Z_i^BI_i^B.",
"$ Figure: (Top) Local electric circuit: each piezoelectric pair is equivalent from an electric point of view to a current generator and an internal capacitance.",
"The current through a patch pair and the voltage at its free electrodes are respectively V i V_i and Q ˙ i \\dot{Q}_i.",
"(Bottom) Boundary conditions for leading edge harvesting, Eq.",
"()."
],
[
"Continous model for the electrical network",
"We follow here the approach presented in [17], by taking $\\mathrm {d}s\\rightarrow 0$ .",
"We define $i_A$ , $i_B$ and $v$ the continuous functions of $s$ such that $i_A(s_i)=I_i^A$ , $i_B(s_i)=I_i^B$ and $v(s_i)=V_i$ .",
"Writing $q(s)$ the lineic charge transfer between the two layers of piezoelectric patches, $c$ , $z_A$ and $z_B$ the lineic internal capacitance and internal impedance, the previous equations can be rewritten as $q&=cv+\\chi \\frac{\\partial \\theta }{\\partial s},\\\\\\frac{\\partial ^2 v}{\\partial s^2}&=z\\cdot \\dot{q},$ with $z=z_A+z_B$ .",
"Equation (REF ) indeed leads to $\\frac{\\partial i_A}{\\partial s}=\\frac{\\partial i_B}{\\partial s}=-\\dot{q}$ , or equivalently $i_A=i_B=i$ provided that the leading and trailing edges of the flag are not connected to each other by an outer circuit (i.e.",
"there is not net current flowing through the flag).",
"In that case, the lineic impedance distribution between the two sides of the flag does not affect the dynamics and only their sum is relevant.",
"In the following, we focus exclusively on a resistive connection between the different piezoelectric pairs, so that $z=r$ is the lineic resistance associated with the piezoelectric connection.",
"The dynamics of the electrical circuit are therefore driven by $\\frac{\\partial v}{\\partial t}-\\frac{1}{rc}\\frac{\\partial ^2v}{\\partial s^2}+\\frac{\\chi }{c}\\frac{\\partial ^2\\theta }{\\partial s\\partial t}=0$"
],
[
"Equations of motion",
"The additional piezoelectric torque applied on the structure now simply writes $-\\chi v(s)$ .",
"An Euler–Bernoulli model is considered here to describe the two-dimensional motion of the piezoelectric plate $\\rho _s\\frac{\\partial ^2\\mathbf {x}}{\\partial s^2}=\\frac{\\partial }{\\partial s}\\left[T\\mathbf {t}-\\mathbf {n}\\frac{\\partial }{\\partial s}\\left(B\\frac{\\partial \\theta }{\\partial s}-\\chi v\\right)\\right]+\\mathbf {F}_\\textrm {fluid}, \\qquad \\frac{\\partial \\mathbf {x}}{\\partial s}=\\mathbf {t}.$ In the previous equation, ($\\mathbf {t}$ ,$\\mathbf {n}$ ) are the local unit tangent and normal vectors in the plane of motion, $T$ is the tension within the structure and $\\mathbf {F}_\\textrm {fluid}$ is the fluid force on the plate.",
"The second equation imposes the inextensibility of the flag.",
"The flag is clamped at the leading edge and free at its trailing edge (the internal tension, torque and shear force vanish).",
"Therefore, $\\textrm {at } s=0,& \\qquad \\mathbf {x}=0,\\quad \\theta =0,\\\\\\textrm {at }s=L,&\\qquad T=B\\frac{\\partial \\theta }{\\partial s}-\\chi v=B\\frac{\\partial ^2\\theta }{\\partial s^2}-\\chi \\frac{\\partial v}{\\partial s}=0.$ Up to this point, the fluid-solid-electric model is completely general, regardless of the method chosen to evaluate the fluid force on the flag $\\mathbf {F}_\\textrm {fluid}$ .",
"Computing this fluid forcing can take many different routes, including direct numerical simulations of the viscous flow field [25], [26], and potential flow simulations using Panel Methods [27], point vortices [13] or vortex sheet models [28].",
"In the limit of a slender flag ($H\\ll L$ ), an asymptotic model can be obtained for the inviscid local flow forces in terms of the local solid velocity using Lighthill's Large Amplitude Elongated Body Theory [29].",
"This result based on the advection of fluid added momentum by the flow along the slender structures can also be interpreted (and proved) as an asymptotic expansion of the potential flow forces in the limit of small aspect ratio [30], [31].",
"For freely-flapping bodies, this purely inviscid model must be complemented by a dissipative drag to account for the effect of lateral flow separation [30].",
"This physical feature of the flow field is described here by a quadratic drag associated with the normal displacement of the plate [32].",
"The result is a purely local formulation of the flow forces $\\mathbf {F}_\\textrm {fluid}$ [8], [33], [9], $\\mathbf {F}_\\textrm {fluid}=-\\frac{\\pi \\rho H^2 m_a}{4}\\left(\\frac{\\partial (u_n\\mathbf {n})}{\\partial t}-\\frac{\\partial }{\\partial s}(u_t u_n\\mathbf {n})+\\frac{1}{2}\\frac{\\partial (u_n^2\\mathbf {t})}{\\partial s}\\right)-\\frac{1}{2}\\rho c_d H |u_n|u_n\\mathbf {n}, $ which is expressed solely in terms of the local relative velocity $\\mathbf {u}_r$ of the solid plate with respect to the background flow: $\\mathbf {u}_r=\\frac{\\partial \\mathbf {x}}{\\partial t}-\\mathbf {U}_\\infty =u_t\\mathbf {t}+u_n\\mathbf {n}.$ In Eq.",
"(REF ), $m_a$ and $c_d$ are the added mass and drag coefficients.",
"For the rectangular cross section considered here, $m_a=1$ and $c_d=1.8$ .",
"A main advantage of this method is that it doesn't require an explicit computation of the flow field which is embedded in Lightill's theory; this provides a strong reduction in the computational time, which is particularly convenient for large parametric or optimization analyses.",
"This feature is also one of its main drawbacks, when dealing with multiple structures or confinement.",
"A generalization of this method to deal with such configuration was recently proposed [34]."
],
[
"Output connection and energy efficiency",
"The connectivity of adjacent piezoelectric pairs leaves two pairs of electrodes free at each end of the flag, that can be connected to an output circuit.",
"In the following, we consider that the output circuit, namely a resistive load $R_\\textrm {ext}$ , is connected at one end of the flag, the other one being shunted (see Figure REF ).",
"As a result, depending on the position of the harvesting circuit, the boundary conditions at the leading and trailing edges of the flag write: $\\textrm {Leading edge harvesting: }v(s=0)&=\\frac{R_\\textrm {ext}}{r}\\frac{\\partial v}{\\partial s}(s=0)\\quad \\textrm { and }\\quad v(s=L)=0,\\\\\\textrm {Trailing edge harvesting: }v(s=0)&=0\\quad \\textrm { and }\\quad v(s=L)=-\\frac{R_\\textrm {ext}}{r}\\frac{\\partial v}{\\partial s}(s=L).$ The output resistance is a proxy for the output circuit that uses the energy produced by the flag, therefore the output power of the system is defined as $\\mathcal {P}=\\left\\langle \\frac{v_e^2}{R_\\textrm {ext}}\\right\\rangle ,$ where $v_e$ is the voltage at the output resistance ($v_e=v(s=0)$ or $v(s=L)$ for a connection at the leading or trailing edge, respectively), and the efficiency $\\eta $ of the system can be defined as $\\eta =\\frac{\\mathcal {P}}{\\mathcal {P}_\\textrm {ref}},\\qquad \\textrm {with }\\mathcal {P}_\\textrm {ref}=\\frac{1}{2}\\rho U_\\infty ^3 H\\mathcal {A},$ namely, the ratio of the output power $\\mathcal {P}$ to the kinetic energy flux $\\mathcal {P}_\\textrm {ref}$ through the surface occupied by the flag (here $\\mathcal {A}$ is the peak-to-peak flapping amplitude at the trailing edge)."
],
[
"Energy transfers along the flag",
"The flapping of a piezoelectric flag induces energy transfers between three different systems: the flowing fluid, the moving structure and the output electrical circuit.",
"The conservation of mechanical energy is obtained by projecting Eq.",
"(REF ) onto the flag's local velocity $\\frac{\\partial E_k}{\\partial t}+\\frac{\\partial E_{el}}{\\partial t}=-\\frac{\\partial \\mathcal {F}_m}{\\partial s}-\\mathcal {T}+\\mathcal {W}_f,$ where $E_k=\\rho _s|\\partial \\mathbf {x}/\\partial t|^2/2$ and $E_{el}=B(\\partial \\theta /\\partial s)^2/2$ are the local kinetic and elastic energy densities on the flag, and $\\mathcal {F}_m&=-\\frac{\\partial \\mathbf {x}}{\\partial t}\\cdot \\left[T\\mathbf {t}-\\frac{\\partial }{\\partial s}\\left(B\\frac{\\partial \\theta }{\\partial s}-\\chi v\\right)\\mathbf {n}\\right]-\\frac{\\partial \\theta }{\\partial t}\\left(B\\frac{\\partial \\theta }{\\partial s}-\\chi v\\right),\\\\\\mathcal {T}&=-\\chi v\\frac{\\partial ^2\\theta }{\\partial t\\partial s},\\\\\\mathcal {W}_f&=\\frac{\\partial \\mathbf {x}}{\\partial t}\\cdot \\mathbf {F}_\\textrm {fluid},$ are respectively the mechanical energy flux along the flag (i.e.",
"the rate of work of internal forces and torques, measured positively from leading to trailing edge), the local rate of energy transfer from the flag to the circuit (solid-to-electric energy transfer), and the rate of work of the fluid forces (fluid-to-solid energy transfer).",
"The local conservation of electrical energy within each piezoelectric pair is obtained by multiplying the time-derivative of Eq.",
"(REF ) by $v$ and writes $\\frac{\\partial E_C}{\\partial t}=\\mathcal {T}-\\mathcal {P}_{el},$ with $E_C=cv^2/2$ the energy stored in the piezoelectric capacitance, and $\\mathcal {P}_{el}=-v\\dot{q}$ the rate of energy transfer from the piezoelectric pairs to the circuit.",
"Finally, for the nonlocal circuits considered here, Eq.",
"() leads to $\\mathcal {P}_{el}=\\mathcal {P}_i+\\frac{\\partial \\mathcal {F}_{el}}{\\partial s},$ with the electrical energy flux along the flag $\\mathcal {F}_{el}$ measured positively from leading to trailing edge, and the rate of dissipation of electrical energy in the internal resistors $\\mathcal {P}_i$ , respectively defined as $\\mathcal {F}_{el}=-\\frac{v}{r}\\frac{\\partial v}{\\partial s}\\quad \\textrm {and}\\quad \\mathcal {P}_i=\\frac{1}{2r}\\left(\\frac{\\partial v}{\\partial s}\\right)^2.$ The mechanical boundary conditions on the flag imposed a fixed trailing edge and a free trailing edge, so that displacement or mechanical load vanishes at either end, in both rotation and translation.",
"Therefore, $\\mathcal {F}_m(s=0,L)=0$ (no flux of mechanical energy out of the flag).",
"The electric boundary conditions, Eqs.",
"(REF ) or (), lead to $\\mathcal {P}=-\\mathcal {F}_{el}(s=0)$ (leading edge harvesting) or $\\mathcal {P}=\\mathcal {F}_{el}(s=L)$ (trailing edge harvesting).",
"The electrical energy flux vanishes at the shunted extremity of the flag ($v=0$ ).",
"Note that it would be the same for an open circuit condition ($\\partial v/\\partial s=0$ )."
],
[
"Non-dimensional equations",
"Equations (REF ), (REF ) and (REF ) together with boundary conditions (REF )–() and (REF )–() form a closed set of equations for the flag's position $\\mathbf {x}$ , the internal tension $T$ and the voltage across the piezoelectric layers $v$ .",
"These equations are made non-dimensional using $L$ , $L/U_\\infty $ , $\\rho HL^2$ and $U_\\infty \\sqrt{\\rho _s}{c}$ as characteristic length, time, mass and voltage.",
"The problem is then completely determined by six non-dimensional parameters, namely $H^*=\\frac{H}{L},\\qquad M^*= \\frac{\\rho H L}{\\rho _s},\\qquad U^*=U_\\infty L\\sqrt{\\frac{\\rho _s}{B}},\\\\\\alpha =\\frac{\\chi }{\\sqrt{B c}},\\qquad \\beta =rcU_\\infty L, \\qquad \\beta _\\textrm {ext}=R_\\textrm {ext}cU_\\infty .$ $H^*$ is the plate's aspect ratio, and $M^*$ denotes the fluid-to-solid mass ratio: for large $M^*$ added mass effects dominate the solid inertia.",
"$U^*$ , the reduced velocity, is a relative measure of the destabilizing effect of flow forces on the flag and of the stabilization by internal rigidity.",
"$\\alpha $ is the coupling coefficient and scales both the direct and reverse coupling between the electrodynamic and mechanical problems.",
"$\\beta $ is the non-dimensional internal resistance of the circuit, and $\\beta _\\textrm {ext}$ the external reduced load of the output circuit."
],
[
"Methods",
"The non-dimensional form of Eqs (REF ),(REF ) and (REF ) and boundary conditions Eqs.",
"(REF )–() and (REF )–() are marched in time numerically using a second-order semi-explicit scheme [28], [9] in order to obtain the dynamical position of the flag $\\mathbf {x}(s,t)$ and of the internal voltage $v(s,t)$ .",
"At a given instant $\\tilde{t}$ , the equations are recast as a set of nonlinear equations $\\mathbf {F}(\\mathbf {X})=0$ , where $\\mathbf {X}$ is a vector containing the discretized version of $\\mathbf {x}$ and $v$ at $\\tilde{t}$ .",
"Integrals and derivatives in space are computed using a Chebyshev collocation method.",
"The non-linear system is solved at each time step iteratively using Broyden's method [35].",
"Initially, the internal piezoelectric capacitance is uncharged ($v=0$ ) and the flag is slightly displaced from its equilibrium position.",
"Beyond a critical flow velocity, this perturbation is exponentially amplified by the fluid-solid-electric interactions and spontaneous flapping develops [9], [19].",
"The system is marched in time until a permanent saturated regime is achieved, for which time-averages can be defined without any ambiguity.",
"The energy harvesting efficiency is a function of six non-dimensional parameters listed in Eqs.",
"(REF )–().",
"Previous publications have focused on the role of the inertia ratio $M^*$ , on the relative importance of flow velocity and bending rigidity measured in $U^*$ , on the coupling coefficient $\\alpha $ and on the aspect ratio $H^*$ [36], [11], [9].",
"The goal of the present publication is to investigate the role of the circuit's structure on the energy harvesting performance, and more specifically the effect of nonlocal electric coupling; in the following, we therefore focus on the influence of the reduced resistances $\\beta $ and $\\beta _\\textrm {ext}$ on the harvesting performance.",
"All simulations are thus performed for $H^*=0.5$ , $\\alpha =0.3$ and $U^*=15$ , a value that is sufficiently above the critical flow velocity in the absence of piezoelectric coupling to avoid any restabilization of the structure due to the fluid-solid-electric interactions."
],
[
"Tuning and harvesting efficiency",
"Previous work on energy harvesting using piezoelectric flags has identified the critical role of the synchronization of the mechanical and electrical systems to maximize the energy transfers to the output resistance, whether for purely resistive circuits (tuning, [9]) or resonant circuits ([19]).",
"In the present case of nonlocal energy harvesting, Figure REF identifies a non-trivial evolution of the efficiency with the internal and output resistances, and two optimal tuning regimes, namely for $\\beta \\sim \\beta _\\textrm {ext}=O(1)$ and for large but finite $\\beta _\\textrm {ext}$ and $\\beta $ .",
"The position of these optimal configurations in the $(\\beta ,\\beta _\\textrm {ext})$ -plane varies only weakly with the fluid-solid parameters (see in Figure REF for the role of $M^*$ which plays a critical role in selecting the flapping mode shape), although the peak efficiency achieved in those configurations and their relative magnitude may change.",
"Figure: (Top) Harvesting efficiency η\\eta as a function of β\\beta and β ext \\beta _\\textrm {ext} for (a) M * =1M^*=1 and (b) M * =10M^*=10, with a harvesting resistor positioned at the leading edge.",
"The dashed line corresponds to the optimal impedance tuning condition, Eq. ().",
"(Bottom) Flapping motion of the piezoelectric flag obtained for (c) M * =1M^*=1, β=1.95\\beta =1.95 and β ext =1.05\\beta _\\textrm {ext}=1.05, and (d) M * =10M^*=10, β=1.210 4 \\beta =1.2\\, 10^4 and β ext =140\\beta _\\textrm {ext}=140.",
"The flapping frequency is measured as (c) ω=1.7U ∞ /L\\omega =1.7U_\\infty /L and ω=6.2U ∞ /L\\omega =6.2U_\\infty /L respectively.",
"For all panels, α=0.3\\alpha =0.3, H * =0.5H^*=0.5 and U * =15U^*=15.The non-dimensional parameters $\\beta $ and $\\beta _\\textrm {ext}$ can be understood as ratios of an electric time-scale to the typical fluid-solid time scale associated with the fluid advection along the flag, and more generally the flapping frequency.",
"When $\\beta $ (resp.",
"$\\beta _\\textrm {ext}$ ) is much lower or much greater than one, the internal (resp.",
"output) resistance behaves as short or open circuit.",
"The existence of an optimal configuration for finite $\\beta $ and $\\beta _\\textrm {ext}$ is therefore expected.",
"When $\\beta _\\textrm {ext}\\ll 1$ or $\\beta _\\textrm {ext}\\gg 1$ , the output circuit effectively behaves as a short-circuit or open-circuit respectively, leading to either no voltage or current through the output circuit and no energy dissipation.",
"Similarly, when $\\beta \\ll 1$ , the internal resistor connecting neighboring piezoelectric patches effectively behave as short circuits, leading to a uniform voltage along the piezoelectric flag.",
"The current powering the output resistance is proportional to $\\partial v/\\partial s$ , therefore $\\beta \\ll 1$ results in negligible energy harvesting.",
"Finally, when $\\beta \\gg 1$ , the internal resistors effectively behave as open circuits, effectively disconnecting the different piezoelectric elements.",
"The output circuit is then only powered by the single closest patch, and for infinitesimal patches, leads to negligible efficiency."
],
[
"Tuning: a simplified model",
"The complexity of the problem comes here from the two-way coupling between the fluid, solid and electric dynamics.",
"To rationalize the results presented above, we analyse a simpler problem, namely that of a prescribed flag kinematics.",
"This is effectively equivalent to neglecting the effect on the flag's kinematics of the feedback coupling, or at least of the change in the feedback coupling introduced by varying the resistance parameters $\\beta $ and $\\beta _\\textrm {ext}$ ; this is a good approximation in the limit of small $\\alpha $ ."
],
[
"Optimal external tuning",
"For simplicity, the flag's deformation is described as a traveling wave $\\theta (s,t)=\\Re \\left[\\Theta _0\\mathrm {e}^{\\mathrm {i}(ks-\\omega t)}\\right],$ with $\\Re [\\zeta ]$ the real part of a complex number $\\zeta $ .",
"The voltage in the circuit satisfies Eq.",
"() together with boundary conditions, Eq (REF ).",
"Writing $v(s,t)=\\Re \\left[f(s)\\mathrm {e}^{-\\mathrm {i}\\omega t}\\right]$ , $f(s)$ is the unique solution of $f^{\\prime \\prime }+\\mathrm {i}\\omega rc f=k\\omega \\chi r\\Theta _0\\mathrm {e}^{\\mathrm {i}ks},\\qquad f(0)=R_\\textrm {ext}/r f^{\\prime }(0),\\qquad f(L)=0.$ Writing $a=\\sqrt{\\mathrm {i}\\omega rc}=\\sqrt{\\omega rc/2}(1+\\mathrm {i})$ , $f(s)$ is obtained as $f(s)=\\frac{k\\omega \\chi r\\Theta _0}{a^2-k^2}\\left[A\\sin (as)+B\\sin (a(s-L))+ \\mathrm {e}^{\\mathrm {i}ks}\\right]$ with $A&=-\\frac{\\mathrm {e}^{\\mathrm {i}kL}}{\\sin (aL)}\\quad \\textrm {and}\\quad B=\\frac{1+\\gamma \\left(\\frac{aL\\mathrm {e}^{\\mathrm {i}kL}}{\\sin (aL)}-\\mathrm {i}kL\\right)}{\\gamma aL \\cos (aL)+\\sin (aL)},$ and $\\gamma =\\beta _\\textrm {ext}/\\beta $ .",
"The total output power is then obtained as $\\mathcal {P}=\\left\\langle \\frac{v(s=0)^2}{2R_\\textrm {ext}}\\right\\rangle =\\frac{|f(0)|^2}{2R_\\textrm {ext}}=\\frac{(k\\omega \\chi r\\Theta _0)^2}{2R_\\textrm {ext}(k^4+\\omega ^2 r^2c^2)}\\left|1-B\\sin (aL)\\right|^2.$ After substitution, $\\mathcal {P}=\\frac{r}{2\\gamma L}\\left(\\frac{(k\\omega \\chi \\Theta _0)^2 }{k^4+\\omega ^2r^2c^2}\\right)\\left|\\frac{\\gamma aL(\\cos (aL)-\\mathrm {e}^{\\mathrm {i}kL})+\\mathrm {i}\\gamma kL\\sin (aL)}{\\gamma aL \\cos (aL)+\\sin (aL)}\\right|^2.$ Maximizing $\\mathcal {P}$ with respect to the output resistance, all other dimensional quantities being held constant, is equivalent to maximizing $\\gamma /|\\gamma aL\\cos (aL)+\\sin (aL)|^2$ with respect to $\\gamma $ .",
"It is easily shown that the optimal value for $\\gamma $ is $\\gamma _\\textrm {opt}=|\\tan (aL)/aL|$ .",
"Recalling that $aL=(1+\\mathrm {i})\\sqrt{\\beta \\bar{\\omega }/2}$ (with $\\bar{\\omega }=\\omega L/U_\\infty $ ), this leads to an optimal relationship between $\\beta _\\textrm {ext}$ and $\\beta $ : $\\left(\\frac{\\beta _\\textrm {ext}}{\\beta }\\right)^2=\\frac{1}{\\beta \\bar{\\omega }}\\left[\\frac{\\cosh (\\sqrt{2\\beta \\bar{\\omega }})-\\cos (\\sqrt{2\\beta \\bar{\\omega }})}{\\cosh (\\sqrt{2\\beta \\bar{\\omega }})+\\cos (\\sqrt{2\\beta \\bar{\\omega }})}\\right]=F(\\beta \\bar{\\omega }).$ For each value of $\\beta $ , this optimal output tuning is shown on Figure REF as a dashed line and coincides with the location of the two optimal configurations identified in the nonlinear simulations.",
"Two regimes can be identified: (i) for $\\beta \\bar{\\omega }\\lesssim 1$ , the optimal tuning of the internal and output impedance corresponds to $\\beta _\\textrm {ext}\\sim \\beta $ ($F\\sim 1$ ), and the total internal resistance and output resistance are similar; (ii) for $\\beta \\bar{\\omega }\\gtrsim 1$ , $\\beta _\\textrm {ext}\\sim \\sqrt{\\beta /\\bar{\\omega }}$ and the internal resistance dominates ($F(\\beta \\bar{\\omega })\\sim 1/\\beta \\bar{\\omega }$ ).",
"This argument explains the existence of an optimal tuning between the output resistance ($\\beta _\\textrm {ext}$ ) and its internal counterpart ($\\beta $ ), and can be understood as an optimal matching of impedance between the continuous piezoelectric layer and the output connection.",
"These results do not explain however why little energy is harvested for intermediate $\\beta $ (regardless of $\\beta _\\textrm {ext}$ )."
],
[
"Avoiding internal dissipation",
"To understand this second feature of Figure REF , we turn back to the non-dimensional form of the electric equation, Eq. ().",
"Its homogeneous part (i.e.",
"without the piezoelectric forcing) reads $\\frac{\\partial ^2v}{\\partial s^2}-\\beta \\frac{\\partial v}{\\partial t}=0, \\qquad v(0)=\\frac{\\beta _\\textrm {ext}}{\\beta }\\frac{\\partial v}{\\partial s}(0),\\qquad v(L)=0.$ which is formally equivalent to the heat equation.",
"The characteristic time of the internal electrical network can be determined by searching for $v= \\mathrm {e}^{-t/\\tau }V(s)$ .",
"After substitution in the equation above, this imposes that $\\tau = \\beta /\\lambda ^2$ with $\\lambda $ solution of $\\frac{\\tan \\lambda }{\\lambda }+\\frac{\\beta _\\textrm {ext}}{\\beta }=0.$ Following [9], we expect the dissipation to be maximum in the internal circuit when $\\omega \\tau \\approx 2\\pi $ .",
"When $\\beta /\\beta _\\textrm {ext}\\ll 1$ or $\\beta /\\beta _\\textrm {ext}\\gg 1$ , $\\lambda \\approx \\pi /2$ or $\\pi $ , respectively, which leads to $\\omega \\beta \\sim \\pi ^3$ .",
"The frequency of flapping $\\omega $ is essentially imposed by the flag motion, and this leads to a region of finite $\\beta $ where dissipation in the internal circuit is maximum, leaving little energy available to the output circuit (Figure REF ).",
"Note that this $\\beta $ -range depends only weakly on $\\beta _\\textrm {ext}$ .",
"Figure: Non-dimensional harvested power 𝒫\\mathcal {P} (a) and internal dissipation 𝒫 i \\mathcal {P}_\\textrm {i} (b)t as a function of varying internal and external resistances β\\beta and β ext \\beta _\\textrm {ext}.",
"For both panels, the dimensional powers are scaled by ρU ∞ 3 HL\\rho U_\\infty ^3HL, and M * =1M^*=1, α=0.3\\alpha =0.3, H * =0.5H^*=0.5 and U * =15U^*=15The optimal harvesting conditions for nonlocal electric circuits can therefore be summarized as follows: An optimal tuning of the internal and external impedances so that energy flowing to the harvesting end is entirely dissipated in the output resistor and only little energy is reflected.",
"A minimization of the internal dissipation by avoiding the perfect tuning condition between the flapping flag and the internal circuit.",
"It should be noted that these conclusions are intrinsically linked to the general flapping pattern of the flag and more specifically the propagation of bending waves that act as a forcing mechanism on the circuit through the electro-mechanical coupling.",
"The detailed fluid dynamics around the flag only plays a secondary role as exemplified by the agreement of the simulations and the results of simplified model.",
"While a more complex representation of the flow field (e.g.",
"using direct numerical simulations of the flow field) is likely to modify the exact details of the flapping pattern and the values of the harvested energy, the main results presented here, in particular the optimal harvesting conditions, would only be marginally modified."
],
[
"Electric energy transfers along the flag",
"The previous results emphasize the critical role of energy transport along the nonlocal electrical circuit.",
"In the analysis of energy transfers proposed in Section REF , this corresponds to the electric flux $\\mathcal {F}_{el}$ which is the rate of electrical energy transfer in the flow direction (left to right) at location $s$ .",
"Because the output resistor can not store electrical energy, the output power $\\mathcal {P}$ is simply $-\\mathcal {F}_{el}(0)$ (resp.",
"$\\mathcal {F}_{el}(L)$ ) for a resistance located upstream (resp.",
"downstream).",
"Figure: Comparison of the electrical energy fluxes ℱ el \\mathcal {F}_{\\text{el}} with no harvesting resistor (blue) and for harvesting resistance at the leading or trailing ends (red and green, respectively) α=0.3\\alpha =0.3, H * =0.5H^*=0.5, M * =1M^*=1, U * =15U^*=15, β=1.95\\beta =1.95, β ext =1.05\\beta _{\\text{ext}}=1.05 (optimal configuration on Figure ).In the absence of any output resistance, the electrical flux must vanish at both ends.",
"Nevertheless, its variations indicate the amount of electrical energy transferred along the flag by the internal circuit (Figure REF ).",
"One easily notes that the downstream half of the flag is characterized by an electrical energy transport in the direction of the flow and of the mechanical bending waves, while the upstream half is characterized by a reverse and lower energy transport against the direction of the flow.",
"At both ends of the flags, the electrical energy flux is therefore directed toward the flag's extremities.",
"Since it must vanish there, energy must be either (i) returned to the mechanical system, and eventually the fluid flow, or (ii) dissipated in the output resistance.",
"The addition of an output resistance does not modify this general direction of transport of electrical energy $\\mathcal {F}_{el}$ , but significantly impacts its quantitative distribution, in particular in the vicinity of the harvesting extremity where $\\mathcal {F}_{el}$ is not zero anymore, as shown on Figure REF .",
"The addition of an output resistance effectively relaxes the constraint $\\mathcal {F}_{el}$ that imposed to dissipate or convert this energy when no output circuit was present: the energy can now be simply transferred to the output circuit.",
"This qualitative picture therefore suggests that an important insight on the optimal harvesting location can be gained from the distribution of electrical energy flux.",
"Indeed, larger electrical energy flux at the boundary is equivalent to a larger output efficiency by definition, and Figure REF suggests that one can determine the optimal location for the output circuit a priori from the distribution of electrical energy flux in the absence of any harvesting: a greater amount of energy transport within the internal circuit in the vicinity of one of the flag's extremity is likely to lead to greater efficiency once a harvesting resistance is added.",
"This amounts to analyzing $\\partial \\mathcal {F}_{el}/\\partial s$ near the boundary in the reference case.",
"For the configuration considered in Figure REF ($M^*=1$ ), this would suggest that trailing edge harvesting is more efficient, which is indeed confirmed by comparing the actual performance of both configurations (Figures REF and REF ).",
"For $M^*=1$ , the maximum efficiency obtained is an order of magnitude larger for trailing edge harvesting than what is obtained with a leading edge output circuit.",
"Results obtained for larger $M^*$ (higher order flapping modes) show the same trend, but the gain is much less pronounced, suggesting a more complex mechanism.",
"For both $M^*$ , a single peak is obtained in the harvested efficiency which lies on the theoretical prediction of the simplified tuning model, Eq.",
"(REF ).",
"Repeating the analysis of section REF indeed shows that the optimal link between $\\beta $ and $\\beta _\\textrm {ext}$ is not modified by moving the harvesting resistance to the trailing edge.",
"The optimal value of $\\beta $ , and its relative position with respect to the region of maximum internal dissipation, is however modified, as well as the magnitude of the efficiency peak.",
"The combination of these effects result in the existence of a single peak of efficiency (in contrast with two different peaks for leading-edge harvesting).",
"Furthermore, the distribution of electrical energy flux (Figure REF ) suggests that alternative strategies may be even more efficient, namely by placing the harvesting resistance in the regions of maximum electrical energy flux.",
"While beyond the scope of this study and modeling framework which focuses on a continuous model of the internal circuit, this opens new opportunities in the optimal design of efficient harvesting systems.",
"Figure: Harvesting efficiency η\\eta as a function of β\\beta and β ext \\beta _\\textrm {ext} for (a) M * =1M^*=1 and (b) M * =10M^*=10, with a harvesting resistor positioned at the trailing edge.",
"The dashed line corresponds to the optimal impedance tuning condition identified in Eq. ().",
"Here, α=0.3\\alpha =0.3, H * =0.5H^*=0.5 and U * =15U^*=15."
],
[
"Conclusions",
"Powering an output external circuit from the flow-induced vibrations of a flexible structure requires dealing with a double complexity.",
"On the mechanical side, flexibility allows for a continuous deformation and the solid's dynamics are characterized by a large number of degrees of freedom.",
"Efficient energy harvesting requires to carefully analyze the effect of the extraction of energy on the flapping dynamics and on the energy transfers along the structure, often requiring a global optimization approach.",
"On the electrical side, the continuous deformation of the structure must be exploited to produce a single electrical forcing to power the useful load.",
"The approach presented here proposes a novel solution to deal with both challenges, by coupling the continous mechanical system to a continuous electrical system and exploit the energy exchanges between mechanical and electrical waves along the flapping structures.",
"A minimal model for an output circuit was analyzed here, namely a single output resistance connected to one end of the flag.",
"Optimal harvesting conditions were determined in terms of the characteristic output and internal impedance.",
"Maximum energy transfer to the output circuit and maximum efficiency were obtained upon satisfying two different conditions: (i) an impedance tuning of the internal and output circuits to avoid reflection of energy, and (ii) an operating regime outside the range leading to maximum internal dissipation.",
"The analysis of the electrical energy transfers along the flag shows that energy harvesting is maximum when the output resistance is positioned near the flag's extremity where large electrical transport are present; in the absence of an output resistance, this energy needs to be either returned to the flow or dissipated internally, but the addition of an output circuit releases this constraint, and the available energy can be dissipated optimally in the harvesting circuit.",
"This analysis suggests potential optimization routes for the positioning of the harvested circuit along the flag.",
"This question is in fact critical for flow energy harvesting, beyond this particular geometry as demonstrated by several recent studies on energy harvesting using Vortex-Induced Vibrations of cables [37], [38], and should be investigated in future work for piezoelectric flags.",
"This work was supported by the French National Research Agency ANR (Grant ANR-2012-JS09-0017)."
]
] | 1612.05482 |
[
[
"Non-cooperative Fisher--KPP systems: traveling waves and long-time\n behavior"
],
[
"Abstract This paper is concerned with non-cooperative parabolic reaction--diffusion systems which share structural similarities with the scalar Fisher--KPP equation.",
"These similarities make it possible to prove, among other results, an extinction and persistence dichotomy and, when persistence occurs, the existence of a positive steady state, the existence of traveling waves with a half-line of possible speeds and a positive minimal speed and the equality between this minimal speed and the spreading speed for the Cauchy problem.",
"Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and su-perlinear competition."
],
[
"Introduction",
"In this paper, we study a large class of parabolic reaction–diffusion systems whose prototype is the so-called Lotka–Volterra mutation–competition–diffusion system: $\\left\\lbrace \\begin{matrix}\\partial _{t}u_{1}-d_{1}\\partial _{xx}u_{1}=r_{1}u_{1}-\\left(\\sum \\limits _{j=1}^{N}c_{1,j}u_{j}\\right)u_{1}-\\mu u_{1}+\\mu u_{2}\\\\\\partial _{t}u_{2}-d_{2}\\partial _{xx}u_{2}=r_{2}u_{2}-\\left(\\sum \\limits _{j=1}^{N}c_{2,j}u_{j}\\right)u_{2}-2\\mu u_{2}+\\mu u_{1}+\\mu u_{3}\\\\\\vdots \\\\\\partial _{t}u_{N}-d_{N}\\partial _{xx}u_{N}=r_{N}u_{N}-\\left(\\sum \\limits _{j=1}^{N}c_{N,j}u_{j}\\right)u_{N}-\\mu u_{N}+\\mu u_{N-1}\\end{matrix}\\right.$ where $N$ is an integer larger than or equal to 2 and the coefficients $d_{i}$ , $r_{i}$ , $c_{i,j}$ (with $i,j\\in \\lbrace 1,\\text{\\dots },N\\rbrace $ ) and $\\mu $ are positive real numbers.",
"This system can be understood as an ecological model, where $\\left(u_{1},\\text{\\dots },u_{N}\\right)$ is a metapopulation density phenotypically structured, $\\mu u_{i-1}-\\mu u_{i}$ and $\\mu u_{i+1}-\\mu u_{i}$ are the step-wise mutations of the $i$ -th phenotype with a mutation rate $\\mu $ , $d_{i}$ is its dispersal rate, $r_{i}$ is its growth rate per capita in absence of mutation, $c_{i,j}$ is the rate of the competition exerted by the $j$ -th phenotype on the $i$ -th phenotype, $\\frac{r_{i}}{c_{i,i}}$ is the carrying capacity of the $i$ -th phenotype in absence of mutation and interphenotypic competition.",
"We are especially interested in spreading properties which describe the invasion of the population in an uninhabited environment and which are expected to involve so-called traveling wave solutions.",
"Such solutions were first studied, independently and both in 1937, by Fisher [28] on one hand and by Kolmogorov, Petrovsky and Piskunov [35] on the other hand for the equation that is now well-known as the Fisher–KPP equation, Fisher equation or KPP equation: $\\partial _{t}u-\\partial _{xx}u=u\\left(1-u\\right).$ While a lot of work has been accomplished about traveling waves and spreading properties for scalar reaction–diffusion equations, the picture is much less complete regarding coupled systems of reaction–diffusion equations.",
"In particular, almost nothing is known about non-cooperative systems like the system above.",
"Before going any further, let us introduce more precisely the problem."
],
[
"Notations",
"Let $\\left(n,n^{\\prime }\\right)\\in \\left(\\mathbb {N}\\cap [1,+\\infty )\\right)^{2}$ .",
"The set of the first $n$ positive integers $\\left[1,n\\right]\\cap \\mathbb {N}$ is denoted $\\left[n\\right]$ (and $\\left[0\\right]=\\emptyset $ by convention)."
],
[
"Typesetting conventions",
"In order to ease the reading, we reserve the italic typeface ($x$ , $f$ , $X$ ) for reals, real-valued functions or subsets of $\\mathbb {R}$ , the bold typeface ($\\mathbf {v}$ , $\\mathbf {A}$ ) for euclidean vectors or vector-valued functions, in lower case for column vectors and in upper case for other matricesThis convention being superseded by the previous one when the dimension is specifically equal to 1., the sans serif typeface in upper case ($\\mathsf {B}$ , $\\mathsf {K}$ ) for subsets of euclidean spacesSame exception.",
"and the calligraphic typeface in upper case (${C}$ , ${L}$ ) for functional spaces and operators."
],
[
"Linear algebra notations",
" The canonical basis of $\\mathbb {R}^{n}$ is denoted $\\left(\\mathbf {e}_{n,i}\\right)_{i\\in \\left[n\\right]}$ .",
"The euclidean norm of $\\mathbb {R}^{n}$ is denoted $\\left|\\bullet \\right|_{n}$ .",
"The open euclidean ball of center $\\mathbf {v}\\in \\mathbb {R}^{n}$ and radius $r>0$ and its boundary are denoted $\\mathsf {B}_{n}\\left(\\mathbf {v},r\\right)$ and $\\mathsf {S}_{n}\\left(\\mathbf {v},r\\right)$ respectively.",
"The space $\\mathbb {R}^{n}$ is equipped with one partial order $\\ge _{n}$ and two strict partial orders $>_{n}$ and $\\gg _{n}$ , defined as $\\mathbf {v}\\ge _{n}\\hat{\\mathbf {v}}\\text{ if }v_{i}\\ge \\hat{v}_{i}\\text{ for all }i\\in \\left[n\\right],$ $\\mathbf {v}>_{n}\\hat{\\mathbf {v}}\\text{ if }\\mathbf {v}\\ge _{n}\\hat{\\mathbf {v}}\\text{ and }\\mathbf {v}\\ne \\hat{\\mathbf {v}},$ $\\mathbf {v}\\gg _{n}\\hat{\\mathbf {v}}\\text{ if }v_{i}>\\hat{v}_{i}\\text{ for all }i\\in \\left[n\\right].$ The strict orders $>_{n}$ and $\\gg _{n}$ coincide if and only if $n=1$ .",
"A vector $\\mathbf {v}\\in \\mathbb {R}^{n}$ is nonnegative if $\\mathbf {v}\\ge _{n}\\mathbf {0}$ , nonnegative nonzero if $\\mathbf {v}>_{n}\\mathbf {0}$ , positive if $\\mathbf {v}\\gg _{n}\\mathbf {0}$ .",
"The sets of all nonnegative, nonnegative nonzero and positive vectors are respectively denoted $\\mathsf {K}_{n}$ , $\\mathsf {K}_{n}^{+}$ and $\\mathsf {K}_{n}^{++}$ .",
"The sets $\\mathsf {K}_{n}^{+}\\cap \\mathsf {S}_{n}\\left(\\mathbf {0},1\\right)$ and $\\mathsf {K}_{n}^{++}\\cap \\mathsf {S}_{n}\\left(\\mathbf {0},1\\right)$ are respectively denoted $\\mathsf {S}_{n}^{+}\\left(\\mathbf {0},1\\right)$ and $\\mathsf {S}_{n}^{++}\\left(\\mathbf {0},1\\right)$ .",
"For any $X\\subset \\mathbb {R}$ , the sets of $X$ -valued matrices of dimension $n\\times n^{\\prime }$ and $n\\times n$ are respectively denoted $\\mathsf {M}_{n,n^{\\prime }}\\left(X\\right)$ and $\\mathsf {M}_{n}\\left(X\\right)$ .",
"If $X=\\mathbb {R}$ and if the context is unambiguous, we simply write $\\mathsf {M}_{n,n^{\\prime }}$ and $\\mathsf {M}_{n}$ .",
"As usual, the entry at the intersection of the $i$ -th row and the $j$ -th column of the matrix $\\mathbf {A}\\in \\mathsf {M}_{n,n^{\\prime }}$ is denoted $a_{i,j}$ and the $i$ -th component of the vector $\\mathbf {v}\\in \\mathbb {R}^{n}$ is denoted $v_{i}$ .",
"For any vector $\\mathbf {v}\\in \\mathbb {R}^{n}$ , $\\text{diag}\\mathbf {v}$ denotes the diagonal matrix whose $i$ -th diagonal entry is $v_{i}$ .",
"Matrices are vectors and consistently we may apply the notations $\\ge _{nn^{\\prime }}$ , $>_{nn^{\\prime }}$ and $\\gg _{nn^{\\prime }}$ as well as the vocabulary nonnegative, nonnegative nonzero and positive to matrices.",
"We emphasize this convention because of the possible confusion with the notion of “positive definite square matrix”.",
"A matrix $\\mathbf {A}\\in \\mathsf {M}_{n}$ is essentially nonnegative, essentially nonnegative nonzero, essentially positive if $\\mathbf {A}-\\min \\limits _{i\\in \\left[n\\right]}\\left(a_{i,i}\\right)\\mathbf {I}_{n}$ is nonnegative, nonnegative nonzero, positive respectively.",
"The identity of $\\mathsf {M}_{n}$ and the element of $\\mathsf {M}_{n,n^{\\prime }}$ whose every entry is equal to 1 are respectively denoted $\\mathbf {I}_{n}$ and $\\mathbf {1}_{n,n^{\\prime }}$ ($\\mathbf {1}_{n}$ if $n=n^{\\prime }$ ) .",
"We recall the definition of the Hadamard product of a pair of matrices $\\left(\\mathbf {A},\\mathbf {B}\\right)^{2}\\in \\left(\\mathsf {M}_{n,n^{\\prime }}\\right)^{2}$ : $\\mathbf {A}\\circ \\mathbf {B}=\\left(a_{i,j}b_{i,j}\\right)_{\\left(i,j\\right)\\in \\left[n\\right]\\times \\left[n^{\\prime }\\right]}.$ The identity matrix under Hadamard multiplication is $\\mathbf {1}_{n,n^{\\prime }}$ .",
"The spectral radius of any $\\mathbf {A}\\in \\mathsf {M}_{n}$ is denoted $\\rho \\left(\\mathbf {A}\\right)$ .",
"Recall from the Perron–Frobenius theorem that if $\\mathbf {A}$ is nonnegative and irreducible, $\\rho \\left(\\mathbf {A}\\right)$ is the dominant eigenvalue of $\\mathbf {A}$ , called the Perron–Frobenius eigenvalue $\\lambda _{PF}\\left(\\mathbf {A}\\right)$ , and is the unique eigenvalue associated with a positive eigenvector.",
"Recall also that if $\\mathbf {A}\\in \\mathsf {M}_{n}$ is essentially nonnegative and irreducible, the Perron–Frobenius theorem can still be applied.",
"In such a case, the unique eigenvalue of $\\mathbf {A}$ associated with a positive eigenvector is $\\lambda _{PF}\\left(\\mathbf {A}\\right)=\\rho \\left(\\mathbf {A}-\\min \\limits _{i\\in \\left[n\\right]}\\left(a_{i,i}\\right)\\mathbf {I}_{n}\\right)+\\min \\limits _{i\\in \\left[n\\right]}\\left(a_{i,i}\\right)$ .",
"Any eigenvector associated with $\\lambda _{PF}\\left(\\mathbf {A}\\right)$ is referred to as a Perron–Frobenius eigenvector and the unit one is denoted $\\mathbf {n}_{PF}\\left(\\mathbf {A}\\right)$ ."
],
[
"Functional analysis notations",
" We will consider a parabolic problem of two real variables, the “time” $t$ and the “space” $x$ .",
"A (straight) parabolic cylinder in $\\mathbb {R}^{2}$ is a subset of the form $\\left(t_{0},t_{f}\\right)\\times \\left(a,b\\right)$ with $\\left(t_{0},t_{f},a,b\\right)\\in \\overline{\\mathbb {R}}^{4}$ , $t_{0}<t_{f}$ and $a<b$ .",
"The parabolic boundary $\\partial _{P}\\mathsf {Q}$ of a bounded parabolic cylinder $\\mathsf {Q}$ is defined classically.",
"A classical solution of some second-order parabolic problem of dimension $n$ set in a parabolic cylinder $\\mathsf {Q}=\\left(t_{0},t_{f}\\right)\\times \\left(a,b\\right)$ is a solution in ${C}^{1}\\left(\\left(t_{0},t_{f}\\right),{C}^{2}\\left(\\left(a,b\\right),\\mathbb {R}^{n}\\right)\\right)\\cap {C}\\left(\\mathsf {Q}\\cup \\partial \\mathsf {Q},\\mathbb {R}^{n}\\right).$ Similarly, a classical solution of some second-order elliptic problem of dimension $n$ set in an interval $\\left(a,b\\right)\\subset \\overline{\\mathbb {R}}$ is a solution in ${C}^{2}\\left(\\left(a,b\\right),\\mathbb {R}^{n}\\right)\\cap {C}\\left(\\left(a,b\\right)\\cup \\partial \\left(a,b\\right),\\mathbb {R}^{n}\\right).$ Consistently with $\\mathbb {R}^{n}$ , the set of functions $\\left(\\mathbb {R}^{n}\\right)^{\\left(\\mathbb {R}^{n^{\\prime }}\\right)}$ is equipped with $\\mathbf {f}\\ge _{\\mathbb {R}^{n^{\\prime }},\\mathbb {R}^{n}}\\hat{\\mathbf {f}}\\text{ if }\\mathbf {f}\\left(\\mathbf {v}\\right)-\\hat{\\mathbf {f}}\\left(\\mathbf {v}\\right)\\in \\mathsf {K}_{n}\\text{ for all }\\mathbf {v}\\in \\mathbb {R}^{n^{\\prime }},$ $\\mathbf {f}>_{\\mathbb {R}^{n^{\\prime }},\\mathbb {R}^{n}}\\hat{\\mathbf {f}}\\text{ if }\\mathbf {f}\\ge _{\\mathbb {R}^{n^{\\prime }},\\mathbb {R}^{n}}\\hat{\\mathbf {f}}\\text{ and }\\mathbf {f}\\ne \\hat{\\mathbf {f}},$ $\\mathbf {f}\\gg _{\\mathbb {R}^{n^{\\prime }},\\mathbb {R}^{n}}\\hat{\\mathbf {f}}\\text{ if }\\mathbf {f}\\left(\\mathbf {v}\\right)-\\hat{\\mathbf {f}}\\left(\\mathbf {v}\\right)\\in \\mathsf {K}_{n}^{++}\\text{ for all }\\mathbf {v}\\in \\mathbb {R}^{n^{\\prime }}.$ We define consistently nonnegative, nonnegative nonzero and positive functionsRegarding functions, some authors use $>$ to denote what is here denoted $\\gg $ .",
"Thus the use of these two functional notations will be as sparse as possible and we will prefer the less ambiguous expressions “nonnegative nonzero” and “positive”..",
"The composition of two compatible functions $\\mathbf {f}$ and $\\hat{\\mathbf {f}}$ is denoted $\\mathbf {f}\\left[\\hat{\\mathbf {f}}\\right]$ , the usual $\\circ $ being reserved for the Hadamard product.",
"If the context is unambiguous, a functional space ${F}\\left(\\mathsf {X},\\mathbb {R}\\right)$ is denoted ${F}\\left(\\mathsf {X}\\right)$ .",
"For any smooth open bounded connected set $\\Omega \\subset \\mathbb {R}^{n^{\\prime }}$ and any second order linear elliptic operator ${L}:{C}^{2}\\left(\\Omega ,\\mathbb {R}^{n}\\right)\\rightarrow {C}\\left(\\Omega ,\\mathbb {R}^{n}\\right)$ with coefficients in ${C}_{b}\\left(\\Omega ,\\mathbb {R}^{n}\\right)$ , the Dirichlet principal eigenvalue of ${L}$ in $\\Omega $ , denoted $\\lambda _{1,Dir}\\left(-{L},\\Omega \\right)$ , is well-defined if ${L}$ is order-preserving in $\\Omega $ .",
"Recall from the Krein–Rutman theorem that $\\lambda _{1,Dir}\\left(-{L},\\Omega \\right)$ is the unique eigenvalue associated with a principal eigenfunction positive in $\\Omega $ and null on $\\partial \\Omega $ .",
"Sufficient conditions for the order-preserving property are: $n=1$ ; $n\\ge 2$ and the system is weakly coupled (the coupling occurs only in the zeroth order term) and fully coupled (the zeroth order coefficient is an essentially nonnegative irreducible matrix).",
"When $n\\ge 2$ , order-preserving operators are also referred to as cooperative operators."
],
[
"Setting of the problem",
"From now on, an integer $N\\in \\mathbb {N}\\cap [2,+\\infty )$ is fixed.",
"For the sake of brevity, the subscripts depending only on 1 or $N$ in the various preceding notations will be omitted when the context is unambiguous.",
"We also fix $\\mathbf {d}\\in \\mathsf {K}^{++}$ , $\\mathbf {D}=\\text{diag}\\mathbf {d}$ , $\\mathbf {L}\\in \\mathsf {M}$ and $\\mathbf {c}\\in {C}^{1}\\left(\\mathbb {R}^{N},\\mathbb {R}^{N}\\right)$ .",
"The semilinear parabolic evolution system under scrutiny is $\\partial _{t}\\mathbf {u}-\\mathbf {D}\\partial _{xx}\\mathbf {u}=\\mathbf {L}\\mathbf {u}-\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u},\\quad \\left(E_{KPP}\\right)$ the unknown being $\\mathbf {u}:\\mathbb {R}^{2}\\rightarrow \\mathbb {R}^{N}$ (although $\\left(E_{KPP}\\right)$ might occasionally be restricted to a parabolic cylinder).",
"The associated semilinear elliptic stationary system is $-\\mathbf {D}\\mathbf {u}^{\\prime \\prime }=\\mathbf {L}\\mathbf {u}-\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u},\\quad \\left(S_{KPP}\\right)$ the unknown being $\\mathbf {u}:\\mathbb {R}\\rightarrow \\mathbb {R}^{N}$ (although $\\left(S_{KPP}\\right)$ might occasionally be restricted to an interval)."
],
[
"Restrictive assumptions",
"The main restrictive assumptions are the following ones.",
"$\\mathbf {L}$ is essentially nonnegative and irreducible.",
"$\\mathbf {c}\\left(\\mathsf {K}\\right)\\subset \\mathsf {K}$ .",
"$\\mathbf {c}\\left(\\mathbf {0}\\right)=\\mathbf {0}$ .",
"There exist $\\left(\\underline{\\alpha },\\delta ,\\underline{\\mathbf {c}}\\right)\\in [1,+\\infty )^{2}\\times \\mathsf {K}^{++}$ such that $\\sum _{j=1}^{N}l_{i,j}n_{j}\\ge 0\\Rightarrow \\alpha ^{\\delta }\\underline{c}_{i}\\le c_{i}\\left(\\alpha \\mathbf {n}\\right)$ for all $\\left(\\mathbf {n},\\alpha ,i\\right)\\in \\mathsf {S}^{+}\\left(\\mathbf {0},1\\right)\\times [\\underline{\\alpha },+\\infty )\\times \\left[N\\right].$ A few immediate consequences of these assumptions deserve to be pointed out.",
"$\\left(E_{KPP}\\right)$ and $\\left(S_{KPP}\\right)$ are not cooperative and do not satisfy a comparison principle.",
"The Perron–Frobenius eigenvalue $\\lambda _{PF}\\left(\\mathbf {L}\\right)$ is well-defined and the system $\\mathbf {u}^{\\prime }=\\mathbf {L}\\mathbf {u}$ is cooperative.",
"For all $\\mathbf {v}\\in \\mathbb {R}^{N}$ , the Jacobian matrix of $\\mathbf {w}\\mapsto \\mathbf {c}\\left(\\mathbf {w}\\right)\\circ \\mathbf {w}$ at $\\mathbf {v}$ is $\\text{diag}\\mathbf {c}\\left(\\mathbf {v}\\right)+\\left(\\mathbf {v}\\mathbf {1}_{1,N}\\right)\\circ D\\mathbf {c}\\left(\\mathbf {v}\\right).$ In particular, at $\\mathbf {v}=\\mathbf {0}$ , this Jacobian is null if and only if $\\left(H_{3}\\right)$ is satisfied.",
"Also, if $D\\mathbf {c}\\left(\\mathbf {v}\\right)\\ge \\mathbf {0}$ for all $\\mathbf {v}\\in \\mathsf {K}$ , then the system $\\mathbf {u}^{\\prime }=-\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u}$ is competitive.",
"This framework contains both the Lotka–Volterra linear competition $\\mathbf {c}\\left(\\mathbf {u}\\right)=\\mathbf {C}\\mathbf {u}$ and the Gross–Pitaevskii quadratic competition $\\mathbf {c}\\left(\\mathbf {u}\\right)=\\mathbf {C}\\left(\\mathbf {u}\\circ \\mathbf {u}\\right)$ (with, in both cases, $\\mathbf {C}\\gg \\mathbf {0}$ )."
],
[
"KPP property",
"The system $\\left(E_{KPP}\\right)$ is, in some sense, a “multidimensional KPP equation”.",
"Let us recall the main features of scalar KPP nonlinearities: $f^{\\prime }\\left(0\\right)>0$ (instability of the null state), $f^{\\prime }\\left(0\\right)v\\ge f\\left(v\\right)$ for all $v\\ge 0$ (no Allee effect), there exists $K>0$ such that $f\\left(v\\right)<0$ if and only if $v>K$ (saturation).",
"Of course, our assumptions $\\left(H_{1}\\right)$ –$\\left(H_{4}\\right)$ aim to put forward a possible generalization of these features.",
"A few comments are in order.",
"Regarding the saturation property, the growth at least linear of $\\mathbf {c}$ $\\left(H_{4}\\right)$ will imply an analogous statement.",
"Ensuring uniform ${L}^{\\infty }$ estimates is really the main mathematical role of the competitive term.",
"Regarding the presence of an Allee effect, $\\mathbf {c}\\left(\\mathsf {K}\\right)\\subset \\mathsf {K}$ $\\left(H_{2}\\right)$ and $\\mathbf {c}\\left(\\mathbf {0}\\right)=\\mathbf {0}$ $\\left(H_{3}\\right)$ clearly yield that $\\partial _{t}\\mathbf {u}-\\mathbf {D}\\partial _{xx}\\mathbf {u}=\\mathbf {L}\\mathbf {u}$ is the linearization at $\\mathbf {0}$ of $\\left(E_{KPP}\\right)$ and moreover that $\\mathbf {f}:\\mathbf {v}\\mapsto \\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}$ satisfies $D\\mathbf {f}\\left(\\mathbf {0}\\right)\\mathbf {v}\\ge \\mathbf {f}\\left(\\mathbf {v}\\right)\\text{ for all }\\mathbf {v}\\in \\mathsf {K}.$ Regarding the instability of the null state, we stress here that the notion of positivity of matrices is somewhat ambiguous and, consequently, finding a natural generalization of $f^{\\prime }\\left(0\\right)>0$ is not completely straightforward.",
"In order to decide which positivity sense is the right one, we offer the following criterion.",
"On one hand, a suitable multidimensional generalization of the KPP equation should enable generalizations of the striking results concerning its scalar counterpart.",
"On the other hand, the most remarkable result about the KPP equation is that the answer to many natural questions (value of the spreading speed, persistence in bounded domains, etc.)",
"only depends on $f^{\\prime }\\left(0\\right)$ (the importance of $f^{\\prime }\\left(0\\right)$ can already be seen in the features above).",
"Thus, in our opinion, a KPP system should also be linearly determinate regarding these questions.",
"With this criterion in mind, let us explain for instance why positivity understood as positive definite matrices (i.e.",
"positive spectrum) is not satisfying.",
"In such a case, Lotka–Volterra competition–diffusion nonlinearities, whose linearization at $\\mathbf {0}$ has the form $\\text{diag}\\mathbf {r}$ with $\\mathbf {r}\\in \\mathsf {K}^{++}$ , would be KPP nonlinearities.",
"Nevertheless, it is known that the spreading speed of a competition–diffusion system is not necessarily linearly determinate (for instance, see Lewis–Li–Weinberger [38]).",
"On the contrary, the main theorems of the present paper will show unambiguously that irreducibility and essential nonnegativity $\\left(H_{1}\\right)$ supplemented with $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ is the right notion.",
"This confirmation of the relevance of $\\left(H_{1}\\right)$ –$\\left(H_{4}\\right)$ will then lead us to a general definition of multidimensional KPP nonlinearity."
],
[
"KPP-type theorems established under $\\left(H_{1}\\right)$ –{{formula:a04c28c2-f564-4c90-82c2-c270d3b5f0d2}}",
"Theorem 1.1 [Strong positivity] For all nonnegative classical solutions $\\mathbf {u}$ of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ , if $x\\mapsto \\mathbf {u}\\left(0,x\\right)$ is nonnegative nonzero, then $\\mathbf {u}$ is positive in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ .",
"Consequently, all nonnegative nonzero classical solutions of $\\left(S_{KPP}\\right)$ are positive.",
"Theorem 1.2 [Absorbing set and upper estimates] There exists a positive function $\\mathbf {g}\\in {C}\\left([0,+\\infty ),\\mathsf {K}^{++}\\right)$ , component-wise non-decreasing, such that all nonnegative classical solutions $\\mathbf {u}$ of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ satisfy $\\mathbf {u}\\left(t,x\\right)\\le \\left(g_{i}\\left(\\sup _{x\\in \\mathbb {R}}u_{i}\\left(0,x\\right)\\right)\\right)_{i\\in \\left[N\\right]}\\text{ for all }\\left(t,x\\right)\\in [0,+\\infty )\\times \\mathbb {R}$ and furthermore, if $x\\mapsto \\mathbf {u}\\left(0,x\\right)$ is bounded, then $\\left(\\limsup _{t\\rightarrow +\\infty }\\sup _{x\\in \\mathbb {R}}u_{i}\\left(t,x\\right)\\right)_{i\\in \\left[N\\right]}\\le \\mathbf {g}\\left(0\\right).$ Consequently, all bounded nonnegative classical solutions $\\mathbf {u}$ of $\\left(S_{KPP}\\right)$ satisfy $\\mathbf {u}\\le \\mathbf {g}\\left(0\\right).$ Theorem 1.3 [Extinction or persistence dichotomy] Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)<0$ .",
"Then all bounded nonnegative classical solutions of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ converge asymptotically in time, exponentially fast, and uniformly in space to $\\mathbf {0}$ .",
"Conversely, assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"Then there exists $\\nu >0$ such that all bounded positive classical solutions $\\mathbf {u}$ of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ satisfy, for all bounded intervals $I\\subset \\mathbb {R}$ , $\\left(\\liminf _{t\\rightarrow +\\infty }\\inf _{x\\in I}u_{i}\\left(t,x\\right)\\right)_{i\\in \\left[N\\right]}\\ge \\nu \\mathbf {1}_{N,1}.$ Consequently, all bounded nonnegative classical solutions of $\\left(S_{KPP}\\right)$ are valued in $\\prod _{i=1}^{N}\\left[\\nu ,g_{i}\\left(0\\right)\\right].$ As will be explained later on, the critical case $\\lambda _{PF}\\left(\\mathbf {L}\\right)=0$ is more challenging than expected and is not solved here, in spite of the following extinction conjecture.",
"Conjecture Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)=0$ and $\\text{span}\\left(\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)\\right)\\cap \\mathsf {K}\\cap \\mathbf {c}^{-1}\\left(\\left\\lbrace \\mathbf {0}\\right\\rbrace \\right)=\\left\\lbrace \\mathbf {0}\\right\\rbrace .$ Then all bounded nonnegative classical solutions of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ converge asymptotically in time and locally uniformly in space to $\\mathbf {0}$ .",
"Although REF proves that the attractor of the induced semiflow is reduced to $\\left\\lbrace \\mathbf {0}\\right\\rbrace $ in the extinction case, in the persistence case the long-time behavior is unclear and might not be reduced to a locally uniform convergence toward a unique stable steady state.",
"This direct consequence of the multidimensional structure of $\\left(E_{KPP}\\right)$ is a major difference with the scalar KPP equation.",
"Still, the following theorem provides some additional information about the steady states of $\\left(E_{KPP}\\right)$ and confirms in some sense the preceding conjecture.",
"Theorem 1.4 [Existence of steady states] If $\\lambda _{PF}\\left(\\mathbf {L}\\right)<0$ , there exists no positive classical solution of $\\left(S_{KPP}\\right)$ .",
"If $\\lambda _{PF}\\left(\\mathbf {L}\\right)=0$ and $\\text{span}\\left(\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)\\right)\\cap \\mathsf {K}\\cap \\mathbf {c}^{-1}\\left(\\left\\lbrace \\mathbf {0}\\right\\rbrace \\right)=\\left\\lbrace \\mathbf {0}\\right\\rbrace ,$ there exists no bounded positive classical solution of $\\left(S_{KPP}\\right)$ .",
"If $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ , there exists a constant positive classical solution of $\\left(S_{KPP}\\right)$ .",
"Due to the unclear long-time behavior of $\\left(E_{KPP}\\right)$ when $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ , it seems inappropriate to consider only traveling wave solutions connecting $\\mathbf {0}$ to some stable positive steady state (as is usually done in the monostable scalar setting).",
"Hence we resort to the following more flexible definition.",
"Definition A traveling wave solution of $\\left(E_{KPP}\\right)$ is a pair $\\left(\\mathbf {p},c\\right)\\in {C}^{2}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)\\times [0,+\\infty )$ which satisfies: $\\mathbf {u}:\\left(t,x\\right)\\mapsto \\mathbf {p}\\left(x-ct\\right)$ is a bounded positive classical solution of $\\left(E_{KPP}\\right)$ ; $\\left(\\liminf \\limits _{\\xi \\rightarrow -\\infty }p_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\in \\mathsf {K}^{+}$ ; $\\lim \\limits _{\\xi \\rightarrow +\\infty }\\mathbf {p}\\left(\\xi \\right)=\\mathbf {0}$ .",
"We refer to $\\mathbf {p}$ as the profile of the traveling wave and to $c$ as its speed.",
"Let us emphasize once and for all that the vector field $\\mathbf {c}$ is not to be confused with the real number $c$ .",
"The former is named after “competition” whereas the latter is traditionally named after “celerity”.",
"Theorem 1.5 [Traveling waves] Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"There exists $c^{\\star }>0$ such that: there exists no traveling wave solution of $\\left(E_{KPP}\\right)$ with speed $c$ for all $c\\in [0,c^{\\star })$ ; if, furthermore, $D\\mathbf {c}\\left(\\mathbf {v}\\right)\\ge \\mathbf {0}\\text{ for all }\\mathbf {v}\\in \\mathsf {K},$ then there exists a traveling wave solution of $\\left(E_{KPP}\\right)$ with speed $c$ for all $c\\ge c^{\\star }$ .",
"All profiles $\\mathbf {p}$ satisfy $\\mathbf {p}\\le \\mathbf {g}\\left(0\\right).$ All profiles $\\mathbf {p}$ satisfy $\\left(\\liminf \\limits _{\\xi \\rightarrow -\\infty }p_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\ge \\nu \\mathbf {1}_{N,1}.$ All profiles are component-wise decreasing in a neighborhood of $+\\infty $ .",
"When traveling waves exist for all speeds $c\\ge c^{\\star }$ , $c^{\\star }$ is called the minimal wave speed.",
"Theorem 1.6 [Spreading speed] Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"For all $x_{0}\\in \\mathbb {R}$ and all bounded nonnegative nonzero $\\mathbf {v}\\in {C}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)$ , the classical solution $\\mathbf {u}$ of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ with initial data $\\mathbf {v}\\mathbf {1}_{\\left(-\\infty ,x_{0}\\right)}$ satisfies $\\left(\\lim _{t\\rightarrow +\\infty }\\sup _{x\\in \\left(y,+\\infty \\right)}u_{i}\\left(t,x+ct\\right)\\right)_{i\\in \\left[N\\right]}=\\mathbf {0}\\text{ for all }c\\in \\left(c^{\\star },+\\infty \\right)\\text{ and all }y\\in \\mathbb {R},$ $\\left(\\liminf _{t\\rightarrow +\\infty }\\inf _{x\\in \\left[-R,R\\right]}u_{i}\\left(t,x+ct\\right)\\right)_{i\\in \\left[N\\right]}\\in \\mathsf {K}^{++}\\text{ for all }c\\in [0,c^{\\star })\\text{ and all }R>0.$ Of course, by well-posedness of $\\left(E_{KPP}\\right)$ , the solution with initial data $x\\mapsto \\mathbf {v}\\left(-x\\right)\\mathbf {1}_{\\left(-x_{0},+\\infty \\right)}$ is precisely $\\left(t,x\\right)\\mapsto \\mathbf {u}\\left(t,-x\\right)$ ($\\mathbf {u}$ being the solution with initial data $\\mathbf {v}\\mathbf {1}_{\\left(-\\infty ,x_{0}\\right)}$ ).",
"This gives the expected symmetrical spreading result (the solution with initial data $x\\mapsto \\mathbf {v}\\left(-x\\right)\\mathbf {1}_{\\left(-x_{0},+\\infty \\right)}$ spreads on the left at speed $-c^{\\star }$ ).",
"Moreover, since these two spreading results with front-like initial data actually cover compactly supported $\\mathbf {v}$ , we also get straightforwardly the spreading result for compactly supported initial data (the solution spreads on the right at speed $c^{\\star }$ and on the left at speed $-c^{\\star }$ ).",
"Consequently, $c^{\\star }$ is also called the spreading speed associated with front-like or compactly supported initial data.",
"We recall that for generic KPP problems these two spreading speeds are different as soon as the spatial domain is multidimensional.",
"In such a case, the spreading speed associated with front-like initial data generically coincides with the minimal wave speed whereas the spreading speed associated with compactly supported initial data is smaller.",
"Theorem 1.7 [Characterization and estimates for $c^{\\star }$ ] Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"We have $c^{\\star }=\\min _{\\mu >0}\\frac{\\lambda _{PF}\\left(\\mu ^{2}\\mathbf {D}+\\mathbf {L}\\right)}{\\mu }$ and this minimum is attained at a unique $\\mu _{c^{\\star }}>0$ .",
"Consequently, if we assume (without loss of generality) $d_{1}\\le d_{2}\\le \\text{\\dots }\\le d_{N},$ the following estimates hold.",
"We have $2\\sqrt{d_{1}\\lambda _{PF}\\left(\\mathbf {L}\\right)}\\le c^{\\star }\\le 2\\sqrt{d_{N}\\lambda _{PF}\\left(\\mathbf {L}\\right)}.$ If $d_{1}<d_{N}$ , both inequalities are strict.",
"If $d_{1}=d_{N}$ , both inequalities are equalities.",
"For all $i\\in \\left[N\\right]$ such that $l_{i,i}>0$ , we have $c^{\\star }>2\\sqrt{d_{i}l_{i,i}}.$ Let $\\mathbf {r}\\in \\mathbb {R}^{N}$ and $\\mathbf {M}\\in \\mathsf {M}$ be given by the unique decomposition of $\\mathbf {L}$ of the form $\\mathbf {L}=\\text{diag}\\mathbf {r}+\\mathbf {M}\\text{ with }\\mathbf {M}^{T}\\mathbf {1}_{N,1}=\\mathbf {0}.$ Let $\\left(\\left\\langle d\\right\\rangle ,\\left\\langle r\\right\\rangle \\right)\\in \\left(0,+\\infty \\right)\\times \\mathbb {R}$ be defined as $\\left\\lbrace \\begin{matrix}\\left\\langle d\\right\\rangle =\\frac{\\mathbf {d}^{T}\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\mathbf {L}\\right)}{\\mathbf {1}_{1,N}\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\mathbf {L}\\right)},\\\\\\left\\langle r\\right\\rangle =\\frac{\\mathbf {r}^{T}\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\mathbf {L}\\right)}{\\mathbf {1}_{1,N}\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\mathbf {L}\\right)}.\\end{matrix}\\right.$ If $\\left\\langle r\\right\\rangle \\ge 0$ , then $c^{\\star }\\ge 2\\sqrt{\\left\\langle d\\right\\rangle \\left\\langle r\\right\\rangle }.$"
],
[
"General definition of multidimensional KPP nonlinearity",
"The set of assumptions $\\left(H_{1}\\right)$ –$\\left(H_{4}\\right)$ supplemented with $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ can be seen as a particular case of the following definition, which we expect to be optimal with respect to the preceding collection of theorems.",
"Definition 1.8 A nonlinear function $\\mathbf {f}\\in {C}^{1}\\left(\\mathbb {R}^{N},\\mathbb {R}^{N}\\right)$ is a KPP nonlinearity if: $\\mathbf {f}\\left(\\mathbf {0}\\right)=\\mathbf {0}$ ; $D\\mathbf {f}\\left(\\mathbf {0}\\right)$ is essentially nonnegative, irreducible and $\\lambda _{PF}\\left(D\\mathbf {f}\\left(\\mathbf {0}\\right)\\right)>0$ ; $D\\mathbf {f}\\left(\\mathbf {0}\\right)\\mathbf {v}\\ge \\mathbf {f}\\left(\\mathbf {v}\\right)$ for all $\\mathbf {v}\\in \\mathsf {K}$ ; the semiflow induced by $\\partial _{t}\\mathbf {u}=\\mathbf {D}\\partial _{xx}\\mathbf {u}+\\mathbf {f}\\left[\\mathbf {u}\\right]$ with globally bounded, sufficiently regular initial data admits an absorbing set bounded in ${L}^{\\infty }\\left(\\mathbb {R}\\right)$ .",
"Let us explain more precisely how this definition differs from $\\left(H_{1}\\right)$ –$\\left(H_{4}\\right)$ supplemented with $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"Defining $\\mathbf {L}=D\\mathbf {f}\\left(\\mathbf {0}\\right),$ $\\mathbf {c}:\\mathbf {v}\\mapsto \\left\\lbrace \\begin{matrix}\\left(\\frac{1}{v_{i}}\\left(\\left(\\mathbf {L}\\mathbf {v}\\right)_{i}-f_{i}\\left(\\mathbf {v}\\right)\\right)\\right)_{i\\in \\left[N\\right]} & \\text{if }\\mathbf {v}\\ne \\mathbf {0}\\\\\\mathbf {0} & \\text{if }\\mathbf {v}=\\mathbf {0}\\end{matrix}\\right.,$ we find $\\mathbf {f}\\left(\\mathbf {v}\\right)=\\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\text{ for all }\\mathbf {v}\\in \\mathbb {R}^{N}.$ On one hand, the irreducibility and essential nonnegativity of $\\mathbf {L}$ $\\left(H_{1}\\right)$ , the positivity of its Perron–Frobenius eigenvalue, as well as the nonnegativity of $\\mathbf {c}$ on $\\mathsf {K}$ $\\left(H_{2}\\right)$ with $\\mathbf {c}\\left(\\mathbf {0}\\right)=\\mathbf {0}$ $\\left(H_{3}\\right)$ follow directly.",
"On the other hand, the ${C}^{1}$ regularity of $\\mathbf {c}$ at $\\mathbf {0}$ and its specific growth at infinity $\\left(H_{4}\\right)$ are not satisfied in general.",
"These two properties are satisfied indeed for the applications we have in mind (which will be exposed in a moment).",
"However it might be mathematically interesting to consider the case where at least one of them fails.",
"For instance, let us discuss briefly $\\left(H_{4}\\right)$ .",
"The only forthcoming result whose proof depends directly on $\\left(H_{4}\\right)$ is REF (which is remarkably one of the main assumptions of a related paper by Barles, Evans and Souganidis [6]).",
"It is easily seen that if $\\mathbf {c}$ grows sublinearly, we cannot hope in general to recover REF (in other words, under some reasonable assumptions, Barles–Evans–Souganidis’s $\\left(\\text{F}3\\right)$ is satisfied if and only if $\\left(H_{4}\\right)$ ; of course this makes $\\left(H_{4}\\right)$ even more interesting).",
"Nevertheless, this lemma is not a result in itself but a tool used for the proofs of REF as well as the existence results of REF and REF .",
"Hence relaxing $\\left(H_{4}\\right)$ mainly means finding new proofs of these results.",
"Now, without entering into too much details, we point out that if there exists $\\eta >0$ such that the following dissipative assumption: $\\left(H_{diss,\\eta }\\right)\\quad \\left\\lbrace \\begin{matrix}\\exists C_{1}\\ge 0\\quad \\forall \\mathbf {v}\\in \\mathbb {R}^{N}\\quad \\left(\\mathbf {f}\\left(\\mathbf {v}\\right)+\\eta \\mathbf {v}\\right)^{T}\\mathbf {v}\\le C_{1}\\\\\\exists C_{2}\\ge 0\\quad \\forall \\mathbf {v}\\in \\mathbb {R}^{N}\\quad D\\mathbf {f}\\left(\\mathbf {v}\\right)+\\eta \\mathbf {I}\\le C_{2}\\mathbf {1}\\\\\\exists \\left(C_{3},p\\right)\\in [0,+\\infty )^{2}\\quad \\forall \\mathbf {v}\\in \\mathbb {R}^{N}\\quad \\left|\\mathbf {f}\\left(\\mathbf {v}\\right)+\\eta \\mathbf {v}\\right|\\le C_{3}\\left(1+\\left|\\mathbf {v}\\right|^{p}\\right),\\end{matrix}\\right.$ holds, then the semiflow induced by $\\partial _{t}\\mathbf {u}=\\mathbf {D}\\partial _{xx}\\mathbf {u}+\\mathbf {f}\\left[\\mathbf {u}\\right]$ admits an attractor in some locally uniform topology which is bounded in ${C}_{b}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)$ (see Zelik [50]).",
"If the semiflow leaves $\\mathsf {K}$ invariant and if we only consider nonnegative initial data, then the quantifiers $\\forall \\mathbf {v}\\in \\mathbb {R}^{N}$ above can all be replaced by $\\forall \\mathbf {v}\\in \\mathsf {K}$ .",
"In particular, $\\mathbf {v}\\mapsto \\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}$ supplemented with $\\left(H_{1}\\right)$ –$\\left(H_{3}\\right)$ and $\\left(H_{4}^{\\prime }\\right)\\quad \\lim _{\\left|\\mathbf {v}\\right|\\rightarrow +\\infty ,\\mathbf {v}\\in \\mathsf {K}}\\left|\\mathbf {c}\\left(\\mathbf {v}\\right)\\right|=+\\infty \\text{ with at most algebraic growth}$ satisfies $\\left(H_{diss,\\eta }\\right)$ for any $\\eta >0$ .",
"(Clearly, $\\left(H_{4}\\right)\\cup \\left(H_{4}^{\\prime }\\right)$ contains every choice of $\\mathbf {c}$ such that $\\lim \\limits _{\\left|\\mathbf {v}\\right|\\rightarrow +\\infty ,\\mathbf {v}\\in \\mathsf {K}}\\left|\\mathbf {c}\\left(\\mathbf {v}\\right)\\right|=+\\infty $ .)",
"Consequently, dissipative theory provides for some slowly decaying KPP nonlinearities a proof of REF .",
"It should also provide a proof of REF , which is the key estimate to derive the existence of traveling waves, as well as a proof of the existence result of REF .",
"With these proofs at hand, all our results would be recovered."
],
[
"Cooperative or almost cooperative systems",
"The bibliography about weakly and fully coupled elliptic and parabolic linear systems is of course extensive.",
"It is possible, for instance, to define principal eigenvalues and eigenfunctions (Sweers et al.",
"[13], [47]), to prove the weak maximum principle (the classical theorems of Protter–Weinberger [42] were refined in the more involved elliptic case by Figueiredo et al.",
"[23], [24] and Sweers [47]) or Harnack inequalities (Chen–Zhao [20] or Arapostathis–Gosh–Marcus [3] for the elliptic caseThey both prove the same type of results but we will refer hereafter only to the latter because the former does not cover, as stated, the one-dimensional space case., Földes–Poláčik [29] for the parabolic case) and to use the super- and sub-solution method to deduce existence of solutions (Pao [45] among others).",
"In some sense, weakly and fully coupled systems form the “right”, or at least the most straightforward, generalization of scalar equations.",
"For (possibly nonlinear) cooperative systems, results analogous to REF REF , REF , REF and REF were established by Lewis, Li and Weinberger [39], [49].",
"Recently, Al-Kiffai and Crooks [1] introduced a convective term into a two-species cooperative system to study its influence on linear determinacy.",
"For non-cooperative systems that can still be controlled from above and from below by weakly and fully coupled systems whose linearizations at $\\mathbf {0}$ coincide with that of the non-cooperative system, Wang [48] recovered the results of Lewis–Li–Weinberger by comparison arguments.",
"Before going any further, let us point out that we will use extensively comparison arguments as well, nevertheless we will not need equality of the linearizations at $\\mathbf {0}$ .",
"This is a crucial difference between the two sets of assumptions.",
"To illustrate this claim, let us present an explicit example of system covered by our assumptions and not by Wang’s ones: take any $N\\ge 3$ , $r>0$ , $\\mu \\in \\left(0,\\frac{r}{2}\\right)$ and define $\\mathbf {L}$ and $\\mathbf {c}$ as follows: $\\mathbf {L}=r\\mathbf {I}+\\mu \\left(\\begin{matrix}-1 & 1 & 0 & \\dots & 0\\\\1 & -2 & \\ddots & \\ddots & \\vdots \\\\0 & \\ddots & \\ddots & \\ddots & 0\\\\\\vdots & \\ddots & \\ddots & -2 & 1\\\\0 & \\dots & 0 & 1 & -1\\end{matrix}\\right),$ $\\mathbf {c}:\\mathbf {v}\\mapsto \\mathbf {1}\\mathbf {v}.$ On one hand, $\\left(H_{1}\\right)$ –$\\left(H_{4}\\right)$ are easily verified, but on the other hand, the function $\\mathbf {f}:\\mathbf {v}\\mapsto \\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left[\\mathbf {v}\\right]\\circ \\mathbf {v}$ is such that, for all $i\\in \\left[N\\right]\\backslash \\left\\lbrace 1,N\\right\\rbrace $ and all $\\mathbf {v}\\in \\mathsf {K}^{++}$ , $\\frac{\\partial \\mathbf {f}_{i}}{\\partial v_{j}}\\left(\\mathbf {v}\\right)=-v_{i}<0\\text{ for all }j\\in \\left[N\\right]\\backslash \\left\\lbrace i-1,i,i+1\\right\\rbrace .$ Consequently, the application $v\\mapsto \\mathbf {f}_{i}\\left(v\\mathbf {e}_{j}\\right)$ is decreasing in $[0,+\\infty )$ .",
"This clearly violates Wang’s assumptions: this instance of $\\left(E_{KPP}\\right)$ cannot be controlled from below by a cooperative system whose linearization at $\\mathbf {0}$ is $\\partial _{t}\\mathbf {u}-\\mathbf {D}\\partial _{xx}\\mathbf {u}=\\mathbf {L}\\mathbf {u}$ .",
"Even if $\\mathbf {L}$ is essentially positive and the cooperative functions $\\mathbf {f}^{-},\\mathbf {f}^{+}$ satisfying $\\left\\lbrace \\begin{matrix}\\mathbf {f}^{-}\\left(\\mathbf {v}\\right)\\le \\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\le \\mathbf {f}^{+}\\left(\\mathbf {v}\\right)\\\\\\mathbf {f}^{-}\\left(\\mathbf {0}\\right)=\\mathbf {f}^{+}\\left(\\mathbf {0}\\right)=\\mathbf {0}\\\\D\\mathbf {f}^{-}\\left(\\mathbf {0}\\right)=D\\mathbf {f}^{+}\\left(\\mathbf {0}\\right)=\\mathbf {L}\\end{matrix}\\right.$ are constructible, in general it is difficult to verify that $\\mathbf {f}^{-}$ and $\\mathbf {f}^{+}$ have each a minimal positive zero (another requirement of Wang).",
"Our setting needs not such a verification.",
"Furthermore, even if these minimal zeros exist, several results presented here are still new.",
"REF REF adds to [48] the existence of a critical traveling wave (Wang obtained the existence of a bounded non-constant nonnegative solution traveling at speed $c^{\\star }$ but the limit at $+\\infty $ of its profile was not addressed).",
"REF , REF , REF and REF as well as REF REF , REF rely more deeply on the KPP structure and are completely new to the best of our knowledge."
],
[
"KPP systems",
"Regarding weakly coupled systems equipped with KPP nonlinearities, as far as we know most related works assume the essential positivity of $\\mathbf {L}$ , some even requiring its positivity.",
"Our results tend to show that this collection of results should be generalizable to the whole class of irreducible and essentially nonnegative $\\mathbf {L}$ $\\left(H_{1}\\right)$ provided $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"Dockery, Hutson, Mischaikow and Pernarowski [25] studied in a celebrated paper the solutions of $\\left(S_{KPP}\\right)$ in a bounded and smooth domain with Neumann boundary conditions.",
"Their matrix $\\mathbf {L}$ had the specific form $a\\left(x\\right)\\mathbf {I}+\\mu \\mathbf {M}$ where $a$ is a non-constant function of the space variable and with minimal assumptions on the constant matrix $\\mathbf {M}$ .",
"They also assumed strict ordering of the components of $\\mathbf {d}$ , explicit and symmetric Lotka–Volterra competition, vanishingly small $\\mu $ .",
"They proved the existence of a unique positive steady state, globally attractive for the Cauchy problem with positive initial data, and which converges as $\\mu \\rightarrow 0$ to a steady state where only $u_{1}$ persists.",
"More recently, the solutions of $\\left(S_{KPP}\\right)$ , still in a bounded and smooth domain with Neumann boundary conditions, were studied under the assumptions of essential positivity of $\\mathbf {L}$ and small Lipschitz constant of $\\mathbf {v}\\mapsto \\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}$ by Hei and Wu [34].",
"They established by means of super- and sub-solutions the equivalence between the negativity of the principal eigenvalue of $-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\mathbf {L}$ and the existence of a positive steady state.",
"Provided the positivity of $\\mathbf {L}$ , the vanishing viscosity limit of $\\left(E_{KPP}\\right)$ is the object of a work by Barles, Evans and Souganidis [6].",
"Although their paper and the present one differ both in results and in techniques, they share the same ambition: describing the spreading phenomenon for KPP systems.",
"Therefore our feeling is that together they give a more complete answer to the problem.",
"For two-component systems with explicit Lotka–Volterra competition, $\\mathbf {D}=\\mathbf {I}_{2}$ and symmetric and positive $\\mathbf {L}$ , REF and REF REF , REF , REF reduce to the results of Griette and Raoul [31] (see Alfaro–Griette [2] for a partial extension to space-periodic media).",
"Their paper uses very different arguments (topological degree, explicit computations involving in particular the sum of the equations, weak mutation limit, phase plane analysis) but was our initial motivation to work on this question: our intent is really to extend their result to a larger setting by changing the underlying mathematical techniques.",
"Let us emphasize that they obtained an algebraic formula for the minimal wave speed, $c^{\\star }=2\\sqrt{\\lambda _{PF}\\left(\\mathbf {L}\\right)}$ , that we are able to generalize (REF ).",
"The case $\\mathbf {D}\\ne \\mathbf {I}_{2}$ has been investigated heuristically and numerically by Elliott and Cornell [26], who considered the weak mutation limit as well and obtained further results.",
"Let us point out that the problem of the spreading speed for the Cauchy problem for the two-component system with explicit Lotka–Volterra competition was formulated but left open by Elliott and Cornell [26] as well as by Cosner [21] and not considered by Griette and Raoul [31].",
"This problem is completely solved here (see REF ).",
"Just after the submission of this paper, a paper by Moris, Börger and Crooks [41] submitted concurrently and devoted to the analytical confirmation of Elliott and Cornell’s numerical observations was brought to our attention.",
"By applying’s successfully Wang’s framework, they obtained the existence of traveling waves as well as the spreading speed for the Cauchy problem.",
"However, in order to apply Wang’s framework, they had to make additional assumptions (roughly speaking, small interphenotypic competition and small mutations) and which are in fact, in view of our results, unnecessary.",
"They also obtained very interesting results regarding the dependency on the mutation rate $\\mu $ of the spreading speed $\\lambda _{PF}\\left(\\mu _{c^{\\star }}\\mathbf {D}+\\mu _{c^{\\star }}^{-1}\\left(\\text{diag}\\mathbf {r}+\\mu \\mathbf {M}\\right)\\right)$ and the associated distribution $\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}\\mathbf {D}+\\mu _{c^{\\star }}^{-1}\\left(\\text{diag}\\mathbf {r}+\\mu \\mathbf {M}\\right)\\right).$"
],
[
"From systems to non-local equations, from mathematics to applications",
"It is well-known that systems can be seen as discretizations of continuous models.",
"In this subsection, we present briefly some equations structured not only in time and space but also with a third variable and whose natural discretizations are particular instances of our system $\\left(E_{KPP}\\right)$ satisfying the criterion $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"Our results bring therefore indirect insight into the spreading properties of these equations.",
"Since these examples provide also examples of biomathematical applications of our results, this subsection gives us the opportunity to present more precisely these applications, to explain how non-cooperative KPP systems arise in modeling situations and finally to comment our results from this application point of view.",
"Several fields of biology are concerned: evolutionary invasion analysis (also known as adaptive dynamics), population dynamics, epidemiology.",
"Applications in other sciences might also exist."
],
[
"The cane toads equation with non-local competition",
"Recall the definition of the discrete laplacian in a finite domain of cardinal $N$ , $\\mathbf {M}_{Lap,N}=\\left(\\begin{matrix}-1 & 1 & 0 & \\dots & 0\\\\1 & -2 & \\ddots & \\ddots & \\vdots \\\\0 & \\ddots & \\ddots & \\ddots & 0\\\\\\vdots & \\ddots & \\ddots & -2 & 1\\\\0 & \\dots & 0 & 1 & -1\\end{matrix}\\right)\\text{ if }N\\ge 3,$ $\\mathbf {M}_{Lap,2}=\\left(\\begin{matrix}-1 & 1\\\\1 & -1\\end{matrix}\\right)\\text{ if }N=2.$ With this notation, the Lotka–Volterra mutation–competition–diffusion system exhibited earlier reads $\\partial _{t}\\mathbf {u}-\\mathbf {D}\\partial _{xx}\\mathbf {u}=\\text{diag}\\left(\\text{$\\mathbf {r}$}\\right)\\mathbf {u}+\\mu \\mathbf {M}_{Lap}\\mathbf {u}-\\left(\\mathbf {C}\\mathbf {u}\\right)\\circ \\mathbf {u}.$ An especially interesting instance of it is the system where: for all $i\\in \\left[N\\right]$ , $d_{N,i}=\\underline{\\theta }+\\left(i-1\\right)\\theta _{N}$ with $\\theta _{N}=\\frac{\\overline{\\theta }-\\underline{\\theta }}{N-1}$ and with some fixed $\\overline{\\theta }>\\underline{\\theta }>0$ ; $\\mathbf {r}_{N}=r\\mathbf {1}_{N,1}$ with some fixed $r>0$ ; $\\mu _{N}=\\frac{\\alpha }{\\theta _{N}^{2}}$ with some fixed $\\alpha >0$ ; $\\mathbf {C}_{N}=\\theta _{N}\\mathbf {1}_{N}$ .",
"Since $\\lambda _{PF}\\left(\\mathbf {M}_{Lap,N}\\right)=0$ (because $\\mathbf {M}_{Lap,N}\\mathbf {1}_{N,1}=\\mathbf {0}$ ), the Perron–Frobenius eigenvalue of $\\mathbf {L}$ is positive indeed: $\\lambda _{PF}\\left(r\\mathbf {I}_{N}+\\frac{\\alpha }{\\theta _{N}^{2}}\\mathbf {M}_{Lap,N}\\right)=r+\\lambda _{PF}\\left(\\frac{\\alpha }{\\theta _{N}^{2}}\\mathbf {M}_{Lap,N}\\right)=r>0.$ As $N\\rightarrow +\\infty $ , this system converges (at least formally) to the cane toads equation with non-local competition and bounded phenotypes: $\\left\\lbrace \\begin{matrix}\\partial _{t}n-\\theta \\partial _{xx}n-\\alpha \\partial _{\\theta \\theta }n=n\\left(t,x,\\theta \\right)\\left(r-\\int _{\\underline{\\theta }}^{\\overline{\\theta }}n\\left(t,x,\\theta ^{\\prime }\\right)\\text{d}\\theta ^{\\prime }\\right)\\\\\\partial _{\\theta }n\\left(t,x,\\underline{\\theta }\\right)=\\partial _{\\theta }n\\left(t,x,\\overline{\\theta }\\right)=0\\text{ for all }\\left(t,x\\right)\\in \\mathbb {R}^{2}\\end{matrix}\\right.$ where $n$ is a function of $\\left(t,x,\\theta \\right)$ , $\\theta \\in \\left[\\underline{\\theta },\\overline{\\theta }\\right]$ is the motility trait, $\\alpha $ is the mutation rate and $\\int _{\\underline{\\theta }}^{\\overline{\\theta }}n\\left(t,x,\\theta ^{\\prime }\\right)\\text{d}\\theta ^{\\prime }$ is the total population present at $\\left(t,x\\right)$ .",
"This equation is named after an invasive species currently invading Australia.",
"A startling ecologic fact is that this invasion is accelerating whereas biological invasions usually occur at a constant speed (as predicted by the KPP equation).",
"However this issue is solved when the phenotypical structure is taken into account and the following spatial sorting phenomenon is understood: the fastest toads lead the invasion, reproduce at the edge of the front, give birth to a new generation of toads among which faster and slower toads can be found (as a result of mutations), and the new fastest toads take the lead of the invasion.",
"The introduction of a motility trait $\\theta $ with a local mutation term $\\alpha \\partial _{\\theta \\theta }n$ into the scalar KPP equation is then a way of verifying this theory: does it lead to accelerating invasions?",
"The answer is positive (transitory acceleration up to a constant asymptotic speed if $\\overline{\\theta }<+\\infty $ , constant acceleration if $\\overline{\\theta }=+\\infty $ ) and this is why the cane toads equation achieved some fame (we refer for instance to [7], [14], [15], [16], where more detailed modeling explanations can also be found).",
"The overcrowding effect, which is nowadays standardly taken into account in population biology modeling, is modeled by the term $-n\\left(t,x,\\theta \\right)\\int _{\\underline{\\theta }}^{\\overline{\\theta }}n\\left(t,x,\\theta ^{\\prime }\\right)\\text{d}\\theta ^{\\prime }$ which basically considers that one given toad competes with all other toads surrounding it, independently of their phenotype, and does not compete with distant toads.",
"Mathematically, this term is the only responsible for the nonlinearity, non-locality and non-cooperativity of the model: it could be tempting to neglect it.",
"However, linear growth models (which go back to Malthus) generically lead to exponential blow-up.",
"The basic idea of the literature about the cane toads equation is then exactly the same as the one we are going to use in the forthcoming proofs: point out and use the KPP nature of the problem.",
"The results of the present paper are consistent with the ones for the cane toads equation with bounded phenotypes.",
"Therefore it might be possible, in a future sequel providing new estimates uniform with respect to $N$ , to rigorously derive the cane toads equation as the continuous limit of a family of KPP systems.",
"Since the discrete version is easier to study, new results might be unfolded by this approach.",
"However, let us stress that the problem of finding these new uniform estimates is not to be underestimated and is expected to be a very difficult one.",
"At least regarding biologists, whose field measurements somehow always produce discrete classes of phenotypes instead of a continuum of phenotypes, our results bring forth an interesting new lead to address the general problem of adaptive dynamics.",
"Let us point out that if, instead of phenotypes of cane toads, the components of $\\mathbf {u}$ model different strains of virus, then we obtain an epidemiological model representing the invasion of a population of sane individuals by a structured population of infected individuals (Griette–Raoul [31]).",
"Notice that this cane toads equation is only the first step of a larger research program: a more realistic model should replace clonal reproduction by sexual reproduction and should take into account the possibility of non-constant coefficients $\\alpha $ and $r$ as well as that of a more general competition term (logistic with a non-constant weight or even non-logistic).",
"It is also interesting to consider non-local spatial or phenotypical dispersion."
],
[
"The cane toads equation with non-local mutations and competition",
"Actually, historically, the cane toads equation comes from a doubly non-local model due to Prévost et al.",
"[4], [46] (see also the earlier individual-based model by Champagnat and Méléard [19]).",
"Since the non-local mutation operator is too difficult to handle mathematically, the cane toads equation with local mutations was favored as a simplified first approach.",
"However it remains unsatisfying from the modeling point of view and non-local kernels, which could take into account large mutations, are the real aim.",
"Defining as above $\\theta _{N}=\\frac{\\overline{\\theta }-\\underline{\\theta }}{N-1}$ and $\\left(\\theta _{i}\\right)_{i\\in \\left[N\\right]}=\\left(\\underline{\\theta }+\\left(i-1\\right)\\theta _{N}\\right)_{\\ddot{\\imath }\\in \\left[N\\right]}$ , the natural discretization of the doubly non-local cane toads equation, $\\partial _{t}n-d\\left(\\theta \\right)\\partial _{xx}n=rn+\\alpha \\left(K\\star _{\\theta }n-n\\right)-n\\int _{\\underline{\\theta }}^{\\overline{\\theta }}n\\left(t,x,\\theta ^{\\prime }\\right)\\text{d}\\theta ^{\\prime }$ with $d\\in {C}\\left(\\left[\\underline{\\theta },\\overline{\\theta }\\right],\\left(0,+\\infty \\right)\\right)$ and $K\\in {C}\\left(\\mathbb {R},[0,+\\infty )\\right)$ , is $\\partial _{t}\\mathbf {u}-\\mathbf {D}_{N}\\partial _{xx}\\mathbf {u}=\\mathbf {L}_{N}\\mathbf {u}-\\left(\\theta _{N}\\mathbf {1}_{N}\\mathbf {u}\\right)\\circ \\mathbf {u},$ with $\\mathbf {d}_{N}=\\left(d\\left(\\theta _{i}\\right)\\right)_{i\\in \\left[N\\right]},$ $\\mathbf {L}_{N} & =r\\mathbf {I}_{N}+\\alpha \\left(\\theta _{N}\\left(K\\left(\\theta _{i}-\\theta _{j}\\right)\\right)_{\\left(i,j\\right)\\in \\left[N\\right]^{2}}-\\mathbf {I}_{N}\\right)\\\\& =\\left(r-\\alpha \\right)\\mathbf {I}_{N}+\\alpha \\theta _{N}\\left(K\\left(\\left(i-j\\right)\\theta _{N}\\right)\\right)_{\\left(i,j\\right)\\in \\left[N\\right]^{2}}.$ The assumptions on $\\mathbf {c}$ $\\left(H_{2}\\right)$ –$\\left(H_{4}\\right)$ are obviously satisfied and, as soon as, say, $K$ is positive, the assumption on $\\mathbf {L}$$\\left(H_{1}\\right)$ is satisfied as well.",
"Subsequently, $\\lambda _{PF}\\left(\\mathbf {L}_{N}\\right)\\ge r-\\alpha $ , whence $r>\\alpha $ is a sufficient condition to ensure $\\lambda _{PF}\\left(\\mathbf {L}_{N}\\right)>0$ for all $N\\in \\mathbb {N}$ .",
"More generally, the system corresponding to the following equation (see Prévost et al.",
"[4], [46]): $\\partial _{t}n-d\\left(\\theta \\right)\\partial _{xx}n & =r\\left(\\theta \\right)n\\left(t,x,\\theta \\right)+\\int _{\\underline{\\theta }}^{\\overline{\\theta }}n\\left(t,x,\\theta ^{\\prime }\\right)K\\left(\\theta ,\\theta ^{\\prime }\\right)\\text{d}\\theta ^{\\prime }\\\\& -n\\left(t,x,\\theta \\right)\\int _{\\underline{\\theta }}^{\\overline{\\theta }}n\\left(t,x,\\theta ^{\\prime }\\right)C\\left(\\theta ,\\theta ^{\\prime }\\right)\\text{d}\\theta ^{\\prime }$ with $d\\in {C}\\left(\\left[\\underline{\\theta },\\overline{\\theta }\\right],\\left(0,+\\infty \\right)\\right)$ , $r\\in {C}\\left(\\left[\\underline{\\theta },\\overline{\\theta }\\right],[0,+\\infty )\\right)$ , $K,C\\in {C}\\left(\\left[\\underline{\\theta },\\overline{\\theta }\\right]^{2},[0,+\\infty )\\right)$ is $\\partial _{t}\\mathbf {u}-\\mathbf {D}_{N}\\partial _{xx}\\mathbf {u}=\\mathbf {L}_{N}\\mathbf {u}-\\left(\\mathbf {C}_{N}\\mathbf {u}\\right)\\circ \\mathbf {u},$ with $\\mathbf {d}_{N}=\\left(d\\left(\\theta _{i}\\right)\\right)_{i\\in \\left[N\\right]},$ $\\mathbf {L}_{N}=\\text{diag}\\left(r\\left(\\theta _{i}\\right)\\right)_{i\\in \\left[N\\right]}+\\theta _{N}\\left(K\\left(\\theta _{i},\\theta _{j}\\right)\\right)_{\\left(i,j\\right)\\in \\left[N\\right]^{2}},$ $\\mathbf {C}_{N}=\\theta _{N}\\left(C\\left(\\theta _{i},\\theta _{j}\\right)\\right)_{\\left(i,j\\right)\\in \\left[N\\right]^{2}}.$ Again, $\\left(H_{3}\\right)$ and $\\left(H_{4}\\right)$ are clearly satisfied, $\\left(H_{2}\\right)$ is satisfied if $C$ is nonnegative and both $\\left(H_{1}\\right)$ and $\\lambda _{PF}\\left(\\mathbf {L}_{N}\\right)>0$ are satisfied if, say, $K$ is positive.",
"In both cases, of course, the positivity of $K$ is a far from necessary condition and might be relaxed.",
"To the best of our knowledge, these doubly non-local equations have been the object of no study apart from [4], [46] and are therefore still very poorly understood.",
"In particular, the traveling wave problem as well as the spreading problem are completely open.",
"Consequently, our results are highly valuable when applied to this system.",
"For mathematicians, they motivate the future work on the limit $N\\rightarrow +\\infty $ .",
"For biologists, they provide new insight into these modeling problems and show for instance how two different mutation strategies can be compared and how the spreading speed can be evaluated."
],
[
"The Gurtin–MacCamy equation with diffusion and overcrowding\neffect",
"In view of the preceding two examples, it is natural to investigate the existence of completely different applications, that is applications not concerned at all with evolutionary biology.",
"Such applications exist indeed, as shown by this third example.",
"Consider the following age-structured equation with diffusion: $\\left\\lbrace \\begin{matrix}\\partial _{t}n+\\partial _{a}n-d\\left(a\\right)\\partial _{xx}n=-n\\left(t,x,a\\right)\\left(r\\left(a\\right)+\\int _{0}^{A}n\\left(t,x,a^{\\prime }\\right)C\\left(a,a^{\\prime }\\right)\\text{d}a^{\\prime }\\right)\\\\n\\left(t,x,0\\right)=\\int _{a_{m}}^{A}n\\left(t,x,a^{\\prime }\\right)K\\left(a^{\\prime }\\right)\\text{d}a^{\\prime }\\text{ for all }\\left(t,x\\right)\\in \\mathbb {R}^{2}\\\\n\\left(t,x,A\\right)=0\\text{ for all }\\left(t,x\\right)\\in \\mathbb {R}^{2}\\end{matrix}\\right.$ where $n$ is a function of $\\left(t,x,a\\right)$ , $a\\in \\left[0,A\\right]$ is the age variable, $a_{m}\\ge 0$ is the maturation age, $A>a_{m}$ is the maximal age, $d\\in {C}\\left(\\left[0,A\\right],\\left(0,+\\infty \\right)\\right)$ is the diffusion rate, $r\\in {C}\\left(\\left[0,A\\right],\\left(0,+\\infty \\right)\\right)$ is the mortality rate, $C\\in {C}\\left(\\left[0,A\\right]^{2},[0,+\\infty )\\right)$ is the competition kernel and $K\\in {C}\\left(\\left[0,A\\right],[0,+\\infty )\\right)$ is the birth rate.",
"This equation is well-known, at least if $C=0$ , and detailed modeling explanations can be found in the classical Gurtin–MacCamy references [32], [33].",
"Defining $a_{N+1}=\\frac{A}{N},$ $\\left(a_{i}\\right)_{i\\in \\left[N\\right]}=\\left(\\left(i-1\\right)a_{N+1}\\right)_{i\\in \\left[N\\right]},$ $j_{m,N}=\\min \\left\\lbrace j\\in \\left[N\\right]\\ |\\ a_{j}\\ge a_{m}\\right\\rbrace ,$ $\\mathbf {u}\\left(t,x\\right)=\\left(n\\left(t,x,a_{i}\\right)\\right)_{i\\in \\left[N\\right]},$ $\\mathbf {d}_{N}=\\left(d\\left(a_{i}\\right)\\right)_{i\\in \\left[N\\right]},$ $\\mathbf {L}_{mortality,N}=-\\text{diag}\\left(r\\left(a_{i}\\right)_{i\\in \\left[N\\right]}\\right),$ $\\mathbf {L}_{birth,N}=a_{N+1}\\left(\\begin{matrix}0 & \\dots & 0 & K\\left(a_{j_{m,N}}\\right) & \\dots & K\\left(a_{N}\\right)\\\\0 & & & \\dots & & 0\\\\\\vdots & & & & & \\vdots \\\\0 & & & \\dots & & 0\\end{matrix}\\right),$ $\\mathbf {L}_{aging,N}=\\frac{1}{a_{N+1}}\\left(\\begin{matrix}0 & 0 & \\dots & \\dots & 0\\\\1 & -1 & \\ddots & & \\vdots \\\\0 & \\ddots & \\ddots & \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & \\ddots & 0\\\\0 & \\dots & 0 & 1 & -1\\end{matrix}\\right),$ $\\mathbf {L}_{N}=\\mathbf {L}_{mortality,N}+\\mathbf {L}_{birth,N}+\\mathbf {L}_{aging,N},$ $\\mathbf {C}_{N}=a_{N+1}\\left(C\\left(a_{i},a_{j}\\right)\\right)_{\\left(i,j\\right)\\in \\left[N\\right]^{2}},$ it follows again that $\\partial _{t}\\mathbf {u}-\\mathbf {D}_{N}\\partial _{xx}\\mathbf {u}=\\mathbf {L}_{N}\\mathbf {u}-\\left(\\mathbf {C}_{N}\\mathbf {u}\\right)\\circ \\mathbf {u}$ is the natural discretization with $\\left(H_{3}\\right)$ and $\\left(H_{4}\\right)$ automatically satisfied.",
"$K$ nonnegative nonzero and $C$ nonnegative are sufficient conditions to enforce $\\left(H_{1}\\right)$ and $\\left(H_{2}\\right)$ .",
"Since we have $\\lambda _{PF}\\left(\\mathbf {L}_{N}\\right)\\ge \\lambda _{PF}\\left(\\mathbf {L}_{birth,N}+\\mathbf {L}_{aging,N}\\right)-\\max _{\\left[0,A\\right]}r$ and since $\\lambda _{PF}\\left(\\mathbf {L}_{birth,N}+\\mathbf {L}_{aging,N}\\right)$ is bounded from below by a positive constant independent of $N$ (the proof of this claim being deliberately not detailed here for the sake of brevity), if $\\max \\limits _{\\left[0,A\\right]}r$ is small enough, then $\\lambda _{PF}\\left(\\mathbf {L}_{N}\\right)>0$ for all $N\\in \\mathbb {N}$ .",
"We point out that this KPP system differs noticeably from the Lotka–Volterra mutation–competition–diffusion system presented up to now as the main instance of KPP system: here, the matrix $\\mathbf {L}$ is highly non-symmetric.",
"This should have important qualitative consequences, numerically observable.",
"It might even be unexpected that these two systems share important properties and this makes our theorems even more interesting.",
"As far as we know, the traveling wave problem and the spreading problem for the continuous age-structured problem are completely open.",
"Therefore the earlier remarks concerning the impact of our results on the doubly non-local cane toads equation apply here as well."
],
[
"Strong positivity",
"REF is mainly straightforward and follows from the following local result.",
"Proposition 2.1 Let $\\mathsf {Q}\\subset \\mathbb {R}^{2}$ be a bounded parabolic cylinder and $\\mathbf {u}$ be a classical solution of $\\left(E_{KPP}\\right)$ set in $\\mathsf {Q}$ .",
"If $\\mathbf {u}$ is nonnegative on $\\partial _{P}\\mathsf {Q}$ , then it is either null or positive in $\\mathsf {Q}$ .",
"Let $K=\\max \\limits _{\\overline{\\mathsf {Q}}}\\left|\\mathbf {u}\\right|$ and observe that, for all $i\\in \\left[N\\right]$ and all $\\left(t,x\\right)\\in \\mathsf {Q}$ , $\\left|l_{i,i}-c_{i}\\left(\\mathbf {u}\\left(t,x\\right)\\right)\\right|\\le \\left|l_{i,i}\\right|+\\max _{\\mathbf {v}\\in \\overline{\\mathsf {B}\\left(\\mathbf {0},K\\right)}}\\left|c_{i}\\left(\\mathbf {v}\\right)\\right|.$ Then, define $\\mathbf {A}:\\left(t,x\\right)\\mapsto \\mathbf {L}-\\text{diag}\\left(\\mathbf {c}\\left(\\mathbf {u}\\left(t,x\\right)\\right)\\right).$ By the irreducibility and the essential nonnegativity of $\\mathbf {L}$ $\\left(H_{1}\\right)$ , $\\mathbf {A}\\left(t,x\\right)$ has these two properties as well for all $\\left(t,x\\right)\\in \\overline{\\mathsf {Q}}$ .",
"By the boundedness of $\\mathbf {u}$ in $\\overline{\\mathsf {Q}}$ , $\\mathbf {A}$ is bounded in $\\overline{\\mathsf {Q}}$ as well.",
"Therefore $\\mathbf {u}$ is a solution of the following linear weakly and fully coupled system with bounded coefficients: $\\partial _{t}\\mathbf {u}-\\mathbf {D}\\partial _{xx}\\mathbf {u}-\\mathbf {A}\\mathbf {u}=\\mathbf {0}.$ By virtue of Protter–Weinberger’s strong maximum principle [42], $\\mathbf {u}$ is indeed either null or positive in $\\mathsf {Q}$ .",
"Actually, noticing that the previous proof remains true without any modification if we add to $\\left(E_{KPP}\\right)$ a diagonal drift term $\\mathbf {b}\\circ \\partial _{x}\\mathbf {u}$ with $\\mathbf {b}\\in \\mathbb {R}^{N}$ , we state right now a corollary that will be quite useful later on.",
"Corollary 2.2 Let $\\left(a,b,c\\right)\\in \\mathbb {R}^{3}$ such that $a<b$ .",
"Let $\\mathbf {u}$ be a nonnegative classical solution of $-\\mathbf {D}\\mathbf {u}^{\\prime \\prime }-c\\mathbf {u}^{\\prime }=\\mathbf {L}\\mathbf {u}-\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u}\\text{ in }\\left(a,b\\right).$ Then $\\mathbf {u}$ is either null or positive in $\\left(a,b\\right)$ .",
"Remark This statement does not establish the non-negativity of all solutions of $-\\mathbf {D}\\mathbf {u}^{\\prime \\prime }-c\\mathbf {u}^{\\prime }=\\mathbf {L}\\mathbf {u}-\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u}$ ; it only enforces the interior positivity of the nonnegative nonzero solutions.",
"Regarding the weak maximum principle, we refer among others to Figueiredo [23], Figueiredo–Mitidieri [24], Sweers [47].",
"In view of what is known in the simpler scalar case, it is to be expected that, for small $\\left|c\\right|$ and large enough intervals $\\left(a,b\\right)$ , sign-changing solutions exist."
],
[
"Absorbing set and upper estimates",
"On the contrary, REF requires some work."
],
[
"Saturation of the reaction term",
"For all $i\\in \\left[N\\right]$ , let $\\mathsf {H}_{i}\\subset \\mathbb {R}^{N}$ be the closed half-space defined as $\\mathsf {H}_{i}=\\left\\lbrace \\mathbf {v}\\in \\mathbb {R}^{N}\\ |\\ \\left(\\mathbf {L}\\mathbf {v}\\right)_{i}\\ge 0\\right\\rbrace .$ Lemma 3.1 There exists $\\mathbf {k}\\in \\mathsf {K}^{++}$ such that, for all $i\\in \\left[N\\right]$ and for all $\\mathbf {v}\\in \\mathsf {K}\\backslash \\mathbf {e}_{i}^{\\perp }$ , $\\left(\\mathbf {L}\\left(\\mathbf {v}+k_{i}\\mathbf {e}_{i}\\right)-\\mathbf {c}\\left(\\mathbf {v}+k_{i}\\mathbf {e}_{i}\\right)\\circ \\left(\\mathbf {v}+k_{i}\\mathbf {e}_{i}\\right)\\right)_{i}<0.$ Let $i\\in \\left[N\\right]$ and let $\\mathsf {F}_{i}=\\left(\\mathsf {S}^{+}\\left(\\mathbf {0},1\\right)\\cap \\mathsf {H}_{i}\\right)\\backslash \\mathbf {e}_{i}^{\\perp }.$ Let $\\begin{matrix}f_{i}: & \\left(0,+\\infty \\right)\\times \\mathsf {S}\\left(\\mathbf {0},1\\right) & \\rightarrow & \\mathbb {R}\\\\& \\left(\\alpha ,\\mathbf {n}\\right) & \\mapsto & \\sum \\limits _{j=1}^{N}l_{i,j}n_{j}-c_{i}\\left(\\alpha \\mathbf {n}\\right)n_{i}.\\end{matrix}$ Notice that for all $\\mathbf {n}\\in \\mathsf {S}^{+}\\left(\\mathbf {0},1\\right)\\backslash \\mathsf {F}_{i}$ , either $\\sum \\limits _{j=1}^{N}l_{i,j}n_{j}<0$ and then $f_{i}\\left(\\alpha ,\\mathbf {n}\\right)<0$ for all $\\alpha >0$ or $n_{i}=0$ and then $f_{i}\\left(\\alpha ,\\mathbf {n}\\right)=\\sum \\limits _{j=1}^{N}l_{i,j}n_{j}\\ge 0$ does not depend on $\\alpha $ .",
"Let $\\mathbf {n}\\in \\mathsf {F}_{i}$ .",
"By virtue of the behavior of $\\mathbf {c}$ as $\\alpha \\rightarrow +\\infty $ $\\left(H_{4}\\right)$ and since $\\mathbf {n}\\notin \\mathbf {e}_{i}^{\\perp }$ , $\\lim _{\\alpha \\rightarrow +\\infty }f_{i}\\left(\\alpha ,\\mathbf {n}\\right)=-\\infty .$ Therefore the following quantity is finite and nonnegative: $\\alpha _{i,\\mathbf {n}}=\\inf \\left\\lbrace \\alpha \\ge 0\\ |\\ \\forall \\alpha ^{\\prime }\\in \\left(\\alpha ,+\\infty \\right)\\quad f_{i}\\left(\\alpha ^{\\prime },\\mathbf {n}\\right)<0\\right\\rbrace .$ Now, the set $\\left\\lbrace \\alpha _{i,\\mathbf {n}}n_{i}\\ |\\ \\mathbf {n}\\in \\mathsf {F}_{i}\\right\\rbrace =\\left\\lbrace \\alpha _{i,\\mathbf {n}}n_{i}\\ |\\ \\mathbf {n}\\in \\mathsf {F}_{i},\\ \\alpha _{i,\\mathbf {n}}>\\underline{\\alpha }\\right\\rbrace \\cup \\left\\lbrace \\alpha _{i,\\mathbf {n}}n_{i}\\ |\\ \\mathbf {n}\\in \\mathsf {F}_{i},\\ \\alpha _{i,\\mathbf {n}}\\le \\underline{\\alpha }\\right\\rbrace $ is bounded if and only if the set $\\left\\lbrace \\alpha _{i,\\mathbf {n}}n_{i}\\ |\\ \\mathbf {n}\\in \\mathsf {F}_{i},\\ \\alpha _{i,\\mathbf {n}}>\\underline{\\alpha }\\right\\rbrace $ is bounded.",
"Recall the definition of $\\underline{\\alpha }\\ge 1$ and $\\delta \\ge 1$ $\\left(H_{4}\\right)$ .",
"For all $\\mathbf {n}\\in \\mathsf {F}_{i}$ such that $\\alpha _{i,\\mathbf {n}}>\\underline{\\alpha }$ , thanks to $\\left(H_{4}\\right)$ , we have by virtue of the discrete Cauchy–Schwarz inequality $\\left|\\alpha _{i,\\mathbf {n}}n_{i}\\right| & =\\alpha _{i,\\mathbf {n}}n_{i}\\\\& \\le \\alpha _{i,\\mathbf {n}}^{\\delta }n_{i}\\\\& \\le \\frac{\\sum _{j=1}^{N}l_{i,j}n_{j}}{\\underline{c}_{i}}\\\\& \\le \\frac{\\left|\\left(l_{i,j}\\right)_{j\\in \\left[N\\right]}\\right|}{\\underline{c}_{i}},$ whence the finiteness of $k_{i}=\\sup \\left\\lbrace \\alpha _{i,\\mathbf {n}}n_{i}\\ |\\ \\mathbf {n}\\in \\mathsf {F}_{i}\\right\\rbrace $ is established.",
"Its positivity follows from the fact that $\\mathbf {c}$ vanishes at $\\mathbf {0}$ $\\left(H_{3}\\right)$ which implies that for all $\\mathbf {n}\\in \\text{int}\\mathsf {F}_{i}$ , $\\alpha _{i,\\mathbf {n}}>0$ .",
"The result about $\\mathbf {v}+k_{i}\\mathbf {e}_{i}$ with $\\mathbf {v}\\in \\mathsf {K}\\backslash \\mathbf {e}_{i}^{\\perp }$ is a direct consequence.",
"Assuming in addition strict monotonicity of $\\alpha \\mapsto c_{i}\\left(\\alpha \\mathbf {n}\\right)$ (which is for instance satisfied if $\\mathbf {c}\\left(\\mathbf {v}\\right)=\\mathbf {C}\\mathbf {v}$ with $\\mathbf {C}\\gg \\mathbf {0}$ , that is in the Lotka–Volterra competition case), we can obtain the following more precise geometric description of the reaction term.",
"The proof is quite straightforward and is not detailed here.",
"Lemma 3.2 Assume in addition that $\\alpha \\mapsto c_{i}\\left(\\alpha \\mathbf {n}\\right)$ is increasing for all $\\mathbf {n}\\in \\mathsf {H}_{i}$ .",
"Then there exists a collection of connected ${C}^{1}$ -hypersurfaces $\\left(\\mathsf {Z}_{i}\\right)_{i\\in \\left[N\\right]}\\subset \\prod \\limits _{i=1}^{N}\\left(\\left(\\mathsf {K}^{+}\\cap \\mathsf {H}_{i}\\right)\\backslash \\mathbf {e}_{i}^{\\perp }\\right)$ such that, for any $i\\in \\left[N\\right]$ and any $\\mathbf {v}\\in \\left(\\mathsf {K}^{+}\\cap \\mathsf {H}_{i}\\right)\\backslash \\mathbf {e}_{i}^{\\perp }$ , $\\left(\\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right)_{i}=0\\text{ if and only if }\\mathbf {v}\\in \\mathsf {Z}_{i}.$ For all $i\\in \\left[N\\right]$ , $\\mathsf {Z}_{i}$ satisfies the following properties.",
"For all $\\mathbf {n}\\in \\left(\\mathsf {S}^{+}\\left(\\mathbf {0},1\\right)\\cap \\mathsf {H}_{i}\\right)\\backslash \\mathbf {e}_{i}^{\\perp }$ , $\\mathsf {Z}_{i}\\cap \\mathbb {R}\\mathbf {n}$ is a singleton.",
"The function $\\mathbf {z}_{i}$ which associates with any $\\mathbf {n}\\in \\left(\\mathsf {S}^{+}\\left(\\mathbf {0},1\\right)\\cap \\mathsf {H}_{i}\\right)\\backslash \\mathbf {e}_{i}^{\\perp }$ the unique element of $\\mathsf {Z}_{i}\\cap \\mathbb {R}\\mathbf {n}$ is continuous and is a ${C}^{1}$ -diffeomorphism of $\\left(\\mathsf {S}^{++}\\left(\\mathbf {0},1\\right)\\cap \\text{int}\\mathsf {H}_{i}\\right)\\backslash \\mathbf {e}_{i}^{\\perp }$ onto $\\text{int}\\mathsf {Z}_{i}$ .",
"For any $\\mathbf {v}\\in \\mathsf {K}^{+}\\backslash \\mathbf {e}_{i}^{\\perp }$ , $\\left(\\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right)_{i}>0$ if and only if $\\mathbf {v}\\in \\mathsf {H}_{i}\\text{ and }\\left|\\mathbf {v}\\right|<\\left|\\mathbf {z}_{i}\\left(\\frac{\\mathbf {v}}{\\left|\\mathbf {v}\\right|}\\right)\\right|.$"
],
[
"Absorbing set and upper estimates",
"Define for all $i\\in \\left[N\\right]$ $\\begin{matrix}g_{i}: & [0,+\\infty ) & \\rightarrow & \\left(0,+\\infty \\right)\\\\& \\mu & \\mapsto & \\max \\left(\\mu ,k_{i}\\right).\\end{matrix}$ The function $g_{i}$ is non-decreasing and piecewise affine (whence Lipschitz-continuous).",
"The following local in space ${L}^{\\infty }$ estimate for the parabolic problem is due to Barles–Evans–Souganidis [6].",
"We repeat its proof for the sake of completeness.",
"Lemma 3.3 Let $\\mathsf {Q}\\subset \\mathbb {R}^{2}$ be a parabolic cylinder bounded in space and bounded from below in time.",
"Let $\\mathbf {u}$ be a nonnegative classical solution of $\\left(E_{KPP}\\right)$ set in $\\mathsf {Q}$ such that $\\mathbf {u}_{|\\partial _{P}\\mathsf {Q}}\\in {L}^{\\infty }\\left(\\partial _{P}\\mathsf {Q},\\mathbb {R}^{N}\\right).$ Then we have $\\left(\\sup _{\\mathsf {Q}}u_{i}\\right)_{i\\in \\left[N\\right]}\\le \\left(g_{i}\\left(\\sup _{\\partial _{P}\\mathsf {Q}}u_{i}\\right)\\right)_{i\\in \\left[N\\right]}.$ Let $t_{0}\\in \\mathbb {R}$ , $T\\in (0,+\\infty ]$ and $\\left(a,b\\right)\\in \\mathbb {R}^{2}$ such that $\\mathsf {Q}=\\left(t_{0},t_{0}+T\\right)\\times \\left(a,b\\right)$ .",
"Let $i\\in \\left[N\\right]$ .",
"Define a smooth convex function $\\eta :\\mathbb {R}\\rightarrow \\mathbb {R}$ which satisfies $\\left\\lbrace \\begin{matrix}\\eta \\left(u\\right)=0 & \\text{if }u\\in (-\\infty ,g_{i}\\left(\\sup \\limits _{\\partial _{P}\\mathsf {Q}}u_{i}\\right)]\\\\\\eta \\left(u\\right)>0 & \\text{otherwise}.\\end{matrix}\\right.$ For all $t\\in \\left(t_{0},t_{0}+T\\right)$ , let $\\Xi _{i}\\left(t\\right)=\\left\\lbrace x\\in \\left(a,b\\right)\\ |\\ u_{i}\\left(t,x\\right)>g_{i}\\left(\\sup \\limits _{\\partial _{P}\\mathsf {Q}}u_{i}\\right)\\right\\rbrace .$ This set is measurable and, by integration by parts, for all $t\\in \\left(t_{0},t_{0}+T\\right)$ , $\\partial _{t}\\left(\\int _{a}^{b}\\eta \\left(u_{i}\\left(t,x\\right)\\right)\\text{d}x\\right) & =\\int _{a}^{b}\\eta ^{\\prime }\\left(u_{i}\\left(t,x\\right)\\right)\\partial _{t}u_{i}\\left(t,x\\right)\\text{d}x\\\\& =-d_{i}\\int _{a}^{b}\\eta ^{\\prime \\prime }\\left(u_{i}\\left(t,x\\right)\\right)\\left(\\partial _{x}u_{i}\\left(t,x\\right)\\right)^{2}\\text{d}x\\\\& +\\int _{a}^{b}\\eta ^{\\prime }\\left(u_{i}\\left(t,x\\right)\\right)\\left(\\sum _{j=1}^{N}l_{i,j}u_{j}\\left(t,x\\right)-c_{i}\\left(\\mathbf {u}\\left(t,x\\right)\\right)u_{i}\\left(t,x\\right)\\right)\\text{d}x\\\\& =-d_{i}\\int _{\\Xi _{i}\\left(t\\right)}\\eta ^{\\prime \\prime }\\left(u_{i}\\left(t,x\\right)\\right)\\left(\\partial _{x}u_{i}\\left(t,x\\right)\\right)^{2}\\text{d}x\\\\& +\\int _{\\Xi _{i}\\left(t\\right)}\\eta ^{\\prime }\\left(u_{i}\\left(t,x\\right)\\right)\\left(\\sum _{j=1}^{N}l_{i,j}u_{j}\\left(t,x\\right)-c_{i}\\left(\\mathbf {u}\\left(t,x\\right)\\right)u_{i}\\left(t,x\\right)\\right)\\text{d}x\\\\& \\le 0$ Since $\\int _{a}^{b}\\eta \\left(u_{i}\\left(t_{0},x\\right)\\right)\\text{d}x=0$ , we deduce $u_{i}\\le g_{i}\\left(\\sup _{\\partial _{P}\\mathsf {Q}}u_{i}\\right)\\text{ in }\\mathsf {Q},$ whence $\\sup _{\\mathsf {Q}}u_{i}\\le g_{i}\\left(\\sup _{\\partial _{P}\\mathsf {Q}}u_{i}\\right).$ As a corollary of this local estimate, we get REF .",
"Proposition 3.4 Let $\\mathbf {u}_{0}\\in {C}_{b}\\left(\\mathbb {R},\\mathsf {K}\\right)$ .",
"Then the unique classical solution $\\mathbf {u}$ of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ with initial data $\\mathbf {u}_{0}$ satisfies $\\left(\\sup _{\\left(0,+\\infty \\right)\\times \\mathbb {R}}u_{i}\\right)_{i\\in \\left[N\\right]}\\le \\left(g_{i}\\left(\\sup _{\\mathbb {R}}u_{0,i}\\right)\\right)_{i\\in \\left[N\\right]}$ and furthermore $\\left(\\limsup _{t\\rightarrow +\\infty }\\sup _{x\\in \\mathbb {R}}u_{i}\\left(t,x\\right)\\right)_{i\\in \\left[N\\right]}\\le \\mathbf {g}\\left(0\\right).$ Consequently, all bounded nonnegative classical solutions of $\\left(S_{KPP}\\right)$ are valued in $\\prod _{i=1}^{N}\\left[0,g_{i}\\left(0\\right)\\right].$ To get the global in space ${L}^{\\infty }$ estimate, apply the local one to the family $\\left(\\mathbf {u}_{R}\\right)_{R>0}$ , where $\\mathbf {u}_{R}$ is the solution of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\left(-R,R\\right)$ with $\\left\\lbrace \\begin{matrix}\\mathbf {u}_{R}\\left(0,x\\right)=\\mathbf {u}_{0}\\left(x\\right) & \\text{for all }x\\in \\left[-R,R\\right],\\\\\\mathbf {u}_{R}\\left(t,\\pm R\\right)=\\mathbf {u}_{0}\\left(\\pm R\\right) & \\text{for all }t\\ge 0,\\end{matrix}\\right.$ and recall that, by classical parabolic estimates (Lieberman [40]) and a diagonal extraction process, $\\left(\\mathbf {u}_{R}\\right)_{R>0}$ converges up to extraction in ${C}_{loc}^{1}\\left(\\left(0,+\\infty \\right),{C}_{loc}^{2}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)\\right)$ to the solution of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ with initial data $\\mathbf {u}_{0}$ .",
"Next, let us prove that the invariant set $\\prod _{i=1}^{N}\\left[0,g_{i}\\left(0\\right)\\right]=\\prod _{i=1}^{N}\\left[0,k_{i}\\right]$ is in fact an absorbing set.",
"Assume by contradiction that there exists a bounded nonnegative classical solution $\\mathbf {u}$ of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ such that there exists $i\\in \\left[N\\right]$ such that $\\limsup _{t\\rightarrow +\\infty }\\sup _{x\\in \\mathbb {R}}u_{i}\\left(t,x\\right)>g_{i}\\left(0\\right).$ Since $\\left[0,g_{i}\\left(0\\right)\\right]$ is invariant, it implies directly $\\sup _{x\\in \\mathbb {R}}u_{i}\\left(t,x\\right)>g_{i}\\left(0\\right)\\text{ for all }t\\ge 0.$ Using the classical second order condition at any local maximum, it is easily seen that at any local maximum in space of $u_{i}$ , the time derivative is negative.",
"At any $t>0$ such that there is no local maximum in space, by ${C}^{1}$ regularity of $u_{i}$ , $x\\mapsto u_{i}\\left(t,x\\right)$ is either strictly monotonic or piecewise strictly monotonic with one unique local minimum and consequently it converges to some constant as $x\\rightarrow \\pm \\infty $ .",
"At least one of these constants is $\\sup \\limits _{x\\in \\mathbb {R}}u_{i}\\left(t,x\\right)$ .",
"For instance, assume it is the limit at $+\\infty $ .",
"By classical parabolic estimates and a diagonal extraction process, there exists $\\left(x_{n}\\right)_{n\\in \\mathbb {N}}\\in \\mathbb {R}^{\\mathbb {N}}$ such that $x_{n}\\rightarrow +\\infty $ and such that the following sequence converges in ${C}_{loc}^{1}\\left(\\left(0,+\\infty \\right),{C}_{loc}^{2}\\left(\\mathbb {R}\\right)\\right)$ : $\\left(\\left(t^{\\prime },x\\right)\\mapsto u_{i}\\left(t+t^{\\prime },x+x_{n}\\right)\\right)_{n\\in \\mathbb {N}}.$ Let $v$ be its limit; by construction, $v\\left(0,x\\right)=\\sup \\limits _{x\\in \\mathbb {R}}u_{i}\\left(t,x\\right)\\text{ for all }x\\in \\mathbb {R},$ so that $\\partial _{xx}v\\left(0,x\\right)=0\\text{ for all }x\\in \\mathbb {R}.$ Using the equation satisfied by $u_{i}$ , we obtain $\\partial _{t}v\\left(0,x\\right)<0\\text{ for all }x\\in \\mathbb {R}.$ Since this argument does not depend on the choice of the sequence $\\left(x_{n}\\right)_{n\\in \\mathbb {N}}$ , we deduce $\\limsup _{x\\rightarrow +\\infty }\\partial _{t}u_{i}\\left(t,x\\right)<0.$ In all cases, $t\\mapsto \\Vert x\\mapsto u_{i}\\left(t,x\\right)\\Vert _{{L}^{\\infty }\\left(\\mathbb {R}\\right)}$ is a decreasing function, and using the global ${L}^{\\infty }$ estimate derived earlier, we deduce that $t\\mapsto \\Vert u_{i}\\Vert _{{L}^{\\infty }\\left(\\left(t,+\\infty \\right)\\times \\mathbb {R}\\right)}$ is a decreasing function as well.",
"Therefore $\\limsup _{t\\rightarrow +\\infty }\\sup _{x\\in \\mathbb {R}}u_{i}\\left(t,x\\right)=\\liminf _{t\\rightarrow +\\infty }\\sup _{x\\in \\mathbb {R}}u_{i}\\left(t,x\\right)=\\lim _{t\\rightarrow +\\infty }\\sup _{x\\in \\mathbb {R}}u_{i}\\left(t,x\\right)>g_{i}\\left(0\\right).$ Now, the sequence $\\left(\\left(t,x\\right)\\mapsto u_{i}\\left(t+n,x\\right)\\right)_{n\\in \\mathbb {N}}$ being uniformly bounded in ${L}^{\\infty }\\left(\\left(0,+\\infty \\right)\\times \\mathbb {R}\\right)$ , by classical parabolic estimates and a diagonal extraction process, it converges up to extraction in ${C}_{loc}^{1}\\left(\\left(0,+\\infty \\right),{C}_{loc}^{2}\\left(\\mathbb {R}\\right)\\right)$ to some limit $u_{\\infty ,i}\\in {C}^{1}\\left(\\left(0,+\\infty \\right),{C}^{2}\\left(\\mathbb {R}\\right)\\right)$ .",
"On one hand, by construction, the function $t\\mapsto \\Vert x\\mapsto u_{\\infty ,i}\\left(t,x\\right)\\Vert _{{L}^{\\infty }\\left(\\mathbb {R}\\right)}$ is constant and larger than $g_{i}\\left(0\\right)$ .",
"But on the other hand, passing also to the limit the other components of $\\left(t,x\\right)\\mapsto \\mathbf {u}\\left(t+n,x\\right)$ and then repeating the argument used earlier to prove the strict monotonicity of $t\\mapsto \\Vert x\\mapsto u_{i}\\left(t,x\\right)\\Vert _{{L}^{\\infty }\\left(\\mathbb {R}\\right)},$ we deduce the strict monotonicity of $t\\mapsto \\Vert x\\mapsto u_{\\infty ,i}\\left(t,x\\right)\\Vert _{{L}^{\\infty }\\left(\\mathbb {R}\\right)},$ which is an obvious contradiction.",
"Quite similarly, we can establish an ${L}^{\\infty }$ estimate for $\\left(S_{KPP}\\right)$ , set in a strip, and with an additional drift.",
"Proposition 3.5 Let $\\left(a,b,c\\right)\\in \\mathbb {R}^{3}$ such that $a<b$ and $\\mathbf {u}$ be a nonnegative classical solution of $-\\mathbf {D}\\mathbf {u}^{\\prime \\prime }-c\\mathbf {u}^{\\prime }=\\mathbf {L}\\mathbf {u}-\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u}\\text{ in }\\left(a,b\\right).$ Then $\\left(\\max _{\\left[a,b\\right]}u_{i}\\right)_{i\\in \\left[N\\right]}\\le \\left(g_{i}\\left(\\max _{\\left\\lbrace a,b\\right\\rbrace }u_{i}\\right)\\right)_{i\\in \\left[N\\right]}.$ Assume by contradiction that there exists $i\\in \\left[N\\right]$ such that $\\max _{\\left[a,b\\right]}u_{i}>g_{i}\\left(\\max _{\\left\\lbrace a,b\\right\\rbrace }u_{i}\\right).$ Then there exists $x_{0}\\in \\left(a,b\\right)$ such that $\\max _{\\left[a,b\\right]}u_{i}=u_{i}\\left(x_{0}\\right)>k_{i}.$ There exists $\\left(x_{1},x_{2}\\right)\\in \\left(a,b\\right)^{2}$ such that $x_{1}<x_{0}<x_{2}$ and $\\left\\lbrace \\begin{matrix}u_{i}\\left(x\\right)>k_{i} & \\text{for all }x\\in \\left(x_{1},x_{2}\\right)\\\\u_{i}\\left(x\\right)=\\frac{1}{2}\\left(k_{i}+u_{i}\\left(x_{0}\\right)\\right) & \\text{for all }x\\in \\left\\lbrace x_{1},x_{2}\\right\\rbrace .\\end{matrix}\\right.$ But then we find the inequality $-d_{i}u_{i}^{\\prime \\prime }-cu_{i}^{\\prime }\\ll 0\\text{ in }\\left(x_{1},x_{2}\\right)$ which contradicts the existence of an interior maximum at $x_{0}\\in \\left(x_{1},x_{2}\\right)$ ."
],
[
"Extinction and persistence",
"This section is devoted to the proof of REF .",
"The extinction case is mainly straightforward but, because of the lack of comparison principle, the persistence case is more involved."
],
[
"Extinction",
"Proposition 4.1 Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)<0$ .",
"Then all bounded nonnegative classical solutions of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ converge asymptotically in time, exponentially fast, and uniformly in space to $\\mathbf {0}$ .",
"It suffices to notice that if $\\mathbf {u}$ is a nonnegative bounded solution of $\\left(E_{KPP}\\right)$ , then $\\mathbf {v}:\\left(t,x\\right)\\mapsto \\text{e}^{\\lambda _{PF}\\left(\\mathbf {L}\\right)t}\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)$ satisfies by virtue of the nonnegativity of $\\mathbf {c}$ on $\\mathsf {K}$ $\\left(H_{2}\\right)$ $\\partial _{t}\\left(\\mathbf {v}-\\mathbf {u}\\right)-\\mathbf {D}\\partial _{xx}\\left(\\mathbf {v}-\\mathbf {u}\\right)-\\mathbf {L}\\left(\\mathbf {v}-\\mathbf {u}\\right)=\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u}\\ge \\mathbf {0}.$ Hence, up to a multiplication of $\\mathbf {v}$ by a large constant, the comparison principle (Protter–Weinberger [42]) applied to the linear weakly and fully coupled operator $\\partial _{t}-\\mathbf {D}\\partial _{xx}-\\mathbf {L}$ in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ implies that $\\mathbf {0}\\le \\mathbf {u}\\le \\mathbf {v}$ .",
"The limit easily follows."
],
[
"Regarding the critical case",
"The proof for the case $\\lambda _{PF}\\left(\\mathbf {L}\\right)<0$ clearly cannot be adapted if $\\lambda _{PF}\\left(\\mathbf {L}\\right)=0$ .",
"In this subsubsection, we briefly explain why the present paper only conjectures the result in this case.",
"Let us recall that for the scalar equation $\\partial _{t}u-\\partial _{xx}u=-u^{2}$ , the comparison principle ensures extinction (by comparison with a solution of $u^{\\prime }\\left(t\\right)=-u\\left(t\\right)^{2}$ with large enough initial data).",
"Since the comparison principle is not satisfied by $\\left(E_{KPP}\\right)$ , we cannot hope to generalize this proof and need to find another method.",
"Still, in view of this scalar result, it is natural to aim for a proof of extinction.",
"As a preliminary observation, if some Perron–Frobenius eigenvectors of $\\mathbf {L}$ are zeros of $\\mathbf {c}$ , then extinction will not occur in general.",
"Therefore, in order to solve the critical case, it is necessary to rule out this somehow degenerated case.",
"This is of course consistent with the critical case for REF .",
"For the non-degenerated non-diffusive system $\\mathbf {u}^{\\prime }=\\mathbf {L}\\mathbf {u}-\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u}$ , we know how to handle two particular cases: if $\\mathbf {L}$ is symmetric, then the classical Lyapunov function $V:\\mathbf {u}\\mapsto \\frac{1}{2}\\left|\\mathbf {u}\\right|^{2}$ ensures extinction; if there exists $\\mathbf {a}\\in \\mathsf {K}^{++}$ such that $\\mathbf {c}\\left(\\mathbf {v}\\right)=\\left(\\mathbf {a}^{T}\\mathbf {v}\\right)\\mathbf {1}_{N,1}$ , the change of unknown $\\mathbf {z}:t\\mapsto \\exp \\left(\\int _{0}^{t}\\left(\\mathbf {a}^{T}\\mathbf {u}\\left(\\tau \\right)\\right)\\text{d}\\tau \\right)\\mathbf {u}\\left(t\\right)$ (exploited for instance by Leman–Méléard–Mirrahimi [37]) ensures extinction.",
"But even in these special cases, the diffusive system cannot be handled (as far as we know).",
"The first idea of proof (which would be in the parabolic setting an entropy proof) would involve an integration by parts of $\\mathbf {u}^{T}\\mathbf {D}\\partial _{xx}\\mathbf {u}$ and therefore would have to deal with the unboundedness of the space domain $\\mathbb {R}$ .",
"In such a situation, the classical trick (multiplication of $\\left(E_{KPP}\\right)$ by $\\text{e}^{-\\varepsilon \\left|x\\right|}\\mathbf {u}\\left(t,x\\right)^{T}$ instead of $\\mathbf {u}\\left(t,x\\right)^{T}$ so that sufficient integrability is recovered) brings forth a new problematic term (see for instance Zelik [50] where this computation is carried on and only leads to the existence of an absorbing set).",
"Hence, apart from some particular cases (space-periodic solutions or solutions vanishing as $x\\rightarrow \\pm \\infty $ ) where we do not have to resort to this trick, the entropy method does not establish the extinction.",
"As for the second idea of proof, it is completely ruined by the space variable: the exponential term now depends also on $x$ and, again, new problematic terms arise in the equation satisfied by $\\mathbf {z}$ .",
"In view of these facts, extinction in the critical case is both a very natural conjecture and a surprisingly challenging problem (which would be way beyond the scope of this article)."
],
[
"Persistence",
"The first step toward the persistence result is giving some rigorous meaning to the statement “if $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ , then $\\mathbf {0}$ is unstable”."
],
[
"Slight digression: generalized principal eigenvalues and eigenfunctions\nfor weakly and fully coupled elliptic systems",
"Theorem 4.2 Let $\\left(n,n^{\\prime }\\right)\\in \\mathbb {N}\\cap [1,+\\infty )\\times \\mathbb {N}\\cap [2,+\\infty )$ and ${L}:{C}^{2}\\left(\\mathbb {R}^{n},\\mathbb {R}^{n^{\\prime }}\\right)\\rightarrow {C}\\left(\\mathbb {R}^{n},\\mathbb {R}^{n^{\\prime }}\\right)$ be a second-order elliptic operator, weakly and fully coupled, with continuous and bounded coefficients.",
"Let $\\lambda _{1}\\left(-{L}\\right)=\\sup \\left\\lbrace \\lambda \\in \\mathbb {R}\\ |\\ \\exists \\mathbf {v}\\in {C}^{2}\\left(\\mathbb {R}^{n},\\mathsf {K}_{n^{\\prime }}^{++}\\right)\\quad -{L}\\mathbf {v}\\ge \\lambda \\mathbf {v}\\right\\rbrace \\in \\overline{\\mathbb {R}}.$ Then $\\lim _{R\\rightarrow +\\infty }\\lambda _{1,Dir}\\left(-{L},\\mathsf {B}_{n}\\left(\\mathbf {0},R\\right)\\right)=\\lambda _{1}\\left(-{L}\\right).$ Furthermore, $\\lambda _{1}\\left(-{L}\\right)$ is in fact a finite maximum and there exists a generalized principal eigenfunction, that is a positive solution of $-{L}\\mathbf {v}=\\lambda _{1}\\left(-{L}\\right)\\mathbf {v}.$ Remark The convergence of the Dirichlet principal eigenvalue to the aforementioned generalized principal eigenvalue as $R\\rightarrow +\\infty $ as well as the existence of a generalized principal eigenfunction are well-known for scalar elliptic equations (see Berestycki–Rossi [12]), but as far as we know these results do not explicitly appear in the literature regarding elliptic systems.",
"Still, the proof of Berestycki–Rossi [12] uses arguments developed in the celebrated article by Berestycki–Nirenberg–Varadhan [11] and which have been generalized to weakly and fully coupled elliptic systems already in order to prove the existence of a Dirichlet principal eigenvalue in non-necessarily smooth but bounded domains by Birindelli–Mitidieri–Sweers [13].",
"Hence we only briefly outline here the proof so that it can be checked that the generalization to unbounded domains is straightforward.",
"It begins with the standard verification of the equality between the generalized principal eigenvalue as defined above and the Dirichlet principal eigenvalue for bounded smooth domains (whose existence was proved for instance by Sweers [47]).",
"Then, since the generalized principal eigenvalue is, by definition, non-increasing with respect to the inclusion of the domains, we get that the limit of the Dirichlet principal eigenvalues as $R\\rightarrow +\\infty $ exists and is larger than or equal to the generalized principal eigenvalue.",
"It remains to prove that it is also smaller than or equal to it.",
"This is done thanks to the family of Dirichlet eigenfunctions $\\left(\\mathbf {v}_{R}\\right)_{R>0}$ associated with the family of Dirichlet principal eigenvalues normalized by $\\min _{i\\in \\left[n^{\\prime }\\right]}v_{i,R}\\left(\\mathbf {0}\\right)=1.$ Thanks to Arapostathis–Gosh–Marcus’s Harnack inequality [3] applied to the operator ${L}$ , we obtain a locally uniform ${L}^{\\infty }$ estimate, whence, by virtue of classical elliptic estimates (Gilbarg–Trudinger [30]) and a diagonal extraction process, the existence of a limit, up to extraction, for the family $\\left(\\mathbf {v}_{R}\\right)_{R>0}$ as $R\\rightarrow +\\infty $ .",
"This limit $\\mathbf {v}_{\\infty }$ is nonnegative nonzero and satisfies $-{L}\\mathbf {v}_{\\infty }=\\left[\\lim _{R\\rightarrow +\\infty }\\lambda _{1,Dir}\\left(-{L},\\mathsf {B}_{n}\\left(\\mathbf {0},R\\right)\\right)\\right]\\mathbf {v}_{\\infty }.$ Thanks again to Arapostathis–Gosh–Marcus’s Harnack inequality, $\\mathbf {v}_{\\infty }$ is in fact positive in $\\mathbb {R}^{n}$ .",
"Thus, by definition of the generalized principal eigenvalue, the limit as $R\\rightarrow +\\infty $ is indeed smaller than or equal to it, and in the end the equality is proved as well as the existence of a generalized principal eigenfunction $\\mathbf {v}_{\\infty }$ ."
],
[
"Local instability and persistence",
"Let $\\gamma \\in \\left[0,1\\right]$ .",
"On one hand, as a direct result of Dancer [22] or Lam–Lou [36], $\\lim _{\\varepsilon \\rightarrow 0}\\lambda _{1,Dir}\\left(-\\varepsilon ^{2}\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\gamma \\lambda _{PF}\\left(\\mathbf {L}\\right)\\mathbf {I}\\right),\\mathsf {B}\\left(\\mathbf {0},1\\right)\\right)=-\\left(1-\\gamma \\right)\\lambda _{PF}\\left(\\mathbf {L}\\right).$ On the other hand, by a standard change of variable, $\\lim _{\\varepsilon \\rightarrow 0}\\lambda _{1,Dir}\\left(-\\varepsilon ^{2}\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\gamma \\lambda _{PF}\\left(\\mathbf {L}\\right)\\mathbf {I}\\right),\\mathsf {B}\\left(\\mathbf {0},1\\right)\\right)=\\lim _{R\\rightarrow +\\infty }\\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\gamma \\lambda _{PF}\\left(\\mathbf {L}\\right)\\mathbf {I}\\right),\\mathsf {B}\\left(\\mathbf {0},R\\right)\\right).$ Therefore, in view of REF , $\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\gamma \\lambda _{PF}\\left(\\mathbf {L}\\right)\\mathbf {I}\\right)\\right)=-\\left(1-\\gamma \\right)\\lambda _{PF}\\left(\\mathbf {L}\\right).$ This equality deserves some attention: the generalized principal eigenvalue of $\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}+\\left(\\mathbf {L}-\\gamma \\lambda _{PF}\\left(\\mathbf {L}\\right)\\mathbf {I}\\right)$ does not depend on $\\mathbf {D}$ .",
"Of course, this is reminiscent of the scalar case, where the equality $\\lambda _{1}\\left(-d\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-r\\right)=-r$ is well-known (and follows for instance from a direct computation of $\\lambda _{1,Dir}\\left(-d\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-r,\\left(-R,R\\right)\\right)$ or from the equality with the periodic principal eigenvalue $\\lambda _{1,per}\\left(-d\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-r\\right)$ ).",
"As a corollary, we get the following lemma.",
"Lemma 4.3 Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"Then there exists $\\left(R_{0},R_{{1}{2}}\\right)\\in \\left(0,+\\infty \\right)^{2}$ such that $\\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\mathbf {L},\\left(-R_{0},R_{0}\\right)\\right)<0,$ $\\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {I}\\right),\\left(-R_{{1}{2}},R_{{1}{2}}\\right)\\right)<0.$ Remark In fact, much more precisely, it can be shown that, for all $\\gamma \\in \\left[0,1\\right]$ , $R\\mapsto \\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\gamma \\lambda _{PF}\\left(\\mathbf {L}\\right)\\mathbf {I}\\right),\\left(-R,R\\right)\\right)$ is a decreasing homeomorphism from $\\left(0,+\\infty \\right)$ onto $\\left(-\\left(1-\\gamma \\right)\\lambda _{PF}\\left(\\mathbf {L}\\right),+\\infty \\right)$ .",
"By continuity of $\\mathbf {c}$ and the fact that it vanishes at $\\mathbf {0}$ $\\left(H_{3}\\right)$ , as soon as $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ , the quantity $\\alpha _{{1}{2}}=\\max \\left\\lbrace \\alpha >0\\ |\\ \\forall \\mathbf {v}\\in \\left[0,\\alpha \\right]^{N}\\quad \\mathbf {c}\\left(\\mathbf {v}\\right)\\le \\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {1}_{N,1}\\right\\rbrace $ is well-defined in $\\mathbb {R}$ and is positive.",
"The pair $\\left(R_{{1}{2}},\\alpha _{{1}{2}}\\right)$ will be used repeatedly up to the end of this section.",
"Lemma 4.4 Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"For all $\\mu \\in \\left(0,\\alpha _{{1}{2}}\\right)$ , let $T_{\\mu }=\\frac{\\ln \\alpha _{{1}{2}}-\\ln \\mu }{-\\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {I}\\right),\\left(-R_{{1}{2}},R_{{1}{2}}\\right)\\right)}>0.$ For all $\\left(t_{0},T,a,b\\right)\\in \\mathbb {R}\\times \\left(0,+\\infty \\right)\\times \\mathbb {R}^{2}$ such that $\\frac{b-a}{2}=R_{{1}{2}}$ and for all nonnegative classical solutions $\\mathbf {u}$ of $\\left(E_{KPP}\\right)$ set in the bounded parabolic cylinder $\\left(t_{0},t_{0}+T\\right)\\times \\left(a,b\\right)$ , if $\\min _{i\\in \\left[N\\right]}\\min _{x\\in \\left[a,b\\right]}u_{i}\\left(t_{0},x\\right)=\\mu ,$ $\\max _{i\\in \\left[N\\right]}\\max _{\\left[t_{0},t_{0}+T\\right]\\times \\left[a,b\\right]}u_{i}\\le \\alpha _{{1}{2}},$ then $T<T_{\\mu }$ .",
"Let $\\Lambda =\\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {I}\\right),\\left(-R_{{1}{2}},R_{{1}{2}}\\right)\\right)<0.$ Let $\\mathbf {n}$ be the principal eigenfunction associated with the preceding Dirichlet principal eigenvalue normalized so that $\\max _{i\\in \\left[N\\right]}\\max \\limits _{\\left[-R_{{1}{2}},R_{{1}{2}}\\right]}n_{i}=1.$ By definition, we have in $\\left(-R_{{1}{2}},R_{{1}{2}}\\right)$ $-\\left(-\\mathbf {D}\\mathbf {n}^{\\prime \\prime }-\\left(\\mathbf {L}-\\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {I}\\right)\\mathbf {n}\\right)=-\\Lambda \\mathbf {n}\\gg \\mathbf {0}.$ By definition of $\\alpha _{{1}{2}}$ and by the nonnegativity of $\\mathbf {c}$ on $\\mathsf {K}$ $\\left(H_{2}\\right)$ , for all $\\mathbf {v}\\in \\left[0,\\alpha _{{1}{2}}\\right]^{N}$ , $\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\le \\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {v},$ whence $-\\left(\\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right)\\le -\\left(\\mathbf {L}-\\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {I}\\right)\\mathbf {v}.$ Now, fix $\\left(t_{0},T,a,b\\right)\\in \\mathbb {R}\\times \\left(0,+\\infty \\right)\\times \\mathbb {R}^{2}$ such that $\\frac{b-a}{2}=R_{{1}{2}}$ and $T\\ge T_{\\mu }$ .",
"Assume by contradiction that there exists a nonnegative solution $\\mathbf {u}:\\left(t_{0},t_{0}+T\\right)\\times \\left(a,b\\right)\\rightarrow \\mathsf {K}$ of $\\left(E_{KPP}\\right)$ such that the following properties hold $\\mu =\\min _{i\\in \\left[N\\right]}\\min _{x\\in \\left[a,b\\right]}u_{i}\\left(t_{0},x\\right)>0,$ $\\max _{i\\in \\left[N\\right]}\\max _{\\left[t_{0},t_{0}+T\\right]\\times \\left[a,b\\right]}u_{i}\\le \\alpha _{{1}{2}}.$ In particular, since $\\mu >0$ , $\\mathbf {u}$ is nonnegative nonzero.",
"To simplify the notations, hereafter we assume that $t_{0}=0$ and $\\frac{a+b}{2}=0$ .",
"The general case is only a matter of straightforward translations.",
"Define the function $\\mathbf {v}:\\left(t,x\\right)\\mapsto \\mu \\text{e}^{-\\Lambda t}\\mathbf {n}\\left(x\\right).$ Clearly $\\mathbf {v}\\left(0,x\\right)\\le \\mathbf {u}\\left(0,x\\right)\\text{ for all }x\\in \\left[a,b\\right].$ It is easily verified as well that $\\mathbf {v}$ satisfies in $\\left(0,T_{\\mu }\\right)\\times \\left(-R_{{1}{2}},R_{{1}{2}}\\right)$ $-\\left(\\partial _{t}\\mathbf {v}-\\mathbf {D}\\partial _{xx}\\mathbf {v}-\\left(\\mathbf {L}-\\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {I}\\right)\\mathbf {v}\\right)\\ge \\mathbf {0},$ whence, by construction of $\\alpha _{{1}{2}}$ , $\\mathbf {w}=\\mathbf {u}-\\mathbf {v}$ satisfies $\\partial _{t}\\mathbf {w}-\\mathbf {D}\\partial _{xx}\\mathbf {w}-\\left(\\mathbf {L}-\\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {I}\\right)\\mathbf {w} & \\ge \\partial _{t}\\mathbf {u}-\\mathbf {D}\\partial _{xx}\\mathbf {u}-\\mathbf {L}\\mathbf {u}+\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u}=\\mathbf {0}.$ Most importantly, since by construction $T_{\\mu }=\\max \\left\\lbrace t>0\\ |\\ \\max _{i\\in \\left[N\\right]}\\max _{x\\in \\left[-R_{{1}{2}},R_{{1}{2}}\\right]}v_{i}\\left(t,x\\right)\\le \\alpha _{{1}{2}}\\right\\rbrace ,$ there exists $t^{\\star }\\le T_{\\mu }\\le T$ and $x^{\\star }\\in \\left(-R_{{1}{2}},R_{{1}{2}}\\right)$ such that $\\mathbf {w}\\gg \\mathbf {0}$ in $[0,t^{\\star })\\times \\left(-R_{{1}{2}},R_{{1}{2}}\\right)$ and $\\mathbf {w}\\left(t^{\\star },x^{\\star }\\right)\\in \\partial \\mathsf {K}$ .",
"The strong maximum principle applied to the weakly and fully coupled linear operator $\\partial _{t}-\\mathbf {D}\\partial _{xx}-\\left(\\mathbf {L}-\\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {I}\\right)$ proves then that $\\mathbf {w}=\\mathbf {0}$ in $[0,t^{\\star })\\times \\left(-R_{{1}{2}},R_{{1}{2}}\\right)$ , which contradicts $\\mathbf {w}\\left(0,\\pm R_{{1}{2}}\\right)\\gg \\mathbf {0}$ .",
"The persistence result follows.",
"Proposition 4.5 Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"There exists $\\nu >0$ such that all bounded nonnegative nonzero classical solutions $\\mathbf {u}$ of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ satisfy, for all bounded intervals $I\\subset \\mathbb {R}$ , $\\left(\\liminf _{t\\rightarrow +\\infty }\\inf _{x\\in I}u_{i}\\left(t,x\\right)\\right)_{i\\in \\left[N\\right]}\\ge \\nu \\mathbf {1}_{N,1}.$ Consequently, all bounded nonnegative classical solutions of $\\left(S_{KPP}\\right)$ are valued in $\\prod _{i=1}^{N}\\left[\\nu ,g_{i}\\left(0\\right)\\right].$ Let $\\mathbf {u}$ be a bounded nonnegative nonzero classical solution of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ .",
"In view of REF , for all $\\varepsilon >0$ there exists $t_{\\varepsilon }\\in \\left(0,+\\infty \\right)$ such that $\\mathbf {u}\\le \\left(\\max _{i\\in \\left[N\\right]}\\left(g_{i}\\left(0\\right)\\right)+\\varepsilon \\right)\\mathbf {1}_{N,1}\\text{ in }\\left(t_{\\varepsilon },+\\infty \\right)\\times \\mathbb {R}.$ Let $I\\subset \\mathbb {R}$ be a bounded interval.",
"Fix temporarily $\\varepsilon >0$ and $x\\in I$ and define $I_{x}=\\left(x-R_{{1}{2}},x+R_{{1}{2}}\\right)$ .",
"A first application of REF establishes that there exists $\\hat{t}_{x}\\in [t_{\\varepsilon },+\\infty )$ such that $\\max _{i\\in \\left[N\\right]}\\max _{y\\in \\overline{I_{x}}}u_{i}\\left(\\hat{t}_{x},y\\right)=\\alpha _{{1}{2}}$ and that there exists $\\tau >0$ such that $\\max _{i\\in \\left[N\\right]}\\max _{y\\in \\overline{I_{x}}}u_{i}\\left(t,y\\right)>\\alpha _{{1}{2}}\\text{ for all }t\\in \\left(\\hat{t}_{x},\\hat{t}_{x}+\\tau \\right).$ Hence the following quantity is well-defined in $\\left[\\hat{t}_{x}+\\tau ,+\\infty \\right]$ : $t_{1}=\\inf \\left\\lbrace t\\ge \\hat{t}_{x}+\\tau \\ |\\ \\max _{i\\in \\left[N\\right]}\\max _{y\\in \\overline{I_{x}}}u_{i}\\left(t,y\\right)<\\alpha _{{1}{2}}\\right\\rbrace .$ Assume first $t_{1}<+\\infty $ .",
"Then by continuity, $\\max _{i\\in \\left[N\\right]}\\max _{y\\in \\overline{I_{x}}}u_{i}\\left(t_{1},y\\right)=\\alpha _{{1}{2}}.$ Let $A_{\\mathbf {L},\\mathbf {c},\\varepsilon }=\\max \\limits _{\\left(i,j\\right)\\in \\left[N\\right]^{2}}\\left|l_{i,j}\\right|+\\max \\limits _{i\\in \\left[N\\right]}\\max \\limits _{\\mathbf {w}\\in \\left[0,\\max \\limits _{i\\in \\left[N\\right]}\\left(g_{i}\\left(0\\right)\\right)+\\varepsilon \\right]^{N}}c_{i}\\left(\\mathbf {w}\\right).$ By virtue of Földes–Poláčik’s Harnack inequality [29], there exists $\\overline{\\kappa }>0$ , dependent only on $N$ , $R_{{1}{2}}$ , $\\min \\limits _{i\\in \\left[N\\right]}d_{i}$ , $\\max \\limits _{i\\in \\left[N\\right]}d_{i}$ and $A_{\\mathbf {L},\\mathbf {c},\\varepsilon }$ such that, for all $\\mathbf {w}\\in {C}_{b}\\left(\\left(0,+\\infty \\right)\\times \\mathbb {R},\\left[0,\\max \\limits _{i\\in \\left[N\\right]}\\left(g_{i}\\left(0\\right)\\right)+\\varepsilon \\right]^{N}\\right),$ all nonnegative classical solutions $\\mathbf {v}$ of the linear weakly and fully coupled system with bounded coefficients $\\partial _{t}\\mathbf {v}-\\mathbf {D}\\partial _{xx}\\mathbf {v}-\\left(\\mathbf {L}-\\text{diag}\\left(\\mathbf {c}\\left[\\mathbf {w}\\right]\\right)\\right)\\mathbf {v}=\\mathbf {0}\\text{ in }I_{x}$ satisfy $\\min _{i\\in \\left[N\\right]}\\min _{y\\in \\overline{I_{x}}}v_{i}\\left(t_{1}+1,y\\right)\\ge \\overline{\\kappa }\\max _{i\\in \\left[N\\right]}\\max _{y\\in \\overline{I_{x}}}v_{i}\\left(t_{1},y\\right).$ We stress that $\\overline{\\kappa }$ does not depend on $\\mathbf {w}$ .",
"In particular, taking $\\mathbf {w}=\\mathbf {v}=\\mathbf {u}$ , we deduce $\\min _{i\\in \\left[N\\right]}\\min _{y\\in \\overline{I_{x}}}u_{i}\\left(t_{1}+1,y\\right)\\ge \\overline{\\kappa }\\alpha _{{1}{2}}.$ Of course, up to a shrink of $\\overline{\\kappa }$ , we can assume without loss of generality $\\overline{\\kappa }\\in \\left(0,1\\right)$ .",
"Then let $T=\\frac{-\\ln \\overline{\\kappa }}{-\\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {I}\\right),I_{x}\\right)}>0.$ $T$ does not depend on the choice of $\\mathbf {u}$ .",
"A second application of REF establishes $\\max _{i\\in \\left[N\\right]}\\max _{y\\in \\overline{I_{x}}}u_{i}\\left(t_{1}+1+T,y\\right)>\\alpha _{{1}{2}}.$ Hence, defining the sequence $\\left(t_{n}\\right)_{n\\in \\mathbb {N}}$ by the recurrence relation $t_{n+1}=\\inf \\left\\lbrace t\\ge t_{n}+1+T\\ |\\ \\max _{i\\in \\left[N\\right]}\\max _{y\\in \\overline{I_{x}}}u_{i}\\left(t,y\\right)<\\alpha _{{1}{2}}\\right\\rbrace $ and repeating by induction the process, we deduce that any connected component of $\\left\\lbrace t\\in \\left(\\hat{t}_{x},+\\infty \\right)\\ |\\ \\max _{i\\in \\left[N\\right]}\\max _{y\\in \\overline{I_{x}}}u_{i}\\left(t,y\\right)<\\alpha _{{1}{2}}\\right\\rbrace $ is an interval of length smaller than $1+T$ .",
"A second application of Földes–Poláčik’s Harnack inequality shows that there exists $\\overline{\\sigma _{\\varepsilon }}>0$ , dependent only on $N$ , $R_{{1}{2}}$ , $T$ , $\\min \\limits _{i\\in \\left[N\\right]}d_{i}$ , $\\max \\limits _{i\\in \\left[N\\right]}d_{i}$ and $A_{\\mathbf {L},\\mathbf {c},\\varepsilon }$ such that, for all $t\\in \\left(\\hat{t}_{x},+\\infty \\right)$ , $\\min _{i\\in \\left[N\\right]}\\min _{y\\in \\overline{I_{x}}}u_{i}\\left(t+T+2,y\\right)\\ge \\overline{\\sigma _{\\varepsilon }}\\max _{i\\in \\left[N\\right]}\\max _{\\left(t^{\\prime },y\\right)\\in \\left[t,t+T+1\\right]\\times \\overline{I_{x}}}u_{i}\\left(t^{\\prime },y\\right),$ whence $\\min _{i\\in \\left[N\\right]}\\min _{y\\in \\overline{I_{x}}}u_{i}\\left(t,y\\right)\\ge \\overline{\\sigma _{\\varepsilon }}\\alpha _{{1}{2}}\\text{ for all }t\\in \\left(\\hat{t}_{_{x}}+T+2,+\\infty \\right).$ Assume next $t_{1}=+\\infty $ .",
"Then $\\max _{i\\in \\left[N\\right]}\\max _{y\\in \\overline{I_{x}}}u_{i}\\left(t,y\\right)\\ge \\alpha _{{1}{2}}\\text{ for all }t\\in \\left(\\hat{t}_{_{x}},+\\infty \\right),$ and consequently $\\min _{i\\in \\left[N\\right]}\\min _{y\\in \\overline{I_{x}}}u_{i}\\left(t,y\\right)\\ge \\overline{\\sigma _{\\varepsilon }}\\alpha _{{1}{2}}\\text{ for all }t\\in \\left(\\hat{t}_{_{x}}+T+2,+\\infty \\right).$ Since $I$ is bounded and $x\\mapsto \\hat{t}_{x}$ can be assumed continuous in $\\mathbb {R}$ without loss of generality, it follows $\\min _{i\\in \\left[N\\right]}\\inf _{y\\in I}u_{i}\\left(t,y\\right)\\ge \\overline{\\sigma _{\\varepsilon }}\\alpha _{{1}{2}}\\text{ for all }t\\in \\left(\\max _{x\\in \\overline{I}}\\left(\\hat{t}_{_{x}}\\right)+T+2,+\\infty \\right),$ whence $\\liminf _{t\\rightarrow +\\infty }\\min _{i\\in \\left[N\\right]}\\inf _{y\\in I}u_{i}\\left(t,y\\right)\\ge \\overline{\\sigma _{\\varepsilon }}\\alpha _{{1}{2}}$ with $\\overline{\\sigma _{\\varepsilon }}\\alpha _{{1}{2}}$ dependent only on $\\varepsilon $ .",
"The conclusion follows of course by setting $\\nu =\\sup _{\\varepsilon >0}\\left(\\overline{\\sigma _{\\varepsilon }}\\right)\\alpha _{{1}{2}}.$ Remark We point out that $\\max \\limits _{x\\in \\overline{I}}\\hat{t}_{_{x}}$ is finite because $I$ is bounded.",
"Of course, in $I=\\mathbb {R}$ , this problem becomes a spreading problem (see REF )."
],
[
"Existence of positive steady states",
"This section is devoted to the proof of REF .",
"Proposition 5.1 Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)<0$ .",
"Then there exists no positive classical solution of $\\left(S_{KPP}\\right)$ .",
"Recall that the Dirichlet principal eigenvalue is non-increasing with respect to the zeroth order coefficient.",
"On one hand, by virtue of the nonnegativity of $\\mathbf {c}$ on $\\mathsf {K}$ $\\left(H_{2}\\right)$ , we have for all $R>0$ and all $\\mathbf {v}\\in {C}_{b}\\left(\\mathbb {R},\\mathsf {K}^{++}\\right)$ , $\\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\text{diag}\\mathbf {c}\\left[\\mathbf {v}\\right]\\right),\\left(-R,R\\right)\\right)\\ge \\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\mathbf {L},\\left(-R,R\\right)\\right),$ whence, as $R\\rightarrow +\\infty $ , $\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\text{diag}\\mathbf {c}\\left[\\mathbf {v}\\right]\\right)\\right)\\ge -\\lambda _{PF}\\left(\\mathbf {L}\\right)>0.$ On the other hand, any positive steady state $\\mathbf {v}$ is also a generalized principal eigenfunction for the generalized principal eigenvalue $\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-\\left(\\mathbf {L}-\\text{diag}\\mathbf {c}\\left[\\mathbf {v}\\right]\\right)\\right)=0.$ Proposition 5.2 Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)=0$ and $\\text{span}\\left(\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)\\right)\\cap \\mathsf {K}\\cap \\mathbf {c}^{-1}\\left(\\left\\lbrace \\mathbf {0}\\right\\rbrace \\right)=\\left\\lbrace \\mathbf {0}\\right\\rbrace .$ Then there exists no bounded positive classical solution of $\\left(S_{KPP}\\right)$ .",
"Remark The forthcoming argument is quite standard in the scalar setting.",
"We detail it for the sake of completeness.",
"Assume by contradiction that there exists a bounded positive classical solution $\\mathbf {v}$ of $\\left(S_{KPP}\\right)$ .",
"By boundedness of $\\mathbf {v}$ , there exists $\\kappa \\in \\left(0,+\\infty \\right)$ such that $\\kappa \\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)-\\mathbf {v}\\ge \\mathbf {0}$ in $\\mathbb {R}$ .",
"Let $\\kappa ^{\\star }=\\inf \\left\\lbrace \\kappa \\in \\left(0,+\\infty \\right)\\ |\\ \\kappa \\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)-\\mathbf {v}\\ge \\mathbf {0}\\text{ in }\\mathbb {R}\\right\\rbrace .$ By positivity of $\\mathbf {v}$ , $\\kappa ^{\\star }>0$ .",
"Let $\\left(\\kappa _{n}\\right)_{n\\in \\mathbb {N}}\\in \\left(0,\\kappa ^{\\star }\\right)^{\\mathbb {N}}$ which converges from below to $\\kappa ^{\\star }$ .",
"For all $n\\in \\mathbb {N}$ , there exists $x_{n}\\in \\mathbb {R}$ such that $\\kappa _{n}\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)-\\mathbf {v}\\left(x_{n}\\right)<\\mathbf {0}.$ Let $\\mathbf {v}_{n}:x\\mapsto \\mathbf {v}\\left(x+x_{n}\\right)\\text{ for all }n\\in \\mathbb {N}.$ By virtue of the global boundedness of $\\mathbf {v}$ , Arapostathis–Gosh–Marcus’s Harnack inequality [3] applied to the linear weakly and fully coupled operator with bounded coefficients $\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}+c\\frac{\\text{d}}{\\text{d}\\xi }+\\left(\\mathbf {L}-\\text{diag}\\left(\\mathbf {c}\\left[\\mathbf {v}_{n}\\right]\\right)\\right)$ and classical elliptic estimates (Gilbarg–Trudinger [30]), $\\left(\\mathbf {v}_{n}\\right)_{n\\in \\mathbb {N}}$ converges up to a diagonal extraction in ${C}_{loc}^{2}$ as $n\\rightarrow +\\infty $ to a nonnegative solution $\\mathbf {v}^{\\star }$ of $\\left(S_{KPP}\\right)$ .",
"Moreover, $\\mathbf {v}^{\\star }$ satisfies $\\mathbf {v}^{\\star }\\le \\kappa ^{\\star }\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)\\text{ in }\\mathbb {R},$ $\\kappa ^{\\star }\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)-\\mathbf {v}^{\\star }\\left(0\\right)\\in \\partial \\mathsf {K},$ $-\\left(\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}+\\mathbf {L}\\right)\\left(\\kappa ^{\\star }\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)-\\mathbf {v}^{\\star }\\right)=\\mathbf {c}\\left[\\mathbf {v}^{\\star }\\right]\\circ \\mathbf {v}^{\\star }\\ge \\mathbf {0}\\text{ in }\\mathbb {R}.$ Applying Arapostathis–Gosh–Marcus’s Harnack inequality [3] to $\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}+\\mathbf {L}$ , we deduce $\\kappa ^{\\star }\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)=\\mathbf {v}^{\\star }\\text{ in }\\mathbb {R}$ and subsequently $\\mathbf {c}\\left(\\kappa ^{\\star }\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)\\right)\\circ \\kappa ^{\\star }\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)=-\\left(\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}+\\mathbf {L}\\right)\\mathbf {0}=\\mathbf {0},$ whence $\\mathbf {c}\\left(\\kappa ^{\\star }\\mathbf {n}_{PF}\\left(\\mathbf {L}\\right)\\right)=\\mathbf {0}$ , which contradicts directly $\\kappa ^{\\star }>0$ .",
"Finally, recall that if $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ , then the following quantity is well-defined and positive: $\\alpha _{{1}{2}}=\\max \\left\\lbrace \\alpha >0\\ |\\ \\forall \\mathbf {v}\\in \\left[0,\\alpha \\right]^{N}\\quad \\mathbf {c}\\left(\\mathbf {v}\\right)\\le \\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\mathbf {1}_{N,1}\\right\\rbrace .$ Proposition 5.3 Assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ .",
"Then there exists a solution $\\mathbf {v}\\in \\mathsf {K}^{++}$ of $\\mathbf {L}\\mathbf {v}=\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}.$ By virtue of the Perron–Frobenius theorem, $\\mathbf {n}_{PF}\\left(\\mathbf {L}^{T}\\right)\\in \\mathsf {K}^{++}$ .",
"There exists $\\eta >0$ such that, for all $\\mathbf {v}\\in \\mathsf {K}$ , if $\\mathbf {n}_{PF}\\left(\\mathbf {L}^{T}\\right)^{T}\\mathbf {v}=\\eta $ , then $\\mathbf {v}\\in \\left[0,\\alpha _{{1}{2}}\\right]^{N}$ .",
"Defining $\\mathsf {A}=\\left\\lbrace \\mathbf {v}\\in \\mathsf {K}\\ |\\ \\mathbf {n}_{PF}\\left(\\mathbf {L}^{T}\\right)^{T}\\mathbf {v}=\\eta \\right\\rbrace ,$ it follows that for all $\\mathbf {v}\\in \\mathsf {A}$ , $\\mathbf {n}_{PF}\\left(\\mathbf {L}^{T}\\right)^{T}\\left(\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right)\\le \\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\eta ,$ whence $\\mathbf {n}_{PF}\\left(\\mathbf {L}^{T}\\right)^{T}\\left(\\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right) & =\\lambda _{PF}\\left(\\mathbf {L}^{T}\\right)\\eta -\\mathbf {n}_{PF}\\left(\\mathbf {L}^{T}\\right)^{T}\\left(\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right)\\\\& \\ge \\frac{\\lambda _{PF}\\left(\\mathbf {L}\\right)}{2}\\eta ,$ which is positive if $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ is assumed indeed.",
"Then, defining the convex compact set $\\mathsf {C}=\\left\\lbrace \\mathbf {v}\\in \\mathsf {K}\\ |\\ \\mathbf {n}_{PF}\\left(\\mathbf {L}^{T}\\right)^{T}\\mathbf {v}\\ge \\eta \\text{ and }\\mathbf {v}\\le \\mathbf {k}+\\mathbf {1}_{N,1}\\right\\rbrace ,$ it can easily be verified that, for all $\\mathbf {v}\\in \\partial \\mathsf {C}$ , $\\mathbf {n}_{\\mathbf {v}}^{T}\\left(\\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right)<0$ where $\\mathbf {n}_{\\mathbf {v}}$ is the outward pointing normal.",
"In particular, there is no solution of $\\mathbf {L}\\mathbf {v}=\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}$ in $\\partial \\mathsf {C}$ .",
"Also, by convexity, for all $\\mathbf {v}\\in \\partial \\mathsf {C}$ , there exists a unique $\\delta _{\\mathbf {v}}>0$ such that $\\mathbf {v}+\\delta _{\\mathbf {v}}\\left(\\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right)\\in \\partial \\mathsf {C}.$ Assume by contradiction that there is no solution of $\\mathbf {L}\\mathbf {v}=\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}$ in $\\text{int}\\mathsf {C}$ .",
"Consequently and by convexity again, for all $\\mathbf {v}\\in \\text{int}\\mathsf {C}$ , there exists a unique $\\delta _{\\mathbf {v}}>0$ such that $\\mathbf {v}+\\delta _{\\mathbf {v}}\\left(\\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right)\\in \\partial \\mathsf {C}.$ The function $\\begin{matrix}\\mathsf {C} & \\rightarrow & \\left(0,+\\infty \\right)\\\\\\mathbf {v} & \\mapsto & \\delta _{\\mathbf {v}}\\end{matrix}$ is continuous and so is the function $\\begin{matrix}\\mathsf {C} & \\rightarrow & \\partial \\mathsf {C}\\\\\\mathbf {v} & \\mapsto & \\mathbf {v}+\\delta _{\\mathbf {v}}\\left(\\mathbf {L}\\mathbf {v}-\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}\\right).\\end{matrix}$ According to the Brouwer fixed point theorem, this function has a fixed point, which of course contradicts the assumption.",
"Hence there exists indeed a solution in $\\text{int}\\mathsf {C}\\subset \\mathsf {K}^{++}$ of $\\mathbf {L}\\mathbf {v}=\\mathbf {c}\\left(\\mathbf {v}\\right)\\circ \\mathbf {v}.$"
],
[
"Traveling waves",
"In this section, we assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ and prove REF .",
"Notice as a preliminary that, for any $\\left(\\mathbf {p},c\\right)\\in {C}^{2}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)\\times [0,+\\infty )$ , $\\mathbf {u}:\\left(t,x\\right)\\mapsto \\mathbf {p}\\left(x-ct\\right)$ is a classical solution of $\\left(E_{KPP}\\right)$ if and only if $\\mathbf {p}$ is a classical solution of $-\\mathbf {D}\\mathbf {p}^{\\prime \\prime }-c\\mathbf {p}^{\\prime }=\\mathbf {L}\\mathbf {p}-\\mathbf {c}\\left[\\mathbf {p}\\right]\\circ \\mathbf {p}\\text{ in }\\mathbb {R}.\\quad \\left(TW\\left[c\\right]\\right)$"
],
[
"The linearized equation",
"As usual in KPP-type problems, the linearized equation near $\\mathbf {0}$ : $-\\mathbf {D}\\mathbf {p}^{\\prime \\prime }-c\\mathbf {p}^{\\prime }=\\mathbf {L}\\mathbf {p}\\text{ in }\\mathbb {R}\\quad \\left(TW_{0}\\left[c\\right]\\right)$ will bring forth the main informations we need in order to construct and study the traveling wave solutions.",
"Hence we devote this first subsection to its detailed study.",
"Lemma 6.1 Let $\\left(c,\\lambda \\right)\\in \\mathbb {R}^{2}$ .",
"If there exists a classical positive solution $\\mathbf {p}$ of $-\\mathbf {D}\\mathbf {p}^{\\prime \\prime }-c\\mathbf {p}^{\\prime }-\\left(\\mathbf {L}+\\lambda \\mathbf {I}\\right)\\mathbf {p}=\\mathbf {0}\\text{ in }\\mathbb {R},\\quad \\left(TW_{0}\\left[c,\\lambda \\right]\\right)$ then there exists $\\left(\\mu ,\\mathbf {n}\\right)\\in \\mathbb {R}\\times \\mathsf {K}^{++}$ such that $\\mathbf {q}:\\xi \\mapsto \\text{e}^{-\\mu \\xi }\\mathbf {n}$ is a classical solution of $\\left(TW_{0}\\left[c,\\lambda \\right]\\right)$ .",
"Remark This is of course to be related with the notions of generalized principal eigenvalue and generalized principal eigenfunction (see REF ).",
"The mere existence of $\\mathbf {p}$ enforces $\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}-c\\frac{\\text{d}}{\\text{d}\\xi }-\\left(\\mathbf {L}+\\lambda \\mathbf {I}\\right)\\right)\\ge 0.$ The following proof is inspired by Berestycki–Hamel–Roques [9].",
"Let $\\mathbf {p}$ be a classical positive solution of $\\left(TW_{0}\\left[c,\\lambda \\right]\\right)$ .",
"Let $\\mathbf {v}=\\left(\\frac{p_{i}^{\\prime }}{p_{i}}\\right)_{i\\in \\left[N\\right]}$ .",
"By virtue of Arapostathis–Gosh–Marcus’s Harnack inequality [3] applied to the operator $\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}+c\\frac{\\text{d}}{\\text{d}\\xi }+\\left(\\mathbf {L}+\\lambda \\mathbf {I}\\right)$ , classical elliptic estimates (Gilbarg–Trudinger [30]) and invariance by translation of $\\left(TW_{0}\\left[c,\\lambda \\right]\\right)$ , $\\mathbf {v}$ is globally bounded.",
"Let $\\Lambda _{i}=\\limsup _{\\xi \\rightarrow +\\infty }v_{i}\\left(\\xi \\right)\\text{ for all }i\\in \\left[N\\right],$ $\\overline{\\Lambda }=\\max \\limits _{i\\in \\left[N\\right]}\\Lambda _{i},$ so that $\\left(\\limsup _{\\xi \\rightarrow +\\infty }v_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\le \\overline{\\Lambda }\\mathbf {1}_{N,1}.$ Let $\\left(\\xi _{n}\\right)_{n\\in \\mathbb {N}}\\in \\mathbb {R}^{\\mathbb {N}}$ such that $\\xi _{n}\\rightarrow +\\infty $ and such that there exists $\\overline{i}\\in \\left[N\\right]$ such that $v_{\\overline{i}}\\left(\\xi _{n}\\right)\\rightarrow \\overline{\\Lambda }.$ On one hand, let $\\hat{\\mathbf {p}}_{n}:\\xi \\mapsto \\frac{\\mathbf {p}\\left(\\xi +\\xi _{n}\\right)}{p_{\\overline{i}}\\left(\\xi _{n}\\right)}\\text{ for all }n\\in \\mathbb {N}.$ Once more by virtue of Arapostathis–Gosh–Marcus’s Harnack inequality, the sequence $\\left(\\hat{\\mathbf {p}}_{n}\\right)_{n\\in \\mathbb {N}}$ is locally uniformly bounded.",
"Since all $\\hat{\\mathbf {p}}_{n}$ solve $\\left(TW_{0}\\left[c,\\lambda \\right]\\right)$ , by classical elliptic estimates, $\\left(\\hat{\\mathbf {p}}_{n}\\right)_{n\\in \\mathbb {N}}$ converges up to a diagonal extraction as $n\\rightarrow +\\infty $ in ${C}_{loc}^{2}$ .",
"Let $\\hat{\\mathbf {p}}_{\\infty }$ be its limit.",
"Notice by linearity of $\\left(TW_{0}\\left[c,\\lambda \\right]\\right)$ that $\\hat{\\mathbf {p}}_{\\infty }$ is in fact smooth and all its derivatives satisfy $\\left(TW_{0}\\left[c,\\lambda \\right]\\right)$ as well.",
"On the other hand, let $\\mathbf {w}_{n}=\\overline{\\Lambda }\\hat{\\mathbf {p}}_{n}-\\hat{\\mathbf {p}}_{n}^{\\prime }\\text{ for all }n\\in \\mathbb {N}\\cup \\left\\lbrace +\\infty \\right\\rbrace .$ Notice the following equality: $\\mathbf {w}_{n}\\left(\\xi \\right)=\\hat{\\mathbf {p}}_{n}\\left(\\xi \\right)\\circ \\left(\\overline{\\Lambda }\\mathbf {1}_{N,1}-\\mathbf {v}\\left(\\xi +\\xi _{n}\\right)\\right)\\text{ for all }n\\in \\mathbb {N}\\text{ and }\\xi \\in \\mathbb {R}.$ Fix $\\xi \\in \\mathbb {R}$ .",
"Recalling $\\left(\\limsup _{n\\rightarrow +\\infty }v_{i}\\left(\\xi +\\xi _{n}\\right)\\right)_{i\\in \\left[N\\right]}\\le \\left(\\limsup _{\\zeta \\rightarrow +\\infty }v_{i}\\left(\\zeta \\right)\\right)_{i\\in \\left[N\\right]}\\le \\overline{\\Lambda }\\mathbf {1}_{N,1},$ it follows that for all $\\varepsilon >0$ there exists $n_{\\xi ,\\varepsilon }\\in \\mathbb {N}$ such that for all $n\\ge n_{\\xi ,\\varepsilon }$ , $\\left(\\overline{\\Lambda }+\\varepsilon \\right)\\mathbf {1}_{N,1}\\ge \\mathbf {v}\\left(\\xi +\\xi _{n}\\right),$ whence, for all $n\\ge n_{\\xi ,\\varepsilon }$ , $\\mathbf {w}_{n}\\left(\\xi \\right) & \\ge -\\varepsilon \\left(\\sup _{m\\ge n_{\\xi ,\\varepsilon }}\\hat{p}_{m,i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\\\& \\ge -\\varepsilon \\left(\\sup _{m\\in \\mathbb {N}}\\hat{p}_{m,i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]},$ and consequently, passing to the limit $n\\rightarrow +\\infty $ and then $\\varepsilon \\rightarrow 0$ , we obtain the non-negativity of $\\mathbf {w}_{\\infty }\\left(\\xi \\right)$ .",
"Hence $\\mathbf {w}_{\\infty }$ is a nonnegative solution of $\\left(TW_{0}\\left[c,\\lambda \\right]\\right)$ satisfying in addition $w_{\\infty ,\\overline{i}}\\left(0\\right)=\\hat{p}_{\\infty ,\\overline{i}}\\left(0\\right)\\left(\\overline{\\Lambda }-\\lim _{n\\rightarrow +\\infty }v_{\\overline{i}}\\left(\\xi _{n}\\right)\\right)=0,$ whence, again by Arapostathis–Gosh–Marcus’s Harnack inequality, $\\mathbf {w}_{\\infty }$ is in fact the null function.",
"Consequently, $\\overline{\\Lambda }\\hat{\\mathbf {p}}_{\\infty }=\\hat{\\mathbf {p}}_{\\infty }^{\\prime }$ , that is $\\hat{\\mathbf {p}}_{\\infty }$ has exactly the form $\\xi \\mapsto \\text{e}^{\\overline{\\Lambda }\\xi }\\mathbf {n}\\text{ with }\\mathbf {n}\\in \\mathbb {R}^{N}.$ Since $\\hat{\\mathbf {p}}_{\\infty }$ is nonnegative with $\\hat{p}_{\\infty ,\\overline{i}}\\left(0\\right)=1$ by construction, $\\mathbf {n}\\in \\mathsf {K}^{+}$ , and since any nonnegative nonzero solution of $\\left(TW_{0}\\left[c,\\lambda \\right]\\right)$ is positive (REF ), $\\mathbf {n}\\in \\mathsf {K}^{++}$ .",
"The proof is ended with $\\mu =-\\overline{\\Lambda }$ .",
"For all $\\mu \\in \\mathbb {R}$ , the matrix $\\mu ^{2}\\mathbf {D}+\\mathbf {L}$ is essentially nonnegative irreducible.",
"Define $\\kappa _{\\mu }=-\\lambda _{PF}\\left(\\mu ^{2}\\mathbf {D}+\\mathbf {L}\\right)$ and $\\mathbf {n}_{\\mu }=\\mathbf {n}_{PF}\\left(\\mu ^{2}\\mathbf {D}+\\mathbf {L}\\right)$ .",
"Of course, the interest of the pair $\\left(\\kappa _{\\mu },\\mathbf {n}_{\\mu }\\right)$ lies in the preceding lemma: for all $\\left(\\mu ,\\mathbf {n}\\right)\\in \\mathbb {R}\\times \\mathsf {K}^{++}$ , $\\xi \\mapsto \\text{e}^{-\\mu \\xi }\\mathbf {n}$ is a solution of $\\left(TW_{0}\\left[c\\right]\\right)$ if and only if $-\\mu ^{2}\\mathbf {D}\\mathbf {n}+\\mu c\\mathbf {n}-\\mathbf {L}\\mathbf {n}=\\mathbf {0},$ that is, thanks to the Perron–Frobenius theorem, if and only if $\\mu c=-\\kappa _{\\mu }$ and $\\frac{\\mathbf {n}}{\\left|\\mathbf {n}\\right|}=\\mathbf {n}_{\\mu }$ .",
"This most important observation leads naturally to the following study of the equation $c=-\\frac{\\kappa _{\\mu }}{\\mu }$ .",
"Lemma 6.2 The quantity $c^{\\star }=\\min _{\\mu >0}\\left(-\\frac{\\kappa _{\\mu }}{\\mu }\\right)$ is well-defined and positive.",
"Let $c\\in [0,+\\infty )$ .",
"In $\\left(-\\infty ,0\\right)$ , the equation $-\\frac{\\kappa _{\\mu }}{\\mu }=c$ admits no solution.",
"In $\\left(0,+\\infty \\right)$ , it admits exactly: no solution if $c<c^{\\star }$ ; one solution $\\mu _{c^{\\star }}>0$ if $c=c^{\\star }$ ; two solutions $\\left(\\mu _{1,c},\\mu _{2,c}\\right)$ if $c>c^{\\star }$ , which satisfy moreover $0<\\mu _{1,c}<\\mu _{c^{\\star }}<\\mu _{2,c}.$ Remark $c^{\\star }$ does not depend on $\\mathbf {c}$ and is entirely determined by $\\mathbf {D}$ and $\\mathbf {L}$ .",
"It will be the minimal speed of traveling waves and this kind of dependency is strongly reminiscent of the scalar Fisher–KPP case, where $c^{\\star }=2\\sqrt{rd}$ .",
"In fact the following proof is mostly a generalization of scalar arguments.",
"Of course, $\\mu \\mapsto -\\frac{\\kappa _{\\mu }}{\\mu }$ is odd in $\\mathbb {R}\\backslash \\left\\lbrace 0\\right\\rbrace $ .",
"It is also positive in $\\left(0,+\\infty \\right)$ : $-\\frac{\\kappa _{\\mu }}{\\mu }=\\frac{1}{\\mu }\\lambda _{PF}\\left(\\mu ^{2}\\mathbf {D}+\\mathbf {L}\\right)>\\frac{1}{\\mu }\\lambda _{PF}\\left(\\mathbf {L}\\right)>0.$ Therefore it is negative in $\\left(-\\infty ,0\\right)$ and in particular there is no solution of $-\\frac{\\kappa _{\\mu }}{\\mu }=c\\ge 0$ in $\\left(-\\infty ,0\\right)$ .",
"We recall Nussbaum’s theorem [44] which proves the convexity of the function $\\mu \\mapsto \\rho \\left(\\mathbf {A}_{\\mu }\\right)$ provided: the matrix $\\mathbf {A}_{\\mu }$ is irreducible, its diagonal entries are convex functions of $\\mu $ , its off-diagonal entries are nonnegative log-convex functions of $\\mu $ .",
"These conditions are easily verified for $\\mu ^{2}\\mathbf {D}+\\mathbf {L}$ and $\\mu \\mathbf {D}+\\frac{1}{\\mu }\\mathbf {L}$ (actually, for all $\\mu ^{-\\gamma }\\left(\\mu ^{2}\\mathbf {D}+\\mathbf {L}\\right)$ provided $\\gamma \\in \\left[0,2\\right]$ ).",
"Their spectral radii being respectively $-\\kappa _{\\mu }$ and $-\\frac{\\kappa _{\\mu }}{\\mu }$ , these are therefore convex functions of $\\mu $ .",
"Moreover, Nussbaum’s result also proves that these convexities are actually strict.",
"Therefore $\\mu \\mapsto -\\kappa _{\\mu }$ and $\\mu \\mapsto -\\frac{\\kappa _{\\mu }}{\\mu }$ are strictly convex functions in $\\left(0,+\\infty \\right)$ .",
"Now, we investigate the behavior of $-\\frac{\\kappa _{\\mu }}{\\mu }$ as $\\mu \\rightarrow 0$ and $\\mu \\rightarrow +\\infty $ .",
"By continuity, $\\kappa _{\\mu }\\rightarrow \\kappa _{0}\\text{ as }\\mu \\rightarrow 0,$ whence $-\\frac{\\kappa _{\\mu }}{\\mu }\\rightarrow +\\infty $ as $\\mu \\rightarrow 0$ .",
"Since $\\mu \\mapsto -\\frac{\\kappa _{\\mu }}{\\mu }$ is convex and positive, either it is bounded in a neighborhood of $+\\infty $ and then it converges to some nonnegative constant, either it is unbounded in a neighborhood of $+\\infty $ and then it converges to $+\\infty $ .",
"Assume that it converges to a finite constant.",
"Notice $\\lim _{\\mu \\rightarrow +\\infty }\\frac{1}{\\mu ^{2}}\\left(\\mu ^{2}\\mathbf {D}+\\mathbf {L}\\right)=\\mathbf {D}.$ There exists a family of Perron–Frobenius eigenvectors of $\\mu \\mathbf {D}+\\frac{1}{\\mu }\\mathbf {L}$ , $\\left(\\mathbf {m}_{\\mu }\\right)_{\\mu >0}$ , normalized so that $\\max \\limits _{i\\in \\left[N\\right]}m_{\\mu ,i}=1$ for all $\\mu >0$ .",
"Thanks to classical compactness arguments in $\\mathbb {R}$ and $\\mathbb {R}^{N}$ , we can extract a sequence $\\left(\\mu _{n}\\right)_{n\\in \\mathbb {N}}$ such that $\\mu _{n}\\rightarrow +\\infty $ , $-\\frac{\\kappa _{\\mu _{n}}}{\\mu _{n}^{2}}$ converges to 0 and $\\mathbf {m}_{\\mu _{n}}$ converges to some $\\mathbf {m}\\in \\mathsf {K}^{+}$ .",
"We point out that we do not know if $\\mathbf {m}\\in \\mathsf {K}^{++}$ , but from the normalizations, we do know that $\\mathbf {m}\\in \\mathsf {K}^{+}$ .",
"Since $\\mathbf {m}$ satisfies $\\mathbf {D}\\mathbf {m}=\\mathbf {0}$ and since $\\mathbf {D}$ is invertible, we get a contradiction.",
"Thus $\\lim _{\\mu \\rightarrow +\\infty }-\\frac{\\kappa _{\\mu }}{\\mu }=+\\infty .$ Hence $\\mu \\mapsto -\\frac{\\kappa _{\\mu }}{\\mu }$ is a strictly convex positive function which goes to $+\\infty $ as $\\mu \\rightarrow 0$ or $\\mu \\rightarrow +\\infty $ : it admits necessarily a unique global minimum in $\\left(0,+\\infty \\right)$ .",
"The quantity $c^{\\star }$ is well-defined.",
"Define $\\mu _{c^{\\star }}>0$ such that $c^{\\star }=-\\frac{\\kappa _{\\mu _{c^{\\star }}}}{\\mu _{c^{\\star }}}.$ The quantity $\\mu _{c^{\\star }}$ is uniquely defined by strict convexity.",
"The function $\\mu \\mapsto -\\frac{\\kappa _{\\mu }}{\\mu }$ is bijective from $\\left(0,\\mu _{c^{\\star }}\\right)$ to $\\left(c^{\\star },+\\infty \\right)$ and from $\\left(\\mu _{c^{\\star }},+\\infty \\right)$ to $\\left(c^{\\star },+\\infty \\right)$ as well.",
"This ends the proof.",
"Putting together REF and REF , we get the following important result.",
"Corollary 6.3 For all $c\\in [0,+\\infty )$ , the set of nonnegative nonzero classical solutions of $\\left(TW_{0}\\left[c\\right]\\right)$ is empty if and only if $c\\in [0,c^{\\star })$ .",
"We can also get the exact values of $c$ for which $\\mathbf {0}$ is an unstable steady state of $\\left(TW_{0}\\left[c\\right]\\right)$ , in the sense of REF .",
"Lemma 6.4 Let $c\\in [0,+\\infty )$ .",
"Then $\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-c\\frac{\\text{d}}{\\text{d}x}-\\mathbf {L}\\right)=\\sup \\limits _{\\mu \\in \\mathbb {R}}\\left(\\kappa _{\\mu }+\\mu c\\right).$ Furthermore: $\\sup \\limits _{\\mu \\in \\mathbb {R}}\\left(\\kappa _{\\mu }+\\mu c\\right)=\\max \\limits _{\\mu \\ge 0}\\left(\\kappa _{\\mu }+\\mu c\\right)$ ; $\\max \\limits _{\\mu \\ge 0}\\left(\\kappa _{\\mu }+\\mu c\\right)<0$ if and only if $c<c^{\\star }$ .",
"Remark Just as in the case $c=0$ , it can be shown that, for all $c\\in [0,+\\infty )$ , $R\\mapsto \\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}-c\\frac{\\text{d}}{\\text{d}\\xi }-\\mathbf {L},\\left(-R,R\\right)\\right)$ is a decreasing homeomorphism from $\\left(0,+\\infty \\right)$ onto $\\left(\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-c\\frac{\\text{d}}{\\text{d}x}-\\mathbf {L}\\right),+\\infty \\right)$ .",
"The fact that $\\sup \\limits _{\\mu \\in \\mathbb {R}}\\left(\\kappa _{\\mu }+\\mu c\\right)$ is finite and actually a maximum attained in $[0,+\\infty )$ is a direct consequence of: the evenness of $\\mu \\mapsto \\kappa _{\\mu }$ (whence, for all $\\mu >0$ , $\\kappa _{-\\mu }+\\left(-\\mu \\right)c<\\kappa _{\\mu }+\\mu c$ ); $\\kappa _{0}<0$ ; $\\frac{\\kappa _{\\mu }}{\\mu }+c\\rightarrow -\\infty $ as $\\mu \\rightarrow +\\infty $ (see the proof of REF ).",
"In addition, the sign of this maximum depending on the sign $c-c^{\\star }$ is given by REF .",
"Hence it only remains to prove $\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-c\\frac{\\text{d}}{\\text{d}x}-\\mathbf {L}\\right)=\\max \\limits _{\\mu \\ge 0}\\left(\\kappa _{\\mu }+\\mu c\\right).$ To do so, we use and adapt a well-known strategy of proof (see for instance Nadin [43]).",
"We recall from REF the definition of the generalized principal eigenvalue: $\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-c\\frac{\\text{d}}{\\text{d}x}-\\mathbf {L}\\right)=\\sup \\left\\lbrace \\lambda \\in \\mathbb {R}\\ |\\ \\exists \\mathbf {n}\\in {C}^{2}\\left(\\mathbb {R},\\mathsf {K}^{++}\\right)\\quad -\\mathbf {D}\\mathbf {n}^{\\prime \\prime }-c\\mathbf {n}^{\\prime }-\\mathbf {L}\\mathbf {n}\\ge \\lambda \\mathbf {n}\\right\\rbrace .$ Also, there exists a generalized principal eigenfunction.",
"We recall from REF that if there exists a generalized principal eigenfunction, then there exists a generalized principal eigenfunction of the form $\\xi \\mapsto \\text{e}^{-\\mu ^{\\star }\\xi }\\mathbf {m}$ with some constant $\\mu ^{\\star }\\ge 0$ and $\\mathbf {m}\\in \\mathsf {K}^{++}$ .",
"Now, $\\left(\\mu ^{\\star },\\mathbf {m}\\right)\\in [0,+\\infty )\\times \\mathsf {K}^{++}$ satisfies $-\\left(\\mu ^{\\star }\\right)^{2}\\mathbf {D}\\mathbf {m}+c\\mu ^{\\star }\\mathbf {m}-\\mathbf {L}\\mathbf {m}=\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}-c\\frac{\\text{d}}{\\text{d}\\xi }-\\mathbf {L}\\right)\\mathbf {m},$ that is $-\\left(\\left(\\mu ^{\\star }\\right)^{2}\\mathbf {D}+\\mathbf {L}\\right)\\mathbf {m}=\\left(\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}-c\\frac{\\text{d}}{\\text{d}\\xi }-\\mathbf {L}\\right)-c\\mu ^{\\star }\\right)\\mathbf {m},$ or in other words $\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}-c\\frac{\\text{d}}{\\text{d}\\xi }-\\mathbf {L}\\right)=\\kappa _{\\mu ^{\\star }}+c\\mu ^{\\star }\\text{ and }\\frac{\\mathbf {m}}{\\left|\\mathbf {m}\\right|}=\\mathbf {n}_{\\mu ^{\\star }}.$ Finally, the suitable test function to verify $\\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}-c\\frac{\\text{d}}{\\text{d}\\xi }-\\mathbf {L}\\right)\\ge \\kappa _{\\mu }+\\mu c\\text{ for all }\\mu \\ge 0$ is of course $\\mathbf {v}_{\\mu }:\\xi \\mapsto \\text{e}^{-\\mu \\xi }\\mathbf {n}_{\\mu }$ itself, which satisfies precisely $-\\mathbf {D}\\mathbf {v}_{\\mu }^{\\prime \\prime }-c\\mathbf {v}_{\\mu }^{\\prime }-\\mathbf {L}\\mathbf {v}_{\\mu }=\\left(\\kappa _{\\mu }+\\mu c\\right)\\mathbf {v}_{\\mu }.$ Corollary 6.5 The quantity $c^{\\star }$ is characterized by $c^{\\star } & =\\sup \\left\\lbrace c\\ge 0\\ |\\ \\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}-c\\frac{\\text{d}}{\\text{d}\\xi }-\\mathbf {L}\\right)<0\\right\\rbrace \\\\& =\\inf \\left\\lbrace c\\ge 0\\ |\\ \\lambda _{1}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}-c\\frac{\\text{d}}{\\text{d}\\xi }-\\mathbf {L}\\right)>0\\right\\rbrace .$"
],
[
"Qualitative properties of the traveling solutions",
"Thanks to REF and REF , we are now in position to establish a few interesting properties that have direct consequences but will also be used at the end of the construction of the traveling waves.",
"Lemma 6.6 Let $c\\in [0,+\\infty )$ and $\\mathbf {p}$ be a bounded nonnegative nonzero classical solution of $\\left(TW\\left[c\\right]\\right)$ .",
"If $\\left(\\liminf \\limits _{\\xi \\rightarrow +\\infty }p_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\in \\partial \\mathsf {K}$ , then $c\\ge c^{\\star }$ .",
"Remark The following proof is analogous to that of Berestycki–Nadin–Perthame–Ryzhik [10] for the non-local KPP equation.",
"Let $\\left(\\zeta _{n}\\right)_{n\\in \\mathbb {N}}\\in \\mathbb {R}^{\\mathbb {N}}$ such that, as $n\\rightarrow +\\infty $ , $\\zeta _{n}\\rightarrow +\\infty $ and at least one component of $\\left(\\mathbf {p}\\left(\\zeta _{n}\\right)\\right)_{n\\in \\mathbb {N}}$ converges to 0.",
"Define $\\mathbf {p}_{n}:\\xi \\mapsto \\mathbf {p}\\left(\\xi +\\zeta _{n}\\right)$ and observe that $\\mathbf {p}_{n}$ satisfies $\\left(TW\\left[c\\right]\\right)$ as well.",
"By virtue of Arapostathis–Gosh–Marcus’s Harnack inequality [3] applied to the linear operator $\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}+c\\frac{\\text{d}}{\\text{d}\\xi }+\\left(\\mathbf {L}-\\text{diag}\\left(\\mathbf {c}\\left[\\mathbf {p}_{n}\\right]\\right)\\right),$ classical elliptic estimates (Gilbarg–Trudinger [30]), $\\left(\\mathbf {p}_{n}\\right)_{n\\in \\mathbb {N}}$ converges up to a diagonal extraction in ${C}_{loc}^{2}$ to $\\mathbf {0}$ .",
"This proves that there is no limit point of $\\mathbf {p}$ at $+\\infty $ in $\\partial \\mathsf {K}\\backslash \\left\\lbrace \\mathbf {0}\\right\\rbrace $ .",
"Next, define $\\tilde{\\mathbf {p}}_{n}:\\xi \\mapsto \\frac{\\mathbf {p}\\left(\\xi +\\zeta _{n}\\right)}{\\left|\\mathbf {p}\\left(\\zeta _{n}\\right)\\right|}$ and notice, again by Arapostathis–Gosh–Marcus’s Harnack inequality, that $\\left(\\tilde{\\mathbf {p}}_{n}\\right)_{n\\in \\mathbb {N}}$ is locally uniformly bounded.",
"Since, for all $n\\in \\mathbb {N}$ , $\\tilde{\\mathbf {p}}_{n}$ solves $-\\mathbf {D}\\tilde{\\mathbf {p}}_{n}^{\\prime \\prime }-c\\tilde{\\mathbf {p}}_{n}^{\\prime }=\\mathbf {L}\\tilde{\\mathbf {p}}_{n}-\\mathbf {c}\\left[\\mathbf {p}_{n}\\right]\\circ \\tilde{\\mathbf {p}}_{n},$ with, thanks to the fact that $\\mathbf {c}$ vanishes at $\\mathbf {0}$ $\\left(H_{3}\\right)$ , $\\mathbf {c}\\left[\\mathbf {p}_{n}\\right]\\rightarrow \\mathbf {0}$ locally uniformly, up to extraction $\\left(\\tilde{\\mathbf {p}}_{n}\\right)_{n\\in \\mathbb {N}}$ converges in ${C}_{loc}^{2}$ to a nonnegative solution $\\tilde{\\mathbf {p}}$ of $\\left(TW_{0}\\left[c\\right]\\right)$ .",
"Since $\\tilde{\\mathbf {p}}_{n}\\left(0\\right)\\in \\mathsf {S}^{++}\\left(\\mathbf {0},1\\right)$ for all $n\\in \\mathbb {N}$ , $\\tilde{\\mathbf {p}}$ is nonnegative nonzero, whence positive (REF ).",
"Now, from REF , we deduce indeed that $c\\ge c^{\\star }$ .",
"This result implies the nonexistence half of REF REF .",
"Corollary 6.7 For all $c\\in [0,c^{\\star })$ , there is no traveling wave solution of $\\left(E_{KPP}\\right)$ with speed $c$ .",
"Now, with REF , $c\\ge c^{\\star }>0$ and the fact that $\\left(t,x\\right)\\mapsto \\mathbf {p}\\left(x-ct\\right)$ solves $\\left(E_{KPP}\\right)$ , we can straightforwardly derive the uniform upper bound REF REF , which is interestingly independent of $c$ .",
"Corollary 6.8 All profiles $\\mathbf {p}$ satisfy $\\mathbf {p}\\le \\mathbf {g}\\left(0\\right)\\text{ in }\\mathbb {R}.$ Subsequently, using REF and again $c\\ge c^{\\star }>0$ and the fact that $\\left(t,x\\right)\\mapsto \\mathbf {p}\\left(x-ct\\right)$ solves $\\left(E_{KPP}\\right)$ , we get REF REF , independent of $c$ as well.",
"Corollary 6.9 All profiles $\\mathbf {p}$ satisfy $\\left(\\liminf _{\\xi \\rightarrow -\\infty }p_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\ge \\nu \\mathbf {1}_{N,1}.$ Now, we establish REF REF .",
"Its proof is actually mostly a repetition of that of REF .",
"Proposition 6.10 Let $\\left(\\mathbf {p},c\\right)$ be a traveling wave solution of $\\left(E_{KPP}\\right)$ .",
"Then there exists $\\overline{\\xi }\\in \\mathbb {R}$ such that $\\mathbf {p}$ is component-wise decreasing in $[\\overline{\\xi },+\\infty )$ .",
"Let $\\mathbf {v}=\\left(\\frac{p_{i}^{\\prime }}{p_{i}}\\right)_{i\\in \\left[N\\right]}$ .",
"By virtue of Arapostathis–Gosh–Marcus’s Harnack inequality [3], classical elliptic estimates (Gilbarg–Trudinger [30]) and invariance by translation of $\\left(TW\\left[c\\right]\\right)$ , $\\mathbf {v}$ is globally bounded.",
"Define for all $i\\in \\left[N\\right]$ $\\Lambda _{i}=\\limsup _{\\xi \\rightarrow +\\infty }v_{i}\\left(\\xi \\right).$ Let $\\overline{\\Lambda }=\\max \\limits _{i\\in \\left[N\\right]}\\Lambda _{i}$ , so that $\\left(\\limsup _{\\xi \\rightarrow +\\infty }v_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\le \\overline{\\Lambda }\\mathbf {1}_{N,1}.$ Let $\\left(\\xi _{n}\\right)_{n\\in \\mathbb {N}}\\in \\mathbb {R}^{\\mathbb {N}}$ such that $\\xi _{n}\\rightarrow +\\infty $ and such that there exists $\\overline{i}\\in \\left[N\\right]$ such that $v_{\\overline{i}}\\left(\\xi _{n}\\right)\\rightarrow \\overline{\\Lambda }\\text{ as }n\\rightarrow +\\infty .$ Let $\\hat{\\mathbf {p}}_{n}:\\xi \\mapsto \\frac{\\mathbf {p}\\left(\\xi +\\xi _{n}\\right)}{p_{\\overline{i}}\\left(\\xi _{n}\\right)}\\text{ for all }n\\in \\mathbb {N}.$ and notice, again by Arapostathis–Gosh–Marcus’s Harnack inequality, that $\\left(\\hat{\\mathbf {p}}_{n}\\right)_{n\\in \\mathbb {N}}$ is locally uniformly bounded.",
"Since, for all $n\\in \\mathbb {N}$ , $\\hat{\\mathbf {p}}_{n}$ solves $-\\mathbf {D}\\hat{\\mathbf {p}}_{n}^{\\prime \\prime }-c\\hat{\\mathbf {p}}_{n}^{\\prime }=\\mathbf {L}\\hat{\\mathbf {p}}_{n}-\\mathbf {c}\\left[p_{\\overline{i}}\\left(\\xi _{n}\\right)\\hat{\\mathbf {p}}_{n}\\right]\\circ \\hat{\\mathbf {p}}_{n},$ and, thanks to the fact that $\\mathbf {c}$ vanishes at $\\mathbf {0}$ $\\left(H_{3}\\right)$ and the asymptotic behavior of $\\mathbf {p}$ at $+\\infty $ , $\\mathbf {c}\\left[p_{\\overline{i}}\\left(\\xi _{n}\\right)\\hat{\\mathbf {p}}_{n}\\right]$ converges locally uniformly to $\\mathbf {0}$ as $n\\rightarrow +\\infty $ , up to a diagonal extraction process, $\\left(\\hat{\\mathbf {p}}_{n}\\right)_{n\\in \\mathbb {N}}$ converges in ${C}_{loc}^{2}$ to a nonnegative solution $\\hat{\\mathbf {p}}_{\\infty }$ of $\\left(TW_{0}\\left[c\\right]\\right)$ .",
"Now we repeat the second part of the proof of REF and we deduce in the end from REF that $\\hat{\\mathbf {p}}_{\\infty }$ has exactly the form $\\xi \\mapsto A\\text{e}^{-\\mu _{c}\\xi }\\mathbf {n}_{\\mu _{c}},$ with $\\mu _{c}\\in \\left\\lbrace \\mu _{1,c},\\mu _{2,c}\\right\\rbrace $ if $c>c^{\\star }$ , $\\mu _{c}=\\mu _{c^{\\star }}$ if $c=c^{\\star }$ , $A>0$ and, most importantly, with $\\mu _{c}=-\\overline{\\Lambda }$ .",
"Thus $\\overline{\\Lambda }<0$ .",
"This implies that there exists $\\overline{\\xi }\\in \\mathbb {R}$ such that, for all $\\xi \\ge \\overline{\\xi }$ , $\\mathbf {v}\\left(\\xi \\right)\\le -\\frac{\\left|\\overline{\\Lambda }\\right|}{2}\\mathbf {1}_{N,1},$ whence, by positivity of $\\mathbf {p}$ , $\\mathbf {p}^{\\prime }\\left(\\xi \\right)\\le -\\frac{\\left|\\overline{\\Lambda }\\right|}{2}\\mathbf {p}\\left(\\xi \\right).$ The right-hand side being negative, $\\mathbf {p}$ is component-wise decreasing indeed.",
"Lemma 6.11 Let $c\\in [0,+\\infty )$ and $\\mathbf {p}$ be a bounded nonnegative nonzero classical solution of $\\left(TW\\left[c\\right]\\right)$ .",
"If $\\left(\\liminf \\limits _{\\xi \\rightarrow +\\infty }p_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\in \\partial \\mathsf {K}$ , then $\\lim \\limits _{\\xi \\rightarrow +\\infty }\\mathbf {p}\\left(\\xi \\right)=\\mathbf {0}$ .",
"Let $\\left(\\zeta _{n}\\right)_{n\\in \\mathbb {N}}\\in \\mathbb {R}^{\\mathbb {N}}$ such that, as $n\\rightarrow +\\infty $ , $\\zeta _{n}\\rightarrow +\\infty $ and at least one component of $\\left(\\mathbf {p}\\left(\\zeta _{n}\\right)\\right)_{n\\in \\mathbb {N}}$ converges to 0.",
"The proof of REF shows that $\\left(\\mathbf {p}_{n}\\right)_{n\\in \\mathbb {N}}$ , defined by $\\mathbf {p}_{n}:\\xi \\mapsto \\mathbf {p}\\left(\\xi +\\zeta _{n}\\right)$ , converges up to extraction in ${C}_{loc}^{2}$ to $\\mathbf {0}$ .",
"Now, defining $\\mathbf {v}_{n}:\\xi \\mapsto \\left(\\frac{p_{n,i}^{\\prime }\\left(\\xi \\right)}{p_{n,i}\\left(\\xi \\right)}\\right)_{i\\in \\left[N\\right]},$ $\\Lambda _{i}=\\limsup _{n\\rightarrow +\\infty }\\max _{\\left[-1,1\\right]}v_{n,i},$ $\\overline{\\Lambda }=\\max _{i\\in \\left[N\\right]}\\Lambda _{i},$ $\\overline{i}\\in \\left[N\\right]\\text{ such that }\\Lambda _{\\overline{i}}=\\overline{\\Lambda },$ and $\\left(n_{m}\\right)_{m\\in \\mathbb {N}}\\in \\mathbb {N}^{\\mathbb {N}}$ an increasing sequence such that $v_{n_{m},\\overline{i}}\\left(0\\right)\\rightarrow \\overline{\\Lambda }$ as $m\\rightarrow +\\infty $ , we can repeat once more the argument of the proof of REF and obtain $\\overline{\\Lambda }\\hat{\\mathbf {p}}_{\\infty }=\\hat{\\mathbf {p}}_{\\infty }^{\\prime }\\text{ in }\\left(-1,1\\right)$ (notice that, contrarily to the proof of REF where this equality was proved in $\\mathbb {R}$ , here it only holds locally).",
"This brings forth $\\overline{\\Lambda }=-\\mu _{c}<0$ , as in the proof of REF , whence $\\mathbf {p}_{n}$ is component-wise decreasing in $\\left[-1,1\\right]$ provided $n$ is large enough.",
"Now, assuming by contradiction $\\left(\\limsup _{\\xi \\rightarrow +\\infty }p_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\in \\mathsf {K}^{+},$ that is $\\left(\\limsup _{\\xi \\rightarrow +\\infty }p_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\in \\mathsf {K}^{++},$ we deduce from the ${C}^{1}$ regularity of $\\mathbf {p}$ that, for any $i\\in \\left[N\\right]$ , there exists a sequence $\\left(\\zeta _{n}^{\\prime }\\right)_{n\\in \\mathbb {N}}\\in \\mathbb {R}^{\\mathbb {N}}$ such that: $\\zeta _{n}^{\\prime }\\rightarrow +\\infty $ as $n\\rightarrow +\\infty $ , $p_{i}\\left(\\zeta _{n}^{\\prime }\\right)$ is a local minimum of $p_{i}$ , $p_{i}\\left(\\zeta _{n}^{\\prime }\\right)\\rightarrow 0$ as $n\\rightarrow +\\infty $ .",
"Since this directly contradicts the preceding argument, we get indeed $\\left(\\limsup _{\\xi \\rightarrow +\\infty }p_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}=\\mathbf {0}=\\left(\\liminf \\limits _{\\xi \\rightarrow +\\infty }p_{i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}.$ Lemma 6.12 Let $c\\in [0,+\\infty )$ .",
"There exists $\\eta _{c}>0$ such that, for all bounded nonnegative classical solutions $\\mathbf {p}$ of $\\left(TW\\left[c\\right]\\right)$ , exactly one of the following properties holds: $\\lim \\limits _{\\xi \\rightarrow +\\infty }\\mathbf {p}\\left(\\xi \\right)=\\mathbf {0}$ ; $\\left(\\inf \\limits _{\\left(0,+\\infty \\right)}p_{i}\\right)_{i\\in \\left[N\\right]}\\ge \\eta _{c}\\mathbf {1}_{N,1}$ .",
"Remark The following proof is again analogous to that of Berestycki–Nadin–Perthame–Ryzhik [10] for the non-local KPP equation.",
"Recall from REF and REF that $\\left(\\inf \\limits _{\\left(0,+\\infty \\right)}p_{i}\\right)_{i\\in \\left[N\\right]}\\in \\partial \\mathsf {K}$ if and only if $\\lim \\limits _{\\xi \\rightarrow +\\infty }\\mathbf {p}\\left(\\xi \\right)=\\mathbf {0}$ .",
"Hence, defining $\\Sigma $ as the set of all bounded nonnegative classical solutions $\\mathbf {p}$ of $\\left(TW\\left[c\\right]\\right)$ such that $\\min _{i\\in \\left[N\\right]}\\inf _{\\left(0,+\\infty \\right)}p_{i}>0,$ this set containing at least one positive constant vector by virtue of REF , it only remains to show the positivity of $\\eta _{c}=\\inf \\left\\lbrace \\min _{i\\in \\left[N\\right]}\\inf _{\\left(0,+\\infty \\right)}p_{i}\\ |\\ \\mathbf {p}\\in \\Sigma \\right\\rbrace .$ We assume by contradiction the existence of a sequence $\\left(\\mathbf {p}_{n}\\right)_{n\\in \\mathbb {N}}\\in \\Sigma ^{\\mathbb {N}}$ such that $\\lim _{n\\rightarrow +\\infty }\\min _{i\\in \\left[N\\right]}\\inf _{\\left(0,+\\infty \\right)}p_{n,i}=0.$ For all $n\\in \\mathbb {N}$ , define $\\beta _{n}=\\min _{i\\in \\left[N\\right]}\\inf _{\\left(0,+\\infty \\right)}p_{n,i}>0,$ fix $\\xi _{n}\\in \\left(0,+\\infty \\right)$ such that $\\min _{i\\in \\left[N\\right]}p_{n,i}\\left(\\xi _{n}\\right)\\in \\left[\\beta _{n},\\beta _{n}+\\frac{1}{n}\\right],$ and define finally $\\mathbf {v}_{n}:\\xi \\mapsto \\frac{1}{\\beta _{n}}\\mathbf {p}_{n}\\left(\\xi +\\xi _{n}\\right).$ By virtue of Arapostathis–Gosh–Marcus’s Harnack inequality [3], classical elliptic estimates (Gilbarg–Trudinger [30]) and invariance by translation of $\\left(TW\\left[c\\right]\\right)$ , $\\left(\\mathbf {v}_{n}\\right)_{n\\in \\mathbb {N}}$ is locally uniformly bounded and, up to a diagonal extraction process, converges in ${C}_{loc}^{2}$ to some bounded limit $\\mathbf {v}_{\\infty }$ .",
"As in the proof of REF , it is easily verified that $\\mathbf {v}_{\\infty }$ is a bounded positive classical solution of $\\left(TW_{0}\\left[c\\right]\\right)$ .",
"Furthermore, by definition of $\\left(\\mathbf {v}_{n}\\right)_{n\\in \\mathbb {N}}$ , $\\mathbf {v}_{\\infty }\\ge \\mathbf {1}_{N,1}\\text{ in }\\left(0,+\\infty \\right).$ Repeating once more the argument of the proof of REF , we deduce that $\\mathbf {v}_{\\infty }$ is component-wise decreasing in a neighborhood of $+\\infty $ .",
"Thus its limit at $+\\infty $ , say $\\mathbf {m}\\ge \\mathbf {1}_{N,1}$ , is well-defined.",
"By classical elliptic estimates, $\\mathbf {m}$ satisfies $\\mathbf {L}\\mathbf {m}=\\mathbf {0}$ , which obviously contradicts $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ ."
],
[
"Existence of traveling waves",
"This whole subsection is devoted to the adaptation of a proof of existence due to Berestycki, Nadin, Perthame and Ryzhik [10] and originally applied to the non-local KPP equation.",
"Remark There is a couple of slight mistakes in the aforementioned proof.",
"Using the notations of [10], the sub-solution is defined as $\\overline{r}_{c}=\\max \\left(0,r_{c}\\right)$ , with $r_{c}$ chosen so that $-cr_{c}^{\\prime }\\le r_{c}^{\\prime \\prime }+\\mu r_{c}-\\mu \\overline{q}_{c}\\left(\\phi \\star \\overline{q}_{c}\\right)$ and it is claimed that $\\overline{r}_{c}$ satisfies as well this inequality, in the distributional sense.",
"This is false: in an interval where $\\overline{r}_{c}=0$ , we have $-c\\overline{r}_{c}^{\\prime }-\\overline{r}_{c}^{\\prime \\prime }-\\mu \\overline{r}_{c}=0>-\\mu \\overline{q}_{c}\\left(\\phi \\star \\overline{q}_{c}\\right).$ As we will show, the correct sub-solution is $\\overline{r}_{c}=\\max \\left(0,r_{c}\\right)$ with $r_{c}$ chosen so that $-cr_{c}^{\\prime }\\le r_{c}^{\\prime \\prime }+\\mu r_{c}-\\mu r_{c}\\left(\\phi \\star \\overline{q}_{c}\\right).$ Fortunately, the function $r_{c}$ constructed by the authors satisfies this inequality as well.",
"Later on, $\\Phi _{a}$ is defined as the mapping which maps $u_{0}$ to the solution of $-cu^{\\prime }=u^{\\prime \\prime }+\\mu u_{0}\\left(1-\\phi \\star u_{0}\\right).$ This mapping does not leave invariant the set of functions $R_{a}$ defined with the correct sub-solution.",
"It is necessary to change $\\Phi _{a}$ and to define it as the mapping which maps $u_{0}$ to the solution of $-cu^{\\prime }=u^{\\prime \\prime }+\\mu u\\left(1-\\phi \\star u_{0}\\right).$ Consequently, in order to establish that the set of functions $R_{a}$ is invariant by $\\Phi _{a}$ , the elliptic maximum principle is applied not to $u\\mapsto -cu^{\\prime }-u^{\\prime \\prime }$ but to $u\\mapsto -u^{\\prime \\prime }-cu^{\\prime }-\\mu u$ on one hand and to $u\\mapsto -u^{\\prime \\prime }-cu^{\\prime }-\\mu \\left(1-\\phi \\star \\overline{q}_{c}\\right)u$ on the other hand.",
"During the first three subsubsections, we fix $c>c^{\\star }$ ."
],
[
"Super-solution",
"We will use $\\overline{\\mathbf {p}}:\\xi \\mapsto \\text{e}^{-\\mu _{1,c}\\xi }\\mathbf {n}_{\\mu _{1,c}}$ as a super-solution (recall from REF that it is a solution of $\\left(TW_{0}\\left[c\\right]\\right)$ )."
],
[
"Sub-solution",
"Proposition 6.13 There exist $\\overline{\\varepsilon }>0$ such that, for any $\\varepsilon \\in \\left(0,\\overline{\\varepsilon }\\right)$ , there exists $A_{\\varepsilon }\\in \\left(0,+\\infty \\right)$ such that the function $\\underline{\\mathbf {p}}:\\xi \\mapsto \\left(\\max \\left(\\text{e}^{-\\mu _{1,c}\\xi }n_{\\mu _{1,c},i}-A_{\\varepsilon }\\text{e}^{-\\left(\\mu _{1,c}+\\varepsilon \\right)\\xi }n_{\\mu _{1,c}+\\varepsilon ,i},0\\right)\\right)_{i\\in \\left[N\\right]},$ satisfies $-\\mathbf {D}\\underline{\\mathbf {p}}^{\\prime \\prime }-c\\underline{\\mathbf {p}}^{\\prime }-\\mathbf {L}\\underline{\\mathbf {p}}\\le -\\mathbf {c}\\left[\\overline{\\mathbf {p}}\\right]\\circ \\underline{\\mathbf {p}}\\text{ in }{H}^{-1}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right).$ Remark Notice that, in the right-hand side of the inequality above, we find $\\mathbf {c}\\left[\\overline{\\mathbf {p}}\\right]$ and not $\\mathbf {c}\\left[\\underline{\\mathbf {p}}\\right]$ .",
"This is of course related to the lack of comparison principle for $\\left(E_{KPP}\\right)$ .",
"During the forthcoming quite technical proof, in order to ease the reading, we denote $\\left\\langle \\bullet ,\\bullet \\right\\rangle _{1}$ and $\\left\\langle \\bullet ,\\bullet \\right\\rangle _{N}$ the duality pairings of ${H}^{1}\\left(\\mathbb {R},\\mathbb {R}\\right)$ and ${H}^{1}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)$ respectively, the latter being of course defined by: $\\left\\langle \\mathbf {f},\\mathbf {g}\\right\\rangle _{{H}^{-1}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)\\times {H}^{1}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)}=\\sum _{i=1}^{N}\\left\\langle f_{i},g_{i}\\right\\rangle _{{H}^{-1}\\left(\\mathbb {R}\\right)\\times {H}^{1}\\left(\\mathbb {R}\\right)}.$ The speed $c$ being fixed, we also omit the subscript $c$ in the notations $\\mu _{1,c}$ and $\\mu _{2,c}$ .",
"For the moment, let $A,\\varepsilon >0$ (they will be made precise during the course of the proof) and define $\\mathbf {v}:\\xi \\mapsto \\text{e}^{-\\mu _{1}\\xi }\\mathbf {n}_{\\mu _{1}}-A\\text{e}^{-\\left(\\mu _{1}+\\varepsilon \\right)\\xi }\\mathbf {n}_{\\mu _{1}+\\varepsilon },$ $\\underline{\\mathbf {p}}:\\xi \\mapsto \\left(\\max \\left(\\text{e}^{-\\mu _{1}\\xi }n_{\\mu _{1},i}-A_{\\varepsilon }\\text{e}^{-\\left(\\mu _{1}+\\varepsilon \\right)\\xi }n_{\\mu _{1}+\\varepsilon ,i},0\\right)\\right)_{i\\in \\left[N\\right]},$ $\\Xi _{+}=\\underline{\\mathbf {p}}^{-1}\\left(\\mathsf {K}^{++}\\right),$ $\\Xi _{0}=\\underline{\\mathbf {p}}^{-1}\\left(\\mathbf {0}\\right),$ $\\Xi _{\\#}=\\mathbb {R}\\backslash \\left(\\Xi _{+}\\cup \\Xi _{0}\\right).$ Notice that $\\Xi _{\\#}$ is a connected compact set.",
"Fix a positive test function $\\varphi \\in {H}^{1}\\left(\\mathbb {R},\\mathsf {K}^{++}\\right)$ .",
"We have to verify that $\\left\\langle -\\mathbf {D}\\underline{\\mathbf {p}}^{\\prime \\prime }-c\\underline{\\mathbf {p}}^{\\prime }-\\mathbf {L}\\underline{\\mathbf {p}},\\varphi \\right\\rangle _{N}\\le \\left\\langle -\\mathbf {c}\\left[\\overline{\\mathbf {p}}\\right]\\circ \\underline{\\mathbf {p}},\\varphi \\right\\rangle _{N}.$ To this end, we distinguish three cases: $\\text{supp}\\varphi \\subset \\Xi _{+}$ , $\\text{supp}\\varphi \\subset \\Xi _{0}$ and $\\text{supp}\\varphi \\cap \\Xi _{\\#}\\ne \\emptyset $ .",
"The case $\\text{supp}\\varphi \\subset \\Xi _{0}$ is trivial, with the inequality above satisfied in the classical sense.",
"Regarding the case $\\text{supp}\\varphi \\subset \\Xi _{+}$ , we only have to verify the inequality in the classical sense in $\\Xi _{+}$ for the regular function $\\mathbf {v}$ .",
"Fix temporarily $\\xi \\in \\Xi _{+}$ .",
"We have $-\\mathbf {D}\\mathbf {v}^{\\prime \\prime }\\left(\\xi \\right)-c\\mathbf {v}^{\\prime }\\left(\\xi \\right)-\\mathbf {L}\\mathbf {v}\\left(\\xi \\right)=A\\text{e}^{-\\left(\\mu _{1}+\\varepsilon \\right)\\xi }\\left(\\left(\\mu _{1}+\\varepsilon \\right)^{2}\\mathbf {D}-c\\left(\\mu _{1}+\\varepsilon \\right)\\mathbf {I}+\\mathbf {L}\\right)\\mathbf {n}_{\\mu _{1}+\\varepsilon },$ $\\left(-\\mathbf {c}\\left[\\overline{\\mathbf {p}}\\right]\\circ \\mathbf {v}\\right)\\left(\\xi \\right)=-\\text{e}^{-\\mu _{1}\\xi }\\mathbf {c}\\left(\\text{e}^{-\\mu _{1}\\xi }\\mathbf {n}_{\\mu _{1}}\\right)\\circ \\left(\\mathbf {n}_{\\mu _{1}}-A\\text{e}^{-\\varepsilon \\xi }\\mathbf {n}_{\\mu _{1}+\\varepsilon }\\right).$ From $\\left(\\left(\\mu _{1}+\\varepsilon \\right)^{2}\\mathbf {D}+\\mathbf {L}\\right)\\mathbf {n}_{\\mu _{1}+\\varepsilon }=-\\kappa _{\\mu _{1}+\\varepsilon }\\mathbf {n}_{\\mu _{1}+\\varepsilon },$ $-c\\left(\\mu _{1}+\\varepsilon \\right)\\mathbf {n}_{\\mu _{1}+\\varepsilon }=\\frac{\\kappa _{\\mu _{1}}}{\\mu _{1}}\\left(\\mu _{1}+\\varepsilon \\right)\\mathbf {n}_{\\mu _{1}+\\varepsilon },$ and the following direct consequence of the nonnegativity of $\\mathbf {c}$ on $\\mathsf {K}$ $\\left(H_{2}\\right)$ , $-\\mathbf {c}\\left(\\text{e}^{-\\mu _{1}\\xi }\\mathbf {n}_{\\mu _{1}}\\right)\\circ \\left(\\mathbf {n}_{\\mu _{1}}-A\\text{e}^{-\\varepsilon \\xi }\\mathbf {n}_{\\mu _{1}+\\varepsilon }\\right)\\ge -\\mathbf {c}\\left(\\text{e}^{-\\mu _{1}\\xi }\\mathbf {n}_{\\mu _{1}}\\right)\\circ \\mathbf {n}_{\\mu _{1}},$ it follows that it suffices to find $A$ and $\\varepsilon $ such that $A\\text{e}^{-\\varepsilon \\xi }\\left(\\mu _{1}+\\varepsilon \\right)\\left(-\\frac{\\kappa _{\\mu _{1}+\\varepsilon }}{\\mu _{1}+\\varepsilon }+\\frac{\\kappa _{\\mu _{1}}}{\\mu _{1}}\\right)\\mathbf {n}_{\\mu _{1}+\\varepsilon }\\le -\\mathbf {c}\\left(\\text{e}^{-\\mu _{1}\\xi }\\mathbf {n}_{\\mu _{1}}\\right)\\circ \\mathbf {n}_{\\mu _{1}}.$ The right-hand side above being nonnegative ($\\mu \\mapsto \\frac{\\kappa _{\\mu }}{\\mu }$ is positive and convex in $\\left(0,+\\infty \\right)$ , as detailed in the proof of REF ), it follows clearly that such an inequality is never satisfied if $\\mu _{1}+\\varepsilon >\\mu _{2}$ , whence a first necessary condition on $\\varepsilon $ is $\\varepsilon \\le \\mu _{2}-\\mu _{1}$ (notice that if $\\varepsilon =\\mu _{2}-\\mu _{1}$ , then the inequality above holds if and only if $\\mathbf {c}\\left(\\text{e}^{-\\mu _{1}\\xi }\\mathbf {n}_{\\mu _{1}}\\right)=\\mathbf {0}$ , which is in general not true).",
"Thus from now on we assume $\\varepsilon <\\mu _{2}-\\mu _{1}$ .",
"This ensures that $\\frac{\\kappa _{\\mu _{1}+\\varepsilon }}{\\mu _{1}+\\varepsilon }-\\frac{\\kappa _{\\mu _{1}}}{\\mu _{1}}>0$ , whence we now search for $A$ and $\\varepsilon $ such that $A\\mathbf {n}_{\\mu _{1}+\\varepsilon }>\\frac{\\text{e}^{\\varepsilon \\xi }}{\\left(\\mu _{1}+\\varepsilon \\right)\\left(\\frac{\\kappa _{\\mu _{1}+\\varepsilon }}{\\mu _{1}+\\varepsilon }-\\frac{\\kappa _{\\mu _{1}}}{\\mu _{1}}\\right)}\\mathbf {c}\\left(\\text{e}^{-\\mu _{1}\\xi }\\mathbf {n}_{\\mu _{1}}\\right)\\circ \\mathbf {n}_{\\mu _{1}}.$ Define $\\overline{\\xi }=\\min \\Xi _{\\#}$ , so that any $\\xi \\in \\Xi _{+}$ satisfies necessarily $\\xi >\\overline{\\xi }$ .",
"Remark that there exists $\\overline{i}\\in \\left[N\\right]$ such that $\\overline{\\xi }=\\frac{1}{\\varepsilon }\\left(\\ln A+\\ln \\left(\\frac{n_{\\mu _{1}+\\varepsilon ,\\overline{i}}}{n_{\\mu _{1},\\overline{i}}}\\right)\\right).$ Now, defining $\\alpha :\\xi \\mapsto \\text{e}^{-\\mu _{1}\\xi }$ , if $A\\ge \\max _{i\\in \\left[N\\right]}\\left(\\frac{n_{\\mu _{1}+\\varepsilon ,i}}{n_{\\mu _{1},i}}\\right),$ then $\\overline{\\xi }\\ge 0$ and $\\alpha \\left(\\xi \\right)\\le 1$ in $\\left(\\overline{\\xi },+\\infty \\right)$ .",
"Moreover, we have $\\text{e}^{\\varepsilon \\xi }=\\left(\\alpha \\left(\\xi \\right)\\right)^{-\\frac{\\varepsilon }{\\mu _{1}}},$ whence, for all $i\\in \\left[N\\right]$ , $\\text{e}^{\\varepsilon \\xi }c_{i}\\left(\\text{e}^{-\\mu _{1}\\xi }\\mathbf {n}_{\\mu _{1}}\\right)=\\frac{c_{i}\\left(\\alpha \\left(\\xi \\right)\\mathbf {n}_{\\mu _{1}}\\right)}{\\left(\\alpha \\left(\\xi \\right)\\right)^{\\frac{\\varepsilon }{\\mu _{1}}}},$ and from the ${C}^{1}$ regularity of $\\mathbf {c}$ as well as the fact that it vanishes at $\\mathbf {0}$ $\\left(H_{3}\\right)$ , the above function of $\\xi $ is globally bounded in $\\left(\\overline{\\xi },+\\infty \\right)$ , provided $\\frac{\\varepsilon }{\\mu _{1}}\\le 1$ , by the positive constant $M_{i} & =\\sup _{\\xi \\in \\left(\\overline{\\xi },+\\infty \\right)}\\frac{c_{i}\\left(\\alpha \\left(\\xi \\right)\\mathbf {n}_{\\mu _{1}}\\right)}{\\alpha \\left(\\xi \\right)}\\\\& =\\sup _{\\alpha \\in \\left(0,1\\right)}\\frac{c_{i}\\left(\\alpha \\mathbf {n}_{\\mu _{1}}\\right)}{\\alpha }.$ Subsequently, if $A$ and $\\varepsilon $ satisfy also $\\varepsilon \\le \\mu _{1},$ $A\\ge \\max _{i\\in \\left[N\\right]}\\left(\\frac{M_{i}n_{\\mu _{1},i}}{\\left(\\mu _{1}+\\varepsilon \\right)\\left(\\frac{\\kappa _{\\mu _{1}+\\varepsilon }}{\\mu _{1}+\\varepsilon }-\\frac{\\kappa _{\\mu _{1}}}{\\mu _{1}}\\right)n_{\\mu _{1}+\\varepsilon ,i}}\\right),$ then the inequality is established indeed in $\\Xi _{+}$ .",
"Hence we define $\\overline{\\varepsilon }=\\min \\left(\\mu _{2}-\\mu _{1},\\mu _{1}\\right)$ and, for any $\\varepsilon \\in \\left(0,\\overline{\\varepsilon }\\right)$ , $A_{\\varepsilon }=\\max _{i\\in \\left[N\\right]}\\max \\left(\\frac{n_{\\mu _{1}+\\varepsilon ,i}}{n_{\\mu _{1},i}},\\frac{M_{i}n_{\\mu _{1},i}}{\\left(\\mu _{1}+\\varepsilon \\right)\\left(\\frac{\\kappa _{\\mu _{1}+\\varepsilon }}{\\mu _{1}+\\varepsilon }-\\frac{\\kappa _{\\mu _{1}}}{\\mu _{1}}\\right)n_{\\mu _{1}+\\varepsilon ,i}}\\right)$ and we assume from now on $\\varepsilon \\in \\left(0,\\overline{\\varepsilon }\\right)$ and $A=A_{\\varepsilon }$ .",
"Let us point out here a fact which is crucial for the next step: choosing $\\overline{\\xi }=\\min \\Xi _{\\#}$ instead of $\\overline{\\xi }=\\max \\Xi _{\\#}$ (which might seem more natural at first view) implies that the differential inequality $-\\mathbf {D}\\mathbf {v}^{\\prime \\prime }-c\\mathbf {v}^{\\prime }-\\mathbf {L}\\mathbf {v}\\le -\\mathbf {c}\\left[\\overline{\\mathbf {p}}\\right]\\circ \\mathbf {v}$ holds classically in $\\Xi _{\\#}\\cup \\Xi _{+}$ .",
"To conclude, let us verify the case $\\text{supp}\\varphi \\cap \\Xi _{\\#}\\ne \\emptyset $ .",
"In order to ease the following computations, we actually assume $\\varphi \\in {D}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)$ (the result with $\\varphi \\in {H}^{1}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)$ can be recovered as usual by density).",
"By definition, $\\left\\langle -\\mathbf {D}\\underline{\\mathbf {p}}^{\\prime \\prime }-c\\underline{\\mathbf {p}}^{\\prime }-\\mathbf {L}\\underline{\\mathbf {p}}+\\mathbf {c}\\left[\\overline{\\mathbf {p}}\\right]\\circ \\underline{\\mathbf {p}},\\varphi \\right\\rangle _{N}=\\sum _{i=1}^{N}\\left\\langle -d_{i}\\underline{p}_{i}^{\\prime \\prime }-c\\underline{p}_{i}^{\\prime }-\\sum _{j=1}^{N}l_{i,j}\\underline{p}_{j}+c_{i}\\left[\\overline{\\mathbf {p}}\\right]\\underline{p}_{i},\\varphi _{i}\\right\\rangle _{1}.$ Fix $i\\in \\left[N\\right]$ and define $\\xi _{0,i}$ as the unique element of $v_{i}^{-1}\\left(\\left\\lbrace 0\\right\\rbrace \\right)$ and $\\Psi _{i}=\\left\\langle -d_{i}\\underline{p}_{i}^{\\prime \\prime }-c\\underline{p}_{i}^{\\prime }-\\sum _{j=1}^{N}l_{i,j}\\underline{p}_{j}+c_{i}\\left[\\overline{\\mathbf {p}}\\right]\\underline{p}_{i},\\varphi _{i}\\right\\rangle _{1}.$ Classical integrations by parts yield $\\int _{\\mathbb {R}}\\underline{p}_{i}^{\\prime \\prime }\\varphi _{i}=\\int _{\\xi _{0,i}}^{+\\infty }v_{i}^{\\prime \\prime }\\varphi _{i}+v_{i}^{\\prime }\\left(\\xi _{0,i}\\right)\\varphi _{i}\\left(\\xi _{0,i}\\right)\\ge \\int _{\\xi _{0,i}}^{+\\infty }v_{i}^{\\prime \\prime }\\varphi _{i},$ $\\int _{\\mathbb {R}}\\underline{p}_{i}^{\\prime }\\varphi _{i}=\\int _{\\xi _{0,i}}^{+\\infty }v_{i}^{\\prime }\\varphi _{i},$ whence $\\Psi _{i}\\le \\int _{\\xi _{0,i}}^{+\\infty }\\left(-d_{i}v_{i}^{\\prime \\prime }-cv_{i}^{\\prime }+c_{i}\\left[\\overline{\\mathbf {p}}\\right]v_{i}\\right)\\varphi _{i}-\\sum _{j=1}^{N}l_{i,j}\\int _{\\xi _{0,j}}^{+\\infty }v_{j}\\varphi _{i}.$ As was pointed out previously, from the construction of $\\varepsilon $ and $A$ , we know that $-\\mathbf {D}\\mathbf {v}^{\\prime \\prime }-c\\mathbf {v}^{\\prime }+\\mathbf {c}\\left[\\overline{\\mathbf {p}}\\right]\\circ \\mathbf {v}\\le \\mathbf {L}\\mathbf {v}\\text{ in }\\Xi _{\\#},$ whence, with $J_{i}=\\left\\lbrace j\\in \\left[N\\right]\\ |\\ \\xi _{0,j}<\\xi _{0,i}\\right\\rbrace $ , $\\Psi _{i}\\le -\\sum _{j\\in J_{i}}\\int _{\\xi _{0,j}}^{\\xi _{0,i}}l_{i,j}v_{j}\\varphi _{i}+\\sum _{j\\in \\left[N\\right]\\backslash J_{i}}\\int _{\\xi _{0,i}}^{\\xi _{0,j}}l_{i,j}v_{j}\\varphi _{i}.$ Finally, recalling that $v_{j}\\left(\\xi \\right)>0$ if $\\xi >\\xi _{0,j}$ and $v_{j}\\left(\\xi \\right)<0$ if $\\xi <\\xi _{0,j}$ , the inequality above yields $\\Psi _{i}\\le 0$ , which ends the proof."
],
[
"The finite domain problem",
"Let $R>0$ and define the following truncated problem: $\\left\\lbrace \\begin{matrix}-\\mathbf {D}\\mathbf {p}^{\\prime \\prime }-c\\mathbf {p}^{\\prime }=\\mathbf {L}\\mathbf {p}-\\mathbf {c}\\left[\\mathbf {p}\\right]\\circ \\mathbf {p} & \\text{ in }\\left(-R,R\\right),\\\\\\mathbf {p}\\left(\\pm R\\right)=\\underline{\\mathbf {p}}\\left(\\pm R\\right).\\end{matrix}\\right.\\quad \\left(TW\\left[R,c\\right]\\right)$ Lemma 6.14 Assume $D\\mathbf {c}\\left(\\mathbf {v}\\right)\\ge \\mathbf {0}\\text{ for all }\\mathbf {v}\\in \\mathsf {K}.$ Then there exists a nonnegative nonzero classical solution $\\mathbf {p}_{R}$ of $\\left(TW\\left[R,c\\right]\\right)$ .",
"Remark The new assumption made here ensures that the vector field $\\mathbf {c}$ is non-decreasing in $\\mathsf {K}$ , in the following natural sense: if $\\mathbf {0}\\le \\mathbf {v}\\le \\mathbf {w}$ , then $\\mathbf {0}\\le \\mathbf {c}\\left(\\mathbf {v}\\right)\\le \\mathbf {c}\\left(\\mathbf {w}\\right)$ .",
"Fix arbitrarily $\\varepsilon \\in \\left(0,\\overline{\\varepsilon }\\right)$ , define consequently $\\underline{\\mathbf {p}}$ and then define the following convex set of functions: ${F}=\\left\\lbrace \\mathbf {v}\\in {C}\\left(\\left[-R,R\\right],\\mathbb {R}^{N}\\right)\\ |\\ \\underline{\\mathbf {p}}\\le \\mathbf {v}\\le \\overline{\\mathbf {p}}\\right\\rbrace .$ Recall that Figueiredo–Mitidieri [24] establishes that the elliptic weak maximum principle holds for a weakly and fully coupled elliptic operator with null Dirichlet boundary conditions if this operator admits a positive strict super-solution.",
"Since, for all $\\mathbf {v}\\in {C}\\left(\\left[-R,R\\right],\\mathbb {R}^{N}\\right)$ such that $\\mathbf {0}\\le \\mathbf {v}\\le \\overline{\\mathbf {p}}$ , we have by the nonnegativity of $\\mathbf {c}$ on $\\mathsf {K}$ $\\left(H_{2}\\right)$ $-\\mathbf {D}\\overline{\\mathbf {p}}^{\\prime \\prime }-c\\overline{\\mathbf {p}}^{\\prime }-\\mathbf {L}\\overline{\\mathbf {p}}+\\mathbf {c}\\left[\\mathbf {v}\\right]\\circ \\overline{\\mathbf {p}}\\ge -\\mathbf {D}\\overline{\\mathbf {p}}^{\\prime \\prime }-c\\overline{\\mathbf {p}}^{\\prime }-\\mathbf {L}\\overline{\\mathbf {p}}\\ge \\mathbf {0},$ $\\overline{\\mathbf {p}}\\left(\\pm R\\right)\\gg \\mathbf {0},$ it follows that every operator of the family $\\left(\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}+c\\frac{\\text{d}}{\\text{d}\\xi }+\\left(\\mathbf {L}-\\text{diag}\\mathbf {c}\\left[\\mathbf {v}\\right]\\right)\\right)_{\\mathbf {0}\\le \\mathbf {v}\\le \\overline{\\mathbf {p}}}$ supplemented with null Dirichlet boundary conditions at $\\pm R$ satisfies the weak maximum principle in $\\left(-R,R\\right)$ .",
"Define the map $\\mathbf {f}$ which associates with some $\\mathbf {v}\\in {F}$ the unique classical solution $\\mathbf {f}\\left[\\mathbf {v}\\right]$ of: $\\left\\lbrace \\begin{matrix}-\\mathbf {D}\\mathbf {p}^{\\prime \\prime }-c\\mathbf {p}^{\\prime }=\\mathbf {L}\\mathbf {p}-\\mathbf {c}\\left[\\mathbf {v}\\right]\\circ \\mathbf {p} & \\text{ in }\\left(-R,R\\right)\\\\\\mathbf {p}\\left(\\pm R\\right)=\\underline{\\mathbf {p}}\\left(\\pm R\\right).\\end{matrix}\\right.$ The map $\\mathbf {f}$ is compact by classical elliptic estimates (Gilbarg–Trudinger [30]).",
"Let $\\mathbf {v}\\in {F}$ .",
"By monotonicity of $\\mathbf {c}$ , the function $\\mathbf {w}=\\mathbf {f}\\left[\\mathbf {v}\\right]-\\underline{\\mathbf {p}}$ satisfies $-\\mathbf {D}\\mathbf {w}^{\\prime \\prime }-c\\mathbf {w}^{\\prime }-\\mathbf {L}\\mathbf {w} & \\ge -\\mathbf {c}\\left[\\mathbf {v}\\right]\\circ \\mathbf {f}\\left[\\mathbf {v}\\right]+\\mathbf {c}\\left[\\overline{\\mathbf {p}}\\right]\\circ \\underline{\\mathbf {p}}\\\\& \\ge -\\mathbf {c}\\left[\\mathbf {v}\\right]\\circ \\mathbf {f}\\left[\\mathbf {v}\\right]+\\mathbf {c}\\left[\\mathbf {v}\\right]\\circ \\underline{\\mathbf {p}}\\\\& \\ge -\\mathbf {c}\\left[\\mathbf {v}\\right]\\circ \\mathbf {w}$ with null Dirichlet boundary conditions at $\\pm R$ .",
"Therefore, by virtue of the weak maximum principle applied to $\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}+c\\frac{\\text{d}}{\\text{d}\\xi }+\\left(\\mathbf {L}-\\text{diag}\\mathbf {c}\\left[\\mathbf {v}\\right]\\right)$ , $\\mathbf {f}\\left[\\mathbf {v}\\right]\\ge \\underline{\\mathbf {p}}$ in $\\left(-R,R\\right)$ .",
"Next, since it is now established that $\\mathbf {f}\\left[\\mathbf {v}\\right]\\ge \\mathbf {0}$ , we also have by $\\left(H_{2}\\right)$ $-\\mathbf {D}\\overline{\\mathbf {p}}^{\\prime \\prime }-c\\overline{\\mathbf {p}}^{\\prime }-\\mathbf {L}\\overline{\\mathbf {p}} & =\\mathbf {0}\\\\& \\ge -\\mathbf {c}\\left[\\mathbf {v}\\right]\\circ \\mathbf {f}\\left[\\mathbf {v}\\right]\\\\& =-\\mathbf {D}\\mathbf {f}\\left[\\mathbf {v}\\right]^{\\prime \\prime }-c\\mathbf {f}\\left[\\mathbf {v}\\right]^{\\prime }-\\mathbf {L}\\mathbf {f}\\left[\\mathbf {v}\\right],$ $\\overline{\\mathbf {p}}\\left(\\pm R\\right)\\ge \\underline{\\mathbf {p}}\\left(\\pm R\\right)=\\mathbf {f}\\left[\\mathbf {v}\\right]\\left(\\pm R\\right),$ whence $\\overline{\\mathbf {p}}\\ge \\mathbf {f}\\left[\\mathbf {v}\\right]$ follows from the weak maximum principle applied this time to $\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}+c\\frac{\\text{d}}{\\text{d}\\xi }+\\mathbf {L}$ .",
"Thus $\\underline{\\mathbf {p}}\\le \\mathbf {f}\\left[\\mathbf {v}\\right]\\le \\overline{\\mathbf {p}}$ and consequently $\\mathbf {f}\\left({F}\\right)\\subset {F}$ .",
"Finally, by virtue of the Schauder fixed point theorem, $\\mathbf {f}$ admits a fixed point $\\mathbf {p}_{R}\\in {F}$ , which is indeed a classical solution of $\\left(TW\\left[R,c\\right]\\right)$ by elliptic regularity."
],
[
"The infinite domain limit and the minimal wave speed",
"The speed $c$ is not fixed anymore.",
"The following uniform upper estimate is a direct consequence of REF .",
"Corollary 6.15 There exists $R^{\\star }>0$ such that, for any $c>c^{\\star }$ , any $R\\ge R^{\\star }$ and any nonnegative classical solution $\\mathbf {p}$ of $\\left(TW\\left[R,c\\right]\\right)$ , $\\left(\\max _{\\left[-R,R\\right]}p_{i}\\right)_{i\\in \\left[N\\right]}\\le \\mathbf {g}\\left(0\\right).$ We are now in position to prove the second half of REF REF .",
"Proposition 6.16 Assume $D\\mathbf {c}\\left(\\mathbf {v}\\right)\\ge \\mathbf {0}\\text{ for all }\\mathbf {v}\\in \\mathsf {K}.$ Then for all $c\\ge c^{\\star }$ , there exists a traveling wave solution of $\\left(E_{KPP}\\right)$ with speed $c$ .",
"Remark Of course, it would be interesting to exhibit other additional assumptions on $\\mathbf {c}$ sufficient to ensure existence of traveling waves for all $c\\ge c^{\\star }$ .",
"In view of known results about scalar multistable reaction–diffusion equations (we refer for instance to Fife–McLeod [27]), some additional assumption should in any case be necessary.",
"Hereafter, for all $c>c^{\\star }$ and all $R>0$ , the triplet $\\left(\\overline{\\mathbf {p}},\\underline{\\mathbf {p}},\\mathbf {p}_{R}\\right)$ constructed in the preceding subsections is denoted $\\left(\\overline{\\mathbf {p}}_{c},\\underline{\\mathbf {p}}_{c},\\mathbf {p}_{R,c}\\right)$ .",
"For all $c>c^{\\star }$ , thanks to REF , the family $\\left(\\mathbf {p}_{R,c}\\right)_{R>0}$ is uniformly globally bounded.",
"By classical elliptic estimates (Gilbarg–Trudinger [30]) and a diagonal extraction process, we can extract a sequence $\\left(R_{n},\\mathbf {p}_{R_{n},c}\\right)_{n\\in \\mathbb {N}}$ such that, as $n\\rightarrow +\\infty $ , $R_{n}\\rightarrow +\\infty $ and $\\mathbf {p}_{R_{n},c}$ converges to some limit $\\mathbf {p}_{c}$ in ${C}_{loc}^{2}$ .",
"As expected, $\\mathbf {p}_{c}$ is a bounded nonnegative classical solution of $\\left(TW\\left[c\\right]\\right)$ .",
"The fact that its limit as $\\xi \\rightarrow +\\infty $ is $\\mathbf {0}$ , as well as the fact that $\\mathbf {p}_{c}$ is nonzero whence positive (REF ), are obvious thanks to the inequality $\\underline{\\mathbf {p}}_{c}\\le \\mathbf {p}_{c}\\le \\overline{\\mathbf {p}}_{c}$ .",
"At the other end of the real line, REF clearly enforces $\\left(\\liminf \\limits _{\\xi \\rightarrow -\\infty }p_{c,i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\in \\mathsf {K}^{++}\\subset \\mathsf {K}^{+}.$ Thus $\\left(\\mathbf {p}_{c},c\\right)$ is a traveling wave solution.",
"In order to construct a critical traveling wave $\\left(\\mathbf {p}_{c^{\\star }},c^{\\star }\\right)$ , we consider a decreasing sequence $\\left(c_{n}\\right)_{n\\in \\mathbb {N}}\\in \\left(c^{\\star },+\\infty \\right)^{\\mathbb {N}}$ such that $c_{n}\\rightarrow c^{\\star }$ as $n\\rightarrow +\\infty $ and intend to apply a compactness argument to a normalized version of the sequence $\\left(\\mathbf {p}_{c_{n}}\\right)_{n\\in \\mathbb {N}}$ .",
"By REF , $\\liminf _{\\xi \\rightarrow -\\infty }\\min _{i\\in \\left[N\\right]}p_{c_{n},i}\\left(\\xi \\right)\\ge \\nu \\text{ for all }n\\in \\mathbb {N}.$ Recall from REF the definition of $\\eta _{c}>0$ .",
"For all $n\\in \\mathbb {N}$ the following quantity is well-defined and finite: $\\xi _{n}=\\inf \\left\\lbrace \\xi \\in \\mathbb {R}\\ |\\ \\min _{i\\in \\left[N\\right]}p_{c_{n},i}\\left(\\xi \\right)<\\min \\left(\\frac{\\nu }{2},\\frac{\\eta _{c^{\\star }}}{2}\\right)\\right\\rbrace .$ We define then the sequence of normalized profiles $\\tilde{\\mathbf {p}}_{c_{n}}:\\xi \\mapsto \\mathbf {p}_{c_{n}}\\left(\\xi +\\xi _{n}\\right)\\text{ for all }n\\in \\mathbb {N}.$ A translation of a profile of traveling wave being again a profile of traveling wave, $\\left(\\tilde{\\mathbf {p}}_{c_{n}},c_{n}\\right)_{n\\in \\mathbb {N}}$ is again a sequence of traveling wave solutions.",
"Notice the following two immediate consequences of the normalization: $\\min _{i\\in \\left[N\\right]}\\tilde{p}_{c_{n},i}\\left(0\\right)=\\min \\left(\\frac{\\nu }{2},\\frac{\\eta _{c^{\\star }}}{2}\\right)\\text{ for all }n\\in \\mathbb {N},$ $\\inf _{\\xi \\in \\left(-\\infty ,0\\right)}\\min _{i\\in \\left[N\\right]}\\tilde{p}_{c_{n},i}\\left(\\xi \\right)\\ge \\min \\left(\\frac{\\nu }{2},\\frac{\\eta _{c^{\\star }}}{2}\\right)\\text{ for all }n\\in \\mathbb {N}.$ We are now in position to pass to the limit $n\\rightarrow +\\infty $ .",
"The sequence $\\left(\\tilde{\\mathbf {p}}_{c_{n}}\\right)_{n\\in \\mathbb {N}}$ being globally uniformly bounded, it admits, up to a diagonal extraction process, a bounded nonnegative limit $\\mathbf {p}_{c^{\\star }}$ in ${C}_{loc}^{2}$ .",
"Since $c_{n}\\rightarrow c^{\\star }$ , $\\mathbf {p}_{c^{\\star }}$ satisfies $\\left(TW\\left[c^{\\star }\\right]\\right)$ .",
"The normalization yields $\\min _{i\\in \\left[N\\right]}p_{c^{\\star },i}\\left(0\\right)=\\min \\left(\\frac{\\nu }{2},\\frac{\\eta _{c^{\\star }}}{2}\\right),$ $\\inf _{\\xi \\in \\left(-\\infty ,0\\right)}\\min _{i\\in \\left[N\\right]}p_{c^{\\star },i}\\left(\\xi \\right)\\ge \\min \\left(\\frac{\\nu }{2},\\frac{\\eta _{c^{\\star }}}{2}\\right).$ Consequently, $\\left(\\liminf _{\\xi \\rightarrow -\\infty }p_{c^{\\star },i}\\left(\\xi \\right)\\right)_{i\\in \\left[N\\right]}\\in \\mathsf {K}^{++}$ and, according to REF , $\\lim _{\\xi \\rightarrow +\\infty }\\mathbf {p}_{c^{\\star }}\\left(\\xi \\right)=\\mathbf {0}.$ The pair $\\left(\\mathbf {p}_{c^{\\star }},c^{\\star }\\right)$ is a traveling wave solution indeed and this ends the proof."
],
[
"Spreading speed",
"In this section, we assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ and prove REF .",
"In order to do so, we fix $\\mathbf {u}_{0}\\in {C}_{b}\\left(\\mathbb {R},\\mathbb {R}^{N}\\right)$ of the form $\\mathbf {u}_{0}=\\mathbf {v}\\mathbf {1}_{\\left(-\\infty ,x_{0}\\right)}$ with $x_{0}\\in \\mathbb {R}$ and $\\mathbf {v}$ nonnegative nonzero and we define $\\mathbf {u}$ as the unique classical solution of $\\left(E_{KPP}\\right)$ set in $\\left(0,+\\infty \\right)\\times \\mathbb {R}$ with initial data $\\mathbf {u}_{0}$ .",
"Remark This type of spreading result, as well as its proof by means of super- and sub-solutions, is quite classical (we refer to Aronson–Weinberger [5] and Berestycki–Hamel–Nadin [8] among others).",
"Still, we provide it to make clear that the lack of comparison principle for $\\left(E_{KPP}\\right)$ is not really an issue.",
"Of course, for the scalar KPP equation, much more precise spreading results exist (for instance the celebrated articles by Bramson [18], [17] using probabilistic methods).",
"Here, our aim is not to give a complete description of the spreading properties of $\\left(E_{KPP}\\right)$ but rather to illustrate that it is, once more, very similar to the scalar situation and that further generalizations should be possible."
],
[
"Upper estimate",
"Proposition 7.1 Let $c>c^{\\star }$ and $y\\in \\mathbb {R}$ .",
"We have $\\left(\\lim _{t\\rightarrow +\\infty }\\sup _{x\\in \\left(y,+\\infty \\right)}u_{i}\\left(t,x+ct\\right)\\right)_{i\\in \\left[N\\right]}=\\mathbf {0}.$ By definition of $\\mathbf {u}_{0}$ , there exists $\\xi _{1}\\in \\mathbb {R}$ such that $\\overline{\\mathbf {p}}:\\xi \\mapsto \\text{e}^{-\\mu _{c^{\\star }}\\left(\\xi -\\xi _{1}\\right)}\\mathbf {n}_{\\mu _{c^{\\star }}}$ (which is a positive solution of $\\left(TW_{0}\\left[c^{\\star }\\right]\\right)$ by REF ) satisfies $\\overline{\\mathbf {p}}\\ge \\mathbf {u}_{0}$ .",
"Then, defining $\\overline{\\mathbf {u}}:\\left(t,x\\right)\\mapsto \\overline{\\mathbf {p}}\\left(x-c^{\\star }t\\right)$ , we obtain by the nonnegativity of $\\mathbf {c}$ on $\\mathsf {K}$ $\\left(H_{2}\\right)$ $\\partial _{t}\\overline{\\mathbf {u}}-\\mathbf {D}\\partial _{xx}\\overline{\\mathbf {u}}-\\mathbf {L}\\overline{\\mathbf {u}} & =\\mathbf {0}\\\\& \\ge -\\mathbf {c}\\left[\\mathbf {u}\\right]\\circ \\mathbf {u}\\\\& =\\partial _{t}\\mathbf {u}-\\mathbf {D}\\partial _{xx}\\mathbf {u}-\\mathbf {L}\\mathbf {u}$ and then, applying the parabolic strong maximum principle to the operator $\\partial _{t}-\\mathbf {D}\\partial _{xx}-\\mathbf {L}$ , we deduce that $\\overline{\\mathbf {u}}-\\mathbf {u}$ is nonnegative in $[0,+\\infty )\\times \\mathbb {R}$ .",
"Consequently, for all $x\\in \\mathbb {R}$ , $t>0$ and $c>c^{\\star }$ , $\\mathbf {0}\\le \\mathbf {u}\\left(t,x+ct\\right)\\le \\overline{\\mathbf {p}}\\left(x+\\left(c-c^{\\star }\\right)t\\right),$ and by component-wise monotonicity of $\\overline{\\mathbf {p}}$ , for all $y\\in \\mathbb {R}$ and all $x\\ge y$ , $\\mathbf {0}\\le \\mathbf {u}\\left(t,x+ct\\right)\\le \\overline{\\mathbf {p}}\\left(y+\\left(c-c^{\\star }\\right)t\\right),$ which gives the result."
],
[
"Lower estimate",
"Proposition 7.2 Let $c\\in [0,c^{\\star })$ and $I\\subset \\mathbb {R}$ be a bounded interval.",
"We have $\\left(\\liminf _{t\\rightarrow +\\infty }\\inf _{x\\in I}u_{i}\\left(t,x+ct\\right)\\right)_{i\\in \\left[N\\right]}\\in \\mathsf {K}^{++}.$ Recall REF and define $\\lambda _{c}=-\\max \\limits _{\\mu \\ge 0}\\left(\\kappa _{\\mu }+\\mu c\\right)>0$ ($-\\lambda _{c}$ being the generalized principal eigenvalue of $-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}x^{2}}-c\\frac{\\text{d}}{\\text{d}x}-\\mathbf {L}$ ) and, using the fact that $\\mathbf {c}$ vanishes at $\\mathbf {0}$ $\\left(H_{3}\\right)$ , $\\alpha _{c}=\\max \\left\\lbrace \\alpha >0\\ |\\ \\forall \\mathbf {v}\\in \\left[0,\\alpha \\right]^{N}\\quad \\mathbf {c}\\left(\\mathbf {v}\\right)\\le \\frac{\\lambda _{c}}{2}\\mathbf {1}_{N,1}\\right\\rbrace .$ Let $R_{c}$ be a sufficiently large radius satisfying $\\lambda _{1,Dir}\\left(-\\mathbf {D}\\frac{\\text{d}^{2}}{\\text{d}\\xi ^{2}}-c\\frac{\\text{d}}{\\text{d}\\xi }-\\left(\\mathbf {L}-\\frac{\\lambda _{c}}{2}\\mathbf {I}\\right),\\left(-R_{c},R_{c}\\right)\\right)<0.$ Let $\\mathbf {u}_{c}:\\left(t,y\\right)\\mapsto \\mathbf {u}\\left(t,y+ct\\right)$ .",
"It is a solution of $\\partial _{t}\\mathbf {u}_{c}-\\mathbf {D}\\partial _{yy}\\mathbf {u}_{c}-c\\partial _{y}\\mathbf {u}_{c}=\\mathbf {L}\\mathbf {u}_{c}-\\mathbf {c}\\left[\\mathbf {u}_{c}\\right]\\circ \\mathbf {u}_{c}\\text{ in }\\left(0,+\\infty \\right)\\times \\mathbb {R}$ with initial data $\\mathbf {u}_{0}$ .",
"Just as in the proof of REF , we can use $R_{c}$ , $\\alpha _{c}$ and Földes–Poláčik’s Harnack inequality [29] to deduce the existence of $\\nu _{c}>0$ such that $\\left(\\liminf _{t\\rightarrow +\\infty }\\inf _{x\\in I}u_{i}\\left(t,x+ct\\right)\\right)_{i\\in \\left[N\\right]}\\ge \\nu _{c}\\mathbf {1}_{N,1}.$ This ends the proof.",
"Remark We point out that $R_{c}\\rightarrow +\\infty $ as $c\\rightarrow c^{\\star }$ .",
"Hence the proof above cannot be used directly to obtain a lower bound uniform with respect to $c$ .",
"Although we expect indeed the existence of such a bound, we do not know how to obtain it."
],
[
"Estimates for the minimal wave speed",
"In this section, we assume $\\lambda _{PF}\\left(\\mathbf {L}\\right)>0$ , $d_{1}\\le d_{2}\\le \\text{\\dots }\\le d_{N},$ and prove the estimates provided by REF .",
"Recall the equality $c^{\\star }=\\min _{\\mu >0}\\left(-\\frac{\\kappa _{\\mu }}{\\mu }\\right).$ Recall as a preliminary that for all $r>0$ and $d>0$ , the following equality holds: $2\\sqrt{rd}=\\min _{\\mu >0}\\left(\\mu d+\\frac{r}{\\mu }\\right).$ Proposition 8.1 We have $2\\sqrt{d_{1}\\lambda _{PF}\\left(\\mathbf {L}\\right)}\\le c^{\\star }\\le 2\\sqrt{d_{N}\\lambda _{PF}\\left(\\mathbf {L}\\right)}.$ If $d_{1}<d_{N}$ , both inequalities are strict.",
"If $d_{1}=d_{N}$ , both inequalities are equalities.",
"Since $d_{1}\\mathbf {1}_{N,1}\\le \\mathbf {d}\\le d_{N}\\mathbf {1}_{N,1}$ , we have, for all $\\mu >0$ , $\\mu d_{1}+\\frac{1}{\\mu }\\lambda _{PF}\\left(\\mathbf {L}\\right)\\le \\lambda _{PF}\\left(\\mu \\mathbf {D}+\\frac{1}{\\mu }\\mathbf {L}\\right)\\le \\mu d_{N}+\\frac{1}{\\mu }\\lambda _{PF}\\left(\\mathbf {L}\\right),$ whence we deduce $2\\sqrt{d_{1}\\lambda _{PF}\\left(\\mathbf {L}\\right)}\\le c^{\\star }\\le 2\\sqrt{d_{N}\\lambda _{PF}\\left(\\mathbf {L}\\right)}.$ On one hand, it is well-known that if $d_{1}<d_{N}$ , then the above inequalities are strict.",
"On the other hand, if $d_{1}=d_{N}$ , we have $\\lambda _{PF}\\left(\\mu \\mathbf {D}+\\frac{1}{\\mu }\\mathbf {L}\\right)=\\mu d_{1}+\\frac{1}{\\mu }\\lambda _{PF}\\left(\\mathbf {L}\\right),$ whence the equality.",
"Recall from REF that $\\mathbf {n}_{\\mu _{c^{\\star }}}=\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\mathbf {L}\\right)$ .",
"Proposition 8.2 For all $i\\in \\left[N\\right]$ such that $l_{i,i}>0$ , we have $c^{\\star }>2\\sqrt{d_{i}l_{i,i}}.$ Let $i\\in \\left[N\\right]$ .",
"The characterization of $c^{\\star }$ (see REF ) yields $\\mu _{c^{\\star }}d_{i}+\\frac{l_{i,i}}{\\mu _{c^{\\star }}}=c^{\\star }-\\frac{1}{\\mu _{c^{\\star }}}\\sum _{j\\in \\left[N\\right]\\backslash \\left\\lbrace i\\right\\rbrace }l_{i,j}\\frac{n_{\\mu _{c^{\\star }},j}}{n_{\\mu _{c^{\\star }},i}},$ whence, if $l_{i,i}>0$ , $c^{\\star }\\ge 2\\sqrt{d_{i}l_{i,i}}+\\frac{1}{\\mu _{c^{\\star }}}\\sum _{j\\in \\left[N\\right]\\backslash \\left\\lbrace i\\right\\rbrace }l_{i,j}\\frac{n_{\\mu _{c^{\\star }},j}}{n_{\\mu _{c^{\\star }},i}}.$ From the irreducibility and essential nonnegativity of $\\mathbf {L}$ $\\left(H_{1}\\right)$ , there exists $j\\in \\left[N\\right]\\backslash \\left\\lbrace i\\right\\rbrace $ such that $l_{i,j}>0$ , whence $c^{\\star }>2\\sqrt{d_{i}l_{i,i}}$ .",
"Recall the existence of a unique decomposition of $\\mathbf {L}$ of the form $\\mathbf {L}=\\text{diag}\\mathbf {r}+\\mathbf {M}\\text{ with }\\mathbf {r}\\in \\mathbb {R}^{N}\\text{ and }\\mathbf {M}^{T}\\mathbf {1}_{N,1}=\\mathbf {0}.$ Remark Regarding the Lotka–Volterra mutation–competition–diffusion ecological model, the decomposition $\\mathbf {L}=\\text{diag}\\mathbf {r}+\\mathbf {M}$ is ecological meaningful: $\\mathbf {r}$ is the vector of the growth rates of the phenotypes whereas $\\mathbf {M}$ describes the mutations between the phenotypes.",
"Proposition 8.3 Let $\\left(\\left\\langle d\\right\\rangle ,\\left\\langle r\\right\\rangle \\right)\\in \\left(0,+\\infty \\right)\\times \\mathbb {R}$ be defined as $\\left\\lbrace \\begin{matrix}\\left\\langle d\\right\\rangle =\\frac{\\mathbf {d}^{T}\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\mathbf {L}\\right)}{\\mathbf {1}_{1,N}\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\mathbf {L}\\right)},\\\\\\left\\langle r\\right\\rangle =\\frac{\\mathbf {r}^{T}\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\mathbf {L}\\right)}{\\mathbf {1}_{1,N}\\mathbf {n}_{PF}\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\mathbf {L}\\right)}.\\end{matrix}\\right.$ If $\\left\\langle r\\right\\rangle \\ge 0$ , then $c^{\\star }\\ge 2\\sqrt{\\left\\langle d\\right\\rangle \\left\\langle r\\right\\rangle }.$ Using $\\left(\\mathbf {r},\\mathbf {M}\\right)$ , the characterization of $c^{\\star }$ (see REF ) is rewritten as $\\left(\\mu _{c^{\\star }}^{2}\\mathbf {D}+\\text{diag}\\mathbf {r}\\right)\\mathbf {n}_{\\mu _{c^{\\star }}}+\\mathbf {M}\\mathbf {n}_{\\mu _{c^{\\star }}}=\\mu _{c^{\\star }}c^{\\star }\\mathbf {n}_{\\mu _{c^{\\star }}}.$ Summing the lines of this system, dividing by $\\sum \\limits _{i=1}^{N}n_{\\mu _{c^{\\star }},i}$ and defining $\\left\\langle d\\right\\rangle $ and $\\left\\langle r\\right\\rangle $ as in the statement, we find $\\mu _{c^{\\star }}^{2}\\left\\langle d\\right\\rangle +\\left\\langle r\\right\\rangle =\\mu _{c^{\\star }}c^{\\star }.$ The equation $\\left\\langle d\\right\\rangle \\mu ^{2}-c^{\\star }\\mu +\\left\\langle r\\right\\rangle =0$ admits a real positive solution $\\mu $ if and only if $\\left(c^{\\star }\\right)^{2}-4\\left\\langle d\\right\\rangle \\left\\langle r\\right\\rangle \\ge 0$ ."
],
[
"Acknowledgments",
"The author thanks Grégoire Nadin for the attention he paid to this work, Vincent Calvez for fruitful discussions on the cane toads equation, Cécile Taing for pointing out the related work by Wang and an anonymous reviewer for a detailed report thanks to which the original manuscript was largely improved."
]
] | 1612.05774 |
[
[
"Dust Scattering from the Taurus Molecular Cloud"
],
[
"Abstract We present an analysis of the diffuse ultraviolet (UV) emission near the Taurus Molecular Cloud based on observations made by the Galaxy Evolution Explorer (GALEX).",
"We used a Monte Carlo dust scattering model to show that about half of the scattered flux originates in the molecular cloud with 25% arising in the foreground and 25% behind the cloud.",
"The best-fit albedo of the dust grains is 0.3 but the geometry is such that we could not constrain the phase function asymmetry factor (g)."
],
[
"Introduction",
"The Taurus Molecular Cloud (TMC) is part of the larger Taurus-Auriga complex of molecular clouds which extends over more than 100 square degrees in the sky.",
"It is one of the nearest star forming regions at a distance of only 140 pc from the Sun with a thickness of around 20 pc [12], [16], [17].",
"Although it is near the Galactic Plane (b $\\sim $ -14), there is little dust either in front of or behind the molecular cloud [25], [17].",
"The TMC has been studied intensively at many different wavelengths and has provided a laboratory for the study of molecular cloud evolution and star formation in fine spatial and spectral detail.",
"High resolution optical maps show that the TMC is comprised of loosely associated diffuse filaments mixed with highly clumpy molecular cores [27].",
"The clumpy molecular gas cores in TMC have a density of about few hundred $cm^{-3}$ [29] and produce very few low mass stars [34], [26] with a stellar density of about 1 - 10 stars pc$^{-3}$ [18].",
"In contrast to other star-forming regions such as Orion or Ophiuchus, the TMC is quiet with a low rate of star formation [13] and no massive stars and hence no ionizing radiation [32].",
"The first diffuse UV studies of the TMC were made by [11] using the BEST (Berkeley Extreme Ultraviolet Shuttle Telescope) instrument, which flew on the Space Shuttle (STS-61C) in 1986 as part of the UVX payload.",
"He observed both continuum emission due to starlight scattered by interstellar dust and emission lines from H${_2}$ fluorescence.",
"The continuum emission was inversely correlated with the optical depth suggesting that the diffuse background originated behind and was shadowed by the dense molecular cloud; a conclusion later supported by observations from the SPEAR/FIMS instrument [14].",
"We have used observations made by the Galaxy Evolution Explorer (GALEX) in two bands (far-ultraviolet (FUV: 1350 - 1750 Å) and near-ultraviolet (NUV: 1750 - 2850 Å)) to study the diffuse UV radiation in and around the TMC.",
"We will describe the observations and our modelling below.",
"[15] performed a similar analysis for a much larger region (the entire Taurus-Perseus-Auriga complex of molecular clouds).",
"However, that included areas with quite different physical properties and we have chosen to focus on the TMC."
],
[
"Data",
"GALEX, a NASA Small Explorer mission, was launched on April 28, 2003 with the primary goal of studying galaxy evolution at low red-shift [20] and completed its mission on June 28, 2013 [2].",
"The final data set of GALEX includes more than 44,000 observations [23] in two ultraviolet bands.",
"The data products from each observation have been described by [21] and are archived at the Mikulski Archive for Space Telescopes (MAST).",
"We have shown an exposure time map of the observed region in the FUV and the NUV bands in Fig.",
"REF .",
"There were a total of 159 FUV and 258 NUV visits near the TMC region taken over a range of 9 years from 2003 to 2012 (2003 – 2007 for the FUV).",
"Most of the observations were taken as part of the AIS (All-sky Imaging Survey) with an exposure time of about 100 seconds per visit but with a few observations for specific science programs [5] with correspondingly greater exposure times.",
"[23] has calculated the diffuse Galactic background (DGL) values over the entire sky by masking out point sources and binning to a resolution of $2^{\\prime }$ and these data are available from the MAST archivehttps://archive.stsci.edu/prepds/uv-bkgd/.",
"We have downloaded all the data within 8 degrees of the nominal centre of the TMC (170.0, -14.0) and binned into 6$$ bins, weighting each observations by its exposure time.",
"Edge effects contaminate the outer part of the GALEX detector and we have only used the central $0.5$ (radius) of each observation.",
"The resultant maps of the TMC in the FUV and the NUV are shown in Fig.",
"REF along with the extinction map derived from Planck observations [30].",
"There were no observations over significant parts of the TMC, partly because of safety concerns for the instrument [5] and partly because of the failure of the FUV power supply.",
"Figure: Correlation between GALEX FUV and NUV intensity.Figure: Regions with different relationship between FUV and E(B-V).",
"The dotted region in red colour on the right hand side of the image, is brighter in FUV and have little dust.Figure: GALEX FUV (top) and NUV (bottom) intensity plotted against Planck E(B-V).The black line separates the regions having different relationship between UV and E(B-V).There is a strong correlation (r $\\sim $ 0.9) between the FUV and the NUV (Fig.",
"REF ).",
"However, it is readily apparent that there is no correlation between the extinction and the diffuse UV background; rather, the UV images appear to be brighter to the right of the field where the extinction is least.",
"This is borne out by a weak anti-correlation between the E(B-V) and the FUV (r = -0.25) and the E(B-V) and the NUV (r = -0.24).",
"A more detailed examination of the data suggests that there exist two separate regions, demarcated in Fig.",
"REF where the points in red are those that fall above the line in Fig.",
"REF .",
"There is a positive correlation between the E(B - V) and the UV (FUV: r = 0.69; NUV: r = 0.51) in this region.",
"The remaining region, which encompasses most of the TMC, has a negative correlation of r=-0.36 between the E(B - V) and the FUV and r=-0.46 between the NUV and the E(B-V), as might be expected if the radiation were shadowed by the dust in the TMC.",
"The first observations of the diffuse UV in the TMC was done by [11] using the UVX instrument on the Space Shuttle followed by observations made by using the SPEAR/FIMS [14], [15].",
"They observed the same anti-correlation between the reddening and the UV flux and interpreted this as due to the shadowing of a distant diffuse background by the molecular cloud.",
"We will explore this in the next section."
],
[
"Modelling",
"[24] has implemented a Monte Carlo model to predict the level of dust scattered radiation throughout the Galaxy.",
"He used stars from the Hipparcos catalog [28] along with model spectra for each spectral type [3] as sources for the stellar photons.",
"Although we are interested only in the emission from the vicinity of the TMC, we have found that many of the scattered photons are from multiply scattered photons from outside the TMC and have therefore modelled the entire Galaxy, but with more detail in the vicinity of the TMC.",
"We have modelled the Galaxy as a 500x500x500 grid with a bin size of 2 pc on a side and filled with a hydrogen density of 1 cm$^{-3}$ at the Galactic Plane falling off with a scale height of 125 pc [19] from the plane.",
"We calculated an optical depth for each bin using the cross-section per hydrogen atom tabulated by [4] for “Milky Way” dust.",
"[33] have found that there is a cavity of radius $\\sim $ 80 pc (the Local Bubble) around the Sun and we have incorporated this in the dust model.",
"However, this does not account for the complex structure in the TMC and we have separately modelled the dust in that region.",
"Table: Properties of brightest stars near the TMC region with stellar flux contribution at three g values.Figure: Correlation between Planck and GSF reddening.Figure: Left: Planck dust distribution in terms of E(B-V) per 2 pc bin in units of `mag', as a function of distance along a representative line of sight.",
"Right: Modelled spatial distribution of the dust in the region.We first used the 3-dimensional extinction map of [7] (GSF hereafter) to characterize the dust distribution in the region finding that the bulk of the dust is found between 140 and 200 pc in reasonable agreement with the 140 – 160 pc from earlier results [12], [31], [17].",
"However, the total column density is greater than that derived from the Planck data [30] (Fig.",
"REF ).",
"The GSF measurements are along specific lines of sight while the Planck measurements average over the $5^{\\prime }$ beam of the instrument and the larger values may be an indicator of clumping in the TMC.",
"We have therefore used a second model where we have taken the reddening along any line of sight from the Planck data [30] and distributed the dust uniformly between 140 - 160 pc (Fig.",
"REF ).",
"Although this gives somewhat lower column densities than the GSF data, it does match the extinction measurements of [1].",
"We tested the effects of both models on our derived diffuse flux finding little difference between the two.",
"We therefore continue with only using the dust distribution as given by the Planck model.",
"As has been standard in studies of interstellar dust, we used the Henyey-Greenstein scattering function [10] with the albedo (a) and the phase function asymmetry factor(g) as free parameters to determine each scattering."
],
[
"Results",
"We have run a series of Monte Carlo models to predict the diffuse background in both the FUV and the NUV bands for different combinations of a and g. We used a baseline of 100 million ($10^{8}$ ) photons for the entire grid but used simulations with 1 billion ($10^{9}$ ) photons for values of $a$ and $g$ near the best fit values.",
"We cannot compare the models directly to the GALEX data on a pixel by pixel basis because of the noise intrinsic to the Monte Carlo procedure and therefore used the root-mean-square (R.M.S) deviation of the flux as a function of Galactic longitude as a metric of the goodness of fit of the model (Fig.",
"REF ).",
"We have plotted the R.M.S.",
"deviations for both the FUV and the NUV as a function of $a$ and $g$ in Fig.",
"REF finding that the best fit occurs for an albedo of about 0.3 in both bands.",
"Although this is in agreement with [11] and [14] in the same region, it is lower than the albedo found in other parts of the Galaxy (compiled in Table 4 of [24]).",
"[35] showed that a clumpy medium, as in the TMC [27], gives rise to a lower effective albedo and it is likely that the “true” albedo is greater than our derived value.",
"We have listed the 20 stars which contribute the most to the diffuse flux at $g = 0$ in Table REF .",
"The two stars which contribute the most flux at any value of $g$ are near the right edge of Fig.",
"REF accounting for part of the rise in the UV fluxes to lower longitudes.",
"However, there are enough stars contributing with a range of distances and positions that the dependence on $g$ is washed out, as was found by [6] in the upper Scorpius region.",
"We have adopted $g = 0.7$ which is consistent with most observations in the UV [4].",
"Figure: Modelled FUV and NUV emission as a function of galactic longitude for aa = 0.3 and gg = 0.7, over-plotted with the original FUV and NUV emission.Figure: Root mean square deviations as a function of albedo (a) and as a function of phase function asymmetry factor (g).Figure: Correlation plot between the modelled and the observed flux.",
"The best fit lines are shown.Figure: Comparison of absolute values between the UVX values from and the GALEX intensity.",
"300 ph cm -2 ^{-2} s -1 ^{-1} sr -1 ^{-1} Å -1 ^{-1} have been subtracted from the UVX data to account for molecular hydrogen emission.Figure: Bright stars are marked as circles over the Planck dust reddening map.",
"The centre of the circles represents the location of the stars while its radius proportional to contribution towards total flux.Figure: Mean E(B-V) as function of galactic longitude in both GSF and Planck cases.Figure: Fraction of total FUV scattered flux in the TMC as a function of distance from the Sun.We have run our model with $10^{10}$ photons for the best fit optical constants ($a = 0.3; g = 0.7$ ) to improve the signal-to-noise and plotted the modelled versus the observed fluxes in Fig.",
"REF , where we have binned the data into $0.2^{\\circ }$ bins.",
"The correlation between the observed and modelled FUV is 0.55 with a slope of 0.68 and an offset of 980 ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ .",
"The slope for the NUV is 0.48 with an offset of 1360 ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ and a correlation coefficient of 0.51.",
"If we look at these values in the broader context of the diffuse radiation at low latitudes, [24] found that there was considerable scatter between the observed reddening and the UV flux, which he attributed to an uncertain dust distribution.",
"We believe that the main source of scatter between our modelled fluxes and the data is the complex structure of the TMC.",
"However, we do find offsets similar to the 700 — 900 ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ found by [24] including 200 — 300 ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ from airglow [22], molecular hydrogen fluorescence in the FUV [11], [15] and a possibly new, unknown component [9], [8].",
"Finally, we return to the anti-correlation between the diffuse flux and the reddening observed by [11] and [14], which they interpreted as scattering from a more distant diffuse source extincted by the TMC.",
"Our data agree with both the UVX observations of [11] (Fig.",
"REF ) and the SPEAR observations of [14] but we believe that the actual situation is more complex.",
"At least part of the brightening to lower longitudes is coincidental because the two strongest contributing stars (Table REF ) are near a longitude of 160$^{\\circ }$ (Fig.",
"REF ) where the reddening is less (Fig.",
"REF ).",
"These two B stars (HIP 18246 and HIP 18532) contribute 45% of the flux at a Galactic longitude less than 165$$ .",
"Our model predicts that about 25% of the total scattered flux is from dust in front of the TMC and another 25% behind the cloud with the remaining 50% arising in the TMC itself (Fig.",
"REF ), as might be expected given that almost all of the dust is in the TMC.",
"However, our model is not well-suited to explore the scattering within a dense molecular cloud where clumping may be important.",
"HIP 18246 is behind the TMC, as well as the other bright stars which make much of the field unobservable by GALEX [5], and much of the flux from within the molecular cloud is due to the scattering of the light from those stars by dust within the cloud.",
"Self-shielding within the cloud, itself, will yield the observed anti-correlation, exacerbated by the coincidental position of the two bright stars at one edge of the field."
],
[
"Conclusions",
"We have modelled the diffuse radiation in the direction of the TMC using GALEX data with a Monte Carlo model finding an albedo of $\\approx 0.3$ , but with no constraints on $g$ .",
"The albedo is likely to be an underestimate because we do not take clumping into effect.",
"About half of the scattered radiation originates in the body of the cloud with self-shielding giving rise to an anti-correlation between the observed flux and the reddening.",
"The two brightest stars in the field are located in the direction of least reddening which adds to the anti-correlation.",
"We find offsets of about 1000 ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Å$^{-1}$ in both bands of which some part may be due to foreground contributors such as airglow and some to an unknown component [9]."
],
[
"Acknowledgements",
"This work is based on the data from NASA's GALEX spacecraft.",
"GALEX is operated for NASA by the California Institute of Technology under NASA contract NAS5-98034.",
"The dust extinction map obtained from PLANCK (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.",
"We thank the anonymous referee for constructive suggestions which has resulted in a better paper.",
"Sathyanarayan would like to thank Dr. Sujatha and Ms. Jyothi for their help during the early stages of this work."
]
] | 1612.05783 |
[
[
"Deep Residual Hashing"
],
[
"Abstract Hashing aims at generating highly compact similarity preserving code words which are well suited for large-scale image retrieval tasks.",
"Most existing hashing methods first encode the images as a vector of hand-crafted features followed by a separate binarization step to generate hash codes.",
"This two-stage process may produce sub-optimal encoding.",
"In this paper, for the first time, we propose a deep architecture for supervised hashing through residual learning, termed Deep Residual Hashing (DRH), for an end-to-end simultaneous representation learning and hash coding.",
"The DRH model constitutes four key elements: (1) a sub-network with multiple stacked residual blocks; (2) hashing layer for binarization; (3) supervised retrieval loss function based on neighbourhood component analysis for similarity preserving embedding; and (4) hashing related losses and regularisation to control the quantization error and improve the quality of hash coding.",
"We present results of extensive experiments on a large public chest x-ray image database with co-morbidities and discuss the outcome showing substantial improvements over the latest state-of-the art methods."
],
[
"Introduction",
"Content-based image retrieval (CBIR) aims at effectively indexing and mining large image databases such that given an unseen query image we can effectively retrieve images that are similar in content.",
"With the deluge in medical imaging data, there is a need to develop CBIR systems that are both fast and efficient.",
"However, in practice, it is often infeasible to exhaustively compute similarity scores between the query image and each image within the database.",
"Adding to the challenge of scalability of CBIR systems is the less understood semantic gap between the visual content of the image and the associated expert annotations [1].",
"To address these challenges, hashing based CBIR systems have come to a forefront where the system indexes each image with a compact similarity preserving binary code that could be potentially leveraged for very fast retrieval.",
"Towards this end, we propose an end-to-end one-stage deep residual hashing (DRH) network to directly generate hash codes from input images.",
"Specifically, the DRH model constitutes of a sub-network with multiple residual convolutional blocks for learning discriminative image representations followed by a fully-connected hashing layer to generate compact binary embeddings.",
"Through extensive validation, we demonstrate that DRH learns discriminative hash codes in an end-to-end fashion and demonstrates high retrieval quality on standard chest x-ray image databases.",
"The existing hashing methods proposed for efficient encoding and searching approaches have been proposed for large scale retrieval in machine learning and medical image computing can be categorised into: (1) shallow learning based hashing methods like Locality Sensitive Hashing (LSH) [2]), data-driven methods e.g.",
"Iterative Quantization (ITQ) [3], Kernel Sensitive Hashing [4], Circulent Binary Embedding (CBE) [5], Metric Hashing Forests (MHF) [6]; (2) hashing using deep architectures (only binarization without feature learning) including Restricted Boltzmann Machines in semantic hashing [7], autoencoders in supervised deep hashing [8] etc.",
"and (3) application-specific hashing methods including weighted hashing for histopathological image search [9], binary code tagging for chest X-ray images [10], forest based hashing for neuron images [11], to name a few.",
"Figure: tSNE embeddings of the hash codes generated by the proposed and comparative methods.",
"Color indicates different classes.",
"The figure needs to be viewed in color."
],
[
"Motivation and Contributions",
"The ultimate objective of earning similarity preserving hashing functions is to generate embeddings in a latent Hamming space such that the class-separability is preserved while embedding and local neighborhoods are well defined and semantically relevant.",
"This can be visualized in 2D by generating the t - Stochastic Neighborhood Embedding (t-SNE) [12] of unseen test data post learning like shown in Fig.",
"REF .",
"Starting from Fig.. REF (a) which is generated by a purely un-superivsed setting we aim at moving towards Fig.. REF (d) which is closer to an ideal embedding.",
"In fact, Fig.",
"REF represents the results of our proposed DRH approach in comparison to other methods and baselines.",
"Hand-crafted features: Conventional hashing methods including LSH, ITQ, KSH, MHF etc.",
"perform encoding in two stages: firstly, generating a vector of hand-crafted descriptors and a second stage involving hashing learning to preserve the captured semantics in a latent Hamming space.",
"These two independent stages may lead to sub-optimal results as the image descriptors may not be tailored for hashing.",
"Moreover, hand-crafting requires significant domain knowledge and extensive parameter tuning which is particularly undesirable.",
"Conventional deep learning: Using point-wise loss-functions like cross-entropy, hinge loss etc.",
"for training (/ finetuning) deep networks may not lead to feature representations that are sufficiently optimal for the task of retrieval as they do not consider crucial pairwise relationships between instances [13].",
"Simultaneous feature learning and hashing: Recently, with the advent of deep learning for hashing we are able to perform effective end-to-end learning of binary representations directly from input images.",
"These include deep hashing for compact binary code learning [8], deep hashing network for effective similarity retrieval [13], simultaneous feature learning and hashing [1] etc.",
"to name a few.",
"However, a crucial disadvantage of these deep learning for hashing methods is that with very deep versions of these networks accuracy gets saturated and often degrades [14].",
"In addition to this, the continuous relaxation of hash codes to train deep networks to be able to learn with more viable continuous optimisation methods (gradient-descent based methods) could potentially lead to uncontrolled quantization and distance approximation errors during binarization.",
"In an attempt to redress the above short-comings of the existing approaches, we make the following contributions with our work: 1) We, for the first lime, design a novel deep hash function learning framework using deep residual networks for representation learning; 2) We introduced a neighborhood component analysis-inspired loss suitably tailored for learning discriminative hash codes; 3) We leverage multiple hashing related losses and regularizations to control the quantization error while binarization of hash codes and to encourage hash codes to be maximally independent of each other; and 4) Clinically, to the best of our knowledge, this is the first retrieval work on medical images (specifically, chest x-ray images) to discuss co-morbidities i.e.",
"co-occuring manifestations of multiple diseases.",
"The paper also aims at encouraging further discussion on the following aspects of CBIR through DRH: Trainability: How do we train very deep neural networks for hashing?",
"Does introducing residual connections aid in this process?",
"Representability: Do networks tailored for the dataset at hand learn better representations over transfer learning ?",
"Compactness: Do highly compact binary representations effectively compress the desired semantic content within an image?",
"Do loss functions to control quantization error while binarzing aid in improved hash coding?",
"Semantic-similarity preservation: Do we learn hash codes such that neighbourhoods in the Hamming space comprise of semantically similar instances?",
"Joint Optimisation: Does end-to-end implicit learning of hash codes work better than a two stage learning process where the images are embedded to a latent space and then quantized explicitly via hashing?",
"Figure: Network architecture for deep residual hashing (DRH) with a hash layer.",
"For a 18 - layer network, the number stacked residual blocks are: P = 2, Q = 2, R = 2 and S = 2.",
"Likewise, for a 34 - layer network, P = 3, Q = 4, R = 6 and S = 3.",
"The inset image on the left corner is a schematic illustration of a residual block."
],
[
"Methodology",
"An ideal hashing method should generate codes that are compact, similarity preserving and easy to compute representations (typically, binary in nature), which can be leveraged for accurate search and fast retrieval [2].",
"The desired similarity preserving aspect of the hashing function implies that semantically similar images are encoded with similar hash codes.",
"Mathematically, hashing aims at learning a mapping $\\mathcal {H}: \\mathcal {I} \\rightarrow \\left\\lbrace -1, 1 \\right\\rbrace ^{K}$ , such that an input image $\\mathcal {I}$ can be encoded into a $K$ bit binary code $\\mathcal {H}(\\mathcal {I})$ .",
"In hashing for image retrieval, we typically define a similarity matrix $\\mathcal {S} = \\left\\lbrace s_{ij} \\right\\rbrace $ , where $s_{ij} = 1$ implies images $\\mathcal {I}_{i}$ and $\\mathcal {I}_{j}$ are similar and $s_{ij} = 0$ indicates they are dissimilar.",
"Similarity preserving hashing aims at learning an encoding function $\\mathcal {H}$ such that the similarity matrix $\\mathcal {S}$ is maximally-preserved in the binary hamming space."
],
[
"Architecture for deep residual hashing",
"We start with a deep convolutional neural network architecture inspired in part by the seminal ResNet architecture proposed for image classification by He et al. [14].",
"As shown in Fig.",
"REF , the proposed architecture consists of the a convolutional layer (Conv 1) followed by a sequence of residual blocks (Conv 2-5) and terminating in a final fully connected hashing (FCH) layer for hash code-generation.",
"The unique advantages offered by the proposed ResNet architecture for hashing over a typical convolutional neural network are as follows: [leftmargin=*] Training of very deep networks: The representational power of deep networks should ideally increase with increased depth.",
"It is empirically observed that in deep feed-forward nets beyond a certain depth, adding additional layers results in higher training and validation error (despite using batch normalization) [14].",
"Residual networks seamlessly solves this via adding short cut connections that are summed with the output of the convolutional blocks.",
"Ease of Optimization: A major issue to training deep architectures is the problem of vanishing gradients during training (this is in part mitigated with the introduction of rectified linear units (ReLU), input batch normalisation and layer normalisation).",
"Residual connections offer additional support via a no-resistance path for the flow of gradients along the shortcut connections to reach the shallow learning layers."
],
[
"Supervised Retrieval Loss Function",
"In order to learn feature embeddings tailored for retrieval and specifically for the scenario at hand where the pairwise similarity matrix $\\mathcal {S}$ should be preserved, we propose our supervised retrieval loss drawing inspiration from the neighbourhood component analysis [15].",
"To encourage the learnt embedding to be binary in nature, we squash the output of the residual layers to be within $\\left[-1, 1 \\right]$ by passing it through a hyperbolic tangent (tanh) activation function.",
"The final binary hash codes $(\\mathbf {b}_{i})$ are generated by quantizing the output of the tanh activation function (say, $\\mathbf {h}_{i}$ ) as follows: $\\mathbf {b}_{i} = \\text{sgn}\\left( \\mathbf {h}_{i} \\right)$ .",
"Given $N$ instances and the corresponding similarity matrix is defined as $\\mathcal {S} = \\left\\lbrace s_{ij} \\right\\rbrace _{i,j=1}^{N} \\in \\left\\lbrace 0,1 \\right\\rbrace ^{N \\times N}$ , the proposed supervised retrieval loss is formulated as: $J_{S} = 1 - \\frac{1}{N}\\sum _{i,j = 1}^{N}p_{ij}s_{ij}$ where $p_{ij}$ is the probability that any two instances ($i$ and $j$ ) can be potential neighbours.",
"Inspired by kNN classification, where the decision of an unseen test sample is determined by the semantic context of its local neighbourhood in the embedding space, we define $p_{ij}$ as a softmax function of the hamming distance (indicated as $\\oplus $ ) between the hash codes of two instances and is derived as: $p_{ij} = \\frac{e^{-\\left( \\mathbf {b}_{i} \\oplus \\mathbf {b}_{j} \\right)}}{\\sum _{l\\ne i}e^{-\\left( \\mathbf {b}_{i} \\oplus \\mathbf {b}_{l} \\right)}} \\text{ where } \\mathbf {b}_{\\left( \\cdot \\right)} = \\text{sgn}\\left( \\mathbf {h}_{\\left( \\cdot \\right)} \\right)$ As gradient based optimisation of $J_{s}$ in a binary embedding space is infeasible due to its non-differentiable nature, we use a continuous domain relaxation and substitute non-quantized embeddings $\\mathbf {h}_{\\left( \\cdot \\right)}$ in place of hash code $\\mathbf {b}_{\\left( \\cdot \\right)}$ and Euclidean distance as as surrogate of Hamming distance between binary codes.",
"This is derived as: $p_{ij} = e^{-\\left\\Vert \\mathbf {h}_{i} - \\mathbf {h}_{j} \\right\\Vert ^{2}} / \\sum _{i \\ne l}e^{-\\left\\Vert \\mathbf {h}_{i} - \\mathbf {h}_{l} \\right\\Vert ^{2}}$ .",
"It must be noted that such an continuous relaxation could potentially result in uncontrollable quantization error and large approximation errors in distance estimation.",
"With continuous relaxation, Eq.",
"(REF ) is now differentiable and continuous thus suited for backpropagation of gradients during training."
],
[
"Hashing related Loss Functions and Regularization",
"Generation of high quality hash codes requires us to control this quantization error and bridge the gap between the Hamming distance and its continuous surrogate.",
"In this paper, we jointly optimise for $J_{s}$ and improve hash code generation by imposing additional loss functions as follows: Quantization Loss: In the seminal work on iterative quantization (ITQ) for hashing [3], Gong and Lazebnik introduced the notion of quantization error $J_{Q-\\text{ITQ}}$ as $J_{Q-\\text{ITQ}} = \\left\\Vert \\mathbf {h}_{i} - \\text{sgn}\\left( \\mathbf {h}_{i} \\right) \\right\\Vert _{2}$ .",
"Optimising for $J_{Q-\\text{ITQ}}$ required a computation intensive alternating optimisation procedure and is not compatible with back propagation which is used to train deep neural nets (due to non-differentiable sgn function within the formulation).",
"Towards this end, we use a modified point-wise quantization loss function proposed by Zhu et al.",
"sans the sgn function as $J_{Q -\\text{Zhu}} = \\left\\Vert \\left| \\mathbf {h}_{i} \\right| - \\mathbf {1} \\right\\Vert _{1}$ [13].",
"They establish that $J_{Q -\\text{Zhu}}$ is an upper bound over $J_{Q-\\text{ITQ}}$ , therefore can be deemed as a reasonable loss function to control quantization error.",
"For ease of back-propagation, we propose to use a differentiable smooth surrogate to $L_{1}$ norm $\\left| \\left( \\cdot \\right) \\right|_{1} \\approx \\text{log cosh}\\left( \\cdot \\right)$ and derived the proposed quantization loss function as:$J_{Q} = \\sum _{i=1}^{N} \\left( \\text{log cosh}\\left(\\left| \\mathbf {h}_{i} \\right| - \\mathbf {1} \\right) \\right)$ .",
"With the incorporation of the quantization loss, we hypothesise that the final binarization step would incur significantly less quantization error and the loss of retrieval quality (also empirically validated in Section ).",
"Bit Balance Loss: In addition to $J_{Q}$ , we introduce an additional bit balance loss $J_{B}$ to maximise the entropy of the learnt hash codes and in effect create balanced hash codes.",
"Here, $J_{B}$ is derived as: $J_{B} = -\\frac{1}{2N}\\text{tr}\\left( \\mathbf {H}\\mathbf {H}^{T} \\right)$ .",
"This loss aims at encouraging maximal information storage within each hash bit.",
"Regularisation: Inspired by ITQ [3], we also introduce a relaxed orthogonality regularisation constraint $R_{O}$ on the convolutional weights (say, $\\mathbf {W}_{h}$ ) connecting the output of the final residual block of the network to the hashing block.",
"This weakly enforces that the generated codes are not correlated and each of the hash bits are independent.",
"Here, $R_{O}$ is formulated as: $R_{O} = \\frac{1}{2} \\left\\Vert \\mathbf {W}_{h}\\mathbf {W}_{h}^{T} - \\mathbf {I} \\right\\Vert _{F}^{2}$ .",
"In additon to $R_{O}$ , we also impose weight decay regularization $R_{W}$ to control the scale of learnt weights and biases."
],
[
"Model Learning",
"In this section, we detail on the training procedure for the proposed DRH network with respect to the supervised retrieval and hashing related loss functions.",
"We learn a single-stage end-to-end deep network to generate hash codes directly given an input image.",
"We formulate the optimisation problem to learn the parameters of our network (say, $\\Theta : \\left\\lbrace \\mathbf {W}^{\\left( \\cdot \\right)}, b^{\\left( \\cdot \\right)} \\right\\rbrace $ ): $\\underset{\\Theta : \\left\\lbrace \\mathbf {W}^{\\left( \\cdot \\right)}, b^{\\left( \\cdot \\right)} \\right\\rbrace }{\\text{arg}\\text{min}} J = J_{S} + \\underbrace{ \\lambda _{q}J_{Q} + \\lambda _{b}J_{B} }_{\\text{Hashing Losses}} + \\underbrace{ \\lambda _{o}R_{O} + \\lambda _{w}R_{W}}_{\\text{Regularisation}}$ where $\\lambda _{q}$ , $\\lambda _{b}$ , $\\lambda _{o}$ and $\\lambda _{w}$ are four parameters to balance the effect of different contributing terms.",
"To solve this optimisation problem, we employ stochastic gradient descent to learn optimal network parameters.",
"Differentiating $J$ with respect to $\\Theta $ and using chain rule, we derive: $\\frac{\\partial J}{\\partial \\Theta } = \\frac{\\partial J}{\\partial \\mathbf {H}}\\frac{\\partial \\mathbf {H}}{\\partial \\Theta } = \\frac{1}{N}\\sum _{i =1}^{N}\\frac{\\partial J}{\\partial \\mathbf {h}_{i}}\\frac{\\partial \\mathbf {h}_{i}}{\\partial \\Theta }$ The second term $\\partial \\mathbf {h}_{i}/\\partial \\Theta $ is computed through gradient back-propagation.",
"The first term ($\\partial J / \\partial \\mathbf {h}_{i}$ ) is the gradient of the composite loss function $J$ with respect to the output hash codes of the DRH network.",
"We differentiate the continuous relaxation of the supervised retrieval loss function with respect to the hash code of a single example ($ \\mathbf {h}_{i}$ ) as follows [15]: JShi = 2( l:sli > 0 plidli - l i( q:slq > 0 plq ) plidli ) - 2( j:sij > 0 pijdij - j:sij > 0 pij( z i pizdiz ) ) where $d_{ij} = \\mathbf {h}_{i} - \\mathbf {h}_{j}$ .",
"The derivatives of hashing related loss functions ($J_{Q}$ and $J_{B}$ ) are derived as: $\\frac{\\partial J_{Q}}{\\partial \\mathbf {h}_{i}} = \\text{tanh}\\left( \\left| \\mathbf {h}_{i} \\right| - \\mathbf {1} \\right)\\text{sgn}\\left( \\mathbf {h}_{i} \\right)$ and $\\frac{\\partial J_{B}}{\\partial \\mathbf {h}_{i}} = -\\mathbf {h}_{i}$ The regularisation function $R_{O}$ acts on the convolutional weights corresponding to the hash layer ($\\mathbf {W}_{h}$ ) and its derivative with respect to $\\mathbf {W}_{h}$ is derived as follows: $\\frac{\\partial R_{O}}{\\partial \\mathbf {W}_{h}} = \\mathbf {W}_{h}\\left( \\mathbf {W}_{h}\\mathbf {W}_{h}^{T} - \\mathbf {I} \\right)$ .",
"Having computed the gradients of the individual components of the loss function with respect to the parameters of DRH, we apply gradient-based learning rule to update $\\Theta $ .",
"We use mini-batch stochastic gradient descent (SGD) with momentum.",
"SGD incurs limited memory requirements and reduces the variance of parameter updates.",
"The addition of the momentum term $\\gamma $ leads to stable convergence.",
"The update rule for the weights of the hash layer is derived as: $\\mathbf {W}_{h}^{t} = \\mathbf {W}_{h}^{t-1} - \\nu ^{t} \\text{ where }\\nu ^{t} = \\gamma \\nu ^{t-1} + \\eta \\left( \\frac{\\partial J}{\\partial \\mathbf {W}_{h}^{t-1} } + \\lambda _{o} \\frac{\\partial R_{O}}{\\partial \\mathbf {W}_{h}^{t-1}} +\\lambda _{w}\\frac{\\partial R_{W}}{\\partial \\mathbf {W}_{h}^{t-1}} \\right)$ The convolutional weights and biases of the other layers are updated similarly.",
"It must be noted that the learning rate $\\eta $ in Eq REF is an important hyper-parameter.",
"For faster learning, we initialise it the largest learning rate that stably decreases the objective function (typically, at $10^{-2}$ or $10^{-3}$ ).",
"Upon convergence at a particular setting of $\\eta $ , we scale the learning rate multiplicatively by a factor of $0.1$ and resume training.This is repeated until convergence or reaching the maximum number of epochs.",
"Database: We conducted empirical evaluations on the publicly available Indiana University Chest X-rays (CXR) dataset archived from their hospital's picture archival systems [16].",
"The fully-anonymized dataset is publicly available through the OpenI image collection system [19].",
"For this paper, we use a subset of 2,599 frontal view CXR images that have matched radiology reports available for different patients.",
"Following the label generation strategy published in [17] for this dataset, we extracted nine most frequently occurring unique patterns of Medical Subject Headings (MeSH) terms related to cardiopulmonary diseases from these expert-annotated radiology report [18].",
"These include normal, opacity, calcified granuloma, calcinosis, cardiomegaly, granulomatous disease, lung hyperdistention, lung hypoinflation and nodule.",
"The dataset was divided into non-overlapping subsets for training (80%) and testing (20%) with patient-level splits.",
"The semantic similarity matrix $\\mathcal {S}$ is contructed using the MeSH terms i.e.",
"a pair of images are considered similar if they share atleast one MeSH term.",
"Comparative Methods and Baselines: We evaluate and compare the retrieval performance of the proposed DRH network with nine state-of-the art methods including five unsupervised shallow-learning methods: LSH [2], ITQ [3], CBE [5]; two supervised shallow-learning methods: KSH [4] and MHF [6] and two deep learning based methods: AlexNet - KSH (A - KSH) [20] and VGGF - KSH (V - KSH) [21].",
"To justify the proposed formulation, we include simplified four variants of the proposed DRH network as baselines: DPH (Deep Plain Net Hashing) by removing the residual connections, DRHNQ (Deep Residual Hashing without Quantization) by removing the hashing related losses and generating binary codes only through tanh activation, DRN - KSH by training a deep residual network with only the supervised retrieval loss and quantizing through KSH post training and DRH - NB which is a variant of DRH where continuous embeddings are used sans quantization, which may act as an upper bound on performance.",
"We used the standard metrics for evaluating retrieval quality as proposed by Lai et al.",
"[1]: Mean Average Precision (MAP) and Precision - Recall Curves varying the code size(16, 32, 48 and 64 bits).",
"For fair comparison, all the methods were trained and tested on identical data folds.",
"The retrieval performance of methods involving residual learning and baselines is evaluated for two variants varying the number of layers: $\\left( \\cdot \\right) - 18$ and $\\left( \\cdot \\right) - 34$ .",
"For the shallow learning methods, we represent each image as a 512 dimensional GIST vector [22].",
"For the DRH and associated baselines, the input image is resized to $224 \\times 224$ and normalized to a dynamic range of 0-1 using the pre-processing steps discussed in [17].",
"For A-KSH and V-KSH, the image normalization routines were identical to that reported in the original works [20] [21].",
"We implement all our deep learning networks ( including DRH) on the open-source MatConvNet framework [23].",
"The hyper-parameters $\\lambda _{q}$ , $\\lambda _{b}$ and $\\lambda _{0}$ were set at 0.05, 0.025 and 0.01 empirically.",
"The momentum term $\\gamma $ was set at 0.9, the initial learning rate $\\eta $ at $10^{-2}$ and batchsize at 128.",
"The training data was augmented on-the-fly extensively through jittering, rotation and intensity augmentation by matching histograms between images sharing similar co-morbidities.",
"All the comparative deep learning methods were also trained with similar augmentation.",
"Furthermore, for A - KSH and V - KSH variants, we pre-initialized the network parameters from the pre-trained models by removing the final probability layer [20] [21].",
"These network learnt a 4096-dimensional embedding by fine-tuning it with cross-entropy loss.",
"The hashing was performed explicitly through KSH upon convergence of the network.",
"Results: The results of the MAP of the Hamming ranking for varying code sizes of all the comparative methods are listed in Table REF .",
"We report the precision-recall curves for the comparative methods at a code size of 64 bits in Fig. .",
"To justify the proposed formulation for DRH, several variants of DRH (namely, DRN - KSH, DPH, DRH - NQ and DRH - NB) were investigated and compare their retrieval results are tabulated in Table REF .",
"In addition to MAP, we also report the retrieval precision withing Hamming radius of 2 (P @ H2).",
"The associated precision-recall curves are shown in Fig. .",
"Figure: NO_CAPTION Figure: NO_CAPTION"
],
[
"Discussion",
"Within this section, we present our discussion answering the questions posed in Section , w.r.t.",
"to the results and observations we reported in Section .",
"Trainability: The introduction of residual connections offers short-cut connections which act as zero-resistance paths for gradient flow thus effectively mitigating vanishing of gradients as network depth increases.",
"This is strongly substantiated by comparing the performance of DRH - 34 to DRH - 18 vs. the plain net variants of the same depth DPH - 34 to DPH - 18.",
"There is a strong improvement in MAP with increasing depth for DRH of about 9.3%.",
"On the other hand, we observe a degradation of 2.2% MAP performance on increasing layer depth in DPH.",
"The performance of DRH-18 is fractionally better than DPH - 18 indicating that DRH exhibits better generalizability and the degradation problem is addressed well as we have significant MAP gains from increased depth.",
"With the introduction of batch normalisation and residual connections, we ensure that the signals during forward pass have non-zero variances and that the back propagated gradients exhibit healthy norms.",
"Therefore, neither forward nor backward signals vanish within the network.",
"This is substantiated by the differences in MAP observed in Table REF between methods using BN (DRH, DPH and V-KSH) in comparison to A-KSH which does not use BN.",
"Representability: Ideally, the latent embeddings in the Hamming space should be such that similar samples are mapped closer while simultaneously mapping dissimilar samples further apart.",
"We plot the t-Stochastic Neighbourhood Embeddings (t-SNE) [12] of the hash codes for four comparative methods ( GIST - ITQ, VGGF - KSH, DPH - 18 and DRH - 34) in Fig.",
"REF to visually assess the quality of the hash codes generated.",
"Visually, we observe that hand-crafted GIST features with unsupervised hashing method ITQ fail to sufficiently induce semantic separability.",
"In comparison, though VGGF-KSH improves significantly owing to network fine-tuning, better embedding results from DRH - 34 (DPH-18 is highly comparable to DRH-34).",
"Additionally, the significant differences in MAP reported in Table REF between these methods substantiates our hypothesis that in scenarios of limited training data it is better to train smaller models from scratch over finetuning to avoid overfitting (DRH - 34 has 0.183M in comparison to VGGF with 138M parameters).",
"Also the significant domain shift between natural images (ImageNet - VGGF) and CXR poses a significant challenge for generalizability of networks finetuned from pre-trained nets.",
"Table: MAP and P @ H2 of the Hamming ranking w.r.t.",
"varying network depths for baseline variants of DRH at a fixed code size of 64 bits.Compactness: Hashing aims at generating compact representations preserving the semantic relevance to the maximal extent.",
"Varying the code sizes, we observe from Table REF that the MAP performance of majority of the supervised hashing methods improves significantly.",
"In particular for DRH - 34, we observe that the improvement in the performance from 48 bits to 64 bits is only fractional.",
"The performance of DRH - 34 at 32 bits is highly comparable to DRH - 18 at 64 bits.",
"This testifies that with increasing layer depth DRH learns more compact binary embeddings such that shorter codes can already result in good retrieval quality.",
"Semantic Similarity Preservation: Visually assessing the t-SNE representation of GIST - ITQ (Fig.",
"REF (a)) we can observe that it fails to sufficiently represent the underlying semantic relevance within the CXR images in the latent hamming space, which retestifies the concerns over hand-crafted features that were raised in Section .",
"VGGF - KSH (Fig.",
"REF (b)) improves over GIST - ITQ substantially, however it fails to induce sufficient class-separability.",
"Despite KSH considering pair-wise relationships while learning to hash, the feature representation generated by fine-tuned VGG-F is limited in representability as the cross-entropy loss is evaluated point-wise.",
"Finally, the tSNE embedding of DRH - 34 shown in Fig.",
"REF visually reaffirms that semantic relevance remains preserved upon embedding and the method generates clusters well separated within the hamming space.",
"The high degree of variance associated with the tSNE embedding of normal class (red in color) is conformal with the high population variability expected within that class.",
"Fig.",
"REF demonstrates the first five retrieval results sorted according to their Hamming rank for four randomly selected CXR images from the testing set.",
"In particular, for Case (d), where we observe that the top neighbours (d 1-5) share atleast one co-occurring pathology.",
"For cases (a), (b) and (c), all the top five retrieved neighbours share the same class.",
"Figure: Retrieval results for DRH-34.Joint Optimisation: The main contribution of the work hinges on the hypothesis that performing an end-to-end learning of hash codes is better than a two stage learning process.",
"Comparative validations against the two-stage deep learning methods (A - KSH, V - KSH and baseline variant DRN - KSH) strongly support this hypothesis.",
"In particular, we observe over 14.2% improvement in MAP comparing DRN - KSH (34 - L) to DRH - 34.",
"This difference in performance may be owed to a crucial disadvantage of DRN - KSH that the generated feature representation is not optimally compatible to binararization.",
"We can also observe that, DRH - 18 and DRH - 34 incur very small average MAP decrease fo 1.8% and 0.7% when binarizing hash codes against non-binarized continuous embeddings in DRH - B- 18 and DRH - B - 34 respectively.",
"In contrast, DRH - NQ suffers from very large MAP decreases of 6.6% and 10.8% in comparison to DRH - B.",
"These observations validate the need for the proposed quantization loss as it leads to nearly lossless binarization."
],
[
"Conclusions and Open Questions",
"In this paper, we have presented a novel deep learning based hashing approach leveraging upon residual learning, termed as Deep Residual Hashing (DRH).",
"DRH integrates representation learning and hash coding into a joint optimisation framework with dedicated losses for improving retrieval performance and hashing related losses to control the quantization error and improve the hash code quality.",
"Our approach demonstrated very promising results on a challenging chest x ray dataset with co-occurring morbidities.",
"Taking insights from this pilot study on retrieval of CXR images with cardiopulmonary diseases, we believe gives rise to the following open questions for further discussion: How deep is deep enough?",
"How does DRH extend to include an additional anatomical view ( like the dorsal view for CXR) improve retrieval performance?",
"Does DRH generalize to unseen disease manifestations?",
"; and Can we visualize what DRH learns?",
"In conclusion, we believe that our paper strongly supports our initial premise of using DRH for retrieval but also opens up questions for future discussions."
]
] | 1612.05400 |
[
[
"Cross-correlating the gamma-ray sky with catalogs of galaxy clusters"
],
[
"Abstract We report the detection of a cross-correlation signal between {\\it Fermi} Large Area Telescope diffuse gamma-ray maps and catalogs of clusters.",
"In our analysis, we considered three different catalogs: WHL12, redMaPPer and PlanckSZ.",
"They all show a positive correlation with different amplitudes, related to the average mass of the objects in each catalog, which also sets the catalog bias.",
"The signal detection is confirmed by the results of a stacking analysis.",
"The cross-correlation signal extends to rather large angular scales, around 1 degree, that correspond, at the typical redshift of the clusters in these catalogs, to a few to tens of Mpc, i.e.",
"the typical scale-length of the large scale structures in the Universe.",
"Most likely this signal is contributed by the cumulative emission from AGNs associated to the filamentary structures that converge toward the high peaks of the matter density field in which galaxy clusters reside.",
"In addition, our analysis reveals the presence of a second component, more compact in size and compatible with a point-like emission from within individual clusters.",
"At present, we cannot distinguish between the two most likely interpretations for such a signal, i.e.",
"whether it is produced by AGNs inside clusters or if it is a diffuse gamma-ray emission from the intra-cluster medium.",
"We argue that this latter, intriguing, hypothesis might be tested by applying this technique to a low redshift large mass cluster sample."
],
[
"Introduction",
"Galaxy clusters are the largest virialized objects in the Universe formed by the gravitational instability-driven hierarchical structure formation process.",
"They are also unique astrophysical laboratories hosting galaxies, highly ionised hot gas in thermal equilibrium, dark matter (DM) and a population of relativistic cosmic rays (CRs).",
"The last two components can provide the conditions in which a diffuse $\\gamma $ -ray emission can be produced.",
"CRs can lead to the the emission of $\\gamma $ -ray photons via three channels: inverse Compton of relativistic electrons with the cosmic microwave background, non-thermal bremsstrahlung, and decay of $\\pi ^0$ produced through collision of relativistic protons with thermal protons (see, e.g., [37] and [49] for recent simulations of $\\gamma $ -ray emission in galaxy clusters).",
"DM can also directly or indirectly produce $\\gamma $ rays through annihilation or decay, and clusters are promising targets in the particle DM search, due to their large DM content.",
"Clusters of galaxies are not isolated objects.",
"They are located at the node of a complex cosmic web, surrounded by a network of filamentary structures populated by astrophysical sources, like the AGNs, that can contribute to the $\\gamma $ -ray emission also from within the cluster itself.",
"The discovery and characterization of cluster-wide $\\gamma $ -ray emission is therefore important in several ways.",
"If the signal is induced by CRs, then it can be used to discriminate between different models for the observed radio emission and clarify the nature of radio halos (see, e.g., [27], [26] and [17] for recent reviews on extended radio emissions in galaxy clusters).",
"A detection of a signal produced by DM would be its final discovery.",
"In this case, a detailed estimate and understanding of the contribution of all astrophysical sources to the cluster $\\gamma $ -ray emission is fundamental to unambiguously detect this more exotic signal.",
"For these reasons, clusters of galaxies have been primary targets for $\\gamma $ -ray observatories.",
"Yet, despite numerous efforts, unambiguous detection of extended $\\gamma $ -ray signal from the intra-cluster medium is lacking.",
"Upper limits on the emission from individual galaxy clusters have been obtained from the analysis of space-based observations, including the EGRET data [43] and, subsequently, the first 18 months of Fermi Large Area Telescope (LAT) data [4], and from ground-based observations in the energy band above 100 GeV (for a complete list of references, see the Introduction in [10]).",
"The lack of detection has paved the way for the stacking approach that has been adopted in the analyses of the most recent Fermi-LAT data releases [59].",
"In [23], they have stacked $\\gamma $ -ray data at positions taken from an X-ray flux-limited sample of clusters, further selecting objects with a core-dominated brightest cluster galaxy with high radio flux.",
"[33] have performed a stacking considering 53 clusters selected from the HIFLUGCS catalog [44].",
"In [31], they have analyzed 78 richest nearby clusters in the Two Micron All-Sky Survey cluster catalog.",
"These searches found no evidence for a signal in the stacked data.",
"[41] have analyzed 52.5-month Fermi-LAT data at the positions of the 55 X-ray galaxy clusters from the HIFLUGCS sample.",
"Only the brightest objects have been considered in the analysis to maximise the chance of detecting signatures of the neutral pion decay which should scale with the X-ray flux.",
"An excess has been detected with a statistical significance of 4.3 $\\sigma $ within a radius of $\\sim 0.25$ deg.",
"However, several arguments suggest that the signal is mainly produced by AGN, with no evidence of a contribution from the intra-cluster material.",
"[6] have similarly searched for a spatially extended $\\gamma $ -ray emission at the locations of 50 HIFLUGS X-ray galaxy clusters in the 4-year Fermi-LAT data, employing an improved LAT data selection (P7REP).",
"They have detected a 2.7 $\\sigma $ significant excess in a joint-likelihood analysis of stacked data.",
"This signal, however, seems to be produced by three objects, Abell 400, Abell 1367 and Abell 311, and has been conservatively attributed to individual sources (radio galaxies) within the cluster rather than to genuine diffuse emission.",
"The $\\gamma $ -ray spectra of galaxy clusters have also been searched for monochromatic $\\gamma $ -ray features, but this characteristic signal has not been revealed in the joint likelihood analysis of different samples of clusters [13], [9].",
"These results have been used to place upper limits on the velocity-averaged DM cross section for self-annihilation into $\\gamma $ rays.",
"More specialized analyses have targeted nearby, individual objects such as the Coma and the Virgo clusters [8], [7], [57].",
"The analysis of the Coma cluster in [8] has revealed an excess emission within the cluster virial radius.",
"However, its statistical significance is well below the threshold to claim detection of $\\gamma $ -ray produced by CRs interactions in the cluster.",
"The analysis of the Virgo cluster in [7] was mainly aimed at indirect DM detection.",
"It has revealed an extended emission within a radius of 3 deg, which has been regarded as an artefact due to the incompleteness of the interstellar emission model.",
"These results were used to set an upper limit on the $\\gamma $ -ray flux produced by both CRs interaction with the intra-cluster medium and DM annihilation.",
"The goal of this work is to expand the galaxy clusters analyses described above along three directions.",
"Firstly we consider three different, non X-ray selected galaxy cluster samples.",
"Namely: i) the redMaPPer catalog consisting of clusters identified through the red-sequence matched-filter Probabilistic Percolation cluster finder applied to the Sloan Digital Sky Survey (SDSS) DR8 [45], ii) the WHL12 catalog [53], [51] obtained from the SDSS DR12 [12] and iii) the Planck catalog of Sunyaev-Zeldovich (PlanckSZ) clusters [39].",
"Secondly, instead of stacking signals, we cross-correlate cluster positions in the three catalogs with Fermi-LAT data and compute 2-point statistics both in configuration and Fourier space.",
"Combining the information from the 2-point angular cross-correlation function (CCF) and the cross angular power spectrum (CAPS) allows one to reduce the impact of systematic errors that may affect the stacking analysis which, in any case, we also perform to corroborate our results.",
"We will follow [55] and [56] for the cross-correlation measurements.",
"Thirdly, we shall interpret the detected signal in the framework of the halo model along the line pursued in [42], [22] (see, e.g., [21] for a review).",
"The paper is organized as follows.",
"Section describes the Fermi-LAT data and the catalogs of clusters employed in the analysis.",
"The measurement of the angular cross-correlation and of the stacked profiles is presented in Section .",
"Section introduces the theoretical models which are then compared to data in Section .",
"We draw our conclusions in Section .",
"Figure: Sky-maps of the cluster counts per pixel for the WHL12 (upper panel), redMaPPer (central panel), and PlanckSZ (lower panel) catalogs of clusters.",
"The images have been smoothed to a resolution of 1 ∘ 1^\\circ to improve the visualization.The maps are shown in Mollweide projection and in Galactic coordinates."
],
[
"Data",
"In this work we use four different datasets: measurements of high-latitude diffuse $\\gamma $ -ray emission, and three different galaxy cluster catalogs.",
"All of them are described below.",
"Figure: Distributions of redshift (left), mass (central) and angular size (right) of the clusters considered in this work.",
"Each histogram, obtained from the catalogs listed in the insets, is normalized to unity.",
"The mass of WHL12 and redMaPPer objects is derived from the reported richness and applying the mass-richness relation in (redMaPPer) and (WHL12).",
"For the PlanckSZ objects, we use the mass estimate present in the catalog.",
"The angular size shown in the right panel corresponds to the scale radius of the cluster, namely θ s =r s /d A \\theta _s=r_s/d_A, where d A d_A is the angular diameter distance and r s =R 500 /c 500 r_s=R_{500}/c_{500} with R 500 =[M 500 /(4/3π500ρ ¯ m (z))] 1/3 R_{500}=[M_{500}/(4/3\\,\\pi \\,500\\,\\bar{\\rho }_m(z))]^{1/3} and c 500 c_{500} being the concentration parameter ."
],
[
"Fermi-LAT",
"Fermi-LAT is the primary instrument onboard the Fermi Gamma-ray Space Telescope launched in June 2008 [15].",
"It is a $\\gamma $ -ray pair-conversion telescope covering the energy range between 20 MeV and $\\sim $ 1 TeV.",
"Due to its excellent angular resolution ($\\sim 0.1^{\\circ }$ above 10 GeV), large field of view ($\\sim 2.4$ sr), and very efficient rejection of background from charged particles, it is currently the best experiment to investigate the nature of the extra-galactic $\\gamma $ -ray background [2] in the GeV energy range.",
"For our analysis, we have used 78 months of data from August 4th 2008 to January 31th 2015 (Fermi Mission Elapsed Time 239557418 s - 444441067 s), considering the Pass 8 event selectionFor a definition of the Pass 8 event selections and their characteristics, see http://www.slac.stanford.edu/exp/glast/groups/canda/ lat_Performance.htm.",
".",
"Furthermore, to reduce the contamination from the bright Earth limb emission, we exclude photons detected with measured zenith angle larger than 100$^{\\circ }$ .",
"In order to generate the final flux maps we have produced the corresponding exposure maps using the standard routines from the LAT Science Toolshttp://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/ Cicerone/ version 10-01-01, and the Pass 8 CLEAN event class, namely the P8R2_CLEAN_V6 instrument response functions (IRFs).",
"We use both back-converting and front-converting events.",
"The GaRDiAn package [5], [3] was adopted to pixelize both photon count and exposure maps in HEALPixhttp://healpix.jpl.nasa.gov/ format [30].",
"The maps contain $N_{\\rm pix} = 12, 582, 912 $ pixels with mean spacing of $\\sim 0.06^{\\circ }$ corresponding to the HEALPix resolution parameter $N_{\\rm side}=1024$ .",
"Finally, the flux maps are obtained by dividing the count maps by exposure maps.",
"We perform the bulk of our analysis in the three separate energy intervals: $0.5<E<1$ GeV, $1<E<10$ GeV and $10<E<100$ GeV.",
"For a more accurate study of the spectral dependence we will also use a finer energy binning, with 8 bins cut at 0.25, 0.5, 1, 2 , 5, 10, 50, 200, 500 GeV.",
"Our analysis is focused on the unresolved $\\gamma $ -ray background (UGRB), i.e., the unresolved EGB emission left after subtracting resolved point sources [2].",
"To obtain such maps we mask out the $\\gamma $ -ray point sources listed in the 3FGL catalog [1].",
"More precisely, we mask the 500 brightest point sources (in terms of the integrated photon flux in the 0.1-100 GeV energy range) with a disc of radius $2^\\circ $ , and the remaining ones with a disc of $1^\\circ $ radius.",
"The Small and Large Magellanic Clouds, which are extended sources, are masked with discs of $3^{\\circ }$ and $5^{\\circ }$ radius respectively.",
"To reduce the impact of the Galactic emission we apply a Galactic latitude cut masking the region with $|b|< 30^{\\circ }$.",
"In [55] we have experimented with different latitude cuts and found that $|b|>30^{\\circ }$ represents the best compromise between pixels statistics and Galactic contamination.",
"We also exclude the regions associated to the Fermi Bubbles and the Loop I structures as in [55].",
"The Galactic diffuse emission can be still significant at the high Galactic latitude used in our analysis and needs to be removed.",
"For this purpose, we use the model of Galactic diffuse emission gll_iem_v06.fitshttp://fermi.gsfc.nasa.gov/ssc/data/access/lat/ BackgroundModels.html , which we subtract from the observed emission to obtain the cleaned $\\gamma $ -ray maps.",
"The model, together with an isotropic template, is convolved with the IRFs and fitted to the photon data in each energy bin and in our region of interest using GaRDiAn.",
"The best fit diffuse plus isotropic model is subtracted from the count map and this residual count map is further divided by the exposure to give the final residual flux map to be analyzed for the given energy bin.",
"As the Galactic diffuse emission model is not exact, cleaning is not perfect and the residual flux maps are still contaminated by spurious signals, especially on large angular scales.",
"However, and this is the main advantage of our analysis, the cross-correlation analyses are expected to be almost immune to these contaminations since Galactic foreground emission is not expected to correlate with the extragalactic signal that we want to investigate.",
"Nonetheless, to minimize the chance of systematic errors we adopt a conservative approach and, following [55] and [56] we apply a further cleaning procedure that, using HEALPix tools, removes all contributions from multipoles up to $\\ell =10$ .",
"Figure: Observed CAPS (PSF deconvolved) between the Fermi-LAT γ\\gamma -ray map in three different energy bins and the redMaPPer (red) and WHL12 (green) catalogs of clusters.",
"Left panel refers to 500 MeV <E γ <1 GeV 500\\,{\\rm MeV}<E_\\gamma <1\\,{\\rm GeV}, central panel to 1 GeV <E γ <10 GeV 1\\,{\\rm GeV}<E_\\gamma <10\\,{\\rm GeV}, and right panel to 10 GeV <E γ <100 GeV 10\\,{\\rm GeV}<E_\\gamma <100\\,{\\rm GeV}.Figure: Observed CAPS (PSF deconvolved) between the Fermi-LAT γ\\gamma -ray map at E γ >1E_\\gamma >1 GeV and the PlanckSZ catalog."
],
[
"Galaxy cluster catalogs",
"The three catalogs of galaxy clusters considered in our analysis are: 1) SDSS redMaPPer [45], 2) WHL12 [53], [51] and 3) PlanckSZ [39].",
"Clusters in the redMaPPer catalog are identified using red-sequence galaxies.",
"Adopting these objects as clusters signposts increases the contrast between cluster and background galaxies in color space, thus enabling accurate and precise photometric redshift estimates.",
"In our analysis we consider all 26,350 clusters detected in the redshift range $0.08 < z < 0.55$ over the $\\sim $ 10,400 sq deg area of the SDSS Data Release 8.",
"Photo-$z$ errors are nearly Gaussian with an amplitude $\\sigma _z \\sim 0.006$ at $z \\sim 0.1$ , that increases to $\\sigma _z \\sim 0.02$ at $z \\sim 0.5$ .",
"Details on the cluster detection procedure and on the iterative method to estimate photometric redshifts can be found in [45].",
"WHL12 updates and refines on the [52] galaxy cluster catalog from SDSS III [11].",
"The original catalog contained 39,668 clusters with photometric redshifts.",
"The new catalog was obtained by [53] by applying an improved cluster detection method [50] to SDSS-III galaxies, exploiting $\\sim $ 1.35 millions Large Red Galaxies with spectroscopic redshifts in the SDSS 12th Data Release (DR12).",
"The updated WHL12 catalog [51] that we considered in our analysis has 158,103 clusters in the the range $0.05<z<0.8$ .",
"Its completeness is larger than 95% for objects with mass $M_{200} > 1.0 \\times 10^{14}\\, M_\\odot $ in the redshift range of $0.05 < z < 0.42$ , decreasing at higher redshifts.",
"The PSZ2 Second Planck Sunyaev-Zed'dovich catalog contains SZ-selected clusters.",
"It is based on the full 29 month mission data [39].",
"The methodology employed to detect clusters refines the one used to produce the first Planck SZ cluster catalog.",
"PSZ2's 1,653 clusters, distributed across 83.6% of the sky, are the union of outcomes from three cluster detection codes [38].",
"It contains 1653 detections, of which 1203 are confirmed to be clusters with identified counterparts in external datasets and with a purity larger than 83%.",
"PSZ2 probes clusters at relatively low redshift with the distribution peaking at $z\\sim 0.2$ and extending up to $z\\lesssim 0.5$ .",
"The catalog also provides estimates for the mass of the SZ clusters as a function of redshift.",
"For each of the three cluster catalogs we built a HEALPix skymap with resolution parameter $N_{\\rm side}=1024$ specifying the cluster counts per pixel $n(\\hat{\\Omega }_i)$ .",
"The three maps are shown in Fig.",
"REF .",
"Clusters are counted as single objects, i.e.",
"no statistical weight has been used to account for the cluster mass or selection effects.",
"The cross-correlation analysis is then performed between the normalized count maps $n(\\hat{\\Omega }_i)/\\bar{n}$ , where $\\bar{n}$ is the mean cluster density in the unmasked area, and the Fermi-LAT residual flux sky-maps.",
"For the two SDSS-based cluster catalogs we use the standard SDSS mask, i.e., the contour of the sky region covered by RedMaPPer which can be seen in Fig.",
"REF .",
"Further, we conservatively mask also the disconnected south-galactic region.",
"For the PSZ2 catalog no mask is used since the masked area is included in the one applied to the Fermi maps.",
"To better understand the catalogs properties, we show in Fig.",
"REF the histograms of the distributions of redshift, mass and angular size of the clusters in the three catalogs.",
"We note already here that the vast majority of the considered clusters are effectively point-like for the Fermi-LAT telescope, namely, their angular size is smaller than the point-spread function (PSF) of Fermi-LAT (see right panel).",
"Figure: Observed CCF (filled points, without PSF deconvolution) between the Fermi-LAT γ\\gamma -ray map at 1<E γ <101 < E_\\gamma < 10 GeV and catalogs of clusters, as a function of the angular separation in the sky, compared to the observed cluster γ\\gamma -ray stacked profiles (open points), as a function of the angle from the center of the stacking.",
"The WHL12 case is shown in the left panel, redMaPPer in the central panel, and PlanckSZ in the right panel.Even though the stacked profiles agree well with the CCF, they are not used for quantitative analyses since the profile andits error bars rely on the assumption that all stacked fields are independent, while in reality they are likely not (see text for details)."
],
[
"Cross-correlation APS",
"In our analysis we estimate both the 2-point CCF and the cross APS.",
"In both cases we used PolSpice,http://www2.iap.fr/users/hivon/software/PolSpice/, a publicly available toolkit to estimate the angular $CCF^{(\\gamma c)}(\\theta )$ and the CAPS $\\bar{C}_\\ell ^{(\\gamma c)}$ of any two datasets pixelized in HEALPix format [48], [20], [24], [19].",
"[56] have tested the reliability and robustness of the PolSpice estimator in similar analyses.",
"PolSpice also estimates the covariance matrix $\\bar{V}_{\\ell \\ell ^{\\prime }}$ of the different multipoles taking into account the correlation effect induced by the mask.",
"In general, other, and possibly more rigorous, statistical techniques (see, e.g., [16], [36]) can be applied.",
"However, for the specific problem under investigation (affected by large uncertainties), they are expected not to affect significantly the results as they could do in the case of a precision test.",
"The CAPS estimated from PolSpice include the effects of the instrument PSF and pixelization.",
"We deconvolve the results from these effects as in [56]: Firstly we compute the beam window function $W_\\ell ^{B}$ associated to the PSF, and the pixel window function $W_\\ell ^{\\rm pixel}$ associated to the pixelization.",
"Then, we derive the deconvolved CAPS $C_\\ell ^{(\\gamma c)}$ from the measured ones $\\bar{C}_\\ell ^{(\\gamma c)}$ as $C_\\ell ^{(\\gamma c)}=(W_\\ell )^{-1}\\,\\bar{C}_\\ell ^{(\\gamma c)}$ , where $W_\\ell = W_\\ell ^{B} W_\\ell ^{\\rm pixel} $ is the global window function.",
"The covariance matrix of the deconvolved $C_\\ell ^{(\\gamma c)}$ is then given by $V_{\\ell \\ell ^{\\prime }} = \\bar{V}_{\\ell \\ell ^{\\prime }} W_\\ell ^{-2}W_{\\ell ^{\\prime }}^{-2}$ .",
"Finally, since the mask induces a strong correlation in nearby multipoles we bin the CAPS measurements into 12 equally spaced logarithmic intervals in the range $\\ell \\in [10,2000]$ .",
"We choose logarithmic bins to account for the rapid loss of power at high $\\ell $ induced by the PSF.",
"In what follows we omit the square bracket in the $[\\ell ]$ subscript and use $C_\\ell ^{(\\gamma c)}$ also to indicate the binned CAPS.",
"In our analysis we mainly focus on the binned quantity and should be clear from the context when, instead, we consider the un-binned CAPS.",
"The $C_\\ell ^{(\\gamma c)}$ in each bin is given by the simple unweighted average of the $C_\\ell ^{(\\gamma c)}$ within the bin.",
"For these binned $C_\\ell ^{(\\gamma c)}$ it is also possible to build the corresponding block covariance matrix as $\\sum _{\\ell \\ell ^{\\prime }} V_{\\ell \\ell ^{\\prime }}/\\Delta \\ell /\\Delta \\ell ^{\\prime }$ , where $\\Delta \\ell , \\Delta \\ell ^{\\prime }$ are the width of the two multipoles bins, and $\\ell , \\ell ^{\\prime }$ run within the multipoles of the first and second multipole bin.",
"We verified that the binning is very efficient in removing correlation among nearby multipoles, resulting in a block covariance matrix that is, to a good approximation, diagonal.",
"For this reason we have neglected the off-diagonal terms in our analysis.",
"The results of the various CAPS measurements are shown in Figs.",
"REF and REF .",
"In Fig.",
"REF we show the PSF-deconvolved and binned CAPS of the redMaPPer (red) and WHL12 (green) catalogs in threee energy bins: $500\\,{\\rm MeV}<E_\\gamma <1\\,{\\rm GeV}$ (left panel), $1\\,{\\rm GeV}<E_\\gamma <10\\,{\\rm GeV}$ (central) and $10\\,{\\rm GeV}<E_\\gamma <100\\,{\\rm GeV}$ (right).",
"The bars represent 1 $\\sigma $ errors from the diagonal elements of the covariance matrix.",
"We clearly detect a non-zero cross-correlation signal.",
"We quantify the local statistical significance of the detection in each energy bin in terms of number of sigmas $N_\\sigma =\\sqrt{\\sum _j (C_{\\Delta \\ell _j}^{(\\gamma c)}/\\delta C_{\\Delta \\ell _j}^{(\\gamma c)})^2}$ with $C_{\\Delta \\ell _j}^{(\\gamma c)}$ the measured CAPS in the $j$ bin and $\\delta C_{\\Delta \\ell _j}^{(\\gamma c)}$ its uncertainty quoted in the figure.",
"For WHL12 we find: $N_\\sigma =3.7, \\, 4.4$ and 2.9 in the three energy bins.",
"The significance for redMaPPer is very similar: $N_\\sigma =3.3, \\, 5.0$ and 2.7.",
"The results for PlanckSZ are shown in Fig.",
"REF .",
"Given the limited number of clusters in this sample the statistics is too poor to divide results into energy bins.",
"Therefore, we have considered a single bin $E_\\gamma >1$ GeV.",
"Also in this case we find a non-negligible cross correlation signal with $N_\\sigma =3.7$ .",
"We note that the amplitude of the measured CAPS is the highest for PlanckSZ and the lowest for WHL12, with redMaPPer being in between.",
"This was somehow expected since the average mass of the clusters in the catalog is the largest in PlanckSZ and the smallest in WHL12 (see Fig.",
"REF b).",
"Since the redshift distribution of the clusters is not dramatically different, larger mass (typically) implies higher $\\gamma $ -ray emission and higher catalog bias, and therefore higher CAPS amplitude."
],
[
"CCF and Stacked signals",
"The CCF statistics provides a complementary information to CAPS.",
"Here we estimate CCF from the CAPS as follows: $CCF^{(\\gamma c)}(\\theta ) = \\sum _\\ell \\frac{2\\ell +1}{4\\pi }\\bar{C}_\\ell ^{(\\gamma c)} P_\\ell [\\cos (\\theta )] \\;,$ where $\\theta $ is the angular separation in the sky and $P_\\ell $ are the Legendre polynomials.",
"The resulting CCFs from the three cluster catalogs are shown Fig.",
"REF .",
"We show the $1 < E_\\gamma < 10$ GeV case only.",
"Unlike the CAPS case, here the CCFs are not de-convolved for the PSF and angular pixels.",
"The reason for this is that unlike the Fourier space case, in which deconvolution is a simple multiplication, deconvolution in configuration space is more unstable if, like in our case, CAPS has large power at high multipoles $\\ell $ .",
"The CCF analysis is quite similar to stacking the $\\gamma $ -ray signal at clusters' locations (as we will show at the end of the Section).",
"Since stacking analyses have been popular in previous studies we decided to also perform this type of analysis.",
"In our procedure we first select a region of 4 degrees of radius around the position of each cluster in the three catalogs.",
"Then, we sum the $\\gamma $ -ray flux in the circular areas with no attempt to rescale the signal to the object's properties (e.g.",
"richness).",
"Each image in the stacking is randomly rotated, in order to better investigate the impact of the region around the centered cluster.",
"A potential issue is that the stacked images might not be independent, since they come from partially overlapping fields.",
"This is most severe in the WHL12 catalog in which the mean angular separation of clusters is $\\sim 0.25^\\circ \\ll 4 ^\\circ $ radius size.",
"For this reason we will not attempt to perform a quantitative statistical comparison with the results of the CCF analysis.",
"The stacked images for $\\gamma $ -rays in the energy range $1 < E_\\gamma < 10$ GeV are shown in Fig.",
"REF .",
"We see a clear $\\gamma $ -ray excess at the clusters' center.",
"From each image we have obtained a stacked $\\gamma $ -ray emission profile $y^{(\\gamma c)}(\\theta )$ by averaging the stacked flux in logarithmically-spaced circular annuli ignoring possible small anisotropies that may survive the stacking procedure.",
"The $\\gamma $ -ray stacked profiles for the clusters in the three catalogs are shown in Fig.",
"REF .",
"The reported error bars are the image-by-image scatter around the stacked flux in each annulus.",
"With the same definition of errors, we show how the stacking signal builds up as the number of clusters increases in Fig.",
"REF .",
"We defined an approximate significance given by $\\sqrt{\\sum _i\\,(y_i/\\sigma _i)^2}$ , where $y_i$ is the measured stacked emission in the annulus $i$ and $\\sigma _i$ is the scatter.",
"On the other hand, these approximations are likely to underestimate the real errors and to overestimate the significance, since rely on the assumption that all stacked fields are independent, while in reality they are not, as already mentioned.",
"Therefore both Fig.",
"REF and the insets of Fig.",
"REF are shown only to illustrate the trends, and will not be used for any quantitative analyses.",
"If the $\\gamma $ -ray emission is circularly symmetric, the stacked profile $y^{(\\gamma c)}(\\theta )$ (derived with the procedure outlined above) is an estimator of $\\langle \\delta _c(0)\\,I_\\gamma (\\theta )\\rangle $ , where $\\delta _c$ is the cluster fluctuation field.",
"This is actually the definition of the CCF, i.e.",
"$CCF^{(\\gamma c)}(\\mathbf {\\theta })= \\langle \\delta _c(\\mathbf {\\theta }^{\\prime })\\,I_\\gamma (\\mathbf {\\theta }^{\\prime }+\\mathbf {\\theta })\\rangle =\\langle \\delta _c(0)\\,I_\\gamma (\\theta )\\rangle $ .",
"Therefore, in the small angle limit, where one can assume the dominant contribution to come from a single object and the circular symmetry to hold, the two quantity $CCF^{(\\gamma c)}$ and $y^{(\\gamma c)}(\\theta )$ are perfectly equivalent.",
"In the literature, the relation between the stacked profile and the CAPS is typically reported using the Fourier transform (see, e.g., [25]): $y^{(\\gamma c)}(\\theta ) = \\int \\frac{d\\ell \\,\\ell }{2\\pi }\\bar{C}_\\ell ^{(\\gamma c)} J_0(\\ell \\theta ) \\;,$ where $J_0$ is the is the zeroth order Bessel function.",
"In the small angle limit $\\ell \\gg 1$ , $\\theta \\ll 1$ we have $J_0(\\ell \\theta )\\simeq P_\\ell [\\cos (\\theta )]$ and thus Eq.",
"(REF ) becomes the discrete form of Eq.",
"(REF ).",
"The similarity between the two quantities is indeed evident from Fig.",
"REF .",
"The nice agreement between CCFs and stacking profiles is an important cross-check for our analysis, since the two measurements have been obtained employing two completely different methods."
],
[
"Models",
"In the Limber approximation [34], the CAPS between $\\gamma $ -ray emitters and galaxy clusters can be written as: $C_\\ell ^{(\\gamma c)}=\\int \\frac{d\\chi }{\\chi ^2} W_{\\gamma }(\\chi )\\, W_{c}(\\chi )\\,P_{\\gamma c}\\left(k=\\ell /\\chi ,\\chi \\right)\\;,$ where $\\chi (z)$ denotes the radial comoving distance.",
"$W_c(\\chi )$ and $W_{\\gamma }(\\chi )$ are the window functions that characterize the distribution of clusters and $\\gamma $ -ray emitters along the line of sight, respectively.",
"$P_{\\gamma c}(k,z)$ is the 3D cross power spectrum (PS) at the redshift $z$ , $k$ is the modulus of the wavenumber and $\\ell $ denotes the angular multipole.",
"The relation $\\chi (z)$ is fully specified by the expansion history of the Universe $H(z)$ : $d\\chi =c\\,dz/H(z)$ .",
"In the following, galactic $\\gamma $ -ray astrophysical emitters are denoted by the symbol $\\gamma _i$ , where $i$ indicates the type of source, and we consider the ones that are known to significantly contribute to the UGRB signal: blazars, misaligned AGN [mAGN] and star forming galaxies [SFGs].",
"The symbol $\\gamma _c$ refers to $\\gamma $ -ray emission from the intra-cluster medium [ICM], while $c_j$ denotes cluster catalogs ($j=$ redMaPPer, WHL12 or PlanckSZ).",
"The ingredients of our model are the cross-PS and the window functions entering in the computation of Eq.",
"(REF ).",
"They are described in the next two subsections.",
"Predictions of Eq.",
"(REF ) will be then compared with the measured CAPS shown in Figs.",
"REF and REF ."
],
[
"Window functions",
"The window functions of the three galactic $\\gamma $ -ray sources considered here (blazars, mAGNs and SFGs) are presented in [22] (Appendix).",
"The collective $\\gamma $ -ray emission from ICM in the Universe gives raise to the window function: $W_{\\gamma _c}(z)=\\int _{M_{c,min}} \\,\\mathrm {d}M\\,\\frac{d^2n}{dM\\,dV}\\,\\frac{\\mathcal {L}_{\\gamma _c}(M_{500}(M),z)}{4\\pi \\,(1+z)}\\;,$ where $d^2n/dM\\,dV$ is the cluster mass function predicted by the ellipsoidal collapse model [47] and $\\mathcal {L}_{\\gamma _c}$ is the cluster $\\gamma $ -ray luminosity per unit energy range.",
"The integral is over the cluster masses above $10^{13.8}\\,h^{-1}\\,M_\\odot $ [58].",
"The relation between the halo and the cluster virial mass $M_{500}$ is specified in [32] (Appendix A).",
"For the luminosity we adopt the empirical relation of [58]: $L_{\\gamma _c}(100\\,{\\rm MeV})=A_{\\gamma _c}\\,\\left(\\frac{M_{500}}{M_\\odot }\\right)^{5/3}\\,{\\rm s^{-1}\\,GeV^{-1}}\\;,$ where $L_{\\gamma _c}=\\mathcal {L}_{\\gamma _c}/E$ and $A_{\\gamma _c} \\lesssim 10^{21}$ is a normalization parameter constrained by the non-detection of $\\gamma $ -rays from nearby clusters [58].",
"Finally, we assume a power-law ICM energy spectrum with index $\\alpha _{\\gamma _c}=2.2$ .",
"The window function of cluster counts is: $W_{c_j}(z)=\\frac{4\\pi \\,\\chi (z)^2}{N_{c_j}}\\int \\,\\mathrm {d}M\\,\\frac{d^2n_{c_j}}{dM\\,dV}\\;,$ where: $N_{c_j}=\\int \\,\\mathrm {d}M\\,dV\\,\\frac{d^2n_{c_j}}{dM\\,dV}$ is the number of object in the $j$ -th cluster catalog.",
"To model the cluster mass function we adopted two different approaches.",
"For the redMaPPer and WHL12 clusters we measure the cluster richness distribution $d^2n_{c_j}/{d\\lambda dz}$ from the catalogs, assume a lognormal distribution for the cluster richness $\\lambda $ , $P(\\lambda |M_{500})$ , with $\\lambda (M_{500})$ taken from [46] (redMaPPer) and [51] (WHL12) and derive the cluster mass function as a function of redshift as: $\\frac{d^2n_{c_j}}{dz\\,dM_{500}}=\\int d\\lambda \\frac{d^2n_{c_j}}{d\\lambda dz} P(\\lambda |M_{500}) \\, .$ Finally, we obtain the mass function per unit volume $d^2n_{c_j}/{dV dM}$ by specifying the cosmology-dependent relation: $\\frac{d^2V}{d\\Omega dz}=\\frac{\\chi (z)^2}{H(z)}\\, .$ We checked that the derived cluster mass function is in good agreement with the theoretical model of [47], in the relevant mass and redshift ranges, once selection effects and completeness are taken into account.",
"For the PlanckSZ clusters we estimate the mass function using directly the masses and the redshifts of the objects in the catalog.",
"Figure: Measured (points) and predicted (lines) autocorrelation APS for the catalogs redMaPPer (red), WHL12 (green) and PlanckSZ (blue)."
],
[
"Three dimensional power spectrum",
"To model the three dimensional cross power spectrum we use the halo model and write the PS as a sum of the one-halo and two-halo terms $P=P^{1h}+P^{2h}$ .",
"Below we provide the expressions for $P^{1h}$ and $P^{2h}$ .",
"The interested reader can find a detailed discussion in [29].",
"We also notice that in some equation the redshift dependence is not explicitly reported.",
"For the cross correlation between point-like astrophysical $\\gamma $ -ray emitters and clusters we have: $& & P_{c_j,\\gamma _i}^{1h}(k,z) = \\int _{\\mathcal {L}_{\\rm min,i}(z)}^{\\mathcal {L}_{\\rm max,i}(z)} \\mathrm {d}\\mathcal {L}\\,\\Phi _i(\\mathcal {L},z)\\,\\times \\nonumber \\\\& & \\qquad \\qquad \\qquad \\frac{\\mathcal {L}}{\\langle f_{\\gamma _i} \\rangle } \\,\\frac{\\langle N_{c_j}\\!",
"(M(\\mathcal {L}))\\,\\rangle }{\\bar{n}_{c_j}} \\\\& & P_{c_j,\\gamma _i}^{2h}(k,z) = \\left[\\int _{\\mathcal {L}_{\\rm min,i}(z)}^{\\mathcal {L}_{\\rm max,i}(z)} \\mathrm {d}\\mathcal {L}\\,\\Phi _i(\\mathcal {L},z)\\, b_{\\gamma _i}(\\mathcal {L})\\,\\frac{\\mathcal {L}}{\\langle f_{\\gamma _i} \\rangle } \\right]\\nonumber \\times \\\\& &\\qquad \\left[\\int _{M_{\\rm min}}^{M_{\\rm max}} \\mathrm {d}M\\,\\frac{dn}{dM} b_h(M)\\,\\frac{\\langle N_{c_j}\\,\\rangle }{\\bar{n}_{c_j}} \\right] \\,P^{\\rm lin}(k)\\;.$ Both terms depend on the luminosity function of the emitter, $\\Phi _i$ and the mean luminosity density $\\langle f_{\\gamma _i}\\rangle = \\int \\mathrm {d}\\mathcal {L} \\, \\mathcal {L} \\, \\Phi _i(\\mathcal {L},z)$ , whereas the linear mass power spectrum $P^{\\rm lin}(k)$ and the bias $b_{\\gamma _i}$ only enter the two-halo term.",
"For the bias we adopt a simple linear model and assume that the bias of the emitter is equal to that of its halo host $b_{\\gamma _i}(\\mathcal {L})=b_h(M(\\mathcal {L}))$ modeled according to [47].",
"For the relation between the mass of the halo host and the luminosity of the emitter, $M(\\mathcal {L})$ , we adopt the one derived by [18].",
"The effective halo occupation of clusters $\\langle N_{c_j}\\rangle =(dn_{c_j}/dM)/(dn/dM)$ is obtained from the cluster mass functions $dn_{c_j}/dM$ used in Section REF .",
"In this way, we account for selection effects and completeness of the catalogs.",
"The average number density of clusters at a given redshift is given by $\\bar{n}_{c_j}(z)=\\int dM\\, \\langle N_{c_j}\\rangle \\,dn/dM$ .",
"Note that Eq.",
"(REF ) does not depend on the wavenumber $k$ .",
"It describes the picture of point-like $\\gamma $ -ray emitters located at the center of the clusters.",
"Being flat, it acts as a shot-noise-like term.",
"[22] have shown that this halo model is not sufficient to describe the effect of the Fermi-LAT PSF that creates an additional shot-noise-like term on small-scales, which is not captured by the above equations.",
"Quantifying the amplitude of this effect is not straightforward.",
"However, since we know it is scale-independent, we can model it empirically by adding an extra, shot-noise-like constant term in the fit of the measured $C_{\\ell }^{(\\gamma c)}$ .",
"Therefore, following [14] and [22] we include one additional free parameter for each combination of cluster catalog and $\\gamma $ -ray source.",
"We note also that the 1-halo term model above assumes that the relation $M(\\mathcal {L})$ is deterministic.",
"We argue that ignoring the scatter in the relation does not significantly affect our results since the 1-halo term is small (blazars, mAGN and SFG reside in halos typically smaller than the cluster size) and subdominant with respect to the shot-noise term.",
"For the cross correlation between $\\gamma $ -ray emission from the ICM and clusters we have: $& & P_{c_j,\\gamma _c}^{1h}(k,z) = \\int _{M_{\\rm c,min}}^{M_{\\rm max}} \\mathrm {d}M\\ \\frac{dn}{dM} \\frac{\\langle N_{c_j}\\,\\rangle }{\\bar{n}_{c_j}} \\frac{\\mathcal {L}_{\\gamma _c}(M)}{\\langle f_{\\gamma _c} \\rangle }\\, \\frac{\\tilde{v}_\\delta (k|M)}{M} \\nonumber \\\\ & & P_{c_j,\\gamma _c}^{2h}(k,z) = \\left[\\int _{M_{\\rm min}}^{M_{\\rm max}} \\mathrm {d}M\\,\\frac{dn}{dM} b_h(M) \\frac{\\langle N_{c_j}\\rangle }{\\bar{n}_{c_j}} \\right]\\times \\\\& & \\left[\\int _{M_{\\rm c,min}}^{M_{\\rm max}} \\mathrm {d}M \\,\\frac{dn}{dM} b_h(M)\\,\\frac{\\mathcal {L}_{\\gamma _c}(M)}{\\langle f_{\\gamma _c} \\rangle }\\,\\frac{\\tilde{v}_\\delta (k|M)}{M} \\right]\\,P^{\\rm lin}(k) \\;,\\nonumber $ where now the luminosity density is $\\langle f_{\\gamma _c}\\rangle = \\int \\mathrm {d}M \\,dn/dM\\, \\mathcal {L}/\\bar{\\rho }$ , and $\\tilde{v}_\\delta (k|M)$ is the Fourier transform of the normalized halo density profile $\\rho _h(\\mathbf {x}|M)/\\bar{\\rho }_{DM}$ , that we assume to have a NFW shape [35].",
"The underlying assumption is that the $\\gamma $ -ray emission from the ICM has the same profile of the host halo (in practice, this is not a crucial assumption, since in the current analysis we do not probe scales smaller than the typical size of a cluster).",
"Unlike in the previous case, uncertainties in the 1-halo term cannot be ignored.",
"They stem from the fact that no extended $\\gamma $ -ray emission from clusters has been unambiguously detected and, consequently, no observational constraint exists for the $\\mathcal {L}_{\\gamma _c}(M)$ relation.",
"To account for this potential source of systematic error, we again include an additional constant term when we fit the cross-correlation model to the data.",
"Figure: CAPS between redMaPPer clusters and γ\\gamma -ray emitters (convolved with beam window function W ℓ B W_\\ell ^{B}) compared to the measurement at E γ >1E_\\gamma >1 GeV.",
"Dotted line shows a noise term at the level of C ℓ (γc) =2·10 -12 cm -2 s -1 C_\\ell ^{(\\gamma c)}=2\\cdot 10^{-12}{\\rm cm^{-2}s^{-1}}.Figure: CAPS between WHL12 clusters and γ\\gamma -ray emitters (convolved with beam window function W ℓ B W_\\ell ^{B}) compared to the measurement at E γ >1E_\\gamma >1 GeV.",
"Dotted line shows a noise term at the level of C ℓ (γc) =7·10 -13 cm -2 s -1 C_\\ell ^{(\\gamma c)}=7\\cdot 10^{-13}{\\rm cm^{-2}s^{-1}}."
],
[
"Results",
"Before comparing model and data we performed a sanity check in which we use our model to predict the angular cluster-cluster power spectra and compare it with the measured auto angular power spectrum for each of the three cluster catalogs.",
"The results are shown in Fig.",
"REF .",
"The agreement is remarkably good except at very large or very small angular scales where, respectively, selection effects and model uncertainties are larger.",
"The results of the comparisons between measured and predicted cross-correlation between $\\gamma $ -ray emitters (blazars, mAGN, SFG and ICM) and clusters (redMaPPer, WHL12 and PlanckSZ) are shown in Figs.",
"REF , REF and REF .",
"Dots with errorbars refer to the data whereas the different curves represent predictions obtained using the models described in Section .",
"The latter have been normalized (without changing neither spectral nor spatial shapes) such that blazars, mAGNs and SFGs individually contribute to 100% of the UGRB above 1 GeV.",
"For the ICM case we consider a 1 % contribution to meet the observational constraints described in Section REF .",
"At $\\ell \\lesssim 100$ , the data are well fitted by a model in which the $\\gamma $ -ray emission is produced by blazars, mAGNs or SFGs (or a combination of them), provided that they contribute to 100 % of the UGRB.",
"On the contrary, the ICM contribution to the cross correlation is highly subdominant, largely because of the 1 % UGRB contribution constraint.",
"Note however that the ICM would account for much more than 1 % of the CAPS (while blazars, mAGNs or SFGs providing 100 % of the UGRB contribute to a similar fraction, roughly 100 %, of the CAPS).",
"This is because, when compared to galactic $\\gamma $ -ray emitters, the ICM contribution has a relatively larger non-linear term and its window function have a better overlapping in redshift with the catalogs window functions.",
"Though small, the amplitude of the ICM CAPS is not the same in all panels.",
"Instead it correlates with the average mass of the clusters in the catalogs.",
"Indeed, one can notice how it increases going from WHL12 to PlanckSZ (the catalogs with, respectively, the smallest and largest average mass, see Fig.",
"REF b), in particular in comparison with the milder increase expected in the amplitudes associated to the other $\\gamma $ -ray emitters.",
"In the case of PlanckSZ catalog, that contains massive clusters, the ICM contribution is larger than that of the other astrophysical sources at $\\ell \\gtrsim 100$ .",
"The similarity in the theoretical CAPS of all the above $\\gamma $ -ray emitters, except the ICM, guarantees the fact that all of them provide a good fit to the data and reflects the similarities in the $\\gamma $ -ray window functions.",
"The small differences originates from two effects: the (relatively small) differences in the shapes of the $\\gamma $ -ray window functions (see, e.g., the Supplemental Material in [42]) and the different redshift dependences of the bias factors of the various emitters (see, e.g., Appendix of [22]).",
"Figure: CAPS between PlanckSZ clusters and γ\\gamma -ray emitters (convolved with beam window function W ℓ B W_\\ell ^{B}) compared to the measurement at E γ >1E_\\gamma >1 GeV.",
"Dotted line shows a noise term at the level of C ℓ (γc) =5·10 -12 cm -2 s -1 C_\\ell ^{(\\gamma c)}=5\\cdot 10^{-12}{\\rm cm^{-2}s^{-1}}.At $\\ell \\gtrsim 100$ the models systematically underestimate the measured CAPS.",
"As we have discussed, the validity of our model is expected to break down at small scale.",
"For this reason we included a shot-noise-like term that restores the missing power at high $\\ell $ .",
"This term, convolved with the Fermi-LAT beam window function, is shown with black dot-dashed lines in Figs.",
"REF , REF and REF .",
"Its introduction clearly improves the quality of the model that now fits the data all the way out to $\\ell =1000$ .",
"In principle, one could account for this term by increasing the normalization of the ICM window function by one order of magnitude.",
"Indeed, the shape of the CAPS ICM is similar to the signal shape over the whole multiple range.",
"On the other hand, this increase would make the ICM contribution to the UGRB to grow to $\\sim $ 10%, something which conflicts with current bounds.",
"The different $\\gamma $ -ray emitters have different energy spectra, and we can have a better insight on the origin of the cross-correlation signal by repeating the correlation analysis in finer energy bins.",
"We considered eight of them: $[0.25,0.5] \\; [0.5,1.0] \\; [1.0,2.0] \\; [2.0,5.0] \\; [5.0,10.0] \\; [10.0,50.0] $ $\\; [50.0,200.0] \\; [200.0,500.0] $ GeV.",
"Building upon the results of the previous analysis we now move on from the specific benchmark models adopted above and consider a simpler CAPS model: $C_{\\ell ,a}^{(\\gamma c)}=C^{1h}_a+A^{2h}_a C_{\\ell ,a}^{2h} \\;,$ with $a=1,..,8$ running over the energy binsIn the correlation with the PlanckSZ catalog, we discard the energy bins 1 and 8 since they are meaningless due to their low statistics., $C^{1h}_a$ is the parameter of the fit related to the 1-halo contribution, $C_{\\ell ,a}^{2h}$ is the 2-halo model term that we have modeled according to the blazar case since, as we have seen above, the other $\\gamma $ -ray emitters produce a similar cross-correlation signal apart from the ICM that, again according to the previous analysis, is subdominant.",
"$A^{2h}_a$ is the second free parameter of the fit and sets the amplitude of the 2-halo term.",
"We remind the reader that, in the halo model, the CAPS is generically given by the sum of a 1-halo and a 2-halo terms, as described in Section ; in Eq.",
"(REF ), the 1-halo CAPS is constant (since the source is assumed to be point-like) and the 2-halo CAPS is computed plugging Eq.",
"() into Eq.",
"(REF ).",
"The null hypothesis (no signal) can be discarded at high significance.",
"Indeed, the $\\Delta \\chi ^2$ with respect to this model is 43.2 (WHL12), 53.5 (redMaPPer), and 14.4 (PlanckSZ).",
"Considering the number of parameters of the model (16 for WHL12 and redMaPPer and 12 for PlanckSZ) and following Wilks' theorem [54], this leads to $p$ -values of $2.6\\times 10^{-4}$ (WHL12), $6.3\\times 10^{-6}$ (redMaPPer), and 0.27 (PlanckSZ).",
"Note that these are conservative estimates of the significance of the signals, since one could reduce the degrees of freedom by fixing the energy spectrum of the model (that, as we will see in the following, is in fair agreement with expectations for blazars).",
"We show the derived fitting parameters in Figs.",
"REF and REF .",
"In Fig.",
"REF , we plot the $C^{1h}$ term for the different energy bins.",
"The global statistical evidence (i.e.",
"adding up in quadrature the evidences of single energy bins) for the 1-halo component is $3.9\\sigma $ (WHL12), $4.7\\sigma $ (redMaPPer), and $2.3\\sigma $ (PlanckSZ).",
"Fig.",
"REF instead shows the 2-halo term $A^{2h}C_{\\ell =80}^{2h}$ whose statistical evidence turns out to be $2.6\\sigma $ (WHL12), $2.1\\sigma $ (redMaPPer), and $1.8\\sigma $ (PlanckSZ).",
"The evidence in each energy bin for the 1-halo (2-halo) term (and in turn the error bars in Fig.",
"REF (Fig.",
"REF )) have been derived by evaluating the likelihood ratio between the model in Eq.",
"(REF ) and the same model but with the 1-halo (2-halo) term set to zero.",
"The energy dependence of the 1-halo term in the redMaPPer case shows a possible break at $E=10$ GeV which suggests the presence of two contributions: a hard component with spectral index close to $-2$ above 10 GeV and a softer component at lower energies.",
"The analysis of the other two catalogs displays a similar trend.",
"However, the quality of the data does not allow us to draw statistically significant conclusions.",
"A fit to the energy spectrum of the redMapper 1-halo CAPS with the sum of two power-laws is preferred over the single power-law case at 85% C.L., with the the best-fit spectral indexes being $-2.9$ and $-2.0$ (while $-2.7$ is the best-fit spectral index in the case of a single power-law).",
"No break is seen in the energy-dependence of the 2-halo term with a spectral index of $\\sim -2$ .",
"This is consistent with being produced by the same sources responsible for the 1-halo term at $E > 10$ GeV.",
"Knowing that BL Lac-type of blazar are characterized by hard energy spectra in contrast with the softer spectra of all other galactic $\\gamma $ -ray emitters (mAGN, FSRQ, SFG) considered in our model, the emerging picture is that of a cross correlation signal dominated by BL Lacs on large scale (where the 2-halo term dominates) and on small scales, but possibly only at high energy.",
"The soft component, seen in the 1-halo term only, might indicate that the small-scale cross-correlation signal is also contributed by a different type of $\\gamma $ -ray emitters that takes over at $E < 10$ GeV.",
"These can be non-BL Lac AGNs or SFGs hosted in the cluster halo, or the ICM itself (or a combination of them).",
"Figure: Energy spectrum of a constant 1-halo CAPS.",
"This has been derived by fitting the measurements in eight energy bins from 0.25 to 500 GeV with C 1h +A 2h C ℓ 2h C^{1h}+A_{2h}C^{2h}_\\ell , where C 1h C^{1h} and A 2h A_{2h} are the fitting parameters and C ℓ 2h C^{2h}_\\ell is taken to follow the blazar case.Figure: Energy spectrum of the 2-halo CAPS term at ℓ=80\\ell =80.",
"This has been derived by fitting the measurements in eight energy bins from 0.25 to 500 GeV with C 1h +A 2h C ℓ 2h C^{1h}+A_{2h}C^{2h}_\\ell , where C 1h C^{1h} and A 2h A_{2h} are the fitting parameters and C ℓ 2h C^{2h}_\\ell is taken to follow the blazar case.",
"The plot shows A 2h C 80 2h A_{2h}C^{2h}_{80}."
],
[
"Conclusions",
"In this work, we have analyzed the 2-point angular cross-correlation between the unresolved EGB observed by Fermi-LAT and the number of galaxy clusters in three different catalogs: WHL12, redMaPPer and PlanckSZ.",
"The main results are: For the first time, we detect a cross-correlation signal in both configuration and Fourier space.",
"The measurement has a strong statistical significance for WHL12 and redMaPPer, while it is at the level of a hint for PlanckSZ.",
"An alternative, more conventional, stacking analysis was performed that confirms these results.",
"As expected, the $\\gamma $ -ray emission profile that emerges from the stacked images turns out to be consistent with that of the angular CCF.",
"The cross-correlation measurement confirms that the unresolved EGB observed by Fermi-LAT correlates with the large scale clustering of matter in the Universe (traced by clusters), as found also in [28] and [56], employing different tracers.",
"We model the cross correlation signal in the framework of the halo model and consider $\\gamma $ -ray emission from four types of astrophysical sources: blazars, mAGN, SFG and ICM.",
"Our model provides a good fit to the measured CAPS on large scales.",
"At $\\ell \\lesssim 100$ all these sources but the ICM generate a cross-correlation compatible with the observations.",
"The ICM contribution is subdominant.",
"Its small amplitude is constrained by the fact that no $\\gamma $ -ray emission from clusters, including the nearby ones, has been detected so far.",
"Only in the PlanckSZ cluster catalog case the ICM may provide a non negligible contribution at intermediate scales.",
"On small scales our model underestimates the observed CAPS.",
"In this range the angular spectrum is approximately flat and can be fitted by a constant, shot-noise-like term, that we included in our model.",
"Its physical interpretation is as follows.",
"Fermi-LAT has a rather large PSF ($\\gtrsim 0.1^\\circ $ ).",
"The recorded arrival direction of photons is “randomized” with respect to the true direction on scales below (comparable to) the PSF.",
"This creates a shot-noise-like term and prevents to fully characterize the correlation among sources beyond the PSF angular scale.",
"Noting that the mean cluster redshift in the considered catalogs is $\\bar{z}\\sim 0.2-0.4$ , any population of sources with physical sizes below a few Mpc could contribute to the small-scale CAPS.",
"The cross correlation signal is significantly detected out to $\\sim 1$ degree, which is beyond the PSF extension (even though the statistical significance of the 2-halo term is not totally conclusive and amounts to $>2\\sigma $ ).",
"At the typical redshifts of the clusters in the considered catalogs, $\\sim 1$ degree corresponds to a linear scale of $\\sim 10$ Mpc.",
"This means that a large fraction of the correlation signal seems to be not physically associated to the clusters.",
"Instead, it can be produced by AGNs or SFGs residing in the larger scale structures that surrounds the high density peaks where clusters reside.",
"Finally, we have investigated the energy dependence of the cross-correlation signal.",
"It turns out that on large scale, where the 2-halo term dominates, the signal is contributed by sources with hard energy spectra, consistent with that of the BL Lacs.",
"On small scales, where the 1-halo term dominates, the correlation signal could be contributed by different types of sources.",
"At high ($E>10$ GeV) energies the dominant sources have hard spectra, i.e.",
"they are probably the same BL Lac population.",
"At smaller energies, the correlation signal shows a hint of contribution by sources with softer spectra.",
"These can be non-BL Lac AGNs, SFGs and/or the ICM.",
"In conclusion, our measurement combined with theoretical expectations suggests that the detected cross-correlation signal is largely contributed by compact sources like AGNs or SFGs.",
"A possible contribution from the ICM, associated to the cluster itself, is however not excluded at small scales.",
"Since its amplitude is expected to increase with the mass of the clusters, a cross-correlation analysis with a wide-field catalog containing a large number of nearby massive clusters can be therefore a suitable way to attempt the detection of the so far elusive $\\gamma $ -ray emission from the ICM."
],
[
"Acknowledgements",
"The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis.",
"These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.",
"Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France.",
"We would like to thank F. Zandanel for discussions.",
"This work is supported by the research grant Theoretical Astroparticle Physics number 2012CPPYP7 under the program PRIN 2012 funded by the Ministero dell'Istruzione, Università e della Ricerca (MIUR); by the research grants TAsP (Theoretical Astroparticle Physics) and Fermi funded by the Istituto Nazionale di Fisica Nucleare (INFN); by the Excellent Young PI Grant: The Particle Dark-matter Quest in the Extragalactic Sky.",
"MV and EB are supported by PRIN MIUR and IS PD51 INDARK grants.",
"MV is also supported by ERC-StG cosmoIGM, PRIN INAF.",
"JX is supported by the National Youth Thousand Talents Program, the National Science Foundation of China under Grant No.",
"11422323, and the Strategic Priority Research Program, The Emergence of Cosmological Structures of the Chinese Academy of Sciences, Grant No.",
"XDB09000000.",
"SC is supported by ERC Starting Grant No.",
"280127.",
"Some of the results in this paper have been derived using the HEALPixhttp://healpix.sourceforge.net/downloads.php package."
],
[
"Validation and cross-checks",
"To assess the robustness of our analyses and results, we performed a few different tests, along the lines described in Section 6 of [56].",
"No unexpected behavior has been found.",
"In this Appendix, we report the results for two tests and, for the sake of definiteness, we focus on the redMaPPer catalog only.",
"First, we considered a different selection of $\\gamma $ -ray events.",
"The analysis in the main text has been conducted using the P8R2_CLEAN class of Fermi-LAT events.",
"This is a common choice for diffuse studies since it provides a good compromise between having a clean event class and sufficiently high statistics.",
"Here we analyze also the P8R2_ULTRACLEANVETO photons, namely the cleanest Pass 8 event class.",
"Moreover, we considered a more conservative zenith angle cut, excluding photons detected with measured zenith angle larger than 90$^{\\circ }$ (instead of 100$^{\\circ }$ , as in the analysis presented in the main text).",
"The comparison is shown in Fig.",
"REF (red versus violet points).",
"It is clear that the results are fully compatible.",
"Also the significances of detection are not dramatically affected.",
"For the more conservative choice of photon events, the $p$ -values of the statistical analysis described in Section would become $8.8\\times 10^{-3}$ (WHL12), $3.7\\times 10^{-5}$ (redMaPPer), and 0.18 (PlanckSZ).",
"To test the robustness of the detection we built a mock realization of the redMaPPer catalog by performing the transformation on the Galactic latitude $b\\rightarrow -b$ for each cluster of the sample.",
"This realization preserves the intrinsic clustering of the catalog (i.e., it provides the same autocorrelation signal), but should remove the cross-correlation (for more details, see Section 6.6 in [56]).",
"Indeed, as clear also from Fig.",
"REF (orange open points), the derived CAPS is compatible with no signal, with the $p$ -value now becoming $0.994$ , meaning no preference over the null hypothesis.",
"Figure: Observed CAPS (PSF deconvolved) between the Fermi-LAT γ\\gamma -ray map in three different energy bins and the redMaPPer catalog.Filled red points show the measurement with the P8R2_CLEAN Fermi-LAT event class and zenith angle cut at 100 ∘ ^\\circ .Filled violet points show instead the measurement with the P8R2_ULTRACLEANVETO Fermi-LAT event class and zenith angle cut at 90 ∘ ^\\circ .Open orange points show the CAPS between the Fermi-LAT γ\\gamma -ray maps and a mock realization of the catalog where we perform the transformation b→-bb\\rightarrow -b for each redMaPPer cluster.Left panel refers to 500 MeV <E γ <1 GeV 500\\,{\\rm MeV}<E_\\gamma <1\\,{\\rm GeV}, central panel to 1 GeV <E γ <10 GeV 1\\,{\\rm GeV}<E_\\gamma <10\\,{\\rm GeV}, and right panel to 10 GeV <E γ <100 GeV 10\\,{\\rm GeV}<E_\\gamma <100\\,{\\rm GeV}."
],
[
"Consistency among different samples of clusters",
"The WHL12 catalog is, among the three samples we considered, the one with the largest number of clusters.",
"This is due to the fact that it has the lowest richness (and in turn mass) threshold, see also Fig.",
"REF b.",
"The redshift range is instead not dramatically different, especially between WHL12 and redMaPPer (that also share the fraction of sky probed).",
"In this Appendix, we test the dependency of the WHL12 cross-correlation signal from the cluster richness (that can be translated into cluster mass, see discussion in Section REF ).",
"We also check the consistency of our findings for the different catalogs, by comparing the signal of a high-richness subsample of WHL12 with the redMaPPer case.",
"To this aim we split the WHL12 into three samples of richness $\\lambda <23$ , $23<\\lambda <35$ and $\\lambda >35$ .",
"The last bin contains 24,903 clusters, similarly to the redMaPPer catalog.",
"Results are shown in Fig.",
"REF , where we focus on the $\\gamma $ -ray energy bin $1\\,{\\rm GeV}<E_\\gamma <10\\,{\\rm GeV}$ for definiteness.",
"Even though the statistics is not very high, left panel shows that the amplitude increases with richness, as expected.",
"The right panel compares the CAPS of the high-richness subsample of WHL12 with the redMaPPer one.",
"The two cross-correlation measurements are fully compatible.",
"Figure: Left panel: Observed CAPS (PSF deconvolved) between the Fermi-LAT γ\\gamma -ray map at 1 GeV <E γ <10 GeV 1\\,{\\rm GeV}<E_\\gamma <10\\,{\\rm GeV} and the WHL12 catalog of clusters for three different richness bins: λ<23\\lambda <23 (cyan), 23<λ<3523<\\lambda <35 (orange) and λ>35\\lambda >35 (dark green).",
"Right panel: Comparison between the CAPS obtained from the cross-correlation with redMaPPer (red) and the clusters with λ>35\\lambda >35 in WHL12 (dark green)."
]
] | 1612.05788 |
[
[
"Mutual information for fitting deep nonlinear models"
],
[
"Abstract Deep nonlinear models pose a challenge for fitting parameters due to lack of knowledge of the hidden layer and the potentially non-affine relation of the initial and observed layers.",
"In the present work we investigate the use of information theoretic measures such as mutual information and Kullback-Leibler (KL) divergence as objective functions for fitting such models without knowledge of the hidden layer.",
"We investigate one model as a proof of concept and one application of cogntive performance.",
"We further investigate the use of optimizers with these methods.",
"Mutual information is largely successful as an objective, depending on the parameters.",
"KL divergence is found to be similarly succesful, given some knowledge of the statistics of the hidden layer."
],
[
"Introduction",
"There is extensive literature on the effects of sustained activity on human cognitive performance, which suggests that performance is determined by some finite resource that causes a decline in performance when depleted[1], [2], [10].",
"However, the nature of this resource remains poorly understood.",
"In other work (to be submitted) we propose a model in which standardized test performance depends on cognitive resources that are depleted as questions are answered.",
"In this work, we present a method for fitting ODE's to data when the mapping between the observables and underlying hidden variables is unknown.",
"For the cognitive performance example, a researcher may propose a microscopic mechanistic model for resource variation in the brain but not know how the variations in the physiological resources translate into variation in the observed performance.",
"We can characterize the relation of cognitive performance to physiological resource usage as a deep nonlinear system.",
"We consider a nonlinear system to be deep if it can be represented as a composition of multiple nonlinear transformations.",
"There is an initial hidden layer, followed by a transformation layer, and an output layer that corresponds to what is actually observed.",
"In short, complex nonlinear dynamics on the microscopic scale ultimately translate to emergent behavior on the macroscale via multiple scales, spatially and temporally.",
"Many types of objective functions may be used to fit equations to hidden variables, such as Kalman filters, Bayesian models, and simple root-mean-squared error (RMSE) regression via supervised learning[8], [9], [12].",
"In general, these models are parametric, and their success requires some fundamental assumptions about the dynamics producing data.",
"For example, to minimize RMSE, one needs to be able to regress to the final observed variable, so the hidden layer needs to also be known or learned.",
"One approach to understanding a deep nonlinear system would be to propose a model for the hidden variables, construct a function approximator (such as a neural network) as the transformation layer, and then use a fitting procedure such as backpropagation[8] to jointly regress the generic transformation layer and fit the underlying hidden model.",
"The viability of this approach depends on the fidelity of the function approximator, which may require an extremely large amount of data to properly determine, akin to traditional deep learning.",
"Here, we propose a method that is agnostic to the transformation layer by seeking to maximize explanatory power of the underlying hidden model to the ultimate observables.",
"We investigate information theoretic measures, such as mutual information and Kullback-Leibler divergence, as objective functions for fitting the model to generated data.",
"We construct a number of example nonlinear systems to validate the ability for these information theoretic measures to `peer through' the intervening nonlinear unknown layers and directly fit observations to the hidden variables we are attempting to model.",
"We confirm that mutual information may be a valuable tool for fitting models to deep nonlinear systems."
],
[
"Deep Nonlinear Models",
"In the present work, we consider systems that we may call “deep nonlinear models,\" in that we may have a clear hypothesis for what the dynamics of the initial layer may be, but there remains an unknown layer that transforms the output of the dynamics into what is observed.",
"We may consider deep nonlinear models to take the form $\\dot{x} &= f(x,t) &\\text{Initial Layer} \\\\\\dot{y} &= g(x,y,t) &\\text{Hidden Layer} \\\\z &= h(y,t).",
"&\\text{Observable/Measurement Layer}$ where $f,g,h$ are arbitrary transformations.",
"This hidden layer may be due to some unknown dynamics, stochasticity, etc.",
"For example, consider the task of using the number of people at the beach to infer the weather.",
"We may know how to predict the weather (initial layer), and we may separately have an empirical count of beach-goers (observation), but the transformation from the weather to how people decide when to visit the beach remains hidden.",
"Because we only care about making estimates of the weather, we would prefer to not have to model people's decision making processes explicitly.",
"Frequently, we may only have the observations – and not even know the initial layer.",
"For example, we may wonder if we can count people at the beach to estimate the weather at different locations.",
"More generally, we may attempt some sort of model identification given some limited sensor activity.",
"To be successful, we need to `see through' the hidden layers to directly fit the parameters of the initial layer.",
"Many techniques exist to do this.",
"Here, we present an information theoretic technique based on mutual information that allows us to non-parametrically conduct fitting under weak assumptions about the dynamics of the hidden layer."
],
[
"Mutual Information",
"Mutual information ($MI$ ) quantifies how many bits of entropy we may reduce our uncertainty in $X$ by knowing another variable $Y$ and vice-versa.",
"For example, how much information about an individual's performance ($X$ ) do we obtain by knowing the amount of cognitive resources available to him or her ($Y$ )?",
"$MI$ is defined as $MI(X,Y) = H(X) + H(Y) - H(XY)$ , where $H(X)$ is the entropy of the random variable $X$ , and $H(XY)$ is the entropy of the joint distribution of $X$ and $Y$ .",
"In the present work, we leverage the property that $MI(X,Y) = MI(f(X),g(Y))$ for invertible functions $f$ and $g$ [7], so the underlying initial layer of the nonlinear system may be fit from the observations as long as the two can be related via an isomorphism.",
"If the relations between $X$ and $Y$ are not isomorphic, we may still utilize mutual information, but we will only be able to explain limited portions of the dynamics.",
"In contrast, optimizing a regression model or Pearson correlation for parameter estimation requires not only correctly modeling the initial layer but also knowing how those resources quantitatively translate into the observations, because $R^2$ is only maximized when predicted and observed results have an affine relation.",
"For example, in Fig.",
"REF , we demonstrate how the mutual information between two variables remains unchanged under isomorphisms.",
"However, the correlation between the transformed variables may go from positive to zero to negative, depending on the transformation.",
"Only the cosine transformation, which is not an isomorphism, shows a decrease in mutual information.",
"This ability to quantitatively relate the explanatory power of two variables, regardless of the isomorphic transformation between them, allows this method to relate the dynamics of the initial layer with the observation in the final layer.",
"In addition, mutual information is well defined for discrete, continuous, or categorical data – or some combination thereof.",
"In this work, we present a method for using mutual information to fit dynamics of the initial layer to discrete observations, without having to know the intervening hidden layer.",
"Figure: Mutual information does not vary under invertible transformations, while correlation is only maximize with a perfect linear relationship."
],
[
"KL Divergence",
"In addition to mutual information, we tested the Kullback-Leibler (KL) divergence as an objective function.",
"KL divergence is defined as the amount of information lost when one distribution is approximated by a second; it can thus be thought of as a distance between the two distributions, although it is not symmetric.",
"The KL divergence of continuous random variables from $Q$ to $P$ is given by $D_{KL}(P||Q) = \\int _{-\\infty }^{\\infty }p(x)\\ln \\frac{p(x)}{q(x)} dx$ When testing KL divergence as an objective function, we considered the distribution of observations against our initial model under a variety of comparisons.",
"First, we considered the divergence from the fitted model to the resource values used to generate the outcome data, given the outcome.",
"In this case, the objective is a loss function, as the goal is for the model to closely approximate reality.",
"Finally, we considered the divergence from the distribution of resource values given one outcome to the distribution given the other outcome; when this divergence is maximized, it forces the distributions to be as different as possible, which should increase the explanatory power of the model."
],
[
"Example with Mutual Information",
"As an example and proof of concept of our fitting with mutual information, we construct a family of deep nonlinear models: $x &= e^{-\\lambda t} \\\\y &= x +y_ \\text{err}\\\\z &= f(y),$ with $y_ \\text{err}$ being Gaussian noise and $z = f(y)$ having one of three functional forms: $f_1(y)& = a y\\\\f_2(y) &= e^{ay}\\\\f_3(y) &= \\sin (a y).$ In each case, we wish to fit $\\lambda $ to the generated data given the time series of $x$ and $z$ .",
"To generate the data, we chose a value of $\\lambda $ and a range for $t$ .",
"With a fixed $\\lambda $ and a given $t$ , we can then easily determine $f_1$ , $f_2$ , and $f_3$ .",
"To evaluate if a maximum in mutual information occurs with the true $\\lambda $ , we simply plotted the $MI$ between data generated from $\\lambda $ and different values of $x$ generated from a series of values of guesses $\\hat{\\lambda }$ : we varied $\\hat{\\lambda }$ in increments of 0.01 in the range (-1, 4) and calculated the mutual information $MI(z, e^{-\\hat{\\lambda } x})$ .",
"This entire process was repeated for several values of $\\lambda $ and $a$ .",
"Because of the highly deterministic relationship between $x$ , $y$ and $z$ , we estimated mutual information using the Non-parametric Entropy Estimation Toolbox (NPEET) with Local Non-uniformity Correction (LNC)https://github.com/BiuBiuBiLL/NPEET_LNC [4], [6].",
"This produced far more robust, less biased estimates of mutual information, by correcting for systematic errors that can emerge for deterministic data for the Kraskov estimator.",
"Fitting the example model using mutual information correctly produced a maximum of the objective function at or very close to the true value for $\\lambda $ (Fig.",
"REF ).",
"In particular, Fig.",
"REF shows a sharp maximum at the true value of $\\lambda = 2$ for all forms of $z$ .",
"While the mutual information curves for the exponential and sinusoidal $z$ in Figs.",
"REF and REF show smoother maxima that are not perfectly at the true value, they provide a fairly accurate estimate of the true value.",
"The dip in mutual information to 0 at $\\lambda = 0$ is due to $y$ taking the form $e^{0x}=1$ ; in practice, this would also prevent a local optimizer from transitioning over regimes when a particular parameter completely decouples the hidden model from the observations.",
"It would require a more global optimization strategy.",
"Figure: a=3,λ=2a=3,\\lambda =2"
],
[
"Application: Cognitive Performance",
"A particular system we consider in the present work is cognitive performance over time.",
"Previous work established a model for cognitive performance based on lactate metabolism [3], [15], [16].",
"In this model, performance is determined by a primary resource, $A$ , which is low during normal resting conditions.",
"Work on a task consumes $A$ .",
"However, $A$ is replenished by conversion of a secondary resource, $B$ , into $A$ ; $B$ then recovers during resting conditions.",
"These dynamics are modeled as: $\\dot{A}(t) &= w(A,B,t) - \\frac{k_w}{t^\\rho }\\frac{A(t)}{K_A + A(t)}\\\\\\dot{B}(t) &= -w(A,B,t) + \\frac{k_r}{t^\\rho }\\left(B_{max} - B(t)\\right)(1 - \\delta (t))\\\\w(A,B,t) &= \\frac{k_b}{t^\\rho }\\frac{\\left(1 - A\\left(t\\right)\\right)B(t)}{K_B + B(t)} \\delta (t),$ where $w$ is the rate of conversion of $B$ into $A$ and $\\delta (t)$ is given by 1 when on task (working) and 0 when off task (resting).",
"Although this model may be difficult to understand outside the context of modeling cognitive performance, it captures many of the properties that make fitting difficult for many methods.",
"Clearly, this model is complex, nonlinear and time-heterogeneous.",
"In addition, it is non-differentiable at the boundaries between tasks.",
"Furthermore, although this may be a proposed model of cognitive resources, we have no hypothesis for how cognitive resources translate into correct/incorrect answers on a standardized test.",
"All we can assume is that fewer resources decreases the likelihood of correct answers.",
"Thus, to fit the model, we need to be ambivalent to the intervening hidden layer."
],
[
"Generated Performance Data",
"To test fitting the model, we generated data similar to data from online practice standardized tests, obtained from the Kaggle “What do you know\" competition$\\endcsname $http://www.kaggle.com/c/WhatDoYouKnow and provided by grockit.com.",
"Problem times were generated stochastically from an exponential distribution, $p(x) = \\lambda e^{-\\lambda x},$ with $x$ the time spent on or off task, in minutes.",
"For time on task, we set $\\lambda = \\frac{1}{4}$ ; for time off task, $\\lambda = \\frac{1}{4}$ , except for every tenth task, when $\\lambda = \\frac{1}{40}$ .",
"Our fictional participant thus spent an average of four minutes on or off each task, with a longer break every ten tasks.",
"To generate the dataset, 25 time series with 3000 event times were generated in this fashion.",
"From the generated time series, we used a numerical ODE solver from the scipy.integrate library to get the primary and secondary resources from the model using the previously obtained parameters.",
"Outcome of a task was determined randomly based on the primary resource, $A$ , at the end of each question; the probability of success was given by the logistic sigmoid function, $P(\\text{correct}|A) = (1+e^{-\\alpha (A-A_0)})^{-1},$ with $\\alpha $ and $A_0$ chosen as 169 and 0.204, respectively, such that typical high values of $A$ would give a success rate of 70% and typical low values whould give a success rate of 30%.",
"Using this generated data, we sought to refit the resource model to determine whether our fitting methods are appropriate.",
"To start, we fit each parameter individually to get a sense for the shape of the objective function.",
"To fit a parameter, we varied around the true value and solved the model ODEs for the resources at each time.",
"We then used the newly calculated model resources and the outcome data to calculate the objective function at each parameter step."
],
[
"Objective Functions",
"To construct an objective function using mutual information, we estimated mixed continuous-discrete mutual information using the implementation based on Kraskov et.",
"al.",
"and Kozachenko and Leonenko in NPEET$\\endcsname $http://www.isi.edu/gregv/npeet.html [6], [5], .",
"We calculated the mutual information of the series of task outcomes (with 1 standing for correct/success and 0 for incorrect/failure) with the primary resource at the end of each task.",
"KL divergence was also estimated using the implementation in NPEET [14], [5].",
"We used two methods for using KL divergence as an objective function.",
"In both cases, we considered the frequency distribution of the primary resource at the end of a task given the outcome.",
"The first use of KL divergence relied upon prior knowledge of the underlying resources used to generate the data.",
"We again used the distribution of fitted resources given outcome, but now compared to the distribution of the underlying resources: $D_{KL}\\left(p(A_{\\text{fitted}}|\\text{correct}) || p(A_\\text{underlying}|\\text{correct})\\right)$ and $D_{KL}\\left(p(A_\\text{fitted}|\\text{incorrect}) || p(A_\\text{underlying}|\\text{incorrect})\\right).$ In the second method, we compared the frequency distribution of the primary resource given a successful outcome with that given an unsuccessful outcome: $D_{KL}(p(A|\\text{correct}) || p(A|\\text{incorrect}))$ and $D_{KL}(p(A|\\text{incorrect}) || p(A|\\text{correct})).$ By maximizing either of these divergences or a combination thereof, the model should be as distinct as possible within a regime of cognitive depletion compared to a regime of plentiful cognitive resources."
],
[
"Optimizers",
"Because mutual information does not require the random variables to be positively correlated for it to be maximized, using it as a metric can result in model resource values that are lower on average for successful outcomes than for unsuccessful outcomes.",
"While this could be interpreted as the model representing waste products instead of resources, it goes against the assumptions that base the model on lactate metabolism.",
"Thus, we added a constraint that the average resource given success be greater than that given failure: $f_c = \\langle A|\\text{correct}\\rangle - \\langle A|\\text{incorrect}\\rangle \\ge 0,$ where $A$ is the primary resource modeled in Eq.",
"REF .",
"We ran tests on optimizing all parameters together over the mutual information objective using constrained optimizers implemented in NLOPTSteven G. Johnson, The NLopt nonlinear-optimization package, http://ab-initio.mit.edu/nlopt.",
"However, the optimizers here posed problems for this task.",
"Because the objective function requires solving the model ODEs for the time series and parameters, it takes a long time to evaluate.",
"Additionally, because the Kraskov mutual information is non-differentiable, the gradient is unknown and must be estimated; therefore, each iteration of the optimizers required multiple evaluations of a computationally expensive objective.",
"Thus, we use Simultaneouse Perturbation Stochastic Approximation (SPSA).",
"SPSA lends itself to objective functions requiring a simulation or other computationally expensive task by requiring only two calls of the objective function to estimate the gradient[11], [13]."
],
[
"Discussion",
"When fitting one parameter from the true value and holding the rest constant, mutual information as an objective function produced a maximum at or near the correct parameter values for $k_w$ , $k_r$ , and $B_{max}$ ; the landscapes for $\\rho $ and $k_b$ featured a plateau-like range of values at the respective maxima (Fig.",
"REF ).",
"Figure: Mutual information of primary resource with task outcomes.",
"True parameters are given by dashed lines.KL divergence comparing distribution of resource values to an a priori distribution produced a minimum at the true parameter value for most of the parameters.",
"However, $\\rho $ did not have its minimum near the true value, but at a greater value.",
"Similarly to the mutual information analysis, $k_b$ had an optimum value for a range of values with the true value near the lower end (Fig.",
"REF ).",
"Figure: KL divergence of frequency distributions of resource from the fitted model with that of the original hidden resources.",
"True parameters are given by the vertical lines.KL divergence comparing the distributions of resource values given outcome fails to produce a maximum at the true parameter; in fact, it produces a local minimum near the true value for $k_w$ , $k_r$ , and $B_{max}$ .",
"The parameter $k_b$ was the only parameter to produce a maximum at the true value, but similarly to its behavior with other objective functions, it also produced optima at larger values (Fig.",
"REF ).",
"Figure: KL divergence of frequency distributions of resource from the fitted model with that of the original hidden resources.",
"True parameters are given by the vertical lines.While tests with KL divergence comparing the model to the original resource values produced the clearest optimum at the true parameter values, it poses a problem that the distribution of resource values at the end of each task must be known a priori.",
"In the case of the generated data, the distribution of resource values was found to be Gaussian, allowing the divergence to be approximated using an approximation for the true distribution; however, even if the functional form of the true distribution is known a priori, its parameters may not be.",
"On the other hand, using KL divergence as an objective function to encourage disjointness of resource probability distributions given outcome failed to recover the underlying parameters of the model.",
"It is worth noting that both successful fitting methods have poor sensitivity for fitting $\\rho $ .",
"The unusual behavior in the fitting of $\\rho $ and $k_b$ seems to indicate that they are less important to the overall fit of the model, except when $k_b$ is too low.",
"This may indicate that establishing a lower bound may be more important when fitting this parameter."
],
[
"Conclusion",
"In this work we investigated fitting the hidden layer of deep nonlinear models using mutual information and KL divergence.",
"Although this was a preliminary investigation, we demonstrated the potential value of using mutual information to fit deep nonlinear models.",
"Although it may be sensitive to biases inherent in the underlying mutual information estimators, the property that mutual information is invariant under isomorphic transformations allows it to estimate the sensitivity of an observation to a model when there is an unknown hidden layer between them.",
"When using KL divergence to compare a model to prior-known statistics of the hidden variables, true parameter values are similarly recovered.",
"However, using KL divergence to maximize explainability by maximizing the divergence between frequency distributions of the hidden layer given the observed layer completely fails in recovering the original parameter values.",
"Future work will further explore the formal relationship between mutual information and variations in parameters in the initial, hidden, and observation layers of deep nonlinear models.",
"The present work provides a promising proof of principle in simple and complex models.",
"The applications for this work include cognitive science, system identification, and machine learning."
],
[
"Acknowledgments",
"This work was funded by the Analysis in Motion Initiative at Pacific Northwest National Laboratory."
]
] | 1612.05708 |
[
[
"Truncated linear statistics associated with the eigenvalues of random\n matrices II. Partial sums over proper time delays for chaotic quantum dots"
],
[
"Abstract Invariant ensembles of random matrices are characterized by the distribution of their eigenvalues $\\{\\lambda_1,\\cdots,\\lambda_N\\}$.",
"We study the distribution of truncated linear statistics of the form $\\tilde{L}=\\sum_{i=1}^p f(\\lambda_i)$ with $p<N$.",
"This problem has been considered by us in a previous paper when the $p$ eigenvalues are further constrained to be the largest ones (or the smallest).",
"In this second paper we consider the same problem without this restriction which leads to a rather different analysis.",
"We introduce a new ensemble which is related, but not equivalent, to the \"thinned ensembles\" introduced by Bohigas and Pato.",
"This question is motivated by the study of partial sums of proper time delays in chaotic quantum dots, which are characteristic times of the scattering process.",
"Using the Coulomb gas technique, we derive the large deviation function for $\\tilde{L}$.",
"Large deviations of linear statistics $L=\\sum_{i=1}^N f(\\lambda_i)$ are usually dominated by the energy of the Coulomb gas, which scales as $\\sim N^2$, implying that the relative fluctuations are of order $1/N$.",
"For the truncated linear statistics considered here, there is a whole region (including the typical fluctuations region), where the energy of the Coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect.",
"Because the entropy scales as $\\sim N$, the relative fluctuations are of order $1/\\sqrt{N}$.",
"Our analysis relies on the mapping on a problem of $p$ fictitious non-interacting fermions in $N$ energy levels, which can exhibit both positive and negative effective (absolute) temperatures.",
"We determine the large deviation function characterizing the distribution of the truncated linear statistics, and show that, for the case considered here ($f(\\lambda)=1/\\lambda$), the corresponding phase diagram is separated in three different phases."
],
[
"Introduction",
"The study of linear statistics of eigenvalues of random matrices has played a major role for the applications of random matrix theory to physical problems, like quantum transport [1], [29], [39], [54], [55], quantum entanglement [18], [41], [42], etc, as it was emphasized in our recent papers [25], [28].",
"Whereas previous work has considered unrestricted sums of the form $L=\\sum _{i=1}^Nf(\\lambda _i)$ , where $\\lbrace \\lambda _1,\\cdots ,\\lambda _N\\rbrace $ is the spectrum of eigenvalues of a random matrix, the question of truncated linear statistics (TLS) of the form $\\tilde{L}=\\sum _{i=1}^pf(\\lambda _i)\\hspace{14.22636pt}\\mbox{with}\\hspace{14.22636pt}p<N$ was recently raised in our previous paper [25].",
"There, we have considered the case where the eigenvalues contributing to the partial sum are further constrained to be the $p$ largest (or smallest) ones, which corresponds to interpolating between two well studied types of problems : for $p=N$ , the statistical analysis of the linear statistics $L=\\sum _{i=1}^Nf(\\lambda _i)$ [1], [13], [27], [28], [29], [41], [42], [48], [54], [55] (and many other references), and for $p=1$ , the study of the distribution of the largest (or smallest) eigenvalue [4], [19], [20], [36], [35], [34], [50], [51], [53].",
"The aim of the present paper is to analyse the statistical properties of TLS (REF ) in the absence of further restriction concerning the ordering of the $p<N$ eigenvalues ; we will see that lifting this restriction has strong implications on the analysis and the results.",
"A related question was introduced by Bohigas and Pato [3] who studied the effect of removing randomly chosen eigenvalues in an invariant matrix ensemble, leading to consider a fraction of the initial eigenvalues.",
"Initially motivated to model the transition from the random matrix spectrum (correlated eigenvalues) to the Poisson spectrum (uncorrelated eigenvalues), this question has shown a renewed interest recently, under the name of “thinned ensembles” [2], [9], [32].",
"Here we study the linear statistics in an ensemble, which is similar to, but different from the thinned ensembles.",
"The relation between the two ensembles is discussed below, see Eqs.",
"(REF ,REF ,).",
"For convenience, we rescale the TLS as $s = N^{-\\eta } \\sum _{i=1}^{p} f(\\lambda _i)\\:,$ where the exponent $\\eta $ is chosen such that $s$ remains of order $N^0$ as $N \\rightarrow \\infty $ .",
"The precise value of $\\eta $ depends on the function $f$ and the matrix ensemble under consideration.",
"The general scenario presented in the paper is valid for any choice of function $f$ .",
"Moreover it applies to different matrix ensembles, although we will focus on Wishart matrices (i.e.",
"the Laguerre ensemble of random matrices).",
"In the sequel, for reasons explained later, we will consider the case $f(\\lambda )=1/\\lambda $ , leading to $\\eta =0$ .",
"Denoting by $P_N(\\lambda _1, \\cdots , \\lambda _N)$ the joint probability distribution function for the eigenvalues (for the Laguerre ensemble, Eq.",
"(REF ) below), we can write the distribution of $s$ as : $P_{N,\\kappa }(s) = \\int {\\rm d}\\lambda _1 \\int {\\rm d}\\lambda _2 \\cdots \\int {\\rm d}\\lambda _N \\, P_N(\\lambda _1, \\cdots , \\lambda _N)\\:\\delta \\bigg (s - N^{-\\eta } \\sum _{i=1}^{p} f(\\lambda _i)\\bigg )\\:.$ Here, $P_N(\\lambda _1, \\cdots , \\lambda _N)$ is a symmetric function of its $N$ arguments.",
"Due to the restriction to a fraction of the eigenvalues in (REF ), the only avalaible approaches seem to be the orthogonal polynomial technique, which provides naturally the $p$ -point correlation functions, and the Coulomb gas method.",
"However the former is restricted to the unitary class and does not permit a simple analysis of the large deviations.",
"For this reason, we will use the Coulomb gas technique.",
"Because the method deals with the eigenvalue density, the restriction to a fraction of eigenvalues requires to introduce “occupation numbers” $\\lbrace n_i\\rbrace _{i=1,\\cdots ,N}$ and write the TLS as : $s = N^{-\\eta } \\sum _{i=1}^{N} n_i f(\\lambda _i)\\:,\\hspace{14.22636pt}\\lambda _1 > \\lambda _2 > \\cdots > \\lambda _N\\:,$ where $n_i=1$ if the eigenvalue $\\lambda _i$ contributes to the sum and $n_i = 0$ otherwise.",
"Note that subset sums $\\sum _in_i\\lambda _i$ of eigenvalues of certain random matrices recently appeared in the spectral analysis of fermionic systems [14].",
"Since the sum should contain $p$ eigenvalues, the $N!/\\big [p!",
"(N-p)!\\big ]$ acceptable configurations $\\lbrace n_i \\rbrace $ verify $\\sum _{i=1}^{N} n_i = p\\:.$ Recently, the case where the summation is restricted to the $p$ largest (or smallest) eigenvalues was considered in Ref. [25].",
"This corresponds to considering the configuration $n_1 = \\cdots = n_{p} = 1$ and $n_{p+1} = \\cdots = n_{N} = 0$ .",
"Here, we consider the same problem in the absence of this restriction, which leads us to introduce a new ensemble described by the joint distribution for the two sets of random numbers $\\boxed{{P}_{N,p}(\\lbrace \\lambda _i\\rbrace ,\\lbrace n_i\\rbrace )= p!",
"(N-p)!\\, P_N(\\lambda _1, \\cdots , \\lambda _N)\\,\\mathbf {1}_{\\lambda _1>\\cdots >\\lambda _N}\\,\\delta _{p,\\sum _in_i}}$ where $\\mathbf {1}_{\\lambda _1>\\cdots >\\lambda _N}=\\prod _{i=1}^{N-1}\\mathop {\\theta _\\mathrm {H}}\\nolimits (\\lambda _{i}-\\lambda _{i+1})$ and $\\mathop {\\theta _\\mathrm {H}}\\nolimits (x)$ the Heaviside step function.",
"Eq.",
"(REF ) is obviously normalised when integrated over all $\\lambda _i$ 's and with summation over the occupations.",
"We can now establish the precise relation with the distribution describing the “thinned” random matrix ensembles [2], [3], [9], [32], which is obtained by relaxing the constraint on the number $p$ .",
"Denoting $B_N(p)=\\begin{pmatrix}N\\\\p\\end{pmatrix}\\kappa ^p(1-\\kappa )^{N-p}$ the binomial distribution, i.e.",
"the probability to select $p$ eigenvalues among the $N$ , where $\\kappa $ is the probability to select an eigenvalue, we have ${P}_{N,\\kappa }^\\mathrm {(thinned)}(\\lbrace \\lambda _i\\rbrace ,\\lbrace n_i\\rbrace )&=\\sum _{p=0}^NB_N(p)\\,{P}_{N,p}(\\lbrace \\lambda _i\\rbrace ,\\lbrace n_i\\rbrace )\\\\&\\hspace{-35.56593pt}= N!\\, P_N(\\lambda _1, \\cdots , \\lambda _N)\\,\\mathbf {1}_{\\lambda _1>\\cdots >\\lambda _N}\\,\\kappa ^{\\sum _in_i} \\, (1-\\kappa )^{N-\\sum _in_i}\\:,$ ($p$ then fluctuates, its average being $\\overline{p}=\\kappa \\,N$ ).",
"The distribution we aim to determine in the paper can be written in terms of (REF ) as $\\nonumber P_{N,\\kappa }(s) = \\sum _{ \\lbrace n_i \\rbrace }\\int {\\rm d}\\lambda _1 \\int ^{\\lambda _1} {\\rm d}\\lambda _2& \\cdots \\int ^{\\lambda _{N-1}} {\\rm d}\\lambda _N \\,{P}_{N,p}(\\lbrace \\lambda _i\\rbrace ,\\lbrace n_i\\rbrace )\\\\ \\times & \\delta \\bigg (s - N^{-\\eta } \\sum _{i=1}^{N} n_i f(\\lambda _i)\\bigg )\\:.",
"$ where $\\kappa =p/N$ in the subscript of $P_{N,\\kappa }(s)$ .",
"Figure: Phase diagram in the (κ,s)(\\kappa ,s) plane,where κ=p/N\\kappa =p/N and s=N -η ∑ i=1 p f(λ i )s=N^{-\\eta }\\sum _{i=1}^{p} f(\\lambda _i)(here for the case f(λ)=1/λf(\\lambda )= 1/\\lambda in the Laguerre ensemble, thus with η=0\\eta =0).Insets show the corresponding optimal “charge” density profiles.Shaded part of the density (insets on the right) shows the fraction of (largest or smallest) pp eigenvalues contributing to ss.Dashed profiles (Phases II and III) mean fluctuations in the “occupations” {n i }\\lbrace n_i\\rbrace (cf.",
"text).In the central part of the diagram (Phase II),the distribution is dominated by entropy,while in Phases I and III it is dominated by the energy.Considering the Laguerre ensemble, Eq.",
"(REF ) below, with $f(\\lambda )=1/\\lambda $ , we have obtained the distribution $P_{N,\\kappa }(s)$ in the limit of large matrix size, $N\\rightarrow \\infty $ , when both $\\kappa $ and $1-\\kappa $ are of order one ; in particular this excludes the case $\\kappa \\sim 1/N$ which is not covered by our approach (the limit $\\kappa \\rightarrow 1$ was also identified as a singular limit in Ref. [25]).",
"In this limit the multiple integral (REF ) is dominated by the optimal configuration for the two sets of random numbers.",
"As the two parameters $\\kappa $ and $s$ are tuned, we obtain different types of optimal configurations, which are interpreted as different “phases”.",
"The phase diagram can be drawn in the $(\\kappa ,s)$ plane, where the physical available phases are located in the stripe $[0,1]\\times [0,\\infty [$ .",
"As one increases $s$ for fixed $\\kappa $ , one encounters three phases described below, the phase transitions taking places at $s=s_0(\\kappa )$ and $s=s_1(\\kappa )$ , which defines two lines in the $(\\kappa ,s)$ plane (see also Fig.",
"REF ) : Phase I : the optimal configuration $\\lbrace \\lambda _i \\rbrace $ , that dominates the integrals (REF ), varies with $(\\kappa ,s)$ while the occupations $\\lbrace n_i \\rbrace $ remain frozen.",
"This case is conveniently analysed within the conventional Coulomb gas interpretation where the eigenvalues are considered as the positions of a one-dimensional gas of $N$ particles with logarithmic interactions.",
"Phase II : $\\lbrace \\lambda _i \\rbrace $ are frozen while $\\lbrace n_i \\rbrace $ fluctuate.",
"To get some insight, we interpret the problem as a gas of $p$ fictitious fermions occupying $N$ fixed “energy levels” $\\varepsilon _i=f(\\lambda _i)$ 's.",
"This phase includes the typical fluctuations (the dashed line in Fig.",
"REF indicates where $P_{N,\\kappa }(s)$ is maximum).",
"Phase III : the last phase is described in terms of a mixed picture where one single eigenvalue splits off the bulk, while the density of the remaining eigenvalues is frozen and the occupation numbers fluctuate (they freeze as $s\\rightarrow \\infty $ ).",
"The different phases are related to the following behaviours for the distribution of the TLS Expressions of the type $P_{N,\\kappa }(s) \\underset{N\\rightarrow \\infty }{\\sim } \\exp (-N^q \\Phi )$ must be understood as $\\lim _{N \\rightarrow \\infty } (-1/N^q) \\ln P_{N,\\kappa }(s) = \\Phi $ .",
": $&P_{N,\\kappa }(s)\\\\\\nonumber &\\underset{N \\rightarrow \\infty }{\\sim }{\\left\\lbrace \\begin{array}{ll}\\mathrm {e}^{- N\\,C_\\kappa }\\exp \\left\\lbrace - \\frac{\\beta N^2}{2} \\Phi _-(\\kappa ;s) \\right\\rbrace & \\text{for } s < s_0(\\kappa )\\\\[0.2cm]\\exp \\left\\lbrace - N \\Phi _0(\\kappa ;s) \\right\\rbrace \\hspace{14.22636pt}& \\text{for } s_0(\\kappa )<s<s_1(\\kappa )\\\\[0.2cm]N^{-\\frac{\\beta N}{2}}\\exp \\left\\lbrace -N\\left[ \\frac{\\beta }{2}\\Phi _+(\\kappa ;s) + \\Phi _0(\\kappa ; \\tilde{s}(s)) \\right] \\right\\rbrace & \\text{for } s > s_1(\\kappa )\\end{array}\\right.",
"}$ where $\\tilde{s}(s)$ smoothly interpolates between $s_1(\\kappa )$ and $\\kappa $ (it will be studied in Section ).",
"The large deviation function in the last expression combines two different functions with different origins (energy versus entropy).",
"A sketch of the distribution is represented in Fig.",
"REF .",
"The different scalings with $N$ in the exponential arise from the fact that entropy $\\sim N$ dominates in Phase II, including the typical fluctuations, while energy $\\sim N^2$ dominates in Phase I (the scaling for Phase III is due to a subleading contribution to the energy).",
"The scaling in Phase II implies in particular that the fluctuations of the TLS scale as $\\sim 1/\\sqrt{N}$ (and are independent of the symmetry index $\\beta $ ).",
"The limiting behaviours of the large deviation functions for the three regimes are : $&\\Phi _-(\\kappa ;s) \\simeq {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{\\kappa ^2}{s} + \\frac{3 \\kappa (2-\\kappa )}{2} \\ln s+ \\mathrm {cste}& \\text{as } s \\rightarrow 0\\\\[0.2cm]\\displaystyle \\omega _\\kappa \\, (s - s_0(\\kappa ))^2& \\text{as } s \\rightarrow s_0(\\kappa )^-\\end{array}\\right.",
"}\\\\&\\Phi _0(\\kappa ;s) \\simeq {\\left\\lbrace \\begin{array}{ll}C_{\\kappa }- c_+\\sqrt{s - s_{0}(\\kappa ) }& \\text{as } s \\rightarrow s_0(\\kappa )^+\\\\[0.2cm]\\displaystyle \\frac{(s-\\kappa )^2}{2 \\kappa (1-\\kappa )}& \\text{as } s \\rightarrow \\kappa \\\\[0.2cm]C_{\\kappa }- \\tilde{c}_+\\sqrt{ s_{1}(\\kappa ) - s }& \\text{as } s \\rightarrow s_1(\\kappa )^-\\end{array}\\right.",
"}\\\\&\\Phi _+(\\kappa ;s) + \\frac{2}{\\beta } \\Phi _0(\\kappa ;\\tilde{s}(s)) \\simeq {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\ln (s - s_1(\\kappa ) ) + \\mathrm {cste}& \\text{as } s \\rightarrow s_1(\\kappa )^+\\\\[0.2cm]\\displaystyle \\ln (s - \\kappa )+ \\mathrm {cste}& \\text{as } s \\rightarrow +\\infty \\end{array}\\right.",
"}$ where $\\omega _\\kappa $ , $C_{\\kappa }=-\\kappa \\ln \\kappa - (1-\\kappa ) \\ln (1-\\kappa )$ , $c_+$ and $\\tilde{c}_+$ are positive constants.",
"Figure: Sketch of the distribution P N,κ (s)P_{N,\\kappa }(s) obtained in the paper, Eq.",
"()."
],
[
"Plan",
"Section exposes the motivations which have led us to the study of the distribution of the TLS for $f(\\lambda )=1/\\lambda $ .",
"In Section , we will describe Phase I, which will be analysed within the usual Coulomb gas picture.",
"We will introduce the notations.",
"The derivation of the results has been made as short as possible, as the problem is similar to the one studied in our previous article [25], although the case considered here is more complicated (in Ref.",
"[25], we have made use of simplifications arising from the specific choice $f(\\lambda )=\\sqrt{\\lambda }$ ).",
"Section will discuss Phase II, analysed within the picture of fictitious non interacting fermions.",
"Phase III is studied in Section .",
"The paper is closed by some concluding remarks."
],
[
"Chaotic scattering in quantum dots and random matrices",
"The problem studied in this article is motivated by the statistical analysis of partial sums of proper time delays in chaotic scattering.",
"Proper time delays are characteristic times capturing temporal aspects of quantum scattering (for a review on time delays, cf.",
"[17] ; aspects related to chaotic scattering are reviewed in [23], [30], [31], [47]).",
"For the sake of concreteness, let us consider a quantum dot (QD) connected to external contacts playing the role of wave guides through which electrons can be injected (Fig.",
"REF ).",
"An electronic wave of energy $\\varepsilon $ injected in the conducting channel $a$ is scattered in channel $b$ with probability amplitude given by the matrix element $\\mathcal {S}_{ba}(\\varepsilon )$ of the $N\\times N$ on-shell scattering matrix $\\mathcal {S}(\\varepsilon )$ .",
"Probing the energy structure of $\\mathcal {S}(\\varepsilon )$ allows one to define several characteristic times.",
"Ref.",
"[47] has emphasized the difference between the partial time delays $\\lbrace \\tilde{\\tau }_i\\rbrace $ , which are derivatives of the scattering phase shifts, and the proper time delays $\\lbrace \\tau _i\\rbrace $ , which are eigenvalues of the Wigner-Smith matrix $\\mathcal {Q}=-{\\rm i}\\mathcal {S}^\\dagger \\partial _\\varepsilon \\mathcal {S}\\:.$ In other words, these two sets of characteristic times are obtained by performing derivation and diagonalisation in different orders.",
"They however satisfy the sum rule $\\sum _i\\tilde{\\tau }_i=\\sum _i\\tau _i$ .",
"Figure: A chaotic quantum dot with four perfect contacts.",
"Contact α\\alpha has N α N_\\alpha open channels.The Wigner-Smith matrix is a central concept allowing to characterize the amount of injected charge in the QD, which can be understood from Krein-Friedel type formulae relating the scattering matrix to the density of states.",
"This observation has played a major role in Büttiker's quantum transport theory for coherent devices beyond DC linear transport : non-linear transport [10], AC transport [8], [11], [27], pumping, frequency dependent noise, etc (see [47] for a review and further references).",
"An important observation is that, in order to address the out-of-equilibrium situation, the contribution of the electrons from a given contact must be identified.",
"This has led to the concept of injectance, $\\overline{\\nu }_\\alpha =(2\\pi )^{-1}\\sum _{i\\in \\mathrm {contact}\\:\\alpha }\\mathcal {Q}_{ii}$ , measuring the contribution to the density of states (DoS) inside the device of the scattering states incoming from the channels associated with the contact $\\alpha $ ; hence the total DoS is $\\sum _\\alpha \\overline{\\nu }_\\alpha =(2\\pi )^{-1} \\mathop {\\mathrm {tr}}\\nolimits \\left\\lbrace \\mathcal {Q} \\right\\rbrace $ , which is related to the Wigner time delay : $\\tau _W = \\frac{1}{N}\\sum _i \\mathcal {Q}_{ii} = \\frac{1}{N}\\sum _i\\tau _i\\:.$ The symmetric concept of emittance, $\\underline{\\nu }_\\alpha =(2\\pi )^{-1}\\sum _{i\\in \\mathrm {contact}\\:\\alpha }[\\mathcal {S}\\mathcal {Q}\\mathcal {S}^\\dagger ]_{ii}$ , was also introduced in order to allow for a postselection of the contact, whereas the injectance $\\overline{\\nu }_\\alpha $ corresponds to the preselection (see Ref.",
"[24] and the review [47]).",
"In systems with complex dynamics, such as chaotic quantum dots (Fig.",
"REF ), the statistical approach is the most efficient, which leads to assume that the scattering matrix $\\mathcal {S}$ belongs to a random matrix ensemble [1], [39].",
"In chaotic QDs with perfect contacts, it is quite natural to choose the uniform distribution over the unitary group, up to some additional constraints related to the symmetry (time reversal symmetry and/or spin rotation symmetry) : this corresponds to the so-called circular matrix ensembles.",
"This approach however does not provide any information on the energy structure of the scattering matrix, what is required in order to characterize the statistical properties of the matrix $\\mathcal {Q}$ .",
"An “alternative stochastic approach” has been introduced in [5] and applied in [6], [7] where the Wigner-Smith matrix distribution was shown to be related to the Laguerre ensemble of random matrices [6], [7]. Refs.",
"[6], [7] has introduced the symmetrized Wigner-Smith matrix $\\mathcal {Q}_s=-{\\rm i}\\mathcal {S}^{-1/2}\\,\\partial _\\varepsilon \\mathcal {S}\\,\\mathcal {S}^{-1/2}$ , with the same spectrum of eigenvalues than $\\mathcal {Q}$ .",
"The precise statement of these references is that $1/\\mathcal {Q}_s$ is a Wishart matrix.",
"Because $\\mathcal {S}$ and $\\mathcal {Q}$ are only independent in the unitary case, $1/\\mathcal {Q}$ is a Wishart matrix only in this case, strictly speaking.",
"However its eigenvalues are always given by the Laguerre distribution (REF ).",
"The joint distribution of the inverse of the proper times $\\lambda _i=\\tau _\\mathrm {H}/\\tau _i$ is $P_N(\\lambda _1, \\cdots , \\lambda _N) \\propto \\prod _{i<j} \\left| \\lambda _i - \\lambda _j \\right|^\\beta \\prod _{n=1}^N \\lambda _n^{\\beta N/2} \\mathrm {e}^{-\\beta \\lambda _n/2}\\:,$ where $\\beta \\in \\lbrace 1,\\,2,\\,4\\rbrace $ is the Dyson index corresponding to orthogonal, unitary and symplectic classes.",
"The time scale denotes the Heisenberg time $\\tau _\\mathrm {H}=2\\pi /\\Delta $ , where $\\Delta $ is the mean level spacing (or mean resonance spacing).",
"The DoS interpretation thus leads to $(2\\pi )^{-1}\\langle \\mathop {\\mathrm {tr}}\\nolimits \\left\\lbrace \\mathcal {Q} \\right\\rbrace \\rangle =1/\\Delta $ .",
"Combined with the sum rule (REF ), we have $\\langle \\tau _W\\rangle =\\langle \\mathcal {Q}_{ii}\\rangle =\\langle \\tau _i\\rangle = \\frac{\\tau _\\mathrm {H}}{N}\\:.$ This scale is also known as the dwell time and measures the average time spent by a wave packet inside the quantum dot.",
"Later we will set $\\tau _\\mathrm {H}=1$ for simplicity.",
"The variance and the correlations of proper times are also known (see updated arXiv version of [47] and references therein) [45] : $\\mathrm {Var}(\\tau _i) &= \\frac{N[\\beta (N-1)+2] + 2}{N^2(N+1)(\\beta N - 2)}\\:,\\\\\\mathrm {Cov}(\\tau _i, \\tau _j) &= - \\frac{1}{N^2(N+1)}\\hspace{28.45274pt}\\mbox{for } i\\ne j\\:.$ (other references on time delay correlations are [31], [43], [16], and can also be found in the review papers quoted above).",
"The joint distribution for the partial time delays $\\tilde{\\tau }_i$ 's and the diagonal matrix elements $\\mathcal {Q}_{ii}$ 's are still unknown.",
"An interesting connection between them was pointed out in [46] (see also [31]) : introducing the unitary matrix $\\mathcal {V}$ which diagonalizes the scattering matrix, these authors have obtained the non trivial relation $\\tilde{\\tau }_i=\\big [\\mathcal {V}^\\dagger \\mathcal {Q}\\mathcal {V}\\big ]_{ii}$ .",
"In the unitary case, $\\mathcal {V}$ is expected to be independent of $\\mathcal {Q}$ .",
"Because $\\mathcal {Q}$ is invariant under unitary transformations in this case, we have $\\tilde{\\tau }_i=\\mathcal {Q}_{ii}$ .",
"Let us emphasize few properties of the $\\mathcal {Q}_{ii}$ 's, which are important as they are involved in the injectances which should be accessible experimentally, and emphasize their difference with the proper times.",
"$\\mathcal {Q}_{ii}$ 's and $\\tau _i$ 's can be related by writing $\\mathcal {Q}=\\mathcal {U}\\,\\mathrm {diag}(\\tau _1,\\,\\cdots ,\\,\\tau _N)\\,\\mathcal {U}^\\dagger $ (the unitary matrices $\\mathcal {V}$ and $\\mathcal {U}$ differ in general), so that $\\mathcal {Q}_{ii} = \\sum _{j=1}^N \\left|\\mathcal {U}_{ij}\\right|^2\\tau _j\\:.$ In the unitary case ($\\beta =2$ ) the matrix $\\mathcal {U}$ and the eigenvalues are uncorrelated, which allows to go further in the statistical analysis of the $\\mathcal {Q}_{ii}$ 's (this is not the case in the orthogonal and symplectic cases [7]).",
"Thus we will focus on the unitary case for a moment, and will come back to the general case later.",
"Correlations of the $\\mathcal {U}$ matrix elements are controlled by Weingarten functions [12].",
"Some algebra gives We have used $\\int _{U(N)} {\\rm d}U\\,U_{i_1 j_1} U_{i_2 j_2} U^*_{k_1 l_1} U^*_{k_2 l_2}&= W(N,1^2)\\left(\\delta _{i_1 k_1} \\delta _{i_2 k_2} \\delta _{j_1 l_1} \\delta _{j_2 l_2}+ \\delta _{i_1 k_2} \\delta _{i_2 k_1} \\delta _{j_1 l_2} \\delta _{j_2 l_1}\\right)\\\\ \\nonumber & + W(N,2)\\left(\\delta _{i_1 k_1} \\delta _{i_2 k_2} \\delta _{j_1 l_2} \\delta _{j_2 l_1}+ \\delta _{i_1 k_2} \\delta _{i_2 k_1} \\delta _{j_1 l_1} \\delta _{j_2 l_2}\\right)\\:,$ where the Weingarten functions $W(N,\\sigma )$ is a function of the matrix size and the permutation of indices.",
"In particular : $W(N,1^2) = \\frac{1}{N^2-1}$ and $W(N,2) = -\\frac{1}{N(N^2-1)}$ .",
"$\\mathrm {Cov}\\left( \\mathcal {Q}_{ii} , \\mathcal {Q}_{jj} \\right) =\\frac{1}{N(N^2-1)}\\left(\\frac{1}{N} + \\delta _{i,j}\\right)\\:.$ In the particular case $i=j$ , this result coincides with the one first obtained in [23] (note that this reference has also obtained the variance for non ideal leads).",
"Eq.",
"(REF ) matches with the covariances of the partial times given in the updated arXiv version of [47], $\\mathrm {Cov}\\left( \\mathcal {Q}_{ii} , \\mathcal {Q}_{jj} \\right) = \\mathrm {Cov}(\\tilde{\\tau }_i, \\tilde{\\tau }_j)$ (see the discussion in Appendices A and E of [31]).",
"Both the scaling with $N$ and the sign of the correlations for $\\mathcal {Q}_{ii}=\\tilde{\\tau }_i$ 's and $\\tau _i$ 's differ : $\\mathrm {Var}\\big ( \\mathcal {Q}_{ii} \\big )\\simeq (1/N)\\,\\mathrm {Var}(\\tau _i)\\simeq 1/N^3$ and $\\mathrm {Cov}\\big ( \\mathcal {Q}_{ii} , \\mathcal {Q}_{jj} \\big )\\simeq -(1/N)\\,\\mathrm {Cov}(\\tau _i, \\tau _j)\\simeq +1/N^4$ , where the factor $1/N$ can be understood as the contribution of the unitary matrix elements in (REF )."
],
[
"Partial sums of $\\tau _i$ 's or {{formula:a412543d-fab5-4db6-aebe-2664f9ed28db}} 's : different scaling with {{formula:42ca89e0-5ca8-47ee-ac92-92461d33788e}}",
"These relations lead to important differences bewteen partial sums of $\\mathcal {Q}_{ii}$ 's and $\\tau _i$ 's.",
"Considering a contact with $p$ conducting channels, the two first cumulants of the related injectance are $\\Big \\langle \\sum _{i=1}^p \\mathcal {Q}_{ii} \\Big \\rangle &= \\frac{p}{N} \\equiv \\kappa \\\\\\mathrm {Var}\\Big ( \\sum _{i=1}^p \\mathcal {Q}_{ii} \\Big )&= \\frac{\\kappa (1+\\kappa )}{N^2-1}$ (the leading order to the second expression was given in [47], Eq.",
"85 of the arXiv version ; it was deduced from the results of [5]).",
"On the other hand, the fluctuations of the partial sum of proper time delays can be deduced from (REF ,) by writing $\\mathrm {Var}\\big ( \\sum _{i=1}^p \\tau _{i} \\big )=p\\, \\mathrm {Var}(\\tau _i) + p(p-1)\\, \\mathrm {Cov}(\\tau _i,\\tau _j)$ , explicitly $\\mathrm {Var}\\Big ( \\sum _{i=1}^p \\tau _{i} \\Big )= \\frac{\\kappa (1-\\kappa )}{N+1} + \\frac{2\\kappa }{N^2-1}\\:.$ Comparing (REF ) and (REF ) leads to two interesting observations : For $\\kappa <1$ , the variance of the injectance is $\\sim (1/N)$ smaller than the variance of partial sum of $\\tau _i$ 's : $\\mathrm {Var}\\big ( \\sum _{i=1}^p\\mathcal {Q}_{ii} \\big )\\sim (1/N)\\,\\mathrm {Var}\\big ( \\sum _{i=1}^p \\tau _{i} \\big )$ .",
"For $\\kappa =1$ , they are both equal to $\\mathrm {Var}\\big ( \\mathop {\\mathrm {tr}}\\nolimits \\left\\lbrace \\mathcal {Q} \\right\\rbrace \\big )=2/(N^2-1)$ as the two quantities satisfy the sum rule (REF ).",
"This is a consequence of the fact that $\\mathrm {Var}\\big ( \\sum _{i=1}^p \\tau _{i} \\big )$ drops by a factor $1/N$ as $\\kappa \\rightarrow 1$ .",
"In other words, $\\mathrm {Var}\\big ( \\sum _{i=1}^p \\tau _{i} \\big )/\\langle \\sum _{i=1}^p \\tau _{i} \\rangle ^2$ crosses over from $\\simeq (1-\\kappa )/(\\kappa N)\\sim 1/p$ , as in the case of $p$ independent variables, to $\\simeq 2/N^2$ , characteristic of the $N$ highly correlated eigenvalues of random matrices.",
"A similar change of scaling has been recently identified in the “thinned circular ensemble” in Ref. [2].",
"These intringuing features have led us to analyse the full distribution of these two partial sums.",
"The distribution of the injectance $\\overline{\\nu }=(2\\pi )^{-1}\\sum _{i=1}^p \\mathcal {Q}_{ii}$ is obtained in Appendix for $\\beta =2$ .",
"This derivation is based on a result of Savin, Fyodorov and Sommers [46], stating that, for $\\beta =2$ , the distribution of a sub-block of the matrix $\\mathcal {Q}$ is also related to the Laguerre distribution.",
"The more general statement of Ref.",
"[46] concerns sub-block of matrix $\\mathcal {Q}_s$ , introduced in the previous footnote.",
"The distributions of the two matrices $\\mathcal {Q}$ and $\\mathcal {Q}_s$ coincide for $\\beta =2$ .",
"This makes this study a variant of the Wigner time delay distribution analysis of Ref.",
"[48] : the distribution of the partial sum $\\sum _{i=1}^p\\mathcal {Q}_{ii}$ is studied in Appendix .",
"On the other hand, the question of the distribution of the truncated linear statistics $s = \\sum _{i=1}^p \\tau _i= \\tau _\\mathrm {H}\\sum _{i=1}^p \\frac{1}{\\lambda _i}\\:,$ where $\\lambda _i$ 's are eigenvalues of a Wishart matrix, Eq.",
"(REF ), is much more challenging and is the main object of investigation of the present article."
],
[
"Variance for arbitrary symmetry class",
"The discussion of the statistical properties of $\\mathcal {Q}_{ii}$ 's has led us to restrict ourselves to the unitary case.",
"However, as we will study the full distribution of $s$ for arbitrary symmetry class, it is useful to express the two cumulants of $s$ for arbitrary $\\beta $ .",
"Using Eqs.",
"(REF ,) we get $\\left\\langle s\\right\\rangle &= \\sum _{i=1}^p\\left\\langle \\tau _i\\right\\rangle = \\frac{p}{N} = \\kappa \\:,\\\\\\mathrm {Var}(s)&= \\frac{\\beta pN(N - p) + 2 p(p+N)}{N^2(N+1)(\\beta N - 2)}\\\\&= \\frac{\\kappa (1-\\kappa )}{N}+ \\frac{4 \\kappa }{\\beta N^2} \\left( 1 - \\beta \\frac{1-\\kappa }{4} \\right) + \\mathcal {O}(N^{-3})\\:.$ Setting $\\kappa = 1$ , we check that () gives the result of Ref.",
"[40] : $\\mathrm {Var}(s)= \\frac{4}{(N+1)(N\\beta -2)}\\underset{N \\rightarrow \\infty }{\\simeq }\\frac{4}{\\beta N^2}$ (the leading order was obtained earlier in [33] for $\\beta =1$ and [5] ; see also [48], [47]).",
"Eq.",
"() exhibits another interesting feature as the leading order term is independent of the symmetry class, while the subleading term surviving in the $\\kappa \\rightarrow 1$ limit does depend on $\\beta $ .",
"The origin of this observation will be clarified in Section ."
],
[
"Formalism : Coulomb gas and occupation numbers",
"Let us introduce the general framework used in the paper.",
"For large $N$ , it is convenient to rescale the eigenvalues as [25] $\\lambda _i = N x_i \\:.$ The Coulomb gas method consists in rewriting the joint distribution of eigenvalues (REF ) as a Gibbs measure $\\exp (- \\beta N^2 \\mathcal {E}_\\mathrm {gas}[\\lbrace x_i \\rbrace ]/2)$ , where $\\mathcal {E}_\\mathrm {gas}[\\lbrace x_i \\rbrace ] =-\\frac{1}{N^2} \\sum _{i \\ne j} \\ln \\left| x_i - x_j \\right|+ \\frac{1}{N} \\sum _{i=1}^N (x_i - \\ln x_i)$ is the energy of a gas of particles on a line, at positions $\\lbrace x_i \\rbrace $ , interacting with logarithmic repulsion and trapped in the external potential $V(x) = x - \\ln x$ .",
"We want to compute the distribution of $s = \\frac{1}{N} \\sum _{i=1}^N n_i f(x_i)\\:,\\hspace{14.22636pt}x_1 > x_2 > \\cdots > x_N\\:,$ for any given function $f$ .",
"This is given by Eq.",
"(REF ), which rewrites : $&P_{N,\\kappa }(s) = \\\\&\\frac{ \\displaystyle \\sum _{ \\lbrace n_i \\rbrace }\\int _0^\\infty \\hspace{-7.11317pt} {\\rm d}x_1 \\int _0^{x_1} \\hspace{-7.11317pt}{\\rm d}x_2 \\cdots \\int _0^{x_{N-1}}\\hspace{-7.11317pt} {\\rm d}x_N \\,\\mathrm {e}^{-\\frac{\\beta N^2}{2} \\mathcal {E}_\\mathrm {gas}[\\lbrace x_i \\rbrace ] }\\delta \\left(s - \\frac{1}{N} \\sum _{i=1}^{N} n_i f(x_i)\\right)\\,\\delta _{p,\\sum _i n_i}}{ \\displaystyle \\sum _{ \\lbrace n_i \\rbrace }\\int _0^\\infty \\hspace{-7.11317pt}{\\rm d}x_1 \\int _0^{x_1} \\hspace{-7.11317pt}{\\rm d}x_2 \\cdots \\int _0^{x_{N-1}}\\hspace{-7.11317pt} {\\rm d}x_N \\,\\mathrm {e}^{-\\frac{\\beta N^2}{2} \\mathcal {E}_\\mathrm {gas}[\\lbrace x_i \\rbrace ] }\\,\\delta _{p,\\sum _i n_i}}\\nonumber \\:.$ Since we are interested in partial sums of proper time delays $\\tau _i = 1/\\lambda _i = 1/(N x_i)$ , we will consider the case $f(x) = \\frac{1}{x}\\:.$ Therefore, Eq.",
"(REF ) gives $s = N^{-1} \\sum _i n_i / x_i$ .",
"The limit $s \\rightarrow 0$ corresponds to $x_i \\rightarrow \\infty $ , for all $p$ eigenvalues such that $n_i = 1$ .",
"The $N - p$ others which do not contribute to the sum are not constrained, so they remain of order 1.",
"Thus, in this case, the $p$ eigenvalues in the sum are the largest.",
"This corresponds to $n_1 = \\cdots = n_{p} = 1 \\: ,\\hspace{14.22636pt}n_{p+1} = \\cdots = n_{N} = 0 \\:.$ Eq.",
"(REF ) reduces to : $s = \\frac{1}{N} \\sum _{i=1}^{p} f(x_i)\\:,\\hspace{14.22636pt}x_1 > x_2 > \\cdots > x_N\\:.",
"$ This problem was analysed in detail in [25], for some specific choices of function $f$ .",
"The general case is discussed in the appendix of this paper.",
"Here we will only sketch briefly the method in order to be self-contained.",
"In the remainder of this section, we will focus on the situation where the occupation numbers $\\lbrace n_i \\rbrace $ are frozen according to (REF ).",
"This will be the case as long as $s < s_0(\\kappa )$ .",
"In Sections and , we will study the case $s > s_0(\\kappa )$ where the $\\lbrace n_i \\rbrace $ 's are allowed to fluctuate."
],
[
"Functional integral formulation",
"Let us introduce the empirical density of eigenvalues $\\rho (x) = \\frac{1}{N} \\sum _{i=1}^N \\delta (x - x_i)\\:.$ In terms of this density, (REF ) reads : $s = \\int _c \\rho (x)f(x) {\\rm d}x \\:,$ where $c \\sim x_{p}$ is a lower bound ensuring that only the $p$ largest eigenvalues contribute to the integral.",
"It is fixed by imposing $\\int _c \\rho (x) {\\rm d}x = \\frac{p}{N} = \\kappa \\:.$ We emphasize that it is only possible to express $s$ in terms of a simple integral over the density of eigenvalues $\\rho (x)$ in the case where only the largest (or the smallest) eigenvalues contribute to $s$ , corresponding to large deviations.",
"In the limit $N \\rightarrow \\infty $ , it is possible to rewrite the multiple integrals in Eq.",
"(REF ) as functional integrals over the density, leading to the measure [21], [20] : $\\mathrm {e}^{-\\frac{\\beta N^2}{2} \\mathcal {E}_\\mathrm {gas}[\\lbrace x_i \\rbrace ] }{\\rm d}x_1 \\cdots {\\rm d}x_N\\rightarrow \\mathrm {e}^{- \\frac{\\beta N^2}{2} {E}[\\rho ] + N (1 - \\frac{\\beta }{2}) {S}[\\rho ]}{\\mathcal {D}}\\rho \\:,$ with the energy ${E}[\\rho ] = - \\int {\\rm d}x \\int {\\rm d}y \\: \\ln \\left| x-y \\right|+ \\int {\\rm d}x \\: \\rho (x)(x - \\ln x)$ and the entropy ${S}[\\rho ] = -\\int {\\rm d}x \\: \\rho (x) \\ln \\rho (x)\\:.$ The energy ${E}[\\rho ]$ is obtained by rewriting (REF ) in terms of the density (REF ).",
"The diagonal terms $i=j$ which are not present in (REF ) are removed by adding the term $-\\big [{\\beta }/{(2N)}\\big ] {S}[\\rho ]$ [22], [20].",
"The other entropic term $({1}/{N}) {S}[\\rho ]$ comes from the loss of information when describing the set $\\lbrace x_i \\rbrace $ by the density $\\rho (x)$ .",
"Since we are interested in the limit $N \\rightarrow \\infty $ , we can neglect the subleading entropic term.",
"Eq.",
"(REF ) becomes : $&P_{N,\\kappa }(s) \\simeq \\mathrm {e}^{- N\\,C_\\kappa } \\\\&\\frac{ \\displaystyle \\int {\\rm d}c \\int {\\mathcal {D}}\\rho \\:\\mathrm {e}^{-\\frac{\\beta N^2}{2} {E}[\\rho ] }\\delta \\left(\\int _c \\rho - \\kappa \\right)\\delta \\left(\\int ^c \\rho - (1-\\kappa )\\right)\\delta \\left(s - \\int _c \\rho (x) f(x) \\: {\\rm d}x\\right)}{ \\displaystyle \\int {\\rm d}c \\int {\\mathcal {D}}\\rho \\:\\mathrm {e}^{-\\frac{\\beta N^2}{2} {E}[\\rho ] }\\delta \\left(\\int _c \\rho - \\kappa \\right)\\delta \\left(\\int ^c \\rho - (1-\\kappa )\\right)}\\nonumber \\:,$ where the factor $\\mathrm {e}^{- N\\,C_\\kappa }$ , with $C_\\kappa =-\\kappa \\ln \\kappa -(1-\\kappa )\\ln (1-\\kappa )$ , arises from the sum over occupations in the denominator of (REF ) (cf.",
"discussion in Section ).",
"This additionnal factor would be absent if one would consider the distribution of the TLS restricted to the largest eigenvalue, like in Ref.",
"[25] : (REF ) without $\\mathrm {e}^{- N\\,C_\\kappa }$ would describe the full range and would be clearly normalized.",
"Here, this form only describes the tail of the distribution of the TLS without restriction on the ordering of the eigenvalues, when $s<s_0(\\kappa )$ , hence the expression has no reason to be normalised.",
"These integrals can be estimated using a saddle point method.",
"The saddle point will give $\\rho $ and $c$ as functions of $\\kappa $ and $s$ .",
"The numerator is dominated by the density $\\rho ^\\star (x;\\kappa ,s)$ which minimizes the energy ${E}[\\rho ]$ under the constraints imposed by the Dirac $\\delta $ -functions.",
"Similarly, the denominator is dominated by a density $\\rho _0^\\star (x)$ .",
"The minimization takes the form similar to (REF ,REF ), with additional constraints, which leads to two coupled integral equations, instead of one (REF ).",
"The precise equations can be found in Ref.",
"[25], where this procedure was carried out explicitly.",
"Here, we will simply jump to the solution of the coupled integral equations (see below Eq.",
"(REF )).",
"Finally, we obtain the distribution $\\boxed{P_{N,\\kappa }(s) \\underset{N \\rightarrow \\infty }{\\sim }\\mathrm {e}^{- N\\,C_\\kappa }\\exp \\left\\lbrace - \\frac{\\beta N^2}{2} \\Phi _-(\\kappa ;s) \\right\\rbrace \\hspace{14.22636pt}\\text{for}\\hspace{14.22636pt}s < s_0(\\kappa )}$ where we have introduced the large deviation function $\\Phi _-(\\kappa ;s) = {E}[\\rho ^\\star (x;\\kappa ,s)] - {E}[\\rho _0^\\star (x)]\\:.$ In (REF ), the scaling as $N^2$ is a manifestation of energy of the Coulomb gas and reflects the long range nature of the interaction between the $N$ particles."
],
[
"Optimal density",
"The density $\\rho ^\\star $ is given explicitly in the general case in the appendix of Ref. [25].",
"For any monotonic function $f$ , it reads : $\\rho ^\\star (x;\\kappa ,s) = & \\frac{1}{2\\pi } \\sqrt{\\frac{(b-x)(d-x)}{(x-a)(c-x)}}\\\\& \\times \\left\\lbrace 1 - \\frac{1}{x} \\sqrt{\\frac{ac}{bd}} + \\mu _1\\mathchoice{{\\displaystyle {\\textstyle -}{\\int }}\\hbox{$\\textstyle -$}}{\\hspace{0.0pt}}{-}{.",
"}5\\right.$ cd dt f'(t)t-x (t-a)(t-c)(t-b)(d-t) , where the principal value is needed only if $x \\in [c,d]$ .",
"This density is supported on two disjoint intervals $[a,b]$ and $[c,d]$ (see Fig.",
"REF ), where $c$ is the boundary introduced above in Eqs.",
"(REF ,REF ).",
"The boundaries of these intervals and the parameter $\\mu _1$ are fixed by the conditions that the density vanishes at these points : $1 - \\sqrt{\\frac{a}{bcd}} + \\mu _1 \\int _c^d \\frac{{\\rm d}t}{\\pi } f^{\\prime }(t) \\sqrt{\\frac{t-a}{(t-b)(t-c)(d-t)}} &=0\\:,\\\\3 + \\frac{a+c-b-d}{2} - \\sqrt{\\frac{ac}{bd}}- \\mu _1 \\int _c^d \\frac{{\\rm d}t}{\\pi } f^{\\prime }(t) \\sqrt{\\frac{(t-a)(t-c)}{(t-b)(d-t)}} &= 0\\: ,\\\\1 - \\sqrt{\\frac{c}{a b d}} + \\mu _1 \\int _c^d \\frac{{\\rm d}t}{\\pi } f^{\\prime }(t) \\sqrt{\\frac{t-c}{(t-a)(t-b)(d-t)}} &= 0\\:,$ along with the constraints $\\int _c^d \\rho ^\\star (x;\\kappa ,s) {\\rm d}x = \\kappa \\:,\\hspace{28.45274pt}\\int _c^d \\rho ^\\star (x;\\kappa ,s) f(x) {\\rm d}x = s\\:.$ The parameter $\\mu _1$ is a Lagrange multiplier introduced to handle the constraint on $s$ .",
"We denote $\\mu _1^\\star (\\kappa ;s)$ the solution of (REF ,,,REF ).",
"Its knowledge allows to compute easily the energy thanks to the thermodynamic identity [27], [13], [28] : $\\frac{\\mathrm {d}{E}[\\rho ^{\\star }(x;\\kappa ,s)]}{\\mathrm {d}s} = - \\mu _1^\\star (\\kappa ;s)\\:.$ The density $\\rho _0^\\star $ which dominates the denominator of (REF ) is the well known Marčenko-Pastur distribution [37] : $\\rho _0^\\star (x) = \\frac{1}{2\\pi x} \\sqrt{(x-x_-)(x_+ - x)}\\:,\\hspace{28.45274pt}x_{\\pm } = 3 \\pm 2 \\sqrt{2}\\:.$ This density can be obtained from (REF ) by taking the limit $\\mu _1 \\rightarrow 0+$ , which corresponds to $c-b \\rightarrow 0$ .",
"Figure: Optimal density ρ ☆ \\rho ^\\star (solid line), for f(x)=1/xf(x)=1/x, compared to the density ρ 0 ☆ \\rho _0^\\star (dashed).The shaded area correspond the fraction κ\\kappa of the largest eigenvalues.The support of the density ρ ☆ \\rho ^\\star is [a,b]∪[c,d][a,b]\\cup [c,d].This derivation is valid when only the largest eigenvalues contribute to $s$ .",
"This is the case as long as the gap between the two intervals of the support of $\\rho ^\\star $ is non-zero.",
"The limit of validity is therefore given by $b=c$ , which corresponds to $\\mu _1 = 0$ .",
"The distribution is then $\\rho _0^\\star $ , thus Eq.",
"(REF ) gives the value of $s$ : $s_0(\\kappa ) = \\int _{c_0}^{x_+} \\rho _0^\\star (x) f(x) {\\rm d}x\\:,$ where $c_0$ is fixed by $\\kappa = \\int _{c_0}^{x_+} \\rho _0^\\star (x) {\\rm d}x\\:.$ Solving the second equation for $c_0$ and plugging the result into the first equation gives the maximal value $s_0(\\kappa )$ allowed for $s$ .",
"This defines a line $s = s_0(\\kappa )$ in the $(\\kappa ,s)$ plane which delimits the region where the assumption that only the largest eigenvalues contribute to $s$ is true.",
"This is the lower solid line in Fig.",
"REF .",
"Exactly on this line, the density of eigenvalues is the Marčenko-Pastur distribution $\\rho _0^\\star $ ."
],
[
"Tail $s \\rightarrow 0$",
"For $f(x)=1/x$ , the limit $s \\rightarrow 0$ , corresponds to push the fraction $\\kappa $ of the rightmost eigenvalues towards infinity.",
"This means to let $d > c \\rightarrow \\infty $ in the previous equations.",
"Expanding Eqs.",
"(REF ,,,REF ) in this limit yields : $a &= 3 - 2 \\sqrt{(1-\\kappa )(2-\\kappa )} - 2 \\kappa + \\mathcal {O}(\\sqrt{s}) \\:,\\\\b &= 3 + 2 \\sqrt{(1-\\kappa )(2-\\kappa )} - 2 \\kappa + \\mathcal {O}(\\sqrt{s}) \\:,\\\\c &= \\frac{\\kappa }{s} \\left(1 - \\sqrt{2s} + \\frac{5s}{4} + \\mathcal {O}(s^{3/2}) \\right) \\:,\\\\d &= \\frac{\\kappa }{s} \\left(1 + \\sqrt{2s} + \\frac{5s}{4} + \\mathcal {O}(s^{3/2}) \\right) \\:,\\\\\\mu _1^\\star &= \\frac{\\kappa ^2}{s^2} - \\frac{3\\kappa (2-\\kappa )}{2s} + \\mathcal {O}(s^{-1/2}) \\:.$ Using the thermodynamic identity (REF ), a simple integration of this last relation gives the behaviour of the energy, therefore of the large deviation function, for $s \\rightarrow 0$ : $\\Phi _-(\\kappa ;s) = {E}[\\rho ^\\star (x;\\kappa ,s)] - {E}[\\rho _0^\\star (x)]= \\frac{\\kappa ^2}{s} + \\frac{3 \\kappa (2-\\kappa )}{2} \\ln s + \\mathcal {O}(1)\\:.$ This expression gives the left tail of the distribution : $P_{N,\\kappa }(s) \\underset{s \\rightarrow 0}{\\sim }s^{- 3 \\beta \\kappa (2-\\kappa ) N^2/4} \\: \\mathrm {e}^{-\\beta (N \\kappa )^2/(2s)}\\:.$ Note that we recover the tail computed in Ref.",
"[48] simply by setting $\\kappa =1$ : $P_{N,\\kappa =1}(s) \\underset{s \\rightarrow 0}{\\sim } s^{-3 \\beta N^2/4} \\: \\mathrm {e}^{- \\beta N^2/(2s)}\\:.$ Interestingly, the leading term in (REF ) can be obtained easily by a heuristic argument.",
"For small $s$ , the energy is dominated by the potential energy of the eigenvalues pushed to infinity.",
"The typical value of these eigenvalues is given by $s \\sim \\kappa /x_\\mathrm {typ}$ , corresponding to $x_\\mathrm {typ} \\sim \\kappa /s$ .",
"The energy is estimated as ${E}[\\rho ^\\star (x;\\kappa ,s)] \\sim \\int _c \\rho ^\\star V \\sim \\kappa V(x_\\mathrm {typ}) \\sim \\kappa ^2/s$ .",
"Qed."
],
[
"Limit $s\\rightarrow s_0(\\kappa )$",
"In the limit $s\\rightarrow s_0(\\kappa )$ , the two bulks of the density $\\rho ^\\star $ merge.",
"Hence, it corresponds to $c-b \\rightarrow 0$ .",
"The behaviour of $\\Phi _-(\\kappa ;s)$ for $s$ close to $s_0(\\kappa )$ is obtained by expanding Eqs.",
"(REF ,...,REF ) in this limit.",
"A straightforward but cumbersome computation gives, for any monotonic function $f$ : $\\mu _1^\\star (\\kappa ;s) \\simeq -2 \\omega _\\kappa \\, (s-s_0(\\kappa )) \\:,$ where $\\omega _\\kappa ^{-1} =\\int _{c_0}^{x_+} \\frac{{\\rm d}x}{\\pi } \\frac{f(x)-f(c_0)}{\\sqrt{(x-x_-)(x_+-x)}}\\mathchoice{{\\displaystyle {\\textstyle -}{\\int }}\\hbox{$\\textstyle -$}}{\\hspace{0.0pt}}{-}{.",
"}5$ c0x+ dt f'(t) (t-x-)(x+-t)x-t , where $c_0$ is fixed by (REF ).",
"The large deviation function is deduced from the thermodynamic identity (REF ) : $\\Phi _-(\\kappa ;s) \\simeq \\omega _\\kappa \\, (s-s_0(\\kappa ))^2 \\:.$ We have performed a numerical simulation in order to check the quadratic behaviour and the value of the coefficient $\\omega _\\kappa $ , in the case $f(x) = 1/x$ .",
"The energy ${E}[\\rho ^\\star (x;\\kappa ,s)]$ is computed using a Monte Carlo method, see Ref.",
"[25] for details on the procedure.",
"Fitting the numerics must be performed with caution : for $\\kappa =0.5$ , Eq.",
"(REF ) gives $\\omega _{1/2} \\simeq 27.1$ , however a fit of the numerical data with a purely quadratic behaviour leads to $\\omega _{1/2}^\\mathrm {num} \\simeq 32.4$ .",
"This apparent discrepancy is explained by the fact that (REF ) is only the leading term of an expansion near $s_0(\\kappa )$ : $\\Phi _-(\\kappa ;s) \\simeq \\omega _\\kappa \\, (s-s_0(\\kappa ))^2 + \\omega _{\\kappa ,3} (s-s_0(\\kappa ))^3 + \\cdots \\:,$ where the higher order term cannot be neglected in practice as it involves a large coefficient.",
"Indeed, fitting the numerics with both the quadratic and the cubic terms, we find $\\omega _{1/2}^\\mathrm {num} \\simeq 27.0 $ and $\\omega _{1/2,3}^\\mathrm {num} \\simeq -239$ , now in excellent agreement with the prediction (see fit in Fig.",
"REF ).",
"We have checked that the accuracy on $\\omega _{1/2}^\\mathrm {num}$ extracted from the fit is improved by increasing the order of the polynomial used to fit the data.",
"Interestingly, the simulation shows that the energy is frozen ${E}[\\rho ^\\star (x;\\kappa ,s)] = {E}[\\rho _0^\\star (x)]$ for $s > s_0(\\kappa )$ .",
"This will be explained in the next section.",
"Figure: Large deviation function Φ - (κ;s)=E[ρ ☆ (x;κ,s)]-E[ρ 0 ☆ (x)]\\Phi _-(\\kappa ;s) = {E}[\\rho ^\\star (x;\\kappa ,s)] - {E}[\\rho _0^\\star (x)] (energy of the Coulomb gas) determined by a Monte Carlo simulation for κ=0.5\\kappa =0.5 and N=1500N=1500 (dots).",
"The results are compared to the quadratic behavior () (dashed line) and a polynomial fit of degree 3, Eq.",
"(), (solid line).The dashed vertical line is s 0 (κ)s_0(\\kappa )."
],
[
"Phase II ($s_0(\\kappa ) < s < s_1(\\kappa )$ ) : frozen Coulomb gas and fictitious fermions",
"In the previous section, we have seen that when $s\\in ]0,s_0(\\kappa )]$ , the density $\\rho ^\\star $ changes with $s$ while the occupation numbers $\\lbrace n_i \\rbrace $ remain fixed.",
"When $s$ reaches $s_0(\\kappa )$ , the density is given by the Marčenko-Pastur distribution $\\rho _0^\\star $ , which minimizes the energy ${E}[\\rho ]$ .",
"We now enter the domain $s > s_0(\\kappa )$ where it is possible to vary $s$ by changing the occupation numbers, while keeping the density fixed.",
"The range of accessible values of $s$ is given by two extreme cases : selecting either the largest eigenvalues ($s = s_0(\\kappa )$ ) or the smallest ones (Fig.",
"REF , right).",
"This latter case corresponds to the value $s = s_1(\\kappa )$ given by : $s_1(\\kappa ) = \\int _{x_-}^{c_1} \\rho _0^\\star (x)f(x) {\\rm d}x\\:,$ where $c_1$ is fixed by $\\kappa = \\int _{x_-}^{c_1} \\rho _0^\\star (x) {\\rm d}x\\:;$ we recall that $x_\\pm =3\\pm 2\\sqrt{2}$ are the boundaries of the support of the Marčenko-Pastur distribution introduced above, Eq.",
"(REF ).",
"This defines another line in the $(\\kappa ,s)$ plane.",
"It is the upper solid line shown in Fig.",
"REF .",
"The two lines are simply related via the relation : $s_1(\\kappa ) = \\bar{s}- s_0(1-\\kappa ) \\:,\\hspace{14.22636pt}\\text{where}\\hspace{14.22636pt}\\bar{s}= \\int _{x_-}^{x_+} \\rho _0^\\star (x) f(x) {\\rm d}x\\:.$ In the Laguerre ensemble considered here, with $f(x) = 1/x$ , we simply have $\\bar{s}= 1$ .",
"The goal of this section is to describe what occurs for $s$ between $s_0(\\kappa )$ and $s_1(\\kappa )$ , and compute the distribution in this domain.",
"Note that the procedure exposed in this section is valid for a monotonic function $f$ ."
],
[
"Distribution of $s$ between {{formula:67ff9518-cb82-486c-bb62-6a98364e92a7}} and {{formula:30f34e4f-e29d-414b-8c35-9fbf7e81c2a5}} : an entropic contribution",
"In order to determine the distribution $P_{N,\\kappa }(s)$ on the interval $[s_0(\\kappa ),s_1(\\kappa )]$ , let us go back to the multiple integrals in Eq.",
"(REF ).",
"For large $N$ , these integrals are dominated by the set of eigenvalues $\\lbrace x_i^\\star \\rbrace $ which minimizes the energy $\\mathcal {E}_\\mathrm {gas}[\\lbrace x_i \\rbrace ]$ given by Eq.",
"(REF ), i.e.",
"$\\left.",
"\\frac{\\partial \\mathcal {E}_\\mathrm {gas}[\\lbrace x_i \\rbrace ]}{\\partial x_i}\\right|_{ \\lbrace x_i^\\star \\rbrace } = 0\\:,\\hspace{14.22636pt}\\forall i \\in \\lbrace 1, \\cdots , N \\rbrace \\:.$ Without changing this set of eigenvalues, it is possible to construct $\\binom{N}{p}$ different values, given by $\\frac{1}{N} \\sum _{i=1}^{N} n_i f(x_{i}^\\star )\\:.$ However, several configurations may give a similar value of $s$ .",
"This shows that the distribution of $s$ comes from an entropic contribution $S(\\kappa ;s)$ : the same “macroscopic” value $s$ may be associated to many “microscopic” configurations $\\lbrace n_i \\rbrace $ .",
"We stress that this should not be confused with the entropy ${S}[\\rho ]$ defined by Eq.",
"(REF ).",
"Indeed, this latter measures the loss of information when describing the set of eigenvalues $\\lbrace x_i \\rbrace $ by a density $\\rho (x)$ .",
"The entropy $S(\\kappa ;s)$ discussed here is hidden in the sums over the configurations $\\lbrace n_i \\rbrace $ in (REF ).",
"It is more convenient to first compute the moment generating function, which is the Laplace transform of $P_{N,\\kappa }(s)$ , using standard tools from statistical physics.",
"In a second step, we will come back to the distribution.",
"Figure: Marčenko-Pastur distribution ρ 0 ☆ \\rho _0^\\star , with either the largest eigenvalues selected (left), corresponding to s=s 0 (κ)s=s_0(\\kappa ), or the smallest eigenvalues (right), corresponding to s=s 1 (κ)s=s_1(\\kappa )."
],
[
"Moment generating function : mapping to free fermions",
"The moment generating function is given by the Laplace transform of $P_{N,\\kappa }$ : $G_{N,\\kappa }(\\tilde{\\beta }) &= \\int e^{- \\tilde{\\beta }N s} P_{N,\\kappa }(s) {\\rm d}s = \\left\\langle \\exp \\left( - \\tilde{\\beta }\\sum _i n_i f(x_i^\\star ) \\right) \\right\\rangle \\:, \\nonumber \\\\&= \\frac{p!",
"(N-p)!}{N!}",
"\\sum _{ \\lbrace n_i \\rbrace } \\prod _{i=1}^N \\mathrm {e}^{- \\tilde{\\beta }n_i f(x_i^\\star )}\\,\\delta _{p,\\sum _i n_i}\\:,$ where the sum runs over the $\\binom{N}{p}$ configurations $\\lbrace n_i \\rbrace $ .",
"The problem of choosing $p$ eigenvalues among $N$ , with the constraint that $s$ is fixed, is equivalent to placing $p$ particles on $N$ “energy levels” $\\varepsilon _i = f(x_i^\\star )$ , with the constraint that the total “energy” is fixed.",
"Since an eigenvalue can be chosen only once, an “energy level” can host only one particle.",
"Therefore these particles behave like $p$ fermions.",
"The parameters $\\kappa $ and $s$ are thus related to two fundamental properties of the fermion gas : $\\kappa \\leftrightarrow \\text{number of fermions}\\hspace{28.45274pt}s \\leftrightarrow \\text{energy of the fermions}$ The set $\\lbrace n_i \\rbrace $ coincides with the occupation numbers of the energy levels : $n_i = 1$ if level $\\varepsilon _i$ is occupied and $n_i = 0$ otherwise.",
"With this interpretation, (REF ) is simply related to the canonical partition function $Z_{N,p}(\\tilde{\\beta }) = \\sum _{ \\lbrace n_i \\rbrace } \\mathrm {e}^{- \\tilde{\\beta }E_\\mathrm {ferm}[\\lbrace n_i \\rbrace ] }\\,\\delta _{p,\\sum _i n_i}\\hspace{8.5359pt}\\mbox{where}\\hspace{8.5359pt}E_\\mathrm {ferm}[\\lbrace n_i \\rbrace ] = \\sum _{i=1}^N n_i \\varepsilon _i\\:,$ via the relation $G_{N,\\kappa }(\\tilde{\\beta }) = \\frac{Z_{N, p}(\\tilde{\\beta })}{Z_{N, p}(0)}\\:.$ The parameter $\\tilde{\\beta }$ now plays the role of an inverse temperature for the fermions, not to be confused with the Dyson index $\\beta \\in \\lbrace 1, 2, 4 \\rbrace $ .",
"It is natural to introduce the grand-canonical partition function, which can be calculated straightforwardly : $\\Xi _N(z,\\tilde{\\beta }) = \\sum _{p= 0}^N z^{p} Z_{N,p}(\\tilde{\\beta })= \\prod _{i=1}^N \\left( 1 + z \\, \\mathrm {e}^{-\\tilde{\\beta }\\varepsilon _i} \\right)\\:,$ where $z$ is the fugacity.",
"For large $N$ , the distribution of the eigenvalues $\\lbrace x_i^\\star \\rbrace $ is the Marčenko-Pastur law $\\rho _0^\\star $ .",
"Therefore in this limit we can write : $\\frac{1}{N} \\ln \\Xi _N(z,\\tilde{\\beta })&= \\frac{1}{N} \\sum _{i=1}^N \\ln \\left( 1 + z\\, \\mathrm {e}^{-\\tilde{\\beta }\\varepsilon _i} \\right) \\nonumber \\\\& \\simeq \\int _{x_-}^{x_+} \\rho _0^\\star (x) \\ln \\left( 1 + z\\, \\mathrm {e}^{-\\tilde{\\beta }f(x)} \\right) {\\rm d}x\\:.$ We can go back to the canonical partition function using the relation $Z_{N,p}(\\tilde{\\beta }) = \\frac{1}{2 {\\rm i}\\pi } \\oint \\frac{\\Xi _N(z,\\tilde{\\beta })}{z^{p+1}} {\\rm d}z\\:,$ where the integral runs over a contour which encloses the origin once in the counter-clockwise direction.",
"For large $N$ , this integral can be estimated using a saddle point method.",
"In the thermodynamic limit $N \\rightarrow \\infty $ , this corresponds to use the equivalence between the canonical and grand-canonical ensembles through a Legendre transform.",
"The value $z_\\mathrm {can}(\\kappa ;s)$ of the fugacity is fixed by imposing that the mean number of fermions is $p$ : $z \\left.",
"\\frac{\\partial \\ln \\Xi }{\\partial z} \\right|_{z_\\mathrm {can}} = p\\hspace{14.22636pt}\\Rightarrow \\hspace{14.22636pt}\\int _{x_-}^{x_+} \\frac{\\rho _0^\\star (x)}{\\exp (\\tilde{\\beta }f(x))/z_\\mathrm {can}+ 1} {\\rm d}x = \\kappa \\:.$ Then Eq.",
"(REF ) yields the free energy per fermion in the thermodynamic limit : $F(\\kappa ; \\tilde{\\beta }) =- \\lim _{N \\rightarrow \\infty }\\frac{1}{N} \\ln Z_{N,p}(\\tilde{\\beta })= \\kappa \\ln z_\\mathrm {can}- \\lim _{N \\rightarrow \\infty } \\frac{1}{N} \\ln \\Xi _N(z_\\mathrm {can},\\tilde{\\beta })\\:.$ Note that we dropped the factor $\\tilde{\\beta }$ in the usual definition of the free energy for convenience.",
"We get : $F(\\kappa ; \\tilde{\\beta }) = -\\int _{x_-}^{x_+} \\rho _0^\\star (x) \\ln \\left( 1 + z_\\mathrm {can}(\\kappa ;\\tilde{\\beta }) \\, \\mathrm {e}^{-\\tilde{\\beta }f(x)} \\right) {\\rm d}x+ \\kappa \\ln z_\\mathrm {can}(\\kappa ;\\tilde{\\beta })\\:.$ Eq.",
"(REF ) shows that the logarithm of the moment generating function, namely the cumulant generating function, is given by the free energy of the fermions : $\\frac{1}{N} \\ln G_{N,\\kappa }(\\tilde{\\beta }) \\simeq F(\\kappa ; 0) -F(\\kappa ; \\tilde{\\beta })\\:.$"
],
[
"Cumulants of the truncated linear statistics $s$ , for {{formula:b077bce5-9356-4e24-b04e-e38ac5dcd416}}",
"The cumulants of (REF ) can be obtained by expanding $\\ln G_{N,\\kappa }$ as a power series : $\\ln G_{N,\\kappa }(\\tilde{\\beta }) = \\sum _{n \\geqslant 1} \\left\\langle s^n\\right\\rangle _c \\frac{(\\tilde{\\beta }N)^n}{n!",
"}\\:.$ The procedure is as follows : first determine $z_\\mathrm {can}$ as a power series in $\\tilde{\\beta }$ from (REF ).",
"Then plug this expression into (REF ) to get the expansion of $F$ in powers of $\\tilde{\\beta }$ , and read the cumulants from the coefficients of this expansion.",
"In particular, for $f(x)=1/x$ , the first six cumulants are given by : $\\left\\langle s\\right\\rangle &= \\kappa \\:, \\\\\\left\\langle s^2\\right\\rangle _c &\\simeq \\frac{\\kappa (1-\\kappa )}{N} \\:, \\\\\\left\\langle s^3\\right\\rangle _c &\\simeq \\frac{2 \\kappa (1-\\kappa )(1-2\\kappa )}{N^2}\\:, \\\\\\left\\langle s^4\\right\\rangle _c &\\simeq \\frac{2 \\kappa (1-\\kappa )(2-15\\kappa +15\\kappa ^2)}{N^3} \\:, \\\\\\left\\langle s^5\\right\\rangle _c &\\simeq \\frac{4\\kappa (1-\\kappa )(1-2\\kappa )(1-42\\kappa +42\\kappa ^2)}{N^4} \\:, \\\\\\left\\langle s^6\\right\\rangle _c &\\simeq -\\frac{2\\kappa (1-\\kappa )(13 + 420\\kappa -2940\\kappa ^2 +5040\\kappa ^3 - 2520\\kappa ^4)}{N^5} \\:.$ The first two cumulants coincide with those obtained before, Eqs.",
"(REF ,), as it should.",
"In addition, we have performed an orthogonal polynomial calculation in the case $\\beta =2$ [38].",
"This provides another check of our expressions as we have obtained that the first four cumulants given by this second method perfectly match the above expressions.",
"We also stress the $\\kappa \\leftrightarrow 1-\\kappa $ symmetry of the cumulants : under this change, the even order cumulants are unchanged, while the odd ones change sign.",
"Consequently, for $\\kappa = 1/2$ , all odd order cumulants vanish : the distribution is symmetric around its mean $s=1/2$ .",
"This can be understood as the symmetry between fermions and holes.",
"It is interesting to compare these cumulants with the cumulants of the Wigner time delay [15], corresponding to $\\kappa =1$ : $\\left\\langle s^2\\right\\rangle _c &\\simeq \\frac{4}{\\beta N^2} \\:, \\\\\\left\\langle s^3\\right\\rangle _c &\\simeq \\frac{96}{\\beta ^2 N^4}\\:, \\\\\\left\\langle s^4\\right\\rangle _c &\\simeq \\frac{5088}{\\beta ^3 N^6} \\:, \\\\\\left\\langle s^5\\right\\rangle _c &\\simeq \\frac{437760}{\\beta ^4 N^8}\\:,$ (the variance was also found in [33], [5], [48] ; Ref.",
"[40] has given the four first cumulants explicitly).",
"The vanishing of the cumulants (REF ,...,) for $\\kappa =1$ can be straightforwardly understood from the fermionic nature of the problem : when $p= N$ , the occupation numbers are all equal to one, and therefore do not fluctuate.",
"However, (REF ,...,) correspond to the dominant terms of expansions in powers of $1/N$ .",
"When $\\kappa =1$ the leading terms vanish, and only subleading contributions remain.",
"This explains why the cumulants do no present the same scaling with $N$ : $\\left\\langle s^n\\right\\rangle _c \\sim N^{-n+1} \\mbox{ for }\\kappa < 1\\hspace{14.22636pt}\\mbox{\\textit {versus}}\\hspace{14.22636pt}\\left\\langle s^n\\right\\rangle _c \\sim \\beta ^{-n+1} N^{-2n+2} \\mbox{ for }\\kappa =1\\:.$ The fluctuations are much larger when $\\kappa < 1$ .",
"This observation on the scaling of the moments has a simple interpretation : the energy of the $N$ interacting particles scales as $N^2 \\mathcal {E}_\\mathrm {gas} \\sim N^2$ due to the long range nature of the interaction, and the entropy of the $p$ fictitious fermions scales as $N S\\sim N$ .",
"In usual studies of linear statistics, the energy dominates, and the entropy $S$ is zero because all the occupation numbers are fixed to $n_i = 1$ .",
"Hence the fluctuations scale as $\\delta s \\sim 1/N$ .",
"But here, for $\\kappa <1$ , the energy is frozen and the additional entropy dominates.",
"Therefore the fluctuations scale as $\\delta s \\sim 1/\\sqrt{N}$ .",
"This interpretation also explains why the cumulants do not depend on the Dyson index $\\beta $ when $\\kappa <1$ : this index is present in the Gibbs measure $\\exp [-(\\beta /2)N^2\\mathcal {E}_\\mathrm {gas}]$ , but not in the measure of entropic nature $\\exp [NS]$ (as stressed in § REF , the entropy of the fermions should not be confused with the entropy of the density of the Coulomb gas)."
],
[
"Numerics",
"We have also performed numerical simulations to check our results.",
"Figure REF shows histograms obtained by diagonalizing $200\\,000$ complex Hermitian matrices ($\\beta = 2$ ) of various sizes.",
"These histograms are compared to the approximate distribution reconstructed using a Edgeworth series, from the large $N$ expressions of the cumulants (REF ,...,).",
"On purpose, we first consider rather small matrices, of size $N=20$ (Fig.",
"REF , left) : we can see a significant deviation, which we mostly attribute to the large $N$ approximation of the cumulants.",
"For still moderate size $N=40$ (Fig.",
"REF , right), we see that the agreement is already very good.",
"The plots of Figure REF correspond to $\\kappa =0.25$ .",
"Note that, increasing $\\kappa $ with fixed $N$ , we have observed that the agreement becomes less good, although cumulants are symmetric under $\\kappa \\leftrightarrow 1-\\kappa $ at leading order in $N$ ; this is explained by the fact that finite $N$ corrections increase with $\\kappa $ , cf.",
"Eq. ().",
"We have also performed the calculation up to $N=100$ where the histogram is almost indistinguishable from the simple Gaussian approximation.",
"Figure: Histogram of P N,κ (s)P_{N,\\kappa }(s), for f(x)=1/xf(x)=1/x, obtained from 200000200\\,000 complex matrices (β=2\\beta =2) of various sizes, for κ=0.25\\kappa = 0.25.The lines correspond to a reconstruction of the distribution using a Edgeworth series, using the first two (dashed) and four (solid) asymptotic forms of the cumulants."
],
[
"Large deviations : fermions with positive or negative absolute temperature",
"The distribution of the truncated linear statistics (REF ) can be obtained in the interval $[s_0(\\kappa ), s_1(\\kappa )]$ by Laplace inversion : $P_{N,\\kappa }(s) = \\frac{N}{2 {\\rm i}\\pi } \\int _{{\\rm i}\\mathbb {R}} \\mathrm {e}^{N \\tilde{\\beta }s} G_{N,\\kappa }(\\tilde{\\beta }) {\\rm d}\\tilde{\\beta }\\:.$ For large $N$ , we can estimate this integral with a saddle point method.",
"This corresponds to performing another Legendre transform on the free energy $F$ .",
"In section REF we already performed a Legendre transform to go from grand-canonical to canonical ensemble.",
"Thus, here we perform a second one to reach the microcanonical ensemble where the energy (i.e.",
"$s$ ) and the fermion number (i.e.",
"$\\kappa $ ) are fixed.",
"Using (REF ) for the expression of $G_{N,p}$ , we obtain that the saddle point $\\tilde{\\beta }_\\mathrm {mic}(\\kappa ;s)$ is given by $\\left.",
"\\frac{\\mathrm {d}}{\\mathrm {d}\\tilde{\\beta }} F(\\kappa ;\\tilde{\\beta }) \\right|_{\\tilde{\\beta }_\\mathrm {mic}} - s = 0\\:.$ Using the expression of $F$ , this equation reads : $\\int _{x_-}^{x_+} \\frac{\\rho _0^\\star (x)}{\\exp (\\tilde{\\beta }_\\mathrm {mic}f(x))/z_\\mathrm {mic}+ 1} f(x) {\\rm d}x= s \\:,$ where we denoted $z_\\mathrm {mic}(\\kappa ;s) = z_\\mathrm {can}(\\kappa ; \\tilde{\\beta }_\\mathrm {mic}(\\kappa ;s))$ .",
"Then the distribution becomes : $\\boxed{P_{N,\\kappa }(s)\\underset{N\\rightarrow \\infty }{\\sim }\\exp \\left\\lbrace - N \\Phi _0(\\kappa ;s)\\right\\rbrace \\hspace{14.22636pt}\\text{for}\\hspace{14.22636pt}s \\in [s_0(\\kappa ),s_1(\\kappa )]} $ where we introduced the large deviation function $\\Phi _0$ , which is given by : $\\Phi _0(\\kappa ;s) = F(\\kappa ; \\tilde{\\beta }_\\mathrm {mic}(\\kappa ;s)) - s \\tilde{\\beta }_\\mathrm {mic}(\\kappa ;s) - F(\\kappa ;0)\\:.$ Explicitly, it reads : $\\Phi _0(\\kappa ;s) = \\kappa \\ln z_\\mathrm {mic}(\\kappa ;s) \\nonumber &- \\int _{x_-}^{x_+} \\rho _0^\\star (x) \\ln \\left( 1 + z_\\mathrm {mic}(\\kappa ;s) \\mathrm {e}^{-\\tilde{\\beta }_\\mathrm {mic}(\\kappa ;s)f(x)} \\right) {\\rm d}x\\\\& - s \\tilde{\\beta }_\\mathrm {mic}(\\kappa ;s)- \\kappa \\ln \\kappa - (1-\\kappa ) \\ln (1-\\kappa )\\:,$ where $\\tilde{\\beta }_\\mathrm {mic}$ and $z_\\mathrm {mic}$ are fixed by Eq.",
"(REF ) and $\\int _{x_-}^{x_+} \\frac{\\rho _0^\\star (x)}{\\exp (\\tilde{\\beta }_\\mathrm {mic}f(x))/z_\\mathrm {mic}+ 1} {\\rm d}x= \\kappa \\:.$ The observations we made earlier discussing the cumulants are also valid for the full distribution of $s$ , Eqs.",
"(REF ,REF ).",
"For $s \\in [s_0(\\kappa ),s_1(\\kappa )]$ the distribution $P_{N,\\kappa }(s)$ does not depend on the Dyson index $\\beta $ .",
"Moreover, the logarithm of the probability scales as $N$ whereas it scales as $N^2$ for $s < s_0(\\kappa )$ , indicating a distribution of width $\\sim 1/\\sqrt{N}$ , much broader that $1/N$ for $\\kappa =1$ .",
"The physical interpretation of $\\Phi _0$ is made clear by Eq.",
"(REF ) : $s$ is the energy of the fermions, $F$ their free energy (up to a factor $\\tilde{\\beta }$ ) and $\\tilde{\\beta }_\\mathrm {mic}$ is the inverse temperature.",
"Therefore, $\\Phi _0$ is a difference of entropy : $\\Phi _0(\\kappa ;s) = S(\\kappa ; s^\\star ) - S(\\kappa ; s)\\:,$ where $S= \\tilde{\\beta }s - F$ is the entropy of the $p= \\kappa N$ fermions with total energy $s$ .",
"The condition $\\tilde{\\beta }_\\mathrm {mic}(\\kappa ;s^\\star ) = 0$ determines $s^\\star $ .",
"The term $S(\\kappa ; s^\\star )$ comes from the normalization of $P_{N,\\kappa }(s)$ , namely the denominator in Eq.",
"(REF ).",
"In particular, the number of configurations $\\lbrace n_i \\rbrace $ associated to the value $s$ is exactly $\\exp [N\\, S(\\kappa ;s)]$ .",
"The values of the inverse temperature $\\tilde{\\beta }_\\mathrm {mic}$ and the fugacity $z_\\mathrm {mic}$ are obtained in terms of $\\kappa $ and $s$ by Eqs.",
"(REF ,REF ).",
"This corresponds to summing over all the eigenvalues (equivalently, the energy levels), with a weight given by the Fermi-Dirac distribution $\\left\\langle n_i\\right\\rangle = \\frac{1}{\\exp (\\tilde{\\beta }f(x_i))/z + 1}\\:.$ In addition, the eigenvalues are distributed according to the optimal distribution in the absence of constraint, i.e.",
"the Marčenko-Pastur distribution $\\rho _0^\\star $ .",
"Note that the parameter $\\tilde{\\beta }$ , which represents the inverse temperature of the fermions, can be either positive or negative.",
"This quite unusual situation occurs because the spectrum $\\lbrace \\varepsilon _i = f(x_i^\\star ) \\rbrace $ is bounded from below and above.",
"The Fermi-Dirac distribution (REF ) interpolates between two step-functions : in the limit $\\tilde{\\beta }\\rightarrow + \\infty $ , it selects only the lowest energy levels (in our case, the largest eigenvalues since $f$ is decreasing, as shown in Fig REF , left).",
"The opposite limit $\\tilde{\\beta }\\rightarrow -\\infty $ corresponds to select only the highest energy levels (smallest eigenvalues, see Fig REF , right).",
"Figure: Fermi-Dirac distribution () in the case f(x)=1/xf(x)=1/x, for β ˜>0\\tilde{\\beta }> 0 (left) and β ˜<0\\tilde{\\beta }< 0 (right), superimposed to the eigenvalue distribution ρ 0 ☆ \\rho _0^\\star .",
"Shaded areas indicate the contribution of the eigenvalues to the linear statistics.Note that the case β ˜>0\\tilde{\\beta }> 0 corresponds to select the largest eigenvalues, hence the smallest “energies” ε n =f(x n )=1/x n \\varepsilon _n = f(x_n) = 1/x_n, as usual.The derivation we followed in this section can be generalized straightforwardly to other matrix ensembles.",
"For example in the Gaussian ensemble, one should replace the Marčenko-Pastur distribution $\\rho _0^\\star $ by the Wigner semicircle law.",
"Because the density is given by the density obtained in the absence of constraint, the scenario described here is completely universal and valid for any monotonic function $f$ (the case of $f$ non monotonic is discussed in Section REF )."
],
[
"Expansion around the typical value : infinite temperature ($\\tilde{\\beta }\\rightarrow 0^{\\pm }$ )",
"The large deviation function $\\Phi _0$ being obtained from a saddle point estimate, Eq.",
"(REF ) can be used to prove an identity similar to (REF ) : $\\boxed{\\frac{\\mathrm {d}S(\\kappa ;s)}{\\mathrm {d}s} = \\tilde{\\beta }_\\mathrm {mic}(\\kappa ;s)}$ From this relation, it is clear that the maximum of $S$ (minimum of $\\Phi _0$ ) is given by $\\tilde{\\beta }_\\mathrm {mic}= 0$ , corresponding to $s = \\kappa $ .",
"This means an infinite temperature for the fermions : the Fermi-Dirac distribution (REF ) is flat, and all the energy levels are occupied with probability $p/N$ .",
"Expanding Eqs.",
"(REF ,REF ) for $\\tilde{\\beta }$ near 0 allows to compute $\\tilde{\\beta }_\\mathrm {mic}$ near $s = \\kappa $ .",
"In the case $f(x)=1/x$ , this gives : $\\tilde{\\beta }_\\mathrm {mic}(\\kappa ;s) \\simeq - \\frac{s-\\kappa }{\\kappa (1-\\kappa )}\\:.$ Using (REF ,REF ), we obtain : $\\Phi _0(\\kappa ;s) \\simeq \\frac{(s-\\kappa )^2}{2 \\kappa (1-\\kappa )} \\:,\\hspace{14.22636pt}\\text{for } s \\rightarrow \\kappa \\:.$ As we have already seen, the distribution $P_{N,\\kappa }(s)$ is dominated by a Gaussian peak located at $s=\\kappa $ , with variance given by (REF ).",
"This expression of $\\Phi _0$ is singular when $\\kappa = 1$ .",
"But as we discussed in section REF , this corresponds only to the leading term of an expansion in powers of $1/N$ of the variance.",
"Eq.",
"() shows that the distribtuion is regular for $\\kappa \\rightarrow 1$ : $-\\frac{1}{N}\\ln P_{N,\\kappa }(s)\\underset{s\\sim \\kappa }{\\simeq }\\frac{(s-\\kappa )^2}{ \\displaystyle 2 \\left[ \\kappa (1-\\kappa )+ \\frac{4\\kappa }{\\beta N} \\left(1 - \\beta \\frac{(1-\\kappa )}{4} \\right)+ \\mathcal {O}(N^{-2}) \\right]} \\:.$ This expression describes correctly the limit $\\kappa \\rightarrow 1$ for $s$ close to $\\kappa $ and account for the transition between fluctuations $\\sim 1/\\sqrt{N}$ for $\\kappa <1$ and fluctuations $\\sim 1/N$ for $\\kappa =1$ (the crossover between the two regimes obviously occurs at $1-\\kappa \\sim 1/{\\beta N}$ )."
],
[
"Limiting behaviours at the edges : zero temperature ($\\tilde{\\beta }\\rightarrow \\pm \\infty $ )",
"The limits $s \\rightarrow s_0(\\kappa )$ and $s \\rightarrow s_1(\\kappa )$ correspond to the inverse temperature $\\tilde{\\beta }_\\mathrm {mic}\\rightarrow + \\infty $ and $\\tilde{\\beta }_\\mathrm {mic}\\rightarrow -\\infty $ , respectively.",
"This means zero temperature for the fermions : the Fermi-Dirac distribution (REF ) becomes a step function, selecting either the largest or the smallest eigenvalues, as shown in Fig.",
"REF .",
"Therefore, Eqs.",
"(REF ,REF ) involving integrals of this distribution can be approximated via a standard Sommerfeld expansion [44], [49] : for any function $H$ and fixed chemical potential $\\mu $ , one has $\\int \\frac{H(\\varepsilon ) {\\rm d}\\varepsilon }{\\mathrm {e}^{\\tilde{\\beta }(\\varepsilon - \\mu )} + 1}= \\int _{-\\infty }^\\mu H(\\varepsilon ) {\\rm d}\\epsilon + \\frac{1}{\\tilde{\\beta }^2} \\frac{\\pi ^2}{6} H^{\\prime }(\\mu ) + \\mathcal {O}(\\tilde{\\beta }^{-4})\\text{ for } \\tilde{\\beta }\\rightarrow +\\infty \\:,$ and a similar expression holds for $\\tilde{\\beta }\\rightarrow -\\infty $ .",
"In our case, since $\\varepsilon = f(x) = 1/x$ , it is convenient to introduce $l= 1/\\mu $ , which delimits the domain of the spectrum of eigenvalues which is “occupied”, as shown in Fig.",
"REF .",
"The fugacity is simply related to the chemical potential $\\mu $ by $z_\\mathrm {mic}= \\mathrm {e}^{\\tilde{\\beta }_\\mathrm {mic}\\mu } = \\mathrm {e}^{\\tilde{\\beta }_\\mathrm {mic}/l}$ .",
"For $\\tilde{\\beta }_\\mathrm {mic}\\rightarrow + \\infty $ (corresponding to $s \\rightarrow s_0(\\kappa )$ ), Eq.",
"(REF ) allows to compute $l$ as a power series $l= l_0 + l_1 \\tilde{\\beta }_\\mathrm {mic}^{-2} + \\mathcal {O}(\\tilde{\\beta }_\\mathrm {mic}^{-4})$ using the Sommerfeld expansion (REF ).",
"This gives $\\int _{l_0}^{x_+} \\rho _0^\\star (x) {\\rm d}x = \\kappa \\hspace{14.22636pt}\\text{and}\\hspace{14.22636pt}l_1 = \\frac{ l_0^3 \\pi ^2 }{6} \\left(2 + l_0 \\frac{\\rho _0^\\star {}^{\\prime }(l_0)}{\\rho _0^\\star {}(l_0)}\\right)\\:.$ Then, following the same procedure with Eq.",
"(REF ) yields $s = s_0(\\kappa ) + \\frac{1}{\\tilde{\\beta }_\\mathrm {mic}^2} \\frac{\\pi ^2 l_0^2 \\rho _0^\\star (l_0)}{6}+ \\mathcal {O}(\\tilde{\\beta }_\\mathrm {mic}^{-4})\\:,$ thus $\\tilde{\\beta }_\\mathrm {mic}\\simeq \\pi l_0 \\sqrt{\\frac{\\rho _0^\\star (l_0)}{6 (s - s_{0}(\\kappa )) }}\\quad \\text{as} \\quad s \\rightarrow s_{0}(\\kappa )\\:.$ A similar computation allows to estimate the integral in Eq.",
"(REF ).",
"Finally, we obtain : $\\Phi _0(\\kappa ;s) \\underset{s \\rightarrow s_0}{\\simeq }C_\\kappa - \\pi l_0 \\sqrt{\\frac{2}{3} \\rho _0^\\star (l_0) (s - s_{0}(\\kappa )) }\\:,$ where $C_\\kappa =-\\kappa \\ln \\kappa - (1-\\kappa ) \\ln (1-\\kappa )>0$ .",
"The case $\\tilde{\\beta }_\\mathrm {mic}\\rightarrow -\\infty $ (corresponding to $s \\rightarrow s_1(\\kappa )$ ) can be treated in the same way.",
"We obtain : $\\Phi _0(\\kappa ;s) \\underset{s \\rightarrow s_1}{\\simeq }C_\\kappa - \\pi \\tilde{l}_0 \\sqrt{\\frac{2}{3} \\rho _0^\\star (\\tilde{l}_0) (s_{1}(\\kappa ) - s) }\\:,$ where $\\tilde{l}_0$ is now fixed by $\\int _{x_-}^{\\tilde{l}_0} \\rho _0^\\star (x) {\\rm d}x = \\kappa \\:.$ Figure: Large deviation function Φ 0 \\Phi _0 (entropy of the fermions) describing in particular the typical fluctuations of the truncated linear statistics.The cases κ=0.5\\kappa =0.5 and κ=0.7\\kappa =0.7 are both represented.The dashed horizontal lines delimit the maximum -κlnκ-(1-κ)ln(1-κ)-\\kappa \\ln \\kappa - (1-\\kappa ) \\ln (1-\\kappa ) of these functions.Note that $\\Phi _0$ reaches its maximal value $C_\\kappa $ for $s = s_{0}(\\kappa )$ and $s_1(\\kappa )$ .",
"This particular value has a very simple meaning : $P_{N,\\kappa }(s)$ is the (normalized) number of configurations among the $\\binom{N}{p}$ which give the value $s$ .",
"For $s = s_{0}(\\kappa )$ or $s_1(\\kappa )$ , only one configuration exists, corresponding either to selecting the largest or the smallest eigenvalues.",
"Therefore, $P_{N,\\kappa }(s_{0}(\\kappa )) \\simeq 1/\\binom{N}{p}$ .",
"In the large $N$ limit, $-\\frac{1}{N} \\ln P_{N,\\kappa }(s_{0}(\\kappa )) \\simeq \\frac{1}{N} \\ln \\binom{N}{p}\\simeq -\\kappa \\ln \\kappa - (1-\\kappa ) \\ln (1-\\kappa )\\equiv C_\\kappa \\:,$ and similarly for $s_1(\\kappa )$ .",
"The function $\\Phi _0(\\kappa ;s)$ is plotted in Fig.",
"REF ."
],
[
"Phase III ($s>s_1(\\kappa )$ ) : mixed picture",
"The distribution of the truncated linear statistics (REF ) for $s < s_1(\\kappa )$ was determined by considering two different situations.",
"First, for $s < s_0(\\kappa )$ , the fraction $\\kappa $ of the largest eigenvalues detach from the others, as we have seen in section .",
"When $s$ is increased, these two bulks of eigenvalues move closer, until they merge for $s = s_0(\\kappa )$ (Fig.",
"REF , left).",
"Then, when $s$ is further increased in the interval $[s_0(\\kappa ),s_1(\\kappa )]$ , the density of eigenvalues is frozen and the occupation numbers $\\lbrace n_i \\rbrace $ fluctuate, as it was shown in section .",
"When $s$ reaches $s_1(\\kappa )$ , these occupation numbers no longer fluctuate and only the smallest eigenvalues are selected (Fig.",
"REF , right).",
"In the domain $s > s_1(\\kappa )$ we could naively expect a similar scenario as for the tail $s \\rightarrow 0$ , namely to detach the fraction $\\kappa $ of the smallest eigenvalues.",
"However, in the case $f(x) = 1/x$ , there is another scenario more favorable energetically.",
"It is enough to detach one eigenvalue (the smallest, $x_N$ ) to obtain large values of $s$ : a “bulk” of $N-1$ eigenvalues can remain frozen while only one eigenvalue contributes to the $s$ -dependent part of the energy, which thus scales as $\\sim N$ instead of $\\sim N^2$ as in Phase I.",
"We write $s = \\frac{1}{N x_N} + \\frac{1}{N} \\sum _{i=1}^{N-1} \\frac{n_i}{x_i}\\:.$ If the smallest eigenvalue $x_N$ scales as $N^{-1}$ , it gives a “macroscopic” contribution of the same order as the other $N-1$ eigenvalues which remain of order $N^0$ .",
"Because $x_N$ is much smaller that the other eigenvalues in the limit $N \\rightarrow \\infty $ , the constraint $x_N < x_{N-1}$ in the multiple integrals (REF ) (for $f(\\lambda )=1/\\lambda $ ) plays no role.",
"We estimate the integral over $\\lbrace x_1, \\cdots , x_{N-1} \\rbrace $ with a saddle point method.",
"The saddle point denoted $\\lbrace x_i^\\star \\rbrace $ , corresponds to the minimum of the energy $ \\mathcal {E}_\\mathrm {gas}[\\lbrace x_i \\rbrace ]$ , Eq.",
"(REF ).",
"Thus, these eigenvalues are frozen and distributed according to the Marčenko-Pastur density $\\rho _0^\\star $ .",
"The corresponding energy is : $\\mathcal {E}_\\mathrm {gas}[\\lbrace x_1^\\star , \\cdots x_{N-1}^\\star , x_N \\rbrace ] \\simeq {E}[\\rho _0^\\star ]- \\frac{1}{N} \\ln x_N+ \\frac{{C}}{N}\\:,$ where the constant ${C} = -1 - 2 \\ln 2$ arises from a careful treatment of $1/N$ corrections [26].",
"Taking into account the contribution of the denominator in (REF ), we obtain : $P_{N,\\kappa }(s) \\sim \\sum _{ \\lbrace n_i \\rbrace }\\int {\\rm d}x_N \\,\\mathrm {e}^{\\frac{\\beta N}{2} (\\ln x_N - {C}) }\\delta \\left(s - \\frac{1}{N x_N} - \\frac{1}{N} \\sum _{i=1}^{N-1} \\frac{n_i}{x_i^\\star }\\right)\\,\\delta _{p,\\sum _i n_i}\\:.$ Note that the substitution (REF ) in the multiple integrals in (REF ) give contributions of the entropy of the Coulomb gas $(1-\\beta /2){S}[\\rho ]$ also of order $N^1$ , however the leading term of such contributions are the same in the numerator and the denominator and thus cancel.",
"Integration over the last eigenvalue is straightforward : $P_{N,\\kappa }(s) \\sim \\sum _{\\lbrace n_i \\rbrace } \\exp \\left\\lbrace -\\frac{\\beta N}{2}\\left[\\ln \\left(N s - \\sum _{i=1}^{N-1} \\frac{n_i}{x_i^\\star }\\right)+ {C} \\right]\\right\\rbrace \\delta _{p,\\sum _i n_i}\\:.$ In order to compute this sum, let us rewrite it as $P_{N,\\kappa }(s) \\sim \\int {\\rm d}u \\:\\mathrm {e}^{-\\frac{\\beta N}{2} (\\ln [N(s - u)] + {C} )}\\sum _{\\lbrace n_i \\rbrace }\\delta \\left(u - \\frac{1}{N} \\sum _{i=1}^{N-1} \\frac{n_i}{x_i^\\star }\\right)\\delta _{p,\\sum _i n_i}\\:.$ We now recognize the distribution of $s$ computed in section : $\\sum _{\\lbrace n_i \\rbrace }\\delta \\left(u - \\frac{1}{N} \\sum _{i=1}^{N-1} \\frac{n_i}{x_i^\\star }\\right)\\delta _{p,\\sum _i n_i}\\sim e^{- N \\Phi _0(\\kappa ;u)}\\:,$ where $\\Phi _0$ is given by (REF ).",
"Therefore, (REF ) reduces to $P_{N,\\kappa }(s) \\sim \\int {\\rm d}u \\:\\exp \\left\\lbrace -\\frac{\\beta N}{2} ( \\ln [N(s - u)] + {C} ) - N \\Phi _0(\\kappa ;u)\\right\\rbrace \\:.",
"$ This last integral can be computed using a saddle point method.",
"The saddle point $\\tilde{s}(s)$ is given by : $\\frac{\\beta }{2} \\frac{1}{\\tilde{s} - s}+ \\left.",
"\\frac{\\mathrm {d}\\Phi _0(\\kappa ;u)}{\\mathrm {d}u} \\right|_{\\tilde{s}(s)}= 0\\:.$ The quantity $\\tilde{s}$ represents the contribution of the frozen bulk to the TLS.",
"Its limiting behaviours are found by using (REF ) and (REF ), where the latter is rewritten $\\Phi _0(\\kappa ;s)\\simeq C_\\kappa -\\tilde{c}_+\\sqrt{s_1(\\kappa )-s}$ .",
"Some elementary algebra gives $\\tilde{s}(s) \\simeq {\\left\\lbrace \\begin{array}{ll}s_1(\\kappa ) - \\left(\\frac{\\tilde{c}_+}{\\beta }\\right)^2 (s-s_1(\\kappa ))^2 & \\mbox{for } s\\rightarrow s_1(\\kappa )^+\\\\[0.25cm]\\kappa + \\frac{\\beta \\kappa (1-\\kappa )}{2(s-\\kappa )} + \\mathcal {O}(s^{-3})& \\mbox{for } s\\rightarrow \\infty \\end{array}\\right.",
"}$ Finally, the probability is given by : $\\boxed{P_{N,\\kappa }(s)\\underset{N\\rightarrow \\infty }{\\sim }N^{-\\beta N/2}\\exp \\left\\lbrace - N \\left[\\frac{\\beta }{2} \\Phi _+(\\kappa ;s) + \\Phi _0(\\kappa ,\\tilde{s}(s)) \\right]\\right\\rbrace \\hspace{7.11317pt}\\text{for}\\hspace{7.11317pt}s > s_1(\\kappa )}$ where the large deviation function is $\\Phi _+(\\kappa ;s) = N( \\mathcal {E}_\\mathrm {gas}[\\lbrace x_n \\rbrace ] - \\mathcal {E}[\\rho _0^\\star (x)])-\\ln N= \\ln (s - \\tilde{s}(s)) +{C}\\:,$ where ${C}=- 1 - 2 \\ln 2$ .",
"There are two different contributions to the distribution of $s$ : the energy of the isolated eigenvalue $x_N$ , encoded in $\\Phi _+$ ; the entropy associated to the possible configurations $\\lbrace n_i \\rbrace $ associated to the $N-1$ frozen eigenvalues, encoded in $\\Phi _0$ .",
"The latter was not present in the case $\\kappa =1$ studied in Ref.",
"[48] because all the occupation numbers were fixed to $n_i = 1$ and do not bring any additional entropy."
],
[
"Behaviour near the edge $s_1(\\kappa )$",
"Using (REF ), $\\tilde{s} \\simeq s_1(\\kappa )$ , gives $\\Phi _+(\\kappa ;s) \\simeq \\ln (s - s_1(\\kappa ) )+ \\mathrm {cste}\\:,$ and $\\Phi _0(\\kappa ; \\tilde{s}(s)) \\rightarrow C_\\kappa \\:,$ where $C_\\kappa >0$ was defined in Eq.",
"(REF ).",
"Therefore, the leading order term $\\Phi _+(\\kappa ;s) + \\frac{2}{\\beta } \\Phi _0(\\kappa ;\\tilde{s}(s))\\simeq \\ln (s - s_1(\\kappa ))+ \\mathrm {cste}\\quad \\text{for}\\quad s \\rightarrow s_1(\\kappa )$ is independent of the Dyson index $\\beta $ , which only appears in the constant."
],
[
"Tail $s \\rightarrow \\infty $",
"The limiting behaviour (REF ), $\\tilde{s} \\simeq \\kappa $ , shows that the entropy no longer plays a role and the contribution of the energy reads : $\\Phi _+(\\kappa ;s) \\simeq \\ln (s - \\kappa ) + \\mathcal {O}(s^{-2})\\:,\\hspace{14.22636pt}\\text{for } s \\rightarrow \\infty \\:,$ which is again independent of $\\beta $ .",
"This gives the right tail of the distribution : $P_{N,\\kappa }(s) \\underset{s \\rightarrow \\infty }{\\sim } s^{-\\beta N/2}\\:.$ We have recovered the tail obtained in Ref.",
"[48] for $\\kappa =1$ : $P_{N,1}(s) \\sim (s-1)^{-\\beta N/2}$ for $s-1 \\gg \\sqrt{\\ln N/N}$ , as expected since this limit is described within the same scenario (one isolated charge and a frozen bulk).",
"From the physical point of view, this power law tail is interpreted as the manifestation of a very narrow resonance which dominates the sum of proper times."
],
[
"A new universal scenario",
"As briefly reviewed in the introduction, the study of linear statistics has played a very important role in random matrix theory.",
"In this article we have introduced a new type of question by considering the statistical analysis of partial sums of eigenvalues $\\sum _{i=1}^p f(\\lambda _i)$ , so-called “truncated linear statistics” (TLS), with $\\kappa =p/N<1$ , where $N$ is the size of the matrices.",
"This question has been set within the Laguerre ensemble for $f(\\lambda )=1/\\lambda $ .",
"The study of this particular model was motivated by the analysis of partial sums of proper time delays in chaotic scattering, which are characteristic times of the scattering process playing an important role in electronic transport.",
"The specificity of the problem has led us to introduce two sets of random variables, the ordered eigenvalues $\\lambda _1>\\lambda _2>\\cdots >\\lambda _N$ and the “occupation numbers” $\\lbrace n_i\\rbrace _{i=1,\\cdots ,N}$ , with $n_i=0$ or 1, leading to express the TLS as $s=\\sum _{i=1}^Nn_i\\,f(\\lambda _i)$ .",
"Correspondingly, we have introduced two different physical interpretations : the eigenvalues $\\lbrace \\lambda _i\\rbrace $ can be interpreted as the positions of a one-dimensional gas of particles with long range (logarithmic) interactions, as usual in such problems.",
"The variables $\\lbrace n_i\\rbrace $ can be viewed as occupation numbers for $p$ fictitious non-interacting fermions in $N$ “energy levels” $\\lbrace \\varepsilon _i=f(\\lambda _i) \\rbrace $ .",
"We have determined the large $N$ behaviour of the distribution $P_{N,\\kappa }(s)$ of the TLS, The numerical analysis of § REF has shown that the large $N$ result describes very well the distribution already for $N\\gtrsim 50$ .",
"which was shown to be controlled by an optimal configuration (solution of saddle point equations), characterised by the density $\\rho ^\\star (x;\\kappa ,s)$ of eigenvalues (with $\\lambda =N x$ for the Laguerre ensemble), while the optimal occupations are given by the Fermi-Dirac distribution for an effective temperature, Eq.",
"(REF ).",
"We have shown that varying the two parameters $(\\kappa ,s)$ drives phase transitions in this optimal configuration.",
"In the absence of any constraint on the eigenvalue density, the (most probable) density is the Marčenko-Pastur distribution (REF ) (Fig.",
"REF ).",
"For a fixed $\\kappa $ , this density can be associated to a whole spectrum of values of the TLS $s\\in [s_0(\\kappa ),s_1(\\kappa )]$ , corresponding to different choices for the occupation numbers $\\lbrace n_i\\rbrace $ .",
"The two extreme situations correspond to choosing the $p$ largest or the $p$ smallest eigenvalues : cf.",
"Fig.",
"REF .",
"This phase exists in a domain of the phase diagram bounded by two lines $s_1(\\kappa )$ and $s_0(\\kappa )$ (Fig.",
"REF ).",
"The energy of the Coulomb gas is frozen and the structure of the distribution is entirely due to the entropy of the $p$ fictitious non-interacting fermions, which scales as $\\sim N$ and is independent of the symmetry index $\\beta $ , thus $P_{N,\\kappa }(s)\\sim \\exp \\left\\lbrace - N \\Phi _0(\\kappa ;s) \\right\\rbrace $ , where the large deviation function $\\Phi _0(\\kappa ;s)$ has been determined in Section .",
"As a consequence of this scaling, the relative fluctuations of the TLS are of order $\\sim 1/\\sqrt{N}$ (whereas full linear statistics usually present relative fluctuations $\\sim 1/N$ as a consequence of the strong correlations in the Coulomb gas).",
"As the limits $s\\rightarrow s_0$ and $s\\rightarrow s_1$ correspond to select the contributions of the largest or the smallest eigenvalues (Fig.",
"REF ), they are associated to effective (absolute) fermionic temperature $1/\\tilde{\\beta }$ positive or negative, respectively (see also Fig.",
"REF ).",
"This new scenario is completly universal (independent of the function $f$ and the specific matrix ensemble)."
],
[
"Phase I.—",
"When $s$ reaches the lower boundary $s_0(\\kappa )$ of the central domain of the phase diagram (Fig.",
"REF ), the effective temperature is fixed to $0^+$ and only the largest eigenvalues contribute to the TLS (Fig.",
"REF ).",
"Smaller value $s<s_0(\\kappa )$ can only be realised by a deformation of the optimal density $\\rho ^\\star (x;\\kappa ,s)$ , which is split in two bulks, while occupations are frozen ($1/\\tilde{\\beta }=0^+$ ).",
"The weight of the optimal configuration is now controlled by the energy of the Coulomb gas.",
"In the $s=\\sum _{i=1}^p\\lambda _i^{-1}\\rightarrow 0$ limit, we have $\\lambda _i\\sim p/s$ , hence the energy is dominated by the confinment energy : $E_N=N^2\\mathcal {E}_\\mathrm {gas}\\sim p^2/s$ .",
"This corresponds to the behaviour $P_{N,\\kappa }(s)\\sim \\exp [-(\\beta /2)N^2\\Phi _-(\\kappa ;s)]$ with $\\Phi _-(\\kappa ;s)\\simeq \\kappa ^2/s$ (see Section )."
],
[
"Phase III.—",
"The case where $s>s_1(\\kappa )$ was analysed with a mixed scenario.",
"As for $\\kappa =1$ [48], one remarks that it is sufficient to split off a single eigenvalue from the bulk, the smallest one $\\lambda _N$ , in order to generate large values of the TLS $s=1/\\lambda _N+\\sum _{i=1}^{N-1}n_i/\\lambda _i$ : i.e.",
"it is more favorable energetically to send one charge towards 0 as $s$ grows, which generates an energy cost $E_N\\sim N$ , than move the full bulk, which would generate a cost $E_N\\sim N^2$ .",
"When the charge is split off the bulk, there is also an entropic contribution $\\sim N$ due to the fluctuating $N-1$ occupation numbers related to the remaining $N-1$ eigenvalues, of the same order as the interaction energy between the isolated eigenvalue and the frozen $N-1$ ones.",
"This part of the scenario is specific to the function $f(\\lambda )=1/\\lambda $ that we have considered.",
"For the case considered in the paper, we have obtained three phases (Fig.",
"REF ), however one can imagine that for another function $f$ , the phase diagram would involve more than three phases : the exhaustive study of Ref.",
"[55] for several linear statistics illustrates than the number of different phases depend on the function $f$ (see also the Table in the conclusion of Ref.",
"[25])."
],
[
"Crossover across $s_0(\\kappa )$ and second order freezing transition",
"As the two phases I and II involve two different scalings with $N$ , see (REF ), it is interesting to discuss further the transition at $s=s_0$ , in the vicinity of which the distribution presents the limiting behaviours $P_{N,\\kappa }(s) \\underset{s \\rightarrow s_0}{\\sim }\\mathrm {e}^{-N C_\\kappa }\\left\\lbrace \\begin{array}{lc}\\mathrm {e}^{-(\\beta /2)N^2\\Phi _-(\\kappa ;s)}\\simeq \\mathrm {e}^{- N^2 c_-(s - s_0(\\kappa ))^2}& \\text{ for } s < s_0(\\kappa )\\\\\\mathrm {e}^{-N[\\Phi _0(\\kappa ;s)-C_\\kappa ]}\\simeq \\mathrm {e}^{+N c_+\\sqrt{s - s_0(\\kappa )}}& \\text{ for } s > s_0(\\kappa )\\end{array}\\right.\\:,$ where $c_-=(\\beta /2)\\omega _\\kappa $ , cf.",
"Eq.",
"(REF ), and $c_+$ is defined in Eq.",
"(REF ).",
"From the point of view of the Coulomb gas, $s=s_0(\\kappa )$ corresponds to a freezing transition : below $s_0(\\kappa )$ , the energy scales as $E_N\\sim N^2 (s - s_0(\\kappa ))^2$ , whereas the energy is frozen above the transition.",
"This corresponds to a second order phase transition (a second order freezing transition of different nature was identified in [48], and also in [41], [42] although the transition was not interpreted along these lines in this latter reference).",
"Note that the energy of the Coulomb gas, Eq.",
"(REF ), is $\\mathcal {E}_\\mathrm {gas}\\sim 1/N$ everywhere above the line $s_0(\\kappa )$ .",
"In the thermodynamic limit $N\\rightarrow \\infty $ , $\\mathcal {E}_\\mathrm {gas}$ becomes flat above this line and all derivatives of the energy are trivially continuous across $s_1(\\kappa )$ .",
"A finer description of the transition (beyond the thermodynamic limit) requires to identify the non trivial scaling with $N$ for the crossover, given by $N^2 (s - s_0(\\kappa ))^2 \\sim N \\sqrt{s - s_0(\\kappa )}\\hspace{14.22636pt}\\Rightarrow \\hspace{14.22636pt}s - s_0(\\kappa ) \\sim N^{-2/3}\\:.$ This leads to introduce a scaling variable $z$ and rewrite the crossover between the two behaviours (REF ) in terms of a unique function $P_{N,\\kappa }(s) \\underset{s \\rightarrow s_0(\\kappa )}{\\sim } \\mathrm {e}^{- N^{2/3} \\Psi (z)}\\hspace{28.45274pt}\\mbox{with }s = s_0(\\kappa ) + N^{-2/3} z\\:.$ The scaling function $\\Psi (z)$ describes the detail of the crossover for finite $N$ , at a scale $\\delta s\\sim N^{-2/3}$ , and thus presents the limiting behaviours $\\Psi (z) \\simeq \\left\\lbrace \\begin{array}{cc}c_-z^2 & \\text{ for } z \\rightarrow -\\infty \\:,\\\\-c_+\\sqrt{z} & \\text{ for } z \\rightarrow +\\infty \\:.\\end{array}\\right.$ Similar considerations for the distribution of the largest eigenvalue in various matrix ensembles (see the review [34]) have shown that the corresponding function is universal (Tracy Widom distribution given by Painlevé transcendents).",
"An interesting challenging open question would be to determine whether the function $\\Psi (z)$ has also a universal character and find its precise nature."
],
[
"Other open questions",
"The discussion concerning the crossover around $s_0(\\kappa )$ cannot be extended to the other phase boundary at $s_1(\\kappa )$ , as it is not clear how the limiting behaviours obtained above, $-(1/N)\\ln P_{N,\\kappa }(s)\\simeq C_\\kappa -c_+\\sqrt{s_1-s}$ for $s<s_1$ and $-(1/N)\\ln P_{N,\\kappa }(s)\\simeq C_\\kappa +(\\beta /2)\\ln \\big [N(s-s_1)\\big ]$ for $s>s_1$ , can be matched.",
"Due to the $N$ in the argument of the logarithm, it does not seem that the crossover can be described by a universal function as for $s\\sim s_0$ .",
"A precise description of the crossover seems therefore even more challenging in this case.",
"Another obvervation concerning finite $N$ corrections to the thermodynamics properties is the following : our approach is suitable to obtain results in the thermodynamic limit, when both $N$ and $p$ are of the same order (in particular the case $p=1$ would require to adapt the method like it was done in Refs.",
"[19], [20], [53], [36], [35], [34]).",
"Also the limit $\\kappa \\rightarrow 1$ is non trivial and can lead to singular behaviours, as it was illustrated in the conclusion of [25].",
"This is clear from our fermionic picture as fermions and holes play symmetric role, so that the two limits $\\kappa \\sim 1/N$ and $1-\\kappa \\sim 1/N$ should be equally difficult to analyse.",
"The new universal scenario introduced in the paper was illustrated for a particular ensemble (Laguerre) and a specific monotonic function $f$ .",
"We argue now that the same main features would also occur for a non monotonic function.",
"Consider for concreteness the Jacobi case, $\\lambda _i\\in [0,1]$ and the function $f(\\lambda )=\\lambda \\,(1-\\lambda )$ (when $\\sum _{i=1}^N\\lambda _i(1-\\lambda _i)$ is the shot noise power of chaotic quantum dots [1]).",
"In Phase II, due to the fact that the “energy levels” $\\varepsilon _i=f(\\lambda _i)$ are non monotonic as a function of the eigenvalues of the random matrix, the occupations controlled by the Fermi-Dirac distribution $\\left\\langle n_i\\right\\rangle $ would also be represented by a non monotonic function of $\\lambda $ (a similar feature occurs when representing the occupation of plane waves by one-dimensional non-interacting fermions : occupation is a monotonic function of the energy $\\varepsilon _k=k^2$ but a non monotonic function of the wave vector $k$ ).",
"The limit $s=(1/N)\\sum _in_if(\\lambda _i)\\rightarrow s_0^+$ (lower boundary of Phase II) should correspond to “cold fermions” with $\\tilde{\\beta }\\rightarrow +\\infty $ where the occupation function $\\left\\langle n_i\\right\\rangle $ selects both the smallest and the largest eigenvalues.",
"The large deviations for $s<s_0$ (Phase I) should correspond then to the opening of two gaps.",
"The upper boundary of Phase II, $s\\rightarrow s_1^-$ , corresponds to “hot fermions” with negative temperature (i.e.",
"$\\tilde{\\beta }\\rightarrow -\\infty $ ) involving the eigenvalues in the middle of the interval $[0,1]$ ; the large deviations for $s>s_1$ should also correspond to the opening of two gaps.",
"Such a situation for a non monotonic $f$ could certainly be worth studying in details.",
"We have pointed out the connection between the problem studied in the paper and the “thinned ensembles” [3], [9], [2], [32], Eq.",
"(REF ).",
"It is clear that the correspondence between the two situations (fixed versus fluctuating $p$ ) requires $\\kappa =p/N$ being the probability of the thinned ensemble.",
"Because the large deviation function controlling the distribution of the truncated linear statistics has the interpretation of a thermodynamic potential and has been computed in the thermodynamic limit, one could naively expect that the two situations are equivalent.",
"However, the equivalence of the statistical physics ensembles is far from being obvious in these log correlated gases and may be worth to explore more into detail (note that, in our paper, the equivalence between grand canonical, canonical and microcanonical ensembles has been extensively used only for the non interacting fictitious fermions, i.e.",
"only in the central region of the phase diagram (Fig.",
"REF ) when eigenvalues are frozen).",
"Coming back to the initial motivation within chaotic scattering, the paper has mostly focused on the question of the partial sums of proper time delays.",
"We have also emphasized the role of the Wigner-Smith matrix' diagonal elements $\\mathcal {Q}_{ii}$ , which are important for applications in quantum transport : the statistical properties of the partial sum of these matrix elements were analysed in the appendix for $\\beta =2$ , based on the relation between $\\mathcal {Q}_{ii}$ and partial time delays $\\tilde{\\tau }_i$ [46].",
"Because the joint distribution of the $\\mathcal {Q}_{ii}$ 's is still unknown, a general analysis of their partial sums in the general case, beyond the two first moments, remains a challenging question."
],
[
"Acknowledgements",
"We are indebted to Dmitry Savin for many useful discussions on chaotic scattering."
],
[
"Distribution of the injectance for $\\beta =2$",
"In this section, we derive the distribution of the injectance of a quantum dot defined in section REF : $\\overline{\\nu }_\\alpha = \\frac{1}{2\\pi } \\sum _{i\\in \\mathrm {contact}\\:\\alpha }\\mathcal {Q}_{ii}\\:.$ We will restrict our study to the situation where $\\beta =2$ because the joint distribution of the $\\mathcal {Q}_{ii}$ 's is known only in this case.",
"We will derive the distribution of $\\overline{\\nu }_\\alpha $ using a Coulomb gas approach.",
"The computation is very similar to the one described in [48] where the Wigner time delay (case where all contacts contribute) is studied.",
"We start from the distribution of a subblock $\\mathcal {Q}_{p}$ of size $ p\\times p$ on the diagonal of $\\mathcal {Q}$ given in [46] for $\\beta =2$ : $P(\\mathcal {Q}_{p}^{-1}) \\propto (\\det \\mathcal {Q}_{p})^{-N} \\mathrm {e}^{- \\mathop {\\mathrm {tr}}\\nolimits \\left\\lbrace \\mathcal {Q}_{p}^{-1} \\right\\rbrace } \\:.$ If we suppose that the contact $\\alpha $ has $p$ open channels, we can rewrite the injectance as : $\\overline{\\nu }_\\alpha = \\frac{1}{2\\pi } \\mathop {\\mathrm {tr}}\\nolimits \\left\\lbrace \\mathcal {Q}_{p} \\right\\rbrace \\:.$ Denoting $\\lbrace \\lambda _i \\rbrace $ the eigenvalues of $\\mathcal {Q}_{p}^{-1}$ , we can write $\\overline{\\nu }_\\alpha = \\frac{1}{2\\pi } \\sum _{i=1}^{p} \\frac{1}{\\lambda _i} \\:,$ where the joint probability density function of the $\\lbrace \\lambda _i \\rbrace $ is given by $P(\\lambda _1 ,\\cdots , \\lambda _{p}) \\propto \\prod _{i<j} \\left| \\lambda _i - \\lambda _j \\right|^2 \\prod _{i=1}^{p} \\lambda _i^{N} \\mathrm {e}^{-\\lambda _i} \\:.$ Rescale the eigenvalues as $\\lambda _i = px_i$ and introduce the empirical density $\\rho (x) = \\frac{1}{p} \\sum _{i=1}^{p} \\delta (x - x_i) \\:.$ The distribution of $s = 2\\pi \\overline{\\nu }_\\alpha = \\frac{1}{p} \\sum _{i=1}^{p} \\frac{1}{x_i}= \\int \\frac{\\rho (x)}{x} {\\rm d}x$ is given by $P_{N,\\kappa }(s) \\simeq \\frac{ \\displaystyle \\int \\mathcal {D}\\rho \\: \\mathrm {e}^{-\\kappa ^2 N^2 {E}[\\rho ]} \\:\\delta \\left(\\int \\rho (x) {\\rm d}x - 1 \\right)\\delta \\left(\\int \\frac{\\rho (x)}{x} {\\rm d}x - s \\right)}{ \\displaystyle \\int \\mathcal {D}\\rho \\: \\mathrm {e}^{- \\kappa ^2 N^2 {E}[\\rho ]} \\:\\delta \\left(\\int \\rho (x) {\\rm d}x - 1 \\right)} \\:,$ where we denoted $\\kappa = p/N$ and introduced the energy (see section ) ${E}[\\rho ] = - \\int {\\rm d}x \\int {\\rm d}y \\: \\rho (x) \\rho (y) \\ln \\left| x-y \\right|+ \\int {\\rm d}x \\left(x - \\frac{1}{\\kappa } \\ln x \\right) \\rho (x)\\:.$ The integrals in (REF ) are dominated by the minimum of the energy under the constraints imposed by the $\\delta $ -functions.",
"Therefore, we introduce the functional ${F}[\\rho ,\\mu _0,\\mu _1] = {E}[\\rho ]+ \\mu _0 \\left( \\int \\rho - 1 \\right)+ \\mu _1 \\left( \\int \\frac{\\rho (x)}{x} {\\rm d}x - s \\right)\\:,$ where $\\mu _0$ and $\\mu _1$ are Lagrange multipliers.",
"Denote $\\rho ^\\star (x;\\kappa ,s)$ the density which dominates the numerator of (REF ).",
"It is given by $\\left.",
"\\frac{\\delta {F}}{\\delta \\rho } \\right|_{\\rho ^\\star } = 0$ , which reads : $2 \\int \\rho ^\\star (y;\\kappa ,s) \\ln \\left| x-y \\right| {\\rm d}y= x - \\frac{1}{\\kappa } \\ln x + \\mu _0 + \\frac{\\mu _1}{x}\\quad \\text{for} \\quad x \\in \\mathrm {Supp}(\\rho ^\\star )\\:.$ It is more convenient to derive this relation with respect to $x$ : $2 \\mathchoice{{\\displaystyle {\\textstyle -}{\\int }}\\hbox{$\\textstyle -$}}{\\hspace{0.0pt}}{-}{.",
"}5$ (x ;,s)x-y dy = 1 - 1 x - 1x2 for x Supp() .",
"The values $\\mu _0^\\star $ and $\\mu _1^\\star $ taken by the Lagrange multipliers are fixed by imposing the constraints $\\int \\rho ^\\star (x;\\kappa ,s) {\\rm d}x = 1 \\:,\\hspace{28.45274pt}\\int \\frac{\\rho ^\\star (x;\\kappa ,s)}{x} {\\rm d}x = s \\:.$ Similarly, denote $\\rho _0^\\star (x)$ the density which dominates the denominator.",
"The distribution of $s$ is then given by : $P_{N,\\kappa }(s) \\sim \\exp \\left\\lbrace - \\kappa ^2 N^2 \\left({E}[\\rho ^\\star (x;\\kappa ,s)]-{E}[\\rho _0^\\star (x)]\\right)\\right\\rbrace \\:.$ The thermodynamic identity (REF ) mentioned in the body of the paper allows a direct computation of the energy via the relation : $\\frac{\\mathrm {d}{E}[\\rho ^\\star (x;\\kappa ,s)]}{\\mathrm {d}s} = - \\mu _1^\\star (\\kappa ;s) \\:.$ Our aim is now to compute the density $\\rho ^\\star $ .",
"As in the body of the paper, depending on the values of the parameters $\\kappa $ and $s$ , we will find different types of densities $\\rho ^\\star $ , which we interpret as different “phases”."
],
[
"Phase I : Solution with one compact support",
"Let us assume that the solution $\\rho ^\\star $ has one compact support $[a,b]$ .",
"Then, Eq.",
"() can be solved using Tricomi's theorem [52].",
"We obtain : $\\rho ^\\star (x;\\kappa ,s) = \\frac{x+c}{2\\pi x^2} \\sqrt{(x-a)(b-x)}\\:.$ It is convenient to parametrize this solution in terms of $u = \\sqrt{a/b}$ .",
"Then, imposing $\\rho ^\\star (a;\\kappa ,s)= \\rho ^\\star (b;\\kappa ,s) = 0$ , along with the constraints (REF ) yields : $v = \\sqrt{a b} = \\frac{2u}{\\kappa } \\frac{(2\\kappa +1)u^2 -2u + (2\\kappa +1)}{(1-u^2)^2}\\:,$ $\\mu _1^\\star =- \\frac{4u^2}{\\kappa ^2}\\frac{((2\\kappa +1)u^2 -2u + (2\\kappa +1))(u^2 -2(2\\kappa +1)u +1)}{(1-u^2)^4}\\:,$ $c = \\frac{\\mu _1^\\star }{v}\\:,$ $s = \\sigma _\\kappa (u) =(1-u)^2\\frac{-u^4 + 4 (3\\kappa +1)u^3 + 2(4\\kappa -3)u^2 + 4(3\\kappa +1)u-1}{16u^2((2\\kappa +1)u^2 -2u + (2\\kappa +1))}\\:.$ Given $\\kappa $ and $s$ , the last equation allows to compute $u$ , from which all the other parameters are deduced.",
"One can check that in the limit $\\kappa \\rightarrow 1$ , one recovers the equations given in [48].",
"The optimal density $\\rho _0^\\star $ for the denominator can be deduced from $\\rho ^\\star $ by releasing the constraint (setting $\\mu _1^\\star = 0$ ).",
"This solution exists as long as $\\rho ^\\star $ is positive, which corresponds to $x+c \\geqslant 0$ .",
"This gives the condition $\\kappa \\geqslant \\frac{1 - 3 u + 3 u^2 - u^3}{2 u (3 + u^2)}\\:,$ which can be rewritten as $s < s_{I}(\\kappa )$ .",
"This corresponds to the lower domain delimited by the upper solid line on figure REF (left)."
],
[
"Typical fluctuations",
"The typical fluctuations are controlled by the minimum of the energy, which is given by $\\mu _1^\\star =0$ .",
"The density is then $\\rho _0^\\star (x) = \\frac{\\sqrt{(x-a_0)(b_0-x)}}{2\\pi x} \\:,$ where $a_0 = \\frac{2 \\kappa +1 - 2\\sqrt{\\kappa (\\kappa +1)}}{\\kappa } \\:,\\quad b_0 = \\frac{2 \\kappa +1 + 2\\sqrt{\\kappa (\\kappa +1)}}{\\kappa } \\:.$ The corresponding value of $s$ is $\\kappa $ , and expanding Eqs.",
"(REF ,REF ) for $s$ close to $\\kappa $ yields : $\\mu _1^\\star = - \\frac{s-\\kappa }{\\kappa ^3(\\kappa +1)} + O((s-\\kappa )^2) \\:.$ The energy is obtained by simple integration, via Eq.",
"(REF ) : ${E}[\\rho ^\\star (x;\\kappa ,s)]-{E}[\\rho _0^\\star (x)]= \\frac{(s-\\kappa )^2}{2\\kappa ^3(\\kappa +1)} + \\mathcal {O}((s-\\kappa )^3) \\:.$ Thus, the distribution of $s$ near $\\kappa $ is given by : $P_{N,\\kappa }(s) \\sim \\exp \\left\\lbrace - N^2 \\frac{(s-\\kappa )^2}{2\\kappa (\\kappa +1)}\\right\\rbrace \\:.$ We recover the leading term of the variance given in the introduction, Eq.",
"(REF ) : $\\mathrm {Var}(s) \\simeq \\frac{\\kappa (1+\\kappa )}{N^2}\\:.$"
],
[
"Limiting behaviour $s \\rightarrow 0$",
"The limit $s \\rightarrow 0$ corresponds to $\\mu _1^\\star \\rightarrow +\\infty $ .",
"Expanding Eqs.",
"(REF ,REF ,REF ,REF ) in this limit gives : $\\mu _1^\\star = \\frac{1}{s^2} - \\frac{\\kappa +2}{2\\kappa } \\frac{1}{s} + \\mathcal {O}(1) \\:.$ Using again Eq.",
"(REF ), we obtain : ${E}[\\rho ^\\star (x;\\kappa ,s)] =\\frac{1}{s} + \\frac{\\kappa +2}{2\\kappa } \\ln s + \\mathcal {O}(1) \\:,$ thus : $P_{N,\\kappa }(s) \\sim s^{-N^2\\kappa (\\kappa +2)/2 } e^{- \\kappa ^2 N^2/s}\\hspace{14.22636pt} \\text{for} \\hspace{14.22636pt}s \\rightarrow 0 \\:.$"
],
[
"Phase II : Solution with an isolated eigenvalue",
"As for the Wigner time delay [48] ($\\kappa =1$ ) and in Section , we look for a solution with an isolated eigenvalue : $\\rho ^\\star (x;\\kappa ,s) = \\frac{1}{p}\\, \\delta (x-x_1) + \\tilde{\\rho }(x)\\:,$ with now $\\int \\tilde{\\rho } = 1-1/p$ .",
"The minimization of ${F}$ with respect to $\\tilde{\\rho }$ reads : $2 \\mathchoice{{\\displaystyle {\\textstyle -}{\\int }}\\hbox{$\\textstyle -$}}{\\hspace{0.0pt}}{-}{.",
"}5$ (y)x-y dy = 1 - 1x - 1x2 - 2p 1x-x1 , for x Supp() .",
"And minimization with respect to $x_1$ gives $2 \\int \\frac{\\tilde{\\rho }(y)}{x_1 - y} = 1 - \\frac{1}{\\kappa x_1} - \\frac{\\mu _1}{x_1^2} \\:,\\hspace{14.22636pt} \\text{for} \\hspace{14.22636pt} x \\in \\mathrm {Supp}(\\tilde{\\rho }) \\:.$ In addition, the constraint becomes : $s = \\frac{1}{px_1} + \\int \\frac{\\tilde{\\rho }(x)}{x} {\\rm d}x \\:.$ For $x_1$ to give a “macroscopic” contribution to $s$ , we must have $x_1 = \\mathcal {O}(N^{-1})$ .",
"Then equation (REF ) imposes $\\mu _1 = -x_1/\\kappa + \\mathcal {O}(p^{-2})$ .",
"Finally, at leading order in $p$ , we get : $\\tilde{\\rho }(x) = \\rho _0^\\star (x) + \\mathcal {O}(p^{-1}) \\:,$ $x_1 = \\frac{1}{p(s-\\kappa )} + \\mathcal {O}(p^{-2}) \\:,$ $\\mu _1^\\star = - \\frac{1}{\\kappa p} \\frac{1}{s - \\kappa } + \\mathcal {O}(p^{-2}) \\:.$"
],
[
"Domain of validity",
"This solution remains valid as long as the separate eigenvalue is away from the bulk, namely $x_1 < a_0$ .",
"This gives the condition $s > \\kappa + \\frac{1}{pa_0} = s_{II}(\\kappa ) \\:.$ This corresponds to the upper domain represented on figure REF (left).",
"Note that $s_{II}(\\kappa ) \\rightarrow \\kappa $ as $N \\rightarrow \\infty $ ."
],
[
"Large deviation function",
"The energy can be computed analytically at leading order : $\\mathcal {E}[\\rho ^\\star (x;\\kappa ,s)] - \\mathcal {E}[\\rho _0^\\star ]= \\frac{1}{\\kappa p} \\ln [p(s - \\kappa )]+ \\frac{{C}}{p}+ \\mathcal {O}(p^{-2}).$ where ${C}=-1 - 2 \\ln 2$ was introduced above.",
"The constant term is the same as the one appearing in section and Ref. [48].",
"It arises from corrections of order $p^{-1}$ to the density $\\rho ^\\star $ [26].",
"From the expression of the energy, we deduce the expression of the distribution of $s$ , for $s > s_c$ : $P_{N,\\kappa }(s) \\sim (s-\\kappa )^{- N}\\:.$ The simplification $p^2/(\\kappa p)=N$ has thus produced the same exponent as for the tail of the distribution of the sum (truncated or not) of proper times.",
"This is explained from the interpretation of Ref.",
"[46], where it was shown that $\\mathcal {Q}_{ii}$ coincides with the partial time delay for $\\beta =2$ .",
"Figure: Left : Domains of existence of each solution, for N=200N=200.The upper solid line corresponds to s=s I (κ)s = s_{I}(\\kappa ) and delimits the domain of existence of phase I.The lower solid line is s=s II (κ)s = s_{II}(\\kappa ).",
"Phase II exists above this line.As N→∞N \\rightarrow \\infty , it goes to the dashed line s=κs = \\kappa , where occurs the transition in the thermodynamic limit.Right : phase diagram for the Coulomb gas."
],
[
"Summary",
"We obtained the following scalings for the distribution : $P_{N,\\kappa }(s) \\underset{N \\rightarrow \\infty }{\\sim }{\\left\\lbrace \\begin{array}{ll}\\exp \\left\\lbrace - \\kappa ^2 N^2 \\Psi _-(\\kappa ;s) \\right\\rbrace \\hspace{14.22636pt}& \\text{for } s < s_{I}(\\kappa )\\\\[0.2cm](\\kappa N)^{-N}\\exp \\left\\lbrace - N \\Psi _+(\\kappa ;s) \\right\\rbrace \\hspace{14.22636pt}& \\text{for } s > s_{II}(\\kappa )\\end{array}\\right.",
"}$ where the large deviation $\\Psi _+$ is given by : $\\Psi _+(\\kappa ;s) = \\ln (s - \\kappa ) - \\kappa (1 + 2 \\ln 2)\\:,$ and $\\Psi _-$ has the following limiting behaviours : $\\Psi _-(\\kappa ;s) \\simeq {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{1}{s} + \\frac{\\kappa +2}{2\\kappa } \\ln s + \\mathcal {O}(1)\\hspace{14.22636pt}& \\text{as } s \\rightarrow 0\\\\[0.2cm]\\displaystyle \\frac{(s-\\kappa )^2}{2\\kappa ^3(\\kappa +1)} + \\mathcal {O}((s-\\kappa )^3)\\hspace{14.22636pt}& \\text{as } s \\rightarrow \\kappa \\end{array}\\right.",
"}$ The precise point $s_t$ where the transition between the two phases occurs can be obtained by matching the two large deviation functions.",
"One can show that, for $N \\rightarrow \\infty $ , $s_t \\rightarrow \\kappa $ .",
"Therefore for large $N$ we have, for $s$ close to $\\kappa $ : $\\lim _{N \\rightarrow \\infty } - \\frac{1}{N^2} \\ln P_{N,\\kappa }(s) \\simeq \\left\\lbrace \\begin{array}{cl}\\displaystyle \\frac{(s-\\kappa )^2}{2\\kappa ^3(\\kappa +1)}&\\text{ for } s < \\kappa \\\\[0.3cm]0 & \\text{ for } s > \\kappa \\end{array}\\right.$ This corresponds to a second order phase transition.",
"This was already the case in Ref.",
"[48] in the study of the full linear statistics ($\\kappa =1$ )."
]
] | 1612.05469 |
[
[
"Divergences in Holographic Complexity"
],
[
"Abstract We study the UV divergences in the action of the \"Wheeler-de Witt patch\" in asymptotically AdS spacetimes, which has been conjectured to be dual to the computational complexity of the state of the dual field theory on a spatial slice of the boundary.",
"We show that including a surface term in the action on the null boundaries which ensures invariance under coordinate transformations has the additional virtue of removing a stronger than expected divergence, making the leading divergence proportional to the proper volume of the boundary spatial slice.",
"We compare the divergences in the action to divergences in the volume of a maximal spatial slice in the bulk, finding that the qualitative structure is the same, but subleading divergences have different relative coefficients in the two cases."
],
[
"Introduction",
"In principle, holography provides a well-defined non-perturbative formulation of quantum gravity.",
"But to really use it to address questions about the nature of spacetime, we need to understand the emergence of the bulk spacetime from the dual field theory description.",
"In [1], Susskind conjectured a new relation between the bulk geometry and the complexity of the dual boundary state.",
"The quantum computational complexity is a measure of the minumum number of elementary gates needed in a quantum circuit which constructs a given state starting from a specified simple reference state (see e.g.",
"[2]).",
"This proposal was refined in [3] into the conjecture that the computational complexity of the boundary state at a given time (on some spacelike slice of the boundary) could be identified with the volume of a maximal volume spacelike slice in the bulk, ending on the given boundary slice.",
"This will be referred to as the CV conjecture.",
"This was further developed in [4], [5].",
"More recently, it was conjectured that the complexity is related instead to the action of a Wheeler-de Witt patch in the bulk bounded by the given spacelike surface [6], [7].",
"This is referred to as the CA conjecture.",
"An appropriate prescription for calculating the action for a region of spacetime bounded by null surfaces was obtained in [8].",
"Further related work is [9], [10], [11].",
"The evidence supporting this conjecture comes so far from the study of black hole spacetimes.",
"Both the CV and CA conjectures produce results for the complexity that grow linearly in time, with $ \\frac{d \\mathcal {C}}{dt} \\propto M.$ This is consistent with general expectations for the behaviour of the complexity for excited states in the field theory.",
"In these investigations, questions about the UV structure of the complexity were avoided, as the contributions from the asymptotic region of the spacetime cancel out in considering the time derivative.Another way to cancel UV contributions is to consider the difference between two spacetimes with the same asymptotic structure, as in [11].",
"However, it is interesting to understand the divergences in the holographic complexity.",
"In both the CV and CA conjectures there will be UV divergences, as the volume or action of the spacetime region in the bulk is divergent near the boundary.",
"We would expect that as for the holographic entanglement entropy [12], [13], these divergences are physical, signalling divergent contributions to the complexity associated with the UV degrees of freedom in the field theory.",
"For the entanglement entropy, the leading divergence is proportional to the area of the entangling surface, and this can be understood as reflecting entanglement of UV modes across this boundary [14], [15].",
"While a detailed understanding of the divergences of the complexity from the field theory perspective does not yet exist, we can study the divergences in the holographic calculation, and see if they have a reasonable form.",
"It is also interesting to compare the divergences between the CV and CA prescriptions, and see to what extent they compute different versions of complexity.",
"While this paper was in preparation, a preprint appeared studying these divergences [16].",
"The purpose of the present note is to add a simple observation to that work.",
"There is a term identified in [8] which can be added to the action which cancels a coordinate-dependence in that prescription.",
"If we add this contribution, it cancels the leading divergence in the CA prescription, so that the divergence structure of this action is the same as in the CV prescription.",
"The leading divergence in both cases is proportional to the volume of the boundary time slice.",
"Such a divergence appears reasonable from a field theory perspective.",
"Considering subleading contributions, we find that in both cases they can be expressed in terms of the geometry of the slice, but the CV and CA prescriptions differ.",
"In section , we review the CV and CA conjectures, and their application to the black hole examples.",
"We discuss the coordinate-dependence of the action proposed in [8], and introduce the term cancelling it.",
"In section , we consider the UV divergences in the CV and CA calculations, and show that including this term cancels the leading divergence in the CA calculation.",
"We consider subleading contributions and show that they have similar structures, depending on local geometric invariants of the boundary geometry, but note that the two prescriptions will differ in general.",
"We study the computation on global AdS to illustrate this difference."
],
[
"Review of CV and CA",
"In the CV conjecture of [1], the complexity $\\mathcal {C}$ of the state $|\\Psi \\rangle $ of a holographic field theory on some spatial slice $\\Sigma $ on the boundary of an asymptotically AdS spacetime is identified with the volume $V$ of the maximal volume codimension one slice $B$ in the bulk having its boundary on $\\Sigma $ , ${\\mathcal {C}_V} = \\frac{V(B)}{G_N l_{AdS}},$ This has a UV divergence proportional to the volume of $\\Sigma $ .",
"If we interpret this as part of the physical complexity, it could be interpreted as reflecting the operations required to set up the appropriate short-distance structure of the state $|\\Psi \\rangle $ starting from some reference state.",
"Qualitatively, this is reasonable; if we imagine modelling the field theory as a lattice, the reference state could be a simple product state on the lattice sites.",
"A Hadamard state in the field theory will not have such a product structure; the absence of high energy excitations implies short-range entanglement/correlation in the state.",
"Setting up this entangled state from the reference product state would require a number of elementary operations which will grow proportional to the volume of the field theory.",
"In [6], [7], an alternative CA conjecture was proposed.",
"This identifies the complexity of $|\\Psi \\rangle $ with the action of the “Wheeler-de Witt patch\", the domain of development of the slice $B$ considered previously.",
"The proposal is that ${\\mathcal {C}_A} = \\frac{S_W}{\\pi \\hbar },$ where $S_W$ is the action of the Wheeler-de Witt patch.",
"This proposal has the advantage that the formula is more universal, containing no explicit reference to a bulk length scale.",
"It also turns out to be easier to calculate, as we don't have a maximisation problem to solve.",
"Finding the Wheeler-de Witt patch for a given boundary slice is easier than finding the maximal volume slice.",
"The Wheeler-de Witt patch has null boundaries, for which the appropriate boundary terms needed for the Einstein-Hilbert action were not yet known.",
"In [8], inspired by the CA conjecture, a prescription for the action of a region of spacetime containing null boundaries was constructed (see also [17], [18]).",
"The prescription was obtained by requiring that the variation of the action vanish on-shell when the variation of the metric vanishes on the boundary of the region.",
"The resulting form for the action is $ S_{\\mathcal {V}} &=& \\int _{\\mathcal {V}} (R- 2 \\Lambda ) \\sqrt{-g} dV + 2 \\sum _{T_i} \\int _{T_i} K d\\Sigma + 2 \\sum _{S_i} \\mathrm {sign}(S_i) \\int _{S_i} K d \\Sigma \\\\ \\nonumber && - 2 \\sum _{N_i} \\mathrm {sign}(N_i) \\int _{N_i} \\kappa dS d\\lambda + 2 \\sum _{j_i} \\mathrm {sign}(j_i) \\oint \\eta _{j_i} dS + 2 \\sum _{m_i} \\mathrm {sign}(m_i) \\oint a_{m_i} dS.$ In this expression $T_i$ and $S_i$ are respectively timelike and spacelike components of the boundary of the region $\\mathcal {V}$ , and $K$ is the trace of the extrinsic curvature of the boundary.",
"For $T_i$ , the normal is taken outward-directed from $\\mathcal {V}$ .",
"For $S_i$ , the normal is always taken future-directed, and sign$(S_i) = 1(-1)$ if $\\mathcal {V}$ lies to the future (past) of $S_i$ , that is if the normal vector points into (out of) the region of interest.",
"$N_i$ are null components of the boundary of $\\mathcal {V}$ , $\\lambda $ is a parameter on null generators of $N_i$ , increasing to the future, $dS$ is an area element on the cross-sections of constant $\\lambda $ , and $k^\\alpha \\nabla _\\alpha k^\\beta = \\kappa k^\\beta $ , where $k^\\alpha = \\partial x^\\alpha /\\partial \\lambda $ is the tangent to the generators.",
"sign$(N_i) = 1(-1)$ if $N_i$ lies to the future (past) of ${\\mathcal {V}}$ .",
"$j_i$ are junctions between non-null boundary components, where $\\eta $ is the logarithm of the dot product of normals.",
"We do not give the rules in detail as such junctions do not occur for Wheeler-de Witt patches; see [8] for full detail.",
"$m_i$ are junctions where one or both of the boundary components are null.",
"We have a null surface with future-directed tangent $k^\\alpha $ and either a spacelike surface with future directed unit normal $n^\\alpha $ , a timelike surface with outward directed unit normal $s^\\alpha $ , or another null surface with future-directed tangent $\\bar{k}^\\alpha $ , and $a = \\left\\lbrace \\begin{array}{c} \\ln | k \\cdot n| \\\\ \\ln | k \\cdot s| \\\\ \\ln |k \\cdot \\bar{k}/2| \\end{array} \\right.$ respectively.",
"sign$(m_i) = +1$ if $\\mathcal {V}$ lies to the future (past) of the null boundary component and $m_i$ is at the past (future) end of the null component, and sign$(m_i) = -1$ otherwise.",
"While this action is diffeomorphism invariant under changes of coordinates in the bulk and on the timelike and spacelike boundaries, [8] show that it depends on the choice of coordinate $\\lambda $ on the null boundary components.",
"This coordinate dependence seems a highly undesirable feature.",
"Coordinate independence on the timelike and spacelike boundaries was incorporated as an assumption in constructing the form of the action.",
"This was built in, as it was possible to work with covariant tensors throughout the calculation.",
"On the null boundaries, such a manifestly coordinate independent formalism does not seem to exist, but one would still like to require that the final expression exhibit coordinate independence as a fundamental feature.",
"Fortunately, [8] found that the coordinate dependence could be eliminated by adding to the action a term $ \\Delta S = -2 \\sum _{N_i} \\mathrm {sign}(N_i) \\int _{N_i} \\Theta \\ln |\\ell \\Theta | dS d\\lambda ,$ where $\\Theta $ is the expansion of the null generators of $N_i$ , $\\Theta = \\frac{1}{\\sqrt{\\gamma }} \\frac{\\partial \\sqrt{\\gamma }}{\\partial \\lambda },$ where $\\gamma $ is the metric on the cross-sections of constant $\\lambda $ .",
"We will henceforth adopt the action $S = S_{\\mathcal {V}} + \\Delta S$ as our definition of the action for a region with null boundaries.",
"There is a further ambiguity noted in [8], which is the freedom to add an arbitrary function independent of the bulk metric to $a_{m_i}$ .",
"We see this as a subcase of a general freedom in the action: the requirement that the variation of the action vanish fixes the form of the boundary terms only up to contributions whose variations vanish under variations of the metric.",
"Since the metric variation vanishes on the boundary, this includes the freedom to add arbitrary functions of the intrinsic geometry of the boundary.",
"If we ignored the requirement of coordinate independence, this freedom would include the freedom to add terms like (REF ), as its variation under metric variations (with the metric fixed on the boundary) vanishes.",
"Since we want to insist on coordinate independence, the coefficient in (REF ) is fixed, but we still have freedom to add terms which are scalars on the null boundary, such as $ \\int _{N_i} \\Theta f(\\gamma ) dS d\\lambda $ , where $f(\\gamma )$ is any scalar function of the cross-section metric $\\gamma $ and curvature invariants built from this metric such as its Ricci scalar.",
"Also we have the freedom to add such scalar terms at the corners."
],
[
"UV divergences",
"We now turn to the consideration of the UV divergences in the action for the Wheeler-de Witt patch.",
"The simplest case to consider is AdS$_{d+1}$ in Poincare coordinates, $ ds^2 = \\frac{\\ell ^2}{z^2} ( dz^2 - dt^2 + d\\vec{x}^2),$ which is dual to the field theory in flat space.",
"We consider a $d+1$ dimensional AdS space, with a $d$ dimensional boundary.",
"If we ask for the complexity of the field theory on the $t=0$ surface, cut off at $z = \\epsilon $ , the Wheeler de Witt patch lies between $t = z-\\epsilon $ and $t = -(z-\\epsilon )$ .",
"Note that although these coordinates do not cover the full spacetime, the Wheeler-de Witt patch lies inside the region covered by this coordinate patch, as shown in figure REF , so we can calculate its action in these coordinates.",
"Figure: AdS, showing the region covered by Poincare coordinates and the Wheeler-de Witt patch of the t=0t=0 surface.For the CV conjecture, the maximal volume slice with boundary at $t=0$ is simply the $t=0$ surface in the bulk, whose volume is $ V(B) = \\int dz d^{d-1} x \\sqrt{h} = \\ell ^d V_x \\int _\\epsilon ^\\infty \\frac{dz}{z^d} = \\frac{\\ell ^d V_x}{(d-1) \\epsilon ^{d-1}},$ where $V_x$ is the IR divergent coordinate volume in the $\\vec{x}$ directions.",
"Thus, the complexity calculated according to the CV prescription is, up to an overall constant, $ \\mathcal {C}_V = \\frac{\\ell ^{d-1} V_x}{(d-1) G_N \\epsilon ^{d-1}}.$ This is proportional to the volume of the space the field theory lives in, in units of the cutoff.",
"This has both an IR and a UV divergence, which is physically reasonable if we think of the complexity as defined with respect to some product lattice state, as previously discussed.",
"Turning to the CA conjecture, consider the Wheeler-de Witt patch of this cutoff surface.We could alternatively take the original Wheeler-de Witt patch of the surface at $t=0,z=0$ and cut off the corner at $z= \\epsilon $ , producing a small timelike boundary component.",
"This would produce a different set of coefficients for subleading divergences [16].",
"The action of the Wheeler-de Witt patch with the prescription of [8] is $ S_W = \\int _{W} (R- 2 \\Lambda ) \\sqrt{-g} dV - 2 \\int _{F} \\kappa dS d\\lambda + 2 \\int _{P} \\kappa dS d\\lambda - 2 \\oint _{\\Sigma } a dS,$ where $F$ ($P$ ) is the future (past) null boundary of the Wheeler-de Witt patch, and $\\Sigma $ is the surface at $t=0$ , $z=\\epsilon $ .",
"The light cones of the boundary surface are at $t = \\pm (z- \\epsilon )$ , and $R - 2 \\Lambda = -2d/\\ell ^2$ , so the volume integral is $S_{\\mathit {Vol}} = - 2 \\frac{d}{\\ell ^2} \\int _\\epsilon ^\\infty dz \\int _{-(z-\\epsilon )}^{z- \\epsilon } dt \\frac{\\ell ^{d+1}}{z^{d+1}} V_x = -4 \\frac{\\ell ^{d-1} V_x}{(d-1) \\epsilon ^{d-1}}.$ This has a very similar structure to the volume in (REF ), but this term is negative, so it is clearly important to include the boundary contributions identified in [8] to obtain a sensible result for the complexity.",
"In calculating (REF ), it is convenient to adopt an affine parametrization of the null surfaces, so that the integrals over the future and past boundaries do not contribute.",
"Let us take the affine parameters along the null surfaces to be $ \\lambda = - \\frac{\\ell ^2}{\\alpha z} \\mbox{ on F}, \\quad \\lambda = \\frac{\\ell ^2}{\\beta z} \\mbox{ on P},$ where we introduce the arbitrary constants $\\alpha , \\beta $ to exhibit explicitly the remaining coordinate dependence.",
"This gives $k = \\alpha z^2/\\ell ^2 (\\partial _t + \\partial _z)$ , $\\bar{k} = \\beta z^2/\\ell ^2 (\\partial _t - \\partial _z)$ .",
"The boundary corner term is thus $S_{\\Sigma } = - 2 \\frac{\\ell ^{d-1} V_x}{\\epsilon ^{d-1}} \\ln (\\alpha \\beta \\epsilon ^2/\\ell ^2).$ Thus, the action calculated according to (REF ) is $S_W = \\frac{\\ell ^{d-1} V_x}{\\epsilon ^{d-2}}[ - 4 \\ln ( \\epsilon /\\ell ) - 2 \\ln (\\alpha \\beta ) - \\frac{1}{d-1} ].$ This has two undesirable features: it depends on the normalization $\\alpha $ , $\\beta $ of the affine parameters on the two null surfaces, and it diverges like $\\epsilon ^{-(d-2)} \\ln \\epsilon $ , which is faster than the volume of the space the field theory lives in.",
"These effects drop out if we consider the time-dependence as in (REF ), but they are both problematic if we want to consider the action as dual to the actual complexity of the state.",
"The first implies that the identification will require some choice of normalization for the affine parameters, which seems strange; these are just coordinates and should have no physical content.",
"The second implies the complexity would have a stronger than volume divergence, which seems not so easy to understand in terms of a simple lattice model.",
"Fortunately, both these problems are removed once we include the additional contribution (REF ).",
"The metric on F has $\\sqrt{\\gamma } = \\ell ^{d-1}/z^{d-1}$ , so the expansion is $\\Theta = \\frac{1}{\\sqrt{\\gamma }} \\frac{\\partial \\sqrt{\\gamma }}{\\partial \\lambda } = - \\frac{1}{\\sqrt{\\gamma }} \\alpha \\frac{z^2}{\\ell ^2} \\frac{\\partial \\sqrt{\\gamma }}{\\partial z} = (d-1) \\alpha \\frac{z}{\\ell ^2},$ so the surface term is $S_{F} &=& -2 (d-1) \\ell ^{d-1} V_x \\int z^{-(d-2)} \\ln (\\alpha (d-1) z/\\ell ) \\alpha d\\lambda \\\\ &=& 2 (d-1) \\ell ^{d-1} V_x \\int _\\epsilon ^\\infty z^{-d} \\ln (\\alpha (d-1) z/\\ell ) dz \\nonumber \\\\ &=& 2\\frac{ \\ell ^{d-1}}{\\epsilon ^{d-1}} V_x \\left( \\ln ( \\alpha (d-1) \\epsilon /\\ell ) + \\frac{1}{d-1} \\right), \\nonumber $ and $S_P = 2\\frac{ \\ell ^{d-1}}{\\epsilon ^{d-1}} V_x \\left( \\ln (\\beta (d-1) \\epsilon /\\ell ) + \\frac{1}{d-1} \\right)$ , so $S = S_W + \\Delta S = S_{\\mathit {Vol}} + S_\\Sigma + S_F + S_P = 4 \\frac{ \\ell ^{d-1}}{\\epsilon ^{d-1}} V_x \\ln (d-1).$ The dependence on $\\alpha $ , $\\beta $ cancels out by construction, as the additional terms were introduced to eliminate the coordinate dependence in (REF ).",
"The surprise is that this also leads to the cancellation of the logarithmic divergence.For the case $d=1$ , that is AdS$_2$ , the null surfaces are one-dimensional, and there is no expansion, so we cannot define a term analogous to (REF ) to cancel the logarithmic divergence.",
"In this case the CV calculation is also logarithmically divergent.",
"It would be interesting to understand this better, as this case will emerge if we want to apply these complexity ideas to near-horizon geometries of near-extremal black holes.",
"This provides a strong additional support for the idea that (REF ) should be included in the calculation of the action.",
"The result now has the same structure as that obtained in the CV calculation (REF ); since we do not understand the relation between the complexity and spacetime very precisely, the difference in the overall coefficient is not particularly significant."
],
[
"Subleading contributions",
"If we consider asymptotically AdS spacetimes, there will also be subleading divergences.",
"It is interesting to consider these contributions and investigate whether the cancellation of the leading logarithmic divergence in (REF ) obtained on adding (REF ) extends to subleading terms.",
"It is also interesting to compare the structure of divergences in the CV and CA calculations.",
"We consider an asymptotically AdS$_{d+1}$ solution of the vacuum Einstein equations.",
"The metric in the asymptotic region can then be written in the Fefferman-Graham gauge [19], [20] $ ds^2 = \\frac{\\ell ^2}{z^2} (dz^2 + g_{\\mu \\nu }(x^\\mu ,z) dx^\\mu dx^\\nu ),$ where the metric along the boundary directions has a power series expansion in $z$ , $g_{\\mu \\nu }(x^\\mu ,z) = g_{\\mu \\nu }^{(0)}(x^\\mu ) + z^2 g_{\\mu \\nu }^{(1)}(x^\\mu ) + \\ldots .$ We can give a simple general argument which shows that the cancellation of the leading logarithmic term extends to all the terms of the form $\\epsilon ^{-n} \\log \\epsilon $ .",
"Logarithmic divergences come from the corner contribution, $S_\\Sigma = - 2 \\oint _{\\Sigma } \\ln | k \\cdot \\bar{k}/2| \\sqrt{\\gamma } d^{d-1} \\sigma ,$ and from the additional contribution on the two null surfaces.",
"Considering the future, $S_F = -2 \\int \\Theta \\ln |\\ell \\Theta | \\sqrt{\\gamma } d^{d-1}x d\\lambda ,$ Now using the fact that the expansion is $\\Theta = \\frac{1}{\\sqrt{\\gamma }} \\partial \\sqrt{\\gamma }/\\partial \\lambda $ , we can rewrite this as $S_F = -2 \\int \\partial _\\lambda \\sqrt{\\gamma } \\ln |\\ell \\Theta | d^{d-1}x d\\lambda ,$ and integrate by parts on $\\lambda $ .",
"Since $\\Sigma $ is a past endpoint of the future surface, we obtain $ S_F = 2 \\oint _{\\Sigma } \\sqrt{\\gamma } \\ln |\\ell \\Theta | d^{d-1}\\sigma +2 \\int \\sqrt{\\gamma } \\frac{ \\partial _\\lambda \\Theta }{\\Theta } d^{d-1}x d\\lambda ,$ dropping a boundary term at the other boundary of the null surface which is irrelevant to the asymptotic calculation.",
"The second term will only contribute power-law divergences, so the logarithmic divergences will come solely from the integral over $\\Sigma $ .",
"Note also that it is this integral over $\\Sigma $ which cancels the coordinate-dependence in (REF ); the second term is coordinate-independent.In fact, one could take an alternative prescription for resolving the issues in (REF ) where one just added the first term in (REF ), rather than the whole expression (REF ).",
"There is a similar contribution from the past surface; $S_P = 2 \\int \\partial _\\lambda \\sqrt{\\gamma } \\ln |\\ell \\Theta | d^{d-1}x d\\lambda ,$ and the boundary term has the opposite sign because $\\Sigma $ is a future boundary of the past surface, so $S_P = 2 \\oint _{\\Sigma } \\sqrt{\\gamma } \\ln |\\ell \\Theta | d^{d-1}\\sigma -2 \\int \\sqrt{\\gamma } \\frac{ \\partial _\\lambda \\Theta }{\\Theta } d^{d-1}x d\\lambda .$ The logarithmic divergences in the full action are then contained in the terms involving integrals on $\\Sigma $ , $S = \\ldots + 2 \\oint _{\\Sigma } (\\ln |\\ell \\Theta _F| + \\ln |\\ell \\Theta _P| - \\ln | k \\cdot \\bar{k}/2|) \\sqrt{\\gamma } d^{d-1} \\sigma $ We know from the previous calculation that the leading order logarithmic term cancels.",
"Subleading terms coming from the expansion of $\\sqrt{\\gamma }$ will then also cancel.",
"Subleading terms in the argument of the logarithm will give power law divergences, once we expand $\\ln (\\epsilon + B\\epsilon ^2 + \\ldots ) = \\ln \\epsilon + \\ln (1 + B \\epsilon + \\ldots ) \\approx \\ln \\epsilon + B\\epsilon + \\ldots $ .",
"Thus, there are no subleading terms of the form $\\epsilon ^{-n} \\ln \\epsilon $ ; once we include (REF ) the divergences are a power series expansion in $\\epsilon $ .This argument is valid for all the terms of the form $\\epsilon ^{-n} \\log \\epsilon $ for $n>0$ ; once we reach the order in the Fefferman-Graham expansion where we encounter the free data in the asymptotic expansion, there may be contributions to either CV or CA calculations at order $\\log \\epsilon $ .",
"We will now extend the explicit calculation of the first subleading corrections to the action in [16] to include the additional contribution along the null surfaces.",
"We will see explicitly that the logarithmic terms cancel, as predicted by the general argument above.",
"We assume we are in $d > 2$ where the term of order $z^2$ is determined locally by the boundary metric $g_{\\mu \\nu }^{(0)}$ , [21], [22] $g_{\\mu \\nu }^{(1)} (x^\\mu ) = - \\frac{\\ell ^2}{(d-2)} \\left( R_{\\mu \\nu }[ g^{(0)}] - \\frac{g^{(0)}_{\\mu \\nu }}{2(d-1)} R[g^{(0)}] \\right),$ where $R_{\\mu \\nu }$ and $R$ are the Ricci tensor and Ricci scalar for the boundary metric.",
"It should be straightforward to extend the analysis to further subleading orders, but we will see interesting differences already at the first subleading order.",
"We consider a boundary slice at $t=0$ , in the cutoff surface at $z= \\epsilon $ , and calculate subleading divergences in the complexity.",
"As in [16], we restrict consideration to cases where the boundary metric is $ g_{\\mu \\nu }^{(0)} dx^\\mu dx^nu = - dt^2 + h_{ab} (t,\\sigma ^a) d\\sigma ^a d \\sigma ^b.$ This is general enough to include many cases of interest, and considerably simplifies the determination of the Wheeler-de Witt patch.",
"In [16], the subleading contributions to the volume of the maximal volume slice were determined, finding $ \\mathcal {C}_V = \\frac{\\ell ^{d-1}}{(d-1) G_N \\epsilon ^{d-1}} \\int d^{d-1} \\sigma \\sqrt{h} \\left[ 1 - \\frac{d-1}{2(d-2)(d-3)} \\epsilon ^2 \\left( R_a^a - \\frac{1}{2} R - \\frac{(d-2)^2}{(d-1)^2} K^2 \\right) + \\ldots \\right],$ where $h$ is the determinant of the metric $h_{ab}$ in (REF ) at $t=0$ , $R^a_a = h^{ab} R_{ab}$ is the trace of the projection of the boundary Ricci tensor into the $t=0$ surface, and $K$ is the trace of the extrinsic curvature of the $t=0$ surface in the boundary metric (REF ).",
"Thus, the first subleading divergence can be expressed in terms of local geometric features of the boundary metric.",
"The first subleading contributions to the action (REF ) were also evaluated in [16], obtaining $ \\mathcal {C}_A(S_W) &=& - \\frac{\\ell ^{d-1}}{4\\pi ^2 G_N (d-1) \\epsilon ^{d-1}} \\int d^{d-1} \\sigma \\sqrt{h} \\left[ 1 \\phantom{\\frac{1}{2}} \\right.",
"\\\\ && \\left.",
"+ \\frac{\\epsilon ^2}{4 (d-2)(d-3)} \\left( 4 K^2 + 4 K_{ab} K^{ab} + (d-7) R - 2 (d-3) R_a^a \\right) \\right] \\nonumber \\\\ && + \\frac{\\ell ^{d-1}}{4\\pi ^2 G_N \\epsilon ^{d-1}} \\log \\left( \\frac{\\ell }{\\sqrt{\\alpha \\beta } \\epsilon } \\right) \\int d^{d-1} \\sigma \\sqrt{h} \\left[ 1 - \\epsilon ^2 \\frac{1}{2(d-2)} (R^a_a - \\frac{1}{2} R ) \\right] + \\ldots \\nonumber $ We want to consider the effect of adding (REF ).",
"A key feature of the calculation in [16] is that the assumption that the boundary metric has the form (REF ) implies that at first subleading order, the tangents to the null generators take the form $k = \\frac{\\alpha }{\\ell ^2} (z^2 \\partial _z + k^t \\partial _t), \\quad \\bar{k} = \\frac{\\beta }{\\ell ^2} (-z^2 \\partial _z + k^t \\partial _t),$ where $k^t$ is determined by requiring these to be null vectors, $k^\\mu k_\\mu =0$ , which gives $k^t = z^2 (g_{tt})^{-1/2}$ .",
"This implies that the form of the affine parameter in (REF ) is unchanged to first subleading order.",
"Near the boundary, the induced metric on the surfaces of constant $\\lambda $ in the null surfaces is thus $\\gamma _{ab} = z^{-2} h_{ab} + g^{(1)}_{ab} + \\ldots $ .",
"Following [16], we write $\\sqrt{\\gamma } = \\frac{\\ell ^{d-1}}{z^{d-1}} \\sqrt{h} ( [1 + q_0^{(2)} z^2 + \\ldots ] + [q_1^{(0)} + \\ldots ] t + [q_2^{(0)} + \\ldots ] t^2 + \\ldots ),$ keeping the first terms in an expansion for small $z$ and $t$ , where $h$ is the determinant of $h_{ab}(\\sigma ^a, t=0)$ .",
"Along the null surface $t = (z-\\epsilon ) + O(z^3)$ , so $\\sqrt{\\gamma } = \\frac{\\ell ^{d-1}}{z^{d-1}} \\sqrt{h} ( 1 + q_1^{(0)} (z-\\epsilon ) + q_0^{(2)} z^2 + q_2^{(0)} (z-\\epsilon )^2 + \\ldots ),$ $\\partial _z \\sqrt{\\gamma } &=& -\\frac{\\ell ^{d-1}}{z^{d}} \\sqrt{h} ( (d-1) + q_1^{(0)} ((d-2) z- (d-1)\\epsilon ) + q_0^{(2)} (d-3) z^2 \\\\ &&+ q_2^{(0)}((d-3)z^2 -2(d-2) z \\epsilon + (d-1) \\epsilon ^2) + \\ldots ), \\nonumber $ so the expansion is $\\Theta = \\frac{\\alpha z}{\\ell ^2} \\left[ (d-1) - q_1^{(0)} z - 2 q_0^{(2)} z^2 -2 q_2^{(0)} z(z-\\epsilon ) + q_1^{(0)2} z (z-\\epsilon ) + \\ldots \\right]$ Performing the integral over $z$ , one finds $S_F &=& 2 \\frac{\\ell ^{d-1}}{\\epsilon ^{d-1}} \\int _\\Sigma d^{d-1} \\sigma \\sqrt{h} \\left[ \\ln (\\alpha (d-1) \\epsilon /\\ell ) ( 1 + q_0^{(2)} \\epsilon ^2) \\right.",
"\\\\ && \\left.",
"+\\frac{1}{(d-1)} \\left( 1 - q_1^{(0)} \\epsilon - q_0^{(2)} \\frac{(d-1)}{(d-3)} \\epsilon ^2 - 2 \\frac{q_2^{(0)}}{(d-3)} \\epsilon ^2 + q_1^{(0)2} \\frac{1}{2(d-3)} \\epsilon ^2 \\right) \\right] .",
"\\nonumber $ $S_P$ will have the same form, but with the sign of $q_1^{(0)}$ reversed, as the past surface is $t = - (z-\\epsilon ) + O(z^3)$ .",
"Thus, the correction to the action is $S_F + S_P &=& 4 \\frac{\\ell ^{d-1}}{\\epsilon ^{d-1}} \\int _\\Sigma d^{d-1} \\sigma \\sqrt{h} \\left[ \\ln (\\sqrt{\\alpha \\beta } (d-1) \\epsilon /\\ell ) ( 1 + q_0^{(2)} \\epsilon ^2) \\right.",
"\\\\ && \\left.",
"+ \\frac{1}{(d-1)} \\left( 1 - q_0^{(2)} \\frac{(d-1)}{(d-3)} \\epsilon ^2 - 2 \\frac{q_2^{(0)}}{(d-3)} \\epsilon ^2 + q_1^{(0)2} \\frac{1}{2(d-3)} \\epsilon ^2 \\right) \\right] .",
"\\nonumber $ Using the geometric expressions from [16], $q_1^{(0)} = K, \\quad q_2^{(0)} = \\frac{1}{2} (K^2 + K_{ab} K^{ab} + R^a_a -R), \\quad q_0^{(2)} = - \\frac{1}{2(d-2)} (R^a_a - \\frac{1}{2} R),$ we can see that the logarithmic term will cancel with the contribution in (REF ), as expected, including the subleading correction.",
"The power law terms will combine with those in (REF ) to give us a result for the complexity $ \\mathcal {C}_A(S) &=& \\frac{\\ell ^{d-1}}{4\\pi ^2 G_N \\epsilon ^{d-1}} \\int d^{d-1} \\sigma \\sqrt{h} \\left[ \\ln (d-1) \\left( 1 - \\frac{\\epsilon ^2}{2(d-2)} (R_a^a - \\frac{1}{2} R) \\right) \\right.",
"\\\\ && \\left.",
"- \\frac{\\epsilon ^2d}{2(d-1)(d-2) (d-3)} K^2 - \\frac{\\epsilon ^2}{(d-2)(d-3)} K_{ab} K^{ab} + \\frac{\\epsilon ^2d}{2(d-1)(d-2)(d-3)} R \\right].",
"\\nonumber $ We see that this has a similar structure to the CV result (REF ), but with different coefficients for the subleading terms."
],
[
"Global AdS",
"A simple example which illustrates the difference between CV and CA is to consider pure AdS in global coordinates, $ ds^2 = \\frac{\\ell ^2}{\\cos ^2 \\theta } (-dt^2 + d\\theta ^2 + \\sin ^2 \\theta d\\Omega _{d-1}^2).$ We consider a slice of the boundary at $t=0$ , cutoff at $\\theta = \\theta _{cut}= \\frac{\\pi }{2} - \\epsilon $ .",
"The maximal volume slice is again $t=0$ in the bulk, and the volume is simply $ V(B) = \\int d\\theta d\\Omega \\sqrt{h} = \\ell ^d \\Omega _{d-1} \\int _0^{\\theta _{cut}} \\frac{d\\theta }{\\cos \\theta } \\tan ^{d-1} \\theta ,$ where $\\Omega _{d-1}$ is the volume of a unit $S^{d-1}$ .",
"Figure: The Wheeler-de Witt patch in global AdS.We again calculate the action of the Wheeler-de Witt patch of the cutoff boundary at $\\theta = \\theta _{cut}$ , as depicted in figure REF .",
"The future boundary is at $t= \\theta _{cut} - \\theta $ , while the past boundary is at $t = \\theta - \\theta _{cut}$ .",
"The volume term in (REF ) is $S_{\\mathit {Vol}} &=& -\\frac{4d}{\\ell ^2} \\int _{0}^{\\theta _{cut}} dt^{\\prime } \\int _0^{t^{\\prime }} d\\theta \\int d\\Omega _{d-1} \\sqrt{-g} \\\\ &=& - 4d\\Omega _{d-1} \\ell ^{d-1} \\int _{0}^{\\theta _{cut}} dt^{\\prime } \\int _0^{t^{\\prime }} d\\theta \\frac{\\sin ^{d-1} \\theta }{\\cos ^{d+1} \\theta } \\nonumber \\\\ &=& - 4\\Omega _{d-1} \\ell ^{d-1} \\int _{0}^{\\theta _{cut}} dt^{\\prime } \\tan ^d t^{\\prime }.",
"\\nonumber $ In the first step, we wrote the volume term as twice the integral over the future half of the Wheeler-de Witt patch.",
"We choose an affine parameter $\\lambda $ , so that the integrals over the future and past boundaries in (REF ) do not contribute.",
"An appropriate parameter is $\\lambda = -\\alpha ^{-1} \\ell \\tan \\theta $ on $F$ and $\\lambda = \\beta ^{-1} \\ell \\tan \\theta $ on $P$ , where we have introduced the arbitrary parameters $\\alpha $ , $\\beta $ purely so that we can see that they will cancel out once we add the term (REF ).",
"The future-pointing tangent is then $k = \\frac{\\alpha }{\\ell } \\cos ^2 \\theta (\\partial _t - \\partial _\\theta )$ on $F$ and $\\bar{k} = \\frac{\\beta }{\\ell } \\cos ^2 \\theta (\\partial _t + \\partial _\\theta )$ on $P$ , so the boundary corner term is $S_\\Sigma = -2 \\Omega _{d-1} \\ell ^{d-1} \\tan ^{d-1} \\theta _{cut} \\ln ( \\alpha \\beta \\cos ^2 \\theta _{cut}).$ Adding the contribution (REF ), the expansion on $F$ is $\\Theta = \\frac{1}{\\sqrt{\\gamma }} \\frac{\\partial \\sqrt{\\gamma }}{\\partial \\lambda } = - \\frac{1}{\\sqrt{\\gamma }} \\frac{\\alpha }{\\ell } \\cos ^2 \\theta \\frac{\\partial \\sqrt{\\gamma }}{\\partial \\theta } = - \\frac{\\alpha }{\\ell } \\cos ^2 \\theta \\cot ^{d-1} \\theta \\partial _\\theta (\\tan ^{d-1} \\theta ) = - \\frac{ \\alpha }{\\ell } (d-1) \\cot \\theta ,$ and similarly on $P$ $\\Theta = \\beta /\\ell (d-1) \\cot \\theta $ .",
"Thus the surface term is $S_F &=&- 2 \\Omega _{d-1} \\ell ^{d-1} \\int \\Theta \\ln |\\Theta | \\tan ^{d-1} \\theta d\\lambda \\\\ &=& 2 (d-1) \\Omega _{d-1} \\ell ^{d-1} \\int _0^{\\theta _{cut}} \\frac{\\tan ^{d-2} \\theta }{\\cos ^2 \\theta } \\ln ( \\alpha (d-1) \\cot \\theta ) d \\theta , \\nonumber \\\\ &=& 2 \\Omega _{d-1} \\ell ^{d-1} [ \\tan ^{d-1} \\theta _{cut} \\ln (\\alpha (d-1) \\cot \\theta _{cut}) + \\frac{1}{(d-1)} \\tan ^{d-1} \\theta _{cut}] \\nonumber $ and similarly $S_P& = &2 \\Omega _{d-1} \\ell ^{d-1} \\int \\Theta \\ln |\\Theta | \\tan ^{d-1} \\theta d\\lambda \\\\ & = & \\Omega _{d-1} \\ell ^{d-1} [ \\tan ^{d-1} \\theta _{cut} \\ln (\\beta (d-1) \\cot \\theta _{cut}) + \\frac{1}{(d-1)} \\tan ^{d-1} \\theta _{cut}]\\nonumber $ so in total $ S &=& S_V + \\Delta S = S_{\\mathit {Vol}} + S_\\Sigma + S_F + S_P \\\\ &=& - 4\\Omega _{d-1} \\ell ^{d-1} \\int _{0}^{\\theta _{cut}} dt^{\\prime } \\tan ^{d} t^{\\prime } +4 \\Omega _{d-1} \\ell ^{d-1} \\tan ^{d-1} \\theta _{cut} ( \\ln (d-1) + \\frac{1}{d-1}).\\nonumber $ We see that while the leading UV divergence is the same as for the volume (REF ), the integrals are different, so the functional dependence on $\\theta _{cut}$ is different for CV and CA.",
"The two conjectures for the complexity are inequivalent.",
"However, as noted in [16] (appendix C), the form of the subleading contributions in the CA calculation here depends on how we choose to cut off the Wheeler-de Witt patch, so it is not clear how much physical meaning it carries."
],
[
"Acknowledgements",
"AR is supported by an STFC studentship.",
"SFR is supported in part by STFC under consolidated grant ST/L000407/1."
]
] | 1612.05439 |
[
[
"A Message Passing Algorithm for the Minimum Cost Multicut Problem"
],
[
"Abstract We propose a dual decomposition and linear program relaxation of the NP -hard minimum cost multicut problem.",
"Unlike other polyhedral relaxations of the multicut polytope, it is amenable to efficient optimization by message passing.",
"Like other polyhedral elaxations, it can be tightened efficiently by cutting planes.",
"We define an algorithm that alternates between message passing and efficient separation of cycle- and odd-wheel inequalities.",
"This algorithm is more efficient than state-of-the-art algorithms based on linear programming, including algorithms written in the framework of leading commercial software, as we show in experiments with large instances of the problem from applications in computer vision, biomedical image analysis and data mining."
],
[
"Introduction",
"Decomposing a graph into meaningful clusters is a fundamental primitive in computer vision, biomedical image analysis and data mining.",
"In settings where no information is given about the number or size of clusters, and information is given only about the pairwise similarity or dissimilarity of nodes, a canonical mathematical abstraction is the minimum cost multicut (or correlation clustering) problem [14].",
"The feasible solutions of this problem, multicuts, relate one-to-one to the decompositions of the graph.",
"A multicut is the set of edges that straddle distinct clusters.",
"The cost of a multicut is the sum of costs attributed to its edges.",
"In the field of computer vision, the minimum cost multicut problem has been applied in [3], [4], [39], [6] to the task of unsupervised image segmentation defined by the BSDS data sets and benchmarks [30] .",
"In the field of biomedical image analysis, the minimum cost multicut problem has been applied to an image segmentation task for connectomics [5].",
"In the field of data mining, applications include [7], [33], [12], [13].",
"As the minimum cost multicut problem is np-hard [9], [16], even for planar graphs [8] large and complex instances with millions of edges, especially those for connectomics, pose a challenge for existing algorithms.",
"Related Work.",
"Due to the importance of multicuts for applications, many algorithms for the minimum cost multicut problem have been proposed.",
"They are grouped below into three categories: primal feasible local search algorithms, linear programming algorithms and fusion algorithms.",
"Primal feasible local search algorithms [35], [31], [20], [18], [19] attempt to improve an initial feasible solution by means of local transformations from a set that can be indexed or searched efficiently.",
"Local search algorithms are practical for large instances, as the cost of all operations is small compared to the cost of solving the entire problem at once.",
"On the downside, the feasible solution that is output typically depends on the initialization.",
"And even if a solution is found, optimality is not certified, as no lower bound is computed.",
"Linear programming algorithms [24], [25], [27], [32], [38] operate on an outer polyhedral relaxation of the feasible set.",
"Their output is independent of their initialization and provides a lower bound.",
"This lower bound can be used directly inside a branch-and-bound search for certified optimal solutions.",
"Alternatively, the LP relaxation can be tightened by cutting planes.",
"Several classes of planes are known that define a facet of the multicut polytope and can be separated efficiently [14].",
"On the downside, algorithms for general LPs that are agnostic to the structure of the multicut problem scale super-linearly with the size of the instance.",
"Fusion algorithms attempt to combine feasible solutions of subproblems obtained by combinatorial or random procedures into successively better multicuts.",
"The fusion process can either rely on column generation [39], binary quadratic programming [11] or any algorithm for solving integer LPs [10].",
"In particular, [39] provides dual lower bounds but is restricted to planar graphs.",
"[11], [10] explore the primal solution space in a clever way, but do not output dual information.",
"Outline.",
"Below, a discussion of preliminaries (Sec. )",
"is followed by the definition of our proposed decomposition (Sec. )",
"and algorithm (Sec. )",
"for the minimum cost multicut problem.",
"Our approach combines the efficiency of local search with the lower bounds of LPs and the subproblems of fusion, as we show in experiments with large and diverse instances of the problem (Sec. ).",
"All code and data will be made publicly available upon acceptance of the paper."
],
[
"Minimum Cost Multicut Problem",
"A decomposition (or clustering) of a graph $G = (V,E)$ is a partition $V_1 \\dot{\\cup }\\ldots \\cup V_k$ of the node set $V$ such that $V_i \\cap V_j = \\varnothing $ $\\forall i\\ne j$ and every cluster $V_i$ , $i=1,\\ldots ,k$ is connected.",
"The multicut induced by a decomposition is the subset of those edges that straddle distinct clusters (cf. Fig.",
"REF ).",
"Such edges are said to be cut.",
"Every multicut induced by any decomposition of $G$ is called a multicut of $G$ .",
"We denote by $\\mathcal {M}_G$ the set of all multicuts of $G$ .",
"Given, for every edge $e \\in E$ , a cost $c_e \\in \\mathbb {R}$ of this edge being cut, the instance of the minimum cost multicut problem w.r.t.",
"these costs is the optimization problem (REF ) whose feasible solutions are all multicuts of $G$ .",
"For any edge $\\lbrace v,w\\rbrace = e \\in E$ , negative costs $\\theta _e < 0$ favour the nodes $v$ and $w$ to be in distinct components.",
"Positive costs $\\theta _e > 0$ favour these nodes to lie in the same component.",
"$\\min _{M \\in \\mathcal {M}_G} \\sum _{e \\in M} \\theta _e$ Figure: Depicted above is a decomposition of a graph into three components (green).The multicut induced by this decomposition consists of the edges that straddle distinct components (red).This problem is np-hard [9], [16], even for planar graphs [8].",
"Below, we recapitulate its formulation as a binary LP and then turn to LP relaxations: For any 01-labeling $x \\in \\lbrace 0,1\\rbrace ^E$ of the edges of $G$ , the subset $x^{-1}(1)$ of those edges labeled 1 is a multicut of $G$ if and only if $x$ satisfies the system () of cycle inequalities [14].",
"Hence, (REF ) can be stated equivalently in the form of the binary LP (REF )–().",
"$\\min _{x \\in \\mathbb {R}^E} \\quad & \\sum _{e \\in E} \\theta _e x_e \\\\\\textnormal {subject to} \\quad & \\forall C \\in \\textnormal {cycles}(G):\\quad x_e \\le \\sum _{\\mathchoice{\\hbox{t}o 0pt{\\hss \\displaystyle {e^{\\prime } \\in C \\setminus \\lbrace e\\rbrace }\\hss }}{\\hbox{t}o 0pt{\\hss \\textstyle {e^{\\prime } \\in C \\setminus \\lbrace e\\rbrace }\\hss }}{\\hbox{t}o 0pt{\\hss \\scriptstyle {e^{\\prime } \\in C \\setminus \\lbrace e\\rbrace }\\hss }}{\\hbox{t}o 0pt{\\hss \\scriptscriptstyle {e^{\\prime } \\in C \\setminus \\lbrace e\\rbrace }\\hss }}} x_{e^{\\prime }} \\\\& x \\in \\lbrace 0,1\\rbrace ^E$ An LP relaxation is obtained by replacing the integrality constraints () by $x \\in P$ with $P \\subseteq [0,1]^E$ .",
"This results in an outer relaxation of the multicut polytope, which is the convex hull of the characteristic functions of all multicuts of $G$ .",
"The LP relaxation obtained for $P := [0,1]^E$ , i.e., with only the cycle inequalities, will not in general be tight.",
"A tighter LP relaxation is obtained by enforing also the odd wheel inequalities [14].",
"A $k$ -wheel is a cycle in $G$ with $k$ nodes all of which are connected to an additional node $u \\in V$ that is not in the cycle and is called the center of the $k$ -wheel (cf. Fig.",
"REF ).",
"For any odd number $k \\in \\mathbb {N}$ , any $k$ -wheel of $G$ , the cycle $C = (v_1 v_2, \\ldots , v_k v_1)$ and the center $u$ of the $k$ -wheel, every characteristic function $x \\in \\lbrace 0,1\\rbrace ^E$ of a multicut $x^{-1}(1)$ of $G$ satisfies the odd wheel inequality $\\sum _{i=1}^k x_{v_i v_{i+1}} - \\sum _{i=1}^k x_{u v_i} \\le \\left\\lfloor \\tfrac{k}{2} \\right\\rfloor \\quad \\textnormal {with}\\quad v_{k+1} := v_1\\hspace{5.0pt}.$ For completeness, we note that other inqualities known to further tighten the LP relaxation can be included in our algorithm, e.g., the bicycle inequalities [14] defind on graphs as in Fig.",
"REF .",
"We, however, do not consider inequalities other than cycles and odd wheels in the algorithm we propose.",
"every node=[circle,draw,minimum width=3ex,inner sep=0ex] Figure: Odd Bicycle Wheel"
],
[
"Integer relaxed pairwise separable LPs",
"LP relaxations of the multicut problem can in principle be solved with algorithms for general LPs which are available in excellent software such as CPlex [2] and Gurobi [22].",
"However, these algorithms scale super-linearly with the size of the problem and are hence impractical for large instances.",
"We define in Sec.",
"an LP relaxation of the multicut problem in form of an IRPS-LP (Def.",
"REF ).",
"IRPS-LPs are a special case of dual decomposition [21].",
"In Def.",
"REF , every $i \\in \\mathbb {V}$ defines a subproblem, and every edge $ij \\in \\mathbb {E}$ defines a dependency of subproblems.",
"Def.",
"REF is more specific in that, firstly, the subproblems are binary and, secondly, the linear constraints () that describe the dependence of subproblems are defined by 01-matrices that map 01-vectors to 01-vectors.",
"IRPS-LPs are amenable to efficient optimization by message passing in the framework of [36].",
"Definition 1 (IRPS-LP [36]) Let $N \\in \\mathbb {N}$ and let $\\mathbb {G}= (\\mathbb {V}, \\mathbb {E})$ be a graph with $\\mathbb {V}= \\lbrace 1, \\ldots , N\\rbrace $ .",
"For every $j \\in \\mathbb {V}$ , let $d_j \\in \\mathbb {N}$ , let $X_j \\subseteq \\lbrace 0,1\\rbrace ^{d_j}$ , and let $\\theta _j \\in \\mathbb {R}^{d_j}$ .",
"Let $\\Lambda := \\operatornamewithlimits{conv}(X_1) \\times \\cdots \\times \\operatornamewithlimits{conv}(X_N)$ .",
"For every $\\lbrace j,k\\rbrace = e \\in \\mathbb {E}$ , let $m_e \\in \\mathbb {N}$ , $A_{(j,k)} \\in \\lbrace 0,1\\rbrace ^{m_e \\times d_j}$ and $A_{(k,j)} \\in \\lbrace 0,1\\rbrace ^{m_e \\times d_k}$ such that $\\forall x \\in X_j: \\quad A_{(j,k)} x \\in \\lbrace 0,1\\rbrace ^{m_e} \\\\\\forall x \\in X_k: \\quad A_{(k,j)} x \\in \\lbrace 0,1\\rbrace ^{m_e}$ Then, the LP written below is called integer relaxed pairwise separable w.r.t.",
"the graph $\\mathbb {G}$ .",
"$\\min _{\\mu \\in \\Lambda } \\quad & \\sum _{j \\in \\mathbb {V}} \\sum _{k = 1}^{d_j} \\theta _{jk} \\mu _{jk} \\\\\\textnormal {subject to} \\quad & \\forall \\lbrace j,k\\rbrace \\in E: \\quad A_{(j,k)} \\mu _j = A_{(k,j)} \\mu _k $"
],
[
"Dual Decomposition",
"A straight-forward decomposition of the minimum cost multicut problem (REF )–() in the form of an IRPS-LP (Def.",
"REF ) consists of one subproblem for every edge, one subproblem for every cycle inequality and one subproblem for every odd-wheel inequality.",
"From a computational perspective, it is however advantageous to triangulate cycles and odd wheels, and to consider the resulting smaller subproblems.",
"Below, three classes of subproblems are defined rigorously."
],
[
"Edge Subproblems.",
"For every edge $e \\in E$ , we consider a subproblem $e \\in \\mathbb {V}$ with the feasible set $X_e := \\lbrace 0,1\\rbrace $ , encoding whether edge $e$ is cut (1) or uncut (0)."
],
[
"Triangle Subproblems",
"For every cycle $C = \\lbrace v_1 v_2, v_2 v_3,\\ldots v_k v_1\\rbrace \\subseteq E$ , we consider the triangles $v_1 v_2 v_3$ to $v_{k-1} v_k v_1$ , as depicted in Fig.",
"REF .",
"If some edge $uv$ of a triangle $C_i$ is not in $E$ , we add it to $E$ with cost zero, i.e., we triangulate the cycle in $G$ .",
"For each triangle $uvw$ , we introduce a subproblem $uvw \\in \\mathbb {V}$ whose feasible set consists of the five feasible multicuts of the triangle, i.e., $X_{uvw} := \\lbrace (0,0,0),(0,1,1),(1,0,1),(1,1,0),(1,1,1)\\rbrace $ ."
],
[
"Lollipop Subproblems",
"For every odd number $k \\in \\mathbb {N}$ and every $k$ -wheel of $G$ consisting of a center node $u$ and cycle nodes $v_1, \\ldots , v_k$ , we introduce two classes of subproblems.",
"For the 5-wheel depicted in Fig.",
"REF , these subproblems are depicted in Fig.",
"REF .",
"For every $j \\in \\lbrace 2, \\ldots , k\\rbrace $ , we add the triangle subproblem $u v_1 v_j \\in \\mathbb {V}$ , as described in the previous section.",
"For every $j \\in \\lbrace 2, \\ldots , k-1\\rbrace $ , we add the subproblem $u v_j v_{j+1}, v_1 \\in \\mathbb {V}$ for the lollipop graph that consists of the triangle $u v_j v_{j+1}$ and the additional edge $u v_1$ .",
"The feasible set $X_{uvw,s}$ of a lollipop graph $uvw,s$ has ten elements, five feasible multicuts of the triangle times two for accounting for the additional edge."
],
[
"Dependencies",
"The dependency between triangle subproblems and edge subproblems are expressed below in the form of a linear system.",
"It fits into thee form () of an IRPS-LP.",
"$\\mu _{uv} & = \\mu _{uvw}(1,1,0) + \\mu _{uvw}(1,0,1) + \\mu _{uvw}(1,1,1)\\\\\\mu _{uw} & = \\mu _{uvw}(1,1,0) + \\mu _{uvw}(0,1,1) + \\mu _{uvw}(1,1,1)\\\\\\mu _{vw} & = \\mu _{uvw}(1,0,1) + \\mu _{uvw}(0,1,1) + \\mu _{uvw}(1,1,1)$ The dependency between a lollipop subproblem with edge set $L = \\lbrace e_1,e_2,e_3,e_4\\rbrace $ and a triangle subproblem with edge set $T = \\lbrace e^{\\prime }_1,e^{\\prime }_2,e^{\\prime }_3\\rbrace $ is stated below as a linear system with sums over edges not shared between $L$ and $T$ .",
"This linear system has the form () of an IRPS-LP.",
"$\\forall x_{L \\cap T}:\\sum _{x_{L \\backslash T}} \\mu _{L}(x_{L \\cap T}, x_{L \\backslash T}) = \\sum _{x_{T \\backslash L}} \\mu _{T}(x_{T \\cap L}, x_{T \\backslash C})$"
],
[
"Remarks",
"Remark 1.",
"The triangulation of cycles can be understood as the constructing of a junction tree [37] in such a way that the minimum cost multicut problem over the cycle can be solved by dynamic programming.",
"The triangulation of cycles can also be understood as a tightening of an outer polyhedral relaxation of the multicut polytope: A cycle inequality () defines a facet of the multicut polytope if and only if the cycle is chordless [14].",
"By triangulating a cycle, we obtain a set of minimal chordless cycles (triangles) whose cycle inequalities together imply that of the entire cycle.",
"Remark 2.",
"Technically, we would not have needed to include triangle subproblems for odd wheels.",
"Instead, we could have introduced dependencies between lollipops directly in the form of an IRPS-LP.",
"However, by introducing triangle factors in addition and by expressing dependencies between lollipops and triangles, we couple lollipop factors from different odd wheels more tightly whenever they share the same triangles.",
"Figure: Depicted above is a triangulation of the odd wheel from Figure .",
"It consists of the triangles uv 1 v 2 ,uv 1 v 3 ,uv 1 v 4 ,uv 1 v 5 uv_1v_2, uv_1v_3, uv_1v_4, uv_1v_5 and the lollipop graphs (uv 2 v 3 ,v 1 ),(uv 3 v 4 ,v 1 ),(uv 4 v 5 ,v 1 )(uv_2v_3,v_1), (uv_3v_4,v_1), (uv_4v_5,v_1)."
],
[
"Algorithm",
"We now define an algorithm for the minimum cost multicut problem (REF )–().",
"This algorithm takes an instance of the problem as input and alternates for a fixed number of iterations between two main procedures.",
"The first procedure, defined in Sec.",
"REF , solves an instance of a dual of the IRPS-LP relaxation defined in the previous section.",
"The output consists in a lower bound and a re-parameterization of the instance of the minimum cost multicut problem given as input.",
"The second procedure tightens the IRPS-LP relaxation by adding subproblems for cycle inequalities () and odd wheel inequalities (REF ) violated by the current solution.",
"Separation procedures for finding such violated inequalities, more efficiently than in cutting plane algorithms for the primal [24], [25], [27], are defined in Sec.",
"REF .",
"To find feasible solutions of the instance of the minimum cost multicut problem given as input, we apply a state-of-the-art local search algorithm on the computed re-parameterizations, a procedure commonly referred to as rounding (Sec.",
"REF )."
],
[
"Message Passing",
"Like other algorithms based on dual decomposition, the algorithm we propose does not solve the IRPS-LP directly, in the primal domain, but optimizes a dual of (REF )–().",
"Specifically, it operates on a space of re-parametrizations of the problem defined below: For any two dependent subproblems $jk \\in \\mathbb {E}$ , we can change the costs $\\theta _j$ and $\\theta _k$ by an arbitrary vector $\\Delta $ according to the update rules $\\theta ^{\\prime }_j & := \\theta _j + A_{(j,k)}^\\\\\\theta ^{\\prime }_k & := \\theta _k - A_{(k,j)}^\\hspace{5.0pt}.$ We refer to any update of $\\theta $ according to the rules (REF )–() as message passing.",
"Message passing does not change the cost of any primal feasible solution, as $& \\langle \\theta ^{\\prime }_j, \\mu _j \\rangle + \\langle \\theta ^{\\prime }_k, \\mu _k \\rangle \\nonumber \\\\= \\, & \\langle \\theta _j + A_{(j,k)}^, \\mu _j \\rangle + \\langle \\theta _k - A_{(k,j)}^, \\mu _k \\rangle \\\\= \\, & \\langle \\theta _j, \\mu _j \\rangle + \\langle \\theta _k, \\mu _k \\rangle + \\langle \\Delta , A_{(j,k)} \\mu _j - A_{(k,j)} \\mu _k \\rangle \\\\\\overset{(\\ref {eq:IPSLP-constraints})}{=} \\, & \\langle \\theta _j, \\mu _j \\rangle + \\langle \\theta _k, \\mu _k \\rangle \\hspace{5.0pt}.$ Message passing does, however, change the dual lower bound $L(\\theta )$ to (REF ) given by $L(\\theta ) := \\sum _{j \\in \\mathbb {V}} \\min _{x \\in X_i} \\langle \\theta _j, x_j \\rangle \\hspace{5.0pt}.$ The maximum of $L(\\theta )$ over all costs obtainable by message passing is equal to the minimum of (REF ), by linear programming duality.",
"We seek to alter the costs $\\theta $ by means of message passing so as to maximize the lower bound $L(\\theta )$ .",
"For the general IRPS-LP, a framework of algorithms to achieve this goal is defined in [36].",
"For the minimum cost multicut problem, we define and implement Alg.",
"REF within this framework.",
"The specifics of this algorithm for the minimum cost multicut problem are discussed below.",
"General properties of message passing for IRPS-LP s are discussed in [36].",
"[t] $\\lbrace i_1,\\ldots ,i_k\\rbrace = \\mathbb {V}$ , $(\\theta _{i})_{i \\in \\mathbb {V}}$ , $(A_{(j,i)},A_{(i,j)})_{ij \\in \\mathbb {E}}$ $i = i_1,\\ldots ,i_k$ $i$ is an edge subproblem $uv$ : Receive messages: $w \\in \\mathsf {V}: uvw \\in T$ $\\delta := \\min _{x_{uw},x_{vw}} \\theta _{uvw}(1,x_{uw},x_{vw})$ $ - \\min _{x_{uw},x_{vw}} \\theta _{uvw}(0,x_{uw},x_{vw}) $ $\\theta _{uv} \\,\\textnormal {+=}\\, \\delta $ $\\forall x_{uw},x_{vw}: \\, \\theta _{uvw}(1,x_{uw},x_{vw}) \\,\\textnormal {-=}\\, \\delta $ Send messages: $\\delta := \\vert \\lbrace w \\in \\mathsf {V}: uvw \\in T\\rbrace \\vert ^{-1} \\theta _{uv}$ $\\theta _{uv} := 0$ $w \\in \\mathsf {V}: uvw \\in T$ $\\forall x_{uw},x_{vw}: \\, \\theta _{uvw}(1,x_{uw},x_{vw}) \\,\\textnormal {+=}\\, \\delta $ $i$ is a triangle subproblem $uvw$ with edges $C$ : Receive messages: lollipops $L$ with $L\\cap C \\ne \\varnothing $ $\\delta (x_{L \\cap C}) := \\min _{x_{L \\backslash C}} \\theta _{L}(x_{L \\cap C},x_{L \\backslash C})$ $\\theta _{C}(x_{L \\cap C},x_{C \\backslash L}) \\,\\textnormal {+=}\\, \\delta (x_{L\\cap C})$ $\\theta _{L}(x_{L \\cap C},x_{L \\backslash C}) \\,\\textnormal {+=}\\, \\delta (x_{L\\cap C}$ Send messages: $\\alpha := \\vert \\lbrace L \\text{ a lollipop} : L \\cap C \\ne \\varnothing \\rbrace \\vert $ lollipops $L$ with $L\\cap C \\ne \\varnothing $ $\\delta _{L}(x_{L \\cap C}) := \\min _{x_{C \\backslash L}} \\theta _{uvw}(x_{L \\cap C},x_{C \\backslash L})$ $\\theta _{L}(x_{L \\cap C},x_{L \\backslash C}) \\,\\textnormal {+=}\\, \\frac{1}{\\alpha } \\delta _L(x_{L\\cap C})$ lollipops $L$ with $L\\cap C \\ne \\varnothing $ $\\theta _{C}(x_{L\\cap C}, x_{C \\backslash L}) \\,\\textnormal {+=}\\, \\frac{1}{\\alpha } \\delta _L(x_{L \\cap C})$ Message passing for the multicut problem"
],
[
"Factor Order.",
"Alg.",
"REF iterates through all edge and triangle subproblems.",
"The order is specified as follows: We assume that a node order is given.",
"With respect to this node order, edges $uv \\in E$ are ordered lexicographically.",
"For every triangle and its edge set $C = \\lbrace e_1,e_2,e_3\\rbrace \\subseteq E$ with $e_1 < e_2 < e_3$ , we define the ordering constraint $e_1 < C < e_3$ .",
"For every lollipop graph and its edge set $L = \\lbrace e_1, e_2, e_3, e_4\\rbrace $ with $e_1 < e_2 < e_3 < e_4$ , we define the ordering constraint $e_1 < L < e_4$ .",
"The strict partial order defined by these constraints is extended to a total order by topological sorting."
],
[
"Message Passing Description.",
"When an edge subproblem $uv \\in \\mathbb {E}$ is visited, Alg.",
"REF receives messages from all dependent triangle subproblems.",
"Having received a message from triangle $uvw \\in \\mathbb {E}$ , the costs $\\theta _{uvw}$ satisfy the condition $\\min _{x_{uw},x_{vw}} \\!\\!",
"\\theta _{uvw}(0,x_{uw},x_{vw})= \\min _{x_{uw},x_{vw}} \\!\\!",
"\\theta _{uvw}(1,x_{uw},x_{vw})\\hspace{5.0pt}.$ In other words, the cost of the triangle factor $\\theta _{uvw}$ has no preference for either $x_{uv} = 0$ or $x_{uv} = 1$ .",
"Sending messages from $\\theta _{uv}$ is analoguous: Having sent messages from $uv$ , we have $\\theta _{uv} = 0$ , i.e., there is again no preference for either $x_{uv} = 0$ or $x_{uv} = 1$ .",
"When we visit a triangle subproblem $uvw$ , we do the analogous with all dependent lollipop subproblems: Once messages have been received, lollipop subproblems have no preference for incident edges.",
"Once messages have been sent, this holds true for the triangle subproblems.",
"Once Alg.",
"REF has visited all subproblems and terminates, we reverse the order of subproblems and invoke Alg.",
"REF again.",
"This double call of Alg.",
"REF is repeated for a fixed number of iterations that is a parameter of our algorithm."
],
[
"Separation",
"Applying Alg.",
"REF with all cycles and all odd wheels of a graph $G$ is impractical, as the number of triangles for cycle inequalities () is cubic, and the number of lollipop graphs for odd wheels (REF ) is quartic in $|E|$ .",
"In order to arrive at a practical algorithm, we take a cutting plane approach in which we separate and add subproblems for violated cycle and odd wheel inequalities periodically.",
"Initially, $\\mathbb {V}$ contains only one element for every edge $e \\in E$ , and $\\mathbb {E}$ is empty.",
"In the primal, given some fractional $x \\in [0,1]^E$ , it is common to look for maximally violated inequalities () and (REF ).",
"This is possible in polynomial time via shortest path computations [14], [17].",
"In our dual formulation, we have no primal solution $x$ to search for violated inequalities.",
"Here, a suitable criterion is to consider those additional triangle or lollipop subproblems that necessarily increase the dual lower bound $L(\\theta )$ by some constant $\\epsilon > 0$ .",
"Among these subproblems, we choose those for which the increase is maximal and add them to the graph $(\\mathbb {V},\\mathbb {E})$ .",
"A similar dual cutting plane approach has shown to be useful for graphical models in [34].",
"As we discuss below, separation is more efficient in the dual than in the primal."
],
[
"Cycle Inequalities",
"[t] $G = (V,E), \\, \\, \\epsilon \\ge 0, \\, \\theta _e \\in \\mathbb {R}$ $l := 1$ $uv \\in E$ $\\theta _{uv} \\ge \\epsilon $ $\\textnormal {union}(u,v)$ $uv \\in E$ $\\theta _{uv} \\le -\\epsilon $ and find(u) = find(v) $C_l := \\textnormal {shortest-path}(u,v,\\epsilon )$ $l := l + 1$ Separation of cycle inequalities () We characterize those cycles whose subproblem increases the dual lower bound $L(\\theta )$ by at least $\\epsilon $ .",
"Proposition 1 Let $C = \\lbrace e_1,\\ldots ,e_k\\rbrace $ be a cycle with $\\theta _{e_1} \\le -\\epsilon $ and $\\theta _{e_l} \\le \\epsilon $ for $l > 1$ .",
"Then, the dual lower bound $L(\\theta )$ can be increased by $\\epsilon $ by including a triangulation of $C$ .",
"In order to find such cycles, we apply Alg.",
"REF .",
"This algorithm first records in a disjoint set data structure whether distinct nodes $u,v \\in V$ are connected via edges with weight $\\ge \\epsilon $ .",
"Then, it visits all edges $e \\in E$ with $\\theta _e \\le - \\epsilon $ .",
"If the endpoints of $e$ are connected by a path along which all edges have weight at least $\\epsilon $ , it searches for a shortest such path by means of breadth first search.",
"In the primal, finding a maximally violated cycle inequality () is more expensive, requiring, for every edge $uv \\in E$ , the search for a $uv$ -path with minimum cost $x$ [14] by, e.g., Dijkstra's algorithm."
],
[
"Odd Wheel Inequalities",
"[t] Triangles $uvw$ , costs $\\theta _{uvw}$ , $\\epsilon \\ge 0$ $l := 0$ $u \\in V$ $G^{\\prime } = (V^{\\prime }, E^{\\prime }), V^{\\prime } = \\varnothing $ , $E^{\\prime } = \\varnothing $ , $Connect = \\varnothing $ triangles $uvw$ (REF ) holds true $V^{\\prime } := V^{\\prime } \\cup \\lbrace v,v^{\\prime },w,w^{\\prime }\\rbrace $ $E^{\\prime } := E^{\\prime } \\cup \\lbrace vw^{\\prime }, v^{\\prime }w\\rbrace $ union$(v,v^{\\prime })$ union$(w,w^{\\prime })$ $v \\in V^{\\prime } \\cap V$ $\\textnormal {find}(u) = \\textnormal {find}(v)$ $P^{\\prime } := \\textnormal {shortest-path}_{G^{\\prime }}(u,v,\\epsilon )$ $C = \\lbrace uv \\in E \\,|\\, uv^{\\prime } \\in P^{\\prime } \\vee u^{\\prime }v \\in P^{\\prime }\\rbrace $ $C$ is a simple cycle in $G$ $O_l := \\lbrace u,P\\rbrace $ $l := l + 1$ Separation of odd wheel inequalities (REF ) We characterize those odd wheels whose lollipop subproblem increases the lower bound $L(\\theta )$ by at least $\\epsilon $ .",
"Proposition 2 Let $O$ an odd wheel with center node $u$ and cycle nodes $v_1,\\ldots ,v_k$ .",
"Adding the lollipop subproblems for $O$ increases $L(\\theta )$ by at least $\\epsilon $ if the costs $\\theta _{u v_i v_{i+1}}$ of each triangle $uv_iv_{i+1}$ are such that the minimal cost of any edge labeling of the triangle cutting precisely one edge incident to $u$ is smaller by $\\epsilon $ than the minimal cost of any edge labeling of the triangle cutting 0 or 2 edges incident to $u$ .",
"That is: $& \\min _{\\lbrace x: \\, x_{uv_i} +x_{uv_{i+1}} = 1\\rbrace } \\theta _{u v_i v_{i+1}}(x) + \\epsilon \\nonumber \\\\\\le \\quad & \\min _{\\lbrace x: \\, x_{uv_i} +x_{uv_{i+1}} \\ne 1\\rbrace } \\theta _{u v_i v_{i+1}}(x)\\hspace{5.0pt}.$ In order to find such odd wheels, we apply Alg.",
"REF .",
"This algorithm builds on our observation that we need to look only at triangles whose subproblem has already been added.",
"Hence, Alg.",
"REF visits each node $u \\in V$ and builds a bipartite graph $G^{\\prime } = (V^{\\prime },E^{\\prime })$ as follows.",
"(An example is depicted in Fig.",
"REF for a 5-wheel and (REF ) holding true for all triangles of the wheel.)",
"For each triangle $u v k$ such that (REF ) holds true, four nodes $v, v^{\\prime }, k, k^{\\prime } \\in V^{\\prime }$ are added to $V^{\\prime }$ , two copies of each original node.",
"These are joined by edges $uv^{\\prime }, u^{\\prime }v \\in E^{\\prime }$ .",
"If a path from $u$ to $u^{\\prime }$ exists in $G^{\\prime }$ , we have found a violated odd wheel inequality (REF ).",
"As $G^{\\prime }$ is bipartite, a $uu^{\\prime }$ -path in $G^{\\prime }$ corresponds to an odd cycle in $G$ .",
"As before, the search for paths is accelerated by connectivity tests via a disjoint set data structure and is carried out by breadth first search.",
"In the primal, finding a maximally violated odd wheel inequality (REF ) entails the same construction of the bipartite graph $G^{\\prime }$ for each node $u \\in V$ [17].",
"However, a shortest path search w.r.t.",
"edge costs $\\frac{1}{2} - x_{vw} + \\frac{1}{2}(x_{uv} + x_{uw})$ needs to be carried out by Dijkstra's algorithm instead of breadth first search.",
"Further complication in the primal comes from the fact that a separation algorithm needs to visit all $v \\in V^{\\prime }$ in order to compute the shortest $vv^{\\prime }$ -path in $G^{\\prime }$ .",
"Figure: Depicted above is the bipartite graph G ' G^{\\prime } constructed by Alg.",
"for separating the 5-wheel depicted in Fig.",
"."
],
[
"Rounding",
"Our message passing Alg.",
"REF improves a dual lower bound on (REF ), but does not provide a feasible solution of (REF )–().",
"In order to obtain a feasible multicut, we apply a local search algorithm defined in [26], namely greedy additive edge contraction (GAEC), followed by Kernighan-Lin with joins (KLj).",
"GAEC computes a multicut by greedily contracting those edges for which the join decreases the cost maximally.",
"It stops as soon as no contraction of any edge strictly decreases the cost.",
"KLj attempts to improve a given multicut recursively by applying transformations from three classes: (1) moving nodes between two components, (2) moving nodes from a given component to a newly forming one or (3) joining two components.",
"GAEC and KLj are local search algorithms that output a feasible multicut that need not be optimal.",
"We apply GAEC and KLj not only to the instance of the minimum cost multicut problem given as input but also to the re-parameterization of this instance output by Alg.",
"REF .",
"The rationale for doing so comes from LP duality: Proposition 3 Assume $\\theta $ maximizes the dual lower bound $L(\\theta )$ and the relaxation is tight, i.e.",
"$L(\\theta )= \\min _{\\lbrace x \\in \\lbrace 0,1\\rbrace ^E \\,|\\, x^{-1}(1) \\in \\mathcal {M}_G\\rbrace } \\langle \\theta , x \\rangle \\hspace{5.0pt}.$ Moreover, let $\\hat{x} \\in \\lbrace 0,1\\rbrace ^E$ such that $\\hat{x}^{-1}(1)$ is an optimal multicut of $G$ .",
"Then, $\\theta _e {\\left\\lbrace \\begin{array}{ll}\\le 0 & \\textnormal {if}\\ \\hat{x}_e = 1 \\\\\\ge 0, & \\textnormal {if}\\ \\hat{x}_e = 0\\end{array}\\right.",
"}$ Having run Alg.",
"REF for a while, we expect $\\theta $ to fulfill the sign condition of Prop.",
"REF approximately.",
"Therefore, the sign of $\\theta _e$ will be a good hint of the edge $e$ being cut.",
"Thus, informally, we expect local search algorithms operating on the re-parameterized instance of the problem to yield better feasible multicuts than local search algorithms operating on the given instance.",
"For MAP-inference in discrete graphical models, it is known from [28], [29] that primal rounding can be improved greatly when applied to cost functions re-parameterized by message passing."
],
[
"Solvers",
"We compare against several state of the art algorithms.",
"The algorithm MC-ILP [25] is an efficient implementation of a cutting plane algorithm solving (REF ) using cycle inequalities () in a cutting plane fashion.",
"CPlex [2] is used to solve the underlying ILP problems.",
"The integrality conditions in () are directly given to the solver.",
"According to [25] this is beneficial due to the excellent branch and cut capabilities of CPlex [2].",
"Cut, Glue & Cut [11], abbreviated as CGC, is a move making algorithm using planar max-cut subproblems to improve multicuts.",
"Fusion moves for correlation clustering [10], abbreviated as CC-Fusion, fuses multicuts generated by various proposal generator with the help of auxiliary multicut problems, solved in turn by MC-ILP.",
"We use randomized hierarchical clustering and randomized watersheds as proposal generators, identified by the suffixes-RHC and -RWS.",
"We use parameters for the proposal generators as recommended by the authors [10].",
"MP-C denotes Algorithm REF when we only separate for cycle inequalities () by Algorithm REF , while MP-COW denotes that we additionally separate for odd wheel inequalities (REF ) by Algorithm REF .",
"We search for triangles and lollipops to add every 10th iteration.",
"KL is the GAEC and KLj implementation [26] described in Section REF for computing multicuts.",
"We let KL run every 100th iteration of MP-C and MP-COW on the current reparametrized edge costs.",
"MC-ILP, CGC and CC-Fusion are implemented as part of the OpenGM suite [23].",
"Only MC-ILP and our solvers MP-C and MP-COW generate dual lower bounds.",
"CGC also outputs dual lower bounds, but these are equivalent to the trivial lower bound $\\sum _{e \\in \\mathsf {E}} \\min (0,\\theta _e)$ , where edge weights $\\theta _e$ are as given by the problem.",
"It has been shown that CGC, CC-Fusion and KL outperform other primal heuristics [10], hence we do not compare to any other heuristic algorithm.",
"Also MC-ILP outperforms the LP-based solver [32], due to the latter using the slower COIN-OR CLP [15] solver internally, hence we exclude it from the comparison as well.",
"All solvers were run on a laptop computer with a i5-5200 CPU with 2.2 GHz and 8GB RAM."
],
[
"Datasets",
"We compare on 8 datasets of diverse origin.",
"image-seg consists of images of the Berkeley segmentation dataset [30], presegmented with superpixels, for which pairwise affinity values have been computed as in [4].",
"The knott-3d-{150|300|450|550} datasets come from a neural circuit reconstruction problem of tissue [5] with $[150]^3$ , $[300]^3$ , $[450]^3$ and $[900]^3$ voxels.",
"The data is presegmented into supervoxels.",
"modularity clustering aims to cluster a social network into subgroups based on affinity between individual persons.",
"CREMI-{small|large} datasets were constructed as part of the CREMI [1] challenge, which aims to reconstruct neural circuits of the adult fly brain.",
"The images are taken by electron microscopy.",
"The -small instances are cropped versions of the -large ones.",
"To our knowledge, the CREMI-large dataset contain the largest multicut problems approached with LP-based methods.",
"The image-seg, knott-3d and modularity clustering datasets were taken from the OpenGM benchmark [23], while the CREMI datasets were kindly provided by their authors and are not yet published.",
"The dataset consists of 100, 8, 8, 8, 8, 6, 3 and 3 instances, in total 144.",
"Dataset details can be found in Table REF ."
],
[
"Evaluation",
"We have set a timelimit of one hour for all algorithms.",
"In Table REF results averaged over all instances in specific datasets are reported.",
"In Figure REF primal solution energy and dual lower bound (where applicable) averaged over all instances in specific datasets are drawn against runtime.",
"As can be seen from Table REF , except for dataset CREMI-large, our solver MP-COW gives dual bounds that are within 0.0045%, 1.9%, 0.0061%, 0.0068%, 0.0017%, 0.0007% and 0.0083% of the dual lower bound obtained by MC-ILP, which uses the advanced branch-and-cut facilities CPlex [2] provides.",
"For CREMI-large only our solvers MP-C and MP-COW output dual lower bounds, as MC-ILP did not finish a single iteration after one hour.",
"As can be seen from Fig.",
"REF our lower bound usually converges faster than MC-ILP's.",
"We conjecture that MC-C and MP-COW inside a branch-and-bound solver can significantly extend the reach of exact methods for the multicut problem.",
"Strangely, KL does not perform well on image-seg, even though the lower bound we achieve with MP-C and MP-COW are not far from the optimal lower bounds computed by MC-ILP.",
"On the other hand, MP-C and MP-COW give much better dual and primal results for modularity-clustering early on.",
"Generally, when compared to MC-ILP's primal convergence, we give much lower values early on, and for the large-scale datasets knott-3d-550, CREMI-small, CREMI-large, MCI-ILP's primal solutions are not useful anymore.",
"Unlike MC-ILP, our reparametrized costs can be used to improve heuristic primal algorithms.",
"An example of this can be seen in Fig.",
"REF , where reparametrized costs improve KL's solutions.",
"Table: Primal solution energy (UB)/dual lower bound (LB)/runtime in seconds averaged over all instances of datasets.#I means number of instances in dataset, #V and #E mean number of vertices and edges in multicut instances.†\\dagger signifies method did not finish one iteration after one hour, so was excluded from comparison.‡\\ddagger means method does not output dual lower bound.Bold numbers signify lowest primal solution energy, highest lower bound, fastest runtime.Figure: Averaged runtime plots for image-seg, modularity clustering, knott-3d-150, knott-3d-300, knott-3d-450, knott-3d-550, CREMI-small and CREMI-large datasets.Continuous lines denote dual lower bounds and dashed ones primal energies.Values are averaged over all instances of the dataset.The x-axis is logarithmic.Figure: NO_CAPTIONInstance gm_knott_3d_072 from dataset knott-3d-300 where reparametrized costs improve KL's solutions."
],
[
"Conclusion",
"We have shown that LP-based methods are feasible for solving large scale multicut problems on commodity hardware and one does not have to resort to heuristic primal algorithms.",
"We achieve dual bounds very close to those computed by state-of-the-art branch-and-cut solvers.",
"Additionally, our method usually gives much faster dual bound convergence, resulting in superior solutions when terminated early.",
"Also the primal heuristic GAEC + KLj can be improved when run on costs as computed by our method.",
"It remains an interesting task to integrate primal heuristics more tightly into our message passing approach and further improve the dual lower bound by e.g.",
"embedding our solver into branch and cut."
],
[
"Acknowledgments",
"The authors would like to thank Vladimir Kolmogorov for helpful discussions.",
"This work is partially funded by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 616160."
],
[
"Proof of Proposition ",
"Proposition Let $C = \\lbrace e_1,\\ldots ,e_k\\rbrace $ be a cycle with $\\theta _{e_1} \\le -\\epsilon $ and $\\theta _{e_l} \\ge \\epsilon $ for $l > 1$ .",
"Then, the dual lower bound $L(\\theta )$ can be increased by $\\epsilon $ by including a triangulation of $C$ .",
"Let cycle $C$ have vertices $\\lbrace v_1,\\ldots ,v_k\\rbrace $ and assume that $e_i \\ v_i v_{i+1}$ with $k+1 = 1$ for notational purposes.",
"After triangulation, triangle factors on vertices $v_1v_2v_3,\\ldots v_1v_{k-1}v_k$ will be present in the model.",
"Let the current reparametrization be $\\theta $ .",
"The triangle factors corresponding to cycle $C$ will enforce the cycle inequality () $x_{e_1} \\le \\sum _{i=1,\\ldots ,k} x_{e_i}\\,.$ It holds that $- \\theta _{e_1} = \\min _{x_{e_1},\\ldots ,x_{e_k} \\in [0,1]} \\sum _{i=1}^k \\theta _{e_i} x_{e_i} \\\\\\le -\\epsilon +\\min _{x_{e_1},\\ldots ,x_{e_k} \\in [0,1]} \\sum _{i=1}^k \\theta _{e_i} x_{e_i} \\text{ s.t.~(\\ref {eq:ViolatedCycle})} \\\\\\le \\epsilon + \\max _{\\left\\lbrace \\begin{array}{c}\\theta _{e_1},\\ldots ,\\theta _{e_k} \\\\ \\theta _{v_1v_2v_3},\\ldots ,\\theta _{v_1v_{k-1}v_k} \\\\ \\text{a reparametrization} \\end{array}\\right\\rbrace } L_C(\\theta )$ where $L_C(\\theta ) = \\sum _{i=1}^k \\min (0,\\theta _{e_i}) + \\sum _{i=2}^{k-1} \\min \\lbrace \\theta _{v_1v_iv_{i+1}}\\rbrace $ the dual lower bound on cycle $C$ .",
"The first inequality above is due to either $x_{e_1} = 0$ in the optimal solution or one $x_{e_2},\\ldots ,x_{e_k}$ being one due to (REF ).",
"The second inequality is due to the fact that (i) $\\max _{\\theta \\text{ a reparametrization}} L(\\theta ) =\\ min_{\\mu \\in \\Lambda } \\langle \\theta ,\\mu \\rangle $ by linear programming duality and (ii) the triangle factors enforce more inequalities than only (REF )."
],
[
"Proof of Proposition ",
"Proposition Let $O$ an odd wheel with center node $u$ and cycle nodes $v_1,\\ldots ,v_k$ .",
"Adding the lollipop subproblems for $O$ increases $L(\\theta )$ by at least $\\epsilon $ if the costs $\\theta _{u v_i v_{i+1}}$ of each triangle $uv_iv_{i+1}$ are such that the minimal cost of any edge labeling of the triangle cutting precisely one edge incident to $u$ is smaller by $\\epsilon $ than the minimal cost of any edge labeling of the triangle cutting 0 or 2 edges incident to $u$ .",
"That is: $& \\min _{\\lbrace x: \\, x_{uv_i} +x_{uv_{i+1}} = 1\\rbrace } \\theta _{u v_i v_{i+1}}(x) + \\epsilon \\nonumber \\\\\\le \\quad & \\min _{\\lbrace x: \\, x_{uv_i} +x_{uv_{i+1}} \\ne 1\\rbrace } \\theta _{u v_i v_{i+1}}(x)\\hspace{5.0pt}.$ Condition (REF ) means that in all triangles in the odd wheel $O$ , the minimal assignment with regard to the current reparametrization, has exactl one edge incident to $u$ .",
"All other assignment have cost greater by at least $\\epsilon $ .",
"As $k$ is odd, there is no possiblity to combine those local assignments to a global assignment on $O$ .",
"On the other hand, our construction of lollipop factors ensures exactness on odd wheels.",
"As at least one triangle must then be assigned costs that are not locally optimal and which is larger by $\\epsilon $ than its minimal reparametrized cost, the result follows."
],
[
"Proof of Proposition ",
"Proposition Assume $\\theta $ maximizes the dual lower bound $L(\\theta )$ and the relaxation is tight, i.e.",
"$L(\\theta )= \\min _{\\lbrace x \\in \\lbrace 0,1\\rbrace ^E \\,|\\, x^{-1}(1) \\in \\mathcal {M}_G\\rbrace } \\langle \\theta , x \\rangle \\hspace{5.0pt}.$ Moreover, let $\\hat{x} \\in \\lbrace 0,1\\rbrace ^E$ such that $\\hat{x}^{-1}(1)$ is an optimal multicut of $G$ .",
"Then, $\\theta _e {\\left\\lbrace \\begin{array}{ll}\\le 0 & \\textnormal {if}\\ \\hat{x}_e = 1 \\\\\\ge 0, & \\textnormal {if}\\ \\hat{x}_e = 0\\end{array}\\right.",
"}$ Follows from the complementary slackness conditions in linear programming duality."
],
[
"Detailed experimental evaluation",
"In Table REF a detailed per instance evaluation of all algorithms considered in the experimental section can be found.",
"|llcccccc| Per instance evaluation of the considered eight datasets.",
"UB means primal solution energy, LB dual lower bound and runtime(s) the runtime in seconds.",
"Bold numbers indicate lowest primal energy, highest lower bound and smallest runtime.",
"$\\dagger $ means method not applicable.",
"Instance MP-C MC-COW CGC MC-ILP CC-Fusion-RWS CC-Fusion-RHC 8 | c | image-seg 3* 101087.bmp UB 2906.16 2906.16 2853.56 2789.90 2800.22 2789.90 LB 2788.69 2788.95 2622.38 2789.90 $\\dagger $ $\\dagger $ runtime(s) 0.31 0.59 0.02 5.11 0.63 0.81 3* 102061.bmp UB 3017.57 3017.57 3090.33 2943.77 2963.42 2944.46 LB 2932.80 2933.39 2750.99 2943.77 $\\dagger $ $\\dagger $ runtime(s) 3.26 5.83 0.05 8.75 0.61 0.96 3* 103070.bmp UB 4437.16 4444.27 4457.13 4199.38 4205.03 4200.64 LB 4196.88 4196.94 3842.84 4199.38 $\\dagger $ $\\dagger $ runtime(s) 11.84 15.93 0.15 6.97 1.32 0.98 3* 105025.bmp UB 6332.73 6333.88 6290.71 6055.33 6070.84 6061.05 LB 6045.21 6046.08 5506.01 6055.33 $\\dagger $ $\\dagger $ runtime(s) 33.75 55.69 0.35 32.15 1.91 1.59 3* 106024.bmp UB 1832.16 1832.16 1654.27 1599.25 1618.18 1599.83 LB 1597.07 1597.21 1466.60 1599.25 $\\dagger $ $\\dagger $ runtime(s) 0.64 4.09 0.02 4.76 0.18 0.25 3* 108005.bmp UB 6839.76 6841.08 6855.33 6578.03 6584.62 6578.18 LB 6567.48 6569.63 6151.29 6578.03 $\\dagger $ $\\dagger $ runtime(s) 4.58 22.38 0.26 10.86 2.49 1.54 3* 108070.bmp UB 9082.04 9083.42 8612.32 8422.24 8445.73 8424.36 LB 8414.64 8414.99 7818.70 8422.24 $\\dagger $ $\\dagger $ runtime(s) 28.23 52.32 0.41 26.67 3.35 1.75 3* 108082.bmp UB 5125.72 5127.99 5090.88 4800.15 4815.78 4806.04 LB 4786.46 4789.19 4380.66 4800.15 $\\dagger $ $\\dagger $ runtime(s) 6.85 25.07 0.16 13.51 1.20 1.89 3* 109053.bmp UB 4575.76 4579.97 4616.61 4421.13 4424.05 4421.13 LB 4412.45 4413.08 4021.22 4421.13 $\\dagger $ $\\dagger $ runtime(s) 20.74 71.75 0.17 9.64 1.50 0.84 3* 119082.bmp UB 4512.04 4512.04 4642.96 4530.71 4535.85 4532.29 LB 4526.91 4526.91 4346.24 4530.71 $\\dagger $ $\\dagger $ runtime(s) 0.10 0.10 0.02 0.48 1.53 1.57 3* 12084.bmp UB 7502.34 7502.34 7443.28 7284.45 7301.17 7287.68 LB 7276.26 7277.38 6941.02 7284.45 $\\dagger $ $\\dagger $ runtime(s) 3.48 8.10 0.15 2.47 2.20 3.99 3* 123074.bmp UB 4200.21 4204.24 4031.03 3842.74 3856.82 3847.83 LB 3829.51 3829.04 3439.47 3842.74 $\\dagger $ $\\dagger $ runtime(s) 35.27 24.55 0.14 23.01 0.47 0.86 3* 126007.bmp UB 2756.86 2760.64 2747.72 2684.83 2706.76 2685.26 LB 2677.02 2677.08 2512.08 2684.83 $\\dagger $ $\\dagger $ runtime(s) 0.23 0.53 0.01 0.79 0.38 0.82 3* 130026.bmp UB 6066.91 6066.91 5580.58 5350.83 5369.95 5354.31 LB 5331.00 5336.93 4828.82 5350.83 $\\dagger $ $\\dagger $ runtime(s) 36.91 167.54 0.26 19.66 0.99 1.40 3* 134035.bmp UB 6840.69 6840.69 6679.89 6578.98 6595.87 6579.62 LB 6562.20 6565.03 6166.95 6578.98 $\\dagger $ $\\dagger $ runtime(s) 8.44 43.34 0.19 28.82 1.81 1.33 3* 14037.bmp UB 1566.34 1582.13 1431.56 1383.14 1393.66 1383.14 LB 1375.55 1375.55 1274.27 1383.14 $\\dagger $ $\\dagger $ runtime(s) 0.09 0.17 0.01 0.25 0.13 0.22 3* 143090.bmp UB 1728.22 1728.22 1807.41 1714.38 1725.88 1715.76 LB 1712.44 1712.26 1595.54 1714.38 $\\dagger $ $\\dagger $ runtime(s) 0.43 2.49 0.01 0.56 0.44 0.34 3* 145086.bmp UB 3806.23 3808.75 3407.83 3322.21 3329.14 3322.59 LB 3319.36 3319.34 3197.53 3322.21 $\\dagger $ $\\dagger $ runtime(s) 0.32 0.45 0.01 0.83 0.41 1.59 3* 147091.bmp UB 4092.25 4092.25 4129.72 3973.71 3982.30 3975.15 LB 3968.35 3970.58 3734.67 3973.71 $\\dagger $ $\\dagger $ runtime(s) 1.50 10.96 0.10 13.24 0.90 0.83 3* 148026.bmp UB 8411.55 8411.55 8436.70 8205.98 8226.20 8207.72 LB 8198.62 8199.84 7780.68 8205.98 $\\dagger $ $\\dagger $ runtime(s) 6.48 29.65 0.15 4.49 3.10 2.73 3* 148089.bmp UB 6680.96 6682.82 6666.83 6439.58 6455.48 6440.33 LB 6432.06 6431.52 6030.94 6439.58 $\\dagger $ $\\dagger $ runtime(s) 10.44 15.57 0.17 18.64 2.00 1.53 3* 156065.bmp UB 5798.42 5801.59 5429.45 5234.15 5248.18 5234.76 LB 5224.22 5225.10 4857.14 5234.15 $\\dagger $ $\\dagger $ runtime(s) 10.19 35.24 0.15 18.51 1.49 1.12 3* 157055.bmp UB 4797.86 4798.32 4768.87 4685.17 4696.42 4685.17 LB 4679.04 4679.39 4472.39 4685.17 $\\dagger $ $\\dagger $ runtime(s) 0.49 3.10 0.02 2.60 1.37 1.48 3* 159008.bmp UB 4688.87 4694.40 4814.85 4540.87 4569.12 4541.22 LB 4534.88 4535.76 4217.95 4540.87 $\\dagger $ $\\dagger $ runtime(s) 3.53 21.00 0.10 16.64 0.83 1.29 3* 160068.bmp UB 3263.91 3265.02 3264.31 3089.32 3103.23 3089.32 LB 3088.23 3088.17 2866.45 3089.32 $\\dagger $ $\\dagger $ runtime(s) 1.18 2.28 0.05 1.80 0.61 1.07 3* 16077.bmp UB 4440.51 4443.05 4408.62 4227.88 4236.46 4228.78 LB 4224.10 4224.60 3921.75 4227.88 $\\dagger $ $\\dagger $ runtime(s) 4.37 13.81 0.07 3.22 1.18 1.63 3* 163085.bmp UB 4493.17 4493.17 4577.62 4381.13 4406.98 4384.59 LB 4370.36 4371.00 3983.52 4381.13 $\\dagger $ $\\dagger $ runtime(s) 21.91 35.30 0.15 9.82 0.91 1.30 3* 167062.bmp UB 1623.20 1623.20 1281.48 1273.72 1275.87 1275.67 LB 1273.01 1273.29 1233.39 1273.72 $\\dagger $ $\\dagger $ runtime(s) 0.10 0.66 0.00 1.00 0.11 0.17 3* 167083.bmp UB 8979.42 8979.42 8545.37 8331.63 8344.35 8331.90 LB 8325.80 8328.03 7921.06 8331.63 $\\dagger $ $\\dagger $ runtime(s) 9.83 53.28 0.23 11.74 2.27 1.80 3* 170057.bmp UB 3602.31 3602.31 3355.95 3266.17 3273.20 3266.73 LB 3260.19 3260.98 2989.38 3266.17 $\\dagger $ $\\dagger $ runtime(s) 21.07 26.30 0.07 18.20 0.53 0.67 3* 175032.bmp UB 11863.40 11863.40 11926.00 11542.63 11566.74 11547.67 LB 11525.63 11526.57 10543.16 11542.63 $\\dagger $ $\\dagger $ runtime(s) 623.66 632.98 1.27 165.83 3.25 3.98 3* 175043.bmp UB 8022.65 8033.50 8224.34 7816.92 7844.49 7822.01 LB 7809.60 7811.17 7136.44 7816.92 $\\dagger $ $\\dagger $ runtime(s) 4.74 29.83 0.39 13.17 4.30 3.46 3* 182053.bmp UB 3700.96 3700.96 3714.59 3579.24 3602.74 3582.99 LB 3575.85 3576.12 3321.74 3579.24 $\\dagger $ $\\dagger $ runtime(s) 4.05 12.83 0.07 14.11 0.65 0.92 3* 189080.bmp UB 1152.76 1155.19 1095.38 1077.47 1086.93 1078.41 LB 1072.82 1073.00 972.41 1077.47 $\\dagger $ $\\dagger $ runtime(s) 0.07 0.11 0.00 0.25 0.11 0.49 3* 19021.bmp UB 4717.39 4718.32 4693.10 4515.08 4521.63 4515.27 LB 4498.74 4499.79 4178.50 4515.08 $\\dagger $ $\\dagger $ runtime(s) 1.67 4.36 0.08 12.93 1.16 1.41 3* 196073.bmp UB 592.63 594.41 572.73 545.47 548.39 545.47 LB 544.89 544.89 508.04 545.47 $\\dagger $ $\\dagger $ runtime(s) 0.12 0.55 0.00 0.76 0.07 0.24 3* 197017.bmp UB 3207.33 3207.33 2857.86 2798.77 2801.34 2798.77 LB 2792.60 2792.59 2663.98 2798.77 $\\dagger $ $\\dagger $ runtime(s) 0.82 1.01 0.01 0.77 0.38 0.77 3* 208001.bmp UB 6533.30 6533.30 6605.51 6272.68 6277.73 6275.37 LB 6254.97 6256.22 5773.37 6272.68 $\\dagger $ $\\dagger $ runtime(s) 22.20 63.32 0.25 78.10 2.12 1.69 3* 210088.bmp UB 1954.78 1956.20 2034.65 1895.44 1901.54 1896.58 LB 1891.95 1891.95 1726.75 1895.44 $\\dagger $ $\\dagger $ runtime(s) 0.26 0.38 0.02 1.26 0.49 0.51 3* 21077.bmp UB 3031.01 3032.66 3001.74 2946.71 2966.82 2946.71 LB 2944.67 2944.60 2793.10 2946.71 $\\dagger $ $\\dagger $ runtime(s) 1.42 3.86 0.01 4.60 0.44 1.13 3* 216081.bmp UB 4462.75 4463.13 4324.17 4158.73 4176.92 4158.73 LB 4155.98 4156.01 3952.38 4158.73 $\\dagger $ $\\dagger $ runtime(s) 0.34 0.60 0.03 1.37 1.13 1.08 3* 219090.bmp UB 2810.16 2810.16 2624.72 2501.27 2507.80 2501.27 LB 2499.72 2499.93 2379.46 2501.27 $\\dagger $ $\\dagger $ runtime(s) 0.16 0.55 0.02 0.14 0.48 0.97 3* 220075.bmp UB 3265.75 3269.41 3155.80 3115.95 3124.04 3115.95 LB 3110.83 3111.22 2939.67 3115.95 $\\dagger $ $\\dagger $ runtime(s) 0.16 1.46 0.01 0.38 0.75 1.65 3* 223061.bmp UB 6812.98 6818.95 6751.72 6576.83 6591.07 6578.02 LB 6565.07 6566.44 5983.87 6576.83 $\\dagger $ $\\dagger $ runtime(s) 89.00 248.06 0.34 37.60 1.67 1.53 3* 227092.bmp UB 2178.48 2180.95 2051.77 1998.46 2001.58 2001.19 LB 1986.49 1986.11 1816.59 1998.46 $\\dagger $ $\\dagger $ runtime(s) 13.93 26.75 0.03 4.25 0.37 0.34 3* 229036.bmp UB 6681.13 6684.62 6250.80 6125.73 6153.41 6126.61 LB 6115.98 6116.08 5729.80 6125.73 $\\dagger $ $\\dagger $ runtime(s) 4.32 12.41 0.09 4.83 1.53 2.00 3* 236037.bmp UB 9324.06 9325.93 9507.86 9060.84 9071.82 9060.84 LB 9047.32 9048.52 8261.54 9060.84 $\\dagger $ $\\dagger $ runtime(s) 62.98 166.69 0.75 20.14 5.02 4.38 3* 24077.bmp UB 4846.51 4847.62 4892.44 4761.98 4779.26 4761.98 LB 4760.74 4760.88 4531.31 4761.98 $\\dagger $ $\\dagger $ runtime(s) 3.06 5.78 0.03 5.10 1.14 2.40 3* 241004.bmp UB 1271.42 1272.99 1077.34 1057.14 1060.90 1057.42 LB 1056.38 1056.34 984.58 1057.14 $\\dagger $ $\\dagger $ runtime(s) 0.18 0.92 0.00 0.15 0.13 0.26 3* 241048.bmp UB 4972.81 4972.81 4945.52 4730.95 4750.17 4731.19 LB 4719.03 4719.99 4343.63 4730.95 $\\dagger $ $\\dagger $ runtime(s) 4.69 26.23 0.10 17.26 0.79 2.00 3* 253027.bmp UB 6839.82 6841.19 6950.82 6606.62 6614.15 6606.62 LB 6603.97 6604.66 6371.53 6606.62 $\\dagger $ $\\dagger $ runtime(s) 18.42 49.86 0.32 7.39 2.90 2.36 3* 253055.bmp UB 1684.63 1684.63 1549.84 1502.16 1518.91 1502.16 LB 1497.70 1497.78 1408.65 1502.16 $\\dagger $ $\\dagger $ runtime(s) 0.30 0.99 0.01 1.31 0.09 0.19 3* 260058.bmp UB 1458.96 1458.96 1091.96 1084.26 1085.01 1084.26 LB 1082.23 1082.23 1017.59 1084.26 $\\dagger $ $\\dagger $ runtime(s) 0.03 0.05 0.00 0.18 0.10 0.15 3* 271035.bmp UB 3706.68 3707.59 3875.99 3621.00 3657.57 3621.48 LB 3613.78 3614.44 3326.25 3621.00 $\\dagger $ $\\dagger $ runtime(s) 0.65 3.19 0.10 12.45 0.78 1.75 3* 285079.bmp UB 6105.56 6105.56 5773.41 5610.12 5640.22 5612.68 LB 5603.39 5603.74 5246.17 5610.12 $\\dagger $ $\\dagger $ runtime(s) 9.30 28.78 0.10 26.96 1.30 2.02 3* 291000.bmp UB 10554.63 10557.44 10401.10 10208.87 10222.91 10210.47 LB 10199.14 10200.78 9626.61 10208.87 $\\dagger $ $\\dagger $ runtime(s) 13.41 44.85 0.56 39.32 2.46 2.24 3* 295087.bmp UB 4562.91 4571.66 4509.08 4290.54 4299.90 4291.66 LB 4288.00 4287.83 3985.98 4290.54 $\\dagger $ $\\dagger $ runtime(s) 3.71 12.35 0.11 6.34 1.43 1.83 3* 296007.bmp UB 2503.63 2507.50 2384.77 2293.13 2306.24 2293.83 LB 2290.91 2290.91 2115.21 2293.13 $\\dagger $ $\\dagger $ runtime(s) 0.32 0.63 0.03 0.34 0.28 0.47 3* 296059.bmp UB 2240.64 2241.78 2160.60 2044.71 2045.98 2044.71 LB 2039.19 2039.18 1891.55 2044.71 $\\dagger $ $\\dagger $ runtime(s) 0.57 0.99 0.02 1.14 0.27 0.24 3* 299086.bmp UB 1570.66 1570.84 1650.07 1557.24 1561.49 1557.24 LB 1550.54 1550.56 1450.72 1557.24 $\\dagger $ $\\dagger $ runtime(s) 0.47 2.10 0.02 0.11 0.29 0.41 3* 300091.bmp UB 1839.69 1839.69 1508.08 1495.10 1498.51 1495.10 LB 1487.72 1487.90 1426.88 1495.10 $\\dagger $ $\\dagger $ runtime(s) 0.16 0.49 0.01 1.91 0.21 0.16 3* 302008.bmp UB 2616.47 2616.47 2583.88 2543.23 2557.22 2543.23 LB 2539.89 2539.89 2481.98 2543.23 $\\dagger $ $\\dagger $ runtime(s) 0.21 0.54 0.01 0.23 0.17 0.46 3* 304034.bmp UB 8090.64 8090.64 8159.63 7835.47 7850.78 7836.49 LB 7824.86 7827.27 7212.50 7835.47 $\\dagger $ $\\dagger $ runtime(s) 9.24 54.06 0.40 24.88 3.14 2.05 3* 304074.bmp UB 4490.66 4497.86 4138.04 3891.88 3897.33 3891.88 LB 3883.34 3882.52 3543.87 3891.88 $\\dagger $ $\\dagger $ runtime(s) 3.48 6.21 0.11 0.51 0.83 0.79 3* 306005.bmp UB 4683.24 4690.02 4497.63 4290.25 4305.78 4290.25 LB 4286.33 4286.33 4004.02 4290.25 $\\dagger $ $\\dagger $ runtime(s) 5.25 14.67 0.11 17.38 0.91 0.98 3* 3096.bmp UB 295.31 295.31 396.90 396.90 411.23 396.90 LB 388.89 388.89 389.83 396.90 $\\dagger $ $\\dagger $ runtime(s) 0.00 0.00 0.00 0.02 0.00 0.06 3* 33039.bmp UB 8286.04 8286.04 8555.09 8069.67 8082.04 8070.48 LB 8061.19 8061.78 7384.06 8069.67 $\\dagger $ $\\dagger $ runtime(s) 27.46 67.13 0.44 20.06 3.61 3.29 3* 351093.bmp UB 6152.76 6155.04 6342.60 6105.28 6111.45 6107.08 LB 6096.66 6097.34 5679.10 6105.28 $\\dagger $ $\\dagger $ runtime(s) 19.48 64.42 0.20 19.54 2.10 2.30 3* 361010.bmp UB 3500.13 3503.88 3459.05 3361.02 3368.12 3364.98 LB 3356.03 3356.01 3189.41 3361.02 $\\dagger $ $\\dagger $ runtime(s) 0.42 1.06 0.02 0.39 0.91 0.79 3* 37073.bmp UB 1991.36 1991.36 2044.68 1975.00 1982.24 1975.00 LB 1972.11 1972.11 1904.57 1975.00 $\\dagger $ $\\dagger $ runtime(s) 0.14 0.30 0.01 0.15 0.31 0.49 3* 376043.bmp UB 6549.11 6557.74 6054.26 5863.83 5872.57 5864.00 LB 5859.62 5860.01 5433.92 5863.83 $\\dagger $ $\\dagger $ runtime(s) 7.34 31.66 0.17 9.80 1.32 0.94 3* 38082.bmp UB 8324.62 8324.62 8492.54 8060.34 8065.57 8066.62 LB 8047.84 8048.69 7359.93 8060.34 $\\dagger $ $\\dagger $ runtime(s) 439.50 702.48 0.62 26.23 2.41 2.49 3* 38092.bmp UB 4501.49 4511.20 4213.16 4071.86 4085.14 4071.86 LB 4067.32 4067.13 3814.35 4071.86 $\\dagger $ $\\dagger $ runtime(s) 1.24 2.30 0.09 0.46 0.91 0.99 3* 385039.bmp UB 3995.35 3995.35 3876.12 3745.53 3752.81 3745.53 LB 3741.30 3742.39 3565.10 3745.53 $\\dagger $ $\\dagger $ runtime(s) 0.42 3.10 0.06 2.51 1.20 0.81 3* 41033.bmp UB 2585.71 2585.71 2050.50 1994.24 2001.06 1997.55 LB 1988.64 1989.20 1841.58 1994.24 $\\dagger $ $\\dagger $ runtime(s) 0.54 4.62 0.01 0.94 0.28 0.40 3* 41069.bmp UB 6685.12 6685.12 5182.46 5110.96 5115.12 5122.63 LB 5091.16 5092.47 4896.23 5110.96 $\\dagger $ $\\dagger $ runtime(s) 24.57 59.03 0.07 35.77 0.36 0.54 3* 42012.bmp UB 3561.54 3561.54 3524.18 3248.70 3252.28 3251.04 LB 3238.97 3240.27 3005.31 3248.70 $\\dagger $ $\\dagger $ runtime(s) 3.96 27.65 0.08 8.95 0.59 0.73 3* 42049.bmp UB 970.16 970.16 1098.96 1069.22 1076.54 1069.22 LB 996.85 996.85 997.53 1069.22 $\\dagger $ $\\dagger $ runtime(s) 0.00 0.00 0.00 0.22 0.10 0.31 3* 43074.bmp UB 2864.46 2871.95 2374.97 2332.83 2340.46 2333.36 LB 2329.22 2329.38 2166.88 2332.83 $\\dagger $ $\\dagger $ runtime(s) 1.82 6.38 0.03 5.82 0.32 0.28 3* 45096.bmp UB 1088.06 1092.36 1024.96 977.78 1031.06 977.78 LB 975.97 975.97 911.19 977.78 $\\dagger $ $\\dagger $ runtime(s) 0.06 0.12 0.01 0.10 0.05 0.25 3* 54082.bmp UB 4075.17 4075.17 3914.51 3796.36 3806.52 3797.18 LB 3785.04 3785.94 3486.46 3796.36 $\\dagger $ $\\dagger $ runtime(s) 4.80 36.47 0.10 5.80 0.43 1.03 3* 55073.bmp UB 8445.42 8449.44 8179.32 7835.96 7844.37 7838.99 LB 7820.19 7822.05 7189.03 7835.96 $\\dagger $ $\\dagger $ runtime(s) 54.41 159.67 0.39 13.66 2.54 1.67 3* 58060.bmp UB 10196.99 10197.09 10133.69 9881.86 9891.88 9882.77 LB 9877.23 9878.45 9389.50 9881.86 $\\dagger $ $\\dagger $ runtime(s) 35.63 155.56 0.27 20.02 4.15 6.54 3* 62096.bmp UB 3915.47 3915.47 3485.31 3419.40 3421.38 3420.03 LB 3413.39 3412.83 3233.59 3419.40 $\\dagger $ $\\dagger $ runtime(s) 1.37 5.87 0.05 5.70 0.54 0.56 3* 65033.bmp UB 7766.31 7767.46 7647.20 7364.57 7372.11 7365.30 LB 7360.45 7360.57 6865.71 7364.57 $\\dagger $ $\\dagger $ runtime(s) 5.16 15.77 0.18 11.87 2.04 2.05 3* 66053.bmp UB 4871.59 4871.59 4522.25 4427.25 4434.10 4427.25 LB 4417.69 4418.10 4172.50 4427.25 $\\dagger $ $\\dagger $ runtime(s) 3.11 7.20 0.07 4.29 0.65 0.80 3* 69015.bmp UB 4378.98 4378.98 4183.41 4024.45 4032.54 4025.34 LB 4019.16 4019.52 3778.35 4024.45 $\\dagger $ $\\dagger $ runtime(s) 1.66 4.08 0.06 6.14 1.00 1.27 3* 69020.bmp UB 5828.19 5831.93 5527.82 5179.29 5183.07 5179.29 LB 5170.40 5170.63 4796.09 5179.29 $\\dagger $ $\\dagger $ runtime(s) 45.51 84.91 0.20 12.34 1.16 0.97 3* 69040.bmp UB 8240.10 8242.44 8255.62 7974.58 7994.72 7983.83 LB 7955.22 7958.22 7213.49 7974.58 $\\dagger $ $\\dagger $ runtime(s) 230.92 371.04 0.55 31.49 2.16 3.76 3* 76053.bmp UB 4625.92 4625.92 4823.04 4514.99 4527.26 4516.03 LB 4504.16 4505.47 4073.68 4514.99 $\\dagger $ $\\dagger $ runtime(s) 13.42 35.32 0.14 18.85 1.59 1.21 3* 78004.bmp UB 3394.42 3394.80 3380.61 3254.61 3271.58 3254.85 LB 3248.43 3248.75 3109.85 3254.61 $\\dagger $ $\\dagger $ runtime(s) 0.41 1.67 0.01 2.70 0.34 0.64 3* 8023.bmp UB 4385.74 4388.92 4108.08 4023.38 4032.92 4026.67 LB 4019.49 4019.76 3679.77 4023.38 $\\dagger $ $\\dagger $ runtime(s) 16.96 56.59 0.14 32.67 0.84 0.60 3* 85048.bmp UB 6056.07 6056.07 6186.51 5851.38 5863.69 5852.33 LB 5844.50 5845.08 5452.57 5851.38 $\\dagger $ $\\dagger $ runtime(s) 6.40 18.79 0.16 8.07 2.03 1.91 3* 86000.bmp UB 4628.18 4628.18 4769.21 4633.86 4643.13 4633.96 LB 4628.53 4628.53 4414.10 4633.86 $\\dagger $ $\\dagger $ runtime(s) 0.31 0.33 0.03 4.90 1.49 1.76 3* 86016.bmp UB 7930.26 7930.26 6654.84 6618.85 6619.75 6620.36 LB 6617.36 6617.46 6502.79 6618.85 $\\dagger $ $\\dagger $ runtime(s) 1.96 5.10 0.02 2.62 0.36 0.53 3* 86068.bmp UB 5870.65 5876.08 5289.20 5198.87 5207.65 5205.68 LB 5186.42 5185.40 4731.74 5198.87 $\\dagger $ $\\dagger $ runtime(s) 28.82 75.20 0.23 9.72 1.10 0.60 3* 87046.bmp UB 4641.36 4641.36 4470.22 4315.53 4321.55 4315.53 LB 4304.42 4305.38 3985.82 4315.53 $\\dagger $ $\\dagger $ runtime(s) 4.69 9.11 0.09 11.52 1.14 0.60 3* 89072.bmp UB 4096.15 4098.40 4159.28 3933.75 3948.57 3934.47 LB 3925.22 3925.52 3707.83 3933.75 $\\dagger $ $\\dagger $ runtime(s) 0.93 3.41 0.06 3.69 1.07 1.00 3* 97033.bmp UB 4594.50 4594.50 4583.48 4320.69 4336.56 4322.76 LB 4311.08 4312.65 3996.62 4320.69 $\\dagger $ $\\dagger $ runtime(s) 2.36 9.46 0.06 1.62 0.83 1.31 8 | c | modularity clustering 3* adjnoun UB -0.31 -0.31 -0.17 0.00 0.00 -0.29 LB -0.46 -0.46 -0.79 -0.45 $\\dagger $ $\\dagger $ runtime(s) 1.10 1.11 0.29 7288.07 0.01 92.48 3* dolphins UB -0.53 -0.53 -0.34 -0.53 0.00 -0.52 LB -0.55 -0.55 -0.83 -0.53 $\\dagger $ $\\dagger $ runtime(s) 0.24 0.22 0.03 44.61 0.00 0.79 3* football UB -0.60 -0.60 -0.34 -0.60 0.00 -0.49 LB -0.62 -0.62 -0.90 -0.60 $\\dagger $ $\\dagger $ runtime(s) 1.99 1.98 0.37 71.91 0.01 7.63 3* karate UB -0.42 -0.42 -0.28 -0.42 0.00 -0.32 LB -0.43 -0.43 -0.66 -0.42 $\\dagger $ $\\dagger $ runtime(s) 0.03 0.06 0.00 0.40 0.00 0.08 3* lesmis UB -0.56 -0.56 -0.37 -0.56 0.00 -0.50 LB -0.57 -0.57 -0.72 -0.56 $\\dagger $ $\\dagger $ runtime(s) 0.18 0.83 0.03 3.77 0.00 0.72 3* polbooks UB -0.52 -0.52 -0.33 -0.53 0.00 -0.51 LB -0.56 -0.56 -0.83 -0.54 $\\dagger $ $\\dagger $ runtime(s) 0.86 0.92 0.17 10057.85 0.01 4.97 8 | c | knott-3d-150 3* gm_knott_3d_032 UB -5811.47 -5811.47 -5365.34 -5811.47 -5745.79 -5767.34 LB -5812.64 -5811.49 -6052.81 -5811.47 $\\dagger $ $\\dagger $ runtime(s) 0.47 2.68 0.03 2.70 0.17 0.48 3* gm_knott_3d_033 UB -2545.84 -2545.84 -2536.26 -2545.84 -2517.65 -2545.84 LB -2545.90 -2545.90 -3029.52 -2545.84 $\\dagger $ $\\dagger $ runtime(s) 0.41 0.88 0.02 1.28 0.22 0.35 3* gm_knott_3d_034 UB -4064.87 -4064.87 -3921.65 -4064.87 -3971.60 -3972.66 LB -4066.65 -4066.31 -4337.06 -4064.87 $\\dagger $ $\\dagger $ runtime(s) 0.51 1.26 0.02 3.69 0.47 0.58 3* gm_knott_3d_035 UB -4595.84 -4595.84 -4238.86 -4595.84 -4568.88 -4595.84 LB -4595.84 -4595.84 -4916.47 -4595.84 $\\dagger $ $\\dagger $ runtime(s) 0.36 0.49 0.08 1.18 0.27 0.46 3* gm_knott_3d_036 UB -5192.26 -5192.26 -4678.23 -5198.37 -5159.18 -5198.37 LB -5199.32 -5198.80 -5440.26 -5198.37 $\\dagger $ $\\dagger $ runtime(s) 0.35 1.01 0.09 1.32 0.32 0.62 3* gm_knott_3d_037 UB -4636.50 -4633.67 -4312.24 -4638.99 -4616.39 -4638.03 LB -4642.64 -4639.28 -4928.85 -4638.99 $\\dagger $ $\\dagger $ runtime(s) 3.53 18.53 0.03 6.27 0.23 0.76 3* gm_knott_3d_038 UB -4625.80 -4625.80 -4235.69 -4625.80 -4616.99 -4619.01 LB -4625.80 -4625.80 -4818.46 -4625.80 $\\dagger $ $\\dagger $ runtime(s) 0.35 1.52 0.05 0.65 0.19 0.46 3* gm_knott_3d_039 UB -5092.32 -5092.32 -4476.96 -5092.32 -5081.58 -5082.97 LB -5092.45 -5092.32 -5318.04 -5092.32 $\\dagger $ $\\dagger $ runtime(s) 0.73 4.11 0.03 1.88 0.23 0.58 8 | c | knott-3d-300 3* gm_knott_3d_072 UB -32986.39 -32986.39 -29632.23 -32999.85 -32883.17 -32875.45 LB -33006.26 -33006.26 -34512.56 -32999.85 $\\dagger $ $\\dagger $ runtime(s) 1466.31 1574.69 4.60 49.78 4.26 3.40 3* gm_knott_3d_073 UB -25863.15 -25863.15 -23433.69 -25863.38 -25738.25 -25740.05 LB -25866.11 -25863.69 -27464.92 -25863.38 $\\dagger $ $\\dagger $ runtime(s) 58.50 81.89 3.24 55.72 2.72 8.88 3* gm_knott_3d_074 UB -25685.56 -25685.03 -23513.94 -25721.90 -25625.74 -25627.65 LB -25726.98 -25723.03 -27196.88 -25721.90 $\\dagger $ $\\dagger $ runtime(s) 172.42 229.99 1.19 39.63 2.18 10.73 3* gm_knott_3d_075 UB -30456.95 -30455.38 -27294.35 -30478.37 -30429.06 -30471.30 LB -30480.69 -30478.37 -31854.88 -30478.37 $\\dagger $ $\\dagger $ runtime(s) 26.07 46.55 2.89 14.29 2.18 3.57 3* gm_knott_3d_076 UB -27000.93 -27000.93 -24789.93 -27056.99 -27004.21 -27031.94 LB -27065.92 -27060.80 -28550.56 -27056.99 $\\dagger $ $\\dagger $ runtime(s) 1456.89 2236.72 2.62 50.33 3.32 10.79 3* gm_knott_3d_077 UB -29482.24 -29482.24 -27122.85 -29482.24 -29476.76 -29481.33 LB -29482.55 -29482.26 -31159.83 -29482.24 $\\dagger $ $\\dagger $ runtime(s) 170.01 155.43 2.40 47.23 4.12 10.59 3* gm_knott_3d_078 UB -20206.78 -20206.78 -19374.02 -20211.55 -20189.82 -20157.27 LB -20217.62 -20214.30 -22015.47 -20211.55 $\\dagger $ $\\dagger $ runtime(s) 350.20 1193.95 1.56 1451.04 2.44 10.97 3* gm_knott_3d_079 UB -26601.32 -26601.32 -23755.71 -26607.98 -26589.21 -26593.33 LB -26612.76 -26608.39 -28457.54 -26607.98 $\\dagger $ $\\dagger $ runtime(s) 104.62 463.29 3.37 110.62 2.46 6.24 8 | c | knott-3d-450 3* gm_knott_3d_096 UB -89941.58 -89941.58 -80280.68 -89959.41 -89779.76 -89786.75 LB -89996.73 -89996.38 -94633.08 -89959.41 $\\dagger $ $\\dagger $ runtime(s) 3643.00 3614.63 48.66 1438.60 16.89 153.09 3* gm_knott_3d_097 UB -73473.47 -73473.47 -67333.36 -73477.55 -73386.50 -73364.35 LB -73516.42 -73516.60 -78026.62 -73477.55 $\\dagger $ $\\dagger $ runtime(s) 3678.62 3674.28 26.83 1075.22 15.59 177.63 3* gm_knott_3d_098 UB -86499.41 -86499.41 -78396.23 -86593.97 -86470.44 -86498.10 LB -86633.78 -86632.24 -91051.86 -86593.97 $\\dagger $ $\\dagger $ runtime(s) 3648.34 3623.29 11.39 1507.50 15.15 69.82 3* gm_knott_3d_099 UB -86177.37 -86177.37 -78712.34 -85956.93 -86184.26 -86180.01 LB -86320.89 -86309.52 -91020.66 -86449.40 $\\dagger $ $\\dagger $ runtime(s) 3634.38 3639.38 34.96 3732.50 11.12 90.11 3* gm_knott_3d_100 UB -76590.45 -76590.45 -68324.61 -76699.37 -76561.72 -76523.90 LB -76761.06 -76758.81 -81763.95 -76699.37 $\\dagger $ $\\dagger $ runtime(s) 3628.67 3668.23 40.53 1076.28 22.65 141.63 3* gm_knott_3d_101 UB -74508.29 -74508.29 -66796.31 -74529.51 -74500.99 -74495.74 LB -74544.89 -74544.24 -79463.73 -74529.51 $\\dagger $ $\\dagger $ runtime(s) 3616.45 3620.81 33.62 1149.05 19.52 110.02 3* gm_knott_3d_102 UB -66423.87 -66423.87 -60651.60 -66482.68 -66454.83 -66455.86 LB -66525.90 -66524.32 -71160.72 -66482.68 $\\dagger $ $\\dagger $ runtime(s) 3680.72 3617.03 19.75 907.64 14.44 131.82 3* gm_knott_3d_103 UB -73799.17 -73799.17 -66427.02 -73431.17 -73750.61 -73743.79 LB -73918.13 -73909.00 -79062.14 -73988.21 $\\dagger $ $\\dagger $ runtime(s) 3662.18 3667.64 36.71 3836.96 16.82 79.70 8 | c | knott-3d-550 3* gm_knott_3d_112 UB -152854.11 -152854.11 -136448.39 -153021.45 -152908.44 -152675.73 LB -153124.66 -153124.66 -160981.94 -153024.89 $\\dagger $ $\\dagger $ runtime(s) 3686.25 3692.63 151.77 3716.74 88.55 462.34 3* gm_knott_3d_113 UB -135567.04 -135567.04 -122181.33 -134820.65 -135466.46 -135386.34 LB -135824.78 -135824.78 -144181.42 -135924.12 $\\dagger $ $\\dagger $ runtime(s) 3657.02 3650.21 105.96 3614.44 57.52 540.01 3* gm_knott_3d_114 UB -149545.82 -149545.82 -134889.26 -149716.68 -149683.60 -149526.15 LB -149831.75 -149831.31 -157228.92 -149722.18 $\\dagger $ $\\dagger $ runtime(s) 3651.93 3627.15 108.75 3614.56 99.81 384.54 3* gm_knott_3d_115 UB -149769.24 -149769.24 -135760.55 -148726.82 -149736.65 -149777.01 LB -150320.76 -150285.41 -158348.78 -150325.94 $\\dagger $ $\\dagger $ runtime(s) 3647.68 3746.48 111.32 3747.40 47.97 384.89 3* gm_knott_3d_116 UB -130577.35 -130577.35 -118822.35 -130757.57 -130720.10 -130580.01 LB -130910.50 -130894.84 -138934.59 -130761.25 $\\dagger $ $\\dagger $ runtime(s) 4152.93 3854.59 88.56 3688.88 92.77 976.88 3* gm_knott_3d_117 UB -123419.07 -123419.07 -112948.58 -122646.08 -123368.23 -123448.71 LB -123849.34 -123819.48 -131937.12 -123810.61 $\\dagger $ $\\dagger $ runtime(s) 3860.97 3778.57 90.49 3617.71 54.49 582.92 3* gm_knott_3d_118 UB -123467.39 -123467.39 -112812.30 -122526.33 -123520.61 -123483.58 LB -123720.59 -123709.54 -131313.86 -123538.09 $\\dagger $ $\\dagger $ runtime(s) 3720.24 3824.32 77.63 3777.86 83.73 869.68 3* gm_knott_3d_119 UB -126318.25 -126318.25 -116868.98 -123919.61 -126308.30 -126289.59 LB -126936.67 -126936.67 -134702.52 -126935.77 $\\dagger $ $\\dagger $ runtime(s) 3889.64 3792.14 84.92 3688.17 58.68 555.56 8 | c | CREMI-small 3* gm_small_1 UB -301663.25 -301663.25 -278423.74 -301674.02 0.00 -301674.02 LB -301678.18 -301677.98 -302379.69 -301673.92 $\\dagger $ $\\dagger $ runtime(s) 3638.89 3686.59 56.18 998.95 0.00 191.08 3* gm_small_2 UB -127448.90 -127448.90 -114182.56 -116678.57 -127414.03 -127292.54 LB -127545.97 -127545.60 -131075.52 -127520.30 $\\dagger $ $\\dagger $ runtime(s) 3623.23 3679.25 369.68 3722.33 3607.92 3752.86 3* gm_small_3 UB -210390.21 -210390.21 -191243.51 -210430.88 -210396.31 -210386.96 LB -210452.80 -210453.44 -212966.73 -210432.60 $\\dagger $ $\\dagger $ runtime(s) 3674.40 3619.29 531.16 3606.16 3479.31 3722.51 8 | c | CREMI-large 3* gm_large_1 UB -5647807.76 -5647807.76 0.00 0.00 -5628646.55 -5524856.64 LB -5648791.54 -5648791.54 $\\dagger $ $\\dagger $ 0.00 0.00 runtime(s) 3672.44 3629.77 0.00 0.00 3863.35 10197.66 3* gm_large_2 UB -2368830.08 -2368830.08 0.00 0.00 -1916548.20 -1713523.77 LB -2382103.57 -2382103.57 $\\dagger $ $\\dagger $ 0.00 0.00 runtime(s) 3613.51 4076.68 0.00 0.00 8092.81 36081.14 3* gm_large_3 UB -3643885.09 -3643885.09 0.00 0.00 0.00 0.00 LB -3648930.86 -3648930.86 $\\dagger $ $\\dagger $ 0.00 0.00 runtime(s) 3717.06 3712.53 0.00 0.00 0.00 0.00"
]
] | 1612.05441 |
[
[
"SonoNet: Real-Time Detection and Localisation of Fetal Standard Scan\n Planes in Freehand Ultrasound"
],
[
"Abstract Identifying and interpreting fetal standard scan planes during 2D ultrasound mid-pregnancy examinations are highly complex tasks which require years of training.",
"Apart from guiding the probe to the correct location, it can be equally difficult for a non-expert to identify relevant structures within the image.",
"Automatic image processing can provide tools to help experienced as well as inexperienced operators with these tasks.",
"In this paper, we propose a novel method based on convolutional neural networks which can automatically detect 13 fetal standard views in freehand 2D ultrasound data as well as provide a localisation of the fetal structures via a bounding box.",
"An important contribution is that the network learns to localise the target anatomy using weak supervision based on image-level labels only.",
"The network architecture is designed to operate in real-time while providing optimal output for the localisation task.",
"We present results for real-time annotation, retrospective frame retrieval from saved videos, and localisation on a very large and challenging dataset consisting of images and video recordings of full clinical anomaly screenings.",
"We found that the proposed method achieved an average F1-score of 0.798 in a realistic classification experiment modelling real-time detection, and obtained a 90.09% accuracy for retrospective frame retrieval.",
"Moreover, an accuracy of 77.8% was achieved on the localisation task."
],
[
"Introduction",
"Abnormal fetal development is a leading cause of perinatal mortality in both industrialised and developing countries [27].",
"Overall early detection rates of fetal abnormalities are still low and are hallmarked by large variations between geographical regions [1], [32], [14].",
"The primary modality for assessing the fetus' health is 2D ultrasound due to its low cost, wide availability, real-time capabilities and the absence of harmful radiation.",
"However, the diagnostic accuracy is limited due to poor signal to noise ratio and image artefacts such as shadowing.",
"Furthermore, it can be difficult to obtain a clear image of a desired view if the fetal pose is unfavourable.",
"Currently, most countries offer at least one routine ultrasound scan at around mid-pregnancy between 18 and 22 weeks of gestation [27].",
"Those scans typically involve imaging a number of standard scan planes on which biometric measurements are taken (e.g.",
"head circumference on the trans-ventricular head view) and possible abnormalities are identified (e.g.",
"lesions in the posterior skin edge on the standard sagittal spine view).",
"In the UK, guidelines for selecting and examining these planes are defined in the fetal abnormality screening programme (FASP) handbook [21].",
"Guiding the transducer to the correct scan plane through the highly variable anatomy and assessing the often hard-to-interpret ultrasound data are highly sophisticated tasks, requiring years of training [19].",
"As a result these tasks have been shown to suffer from low reproducibility and large operator bias [6].",
"Even identifying the relevant structures in a given standard plane image can be a very challenging task for certain views, especially for inexperienced operators or non-experts.",
"At the same time there is also a significant shortage of skilled sonographers, with vacancy rates reported to be as high as 18.1% in the UK [30].",
"This problem is particularly pronounced in parts of the developing world, where the WHO estimates that many ultrasound scans are carried out by individuals with little or no formal training [27].",
"Figure: Overview of proposed SonoNet: (a) 2D fetal ultrasound data can be processed in real-time through our proposed convolutional neural network to determine if the current frame contains one of 13 fetal standard views (here the 4 chamber view (4CH) is shown); (b) if a standard view was detected, its location can be determined through a backward pass through the network.With this in mind, we propose a novel system based on convolutional neural networks (CNNs) for real-time automated detection of 13 fetal standard scan planes, as well as localisation of the fetal structures associated with each scan plane in the images via bounding boxes.",
"We model all standard views which need to be saved according to the UK FASP guidelines for mid-pregnancy ultrasound examinations, plus the most commonly acquired cardiac views.",
"The localisation is achieved in a weakly supervised fashion, i.e.",
"with only image-level scan plane labels available during training.",
"This is an important aspect of the proposed work as bounding box annotations are not routinely recorded and would be too time-consuming to create for large datasets.",
"Fig.",
"REF contains an overview of the proposed method.",
"Our approach achieves real-time performance and very high accuracy in the detection task and is the first in the literature to tackle the weakly-supervised localisation task on freehand ultrasound data.",
"All evaluations are performed on video data of full mid-pregnancy examinations.",
"The proposed system can be used in a number of ways.",
"It can be employed to provide real-time feedback about the content of a image frame to the operator.",
"This may reduce the number of mistakes made by inexperienced sonographers and could also be applied to automated quality control of acquired images.",
"We also demonstrate how this system can be used to retrospectively retrieve standard views from very long videos, which may open up applications for automated analysis of data acquired by operators with minimal training and make ultrasound more accessible to non-experts.",
"The localisation of target structures in the images has the potential to aid non-experts in the detection and diagnosis tasks.",
"This may be particularly useful for training purposes or for applications in the developing world.",
"Moreover, the saliency maps and bounding box predictions improve the interpretability of the method by visualising the hidden reasoning of the network.",
"That way we hope to build trust into the method and also provide an intuitive way to understand failure cases.",
"Lastly, automated detection and, specifically, localisation of fetal standard views are essential preprocessing steps for other automated image processing such as measurement or segmentation of fetal structures.",
"This work was presented in preliminary form in [2].",
"Here, we introduce a novel method for computing category-specific saliency maps, provide a more in-depth description of the proposed methods, and perform a significantly more thorough quantitative and qualitative evaluation of the detection and localisation on a larger dataset.",
"Furthermore, we significantly outperform our results in [2] by employing a very deep network architecture."
],
[
"Related work",
"A number of papers have proposed methods to detect fetal anatomy in videos of fetal 2D ultrasound sweeps (e.g.",
"[19], [20]).",
"In those works the authors have been aiming at detecting the presence of fetal structures such as the skull, heart or abdomen rather than specific standardised scan planes.",
"Yaqub et al.",
"[34] have proposed a method for the categorisation of fetal mid-pregnancy 2D ultrasound images into seven standard scan planes using guided random forests.",
"The authors modelled an “other” class consisting of non-modelled standard views.",
"Scan plane categorisation differs significantly from scan plane detection since in the former setting it is already known that every image is a standard plane.",
"In standard plane detection on a real-time data stream or video data, standard views must be distinguished from a very large amount of background frames.",
"This is a very challenging task due to the vast amount of possible appearances of the background class.",
"Automated fetal standard scan plane detection has been demonstrated for 1–3 standard planes in short videos of 2D fetal ultrasound sweeps [7], [9], [23], [22].",
"The earlier of those works rely on extracting Haar-like features from the data and training a classifier such as AdaBoost or random forests on them [22], [23], [35].",
"Motivated by advances in computer vision, there has recently been a shift to analyse ultrasound data using CNNs.",
"The most closely related work to ours is that by Chen et al.",
"[9] who employed a classical CNN architecture with five convolutional and two fully-connected layers for the detection of the standard abdominal view.",
"During test time, each frame of the input video was processed by evaluating the classifier multiple times for overlapping image patches.",
"The drawback of this approach is that the classifier needs to be applied numerous times, which precludes the system from running in real-time.",
"In [7], the same authors extended the above work to three scan planes and a recurrent architecture which took into account temporal information, but did not aim at real-time performance.",
"An important distinction between the present study and all of the above works is that the latter used data acquired in single sweeps while we use freehand data.",
"Sweep data are acquired in a fixed protocol by moving the ultrasound probe from the cervix upwards in one continuous motion [9].",
"However, not all standard views required to determine the fetus' health status are adequately captured using a sweep protocol.",
"For example, imaging the femur or the lips normally requires careful manual scan plane selection.",
"Furthermore, data obtained using the sweep protocol are typically only 2–5 seconds long and consist of fewer than 50 frames [9].",
"In this work, we consider data acquired during real clinical abnormality screening examinations in a freehand fashion.",
"Freehand scans are acquired without any constraints on the probe motion and the operator moves from view to view in no particular order.",
"As a result such scans can last up to 30 minutes and the data typically consists of over 20,000 individual frames for each case.",
"To our knowledge, automated fetal standard scan plane detection has never been performed in this challenging scenario.",
"A number of works have been proposed for the supervised localisation of structures in ultrasound.",
"Zhang et al.",
"[35] developed a system for automated detection and fully supervised localisation of the gestational sac in first trimester sweep ultrasound scans.",
"Bridge et al.",
"[5] proposed a method for the localisation of the heart in short videos using rotation invariant features and support vector machines for classification.",
"In more recent work, the same authors have extended the method for the supervised localisation of three cardiac views taking into account the temporal structure of the data [4].",
"The method was also able to predict the heart orientation and cardiac phase.",
"To our knowledge, the present work is the first to perform localisation in fetal ultrasound in a weakly supervised fashion.",
"Although, weakly supervised localisation (WSL) is an active area of research in computer vision (e.g.",
"[26]) we are not aware of any works which attempt to perform WSL in real-time."
],
[
"Data",
"Our dataset consisted of 2694 2D ultrasound examinations of volunteers with gestational ages between 18–22 weeks which have been acquired and labelled during routine screenings by a team of 45 expert sonographers according to the guidelines set out in the UK FASP handbook [21].",
"Those guidelines only define the planes which need to be visualised, but not the sequence in which they should be acquired.",
"The large number of sonographers involved means that the dataset contains a large number of different operator-dependent examination “styles” and is therefore a good approximation of the normal variability observed between different sonographers.",
"In order to reflect the distribution of real data, no selection of the cases was made based on normality or abnormality.",
"Eight different ultrasound systems of identical make and model (GE Voluson E8) were used for the acquisitions.",
"For each scan we had access to freeze-frame images saved by the sonographers during the exam.",
"For a majority of cases we also had access to screen capture videos of the entire fetal exam.",
"A large fraction of the freeze-frame images corresponded to standard planes and have been manually annotated during the scan allowing us to infer the correct ground-truth (GT) label.",
"Based on those labels we split the image data into 13 standard views.",
"In particular, those included all views required to be saved by the FASP guidelines, the four most commonly acquired cardiac views, and the facial profile view.",
"An overview of the modelled categories is given in Table REF and examples of each view are shown in Fig.",
"REF .",
"Table: Overview of the modelled categories.Additionally, we modelled an “other” class using a number of views which do not need to be saved according to the FASP guidelines but are nevertheless often recorded at our partner hospital.",
"Specifically, the “other” class was made up from the arms, hands and feet views, the bladder view, the diaphragm view, the coronal face view, the axial orbits view, and views of the cord-insert, cervix and placenta.",
"Overall, our dataset contained 27731 images of standard views and 6856 of “other” views.",
"The number of examples for each class ranged from 543 for the profile view to 4868 for the brain (tv.)",
"view.",
"Note that a number of the cases were missing some of the standard planes while others had multiple instances of the same view acquired at different times."
],
[
"Video data",
"In addition to the still images, our dataset contained 2638 video recordings of entire fetal exams, which were on average over 13 minutes long and contained over 20000 frames.",
"2438 of those videos corresponded to cases for which image data was also available.",
"Even though in some examinations not all standard views were manually annotated, we found that normally all standard views did appear in the video.",
"It was possible to find each freeze-frame image in its corresponding video if the latter existed.",
"As will be described in more detail in Sec.",
"REF we used this fact to augment our training dataset in order to bridge the small domain gap between image and video data.",
"Specifically, the corresponding frames could be found by iterating through the video frames and calculating the image distance of each frame to the freeze-frame image.",
"The matching frame was the one with the minimum distance to the freeze-frame.",
"As is discussed in detail in Sec.",
", all evaluations were performed on the video data in order to test the method in a realistic scenario containing motion and a large number of irrelevant background frames.",
"The image and video data were preprocessed in five steps which are summarised in Table REF and will be discussed in detail in the following.",
"Since, in this study, we were only interested in structural images we removed all freeze-frame images and video frames containing colour Doppler overlays from the data.",
"We also removed video frames and images which contained split views showing multiple locations in the fetus simultaneously.",
"To prevent our algorithm from learning the manual annotations placed on the images by the sonographers rather than from the images themselves, we removed all the annotations using the inpainting algorithm proposed in [33].",
"We rescaled all image and frame data and cropped a 224x288 region containing most of the field of view but excluding the vendor logo and ultrasound control indicators.",
"We also normalised each image by subtracting the mean intensity value and dividing by the image pixel standard deviation.",
"In order to tackle the challenging scan plane detection scenario in which most of the frames do not show any of the standard scan planes, a large set of background images needed to be created.",
"The data from the “other” classes mentioned above were not enough to model this highly varied category.",
"Note that our video data contained very few frames showing standard views and the majority of frames were background.",
"Thus, it was possible to create the background class by randomly sampling frames from the available video recordings.",
"Specifically, we sampled 50 frames from all training videos and 200 frames from all testing videos.",
"While we found that 50 frames per case sufficed to capture the full variability of the background class during training, we opted for a larger number of background frames for the test set in order to evaluate the method in a more challenging and realistic scenario.",
"This resulted in a very large background class with 110638 training images and 105611 testing images.",
"Note that operators usually hold the probe relatively still around standard planes, while the motion is larger when they are searching for views.",
"Thus, in order to decrease the chance of randomly sampling actual standard planes, frames were only sampled where the probe motion, i.e.",
"image distance to previous video frame, was above a small threshold.",
"Note that the location in the video of some of the standard scan planes could be determined by comparing image distances to the freeze frames as described earlier (see Sec.",
"REF ).",
"However, this knowledge could not be used to exclude all standard views for the background class sampling because it only accounted for a very small fraction of standard views in the video.",
"The videos typically contained a large number of unannotated standard views in the frames before and after the freeze frame, and also in entirely different positions in the video.",
"The images from the “other” category were also added to the background class.",
"Overall the dataset including the background class had a substantial (and intentional) class imbalance between standard views and background views.",
"For the test set the standard view to background ratios were between 1:138 and 1:1148, depending on the category.",
"In the last step, we split all of the cases into a training set containing 80% of the cases and test set containing the remaining 20%.",
"The split was made on the case level rather than the image level to guarantee that no video frames originating from test videos were used for training.",
"Note that not all cases contained all of the standard views and as a result the ratios between test and training images were not exactly 20% for each class."
],
[
"Network architecture",
"Our proposed network architecture, the sonography network or SonoNet, is inspired by the VGG16 model which consists of 13 convolutional layers and 3 fully-connected layers [29].",
"However, we introduce a number of key changes to optimise it for the real-time detection and localisation tasks.",
"The network architectures explored in this work are summarised in Fig.",
"REF .",
"Generally, the use of fully-connected layers restricts the model to fixed image sizes which must be decided during training.",
"In order to obtain predictions for larger, rectangular input images during test time, typically the network is evaluated multiple times for overlapping patches of the training image size.",
"This approach was used, for example, in some related fetal scan plane detection works [7], [9].",
"Fully convolutional networks, in which the fully-connected layers have been replaced by convolutions, can be used to calculate the output to arbitrary images sizes much more efficiently in a single forward pass.",
"The output of such a network is no longer a single value for each class, but rather a class score map with a size dependent on the input image size [18].",
"In order to obtain a fixed-size vector of class scores, the class score maps can then be spatially aggregated using the sum, mean or max function to obtain a single prediction per class.",
"Fully convolution networks have been explored in a number of works in computer vision (e.g.",
"[24], [16]), and in medical image analysis, for example for mitosis detection [8].",
"Simonyan et al.",
"[29] proposed training a traditional model with fully-connected layers, but then converting it into a fully convolutional architecture for efficient testing.",
"This was achieved by converting the first fully-connected layer to a convolution over the full size of the last class score map (i.e.",
"a 7x7 convolution for the VGG16 network), and the subsequent ones to 1x1 convolutions.",
"In the case of 224x288 test images this would produce 1x14 class score maps for each category.",
"In this work we use the spatial correspondence between class score maps with the input image to obtain localised category-specific saliency maps (see Sec.",
"REF ).",
"Consequently, it is desirable to design the network such that it produces class score maps with a higher spatial resolution.",
"To this end, we forgo the final max-pooling step in the VGG16 architecture and replace all the fully-connected layers with two 1x1 convolution layers.",
"Following the terminology introduced by Oquab et al.",
"[24], we will refer to those 1x1 convolutions as adaptation layers.",
"The output of those layers are $K$ class score maps $F_k$ , where $K$ is the number of modelled classes (here $K=14$ , i.e.",
"13 standard views plus background).",
"We then aggregate them using the mean function to obtain a prediction vector which is fed into the final softmax.",
"In this architecture the class score maps $F_k$ have a size of 14x18 for an 224x288 input image.",
"Note that each neuron in $F_k$ corresponds to a receptive field in the original image creating the desired spatial correspondence with the input image.",
"During training, each of the neurons learns to respond to category-specific features in its receptive field.",
"Note that the resolution of the class score maps is not sufficient for accurate localisation.",
"In Sec.",
"REF we will show how $F_k$ can be upsampled to the original image resolution using a backpropagation step to create category-specific saliency maps.",
"The design of the last two layers of the SonoNet is similar to work by Oquab et al. [24].",
"However, in contrast to that work, we aggregate the final class score maps using the mean function rather than the max function.",
"Using the mean function incorporates the entire image context for the classification while using the max function only considers the receptive field of the maximally activated neuron.",
"While max pooling aggregation may be beneficial for the localisation task [25], [24], we found the classification accuracy to be substantially lower using that strategy.",
"Since we are interested in operating the network in real-time, we explore the effects of reducing the complexity of the network on inference times and detection accuracy.",
"In particular, we investigate three versions of the SonoNet.",
"The SonoNet-64 uses the same architecture for the first 13 layers as the VGG16 model, with 64 kernels in the first convolutional layer.",
"We also evaluate the SonoNet-32 and the SonoNet-16 architectures, where the number of all kernels in the network is halved and quartered, respectively.",
"In contrast to the VGG16 architecture, we include batch normalisation in every convolutional layer [15].",
"This allows for much faster training because larger learning rates can be used.",
"Moreover, we found that for all examined networks using batch normalisation produced substantially better results.",
"In addition to the three versions of the SonoNet, we also compare to a simpler network architecture which is loosely inspired by the AlexNet [17], but has much fewer parameters.",
"This is also the network which we used for our initial results presented in [2].",
"Due to the relatively low complexity of this network compared to the SonoNet, we refer to it as SmallNet."
],
[
"Training",
"We trained all networks using mini-batch gradient descent with a Nesterov momentum of 0.9, a categorical cross-entropy loss and with an initial learning rate of $0.1$ .",
"We subsequently divided the learning rate by 10 every time the validation error stopped decreasing.",
"In some cases we found that a learning rate of $0.1$ was initially too aggressive to converge immediately.",
"Therefore, we used a warm-up learning rate of $0.01$ for 500 iterations [13].",
"Since the SmallNet network did not have any batch normalisation layers it had to be trained with a lower initial learning rate of $0.001$ .",
"Note that there is a small domain gap between the annotated image data and the video data we use for our real-time detection and retrospective retrieval evaluations.",
"Specifically, the video frames are slightly lower resolution and have been compressed.",
"In order to overcome this, we automatically identified all frames from the training videos which corresponded to the freeze-frame images in our training data.",
"However, as mentioned in Sec.",
"REF not all cases had a corresponding video, such that the frame dataset consisted of fewer instances than the image dataset.",
"To make the most of our data while ensuring that the domain gap is bridged, we combined all of the images and the corresponding video frames for training.",
"We used 20% of this combined training dataset for validation.",
"In order to reduce overfitting and make the network more robust to varying object sizes we used scale augmentation [29].",
"That is, we extracted square patches of the input images for training by randomly sampling the size of the patch (between 174x174 and 224x224) and then scaling it up to 224x224 pixels.",
"To further augment the dataset, we randomly flipped the patches in the left-right direction, and rotated them with a random angle between $-25^\\circ $ and $25^\\circ $ .",
"The training procedure needed to account for the significant class imbalance introduced by the randomly sampled background frames.",
"Class imbalance can be addressed either by introducing an asymmetric cost-function, by post-processing the classifier output, or by sampling techniques [36], [12].",
"We opted for the latter approach which can be neatly integrated with mini-batch gradient descent.",
"We found that the strategy which produced the best results was randomly sampling mini-batches that were made up of the same number of standard planes and background images.",
"Specifically, we used 2 images of each of the 13 standard planes and 26 background images per batch.",
"The optimisation typically converged after around 2 days of training on a Nvidia GeForce GTX 1080 GPU."
],
[
"Frame annotation and retrospective retrieval",
"After training we fed the network with cropped video frames with a size of 224x288.",
"This resulted in $K$ class score maps $F_k$ with a size of 14x18.",
"Those where averaged in the mean pooling layer to obtain a single class score $a_k$ for each category $k$ .",
"The softmax layer then produced the class confidence $c_k$ of each frame.",
"The final prediction was given by the output with the highest confidence.",
"For retrospective frame retrieval we calculated and recorded the confidence $c_k$ for each class over the entire duration of an input video.",
"Subsequently, we retrieved the frame with the highest confidence for each class."
],
[
"Weakly supervised localisation",
"After determining the 14x18 class score maps $F_k$ and the image category in a forward pass through the network, the fetal anatomical structures corresponding to that category can then be localised in the image.",
"A coarse localisation could already be achieved by directly relating each of the neurons in $F_k$ to its receptive field in the original image.",
"However, it is also possible to obtain pixel-wise maps containing information about the location of class-specific target structures at the resolution of the original input images.",
"This can be achieved by calculating how much each pixel influences the activation of the neurons in $F_k$ .",
"Such maps can be used to obtained much more accurate localisation.",
"Examples of $F_k$ and corresponding saliency maps are shown in Fig.",
"REF .",
"In the following we will show how category-specific saliency and confidence maps can be obtained through an additional backward pass through the network.",
"Secondly, we show how to post-process the saliency maps to obtain confidence maps from which we then extract a bounding box around the detected structure."
],
[
"Category-specific saliency maps",
"Generally, category-specific saliency maps $S_k$ can be obtained by computing how much each pixel in the input image $X$ influences the current prediction.",
"This is equivalent to calculating the gradient of the last activation before the softmax $a_k$ with respect to the pixels of the input image $X$ .",
"$S_k = \\frac{\\partial a_k}{\\partial X}$ The gradient can be obtained efficiently using a backward pass through the network [28].",
"Springenberg et al.",
"[31] proposed a method for performing this back-propagation in a guided manner by allowing only error signals which contribute to an increase of the activations in the higher layers (i.e.",
"layers closer to the network output) to back-propagate.",
"In particular, the error is only back-propagated through each neuron's ReLU unit if the input to the neuron $x$ , as well as the error in the next higher layer $\\delta _n$ are positive.",
"That is, the back-propagated error $\\delta _{n-1}$ of each neuron is given by $\\delta _{n-1}=\\delta _n\\sigma (x)\\sigma (\\delta _n),$ where $\\sigma (\\cdot )$ is the unit step function.",
"Examples of saliency maps obtained using this method are shown in Fig.",
"REF b.",
"It can be observed that those saliency maps, while highlighting the fetal anatomy, also tend to highlight background features, which adversely affects automated localisation.",
"Figure: Examples of saliency maps.",
"Column (a) shows three different input frames, (b) shows the corresponding class score maps F k F_k obtained in the forward pass of the network, (c) shows saliency maps obtained using the method by Springenberg et al.",
"and (d) shows the saliency maps resulting from our proposed method.",
"Some of the unwanted saliency artefacts are highlighted with arrows in (c).In this work, we propose a method to generate significantly less noisy, localised saliency maps by taking advantage of the spatial encoding in the class score maps $F_k$ .",
"As can be seen in Fig.",
"REF a, the class score maps can be interpreted as a coarse confidence map of the object's location in the input frame.",
"In particular, each neuron $h_k^{n}(X)$ in $F_k$ has a receptive field in the original image $X$ .",
"In our preliminary work [2], we backpropagated the error only from a fixed percentile $P$ of the most highly activated neurons in $F_k$ to achieve a localisation effect.",
"However, this required heuristic selection of $P$ .",
"In this paper, we propose a more principled approach.",
"Note that very high or very low values in the saliency map mean that a change in that pixel will have a large effect on the classification score.",
"However, those values do not necessarily correspond to high activations in the class score map.",
"For example, an infinitesimal change in the input image may not have a very large impact if the corresponding output neuron is already very highly activated.",
"Conversely, another infinitesimal change in the input image may have a big impact on a neuron with low activation, for example by making the image look less like a competing category.",
"To counteract this, we preselect the areas of the images which are likely to contain the object based on the class score maps and give them more influence in the saliency map computation.",
"More specifically, we use the activations $h_k^{n}(X)$ in $F_k$ to calculate the saliency maps as a weighted linear combination of the influence of each of the receptive fields of the neurons in $F_k$ .",
"In this manner, regions corresponding to highly activated neurons will have more importance than neurons with low activations in the resulting saliency map.",
"In the following, we drop the subscripts for the category $k$ for conciseness.",
"We calculate the saliency map $S$ as $S = \\sum _n h_{>0}^n(X) \\frac{\\partial h^n(X)}{\\partial X},$ where $h_{>0}^n$ are the class score map activations thresholded at zero, i.e.",
"$h_{>0}^n = h^n\\sigma (h^n)$ .",
"By thresholding at zero we essentially prevent negative activations from contributing to the saliency maps.",
"Note that it is not necessary to back-propagate for each neuron $h^n$ separately.",
"In fact, the saliency can still be calculated in a single back-propagation step, which can be seen by rewriting Eq.",
"REF as $S = \\sum _n \\frac{1}{2}\\frac{\\partial (h_{>0}^n(X))^2}{\\partial X} = \\frac{1}{2}\\frac{\\partial \\mathbf {e}^T F_{>0} \\circ F_{>0} \\mathbf {e}}{\\partial X},$ where $F_{>0}$ is the class score map thresholded at zero, $\\circ $ is the element-wise matrix multiplication and $\\mathbf {e}$ is a vector with all ones.",
"The first equality stems from the chain-rule and the observation that $h_{>0}^n h^n=h_{>0}^n h_{>0}^n$ , and the second equality stems from rewriting the sum in matrix form.",
"Examples of saliency maps obtained using this method are shown in Fig.",
"REF c. It can be seen that the resulting saliency maps are significantly less noisy and the fetal structures are easier to localise compared to the images obtained using the approach presented in [31]."
],
[
"Bounding box extraction",
"Next, we post-process saliency maps obtained using Eq.",
"REF to obtain confidence maps from which we then calculate bounding boxes.",
"In a first step, we take the absolute value of the saliency map $S$ and blur it using a $5x5$ Gaussian kernel.",
"This produces confidence maps of the location of the structure in the image such as the ones shown in Fig.",
"REF b.",
"Note that even though both structures are challenging to detect on those views, the confidence maps localise them very well, despite artefacts (shadows in row 1) and similar looking structures (arm in row 2).",
"Due to the way the gradient is calculated structures that appear dark in the images (such as cardiac vessels) will usually have negative saliencies and structures that appear bright (bones) will usually have positive saliencies in $S_k$ .",
"We exploit this fact to introduce some domain knowledge into the localisation procedure.",
"In particular, we only consider positive saliencies for the femur, spine and lips, and we only consider negative saliencies for all cardiac views.",
"We use both positive and negative for the remainder of the classes.",
"Next, we threshold the confidence maps using the Isodata thresholding method proposed in [10].",
"In the last step, we take the largest connected component of the resulting mask and fit the minimum rectangular bounding box around it.",
"Two examples are shown in Fig.",
"REF c."
],
[
"Real-time scan plane detection",
"In order to quantitatively assess the detection performance of the different architectures we evaluated the proposed networks on the video frame data corresponding to the freeze-frames from the test cohort including the large amount of randomly sampled background frames.",
"We measured the algorithm's performance using the precision (TP / (TP + FP)) and recall (TP / (TP + FN)) rates as well as the F1-score, which is defined as the harmonic mean of the precision and recall.",
"In Table REF we report the average scores for all examined networks.",
"Importantly, the average was not weighted by the number of samples in each category.",
"Otherwise, the average scores would be dominated by the massive background class.",
"In Table REF we furthermore report the frame rates achieved on a Nvidia Geforce GTX 1080 GPUThe system was furthermore comprised of an Intel Xeon CPU E5-1630 v3 at 3.70GHz and 2133 MHz DDR4 RAM.",
"for the detection task alone, the localisation task alone and both of them combined.",
"There is no consensus in literature over the minimum frame rate required to qualify as real-time, however, a commonly used figure is 25 frames per second (fps), which coincides with the frame rate our videos were recorded at.",
"Table: Classification scores for the four examined network architectures.Table: Frame rates in fps for the detection (forward pass), localisation (backward pass) and the two combined.From Tables REF and REF it can be seen that SonoNet-64 and SonoNet-32 performed very similarly on the detection task with SonoNet-64 obtaining slightly better F1-scores, but failing to perform the localisation task at more than 25 fps.",
"The SonoNet-32 obtained classification scores very close to the SonoNet-64 but at a substantially lower computational cost, achieving real-time in both the detection and localisation tasks.",
"Further reducing the complexity of the network led to more significant deteriorations in detection accuracy as can be seen from the SonoNet-16 and the SmallNet network.",
"Thus, we conclude that the SonoNet-32 performs the best out of the examined architectures which achieve real-time performance and we use that architecture for all further experiments and results.",
"In Table REF we show the detailed classification scores for the SonoNet-32 for all the modelled categories.",
"The right-most column lists the number of test images in each of the classes.",
"Additionally, the class confusion matrix obtained with SonoNet-32 is shown in Fig.",
"REF .",
"The results reported for this classification experiment give an indication of how the method performs in a realistic scenario.",
"The overall ratio of standard planes to background frames is approximately 1:24 meaning that in a video on average 1 second of any of the standard views is followed by 24 seconds of background views.",
"This is a realistic reflection of what we observe in clinical practice.",
"Table: Detailed classification scores for SonoNet-32Figure: Class confusion matrix for SonoNet-32.Some of the most important views for taking measurements and assessing the fetus' health (in particular the brain views, the abdominal view and the femur view) were detected with F1-scores of equal to or above 0.9, which are very high scores considering the difference in number of images for the background and foreground classes.",
"The lowest detection accuracies were obtained for the profile view, the right-ventricular outflow tract (RVOT) and the three vessel view (3VV).",
"The two cardiac views – which are only separated from each other by a slight change in the probe angle and are very similar in appearance – were often confused with each other by the proposed network.",
"This can also be seen in the confusion matrix in Fig.",
"REF .",
"We also noted that for some views the method produced very high recall rates with relatively low precision.",
"The Spine (sag.)",
"view and the profile view were particularly affected by this.",
"We found that for a very large fraction of those false positive images, the prediction was in fact correct, but the images had an erroneous background ground-truth label.",
"This can be explained by the fact that the spine and profile views appear very frequently in the videos without being labelled and thus many such views were inadvertently sampled in the background class generation process.",
"Examples of cases with correct predictions but erroneous ground-truth labels for the profile and spine (sag.)",
"classes are shown in the first three columns of Fig.",
"REF .",
"We observed the same effect for classes which obtained higher precision scores as well.",
"For instance, we verified that the majority of background frames classified as Brain (Cb.)",
"are actually true detections.",
"Examples are also shown in Fig.",
"REF .",
"All of the images shown in the first three columns of Fig.",
"REF are similar in quality to our ground-truth data and could be used for diagnosis.",
"Unfortunately, it is infeasible to manually verify all background images.",
"We therefore conclude that the precision scores (and consequently F1-scores) reported in Tables REF and REF can be considered a lower bound of the true performance.",
"Figure: Examples of video frames labelled as background but classified as one of three standard views.",
"The first three columns were randomly sampled from the set of false positives and are in fact correct detections.",
"The last column shows manually selected true failure cases.For a qualitative evaluation, we also annotated a number of videos from our test cohort using the SonoNet-32.",
"Two example videos demonstrating the SonoNet-32 in a real clinical exam are available at https://www.youtube.com/watch?v=4V8V0jF0zFc and https://www.youtube.com/watch?v=yPCvAdOYncQ."
],
[
"Retrospective scan plane retrieval",
"We also evaluated the SonoNet-32 for retrospective retrieval of standard views on 110 random videos from the test cohort.",
"The average duration of the recordings was 13 min 33 sec containing on average 20321 frames.",
"The retrieved frames were manually validated by two clinical experts in obstetrics with 11 years and 3 years of experience, respectively.",
"The time-consuming manual validation required for this experiment precluded using a larger number of videos.",
"Table REF summarises the retrieval accuracy (TP / (P + N)) for 13 standard planes.",
"We achieved an average retrieval accuracy of 90.09%.",
"As above, the most challenging views proved to be the cardiac views for which the retrieval accuracy was 82.12%.",
"The average accuracy for all non-cardiac views was 95.52%.",
"In contrast to the above experiment, the results in this section were obtained directly from full videos, and thus reflect the true performance of the method in a real scenario.",
"Table: Retrieval accuracy for SonoNet-32Figure: Results of retrospective retrieval for two example subjects.",
"The respective top rows show the ground truth (GT) saved by the sonographer.",
"The bottom rows show the retrieved (RET) frames.",
"For subject (a) all frames have been correctly retrieved.",
"For subject (b) the frames marked with red have been incorrectly retrieved.The retrieved frames for two cases from the test cohort are shown in Fig.",
"REF along with the ground truth (GT) frames saved by the sonographers.",
"In the case shown in Fig.",
"REF a, all views have been correctly retrieved.",
"It can be seen that most of the retrieved frames either matched the GT exactly or were of equivalent quality.",
"We observed this behaviour through-out the test cohort.",
"However, a number of wrong retrievals occasionally occurred.",
"In agreement with the quantitative results in Tab.",
"REF , we noted that cardiac views were affected the most.",
"Fig.",
"REF b shows a case for which two cardiac views have been incorrectly retrieved (marked in red)."
],
[
"Weakly supervised localisation",
"We quantitatively evaluated the weakly supervised localisation using SonoNet-32 on 50 images from each of the 13 modelled standard scan planes.",
"The 650 images were manually annotated with bounding boxes which were used as ground truth.",
"We employed the commonly used intersection over union (IOU) metric to measure the similarity of the automatically estimated bounding box to the ground truth [11].",
"Table REF summarises the results.",
"As in [11], we counted a bounding box as correct if its IOU with the ground truth was equal to or greater than 0.5.",
"Using this metric we found that on average 77.8% of the automatically retrieved bounding boxes were correct.",
"Cardiac views were the hardest to localise with an average accuracy of 62.0%.",
"The remaining views obtained an average localisation accuracy of 84.9%.",
"Table: Localisation evaluation: IOU and accuracy for all modelled standard views.In Fig.",
"REF we show examples of retrieved bounding boxes for each of the classes.",
"From these examples, it can be seen that our proposed method was able to localise standard planes which are subject to great variability in scale and appearance.",
"Qualitatively very good results were achieved for small structures such as the lips or the femur.",
"The reason why this was not reflected in the quantitative results in Table REF was that the IOU metric more is more sensitive to small deviations in small boxes than in large ones.",
"We noted that the method was relatively robust to artefacts and performed well in cases where it may be hard for non-experts to localise the fetal anatomy.",
"For instance, the lips view in the third column of Fig.",
"REF and the RVOT view in the second column were both correctly localised.",
"The last column for each structure in Fig.",
"REF shows cases with incorrect ($IOU < 0.5$ ) localisation.",
"It can be seen that the method almost never failed entirely to localise the view.",
"Rather, the biggest source of error was inaccurate bounding boxes.",
"In many cases the saliency maps were dominated by the most important feature for detecting this view, which caused the method to focus only on that feature at the expense of the remainder of the view.",
"An example is the stomach in the abdominal view shown in the fourth column of Fig.",
"REF .",
"Another example is the brain (tv.)",
"view, for which the lower parts – where the ventricle is typically visualised – was much more important for the detection.",
"In other cases, regions outside of the object also appeared in the saliency map, which caused the bounding box to overestimate the extent of the fetal target structures.",
"An example is the femur view, where the other femur also appeared in the image and caused the bounding box to cover both.",
"An example video demonstrating the real-time localisation for a representative case can be viewed at https://www.youtube.com/watch?v=yPCvAdOYncQ.",
"Figure: Examples of weakly supervised localisation using the SonoNet-32.",
"The first three columns for each view show correct bounding boxes marked in green (IOU≥0.5IOU \\ge 0.5), the respective last columns shows an example of an incorrect localisation marked in red (IOU<0.5IOU < 0.5).",
"The ground truth bounding boxes are shown in white."
],
[
"Discussion and Conclusion",
"In this paper, we presented the first real-time framework for the detection and bounding box localisation of standard views in freehand fetal ultrasound.",
"Notably, the localisation task can be performed without the need for bounding boxes during training.",
"Our proposed SonoNet employs a very deep convolutional neural network, based on the widely used VGG16 architecture, but optimised for real-time performance and accurate localisation from category-specific saliency maps.",
"We showed that the proposed network achieves excellent results for real-time annotation of 2D ultrasound frames and retrospective retrieval on a very challenging dataset.",
"Future work will focus on including the temporal dimension in the training and prediction framework as was done for sweep data in [7] and for fetal cardiac videos in [4].",
"We expect that especially the detection of cardiac views may benefit from motion information.",
"We also demonstrated the method's ability for real-time, robust localisation of the respective views in a frame.",
"Currently, the localisation is based purely on the confidence maps shown in Fig.",
"REF .",
"Although, this already leads to very accurate localisation, we speculate that better results may be obtained by additionally taking into account the pixel intensities of the original images.",
"Potentially, the proposed localisation method could also be combined using a multi-instance learning framework in order to incorporate the image data into the bounding box prediction [26].",
"We also note that the confidence maps could potentially be used in other ways, for instance, as a data term for a graphical model for semantic segmentation [3].",
"The pretrained weights for all of the network architectures compared in this paper are available at https://github.com/baumgach/SonoNet-weights."
]
] | 1612.05601 |
[
[
"On the impact of the magnitude of Interstellar pressure on physical\n properties of Molecular Cloud"
],
[
"Abstract Recently reported variations in the typical physical properties of Galactic and extra-Galactic molecular clouds (MCs), and in their ability to form stars have been attributed to local variations in the magnitude of interstellar pressure.",
"Inferences from these surveys have called into question two long-standing beliefs that the MCs : 1 are Virialised entities and (2) have approximately constant surface density i.e., the validity of the Larson's third law.",
"In this work we invoke the framework of cloud-formation via collisions between warm gas flows.",
"Post-collision clouds forming in these realisations cool rapidly and evolve primarily via the interplay between the Non-linear Thin Shell Instability (NTSI), and the self-gravity.",
"Over the course of these simulations we traced the temporal evolution of the surface density of the assembled clouds, the fraction of dense gas, the distribution of gas column density (NPDF), and the Virial nature of the assembled clouds.",
"We conclude, these physical properties of MCs not only exhibit temporal variation, but their respective peak-magnitude also increases in proportion with the magnitude of external pressure, $P_{ext}$.",
"The velocity dispersion in assembled clouds appears to follow the power-law, $\\sigma_{gas}\\propto P_{ext}^{0.23}$.",
"Also, the power-law tail at higher densities becomes shallower with increasing magnitude of external pressure, for magnitudes, $P_{ext}/k_{B}\\lesssim 10^{7}$ K cm$^{-3}$, at higher magnitudes such as those typically found in the Galactic CMZ ($P_{ext}/k_{B} > 10^{7}$ K cm$^{-3}$), the power-law shows significant steepening.",
"Thus while our results are broadly consistent with inferences from various recent observational surveys, it appears, MCs hardly exhibit a unique set of properties, but rather a wide variety, that can be reconciled with a range of magnitudes of pressure between 10$^{4}$ K cm$^{-3}$ - 10$^{8}$ K cm$^{-3}$."
],
[
"Introduction",
"Recently reported results by Rice et al.",
"2016 (hereafter, Rice et al.",
"), after a fresh analysis of the ${}^{12}$ CO data from the CfA-Chile survey of the Galactic Giant molecular clouds (GMCs), lends further credence to the hypothesis that the prevalent ambient conditions in the Galactic disk likely influence the physical properties of GMCs.",
"As in an earlier study of physical properties of GMCs in the Galactic Ring Survey by Heyer et al.",
"(2009), Rice et al.",
"also report a variation in the coefficient associated with the Larson's first law (Larson 1981), the so-called size-linewidth scaling relation ($\\sigma \\equiv b\\mathcal {L}^{a}; a=0.38, b=1.1$ ), as a function of the position of a GMC in the Galactic disk.",
"In particular, Rice et al.",
"reported, $b\\equiv 0.5\\pm 0.05$ for clouds located in the inner Galactic disk and $b\\equiv 0.38\\pm 0.05$ for those located farther out; the exponent, $a$ , for all the clouds in their sample was close to $\\sim 0.5$ .",
"This observed variation in the magnitude of the coefficient, $b$ , has been inferred as evidence suggesting a corresponding variation in the linewidth; the larger magnitude of $b$ for clouds located in the inner Galactic ring has been attributed to those clouds having a larger velocity dispersion.",
"Heyer et al.",
"(2009), on the other hand, reported that $b\\equiv \\Big (\\frac{\\sigma }{\\mathcal {L}^{a}}\\Big )$ , in fact varied in proportion with the square-root of the gas column density, $\\Sigma _{gas}$ .",
"While the recent findings reported by Rice et al., like one of the earliest studies of Galactic clouds reported by Solomon et al.",
"(1987), still imply that clouds in the Galactic disk are approximately consistent with the Simple Virial equilibrium ($\\sigma \\propto \\mathcal {L}^{0.5}$ ; SVE), their conclusion is inconsistent with that deduced by Heyer et al.",
"(2009).",
"The reason for this inconsistency is probably the difference in the tracer used by the respective authors : Rice et al.",
"used the more common ${}^{12}$ CO emission line to detect clouds, on the contrary, Heyer et al.",
"(2009) used the isotopologue ${}^{13}$ CO which is known to be a more reliable tracer at higher column densities.",
"In spite of this difference, conclusions from these two surveys unambiguously demonstrate the variation in the linewidth across clouds located in different regions of the Galactic disk.",
"In other related work, Hughes et al.",
"(2010) reported a size-linewidth relation, $\\sigma \\equiv 0.18\\mathcal {L}^{0.74}$ , for clouds in the LMC.",
"The obvious conclusion being that these clouds have significantly smaller line-widths as compared with the Galactic clouds.",
"Furthermore, these authors also point to the fact that the H$_{2}$ mass surface density appears to increase with increasing magnitude of the interstellar pressure, $P_{ext}$ .",
"These reported findings were further corroborated by Hughes et al.",
"(2013b) in their study of GMCs in other MW-like galaxies such as the M51 and M33.",
"Not only did Hughes et al.",
"(2013b) find variations in the H$_{2}$ surface density across clouds, but also reported variation of the coefficients $(a,b)$ in the size-linewidth relation; the inferred line-widths for GMCs in the M51 were higher than those for clouds in the M33 and the LMC.",
"These results challenge our extant beliefs and understanding about cloud-properties.",
"Previous studies, for instance those by Rosolowsky et al.",
"(2003; 2007) and Bolatto et al.",
"(2008) argued that physical properties of Galactic clouds and indeed of those in other nearby galaxies are approximately uniform, or rather, universal.",
"These conclusions were likely a consequence of GMC properties being derived only from ${}^{12}$ CO observations and were therefore a mere reflection of the physical conditions necessary to produce this emission.",
"These emission peaks only identified the high density regions that are likely to be immune to the environmental conditions.",
"Observational evidence pointing to variations in GMC properties have been variously interpreted : (i) Dobbs et al.",
"(2011) argued that the deviation of observationally deduced size-linewidth relation from the canonical relation possibly implied that clouds are largely unbound entities, (ii) Heyer et al.",
"(2009) suggested that cloud masses in their survey could have been systematically underestimated leading to the reported deviation, and (iii) the reported data could have been influenced by the external pressure, $P_{ext}$ , a rather old idea (e.g.",
"Bonnor 1956, Keto & Myers 1986, Elmegreen 1989, Bertoldi & McKee 1992, Field et al.",
"2011).",
"In fact, Elmegreen (1989), having argued that clouds were entities that probably marked an approximate equipartition between self-gravity, external pressure and their internal energy, deduced the relation $\\sigma \\propto \\Big (\\frac{P_{ext}/k_{B}}{10^{4} \\mathrm {K\\ cm^{-3}}}\\Big )^{1/4}\\Big (\\frac{\\mathcal {L}}{\\mathrm {pc}}\\Big )^{1/2}.$ Alternatively, Field et al.",
"(2011) proposed that the observed variations in the size-linewidth relation could be reconciled if the clouds obeyed the (external)pressure-modified Virial equilibrium (PVE), instead of the usual Simple Virial Equilibrium (SVE).",
"The implication being, a single magnitude of the external pressure, $P_{ext}$ , may be insufficient to explain the variations reported by Heyer et al.",
"(2009).",
"Instead, their data could be reconciled only if different clouds in the sample experienced different magnitudes of pressure in the range 10$^{4}$ K cm$^{-3}$ - 10$^{7}$ K cm$^{-3}$ .",
"Equivalently, it meant, the size-linewidth relation would be modified by the magnitude of external pressure, $P_{ext}$ , experienced by a GMC.",
"In an earlier work, Ballesteros-Paredes (2006) strongly argued against GMCs obeying the SVE because : (i) clouds being the consequence of turbulent fragmentation in the interstellar medium (ISM), the fragmentation process must necessarily induce a flux of mass, momentum and energy between clouds and the ISM, and (ii) clouds often exhibit asymmetries in their respective line-profiles which is inconsistent with clouds obeying the SVE.",
"Instead, the Larson's scaling relations should, at best, be viewed as evidence for energy-equipartition.",
"The possibility of bound, turbulence-supported clouds as suggested by Krumholz & McKee (2005) is difficult to reconcile, for it is unlikely there will be no redistribution of mass within such clouds over their lifetime (Ballesteros-Paredes 2006).",
"Gas dynamics within a typical GMC is likely governed by collisions between smaller clumps within them (e.g.",
"Anathpindika 2009 a,b and references there-in), and/or feedback from existing stellar populations (see e.g.",
"MacLow & Klessen 2004 and references there-in).",
"In a more recent work, Ballesteros-Paredes et al.",
"(2011) argued that the variations in the size-linewidth relation reported by Heyer et al.",
"(2009) is actually consistent with the scenario of hierarchical fragmentation in which a cloud does not collapse as a whole, but only isolated pockets in it collapse to form stars.",
"In this paradigm, although turbulence in the ISM possibly plays a role in assembling the cloud, post its formation, when the contribution due to self-gravity becomes significant, the cloud no longer need be in equilibrium with the external medium.",
"It is therefore unnecessary to resort to mechanisms that can actually hold clouds with complex geometries and density distributions in approximate equilibrium (see also Burkert & Hartmann 2004).",
"Simulations of unbound clouds leading to inefficient star-formation were discussed by e.g.",
"Clark et al.",
"(2005 & 2008).",
"The Virial state of GMCs has also been interrogated numerically.",
"Dobbs et al.",
"(2008) and Tasker & Tan (2009) for instance, reported that clouds forming in a galactic disk typically have a Virial coefficient between 0.2 - 10.",
"Furthermore, Dobbs et al.",
"(2011) argued that intercloud collisions in the disk of a galaxy was a viable mechanism to render clouds unbound.",
"Alternatively, several other authors examined the formation of clouds out of collisions between large-scale flows (e.g.",
"Vázquez-Semadeni et al.",
"1995; 2007, Joung & Mac Low 2006).",
"Hennebelle et al.",
"(2008), for instance, developed magnetohydrodynamic realisations of moderately supersonic atomic flows to demonstrate the formation of clouds.",
"They demonstrated that the density PDFs of the resulting cloud were consistent with those reported observationally.",
"Klessen & Hennebelle (2010) argued that turbulence in the ISM could be driven during the early phases of Galactic evolution when the disk was still probably accreting gas.",
"They demonstrated this by setting-up a system of colliding-flows with typical accretional velocities of $\\sim $ 15 km/s to $\\sim $ 20 km/s.",
"Their simulations showed that the collision induced a velocity dispersion $\\sim $ 4 km/s - 5 km/s in the diffuse ($\\lesssim $ 100 cm$^{-3}$ ) post-collision gas and a relatively weaker velocity dispersion, $\\lesssim 1$ km/s, in dense gas ($\\gtrsim 10^{4}$ cm$^{-3}$ ).",
"In other similar work, Heitsch et al.",
"(2008 a) demonstrated that the post-collision cloud was never in any equilibrium though there was always an approximate equipartition of energy.",
"The post-collision cloud, however, did develop localised centres of gravitational collapse.",
"More recently, Carroll-Nellenback et al.",
"(2014) argued that collision between fractal flows was more likely to delay core-formation in the post-collision cloud.",
"Likewise, Stanchev et al.",
"(2015) simulated a collision between uniform flows of atomic gas to reconcile the physical properties of the Perseus MC.",
"The principal objective of their exercise was to investigate the length-scale over which gravity was likely to dominate turbulence.",
"These efforts, however, neither address the issue about the observed variations in physical properties of clouds nor do they shed much light on the possibility that variations in the magnitude of interstellar pressure could possibly reconcile these observed variations in cloud-properties; see for instance Hughes et al.",
"(2013b).",
"Interaction between converging flows can be envisaged within the classic density-wave paradigm in which the arms of a typical spiral galaxy are believed to be patterns generated due to the propagation of a disturbance in the density field of a galactic-disk.",
"In this paradigm molecular clouds are assembled in crests/troughs of the propagating wave where gas-flows converge (see for instance the recent review by Dobbs & Baba 2014).",
"In this work we therefore investigate the dynamical evolution of the post-collision cloud that is assembled via collision between such flows.",
"We will investigate this problem numerically by developing self-gravitating realisations of flows having initially uniform density and colliding head-on; the case of non-headon collision between flows will be investigated in a sequel to this paper.",
"In particular, we will address the issue about the possible dependence of various physical properties and especially the size-linewidth relation, column density N-PDF, the gas surface density, $\\Sigma _{gas}$ , the magnitude of internal pressure, $P_{int}$ , in a cloud and the time-scale on which gas in a cloud is assembled into the dense phase (the gas-depletion timescale), on the magnitude of the external pressure, $P_{ext}$ , or equivalently, on the magnitude of the precollision velocity.",
"The layout of the paper is as follows : In §2 we discuss the numerical method and the initial conditions for these simulations and present the results in §3.",
"These results are discussed in §4 and we conclude in §5.",
"Figure: Cartoon showing a schematic representation of a head-on collision between identical cylindrical gas-flows.",
"Each flow has length, L cyl L_{cyl}, radius, R cyl =0.5*dR_{cyl}=0.5*d, and initial velocity magnitude, V inf V_{inf}.",
"See Table 1 for other physical details.The set-up, as shown in Fig.",
"1, involves merely two identical cylindrical gas-flows of uniform density directed head-on towards each other and each having an equal magnitude of initial velocity, $V_{inf}$ .",
"Individual flows are characterised by their mean density, $\\bar{n}$ , the pre-collision temperature, $T_{gas}$ , while $M_{gas}$ is the total mass of gas in the computational box.",
"Listed in Table 1 are details of the realisations developed in this work.",
"The mass of the post-collision cloud, $M_{cld}$ , which we nominally define as the volume of gas having density in excess of 50 cm$^{-3}$ will be $\\lesssim M_{gas}$ .",
"We performed 11 realisations of the problem to produce a magnitude of pressure, $P_{ext}$ , confining the post-collision cloud in the range 10$^{3}$ K cm$^{-3}$ - 10$^{8}$ K cm$^{-3}$ to mimic the ambient conditions that prevail at different radial locations in the Galactic disk.",
"The least massive clouds and the lowest magnitude of the interstellar pressure, $P_{ext}$ (typically, 10$^{4}$ K cm$^{-3}$ ), is found in the outermost regions of the Galactic disk.",
"On the contrary, the most massive clouds appear to be preferably located in the inner regions of the Galactic disk where the magnitude of $P_{ext}$ is comparatively higher ($\\gtrsim 10^{5}$ K cm$^{-3}$ ) (Rice et al.",
"2016; Kasparova & Zasov 2008).",
"The most massive clouds ($\\gtrsim 10^{5}$ M$_{\\odot }$ ), confined by a relatively large magnitude of interstellar pressure ($P_{ext}\\gtrsim 10^{6}$ K cm$^{-3}$ ), are found in the Galactic Central molecular Zone (CMZ)(e.g.",
"Ao et al.",
"2013, Rathborne et al.",
"2014).",
"Listed in column 4 of Table 1 is the magnitude of $P_{ext}$ in each realisation of this numerical exercise.",
"Table: Physical details of realisations.Figure: Shown here are the rendered density images of the shocked-slab in realisation 6 at different epochs; time in Myrs has been marked on the top right-hand corner of each panel.",
"Pictures on the upper-panel show the post-collision slab in the plane of collision while those on the lower-panel show its transverse section taken through the mid-plane.",
"That the growth of the shell-instability causes the slab to buckle and fragments it rapidly is evident from these images."
],
[
"Numerical Method",
"Realisations discussed in this work were developed using the well tested SPH code SEREN (Hubber et al.",
"2011).",
"We used the Monaghan-Riemann viscosity (Monaghan 1997), to capture shocks in the simulations discussed below.",
"Signal velocity in the viscous dissipation term was calculated as $v_{sig}(i,j) = (c_{i}^{2} + \\beta (\\textbf {v}_{ij}\\cdot \\hat{\\textbf {r}}_{ij})^{2})^{1/2} + (c_{j}^{2} + \\beta (\\textbf {v}_{ij}\\cdot \\hat{\\textbf {r}}_{ij})^{2})^{1/2} - \\textbf {v}_{ij}\\cdot \\hat{\\textbf {r}}_{ij},$ where $(c_{i},c_{j})$ are the respective sound-speeds for particles $(i,j)$ , $\\textbf {v}_{ij}\\equiv \\vert \\textbf {v}_{i}- \\textbf {v}_{j}\\vert $ and $\\hat{\\textbf {r}}_{ij}$ is the unit vector along the direction $\\textbf {r}_{ij}$ , connecting the particles $(i,j)$ .",
"This expression for $v_{sig}(i,j)$ , though similar to the conventional prescription, performs better for stronger shocks (Monaghan 1997; Toro 1992).",
"In other words, the resulting shocks are sharper.",
"Dynamical cooling of gas, as in some of our earlier work, was implemented with the aid of a parametric cooling-curve for the interstellar medium (e.g.",
"Koyama & Inutsuka 2002; Vazquez-Semadeni et al.",
"2007).",
"Resolution The number of gas particles, $N_{gas}$ , used in a realisation were calculated such that the Bate-Burkert criterion for resolution was satisfied at the minimum gas temperature ($\\sim $ 10 K), set for each realisation.",
"The minimum resolvable mass in an SPH realisation is $M_{min}\\sim \\Big (\\frac{2N_{neibs}}{N_{gas}}\\Big )\\cdot M_{gas}$ Bate & Burkert (1997); here $N_{neibs}$ =50, is the fixed number of neighbours that each SPH particle has and, $M_{gas}$ , the mass of gas in the computational domain.",
"We note that the minimum resolvable mass, $M_{min}$ , in this exercise has been varied between 0.2 M$_{\\odot }$ and 6 M$_{\\odot }$ for the three choices of initial gas density.",
"This was done to merely keep the number of gas particles, $N_{gas}$ , within manageable limits.",
"In this work we used $N_{gas}\\sim 2.15\\times 10^{6}$ and $N_{icm}=0.85\\times 10^{6}$ (those representing the intercloud medium, ICM, which like the live gas-particles exert hydrodynamic force, but unlike them do not posses self-gravity), particles to develop each realisation.",
"Consequently, the initial average smoothing length, $h_{avg}^{init}$ , that determines the extent of the smallest resolvable spatial-region, also varies.",
"It is defined as $h_{avg}^{init} = \\Big (\\frac{N_{neibs}}{N_{gas}}\\Big )\\Big (\\frac{3d^{2}L_{cyl}}{128}\\Big ),$ $d$ , and, $L_{cyl}$ , being respectively the diameter and the length of individual flow; see cartoon in Fig.",
"1.",
"The ICM particles were assembled to jacket the cylindrical flows and were set-up such that there was no density contrast across the gas-ICM interface.",
"The entire assembly was then placed in a periodic-box meant only to ghost particles in the box, i.e.",
"particles leaving from one face were allowed to re-enter from the opposite face.",
"Finally, listed in column 2 of Table 1 are the physical details of the pre-collision flows in each realisation.",
"We reiterate, our extant interest lies in investigating the physical properties of the assembled cloud and the variation of their magnitude as a function of the magnitude of external pressure, $P_{ext}$ , and on the efficiency with which gas in these clouds is cycled into potentially star-forming pockets.",
"Consequently, we do not follow the actual formation of prestellar cores in these realisations and defer this aspect of the question to a sequel to this article.",
"In spite of the relatively coarse numerical resolution, these realisations are sufficiently well resolved as to render the deduced properties of assembled clouds robust."
],
[
"Evolution of the shocked-slab",
"Precollision flows were supersonic in all the simulations listed in Table 1, care for the first realisation where the individual gas-flows were sub-sonic with a precollision Mach number of 0.7.",
"In an earlier work (Anathpindika 2009a,b), we have demonstrated the difference in evolution of a post-collision slab confined by shocks generated by a a super-sonic collision, as against one confined by ram-pressure, as in the case of a sub-sonic collision.",
"The slab in the former case is thin and soon after its assembly, develops corrugations on its surface which marks the onset of the Thin Shell Instability (TSI).",
"The amplitude of the associated crests and troughs grows rapidly due to the transfer of momentum between them and causes the shocked-slab to buckle.",
"Soon after it is triggered, the TSI grows non-linearly, a phase better known as the the Non-linear Thin Shell Instability (NTSI), and rapidly fragments the slab.",
"Shortly thereafter, the amplitude of slab-oscillations becomes comparable to the thickness of the slab as the NTSI saturates (Vishniac 1994, Heitsch et al.",
"2008a, Anathpindika 2009a).",
"Importantly, growth of the NTSI triggers a strong shearing motion between slab-layers which leads to mixing between them and dissipates turbulent energy that is crucial towards supporting the slab against self-gravity.",
"The upshot being, the shocked-slab soon puffs-up as the growth of the NTSI approaches saturation and the slab eventually loses support against self-gravity.",
"Consequently, the slab collapses globally to form a dense elongated globule in the plane of collision.",
"By contrast, the ram-pressure confined post-collision slab evolves in quasi-static fashion, via the interplay between self-gravity and the kinetic energy injected by the collision (Anathpindika 2009b).",
"Besides, in either case, the slab is also attended to by the cooling instability.",
"Heitsch et al.",
"(2008b) for instance, discussed the parameter regimes over which the respective instabilities are likely to dominate.",
"Within the context of this work it is important to underline the ability of the SPH to model the TSI and its non-linear excursion, the NTSI.",
"Anathpindika (2009 a) and Hubber et al.",
"(2013) demonstrated that the SPH can indeed reproduce the analytically predicted growth-rate by Vishniac (1994) for this instability.",
"By choice, all but one realisation in this work involve initially supersonic gas-flows suggesting that the post-collision slab in these slabs would be attended by the NTSI, albeit the triggering of this instability is delayed for relatively lower pre-collision Mach numbers.",
"Here we therefore present images of the shocked slab from only one realisation.",
"Shown in Fig.",
"2 is the observed temporal evolution of the shocked slab in realisation 6.",
"Buckling along the slab-surface and the associated temporal amplification of the corrugations on its surface, both classic identifying features of the NTSI, are visible in the rendered plots shown on the upper-panel of Fig.",
"2.",
"Subsequently, as it is evident from the picture in the right-hand frame ($t\\sim $ 2.81 Myrs), on the upper-panel of Fig.",
"2, the amplitude of perturbations on slab-surface becomes comparable to its thickness when growth of the NTSI saturates.",
"Similarly, pictures on the lower-panel of this figure show the fragmentation in the plane of the shocked-slab.",
"With passage of time the slab becomes dominated by a network of dense filaments.",
"However, we did not follow the calculations in this realisation to the stage where the shocked-slab collapses, though we did so for a latter case, listed 10 in Table 1 simply because a shocked-slab such as the one in this latter case is known from our earlier works (Anathpindika 2009 a), to evolve on a relatively shorter timescale.",
"Figure: Velocity-dispersion for gas in the shocked-slab for each set of simulations with initial choice of gas density, n ¯\\bar{\\mathrm {n}}=1 cm -3 ^{-3}, 50 cm -3 ^{-3} and 100 cm -3 ^{-3} is shown respectively on the upper, middle and the lower-hand panel."
],
[
"Typical diagnostics of gas in a post-collision slab",
"Gas-velocity dispersion Growth of dynamic instabilities injects a velocity field in the slab-layers and shown in various panels of Fig.",
"3 is the temporal variation of velocity dispersion, $\\sigma _{gas}$ , of gas in the post-collision slab for the three choices of initial gas-density viz., $\\bar{\\mathrm {n}} = $ 1 cm$^{-3}$ (realisations 1-3), 50 cm$^{-3}$ (realisations 4-6) and 100 cm$^{-3}$ (realisations 7-11).",
"Irrespective of the choice of the initial gas-density, the plots exhibit two common features : (i) the velocity dispersion acquires a maxima as the post-collision slab steadily accretes gas soon after its formation.",
"Thereafter, as the shell-instability begins to grow on the slab-surface, it causes kinetic energy within the slab-layers to dissipate as manifested by a gradual tapering-off of the velocity-dispersion at later times of evolution of the slab, and (ii) despite this dissipation, the magnitude of velocity-dispersion, $\\sigma _{gas}$ , in general is proportional to the magnitude of the precollision velocity or equivalently, to the magnitude of the external pressure, $P_{ext}$ ; see column 4 of Table 1.",
"Here we also draw the attention of our reader to the fact that the realisations 1-3, as can be seen from the plot in the upper-panel of this figure, were allowed to run for significantly longer than other cases simply to allow the gas time to cycle into the dense phase i.e., to become potentially star-forming.",
"Consequently, we also see the steep decline in magnitude of the gas velocity dispersion, $\\sigma _{gas}$ , at later epochs in these plots.",
"Similar steep decline is also visible in cases where the respective realisations were allowed to run to the point where the post-collision slab had begun collapsing.",
"At earlier epochs though, the respective curves in all the sets look mutually similar.",
"Figure: Shown respectively on the left and the right-hand panels here are plots showing the maximum magnitude of velocity dispersion, σ gas \\sigma _{gas}, for each realisation against the respective magnitude of external pressure, P ext P_{ext}, and, σ gas \\sigma _{gas}, against the magnitude of the inflow velocity,V inf _{inf}, of gas.Furthermore, shown on the left-hand panel of Fig.",
"4 is a plot of the magnitude of maximum velocity dispersion measured for each realisation i.e., the peak of each characteristic shown on the upper, middle and the lower-panel of Fig.",
"3, against the respective magnitude of external pressure, $P_{ext}$ .",
"Here one can readily see that a larger magnitude of external pressure, $P_{ext}$ , induces a higher magnitude of velocity dispersion within the slab layers.",
"The magnitude of velocity dispersion increases steadily for a given choice of the initial gas-density.",
"In general, the data over the entire range of pressure can be fitted reasonably well with a power-law of the kind $\\sigma _{gas}\\propto (P_{ext}/[10^{4} \\mathrm {K\\ cm}^{-3}])^{0.23}$ , derived by the technique of regression and which the reader will easily recollect, is roughly consistent with the power-law suggested by Elmegreen (1989); see Eqn.",
"(1) above.",
"This similarity in the exponent lends credence to the hypothesis that clouds merely represent energy-equipartition (e.g.",
"Elmegreen 1989, Ballesteros-Paredes 2006, Heitsch et al.",
"2008 a).",
"Similarly, shown on the right-hand panel of this figure is a plot showing $\\sigma _{gas}$ for different choices of the inflow velocity, V$_{inf}$ , across the realisations developed in this work.",
"This plot is interesting because not only does it show that the magnitude of velocity dispersion, $\\sigma _{gas}$ , increases approximately linearly with increasing $V_{inf}$ , but also helps reconcile the discontinuities in the magnitude of velocity-dispersion across density regimes in the $\\sigma _{gas}-P_{ext}$ plot.",
"The $\\sigma _{gas} - V_{inf}$ plot reinforces the conclusion from our earlier work (Anathpindika 2009 a,b), that the magnitude of $\\sigma _{gas}$ is fundamentally sensitive to the magnitude of V$_{inf}$ .",
"Pressure, on the other hand being a derived physical quantity, also depends on the mean density of the in-flowing gas so that the magnitude of pressure experienced by a cloud need not be the result of an interaction between gas-flows having a unique mean density and inflow velocity.",
"Thus, in view of the plot on the left-hand panel of Fig.",
"4 we suggest, clouds experiencing relatively low to intermediate magnitudes of external pressure, $P_{ext} \\sim 10^{3.5}$ K cm$^{-3}$ - 10$^{5.5}$ K cm$^{-3}$ , may not show an unambiguous trend of increasing magnitude of gas-velocity dispersion, $\\sigma _{gas}$ , corresponding to an increasing magnitude of external pressure.",
"Albeit, the trend is clearer at significantly higher magnitudes of $P_{ext}$ , typically in excess of 10$^{6}$ K cm$^{-3}$ , usually found in clouds closer to the Galactic centre (e.g.",
"Rice et.",
"al.",
"2016).",
"Figure: Column density distributions (N-PDFs) at different epochs of the post-collision slab in realisations 1-3 (upper-panel and central-panel left-hand), and realisations 4-6 (central-panel right-hand and lower-panel).",
"The constituting lognormal distributions of a N-PDF have been shown on some plots using a black dashed line.",
"The development of a power-law tail at higher densities, especially at latter epochs, can be readily seen in the N-PDFs for the latter realisations where the magnitude of external pressure was an order of magnitude higher in comparison to the former set of realisations.Column density probability distribution function (N-PDF) Shown on the panels of Fig.",
"5 are respectively the PDFs for gas column density (hereafter referred to as N-PDFs), at different epochs, for each choice of the magnitude of external pressure, $P_{ext}$ , and the average initial density, $\\bar{\\mathrm {n}}$ .",
"Following extensive numerical work, it is now well established that gas dominated by turbulence is characterised by a lognormal distribution and develops a power-law extension towards higher column densities once self-gravity becomes dominant (e.g.",
"Padoan et al.",
"1997; Federrath et al.",
"2008; Stanchev et al.",
"2015).",
"Let us begin by examining the N-PDFs for realisations 1, 2 and 3, where the individual pre-collision flows had a density of $\\bar{\\mathrm {n}}=1$ cm$^{-3}$ and a relatively low magnitude of external pressure, $P_{ext}/k_{B}\\sim 10^{3}-10^{4}$ K cm$^{-3}$ , in Fig.",
"5.",
"Evidently, the collision between the initially warm atomic flows in each of these three cases appears to have culminated in nothing more than a nebula of diffuse HI.",
"Albeit, the nebula in the third case, at least to some extent, appears to have some gas above the column-density threshold ($\\gtrsim 10^{20}$ cm$^{-2}$ ; e.g.",
"Stecher & Williams 1967, Hollenbach et.",
"al.",
"1971, Federman et.",
"al.",
"1979), required for the formation of molecular hydrogen.",
"The N-PDFs for these realisations appears to be a combination of lognormal distributions as has been shown by overlaying the constituent distributions on the N-PDF for realisation 1.",
"Thus with little gas in the dense phase, it appears that an environment of low interstellar pressure is likely inconducive to trigger star-formation.",
"And if at all star-formation does eventually commence, it will probably be sluggish.",
"We next examine the N-PDFs for realisations 4, 5 and 6 in Fig.",
"5.",
"The corresponding magnitude of external pressure, $P_{ext}$ , as can be seen from Table 1, was an order of magnitude higher in comparison with that for the first three realisations.",
"In contrast with the N-PDFs for realisations 1-3, these latter N-PDFs appear to be a combination of lognormal-distribution at relatively lower densities and a power-law tail at higher densities that has a distinct break, especially at latter epochs.",
"That the power-law tail for these realisations has two components is readily visible for the respective N-PDFs.",
"For illustrative purposes the constituent lognormal distribution at the low-density end of the distribution has been overlaid with a dashed black line on the plot corresponding to realisation 4.",
"Also, the power-laws fitting the high-density end of the N-PDF for each realisation in this set has been shown on the individual plots.",
"Please note, power-law fits have been shown for the N-PDF deduced at the latest epoch of the respective realisation.",
"Thus, for instance, in the case of realisation 4, the power-law fits have been shown for the N-PDF corresponding to $t=4.2$ Myrs.",
"Furthermore, the power-law fits shown for this realisation and those that follow later are good enough to be accepted at the 0.01 significance level of the simple Kolmogorov-Smirnov test.",
"Evidently, with increasing magnitude of the external pressure, $P_{ext}$ , both power-laws fitting the high-density end of the individual N-PDF become shallower or in other words, gas is cycled to increasingly higher densities.",
"It is also interesting to note that the double power-law tail generated in this set of realisations (i.e.",
"4, 5 & 6), is similar to the kind reported by Pokhrel et al.",
"(2016) for some regions within the cloud Mon R2.",
"Similarly, shown on the various panels of Fig.",
"6 are the N-PDFs for the remaining five realisations listed 7-11 in Table 1.",
"As with the N-PDFs for the previous set of realisations (listed 4-6), the N-PDF for the post-collision slab in these realisations is also lognormal at early epochs before subsequently developing a power-law tail at higher column densities.",
"We note, in this set of realisations the N-PDF for realisations 7, 9 and 11 developed a double power-law tail at its high density end.",
"As before, the power-law fitting the high density end of the N-PDF has been shown for each realisation.",
"In general, the N-PDFs for $P_{ext}\\lesssim 2.56\\times 10^{7}$ K cm$^{-3}$ (i.e., realisations 7-10), develop tails at their high-density end with relatively shallow slopes and the slope of this power-law is -0.6 for realisation 10.",
"In the somewhat extreme case such as in realisation 11, however, where $P_{ext}/k_{B}\\sim 10^{8}$ K cm$^{-3}$ , the N-PDF exhibits considerable steepening at the high density end; see lower panel of Fig.",
"6.",
"We will revisit this point in §4.",
"We also note, the power-law extension in these latter realisations develops over a relatively shorter timescale in comparison with that for the earlier three realisations.",
"Figure: Same as the plots in Fig.",
"5, but now for realisations listed 7-11 in Table 1.Figure: Plots showing the temporal variation of the fractional mass of cold, dense gas; Upper-panel : realisations 4-6, Central-panel : realisations 7-11.",
"Observe that the gas-depletion timescale becomes progressively smaller with increasing magnitude of P ext /k B P_{ext}/k_{B}; Lower-panel : plot showing temporal variation of fractional mass of gas having column density greater than 10 21 ^{21} cm -2 ^{-2} in realisations 6-11.Dense gas-fraction ($f_{Mass}\\equiv \\frac{\\mathrm {M}_{thresh}}{\\mathrm {M}_{gas}}$ ) Presently by dense, cold gas we imply the volume of putative star-forming gas and adopt a working threshold of gas denser than $\\sim 10^{3}$ cm$^{-3}$ and colder than $\\sim 50$ K. This is the typical density of a potential star-forming clump though denser regions within such clumps have been mapped with the aid of the emission due to transitions of molecules such as HCN, HCO$^{+}$ and HNC.",
"These emissions trace regions having density upwards of $\\sim $ 10$^{4}$ cm$^{-3}$ (e.g.",
"Gao & Solomon 2004, Usero et al.",
"2015 and Bigiel et al.",
"2016).",
"While such a steep density threshold could, in principal, have been adopted here to investigate the variation of the dense gas fraction, it would have picked only the more strongly self-gravitating pockets in the post-collision slab.",
"Instead, we are presently interested in studying the variation in the fraction of gas potentially available for core-formation and have therefore adopted a threshold that is an order of magnitude lower, but consistent with that used by Ragan et al.",
"(2016) to quantify the star-forming fraction in the Galactic disk.",
"The first three realisations are obviously eliminated from this exercise, for the post-collision gas body in those respective cases remained diffuse and therefore warm at the time of terminating calculations.",
"Shown on the upper and the central panel of Fig.",
"7 are respective plots showing the temporal evolution of this fraction in realisations 4-6, and 7-11.",
"In general, the more energetic a collision, or larger the inflow velocity, $V_{inf}$ , the higher the fraction of gas in the post-collision slab cycled into the dense, cold phase and shorter is the timescale of this cycling i.e., the timescale of gas-depletion in the cloud reduces progressively for an increasing magnitude of the external pressure.",
"For the largest magnitude of external pressure, $P_{ext}/k_{B}\\sim 10^{8}$ K cm$^{-3}$ , up to 80% of the gas appears in the dense, cold phase which in our realisation translates into a mass typically $\\sim 9\\times 10^{4}$ M$_{\\odot }$ , which is within a factor of two of the mass of the so-called 'Brick' in the Galactic CMZ (e.g.",
"Longmore et.",
"al.",
"2014).",
"There are, however, a few differences in the characteristics of this fraction as a function of external pressure.",
"At intermediate magnitudes of pressure such as, $P_{ext}/k_{B}\\sim (10^{5} - 10^{6})$ K cm$^{-3}$ , the fraction, $f_{Mass}$ , appears to gradually asymptote to $\\sim 50\\%$ .",
"By contrast, for larger magnitudes such as, $P_{ext}/k_{B}\\gtrsim $ 10$^{6}$ K cm$^{-3}$ , $f_{Mass}$ attains a peak before eventually petering-out.",
"We attribute this difference to that in the difference in the strength of growth of the NTSI.",
"As discussed in §3.1 earlier, the NTSI grows more vigorously in cases where the inflows are highly supersonic which presently is the case with realisations 10 and 11.",
"The growth of this instability in the shocked-slab, as noted above, is associated with its rapid fragmentation and a strong shearing interaction between slab-layers.",
"The observed decline in the dense-mass fraction in these latter realisations with a higher inflow velocity is the result of the NTSI-induced mixing between slab-layers.",
"Then as dense filaments begin to form in the collapsed globule in realisation 10, the dense-gas fraction starts rising again.",
"Note, however, this fraction for realisation 11 was still comparatively smaller at the time calculations were terminated.",
"The difference in these characteristics becomes clearer from the plot shown on the lower-panel of Fig.",
"7.",
"Shown in this plot is the temporal variation of the mass of gas having column density in excess of 10$^{21}$ cm$^{-2}$ in realisations 6-11.",
"At intermediate magnitudes of pressure i.e., ($10^{5}\\lesssim (P_{ext}/k_{B})\\lesssim 10^{6}$ ) K cm$^{-3}$ , this fraction increases steadily.",
"At larger magnitudes of pressure i.e., ($10^{6}\\lesssim (P_{ext}/k_{B})\\lesssim 10^{7}$ ) K cm$^{-3}$ , where the collision is relatively stronger and initially gas is quickly transferred into the dense-phase, but thereafter, as the NTSI dominates the evolution of the post-collision slab, this fraction gradually declines.",
"Finally, in the extreme case where $(P_{ext}/k_{B})\\gtrsim 10^{7}$ K cm$^{-3}$ , this fraction oscillates between $\\sim $ 1% - 3% which is due to the stronger NTSI-induced mixing between slab-layers and buckling of the slab-surface that rapidly creates and ruptures pockets of dense gas.",
"Consequently, only a tiny fraction of gas is assembled into pockets with potential to spawn stars."
],
[
"Physical properties of the assembled cloud",
"So far having examined the physical properties of gas in the post-collision slab, we now examine the typical physical properties of this composite gas-body.",
"For the purpose, we neglect any gas particles having density smaller than $\\sim 50$ cm$^{-3}$ , the lower density threshold for typical Galactic clouds.",
"The object so identified will hereafter be referred to as a cloud.",
"The mass, $M_{cld}$ , size, $L$ , and the surface density, $\\Sigma _{gas}$ , of this cloud was calculated as follows - $M_{cld} &= \\sum _{i} m_{i}, \\\\L &= 2\\cdot \\mathrm {max}<\\textbf {r}_{c} - \\textbf {r}_{i}>, \\mathrm {and} \\\\\\Sigma _{gas} &= \\frac{M_{cld}}{L^{2}},$ $m_{i}$ , being the mass of individual particles, $i$ , comprising this cloud; $\\textbf {r}_{c}$ , and $\\textbf {r}_{i}$ respectively the centre of this cloud, and position of particles in this cloud.",
"As noted in the Introduction, the observationally deduced magnitude of average density for clouds suffers from biases introduced by the specific choice of gas-tracers.",
"The average density of clouds ($\\sim $ 100 cm$^{-3}$ ), deduced by for instance, Solomon et.",
"al.",
"(1987), is largely based on CO-surveys which is a good tracer of relatively dense agglomerations of gas.",
"A similar threshold has been used by several authors to identify clouds in their respective simulations (e.g.",
"Fujimoto et al.",
"2014).",
"However, as has been discussed in §3.1 above, growth of the NTSI and the thermal instability in the post-collision slab segregate the dense and the rarefied phase in the post-collision slab.",
"As a consequence, raising the density threshold for cloud-identification to one that would be comparable with the observationally deduced magnitude could, to some extent, blight the variations in the physical properties of the clouds assembled in our realisations.",
"Surface density of the post-collision cloud Shown on the upper left-hand panel of Fig.",
"8 is the temporal variation of the surface density, $\\Sigma _{gas}$ , of this cloud in four of our realisations.",
"Here we use two realisations each as representative cases of the set of realisations corresponding to the choices of the initial density viz., $\\bar{n}$ = 50 cm$^{-3}$ and 100 cm$^{-3}$ .",
"First, evident from this plot is a distinct trend where the peak magnitude of surface-density increases with increasing magnitude of external pressure, $P_{ext}$ .",
"Shown on the neighbouring plot on the right-hand panel of this figure is the surface-density of the cloud at different epochs in a realisation as a function of the magnitude of external pressure for that realisation.",
"Data points marked with green crosses are those that correspond to the peak magnitude of the surface density for the respective realisation.",
"This plot has been made for the same four realisations as before and the bold, black line signifying a proportionality between $\\Sigma _{gas}$ and $P_{ext}$ appears to fit the green crosses reasonably well.",
"Second, as with the dense gas fraction, $f_{Mass}$ , the cloud surface density, $\\Sigma _{gas}$ , for realisations 10 and 11, as is visible in the plot on the left-hand panel of Fig.",
"8, also behaves differently in comparison with that for realisations 6 and 8 where the magnitude of $P_{ext}$ was relatively lower.",
"Thus while $\\Sigma _{gas}$ tends to asymptote in the latter two realisations, in the former two, however, it achieves a peak magnitude before eventually falling-off as the cloud buckles strongly and finally collapses as the NTSI saturates at the latter stages of its growth.",
"This can be seen in the pictures on the lower-panel of Fig.",
"8.",
"Shown here are the rendered density images of the shocked slab in realisation 10 at an early epoch ($t\\sim 0.65$ Myr), when the slab was buckling with the NTSI in attendance and then at a later epoch ($t\\sim 1.8$ Myrs), when it had collapsed to form an elongated globule.",
"This picture is also interesting because it shows not only the formation of a network of filaments, but also two other more prominent filaments that are likely to collide close to the centre of this globule.",
"Colliding and/or interacting filaments have also been reported in recent literature.",
"For instance, according to Nakamura et al.",
"(2014), star-formation in the Serpens South was likely triggered by a collision between filamentary clouds.",
"Similarly, the filaments in the Cygnus OB7 MC appear to be colliding and which according to numerical simulations developed by Dobashi et al.",
"(2014), cannot possibly be reconciled without invoking an external velocity gradient in the host cloud.",
"The picture in the lower right-hand panel of Fig.",
"8 is consistent with the argument presented by these authors.",
"The required velocity gradient in this realisation is invoked naturally by virtue of the collapse of the post-collision slab.",
"Furthermore, there is also evidence of colliding filaments triggering the formation of dense clumps in the Perseus MC (e.g.",
"Frau et al.",
"2015).",
"Earlier we noted that the surface density for realisation 6 appeared to grow asymptotically unlike that for cases 10 and 11.",
"It must be remembered that realisation 6 was developed with a relatively lower magnitude of the inflow velocity, $V_{inf}$ , and therefore a proportionally smaller magnitude of external pressure, $P_{ext}$ .",
"Now, although the evolution of the slab in this case was qualitatively similar to that in realisations 10 or 11, and as is visible from the rendered images shown in Fig.",
"2, the NTSI in this case grew on a relatively longer timescale.",
"The picture on the right-hand panel of Fig.",
"2 shows that at $t\\sim 2.81$ Myrs the growth of this instability was yet to acquire its saturation.",
"At this epoch, before the slab could collapse, the realisation in this case had been terminated.",
"As remarked earlier, the NTSI grows faster for higher inflow velocities so that in the present work it grew the fastest in realisations 10 and 11, those with the highest inflow velocity.",
"Figure: Upper left-hand panel : Temporal variation of the surface density, Σ gas \\Sigma _{gas}, of the cloud in each of the four representative realisations; Upper right-hand panel : Surface density, Σ gas \\Sigma _{gas}, of the cloud in these four realisations at different epochs of its evolution as a function of the corresponding external pressure, P ext /k B P_{ext}/k_{B}.",
"The bold line represents direct proportionality between Σ gas \\Sigma _{gas} and P ext /k B P_{ext}/k_{B}.",
"Lower left-hand panel : Rendered density image showing the slab in realisation 10 at an earlier epoch (t∼t\\sim 0.65 Myr), when it had its peak surface density;Lower right-hand panel : Rendered density image of the same slab at a latter epoch (t∼t\\sim 1.8 Myr), when its surface density had fallen significantly.",
"At this latter epoch, the slab has collapsed to form an elongated globule and in which a plethora of thin filaments have started forming; close to the middle of the collapsed globule, two large filaments appear to be interacting.Figure: Test of Larson's third-law : Shown in this plot is the surface density, Σ gas \\Sigma _{gas}, of the cloud as a function of its size at different epochs of its evolution.Larson's Third Law : Do clouds indeed have uniform surface density ?",
"We now come to the next important point, that about the validity of Larson's third law.",
"An important conclusion that follows from the Larson's scaling relations (Larson 1981), is that clouds are entities having approximately uniform surface densities and specifically, the surface density varies weakly with cloud-size ($\\propto L^{-0.1}$ ).",
"We test this conclusion by generating a surface density-size plot for clouds at different epochs in the same four realisations as above.",
"Shown in Fig.",
"9 is the resulting plot from which it is evident that the surface density of clouds is far from uniform and in fact, can be fit reasonably well with a power-law of the type $L^{-1.3}$ .",
"So, evidently the clouds simulated in this work are inconsistent with the Larson's third law.",
"This is hardly surprising, for as shown above, physical dimensions of the assembled clouds undergo significant changes over the course of their evolution by virtue of the growth of the NTSI.",
"Internal pressure vis-a-vis the external pressure The calculation of the magnitude of internal pressure, $P_{int}/k_{B}$ , for the cloud assembled in these realisations includes contribution from the thermal ($P_{th}/k_{B}\\equiv \\bar{n}_{gas}\\bar{T}_{gas}$ ; $\\bar{n}_{gas}$ and $T_{gas}$ are respectively the mean gas density, and temperature of the cloud), and the non-thermal component ($P_{NT}/k_{B}\\equiv \\mu \\bar{n}_{gas}\\sigma _{gas}^{2}/k_{B}; \\mu $ , being the mean molecular weight of the gas).",
"The corresponding plots on the left-hand panel of Fig.",
"10 show that like the surface density, the peak magnitude of the internal pressure also increases with increasing magnitude of the external pressure, $P_{ext}/k_{B}$ .",
"In each of these realisations the observed peak in $P_{int}/k_{B}$ was largely due to a higher contribution from the non-thermal component.",
"Note also that the magnitude of internal pressure tends to rise spectacularly at later times for realisations 8 and 10.",
"This observed increase in $P_{int}/k_{B}$ was primarily due to a higher contribution from the thermal component of the gas which in the collapsed globule in realisation 10 for instance, had become strongly self-gravitating.",
"The dominance of self-gravity manifested itself in the form of dense filaments visible in the lower right-hand panel of Fig.",
"8.",
"The calculations for realisation 11 were terminated soon after the shocked-slab consumed the inflows and began to exhibit strong buckling.",
"It is for this reason that a similar sharp increase in $P_{int}/k_{B}$ is not visible in the characteristic for this latter realisation.",
"The plot on the right-hand panel of Fig.",
"10 demonstrates that the observed increase in the magnitude of internal pressure, $P_{int}/k_{B}$ , remains within about an order of magnitude of the external pressure, $P_{ext}/k_{B}$ , for the respective realisation.",
"At no instance does the cloud in any realisation ever become over-pressured, not even for a relatively large magnitude of external pressure, $P_{ext}/k_{B}\\gtrsim 10^{7}$ K cm$^{-3}$ .",
"In all these realisations the assembled cloud at best acquired a configuration where $P_{ext}/k_{B}\\sim P_{int}/k_{B}$ , i.e., it tended toward a configuration obeying the pressure-modified Virial equilibrium (PVE), and thereafter, it became pressure-confined.",
"In fact, at the epoch when the respective clouds acquired their peak magnitude of $P_{int}$ , marked with green crosses on the plot shown on the right-hand panel of Fig.",
"10, which is also the epoch when the respective clouds had acquired their peak density, $\\Sigma _{gas}$ , as shown in Fig.",
"8 above, the clouds were in approximate pressure-equilibrium.",
"It therefore appears that after all molecular clouds may, at some stage of their evolution, appear to be obeying the PVE.",
"Strictly speaking, it may not even be necessary to treat clouds as objects in pressure equilibrium, for they appear to be so only briefly during their evolutionary sequence; see also Heitsch et al.",
"(2008a) for a discussion on the matter.",
"Figure: Left-hand panel : Shown in this plot is the temporal variation of the internal pressure, P int /k B P_{int}/k_{B}, of the cloud for these four representative realisations.",
"Right-hand panel : The magnitude of internal pressure, P int /k B P_{int}/k_{B}, at different epochs of the cloud in these four realisations as a function of the respective external pressure, P ext /k B P_{ext}/k_{B}.",
"The bold black line corresponds to a direct proportionality between P int /k B P_{int}/k_{B} and P ext /k B P_{ext}/k_{B}; the dotted lines on either side of this line represent an order of magnitude variation in pressure.Are Molecular clouds in Virial equilibrium ?",
"Following the convention adopted by Field et al.",
"(2011), we wish to distinguish between the Simple Virial equilibrium (SVE), and the pressure modified Virial equilibrium (PVE).",
"The latter accounts for the contribution due to the external pressure.",
"The velocity dispersion, $\\sigma _{gas}$ , of gas in a cloud of size, $L$ , is related to its column density, $\\Sigma _{gas}$ , and the external pressure according to - $\\frac{\\sigma _{gas}^{2}}{L} = \\frac{1}{6}\\Big (\\pi \\Gamma G\\Sigma _{gas} + \\frac{4P_{ext}}{\\Sigma _{gas}}\\Big )$ Field et.",
"al.",
"(2011); $\\Gamma $ =0.6 for a cloud having uniform density.",
"In the corresponding expression for the SVE there is no contribution due to the term from external pressure.",
"The plots shown in Fig.",
"11 here are largely the same as those presented by Field et al.",
"(2011) in their Fig.",
"3.",
"Overlaid on top of the V-shaped characteristics for different magnitudes of $P_{ext}$ is the data for clouds from Heyer et al.",
"(2009), as well as those for clouds from the LMC and the SMC adopted from Bolatto et al.",
"(2008).",
"The bold black line represents the SVE.",
"The distribution of these cloud data about the PVE characteristics was interpreted by Field et.",
"al.",
"(2011) as evidence suggesting that the size-linewidth relation for clouds is modified by external pressure and in fact, its proportionality is not constant but dependant on the magnitude of external pressure.",
"In order to examine if the clouds assembled in our simulations do obey the SVE or indeed the PVE, we calculated the ratio on the left-hand side of Eqn.",
"(7), the square of the size-linewidth coefficient, for them and plotted them along with the solutions for the PVE and the Heyer et al.",
"data in Fig.",
"11.",
"It can be readily seen that clouds in these realisations do not obey the SVE.",
"The size-linewidth coefficient is progressively higher for larger magnitudes of external pressure, $P_{ext}$ , and that the characteristic for a single magnitude of $P_{ext}$ cannot reconcile the dynamical state of the assembled cloud.",
"In other words, the cloud assembled in a realisation, as it evolves with time, cuts across the PVE solution for that magnitude of external pressure, $P_{ext}$ .",
"For instance, the cloud assembled in realisation 10 with $P_{ext}/k_{B}\\sim 2.56\\times 10^{7}$ K cm$^{-3}$ makes its initial appearance below the PVE solution for $P_{ext}/k_{B}$ = 10$^{6}$ K cm$^{-3}$ , thereafter as it continues to accrete gas, it makes an excursion beyond the PVE solution for $P_{ext}/k_{B}$ = 10$^{7}$ K cm$^{-3}$ and eventually arrives at a point between the PVE solutions for 10$^{6}$ K cm$^{-3}$ and 10$^{7}$ K cm$^{-3}$ at a later epoch.",
"Here we note that Eqn.",
"(7) is a relatively simple expression that does not account for the complex dynamics of a shocked-slab and so, it overestimates the size-linewidth coefficient for a given magnitude of surface density, $\\Sigma _{gas}$ , and external pressure, $P_{ext}$ .",
"As has been shown above, both the gas velocity-dispersion and the surface density of the post-collision cloud changes continuously as the NTSI grows and attains saturation.",
"Consequently, one sees an excursion of the post-collision cloud about the PVE characteristics shown in Fig.",
"11 for a given magnitude of $P_{ext}$ .",
"Thus there is only brief period when the cloud obeys the PVE as it intersects the PVE characteristic corresponding to the magnitude of $P_{ext}$ for a realisation.",
"This was also reflected in the plots on the left-hand panel of Fig.",
"10 when briefly there was an approximate equality between $P_{int}$ and $P_{ext}$ .",
"This observation leads us to the inference that we are presently observing field clouds at different stages of their evolution and which therefore occupy different locations on this plot.",
"It also lends further credence to the hypothesis proposed by Field et al.",
"(2011) that the observed dynamical properties of clouds could possibly be reconciled with not a single magnitude of $P_{ext}$ , but a range of magnitudes typically between 10$^{4}$ K cm$^{-3}$ and 10$^{8}$ K cm$^{-3}$ .",
"Figure: The variation of the size-linewidth coefficient for a cloud with its surface density.",
"This is the same plot as Fig.",
"3 in Field et al.",
"(2011), but now overlaid with the temporal locations of clouds in four of our realisations.",
"The 'V'-shaped characteristics are the solutions of the PVE given by Eqn.",
"7 for specific magnitudes of P ext /k B P_{ext}/k_{B}."
],
[
"Discussion",
"There are largely two schools of thought as regards the nature of molecular clouds (MCs).",
"On the one hand, Krumholz & McKee (2005) argued that MCs are objects in Virial equilibrium while on the other, Ballesteros-Paredes (2006) and other authors argued that clouds are unlikely to be in Virial equilibrium, but rather objects which at best represent equipartition between self-gravity, the kinetic energy and the magnetic energy.",
"See also the review by Hennebelle & Falgarone (2012) for a further discussion of this point.",
"The Krumholz-McKee hypothesis is supported by a number of older cloud surveys such as those by Solomon et al.",
"(1987), Rosolowsky et al.",
"(2003, 2005) and Bolatto et al.",
"(2008).",
"However, this hypothesis is inconsistent with the more recent findings such as those reported by Heyer et al.",
"(2009), Hughes et al.",
"(2010, 2013b), Meidt et al.",
"(2013) and Rice et al.",
"(2016).",
"These latter studies and in particular Heyer et al.",
"(2009) argue that cloud masses in a number of earlier surveys were significantly overestimated leading them to being classified as Virialised objects.",
"These authors also reported a variation in the size-linewidth coefficient in proportion with the surface-density of clouds in their data sample.",
"This is inconsistent with the principal conclusion of Larson (1981) that column density of clouds is approximately constant and as a consequence, the size-linewidth coefficient must not vary.",
"Recent surveys of both Galactic, and extra-Galactic clouds show that this is no longer the case.",
"Hughes et al.",
"(2010 & 2013b) and Meidt et al.",
"(2013) have attributed the observed variations in physical properties, including variations in the size-linewidth coefficient for clouds in the M51, M33 and the LMC to variations in the ambient environment i.e., variation in the magnitude of interstellar pressure in the disk of a galaxy.",
"In a related work Field et al.",
"(2011) argued that the Heyer et al.",
"data could be reconciled if clouds were to obey the Virial equilibrium modified by the external pressure.",
"Furthermore, they argued that a cloud could possibly experience a range of pressures between 10$^{4}$ K cm$^{-3}$ - 10$^{7}$ K cm$^{-3}$ .",
"Similarly, Dobbs et al.",
"(2011) reported that most of the clouds that formed in their simulated galactic disk were unbound.",
"In fact, Clark et al.",
"(2005) demonstrated that individual star-forming clouds need not be Virially bound.",
"In view of these recent suggestions about the possible dependence of physical properties of clouds on their ambient environment, here we examined this question numerically.",
"To this end we developed hydrodynamic simulations to study formation of clouds via head-on collisions of cylindrical gas-flows having initially uniform density.",
"We developed 11 realisations spanning a range of external pressures between 10$^{3.5}$ K cm$^{-3}$ - 10$^{8}$ K cm$^{-3}$ , commensurate with that reported for different regions of the Galaxy - from the outermost to the innermost radius (e.g.",
"Kasparova & Zasov 2008, Ao et al.",
"2013).",
"In general we found that clouds experiencing relatively small magnitudes of external pressure, typically lower than $\\sim 10^{4}$ K cm$^{-3}$ and therefore likely to be found in the outermost regions of the Galactic disk are unlikely to be of much interest from the perspective of star-formation, despite the small magnitude of gas velocity dispersion; see e.g.",
"Figs.",
"3 and 4.",
"However, a more definite statement can be made only after developing realisations that include details of the relevant atomic/molecular chemistry which presently has been substituted with a relatively simple cooling curve for the ISM.",
"From the respective plots shown in Figs.",
"3 and 4 it can also be readily inferred that the magnitude of velocity dispersion steadily increases with increasing magnitude of external pressure, though this rise is rather drastic for $P_{ext}/k_{B}\\gtrsim 10^{6}$ K cm$^{-3}$ .",
"In general, the velocity dispersion, $(\\sigma _{gas}/[\\mathrm {km/s}]) \\propto ((P_{ext}/k_{B})/[\\mathrm {K \\ cm^{-3}}])^{0.23}$ , which is roughly consistent with the analytic prediction made by Elmegreen (1989).",
"This is also qualitatively consistent with the trend recently reported for Galactic clouds (e.g.",
"Rice et al.",
"2016), and for extra-Galactic clouds (e.g.",
"Hughes et al.",
"2013b).",
"Now, although there is a dearth of literary evidence comparing the N-PDFs of clouds as a function of the magnitude of interstellar pressure, $P_{ext}$ , across a wide range of environments, a few studies have reported physical properties of the dense cloud G0.253+0.016, better known as the Brick, located close to the Galactic centre in the Central Molecular Zone (CMZ).",
"Although this cloud has an average density $\\sim 10^{4}$ cm$^{-3}$ , its N-PDF does not exhibit a power-law tail at the high-density end (e.g.",
"Rathborne et al.",
"2014).",
"As is well-known, the power-law tail is a good proxy for potentially star-forming gas in a cloud (see e.g.",
"Kainulainen et al.",
"2009; Lombardi et al.",
"2015).",
"Plots of N-PDFs shown in Figs.",
"5 and 6 reveal a trend where-in clouds experiencing a relatively small magnitude of external pressure, typically $P_{ext}/k_{B}\\lesssim 10^{4}$ K cm$^{-3}$ , tend to have a lognormal distribution and develop a distinct power-law tail at the high-density end for higher magnitudes of external pressure, $P_{ext}/k_{B}\\gtrsim 10^{5}$ K cm$^{-3}$ .",
"In fact, this power-law tail often appears to be composed of two components.",
"For extreme magnitudes of pressure typically upwards of $10^{7}$ K cm$^{-3}$ , however, the N-PDF shows further steepening at the high density end which is consistent with that reported for the Brick.",
"This reinforces the inference that in spite of the relatively large volume density of the assembled cloud, the fraction of putative star-forming gas in such clouds is considerably small.",
"These simulations also demonstrate the evolution of N-PDFs for different magnitudes of external pressure, $P_{ext}$ , from a lognormal distribution to one where a power-law begins to appear at higher densities.",
"The timescale of evolution, however, appears to depend on the magnitude of $P_{ext}$ , or equivalently, on the magnitude of the inflow velocity, $V_{inf}$ .",
"We also observe that the power-law at the high-density end becomes shallower with the slope in the range (-1, -0.6), for increasing magnitude of $P_{ext}$ , but less than $\\sim 10^{7}$ K cm$^{-3}$ .",
"Fortunately, there is some corroborative evidence from recent studies of different regions of the Orion A cloud.",
"Stutz & Kainulainen (2015) reported variations in the slope of the high-density end of the N-PDF for different regions of this cloud.",
"This slope was shallow ($\\sim $ -0.9), for the region having the highest concentration of YSOs.",
"This is also the region that apparently is most affected by feedback from the nearby population of O-stars.",
"By contrast, regions with little star-formation activity and therefore with fewer YSOs had significantly steeper power-law tails with a slope in excess of $\\sim -2$ .",
"Similar conclusions were also deduced by Pokhrel et al.",
"(2016) in their study of the different regions of the GMC Mon R2.",
"In fact, these authors also report that the tail at the high-density end of the N-PDF for some regions shows a break so that the tail could be fitted with two power-laws.",
"This is also true of the N-PDFs deduced for some clouds simulated in this work as is evident from the plots for realisations 4-6 shown in Fig.",
"5.",
"The correlation between star-formation activity and a shallower tail of the N-PDF appears universal.",
"Abreu Vicente et al.",
"(2015), for instance, in their extensive study of 330 MCs in the first Galactic quadrant reported similar conclusions.",
"The same also appears true with the N-PDFs for extra-Galactic clouds as shown by Hughes et al.",
"(2013a) in their study of clouds in the M51.",
"Another feature visible from the plots in Figs.",
"5 and 6 showing the N-PDFs is the shift in the position of the peak of a N-PDF as a function of the external pressure.",
"For magnitudes of pressure, $P_{ext}/k_{B}\\lesssim 10^{6}$ K cm$^{-3}$ , this shift is not very significant, but a leftward shift towards a lower column density is readily visible in N-PDFs for higher magnitudes of pressure shown in Fig.",
"6.",
"Thus, while on the one hand a stronger collision i.e., a larger magnitude of external pressure, $P_{ext}$ , cycles more gas into the dense-phase, the peak of the N-PDF appears to shift towards lower column densities which is consistent with the N-PDF for the typical Galactic centre cloud such as the Brick.",
"Equivalently this means, a larger magnitude of external pressure causes greater segregation of gas between the dense and the rarefied phase.",
"The upshot being that although a large magnitude of external pressure might easily cycle gas into the dense-phase (see plots shown on the upper and central-panel of Fig.",
"7), the fraction of gas at large column densities that can potentially form stars would diminish.",
"This is indeed the case as is evident from the plot on the lower-panel of Fig.",
"7.",
"This latter plot shows that for intermediate magnitude of external pressure, $P_{ext}$ , the fraction of cloud-mass assembled in the dense-phase steadily increased to 1% at the time of terminating calculations.",
"By contrast, for higher magnitudes of pressure, but $P_{ext}< 10^{7}$ K cm$^{-3}$ , this fraction was as high as 3% and thereafter, it steadily declined.",
"In the extreme cases of $P_{ext}/k_{B}\\gtrsim 10^{7}$ K cm$^{-3}$ , however, it appears, gas in the shocked-slab is rapidly cycled between the dense and rarefied phases.",
"These inferences are also consistent with deductions from some of the more recent surveys of Galactic clouds.",
"Ragan et al.",
"(2016), for instance, using data from the Hi-GAL survey, suggest that the dense mass fraction (which they refer to as the star-forming fraction), shows a statistically significant decline with increasing Galactic radius, or equivalently, with decreasing magnitude of the interstellar pressure.",
"This conclusion is also consistent with the one presented by Roman-Duval et al.",
"(2010) who had also shown a decrease in the dense gas-fraction i.e., the fraction of putative star-forming gas, with increasing Galactic radius.",
"Furthermore, Koda et al.",
"(2016) also showed that the dense gas-fraction steadily declined towards the Galactic centre, although the efficiency of converting neutral gas into its dense molecular counterpart is almost 100% at small Galactic radii where the magnitude of interstellar is significantly large, $P_{ext}/k_{B}\\gtrsim 10^{7}$ K cm$^{-3}$ .",
"Results presented in the plots shown on the central and the lower-panel of Fig.",
"7 above are also consistent with these observational deductions.",
"Now, although the dense mass fraction shows a steady decline for the latter set of realisations at the time of terminating calculations, this fraction could increase somewhat as dense filamentary structure might start forming in the post-collapse globule as was seen to be the case in realisation 10; see upper-panel of Fig.",
"8.",
"This image is interesting as it shows not only the formation of a filamentary network, but also two other prominent filaments that are likely to collide close to the centre of this globule.",
"The foregoing discussion demonstrates that star-formation in a pristine cloud is likely to be terribly inefficient with only a small fraction of gas being cycled into the dense-phase.",
"This conclusion is consistent with observational inferences; Murray (2011), for instance, estimated that only about 8% of the gas in a typical MW GMC is converted into stars over its lifetime.",
"In one of their earlier works, Fukui & Mizuno (1991) also showed that star-formation in nearby MCs is sluggish.",
"The situation with clouds in the Galactic CMZ is even worse as a typical CMZ cloud such as the Brick shows very little evidence of star-formation (e.g.",
"Kauffman et al.",
"2013).",
"Although a more recent study by Marsh et al.",
"(2016) provides further evidence about the possible onset of star-formation in this cloud.",
"There is, however, no immediate estimate about the fraction of gas that is possibly involved in forming stars.",
"We note, our models in no way refute the possibility of star-formation in clouds confined by a relatively large magnitude of pressure, and in fact, do corroborate the formation of putative star-forming pockets in such clouds at a later stage of their evolution.",
"But we also suggest that the fraction of this gas is likely to be relatively small.",
"However, our inferences contradict those reported by Bertram et al.",
"(2015).",
"These latter authors reported that irrespective of the hostile nature of ambient environment, their model clouds yielded star-formation efficiency significantly higher than the rather sluggish rate reported for the Brick by Kauffman et al.",
"Bertram et al.",
"therefore concluded that the merely unbounded nature of clouds and a stronger interstellar radiation field were not by themselves sufficient to arrest and/or inhibit star-formation in their model clouds, although increasing the unboundedness of their clouds did significantly reduce the star-formation efficiency in their simulations.",
"However, despite this, it remained considerably higher than the one inferred for typical clouds in the Galactic CMZ.",
"Results from their simulations led the authors to infer that star-formation in the CMZ clouds would eventually pick-up and that we are probably observing only the earliest stages of stellar-birth in these clouds.",
"It is indeed true that star-formation activity in the Brick is only recent (e.g.",
"Longmore et al.",
"2013, Marsh et al.",
"2016), but it is also true that the N-PDF for this cloud, as discussed above, appears clipped at large column densities suggesting that star-formation will likely remain sluggish even in the future.",
"On the contrary, our models here show that while a higher magnitude of external pressure, typically, $P_{ext}/k_{B}\\gtrsim 10^{6}$ K cm$^{-3}$ , undoubtedly cycles a larger fraction of gas into the dense-phase (gas-density in excess of 10$^{3}$ cm$^{-3}$ ), the fraction of putative star-forming gas i.e., the fraction of gas at relatively large column densities (N$_{gas}\\gtrsim $ 10$^{21}$ cm$^{-2}$ ), however, does not show any appreciable increase.",
"This could possibly explain why clouds close to the Galactic centre form stars sluggishly despite their relatively large volume density and richness in molecular gas.",
"We now proceed to examine the effect of the magnitude of external pressure on the physical properties of the cloud assembled in respective realisations.",
"As defined in the earlier section, the gas-body comprising of particles denser than $\\sim $ 50 cm$^{-3}$ in a realisation was described as a cloud, the physical properties of which were calculated using Eqns.",
"4 - 6 above.",
"Plots in the upper-panel of Fig.",
"8 made for four representative realisations show that the surface density, $\\Sigma _{gas}$ , of the cloud so identified at different epochs of a realisation is indeed sensitive to the magnitude of the external pressure.",
"Not only does $\\Sigma _{gas}$ for a cloud vary temporally for a given magnitude of external pressure, as can be seen from the plot on the upper left-hand panel of this figure; the plot on the upper right-hand panel of this figure shows that $\\Sigma _{gas}$ also varies proportionally with $P_{ext}$ .",
"This temporal variation of $\\Sigma _{gas}$ is the result of the structural changes to the post-collision slab induced by the growth of the NTSI in it.",
"The crosses on this latter plot represent the magnitude of $\\Sigma _{gas}$ for the clouds at different epochs as a function of $P_{ext}$ for these respective realisations, where-as the solid-line corresponds to $\\Sigma _{gas}\\propto P_{ext}$ .",
"Although the plot on the right-hand panel of Fig.",
"8 appears to suggest that the surface density of clouds must typically increase with increasing magnitude of the interstellar pressure, we note that the trend is visible only with regard to the peak magnitude of $\\Sigma _{gas}$ that the cloud acquires briefly during its evolutionary cycle.",
"Thereafter the cloud surface-density diminished rapidly.",
"Ragan et al.",
"(2016) also argued that the surface density of mass in dense clumps falls-off at small Galactic radii.",
"This finding can be reconciled with for example, the results from simulations 10 and 11 presented in this work.",
"In these latter realisations the surface density, $\\Sigma _{gas}$ , of the assembled cloud did in fact show a decline shortly after having acquired its peak magnitude as the cloud was attended to by the NTSI.",
"This leads to the suggestion that the clouds detected at small Galactic radii may well have had larger surface densities at an earlier epoch, soon after they were assembled and presently are at a relatively advanced stage of their evolution.",
"Evidently, the observed variation in the magnitude of $\\Sigma _{gas}$ for the simulated clouds in this work is inconsistent with one of the principle conclusions of Larson (1981), that was also reinforced by observational findings reported by, for e.g.",
"Solomon et al.",
"(1987) and Bolatto et al.",
"(2008), that clouds have an approximately constant surface density, typically $\\sim $ 100 M$_{\\odot }$ pc$^{-2}$ .",
"On the contrary, more recent surveys of Galactic clouds show that $\\Sigma _{gas}$ is indeed sensitive to their ambient environment i.e., to the magnitude of interstellar pressure, $P_{ext}$ , experienced by them (see review by Heyer & Dame 2015).",
"This is also true of extra-Galactic clouds (e.g.",
"Hughes et al.",
"2013b).",
"The inference from these respective data being that clouds experiencing a higher magnitude of external pressure would have a larger surface density.",
"On the other hand, if indeed the clouds obeyed the SVE then, $\\Sigma _{gas}\\propto P_{ext}^{1/2}$ , and furthermore, if the magnitude of interstellar pressure, $P_{ext}$ , in the disk of a galaxy were to be uniform then clouds would naturally be objects having a roughly uniform surface density (e.g.",
"Elmegreen 1989).",
"In such eventuality the Larson's third law would continue to hold.",
"But we now know that the magnitude of pressure in the Galactic disk and indeed in the disks of other MW-like galaxies is not uniform and that physical properties of clouds are sensitive to the magnitude of interstellar pressure.",
"The plot shown in Fig.",
"9, the surface density-size relation for the clouds assembled in these realisations reinforces our conclusion that clouds are unlikely to be entities having uniform surface density.",
"Contrary to earlier suggestions based on a number of surveys discussed above, we observe here that the surface density-size relationship for clouds is significantly steeper (than Larson's), and in fact, $\\Sigma _{gas}\\propto L^{-1.3}$ .",
"Similarly, a number of authors have argued that the inference that clouds have an approximately uniform surface density is likely an artefact of the specific choice of the molecular tracer (the CO) used to identify clouds (e.g.",
"Scalo 1990, Ballesteros-Paredes & MacLow 2002, Feldman et al.",
"2012).",
"Next, the plots in respectively the left and right-hand panels of Fig.",
"10 show the temporal variation in the magnitude of internal pressure, $P_{int}/k_{B}$ , in four of our representative realisations and the magnitude of internal pressure, $P_{int}/k_{B}$ , as a function of the corresponding external pressure, $P_{ext}/k_{B}$ , at different epochs of cloud evolution in these realisations.",
"We remind the reader that for any given realisation the magnitude of $P_{ext}/k_{B}$ remains unchanged over the entire course of a realisation.",
"As with the plots of the gas surface-density, the magnitude of internal pressure within a cloud also changes over the course of its evolution.",
"It attains a peak soon after the slab is assembled and thereafter it cools rapidly, though in the process the growth of the NTSI induces a velocity dispersion within the layers of the shocked-slab.",
"Then subsequently, as this instability grows into saturation and the velocity dispersion in the slab steadily decreases, the slab collapses to form an elongated globule within which dense filaments start forming as is visible in the lower-panel of Fig.",
"8 for realisation 10.",
"The strongly self-gravitating gas that is forming filaments causes localised thermal heating which manifests itself in the form of increasing magnitude of $P_{int}$ in realisations 8 and 10, as is also visible in the characteristics shown on the left-hand panel of Fig.",
"10.",
"However, the nature of this variation in the magnitude is such that clouds, even at relatively large magnitudes of external pressure, $P_{ext}/k_{B}\\gtrsim 10^{7}$ K cm$^{-3}$ , do not ever become over-pressured.",
"At best, the clouds in these realisations only briefly acquired a configuration where $P_{int}/k_{B}\\sim P_{ext}/k_{B}$ i.e., one that was closer to the PVE, before becoming pressure-confined.",
"The timescale on which a cloud evolved was shorter the higher the magnitude of $P_{ext}$ .",
"This is interesting especially in view of the traditional belief that clouds experiencing higher magnitudes of external pressure and therefore located preferably in the inner regions of the Galactic disk, must be gravitationally bound, as against those located in outer regions of the disk which must be pressure-confined.",
"On the contrary, we observe that clouds at higher pressure do in fact become pressure-confined at later stages of their evolution which is dominated by a strong growth of the NTSI.",
"Unfortunately we cannot independently corroborate this inference observationally since the results reported by for instance, Hughes et al.",
"(2010, 2013b) for their survey of clouds (MAGMA and PAWS) only cover the range of interstellar pressure up to $\\sim 10^{6}$ K cm$^{-3}$ .",
"Although our inference is consistent with the dynamic state of the Brick for which the magnitude of the internal pressure, $P_{int}/k_{B}\\sim 10^{8}$ K cm$^{-3}$ , using the physical properties deduced by Rathborne et al.",
"(2014), which is about an order of magnitude smaller than the estimated magnitude of external pressure, $P_{ext}/k_{B}\\sim 10^{9}$ K cm$^{-3}$ .",
"Thus in view of the observed behaviour of various physical properties of clouds in this work as a function of $P_{ext}$ , a simple bi-modal classification of clouds as either gravitationally bound or pressure-confined appears far-fetched.",
"We have seen, the dynamical state of a cloud varies temporally and the extent of variation appears to depend on the magnitude of external pressure it experiences.",
"It may therefore be appropriate to argue that clouds rather display a spectrum of properties depending on their ambient conditions (see also, Rosolowsky 2007).",
"An important consequence of the sensitivity of various cloud properties to the magnitude of ambient pressure, $P_{ext}$ , is the corresponding variation in the size-linewidth relation.",
"Recent work by Rice et al.",
"(2016) and indeed that by Heyer et al.",
"(2009) has conclusively demonstrated the variation in the coefficient of the size-linewidth relation for clouds varies as a function of their respective position in the Galactic disk.",
"While the former authors showed that the said coefficient was significantly larger, implying a higher velocity-dispersion, for clouds in the inner Galactic disk, Heyer et al.",
"(2009) showed, it varied proportionally with the surface density of clouds which as we know, is itself sensitive to the magnitude of interstellar pressure.",
"Variations in the coefficient of the size-linewidth relation have also been reported for molecular structures identified in the LMC, M33 and M51 (e.g.",
"Hughes et al.",
"2013b, Meidt et al.",
"2013), which the authors attributed to variations in the local environment.",
"This is also broadly consistent with the conclusions reported by Dobbs & Bonnell (2007), who observed considerable steepening of the size-linewidth relation for dense structures in their numerically simulated disk.",
"We observe a similar trend in Fig.",
"11 which is a plot showing the variation of the square of the size-linewidth coefficient for a cloud against its surface-density.",
"In general, we observe that the maximum magnitude of size-linewidth coefficient increases with the increasing magnitude of $P_{ext}/k_{B}$ .",
"Furthermore, this plot also demonstrates that each cloud evolves differently depending on its ambient environment i.e., the magnitude of external pressure experienced by it.",
"This conclusion is equivalent to the interpretation of this plot (i.e., Fig.",
"11), by Field et al.",
"(2011) that observed cloud properties could likely be explained by not one unique magnitude of pressure, but a range between 10$^{4}$ K cm$^{-3}$ and 10$^{7}$ K cm$^{-3}$ ."
],
[
"Limitations of this work",
"The simple assumption of cylindrical flows having initially uniform density and colliding head-on is ideal since turbulent flows in a galaxy are more likely to be fractal.",
"Also, such flows are more likely to interact at a certain impact-factor.",
"Carrol-Nellenback et.",
"al.",
"(2014), for instance, have demonstrated that the slab resulting from a collision between uniform flows fragments rapidly due to the onset of various hydrodynamic instabilities.",
"Consequently, such realisations typically showed a relatively lower core-formation rate in comparison with those in which fractal flows were allowed to collide.",
"Similarly, the former genre of realisations showed a greater proclivity to form low-mass cores.",
"In spite of this, conclusions presented in this work are unlikely to be altered significantly even if fractal flows were used, for the shocked slab in either case must evolve in a mutually similar fashion.",
"However, the effect of the magnetic field towards stabilising the post-collision slab against various dynamic instabilities and the subsequent effect on the various physical properties investigated in this work must be examined.",
"Even in this latter case we do not foresee any qualitative change to the inferences deduced in this work."
],
[
"Conclusions",
"Numerical simulations reported in this work demonstrate that assembled clouds primarily evolve via an interplay between the NTSI and self-gravity.",
"The NTSI grows rapidly and its growth is more vigorous in cases where the inflow velocity is relatively large, or equivalently, the inflows are highly supersonic.",
"Over the course of its growth, the NTSI induces significant structural changes to the post-collision cloud.",
"Consequently, physical properties of clouds and indeed those of the gas that constitutes these clouds show significant variation with time and as a function of the magnitude of external pressure, or in other words, the magnitude of the inflow velocity.",
"We observed that clouds in low-pressure environment show a greater propensity towards remaining diffuse and so, are not of much interest from the perspective of star-formation, in spite of their low gas velocity-dispersion.",
"In general, we observe that the magnitude of velocity-dispersion, $\\sigma _{gas}$ , induced within the assembled clouds in this exercise varies as $(\\frac{P_{ext}/k_{B}}{[\\mathrm {K\\ cm}^{-3}]})^{0.23}$ , which is roughly consistent with the hypothesis of clouds following energy-equipartition as against the SVE or indeed the PVE.",
"While a larger fraction of gas is cycled into the cold, dense phase with an increasing magnitude of external pressure, for magnitudes of pressure in excess of $\\sim 10^{6}$ K cm$^{-3}$ , the fraction of potentially star-forming gas, usually characterised by a column density in excess of $\\sim 10^{21}$ cm$^{-2}$ , was hardly ever seen to exceed $\\sim 3$ % and in fact, this fraction showed a steady decline as the assembled clouds evolved.",
"On the other hand, in the clouds confined by an intermediate range of pressure between $\\sim 10^{5}$ K cm$^{-3}$ - $\\sim 10^{6}$ K cm$^{-3}$ , the fraction of this putative star-forming gas showed a steady rise as the clouds evolved.",
"This suggests, clouds of the latter type are likely to exhibit greater propensity towards forming stars.",
"This is interesting as it implies, clouds in high-pressure environments such as those close to the Galactic centre, in spite of their relatively large average volume density, must be sluggish in forming stars as is indeed the case with the well-known cloud, Brick, in the Galactic CMZ.",
"An interesting diagnostic of potentially star-forming clouds is the appearance of the power-law tail in the N-PDF.",
"We observe that clouds experiencing pressure magnitude, $P_{ext}/k_{B} \\lesssim 10^{4}$ K cm$^{-3}$ , are likely to remain diffuse so that their N-PDF always remains lognormal, or a combination of several lognormal distributions.",
"For intermediate magnitude of pressure, $10^{5}\\lesssim P_{ext}/k_{B}\\lesssim 10^{6}$ K cm$^{-3}$ , on the other hand, the N-PDFs steadily evolve from a purely lognormal form to one with a power-law tail at the high-density end.",
"Furthermore, there is a distinct trend where-in the slope of this power-law tail becomes shallower (slope in the range -1 to -0.9), with increasing magnitude of external pressure, $P_{ext}$ .",
"Also, in some cases we observe, this power-law tail in fact, has a break and is composed of two constituents.",
"Finally, for magnitudes of pressure $\\gtrsim 10^{7}$ K cm$^{-3}$ , however, the power-law tail shows considerable steepening.",
"Furthermore, the outcome from simulations reported in this work also demonstrates that over their course of evolution, clouds evolve from a state where they obey the pressure-modified Virial equilibrium (PVE) to one where they could become pressure-confined; the latter is especially true for clouds in high-pressure environments at a magnitude of pressure upwards of $\\sim 10^{7}$ K cm$^{-3}$ .",
"This appears consistent with the recently deduced physical properties for the Brick.",
"Finally, these simulations reconcile the observationally reported variation in the size-linewidth coefficient.",
"Indeed, we find that its magnitude increases with an increasing magnitude of external pressure which is consistent with various recent observational inferences.",
"In a follow-up work we will further investigate the impact of Galactic shear on cloud-properties and the effect of ambient environment on the ability of dense gas to form pre-stellar cores and eventually stars."
],
[
"Acknowledgements",
"SVA is supported by the grant YSS/2014/000304 of the Department of Science & Technology, Government Of India, under the Young Scientist Scheme.",
"This work would not have been possible without a generous grant of the Royal Astronomical Society awarded October 2015.",
"SVA wishes to gratefully thank the Institute of Astronomy & Astrophysics, T$\\ddot{\\mathrm {u}}$ bingen, and the Sternwarte, Ludwig Maximilians Universit$\\ddot{\\mathrm {a}}$ t, for extending their warm hospitality.",
"Simulations reported in this work were developed on the bwGRiD computing cluster, a member of the German D-Grid initiative, funded by the Ministry for Education and Research (Bundesministerium f$\\ddot{\\mathrm {u}}$ r Bildung und Forschung) and the Ministry for Science, Research and Arts Baden-W$\\ddot{\\mathrm {u}}$ rttemberg (Ministerium f$\\ddot{\\mathrm {u}}$ r Wissenschaft, Forschung und Kunst Baden-W$\\ddot{\\mathrm {u}}$ rttemberg).",
"RK acknowledges financial support within the Emmy Noether research group on \"Accretion Flows and Feedback in Realistic Models of Massive Star-formation\" funded by the German Research Foundation under grant no.",
"KU 2849/3-1."
]
] | 1612.05690 |
[
[
"The fundamental theorem of affine geometry on tori"
],
[
"Abstract The classical Fundamental Theorem of Affine Geometry states that for $n\\geq 2$, any bijection of $n$-dimensional Euclidean space that maps lines to lines (as sets) is given by an affine map.",
"We consider an analogous characterization of affine automorphisms for compact quotients, and establish it for tori: A bijection of an n-dimensional torus ($n\\geq 2$) is affine if and only if it maps lines to lines."
],
[
"Main result",
"A map $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}^n$ is affine if there is an $n\\times n$ -matrix $A$ and $b\\in \\mathbb {R}^n$ such that $f(x)=Ax+b$ for all $x\\in \\mathbb {R}^n$ .",
"The classical Fundamental Theorem of Affine Geometry (hereafter abbreviated as FTAG), going back to Von Staudt's work in the 1840s [8], characterizes invertible affine maps of $\\mathbb {R}^n$ for $n\\ge 2$ : Namely, if $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}^n$ is a bijection so that for any line $\\ell $ , the image $f(\\ell )$ is also a line, then $f$ is affine.",
"Since then, numerous generalizations and variations have been proven, both algebraic and geometric.",
"For example, on the algebraic side, there is a version for for projective spaces (see e.g.",
"[2]), general fields $k$ , and for noninvertible maps (by Chubarev-Pinelis [3]).",
"On the geometric side, an analogue of FTAG holds in hyperbolic geomety (see e.g.",
"[5]), as well as in Lorentzian geometry, where the role of lines is played by lightcones (by Alexandrov [1]).",
"All of these versions have in common that they are characterizations of self-maps of the model space of the corresponding geometry (e.g.",
"$\\mathbb {E}^n, \\mathbb {H}^n$ , or $\\mathbb {P}^n$ ).",
"On the other hand, if $X$ is some topological space, and $\\mathcal {C}$ is a sufficiently rich family of curves on $X$ , then one can expect rigidity for self-maps of $X$ preserving $\\mathcal {C}$ .",
"In this light it seems reasonable to ask for the following generalized version of the FTAG: Question 1.1 (Generalized FTAG) Let $M$ be an affine manifold of dimension $>1$ .",
"Suppose that $f:M\\rightarrow M$ is a bijection such that for any affine line $\\ell $ in $M$ , the image $f(\\ell )$ is also an affine line (as a set).",
"Is $f$ affine?",
"Remark 1.2 Here by affine manifold and affine line, we use the language of geometric structures: If $M$ is a smooth manifold of dimension $n$ , then an affine structure on $M$ is a covering of $M$ by $\\mathbb {R}^n$ -valued charts whose transition functions are locally affine.",
"An affine line in an affine manifold $M$ is a curve in $M$ that coincides with an affine line segment in affine charts.",
"A map is affine if in local affine coordinates, the map is affine.",
"See [4] for more information on affine manifolds and geometric structures.",
"Remark 1.3 Of course one can formulate analogues of Question REF for the variations of FTAG cited above, e.g.",
"with affine manifolds (affine lines) replaced with projective manifolds (projective lines) or hyperbolic manifolds (geodesics).",
"The only case in which Question REF known is the classical FTAG (i.e.",
"$M=\\mathbb {A}^n$ is affine $n$ -space).",
"Our main result is that Question REF has a positive answer for the standard affine torus: Theorem 1.4 Let $n\\ge 2$ and let $T=\\mathbb {R}^n\\slash \\mathbb {Z}^n$ denote the standard $n$ -torus.",
"Let $f:T\\rightarrow T$ be any bijection that maps lines to lines (as sets).",
"Then $f$ is affine.",
"Remark 1.5 Note that $f$ is not assumed to be continuous!",
"Therefore it is not possible to lift $f$ to a map of $\\mathbb {R}^n$ and apply the classical FTAG.",
"In addition, the proof of FTAG does not generalize to the torus setting: The classical proof starts by showing that $f$ maps midpoints to midpoints.",
"Iteration of this property gives some version of continuity for $f$ , which is crucial to the rest of the proof.",
"The reason that $f$ preserves midpoints is that the midpoint of $P$ and $Q$ is the intersection of the diagonals of any parallellogram that has $PQ$ as a diagonal.",
"This property is clearly preserved by $f$ .",
"There are several reasons this argument fails on the torus: First, any two lines may intersect multiple, even infinitely many, times.",
"Hence there is no hope of characterizing the midpoint as the unique intersection of two diagonals.",
"And second, any two points are joined by infinitely many distinct lines.",
"Therefore there is no way to talk about the diagonals of a parallellogram.",
"In fact, these two geometric differences (existence of multiple intersections and multiple lines between points) will play a crucial role in the proof of Theorem REF .",
"Remark 1.6 The standard affine structure on the torus is the unique Euclidean structure.",
"However, there are other affine structures on the torus, both complete and incomplete.",
"We refer to [4] for more information.",
"We do not know whether a similar characterization of affine maps holds for these other, non-Euclidean structures.",
"Finally, let us mention some related questions in (pseudo-)Riemannian geometry.",
"If $M$ is a smooth manifold, then two metrics $g_1$ and $g_2$ on $M$ are called geodesically equivalent if they have the same geodesics (as sets).",
"Of course if $M$ is a product, then scaling any factor will not affect the geodesics.",
"Are any two geodesically equivalent metrics isometric up to scaling on factors?",
"This is of course false for spheres (any projective linear map preserves great circles), but Matveev essentially gave a positive answer for Riemannian manifolds with infinite fundamental group [6]: If $M$ admits two Riemannian metrics that are geodesically equivalent but not homothetic, and $\\pi _1(M)$ is infinite, then $M$ supports a metric such that the universal cover of $M$ splits as a Riemannian product.",
"Of course Matveev's result makes no reference to maps that preserve geodesics.",
"The related problem for maps has been considered with a regularity assumption, and has been called the “Projective Lichnerowicz Conjecture\" (PLC): First we say that a smooth map $f:M\\rightarrow M$ of a closed (pseudo-)Riemannian manifold $M$ is affine if $f$ preserves the Levi-Civita connection $\\nabla $ .",
"Further $f$ is called projective if $\\nabla $ and $f^\\ast \\nabla $ have the same (unparametrized) geodesics.",
"PLC then states that unless $M$ is covered by a round sphere, the group of affine transformations has finite index in the group of projective transformations.",
"For Riemannian manifolds, Zeghib has proven PLC [9].",
"See also [7] for an earlier proof by Matveev of a variant of this conjecture.",
"In view of these results, and Question REF , let us ask: Question 1.7 Let $M$ be a closed nonpositively curved manifold of dimension $>1$ and let $\\nabla $ be the Levi-Civita connection.",
"Suppose $f:M\\rightarrow M$ maps geodesics to geodesics (as sets).",
"Is $f$ affine (i.e.",
"smooth with $f^\\ast \\nabla =\\nabla $ )?",
"As far as we are aware, the answer to Question REF is not known for any choice of $M$ .",
"Theorem REF is of course a positive answer to Question REF for the case that $M=T^n$ is a flat torus."
],
[
"Outline of the proof",
"Let $f:T^n\\rightarrow T^n$ be a bijection preserving lines.",
"The engine of the the proof is that $f$ preserves the number of intersections of two objects.",
"A key observation is that affine geometry of $T^n$ allows for affine objects (lines, planes, etc.)",
"to intersect in interesting ways (e.g.",
"unlike in $\\mathbb {R}^n$ , lines can intersect multiple, even infinitely many, times).",
"In Section , we give a characterization of rational subtori in terms of intersections with lines.",
"This is used to prove that $f$ maps rational subtori to rational subtori.",
"Then we start the proof of Theorem REF .",
"The proof is by induction on dimension, and the base case (i.e.",
"dimension 2) is completed in Sections and .",
"In Section , we start by recalling a characterization of the homology class of a rational line in terms of intersections with other lines.",
"This allows us to associate to $f$ an induced map $A$ on $H_1(T^2)$ , even though $f$ is not necessarily continuous.",
"We regard $A$ as the linear model for $f$ , and the rest of the section is devoted to proving $f=A$ (up to a translation) on the rational points $\\mathbb {Q}^2\\slash \\mathbb {Z}^2$ .",
"In Section , we finish the proof by showing that $A^{-1}\\circ f$ is given by a (group) isomorphism on each factor of $T^2$ .",
"We show this isomorphism lifts to a field automorphism of $\\mathbb {R}$ , and hence is trivial.",
"This completes the proof of the 2-dimensional case Finally in Section , we use the base case and the fact that $f$ preserves rational subtori (proven in Section ), to complete the proof in all dimensions."
],
[
"Notation",
"We will use the following notation for the rest of the paper.",
"Let $n\\ge 1$ .",
"Then $T=\\mathbb {R}^n\\slash \\mathbb {Z}^n$ will be the standard affine $n$ -torus.",
"If $x\\in \\mathbb {R}^n$ , then $[x]$ denotes the image of $x$ in $T$ .",
"Similarly, if $X\\subseteq \\mathbb {R}^n$ is any subset, then $[X]$ denotes the image of $X$ in $T$ ."
],
[
"Acknowledgments",
"WvL would like to thank Ralf Spatzier for interesting discussions on the geodesic equivalence problem, which sparked his interest in Question REF .",
"In addition we would like to thank Kathryn Mann for conversations about the structure of homomorphisms of $S^1$ .",
"This work was completed as part of the REU program at the University of Michigan, for the duration of which JS was supported by NSF grant DMS-1045119.",
"We would like to thank the organizers of the Michigan REU program for their efforts."
],
[
"Rational subtori are preserved",
"Definition 2.1 Let $n\\ge 1$ and $1\\le k\\le n$ .",
"Then a $k$ -plane $W$ in $T=\\mathbb {R}^n\\slash \\mathbb {Z}^n$ is the image in $T$ of an affine $k$ -dimensional subspace $V\\subseteq \\mathbb {R}^n$ .",
"Let $V_0$ be the translate of $V$ containing the origin.",
"If $V_0$ is spanned by $V_0\\cap \\mathbb {Q}^n$ , we say $W$ is rational.",
"In this case we also say $W$ is a rational subtorus of dimension $k$ .",
"If $k=1$ , we also say $W$ is a rational line.",
"The goal of this section is to show: Proposition 2.2 Let $n\\ge 2$ and $1\\le k\\le n$ .",
"Write $T=\\mathbb {R}^n\\slash \\mathbb {Z}^n$ and let $f:T\\rightarrow T$ be a bijection that maps lines to lines.",
"Then the image under $f$ of any rational $k$ -dimensional subtorus is again a rational $k$ -dimensional subtorus.",
"For the rest of this section, we retain the notation of Proposition REF .",
"We start with the following criterion for subtori to be rational: Lemma 2.3 Let $V$ be a plane in $T$ .",
"Then the following are equivalent: $V$ is rational.",
"$V$ is compact.",
"Every rational line not contained in $V$ intersects $V$ at most finitely many times.",
"The equivalence of (i) and (ii) is well-known.",
"Let us prove (ii)$\\Rightarrow $ (iii).",
"First suppose that $V$ is compact and let $\\ell $ be any rational line not contained in $V$ .",
"If $\\ell \\cap V=\\emptyset $ , we are done.",
"If $v_0\\in \\ell \\cap V$ , we can translate $\\ell $ and $V$ by $-x_0$ , so that we can assume without loss of generality that $\\ell $ and $V$ intersect at 0.",
"Then $\\ell \\cap V$ is a compact subgroup of $T$ and not equal to $\\ell $ .",
"Because $\\ell $ is 1-dimensional, any proper compact subgroup of $\\ell $ is finite.",
"This proves $\\ell $ and $V$ intersect only finitely many times.",
"Finally we prove (iii)$\\Rightarrow $ (ii).",
"Suppose that $V$ is not compact.",
"Then $\\overline{V}$ is a compact torus foliated by parallel copies of $V$ .",
"Let $\\psi :U\\rightarrow \\mathbb {R}^k\\times \\mathbb {R}^{n-k}$ be a chart near $0\\in T$ such that the slices $\\mathbb {R}^k\\times \\lbrace y\\rbrace , \\, y\\in \\mathbb {R}^{n-k}$ , correspond to the local leaves of the foliation.",
"Since $V$ is dense in $\\overline{V}$ , there are infinitely many values of $y$ such that $\\mathbb {R}^k\\times y\\subseteq \\psi (V\\cap U)$ .",
"Now choose a rational line $\\ell $ in $\\overline{V}$ that is not contained in $V$ but with $0\\in \\ell $ .",
"If the neighborhood $U$ above is chosen sufficiently small, then $\\psi (\\ell \\cap U)$ intersects all leaves $\\mathbb {R}^k\\times y$ .",
"Since there are infinitely many values of $y$ such that $\\mathbb {R}^k\\times y\\subseteq \\psi (V)$ , it follows that $\\ell \\cap V$ is infinite.",
"Recall that we are trying to show that the image under $f$ of any rational subtorus is again a rational subtorus.",
"We first show the image is a plane.",
"Claim 2.4 Let $S$ be any rational subtorus of dimension $k$ .",
"Then $f(S)$ is a $k$ -plane.",
"We induct on $k=\\dim (S)$ .",
"The base case $k=1$ is just the assertion that $f$ maps lines to lines.",
"Suppose now $k\\ge 2$ and that the claim holds for $l$ -planes where $l<k$ , and let $S$ be a rational $k$ -dimensional subtorus.",
"Choose some $x_0\\in S$ and let $S_0:=S-x_0$ be the translate of $S$ passing through 0.",
"Let $V_0\\subseteq \\mathbb {R}^n$ be the subspace that projects to $S_0$ .",
"Since $S$ is rational, we can choose a basis $v_1,\\dots ,v_k$ of $V$ with $v_i\\in \\mathbb {Q}^n$ for every $1\\le i\\le k$ .",
"To use the inductive hypothesis, consider the $(k-1)$ -plane $V_{1}=\\operatorname{span}\\lbrace v_{1},...,v_{k-1}\\rbrace $ and similarly $V_{2}=\\operatorname{span}\\lbrace v_{2},...,v_{k}\\rbrace $ .",
"For $i=1,2$ , set $S_i:=[V_i+x_0]$ .",
"Since $v_i$ are rational vectors for each $i$ , we clearly have that $S_1$ and $S_2$ are rational $(k-1)$ -dimensional subtori.",
"Also let $S_{12}:=S_1\\cap S_2$ be the intersection, which is a $(k-2)$ -dimensional rational subtorus.",
"The inductive hypothesis implies that $f(S_1)$ and $f(S_2)$ and $f(S_{12})$ are all rational subtori of $T$ .",
"Since $f(S_1)$ and $f(S_2)$ are $(k-1)$ -planes that intersect in a $(k-2)$ -plane, they span a $k$ -dimensional plane.",
"More precisely, let $W_1$ (resp.",
"$W_2$ ) be the subspace of $\\mathbb {R}^n$ that projects to $f(S_1)-f(x_0)$ (resp.",
"$f(S_2)-f(x_0)$ ).",
"Then $W_1$ and $W_2$ are $(k-1)$ -dimensional subspaces of $\\mathbb {R}^n$ that intersect in a $(k-2)$ -dimensional subspace, and hence span a $k$ -dimensional subspace $W_0$ .",
"Then the $k$ -plane $W:=[W_0]+f(x_0)$ contains both $f(S_1)$ and $f(S_2)$ .",
"We claim that $f(S)=W$ .",
"We start by showing the inclusion $f(S)\\subseteq W$ .",
"Let $x\\in S$ .",
"If $x\\in S_1\\cup S_2$ , then clearly $f(x)\\in W$ , so we will assume that $x\\in S$ but $x\\notin S_1\\cup S_2$ .",
"Figure: A ball B⊆S 2 B\\subseteq S_2 such that lines from xx to BB meet S 1 S_1.Since $S_1$ and $S_2$ are closed codimension 1 submanifolds of $S$ , there is an open ball $B\\subseteq S_2$ such that for any $y\\in B$ , there is a line joining $x$ and $y$ that intersects $S_1$ (see Figure REF ).",
"Set $Y:=f(B)$ .",
"Since $f$ sends lines to lines and preserves intersections, $f(x)$ is a point with the property that for any $y\\in Y$ , there is a line $\\ell _y$ joining $f(x)$ and $y$ that intersects $f(S_1)$ .",
"We claim that this implies that $f(x)\\in W$ .",
"Write $U_1:=f(S_1)-f(x_0)$ for the translate of $f(S_1)$ that passes through the origin.",
"We can regard $U_1$ as a subgroup of $T$ , and consider the projection $\\pi : T\\rightarrow T\\slash U_1.$ The image $\\pi (W)$ of $W$ is a line because $f(S_1)$ has codimension 1 in $W$ .",
"Further $\\pi (f(x))$ is a point with the property that for any $y\\in Y$ , the line $\\pi (\\ell _y)$ joins the point $\\pi (f(S_1))=\\pi (f(x_0))$ and $\\pi (f(x))$ , and intersects $\\pi (W)$ at the point $\\pi (y)$ .",
"Note that there are only countably many lines from $\\pi (f(x_0))$ to $\\pi (f(x))$ , and unless one of them is contained in $\\pi (W)$ , each one has at most countably many intersections with the line $\\pi (W)$ .",
"However, since $\\pi (Y)$ is uncountable, we conclude that not every point $\\pi (y)$ can lie on a line that passes through both $\\pi (f(x_0))$ and $\\pi (f(x))$ , unless $\\pi (f(x))\\in \\pi (W)$ .",
"Therefore we must have that $\\pi (f(x))\\in \\pi (W)$ , so that $f(x)\\in W$ .",
"This proves that $f(S)\\subseteq W$ .",
"To establish the reverse inclusion, we just apply the same argument to $f^{-1}$ .",
"The above argument then yields that $f^{-1}(W)\\subseteq S$ .",
"Applying $f$ gives $W\\subseteq f(S)$ , as desired.",
"Actually the above proof also shows the following more technical statement, which basically states that linearly independent lines are mapped to linearly independent lines.",
"We will not need this until Section , but it is most convenient to state it here.",
"Lemma 2.5 Let $S=[V]$ be a rational subtorus containing 0, where $V\\subseteq \\mathbb {R}^n$ is a subspace.",
"For $1\\le i\\le k:=\\dim (S)$ , let $v_i\\in \\mathbb {Q}^n$ such that $v_1,\\dots ,v_k$ is a basis for $V$ .",
"Set $\\ell _i:=[\\mathbb {R}v_i]$ and choose $w_i\\in \\mathbb {Q}^n$ such that $f(\\ell _i)=[\\mathbb {R}w_i]$ .",
"Then $f(S)=[\\operatorname{span}(w_1,\\dots ,w_k)].$ Recall that (with the notation of the proof of Claim REF ), we have $W=\\operatorname{span}(w_1,\\dots ,w_k),$ and we have shown $f(S)=[W]$ , as desired.",
"It remains to show that if $S$ is a rational subtorus, then $f(S)$ is also rational.",
"We first show this for $S$ of codimension 1.",
"Claim 2.6 Let $S \\subseteq T$ be a rational codimension 1 subtorus.",
"Then $f(S)$ is also rational.",
"Let $S\\subseteq T$ be a codimension 1 rational subtorus and let $\\ell $ be any rational line not contained in $S$ .",
"By Lemma REF applied to $S$ , we see that $\\ell \\cap S$ is finite.",
"Then $f(\\ell )$ and $f(S)$ also intersect only finitely many times.",
"Suppose now that $f(S)$ is not rational.",
"Then $\\overline{f(S)}$ is a connected compact torus properly containing the codimension 1 plane $f(S)$ , and therefore $\\overline{f(S)}=T$ , i.e.",
"$f(S)$ is dense in $T$ .",
"But if $f(S)$ is dense, then it intersects any line that is not parallel to $S$ infinitely many times.",
"We know that $f(\\ell )$ is not parallel to $f(S)$ , because $f(\\ell )$ and $f(S)$ intersect at least once.",
"On the other hand, $f(\\ell )$ and $f(S)$ intersect finitely many times.",
"This is a contradiction.",
"Finally we can finish the proof of Proposition REF .",
"First note that if $S_1$ and $S_2$ are rational subtori, then any component of $S_1\\cap S_2$ is also rational (e.g.",
"by using that rationality is equivalent to compactness).",
"Now let $S$ be any rational subtorus of codimension $l$ .",
"Then we can choose $l$ rational codimension 1 subtori $S_1,\\dots ,S_l$ such that $S$ is a component of $\\cap _i S_i$ .",
"Since $f$ is a bijection, we have $f(S)\\subseteq f\\left(\\bigcap _{1\\le i\\le l} S_i\\right)=\\bigcap _{1\\le i\\le l} f(S_i)$ and by Claim REF , we know that $f(S_i)$ are rational.",
"Therefore $f(S)$ is a codimension $l$ -plane contained in $\\cap _i f(S_i)$ .",
"The components of $\\cap _i f(S_i)$ have codimension $l$ , so we must have that $f(S)$ is a component, and hence is rational."
],
[
"The 2-dimensional case: Affinity on rational points",
"The goal of this section is to prove a rational version Theorem REF in the two-dimensional case.",
"More precisely, under the assumptions of Theorem REF , we will prove that there is a linear automorphism $A$ of $T^2$ such that, up to a translation of $T^2$ , we have $f=A$ on $\\mathbb {Q}^2\\slash \\mathbb {Z}^2$ .",
"In the next section, we will complete the proof of the two-dimensional case by proving that, up to a translation, $f=A$ on all of $T^2$ .",
"We start by recalling the following elementary computation of the number of intersections of a pair of rational lines.",
"Proposition 3.1 Let $\\ell _1$ and $\\ell _2$ be two affine rational lines in the torus.",
"For $i=1,2$ , let $v_i \\in \\mathbb {Z}^2$ be a primitive tangent vector to the translate of $\\ell _i$ passing through $0\\in T^2$ .",
"Then the number of intersections of $\\ell _1$ and $\\ell _2$ is given by $|\\ell _1\\cap \\ell _2|= \\left| \\det \\begin{pmatrix}| & | \\\\v_1 & v_2 \\\\| & |\\end{pmatrix}\\right|.$ We now turn towards proving Theorem REF in the two-dimensional case.",
"For the rest of this section, suppose $f: T^2\\rightarrow T^2$ is a bijection that maps lines to lines.",
"Also let us fix the following notation: For $i=1,2$ , set $\\ell _i:=\\begin{bmatrix} \\mathbb {R}e_i\\end{bmatrix}$ .",
"We first make some initial reductions: By replacing $f$ with $x\\mapsto f(x)-f(0),$ we can assume that $f(0)=0$ .",
"For the next reduction, we need the following claim.",
"Claim 3.2 There is a linear automorphism $A:T^2\\rightarrow T^2$ such that $A \\ell _i = f(\\ell _i)$ .",
"Since $f(0)=0$ , the lines $f(\\ell _i)$ pass through 0.",
"In addition, because $\\ell _i$ are rational, so are $f(\\ell _i)$ (see Proposition REF ).",
"Therefore there are coprime integers $p_i$ and $q_i$ such that $f(\\ell _i)=\\begin{bmatrix}\\mathbb {R}\\begin{pmatrix} p_i \\\\ q_i\\end{pmatrix}\\end{bmatrix}.$ Note that $\\ell _1$ and $\\ell _2$ intersect exactly once, and hence so do $f(\\ell _1)$ and $f(\\ell _2)$ .",
"By Proposition REF , the number of intersections is also given by $|f(\\ell _1)\\cap f(\\ell _2)|=\\left| \\det \\begin{pmatrix} p_1 & p_2 \\\\ q_1 & q_2\\end{pmatrix}\\right|,$ so the linear transformation $A:T^2\\rightarrow T^2$ with matrix $A=\\begin{pmatrix} p_1 & p_2 \\\\ q_1 & q_2 \\end{pmatrix}$ is an automorphism, and clearly satisfies $A\\ell _i = f(\\ell _i)$ .",
"Let $A$ be as in Claim REF .",
"Then by replacing $f$ by $A^{-1}\\circ f$ , we can assume that $f(\\ell _i)=\\ell _i$ for $i=1,2$ .",
"Note that because each $v_i$ is only unique up to sign, the matrix $A$ is not canonically associated to $f$ .",
"Therefore we cannot expect that $A^{-1}\\circ f=\\text{id}$ , and we have to make one further reduction to deal with the ambiguity in the definition of $A$ .",
"Let us introduce the following notation: For a rational line $\\ell =\\begin{bmatrix}\\mathbb {R}\\begin{pmatrix} p \\\\ q\\end{pmatrix}\\end{bmatrix}$ , let $\\ell $ be the line obtained by reflecting $\\ell $ in the $x$ -axis, i.e.",
"$\\ell $ =$\\begin{bmatrix} \\mathbb {R}\\begin{pmatrix} p \\\\ -q\\end{pmatrix} \\end{bmatrix}$ .",
"Claim 3.3 Let $\\ell $ be a rational line in $T^2$ passing through the origin.",
"Then $f(\\ell )=\\ell $ or $f(\\ell )=\\scalebox {-1}[1]{\\rotatebox {180}{\\XMLaddatt {origin}{c}\\ell }}$ .",
"Let $p,q$ be coprime integers such that $\\ell =\\begin{bmatrix} \\mathbb {R}\\begin{pmatrix} p \\\\ q\\end{pmatrix} \\end{bmatrix}$ .",
"Also choose coprime integers $r,s$ such that $f(\\ell )=\\begin{bmatrix}\\mathbb {R}\\begin{pmatrix} r\\\\ s\\end{pmatrix}\\end{bmatrix}$ .",
"We can compute $p,q,r,s$ as suitable intersection numbers.",
"Indeed, we have $|p|=\\left| \\det \\begin{pmatrix}p & 0 \\\\ q & 1\\end{pmatrix}\\right| = |\\ell \\cap \\ell _2|.$ Since $f(\\ell _2)=\\ell _2$ and $f$ preserves the number of intersections of a pair of lines, we see that $|p|=|\\ell \\cap \\ell _2|=|f(\\ell )\\cap \\ell _2| = |r|,$ and similarly $|q|=|s|$ .",
"Therefore we find that $\\mathbb {R}\\begin{pmatrix} p \\\\ q\\end{pmatrix} = \\mathbb {R}\\begin{pmatrix} r \\\\ s \\end{pmatrix} \\hspace{28.45274pt} \\text{or} \\hspace{28.45274pt} \\mathbb {R}\\begin{pmatrix} p \\\\ -q\\end{pmatrix} = \\mathbb {R}\\begin{pmatrix} r \\\\ s \\end{pmatrix},$ which exactly corresponds to $\\ell =f(\\ell )$ or $\\scalebox {-1}[1]{\\rotatebox {180}{\\XMLaddatt {origin}{c}\\ell }}=f(\\ell )$ .",
"We will now show that whichever of the two alternatives of Claim REF occurs does not depend on the line $\\ell $ chosen.",
"Claim 3.4 Let $\\ell $ be a rational line passing through the origin that is neither horizontal nor vertical (i.e.",
"$\\ell \\ne \\ell _i$ for $i=1,2$ ).",
"Suppose that $f(\\ell )=\\ell $ .",
"Then for any rational line $m$ passing through the origin, we have $f(m)=m$ .",
"Let $p,q$ coprime integers such that $\\ell =\\begin{bmatrix}\\mathbb {R}\\begin{pmatrix} p \\\\ q\\end{pmatrix}\\end{bmatrix}$ .",
"Since $\\ell $ is neither vertical nor horizontal, we know that $p$ and $q$ are nonzero.",
"Now let $m$ be any other rational line, and choose coprime integers $r,s$ such that $m=\\begin{bmatrix}\\mathbb {R}\\begin{pmatrix} r\\\\ s\\end{pmatrix}\\end{bmatrix}$ .",
"The number of intersections of $\\ell $ and $m$ is given by $|\\ell \\cap m| = \\left| \\det \\begin{pmatrix} p & r \\\\ q & s\\end{pmatrix}\\right| = |ps-qr|.$ We will argue by contradiction, so suppose that $f(m)\\ne m$ .",
"Using the other alternative of Claim REF for $m$ , we have that $|f(\\ell )\\cap f(m)|= \\left| \\det \\begin{pmatrix} p & r \\\\ q & -s\\end{pmatrix}\\right| = |ps+qr|.$ Since $f$ preserves the number of intersections, we therefore have $|ps-qr|=|ps+qr|$ which can only happen if $ps=0$ or $qr=0$ .",
"Since we know that $p$ and $q$ are nonzero, we must have $r=0$ or $s=0$ .",
"This means that $m$ is horizontal or vertical, but in those cases the alternatives provided by Claim REF coincide.",
"We can now make the final reduction: If $f(\\ell )=\\ell $ for any rational line $\\ell $ , then we leave $f$ as is.",
"In the other case, i.e.",
"$f(\\ell )=\\scalebox {-1}[1]{\\rotatebox {180}{\\XMLaddatt {origin}{c}\\ell }}$ for any rational line $\\ell $ , we replace $f$ by $\\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix} \\circ f,$ after which we have $f(\\ell )=\\ell $ for any rational line $\\ell $ passing through the origin.",
"After making these initial reductions, our goal is now to prove that $f=\\text{id}$ on the rational points $\\mathbb {Q}^2\\slash \\mathbb {Z}^2$ .",
"We start with the following easy observation: Claim 3.5 If $\\ell $ and $m$ are parallel lines in $T$ , then $f(\\ell )$ and $f(m)$ are also parallel.",
"Since $\\dim (T)=2$ , two lines in $T$ are parallel if and only if they do not intersect.",
"This property is obviously preserved by $f$ .",
"Combining our reductions on $f$ so far with Claim REF yields the following quite useful property for $f$ : Given any rational line $\\ell $ , the line $f(\\ell )$ is parallel to $\\ell $ .",
"However, we still do not know the behavior of $f$ in the direction of $\\ell $ or transverse to $\\ell $ .",
"In particular we do not know whether or not $f$ leaves invariant every line $\\ell $ ; we only know this for $\\ell $ passing through the origin, and whether or not $f=\\text{id}$ on rational lines $\\ell $ passing through the origin.",
"We can now prove: Theorem 3.6 For any rational point $x\\in \\mathbb {Q}^2/\\mathbb {Z}^2 $ we have $f(x)=x$ .",
"For $n>2$ , set $G_{n}=\\begin{bmatrix} \\dfrac{1}{n} \\, \\mathbb {Z}^2\\end{bmatrix}.$ Clearly we have $\\mathbb {Q}^2\\slash \\mathbb {Z}^2=\\cup _n G_n$ , so it suffices to show that $f=\\text{id}$ on $G_n$ for every $n> 2$ .",
"We purposefully exclude the case $n=2$ because $G_2$ only consists of 4 points, and this is not enough for the argument below.",
"However, since we have $G_2\\subseteq G_6$ , this does not cause any problems.",
"Fix $n\\ge 2$ .",
"Set $Q:=\\begin{bmatrix} \\dfrac{1}{n}\\begin{pmatrix} 1 \\\\ 1\\end{pmatrix}\\end{bmatrix}.$ Claim 3.7 If $f(Q)=Q$ , then $f=\\text{id}$ on $G_n$ .",
"Suppose that $x=\\begin{bmatrix} x_1 \\\\ y_1 \\end{bmatrix}$ is any point of $G_n$ with $f(x)\\ne x$ .",
"We can choose $x_1$ and $y_1$ that lie in $[0,1)$ .",
"We will show that $f(Q)\\ne Q$ .",
"Let $x_2, y_2\\in [0,1)$ be such that $f(x)=\\begin{bmatrix} x_2 \\\\ y_2\\end{bmatrix}$ .",
"Then there are two cases: Either $x_1\\ne x_2$ or $y_1\\ne y_2$ .",
"Clearly the entire setup is symmetric in the two coordinates, so we will just consider the case $x_1\\ne x_2$ .",
"See Figure REF for an illustration of the various lines introduced below.",
"For $a\\in [0,1)$ , let $v_a$ (resp.",
"$h_a$ ) denote the vertical (resp.",
"horizontal) line $\\begin{bmatrix} x=a\\end{bmatrix}$ (resp.",
"$\\begin{bmatrix} y=a\\end{bmatrix}$ ).",
"Since $f$ maps parallel lines to parallel lines by Claim REF and $f(\\ell _i)=\\ell _i$ by assumption, we know that $f$ maps vertical (resp.",
"horizontal) lines to vertical (resp.",
"horizontal) lines.",
"In particular we have that $f(v_{x_1})=v_{x_2}$ and $f(h_{y_1})=h_{y_2}$ .",
"We will show that $f(Q)\\ne Q$ by showing that $f(h_{1/n}) \\ne h_{1/n}$ .",
"Figure: Illustration of the proof of Claim .",
"The points in aquaaqua are the possibilities for f(z)f(z).",
"The proof amounts to showing all of these lie at different heights than zz does.Consider the point $z:=\\begin{bmatrix} x_1 , \\dfrac{1}{n} \\end{bmatrix}\\in T^2$ .",
"Since $x \\in G_n$ , there is an integer $0\\le a<n$ such that $x_1 = \\dfrac{a}{n}$ .",
"First note that $a>0$ : For if $a=0$ , then $x\\in \\ell _2$ .",
"Since $f(\\ell _2)=\\ell _2$ , we see that $x_2=0$ as well.",
"This contradicts the initial assumption that $x_1\\ne x_2$ .",
"Since $a>0$ , $z$ lies on the line $\\ell $ of slope $\\dfrac{1}{a}$ going through the origin.",
"By Claim REF we have $f(\\ell )=\\ell $ , so $f(z)\\in \\ell $ .",
"In addition, since $z\\in v_{x_1}$ , we have $f(z)\\in v_{x_2}$ .",
"An easy computation yields $\\ell \\cap v_{x_2} = \\begin{bmatrix} x_2 \\\\ \\frac{x_2}{a} \\end{bmatrix} + \\frac{1}{a}\\begin{bmatrix} 0 \\\\ \\mathbb {Z}\\end{bmatrix}.$ Therefore we automatically have $f(h_{\\frac{1}{n}})\\ne h_{\\frac{1}{n}}$ unless $\\frac{1}{a} x_2 + \\frac{k}{a} \\equiv \\frac{1}{n} \\mathbb {}\\mod {\\mathbb {Z}}$ for some integer $k$ with $0\\le k<a$ .",
"Solving for $k$ yields $k\\equiv \\frac{a}{n}-x_2 \\mod {a}\\unknown.",
"\\mathbb {Z}.$ Since $0\\le k<a$ and $0<a<n$ and $0\\le x_2<1$ , we see that the only option is that $k=0$ .",
"Hence $x_2\\equiv \\frac{a}{n}\\equiv x_1\\mathbb {}\\mod {\\mathbb {Z}}$ , which contradicts our initial assumption that $x_1\\ne x_2$ .",
"It remains to show that $f(Q)=Q$ .",
"Consider the points $P, R, S$ as shown in Figure REF .",
"Let us give a brief sketch of the idea so that the subsequent algebraic manipulations are more transparent.",
"Figure: Quadrilateral in T 2 T^2 with vertices P,Q,R,SP,Q,R,S.Because $P$ and $Q$ lie on a vertical line, and $f$ maps vertical lines to vertical lines, $f(P)$ and $f(Q)$ have to lie on the same vertical line.",
"Because $P$ lies on the line of slope $n-1$ through the origin, $f(P)$ also has to lie on this line.",
"Therefore however much $P$ and $f(P)$ differ in the horizontal direction determines how much they differ in the vertical direction.",
"Finally since $P$ and $S$ lie on a horizontal line, the vertical difference between $P$ and $f(P)$ equals the vertical difference between $S$ and $f(S)$ .",
"The upshot of this is that any difference between $Q$ and $f(Q)$ translates into information about the vertical difference between $S$ and $f(S)$ .",
"The same reasoning with $P$ replaced by $R$ yields information about the horizontal difference between $S$ and $f(S)$ .",
"Finally because $S$ lies on a line of slope 1 through the origin, any horizontal difference between $S$ and $f(S)$ matches the vertical difference.",
"Therefore the information gained about the difference between $S$ and $f(S)$ using the two methods (once using $P$ and once using $R$ ) has to match.",
"We will see that this forces $Q=f(Q)$ .",
"Let us now carry out the calculations.",
"Choose $\\delta \\in \\left(\\dfrac{-1}{n},\\dfrac{n-1}{n}\\right)$ such that $f(v_{\\frac{1}{n}})=v_{\\frac{1}{n}}+\\begin{bmatrix} \\delta \\\\ 0\\end{bmatrix},$ so that $f(P)$ and $f(Q)$ have first coordinate $\\frac{1}{n}+\\delta $ .",
"Note that the boundary cases $\\delta =-\\frac{1}{n}$ and $\\delta =\\frac{n-1}{n}$ are impossible since these correspond to $f(v_{\\frac{1}{n}})=\\ell _2$ .",
"Since $Q$ lies on the line of slope 1 through the origin, we have $ f(Q)=\\begin{bmatrix} \\dfrac{1}{n}+\\delta \\\\ \\dfrac{1}{n}+\\delta \\end{bmatrix}$ Since $P$ lies on the line of slope $n-1$ through the origin, there is some integer $k_P$ such that $ f(P)=\\begin{bmatrix} \\dfrac{1}{n}+\\delta \\\\ \\dfrac{n-1}{n}+\\delta (n-1)-k_P\\end{bmatrix}.$ Here we can choose $k_P$ such that $0\\le \\frac{n-1}{n}+\\delta (n-1)-k_P < 1.$ Using that $R$ lies on the line of slope $\\frac{1}{n-1}$ through the origin, we similarly find an integer $k_R$ such that $f(R)=\\begin{bmatrix} \\dfrac{n-1}{n}+\\dfrac{\\delta }{n-1}-k_R \\\\ \\dfrac{1}{n}+\\delta \\end{bmatrix}.$ Finally, using that $S$ lies on the same horizontal line as $P$ , and on the same vertical line as $R$ , we find $ f(S)=\\begin{bmatrix} \\dfrac{n-1}{n}+\\dfrac{\\delta }{n-1}-k_R \\\\ \\dfrac{n-1}{n}+\\delta (n-1)-k_P\\end{bmatrix}.$ Since $S$ lies on the line of slope 1 through the origin, so does $f(S)$ .",
"Setting the two coordinates of $f(S)$ given by Equation REF equal to each other, we find (after some simple algebraic manipulations): $ k_P - k_R = \\delta \\left(n-1-\\dfrac{1}{n-1}\\right).",
"$ We can obtain one more equation relating $k_P$ and $\\delta $ by noting that $P$ lies on the line of slope $-1$ through the origin, and hence so must $f(P)$ .",
"This means that $f(P)$ is of the form $(x,1-x)$ .",
"Using Equation REF , this gives after some algebraic manipulations: $ k_P=n\\delta $ We can use Equation REF to eliminate $k_P$ from Equation REF to obtain: $ k_R = \\delta \\left(1+\\frac{1}{n-1}\\right).",
"$ Using that $-\\frac{1}{n}<\\delta <\\frac{n-1}{n},$ and $n>2$ one easily sees that $-1<k_R<1.$ Since $k_R$ is also an integer, this forces $k_R=0$ .",
"Combining this with Equation REF , we see that $\\delta =0$ .",
"Therefore $f(Q)=Q$ , as desired.",
"This finishes the proof that $f$ is affine on $\\mathbb {Q}^2\\slash \\mathbb {Z}^2$ .",
"In the next section, we will promote this to the entire torus."
],
[
"The 2-dimensional case: End of the proof",
"At this point we know that the map $f:T^2\\rightarrow T^2$ is a bijection sending lines to lines, with the additional properties that $f=\\text{id}$ on $\\mathbb {Q}^2\\slash \\mathbb {Z}^2$ , if $\\ell $ is a line with rational slope $\\alpha $ , then $f(\\ell )$ is also a line with slope $\\alpha $ , and if $\\ell $ has irrational slope, then so does $f(\\ell )$ .",
"In this section we will prove that $f=\\text{id}$ on all of $T^2$ .",
"We start with the following observation.",
"Write $T^2=S^1\\times S^1$ and for $x,y\\in S^1$ , write $f(x,y)=(f_1(x,y),f_2(x,y)).$ Since $f$ maps vertical lines to vertical lines, the value of $f_1(x,y)$ does not depend on $y$ .",
"Similarly, the value of $f_2(x,y)$ does not depend on $x$ .",
"Hence for $i=1,2$ , there are functions $f_i:S^1\\rightarrow S^1$ such that $f(x,y)=(f_1(x),f_2(y)).$ We actually have $f_1=f_2$ : Indeed, for any $x\\in S^1$ , consider the point $p:=(x,x)\\in T^2$ .",
"Since $p$ lies on the line of slope 1 through the origin, so does $f(p)=(f_1(x),f_2(x))$ .",
"Hence $f_1=f_2$ .",
"We will write $\\sigma :S^1\\rightarrow S^1$ for this map, so that $f(x,y)=(\\sigma (x),\\sigma (y))$ for any $x,y\\in S^1$ .",
"Let us now outline the proof that $f=\\text{id}$ .",
"We first show that $\\sigma $ is a homomorphism $S^1\\rightarrow S^1$ (Claim REF ), and then that $\\sigma $ lifts to a map $\\widetilde{\\sigma }:\\mathbb {R}\\rightarrow \\mathbb {R}$ (Claim REF ).",
"We finish the proof by showing that $\\widetilde{\\sigma }$ is a field automorphism of $\\mathbb {R}$ and hence trivial (Claims REF -REF ).",
"We start with the observation that besides collinearity of points, $f$ preserves another geometric configuration: Definition 4.1 A set of 4 points in $T^2$ is a block $B$ if the points are the vertices of a square all of whose sides are either horizontal or vertical, i.e.",
"if we can label the points $P,Q,R,S$ such that $P$ and $Q$ lie on a horizontal line $h_{x_0}$ , and $R$ and $S$ lie on a horizontal line $h_{x_1}$ , $P$ and $R$ lie on a vertical line $v_{x_0}$ , and $Q$ and $S$ lie on a vertical line $v_{x_1}$ , and $P$ and $S$ lie on a line of slope 1, and $Q$ and $R$ lie on a line of slope -1.",
"It is immediate from the fact that $f$ preserves horizontal lines, vertical lines, and lines of slope $\\pm 1$ , that $f$ preserves blocks.",
"If $B$ is a block, we will denote the block obtained by applying $f$ to the vertices of $B$ by $f(B)$ .",
"We have the following useful characterization of blocks: Lemma 4.2 Let $x_0,x_1,y_0,y_1\\in S^1$ .",
"Then the points $(x_i,y_j)$ where $i,j\\in \\lbrace 1,2\\rbrace $ form a block if and only if either $x_1-x_0=y_1-y_0$ , or $x_1-x_0=y_0-y_1$ .",
"To be done.",
"We have the following application of this characterization.",
"Claim 4.3 $\\sigma :S^1\\rightarrow S^1$ is a (group) isomorphism.",
"Since $\\sigma (0)=0$ , it suffices to show that $\\sigma (x+y)=\\sigma (x)+\\sigma (y)$ for all $x,y\\in S^1$ .",
"We will first show that this holds if $y$ is rational.",
"Consider the block $B$ consisting of the vertices $(0,y),(0,x+y), (x,y),$ and $(x,x+y)$ .",
"The two alternatives of Lemma REF applied to the block $f(B)$ yield $\\sigma (x)=\\sigma (x+y)-\\sigma (y),$ or $\\sigma (x)=\\sigma (y)-\\sigma (x+y).$ If (1) holds we are done.",
"If (2) holds we reverse the roles of $x,y$ (note that we have not used yet that $y$ is rational).",
"Again we obtain either $\\sigma (y)=\\sigma (x+y)-\\sigma (x)$ , in which case we are done, or $\\sigma (y)=\\sigma (x)-\\sigma (x+y)$ .",
"If the latter holds, we find that $2(\\sigma (x)-\\sigma (y))=0$ , so that either $\\sigma (x)=\\sigma (y)$ or $\\sigma (x)=\\sigma (y)+\\frac{1}{2}$ .",
"In either case, using that $y=\\sigma (y)$ is rational, we see that $\\sigma (x)$ is rational as well, and hence we have $x=\\sigma (x)$ .",
"This finishes the proof under the additional assumption that $y$ is rational.",
"We will now show that $\\sigma (x+y)=\\sigma (x)+\\sigma (y)$ for all $x,y$ .",
"Note that we only used that $y$ is rational in the above argument to deal with the final two cases, where either $\\sigma (x)=\\sigma (y)$ or $\\sigma (x)=\\sigma (y)+\\frac{1}{2}$ .",
"We consider these two cases separately.",
"Case 1 ($\\sigma (x)=\\sigma (y)$ ): Since $\\sigma $ is a bijection, we have $x=y$ .",
"Consider the block $B$ with vertices $(0,x),(x,x),(0,2x),(x,2x)$ .",
"The two alternatives of Lemma REF for the block $f(B)$ yield that either $\\sigma (x)=\\sigma (2x)-\\sigma (x)$ , or $\\sigma (x)=\\sigma (x)-\\sigma (2x).$ In the first case we obtain that $\\sigma (2x)=2\\sigma (x)$ , so that $\\sigma (x+y)=\\sigma (2x)=2\\sigma (x)=\\sigma (x)+\\sigma (y).$ If (2) holds, we have that $\\sigma (2x)=0$ and hence $2x=0$ , so that $x$ is rational (in which case the claim is already proven).",
"Case 2 ($\\sigma (x)=\\sigma (y)+\\frac{1}{2}$ ): In this case we had $\\sigma (x+y)=\\sigma (x)-\\sigma (y)=\\frac{1}{2}.$ Now by using that $\\sigma $ is additive if one of the variables is rational, we see that $\\sigma (x)=\\sigma (y)+\\frac{1}{2}=\\sigma \\left(y+\\frac{1}{2}\\right).$ Since $\\sigma $ is a bijection, we must have $x=y+\\frac{1}{2}$ .",
"Hence $\\sigma (x+y) &=\\sigma \\left(2y+\\frac{1}{2}\\right)=\\sigma (2y)+\\frac{1}{2}\\\\&=2\\sigma (y)+\\frac{1}{2}=\\sigma \\left(y+\\frac{1}{2}\\right)+\\sigma (y)\\\\&=\\sigma (x)+\\sigma (y),$ where on the second line we also use that $\\sigma (2z)=2\\sigma (z)$ for all $z$ , which was proven in the solution of Case 1.",
"This completes the proof.",
"For the remainder of this section, we introduce the following notation: If $\\alpha \\in \\mathbb {R}$ , let $\\ell _\\alpha $ be the line in $T^2$ of slope $\\alpha $ through 0.",
"Also recall that $h_\\alpha $ (resp.",
"$v_\\alpha $ ) denotes the horizontal line $[y=\\alpha ]$ (resp.",
"the vertical line $[x=\\alpha ]$ ).",
"We use the homomorphism property of $\\sigma $ to provide a link between the $\\sigma $ and the image under $f$ of any line: Claim 4.4 Let $x_0\\in \\mathbb {R}$ and let $y_0$ be the slope of $f(\\ell )$ .",
"Then $y_0\\equiv \\sigma (x_0)\\mathbb {}\\mod {\\mathbb {Z}}$ .",
"Consider the intersections of $\\ell $ with the line $v_0:=[x=0]$ .",
"We have $\\ell \\cap v_0=\\lbrace [0,k x_0]\\mid k\\in \\mathbb {Z}\\rbrace .$ Let $f(\\ell )$ have slope $y_0$ .",
"Then $[0,\\sigma (x_0)]\\in f(\\ell )\\cap v_0$ , so we can write $\\sigma (x_0)=ky_0 \\mathbb {}\\mod {\\mathbb {Z}}$ for some $k\\in \\mathbb {Z}$ .",
"On the other hand $[0,y_0]\\in f(\\ell )\\cap v_0=f(\\ell \\cap v_0)$ , so there exists $l\\in \\mathbb {Z}$ such that $y_0\\equiv l\\sigma (x_0)$ mod $\\mathbb {Z}$ .",
"Hence we have $y_0\\equiv l\\sigma (x_0)\\equiv lk y_0 \\mathbb {}\\mod {\\mathbb {Z}}.$ Since images under $f$ of lines with irrational slope are irrational, and $x_0$ is irrational, we must have that $y_0$ is irrational.",
"Therefore we have $lk=1$ , so that $l=\\pm 1$ .",
"It remains to show that $k=1$ .",
"Suppose not, so that we have $y_0\\equiv -\\sigma (x_0)\\mathbb {}\\mod {\\mathbb {Z}}$ .",
"Write $y_0=-\\sigma (x_0)+k_0$ for some $k_0\\in \\mathbb {Z}$ .",
"Consider the intersections of $\\ell $ with the line $v_{\\frac{1}{3}}$ .",
"These occur at the points $[\\frac{1}{3},\\frac{1}{3}x_0+l x_0]$ for $l\\in \\mathbb {Z}$ .",
"Now application of $f$ maps these points to intersections of $f(\\ell )$ and $v_{\\frac{1}{3}}$ , which are given by $[\\frac{1}{3},\\frac{1}{3}y_0+l y_0]$ for $l\\in \\mathbb {Z}$ .",
"Consider the image of the point $[\\frac{1}{3},\\frac{1}{3}x_0]\\in \\ell \\cap v_{\\frac{1}{3}}$ .",
"Then we can write $\\sigma \\left(\\frac{1}{3}x_0\\right)\\equiv \\frac{1}{3}y_0+k_3 y_0\\mathbb {}\\mod {\\mathbb {Z}}$ for some $k_3\\in \\mathbb {Z}$ .",
"Now using that $y_0=-\\sigma (x_0)+k_0$ , we have $\\sigma \\left(\\frac{1}{3}x_0\\right)\\equiv -\\frac{1}{3}\\sigma (x_0)+\\frac{k_0}{3}+k_3\\sigma (x_0) \\mathbb {}\\mod {\\mathbb {Z}}.$ Hence $\\sigma (x_0)\\equiv 3\\sigma \\left(\\frac{1}{3}x_0\\right)\\equiv (3k_3-1)\\sigma (x_0)\\mathbb {}\\mod {\\mathbb {Z}}.$ Since $\\sigma (x_0)\\notin \\mathbb {Q}$ , we must have $3k_3-1=1$ , which is a contradiction.",
"Consider now the map $\\widetilde{\\sigma }:\\mathbb {R}\\rightarrow \\mathbb {R}$ defined by $\\widetilde{\\sigma }(x_0):=y_0$ , where the image under $f$ of a line with slope $x_0$ has slope $y_0$ .",
"The claim above establishes that $\\widetilde{\\sigma }$ is a lift of the map $\\sigma :S^1\\rightarrow S^1$ .",
"In the claims below we will establish that $\\widetilde{\\sigma }$ is a field automorphism of $\\mathbb {R}$ and hence $\\widetilde{\\sigma }=\\text{id}$ .",
"Claim 4.5 For any integer $a$ and $x\\in \\mathbb {R}$ , we have $\\widetilde{\\sigma }(ax)=a\\widetilde{\\sigma }(x)$ .",
"Let $a\\in \\mathbb {Z}$ and $x_0\\in \\mathbb {R}$ .",
"If $x_0$ is rational then the claim is already proven, so we will assume that $x_0$ is irrational.",
"Since we have $\\widetilde{\\sigma }(ax_0)\\equiv \\sigma (ax_0)\\equiv a\\sigma (x_0)\\equiv a\\widetilde{\\sigma }(x_0) \\mathbb {}\\mod {\\mathbb {Z}},$ we can choose $k\\in \\mathbb {Z}$ such that $\\widetilde{\\sigma }(ax_0)=a\\widetilde{\\sigma }(x_0)+k$ .",
"Choose $p>1$ prime and let $\\ell _{ax_0}$ be the line in $T^2$ through 0 with slope $ax_0$ .",
"The intersections of $\\ell _{ax_0}$ and $v_{\\frac{1}{p}}$ occur at heights $\\frac{a}{p}x_0+a\\mathbb {Z}x_0$ mod $\\mathbb {Z}$ .",
"Under $f$ these are bijectively mapped to the intersections of $v_{\\frac{1}{p}}$ and $\\ell _{\\widetilde{\\sigma }(ax_0)}$ , which occur at heights $\\frac{1}{p}\\widetilde{\\sigma }(ax_0)+\\mathbb {Z}\\widetilde{\\sigma }(ax_0)$ .",
"In particular there exists $l\\in \\mathbb {Z}$ such that $ \\sigma \\left(\\frac{a}{p}x_0+lax_0\\right)\\equiv \\frac{1}{p}\\widetilde{\\sigma }(ax_0)\\mathbb {}\\mod {\\mathbb {Z}}.",
"$ For the left-hand side we have $\\sigma \\left(\\frac{a}{p}x_0+lax_0\\right)\\equiv a \\sigma \\left(\\frac{1}{p}x_0+lx_0\\right)\\mathbb {}\\mod {\\mathbb {Z}}.$ Note that $[\\frac{1}{p},\\frac{1}{p}x_0+lx_0]$ is an intersection point of $\\ell _{x_0}$ and $v_{\\frac{1}{p}}$ so that $[\\frac{1}{p},\\sigma (\\frac{1}{p}x_0+lx_0)]$ is an intersection point of $\\ell _{\\widetilde{\\sigma }(x_0)}$ and $v_{\\frac{1}{p}}$ .",
"Hence there exists $n\\in \\mathbb {Z}$ such that $\\sigma \\left(\\frac{1}{p}x_0+lx_0\\right)\\equiv \\frac{1}{p}\\widetilde{\\sigma }(x_0)+n\\widetilde{\\sigma }(x_0)\\mathbb {}\\mod {\\mathbb {Z}}.$ Combining Equations REF , REF and REF , we see that $\\frac{a}{p}\\widetilde{\\sigma }(x_0)+an\\widetilde{\\sigma }(x_0)\\equiv \\frac{1}{p}\\widetilde{\\sigma }(ax_0) \\mathbb {}\\mod {\\mathbb {Z}}.$ Further using that $\\widetilde{\\sigma }(ax_0)=a\\widetilde{\\sigma }(x_0)+k$ , we have $\\frac{a}{p}\\widetilde{\\sigma }(x_0)+an\\widetilde{\\sigma }(x_0)\\equiv \\frac{a}{p}\\widetilde{\\sigma }(x_0)+\\frac{k}{p} \\mathbb {}\\mod {\\mathbb {Z}},$ so that $an\\widetilde{\\sigma }(x_0)\\equiv \\frac{k}{p}\\mathbb {}\\mod {\\mathbb {Z}}.$ Because $\\widetilde{\\sigma }(x_0)$ is irrational, this is impossible unless $an=0$ and $\\frac{k}{p}\\in \\mathbb {Z}$ .",
"Since $p$ was an arbitrary prime number, we must have $k=0$ .",
"Claim 4.6 $\\widetilde{\\sigma }$ is a homomorphism of $\\mathbb {R}$ (as an additive group).",
"The previous claim with $a=-1$ shows that $\\widetilde{\\sigma }(-x)=-\\widetilde{\\sigma }(x)$ .",
"Therefore it remains to show that $\\widetilde{\\sigma }$ is additive.",
"Let $x,y\\in \\mathbb {R}$ be arbitrary.",
"Since $\\widetilde{\\sigma }$ is a lift of $\\sigma $ , we have for any $n\\ge 1$ : $\\widetilde{\\sigma }\\left(\\frac{x+y}{n}\\right)&\\equiv \\sigma \\left(\\frac{x}{n}+\\frac{y}{n}\\right)\\\\&\\equiv \\sigma \\left(\\frac{x}{n}\\right)+\\sigma \\left(\\frac{y}{n}\\right)\\\\&\\equiv \\widetilde{\\sigma }\\left(\\frac{x}{n}\\right)+\\widetilde{\\sigma }\\left(\\frac{y}{n}\\right)\\mathbb {}\\mod {\\mathbb {Z}}.$ Multiplying by $n$ and using Claim REF with $a=n$ , we have $\\widetilde{\\sigma }(x+y)\\equiv \\widetilde{\\sigma }(x)+\\widetilde{\\sigma }(y)\\mod {n}\\unknown.",
"\\mathbb {Z}.$ Since $n$ was arbitrary, we must have $\\widetilde{\\sigma }(x+y)=\\widetilde{\\sigma }(x)+\\widetilde{\\sigma }(y)$ as desired.",
"Claim 4.7 For any $x\\ne 0$ , we have $\\widetilde{\\sigma }(\\frac{1}{x})=\\frac{1}{\\widetilde{\\sigma }(x)}$ .",
"We will first show that $\\widetilde{\\sigma }(\\frac{1}{x})\\equiv \\frac{1}{\\widetilde{\\sigma }(x)}\\mathbb {}\\mod {\\mathbb {Z}}$ for $x\\ne 0$ .",
"Let $x_0\\in \\mathbb {R}^\\times $ and let $\\ell _{x_0}$ again be the line with slope $x_0$ through 0.",
"We can assume that $x_0$ is irrational.",
"Note that the intersections of $\\ell _{x_0}$ with the horizontal line $h_0$ occur at the points $[\\frac{k}{x_0}, 0]$ .",
"Under $f$ these are bijectively mapped to the intersections of $\\ell _{\\widetilde{\\sigma }(x_0)}$ with $h_0$ .",
"In particular there are integers $k,l\\in \\mathbb {Z}$ such that $\\sigma \\left(\\frac{1}{x_0}\\right)\\equiv \\frac{k}{\\widetilde{\\sigma }(x_0)}\\mathbb {}\\mod {\\mathbb {Z}}$ and $\\sigma \\left(\\frac{l}{x_0}\\right)\\equiv \\frac{1}{\\widetilde{\\sigma }(x_0)}\\mathbb {}\\mod {\\mathbb {Z}}.$ Hence $\\frac{1}{\\widetilde{\\sigma }(x_0)}\\equiv \\sigma \\left(\\frac{l}{x_0}\\right)\\equiv l\\sigma \\left(\\frac{1}{x_0}\\right)\\equiv kl\\frac{1}{\\widetilde{\\sigma }(x_0)}\\mathbb {}\\mod {\\mathbb {Z}}.$ Since $\\widetilde{\\sigma }(x_0)$ is irrational, we must have $kl=1$ so that $k=\\pm 1$ .",
"We claim that $k=1$ .",
"To see this, consider the intersections of $\\ell _{x_0}$ with the horizontal line $h_{\\frac{1}{3}}$ .",
"These occur at $[\\frac{1}{3x_0}+\\frac{n}{x_0},\\frac{1}{3}]$ for $n\\in \\mathbb {Z}$ .",
"Under $f$ these are mapped to the intersections of $\\ell _{\\widetilde{\\sigma }(x_0)}$ with $h_{\\frac{1}{3}}$ .",
"Hence there exists $n\\in \\mathbb {Z}$ such that $ \\sigma \\left(\\frac{1}{3x_0}\\right)\\equiv \\frac{1}{3\\widetilde{\\sigma }(x_0)}+\\frac{n}{\\widetilde{\\sigma }(x_0)} \\mathbb {}\\mod {\\mathbb {Z}}.",
"$ For the left-hand side, we have $ \\sigma \\left(\\frac{1}{3x_0}\\right)\\equiv \\widetilde{\\sigma }\\left(\\frac{1}{3x_0}\\right)\\equiv \\frac{1}{3}\\widetilde{\\sigma }\\left(\\frac{1}{x_0}\\right)\\equiv \\frac{k}{3}\\frac{1}{\\widetilde{\\sigma }(x_0)} \\mathbb {}\\mod {\\mathbb {Z}},$ where we used that $\\widetilde{\\sigma }$ is $\\mathbb {Q}$ -linear (because it is a homomorphism $\\mathbb {R}\\rightarrow \\mathbb {R}$ ).",
"Combining Equations REF and REF , we find $(3n+1-k)\\frac{1}{\\widetilde{\\sigma }(x_0)}\\equiv 0\\mod {3}\\unknown.",
"\\mathbb {Z}.$ Since $\\widetilde{\\sigma }(x_0)$ is not rational, we must have $3n+1-k=0$ so that $k\\equiv 1\\mod {3}$ .",
"Since we already found that $k=1$ or $-1$ , we must have $k=1$ , as desired.",
"At this point we know that $\\widetilde{\\sigma }(\\frac{1}{x})\\equiv \\frac{1}{\\widetilde{\\sigma }(x)}\\mathbb {}\\mod {\\mathbb {Z}}$ for any $x\\ne 0$ .",
"It remains to show that $\\widetilde{\\sigma }(\\frac{1}{x})=\\frac{1}{\\widetilde{\\sigma }(x)}$ .",
"To see this, let $x\\in \\mathbb {R}^\\times $ be arbitrary.",
"We can assume that $x$ is irrational.",
"Let $n\\in \\mathbb {Z}$ such that $\\widetilde{\\sigma }\\left(\\frac{1}{x}\\right)=\\frac{1}{\\widetilde{\\sigma }(x)}+n.$ Set $N=|n|+1$ , and note that $\\widetilde{\\sigma }\\left(\\frac{1}{Nx}\\right)=\\frac{1}{N}\\widetilde{\\sigma }\\left(\\frac{1}{x}\\right)=\\frac{1}{\\widetilde{\\sigma }(NX)}+\\frac{n}{N}.$ Hence if $n\\ne 0$ , we have $\\widetilde{\\sigma }\\left(\\frac{1}{Nx}\\right)\\lnot \\equiv \\frac{1}{\\widetilde{\\sigma }(Nx)}\\mathbb {}\\mod {\\mathbb {Z}}$ , which is a contradiction.",
"In the final two claim we will show that $\\widetilde{\\sigma }$ is multiplicative.",
"Claim 4.8 For any $x,y\\in \\mathbb {R}$ , we have $\\widetilde{\\sigma }(xy)=\\widetilde{\\sigma }(x)\\widetilde{\\sigma }(y)$ .",
"Let $x_0,y_0\\in \\mathbb {R}$ .",
"Note that since $\\widetilde{\\sigma }$ is an isomorphism of $\\mathbb {R}$ as an additive group, $\\widetilde{\\sigma }$ is $\\mathbb {Q}$ -linear.",
"Hence without loss of generality, we assume that $x_0$ is irrational.",
"Consider the intersections of $\\ell _{x_0}$ with the vertical line $v_{y_0}$ .",
"These occur at the points $[y_0,x_0 y_0+k x_0]$ for $k\\in \\mathbb {Z}$ .",
"Under $f$ these intersection points are mapped to the points of intersection of $\\ell _{\\widetilde{\\sigma }(x_0)}$ with $v_{\\widetilde{\\sigma }(y_0)}$ .",
"In particular, by considering the image of $[y_0,x_0 y_0]$ , we see that there is $k\\in \\mathbb {Z}$ such that $\\sigma (x_0 y_0)\\equiv \\widetilde{\\sigma }(x_0)\\widetilde{\\sigma }(y_0)+k\\widetilde{\\sigma }(x_0)\\mathbb {}\\mod {\\mathbb {Z}}.$ Since $\\sigma (x_0 y_0)\\equiv \\widetilde{\\sigma }(x_0 y_0)\\mathbb {}\\mod {\\mathbb {Z}}$ , we see that $\\widetilde{\\sigma }(x_0 y_0)-\\widetilde{\\sigma }(x_0)\\widetilde{\\sigma }(y_0)\\in \\mathbb {Z}+\\mathbb {Z}\\widetilde{\\sigma }(x_0).$ For $x\\in \\mathbb {R}$ , let $\\mu _x:\\mathbb {R}\\rightarrow \\mathbb {R}$ denote multiplication by $x$ .",
"Then the above argument shows that the homomorphism $(\\widetilde{\\sigma }\\circ \\mu _{x_0})-\\left(\\mu _{\\widetilde{\\sigma }(x_0)}\\circ \\widetilde{\\sigma }\\right):\\mathbb {R}\\rightarrow \\mathbb {R}$ has image contained in $\\mathbb {Z}+\\mathbb {Z}\\widetilde{\\sigma }(x_0)$ .",
"Since $\\widetilde{\\sigma }(x_0)$ is irrational, we have $\\mathbb {Z}+\\mathbb {Z}\\widetilde{\\sigma }(x_0)\\cong \\mathbb {Z}^2$ as additive groups.",
"But any homomorphism $\\mathbb {R}\\rightarrow \\mathbb {Z}^2$ is trivial (because $\\mathbb {R}$ is divisible), so that $\\widetilde{\\sigma }\\circ \\mu _{x_0}=\\mu _{\\widetilde{\\sigma }(x_0)}\\circ \\widetilde{\\sigma }$ , as desired.",
"As commented at the beginning of this section, the above results finish the 2-dimensional case of Theorem REF .",
"Indeed, up to precomposition by an affine automorphism, any bijection $f:T^2\\rightarrow T^2$ that maps lines to lines, is of the form $(\\sigma ,\\sigma )$ , where $\\sigma $ is a homomorphism $S^1\\rightarrow S^1$ that lifts to a field automorphism $\\widetilde{\\sigma }$ of $\\mathbb {R}$ .",
"Since any field automorphism of $\\mathbb {R}$ is trivial, it follows that $\\sigma =\\text{id}$ and hence the original map is affine.",
"The $n$ -dimensional case We finish the proof of Theorem REF that any bijection of $T=\\mathbb {R}^n\\slash \\mathbb {Z}^n, \\, n\\ge 2,$ that maps lines to lines, is an affine map.",
"We argue by induction on $n$ .",
"The base case $n=2$ has been proven in the previous section.",
"Let $f:T\\rightarrow T$ be a bijection that maps lines to lines.",
"We recall that in Section , we showed that for any rational subtorus $S\\subseteq T$ , the image $f(S)$ is also a rational subtorus.",
"Without loss of generality, we assume $f(0)=0$ .",
"For $1\\le i\\le n$ , we let $\\ell _i:=\\begin{bmatrix} \\mathbb {R}e_i \\end{bmatrix}$ denote the (image of the) coordinate line.",
"Each $\\ell _i$ is a rational line, so $f(\\ell _i)$ is also a rational line.",
"Choose primitive integral vectors $v_i\\in \\mathbb {Z}^n$ such that $f(\\ell _i)=\\begin{bmatrix} \\mathbb {R}v_i\\end{bmatrix}.$ Note that $v_i$ is unique up to sign.",
"Let $A$ be the linear map of $\\mathbb {R}^n$ with $A e_i = v_i$ for every $i$ .",
"Our goal is to show that $f=A$ as maps of the torus.",
"Claim 5.1 $A$ is invertible with integer inverse.",
"To show that $A$ is invertible, we need to show that $v_1,\\dots , v_n$ span $\\mathbb {R}^n$ .",
"To show that $A^{-1}$ has integer entries, we need to show that in addition, $v_1,\\dots ,v_n$ generate $\\mathbb {Z}^n$ (as a group).",
"For $1\\le j\\le n$ , set $U_j:=\\operatorname{span}\\lbrace e_i \\mid i\\le j\\rbrace ,$ and $V_j:=\\operatorname{span}\\lbrace v_i \\mid i \\le j\\rbrace .$ Note that $f [U_j]=[V_j]$ by Lemma REF .",
"Taking $j=n$ , this already shows that $v_1,\\dots ,v_n$ span $\\mathbb {R}^n$ , so $A$ is invertible.",
"We argue by induction on $j$ that $\\lbrace v_1,\\dots ,v_j\\rbrace $ generate $\\pi _1[V_j]\\subseteq \\pi _1 T$ .",
"For $j=n$ , this exactly means that $v_1,\\dots ,v_n$ generate $\\pi _1 T$ , which would finish the proof.",
"The base case $j=1$ is exactly the assertion that $v_1$ is a primitive vector.",
"Now suppose the statement is true for some $j$ .",
"Consider the composition $ f(\\ell _j) \\hookrightarrow V_j \\rightarrow V_j\\slash V_{j-1}.",
"$ This composition is a homomorphism of $f(\\ell _j)\\cong S^1$ to $V_j\\slash V_{j-1}\\cong S^1$ .",
"The kernel is given by $f(\\ell _j)\\cap V_{j-1}= f(\\ell _j \\cap U_{j-1})=f(0)=0.$ Therefore the map $f(\\ell _j)\\rightarrow V_j\\slash V_{j-1}$ is an isomorphism, and $\\pi _1(V_j)\\cong \\pi _1(V_{j-1})\\oplus \\pi _1(f(\\ell _j)).$ Since we know by the inductive hypothesis that $\\pi _1(V_{j-1})$ is generated by $v_1,\\dots ,v_{j-1}$ , and that $\\pi _1(f(\\ell _j))$ is generated by $v_j$ (again because $v_j$ is primitive), we find that $\\pi _1(V_j)$ is generated by $v_1,\\dots ,v_j$ , as desired.",
"For the remainder of the proof we set $g:=A^{-1}\\circ f$ .",
"Our goal is to show that $g=\\text{id}$ .",
"Note that $g$ is a bijection of $T$ with $g(0)=0$ and $g(\\ell _i)=\\ell _i$ for every $i$ .",
"Let $H_{i}:=\\begin{bmatrix} \\operatorname{span}(e_j \\mid j\\ne i)\\end{bmatrix}$ denote the (image of the) coordinate hyperplane.",
"Since $g(\\ell _i)=\\ell _i$ for every $i$ , we know (using Lemma REF again) that $g(H_i)=H_i$ .",
"By the inductive hypothesis, $g$ is given by a linear map $A_i$ on $H_i$ .",
"But any linear automorphism of $H_i$ that leaves the coordinate lines $\\ell _j, \\, j\\ne i,$ invariant, must be of the form $\\begin{pmatrix} \\pm 1 & & \\\\ & \\ddots & \\\\ & & \\pm 1 \\end{pmatrix}.$ Now recall that $v_i$ was unique up to sign.",
"Replace $v_i$ by $-v_i$ whenever $g|_{\\ell _i} = -\\text{id}$ .",
"After this modification, we have that $g=\\text{id}$ on every coordinate hyperplane $H_i$ .",
"Our goal is to show that $g=\\text{id}$ on $T$ .",
"Claim 5.2 Fix $1\\le i\\le n$ and let $\\ell $ be any line parallel to $\\ell _i$ .",
"Then $g(\\ell )=\\ell $ .",
"First recall the following elementary general fact about the number of intersections of a line with a coordinate hyperplane: If $m$ is any rational line and $v=\\begin{pmatrix} v_1 \\\\ \\vdots \\\\ v_n\\end{pmatrix}$ is a primitive integer vector with $m=[\\mathbb {R}v]$ , then $|v\\cap H_i|=|v_i|.$ Now let $\\ell $ be parallel to $\\ell _i$ .",
"Then $|\\ell \\cap H_j|=\\delta _{ij},$ and hence also $|g(\\ell )\\cap H_j|=|g(\\ell )\\cap g(H_j)|=|\\ell \\cap H_j|=\\delta _{ij}.$ Therefore we see that if $v$ is a primitive integer vector with $g(\\ell )=[\\mathbb {R}v]$ , then $v=e_i$ .",
"This exactly means that $\\ell $ and $g(\\ell )$ are parallel.",
"Hence to show that $\\ell =g(\\ell )$ , it suffices to show that $\\ell \\cap g(\\ell )$ is nonempty.",
"Since $\\ell $ is parallel to $\\ell _i$ , there is a unique point of intersection $x_\\ell :=\\ell \\cap H_i$ .",
"Since $x_\\ell \\in H_i$ , we have $g(x_\\ell )=x_\\ell $ , so $x_\\ell \\in \\ell \\cap g(\\ell )$ , as desired.",
"We are now able to finish the proof of the main theorem.",
"Let $x\\in T$ and let $\\ell _i(x)$ be the unique line parallel to $\\ell _i$ that passes through $x$ .",
"For any two distinct indices $i\\ne j$ , the point $x$ is the unique point of intersection of $\\ell _i(x)$ and $\\ell _j(x)$ .",
"On the other hand, $g(x)\\in g(\\ell _i(x))\\cap g(\\ell _j(x))=\\ell _i(x)\\cap \\ell _j(x),$ so we must have $g(x)=x$ ."
],
[
"The $n$ -dimensional case",
"We finish the proof of Theorem REF that any bijection of $T=\\mathbb {R}^n\\slash \\mathbb {Z}^n, \\, n\\ge 2,$ that maps lines to lines, is an affine map.",
"We argue by induction on $n$ .",
"The base case $n=2$ has been proven in the previous section.",
"Let $f:T\\rightarrow T$ be a bijection that maps lines to lines.",
"We recall that in Section , we showed that for any rational subtorus $S\\subseteq T$ , the image $f(S)$ is also a rational subtorus.",
"Without loss of generality, we assume $f(0)=0$ .",
"For $1\\le i\\le n$ , we let $\\ell _i:=\\begin{bmatrix} \\mathbb {R}e_i \\end{bmatrix}$ denote the (image of the) coordinate line.",
"Each $\\ell _i$ is a rational line, so $f(\\ell _i)$ is also a rational line.",
"Choose primitive integral vectors $v_i\\in \\mathbb {Z}^n$ such that $f(\\ell _i)=\\begin{bmatrix} \\mathbb {R}v_i\\end{bmatrix}.$ Note that $v_i$ is unique up to sign.",
"Let $A$ be the linear map of $\\mathbb {R}^n$ with $A e_i = v_i$ for every $i$ .",
"Our goal is to show that $f=A$ as maps of the torus.",
"Claim 5.1 $A$ is invertible with integer inverse.",
"To show that $A$ is invertible, we need to show that $v_1,\\dots , v_n$ span $\\mathbb {R}^n$ .",
"To show that $A^{-1}$ has integer entries, we need to show that in addition, $v_1,\\dots ,v_n$ generate $\\mathbb {Z}^n$ (as a group).",
"For $1\\le j\\le n$ , set $U_j:=\\operatorname{span}\\lbrace e_i \\mid i\\le j\\rbrace ,$ and $V_j:=\\operatorname{span}\\lbrace v_i \\mid i \\le j\\rbrace .$ Note that $f [U_j]=[V_j]$ by Lemma REF .",
"Taking $j=n$ , this already shows that $v_1,\\dots ,v_n$ span $\\mathbb {R}^n$ , so $A$ is invertible.",
"We argue by induction on $j$ that $\\lbrace v_1,\\dots ,v_j\\rbrace $ generate $\\pi _1[V_j]\\subseteq \\pi _1 T$ .",
"For $j=n$ , this exactly means that $v_1,\\dots ,v_n$ generate $\\pi _1 T$ , which would finish the proof.",
"The base case $j=1$ is exactly the assertion that $v_1$ is a primitive vector.",
"Now suppose the statement is true for some $j$ .",
"Consider the composition $ f(\\ell _j) \\hookrightarrow V_j \\rightarrow V_j\\slash V_{j-1}.",
"$ This composition is a homomorphism of $f(\\ell _j)\\cong S^1$ to $V_j\\slash V_{j-1}\\cong S^1$ .",
"The kernel is given by $f(\\ell _j)\\cap V_{j-1}= f(\\ell _j \\cap U_{j-1})=f(0)=0.$ Therefore the map $f(\\ell _j)\\rightarrow V_j\\slash V_{j-1}$ is an isomorphism, and $\\pi _1(V_j)\\cong \\pi _1(V_{j-1})\\oplus \\pi _1(f(\\ell _j)).$ Since we know by the inductive hypothesis that $\\pi _1(V_{j-1})$ is generated by $v_1,\\dots ,v_{j-1}$ , and that $\\pi _1(f(\\ell _j))$ is generated by $v_j$ (again because $v_j$ is primitive), we find that $\\pi _1(V_j)$ is generated by $v_1,\\dots ,v_j$ , as desired.",
"For the remainder of the proof we set $g:=A^{-1}\\circ f$ .",
"Our goal is to show that $g=\\text{id}$ .",
"Note that $g$ is a bijection of $T$ with $g(0)=0$ and $g(\\ell _i)=\\ell _i$ for every $i$ .",
"Let $H_{i}:=\\begin{bmatrix} \\operatorname{span}(e_j \\mid j\\ne i)\\end{bmatrix}$ denote the (image of the) coordinate hyperplane.",
"Since $g(\\ell _i)=\\ell _i$ for every $i$ , we know (using Lemma REF again) that $g(H_i)=H_i$ .",
"By the inductive hypothesis, $g$ is given by a linear map $A_i$ on $H_i$ .",
"But any linear automorphism of $H_i$ that leaves the coordinate lines $\\ell _j, \\, j\\ne i,$ invariant, must be of the form $\\begin{pmatrix} \\pm 1 & & \\\\ & \\ddots & \\\\ & & \\pm 1 \\end{pmatrix}.$ Now recall that $v_i$ was unique up to sign.",
"Replace $v_i$ by $-v_i$ whenever $g|_{\\ell _i} = -\\text{id}$ .",
"After this modification, we have that $g=\\text{id}$ on every coordinate hyperplane $H_i$ .",
"Our goal is to show that $g=\\text{id}$ on $T$ .",
"Claim 5.2 Fix $1\\le i\\le n$ and let $\\ell $ be any line parallel to $\\ell _i$ .",
"Then $g(\\ell )=\\ell $ .",
"First recall the following elementary general fact about the number of intersections of a line with a coordinate hyperplane: If $m$ is any rational line and $v=\\begin{pmatrix} v_1 \\\\ \\vdots \\\\ v_n\\end{pmatrix}$ is a primitive integer vector with $m=[\\mathbb {R}v]$ , then $|v\\cap H_i|=|v_i|.$ Now let $\\ell $ be parallel to $\\ell _i$ .",
"Then $|\\ell \\cap H_j|=\\delta _{ij},$ and hence also $|g(\\ell )\\cap H_j|=|g(\\ell )\\cap g(H_j)|=|\\ell \\cap H_j|=\\delta _{ij}.$ Therefore we see that if $v$ is a primitive integer vector with $g(\\ell )=[\\mathbb {R}v]$ , then $v=e_i$ .",
"This exactly means that $\\ell $ and $g(\\ell )$ are parallel.",
"Hence to show that $\\ell =g(\\ell )$ , it suffices to show that $\\ell \\cap g(\\ell )$ is nonempty.",
"Since $\\ell $ is parallel to $\\ell _i$ , there is a unique point of intersection $x_\\ell :=\\ell \\cap H_i$ .",
"Since $x_\\ell \\in H_i$ , we have $g(x_\\ell )=x_\\ell $ , so $x_\\ell \\in \\ell \\cap g(\\ell )$ , as desired.",
"We are now able to finish the proof of the main theorem.",
"Let $x\\in T$ and let $\\ell _i(x)$ be the unique line parallel to $\\ell _i$ that passes through $x$ .",
"For any two distinct indices $i\\ne j$ , the point $x$ is the unique point of intersection of $\\ell _i(x)$ and $\\ell _j(x)$ .",
"On the other hand, $g(x)\\in g(\\ell _i(x))\\cap g(\\ell _j(x))=\\ell _i(x)\\cap \\ell _j(x),$ so we must have $g(x)=x$ ."
]
] | 1612.05819 |
[
[
"Nonlinear adiabatic optical isolator"
],
[
"Abstract We theoretically propose a method for optical isolation based on adiabatic nonlinear sum frequency generation in a chirped quasi-phase-matching crystal with strong absorption at the generated sum frequency wave.",
"The method does not suffer from limitations of dynamic reciprocity found in other nonlinear optical isolation methods, and can provide tunable optical isolation with ultrafast all-optical switching capability.",
"Moreover, as an adiabatic technique it is robust to variations in the optical design and is relatively broadband."
],
[
"[ $^1$ Department of Physics, Sofia University, James Bourchier 5 blvd., 1164 Sofia, Bulgaria $^2$ Dipartimento di Fisica, Politecnico di Milano and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Piazza L. da Vinci 32, I-20133 Milano, Italy $^*$ Corresponding author: [email protected] We theoretically propose a method for optical isolation based on adiabatic nonlinear sum frequency generation in a chirped quasi-phase-matching crystal with strong absorption at the generated sum frequency wave.",
"The method does not suffer from limitations of dynamic reciprocity found in other nonlinear optical isolation methods, and can provide tunable optical isolation with ultrafast all-optical switching capability.",
"Moreover, as an adiabatic technique it is robust to variations in the optical design and is relatively broadband.",
"190.0190, 190.4223, 190.2620, 260.1180. ]",
"An optical isolator (optical diode) is the optical correspondent of electronic diode, allowing unidirectional non-reciprocal light transmission.",
"These devices are widely used in optical telecommunications and laser applications to prevent the unwanted feedback that might be harmful to optical instruments and devices.",
"Moreover, the use of an isolator generally improves the performance of an optical circuit as it suppresses spurious interferences, interactions between different devices and undesired light routing [1].",
"Currently, optical diodes rely almost exclusively on the Faraday effect where external magnetic fields are used to break time reversal symmetry [2].",
"However, optical isolators based on the Faraday effect are typically large-size devices, they cannot be implemented easily in on-chip integrated systems [3], and do not provide dynamic optical isolation with all-optical switching capability, which is desirable in advanced optical signal processing.",
"Several recent works have suggested and experimentally demonstrated new ways to realize optical isolators that do not rely on magneto-optical effects [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].",
"Dynamic modulation methods, which enable tunable optical isolation with all-optical switching capabilities, have been recently proposed and experimentally demonstrated using liquid-crystal heterojunctions [7], phase modulators in InP and silicon photonics [9], [10], opto-acoustic photonic crystal fibers [11], and traveling-wave Mach-Zehnder modulators [12].",
"A broad class of optical isolators is the one based on nonlinear interaction of light in a nonlinear medium, which breaks Lorentz reciprocity.",
"Nonlinear optical isolators have raised great attention since more than two decades [4], [5], [6], [13], [14], [15], [16], [17], [18], [19], mainly because of their all-optical switching capability exploiting the ultrafast response of the nonlinear medium and for on-chip integration possibilities.",
"However, their effectiveness in optical isolation has been recently questioned owing of the appearance of so-called dynamical reciprocity [20].",
"In a nonlinear optical isolator, non-reciprocal transmission contrast is observed when strong waves are injected in forward or backward directions, however optical isolation is constrained by a reciprocity relation for a class of small-amplitude additional waves that can be transmitted.",
"In this Letter we suggest a different route toward nonlinear optical isolation, which does not suffer from dynamical reciprocity limitations.",
"The method is based on three wave mixing process in a second-order nonlinear medium with a strong control pump wave at frequency $\\omega _1$ , a weak signal wave at frequency $\\omega _2$ , and large absorption at the generated frequency $\\omega _3= \\omega _1 \\pm \\omega _2$ .",
"The undepleted pump approximation linearizes the three wave mixing process and thus our nonlinear optical diode does not suffer from dynamical reciprocity.",
"Furthermore since the nonlinear response of the medium is very fast (instantaneous), switching on and off the pump wave results in on/off diode action, allowing for ultrafast all-optical switching isolation capability.",
"Finally to make the optical diode robust and relatively broadband we use adiabatic frequency conversion in aperiodically-poled quasi-phase-matched (QPM) crystals, a technique recently demonstrated by Suchowski et al.",
"[21], [22], [23], [24], [25].",
"The feasibility of the method is illustrated for sum frequency generation in potassium titanyl phosphate (KTP) crystals, showing the possibility of achieving an optical isolation up to $\\sim 40$ dB over more than 50 nm in the near-infrared.",
"Figure: (Color online) Principle of a nonlinear adiabatic opticaldiode.",
"Adiabatic SFG schemes with no absorption at the generatedfrequency (a) and with absorption at the generated frequency (b) and(c).",
"(a) Slowly changing the poling period along the crystal lengthensures broadband and robust generation of the SFG wave.",
"(b) For aforward propagating signal wave, the broadband SFG wave is absorbedin the crystal, and no transmission occurs.",
"(c) Reversing thepropagation direction of the signal breaks the symmetry and as aresult there is no interaction of the contra propagating pump andsignal waves.",
"Therefore (b) together with (c) work as a broadbandoptical diode.The starting point of our analysis is provided by a standard model of sum frequency generation (SFG) or frequency difference generation (DFG) in a nonlinear second-order crystal with a chirped QPM grating, in which the SFG (or DFG) wave experiences strong linear absorption during propagation along the crystal.",
"For the sake of definiteness, we will consider here the SFG case, however a similar analysis holds for the DFG scheme.",
"In the undepleted pump approximation, SFG is described by the linear coupled equations [26], [27] $i\\frac{d}{dx}\\tilde{A}_{2} &=&\\frac{1}{2}\\Omega _{2}\\tilde{A}_{3}\\exp \\left(-i\\int _{0}^{x}\\Delta k(\\xi )d\\xi \\right) , \\\\i\\frac{d}{dx}\\tilde{A}_{3} &=&\\frac{1}{2}\\Omega _{3}\\tilde{A}_{2}\\exp \\left( i\\int _{0}^{x}\\Delta k(\\xi )d\\xi \\right)-i\\frac{\\Gamma }{2}\\tilde{A}_{3} \\;\\;\\;$ where $\\tilde{A}_{2}$ , $\\tilde{A}_{3}$ are the slowly-varying amplitudes of signal and SFG waves at frequencies $\\omega _{2}$ and $\\omega _{3}=\\omega _{1}+\\omega _{2}$ , respectively, $\\omega _{1}$ is the frequency of the pump beam with undepleted amplitude $\\tilde{A}_{1}$ , $\\Delta k(x)$ is the effective residual local phase mismatch that accounts for the chirped QPM grating, ${\\Omega }_{m}=\\chi ^{\\left( 2\\right) }\\omega _{m}\\tilde{A}_{1}/(2cn_{\\omega _{m}})$ are the coupling coefficients, $n_{\\omega _{l}}$ ($l=1,2,3$ ) is the refractive indices of the crystal at frequency $\\omega _{l}$ ($l=1,2,3$ ), $\\chi ^{\\left( 2\\right) }$ is the effective second order susceptibility of the crystal, $x$ is the position along the propagation axis, $c$ is the speed of light in vacuum, and $\\Gamma $ is the absorption coefficient of the SFG wave.",
"After the substitution $\\tilde{A}_{2}=A_{2}\\sqrt{\\Omega _{2}}\\exp \\left[ -i\\int _{0}^{x}d\\xi \\Delta k(\\xi )/2\\right] $ , $\\tilde{A}_{3}=A_{3}\\sqrt{\\Omega _{3}}\\exp \\left[ i\\int _{0}^{x}d\\xi \\Delta k(\\xi )/2\\right] $ Eq.",
"(1) can be cast in the form $i\\frac{d}{dx}\\left(\\begin{array}{c}A_{2} \\\\A_{3}\\end{array}\\right) =\\frac{1}{2}\\left(\\begin{array}{cc}-\\Delta k & \\Omega \\\\\\Omega & \\Delta k-i\\Gamma /2\\end{array}\\right) \\left(\\begin{array}{c}A_{2} \\\\A_{3}\\end{array}\\right) , $ where $\\Omega =\\sqrt{\\Omega _{2}\\Omega _{3}}$ .",
"Interestingly, Eq.",
"(REF ) can be viewed as a photonic analogue of a two-level atomic system which interacts with an external chirped pulsed field, with the excited state decaying out of the system with a decay rate $\\Gamma $ [28], [29], [30].",
"For a linear chirp $\\Delta k(x)=\\alpha x$ , Eq.",
"(REF ) describes the dissipative Landau-Zener model which admits of an exact solution in terms of parabolic cylinder functions [29].",
"The non-vanishing absorption $\\Gamma $ at the SFG wave basically annihilates the signal wave while being converted in the nonlinear crystal, thus preventing forward propagation, while the chirped QPM grating ensures broadband and robust optical isolation under adiabatic operation.",
"The limiting case $\\Gamma =0$ , previously considered in Refs.",
"[21], [22], [26], realizes broadband SFG via adiabatic rapid passage under the adiabatic condition $\\left|\\Omega \\frac{d}{dx}\\Delta k\\right|\\ll \\left( \\Omega ^{2}+\\Delta k^{2}\\right) ^{3/2}.",
"$ Such a condition requires a smooth $x$ variation of the phase mismatch $\\Delta k(x)$ , i.e.",
"a sufficiently small gradient $\\alpha $ , and large coupling $\\Omega $ .",
"In this way broadband, robust and almost $\\sim 100\\%$ conversion efficiency of the injected signal wave into the SFG wave can be obtained; see Fig.REF (a).",
"To realize an optical diode, a relatively strong absorption at the generated SFG wave should be considered.",
"Absorption together with the pump field direction break mirror symmetry and the optical transmission of the signal wave at frequency $\\omega _{2}$ becomes non reciprocal, as schematically shown in Figs.REF (b) and (c).",
"In fact, in the forward propagation direction phase matching among signal, pump and SFG wave is realized, the signal wave is converted into the SFG via rapid adiabatic passage and the SFG is fully absorbed [Fig.REF (b)].",
"On the other hand, for backward propagation of the signal wave the phase matching for frequency conversion is not realized and the crystal turns out to fully transmit the signal wave [Fig.REF (c)].",
"To realize the nonlinear optical diode, some constraints should be met for the absorption coefficient, crystal length and adiabatic rate of the the QPM grating in order to satisfy the adiabatic condition (3) and to avoid the so-called overdamping problem, i.e.",
"phase mismatch induced by too strong absorption.",
"Such constraints were investigated in details in the study of the dissipative Landau-Zener model [28], [29].",
"In addition to the adiabatic condition (REF ) the absorption coefficient should be strong enough at the generated SFG wave in order to suppress it, but too strong absorption coefficient will have opposite effect as pointed by Vitanov and Stenholm [29]: a too strong absorption freezes the dynamics like in an overdamped oscillator.",
"One can view the effect of the large absorption rate as similar to that of large phase mismatching: both effectively decouple the interaction in three wave mixing.",
"The optimum regime for the optical diode is realized when the absorption rate is of the same order as the coupling, i.e.",
"$\\Gamma \\approx \\Omega $ [29].",
"Figure: (Color online) Schematic of the nonlinear adiabatic opticaldiode realized in a KTP waveguide with a linearly-chirped QPMgrating.",
"The strong pump wave at frequency ω 1 \\protect \\omega _{1} andthe weak signal wave at frequency ω 2 \\protect \\omega _{2} arepolarized in zz direction and phase matched in the KTP crystal togenerate the SFG wave at frequency ω 3 =ω 1 +ω 2 \\protect \\omega _{3}=\\protect \\omega _{1}+\\protect \\omega _{2}, which is stronglyabsorbed by the crystal.Figure: (Color online) Numerically-computed isolation spectra ofthe nonlinear adiabatic isolator for three different pumpintensities I 1 =200,300I_1=200, 300 and 500 MW/cm 2 ^2 versus wavelength ofthe signal wave for a 5-cm-long KTP crystal.",
"The isolation of a5-cm-long phase-matched crystal at 930 nm signal with constantpoling period of 1.79 μ\\protect \\mu m is plotted by blackdashed-dotted curve for easy reference.To illustrate the feasibility of the proposed method and to provide design parameter of optimized optical isolation based on the phootnic analogue of the dissipative Landau-Zener model, let us consider SFG in potassium titanyl phosphate KTiOPO$_{4}$ (KTP) crystal.",
"KTP crystals are commonly used in nonlinear optics applications, showing high damage threshold, a high nonlinear optical coefficient and strong absorption in the near ultraviolet spectral range [31], [32].",
"In waveguide configuration, relatively long crystals can be manufactured with small mode area to ensure high intensity and diffraction-free long interaction lengths [33], [34].",
"We consider specifically type-0 SFG at room temperature, with a strong pump wave at wavelength $\\lambda _1=532$ nm and a weak signal wave at around $\\lambda _2=930$ nm, corresponding to a SFG wave at $\\lambda _3=338$ nm.",
"The optical waves propagate along the $x$ optical axis and all electric fields are polarized in $z$ direction of the crystal (Fig.REF ), while phase matching over a bandwidth of more than 50 nm is achieved using a linearly-chirped QPM grating.",
"The absorption coefficient of KTP at the SFG wave $\\lambda _3 \\sim 340$ nm is $\\Gamma \\sim 229$ cm$^{-1}$ [35].",
"In our simulations, we assume a strong pump wave at $\\lambda _1=532$ nm with an intensity $I_1$ up to 300-500 MW/cm$^{2}$ , which provides high efficiency frequency conversion avoiding the overdamping problem discussed above.",
"For a KTP waveguide with an effective mode area $A_1 \\sim 10 \\; \\mu $ m$^2$ , a pump intensity of 300 MW/cm$^2$ corresponds to an optical pump power $P_1 =A_1 I_1 \\sim 30$ W, which is available both in continuous-wave or pulsed regimes from frequency-doubled high-power Nd:YAG lasers [36].",
"A crystal length of 3-5 cm is typically assumed, with poling period of the chirped grating varying from 1.7 $\\mu $ m to 1.9 $\\mu $ m along the sample for first-order QPM.",
"Such grating periods are feasible with current poling techniques in KTP [37].",
"The performance of the nonlinear optical diode is provided by the isolation parameter, defined as $dB=10 \\times {\\rm Log}_{10}\\left( \\frac{T_{f}}{T_{b}}\\right) ,$ where $T_{f}$ and $T_{b}$ are the transmitted electric field intensities in the forward and backward directions, respectively.",
"Figures REF and REF show the numerically-computed isolation parameter of the adiabatic nonlinear diode for 5-cm and 3-cm-long KTP crystals, respectively.",
"As can be seen from the figure, a maximum isolation of $\\sim 40$ dB and good isolation $>$ 35dB over a spectral region of 60 can be obtained in the longer crystal configuration.",
"Figure: (Color online) Same ad Fig.3, but for a 3-cm-long KTPcrystal.In conclusion, we have presented a novel route toward the realization of broadband and tunable nonlinear optical isolators with ultrafast all-optical switching capabilities, that do not suffer from dynamical reciprocity commonly found in nonlinear optical isolation schemes [20].",
"Our method is based on adiabatic frequency conversion (SFG) in aperiodically-poled quasi-phase-matched crystals with strong absorption at the generated SFG wave.",
"Ultrafast tunability of optical isolation can be simply achieved by changing the intensity level of the strong pump wave.",
"Frequency conversion in the chirped nonlinear crystal is described by an effective dissipative Landau-Zener model [28], [29], [30], efficient frequency conversion and absorption of the SFG wave requiring a balance between absorption coefficient and nonlinear coupling.",
"The feasibility of the method has been discussed by considering as an example optical isolation in the near-infrared ($ \\sim 900$ nm) using a KTP crystal with a chirped QPM grating pumped at 532 nm (frequency-doubled Nd:YAG laser).",
"Optical isolation up to $\\sim 40$ dB over more that 50 nm bandwidth has been obtained in numerical simulations.",
"Such a relatively strong and broadband optical isolation indicates that nonlinear optical schemes can provide viable routes toward tunable optical isolation with ultrafast switching capabilities, without being limited by dynamical reciprocity [20].",
"We acknowledge support by program DRILA."
]
] | 1612.05607 |
[
[
"Entropic Lattice Boltzmann Model for Charged Leaky Dielectric Multiphase\n Fluids in Electrified Jets"
],
[
"Abstract We present a lattice Boltzmann model for charged leaky dielectric multiphase fluids in the context of electrified jet simulations, which are of interest for a number of production technologies including electrospinning.",
"The role of non-linear rheology on the dynamics of electrified jets is considered by exploiting the Carreau model for pseudoplastic fluids.",
"We report exploratory simulations of charged droplets at rest and under a constant electric field, and we provide results for charged jet formation under electrospinning conditions."
],
[
"Introduction",
"The dynamics of charged jets presents a major interest, both as an outstanding problem in non-equilibrium thermodynamics, as well as for its numerous applications in science and engineering [1], [2], [3].",
"In particular, the recent years have witnessed a surge of interest towards the manufacturing of electrospun nanofibers, mostly on account of their prospective applications, such as tissue engineering, air and water filtration, optoelectronics, drug delivery and regenerative medicine [3], [4], [5].",
"As a consequence, several experimental studies have focused on the characterization and production of one-dimensional elongated nanostructures [6], [7], [8], [5], [9].",
"Electrospun nanofibers are typically produced at laboratory scale via the uniaxial stretching of a jet, which is ejected at a nozzle from an electrified charged polymer solution.",
"The charged jet elongates under the effect of an external electrostatic field applied between the spinneret and a conductive collector and eventually undergoes electromechanical (e.g., whipping) instabilities due to various sources of disturbance, such as mechanical vibrations at the spinneret, hydrodynamic friction with the surrounding fluid and others [10].",
"While such instabilities can be detrimental in some respect, making an accurate position of individual fibers on target substrates very hard, in other experiments they are sought for, since they result in thinner cross sections, hence finer electrospun fibres, as they hit the collector [4].",
"This follows from a plain argument of mass conservation: whipping instabilities generate longer jets, hence thinner cross sections [11].",
"The computational modelling of the electrospinning process is based on two main families of techniques: particle methods and Lagrangian fluid methods.",
"The former is based on the representation of the polymer jet as a discrete collection of discrete particles (beads) connected via elastic springs with frictional coupling (dissipative dashpots) and interacting via long-range Coulomb electrostatics [10], [12], [13], [14].",
"The latter, on the other hand, describe the jet as a continuum media, obeying the Navier-Stokes equations for a charged fluid in Lagrangian form [15], [16], [17].",
"Both methods are grid-free, hence well suited to describe abrupt changes of the jet morphology without taxing the grid resolution, as it is the case for Eulerian grid methods.",
"In this respect, grid-based methods, such as Lattice Boltzmann (LB), are not expected to offer a competitive alternative to the two aforementioned class of methods.",
"Nevertheless, owing to its efficiency, especially on parallel computers, and its flexibility towards the inclusion of physical effects beyond single-phase hydrodynamics, it appears worth exploring the possibility of using LB also in the framework of electrified fluids and jets.",
"For instance, in the last decade significant improvements in LB methods for modelling microfluidic flows containing electrostatic interactions have been achieved [18], [19], opening new applications of LB methods in electrohydrodynamic problems [20], [21], [22], [23], [24].",
"In particular, LB methods were successfully employed to simulate deformations and breakup of conductive vapor bubbles, bubble deformation due to electrostriction, dynamics of drops with different electric permittivity.",
"All these investigations usually exploit the approach originally introduced by A. L. Kupershtokh and D. A. Medvedev [19], where dielectric liquids are assumed with zero free charge density, so that the charge carriers are essentially locally bounded to the material [25].",
"Within this assumption, charge carriers are explicitly modeled by a convective transport equation solved by a second LB solver, taking into account the rates of ionization and recombination of charge carriers fluctuating around a local value (distribution) of equilibrium.",
"In the 1960s, G. I. Taylor provided several considerations for dealing with electrified fluid in a series of papers [26], [27], [28].",
"In particular, Taylor discovered that a moving charged fluids cannot be considered either as a perfect dielectric or as a perfect conductor.",
"Instead, the fluid acts as a \"leaky dielectric liquid\", where a non-zero free charge is mainly accumulated on the interface between the charged liquid and the gaseous phase [29].",
"As a consequence, the charge produces electric stresses different from those observed in perfect conductors or perfect dielectrics.",
"Indeed, in the last cases the charge induces a stress which is perpendicular to the interface, altering the interface shape to balance the extra stress.",
"In the electrospinning process, the non-zero electrostatic field tangent to the liquid interface produces a non-zero tangential stress on the interface which is balanced from the viscous force [16].",
"The present work introduces a LB method for simulating charged leaky dielectric liquids, which is of main interest for modeling the electrospinning process.",
"In this context, the largest part of the charge is modeled to lie along the interface between the liquid and the gaseous phase in similarity with previous works [30], [31].",
"Further, the present LB approach is generalized to the case of non-Newtonian flows with shear-thinning viscosity in order to account the rheological properties of electrospun jets.",
"In this context, we adopt an entropic approach [32], [33] in order to preserve locally the second principle (H- theorem) also in presence of sharp changes in the fluid viscosity and structure.",
"The paper is organised as follows.",
"In section II we present the basic features of the LB extension to the case of charged multiphase fluids.",
"In sections III, we present results for the case of charged multiphase fluids at rest, and we report on preliminary results for charged multiphase jets under conditions related to electrospinning experiments."
],
[
"Model",
"We consider a single species, charged fluid as composed of point-like particles and neglect correlations stemming from excluded volume interactions.",
"Following Boltzmann's description, the state of the fluid is determined by the distribution function $f_{p}(\\vec{r},t)$ being the probability of finding at time $t$ the fluid at position $\\vec{r}$ and moving with discrete velocity $\\vec{c}_{p}$ , with $p=1,b$ given $b$ the number of lattice directions.",
"Here, the velocities $\\vec{c}_{p}$ are also viewed as vectors connecting a lattice site $\\vec{r}$ to its lattice neighbors.",
"The LB equation reads $f_{p}(\\vec{r}+\\vec{c}_{p} \\Delta t,t+\\delta t)=f_{p}(\\vec{r},t)-\\alpha \\beta (f_p - f^{eq}_p(\\rho , \\vec{u})) + S_{p}(\\vec{r},t),$ where the product $\\alpha \\beta $ plays the role of a collision frequency, $S_p$ is a source term (see below), and $f^{eq}_p$ is the continuum Maxwell-Boltzmann distribution computed at density $\\rho $ and velocity $\\vec{u}$ [32].",
"The macroscopic variables are given by the density $\\rho =\\sum _{p}f_{p}$ and the fluid velocity $\\vec{u}=1/\\rho \\sum _{p}\\vec{c}_{p}f$ .",
"In the following, we refer to lattice units where the mesh spacing and timestep $\\Delta t$ are conveniently set to unity.",
"Also, we adopt the so-called D2Q9 scheme, composed by 8 discrete speeds (connecting first and second lattice neighbors) and one extra null vectors accounting for particles at rest.",
"In this scheme, Here, the $f^{eq}_p$ are chosen as a second-order Mach-number expansion $f_{p}^{eq}=w_{p}\\rho \\left[1+\\frac{\\vec{u}\\cdot \\vec{c}_{p}}{c^{2}_s}+\\frac{(\\vec{u}\\cdot \\vec{c}_{p})^{2}-c^{2}_s u^{2}}{2c^{4}_s}\\right]$ where the $w_p$ are weights equal to 4/9 for the rest particles, 1/9 and 1/36 respectively for the smallest and largest velocities $\\vec{c}_{p}$ , and $c_s$ is the speed of sound that in lattice units is equal to $1/\\sqrt{3}$ .",
"At the same time, we consider a unit fluid molecular mass, so that the thermal energy is equal to $k_B T=c^{2}_s$ with $k_B$ the Boltzmann constant and $T$ the temperature.",
"Following the approach of Refs.",
"[34], [32], [33], the factor $\\beta $ in Eq.",
"REF depends on the kinematic viscosity $\\nu $ by the relation $\\beta =\\frac{2c^2_s}{2\\nu +c^2_s},$ while $\\alpha $ is the largest value of the over-relaxation parameter so that the local entropy reduction can be avoided, ensuring the H- theorem.",
"In particular, $\\alpha $ is computed as the root of the scalar nonlinear equation [33], [35] $H\\left(f+\\alpha (f^{eq}-f) \\right)=H\\left(f \\right),$ where $H$ denotes the Boltzmann's entropy function, defined in discrete form [36] as $H\\left(f \\right) \\equiv \\sum _p f_{p} \\ln \\left(\\frac{f_{p}}{w_{p}} \\right).$ In Eq.",
"REF , the source term $S_p$ takes into account the global effect of all the internal and external forces $\\vec{F}$ .",
"This is assessed by the exact difference method proposed by Kupershtokh et al.",
"[37], which reads $S_{p}= f_{p}^{eq} \\left(\\rho ,\\vec{u} + \\Delta \\vec{u} \\right)-f_{p}^{eq} \\left(\\rho ,\\vec{u} \\right),$ where $\\Delta \\vec{u}=\\vec{F} /\\rho $ .",
"In bulk conditions, the LB method is intrinsically second-order accurate in space and time, and, in order to ensure the same accuracy in presence of forces, the local velocity is taken at half time step $\\rho \\vec{u}=\\sum _{p}f_{p}(\\vec{r},t)\\vec{c}_{p} + \\frac{1}{2}\\vec{F}.$ Here, the total body force $\\vec{F}=\\vec{F}_{int}+\\vec{F}_{el}$ includes the inter-particle force $\\vec{F}_{int}$ and the electric force $\\vec{F}_{el}$ .",
"The electric force acting on the boundary point $\\vec{r}$ between a gas and a fluid with the local non-uniform permittivity $\\varepsilon (\\vec{r})$ in an electric field $\\vec{E}$ reads [25], [29] $\\begin{aligned}\\vec{F}_{el} &= q \\vec{E} -\\frac{1}{2} |E|^2 \\nabla \\varepsilon +\\frac{1}{2} \\nabla \\left(|E|^2 \\rho \\frac{\\partial \\varepsilon }{\\partial \\rho } \\right) \\\\&= q \\vec{E} + \\frac{1}{2} \\rho \\frac{\\partial \\varepsilon }{\\partial \\rho } \\nabla |E|^2 ,\\end{aligned}$ where $q$ is the local free charge carried on the fluid.",
"In the last Eq., the vacuum permittivity $\\varepsilon _0$ was assumed equal to 1 as in the Gaussian centimetre-gram-second (cgs) unit system, so that the charge in lattice units is $length^{3/2} \\, mass^{1/2} \\,time^{-1}$ in similarity with the statcoulomb definition (note that Coulomb's constant is also 1).",
"For the sake of convenience, we report in Appendix the units conversion Table REF in cgs dimensions from lattice units for several physical quantities shown in the following.",
"As in Ref.",
"[19], we consider a fluid with permittivity $\\varepsilon =1+\\rho /\\rho _0$ with $\\rho _0$ an arbitrary constant (in the following taken for simplicity equal to 1) so that it is $\\rho (\\partial \\varepsilon / \\partial \\rho )=\\varepsilon -1$ .",
"As a consequence, Eq.",
"REF reduces to $\\vec{F}_{el} = q \\vec{E} + \\frac{ \\varepsilon -1 }{2} \\nabla |E|^2 .$ In the following, we assume that the magnetic induction effects can be neglected so $\\nabla \\times \\vec{E} = 0$ , and the system follows the Gauss law $\\nabla \\cdot (\\varepsilon \\vec{E})=q$ .",
"Since $\\vec{E}=-\\nabla \\phi $ with $\\phi $ the electric potential, the Poisson equation $div(\\varepsilon (\\vec{r}) \\nabla \\phi )=-q(\\vec{r})$ can be solved at each lattice node $\\vec{r}$ , given the boundary conditions of the system and the local charge $q(\\vec{r})$ at the node (specified below).",
"In particular, we determine the electric potential by solving numerically the two-dimensional Poisson equation by means of a SOR (Successive Over-Relaxation) algorithm and the Gauss-Seidel method [38].",
"Note that the Poisson equation includes the non-uniformity of the permittivity $\\varepsilon (\\vec{r})$ , and it is solved on-the-fly during the simulation.",
"Hence, the electric force $\\vec{F}_{el}=q (- \\nabla \\phi )$ is added into the LB scheme by Eq.",
"REF .",
"Since we are modeling a leaky dielectric fluid, we assume that the free charge in the system is mainly distributed over the liquid-gaseous interface.",
"Further, in similarity to previous electrospinning models [16], [17], the relaxation time of free charge in the system is assumed to be irrelevant.",
"In other words, the free charge in bulk liquid relaxes to the liquid interface in a smaller time than any other characteristic time in the system [39].",
"This is a well-established assumption of a leaky dielectric fluid (for further details see Ref.",
"[29]).",
"The liquid charge in the point $\\vec{r}$ is given as $q=q_b+q_s,$ which is the sum of a surface charge $q_s$ and a small bulk term $q_b$ .",
"The bulk term $q_b$ is taken as $q_b(\\vec{r})=Q_b \\frac{ \\rho (\\vec{r}) \\, \\theta (\\rho (\\vec{r});\\rho _0)}{\\int \\rho (\\vec{r}) \\, \\theta (\\rho (\\vec{r});\\rho _0) d\\vec{r}},$ where $Q_b$ denotes the total charge in the bulk, the denominator acts to keep constant the charge due to the charge conservation principle, and $\\theta (\\rho ;\\rho _0)$ denotes a smoothed version of the Heaviside step function switching from zero to one at $\\rho _0$ (equal to 1 in all the following simulations) in order to select only the liquid phase.",
"The term $q_s$ is modeled as a proportional to the absolute density gradient $q_s(\\vec{r})= Q_s \\frac{|\\nabla \\rho (\\vec{r})|^2}{\\int |\\nabla \\rho (\\vec{r})|^2 d\\vec{r}},$ where $Q_s$ denotes the total charge over the surface, and the denominator ensures the charge conservation principle as in the previous case.",
"This approach is usually referred to as the constant surface charge model, and it was already adopted in Refs [30], [31] as a strategy to simplify the charge transport and distribution on the droplet interface.",
"Nonetheless, the constant surface charge model fails in describing a distributed charge on the drop interface whenever the charge density is high, since the curvature surface alters the local charge density [16].",
"In order to address the issue, we assume that the curvature biases the surface charge density as in a conductive liquid, following the power-law introduced by I. W. McAllister [40], which states $q_s=q_{s,max} (K/K_{max})^{\\frac{1}{4}}.$ Here, $K$ denotes the mean curvature $ K=\\nabla \\cdot \\widehat{n}$ with the local interface normal $\\widehat{n}=\\nabla \\rho (\\vec{r})/ |\\nabla \\rho (\\vec{r})|$ [41], while $q_{s,max}$ is the maximum surface charge at the maximum curvature $K_{max}$ chosen as a reference value for the system under investigation.",
"It is worth to emphasize that treating a leaky dielectric as a conductive liquid is a simplification already made by several authors (e.g., G. Taylor [26], A. Yarin et al.",
"[42] , etc.).",
"For the sake of simplicity, we take in the following the maximum curvature $K_{max}$ equal to $K_{d}$ value, defined as the curvature doubling the local surface charge density.",
"Thus, we rewrite Eq.",
"REF as $q(\\vec{r})=Q_b \\frac{ \\rho (\\vec{r}) \\, \\theta (\\rho (\\vec{r});\\rho _0)}{\\int \\rho (\\vec{r}) \\, \\theta (\\rho (\\vec{r});\\rho _0) d\\vec{r}}+ Q_s \\frac{|\\nabla \\rho (\\vec{r})|^2 [1+(K/K_{d})^{\\frac{1}{4}}]}{\\int |\\nabla \\rho (\\vec{r})|^2 [1+(K/K_{d})^{\\frac{1}{4}}] d\\vec{r}}.$ The total charge of the system is conserved and equal to $Q=Q_b + Q_s$ .",
"In addition, the fluid is subjected to an internal thermodynamic force $\\vec{F}_{int}$ promoting a phase separation.",
"The phase separation force is accounted for by means of the Shan-Chen method [43].",
"We construct the local force as $\\vec{F}_{int}(\\vec{r},t)=-\\left[G\\sum _{p\\in fluid}w_{p}\\psi (\\rho (\\vec{r}+\\vec{c}_{p},t))\\vec{c}_{p}+G_{w}\\sum _{p\\in wall}w_{p} \\psi (\\rho (\\vec{r},t)) \\vec{c}_{p}\\right]\\psi (\\rho (\\vec{r},t))$ with the sum $\\sum _{p\\in fluid}$ running over lattice nodes where the fluid is allowed, that is, not belonging to the wall, and $\\sum _{p\\in wall}$ runs over nodes belonging to the wall.",
"$G$ and $G_{w}$ are fluid-fluid and fluid-wall interaction strengths, respectively.",
"In Eq REF , $\\psi $ is an effective number density, which is taken for simplicity $\\psi (\\rho )=\\rho _0 [1- \\exp (-\\rho / \\rho _{0})]$ , being $\\rho _{0}$ an arbitrary constant [44] (in the following assumed equal to 1)."
],
[
"Extension to Non-Newtonian flows",
"In the electrospinning process, the rheological behavior of polymeric liquid with shear-rate-dependent viscosity is expected to play a significant role on jet dynamics.",
"As a consequence, we now generalize the present model to the case of non-Newtonian flows, in similarity with the approach reported in Refs [45], [46], [47].",
"The shear-rate $\\dot{\\gamma }$ is a functional of the density distribution function $f$ .",
"In particular, the strain tensor $\\mathbf {\\Gamma }_{\\eta ,\\delta }$ reads [45], [48] $\\mathbf {\\Gamma }_{\\eta ,\\delta }=-\\frac{1}{2\\rho \\tau c^2_s}\\mathbf {\\Pi }_{\\eta ,\\delta },$ where $\\mathbf {\\Pi }_{\\eta ,\\delta }=\\sum _{p}\\left(f_p - f_p^{eq} \\right)\\vec{c}_{p \\eta } \\vec{c}_{p \\delta },$ is the the stress tensor with $\\eta $ and $\\delta $ running over the spatial dimensions.",
"Note that $\\tau $ in Eq.",
"REF is defined as the inverse of the product $\\alpha \\beta $ , where $\\alpha $ was computed by Eq.",
"REF , and $\\beta $ depends by Eq.",
"REF on the kinematic viscosity $\\nu $ .",
"We now rewrite Eq.",
"REF as $\\dot{\\gamma }=\\frac{\\Pi }{\\rho \\tau (\\dot{\\gamma }) c^2_s} ,$ where $\\dot{\\gamma }$ and $\\Pi $ are computed as matrix 2-norm $\\dot{\\gamma }=2||\\mathbf {\\Gamma }||_2$ and $\\Pi =||\\mathbf {\\Pi }||_2$ of the shear and stress tensor [45], respectively.",
"Note that in the last Eq.",
"we exploit a constitutive relation between the kinematic viscosity $\\nu $ and the shear-rate $\\dot{\\gamma }$ , so that $\\tau =\\tau (\\dot{\\gamma })$ .",
"As a consequence, $\\dot{\\gamma }$ is computed as the root of the scalar nonlinear Eq.",
"REF .",
"We should now consider the general trend observed in electrospun polymeric filaments [49].",
"As main features, we highlight that a polymeric spinning solution at low shear rate behaves as a quasi-Newtonian fluid with viscosity $\\nu _0$ , since the initial condition can be recovered, while at high shear rate a non-reversible disentanglement is present.",
"In particular, it is possible to identify a retardation time $\\lambda $ at which the shear-thinning starts, which is equal to the inverse value of the shear rate at that instant.",
"At very high shear rate, a quasi-Newtonian behavior is again observed as soon as the alignment of the polymer chains is extremely high (almost complete).",
"The last region is characterized by a final viscosity value (infinite viscosity $\\nu _{\\infty }$ ), which is lower than $\\nu _0$ .",
"In the present investigation, we exploit the Carreau model [50], which is able to describe all the mentioned rheological properties.",
"The Carreau model states that $\\nu (\\dot{\\gamma })=\\nu _{\\infty }+\\left(\\nu _{0}-\\nu _{\\infty } \\right) \\left[ 1 + \\left(\\lambda \\dot{\\gamma } \\right)^2 \\right]^{(n-1)/2} ,$ where $n$ is the flow index ($n<1$ for a pseudoplastic fluid).",
"Obtained $\\dot{\\gamma }$ by resolving Eq.",
"REF and $\\nu (\\dot{\\gamma })$ by Eq.",
"REF , and assuming a slow variation of $\\nu (\\dot{\\gamma })$ over the time $\\Delta t$ , the local parameter $\\beta $ is finally estimated by Eq.",
"REF .",
"Note that a validation of a similar implementation in LB scheme of the Carreau model was given in Ref.",
"[45].",
"In order to assess the properties of our implementation for charged multiphase systems, we have initially run a set of simulations modeling a charged leaky dielectric fluid system obeying the Shan-Chen equation of state [44].",
"In order to assess the static behavior, we take a two-dimensional periodic mesh made of 320 x 320 nodes, and prepare the system by creating a circular drop of density $\\rho =2.0$ and radius $R=40$ in lattice units, immersed in the second background phase at lower density $\\rho _b=0.16$ .",
"Further, the strength of non-ideal interactions was set equal to $G = -5$ , $G/G^o_{crit} = 1.25 $ where $G^o_{crit}=-4$ is the critical Shan-Chen coupling at the critical density $\\rho _{crit}= \\ln 2$ in the absence of electric fields.",
"Since we aim to model a leaky dielectric fluid, the ratio $Q_s/Q_b$ is taken equal to 10, so that the largest part of the charge lies over the surface.",
"The total charge $Q=Q_s+Q_b$ was set equal to $2.13$ , and $q(\\vec{r})$ computed by Eq.",
"REF .",
"Whenever the Poisson equation is solved, a uniform negative charge is added to obtain a system with net charge zero.",
"Hence, the electric force $\\vec{F}_{el}=q(-\\nabla {\\phi })$ is added into the LB scheme by Eq.",
"REF , where $q$ is computed by Eq.",
"REF with $K_d$ equal to 1.",
"The liquid is Newtonian with kinematic viscosity $\\nu =1/6$ .",
"The stationary configuration of the described system is obtained after 1000 time steps.",
"Hence, we inspect the electric field (see Fig.",
"REF ) at rest conditions in order to analyze the balance of forces acting at the interface, including Shan-Chen pressure and capillary and electrostatic forces, the latter pointing normal to the interface (see panel b of Fig.",
"REF ).",
"Here, we observe that the largest part of the electric field is located over the liquid surface where the charge distribution is higher.",
"In particular, at the boundary of the drop the magnitude of the electric field $|\\vec{E}|$ is equal to $3.5 \\cdot 10 ^{-3}$ .",
"Figure: Profile of the electric field magnitude |E →||\\vec{E}| in(panel a),and its vectorial representation (panel b).",
"Both quantities refer to the charged drop at rest.It is of interest to estimate the various forces which concur to provide a stable configuration of the droplet.",
"The mechanical balance reads as follows: $p_L + p_{el} = p_V + p_{cap}$ where $p_L$ and $p_V$ are the liquid and vapour pressure, respectively, $p_{cap}=\\sigma /R$ is the capillary pressure (given the surface tension $\\sigma $ and the drop radius $R$ ), and $p_{el}$ is the repulsive electrostatic pressure.",
"The latter can be estimated by standard considerations in electrostatics, namely: $p_{el} = \\frac{Q_s \\vec{E_s} \\cdot \\vec{n}}{2 \\pi R}$ where $E_s$ is the electric field at the surface.",
"$\\frac{p_{el}}{p_{cap}} = \\frac{Q_s E_s}{2\\pi \\sigma }$ In actual numbers with $\\sigma =5.8 \\cdot 10^{-2}$ , this ratio is equal to $1/50$ .",
"This shows that electrostatic forces act as a small perturbation on top of the neutral multiphase physics.",
"Next, we investigate the effects of a uniform external electric field $E_{ext}$ of magnitude pointing along the $x$ axis.",
"Using the previous configuration at the equilibrium as starting point of our simulation, we set $E_{ext}$ at two different values equal to 0.1 and $0.5$ .",
"For each one of the two cases, we report a snapshot of the fluid density $\\rho $ taken as soon as the liquid drop touches the point of coordinates (280,160) in lattice units.",
"The set in Fig.",
"REF highlights the significant motion of the charged drop towards right in accordance with the direction of the electric field.",
"In the figure, we note that a sizable change in the drop shape is present only for the case $E_{ext}=0.5$ .",
"In order to elucidate this effect, it is instructive to assess the strength of the electrostatic field in units of capillary forces, namely: $\\tilde{E} \\equiv \\frac{Q E_{ext}}{2 \\pi R} \\frac{R}{\\sigma }= \\frac{Q E_{ext}}{2 \\pi \\sigma }$ In actual numbers, this ratio is equal to 0.5 and 3 for the case at lower and higher $E_{ext}$ , respectively.",
"This shows that the electric force magnitude is sufficiently large to provide an alteration of its shape only in the case at higher $E_{ext}$ .",
"In particular, the shape shows an elongation towards the direction of the electric field $E_{ext}$ , which results from the effect of the curvature on the surface charge.",
"In Fig.",
"REF , we report the alteration of charge density due to the mean curvature term $K$ of Eq.",
"REF .",
"The alteration is estimated as $\\delta _q=q-q_{K=0}$ , where $q_{K=0}$ is computed with Eq.",
"REF with $K=0$ everywhere, corresponding to a constant surface charge model without curvature effect correction.",
"Here, we note an accumulation of charge on the rightest part of the drop, where the mean curvature $K$ shows a maximum value equal to $5.17 \\cdot 10^{-2}$ , which corresponds to a charge accumulation $\\delta _q$ equal to $1 \\cdot 10^{-3}$ , in the following denoted $\\delta _q^{+}$ .",
"The accumulation of charge is counterbalanced by a negative charge $\\delta _q^{-}$ distributed over the almost straight surface part of the drop (just behind the rightest protrusion in Fig.",
"REF ).",
"Both partial charges $\\delta _q^{+}$ and $\\delta _q^{-}$ favor the presence of a protrusion in the drop shape.",
"In order to analyze this effect, we report in Fig.",
"REF the mean curvature computed at the same time $t=1250 \\delta t$ for two simulations, both at $E_{ext}=0.5$ differing for the inclusion of the curvature effects in the constant surface charge model of Eq.",
"REF .",
"Even though the circular shape of the drop is deformed by the external electric field in both cases, the curvature effects increases the protrusion on the drop shape (see Fig.",
"REF panel b).",
"Further, the charge differences provide a shift in the electric force acting on the drop surface, the effect of which accumulates in time, so that the alteration in the drop shape increases in time.",
"Figure: Two snapshots of density ρ\\rho for a charged drop under an externalelectric field E ext E_{ext} equal to 0.10.1 (panel a) and 0.50.5 (panel b) taken as soon as thedrop reaches the point of coordinates (280,160)(280,160).Figure: Alteration of charge density qq due to the mean curvature KK.",
"The alteration is estimated as δ q =q-q K=0 \\delta _q=q-q_{K=0},where q K=0 q_{K=0} is computed with Eq.",
"with K=0K=0 everywhere.Figure: Mean curvature K(x,y,)K(x,y,) computed at the same time t=1250δtt=1250 \\; \\delta t with same surface charge Q s Q_s at E ext =0.5E_{ext}=0.5 with two constant surface charge models:without the curvature effect (panel a), and with the curvature effect (panel b)."
],
[
"Charged multiphase jet in electrospinning setup",
"We set up a system modeling the electrospinning process, containing a charged Shan-Chen fluid.",
"The system is a mesh made of 320 x 320 nodes (see Fig.",
"REF ).",
"The system geometry presents, on the left side, a nozzle of diameter $D=40$ that reproduces the needle of the actual electrospinning apparatus where the charged fluid is injected, while on the up and bottom sides we impose the bounce back boundary condition.",
"As a consequence, the system is open with the inlet nodes located inside (left side) the nozzle (at $x=1$ ).",
"Similarly, we set outlet nodes on the right side (at $x=320$ ) where the jet will impinge under the effect of the external electric field.",
"Such electric field is chosen to mimic the potential difference that is normally applied between the nozzle and a conductive collector in the real electrospinning setup [51].",
"The computational setup is quite sensitive to the choice of the simulation parameters, and numerical stability has to be guaranteed by finely tuning several parameters, in particular: the density and velocity of inlet and outlet nodes, the Shan-Chen coupling constants of fluid-fluid and fluid-wall interactions, the charge constant and the magnitude of the external electric field.",
"After preliminary simulations, we obtain a consistent set of parameters that guarantees a stable and well-shaped charged jet ejected from the nozzle.",
"The initial density of the two phases are $2.0$ and $0.16$ for the liquid and gaseous phase, respectively.",
"The initial configuration consists of the liquid phase filling the inner space of nozzle with a liquid drop just outside the needle (see Fig.",
"REF panel a).",
"All remaining fluid nodes are initialized to gaseous density.",
"Figure: In panel a, the density distribution ρ\\rho of the initial configuration for the electrospinning setup.In panel b, the electric field magnitude |E →||\\vec{E}|.Both the Shan-Chen constants for the fluid-fluid $G$ and fluid-wall $G_w$ interaction are set to $-5$ .",
"As in previous section, for the resolution of the Poisson equation a uniform negative charge is added to the system in order to counterbalance the positive charge and obtain a system with net charge zero.",
"Further, we impose Dirichlet boundary condition in the following form: we impose the electric potential $\\phi _l=0$ on the left side ($x=0$ ), while the electric potential on the right side ($x=321$ ) was set equal to $\\phi _r=-32.2$ , providing a background electric field $E_{back}=(\\phi _r-\\phi _l)/322$ , which is imposed between the two opposite sides (left-right) of the system.",
"On the upper and bottom sides, the Dirichlet boundary conditions are set equal to the $\\phi _u(x)=\\phi _b(x)=x(\\phi _r-\\phi _l)/322$ .",
"Note that the last condition is equivalent to impose an electric field $\\vec{E}$ of magnitude $0.1$ oriented along the x-axis.",
"Since in a typical electrospinning setup the liquid phase is always connected to a generator addressing a charge, it is reasonable to assume that, whenever the stretching of the liquid jet increases the jet interface, extra charge rapidly reaches the liquid boundary in order preserve the value of charge surface density.",
"As a consequence, the charge conservation condition can not be applied (the liquid jet is not isolated).",
"Instead, we assume the conservation condition of the surface charge density value for the same mean curvature, so that Eq.",
"REF is rewritten as $q(\\vec{r})=\\xi _b \\rho (\\vec{r}) \\, \\theta (\\rho (\\vec{r});\\rho _0)+ \\xi _s |\\nabla \\rho (\\vec{r})|^2 [1+(K/K_{d})^{\\frac{1}{4}}],$ where we have adopted a similar condition also for the bulk charge term, being $\\xi _b$ and $\\xi _s$ two proportionality constants, in the following taken equal to $1\\cdot 10^{-4}$ and $6\\cdot 10^{-2}$ , respectively.",
"Note that the two proportionality constants were tuned in order to obtain a mean ratio $Q_s/Q_b$ between the surface and bulk charge close to the target value 10, as in the previous case.",
"At the inlet we set the fluid velocity in accordance with the Poiseuille velocity profile, while the density is set to $2.0$ .",
"In particular, at each time step we compute the mean velocity of the fluid inside the nozzle, then we used this value to set up the Poiseuille profile.",
"As a consequence, the velocities at the inlet nodes are not fixed but can change during the simulation according to the actual mean velocity measured inside the nozzle.",
"The outlet nodes (on the right edge) are put in contact with a gas reservoir with $\\rho =0.16$ , so that the liquid exits by diffusion/advection.",
"We run three different simulations, all starting from the same initial configuration.",
"In the first simulation the liquid is Newtonian with kinematic viscosity $\\nu =1/6$ , in the following denoted $case\\;1$ .",
"In the other two, we employ the Carreau model (see Subsec.",
"REF ) with zero shear kinematic viscosity $\\nu _0=1/6$ , and infinite viscosity $\\nu _{\\infty }=0.001$ .",
"The flow index $n$ is taken equal to 0.75, and 0.5, for the case labeled $b$ , and $c$ , respectively, while the retardation time $\\lambda $ was set equal to 1000 for all the two last cases.",
"The internal electric field computed by the Poisson solver (see Section 2) is computed on-the-fly during the simulation.",
"In panel b of Fig.",
"REF we report the electric field magnitude $|\\vec{E}|$ for the initial configuration.",
"Here, we note a maximum value in $|\\vec{E}|$ close to the drop interface, which is due to the higher surface charge density.",
"Further, a lower value of $|\\vec{E}|$ is observed in the nozzle as a consequence of the larger dielectric constant $\\varepsilon \\simeq 3$ in the liquid phase (versus $\\varepsilon \\simeq 1$ in the gaseous phase).",
"Figure: Series of snapshots of the fluid density ρ\\rho for the electrospinning simulation casebcase \\; b with index flow n=0.75n=0.75 and retardation time λ=1000\\lambda = 1000 taken at timesteps 2500(a), 5000(b), 7500(d), and 8800(d).We now report in Fig.",
"REF several snapshots taken over the time evolution of the system labeled $case \\; b$ .",
"In all the cases, we observe the formation of a liquid charged jet, which is ejected from the nozzle.",
"Further, we report in Fig.",
"REF the velocity component $u_x$ measured at the extreme point (rightest point) of the drop surface versus time $t$ .",
"Here, for all cases under investigation we note that the velocity trend show the presence of a quasi-stationary point, where the viscous forces balances the external electric force in agreement with previous theoretical investigations [52], [10].",
"After the jet touches the collector, the jet shape fluctuates around a mean profile, providing a stationary regime.",
"In particular, at this stage the jet shows an hyperbolic profile (see Fig.",
"REF panel b) which appears to be in qualitative agreement with the characteristic shape of the jet experimentally observed close to the injecting nozzle by the Rafailovich and Zussman groups [53] (see Fig.",
"REF panel a) and in consistency with previous theoretical results on the jet conical shape [54], [11], [55], [56], [57], [58].",
"Figure: Velocity component u x u_x registered at the extreme point (rightest point) of the drop surface versus timefor all the three cases under investigation.Figure: In panel (a), a rectilinear section of a jet in an electrospinning experiment ofa solution of 5 wt% polyethylene oxide in water.",
"Figure adapted with permission from Ref.",
".Copyrighted by the American Physical Society.In panel (b), a snapshot of the fluid density ρ\\rho in the stationary regimeafter the jet has touched the right side of the simulation boxin the casebcase \\; b.",
"In panel (c), the corresponding velocity field magnitude |u(x,y)||u(x,y)| and the LIC representation of the velocity field.In order to characterize the stationary regime, we report in panel c of Fig.",
"REF the magnitude of the velocity field, and the line integral convolution (LIC) visualization technique [59], highlighting the fine details of the flow field.",
"As expected, we observe a higher value of $u(x,z)$ along the jet towards the collector.",
"In particular, we analyze the profile of the velocity in a jet cross section along what is generally observed in the experimental process.",
"We investigate the effect of pseudoplastic rheology on the stress tensor $\\mathbf {\\Pi }$ .",
"In panel a of Fig.",
"REF we report the mean value of the stress $\\Pi $ measured along the central axis of the jet $y=160$ , and averaged over a time interval of 15000 steps in the stationary regime for the three cases under investigation.",
"Here, we note a decreasing trend of the the stress by increasing the pseudoplasticity of the fluid (decreasing the flow index $n$ ).",
"Nonetheless, we observe a small shift in the stress magnitude.",
"This is essentially due to the low value of retardation time $\\lambda $ in the Carreau model adopted in our simulation, which provides a small decrease in the kinematic viscosity $\\nu $ .",
"In order to clarify the this point, we report in panel b of Fig.",
"REF the mean value of the kinematic viscosity $<\\nu >$ again assessed along the central axis of the jet $y=160$ , and averaged over the stationary regime time, where we observe a small decrease of $<\\nu >$ along the stretching direction as a function of the pseudoplastic behavior in the fluid.",
"These results look promising, and we plan to investigate systematically the effect of the rheological parameters on the jet dynamics in a future work.",
"Figure: In panel (a), the mean value of the stress Π\\Pi computed as matrix 2-norm <||Π|| 2 > t <||\\mathbf {\\Pi }||_2>_t of the stress tensor measured along thecentral axis of the jet y=160y=160 averaged over a time interval of 15000 steps.In panel (b), the mean value of the kinematic viscosity ν\\nu computed with Eq.",
"along thecentral axis of the jet y=160y=160, and averaged over a time interval of 15000 steps."
],
[
"Summary",
"Summarizing, we developed a Shan-Chen model for charged leaky dielectric fluids mainly aimed at modeling the electrospinning process.",
"The curvature effects on the charge surface were included in our theoretical treatment, and we generalized the model to non-Newtonian flows in order to account for the peculiar rheological behavior.",
"Different scenarios were investigated to test the model.",
"We initially investigated the effect of strong electric fields on the droplet shape evolution.",
"We also probed the jet formation under electrospinning-like conditions, obtaining a good agreement with both experimental results and previous theoretical works present in literature.",
"At first glance, the pseudoplastic behavior alters the jet dynamics, although a more systematic investigation requires an extensive test of the rheological parameters.",
"Work along these lines is currently underway.",
"The preliminary applications of the presented LB model look promising, although more systematic numerical investigations, as well as theoretical analysis need to be undertaken.",
"Nonetheless, the actual effort can be regarded as a significant forward step to extend the applicability of the LB method to the context of electrospinning systems, providing a useful computational tool in completion of the others presently available in literature."
],
[
"Acknowledgments",
"The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 306357 (NANO-JETS), and under the European Union's Horizon 2020 Framework Programme (No.",
"FP/2014- 2020)/ERC Grant Agreement No.",
"739964 (COPMAT).",
"*"
]
] | 1612.05404 |
[
[
"Jet transverse fragmentation momentum from h-h correlations in pp and\n p-Pb collisions"
],
[
"Abstract QCD color coherence phenomena, like angular ordering, can be studied by looking at jet fragmentation.",
"As the jet is fragmenting, it is expected to go through two different phases.",
"First, there is QCD branching that is calculable in perturbative QCD.",
"Next, the produced partons hadronize in a non-perturbative way later in a hadronization process.",
"The jet fragmentation can be studied using the method of two particle correlations.",
"A useful observable is the jet transverse fragmentation momentum $j_{\\mathrm{T}}$, which describes the angular width of the jet.",
"In this contribution, a differential study will be presented in which separate $j_{\\mathrm{T}}$ components for branching and hadronization will be distinguished from the data measured by the ALICE experiment.",
"The $p_{\\mathrm{Tt}}$ dependence of the hadronization component $\\sqrt{\\left<j_{\\mathrm{T}}^{2}\\right>}$ is found to be rather flat, which is consistent with universal hadronization assumption.",
"However, the branching component shows slightly rising trend in $p_{\\mathrm{Tt}}$.",
"The $\\sqrt{s} = 7\\,\\mathrm{TeV}$ pp and $\\sqrt{s_{\\mathrm{NN}}} = 5.02\\,\\mathrm{TeV}$ p-Pb data give the same results within error bars, suggesting that this observable is not affected by cold nuclear matter effects in p-Pb collisions.",
"The measured data will also be compared to the results obtained from PYTHIA8 simulations."
],
[
"Introduction",
"0.95 In this work we study the jet transverse fragmentation momentum $j_{\\rm T}$ , which is defined as the transverse momentum component of the jet fragment with respect to the jet axis.",
"An illustration of $j_{\\rm T}$ is shown in Figure REF .",
"This quantity can be connected to two particle correlations by requiring that the trigger particle is the particle with the highest transverse momentum in an event (leading particle) and that this transverse momentum is sufficiently high.",
"In this case the trigger particle momentum vector approximates the jet axis.",
"Identifying the associated particle as the jet fragment, $\\vec{j}_{\\rm T}$ becomes the transverse momentum component of the associated particle momentum $\\vec{p}_{\\mathrm {a}}$ with respect to the trigger particle momentum $\\vec{p}_{\\mathrm {t}}$ .",
"The length of the $\\vec{j}_{\\rm T}$ vector is $j_{\\rm T} = \\frac{|\\vec{p}_{\\mathrm {t}} \\times \\vec{p}_{\\mathrm {a}}|}{|\\vec{p}_{\\mathrm {t}}|} \\,.$ In the analysis, the results are obtained as a function of the fragmentation variable $x_{\\parallel }$ .",
"This is defined as the projection of the associated particle momentum to the trigger particle divided by the trigger particle momentum: $x_{\\parallel }= \\frac{\\vec{p}_{\\mathrm {t}} \\cdot \\vec{p}_{\\mathrm {a}}}{\\vec{p}_{\\mathrm {t}}^{2}} \\,.$ Binning in $x_{\\parallel }$ rather than $p_{\\rm Ta}$ is chosen because $x_{\\parallel }$ scales with the trigger $p_{\\rm T}$ .",
"We measure bins where the associated particles have similar momentum fraction relative to trigger.",
"Because $x_{\\parallel }$ follows the jet axis by construction, it is intuitive to define the near side with respect to this axis rather than using only azimuthal angle difference.",
"The associated particle is defined to be in the near side if it is in the same hemisphere as the trigger particle: $\\vec{p}_{\\mathrm {t}} \\cdot \\vec{p}_{\\mathrm {a}} > 0 \\;.$ Figure: Illustration of j → T \\vec{j}_{\\rm T} and x ∥ x_{\\parallel }.",
"When the trigger particle is a high-p T p_{\\rm T} particle that approximates the jet axis sufficiently well, the j → T \\vec{j}_{\\rm T} can be written as the transverse momentum component of the associated particle momentum p → a \\vec{p}_{\\mathrm {a}} with respect to the trigger particle momentum p → t \\vec{p}_{\\mathrm {t}}.",
"The fragmentation variable x ∥ x_{\\parallel } is the projection of p → a \\vec{p}_{\\mathrm {a}} to p → t \\vec{p}_{\\mathrm {t}} divided by p t p_{\\mathrm {t}}.Previously $j_{\\rm T}$ has been measured for example by correlating the particles inside a jet cone with the reconstructed jet axis [1], [2], [3] or by calculating it from the azimuthal correlation function [4].",
"Only one component for $j_{\\rm T}$ is extracted in these studies, describing the whole time evolution of the jet.",
"In our study we want to be able to isolate different components for parton shower and hadronization.",
"To do this, we follow a similar approach as is taken in PYTHIA [5].",
"In this model jet fragmentation consists of two phases, a perturbative showering phase where partons lose their virtuality by emitting gluons followed by a non-perturbative hadronization phase where the partons combine to hadrons.",
"Our working hypothesis is illustrated in Figure REF .",
"If there are no sufficiently high $p_{\\rm T}$ particles in an event, the showering and hadronization components are folded together, as illustrated by the upper half of the figure.",
"But if we restrict ourselves to events with a high $p_{\\rm T}$ leading particle, the folding is weaker and we are able to separate these two.",
"Based on pQCD predictions like angular ordered parton cascades [6], we expect that the leading parton emits soft gluons preferably to rather wide angles.",
"Also the leading parton is not much affected by this soft emission.",
"On the other hand, based on Lund string model [5], we assume that the non-perturbative hadronization produces particles in relatively narrow angles with respect to the hadronizing parton.",
"Thus, if the leading particle is a good approximation for the leading parton, the particles near the leading particle are mainly coming from the hadronization of the leading parton.",
"As the showering produces partons to wide angles and their direction is not changed much by the hadronization, the orientation of the produced particles with respect to the trigger particle is mainly determined by the showering phase.",
"A two component fit to the $j_{\\rm T}$ distribution could allow us to separate the two phases.",
"Figure: Up: If there is no high p T p_{\\rm T} particle in the event, the showering and hadronization components of jet fragmentation are folded together.",
"Down: In the presence of a high p T p_{\\rm T} particle, the showering and hadronization components can be separated from the data.To see if the separation of two components is justifiable, a PYTHIA8 study was conducted.",
"In this study a two gluon initial state was created to get as clean dijet samples as possible.",
"Then the final state particles were produced controlling the presence of the final state radiation.",
"Without final state radiation, the final state particles come purely from the hadronization of the leading parton.",
"When the final state radiation is allowed, the partons go through both showering and hadronization before becoming final state particles.",
"The results of this study are presented in Figure REF .",
"When the final state radiation is turned off, the component from hadronization appears as a narrow, Gaussian like distribution with a short tail.",
"When the final state radiation is turned back on, a long tail appears after this peak.",
"These observations support the idea behind the two component model.",
"Figure: Results from PYTHIA8 study with a di-gluon initial state.",
"Black distribution is obtained when the final state radiation is on.",
"When the final state radiation is turned off, the red distribution emerges.",
"Blue one is calculated by subtracting the red from the black."
],
[
"Analysis methods",
"0.95 The analysis is done using $\\sqrt{s}= 7\\,\\mathrm {TeV}$ $\\mathrm {p\\hspace{-0.50003pt}p}$ ($3.0 \\cdot 10^{8}$ events) and $\\sqrt{s_{\\mathrm {NN}}}= 5.02\\,\\mathrm {TeV}$ $\\mbox{p--Pb}$ ($1.3 \\cdot 10^{8}$ events) data recorded by the ALICE detector [7].",
"The tracks are measured by the Inner Tracking System (ITS) and the Time Projection Chamber (TPC).",
"One out of six possible hits is required in ITS and 70/159 in TPC.",
"The innermost layer of ITS ($|\\eta | < 2$ ) and the V0 detector ($2.8 < \\eta < 5.1$ and $-3.7 < \\eta < -1.7$ ) are used for triggering.",
"Charged tracks with $p_{\\rm T} > 0.3\\,\\mathrm {GeV}\\hspace{-0.50003pt}/\\hspace{-0.20004pt}c$ in TPC acceptance ($|\\eta | < 0.8$ ) are selected for the analysis.",
"The chosen analysis method is two particle correlations.",
"All charged particles inside each $x_{\\parallel }$ bin are paired with a leading particle trigger and $j_{\\rm T}$ is calculated for each of these pairs.",
"An example of a measured $j_{\\rm T}$ distribution is presented in Figure REF .",
"Here the black histogram contains both signal and background.",
"The background is mostly coming from the underlying event.",
"We use $\\eta $ -gap method to estimate the background contribution.",
"The pairs with $\\Delta \\eta > 1.0$ are regarded as background pairs, since the jet correlation is expected to be a small angle correlation.",
"We extrapolate the background to the signal region by generating new pairs from each background pair by randomizing $\\eta $ for trigger and associated particles from inclusive $\\eta $ distributions in corresponding $p_{\\rm Tt}$ and $p_{\\rm Ta}$ bins.",
"This will give a background template for the analysis.",
"This template, generated separately for each $p_{\\rm Tt}$ and $x_{\\parallel }$ bin, is then fitted to the $j_{\\rm T}$ distribution together with a Gaussian function for the narrow (hadronization) component and an Inverse Gamma function for the wide (showering) component.",
"It can be seen from Figure REF that the fit works very well, except in the region around $j_{\\rm T} \\sim {0.4}{GeV}$ , where a small bump appears.",
"Based on PYTHIA studies, this bump comes from events where the leading particle is a neutral meson decay product.",
"When the other decay product is correlated with this trigger, the invariant mass of the pair brings an additional correlation component on top of the jet correlations.",
"This is taken into account in the evaluation of systematic uncertainties.",
"Figure: Measured j T j_{\\rm T} distribution with a three component fit into it.",
"The three components describe the background (blue), hadronization (black dashed) and showering (magenta)."
],
[
"Results",
"0.94 The final $\\sqrt{\\left< j_{\\rm T}^2\\right> }$ results are calculated from the obtained fit parameters for narrow and wide components.",
"Based on earlier results [1], [4] the hadronization component is expected to be universal, meaning that it is $\\sqrt{s}$ independent and that similar jets in different $p_{\\rm Tt}$ (same $x_{\\parallel }$ bin) give the same $\\sqrt{\\left< j_{\\rm T}^2\\right> }$ results.",
"It can be seen from Figure REF that the narrow component results show a flat trend as a function of $p_{\\rm Tt}$ and that there is no difference between results from $\\mathrm {p\\hspace{-0.50003pt}p}$ and $\\mbox{p--Pb}$ data.",
"Both of these observations support the universality expectation.",
"Also PYTHIA8 seems to describe the data.",
"The completely new result in this work is the $j_{\\rm T}$ for the showering part of the jet fragmentation.",
"This is the wide component in Figure REF .",
"Here we can see that there is a rising trend in $p_{\\rm Tt}$ .",
"This can be explained by the fact that higher $p_{\\rm T}$ partons are likely to have higher virtuality.",
"Higher virtuality leads to stronger gluon emission in the showering phase.",
"When the emission is stronger, it is likely to increase the RMS of the distribution since more splittings are allowed in the shower.",
"The fact that $\\mathrm {p\\hspace{-0.50003pt}p}$ and $\\mbox{p--Pb}$ agree within the error bars suggests that there are no significant cold nuclear matter effects.",
"Again, PYTHIA8 describes the data well.",
"Figure: RMS values of narrow (hadronization) and wide (showering) j T j_{\\rm T} components.",
"Results from pp\\mathrm {p\\hspace{-0.50003pt}p} and p--Pb\\mbox{p--Pb} data agree very well together and with PYTHIA8.The final $j_{\\rm T}$ yield results are presented in Figure REF .",
"The narrow yield shows a flat trend in $p_{\\rm Tt}$ , the wide yield shows some hints of rising as a function of $p_{\\rm Tt}$ , but it could also be flat within current error bars.",
"PYTHIA8 seems to be overestimating the yield of the narrow component, but manages to describe the wide component rather well.",
"Similar behavior was earlier observed in an underlying event analysis in $\\mathrm {p\\hspace{-0.50003pt}p}$ collisions at $\\sqrt{s}= 0.9$ and ${7}{TeV}$ [8].",
"Figure: Yields of narrow (hadronization) and wide (showering) j T j_{\\rm T} components.",
"Results from pp\\mathrm {p\\hspace{-0.50003pt}p} and p--Pb\\mbox{p--Pb} data agree very well together but PYTHIA8 seems to overestimate the narrow component yields somewhat."
],
[
"Conclusions",
"0.95 Using two particle correlations, we extracted the RMS and yield for two different $j_{\\rm T}$ components, hadronization and showering.",
"Our main observations are the following: The narrow component RMS shows a flat $p_{\\rm Tt}$ trend and the results between $\\mathrm {p\\hspace{-0.50003pt}p}$ and $\\mbox{p--Pb}$ datasets agree with each other.",
"This supports the universal hadronization expectation.",
"The wide component RMS shows a rising $p_{\\rm Tt}$ trend.",
"This could be caused by increased parton emission allowed by increased leading parton virtuality.",
"There is no difference in wide component RMS between $\\mathrm {p\\hspace{-0.50003pt}p}$ and $\\mbox{p--Pb}$ .",
"No cold nuclear matter effects are seen.",
"PYTHIA8 seems to describe all the RMS results and showering yield well.",
"However, it overestimates the hadronization yield by up to 50 %.",
"1.00"
]
] | 1612.05475 |
[
[
"Analyzing Web Archives Through Topic and Event Focused Sub-collections"
],
[
"Abstract Web archives capture the history of the Web and are therefore an important source to study how societal developments have been reflected on the Web.",
"However, the large size of Web archives and their temporal nature pose many challenges to researchers interested in working with these collections.",
"In this work, we describe the challenges of working with Web archives and propose the research methodology of extracting and studying sub-collections of the archive focused on specific topics and events.",
"We discuss the opportunities and challenges of this approach and suggest a framework for creating sub-collections."
],
[
"Introduction",
"Web archives such as the Internet Archivehttp://www.archive.org or the archives collected by national libraries allow researchers in Web Science and the Digital Humanities to look back at the past of the Web and trace its development over time.",
"These archives are created by regularly crawling the Web (in the case of the Internet Archive) or selected subsets (typically national sub-domains) to create snapshots of Web sites at different points in time.",
"Researchers can look up any of the crawled versions to look back at specific points in the past or compare different versions.",
"An important challenge when using Web archives is the access to the collected data.",
"As an example, the Internet Archive has a size of several petabytes over a time span of 20 years.",
"A researcher working with such an archive needs efficient and effective tools to scope relevant documents for further research.",
"In contrast to typical use cases where only individual pages are considered (e.g.",
"in legal disputes or to provide persistent links) or the entire archive is analyzed using automatic methods (e.g.",
"using text mining) [12], many research questions in Web Science and the Digital Humanities require an analysis of documents related to specific topics and events.",
"The analysis is often performed manually, therefore the documents to be analyzed in detail have to be carefully selected.",
"We call a collection of documents related to a specific topic or event a topic and event focused sub-collection of the archive.",
"Current tools do not support the researcher enough in creating such a sub-collection (cf.",
"for example [7]).",
"Current approaches use browsing and searching as access methods for Web archives.",
"Browsing is done by entering URLs in a Web interface such as the Wayback Machine [8] and navigating the archived pages using hyperlinks.",
"This requires that the URLs of relevant pages or at least pages linking to them are known in advance.",
"As the entire process is done manually, the researcher will typically have to try many hyperlinks that are not available in the archive.",
"The researcher also has to be aware of temporal drift while navigating the archive [1].",
"Temporal drift occurs because the linking and linked page were usually crawled at different times and therefore each navigation step moves the analyzed point in time.",
"Iterated navigation can even lead to page versions that were crawled outside of the relevant time window.",
"The browsing approach is therefore inefficient and error-prone.",
"An alternative approach is to use keyword search over the entire archive.",
"Here the user only needs to enter keywords into a search interface and can see all pages matching the query.",
"Many search interfaces also provide faceted browsing, where the search can be narrowed down further by e.g.",
"the crawl time, the document content type or the domain name.",
"It is much easier for users to get started using this approach, as they do not have to know about relevant documents in advance.",
"They can also combine this approach with the browsing approach by starting the navigation from a search result document.",
"Search has however many technical challenges.",
"First of all, the archive needs to create and keep up to date a full text index of its entire content and execute queries over this index.",
"Because of the typically large size of the archives this necessitates distributed indexing and retrieval architectures [5] that require many computational resources as well as a lot of technical expertise to set up and maintain.",
"An additional issue is the ranking of documents matching a query.",
"On the one hand, the archive contains many snapshots of the same document, which can negatively impact traditional ranking methods that use for example link-based measures [4].",
"On the other hand, the information need of Web archive users is different than in standard search applications because it is usually focused around specific time periods and therefore requires different ranking measures [13].",
"Therefore the searching of Web archive using current tools still requires a lot of manual effort from the user to tune queries and go through result lists.",
"We therefore propose an alternative approach for the access to Web archives through the automatic extraction of topic and event focused sub-collections.",
"Such sub-collections contain documents from the archive that have been automatically classified as referring to a given topic or event.",
"We may additionally pose the constraints that documents in the collection are connected through hyperlinks or that the temporal distance between any two documents in the collection is minimal.",
"By extracting such sub-collections we provide researchers with relevant sets of documents, which can be further analyzed in their appropriate context.",
"In this work we define the concept of topic and event focused sub-collections, describe a framework for extracting them and discuss challenges."
],
[
"Web Archive Sub-Collections",
"In this section we define the concept of topic and event focused sub-collections and describe several important variants of such sub-collections.",
"A Web archive is a collection of Web document snapshots.",
"Web document snapshot refers to the content retrieved from a given URL (document URL) at a given time (crawl time).",
"In addition to the content, the archive typically also stores metadata about the snapshot such as the software used for retrieval, the HTTP headers or the document content type.",
"An topic and event focused sub-collection is a set of Web document snapshots, where each snapshot is available in the given Web archive.",
"It is defined in a sub-collection specification that describes the scope of the sub-collection.",
"The format of the sub-collection specification depends on the extraction algorithm used to create the sub-collection.",
"Each type of algorithm defines a number of scopes that describe relevant documents.",
"A list of exemplary scopes is given in Tab.",
"REF .",
"Scopes can be combined to narrow down the sub-collection.",
"For example, a simple algorithm that supports the URL and time scopes can be used to extract all snapshots of a given URL in a specific time frame.",
"A scope does not need to be exact.",
"For example, a topic scope can be implemented using a machine learning algorithm, that classifies a snapshot as relevant based e.g.",
"on the similarity to a given set of topic keywords.",
"In this case evaluation metrics like precision and recall can be used to analyze the quality of a given algorithm.",
"Given the large size of typical Web archives it is often neither feasible nor desired to find all snapshots matching a scope.",
"Therefore an algorithm should have a high precision in matching the scopes but may have a lower recall.",
"A good algorithm should however aim to find a representative sub-collection, i.e.",
"one that has a similar diversity as the original archives in terms of domains, crawl times or types of sources.",
"We further distinguish between a connected and a disconnected sub-collection.",
"A connected sub-collection needs to contain for any snapshot $s$ contained in the sub-collection also at least one snapshot $t$ for each document that is linked from $s$ , if one is available in the archive.",
"In contrast, a disconnected sub-collection can consist only of isolated snapshots.",
"A connected sub-collection is needed to perform e.g.",
"link graph analyses, whereas e.g.",
"content-based analyses can also be performed on a disconnected sub-collection.",
"An additional distinction is between snapshot and timeline sub-collections.",
"In a snapshot sub-collection, each document URL should occur only once, a timeline sub-collection should however have all snapshots of an in scope URL that are also in scope.",
"A snapshot sub-collection is useful in synchronic analyses, where the researcher is looking at a specific point in time and does not want to deal with multiple versions of the same URL.",
"In contrast, a timeline sub-collection is needed to perform diachronic analyses where we want to track a development over time."
],
[
"Sub-Collection Extraction Framework",
"In this section we will describe the framework, in which the sub-collection extraction process takes place and describe measures to evaluate extraction algorithms.",
"To create a Web archive sub-collection $C$ the researcher first has to choose a base Web archive $W$ and create a sub-collection specification $CS$ that describe their collection need.",
"Then they need to select an algorithm $A$ that supports the scopes specified in $CS$ and run it over the archive $W$ , using the sub-collection specification $CS$ as a parameter.",
"The result of this process is the extracted sub-collection $C$ .",
"We expect that the extraction process will typically be iterative, i.e.",
"that the research will create a modified specification $CS^{\\prime }$ after analyzing the sub-collection $C$ and create a new sub-collection $C^{\\prime }$ , maybe even using a different extraction algorithm $A^{\\prime }$ .",
"To compare different extraction algorithms, we can use the following evaluation measures: Precision As there is no metadata about what topics and events a snapshot in a Web archive is relevant for, the relevance calculation needs to be done using automated methods such as machine learning, e.g.",
"Support Vector Machines (SVMs) for topic classification [9].",
"These methods will however mistake some irrelevant documents as relevant and vice versa.",
"The precision measures the rate of such errors (cf.",
"[11]): $\\text{precision} := \\frac{|\\text{retrieved relevant snapshots}|}{|\\text{retrieved snapshots}|} $ Recall An extraction algorithm can simply iterate over the entire archive and evaluate the specified scopes on each snapshot.",
"Given the large size of Web archives, this is often prohibitively expensive even if parallel processing facilities are available.",
"Therefore it is desirable that the extraction algorithm can use indexes or heuristics to speed up the execution.",
"The recall measures the fraction of relevant snapshots that were extracted from the archive (cf.",
"[11]): $\\text{recall} := \\frac{|\\text{retrieved relevant snapshots}|}{|\\text{relevant snapshots}|} $ In this way it quantifies the expected loss of using a more efficient algorithm.",
"Note that we also need to consider the precision of the relevance estimation method when computing the recall as it may perform differently on the selected subset, for example because the extraction algorithm will preferentially select pages from certain domains or having a certain link structure.",
"In this case also the recall on the entire collection needs to be examined.",
"Diversity As described above, the goal of extracting sub-collections is to find a set of documents that help in answering a research question.",
"While the collection needs to have a manageable size so that it can be analyzed, it also needs to be representative of the entire archive.",
"This can be measured using diversity measures that describe how well different aspects of the given topic or event are represented [3].",
"Link completeness When we analyze the context of a snapshot, e.g.",
"using link graph analysis, it is important to also have all relevant linked pages in the sub-collection.",
"We measure the link completeness of a collection $C$ as follows: $ lc(C) = {\\sum _{s \\in C}}\\frac{|\\text{retrieved relevant outlinks of $s$}|}{|\\text{relevant outlinks of $s$}|} $ Temporal coherence Snapshots in a Web archive are typically crawled at different points in time, even if they refer to the same event.",
"Additionally, a given URL may have been crawled several times in a relevant time frame, providing an algorithm with multiple snapshots to choose from.",
"The selection of snapshots can however introduce errors in the downstream analyses when selecting snapshots that are from distant points in time.",
"A way to reduce this risk is to optimize the selection of snapshot such that the time between any pair of snapshots is minimized.",
"A similar measure is the blur of a Web archive which also considers the expected number of changes to retrieved pages [6].",
"Run time To allow for a fast iteration of refined sub-collection specifications, it is important that extraction algorithms can produce their results fast.",
"As operations on Web archives are often executed in parallel on large clusters, typical run time measures such as the elapsed time in seconds are less useful because they are heavily influenced by the size and current load of the cluster.",
"A better evaluation metric is instead the number of disk accesses to retrieve and evaluate snapshots, as this is typically the dominant cost in the extraction process.",
"Figure: Average number occurrences of HTML tags per document.",
"Eachvalue is normalized to its maximum value.Figure: Average number occurrences of HTML tags per document.Figure: Average number of outlinks per document.",
"Internal linkshave the same domain name as the linking page, external links goto different domains."
],
[
"Challenges",
"In this section we discuss several challenges when working with Web archives, especially when extracting sub-collections.",
"We illustrate the challenges using data extracted from the German Web archive, a collection of all Web pages from the .de top level domain crawled by the Internet Archive between 1994 and 2013.",
"All values except the total number of snapshots (Fig.",
"REF ) were calculated using a random sample of 40K snapshots.",
"Temporal Scope The number of snapshots per year can have strong fluctuations (see Fig.",
"REF ).",
"In general there are more snapshots for recent years, which is consistent with the general trend of a growing web.",
"The exact number of snapshots per year can however vary due to different crawl strategies, intermittent errors and other factors.",
"In the context of Web archive sub-collections this means that a temporal scope can only be used effectively for some time periods, whereas e.g.",
"in the first years of the archive's time span it may exclude too many documents.",
"This also means that diversification may be necessary to avoid that snapshots from sparse time periods get lost while more active time periods are over-represented.",
"Archive Completeness Most archives do not have complete snapshots of the Web due to limited resources, legal restrictions and technical challenges [10] that restrict the collection of the archives.",
"Fig.",
"REF gives estimates of rate of missing content.",
"For each snapshot in our sample we extracted the outgoing links and tried to retrieve them from the archive.",
"We see that about 50-80% of all links within the same domain (internal links) can be retrieved, whereas for external links this rate is much lower at 20-60%.",
"A possible reason for the lower ratio for external links is the restriction of the archive to the .de domain.",
"This is however a realistic scenario, as many current Web archives are run by national libraries and have similar restrictions to Web sites from specific countries or top level domains.",
"For the sub-collection extraction this means that there is an inherent upper bound on the achievable link completeness.",
"It also suggests that optimizing for link completeness may bias the sub-collection towards sites with many relevant internal links and away from hub pages with many relevant external links, as the former are likely to be more complete.",
"Content Diversity The content in our archive spans 20 years, which means that it reflects many developments of Web technology.",
"Fig.",
"REF shows the relative frequency of representative HTML tags over time.",
"Each curve has been normalized such that it reflects the prevalence of the tag in comparison to its peak value.",
"We can easily see the decline of table-based layouts in the 2000s and the growth of layout techniques using div tags styled by CSS style sheets.",
"Similarly, Fig.",
"REF shows the absolute average number of scripts and style sheets per page.",
"We see after 2005 a dramatic increase in the number scripts and to a lesser extent of linked style sheets.",
"The former may be a reflection of the increased use of advertising networks and tracking services.",
"Similarly, we can look at the number of links per page over time (Fig.",
"REF ).",
"Whereas the number of external links stays relatively stable, the number of internal links increases continuously.",
"This may be due to the maturing of Web sites that accumulate content over time, but it may also reflect the changed Web environment where e.g.",
"search engine optimization (SEO) through specific link strategies becomes more common.",
"For the extraction of sub-collections this means that our algorithms must be adaptive to different types of Web content: Over time the format of Web pages has changed dramatically, therefore also the position of relevance cues on the pages may have changed.",
"Furthermore, the value of links has changed over time, such that we have to be more selective when selecting links on more recent pages."
],
[
"Conclusion",
"In this work we have presented a new approach for the access to Web archives through the automatic extraction of topic and event focused sub-collections.",
"In contrast to existing approaches, sub-collection decrease the amount of manual effort required to find a reasonably-sized collection of documents for further research, while increasing the value of the collection through better extraction of link graphs and the avoidance of temporal drift.",
"We have defined the problem of extracting sub-collections and have described several typical extensions to the problem.",
"Additionally, we have described the framework in which this approach is applied and have shown several evaluation metrics to compare algorithms for this problem.",
"Based on data from a real-world Web archive we have demonstrated several issues for algo- rithms trying to solve this problem and have discussed approaches for dealing with them.",
"We hope that this work sparks interest in the extraction and use of topic and event focused sub-collections.",
"In future work, we will present algorithms that create sub-collections following this framework."
],
[
"Acknowledgments",
"This work was partially funded by the European Research Council under ALEXANDRIA (ERC 339233) and the European Commission under SoBigData (RIA 654024) and H2020-MSCA-ITN-2014 WDAqua (grant agreement 64279)."
]
] | 1612.05413 |
[
[
"Nuclear matter properties with nucleon-nucleon forces up to fifth order\n in the chiral expansion"
],
[
"Abstract The properties of nuclear matter are studied using state-of-the-art nucleon-nucleon forces up to fifth order in chiral effective field theory.",
"The equations of state of symmetric nuclear matter and pure neutron matter are calculated in the framework of the Brueckner-Hartree-Fock theory.",
"We discuss in detail the convergence pattern of the chiral expansion and the regulator dependence of the calculated equations of state and provide an estimation of the truncation uncertainty.",
"For all employed values of the regulator, the fifth-order chiral two-nucleon potential is found to generate nuclear saturation properties similar to the available phenomenological high precision potentials.",
"We also extract the symmetry energy of nuclear matter, which is shown to be quite robust with respect to the chiral order and the value of the regulator."
],
[
"Introduction",
"The nuclear force, a residual strong force between colorless nucleons, lies at the very heart of nuclear physics.",
"Enormous progress has been made towards its quantitative understanding since the seminal work by Yukawa on the one-pion-exchange mechanism, which has been published more than eight decades ago [1].",
"Already in the fifties of the last century, Taketani et al.",
"have pointed out that the range of nucleon-nucleon ($NN$ ) potential can be divided into three distinct regions [2].",
"While the long-distance interaction is dominated by one-pion exchange, the two-pion exchange mechanism plays an important role in the intermediate region of $r \\sim 1 \\ldots 2~$ fm.",
"Multi-pion exchange interactions are most essential in the core region.",
"After the discovery of heavy mesons, the $NN$ potential was successfully modeled using the one-boson-exchange (OBE) picture [3], [4] with multi-pion exchange potentials being effectively parametrized by single exchanges of heavy mesons like $\\sigma $ -, $\\omega $ - and, $\\rho $ -mesons.",
"With a fairly modest number of adjustable parameters, the OBE potential models such as the Bonn [5], [6] and Nijmegen 93 [7] models were able to achieve a semi-quantitative description of $NN$ scattering data.",
"Furthermore, based on the general operator structure of the two-nucleon interaction in coordinate space, a phenomenological $NN$ potential model was also developed by the Argonne group [8].",
"In the 1990s, high-precision charge-dependent $NN$ potential models such as e.g.",
"the Reid93 and Nijmegen I, II [7], AV18 [9] and the CD Bonn [10] potentials have been developed, which describe the available proton-proton and neutron-proton elastic scattering data with $\\chi ^2/$ datum$\\sim 1$ .",
"While phenomenologically successful, the above mentioned high-precision $NN$ potentials have no clear relation to quantum chromodynamics (QCD), the underlying theory of the strong interactions.",
"Further, they do not provide a straightforward way to generate consistent and systematically improvable many-body forces and exchange currents and do not allow to estimate the theoretical uncertainty.",
"In this sense, a more promising and systematic approach to nuclear forces and current operators has been proposed by Weinberg in the framework of chiral effective field theory (EFT) based on the most general effective chiral Lagrangian constructed in harmony with the symmetries of QCD [11], [12], [13].",
"The first quantitative studies of $NN$ scattering up to next-to-next-to-leading order (N$^2$ LO) in the chiral expansion have been carried out by Ordóñez et al.",
"[14], [15] using time-ordered perturbation theory, see also [16], [17] where the calculations were done using the method of unitary transformations.",
"In the early 2000s, the $NN$ potential has been worked out to fourth order in the chiral expansion (N$^3$ LO) by Epelbaum, Glöckle and Meißner [18] and by Entem and Machleidt [19] based on the expressions for the pion exchange contributions derived by Kaiser [20], [21], [22].",
"The corresponding three- and four-nucleon forces have also been worked out to N$^3$ LO [23], [24], [25], [26], [27], see [28], [29] for review articles and [30], [31], [32] for calculations beyond N$^3$ LO.",
"Recently, fifth- (N$^4$ LO) and even some of the sixth-order contributions to the two-nucleon force have been worked out in [33], [34], and a new generation of chiral $NN$ potentials up to N$^4$ LO utilizing a local coordinate-space regulator for the long-range terms has been introduced in [35], [36].",
"In parallel, a novel simple approach for estimating the theoretical uncertainty from the truncation of the chiral expansion has been proposed in [35] and successfully validated for two-nucleon observables [35], [36].",
"The algorithm makes use of the explicit knowledge of the contributions to an observable of interest at various orders in the chiral expansion without relying on cutoff variation.",
"The new state-of-the-art $NN$ potentials confirm a good convergence of the chiral expansion for nuclear forces and lead to accurate description of Nijmegen phase shifts [37].",
"For related recent developments see Refs.",
"[38], [39].",
"Currently, work is in progress by the recently established Low Energy Nuclear Physics International Collaboration (LENPIC) [40] towards including the consistently regularized three-nucleon force (3NF) at N$^3$ LO in ab initio calculations of light- and medium-mass nuclei.",
"In parallel, the novel chiral $NN$ potentials have been tested in nucleon-deuteron elastic scattering and properties of $^3$ H, $^4$ He, and $^6$ Li [41] and selected electroweak processes [42], where special focus has been put on estimating the theoretical uncertainty at each order of the expansion.",
"These studies have revealed the important role of the 3NF, whose expected contributions to various bound and scattering state observables appear to be in good agreement with the expectation based on the power counting.",
"Light- and medium-mass nuclei can nowadays be studied using various ab initio methods such as the Green's function Monte Carlo method [43], the self-consistent Green's function method [44], the coupled-cluster approach [45], nuclear lattice simulations [46], [47], [48] or the no-core-shell model [49], see also Ref.",
"[50] for a first application of the relativistic Brueckner-Hartree-Fock theory to finite nuclei.",
"Infinite nuclear matter has also been widely studied based on various versions of the chiral potentials using e.g.",
"the quantum Monte Carlo approach [38], self-consistent Green's function method [51], [52], the coupled-cluster method [53], many-body perturbation theory [54], functional renormalization group (FRG) method [55], [56] and the Brueckner-Hartree-Fock (BHF) theory [57], [58].",
"Recently, Sammarruca et al.",
"have discussed the convergence of chiral EFT in infinite nuclear matter using the nonlocal $NN$ potentials up to N$^3$ LO [19] and including the 3NF at the N$^2$ LO (i.e.",
"$Q^3$ ) level [59].",
"Fairly large deviations between the results at different chiral orders as compared with the spread in predictions due to the employed cutoff variation have been reported in that paper.",
"This suggests that cutoff variation does not represent a reliable approach to uncertainty quantification, which is fully in line with the conclusions of [35].",
"Regulator artifacts in uniform matter have also been addressed in Ref. [61].",
"In this letter we calculate, for the first time, the properties of symmetric nuclear matter (SNM) and pure neutron matter (PNM) based the latest generation of chiral $NN$ potentials up to N$^4$ LO of Refs.",
"[35], [36] using the BHF theory.",
"The purpose of our study is twofold.",
"First, we explore the suitability of the most recent generation of the chiral forces for microscopic description of the equation of state (EOS) of SNM and PNM.",
"Second, by performing an error analysis along the lines of Refs.",
"[35], [36], [41] without relying on cutoff variation, we estimate the theoretical accuracy in the description of the nuclear EOS achievable at various orders of the chiral expansion.",
"Our paper is organized as follows.",
"In section we briefly outline our calculation approach based on the BHF theory.",
"The results of our calculations are presented in section for all available cutoff values, while the theoretical uncertainty from the truncation of the chiral expansion is quantified in section .",
"Finally, the main conclusions of our paper are summarized in section ."
],
[
"Brueckner-Hartree-Fock theory",
"In the BHF theory of nuclear matter, the underlying $NN$ potential, determined by the $NN$ scattering data, is replaced by an effective $NN$ interaction, i.e.",
"the $G$ -matrix, which can be calculated by solving the Bethe-Goldstone equation [57], [62], $G[\\omega ,\\rho ]=V+\\sum _{k_a,k_b>k_F}V\\frac{|k_ak_b\\rangle \\langle k_ak_b|}{\\omega -e(k_a)-e(k_b)+i\\epsilon }G[\\omega ,\\rho ],$ where $V$ is the underlying $NN$ potential provided by chiral EFT, $\\rho $ is the nucleon number density, and $\\omega $ the starting energy.",
"The single-particle energy is $e(k)=e(k;\\rho )=\\frac{k^2}{2m}+U(k,\\rho ).$ The continuous choice for the single-particle potential $U(k,\\rho )$ used in the present BHF theory [62] has the form $U(k;\\rho )={\\rm Re} \\sum _{k^{\\prime }<k_F}\\langle kk^{\\prime }|G[e(k)+e(k^{\\prime });\\rho ]|kk^{\\prime }\\rangle _a,$ where the subscript $a$ indicates antisymmetrization of the matrix elements.",
"These coupled equations are solved in a self-consistent way.",
"Finally, in the BHF theory, we obtain the energy per nucleon as $\\frac{E}{A}=\\frac{3}{5}\\frac{k^2_F}{2m}+\\frac{1}{2\\rho }{\\rm Re} \\sum _{k,k^{\\prime }<k_F}\\langle kk^{\\prime }|G[e(k)+e(k^{\\prime });\\rho ]|kk^{\\prime }\\rangle _a.$"
],
[
"Results",
"In Fig.",
"REF , we show our results for the density dependence of the energy per nucleon of symmetric nuclear matter and pure neutron matter for all available chiral orders and cutoff values, where the $G-$ matrices are solved up to the partial waves $J=6$ .",
"We remind the reader that the long-range contributions are regularized in the newest chiral $NN$ potentials by multiplying the corresponding coordinate-space expressions with the function $f (r) = \\bigg [ 1 - \\exp \\bigg ( -\\frac{r^2}{R^2} \\bigg ) \\bigg ]^n\\,,\\quad n=6\\,, \\quad R = 0.8 \\ldots 1.2~\\mbox{fm}.$ For contact interactions, a non-local Gaussian regulator in momentum space is employed with the cutoff $\\Lambda $ being related to $R$ via $\\Lambda =2/R$ .",
"We emphasize that the calculations reported in this paper do not include the contributions of three- and four-nucleon forces and are thus incomplete starting from N$^2$ LO.",
"For SNM, the LO (i.e.",
"$Q^0$ ), NLO (i.e.",
"$Q^2$ ) and N$^4$ LO $NN$ potentials yield lager binding energies for softer interactions (i.e.",
"for larger cutoffs $R$ ), while the situation is opposite at N$^2$ LO and N$^3$ LO.",
"For PNM, the harder (softer) interactions yield more (less) attraction at LO$\\ldots $ N$^3$ LO (N$^4$ LO).",
"This complicated pattern suggests that the EOS is rather sensitive to the details of the nuclear force and especially to the interplay between its intermediate and short-range components which is expected to be strongly regulator dependent.",
"Our results at NLO agree well with the ones reported in [59] both for SNM and PNMWe cannot compare our N$^2$ LO and N$^3$ LO predictions with those of [59] since no results based on $NN$ interactions only are provided in that work.",
"and with the Quantum Monte Carlos calculation of [38] for PNM.",
"Interestingly, the cutoff dependence of the energy per particle of PNM at NLO is qualitatively different from the one found in [59] which demonstrates that the form of the regulator does significantly affect the properties of the resulting potentials.",
"Generally, our results for both SNM and PNM show an increasing attraction in the $NN$ force when going from LO to N$^2$ LO, that can probably be traced back to the two-pion exchange potential (TPEP), which has a very strong attractive central isoscalar piece.",
"At N$^3$ LO, the chiral TPEP receives further attractive contributions but also develops a repulsive short-range core.",
"The additional repulsion at N$^4$ LO comes from the contributions to the TPEP at this order.",
"The EOSs based on the N$^3$ LO and N$^4$ LO potentials alone show saturation points below $\\rho =0.4~$ fm$^{-3}$ except for N$^3$ LO at $R=0.8~$ fm and $R=0.9~$ fm.",
"Table: Saturation properties of SNM based on the AV18 potentialand the N 4 ^4LO chiral NNNN potentials for all available cutoffvalues.Table: Contributions of the various partial waves (in units of MeV) to the binding energies of SNM at the corresponding saturation densities for the AV18 and chiral N 4 ^4LO NNNN potentials for all available cutoffvalues.Table: Contributions of the various partial waves (in units ofMeV) to the binding energies of SNM at the empirical saturation density, ρ=0.16\\rho =0.16 fm -3 ^{-3},for the AV18 and chiral N 4 ^4LO NNNN potentials for all available cutoffvalues.It is instructive to compare the results based on the most accurate chiral potentials at N$^4$ LO with the ones from high-precision phenomenological interactions such as the AV18 potential [9].",
"In Table.",
"REF , we list the saturation properties, saturation densities and saturation binding energies per particle, and the effective mass of the nucleon [60]: $\\frac{M^*}{M}=1-\\frac{dU(k;e(k))}{de(k)},$ at the saturation point for the AV18 and N$^4$ LO potentials, while the contributions of the various partial waves up to $J=4$ to the potential energy per nucleon at the saturation density are given in Table REF .",
"Notice that the listed saturation properties are still far from the empirical data ($\\rho _{\\text{sat}}\\sim 0.16$ fm$^{-3}$ and $E/A\\sim 16$ MeV) due to the missing 3NF contributions [57], [62].",
"Naturally, we observe that the results based on the hardest version of the N$^4$ LO potential with $R=0.8~$ fm are rather similar to those based on AV18.",
"Interestingly, we find that the partial wave contributions to the energy increase when the N$^4$ LO potentials are softened by increasing the coordinate-space cutoff $R$ (except for the $^3D_3$ -$^3G_3$ channel).",
"In Table REF , the partial wave contributions to potential energy at the empirical saturation density, $\\rho =0.16$ fm$^{-3}$ for different $NN$ potentials are listed from $^1S_0$ to $^3G_4$ states.",
"It is found that all contributions are nearly cutoff-independent expect the ones from $^1S_0$ , $^3S_1$ -$^3D_1$ , and $^3D_3$ -$^3G_3$ states, which are decreasing with the cutoffs $R$ .",
"Actually, the size of these contributions is strongly dependent on the central and tensor components in the $NN$ potential.",
"The larger cutoff $R$ corresponds to stronger short-range correlations and removes more repulsive contribution on the $NN$ potential at short distance.",
"It will generate more attractive binding energy.",
"Our results for the saturation density and binding energy confirm the linear correlation between these two quantities known as the Coester line [63], see also [57].",
"Calculations within the BHF theory using phenomenological potentials have revealed that the position on the Coester line is correlated with the deuteron $D$ -state probability $P_D$ with smaller values of $P_D$ typically resulting in smaller saturation energy and density [6], [57].",
"We do not observe this correlation for the chiral N$^4$ LO potentials with $P_D=4.28\\%$ ($P_D=5.12\\%$ ) for $R=0.8~$ fm ($R=1.2~$ fm).",
"This is similar to the lack of correlation between $P_D$ and the triton binding energy for the novel chiral potentials [41].",
"We remind the reader that the $D$ -state probability is not an observable.",
"We have also extracted the symmetry energy of nuclear matter $a_\\text{symm} (\\rho )$ , the quantity which describes the response of the nuclear force on excess neutrons or protons and plays an important role in understanding the properties of nuclei and astrophysical objects.",
"The symmetry energy $a_\\text{symm} (\\rho ) $ is defined in terms of the expansion of the asymmetric nuclear matter in powers of the asymmetry parameter $\\delta \\equiv (\\rho _n - \\rho _p )/\\rho $ , with $\\rho _n$ and $\\rho _p$ referring to the neutron and proton number densities, via $\\frac{E}{A} (\\rho , \\delta ) = \\frac{E}{A} (\\rho , 0 ) + a_\\text{symm}(\\rho ) \\, \\delta ^2 + \\ldots \\,.$ The terms beyond the quadratic one are known to be very small [64], so that the symmetry energy can be well approximated by $a_\\text{symm} (\\rho ) = \\left( \\frac{E}{A} \\right)_{\\rm PNM} - \\left( \\frac{E}{A} \\right)_{\\rm SNM} \\,,$ where $E/A$ is viewed as a function of $\\rho $ and $\\delta $ .",
"While the calculated symmetry energies show significant cutoff dependence at LO and NLO, which is comparable to that of $(E/A)_{\\rm SNM}$ and $(E/A)_{\\rm PNM}$ , the results at higher orders are almost insensitive to the values of $R$ and show a little variation with the order of the chiral expansion.",
"The resulting value of $a_\\text{symm}=27.9-30.5~$ MeV at the empirical saturation density, calculated using the N$^4$ LO potentials, is consistent with the empirical constraints and the results from the phenomenological high-precision $NN$ potentials [57] with $a_\\text{symm}=28.5-32.6~$ MeV at $\\rho =0.17$ fm$^{-3}$ and the ones from the functional renormalization group method with $a_\\text{symm}=29.0-33.0~$ MeV at $\\rho =0.16$ fm$^{-3}$ [55].",
"Furthermore, Vidaña et al.",
"also studied the properties of the symmetry energy with AV 18 potential plus a phenomenological three-body force as Urbana type [66].",
"However, it is found that the isovector properties of nuclear matter are not affected by the three-body force too much."
],
[
"Uncertainty quantification",
"We now turn to the important question of uncertainty quantification from the truncation of the chiral expansion.",
"Actually, Baldo et al.",
"attempted to quantify the theoretical uncertainties of the EOSs with the family of Argonne $NN$ potential through comparing the BHF theory to other many-body approaches [65].",
"These uncertainties are strongly dependent on the methodologies of nuclear many-body approximation to treat the spin structures of potentials.",
"Here we follow the approach formulated in Ref.",
"[35], which makes use of the explicitly known contributions to an observable of interest at various chiral orders to estimate the size of truncated terms without relying on cutoff variation.",
"The algorithm proposed in [35] has been adjusted in Ref.",
"[41] to enable applications to incomplete few- and many-nucleon calculations based on two-nucleon forces only.",
"Here and in what follows, we use the method as formulated in that paper, which was also employed in [42].",
"Specifically, for an observable $X (p)$ with $p$ referring to the corresponding center-of-mass momentum scale, the theoretical uncertainty $\\delta X^{(i)}$ of the $i$ -th chiral order prediction $X^{(i)}$ is estimated via $\\delta X^{(0)} &=& \\max (Q^2 | X^{(0)}|, \\, |X^{(\\ge 0)} -X^{(\\ge 0)}| ), \\nonumber \\\\\\delta X^{(2)} &=& \\max (Q^3 | X^{(0)}|, \\,Q | \\Delta X^{(2)} |, \\, Q\\delta X^{(0)}, \\, |X^{(\\ge 2)} -X^{(\\ge 2)}|), \\nonumber \\\\\\delta X^{(i)} &=& \\max (Q^{i+1} | X^{(0)}|, \\,Q^{i-1} | \\Delta X^{(2)} |, \\, Q^{i-2} | \\Delta X^{(3)} |, Q \\delta X^{(i-1)}) \\quad \\mbox{for} \\; i \\ge 3\\,,$ where $Q = \\max ( p/\\Lambda _{\\rm b}, \\; M_\\pi /\\Lambda _{\\rm b} )$ is the estimated expansion parameter while $\\Delta X^{(2)} \\equiv X^{(2)}- X^{(0)}$ and $\\Delta X^{(i)} \\equiv X^{(i)}- X^{(i-1)}$ , $i> 2$ , denote the chiral-order $Q^2$ and $Q^i$ contributions to $X(p)$ .",
"The breakdown scale of the nuclear chiral EFT was estimated to be $\\Lambda _{\\rm b} \\simeq 600~$ MeV [35].To account for increasing finite-cutoff artefacts using softer versions of the chiral forces, the lower values of $\\Lambda _{\\rm b} = 500~$ MeV and $400~$ MeV were employed in calculations based on $R=1.1~$ fm and $R=1.2~$ fm, respectively.",
"The Bayesian analysis of the chiral EFT predictions for the $NN$ total cross section of Ref.",
"[67] has revealed, that the actual breakdown scale may even be a little higher than $\\Lambda _{\\rm b} \\simeq 600~$ MeV for $R=0.9~$ fm.",
"Figure: (Color online) Chiral expansion of the symmetryenergy a symm a_\\text{symm} (left panel) and the slope parameterLL (right panel) at the empirical saturation density of ρ=0.16\\rho =0.16 fm -3 ^{-3} for the cutoff values of R=0.9R=0.9~fm (upperraw) and R=1.0R=1.0~fm (lower raw) along with the estimated theoreticaluncertainty.",
"Solid circles (open rectangles) show the complete results at a given chiral order(incomplete results based on NNNN interactions only).",
"Solid triangles show the current experimental constraints ona symm a_\\text{symm} and LL as described in the text.In Fig.",
"REF , we show the results for the EOS for SNM and PNM including the estimated theoretical uncertainties at various orders of the chiral expansion for the most accurate versions of the $NN$ potentials with $R=0.9~$ fm and $R=1.0$ fm [35], [36].",
"The expansion parameter $Q$ at a given density is estimated by identifying the momentum scale $p$ with the Fermi momentum $k_{\\rm F}$ , which is related to the density $\\rho $ via $\\rho = 2 k_{\\rm F}^3 /(3 \\pi ^2)$ ($\\rho = k_{\\rm F}^3 /(3 \\pi ^2)$ ) for SNM (PNM), and assuming $\\Lambda _{\\rm b} = 600~$ MeV.",
"At the saturation density, the achievable accuracy of the chiral EFT predictions for the energy per particle may be expected to be about $\\pm 1.5~$ MeV ($\\pm 0.3~$ MeV) for SNM and $\\pm 2~$ MeV ($\\pm 0.7~$ MeV) for PNM at N$^2$ LO (N$^4$ LO).",
"Notice that the expected accuracy at N$^4$ LO is significantly smaller than the current model dependence for these quantities.",
"We further emphasize that the presented estimations should be taken with some care due to the non-availability of complete calculations beyond NLO.",
"More reliable estimations of the theoretical uncertainty using the approach of [35] will be possible once the corresponding three- and four-nucleon forces are included.",
"Our results confirm the conclusions of [59] that cutoff variation does not provide an adequate way for estimating the uncertainties in the calculations of the nuclear EOS.",
"As discussed in [35], the residual cutoff-dependence of observables may generally be expected to underestimate the theoretical uncertainty at NLO and N$^3$ LO, which is consistent with our results.",
"Further, the spread of results for different values of $R$ at N$^4$ LO appears to be roughly of a similar size as the estimated uncertainty at this order.",
"We, however, refrain from drawing more definite conclusions on the cutoff dependence based on the incomplete calculations.",
"Finally, we have also quantified the achievable accuracy of the theoretical determination of the symmetry energy $a_\\text{symm}$ and the slope parameter $L$ , defined as $L = 3 \\rho \\, \\partial (E/A)_{\\rm SNM}/\\partial \\rho $ , at the empirical saturation density.",
"These important quantities have been constrained by the available experimental information on e.g.",
"neutron skin thickness, heavy ion collisions and dipole polarizabilities leading to the ranges of $29\\mbox{ MeV} \\lesssim a_\\text{symm} \\lesssim 33\\mbox{ MeV}$ and $40\\mbox{ MeV} \\lesssim L \\lesssim 62\\mbox{ MeV}$ [68], [69], [70].",
"In Fig.",
"REF , we show our results for these quantities using the $NN$ potentials from LO to N$^4$ LO along with the estimated theoretical uncertainties.",
"Especially for the slope parameter, a complete calculation at N$^4$ LO would yield a theoretical prediction much more accurate than the current experimental data."
],
[
"Summary and conclusions",
"In summary, we calculated the equations of state (EOSs) of SNM and PNM with the state-of-the-art chiral $NN$ potentials from LO to N$^4$ LO in the framework of Brueckner-Hartree-Fock theory.",
"At N$^4$ LO, the EOS of SNM has saturation points for all employed cutoff values with the corresponding saturation densities and binding energies per particle being within the range of $0.28\\ldots 0.40$ fm$^{-3}$ and $-17.14 \\ldots -23.28~$ MeV, respectively.",
"These values are compatible with the ones based on the phenomenological high-precision potentials like e.g.",
"the AV18 potential.",
"The symmetry energy and the slope parameter at the saturation density are found to be in the range of $a_\\text{symm} = 27.9 \\ldots 30.5~$ MeV and $L = 49.4 \\ldots 55.0~$ MeV, respectively, using the N$^4$ LO potentials with the cutoff in the range of $R=0.8\\ldots 1.2~$ fm.",
"We have also estimated the achievable theoretical accuracy at various orders in the chiral expansion using the novel approach formulated in Refs.",
"[35], [41] and discussed the convergence of the chiral expansion.",
"Similar to [59], we find that the residual cutoff dependence of the energy per particle does not allow for a reliable estimation of the theoretical uncertainty, see also the discussion in Ref. [35].",
"We find that chiral EFT may be expected to provide an accurate description of SNM and PNM at the saturation density, with the expected accuracy of a few percent at N$^4$ LO.",
"At this order, a semi-quantitative description of the EOS should be possible up to about twice the saturation density of nuclear matter.",
"Clearly, this will require a consistent inclusion of the corresponding many-body forces.",
"Work along these lines is in progress to compare with the existing calculations with two-body and three-body chiral force [52], [59]."
],
[
"Acknowledgments",
"We would like to thank Arnau Rios Huguet for sharing his insights into the topics discussed here.",
"UGM thanks the ITP (CAS, Beijing) for hospitality, where part of this work was done.",
"This work was supported in part by the National Natural Science Foundation of China (Grant Nos.",
"11335002, 11405090, 11405116 and 11621131001), DFG (SFB/TR 110, “Symmetries and the Emergence of Structure in QCD”) and BMBF (contract No.",
"05P2015 -NUSTAR R&D).",
"The work of UGM was supported in part by The Chinese Academy of Sciences (CAS) President's International Fellowship Initiative (PIFI) grant no.",
"2015VMA076."
]
] | 1612.05433 |
[
[
"Multifractional theories: an unconventional review"
],
[
"Abstract We answer to 72 frequently asked questions about theories of multifractional spacetimes.",
"Apart from reviewing and reorganizing what we already know about such theories, we discuss the physical meaning and consequences of the very recent flow-equation theorem on dimensional flow in quantum gravity, in particular its enormous impact on the multifractional paradigm.",
"We will also get some new theoretical results about the construction of multifractional derivatives and the symmetries in the yet-unexplored theory $T_\\gamma$, the resolution of ambiguities in the calculation of the spectral dimension, the relation between the theory $T_q$ with $q$-derivatives and the theory $T_\\gamma$ with fractional derivatives, the interpretation of complex dimensions in quantum gravity, the frame choice at the quantum level, the physical interpretation of the propagator in $T_\\gamma$ as an infinite superposition of quasiparticle modes, the relation between multifractional theories and quantum gravity, and the issue of renormalization, arguing that power-counting arguments do not capture the exotic properties of extreme UV regimes of multifractional geometry, where $T_\\gamma$ may indeed be renormalizable.",
"A careful discussion of experimental bounds and new constraints are also presented."
],
[
"Introduction",
"The unprecedented convergence of experiments in particle physics (LHC), astrophysics (LIGO) and cosmology (Planck) has led to discoveries that confirmed the standard knowledge of quantum interactions and classical gravity, either through the observation of phenomena predicted by the theories (the Higgs boson of the Standard Model [1], [2], [3] and general-relativistic gravitational waves from black-hole binary systems [4], [5], [6]) or the gradual refinement of models of the early universe [7], [8].",
"New physics involving supersymmetry, effects of quantum gravity, or an explanation of the cosmological constant are the next desirata, which many scenarios beyond standard predict to be in the range of our current or next-generation instruments.",
"Some of these scenarios, such as string theory [9], [10], [11], loop quantum gravity (LQG) [12], [13], spin foams [14], noncommutative spacetimes [15], [16], [17], [18], and effective quantum gravity [19], [20], are very well known by theoreticians and phenomenologists of various extractions.",
"Others, which include asymptotic safety [21], [22], [23], causal dynamical triangulations (CDT) [24], causal sets [25], [26], and group field theory (GFT) [27], [28], [29], are perhaps that famous in the more restricted community of quantum gravity, while nonlocal quantum gravity [31], [32], [33], [34], [35], [36], [37], [38] and multifractional spacetimes [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63] have just begun to make their appearance on the scene (despite some older precedents), both as theoretical foundations of new paradigms of exotic geometry and as producers of novel phenomenology.",
"It is part of the game that new proposals may meet some resistance at first and, in fact, multifractional theories have been considered in two rather radical ways: either welcomed as a fresh insight into several aspects of quantum gravity or rejected tout court with a wide range of qualifications, from trivial to uninteresting to outright inconsistent.",
"The first purpose of this paper is to collect the most frequent questions and criticism the author came across in the last few years and to give them a hopefully clear answer.",
"Rather than concluding the debate, this contribution will probably fuel it further, either because some of the answers might not satisfy everybody or because new questions or objections can arise.",
"The reader is free to make their own judgment on the matter or even to contribute to the debate actively in the appropriate channels.",
"The recent formulation of two theorems [60] showing how a universal multiscale measure of geometry naturally emerges whenever the dimension of spacetime changes with the scale (as in all quantum gravities) provides the perhaps most powerful justification to the choice of measure in multifractional theories, and an answer to many of the questions we will see below.",
"The remarks are presented in an order that permits to introduce the basic ingredients of multifractional theories in a self-contained way.",
"Therefore, the present work is an updated review on the subject, which was long due.",
"The most recent one [64] dates back to 2012 and it does not cover any of the major advancements regarding the motivations of the theory, several conceptual points about the measure, the field-theory and cosmological dynamics, and observational constraints.",
"We divide the topics in a preliminary but necessary setting of the terminology (section , 3 items), general motivations (section , 3 items), basic aspects of the geometry and symmetry of multifractional spacetimes (section , 15 items), frames and their physical interpretation (section , 9 items), field theory (section , 9 items), classical gravity and cosmology (section , 6 items), quantum gravity (section , 5 items), observational and experimental constraints (section , 18 items), and a final perspective (section , 4 items).",
"See table REF .",
"Table: Summary of the questions per topic.The questions are the subsection titles in the table of content (actual text of the questions adapted).",
"For each answer, bibliography is given where one can find more technical details.",
"The question-answer format should both facilitate the search for specific topics and make an easier reading than the traditional review article.",
"We also note that this is a “review plus plus” because it contains a number of novel results that augment the theory by new elements: a more thorough discussion about the physical meaning and consequences of the very recent flow-equation theorems, succinctly presented in ref.",
"[60], which have repercussions both in general quantum gravity and on the theories of multifractional spacetimes (questions 0, 0, , , , , , , and ); advances in the theory $T_\\gamma $ with fractional derivatives, incompletely formulated in refs.",
"[41], [45], regarding its symmetries (question ), a proposal for a multiscale fractional derivative (question ), a multiscale line element generalizing the no-scale one of ref.",
"[40] (question ), the recasting of the propagator as a superposition of quasiparticle modes with a characteristic mass distribution (question ), and its renormalizability (question ); a clarification of the unit conversion of the scales of these geometries, previously assumed without an explanation (question 0); the formulation of an important approximation of $T_\\gamma $ , that we will denote by $T_{\\gamma =\\alpha }\\cong T_q$ , with the theory with $q$ -derivatives, carried through a comparison of their critical behavior (question 0), a comparison and mutual approximation of their differential calculus (question ), a comparison of their propagators (question ) and of their renormalization properties (question ); the dissipation of some ambiguities [49] in the calculation of the spectral dimension (question ); a discussion on complex dimensions in quantum gravity and fractal geometry (question ); some remarks clarifying that the frame choice in multifractional theories and in scalar-tensor theories is made, respectively, at the classical and at the quantum level (question ); the recognition, of utmost importance for this class of theories, that the second flow-equation theorem fixes the presentation of the geometry measure in an elegant way, which eventually leads to an unexpected solution of the presentation problem (question ); a detailed summary of results on the renormalization in multifractional theories and discussions on the new perspectives opened by the stochastic view and on the inadequacy of the usual power-counting argument (question ); new experimental bounds on the theory with $q$ -derivatives, approximated in the stochastic view, coming from general dispersion relations (questions and ) and from vacuum Cherenkov radiation (questions and )."
],
[
"Terminology",
"$\\blacktriangleright $01$\\blacktriangleleft $ What is the dimension of spacetime?tocsubsection0 What is the dimension of spacetime?",
"There are several definitions of dimension.",
"The most used in theoretical physics is that of topological dimension $D$ , which is simply the total number of spatial and time directions.",
"In a spacetime with Lorentzian signature and one time direction, $D=4$ means that there are three space directions.",
"Other important geometric indicators are the Hausdorff dimension $d_\\textsc {h}$ , the spectral dimension $d_\\textsc {s}$ , and the walk dimension $d_\\textsc {w}$ .",
"In all these cases and by a convention accepted by physicists and mathematicians, the dimension of spacetime is defined after Euclideanizing the time direction.The reader uneasy with this convention can limit the discussion in the text to spatial slices and time separately.",
"Little changes about the main results.",
"The Hausdorff dimension is defined as the scaling of the Euclideanized volume $\\mathcal {V}(\\ell )$ of a $D$ -ball of radius $\\ell $ or of a $D$ -hypercube of edge size $\\ell $ .",
"There is no difference in scaling between the ball and the hypercube.",
"On a classical continuum spacetime, this reads $d_\\textsc {h}(\\ell ):=\\frac{d\\ln \\mathcal {V}(\\ell )}{d\\ln \\ell }\\,.$ Since the volume is the integral $\\mathcal {V}=\\int d\\varrho (x)$ of the spacetime measure $\\varrho (x)=\\varrho (x^0,x^1,$ $\\dots ,x^{D-1})$ in a given region, an approximately constant $d_\\textsc {h}$ is nothing but the scaling of the measure under dilations of the coordinates, $\\varrho (\\lambda x)=\\lambda ^{d_\\textsc {h}}\\varrho (x)$ .",
"On a quantum geometry, the volume $\\mathcal {V}$ may be replaced by the expectation value $\\langle \\hat{\\mathcal {V}}\\rangle $ of the volume operator $\\hat{\\mathcal {V}}$ on a superposition of quantum states of geometry [65], [66].",
"By using an embedding space, $D$ -balls can be defined also on a discrete geometry or on a pre-geometric combinatorial structure (for instance, LQG and GFT), as well as on totally disconnected or highly irregular sets such as fractals [67].",
"In the latter case, the definition of $d_\\textsc {h}$ is more complicated than eq.",
"(REF ) but it conveys essentially the same information, in particular about the scaling of the measure defining the set [41].",
"Moreover, a continuous parameter $\\ell $ exists in all discrete settings or quantum gravities with a notion of distance, even in the absence of a fundamental notion of continuous spacetime [68], [65].",
"In such settings, $\\ell $ is measured in units of a lattice spacing or of the labels of combinatorial complexes.",
"The spectral dimension $d_\\textsc {s}$ is the scaling of the return probability in a diffusion process (see [30] for a review).",
"Let $\\bar{\\mathcal {K}}(\\partial )$ be the Laplacian on a smooth manifold.",
"Placing a pointwise test particle at point $x^{\\prime }$ on the manifold and letting it diffuse, its motion will obey the nonrelativistic diffusion equation $(\\partial _\\sigma -\\kappa _1\\bar{\\mathcal {K}})P(x,x^{\\prime };\\sigma )=0$ with initial condition $P(x,x^{\\prime };0)=\\delta (x-x^{\\prime })/\\sqrt{g}$ , where $\\kappa _1$ is a diffusion coefficient, $\\sigma $ is an abstract diffusion time parametrizing the process, and $g$ is the determinant of the metric.",
"Integrating the heat kernel $P$ for coincident points over all points of the geometry, one obtains a function $\\mathcal {P}(\\sigma ):=Z/\\mathcal {V}=\\int d^D x\\sqrt{g}P(x,x;\\sigma )/\\mathcal {V}$ called return probability (the volume factor makes the normalization finite).",
"In an alternative interpretation [30], the diffusion process is replaced by a probing of the geometry with a resolution $\\sim 1/\\ell $ , where $\\ell =\\sqrt{\\kappa _1\\sigma }$ is the characteristic length scale detectable by the apparatus.",
"Adding also a quantum-field-theory twist to the story, the diffusion equation is reinterpreted as the running equation of the transition amplitude $P$ defined by the Green function $G(x,x^{\\prime })=-\\int _0^{+\\infty }d(L^2)\\,P(x,x^{\\prime };L)$ , corresponding in momentum space to the Schwinger representation $\\tilde{G}(k)=-\\frac{1}{\\tilde{\\mathcal {K}}(k)}=-\\int _0^{+\\infty }d(L^2)\\,\\exp [-L^2\\tilde{\\mathcal {K}}(k)]\\,.$ Here $L$ is a parameter related to the probed scale $\\ell $ and $\\tilde{\\mathcal {K}}$ is the Fourier transform of the kinetic operator $\\mathcal {K}(\\partial )$ in the field action (not necessarily equal to $\\bar{\\mathcal {K}}$ , in general; see question ).",
"The propagator $G$ governs the quantum propagation of a particle from $x^{\\prime }$ to $x$ and $\\mathcal {P}[L(\\ell )]$ is the probability of finding the particle in a neighborhood of $x$ of size $\\ell $ .",
"Whatever the interpretation of $\\mathcal {P}$ , the spectral dimension is the scaling of the return probability: $d_\\textsc {s}(\\ell ):=-\\frac{d\\ln \\mathcal {P}(\\ell )}{d\\ln \\ell }\\,.$ Using $\\sigma $ instead, one gets the more common form $d_\\textsc {s}=-2d\\ln \\mathcal {P}(\\sigma )/d\\ln \\sigma $ .",
"For a set with approximately constant spectral dimension, $\\mathcal {P}(\\ell )\\sim \\ell ^{-d_\\textsc {s}}$ .",
"As in the case of the Hausdorff dimension, a continuous parameter $\\ell $ can always be defined.",
"In quantum geometries, the return probability in eq.",
"(REF ) may be replaced by the expectation value $\\langle \\hat{\\mathcal {P}}\\rangle $ of a certain operator $\\hat{\\mathcal {P}}$ on a superposition of quantum states of geometry [65].",
"The walk dimension is the scaling of the mean-square displacement of a random walker $X(\\sigma )$ (a stochastic motion $X$ over the manifold): $d_\\textsc {w}:=2\\left(\\frac{d\\ln \\langle X^2(\\sigma )\\rangle }{d\\ln \\sigma }\\right)^{-1},$ where $\\langle X^2(\\sigma )\\rangle =\\int d^Dx\\,\\sqrt{g}\\,x^2\\,P(x,0;\\sigma )$ .",
"For a set with approximately constant walk dimension, $\\langle X^2(\\sigma )\\rangle \\sim \\sigma ^{2/d_\\textsc {w}}$ .",
"More information on $d_\\textsc {w}$ can be found in refs.",
"[49], [56].",
"In a continuous space, there is a relation between the three dimensions we just introduced.",
"Simply by scaling arguments, one notes thatIn this chain of relations, a small typo in ref.",
"[56] is corrected.",
"$\\sigma ^{-d_\\textsc {s}/2}\\sim \\mathcal {P}=Z/\\mathcal {V}\\sim \\mathcal {V}^{-1} \\sim \\ell ^{-d_\\textsc {h}} \\sim X^{-d_\\textsc {h}} \\sim \\sigma ^{-d_\\textsc {h}/d_\\textsc {w}}$ , hence $d_\\textsc {s}=2d_\\textsc {h}/d_\\textsc {w}$ .",
"We will comment on this equation in the next question.",
"For Euclidean space or imaginary-time Minkowski spacetime ($\\mathcal {K}=\\nabla ^2$ ), it is immediate to check that $d_\\textsc {h}=D=d_\\textsc {s}$ and $d_\\textsc {w}=2$ .",
"Other definitions of dimension, much less frequently used in theoretical physics, can be found in refs.",
"[41], [67].",
"In footnote REF and questions 0 and 0, we will invoke one such definition, called capacity of a set.",
"For continuous manifolds and in the presence of very simple but nontrivial dispersion relations $\\mathcal {K}(\\partial )\\rightarrow \\tilde{\\mathcal {K}}(k)\\ne -k^2$ , it is easy to show that the spectral dimension $d_\\textsc {s}$ is nothing but the Hausdorff dimension $d_\\textsc {h}^{(k)}$ of momentum space [69], [70].",
"For fractals, this identification is conjectured but not yet proved [71], [72].",
"In general, it is not true that $d_\\textsc {s}=d_\\textsc {h}^{(k)}$ for the most general multiscale geometry, as already recognized in ref.",
"[69].",
"Consider the case where $\\tilde{\\mathcal {K}}(k)$ [a function almost always such that $\\tilde{\\mathcal {K}}(0)=0$ and $\\tilde{\\mathcal {K}}(\\infty )=\\infty $ ] depends on $k=\\sqrt{k_\\mu k^\\mu }$ and the measure in $k$ -momentum space is the usual one, $d^Dk=dk\\,k^{D-1}d\\Omega _D$ , where $d\\Omega _D$ is the angular measure.",
"All the other cases, including multifractional spacetimes, can be derived from this straightforwardly.",
"Calling $K^2:=\\tilde{\\mathcal {K}}(k)$ , we have $2KdK=\\tilde{\\mathcal {K}}^{\\prime }(k)dk$ , where a prime denotes a derivative with respect to $k$ .",
"Therefore, up to an angular prefactor the measure in $K$ -momentum space is $dk\\,k^{D-1}=dK\\,w(K)$ , where $w(K)=\\frac{2K[\\tilde{\\mathcal {K}}^{-1}(K^2)]^{D-1}}{\\tilde{\\mathcal {K}}^{\\prime }(k)|_{k=\\tilde{\\mathcal {K}}^{-1}(K^2)}}\\,,$ where we assumed that we can invert $K(k)$ as $k=\\tilde{\\mathcal {K}}^{-1}(K^2)$ .",
"Since a momentum volume of linear size $K$ is $\\mathcal {V}^{(K)}=\\int dK\\, w(K)$ , the Hausdorff dimension of the $K$ -momentum space is $d_\\textsc {h}^{(k)}=\\frac{d\\ln \\mathcal {V}^{(K)}}{d\\ln K}=\\frac{K w(K)}{\\int dK w(K)}\\,.$ On the other hand, the spectral dimension is $d_\\textsc {s}=\\frac{\\ell ^2\\int _0^{+\\infty }dk\\,k^{D-1}\\,\\tilde{\\mathcal {K}}(k)\\,e^{-\\ell ^2\\tilde{\\mathcal {K}}(k)}}{\\int _0^{+\\infty }dk\\,k^{D-1}\\,e^{-\\ell ^2\\tilde{\\mathcal {K}}(k)}}=\\frac{\\ell ^2\\int _0^{+\\infty }dK\\,w(K)K^2\\,e^{-\\ell ^2K^2}}{\\int dK w(K)\\,e^{-\\ell ^2K^2}}\\,.$ For simple dispersion relations, we know that $d_\\textsc {s}=d_\\textsc {h}^{(k)}$ .",
"For instance, taking the power law $\\tilde{\\mathcal {K}}(k)=k^{2\\gamma }$ , we have $w(K)=K^{D/\\gamma -1}/\\gamma $ , $\\mathcal {V}^{(K)}=K^{D/\\gamma }/D$ , and $d_\\textsc {s}=D/\\gamma =d_\\textsc {h}^{(k)}$ .",
"Already for a binomial dispersion relation $\\tilde{\\mathcal {K}}(k)=k^{2\\gamma _1}+ a k^{2\\gamma _2}$ , one cannot get an exact result.",
"Asymptotically, $d_\\textsc {s}\\simeq D/\\gamma _{1,2}$ [73], and clearly one also has $d_\\textsc {h}^{(k)}\\simeq D/\\gamma _{1,2}$ ; transient regimes of $d_\\textsc {s}$ and $d_\\textsc {h}^{(k)}$ differ.",
"Therefore, one should take eq.",
"(REF ) as yet another definition of spacetime dimension.",
"$\\blacktriangleright $02$\\blacktriangleleft $ Are “multiscale,” “multifractional,” and “multifractal” synonyms?tocsubsection0 Are “multiscale,” “multifractional,” and “multifractal” synonyms?",
"No.",
"Although there has been, in quantum gravity, a lot of confusion about “fractal” and “multiscale” geometries before the appearance of this proposal, and between “multiscale” and “multifractional” after that, now the terminology has been clarified [56].",
"A geometry is multiscale if the dimension of spacetime ($d_\\textsc {h}$ , $d_\\textsc {s}$ , and/or $d_\\textsc {w}$ ) changes with the probed scale.",
"By this, we mean that experiments performed at different energy or length scales are affected by different spacetime dimensionalities.",
"In a multiscale geometry, at different length scales $\\ell _1>\\ell _2>\\ell _3>\\dots \\,,$ one experiences different properties of the geometry.",
"This is called dimensional flow.",
"In the infrared (IR, large scales $\\ell >\\ell _1$ ), the dimension of spacetime is known to be equal to the topological dimension $D$ .",
"In our case $D=4$ , there are three spatial dimensions and one time dimension.",
"The scales of the hierarchy (REF ) are intrinsic to the geometry and appear in many (not necessarily all) physical observables.",
"More precisely, a multiscale spacetime is such that dimensional flow occurs with three properties: [A1] at least two of the dimensions $d_\\textsc {h}$ , $d_\\textsc {s}$ , and $d_\\textsc {w}$ vary; [A2] the flow is continuous from the IR down to an ultraviolet (UV) cutoff (possibly trivial, in the absence of any minimal length scale); [A3] the flow occurs locally, i.e., curvature effects are ignored (this is to prevent a false positive).",
"[B] As a byproduct of A, a noninteger dimension ($d_\\textsc {h}$ , $d_\\textsc {s}$ , $d_\\textsc {w}$ , or all of them) is observed during dimensional flow, except at a finite number of points (e.g., the UV and the IR extrema).",
"On the other hand, multifractional geometries are a special case of multiscale spacetimes.",
"Their measure in position and momentum space and their Laplace–Beltrami operator are all factorizable in the coordinates: $d^Dq(x) &:=& dq^0(x^0)\\,dq^1(x^1)\\cdots dq^{D-1}(x^{D-1})\\,,\\\\d^Dp(k) &:=& dp^0(k^0)\\,dp^1(k^1)\\cdots dp^{D-1}(k^{D-1})\\,,\\\\\\mathcal {K}_x &=& \\sum _\\mu \\mathcal {K}(x^\\mu )\\,,$ in $D$ topological dimensions.",
"Weakly multifractal spacetimes are multiscale spacetimes with the following property (inherited from fractal geometry, a standard branch of mathematics) in addition of A and B: [C] the relations $d_\\textsc {w}=2\\frac{d_\\textsc {h}}{d_\\textsc {s}}\\,,\\qquad d_\\textsc {s}\\leqslant d_\\textsc {h}$ hold at all scales in dimensional flow.",
"Strongly multifractal geometries satisfy A, B, C, and [D] are nowhere differentiable in the sense of integer-order derivatives, at all scales except at a finite number of points (e.g., the UV and the IR extrema).",
"For the traditional definition of fractal set, which we will not use in this context of spacetime models, see [56], [67] and references therein.",
"$\\blacktriangleright $03$\\blacktriangleleft $ How many multiscale, multifractional, and multifractal theories are there?tocsubsection0 How many multiscale, multifractional, and multifractal theories are there?",
"There are as many multiscale theories as the number of proposals in quantum gravity, plus some more.",
"In fact, dimensional flow (mainly in $d_\\textsc {s}$ , but in some cases also in $d_\\textsc {h}$ ) is a universal phenomenon [74], [75], [76] found in all the main scenarios beyond general relativity: string theory [77], asymptotically-safe gravity ($d_\\textsc {s}\\simeq D/2$ in $D$ topological dimensions at the UV non-Gaussian fixed point; analytic results) [78], [23], [79]; CDT (for phase-C geometries, $d_\\textsc {s}\\simeq D/2$ in the UV [80], [81], [82], [83] or, more recently, $d_\\textsc {s}\\simeq 3/2$ [84]; numerical results) and the related models of random combs [85], [86] and random multigraphs [87], [88]; causal sets [89]; noncommutative geometry [90], [91], [92] and $\\kappa $ -Minkowski spacetime [42], [59], [93], [94], [95], [96]; Stelle higher-order gravity ($d_\\textsc {s}=2$ in the UV for any $D$ [30]); nonlocal quantum gravity ($d_\\textsc {s}<1$ in the UV in $D=4$ ) [33].",
"In LQG, while there is no conclusive evidence of variations of the spectral dimension for individual quantum-geometry states based on given graphs or complexes [68], genuine dimensional flow has been encountered in nontrivial superpositions of spin-network states [65], as an effect of quantum discreteness of geometry.",
"These states appear also in spin foams (where there were preliminary results [97], [98]) and GFT, so that both theories inherit the same feature.",
"It must be said, however, that not all possible quantum states may correspond to multiscale geometries.",
"Other examples, all based on analytic results, are Hořava–Lifshitz gravity ($d_\\textsc {s}\\simeq 2$ in the UV for any $D$ ) [79], [83], [99], spacetimes with black holes [100], [101], [102], fuzzy spacetimes [103], and multifractional spacetimes (variable model-dependent $d_\\textsc {h}$ and $d_\\textsc {s}$ ).",
"With the exception of noncommutative spacetimes, all these multiscale examples have factorizable measures in position and momentum space, either exactly or in certain effective limits (for instance, the low-energy limit in string field theory, or the continuum limit of discretized or discrete combinatorial approaches such as CDT, spin foams, and GFT).",
"However, only multifractional geometries are characterized by factorizable Laplace–Beltrami operators (hence their name).",
"There are one multifractional toy model and three theories in total, depending on the differential operators appearing in the action: the model $T_1$ with ordinary derivatives [40], [41], [49], [53] (a special case of the original nonfactorizable model $\\tilde{T}_1$ of refs.",
"[76], [104], [105]) and the theories $T_v$ , $T_q$ , and $T_\\gamma $ with, respectively, weighted derivatives [43], [45], [48], [49], [51], [53], [55] (fixing the problems of $T_1$ ), $q$ -derivatives [41], [49], [51], [53], [54], [57], [58], and fractional derivatives [40], [41], [44], [45].",
"We will explain their differences later.",
"Finally, only a few of these theories have been explicitly checked to be weakly multifractal: asymptotic safety, certain multiscale states in LQG/spin foams/GFT, and the multifractional theory with $q$ -derivatives.",
"The multifractional theory with fractional derivatives is strongly multifractal.",
"Noncommutative spacetimes where $d_\\textsc {s}>d_\\textsc {h}$ in the UV (as in most realizations of $\\kappa $ -Minkowski) and black-hole geometries described by a nonlocal effective field theory violate the inequality in (REF ), hence they are not multifractal.",
"In the other cases, one should calculate the walk dimension $d_\\textsc {w}$ to verify whether spacetime is multifractal or only multiscale.",
"We should also mention some early studies of field theories on fractal sets [106], [107], [108]; by construction, these spacetimes are fractal but they are not multifractal (there is no change of spacetime dimensionality), hence they are not physical models.A yet older attempt [109] defines a spacetime with fixed noninteger dimension but we do not know whether this can be considered a fractal.",
"On the other hand, Nottale's scale relativity [110], [111], [113] is multiscale and presumably also multifractal.",
"A proposal for “fractal manifolds” [112] is multifractal but, like scale relativity, it is a principle rather than a physical theory, since the field dynamics is not defined systematically for matter sectors and gravity.",
"Table REF summarizes the cases.",
"Table: Multiscale, multifractional, and (multi)fractal theories and models."
],
[
"Motivations",
"$\\blacktriangleright $04$\\blacktriangleleft $ What are the motivations of multifractional theories?tocsubsection0 What are the motivations of multifractional theories?",
"There are at least four motivations to consider these theories.",
"We call them the quantum-gravity-candidate argument, the flow-versus-finiteness argument, the uniqueness argument, and the phenomenology argument.",
"(i) Quantum-gravity candidate.",
"Multifractional spacetimes were originally proposed as a class of theories where the renormalization properties of perturbative quantum field theory (QFT) could be improved, including in the gravity sector.",
"The objective of obtaining a renormalizable quantum gravity was supported by a power-counting argument calculating the superficial degree of diverge of Feynman graphs for fields living on a multiscale geometry [41].",
"Later on, it was shown that the theory $T_1$ with ordinary derivatives is only a toy modelHere and there in the text, we will make a small abuse of terminology and call $T_1$ a “theory.” due to the lack of a direct definition of a self-adjoint momentum operator [46] (in other words, one has to prescribe an operator ordering in the field action [49]) and to issues with microcausality [41].",
"Also, explicit loop calculations and the general scaling of the Green function showed that renormalizability is not improved in the theories $T_v$ and $T_q$ with, respectively, weighted and $q$ -derivatives [51].",
"However, the theory $T_\\gamma $ with fractional derivatives is likely to fulfill the original expectations (to see why, check question ), but its study involves a number of technical challenges.",
"Nevertheless, massive evidence has been collected that all multifractional models share very similar properties [41], [45], [52], [55], [58], especially $T_q$ and $T_\\gamma $ (questions and ).",
"In preparation of dealing with the theory with fractional derivatives and to orient future research on the subject, it is important to understand in the simplest cases what type of phenomenology one has on a multiscale spacetime.",
"In particular, $T_v$ and $T_q$ are simple enough to allow for a fully analytic treatment of the physical observables, while having all the features of multiscale geometries.",
"Therefore, they are the ideal testing ground for these explorations.",
"A better knowledge about the typical phenomenology occurring in multifractional spacetimes will be of great guidance for the study of the case with fractional derivatives.",
"(ii) Flow versus finiteness.",
"As soon as dimensional flow was recognized as a universal property of effective spacetimes emerging in quantum gravity [74], the possibility was considered that such property is related to the UV finiteness of a theory.",
"This suspicion was mainly fueled by the fact that $d_\\textsc {s}\\simeq 2$ in the UV of many different models: having two effective dimensions would imply that two-point correlation functions (propagators, potentials, and so on) diverge logarithmically with the distance rather than as an inverse power law in the UV.",
"Multifractional spacetimes are a class of theories where dimensional flow is under complete analytic control and where one can test the conjecture that multiscale geometries are related to UV finiteness.",
"The counterexamples offered by the multifractional paradigm [51], regardless of the value of the spectral dimension in the UV, disproved this conjecture and reappraised the relative importance of dimensional flow with respect to UV finiteness.",
"In parallel, the supposed universality of the magic number $d_\\textsc {s}=2$ was later recognized as fictitious because based on a poor statistics; many models, supposedly UV finite, were in fact found where $d_\\textsc {s}\\ne 2$ at short scales, including some already considered in the past (such as CDT [84]).",
"(iii) Uniqueness.",
"Although renormalizability is a strongly model-dependent feature, it remains to understand why dimensional flow is so similar in so different and so many theories.",
"A recent theorem explains why [60].",
"Let dimensional flow of spacetime in the Hausdorff or spectral dimension $d=d_\\textsc {h},d_\\textsc {s}$ be described by a continuous scale parameter $\\ell $ (this is always the case, as stated in 0).",
"Let also effective spacetime be noncompact, so that $d\\simeq D$ in the IR and there are no undesired topology effects.",
"As a further very general requirement, we also ask that dimensional flow is slow at large scales, meaning that the dimension $d$ forms a plateau in the IR (figure REF ).",
"Since the IR limit $\\ell \\rightarrow +\\infty $ is asymptotic, this flatness of $d(\\ell )$ in the IR is always guaranteed.",
"IR flatness can be encoded perturbatively by requiring that $d\\simeq d^{\\rm IR}$ approximately at large scales.",
"The accuracy of the approximation is governed by an order-by-order estimate of the logarithmic derivatives of $d$ with respect to the scale $\\ell $ , via the linear flow equation $\\sum _{j=0}^n c_j\\frac{d^j}{(d\\ln \\ell )^j}[d^{(n)}(\\ell )-d^{(n-1)}(\\ell )]=0\\,,\\qquad d^{(0)}:=d^{\\rm IR}\\,,$ where $c_j$ are constants.",
"Then, given the three assumptions above (obeyed by all known quantum gravities) and eq.",
"(REF ), we can completely determine the profile $d(\\ell )$ at large and mesoscopic scales once we also specify the symmetries of the measures in position and momentum space.",
"The first flow-equation theorem states that $d(\\ell )\\simeq D+b\\left(\\frac{\\ell _*}{\\ell }\\right)^c+\\text{\\rm (log oscillations)},$ where $b$ and $c$ are constants fixed by the dynamics of the specific theory, $\\ell _*$ is the largest characteristic scale of the geometry, and the omitted part is a combination of logarithmic oscillations in $\\ell $ .",
"Using eqs.",
"(REF ) and (REF ), for $d=d_\\textsc {h}$ (slow flow in position space) one can specify the scaling of spacetime volumes $\\mathcal {V}(\\ell )$ with their linear size $\\ell $ , while for $d=d_\\textsc {s}$ (slow flow in momentum space) one can derive the return probability $\\mathcal {P}(\\ell )$ from (REF ).",
"The proof of (REF ) is independent of the dynamics of the theory and of the geometrical background, except for the requirement that dimensional flow exists [obviously, this implies that spacetime geometry is characterized by a hierarchy of fundamental scales (REF )].",
"The dynamics, and thus the details of the theory, determines the numerical value of the constants $b$ and $c$ and the identification of $\\ell _*$ within the scale hierarchy of the theory.",
"Many quantum-gravity examples are given in question .",
"Figure: The central hypothesis of the theorems on dimensional flow described in the text.Now, if the measures in position and momentum space are not Lorentz invariant but factorizable, and if the Laplace–Beltrami operator is also factorizable, we hit precisely the case of multifractional theories.",
"Then, eq.",
"(REF ) ceases to be valid.",
"In its stead, one has $D$ copies of it with $D=1$ , one for each spacetime direction: $d(\\ell )\\simeq \\sum _{\\mu =0}^{D-1}d_\\mu (\\ell ):=\\sum _{\\mu =0}^{D-1}\\left[1+b_\\mu \\left(\\frac{\\ell _*^\\mu }{\\ell }\\right)^{c_\\mu }+\\text{\\rm (log oscillations)}\\right],$ where $b_\\mu $ and $c_\\mu $ are constant.",
"This is the second flow-equation theorem.",
"Since a factorizable measure in position space can be written as eq.",
"(REF ) for $D$ independent profiles $q^\\mu (x^\\mu )$ (called geometric coordinates), in multifractional spacetimes volumes (of same linear size $\\ell $ in all directions) are of the form $\\mathcal {V}(\\ell )\\sim \\int _{\\rm vol}d^Dq(x)=\\prod _\\mu q^\\mu (\\ell )$ .",
"Plugging this expression into eq.",
"(REF ) and integrating using eq.",
"(REF ), we get an approximate $q^\\mu (\\ell )$ for each direction.",
"The theorem determines the profiles $q^\\mu (x^\\mu )$ exactly.",
"In this paragraph, we focus our attention on real solutions to the flow equation (REF ), postponing the case of complex solutions to question .",
"At leading order in the perturbative expansion (REF ) of $d(\\ell )$ centered at the IR point, one has $q^\\mu (x^\\mu )\\simeq x^\\mu +\\frac{\\ell _*^\\mu }{\\alpha _\\mu }{\\rm sgn}(x^\\mu )\\left|\\frac{x^\\mu }{\\ell _*^\\mu }\\right|^{\\alpha _\\mu } F_\\omega (x^\\mu )\\,,$ where $F_\\omega (x^\\mu )= 1+A_\\mu \\cos \\left(\\omega _\\mu \\ln \\left|\\frac{x^\\mu }{\\ell _\\infty ^\\mu }\\right|\\right)+B_\\mu \\sin \\left(\\omega _\\mu \\ln \\left|\\frac{x^\\mu }{\\ell _\\infty ^\\mu }\\right|\\right)\\,,$ all indices $\\mu $ are inert (there is no Einstein summation convention), the first factor 1 in (REF ) is optional [60], $\\ell _*^\\mu $ and $\\ell _\\infty ^\\mu $ are $D+D$ length scales, and $\\alpha _\\mu $ , $A_\\mu $ , $B_\\mu $ , and $\\omega _\\mu $ are $D+D+D+D$ real constants.",
"Going beyond leading order in the perturbative expansion of the dimension at the IR, one gets the even more general form, valid at all scales, $q^\\mu (x^\\mu ) = x^\\mu +\\sum _{n=1}^{+\\infty } \\frac{\\ell _n^\\mu }{\\alpha _{\\mu ,n}}{\\rm sgn}(x^\\mu )\\left|\\frac{x^\\mu }{\\ell _n^\\mu }\\right|^{\\alpha _{\\mu ,n}} F_n(x^\\mu )\\,,$ where $F_n(x^\\mu )$ is $F_\\omega (x^\\mu )$ with all real constants $\\ell _\\infty ^\\mu ,\\alpha _\\mu ,A_\\mu ,B_\\mu ,\\omega _\\mu $ labeled by the sum index $n$ .",
"Equation (REF ) describes the most general real-valued multifractional geometry along the direction $\\mu $ , characterized by an infinite hierarchy of scales $\\lbrace \\ell _n^\\mu ,\\ell _{\\infty ,n}^\\mu \\rbrace $ .",
"Remarkably, exactly the same form of the geometric coordinates (REF ) can be obtained in a totally independent way by asking a priori that the measure (REF ) on the continuum represent the integration measure on a multifractal [41].",
"For each direction, one first considers a deterministic fractal setA deterministic fractal $\\mathcal {F}=\\bigcup _i \\mathcal {S}_i(\\mathcal {F})$ is the union of the image of some maps $\\mathcal {S}_i$ which take the set $\\mathcal {F}$ and produce smaller copies of it (possibly deformed, if the $\\mathcal {S}_i$ are affinities).",
"Not all fractals are deterministic.",
"Sets with similarity ratios randomized at each iteration are called random fractals.",
"Cantor sets are popular examples of deterministic and random fractals.",
"Let $\\mathcal {S}_1(x)=a_1 x+b_1$ and $\\mathcal {S}_2(x)=a_2 x+b_2$ be two similarity maps, where $a_{1,2}$ (called similarity ratios) and $b_{1,2}$ (called shift parameters) are real constants and $x\\in I$ is a point in the unit interval $I=[0,1]$ .",
"The image $\\mathcal {S}_i(A)$ of a subset $A\\subset I$ is the set of all points $\\mathcal {S}_1(x)$ where $x\\in A$ .",
"A Cantor set or Cantor dust $\\mathcal {C}$ is given by the union of the image of itself under the two similarity maps, $\\mathcal {C}=\\mathcal {S}_1(\\mathcal {C})\\cup \\mathcal {S}_2(\\mathcal {C})$ .",
"For instance, the ternary (or middle-third) Cantor set $\\mathcal {C}_3$ has $a_1=1/3=a_2$ , $b_1=2/3$ , and $b_2=0$ : $\\mathcal {S}_1(x)=\\tfrac{1}{3} x+\\tfrac{2}{3}$ , $\\mathcal {S}_2(x)=\\tfrac{1}{3} x$ .",
"At the first iteration, the interval $[0,1]$ is rescaled by $1/3$ and duplicated in two copies: one copy (corresponding to the image of $\\mathcal {S}_2$ ) at the leftmost side of the unit interval and the other one (corresponding to $\\mathcal {S}_1$ ) at the rightmost side.",
"In other words, one removes the middle third of the interval $I$ .",
"In the second iteration, each small copy of $I$ is again contracted by $1/3$ and duplicated, i.e., one removes the middle third of each copy thus producing four copies 9 times smaller than the original; and so on.",
"Iterating infinitely many times, one obtains $\\mathcal {C}_3$ , a dust of points sprinkling the line.",
"The set is self-similar inasmuch as, if we zoom in by a multiple of 3, we observe exactly the same structure.",
"It is easy to determine the dimensionality of the Cantor set $\\mathcal {C}$ .",
"Since this dust does not cover the whole line, it has less than one dimension.",
"Naively, one might expect that the dimension of $\\mathcal {C}$ is zero, since it is the collection of disconnected points (which are zero-dimensional).",
"However, there are “too many” points of $\\mathcal {C}$ on $I$ and, as it turns out, the dimension of the set is a real number between 0 and 1.",
"In particular, given $N$ similarity maps all with ratio $a$ , the similarity dimension or capacity of the set is $d_\\textsc {c}(\\mathcal {C}):=-\\ln N/\\ln a$ , a formula valid for an exactly self-similar set made of $N$ copies of itself, each of size $a$ .",
"Note that $a= N^{-1/d_\\textsc {c}}$ : the smaller the size $a$ , the smaller the copies at each iteration and the smaller the dimensionality of the set.",
"In the case of the middle-third Cantor set, $N=2$ and $a=1/3$ , so that $d_\\textsc {c}=\\ln 2/\\ln 3\\approx 0.63$ .",
"living on a line and obtains the typical (power law)$\\times $ (log oscillations) structure [114], [115], [116]; summing over different scales, one obtains the multifractal profile (REF ).",
"The independence of this derivation of the measure is important because it yields information not apparently available in the flow-equation theorems (see question 0).",
"In this review, we will not insist too much upon the beautiful formalism of fractal geometry implemented into multifractional spacetimes; a concise presentation can be found in refs.",
"[41] and [56].",
"To summarize, the measure (REF ) used in multifractional theories is the most general one when spacetime geometry is multiscale and factorizable [60].",
"It also happens to coincide with the integration measure of a multifractal [41].This is not inconsistent with what said in question 0.",
"Even if the measure is multifractal, the geometry of spacetime may be nonmultifractal, depending on the symmetries enforced on the dynamics (i.e., type of derivatives).",
"Thus, multifractional theories are the most general factorizable framework wherein to study the phenomenon of dimensional flow.",
"This can (and did) help to better understand the flow properties of other quantum gravities (even despite their nonfactorizability), either by recasting the dynamics of such theories as a multifractional effective model [47], [42], [59] or by employing the same mathematical tools endemic in multifractional theories [77], [79], [65], [44].",
"The geometrical and physical reason beyond the existence of the flow-equation theorems and of a unique (in the sense of being described by the same multiparametric function) dimensional flow in all quantum gravities is the fact that the IR is reached as an asymptote where the dimension varies slowly.",
"There is also a perhaps deeper physical reason, more delicate to track, that also sheds light into the flow-versus-finiteness issue.",
"It consists in the fact that dimensional flow is the typical outcome of the combination of general relativity with quantum mechanics [61], [62].",
"(iv) Phenomenology.",
"The search for experimental constraints on fractal spacetimes dates back to the 1980s [117], [118], [119], [120].",
"Since early proposals of fractal spacetimes were quite difficult to handle [106], [107], [108], [109], toy models of dimensional regularization were used and several bounds on the deviation $\\epsilon =D-4$ of the spacetime dimension from 4 were obtained.",
"However, these models were not backed by any theoretical framework and they were not even multiscale.",
"Multifractional theories are genuine realizations of multiscale geometries based on much more solid foundations, i.e., all the sectors one would possibly like to investigate are under theoretical control (classical and quantum mechanics, classical and quantum field theory, gravity, cosmology, and so on) and they give rise to well-defined physical predictions that can be (and actually have been) tested experimentally.",
"Most notably, all the phenomenology extracted from multifractional scenarios comes directly from the full theory, with very few or no approximations.",
"We will always use the term “phenomenology” in this sense, in contrast with its other use as a synonym of “heuristic” (i.e., inspired by a theory rather than derived from it rigorously) in some literature of quantum gravity.",
"The questions left unanswered by the dimensional-regularization toy models can now receive proper attention; see section .",
"In the same section, we will see that multifractional theories make it possible to explore observable consequences of dimensional flow, which is not just a mathematical property.",
"$\\blacktriangleright $05$\\blacktriangleleft $ I understand that spacetimes endowed with a structure of weighted derivatives or $q$ -derivatives are analyzed more in detail because they are simpler than the theory with fractional derivatives, which is most promising especially as far as renormalization is concerned.",
"However, what is the physical reason why such extensions $T_v$ and $T_q$ should be of interest and relevance to particle-physics phenomenology?",
"They are only distant relatives of a theory supposed to describe geometry (dimensional flow) and quantum gravity, with no connection to the Standard Model.tocsubsection0 Why should multifractional theories with weighted and $q$ -derivatives be of interest for particle-physics phenomenology?",
"A first answer is given by the quantum-gravity-candidate argument of 0.",
"All multifractional theories share similar phenomenology, as far as we can see.",
"In the context of particle physics, it was shown that the scale hierarchy of $T_v$ is quite similar to the scale hierarchy of $T_q$ , even if individual experiments may be sensitive to such scales in different ways [for instance, variations of the fine-structure constant in quantum electrodynamics (QED) are detectable only in the case with $q$ -derivatives but not in the other] [54], [55].",
"In questions 0, , and , we will find a striking similarity between $T_q$ and $T_\\gamma $ when $\\gamma =\\alpha $ , based on the dimensionality of the Laplace–Beltrami operator [41], on the form of the propagator in the UV, and on approximations of the integrodifferential calculi of the theories.",
"Because of this approximate but crucial matching $T_{\\gamma =\\alpha }\\cong T_q\\,,$ we expect the phenomenology with $q$ -derivatives to be very similar to that with fractional derivatives.",
"Thus, it is useful to understand what type of experiments would be capable of constraining $T_\\gamma $ .",
"Apart from this goal, it is important to recognize the impact of dimensional flow on physical observables.",
"The quest for an observable imprint of quantum gravity is more feverish than ever and it is natural to look at possible effects of the most evident feature all these competitors have in common.",
"The theories $T_v$ and $T_q$ are not mere toy models of a “better” theory: they represent autonomous realizations of physics on a geometry with dimensional flow.",
"Even if their renormalizability is not better than in standard field theories, they display a full set of testable physical observables from particle and atomic physics to cosmology.",
"Since the geometry described by the multifractional measures (REF ) and (REF ) is the most general factorizable one if $d_\\textsc {h}$ varies with the scale, the constraints on the scale hierarchy obtained in multifractional theories possibly have a much wider scope of validity, being somewhat prototypical of the whole class of multiscale theories; thus, including quantum gravities.",
"$\\blacktriangleright $06$\\blacktriangleleft $ These theories have been developed mainly by the author himself and hence their impact on the community at large might be limited.",
"Will this line of research illuminate anything about quantum gravity?tocsubsection0 Will this line of research illuminate anything about quantum gravity?",
"Yes, mainly for the reason spelled out in 0.",
"Multifractional theories did receive attention by the quantum-gravity community and have been actively studied not just by the author but also by researchers working in different fields such as quantum field theory [48], [49], [51], [54], [55], noncommutative spacetimes [42], [59], quantum cosmology and supergravity [42], group field theory [42], classical cosmology [50], [58], and numerical relativity [58].",
"As mentioned in 0, interest has not been limited to multifractional theories per se, but extended to the possibility to use their machinery in different, quantum-gravity-related contexts [77], [79], [65].",
"However, despite the ongoing collaborative effort, the limited number of people involved is sometimes perceived as a signal that multifractional theories are not as interesting and useful as advertized.",
"There were two causes that led to this opinion.",
"The first is the type of development that multifractional theories have undergone since the beginning [76].",
"Many of their aspects have evolved slowly and heterogeneously from paper to paper and this has hindered a coherent exposition of the main ideas from the start.",
"The present manifesto, with its overview and active integration of different elements, should help to clarify the context, advantages, and status of these theories.",
"The second cause is that multifractional theories had to talk with a number of communities widely different from one another.",
"On one hand, the original proposal was directed to the quantum-gravity sector, which is not at all annoyed by the breaking of Lorentz symmetries but is fragmented into, and busy with, a number of independent and very strong agendæ based on elegant mathematical structures and convincing evidence (or proofs) of UV finiteness.",
"Since there are hints that it is possible to quantize multifractional gravity but there is no proof yet, the present proposal is understandably regarded as unripe.",
"On the other hand, the study of the multiscale Standard Models left gravity aside and was of more interest for the traditional QFT community, for which Lorentz invariance is a cornerstone and dimensional flow is an unnecessary concept.",
"Consequently, the main motivation of the theories was lost (question ).",
"The intrinsic difficulty in changing spacetime paradigm (a change of measure is relatively alien to “usual” quantum-gravity scenarios, with the exception of noncommutative spacetimes) and the lack of contact with observations have partially limited the reception of this proposal until now.",
"However, the important conceptual clarifications and simplifications carried out in the last year (mainly in refs.",
"[55], [56], [60]) and the obtainment of the first observational constraints ever on the theory [54], [55], [57], [58] are already contributing to boost its visibility.",
"It may also be relevant to recall that, contrary to popular quantum-gravity candidates, the case with $q$ -derivatives is the first and only known example of a theory of exotic geometry that is efficiently constrained by gravitational waves alone [57].",
"In this respect, as far as gravity waves are concerned, and until further notice, multifractional theories are proving themselves to be observationally as competitive as the usual quantum-gravity scenarios.",
"This is the type of result one might like to find in the context of quantum gravity at the interface between theory and experiment.",
"This and other results on phenomenology, together with the universality traits described in 0, make the multifractional paradigm not only a useful and general tool of comparison of different features in the landscape of quantum gravity, but also an independent theory that is legitimate to study separately.",
"In this sense, it is not strictly subordinate to the problem of quantum gravity at large.",
"It is also relevant to recall that the idea underlying multifractional theories is not a prerogative of this framework.",
"In other proposals [106], [107], [108], [109], [110], [111], [113], [112], an Ansatz for geometry and symmetries was made but no field-theory action thereon was given.",
"The multifractional paradigm not only makes the “fractal spacetimes” idea systematic, but it also provides an explicit form for the dynamics (questions and ).",
"In particular, the “fractal coordinates” of scale relativity correspond to our binomial geometric coordinates but written as a power-law profile with a scale-dependent exponent, $q\\sim x^{\\alpha (\\ell )}$ with $\\alpha (\\ell )=1+(\\alpha -1)/[1+(\\ell /\\ell _*)^{\\alpha -1}]$ [41]."
],
[
"Geometry and symmetries",
"$\\blacktriangleright $07$\\blacktriangleleft $ The choice of measure (REF ) with eq.",
"(REF ) and $\\alpha _\\mu =\\alpha _0,\\alpha \\,,\\qquad \\ell _*^\\mu =t_*,\\ell _*\\,,\\qquad \\ell _\\infty ^\\mu =t_\\infty ,\\ell _\\infty \\,,$ so often used in multifractional models, is completely ad hoc.",
"On one hand, why should we limit our attention to factorizable measures (REF )?",
"On the other hand, why should one choose the specific profile $q(x)$ in eq.",
"(REF )?tocsubsection0 Why should we limit our attention to factorizable and binomial measures?",
"A highly irregular geometry such as multidimensional fractals is generically described by a nonfactorizable measure $\\varrho (x^0,x^1,\\dots ,x^{D-1})$ .",
"There have been attempts to place a field theory on such geometries in the past [106], [107], [108] and even recently [76], [104], [105] but, unfortunately, and regardless of their level of rigorousness, their range of applicability to physical situations was severely restricted.",
"This was due to purely technical reasons, which include, for instance, the difficulty in finding a self-adjoint momentum operator and a self-adjoint Laplace–Beltrami operator compatible with the momentum transform.",
"In order to make progress, factorizable measures $d\\varrho (x^0,x^1,\\dots ,x^{D-1})=\\prod _\\mu dq^\\mu (x^\\mu )$ [eq.",
"(REF )] were considered starting from ref.",
"[40].",
"This choice has been successful in fully constructing a whole class of theories, in extracting observational constraints thereon, and in connecting efficiently with quantum-gravity frameworks.",
"If we compare the 25-year stalling of nonfactorizable models with the 5-year advancement of factorizable models from theory to experiments, the practical justification of (REF ) is evident.",
"Also, from the point of view of the phenomenology of dimensional flow, there is nothing wrong with factorizable measures: they have exactly the same scaling properties of nonfactorizable measures, which is a necessary and sufficient condition to have the same change in dimensionality.",
"Of course, it may be that Nature, if multiscale, is not represented by factorizable geometries, in which case we have to look into other proposals.",
"As discussed in ref.",
"[59], the natural generalization of multifractional geometries to nonfactorizable measures are, arguably, noncommutative spacetimes, which overcome the problems associated with nonfactorizability with the introduction of a noncommutative product.",
"The utility of factorizable multifractional theories is not exhausted even in that case because, although the mathematical and practical language describing noncommutative systems is different from the one employed in multiscale or fractal geometries, many contact points between these two frameworks are possible nonetheless [42], [59].",
"Once accepted the use of factorizable measures, according to the second flow-equation theorem the only possible choice is (REF ).",
"We can walk the logical path (REF )$\\rightarrow $ (REF )$\\rightarrow $ (REF ) as follows.",
"Equation (REF ) is an IR expansion with $D$ copies of an infinite number of free parameters (fractional exponents $\\alpha _{n,\\mu }$ , frequencies $\\omega _{n,\\mu }$ , amplitudes, and the scales $\\ell _n^\\mu $ and $\\ell _{\\infty ,n}^\\mu $ ), which means that one can fit any wished profile when no dynamical input on the values of such parameters is given (it is given in quantum gravities).",
"The first step in reducing this ambiguity in multifractional theories comes from the scale hierarchy itself, which is divided in two sets.",
"Omitting the index $\\mu $ from now on, the first is the set of scales $\\lbrace \\ell _n\\rbrace =\\lbrace \\ell _1\\geqslant \\ell _2\\geqslant \\dots \\rbrace $ characterizing regimes where the dimension of spacetime takes different values (we will see which values in question ); it is the scale hierarchy par excellence, the one defining dimensional flow via the polynomials of (REF ).",
"Superposed to that is the set of scales $\\lbrace \\ell _{\\infty ,n}\\rbrace $ , called harmonic structure in fractal geometry [41].",
"The harmonic structure does not govern the main traits of dimensional flow but it modulates it with a superposition of $n$ patterns of logarithmic oscillations; such modulation affects even scales much larger than $\\ell _\\infty $ , as cosmology shows [53], [58].",
"The scale hierarchies $\\lbrace \\ell _n\\rbrace $ and $\\lbrace \\ell _{\\infty ,n}\\rbrace $ are mutually independent but, from the derivation of eq.",
"(REF ), it is easy to convince oneself that $\\ell _n\\geqslant \\ell _{\\infty ,n}$ for each $n$ [60].",
"Thus, the long-range modulation of the harmonic structure and the theoretical “coupling” $\\ell _n\\leftrightarrow \\ell _{\\infty ,n}$ leads to the conclusion that the first multiscale effects we could observe in experiments would be at scales $\\gtrsim \\ell _*\\equiv \\ell _1$ , possibly modulated by log oscillations with scale $\\ell _\\infty \\equiv \\ell _{\\infty ,1}$ .",
"In other words, eq.",
"(REF ) is the approximation of (REF ) at scales $\\gtrsim \\ell _*$ .",
"But this is already sufficient to extract all relevant phenomenology.",
"Scales below $\\ell _*$ are too small to be constrained by experiments, and $\\ell _*$ acts as a sort of “screen” hiding the yet-unreachable microscopic structure of the measure at smaller scales.",
"Whatever happens at smaller scales, no matter the number of transient regimes with different dimensionalities from $\\ell _*$ down to Planck scales, from the point of view of a macroscopic observer the first transition to an anomalous geometry will occur near $\\ell _*$ .",
"Experiments constrain just this scale, the end of the multiscale hierarchy.",
"Thus, for all practical purposes there is no loss of generality in considering eq.",
"(REF ) instead of the too formal (REF ).",
"The further simplification from (REF ) to (REF ) is an isotropization of the scale hierarchies and dimensions to all spatial directions, while the time direction is left free to evolve independently.",
"Full isotropization is achieved when $\\alpha _0=\\alpha $ , but this is almost never needed in calculations.",
"If one wishes to consider geometries which are multiscale only in the time or space directions, it is sufficient to set $\\alpha _0\\ne 1,\\alpha =1$ or $\\alpha _0=1,\\alpha \\ne 1$ , respectively.",
"Having an isotropic spatial hierarchy (one scale $\\ell _*^i=\\ell _*$ for all directions) partially compensates for the restrictions of factorizability and makes observables easier to compute.",
"One can even invoke this choice as a symmetry principle defining the theory, since there is no reason a priori to have a strongly different dimensional flow along different spatial directions.",
"One can consider this as part of a multiscale version of the principle of special relativity.",
"$\\blacktriangleright $08$\\blacktriangleleft $ What is the parameter space of these theories?tocsubsection0 What is the parameter space?",
"There are severe theoretical priors on $(\\alpha _\\mu ,t_*,\\ell _*,t_\\infty ,\\ell _\\infty ,A,B,\\omega )$ .",
"– The fractional exponents $\\alpha _0$ and $\\alpha $ are taken within the interval $0\\leqslant \\alpha _\\mu \\leqslant 1\\,.$ The lower bound $\\alpha _\\mu \\geqslant 0$ guarantees that the UV Hausdorff dimension $\\alpha _\\mu $ of each direction in spacetime be non-negative, a minimal requirement if we want to be able to probe the geometry with conventional rules.",
"The upper bound $\\alpha _\\mu \\leqslant 1$ guarantees that the dimension in the UV is always smaller than the topological dimension $D$ .",
"Neither bound can be easily extended in the theories $T_1$ , $T_v$ , and $T_q$ .",
"The lower limit $\\alpha _\\mu \\geqslant 0$ can be replaced by $\\sum \\alpha _\\mu \\geqslant 0$ (e.g., ref.",
"[51]); in general, not all $\\alpha _\\mu $ can be negative, lest $d_\\textsc {h}\\simeq \\sum _\\mu \\alpha _\\mu <0$ [see eq.",
"(REF )].",
"However, this would lead to a negative-definite dimension either of time or of spatial slices, and it is not clear whether such a configuration would make sense physically.",
"On the other hand, if we took the upper limit arbitrarily large, we would get a dimensionally larger UV geometry that has very few examples in quantum gravity; still, there exist a minority of cases where $d_\\textsc {s}>D$ in the UV, as in $\\kappa $ -Minkowski spacetime [92], [94] o near a black hole [102].",
"However, multiscale corrections of physical observables are always of the form $v_\\mu (x^\\mu ):=\\partial _\\mu q^\\mu (x^\\mu )=1+O(|x^\\mu /\\ell _*|^{\\alpha _\\mu -1}).$ Therefore, an $\\alpha _\\mu >1$ always leads to a wrong IR limit, which is defined by the largest fractional charge $\\alpha _{\\mu ,n}$ in eq.",
"(REF ).",
"By definition, this is equal to 1 (nonanomalous scaling of the coordinates).",
"The special value $\\alpha _\\mu =\\frac{1}{2}$ at the center of the interval (REF ) plays a unique role, not only because it is the average representative of this class of geometries (it is typical and instructive to compare experimental constraints with $\\alpha _\\mu \\ll 1$ , $\\alpha _\\mu =1/2$ , and the standard geometry $\\alpha _\\mu =1$ ), but also because it signals a phase transition across a critical point in the theory [41].",
"Take, for instance, a scalar field in flat multifractional Minkowski space: $S_\\phi =\\int d^Dq(x)\\left[\\frac{1}{2}\\phi \\mathcal {K}\\phi -V(\\phi )\\right],$ where the signature of the Minkowski metric is $\\eta _{\\mu \\nu }=(-,+,\\cdots ,+)_{\\mu \\nu }$ and $\\mathcal {K}$ is the Laplace–Beltrami operator.",
"The engineering (scaling) dimension of the scalar field is $[\\phi ]=\\frac{d_\\textsc {h}-[\\mathcal {K}]}{2}\\,,$ where $d_\\textsc {h}$ is the scaling of the coordinate-dependent part of the measure $d^Dq$ .Note that $[d^Dq]=-D$ for the measure given by (REF ), (REF ), and (REF ) (or in the general case (REF )) because all elements in the sum scale in the same way.",
"However, what matters here is the scaling of the nonconstant terms of the measure, which is $-\\alpha _\\mu $ for each direction.",
"From eq.",
"(REF ) ($\\alpha _\\mu =\\alpha $ for all $\\mu $ ), it follows that $\\phi $ becomes dimensionless when $\\alpha =[\\mathcal {K}]/D$ .",
"In the model $T_1$ and in the theory $T_v$ , the Laplace–Beltrami operator is $T_1\\,:\\quad \\mathcal {K}=\\Box \\,,\\qquad T_v\\,:\\quad \\mathcal {K}=\\mathcal {D}_\\mu \\mathcal {D}^\\mu =\\frac{1}{\\sqrt{v}}\\Box \\left(\\sqrt{v}\\,\\cdot \\,\\right),\\qquad \\mathcal {D}_\\mu :=\\frac{1}{\\sqrt{v}}\\partial _\\mu \\left(\\sqrt{v}\\,\\cdot \\,\\right),$ where $v=v(x):=v_0(x^0)\\,v_1(x^1)\\cdots v_{D-1}(x^{D-1})\\geqslant 0\\,.$ Thus, $[\\mathcal {K}]=2$ at all scales and the critical value of the isotropic fractional exponent is $\\alpha =2/D$ .",
"This is precisely $1/2$ in $D=4$ dimensions.",
"Thus, in $T_1$ and $T_v$ the value (REF ) is somewhat preferred because the critical point is interpreted (as in Hořava–Lifshitz gravity) as a UV fixed point.",
"In the theories $T_q$ and $T_\\gamma $ on Minkowski spacetime, the Laplace–Beltrami operator is (Einstein's sum convention is used) [41], [45] $T_q\\,:\\quad \\mathcal {K}=\\Box _q=\\eta ^{\\mu \\nu }\\frac{\\partial }{\\partial q^\\mu }\\frac{\\partial }{\\partial q^\\nu }\\,,\\qquad T_\\gamma \\,:\\quad \\mathcal {K}=\\mathcal {K}_\\gamma \\,,$ where $\\mathcal {K}_\\gamma $ is composed by the operators ${}_\\infty \\partial ^{2\\gamma }$ and ${}_\\infty \\bar{\\partial }^{2\\gamma }$ , respectively, the Liouville and Weyl fractional derivative of order $2\\gamma $ [40], [121] (see question for the explicit expression).",
"The varying part of the Laplacian scales as $[\\Box _q]\\simeq 2\\alpha $ and $[\\mathcal {K}_\\gamma ]\\simeq 2\\gamma $ (in the UV) for the isotropic choices $\\alpha _\\mu =\\alpha $ and $\\gamma _\\mu =\\gamma $ , and the scalar field scales as $[\\phi ]=(D\\alpha -2\\gamma )/2$ .",
"For $\\alpha =\\gamma $ , there is no UV critical point but the behaviour of $T_q$ and $T_\\gamma $ is quite similar.",
"In the case with fractional derivatives $T_{\\gamma =\\alpha }$ , the range (REF ) is further shrunk to $1/2\\leqslant \\alpha _\\mu \\leqslant 1$ by requiring multifractional spacetime to be normed (that is, distances obey the triangle inequality) [40].There is no such restriction in $T_q$ , which has a well-defined norm for any positive value of $\\alpha $ [53].",
"Then, the value $\\alpha _\\mu =1/2$ is even more special being it the lowest possible in the theory.",
"Equation (REF ) is also supported independently by a recent connection with a heuristic estimate of quantum-gravity effects on measurement uncertainties [61], [62].",
"A parallel estimate, however, selects $\\alpha _\\mu =\\frac{1}{3}$ as an alternative preferred value [61], [62].",
"This lies in the region of parameter space where $T_{\\gamma =\\alpha }$ is not normed, but in questions and we will reconsider the restriction (REF ).",
"Last, the arguments of [61], [62] also fix the fractional exponents to the fully isotropic configuration $\\alpha _0=\\alpha \\,,$ although in general we will not enforce this relation.",
"– There is no prior on $t_*$ , $\\ell _*$ , $t_\\infty $ , and $\\ell _\\infty $ , except that they are positive; there are also other free parameters $E_*$ , $k_*$ , $E_\\infty $ , and $k_\\infty $ in the momentum-space measure.",
"One can reduce the number of free parameters by relating time and length scales by a unit conversion.",
"In a standard setting, one would make such conversion using Planck units.",
"Here, the most fundamental scale of the system is the one appearing in the full measure with logarithmic oscillations, denoted above as $\\ell _\\infty $ .",
"For the time direction one has a scale $t_\\infty $ , while in the measure in momentum space the fundamental energy $E_\\infty $ and momentum $p_\\infty $ appear.",
"Then, one may postulate that the scales $\\ell _*\\geqslant \\ell _\\infty $ , $t_*\\geqslant t_\\infty $ and $E_*\\leqslant E_\\infty $ are related by $E_*=\\frac{t_\\infty E_\\infty }{t_*}\\,,\\qquad t_*=\\frac{t_\\infty \\ell _*}{\\ell _\\infty }\\,,$ and so on with momenta.",
"The origin of these formulæ was left unexplained in [54], [55], but we can understand them better by a simple observation [61], [62].",
"The origin of $\\ell _\\infty ^\\mu $ is a partition of the scales in fractional complex measures.",
"As we will see in , the general real-valued leading-order solution of the flow equation has terms of the form $|x^\\mu /\\ell _*^\\mu |^{\\alpha +i\\omega }\\pm |x^\\mu /\\ell _*^\\mu |^{\\alpha -i\\omega }$ .",
"Splitting $|x^\\mu /\\ell _*^\\mu |^{\\alpha \\pm i\\omega }=\\lambda _{(\\mu )}|x^\\mu /\\ell _*^\\mu |^\\alpha |x^\\mu /\\ell _\\infty ^\\mu |^{\\pm i\\omega }$ , where $\\lambda _{(\\mu )}=(\\ell _\\infty ^\\mu /\\ell _*^\\mu )^{\\pm i\\omega }$ is purely imaginary and $\\ell _\\infty ^\\mu $ is an arbitrary length, we get the log-oscillating measure (REF ).",
"If $\\lambda _{(\\mu )}=\\lambda $ for all $\\mu $ (same partition in all directions) and a space-isotropic hierarchy, we get $(t_*/t_\\infty )^{\\pm i\\omega }=\\lambda _{(0)}=\\lambda _{(i)}=(\\ell _*/\\ell _\\infty )^{\\pm i\\omega }$ , hence the second equation in (REF ).",
"On the other hand, the scales $k_*^\\mu $ and $k_\\infty ^\\mu $ in momentum space are conjugate to $\\ell _*^\\mu $ and $\\ell _\\infty ^\\mu $ , in the sense that $k_*^\\mu \\propto 1/\\ell _*^\\mu $ and $k_\\infty ^\\mu \\propto 1/\\ell _\\infty ^\\mu $ with the same proportionality coefficient.",
"This is clear from dimensional arguments but it is made especially rigorous in $T_q$ , where the momentum measure () is completely determined by asking that the geometric momentum coordinate $p^\\mu (k^\\mu )$ be conjugate to $q^\\mu (x^\\mu )$ .",
"For each direction, one has $p^\\mu (k^\\mu )=\\frac{1}{q^\\mu (1/k^\\mu )}\\,,$ where all scales $\\ell _n^\\mu $ in (REF ) are replaced by energy-momentum scales $k_n^\\mu $ [53], [58].",
"Therefore, in the case of a binomial space-isotropic measure we have $k_*^\\mu \\ell _*^\\mu =k_\\infty ^\\mu \\ell _\\infty ^\\mu $ , which reduces to the first equation in (REF ) for $\\mu =0$ .",
"Having understood eq.",
"(REF ), one recalls that log-oscillating measures provide an elegant extension of noncommutative $\\kappa $ -Minkowski spacetime and explain why the Planck scale does not appear in the effective measure thereon [42] (see also question ).",
"In turn, this connection suggests that the fundamental scales in the log oscillations coincide with the Planck scales: $t_\\infty =t_{\\rm Pl}\\,,\\qquad \\ell _\\infty =\\ell _{\\rm Pl}\\,,\\qquad E_\\infty =k_\\infty =E_{\\rm Pl}=m_{\\rm Pl}c^2\\,.$ In four dimensions, $t_{\\rm Pl}=\\sqrt{\\hbar G/c^3}\\approx 5.3912 \\times 10^{-44}\\,\\text{s}$ , $\\ell _{\\rm Pl}=\\sqrt{\\hbar G/c^5} \\approx 1.6163 \\times 10^{-35}\\,\\text{m}$ , and $m_{\\rm Pl}=\\sqrt{\\hbar c/G} \\approx 1.2209\\times 10^{19}\\,\\text{GeV}c^{-2}$ .",
"Remarkably, eq.",
"(REF ) connects, via Newton constant, the prefixed multiscale structure of the measure with the otherwise independent dynamical part of the geometry.",
"Also, it makes the log-oscillating part of multiscale measures “intrinsically quantum” in the sense that Planck constant $\\hbar =h/(2\\pi )$ appears in the geometry.",
"An interesting follow-up of this concept will be seen in .",
"With eqs.",
"(REF ) and (REF ), one reduces the number of free scales of the binomial measure (REF ) with (REF ) to one: $t_*$ or $\\ell _*$ or $E_*$ .",
"– The real amplitudes $A$ and $B$ can be set to be non-negative, since they multiply trigonometric functions.",
"Also, they must be no greater than 1 in order to avoid negative distances [56].",
"Therefore, $0\\leqslant A,B\\leqslant 1\\,.$ – The frequency $\\omega $ stands out with respect to the other parameters because it takes a discrete set of values.",
"As mentioned in 0, the measures (REF ) and (REF ) can be derived from a pure calculation in fractal geometry.",
"The geometry of the measure without log oscillations is a random fractal, namely, a fractal endowed with symmetries whose parameters are randomized each time they are applied over the set [41], [115].",
"On the other hand, the measure with logarithmic oscillations corresponds to a deterministic fractal where the symmetry parameters are fixed (see footnote REF ).",
"For the binomial measure (REF ) with (REF ), $\\alpha _0=\\alpha $ , and only one frequency $\\omega >0$ , the underlying fractal $\\mathcal {F}=\\otimes _\\mu \\mathcal {F}_\\mu $ is given, for each direction, by the union of $N$ copies of itself rescaled by a factor $\\lambda _\\omega =\\exp (-2\\pi /\\omega )$ at each iteration.",
"Since the capacity of $\\mathcal {F}_\\mu $ is equal to the Hausdorff dimension and reads $d_\\textsc {c}=-\\ln N/\\ln \\lambda _\\omega =d_\\textsc {h}=\\alpha $ , the number of copies is $N=\\exp (-\\alpha \\ln \\lambda _\\omega )=\\exp (2\\pi \\alpha /\\omega )$ .",
"$N$ is a positive integer, so that $\\omega $ can only take the irrational valuesHere we discover that, for consistency, we can have $\\omega _\\mu =\\omega $ for all $\\mu $ only if the measure (REF ) is isotropic, $\\alpha _\\mu =\\alpha $ for all $\\mu $ .",
"This piece of information has never been used in the literature but it does not affect observations much.",
"$\\omega =\\omega _N:=\\frac{2\\pi \\alpha }{\\ln N}\\,.$ For $\\alpha =1/2$ and $N=2,3,\\ldots $ , we have $\\lambda _\\omega =1/N^2$ and $\\omega _2\\approx 4.53> \\omega _3\\approx 2.86>\\dots $ .",
"The case $N=1$ is not a fractal [eq.",
"(REF ) is ill defined then], while for each $N$ one has a different fractal in the same class [56].",
"Overall, the prior on $\\omega $ is $0<\\omega <\\omega _2=\\frac{2\\pi \\alpha }{\\ln 2}\\,,$ with irrational values picked in between.",
"$\\blacktriangleright $09$\\blacktriangleleft $ Are theories on multifractional spacetimes predictive and falsifiable?",
"The reason of this concern is the presence of a largely arbitrary element, the measure profiles $q^\\mu (x^\\mu )$ .",
"Their choice is dictated only by mathematics (in particular, by multifractal geometry) and by very general properties of dimensional flow, but not by physics and physical observations or experiments.",
"In most papers of the subject, the simplest form (REF ) with (REF ) of the measure is chosen, but still mathematically infinitely many other measures are possible which satisfy criteria of fractal geometry.",
"The ambiguity in the selection of the measure is equivalent to having infinitely many parameters of the theory and this renders the theory nonpredictive.",
"Nothing prevents one from using polynomial distributions or multiple logarithmic oscillations, such as in the measure (REF ).",
"The criterion of subjective simplicity should never be used to substitute the requirement of physical predictability.",
"Since the measure $q(x)$ is part of the definition of multifractional spacetimes, it cannot be verified and tested physically.",
"Or, in other words, it can always be fine-tuned to correctly reproduce any phenomenological data.",
"This means that these theories are not falsifiable.tocsubsection0 Are multifractional theories predictive and falsifiable?",
"We already answered to this in 0.",
"Theories with the binomial measure (REF ) are representative for the derivation of phenomenological consequences of the whole class of theories on multifractional spacetimes.",
"No matter what the detailed behaviour of the most general measure (REF ) is, the physical consequences are universal and the theory is back-predictive.",
"Furthermore, the ranges (REF ), (REF ), and (REF ) of the values of the free parameters in (REF ) with (REF ) is so limited that it is extremely easy to falsify the theory and exclude large portions of the parameter space $(\\alpha _0,\\alpha ,t_*,\\ell _*,t_\\infty ,\\ell _\\infty ,A,B,\\omega )$ , as done by observations of gravitational waves [57] and of the cosmic microwave background (CMB) [58].",
"$\\blacktriangleright $10$\\blacktriangleleft $ What is the physical motivation of the choice of measure?",
"I agree that, once the measure is chosen, the theory is fully predictive and experimental consequences can be derived.",
"The problem, however, is how to predict such measure in the first place, based on physical considerations.",
"If a measure $q(x)$ is fixed, then predictability and falsifiability are recovered, but then the new question is to physically motivate the choice of $q(x)$ .",
"I view its lack as the big drawback of this class of theories.tocsubsection What is the physical motivation of the choice of measure?",
"This type of remark, redundant with 0 and 0, used to arise before the formulation of the flow-equation theorems [60].",
"It is true that general theories of multifractional spacetimes with measure $q(x)$ unspecified lack predictability and falsifiability, but the same could be said about the general framework of “quantum field theory” with arbitrary interactions as opposed to the very concrete Standard Model.",
"In our case, the measure $q(x)$ is given by (REF ) or its approximation (REF ), which is the general factorizable solution of eq.",
"(REF ).",
"Any other measure corresponds to different regimes of the general expression (REF ).",
"The physical mechanism which determines the measure is precisely this flow equation (almost constant dimension in the IR) and it agrees completely with the arguments and calculations in fractal geometry invoked in early papers.",
"We say “physical” rather than “geometric” because the geometry expressed by dimensional flow has a direct impact on physical observables.",
"$\\blacktriangleright $11$\\blacktriangleleft $ You said that the binomial measure captures all the main properties of a multifractal geometry.",
"Can you illustrate that in a pedagogical way?tocsubsection Can you illustrate the multifractal properties of the binomial measure in a pedagogical way?",
"Consider the theory $T_q$ with binomial measure (REF ) with $F_\\omega =1$ .",
"From eq.",
"(REF ), we get the measure in momentum space $p^\\mu (k^\\mu )=k^\\mu \\left[1+\\frac{1}{\\alpha _\\mu }\\left|\\frac{k^\\mu }{k_*^\\mu }\\right|^{1-\\alpha _\\mu }\\right]^{-1}\\,.$ The eigenvalue equation of the Laplace–Beltrami operator $\\Box _q$ in eq.",
"(REF ) is $\\Box _q\\mathbb {e}(k,x)=-p^2(k)\\,\\mathbb {e}(k,x)$ , where $\\mathbb {e}(k,x)=\\exp [iq_\\mu (x_\\mu ) p^\\mu (k^\\mu )]$ and $p^2=p_\\mu p^\\mu $ .",
"In one dimension, this means that the spectrum of $-\\partial _q^2$ follows the distribution $p^2(k)=k^2\\left[1+\\frac{1}{\\alpha }\\left|\\frac{k}{k_*}\\right|^{1-\\alpha }\\right]^{-2}.$ (Including log oscillations, we would get the same spectrum but with a periodic modulation.)",
"We can compare this distribution with the ordinary spectrum $k^2$ and with the distribution $|k|^{2\\alpha }$ of a purely fractional measure (obtained by removing the factor 1 in eq.",
"(REF ) or by taking a fractional Laplacian [45]).",
"As one can appreciate from figure REF , the binomial profile (REF ) interpolates between the fractional and the integer spectra.",
"Figure: The binomial distribution () of the Laplacian eigenvalues (solid curve) corresponding to a bifractal, compared with the ordinary distribution k 2 k^2 (usual Laplacian, standard geometry, dotted line) and with the fractional distribution -|k| 2α -|k|^{2\\alpha } corresponding to a monofractal (dashed line).",
"Here k * =1k_*=1 and α=1/2\\alpha =1/2.The spectral distribution (REF ) plotted in the figure is an idealization (but a faithful one) of what is found in actual experiments or observations involving multifractals, not only in physics, but also in fields of research as diverse as geology, ethology, and human cognition [122], [123].",
"Adding an extra power law to the binomial measure (i.e., considering a trinomial measure with two scales $\\ell _1>\\ell _2$ ), one would bend the right part of the solid curve in the figure towards a different asymptote.",
"And so on.",
"$\\blacktriangleright $12$\\blacktriangleleft $ Are multifractional theories Lorentz invariant?tocsubsection Are multifractional theories Lorentz invariant?",
"No, they are not because factorizable measures (REF ) explicitly break rotation and boost invariance.",
"They are not Poincaré invariant either, because they also break translations.",
"An early nonfactorizable version $\\tilde{T}_1$ of multifractal theories proposed a Lorentz-invariant measure, working on the assumption that keeping as many Lorentz symmetries as possible would lead to viable phenomenology [76], [104], [105].",
"However, problems in finding an invertible Fourier transform associated with a self-adjoint momentum operator soon paved the way to the factorizable Ansatz (REF ), as described in 0.",
"As a consequence, the Poincaré symmetries ${x^{\\prime }}^\\mu =\\Lambda _\\nu ^{\\ \\mu }x^\\nu +b^\\mu $ of standard field theory on Minkowski spacetime are not enjoyed by multifractional field theories.",
"$\\blacktriangleright $13$\\blacktriangleleft $ Then, what are their local symmetries?tocsubsection What are their local symmetries?",
"The symmetries of the dynamics depend on the structure of the action.",
"Consider first the case without gravity (gravity will be included in question ).",
"All multifractional theories have the same measure $d^Dq(x)$ invariant under the nonlinear $q$ -Poincaré transformations $d^Dq(x)\\rightarrow d^Dq(x^{\\prime })$ , where for each individual $q^\\mu (x^\\mu )$ ${q}^\\mu ({x^{\\prime }}^\\mu )=\\Lambda _\\nu ^{\\ \\mu }q^\\nu (x^\\nu )+a^\\mu \\,,$ $\\Lambda _\\nu ^{\\ \\mu }$ are the usual Lorentz matrices, and $a^\\mu $ is a constant vector.",
"Seen as a change on the geometric coordinates $q^\\mu $ , this looks like a standard Poincaré transformation.",
"Seen as a transformation on the coordinates $x^\\mu $ , it is highly nonlinear and, in general, noninvertible.",
"Looking at eq.",
"(REF ), we cannot write $x^\\mu (q^\\mu )$ explicitly, unless we ignore log oscillations.",
"The $q$ -Poincaré transformations (REF ) are a symmetry of the measure but, in general, not of the dynamics.",
"Multifractional theories may still be invariant under other types of symmetries, which typically are a deformation of classical Poincaré and diffeomorphism symmetries.",
"Before discussing that, it is useful to recall a few basic facts on symmetry algebras.",
"Ordinary Poincaré symmetries are defined in three mutually consistent manners: as coordinate transformations, as an algebra of operators on a vector space, and as an algebra of field operators.",
"Meant as coordinate transformations, they are defined by eq.",
"(REF ).",
"At the level of operators on a vector space, they are defined by an infinite-dimensional representation of operators $\\hat{P}_\\mu =\\hat{p}$ and $\\hat{J}=\\hat{\\jmath }$ satisfying the undeformed Poincaré algebra $&& [\\hat{P}_\\mu ,\\hat{P}_\\nu ]=0\\,,\\\\&& [\\hat{P}_\\mu ,\\hat{J}_{\\nu \\rho }]=i(\\eta _{\\mu \\rho }\\hat{P}_\\nu -\\eta _{\\mu \\nu }\\hat{P}_\\rho )\\,,\\\\&& [\\hat{J}_{\\mu \\nu },\\hat{J}_{\\sigma \\rho }]=i(\\eta _{\\mu \\rho }\\hat{J}_{\\nu \\sigma }-\\eta _{\\nu \\rho }\\hat{J}_{\\mu \\sigma }+\\eta _{\\nu \\sigma }\\hat{J}_{\\mu \\rho }-\\eta _{\\mu \\sigma }\\hat{J}_{\\nu \\rho })\\,.\\nonumber \\\\ $ Ordinary time and spatial translations are generated by $\\hat{p}_\\mu :=-i\\partial _\\mu $ , while ordinary rotations and boosts are generated by $\\hat{\\jmath }_{\\mu \\nu }:=x_\\mu \\hat{p}_\\nu -x_\\nu \\hat{p}_\\mu $ .",
"The mass and spin of a particle field can be defined by finding first a vector space where the operators $\\hat{P}$ and $\\hat{J}$ act upon, and then the eigenstates of $\\hat{P}^2$ and $\\hat{W}^2$ (where $\\hat{W}^\\mu =\\epsilon ^{\\mu \\nu \\rho \\sigma }\\hat{P}_\\nu \\hat{J}_{\\rho \\sigma }/2$ is the Pauli–Lubanski pseudovector).",
"For a local relativistic theory, there is the further requirement that such vector space be invariant under representations of $\\hat{P}$ and $\\hat{J}$ .",
"At the level of field operators, ordinary Poincaré symmetries are encoded in some operators (without hat) $P_\\mu =P_\\mu [\\phi ^i]$ and $J_{\\mu \\nu }=J_{\\mu \\nu }[\\phi ^i]$ obeying the algebra () where commutators $[\\,\\cdot \\,,\\,\\cdot \\,]$ are replaced by Poisson brackets $\\lbrace \\,\\cdot \\,,\\,\\cdot \\,\\rbrace $ : $&& \\lbrace P_\\mu ,\\hat{P}_\\nu \\rbrace =0\\,,\\\\&& \\lbrace P_\\mu ,\\hat{J}_{\\nu \\rho }\\rbrace =i(\\eta _{\\mu \\rho }P_\\nu -\\eta _{\\mu \\nu }P_\\rho )\\,,\\\\&& \\lbrace J_{\\mu \\nu },\\hat{J}_{\\sigma \\rho }\\rbrace =i(\\eta _{\\mu \\rho }J_{\\nu \\sigma }-\\eta _{\\nu \\rho }J_{\\mu \\sigma }+\\eta _{\\nu \\sigma }J_{\\mu \\rho }-\\eta _{\\mu \\sigma }J_{\\nu \\rho })\\,.\\nonumber \\\\ $ In quantum gravity (including noncommutative spacetimes) and in multifractional classical theories, quantum and/or multiscale effects (in quantum gravity, multiscale effects are quantum by definition) can deform the above algebra of generators $A_i=\\hat{p}_\\mu ,\\hat{\\jmath }_{\\mu \\nu }$ in two ways.",
"One is by deforming the generators $A_i\\rightarrow A_i^{\\prime }$ , which corresponds to a deformation of classical symmetries.",
"For instance, one can have a momentum operator $A_i^{\\prime }=\\hat{P}_\\mu $ which generates a symmetry $x^\\mu \\rightarrow f(x^\\mu )$ analogous to the usual spatial and time translations $x^\\mu \\rightarrow x^\\mu +b^\\mu $ generated by $\\hat{p}_\\mu $ , such that $f(x_i)\\simeq x_i+b_i$ when quantum corrections are negligible.",
"In this case, one regards $\\hat{P}^\\mu $ as the generator of “deformed translations.” The other way in which an algebra is deformed is by a change in its structure.",
"For instance, given an algebra $\\lbrace A_i,A_j\\rbrace =f_{ij}^kA_k$ in ordinary spacetime or in a classical gravitational theory, one might end up with an algebra $\\lbrace A_i^{\\prime },A_j^{\\prime }\\rbrace =F(A_k^{\\prime })$ in a multifractional or quantum theory, which can be written also in terms of the generators of the classical symmetries, $\\lbrace A_i,A_j\\rbrace =G(A_k)$ , for some $G\\ne F$ .",
"Depending on the specific multifractional theory, we can have no symmetry algebra at all (case $T_1$ ), a symmetry algebra with deformed operators and deformed structure (case $T_v$ ), or a symmetry algebra with deformed operators but undeformed structure (case $T_q$ ).",
"Question retakes the topic of deformed algebras in the context of gravity.",
"Let $\\phi ^i$ be a generic family of matter fields (scalars, gauge vectors, bosons, and so on) and let $S[{\\rm weight},{\\rm derivatives},\\phi ^i]$ be the action of the theory with a specific choice of measure weight (REF ) and of derivatives in kinetic terms.",
"– In the model $T_1$ with ordinary derivatives, the Lagrangian is defined exactly as the usual one, for a scalar, for the Standard Model, and so on.",
"As an example, for a scalar field the action $S_1[v,\\partial ,\\phi ^i]=\\int d^Dx\\,v(x)\\,\\mathcal {L}_1[\\partial ,\\phi ^i]$ is eq.",
"(REF ) with $\\mathcal {K}=\\Box $ .",
"Therefore, the Lagrangian $\\mathcal {L}_1$ is invariant under ordinary Lorentz transformations but the action $S_1$ is not.",
"Since the profiles $q^\\mu (x^\\mu )$ are given a priori by the flow equation (or by fractal geometry), the dynamics will not enjoy any symmetry at all.",
"In other words, the structure of the geometric coordinates $q^\\mu (x^\\mu )$ is irreconcilable with that of the differential operators $\\partial _\\mu $ .",
"Said in a more formal way, the operator $\\hat{p}_\\mu $ generating ordinary translations is not self-adjoint [49] with respect to the natural inner product on the space of test functions defined on multifractional Minkowski space: $(f_1,\\hat{p}_\\mu f_2):=\\int _{-\\infty }^{+\\infty }d^Dq(x)\\,f_1(x)\\,\\hat{p}_\\mu \\,f_2(x)\\ne (\\hat{p}_\\mu f_1,f_2)\\,.$ Consequently, the system is not translation invariant and ordinary momentum is not conserved.",
"The proof for rotations and boosts is similar.",
"This absence of symmetries is clearly a problem of this theory.",
"Notice that one can define a self-adjoint momentum operator $\\hat{P}_\\mu :=-\\frac{i}{2}\\,\\left[\\partial _\\mu +\\frac{1}{v}\\partial _\\mu (v\\,\\cdot \\,)\\right]=-i\\left(\\partial _\\mu +\\frac{\\partial _\\mu v}{2v}\\right)\\,,$ but this is equivalent to the momentum operator in $T_v$ .",
"As a matter of fact, $T_v$ was born as the “upgrade” of $T_1$ and we should talk about three multifractional theories ($T_v$ , $T_q$ , and $T_\\gamma $ ) rather than four.",
"For this reason, we regard $T_1$ only as a phenomenological model, in the sense of being inspired by the multiscale principle without the pretension of being a rigorous theoretical construct.",
"– In the theory $T_v$ with weighted derivatives, field Lagrangians are defined by replacing standard operators $\\partial _\\mu $ with the weighted derivatives defined in eq.",
"(REF ): $S_v[v,\\mathcal {D},\\phi ^i]=\\int d^Dx\\,v(x)\\,\\mathcal {L}_v[\\mathcal {D},\\phi ^i]\\,.$ The scalar-field example is eq.",
"(REF ) with $\\mathcal {K}=\\mathcal {D}_\\mu \\mathcal {D}^\\mu $ .",
"The action of the Standard Model of electroweak and strong interactions can be found in ref.",
"[55] and in question .",
"Just like in the case with ordinary derivatives, the symmetry structure of the measure and of the operators $\\mathcal {D}_\\mu $ is different and $S_v$ is not invariant under standard Poincaré symmetries.",
"However, contrary to $T_1$ , the theory $T_v$ is invariant under new symmetries encoded in deformed Poincaré transformations, but only in the absence of nonlinear interactions of third or higher order in at least one field.",
"Let us explain.",
"The weighted derivative defines the operator $\\hat{P}_\\mu :=-i\\mathcal {D}_\\mu $ [clearly equivalent to eq.",
"(REF )], which is self-adjoint: $(f_1,\\hat{P}_\\mu f_2)&:=&-i\\int _{-\\infty }^{+\\infty }d^Dq\\,f_1\\,\\mathcal {D}_\\mu \\,f_2=-i\\int _{-\\infty }^{+\\infty }d^Dx\\,v\\,f_1\\,\\mathcal {D}_\\mu \\,f_2\\nonumber \\\\&=&-i\\int _{-\\infty }^{+\\infty }d^Dx\\,\\sqrt{v}\\,f_1\\,\\partial _\\mu (\\sqrt{v}f_2)=i\\int _{-\\infty }^{+\\infty }d^Dx\\,\\sqrt{v}\\,\\partial _\\mu (\\sqrt{v}f_1)\\,f_2\\nonumber \\\\&=&(\\hat{P}_\\mu f_1,f_2)\\,.$ This operator generates “fractional” translations rather than ordinary ones.",
"The transformation law of fields can be worked out explicitly [48], [55], but here suffice it to note that a field redefinition $\\varphi ^i:=\\sqrt{v}\\,\\phi ^i$ permits to write fractional expressions as ordinary ones, e.g., $\\hat{P}_\\mu \\phi ^i=v^{-1/2}\\hat{p}_\\mu \\varphi ^i$ .",
"The same holds for the generators of rotations and boosts.",
"Thus, it is possible for the theory to be invariant under weighted Poincaré transformations (deformed translations, rotations, and boosts) generated by $T_v\\,:\\qquad \\hat{P}_\\mu :=-i\\mathcal {D}_\\mu =\\frac{1}{\\sqrt{v}}\\,\\hat{p}_\\mu \\,\\sqrt{v}\\,,\\qquad \\hat{J}_{\\nu \\rho }:=x_\\nu \\hat{P}_\\rho -x_\\rho \\hat{P}_\\nu = \\frac{1}{\\sqrt{v}}\\,\\hat{\\jmath }_{\\nu \\rho }\\,\\sqrt{v}\\,,$ which satisfy the undeformed algebra () or its field-operator equivalent (), but only if the action has no third- or higher-order terms in one or more fields.",
"If it does, then the algebraic structure () and () is deformed.",
"In the scalar-field example, it is easy to show that, given the Hamiltonian and spatial momentum (here $i=1,\\dots ,D-1$ are spatial directions) $H &:=& P^0 = \\int d^{D-1}{\\bf x}\\,v({\\bf x})\\,\\left[\\frac{1}{2}(\\mathcal {D}_t\\phi )^2+\\frac{1}{2}\\mathcal {D}_i\\phi \\mathcal {D}^i\\phi +V(\\phi )\\right]\\,,\\\\P^i &=&-\\int d^{D-1}{\\bf x}\\,v({\\bf x})\\,\\mathcal {D}_t\\phi \\mathcal {D}^i\\phi \\,,$ one has [48] $\\lbrace P^i,H\\rbrace =\\int d^{D-1}{\\bf x}\\,\\partial ^i v({\\bf x})\\left[\\frac{1}{2} \\phi \\,V_{,\\phi }(\\phi )-V(\\phi )\\right]\\,,$ which vanishes only if $V\\propto \\phi ^2$ .",
"Therefore, eq.",
"(REF ) is violated and the Poincaré algebra is deformed not only in the form of the generators, but also in its structure.",
"Similar violations occur in eqs.",
"() and ().",
"It is important to distinguish between the symmetries of a generic field theory with weighted derivatives and the specific field theory describing natural phenomena.",
"In the second case, the theory $T_v$ is invariant under weighted Poincaré transformations.",
"The ${\\rm SU}(3)\\otimes {\\rm SU}(2)\\otimes {\\rm U}(1)$ Standard Model of electroweak and strong interactions has been constructed in ref.",
"[55] for the theory $T_v$ .",
"The only nonlinear terms arising are those of gauge derivatives and in the Higgs potential.",
"The first type is of the form $\\bar{\\psi }A_\\mu \\gamma ^\\mu \\psi $ , linear in gauge vectors and quadratic in fermions (see ), so that all spacetime dependence can be reabsorbed in field and couplings redefinitions.",
"Since there is no $O[(\\phi ^i)^3]$ or higher-order term, the structure of the Poincaré algebra is undeformed (although the generators are).",
"The Higgs potential does have third- and fourth-order terms (again, see ), but their measure dependence is reabsorbed in the fields and in the couplings.",
"The crucial point here, which solves the apparent contradiction with eq.",
"(REF ), is that not only fields can be redefined, but also the physical couplings [55].",
"– In the theory $T_q$ with $q$ -derivatives, the action is defined by taking the ordinary action (of a scalar field, of the Standard Model, of gravity, and so on) and replacing all coordinates $x^\\mu $ therein with the geometric coordinates $q^\\mu (x^\\mu )$ : $S_q[v,\\partial _q,\\phi ^i]=\\int d^Dx\\,v(x)\\,\\mathcal {L}_q[\\partial _{q(x)},\\phi ^i]=S[v,v^{-1}\\partial _x,\\phi ^i]\\,.$ Clearly, the theory is invariant under $q$ -Poincaré transformations (REF ).",
"The symmetry algebra is undeformed, eq.",
"() with (obviously, no Einstein summation) $T_q\\,:\\qquad \\hat{P}_\\mu :=-i\\partial _{q^\\mu }=\\frac{1}{v_\\mu }\\,\\hat{p}_\\mu \\,,\\qquad \\hat{J}_{\\nu \\rho }:=x_\\nu \\hat{P}_\\rho -x_\\rho \\hat{P}_\\nu \\,,$ where $\\partial _{q^\\mu }=\\partial /\\partial q^\\mu (x^\\mu )=[v_\\mu (x^\\mu )]^{-1}\\partial _\\mu $ .",
"These operators are quite different from the $T_v$ case (REF ) but, just like that, they describe deformed translations, rotations, and boosts.",
"– In the theory $T_\\gamma $ with fractional derivatives, the action sports fractional derivatives (or differintegrals) “$\\partial ^\\gamma $ ,” for which there are many available definitions in the literature (see [40] for a review and [121] for a textbook on the subject).",
"For example, in refs.",
"[40], [41] the left and right Caputo derivatives were preferred among other choices to define $T_{\\gamma =\\alpha }$ , because of the possibility to define geometric coordinates such that $\\partial ^\\alpha _\\mu q^\\nu =\\delta _\\mu ^\\nu $ ; later on, the Liouville and Weyl derivatives were chosen in the second definition of eq.",
"(REF ), since they are one the adjoint of the other [45].",
"Omitting the $\\mu $ index everywhere, along the $\\mu $ direction the Liouville and Weyl derivatives are ${}_\\infty \\partial ^\\alpha _x f(x) &:=& \\frac{1}{\\Gamma (1-\\alpha )}\\int _{-\\infty }^{+\\infty }dx^{\\prime }\\, \\frac{\\theta (x-x^{\\prime })}{(x-x^{\\prime })^\\alpha }\\partial _{x^{\\prime }}f(x^{\\prime })\\,,\\qquad 0\\leqslant \\alpha <1\\,,\\\\{}_\\infty \\bar{\\partial }^\\alpha _x f(x) &:=& \\frac{1}{\\Gamma (1-\\alpha )}\\int _{-\\infty }^{+\\infty }dx^{\\prime }\\, \\frac{\\theta (x^{\\prime }-x)}{(x^{\\prime }-x)^\\alpha }\\partial _{x^{\\prime }}f(x^{\\prime })\\,,\\qquad 0\\leqslant \\alpha <1\\,,$ where $\\theta $ is the Heaviside step function.",
"In particular, one can consider the combination $\\tilde{\\mathcal {D}}^\\alpha _x:=\\frac{1}{2}({}_\\infty \\partial ^\\alpha _x+{}_\\infty \\bar{\\partial }^\\alpha _x) &=&\\frac{1}{2\\Gamma (1-\\alpha )}\\int _{-\\infty }^{+\\infty }dx^{\\prime }\\,\\left[\\frac{\\theta (x-x^{\\prime })}{(x-x^{\\prime })^\\alpha }+\\frac{\\theta (x^{\\prime }-x)}{(x^{\\prime }-x)^\\alpha }\\right]\\partial _{x^{\\prime }}\\nonumber \\\\&=&\\frac{1}{2\\Gamma (1-\\alpha )}\\int _{-\\infty }^{+\\infty }\\frac{dx^{\\prime }}{|x-x^{\\prime }|^\\alpha }\\partial _{x^{\\prime }}\\,.$ Since the definition of ${}_\\infty \\partial ^\\alpha =I^{1-\\alpha }\\partial $ is inspired by the Cauchy formula for the $n$ -repeated integration $I^n$ , when $\\alpha \\rightarrow 1$ one obtains the ordinary derivative $\\partial _x$ in both the Liouville and Weyl case; this explains the prefactor $1/2$ in eq.",
"(REF ), $\\tilde{\\mathcal {D}}^1_x=\\partial _x$ .",
"Caputo left and right derivatives are defined as in eq.",
"(REF ) but with integration domains $(0,+\\infty )$ and $(-\\infty ,0)$ , respectively.",
"The theory $T_\\gamma $ is invariant under $q$ -Lorentz transformations [eq.",
"(REF ) with $a^\\mu =0$ ] but, contrary to the $T_q$ case, only up to boundary terms and only at individual plateaux in dimensional flow (i.e., in the toy-model limit of no-scale fractional geometries, pure power-law measure $q\\propto |x|^\\alpha $ [40], [41]).",
"Therefore, $q$ -Lorentz invariance is exact in $T_q$ (and extendable to $q$ -Poincaré invariance) but approximate in $T_\\gamma $ , and the phenomenology of the two theories is thus expected to be more similar than between $T_\\gamma $ and $T_v$ or $T_1$ .",
"To show this, one must first extend the definition of fractional derivatives to multiscale geometries.",
"This can be done in two ways, which we will explore in greater detail in the future.",
"One is to implement multiscaling “externally” with respect to the definition of fractional derivatives.",
"In this case, one defines a superposition of fractional derivatives (indices $\\mu $ inert, as usual) $\\tilde{\\mathcal {D}}_\\mu :=\\sum _n g_{\\mu ,n}\\tilde{\\mathcal {D}}^{\\alpha _{\\mu ,n}}_\\mu ,$ where the scale hierarchy appears in the coefficients $g_{\\mu ,n}(\\ell _n^\\mu )$ .",
"This definition of multifractional derivative is self-adjoint in the scalar product with integration $dx^\\mu $ .",
"To have it self-adjoint with integration measure $dq^\\mu =dx^\\mu v_\\mu (x^\\mu )$ , it is sufficient to decorate eq.",
"(REF ) with weight factors, so that the operator $\\mathcal {K}_\\gamma $ in eq.",
"(REF ) becomes $\\mathcal {K}_\\gamma =\\frac{1}{\\sqrt{v}}\\tilde{\\mathcal {D}}_\\mu \\tilde{\\mathcal {D}}^\\mu \\left(\\sqrt{v}\\,\\cdot \\,\\right)\\,.$ Expression (REF ) is similar to the so-called distributed-order fractional derivatives D [124], [125], [126], [127], [128], [129], [130], [131], where an integration over the parameter $\\alpha $ is performed instead of the sum: ${\\rm D}:=\\int _0^1d\\alpha \\,m(\\alpha )\\,\\partial ^\\alpha $ , where $m(\\alpha )$ is a distribution on the interval $[0,1]$ .",
"We do not know whether a continuous distribution would be more convenient that the discrete sum (REF ).",
"In either case, a global notion of $q$ -Lorentz invariance does not exist, and one can only count on a forcefully approximated “local” version of the symmetries (REF ).",
"It may be that other “fractional Poincaré” symmetries are enforced, but we have not checked it yet.",
"The other possibility is to have an “internal” notion of multiscaling, that is, we can modify the definition of fractional derivatives so that to include the scale hierarchy within.",
"To generalize eq.",
"(REF ) to multiscale profiles such as (REF ) or (REF ), we define (index $\\mu $ restored and not summed over) ${}_q\\mathcal {D}_\\mu :=\\int _{-\\infty }^{+\\infty }\\frac{d{x^{\\prime }}^\\mu }{q^\\mu (x^\\mu -{x^{\\prime }}^\\mu )}\\frac{\\partial }{\\partial {x^{\\prime }}^\\mu }\\,.$ This expression is strikingly similar to a not much known proposal for so-called variable-order fractional derivatives [129].",
"The main difference is that, for us, the distribution $q(x-x^{\\prime })$ is determined from the start by the second flow-equation theorem.",
"At plateaux in dimensional flow, eq.",
"(REF ) reduces to the mixed Liouville–Weyl derivative (REF ) and there is a manageable fractional calculus in all regions of interest, i.e., in the deep UV, in the IR, and in whatever intermediate region the dimension of spacetime is approximately constant (we expect dimension gradients to be difficult or even impossible to detect).",
"Equation (REF ) should be further explored to clarify the role of boundary terms and discontinuities.The junction of the left and right derivatives in eq.",
"(REF ) masks a potentially tricky point at $x=0$ [40].",
"In particular, one will have to show that ${}_q\\mathcal {D}_\\mu q^\\mu \\simeq 1$ at least at each plateau in dimensional flow.",
"The constant coefficients inside $q^\\mu $ should be chosen so that ${}_q\\mathcal {D}\\rightarrow \\partial $ at large scales (roughly speaking, in the $q\\rightarrow \\mathbb {1}$ limit) and ${}_q\\mathcal {D}\\simeq \\tilde{\\mathcal {D}}^{\\alpha _n}$ at the $n$ -th plateau.",
"Weight factors can appear to the left and to the right of (REF ) to guarantee self-adjointness with respect to the measure $d^D q(x)$ [45].",
"An alternative to eq.",
"(REF ) is thus $\\mathcal {K}_\\gamma =\\frac{1}{\\sqrt{v}}\\,{}_q\\mathcal {D}_\\mu \\,{}_q\\mathcal {D}^\\mu \\left(\\sqrt{v}\\,\\cdot \\,\\right),$ and the general action of the theory $T_\\gamma $ in the absence of gravity is $S_\\gamma [v,{}_q\\mathcal {D},\\phi ^i]=\\int d^Dq(x)\\,\\mathcal {L}_\\gamma [{}_q\\mathcal {D},\\phi ^i]\\,,$ which has no integer picture associated.",
"A tentative proposal for the Lagrangian of a scalar field is, modulo weight factors, $\\mathcal {L}_\\gamma [\\phi ]=-(1/2) {}_q\\mathcal {D}_\\mu \\phi {}_q\\mathcal {D}^\\mu \\phi -V(\\phi )$ .",
"Fractional derivatives, either of fixed order as eq.",
"(REF ) or multiscale as in eqs.",
"(REF ) and (REF ), have a technical complication which is one of the reasons why the dynamics of $T_\\gamma $ has not been studied adequately so far: the Leibniz rule $\\partial ^\\gamma (fg)=(\\partial ^\\gamma f) g+f(\\partial ^\\gamma g)+\\dots $ is rather messy in the “...” part and it complicates the equations of motion (after integrating by parts to calculate the field variations).",
"Therefore, the kinetic term $-({}_q\\mathcal {D}\\phi )^2/2$ is not equivalent to $\\phi \\mathcal {K}_\\gamma \\phi /2$ .",
"This issue will be tackled in a separate publication.",
"Another, more formal way to get a multifractional derivative is via the differentials of the theory.",
"In $T_\\gamma $ , the exterior derivative $d$ can be replaced by a new definition $\\mathbb {d}$ which was proposed in ref.",
"[40] for a fractional measure $q\\propto |x|^\\alpha $ .",
"Instead of repeating that discussion, we extend it directly to multiscale geometries and define $\\mathbb {d}$ implicitly by $\\mathbb {d}q^\\mu (x^\\mu )=q^\\mu (dx^\\mu )\\,,$ so that $\\mathbb {d}q\\sim \\mathbb {d}x+\\mathbb {d}x^\\alpha +\\ldots =dx+(dx)^\\alpha +\\dots $ .",
"The line element in geometric notation is $\\mathbb {d}q(s)=\\sqrt{g_{\\mu \\nu } \\mathbb {d}q^\\mu (x^\\mu )\\otimes \\mathbb {d}q^\\nu (x^\\nu )}\\,,$ or, in fractional notation with ordinary differential, $q(ds)= \\sqrt{g_{\\mu \\nu } q^\\mu (dx^\\mu )\\otimes q^\\nu (dx^\\nu )}\\,.$ The most natural multifractional derivative in this formalism is $\\mathbb {D}_\\mu :=\\frac{\\mathbb {d}}{\\mathbb {d}q^\\mu }\\,.$ The $T_{\\gamma =\\alpha }\\cong T_q$ approximation corresponds to $\\mathbb {d}\\simeq d$ and $\\mathbb {D}_\\mu \\simeq d/dq^\\mu (x^\\mu )$ .",
"In the case of eq.",
"(REF ), there is no need to insert weight factors and the Laplace–Beltrami operator in flat space is $\\mathcal {K}_\\gamma =\\mathbb {D}_\\mu \\mathbb {D}^\\mu \\,.$ The general action in the absence of gravity is $S_\\gamma [v,\\mathbb {D},\\phi ^i]=\\int d^Dq(x)\\,\\mathcal {L}_\\gamma [\\mathbb {D},\\phi ^i]\\,,$ where for a scalar field $\\mathcal {L}_\\gamma [\\phi ]=-(1/2)\\mathbb {D}_\\mu \\phi \\mathbb {D}^\\mu \\phi -V(\\phi )$ or $\\mathcal {L}_\\gamma [\\phi ]=-(1/2)\\phi \\mathcal {K}_\\gamma \\phi -V(\\phi )$ ; operator ordering issues will have to be studied carefully.",
"The integral $\\int $ can be replaced by a multiscale “geometric” integral $$ generalizing the fractional operator of [40], so that $\\int d^Dq(x)=\\mathbb {d}^Dq(x)$ and one can completely recast the system in geometric notation.",
"Part of future work will also be to see if we can identify ${}_q\\mathcal {D}_\\mu $ with eq.",
"(REF ) at the plateaux of dimensional flow, but we anticipate a positive answer provided ${}_q\\mathcal {D}\\simeq \\tilde{\\mathcal {D}}^{\\alpha _n}$ at the $n$ -th plateau.",
"In fact, there one has $\\mathbb {d}q\\simeq (dx)^{\\alpha _n}$ , so that $\\mathbb {D}\\simeq \\mathbb {d}/(dx)^{\\alpha _n}=\\tilde{\\mathcal {D}}^{\\alpha _n}$ is the fractional derivative (REF ) of $\\alpha _n$ -th order.",
"Moreover, notice the invariance of definitions (REF ) and (REF ) under translations, ${}_q\\mathcal {D}_{x-\\bar{x}}={}_q\\mathcal {D}_x\\,,\\qquad \\mathbb {D}_{x-\\bar{x}}=\\mathbb {D}_x\\,,$ for which the integration domain on the whole real line in (REF ) is crucial.",
"Equations (REF )–(REF ) are given here for the first time.",
"To summarize, with the derivative (REF ) the Lagrangian is invariant under $q$ -Lorentz transformation, but clearly not under $q$ -Poincaré (REF ): ${}_q\\mathcal {D}_x$ is invariant under a translation in $x$ but not under a translation in $q$ .",
"On the other hand, a preliminary inspection seems to find that $\\mathbb {D}_x$ is $q$ -Poincaré invariant.",
"Poincaré (in the absence of gravity) or local Lorentz symmetry (with gravity, in inertial frames) are restored at large scales and late times, where $q^\\mu (x^\\mu )\\simeq x^\\mu $ and the geometry measure becomes the standard Lebesgue measure on a smooth manifold.",
"Whether a residual violation of Lorentz invariance is observable and what constraints on it are, will be the subject of section .",
"Other local symmetries of multifractional theories are the gauge symmetries of QFT, which are deformations of the usual gauge invariance in ordinary Minkowski spacetime.",
"These are discussed in ref.",
"[55].",
"$\\blacktriangleright $14$\\blacktriangleleft $ Is diffeomorphism invariance respected in multifractional theories?tocsubsection Is diffeomorphism invariance respected in multifractional theories?",
"No, except in $T_v$ in the absence of matter and in $T_q$ .",
"The reason is that the measure weight (REF ) is not a scalar field but a fixed coordinate profile.",
"Therefore, any coordinate transformation would change $v(x)$ , which is not allowed by the flow equation (REF ) if the measure is imposed to be factorizable.",
"The lack of diffeomorphism invariance in most multifractional theories is not in contradiction with the fact that all of them are covariant.",
"The reason is that covariance and diffeomorphism (in short, diffeo) invariance can be confused without damage in the absence of a nondynamical structure, while they are clearly separated concepts in the presence of such a structure (the measure, in the multifractional case).To illustrate the point, we report the general discussion made in ref.",
"[53] and inspired by [132].",
"Let $\\mathcal {M}$ be a manifold endowed with some nondynamical structure $\\Sigma $ , and obeying the equations of motion $F[\\phi ^i,\\Sigma ]=0$ .",
"Covariance determines that, under a diffeomorphism $f$ , the transformed fields $f\\cdot \\phi ^i$ obey equations of motions with transformed nondynamical structure: $F[\\phi ^i,\\Sigma ]=0=F[f\\cdot \\phi ^i,f\\cdot \\Sigma ]$ .",
"On the other hand, diffeomorphism invariance limits the amount of nondynamical structure: it requires that the same equation of motion be satisfied by the fields and their transforms, $F[\\phi ^i,\\Sigma ]=0=F[f\\cdot \\phi ^i,\\Sigma ]$ (active diffeomorphism), or, equivalently, that any solution $\\phi ^i$ of the equations of motion is also solution of a different set of equations parametrized by a transformed nondynamical structure, $F[\\phi ^i,\\Sigma ]=0=F[\\phi ^i,f\\cdot \\Sigma ]$ (passive diffeomorphism).",
"To answer in more detail, we have to turn gravity on and consider a curved embedding manifold (so far in this review, we have discussed only field theories on flat Minkowski spacetime).",
"For instance, the multifractional action of gravity with a minimally coupled matter scalar field is of the form $S=S[g]+\\int \\,d^Dx\\,v\\,\\sqrt{-g}\\,\\mathcal {L}[\\phi ]\\,,$ where $S[g]$ is the action for the metric (which can be found in ref.",
"[53] and in question for $T_1$ , $T_v$ , and $T_q$ ), $g$ is the determinant of the metric, $\\mathcal {L}[\\phi ]$ is the Lagrangian in (REF ) of the scalar and, everywhere in the total action, indices are contracted with the metric $g_{\\mu \\nu }$ .",
"In $T_1$ and $T_v$ , there is no field or metric redefinition absorbing completely the dependence on the trivial measure.",
"Even if one can do so in a Standard-Model matter sector, measure factors pop back in the gravitational action and in any non-Standard-Model matter sector with nonlinear interactions [53].",
"Still, in the case of $T_v$ we can identify the matter sector as the responsible for violating diffeo invariance: in the absence of matter, the algebra of the canonical constraints of gravity is preserved (see ) [59].",
"On the other hand, one immediately recognizes that the theory $T_q$ is diffeo invariant under active diffeomorphisms with respect to the geometric coordinates $q^\\mu (x^\\mu )$ , but only in the absence in $S_q[g]$ of geometric Lagrangian terms made purely by the measure weight (question ).",
"The gravitational sector of the theory $T_\\gamma $ has not been built yet, and presently we cannot comment on that.",
"However, by analogy with the theory with $q$ -derivatives and encouraged by the $T_{\\gamma =\\alpha }\\cong T_q$ approximation, it should be possible to generalize the notion of diffeo invariance, at least approximately at the scales corresponding to plateaux in dimensional flow.",
"$\\blacktriangleright $15$\\blacktriangleleft $ What is the dimension of multifractional spacetimes?tocsubsection What is the dimension of multifractional spacetimes?",
"It is not difficult to compute the dimensions $d_\\textsc {h}$ , $d_\\textsc {s}$ , and $d_\\textsc {w}$ [41], [45], [49].",
"The volume of a $D$ -hypercube of size $\\ell $ oriented along the Cartesian axes with a corner at $x^\\mu =0$ is $\\mathcal {V}(\\ell )\\propto \\prod _\\mu q^\\mu (\\ell )$ .",
"Centering the hypercube elsewhere, with a corner at $x^\\mu =\\bar{x}^\\mu $ would only bring the change $q^\\mu (\\ell )\\rightarrow q^\\mu (\\ell -\\bar{x}^\\mu )- q^\\mu (\\bar{x}^\\mu )$ , which does not change the $\\ell $ scaling of $\\mathcal {V}$ .",
"Using a $D$ -ball instead of the cube would lead (up to immaterial centering effects) to $\\mathcal {V}(\\ell )\\propto \\sqrt{\\sum _\\mu [q^\\mu (\\ell )]^2}$ , again with no new impact on the overall scaling.",
"Writing eq.",
"(REF ) evaluated at $x^\\mu =\\ell $ for all $\\mu $ as $q^\\mu (\\ell )=\\ell [1+\\sum _n b_{\\mu ,n}(\\ell /\\ell _n^\\mu )^{\\alpha _{\\mu ,n}-1}F_n(\\ell )]$ , the Hausdorff dimension (REF ) is (no index contraction, of course) $d_\\textsc {h}(\\ell )=\\sum _\\mu \\frac{1+\\sum _n b_{\\mu ,n}(\\ell /\\ell _n^\\mu )^{\\alpha _{\\mu ,n}-1}[\\alpha _{\\mu ,n}+F_n^{\\prime }(\\ell )]}{1+\\sum _n b_{\\mu ,n}(\\ell /\\ell _n^\\mu )^{\\alpha _{\\mu ,n}-1}F_n(\\ell )}\\,,$ where $F^{\\prime }_n=dF_n(\\ell )/d\\ln \\ell $ .",
"This result is independent of the dynamics and is therefore valid for all multifractional theories.",
"It is easy to convince oneself that this expression has all the properties we would expect in dimensional flow.",
"In the IR, $d_\\textsc {h}\\simeq D$ , while at the $n$ -th plateau $d_\\textsc {h}\\simeq \\sum _\\mu \\alpha _{\\mu ,n}$ .",
"Taking only $n=1$ and one scale $\\ell _1=\\ell _*$ for all directions [binomial measure (REF ) with (REF )], we have $d_\\textsc {h}(\\ell )=\\sum _\\mu \\frac{1+b_\\mu (\\ell /\\ell _*)^{\\alpha _\\mu -1}[\\alpha _\\mu +F_\\omega ^{\\prime }(\\ell )]}{1+b_\\mu (\\ell /\\ell _*)^{\\alpha _\\mu -1}F_\\omega (\\ell )}\\,.$ Near the IR, an expansion of (REF ) for $\\ell /\\ell _*\\gg 1$ yields eq.",
"(REF ) with $c_\\mu =1-\\alpha _\\mu $ .",
"Thus, $d_\\textsc {h}\\simeq D$ at large spacetime scales.",
"Near the UV ($\\ell /\\ell _*\\ll 1$ ), $d_\\textsc {h}\\stackrel{\\rm UV}{\\simeq } \\sum _\\mu \\frac{\\alpha _\\mu +F_\\omega ^{\\prime }(\\ell )}{F_\\omega (\\ell )}\\simeq \\sum _\\mu \\alpha _\\mu +\\text{(log oscillations)}\\,.$ Ignoring logarithmic oscillations, the spacetime UV Hausdorff dimension is $d_\\textsc {h}\\simeq \\sum _\\mu \\alpha _\\mu $ , as anticipated in 0.",
"For an isotropic measure, $d_\\textsc {h}\\stackrel{\\rm UV}{\\simeq } D\\alpha \\,.$ The spectral dimension is calculated from the diffusion equation and the latter can be derived from the microscopic stochastic dynamics of the diffusing particle, governed by the Langevin equation [49].",
"If the Laplacian $\\bar{\\mathcal {K}}$ appearing in the diffusion equation is not self-adjoint (as it may happen in transport phenomena), then it does not necessarily coincide with the Laplace–Beltrami operator $\\mathcal {K}$ of theory.",
"This is the case of the theories $T_1$ and $T_v$ , whose diffusion equations are one the adjoint of the other.",
"In both cases, one can show that $T_1,T_v\\,:\\quad d_\\textsc {s}=D\\frac{d\\ln L^2(\\sigma )}{d\\ln \\sigma }\\,,\\qquad L^2(\\sigma ):=\\int ^\\sigma \\frac{d\\sigma ^{\\prime }}{v(\\sigma ^{\\prime })}\\,,$ where $\\sigma $ is the diffusion scale and, if it is anomalous, it is weighted by a distribution $v(\\sigma )$ .",
"In the diffusion interpretation, there is no guiding principle telling us what $v(\\sigma )$ should be, but assuming that it behaves like the multifractional measure weight of spacetime, we can take the profile $v(\\sigma )=1+\\sum _n \\tilde{b}_n(\\sigma /\\sigma _n)^{\\beta _n-1}F_n(\\sigma )$ .",
"At the $n$ -th plateau of dimensional flow, $d_\\textsc {s}\\simeq D(2-\\beta _n)$ , while for a binomial profile and $0<\\beta \\equiv \\beta _1<1$ one obtains [49] $T_1,T_v\\,:\\quad d_\\textsc {s}\\stackrel{\\rm IR}{\\simeq } D\\,,\\qquad d_\\textsc {s}\\stackrel{\\rm UV}{\\simeq }D(2-\\beta )\\,.$ A fact gone unnoticed in previous works is that the QFT interpretation of the spectral dimension [30] does not have any of the ambiguities of the diffusion interpretation and fixes $d_\\textsc {s}$ for this class of theories.",
"Both $T_1$ and $T_v$ have standard propagator in position space and, for a massless scalar particle, $\\tilde{G}(k^2)=-1/\\tilde{\\mathcal {K}}(k)=-1/k^2$ [41], [48], [55].",
"From the Schwinger representation (REF ) of this expression, one derives the running equation in momentum space $(\\partial _{L^2}+k^2)\\tilde{P}(k^2,\\ell )=0$ .",
"Seeing $L$ just as in integration parameter of the Schwinger representation, there is no reason to give it a nontrivial measure weight.",
"Then, $\\beta =1$ and $d_\\textsc {s}=D>d_\\textsc {h}$ at all scales and eq.",
"(REF ) is violated: these geometries are not multifractal.",
"Changing the initial condition of the solution of the diffusion equation, one can even produce dimensional flows from 0 to $D$ .",
"The diffusion equation for the theory $T_q$ is straightforward: $[\\partial _{L^2(\\ell )}-\\nabla _{q(x)}^2]P(x,x^{\\prime };\\ell )=0$ .",
"Both in the diffusion and QFT interpretation, one considers the multiscale version of diffusion time or Schwinger parameter and a profile $L(\\ell )$ .",
"In the QFT interpretation of the running equation, $L$ is a length or a time, whose inverse gives the spatial and temporal resolution of the measurement.",
"In these geometries, $L$ is not the scale $\\ell $ directly measured but it is related to that by a scale-dependent relation $L(\\ell )=\\ell [1+\\sum _n b_n(\\ell /\\ell _n)^{\\beta _n-1}F_n(\\ell )]$ .",
"If we chose $L$ to be on a specific space or time direction, we would have $\\beta _n=\\alpha _{i,n}$ or $\\beta _n=\\alpha _{0,n}$ , and the spectral dimension at the $n$ -th plateau would be insensitive to the geometry of the other directions.",
"Therefore, it is more sensible to identify $\\beta _n$ with the average $n$ -th fractional charge of the measure, $\\beta _n=\\frac{\\sum _\\mu \\alpha _{\\mu ,n}}{D},$ which corresponds to $\\beta _n=\\alpha _n$ in the isotropic case.",
"The spectral dimension in the theory $T_\\gamma $ is more difficult to calculate than for $T_q$ [45] but, at the end of the day, both cases agree: $T_q,T_\\gamma \\,:\\quad d_\\textsc {s}=D\\frac{1+\\sum _n b_n(\\ell /\\ell _n)^{\\beta _n-1}[\\beta _n+F_n^{\\prime }(\\ell )]}{1+\\sum _n b_n(\\ell /\\ell _n)^{\\beta _n-1}F_n(\\ell )}\\,.$ In the IR and ignoring log oscillations, $d_\\textsc {s}\\simeq D$ , while at the $n$ -th plateau $d_\\textsc {s}\\simeq D\\beta _n$ .",
"In the binomial case, $T_q,T_\\gamma :\\quad d_\\textsc {s}\\stackrel{\\rm IR}{\\simeq } D\\,,\\qquad d_\\textsc {s}\\stackrel{\\rm UV}{\\simeq }D\\beta \\,.$ Taking eq.",
"(REF ), $d_\\textsc {s}=d_\\textsc {h}$ at all scales.",
"This is the first case to our knowledge that agreement between the two interpretations of $d_\\textsc {s}$ (diffusive or QFT) fixes a free parameter in one of them ($\\beta $ in the diffusion case).",
"Finally, the walk dimension (REF ) of spacetime in $T_1$ and $T_v$ is $d_\\textsc {w}=2D/d_\\textsc {s}$ (confirming that this is not a multifractal), while in $T_q$ it is $d_\\textsc {w}=2d_\\textsc {h}/d_\\textsc {s}$ , independently of the use of (REF ) [49].",
"We have not calculated yet $d_\\textsc {w}$ for $T_\\gamma $ .In [45], the definition of $d_\\textsc {w}$ was naively assumed to be eq.",
"(REF ) rather than eq.",
"(REF ).",
"$\\blacktriangleright $16$\\blacktriangleleft $ Can the dimension of spacetime become complex or imaginary?tocsubsection Can the dimension of spacetime become complex or imaginary?",
"Yes it can, in multiscale setups such as quantum gravities [68], [60], [133], in multifractional theories [60], and in fractal geometry [134], [135], [136].",
"In multiscale theories (including quantum gravity at large), the flow-equation theorems establish that the most general iterative solution of eq.",
"(REF ) at infinite order is the dimension (Hausdorff and/or spectral) $d(\\ell ):=\\lim _{n\\rightarrow +\\infty }d^{(n)}$ , where [60] $d^{(n)}(\\ell )-d^{(n-1)}(\\ell )=\\sum _{i=0}^{n-1} b_{i,n}\\,\\ell ^{\\alpha _{i,n}+i\\omega _{i,n}}\\,,\\qquad \\sum _{j=0}^n c_j (\\alpha _{i,n}+i\\omega _{i,n})^j=0\\,.$ The complex exponents $\\alpha _{i,n}+i\\omega _{i,n}$ satisfy a characteristic equation for all $i$ .",
"All quantum gravities have dimensional flow but, formally, all dimensional flows follow the same universal profile, which can vary from case to case depending on how the dynamics determines the free parameters $b_{i,n}$ , $\\alpha _{i,n}$ , $\\omega _{i,n}$ , and $c_j$ within.",
"Some quantum gravities may just have real-valued dimensions either because $\\omega _{i,n}=0$ for all $i$ and $n$ or because conjugate powers $\\pm i\\omega _{i,n}$ combine to give the log oscillations we discussed so far.",
"Other quantum gravities, however, can display complex dimensionalities $d(\\ell )\\in \\mathbb {C}$ because conjugate powers do not combine.",
"The question now is whether this feature is only an abstract mathematical property of the solution (REF ) or is realized in concrete scenarios.",
"There is evidence that such is indeed the case in spin foams [68], [133].",
"In contrast with kinematical states, spin-foam sums of dynamical states generically contain degenerate geometries (i.e., some component of the tetrad $e_\\mu ^I$ vanish identically), where the volume operator is not densely defined and has 0 as an eigenvalue.",
"Preliminary calculations of the spectral dimension on small combinatorial complexes, using the graviton propagator in $(2+1)$ -dimensional spin foams, show that the heat kernel $P$ acquires an imaginary part, from which it stems that also the return probability $\\mathcal {P}$ and the spectral dimension $d_\\textsc {s}$ are complex-valued.",
"These results were reported in ref.",
"[68] without giving the details; work in progress [133] on a recent model of spin foams on a hypercubic lattice [137] finds similar results.",
"The general solution (REF ) of the flow equation affects also multifractional theories; therefore, they too can have complex dimensions.",
"However, since the beginning [39] and to date, attention has been limited to real-valued measures (i.e., with log oscillations rather than imaginary powers).",
"As a further guarantee of avoiding “unphysical” situations with negative dimensions due to large oscillation amplitudes, the spacetime dimensions $d_\\textsc {h}$ and $d_\\textsc {s}$ have also been defined to be calculated after averaging out log oscillations, which is easily done by replacing $\\mathcal {V}$ and $\\mathcal {P}$ in eqs.",
"(REF ) and (REF ) with their log average [41].",
"These conditions are sufficient to have $d_\\textsc {h},d_\\textsc {s}\\geqslant 0$ but, after a few years of investigation, they might turn out to be too restrictive inasmuch as they exclude geometries that are physical despite their highly unconventional features.",
"Abandoning the averaging procedure (as done here) is not particularly dangerous: in practice, and in all known examples, log oscillations have a very small amplitude [58], [136] and they reduce to tiny ripples around the average.",
"Relaxing also the reality condition, we get access to complex dimensions (REF ) and have to face the task of interpreting the ensuing spacetimes.",
"The spin-foam results mentioned above could shed some light on this interesting subject and hint to an association between complex dimensions and degenerate geometries, for which the metric $g_{\\mu \\nu }=\\eta _{IJ}e_\\mu ^I e_\\nu ^J$ has some ill-defined components.",
"We argue here that fractal geometry supports this view and, therefore, that we might be on the right track.",
"Given the Laplacian $\\mathcal {K}$ on a deterministic fractal, one can compute the Mellin–Laplace transform of the associated heat kernel $P$ , which is a function $\\zeta _\\mathcal {K}(s)$ of the Laplace momentum $s$ called spectral zeta function (usually proportional to the Riemann $\\zeta $ function).",
"The spectral zeta function is given by $\\zeta _\\mathcal {K}(s)=\\sum _j\\lambda _j^{-s}\\,,$ where $\\lambda _j$ are the nonzero eigenvalues of $\\mathcal {K}$ .",
"The poles of $\\zeta _\\mathcal {K}(s)$ are complex-valued and of the form $s_m =\\frac{1}{2}(d_\\textsc {s}+ id_{\\textsc {cs},m})\\,,\\qquad m\\in \\mathbb {Z}\\,,$ where $d_{\\textsc {cs},m}\\propto 4\\pi m$ are called complex spectral dimensions [134], [135] and accompany the usual spectral dimension, which is the real-valued pole of $\\zeta _\\mathcal {K}(s)$ .",
"The complex poles (REF ) are a typical feature of fractals (even of popular examples such as the Sierpiński gasket, the Julia sets, diamond fractals, and the Cantor string, the complement of the middle-third Cantor set [135], [136]; see also [71], [72]), and their “physical” origin can be understood from eq.",
"(REF ).",
"The infinite number of poles $m$ is due to the presence of an exponentially large degeneracy of some special eigenvalues of the Laplacian called iterated (in contrast, in ordinary manifolds this degeneracy factor is power-law) [136].",
"In turn, nonmetric geometries or labels on combinatorial graphs have spectral features that could easily lead to exponential degeneracies in the Laplacian eigenvalues, and hence could acquire complex dimensions.",
"This was briefly commented upon in ref.",
"[136] and agrees intriguingly with what found later in ref.",
"[68].",
"The relation between metric degeneracy and Laplacian eigenvalue degeneracy has not been clarified to date, but these few fragments we collected here are suggestive of a coherent picture awaiting further study.",
"$\\blacktriangleright $17$\\blacktriangleleft $ Do multifractional theories really have dimensional flow?",
"Regardless of the choice of derivatives, the measure (REF ) is mathematically equivalent to the standard Lebesgue measure $d^Dx$ , where one uses the symbol “$q$ ” instead of “$x$ .” If we compute the volume of a hypercube or of a $D$ -ball, we find $\\nonumber \\mathcal {V}\\sim \\int _0^\\ell d^D q(x)=\\int _0^Ld^D q=L^D,$ where $L=q(\\ell )$ (here we are ignoring $\\mu $ indices for simplicity) is the edge size of the hypercube or the radius of the ball.",
"Then, the Hausdorff dimension coincides with the topological dimension: $d_\\textsc {h}(L)=\\frac{\\partial \\ln \\mathcal {V}}{\\partial \\ln L}=D\\,.$ One could make a similar calculation for the spectral dimension and show that $d_\\textsc {s}=D$ .tocsubsection Do multifractional theories really have dimensional flow?",
"The above calculation is mathematically correct but it neglects the physics.",
"The step $x\\rightarrow q(x)$ is not a coordinate transformation in multifractional theories, which break Lorentz invariance (see question ).",
"An absolutely indispensable ingredient of the multifractional recipe is the establishing of measurement units or, in other words, of a coordinate frame where all physical measurements must be carried out.",
"This step is necessary because the profiles $q^\\mu (x^\\mu )$ are noninvariant under coordinate transformations, and one must fix the frame where the form (REF ) is valid.",
"By definition from the onset, the coordinates $x^\\mu $ have the scaling dimension of lengths and time, $[x^\\mu ]=-1\\,,$ and are called fractional coordinates.",
"The frame $\\lbrace x^\\mu \\rbrace $ is called fractional frame or picture.",
"The geometric coordinates $q^\\mu $ , which define the integer frame or picture in the theory with $q$ -derivatives, also have the dimension of lengths and time, $[q^\\mu ]=-1$ exactly, but their $x$ -dependent part does not.",
"At the $n$ -th plateau in dimensional flow, i.e., at distances or times $\\sim \\ell _n^\\mu $ , this varying part scales as $[|q^\\mu |]\\stackrel{x\\sim \\ell _n}{\\sim } [|x^\\mu |^{\\alpha _{\\mu ,n}}]=-\\alpha _{\\mu ,n}\\ne -1\\,.$ This is what is meant in the literature by anomalous scaling.",
"The physical meaning of eqs.",
"(REF ) and (REF ) is that, in the fractional picture constituted by the fractional coordinates $x^\\mu $ , measurements are taken by clocks and rods that do not change with the probed scale [there is no scale dependence in (REF )], while in the integer picture made of the geometric coordinates $q^\\mu $ measurements are taken by clocks and rods that adapt with the probed scale (there is a scale dependence in (REF )).",
"By definition, physical measurements in multifractional theories are performed in the fractional picture: clocks and rods are nonadaptive, rigid, not multiscale.In asymptotic safety, precisely the opposite holds and physical rods are adaptive [47].",
"We will comment on this in question .",
"The reason beyond this choice instead of its complementary is simple.",
"Measurement apparatus created by humans are local objects with definite size probing length, time, or energy scales in a definite range.",
"Rods measuring the length of a goldfish are the same rods measuring a whale, only shorter.",
"When we probe lengths at very different scales, such as of goldfish or atomic or elementary-particle size, we do not have one general “rod” marking centimeters and Compton lengths: we have to construct new “rods” for each scale, based on different principles.",
"Having thus established nonadaptive rods (i.e., the fractional picture) as the measuring tool of physics, it is clear that the radius of the ball we measure is $\\ell $ , not $L$ , so that its volume scales as $\\ell ^{d_\\textsc {h}}$ , not as $L^{d_\\textsc {h}}$ .",
"Consequently, the Hausdorff dimension is (REF ), not (REF ).",
"A similar reasoning holds for $d_\\textsc {s}$ .",
"$\\blacktriangleright $18$\\blacktriangleleft $ Is prescribing measurement units in this way scientific?",
"We all know that any theory of physics is based upon some principles or axioms, but we could obtain everything just by changing well-established axioms or by replacing them by something else, as you do in multifractional theories.tocsubsection Is prescribing measurement units in this way scientific?",
"And as done in scalar-tensor theories [138], [139] or in varying-speed-of-light (VSL) models [140], [141].",
"The selection of special frames where physical observables are measured is not a novelty.",
"There is nothing wrong in modifying well-established axioms, as long as the resulting theory is motivated from above, internally consistent, and testable by experiments.",
"In scalar-tensor theories, the change from the Jordan to the Einstein frame corresponds to a change of measurement units.",
"In VSL theories, we are dealing with units adapted with the scales in the dynamics and, in particular, chosen such that the speed of light $c(x)$ varies in space and time.",
"Time and space units are redefined so that the differentials scale as $dt\\rightarrow [f(x)]^a dt$ , $dx^i\\rightarrow [f(x)]^b dx^i$ , where $f$ is a function, $a$ and $b$ are constants, and local Lorentz invariance of the line element requires $c(x)\\propto [f(x)]^{b-a}$ .",
"We recognize here a particular form of anisotropic multiscaling (one that distinguishes between space and time variables).",
"In particular, when $b=0$ one formally reabsorbs $c$ in the coordinate $x^0=\\int dt\\, c(t)$ , which scales as a length.",
"With this coordinate, all equations can be made formally identical to the usual ones provided some conditions are met.",
"Models where the electric charge $e$ or the speed of light $c$ varies can be recast in new units such that, respectively, the electric charge and the speed of light become constant, but in both cases the dynamics can become substantially more complicated.",
"This criterion of simplicity is not the only one which attaches one label or the other (varying-$e$ or varying-$c$ ) to these models: experiments are able to distinguish between them.",
"The change of units at the base of scalar-tensor theories, VSL models, and varying-electric-charge models all map in one way or another [50] to the Manichaean notion of “adapting” versus “nonadapting” rods in multifractional models [47], [56].",
"Furthermore, the multifractional paradigm can be discriminated from scalar-tensor, VSL, and other changing-unit proposals both by experiments and by their theoretical structure.",
"Despite the striking similarity of the Einstein equations of scalar-tensor theories [53], the measure weight is not a Lorentz scalar and it heavily affects the gravitational dynamics (for instance in cosmology) in a way irreproducible by scalar-tensor models.",
"The presence of a nontrivial measure consistently affects the definition of functional variations, Poisson brackets and Dirac distribution, in turn leading to a deformation of the Poincaré symmetries (see ) not realized in varying-$e$ and varying-$c$ scenarios.",
"In question , we will see an example of how one can measure departure from a standard space in a multiscale geometry.",
"$\\blacktriangleright $19$\\blacktriangleleft $ Is the volume density $\\sqrt{-g}$ from the metric implemented consistently?",
"Therein, I do not see any change of anomalous geometry with the scale.tocsubsection Is the volume density $\\sqrt{-g}$ from the metric implemented consistently?",
"This somewhat vague question arises because in the majority of papers gravity is ignored and the measure is $d^Dq(x)$ (with trivial volume density factor $\\sqrt{-\\eta }=1$ ), while when gravity is triggered the volume measure is $d^Dq(x)\\,\\sqrt{-g}$ [53].",
"This creates confusion because no show of dimensional flow seems to emanate from the volume density.",
"The point is that the calculus structure and the metric structure are totally independent at the level of the action.",
"On one hand, there is the calculus structure embodied by the integral measure $d^Dq(x)$ and the choice of derivatives.",
"On the other hand, there is the metric structure expressed by the volume density $\\sqrt{-g}$ , curvature terms, and covariant derivatives.",
"When the calculus structure reduces to the usual one and $q^\\mu \\simeq x^\\mu $ , then standard general relativity is recovered (by construction).",
"This limit is independent of the curvature of spacetime, so that to preserve covariance and diffeo invariance in the IR the factor $\\sqrt{-g}$ must be there.",
"The mutual independence of the integrodifferential and the metric structures does not imply that they do not talk to each other.",
"The multiscaling of the geometric coordinates $q^\\mu (x^\\mu )$ strongly affects the dynamics and, hence, the solutions to the Einstein equations.",
"In particular, the background metric $g_{\\mu \\nu }(x)$ solving the dynamical equations is multiscale [53], [58].",
"$\\blacktriangleright $20$\\blacktriangleleft $ Is geometry discrete at the smallest scales?tocsubsection Is geometry discrete at the smallest scales?",
"Yes, it is.",
"Take for simplicity the binomial measure (REF ) in one dimension: $q(x) = x+(\\ell _*/\\alpha ){\\rm sgn}(x)|{x}/{\\ell _*}|^\\alpha F_\\omega (x)$ , where $F_\\omega (x)= 1+A\\cos (\\omega \\ln |{x}/{\\ell _\\infty }|)+B\\sin (\\omega \\ln |{x}/{\\ell _\\infty }|)$ .",
"The distribution $F_\\omega $ is invariant under the discrete scale invariance (DSI) $x\\,\\rightarrow \\, \\lambda _\\omega ^n x\\,,\\qquad \\lambda _\\omega :=\\exp \\left(-\\frac{2\\pi }{\\omega }\\right)\\,,\\qquad n\\in \\mathbb {Z}\\,.$ This symmetry, often found in chaotic systems [142], [143], [144], is a dilation transformation under integer powers of a prefixed scaling ratio $\\lambda _\\omega $ .",
"Although $F_\\omega (\\lambda _\\omega x)=F_\\omega (x)$ , the measure $q(x)$ is not invariant (up to an overall constant factor), since $q(\\lambda _\\omega x)= \\lambda _\\omega ^\\alpha q(x)+(\\lambda _\\omega -\\lambda _\\omega ^\\alpha )x\\,.$ The last term never vanishes.",
"However, at scales $\\lesssim \\ell _*$ the overall scaling is determined by $\\alpha $ and the dominant piece of the measure is DSInvariant.",
"In the IR, the usual dilation symmetry $x\\rightarrow \\lambda x$ with arbitrary $\\lambda $ is recovered, while a natural discrete-to-continuum transition happens at intermediate scales (for a detailed description, see [39], [41]).",
"At one extremum of this transition, UV spacetime is effectively discrete and described by a lattice of size $\\lambda _\\omega \\ell _\\infty $ , even if the full integration measure is defined on a continuum.Discreteness of a geometry can be encoded either in continuum models $\\int _{\\rm lattice}dx\\,v(x)\\,\\mathcal {L}(x)$ with discrete integration domain (integrals in a continuous embedding weighted by measures with discrete support), or by a setting with discrete calculus, $\\sum _n \\mathcal {L}(x_1,\\dots ,x_n)$ .",
"Multifractional theories adopt the first option, while CDT (as an artifact), GFT, LQG, and spin foams realize the second.",
"$\\blacktriangleright $21$\\blacktriangleleft $ Is $D=4$ assumed or predicted?tocsubsection Is $D=4$ assumed or predicted?",
"In general, it is assumed, just like in any other theory except string theory.",
"However, in question 0 we mentioned that there is a phase transition in the theories $T_1$ and $T_v$ for the special value $\\alpha =2/D$ of the fractional exponent in the measure, so that $d_\\textsc {h}\\simeq D\\alpha =2$ in the UV.",
"Only in $D=4$ does this exponent $\\alpha =1/2$ lie at the middle of the allowed interval (REF ).",
"Intriguingly, the value of the Hausdorff dimension in the UV ($d_\\textsc {h}\\simeq 2$ ) and in the IR ($d_\\textsc {h}\\simeq 4$ ) are mutually related rather than being independent as in many multiscale quantum gravities.",
"Thus, $D=4$ is special among any other possibility, but only in $T_1$ and $T_v$ and only in relation with the UV value: in this sense, the above argument is circular and does not allow to make separate claims about the uniqueness of the UV and the IR dimension separately.",
"In $T_v$ , however, there is an independent argument selecting $D=4$ as the only case where the gravitational action simplifies (see question ) and the metric $g_{\\mu \\nu }$ has the natural structure of a bilinear field with measure weight $-1$ [53].",
"In the theory $T_q$ , there is no phase transition relating the UV and the IR dimensions.",
"In the theory $T_\\gamma $ , there is a stronger argument to select $\\alpha =1/2$ (it is the lowest possible value to have a normed space), but it is not related to the IR dimension.",
"In these cases, we are not aware of any robust argument to select $D=4$ ."
],
[
"Frames and physics",
"$\\blacktriangleright $22$\\blacktriangleleft $ The theory with weighted derivatives is trivial.",
"Consider for instance the scalar-field action (REF ) with polynomial interactions: $S_\\phi =-\\int d^Dx\\,v\\left(\\frac{1}{2}\\mathcal {D}_\\mu \\phi \\mathcal {D}^\\mu \\phi +\\sum _{n=2}^N\\frac{\\sigma _n}{n}\\phi ^n\\right).$ After the field redefinition (REF ), $\\varphi =\\sqrt{v}\\,\\phi $ , the action becomes $S_\\phi =-\\int d^Dx\\left(\\frac{1}{2}\\partial _\\mu \\varphi \\partial ^\\mu \\varphi +\\sum _n\\frac{\\tilde{\\sigma }_n}{n}\\varphi ^n\\right),\\qquad \\tilde{\\sigma }_n=\\sigma _nv^{1-\\frac{n}{2}}\\,.$ If we also assume that, originally, the $\\sigma _n$ were spacetime dependent and such that $\\sigma _n(x)\\propto [v(x)]^{n/2-1}$ for all $n$ , then the couplings $\\tilde{\\sigma }_n$ are constant (the mass $\\sigma _2=m^2$ is constant also in the fractional picture) and eq.",
"(REF ) is the usual action in standard Minkowski spacetime.",
"In [48], [51], the $\\sigma _n$ were assumed to be constant, but in the case of the Standard Model [50], [55] all the effective couplings $\\tilde{\\lambda }_i$ after the field transformation (REF ) were found to be constant.",
"Nevertheless, it was concluded that the theory was nontrivial.",
"I do not see how, since the actions (REF ) and (REF ) are equivalent.tocsubsection Is the theory with weighted derivatives trivial?",
"(i) As in the case of $T_q$ , $T_v$ can be written in two different ways or frames (question ).",
"The one defining the theory, and where physical measurements have to take place, is called the fractional picture or fractional frame and corresponds to eq.",
"(REF ) and to the general action on flat space (REF ).",
"After the field redefinition (REF ), the theory is simplified and takes exactly the form of a field theory on ordinary Minkowski space, provided all couplings $\\lambda $ in the fractional picture have a spacetime dependence such that the couplings $\\tilde{\\lambda }$ in the integer frame or integer picture are constant.Note that the integer picture in the theory $T_q$ is defined differently and does not involve field transformations (see ).",
"In general and in the absence of gravity, the two frames are related by $S_v[v,\\mathcal {D},\\phi ^i,\\lambda _i]=S_1[1,\\partial ,\\varphi ^i,\\tilde{\\lambda }_i]\\,,\\qquad \\text{$v(x)$ in the left-hand side fixed}.$ The claim in the question is that the right-hand side is the standard action of a QFT on Minkowski spacetime, hence the theory is trivial.",
"However, there are three elements that should be taken into account: a general remark about the physical frame, information from non-QFT physics, and inclusion of gravity.",
"The general fact is that, by definition, physical observables must be evaluated in the fractional picture, which is the frame where physical measurements take place.",
"This was stated in .",
"The integer picture is a frame where the theory is simplified in such a way that all calculations in QFT can be carried out easily, that is, with standard perturbative QFT techniques.",
"These techniques are not applicable in the fractional picture: the field theory described by (REF ) or (REF ) has spacetime-dependent kinetic terms and couplings, which make Feynman rules difficult or practically impossible [51], [55].",
"The QFT in the integer picture is the usual one and we can calculate effective observables easily.",
"However, the effective observables in the integer picture must be converted into the physical observables in the fractional picture, which are those to be compared with experiments.",
"Therefore, the integer picture is only a convenient way to recast the theory and make calculations, but it is not physically equivalent to the fractional picture.",
"Several observables have been computed and constrained experimentally which illustrate the point [50], [55], [58].",
"A similar situation happens in scalar-tensor theories, although in that case the frame dilemma is shifted to the quantum level (see question ).",
"Another general argument [55] is that QFT is only part of the whole story.",
"The QFT couplings in the theory $T_v$ are constant in the integer picture not only for necessity (masses are constant to allow for a manageable quantum perturbative treatment), but also as a requirement of gauge invariance [55].",
"Such restrictions do not exist in the realm of statistical and particle mechanics.",
"Examples are the random motion of a molecule [49], the dynamics of a relativistic particle [52], and the black-body radiation spectrum [58], all processes with a characteristic energy much smaller than that in the center of mass of subatomic scattering events.",
"On one hand, the form of the couplings in QFT is constrained by the way we are able to deal with interacting quantum fields.",
"On the other hand, statistical and particle mechanics are intrinsically nonlinear, either through the stochastic interaction of a degree of freedom with the environment (as in the multifractional Brownian motion of a particle [49]), or by definition of the action (as for the relativistic particle [52]), or via the collective description of microscopic degrees of freedom (as in the frequency distribution of a thermal bath of photons [58]).",
"These systems yield nontrivial predictions because they are not subject to requirements as severe as those we imposed on a quantum field theory.",
"Therefore, one should not identify the theory $T_v$ with QFT alone, just like standard QFT cannot describe all possible systems of physics.",
"A third consideration to make is about gravity.",
"On a curved background, the equivalence of frames after field and metric redefinitions is broken.",
"In the integer picture, the theory $T_v$ is not general relativity with minimally coupled matter, and one can never trivialize the theory to the ordinary one as in the flat case [53].",
"The gravitational dynamics of the theory with weighted derivatives was studied in ref.",
"[53].",
"The metric is not covariantly conserved and the geometry corresponds to a Weyl-integrable spacetime.",
"The total action reads $S_v[g,\\phi ^i] =\\frac{1}{2\\kappa ^2}\\int d^Dx\\,v\\,\\sqrt{-g}\\left[{\\cal R}-\\omega \\mathcal {D}_\\mu v\\mathcal {D}^\\mu v-U(v)\\right]+S_v[\\phi ^i]\\,,$ where ${\\cal R}$ is the Ricci scalar constructed with weighted derivatives of different weight [53] (see question ), $\\omega $ and $U$ are functions of the weight $v$ , and in $S_v[\\phi ^i]$ the metric is minimally coupled.",
"Absorbing weight factors into the matter fields $\\phi ^i$ with the picture change (REF ) requires a redefinition of the metric $g_{\\mu \\nu }\\rightarrow \\tilde{g}_{\\mu \\nu }$ .",
"Indeed, one can go to the integer picture (Einstein frame) where the gravitational action is $\\propto \\int d^Dx\\,\\sqrt{-\\tilde{g}}\\,\\tilde{R}$ but not without introducing nontrivial measure-dependent terms.",
"These terms affect the cosmic evolution.",
"Thus, a change of picture does not lead to standard general relativity plus matter and the dynamics is different from (and much more constrained than) that of scalar-tensor scenarios in both frames.",
"In general, one should be careful about the issue of the physical inequivalence between the fractional and the integer picture.",
"As for scalar-tensor models, from a simple visual inspection of the actions one cannot conclude that the Jordan and Einstein frames define different physics.",
"What matters are the physical observables.",
"The homogeneous classical cosmology of multifractional theories is physically distinguishable from the usual one even in the integer picture (Einstein frame), since $\\tilde{\\omega }\\ne 0\\ne \\tilde{U}$ .",
"$\\blacktriangleright $23$\\blacktriangleleft $ In the so-called fractional picture, the theory with weighted derivatives appears to violate Poincaré invariance explicitly, as also stated in and .",
"But if Poincaré violations can be eliminated by redefining the fields (in the so-called integer picture), then where is the new physics?",
"Fields are auxiliary concepts and redefining them should not change the physical content (for instance, the S-matrix) of the theory.tocsubsection Is the theory with weighted derivatives trivial?",
"(ii) The fractional and integer pictures are not related only by the field redefinition (REF ) (together with redefinitions of couplings [55]).",
"When an observable is computed (for convenience) in the integer picture, it must be mapped back to the measurement units of the fractional pictures, which is the frame where physical measurements take place with nonadaptive clocks, rods, and particle detectors.",
"For instance, the observed electron charge $\\tilde{e}=e_0$ is constant in the integer picture, but it is a time-dependent quantity $Q(t)$ in the fractional picture [50].This property tells the electric charge apart from all other gauge couplings of the Standard Model [55].",
"See question .",
"This means than all the phenomenology associated with the fine-structure constant will be standard in the integer picture but time-dependent in the fractional picture.",
"What we constrain by observations is the second.",
"In general, the symmetries enjoyed in the integer frame (such as Poincaré invariance) can be violated in the physical frame, and observables are affected consequently.",
"We postpone to question a discussion on the S-matrix.",
"$\\blacktriangleright $24$\\blacktriangleleft $ The theory with $q$ -derivatives is trivial.",
"Consider for instance the scalar-field action (REF ) with a mass term and a higher-order interaction: $S_\\phi =-\\int d^Dx\\,v\\left[\\frac{1}{2}\\eta ^{\\mu \\nu }\\frac{\\partial \\phi }{\\partial q^\\mu (x^\\mu )}\\frac{\\partial \\phi }{\\partial q^\\nu (x^\\nu )}+\\sum _{n=2}^N\\frac{\\sigma _n}{n}\\phi ^n\\right].$ Here there is no field redefinition available but one can consider $x^\\mu \\rightarrow q^\\mu (x^\\mu )$ simply as a change of coordinates.",
"Since the physics should be invariant under such coordinate transformations, then the theory is equivalent to the usual one.tocsubsection Is the theory with $q$ -derivatives trivial?",
"(i) In general, the mapping between the fractional and the integer picture is $S_q[v,v^{-1}\\partial _x,\\phi ^i,\\lambda _i]=S_q[1,\\partial _q,\\phi ^i,\\lambda _i]\\,,\\qquad \\text{$v(x)$ in the left-hand side fixed}.$ The fractional picture is the frame where the $x$ -dependence of the composite coordinates $q(x)$ is manifest [left-hand side of (REF )], while the integer picture is the frame described by the geometric coordinates $q$ [right-hand side of (REF )].",
"Contrary to the mapping (REF ) for the theory $T_v$ , there is no redefinition of the couplings.",
"As in the theory $T_v$ , the difference between the fractional and the integer picture is in the way geometry is perceived by the dynamical degrees of freedom: as standard Minkowski spacetime in the integer picture, as an anomalous geometry with a fixed integrodifferential structure in the fractional picture.",
"The presence of this predetermined structure does affect the physics because it prescribes the existence of a preferred frame where physical observables should be compared with experiments.",
"As we already said, by definition of the theory, this frame is the fractional picture.",
"This is an important conceptual novelty with respect to theories with an ordinary integrodifferential structure: a choice of frame is a mandatory step in the definition of multifractional spacetimes.",
"In the case with $q$ -derivatives, time intervals, lengths and energies are physically measured in the fractional picture where coordinate transformations are described by the nonlinear law (REF ).",
"We stress that eq.",
"(REF ) is not a coordinate transformation.",
"It governs the formal passage between the fractional picture described by the composite coordinates $q^\\mu (x^\\mu )$ and the integer picture described by coordinates $q^\\mu $ .",
"The integer picture is a convenient frame for calculations, but it is no more than that, since eq.",
"(REF ) is not even invertible except in the simple case of a binomial measure without oscillations.",
"To illustrate in what sense the integer frame is “convenient,” we write down eq.",
"(REF ) in $D=1+1$ dimensions: $S_\\phi &=&\\int d^2 q\\,\\left\\lbrace \\frac{1}{2}[\\partial _{q_0(t)}\\phi ]^2-\\frac{1}{2}[\\partial _{q_1(x)}\\phi ]^2-\\sum _n\\frac{\\sigma _n}{n}\\phi ^n\\right\\rbrace \\nonumber \\\\&=&\\int d^2 x\\,\\left[\\vphantom{\\sum _n}\\frac{v_1(x)}{2v_0(t)}\\dot{\\phi }^2-\\frac{v_0(t)}{2v_1(x)}(\\partial _x\\phi )^2-\\sum _n\\frac{v_0(t)v_1(x)\\sigma _n}{n}\\phi ^n\\right].$ Since we do not know how to define a quantum field theory with varying couplings and nonhomogeneous kinetic terms, it is necessary to perform all calculations in geometric coordinates.",
"Therefore, we transform to the integer picture via (REF ) where the theory looks trivial and one can borrow all the known calculations in standard QFT.",
"Any “time” or “spatial” interval or “energy” predicted in the integer picture are not a physical time or spatial interval or energy, since they are measured with $q$ -clocks, $q$ -rods, or $q$ -detectors.",
"The results must be reconverted to the fractional picture in order to interpret them correctly.",
"QFT examples of this inequivalence of observables are the muon decay rate, the Lamb shift, and the variation of the fine-structure constant [54], [55], while cosmological and astrophysical examples are given in refs.",
"[57], [58].",
"$\\blacktriangleright $25$\\blacktriangleleft $ I am still not convinced, so let me rephrase my criticism.",
"The theory $T_q$ tries to incorporate the effects of new fundamental energy, time and length scales at a microscopic scales while getting the standard physics at mesoscopic distances.",
"This is done through a particular replacement of coordinates.",
"As it is, it is unclear what this replacement actually is.",
"I see two possibilities, it is either a change in the description or a change in the physical behavior.",
"I will argue against any of these possibilities.",
"Let us first assume that the replacement (REF ) is a change of the description.",
"This corresponds to a coordinate change but, as we know, the theory of relativity is built in such a way that a change of coordinates does not change the physics.",
"The new effects claimed to be found are spurious and unphysical because the coordinate change is ill defined, since it is not invertible in general.",
"To avoid the problems associated with invertibility, one would need to focus on a single chart where the $q^\\mu (x^\\mu )$ were invertible, but this restriction is not considered.",
"In fact, this omission is the root of the DSI of the function $F_\\omega $ , and ultimately of the supposed “fractal” nature of the theory.",
"Therefore, this is not a valid mechanism to introduce new scales.tocsubsection Is the theory with $q$ -derivatives trivial?",
"(ii) The theory of general relativity is built in such a way that a change of coordinates does not change the physics, but multifractional theories are not.",
"It is a mistake to impose the principles of Einstein gravity to a multiscale geometry.",
"Noninvertibility, which is a consequence of the flow-equation theorems having nothing to do with ill-defined coordinate changes, is rather one of the reasons why eq.",
"(REF ) cannot be regarded as a coordinate transformation; the other reason is that different frames correspond to different measurement units and one must make a choice (see and ).",
"Making a frame/unit choice is not particularly exotic, as recalled in .",
"$\\blacktriangleright $26$\\blacktriangleleft $ The change of coordinates (REF ) is badly implemented inasmuch as the volume form is not corrected with the square root of the metric determinant $\\sqrt{-g}$ , nor is the inverse metric corrected in the kinetic term of the scalar field.",
"Again, if these issues were considered, no new physics would arise.tocsubsection Is the theory with $q$ -derivatives trivial?",
"(iii) We just argued against the interpretation of eq.",
"(REF ) as a change of coordinates.",
"Letting aside this abuse of terminology, the volume density $\\sqrt{-g}$ does appear in the theory as soon as gravity is switched on, and derivatives are made covariant accordingly [53].",
"New physics does arise in that case [57], [58], simply because the dichotomy between fractional and integer frame persists also when the embedding manifold is curved.",
"We can even say more: the theory in the integer frame is invariant under a change of geometric coordinates $q^\\mu \\rightarrow {q^{\\prime }}^\\mu $ [53], as stated in .",
"This is not a symmetry of physical observables, since it is broken in the fractional picture where the form of the geometric coordinates is given by the second flow-equation theorem.",
"$\\blacktriangleright $27$\\blacktriangleleft $ Let me give you a third argument against the interpretation of eq.",
"(REF ) as a change of description.",
"In order to get a dispersion relation for a particle, the physical meaning for the $x$ coordinates should be specified.",
"Interpreting them as the position of a particle (i.e., its worldline in an arbitrary parametrization), one immediately notes that the profile $q^0(t)$ must be monotonic in time, something that is not fulfilled by eqs.",
"(REF ) or (REF ).",
"Hence, in terms of the composite coordinates $q$ particles do not follow proper worldlines.",
"This should be enough to understand that no new physics can be obtained in the $q$ -theory, outside a single chart.tocsubsection Is the theory with $q$ -derivatives trivial?",
"(iv) The profile $q^0(t)$ is not monotonic due to log oscillations, but this does not mean that time $t$ for the particle goes back and forth.",
"Again, here one is confusing geometric coordinates with physical ones.",
"Moreover, worldlines in a multiscale spacetimes are certainly not expected to behave as usual and, in fact, they do not, as was shown in the theory $T_v$ for a nonrelativistic and a relativistic particle [46], [52].",
"The case of the point particle in $T_q$ is straightforward; in this theory, the physical inequivalence of the fractional and integer pictures is further shown by the fact that dispersion relations are modified (question ).",
"$\\blacktriangleright $28$\\blacktriangleleft $ Even granting that the measure (REF ) with (REF ) comes from some different paradigm we are not accustomed to in general relativity, it breaks Poincaré invariance and, as any theory with Lorentz violation, fixes a preferred frame.",
"While in general relativity frames are equivalent at least at the classical level, here one must make a frame choice.",
"With what criteria?",
"What exactly is the preferred frame in physical terms?tocsubsection What are the criteria to choose the physical frame?",
"We already answered in , , –.",
"Here we make a couple of remarks on the similar problem of choice between the Einstein and the Jordan frame in scalar-tensor theories.",
"After several years of debate, it has by now become accepted that the two frames are physically equivalent both classically [145], [146] and at the quantum level to first order in perturbation theory (both in a QFT and a cosmological sense), but they differ in a nonlinear quantum regime [147], [148], [149], [150].",
"At that point, a choice of frame is necessary according to some criterion.",
"For instance, one might regard the Jordan frame as the fundamental one because it is the frame where matter follows the geodesics.",
"A choice of frame is a choice of measurement units [138].",
"In the case of the VSL models mentioned in , the criterion for the choice of units is simplicity of the dynamics.",
"In the case of multifractional theories, it is to have nonanomalous clocks and rods at all scales in a multiscale spacetime (see ).",
"A small caveat about quantum inequivalence of frames will conclude the discussion.",
"Let us recall an argument by Duff against having quantum fields on a classical gravitational background [151].",
"Consider an ordinary (nonmultiscale) spacetime and an action $S[g,\\phi ^i]$ dependent on the metric and on some matter fields.",
"Consider also a suitably regular field redefinition $\\bar{g}_{\\mu \\nu }=\\bar{g}_{\\mu \\nu } (g_{\\mu \\nu },\\phi ^i)$ , $\\bar{\\phi }^i=\\bar{\\phi }(g_{\\mu \\nu },\\phi ^i)$ , so that the actions $\\bar{S}[\\bar{g},\\bar{\\phi }^i]=S[g,\\phi ^i]$ describe the same physics at the classical level.",
"At the quantum level, if all fields (including $g_{\\mu \\nu }$ ) are quantized, then the two theories are equivalent on shell order by order in perturbation theory, although they differ as far as individual Feynman diagrams and off-shell S-matrix elements are concerned.",
"This is because the on-shell S-matrix is invariant under field redefinitions.",
"However, if gravity is purely classical only matter fields are quantized and the two theories are physically inequivalent.",
"The intuitive reason is that internal graviton lines, which are essential to maintain the on-shell equivalence, are now absent.",
"An example is the minimally-coupled massless scalar field theory $S[g,\\phi ]=\\int d^4x\\sqrt{-g}\\left(\\frac{R}{2\\kappa ^2}-\\frac{1}{2}g^{\\mu \\nu }\\partial _\\mu \\phi \\partial _\\nu \\phi \\right)\\,.$ At the one-loop level, UV divergences are removed if one adds a certain counterterm $\\Delta S$ [152].",
"The classical theory (REF ) is equivalent to the nonminimal action $\\bar{S}[\\bar{g},\\bar{\\phi }]=\\int d^4x\\sqrt{-\\bar{g}}\\left[\\bar{R}\\left(\\frac{1}{2\\kappa ^2}-\\frac{\\bar{\\phi }^2}{12}\\right)-\\frac{1}{2}\\bar{g}^{\\mu \\nu }\\partial _\\mu \\bar{\\phi }\\partial _\\nu \\bar{\\phi }\\right]$ via a conformal transformation.",
"One-loop finiteness of this theory requires a counterterm $\\Delta \\bar{S}$ .",
"When graviton internal lines are taken into account, on shell we have $\\Delta S=\\Delta \\bar{S}$ .",
"However, when only the scalar field is quantized one finds that $\\Delta S\\ne \\Delta \\bar{S}$ [151].",
"Therefore, the same classical theory could be written in infinitely many different ways and one would have to invoke a criterion selecting one frame among all the others.",
"This may be problematic, but the existence of such a criterion is not altogether unreasonable: for instance, one could impose positivity of energy and choose the Einstein frame $\\bar{g}_{\\mu \\nu }$ as the frame where the fundamental theory is defined [153].",
"Duff's example illustrates why two classically equivalent frames can differ at the quantum level and a frame choice must be made.",
"In multifractional theories, the situation is different because in the fractional frame we do not know how to deal with the quantum theory [51], [55] (see and ).",
"In the multifractional case, the choice where to do QFT is somewhat mandatory: we move to the integer frame to do all intermediate QFT calculations before getting physical observables.",
"The latter are obtained in the end in the fractional frame, which was selected as preferred already at the classical level.",
"This marks a difference with respect to the scalar-tensor case, where the frame choice dictated by some principle is necessary only at the quantum level.",
"$\\blacktriangleright $29$\\blacktriangleleft $ Even accepting that it is part of the definition of these theories to establish a frame choice, what is the meaning of the point $x=0$ in eqs.",
"(REF ) and (REF )?",
"If we write the measure in one direction as $dq(x)=dx\\,v(x)$ , then the measure weight $v(x)\\sim 1+|x/\\ell _*|^{\\alpha -1}$ is singular at $x=0$ because $\\alpha <1$ .",
"So where are we with respect to this singularity?",
"What are the physical consequences of having this uniquely special spacetime point?tocsubsection What is the meaning of the singularity in the measure?",
"This question hits one of the most peculiar aspects of multifractional theories, known as the presentation problem.",
"Let us explain it in detail for the theory with $q$ -derivatives, following [56] up to some point but greatly improving on the interpretation and on the physics thanks to the second flow-equation theorem.",
"The model $T_1$ and the theory $T_v$ face a similar issue, while the theory $T_\\gamma $ is a separate case.",
"Suppose we wish to measure the distance $\\Delta x$ of two points A and B on a sheet of paper.",
"If the paper is charted by a Cartesian system, then the distance is given by the two-dimensional Euclidean norm $\\Delta x:=\\sqrt{|x_{\\rm B}^1-x_{\\rm A}^1|^2+|x_{\\rm B}^2-x_{\\rm A}^2|^2}$ .",
"Then we make a coordinate transformation $x^i\\rightarrow {x^{\\prime }}^i$ such that $\\Delta x=F({x_{\\rm A}^{\\prime }}^i,{x_{\\rm B}^{\\prime }}^i)$ is a function of the new coordinates.",
"For instance, going to polar coordinates $\\lbrace x^1,x^2\\rbrace \\rightarrow \\lbrace \\varrho ,\\theta \\rbrace $ conveniently centered at $x_A$ , one has $\\Delta x=r$ .",
"The observed value of the distance is insensitive to the coordinates we choose to represent $\\Delta x$ with.",
"In the theory $T_q$ , we can try to do the same in the fractional picture, which is one of the coordinate frames $\\lbrace x^1,x^2\\rbrace $ where the distance $\\Delta x$ is calculated.",
"However, to each of these fractional frames we must associate an integer frame described by geometric coordinates.",
"Thus, the Cartesian fractional frame $\\lbrace x^1,x^2\\rbrace $ is mapped into the integer frame $\\lbrace q^1(x^1),q^2(x^2)\\rbrace $ and, after inverting to $x^i=x^i(q^i)$ (assuming it possible, which is not always the case) the Euclidean norm $\\Delta x$ is mapped into some complicated expression $\\Delta x(q_{\\rm A}^i,q_{\\rm B}^i)$ differing from the geometric Euclidean norm $\\Delta q:=\\sqrt{\\sum _{i=1}^2|q_{\\rm B}^i-q_{\\rm A}^i|^2}$ by correction terms $\\mathcal {X}$ and $\\mathcal {T}$ we will calculate below.",
"If we redo the mapping to geometric coordinates starting from polar fractional coordinates, we get another integer frame $\\lbrace q_r(r),q_\\theta (\\theta )\\rbrace $ , where the relations between $q_r$ and the $q^i$ are $q^1=q_r\\cos q_\\theta $ and $q^2=q_r\\sin q_\\theta $ .",
"Thus, on which chart is eq.",
"(REF ) or (REF ) represented?",
"In the example of the paper sheet, is eq.",
"(REF ) the form of $q$ in the integer frame $\\lbrace q^1(x^1),q^2(x^2)\\rbrace $ based on Cartesian coordinates $\\lbrace x^1,x^2\\rbrace $ or the form of $q$ in the integer frame $\\lbrace q^1(r),q^2(\\theta )\\rbrace $ based on polar coordinates $\\lbrace r,\\theta \\rbrace $ [so that $q^1(r)=r+(\\ell _*/\\alpha )(r/\\ell _*)^{\\alpha }$ ], or something else?",
"Ordinary Poincaré invariance is violated by factorizable measures (REF ).",
"A change of presentation such as a translation, a rotation of the coordinates or an ordinary Lorentz transformation modifies the size of the multiscale corrections to the measure.",
"One realizes that different choices of the fractional frame lead to different theories in the integer frame.",
"Clearly, $q^1(r)\\ne \\sqrt{[q^1(x^1)]^2+[q^2(x^2)]^2}$ due to the nonlinear terms in the geometric coordinates.",
"In factorizable measures (REF ), coordinates never mix together due to the absence of rotation and boost invariance.",
"The only transformations preserving this structure are translations, which encode the ambiguity of presentation: $q^\\mu (x^\\mu )\\rightarrow \\bar{q}^\\mu (x^\\mu )=q^\\mu (x^\\mu -\\bar{x}^\\mu )\\,.$ Given an interval $\\Delta x^\\mu =|x_{\\rm B}^\\mu -x_{\\rm A}^\\mu |$ between two points A and B lying on the $\\mu $ -th direction, its geometric analogue $\\Delta \\bar{q}^\\mu = |\\bar{q}(x_{\\rm B}^\\mu )-\\bar{q}(x_{\\rm A}^\\mu )|$ for a binomial measure is $\\Delta \\bar{q}^\\mu = \\Delta x^\\mu |1\\pm \\mathcal {X}^\\mu |\\,,$ where $\\nonumber \\mathcal {X}^\\mu :=\\pm \\frac{1}{\\alpha _\\mu }\\frac{\\ell _*^\\mu }{\\Delta x^\\mu }\\left[\\left|\\frac{x_{\\rm B}^\\mu -\\bar{x}^\\mu }{\\ell _*^\\mu }\\right|^{\\alpha _\\mu } F_\\omega (x_{\\rm B}^\\mu -\\bar{x}^\\mu )-\\left|\\frac{x_{\\rm A}^\\mu -\\bar{x}^\\mu }{\\ell _*^\\mu }\\right|^{\\alpha _\\mu }F_\\omega (x_{\\rm A}^\\mu -\\bar{x}^\\mu )\\right].$ We define four different presentations characterized by special values of $\\bar{x}^\\mu $ : null presentation $\\bar{x}^\\mu =0$ , initial-point presentation $\\bar{x}^\\mu =x_{\\rm A}^\\mu $ , final-point presentation $\\bar{x}^\\mu =x_{\\rm B}^\\mu $ , and symmetrized presentation $\\bar{x}^\\mu =(x_{\\rm B}^\\mu +x_{\\rm A}^\\mu )/2$ .",
"At a first sight, one might want to discard all but the null presentation, which is the only one where the $\\bar{x}^\\mu $ do not depend on the “beginning” or “end” of the experiment (the measure of the geometry should be the same for all experiments).",
"However, we now show that the most natural choice is quite the contrary, the initial- and final-point presentations!",
"The origin of the multiscale measure of the theory has been recently clarified by the second flow-equation theorem and it reveals an important omission in eqs.",
"(REF ) and (REF ), which we correct here for the first time.",
"There, we interpreted $q^\\mu (\\ell ^\\mu )$ as the integral (indices $\\mu $ inert) $q^\\mu (\\ell ^\\mu )=\\int _0^{q^\\mu (\\ell ^\\mu )}dq^\\mu (x^\\mu )\\stackrel{?",
"}{=}\\int _0^{\\ell ^\\mu }dx^\\mu \\,v_\\mu (x^\\mu )\\qquad \\forall \\,\\mu \\,,$ but the following alternative is equally valid and based on the fact that the scales $\\ell ^\\mu =|x_{\\rm B}^\\mu -x_{\\rm A}^\\mu |$ are distances: $q^\\mu (\\ell ^\\mu )=\\int _{x_{\\rm A}^\\mu }^{x_{\\rm B}^\\mu }dx^\\mu \\,v_\\mu (x^\\mu -\\bar{x}^\\mu )\\qquad \\forall \\,\\mu \\,,\\qquad \\bar{x}^\\mu =x_{\\rm A}^\\mu , x_{\\rm B}^\\mu \\,.$ Equation (REF ) corresponds to the null presentation or, in other words, the null presentation is the choice of integration interval $[0,\\ell ^\\mu ]$ .",
"With posterior wisdom, it is almost obvious that this choice is not particularly happy.",
"On one hand, it takes both coordinate extrema $x_{\\rm A}^\\mu $ and $x_{\\rm B}^\\mu $ on the upper limit of the integral, which should already sound an alarm bell because it fixes the edge origin.",
"On the other hand, it eventually leads to corrections $\\mathcal {X}^\\mu (x_{\\rm A},x_{\\rm B})$ that depend on the initial and final coordinate separately.",
"The symmetrized presentation relies on an even more unnatural choice of integration domain and it leads to a trivial theory with $\\mathcal {X}^\\mu =0$ .",
"In contrast, eq.",
"(REF ) is valid both in the initial-point presentation [its right-hand side is $q^\\mu (\\ell ^\\mu )-q^\\mu (0)=q^\\mu (\\ell ^\\mu )$ , since $q^\\mu (0)=0$ ] and in the final-point presentation [the right-hand side is $q^\\mu (0)-q^\\mu (-\\ell ^\\mu )=q^\\mu (\\ell ^\\mu )$ , since the $q^\\mu $ are odd in their argument].",
"The integration domain is now the natural one $[x_{\\rm A}^\\mu ,x_{\\rm B}^\\mu ]$ and the corrections $\\mathcal {X}^\\mu (x_{\\rm B}^\\mu -x_{\\rm A}^\\mu )$ now depend only on the spatial distance or time interval, but not on the initial and final coordinates separately.",
"The desirability of this feature for physical predictions is evident and it was implicitly used in all phenomenology-oriented papers [50], [54], [55], [57], [58].Although incorrectly associated with a measure in null presentation.",
"In particle-physics experiments, one regards the point $\\bar{t}$ as the beginning of the observation or the moment when a certain collision occurs or a certain particle is created, while $t_*$ is the time, measured from $\\bar{t}$ , before which multiscale effects are important [55].",
"In cosmology, $\\bar{t}$ is the discriminator between “early” times $\\Delta t=t-\\bar{t}\\lesssim t_*$ and “late” times $\\Delta t\\gg t_*$ ; $\\Delta t$ represents the moment when a cosmological phenomenon takes place with respect to some special instant $\\bar{t}$ in the history of the universe, which may be the big bang [50], [53], [58].",
"And so on.",
"Therefore, we supersede the discussion of [56] and rule out the null and symmetrized presentations from the game, leaving only the initial- and final-point presentations.",
"The correction in eq.",
"(REF ) reads $\\mathcal {X}^\\mu =\\frac{1}{\\alpha _\\mu }\\left|\\frac{\\ell _*^\\mu }{\\ell ^\\mu }\\right|^{1-\\alpha _\\mu }F_\\omega (\\ell ^\\mu ).$ The sign in eq.",
"(REF ) depends on the choice between initial-point presentation ($+$ ) and final-point presentation ($-$ ).",
"Thus, the presentation (i.e., the value of $\\bar{x}^\\mu $ ) affects the output of physical measurements via the sign in front of multiscale corrections.",
"But how can we reconcile the initial- and final-point presentations with the requirement that the constant $\\bar{x}^\\mu $ , fixed in the measure of geometry, be the same for all observers?",
"In Minkowski spacetime for the theories $T_1$ , $T_v$ , and $T_q$ , we cannot because the weights $v(x-\\bar{x})$ appearing in the derivatives in the equations of motion break translation invariance.",
"On a curved background, however, the chart where the measure $q(x-\\bar{x})$ is defined is the local inertial frame of an observer, and the multiscale version of such frames exists for $T_v$ and $T_q$ (we do not know about $T_1$ and $T_\\gamma $ , but they probably exist at least for $T_\\gamma $ ).",
"Thus, a local observer is at liberty to choose $\\bar{x}$ in such a way that it coincides with $x_{\\rm A}$ or $x_{\\rm B}$ .",
"We will stress on this point also in question .",
"In the theory with multifractional derivatives (REF ) or (REF ), the problem is solved [if eq.",
"(REF ) satisfies a set of requirements yet to be checked] without invoking gravity, already in the case of a Minkowski embedding: the derivatives appearing in the equations of motion are translation invariant [eq.",
"(REF )], independently of the choice of $q(x-x^{\\prime })$ .",
"This is the presentation problem.",
"We differentiate between two possible views of it.",
"One, which we dub “deterministic,” has been advocated consistently from [40] until the appearance of [61], [62].",
"The other, which we call “stochastic,” has been proposed in ref.",
"[61], [62].",
"Although the deterministic view works, the stochastic view may work even better because it solves the presentation problem not by the brute force of Aristotelian logic (either one presentation or the other, tertium non datur), but by accepting both presentations at the same time.",
"– Deterministic view.",
"The tenet of this view is that a change of presentation changes the theory, i.e., the sign and magnitude of the corrections $\\mathcal {X}^\\mu $ .",
"Due to the smallness of these corrections, all qualitative features are unaffected [56].",
"It is well known that inequivalent presentations leave the anomalous scaling of the measure and the dimension of spacetime untouched [40], [41], basically for the same scaling argument by which the volume of a hypercube or of a $D$ -ball scale in the same way (question 0).",
"Therefore, multifractional scenarios are robust across different presentations, including those that we disfavoured above.",
"Picking a presentation corresponds to defining the theory and allows us to make predictions which will change in another presentation (i.e., another theory connected to the first by a one-parameter transformation), but not by much.",
"– Stochastic view.",
"Instead of making a choice between two inequivalent but equally valid theories, we can can try to have both theories coexist.",
"Since it is impossible to choose between the initial- and final-point presentation without an external input, we conceive a “macrotheory” with an intrinsic uncertainty in the presentation, so that the term $\\pm \\chi ^\\mu $ in eq.",
"(REF ) is interpreted as an irresoluble uncertainty in distance and time measurements [61], [62].",
"The mechanism to do so is not quantum mechanics but a stochastic reinterpretation of the coordinates of multifractional spacetimes [56], [61], [62].",
"The stochastic view could be realized in two ways, which are still under study.",
"One is by using log oscillations as a direct source of fluctuations, mimicking a stochastic effect when they average to zero [61], [62]; this possibility applies exactly to all multifractional theories.",
"The other way, which we will consider here, goes through the integration and differential structure as a whole [56], in which case this view is naturally implemented in $T_\\gamma $ (where the restriction to having a normed space is naturally lifted [61], [62]), while it is “superposed” to the structure of $T_1$ , $T_v$ , and $T_q$ .",
"Since the theory closer to $T_\\gamma $ is $T_q$ , we can apply this view successfully only in these two cases, the second being an approximation.",
"For $T_1$ and $T_v$ , we have to adopt the usual deterministic view, so that the presentation problem persists in the absence of gravity and is reduced to two presentations (determining the maximal uncertainty) in its presence.",
"Whenever we can choose either the initial- or the final-point presentation, in both views there is no reference to any special point in the coordinate chart $\\lbrace x^\\mu \\rbrace $ defining the fractional frame.",
"Geometry becomes a pure relativity of scales.",
"To summarize: – $T_\\gamma $ : the number of allowed presentations is two (initial- and final-point) in both flat and curved space.",
"The deterministic view holds and the two presentations define inequivalent theories that, in principle, can be discriminated by experiments sensitive enough to detect a deviation from standard physics.",
"In alternative, the stochastic view holds exactly and the presentation problem is replaced by an uncertainty on distance and time measurements.",
"– $T_q$ : the number of allowed presentations is infinite (a one-parameter family) in flat space and is reduced to two in the presence of gravity.",
"The deterministic view holds and the two presentations define inequivalent theories.",
"However, one can also adopt the stochastic view as an approximation and regard the two presentations as an intrinsic uncertainty effect.",
"– $T_v$ : the number of allowed presentations is infinite in flat space and is reduced to two in the presence of gravity.",
"The deterministic view holds and the two presentations define inequivalent theories.",
"There is no stochastic view.",
"– $T_1$ : the number of allowed presentations could be reduced to two only in the presence of gravity, provided multiscale local inertial frames existed.",
"The deterministic view holds and different presentations (two or infinitely many) define inequivalent theories.",
"There is no stochastic view.",
"$\\blacktriangleright $30$\\blacktriangleleft $ To show that the presentation problem signals an inconsistency, let us just confine ourselves to classical physics.",
"The fundamental principle governing classical dynamics is that the classical trajectories minimize a quantity that we call the action.",
"While, as we go from one frame to another, the action may look different written in terms of the fields, it is the same quantity that we must calculate in every frame.",
"In other words, once we decide that the action looks a certain way in a given frame, in any other frame its functional form must completely be determined via the usual Lorentz field transformations.",
"This property just follows from the requirement of the invariance of the action.",
"This functional form will not be preserved in multifractional theories, as the nonscalar $v(x)$ changes from one frame to another.",
"So, in essence, one has only one opportunity to choose a unique spacetime point in the universe, and once chosen one does not have the luxury to keep changing it to suit one's needs just because one is conducting different experiments.",
"That would mean that one is changing the action depending upon what experiment one is doing, when and where one is doing.",
"Also, from the point of view of plain diffeomorphisms, the zeros or the singularities of $v(x)$ are special points which have an independent meaning, contrary to diffeo-invariant theories where a point acquires meaning only in relation to the happening of a physical event.tocsubsection Does the presentation problem make the theory inconsistent?",
"Tensor fields in multifractional spacetimes transform with different laws with respect to the standard case [48], [55], and such laws replace usual Poincaré transformations as detailed in .",
"Advocating arguments based on symmetries that cannot be valid for actions with factorizable measures can only mislead to dead ends.",
"Moreover, when gravity is turned on the singularity point $\\bar{x}$ in the measure is no longer a unique point in the universe.",
"Rather, it is replicated at every local inertial frame (which exist both in $T_v$ and $T_q$ ), each with its own measure weight $v(x)$ attached.",
"This realizes the intuitive characteristic of self-similar fractals that geometry is anomalous at any point of the set and with the same scaling law everywhere [53], [56].",
"Although these arguments suffice, the core of this criticism affects the just-old version of multiscale theories where there was no superselection criterion for the choice of one of the four available presentations.",
"The justification then was that such a choice is simply part of the definition of the theory.",
"In [56], it was also suggested that the theory with fractional derivatives could realize a local notion of anomalous geometry even in the absence of gravity.",
"In that case, fractional calculus is shown to introduce a probabilistic character to spacetime: spacetime points $x$ become stochastic processes $X$ ; different presentations would simply amount to inequivalent prescriptions of integrodifferential calculus and, in turn, of stochastic integration.",
"With the advances made in , where we restricted the number of choices to two and removed any reference to a preferred point, we went a long way in giving a more satisfactory answer.",
"The arbitrariness in the presentation has been reduced to only two options, and what was previously interpreted as an integrable singularity of the measure, to be found somewhere in the universe, corresponds in fact to local measurements with no spatial or time extension, $\\ell ^\\mu =0$ .",
"This is the physical interpretation we were looking for."
],
[
"Field theory",
"$\\blacktriangleright $31$\\blacktriangleleft $ What is the action of the multifractional Standard Model?tocsubsection What is the action of the multifractional Standard Model?",
"Let $S_{\\rm SM}=\\int d^Dq(x)\\,\\mathcal {L}_{\\rm SM}$ be the Standard-Model action, where we split the Lagrangian into an electroweak bosonic, an electroweak leptonic, a quark, a Yukawa, and a Higgs sector: $\\mathcal {L}_{\\rm SM}=\\mathcal {L}_\\text{ew-bos}+\\mathcal {L}_\\text{ew-lep}+\\mathcal {L}_{\\rm quark}+\\mathcal {L}_{\\rm Yuk}+\\mathcal {L}_\\Phi $ .",
"The Standard Model Lagrangian with ordinary derivatives is $\\mathcal {L}_\\text{ew-bos} &=&-\\frac{1}{4}F^a_{\\mu \\nu }F_a^{\\mu \\nu }-\\frac{1}{4}B_{\\mu \\nu }B^{\\mu \\nu }\\,,\\\\\\mathcal {L}_\\text{ew-lep} &=&i\\overline{e_{\\rm R}}\\gamma ^\\mu \\nabla _\\mu e_{\\rm R} +i\\overline{L}\\gamma ^\\mu \\nabla _\\mu L\\,,\\\\\\mathcal {L}_{\\rm quark} &=&i{\\rm q}^{\\dagger \\alpha i} \\bar{\\sigma }^\\mu (\\nabla _\\mu {\\rm q})_{\\alpha i} +i\\bar{u}^{\\dagger }_\\alpha \\bar{\\sigma }^\\mu (\\nabla _\\mu \\bar{u})^\\alpha +i\\bar{d}^{\\dagger }_\\alpha \\bar{\\sigma }^\\mu (\\nabla _\\mu \\bar{d})^\\alpha +\\mathcal {L}[t,b,c,s]\\,,\\\\\\mathcal {L}_{\\rm Yuk} &=&-G_e\\overline{L}\\Phi \\,e_{\\rm R}+y^{\\prime }\\epsilon ^{ij}\\Phi _i {\\rm q}_{\\alpha j}\\bar{u}^\\alpha -y^{\\prime \\prime }\\Phi ^{\\dagger i} {\\rm q}_{\\alpha i}\\bar{d}^\\alpha +\\text{H.c.}\\,,\\\\\\mathcal {L}_\\Phi &=&-\\left(\\nabla _\\mu \\Phi \\right)^\\dagger \\left(\\nabla ^\\mu \\Phi \\right)+V(\\Phi )\\,,\\\\V(\\Phi ) &=& \\frac{\\lambda }{4}\\left(\\Phi ^\\dagger \\Phi -\\frac{1}{2} w^2\\right)^2,$ where the field strengths of the ${\\rm SU}(2)$ and ${\\rm U}(1)$ gauge fields $A_\\mu ^a$ and $B_\\mu $ are $F^a_{\\mu \\nu } &=& \\partial _\\mu A^a_\\nu -\\partial _\\nu A^a_\\mu - g^{\\prime }\\epsilon ^a_{\\ bc} A_\\mu ^b A_\\nu ^c\\,,\\\\B_{\\mu \\nu } &=& \\partial _\\mu B_\\nu -\\partial _\\nu B_\\mu \\,,$ the gauge covariant derivatives are $\\nabla _\\mu L &=& \\left(\\partial _\\mu + \\frac{i}{2}g^{\\prime }\\sigma _a A^a_\\mu +\\frac{i}{2}g B_\\mu \\right)L\\,,\\\\\\nabla _\\mu e_{\\rm R} &=& (\\partial _\\mu +ig B_\\mu ) e_{\\rm R}\\,,\\\\(\\nabla _\\mu {\\rm q})_{\\alpha i} &=& \\partial _\\mu {\\rm q}_{\\alpha i} + ig_s C^a_\\mu (\\lambda ^a)_\\alpha ^{\\ \\beta } {\\rm q}_{\\beta i}+ \\frac{i}{2}g^{\\prime }{A}^a_\\mu (\\sigma _a)_{i}^{\\ j} {\\rm q}_{\\alpha j}+\\frac{i}{6}g B_\\mu {\\rm q}_{\\alpha i}\\,,\\\\(\\nabla _\\mu \\bar{u})^\\alpha &=&\\partial _\\mu \\bar{u}^\\alpha + ig_s C^a_\\mu (\\lambda ^a)^\\alpha _{\\ \\beta } \\bar{u}^\\beta -\\frac{2i}{3}g B_\\mu \\bar{u}^\\alpha \\,,\\\\(\\nabla _\\mu \\bar{d})^\\alpha &=&\\partial _\\mu \\bar{d}^\\alpha + ig_s C^a_\\mu (\\lambda ^a)^\\alpha _{\\ \\beta } \\bar{d}^\\beta +\\frac{i}{3}g B_\\mu \\bar{d}^\\alpha \\,,\\\\\\nabla _\\mu \\Phi &=& \\text{same as $\\nabla _\\mu L$}\\,,$ the $\\sigma _a$ are the $2\\times 2$ Pauli matrices [generators of ${\\rm SU}(2)$ ], $\\gamma ^\\mu $ are the Dirac matrices, $\\lambda ^a$ , $a=1,\\ldots ,8$ , are the $3\\times 3$ Gell-Mann matrices [generators of ${\\rm SU}(3)$ ], $\\bar{\\sigma }^\\mu = (\\mathbb {1},-\\sigma ^a)$ , $C^a_\\mu $ are the color gauge potentials, $L=\\left(\\begin{matrix} \\nu _e\\\\ e_{\\rm L}\\\\ \\end{matrix}\\right)$ is the left weak isospin doublet, $e_{\\rm R}$ is the right isospin singlet, in $\\mathcal {L}_{\\rm quark}$ we wrote only the first quark family $(u,d)$ , ${\\rm q}_i$ , $i=1,2=u,d$ is a left-handed Weyl spinor under ${\\rm SU}(2)$ , $\\bar{u}$ and $\\bar{d}$ are antiquarks [singlets under ${\\rm SU}(2)$ ], $\\mathcal {L}[t,b,c,s]$ is the Lagrangian for the other quarks, “H.c.” means Hermitian conjugate, $\\Phi $ is the Higgs doublet, and $V(\\Phi )$ is its potential.",
"In standard Minkowski spacetime, $S_{\\rm SM}=\\int d^Dx\\,\\mathcal {L}_{\\rm SM}$ .",
"– The Lagrangian $\\mathcal {L}_{\\rm SM}$ in $T_1$ is the usual one ().",
"The couplings $g$ , $g^{\\prime }$ , $g_s$ , $G_e$ , $y^{\\prime }$ , $y^{\\prime \\prime }$ , $\\lambda $ , and $w$ are all constant.",
"– In $T_v$ [55], we have eqs.",
"() with $\\partial _\\mu $ replaced by $\\mathcal {D}_\\mu $ everywhere.",
"The couplings $g$ , $g^{\\prime }$ , $g_s$ , $G_e$ , $y^{\\prime }$ , $y^{\\prime \\prime }$ , $\\lambda $ , and $w$ are all measure dependent with the following form: $C(x) &=&\\sqrt{v(x)}\\,C_0\\,,\\qquad C=g,g^{\\prime },g_s,G_e,y^{\\prime },y^{\\prime \\prime }\\,,\\nonumber \\\\ C_0&=&g_0,g_0^{\\prime },g_{s0},G_{e0},y_0^{\\prime },y_0^{\\prime \\prime }={\\rm const}\\,,\\\\\\lambda (x) &=&v(x)\\,\\lambda _0\\,,\\qquad w(x)=\\frac{w_0}{\\sqrt{v(x)}}\\,,\\qquad \\lambda _0,w_0={\\rm const}\\,.$ – In $T_q$ [54], [55], we have eqs.",
"() with $\\partial _\\mu $ replaced by $\\partial /\\partial q^\\mu (x^\\mu )$ everywhere.",
"The couplings $g$ , $g^{\\prime }$ , $g_s$ , $G_e$ , $y^{\\prime }$ , $y^{\\prime \\prime }$ , $\\lambda $ , and $w$ are all constant.",
"– The Standard Model in $T_\\gamma $ has never been written down since it requires more study of the fundamentals of the theory.",
"We do not know whether it can be defined simply by replacing $\\partial _\\mu $ in () with ${}_q\\mathcal {D}_\\mu $ or $\\mathbb {D}_\\mu $ everywhere.",
"$\\blacktriangleright $32$\\blacktriangleleft $ Why to extend a so well functioning Standard Model with a multiscale version of it?tocsubsection Why to extend a so well functioning Standard Model with a multiscale version of it?",
"As discussed in point 0, the main motivation of multifractional theories is not phenomenological, it is to address two fundamental problems of quantum gravity: the physical meaning and consequences of dimensional flow and whether it is possible to carry the quantization program in a perturbative framework.",
"Once dimensional flow is implemented via the second flow-equation theorem in the measure of the theory, it affects virtually all sectors of physics, including that of fundamental quantum interactions.",
"Then, in order to assess the viability of multifractional theories, it is mandatory to explore all such sectors, in particular the consequences of a multiscale spacetime on the theoretical and observational characteristics of QFT.",
"The extension of the Standard Model is not an objective per se; certainly, it is an occasion to place strong constraints on the free parameters of the measure, hence our interest in it.",
"$\\blacktriangleright $33$\\blacktriangleleft $ In the theory with weighted derivatives, constants are promoted to fields, sometimes only time-dependent (for instance, the electric charge mentioned in question ), sometimes not.",
"What is the rationale behind these choices?tocsubsection What is the origin of the spacetime-dependent couplings in the theory with weighted derivatives?",
"There are four specifications to make from the start.",
"First, all the couplings in gauge covariant derivatives and field interactions in the fractional-picture action of the Standard Model depend on the measure weight (REF ), which is a fixed profile of spacetime coordinates.",
"Thus, according to eqs.",
"(REF ) and (), they depend on both time and space.",
"Second, constants are not promoted to fields because the measure weight (REF ) is not a scalar field.",
"Third, the spacetime-dependent couplings (REF ) and () are not ad hoc but originate from gauge invariance and the requirement of being able to do free field theory.",
"Fourth, one must distinguish between the couplings in the Lagrangian and observable couplings.",
"Let us clarify the origin of eqs.",
"(REF ) and ().",
"Consider a generic Yang–Mills theory $S=\\int d^Dx\\,v\\,\\mathcal {L}$ with a gauge bosonic vector field $A_\\mu ^a$ (Abelian in the case of electromagnetism, non-Abelian in general) and fermionic matter $\\Psi $ [55]: $\\mathcal {L}=-\\frac{1}{2}{\\rm tr}(\\mathcal {F}_{\\mu \\nu }\\mathcal {F}^{\\mu \\nu })+i\\overline{\\Psi }\\gamma _\\mu \\nabla _\\mu \\Psi -m\\overline{\\Psi }\\Psi \\,,$ where $\\mathcal {F}_{\\mu \\nu }:=\\mathcal {F}^a\\!_{\\mu \\nu } t_a$ , $F^a_{\\mu \\nu } =g^{-1}[\\partial _\\mu (g A^a_\\nu )-\\partial _\\nu (g A^a_\\mu )] - g f^a_{\\ bc} A_\\mu ^b A_\\nu ^c$ is the field strength of $A$ , $g=g(x)$ is the gauge coupling, and $t^a$ are the matrix representations of the Lie algebra $[t_a ,t_b]=if_{abc}t^c$ associated with the gauge group.",
"The covariant derivative in (REF ) is $\\nabla _\\mu =\\mathcal {D}_\\mu +ig A_\\mu ^a t_a\\,,$ where $\\mathcal {D}_\\mu $ is defined in (REF ).",
"A priori, the coupling $g(x)$ can be spacetime dependent; to see what this dependence is, one defines the gauge-invariant matter current $J^\\mu _a:=-g\\,\\overline{\\Psi }\\gamma ^\\mu t_a\\Psi $ , which is covariantly conserved: $\\nabla _\\mu J^\\mu _a=0\\,.$ Also, the Lagrangian density (REF ) is invariant under a ${\\rm U}(1)$ symmetry whose Noether current obeys the generalized conservation law $\\check{\\mathcal {D}}_\\mu (\\overline{\\Psi }\\gamma ^\\mu \\Psi )=0$ , where $\\check{\\mathcal {D}}_\\mu =v^{-1}\\partial _\\mu (v\\,\\cdot \\,)$ (in the theory $T_v$ , this type of derivative appears often in some conservation laws and in gravity [53], [55]).",
"Since $\\nabla _\\mu J^\\mu _a=0$ and $\\check{\\mathcal {D}}_\\mu (\\overline{\\Psi }\\gamma ^\\mu \\Psi )=0$ must agree when $f_{abc}=0$ , this implies $g(x)=\\sqrt{v(x)}g_0\\,,$ where $g_0$ is a constant.",
"Therefore, all couplings in the fractional picture have the spacetime dependence given by eq.",
"(REF ), where $v(x)$ is determined by the second flow-equation theorem.",
"In particular, the ${\\rm U}(1)$ charge of electromagnetism in the fractional picture is $e(x)=\\sqrt{v(x)}e_0$ .",
"The same dependence is found in Yukawa interactions.",
"In the Higgs sector, the scalar potential is (), which gives a nonzero vacuum expectation value to the Higgs doublet.",
"To obtain a Standard Model whose free sector is stable in the integer picture (a necessary requirement, if we want to have a manageable perturbation theory), both $\\lambda $ and $w$ must acquire a specific dependence on the measure weight $v(x)$ , given by eq.",
"() [55].",
"Then, the Higgs mass is the same in both the fractional and the integer picture.",
"None of the above couplings is a physical observable.",
"In the case of weak interactions, all observable couplings (for example, the Fermi constant or the masses of the $W$ and $Z$ gauge bosons) are a combination of Lagrangian couplings, and it turns out that the measure dependence cancels out in such combinations [55].",
"As a consequence, no exotic signatures are predicted in the weak sector alone.",
"The electromagnetic sector is more interesting.",
"The deformed conservation law for ${\\rm U}(1)$ is a special case of eq.",
"(REF ), $\\mathcal {D}_\\mu J^\\mu =0$ , which leads to the nonconservation equation of the electric charge [50] $Q(t):=\\int d^{D-1}{\\bf x}\\,v({\\bf x})\\,J^0(t,{\\bf x})\\simeq \\frac{e_0}{\\sqrt{v_0(t)}}\\,,$ where $v({\\bf x})$ is the spatial part of (REF ) and the last approximated expression was found in ref.",
"[50].",
"Therefore, this particular observable coupling is only time-dependent because it comes from the usual definition of Noether charges.",
"All the other Lagrangian couplings (REF ) in the strong and weak sectors are spacetime-dependent but this property is not seen in the observable couplings of strong and weak interactions.",
"In the other multifractional theories where the Standard Model is obvious ($T_1$ ) or has been constructed ($T_q$ [54], [55]), there is no effective definition of spacetime-dependent couplings and they are all constant.",
"The particle phenomenology is different from the case $T_v$ ; in particular, the weak sector of $T_q$ is nontrivial observationally.",
"$\\blacktriangleright $34$\\blacktriangleleft $ Can multiscale effects be mimicked by more traditional extensions of the Standard Model such as effective field theories?",
"There may be a strong call, from particle physicists, for some simplified exposition of the main ideas of multifractional theories.",
"For instance, effective field theories speak the language in which most extensions of the Standard Model are usually formulated.",
"It would be of big value to derive which higher-dimensional operators should be added to the Standard-Model Lagrangian to mimic multiscale effects.tocsubsection Can multiscale effects be mimicked by more traditional extensions of the Standard Model such as effective field theories?",
"To explain the QFT results in $T_v$ intuitively to a bigger circle of phenomenologists, it may be useful to make a link with more familiar formulations of physics beyond the Standard Model (this answer is taken from [55]).",
"What discussed in can be summarized by saying that the presence of an underlying multiscale geometry affects field theory in such a way that interaction terms (in gauge derivatives or in nonlinear potentials) acquire an explicit spacetime dependence of the form $[1+f(x)]\\phi ^i\\phi ^j\\cdots \\,,$ where $f(x)=f[v(x)]$ depends on the measure weight $v(x)$ and $\\phi ^i$ are some generic fields.",
"Terms such as (REF ) have a naive interpretation of “having promoted coupling constants to fields” and, in some sense, some of the physical effects we encounter are similar to those in models with varying couplings.",
"Another possibility to mimic effects of the form (REF ) is to add higher-dimensional operators to a traditional Lagrangian.",
"For instance, in a scalar-field theory one would have $\\nonumber V(\\phi )\\rightarrow [1+f(x)]V(\\phi )\\sim [1+\\phi ^m+\\phi ^n+\\cdots ]V(\\phi )$ for some exponents $m$ and $n$ , and one would fall into the context of effective field theories.",
"These are only superficial analogies not capturing the real nature of the multiscale paradigm.",
"The most evident departure is that $v(x)$ is not a scalar field and none of the above interpretations based on ordinary field theories has any such premade, nontrivial integrodifferential structure.",
"Since $v(x)$ [hence $f(x)$ ] is fixed by the geometry, it cannot be interpreted as a field and the higher-order-operators comparison dies as soon as one writes down the classical or quantum dynamics [classically, one does not vary with respect to $v(x)$ ; at the quantum level, $v(x)$ does not propagate].",
"The varying-coupling analogy is also of limited utility in the long run, since it does not explain why only certain couplings, but not others, depend on spacetime.",
"$\\blacktriangleright $35$\\blacktriangleleft $ Is field theory unitary?tocsubsection Is field theory unitary?",
"No, it is not, but it does not lead to problems.",
"To accept this paradoxical answer, we should examine its grounds.",
"In a general multiscale geometry, the usual symmetries are deformed and Noether currents are modified accordingly.",
"In the theory $T_v$ , these currents obey conservation laws such as (REF ), where the gradient operator encodes the multiscale nature of the geometry.",
"In the theory $T_q$ , one has conservation laws with $q$ -derivatives, $\\nabla _{q^\\mu (x^\\mu )}J^\\mu _a=0$ .",
"Thus, once written according to the differential structure typical of the theory, there is a notion of current conservation that implies unitarity.",
"More precisely, both $T_v$ and $T_q$ admit an integer picture where we have a standard unitary QFT, which is necessary and sufficient to compute the QFT observables of the theory.",
"On the other hand, however, the conservation laws with nonstandard derivatives are equivalent, in $T_v$ and $T_q$ [50], [55], to nonconservation laws with standard derivatives, which means that these theories in the fractional picture, where the weighted and $q$ -derivatives are written as standard derivatives multiplied by measure factors, are classically dissipative, i.e., nonunitary at the quantum level [41], [48].",
"Therefore, although the auxiliary QFT developed in the integer picture is unitary, the QFT in the fractional picture is nonunitary.",
"Also the study of quantum mechanics indicate that unitarity is violated but in a controllable way [46].",
"The model $T_1$ has no integer picture and the system is manifestly nonconservative.",
"It was in this context, similar to the more general case of multiscale field theories with nonfactorizable measures, that violation of unitarity was first predicted [104].",
"However, even if field theory can never be trivialized, nonconservation laws can be interpreted as governing an exchange of probability densities between the multiscale world and its $D$ -dimensional topological bulk [104].",
"Again, nonunitarity is there but under check.",
"Presumably, in the theory $T_\\gamma $ there will be conservation laws of the form “$\\partial ^\\alpha _\\mu J^\\mu =0$ ,” where the gradient is made of multifractional derivatives such as (REF ), (REF ), or (REF ).",
"Then, conservation in terms of first-order ordinary gradients will appear only in the IR asymptotic regime (it cannot be exact: fractional derivatives reduce to ordinary ones only asymptotically).",
"Or, nonconservation equations as in the $T_1$ case could appear.",
"$\\blacktriangleright $36$\\blacktriangleleft $ What is the propagator in multifractional theories?tocsubsection What is the propagator in multifractional theories?",
"We give the example of a real massive scalar in flat space, which captures all the main features of propagators.",
"Also, we omit the causal prescription of the propagator and consider a generic Green function solving $(\\mathcal {K}_x-m^2)\\,G(x,x^{\\prime })=\\delta _q(x,x^{\\prime })$ , where $m$ is the mass of the scalar and $\\delta _q(x,x^{\\prime })$ is the equivalent of the Dirac distribution in a multifractional geometry.",
"In general, the structure of the Green function is $G(x,x^{\\prime })=\\int \\frac{d^Dp(k)}{(2\\pi )^D}\\,\\mathbb {e}(k,x)\\mathbb {e}^*(k,x^{\\prime })\\,\\tilde{G}(k)\\,,\\qquad \\tilde{G}(k)=-\\frac{1}{-\\tilde{\\mathcal {K}}(k)+m^2}\\,,$ where $d^D p(k)=\\prod _\\mu dp^\\mu (k^\\mu )=:d^Dk\\,w(k)$ is the measure in momentum space [$w(k)=\\prod _\\mu w_\\mu (k^\\mu )$ is the measure weight], $k^\\mu $ are the momentum coordinates in the fractional frame, and $\\mathbb {e}(k,x)$ are the “plane waves” of the theory, i.e., the eigenfunctions of the Laplace–Beltrami operator $\\mathcal {K}$ : $\\mathcal {K}_x \\mathbb {e}(k,x)=\\tilde{\\mathcal {K}}(k)\\,\\mathbb {e}(k,x)$ .",
"– In $T_1$ , the fact that $\\mathcal {K}^\\dagger \\ne \\mathcal {K}=\\Box $ implies that the action (REF ) is physically inequivalent to an action with kinetic term $-(1/2)\\partial _\\mu \\phi \\partial ^\\mu \\phi $ .",
"In the first case, the Green function in momentum space is $\\tilde{G}_1(k)=-\\frac{1}{k^2+m^2}\\,,$ where $k^2=k_\\mu k^\\mu =-(k^0)^2+|{\\bf k}|^2$ .",
"There are two poles at ${\\rm Re} k^0=\\pm \\sqrt{m^2+|{\\bf k}|^2}$ and the usual interpretation of fields as particles.",
"In the case of the kinetic term $-(1/2)\\partial _\\mu \\phi \\partial ^\\mu \\phi $ , the structure of $\\tilde{G}(k)$ is completely different and branch cuts may arise for $\\alpha =2/D$ (this conclusion is reached by adapting the findings of ref.",
"[104] for $\\tilde{T}_1$ to the factorizable measure of $T_1$ ).",
"These problems disappear in $T_v$ , the natural upgrade of $T_1$ .",
"– In $T_v$ , the Dirac distribution is $\\delta _q(x,x^{\\prime })=\\delta (x-x^{\\prime })/\\sqrt{v(x)v(x^{\\prime })}$ , the plane waves are the weighted phases $\\mathbb {e}(k,x)=\\exp (ix_\\mu k^\\mu )/\\sqrt{w(k)v(x)}$ [43], $\\tilde{\\mathcal {K}}(k)=-k^2$ , and the Green function in momentum space is eq.",
"(REF ) [41], [48].",
"Again, we have two mass poles and fields are associated with particles.",
"– In $T_q$ , the momentum measure is eq.",
"(REF ).",
"The delta distribution is $\\delta _q(x,x^{\\prime })=\\delta [q(x)-q(x^{\\prime })]$ , plane waves are $\\mathbb {e}(k,x)=\\exp [iq_\\mu (x_\\mu ) p^\\mu (k^\\mu )]$ , and the Green function is attractively simple in geometric coordinates: $\\tilde{G}_q(k)=-\\frac{1}{p^2(k)+m^2}\\,,\\qquad p^2(k):=\\sum _\\mu [p^\\mu (k^\\mu )]^2=k^2+\\dots \\,.$ The usual poles are replaced by branch points, which cannot be determined analytically in general.",
"– In $T_\\gamma $ , we have not calculated the full multiscale propagator yet, but we can guess its general structure at any plateau of $d_\\textsc {h}$ , where (up to weight factors) $\\mathcal {K}_\\gamma \\sim \\partial ^{2\\gamma }$ is a fractional derivative and $\\mathcal {K}_\\gamma \\mathbb {e}(k,x)\\sim |k|^{2\\gamma } \\mathbb {e}(k,x)$ for each direction and up to a constant (to be determined by the type of derivative [40], [45]) [40], [41], [45].",
"Then, the Green function is something of the form (mass term rescaled) [41] $\\tilde{G}_\\gamma (k)\\sim -\\frac{1}{F^{2\\gamma }(k)+m^{2\\gamma }}\\,,\\qquad F^{2\\gamma }(k):=\\sum _\\mu |k^\\mu |^{2\\gamma }\\,,$ which, for $2\\gamma \\notin \\mathbb {N}$ , has a branch point at ${\\rm Re} k^0=\\pm (m^{2\\gamma }+\\sum _i |k^i|^{2\\gamma })^{1/(2\\gamma )}$ and a branch cut corresponding to a continuum of modes of rest mass $>m$ .",
"$\\blacktriangleright $37$\\blacktriangleleft $ The multiscale idea is quite exotic because it involves a nonstandard algebra of derivatives, and it may be difficult to understand its consequences for a field theory.",
"What is the physics behind perturbation theory?tocsubsection What is the physics behind perturbation theory?",
"This question is difficult to answer because QFT is yet unknown in $T_\\gamma $ (and in the less interesting case $T_1$ ), while in $T_v$ and $T_q$ it is under full control only in the integer picture.",
"We do not know much about the physical interpretation, i.e., about what happens in the fractional picture.",
"The following descriptions are an orientative start.",
"– The absence of symmetries and of a self-adjoint Laplacian has fatally stalled progress in the case of $T_1$ .",
"An example of the problems one may incur into is in the form of the propagator, which changes with the prescription made on the kinetic term (see the previous question).",
"Therefore, we directly move to its upgrade $T_v$ .",
"– In the theory with weighted derivatives $T_v$ , we have point particles but a perturbative treatment of their interactions does not follow conventional Feynman rules.",
"The main problem is that ordinary momentum is not conserved, as remarked in and .",
"Vertices in anomalous geometries do not combine like delta distributions as in ordinary QFT, since the Dirac delta is smeared to a sort of landscape of volcanoes [one for each term $n$ in eq.",
"(REF )].",
"Each external momentum brings a distribution $\\sim |k|^{-\\beta }$ (where $\\beta $ depends on $\\alpha _n$ ) which does not combine into a vertex distribution $\\sim |k_{\\rm tot}|^{-\\beta }$ .",
"– In the theory with $q$ -derivatives $T_q$ , we do not even have a notion of particle in the fractional picture, due to the form of (REF ).",
"Once recast the system into the integer picture both in position and in momentum space, we have effective particle fields in an effective ordinary QFT [mass poles at $p^2=-m^2$ in eq.",
"(REF )], and everything goes through smoothly.",
"But in the physical frame, none of that holds.",
"As in the case of $T_v$ , it seems that quantum interactions are heavily altered by taking place in a multiscale anomalous geometry, which dissipates energy and momentum into the embedding bulk.",
"In other words, quantum physics cannot be described by the nonadaptive measurements units of the fractional picture but, as soon as we consider adaptive units [i.e., multiscale coordinates $q(x)$ ] and move to the integer picture, a standard QFT emerges.",
"The resulting “observables” must be reconverted to nonadaptive measurements, which is all we have in the real world.",
"The surprising thing is that this procedure works and the final physical observables are well defined.",
"It may be that some deep mechanism is in action such that the scale hierarchy of the geometry and the measurement of quantum phenomena by a macroscopic apparatus of size $s$ affect each other in some yet poorly understood way.",
"In some still mysterious sense, the presence of yet another scale $s\\gg \\ell $ in the system, determined by the measurement apparatus and represented by the final conversion from the integer to the fractional picture, alters the multiscale hierarchy in quantum interactions and tames it to a finite result.",
"The appearance of such a scale in a recent comparison of the multifractional paradigm (with $\\alpha _\\mu =1/2$ ) with quantum-gravity uncertainties [61], [62] may be especially informative.",
"– In the case of the theory $T_\\gamma $ , the branch cut in eq.",
"(REF ) signals the presence of an infinite number of unstable quasiparticles for which we do not have a representation by Feynman diagrams.",
"We hereby recast the propagator (REF ) explicitly as such a superposition of pseudoparticle modes.",
"Ignoring the index $\\mu $ everywhere and taking $k>0$ , we have $-\\frac{1}{k^{2\\gamma }+m^{2\\gamma }}=-\\int _0^{k}d\\kappa \\frac{f(\\kappa )}{\\kappa ^2+m^2}\\,,\\qquad f(\\kappa )=2\\gamma \\kappa ^{2\\gamma -1}\\frac{\\kappa ^2+m^2}{(\\kappa ^{2\\gamma }+m^{2\\gamma })^2}\\,.$ This continuum of quasiparticles of mass $>m$ is equivalent to the superposition of massive particle modes of momenta $\\kappa $ smaller than $k$ , weighted by the distribution $f(\\kappa )$ .",
"The momentum distribution is plotted in figure REF for some values of $1/2\\leqslant \\gamma <1$ .",
"For $\\gamma =1/2$ and $m\\ne 0$ , $f(0)=1$ and $f(\\kappa )$ tends to 1 asymptotically at large $\\kappa $ ; at $\\kappa =m$ , there is a global minimum.",
"This case does not correspond to a continuum of quasiparticles since the propagator has a simple pole at $k=-m$ in this case.",
"For $1/2<\\gamma <1$ , $f(\\kappa )$ vanishes both at $\\kappa =0$ and asymptotically at large $\\kappa $ , with in between a local maximum at some $0<\\kappa <m$ and a minimum at $\\kappa =m$ .",
"As $\\gamma $ increases, the maximum gets closer to the minimum until the latter disappears at some critical value $\\gamma =\\gamma _*$ ; for $\\gamma >\\gamma _*$ , the distribution has a global maximum at $\\kappa =m$ .",
"In the massless limit for $1/2<\\gamma <1$ , the monotonic profile $f(\\kappa )= 2\\gamma \\kappa ^{1-2\\gamma }$ diverges at $\\kappa =0$ and vanishes asymptotically at large $\\kappa $ .",
"Therefore, for massless fields the main contribution in $f(\\kappa )$ comes from the $\\kappa =k$ mode, while for massive fields it comes from the branch point $\\kappa =m$ for sufficiently large fractional exponent $\\gamma $ .",
"Figure: The distribution f(κ)f(\\kappa ) in eq.",
"() for m=1m=1 and some values of γ\\gamma : with increasing thickness, γ=0.5,0.6,0.8\\gamma =0.5, 0.6, 0.8.",
"The dashed curve is the m=0m=0 case for γ=0.6\\gamma =0.6.This rewriting of the fractional Green function in terms of a superposition of ordinary propagators clarifies the physical interpretation of field theory in $T_\\gamma $ and it could help in the construction of perturbative QFT therein.",
"$\\blacktriangleright $38$\\blacktriangleleft $ Does Lorentz violation lead to a fine tuning in loop corrections, as predicted by the general argument of Collins et al.",
"[154], [155]?tocsubsection Does Lorentz violation lead to a fine tuning in loop corrections?",
"In scenarios with modified dispersion relations breaking Lorentz symmetry, the extra terms lead to large fine tunings at the quantum level [154], [155].",
"More precisely, loop corrections to the propagator generally lead to Lorentz violations several orders of magnitude larger than the tree-level estimate, unless the bare parameters of the model are fine-tuned.",
"Thus, even if one starts with a classical theory with tiny Lorentz-symmetry violations, one may end up with an observationally unacceptable enhancement of order of percent.",
"Usually, this happens in models where the dispersion relation acquires terms which dominate at small scales, as for instance in Lifshitz-type field theories [156] and, presumably, in Hořava–Lifshitz gravity.",
"There may exist quantum-gravity models which can bypass that argument [157] (but see ref.",
"[158]), and it was checked in ref.",
"[51] that also the multifractional theories $T_v$ and $T_q$ avoid this problem.",
"$T_1$ and $T_\\gamma $ have not been explored.",
"In both $T_v$ and $T_q$ , the key reason is the existence of an integer frame hosting an ordinary, formally Lorentz invariant field theory.",
"After field and coupling redefinitions (in $T_v$ ) or after moving to geometric coordinates (in $T_q$ ), loop calculations in the integer frame disclose no bad news.",
"In the case of $T_v$ , there also is an explicit calculation in the integer frame with nontrivial measure-dependent interactions [51].",
"The Dyson series for the full quantum propagator $\\tilde{G}$ of a scalar field in momentum space can be formally resummed to $\\tilde{G} = \\tilde{G}_1+A \\tilde{G}_1+A(A \\tilde{G}_1)+\\dots =[1-A]^{-1}\\tilde{G}_1$ , where $\\tilde{G}_1$ is given by eq.",
"(REF ), $A:=\\tilde{G}_1\\partial ^2(\\tilde{\\Pi }\\partial ^2\\,\\cdot \\,)$ , and $\\tilde{\\Pi }(k^2) \\sim (k^2+m^2) \\ln (k^2/m^2)$ is the self-energy function for large $|k^2|$ .",
"In a coupling expansion up to quadratic order and keeping only the first two terms of the Dyson series, In the large-$k$ limit one has $\\tilde{G} \\sim 1/(k^2-C\\ln k^2/k^2)$ , where $C$ is a constant.",
"Thus, in the ultraviolet ($k\\rightarrow \\infty $ ) the correction term is subdominant with respect to the usual one, and the fine-tuning problem is avoided [154], [155].",
"These modifications to the propagator are not introduced by hand, they are derived from the theory.",
"Thus, they also bypass some other general arguments related to Collins et al.",
"'s Lorentz violations [158].",
"$\\blacktriangleright $39$\\blacktriangleleft $ What about CPT symmetry?tocsubsection What about CPT symmetry?",
"Having discussed the transformation properties of the fields and the violation of local Poincaré symmetries in , we consider discrete Lorentz transformations: charge conjugation C, parity P, and time reversal T. The requirement of having a positive-semidefinite measure weight implies that the geometric coordinates are odd under reflection $q^\\mu (-x^\\mu )=-q^\\mu (x^\\mu )$ .",
"Since the measure weight (REF ) is even in the coordinates, classical multifractional theories are invariant under parity and time-reversal transformations PT (see [41] for $T_\\gamma $ and [55] for $T_v$ and $T_q$ ; the case of $T_1$ is obvious).",
"The presence of measure weights in the action does not affect the charge properties of spinors [55], so that also C is preserved classically.",
"Since QFT is performed in the integer picture, where $T_v$ and $T_q$ look the same as the ordinary Standard Model, the fate of CPT symmetry at the quantum level is the same as in the usual case, although differences in quantitative predictions may arise by the mechanisms detailed in ."
],
[
"Classical gravity and cosmology",
"$\\blacktriangleright $40$\\blacktriangleleft $ What is the gravitational action?tocsubsection What is the gravitational action?",
"– In $T_1$ , the action of gravity is [104], [53] $S_1[g,\\phi ^i] =\\frac{1}{2\\kappa ^2}\\int d^Dx\\,v\\,\\sqrt{-g}\\,\\left[R-\\omega (v)\\partial _\\mu v\\partial ^\\mu v-U(v)\\right]+S_1[\\phi ^i]\\,,$ where $\\omega $ and $U$ are functions of the weight $v$ , $R$ is the ordinary Ricci scalar and $S[v,\\phi ^i]$ is the matter contribution minimally coupled with the metric.",
"Apart from the dependence on the measure, the system is nonautonomous (i.e., the Lagrangian depends explicitly on the coordinates) unless $\\omega =0=U$ .",
"Even setting $\\omega =0=U$ , the gravitational sector is not the Einstein–Hilbert action, due to the presence of $v$ in the measure.",
"The equations of motion are different from those in an ordinary scalar-tensor theory, since $v$ is not a scalar field and the action is not varied with respect to it.",
"– In $T_v$ , the weighted derivatives in eq.",
"(REF ) are not the only ones appearing in the action of this theory.",
"Derivatives with more general weights ${}_\\beta \\mathcal {D}:=\\frac{1}{v^\\beta }\\partial (v^\\beta \\,\\cdot \\,)$ are necessary when tensor fields of rank greater than 1 enter the dynamics.",
"In the case of gravity, eq.",
"(REF ) is used to define the metric connection ${}^\\beta \\Gamma ^\\rho _{\\mu \\nu }:= \\tfrac{1}{2} g^{\\rho \\sigma }\\left({}_\\beta \\mathcal {D}_{\\mu } g_{\\nu \\sigma }+{}_\\beta \\mathcal {D}_{\\nu } g_{\\mu \\sigma }-{}_\\beta \\mathcal {D}_\\sigma g_{\\mu \\nu }\\right)\\,.$ It turns out that the only nontrivial covariant derivative among several consistent possibilities is $\\nabla _\\sigma ^- g_{\\mu \\nu } := \\partial _\\sigma g_{\\mu \\nu }-{}^\\beta \\Gamma _{\\sigma \\mu }^\\tau g_{\\tau \\nu }-{}^\\beta \\Gamma _{\\sigma \\nu }^\\tau g_{\\mu \\tau }$ .",
"Covariant conservation $\\nabla ^-_\\sigma g_{\\mu \\nu }=0$ of the metric translates into the Weyl-integrable condition $\\nabla _\\sigma g_{\\mu \\nu }=(\\beta \\partial _\\sigma \\ln v)\\,g_{\\mu \\nu }$ with respect to the standard covariant derivative.",
"Defining the fractional Riemann tensor $\\mathcal {R}^\\rho _{~\\mu \\sigma \\nu }:= \\partial _\\sigma {}^\\beta \\Gamma ^\\rho _{\\mu \\nu }-\\partial _\\nu {}^\\beta \\Gamma ^\\rho _{\\mu \\sigma }+{}^\\beta \\Gamma ^\\tau _{\\mu \\nu }{}^\\beta \\Gamma ^\\rho _{\\sigma \\tau }-{}^\\beta \\Gamma ^\\tau _{\\mu \\sigma }{}^\\beta \\Gamma ^\\rho _{\\nu \\tau }$ and the curvature invariants $\\mathcal {R}_{\\mu \\nu }:= \\mathcal {R}^\\rho _{~\\mu \\rho \\nu }$ and $\\mathcal {R}:= g^{\\mu \\nu }\\mathcal {R}_{\\mu \\nu }$ , the gravitational action is eq.",
"(REF ) [53].",
"For $\\beta =0$ , one recovers the case (REF ).",
"In general, one can obtain a relatively simple integer frame (not equivalent to standard general relativity) only when the gauge invariance of Weyl-integrable spacetimes is implemented (exactly if $\\omega =0=U$ , approximately if $\\omega $ or $U$ are nonvanishing), which results in fixing $\\beta =2/(D-2)$ .",
"In $D=4$ , $\\beta =1$ and the metric is a bilinear field density of weight $-1$ .",
"– In $T_q$ , the metric connection and the Riemann tensor are defined from the ordinary expressions, with the replacement (REF ): ${}^q\\Gamma ^\\rho _{\\mu \\nu } &:=& \\tfrac{1}{2} g^{\\rho \\sigma }\\left(\\frac{1}{v_\\mu }\\partial _{\\mu } g_{\\nu \\sigma }+\\frac{1}{v_\\nu }\\partial _{\\nu } g_{\\mu \\sigma }-\\frac{1}{v_\\sigma }\\partial _\\sigma g_{\\mu \\nu }\\right)\\,,\\\\{}^q R^\\rho _{~\\mu \\sigma \\nu } &:=& \\frac{1}{v_\\sigma }\\partial _\\sigma {}^q\\Gamma ^\\rho _{\\mu \\nu }-\\frac{1}{v_\\nu }\\partial _\\nu {}^q\\Gamma ^\\rho _{\\mu \\sigma }+{}^q\\Gamma ^\\tau _{\\mu \\nu }\\,{}^q\\Gamma ^\\rho _{\\sigma \\tau }-{}^q\\Gamma ^\\tau _{\\mu \\sigma }\\,{}^q\\Gamma ^\\rho _{\\nu \\tau }\\,,$ plus the curvature invariants ${}^q R_{\\mu \\nu }:= {}^q R^\\rho _{~\\mu \\rho \\nu }$ and ${}^q R:= g^{\\mu \\nu } {}^q R_{\\mu \\nu }$ .",
"The action of gravity and matter is $S_q[g,\\phi ^i] =\\frac{1}{2\\kappa ^2}\\int d^Dx\\,v\\,\\sqrt{-g}\\,({}^q R-2\\Lambda )+S_q[\\phi ^i]\\,,$ where in $S_q[\\phi ^i]$ the metric is minimally coupled.",
"– Gravity with multifractional derivatives is still under construction.",
"$\\blacktriangleright $41$\\blacktriangleleft $ What are the main features of cosmological dynamics in multifractional spacetimes?tocsubsection What are the main features of cosmological dynamics in multifractional spacetimes?",
"Despite the full dynamical equations having been laid down already, cosmological solutions have not been discussed in detail.",
"The little we know shows signs of an exotic evolution.",
"Here we write only the $D=4$ Friedmann equations (00 component of Einstein equations and the trace equation) for a homogeneous and isotropic background evolving with scale factor $a(t)$ and Hubble parameter $H=\\dot{a}/a$ .",
"These expressions were derived in ref.",
"[53] from the full Einstein equations of the theories (REF ), (REF ), and (REF ).",
"– In $T_1$ , the Friedmann equations with curvature $\\textsc {k}=0,\\pm 1$ and a perfect fluid with energy density $\\rho $ and pressure $P$ are $H^2&=&\\frac{\\kappa ^2}{3}\\rho -\\frac{\\textsc {k}}{a^2}-H\\frac{\\dot{v}}{v}+f_1(v,\\dot{v})\\,,\\\\\\frac{\\ddot{a}}{a}&=&H^2+\\dot{H}=-\\frac{\\kappa ^2}{6}(\\rho +3P)+H\\frac{\\dot{v}}{v}+f_2(v,\\dot{v})\\,,$ where possible measure-dependent terms have been collected into two contributions $f_1$ and $f_2$ .",
"The dynamics of this model has not been studied beyond a preliminary inspection [104].The homogenous cosmology of $\\tilde{T}_1$ is the same of $T_1$ , since the two models have the same type of derivatives and they differ only in the factorizability of the measure.",
"– In $T_v$ , one has a non-Riemannian geometry ($g_{\\mu \\nu }$ not conserved) that strongly resembles a Weyl-integrable spacetime.",
"With some manipulation, the action (REF ) looks like that of a scalar-tensor theory, with the difference that the scalar field is here replaced by a function of the measure weight $v$ .",
"After a conformal transformation to a frame (the integer picture) where the metric is conserved, the Friedmann equations read $H^2&=&\\frac{\\kappa ^2}{3}\\,\\rho +\\frac{\\Omega }{2}\\frac{\\dot{v}^2}{v^2}+\\frac{U(v)}{6v}-\\frac{\\textsc {k}}{a^2}\\,,\\\\\\frac{\\ddot{a}}{a}&=&-\\frac{\\kappa ^2}{6}\\,(\\rho +3P)+\\frac{U(v)}{6v}\\,,$ where $U(v)$ is a “potential” term for $v$ determined by the geometry (in general, solutions require $U\\ne 0$ ) and $\\Omega =-3/2+f(v)$ , where $f$ is a function of $v$ that, just like $f_{1,2}$ in eqs.",
"(REF ) and (), is not necessary usually and can be set to zero.",
"– In $T_q$ , the dynamics is $\\frac{H^2}{v^2}&=&\\frac{\\kappa ^2}{3}\\,\\rho -\\frac{\\textsc {k}}{a^2}\\,,\\\\\\frac{\\ddot{a}}{a}&=&-\\frac{\\kappa ^2}{6}\\,v^2(\\rho +3P)+H\\frac{\\dot{v}}{v}\\,.$ A simple power-law solution $a(t)=[q^0(t)]^p$ with a binomial measure illustrates the typical evolution [53].",
"The log oscillations of the measure give rise to a cyclic universe characterized by epochs of contraction and expansion, which progressively evolve to a monotonic scale factor at times $t\\gg t_*$ .",
"The average or coarse-grained scale factor is given by the zero-mode contribution only, i.e., setting $F_\\omega =1$ in the measure.",
"At early times $t\\lesssim t_*$ , the coarse-grained particle horizon expands faster than in standard cosmology.",
"In this theory, we also know what happens when inhomogeneities are included [58] (see question ).",
"– The cosmology of $T_\\gamma $ is unknown.",
"In none of the theories the evolution in the presence of radiation and dust matter has been considered yet and it would be important to check whether multifractional cosmologies are realistic.",
"The extreme rigidity of the dynamics, where the evolution of curvature is governed by that of the measure, should make all these cosmological models easily verifiable.",
"$\\blacktriangleright $42$\\blacktriangleleft $ Can you get acceleration from geometry without slow-rolling fields?tocsubsection Can you get acceleration from geometry without slow-rolling fields?",
"Yes, you can.",
"In $T_1$ , the term $H\\dot{v}/v+f_2$ in the right-hand side of eq.",
"() can give a positive contribution (averaging over log oscillations).",
"In $T_v$ , the term $\\propto \\Omega (\\dot{v}/v)^2<0$ in the right-hand side of eq.",
"(REF ) acts like the kinetic term of a phantom field (without having the theoretical problems associated with it), while the term $\\propto U/v$ is like a potential or a time-varying cosmological constant and, since $U>0$ for self-consistency of the solutions (it is not imposed by hand [53]), it gives a positive contribution to the right-hand side of eq.",
"().",
"Phantoms typically trigger super-acceleration; the bouncing vacuum solution found in ref.",
"[53] confirms this expectation.",
"The theory $T_q$ is less transparent.",
"Since $v\\simeq 1+|t/t_*|^{\\alpha -1}$ , for an expanding universe one has $H\\dot{v}/v\\propto \\dot{v} \\propto (\\alpha -1)|t/t_*|^{\\alpha -2}<0$ and the right-hand side of eq.",
"() can vanish for an equation of state $w=P/\\rho <-1/3$ .",
"Thus, it would seem that one needs a strong slow roll to get acceleration.",
"However, measure factors $1/v^2<1$ are hidden in $\\rho $ and $P$ , inside the kinetic term of fields, and they suppress it.",
"By this mechanism, potentials can dominate even if fields are not in very-slow roll.",
"$\\blacktriangleright $43$\\blacktriangleleft $ Can you explain inflation with this mechanism?tocsubsection Can you explain inflation with this mechanism?",
"Not in $T_q$ , because the flatness problem is not solved [53].",
"However, the slow-roll condition is relaxed.",
"In standard inflation, the primordial spectrum of scalar and tensor perturbations is described, as a first approximation, by the power spectrum $\\mathcal {P}_{\\rm s,t}=\\mathcal {A}_{\\rm s,t}(k/k_0)^n$ , where $k=|{\\bf k}|$ is the comoving wavenumber, $k_0$ is an experiment-dependent pivot scale and $n=n_{\\rm s}-1,\\,n_{\\rm t}$ are, respectively, the scalar and tensor spectral index.",
"In the theory $T_q$ with a binomial measure, this power law is deformed by the multifractional geometry according to the following expression [58]: $\\mathcal {P}_{\\rm s,t}(k) &\\simeq &\\mathcal {A}_{\\rm s,t}\\left(\\frac{k}{k_0}\\frac{\\alpha +\\left|\\frac{k_0}{k_*}\\right|^{1-\\alpha }}{\\alpha +\\left|\\frac{k}{k_*}\\right|^{1-\\alpha }}\\right)^n\\left[1+ A n\\cos \\left(\\omega \\ln \\frac{k_\\infty }{k}\\right)+B n\\sin \\left(\\omega \\ln \\frac{k_\\infty }{k}\\right)\\right.\\nonumber \\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\left.~~-A n\\cos \\left(\\omega \\ln \\frac{k_\\infty }{k_0}\\right)-B n\\sin \\left(\\omega \\ln \\frac{k_\\infty }{k_0}\\right)\\right].$ In the limit $k_*\\ll k< k_\\infty $ and averaging on log oscillations, one gets an effective power law $\\mathcal {P}_{\\rm s,t}(k)\\sim (k/k_0)^{n_{\\rm eff}}$ , where $n_{\\rm eff}=\\alpha n$ .",
"In particular, the effective spectral index of the primordial scalar spectrum is $n_{\\rm eff}-1 \\simeq \\alpha (n_{\\rm s}-1)$ asymptotically.",
"If $\\alpha $ is sufficiently small, one can soften the slow-roll condition [53] and get viable inflation, even when $n_{\\rm s}$ deviates from 1 by more than $10\\%$ [58].",
"One can see this intuitively by noting that the factor $1/v^2$ in the left-hand side of eq.",
"(REF ) can match a nontrivial time evolution of the right-hand side even when $H$ is approximately constant, while in standard cosmology a quasi-de Sitter evolution requires a matter energy density $\\rho >-3P$ .",
"However, one still needs a scalar field in slow roll in order to have a shrinking horizon during acceleration.",
"For the cosmological toy model $T_1$ and the theory $T_v$ , the Friedmann equations are known [53] but they have not been studied, nor have cosmological perturbations been considered.",
"Nothing is known about the cosmology of $T_\\gamma $ .",
"$\\blacktriangleright $44$\\blacktriangleleft $ Can you explain dark energy with this mechanism?tocsubsection Can you explain dark energy with this mechanism?",
"We do not know, but work is in progress and preliminary results are encouraging.",
"A cosmological constant term of purely geometric origin arises in $T_v$ , both in homogeneous cosmological solutions [53] and in black holes [63]; however, it is not clear whether it can act as dark energy in a fully realistic evolution of the universe.",
"$\\blacktriangleright $45$\\blacktriangleleft $ Are the big-bang and black-hole singularities resolved?tocsubsection Are the big-bang and black-hole singularities resolved?",
"The answer depends on the theory.",
"We do not know in the case of $T_1$ and $T_\\gamma $ , but there are some results for the other two theories.",
"In $T_v$ , the vacuum solution $a(t)$ of the dynamics (REF )–() with $\\textsc {k}=0$ is a bouncing universe that tends to de Sitter asymptotically in the future [53].",
"If one could show that general stable solutions with $\\rho \\ne 0$ have the same feature, there would be a concrete possibility to solve the big-bang problem in this theory.",
"Regarding black holes, it was recently shown that spherically-symmetric solutions to the Einstein equations are of Schwarzschild–de Sitter type, hence the pointwise singularity at the center persists [63].",
"Thus, the fate of singularities in $T_v$ is not clear.",
"In $T_q$ , an original reinterpretation of the big-bang problem was proposed [53].",
"Since a shift $q^\\mu (x^\\mu )\\rightarrow q^\\mu (x^\\mu )+x^\\mu _{\\rm bb}$ does not change the measure, an arbitrary constant $x^\\mu _{\\rm bb}$ can be added which would leave the gravitational action formally unchanged but would regularize the scale factor $a[q(t)]\\rightarrow a[q(t)+t_{\\rm bb}]$ at $t=0$ .",
"In the light of the second flow-equation theorem [60], we can now exclude this shift: the constant vector $x^\\mu _{\\rm bb}$ can be assimilated to the presentation label $\\bar{x}^\\mu $ , but we already have fixed that in the final- or initial-point prescriptions in the deterministic view of the theory and in the $T_{\\gamma =\\alpha }\\cong T_q$ approximation (see ).",
"Also, the arguments below eq.",
"(REF ) clearly show that what is really special is the null-distance configuration $\\ell ^\\mu =0$ , not the coordinate point $x^\\mu =0$ .",
"Therefore, the shift regularization cannot be implemented consistently in the theory.",
"The same effect could be achieved without any shift in the geometric coordinates, setting $\\alpha =0$ ; then, the constant term would come from the modulation factor in the measure $q(t)=t+F_\\omega (t)$ [53].",
"This geometric configuration has not been considered much in the past, since it does not correspond to a traditional dimensional flow where the spacetime dimension changes at large scale excursions $\\Delta \\ell $ : in this case, the dimension is constant in average but it is modulated by log oscillations.",
"An alternative that capitalizes on the stochastic view of [61], [62] is that, due to the intrinsic microscopic uncertainty in the geometry, we cannot probe the zero scale of the big bang, which is thus screened by this most peculiar effect.",
"Notice, however, that this mechanism does not work in the case of black holes: the singularity oscillates between a point and a ring topology (the two extrema of the initial- and final-point presentations) without ever disappearing [63].",
"We leave all these possibilities open for future study."
],
[
"Quantum gravity",
"$\\blacktriangleright $46$\\blacktriangleleft $ Is dimensional flow really so important in quantum gravity, where there may not even be a notion of spacetime?",
"The claim that one of the most striking phenomena we come across the landscape of quantum-gravity models is dimensional flow might be true in some abstract sense.",
"However, some would say that the very concept of spacetime might not make sense and thus the theory of multifractional spacetimes might not be of any interest to them.",
"Thus, multifractional models are addressed to a very particular sub-community of the overall quantum-gravity community.tocsubsection Is dimensional flow really so important in quantum gravity, where there may not even be a notion of spacetime?",
"Some theories of quantum gravity do not admit a notion of spacetime at the fundamental level.",
"The most clear example of that is the group of GFT-spin foams-LQG, where geometry emerges from a combinatorial structure (e.g., ref.",
"[27]).",
"Even in CDT, where the path integral over geometries is regularized by a discretization procedure and the continuum limit is eventually taken, a smooth spacetime arises only in the so-called phase C in the phase diagram of the theory, while all the other phases correspond to non-Riemannian geometries (mutually disconnected lumps of space in the branched-polymeric phase A, large-volume configurations of vanishing time duration in phase B, and signature changes in phase ${\\rm D}={\\rm C}_{\\rm b}$ [81], [159], [160]).",
"Nevertheless, the Hausdorff, spectral, and walk dimensions are indicators valid also in non-Riemannian geometries, as discussed in question 0 and showed in refs.",
"[68], [65] for the GFT-spin foams-LQG group of theories and in the typical sets describing the non-Riemannian CDT phases [81], [161], [162], [163], [164], [85], [165].",
"Fractal geometry by itself is proof that we do not need a smooth manifold to have dimensional flow [67].",
"Whether one sees them as independent theories of geometry or as effective models describing the flow of other theories in certain regimes, multifractional spacetimes are not addressed to a restricted audience.",
"They do not lack personality since they are based on a characteristic paradigm, they are a top-down approach from theory to experiments, and they offer their own predictions about physical observables.",
"More popular quantum-gravity approaches on the market have better or clearer results about the UV finiteness, but in some cases the phenomenology and contact with experiment is still underdeveloped or even absent.",
"The uniqueness argument in 0 guarantees that multifractional theories have a certain degree of universality in dimensional flow, so that placing constraints on this proposal can help to assess the phenomenology of other theories with dimensional flow.",
"If anything, one of the main messages of multifractional theories is that dimensional flow can be a testable property of exotic geometries, rather than an abstract property disconnected from physics.",
"$\\blacktriangleright $47$\\blacktriangleleft $ Let me reformulate the question.",
"Even accepting that dimensional flow exists for all quantum gravities, are dimensions really measurable?tocsubsection Are dimensions really measurable?",
"If $dh$ , $d_\\textsc {s}$ , and $d_\\textsc {w}$ were not physical observables, dimensional flow would be only a mathematical feature useful to classify multiscale spacetimes.",
"Some believe that these dimensions are not measurable and that they are just mathematical labels.",
"Others recognize that the Hausdorff dimension is measurable but they do not acknowledge the same status for the spectral dimension; consequently (but this has never been said explicitly), also the walk dimension cannot be measured.",
"Still others, like the author, are firm proponents of the measurability of all three dimensions.",
"That the spatial Hausdorff dimension is an observable is made clear by the following example [56].",
"Consider an observer in a space with $d_\\textsc {h}=D-1$ at large scales $\\ell \\gg \\ell _*$ and $0<d_\\textsc {h}<D-1$ at small scales $\\ell \\ll \\ell _*$ .",
"They can make several balls of radius $\\ell _1+\\delta \\ell $ close to some average value $\\ell _1\\gg \\ell _*$ (where $\\delta \\ell \\ll \\ell _*$ ), submerge each ball in a container of water and measure the volume of displaced liquid, noting a distribution of volumes with average $\\ell _1^{D-1}$ and width $\\sim \\ell _1^{D-2}\\delta \\ell $ .",
"Making another set of balls of average radius $\\delta \\ell <\\ell _2\\ll \\ell _*$ with the same fluctuation $\\delta \\ell $ , they find an average volume $\\ell _2^{d_\\textsc {h}}$ and (for $D\\geqslant 3$ and $d_\\textsc {h}\\geqslant 1$ ) a narrower distribution, since $1\\ll (\\ell _1/\\ell _*)^{D-2} > (\\ell _2/\\ell _*)^{d_\\textsc {h}-1}\\ll 1$ .",
"The inequality may change direction for $d_\\textsc {h}<1$ but, in any case, by comparing these dimensionless observables the experimenter realizes that they are living in a space with dimensional flow.",
"Ideally, the spectral and walk dimensions are measurable by placing a particle in a spacetime and let it diffuse.",
"Literally.",
"In practice, this procedure does not work when the scales we want to probe are much smaller than those covered by a molecule in Brownian motion.",
"For that reason, and also to solve the negative-probabilities problem in quantum gravity, it may be better to adopt the QFT perspective that the diffusing probe is a quantum particle in a scattering process [30].",
"However, it is not yet clear how this would help since $d_\\textsc {s}$ is the scaling of self-energy diagrams and, moreover, experiments with particle interactions cannot reach quantum-gravity scales.",
"This does not mean that the spectral dimension is not a physical observable, since its relation (or even identification) with the dimension of momentum space (see 0) opens up several possibilities of measurement [69].",
"When dealing with microscopic or very large scales, we cannot construct balls and submerge them in a liquid, or have ideal particles diffuse in spacetime, but appropriate experiments on high-energy physics or observations of cosmological scales can constrain both $d_\\textsc {h}$ and $d_\\textsc {s}$ with their characteristic tools.",
"In the case of multifractional theories, this is illustrated by the many examples reported in section .",
"$\\blacktriangleright $48$\\blacktriangleleft $ There are many approaches to quantum gravity, some of which were listed in section .",
"Can you compare multifractional theories with these other scenarios?tocsubsection Can you compare multifractional theories with other approaches to quantum gravity?",
"We can make this comparison at five levels: (i) in the characteristics of dimensional flow, (ii) in other characteristics of the geometry, (iii) in terms of the UV properties of renormalizability or finiteness, (iv) in the way the multifractional paradigm, seen as an effective framework, captures the geometry of other theories, and (v) at the level of phenomenology and observational constraints.",
"(i) The flow-equation theorems predict the general dimensional flow near the IR for any quantum gravity with nonfactorizable Laplacians [eq.",
"(REF )] and for multifractional theories where Laplacians are factorizable in the coordinates [eq.",
"(REF )].",
"The coefficients $b$ and $c$ in eq.",
"(REF ) are determined by the dynamics of the theory, while $b_\\mu $ and $c_\\mu =1-\\alpha _\\mu $ in eq.",
"(REF ) are free parameters with a restricted range (question 0).",
"The spacetime dimensions in multifractional theories have been computed in .",
"We compare the coefficients $b$ and $c$ in different theories of quantum gravity, expanding on the discussion of [60].",
"The results are summarized in table REF .",
"Table: Parameters of the IR Hausdorff and spectral dimension of spacetime () (subscript H and S, respectively) in quantum gravities.“Only space” (o.s.)",
"cases means that ℓ\\ell in eq.",
"() is a spatial scale (time dimension does not flow).",
"Question marks indicate cases not studies in the literature.The Hausdorff dimension $d_\\textsc {h}$ is the easiest to discuss: – Asymptotic safety [78], [23], [79], CDT [80], [81], [82], [83], spacetimes near black holes [100], [101], [102], fuzzy spacetimes [103], and string field theory and nonlocal gravity [77], [33] all have trivial dimensional flow in the Hausdorff dimension ($d_\\textsc {h}=D$ , where $D=4$ is some cases).",
"– Noncommutative spacetimes usually have $d_\\textsc {h}=D$ [92], [93], [94], but in the case of $\\kappa $ -Minkowski with cyclic-invariant action [166] $b<0$ ($d_\\textsc {h}$ increases from below) and $c=1$ [42], [59].",
"– Hořava–Lifshitz gravity has Lebesgue measure $dt\\,d^{D-1}x$ but with a time coordinate with anomalous scaling $[t]=-z<-1$ .",
"One can reabsorb this scaling into the spatial gradients $\\nabla ^{2z}$ of the theory, so that $d_\\textsc {h}=D$ [47].",
"– States of LQG and GFT describing general discrete quantum geometries display the kink profile of the binomial measure (REF ) without log oscillations [65], [66] (see figure 6 of ref.",
"[65]).",
"In the analytic example of the lattice $\\mathcal {C}_\\infty =\\mathbb {Z}^{D-1}$ , the Hausdorff dimension reads $d_\\textsc {h}-1=\\ell [\\psi (\\ell +D-1)-\\psi (\\ell )]$ , where $\\psi $ is the digamma function: $d_\\textsc {h}=2+O(\\ell )$ in the UV ($\\ell $ is measured in units of the lattice spacing), while in the IR $d_\\textsc {h}=D-(D-1)(D-2)/(2\\ell )+O(\\ell ^{-2})$ , giving $b<0$ and $c=1$ .",
"Concerning the spectral dimension $d_\\textsc {s}$ near the IR: – In asymptotic safety, $\\ell $ is the IR cutoff governing the renormalization-group equation of the metric [78], [23], [79].",
"The multiscale profile of the spectral dimension is calculated analytically at each plateau and numerically in transition regions.",
"The author is unaware of any semianalytic approximation giving $b$ and $c$ in (REF ).",
"– In Hořava–Lifshitz gravity, $d_\\textsc {s}\\simeq 1+(D-1)/z<D$ in the UV and $d_\\textsc {s}\\simeq D$ in the IR [99], so that $b<0$ .",
"No semianalytic profile connecting the UV to the IR has been computed, so that we cannot say much about $c$ apart that it is positive.",
"Using anomalous transport theory, it should be possible to find such profile with the multiscale tools of [45].",
"– The rest of the models have $c=2$ , without exception.",
"In CDT, $b<0$ is found numerically [80], [82], [83].",
"In a nonlocal field-theory model near a black hole, $b=(D+1)/2$ [102].",
"In fuzzy spacetimes, $b=-D$ [103].",
"In nonlocal gravity with $e^\\Box $ operators as in string field theory, $b<0$ (one can show that $b=-36$ in $D=4$ ) [77].",
"In LQG and GFT, one can check numerically that $b>0$ for all the classes of states inspected, that is, dimensional flow occurs from a UV with low dimensionality, reaches a local maximum $>D$ , and then drops down to the IR limit from above [65], [66].",
"Effective bottom-up approaches to LQG confirm dimensional flow to an UV spectral dimension smaller than $D$ , although they do not make an analysis of quantum states [70], [167].",
"– To date, the spectral dimension for $\\kappa $ -Minkowski with cyclic-invariant measure has not been calculated.",
"The other noncommutative examples, all with $c=2$ , are the following: in $D=3$ Einstein gravity with quantized relativistic particles, $b=-21/16$ [92]; in Euclidean $\\kappa $ -Minkowski space with bicovariant Laplacian and AN(3) momentum group manifold, $D=4$ and $b=-2$ [93], [94]; with AN(2) momentum group manifold, $D=3$ and $b=-3/2$ [94]; with bicrossproduct Laplacian, $D=4$ and $b=1$ [94].",
"The bicovariant-Laplacian results are compatible with an independent calculation in generic $D$ , where $b<0$ and $c=2$ [95], [96].",
"In none of these cases, except hints in the GFT-spin foams-LQG group [68], [133], complex dimensions preluding to log oscillations have been detected.",
"In the cases with discrete structures, this may be due to technical limitations in the way the spectral dimension has been computed, while in asymptotic safety the cutoff identification or the truncation of the action may play a role.",
"(ii) Asymptotic safety, phase C of CDT (after sending the triangulation size to zero), nonlocal gravity, string theory, and Hořava–Lifshitz gravity are defined on a continuum and spacetime, no matter how irregular it looks like at small scales due to quantum effects, can be described by a fundamental or effective metric $g_{\\mu \\nu }$ .",
"Phases A, B, and D of CDT do not correspond to metric manifolds but they admit a continuum description.",
"$\\kappa $ -Minkowski and other noncommutative spacetimes are defined in a continuous embedding, but noncommutativity of the coordinates makes the spacetime structure highly non-Riemannian.",
"GFT, spin foams, and LQG are all defined on pre-geometric structures such as group manifolds and combinatorial graphs; a continuous-spacetime approximation is reached in certain regimes (not only limited to the obvious semiclassical limit).",
"In LQG and spin foams, the continuum limit is subject to a number of delicate technicalities, while in GFT its realization is perhaps more transparent [168], [169], [170].",
"Multifractional spacetimes are fundamentally discrete in the sense that there is a DSI at ultrasmall scales (see ).",
"This symmetry is not exact and at larger scales it gives way to a continuum.",
"This transition happens via a natural coarse-graining procedure on the harmonic structure of the geometry [39], [41].",
"(iii) In nonperturbative approaches such as asymptotic safety, CDT, LQG, and spin foams, UV finiteness is achieved by other means than perturbative renormalizability.",
"In asymptotic safety, via the functional renormalization approach [171], [21], [22], [172].",
"In CDT [173], [174], [28], [24] and spin foams [175], [176], [177], [14], via the well-definiteness of the path integral of (pre)geometries.",
"In LQG, via canonical quantization of the gravitational constraints on a Hilbert space of (pre)geometric states (the spin networks) [12], [13].",
"GFT includes spin foams and LQG but it is a Lagrangian field theory on a group manifold; therefore, its renormalization properties can be tested either perturbatively (which constrains the forms of the kinetic term allowed by renormalizability) [178], [179], [180], [181] or nonperturbatively via the functional renormalization approach [182], [183], [184], [185].",
"Also the other major theories discussed in this review are based on perturbative field-theory renormalization, although in very different forms: examples are perturbative superstring theory (genus-expansion series of Riemann surfaces) [186], [187], [188], [189], [190], [191], [192], [193], [194], [195], [196], noncommutative field theory (nonplanar graphs) [197], [198], [199], [200], [201], [202], [203], [204], [205], nonlocal gravity (traditional QFT but with nonlocal operators) [31], [33], [37], [38], and Hořava–Lifshitz gravity (traditional QFT but with higher-order Laplacians) [206].",
"In nonperturbative formulations, UV finiteness is achievable but subject to a number of technical challenges or assumptions.",
"For instance, in the functional renormalization approach used in asymptotic safety and in GFT a truncation of the effective action is performed.",
"Still in GFT nonperturbative renormalization, all models considered so far are “toy” in the sense that they are with an Abelian group and without simplicity constraints (gravity requires a non-Abelian group and the implementation of simplicity constraints).",
"In perturbative formulations, renormalization has been proven only at a finite order (as in perturbative string theory and Hořava–Lifshitz gravity), or at all orders but for a scalar field or other toy models (such is the case of GFT and noncommutative QFT), or modulo important technical or phenomenological issues (as in nonlocal gravity and Hořava–Lifshitz gravity).",
"The case of multifractional theories will be discussed in question .",
"(iv) Some quantum gravities have been connected directly with multifractional spacetimes.",
"– The renormalization-group flow of asymptotic safety admits a complementary description in terms of a multifractional geometry [47], based on the observation that in the renormalization-group flow the physical momentum carries a scale dependence by the identification of the momenta $p_\\textsc {as}$ with the cutoff scale $L$ of the renormalization-group flow.",
"In the simplest case, $p_\\textsc {as}(L)=1/L$ .",
"These momenta are scale-dependent and, by requiring the same dimensional flow of asymptotic safety, they can be matched by the geometric coordinates $p(k)$ in the momentum space of the theory $T_q$ .",
"Thus, in asymptotic safety physical rods are adaptive and momenta are implicitly scale-dependent, while in multifractional theories physical rods are nonadaptive and momenta are scale-independent, but one establishes a mapping by using geometric coordinates in position and momentum space, corresponding to adaptive “mathematical” rods and explicitly scale-dependent momenta.",
"This direction-by-direction mapping is exact (Laplacians are factorizable) and also explains the reason why these two theories predict the same spectral dimension of spatial slices, $d_\\textsc {s}^{\\rm space}\\simeq 3/2$ in the deep UV, when $D=4$ and $\\alpha =1/2$ .",
"This should be contrasted with nonfactorizable theories such as Hořava–Lifshitz gravity, for which $d_\\textsc {s}^{\\rm space}\\simeq 1$ in the deep UV.",
"– The phase structure of CDT may find a counterpart in multifractional geometries [41].",
"The branched polymers of phase A might be describable by a UV multifractional regime at scales $\\ell _\\infty <\\ell \\ll \\ell _*$ where log oscillations modulate a highly nontrivial $d_\\textsc {h}\\simeq 2$ disconnected geometry.",
"In phase B, the concepts of dimension, metric and volume seem not to play a major role, since a phase-B universe has no extension in the time direction and topology becomes important.",
"This is akin to the most extreme limit of the multifractional measure, the so-called “boundary-effect” or “near-boundary” regime at scales $\\ell \\sim \\ell _\\infty $ , where the binomial measure (REF ) (indices $\\mu $ omitted here) is expanded at $|x/\\ell _\\infty |\\sim 1$ and becomes $q(x)\\sim \\ln |x|$ [39], [41].",
"The name of this regime stems from its relation with an approximation of fractional derivatives near the extrema of integration in their definition and it signals a central role for topology, just like in phase B.",
"This correspondence has not been formalized anywhere but it should not be hard to do so.",
"It would be worth doing it not only for its simplicity, but also for the payback it brings.",
"In particular, it immediately explains why random combs [85], [86], [165] cannot be associated with phase B: log oscillations are washed away in random structures.",
"– The anomalous scaling of the coordinates in Hořava–Lifshitz gravity can be easily interpreted in terms of binomial geometric coordinates [47].",
"In these anisotropic critical systems [99], [206], coordinates scale as $t\\rightarrow \\lambda ^z t$ and ${\\bf x}\\rightarrow \\lambda {\\bf x}$ for constant $\\lambda $ , so that time and space directions have dimensions $[t]=-z$ and $[x^i]=-1$ in momentum units.",
"This UV scaling is reproduced by an anisotropic multifractional model with $\\alpha _0=1$ and $\\alpha _i=1/z=1/(D-1)$ .",
"In particular, in four dimensions $\\alpha _i=1/3$ , the special value (REF ) recently come to the fore [61], [62].",
"The correspondence of coordinates is $q^0(x^0)=x^0=x^0_{\\rm HL}=t$ , $q^i(x^i)=x^i_{\\rm HL}$ , and physical momenta are defined consequently, $p^0(k^0)=p^0_{\\rm HL}$ , $p^i(k^i)\\sim p^i_{\\rm HL}$ .",
"To get a multiscale geometry, one builds the total action with a hierarchy of differential Laplacian operators, from order $2z$ (UV) to 2 (IR).",
"Of course, symmetries and action differ in these two theories: while in Hořava–Lifshitz gravity the UV spectral dimension is anomalous due to the higher-order Laplacian, in multifractional theories it is so because of the nontrivial measure appearing in integrals and derivatives.",
"– Noncommutative and multifractional spacetimes enjoy different symmetries and are therefore physically inequivalent.",
"Also, while we can devise noncommutative versions of multifractional theories and all noncommutative theories have nontrivial multiscale measures, we cannot interpret commutative multifractional theories as noncommutative theories.",
"The ultimate cause of these discrepancies is the fact that noncommutative theories have nonfactorizable measures, while the measure of multifractional theories is factorizable [59].",
"Nevertheless, these two classes of models have much in common, to the point where noncommutative geometry seems the natural candidate to generalize multifractional spacetimes to nonfactorizable measures [59].",
"In particular, the spacetime algebra of $\\kappa $ -Minkowski spacetime is obtained by a noncommutative $q$ -theory where geometric coordinates obey the Heisenberg algebra [42], [59], with a measure weight $v(x)$ reproducing the nontrivial measure found in the cyclic-invariant action of field theory on $\\kappa $ -Minkowski [42].",
"This correspondence is valid in the near-boundary regime discussed above and it yields eq.",
"(REF ) as an important bonus: one can identify the scale in log oscillations with the Planck scale.",
"Remarkably, the same identification is supported by a completely independent argument on distance uncertainties [61], [62], but only for $\\alpha =1/3$ .",
"– Motivated by the contact points between multifractional and noncommutative spacetimes on one hand [42], [61], [62], and the compatibility between the deformed Poincaré symmetries of $\\kappa $ -Minkowski spacetime and those of the effective-dynamics approach to LQG on the other hand [207], the constraint algebra of gravity in the multifractional theories $T_v$ and $T_q$ has been compared [59] with the deformed algebra of LQG models of effective dynamics [208], [209], [210], [211].",
"Although differences were expected for the reasons explained in the previous item, the types of deformation have been discussed in some detail [59].",
"See question .",
"– A comparison of multifractional theories with models beyond general relativity at the border with quantum gravity, such as varying-$e$ models [212], [213], [214], VSL models [140], [141], [215], doubly special relativity [216], [217], [218], [219], [220], and fuzzy spacetimes [103] can be found in refs.",
"[41], [50] (see references therein for a more exhaustive bibliography).",
"Another but less direct connection is that the spacetime dimensions in asymptotic safety, Hořava–Lifshitz gravity, and GFT-spin foams-LQG have been reconsidered or found anew with the tools of anomalous transport theory [79], [68], [65], which are heavily used in multifractional theories and have been proposed as a sharp instrument of analysis for quantum gravity in general [44], [45].",
"(v) See question .",
"$\\blacktriangleright $49$\\blacktriangleleft $ Some quantum gravities predict a deformation of the algebra of gravitational constraints.",
"What is the constraint algebra for multifractional gravities?tocsubsection What is the constraint algebra for gravity?",
"It should not come as a surprise that the only available results are, once again, for $T_v$ and $T_q$ [59].",
"We limit our attention to the classical algebra.",
"In the case of $T_v$ in $D=4$ , the super-Hamiltonian constraint in the integer frame and in ADM variables can be written as $H[N] = H_0[N] + H_v[N]=\\int d^3x\\,N(\\mathcal {H}_0+\\sqrt{\\tilde{h}}\\mathcal {H}_v)$ , where $N$ is the lapse function, $\\tilde{h}$ is the determinant of the spatial metric, $\\mathcal {H}_0=\\pi _{ij}\\pi ^{ij}/\\sqrt{\\tilde{h}}-\\pi ^2/(2\\sqrt{\\tilde{h}})-{}^{(3)}\\tilde{R}\\sqrt{\\tilde{h}}$ is only metric dependent, $\\pi _{ij}=\\delta S_v[\\tilde{g}]/\\delta \\dot{\\tilde{g}}^{ij}$ , and the density $\\mathcal {H}_v$ is both metric and measure dependent.",
"The diffeomorphism constraint is the usual one, $d[N^i]=-2\\int d^3x\\,N^i \\tilde{h}_{ij}d_{k}\\pi ^{kj}$ , where $N^i$ is the shift vector.",
"Since there are no dynamical degrees of freedom associated with $v$ , there is no conjugate momentum $\\pi _v$ .",
"Also, the $v$ -dependent correction term $\\mathcal {H}_v$ is not affected by diffeomorphisms.",
"Overall, $&&\\lbrace d[M^{k}],d[N^{j}]\\rbrace =d[\\mathcal {L}_{\\vec{M}}N^{k}],\\nonumber \\\\&&\\lbrace d[N^{k}],H[M]\\rbrace =\\lbrace d[N^{k}],H_0[M]\\rbrace =H_0[\\mathcal {L}_{\\vec{N}}M],\\\\&&\\lbrace H[N],H[M]\\rbrace = \\lbrace H_0[N],H_0[M]\\rbrace =d[\\tilde{h}^{jk}(N\\partial _{j} M-M\\partial _{j} N)],\\nonumber $ where $\\mathcal {L}$ is the Lie derivative.",
"As claimed in question , standard diffeomorphism invariance is preserved in the integer frame of $T_v$ in the absence of matter; when interacting matter fields are present, diffeomorphism invariance is broken.",
"In the case of $T_q$ , the algebra of first-class constraints is $&&\\lbrace d^q[M^{k}],d^q[N^{j}]\\rbrace =d^q\\left[\\frac{1}{v_j(x^j)}(M^j\\partial _j N^{k}-N^j \\partial _j M^k)\\right],\\nonumber \\\\&&\\lbrace d^q[N^{k}],H^q[M]\\rbrace =H^q\\left[\\frac{1}{v_j(x^j)} N^j\\partial _j M\\right],\\\\&&\\lbrace H^q[N],H^q[M]\\rbrace =d^q\\left[\\frac{h^{jk}}{v_j(x^j)}(N\\partial _{j} M-M\\partial _{j} N)\\right],\\nonumber $ where the index of the deformed measure weight $v_j$ is inert as usual.",
"The constraints $H^q[N]$ and $d^q[N^{k}]$ generate time translations and spatial diffeomorphisms of the geometric coordinates $q^\\mu (x^\\mu )$ , which means that these are not the usual time translation and diffeomorphisms.",
"A deformed constraint algebra appears in LQG when cancellation of quantum anomalies is imposed [208], [209], [210], [211].",
"The only constraint deformed is the bracket of the super-Hamiltonian, $\\lbrace H[N],H[M]\\rbrace = d[\\beta h^{ij}(N\\partial _i M-M\\partial _j N)]\\,,$ where $\\beta $ is a function; in general relativity and in other quantization schemes of LQG [221], [222], $\\beta =+1$ .",
"From eqs.",
"(REF ) and (REF ), we can see that the constraint algebra of LQG, independently of the quantization scheme, differs from the algebras of $T_v$ and $T_q$ .",
"In the case of $T_v$ , $\\lbrace H,H\\rbrace $ is untouched but deformations different from eq.",
"(REF ) appear when matter is turned on.",
"In the case of $T_q$ , both $\\lbrace d,H\\rbrace $ and $\\lbrace d,d\\rbrace $ are modified (more precisely, the algebra is not deformed but the generators $d\\rightarrow d^q$ and $H\\rightarrow H^q$ are), contrary to what happens in LQG.",
"Also, we cannot naively identify the LQG deformation function with $\\beta = 1/v_i(x^i)$ , since the left-hand side is a background-dependent function of the phase-space variables that can change sign, while the right-hand side is always positive and independent of the metric structure.",
"$\\blacktriangleright $50$\\blacktriangleleft $ Are multifractional field theories renormalizable?tocsubsection Are multifractional field theories renormalizable?",
"This question is general but its implicit target is quantum gravity.",
"A power-counting argument [76], [41] was at the origin of the multiscale paradigm.",
"According to eq.",
"(REF ), a scalar theory becomes super-renormalizable if $[\\mathcal {K}]=d_\\textsc {h}$ , that is to say, if the Laplace–Beltrami operator $\\mathcal {K}$ scales as a momentum to the power of the Hausdorff dimension of spacetime.",
"For a polynomial potential $V=\\sum _{n=0}^N \\sigma _n\\phi ^n$ , the coupling $\\sigma _N$ of the highest power has engineering dimension $[\\sigma _N]=d_\\textsc {h}-N(d_\\textsc {h}-[\\mathcal {K}])/2$ and the theory is power-counting renormalizable if $[\\sigma _N]\\geqslant 0$ , i.e., $&&N\\leqslant \\frac{2d_\\textsc {h}}{d_\\textsc {h}-[\\mathcal {K}]}\\qquad {\\rm if}\\quad [\\mathcal {K}]<d_\\textsc {h}\\,,\\\\&& N\\leqslant +\\infty \\qquad \\qquad {\\rm if}\\quad [\\mathcal {K}]\\geqslant d_\\textsc {h}\\,.$ When $[\\mathcal {K}]>d_\\textsc {h}$ , the theory is super-renormalizable.",
"Concentrating on eq.",
"(), if $d_\\textsc {h}=D$ , we need higher-order derivative operators, which introduce ghosts (Stelle gravity is a masterpiece example of this [223], [224]).",
"If $d_\\textsc {h}\\simeq D\\alpha $ in the UV, then we need either a second-order $\\mathcal {K}$ for $\\alpha =2/D$ (as in the original multiscale proposal $\\tilde{T}_1$ [76], [104], [105], in $T_1$ , and in $T_v$ ) or a $\\mathcal {K}$ with anomalous scaling for general $\\alpha <1$ (as in $T_q$ and $T_\\gamma $ ).",
"In the second case, however, if $\\gamma =\\alpha $ one has $[\\Box _q]=2\\alpha =[\\mathcal {K}_\\alpha ]$ in the UV and, from eq.",
"(REF ), one obtains the usual condition $N\\leqslant 2D/(D-2)$ .",
"Therefore, the theories $T_q$ and $T_{\\gamma =\\alpha }$ are not more renormalizable than in standard spacetime.",
"If $\\gamma \\ne \\alpha $ , we have $[\\mathcal {K}]\\geqslant d_\\textsc {h}$ only if $\\gamma \\geqslant \\frac{D\\alpha }{2}\\,.$ The limiting case is $\\alpha =2/D$ , where $\\gamma =1$ and one recovers either $T_1$ or $T_v$ .",
"In $D=4$ , having $\\gamma <1$ and asking for power-counting renormalizability corresponds to having a non-normed spacetime.",
"However, the condition for a norm was found in the absence of log oscillations [40] and the latter disrupt the standard properties of spacetime anyway.",
"Moreover, the presence of an intrinsic distance uncertainty in the deep UV of $T_\\gamma $ (question ) further indicates that having a norm is bound to become, sooner or later in the UV, an obsolete requirement.",
"Cognitive estrangement is thus generally expected in the extreme regimes of multifractional spacetimes.",
"The question is whether it is due to physically acceptable mechanisms.",
"Therefore, the power-counting argument gives good news for $T_1$ and $T_v$ , bad news for $T_q$ , and unclear news for $T_\\gamma $ .",
"To check whether renormalizability is actually improved (or not) on a multifractional spacetime, one must go beyond the power-counting argument and employ either perturbative or nonperturbative QFT techniques.",
"The only clear results we have so far are perturbative and only for $T_v$ and $T_q$ in the deterministic view.",
"Here we review them and provide a new insight in $T_q$ and $T_\\gamma $ .",
"The bottom line is that we have no news for $T_1$ (but, as we said, we do not care too much about that, since the upgrade of $T_1$ is $T_v$ ), bad news for $T_v$ (against the power-counting argument), bad news for $T_q$ in the deterministic view (in line with the power-counting argument), and intriguing news both for $T_q$ in the stochastic view and for $T_\\gamma $ in either view.",
"In the theory with weighted derivatives, the degree of divergence of Feynman graphs in a scalar field theory does not improve with respect to standard QFT [51].",
"An easy argument showing that the renormalizability of this theory is basically the same as that of the standard theory is the following.",
"In the fractional frame, the measure in the momentum integration in loops is $d^Dk\\,w(k)$ , where the weight $w(k)$ is such that the scaling dimension of the measure is smaller than $D$ .",
"However, when coupled with the full expression with two fractional phases $\\mathbb {e}(k,x)=e^{ik\\cdot x}/\\sqrt{w(k)v(x)}$ (such as in propagators), the latter include two factors $w^{-1/2}$ , which cancel the weight in the measure.",
"Thus, the degree of divergence of momentum integrals remains the same as in the integer field theory.",
"The actual degree of divergence of some diagrams differ with respect to the power-counting argument [51] but essentially agrees with its main conclusion.",
"Yet another, more intuitive way to understand this point is to notice that the free multifractional propagator in position space is of the form $G_v(x,y)=\\frac{G_1(y-x)}{\\sqrt{v(y-\\bar{x})v(x-\\bar{x})}}$ for any factorizable positive semidefinite measure $v$ in presentation $\\bar{x}$ [48] (see question ).",
"Therefore, the divergence of $G_v(x,y)$ at coincident points $x\\sim y$ is solely determined by the usual propagator $G_1(x-y)$ and not by the prefactor $\\sim 1/v(y)$ .",
"The theory with $q$ -derivatives in the deterministic view does not work, either.",
"Its basic renormalization properties can be inferred from position space, according to the following scaling argument.",
"In the rest of the answer, we omit spacetime indices and also avoid cumbersome expressions in geometric polar coordinates; a rigorous calculation could easily fill the gaps in this heuristic reasoning without major surprises.",
"The free propagator is $G_q(x,y)=G_1[q(y)-q(x)]$ and its behaviour at $x\\sim y$ is the same as the standard theory.",
"For instance, in the massless case $G_q(x,y)\\sim \\frac{1}{|q(y)-q(x)|^{D-2}}\\sim \\frac{1}{|v(y-\\bar{x})(y-x)|^{D-2}}$ upon Taylor expanding around $x=y$ , and at coincident points inverse powers of $q(x)-q(y)$ diverge as inverse powers of $x-y$ .",
"Here $|q(y)-q(x)|=\\sqrt{\\sum _\\mu [q^\\mu (y^\\mu )-q^\\mu (x^\\mu )]^2}$ .",
"The only points where these arguments fail are those corresponding to the measure singularity at $y=\\bar{x}=x$ , where the above expressions vanish.",
"The main conclusion is not modified in the deterministic view, but something interesting may happen in the stochastic view.",
"As said in , we can adopt this view in $T_q$ when regarded as an approximation of $T_{\\gamma =\\alpha }$ .",
"We can see here how by computing the Green function both in $T_\\gamma $ and in $T_q$ ; for $T_\\gamma $ , we only sketch a back-of-the-envelope calculation.",
"The second flow-equation theorem selects the initial-point and the final-point presentation as special among all the others, and in $T_\\gamma $ one can always choose either presentation thanks to translation invariance.",
"Therefore, the propagator will be of the form $G_\\gamma (x,y)=G_\\gamma (y-x)$ .",
"Calling $r=|y-x|$ , recall that the Fourier transform $\\mathcal {F}$ of a power law $r^\\beta $ is proportional to $k^{-(\\beta +1)}$ .",
"In $D$ -dimensional Euclidean space, in polar coordinates we have $k^{-2}\\propto \\mathcal {F}[r]=\\mathcal {F}[r^{D-1} r^{2-D}]$ .",
"The factor $r^{D-1}$ is the Jacobian in polar coordinates, which leads to $G_1(r)\\sim r^{2-D}$ as the Green function in position space.",
"Similarly, from the propagator (REF ) we get $k^{-2\\gamma }\\propto \\mathcal {F}[r^{2\\gamma -1}]=\\mathcal {F}[r^{D\\alpha -1} r^{2\\gamma -D\\alpha }]$ , and identifying $r^{D\\alpha -1}$ with the Jacobian in a space with UV Hausdorff dimension $d_\\textsc {h}\\simeq D\\alpha $ , we get the free propagator $G_\\gamma (r)\\sim r^{2\\gamma -D\\alpha }$ in the theory $T_\\gamma $ at the plateau $d_\\textsc {h}\\simeq \\alpha $ , which can be generalized to the whole dimensional flow and to the presence of log oscillation.",
"When $\\gamma =\\alpha $ , $G_\\alpha (x-y)\\sim \\frac{1}{|q(y-x)|^{D-2}}\\stackrel{{\\rm UV}}{\\sim } \\frac{1}{|y-x|^{\\alpha (D-2)}F_\\omega ^{D-2}(y-x)}\\,,$ where we have taken an isotropic binomial measure to illustrate the typical UV behaviour.",
"On the other hand, we cannot use the initial- or final-point presentations in $T_q$ because we cannot conveniently fix $\\bar{x}$ case by case.",
"However, if we did, from eq.",
"(REF ) we would obtain exactly the same behaviour as in $T_{\\gamma =\\alpha }$ : $G_q(x,y)\\simeq G_1[q(y-x)-q(0)]\\simeq G_\\alpha (x-y)\\,.$ Thus, eq.",
"(REF ) is the typical Green function of the theory $T_\\gamma $ with fractional derivatives of order $\\alpha $ , approximated by the theory $T_q$ with $q$ -derivatives.",
"Let us discuss its main properties, beginning with the deterministic view.",
"For $F_\\omega =1$ (coarse-grained or no log oscillations), the singularity of the propagator (or of the Newtonian potential, to cite another example) is softened but, in accordance with the power-counting argument, not removed.",
"Nevertheless, in the limit $\\alpha \\rightarrow 0$ , we reach the $\\alpha =0$ geometric configuration already met in and the propagator (or the potential) tends to a constant.",
"This phenomenon is very similar to what found in nonlocal theories and is related to asymptotic freedom [225], [226].",
"It signals the possibility that interactions, including gravity, become weak in the deep UV.",
"The limit $\\alpha _\\mu \\rightarrow 0$ cannot be reached in $T_\\gamma $ if we require spacetime to be normed (question 0), but if we regard $\\alpha $ in the Hausdorff dimension in eq.",
"(REF ) as the average fractional charge we can get $\\alpha =0$ by setting some of the charges $\\alpha _\\mu $ to negative values.",
"As said in 0, these geometries are strange (or even unphysical) because they have no norm along some or all directions [40] and, in general, the dimension of time or of spatial slices become negative.",
"However, this is not the end of the story.",
"If $F_\\omega \\ne 1$ is nontrivial, then at scales $\\sim \\ell _\\infty $ the divergence becomes of the form $\\sim (\\ln 1)^{2-D}$ for any $\\alpha $ , since $q(y-x)\\sim \\ln (y-x)$ in that case; this is the near-boundary regime described in .",
"Going at even smaller scales, $G_\\gamma $ diverges periodically at the zeros of $F_\\omega (y-x)$ .",
"This behaviour, induced by the discreteness of the geometry at these scales, is totally different from what we would expect in a traditional resolution of singularities or in a renormalization scheme in a continuum.",
"In the absence of a better name and of an explanation, we call this a DSI divergence or DSI singularity.In [53], the DSI approach to the big bang was compared at first sight to the BKL singularity.",
"A quantitative comparison is still missing.",
"Notice that this possibility is realized only if the amplitudes of the log oscillations are large enough.",
"The most negative contribution to $F_\\omega $ is given by an angle of $5\\pi /4$ , where $F_\\omega =1-\\sqrt{2}(A+B)$ .",
"Assuming $A=B$ , $F_\\omega $ vanishes for as small an amplitude as $A=B=\\frac{1}{2\\sqrt{2}}\\approx 0.35\\,,$ which is not excluded by CMB observations [58] (see question ).",
"In the stochastic view, the propagator is $G_\\alpha \\sim G_q\\stackrel{{\\rm UV}}{\\sim } \\frac{1}{|r(1\\pm \\mathcal {X})|^{D-2}},$ where $\\mathcal {X}\\sim |\\ell _*/r|^{1-\\alpha }$ is an adaptation to polar coordinates of the correction (REF ).",
"The DSI oscillations are just a blurring out of spacetime below the intrinsic uncertainty $\\mathcal {X}$ , which grows as we approach the singular point $r=0$ .",
"This stochastic noise is the authentic texture of spacetime at these scales and it screens the observer from singularities: the point $r=0$ cannot be reached physically.",
"Since we cannot measure lengths smaller than $r\\mathcal {X}$ , the existence of a norm at these scales is irrelevant and we can contemplate exponents $\\alpha <1/2$ .",
"So, are infinities tamed in $T_\\gamma $ or not?",
"We do not know, but the quest for an answer promises to be stimulating both in the deterministic view (where we have the mysterious DSI singularity or a very exotic non-normed geometry) and in the stochastic view just described."
],
[
"Observations",
"$\\blacktriangleright $51$\\blacktriangleleft $ Can a multifractal observer be aware of being in a multifractal spacetime?tocsubsection Can a multifractal observer be aware of being in a multifractal spacetime?",
"Yes, they can.",
"An observer can recognize whether the underlying geometry is standard or multiscale (in particular, multifractal or multifractional) by measuring dimensionless quantities such as the ratio of two observables of the same kind [56].",
"We saw an example of this procedure in question for the measurement of volumes.",
"Another instance is the following.",
"Consider $T_q$ in $D=1+1$ dimensions and suppose that two nonrelativistic objects a and b of very different size move with velocities $V_{x,{\\rm a}}=\\Delta x_{\\rm a}/\\Delta t$ and $V_{x,{\\rm b}}=\\Delta x_{\\rm b}/\\Delta t$ in the fractional picture.",
"In the integer picture, one can compute the geometric velocity $V_q=\\frac{\\Delta q(x)}{\\Delta q(t)}=\\frac{\\Delta x|1\\pm \\mathcal {X}|}{\\Delta t|1\\pm \\mathcal {X}^0|}=V_x \\left|\\frac{1\\pm \\mathcal {X}}{1\\pm \\mathcal {X}^0}\\right|\\,,$ where we used eq.",
"(REF ).",
"Clearly, the ratio of the velocities of a and b will be different in a multifractional spacetime with respect to an ordinary spacetime, $V_{x,{\\rm a}}/V_{x,{\\rm b}}\\ne V_{q,{\\rm a}}/V_{q,{\\rm b}}$ , and a discrimination between the two spaces is possible when we measure the ratios of several objects, or when a and b are related to each other and some physical law predicts the value of such ratio.",
"This naive example is obviously inapplicable to the real world where, if multiscale geometry were true, one would find exotic effects at the scales of relativistic quantum physics or smaller.",
"However, the main mechanism can be adapted to more realistic experiments.",
"$\\blacktriangleright $52$\\blacktriangleleft $ Have these theories been constrained by observations?",
"What are the constraints?tocsubsection Have these theories been constrained by observations?",
"What are the constraints?",
"Yes.",
"The multifractional theories $T_v$ and $T_q$ with a binomial measure have been confronted with experiments and observations, and bounds have been placed on the scale $\\ell _*$ , on the fractional exponents $\\alpha _\\mu =\\alpha _0,\\alpha $ , and on the amplitudes $A$ and $B$ of log oscillations.",
"The first datum that was considered was the variation of the fine-structure constant $\\alpha _\\textsc {qed}$ in quasars, but the bound on $T_v$ thus found was poor [50].",
"The construction of the multifractional Standard Model permitted to use known constraints on electroweak interactions, including the estimate of the muon lifetime and of $\\alpha _\\textsc {qed}$ , and the Lamb shift [54], [55].",
"Astrophysical processes such as the first black-hole merger observed by LIGO and gamma-ray bursts (GRB) from distant objects placed the strongest bounds on $\\ell _*$ [57], while the main contribution of cosmology comes from the CMB black-body and inflationary spectra [58].",
"The latter do not constrain $\\ell _*$ efficiently but do allow to constrain the fractional charge (hence, the dimension of spacetime) and the log oscillations.",
"Tables REF –REF summarize these results.In the line “CMB black-body spectrum ” in table REF , we correct a typo of table 2 of ref.",
"[58]; compare eq.",
"(3.17) therein.",
"Table: Absolute bounds on the hierarchy of multifractional spacetimes with weighted derivatives (obtained for α 0 ,α≪1\\alpha _0,\\alpha \\ll 1).",
"All figures are rounded.",
"Items “—” are cases where the theory gives the standard result or where the experiments listed in the table are unable to place significant constraints.",
"Empty cells are cases not explored yet.Table: Bounds on the hierarchy of multifractional spacetimes with weighted derivatives for α 0 =1/2=α\\alpha _0=1/2=\\alpha .Table: Absolute bounds on the hierarchy of multifractional spacetimes with qq-derivatives (obtained for α 0 ,α≪1\\alpha _0,\\alpha \\ll 1 in all cases but for the last one, where a likelihood analysis has been used).",
"“Pseudo” indicates bounds obtainable in the stochastic view, where T q T_q is regarded as an approximation of T γ=α T_{\\gamma =\\alpha }.Table: Bounds on the hierarchy of multifractional spacetimes with qq-derivatives for α 0 =1/2=α\\alpha _0=1/2=\\alpha .",
"“Pseudo” indicates bounds obtainable in the stochastic view, where T q T_q is regarded as an approximation of T γ=α T_{\\gamma =\\alpha }.The results of [60] stimulates us to review these bounds critically.",
"They all arise from the binomial measure (REF ) with (REF ), with or without log oscillations.",
"However, the second flow-equation theorem does not really fix the coefficient $\\ell _*/\\alpha _\\mu $ but, rather, it treats it as an arbitrary constant $\\ell _* u_\\mu $ not necessarily $\\alpha _\\mu $ -dependent.",
"The original motivation for the coefficient $\\ell _*/\\alpha _\\mu $ is that the equations of motion and the physical observables in the theory $T_v$ depend only on the measure weight $v(x)=1+|x/\\ell _*|^{\\alpha -1}$ (log modulation is ignored), not on $q(x)$ .",
"The anomalous correction in $v(x)$ depends on the arbitrary scale $\\ell _*$ and there is no need to introduce a new parameter $u_\\mu $ .",
"However, in the theory $T_q$ having an $\\alpha $ -dependent or $\\alpha $ -independent constant $u_\\mu $ can weaken some of the bounds in tables REF and REF .",
"These new bounds with (index or label $\\mu $ omitted everywhere) $q(x)=x+\\ell _* u\\left|\\frac{x}{\\ell _*}\\right|^{\\alpha }\\,,\\qquad u=O(1)\\,,$ are shown in table REF and commented upon in .",
"We will compute one of them explicitly in .",
"As one can see by comparing tables REF –REF and REF , the new bounds are slightly weaker than the previous ones, except that from the CMB black-body spectrum which is almost unchanged.",
"Table: Absolute bounds (obtained for α 0 ,α≪1\\alpha _0,\\alpha \\ll 1, upper part) and bounds for α 0 =1/2=α\\alpha _0=1/2=\\alpha (lower part) on the hierarchy of multifractional spacetimes with qq-derivatives with measure ().",
"The key formulæ used to compute the constraints in the table are: for the muon lifetime, t * <(uδτ/τ 0 α 0 ) 1/(1-α 0 ) t_*< (u\\delta \\tau /\\tau _0^{\\alpha _0})^{1/(1-\\alpha _0)} replacing eq.",
"(139) of , with u=1u=1; for the Lamb shift, E * >{uδE/[(2-α 0 )ΔE]} 1/(α 0 -1) |E 2S |E_*>\\lbrace u\\delta E/[(2-\\alpha _0)\\Delta E]\\rbrace ^{1/(\\alpha _0-1)}|E_{2S}| replacing eq.",
"(142) of , with u=2-α 0 u=2-\\alpha _0; for gravitational waves, eq.",
"() with u μ ∝C μ /(3-α μ )u_\\mu \\propto C_\\mu /(3-\\alpha _\\mu ) and 2C μ =12C_\\mu =1; for GRBs, eq.",
"() [u=O(1)u=O(1)]; for Cherenkov radiation, eq.",
"() with 2C μ =12C_\\mu =1 [u=O(1)u=O(1)]; for the CMB black-body spectrum, a data fit with eq.",
"(3.5) of with the factor 1/α 0 1/\\alpha _0 in the denominator replaced by u=1u=1.",
"“Pseudo” indicates bounds obtainable in the stochastic view, where T q T_q is regarded as an approximation of T γ=α T_{\\gamma =\\alpha }.$\\blacktriangleright $53$\\blacktriangleleft $ Measurements of the anomalous magnetic moment of the electron tests QED to a much higher level of accuracy than the Lamb shift or the muon lifetime.",
"Why not to use this datum?tocsubsection Why not to use constraints on the anomalous magnetic moment of the electron?",
"Indeed, the $g-2$ factor can constrain $T_v$ efficiently [55].",
"From the triangular vertex in the integer picture, at one loop it is known that $g-2=\\tilde{\\alpha }_\\textsc {qed}/\\pi $ .",
"The fine-structure constant is measured with an accuracy of $\\delta \\alpha _\\textsc {qed}/\\alpha _\\textsc {qed}\\sim 10^{-10}$ .",
"Since, from eq.",
"(REF ), the measured fine-structure constant in the fractional picture is (in $c=1=\\hbar $ units) $\\alpha _\\textsc {qed}(t)\\simeq Q^2(t)=\\tilde{\\alpha }_\\textsc {qed}/v_0(t)$ , for the binomial measure (REF ) with (REF ) the difference between the integer and fractional constant is $\\Delta \\alpha _\\textsc {qed}=\\alpha _\\textsc {qed}(t)|t_*/t|^{1-\\alpha _0}$ .",
"Demanding $\\Delta \\alpha _\\textsc {qed}<\\delta \\alpha _\\textsc {qed}$ and setting $t=t_\\textsc {qed}=10^{-16}\\,{\\rm s}$ , one obtains $t_*< 10^{-16-10/(1-\\alpha _0)}\\,{\\rm s}$ .",
"The bounds from $\\alpha _\\textsc {qed}$ are reported in tables REF and REF , and they are several orders of magnitude stronger than the Lamb-shift bounds.",
"The theory $T_q$ is immune to similar constraints because it predicts the same $g-2$ factor and fine-structure constant as in the ordinary Standard Model.",
"It is easy to understand why.",
"The way the $q$ -theory conveys multiscale effects to physical observables is via a transition from adaptive measurement units (integer picture) to nonadaptive ones (fractional picture).",
"In the case of the Lamb shift, one borrows the standard QED result for the shift in the energy levels and applies it to the difference $\\Delta p(E)$ between geometric energies; then, from $\\Delta p(E)$ one extracts the actual Lamb shift $\\Delta E$ and proceeds with the comparison with experiments [54], [55].",
"One could do essentially the same thing by looking at the hydrogen spectrum on a photographic plate, measuring the separation between two spectral lines; in either case, we are measuring dimensionful quantities.",
"However, dimensionless quantities such as $\\alpha _\\textsc {qed}$ and $g-2$ are unaffected by having worked with composite momentum or position coordinates in the integer picture.",
"Therefore, these fundamentalBy fundamental, we mean that they are not obtained from the composition of other directly measurable quantities; see ref.",
"[56] and question for examples of nonfundamental observables that can discriminate the theory.",
"dimensionless observables remain the same in both frames of $T_q$ .",
"Curiously, this situation is complementary to the one for the muon lifetime, where $T_q$ is sensitive to changes in the geometry while $T_v$ is not [55].",
"$\\blacktriangleright $54$\\blacktriangleleft $ What are the motivations and the gains of the bounds found for the multifractional Standard Models?",
"None of the exotic realizations of the Standard Model contains any virtue with respect to the ordinary Standard Model.",
"The constraints on the energy and length obtained appear to be irrelevant in view of the same quantities in renormalized QFT.tocsubsection What are the motivations and the gains of the bounds found for the multifractional Standard Models?",
"This criticism echoes question and we can only answer in the same way: the motivations of multifractional theories lie in quantum gravity (section ), not in the desire of modifying the celebrated Standard Model.",
"If changing spacetime geometry carries consequences also for the fundamental particle interactions, then it becomes both interesting and necessary to verify whether these changes are compatible with extant experimental constraints.",
"$\\blacktriangleright $55$\\blacktriangleleft $ Setting experimental limits on an ad hoc proposal is not interesting.",
"Setting limits on effective higher-dimensional operators is more systematic and model-independent, but it has already been done in the past.tocsubsection Do these experimental limits come from an ad hoc proposal?",
"Sections and (questions 0–0, –, and ) bring a number of arguments on the fact that the multifractional proposal is neither ad hoc nor sterile in its theoretical and phenomenological structure.",
"On top of that, in we saw that expansions in higher-dimensional operators can mimic, and not even fully reproduce, only some aspects of the multifractional QFT phenomenology.",
"$\\blacktriangleright $56$\\blacktriangleleft $ Even granting that these theories are not ill defined and have some physical motivation, it is not possible to reach any conclusion about their phenomenology because they have not been developed rigorously.",
"In particular, there is no top-down construction of a quantum field theory, let alone that of a Standard Model of electroweak and strong interactions.tocsubsection Have these theories been developed rigorously to the point of being able to reach any robust conclusion about phenomenology?",
"We hope that this review, and in particular section (questions –), has convinced the reader that a top-down construction exists for the multifractional Standard Model in $T_v$ and $T_q$ [55], hence the phenomenology of these two theories comes directly from their foundations.",
"We have not yet constructed the Standard Model for $T_\\gamma $ , but the resemblance of $T_q$ with $T_{\\gamma =\\alpha }$ (questions , , and ) justifies the hope that the phenomenology of $T_\\gamma $ be very similar to that found for $T_q$ in refs.",
"[54], [57].",
"$\\blacktriangleright $57$\\blacktriangleleft $ Is it true that, in these theories, and despite the fact that the spacetime structure itself has been changed, it is assumed that only gravity is altered while the electromagnetic field is the usual one?",
"What is the justification behind that?tocsubsection Is it true that it is assumed that only gravity is altered while the electromagnetic field is the usual one?",
"No, it is not true.",
"The multiscale geometry of these spacetimes affect all fundamental interactions [50], [55], including electromagnetism.",
"The speed of light is modified accordingly in the theory $T_q$ [56], while in $T_v$ it is the usual $c=1$ .",
"See and .",
"Amusingly, this question is somewhat “complementary” to .",
"$\\blacktriangleright $58$\\blacktriangleleft $ Since, as claimed in and , multifractional theories lead to violations of Lorentz symmetries, then what are the constraints?tocsubsection What are the constraints on violations of Lorentz symmetries?",
"In general, all constraints coming from the Standard Model explicitly limit deviations from Lorentz invariance, but the strongest bounds to date are based on classical deformed dispersion relations (at the quantum level, we avoid problems; see ).",
"The theory $T_\\gamma $ has not been tested with these observations, due to its underdevelopment.",
"We will not fill this gap here.",
"The theory $T_v$ and the model $T_1$ have standard dispersion relations [see eqs.",
"(REF ) and (REF )] and do not predict any change in the propagation speed of particles.",
"The remaining case is the theory $T_q$ , for which constraints have been obtained from gravitational waves and GRBs [57].",
"Let us begin with a general analysis of dispersion relations in $T_q$ .",
"From eq.",
"(REF ), one finds the massive dispersion relation $[p^0(E)]^2=|{\\bf p}|^2+m^2=\\sum _i [p^i(k^i)]^2+m^2$ , where $E=k^0$ .",
"This expression, valid for a scalar field, may be regarded as the general representative of dispersion relations for this theory.",
"From now on, we drop the mass term, which plays no role in the main argument.",
"Also, we approximate the dispersion relation for small multifractional corrections and combining the spatial momentum components into the absolute value $k=|{\\bf k}|$ .",
"The latter approximation can be done in different ways that all give very similar results, modulo a prefactor $C_\\mu $ in front of the correction which is always $O(1)$ .",
"Taking the binomial measure (REF ) with isotropic spatial hierarchy (which is all we need, according to 0), setting $k_i\\simeq k/\\sqrt{3}$ and defining $E_*=k_*$ [energy scale $E_*$ identified with the inverse of the time and length scales $t_*$ and $\\ell _*=1/k_*$ in Planck units, eqs.",
"(REF ) and (REF )], we get the full dispersion relation $E^2\\simeq k^2+2E_*^2\\left[\\frac{1}{\\alpha _0}\\left(\\frac{k}{E_*}\\right)^{3-\\alpha _0} F_\\omega (k)-\\frac{3}{\\alpha }\\left(\\frac{k}{\\sqrt{3} E_*}\\right)^{3-\\alpha } F_\\omega \\left(\\frac{k}{\\sqrt{3}}\\right)\\right].$ This expression is simplified to $E^2\\simeq k^2+\\frac{2E_*^2C_\\mu }{3-\\alpha _\\mu }\\left(\\frac{k}{E_*}\\right)^{3-\\alpha _\\mu }\\,,$ when log oscillations are averaged or absent ($F_\\omega =1$ ).",
"For timelike fractal geometries ($\\mu =0$ , trivial measure in spatial directions) with ${\\bf p}={\\bf k}$ , one has $C_0=(3-\\alpha _0)/\\alpha _0$ ; the correction is positive.",
"For spacelike fractal geometries ($\\mu =i$ , trivial measure in the time-energy direction) with $p^0=E$ , one has $C_i=-(3-\\alpha )/[3^{(1-\\alpha )/2}\\alpha ]$ and the correction is negative.",
"Generic configurations with multifractional time and space directions can produce corrections of either sign, periodically suppressed by the log oscillations.",
"The timelike and spacelike cases without oscillations are extreme representatives of this spectrum of possibilities, both corresponding to corrections with a unique sign and maximal amplitude.",
"Given a dispersion relation $E^2=E^2(k)$ , the velocity of propagation of a wave front for a particle p is given by the group velocity $V_{\\rm p}:=\\frac{dE}{dk}\\,.$ In this and the next question, we reserve the symbol $V$ for velocities and the reader should not confuse it with potentials.",
"For the usual Lorentz-invariant dispersion relation $E^2=k^2+m^2$ , in the small-mass limit one gets the difference $\\Delta V:= V_{\\rm p}-1\\simeq -m^2/(2E^2)$ between the propagation speed of the particle and the speed of light.",
"Plugging the timelike or spacelike approximations of eq.",
"(REF ) into (REF ) and replacing $k\\rightarrow E$ in the right-hand side consistently with the small-correction approximation, we get $\\Delta V \\simeq C_\\mu (E/E_*)^{1-\\alpha _\\mu }$ .",
"This correction is less suppressed than those in usual modified dispersion relations in quantum gravity [227], [228], where $0<1-\\alpha _\\mu <1$ is replaced by some exponent $n\\geqslant 1$ .",
"This determines a stronger and more sensitive bound on the characteristic energy $E_*$ , via $E_* =\\left|\\frac{\\Delta V}{C_\\mu }\\right|^{-\\frac{1}{1-\\alpha _\\mu }}E\\,.$ We pause for a moment and highlight a caveat.",
"The propagation speed (REF ) does not depend on the species of the particle.",
"This is clear from eq.",
"(REF ), which is derived from the pole structure of a generic propagator in the massless limit, regardless of its tensorial structure.",
"The effect of multiscale geometry on the propagation of particles is, thus, universal and the difference $\\Delta V_{12}=V_1-V_2$ between the velocity of two species 1 and 2 is theoretically zero.",
"In particular, the dispersion relation of photons acquire the same corrections (REF ) as any other particle and the speed of light is not $c=1$ [56].",
"Therefore, in the deterministic view gravitons propagate at the speed of light and $\\Delta V_{12}=0$ .",
"However, in the stochastic view the correction in the right-hand side of (REF ) represents a fluctuation of the geometry.",
"Maximizing this fluctuation one finds eq.",
"(REF ) and, taking opposite signs for particles 1 and 2, one obtains eq.",
"(REF ) with $\\Delta V\\rightarrow \\Delta V_{12}/2$ : $E_* =\\left|\\frac{\\Delta V_{12}}{2C_\\mu }\\right|^{-\\frac{1}{1-\\alpha _\\mu }}E\\,.$ Little or nothing changes for phenomenology because the extra factor $1/2$ can modify the order of magnitude of $E_*$ at most by one.Compare tables REF and REF with the numbers found at the end of ref.",
"[57] in the main body, where the factor $1/2$ is absent.",
"The values in the table in ref.",
"[57] (also reported in ref.",
"[58]) use a different frequency peak.",
"If ignored, this delicate point may trigger question , since in eq.",
"(REF ) $\\Delta V$ is the difference between the particle propagation velocity and a constant speed of light $c=1$ .",
"From eq.",
"(REF ) and similar others, one usually extracts two types of bounds, an “absolute” one giving the most conservative estimate of multifractional effects (typically obtained for $\\alpha _0,\\alpha \\ll 1/2$ or zero) and one for a specific choice of $\\alpha _0$ or $\\alpha $ , as in tables REF –REF .",
"Here we consider the bounds on the propagation speed of gravitational waves from the LIGO observation of the black-hole merger GW150914 [4].",
"Following [229], we take the gravitational-wave signal to peak at frequencies $f=\\omega /(2\\pi )\\sim 100\\,{\\rm Hz}$ , corresponding to $\\omega \\approx 630\\,{\\rm Hz}$ , an energy $E=\\hbar \\omega \\approx 4.1\\times 10^{-13}\\,{\\rm eV}$ , and a velocity difference $|\\Delta V_{12}|<4.2\\times 10^{-20}\\,.$ The bounds for $C_\\mu $ fixed as in the text below eq.",
"(REF ) are shown in the line “Gravitational waves (pseudo)” of tables REF and REF (there is no detectable difference between the timelike and the spacelike cases), while for $2C_\\mu =1$ they are in table REF .",
"Bounds on $E_*$ are converted to bounds on $t_*$ and $\\ell _*$ via eqs.",
"(REF ) and (REF ).",
"The bounds from photon time delays in GRBs are more severe but obtained in a more heuristic way [57].",
"The difference in the velocities of two photons with different energies emitted in a GRB at the same time is $|\\Delta V_{12}|\\propto (E_2^{1-\\alpha _\\mu }-E_1^{1-\\alpha _\\mu })/E_*^{1-\\alpha _\\mu }$ .",
"Taking $E_2\\gg E_1$ (highly-energetic photons), one gets eq.",
"(REF ) with $\\Delta V\\rightarrow \\Delta V_{12}$ (and no $1/2$ factor).",
"Letting $d$ be the luminosity distance between the source and us and $\\Delta t=t_1-t_2$ the time delay in the arrival of the photons, we also have $1\\gg \\Delta V_{12}\\sim d/t_1-d/t_2\\simeq d\\Delta t/t_1^2\\simeq V_2^2\\Delta t/d\\sim \\Delta t/d$ .",
"The observed sources of bright GRBs are in the range of redshift $z=0.16-3.37$ (i.e., [230]), corresponding to $d\\sim 10^{25}-10^{27}\\,{\\rm m}$ .",
"For typical photon emissions, $\\Delta t\\sim 10^{-2}-10^{-1}\\,{\\rm s}$ , so that $\\Delta V_{12}\\sim 10^{-20}-10^{-18}$ .",
"Taking $E_2\\sim 10^{-4}\\,{\\rm GeV}$ and the most conservative value $\\Delta V_{12}\\sim 10^{-18}$ , we get $E_*>10^{-4+\\frac{18}{1-\\alpha _\\mu }}\\,{\\rm GeV}\\,.$ This bound [57], shown in table REF , is much tighter with respect to the other constraints, even discounting a few orders of magnitude with respect to a rigorous estimate.",
"$\\blacktriangleright $59$\\blacktriangleleft $ But there are much stricter constraints in particle physics, for instance those of refs.",
"[231], [232].",
"One derives limits on coefficients of effective operators, which are typically more stringent than those quoted above.",
"Even for Lorentz-invariant operators, current limits are mostly in the TeV range or higher.tocsubsection Are there stricter constraints in particle physics?",
"Good point.",
"The constraints reviewed in ref.",
"[231] are on the difference $\\Delta V$ of the maximal attainable velocity of (i) photons and electrons (from photon decay) [233], (ii) muons and electrons (from muon decay) [234], (iii) muon and electron neutrinos (from neutrino oscillations) [235], (iv) neutral kaons K-long and K-short [236], (v) photons and atoms [237], and (vi) photons and cosmic-ray protons (via vacuum Cherenkov radiation) [233].",
"Lorentz-violating effects combined with CPT violation were discussed in ref.",
"[232].",
"Just like the propagation speed, the maximal attainable velocity of a particle is independent of its species in multifractional theories at the classical level, but the constraints (i)–(v) are calculated in quantum field theory and they are nontrivial also in $T_v$ (only when charged particles are involved, since the theory is nontrivial only in the QED sector [55]) and in the deterministic view of $T_q$ and $T_\\gamma $ .",
"Even at the classical level, the microscopic stochastic fluctuations of geometry in $T_q$ and $T_\\gamma $ can induce a relative excursion between velocities which cannot exceed the experimental bounds.",
"This mechanism is very different from the Lorentz violation from CPT-even renormalizable rotationally invariant interactions in ordinary spacetime [231].",
"The bound from Cherenkov radiation (vi) is stronger than the others but it requires energies much greater than those accessible in colliders.",
"To see whether we can use it to constrain multifractional theories, let us first review its origin in ordinary spacetimes with modified dynamics.",
"Primary cosmic rays (i.e., originated outside the Solar System) are made of protons and atomic nuclei; ultra-high-energy cosmic rays (UHECR) carry energies greater that $10^{18}\\,\\text{eV}$ .",
"Cosmic rays with energies above $10^{19}\\,\\text{eV}$ have been observed systematically [238], [239], but isolated events associated with primary protons of energy $E_{\\rm UHECR}\\approx 1-3\\times 10^{20}\\,\\text{eV}$ have also been detected [240], [241], [242].",
"Assume to be in a spacetime where the speed of light $c_x$ is smaller than the usual $c$ (the reason of the symbol $c_x$ will become clear soon).",
"A proton travelling faster than light would rapidly release energy via photon emission, $p\\rightarrow p+\\gamma $ , until its speed drops below luminal.",
"While travelling a distance $V_p t$ with velocity $V_p$ , the particle produces a shock wave of photons travelling at speed $c_x$ .",
"At time $t$ , the electromagnetic wave produced at $t=0$ has traveled a distance $c_x t$ and the angle of the shock wave with respect to the proton trajectory has $|\\cos \\theta |=(c_x t)/(V_p t)=c_x/V_p\\leqslant 1$ .",
"The threshold for the production of Cherenkov radiation is thus reached when the particle travels at the same speed of the wave front, $V_p=c_x$ .",
"Therefore, restoring $c=1$ units temporarily, from the special-relativistic energy of the proton $E=m_p c^2/\\sqrt{1-(V_p/c)^2}$ , where $m_p c^2\\approx 938.28\\,\\text{MeV}$ is the proton rest mass, one gets the threshold energy $E_{\\rm min}=m_p c^2/\\sqrt{1-(c_x/c)^2}$ .",
"Since superluminal UHECRs must have become subluminal well before reaching us, their energy must be smaller than the threshold energy, $E_{\\rm UHECR}<E_{\\rm min}$ .",
"Taking $E_{\\rm UHECR}\\approx 10^{11}\\,\\text{GeV}$ , one gets the bound (back to $c=1$ units) [233] $1-c_x^2<\\left(\\frac{m_p}{E_{\\rm UHECR}}\\right)^2\\approx 10^{-22}\\,.$ Vacuum Cherenkov radiation can be realized in Lorentz-violating extensions of the Standard Model [243], [244], [245] and we now ask whether it happens also in multifractional theories.",
"For $T_v$ , the answer is negative.",
"As in the case of gravitational waves examined in question , the theory $T_v$ is left unscathed because the speed of light is $c_x/c=1$ there, and the bound (REF ) has nothing to say.",
"The case of $T_q$ is more interesting.",
"As pointed out in ref.",
"[56], in the fractional frame particles can travel at speed slightly higher than light, and vacuum Cherenkov radiation can occur.",
"We can make a crude estimate of the effect from eq.",
"(REF ).",
"To measure the maximal departure $\\Delta c=c_q-c_x$ of the speed of light $c_x$ in the fractional frame from the standard speed of light $c_q=c=1$ (the geometric velocity of photons in the integer frame), we combine eqs.",
"(REF ) [not (REF ); see the discussion above] and (REF ), noting that $1-c_x^2=(1+c_x)(1-c_x)\\simeq 2\\Delta c$ when $\\Delta c$ is small: $E_* >\\left|\\frac{1}{2C_\\mu }\\left(\\frac{m_p}{E_{\\rm UHECR}}\\right)^2\\right|^{-\\frac{1}{1-\\alpha _\\mu }}E_{\\rm UHECR}\\,.$ If we use the binomial measure (REF ) with $\\alpha $ -dependent coefficients, then $\\Delta c\\propto -C_\\mu >0$ in a spacelike fractal geometry and $C_\\mu =C_i=-(3-\\alpha )/[3^{(1-\\alpha )/2}\\alpha ]$ .",
"For this choice, one finds the absolute and $\\alpha =1/2$ bounds of tables REF and REF , respectively.",
"For a generic $2C_\\mu =1$ , one gets $E_* >10^{11+\\frac{22}{1-\\alpha _\\mu }}\\,{\\rm GeV}\\,,$ and the weaker bounds reported in table REF .",
"Comparing eq.",
"(REF ) with (REF ), we see two factors that improve the GRB bound.",
"One is in the velocity difference (REF ), which is 4 orders of magnitude smaller than in the GRB case.",
"The other, and most important, is the reference energy $E_{\\rm UHECR}$ , 15 orders of magnitude larger than that of typical GRB photons.",
"It is no wonder that the values reported in the “Cherenkov radiation” line of tables REF –REF are much tighter than those from GRBs.",
"$\\blacktriangleright $60$\\blacktriangleleft $ Is the dispersion relation (REF ), which is claimed to affect the propagation of gravitons, photons or other particles, physical?",
"It was derived from the propagator (REF ), which has the conventional form in terms of the $p$ 's.",
"However, any dispersion relation in which one mixes momentum components in two or more coordinates, or where one calls “$p(k)$ ” momentum $p$ , will take an unconventional form without having unconventional physics.tocsubsection Is the theory with $q$ -derivatives trivial?",
"(v) This is questions , , and – disguised in another form.",
"Once we choose the time and length units of our devices as the scaling units of the fractional coordinates $x^\\mu $ in position space, we also automatically fix the momentum and energy units as the scaling units of the fractional coordinates $k^\\mu $ in momentum space: $[k^\\mu ]=1\\,.$ In the case of the theory with $q$ -derivatives, the measure () in momentum space is fixed uniquely by eq.",
"(REF ), so that the momentum-space analogue of eq.",
"(REF ), $k^\\mu \\rightarrow p^\\mu (k^\\mu )\\,,$ is not a change of coordinates but a mapping from the fractional frame where observables are computed and the integer frame where the theory looks simpler.",
"In particular, the propagator (REF ) is a highly nontrivial and rigid function of $k^\\mu $ , even if it has the usual form in terms of $p=p(k)$ .",
"All these properties are determined by the symmetries of the theory.",
"We can obtain any dispersion relation without unconventional physics only in a theory admitting eq.",
"(REF ) as a coordinate transformation leaving physical observables invariant.",
"This is not the case of $T_q$ , as we discussed at length in section .",
"$\\blacktriangleright $61$\\blacktriangleleft $ Even if the replacements $x\\rightarrow q(x)$ and $k\\rightarrow p(k)$ were somehow physical, they are not done at the required level of rigor.",
"In particular, one would need to follow a first-principle approach where one starts with a field action and performs the well-known procedure to get the Hamiltonian density.",
"Until such a rigorous analysis is done, it is not justified to assume that the symbols that are used such as $p$ , $k$ , $E$ , and so on, have the meaning of momentum and energy.tocsubsection Is the phenomenology of the theory with $q$ -derivatives robust?",
"(i) A first-principle approach is followed.",
"Quantum-gravity motivations aside, we have a spacetime measure dictated by the second flow-equation theorem (question 0) and a momentum space measure determined by that automatically (question 0).",
"We have a field action, both for the Standard Model [54], [55] and for gravity [53] (questions and ; the general structure of field actions in $T_q$ are discussed in and ).",
"The Hamiltonian analysis could not be easier than in $T_q$ : it follows all the steps of the standard case with the replacements $x\\rightarrow q(x)$ and $k\\rightarrow p(k)$ , and it is not necessary to repeat it here in detail.In [59], the algebra of first-class constraints of gravity plus matter for the theory with $q$ -derivatives has been written down (question ).",
"Instances of Hamiltonian analyses of $T_1$ , $T_v$ , and other multiscale theories can be found in refs.",
"[104], [46], [48], [52], [59].",
"Equation (REF ) is an example in $T_v$ .",
"The example of a classical real scalar field in flat space will suffice.",
"From the action (REF ), one obtains the momentum $\\Pi _\\phi =\\partial \\mathcal {L}/\\partial q^0(t)=\\partial _{q(t)}\\phi $ , the super-Hamiltonian density $\\mathcal {H}$ , and the supermomentum density $\\mathcal {H}_i$ : $\\mathcal {H}=\\Pi _\\phi \\partial _{q(t)}\\phi -\\mathcal {L}=\\frac{1}{2} \\Pi _\\phi ^2+\\frac{1}{2}\\sum _{i=1}^{D-1}[\\partial _{q^i(x^i)}\\phi ]^2+V(\\phi )\\,,\\qquad \\mathcal {H}_i=\\Pi _\\phi \\partial _{q^i(x^i)}\\phi \\,.$ The Hamiltonian is $H=\\int d^{D-1}q({\\bf x})\\,\\mathcal {H}$ , where one integrates only on spatial coordinates.",
"For $V(\\phi )=m^2\\phi ^2/2$ and using the Fourier transform $\\phi (x)=\\int \\frac{d^Dp(k)}{(2\\pi )^D}\\,e^{i\\eta _{\\mu \\nu }p^\\mu (k^\\mu ) q^\\nu (x^\\nu )}\\phi _k\\,,$ it is not difficult to quantize canonically and to identify $H$ as the charge conserved under fractional time translations.",
"At the classical level, $p^0(E)$ is the geometric energy in the integer picture and, hence, $k^0=E$ is the energy in the fractional picture.",
"All of this stems from the fact that $p^\\mu (k^\\mu )$ is Fourier conjugate to $q^\\mu (x^\\mu )$ .",
"$\\blacktriangleright $62$\\blacktriangleleft $ Granting that a given action describes this framework, it is a fact that there would not be two types of momenta $p$ and $k$ for the same field (for instance, gravity), as they appear in the modified dispersion relation (REF ).",
"Therefore, at the level in which the theory currently stands, it is impossible to claim that one can make contact with experiments and observations.tocsubsection Is the phenomenology of the theory with $q$ -derivatives robust?",
"(ii) As said in , there is only one momentum for a field, which is $k^\\mu $ .",
"The geometric momentum $p^\\mu =p^\\mu (k^\\mu )$ is only a convenient tool to cast the theory $T_q$ in the integer picture.",
"$\\blacktriangleright $63$\\blacktriangleleft $ Experimental constraints of multifractional models are typically based on equations which show an extreme sensitivity to the value chosen for the parameters $\\alpha _0$ or $\\alpha $ .",
"Does this indicate that the domain of validity of these formulæ is limited and that a more refined analysis is required?tocsubsection Does the extreme sensitivity to the value of $\\alpha _\\mu $ of the formulæ used for experimental constraints indicate that their domain of validity is limited and that a more refined analysis is required?",
"Tables REF –REF show that the bounds on the scales of the binomial measure can change by a few orders of magnitude when varying the fractional exponents $\\alpha _\\mu $ in the range $[0,1)$ ; the results for values close to zero and for $\\alpha _\\mu =1/2$ are compared.",
"This sensitivity on a fundamental parameter of the theory with a clear-cut geometric interpretation should not be regarded as a drawback.",
"In fact, this feature is an invaluable bonus: it guarantees that these theories can be easily falsified.",
"Already the estimates from GRBs are an example of this: they exclude the values $\\geqslant 1/2$ for $T_q$ in the absence of log oscillations and they limit the parameter space of this theory in an unprecedented way, the characteristic energy of the momentum measure being pushed very close to grand-unification and Planck scales.",
"A key difference with respect to other quantum-gravity-inspired dispersion-relation bounds [227], [228] is that our constraints are obtained directly from a full theory, without invoking any generic assumption encoding uncontrolled effects in heuristic umbrella constants.",
"We do have free parameters but they are fundamental, intrinsic to the theory.",
"In this respect, our approach is less qualitative, more rigid and, therefore, more sensitive to the strength of the observational constraints [57].",
"$\\blacktriangleright $64$\\blacktriangleleft $ Are there constraints from tests of the equivalence principle?tocsubsection Are there constraints from tests of the equivalence principle?",
"Not yet, but it is an interesting question.",
"$\\blacktriangleright $65$\\blacktriangleleft $ Are there constraints on the dimension of spacetime?tocsubsection Are there constraints on the dimension of spacetime?",
"Yes, there are for the theory with $q$ -derivatives.",
"A likelihood analysis of the primordial CMB scalar spectrum excludes portions in the parameter space of $T_q$ , due to the fact that CMB data disfavor the logarithmic oscillations of the spectrum (REF ).",
"The marginalized likelihood for the spatial fractional exponent $\\alpha $ , when $N$ in eq.",
"(REF ) is fixed, indicates that $\\alpha \\lesssim 10^{-1},10^{-0.2},10^{-0.25}$ at the 95% confidence level for, respectively, $N=2,3,4$ .",
"From eqs.",
"(REF ) and (REF ), $\\begin{matrix}N=2:\\qquad & d_\\textsc {s}^{\\rm \\,space}=d_\\textsc {h}^{\\rm \\,space}\\lesssim 0.3\\qquad \\text{(UV)}\\,,\\\\N=3:\\qquad & d_\\textsc {s}^{\\rm \\,space}=d_\\textsc {h}^{\\rm \\,space}\\lesssim 1.9\\qquad \\text{(UV)}\\,,\\\\N=4:\\qquad & d_\\textsc {s}^{\\rm \\,space}=d_\\textsc {h}^{\\rm \\,space}\\lesssim 1.7\\qquad \\text{(UV)}\\,.\\end{matrix}$ Higher $N$ should give similar constraints.",
"This result is somewhat surprising, as it forces an upper bound on the dimension of space in the UV.",
"Therefore, the primordial universe is very well described by the standard inflationary model in a smooth spacetime with four topological dimensions but, as soon as one assumes that spacetime geometry undergoes dimensional flow, this flow must be nontrivial to fit data.",
"During this flow, the effective dimension of space is reduced at least by 1 ($N=3$ case) in the UV.",
"There are no analogous results for the other multifractional theories, although it is possible that eq.",
"(REF ) could apply also to the case with fractional derivatives thanks to the $T_{\\gamma =\\alpha }\\cong T_q$ approximation.",
"$\\blacktriangleright $66$\\blacktriangleleft $ If the length scales of these theories are so small, how is it possible to test them at cosmological scales?",
"Modifications to gravity are strongly suppressed during inflation.",
"The reason is that the ratio between the inflationary energy density and Planck density (at which classical gravity is believed to break down) is very small, $\\rho _{\\rm infl}/\\rho _{\\rm Pl}\\sim (\\ell _{\\rm Pl}H)^2 \\sim 10^{-8}$ , where we estimated the typical energy scale during inflation to be about the grand-unification scale, $H\\sim 10^{15}\\,\\mbox{GeV}$ .",
"Thus, quantum corrections or corrections from exotic geometries are expected to be well below any reasonable experimental sensitivity threshold.tocsubsection If the length scales of these theories are so small, how is it possible to test them at cosmological scales?",
"This type of argument holds only when corrections to general relativity are limited to higher-order curvature corrections.",
"As is known in quantum gravity (and, in particular, in string cosmology and in loop quantum cosmology), the effective dynamics of gravity in the early universe can be modified by far more sophisticated mechanisms than curvature corrections to the Einstein–Hilbert action.",
"The case of multifractional spacetimes illustrates the point in a rather unique fashion.",
"By definition of these theories, geometry is characterized by a hierarchy of fundamental scales.",
"The main features of this configuration are exemplified to the bone by the binomial measure (REF ) with (REF ).",
"Here, we have two characteristic length scales $\\ell _\\infty \\leqslant \\ell _*$ .",
"At scales above $\\ell _*$ , spacetime looks smooth and the usual description of general relativity holds.",
"However, when inspected at scales $\\lesssim \\ell _*$ , in the deterministic view geometry changes properties smoothly and, in particular, the spacetime dimension decreases to some asymptotic value smaller than 4.",
"If one further zooms in, at scales $\\sim \\ell _\\infty $ a discrete symmetry emerges and the notion of smooth spacetime with well-defined dimensionality is lost.",
"The length $\\ell _\\infty $ can be identified with the Planck length (see question 0), while $\\ell _*$ is constrained to be at least as small as the grand unification scale (table REF ).",
"Therefore, it might seem difficult that multifractional geometries could leave an observable imprint anywhere.",
"However, primordial inflation expands Planckian scales to cosmological size.",
"If geometry is modified at Planck scales, then we can expect that multiscale effects are magnified by the early-universe expansion up to the size of the visible sky.",
"Such is indeed the case and CMB observations are capable of placing strong constraints on multifractional geometries [58].",
"This cosmological mechanism is in action in most models of quantum gravity, but in the case of multifractional spacetimes there is also a subtler effect.",
"Log oscillations are a manifestation of discrete UV symmetries and of the long-range correlations typical of complex systems, anomalous stochastic processes (see, e.g., ref.",
"[123] for a pedagogical review), and multifractals (via the so-called harmonic structure, reviewed in refs.",
"[40], [41]).",
"This long-range effect is clearly visible both in theoretical cosmology (where the oscillatory modulation of the scale factor dies out at scales much larger than $\\ell _\\infty $ and larger even than $\\ell _*$ [53]) and in observations, as we just remarked (see also ).",
"It is a most unusual phenomenon from the point of view of standard QFT, because it entails a symmetry (discrete scale invariance) that, despite being explicitly broken already near the UV, propagates to the IR and governs the physics at large scales.",
"$\\blacktriangleright $67$\\blacktriangleleft $ How would the discrete spacetime at scales $\\sim \\ell _\\infty $ look like to an observer?tocsubsection How would the discrete spacetime at scales $\\sim \\ell _\\infty $ look like to an observer?",
"If spacetime is discrete at scales $\\sim \\ell _\\infty $ , then we could picture it as a totally disconnected set of points.",
"How would an observer therein perceive this geometry?",
"Certainly not as “holes” in the fabric of spacetime, since signals propagate only within the set; the holes picture would best suit an ideal observer living outside our universe, in the $D$ -dimensional embedding space where the theory is defined.",
"At the cosmological level, the visible effect of this spacetime geometry is a long-range logarithmic modulation of the power spectrum of primordial fluctuations [58] as we discussed in the previous question.",
"At the microscopic level, the stochastic view advanced here and in refs.",
"[61], [62] predicts a fuzziness where measurements cannot be performed with arbitrary precision, and that get worse when trying to probe scales deeper in the UV.",
"$\\blacktriangleright $68$\\blacktriangleleft $ Are multifractional theories ruled out?tocsubsection Are multifractional theories ruled out?",
"A multifractional theory is ruled out observationally if the length scale $\\ell _*$ in the binomial measure is much smaller than the Planck scale, $\\ell _*\\ll \\ell _{\\rm Pl}$ .",
"In momentum space, this corresponds to $E_*\\gg E_{\\rm Pl}$ .",
"The phenomenology of $T_1$ has not been studied and we cannot say much about it.",
"The spectral dimension of $T_1$ is the same as $T_v$ because the diffusion equation in these theories is one the adjoint of the other [49].",
"Therefore, the observable consequences of their dimensional flow should be about the same.",
"This is a non-issue, since $T_1$ was useful as a first exploration of the multifractional paradigm but nowadays it has been replaced by the more rigorous $T_v$ .",
"The most conservative bounds on $T_v$ (table REF ) are very weak because the theory bypasses all the strongest tests.",
"The $\\alpha _\\mu =1/2$ case is better constrained and measurements of the fine-structure constant require $E_*>10^{-8}\\,E_{\\rm Pl}$ .",
"Until now, the strongest bound on $T_q$ came from a crude estimate of the arrival time of photons with different energies emitted by GRBs [57].",
"For $\\alpha _\\mu \\ll 1/2$ , this bound is $E_*>10^{-5}\\,E_{\\rm Pl}$ , while for $\\alpha _\\mu =1/2$ the theory is ruled out, since $E_*>10^{13}\\,E_{\\rm Pl}$ (table REF ).",
"Inclusion of log oscillation could lead to an accidental erasure of corrections to dispersion relations, but not without fine tuning [57].",
"The only chance to avoid the GRB bound would be to disprove the estimate reported in by a precise calculation.",
"However, the constraints from emission of Cherenkov radiation by cosmic rays, which are several orders of magnitude stronger, rely only on the multifractional modification of special relativity, and they look much harder to evade.",
"All these constraints are valid in the deterministic view and could be avoided by invoking the stochastic view and considering the possibility that stochastic fluctuations cancel out when integrated along the photon or cosmic-ray paths [61], [62].",
"In fact, eq.",
"(REF ) assumed that the two particles for which one is measuring the velocity difference experience maximal and opposite fluctuations.",
"On the other hand, in average the effect could be just zero and all constraints from gravitational waves, GRBs, and UHECRs would evaporate.",
"Like in the case of $T_1$ , we do not have direct calculations of physical observables in $T_\\gamma $ and we conjectured that the $T_{\\gamma =\\alpha }\\cong T_q$ approximation allows one to apply the constraints found for $T_q$ to $T_\\gamma $ .",
"In that case, what said for $T_q$ would hold also here: the stochastic view bypasses all the strongest tests (gravitational waves, GRBs, and UHECRs) only if stochastic fluctuations are averaged out, while the deterministic view is constrained much more severely.",
"There are three possible ways in which the theory $T_\\gamma $ can be rescued: (i) giving up the deterministic view and adopting only the stochastic view, which is more justified here than in $T_q$ (where it is a juxtaposed approximation when meant in the sense of [56]); (ii) finding that, despite their similarities, $T_q$ and $T_\\gamma $ are essentially different in some key physical consequences and that some bounds do not apply after a closer scrutiny; (iii) finding that the fractional derivatives in $T_\\gamma $ must or can be taken with an order $\\gamma $ smaller than the fractional exponent $\\alpha $ in the measure.",
"Case (ii) is particularly interesting.",
"As discussed in 0, the value $\\alpha _\\mu =1/2$ is special according to some rigorous arguments advanced for $T_\\gamma $ , which is closely similar to $T_q$ when $\\gamma =\\alpha $ .",
"However, these two theories are mathematically different and arguments rigorously valid for $T_\\gamma $ can be taken only as suggestions in $T_q$ , and we do not expect that any theoretical argument in the future will fix $\\alpha _\\mu $ uniquely for $T_q$ (or $T_v$ ).",
"Vice versa, the $\\alpha $ -dependent observational constraints obtained for $T_q$ are robust for that theory, but only indicative for the yet-unexplored case of $T_\\gamma $ .",
"In particular, we cannot conclude that GRBs rule out $T_\\gamma $ just because they rule out $T_q$ for the range $\\alpha _\\mu \\geqslant 1/2$ for which $T_\\gamma $ is normed.",
"However, the UHECR bound of $T_q$ is strong for all $0\\leqslant \\alpha _\\mu <1$ and it could be avoided in $T_\\gamma $ only with a radical departure from $T_q$ in special relativity.",
"Again, the explicit construction of $T_\\gamma $ and calculations of its predictions will settle the question."
],
[
"Perspective",
"$\\blacktriangleright $69$\\blacktriangleleft $ In a nutshell, what are the main virtues of multifractional theories?tocsubsection In a nutshell, what are the main virtues of multifractional theories?",
"– They are a novel paradigm because, contrary to many other effective models, what one modifies here is not the dynamics but the integrodifferential structure describing how we measure the geometry.",
"Dynamics is modified as a byproduct of having a spacetime that can be conveniently treated with multidisciplinary tools of fractal geometry, anomalous transport theory, and complex systems.",
"This framework is different from much of the mainstream in theoretical physics, quantum gravity, and cosmology, and some researchers find this intellectually stimulating.",
"– They are simple without being simplistic.",
"It is the first attempt to control the most generic profile of dimensional flow in a purely analytic way.",
"All the usual techniques employed in quantum field theory and classical gravity can be adapted, with caution.",
"This allowed us to extract the first serious experimental constraints [54] not long since the original proposal [39].",
"– Their phenomenology is rich and spreads across all scales, from elementary particle interactions to cosmology.",
"It is also rigid enough to allow to exclude large portions of the parameter space.",
"– They have much potential yet untapped, especially regarding observational constraints and major open issues in cosmology and quantum gravity (see question ).",
"$\\blacktriangleright $70$\\blacktriangleleft $ And their problems?tocsubsection And their problems?",
"– The novelty of the paradigm carries some difficulties such as the breaking of symmetries (but the emergence of others ...) in the UV and the consequent need to choose a frame in position space.",
"This is unattractive for someone accustomed to work in Lorentz-invariant theories, not only because Lorentz invariance is a powerful theoretical asset making life simpler, but also because preferred frames are usually more difficult to justify scientifically and epistemologically, and can be much trickier when it comes to extract physical observables.",
"These are not the first models of gravity and matter breaking Lorentz invariance, and surely they will not be the last; however, their foundations are so different with respect to other, more conventional proposals that it is natural to find resistance.",
"Many of the questions collected here were actually raised during interactions between the author and colleagues.",
"Some of these questions had already been answered in the literature at the moment of their formulation, while others triggered more thinking.",
"One of the goals of this work was to gather all these issues in one basket and address them in a unified systematic way.",
"– The most interesting among the proposals, the theory with multifractional derivatives, has not been developed much.",
"A top priority will be to make it progress.",
"$\\blacktriangleright $71$\\blacktriangleleft $ What is the agenda for the future?tocsubsection What is the agenda for the future?",
"The recent proposal of the stochastic view [61], [62] confirms that dimensional flow is a solid manifestation of quantum gravity, while the original motivation of the multifractional paradigm was to quantize gravity successfully precisely because of dimensional flow [76].",
"This is the usual dualism of multifractional spacetimes viewed as effective models or as fundamental theories.",
"Both possibilities are viable and can be pursued in parallel and in several distinct ways.",
"In order of importance: To complete the formulation of the theory with multifractional derivatives, starting from a coherent and useful definition of multiscale fractional calculus (question ), the construction of perturbative QFT thereon, the study of its renormalizability, and the study of its cosmology.",
"Quantizing gravity consistently will be one of the main goals.",
"To verify the viability of the $T_{\\gamma =\\alpha }\\cong T_q$ approximation explicitly by extracting experimental constraints directly from $T_\\gamma $ .",
"If these bounds turned out to be close to those obtained in $T_q$ , then the $T_{\\gamma =\\alpha }\\cong T_q$ approximation would be confirmed and one could use the simple $T_q$ setting to explore more features of $T_\\gamma $ in advance.",
"If, on the contrary, the direct bounds on $T_\\gamma $ departed from those on $T_q$ , we would have to treat these theories separately.",
"To study the late-time cosmology of all theories, in order to check whether we can explain late-time acceleration with multiscale geometry (question ).",
"To check whether the $\\alpha =0$ configuration can help to address the big-bang problem (question ).",
"To study the role of complex dimensions and degenerate geometries, their theoretical viability, and their physical consequences (question ).",
"To investigate the relation between the near-boundary regime of multifractional spacetimes and phase B of CDT [41] (question ).",
"The multiscale model $\\tilde{T}_1$ of refs.",
"[76], [104], [105] has a Lorentz-invariant measure $d^Dx\\,v(s)$ but its Laplace–Beltrami operator is not self-adjoint.",
"For this reason, it was abandoned in favor of the multifractional paradigm [39].",
"However, $\\tilde{T}_1$ is an example of geometry obeying the first flow-equation theorem that could be used for phenomenology, without the ambition of defining a theory with a rigorous top-down construction.",
"The status of each theory, together with the discontinued model $T_1$ , is summarized in table REF .",
"Table: Status of the multifractional model T 1 T_1 and of the three multifractional theories T v,q,γ T_{v,q,\\gamma }.",
"Empty cells correspond to topics not studied yet.",
"Items with a question mark “?” indicate partial results.",
"If an nonempty item (with or without question mark) has no references given, the result is either obvious (no question mark) or easily doable (with question mark).",
"The items “—” for T 1 T_1 are not of interest for the future since T 1 T_1 is a toy model replaced by T v T_v; however, one could still do some cosmological phenomenology with it.$\\blacktriangleright $72$\\blacktriangleleft $ To conclude with a motivational appeal, why would I want to work on multifractional theories?tocsubsection Why would I want to work on multifractional theories?",
"Because they are based on a guiding principle whose implementation is gradually improving in rigorousness, their UV geometry is extremely interesting and affects all sectors in physics, they yield characteristic phenomenological predictions, they are rigid enough to be easily falsifiable by experiments, and they may contribute to the big-bang, the cosmological constant, and the quantum-gravity problems.",
"The guiding principle is the second flow-equation theorem [60], supported by multifractal geometry [39], [40], [41] and motivated by quantum gravity (section ).",
"In the UV, logarithmic oscillations can give rise to some esoteric form of propagation of quantum degrees of freedom (questions , , and ) or else melt away in a stochastic structure not allowing for precise measurements (questions and ).",
"The effects of the multiscale geometry of these scenarios is not confined to the UV limit of gravity.",
"On one hand, it propagates to large scales via the long-range modulation of log oscillations [53], [58], which are a manifestation of microscopic discrete scale invariance [43], [41] (question ).",
"On the other hand, the nontrivial integrodifferential structure of multifractional theories modifies not just the gravitational sector [sections and ; compare, in contrast, changes of the dynamics as in, say, $f(R)$ gravity] but also the Standard Model of particles (section ), thus opening up the possibility to constrain the theories with a great variety of experiments (section ).",
"The wealth of bounds that have been obtained from atomic and particle physics, astrophysics, and cosmology are sensitive to the free parameters of the measure, in particular to the fractional exponents determining the dimension of spacetime.",
"This property, together with the rigid theoretical structure of each proposal (especially $T_q$ and $T_\\gamma $ , the models with more symmetries), make multifractional scenarios easily falsifiable.",
"Much still needs to be done in order to get control over $T_\\gamma $ and the new developments on the stochastic view, but it can be done in a very reasonable time span.",
"Finally, throughout this review-plus-plus we stumbled across many unsolved problems of modern theoretical physics, including the resolution of singularities such as the big bang or in black holes (question ), the cosmological constant problem or the nature of dark energy (questions and ), the nature of inflation (questions and ), and the problem of quantum gravity (questions 0, 0, and ).",
"We cannot and do not claim that multifractional theories have the final answer to any of these topics, but they are contributing to the debate in an alternative way and there is a lot of potential to be uncovered from preliminary results.",
"We hope to report on, or to see news about, all this in the near future and, as paradoxical as it may sound, to come back with more frequently asked questions than now."
],
[
"Acknowledgments",
"The author is under a Ramón y Cajal contract and is supported by the I+D grant FIS2014-54800-C2-2-P.",
"He thanks L. Modesto for the discussion that led to eq.",
"(REF ), and M. Arzano, D. Oriti, and M. Ronco for other discussions.",
"Open Access.",
"This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited."
]
] | 1612.05632 |
[
[
"Best Friends Forever (BFF): Finding Lasting Dense Subgraphs"
],
[
"Abstract Graphs form a natural model for relationships and interactions between entities, for example, between people in social and cooperation networks, servers in computer networks, or tags and words in documents and tweets.",
"But, which of these relationships or interactions are the most lasting ones?",
"In this paper, we study the following problem: given a set of graph snapshots, which may correspond to the state of an evolving graph at different time instances, identify the set of nodes that are the most densely connected in all snapshots.",
"We call this problem the Best Friends For Ever (BFF) problem.",
"We provide definitions for density over multiple graph snapshots, that capture different semantics of connectedness over time, and we study the corresponding variants of the BFF problem.",
"We then look at the On-Off BFF (O^2BFF) problem that relaxes the requirement of nodes being connected in all snapshots, and asks for the densest set of nodes in at least $k$ of a given set of graph snapshots.",
"We show that this problem is NP-complete for all definitions of density, and we propose a set of efficient algorithms.",
"Finally, we present experiments with synthetic and real datasets that show both the efficiency of our algorithms and the usefulness of the BFF and the O^2BFF problems."
],
[
"Introduction",
"Graphs offer a natural model for capturing the interactions and relationships among entities.",
"Oftentimes, multiple snapshots of a graph are available; for example, these snapshots may correspond to the states of a dynamic graph at different time instances, or the states of a complex system at different conditions.",
"We call such sets of graph snapshots, a graph history.",
"Analysis of the graph history finds a large spectrum of applications, ranging from social-network marketing, to virus propagation and digital forensics.",
"A central question in this context is: which interactions, or relationships in a graph history are the most lasting ones?",
"In this paper, we formalize this question and we design algorithms that effectively identify such relationships.",
"In particular, given a graph history, we introduce the problem of efficiently finding the set of nodes, that remains the most tightly connected through history.",
"We call this problem the Best Friends For Ever ($\\textsc {Bff}$ ) problem.",
"We formulate the $\\textsc {Bff}$ problem as the problem of locating the set of nodes that have the maximum aggregate density in the graph history.",
"We provide different definitions for the aggregate density that capture different notions of connectedness over time, and result in four variants of the $\\textsc {Bff}$ problem.",
"We then extend the $\\textsc {Bff}$ problem to capture the cases where subsets of nodes are densely connected for only a subset of the snapshots.",
"Consider for example, a set of collaborators that work intensely together for some years and then they drift apart, or, a set of friends in a social network that stop interacting for a few snapshots and then, they reconnect with each other.",
"To identify such subsets of nodes, we define the On-Off $\\textsc {Bff}$ problem, or $\\textsc {O$ 2$Bff}$ for short.",
"In the $\\textsc {O$ 2$Bff}$ problem, we ask for a set of nodes and a set of $k$ snapshots such that the aggregate density of the nodes over these snapshots is maximized.",
"Identifying $\\textsc {Bff}$ nodes finds many applications.",
"For example, in collaboration and social networks such nodes correspond to well-acquainted individuals, and they can be chosen to form teams, or organize successful professional or social events.",
"In a protein-interaction network, we can locate protein complexes that are densely interacting at different states, thus indicating a possible underlying regulatory mechanism.",
"In a network where nodes are words or tags and edges correspond to their co-occurrences in documents or tweets published in a specific period of time, identifying $\\textsc {Bff}$ nodes may serve as a first step in topic identification, tag recommendation and other types of analysis.",
"In a computer network, locating servers that communicate heavily over time may be useful in identifying potential attacks, or bottlenecks.",
"The problem of identifying a dense subgraph in a static (i.e., single-snapshot) graph has received a lot of attention (e.g., [1], [2], [3]).",
"There has been also work on finding dense subgraphs in dynamic graphs (e.g., [4]).",
"However, in this line of work, the goal is to efficiently locate the densest subgraph in the current graph snapshot, whereas we are interested in locating subgraphs that remain dense in the whole graph history.",
"To the best of our knowledge, we are the first to systematically introduce and study density in a graph history, and define the $\\textsc {Bff}$ and $\\textsc {O$ 2$Bff}$ problems.",
"The most related work to ours is [5] where the authors study just one of the four variants of the $\\textsc {Bff}$ problem in the context of graph databases.",
"We compare the performance of our algorithms for this variant with the algorithm proposed in [5] experimentally.",
"We study the complexity of the different variants of the $\\textsc {Bff}$ and $\\textsc {O$ 2$Bff}$ problems.",
"Two of the $\\textsc {Bff}$ variants can be solved optimally, while the $\\textsc {O$ 2$Bff}$ is NP-hard.",
"We propose a generic algorithmic framework for solving our problems, that works in linear time.",
"Experimental results with real and synthetic datasets show the efficiency and effectiveness of our algorithms in discovering lasting dense subgraphs.",
"Two case studies on bibliographic collaboration networks, and hashtag co-occurrence networks in Twitter validate our approach.",
"To summarize, the main contributions of this work are the following: $\\bullet $ We introduce the novel $\\textsc {Bff}$ and $\\textsc {O$ 2$Bff}$ problems of identifying a subset of nodes that define dense subgraphs in a graph history.",
"To this end, we extend the notion of density for graph histories, and provide definitions that capture different semantics of density over time leading to four variants of our problems.",
"$\\bullet $ We study the complexity of the variants of the $\\textsc {Bff}$ and $\\textsc {O$ 2$Bff}$ problems and propose appropriate algorithms.",
"We prove the optimality, or the approximation factor of our algorithms whenever possible.",
"$\\bullet $ We extend our definitions and algorithms to identify the $\\textsc {Bff}$ s of an input set of query nodes.",
"$\\bullet $ We perform experiments with both real and synthetic datasets and demonstrate that our problem definitions are meaningful, and that our algorithms work well in identifying dense subgraphs in practice.",
"Roadmap: In Section , we provide definitions of aggregate density.",
"We introduce the $\\textsc {Bff}$ problem and its algorithms in Section , and the $\\textsc {O$ 2$Bff}$ problem and its algorithms in Section , while in Section we study extensions to the original problem.",
"Our experimental evaluation is presented in Section and comparison with related work in Section .",
"Section concludes the paper."
],
[
"Aggregate density",
"We assume that we are given as input multiple graph snapshots over the same set of nodes.",
"Snapshots may be ordered, for example, when the snapshots correspond to the states of a dynamic graph.",
"We may also have an unordered collection of graphs, for example, when the snapshots correspond to graphs collected as a result of some scientific experiments.",
"We refer to such graph collections as a graph history.",
"Definition 1 (Graph History) A graph history ${{\\mathcal {G}}}$ = $\\lbrace G_1$ , $G_2$ , $\\dots ,$ $G_{\\tau }\\rbrace $ is a collection of $\\tau $ graph snapshots, where each snapshot $G_t = (V, E_t)$ , $t$ $\\in $ $[1, \\tau ]$ , is defined over the same set of nodes $V$ .",
"An example of a graph history with four snapshots is shown in Figure REF .",
"Note that our definition is applicable to graph snapshots with different set of nodes by considering $V$ as their union.",
"Figure: A graph history 𝒢={G 1 ,...,G 4 }{\\mathcal {G}}=\\lbrace G_1,\\ldots ,G_4\\rbrace consisting of four snapshots.We will now define the notion of density for a graph history.",
"We start by reviewing two basic definitions of graph density for a single graph snapshot.",
"Given an undirected graph $G = (V, E)$ and a node $u$ in $V$ , we use $\\textit {degree}(u,G)$ to denote the degree of $u$ in $G$ .",
"The average density, $d_a(G)$ , of the graph $G$ is the average degree of the nodes in $V$ : $d_a(G)=\\frac{1}{|V|}\\sum _{u\\in V}\\textit {degree}(u,G) = \\frac{2|E|}{|V|}$ while the minimum density, $d_m(G)$ , of the graph is the minimum degree of any node in $V$ : $d_m(G) = \\min _{u \\in V}degree(u, G).$ Intuitively, for a given graph, $d_m$ is defined by a single node, the one with the minimum degree, while $d_a$ accounts for the degrees and thus the connectivity of all nodes.",
"For example, in Figure (REF ), $d_m(G_1)=2$ , while $d_a(G_1)=10/3$ .",
"Clearly, $d_m$ is a lower bound for $d_a$ .",
"From now on, when the subscript of $d$ is ignored, density can be either $d_a$ or $d_m$ .",
"We also define the density of a subset of nodes $S \\subseteq V$ in the graph $G = (V,E)$ .",
"To this end, we use the induced subgraph $G[S] = (S, E(S))$ in $G$ , where $E(S) = \\lbrace (u,v) \\in E: u \\in S, \\, v \\in S\\rbrace $ .",
"We define the density $d(S,G)$ of $S$ in $G$ as $d(G[S])$ .",
"For example, again for snapshot $G_1$ in Figure REF , for $S_x=\\lbrace x_1,x_2,x_3,x_4\\rbrace $ , $d_m(S_x,G_1) = d_a(S_x,G_1)=3$ , while for $S_y=\\lbrace y_1,y_2,y_3,y_4,y_5\\rbrace $ , $d_m(S_y,G_1) =2$ and $d_a(S_y,G_1)=16/5$ .",
"Between $S_x$ and $S_y$ , $S_x$ has the highest minimum density, whereas $S_y$ the highest average density.",
"We now define the density of a set of nodes $S$ on a graph history.",
"For this, we need a way to aggregate the density of a set of nodes over multiple graph snapshots.",
"Aggregating density sequences: Given a graph history ${\\mathcal {G}}=\\lbrace G_1,\\ldots ,$ $G_\\tau \\rbrace $ , we will use $d(S,{\\mathcal {G}})$ $= \\lbrace d(S,G_1),\\dots ,d(S,G_\\tau )\\rbrace $ to denote the sequence of density values for the graph induced by the set $S$ in the graph snapshots.",
"We consider two definitions for an aggregation function $g(d(S,{\\mathcal {G}}))$ that aggregates the densities over snapshots: the first, $g_m$ , computes the minimum density over all snapshots: $g_m(d(S,{\\mathcal {G}})) = \\min _{G_t\\in {\\mathcal {G}}} d(S,G_t),$ while the second, $g_a$ , computes the average density over all snapshots: $g_a(d(S,{\\mathcal {G}})) = \\frac{1}{\\left|{\\mathcal {G}}\\right|} \\sum _{G_t\\in {\\mathcal {G}}} d(S,G_t).$ Intuitively, the minimum aggregation function requires high density in each and every snapshot, while the average aggregation function looks at the snapshots as a whole.",
"Again, we use $g$ to collectively refer to $g_m$ or $g_a$ .",
"We can now define the aggregate density $f$ .",
"Definition 2 (Aggregate Density) Given a graph history ${\\mathcal {G}}=\\lbrace G_1,\\ldots ,G_\\tau \\rbrace $ defined over a set of nodes $V$ and $S\\subseteq V$ , we define the aggregate density $f(S,{\\mathcal {G}})$ to be $f(S,{\\mathcal {G}})=g(d(S,{\\mathcal {G}}))$ .",
"Depending on the choice of the density function $d$ and the aggregation function $g$ , we have the following four versions of $f$ : (a) $ f_\\textit {mm}(S,{\\mathcal {G}}) = g_m(d_m(S,{\\mathcal {G}}))$ , (b) $f_\\textit {ma}(S,{\\mathcal {G}}) = g_m(d_a(S,{\\mathcal {G}}))$ , (c) $f_\\textit {am}(S,{\\mathcal {G}}) = g_a(d_m(S,{\\mathcal {G}}))$ , and (d) $ f_\\textit {aa}(S,{\\mathcal {G}}) = g_a(d_a(S,{\\mathcal {G}}))$ .",
"Each density definition associates different semantics with the meaning of density among nodes in a graph history.",
"Large values of $f_\\textit {mm}(S,{\\mathcal {G}})$ correspond to groups of nodes $S$ where each member of the group is connected with a large number of other members of the group at each snapshot.",
"A node ceases to be considered a member of the group, if it loses touch with the other members even in a single snapshot.",
"Large values of $f_\\textit {ma}(S,{\\mathcal {G}})$ are achieved for groups with high average density at each snapshot $G\\in {\\mathcal {G}}$ .",
"As opposed to $f_\\textit {mm}(S,{\\mathcal {G}})$ , where the requirement is placed at each member of the group, large values of $f_\\textit {ma}(S,{\\mathcal {G}})$ are indicative that the group $S$ has persistently high density as a whole.",
"The $f_\\textit {aa}(S,{\\mathcal {G}})$ metric takes large values when the group $S$ has many connections on average; thus, $f_\\textit {aa}$ is more “loose\" both in terms of consistency over time and in terms of requirements at the individual group member level.",
"Lastly, $f_\\textit {am}(S,{\\mathcal {G}})$ takes the average of the minimum degree node at each snapshot, thus is less sensitive to the density of $S$ at a single instance.",
"For example, in the graph history ${\\mathcal {G}}$ in Figure REF , all aggregate densities for $S_x$ are equal to 3.",
"However, for $S_y$ $f_{\\textit {aa}}(S_y,{\\mathcal {G}})$ = $31/10$ , while $f_{\\textit {ma}}(S_y,{\\mathcal {G}})$ = $12/5$ .",
"That is, while $f_{\\textit {aa}}(S_y,{\\mathcal {G}}) > f_{\\textit {aa}}(S_x,{\\mathcal {G}})$ , $f_{\\textit {ma}}(S_y,{\\mathcal {G}}) < f_{\\textit {ma}}(S_x,{\\mathcal {G}})$ due to the last instance.",
"Note also that $f_{\\textit {mm}}(S_y,{\\mathcal {G}})$ = 1 and that this value is determined by just one node in just one snapshot, i.e., node $y_4$ in the last snapshot, while $f_{\\textit {am}}(S_y,{\\mathcal {G}})$ = 2.",
"The average graph: Finally, let us define the average graph of a graph history ${\\mathcal {G}}$ which is an edge-weighted graph where the weight of an edge is equal to the fraction of snapshots in ${\\mathcal {G}}$ where the edge appears.",
"Definition 3 (Average Graph) Given a graph history ${\\mathcal {G}}=\\lbrace G_1,\\dots ,G_\\tau \\rbrace $ on a set of nodes $V$ , the average graph $\\widehat{H}_{\\mathcal {G}}=(V,\\widehat{E},\\widehat{w})$ is a weighted, undirected graph on the set of nodes $V$ , where $\\widehat{E}$ = $V\\times V$ , and for each $(u,v)\\in \\widehat{E}$ , $\\widehat{w}(u,v) = \\frac{\\left|G_t=(V,E_t)\\in {\\mathcal {G}}\\mid (u,v)\\in E_t\\right|}{\\left|{\\mathcal {G}}\\right|}.$ As usual, the degree of a node $u$ in a weighted graph is defined as: $\\textit {degree}(u,\\widehat{H}_{\\mathcal {G}}) =\\sum _{(u,v)\\in \\widehat{E}}\\widehat{w}(u,v)$ .",
"The average graph performs aggregation on a per-node basis, in that, the degree of each node $u$ in $\\widehat{H}_{\\mathcal {G}}$ is the average degree of $u$ in time.",
"With the average graph.",
"we lose information regarding density at individual snapshots.",
"With some algebraic manipulation, we can prove the following lemma that shows a connection between the average graph and the $f_\\textit {aa}$ density function: Lemma 1 Let ${\\mathcal {G}}=\\lbrace G_1,\\ldots ,G_\\tau \\rbrace $ be a graph history over a set of nodes $V$ and $S$ a subset of nodes in $V$ , it holds: $f_\\textit {aa}(S,{\\mathcal {G}})$ $=$ $d_a\\left(\\widehat{H}_{\\mathcal {G}}\\left[S\\right]\\right)$ ."
],
[
"The ",
"In this section, we introduce the $\\textsc {Bff}$ problem, we study its hardness and propose appropriate algorithms."
],
[
"Problem definition",
"Given the snapshots of a graph history ${{\\mathcal {G}}}$ , our goal is to locate the Best Friends For Ever ($\\textsc {Bff}$ ), that is, to identify a subset of nodes of $V$ such that these nodes remain densely connected with each other in all snapshots of ${\\mathcal {G}}$ .",
"Formally: Problem 1 (The Best Friends Forever ($\\textsc {Bff}$ ) Problem) Given a graph history ${{\\mathcal {G}}}$ and an aggregate density function $f$ , find a subset of nodes $S\\subseteq V$ , such that $f(S,{\\mathcal {G}})$ is maximized.",
"By considering the four choices for the aggregate density function $f$ , we have four variants of the $\\textsc {Bff}$ problem.",
"Specifically, $f_\\textit {mm}$ , $f_\\textit {ma}$ , $f_\\textit {am}$ and $f_\\textit {aa}$ give rise to problems: $\\textsc {Bff-mm}$ , $\\textsc {Bff-ma}$ , $\\textsc {Bff-am}$ and $\\textsc {Bff-aa}$ respectively."
],
[
"We now introduce a generic algorithm for the $\\textsc {Bff}$ problem.",
"The algorithm (shown in Algorithm REF ) is a “greedy-like\" algorithm inspired by a popular algorithm for the densest subgraph problem on a static graph [6], [1].",
"We use ${\\mathcal {G}}[S]$ $= \\lbrace G_1[S],\\ldots ,G_\\tau [S]\\rbrace $ to denote the sequence of the induced subgraphs of the set of nodes $S$ .",
"The algorithm starts with a set of nodes $S_0$ consisting of all nodes $V$ , and then it performs $n-1$ steps, where at each step $i$ it produces a set $S_i$ by removing one of the nodes in the set $S_{i-1}$ .",
"It then finds the set $S_i$ with the maximum aggregate density $f(S,{\\mathcal {G}})$ .",
"The $\\textsc {FindBff}$ algorithm.",
"[1] Input: Graph history ${\\mathcal {G}}= \\lbrace G_1,\\dots ,G_\\tau \\rbrace $ ; aggregate density function $f$ Output: A subset of nodes $S$ $S_0 = V$ $i = 1,\\dots , n-1$ $v_i=\\displaystyle \\arg \\min _{v\\in S_{i-1}}\\textit {score}\\left(v,{\\mathcal {G}}\\left[S_{i-1}\\right]\\right)$ $S_i = S_{i-1} \\setminus \\lbrace v_i\\rbrace $ return $\\arg \\displaystyle \\max _{i=0\\ldots n-1}f(S_i,{\\mathcal {G}})$ The $\\textsc {FindBff}$ algorithm forms the basis for the algorithms we propose for the four variants of the $\\textsc {Bff}$ problem.",
"Interestingly, by defining appropriate scoring functions, $\\textit {score}\\left(v,{\\mathcal {G}}\\left[S\\right]\\right)$ , (used in line REF to select which node to remove), we can get efficient algorithms for each of the variants.",
"[ht!]",
"The score$_m$ algorithm.",
"[1] Input: Graph history ${\\mathcal {G}}= \\lbrace G_1,\\dots ,G_\\tau \\rbrace $ Output: Node with the minimum score$_m$ ${\\cal L}_{t}$ [$d$ ] $\\leftarrow $ list of nodes with degree $d$ in $G_t$ ScoreAndUpdate$ $ $t = 1,\\dots ,\\tau $ $dmin_t$ $\\leftarrow $ smallest $d$ s.t.",
"${\\cal L}_t[d] \\ne \\emptyset $ $score_m$ = $\\displaystyle \\min _{t = 1,\\dots ,\\tau }{dmin_t}$ $t^{\\prime }$ = $\\displaystyle \\arg \\min _{t = 1,\\dots ,\\tau }{dmin_t}$ $u$ = ${\\cal L}_{t^{\\prime }}[score_m]$ .get() each $G_t \\in {\\mathcal {G}}$ ${\\cal L}_{t}$ [$score_m$ ].remove($u$ ) each $(u, v) \\in E_t$ ${\\cal L}_{t}$ [degree($v$ , $G_t$ )].remove($v$ ) $E_t = E_t - (u, v)$ // update $\\displaystyle degree_{v \\in V}(v, G_t)$ ${\\cal L}_{t}$ [degree($v$ , $G_t$ )].add($v$ ) $V = V\\setminus \\lbrace u\\rbrace $ return u"
],
[
"Solving ",
"For the $\\textsc {Bff-mm}$ problem, we define the score for a node $v$ in $S$ , $\\textit {score}_m$ , as the minimum degree of $v$ in the sequence ${\\mathcal {G}}\\left[S\\right]$ .",
"That is, $\\textit {score}_m\\left(v,{\\mathcal {G}}\\left[S\\right]\\right) = \\min _{G_t\\in {\\mathcal {G}}}\\textit {degree}\\left(v,G_t\\left[S\\right]\\right).$ At the $i$ -th iteration of the $\\textsc {FindBff}$ algorithm, we select the node $v_i$ with the minimum $\\textit {score}_m$ value.",
"We call this instantiation of the $\\textsc {FindBff}$ algorithm $\\textsc {FindBff}_\\textsc {M}$ .",
"Below we prove that $\\textsc {FindBff}_\\textsc {M}$ provides the optimal solution to the $\\textsc {Bff-mm}$ problem.",
"Proposition 1 The $\\textsc {Bff-mm}$ problem can be solved optimally in polynomial time using the $\\textsc {FindBff}_\\textsc {M}$ algorithm.",
"Let $i$ be the iteration of the $\\textsc {FindBff}_\\textsc {M}$ algorithm, where for the first time, a node that belongs to an optimal solution $S^\\ast $ is selected to be removed.",
"Let $v_i$ be this node.",
"Then clearly, $S^\\ast \\subseteq S_{i-1}$ and by the fact that at every iteration we remove edges from the graphs we have that $\\textit {score}_m\\left(v_i,{\\mathcal {G}}\\left[S_{t-1}\\right]\\right)\\ge \\textit {score}_m\\left(v_i,{\\mathcal {G}}\\left[S^\\ast \\right]\\right).$ Since $v_i$ is the node we pick at iteration $i$ , every node $u\\in S_{i-1}$ satisfies: $\\min _{G_t\\in {\\mathcal {G}}}\\textit {degree}(u,G_t[S_{i-1}]) $ = $ \\textit {score}_m\\left(u,{\\mathcal {G}}\\left[S_{i-1}\\right]\\right)$ $\\ge $ $\\textit {score}_m\\left(v_i,{\\mathcal {G}}\\left[S_{i-1}\\right]\\right) $ $ \\ge \\textit {score}_m\\left(v_i,{\\mathcal {G}}\\left[S^\\ast \\right]\\right)$ .",
"Since this is true for every node $u$ , this means that $S_{i-1}$ is indeed optimal and that our algorithm will find it.",
"The running time of $\\textsc {FindBff}_\\textsc {M}$ is $O(n\\tau + M)$ , where $n=|V|$ , $\\tau $ the number of snapshots in the history graph and $M=m_1+m_2+\\ldots + m_\\tau $ the total number of edges that appear in all snapshots.",
"[ht!]",
"The score$_a$ algorithm.",
"[1] Input: Graph history ${\\mathcal {G}}= \\lbrace G_1, \\dots G_\\tau \\rbrace $ Output: Node with the minimum score$_a$ $\\widehat{H}_{\\mathcal {G}}\\leftarrow $ construct the average graph of ${\\mathcal {G}}$ ${\\cal L}[d]$ $\\leftarrow $ list of nodes with degree $d$ in $\\widehat{H}_{\\mathcal {G}}$ ScoreAndUpdate$ $ $score_a$ $\\leftarrow $ smallest $d$ s.t.",
"${\\cal L}[d] \\ne \\emptyset $ $u$ = ${\\cal L}[score_a]$ .get() ${\\cal L}$ [$score_a$ ].remove($u$ ) each $(u, v) \\in \\widehat{E}$ ${\\cal L}$ [degree($v$ , $\\widehat{H}_{\\mathcal {G}})$ ].remove($v$ ) $\\widehat{E} = \\widehat{E} - (u, v)$ // update $\\displaystyle {degree_{v \\in V}(v, \\widehat{H}_{\\mathcal {G}})}$ ${\\cal L}$ [degree($v$ , $\\widehat{H}_{\\mathcal {G}}$ )].add($v$ )$V = V \\setminus \\lbrace u\\rbrace $ return u The node with the minimum $\\textit {score}_m$ value is computed by the procedure ScoreAndUpdate shown in Algorithm REF , which also removes the node and its edges from all snapshots.",
"For each snapshot $G_t$ , we keep the list of nodes with degree $d$ (line 1 in Algorithm REF ); these lists can be constructed in time $O(n\\tau )$ .",
"Given these lists, the time required to find the node with the minimum $\\textit {score}_m$ is $O(\\tau )$ (lines 4–8).",
"Now in all snapshots, the neighbors of the removed node need to be moved from their position in the $\\tau $ lists (lines 9–14); the degree of every neighbor of the removed node is decreased by one.",
"Throughout the execution of the algorithm at most $O(M)$ such moves can happen.",
"Therefore, the total running time of $\\textsc {FindBff}_\\textsc {M}$ is $O(n\\tau + M)$ .",
"Note that an algorithm that iteratively removes from a graph $G$ the node with the minimum degree was first studied in [6] and shown to compute a 2-approximation of the densest subgraph problem for the $d_a(G)$ density in [1] and the optimal for the $d_m(G)$ density in [7]."
],
[
"Solving ",
"To solve the $\\textsc {Bff-aa}$ problem, we shall use the average graph $\\widehat{H}_{{\\mathcal {G}}}$ of ${\\mathcal {G}}$ .",
"Lemma REF shows that $f_\\textit {aa}(S,{\\mathcal {G}}) = d_a\\left(\\widehat{H}_{\\mathcal {G}}\\left[S\\right]\\right)$ .",
"Thus, based on the results of Charikar [1] and Goldberg [2], we conclude that: Proposition 2 The $\\textsc {Bff-aa}$ problem can be solved optimally in polynomial time.",
"Although there exists a polynomial-time optimal algorithms for $\\textsc {Bff-aa}$ , the computational complexity of these algorithms (e..g., $O(|V||\\widehat{E}|^2)$ for the case of the max-flow algorithm in [2]), makes them hard to use for large-scale real graphs.",
"Therefore, instead of these algorithm we use the $\\textsc {FindBff}$ algorithm, where we define the score of a node $v$ in $S$ , $\\textit {score}_a$ , to be equal to its average degree of $v$ in graph history ${{\\mathcal {G}}}[S]$ .",
"That is, $\\textit {score}_a\\left(v,{\\mathcal {G}}\\left[S\\right]\\right) = \\frac{1}{\\left|{\\mathcal {G}}\\right|}\\sum _{G_t\\in {\\mathcal {G}}}\\textit {degree}\\left(v,G_t\\left[S\\right]\\right).$ At the $i$ -th iteration, we select the node $v_i$ with the minimum average degree in ${\\mathcal {G}}$.",
"We will refer to this instantiation of the $\\textsc {FindBff}$ , as $\\textsc {FindBff}_\\textsc {A}$ .",
"Using Lemma REF and the results of Charikar [1] we have the following: Proposition 3 $\\textsc {FindBff}_\\textsc {A}$ is a $\\frac{1}{2}$ -approximation algorithm for the $\\textsc {Bff-aa}$ problem.",
"It is easy to see that $\\textsc {FindBff}_\\textsc {A}$ removes the node with the minimum density in $\\widehat{H}_{{\\mathcal {G}}}$ .",
"Charikar [1] has shown that an algorithm that iteratively removes from a graph the node with minimum density provides a $\\frac{1}{2}$ -approximation for finding the subset of nodes that maximizes the average density on a single (weighted) graph snapshot.",
"Given the equivalence we established in Lemma REF , $\\textsc {FindBff}_\\textsc {A}$ is also a $\\frac{1}{2}$ -approximation algorithm for $\\textsc {Bff-aa}$ .",
"We show the steps for finding the node with the minimum $\\textit {score}_a$ value in Algorithm REF that uses lists of nodes with degree $d$ in the average graph to achieve an $O(n\\tau + M)$ total running time for $\\textsc {FindBff}_\\textsc {A}$ ."
],
[
"Solving ",
"We consider the application of $\\textsc {FindBff}_\\textsc {M}$ and $\\textsc {FindBff}_\\textsc {A}$ algorithms for the two problems.",
"In the following propositions, we prove that the two algorithms give a poor approximation ratio for both problems.",
"Recall that all our problems are maximization problems, and, therefore, the lower the approximation ratio, the worse the performance of the algorithm.",
"Proposition 4 The approximation ratio of algorithm $\\textsc {FindBff}_\\textsc {M}$ for the $\\textsc {Bff-am}$ problem is $O\\left(\\frac{1}{n}\\right)$ where $n$ is the number of nodes.",
"In the Appendix.",
"Proposition 5 The approximation ratio of algorithm $\\textsc {FindBff}_\\textsc {A}$ for the $\\textsc {Bff-am}$ problem is $O\\left(\\frac{1}{n}\\right)$ where $n$ is the number of nodes.",
"In the Appendix.",
"Proposition 6 The approximation ratio of algorithm $\\textsc {FindBff}_\\textsc {M}$ for the $\\textsc {Bff-ma}$ problem is $O\\left(\\frac{1}{\\sqrt{n}}\\right)$ where $n$ is the number of nodes.",
"In the Appendix.",
"We also consider applying the $\\textsc {FindBff}_\\textsc {A}$ algorithm that selects to remove the node with the minimum average degree.",
"We can show that $\\textsc {FindBff}_\\textsc {A}$ has a poor approximation ratio for the $\\textsc {Bff-am}$ problem.",
"Proposition 7 The approximation ratio of algorithm $\\textsc {FindBff}_\\textsc {A}$ for the $\\textsc {Bff-ma}$ problem is $O\\left(\\frac{1}{\\sqrt{n}}\\right)$ where $n$ is the number of nodes.",
"In the Appendix.",
"The complexity of $\\textsc {Bff-ma}$ and $\\textsc {Bff-am}$ is an open problem.",
"Jethava and Beerenwinkel [5] conjecture that the $\\textsc {Bff-ma}$ problem is NP-hard, yet they do not provide a proof.",
"Given that $\\textsc {FindBff}_\\textsc {A}$ and $\\textsc {FindBff}_\\textsc {M}$ have no theoretical guarantees, we also investigate a greedy approach, which selects which node to remove based on the objective function of the problem at hand.",
"This greedy approach is again an instance of the iterative algorithm shown in Algorithm REF .",
"More specifically, for a target function $f$ (either $f_\\textit {am}$ or $f_\\textit {ma}$ ), given a set $S_{i-1}$ , we define the score $\\textit {score}_g(v,{\\mathcal {G}}[S_i])$ of node $v\\in S_i$ as follows: $\\textit {score}_g(v,{\\mathcal {G}}[S_{i-1}]) = f\\left(S_{i-1},{\\mathcal {G}}\\right) - f\\left(S_{i-1}\\setminus \\lbrace v\\rbrace ,{\\mathcal {G}}\\right).$ At iteration $i$ , the algorithm selects the node $v_i$ that causes the smallest decrease, or the largest increase in the target function $f$ .",
"We refer to this algorithm as $\\textsc {FindBff}_\\textsc {G}$ .",
"$\\textsc {FindBff}_\\textsc {G}$ requires to check all nodes when choosing which node to remove at each step (shown in Algorithm REF ), thus leading to complexity $O(n^2\\tau + nM)$ .",
"The score$_g$ algorithm.",
"[1] Input: Graph history ${\\mathcal {G}}= \\lbrace G_1, \\dots G_\\tau \\rbrace $ ; an aggregate density function $f$ Output: Node with the minimum score$_g$ ScoreAndUpdate$ $ $score_{g}[u] = \\emptyset $ for $u \\in V$ each $u \\in V$ $V^{\\prime } = V \\setminus \\lbrace u\\rbrace $ $score_{g}[u] = f(V^{\\prime },{\\mathcal {G}})$ $u$ = $arg\\displaystyle \\min _{v \\in V}{score_{g}[v]}$ each $G_t \\in {\\mathcal {G}}$ each $(u, v) \\in E_t$ $E_t = E_t - (u, v)$ $V = V \\setminus \\lbrace u\\rbrace $ return u"
],
[
"The ",
"In this section, we relax the requirement that the nodes are connected in all snapshots of a graph history.",
"Instead, we ask to find the subset of nodes with the maximum aggregate density in at least $k$ of the snapshots.",
"We call this problem On-Off $\\textsc {Bff}$ ($\\textsc {O$ 2$Bff}$ ) problem.",
"We formally define $\\textsc {O$ 2$Bff}$ , we show that it is NP-hard and develop two general types of algorithms for efficiently solving it in practice."
],
[
"Problem definition",
"In the $\\textsc {O$ 2$Bff}$ problem, we seek to find a collection ${\\mathcal {C}}_k$ of $k$ graph snapshots, and a set of nodes $S\\subseteq V$ , such that the subgraphs induced by $S$ in ${\\mathcal {C}}_k$ have high aggregate density.",
"Formally, the $\\textsc {O$ 2$Bff}$ problem is defined as follows: Problem 2 (The On-Off $\\textsc {Bff}$ ($\\textsc {O$ 2$Bff}$ ) Problem) Given a graph history ${{\\mathcal {G}}}$ = $\\lbrace G_1$ , $G_2$ , $\\dots ,$ $G_\\tau \\rbrace $ , an aggregate density function $f$ , and an integer $k$ , find a subset of nodes $S\\subseteq V$ , and a subset ${\\mathcal {C}}_k$ of ${\\mathcal {G}}$ of size $k$ , such that $f\\left(S,{\\mathcal {C}}_k\\right)$ is maximized.",
"As for Problem REF , depending on the choice of the aggregate density function $f$ , we have four variants of $\\textsc {O$ 2$Bff}$ .",
"Thus, $f_\\textit {mm}$ , $f_\\textit {ma}$ , $f_\\textit {am}$ and $f_\\textit {aa}$ give rise to problems $\\textsc {O$ 2$Bff-mm}$ , $\\textsc {O$ 2$Bff-ma}$ , $\\textsc {O$ 2$Bff-am}$ and $\\textsc {O$ 2$Bff-aa}$ respectively.",
"Note that the subcollection of graphs ${\\mathcal {C}}_k \\subset {\\mathcal {G}}$ does not need to consist of contiguous graph snapshots.",
"If this were the case, then the problem could be solved easily by considering all possible contiguous subsets of $[1,\\tau ]$ and outputting the one with the highest density.",
"However, all the four variants of the $\\textsc {O$ 2$Bff}$ become NP-hard if we drop the constraint for consecutive graph snapshots.",
"Theorem 1 Problem REF is NP-hard for any definition of the aggregate density function $f$ .",
"We will prove that there exists a clique of size at least $k$ in graph $G$ if and only if there exists a set of nodes $S$ and a subset ${\\mathcal {C}}_k \\subseteq {\\mathcal {G}}$ of $k$ snapshots, with $f(S,{\\mathcal {C}}_k) \\ge 1$ .",
"The forward direction is easy; if there exists a subset of nodes $S$ in $G$ , with $|S| \\ge k$ , that form a clique, then selecting this set of nodes $S$ , and a subset ${\\mathcal {C}}_k$ of $k$ snapshots that correspond to nodes in $S$ will wield $f_\\textit {mm}(S,{\\mathcal {C}}_k) = f_\\textit {am}(S,{\\mathcal {C}}_k) = 1$ .",
"This follows from the fact that every snapshot is a complete star where $d_m(S,G_i) = 1$ for all $G_i \\in {\\mathcal {C}}_k$ .",
"To prove the other direction, we observe that all our snapshots consist of a star graph, and a collection of disconnected nodes.",
"Given a set $S$ , $d_m(S,G_i) = 1$ , if $i\\in S$ and all nodes in $S$ are connected to the center node $i$ , and zero otherwise.",
"Therefore, if $f_\\textit {mm}(S,{\\mathcal {C}}_k) = 1$ or $f_\\textit {am}(S,{\\mathcal {C}}_k) = 1$ , then this implies that $d_m(S,G_i) = 1$ for all $G_i \\in {\\mathcal {C}}_k$ , which means that the $k$ centers of the graph snapshots in ${\\mathcal {C}}_k$ are connected to all nodes in $S$ , and hence to each other.",
"Therefore, they form a clique of size $k$ in the graph $G$ .",
"In the case of $f_\\textit {aa}$ and $f_\\textit {ma}$ the construction proceeds as follows: given the graph $G=(V,E)$ , with $|E| = m$ edges, we construct a graph history ${\\mathcal {G}}=\\lbrace G_1,\\ldots , G_\\tau \\rbrace $ with $\\tau = m$ snapshots.",
"All snapshots are defined over the vertex set $V$ .",
"There is a snapshot $G_e$ for each edge $e\\in E$ , consisting of the single edge $e$ .",
"We can prove that there exists a clique of size at least $k$ in graph $G$ if and only if there exists a set of nodes $S$ and a subset ${\\mathcal {C}}_K \\subseteq {\\mathcal {G}}$ of $K = k(k-1)/2$ snapshots, with $f(S,{\\mathcal {C}}_K) \\ge 1/k$ .",
"We will prove that there exists a clique of size at least $k$ in graph $G$ if and only if there exists a set of nodes $S$ and a subset ${\\mathcal {C}}_K \\subseteq {\\mathcal {G}}$ of $K = k(k-1)/2$ snapshots, with $f(S,{\\mathcal {C}}_K) \\ge 1/k$ .",
"The forward direction is easy.",
"If there exists a subset of nodes $S$ in $G$ , with $|S| = k$ , that form a clique, then selecting this set of nodes $S$ , and the ${k \\atopwithdelims ()2}$ snapshots ${\\mathcal {C}}_K$ in ${\\mathcal {G}}$ that correspond to the edges between the nodes in $S$ will yield $f_\\textit {aa}(S,{\\mathcal {C}}_K) = f_\\textit {ma}(S,{\\mathcal {C}}_K) = 1/k$ .",
"To prove the other direction, assume that there is no clique of size greater or equal to $k$ in $G$ .",
"Let ${\\mathcal {C}}_K$ be any subset of $K = k(k-1)/2$ snapshots, and let $S$ be the union of the endpoints of the edges in ${\\mathcal {C}}_K$ .",
"Since $S$ cannot be a clique, it follows that $|S| = \\ell > k$ .",
"Therefore, $f_\\textit {aa}(S,{\\mathcal {C}}_K) = f_\\textit {ma}(S,{\\mathcal {C}}_K)= 1/\\ell < 1/k$ .",
"[t] The Iterative (ITR) $\\textsc {FindO$ 2$Bff}$ algorithm.",
"[1] Input: Graph history ${\\mathcal {G}}= \\lbrace G_1,\\ldots G_\\tau \\rbrace $ ; an aggregate-density function $f$ ; integer $k$ Output: A subset of nodes $S$ and a subset of snapshots ${\\mathcal {C}}_k\\subseteq {\\mathcal {G}}$.",
"converged $=$ False $({\\mathcal {C}}_k^0, S^0) = \\textsc {Initialize}\\left({\\mathcal {G}},f\\right)$ $ds^0 = 0$ not converged ${\\mathcal {C}}_k = \\textsc {BestSnapshots}(S^0,f)$ $S = \\textsc {FindBff}({\\mathcal {C}}_k, f)$ $ds = f(S,{\\mathcal {C}}_k)$ $ds<ds^0$ Converged $=$ True $ds^0=ds$ , $S^0=S$ return $S,{\\mathcal {C}}_k$"
],
[
"We consider two general types of algorithms: iterative and incremental algorithms.",
"The iterative algorithm starts with an initial size $k$ collection ${\\mathcal {C}}_k$ of graph snapshots and improves it, whereas the incremental algorithm builds the collection incrementally, adding one snapshot at a time.",
"Next, we describe these two types of algorithms in detail.",
"Note that depending on whether we are solving the $\\textsc {O$ 2$Bff-mm}$ , $\\textsc {O$ 2$Bff-ma}$ , $\\textsc {O$ 2$Bff-am}$ or $\\textsc {O$ 2$Bff-aa}$ problem, we use the appropriate version of the $\\textsc {FindBff}$ algorithm in each of these algorithms."
],
[
"Iterative Algorithm",
"The iterative ($\\textsc {ITR}$ ) algorithm (shown in Algorithm REF ) starts with an initial collection of snapshots ${\\mathcal {C}}^0_k$ and set of nodes $S^0$ (routine Initialize).",
"At each iteration, given a set $S$ , it finds the $k$ graph snapshots with the highest $d(S, G_i)$ score; this is done by BestSnapshots.",
"BestSnapshots computes the density $d(S, G_i)$ of $S$ in each snapshot $G_i$ $\\in $ ${\\mathcal {G}}$ and outputs the $k$ snapshots ${\\mathcal {C}}_k$ with the highest density.",
"Given ${\\mathcal {C}}_k$ , the algorithm then finds the set $S\\subseteq V$ such that $f\\left(S,{\\mathcal {C}}_k\\right)$ is maximized.",
"This step essentially solves Problem REF on input ${\\mathcal {C}}_k$ for aggregate density function $f$ using the $\\textsc {FindBff}$ algorithm.",
"The $\\textsc {ITR}$ algorithm keeps iterating between collections ${\\mathcal {C}}_k$ and dense sets of nodes $S$ until no further iterations can improve the score $f\\left(S,{\\mathcal {C}}_k\\right)$ .",
"An important step of the Iterative $\\textsc {FindO$ 2$Bff}$ is the initialization of ${\\mathcal {C}}^0_k$ and $S^0$ .",
"We consider three different alternatives for this initialization: random, contiguous, and at least-k. Random initialization ($\\textsc {ITR}_{R}$ ): In this initialization, we randomly pick $k$ snapshots ${\\mathcal {C}}^0_k$ from ${\\mathcal {G}}$ .",
"These snapshots are then used for solving the corresponding $\\textsc {Bff}$ problem on input ${\\mathcal {C}}^0_k$ and produce $S^0 = \\textsc {FindBff}({\\mathcal {C}}^0_k, f)$ .",
"Contiguous initialization ($\\textsc {ITR}_{C}$ ): In this initialization, we first find an $S^0$ that consists of the best $k$ contiguous graph snapshots.",
"Given ${\\mathcal {G}}=\\lbrace G_1,\\dots ,G_\\tau \\rbrace $ , we go over all the $O(\\tau )$ contiguous sets of $k$ snapshots from ${\\mathcal {G}}$ , and find the set of $k$ snapshots ${\\mathcal {C}}^0_k$ and corresponding set of nodes $S^0$ that maximize $f(S^0,{\\mathcal {C}}^0_k)$ .",
"The intuition behind this initialization technique is that it assumes that the best $k$ snapshots of ${\\mathcal {G}}$ are going to be contiguous.",
"Our experiments demonstrate that in practice this is true in many datasets – e.g., in collaboration networks that evolve over time and we expect to see some temporal locality.",
"At least-$k$ initialization ($\\textsc {ITR}_{K}$ ): In this initialization, we solve the $\\textsc {Bff}$ problem independently in each snapshot $G_i\\in {\\mathcal {G}}$ .",
"This results in $\\tau $ different sets $S_i\\subseteq V$ , one for each solution of $\\textsc {Bff}$ on $G_i$ .",
"$S^0$ includes the nodes that appear in at least $k$ of the $\\tau $ sets $S_i$ .",
"The intuition behind this initialization is to include in the initial solution those nodes that appear to be densely connected in many snapshots.",
"We also experimented with other natural alternatives, such as the union: $S^0 = \\cup _{i=1\\ldots \\tau }S_i$ and the intersection: $S^0 = \\cap _{i=1\\ldots \\tau }S_i$ ; the at least-$k$ approach seems to strike a balance between the two.",
"The running time of the iterative $\\textsc {FindO$ 2$Bff}$ algorithm is $O\\left(I\\left(n\\tau +M\\right)\\right)$ , where $I$ is the number of iterations required until convergence, and the $O(n\\tau +M)$ comes from the running time of $\\textsc {FindBff}$ .",
"In practice, we observed that the algorithm converges in at most 6 iterations.",
"[t] The Incremental Density ($\\textsc {INC}_{D}$ ) $\\textsc {FindO$ 2$Bff}$ algorithm.",
"[1] Input: Graph history ${\\mathcal {G}}= \\lbrace G_1,\\ldots G_\\tau \\rbrace $ ; aggregate-density function $f$ ; integer $k$ Output: A subset of nodes $S$ and a subset of snapshots ${\\mathcal {C}}_k\\subseteq {\\mathcal {G}}$.",
"$S_{ij} = \\textsc {FindBff}(\\lbrace G_i,G_j\\rbrace , f)$ , $\\forall G_i,G_j \\in {{\\mathcal {G}}}$ ${\\mathcal {C}}_2 = \\arg \\displaystyle \\max _{G_i,G_j \\in {{\\mathcal {G}}}}{f(S_{ij},\\lbrace G_i,G_j\\rbrace )}$ $i=3$ ; $i \\le k$ each $G_t \\in {{\\mathcal {G}}} \\setminus {{\\mathcal {C}}}_{i-1}$ $S_t = \\textsc {FindBff}({\\mathcal {C}}_{i-1} \\cup \\lbrace G_t\\rbrace ,f)$ $G_m = \\arg \\displaystyle \\max _{G_t}{f(S_t,{\\mathcal {C}}_{i-1} \\cup \\lbrace G_t\\rbrace )}$ ${{\\mathcal {C}}}_i = {\\mathcal {C}}_{i-1} \\cup \\lbrace G_m\\rbrace $ $S = \\textsc {FindBff}({\\mathcal {C}}_k,f)$ return $S,{\\mathcal {C}}_k$ [t] The Incremental Overlap ($\\textsc {INC}_{O}$ ) $\\textsc {FindO$ 2$Bff}$ algorithm.",
"[1] Input: Graph history ${\\mathcal {G}}= \\lbrace G_1,\\ldots G_\\tau \\rbrace $ ; aggregate-density function $f$ ; integer $k$ Output: A subset of nodes $S$ and a subset of snapshots ${\\mathcal {C}}_k\\subseteq {\\mathcal {G}}$.",
"$S_i = \\textsc {FindBff}(G_i, f)$ , $\\forall G_i \\in {{\\mathcal {G}}}$ ${\\mathcal {C}}_2 = \\arg \\displaystyle \\max _{\\begin{array}{c}G_i,G_j \\in {\\cal G}\\end{array}}{\\frac{|S_i \\cap S_j|}{|S_i \\cup S_j|}}$ $i=3$ ; $i \\le k$ $S_C = \\textsc {FindBff}({\\mathcal {C}}_{i-1}, f)$ $G_m = \\arg \\displaystyle \\max _{G_t}{\\frac{|S_t \\cap S_C|}{|S_t \\cup S_C|}}$ ${{\\mathcal {C}}}_i = {\\mathcal {C}}_{i-1} \\cup \\lbrace G_m\\rbrace $ $S = \\textsc {FindBff}({\\mathcal {C}}_k,f)$ return $S,{\\mathcal {C}}_k$"
],
[
"Incremental Algorithm",
"The incremental algorithm starts with a collection ${\\mathcal {C}}_2$ with two snapshots and incrementally adds snapshots to it until a collection ${\\mathcal {C}}_k$ with $k$ snapshots is formed.",
"Then, the appropriate $\\textsc {FindBff}$ algorithm is used to compute the most dense subset of nodes $S$ in ${\\mathcal {C}}_k$ .",
"We use two different policies for selecting snapshots.",
"The first one, termed incremental density ($\\textsc {INC}_{D}$ ) algorithm (shown in Algorithm REF ), selects graph snapshots so as to maximize density, whereas the second one, termed incremental overlap ($\\textsc {INC}_{O}$ ) algorithm (shown in Algorithm REF ), selects graph snapshots so as to maximize the overlap among nodes in the dense subsets.",
"Incremental density ($\\textsc {INC}_{D}$ ): To select the pair of snapshots to form the initial collection ${\\mathcal {C}}_2$ , we solve the $\\textsc {Bff}$ problem independently for each pair of snapshots $G_i,G_j\\in {\\mathcal {G}}$ .",
"This gives us ${\\tau }\\atopwithdelims (){2}$ dense sets $S_{ij}$ as solutions.",
"We select the pair of snapshots whose dense subgraph $S_{ij}$ has the largest density (lines 1–2).",
"The algorithm then builds the solution incrementally in iterations.",
"In iteration $i$ , we construct the solution ${\\mathcal {C}}_i$ by adding to solution ${\\mathcal {C}}_{i-1}$ the graph snapshot $G_m$ that maximizes the density function $f$ .",
"That is if $S_t$ is the densest subset in the sequence ${\\mathcal {C}}_{i-1}\\cup \\lbrace G_t\\rbrace $ , $G_m = \\displaystyle \\arg \\max _{G_t} f(S_t,{\\mathcal {C}}_{i-1}\\cup \\lbrace G_t\\rbrace )$ (lines 3–6).",
"The running time of the $\\textsc {INC}_{D}$ algorithm is $O\\left(\\tau ^2(n+M) + k\\tau \\left(kn\\tau +M\\right)\\right)$ .",
"The first term is due to the initialization step in line 1, where we look for the best pair of snapshots.",
"If efficiency is important we can initialize the algorithm with a random pair to save time.",
"Incremental overlap ($\\textsc {INC}_{O}$ ): To form the initial collection ${\\mathcal {C}}_2$ , we first solve the $\\textsc {Bff}$ problem independently in each snapshot $G_i\\in {\\mathcal {G}}$ .",
"This gives us $\\tau $ different sets $S_i\\subseteq V$ , where $S_i$ is the most dense subgraph in $G_i$ .",
"The algorithm selects from these $\\tau $ sets the two most similar ones, $S_i$ and $S_j$ , and initializes $C_2$ with the corresponding snapshots $G_i$ and $G_j$ (lines 1–2).",
"For defining similarity between sets of nodes, we use the Jaccard similarity.",
"To form $C_i$ from $C_{i-1}$ , the algorithm first solves the $\\textsc {Bff}$ problem in $C_{i-1}$ .",
"Let $S_C$ be the solution.",
"Then, it selects from the remaining snapshots and adds to $C_{i-1}$ the snapshot $G_m$ whose dense set $S_t$ is the most similar with $S_C$ (lines 3–6).",
"The running time of the $\\textsc {INC}_{O}$ algorithm is $O(\\tau ^2n + k\\left(n\\tau +M\\right))$ , where the first term is the time for the initialization and the second term for the for-loop."
],
[
" ",
"The definitions of the $\\textsc {Bff}$ and $\\textsc {O$ 2$Bff}$ problem focus on the identification of a set of nodes $S$ such that their aggregate density is maximized.",
"We now consider natural extensions of the $\\textsc {Bff}$ problem by placing additional constraints on the dense subgraphs.",
"Query-node constraint: An interesting extension is introducing a set $Q$ of seed query nodes and requiring that the output set of nodes $S$ has high density and also contains the input seed nodes.",
"A similar extension was introduced for static (e.g., single snapshots) graphs in [7].",
"In practice, this variant of $\\textsc {Bff}$ identifies the lasting “best friends\" of the query nodes.",
"We call this the $\\textsc {Qr-Bff}$ problem.",
"We can modify the $\\textsc {FindBff}$ algorithms appropriately so that they take into consideration this additional constraint.",
"In particular, $\\textsc {FindBff}_\\textsc {M}$ stops when a query node in $Q$ is selected to be removed.",
"Let us call this modified algorithm, $\\textsc {Qr-FindBff}_\\textsc {M}$ .",
"We can prove the following proposition.",
"(We omit the proof due to space constraints.)",
"Proposition 8 $\\textsc {Qr-FindBff}_\\textsc {M}$ solves the $\\textsc {Qr-Bff-mm}$ problem optimally in polynomial time.",
"We also modify $\\textsc {FindBff}_\\textsc {A}$ so that it does not remove seed nodes as follows: If at any step, the node with the minimum average degree happens to be a seed node, the algorithm selects to remove the node with the next smallest degree that is not a seed node.",
"The algorithm stops when the only remaining nodes are seed nodes.",
"Let us call this modified algorithm, $\\textsc {Qr-FindBff}_\\textsc {A}$ .",
"Proposition 9 Let $S^*$ be an optimal solution for the $\\textsc {Qr-Bff-aa}$ problem and $S_A$ be the solution of the $\\textsc {Qr-FindBff}_\\textsc {A}$ algorithm.",
"It holds: $f_\\textit {aa}(S_A)$ $\\ge $ $\\frac{s \\, f_\\textit {aa}(S^*) + 2 \\, \\omega }{2(s+q)}$ , where $q$ = $|Q|$ , $s$ = $|S^* \\setminus Q|$ and $\\omega $ $=$ $\\sum _{u \\in Q}degree(u, S^*)$ .",
"By Lemma REF , it suffices to show that the $\\textsc {Qr-FindBff}_\\textsc {A}$ algorithm provides an approximation of the average density of a single graph $G$ .",
"Let $S^*$ be the optimal solution for $G$ .",
"Let $G^{\\prime }$ be the graph that results from $G$ when we delete all edges between two query nodes in $G$ .",
"Clearly, $S^*$ is also an optimal solution for $G^{\\prime }$ .",
"Assume that we assign each edge $(u, v)$ to either $u$ or $v$ .",
"For each node $u$ , let $a(u)$ be the number of edges assigned to it and let $a_{max}$ = $max_u\\lbrace a(u)\\rbrace $ .",
"It is easy to see that $f_\\textit {aa}(S^*)$ $\\le $ $\\frac{1}{2}$ $a_{max}$ , since each edge in the optimal solution must be assigned to a node in it.",
"Now assume that the assignment of edges to nodes is performed as the $\\textsc {Qr-FindBff}_\\textsc {A}$ algorithm proceeds.",
"Initially, all edges are unassigned.",
"When at step $i$ , a node $u$ is deleted, we assign to $u$ all the edges that go from $S_{i-1}$ to $u$ .",
"Note that this assignment maintains the invariant that at each step, all edges between two nodes in the current set $S$ are unassigned, while all other edges are assigned.",
"When the algorithm stops, all edges have been assigned.",
"Consider a single iteration of the algorithm when a node $u_{min}$ is selected to be removed and let $S$ be the current set.",
"Let $s$ be the number of non-query nodes in $S$ , and $q$ = $|Q|$ be the number of query nodes.",
"It holds: $f_\\textit {aa}(S)$ $=$ $\\frac{1}{s+q}$$\\sum _{u \\in S}$$degree(u)$ = $\\frac{1}{s+q}$$\\sum _{v \\in S \\setminus Q }$$degree(u)$ +$\\frac{1}{s+q}$$\\sum _{u \\in Q}$$degree(u)$ .",
"Let $\\Omega $ $=$ $\\frac{1}{s+q}$$\\sum _{u \\in Q}$$degree(u)$ .",
"Since $u_{min}$ has the smallest degree among all nodes in $S$ but the seed nodes, we have $f_\\textit {aa}(S)$ $\\ge $ $\\frac{1}{s+q}$ $s$ $a(u_{min})$ + $\\Omega $ .",
"Since all edges are assigned and edges are assigned to a node only when this node is removed, at some step of the execution of the algorithm $a(u_{min})$ = $a_{max}$ .",
"Thus, for some $S$ , $f_\\textit {aa}(S)$ $\\ge $ $\\frac{s}{s+q}$$a_{max}$$+$$\\Omega $ $\\ge $ $\\frac{s}{s+q}$ $\\frac{1}{2}$$f_\\textit {aa}(S^*)$$+$$\\Omega $ .",
"Connectivity constraint: Another meaningful extension is to impose restrictions on the connectivity of $S$ .",
"The connectivity of $S$ in a graph history ${\\mathcal {G}}=\\lbrace G_1,\\ldots , G_\\tau \\rbrace $ may have many different interpretations.",
"One may consider a version where all the induced subgraphs $G_t[S]$ for $t\\in \\lbrace 1,\\ldots ,\\tau \\rbrace $ are connected.",
"Another alternative is that at least $m$ $>$ 0 of the $\\tau $ $G_t[S]$ 's are connected.",
"Here, we assume that a definition of connectivity for $S$ is given in the form of a predicate $connected(S, {\\mathcal {G}})$ which is true if $S$ is connected and false otherwise.",
"Our problem now becomes: given a graph history ${\\mathcal {G}}$ , a set of query nodes $Q$ $\\subset $ $V$ and an aggregate density function $f$ , find a subset of nodes $S\\subseteq V$ , such that (1) $f(S,{\\mathcal {G}})$ is maximized, (2) $Q$ $\\subseteq $ $S$ , and (3) $connected(S, {\\mathcal {G}})$ is true.",
"To solve this problem, we can modify $\\textsc {FindBff}$ so that it tests for the connectivity predicate $connected(S, {\\mathcal {G}})$ and stops when the connectivity constraint no longer holds.",
"In our experiments, we apply a simplest test, just running the algorithms on the connected components of the query nodes.",
"Size constraint: Finally, note that the definition of $\\textsc {Bff}$ does not impose a constraint on the size of the output set of nodes $S$ .",
"In that respect the problem is parameter-free.",
"If necessary, one can add an additional constraint to the problem definition by imposing a cardinality constraint on the output $S$ .",
"However the cardinality constraint makes the subgraph-discovery problem computationally hard [7].",
"This also holds for the $\\textsc {Bff}$ problem; simply consider a graph history with replicas of the same single-snapshot graph."
],
[
"Experimental Evaluation",
"The goal of our experimental evaluation is threefold.",
"First, we want to evaluate the performance of our algorithms for the $\\textsc {Bff}$ and the $\\textsc {O$ 2$Bff}$ problems in terms of the quality of the solutions and running time.",
"Second, we want to compare the different variants of the aggregate density functions.",
"Third, we want to show the usefulness of the problem, by presenting results of $\\textsc {Bff}$ 's and $\\textsc {O$ 2$Bff}$ 's in two real datasets, namely research collaborators in DBLP and hashtags in Twitter.",
"Table: Real dataset characteristicsDatasets and setting.",
"To evaluate our algorithms, we use a number of real graph histories, where the snapshots correspond to collaboration, computer, and concept networks.",
"$\\bullet $ The DBLP$_{10}$http://dblp.uni-trier.de/ dataset contains yearly snapshots of the co-authorship graph in the 2006-2015 interval, for 11 top database and data mining conferences.",
"There is an edge between two authors in a graph snapshot, if they co-authored a paper in the corresponding year and more than two papers in the corresponding interval.",
"$\\bullet $ The Oregon$_1$https://snap.stanford.edu/data/oregon1.html dataset consists of nine graph snapshots of AS peering information inferred from Oregon route-views between March 31 2001 and May 26 2001 (one snapshot per week).",
"$\\bullet $ The Oregon$_2$https://snap.stanford.edu/data/oregon2.html dataset consists of nine weekly snapshots of AS graphs, between March 31, 2001 and May 26, 2001.",
"$\\bullet $ The Caidahttp://www.caida.org/data/as-relationships/ dataset, contains 122 CAIDA autonomous systems (AS) graphs, derived from a set of route views BGP-table instances.",
"$\\bullet $ In the Twitter dataset [8], nodes are hashtags of tweets and edges represent the co-appearance of hashtags in a tweet.",
"The dataset contains 15 daily snapshots from October 27, 2013 to November 10, 2013.",
"$\\bullet $ The AShttps://snap.stanford.edu/data/as.html dataset represents a communication network of who-talks-to-whom from the BGP (Border Gateway Protocol) logs.",
"The dataset contains 733 daily snapshots which span an interval of 785 days from November 8, 1997 to January 2, 2000.",
"The dataset characteristics are summarized in Table REF .",
"Since we do not have any ground truth information for the real datasets, we also use synthetic datasets.",
"In particular, we create graph snapshots using the forest fire model [9], a well-known model for creating evolving networks, using the default forward and backward burning probabilities of 0.35.",
"Then, we plant dense subgraphs in these snapshots, by randomly selecting a set $X$ $\\subset $ $V$ of the nodes and creating additional edges between them, different at each snapshot.",
"We ran our experiments on a system with a quad-core Intel Core i7-3820 3.6 GHz processor, with 64 GB memory.",
"We only used one core in all experiments."
],
[
"Since, as shown in Section REF , $\\textsc {FindBff}_\\textsc {M}$ and $\\textsc {FindBff}_\\textsc {A}$ are provably good for the $\\textsc {Bff-mm}$ and $\\textsc {Bff-aa}$ problems respectively, we only consider these algorithms for these problems.",
"For the $\\textsc {Bff-ma}$ and $\\textsc {Bff-am}$ problems, we use all three algorithms, i.e., $\\textsc {FindBff}_\\textsc {M}$ , $\\textsc {FindBff}_\\textsc {A}$ , and $\\textsc {FindBff}_\\textsc {G}$ .",
"For the $\\textsc {Bff-ma}$ problem, we also use the DCS algorithm proposed in [5] for a problem similar to $\\textsc {Bff-ma}$ .",
"The DCS algorithm is also an iterative algorithm that removes nodes, one at a time.",
"At each step, DCS finds the subgraphs with the largest average density for each of the snapshots.",
"Then, it identifies the subgraph with the smallest average density among them and removes the node that has the smallest degree in this subgraph.",
"Accuracy of $\\textsc {FindBff}$ and comparison of the density definitions: We start by an evaluation of the accuracy of our algorithms along with a comparison of the different aggregate densities.",
"Since we do not have any ground truth information for the real data, we use first the synthetic datasets.",
"First, we create 10 graph snapshots with $4,000$ nodes each using the forest fire model [9].",
"Then, in each one of the 10 snapshots we plant a dense random subgraph $A$ with 100 nodes by inserting extra edges with probability $p_A$ .",
"We vary the edge probabilities from $p_A = 0.1$ to $p_A = 0.9$ , and in Fig.",
"REF (a), we report the $F$ measure achieved for the four density definitions, when trying to recover subgraph $A$ .",
"Recall that the $F$ takes values in $[0,1]$ and the larger the value the better the recall and precision of the solution with respect to the ground truth (in this case $A$ ).",
"$\\textsc {Bff-mm}$ is the most sensitive measure, since it reports $A$ as a dense subgraph even for the smallest edge probability.",
"$\\textsc {Bff-ma}$ and $\\textsc {Bff-am}$ achieve a perfect $F$ value, for an edge probability larger than $p_A = 0.1$ and $\\textsc {Bff-aa}$ for an edge probability at least $p_A = 0.3$ .",
"For smaller values, these three density definitions locate supersets of $A$ , due to averaging.",
"All variations of the $\\textsc {FindBff}$ algorithms produce the same results.",
"We now study how the various density definitions behave when there is a second dense subgraph.",
"In this case, we plant a subgraph $A$ with edge probability $p_A = 0.5$ in all snapshots and a second dense subgraph $B$ with the same number of nodes as $A$ and edge probability $p_B = 0.9$ in a percentage $\\ell $ of the snapshots, for different values of $\\ell $ .",
"Fig.",
"REF (b) depicts which of two graphs, graph $A$ (shown in blue), or graph $B$ (shown in red), is output by the $\\textsc {FindBff}$ algorithms for the different density definitions.",
"$\\textsc {Bff-mm}$ and $\\textsc {Bff-ma}$ report $A$ as the densest subgraph, since these measures ask for high density at each and every snapshot.",
"However, $\\textsc {Bff-am}$ and $\\textsc {Bff-aa}$ report $B$ , when the very dense subgraph $B$ appears in a sufficient number (more than half) of the snapshots.",
"All density definitions and algorithms, recover the exact set $A$ , or $B$ , at each case.",
"Figure: Accuracy and density definition comparison for Bff\\textsc {Bff}Table: Results of the different algorithms for the Bff\\textsc {Bff} problem on the real datasets.Table: Execution time (sec) of the different algorithms for the Bff\\textsc {Bff} problem on the real datasets.We also run all algorithms using the real datasets and present the results in Table REF , where we report the value of the objective function and the size of the solution.",
"A first observation is that as expected, the value of the aggregate density of the reported solution (independently of the problem variant) increases with the density of the graphs.",
"For $\\textsc {Bff-mm}$ problem we observe that the solutions usually have small cardinality compared to the solutions for other problems, since the $f_{\\textit {mm}}$ objective is rather strict (the solution for Twitter was empty).",
"The solutions for $\\textsc {Bff-mm}$ problem in the autonomous-system datasets appear to have higher $f_\\textit {mm}$ scores.",
"This may be due to the fact that there are larger groups of nodes with lasting connections in these datasets, e.g., nodes that communicate intensely between each other during the observation period.",
"Comparison of $\\textsc {FindBff}$ alternatives for $\\textsc {Bff-ma}$ and $\\textsc {Bff-am}$ : As shown in Table REF , for the $\\textsc {Bff-ma}$ problem, $\\textsc {FindBff}_\\textsc {G}$ and $\\textsc {FindBff}_\\textsc {A}$ perform overall the best in all datasets producing subgraphs with large $f_{\\textit {ma}}$ values.",
"$\\textsc {FindBff}_\\textsc {A}$ performs slightly worse than $\\textsc {FindBff}_\\textsc {G}$ only in the Caida dataset.",
"In the Caida dataset, due probably to the large number of snapshots, $\\textsc {FindBff}_\\textsc {A}$ – which is based on the average degree – returns a set with the smallest density.",
"$\\textsc {FindBff}_\\textsc {M}$ and DCS have comparable performance, since they both remove nodes with small degrees in individual snapshots.",
"They are both outperformed by $\\textsc {FindBff}_\\textsc {A}$ and $\\textsc {FindBff}_\\textsc {G}$ .",
"For the $\\textsc {Bff-am}$ problem, $\\textsc {FindBff}_\\textsc {A}$ outperforms both $\\textsc {FindBff}_\\textsc {M}$ and $\\textsc {FindBff}_\\textsc {G}$ .",
"Our deeper analysis of the inferior performance of $\\textsc {FindBff}_\\textsc {G}$ for this problem revealed that $\\textsc {FindBff}_\\textsc {G}$ often gets trapped in local maxima after removing just a few nodes of the graph and it cannot find good solutions.",
"Running time: In Table REF , we report execution times.",
"As expected, the response time of $\\textsc {FindBff}_\\textsc {G}$ algorithm is the slowest in all datasets, due to its quadratic complexity.",
"For the $\\textsc {Bff-ma}$ problem, $\\textsc {FindBff}_\\textsc {A}$ is in general faster than DCS.",
"The difference in times in $\\textsc {FindBff}_\\textsc {M}$ algorithms are due to differences in the computation of the density functions.",
"Additional experiments including ones with synthetic datasets with larger graphs and more intervals that show similar behavior are depicted in Figs.",
"REF (a)(b).",
"In particular the Fig.",
"REF (a) show the execution time of the different algorithms for the $\\textsc {Bff-ma}$ problem for varying nodes with $\\tau = 10$ , whereas Fig.",
"REF (b) shows the execution time for varying snapshots.",
"Summary: In conclusion, our algorithms successfully discovered the planted dense subgraphs even when their density is small, with $\\textsc {Bff-mm}$ being the most sensitive measure.",
"Minimum aggregation over densities (i.e., $\\textsc {Bff-mm}$ , $\\textsc {Bff-ma}$ ) requires a dense subgraph to be present at all snapshots, whereas average aggregation over densities (i.e., $\\textsc {Bff-am}$ , $\\textsc {Bff-aa}$ ) asks that the nodes are sufficiently connected with each other on average.",
"For the $\\textsc {Bff-ma}$ and $\\textsc {Bff-am}$ problems, $\\textsc {FindBff}_\\textsc {A}$ returns in general more dense subgraphs than the alternatives (including DCS).",
"Both $\\textsc {FindBff}_\\textsc {A}$ and $\\textsc {FindBff}_\\textsc {M}$ scale well.",
"They perform similarly for the different density functions with the differences in running time attributed to the complexity of calculating the respective functions.",
"Figure: Synthetic dataset (p A =0.5)p_A =0.5): execution time of the different algorithms for the Bff-ma\\textsc {Bff-ma} problem for varying number of (a) nodes, and (b) snapshots.Figure: Synthetic dataset (p A =0.9p_A = 0.9): FF-measure values for the O\\textsc {O2BffBff} problemsFigure: Synthetic dataset (p A =0.5p_A = 0.5, p B =0.9p_B = 0.9): FF-measure values for the O\\textsc {O2BffBff} problemsFigure: DBLP 10 _{10} dataset: scores of aggregate density functions ffFigure: Oregon 1 _1 dataset: scores of aggregate density functions ffFigure: Oregon 2 _2 dataset: scores of aggregate density functions ff"
],
[
"In this set of experiments, we evaluate the performance of the iterative and incremental $\\textsc {FindO$ 2$Bff}$ algorithms.",
"Comparison of the algorithms in terms of solution quality: Similar to before, we plant a dense random graph $A$ in $k$ snapshots.",
"We then run the $\\textsc {FindO$ 2$Bff}$ algorithms with the same value of $k$ .",
"In Fig.",
"REF , we report the $F$ measure for the different values of $k$ expressed as a percentage of the total number of snapshots.",
"For the iterative $\\textsc {FindO$ 2$Bff}$ algorithm, the at-least-k initialization ($\\textsc {ITR}_{K}$ ) outperforms the other two, and it successfully locates $A$ for all four density definitions, when $A$ appears in a sufficient number of snapshots.",
"Non-surprisingly, all initializations work equally well for average aggregation over time (i.e., $\\textsc {O$ 2$Bff-am}$ and $\\textsc {O$ 2$Bff-aa}$ ).",
"For the incremental $\\textsc {FindO$ 2$Bff}$ algorithm, density ($\\textsc {INC}_{D}$ ) slightly outperforms overlap ($\\textsc {INC}_{O}$ ).",
"Overall, the incremental algorithms achieve highest $F$ , when compared with the iterative ones.",
"We also conduct a second experiment in which we plant a dense random graph $A$ with edge probability $p_A$ = 0.5 in all snapshots and a dense random graph $B$ with edge probability $p_B$ = 0.9 in $k$ snapshots.",
"In Fig.",
"REF , we report the $F$ measure assuming that $B$ is the correct output for the $\\textsc {O$ 2$Bff}$ problem for different values of $k$ expressed as a percentage of the total number of snapshots.",
"Again, by comparing the different initializations for the iterative $\\textsc {FindO$ 2$Bff}$ algorithm, we observe that among the iterative algorithms, $\\textsc {ITR}_{K}$ successfully locates $B$ for all four density definitions, when $B$ appears in a sufficient number of snapshots.",
"As in the previous experiment, all initializations work equally well for average aggregation over time.",
"The incremental algorithms outperform the iterative ones with $\\textsc {INC}_{D}$ being the champion, since they achieve higher $F$ measure values even when $B$ appears in a few snapshots.",
"We also apply the $\\textsc {FindO$ 2$Bff}$ algorithms on all real datasets for various values of $k$ .",
"In Figs.",
"REF – REF , we report the value of the aggregate density for DBLP$_{10}$, Oregon$_1$, and Oregon$_2$ for different values of $k$ , again expressed as a percentage of the total number of snapshots of the input graph history.",
"Overall, we observed that, in contradistinction to the experiments with real datasets, the contiguous initialization ($\\textsc {ITR}_{C}$ ) of the iterative $\\textsc {O$ 2$Bff-aa}$ algorithm emerges as the best algorithm in many cases, slightly outperforming $\\textsc {INC}_{D}$ .",
"This is indicative of temporal locality of dense subgraphs in these datasets, i.e., in these datasets dense subgraphs are usually alive in a few contiguous snapshots.",
"This is especially evident in datasets from collaboration networks such as the DBLP datasets.",
"We also notice that the incremental algorithms find solutions with density very close to that of the iterative algorithms.",
"Finally, we also observe that as $k$ increases the aggregate density of the solutions decrease.",
"This again is explained by the fact that often dense subgraphs are only “alive\" in a few snapshots.",
"Figure: Execution time of the different algorithms for the O\\textsc {O2Bff-mmBff-mm} problem in (a) Synthetic (p A =0.5p_A = 0.5, p B =0.9p_B = 0.9), and (b) Oregon 2 _2 datasets.Convergence and running time: In terms of convergence, iterative $\\textsc {FindO$ 2$Bff}$ requires 2-6 iterations to converge in all datasets.",
"In Fig.",
"REF we report the execution time of $\\textsc {O$ 2$Bff}$ algorithms for the $\\textsc {Bff-mm}$ problem in synthetic ($p_A = 0.5$ , $p_B = 0.9$ ), and (b) Oregon$_2$ datasets.",
"As we observed, iterative and incremental $\\textsc {INC}_{O}$ algorithms scale well with $k$ .",
"Comparing incremental algorithms, $\\textsc {INC}_{O}$ is up to 6x and 3.5x faster than $\\textsc {INC}_{D}$ in synthetic and Oregon$_2$ datasets respectively due to the quadratic complexity of the latter.",
"Additional experiments including ones with synthetic datasets with larger graphs and more intervals are depicted in Fig.",
"REF (a) and Fig.",
"REF (b) respectively.",
"In particular, Fig.",
"REF (a) shows the execution time of the different algorithms for the $\\textsc {O$ 2$Bff-mm}$ problem for varying number of nodes, with $\\tau = 10$ and $k = 6$ whereas Fig.",
"REF (b) shows the execution time for varying number of snapshots with $k = \\frac{1}{6}~\\tau $ .",
"Summary: In conclusion, all algorithms successfully discovered the planted dense subgraphs that lasted a sufficient percentage (much less than half) of the snapshots with the incremental ones being more sensitive.",
"Among the $\\textsc {FindO$ 2$Bff}$ algorithms, incremental algorithms outperform the iterative ones in most cases.",
"Among the incremental algorithms, $\\textsc {INC}_{D}$ is slightly better than $\\textsc {INC}_{O}$ .",
"However, given the slow running time of $\\textsc {INC}_{D}$ , $\\textsc {INC}_{O}$ is a more preferable choice.",
"Finally, in datasets consisting of dense subgraphs with temporal locality, $\\textsc {ITR}_{C}$ is a good choice for detecting such graphs.",
"Figure: Synthetic dataset (p A =0.5)p_A =0.5): execution time (log scale) of the different algorithms for the O\\textsc {O2Bff-mmBff-mm} problem for varying number of (a) nodes, and (b) snapshots.Table: The Bff\\textsc {Bff} solutions for DBLP 10 _{10} (in parenthesis dense author subgroups)."
],
[
"Case studies",
"In this section, we report indicative results we obtained using the DBLP$_{10}$ and the Twitter datasets.",
"These results identify lasting dense author collaborations and hashtag co-occurrences respectively.",
"Lasting dense co-authorships in DBLP$_{10}$: In Table REF , we report the set of nodes output as solutions to the different $\\textsc {Bff}$ problem variants, on the DBLP$_{10}$ dataset.",
"First, observe that three authors “Wei Fan”, “Philip S. Yu”, and “Jiawei Han” are part of all four solutions.",
"These three authors have co-authored only two papers together in our dataset, but pairs of them have collaborated very frequently over the last decade.",
"The solutions for $\\textsc {Bff-am}$ and $\\textsc {Bff-aa}$ contain additional collaborators of these authors.",
"For $\\textsc {Bff-aa}$ we obtain a solution of 8 authors.",
"Although, this group has no paper in which they are all co-authors, subsets of the authors have collaborated with each other in many snapshots, resulting in high value of $f_\\textit {aa}$ .",
"The solutions for $\\textsc {Bff-mm}$ and $\\textsc {Bff-ma}$ contain the aforementioned three authors and some of their collaborators, but also some new names.",
"These are authors that have scarce or no collaborations with the former group.",
"Thus, in this case, the solutions consist of more than one dense subgroups of authors (grouped in parentheses), that are densely connected within themselves, but sparsely or not connected with others, while this is not the case for $\\textsc {Bff-am}$ and $\\textsc {Bff-aa}$ .",
"In Table REF , we report results for $\\textsc {O$ 2$Bff-mm}$ , $\\textsc {O$ 2$Bff-ma}$ , $\\textsc {O$ 2$Bff-am}$ and $\\textsc {O$ 2$Bff-aa}$ on the same dataset.",
"These authors are the most dense collaborators for $k$ = 2, 4, 6, and 8 (recall there are 10 years in the dataset).",
"We also report the corresponding years of their dense collaborations.",
"Many new groups of authors appear.",
"For example, we have new groups of collaborators from Tsinghua University, CMU and RPI among others.",
"The authors appeared in the solutions of $\\textsc {Bff}$ also appear here for large values of $k$ .",
"Table: The authors output as solutions to the O\\textsc {O2BffBff} problemon DBLP 10 _{10}.We also studied experimentally the $\\textsc {Qr-Bff}$ problem.",
"In Table REF , we show indicative results for three of the authors of this paper as seed nodes.",
"For E. Pitoura, we retrieve a group of ex-graduate students with whom she had a lasting and prolific collaboration; for E. Terzi close collaborators from BU University, and for P. Tsaparas, a group of collaborators from his time at Microsoft Research.",
"Note that in the last case, the selected set consists of researchers with whom P. Tsaparas has co-authored several papers in the period recorded in our dataset, but these authors are also collaborating amongst themselves.",
"Finally, we use one of the authors appearing in the dense subgraphs of the $\\textsc {O$ 2$Bff}$ , namely C. Faloutsos as seed node.",
"In this case, we obtain a dense subgraph similar to the one we have reported in Table REF .",
"Finally, we consider a query with two authors: C. Faloutsos and his student D. Koutra.",
"Adding D. Koutra to the query set changes the consistency of the result, focusing more on authors that are collaborators of both query nodes.",
"Lasting dense hashtag appearances in Twitter: In Table REF , we report results of the $\\textsc {O$ 2$Bff}$ problem on the Twitter dataset.",
"Note that the results of the $\\textsc {Bff}$ problem on this dataset (as shown in Table REF ) are very small graphs, since very few hashtags appear together in all 15 days of the dataset.",
"As seen in Table REF , we were able to discover interesting dense subgraphs of hashtags appearing in $k$ = 3, 6, and 9 of these days.",
"These hashtags correspond to actual events (including f1 races and wikileaks) that were trending during that period.",
"For each solution, we also report the selected snapshot dates.",
"As expected there is time-contiguity in the selected dates, but our approach also captures the interest fluctuation over time.",
"For example, for the wikileaks topic that is captured in the dense hashtag set {“wikileaks”, “snowden”, “nsa”, “prism”}, the best snapshots are collections of contiguous intervals, rather than a single contiguous interval.",
"Note also, that for large values of $k$ , we do not get interesting results which is a fact consistent with the ephemeral nature of Twitter, where hashtags are short-lived.",
"This is especially true for $f_\\textit {mm}$ and $f_\\textit {ma}$ that impose strict density constraints and as a result the solutions consist of disconnected edges.",
"Table: An example of authors output as solutions to the Qr-Bff\\textsc {Qr-Bff} problem on DBLP 10 _{10}.Table: The hashtags and the chosen snapshot dates output as solutions to the O\\textsc {O2BffBff} problem on Twitter."
],
[
"Related Work",
"To the best of our knowledge, we are the first to systematically study all the variants of the $\\textsc {Bff}$ , and $\\textsc {O$ 2$Bff}$ problems.",
"The research most related to ours is the recent work of Jethava and Beerenwinkel [5] and Rozenshtein et al. [10].",
"To the best of our understanding, the authors of [5] introduce one of the four variants of the $\\textsc {Bff}$ problem we studied here, namely, $\\textsc {Bff-ma}$ .",
"In their paper, the authors conjecture that the problem is NP-hard and they propose a heuristic algorithm.",
"Our work performs a rigorous and systematic study of the general $\\textsc {Bff}$ problem for multiple variants of the aggregate density function.",
"Additionally, we introduce and study the $\\textsc {O$ 2$Bff}$ problem, which is not studied in [5].",
"The authors of [10] study a problem that can be considered a special case of the $\\textsc {O$ 2$Bff}$ problem.",
"In particular, their goal is to identify a subset of nodes that are dense in the graph consisting of the union of edges appearing in the selected snapshots, which is a weak definition of aggregate density.",
"Furthermore, they focus on finding collections of contiguous intervals, rather than arbitrary snapshots.",
"They propose an algorithm similar to the iterative algorithm we consider, which we have shown to be outperformed by the incremental algorithms.",
"There is a huge literature on extracting “dense” subgraphs from a single graph snapshot.",
"Most formulations for finding subgraphs that define near-cliques are often NP-hard and often hard to approximate due to their connection to the maximum-clique problem [11], [12], [13], [14], [15].",
"As a result, the problem of finding the subgraph with the maximum average or minimum degree has become particularly popular, due to its computational tractability.",
"Specifically, the problem of finding a subgraph with the maximum average degree can be solved optimally in polynomial time [1], [2], [3], and there exists a practical greedy algorithm that gives a 2-approximation guarantee in time linear to the number of edges and nodes of the input graph [1].",
"The problem of identifying a subgraph with the maximum minimum degree, can be solved optimally in polynomial time [7], using again the greedy algorithm proposed by Charikar [1].",
"In our work, we use the average and minimum degree to quantify the density of the subgraph in a single graph snapshot, and we extend these definitions to sets of snapshots.",
"The algorithmic techniques we use for the $\\textsc {Bff}$ problem are inspired by the techniques proposed by Charikar [1], and by Sozio and Gionis [7]; however, adapting them to handle multiple snapshots is non-trivial.",
"Existing work also studies the problem of identifying a dense subgraph on time-evolving graphs [4], [16], [17]; these are graphs where new nodes and edges may appear over time and existing ones may disappear.",
"The goal in this line of work is to devise a streaming algorithm that at any point in time it reports the densest subgraph for the current version of the graph.",
"In our work, we are not interested in the dynamic version of the problem and thus the algorithmic challenges that our problem raises are orthogonal to those faced by the work on streaming algorithms.",
"Other recent work [18] focuses on detecting dense subgraphs in a special class of temporal weighted networks with fixed nodes and edges, where edge weights change over time and may take both positive and negative values.",
"This is a different problem, since we consider graphs with changing edge sets.",
"Furthermore, density in the presence of edges with negative weights is different than density when edges have only positive weights.",
"Finally, another line of research focuses on processing queries e.g., reachability, path distance, graph matching, etc.",
"over multiple graph snapshots [19], [20], [21], [22], [23].",
"The main goal of this work is to devise effective storage, indexing and retrieving techniques so that queries over such sequences of graphs are answered efficiently.",
"In this paper, we propose a novel problem that of finding dense subgraphs."
],
[
"Summary",
"In this paper, we introduced and systematically studied the problem of identifying dense subgraphs in a collection of graph snapshots defining a graph history.",
"We showed that for many definitions of aggregate density functions the problem of identifying a subset of nodes that are densely-connected in all snapshots (i.e., the $\\textsc {Bff}$ problem) can be solved in linear time.",
"We also demonstrated that other versions of the $\\textsc {Bff}$ problem (i.e., $\\textsc {Bff-ma}$ and $\\textsc {Bff-am}$ ) cannot be solved with the same algorithm.",
"To identify dense subgraphs that occur in $k$ , yet not all, the snapshots of a graph history we also defined the $\\textsc {O$ 2$Bff}$ problem.",
"For all variants of this problem we showed that they are NP-hard and we devised an iterative and an incremental algorithm for solving them.",
"Our extensive experimental evaluation with datasets from diverse domains demonstrated the effectiveness and the efficiency of our algorithms.",
"Appendix In this section we present counter-examples that demonstrate that the $\\textsc {FindBff}_\\textsc {M}$ and $\\textsc {FindBff}_\\textsc {A}$ when applied to the $\\textsc {Bff-ma}$ and $\\textsc {Bff-aa}$ yield a solution that is a poor approximation of the optimal solution.",
"For the following, we use $n = |V|$ to denote the number of nodes in the different snapshots, and $\\tau = |{\\mathcal {G}}|$ to denote the number of snapshots.",
"Proof of Proposition 4 In order to prove our claim we need to construct an instance of the $\\textsc {Bff-am}$ problem where the $\\textsc {FindBff}_\\textsc {M}$ algorithm produces a solution with approximation ratio $O\\left(\\frac{1}{n}\\right)$ .",
"We construct the graph history ${\\mathcal {G}}= \\lbrace G_1,...,G_\\tau \\rbrace $ as follows.",
"The first $\\tau -1$ snapshots consist of a full clique with $n-1$ nodes, plus an additional node $v$ that is connected to a single node $u$ from the clique.",
"The last snapshot $G_\\tau $ consists of just the edge $(v,u)$ .",
"In the first $n-2$ iterations of the $\\textsc {FindBff}_\\textsc {M}$ algorithm, the node with the minimum minimum degree is one of the nodes in the clique (other than the node $u$ ).",
"Thus the nodes in the clique will be iteratively removed, until we are left with the edge $(u,v)$ .",
"Since node $v$ is present in all intermediate subsets $S_i$ , the minimum degree in all snapshots $G_t$ is 1.",
"Therefore, the solution $S$ of the $\\textsc {FindBff}_\\textsc {M}$ algorithm has $f_{am}(S) = 1$ .",
"On the other hand clearly the optimal solution $S^*$ consists of the nodes in the clique, where we have minimum degree $n-2$ , except of the last instance where the minimum degree is zero.",
"Therefore, $f(S^*) = (n-2)\\frac{\\tau -1}{\\tau }$ which proves our claim.",
"Proof of Proposition 5 In order to prove our claim we need to construct an instance of the $\\textsc {Bff-am}$ problem where the $\\textsc {FindBff}_\\textsc {A}$ algorithm produces a solution with approximation ratio $O\\left(\\frac{1}{n}\\right)$ .",
"We construct the graph history ${\\mathcal {G}}= \\lbrace G_1,...,G_\\tau \\rbrace $ , where $\\tau $ is even, as follows.",
"Each snapshot $G_t$ contains $n = 2b+3$ nodes.",
"The $2b$ of these nodes form a complete $b\\times b$ bipartite graph.",
"Let $u$ , $v$ , and $s$ denote the additional three nodes.",
"Node $s$ is connected to all nodes in the graph, in all snapshots, except for the last snapshot where $s$ is connected only to $u$ and $v$ .",
"Nodes $u$ and $v$ are connected to each other in all snapshots, and node $u$ is connected to all $2b$ nodes of the bipartite graph in the first $\\tau /2$ snapshots, while node $v$ is connected to all $2b$ nodes of the bipartite graph in the last $\\tau /2$ snapshots.",
"Throughout assume that $\\tau \\ge 2$ .",
"Note that the optimal set $S^*$ for this history graph consists of the $2b$ nodes in the bipartite graph, with $f_\\textit {am}(S^*,{\\mathcal {G}}) = b = \\Theta (n)$ .",
"The score $\\textit {score}_a$ for every node $w$ of the $2b$ nodes in the bipartite graph is $\\textit {score}_a(w,{\\mathcal {G}}) = b+1+\\frac{\\tau -1}{\\tau }$ .",
"For the nodes $u$ and $v$ , we have $\\textit {score}_a(u,{\\mathcal {G}}) = \\textit {score}_a(v,{\\mathcal {G}}) = \\frac{2b\\tau /2 + 2\\tau }{\\tau } = b+2$ .",
"Node $s$ has score $\\textit {score}_a(s,{\\mathcal {G}}) = 2b\\frac{\\tau -1}{\\tau } + 2$ .",
"Therefore, in the first iteration, the algorithm will remove one of the nodes of the bipartite graph.",
"Without loss of generality assume that it removes one of the nodes in the left partition.",
"Now, for a node $w$ in the left partition, we still have that $\\textit {score}_a(w,{\\mathcal {G}}[S_1]) = b+1+\\frac{\\tau -1}{\\tau }$ .",
"For a node $w$ in the right partition we have that $\\textit {score}_a(w,{\\mathcal {G}}[S_1]) = b + \\frac{\\tau -1}{\\tau }$ .",
"For nodes $u$ and $v$ we have $\\textit {score}_a(u,{\\mathcal {G}}[S_1]) = \\textit {score}_a(v,{\\mathcal {G}}[S_1]) = \\frac{(2b-1)\\tau /2 + 2\\tau }{\\tau } = b+\\frac{3}{2}$ .",
"For node $s$ we have that $\\textit {score}_a(s,{\\mathcal {G}}[S_1]) = (2b-1)\\frac{\\tau -1}{\\tau } + 2$ .",
"Therefore, in the second iteration the algorithm will select to remove one of the nodes in the right partition.",
"Note that the resulting graph ${\\mathcal {G}}[S_2]$ is identical in structure with ${\\mathcal {G}}$ , with $n = 2(b-1)+3$ nodes.",
"Therefore, the same procedure will be repeated until all the nodes from the bipartite graph are removed, while nodes $u$ and $v$ will be kept in the set until the last iterations.",
"As a result, the set $S$ returned by $\\textsc {FindBff}_\\textsc {A}$ has $f_\\textit {am}(S,{\\mathcal {G}}) = 2$ (the degree of the nodes $u$ and $v$ ), yielding approximation ratio $O\\left(\\frac{1}{n}\\right)$ .",
"Proof of Proposition 6 In order to prove our claim we need to construct an instance of the $\\textsc {Bff-ma}$ problem where the $\\textsc {FindBff}_\\textsc {M}$ algorithm produces a solution with approximation ratio $O\\left(\\frac{1}{\\sqrt{n}}\\right)$ .",
"We construct the graph history ${\\mathcal {G}}= \\lbrace G_1,...,G_\\tau \\rbrace $ as follows.",
"We have $\\tau =m$ snapshots that are all identical.",
"They consist of two sets of nodes $A$ and $B$ of size $m$ and $m^2$ respectively.",
"The nodes in $B$ form a cycle.",
"The nodes in $A$ in graph snapshot $G_t$ form a clique with all nodes except for one node $v_t$ , different for each snapshot.",
"The optimal set $S^*$ consists of the nodes in $A$ , that have average degree $\\frac{(m-1)(m-2)}{m} = \\Theta (m)$ .",
"The $\\textsc {FindBff}_\\textsc {M}$ starts with the set of all nodes.",
"The average degree of any snapshot is $\\frac{2m^2 + (m-1)(m-2)}{m^2+m} = \\Theta (1)$ , which is also the value of the $f_{ma}(V)$ function.",
"In the first $m$ iterations of the algorithm, the nodes in $A$ have $\\textit {score}_m(v,S_i) = 0$ , so these are the ones to be removed first.",
"Then the nodes in $B$ are removed.",
"In all iterations the average degree in each snapshot remains $O(1)$ .",
"Therefore, the set $S$ returned by the $\\textsc {FindBff}_\\textsc {M}$ has $f_{ma}(S) = \\Theta (1)$ , and the approximation ratio is $\\Theta \\left(\\frac{1}{m}\\right)$ .",
"Since $m = \\sqrt{n}$ , this proves our claim.",
"Proof of Proposition 7 In order to prove our claim we need to construct an instance of the $\\textsc {Bff-ma}$ problem where the $\\textsc {FindBff}_\\textsc {A}$ algorithm produces a solution with approximation ratio $O\\left(\\frac{1}{\\sqrt{n}}\\right)$ .",
"The construction of the proof is very similar to before.",
"We construct the graph history ${\\mathcal {G}}= \\lbrace G_1,...,G_\\tau \\rbrace $ as follows.",
"We have $\\tau =m$ snapshots that are all identical, except for the last snapshot $G_m$ .",
"The snapshots $G_1,...,G_{m-1}$ consist of two sets of nodes $A$ and $B$ that form two complete cliques of size $m$ and $m^2$ respectively.",
"In the last snapshot the nodes in $B$ are all disconnected.",
"The optimal set $S^*$ consists of the nodes in $A$ , that have $f_{ma}(A) = \\frac{m(m-1)}{m} = \\Theta (m-1)$ .",
"The $\\textsc {FindBff}_\\textsc {A}$ starts with the set of all nodes.",
"The value of $f_{ma}(V)$ is determined by the last snapshot $G_m$ that has average degree $\\frac{m(m-1)}{m^2+m} = \\Theta (1)$ .",
"The nodes in $A$ have average degree (over time) $\\frac{m(m-1)}{m} = \\Theta (m)$ , while the nodes in $B$ have average degree $\\frac{(m-1)(m^2-1)}{m} = \\Theta (m^2)$ .",
"Therefore, the algorithm will iteratively remove all nodes in $A$ .",
"In each iteration the resulting set $S_i$ has $f_{ma}(S_i) = O(1)$ .",
"When all the nodes in $A$ are removed, we have that $f_{ma}(S_i) = 0$ .",
"Therefore, the approximation ratio for this instance is $\\Theta (\\frac{1}{m})$ .",
"Our claim follows from the fact that $n = m^2+m$ ."
],
[
"Appendix",
"In this section we present counter-examples that demonstrate that the $\\textsc {FindBff}_\\textsc {M}$ and $\\textsc {FindBff}_\\textsc {A}$ when applied to the $\\textsc {Bff-ma}$ and $\\textsc {Bff-aa}$ yield a solution that is a poor approximation of the optimal solution.",
"For the following, we use $n = |V|$ to denote the number of nodes in the different snapshots, and $\\tau = |{\\mathcal {G}}|$ to denote the number of snapshots."
],
[
"Proof of Proposition 4",
"In order to prove our claim we need to construct an instance of the $\\textsc {Bff-am}$ problem where the $\\textsc {FindBff}_\\textsc {M}$ algorithm produces a solution with approximation ratio $O\\left(\\frac{1}{n}\\right)$ .",
"We construct the graph history ${\\mathcal {G}}= \\lbrace G_1,...,G_\\tau \\rbrace $ as follows.",
"The first $\\tau -1$ snapshots consist of a full clique with $n-1$ nodes, plus an additional node $v$ that is connected to a single node $u$ from the clique.",
"The last snapshot $G_\\tau $ consists of just the edge $(v,u)$ .",
"In the first $n-2$ iterations of the $\\textsc {FindBff}_\\textsc {M}$ algorithm, the node with the minimum minimum degree is one of the nodes in the clique (other than the node $u$ ).",
"Thus the nodes in the clique will be iteratively removed, until we are left with the edge $(u,v)$ .",
"Since node $v$ is present in all intermediate subsets $S_i$ , the minimum degree in all snapshots $G_t$ is 1.",
"Therefore, the solution $S$ of the $\\textsc {FindBff}_\\textsc {M}$ algorithm has $f_{am}(S) = 1$ .",
"On the other hand clearly the optimal solution $S^*$ consists of the nodes in the clique, where we have minimum degree $n-2$ , except of the last instance where the minimum degree is zero.",
"Therefore, $f(S^*) = (n-2)\\frac{\\tau -1}{\\tau }$ which proves our claim."
],
[
"Proof of Proposition 5",
"In order to prove our claim we need to construct an instance of the $\\textsc {Bff-am}$ problem where the $\\textsc {FindBff}_\\textsc {A}$ algorithm produces a solution with approximation ratio $O\\left(\\frac{1}{n}\\right)$ .",
"We construct the graph history ${\\mathcal {G}}= \\lbrace G_1,...,G_\\tau \\rbrace $ , where $\\tau $ is even, as follows.",
"Each snapshot $G_t$ contains $n = 2b+3$ nodes.",
"The $2b$ of these nodes form a complete $b\\times b$ bipartite graph.",
"Let $u$ , $v$ , and $s$ denote the additional three nodes.",
"Node $s$ is connected to all nodes in the graph, in all snapshots, except for the last snapshot where $s$ is connected only to $u$ and $v$ .",
"Nodes $u$ and $v$ are connected to each other in all snapshots, and node $u$ is connected to all $2b$ nodes of the bipartite graph in the first $\\tau /2$ snapshots, while node $v$ is connected to all $2b$ nodes of the bipartite graph in the last $\\tau /2$ snapshots.",
"Throughout assume that $\\tau \\ge 2$ .",
"Note that the optimal set $S^*$ for this history graph consists of the $2b$ nodes in the bipartite graph, with $f_\\textit {am}(S^*,{\\mathcal {G}}) = b = \\Theta (n)$ .",
"The score $\\textit {score}_a$ for every node $w$ of the $2b$ nodes in the bipartite graph is $\\textit {score}_a(w,{\\mathcal {G}}) = b+1+\\frac{\\tau -1}{\\tau }$ .",
"For the nodes $u$ and $v$ , we have $\\textit {score}_a(u,{\\mathcal {G}}) = \\textit {score}_a(v,{\\mathcal {G}}) = \\frac{2b\\tau /2 + 2\\tau }{\\tau } = b+2$ .",
"Node $s$ has score $\\textit {score}_a(s,{\\mathcal {G}}) = 2b\\frac{\\tau -1}{\\tau } + 2$ .",
"Therefore, in the first iteration, the algorithm will remove one of the nodes of the bipartite graph.",
"Without loss of generality assume that it removes one of the nodes in the left partition.",
"Now, for a node $w$ in the left partition, we still have that $\\textit {score}_a(w,{\\mathcal {G}}[S_1]) = b+1+\\frac{\\tau -1}{\\tau }$ .",
"For a node $w$ in the right partition we have that $\\textit {score}_a(w,{\\mathcal {G}}[S_1]) = b + \\frac{\\tau -1}{\\tau }$ .",
"For nodes $u$ and $v$ we have $\\textit {score}_a(u,{\\mathcal {G}}[S_1]) = \\textit {score}_a(v,{\\mathcal {G}}[S_1]) = \\frac{(2b-1)\\tau /2 + 2\\tau }{\\tau } = b+\\frac{3}{2}$ .",
"For node $s$ we have that $\\textit {score}_a(s,{\\mathcal {G}}[S_1]) = (2b-1)\\frac{\\tau -1}{\\tau } + 2$ .",
"Therefore, in the second iteration the algorithm will select to remove one of the nodes in the right partition.",
"Note that the resulting graph ${\\mathcal {G}}[S_2]$ is identical in structure with ${\\mathcal {G}}$ , with $n = 2(b-1)+3$ nodes.",
"Therefore, the same procedure will be repeated until all the nodes from the bipartite graph are removed, while nodes $u$ and $v$ will be kept in the set until the last iterations.",
"As a result, the set $S$ returned by $\\textsc {FindBff}_\\textsc {A}$ has $f_\\textit {am}(S,{\\mathcal {G}}) = 2$ (the degree of the nodes $u$ and $v$ ), yielding approximation ratio $O\\left(\\frac{1}{n}\\right)$ ."
],
[
"Proof of Proposition 6",
"In order to prove our claim we need to construct an instance of the $\\textsc {Bff-ma}$ problem where the $\\textsc {FindBff}_\\textsc {M}$ algorithm produces a solution with approximation ratio $O\\left(\\frac{1}{\\sqrt{n}}\\right)$ .",
"We construct the graph history ${\\mathcal {G}}= \\lbrace G_1,...,G_\\tau \\rbrace $ as follows.",
"We have $\\tau =m$ snapshots that are all identical.",
"They consist of two sets of nodes $A$ and $B$ of size $m$ and $m^2$ respectively.",
"The nodes in $B$ form a cycle.",
"The nodes in $A$ in graph snapshot $G_t$ form a clique with all nodes except for one node $v_t$ , different for each snapshot.",
"The optimal set $S^*$ consists of the nodes in $A$ , that have average degree $\\frac{(m-1)(m-2)}{m} = \\Theta (m)$ .",
"The $\\textsc {FindBff}_\\textsc {M}$ starts with the set of all nodes.",
"The average degree of any snapshot is $\\frac{2m^2 + (m-1)(m-2)}{m^2+m} = \\Theta (1)$ , which is also the value of the $f_{ma}(V)$ function.",
"In the first $m$ iterations of the algorithm, the nodes in $A$ have $\\textit {score}_m(v,S_i) = 0$ , so these are the ones to be removed first.",
"Then the nodes in $B$ are removed.",
"In all iterations the average degree in each snapshot remains $O(1)$ .",
"Therefore, the set $S$ returned by the $\\textsc {FindBff}_\\textsc {M}$ has $f_{ma}(S) = \\Theta (1)$ , and the approximation ratio is $\\Theta \\left(\\frac{1}{m}\\right)$ .",
"Since $m = \\sqrt{n}$ , this proves our claim."
],
[
"Proof of Proposition 7",
"In order to prove our claim we need to construct an instance of the $\\textsc {Bff-ma}$ problem where the $\\textsc {FindBff}_\\textsc {A}$ algorithm produces a solution with approximation ratio $O\\left(\\frac{1}{\\sqrt{n}}\\right)$ .",
"The construction of the proof is very similar to before.",
"We construct the graph history ${\\mathcal {G}}= \\lbrace G_1,...,G_\\tau \\rbrace $ as follows.",
"We have $\\tau =m$ snapshots that are all identical, except for the last snapshot $G_m$ .",
"The snapshots $G_1,...,G_{m-1}$ consist of two sets of nodes $A$ and $B$ that form two complete cliques of size $m$ and $m^2$ respectively.",
"In the last snapshot the nodes in $B$ are all disconnected.",
"The optimal set $S^*$ consists of the nodes in $A$ , that have $f_{ma}(A) = \\frac{m(m-1)}{m} = \\Theta (m-1)$ .",
"The $\\textsc {FindBff}_\\textsc {A}$ starts with the set of all nodes.",
"The value of $f_{ma}(V)$ is determined by the last snapshot $G_m$ that has average degree $\\frac{m(m-1)}{m^2+m} = \\Theta (1)$ .",
"The nodes in $A$ have average degree (over time) $\\frac{m(m-1)}{m} = \\Theta (m)$ , while the nodes in $B$ have average degree $\\frac{(m-1)(m^2-1)}{m} = \\Theta (m^2)$ .",
"Therefore, the algorithm will iteratively remove all nodes in $A$ .",
"In each iteration the resulting set $S_i$ has $f_{ma}(S_i) = O(1)$ .",
"When all the nodes in $A$ are removed, we have that $f_{ma}(S_i) = 0$ .",
"Therefore, the approximation ratio for this instance is $\\Theta (\\frac{1}{m})$ .",
"Our claim follows from the fact that $n = m^2+m$ ."
]
] | 1612.05440 |
[
[
"An analytical solution of the stationary fully-compressible linear Euler\n equations over orography"
],
[
"Abstract An analytical linear solution of the fully compressible Euler equations is found, in the particular case of a stationary two dimensional flow that passes over an orographic feature with small height-width ratio.",
"A method based on the covariant formulation of the Euler equations is used, and the analytical vertical velocity as well as the horizontal velocity, density and pressure, are obtained.",
"The analytical solution is tested against a numerical model in three different regimes, hydrostatic, non-hydrostatic and potential flow.",
"The model used is a non-hydrostatic spectral semi-implicit model, with a height-based vertical coordinate.",
"It is shown that there is a clear and consistent convergence of the numerical solution towards the analytical solution, when the resolution increases.",
"The method described is intended to be used as an idealized test for numerical weather models."
],
[
"Introduction",
"During the design and development of a weather numerical model, it is customary to perform a number of tests.",
"The tests found in the literature are suited for different purposes, and there are specific tests for examining vertical discretization schemes.",
"The vertical slice test designed by [16], which consists in non-stationary gravity waves, is used for checking the spatial and time discretization schemes in a linear regime without orography.",
"Another vertical slice test, described in [19], consists in a density current, and serves to check non-linear non-hydrostatic regimes with diffusion.",
"The vertical slice tests in [5] include condensation and evaporation processes, and simplified cloud microphysics.",
"Other interesting vertical slice tests are those found in [13], [10] and [8].",
"We are interested in vertical slice tests for checking a stationary flow over an orographic feature with small height-width ratio, where a set of gravity waves are generated.",
"Usually, this kind of tests are performed and checked against analytical solutions that come from a simplified version of the Euler equations, for instance, the Boussinesq equations.",
"Our intention is to find an analytical solution of the fully compressible Euler equations, in the particular case of a stationary flow that passes over an arbitrarily shaped smooth orography.",
"This problem has been studied [18], [12], [11].",
"However, in this work we apply a different method for finding the analytical solution, and we check its consistency using a non-linear non-hydrostatic numerical model.",
"The starting point of the method proposed in this paper is a spatial domain delimited by a flat orography, where a stationary and trivial solution is known.",
"This solution is an isothermal and stratified flow, with constant horizontal velocity.",
"This is the background state, which includes a flat orography in its definition.",
"A second stationary state is taken into account, the perturbed state, for which the domain is deformed in the lower limit with an arbitrarily shaped orography.",
"In doing so, a set of stationary waves appears in the flow.",
"The definition of the perturbed state includes not only the stationary waves, but also the deformation of the lower limit of the domain by the orography.",
"The way chosen for solving the stationary Euler equations in a domain deformed by a orography is to write the Euler equations in a coordinate independent formulation, also called covariant formulation, as dictated by the differential geometry [1].",
"In this formulation, the free slip condition in the lower part of the domain is simplified to set the vertical component of the velocity to zero at the surface.",
"That is, the orography is not involved in the lower boundary condition, which is reduced to a trivial condition, and it only appears explicitly in the equations.",
"The idea is that the orography is moved from the boundary conditions into the equations, and this change is achieved using the covariant formulation.",
"We provide the details of this procedure in section .",
"Then, after writing the Euler equations in a covariant form with an arbitrarily shaped orography, the equations are linearized.",
"The linearization procedure includes, not only the velocity components, density and pressure perturbations, but the orographic terms that appears in the covariant version of the Euler equations.",
"It comes into view that the orographic terms act as forcing terms for the momentum equations, and are responsible for the waves generated in the flow.",
"Then, the linearized equations are solved, and the perturbations of the velocity components, density and pressure are found.",
"The analytical solution of the Euler equations for a stationary flow that passes over a hill, following the method outlined in the previous paragraphs, is described in Section .",
"In section the numerical simulations are exposed and compared to the analytical solutions.",
"Three types of flow are considered, hydrostatic, non-hydrostatic and potential.",
"Finally, the conclusions are pointed out in section ."
],
[
"Analytical solution",
"We consider a two dimensional channel of length $L$ .",
"In the vertical dimension, the upper boundary is open and the lower boundary is limited by the orography, $B(x)$ .",
"The Euler equations are $\\frac{d u}{d t} + \\frac{1}{\\rho }\\frac{\\partial p}{\\partial x} & = 0, \\\\\\frac{d w}{d t} + \\frac{1}{\\rho }\\frac{\\partial p}{\\partial z} + g & = 0, \\\\\\frac{\\partial \\rho }{\\partial t} + \\nabla ({\\bf u} \\rho ) & = 0, \\\\\\frac{d p}{d t} -c_s^2 \\frac{d \\rho }{d t} & = 0,$ where ${\\bf u} = (u, w)$ is the velocity vector, $\\rho $ the density, $p$ the pressure, $g$ the acceleration due to gravity, and $c_s^2$ the speed of sound, given by $c_s & = \\sqrt{\\frac{c_p}{c_v} RT}, \\\\T & = \\frac{p}{R \\rho }.$ As we consider stationary states, the partial time derivative vanishes.",
"The total time derivative is reduced to the advective part, that is $\\frac{d}{dt} = {\\bf {u}} \\cdot \\nabla .$ The boundary conditions imposed to the solution are the following.",
"The free slip boundary condition is used at the bottom, meaning that there is no flux of mass through that surface.",
"It is $w(x,z=B(x)) = u(x,z=B(x)) \\, \\frac{\\partial B}{\\partial x}.$ In the upper limit, considered to be at the infinity, density, and therefore pressure, tends to vanish.",
"The mathematical condition for the density is then $\\lim _{z \\rightarrow +\\infty } \\rho (z) = 0.$ On the other hand, the lateral boundaries are periodic, and consequently we impose the condition $\\psi (0, z) = \\psi (L, z)$ for any function $\\psi (x,z)$ involved in the problem.",
"As it is shown later, this periodicity will permit to solve the problem by Fourier transforming the equations in the horizontal dimension.",
"The boundary condition for the velocity at the lower limit, given by (REF ), involves both the horizontal and vertical components of the velocity, as well as the orography, in a non linear way.",
"Imposing this condition can be cumbersome.",
"We propose a method based on a coordinate transformation to circumvent this problem, so that the boundary condition is transformed into a trivial condition in the new coordinate system.",
"The proposed change of coordinates is simple.",
"The new coordinates, named $(X,Z)$ are related to the original euclidean coordinates $(x,z)$ by $x & = X, \\\\z & = Z + B(X),$ where $B(X)$ is the orography.",
"With this change of coordinates, the lower limit is the coordinate line $Z=0$ .",
"The Jacobian of this transformation is $J = \\frac{\\partial (x,z)}{\\partial (X,Z)} =\\left( \\begin{array}{ccc}1 & 0 \\\\B_X & 1 \\end{array} \\right),$ where the subindex in $B_X$ means partial derivative with respect to coordinate $X$ .",
"The contravariant components of the velocity in the original and new coordinates, named respectively $(u,w)$ and $(U,W)$ , are related by $u & = U, \\\\w & = B_X U + W,$ which implies that the boundary condition for the velocity given by (REF ) is reduced to a very simple and convenient condition, which is $W(X,Z=0) = 0.$ The relations (REF ) and (), are valid for any contravariant vector, not only for the velocity.",
"From them, we observe that the acceleration due to gravity, the contravariant vector $(0, g)$ , has the same components in the new coordinate system.",
"The Euler equations must be transformed into the new coordinate system.",
"To this end, the metric tensor must be found, as well as the Christoffel symbols.",
"The euclidean metric, $\\eta _{xz}$ , is the identity in the original coordinate system, whereas in the new coordinate system it is $\\eta _{XZ} =\\left( \\begin{array}{ccc}1 + B_X^2 & B_X \\\\B_X & 1 \\end{array} \\right).$ The determinant is $|\\eta _{XZ}| = 1$ , and the inverse of the metric tensor is $\\eta ^{-1}_{XZ} =\\left( \\begin{array}{ccc}1 + B_X^2 & -B_X \\\\-B_X & 1 \\end{array} \\right).$ Finally, we need the Christoffel symbols for the advection term of the contravariant components of the velocity.",
"The only non-zero symbol is $\\Gamma ^Z_{XX} = B_{XX}.$ The advection of a contravariant vector $\\bf V$ by the velocity field $\\bf U$ is $\\left({\\bf U} \\cdot \\nabla ) \\, {\\bf V} \\right)^i= U^j \\frac{\\partial }{\\partial X^j} V^i + \\Gamma ^i_{jk} U^j U^k,$ where the indexes, running over 1 and 2, are referencing the coordinates $X$ and $Z$ respectively (that is, $X^1 = X$ , $X^2 = Z$ , and $U^1 = U$ , $U^2 = W$ ).",
"Then, from (REF ) and (REF ) the advection of the contravariant velocity components by the velocity itself is $({\\bf U} \\cdot \\nabla ) \\, U & = (U \\, \\frac{\\partial }{\\partial X} + W \\, \\frac{\\partial }{\\partial Z}) \\, U , \\\\({\\bf U} \\cdot \\nabla ) \\, W & = (U \\, \\frac{\\partial }{\\partial X} + W \\, \\frac{\\partial }{\\partial Z}) \\, W + B_{XX} \\, U^2,$ where the new term $B_{XX} \\, U^2$ , appearing in the advection of the vertical component of the velocity, is due to the non-zero Christoffel symbol (REF ).",
"It is usual to interpret those terms as inertial forces.",
"The pressure gradient is $(\\nabla p)^i = (\\eta ^{ij} \\, \\frac{\\partial }{\\partial X^j}) \\, p,$ and then, from the inverse of the metric tensor given in (REF ), the contravariant pressure gradient writes $(\\nabla p)^X & = ((1 + B_X^2) \\frac{\\partial }{\\partial X} - B_X \\frac{\\partial }{\\partial Z}) \\, p, \\\\(\\nabla p)^Z & = (-B_X \\frac{\\partial }{\\partial X} + \\frac{\\partial }{\\partial Z}) \\, p.$ The divergence term of the continuity equation (REF ) is written in the new coordinate system as $\\nabla \\cdot (\\rho {\\bf {U}}) =\\frac{1}{\\sqrt{|g|}} \\, \\frac{\\partial }{\\partial X^i} ( \\sqrt{|g|} \\, \\rho \\, U^i),$ and, taking into account that $|g| = 1$ the divergence remains written in the same form as in the euclidean coordinate system, that is $\\nabla \\cdot (\\rho {\\bf {U}}) = \\frac{\\partial }{\\partial X} \\left( \\rho \\, U \\right) + \\frac{\\partial }{\\partial Z} \\left( \\rho \\, W \\right).$ Finally, the advection of the pressure and density in the equation (), as well as the advection of any scalar function $\\psi $ , is simply $\\left( {\\bf {U}} \\cdot \\nabla \\right) \\, \\psi = ( U \\frac{\\partial }{\\partial X} + W \\frac{\\partial }{\\partial Z}) \\, \\psi .$ Using the previous results, the non linear stationary Euler equations are written in the new coordinate system $(X,Z)$ as $\\frac{DU}{Dt} + \\frac{1}{\\rho } \\,((1 + B_X^2) \\frac{\\partial p}{\\partial X} - B_X \\frac{\\partial p}{\\partial Z} ) & = 0, \\\\\\frac{DW}{Dt} + B_{XX} \\, U^2 +\\frac{1}{\\rho } \\, (-B_X \\frac{\\partial p}{\\partial X} + \\frac{\\partial p}{\\partial Z}) + g & = 0, \\\\\\frac{\\partial }{\\partial X} \\left( \\rho \\, U \\right) +\\frac{\\partial }{\\partial Z} \\left( \\rho \\, W \\right) & = 0, \\\\\\frac{Dp}{Dt} - c_s^2 \\, \\frac{D \\rho }{Dt} & = 0,$ where $\\frac{D \\psi }{Dt} \\equiv U \\, \\frac{\\partial \\psi }{\\partial X} + W \\, \\frac{\\partial \\psi }{\\partial Z}.$ At this point, we stress that the Euler equations for the horizontal and vertical momentum (REF ) and () contains orographic terms, that is, terms with the orography function $B(X)$ , whereas the lower boundary condition is independent of the orographic features, reduced to be $W(X,Z=0)=0.$ This is why we can say that, by writing the equations in a covariant form, we move the orography $B(X)$ from the boundary condition into the Euler equations.",
"As has been already mentioned, we are interested in finding out a linear solution of the stationary Euler equations.",
"We have been inspired by the methods used in [2], were the authors found a non stationary solution of the linear Euler equations in a two dimensional atmosphere, with a flat lower boundary and an upper boundary placed at a fixed and finite height.",
"In this work, instead, we look for a stationary solution, with a non flat bottom boundary and without an upper boundary.",
"As already said, we consider an arbitrary orographic obstacle of small amplitude given by the function $B(X)$ .",
"That is, because $B(X)$ is small, we expect that the waves that are produced in a flow that passes over this obstacle will be of small amplitude.",
"In the linearization procedure, we will only retain the orographic terms that are at most first order, rejecting higher order terms.",
"We rewrite the Euler equations (REF ) to (), placing in the right hand side the linear orographic terms, which will be treated as forcing terms of the equations $\\frac{DU}{Dt} + \\frac{1}{\\rho } \\, \\frac{\\partial p}{\\partial X} & =\\frac{B_X}{\\rho } \\, \\frac{\\partial p}{\\partial Z}, \\\\\\frac{DW}{Dt} + \\frac{1}{\\rho } \\, \\frac{\\partial p}{\\partial Z} + g & = \\frac{B_X}{\\rho } \\, \\frac{\\partial p}{\\partial X} -B_{XX} \\, U^2 , \\\\\\frac{\\partial }{\\partial X} \\left( \\rho \\, U \\right) +\\frac{\\partial }{\\partial Z} \\left( \\rho \\, W \\right) & = 0, \\\\\\frac{Dp}{Dt} - c_s^2 \\, \\frac{D \\rho }{Dt} & = 0.$ Linearising the Euler equations implies the existence of a steady reference state, being the solution a small perturbation around it.",
"The reference unperturbed state is a steady hydrostatic solution of the Euler equations (REF ) to (), with a flat lower boundary, that is $B(X) = 0$ .",
"Moreover, the reference state is isothermal ($T_0$ ) with a constant horizontal velocity ($U_0$ ), and it is completely determined if the surface pressure is given ($p_s$ ).",
"Then, the background state has a flat orography $B(X)=0$ , a constant velocity field defined by the contravariant components $(U_0, 0)$ , and a pressure and density distribution which depends on the vertical coordinate $Z$ through $p_0(Z) & = p_s \\, e^{-\\delta Z}, \\\\\\rho _0(Z) & = \\rho _s \\, e^{-\\delta Z},$ where $\\rho _s & \\equiv \\frac{p_s \\delta }{g}, \\\\\\delta & \\equiv \\frac{g}{R T_0}.$ For later use, we mention that the Brunt-Väisälä frequency of the background state is constant, equal to $N_0 = \\frac{g}{\\sqrt{c_p T_0}}.$ The linear version of the Euler equations (REF ) to () are found to be $U_0 \\frac{\\partial U^{\\prime }}{\\partial X} + \\frac{1}{\\rho _0} \\frac{\\partial p^{\\prime }}{\\partial X} & =-B_X \\, g, \\\\U_0 \\frac{\\partial W^{\\prime }}{\\partial X} + \\frac{g}{\\rho _0} \\, \\rho ^{\\prime } + \\frac{1}{\\rho _0} \\frac{\\partial p^{\\prime }}{\\partial Z} & =-B_{XX} \\, U_0^2, \\\\\\frac{\\partial U^{\\prime }}{\\partial X} + (\\frac{\\partial }{\\partial Z} - \\delta ) \\, W^{\\prime } + \\frac{U_0}{\\rho _0} \\frac{\\partial \\rho ^{\\prime }}{\\partial X} & = 0, \\\\\\delta ( p_0 - c_s^2 \\rho _0) \\, W^{\\prime } + U_0 \\frac{\\partial }{\\partial X} \\, (c_s^2 \\, \\rho ^{\\prime }- p^{\\prime }) & = 0,$ where the perturbed quantities are $\\psi ^{\\prime } = \\psi - \\psi _0$ .",
"Observe that the horizontal and vertical momentum equations have orographic forcing terms, whereas the continuity and the thermodynamic equations are free of them.",
"In order to solve this linear system, following [2], we apply the [4] transformation to the linear system (REF ) to ().",
"In doing so, as we show below, we obtain a new liner system where the coefficients for the variables are constants.",
"The perturbations $\\psi ^{\\prime }$ are transformed to the Bretherton variables $\\hat{\\psi }$ in this way $\\psi ^{\\prime }(X,Z) & \\equiv \\gamma ^\\pm (Z) \\, \\hat{\\psi }(X,Z), \\\\\\gamma ^\\pm (Z) & \\equiv e^{\\pm \\frac{\\delta }{2} Z},$ being $U^{\\prime } = \\gamma ^+ \\, \\hat{U}$ and $W^{\\prime } = \\gamma ^+ \\, \\hat{W}$ , whereas $p^{\\prime } = \\gamma ^- \\, \\hat{p}$ and $\\rho ^{\\prime } = \\gamma ^- \\, \\hat{\\rho }$ .",
"The linear system (REF ) to () is written, in terms of the Bretherton variables, as $U_0 \\frac{\\partial \\hat{U}}{\\partial X} + \\frac{1}{\\rho _s} \\frac{\\partial \\hat{p}}{\\partial X} & =-B_X \\, g \\, \\gamma ^-, \\\\U_0 \\frac{\\partial \\hat{W}}{\\partial X} + \\frac{g}{\\rho _s} \\, \\hat{\\rho } + \\frac{1}{\\rho _s} ( \\frac{\\partial }{\\partial Z} - \\frac{\\delta }{2}) \\, \\hat{p} & =-B_{XX} \\, U_0^2 \\, \\gamma ^-, \\\\\\frac{\\partial \\hat{U}}{\\partial X} + (\\frac{\\partial }{\\partial Z} - \\delta ) \\, \\hat{W} + \\frac{U_0}{\\rho _s} \\frac{\\partial \\hat{\\rho }}{\\partial X} & = 0, \\\\\\delta \\rho _s (\\frac{g}{\\delta } - c_s^2) \\, \\hat{W} - U_0 \\frac{\\partial }{\\partial X} \\, (\\hat{p} - c_s^2 \\, \\hat{\\rho }) & = 0.$ In the case of a finite horizontal domain of length $L$ , the variables can be expanded in Fourier series (a Fourier transformation would be used for an infinite horizontal domain).",
"Then, any variable $\\hat{\\psi }$ is transformed to $\\tilde{\\psi }$ following $\\hat{\\psi }(X,Z) = \\sum _{k \\in \\frac{2\\pi }{L} \\mathbb {Z}} \\tilde{\\psi }(k,Z) \\, e^{ikX}.$ Finally, the linear system to solve given in equations (REF ) to () is $i k U_0 \\, \\tilde{U} + \\frac{ik}{\\rho _s} \\tilde{p} & =-g \\, ik \\tilde{B}(k) \\, \\gamma ^-(Z), \\\\ik U_0 \\, \\tilde{W} + \\frac{g}{\\rho _s} \\, \\tilde{\\rho } + \\frac{1}{\\rho _s} ( \\frac{\\partial }{\\partial Z} - \\frac{\\delta }{2}) \\, \\tilde{p} & = U_0^2 \\, k^2 \\tilde{B}(k) \\, \\gamma ^-(Z), \\\\ik \\tilde{U} + (\\frac{\\partial }{\\partial Z} - \\delta ) \\, \\tilde{W} + \\frac{ik U_0}{\\rho _s} \\tilde{\\rho } & = 0, \\\\\\delta \\rho _s (\\frac{g}{\\delta } - c_s^2) \\, \\tilde{W} - ik U_0 \\, (\\tilde{p} - c_s^2 \\, \\tilde{\\rho }) & = 0,$ where $\\tilde{B}(k)$ are the Fourier coefficients of the orography function $B(X)$ .",
"The linear system (REF ) to (), with a vertical derivative operator, is solved in the following way.",
"The system is manipulated in order get to an equation where the only variable is the vertical component of the velocity.",
"The result is a second order differential equation, which is solved to obtain $\\tilde{W}(k,Z) & = ik U_0 \\tilde{B}(k) (e^{\\beta Z} - e^{-\\frac{\\delta }{2} Z}),$ where $\\beta ^2 & \\equiv {k^2 \\alpha _0 - \\frac{N_0^2}{U_0^2} + \\frac{\\delta ^2}{4}}, \\\\\\alpha _0 & \\equiv 1 - \\frac{U_0^2}{c_s^2}.$ We observe, from (REF ), that $\\beta $ can be a real or an imaginary number, depending on the value of the wave number $k$ .",
"For values of $k$ such that $k^2 < \\frac{1}{\\alpha _0} (\\frac{N_0^2}{U_0^2} - \\frac{\\delta ^2}{4}),$ $\\beta ^2$ is a negative real number, and therefore $\\beta $ is an imaginary number, the solution leads to a contribution that is a wave in the vertical.",
"On the other hand, for $k$ values that do not satisfy this condition, $\\beta ^2$ is a positive real number.",
"In this case, $\\beta $ is chosen as the negative root of $\\beta ^2$ , otherwise the density and pressure perturbation would increase exponentially with height leading to a non-physical solution.",
"The other variables, horizontal velocity, pressure and density, can be calculated and are $\\tilde{U} & = -(\\frac{\\delta }{2}+\\beta -\\frac{g}{c_s^2}) \\, \\frac{U_0 \\tilde{B}}{\\alpha _0} \\, e^{\\beta Z}, \\\\\\tilde{p} & = -(\\frac{\\delta }{2}+\\beta -\\frac{g}{c_s^2}) \\, \\frac{\\rho _s U_0^2 \\tilde{B}}{\\alpha _0} \\, e^{\\beta Z} - \\tilde{B} \\delta p_s e^{-\\frac{\\delta }{2} Z}, \\\\\\tilde{\\rho } & = (\\delta + \\frac{1}{c_s^2}(U_0^2(\\beta - \\frac{\\delta }{2})-g)) \\, \\frac{\\rho _s \\tilde{B}}{\\alpha _0} \\, e^{\\beta Z} - \\tilde{B} \\delta \\rho _s e^{-\\frac{\\delta }{2} Z}.$ Finally, the solution in the original coordinate system $(x,z)$ is obtained by undoing the Fourier, the Bretherton and the coordinate change transformations.",
"Table: Hydrostatic, non-hydrostatic and potential flow tests settings for the background horizontal velocity (U 0 U_0), Brunt-Väisälä frequency (N 0 N_0), half width and height of the hill (aa and hh), horizontal and vertical size of the domain (LL and HH).",
"The hill is located in the centre of the domain in all cases.",
"The model runs up to the dimensionless time t * =tU 0 /at^*=t U_0 / a, time at which the numerical and the analytical solutions are compared.Table: The horizontal and vertical resolution (dxdx and dzdz), time step (dtdt), the number horizontal and vertical grid points (N x N_x and N z N_z) and number of time steps (N t N_t), for the hydrostatic (HH), non-hydrostatic (NN) and potential flow (PP) tests.Figure: Hydrostatic test at the higher resolution (H 1 H_1): vertical velocity (WW), horizontal velocity (UU), density (DD) and pressure (PP) perturbations.",
"On the left column it is plotted the numerical solution, and on the right the difference between the numerical and the analytical solutions.Figure: Non-hydrostatic test at the higher resolution (N 1 N_1): vertical velocity (WW), horizontal velocity (UU), density (DD) and pressure (PP) perturbations.",
"On the left column it is plotted the numerical solution, and on the right the difference between the numerical and the analytical solutions.Figure: Potential flow test at the higher resolution (P 1 P_1): vertical velocity (WW), horizontal velocity (UU), density (DD) and pressure (PP) perturbations.",
"On the left column it is plotted the numerical solution, and on the right the difference between the numerical and the analytical solutions."
],
[
"Comparison with model simulations",
"In this section the analytical linear solution found in the previous section is compared to the numerical solution provided by a non-linear non-hydrostatic numerical model.",
"To this end, we are going to use the model described in [15].",
"The model is semi-implict and uses the spectral method in the horizontal discretization.",
"It has a hybrid height-based vertical coordinate.",
"For the experiments presented in this section, it is configured to use fourth order finite differences vertical operators.",
"The analytical solution can be used to test the accuracy and convergence of a numerical model.",
"If fact, this was the motivation of this work: to provide an analytical linear solution of the gravitational waves produced in a flow that passes over a hill, with the intention to use it for testing the accuracy of a non-hydrostatic numerical model.",
"We have selected three different flow regimes for the tests: hydrostatic, non-hydrostatic and a potential flow.",
"The tests configuration are the same as the experiments found in [6].",
"The orography is the Agnesi hill, defined by the half width ($a$ ) and the height ($h$ ) of the hill, whereas the flow is determined by the Brunt-Väisälä frequency ($N_0$ ) and the background horizontal velocity ($U_0$ ).",
"The values of this parameters are given in Table REF .",
"The Agnesi hill, which is located in the centre of the domain, is defined by $B(X) = \\frac{h \\, a^2}{a^2 + X^2}.$ For this type of flow, a constant Brunt-Väisälä frequency and horizontal velocity flow over a localized hill, the type of the waves generated downstream the obstacle depends on the value of the dimensionless parameter $a N_0 / U_0$ .",
"For values much greater than one the flow is hydrostatic, for values near one is non-hydrostatic, whereas for values much less than one it becomes a potential flow.",
"For the values given in the Table REF it is found that this parameter takes the values 40, $~0.7$ and $~0.1$ , so we expect the flow to be hydrostatic, non-hydrostatic and potential respectively.",
"The model is initialized with the analytical solution, and it integrates temporally up to $t^* = t U_0 /a$ equal to 120, 90 and 60 for the hydrostatic, non-hydrostatic and potential flow cases respectively, as summarized in Table REF .",
"These time lengths are equal to those reported in [6], and are supposed to be big enough to let the model to get a quasi-stationary solution.",
"The lateral conditions are cyclic, that is, the flow going out from the right boundary is exactly the same as the flow coming in through the left boundary.",
"In the vertical direction there is a sponge layer, from the model top at $H$ to a height equal to $\\alpha \\, H$ , being $\\alpha = 0.6$ for all the tests.",
"In this layer the numerical solution is damped towards a predefined solution, which is the analytical solution in the lower domain of the sponge layer, and the background solution towards the top of the sponge layer, being this transition smooth.",
"The analytical and numerical solutions are then compared at $t^*$ .",
"The maximum error ($L_\\infty $ ) and the mean squared error ($L_2$ ) are calculated for all the variables, vertical and horizontal velocity components, density and pressure.",
"The numerical model has the prognostic variables $r \\equiv \\log (T)$ and $q \\equiv \\log (p)$ , and therefore, it is necessary to compute the density from them, before doing the comparison.",
"The model has been run at three different resolutions for each type of flow, resulting in a total of nine runs.",
"The vertical and horizontal resolutions, the time step and the number of time steps for each run, are listed in the Table REF .",
"The hydrostatic test, plotted in Figure REF , effectively consists in a hydrostatic wave that stays over the hill and propagates in the vertical.",
"The difference between the numerical and the analytical solutions are small, compared to the perturbations (later in this section, the maximum and mean squared errors are calculated for all the tests, resolutions and variables).",
"The pattern of the error varies from one variable to other (vertical velocity, for instance, has a horizontal oscillation, whereas pressure error is higher in the lower part of the domain).",
"The non-hydrostatic test, plotted in Figure REF , produces a wave that propagates upstream as well as in the vertical.",
"It is remarkable that the errors are quite uniform in all the spatial domain, except for a small area near the hill, where the maximum errors there are located.",
"The pressure, on the other hand, shows a noisy error field, with values that are positive (numerical values higher than the analytical) and bigger in the lower part of the domain.",
"The potential flow, potted in Figure REF , is a nearly irrotational.",
"The disturbance does not have an oscillatory behaviour, and diminishes exponentially with the height for all the variables.",
"The maximum errors are also located in a reduced area near the hill.",
"The density, moreover, shows a small disturbance propagating within the flow, probably from the beginning of the simulation.",
"Every run provide four variables to be compared to their analytical counterparts.",
"Therefore there are 36 comparisons between numerical and analytical results, with their respective maximum ($L_\\infty $ ) and mean squared ($L_2$ ) errors, all of them listed in Table REF .",
"As it is expected, the errors, both maximum and mean squared, increase when the grid becomes coarser.",
"The analytical and numerical solutions are compared in a window, which is the spatial domain plotted in the Figures REF , REF and REF .",
"Those domains correspond exactly to the figures of the tests mentioned in [6].",
"The order of convergence represents how the reduction of the error depends on the increase of the resolution.",
"Theoretically, there is a linear relationship between the logarithm of the spatial resolution and the logarithm of the error, both maximum or mean squared errors.",
"For each of the three types of flows (hydrostatic, non-hydrostatic and potential), and for each variable (the vertical and horizontal velocity, density and pressure), we have got three different resolutions and errors (for both $L_\\infty $ and $L_2$ norms).",
"Therefore we can perform a linear regression between the logarithms of the vertical resolution and the logarithm of the errors, and test that solutions converge towards the analytical solutions.",
"In the Table REF there is a relation of the slopes and correlations for all the three flows, variables and types of errors.",
"The correlations are remarkable high, only two bellow $0.98$ .",
"The convergence rates vary more, and are between $0.872$ (for the maximum error of the vertical velocity in the non-hydrostatic flow) and $4.168$ (for the mean squared error of the vertical velocity in the hydrostatic flow).",
"In all the cases, as expected, the convergence rate of the mean squared error is greater than the respective convergence of the maximum error.",
"In the view of the convergence rates and correlations, we can conclude that the numerical solutions converges towards the analytical solutions."
],
[
"Conclusions",
"The aim of this work has been to find an analytical solution of the waves produced in a flow that passes over an orographic feature with small height-width ratio, in order to test the accuracy and convergence properties of the simulations obtained from a numerical model.",
"Because the solution is stationary, the test is better suited to get insight of those aspects of the model that involve the spatial discretization.",
"For finding the analytical solution we use the covariant formulation of the Euler equations.",
"The coordinate transformation is chosen as the simplest one that makes the lower boundary a coordinate line.",
"After the coordinate transformation, the free slip condition is trivial: the vertical contravariant component of the velocity must be equal to zero.",
"This has the consequence that the orography is not involved in the boundary condition, although it appears in the covariant formulation of the Euler equations in the new coordinates.",
"We could say that the orography has been moved from the boundary condition towards the equations, via a covariant formulation of the Euler equations and a convenient coordinate transformation.",
"Once the orography is not involved in the boundary conditions, and it is only present in the equations, the method goes forward through the linearization of the covariant Euler equations.",
"As the orography is supposed to have low height, the linearization procedure also affects the orographic terms, and only those terms that are linear in the orography are retained.",
"Finally, we obtain and solve a linear system that has forcing terms related to the orography.",
"The solution is written, not only for the vertical velocity, but as well for the other variables, horizontal velocity, density and pressure.",
"We use a non hydrostatic numerical model for verifying the consistency between the analytical and numerical solutions.",
"The experiments are configured so that the domain is cyclic in the horizontal dimension.",
"In the vertical, a sponge layer is placed in the upper part of the domain.",
"The model is initialized with the analytical solution, and a time integration is performed for a long enough period of time.",
"During the time integration, the model evolves the initial condition towards a numerical quasi-stationary solution.",
"The numerical solution at the end of this time integration is compared to the initial analytical solution.",
"The test is configured in three different regimes, hydrostatic and non hydrostatic, and potential flow.",
"It is shown that there is a convergence of the numerical solution towards the analytical solution, when increasing the horizontal and vertical resolutions.",
"The convergence rate varies from one experiment to another, and also from one variable to other.",
"The correlation between the logarithms of the spatial resolution and the errors is very high, showing the consistency of the convergence rates.",
"The use of a covariant formulation of the Euler equations has shown to be a successful method for taking into account the orographic forcing in a stationary flow.",
"This method, which up to our knowledge has not been used before in this context, could be extended for other problems.",
"This exploration is left for a future work."
]
] | 1612.05673 |
[
[
"Formation Scenario of the Progenitor of iPTF13bvn Revisited"
],
[
"Abstract The formation scenario of the progenitor of iPTF13bvn has been revisited.",
"iPTF13bvn is unique in the sense that a corresponding pre-supernova image has been identified.",
"This has enabled us to strongly constrain the nature of its progenitor.",
"From the pre-supernova image, light curve and the latest observations, it is currently widely accepted that the progenitor of iPTF13bvn was in a binary system.",
"The fast decline in the light curve suggests a progenitor mass of $\\sim3.5M_\\odot$, and the upper limit on the remaining companion is $\\sim20M_\\odot$.",
"Recent works suggest that binary evolution models involving common envelope episodes can satisfy all the observational constraints.",
"We have examined the common envelope scenario based on latest knowledges on common envelope evolution.",
"We have found that the common envelope scenario for the progenitor of iPTF13bvn seems not to be suitable since it can not explain the pre-supernova radius.",
"We also propose an alternative model with a black hole companion.",
"Stellar evolution calculations with a large black hole companion were carried out and succeeded in satisfying all observational constraints.",
"However, more studies should be carried out to explain the origin of the large black hole companion."
],
[
"Introduction",
"iPTF13bvn is the only type Ib supernova (SN) so far known to have a corresponding pre-explosion image.",
"Ever since the first detection by [4], this pre-supernova (SN) image combined with information from the light curve has helped us deeply constrain the properties of the progenitor star.",
"Hydrodynamical modelling and analytical fits to the light curve show that the ejecta mass was small, corresponding to a progenitor mass of $\\sim 3$ –$4\\textnormal {M}_\\odot $ [2], , .",
"Such small mass progenitors are difficult to produce with single star evolution models, so it is now widely accepted that the progenitor had undergone binary interactions [2], , , , .",
"There are two main channels of binary interactions; Roche lobe overflow and common envelope (CE) phases.",
"Roche lobe overflow is a form of stable mass transfer, in which a Roche lobe filling star spills some of its mass from the outer envelope to the companion star through the inner Lagrangian point.",
"It is not clear how much of the transferred mass will be retained by the accretor, but the secondary will become massive in general.",
"On the other hand, CE phases are the consequence of unstable mass transfer, where the companion star plunges into the envelope of the evolved primary.",
"If there is enough energy in the system to eject the entire envelope of the primary, it will end up as a close binary consisted of the core of the primary orbiting its companion.",
"If not, the plunged-in star will fall to the centre, and can be regarded as a stellar merger.",
"This process was first introduced to explain the formation of X-ray binaries , but it is now commonly used to explain the formation of many close binaries with compact object components [1], .",
"Earlier works on iPTF13bvn have favoured the stable mass transfer scenario for the formation of the progenitor [2].",
"The initial mass ratio should have been close to unity for the mass transfer to be stable, and in the end the secondary will be much larger as a consequence of the mass accretion.",
"The final companion mass predicted in this scenario was $18\\lesssim m_2/\\textnormal {M}_\\odot \\lesssim 40$ , which will be bright enough so that we will be able to detect it three years after the explosion.",
"But it turned out that the expected companion did not show up, ruling out this scenario .",
"After this observation, the CE scenario has now become the current favourite.",
"has showed some evolutionary models with CE phases which can produce the compact progenitor which is consistent with the pre-explosion image, with a fairly small mass companion.",
"However, the whole process and outcome of CE evolution is still poorly understood, and we should be careful about how we treat CE phases in calculations .",
"In this paper we will revisit the observational constraints on the progenitor of iPTF13bvn, and check the relevance of the CE scenario.",
"We find that the CE scenario has a difficulty in explaining the final radius of the progenitor, which may be critical.",
"We then propose another possible scenario which involves stable mass transfer with a black hole (BH) companion, and show some evolutionary tracks that are consistent with observation.",
"This paper is structured as follows.",
"We will first review the observational constraints on the progenitor of iPTF13bvn in section 2.",
"In section 3 we will reconstrain the progenitor's position on the Hertzsprung-Russel (HR) diagram using the observational data.",
"We will then discuss the relevance of the CE scenario in section 4 and suggest an alternative scenario in section 5.",
"We will summarize and conclude our results in section 6."
],
[
"Summary of Observational data",
"The rich observational data for the SN iPTF13bvn has enabled us to place strong constraints on the properties of the progenitor.",
"In this section we will review and summarize the observational data and the analyses made in previous works."
],
[
"Host Galaxy Properties",
"iPTF13bvn was first discovered by the intermediate Palomar Transient Factory in June 2013, in the galaxy NGC 5806.",
"There is a wide scatter in the estimated host galaxy properties among various groups.",
"For the extinction, [4] suggest a host galaxy colour excess of $E(B-V)$$_\\textrm {host}$$=0.0278$ mag using Na i D absorption lines from their high resolution spectroscopy data.",
"[2] derived a higher reddening value of $E(B-V)$$_\\textrm {host}$$=0.17\\pm 0.03$ mag, assuming an intrinsic colour law based on observational samples by the Carnegie Supernova Project.",
"This was supported by from a different analysis.",
"[2] also measured the Na i D lines and obtained $E(B-V)$$_\\textrm {host}$$=0.07$ or 0.22 depending on the model used.",
"An intermediate value $E(B-V)$$_\\textrm {host}$$=0.08^{+0.07}_{-0.04}$ mag was suggested by , based on an assumption that the intrinsic colour of iPTF13bvn was similar to that of SN2011dh.",
"This is consistent with all other values within the uncertainties.",
"The distance to the galaxy also holds a large uncertainty.",
"Many works in the literature use $22.5_{-3.4}^{+4.0}$ Mpc, $\\mu =31.76\\pm 0.36$ for the distance and distance modulus which are taken from .",
"More recent works use the updated distance of $26.8^{+2.6}_{-2.4}$ Mpc, $\\mu =32.14\\pm 0.20$ by , or the mean value of all estimates $25.8\\pm 2.3$ Mpc, $\\mu =32.05\\pm 0.20$ provided by the NASA/IPAC Extragalactic Database (NED).",
"In this paper, we adopt $E(B-V)$$_\\textrm {host}$$=0.08^{+0.07}_{-0.04}$ mag for the extinction value, and $25.8\\pm 2.3$ Mpc for the distance to the host galaxy."
],
[
"Pre-Explosion Image",
"It was first reported by [4] that they have identified a progenitor candidate at the location of the SN from an observation made by the HST in 2005.",
"Based on their results, various studies were carried out to construct a progenitor model consistent with this pre-SN source and the light curve of the SN itself.",
"Early works showed that they were consistent with a Wolf-Rayet star progenitor [4], , but binary progenitors were also suggested later on [2], , .",
"re-analysed the HST data of the progenitor candidate, and found that the reported magnitude by [4] was lower than that of their analysis by $\\sim 0.7$ mag.",
"Their new magnitude was supported by other following studies , although they seem to have misread the results by .",
"In Fig.",
"REF we compare the fluxes calculated from the reported magnitudes by [4], and .",
"It can be seen that the latter two have some overlaps within the uncertainties, but the three results are rather inconsistent with each other overall.",
"It is not clear why there is such a large discrepancy between the different analyses.",
"It is suggested that the differences of the parameters used in the data reduction may have amplified very small errors .",
"The late-time view of the SN position may help us improve our knowledge of the pre-SN image by refining the background information .",
"Figure: Observed pre-SN flux of the location of iPTF13bvn.",
"Each colour shows the reported flux by Eldridge et al.",
"2015 (red), Folatelli et al.",
"2016 (blue) and Cao et al.",
"2013(black)."
],
[
"Light Curve",
"Another constraint can be placed on the progenitor from the light curve of the SN itself.",
"Although there were very high ejecta mass estimates ($M_\\textrm {ej}\\sim 8\\textnormal {M}_\\odot $ ) in the early works [4], , the relatively fast decline in the light curve of iPTF13bvn showed that the ejected mass should have been small.",
"According to hydrodynamical modelling, the ejecta mass was estimated to be $M_\\textrm {ej}\\approx 2\\textnormal {M}_\\odot $ which indicates that the progenitor was a $M_\\textrm {He}\\approx 3.5\\textnormal {M}_\\odot $ He star [2], .",
"A simple analytical fit also suggested $M_\\textrm {ej}\\sim 1.5$ –$2.2\\textnormal {M}_\\odot $ .",
"These analyses ruled out all single star evolution models.",
"The minimum possible mass achieved by single star models with realistic wind mass-loss rates is $\\sim 8\\textnormal {M}_\\odot $ .",
"The only other way to remove the entire hydrogen envelope up to the observed mass is by binary interactions (in our current understandings)."
],
[
"Other Constraints",
"There are some attempts to infer the zero-age main sequence (ZAMS) mass from late time spectra.",
"The [O I]$\\lambda \\lambda 6300, 6343$ emission lines can be used to estimate the mass of ejected oxygen .",
"By fitting the late time spectrum with ejecta models, the ejected oxygen mass was estimated to be $\\sim 0.3\\textnormal {M}_\\odot $ .",
"associated this mass with a star with ZAMS mass $\\sim 12\\textnormal {M}_\\odot $ based on 1D single star nucleosynthesis calculations by .",
"derived the ZAMS mass to be $\\lesssim 15$ –$17\\textnormal {M}_\\odot $ using nucleosynthesis models by , , .",
"Both values have large uncertainties due to the complexity in modelling the star and explosive nucleosynthesis.",
"For example, all models do not take into account the possible multidimensional effects such as turbulent mixing in the core, that may change the nucleosynthesis yields significantly .",
"Therefore this constraint should be treated carefully when comparing with stellar models.",
"Latest observations by have revealed that the progenitor of iPTF13bvn has disappeared, and also placed an upper limit on the brightness of the possible companion.",
"The magnitudes in each band were $m_\\textrm {F225W}>26.4, m_\\textrm {F435W/F438W}=26.62\\pm 0.14, m_\\textrm {F555W}=26.72\\pm 0.08, m_\\textrm {F814W}=26.03\\pm 0.15$ in June 2016.",
"Especially the strict constraint in the F225W band ruled out most luminous companions as predicted in [2], and they stated that only late-O type stars with masses $\\lesssim 20\\textnormal {M}_\\odot $ are possible assuming that it is not obscured by newly created dust.",
"derived a slightly brighter magnitude from the same data, $m_\\textrm {F438W}=26.48\\pm 0.08$ and $m_\\textrm {F555W}=26.33\\pm 0.05$ ."
],
[
"Methodology",
"Using these constraints, we attempt to pin down the position of the progenitor on the HR diagram.",
"We use the reddening law of [5] for the extinction correction, with the standard coefficient $R_\\textsc {v}=3.1$ and combining the reddening values from the host galaxy and the Milky way foreground .",
"For each combination of luminosity and temperature $(L,T_\\textrm {eff})$ and assuming that the star can be approximated as a black bodyThis is a good approximation for low-mass He star progenitors, since they do not have optically thick winds .",
"We also assume that the flux is dominated by the primary star in the bands considered here.",
"If the binary companion is on the main sequence, it will be optically fainter than the cool envelope of the low-mass progenitor., we can calculate the flux in each band after applying the extinction correction and giving a distance.",
"If there is a consistent combination of $E(B-V)$ and distance within their uncertainties where all three calculated fluxes fit in the error bars of the observation (see Fig.REF ), we consider the combination ($L,T_\\textrm {eff}$ ) is “allowed”.",
"This procedure is similar to the selection process of matching models in , .",
"However, our selection is more strict since we require to find a combination of distance and extinction value that is consistent for all three bands.",
"In the same way we can derive a “forbidden” region for the secondary star.",
"We assume that the data obtained by are upper limits.",
"Then for each combination of ($L,T_\\textrm {eff}$ ), we calculate the flux in each of the four bands assuming that it is a black body.",
"If the flux in any band exceeds the upper limit, we mark the combination as “forbidden”."
],
[
"HR Diagram Constraints",
"Fig.REF shows the calculated allowed regions for the progenitor on the HR diagram, i.e.",
"the progenitor for iPTF13bvn should have been positioned in the shaded region eight years before the explosion.",
"The size and place of the allowed region strongly depends on which observational data are used.",
"It also depends on the assumed host galaxy properties.",
"For example, in Fig.REF we show the same plot but using a smaller distance ($22.5^{+4.0}_{-3.4}$ Mpc) to the host galaxy.",
"Smaller luminosities become allowed obviously because of the closer distance assumed.",
"If we take a larger extinction value, the shape of the region will extend to the upper left direction.",
"It should also be noted that the overlapped region is not particularly favoured because the three reported fluxes are not independent observations, but different analyses performed on the same data.",
"Figure: Allowed regions of the progenitor of iPTF13bvn on the HR diagram.",
"Colours of the shaded regions show the results that fit the observed magnitudes obtained by , and .",
"Lines correspond to constant radii drawn with intervals of 10 R ⊙ _\\odot .Figure: Same as Fig.",
"but calculated using the smaller value (22.5 -3.4 +4.0 22.5^{+4.0}_{-3.4}Mpc) for the distance to the host galaxy.From this analysis only, we can place a stringent constraint on the radius of the progenitor.",
"In Figs.REF and REF , we have overplotted lines of constant radii.",
"Most parts of the allowed region are within 20–70R$_\\odot $ .",
"Since the progenitor mass is constrained to very low masses ($\\sim 3.5\\textnormal {M}_\\odot $ ), the luminosity should not be so high.",
"Therefore, the progenitor had most likely been in the lower right end of the allowed region.",
"This implies that the radius was larger than $\\sim 30\\textrm {R}_\\odot $ .",
"The forbidden regions for the companion calculated from the post-explosion photometry are shown in Fig.REF .",
"With the fiducial set of parameters for the host galaxy ($E(B-V)$$_\\textrm {host}$$=0.08^{+0.07}_{-0.04}$ mag, $25.8\\pm 2.3$ Mpc), main sequence stars larger than $23\\textnormal {M}_\\odot $ can be ruled out.",
"A stricter constraint $m_2<20\\textnormal {M}_\\odot $ can be placed if the host galaxy is closer ($22.5^{+4.0}_{-3.4}$ Mpc), whereas the upper limit goes up to $m_2<29\\textnormal {M}_\\odot $ if the larger extinction value $E(B-V)=0.17\\pm 0.03$ is true.",
"It should be noted that these limits are rather overestimated.",
"The line showed in Fig.REF is the location of ZAMS stars, but the secondary will at least have an age equivalent to the lifetime of the primary.",
"A star on the main sequence evolves slowly to the upper right in the HR diagram, so stars just outside the forbidden region will slide in eventually.",
"Also, suggest that the SN ejecta can drive a shock into the companion star, injecting heat to the outskirts of the envelope.",
"The heat excess will puff up the star to larger radii.",
"This can lower the surface temperature temporarily, taking the star to the right on the HR diagram, which will also strengthen the upper constraint.",
"Having these in mind, we consider that the upper limit $m_2\\lesssim 20\\textnormal {M}_\\odot $ noted by is reasonable.",
"Figure: Forbidden regions of the possible companion star.",
"calculated from the post-explosion photometry combined with the fiducial parameters for the host galaxy (blue+purple), with a larger extinction value (purple) or with a larger distance (light blue+blue+purple).",
"The line indicates the ZAMS stars coloured according to the mass.As we have seen in the previous section, the progenitor of iPTF13bvn was most likely a $\\sim 3.5\\textnormal {M}_\\odot $ He star.",
"Stars that have such a large He core must have had a ZAMS mass of $M_\\textsc {zams}\\gtrsim 10\\textnormal {M}_\\odot $ .",
"This means that the progenitor should have lost at least $\\gtrsim 7\\textnormal {M}_\\odot $ of its matter on the course of its evolution.",
"Single star models have been excluded already because strong stellar winds in the Wolf-Rayet phase is not enough to produce such small progenitors [2], , .",
"This leads us to resort to binary evolution models.",
"If the mass was stripped off via stable mass transfer, the companion star should have grown rather large [2].",
"However, such large companions have been ruled out.",
"The other possible scenario to strip off such a large amount of mass is by experiencing a CE phase.",
"In this way the primary can lose most of its hydrogen envelope, even with relatively small companions.",
"The fact that a CE process is necessary was already suggested by the calculations in .",
"From what we have shown in this section, all the observational facts seem to favour the CE scenario."
],
[
"Common Envelope Scenario",
"We have shown that the progenitor of iPTF13bvn has most likely experienced a CE phase.",
"In this section, we will first briefly review the current status on CE evolution in general.",
"Then we will inspect the CE scenario for the progenitor of iPTF13bvn by modelling the post-CE structures of stars with various ZAMS masses, and checking whether their final position on the HR diagram lies within the allowed region.",
"We use a different treatment for CE evolution from , .",
"Using those results, we also discuss the final separation which is strongly related with the CE efficiency and check the ejectability of the envelope."
],
[
"Common Envelope Evolution",
"The main focus of CE studies is whether or not the system can eject the envelope.",
"In most population synthesis studies, the outcome is estimated by the “energy formalism” or “alpha-formalism”, which is expressed as belows , .",
"$E_\\textrm {env}=\\alpha _\\textsc {ce}\\left(-\\frac{Gm_1m_2}{a_i}+\\frac{Gm_{1,c}m_2}{a_f}\\right)$ $E_\\textrm {env}$ is the binding energy of the envelope, $G$ is the gravitational constant, $m_1, m_2, m_{1,c}$ are the masses of the primary, secondary and core of the primary respectively, $a_i$ and $a_f$ are the initial and final separations respectively.",
"It assumes that as the secondary star plunges into the envelope, the orbital energy is somehow transferred to the envelope to unbind it.",
"The mass of the secondary is assumed to be unchanged before and after the CE phase, because the time-scale of the CE phase is much shorter than the thermal time-scale of the secondary, so there will be almost no accretion , .",
"$\\alpha _\\textsc {ce}$ is a parameter expressing the efficiency of the energy conversion.",
"The value of $\\alpha _\\textsc {ce}$ should be calibrated somehow by observation or theory, but so far there is no guiding principle.",
"Instead, many studies simply take $\\alpha _\\textsc {ce}=1$ or leave it as a free parameter to study the dependences of the resulting populations.",
"The binding energy $E_\\textrm {env}$ is often estimated by $E_\\textrm {env}=\\frac{Gm_1m_\\textrm {1,env}}{\\lambda R_1}$ where $m_\\textrm {1,env}=m_1-m_{1,c}$ is the envelope mass, and $R_1$ is the radius of the primary.",
"$\\lambda $ is another parameter introduced to characterize the structure of the star .",
"Although there are several studies deriving a fitting formula for the value of this parameter , , many studies combine the uncertainties of the two parameters and simply take $\\alpha _\\textsc {ce}\\lambda =1$ with no strong reasoning.",
"Given the masses $m_1$ and $m_2$ , an estimate for the core mass $m_{1,c}$ , the initial separation and a value for the parameters, we can calculate the resulting separation $a_f$ of the binary.",
"The criterion for a successful ejection is that both of the post-CE binary components do not overfill their Roche lobes.",
"However, there are still issues regarding the radius of the post-CE remnant .",
"There are of course some other attempts to understand CE phases such as from observation and simulations.",
"As the secondary plunges into the envelope, it is considered that a small amount of mass is ejected due to the shock created at the interface, and this can be observed as a “luminous red nova”.",
"But the typical ejecta mass is very small, making detections difficult due to the low luminosity.",
"The situation has started to change in the past few years, and now there is a rapidly growing number of candidates for the detection of a CE onset , [6], , [3].",
"However, much more data are required to be able to constrain CE physics from observation.",
"Hydrodynamical simulations have been performed to investigate the internal physics of a CE phase but the huge dynamical range ($\\sim 10^{13}$ ) makes it extremely computationally expensive.",
"Several groups have already attempted large-scale simulations, but it is still hard to extract general features from the small number of models studied , , , , , ."
],
[
"Post-CE Structure",
"The progenitor of iPTF13bvn should have a temperature and luminosity in the allowed region shown in Fig.REF , eight years before the explosion.",
"To see what kinds of stars can end up in this region, we carry out stellar evolution calculations to model the pre-SN state of stars which have experienced CE evolution.",
"All calculations were carried out using the stellar evolution code MESA , , .",
"For convection, we use the mixing length theory, with the Ledoux criterion and a mixing length parameter 1.6.",
"We use the prescription by for the wind mass-loss rate.",
"To create post-CE stellar structures, we follow the procedures taken in .",
"First, we evolve a star until it enters the hydrogen shell burning phase.",
"Once the stellar radius expands up to a certain value, at which we assume the CE phase kicks in, we search for the mass coordinate of the “maximum compression point” in the hydrogen burning shell $m_\\textrm {cp}$ .",
"This is currently assumed to be the best estimate for the bifurcation point of the core and envelope , .",
"After that we give an extremely high mass-loss rate of $0.1\\textnormal {M}_\\odot \\textrm {yr}^{-1}$ artificiallyThis corresponds to a CE timescale of $\\sim 100$ yr, which is the typical CE timescale., and wait until the mass drops to $m_\\textrm {cp}$ .",
"Once the mass has dropped to $m_\\textrm {cp}$ , we switch off the artificial mass-loss and evolve the star until it starts burning neon at the centre.",
"A star burning neon will explode within a few more days.",
"The radius at which we start the artificial mass loss is not so important since the time-scale of the expansion is smaller than the time-scale of the core mass growth.",
"This can be checked in Fig.REF where we show an example of the evolution of the radius and the core mass in the late stages.",
"The core mass increases by only $\\sim 1\\%$ during the expansion.",
"Fig.REF is for a 17$\\textnormal {M}_\\odot $ star, but the same applies to all stars in the mass range we used.",
"Figure: Time evolution of the radius and m cp m_\\textrm {cp} for a 17M ⊙ \\textnormal {M}_\\odot star with metallicity Z=0.02Z=0.02.In Fig.REF we show the evolutionary tracks of our post-CE stars with an initial metallicity $Z=0.02$ .",
"All stars follow similar tracks from ZAMS to the red giant phase (dashed line).",
"After that we switch on the artificial mass-loss, and at the end of the CE phase all stars end up on the left end of the HR diagram.",
"Then the stars evolve towards core-collapse.",
"The lighter stars ($M_\\textsc {zams}=$ 15–16$\\textnormal {M}_\\odot $ ) evolve from left to right, crossing over the allowed region and then follow very complex paths.",
"This complex evolution is probably not real, so we simply stop our calculation after the track has moved away significantly.",
"The heavier stars ($M_\\textsc {zams}=$ 17–19$\\textnormal {M}_\\odot $ ) also evolve with constant luminosity from left to right, but starts to collapse somewhere on the way towards the allowed region.",
"We only plot up to eight years before collapse, since the pre-SN image for iPTF13bvn was taken eight years before explosion.",
"Only the $17\\textnormal {M}_\\odot $ model ended up in the allowed region in our mass range.",
"However, the final temperature – or radius – is very sensitive to the details of the calculation such as the mixing length or overshoot parameters or metallicity, so we can not derive a concrete conclusion about the best mass range.",
"For example in Figs.REF and REF we show the evolutionary tracks for our lower and higher metallicity models.",
"With lower metallicity the expansion of the stars are smaller than in Fig.REF , and somewhere between 15 and 16$\\textnormal {M}_\\odot $ seems to be the matching model.",
"Higher metallicity led to larger expansion, and the mass of the matching models increases.",
"Figure: Evolutionary tracks of stars with a metallicity Z=0.02Z=0.02 on the HR diagram.",
"Dashed lines are for before the CE phase, and solid lines are for after the CE phase up to eight years before collapse.",
"The shaded regions are taken from Fig..Figure: Same as Fig.",
"but with a metallicity Z=0.01Z=0.01.Figure: Same as Fig.",
"but with a metallicity Z=0.04Z=0.04.The ZAMS masses of our matching models are rather heavier than the matching models in .",
"This may be due to the different treatments of the CE evolution.",
"In their BPASS code, they use the usual RLOF rate but limit it by $\\dot{M}=10^{-3}\\textnormal {M}_\\odot $ for the mass-loss rate during CE evolution , and terminate when both stars reside within their Roche lobes.",
"Their choice for the upper limit value is due to numerical reasons, and not motivated physically.",
"A CE phase is a highly dynamical process, and the usual mass-loss rates that were derived assuming nuclear time-scale processes do not describe the dynamical nature of CE evolution well.",
"The lower mass-loss rate will lead to a longer time-scale for the CE phase, giving more time for the core to grow.",
"Together with their different termination criterion, their method will always leave a larger remnant than ours, which may possibly explain the discrepancy of the results.",
"It should also be noticed that the ZAMS masses of our matching models are within the range estimated from the nebular phase oxygen lines.",
"But because in the CE scenario we remove the envelope before the core grows to its full size, the final ejected oxygen may be smaller than what we would expect from a progenitor with that ZAMS mass.",
"Although we have a matching model, the final temperatures in the stellar evolution calculations are not so reliable, so we will not conclude which models are the best.",
"On the other hand, the luminosity is almost constant in the final stages, which is strongly correlated with the core mass.",
"From the lower limit of the luminosity of the allowed region, we can place a rough lower limit $\\sim 2.5\\textnormal {M}_\\odot $ on the core mass of the progenitor."
],
[
"Pre-CE Separation",
"Here we will discuss the upper limit to the initial orbital separation of the progenitor system in the context of the CE scenario.",
"There are two pathways known so far to initiate a CE phase.",
"The first is via unstable mass transfer.",
"Once the primary star fills its Roche lobe, a part of the outer envelope of the star will be transferred to the secondary through the inner Lagrangian point.",
"This is the usual Roche lobe overflow.",
"If the mass transfer is unstable, the star will eventually overfill the second Lagrangian point (only the primary component).",
"Then a part of the envelope material will start trickling away from the system.",
"This flow will take away angular momentum, shrinking the orbit even more, leading to a CE phase.",
"The onset of an unstable mass transfer is usually computed by comparing the volume of the star with the primary component of the volume enclosed within the equipotential surface passing through the second Lagrangian point.",
"The effective radius of this volume $R_{\\textrm {L}_2}$ can be approximated by $\\frac{R_{\\textrm {L}_2}}{a}&\\approx &\\frac{0.49q^{2/3}+0.27q-0.12q^{4/3}}{0.6q^{2/3}+\\ln (1+q^{1/3})},\\;\\;\\; q\\leqq 1\\\\&\\approx & \\frac{0.49q^{2/3}+0.15}{0.6q^{2/3}+\\ln (1+q^{1/3})},\\;\\;\\; q\\geqq 1$ where $a$ is the binary separation, $q\\equiv {m_1}/{m_2}$ and $m_1, m_2$ are the primary and secondary masses .",
"Thus the criterion for unstable mass transfer will be $R>R_{\\textrm {L}_2}$ where $R$ is the radius of the primary.",
"The other path is via Darwin instability , .",
"This occurs when the tidal forces extract orbital angular momentum to spin up the stars, but there is not enough angular momentum left in the orbit to do so.",
"The condition for this instability is $J_\\textrm {spin}>\\frac{1}{3}J_\\textrm {orb}$ where $J_\\textrm {spin}$ is the moment of inertia of the primary star and $J_\\textrm {orb}$ is the moment of inertia of the orbit.",
"$J_\\textrm {orb}$ can be expressed as $J_\\textrm {orb}=\\mu a^2$ where $\\mu =m_1m_2/(m_1+m_2)$ is the reduced mass.",
"In either of the cases, the CE phase will be initiated at the time when the primary star evolves to a red giant, and is rapidly expanding in size.",
"Both the radius and moment of inertia of the star grow rapidly at this stage, and will eventually satisfy one of the criteria above, depending on the secondary mass.",
"If $m_2$ is relatively large, $J_\\textrm {orb}$ will be large, so the system is unlikely to be Darwin unstable and thus enters the CE phase via unstable mass transfer.",
"The maximum possible separation for unstable mass transfer to occur can be estimated by $a_{\\textrm {max,L}_2}\\approx R_\\textrm {max}\\frac{0.6q^{2/3}+\\ln (1+q^{1/3})}{0.49q^{2/3}+0.15}$ where $R_\\textrm {max}$ is the maximum radius achieved in single star evolution.",
"If $m_2$ is relatively small, $J_\\textrm {orb}$ is small and the system will be Darwin unstable before the primary overflows the L$_2$ point.",
"The maximum possible separation to be Darwin unstable $a_\\textrm {max,DI}$ can be estimated by $a_\\textrm {max,DI}\\approx \\sqrt{\\frac{3J_\\textrm {spin,max}}{\\mu }}$ where $J_\\textrm {spin,max}$ is the maximum moment of inertia obtained in the single evolution models.",
"In Fig.REF we show the maximum possible orbital separation as a function of secondary mass.",
"We have used $J_\\textrm {spin,max}$ and $R_\\textrm {max}$ obtained from single star evolution calculations with metallicity $Z=0.02$ .",
"The maximum separation is $\\sim 1000$ –$1800\\textnormal {R}_\\odot $ throughout most of the mass range, which is determined by the L$_2$ overflow criterion.",
"Larger separations would be possible only if the secondary mass was smaller than $\\sim 4\\textnormal {M}_\\odot $ .",
"Figure: Maximum orbital separation as a function of the secondary mass M 2 M_2.",
"Line colours express the primary mass, with the same colours as in Fig."
],
[
"CE Efficiency parameter",
"We will now constrain the $\\alpha _\\textsc {ce}$ parameter in this system to discuss the ejectability of the envelope.",
"In usual population synthesis calculations, $\\alpha _\\textsc {ce}$ is given by hand, to calculate the final separation $a_f$ .",
"We will go the other way round, and use the constraints on $a_f$ to calculate a lower limit to $\\alpha _\\textsc {ce}$ .",
"Eq.REF can be rewritten as $\\alpha _\\textsc {ce}&=&E_\\textrm {env}\\left(-\\frac{Gm_1m_2}{a_i}+\\frac{Gm_\\textrm {cp}m_2}{a_f}\\right)^{-1}\\nonumber \\\\&\\ge &\\frac{E_\\textrm {env}a_f}{Gm_2m_\\textrm {cp}}$ The inequality can almost be regarded as an equality because the initial separation is usually much larger than the final separation, and thus the first term in the parenthesis can be ignored.",
"We have a rough estimate on $m_\\textrm {cp}$ from the observed ejecta mass.",
"Therefore the important values that determines $\\alpha _\\textsc {ce}$ are $E_\\textrm {env}$ and $a_f$ .",
"The binding energy of the envelope is usually estimated by $E_\\textrm {env}=-\\int _{m_\\textrm {cp}}^{m_1}\\left(-\\frac{Gm}{r}+\\epsilon \\right)dm$ where $m_1$ is the total mass of the star and $\\epsilon $ is the specific internal energy.",
"But in order to take into account the relaxation of the core after the mass ejection, it should be calculated by comparing the total binding energies of the star before and after the CE event .",
"$E_\\textrm {env}&=E_{\\textrm {bind},i}-E_{\\textrm {bind},f}\\nonumber \\\\&=-\\int _0^{m_{1,i}}\\left(-\\frac{Gm}{r}+\\epsilon \\right)dm+\\int _0^{m_{1,f}}\\left(-\\frac{Gm}{r}+\\epsilon \\right)dm$ Here the integration is taken over the whole star before (first term) and after (second term) the CE phase.",
"For the model CE calculations in section REF , the values calculated by Eq.REF overestimated the binding energy by $\\sim 10\\%$ .",
"We use the values calculated by Eq.REF in our following discussions.",
"The final separation is quite uncertain.",
"The closest possible separation is when the post-CE primary star (and of course the secondary) does not overfill its Roche lobe.",
"Using the post-CE radius obtained in the model CE simulations, we can calculate the minimum possible separation by assuming that one of the binary components exactly fills its Roche lobe.",
"This can be expressed as $a_{f,\\textrm {min}}=\\max \\left(\\frac{R_f}{f(q)},\\frac{R_2}{f(q^{-1})}\\right)$ where $R_f$ is the post-CE radius of the primary star, $R_2$ is the secondary radius, and $q=m_\\textrm {cp}/m_2$ .",
"$f(q)$ is a function fitted to the approximate Roche lobe radius .",
"$f(q)\\equiv \\frac{0.49q^{2/3}}{0.6q^{2/3}+\\log (1+q^{1/3})}$ We can obtain the lower limit to $\\alpha _\\textsc {ce}$ as a function of the secondary mass by plugging in $E_\\textrm {env}$ and $a_{f,\\textrm {min}}$ into Eq.REF .",
"This is shown in the lower panel of Fig.REF .",
"We have used the ZAMS radius for $R_2$ .",
"The minimum value in our calculations was $\\sim 0.5$ , which means that at least half of the orbital energy should be used to eject the envelope.",
"The limit increases as the secondary mass decreases because of the decrease of the energy reservoir in the orbit.",
"If the secondary was $\\lesssim 6\\textnormal {M}_\\odot $ , the lower limit exceeds unity, which suggests the presence of an extra energy source to eject the envelope.",
"All models were limited by the secondary star radius filling its Roche lobe.",
"If we consider a compact object as the companion, the final separation will be limited by the post-CE radius of the primary, and $\\alpha _{\\textsc {ce},\\textrm {min}}$ will be smaller by a factor of $\\sim 5$ .",
"Figure: Lower limits on the α c e\\alpha _\\textsc {ce} parameter based on the assumption that the secondary never interacted with the primary again (upper panel), or the secondary has already been lost (lower panel)."
],
[
"Deficits of the CE scenario",
"So far the CE scenario seems successful, since the system can eject the envelope with an efficiency smaller than unity $\\alpha _\\textsc {ce}<1$ if the companion was larger than $\\sim 6\\textnormal {M}_\\odot $ .",
"However, this scenario has a difficulty in explaining the post-CE evolution of the binary.",
"Since the final radius of the progenitor should be larger than $\\sim 30\\textnormal {R}_\\odot $ (see Fig.REF and REF ), which is much larger than the values of $a_f$ calculated above, it is almost impossible to avoid a second CE phase.",
"The outcome of a CE phase with a naked helium star is not known.",
"But if the binary can successfully eject the envelope againThis may lead to the ejection of the whole helium envelope, which may change the spectral type of SN to type Ic., it will surely shrink the orbit even more.",
"The primary will not be able to re-expand to $\\sim 30\\textnormal {R}_\\odot $ this way.",
"Therefore the second CE phase should have failed and the secondary star will have been engulfed by the primary before SN explosion.",
"If the secondary was a main sequence star, there will be a substantial amount of fresh hydrogen injected to the core of the primary.",
"This can significantly alter the appearance of the progenitor, taking it away from the allowed region and also may change the spectral type of the SN.",
"The mass of the secondary should also be very small in order to keep the ejecta mass smaller than $\\lesssim 2\\textnormal {M}_\\odot $ .",
"But the first CE phase will not have succeeded in the first place if the mass was so small, unless the $\\alpha _\\textsc {ce}$ parameter is considerably large.",
"Therefore the secondary should avoid the second CE phase or be completely lost before SN.",
"In order to avoid the second CE phase, the post-CE separation should be large enough so that the primary never interacts with the secondary again.",
"In the upper panel of Fig.REF , we show the minimum $\\alpha _\\textsc {ce}$ required to have a large enough post-CE separation so that the Roche lobe radius for the primary becomes $30\\textnormal {R}_\\odot $ .",
"The required value for $\\alpha _\\textsc {ce}$ becomes $\\gtrsim 6$ even for the largest possible secondary masses, which is very unlikely even with the consideration of other energy sources such as recombination energy.",
"The primary may have lost its companion because of a third body encounter, but this may also be difficult considering the very tight post-CE orbit.",
"Unless we resolve this problem, the CE scenario should be refuted.",
"To sum up, the CE scenario is able to reproduce the observed ejecta mass, pre- and post-SN photometry.",
"However, the success of this model requires a significant orbital shrinkage, which will suffer a second CE phase before SN.",
"The second CE phase will ruin the advantages of this model by increasing the ejecta mass and altering the pre-SN photometry.",
"The ejected oxygen mass may also be smaller than the observed amount.",
"Therefore we conclude here that the CE scenario is not suitable to explain the formation of the progenitor of iPTF13bvn.",
"From the previous discussion, the CE scenario seems not to be suitable as the formation scenario of the progenitor of iPTF13bvn.",
"Here we will return to the stable mass transfer scenario again.",
"The reason that we have excluded this scenario in the first place was the non-detection of a companion.",
"A sufficiently large companion is needed to enable stable mass transfer, and the star will also grow due to the accretion of transferred mass.",
"However, this is only problematic if the companion is on the main sequence.",
"The situation will be completely different if the secondary was a BH, since we can not observe a BH whatever the mass is unless it has an accretion disc around it.",
"Here we will demonstrate that a binary with a BH component can evolve up to SN without conflicting with any of the observational constraints.",
"We used the binary module in MESA, and simulated the evolution of a $16\\textnormal {M}_\\odot $ star with a $15\\textnormal {M}_\\odot $ BH companion in an 8 day circular orbit.",
"The metallicity is assumed to be $Z=0.02$ .",
"The mass transfer rate was calculated according to the prescription by and the mass retention on the BH was limited by the Eddington limit.",
"The evolutionary track of the primary is shown in Fig.REF , overplotted on the allowed region again.",
"This system undergoes a case B mass tranfer, losing most of its hydrogen envelope during this phase.",
"When the remaining hydrogen becomes sufficiently small, the star contracts rapidly and detaches from the BH.",
"Most of the remaining hydrogen is burned in the H burning shell and only $\\lesssim 0.04\\textnormal {M}_\\odot $ is left by the time of SN.",
"This small amount of hydrogen may be the origin of the weak H$\\alpha $ lines in the early spectra .",
"The endpoint of the evolution rests in the allowed region, which makes this system a good candidate for the progenitor of iPTF13bvn.",
"There is almost no change in the mass of the BH, only growing by $\\sim 0.017\\textnormal {M}_\\odot $ , meaning that most of the mass has been lost from the system.",
"The overall evolution of the primary does not change largely even if we increase the mass of the BH up to $\\sim 100\\textnormal {M}_\\odot $ .",
"Figure: Evolutionary track of a 16M ⊙ 16\\textnormal {M}_\\odot star with a 15M ⊙ 15\\textnormal {M}_\\odot BH companion.",
"The star symbol marks the position eight years before SN.",
"The red part of the curve indicates the mass transfer phase.This demonstration is only an example of the evolutionary path, and there is a wider range of possible initial parameters.",
"The primary mass should be in the range $\\sim 14$ –17$\\textnormal {M}_\\odot $ to create a He core in the mass range of the observed ejecta mass, create the right amount of oxygen, and have a final luminosity consistent with the pre-SN fluxes.",
"On the other hand, there is no strong constraint on the BH mass because it does not largely affect the evolution of the primary.",
"The only constraint is the lower limit which is roughly $\\sim 0.8$ times the primary mass, to enable stable mass transfer.",
"The initial period range should be roughly 4–20 days for the mass transfer to initiate in case B, although we can not rule out case A mass transferring modelsThe mass range discussed here are quite sensitive to the parameters and assumptions applied to the stellar evolution code such as the mixing length, overshoot parameters or convection criteria..",
"The largest uncertainty in this model is the origin of the BH.",
"For example, the BH could have been an extremely massive star ($\\gtrsim 30\\textnormal {M}_\\odot $ ), in a relatively wide orbit with a $\\sim 14$ –$17\\textnormal {M}_\\odot $ companion.",
"As the more massive star evolves, it develops a $\\gtrsim 15\\textnormal {M}_\\odot $ He core, and expands to $\\sim 2000\\textnormal {R}_\\odot $ .",
"This may initiate a CE episode, and because the core mass is large, the post-CE separation is at moderate distances $\\sim 50\\textnormal {R}_\\odot $ .",
"At some point the massive He core will collapse to a BH and then the system will follow an evolution similar to that in the above demonstration.",
"There is no strong support to this scenario, and deeper investigations should be carried out to check the relevance of this model.",
"We will leave this to future works.",
"Since the ejecta mass is much smaller than the expected BH mass, the system will still be bound after the SN explosion, and the outcome of this model will be a BH-NS binary in a relatively wide orbit ($\\sim 100\\textnormal {R}_\\odot $ ).",
"Thus we expect that we will not find a companion star in any future observations.",
"The orbital separation is too wide to cause a BH-NS merger in a realistic time-scale, leaving no hope for gravitational wave detection.",
"It will be extremely difficult to confirm our scenario, but the non-detection of a companion star in the next few years can strengthen our hypothesis."
],
[
"Conclusions",
"The observational constraints on the progenitor of iPTF13bvn have been revisited.",
"We evaluated the possible position on the HR diagram and constrained the photospheric radius of the progenitor.",
"The radius should have been in the range $\\sim $ 20–70$\\textnormal {R}_\\odot $ .",
"All studies now agree that the progenitor should have been in a binary system, and expect to be able to detect a companion star in the future.",
"We have derived the upper limit on the remaining secondary star based on the latest observational data of the SN and obtained similar results to previous works ($\\sim 20\\textnormal {M}_\\odot $ ).",
"But this is probably much smaller if we consider the effects of SN ejecta-companion interaction as discussed in .",
"We have also reassessed the relevance of the formation scenario of the progenitor via a CE phase.",
"We performed stellar evolution calculations to mimick the post-CE evolutionary tracks just for the donor, and found a model that matches the observational constraints.",
"However, if we consider the energy budget in the CE phase, an extremely large companion or a very high CE efficiency ($\\gtrsim 6$ ) is required to avoid a second CE phase, which is unrealistic.",
"Therefore we conclude that the CE scenario is unlikely to be the formation scenario for the progenitor of iPTF13bvn.",
"As an alternative model, we considered the evolution of a binary with a BH component.",
"Stable mass transfer from the primary star to a BH can strip off most of the hydrogen envelope up to the edge of the He core.",
"We have demonstrated one example evolutionary track that satisfies all observational constraints.",
"We roughly estimate that the mass of the primary should be in the range $\\sim 14$ –$17\\textnormal {M}_\\odot $ and the BH should be heavier than 0.8 times the primary mass.",
"The orbital period should be $\\sim $ 4–20 days.",
"There is still no quantitative support on the origin of the BH.",
"The system may have experienced a CE phase of a much larger star, but it remains a matter of speculation.",
"We will leave the investigation to future works.",
"It is almost impossible to confirm our scenario by future observations because the expected outcome is a wide BH-NS binary.",
"However, if there is no detection of a companion star in the coming few years, we believe it will be a strong support of our model."
],
[
"Acknowledgements",
"The author would like to thank Shoichi Yamada, Yu Yamamoto, Keiichi Maeda, Yuichiro Sekiguchi and Pablo Marchant for useful discussions.",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"This work was supported by JSPS Research fellowship for young scientists (DC2, No 16J07613)."
]
] | 1612.05666 |
[
[
"Quantum simulation of ultrastrongly coupled bosonic modes using\n superconducting circuits"
],
[
"Abstract The ground state of a pair of ultrastrongly coupled bosonic modes is predicted to be a two-mode squeezed vacuum.",
"However, the corresponding quantum correlations are currently unobservable in condensed matter where such a coupling can be reached, since it cannot be extracted from these systems.",
"Here, we show that superconducting circuits can be used to perform an analog simulation of a system of two bosonic modes in regimes ranging from strong to ultrastrong coupling.",
"More importantly, our quantum simulation setup enables us to detect output excitations that are related to the ground-state properties of the bosonic modes.",
"We compute the emission spectra of this physical system and show that the produced state presents single- and two-mode squeezing simultaneously."
],
[
"Introduction",
"Theoretically predicted more than a decade ago for two-dimensional electron gases [1], and later for superconducting circuits [2], ultrastrong coupling is a fascinating regime of light-matter interaction.",
"In strong coupling, quantum systems are coupled at a higher rate than any dissipation process, while in ultrastrong coupling they are coupled at a rate that is non-negligible even compared to the dynamics of each system taken separately.",
"As a consequence, the rotating-wave approximation cannot be performed and all the terms of the coupling Hamiltonian should be, a priori, considered.",
"These terms play an important role in the properties of the system [1], [3], and the number of excitations is not conserved throughout the dynamics.",
"Hence, the ground state is deeply modified.",
"In recent years, this regime has been experimentally achieved in various physical systems: first, in cavity-embedded semiconductor quantum wells [4], [6], [5], where the ultrastrong coupling was originally predicted, as well as in superconducting circuits [7], [8], [9], [10], [11], and in cavities confining molecules [12], [13], [14].",
"Quantum simulation of ultrastrong coupling, or even deep strong coupling [15], has recently received growing interest as the only way to probe dynamics or exotic features that are currently out of reach in genuine physical systems.",
"In analog quantum simulators, the implementation relies on properly driven strongly interacting systems that effectively behave as ultrastrongly coupled modes and exhibit the corresponding characteristic features.",
"Previous theoretical works have focused on simulating the interaction between a qubit and a cavity mode, namely, the quantum Rabi model, in several physical systems, with, for instance, proposals in light transport in photonic superlattices [16], in superconducting circuits [17], [18], [19], in cavity quantum electrodynamics [20], in trapped ions [21], and in ultracold atoms where the first and second Bloch bands in the first Brillouin zone encode the qubit [22].",
"On the experimental side, quantum simulations of the Rabi model in the ultrastrong and deep strong coupling regimes have been reported in photonic superlattices [23], and in superconducting circuits [24], [25].",
"Surprisingly, the important case of a quantum simulation of ultrastrongly coupled bosonic modes is still missing.",
"Remarkably, the ultrastrong interaction between two bosonic modes has the particularity of producing a two-mode squeezed vacuum in the ground state [1], [3].",
"However, this squeezed state cannot lead to actual excitations coming out of the system and, thus, cannot be directly observed.",
"In the case of spin-boson ultrastrong coupling, methods to probe the ground-state properties of the system were proposed in Refs.",
"[26], [27], [28].",
"Nonetheless, in the case of two ultrastrongly coupled bosonic modes, the only studied solution to the problem is to modulate the coupling between the two modes in time [29], [30].",
"However, measuring the corresponding correlations between the two output channels of both bosonic modes seems currently out of reach for physical implementations of light-matter coupling [4], [6], [5] since matter excitations (for instance the electron gas of quantum wells) decay through a nonradiative channel.",
"Because of these serious experimental issues, a quantum simulation of ultrastrongly coupled bosonic modes is timely.",
"In this paper, we propose a way to realize it using the three-wave mixing process of a superconducting device.",
"The Josephson mixer is made of a ring of four Josephson junctions that couples two microwave resonators [31], [32].",
"It has been demonstrated to act as a microwave amplifier near the quantum limit [33], [34], to generate two-mode squeezed vaccum shared between two traveling or stationary microwave modes [35], [36], to realize coherent frequency conversion [37] and to act as a circulator or directional amplifier [38].",
"The versatility of the Josephson mixer and the ease of measuring its two output channels make it an ideal platform to perform quantum simulations.",
"Of particular relevance to our goal, it enables us to fully characterize the squeezing of its two output transmission lines.",
"We propose here to drive this device in such a way that, in a particular rotating frame, its effective Hamiltonian is formally equivalent to the boson-boson ultrastrong coupling Hamiltonian.",
"The peculiar properties of the simulated ground state lead to squeezing of the physically observable output modes in the laboratory frame.",
"We predict the emission of an unusual two-mode output state, where both modes exhibit single-mode squeezing and, additionally, have quantum correlations between them.",
"The paper is organized as follows.",
"First, we briefly characterize the squeezing properties of two ultrastrongly coupled bosonic modes in the ground state.",
"In particular, we show that this ground state differs from a two-mode squeezed vacuum.",
"Next, we make our model explicit, showing how driving the Josephson mixer leads to an effective Hamiltonian that simulates ultrastrong coupling.",
"Then we predict the squeezing in the emission spectrum of the system, with realistic parameters.",
"Finally, we discuss the physical meaning of the emitted squeezing, in particular, by interpreting the results in terms of two equivalent models, each one having a different environment."
],
[
"Ground state squeezing",
"Let us start by studying the squeezing of a pair of ultrastrongly coupled bosonic modes in their ground state.",
"For this we introduce the light-matter Hamiltonian we want to simulate, $\\hat{H} = \\omega _{\\alpha } \\hat{a}^{\\dagger } \\hat{a} + \\omega _{\\beta } \\hat{b}^{\\dagger } \\hat{b} + G (\\hat{a} + \\hat{a}^{\\dagger }) (\\hat{b} + \\hat{b}^{\\dagger }),$ where $\\hat{a}$ ($\\hat{b}$ ) and $\\omega _\\alpha $ ($\\omega _\\beta $ ) are the annihilation operator and the frequency of the light (matter) mode, and $\\hbar =1$ .",
"The two modes are coupled at a rate $G$ .",
"We consider here the Hamiltonian (REF ) in its most elementary form.",
"For instance, we do not include extra terms such as a squared electromagnetic vector potential, as is the case in semiconductors described in the Coulomb gauge [1], [3].",
"Indeed, while the versatility of superconducting circuits would allow us to simulate extra terms, we choose to restrict the simulation to the simplest form of ultrastrong coupling in the present paper [39].",
"In order to identify the ground state of the Hamiltonian (REF ), we first apply the Hopfield method [40] to identify the two eigenmodes of the system, which are called polaritons in case of a genuine light-matter interaction.",
"The annihilation operators $\\hat{p}_1$ and $\\hat{p}_2$ of the two eigenmodes are expressed [1], [3] as linear combinations of $\\hat{a}, \\hat{b}, \\hat{a}^\\dagger $ and $\\hat{b}^\\dagger $ ; this is a Gaussian operation.",
"Their expressions as well as their eigenvalues determine the validity of this model (see Appendix ).",
"We then express $\\hat{a}$ and $\\hat{b}$ as a function of the eigenmode operators $\\hat{p}_1$ and $\\hat{p}_2$ .",
"The ground state $\\vert \\text{GS} \\rangle $ being defined as $\\hat{p}_1 \\vert \\text{GS} \\rangle =\\hat{p}_2 \\vert \\text{GS} \\rangle =0$ , we can fully characterize the squeezing of the original modes $a$ and $b$ in the ground state $\\vert \\text{GS} \\rangle $ by computing the covariance matrix $\\mathcal {V}=\\lbrace \\langle x_ix_j + x_jx_i \\rangle _{\\vert \\text{GS} \\rangle }-2\\langle x_i \\rangle _{\\vert \\text{GS} \\rangle }\\langle x_j \\rangle _{\\vert \\text{GS} \\rangle }\\rbrace _{ij}$ in the basis $\\lbrace x_1,x_2,x_3,x_4\\rbrace =\\lbrace (\\hat{a}^\\dagger +\\hat{a})/\\sqrt{2},(i\\hat{a}^\\dagger -i\\hat{a})/\\sqrt{2},(\\hat{b}^\\dagger +\\hat{b})/\\sqrt{2},(i\\hat{b}^\\dagger -i\\hat{b})/\\sqrt{2}\\rbrace $ [41], [42].",
"Figure: Squeezing of ultrastrongly coupled aa and bb modes in the ground state, as a function of the coupling rate GG.",
"Red dotted line: single mode squeezing of the quadrature X ^ a \\hat{X}_{a}.",
"Here the modes are degenerate ω α =ω β \\omega _\\alpha =\\omega _\\beta , therefore the squeezing of X ^ b \\hat{X}_b is the same as of X ^ a \\hat{X}_{a}.",
"Gray dashed line: two-mode squeezing of the collective quadrature X ^ a -X ^ b \\hat{X}_a-\\hat{X}_b.",
"Inset: EPR variance ΔEPR=Δ(X a -X b ) 2 +Δ(Y a +Y b ) 2 \\Delta \\text{EPR}= \\Delta (X_a-X_b)^2 + \\Delta (Y_a+Y_b)^2.In Fig.",
"REF , we show the single-mode squeezing, the two-mode squeezing, and the EPR variance (a measure of entanglement) in the ground state of a pair of ultrastrongly coupled bosonic modes, as a function of the coupling constant $G$ .",
"Specifically, we show the squeezing in dB using the following logarithmic scale $S_{X_{\\theta }} = 10 \\log _{10}{( \\langle \\Delta \\hat{X}_{\\theta }^2 \\rangle / \\langle \\Delta \\hat{X}_{\\text{vac}}^2 \\rangle )}$ , where $\\langle \\Delta \\hat{X}_{\\text{vac}}^2 \\rangle =1/2$ corresponds to the noise of a vacuum state.",
"We use the definitions $\\langle \\Delta \\hat{X}_{\\theta }^2 \\rangle = \\langle \\hat{X}_{\\theta }^2 \\rangle - \\langle \\hat{X}_{\\theta } \\rangle ^2$ , $\\hat{X}_{\\theta } = (e^{-i \\theta } \\hat{a} + e^{i \\theta } \\hat{a}^\\dagger )/\\sqrt{2}$ , with $\\hat{X}_{\\theta =0}=\\hat{X}$ and $\\hat{X}_{\\theta =\\pi /2}=\\hat{Y}$ .",
"One can note that the ground state shows a significant amount of squeezing in the single-mode picture, as well as in the two-mode picture, enough to be detected by a Gaussian entanglement witness: the EPR variance goes below 1 [43], [44].",
"Note that since here the two modes are at resonance, only $\\Delta \\hat{X}_a^2$ is shown, because $\\Delta \\hat{X}_b^2$ has exactly the same amount of squeezing.",
"We thus verified that there is two-mode squeezing in the ground state, and additionally found single-mode squeezing as well.",
"Note that it is possible to intuitively predict squeezing in the Hamiltonian (REF ), by rewriting it in terms of particular collective operators (see Appendix ).",
"Figure: Scheme of a possible implementation based on a Josephson mixer .",
"A ring of four Josephson junctions is shorted by inductors and couples two λ/2\\lambda /2 microwave resonators of frequency ω a \\omega _a and ω b \\omega _b.",
"Capacitors couple the resonators to transmission lines leading to decay rates γ a +γ L \\gamma _a+\\gamma _L and γ b +γ L \\gamma _b+\\gamma _L, where γ L \\gamma _L corresponds to internal losses of the resonators.",
"This circuit implements three-wave mixing between the nondegenerate modes aa and bb and a mode cc that can be addressed using a signal driven with the same phase on each port of the resonator aa.",
"One may use a 180 ∘ 180^\\circ hybrid coupler (box on the left) to selectively couple aa and cc modes to two separate transmission lines.",
"Circulators ensure that the input modes a in a_\\mathrm {in} and b in b_\\mathrm {in} are prepared in the vacuum state by thermalizing a 50Ω50~\\Omega load at T≪ℏω a,b /k B T\\ll \\hbar \\omega _{a,b}/k_B.",
"When mode cc is driven off resonance by two tones at frequency ω B =ω a +ω b +2δ\\omega _B=\\omega _a+\\omega _b+2\\delta and ω R =ω a -ω b \\omega _R=\\omega _a-\\omega _b, it reproduces the physics of two ultrastrongly coupled bosonic modes of frequency δ\\delta .",
"Signatures of the ultrastrong coupling can be observed in the squeezing properties of the noise in ports a out a_\\mathrm {out} and b out b_\\mathrm {out}."
],
[
"Modeling the quantum simulation",
"In this section we present the model of our quantum simulation, which could be used as a tool to measure the squeezing of the output field extracted from the system in its ground state.",
"As mentioned in previous sections, our model is based on the Josephson mixer (Fig.",
"REF ) [32], where the interaction Hamiltonian of the three-wave mixing process reads, $\\hat{H}_{\\text{int}} = \\chi ( \\hat{c} + \\hat{c}^{\\dagger }) ( \\hat{a} + \\hat{a}^{\\dagger } ) ( \\hat{b} + \\hat{b}^{\\dagger }),$ where $\\hat{c}$ , $\\hat{a}$ , and $\\hat{b}$ are the annihilation operators of the spatially separated pump, $a$ and $b$ microwave modes, respectively.",
"This purely three-wave mixing Hamiltonian is close to what the circuit in Fig.",
"REF can realize for a well chosen value of the magnetic flux threading the inner loops of the Josephson ring.",
"For more details on how to obtain the system interaction Hamiltonian (REF ) from the general Hamiltonian describing the Josephson mixer, as well as on the measurement process of the outputs of modes $a$ and $b$ , we refer the reader to Ref. [45].",
"To generate an effective Hamiltonian that is formally equivalent to Eq.",
"(REF ), we drive the system with a two-tone radiation.",
"A blue pump drives mode $c$ with an amplitude $c_B$ at frequency $\\omega _{B} = \\omega _{a} + \\omega _{b} + 2 \\delta $ , while a red pump drives the same mode $c$ with an amplitude $c_R$ at $\\omega _{R} = \\omega _{a} - \\omega _{b}$ .",
"Here $\\omega _{a}$ and $\\omega _{b}$ are the frequencies of modes $a$ and $b$ , and $2 \\delta $ is a small detuning compared to them.",
"Mode $c$ being driven off resonance, we use the stiff pump approximation and describe its amplitude as a complex number instead of an operator.",
"The interaction Hamiltonian now has two three-wave mixing terms, which result in the following effective Hamiltonian in the rotating frame where mode $a$ rotates at $\\omega _a + \\delta $ and mode $b$ at $\\omega _b + \\delta $ (see Appendix ), $\\hat{H}_{\\text{eff}} = \\delta \\, \\hat{a}^{\\dagger } \\hat{a} + \\delta \\, \\hat{b}^{\\dagger } \\hat{b} + G_{B} ( \\hat{a}^\\dagger \\hat{b}^\\dagger + \\hat{a} \\, \\hat{b}) + G_{R} ( \\hat{a}^{\\dagger } \\hat{b} + \\hat{a} \\, \\hat{b}^\\dagger ),$ where $G_{B,R}=\\chi c_{B,R}$ is time-independent and results from the physical time-dependent coupling rate $\\tilde{G}_{B,R}(t) = G_{B,R} e^{-i \\omega _{B,R} t}$ .",
"The derivation above is valid only for low three-wave mixing rates $\\vert G_{B,R} \\vert \\ll \\omega _a, \\omega _b, \\vert \\omega _a - \\omega _b \\vert $ .",
"In the case when $G_R=0$ , the Hamiltonian describes parametric amplification, which results in two-mode squeezing, while when $G_B=0$ , it describes a beam splitter between modes $a$ and $b$ .",
"Now if $G_B = G_R=G$ , Eq.",
"(REF ) has exactly the same form as Eq.",
"(REF ) if $\\omega _\\alpha =\\omega _\\beta $ .",
"Here, $\\delta $ plays the role of the bosonic mode free oscillation frequency.",
"It is now clear that when the coupling $G$ becomes comparable to $\\delta $ , the doubly pumped Josephson mixer simulates ultrastrongly coupled modes, even if the genuine coupling is much smaller than the genuine free oscillation frequencies of the physical system.",
"It is worthwhile to note that although the simulated coupling rates $G_{B,R}$ are time-independent, as in the case of genuine ultrastrong coupling in semiconductors [4], [5], [6], the actual coupling rate oscillates in the laboratory frame of the output ports of modes $a$ and $b$ .",
"Note that a method to obtain a genuine ultrastrong coupling between two bosonic modes in superconducting circuits was proposed in Ref. [46].",
"There the coupling is mediated not by a third oscillator but by a SQUID, and while the physical coupling could in principle reach the ultrastrong regime, its predicted coupling-to-frequency ratio does not reach the highest values of the coupling–to–effective frequency ratio leading to the interesting squeezing properties that we study here and that are realistically achievable in our quantum simulation."
],
[
"Results: emission spectra of the system",
"We now show the expected results of the quantum simulation, by determining the radiation emitted by the device in the regime where the latter can be described by the effective Hamiltonian (REF ).",
"Each mode $a$ or $b$ is connected to a transmission line at a rate $\\gamma _{a,b}$ and is subject to internal losses at a rate $\\gamma _L$ .",
"In the input-output formalism [3], [47], [48], [49], we are interested in the state of the output modes whose operators are $\\hat{a}_{\\text{out}}$ and $\\hat{b}_{\\text{out}}$ .",
"They are related to the input mode operators by the input-output relations $\\hat{a}_{\\text{out}} = \\hat{a}_{\\text{in}} + \\sqrt{\\gamma _a} \\hat{a}$ and $\\hat{b}_{\\text{out}} = \\hat{b}_{\\text{in}} + \\sqrt{\\gamma _b} \\hat{b}$ .",
"From the known input state, one gets the output state from the above expressions and from the quantum Langevin equations for the intracavity operators $\\dot{\\hat{a}} (t) = & - i \\delta \\hat{a} (t) - \\frac{\\gamma _a + \\gamma _L}{2} \\hat{a} (t) - i G \\big ( \\hat{b} (t) + \\hat{b}^\\dagger (t) \\big ) \\nonumber \\\\& -\\sqrt{\\gamma _a} \\hat{a}_{\\text{in}} (t) -\\sqrt{\\gamma _L} \\hat{f}_{a} (t) \\\\\\dot{\\hat{b}} (t) = & - i \\delta \\hat{b} (t) - \\frac{\\gamma _b + \\gamma _L}{2} \\hat{b} (t) - i G \\big ( \\hat{a} (t) + \\hat{a}^\\dagger (t) \\big ) \\nonumber \\\\& -\\sqrt{\\gamma _b} \\hat{b}_{\\text{in}} (t) -\\sqrt{\\gamma _L} \\hat{f}_{b} (t).$ where $\\hat{f}_a$ and $\\hat{f}_b$ are noise operators modeling the internal losses of the system.",
"It is straightforward to solve these equations in the frequency domain to obtain the expressions of $\\hat{a}_{\\text{out}} [\\omega ]$ and $\\hat{b}_{\\text{out}} [\\omega ]$ , from which we obtain the covariance matrix $\\mathcal {V}$ that fully characterizes the Gaussian output state.",
"The noise properties are directly given by the elements of $\\mathcal {V}$ .",
"In Figs.",
"REF (a)-(f) we show the output noise spectra of single-mode quadratures $\\hat{X}_a$ and $\\hat{Y}_a$ , of two-mode quadratures $\\hat{X}_a-\\hat{X}_b$ and $\\hat{Y}_a+\\hat{Y}_b$ , and the EPR variance [43], [44], as a function of frequency.",
"In the rotating frame, a signal at frequency $\\omega $ corresponds to $\\omega _a + \\delta + \\omega $ for mode $a$ , and to $\\omega _b + \\delta + \\omega $ for mode $b$ in the laboratory frame.",
"We do not show the noise spectra of $\\hat{X}_b$ and $\\hat{Y}_b$ since they are the same as for $\\hat{X}_a$ and $\\hat{Y}_a$ , both modes having the same effective frequency $\\delta $ , and the dissipation rates $\\gamma _{a}$ and $\\gamma _b$ being assumed identical.",
"As expected the output radiation is more squeezed for stronger coupling $G=G_B=G_R$ .",
"Furthermore, the squeezing becomes visible in the figures when ultrastrong coupling is reached for $G \\gtrsim 0.1 \\delta $ .",
"The behavior of the system in the physical implementation picture is illustrated in Fig.",
"REF (g).",
"When modes $a$ and $b$ are in the vacuum state at the input, the output of the system is in an unusual two-mode state, where each mode is squeezed, while the two modes are quantum correlated, similarly to the ground state of the Hamiltonian (REF ), shown in Fig.",
"REF .",
"Interestingly, the squeezing we predict here occurs between two propagating modes that are separated both in space and frequency.",
"Let us now comment on the shape of the spectra.",
"In the rotating frame, we show the positive and negative parts of the frequency spectrum, which correspond to measurable noise powers at positive frequencies in the laboratory frame.",
"In Figs.",
"REF (a)-(c), we can see that for the smallest shown coupling $G=0.01\\delta $ , the spectra develop a resonance at $\\omega =\\pm \\delta $ , symmetrically for positive and negative frequencies.",
"This resonance occurs at the transition frequency $\\delta $ of the effectively degenerate modes $a$ and $b$ in the rotating frame (see Eq.",
"(REF )).",
"As $G$ increases, the resonance splits into two, leading to four dips in the EPR variance Figs.",
"REF (c).",
"This can be understood as the vacuum Rabi splitting of both effective modes, as already observed in a physically ultrastrongly coupled light-matter system [4].",
"As the splitting increases with $G$ , one of the two resonance frequencies resulting from the Rabi splitting shifts towards $\\omega =0$ .",
"In Figs REF (a)-(c), this can be seen as two dips getting closer to the origin, corresponding to the resonance frequency and its image on the negative part of the spectrum.",
"When $G \\approx 0.5 \\delta $ , the dips merge at the origin and the resonance occurs at $\\omega =0$ .",
"This is shown in Figs.",
"REF (d)-(f), where there are no longer four dips but only three, and the one at the origin shows the largest amount of two-mode squeezing.",
"Thereby, the EPR variance almost reaches the lower bound of $0.5$ , which corresponds to an optimal case where $\\hat{Y}_a+\\hat{Y}_b$ is infinitely squeezed, while $\\hat{X}_a-\\hat{X}_b$ is shot noise limited only.",
"Besides, the single mode squeezing in the quadratures $\\hat{Y_a}$ and $\\hat{Y_b}$ reaches almost $-3$ dB.",
"This is in fact the maximal expected single-mode squeezing one can hope for.",
"We note that if the two outputs were combined in a 50:50 beam splitter (with frequency conversion on one arm), one of the output modes would be in an infinitely squeezed state while the other would be in the vacuum state [50]; the reverse has been demonstrated in [51].",
"This can be done using an extra Josephson mixer as in Ref.",
"[35] but in frequency conversion mode.",
"The squeezing amplitudes in Fig.",
"REF (d)-(e) are limited by the realistic internal losses and coupling rates to the transmission lines we use in the model.",
"Their minimal value is set by the need to stay in the regime where the three wave mixing Hamiltonian (REF ) is valid [45].",
"The figures stop at $G\\approx 0.5\\delta $ since beyond that point, the Hamiltonian (REF ) has no stable solution and extra terms should be included to make the Hamiltonian physically sound again (see Appendix ).",
"For instance, in case of the Dicke model modeling a spin ensemble coupled to a bosonic mode, this value for the coupling is a critical point of a quantum phase transition [52], [53], [54].",
"In the proposed simulation using a Josephson mixer, these extra terms arise from a Taylor expansion of the Josephson Hamiltonian beyond second order.",
"It is worthwhile to wonder how realistic are the parameters we chose in Fig.",
"REF .",
"The phenomena we propose to observe require that $\\gamma _L\\ll \\gamma _{a,b}<\\delta $ and that $2G_{B,R}\\lesssim \\delta $ .",
"It is shown in Ref.",
"[45] that $\\frac{2G_{B,R}}{\\sqrt{\\gamma _a\\gamma _b}}\\le \\frac{1}{4} \\sqrt{\\xi _a\\xi _bQ_aQ_b},$ where $\\xi <1$ is the participation ratio of the Josephson junction in the resonator [45] and $Q$ is the quality factor of the resonator.",
"Therefore, in order to reach $2G_{B,R}\\approx \\delta $ , one needs $1<\\frac{\\delta }{\\gamma _{a,b}}\\le \\frac{1}{4} \\sqrt{\\xi _a\\xi _bQ_aQ_b}.$ The condition that $\\sqrt{\\xi _a\\xi _bQ_aQ_b}>4$ sets constraints on the device similar to the ones needed to realize a quantum limited amplifier using the Josephson mixer [32], [55] and is perfectly realistic.",
"The parameters we chose in Fig.",
"REF are thus within reach in standard devices."
],
[
"Discussion",
"So far, we have focused on the observable squeezing contained in the output modes of the physical system that simulates the ultrastrong coupling Hamiltonian (REF ).",
"In this section, we interpret the nature of the output modes in the simulated picture.",
"We summarize the key components of the physical system in Fig.",
"REF (a).",
"The coupling between modes $a$ and $b$ is modulated in time and each mode is coupled to a zero temperature bath at a rate $\\gamma _{a,b}[\\omega ]$ with vanishing contribution from negative frequencies [3].",
"In contrast, in the simulated picture, the modes are coupled at a fixed rate $G$ but interact with an unusual environment, whose coupling rates $\\tilde{\\gamma }_{a,b}[\\omega ]$ are nonzero at negative frequencies (Fig.",
"REF (b)).",
"This results from a shift of the zero frequency in the rotating frame of the simulation.",
"The vacuum squeezing of the ultrastrongly coupled modes in their ground state can be understood as resulting from the excitations corresponding to the nonzero $\\tilde{\\gamma }_{a,b}[\\omega ]$ for $\\omega <0$ .",
"In order to release solely the ground-state photons, in genuine ultrastrongly coupled systems, one should abruptly switch off the interaction [1], [56] [Fig.",
"REF (c)].",
"In this way, the energy contained in the virtual excitations of the ground state is released from the cavity until the system reaches its new ground state, that of a noninteracting system.",
"In our simulation, we can avoid turning off the interaction and still observe squeezing in the output, owing to the peculiarity of the environment [Fig.",
"REF (b)]."
],
[
"Conclusion",
"In conclusion, we propose a superconducting circuit experiment to simulate ultrastrongly coupled bosonic modes.",
"This work contributes to clarify their elusive ground-state properties.",
"Using the high level of control of superconducting circuits enables us to access both modes directly, a feat that is not possible with current light-matter systems.",
"Our proposal will determine the smooth transition from strong to ultrastrong coupling regimes by measuring the squeezing properties of the output modes.",
"Beyond the fundamental interest in observing this transition, the unusual squeezing properties of the proposed device can be used as a resource for bath engineering [57], [58] and nonclassical state generation [59], [60], [61] in complex resonator networks.",
"The authors acknowledge Z. Leghtas, F. Portier, C. Ciuti, and P. Bertet for helpful discussions.",
"This work was supported by the French Agence Nationale de la Recherche (ANR COMB project No.",
"ANR-13-BS04-0014, GEARED project No.",
"ANR-14-CE26-0018), by the Emergences program of Ville de Paris under grant Qumotel, by University Sorbonne Paris Cité EQDOL, and by Paris Science et Lettres Idex PSL."
],
[
"Validity of the model",
"In this Appendix we briefly discuss the validity region of our model used to compute the ground state squeezing shown in Fig.",
"REF .",
"To understand it we need the expressions of $\\hat{p}_{1,2}$ , the eigenmodes of the Hamiltonian (REF ).",
"These operators are linear combinations of $\\hat{a}$ and $\\hat{b}$ , $\\hat{p}_{1,2} = t_{1,2} \\hat{a} + u_{1,2} \\hat{b} + v_{1,2} \\hat{a}^\\dagger + w_{1,2} \\hat{b}^\\dagger ,$ where the coefficients $\\vec{p}_{1,2}=\\lbrace t_{1,2}, u_{1,2}, v_{1,2}, w_{1,2} \\rbrace $ are obtained by diagonalizing the Hopfield matrix [40] for the Hamiltonian (REF ).",
"These coefficients are $\\vec{p}_{1} & =\\frac{1}{\\sqrt{N_1}}\\left(\\begin{array}{ccc}\\frac{\\sqrt{(\\delta - 2 G)\\delta }+ \\delta }{G}-1 \\\\- \\frac{\\sqrt{(\\delta - 2 G)\\delta }+ \\delta }{G}+1 \\\\-1 \\\\1\\end{array} \\right) \\\\\\vec{p}_{2} & =\\frac{1}{\\sqrt{N_2}}\\left(\\begin{array}{ccc}\\frac{\\sqrt{(\\delta + 2 G)\\delta }+ \\delta }{G}+1 \\\\\\frac{\\sqrt{(\\delta + 2 G)\\delta }+ \\delta }{G}+1 \\\\1 \\\\1\\end{array} \\right)$ with eigenvalues $\\omega _{1,2}=\\sqrt{(\\delta \\mp 2G)\\delta } .$ $N_{1,2}$ are the normalization coefficients, such that the condition $\\vert t_{1,2} \\vert ^2 + \\vert u_{1,2} \\vert ^2 - \\vert v_{1,2} \\vert ^2 - \\vert w_{1,2} \\vert ^2 = 1$ is satisfied, imposed by the Bose commutation rule.",
"With Eqs.",
"(REF ) and (REF ) one can see that when $G>\\delta /2$ , the model is not valid anymore."
],
[
"Two-mode squeezing operations",
"Let us show an intuitive picture in which the ground state squeezing is naturally predicted in a system described by the Hamiltonian (REF ).",
"We rewrite this Hamiltonian in terms of the following two collective modes $\\hat{m}=(\\hat{a} + \\hat{b})/\\sqrt{2}$ and $\\hat{n}=(\\hat{a} - \\hat{b})/\\sqrt{2}$ , that are well defined bosonic modes, $\\hat{H} &= (\\omega +G) \\hat{m}^{\\dagger } \\hat{m} + (\\omega -G)\\hat{n}^{\\dagger } \\hat{n} \\nonumber \\\\&+ \\frac{G}{2} (\\hat{m}^2 + (\\hat{m}^{\\dagger })^2) -\\frac{G}{2} (\\hat{n}^2 + (\\hat{n}^{\\dagger })^2),$ where we considered the case $\\omega _\\alpha =\\omega _\\beta =\\omega $ , a condition used in Fig.",
"REF .",
"One can clearly see from Eq.",
"(REF ) that modes $\\hat{m}$ and $\\hat{n}$ are independent and both ruled by a squeezing Hamiltonian.",
"Hence their ground state is expected to be largely squeezed in the ultrastrong coupling regime.",
"Note that the single-mode squeezing of $\\hat{m}$ (resp.",
"$\\hat{n}$ ) implies a two-mode squeezing in the original $\\hat{a},\\hat{b}$ basis along $\\hat{X}_a-\\hat{X}_b$ (resp.",
"$\\hat{Y}_a+\\hat{Y}_b$ ) as shown in Fig.",
"REF .",
"However, while this alternative description in terms of modes $m$ and $n$ clearly shows the correlations between modes $a$ and $b$ , it is less obvious to predict their single-mode squeezing, which can be definitely verified with the covariance matrix."
],
[
"Derivation of the effective Hamiltonian",
"Here we show the derivation of the effective Hamiltonian (REF ).",
"As mentioned in the main text, the interaction in the physical system is a three-wave mixing process between a pump mode $c$ and two microwave modes $a$ and $b$ , described by Eq.",
"(REF ).",
"However, since we drive the pump mode by a two-tone radiation, the interaction Hamiltonian now includes two three-wave mixing terms, and the full system Hamiltonian reads $\\hat{H} & = \\omega _{a} \\, \\hat{a}^{\\dagger } \\hat{a} + \\omega _{b} \\, \\hat{b}^{\\dagger } \\hat{b} + \\omega _{B} \\, \\hat{c}_{B}^{\\dagger } \\hat{c}_{B} + \\omega _{R} \\, \\hat{c}_{R}^{\\dagger } \\hat{c}_{R} \\nonumber \\\\& + \\chi ( \\hat{c}_B + \\hat{c}^{\\dagger }_B) ( \\hat{a} + \\hat{a}^{\\dagger } ) ( \\hat{b} + \\hat{b}^{\\dagger }) \\nonumber \\\\& + \\chi ( \\hat{c}_R + \\hat{c}^{\\dagger }_R) ( \\hat{a} + \\hat{a}^{\\dagger } ) ( \\hat{b} + \\hat{b}^{\\dagger }),$ In the interaction picture, this Hamiltonian reads $\\hat{H}_{\\text{IP}} & = \\chi (\\hat{c}_{B} e^{-i \\omega _{B} t} + \\hat{c}_{R} e^{-i \\omega _{R} t}) ( \\hat{a} \\, \\hat{b} \\, e^{-i (\\omega _{a} + \\omega _{b}) t} \\nonumber \\\\& + \\hat{a} \\, \\hat{b}^\\dagger e^{-i (\\omega _{a} - \\omega _{b}) t} + \\hat{a}^{\\dagger } \\hat{b} \\, e^{i (\\omega _{a} -\\omega _{b} ) t} \\nonumber \\\\& + \\hat{a}^\\dagger \\hat{b}^\\dagger e^{i (\\omega _{a} + \\omega _{b}) t} ) + \\text{h.c.},$ where the frequencies of the two-tone driving are $\\omega _{B} & = \\omega _{a} + \\omega _{b} + 2 \\delta , \\\\\\omega _{R} & = \\omega _{a} - \\omega _{b}, $ where $\\vert \\delta \\vert \\ll \\omega _a, \\omega _b, \\vert \\omega _a - \\omega _b \\vert $ .",
"Mode $c$ being driven off resonance, we use the stiff pump approximation and describe its amplitude as a complex number instead of an operator.",
"Let us call $G_{B,R}=\\chi c_{B,R}$ as the time independent parts of the coupling rates $\\tilde{G}_{B,R}(t) = G_{B,R} e^{-i \\omega _{B,R} t}$ .",
"Using Eqs.",
"(REF ),(REF ) and (), we obtain $\\hat{H}_{\\text{IP}} & = G_{B} ( \\hat{a} \\, \\hat{b} \\, e^{-2 i (\\omega _{a} + \\omega _{b} + \\delta ) t} + \\hat{a} \\, \\hat{b}^\\dagger e^{-2 i (\\omega _{a} + \\delta ) t} \\nonumber \\\\& + \\hat{a}^{\\dagger } \\hat{b} \\, e^{-2 i (\\omega _{b} + \\delta ) t} + \\hat{a}^\\dagger \\hat{b}^\\dagger e^{-2 i \\delta t} ) + \\nonumber \\\\& + G_{R} ( \\hat{a} \\, \\hat{b} \\, e^{-2 i \\omega _{a} t} + \\hat{a} \\, \\hat{b}^\\dagger e^{-2 i (\\omega _{a} - \\omega _{b}) t} \\nonumber \\\\& + \\hat{a}^{\\dagger } \\hat{b} + \\hat{a}^\\dagger \\hat{b}^\\dagger e^{2 i \\omega _{b} t} ) + \\text{h.c.},$ We work in a regime where $\\vert G_{B,R} \\vert \\ll \\omega _a, \\omega _b, \\vert \\omega _a - \\omega _b \\vert $ and $\\vert G_{B,R} \\vert \\lesssim \\vert \\delta \\vert $ , which allows us to perform a rotating wave approximation.",
"Thus, in the interaction picture, the important terms that contribute to the evolution of the system are resonant in this rotating frame, or oscillate at $2\\delta $ , and all the other terms can be fairly neglected, $\\hat{H}_{\\text{IP}} \\approx G_{B} ( \\hat{a}^\\dagger \\hat{b}^\\dagger e^{-2 i \\delta t} + \\hat{a} \\, \\hat{b} \\, e^{2 i \\delta t}) + G_{R} ( \\hat{a}^{\\dagger } \\hat{b} + \\hat{a} \\, \\hat{b}^\\dagger ).$ With a rather simple, yet judiciously chosen unitary transformation we obtain the effective Hamiltonian $\\hat{H}_{\\text{eff}} = \\delta \\, \\hat{a}^{\\dagger } \\hat{a} + \\delta \\, \\hat{b}^{\\dagger } \\hat{b} + G_{B} ( \\hat{a}^\\dagger \\hat{b}^\\dagger + \\hat{a} \\, \\hat{b}) + G_{R} ( \\hat{a}^{\\dagger } \\hat{b} + \\hat{a} \\, \\hat{b}^\\dagger ).$ We are now in a rotating frame where mode $a$ oscillates at $\\omega _{a} + \\delta $ and mode $b$ oscillates at $\\omega _{b} + \\delta $ .",
"Any single mode squeezing or correlations observed in this frame at a frequency $\\omega $ would correspond in the laboratory frame to $\\omega _{a} + \\delta + \\omega $ and $\\omega _{b} + \\delta + \\omega $ for modes $\\hat{a}$ and $\\hat{b}$ respectively."
]
] | 1612.05542 |
[
[
"On interpolations from SUSY to non-SUSY strings and their properties"
],
[
"Abstract The interpolation from supersymmetric to non-supersymmetric heterotic theories is studied, via the Scherk-Schwarz compactification of supersymmetric 6D theories to 4D.",
"A general modular-invariant Scherk-Schwarz deformation is deduced from the properties of the 6D theories at the endpoints, which significantly extends previously known examples.",
"This wider class of non-supersymmetric 4D theories opens up new possibilities for model building.",
"The full one-loop cosmological constant of such theories is studied as a function of compactification radius for a number of cases, and the following interpolating configurations are found: two supersymmetric 6D theories related by a T-duality transformation, with intermediate 4D maximum or minimum at the string scale; a non-supersymmetric 6D theory interpolating to a supersymmetric 6D theory, with the 4D theory possibly having an AdS minimum; a \"metastable\" non-supersymmetric 6D theory interpolating via a 4D theory to a supersymmetric 6D theory."
],
[
"Motivation for studying interpolating models",
"An important question in string phenomenology is how and when supersymmetry (SUSY) is broken.",
"A great deal of effort has been devoted to frameworks in which it is broken non-perturbatively in the supersymmetric effective field theory.",
"Much less effort has been devoted to string theories that are non-supersymmetric by construction.",
"On the face of it, the trade off for the second option, is that non-supersymmetric string models do not have the stability properties of supersymmetric ones.",
"However it can be argued that as long as the SUSY breaking is spontaneous and parametrically smaller than the string scale, the associated instability is under perturbative control [1].",
"There is then no genuine moral, or even practical, advantage to the former more traditional option, since nature is not supersymmetric.",
"Sooner or later, either route to the Standard Model (SM) will lead to runaway potentials for moduli that need to be stabilised.",
"Indeed spontaneous breaking at the string level may even confer advantages in this respect, as discussed in ref.[2].",
"Parametric control over SUSY breaking requires a generic method for passing from a non-superymmetric theory to a supersymmetric counterpart, under certain limiting conditions.",
"The method that was studied in ref.",
"[1] is interpolation via compactification to lower dimensions, with SUSY broken by the Scherk-Schwarz mechanism [3].",
"The two great advantages of interpolating models are that their compactification volumes can be tuned to make the cosmological constant arbitrarily small, and that some of them exhibit enhanced stability due to a one-loop cosmological constant that is exponentially suppressed with respect to the generic SUSY breaking scale [1].",
"They can be viewed as natural and phenomenologically interesting extensions of the original observation in refs.",
"[4], [5] that the $10D$ tachyon-free non-supersymmetric $SO(16)\\times SO(16)$ model interpolates to the heterotic $E_8\\times E_8$ model, via a Scherk-Schwarz compactification to $9D$ .",
"The general properties under interpolation of theories broken by the Scherk-Schwarz mechanism are not known.",
"For example, what determines if the zero radius endpoint theory is supersymmetric?",
"This paper focusses on the properties of 4-dimensional ($4D$ ) theories that interpolate between stable, supersymmetric $6D$ tachyon-free models.",
"Three main results are presented.",
"First, we derive and study the general form of the $6D$ endpoint theories, and show that their modular invariance properties derive directly from the Scherk-Schwarz deformation.",
"This enables us to generalise the construction of modular invariant Scherk-Schwarz deformed theories by beginning with the $6D$ endpoint theory.",
"Second, we determine a simple criterion for whether a SUSY theory, broken by Scherk-Schwarz, will interpolate to a SUSY or a non-SUSY one at zero radius: the zero radius theory is non-supersymmetric, if and only if the Scherk-Schwarz acts on the gauge group as well as the space-time side.",
"Third, we undertake a preliminary survey (in the sense that the models we study only have orthogonal gauge groups) of some representative models that confirm these two properties, by examining their potentials and spectra.",
"The general framework for the interpolations are as shown in Figure REF .",
"Beginning with a supersymmetric $6D$ theory generically referred to as ${\\mathcal {M}}_1$ , the theory is compactified to a non-supersymmetric $4D$ theory ${\\mathcal {M}}$ by adapting the Coordinate Dependent Compactification (CDC) technique first presented in refs.",
"[6], [7], [8], [9].",
"This is the string version of the Scherk-Schwarz mechanism, which spontaneously breaks SUSY in the $4D$ theory with a gravitino mass of ${\\cal O}(1/2r_i)$ where $r_i$ is the largest radius carrying a Scherk-Schwarz twist.",
"(We will use “CDC” and “Scherk-Schwarz” interchangeably.)",
"As usual it is the gravitino mass that is the order parameter for SUSY breaking: it can be continuously dialled to zero at large radius where SUSY is restored and ${\\mathcal {M}}_1$ regained.",
"Figure: The interpolation map between a 6D6D supersymmetric theory at infinite radius, ℳ 1 \\mathcal {M}_1 and its supersymmetric and non-supersymmetric, interpolated 6D6D duals, which are defined in the vanishing radius limit, with the vertical direction representing dimension.",
"Whether or not SUSY is restored in the non-compact r 1 =r 2 =r=0r_1=r_2=r=0, 6D6D model is determined by the structure of the Scherk-Schwarz action.One of the main properties that will be addressed is the nature of the theory as the radius of compactification is taken to zero.",
"This depends upon the precise details of the Scherk-Schwarz compactification, and indeed we will find that the presence or otherwise of SUSY at zero radius depend on the choice of basis vectors and structure constants defining the model.",
"It is possible that the $4D$ theory interpolates to either a supersymmetric or a non-supersymmetric model (${\\mathcal {M}}_{2a}$ or ${\\mathcal {M}}_{2b}$ respectively).",
"Models of the latter kind correspond to a $6D$ theory in which SUSY is broken by discrete torsion [1].",
"We begin in §REF by reviewing the basic formalism for interpolation.",
"Section REF then presents the construction of $4D$ non-supersymmetric models as compactifications of $6D$ supersymmetric ones.",
"The modification of the massless spectra in the decompactification and $r_i\\rightarrow 0$ limits (with the latter corresponding to the decompactification limit of a $6D$ $T$ -dual theory) is analysed, in order to determine the nature of the theories at the small and large radii endpoints.",
"Section REF discusses the technique for rendering the cosmological constant in an interpolating form, allowing it to be calculated across a regime of small and large radii.",
"The modification of the projection conditions and massless spectrum by the choice of basis vectors and structure constants is made explicit, and based on these observations, in particular how the CDC correlates with modified GSO projections in the $6D$ endpoint theories, §REF then derives the general form of deformation within this framework, extending previous constructions.",
"This more general formulation may prove to be useful for future model building.",
"The conditions under which SUSY is preserved or broken at the endpoints of the interpolation are discussed in §.",
"Particular focus is given to the constraints on the appearance of light gravitino winding modes in the zero radius limit.",
"It is found that models in which the CDC acts only on the space-time side, are inevitably supersymmetric at zero radius, while models within which the CDC vector is non-trivial on the gauge side as well yield a non-supersymmetric model in the same limit.",
"This analysis paves the way for a presentation in § of explicit interpolations (in terms of their cosmological constants) in particular models that display various different behaviours: namely we find examples of interpolation between two supersymmetric $6D$ theories via $4D$ theories with negative or positive cosmological constant; interpolation between a non-supersymmetric $6D$ theory and a supersymmetric one, with or without an intermediate $4D$ AdS minima; we also find examples of “metastable” non-supersymmetric $6D$ theories (by which we mean theories that have a positive cosmological constant with an energy barrier) that can decay to supersymmetric ones.",
"As mentioned, this paper follows on from a reasonably large body of work on non-supersymmetric strings that is nonetheless much smaller than the work on supersymmetric theories.",
"Following on from the original studies of the ten-dimensional $SO(16)\\times SO(16)$ heterotic string [10], there were further studies of the one-loop cosmological constants [14], [15], [4], [5], [16], [17], [18], [13], [11], [12], [19], [20], [21], [22], [23], [24], their finiteness properties [11], [12], [25], their relations to strong/weak coupling duality symmetries [26], [27], [28], [29], and string landscape ideas [30], [31].",
"The relationship to finite temperature strings was explored in refs.",
"[32], [33], [34], [6], [35]).",
"Further development of the Scherk-Schwarz mechanism in the string context was made in refs.",
"[36], [37], [38], [39], [40].",
"Progress towards phenomenology within this class has been made in refs.",
"[41], [42], [43], [15], [44], [45], [29], [46], [47], [48], [49].",
"Related aspects concerning solutions to the large-volume “decompactification problem” were discussed in refs.",
"[50], [51], [52], [53], [2].",
"Non-supersymmetric string models have also been explored in a wide variety of other configurations [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], including studies of the relations between scales in various schemes [68], [69], [70], [71], [72], [73], [74].",
"Some aspects of this study are particularly relevant to the recent work in refs.[75].",
"Note that here we will not elaborate on the properties of the non-supersymmetric $4D$ theory at radii of order the string length.",
"As we will see, and as found in ref.",
"[1], often there is a minimum in the cosmological constant at this point which suggests some kind of enhancement of symmetry at a special radius.",
"(Indeed often it is possible to identify gauge boson winding modes that become massless at the minimum.)",
"There is therefore the possibility of establishing connections to yet more non-supersymmetric $4D$ theories.",
"Conversely one can ask if every non-supersymmetric tachyon-free $4D$ theory can be interpolated to a supersymmetric higher dimensional theory.",
"We comment on this and other prospects in the Conclusions in §.",
"In this section, we revisit the calculation of the cosmological constant in the Scherk-Schwarzed theories, and in particular derive a formulation for the partition function of interpolating models, that is useful for the later analysis.",
"The discussion is a natural generalisation of the “compactification-on-a-circle” treatment of ref.",
"[1], and as we shall see it ultimately leads to an improved and more general construction for this class of theory.",
"Let us begin by briefly summarising the implementation of the Scherk-Schwarz mechanism described in that work.",
"As already mentioned, this is incorporated using a Coordinate Dependent Compactification (CDC) [6] of an initially supersymmetric $6D$ theory, namely the ${\\cal M}_1$ model.",
"For our purposes it is useful to define it in the fermionic formulation, although any construction method would be applicable.",
"In this formulation, the initial theory is defined by assigning boundary conditions to worldsheet fermions.",
"If they are real, this is encapsulated in a set of 28-dimensional basis vectors, $V_i$ , containing periodic or antiperiodic phases.",
"The sectors of the theory are given by the set of $\\overline{\\alpha V}\\equiv {\\alpha _i V_i} + \\Delta $ where $\\Delta \\in \\mathbb {Z} $ so that $\\overline{\\alpha V}\\in [ -\\frac{1}{2},\\frac{1}{2}) $ .",
"We follow the usual the convention that $\\alpha _i$ denotes the sum over spin structures on the $\\alpha $ cycle.",
"The spectrum of the theory at generic radius in any sector is determined by imposing the GSO projections governed by the vectors $V_i$ and a set of structure constants $k_{ij}$ , according to the KLST set of rules in refs.",
"[76], [77], [78], [79] (and equivalently ref.",
"[80]), which are summarised in Appendix, .",
"The model is then further compactified down to $4D$ on a ${\\mathbb {T}}_2/{\\mathbb {Z}}_2$ orbifold.",
"In the absence of any CDC the result would simply be an ${\\cal N}=1$ model resulting from an overall $(\\mathbb {K}_3\\times \\mathbb {T}_2)/\\mathbb {Z}_2$ compactification.",
"The $\\mathbb {K}_3$ in question corresponds to the $6D$ $\\mathcal {N}=1$ theory in the fermionic construction in our examples.",
"In theories of the type discussed in [1], in which the orbifold twist preserves SUSY, the twisted sectors have a supersymmetric spectrum, and therefore do not contribute to the cosmological constant, and thus the nature of the orbifold is unimportant.",
"The CDC is implemented by introducing a deformation, described by another vector $\\mathbf {e}$ , of shifts in the charge lattice that depend on the radii $r_{i=1,2}$ of the ${\\mathbb {T}}_2$ ; this will be shown explicitly below.",
"Under the CDC, the Virasoro generators of the theory are modified, yielding an extra effective projection condition, (in addition to the GSO projections associated with the $V_i$ from which the initial $\\mathbb {K}_{3}$ is constructed), governed by ${\\bf e}$ , on the states constituting the massless spectrum of the $4D$ theory.",
"The remaining massless states are then characterized by their charges under the $U(1)$ symmetry associated with ${\\bf e}$ .",
"To qualify as a Scherk-Schwarz mechanism, this $U(1)$ symmetry has to include some component of the $R$ -symmetry in order to distinguish bosons from fermions, thereby projecting out the gravitinos, and breaking spacetime SUSY.",
"The effect of the CDC of course disappears in the strict $r \\rightarrow \\infty $ limit where the Kaluza-Klein (KK) spectrum becomes continuous, and the $6D$ endpoint model ${\\cal M}_1$ is recovered.",
"On the other hand as we shall see the CDC turns into another GSO projection vector in the $r_i \\rightarrow 0$ limit, where states either remain massless or become infinitely massive.",
"Upon $T$ -dualising, $ r_i \\rightarrow \\tilde{r}_i=1/r_i\\, ,$ the $\\tilde{r}_i \\rightarrow \\infty $ model becomes the non-compact theory ${\\cal M}_{2}$ , whose properties depend precisely on the form of ${\\bf e}$ .",
"The theories at the two endpoints can contain a different number of states and charges.",
"Because ${\\bf e}$ can overlap the gauge degrees of freedom, ${\\cal M}_{2}$ will generically have a gauge symmetry that differs from that of ${\\cal M}_1$ , and possibly no SUSY.",
"As we will see the two are in fact linked: if ${\\cal M}_{2}$ is supersymmetric then the gauge group is the same as that of ${\\cal M}_1$ , if it is not, then the gauge group is different."
],
[
"CDC-Modified Virasoro Operators",
"Let us now elaborate on the above description.",
"The conventions for the fermionic construction are as in refs.",
"[76], [77], [78], [79] and for the CDC are as outlined in ref.",
"[1], and summarised in Appendix .",
"That is the unmodified Virasoro operators are defined as ${{L_0}/\\overline{L}_0}=\\frac{1}{2}\\alpha ^{\\prime } p_{L/R}^2 + {\\rm \\text{oscillator contributions}}\\, ,$ where, in terms of the winding and KK numbers, $n_i$ and $m_i$ respectively, the left- and right-moving momenta for a theory compactified on two circles of radii $r_{i = 1,2}$ take the unshifted form $p_{L/R} \\sim \\left(\\frac{m_i}{r_i} +\\!/\\!- n_ir_i \\right)\\, .$ Ultimately we wish to derive the largest possible class of deformations to the Virasoro operators that is compatible with modular invariance.",
"This will turn out to be more general than those considered in refs.",
"[9], [6].",
"In order to achieve this, we will now display the most general modification possible of the Virasoro operators under the Scherk-Schwarz action, along with a free parameter $m_{\\mathbf {e}}$ , which will ultimately be fixed by imposing modular invariance: ${{L_0}^{\\prime }/\\overline{L}^{\\prime }_0}&=~\\frac{1}{2}\\left[\\mathbf {Q}_{L/R}-\\mathbf {e}_{L/R}(n_1+n_2)\\right]^2+\\frac{1}{4}\\left[\\frac{m_1+m_e}{r_1} +\\!/\\!- n_1r_1\\right]^2 \\nonumber \\\\&~~~~~~~~+\\frac{1}{4}\\left[\\frac{m_2+m_e}{r_2} +\\!/\\!- n_2r_2\\right]^2-1/\\frac{1}{2}+ \\mbox{ other oscillator contributions}\\, ,$ where the other oscillator contributions can be deduced from REF , and where $\\mathbf {Q}$ are the vectors of Cartan gauge and $R$ -charges, defined by $\\mathbf {Q} = {\\bf N}_{\\overline{\\alpha V}} + \\overline{\\alpha V}$ .",
"As promised, the parameter $m_e$ will now be determined by modular invariance.",
"The partition function of the modified theory is then expressed in terms of $q=e^{2 \\pi i \\tau }$ (where as usual the real and imaginary parts of $\\tau $ are defined to be $\\tau = \\tau _1 + i \\tau _2$ ): ${\\cal Z}(\\tau )~=~{\\rm Tr}\\, \\sum _{m_{1,2}, n_{1,2}}\\mathtt {g} \\, \\overline{q}^{\\overline{L}^{\\prime }_0}q^{L^{\\prime }_0}\\, .$ Modular invariance requires $L^{\\prime }_0 - \\overline{L}^{\\prime }_0 \\in \\mathbb {Z}$ .",
"Given that the original supersymmetric theory is modular invariant (i.e.",
"$L_0 - \\overline{L}_0 \\in \\mathbb {Z}$ ) this can be used to determine a consistent $m_e$ as follows: $L^{\\prime }_0 - \\overline{L}^{\\prime }_0&=(m_1n_1 + m_2n_2) + \\frac{1}{2}\\left[\\mathbf {Q}_{L}^2-\\mathbf {Q}_{R}^2\\right] + {(n_1+n_2)\\, m_e}- \\mathbf {e} \\cdot \\mathbf {Q}(n_1+n_2) +\\mathbf {e}\\cdot \\mathbf {e}\\frac{(n_1+n_2)^2}{2}\\nonumber \\\\&={{L_0}-\\overline{L}_0}+{(n_1+n_2) m_e}- (n_1+n_2)\\mathbf {e} \\cdot \\left[\\mathbf {Q} - \\mathbf {e}\\frac{(n_1+n_2)}{2}\\right]\\, ,$ where the dot products are Lorentzian.",
"Thus a KK shift of $m_e=\\mathbf {e\\cdot Q}-\\frac{1}{2}(n_1+n_2)\\,\\mathbf {e\\cdot e}\\, ,$ is sufficient to maintain modular invariance in the deformed theory.",
"This matches the result of ref.",
"[6].",
"The vector ${\\mathbf {e}}$ then lifts the masses of states according to their charges under the linear combination $q_e={\\bf e\\cdot Q}$ .",
"Restricting the discussion to half-integer mass-shifts imposes the constraint $\\mathbf {e} \\cdot \\mathbf {e} = 1$ mod(2).",
"Later on the partition function will be reorganised into sums over different values of $4m_e=0\\ldots 3$ (as we restrict the study to $\\frac{1}{2}$ phases in all examples, fractions of at most $\\frac{1}{4}$ can arise in the GSO projections via odd numbers of overlapping $\\frac{1}{2}$ 's).",
"So far these deformations are precisely those of refs.",
"[6], [7], [8], [9]: once we consider the interpolation to the $6D$ theories, it will become clear how they can be made general.",
"Note that level-matching is preserved by the CDC, but the mass spectrum is modified rather than the number of degrees of freedom contained within the theory, as required for a spontaneous breaking of SUSY [6], [7], [8], [9].",
"It is clear from eq.",
"(REF ) that for zero winding modes ($n_i=0$ ), states for which $q_e=\\mathbf {e \\cdot Q} \\ne 0\\,\\,\\,{\\rm mod}(1)$ become massive under the action of the CDC.",
"Conversely all the zero winding states in the NS-NS sector remain unshifted by the CDC since they are chargeless.",
"As described in ref.",
"[1] there may or may not be massless gravitinos depending on whether the effective projection $\\mathbf {e \\cdot Q} = 0\\,\\,\\, {\\rm mod}(1)$ is aligned with the other projections: this is in turn dependent on the choice of structure constant, so that ultimately the breaking of SUSY is associated with breaking by discrete torsion."
],
[
"Details of Cosmological Constant Calculation",
"To evaluate the cosmological constant, at given radii $r_1 = r_2 = r$ , one must integrate each $q^{M} \\bar{q}^N$ term (weighted by its coefficient $c_{MN}$ ) in the total 1-loop partition function over the fundamental domain $\\mathcal {F}$ of the modular group: $\\Lambda ^{(D)} \\equiv -\\frac{1}{2} \\mathcal {M}^{(D)}\\int _{\\mathcal {F}} \\frac{d^2\\tau }{{\\tau _{2}}^{2}}\\, \\mathcal {Z}_{total}(\\tau )\\, ,$ where $D$ is the number of uncompactified spacetime dimensions (equal to 4 at all intermediate radii between the small and large radius $6D$ endpoint theories, along which the cosmological constant will be evaluated), and ${\\cal M}\\equiv M_{\\rm string}/(2\\pi )=1/(2\\pi \\sqrt{\\alpha ^{\\prime }})$ is the reduced string scale.",
"Henceforth ${\\cal M}$ is set to 1; it can be reinserted by dimensional analysis at the end of the calculation if desired.",
"The integral splits into upper ($\\tau _{2}>1$ ) and lower regions of the fundamental domain.",
"Only terms for which $M=N$ can receive contributions from both regions, with the $\\tau _{1}$ integral yielding zero in the upper region when $M \\ne N$ , enforcing level matching in the infra-red (but allowing contributions from unphysical proto-graviton modes in the ultra-violet as described in ref.[1]).",
"At general radius the evaluation of the cosmological constant is complicated immensely by the fact that $M,N$ vary with $r_i$ .",
"In order to make the evaluation tractable, the total partition function, $\\mathcal {Z}_{total}(\\tau )$ , has to be rearranged into separate bosonic and fermionic factors as follows.",
"It is convenient to define $n=(n_{1}+n_{2})$ and $\\ell =(\\ell _{1}+\\ell _{2})$ .",
"Twisted sectors do not need to be considered in this implementation as, being supersymmetric, they do not contribute to the cosmological constant.",
"In other words, the cosmological constant calculated without the orbifolding, is the same up to a factor of two, as the actual cosmological constant, as explained in detail in [6], [7], [8], [9], [1].",
"However, we will make further comments on twisted sectors later when we come to generalise the construction.",
"We have ${\\cal Z}(\\tau )=\\frac{1}{\\tau _{2} \\eta ^{22} \\overline{\\eta }^{10}}\\sum _{\\vec{\\ell },\\vec{n}}{\\cal Z}_{ \\vec{\\ell },\\vec{n}}\\sum _{\\alpha ,\\beta } \\Omega _{{\\ell },{n}}{\\tiny \\begin{bmatrix}\\alpha \\\\\\beta \\end{bmatrix}}\\, ,$ where the Poisson-resummed partition function for the compactified complex boson is given by (see Appendix ) ${\\cal Z}_{\\vec{\\ell },\\vec{n}} =~\\frac{ r_1r_2}{\\tau _2 \\eta ^2 \\overline{\\eta }^2}\\sum _{{\\vec{\\ell },\\vec{n}}}\\exp \\Big \\lbrace -\\frac{\\pi }{\\tau _2}\\left[r_1^2|\\ell _1-n_1\\tau |^2+r_2^2|\\ell _2-n_2\\tau |^2\\right]\\Big \\rbrace \\, ,$ and the theta function products, each of which has characteristics defined by the sectors $\\alpha , \\beta $ , with their respective CDC shifts, are $\\Omega _{{\\ell },{n}}{\\tiny \\begin{bmatrix}\\alpha \\\\\\beta \\\\\\end{bmatrix}}~=~ {\\tilde{C}}^{\\alpha ,-n}_{\\beta ,-\\ell }\\prod _{i_L }\\vartheta {\\tiny \\begin{bmatrix}\\overline{\\alpha \\mathbf {V}_i-n\\mathbf {e}_i}\\\\-\\beta \\mathbf {V}_i +\\ell \\mathbf {e}_i\\\\\\end{bmatrix}}\\prod _{j_R}\\overline{\\vartheta }{\\tiny \\begin{bmatrix}\\overline{\\alpha \\mathbf {V}_j-n\\mathbf {e}_j}\\\\-\\beta \\mathbf {V}_j +\\ell \\mathbf {e}_j\\end{bmatrix}}\\, ,$ where the conventions can be found in Appendix .",
"In the above, the coefficients of the partition function are given by ${\\tilde{C}}^{\\alpha ,-n}_{\\beta ,-\\ell }~=~\\exp \\Big \\lbrace -2\\pi i \\left[n\\mathbf {e}\\cdot \\beta {V}-\\frac{1}{2}n\\ell \\mathbf {e}^2\\right]\\Big \\rbrace \\, C^\\alpha _\\beta \\, ,$ where $C^\\alpha _\\beta $ are the coefficients of the original theory before CDC, expressed in terms of the structure constants $k_{ij}$ , and spin-statistic $s_i = V^{1}_i$ , as in the original notation and Appendix , namely $C_{\\mathbf {\\beta }}^{\\mathbf {\\alpha }}~=~\\exp \\left[2\\pi i\\left(\\alpha s+\\beta s+\\beta _{i}k_{ij}\\alpha _{j}\\right)\\right]\\, .$ It is convenient to use the resummed version of this expression; certainly for the $q$ -expansion this is the preferred method as it makes modular invariance explicit.",
"This removes the $r_1 r_2$ prefactor and adds a factor of $\\tau _2 $ .",
"The bosonic factor in the partition function $\\mathcal {Z}_B(\\tau )$ depend upon the radii of compactification, the winding and resummed KK numbers and the CDC induced shift in the KK levels, $m_e$ , as follows: ${\\cal Z}_{B_{ \\vec{m},\\vec{n},m_e}}=\\frac{1}{\\eta ^{2} \\bar{\\eta }^{2}} \\sum _{\\vec{m},n_{1},k} q^{\\frac{1}{4}\\left(\\frac{m_{1} + m_e}{r_{1}}+n_{1}r_{1}\\right)^{2}+\\frac{1}{4}\\left(\\frac{m_{2} + m_e}{r_{2}}+(n-n_{1}+4k)r_{2}\\right)^{2}} \\times \\bar{q}^{\\frac{1}{4}\\left(\\frac{m_{1} + m_e}{r_{1}}-n_{1}r_{1}\\right)^{2}+\\frac{1}{4}\\left(\\frac{m_{2} + m_e}{r_{2}}-(n-n_{1}+4k)r_{2}\\right)^{2}} \\, .$ The effective shift in the KK number, given by the requisite $m_e \\equiv {\\bf e}\\cdot ( {\\bf Q} - n\\frac{{\\bf e}}{2} )$ , arises from the choice of ${\\tilde{C}}^{\\alpha ,-n}_{\\beta ,-\\ell }$ , which gives an overall phase $e^{2\\pi i \\ell ({\\bf e}\\cdot ({\\bf Q}-n{\\bf e})- n {\\bf e}^2/2)}$ in the partition function; as we shall see this shift in the KK number ultimately amounts to introducing a new vector $V_e\\equiv {\\bf e}$ in the non-compact $T$ -dual theory at zero radius, combined with structure constants $k_{ei}=0$ , $k_{ee}=1/2$ .",
"Note that this means in the 4D spectrum one may find states with 1/4-charges ${\\bf e\\cdot Q}=1/4,3/4$ , that since they have $m_e\\ne 0$ , become infinitely massive in the zero radius limit.",
"In order to reorder the sum to do it efficiently, a projection in the ${\\cal Z}_F$ on $\\mathbf {Q}$ is now introduced to select possible values of $m_e$ .",
"Following the notation that $\\beta _i$ represents the sum over spin structures, the parameter for this projection over the vector $\\mathbf {e}$ will be called $\\beta _e=0\\ldots 3$ .",
"Thus overall, using the results in Appendix , one can write, $\\mathcal {Z}_{total}(\\tau ) = \\frac{1}{4}\\frac{1}{\\tau _{2} \\eta ^{22} \\overline{\\eta }^{10}}\\sum _{\\stackrel{m_e=(0\\ldots 3)/4}{\\mbox{\\tiny {$\\vec{m},\\vec{n}$}}}}{\\cal Z}_{B_{ \\vec{m},\\vec{n},m_e}}\\sum _{\\alpha ,\\beta ,\\beta _e}e^{2\\pi i {\\beta _e m_e} }\\Omega _{{n}}{\\tiny \\begin{bmatrix}\\alpha \\\\\\beta ,\\beta _e\\end{bmatrix}},$ where $\\Omega _{{n}}{\\tiny \\begin{bmatrix}\\alpha \\\\\\beta ,\\beta _e \\\\\\end{bmatrix}}~=~ {\\tilde{C}}^{\\alpha ,-n}_{\\beta ,\\beta _e} \\prod _{i_L }\\vartheta {\\tiny \\begin{bmatrix}\\overline{\\alpha {V}_i-n\\mathbf {e}_i}\\\\{-\\beta V_i - \\beta _e\\mathbf {e}_i}\\\\\\end{bmatrix}}\\prod _{j_R}\\overline{\\vartheta }{\\tiny \\begin{bmatrix}\\overline{\\alpha {V}_j-n\\mathbf {e}_j}\\\\{-\\beta {V}_j -\\beta _e\\mathbf {e}_j}\\end{bmatrix}}\\, .$ Note that the phases in ${\\tilde{C}}^{\\alpha ,-n}_{\\beta ,\\beta _e}$ are precisely what is needed to cancel the contribution coming from the theta functions in $\\Omega _n$ , so that overall the spectrum is merely shifted, with the GSO projections remaining independent of ${\\bf e}$ .",
"The bosonic contribution to the total partition function is independent of the fermionic sectors within the theory so $\\mathcal {Z}_B$ appears as a pre-factor to the sector sum for any given $m_e$ .",
"Conversely, the fermionic partition function is composed of terms that depend upon the boundary conditions of the fermions within the sectors $\\alpha , \\beta $ each of which is independent of the compactification radii.",
"The advantage of this reordering is that one can therefore collect 16 representative factors, $n,\\, 4 m_e={0..3\\,\\,\\, \\mbox{mod(4)}}$ , $ {\\cal Z}_{F,n,m_e} =\\frac{1}{4}\\sum _{\\alpha \\beta \\beta _e}e^{2\\pi i {\\beta _e m_e} }\\Omega _{{n}}{\\tiny \\begin{bmatrix}\\alpha \\\\\\beta ,\\beta _e\\end{bmatrix}},$ which are independent of the radii, and 16 respective ${\\mathbb {T}}_2/{\\mathbb {Z}}_2$ factors ($n,\\, 4 m_e={0..3\\,\\,\\, \\mbox{mod(4)}}$ ), which being independent of the internal degrees of freedom, depend only on the ${\\mathbb {T}}_2$ compactification, ${\\cal Z}_{B_{n,m_e}} =~\\frac{1}{\\eta ^2 \\overline{\\eta }^2}\\sum _{{\\vec{m},n_1,k}}q^{\\frac{1}{4} \\left( \\frac{m_1 + m_e }{r_1} + n_1 r_1 \\right)^2 +\\frac{1}{4} \\left( \\frac{m_2 + m_e}{r_2} + (n-n_1+4k) r_2 \\right)^2}{\\bar{q}}^{\\frac{1}{4} \\left( \\frac{m_1 + m_e }{r_1} - n_1 r_1 \\right)^2+\\frac{1}{4} \\left( \\frac{m_2 + m_e }{r_2} - (n-n_1+4k) r_2 \\right)^2}\\, .$ The latter are radius dependent interpolating functions, analogous to the functions ${\\cal E}_{0,1/2},{\\cal O}_{0,1/2}$ in the simple circular case studied in ref.[1].",
"We refer to the ${\\cal Z}_{F,n,m_e}$ terms as `${\\mathbb {K}}_3$ factors', since they involve only the internal degrees of freedom of the $6D$ theory, and thus can be computed for all radii at the beginning of the calculation.",
"The total partition function is then compiled by summing over the 16 $(n,m_e)$ sectors as ${\\cal Z}(\\tau )=\\frac{1}{4}\\frac{1}{\\tau _{2} \\eta ^{22} \\overline{\\eta }^{10}}\\sum _{n,\\,4m_e = 0..3}{\\cal Z}_{B,n,m_e}{\\cal Z}_{F,n,m_e}\\, .$ To summarise, via the procedure of re-ordering the original sum REF , a projection on to different consistent $m_{e}$ values has been performed, such that a sum over $m_{e}$ can now be taken."
],
[
"The zero radius theory and a more general formulation of Scherk-Schwarz",
"An interesting aspect of the above approach is that in the small radius limit, that part of the spectrum with $m_e\\ne 0$ mod(1) decouples and can be discarded, leaving the partition function of the non-compact $6D$ theory at $r_i=0$ .",
"Indeed, Poisson resumming on $n_1$ and $k$ gives ${\\cal Z}_{B,n,m_e}\\rightarrow \\sum _{{\\vec{m}}}e^{-\\left( \\frac{(m_1+m_e)^2}{r_1^2} +\\frac{ (m_2+m_e)^2}{r_2^2}\\right) \\frac{\\pi |\\tau |^2}{\\tau _2}} \\frac{1}{4\\tau _2 r_1 r_2}+\\ldots \\, ,$ where the ellipsis indicate terms that are further exponentially suppressed.",
"Thus the total untwisted partition function in the small radius limit can be expressed as ${\\cal Z}(\\tau )\\rightarrow \\frac{1}{16r_1r_2 }\\frac{1}{\\tau ^2_{2} \\eta ^{22} \\overline{\\eta }^{10}}\\sum _{n}{\\cal Z}_{F,n,0}\\, .$ Note that $1/(r_1r_2)$ is simply the expected volume factor of the partition function in the $T$ -dual $6D$ theory.",
"In conjunction with the fermionic component of the partition function, this then reproduces a $6D$ model with an additional basis vector ${\\bf e}$ , appearing in the sector definitions as $\\alpha V-n {\\bf e}$ , and with eq.",
"(REF ) providing a new GSO projection, namely $ m_e={\\bf {e}\\cdot {\\bf Q}}-n/2=0$ mod (1).",
"(The mod (1) comes courtesy of the sum over $m_i$ .)",
"Upon inspection therefore, we are finding that eq.",
"(REF ) is actually the GSO projection of an additional vector $V_e\\equiv {\\bf e}$ in the non-compact $6D$ theory.",
"Beginning with the choice of ${\\bf e}\\cdot {\\bf e}=1$ , one can infer that the $6D$ theory at zero radius for the examples we have been considering has structure constants $k_{ei}=0$ and $k_{ee}=1/2$ , consistent with the modular invariance rules of KLST in refs.",
"[76], [77], [78], [79].",
"In fact identifying sectors as ${\\alpha V}={\\alpha _i V_i+\\alpha _e V_e}$ with the sum over the spin structures on the $\\mathbf {e}$ cycle as $\\alpha _e=-n$ mod(2), the entire partition function at zero radius is that of the $6D$ theory with the appropriate corresponding GSO phases, ${\\tilde{C}}^{\\alpha ,-n}_{\\beta ,\\beta _e}~=~ {C}^\\alpha _\\beta e^{2\\pi i ( \\beta _e k_{e j} \\alpha _j - \\beta _i k_{ie} n - \\beta _e k_{ee} n)}\\, .$ Reversing the line of reasoning above, leads us finally to a generalisation of the construction of interpolating models based on the modular invariance of their endpoint $6D$ theories: First, define a $6D$ theory in terms of a set of vectors $V_i$ , and any additional $V_e\\equiv {\\bf e}$ vector that obeys the $6D$ modular invariance rules of ref.",
"[76], [77], [78], [79], together with a set of consistent structure constants $k_{ei}$ and $k_{ee}$ .",
"(The $k_{ei}$ are then fixed by the modular invariance rules in the usual way.)",
"In theories that have an additional $\\mathbb {Z}_2$ orbifold action ${\\bf \\hat{g}}$ on compactification to $4D$ , $V_e\\equiv {\\bf e}$ is still constrained by the need to preserve mutually consistent GSO projections, with the condition $\\left\\lbrace {\\bf e\\cdot Q} ,\\, {\\bf \\hat{g}}\\right\\rbrace =0$ (as in refs.",
"[9], [6] and discussed in ref.[1]).",
"The partition function is then in the form of eqs.",
"(REF ),(REF ) with coefficients as in eq.",
"(REF ).",
"The projection obtained by performing the $\\beta _e$ sum determines the corresponding KK shift to be $m_e={\\bf e}\\cdot {\\bf Q} +(k_{ee}-{\\bf e}^2) \\,n - k_{ei} \\alpha _i\\, ,$ generalizing REF .",
"The last statment, namely that one may simply treat the Scherk-Schwarz action as another basis vector, leading to considerable generalisations, is one of the main results of the paper.",
"In order to prove it, one may first Poisson-resum back to the original expression but retaining $\\beta _e$ , so that entire partition function is $\\mathcal {Z}~=~\\frac{1}{4}\\frac{1}{\\tau _{2} \\eta ^{22} \\overline{\\eta }^{10}}\\sum _{\\stackrel{m_e=(0\\ldots 3)/4}{\\beta _e}}\\sum _{\\alpha ,\\beta ,{\\mbox{\\tiny {$\\vec{\\ell },\\vec{n}$}}}}\\,e^{2\\pi i(\\ell +\\beta _{e})m_{e}}{\\cal Z}_{\\vec{\\ell },\\vec{n}}\\, {\\tilde{C}}^{\\alpha ,-n}_{\\beta ,\\beta _e} \\prod _{i_L }\\vartheta {\\tiny \\begin{bmatrix}\\overline{\\alpha {V}_i-n\\mathbf {e}_i}\\\\{-\\beta V_i - \\beta _e\\mathbf {e}_i}\\\\\\end{bmatrix}}\\prod _{j_R}\\overline{\\vartheta }{\\tiny \\begin{bmatrix}\\overline{\\alpha {V}_j-n\\mathbf {e}_j}\\\\{-\\beta {V}_j -\\beta _e\\mathbf {e}_j}\\end{bmatrix}}\\, .$ Note that the sum over $m_e$ provides a projection that equates $\\beta _e \\equiv -\\ell $ mod(1).",
"Using the modular transformations for theta functions detailed in Appendix , it is then straightforward to show that the partition function is invariant under $\\tau \\rightarrow \\tau +1$ provided that $e^{-i\\pi \\left(\\mathbf {\\overline{\\alpha V}}-n\\mathbf {e}\\right)\\cdot (2V_{0}+\\mathbf {\\overline{\\alpha V}-n\\mathbf {e}})}\\tilde{C}{}_{\\mathbf {\\beta ,\\beta _{e}}}^{\\mathbf {\\alpha },-n}=\\tilde{C}{}_{\\mathbf {\\beta -\\alpha -}\\delta _{i0}\\mathbf {,}\\beta _{e}+n}^{\\mathbf {\\alpha },-n},$ and invariant under $\\tau \\rightarrow -1/\\tau $ provided that $e^{-2\\pi i\\left(\\mathbf {\\overline{\\alpha V}}-n\\mathbf {e}\\right)\\cdot \\left(\\mathbf {\\beta V}+\\beta _{e}\\mathbf {e}\\right)}\\tilde{C}{}_{\\mathbf {\\beta ,\\beta _{e}}}^{\\mathbf {\\alpha },-n}=\\tilde{C}{}_{-\\mathbf {\\alpha ,}n}^{\\beta ,\\beta _{e}}.$ This overall set of conditions is precisely that of KLST [76], [77], [78], [79] with the original theory enlarged to include the vector $V_e\\equiv {\\bf e}$ .",
"$\\square $ Note that these rules are significantly more general than those of refs.",
"[6], [7], [8], [9], in which the choice $\\tilde{C}{}_{\\mathbf {\\beta ,\\beta _{e}}}^{\\mathbf {\\alpha },-n}=C_{\\mathbf {\\beta }}^{\\mathbf {\\alpha }}e^{-2\\pi i\\left(n\\mathbf {e}\\cdot \\beta V+\\beta _{e}n\\frac{\\mathbf {e\\cdot e}}{2}\\right)},$ corresponds to taking $k_{ei}=0$ and $k_{ee}=1/2$ , in REF .",
"Now for example the CDC vectors are no longer restricted to obey ${\\bf e}^2=1 $ mod(1), and moreover the KK shifts have additional sector dependence if $k_{ei}\\ne 0$ .",
"We should add that, as well as being a generalisation, these rules simplify the construction of viable phenomenological models, because the $\\left\\lbrace {\\bf e\\cdot Q} ,\\, {\\bf \\hat{g}}\\right\\rbrace =0$ condition can be implemented independently, with consistency then guaranteed with respect to all the other $V_i$ vectors This is a somewhat subtle point because the basis in which the orbifold action is diagonal is not the same as the basis in which the Scherk-Schwarz action is diagonal.",
"However the two act relatively independently on the partition function.",
"This point is discussed in explicit detail in ref.[81]..",
"One can also conclude that for consistency a theory that is Scherk-Schwarzed on an orbifold should contain additional sectors that are twisted under the action of both the orbifold and the Scherk-Schwarz – i.e.",
"twisted sectors that have non-zero $\\alpha _e$ .",
"Of course $\\alpha _e$ for such sectors has no association with any windings, but one finds that those sectors (which being twisted are supersymmetric) are required for consistency (anomaly cancellation for example)."
],
[
"Is the theory at small radius supersymmetric?",
"Let us now move on to the conditions under which the endpoint theories exhibit SUSY.",
"We will always consider models in which the theory at infinite radius is supersymmetric (as would be evidenced by the vanishing of the cosmological constant there) but we would like to determine whether or not SUSY is restored at zero radius as well.",
"In this section we develop arguments to address this question based on the existence or otherwise of massless gravitinos as $r_i \\rightarrow 0$ .",
"As usual the pure Neveu-Schwarz (NS-NS) sector, $\\bf {0}$ gives rise to the gravity multiplet, $g_{\\mu \\nu }$ (the graviton), $\\phi $ (the dilaton) and $B_{[\\mu \\nu ]}$ (the two index antisymmetric tensor), from the states $\\psi ^{3,4}_{-\\frac{1}{2}} \\left|0\\right\\rangle _{R} \\otimes X^{3,4}_{-1} \\left|0\\right\\rangle _{L}$ in the notation of ref.[1].",
"These states are chargeless under $\\mathbf {e} \\cdot \\mathbf {Q}$ and no projection on them can occur, since the CDC vector is always zero in the $4D$ space-time dimensions $\\psi ^{3,4}$ .",
"Given the inevitable presence of the graviton, the SUSY properties of the theory are then dictated by the presence or absence of the R-NS gravitinos, namely $ \\Psi _\\alpha ^\\mu \\equiv \\left\\lbrace \\psi ^{3,4}_{0}\\chi ^{5,6}_{0}\\chi ^{7,8}_{0}\\chi ^{9,10}_{0} \\right\\rbrace _\\alpha \\left|0\\right\\rangle _{R} \\otimes X^{34}_{-1} \\left|0\\right\\rangle _{L}\\, .$ Their Scherk-Schwarz projections are determined purely by the Scherk-Schwarz action on the right-moving degrees of freedom The spectrum is found from the expressions for the modified Virasoro operators in eq.",
"(REF ).",
"For the non-winding gravitinos, the shifted KK momentum becomes virtually continuous in the $r_i \\rightarrow \\infty $ limit and the full $6D$ gravitino state is inevitably recovered there.",
"The scale at which SUSY is spontaneously broken by the CDC is set by the gravitino mass $1/2r_i$ .",
"As the compactification is turned on, the SUSY of the $6D$ theory is broken, and then towards the $r_i \\rightarrow 0$ end of the interpolation, new gravitinos may or may not appear in the massless spectrum, perhaps heralding the restoration of SUSY at small radius as well.",
"To see if they do, consider how the CDC modifies the theories that sit at the endpoints of the interpolation.",
"We denote by ${\\bf Q}_{\\psi }^0$ the charge of the lightest gravitino state at large radius.",
"SUSY is exact even in the presence of ${\\bf e}$ , with the state ${\\bf Q}_{\\psi }^0$ being exactly massless, if both the first and second terms in the modified Virasoro operators of eq.",
"(REF ), namely $(\\mathbf {Q}_{\\psi }^0 - \\mathbf {e}\\,n)^2$ and $\\left( \\frac{m_i + \\mathbf {e} \\cdot \\mathbf {Q}_{\\psi }^0 -\\frac{1}{2}n\\mathbf {e^2}}{r_i} + n_i r_i \\right)^2 \\, ,$ vanish.",
"(For convenience we continue for this discussion to use the original more restrictive rules of refs.",
"[6], [7], [8], [9]; it would be trivial to extend the discussion to the more general rules of eq.",
"(REF ).)",
"With $n_1 = n_2=0$ , the first term receives no extra contribution due to the CDC.",
"Furthermore, there is no winding contribution to the second term.",
"Therefore gravitinos that have $\\mathbf {e \\cdot Q_{\\psi }^0} = 0$ remain massless and indicate the presence of exact SUSY.",
"Conversely, if the only remaining gravitinos have $\\mathbf {e \\cdot Q_{\\psi }^0} = \\frac{1}{2}\\, ,$ their mass is $\\frac{1}{2}\\sqrt{\\frac{1}{r_1^2} + \\frac{1}{r_2^2}}$ and SUSY is spontaneously broken.",
"Without loss of generality, one can consider SUSY breaking to amount to a conflict between $\\mathbf {e}$ and a single basis vector, denoted by $V_{con}$ .",
"That is, $V_{con}$ constrains the gravitinos, while the remaining $V_i$ cannot project them out of the theory.",
"In order for the above light (but not massless) gravitino to be the one that is left un-projected, the projections due to $\\bf e$ and $V_{con}$ must disagree, that is the massive $\\mathbf {e \\cdot Q_{\\psi }^0} = \\frac{1}{2}$ state is retained by $V_{con}$ while the massless $\\mathbf {e \\cdot Q_{\\psi }^0} = 0$ state is projected out.",
"Again without loss of generality, it is always possible to choose $V_{con}$ so that the conditions are aligned; that is $V_{con}\\cdot {\\bf Q}_{\\psi }^0=\\frac{1}{2} \\Rightarrow {\\bf e}\\cdot {\\bf Q}_{\\psi }^0=\\frac{1}{2}$ .",
"These modes are preserved (and have a mass $\\sim \\frac{1}{2r_i}$ ) while $V_{con}$ projects the massless $\\mathbf {e \\cdot Q_{\\psi }} = 0$ modes out of the theory entirely.",
"Now consider the zero radius end of the interpolation, and denote the new would-be massless gravitino state by $\\widetilde{\\mathbf {Q}}_{\\psi }$ .",
"Although a different state, it can be related to the infinite radius gravitino $\\mathbf {Q}_{\\psi }^{0}$ by a shift in the charge vector, induced by a potentially non-zero winding number; $\\widetilde{\\mathbf {Q}}_{\\psi } = \\mathbf {Q}_{\\psi }^{0} - \\mathbf {e}\\,n\\, .$ As $r_i$ vanish, the spectrum associated with the winding modes becomes continuous, while the KK states become extremely heavy.",
"As described in the previous section, the requirement that the KK term in eq.",
"(REF ) vanishes forms an effective projection that constrains the light states at zero radius, selecting the modes for which $\\mathbf {e}\\cdot \\mathbf {\\widetilde{Q}}_{\\psi }= \\frac{n}{2}~~~ \\text{mod(1),}$ where we will assume that $\\mathbf {e \\cdot e}=1$ .",
"It is clear from the relation between $\\mathbf {\\widetilde{Q}}_{\\psi }$ and $\\mathbf {Q}_{\\psi }^0$ in eq.",
"(REF ) that the projection due to the CDC vector remains unchanged for any gravitino state there, since $\\mathbf {e}^2 n \\in \\mathbb {Z}$ ; that is $\\mathbf {e} \\cdot \\mathbf {\\widetilde{Q}}_{\\psi } = \\mathbf {e} \\cdot \\mathbf {Q}_{\\psi }\\, .$ This equation together with eqs.",
"(REF ) and (REF ), imply that any gravitino of the spontaneously broken theory that becomes light at small radius must be an odd-winding mode.",
"Under the shift in $\\mathbf {Q}$ given by eq.",
"(REF ), the $V_{con}$ projection constraining the gravitinos is ${\\: V}_{con}\\cdot \\widetilde{\\mathbf {Q}}_{\\psi } &=& V_{con} \\cdot \\mathbf {Q}^0_\\psi -n\\, V_{con} \\cdot \\mathbf {e}~~\\mbox{mod}\\:(1)\\, \\nonumber \\\\&=& \\frac{1}{2} -n\\, V_{con} \\cdot \\mathbf {e}~~\\mbox{mod}\\:(1)\\, .$ For the effective projection in eq.",
"($\\ref {eqn:Modified_projection}$ ) to agree with the modified GSO condition in eq.",
"(REF ) for $n=odd$ , we then require that $V_{con} \\cdot {\\bf e}=0 \\qquad {\\mbox{mod~(1)}}.",
"$ Eq.",
"(REF ) is a necessary condition for a model with SUSY spontaneously broken by the Scherk-Schwarz mechanism to have massless gravitino states in both the infinite and zero radius limits.",
"Let us see what it implies in a specific theory.",
"Consider the basis vector set $\\lbrace V_0, V_1, V_2, V_4\\rbrace $ , together with a CDC vector that is empty in its left-moving elements, the standard set up outlined in [1], in which the vectors $\\lbrace V_0, V_1, V_2\\rbrace $ project down to $6D$ SUSY with orthogonal gauge groups: $V_0&=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~11~111~111~ | ~1111~11111~111~111~11~111~]\\nonumber \\\\V_1&=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~011~011~ | ~1111~11111~111~111~11~111~]\\nonumber \\\\V_2&=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~0101~00000~011~111~11~111~]\\nonumber \\\\V_4&=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~0101~00000~011~000~00~000~]\\nonumber \\\\{\\bf e} &=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~0000~00000~000~000~00~000~]\\, .$ A suitable and consistent set of structure constants is $k_{ij} =\\left( \\begin{array}{c}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\end{array}\\right)\\,.$ Gravitinos are found in the $\\overline{V_0 + V_1} = \\frac{1}{2}[11~100~100~|~(0)^{20}]$ sector, with vacuum energies $[\\epsilon _{R}, \\epsilon _{L}] = [0, -1]$ .",
"The charge operator for the non-winding gravitinos in the initial (infinite radius) theory takes the same form as the sector vector itself.",
"They have charges determined by $V_4$ that give the required ${\\bf e\\cdot Q}=1/2$ mod (1) for spontaneous SUSY breaking: the positive helicity states with this choice of structure constants are $\\mathbf {Q}^0_{\\psi } = \\frac{1}{2}[1 \\mbox{\\,\\,-\\vspace{-2.84544pt}1}\\, {\\mbox{\\tiny $\\pm $}}100~{\\mbox{\\tiny $\\pm $}}100~|~(0)^{20}~]\\, ,$ where the $\\pm $ signs on the fermions are co-dependent.",
"It is clear from the vector overlap between $\\mathbf {Q}^0_{\\psi }$ and $V_4$ that the latter is playing the role of $V_{con}$ that constrains the gravitini states.",
"(The structure constants have been chosen such that $V_2$ yields identical constraints.)",
"Whether or not any of the winding modes of the gravitinos are light at zero radius depends upon them satisfying the modified GSO projection condition of eq.",
"(REF ): $V_{4} \\cdot \\mathbf {\\widetilde{Q}}_{\\psi } = V_{4} \\cdot \\mathbf {Q}_{\\psi }^{0} - V_{4} \\cdot \\mathbf {e} \\,(n_1 + n_2)\\, \\;\\;\\;\\; \\text{mod (1)}\\, .$ As we saw the two projections agree for the odd-winding modes of the $\\widetilde{\\mathbf {Q}}_{\\psi }$ states since $V_{4} \\cdot \\mathbf {e} = 0 \\,\\, \\text{mod(1)}$ , and under the CDC, the charge vector for the small radius gravitino is $\\widetilde{\\mathbf {Q}}_{\\psi } = \\frac{1}{2}[1\\mbox{\\,\\,-\\vspace{-2.84544pt}1} ~\\,0\\,0{\\mbox{\\tiny $\\pm $}}1~0\\,0{\\mbox{\\tiny $\\pm $}}1~|~(0)^{20}~]\\, .$ Note that non-zero right-moving charges of the small radius gravitino are on the $\\omega ^{3,4}$ and $\\omega ^{5,6}$ world-sheet degrees of freedom, and they no longer overlap the SUSY charges of the large radius theory.",
"The appearance of gravitino states in the light spectrum in the zero radius limit of this theory reflects a general conclusion.",
"If the left-moving elements of the CDC vector vanish, eq.",
"(REF ) is automatically satisfied.",
"Any theory with a CDC vector acting purely on the space-time side becomes supersymmetric at zero radius since the projection always preserves the odd-winding modes of the gravitinos.",
"The non-supersymmetric $4D$ theory at generic radius is therefore an interpolation between two supersymmetric theories quite generally in these cases, which sit at the zero and infinite radius endpoints.",
"The supersymmetric nature of the zero radius theory will later be verified by the vanishing of the cosmological constant in the $r_i \\rightarrow 0$ limit (Figure REF ), as presented in the following section.",
"Note that the necessary cancellation between thousands of terms is highly non-trivial."
],
[
"Example of a CDC vector with non-zero left-moving entries",
"Consider instead a theory composed of the same basis vector set as in eq.",
"(REF ), but now with a CDC vector containing non-zero left-moving entries: for example ${\\bf e} &=& \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~1011~00000~000~100~01~111~]\\, .$ Under the CDC, and for convenience of presentation dropping the $\\pm $ signs, the charge vector for the odd-winding gravitino modes is modified to $\\widetilde{\\mathbf {Q}}_{\\psi } = \\frac{1}{2}[1 1~001~001~|~1011~00000~000~100~01~111~]\\, .$ As in the previous example the vector contains the same number of non-zero right-moving entries, but lying in different columns, so there is no contribution from eq.",
"(REF ) to the mass squared on the space-time side.",
"However the non-zero left-moving elements now result in a non-zero contribution.",
"Under the shift, $(\\mathbf {Q}^0_{R},\\mathbf {Q}^0_{L})^2 \\rightarrow (\\widetilde{\\mathbf {Q}}_{R},\\widetilde{\\mathbf {Q}}_{L})^2 = (\\mathbf {Q}^0_{R} + \\mathbf {e}_{R},\\mathbf {Q}^0_{L} + \\mathbf {e}_{L})^2 \\, ,$ any non-zero shift in $\\mathbf {Q}^0_{L}$ will inevitably produce massive gravitinos since in the R-NS sector the charges of massless states must be zero mod (1) on the left-moving side.",
"We conclude that SUSY is restored at small as well as large radius if and only if the Scherk-Schwarz mechanism does not act on the gauge-side.",
"Conversely if SUSY is broken at zero radius then so is the gauge symmetry.",
"The nett bose-fermi number appears as the constant term in the parition function $\\mathcal {Z}\\supset (N_b - N_f)q^{0}\\bar{q}^{0}+\\ldots $ .",
"Thus, the dominant terms in the one loop contribution to the cosmological constant are proportional to $(N_b - N_f)$ for the massless states [1], so non-supersymmetric models with an equal number of massless bosonic and fermionic states have an exponentially suppressed one-loop cosmological constant, and hence exhibit an increased degree of stability.",
"Unfortunately it seems to be necessary to determine the full massless spectrum in order to deduce whether or not $N_b = N_f$ .",
"There appears to be no principle, or algebraically feasible generic procedure, for choosing the basis vectors $\\lbrace V_i\\rbrace $ , the CDC vector $\\mathbf {e}$ , and the structure constants $k_{ij}$ , that ensures that $N_b = N_f$ ."
],
[
"Surveying the interpolation landscape",
"We now turn to a survey of the different possible interpolations, in order to verify the rules derived in the previous sections, in particular those that govern the supersymmetry properties of the models.",
"We should remark that in order to make the exercise computationally feasible, we will only use 1/2 phases so that the theories contain only large orthogonal gauge groups.",
"As such, we are not here attempting to construct the SM, and the massless spectrum for each example will not be presented.",
"(They can easily be determined using the rules in Appendix ).",
"Rather, studying the relationship between the cosmological constant and the radii of compactification exemplifies interpolation patterns between different types of model.",
"Following the procedure outlined in Section REF , the total partition function, $Z_{total}(\\tau )$ , truncated at an order $\\mathcal {O}(q^2)$ in the $q$ -expansion, which is computationally manageable while displaying the qualitative behaviour, is input in to the integral in REF , for a range of compactification radii between either ends of the interpolation range."
],
[
"Nb>Nf",
"Consider a theory containing $V_0$ , $V_1$ and $V_2$ as in the above basis vector set in eqs.",
"(REF ), a modified $V_4$ , an additional vector $V_5$ , and a CDC vector that acts only on the space-time side: $V_4&=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~000~000~ | ~0101~00000~000~011~00~000~]\\nonumber \\\\V_5&=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~000~011~ | ~0101~11100~001~000~10~111~]\\nonumber \\\\{\\bf e} &=& \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~0000~00000~000~000~00~000~] \\, .$ A suitable and consistent set of structure constants $k_{ij}$ is $k_{ij} =\\left( \\begin{array}{c}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$}\\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\nonumber \\\\\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\hspace{2.84544pt}\\mbox{$0$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\nonumber \\end{array}\\right)\\, .$ This model can be investigated using the general method presented in the previous section.",
"$V_4$ plays the role of $V_{con}$ , while $V_5$ respects its projections on the gravitinos.",
"As discussed the interpolation is between two supersymmetric endpoints at both small and large radius.",
"The cosmological constant takes a non-zero negative value with a minimum at intermediate values, and returns to zero at the two extremes, displayed in Figure REF .",
"Figure: Cosmological Constant vs. Radius, r 1 =r 2 =r∈[0.1,2.1]r_{1}=r_{2}=r\\in [0.1,2.1] with radius increments of 0.020.02 for a model with 𝐞 L =\\mathbf {e}_{L} = trivial.",
"N b -N f =28N_b-N_f=28."
],
[
"Nb<Nf",
"A theory in which $N_b<N_f$ can be generated by a performing an alternative modification to the vectors $V_4, V_5$ : $V_4&=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~0101~00000~011~000~01~111~]\\nonumber \\\\V_5&=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~000~011~ | ~0101~11100~010~110~00~011~]\\nonumber \\\\{\\bf e} &=& \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~0000~00000~000~000~00~000~] \\, ,$ with the following structure constants: $k_{ij} =\\left( \\begin{array}{c}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\end{array}\\right)\\, .$ Similarly to the $N_b>N_f$ model with a exclusively non-trivial right-moving CDC vector, this model interpolates between two supersymmetric endpoints at both small and large radius, with the cosmological constant now taking a non-zero positive value at intermediate radii, displayed in Figure REF , corresponding to unstable runaway to decompactification at either end of the interpolation.",
"Figure: Cosmological Constant vs. Radius, r 1 =r 2 =r∈[0.1,4.1]r_{1}=r_{2}=r\\in [0.1,4.1] with radius increments of 0.020.02 for a model with 𝐞 L =\\mathbf {e}_{L} = trivial and N b -N f =-228.N_b-N_f=-228."
],
[
"Nb=Nf",
"A theory with Bose-Fermi degeneracy can be achieved with a theory comprised of the basis vector set in eq.",
"(REF ), plus a basis vector $V_5$ and CDC vector of the form $V_5&=& - \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~000~011~ | ~0100~11100~000~111~10~011~] \\, ,\\nonumber \\\\{\\bf e} &=& \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~1011~00000~000~100~01~111~]\\, ,$ with $k_{ij}$ given by $k_{ij} =\\left( \\begin{array}{c}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\\\\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{${\\scriptstyle \\frac{1}{2}}$}\\hspace{2.84544pt}\\mbox{$0$} \\hspace{2.84544pt}\\mbox{$0$} \\nonumber \\end{array}\\right)\\, .$ $N_b$ and $N_f$ are found to be equal despite the fact that the theory is non-supersymmetric (as can be seen by the absence of any massless gravitini in the spectrum).",
"For models in which the CDC vector $\\mathbf {e}$ is non-trivial in both the gauge and the global entries, the cosmological constant takes a non-zero value at small radius, while it vanishes exponentially quickly for large compactification scales, as displayed in Figure REF .",
"Figure: Cosmological Constant vs. Radius, r 1 =r 2 =r∈[0.1,2.1]r_{1}=r_{2}=r\\in [0.1,2.1] with radius increments of 0.020.02 for a model with 𝐞 L =\\mathbf {e}_{L} = non-trivial and N b =N f N_b = N_f."
],
[
"Nb>Nf",
"An interpolation from SUSY to non-SUSY in which $N_b>N_f$ , can be achieved by taking the corresponding set of basis vectors in eqs.",
"(REF ), but now with a CDC vector of the form ${\\bf e} &=& \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~0101~00000~000~110~11~011~]\\, .$ For models in which $N_b>N_f$ , the cosmological constant reduces from a constant positive value at small radius reaching a negative minimum at approximately $r=1.0$ in string units.",
"As the radius increases to $\\infty $ , the cosmological constant tends to zero from negative values, consistent with the restoration of SUSY in the endpoint model, as displayed in Figure REF .",
"In this particular example, the turnover appears to be at precisely 1 string unit, suggesting that a winding mode is becoming massless at this point, enhancing the gauge symmetry.",
"Figure: Cosmological Constant vs. Radius, r 1 =r 2 =r∈[0.1,5.1]r_{1}=r_{2}=r \\in [0.1,5.1 ] with radius increments of 0.020.02 for a model with 𝐞 L =\\mathbf {e}_{L} = non-trivial, and N b -N f =192N_b-N_f=192."
],
[
"Nb<Nf",
"Finally for a non-SUSY to SUSY interpolation with $N_b<N_f$ , we take the model in eqs.",
"(REF ) but now with a CDC vector of the form ${\\bf e} &=& \\mbox{${\\scriptstyle \\frac{1}{2}}$}[~00~101~101~ | ~0101~00000~000~011~11~011~]\\, .$ The cosmological constant increases from a constant negative minimum at small radius, to a non-SUSY 6D theory at small radius and a SUSY 6D theory at infinite radius, as displayed in Figure REF .",
"Figure: Cosmological Constant vs. Radius, r 1 =r 2 =r∈[0.1,3.1]r_{1}=r_{2}=r \\in [0.1,3.1] with radius increments of 0.020.02 for a model with 𝐞 L =\\mathbf {e}_{L} = non-trivial and N b -N f =-64N_b-N_f=-64."
],
[
"Conclusions",
"Following on from ref.",
"[1], the nature of heterotic strings in the context of Scherk-Schwarz compactification has been investigated, with particular emphasis on their properties under interpolation.",
"From the starting point of supersymmetric $6D$ theories in the infinite radius limit, Scherk-Schwarz compactification to $4D$ yields models that have $N_b \\, \\lbrace = \\text{,} < \\text{,} > \\rbrace \\, N_f$ , each possibility exhibiting different behaviours under interpolation.",
"The behaviour of their cosmological constants was studied as a function of compactification radius, and it was found that theories can yield maxima or minima in the cosmological constant at intermediate values, as well as barriers with apparent metastability.",
"The latter feature may have interesting phenomenological and/or cosmological applications.",
"The nature of the Scherk-Schwarz action, in particular whether or not it simultaneously acts to break the gauge group, dictates whether or not SUSY emerges in the $6D$ theory at zero radius.",
"We studied the relation of the interpolating theory to the $6D$ theories that emerge at the end-points of the interpolation, and made the novel observation that the Scherk-Schwarz action descends from an additional GSO projection in the $6D$ zero radius endpoint theory.",
"This allowed us to use the modular invariance constraints of the $6D$ theory to derive a more general class of Scherk-Schwarz compactification.",
"The aim in this work has been to establish the general features of interpolating models, relating higher, $D$ -dimensional models to $(D-d)$ -dimensional compactified models.",
"It is conceivable that very many non-supersymmetric tachyon-free $4D$ models can be interpolated to higher dimensional supersymmetric ones.",
"This would imply the existence of a formal relation between the process of interpolation, and the restoration of SUSY.",
"Looking forward, it may not be possible to show that every non-supersymmetric theory is related to a supersymmetric counterpart via the process of interpolation.",
"However, it seems possible that such a relation may always hold for the particular class of theories in which SUSY is broken by discrete torsion, as in ref.",
"[46] for example.",
"A goal for future work would be to establish relationships, of the type found in this study, between additional lower dimensional, non-supersymmetric models, ideally of greater phenomenological appeal, and their supersymmetric counterparts.",
"If it can be shown that non-supersymmetric models generically relate to supersymmetric theories in this way, interpolation could be used as a tool with which to relate many tachyon-free non-supersymmetric string theories to their supersymmetric siblings.",
"Thus it would be possible to locate non-supersymmetric models within the larger network of string theories extending previous work in this direction.",
"Acknowledgements: We are extremely grateful to Keith Dienes, Emilian Dudas and Hervé Partouche for many interesting discussions and comments.",
"SAA would like to thank the École Polytechnique for hospitality extended during this work."
],
[
" Notation and conventions for partition functions",
"The basic $\\eta $ and $\\vartheta $ functions are as given in [1].",
"For convenience we will here reproduce the required generalizations of these functions.",
"The more general theta functions with characteristics are defined as $\\vartheta {\\tiny \\begin{bmatrix}a\\\\b\\end{bmatrix}}(z,\\tau ) & \\equiv & \\sum _{n=-\\infty }^{\\infty } e^{2\\pi i(n+a)(z+b)} \\, q^{(n+a)^{2}/2}\\nonumber \\\\& = &e^{2\\pi iab}\\,\\xi ^{a}\\,q^{a^{2}/2}\\, \\vartheta (z+a\\tau +b,\\tau )~;$ of course these functions have a certain redundancy, depending on only $z+b$ rather than $z$ and $b$ separately.",
"In general, the functions in Eq.",
"(REF ) have modular transformations $\\vartheta {\\tiny \\begin{bmatrix} a\\\\ b \\end{bmatrix}}(z,-1/\\tau ) &=&\\sqrt{-i\\tau }\\, e^{2\\pi i a b} e^{i \\pi \\tau z^2}\\vartheta {\\tiny \\begin{bmatrix} -b\\\\ a \\end{bmatrix}}(-z\\tau , \\tau )~, \\nonumber \\\\\\vartheta {\\tiny \\begin{bmatrix} a\\\\ b \\end{bmatrix}}(z,\\tau +1) &=&e^{-i\\pi (a^2 + a)}\\vartheta {\\tiny \\begin{bmatrix} a \\\\ a+b+1/2 \\end{bmatrix}}(z,\\tau )~.$ To evaluate the cosmological constant from the partition function in §Section REF , we require the following $q$ -expansions: $\\eta (\\tau ) & \\sim & q^{1/24}+\\ldots \\nonumber \\\\\\vartheta {\\tiny \\begin{bmatrix} 0\\\\ 0 \\end{bmatrix}}(0,\\tau ) & \\sim & 1+2q^{1/2}+\\ldots \\nonumber \\\\\\vartheta {\\tiny \\begin{bmatrix} 0\\\\ 1/2 \\end{bmatrix}}(0,\\tau ) & \\sim & 1-2q^{1/2}+\\ldots \\nonumber \\\\\\vartheta {\\tiny \\begin{bmatrix} 1/2\\\\ 0 \\end{bmatrix}}(0,\\tau ) & \\sim & 2q^{1/8}+\\ldots \\nonumber \\\\\\vartheta {\\tiny \\begin{bmatrix} 1/2\\\\ 1/2 \\end{bmatrix}}(0,\\tau ) & = & 0~.$ Regarding partition functions, the expression for the compactified bosonic component of the partition function is given in [1].",
"Here we will need the expression for the untilted torus in terms of radii $r_1$ , $r_2$ .",
"The Poisson-resummed partition function is given by $\\mathcal {Z}_{\\bf B}{\\tiny \\begin{bmatrix}0\\\\0 \\end{bmatrix}}(\\tau )~=~{\\cal M}^2 \\frac{r_1r_2}{{\\tau _{2}}|\\eta (\\tau )|^{4}}\\sum _{\\mathbf {n},\\mathbf {m}}\\exp \\left\\lbrace -\\frac{\\pi }{\\tau _{2}}r_1^2|m_{1}+n_{1}\\tau |^{2}-\\frac{\\pi }{\\tau _{2}}r_2^2|m_{2}+n_{2}\\tau |^{2}\\right\\rbrace .$ Each internal complex fermion degree of freedom contributes to the partition function depending on its world sheet boundary conditions, $v\\equiv \\overline{\\alpha V}_i $ and $u\\equiv \\beta V_i $ , as $\\mathcal {Z}_{u}^{v} & = & \\mbox{Tr}\\left[q^{\\hat{H}_{v}}e^{-2\\pi iu\\hat{N}_{v}}\\right]\\nonumber \\\\& = & q^{\\frac{1}{2}(v^{2}-\\frac{1}{12})}\\prod _{n=1}^{\\infty }(1+e^{2\\pi i(v\\tau -u)}q^{n-\\frac{1}{2}})(1+e^{-2\\pi i(v\\tau -u)}q^{n-\\frac{1}{2}})\\nonumber \\\\& = & e^{2\\pi iuv}\\,\\vartheta {\\tiny \\begin{bmatrix}v\\\\-u\\end{bmatrix}}(0,\\tau )/ {\\eta (\\tau )}~.$"
],
[
" Conventions and spectrum of the fermionic string",
"In this paper, the free-fermionic construction [76], [80], [79] serves as the anchor underpinning our models.",
"In the free-fermionic construction, all world-sheet conformal anomalies are cancelled through the introduction of free real world-sheet fermionic degrees of freedom.",
"In the particular examples that we will be considering (which begin in $6D$ ), there are 8 right-moving and 20 left-moving complex Weyl fermions on the world-sheet.",
"Models are defined by the phases acquired under parallel transport around non-contractible cycles of the one-loop world-sheet, $\\mathbf {1}: ~~~f_{i_{R/L}}&\\rightarrow ~ -e^{-2\\pi i v_{i_{R/L}}}f_{i_{R/L}} \\nonumber \\\\\\tau : ~~~f_{i_{R/L}}&\\rightarrow ~ -e^{-2\\pi i u_{i_{R/L}}}f_{i_{R/L}}\\, ,$ where $i_R=1,\\dots ,8$ and $i_L=1,\\dots ,20$ , which we collect in vectors written as $v\\equiv \\lbrace v_R; v_L\\rbrace &\\equiv ~\\lbrace v_{i_R}; v_{i_L}\\rbrace \\nonumber \\\\u\\equiv \\lbrace u_R; u_L\\rbrace &\\equiv ~\\lbrace u_{i_R}; u_{i_L}\\rbrace \\, ,$ where $v_{i_R}, v_{i_L},u_{i_R}, u_{i_L} \\in [-\\frac{1}{2},\\frac{1}{2})$ .",
"The spin structure of the model is then given in terms of a set of basis vectors ${V}_i$ [79].",
"In order to define consistent modular invariant models, the basis vectors must obey $m_jk_{ij}&=&0 ~~~~~~~~~~~~~~~ {\\rm mod}\\,(1) \\nonumber \\\\k_{ij}+k_{ji}&=&{V}_i\\cdot {V}_j ~~~~~~~~~ {\\rm mod}\\,(1)\\nonumber \\\\k_{ii}+k_{i0}+s_i &=& \\frac{1}{2}{V}_i\\cdot {V}_i~~~~~~~ {\\rm mod}\\,(1)~,$ where the $k_{ij}$ are otherwise arbitrary structure constants that completely specify the theory, where $m_i$ is the lowest common denominator amongst the components of ${V}_i$ , and where $s_i\\equiv V_i^1$ is the spin-statistics associated with the vector $V_i$ .",
"The basis vectors span a finite additive group $G=\\sum _k\\alpha _k{V}_k$ where $\\alpha _k\\in \\lbrace 0,..., m-1\\rbrace $ , each element of which describes the boundary conditions associated with a different individual sector of the theory.",
"Within each sector $\\overline{\\alpha V}$ , the physical states are those which are level-matched and whose fermion-number operators ${N}_{\\overline{\\alpha V}}$ satisfy the generalized GSO projections ${\\: V}_i\\cdot {N}_{\\overline{\\alpha V}}~=~\\sum _jk_{ij}\\alpha _j+s_i-{\\: V}_i\\cdot \\overline{\\alpha {\\: V}} \\:\\: \\mbox{mod}\\:(1) ~~~~~ {\\rm for~all}~~i~.$ The world-sheet energies associated with such states are given by $M^2_{L,R} ~=~\\sum _{{\\ell }}\\left\\lbrace E_{\\overline{\\alpha V^{\\ell }}} + \\sum _{q=1}^\\infty \\left[(q-\\overline{\\alpha V^{\\ell }})\\overline{n}_q^{\\ell }+(q+\\overline{\\alpha V^{\\ell }}-1)n_q^{\\ell }\\right]\\right\\rbrace -\\frac{(D-2)}{24}+\\sum _{i=2}^D\\sum _{q=1}^\\infty qM_q^i$ where $\\ell $ sums over left- or right world-sheet fermions, where $n_q, \\overline{n}_q$ are the occupation numbers for complex fermions, where $M_q$ are the occupation numbers for complex bosons, and where $E_{\\overline{\\alpha V^{\\ell }}}$ is the vacuum-energy contribution of the $\\ell ^{\\rm th}$ complex world-sheet fermion: $E_{\\overline{\\alpha V^{\\ell }}}~=~\\frac{1}{2}\\left[(\\overline{\\alpha V^{\\ell }})^2-\\frac{1}{12}\\right]\\, .$ Moreover, the vector of $U(1)$ charges for each complex world-sheet fermion is given by $\\mathbf {Q}~=~{N}_{\\overline{\\alpha V}}+{\\overline{\\alpha V}}$ where ${\\overline{\\alpha V}}$ is 0 for an NS boundary condition and $-\\frac{1}{2}$ for a Ramond."
]
] | 1612.05742 |
[
[
"Video Propagation Networks"
],
[
"Abstract We propose a technique that propagates information forward through video data.",
"The method is conceptually simple and can be applied to tasks that require the propagation of structured information, such as semantic labels, based on video content.",
"We propose a 'Video Propagation Network' that processes video frames in an adaptive manner.",
"The model is applied online: it propagates information forward without the need to access future frames.",
"In particular we combine two components, a temporal bilateral network for dense and video adaptive filtering, followed by a spatial network to refine features and increased flexibility.",
"We present experiments on video object segmentation and semantic video segmentation and show increased performance comparing to the best previous task-specific methods, while having favorable runtime.",
"Additionally we demonstrate our approach on an example regression task of color propagation in a grayscale video."
],
[
"Introduction",
"In this work, we focus on the problem of propagating structured information across video frames.",
"This problem appears in many forms (e.g., semantic segmentation or depth estimation) and is a pre-requisite for many applications.",
"An example instance is shown in Fig.",
"REF .",
"Given an object mask for the first frame, the problem is to propagate this mask forward through the entire video sequence.",
"Propagation of semantic information through time and video color propagation are other problem instances.",
"Videos pose both technical and representational challenges.",
"The presence of scene and camera motion lead to the difficult pixel association problem of optical flow.",
"Video data is computationally more demanding than static images.",
"A naive per-frame approach would scale at least linear with frames.",
"These challenges complicate the use of standard convolutional neural networks (CNNs) for video processing.",
"As a result, many previous works for video propagation use slow optimization based techniques.",
"Figure: Video Propagation with VPNs.",
"The end-to-end trained VPN network is composedof a bilateral network followed by a standard spatial network and can be used forpropagating information across frames.",
"Shown here is an example propagationof foreground mask from the 1st frame to other video frames.We propose a generic neural network architecture that propagates information across video frames.",
"The main innovation is the use of image adaptive convolutional operations that automatically adapts to the video stream content.",
"This yields networks that can be applied to several types of information, e.g., labels, colors, etc.",
"and runs online, that is, only requiring current and previous frames.",
"Our architecture is composed of two components (see Fig.",
"REF ).",
"A temporal bilateral network that performs image-adaptive spatio-temporal dense filtering.",
"The bilateral network allows to connect densely all pixels from current and previous frames and to propagate associated pixel information to the current frame.",
"The bilateral network allows the specification of a metric between video pixels and allows a straight-forward integration of temporal information.",
"This is followed by a standard spatial CNN on the bilateral network output to refine and predict for the present video frame.",
"We call this combination a Video Propagation Network (VPN).",
"In effect, we are combining video-adaptive filtering with rather small spatial CNNs which leads to a favorable runtime compared to many previous approaches.",
"VPNs have the following suitable properties for video processing:"
],
[
"General applicability:",
"VPNs can be used to propagate any type of information content i.e., both discrete (e.g., semantic labels) and continuous (e.g., color) information across video frames."
],
[
"Online propagation:",
"The method needs no future frames and can be used for online video analysis."
],
[
"Long-range and image adaptive:",
"VPNs can efficiently handle a large number of input frames and are adaptive to the video with long-range pixel connections."
],
[
"End-to-end trainable:",
"VPNs can be trained end-to-end, so they can be used in other deep network architectures."
],
[
"Favorable runtime:",
"VPNs have favorable runtime in comparison to many current best methods, what makes them amenable for learning with large datasets.",
"Empirically we show that VPNs, despite being generic, perform better than published approaches on video object segmentation and semantic label propagation while being faster.",
"VPNs can easily be integrated into sequential per-frame approaches and require only a small fine-tuning step that can be performed separately."
],
[
"General propagation techniques",
"Techniques for propagating content across image/video pixels are predominantly optimization based or filtering techniques.",
"Optimization based techniques typically formulate the propagation as an energy minimization problem on a graph constructed across video pixels or frames.",
"A classic example is the color propagation technique from [46].",
"Although efficient closed-form solutions [47] exists for some scenarios, optimization tends to be slow due to either large graph structures for videos and/or the use of complex connectivity.",
"Fully-connected conditional random fields (CRFs) [41] open a way for incorporating dense and long-range pixel connections while retaining fast inference.",
"Filtering techniques [40], [15], [30] aim to propagate information with the use of image/video filters resulting in fast runtimes compared to optimization techniques.",
"Bilateral filtering [5], [73] is one of the popular filters for long-range information propagation.",
"A popular application is joint bilateral upsampling [40] that upsamples a low-resolution signal with the use of a high-resolution guidance image.",
"The works of [51], [22], [37], [34], [81], [66] showed that one can back-propagate through the bilateral filtering operation for learning filter parameters [37], [34] or doing optimization in the bilateral space [8], [7].",
"Recently, several works proposed to do upsampling in images by learning CNNs that mimic edge-aware filtering [78] or that directly learn to upsample [49], [32].",
"Most of these works are confined to images and are either not extendable or computationally too expensive for videos.",
"We leverage some of these previous works and propose a scalable yet robust neural network approach for video propagation.",
"We will discuss more about bilateral filtering, that forms the core of our approach, in Section ."
],
[
"Video object segmentation",
"Prior work on video object segmentation can be broadly categorized into two types: Semi-supervised methods that require manual annotation to define what is foreground object and unsupervised methods that does segmentation completely automatically.",
"Unsupervised techniques such as [25], [48], [45], [55], [77], [80], [72], [23] use some prior information about the foreground objects such as distinctive motion, saliency etc.",
"In this work, we focus on the semi-supervised task of propagating the foreground mask from the first frame to the entire video.",
"Existing works predominantly use graph-based optimization that perform graph-cuts [9], [10], [69] on video.",
"Several of these works [64], [50], [61], [76], [39], [33] aim to reduce the complexity of graph structure with clustering techniques such as spatio-temporal superpixels and optical flow [75].",
"Another direction was to estimate correspondence between different frame pixels [4], [6], [44] by using nearest neighbor fields [26] or optical flow [18].",
"Closest to our technique are the works of [60] and [53].",
"[60] proposed to use fully-connected CRF over the object proposals across frames.",
"[53] proposed a graph-cut in the bilateral space.",
"Instead of graph-cuts, we learn propagation filters in the high-dimensional bilateral space.",
"This results in a more generic architecture and allows integration into other deep networks.",
"Two contemporary works [14], [36] proposed CNN based approaches for object segmentation and rely on fine-tuning a deep network using the first frame annotation of a given test sequence.",
"This could result in overfitting to the test background.",
"In contrast, the proposed approach relies only on offline training and thus can be easily adapted to different problem scenarios as demonstrated in this paper."
],
[
"Semantic video segmentation",
"Earlier methods such as [12], [70] use structure from motion on video frames to compute geometrical and/or motion features.",
"More recent works [24], [16], [19], [54], [74], [43] construct large graphical models on videos and enforce temporal consistency across frames.",
"[16] used dynamic temporal links in their CRF energy formulation.",
"[19] proposes to use Perturb-and-MAP random field model with spatial-temporal energy terms and [54] propagate predictions across time by learning a similarity function between pixels of consecutive frames.",
"In the recent years, there is a big leap in the performance of semantic segmentation [52], [17] with the use of CNNs but mostly applied to images.",
"Recently, [67] proposed to retain the intermediate CNN representations while sliding a image CNN across the frames.",
"Another approach is to take unary predictions from CNN and then propagate semantic information across the frames.",
"A recent prominent approach in this direction is of [43] which proposes a technique for optimizing feature spaces for fully-connected CRF."
],
[
"Bilateral Filtering",
"We briefly review the bilateral filtering and its extensions that we will need to build VPN.",
"Bilateral filtering has its roots in image denoising [5], [73] and has been developed as an edge-preserving filter.",
"It has found numerous applications [58] and recently found its way into neural network architectures [81], [27].",
"We will use this filtering at the core of VPN and make use of the image/video-adaptive connectivity as a way to cope with scenes in motion.",
"Let $a,\\mathbf {a},A$ represent a scalar, vector and matrix respectively.",
"Bilateral filtering a vectorized image $\\mathbf {v} \\in \\mathbb {R}^n$ having $n$ image pixels can be viewed as a matrix-vector multiplication with a filter matrix $W \\in \\mathbb {R}^{n\\times n} $ : $\\hat{\\mathbf {v}}^i = \\sum _{j \\in n} W^{i,j} \\mathbf {v}^j,\\vspace{-5.69046pt}$ where the filter weights $W^{i,j}$ depend on features $F^i,F^j \\in \\mathbb {R}^g$ at input pixel indices $i,j$ and $F \\in \\mathbb {R}^{g \\times n}$ for $g$ -dimensional features.",
"For example a Gaussian bilateral filter amounts to a particular choice of $W$ as $W^{i,j} = \\frac{1}{\\eta }\\exp {(-\\frac{1}{2}(F^i-F^j)^{\\top }\\Sigma ^{-1}(F^i-F^j))}$ , where $\\eta $ is a normalization constant and $\\Sigma $ is covariance matrix.",
"The choice of features $F$ define the effect of the filter, the way it adapts to image content.",
"To use only positional features, $F^i=(x,y)^{\\top }$ , the bilateral filter operation reduces to a spatial Gaussian filter, with width controlled by $\\Sigma $ .",
"A common choice for edge-preserving filtering is to choose color and position features $F^i=(x,y,r,g,b)^{\\top }$ .",
"This results in image smoothing without blurring across the edges.",
"The filter values $W^{i,j}$ change for every pixel pairs $i,j$ and depend on the image/video content.",
"And since the number of image/video pixels is usually large, a naive implementation of Eq.",
"REF is prohibitive.",
"Due to the importance of this filtering operation, several fast algorithms [2], [3], [57], [28] have been proposed, that directly computes Eq.",
"REF without explicitly building $W$ matrix.",
"One natural view that inspired several implementations was offered by [57], who viewed the bilateral filtering operation as a computation in a higher dimensional space.",
"Their observation was that bilateral filtering can be implemented by 1. projecting $\\mathbf {v}$ into a high-dimensional grid (splatting) defined by features $F$ , 2. high-dimensional filtering (convolving) the projected signal and 3. projecting down the result at the points of interest (slicing).",
"The high-dimensional grid is also called bilateral space/grid.",
"All these operations are linear and written as: $\\hat{\\mathbf {v}} = S_{slice} B S_{splat} \\mathbf {v},\\vspace{-2.84544pt}$ where, $S_{splat}$ and $S_{slice}$ denotes the mapping to-from image pixels and bilateral grid, and $B$ denotes convolution (traditionally Gaussian) in the bilateral space.",
"The bilateral space has same dimensionality $g$ as features $F^i$ .",
"The problem with this approach is that a standard $g$ -dimensional convolution on a regular grid requires handling of an exponential number of grid points.",
"This was circumvented by a special data structure, the permutohedral lattice as proposed in [2].",
"Effectively permutohedral filtering scales linearly with dimension, resulting in fast execution time.",
"The recent work of [37], [34] then generalized the bilateral filter in the permutohedral lattice and demonstrated how it can be learned via back-propagation.",
"This allowed the construction of image-adaptive filtering operations into deep learning architectures, which we will build upon.",
"See Fig.",
"REF for a illustration of 2D permutohedral lattices.",
"Refer to [2] for more details on bilateral filtering using permutohedral lattice and refer to [34] for details on learning general permutohedral filters via back-propagation.",
"Figure: Schematic of Fast Bilateral Filtering for Video Processing.",
"Mask probabilities from previous frames 𝐯 1,...,t-1 \\mathbf {v}_{1,\\ldots ,t-1} are splatted on to thelattice positions defined by the image features F 1 ,F 2 ,...,F t-1 F_1,F_2,\\ldots ,F_{t-1}.The splatted result is convolved with a 1×11 \\times 1 filter BB, and the filteredresult is sliced back to the original image space to get 𝐯 t \\mathbf {v}_t for the present frame.Input and output need not be 𝐯 t \\mathbf {v}_t, but can also be any intermediate neural network representation.BB is learned via back-propagation through these operations."
],
[
"Video Propagation Networks",
"We aim to adapt the bilateral filtering operation to predict information forward in time, across video frames.",
"Formally, we work on a sequence of $h$ (color or grayscale) images $S = (\\mathbf {s}_1, \\mathbf {s}_2, \\ldots , \\mathbf {s}_h)$ and denote with $V = (\\mathbf {v}_1, \\mathbf {v}_2, \\ldots , \\mathbf {v}_h)$ a sequence of outputs, one per frame.",
"Consider as an example a sequence $\\mathbf {v}_1,\\ldots ,\\mathbf {v}_h$ of foreground masks for a moving object in the scene.",
"Our goal is to develop an online propagation method that can predict $\\mathbf {v}_t$ , having observed the video up to frame $t$ and possibly previous $\\mathbf {v}_{1,\\ldots ,t-1}$ $\\mathcal {F}(\\mathbf {v}_{t-1}, \\mathbf {v}_{t-2}, \\ldots ; \\mathbf {s}_t, \\mathbf {s}_{t-1}, \\mathbf {s}_{t-2},\\ldots ) = \\mathbf {v}_t.\\vspace{-2.84544pt}$ If training examples $\\lbrace (S_i,V_i) | i = 1,\\ldots ,l \\rbrace $ with full or partial knowledge of $\\mathbf {v}$ are available, it is possible to learn $\\mathcal {F}$ and for a complex and unknown input-output relationship, a deep CNN is a natural design choice.",
"However, any learning based method has to face the challenge: the scene/camera motion and its effect on $\\mathbf {v}$ .",
"Since no motion in two different videos is the same, fixed-size static receptive fields of CNN are insufficient.",
"We propose to resolve this with video-adaptive filtering component, an adaption of the bilateral filtering to videos.",
"Our Bilateral Network (Section REF ) has a connectivity that adapts to video sequences, its output is then fed into a spatial Network (Section REF ) that further refines the desired output.",
"The combined network layout of this VPN is depicted in Fig.",
"REF .",
"It is a sequence of learnable bilateral and spatial filters that is efficient, trainable end-to-end and adaptive to the video input."
],
[
"Bilateral Network (BNN)",
"Several properties of bilateral filtering make it a perfect candidate for information propagation in videos.",
"In particular, our method is inspired by two main ideas that we extend in this work: joint bilateral upsampling [40] and learnable bilateral filters [34].",
"Although, bilateral filtering has been used for filtering video data before [56], its use has been limited to fixed filter weights (say, Gaussian).",
"Fast Bilateral Upsampling across Frames The idea of joint bilateral upsampling [40] is to view upsampling as a filtering operation.",
"A high resolution guidance image is used to upsample a low-resolution result.",
"In short, a smaller number of input points are given $\\lbrace \\mathbf {v}_{in}^i,F_{in}^i| i=1,\\ldots ,n_{in}\\rbrace $ , for example a segmentation result $\\mathbf {v}_{in}$ at a lower resolution with the corresponding guidance image features $F_{in}$ .",
"This is then scaled to a larger number of output points $\\mathbf {v}_{out}$ with features $\\lbrace F_{out}^j|j=1,\\ldots ,n_{out}\\rbrace $ using the bilateral filtering operation, that is to compute Eq.",
"REF , where the sum runs over all $n_{in}$ points and the output is computed for all $n_{out}$ positions ($W \\in \\mathbb {R}^{n_{in} \\times n_{out}}$ ).",
"We will use this idea to propagate content from previous frames ($\\mathbf {v}_{in} = \\mathbf {v}_{1,\\ldots ,t-1}$ ) to the current frame ($\\mathbf {v}_{out} = \\mathbf {v}_t$ ).",
"The summation in Eq.",
"REF now runs over all previous frames and pixels.",
"This is illustrated in Fig.",
"REF .",
"We take all previous frame results $\\mathbf {v}_{1,\\ldots ,t-1}$ and splat them into a lattice using the features $F_{1,\\ldots ,t-1}$ computed on video frames $\\mathbf {s}_{1,\\ldots ,t-1}$ .",
"A filtering (described below) is then applied to every lattice point and the result is then sliced back using the features $F_t$ of the current frame $\\mathbf {s}_t$ .",
"This result need not be the final $\\mathbf {v}_t$ , in fact we compute a filter bank of responses and continue with further processing as will be discussed.",
"Standard bilateral features $F^i=(x,y,r,g,b)^{\\top }$ used for images need not be optimal for videos.",
"A recent work of [43] propose to optimize bilateral feature spaces for videos.",
"Instead, we choose to simply add frame index $t$ as an additional time feature yielding a 6 dimensional feature vector $F^i=(x,y,r,g,b,t)^{\\top }$ for every video pixel.",
"Imagine a video where an object moves to reveal some background.",
"Pixels of the object and background will be close spatially $(x,y)^\\top $ and temporally $(t)$ but likely be of different color $(r,g,b)^\\top $ .",
"Therefore they will have no strong influence on each other (being splatted to distant positions in the six-dimensional bilateral space).",
"One can understand the filter to be adaptive to color changes across frames, only pixels that are static and have similar color have a strong influence on each other (end up nearby in the bilateral space).",
"In all our experiments, we used time $t$ as additional feature for information propagation across frames.",
"In addition to adding time $t$ as additional feature, we also experimented with using optical flow.",
"We make use of optical flow estimates (of the previous frames with respect to the current frame) by warping pixel position features $(x,y)^\\top $ of previous frames by their optical flow displacement vectors $(u_x,u_y)^\\top $ to $(x+u_x,y+u_y)^\\top $ .",
"If the perfect flow was available, the video frames could be warped into a common frame of reference.",
"This would resolve the corresponding problem and make information propagation much easier.",
"We refer to the VPN model that uses modified positional features $(x+u_x,y+u_y)^\\top $ as VPN-Flow.",
"Another property of permutohedral filtering that we exploit is that the input points need not lie on a regular grid since the filtering is done in the high-dimensional lattice.",
"Instead of splatting millions of pixels on to the lattice, we randomly sample or use superpixels and perform filtering using these sampled points as input to the filter.",
"In practice, we observe that this results in big computational gains with minor drop in performance (more in Section REF ).",
"Figure: Computation Flow of Video Propagation Network.",
"Bilateral networks (BNN) consist of a series ofbilateral filterings interleaved with ReLU non-linearities.",
"The filtered information from BNN is then passedinto aspatial network (CNN) which refines the features with convolution layers interleaved with ReLUnon-linearities, resulting in the prediction for the current frame.Learnable Bilateral Filters Bilateral filters help in video-adaptive information propagation across frames.",
"But the standard Gaussian filter may be insufficient and further, we would like to increase the capacity by using a filter bank instead of a single fixed filter.",
"We propose to use the technique of [34] to learn a filter bank in the permutohedral lattice using back-propagation.",
"The process works as follows.",
"A input video is used to determine the positions in the bilateral space to splat the input points $\\mathbf {v}^i \\in \\mathbf {v}_{1,\\ldots ,t-1}$ of the previous frames.",
"In a general case, $\\mathbf {v}^i$ need not be a scalar and let us assume $\\mathbf {v}^i \\in \\mathbb {R}^d$ .",
"The features $F_{1,\\ldots ,t}$ (e.g.",
"$(x,y,r,g,b,t)^\\top $ ) define the splatting matrix $S_{splat}$ .",
"This leads to a number of vectors $\\mathbf {v}_{splatted} = S_{splat}\\mathbf {v}$ , that lie on the permutohedral lattice, with dimensionality $\\mathbf {v}^i_{splatted}\\in \\mathbb {R}^d$ .",
"In effect, the splatting operation groups points that are close together, that is, they have similar $F^i,F^j$ .",
"All lattice points are now filtered using a filter bank $B\\in \\mathbb {R}^{k\\times d}$ which results in $k$ dimensional vectors on the lattice points.",
"These are sliced back to the $n_{out}$ points of interest (present video frame).",
"The values of $B$ are learned by back-propagation.",
"General parametrization of $B$ from [34], [37] allows to have any neighborhood size for the filters.",
"Since constructing the neighborhood structure in high-dimensions is time consuming, we choose to use $1 \\times 1$ filters for speed reasons.",
"These three steps of splatting, convolving and slicing makes up one Bilateral Convolution Layer (BCL) which we will stack and concatenate to form a Bilateral Network.",
"See Fig.",
"REF for a BCL illustration.",
"BNN Architecture The Bilateral Network (BNN) is illustrated in the green box of Fig.",
"REF .",
"The input is a video sequence $S$ and the corresponding predictions $V$ up to frame $t$ .",
"Those are filtered using two BCLs (BCL$_a$ , BCL$_b$ ) with 32 filters each.",
"For both BCLs, we use the same features $F^i$ but scale them with different diagonal matrices: $\\Lambda _a F^i,\\Lambda _b F^i$ .",
"The feature scales ($\\Lambda _a,\\Lambda _b$ ) are found by validation.",
"The two 32 dimensional outputs are concatenated, passed through a ReLU non-linearity and passed to a second layer of two separate BCL filters that uses same feature spaces $\\Lambda _a F^i,\\Lambda _b F^i$ .",
"The output of the second filter bank is then reduced using a $1\\times 1$ spatial filter to map to the original dimension $d$ of $\\mathbf {v}$ .",
"We investigated scaling frame inputs with an exponential time decay and found that, when processing frame $t$ , a re-weighting with $(\\alpha \\mathbf {v}_{t-1}, \\alpha ^2 \\mathbf {v}_{t-2}, \\alpha ^3 \\mathbf {v}_{t-3} \\ldots )$ with $0\\le \\alpha \\le 1$ improved the performance a little bit.",
"In the experiments, we also included a simple BNN variant, where no filters are applied inside the permutohedral space, just splatting and slicing with the two layers BCL$_a$ and BCL$_b$ and adding the results.",
"We will refer to this model as BNN-Identity as this is equivalent to using filter $B$ that is identity matrix.",
"It corresponds to an image adaptive smoothing of the inputs $V$ .",
"We found this filtering to already have a positive effect in our experiments."
],
[
"Spatial Network",
"The BNN was designed to propagate information from the previous frames to the present one, respecting the scene and object motion.",
"We then add a small spatial CNN with 3 layers, each with 32 filters of size $3\\times 3$ , interleaved with ReLU non-linearities.",
"The final result is then mapped to the desired output of $\\mathbf {v}_t$ using a $1\\times 1$ convolution.",
"The main role of this spatial CNN is to refine the information in frame $t$ .",
"Depending on the problem and the size of the available training data, other network designs are conceivable.",
"We use the same network architecture shown in Fig.",
"REF for all the experiments to demonstrate the generality of VPNs."
],
[
"Experiments",
"We evaluated VPN on three different propagation tasks: propagation of foreground masks, semantic labels and color in videos.",
"Our implementation runs in Caffe [35] using standard settings.",
"We used Adam [38] stochastic optimization for training VPNs, multinomial-logistic loss for label propagation networks and Euclidean loss for training color propagation networks.",
"We use a fixed learning rate of 0.001 and choose the trained models with minimum validation loss.",
"Runtime computations were performed using a Nvidia TitanX GPU and a 6 core Intel i7-5820K CPU clocked at 3.30GHz machine.",
"The code is available online at http://varunjampani.github.io/vpn/."
],
[
"Video Object Segmentation",
"We focus on the semi-supervised task of propagating a given first frame foreground mask to all the video frames.",
"Object segmentation in videos is useful for several high level tasks such as video editing, rotoscoping etc."
],
[
"Dataset",
"We use the recently published DAVIS dataset [59] for experiments on this task.",
"It consists of 50 high-quality videos.",
"All the frames come with high-quality per-pixel annotation of the foreground object.",
"For robust evaluation and to get results on all the dataset videos, we evaluate our technique using 5-fold cross-validation.",
"We randomly divided the data into 5 folds, where in each fold, we used 35 videos for training, 5 for validation and the remaining 10 for the testing.",
"For the evaluation, we used the 3 metrics that are proposed in [59]: Intersection over Union (IoU) score, Contour accuracy ($\\mathcal {F}$ ) score and temporal instability ($\\mathcal {T}$ ) score.",
"The widely used IoU score is defined as $TP/(TP+FN+FP)$ , where TP: True Positives; FN: False Negatives and FP: False Positives.",
"Refer to [59] for the definition of the other two metrics.",
"Table: 5-Fold Validation on DAVIS Video Segmentation Dataset.",
"Average IoU scores for different models on the 5 folds.Table: Results of Video Object Segmentation on DAVIS dataset.",
"Average IoU score, contour accuracy (ℱ\\mathcal {F}),temporal instability (𝒯\\mathcal {T}) scores, and average runtimes (in seconds)per frame for different VPN models along with recent publishedtechniques for this task.",
"VPN runtimes also include superpixel computation (10ms).Runtimes of other methods are taken from , , which are indicative and are not directly comparable to our runtimes.Runtime of VPN-Stage2 includes the runtime of VPN-Stage1 which in turn includes the runtime of BNN-Identity.Runtime of VPN-DeepLab model includes the runtime of DeepLab."
],
[
"VPN and Results",
"In this task, we only have access to foreground mask for the first frame $\\mathbf {v}_1$ .",
"For the ease of training VPN, we obtain initial set of predictions with BNN-Identity.",
"We sequentially apply BNN-Identity at each frame and obtain an initial set of foreground masks for the entire video.",
"These BNN-Identity propagated masks are then used as inputs to train a VPN to predict the refined masks at each frame.",
"We refer to this VPN model as VPN-Stage1.",
"Once VPN-Stage1 is trained, its refined mask predictions are in-turn used as inputs to train another VPN model which we refer to as VPN-Stage2.",
"This resulted in further refinement of foreground masks.",
"Training further stages did not result in any improvements.",
"Instead, one could consider VPN as a RNN unit processing one frame after another.",
"But, due to GPU memory constraints, we opted for stage-wise training.",
"Figure: Random Sampling of Input Points vs. IoU.",
"The effect of randomly sampling points from input video frames on objectsegmentation IoU of BNN-Identity on DAVIS dataset.The points sampled are out of ≈\\approx 2 million points from the previous 5 frames.Following the recent work of [53] on video object segmentation, we used $F^i=(x,y,Y,Cb,Cr,t)^\\top $ features with YCbCr color features for bilateral filtering.",
"To be comparable with one of the fastest state-of-the-art technique [53], we do not use any optical flow information.",
"First, we analyze the performance of BNN-Identity by changing the number of randomly sampled input points.",
"Figure REF shows how the segmentation IoU changes with the number of sampled points (out of 2 million points) from the previous frames.",
"The IoU levels out after sampling 25% of the points.",
"For further computational efficiency, we used superpixel sampling instead of random sampling.",
"Compared to random sampling, usage of superpixels reduced the IoU slightly (0.5), while reducing the number of input points by a factor of 10.",
"We used 12000 SLIC [1] superpixels from each frame computed using the fast GPU implementation from [63].",
"As an input to VPN, we use the mask probabilities of previous 9 frames as we observe no improvements with more frames.",
"We set $\\alpha = 0.5$ and the feature scales ($\\Lambda _a,\\Lambda _b$ ) are presented in Tab.",
"REF .",
"Table REF shows the IoU scores for each of the 5 folds and Tab.",
"REF shows the overall scores and runtimes of different VPN models along with the best performing techniques.",
"The performance improved consistently across all 5 folds with the addition of new VPN stages.",
"BNN-Identity already performed reasonably well.",
"VPN outperformed the present fastest BVS method [53] by a significant margin on all the performance measures while being comparable in runtime.",
"VPN perform marginally better than OFL method [75] while being at least 80$\\times $ faster and OFL relies on optical flow whereas we obtain similar performance without using any optical flow.",
"Further, VPN has the advantage of doing online processing as it looks only at previous frames whereas BVS processes entire video at once.",
"Figure: Video Object Segmentation.",
"Shown are the different frames in example videos with the correspondingground truth (GT) masks, predictions from BVS ,OFL , VPN (VPN-Stage2) and VPN-DLab (VPN-DeepLab) models."
],
[
"Augmentation of Pre-trained Models",
"One of the main advantages of VPN is that it is end-to-end trainable and can be easily integrated into other deep networks.",
"To demonstrate this, we augmented VPN architecture with standard DeepLab segmentation network [17].",
"We replaced the last classification layer of DeepLab-LargeFOV model to output 2 classes (foreground and background) in our case and bi-linearly upsampled the resulting low-resolution probability map to the original image dimension.",
"5-fold fine-tuning of the DeepLab model on DAVIS dataset resulted in the average IoU of 57.0 and other scores are shown in Tab.",
"REF .",
"To construct a joint model, the outputs from the DeepLab and the bilateral network (in VPN) are concatenated and then passed on to the spatial CNN.",
"In other words, the bilateral network propagates label information from previous frames to the present frame, whereas the DeepLab network does the prediction for the present frame.",
"The results of both are then combined and refined by the spatial network in the VPN.",
"We call this `VPN-DeepLab' model.",
"We trained this model end-to-end and observed big improvements in performance.",
"As shown in Tab.",
"REF , the VPN-DeepLab model has the IoU score of 75.0 which is a significant improvement over the published results.",
"The total runtime of VPN-DeepLab is only 0.63s which makes this also one of the fastest techniques.",
"Figure REF shows some qualitative results with more in Figs.",
"REF , REF and REF .",
"One can obtain better VPN performance with using better superpixels and also incorporating optical flow, but this increases runtime as well.",
"Visual results indicate that learned VPN is able to retain foreground masks even with large variations in viewpoint and object size."
],
[
"Semantic Video Segmentation",
"This is the task of assigning semantic label to every video pixel.",
"Since the semantics between adjacent frames does not change radically, intuitively, propagating semantics across frames should improve the segmentation quality of each frame.",
"Unlike video object segmentation, where the mask for the first frame is given, we approach semantic video segmentation in a fully automatic fashion.",
"Specifically, we start with the unary predictions of standard CNNs and use VPN for propagating semantics across the frames."
],
[
"Dataset",
"We use the CamVid dataset [11] that contains 4 high quality videos captured at 30Hz while the semantically labeled 11-class ground truth is provided at 1Hz.",
"While the original dataset comes at a resolution of 960$\\times $ 720, we operate on a resolution of 640$\\times $ 480 as in [79], [43].",
"We use the same splits as in [70] resulting in 367, 100 and 233 frames with ground truth for training, validation and testing."
],
[
"VPN and Results",
"Since we already have CNN predictions for every frame, we train a VPN that takes the CNN predictions of previous and present frames as input and predicts the refined semantics for the present frame.",
"We compare with a state-of-the-art CRF approach [43] which we refer to as FSO-CRF.",
"We also experimented with optical flow in VPN and refer that model as VPN-Flow.",
"We used the fast DIS optical flow [42] and modify the positional features of previous frames.",
"We used superpixels computed with Dollar et al.",
"[20] as gSLICr [63] has introduced artifacts.",
"Table: Results of Semantic Segmentation on the CamVid Dataset.",
"Average IoU and runtimes (in seconds)per frame of different models on test split.Runtimes exclude CNN computations which are shown separately.VPN and BNN-Identity runtimes include superpixel computationof 0.23s (large portion of runtime).We experimented with predictions from two different CNNs: One is with dilated convolutions [79] (CNN-1) and another one [65] (CNN-2) is trained with the additional video game data, which is the present state-of-the-art on this dataset.",
"For CNN-1 and CNN-2, using 2 and 3 previous frames respectively as input to VPN is found to be optimal.",
"Other parameters of VPN are presented in Tab.",
"REF .",
"Table REF shows quantitative results.",
"Using BNN-Identity only slightly improved the performance whereas training the entire VPN significantly improved the CNN-1 performance by over 1.2 IoU, with both VPN and VPN-Flow.",
"Moreover, VPN is at least 25$\\times $ faster, and simpler to use compared to the optimization based FSO-CRF which relies on LDOF optical flow [13], long-term tacks [71] and edges [21].",
"Replacing bilateral filters with spatial filters in VPN improved the CNN-1 performance by only 0.3 IoU showing the importance of video-adaptive filtering.",
"We further improved the performance of the state-of-the-art CNN-2 [65] with VPN-Flow model.",
"Using better optical flow estimation might give even better results.",
"Figure REF shows some qualitative results with more in Fig.",
"REF .",
"Figure: Semantic Video Segmentation.",
"Input video frames and the corresponding ground truth (GT)segmentation together with the predictions of CNN and withVPN-Flow."
],
[
"Video Color Propagation",
"We also evaluate VPNs on a regression task of propagating color information in a grayscale video.",
"Given the color image for the first video frame, the task is to propagate the color to the entire video.",
"For experiments on this task, we again used the DAVIS segmentation dataset [59] with the first 25 frames from each video.",
"We randomly divided the dataset into 30 train, 5 validation and 15 test videos.",
"We work with YCbCr representation of images and propagate CbCr values from previous frames with pixel intensity, position and time features as guidance for VPN.",
"The same strategy as in object segmentation is used, where an initial set of color propagated results is obtained with BNN-Identity and then used to trained a VPN-Stage1 model.",
"Training further VPN stages did not improve the performance.",
"We use 300K randomly sampled points from previous 3 frames as input to the VPN network.",
"Table REF shows the PSNR results.",
"We also show a baseline result of [46] that does graph based optimization using optical flow.",
"We used fast DIS optical flow [42] in the baseline method [46] and we did not observe significant differences with using LDOF optical flow [13].",
"Figure REF shows a visual result with more in Fig.",
"REF .",
"VPN works reliably better than [46] while being 20$\\times $ faster.",
"The method of [46] relies heavily on optical flow and so the color drifts away with incorrect flow.",
"We observe that our method also bleeds color in some regions especially when there are large viewpoint changes.",
"We could not compare against recent color propagation techniques [31], [68] as their codes are not available online.",
"This application shows general applicability of VPNs in propagating different kinds of information.",
"Table: Results of Video Color Propagation.",
"Average Peak Signal-to-Noise Ratio (PSNR) and runtimes ofdifferent methods for video color propagation on images from DAVIS dataset.Figure: Video Color Propagation.",
"Input grayscale video frames and corresponding ground-truth (GT) color imagestogether with color predictions of Levin et al.",
"and VPN-Stage1 models."
],
[
"Conclusion",
"We proposed a fast, scalable and generic neural network approach for propagating information across video frames.",
"The VPN uses bilateral network for long-range video-adaptive propagation of information from previous frames to the present frame which is then refined by a spatial network.",
"Experiments on diverse tasks show that VPNs, despite being generic, outperformed the current state-of-the-art task-specific methods.",
"At the core of our technique is the exploitation and modification of learnable bilateral filtering for the use in video processing.",
"We used a simple VPN architecture to showcase the generality.",
"Depending on the problem and the availability of data, using more filters or deeper layers would result in better performance.",
"In this work, we manually tuned the feature scales which could be amendable to learning.",
"Finding optimal yet fast-to-compute bilateral features for videos together with the learning of their scales is an important future research direction."
],
[
"Acknowledgments",
"We thank Vibhav Vineet for providing the trained image segmentation CNN models for CamVid dataset.",
"figuresection tablesection Parameters and Additional Results In this appendix, we present experiment protocols and additional qualitative results for experiments on video object segmentation, semantic video segmentation and video color propagation.",
"Table REF shows the feature scales and other parameters used in different experiments.",
"Figures REF , REF show some qualitative results on video object segmentation with some failure cases in Fig.",
"REF .",
"Figure REF shows some qualitative results on semantic video segmentation and Fig.",
"REF shows results on video color propagation.",
"Table: Experiment Protocols.",
"Experiment protocols for the different experiments presented in this work.",
"Feature Types:Feature spaces used for the bilateral convolutions, with position (x,yx,y) and color(R,G,BR,G,B or Y,Cb,CrY,Cb,Cr) features ∈[0,255]\\in [0,255].",
"u x u_x, u y u_y denotes optical flow with respectto the present frame and II denotes grayscale intensity.Feature Scales (Λ a ,Λ b \\Lambda _a, \\Lambda _b): Cross-validated scales for the features used.α\\alpha : Exponential time decay for the input frames.Input Frames: Number of input frames for VPN.Loss Type: Typeof loss used for back-propagation.",
"“MSE” corresponds to Euclidean mean squared error loss and “Logistic” corresponds to multinomial logistic loss.Figure: Video Object Segmentation.",
"Shown are the different frames in example videos with the correspondingground truth (GT) masks, predictions from BVS ,OFL , VPN (VPN-Stage2) and VPN-DLab (VPN-DeepLab) models.Figure: Video Object Segmentation.",
"Shown are the different frames in example videos with the correspondingground truth (GT) masks, predictions from BVS ,OFL , VPN (VPN-Stage2) and VPN-DLab (VPN-DeepLab) models.Figure: Failure Cases for Video Object Segmentation.",
"Shown are the different frames in example videos with the correspondingground truth (GT) masks, predictions from BVS ,OFL , VPN (VPN-Stage2) and VPN-DLab (VPN-DeepLab) models.Figure: Semantic Video Segmentation.",
"Input video frames and the corresponding ground truth (GT)segmentation together with the predictions of CNN and withVPN-Flow.Figure: Video Color Propagation.",
"Input grayscale video frames and corresponding ground-truth (GT) color imagestogether with color predictions of Levin et al.",
"and VPN-Stage1 models."
],
[
"Parameters and Additional Results",
"In this appendix, we present experiment protocols and additional qualitative results for experiments on video object segmentation, semantic video segmentation and video color propagation.",
"Table REF shows the feature scales and other parameters used in different experiments.",
"Figures REF , REF show some qualitative results on video object segmentation with some failure cases in Fig.",
"REF .",
"Figure REF shows some qualitative results on semantic video segmentation and Fig.",
"REF shows results on video color propagation.",
"Table: Experiment Protocols.",
"Experiment protocols for the different experiments presented in this work.",
"Feature Types:Feature spaces used for the bilateral convolutions, with position (x,yx,y) and color(R,G,BR,G,B or Y,Cb,CrY,Cb,Cr) features ∈[0,255]\\in [0,255].",
"u x u_x, u y u_y denotes optical flow with respectto the present frame and II denotes grayscale intensity.Feature Scales (Λ a ,Λ b \\Lambda _a, \\Lambda _b): Cross-validated scales for the features used.α\\alpha : Exponential time decay for the input frames.Input Frames: Number of input frames for VPN.Loss Type: Typeof loss used for back-propagation.",
"“MSE” corresponds to Euclidean mean squared error loss and “Logistic” corresponds to multinomial logistic loss.Figure: Video Object Segmentation.",
"Shown are the different frames in example videos with the correspondingground truth (GT) masks, predictions from BVS ,OFL , VPN (VPN-Stage2) and VPN-DLab (VPN-DeepLab) models.Figure: Video Object Segmentation.",
"Shown are the different frames in example videos with the correspondingground truth (GT) masks, predictions from BVS ,OFL , VPN (VPN-Stage2) and VPN-DLab (VPN-DeepLab) models.Figure: Failure Cases for Video Object Segmentation.",
"Shown are the different frames in example videos with the correspondingground truth (GT) masks, predictions from BVS ,OFL , VPN (VPN-Stage2) and VPN-DLab (VPN-DeepLab) models.Figure: Semantic Video Segmentation.",
"Input video frames and the corresponding ground truth (GT)segmentation together with the predictions of CNN and withVPN-Flow.Figure: Video Color Propagation.",
"Input grayscale video frames and corresponding ground-truth (GT) color imagestogether with color predictions of Levin et al.",
"and VPN-Stage1 models."
]
] | 1612.05478 |
[
[
"Pair correlations and equidistribution"
],
[
"Abstract A deterministic sequence of real numbers in the unit interval is called \\emph{equidistributed} if its empirical distribution converges to the uniform distribution.",
"Furthermore, the limit distribution of the pair correlation statistics of a sequence is called Poissonian if the number of pairs $x_k,x_l \\in (x_n)_{1 \\leq n \\leq N}$ which are within distance $s/N$ of each other is asymptotically $\\sim 2sN$.",
"A randomly generated sequence has both of these properties, almost surely.",
"There seems to be a vague sense that having Poissonian pair correlations is a \"finer\" property than being equidistributed.",
"In this note we prove that this really is the case, in a precise mathematical sense: a sequence whose asymptotic distribution of pair correlations is Poissonian must necessarily be equidistributed.",
"Furthermore, for sequences which are not equidistributed we prove that the square-integral of the asymptotic density of the sequence gives a lower bound for the asymptotic distribution of the pair correlations."
],
[
"Preliminaries",
"Throughout this section, we will use the following notation.",
"Assume that $x_1, \\dots , x_N$ are given.",
"Let $F_N(s)$ be defined as in (REF ).",
"We partition the unit interval $[0,1)$ into subintervals $I_1, \\dots , I_M$ , where $I_m = [m/M,(m+1)/M)$ , and we set $y_m = \\# \\Big \\lbrace 1 \\le n \\le N:~x_n \\in I_m \\Big \\rbrace .$ Then trivially we have $\\sum _{m=1}^M y_m = N.$ For notational convenience, we assume that the sequence $(y_m)_{1 \\le m \\le M}$ and the partition $I_1, \\dots , I_M$ are extended periodically; in other words, we set $y_m = y_{(m~\\textup {mod}~M)}, \\qquad \\textrm {and} \\qquad I_m = I_{(m~\\textup {mod}~M)}, \\qquad m \\in \\mathbb {Z}.$ Let $s \\ge 1$ be an integer.",
"We set $H_{N,M} (s) = \\sum _{m=1}^M~ \\sum _{-s+1 \\le \\ell \\le s-1} y_m y_{m + \\ell }.$ Then by construction we have $H_{N,M}(s) & = & \\sum _{m=1}^M ~~\\sum _{n \\in \\lbrace 1, \\dots , N\\rbrace :~x_n \\in I_m}~ \\# \\left\\lbrace 1 \\le k \\le N:~x_k \\in \\bigcup _{\\ell =-s+1}^{s-1} I_{m+\\ell } \\right\\rbrace \\nonumber \\\\& \\le & \\sum _{n=1}^N \\left\\lbrace 1 \\le k \\le N:~\\Vert x_k - x_n\\Vert \\le \\frac{s}{M} \\right\\rbrace \\nonumber \\\\& = & \\left( \\sum _{n=1}^N \\left\\lbrace 1 \\le k \\le N, ~n \\ne k:~\\Vert x_k - x_n\\Vert \\le \\frac{s}{M} \\right\\rbrace \\right) + N \\nonumber \\\\& = & NF_N\\left( \\frac{sN}{M} \\right) + N. $ Thus a lower bound for $H_{N,M}$ implies a lower bound for $F_N$ .",
"We have the following lemma.",
"Lemma 1 Let $y_1, \\dots , y_M$ be non-negative real numbers whose sum is $N$ , assume that $(y_m)_{1 \\le m \\le M}$ is extended periodically as above, and let $H_{N,M}(s)$ be defined as above.",
"Let $S \\ge 1$ be an integer for which $2S < M$ .",
"Then $\\frac{1}{S} \\sum _{s=1}^S H_{N,M}(s) \\ge \\frac{SN^2}{M}.$ The sum $\\sum _{m=1}^M ~\\sum _{-s+1 \\le \\ell \\le s-1} y_m y_{m + \\ell }$ in the definition of $H_{N,M}$ is a quadratic form which is attached to the matrix $A^{(s)} = \\left(a_{ij}^{(s)}\\right)_{1 \\le i,j \\le M} = \\left\\lbrace \\begin{array}{ll} 1 & \\text{if $\\textup {dist}(i-j) \\le s-1$,}\\\\ 0 & \\text{otherwise,} \\end{array} \\right.$ where $\\textup {dist}$ is the periodic distance such that $\\textup {dist}(i-j) \\le s-1$ whenever $i-j ~\\in ~(-\\infty ,-M+s-1] \\cup [-s+1,s-1] \\cup [M-s+1,\\infty ).$ Thus $A^{(s)}$ is a band matrix which also has non-zero entries in its right upper and left lower corner.",
"This matrix $A^{(s)}$ is symmetric, and it is of a form which is called circulant.",
"Generally, a circulant matrix is a matrix of the form $\\begin{pmatrix}c_0 & c_1 & c_2 & \\dots & c_{M-1} \\\\c_{M-1} & c_0 & c_1 & \\dots & c_{M-2} \\\\c_{M-2} & c_{M-1} & c_0 & \\ddots & c_{M-3} \\\\\\vdots & & \\ddots & \\ddots & \\vdots \\\\c_1 & & \\dots & c_{M-1} & c_0\\end{pmatrix},$ where each row is obtained by a cyclic shift of the previous row.",
"We recall some properties of such matrices; for a reference see for example [5].",
"The eigenvectors of such a matrix are $ v_m = \\left(1, \\omega ^m, \\omega ^{2m}, \\dots , \\omega ^{(M-1)m} \\right), \\qquad m = 0, \\dots , M-1,$ where $\\omega = e^{\\frac{2 \\pi i}{M}}$ .",
"Note that these eigenvectors are pairwise orthogonal, and that they are independent of the coefficients of the matrix (they just depend on the fact that the matrix is circulant).",
"The eigenvalue $\\lambda _m$ to the eigenvector $v_m$ is given by $ \\lambda _m = \\sum _{\\ell =0}^{M-1} c_\\ell \\omega ^m.$ We have already noted that our matrix $A^{(s)}$ is symmetric, which implies that all its eigenvalues are real.",
"Furthermore, if we use the formula (REF ) to calculate the eigenvalues of $A^{(s)}$ then we obtain $\\lambda _m^{(s)} = \\sum _{\\ell =-s+1}^{s-1} \\omega ^m = \\frac{\\sin \\left(\\frac{(2s-1)\\pi m}{M} \\right)}{\\sin \\left(\\frac{\\pi m}{M}\\right)},$ which is the $s-1$ -st order Dirichlet kernel $D_{s-1}$ (with period 1 rather than the more common period $2 \\pi $ ), evaluated at position $m/M$ .",
"Note that the largest eigenvalue is $\\lambda _0^{(s)} = 2s-1$ .",
"Since the eigenvectors of $A^{(s)}$ form an orthogonal basis, we can express our vector $(y_1, \\dots , y_M)$ in this basis.",
"We write $(y_1, \\dots , y_M) = \\sum _{m=0}^{M-1} \\varepsilon _m v_m$ for appropriate coefficients $(\\varepsilon _m)_{1 \\le m \\le M}$ .",
"Note that we have $y_1+\\dots +y_M=N$ , which can be rewritten as $(y_1, \\dots , y_M) v_0 = N$ ; thus we must have $\\varepsilon _0 = N/M$ (since the eigenvectors are orthogonal).",
"Furthermore, we have $H_{N,M}(s) & = & \\left( \\sum _{m=0}^{M-1} \\varepsilon _m v_m \\right)^T A^{(s)} \\left( \\sum _{m=0}^{M-1} \\varepsilon _m v_m \\right) \\nonumber \\\\& = & \\sum _{m=0}^{M-1} \\lambda _m^{(s)} \\varepsilon _m^2 \\Vert v_m\\Vert _2^2 \\nonumber \\\\& = & M \\sum _{m=0}^{M-1} \\lambda _m^{(s)} \\varepsilon _m^2, $ again by orthogonality.",
"However, from this we cannot deduce that $H_{N,M}(s) \\ge M \\lambda _0^{(s)} \\varepsilon _0^2 = (2s-1) N^2/M$ , since (in general) some of the eigenvalues are negative.",
"To solve this problem we will make a transition from the Dirichlet kernel to the Fejér kernel, which is non-negative.",
"We repeat that the eigenvectors of $A^{(s)}$ depend on $M$ , but not on $s$ .",
"Let $S \\ge 1$ be an integer and consider $A^{(\\Sigma )} = \\frac{1}{S} \\sum _{s=1}^S A^{(s)},$ where we assume that $S < 2M$ (to retain the structure of the matrix).",
"Then clearly the eigenvectors of this matrix are also given by $v_0, \\dots , v_{M-1}$ , and the corresponding eigenvalues are $\\lambda _m^{(\\Sigma )} = \\frac{1}{S} \\sum _{s=1}^{S} \\lambda _m^{(s)} = \\frac{1}{S} \\sum _{s=1}^{S} ~\\sum _{\\ell =-s+1}^{s-1} \\omega ^m, \\qquad 0 \\le m \\le M-1.$ Now $\\lambda _m^{(\\Sigma )}$ can be identified as the Fejér kernel of order $S-1$ (with period 1 instead of $2 \\pi $ ), evaluated at position $m/M$ .",
"It is well-known that the Fejér kernel is non-negative, so we have $ \\lambda _m^{(\\Sigma )} \\ge 0, \\qquad m = 0, \\dots , M-1,$ and we also have $\\lambda _0^{(\\Sigma )} = \\frac{1}{S} \\sum _{s=1}^{S} (2s-1) = S.$ Now using again the considerations which led to (REF ) we can show that $\\frac{1}{S} \\sum _{s=1}^S H_{N,M}(s) \\ge M \\sum _{m=0}^{M-1} \\lambda _m^{(\\Sigma )} \\varepsilon _m^2 \\ge M \\lambda _0^{(\\Sigma )} \\varepsilon _0^2 = SN^2/M,$ where (REF ) played a crucial role.",
"This proves the lemma."
],
[
"Proof of Theorem ",
"Let $(x_n)_{n \\ge 1}$ be a sequence of real numbers in $[0,1]$ , and assume that it is not equidistributed.",
"Thus there exists an $a \\in (0,1)$ for which $\\frac{1}{N} \\sum _{n=1}^N {1}_{[0,a)} (x_n) \\lnot \\rightarrow a \\qquad \\text{as $N \\rightarrow \\infty $}$ (here, and in the sequel, ${1}_B$ denotes the indicator function of a set $B$ ).",
"However, for this value of $a$ by the Bolzano–Weierstraß theorem there exists a subsequence $(N_r)_{r \\ge 1}$ of $\\mathbb {N}$ along which a limit exists; that is, there exists a number $b \\ne a$ such that $ \\lim _{r \\rightarrow \\infty } \\frac{1}{N_r} \\sum _{n=1}^{N_r} {1}_{[0,a)} (x_n) = b.$ Let $\\varepsilon >0$ be given, and assume that $\\varepsilon $ is “small”.",
"Choose an integer $S$ (which is “large”).",
"Let $r \\ge 1$ be given, let $N_r$ be from the subsequence in the previous paragraph, and consider the points $x_1, \\dots , x_{N_r}$ .",
"Let $\\mathcal {E}$ denote the union of the sets $\\left[0,\\frac{2S}{N_r} \\right] \\cup \\left[a - \\frac{2S}{N_r}, a + \\frac{2S}{N_r} \\right] \\cup \\left[1-\\frac{2S}{N_r},1 \\right].$ Furthermore, we set $B_1 = [0,a] \\backslash \\mathcal {E}$ and $B_2=[a,1] \\backslash \\mathcal {E}$ .",
"First consider the case that $\\# \\left\\lbrace 1 \\le n \\le N_r:~ x_n \\in \\mathcal {E} \\right\\rbrace \\ge \\varepsilon N_r$ .",
"Then by the pigeon hole principle there exists an interval of length at most $1/N_r$ in $\\mathcal {E}$ which contains at least $\\varepsilon N_r/(8S)$ elements of $\\lbrace x_1, \\dots , x_{N_r}\\rbrace $ .",
"All of these numbers are within distance $1/N_r$ of each other, which implies that $N_r F_{N_r}(1) \\ge \\left(\\frac{\\varepsilon N_r}{8S} \\right)^2 - N_r.$ If this inequality holds for infinitely many $r$ , then $\\limsup _{r \\rightarrow \\infty } F_{N_r}(1) = \\infty ,$ which implies that the pair correlations distribution cannot be asymptotically Poissonian.",
"Thus we may assume that $\\# \\left\\lbrace 1 \\le n \\le N:~ x_n \\in \\mathcal {E} \\right\\rbrace < \\varepsilon N_r$ for all elements of the subsequence $(N_r)_{r \\ge 1}$ .",
"Then $[0,1] \\backslash \\mathcal {E} = B_1 \\cup B_2$ contains at least $(1-\\varepsilon ) N_r$ elements of $\\lbrace x_1, \\dots , x_{N_r}\\rbrace $ .",
"Consequently, if $r$ is sufficiently large, by (REF ) we have $\\# \\lbrace 1 \\le n \\le N_r: x_n \\in B_1\\rbrace \\ge (b-2\\varepsilon ) N_r,$ and $\\# \\lbrace 1 \\le n \\le N_r: x_n \\in B_2\\rbrace \\ge ((1-b)-2\\varepsilon ) N_r.$ We assume that $r$ is so large that we can find positive integers $M_1, M_2$ for which $a/M_1 \\approx (1-a)/M_2 \\approx \\frac{1}{N_r}$ ; more precisely, we demand that $ \\frac{a}{M_1} \\in \\left[\\frac{1-\\varepsilon }{N_r}, \\frac{1}{N_r} \\right], \\qquad \\frac{1-a}{M_2} \\in \\left[ \\frac{1-\\varepsilon }{N_r}, \\frac{1}{N_r} \\right].$ We partition $B_1$ and $B_2$ into $M_1$ and $M_2$ disjoint subintervals of equal length, respectively, and write $y_1, \\dots , y_{M_1}$ and $z_1, \\dots , z_{M_2}$ for the number of elements contained in each of these subintervals (we assume that the subintervals are sorted in the “natural” order from left to right).",
"Next, for $s \\in \\lbrace 1, \\dots , S\\rbrace $ we define $H_{M_1}^* (s) = \\sum _{m=1}^{M_1} ~\\sum _{-s+1 \\le \\ell \\le s-1} y_m y_{m + \\ell }$ and $H_{M_2}^* (s) = \\sum _{m=1}^{M_2} ~\\sum _{-s+1 \\le \\ell \\le s-1} z_m z_{m + \\ell }.$ By construction, $\\sum _{m=1}^{M_1} y_m \\ge (b-2\\varepsilon ) N_r$ and $\\sum _{m=1}^{M_2} z_m \\ge ((1-b)-2\\varepsilon )N$ .",
"Also by construction the cyclic extension is not necessary here, provided that $r$ is sufficiently large; by excluding all the points in $\\mathcal {E}$ we have $y_1= \\dots = y_S = 0$ and $y_{M_1-S+1} = \\dots = y_{M_1}= 0$ , and the same holds for the $z_m$ 's.",
"Then by Lemma REF and by our choice of $M_1,M_2$ we have $ \\frac{1}{S} \\sum _{s=1}^S H_{M_1}^* (s) \\ge \\frac{S ((b-2\\varepsilon )N)^2}{M_1} \\ge \\frac{S (b-2\\varepsilon )^2 N_r (1-\\varepsilon )}{a},$ and $ \\frac{1}{S} \\sum _{s=1}^S H_{M_2}^* (s) \\ge \\frac{S ((1-b-2\\varepsilon )N)^2}{M_2} \\ge \\frac{S (1-b-2\\varepsilon )^2 N_r (1-\\varepsilon )}{1-a}.$ As in the calculation leading to (REF ) we can obtain a lower bound for the pair correlation function $F_{N_r}(s)$ from the lower bounds for $H_{M_1}^*$ and $H_{M_2}^*$ .",
"More precisely, we obtain $N_r F_{N_r} \\left(s\\right) + N_r \\ge H_{M_1}^*(s) + H_{M_2}^*(s),$ and accordingly, by (REF ) and (REF ), we have $& & \\frac{1}{S} \\sum _{s=1}^S \\left( N_r F_{N_r} (s) + N_r \\right) \\nonumber \\\\& \\ge & (1 - \\varepsilon ) S N_r \\left(\\frac{(b-\\varepsilon )^2}{a}+ \\frac{(1-b-\\varepsilon )^2}{1-a}\\right).",
"$ Now note that for $0 \\le a,b \\le 1$ we can only have $b^2/a + (1-b)^2/(1-a)=1$ if $a=b$ ; however, this is ruled out by assumption.",
"For all other pairs $(a,b)$ we have $b^2/a + (1-b)^2/(1-a)>1$ , and thus (REF ) implies that $\\frac{1}{S} \\sum _{s=1}^S N_r F_{N_r}(s) \\ge N_r \\left(S (1+2c_{\\varepsilon }) -1\\right)$ for a positive constant $c_{\\varepsilon }$ depending only on $\\varepsilon $ , provided that $\\varepsilon $ is sufficiently small.",
"This implies $ \\frac{1}{S} \\sum _{s=1}^S F_{N_r} (s) \\ge S (1+2c_{\\varepsilon }) - 1 \\ge S (1+c_{\\varepsilon }) \\left(1 + \\frac{1}{S} \\right),$ where the last inequality holds under the assumption that $S$ is sufficiently large.",
"Consequently there exists an $s \\in \\lbrace 1,\\dots , S\\rbrace $ such that $ F_{N_r}(s) \\ge (1+c_{\\varepsilon }) 2s,$ since otherwise (REF ) is impossible.",
"For every sufficiently large $N_r$ in the subsequence in (REF ) such an $s \\in \\lbrace 1, \\dots , S\\rbrace $ exists; accordingly, there is an $s$ such that for infinitely many $r$ we have (REF ).",
"Thus for this $s$ we have $\\limsup _{r \\rightarrow \\infty } \\frac{F_{N_r}(s)}{2s} \\ge (1+c_{\\varepsilon }) > 1,$ which proves the theorem."
],
[
"Proof of Theorem ",
"First assume that the measure $\\mu _G$ defined by the asymptotic distribution function $G(x)$ is not absolutely continuous with respect to the Lebesgue measure.",
"A function which is not absolutely continuous is not Lipschitz continuous as well.",
"Thus there is an $\\varepsilon >0$ such that for every $\\delta >0$ there exists an interval $I \\subset [0,1]$ such that $\\lambda (I) \\le \\delta , \\qquad \\text{but} \\qquad \\mu _G(I) \\ge \\varepsilon ,$ where $\\lambda $ denotes the Lebesgue measure (that is, the length) of $I$ .",
"Let $\\hat{I}$ denote the interval $I$ after removing a subinterval of length $N^{-1}$ from the left and right end, respectively (to remove the influence of the cyclic “overlap” in Lemma REF ).",
"Then for sufficiently large $N$ the interval $\\hat{I}$ contains at least $\\varepsilon N/2$ elements of $(x_n)_{1 \\le n \\le N}$ .",
"Set $M = \\lceil \\lambda (I) N \\delta ^{-1/2} \\rceil $ , split $I$ into $M$ subintervals, and denote the number of elements of the set $\\lbrace x_1, \\dots , x_N\\rbrace \\cap \\hat{I}$ contained in each of these subintervals by $y_1, \\dots , y_M$ , respectively.",
"Let $\\hat{N} = y_1 + \\dots + y_M$ , and define $H_{M,\\hat{N}}(1) = y_1^2 + \\dots + y_M^2$ .",
"Applying Lemma REF and using a rescaled version of (REF ) we have $N F_N \\left(2 \\delta ^{1/2} \\right) + N & \\ge & N F_N \\left(\\frac{2 N \\lambda (I)}{M} \\right) + N \\\\& \\ge & H_{M,\\hat{N}} (1) \\\\& \\ge & \\frac{(\\varepsilon N/2)^2}{M} \\\\& \\ge & \\frac{\\varepsilon ^2 N}{5 \\delta ^{1/2}}$ for sufficiently large $N$ .",
"Since $\\varepsilon $ is fixed and $\\delta $ can be chosen arbitrarily small, this proves the theorem when $\\mu _G$ is not absolutely continuous.",
"Now assume that the measure $\\mu _G$ defined by $G(x)$ is absolutely continuous with respect to the Lebesgue measure, and thus has a density $g(x)$ .",
"In the sequel we will think of $([0,1],\\mathcal {B}([0,1]), \\lambda )$ as a probability space, and write $\\mathbb {E}$ for the expected value (of a measurable real function) with respect to this space.",
"We split the unit interval into $2^R$ intervals of equal lengths.",
"Let $\\mathcal {F}_R$ denote the $\\sigma $ -field generated by these intervals.",
"Assume for simplicity that $g$ is bounded on $[0,1]$ .",
"Then we can use arguments similar to those above to prove that for given $\\varepsilon >0$ there exist infinitely many values of $s$ such that for each of these values we have $\\frac{F\\left(s\\right)}{2s} \\ge \\mathbb {E} \\left( \\left(\\mathbb {E} \\big ( g | \\mathcal {F}_R \\big ) \\right)^2 \\right) - \\varepsilon ,$ where $\\mathbb {E} \\left( g | \\mathcal {F}_k \\right)$ denotes the conditional expectation of $g$ under the $\\sigma $ -field $\\mathcal {F}_R$ .In the proof of Theorem REF , the role of the second moment of the conditional expectation function is played by the expression $b^2/a + (1-b)^2/(1-a)$ , which appears in line (REF ).",
"Note that a direct generalization of the proof of Theorem REF only guarantees the existence of one such integer $s$ ; however, we can use the fact that $F(s)$ is monotonically increasing to show that there actually must be infinitely many such values of $s$ .",
"The family $(\\mathcal {F}_R)_{R \\ge 1}$ forms a filtration whose limit is $\\mathcal {B}([0,1])$ , in the sense that $\\mathcal {B}([0,1])$ is the sigma-field generated by $\\bigcup _{R \\ge 1} \\mathcal {F}_R$ .",
"Thus by the convergence theorem for conditional expectations (also known as Lévy's zero-one law) we have $\\lim _{R \\rightarrow \\infty } \\mathbb {E} \\left( \\left(\\mathbb {E} \\big ( g | \\mathcal {F}_R \\big ) \\right)^2 \\right) = \\mathbb {E} \\left(g^2 \\right) = \\int _0^1 g(x)^2~dx,$ which proves the theorem in the case when $g$ is bounded.",
"Finally, if $g$ is not bounded then we can apply the argument above to a truncated version $g_{\\textup {trunc}}$ of $g$ and show that in this case $\\limsup _{s \\rightarrow \\infty } \\frac{F(s)}{2s} \\ge \\int _0^1 g_{\\textup {trunc}}(x)^2~dx.$ By raising the level where $g$ is truncated this square-integral can be made arbitrarily close to $\\int _0^1 g(x)^2~dx$ , or arbitrarily large in case we have $\\int _0^1 g(x)^2~dx= \\infty $ .",
"This proves the theorem."
],
[
"Acknowledgements",
"We want to thank Ivan Izmestiev (University of Fribourg) for his very helpful comments during the preparation of this manuscript."
]
] | 1612.05495 |
[
[
"High order numerical schemes for one-dimension non-local conservation\n laws"
],
[
"Abstract This paper focuses on the numerical approximation of the solutions of non-local conservation laws in one space dimension.",
"These equations are motivated by two distinct applications, namely a traffic flow model in which the mean velocity depends on a weighted mean of the downstream traffic density, and a sedimentation model where either the solid phase velocity or the solid-fluid relative velocity depends on the concentration in a neighborhood.",
"In both models, the velocity is a function of a convolution product between the unknown and a kernel function with compact support.",
"It turns out that the solutions of such equations may exhibit oscillations that are very difficult to approximate using classical first-order numerical schemes.",
"We propose to design Discontinuous Galerkin (DG) schemes and Finite Volume WENO (FV-WENO) schemes to obtain high-order approximations.",
"As we will see, the DG schemes give the best numerical results but their CFL condition is very restrictive.",
"On the contrary, FV-WENO schemes can be used with larger time steps.",
"We will see that the evaluation of the convolution terms necessitates the use of quadratic polynomials reconstructions in each cell in order to obtain the high-order accuracy with the FV-WENO approach.",
"Simulations using DG and FV-WENO schemes are presented for both applications."
],
[
"Introduction",
"This paper is concerned with the design of numerical schemes for the one-dimensional Cauchy problem for non-local scalar conservation law of the form ${\\left\\lbrace \\begin{array}{ll}\\rho _t+(f(\\rho )V(\\rho *\\omega _\\eta ))_x=0, & x\\in \\mathbb {R},\\quad t>0, \\\\ \\rho (x,0)=\\rho _0(x) & x\\in \\mathbb {R},\\end{array}\\right.",
"}$ where the unknown density $\\rho $ depends on the space variable $x$ and the time variable $t$ , $\\rho \\rightarrow V(\\rho )$ is a given velocity function, and $\\rho \\rightarrow g(\\rho ) = f(\\rho )V(\\rho )$ is the usual flux function for the corresponding local scalar conservation law.",
"Here, (REF ) is non-local in the sense that the velocity function $V$ is evaluated on a “neighborhood” of $x\\in \\mathbb {R}$ defined by the convolution of the density $\\rho $ and a given kernel function $\\omega _\\eta $ with compact support.",
"In this paper, we are especially interested in two specific forms of (REF ) which naturally arise in traffic flow modelling [9], [17] and sedimentation problems [8].",
"They are given as follows.",
"A non-local traffic flow model.",
"In this context, we follow [9], [17] and consider (REF ) as an extension of the classical Lighthill-Whitham-Richards traffic flow model, in which the mean velocity is assumed to be a non-increasing function of the downstream traffic density and where the flux function is given by $f(\\rho )=\\rho ,\\quad V(\\rho )= {black}{1-\\rho }, \\quad (\\rho *\\omega _\\eta )(x,t)=\\int _x^{x+\\eta }\\omega _\\eta (y-x)\\rho (y,t)dy.$ Four different nonnegative kernel functions will be considered in the numerical section, namely $\\omega _\\eta (x) = 1/\\eta $ , $\\omega _\\eta (x) = 2(\\eta -x)/\\eta ^2$ , $\\omega _\\eta (x) = 3(\\eta ^2-x^2)/(2 \\eta ^3)$ and black$\\omega _\\eta (x) = 2x/\\eta ^2$ with support on $[0,\\eta ]$ for a given value of the real number $\\eta >0$ .",
"Notice that the well-posedness of this model together with the design of a first and a second order FV approximation have been considered in [9], [17].",
"A non-local sedimentation model.",
"Following [8], we consider under idealized assumptions that equation (REF ) represents a one-dimensional model for the sedimentation of small equal-sized spherical solid particles dispersed in a viscous fluid, where the local solids column fraction $\\rho =\\rho (x,t)$ is a function of depth $x$ and time $t$ .",
"In this context, the flux function is given by $f(\\rho )=\\rho (1-\\rho )^\\alpha ,\\quad V(\\rho )=(1-\\rho )^n, \\quad (\\rho *\\omega _\\eta )(x,t)=\\int _{-2\\eta }^{2\\eta }\\omega _\\eta (y)\\rho (x+y,t)dy,$ where $n \\ge 1$ and the parameter $\\alpha $ satisfies $\\alpha =0$ or $\\alpha \\ge 1$ .",
"The function $V$ is the so-called hindered settling factor and the convolution term $\\rho *\\omega _\\eta $ is defined by a symmetric, nonnegative, and piecewise smooth kernel function $\\omega _\\eta $ with support on $[-2\\eta ,2\\eta ]$ for a parameter $\\eta >0$ and $\\int _{\\mathbb {R}}\\omega _\\eta (x)dx=1$ .",
"More precisely, the authors define in [8] a truncated parabola $K$ by $K(x)=\\frac{3}{8}\\left(1-\\frac{x^2}{4}\\right) \\, \\text{ for } |x|<2,\\qquad K(x)=0\\,\\,\\text{ otherwise, }$ and set $\\omega _\\eta (x):=\\eta ^{-1}K(\\eta ^{-1}x).$ Conservation laws with non-local terms arise in a variety of physical and engineering applications: besides the above cited ones, we mention models for granular flows [4], production networks [21], conveyor belts [18], weakly coupled oscillators [3], laser cutting processes [15], biological applications like structured populations dynamics [23], crowd dynamics [10], [12] or more general problems like gradient constrained equations [6].",
"While several analytical results on non-local conservation laws can be found in the recent literature (we refer to [7] for scalar equations in one space dimension, [5], [13], [24] for scalar equations in several space dimensions and [2], [14], [16] for multi-dimensional systems of conservation laws), few specific numerical methods have been developed up to now.",
"Finite volume numerical methods have been studied in [2], [17], [22], [25].",
"At this regard, it is important to notice that the lack of Riemann solvers for non-local equations limits strongly the choice of the scheme.",
"At the best of our knowledge, two main approaches have been proposed in the literature to treat non-local problems: first and second order central schemes like Lax-Friedrichs or Nassyau-Tadmor [2], [7], [8], [17], [22] and Discontinuous Galerkin (DG) methods [19].",
"In particular, the comparative study presented in [19] on a specific model for material flow in two space dimensions, involving density gradient convolutions, encourages the use of DG schemes for their versatility and lower computational cost, but further investigations are needed in this direction.",
"Besides that, the computational cost induced by the presence of non-local terms, requiring the computation of quadrature formulas at each time step, motivate the development of high order algorithms.",
"The aim of the present article is to conduct a comparison study on high order schemes for a class of non-local scalar equations in one space dimension, focusing on equations of type (REF ).",
"In Section we review DG and FV-WENO schemes for classical conservation laws.",
"These schemes will be extended to the non-local case in Sections and .",
"Section is devoted to numerical tests."
],
[
"A review of Discontinuous Galerkin and Finite Volume WENO schemes for local conservation laws",
"The aim of this section is to introduce some notations and to briefly review the DG and FV-WENO numerical schemes to solve the classical local nonlinear conservation law ${\\left\\lbrace \\begin{array}{ll}\\rho _t+g(\\rho )_x=0, & x\\in \\mathbb {R},\\quad t>0, \\\\\\rho (x,0)=\\rho _0(x), & x\\in \\mathbb {R}.\\end{array}\\right.",
"}$ We first consider $\\lbrace I_j\\rbrace _{j\\in \\mathbb {Z}}$ a partition of $\\mathbb {R}$ .",
"The points $x_j$ will represent the centers of the cells $I_j=[x_{j-\\frac{1}{2}},x_{j+\\frac{1}{2}}]$ , and the cell sizes will be denoted by $\\Delta x_j=x_{j+\\frac{1}{2}}-x_{j-\\frac{1}{2}}$ .",
"The largest cell size is $h=\\sup _j \\Delta x_j$ .",
"Note that, in practice, we will consider a constant space step so that we will have $h=\\Delta x$ ."
],
[
"The Discontinuous Galerkin approach",
"In this approach, we look for approximate solutions in the function space of discontinuous polynomials $V_h:=V_h^k=\\lbrace v:v|_{I_j}\\in P^k(I_j)\\rbrace ,$ where $P^k(I_j)$ denotes the space of polynomials of degree at most $k$ in the element $I_j$ .",
"The approximate solutions are sought under the form $\\rho ^h(x,t) = \\sum _{l=0}^k c_j^{(l)}(t) v_j^{(l)}(x), \\quad v_j^{(l)}(x) = v^{(l)}(\\zeta _j(x)),$ where $c_j^{(l)}$ are the degrees of freedom in the element $I_j$ .",
"The subset $\\lbrace v_j^{(l)}\\rbrace _{l=0,...,k}$ constitutes a basis of $P^k(I_j)$ and in this work we will take Legendre polynomials as a local orthogonal basis of $P^k(I_j)$ , namely $v^{(0)}(\\zeta _j)=1,\\qquad v^{(1)}(\\zeta _j)=\\zeta _j ,\\qquad v^{(2)}(\\zeta _j)=\\frac{1}{2}\\left( 3 \\zeta _j^2-1\\right),\\dots , \\quad \\zeta _j:=\\zeta _j(x)=\\frac{x-x_j}{{\\Delta x}/2},$ see for instance black[11], [26].",
"Multiplying (REF ) by $v_h\\in V_h$ and integrating over $I_j$ gives $\\int _{I_{j}}\\rho _{t}v_{h}dx-\\int _{I_{j}}g(\\rho )v_{h,x}dx+g(\\rho (\\cdot ,t))v_{h}(\\cdot )\\bigl \\vert _{x_{j-\\frac{1}{2}}}^{x_{j+\\frac{1}{2}}}=0,$ and the semi-discrete DG formulation thus consists in looking for $\\rho ^{h}\\in V_{h}$ , such that for all $v_{h}\\in V_{h}$ and all $j$ , $\\int _{I_{j}}\\rho ^h_{t}v_{h}dx-\\int _{I_{j}}g(\\rho ^h)v_{h,x}dx+\\hat{g}_{j+\\frac{1}{2}}v^{-}_{h}(x_{j+\\frac{1}{2}})-\\hat{g}_{j-\\frac{1}{2}}v^{+}_{h}(x_{j-\\frac{1}{2}})=0 ,$ where $\\hat{g}_{j+\\frac{1}{2}}=\\hat{g}(\\rho ^{h,-}_{j+\\frac{1}{2}},\\rho ^{h,+}_{j+\\frac{1}{2}})$ is a consistent, monotone and Lipschitz continuous numerical flux function.",
"In particular, we will choose to use the Lax-Friedrichs flux $\\hat{g}(a,b):=\\frac{g(a)+g(b)}{2}+\\alpha \\frac{a-b}{2},\\quad \\quad \\alpha =\\max _u |g^{\\prime }(u)|.$ Let us now observe that if $v_h$ is the $l$ -th Legendre polynomial, we have $v^{+}_{h}(x_{j-\\frac{1}{2}})=v^{(l)}(\\zeta _{j}(x_{j-\\frac{1}{2}}))=(-1)^{l}$ , and $v^{+}_{h}(x_{j+\\frac{1}{2}})=v^{(l)}(\\zeta _{j}(x_{j+\\frac{1}{2}}))=1$ , $\\forall j\\,, l=0,1,\\dots ,k$ .",
"Therefore, replacing $\\rho ^h(x,t)$ by $\\rho _j^{h}(x,t)$ and $v_h(x)$ by $v^{(l)}(\\zeta _j(x))$ in (REF ), the degrees of freedom $c^{(l)}_{j}(t)$ satisfy the differential equations $\\frac{d}{dt}c^{(l)}_{j}(t)+\\frac{1}{a_{l}} \\left(-\\int _{I_{j}}g(\\rho _j^h)\\frac{d}{dx}v^{(l)}(\\zeta _j(x))dx+\\hat{g}(\\rho ^{h,-}_{j+\\frac{1}{2}},\\rho ^{h,+}_{j+\\frac{1}{2}})-(-1)^{l}\\hat{g}(\\rho ^{h,-}_{j-\\frac{1}{2}},\\rho ^{h,+}_{j-\\frac{1}{2}})\\right)=0,$ with ${a_{l}}=\\int _{I_{j}}(v^{(l)}(\\zeta _j(x)))^2dx=\\frac{\\Delta x}{2l+1},\\qquad l=0,1,\\dots ,k.$ On the other hand, the initial condition can be obtained using the $L^2$ -projection of $\\rho _0(x)$ , namely $c^{(l)}_{j}(0)=\\frac{1}{a_{l}}\\int _{I_{j}}\\rho _0(x)v^{(l)}(\\zeta _j(x))dx=\\frac{2l+1}{2}\\int _{-1}^1\\rho _0\\left(\\frac{\\Delta x}{2}y+x_j\\right)v^{(l)}(y)dy,\\qquad l=0,\\dots ,k.$ The integral terms in (REF ) can be computed exactly or using a high-order quadrature technique after a suitable change of variable, namely $\\int _{I_{j}}g(\\rho _j^h)\\frac{d}{dx}v^{(l)}(\\zeta _j(x))dx=\\int _{-1}^1g\\left(\\rho _j^h\\left(\\frac{\\Delta x}{2}y+x_j,t\\right)\\right)(v^{(l)})^{\\prime }(y)dy.$ In this work, we will consider a Gauss-Legendre quadrature with $N_G=5$ nodes for integrals on $[-1,1]$ $\\int _{-1}^{1}g(y)dy=\\sum _{e=1}^{N_G}w_eg(y_e),$ where $y_e$ are the Gauss-Legendre quadrature points such that the quadrature formula is exact for polynomials of degree until $2N_G-1=9$ black[1].",
"black The semi-discrete scheme (REF ) can be written under the usual form $\\frac{d}{dt}C(t)= \\mathcal {L}(C(t)),$ where $\\mathcal {L}$ is the spatial discretization operator defined by (REF ).",
"In this work, we will consider a time-discretisation based on the following total variation diminishing (TVD) third-order Runge-Kutta method [28], $C^{(1)}&=&C^{n}+\\Delta t \\mathcal {L}(C^{n}),\\\\C^{(2)}&=&\\frac{3}{4}C^{n}+\\frac{1}{4}C^{(1)}+ \\frac{1}{4}\\Delta t \\mathcal {L}(C^{(1)}),\\\\C^{n+1}&=&\\frac{1}{3}C^{n}+\\frac{2}{3}C^{(2)}+ \\frac{2}{3}\\Delta t \\mathcal {L}(C^{(2)}).$ Other TVD or strong stability preserving SSP time discretization can be also used [20].",
"The CFL condition is $\\frac{\\Delta t}{\\Delta x}\\max _\\rho |g^{\\prime }(\\rho )|\\le \\mathrm {C}_{\\mathrm {CFL}}= \\frac{1}{2k+1},$ where $k$ is the degree of the polynomial, see black[11].",
"The scheme (REF ) and (REF ) will be denoted RKDG."
],
[
"Generalized slope limiter",
"It is well-known that RKDG schemes like the one proposed above may oscillate when sharp discontinuities are present in the solution.",
"In order to control these instabilities, a common strategy is to use a limiting procedure.",
"We will consider the so-called generalized slope limiters proposed in [11].",
"With this in mind and $\\rho _j^h(x)=\\sum _{l=0}^{k}c^{(l)}_{j}v^{(l)}(\\zeta _j(x))\\in P^k(I_j)$ , we first set $u_{j+\\frac{1}{2}}^{-}:=\\bar{\\rho }_j+\\operatornamewithlimits{\\mathrm {mm}}(\\rho _j^h(x_{j+\\frac{1}{2}})-\\bar{\\rho }_j,\\Delta _{+}\\bar{\\rho }_j,\\Delta _{-}\\bar{\\rho }_{j})$ and $ u_{j-\\frac{1}{2}}^{+}:=\\bar{\\rho }_j-\\operatornamewithlimits{\\mathrm {mm}}(\\bar{\\rho }_j-\\rho _j^h(x_{j-\\frac{1}{2}}),\\Delta _{+}\\bar{\\rho }_j,\\Delta _{-}\\bar{\\rho }_{j}),$ where $\\bar{\\rho }_j$ is the average of $\\rho ^h$ on $I_j$ , $\\Delta _{+}\\bar{\\rho }_j=\\bar{\\rho }_{j+1}-\\bar{\\rho }_j$ , $\\Delta _{-}\\bar{\\rho }_j=\\bar{\\rho }_j-\\bar{\\rho }_{j-1}$ , and where $\\operatornamewithlimits{\\mathrm {mm}}$ is given by the minmod function limiter $\\operatornamewithlimits{\\mathrm {mm}}(a_1,a_2,a_3)={\\left\\lbrace \\begin{array}{ll} s\\cdot \\min _{j} |a_j| & \\text{ if } s=sign(a_1)=sign(a_2)=sign(a_3)\\\\ 0 & \\text{ otherwise,} \\end{array}\\right.",
"}$ or by the TVB modified minmod function ${\\overline{\\operatornamewithlimits{\\mathrm {mm}}}}(a_1,a_2,a_2)={\\left\\lbrace \\begin{array}{ll} a_1 & \\text{ if } |a_1|\\le M_{b} h^2,\\\\ \\operatornamewithlimits{\\mathrm {mm}}(a_1,a_2,a_3) & \\text{ otherwise,} \\end{array}\\right.",
"}$ where $M_{b} >0$ is a constant.",
"According to [11], [26], this constant is proportional to the second-order derivative of the initial condition at local extrema.",
"Note that if $M_b$ is chosen too small, the scheme is very diffusive, while if $M_b$ is too large, oscillations may appear.",
"The values $u_{j+\\frac{1}{2}}^{-}$ and $u_{j-\\frac{1}{2}}^{+}$ allow to compare the interfacial values of $\\rho ^h_j(x)$ with respect to its local cell averages.",
"Then, the generalized slope limiter technique consists in replacing $\\rho _j^h$ on each cell $I_j$ with $\\Lambda \\Pi _h$ defined by $\\Lambda \\Pi _h(\\rho _j^h)={\\left\\lbrace \\begin{array}{ll}\\rho _j^h &\\text{if } u_{j-\\frac{1}{2}}^{+}=\\rho _j^h(x_{j-\\frac{1}{2}})\\text{ and } u_{j+\\frac{1}{2}}^{-}=\\rho _j^h(x_{j+\\frac{1}{2}}),\\\\\\displaystyle \\bar{\\rho }_j+\\frac{(x-x_j)}{\\Delta x/2}\\operatornamewithlimits{\\mathrm {mm}}(c^{(1)}_{j},\\Delta _{+}\\bar{\\rho }_j,\\Delta _{-}\\bar{\\rho }_{j}) &\\text{otherwise.}\\end{array}\\right.",
"}$ Of course, this generalized slope limiter procedure has to be performed after each inner step of the Runge-Kutta scheme (REF )."
],
[
"The Finite Volume WENO approach",
"In this section, we solve the nonlinear conservation law (REF ) by using a high-order finite volume WENO scheme black[27], [28].",
"Let us denote by $\\bar{\\rho }(x_j,t)$ the cell average of the exact solution $\\rho (\\cdot ,t)$ in the cell $I_j$ : $\\bar{\\rho }(x_j,t):=\\frac{1}{\\Delta x}\\int _{I_j}\\rho (x,t)dx.$ The unknowns are here the set of all $\\lbrace \\bar{\\rho }_j(t)\\rbrace _{j\\in \\mathbb {Z}}$ which represent approximations of the exact cell averages $\\bar{\\rho }(x_j,t)$ .",
"Integrating (REF ) over $I_j$ we obtain $\\frac{d}{dt}\\bar{\\rho }(x_j,t)=-\\frac{1}{\\Delta x}\\left( g(\\rho (x_{j+\\frac{1}{2}},t))-g(\\rho (x_{j-\\frac{1}{2}},t))\\right),\\qquad \\forall j \\in \\mathbb {Z}.$ This equation is approximated by the semi-discrete conservative scheme $\\frac{d}{dt}\\bar{\\rho }_j(t)=-\\frac{1}{\\Delta x}\\left( \\hat{g}_{j+\\frac{1}{2}}-\\hat{g}_{j-\\frac{1}{2}}\\right),\\qquad \\forall j \\in \\mathbb {Z},$ where the numerical flux $\\hat{g}_{j+\\frac{1}{2}}:=\\hat{g}(\\rho _{j+\\frac{1}{2}}^l,\\rho _{j+\\frac{1}{2}}^r)$ is the Lax-Friedrichs flux and $\\rho _{j+\\frac{1}{2}}^l$ and $\\rho _{j+\\frac{1}{2}}^r$ are some left and right high-order WENO reconstructions of $\\rho (x_{j+\\frac{1}{2}},t)$ obtained from the cell averages $\\lbrace \\bar{\\rho }_j(t)\\rbrace _{j \\in \\mathbb {Z}}$ .",
"Let us focus on the definition of $\\rho _{j+\\frac{1}{2}}^l$ .",
"In order to obtain a $(2k-1)$ th-order WENO approximation, we first compute $k$ reconstructed values $\\hat{\\rho }_{j+\\frac{1}{2}}^{(r)}=\\sum _{i=0}^{k-1}c_{r}^i\\bar{\\rho }_{i-r+j},\\quad r=0,\\dots ,k-1,$ that correspond to $k$ possible stencils $S_r(j)=\\lbrace x_{j-r},\\dots ,x_{j-r+k-1}\\rbrace $ for $r=0,\\dots ,k-1$ .",
"The coefficients $c_{r}^i$ are chosen in such a way that each of the $k$ reconstructed values is $k$ th-order accurate [27].",
"Then, the $(2k-1)$ th-order WENO reconstruction is a convex combination of all these $k$ reconstructed values and defined by $\\rho _{j+\\frac{1}{2}}^l=\\sum _{r=0}^{k-1}w_{r}\\hat{\\rho }_{j+\\frac{1}{2}}^{(r)},$ where the positive nonlinear weights $w_r$ satisfy $\\sum _{r=0}^{k-1}w_r=1$ and are defined by $w_r=\\frac{\\alpha _r}{ \\sum _{s=0}^{k-1}\\alpha _s},\\quad \\alpha _r=\\frac{d_r}{(\\epsilon +\\beta _r)^2}.$ Here $d_r$ are the linear weights which yield the $(2k-1)$ th-order accuracy, $\\beta _r$ are called the “smoothness indicators” of the stencil $S_r(j)$ , which measure the smoothness of the function $\\rho $ in the stencil, and $\\epsilon $ is a small parameter used to avoid dividing by zero (typically $\\epsilon =10^{-6}$ ).",
"The exact form of the smoothness indicators and other details about WENO reconstructions can be found in [27].",
"The reconstruction of $\\rho _{j-\\frac{1}{2}}^r$ is obtained in a mirror symmetric fashion with respect to $x_{j-\\frac{1}{2}}$ .",
"black The semi-discrete scheme (REF ) is then integrated in time using the (TVD) third-order Runge-Kutta scheme (REF ), under the CFL condition $\\frac{\\Delta t}{\\Delta x}\\max _\\rho |g^{\\prime }(\\rho )|\\le \\mathrm {C}_{\\mathrm {CFL}}<1.$ ."
],
[
"Construction of DG schemes for non-local problems",
"We now focus on the non-local equation (REF ), for which we set $R(x,t):=(\\rho *\\omega _\\eta )(x,t)$ .",
"Since $\\rho ^{h}(x,t)|_{I_j}=\\sum _{l=0}^{k}c^{(l)}_{j}(t)v^{(l)}(\\zeta _j(x))\\in P^k(I_j)$ , is a weak solution of the non-local problem (REF ), the coefficients $c^{(l)}_{j}(t)$ can be calculated by solving the following differential equation, $\\frac{d}{dt}c^{(l)}_{j}(t)+\\frac{1}{a_{l}} \\left(-\\int _{I_{j}}f(\\rho ^h) V(R)\\frac{d}{dx}v^{(l)}(\\zeta _j(x))dx+\\check{f}_{j+\\frac{1}{2}}-(-1)^{l}\\check{f}_{j-\\frac{1}{2}}\\right)=0,$ where $\\check{f}_{j+\\frac{1}{2}}$ is a consistent approximation of $f(\\rho )V(R)$ at interface $x_{j+1/2}$ .",
"Here, we consider again the Lax-Friedrichs numerical flux defined by $\\check{f}_{j+\\frac{1}{2}}=\\frac{1}{2}\\left(f(\\rho _{j+\\frac{1}{2}}^{h,-})V(R_{j+\\frac{1}{2}}^{h,-})+f(\\rho _{j+\\frac{1}{2}}^{h,+})V(R_{j+\\frac{1}{2}}^{h,+})+\\alpha (\\rho _{j+\\frac{1}{2}}^{h,-}-\\rho _{j+\\frac{1}{2}}^{h,+}) \\right),$ with $\\alpha :=\\max _\\rho |\\partial _\\rho (f(\\rho )V(\\rho ))|$ and where $R_{j+\\frac{1}{2}}^{h,-}$ and $R_{j+\\frac{1}{2}}^{h,+}$ are the left and right approximations of $R(x,t)$ at the interface $x_{j+\\frac{1}{2}}$ .",
"Since $R$ is defined by a convolution, we naturally set $R_{j+\\frac{1}{2}}^{h,-}=R_{j+\\frac{1}{2}}^{h,+}=R(x_{j+\\frac{1}{2}},t)=:R_{j+\\frac{1}{2}}$ , so that (REF ) can be written as $\\check{f}_{j+\\frac{1}{2}}:=\\check{f}(\\rho _{j+\\frac{1}{2}}^{h,-},\\rho _{j+\\frac{1}{2}}^{h,+},R_{j+\\frac{1}{2}})=\\frac{1}{2}\\left((f(\\rho _{j+\\frac{1}{2}}^{h,-})+f(\\rho _{j+\\frac{1}{2}}^{h,+}))V(R_{j+\\frac{1}{2}})+\\alpha (\\rho _{j+\\frac{1}{2}}^{h,-}-\\rho _{j+\\frac{1}{2}}^{h,+}) \\right).$ Next, we propose to approximate the integral term in (REF ) using the following high-order Gauss-Legendre quadrature technique, $\\int _{I_{j}}f(\\rho ^h) V(R)\\frac{d}{dx}v^{(l)}(\\zeta _j(x))dx&=&\\int _{-1}^1f(\\rho ^h\\left(\\frac{\\Delta x}{2}y+x_j,t\\right))V\\left(R\\left(\\frac{\\Delta x}{2}y+x_j,t\\right)\\right)(v^{(l)})^{\\prime }(y)dy \\\\&=& \\sum _{e=1}^{N_G}w_ef(\\rho ^{h} \\left(\\hat{x}_e,t\\right))V\\left(R\\left(\\hat{x}_e,t\\right)\\right)(v^{(l)})^{\\prime }(y_e),$ where we have set $\\hat{x}_e=\\frac{\\Delta x}{2}y_e+x_j\\in I_j$ , $y_e$ being the Gauss-Legendre quadrature points ensuring that the quadrature formula is exact for polynomials of order less or equal to $2N_G-1$ .",
"It is important to notice that the DG formulation for the non-local conservation law (REF ) requires the computation of the extra integral terms $R_{j+\\frac{1}{2}}$ in (REF ) and $R\\left(\\hat{x}_e,t\\right)$ in (REF ) for each quadrature point, which increases the computational cost of the strategy.",
"For $\\rho ^{h}(x,t)|_{I_j}\\in P^k(I_j)$ , we can compute these terms as follows for both the LWR and sedimentation non-local models considered in this paper.",
"Non-local LWR model.",
"For the non-local LWR model, we impose the condition $N\\Delta x=\\eta $ for some $N\\in \\mathbb {N}$ , so that we have $R_{j+\\frac{1}{2}}&=&\\int _{x_{j+\\frac{1}{2}}}^{x_{j+\\frac{1}{2}}+\\eta }\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})\\rho ^h(y,t)dy=\\sum _{i=1}^N\\int _{I_{j+i}}\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})\\rho ^h_{j+i}(y,t)dy\\\\&=&\\sum _{i=1}^N \\sum _{l=0}^{k}c^{(l)}_{j+i}(t)\\int _{I_{j+i}}\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})v^{(l)}(\\zeta _{j+i}(y))dy\\\\&=&\\sum _{i=1}^N \\sum _{l=0}^{k}c^{(l)}_{j+i}(t)\\underbrace{\\frac{\\Delta x}{2}\\int _{-1}^1\\omega _\\eta \\left(\\frac{\\Delta x}{2}y+(i-\\frac{1}{2})\\Delta x\\right)v^{(l)}(y)dy}_{\\Gamma _{i,l}}=\\sum _{i=1}^N \\sum _{l=0}^{k}c^{(l)}_{j+i}\\Gamma _{i,l},$ while for each quadrature point $\\hat{x}_e$ we have $R\\left(\\hat{x}_e,t\\right)&=&\\int _{\\hat{x}_e}^{\\hat{x}_e+\\eta }\\omega _\\eta (y-\\hat{x}_e)\\rho ^h(y,t)dy= \\underbrace{\\int _{\\hat{x}_e}^{x_{j+\\frac{1}{2}}}\\omega _\\eta (y-\\hat{x}_e)\\rho _{j}^h(y,t)dy}_{\\Gamma _a}+\\\\&&\\sum _{i=1}^{N-1}\\underbrace{\\int _{I_{j+i}}\\omega _\\eta (y-\\hat{x}_e)\\rho _{j+i}^h(y,t)dy}_{\\Gamma _b^i} +\\underbrace{\\int _{x_{j+N-\\frac{1}{2}}}^{\\hat{x}_e+\\eta }\\omega _\\eta (y-\\hat{x}_e)\\rho _{j+N}^h(y,t)dy}_{\\Gamma _c}.$ The three integrals $\\Gamma _a$ , $\\Gamma _b^i$ and $\\Gamma _c$ can be computed with the same change of variable as before, namely $\\Gamma _a&=&\\frac{\\Delta x}{2}\\int _{y_e}^1\\omega _\\eta \\left(\\frac{\\Delta x}{2}(y-y_e)\\right)\\sum _{l=0}^kc^{(l)}_j(t)v^{(l)}(y)dy\\\\&=&\\sum _{l=0}^kc^{(l)}_j(t)\\underbrace{\\frac{\\Delta x}{2}\\int _{y_e}^1\\omega _\\eta \\left(\\frac{\\Delta x}{2}(y-y_e)\\right)v^{(l)}(y)dy}_{\\Gamma _{0,l}^{(e)}}=\\sum _{l=0}^kc^{(l)}_j \\Gamma _{0,l}^{(e)}, \\\\\\Gamma _b^i&=&\\frac{\\Delta x}{2}\\int _{-1}^1\\omega _\\eta \\left(\\frac{\\Delta x}{2}(y-y_e)+i\\Delta x\\right)\\sum _{l=0}^kc^{(l)}_{j+i}(t)v^{(l)}(y)dy\\\\&=&\\sum _{l=0}^kc^{(l)}_{j+i}\\underbrace{\\frac{\\Delta x}{2}\\int _{-1}^1\\omega _\\eta \\left(\\frac{\\Delta x}{2}(y-y_e)+i\\Delta x\\right)v^{(l)}(y)dy}_{\\Gamma _{i,l}^{(e)}}=\\sum _{l=0}^kc^{(l)}_{j+i} \\Gamma _{i,l}^{(e)}, \\\\\\Gamma _c&=&\\frac{\\Delta x}{2}\\int _{-1}^{y_e}\\omega _\\eta \\left(\\frac{\\Delta x}{2}(y-y_e)+N\\Delta x\\right)\\sum _{l=0}^kc^{(l)}_{j+N}v^{(l)}(y)dy\\\\&=&\\sum _{l=0}^kc^{(l)}_{j+N}(t)\\underbrace{\\frac{\\Delta x}{2}\\int _{-1}^{y_e}\\omega _\\eta \\left(\\frac{\\Delta x}{2}(y-y_e)+N\\Delta x\\right)v^{(l)}(y)dy}_{\\Gamma _{N,l}^{(e)}}=\\sum _{l=0}^kc^{(l)}_{j+N} \\Gamma _{N,l}^{(e)}.$ Finally we can compute $R\\left(\\hat{x}_e,t\\right)$ as $R\\left(\\hat{x}_e,t\\right)=\\sum _{i=0}^N\\sum _{l=0}^kc^{(l)}_{j+i} (t)\\Gamma _{i,l}^{(e)},$ in order to evaluate (REF ).",
"Non-local sedimentation model.",
"Considering now the non-local sedimentation model, we impose $N\\Delta x=2\\eta $ for some $N\\in \\mathbb {N}$ , so that we have $R_{j+\\frac{1}{2}}= \\int _{x_{j+\\frac{1}{2}}-2\\eta }^{x_{j+\\frac{1}{2}}+2\\eta }\\omega _\\eta (y-x_{j+\\frac{1}{2}})\\rho ^h(y,t)dy=\\sum _{i=-N+1}^N\\sum _{l=0}^kc^{(l)}_{j+i}(t)\\Gamma _{i,l},$ and for each quadrature point $\\hat{x}_e$ , $R\\left(\\hat{x}_e,t\\right)=\\int _{\\hat{x}_e-2\\eta }^{\\hat{x}_e+2\\eta }\\omega _\\eta (y-\\hat{x}_e)\\rho ^h(y,t)dy=\\sum _{i=-N}^N\\sum _{l=0}^kc^{(l)}_{j+i}(t) \\Gamma _{i,l}^{(e)},$ with $\\Gamma _{-N,l}^{(e)}&=& \\frac{\\Delta x}{2}\\int _{y_e}^1\\omega _\\eta \\left(\\frac{\\Delta x}{2}(y-y_e)-N\\Delta x\\right)v^{(l)}(y)dy,\\\\\\Gamma _{i,l}^{(e)}&=& \\frac{\\Delta x}{2}\\int _{-1}^1\\omega _\\eta \\left(\\frac{\\Delta x}{2}(y-y_e)+i\\Delta x\\right)v^{(l)}(y)dy,\\\\\\Gamma _{N,l}^{(e)}&=& \\frac{\\Delta x}{2}\\int _{-1}^{y_e}\\omega _\\eta \\left(\\frac{\\Delta x}{2}(y-y_e)+N\\Delta x\\right)v^{(l)}(y)dy.$ Remark.",
"blackIn order to compute integral terms in (REF ) as accurately as possible, the integrals $R_{j+\\frac{1}{2}}$ and $R\\left(\\hat{x}_e,t\\right)$ above, and in particular, the coefficients $\\Gamma _{i,l}$ , must be calculated exactly or using a suitable quadrature formula accurate to at least $\\mathcal {O}(\\Delta x ^{l+p})$ where $p$ is the degree of the convolution term $\\omega _{\\eta }$ .",
"It is important to notice that the coefficients can be precomputed and stored in order to save computational time.",
"black Finally the semi-discrete scheme (REF ) can be discretized in time using the (TVD) third-order Runge-Kutta scheme (REF ), under the CFL condition $\\frac{\\Delta t}{\\Delta x}\\max _\\rho |\\partial _\\rho (f(\\rho )V(\\rho ))|\\le \\mathrm {C}_{\\mathrm {CFL}}= \\frac{1}{2k+1},$ where $k$ is the degree of the polynomial."
],
[
"Construction of FV schemes for non-local conservation laws",
"Let us now extend the FV-WENO strategy of Section REF to the non-local case.",
"We first integrate (REF ) over $I_j$ to obtain $\\frac{d}{dt}\\bar{\\rho }(x_j,t)=-\\frac{1}{\\Delta x}\\left( f(\\rho (x_{j+\\frac{1}{2}},t))V(R(x_{j+\\frac{1}{2}},t))-f(\\rho (x_{j-\\frac{1}{2}},t))V(R(x_{j-\\frac{1}{2}},t))\\right),\\quad \\forall j\\in \\mathbb {Z},$ so that the semi-discrete discretization can be written as $\\frac{d}{dt}\\bar{\\rho }_j(t)=-\\frac{1}{\\Delta x}\\left( \\check{f}_{j+\\frac{1}{2}}-\\check{f}_{j-\\frac{1}{2}}\\right),\\qquad \\forall j\\in \\mathbb {Z},$ where the use of the Lax-Friedrichs numerical flux gives $\\check{f}_{j+\\frac{1}{2}}=\\check{f}(\\rho _{j+\\frac{1}{2}}^l,\\rho _{j+\\frac{1}{2}}^r,R_{j+\\frac{1}{2}})=\\frac{1}{2}\\left((f(\\rho _{j+\\frac{1}{2}}^l)+f(\\rho _{j+\\frac{1}{2}}^r))V(R_{j+\\frac{1}{2}})+\\alpha (\\rho _{j+\\frac{1}{2}}^l-\\rho _{j+\\frac{1}{2}}^r) \\right).$ Recall that $\\rho _{j+\\frac{1}{2}}^l$ and $\\rho _{j+\\frac{1}{2}}^r$ are the left and right WENO high-order reconstructions at point $x_{j+\\frac{1}{2}}$ .",
"At this stage, it is crucial to notice that in the present FV framework, the approximate solution is piecewise constant on each cell $I_j$ , so that a naive evaluation of the convolution terms may lead to a loss of high-order accuracy.",
"Let us illustrate this.",
"Considering for instance the non-local LWR model and using that $\\rho (x,t)$ is piecewise constant on each cell naturally leads to $R_{j+\\frac{1}{2}}&=&\\int _{x_{j+\\frac{1}{2}}}^{x_{j+\\frac{1}{2}}+\\eta }\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})\\rho (y,t)dy=\\sum _{i=1}^N\\int _{I_{j+i}}\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})\\rho (y,t)dy\\\\&=&\\Delta x \\sum _{i=1}^N\\bar{\\rho }_{j+i}\\int _{I_{j+i}}\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})dy,$ which does not account for the high-order WENO reconstruction.",
"In order to overcome this difficulty, we propose to approximate the value of $\\rho (x,t)$ using quadratic polynomials in each cell.",
"This strategy is detailed for each model in the following two subsections."
],
[
"Computation of $R_{j+\\frac{1}{2}}$ for the non-local LWR model",
"In order to compute the integral $R_{j+\\frac{1}{2}}=\\int _{x_{j+\\frac{1}{2}}}^{x+\\eta }\\omega _\\eta (y-x_{j+\\frac{1}{2}})\\rho (y,t)dy,$ we propose to consider a reconstruction of $\\rho (x,t)$ on $I_j$ by taking advantage of the high-order WENO reconstructions $\\rho _{j-\\frac{1}{2}}^r$ and $\\rho _{j+\\frac{1}{2}}^l$ at the boundaries of $I_j$ , as well as the approximation of the cell average $\\bar{\\rho }_j^n$ .",
"More precisely, we propose to define a polynomial $P_j(x)$ of degree 2 on $I_j$ by $P_j(x_{j-\\frac{1}{2}})=\\rho _{j-\\frac{1}{2}}^r,\\quad P_j(x_{j+\\frac{1}{2}})=\\rho _{j+\\frac{1}{2}}^l,\\quad \\frac{1}{\\Delta x}\\int _{I_j}P_j(x)dx=\\bar{\\rho }_j^n,$ which is very easy to handle.",
"In particular, we have $P_j (x):=a_{j,0}+a_{j,1}\\left( \\frac{x-x_j}{\\Delta x/2} \\right)+a_{j,2}\\left(3 \\left( \\frac{x-x_j}{\\Delta x/2} \\right)^2-1\\right)/2,\\qquad x \\in I_j,$ with $a_{j,0}=\\bar{\\rho }_j^n,\\qquad a_{j,1}=\\frac{1}{2}\\left(\\rho _{j+\\frac{1}{2}}^l-\\rho _{j-\\frac{1}{2}}^r \\right), \\qquad a_{j,2}=\\frac{1}{2}\\left(\\rho _{j+\\frac{1}{2}}^l+\\rho _{j-\\frac{1}{2}}^r \\right)-\\bar{\\rho }_j^n.$ Observe that we have used the same polynomials as in the DG formulation, i.e., $P_j(x)=\\sum _{l=0}^2a_{j,l}v^{(l)}(\\zeta _{j}(y))$ .",
"With this, $R_{j+\\frac{1}{2}}$ can be computed as $R_{j+\\frac{1}{2}}&=&\\sum _{i=1}^N\\int _{I_{j+i}}\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})P_{j+i}(y)dy=\\sum _{i=1}^N\\int _{I_{j+i}}\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})\\sum _{l=0}^2a_{j+i,l}v^{(l)}(\\zeta _{j+i}(y))dy \\\\&=&\\sum _{i=1}^N\\sum _{l=0}^2a_{j+i,l}\\int _{I_{j+i}}\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})v^{(l)}(\\zeta _{j+i}(y))dy\\\\ &=&\\sum _{i=1}^N\\sum _{l=0}^2a_{j+i,l}\\underbrace{\\frac{\\Delta x}{2}\\int _{-1}^{1}\\omega _\\eta \\left(\\frac{\\Delta x}{2}y+(i-\\frac{1}{2})\\Delta x\\right)v^{(l)}(y)dy}=\\sum _{i=1}^N\\sum _{l=0}^2a_{j+i,l}{\\Gamma }_{i,l},$ where the coefficients ${\\Gamma }_{l,i}$ are computed exactly or using a high-order quadrature approximation."
],
[
"Computation of $R_{j+\\frac{1}{2}}$ for non-local sedimentation model",
"As far as the non-local sedimentation model (REF )-(REF ) is concerned, we have $R_{j+\\frac{1}{2}}=\\int _{-2\\eta }^{2\\eta }\\omega _\\eta (y)\\rho ({x_{j+\\frac{1}{2}}}+y,t)dy=\\int _{{x_{j+\\frac{1}{2}}}-2\\eta }^{{x_{j+\\frac{1}{2}}}+2\\eta }\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})\\rho (y,t)dy.$ Considering again the assumption $N\\Delta x=2\\eta $ , we get $R_{j+\\frac{1}{2}}&=&\\sum _{i=-N+1}^N\\int _{I_{j+i}}\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})P_{j+i}(y)dy=\\sum _{i=-N+1}^N\\sum _{l=0}^2a_{j+i,l}\\int _{I_{j+i}}\\omega _\\eta (y-{x_{j+\\frac{1}{2}}})v^{(l)}(\\zeta _{j+i}(y))dy \\\\&=&\\sum _{i=-N+1}^N\\sum _{l=0}^2a_{j+i,l}\\underbrace{\\frac{\\Delta x}{2}\\int _{-1}^{1}\\omega _\\eta \\left(\\frac{\\Delta x}{2}y+(i-\\frac{1}{2})\\Delta x\\right)v^{(l)}(y)dy}=\\sum _{i=-N+1}^N\\sum _{l=0}^2a_{j+i,l}{\\Gamma }_{i,l}.$ To conclude this section, let us remark that the coefficients ${\\Gamma }_{i,l}$ are computed for $l=0,\\dots ,k$ in the DG formulation, where $k$ is the degree of the polynomials in $P^k(I_j)$ , while in the FV formulation, the coefficients ${\\Gamma }_{i,l}$ are computed only for $l=0,\\dots ,2$ due to the quadratic reconstruction of the unknown in each cell.",
"black The semi-discrete scheme (REF ) is then integrated in time using the (TVD) third-order Runge-Kutta scheme (REF ), under the CFL condition $\\frac{\\Delta t}{\\Delta x}\\max _\\rho |\\partial _\\rho (f(\\rho )V(\\rho ))|\\le \\mathrm {C}_{\\mathrm {CFL}}<1.$ ."
],
[
"Numerical experiments",
"In this section, we propose several test cases in order to illustrate the behaviour of the RKDG and FV-WENO high-order schemes proposed in the previous Sections and for the numerical approximation of the solutions of the non-local traffic flow and sedimentation models on a bounded interval $I=[0,L]$ with boundary conditions.",
"We consider periodic boundary conditions for the traffic flow model, i.e.",
"$\\rho (0,t)=\\rho (L,t)$ for $t\\ge 0$ and zero-flux boundary conditions for sedimentation model, in order to simulate a batch sedimentation process.",
"More precisely, we assume that $\\rho (x,t)=0$ for $x\\le 0$ and $\\rho (x,t)=1$ for $x\\ge 1.$ Given a uniform partition of $[0,L]$ $\\lbrace I_{j}\\rbrace _{j=1}^{M}$ with $\\Delta x= L/M$ , in order to compute the numerical fluxes $\\check{f}_{j+1/2}$ for $j=0,\\dots ,M+1$ , we define $\\rho _{j}^{n}$ in the ghost cells as follow: for the traffic model, $\\rho _{-1}^{n}:=\\rho _{M-1}^{n}, \\quad \\rho _{0}^{n}:=\\rho _{M}^{n}, \\quad \\rho _{M+i}^{n}:=\\rho _{i}^{n}, \\qquad \\text{ for } i=1,\\dots ,N;$ and for the sedimentation model $\\rho _{i}^{n}:=0, \\text{ for } i=-N,\\dots ,0, \\quad \\text{ and } \\quad \\rho _{M+i}^{n}:=1, \\qquad \\text{ for } i=1,\\dots ,N.$ A key element of this section will be the computation of the Experimental Order of Accuracy (EOA) of the proposed strategies, which is expected to coincide with the theoretical order of accuracy given by the high-order reconstructions involved in the corresponding numerical schemes.",
"Let us begin with a description of the practical computation of the EOA.",
"Regarding the RKDG schemes, if $\\rho ^{\\Delta x}(x,T)$ and $\\rho ^{\\Delta x/2}(x,T)\\in V_h^k$ are the solutions computed with $M$ and $2M$ mesh cells respectively, the L${}^1$ -error is computed by $e(\\Delta x)&=\\Vert \\rho ^{\\Delta x}(x,T)-\\rho ^{\\Delta x/2}(x,T)\\Vert _{L{}^1} \\\\&=\\sum _{j=1}^M \\int _{I_{2j-1}} | \\rho ^{\\Delta x}_{j}(x,T)-\\rho ^{\\Delta x/2}_{2j-1}(x,T)|dx+\\int _{I_{2j}} | \\rho ^{\\Delta x}_j(x,T)-\\rho ^{\\Delta x/2}_{2j}(x,T)|dx,$ where the integrals are computed with a high-order quadrature formula.",
"As far as the FV schemes are concerned, the L${}^1$ -error is computed as $e(\\Delta x)=\\Vert \\rho ^{\\Delta x}(x,T)-\\rho ^{\\Delta x/2}(x,T)\\Vert _{L{}^1}=\\Delta x \\sum _{j=1}^M | \\rho ^{\\Delta x}_{j}(x_j,T)-\\tilde{\\rho }_{j}(x_j)|dx,$ where $\\tilde{\\rho }_{j}(x)$ is a third-degree polynomial reconstruction of $\\rho ^{\\Delta x/2}_{j}(x,T)$ at point $x_j$ , i.e., $\\tilde{\\rho }_{j}(x_{j}) =\\frac{9}{16}\\left( \\rho ^{\\Delta x/2}_{2j}+\\rho ^{\\Delta x/2}_{2j-1}\\right) -\\frac{1}{16}\\left(\\rho ^{\\Delta x/2}_{2j+1}+\\rho ^{\\Delta x/2}_{2j-2} \\right).$ In both cases, the EOA is naturally defined by $\\gamma (\\Delta x)=\\log _2\\left(e(\\Delta x)/e(\\Delta x/2)\\right).$ In the following numerical tests, the CFL number is taken as $\\mathrm {C}_{\\mathrm {CFL}}=0.2, 0.1, 0.05$ for RKDG1, RKDG2 and RKDG3 schemes respectively, and $\\mathrm {C}_{\\mathrm {CFL}}=0.6$ for FV-WENO3 and FV-WENO5 schemes.",
"For RKDG3 and FV-WENO5 cases $\\Delta t$ is further reduced in the accuracy tests."
],
[
"Test 1a: non-local LWR model",
"We consider the Riemann problem $\\rho (x,0)= {\\left\\lbrace \\begin{array}{ll} 0.95 & x\\in [-0.5,0.4], \\\\0.05 & \\hbox{otherwise},\\end{array}\\right.",
"}$ with absorbing boundary conditions and compute the numerical solution of (REF )-(REF ) at time $T=0.1$ with $\\eta =0.1$ and $w_\\eta (x)=3(\\eta ^2-x^2)/(2\\eta ^3)$ .",
"We set $\\Delta x=1/800$ and compare the numerical solutions obtained with the FV-WENO3, FV-WENO5, and for RKDG1, RKDG2 and RKDG3 we use a generalized slope limiter (REF ) with $M_b=35$ .",
"The results displayed in Fig.",
"REF are compared with respect to the reference solution which was obtained with FV-WENO5 scheme and $\\Delta x=1/3200$ .",
"In FigREF (b) we observe that RKDG schemes are much more accurate than FV-WENO3 and even FV-WENO5.",
"Figure: Test 1a.",
"(a) Solution of Riemann problem with w η (x)=3(η 2 -x 2 )/(2η 3 )w_\\eta (x)=3(\\eta ^2-x^2)/(2\\eta ^3) at T=0.1T=0.1.",
"We compare solutions computed with FV-WENO andRKDG schemes using Δx=1/800\\Delta x=1/800.",
"(b) Zoomed view of the left discontinuity in Fig (a).",
"Reference solution is computed with FV-WENO5 with Δx=1/3200\\Delta x=1/3200."
],
[
"Test 1b: non-local LWR model",
"Now, we consider an initial condition with a small perturbation $\\rho _0(x)=0.35-(x-0.5)\\exp {(-2000(x-0.5)^2)}$ and an increasing kernel function $\\omega _{\\eta }=2x/\\eta ^2$ , with $\\eta =0.05$ and periodic boundary conditions.",
"According to [9], [17], the non-local LWR model is not stable with increasing kernels, in the sense that oscillations develop in short time.",
"Fig.",
"REF displays the numerical solution with different RKDG and FV-WENO schemes with $\\Delta x=1/400$ at time $T=0.3$ .",
"The profile provided by the first order scheme proposed in [9] is also included.",
"The reference solution is computed with a FV-WENO5 scheme with $\\Delta x=1/3200$ .",
"We observe that the numerical solutions obtained with the high-order schemes provide better approximations of the oscillatory shape of the solution than the first-order scheme, and that RKDG schemes give better approximations than the FV-WENO schemes.",
"Figure: Test1b.",
"Numerical solution non-local LWR model with ω η =2x/η 2 \\omega _{\\eta }=2x/\\eta ^2, η=0.05\\eta =0.05 and ρ 0 (x)=0.35-(x-0.5)exp(-2000(x-0.5) 2 )\\rho _0(x)=0.35-(x-0.5)\\exp {(-2000(x-0.5)^2)}.We compare solutions computed with FV Lax-Friedrichs, FV-WENO and RKDG schemes, using Δx=1/400\\Delta x=1/400.",
"Reference solution is computed with FV-WENO5 with Δx=1/3200\\Delta x=1/3200."
],
[
"Test 2: non-local sedimentation model",
"We now solve (REF )-(REF )-(REF ) with the piece-wise constant initial datum $\\rho (x,0)= {\\left\\lbrace \\begin{array}{ll} 0 & x\\le 0, \\\\0.5 & 0<x< 1, \\\\1 & x\\ge 1,\\end{array}\\right.",
"}$ with zero-flux boundary conditions in the interval $[0,1]$ , and compute the solution at time $T=1$ , with parameters $\\alpha =1$ , $n=3$ and $a=0.025$ .",
"We set $\\Delta x=1/400$ and compute the solutions with different RKDG and FV-WENO schemes, including the first-order Lax-Friedrichs scheme used in [8].",
"The results displayed in Fig.",
"REF are compared to a reference solution computed with FV-WENO5 and $\\Delta x=1/3200$ .",
"Compared to the reference solution, we observe that RKDG1 is more accurate than FV-WENO3, and FV-WENO5 more accurate than RKDG3 and RKDG2 (Fig.",
"REF ).",
"In particular, we observe that the numerical solutions obtained with the high-order schemes provide better approximations of the oscillatory shape of the solution than the first order scheme.",
"These oscillations can possibly be explained as layering phenomenon in sedimentation [29], which denotes a traveling staircases pattern, looking as several distinct bands of different concentrations.",
"We observe in Fig.",
"REF , where the evolution of $\\rho ^{h}(\\cdot ,t)$ is displayed for $t\\in [0,5]$ , that this layering phenomenon is observed with high order schemes, in this case RKDG1 and FV-WENO5, instead of Lax-Friedrichs scheme.",
"Figure: Test2.",
"Riemann problem for non-local sedimentation model.",
"Comparing numerical solution computed with FV Lax-Friedrichs,FV-WENO schemes and RKDG schemes using limiter function with M b =50M_b=50 using Δx=1/400\\Delta x=1/400.",
"Reference solution is computed with FV-WENO5 with Δx=1/3200\\Delta x=1/3200.Figure: Test 2.",
"Nonlocal sedimentation model.",
"History solution for t∈[0,5]t\\in [0,5] computed with LxF (left) RKDG1 (center) and WENO 5 (right) with Δx=1/400\\Delta x=1/400.We observe that the same kind of layering phenomenon reported in seems to be observed with high order schemes RKDG1 and FV-WENO5 instead of LxF scheme."
],
[
"Test 3: Experimental Order of Accuracy for ",
"In this subsection we compute the EOA for local conservation laws.",
"More precisely, we first consider the advection equation $\\rho _t+\\rho _x=0,$ and the initial datum $\\rho _0(x)=0.5+0.4\\sin (\\pi x)$ for $x\\in [-1,1]$ with periodic boundary conditions.",
"We compare the numerical approximations obtained with $\\Delta x=1/M$ and $M=20,40,80,160,320,640$ at time $T=1$ .",
"RKDG schemes are used without limiting function and $L^1$ -errors are collected in Table REF .",
"We observe that the correct EOA is obtained for the different schemes, moreover, we observe that the $L^{1}$ -errors obtained with FV-WENO3 and RKDG1 are comparable, and also the $L^{1}$ -errors for FV-WENO5 and RKDG3.",
"We also consider the nonlinear local LWR model $\\rho _t+(\\rho (1-\\rho ))_x=0,$ with initial datum $\\rho _0(x)=0.5+0.4\\sin (\\pi x)$ for $x\\in [-1,1]$ and periodic boundary conditions.",
"A shock wave appears at time $T=0.3$ .",
"We compare the numerical approximations obtained with $\\Delta x=1/M$ and $M=20,40,80,160,320,640$ at time $T=0.15$ .",
"RKDG schemes are used without limiters and $L^1$ -errors are given in Table REF .",
"We again observe that the correct EOA is obtained for the different schemes, and the $L^{1}$ -errors obtained with FV-WENO3 and RKDG1 are comparable, and also the $L^{1}$ -errors for FV-WENO5 and RKDG3.",
"Table: Test 3: Advection equation.",
"Experimental order of accuracy for FV-WENO schemes and RKDG schemes without Limiterat T=1T=1 with initial condition ρ 0 (x)=0.5+0.4sin(πx)\\rho _0(x)=0.5+0.4\\sin (\\pi x) on x∈[-1,1]x\\in [-1,1].Table: Test 3: Classical LWR equation.",
"Experimental order of accuracy for FV-WENO schemes and RKDG schemes without Limiterat T=0.15T=0.15 with initial condition ρ 0 (x)=0.5+0.4sin(πx)\\rho _0(x)=0.5+0.4\\sin (\\pi x) on x∈[-1,1]x\\in [-1,1]."
],
[
"Test 4: Experimental Order of Accuracy for the ",
"We now consider the non-local LWR and sedimentation models.",
"Considering the non-local LWR model, we compute the solution of (REF )-(REF ) with initial data $\\rho _0(x)=0.5+0.4\\sin (\\pi x)$ for $x\\in [-1,1]$ , periodic boundary conditions, with $\\eta =0.1$ and $\\Delta x=1/M$ and $M=80,160,320,640,1280$ at final time $T=0.15$ .",
"The results are given for different kernel functions $w_\\eta (x)$ in Table REF .",
"For RKDG schemes, we obtain the correct EOA.",
"For the FV-WENO schemes, the EOA is also correct thanks to the in-cell quadratic reconstructions used to compute the non-local terms.",
"For the non-local sedimentation model, we compute the solution of (REF )-(REF )-(REF ) with initial data $\\rho _0(x)=0.8\\sin (\\pi x)^{10}$ for $x\\in [0,1]$ with $\\eta =0.025$ , $\\alpha =1$ , $V(\\rho )=(1-\\rho )^3$ and $\\Delta x=1/M$ with $M=100,200,400,800,1600,3200$ at final time $T=0.04$ .",
"The results are given in Table REF .",
"Table: Test 4: Non-local LWR model.",
"Experimental order of accuracy.",
"Initial condition ρ 0 (x)=0.5+0.4sin(πx)\\rho _0(x)=0.5+0.4\\sin (\\pi x), η=0.1\\eta =0.1, numerical solution at T=0.15T=0.15 forFV-WENO and RKDG schemes without generalized slope limiter.Table: Test 4: Non-local sedimentation problem.",
"Initial condition ρ 0 (x)=0.8sin(πx) 10 \\rho _0(x)=0.8\\sin (\\pi x)^{10}.Experimental order of accuracy at T=0.04T=0.04 with f(ρ)=ρ(1-ρ)f(\\rho )=\\rho (1-\\rho ), V(ρ)=(1-ρ) 3 V(\\rho )=(1-\\rho )^3, ω η (x):=η -1 K(η -1 x)\\omega _\\eta (x):=\\eta ^{-1}K(\\eta ^{-1}x) with η=0.025\\eta =0.025."
],
[
"Conclusion",
"In this paper we developed high order numerical approximations of the solutions of non-local conservation laws in one space dimension motivated by application to traffic flow and sedimentation models.",
"We propose to design Discontinuous Galerkin (DG) schemes which can be applied in a natural way and Finite Volume WENO (FV- WENO) schemes where we have used quadratic polynomial reconstruction in each cell to evaluate the convolution term in order to obtain the high-order accuracy.",
"The numerical solutions obtained with high-order schemes provide better approximations of the oscillatory shape of the solutions than the first order schemes.",
"We also remark that DG schemes are more accurate but expensive, while FV-WENO are less accurate but also less expensive.",
"The present work thus establishes the preliminary basis for a deeper study and optimum programming of the methods, so that efficiently plots could then be considered.",
"This research was conducted while LMV was visiting Inria Sophia Antipolis Méditerranée and the Laboratoire de Mathématiques de Versailles, with the support of Fondecyt-Chile project 11140708 and “Poste Rouge” 2016 of CNRS -France.",
"The authors thank Régis Duvigneau for useful discussions and advices."
]
] | 1612.05775 |
[
[
"Lidov-Kozai Mechanism in Hydrodynamical Disks: Linear Stability Analysis"
],
[
"Abstract Recent SPH simulations by Martin et al.",
"(2014) suggest a circumstellar gaseous disk may exhibit coherent eccentricity-inclination oscillations due to the tidal forcing of an inclined binary companion, in a manner that resembles Lidov-Kozai oscillations in hierarchical triple systems.",
"We carry out linear stability analysis for the eccentricity growth of circumstellar disks in binaries, including the effects of gas pressure and viscosity and secular (orbital-averaged) tidal force from the inclined companion.",
"We find that the growth of disk eccentricity depends on the dimensionless ratio ($S$) between $c_{\\rm s}^2$ (the disk sound speed squared) and the tidal torque acting on the disk (per unit mass) from the companion.",
"For $S\\ll 1$, the standard Lidov-Kozai result is recovered for a thin disk annulus: eccentricity excitation occurs when the mutual inclination $I$ between the disk and binary lies between $39^\\circ$ and $141^\\circ$.",
"As $S$ increases, the inclination window for eccentricity growth generally becomes narrower.",
"For $S\\gtrsim$ a few, eccentricity growth is suppressed for all inclination angles.",
"Surprisingly, we find that for $S\\sim 1$ and certain disk density/pressure profiles, eccentricity excitation can occur even when $I$ is much less than $39^\\circ$."
],
[
"Introduction",
"When a test particle orbiting a central mass has a distant binary companion, it can undergo eccentricity and inclination oscillations if the initial inclination $I$ between the orbital planes of the test mass and the binary is sufficiently large.",
"This is termed Lidov-Kozai (LK) oscillation, and was originally invoked to explain the dynamics of artificial satellites [21] and asteroids [18].",
"Since then, the LK effect has found a plethora of applications in astrophysics (e.g.",
"[44], [30]), such as the formation of the Jovian irregular satellites [5], [32], mergers of massive black hole binaries [4], formation of short-period stellar binaries [7] and hot Jupiters [45], [8], [39], [1], and Type Ia supernovae from white dwarf binary mergers [16].",
"The simplest LK oscillation involves only the quadrupole potential from the companion.",
"It has been recognized that the high-order perturbation (e.g., [9], [31], [17]) and short-range forces (e.g., [15], [45], [22]) can significantly influence the LK oscillation dynamics.",
"Thus, one may expect that any eccentricity/inclination oscillations of a gaseous disk inside a stellar binary, if occur at all, may be modified or suppressed by hydrodynamic forces.",
"Recently [27] used SPH simulations to show that LK oscillations may be excited in circumstellar disks with distant, inclined binary companions (see also [12]).",
"[13] showed that these disk oscillations can be suppressed by the disk self-gravity when the disk mass is sufficiently large ([3]; see discussion in Sec.",
"4).",
"If real, this may have interesting astrophysical implications due to the ubiquity of misaligned circumstellar accretion disks in binary systems.",
"In this paper we use linear theory of eccentric disks[14], [35], [42] to study the possibility of coherent LK oscillations of circumstellar disks in binaries.",
"Section gives the set-up and formalism of this work.",
"Section contains our results.",
"Section presents the summary and discussion of our work."
],
[
"Setup and Formalism",
"Consider a circumstellar disk around a host star of mass $M$ .",
"The disk has an inner radius $r = r_{\\rm in}$ , outer radius $r = r_\\text{out}$ , and surface density $\\Sigma = \\Sigma (r)$ .",
"The disk warp and eccentricity are specified by the unit angular momentum vector $\\mathbf {\\hat{l}}=\\mathbf {\\hat{l}}(r,t)$ and eccentricity vector ${\\mathbf {e}}={\\mathbf {e}}(r,t)$ .",
"We take the disk to be nearly circular, so $e \\ll 1$ everywhere.",
"We adopt a locally isothermal equation of state, so that the height-integrated pressure at any location in the disk is given by $P = c_{\\rm s}^2 \\Sigma $ , where $c_{\\rm s}= c_{\\rm s}(r)$ is the sound speed.",
"For a thin disk with mass much less than $M$ , the orbital frequency of the disk is given by $n(r) \\simeq \\sqrt{GM/r^3}$ .",
"The host star has a distant external binary companion with semimajor axis $a_{\\rm b}\\gtrsim 3r_\\text{out}$The upper bound on the outer disk radius is set by tidal truncation [2], [28], mass $M_{\\rm b}$ , and orbital angular momentum unit vector $\\mathbf {\\hat{l}}_{\\rm b}$ .",
"We take the binary's orbit to be circular.",
"Because the angular momentum of the binary is much larger than that of the circumstellar disk, we take $\\mathbf {\\hat{l}}_{\\rm b}$ to be fixed in time.",
"The gravitational force of the binary companion drives the eccentricity and angular momentum unit vectors of disk annuli according to [43], [44] $\\left( \\frac{\\partial \\mathbf {\\hat{l}}}{\\partial t} \\right)_{\\rm bin} &= \\omega _{\\rm b}(\\mathbf {\\hat{l}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}}_{\\rm b}) \\mathbf {\\hat{l}}{\\mathbf {\\times }}\\mathbf {\\hat{l}}_{\\rm b}+ \\mathcal {O}(e^2) \\\\\\left(\\frac{\\partial {\\mathbf {e}}}{\\partial t} \\right)_{\\rm bin} &= \\omega _{\\rm b}\\Big [ (\\mathbf {\\hat{l}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}}_{\\rm b}) {\\mathbf {e}}{\\mathbf {\\times }}\\mathbf {\\hat{l}}_{\\rm b}- 5({\\mathbf {e}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}}_{\\rm b}) \\mathbf {\\hat{l}}{\\mathbf {\\times }}\\mathbf {\\hat{l}}_{\\rm b}+ 2 \\mathbf {\\hat{l}}{\\mathbf {\\times }}{\\mathbf {e}}\\Big ] + \\mathcal {O}(e^3),$ where $\\omega _{\\rm b}(r) = \\frac{3 G M_{\\rm b}}{4 a_{\\rm b}^3 n}$ characterizes the precession frequency of a disk annulus around the external binary.",
"Equations (REF ) and () include the effect of the quadrupole potential from the binary and are averaged over the binary period.",
"Internal hydrodynamical forces work to resist the differential nodal precession of the disk annuli, either in the form of bending waves [37], [26] or viscosity [38], [33], and enforce both coplanarity ($|\\partial \\mathbf {\\hat{l}}/\\partial \\ln r| \\ll 1$ ) and rigid body precession [19], [46].",
"Under their influence, the time evolution of the disk's unit angular momentum vector is given by $\\bigg ( \\frac{\\partial \\mathbf {\\hat{l}}}{\\partial t} \\bigg )_{\\rm int} + \\bigg ( \\frac{\\partial \\mathbf {\\hat{l}}}{\\partial t} \\bigg )_{\\rm bin} &= \\bar{\\omega }_{\\rm b}(\\mathbf {\\hat{l}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}}_{\\rm b}) \\mathbf {\\hat{l}}{\\mathbf {\\times }}\\mathbf {\\hat{l}}_{\\rm b}+ \\mathcal {O}(e^2) \\\\\\Rightarrow \\bigg ( \\frac{\\partial \\mathbf {\\hat{l}}}{\\partial t} \\bigg )_{\\rm int} &= (\\bar{\\omega }_{\\rm b}- \\omega _{\\rm b}) (\\mathbf {\\hat{l}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}}_{\\rm b}) \\mathbf {\\hat{l}}{\\mathbf {\\times }}\\mathbf {\\hat{l}}_{\\rm b}+ \\mathcal {O}(e^2),$ where $\\mathbf {\\hat{l}}$ is (nearly) independent of $r$ , and $\\bar{\\omega }_{\\rm b}= \\frac{\\int _{r_{\\rm in}}^{r_\\text{out}} \\Sigma r^3 n \\omega _{\\rm b}\\text{d}r}{\\int _{r_{\\rm in}}^{r_\\text{out}} \\Sigma r^3 n \\text{d}r}$ characterizes the precession frequency of the rigid disk around the binary.",
"The internal force that enforces rigid disk nodal precession must also act on ${\\mathbf {e}}$ , so that ${\\mathbf {e}}$ remains perpendicular to $\\mathbf {\\hat{l}}$ , i.e., $\\left[ \\frac{\\partial ({\\mathbf {e}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}})}{\\partial t} \\right]_{\\rm int} = 0.$ This requirement, together with the assumption that the internal force responsible for Eq.",
"() is perpendicular to the disk, imply that the time evolution of the disk's eccentricity vector is $\\left( \\frac{\\partial {\\mathbf {e}}}{\\partial t} \\right)_{\\rm int} = (\\bar{\\omega }_{\\rm b}- \\omega _{\\rm b}) (\\mathbf {\\hat{l}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}}_{\\rm b}) \\big [ \\mathbf {\\hat{l}}{\\mathbf {\\cdot }}({\\mathbf {e}}{\\mathbf {\\times }}\\mathbf {\\hat{l}}_{\\rm b}) \\big ] \\mathbf {\\hat{l}}+ \\mathcal {O}(e^3).$ We justify Eq.",
"(REF ) in the appendix.",
"Before we proceed, we comment on the validity of the assumption of coplanarity and rigid-body precession.",
"When the dimensionless Shakura-Sunyaev viscosity parameter $\\alpha $ satisfies $\\alpha \\lesssim H/r$ ($H$ is the disk scaleheight), bending waves keep the disk coherent [37], [26].",
"The amount of disk warp in this bending wave regime has been calculated in [10], and assuming $p=1$ and $q=1/2$ [see Eqs.",
"(REF )-(REF ) in next section], is $\\mathbf {\\hat{l}}&(r_\\text{out},t) - \\mathbf {\\hat{l}}({r_\\text{in}},t) \\approx \\nonumber \\\\&0.01\\left( \\frac{ \\alpha }{0.01} \\right) \\left( \\frac{H(r_\\text{out})}{0.1 \\, r_\\text{out}} \\right)^{-2} \\left( \\frac{ M_{\\rm b}}{ M } \\right) \\left( \\frac{ 3 r_\\text{out}}{a_{\\rm b}} \\right)^3 \\frac{ \\mathbf {\\hat{l}}_{\\rm b}{\\mathbf {\\times }}\\mathbf {\\hat{l}}(r_\\text{out},t) }{\\sin I}\\nonumber \\\\&- 0.01 \\left( \\frac{H(r_\\text{out})}{0.1 \\, r_\\text{out}} \\right)^{-2} \\left(\\frac{M_{\\rm b}}{M} \\right)^2 \\left( \\frac{3 r_\\text{out}}{a_{\\rm b}} \\right)^6 \\frac{[\\mathbf {\\hat{l}}_{\\rm b}{\\mathbf {\\times }}\\mathbf {\\hat{l}}(r_\\text{out},t)] {\\mathbf {\\times }}\\mathbf {\\hat{l}}_{\\rm b}}{ \\sin I}.$ Numerical simulations give a similar result (e.g.",
"[19], [46], [40]).",
"On the other hand, when $\\alpha \\gtrsim H/r$ , viscous torques keep the disk coherent [38], [33], and the disk diffusively damps to it's steady-state equilibrium warp profile over the timescale $t_{\\rm visc} \\sim 2 \\alpha r^2/(H^2 n)$ [24], [23], [11].",
"Large warping and sometimes disk breaking is observed when the disk's viscous torque is comparable to or less than the torque exerted on the disk by the distant binary (e.g.",
"[19], [6]).",
"Thus, the following derivation of the LK disk instability will be restricted to the $\\alpha \\lesssim H/r$ regime, which is applicable to protoplanetary disks.",
"For a flat disk, the effect of pressure on the time evolution of the disk's eccentricity is described by [42] $\\bigg ( &\\frac{\\partial {\\mathbf {e}}}{\\partial t} \\bigg )_{\\rm press} = \\mathbf {\\hat{l}}{\\mathbf {\\times }}\\left[ \\frac{1}{\\Sigma r^3 n} \\frac{\\partial }{\\partial r} \\left( \\frac{\\Sigma c_{\\rm s}^2 r^3}{2} \\frac{\\partial {\\mathbf {e}}}{\\partial r} \\right) \\right]\\nonumber \\\\&+ \\frac{1}{2 \\Sigma r n} \\frac{\\text{d}( \\Sigma c_{\\rm s}^2)}{\\text{d}r} \\mathbf {\\hat{l}}{\\mathbf {\\times }}{\\mathbf {e}}- \\mathbf {\\hat{l}}{\\mathbf {\\times }}\\left[ \\frac{1}{2 \\Sigma r^3 n} \\frac{\\partial }{\\partial r} \\left( \\Sigma \\frac{\\text{d}c_{\\rm s}^2}{\\text{d}r} r^3 {\\mathbf {e}}\\right) \\right]\\nonumber \\\\&+ \\frac{3}{2 r^3 n} \\frac{ \\text{d}(c_{\\rm s}^2 r^2)}{\\text{d}r} \\mathbf {\\hat{l}}{\\mathbf {\\times }}{\\mathbf {e}}+ \\mathcal {O}(e^2).$ The last term in Eq.",
"(REF ) arises from the disk's “breathing mode,\" where the fluid displacements are proportional to $z^2$ , where $z$ is the vertical coordinate of the disk [35].",
"Earlier theories of eccentric disks do not include this term [14].",
"Following [42], we also include the effect of bulk viscosity on the disk eccentricity evolution: $\\left( \\frac{\\partial {\\mathbf {e}}}{\\partial t} \\right)_{\\rm visc}= \\frac{1}{2 \\Sigma r^3 n} \\frac{\\partial }{\\partial r} \\left( \\alpha _{\\rm b} \\Sigma c_{\\rm s}^2 r^3 \\frac{\\partial {\\mathbf {e}}}{\\partial r} \\right) + \\mathcal {O}(e^2),$ small kinematic viscosity leads to over-stability, and a small bulk viscosity is needed to stabilize the eccentric disturbance [34], [20].",
"From Equation (REF ), we see that the disk's unit angular momentum vector $\\mathbf {\\hat{l}}(t)$ precesses uniformly around $\\mathbf {\\hat{l}}_{\\rm b}$ with frequency $\\omega _{\\rm prec} = -\\bar{\\omega }_{\\rm b}\\cos I$ , where $I$ is the inclination angle ($\\cos I = \\mathbf {\\hat{l}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}}_{\\rm b}$ ).",
"Indeed, in the linear theory of LK oscillation of a test mass, the inclination stays constant while the eccentricity grows in time [44].",
"To determine the stability of ${\\mathbf {e}}(r,t)$ , it is necessary to consider the evolution equation of ${\\mathbf {e}}$ in the frame co-rotating with $\\mathbf {\\hat{l}}(t)$ [44].",
"Including the gravitational perturbations and hydrodynamical effects, the time evolution of the disk's eccentricity vector ${\\mathbf {e}}$ is given by $\\bigg ( \\frac{\\partial {\\mathbf {e}}}{\\partial t} \\bigg )_{\\rm rot} =&\\left( \\frac{\\partial {\\mathbf {e}}}{\\partial t} \\right)_{\\rm bin} + \\left( \\frac{\\partial {\\mathbf {e}}}{\\partial t} \\right)_{\\rm int} + \\left( \\frac{\\partial {\\mathbf {e}}}{\\partial t} \\right)_{\\rm press}\\nonumber \\\\&+ \\left( \\frac{\\partial {\\mathbf {e}}}{\\partial t} \\right)_{\\rm visc} + (\\bar{\\omega }_{\\rm b}\\cos I) \\mathbf {\\hat{l}}_{\\rm b}{\\mathbf {\\times }}{\\mathbf {e}}.$ We will work in this frame for the rest of the paper, and drop the subscript “rot.\"",
"Define the complex eccentricity $E(r,t) \\equiv {\\mathbf {e}}(r,t) {\\mathbf {\\cdot }}(\\mathbf {\\hat{x}}+ \\textit {i}\\mathbf {\\hat{y}})$ , where $\\mathbf {\\hat{y}}= \\mathbf {\\hat{l}}{\\mathbf {\\times }}\\mathbf {\\hat{l}}_{\\rm b}/\\sin I$ and $\\mathbf {\\hat{x}}= \\mathbf {\\hat{y}}{\\mathbf {\\times }}\\mathbf {\\hat{l}}$ are unit vectors, constant in the rotating frame.",
"Then Equation (REF ) becomes $&\\frac{\\partial E}{\\partial t} = \\textit {i}\\omega _{\\rm b}\\left[ 2 E - \\frac{5 \\sin ^2 I}{2}(E + E^*) \\right]\\nonumber \\\\&+ \\textit {i}(\\bar{\\omega }_{\\rm b}- \\omega _{\\rm b}) \\cos ^2 I E + \\frac{\\textit {i}}{\\Sigma r^3 n} \\frac{\\partial }{\\partial r} \\left( \\frac{\\Sigma c_{\\rm s}^2 r^3}{2} \\frac{\\partial E}{\\partial r} \\right)\\nonumber \\\\&+ \\frac{\\textit {i}}{2 \\Sigma r n} \\frac{\\text{d}(\\Sigma c_{\\rm s}^2)}{\\text{d}r} E - \\frac{\\textit {i}}{2 \\Sigma r^3 n} \\frac{\\partial }{\\partial r} \\left( \\Sigma \\frac{\\text{d}c_{\\rm s}^2}{\\text{d}r} r^3 E \\right)\\nonumber \\\\&+\\frac{3 \\textit {i}}{2 r^3 n} \\frac{ \\text{d}(c_{\\rm s}^2 r^2)}{\\text{d}r} E + \\frac{1}{2 \\Sigma r^3 n} \\frac{\\partial }{\\partial r} \\left( \\alpha _{\\rm b}\\Sigma c_{\\rm s}^2 r^3 \\frac{\\partial E}{\\partial r} \\right),$ where $E^*$ denotes the complex conjugate to $E$ .",
"To find the eigenmodes of Eq.",
"(REF ), we separate $E$ into two “polarizations\": $E(r,t) = E_+(r) \\exp (\\lambda t) + E_-^*(r) \\exp (\\lambda ^* t).$ Here, $E_+$ and $ E_-$ are two complex functions, while $\\lambda $ is a complex eigenvalue.",
"Substituting Eq.",
"(REF ) into Eq.",
"(REF ), we obtain the coupled eigenvalue equations $\\lambda &E_+= \\textit {i}\\omega _{\\rm b}\\left[ 2 E_+- \\frac{5 \\sin ^2 I}{2}(E_++ E_-) \\right]\\nonumber \\\\&+ \\textit {i}(\\bar{\\omega }_{\\rm b}- \\omega _{\\rm b}) \\cos ^2 I E_++ \\frac{\\textit {i}}{\\Sigma r^3 n} \\frac{\\text{d}}{\\text{d}r} \\left( \\frac{\\Sigma c_{\\rm s}^2 r^3}{2} \\frac{\\text{d}E_+}{\\text{d}r} \\right)\\nonumber \\\\&+ \\frac{\\textit {i}}{2 \\Sigma r n} \\frac{\\text{d}(\\Sigma c_{\\rm s}^2)}{\\text{d}r} E_+- \\frac{\\textit {i}}{2 \\Sigma r^3 n} \\frac{\\text{d}}{\\text{d}r} \\left( \\Sigma \\frac{\\text{d}c_{\\rm s}^2}{\\text{d}r} r^3 E_+\\right)\\nonumber \\\\&+\\frac{3 \\textit {i}}{2 r^3 n} \\frac{ \\text{d}(c_{\\rm s}^2 r^2)}{\\text{d}r} E_++ \\frac{1}{2 \\Sigma r^3 n} \\frac{\\text{d}}{\\text{d}r} \\left( \\alpha _{\\rm b}\\Sigma c_{\\rm s}^2 r^3 \\frac{\\text{d}E_+}{\\text{d}r} \\right), \\\\\\lambda & E_-= -\\textit {i}\\omega _{\\rm b}\\left[ 2 E_-- \\frac{5 \\sin ^2 I}{2}(E_++ E_-) \\right]\\nonumber \\\\&- \\textit {i}(\\bar{\\omega }_{\\rm b}- \\omega _{\\rm b}) \\cos ^2 I E_-- \\frac{\\textit {i}}{\\Sigma r^3 n} \\frac{\\text{d}}{\\text{d}r} \\left( \\frac{\\Sigma c_{\\rm s}^2 r^3}{2} \\frac{\\text{d}E_-}{\\text{d}r} \\right)\\nonumber \\\\&- \\frac{\\textit {i}}{2 \\Sigma r n} \\frac{\\text{d}(\\Sigma c_{\\rm s}^2)}{\\text{d}r} E_-+ \\frac{\\textit {i}}{2 \\Sigma r^3 n} \\frac{\\text{d}}{\\text{d}r} \\left( \\Sigma \\frac{\\text{d}c_{\\rm s}^2}{\\text{d}r} r^3 E_-\\right)\\nonumber \\\\&-\\frac{3 \\textit {i}}{2 r^3 n} \\frac{ \\text{d}(c_{\\rm s}^2 r^2)}{\\text{d}r} E_-+ \\frac{1}{2 \\Sigma r^3 n} \\frac{\\text{d}}{\\text{d}r} \\left( \\alpha _{\\rm b}\\Sigma c_{\\rm s}^2 r^3 \\frac{\\text{d}E_-}{\\text{d}r} \\right).$ When $\\alpha _{\\rm b} = 0$ , the eigenvalue $\\lambda $ is either real or imaginary.",
"Imaginary eigenvalues imply the eccentricity vector ${\\mathbf {e}}$ is precessing or librating around $\\mathbf {\\hat{l}}$ , while real eignenvalues imply an exponentially growing or damping eccentricity.",
"For a thin ring (${r_\\text{in}}\\simeq r_\\text{out}$ ) of pressureless particles $(c_{\\rm s}= 0)$ , Eqs.",
"(REF )-() can be easily solved, giving $\\lambda ^2 = - 2 \\omega _{\\rm b}^2(2 - 5 \\sin ^2 I).$ This recovers the standard results: eccentricity grows when $I_{\\rm LK}< I < 180^\\circ - I_{\\rm LK}$ [44], where $I_{\\rm LK}\\equiv \\sin ^{-1} \\sqrt{2/5} \\simeq 39^\\circ .$"
],
[
"Results",
"To analyze the solutions of Eqs.",
"(REF ) and (), we assume the disk surface density and sound-speed profiles of $\\Sigma (r) = \\Sigma (r_\\text{out}) \\left( \\frac{r_\\text{out}}{r} \\right)^p$ and $c_{\\rm s}(r) = c_{\\rm s}(r_\\text{out}) \\left( \\frac{r_\\text{out}}{r} \\right)^q.$ A key dimensionless parameter in our analysis is the ratio $S &\\equiv \\frac{c_{\\rm s}^2(r_\\text{out})}{r_\\text{out}^2 n(r_\\text{out}) \\omega _{\\rm b}(r_\\text{out})}\\nonumber \\\\&\\simeq 0.36 \\left( \\frac{a_{\\rm b}}{3 \\, r_\\text{out}} \\right)^3 \\left( \\frac{M}{M_{\\rm b}} \\right) \\left( \\frac{H(r_\\text{out})}{0.1 \\, r_\\text{out}} \\right)^2,$ where we have approximated $c_{\\rm s}\\simeq H n$ (where $H$ is the disk scale-height), and $\\omega _{\\rm b}$ is defined in Eq.",
"(REF ).",
"Physically, $S^{-1}$ measures the strength of the tidal torque (per unit mass) acting on the outer disc from the external companion $(r^2 n \\omega _{\\rm b})$ relative to the torque associated with gas pressure ($c_{\\rm s}^2$ ).",
"Define the dimensionless radial coordinate $x \\equiv r/r_\\text{out}$ , inner radius parameter $x_{\\rm in}\\equiv {r_\\text{in}}/r_\\text{out}$ , and dimensionless eigenvalue $\\bar{\\lambda }\\equiv \\lambda /\\omega _{\\rm b}(r_\\text{out}).$ We assume that $\\alpha _{\\rm b}= \\text{constant}$ .",
"In terms of these parameters, Equations (REF ) and () become $\\bar{\\lambda }& E_+= \\textit {i}x^{3/2} \\left[ 2 E_+- \\frac{5 \\sin ^2 I}{2} (E_++ E_-) \\right]\\nonumber \\\\&+ \\textit {i}\\left[ \\frac{5/2 - p}{4 - p} \\left( \\frac{1-x_{\\rm in}^{4-p}}{1 - x_{\\rm in}^{5/2-p} } \\right) - x^{3/2} \\right] \\cos ^2 I E_+\\nonumber \\\\& + \\textit {i}\\frac{S x^{3/2 - 2q}}{2} \\left[ \\frac{\\text{d}^2}{\\text{d}x^2} + \\left( \\frac{3 - p}{x} \\right) \\frac{\\text{d}}{\\text{d}x} + \\frac{A(p,q)}{x^2} \\right] E_+\\nonumber \\\\&+ \\alpha _{\\rm b} \\frac{S x^{3/2 - 2q}}{2} \\left[ \\frac{\\text{d}^2}{\\text{d}x^2} + \\left( \\frac{3 - p - 2q}{x} \\right) \\frac{\\text{d}}{\\text{d}x} \\right] E_+, \\\\\\bar{\\lambda }& E_-= - \\textit {i}x^{3/2} \\left[ 2 E_-- \\frac{5 \\sin ^2 I}{2} (E_++ E_-) \\right]\\nonumber \\\\&- \\textit {i}\\left[ \\frac{5/2 - p}{4 - p} \\left( \\frac{1 - x_{\\rm in}^{4-p}}{1 - x_{\\rm in}^{5/2 - p} } \\right) - x^{3/2} \\right] \\cos ^2 I E_-\\nonumber \\\\&- \\textit {i}\\frac{S x^{3/2 - 2q}}{2} \\left[ \\frac{\\text{d}^2}{\\text{d}x^2} + \\left( \\frac{3-p}{x} \\right) \\frac{\\text{d}}{\\text{d}x} + \\frac{A(p,q)}{x^2} \\right] E_-\\nonumber \\\\& + \\alpha _{\\rm b} \\frac{S x^{3/2 - 2q}}{2} \\left[ \\frac{\\text{d}^2}{\\text{d}x^2} + \\left( \\frac{3-p-2q}{x} \\right) \\frac{\\text{d}}{\\text{d}x} \\right] E_-,$ whereIf the breathing mode term is not included [last term in Eq.",
"(REF )], $A(p,q) = 2q - p - 2 p q - 4 q^2$ .",
"Equations (REF )-() otherwise remain unchanged.",
"$A(p,q) = 6 - 4 q - p - 2pq - 4q^2.$ We adopt a free boundary condition, where the eccentricity gradient vanishes on the disk's boundaries: $\\left.",
"\\frac{\\text{d}E_\\pm }{\\text{d}r} \\right|_{r=r_{\\rm in}} = \\left.",
"\\frac{\\text{d}E_\\pm }{\\text{d}r} \\right|_{r=r_\\text{out}} = 0.$ In the following subsections, we calculate the eigenvalues and eigenmodes to Eqs.",
"(REF ) and ().",
"In Section REF , we investigate the limit $|r_\\text{out}-r_{\\rm in}| \\ll r_\\text{out}$ , where $\\lambda $ , $E_+(r)$ , and $E_-(r)$ may be found analytically.",
"In Section REF , we calculate numerically $\\lambda $ , $E_+(r)$ , and $E_-(r)$ for an inviscid ($\\alpha _{\\rm b} = 0$ ) extended ($|r_\\text{out}-{r_\\text{in}}| \\sim r_\\text{out}$ ) disk.",
"In Section REF , we investigate the effect of a non-zero bulk viscosity $\\alpha _{\\rm b}$ on the eigenvalues $\\lambda $ .",
"Figure: Real (solid) and imaginary (dotted) components of eigenvalue λ\\lambda for a thin annulus [see Eqs.",
"() and ()] as functions of inclination I=cos -1 (𝐥 ^·𝐥 ^ b )I = \\cos ^{-1}(\\mathbf {\\hat{l}}\\cdot \\mathbf {\\hat{l}}_{\\rm b}), for values of SS [Eq.",
"()] and qq [Eq.",
"()] as indicated.",
"We take p=1p = 1 [Eq.",
"()].Figure: Real (solid) and imaginary (dotted) components of the eigenvalue λ\\lambda [see Eqs.",
"() and ()] as functions of SS [Eq.",
"()], for values of inclination I=cos -1 (𝐥 ^·𝐥 ^ b )I = \\cos ^{-1}(\\mathbf {\\hat{l}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}}_{\\rm b}) and qq [Eq.",
"()] as indicated.",
"We take p=1p = 1 [Eq.",
"()]."
],
[
"Analytic Result for Thin Annulus",
"When $r_\\text{out}-r_{\\rm in}\\ll r_\\text{out}$ , we may expand all quantities in Equations (REF ) and () in terms of the small parameter $(r_\\text{out}-r)/r_\\text{out}=1-x$ .",
"The boundary condition (REF ) and normalization condition $E_+(r_\\text{out}) = 1$ imply $E_+(r) = 1 + \\mathcal {O}\\left[ \\left( 1-x \\right)^3 \\right]$ and $E_-(r) = E_-(r_\\text{out}) + \\mathcal {O}\\left[ \\left(1-x \\right)^3 \\right].$ Using the form of solutions (REF ) and (REF ), we may solve for the eigenvalue $\\bar{\\lambda }$ [Eq.",
"(REF )] to lowest order in $(r_\\text{out}-{r_\\text{in}})/r_\\text{out}= 1-x_{\\rm in}$ : $\\bar{\\lambda }^2 = \\, - & \\big [ 2 + S A(p,q)/2 \\big ] \\big [(2-5 \\sin ^2 I) + S A(p,q)/2 \\big ].$ The polynomial $A(p,q)$ is defined in Eq.",
"(REF ), and $S$ in Eq.",
"(REF ).",
"Plotted in Figure REF are the real (solid) and imaginary (dashed) components of the eigenvalue $\\lambda $ given by Equation (REF ), as functions of inclination $I$ with values of $S$ as indicated.",
"We always show the solutions with ${\\rm Re}(\\lambda ) > 0$ and ${\\rm Im}(\\lambda ) > 0$ .",
"When $S \\ll 1$ , we recover the classic LK result for a test particle, with $\\lambda ^2 > 0$ when $I$ exceeds the critical inclination angle $I_{\\rm LK}$ [Eq.",
"(REF )].",
"When $S \\gg 1$ , the Lidov-Kozai effect is suppressed by pressure gradients even when $I > I_{\\rm LK}$ .",
"In general, the critical inclination angle for eccentricity growth increases with increasing $S$ .",
"However, we see from Fig.",
"REF that for $S = 1.5$ and $q=3/4$ , the instability sets in when $I \\gtrsim 22^\\circ $ .",
"Figure REF further illustrates the difference in behavior between $q = 1/4$ (top panel) and $q=3/4$ (bottom panel).",
"For $q = 1/4$ , the real growth rate for inclinations $I > I_{\\rm LK}$ [Eq.",
"(REF )] monotonically decreases with increasing $S$ , until $\\lambda $ becomes imaginary.",
"But for $q = 3/4$ , a “window of instability\" opens for inclinations $I < I_{\\rm LK}$ when $S \\sim 1$ .",
"To understand the difference between these two models, consider the test particle limit ($c_{\\rm s}= 0$ ) and some additional pericenter precession $\\omega _{\\rm ext}$ from a source other than the binary companion.",
"In the frame co-rotating with the test particle's orbit normal, the time evolution of the eccentricity vector is given by $\\frac{\\text{d}{\\mathbf {e}}}{\\text{d}t} = \\omega _{\\rm b}\\big [ 2 \\mathbf {\\hat{l}}{\\mathbf {\\times }}{\\mathbf {e}}- 5 ({\\mathbf {e}}{\\mathbf {\\cdot }}\\mathbf {\\hat{l}}_{\\rm b}) \\mathbf {\\hat{l}}{\\mathbf {\\times }}\\mathbf {\\hat{l}}_{\\rm b}\\big ] + \\omega _{\\rm ext}\\mathbf {\\hat{l}}{\\mathbf {\\times }}{\\mathbf {e}}.$ Assuming ${\\mathbf {e}}\\propto \\exp (\\lambda t)$ , we find the eigenvalue $\\lambda ^2 = -(2 \\omega _{\\rm b}+ \\omega _{\\rm ext})(2 \\omega _{\\rm b}+ \\omega _{\\rm ext}- 5 \\omega _{\\rm b}\\sin ^2 I).$ When $\\omega _{\\rm ext}\\ge 0$ , the extra pericenter precession works to suppress the LK instability, decreasing the range of $I$ values for eccentricity growth ($\\lambda ^2 > 0$ ).",
"When $\\omega _{\\rm ext}\\le -2\\omega _{\\rm b}$ or $\\omega _{\\rm ext}\\ge 3 \\omega _{\\rm b}$ , no value of $I$ is capable of exciting eccentricity growth.",
"But when $-2 \\omega _{\\rm b}< \\omega _{\\rm ext}< 0$ , the extra precession works to cancel the pericenter precession induced on the test particle by the distant binary ($2 \\omega _{\\rm b}$ ), thus increases the range of $I$ values for eccentricity growth.",
"Comparing Eq.",
"(REF ) to Eq.",
"(REF ) shows the pressure force in a disk annulus induces precession $\\omega _{\\rm ext}= \\omega _{\\rm b}S A(p,q)/2$ .",
"Since $A(1,1/4)>0$ , the pressure force in the $p = 1$ and $q=1/4$ disk tends to suppress eccentricity growth (Figs.",
"REF -REF , top).",
"But because $A(1,3/4)<0$ , the pressure force in the $p = 1$ and $q=3/4$ disk can lead to eccentricity growth even when $I < I_{\\rm LK}$ (Figs.",
"REF -REF , bottom)."
],
[
"Inviscid Extended Disk",
"We solve eigenvalue equations (REF ) and () using the shooting method [41] for an inviscid ($\\alpha _{\\rm b} = 0$ ) extended ($|r_\\text{out}-{r_\\text{in}}| \\sim r_\\text{out}$ ) disk.",
"In Figure REF , we plot the real (solid) and imaginary (dashed) components of the eigenvalues $\\lambda = \\bar{\\lambda }\\omega _{\\rm b}$ as functions of inclination $I$ .",
"For ${r_\\text{in}}/r_\\text{out}$ close to unity, our numerical result agrees with the analytic expression for a thin annulus [Eq.",
"(REF )].",
"In general, when $S \\gg 1$ , the pressure force suppresses the eccentricity growth for all values of $I$ .",
"When $S \\sim 1$ , Fig.",
"REF displays the importance of the disk's radial extent on the eigenvalues $\\lambda $ .",
"For example, when $S = 0.6$ and $x_{\\rm in}= 0.4$ , eccentricity growth is achieved for $I\\gtrsim 69^\\circ $ , while for $x_{\\rm in}= 0.2$ the LK instability occurs for $I \\gtrsim 27^\\circ $ .",
"In Figure REF , we plot the eigenvalue $\\lambda = \\bar{\\lambda }\\omega _{\\rm b}(r_\\text{out})$ as a function of $S$ , for ${r_\\text{in}}/r_\\text{out}= 0.2$ , $p=1$ , and values of $q$ and $I$ as indicated.",
"Both models ($q = 1/4$ and $q = 3/4$ ) exhibit the suppression of eccentricity growth for $S \\gtrsim 1$ , and both models have a window of instability open when $S \\sim (\\text{few}) \\times 0.1$ .",
"This window of instability is similar to that seen in Figure REF .",
"Figure: Real (solid lines) and imaginary (dashed lines) components of the eigenfunctions E + (r)E_+(r) and E - (r)E_-(r) for an extended disk.",
"The normalization condition is E + (r out )=1E_+(r_\\text{out}) = 1.",
"The disk parameters are α b =0\\alpha _{\\rm b} = 0, p=1p=1, q=3/4q=3/4, r in /r out =0.2{r_\\text{in}}/r_\\text{out}= 0.2, inclination I=70 ∘ I = 70^\\circ , and values of SS [Eq.",
"()] as indicated.",
"The corresponding eigenvalues are λ ¯=1.84\\bar{\\lambda }= 1.84 (top), λ ¯=1.44\\bar{\\lambda }= 1.44 (middle), and λ ¯=5.57𝑖\\bar{\\lambda }= 5.57 \\textit {i} (bottom).Figure REF depicts some examples of the eigenfunctions $E_+(r)$ and $E_-(r)$ for disk models with $S = 0.03, 0.3$ , and 3.",
"We see that for small $S$ (top panel), the amplitudes $|E_+|$ and $|E_-|$ are largest at $r = r_\\text{out}$ and decreases rapidly as $r \\rightarrow {r_\\text{in}}$ .",
"For larger $S$ (middle and lower panels), the variations of $|E_+|$ and $|E_-|$ across the disk become smaller as the larger sound speed “smooths out\" the disk.",
"The bottom panel of Fig.",
"REF shows that when $S = 3$ (for which the disk is stable since $\\lambda $ is imaginary), the eigenfunctions $E_+$ and $E_-$ are both real and satisfy $E_-> E_+$ , implying retrograde precession of the disk's eccentricity."
],
[
"Effect of Viscosity",
"We solve the eigenvalue equations (REF )-() including the viscosity term.",
"In Figure REF , we plot the real parts of the eigenvalues $\\lambda $ for $\\alpha _{\\rm b} = 0, 0.03$ , and $0.1$ .",
"When $S \\lesssim 1$ , we see for a range of inclinations, the growth rates are only slightly modified by viscosity.",
"When $S \\gtrsim 1$ , the addition of a small viscosity begins to be important.",
"However, in this regime, the instability is already suppressed by the disk's pressure, so the additional damping from $\\alpha _{\\rm b}$ when $S \\gtrsim 1$ is not relevant for the LK effect.",
"We conclude that a small bulk viscosity does little to quench the LK instability."
],
[
"Summary of Key Results",
"Using linear theory of eccentric disturbances in hydrodynamical disks, we have shown that circumstellar disks in binary systems may undergo coherent eccentricity growth when the disk is significantly inclined with respect to binary orbital plane.",
"We consider the regime where the disk remains approximately flat and undergoes rigid-body nodal precession around the binary; this requires that bending waves efficiently communicate warps in different regions of the disk within the precession period.",
"We find that the disk's eccentricity response to the secular tidal forcing from the binary companion depends crucially on the dimensionless ratio [see Eq.",
"(REF )], $S=\\left( \\frac{c_{\\rm s}^2}{ 3G M_{\\rm b}r^2/4a_{\\rm b}^3}\\right)_{r=r_\\text{out}},$ where $c_{\\rm s}^2$ (disk sound speed squared) measures the characteristic torque (per unit mass) associated with gas pressure, $3GM_{\\rm b}r^2/4a_{\\rm b}^3$ (with $M_{\\rm b}$ and $a_{\\rm b}$ the companion mass and semi-major axis) measures the tidal torque from the companion.",
"The eccentricity response also depends on the disk's radial extent ($r_{\\rm out}/r_{\\rm in}$ ) and density and sound speed profiles [Eqs.",
"(REF ) and (REF )].",
"When $S \\ll 1$ , the “standard\" Lidov-Kozai effect is reproduced for a thin disk annulus ($r_{\\rm out}/r_{\\rm in}\\rightarrow 1$ ), with exponential eccentricity growth occuring for disk inclination $I$ (with respect to the binary orbital plane) between $39^\\circ $ and $141^\\circ $ .",
"As $S$ increases, the inclination window for disk eccentricity growth generally decreases.",
"When $S \\gg 1$ , eccentricity growth is completely quenched for all disk inclinations.",
"When $S \\sim 1$ , a new “window of instability\" opens up for certain disk parameters, where coherent disk eccentricity growth is observed for inclinations $I$ outside the standard $(39^\\circ ,141^\\circ )$ window.",
"These conclusions are qualitatively robust, shown through both analytic calculations when the disk's radial extent is negligible (thin annulus; Sec.",
"REF ) and numerical eigenmode analyses when the disk has a significant radial extent (Sec.",
"REF ).",
"We find that viscosity does little to quench the Lidov-Kozai instability of the disk (Sec.",
"REF ).",
"The different disk eccentricity responses to the secular tidal forcing can be understood in terms of the apsidal precession produced by gas pressure (i.e.",
"[36], [14], [42]).",
"This precession depends on the $S$ and the disk density/pressure profiles.",
"Unlike the other short-range forces, such as those due to General Relativity and tidal interaction in hierarchical triple systems (e.g.",
"Liu et al.",
"2015), the pressure-induced precession can be either prograde or retrograde, depending on the disk profiles [see Eq.",
"(REF ); see also [42]].",
"This gives rise to the nontrivial behavior of the disk's eccentricity response for $S\\sim 1$ ."
],
[
"Discussion",
"In this paper we have focused on the linear regime of the disk Lidov-Kozai instability, which manifests as the coherent growth of disk eccentricity, with no change in the disk inclination (which enters at the order $e^2$ ).",
"Numerical simulations are necessary to fully understand the nonlinear development of the disk eccentricity-inclination oscillations [27], [12], [13].",
"Nevertheless, our analytic results can be used to determine under what conditions a hydrodynamical circumstellar disk is susceptible to Lidov-Kozai oscillations, without resorting to full 3D numerical simulations.",
"We note that the dynamical behavior of eccentric disturbances in a hydrodynamical disk depends on the disk's equation of state and vertical structure [14], [35], [42].",
"We have adopted the eccentric disk models with locally isothermal equation of state, including the 3D breathing mode term from the disk's vertical structure (see [35] for discussion).",
"Using different models can change the details of our results, but not the general conclusions summarized in Section REF .",
"The disk eccentricity excitation mechanism studied in this paper is distinct from the mechanism that relies on eccentric Lindblad resonance [25].",
"The latter operates on the dynamical timescale and requires that the disk be sufficiently extended relative to the binary separation (i.e., $r_{\\rm out}/a$ is sufficiently larger) so that the resonance resides in the disk.",
"By contrast, the disk Lidov-Kozai mechanism for eccentricity excitation requires an inclined binary companion, and operates on a secular timescale [Eq.",
"(REF )] $&& t_{\\rm LK}\\sim \\omega _{\\rm b}(r_{\\rm out})^{-1}= 5.7\\times 10^3\\,{\\rm years}\\left(\\frac{M}{M_{\\rm b}}\\right)\\left(\\frac{a_{\\rm b}}{3 r_\\text{out}}\\right)^3\\nonumber \\\\&&\\times \\left(\\frac{M}{1 M_\\odot }\\right)^{-1/2}\\left(\\frac{r_\\text{out}}{100 \\,\\text{AU}}\\right)^{3/2}.$ For protoplanetary disks, this timescale is much less than the disk lifetime (a few Myrs).",
"To avoid suppression of the instability by the gas pressure, we also require $S = 0.36 \\left( \\frac{a_{\\rm b}}{3\\,r_\\text{out}}\\right)^3 \\left(\\frac{M}{M_{\\rm b}}\\right)\\left( \\frac{H(r_\\text{out})}{0.1 \\, r_\\text{out}} \\right)^2 \\lesssim 1.$ Thus, a “weaker” companion (large $a_{\\rm b}$ and small $M_{\\rm b}$ ) would not excite eccentricity in a thick (large $H/R$ ) disk.",
"Condition (REF ) is consistent with the SPH simulations of [27] and [12], where $S$ values in the range $8.5\\times 10^{-3}$ to $0.11$ were used.",
"Finally, for a massive disk, the LK instability can be suppressed due to apsidal precession generated by disk self-gravity [3], [13].",
"The apsidal precession rate from the disk's self gravity is roughly $\\omega _{\\rm sg}(r) \\sim \\frac{\\pi G \\Sigma }{r n}.$ Crudely, to avoid suppression of the LK instability, we require $\\omega _{\\rm sg}(r_\\text{out}) \\lesssim \\omega _{\\rm b}(r_\\text{out})$ , or the disk mass $M_\\text{d}\\lesssim M_{\\rm b}\\left(\\frac{r_\\text{out}}{a_{\\rm b}}\\right)^3 \\sim 0.04 \\, M_{\\rm b}\\left( \\frac{3 r_\\text{out}}{a_{\\rm b}} \\right)^3.$"
],
[
"Acknowledgments",
"We thank the anonymous referee for his or her valuable comments.",
"This work has been supported in part by NASA grants NNX14AG94G and NNX14AP31G, and a Simons Fellowship from the Simons Foundation.",
"JZ is supported by a NASA Earth and Space Sciences Fellowship in Astrophysics.",
"This appendix is devoted to the derivation of Eqs.",
"() and (REF ).",
"Our key assumption is that the internal force in the disk acts to enforce coplanarity and rigid body precession of the disk.",
"Consider a disk particle (test mass) with the position vector ${\\mathbf {r}}$ and velocity ${\\mathbf {v}}$ relative to the central star.",
"It's angular momentum is ${\\mathbf {L}}= {\\mathbf {r}}{\\mathbf {\\times }}{\\mathbf {v}}$ , and its eccentricity vector is ${\\mathbf {e}}= \\frac{1}{G M} {\\mathbf {v}}{\\mathbf {\\times }}({\\mathbf {r}}{\\mathbf {\\times }}{\\mathbf {v}}) - \\frac{{\\mathbf {r}}}{r}.$ Under the action of a perturbing force ${\\mathbf {f}}$ , the vectors ${\\mathbf {L}}$ and ${\\mathbf {e}}$ evolve according to $\\frac{\\partial {\\mathbf {L}}}{\\partial t} &= {\\mathbf {r}}{\\mathbf {\\times }}{\\mathbf {f}}, \\\\\\frac{\\partial {\\mathbf {e}}}{\\partial t} &= \\frac{1}{GM} {\\mathbf {f}}{\\mathbf {\\times }}({\\mathbf {r}}{\\mathbf {\\times }}{\\mathbf {v}}) + \\frac{1}{GM} {\\mathbf {v}}{\\mathbf {\\times }}({\\mathbf {r}}{\\mathbf {\\times }}{\\mathbf {f}}).$ The perturbing force ${\\mathbf {f}}= {\\mathbf {f}}_{\\rm b} + {\\mathbf {f}}_{\\rm int}$ consists of the tidal force from the binary companion ${\\mathbf {f}}_{\\rm b}$ and the internal pressure force ${\\mathbf {f}}_{\\rm int}$ .",
"To quadrapole order, the tidal force is given by ${\\mathbf {f}}_{\\rm b} = \\frac{G M_{\\rm b}}{|{\\mathbf {r}}_{\\rm b}|^3} \\left[ {\\mathbf {r}}- 3 \\frac{{\\mathbf {r}}_{\\rm b}({\\mathbf {r}}{\\mathbf {\\cdot }}{\\mathbf {r}}_{\\rm b})}{|{\\mathbf {r}}_{\\rm b}|^2} \\right],$ where $M_{\\rm b}$ and ${\\mathbf {r}}_{\\rm b}$ are the mass and position vectors of the companion.",
"Take the binary to be on a circular orbit with semi-major axis $a_{\\rm b}$ and mean anomaly $\\phi _{\\rm b}$ , and let ${\\mathbf {\\hat{r}}}$ , ${\\mathbf {\\hat{\\phi }}}= \\mathbf {\\hat{l}}{\\mathbf {\\times }}{\\mathbf {\\hat{r}}}$ , and $\\mathbf {\\hat{l}}$ be the radial, azimuthal, and angular momentum unit vectors of the test mass, respectively.",
"Averaging over the binary's orbital motion, we obtain the averaged tidal force $\\bar{{\\mathbf {f}}}_{\\rm b} \\equiv &\\frac{1}{2\\pi } \\int _{0}^{2\\pi } {\\mathbf {f}}_{\\rm b} \\text{d}\\phi _{\\rm b}\\\\= & \\frac{2}{3} r n \\omega _{\\rm b}\\big ( 1 - 3 \\sin ^2\\varphi \\sin ^2 I \\big ) \\hat{\\mathbf {r}}\\nonumber \\\\&- 2 r n \\omega _{\\rm b}\\big ( \\sin \\varphi \\cos \\varphi \\sin ^2 I \\big ) \\hat{\\mathbf {\\phi }}\\nonumber \\\\&- 2 r n \\omega _{\\rm b}\\big ( \\sin \\varphi \\sin I \\cos I \\big ) \\mathbf {\\hat{l}},$ where $\\omega _{\\rm b}$ is defined in Eq.",
"(REF ), and $\\varphi = \\omega + f$ is the azimuthal angle of the test mass measured from the ascending node ($\\omega $ and $f$ are the argument of pericenter and true anomaly).",
"The ${\\mathbf {\\hat{r}}}$ and ${\\mathbf {\\hat{\\phi }}}$ components of $\\bar{{\\mathbf {f}}}_{\\rm b}$ do not change ${\\mathbf {L}}$ , and the $\\mathbf {\\hat{l}}$ component induces precession at a rate $-\\omega _{\\rm b}\\cos I \\mathbf {\\hat{l}}_{\\rm b}$ [see Eq.",
"(REF )].",
"To ensure coplanarity and rigid-body precession of test particles at different radii, we assume that the internal force from disk pressure has the form ${\\mathbf {f}}_{\\rm int} = -2 r n (\\bar{\\omega }_{\\rm b}- \\omega _{\\rm b}) \\big (\\sin \\varphi \\sin I \\cos I \\big ) \\mathbf {\\hat{l}},$ where $\\bar{\\omega }_{\\rm b}$ is given in Eq.",
"(REF ).",
"We now substitute Eq.",
"(REF ) into Eqs.",
"(REF ) and () to obtain the effect of ${\\mathbf {f}}_{\\rm int}$ on $\\mathbf {\\hat{l}}$ and ${\\mathbf {e}}$ .",
"For a disk particle on an eccentric orbit $e \\ll 1$ , we can expand $r$ and $f$ in powers of $e$ [29].",
"Averaging over the mean anomaly of the test particle, we obtain Eqs.",
"() and (REF )."
]
] | 1612.05598 |
[
[
"A proof of the Multijoints Conjecture and Carbery's generalization"
],
[
"Abstract We present a new proof of the Joints Theorem without taking derivatives.",
"Then we generalize the proof to prove the Multijoints Conjecture and Carbery's generalization.",
"All results are in any dimension over an arbitrary field."
],
[
"Introduction",
"A recent major application of the polynomial method in discrete geometry is the proof of the following Joints Theorem.",
"It was initially proved by Guth and Katz [6] in $\\mathbb {R}^3$ and then generalized to $\\mathbb {R}^d$ by Quilodrán [14] and Kaplan-Sharir-Shustin [13].",
"For the arbitrary field case $\\mathbb {F}^d$ it is also known by the work of Carbery-Iliopoulou[3].",
"Theorem 1.1 (Joints Theorem [6][14][13][3]) For $N$ lines in $\\mathbb {F}^d$ ($d \\ge 2$ ), a joint is an intersection of $d$ lines such that the directions of the $d$ lines are linearly independent.",
"Then there are $\\lesssim N^{\\frac{d}{d-1}}$ joints.",
"The joint theorem states an inequality of multilinear flavor and can be viewed as a simplified discrete counterpart of the Multilinear Kakeya theorem proved by Bennett-Carbery-Tao [1] (non-endpoint case) and Guth [7] (full conjecture).",
"Being a central topic in harmonic analysis, multilinear Kakeya states similar bounds to Theorem REF on intersections of 1-tubes rather than lines.",
"However, it says “more” than Theorem REF in the sense that it also takes the multiplicities of tubes into account (hence of a “Kakeya maximal inequality” flavor).",
"Based on the multilinear Kakeya and people's success of proving the Joints Theorem, Carbery brought up the concept of “multijoints”, which is a straight generalization of the concept of joints (see Theorem REF ).",
"He then made a general conjecture on upper bounding the number of multijoints with multiplicity in any finite dimensional vector space over any field as a discrete analogue of the multilinear Kakeya.",
"In this paper we prove this conjecture and prove its important special case known as the Multijoints Conjecture (also brought up by Carbery) along the way.",
"Namely we will prove: Theorem 1.2 [Multijoints Theorem] Let $\\mathbb {F}$ be a field.",
"In $\\mathbb {F}^d$ , assuming we are given $d$ families of lines $L_1, \\ldots , L_d$ such that $N_i = |L_i| > 0$ .",
"A multijoint is an intersection of $d$ lines $l_i \\in L_i, 1\\le i \\le d$ such that the directions of the $d$ lines are linearly independent.",
"Then there are $\\lesssim (\\prod _{i=1}^d N_i)^{\\frac{1}{d-1}}$ multijoints.",
"Theorem 1.3 [Carbery's conjecture holds] In $\\mathbb {F}^d$ , assuming we are given $d$ families of lines $L_1, \\ldots , L_d$ such that $N_i = |L_i| > 0$ .",
"For every point $P \\in \\mathbb {F}^d$ , define the multijoint multiplicity $\\mu (P) = |\\lbrace (l_1, \\ldots , l_d): l_i \\in L_i, \\lbrace l_i\\rbrace \\text{ form a multijoint at } P\\rbrace |$ .",
"Then $\\sum _{P \\in \\mathbb {F}^d} \\mu (P)^{\\frac{1}{d-1}} \\lesssim (\\prod _{i=1}^d N_i)^{\\frac{1}{d-1}}.$ As stated above, Theorem REF (as well as its generalization Theorem REF that we shall see a bit later) can be viewed as a discrete analogue of the Multilinear Kakeya Theorem.",
"Important partial results on Theorem REF and Theorem REF has been made in [9][10][3][4][11][12][8][15].",
"For example in [12], Theorem REF was shown to be true for $\\mathbb {R}^d$ or $\\mathbb {F}^3$ for a general field $\\mathbb {F}$ .",
"The partial results in the paper we list above have various restrictions on the field, the dimension, the type of multijoints (assuming the multijoints are generic, which means whenever $d$ lines from the $d$ families meet they form a multijoint, is usually helpful), the exponent (In [15] Theorems REF and REF in $\\mathbb {R}^d$ up to endpoint on the exponent was shown, among other generalizations).",
"We do not state a comprehensive set of known results here and refer the interested readers to the above references.",
"Our approach, as with all current proofs of the Joints Theorem, uses the polynomial method whose initial application in incidence geometry was in Dvir's proof of the finite field Kakeya Conjecture [5].",
"A somewhat unique feature of our proof can be briefly described in the following way: In the “classical” polynomial method in discrete geometry, we usually force the polynomial to vanish on a lot of lines as a result of the polynomial being heavily incident to the line.",
"We then usually have to take special care to show that a nonzero polynomial could not vanish on too many lines in the problem and hence deduce a contradiction.",
"On the other hand, in our proof we mainly look at the Taylor series of the polynomial around each multijoint rather than caring if it actually vanishes at the point.",
"As a result, the lines where our polynomial vanish do not need to be specially taken care of, and could be handled in exactly the same way as we handle the other lines where the polynomial does not vanish.",
"For example along the way of our proof of Theorems REF and REF , we are able to define a “generalized intersection multiplicity” of a line $l$ and a polynomial hypersurface $Z(Q)$ at a point $P \\in l \\bigcap Z(Q)$ even if $l \\subseteq Z(Q)$ .",
"And for this generalized intersection multiplicity we have an upper bound estimate as good as what Bézout's Theorem gives (see Section 3 for details).",
"The “Taylor series” viewpoint here was already around for some time ([2]), while it seems to me that the “generalized intersection multiplicity” we defined above is a relatively new concept.",
"I believe the results in the current paper shows that both are interesting in their own rights and are nice additions to our current toolbox of the polynomial method.",
"At least we see its use in the multilinear setting of the incidence problems through the problems here.",
"For future directions, it is conceivable that one may generalize Theorems REF and REF to the setting where we have joints formed by general algebraic curves (which was considered in [9][10][11][12][15]).",
"Another more interesting direction is to try to prove the similar upper bounds of joints formed by higher dimensional objects (which should be the endpoint and $\\mathbb {F}^d$ case of the results in [15]).",
"Maybe this can be done by generalizing the tools we have here and probably those we developed in [16] for handling the interaction between multiple hypersurfaces and a high dimensional object.",
"We do not address either topic here and leave them to interested readers.",
"In this paper, we fix the dimension $d>1$ .",
"All the implied constants will depend on $d$ and we hence suppress the dependence on $d$ in our notations.",
"For example, “$\\lesssim $ ” should really be understood as “$\\lesssim _d$ ”."
],
[
"Acknowledgements",
"I was supported by Princeton University and the Institute for Pure and Applied Mathematics (IPAM) during the research.",
"Part of this research was performed while I was visiting IPAM, which is supported by the National Science Foundation.",
"I thank IPAM for their warm hospitality and amazing program.",
"I would like to thank Anthony Carbery, Jordan Ellenberg, Larry Guth, Marina Iliopoulou, Yuchen Liu, Ben Yang and Ziquan Zhuang for helpful discussions."
],
[
"A new proof of the Joints Theorem",
"In this section we present a new proof of the Joints Theorem REF that could be generalized to prove the Multijoints Theorem and Carbery's conjecture as we will see a bit later.",
"In this proof we do not take any kind of derivatives hence it works the same way on linear spaces over every field.",
"Let $\\mathbb {F}$ be an arbitrary field.",
"As a convention, by a nonzero polynomial over $\\mathbb {F}^d$ we mean a polynomial whose coefficients are not all zero, even though it might vanish at every point in the whole $\\mathbb {F}^d$ .",
"We say $Q \\ne 0$ if $Q$ is a nonzero polynomial and shall continue using this notion throughout this paper.",
"First we state the following variant of the well-known parameter counting lemma in linear algebra, which is immediate from the theory of linear equations.",
"Lemma 2.1 (Parameter Counting) Write any polynomial $Q$ of $d$ variables as $Q = \\sum _{\\beta = (\\beta _1, \\ldots , \\beta _d)} c_{\\beta } x^{\\beta }$ where almost all $c_{\\beta } = 0$ .",
"For any finite number $A$ and $A$ homogeneous linear forms $H_1, \\ldots , H_A$ , each supported on finitely many $c_{\\beta }$ as variables, there exists a nonzero polynomial $Q_0$ of degree $\\lesssim A^{\\frac{1}{d}}$ such that $H_1, \\ldots , H_A$ all vanish at the coefficients of $Q_0$ .",
"Assume that $J$ is the set of joints and $L$ the set of given lines.",
"By parameter counting, we can find a nonzero polynomial $Q$ of degree $\\lesssim |J|^{\\frac{1}{d}}$ that vanishes on each joint.",
"Now for each line $l \\in L$ , we define the concept of “ordinary” and “special” joints (with respect to $Q$ ) on it: For any joint $P \\in l$ , there exists an affine linear transform $T$ that sends $P$ to the origin, $l$ to the $x_d$ -axis, and $Q$ to a new function $(T^{-1})^* Q$ which is a polynomial of degree same as $Q$ .",
"We call $P$ an ordinary joint on $l$ , if the lowest homogeneous term of $(T^{-1})^* Q$ is independent of $x_d$ .",
"Otherwise $P$ is called a special joint on $l$ .",
"The above definition is intrinsic, i.e.",
"independent of the transform $T$ .",
"In fact, if there is another such linear transform $T^{\\prime }$ .",
"Then $T^{\\prime } = T_1 \\circ T$ where $T_1$ is a scaling when restricted to the $x_d$ -axis.",
"Such $T_1$ has the form $(T_1^{-1})^* (x_1, \\ldots , x_d) = (h_1 (x_1, \\ldots , x_{d-1}), \\ldots , h_{d-1} (x_1, \\ldots , x_{d-1})), \\lambda x_d + h_d (x_1, \\ldots , x_{d-1})$ where $\\lambda \\ne 0$ , $h_1, \\ldots , h_d$ are linear functions and the transform $(h_1, \\ldots , h_{d-1})$ is invertible.",
"Now $({T^{\\prime }}^{-1})^* Q = (T_1^{-1})^* (T^{-1})^* Q$ .",
"By the explicit form of $(T_1^{-1})^*$ we obtained above, we see that the definition of ordinarity/speciality does not depend on whether we choose the linear transform to be $T$ or $T^{\\prime }$ .",
"Next we notice that a joint cannot be ordinary on all lines passing through it, since we can take a linear transform to transform the $d$ transversal lines passing through it to the coordinate axes and $Q \\ne 0$ .",
"Then the lowest order homogeneous term of the new polynomial has to depend on some variable.",
"Hence $P$ is special with respect to the corresponding line.",
"Let us prove that on any line $l$ there are $\\le \\deg Q$ special joints.",
"We take a linear transform $T$ that sends $l$ to $x_d$ axis.",
"Now the polynomial $Q$ under the new coordinate system have the form $Q = \\sum _{\\alpha } f_{\\alpha } (x_d) x^{(\\alpha , 0)}$ .",
"Here by definition, $\\alpha = (\\alpha _1, \\alpha _2, \\ldots , \\alpha _{d-1})$ is a $(d-1)$ -dimensional multi-index and $x^{(\\alpha , 0)} = x_1^{\\alpha _1} x_2^{\\alpha _2} \\cdots x_{d-1}^{\\alpha _{d-1}}$ .",
"We denote $|\\alpha | = \\sum _{i} \\alpha _i$ .",
"Now we find a minimal $|\\alpha _0|$ such that $ f_{\\alpha _0} (x_d) \\ne 0$ and claim that all special joints mush have their $x_d$ coordinate being a root of $f_{\\alpha _0}$ .",
"Indeed, if $P \\in l$ is a joint such that its $x_d$ coordinate $x_d (P)$ is not a root of $f_{\\alpha _0}$ , then the lowest homogeneous term of $(T^{-1})^* Q$ will include a nonzero monomial $f_{\\alpha _0} (x_d (P)) x^{(\\alpha _0, 0)}$ , since $|\\alpha _0|$ is the smallest among all $|\\alpha |$ .",
"Now all monomials in this lowest homogeneous term have to be independent of $x_d$ , since otherwise the total power of $x_1, \\ldots , x_{d-1}$ in some monomial would be smaller than $|\\alpha _0|$ (a contradiction).",
"Hence $P$ is ordinary on $l$ .",
"Since each joint must be special with respect to at least one line, and on each line there are $\\le \\deg Q$ special joints, we have $|J| \\le N \\deg Q \\lesssim N |J|^{\\frac{1}{d}}$ .",
"Thus $|J| \\lesssim N^{\\frac{d}{d-1}}$ ."
],
[
"Some definitions motivated by the new proof of the Joints Theorem",
"The key idea in the proof of Theorem REF is not to be scared of the vanishing of a polynomial on a line, and use the “ordinarily” of most points on a line to proceed.",
"We will generalize this idea to prove the Multijoints Conjecture (Theorem REF ) and Carbery's generalization (Theorem REF ) for all fields in all dimensions.",
"We prepare some tools for the proofs.",
"In order to deal with multijoints problems, we would like to generalize the concept of “ordinarity/speciality” to take multiplicity into account.",
"The definition we end up using is slightly different from a direct generalization of “ordinarity/speciality” in the last section.",
"Let $\\mathbb {F}$ be an arbitrary field.",
"In $\\mathbb {F}^d$ assuming we have a point $P$ on a line $l$ and a polynomial $Q$ .",
"We define the $(P, l)$ -multiplicity of $Q$ in the following way: We choose a linear transform $T$ that sends $l$ to the $d$ -th coordinate axis.",
"Then we can write $(T^{-1})^* Q = \\sum _{\\alpha = (\\alpha _1, \\ldots , \\alpha _{d-1})} x_1^{\\alpha _1} \\cdots x_{d-1}^{\\alpha _{d-1}} f_{\\alpha } (x_d) = \\sum _{\\alpha = (\\alpha _1, \\ldots , \\alpha _{d-1})} x^{(\\alpha , 0)} f_{\\alpha } (x_d).$ In the above expansion, look at all $(d-1)$ -dimensional indices $\\alpha $ with $f_{\\alpha } \\ne 0$ and $|\\alpha | = \\alpha _1 + \\cdots + \\alpha _{d-1}$ being the smallest possible.",
"Call all such tuples $\\alpha $ to be lowest (with respect to $(P, l, T)$ ).",
"Assuming $T (P) = (0, \\ldots , 0, p_T)$ .",
"Look at all lowest tuples $\\alpha $ and corresponding $f_{\\alpha }$ .",
"Assuming $p_T$ is a root of $f_{\\alpha }$ with multiplicity $m_{\\alpha }$ (which is allowed to be zero).",
"Then we define the $(P, l)$ -multiplicity of $Q$ to be the minimal $m_{\\alpha }$ among all lowest $\\alpha $ .",
"We check that this is a well-defined quantity largely similar to what we did in the last section.",
"In fact, if we have another linear transform $T^{\\prime }$ sending $l$ to the $d$ -th coordinate axis and $P$ to $(0, \\ldots , 0, p_{T^{\\prime }})$ , then $T^{\\prime } = T_1 \\circ T$ where $T_1$ is a non-degenerate linear transform when restricted to the $x_d$ -axis.",
"Such $T_1$ has the form $(T_1^{-1})^* (x_1, \\ldots , x_d) = (h_1 (x_1, \\ldots , x_{d-1}), \\ldots , h_{d-1} (x_1, \\ldots , x_{d-1})), h(x_d) + h_d (x_1, \\ldots , x_{d-1})$ where $h, h_1, \\ldots , h_d$ are linear functions and the transform $(h_1, \\ldots , h_{d-1})$ as well as $h$ is invertible.",
"Moreover we have $h(p_T) = p_{T^{\\prime }}$ .",
"As before we have $({T^{\\prime }}^{-1})^* Q = (T_1^{-1})^* (T^{-1})^* Q$ .",
"Now we have $(T_1^{-1})^* (x^{(\\alpha , 0)} f_{\\alpha } (x_d)) = (h_1 (x_1, \\ldots , x_{d-1}), \\ldots , h_{d-1} (x_1, \\ldots , x_{d-1}))^{\\alpha } f_{\\alpha } (h(x_d) + h_d (x_1, \\ldots , x_{d-1}))$ .",
"Since $(h_1, \\ldots , h_{d-1})$ is non-degenerate, all $(h_1 (x_1, \\ldots , x_{d-1}), \\ldots , h_{d-1} (x_1, \\ldots , x_{d-1}))^{\\alpha }$ are linearly independent.",
"Hence the new lowest $|\\alpha ^{\\prime }|$ (with respect to $(P, l, T^{\\prime })$ ) is the same as the lowest $|\\alpha |$ (with respect to $(P, l, T)$ ).",
"When explicitly computing the $(P, l)$ -multiplicity of $Q$ under $T^{\\prime }$ , we may ignore the contribution of $h_d (x_1, \\ldots , x_{d-1})$ altogether since they only produces $|\\alpha ^{\\prime }|$ larger than the lowest one.",
"It is then straightforward to see that (by the linear independence of all $(h_1 (x_1, \\ldots , x_{d-1}), \\ldots , h_{d-1} (x_1, \\ldots , x_{d-1}))^{\\alpha }$ for lowest $\\alpha $ with respect to $T$ ) the $(P, l)$ -multiplicity of $Q$ does not depend on whether we choose the linear transform to be $T$ or $T^{\\prime }$ .",
"For our convenience denote $m_Q (P, l) \\ge 0$ to be the $(P, l)$ -multiplicity of $Q$ .",
"Remark 3.1 When $Q$ is not identically zero on $l$ , $m_Q (P, l)$ is equal to the intersection multiplicity between $Q$ and $l$ .",
"However, when $Q$ vanishes on $l$ we still have this $m_Q (P, l) < \\infty $ .",
"Note that under this situation it is still true that for almost all $P \\in l$ , $m_Q (P, l)=0$ .",
"In terms of $(P, l)$ -multiplicity, we have an inequality in place of Bézout's Theorem of intersection multiplicity.",
"Lemma 3.2 $\\sum _{P \\in l} m_Q (P, l) \\le \\deg Q.$ Without loss of generality, we may assume $l$ is the $x_d$ -axis.",
"Write $Q = \\sum _{\\alpha = (\\alpha _1, \\ldots , \\alpha _{d-1})} x^{(\\alpha , 0)} f_{\\alpha } (x_d).$ We choose a lowest $\\alpha = \\alpha _0$ .",
"Then $\\sum _{P \\in l} m_Q (P, l) \\le \\sum _{P = (0, \\ldots , 0, p) \\in l} \\text{ the multiplicity of } p \\text{ as a root of } f_{\\alpha _0} \\le \\deg f_{\\alpha _0} \\le \\deg Q.$ In order to obtain a lower bound of $m_Q (P, l)$ we will often invoke the following very strong lemma.",
"Lemma 3.3 Assuming that a linear transform $T$ sends $P$ to $(0, \\ldots , 0, p_T)$ and $l$ to the $x_d$ -axis.",
"We look at the Taylor expansion of $(T^{-1})^* Q$ at $(0, \\ldots , 0, p_T)$ : $(T^{-1})^* Q = \\sum _{\\beta = (\\beta _1, \\ldots , \\beta _d)} c_{\\beta } x_1^{\\beta _1} \\cdots x_{d-1}^{\\beta _{d-1}} \\cdot (x_d-p_T)^{\\beta _d}.$ Assuming some $\\beta _0 = (\\beta _{0, 1}, \\ldots , \\beta _{0, d})$ satisfying that $c_{\\beta _0} \\ne 0$ and that among all $\\beta $ with $c_{\\beta } \\ne 0$ , $|\\beta _0|$ is the smallest possible.",
"Then $m_Q (P, l) \\ge \\beta _{0, d}.$ Look at all possible $\\beta = \\beta ^{\\prime } = (\\beta _1 ^{\\prime }, \\ldots , \\beta _d ^{\\prime })$ in (REF ) such that $c_{\\beta ^{\\prime }} \\ne 0$ and $\\beta _1 ^{\\prime } + \\cdots + \\beta _{d-1} ^{\\prime }$ is smallest possible.",
"We must have $\\sum _{j=1}^{d-1}\\beta _j ^{\\prime } \\le \\sum _{j=1}^{d-1} \\beta _{0, j}$ .",
"But $|\\beta ^{\\prime }| \\ge |\\beta _0|$ by assumption.",
"Thus $\\beta _d ^{\\prime } \\ge \\beta _{0, d}$ .",
"Now merge the terms in (REF ) to have the form (REF ).",
"We see that all the lowest $\\alpha $ will have its $f_{\\alpha }$ , when expanded as Taylor series at $P$ , having every term divisible by $(x_d - p_T)^{\\beta _{0, d}}$ .",
"Hence $f_{\\alpha }$ is divisible by $(x_d - p_T)^{\\beta _{0, d}}$ and by definition (REF ) holds.",
"Remark 3.4 Lemma REF is strong in the following sense: Despite the fact that $m_Q (P, l)$ is independent of the choice of $T$ , by this lemma we can bound it from below when only given some (very incomplete) information of any fixed $T$ ."
],
[
"A proof of the Multijoints Conjecture",
"Theorem REF is a special case of Theorem REF .",
"However we have a very simple proof for Theorem REF by the tools we have developed.",
"Hence we present this proof in a separate section before proving the harder Theorem REF .",
"Assume that $J$ is the set of multijoints.",
"It has finitely many points.",
"For each $P \\in J$ , choose $l_{i, P} \\in L_i (1 \\le i \\le d)$ such that $l_{i, P}$ all pass through $P$ and have their directions span $\\mathbb {F}^d$ .",
"Choose an affine linear transform $T_P$ to transform $P$ to the origin and $l_{i, P}$ to the $i$ -th coordinate axis ($1 \\le i \\le d$ ).",
"By parameter counting we deduce that there exists a nonzero polynomial $Q$ of degree $\\lesssim |J|^{\\frac{1}{d}}\\cdot (\\prod _{i=1}^d |L_i|)^{\\frac{1}{d}}$ such that: For each point $P \\in J$ , $(T_P^{-1})^*Q$ , when expanded as a sum of monomials, has no term $x_1^{\\beta _1} \\cdots x_d^{\\beta _d}$ with $\\beta _i \\le |L_i|$ simultaneously holding.",
"For any $P\\in J$ , assuming $c_{\\beta } x^{\\beta }$ is a nonzero monomial in $(T_P^{-1})^* (Q)$ such that $|\\beta | = \\beta _1 + \\cdots \\beta _d$ is the smallest (here $\\beta = (\\beta _1, \\ldots , \\beta _d)$ ).",
"By the assumption on $Q$ above, at least one of $\\beta _i > |L_i|$ holds.",
"If there is such a $\\beta $ s.t.",
"$\\beta _i > |L_i|$ holds we say that $P$ is of type $i$ .",
"By Lemma REF we have that for any $P \\in J$ of type $i$ , $m_Q (P, l_{i, P}) \\ge \\beta _i > |L_i|$ .",
"Since each $P \\in J$ is of some type $i= i_P \\in \\lbrace 1, 2, \\ldots , d\\rbrace $ , there exists some popular $i_0$ such that at least $\\frac{|J|}{d}$ points in $J$ are of type $i_0$ .",
"Assuming such points form a set $J_{i_0}$ .",
"Then by the discussion above and Lemma REF , $|J||L_{i_0}| \\lesssim |J_{i_0}||L_{i_0}| \\le \\sum _{P \\in J_{i_0}} m_Q (P, l_{i_0, P})\\le \\sum _{P \\in J} m_Q (P, l_{i_0, P}) \\le |L_{i_0}|\\deg Q \\lesssim |J|^{\\frac{1}{d}}\\cdot (\\prod _{i=1}^d |L_i|)^{\\frac{1}{d}} \\cdot |L_{i_0}|$ which is equivalent to $|J| \\lesssim (\\prod _{i=1}^d N_i)^{\\frac{1}{d-1}}$ .",
"Remark 4.1 In the study of multilinear incidence geometry problems such as ones in this paper, it was noted for quite a while ([2]) that it can be good to have the low degree Taylor series, in addition to the polynomial itself, vanishing at given points.",
"This is the approach we take here and in the next section."
],
[
"A proof of Carbery's conjecture",
"In this section we prove Carbery's conjecture on counting multijoints with multiplicity (Theorem REF ), which generalizes the Multijoints Theorem REF we just proved.",
"Our proof will be based on the techniques we have developed so far.",
"Theorem REF is implied by the following theorem on joints with multiplicity.",
"Theorem 5.1 In $\\mathbb {F}^d$ , assuming we are given $N$ lines.",
"Here we allow a same line to show up multiple times and denote the resulting set with multiplicity to be $L = (l_1, \\ldots , l_N)$ .",
"For every point $P \\in \\mathbb {F}^d$ , define the joint multiplicity $M(P) = |\\lbrace (l_{i_1}, \\ldots , l_{i_d}): \\lbrace l_{i_j}\\rbrace \\text{ form a joint at } P\\rbrace |$ .",
"Then $\\sum _{P \\in \\mathbb {F}^d} M(P)^{\\frac{1}{d-1}} \\lesssim N^{\\frac{d}{d-1}}.$ Assuming we have Theorem REF proved already and are given the assumption of Theorem REF .",
"Now consider the collection $L$ of all $d$ families $L_i$ with each line in $L_i$ repeated $\\prod _{j \\ne i} N_j$ times.",
"Then $N = |L| \\sim \\prod _{i=1}^d N_i$ .",
"Moreover, at each $P$ each multijoint contributes $(\\prod _{i=1}^d N_i)^{d-1}$ to the joint multiplicity $M(P)$ of $L$ at $P$ .",
"Apply Theorem REF to $L$ and we have $\\sum _{P \\in \\mathbb {F}^d} \\mu (P)^{\\frac{1}{d-1}}\\cdot \\prod _{i=1}^d N_i \\le \\sum _{P \\in \\mathbb {F}^d} M(P)^{\\frac{1}{d-1}} \\lesssim (\\prod _{i=1}^d N_i)^{\\frac{d}{d-1}}$ and hence (REF ), as desired.",
"In order to prove Theorem REF , we do some preliminary work to understand $M(P)$ better.",
"Definition 5.2 Assuming $L = (l_1, \\ldots , l_N)$ is a set (with multiplicity) of lines in $\\mathbb {F}^d$ .",
"For any $P \\in \\mathbb {F}^d$ and arbitrary integer $1 \\le j \\le d$ , define the $j$ th-minimum of $L$ at $P$ $r_j (P, L)$ to be $r_j (P, L) = \\min _{V \\text{ is a subspace of } \\mathbb {F}^d, \\dim V = j-1} |\\lbrace i: P \\in l_i, l_i \\text{ is not parallel to } V\\rbrace |.$ Hence $r_1 (P, L)$ is simply the number of $l_i$ 's that pass through $P$ .",
"As another example, $r_d (P, L) > 0$ is equivalent to saying that the lines in $L$ form at least one joint at $P$ .",
"It is also trivial that $r_1 (P, L) \\ge \\cdots \\ge r_d (P, L)$ .",
"The reason that we call them $j$ th-minimum is simply that they resemble the successive minima in the geometry of numbers a bit.",
"Lemma 5.3 $M(P) \\sim \\prod _{j=1}^d r_j (P, L).$ Let us count the number of tuples $(l_{i_1}, \\ldots , l_{i_d})$ forming a joint at $P$ .",
"Once $\\lbrace l_{i_k}\\rbrace _{1\\le k < j}$ are fixed we always have at least $r_j (P, L)$ different ways to choose $l_{i_j} \\ni P$ not parallel to the $(j-1)$ -dimensional subspace determined by $\\lbrace l_{i_k}\\rbrace _{1\\le k < j}$ .",
"In this way eventually the $d$ lines we choose will form a joint at $P$ .",
"Hence $M(P) \\ge \\prod _{i=1}^d r_j (P, L)$ .",
"On the other hand, for $1\\le j \\le d$ choose $W_j (P)$ to be a $(j-1)$ -dimensional space such that the set $X_j (P) = \\lbrace i : p \\in l_i, l_i \\text{ not parallel to } W_j (P)\\rbrace $ has cardinality $r_j (P, L)$ .",
"Then for any $l_{i_1}, \\ldots , l_{i_d}$ forming a joint at $P$ , there has to be at least one number among $i_1, \\ldots , i_d$ that belongs to $X_d (P)$ ; at least two numbers among $i_1, \\ldots , i_d$ that belong to $X_{d-1} (P)$ , $\\ldots $ , at least $d$ numbers among $i_1, \\ldots , i_d$ that belong to $X_{1} (P)$ .",
"Hence among $i_1, \\ldots , i_d$ there is at least one number that belongs to $X_d (P)$ ; among the rest there is at least one that belongs to $X_{d-1} (P)$ , $\\ldots $ , in the end the one number left belongs to $X_{1} (P)$ .",
"Hence $M(P) \\lesssim \\prod _{j=1}^d r_j (P, L)$ .",
"The proof of Theorem REF is then a result of a good understanding on the lower bound $m_Q (P, l)$ for an arbitrary $l$ passing through $P$ .",
"We want $m_Q (P, l)$ to be large on most directions and are able to prove that (intuitively) when it is small the line $l$ is usually parallel to certain “bad” subspaces (note that we do not run into more complicated “bad subvarieties situation” in our proof at all, which could be somewhat counterintuitive).",
"This is then good enough to prove Theorem REF .",
"For every point $P$ such that $M(P) > 0$ , we claim that we can find a flag $V_1 (P) \\subseteq V_2 (P) \\subseteq \\cdots \\subseteq V_d (P) \\subseteq V_{d+1} (P) = \\mathbb {F}^d$ , such that (a) $\\dim V_j (P) = j-1, 1\\le j \\le d$ ; (b) the set $L_j (P) = \\lbrace i : p \\in l_i, l_i \\text{ not parallel to } V_j (P)\\rbrace $ has cardinality $\\sim r_j (P, L)$ for $1 \\le j \\le d$ and (c) for any $1 \\le j \\le d$ and any $(j-1)$ -dimensional subspace $V$ of $V_{j+1} (P)$ , we have $|\\lbrace i : p \\in l_i, l_i \\text{ is parallel to } V\\rbrace | \\le |\\lbrace i : p \\in l_i, l_i \\text{ is parallel to } V_j (P)\\rbrace |.$ Note that this is slightly stronger than what we had for $W_j (P)$ 's and $X_j (P)$ 's in the proof of Lemma REF .",
"In this paragraph we prove the above claim.",
"Our strategy is to choose $\\lbrace V_j (P)\\rbrace $ inductively (downwards): By the definition of $r_d (P, L)$ , there exists a $(d-1)$ -dimensional subspace $V_d (P)$ such that the above defined $L_d (P)$ has cardinality $r_d (P, L)$ .",
"Assuming we already have $V_{j+1} (P)\\subseteq \\cdots \\subseteq V_d (P) (1\\le j < d)$ such that (a), (b) and (c) hold for them.",
"Now we look for $V_j (P)$ .",
"By the definition of $r_j (P, L)$ , there exists $V_j ^{\\prime } (P)$ being a $(j-1)$ -dimensional subspace such that the set $\\lbrace i : p \\in l_i, l_i \\text{ not parallel to } V_j ^{\\prime } (P)\\rbrace $ has cardinality $r_j (P, L)$ .",
"We choose $V_j ^{\\prime \\prime } (P)$ to be an arbitrary $(j-1)$ -dimensional subspace of $V_{j+1} (P)$ containing $V_j ^{\\prime } (P) \\bigcap V_{j+1} (P)$ .",
"Now if a line is not parallel to $V_j ^{\\prime \\prime } (P)$ , then it is either not parallel to $V_j ^{\\prime } (P)$ or not parallel to $V_{j+1} (P)$ .",
"The number of the lines of the first type is $\\le r_j (P, L)$ while by induction hypothesis the number of the lines of the second type is $\\lesssim r_{j+1} (P, L) \\le r_j (P, L)$ .",
"Hence the set $\\lbrace i : p \\in l_i, l_i \\text{ not parallel to } V_j ^{\\prime \\prime } (P)\\rbrace $ has cardinality $\\lesssim r_j (P, L)$ .",
"Now choose $V_j (P) \\subseteq V_{j+1} (P)$ such that $\\dim V_j (P) = j-1$ and $|\\lbrace i : P \\in l_i, l_i \\text{ parallel to } V_j (P)\\rbrace |$ is maximal possible.",
"Then it is obvious that $|\\lbrace i : p \\in l_i, l_i \\text{ not parallel to } V_j (P)\\rbrace | \\le |\\lbrace i : p \\in l_i, l_i \\text{ not parallel to } V_j ^{\\prime \\prime } (P)\\rbrace | \\lesssim r_j (P, L)$ and $V_j(P)$ satisfies (c) as well as (a) and (b).",
"This closes the induction and our claim follows.",
"Now for every $P$ such that $M(P) > 0$ , we choose $d$ lines $s_{1, P}, \\ldots , s_{d, P}$ passing through $P$ such that $V_j (P)$ is spanned by the directions of $s_{k, P}, 1 \\le k < j$ .",
"Notice that $s_{j, P}$ may not belong to $L$ anymore.",
"We then fix a linear transform $T_P$ that sends $P$ to the origin and $s_{j, P}$ to the $j$ -th coordinate axis.",
"We look for a nonzero polynomial $Q$ such that $(T_P^{-1})^* Q$ , when expanded as a sum of monomials (Taylor series), has no term of the form $x_1^{\\beta _1} \\cdots x_d^{\\beta _d}$ with $\\beta _j \\le \\frac{100^j B\\cdot \\prod _{k \\ne j} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}}$ simultaneously holding.",
"Here $B$ is a large positive integer depending on $L$ such that all $\\frac{B\\cdot \\prod _{k \\ne j} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}} > 100d^{100}$ , $\\forall P, j$ with $M(P) > 0$ .",
"By Lemma REF and parameter counting, there exists such a nonzero $Q$ with $\\deg Q \\lesssim B\\cdot (\\sum _{P \\in \\mathbb {F}^d} M(P)^{\\frac{1}{d-1}})^{\\frac{1}{d}}$ .",
"Next we prove the following crucial estimate for every $P \\in \\mathbb {F}^d$ satisfying $M(P) > 0$ : $\\sum _{i: P \\in l_i} m_Q (P, l_i) \\gtrsim B\\cdot M(P)^{\\frac{1}{d-1}}.$ To prove (REF ), we fix $P$ .",
"Write $(T_P^{-1})^* Q = \\sum _{\\beta } c_{\\beta } x^{\\beta }$ and then look at all $\\beta $ such that $c_{\\beta } \\ne 0$ and $|\\beta |$ smallest possible.",
"Call all such $\\beta $ to be lowest.",
"By the assumption of $Q$ , there is some $1 \\le j_0 \\le d$ such that (i) for every lowest $\\beta $ and every $j < j_0$ , $\\beta _j \\le \\frac{100^j B\\cdot \\prod _{k \\ne j} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}}$ and (ii) there exists a lowest $\\beta $ such that $\\beta _{j_0} > \\frac{100^{j_0} B\\cdot \\prod _{k \\ne j_0} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}}$ .",
"We fix $j_0$ in the rest of the proof of (REF ).",
"Let $j_1\\le d$ be the largest integer $j$ such that $& |\\lbrace i : P \\in l_i, l_i \\text{ is parallel to } V_j (P) \\text{ but not parallel to }V_{j_0} (P)\\rbrace |\\nonumber \\\\& \\le \\frac{1}{2} |\\lbrace i : P \\in l_i, l_i \\text{ is parallel to } V_{j+1} (P) \\text{ but not parallel to }V_{j_0} (P)\\rbrace |.$ We notice $j = j_0$ satisfies (REF ) as LHS would be 0.",
"Hence $j_1 \\ge j_0$ .",
"Note that $|\\lbrace i : P \\in l_i, l_i \\text{ is parallel to } V_{d+1} (P) \\text{ but not parallel to }V_{j_0} (P)\\rbrace | = |L_{j_0} (P)| \\sim r_{j_0} (P, L)$ .",
"By the opposite of (REF ) from $j= j_1 + 1$ to $j = d$ we deduce $|\\lbrace i : P \\in l_i, l_i \\text{ is parallel to } V_{j_1 + 1} (P) \\text{ but not parallel to }V_{j_0} (P)\\rbrace | \\gtrsim r_{j_0} (P, L)$ which also holds when $j_1 = d$ .",
"Since $V_{j_1 + 1} \\subseteq V_{d+1}$ we also have the other side of the inequality: $|\\lbrace i : P \\in l_i, l_i \\text{ is parallel to } V_{j_1 + 1} (P) \\text{ but not parallel to }V_{j_0} (P)\\rbrace | \\sim r_{j_0} (P, L).$ Moreover by (REF ) for $j = j_1$ we also have $|\\lbrace i : P \\in l_i, l_i \\text{ is parallel to } V_{j_1 + 1} (P) \\text{ but not parallel to } V_{j_1}\\rbrace | \\gtrsim r_{j_0} (P, L).$ By the assumption (c) in the beginning of our current proof of Theorem REF and (REF ), for any $V_{j_0} (P) \\subseteq V \\subseteq V_{j_1+1} (P)$ with $\\dim V =j_1 -1$ , $|\\lbrace i : P \\in l_i, l_i \\text{ is parallel to } V_{j_1 + 1} (P) \\text{ but not parallel to }V\\rbrace | \\gtrsim r_{j_0} (P, L).$ By (REF ) we also have the other side of (REF ): $|\\lbrace i : P \\in l_i, l_i \\text{ is parallel to } V_{j_1 + 1} (P) \\text{ but not parallel to }V\\rbrace | \\sim r_{j_0} (P, L).$ Let $t$ = $j_1 - j_0 +1 > 0$ .",
"For any $t$ lines $l_{i_1}, \\ldots , l_{i_t}$ such that the direction of them and $V_{j_0}$ exactly span $V_{j_1 + 1}$ , we prove there exists $1 \\le h \\le t$ such that for $l = l_{i_h}$ , $m_{Q} (P, l) \\gtrsim \\frac{B\\cdot \\prod _{k \\ne j_0} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}}.$ In order to show (REF ), we choose a linear transform $T_P ^{\\prime }$ that preserves the origin and the directions of the $j$ -th coordinate axis for $1\\le j < j_0$ or $j_1 < j \\le d$ , while sending ${(T_P)}_* l_{i_{j-j_0 + 1}}$ to the $j$ -th coordinate axis for $j_0 \\le j \\le j_1$ (by assumption in the beginning of this paragraph, all $d$ directions we mentioned are linearly independent).",
"Then $T_P ^{\\prime } \\circ T_P$ sends $l_{i_{j-j_0 + 1}}$ to the $j$ -th coordinate axis, $j_0 \\le j \\le j_1$ .",
"Moreover, $((T_P ^{\\prime } \\circ T_P)^{-1})^* Q = ((T_{P} ^{\\prime })^{-1})^* (T_P^{-1})^* Q$ .",
"We assume that $\\delta $ is a positive number such that among all the lowest order terms $\\prod _{k=1}^d x_k^{\\gamma _k}$ of $((T_P ^{\\prime } \\circ T_P)^{-1})^* Q$ , $\\delta $ is the maximal possible sum $\\gamma _{j_0} + \\ldots + \\gamma _{j_1}$ .",
"Now look at the action of $(T_P ^{\\prime })^*$ on this polynomial.",
"By the assumptions, $(T_P ^{\\prime })^* x_j = x_j$ for $j_1 < j \\le d$ , $(T_P ^{\\prime })^* x_j = x_j + H_j (x_{j_0}, \\ldots , x_{j_1})$ for $1 \\le j < j_0$ where $H_j$ is some linear form, and finally $(T_P ^{\\prime })^* x_j = F_j (x_{j_0}, \\ldots , x_{j_1})$ for $j_0 \\le j \\le j_1$ where the linear transformation $(x_{j_0}, \\ldots , x_{j_1}) \\mapsto (F_{j_0} (x_{j_0}, \\ldots , x_{j_1}), \\ldots , F_{j_1} (x_{j_0}, \\ldots , x_{j_1}))$ is invertible.",
"The action of its inverse $((T_P ^{\\prime })^{-1})^*$ on coordinate functions has the same form.",
"Now by assumption (i), for every monomial $x^{\\beta }$ with nonzero coefficient in the lowest homogeneous term of $((T_P)^{-1})^* Q$ and every $j < j_0$ , $\\beta _j \\le \\frac{100^j B\\cdot \\prod _{k \\ne j} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}}$ .",
"Hence this is also true for every monomial $x^{\\beta }$ with nonzero coefficient in the lowest homogeneous term of $((T_P ^{\\prime } \\circ T_P)^{-1})^* Q$ .",
"Apply $(T_P ^{\\prime })^*$ to $((T_P ^{\\prime } \\circ T_P)^{-1})^* Q$ we actually get $(T_P^{-1})^* Q$ .",
"By the bounds on the exponents of $x_j (j \\le j_1)$ in the lowest homogeneous part of $((T_P ^{\\prime } \\circ T_P)^{-1})^* Q$ we have above and the fact that $r_1 (P, L) \\ge \\cdots \\ge r_d (P, L)$ , we deduce that for any lowest order term $\\prod _{k=1}^d x_k^{\\beta _k}$ of $(T_P^{-1})^* Q$ , $\\sum _{j=1}^{j_1} \\beta _j &\\le \\sum _{j=1}^{j_0 -1} \\frac{100^j B\\cdot \\prod _{k \\ne j} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}} + \\delta \\nonumber \\\\& \\le \\frac{2 \\cdot 100^{j_0 -1} B\\cdot \\prod _{k \\ne j_0} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}} + \\delta .$ But by the assumption (ii) there exists such a $\\beta $ satisfying $\\beta _{j_0} > \\frac{100^{j_0} B\\cdot \\prod _{k \\ne j_0} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}}$ .",
"We deduce that $\\delta \\gtrsim \\frac{B\\cdot \\prod _{k \\ne j_0} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}}$ .",
"This means among all the lowest order terms $\\prod _{k=1}^d x_k^{\\gamma _k}$ of $((T_P ^{\\prime } \\circ T_P)^{-1})^* Q$ , there is at least one with $\\gamma _{j_0} + \\ldots + \\gamma _{j_1} \\gtrsim \\frac{B\\cdot \\prod _{k \\ne j_0} r_k(P, L)}{(\\prod _{k=1}^d r_k (P, L))^{\\frac{d-2}{d-1}}}$ .",
"Note that $T_P ^{\\prime } \\circ T_P$ sends $l_{i_{j-j_0 + 1}}$ to the $j$ -th coordinate axis, $j_0 \\le j \\le j_1$ , by Lemma REF we deduce that (REF ) has to hold for some $l = l_{i_h}$ , $1 \\le h \\le t$ .",
"Define $Y(P) = \\lbrace i : P \\in l_i, l_i \\text{ is parallel to } V_{j_1 + 1} (P) \\text{ but not parallel to }V_{j_0} (P)\\rbrace $ Look at all the lines $l = l_i$ violating (REF ) with $i \\in Y(P)$ .",
"By the above discussion, there are never $(j_1 - j_0 + 1)$ such lines satisfying their directions and $V_{j_0}$ span $V_{j_1}$ .",
"Therefore all the directions of such lines have to be contained in a common $(j_1 -1)$ -dimensional subspace $\\widetilde{V} (P)$ .",
"Now by (REF ) with $V = \\widetilde{V} (P) \\supseteq V_{j_0}$ we deduce that there are $\\gtrsim r_{j_0} (P, L)$ different $i\\in Y(P)$ such that $l = l_i$ satisfies (REF ).",
"Summing (REF ) over all of them and note that $M(P) \\sim \\prod _{k=1}^d r_k (P, L)$ , we obtain (REF ).",
"The rest of the proof is simple.",
"Summing (REF ) over all $P \\in \\mathbb {F}^d$ (it certainly holds for those $P$ with $M(P) = 0$ too) and use Lemma REF , we deduce $B\\cdot \\sum _{P \\in \\mathbb {F}^d} M(P)^{\\frac{1}{d-1}} \\lesssim &\\sum _{P\\in \\mathbb {F}^d} \\sum _{i: P \\in l_i} m_Q (P, l_i)\\nonumber \\\\= & \\sum _{1\\le i\\le N} \\sum _{P \\in l_i} m_Q (P, l_i)\\nonumber \\\\\\le & N \\cdot \\deg Q\\nonumber \\\\\\lesssim & B\\cdot N \\cdot (\\sum _{P \\in \\mathbb {F}^d} M(P)^{\\frac{1}{d-1}})^{\\frac{1}{d}}$ which is equivalent to the conclusion of the theorem.",
"Department of Mathematics, Princeton University, Princeton, NJ 08540 [email protected]"
]
] | 1612.05717 |
[
[
"Probing Seesaw with Parity Restoration"
],
[
"Abstract We present a novel way of testing the seesaw origin of neutrino mass in the context of the minimal Left-Right Symmetric Model.",
"It is based on the connection between the leptonic interactions of the doubly charged scalars, whose presence is at the core of the seesaw mechanism, and the neutrino Dirac Yukawa couplings which govern, among other processes, the right-handed neutrino decays into left-handed charged leptons.",
"We prove that any physical quantity depending on these couplings is a function of the hermitian part only which can significantly simplify their future experimental determination."
],
[
"colorlinks, citecolor=nicegreen,linkcolor=nicered Probing Seesaw with Parity Restoration Goran Senjanović Gran Sasso Science Institute, Viale Crispi 7, L'Aquila 67100, Italy International Centre for Theoretical Physics, Trieste 34100, Italy Vladimir Tello Gran Sasso Science Institute, Viale Crispi 7, L'Aquila 67100, Italy We present a novel way of testing the seesaw origin of neutrino mass in the context of the minimal Left-Right Symmetric Model.",
"It is based on the connection between the leptonic interactions of the doubly charged scalars, whose presence is at the core of the seesaw mechanism, and the neutrino Dirac Yukawa couplings which govern, among other processes, the right-handed neutrino decays into left-handed charged leptons.",
"We prove that any physical quantity depending on these couplings is a function of the hermitian part only which can significantly simplify their future experimental determination.",
"I.",
"Introduction.",
"The Standard electroweak Model (SM) started as a gauge theory of weak interactions and over the years, with the advent of the Higgs mechanism, turned into a theory of particle masses.",
"In the case of charged fermions this mechanism can be verified from the correlation between their masses and the Higgs boson decays into fermion antifermion pairs.",
"A key task of present-day particle physics is to achieve similar correlations for neutrino.",
"A natural candidate to address this issue is the minimal left-right (LR) symmetric theory [1], suggested originally to explain parity violation in weak interactions through the spontaneous breaking of LR symmetry, which led to a non-vanishing neutrino mass and eventually to the seesaw mechanism [2], [3], [4] as an explanation of its smallness.",
"The Majorana nature of heavy right-handed (RH) neutrinos leads to the production of same-sign lepton pairs at hadronic colliders [5] and to a new contribution [3], [6] to neutrinoless double beta decay, two deeply interconnected manifestations of lepton number violation [7].",
"This framework provides an ideal setting to address the issue of neutrino mass.",
"Indeed, we find the existence of a novel connection between the neutrino Dirac Yukawa couplings and the leptonic doubly charged scalar interactions which allows to predict a number of physical processes.",
"This way of disentangling the seesaw mechanism emerges as a consequence of generalized parity ($\\mathcal {P}$ ) as the LR symmetry and goes along the same lines as the correlation between the left-handed (LH) and RH quark mixings [8].",
"For the case of generalized charge conjugation as the LR symmetry the above connection is lost.",
"In this case, however, it is possible to determine the neutrino Dirac Yukawa couplings from the knowledge of the LH and RH neutrino masses and mixings [9].",
"Still, in the case of $\\mathcal {P}$ chosen here we show that such a way of unravelling the seesaw mechanism could also be achieved.",
"We demonstrate this explicitly for purely hermitian Dirac Yukawa couplings, although the employed procedure can serve to tackle the general case as well.",
"Needless to say, the central aspect of our work, i.e., the correlation between the Dirac mass matrix and the doubly charged scalar Yukawa couplings, is valid in all of the parameter space.",
"II.",
"The Minimal LR Model.",
"The minimal left-right symmetric model (LRSM) is based on the gauge group $SU(2)_L \\otimes SU(2)_R \\otimes U(1)_{B-L}$ , enlarged by generalized parity $\\mathcal {P}$ .",
"Among other features, this implies the equality of gauge couplings $g_L = g_R \\equiv g$ .",
"Fermions transform as LR symmetric doublet representations $q_{L,R} = (u,d)_{L,R}$ , $\\ell _{L} = (\\nu _L, e_L)$ , $\\ell _{R} = (N_R, e_R)$ and the charged gauge interactions are given by $ & \\mathcal {L}_{gauge} = \\frac{g}{\\sqrt{2}} \\left(\\overline{\\nu }_L V_{L}^\\dag {W}_{\\!L} e_L +\\overline{N}_R V_{R}^\\dag {W}_{\\!R} e_R\\right) + \\text{h.c.}$ where $V_L$ is the PMNS leptonic mixing matrix and $V_R$ its RH analog.",
"Once produced, the RH charged gauge boson $W_R$ decays into RH neutrinos which allows to determine their masses and mixings through the so-called KS process [5].",
"This is tailor made for hadronic colliders, such as the LHC where the experimental reach goes all the way up to a $W_R$ mass of about 6 TeV [10].",
"The Higgs sector consists [2], [3], [6] of a complex bi-doublet $\\Phi (2,2,0)$ and complex triplets $\\Delta _L(3,1,2)$ and $\\Delta _R(1,3,2)$ with quantum numbers referring to $SU(2)_L \\otimes SU(2)_R \\otimes U(1)_{B-L}$ .",
"At the first stage of symmetry breaking, the neutral component of $\\Delta _R$ develops a vev $v_R$ and breaks the original symmetry down to the SM one.",
"The latter is broken through the vevs of the bi-doublet neutral components $\\langle \\Phi \\rangle =v\\, \\text{diag} (\\cos \\beta ,-\\sin \\beta e^{-ia}).$ The small parameter $s_at_{2 \\beta }$ measures the amount of spontaneous CP violation and is bounded from above [8] by $ s_at_{2 \\beta } \\lesssim 2 m_b/m_t$ which makes it a suitable expansion parameter.",
"The electroweak symmetry breaking turns on a small vev for $\\Delta _L$ , $v_L \\propto v^2/v_R$ [6], whose experimental determination is somewhat involved [11].",
"The lepton Yukawa couplings in the minimal theory take the following form $- &\\mathcal {L}_Y= \\overline{\\ell _{L}}\\,\\big (Y_{1} \\Phi - Y_{2}\\, \\sigma _2 \\Phi ^* \\sigma _2)\\, \\ell _R \\nonumber \\\\ &+\\frac{1}{2} \\ell ^T_L Y_{L} i\\sigma _2 \\Delta _L \\ell _L +\\frac{1}{2} \\ell ^T_R Y_R i\\sigma _2 \\Delta _R\\ell _R + \\text{h.c.}$ Under generalized parity the fields transform as $\\ell _L \\leftrightarrow \\ell _R$ , $\\Phi \\rightarrow \\Phi ^{\\dagger }$ , $\\Delta _L \\leftrightarrow \\Delta _R$ , which implies the following relations between Yukawa matrices $Y_{1,2}=Y_{1,2}^{\\dagger },\\qquad Y_{L}=Y_{R}.$ These equalities play an essential basis in what follows.",
"III.",
"Lepton masses and mixings.",
"From (REF ), the hermitian nature of $Y_{1,2}$ implies the following relations $&M_D- U_{e}M_D^{\\dagger }U_{e}=is_at_{2\\beta }(e^{ia}t_{\\beta }M_D+m_{e})\\\\[8pt]&m_{e}-U_{e} m_{e}U_{e}=-is_at_{2\\beta }(M_D+e^{-ia}t_{\\beta }m_{e})$ where $M_D$ is the neutrino Dirac mass matrix.",
"The unitary matrix $U_e$ is given by $U_{e}=E_R^{\\dagger }E_L$ , where $E_{L,R}$ are the left and right-handed unitary rotations of charged leptons obtained from the diagonalization of their mass matrix $M_{e}=E_Lm_{e}E_R^{\\dagger }$ .",
"In the same basis the light neutrino mass matrix $M_{\\nu }$ in the seesaw picture [6] becomes $&M_{\\nu }= \\frac{v_L}{v_R}U_{e}^T M_N^* U_{e} - M_D^T \\frac{1}{M_{N}} M_D,$ where $M_N$ is the mass matrix of heavy RH neutrinos.",
"In terms of masses and mixings one has $M_{\\nu }=V_L^*m_{\\nu }V_L^{\\dagger }, \\,\\,\\,\\,\\,\\,\\, M_N=V_R m_N V_R^T$ The presence of $U_e$ signals the lack of hermiticity of the charged lepton matrix due to the spontaneous breaking of parity.",
"To appreciate better what is going on, it is instructive to take the limit of the so-called type II seesaw [12] in which the first term in the rhs of (REF ) dominates the neutrino mass.",
"This does not necessarily imply a tiny $M_D$ ; it could be simply a consequence of a large enough $v_L$ .",
"It is easy to show that, in this case, the masses of light and heavy neutrinos are proportional to each other $m_N \\propto m_\\nu $ and $U_e = V_R V_L^{\\dagger }$ up to an overall phase.",
"Since in general the charged lepton matrix is not hermitian, the LH and RH lepton mixings are not necessarily equal, the difference of which is contained in the $U_e$ matrix.",
"In order to probe the origin of neutrino mass in a generic seesaw, it would seem that one must determine $M_D$ from (REF ), hard to achieve since in general this matrix is not hermitian [13].",
"We offer here an alternative approach based on the following simple but important observation.",
"It turns out, from (), that $U_e$ is not arbitrary but is actually correlated with the neutrino Dirac mass matrix $U_{e}&=\\frac{1}{m_{e}}\\sqrt{m_{e}^2+is_at_{2\\beta } (t_{\\beta }e^{-ia}m_{e}^2+m_{e}M_D)}$ which allows to disentangle the seesaw mechanism and leads to a number of important phenomenological implications.",
"Since $U_e$ is unitary, $M_D$ is not an arbitrary complex matrix.",
"Instead of $2 n^2$ it has only $n^2$ real elements.",
"Indeed, it can be shown from (REF ) that its anti-hermitian part $M_D^A$ is not independent but can be determined from the knowledge of the hermitian part $M_D^H$ .",
"At leading order in $s_at_{2\\beta }$ it takes the form $M_D^{A}\\!=\\!\\frac{is_at_{2\\beta }}{2}\\!\\Big (\\!",
"m_{e}\\!+\\!2t_{\\beta }M_D^H\\!+\\!\\mathcal {H_D}M_D^H\\!+\\!M_D^H\\mathcal {H_D} \\!\\Big ) \\!$ where we have introduced the hermitian matrix $(\\mathcal {H}_D)_{ij}= \\frac{\\,\\,\\,(M_D^H)_{ij}}{m_{e_i}+m_{e_j}}.$ The fact that generalized parity fixes $M_D^A$ simplifies matters tremendously and opens the possibility for the experimental determination of the full $M_D$ matrix, as we show below.",
"A discussion is in order.",
"First of all, in this work we focus on a directly testable seesaw picture in which the RH neutrinos can actually be produced at the LHC or next generation of hadronic colliders.",
"This requires their masses to lie below 10 TeV or so.",
"From the seesaw formula (REF ), this means that the elements of $M_D$ are at most of the order of MeV.",
"This justifies the validity of the expansion in $s_at_{2\\beta }$ and allows us to work at leading order; for larger values of $M_D$ this would not be feasible a priori.",
"Very large $M_D$ becomes of special interest in grand unified theories where one expects superheavy RH neutrinos.",
"This is beyond the scope of this Letter and will be addressed in a separate publication.",
"We wish to emphasize that our formalism is general and can be used for any value of $M_D$ .",
"In principle, one could also imagine cancellations between type I and type II seesaw contributions, so that $M_D$ could be large even for relatively light $N$ .",
"We do not advocate this, but it can be dealt with in the same way without modification.",
"What matters is that expression (REF ) for $U_e$ is valid for all $M_D$ satisfying (REF ).",
"On the other hand, in the absence of such cancellations, $s_at_{2 \\beta }$ is limited by (3) just as in the quark sector.",
"Next, in the same verifiable seesaw picture, the RH neutrinos born out of the production of $W_R$ lead to a background free final state composed of charged lepton and jet pairs [5].",
"They decay dominantly through a virtual $W_R$ which allows to determine both $m_N$ and $V_R$ [14].",
"This process offers a starting point to probe the nature and origin of neutrino mass.",
"The subdominant decay of RH neutrinos into LH leptons and RH antileptons allows also for the determination of $M_D$ by a careful measurement of the chirality of the outgoing charged lepton [10], [15].",
"Moreover, there are other ways of probing $M_D$ through, for example, the leptonic decay of the heavy scalar doublet $H$ belonging to the bi-doublet representation.",
"Limits from $K$ - and $B$ -meson physics set a stringent limit on its mass of about 20 TeV [16].",
"Once again, only the hermitian part $M_D^H$ is independent leading to a one-to-one correspondence between the number of independent channels of each of these decays and the real elements of the hermitian part.",
"Finally, as we now show, with the experimental knowledge of $M_D^H$ one can obtain the flavor structure of the LH doubly charged scalars and in this manner pave the way towards a verification of the Higgs origin of neutrino mass.",
"This is central to our work.",
"IV.",
"From Dirac to Doubly Charged.",
"The crucial point to realize is that the $U_e$ matrix enters in the Yukawa interaction of the doubly charged scalars $\\delta _{L,R}^{++}$ $-\\mathcal {L}_{\\delta }\\!=\\!\\frac{1}{2} \\, \\delta _L^{^{++}} e_{L}^T \\bigg (\\!U_{e}^T \\frac{M_N^*}{v_R} U_e\\!\\bigg ) e_L\\!+ \\frac{1}{2}\\delta _R^{^{++}} e_{R}^T \\bigg (\\!",
"\\frac{ M_N^*}{v_R}\\!",
"\\bigg ) e_R \\!",
"$ Since both LH and RH triplet Yukawa couplings are proportional to the mass matrix of RH neutrinos, the role of $U_e$ is to account for the mismatch between LH and RH sectors.",
"Doubly charged scalars are produced pairwise and thus expected, unless very light, to be less accessible than $W_R$ .",
"The $\\delta _R ^{++} \\rightarrow e_{iR}^+ e_{jR}^+$ decays provide a way of measuring $m_N$ and $V_R$ , complementary to the KS process.",
"Even more interesting are the decays of $\\delta _L^{++}\\rightarrow e_{iL}^+ e_{jL}^+$ since they offer a physical connection with the seesaw mechanism.",
"In addition to the RH neutrino masses and mixing, their determination requires the knowledge of the $U_{e}$ matrix which probes $M_D$ through (REF ).",
"By studying the flavor structure of these decays and the corresponding low energy rare leptonic processes one can in principle extract the elements of the $U_e$ matrix.",
"We leave aside the low energy aspects and concentrate on high energy decays because colliders have a higher capacity to identify the chirality of outgoing leptons.",
"In particular, the comparison between $\\delta _L^{++}$ and $\\delta _R ^{++}$ decays is rather useful due to the common $m_N$ and $V_R$ dependence.",
"After some deliberation one obtains, at leading order in small $ s_at_{2\\beta }$ , the following prediction for the ratio between $\\delta _L^{++}$ and $\\delta _R^{++}$ decay rates into charged lepton pairs $\\frac{\\Gamma _{\\delta _L\\rightarrow e_{L_i} e_{L_j}}}{\\Gamma _{\\delta _{R}\\rightarrow e_{R_i} e_{R_j}}}\\simeq \\frac{m_{\\delta _L } }{m_{\\delta _R } } \\!\\bigg [1\\!+\\!2 s_at_{2\\beta } \\, \\text{Im} \\frac{ \\big ( \\mathcal {H}_D M_N + M_N \\mathcal {H}_D^T \\big )_{ij} }{(M_N)_{ij} } \\bigg ] $ As we argued before, the result manifestly depends only on the hermitian part of $M_D$ .",
"Moreover, both $U_{e}$ and $M_D$ contribute to the neutrino mass.",
"The correlation between these two quantities, previously thought unrelated, provides a novel way of disengaging the seesaw mechanism and opens a new chapter into the probe of the origin of neutrino masses.",
"Besides the doubly charged states, the triplets contain also singly charged and neutral scalars.",
"Their interactions are however of secondary importance since neutrinos are missing energy at colliders and we do not consider them here.",
"The hermitian case.",
"It is evident from (REF ) that $U_{e}$ becomes the identity matrix in the limit $s_at_{2\\beta } \\rightarrow 0$ and its connection with $M_D$ disappears.",
"In this limit however one can express $M_D$ as a function of light and heavy neutrino masses and mixings.",
"The point is that $M_D$ becomes purely hermitian and by using (REF ) it can be decomposed as $M_D= V_R \\sqrt{m_N}\\, \\left(O \\sqrt{s}EO^{\\dagger }\\right) \\,\\sqrt{m_N} V_R^{\\dagger }$ The orthogonal matrix $O$ and the symmetric normal form $s$ (see [17] for mathematical details) are obtained from the following symmetric matrix $\\frac{v_L^*}{v_R}-\\frac{1}{\\sqrt{m_N}} V_R ^{\\dagger } V_Lm_{\\nu }V_L^{T} V_R^{*}\\frac{1}{\\sqrt{m_N}}= O s O^T$ provided that the following conditions are satisfied $\\text{Im}\\text{Tr} \\left[ \\frac{v_L^*}{v_R}- \\frac{1}{M_N} M_{\\nu }^* \\right]^n= 0,\\quad (n=1,2,3)$ which imply that the phases of light and heavy neutrino mass matrices are not independent.",
"It can be shown that due to the above constraints, excluding pathological situations, the symmetric normal form $s$ takes only two possible diagonal forms $s_{I}=\\text{diag}(s_1,s_0,s_2) ,\\quad s_{II}=\\text{diag}(s,s_0,s^*)$ where $s_{0,1,2}$ are real, whereas $s$ is complex.",
"This in turn fixes uniquely the form of the matrix $E$ which ensures that (REF ) is hermitian.",
"After some thought one getsIt is straightforward to obtain the matrix $E$ for the more complicated degenerate cases in which $s$ is not diagonal.",
"$E_{\\text{I}}= \\left( \\begin{array}{ccc}1& 0&0 \\\\0& 1 & 0 \\\\0 & 0 & 1\\end{array} \\right), \\qquad E_{II}= \\left( \\begin{array}{ccc}0& 0&1 \\\\0& 1 & 0 \\\\1 & 0 & 0\\end{array} \\right)$ corresponding to $s_I$ and $s_{II}$ , respectively.",
"For $s_a t_{2\\beta }=0$ , just as in the case of charge conjugation [9], the Dirac Yukawa matrix depends only on the left- and right-handed neutrino mass matrices, and takes a natural value on the order of $\\sqrt{m_{\\nu }m_N}$ .",
"For the sake of illustration, we exemplify the above procedure for a simplified choice $V_R=V_L$ which from (REF ) implies a real $v_L$ .",
"Using (REF ) the Dirac mass matrix then takes the simple form $M_D= V_L m_N \\sqrt{ \\frac{v_L}{v_R}-\\frac{m_{\\nu }}{m_N}}V_L^{\\dagger }$ In general, for a non-vanishing $s_a t_{2\\beta }$ , there is an additional dependence on the masses of charged leptons as can be seen from (REF ).",
"V. Phenomenological implications.",
"We discuss here how to probe the origin of neutrino mass by determining the neutrino Dirac mass matrix and the doubly charged scalar decay rates.",
"As we already said, the Dirac Yukawa couplings can be obtain from rare heavy $N$ decays into LH charged leptons or RH antileptons.",
"These decays are due to the light-heavy neutrino mixing induced by the Dirac Yukawa couplings [18].",
"The decay rate has the following form $\\begin{split}&\\Gamma (N_i \\rightarrow W_L^+ e^{}_{L_{ j}})\\propto \\frac{ m_{N_i}}{M_{W_L}^2}\\big |(V_R^{\\dagger } M_D)_{ij}\\big |^2\\end{split}$ Due to (REF ) this depends only on $M_D^H$ .",
"In order for this channel to provide an efficient way of measuring the neutrino Dirac Yukawas, it needs the $m_N$ and $V_R$ input from the KS process.",
"In case $N$ were lighter than $W_L$ , one would obviously have the analog $W_L \\rightarrow N e_L$ decays controlled by the same couplings.",
"With enough energy, future colliders can also in principle detect the heavy scalar doublet $H$ and measure its leptonic decays.",
"In particular, the decays of the neutral component $H^0$ into charged lepton pairs are given by $\\begin{split}\\Gamma (H^0\\rightarrow e_i\\bar{e}_j)\\propto \\frac{m_H}{M_{W_L}^2} |(M_D^H+s_{2\\beta }m_{e})_{ij}|^2\\end{split}$ where for the sake of simplicity and illustration we have neglected the phase $a$ .",
"We have also ignored the tiny scalar mixings, suppressed by the electro-weak scale, between the superheavy $H^0$ and other scalars such as the SM Higgs and $\\delta _R^0$ .",
"In this approximation, the real and imaginary components of $H^0$ can be treated as degenerate particles.",
"These decays provide a rather clean test of $M_D^H$ since there is no additional lepton mixing.",
"Moreover, the flavor changing decays are directly proportional to the off-diagonal elements of $M_D^H$ which facilitate their determination.",
"It is important that there are nine different channels for nine different $M_D^H$ elements.",
"Of course, together with the above $N$ decays, one will end up with an over constrained system and be able to determine also $a$ and $\\beta $ .",
"The $a$ and $\\beta $ parameters parameters enter in a number of other physical processes.",
"For example, the mixing $\\xi _{LR}$ between $W_L$ and $W_R$ gauge bosons is given by $|\\xi _{LR}| \\simeq {M_{W_L}^2}/{M_{W_R}^2} \\sin 2\\beta $ , which in principle allows for a measure of $\\beta $ , assuming it is not very small.",
"Not an easy task due to the strong suppression of the small ratio of left to right charged gauge boson masses.",
"Likewise, the couplings of the heavy doublet to quarks are readily seen to depend explicitly on $a$ and $\\beta $ , allowing for their simultaneous determination.",
"They are also a function of the quark mixing matrix $V_R^q$ whose analytic dependance on $s_a t_{2\\beta }$ can be found in [8].",
"The measurement of the RH quark mixing itself from $W_R$ interactions can thus also be used to deduce the essential CP-violating parameter $s_a t_{2\\beta }$ .",
"VI.",
"Conclusions and Outlook.",
"In the SM the knowledge of charged fermion masses uniquely predicts the branching ratios of the Higgs boson decays.",
"As shown in [9], exactly the same happens in the minimal Left-Right Symmetric Model for the masses of light and heavy neutrinos with generalized charge conjugation as the LR symmetry.",
"The case of generalized parity is more complex.",
"We obtained an explicit analogous result for purely hermitian Dirac Yukawa couplings and at the same time we have set the mathematical framework needed to address the general case.",
"Moreover, in this work we devised a novel strategy to deal with this complex issue by exploiting the di-lepton final states in the decay of the LH doubly charged scalar.",
"We showed that the flavor structure of these decays depends on the neutrino Dirac Yukawa couplings, which can be determined from the decays of RH neutrinos into LH charged leptons.",
"There are additional processes such as the leptonic decays of the heavy scalar doublet which probe these couplings as well.",
"In particular, the decay of its neutral component into charged leptons pairs does this in a transparent manner.",
"Furthermore, we have demonstrated that only the hermitian part of the Dirac Yukawa couplings matters.",
"This feature substantially facilitates its experimental determination by effectively halving the degrees of freedom to be measured.",
"The complete determination of the neutrino Dirac mass matrix is admittedly a task for a future collider, but then so is the verification of the Higgs origin of light charged fermion masses.",
"Nonetheless, the point we are making is also one of principle, i.e., the neutrino Dirac Yukawa couplings in this theory correlate seemingly disconnected physical processes.",
"This is what sets this theory apart from the usual seesaw mechanism in the SM.",
"We cannot overemphasize the fact that the LR symmetry used originally to understand parity violation in weak interactions not only requires the existence of RH neutrinos which leads to the seesaw mechanism, but also connects the seesaw with a number of different physical processes and makes the LRSM a self-contained theory of neutrino mass.",
"Acknowledgments.",
"We thank A. Melfo for useful discussions, comments and careful reading of the manuscript.",
"GS acknowledges warm hospitality of the Stermasi establishment on the island of Mljet during the last stages of this work."
]
] | 1612.05503 |
[
[
"Computability of Perpetual Exploration in Highly Dynamic Rings"
],
[
"Abstract We consider systems made of autonomous mobile robots evolving in highly dynamic discrete environment i.e., graphs where edges may appear and disappear unpredictably without any recurrence, stability, nor periodicity assumption.",
"Robots are uniform (they execute the same algorithm), they are anonymous (they are devoid of any observable ID), they have no means allowing them to communicate together, they share no common sense of direction, and they have no global knowledge related to the size of the environment.",
"However, each of them is endowed with persistent memory and is able to detect whether it stands alone at its current location.",
"A highly dynamic environment is modeled by a graph such that its topology keeps continuously changing over time.",
"In this paper, we consider only dynamic graphs in which nodes are anonymous, each of them is infinitely often reachable from any other one, and such that its underlying graph (i.e., the static graph made of the same set of nodes and that includes all edges that are present at least once over time) forms a ring of arbitrary size.",
"In this context, we consider the fundamental problem of perpetual exploration: each node is required to be infinitely often visited by a robot.",
"This paper analyzes the computability of this problem in (fully) synchronous settings, i.e., we study the deterministic solvability of the problem with respect to the number of robots.",
"We provide three algorithms and two impossibility results that characterize, for any ring size, the necessary and sufficient number of robots to perform perpetual exploration of highly dynamic rings."
],
[
"Introduction",
"We consider systems made of autonomous robots that are endowed with visibility sensors and motion actuators.",
"Those robots must collaborate to perform collective tasks, typically, environmental monitoring, large-scale construction, mapping, urban search and rescue, surface cleaning, risky area surrounding, patrolling, exploration of unknown environments, to quote only a few.",
"Exploration belongs to the set of basic task components for many of the aforementioned applications.",
"For instance, environmental monitoring, patrolling, search and rescue, and surface cleaning are all tasks requiring that robots (collectively) explore the whole area.",
"To specify how the exploration is achieved, the so-called “area” is often considered as “zoned area” (e.g., a building, a town, a factory, a mine, etc.)",
"modeled by a finite graph where (anonymous) nodes represent locations that can be sensed by the robots, and edges represent the possibility for a robot to move from one location to the other.",
"To fit various applications and environments, numerous variants of exploration have been studied in the literature, for instance, terminating exploration —the robots stop moving after completion of the exploration of the whole graph [13], [9], [8]—, exclusive perpetual exploration —every node is visited infinitely often, but no two robots collide at the same node [1], [2]—, exploration with return —each robot comes back to its initial location once the exploration is completed [11]—, etc.. Clearly, some of these variants may be mixed (e.g., exclusive perpetual exploration vs. non exclusive terminating exploration) and either weakened or strengthened (weak perpetual exploration —every node is visited infinitely often by at least one robot [3]— vs. strong perpetual exploration —every node is visited infinitely often by each robot—, etc.).",
"Note that all these instances of exploration are different problems in the sense that, in most of the cases, solutions for any given instance cannot be used to solve another instance.",
"Also, some solutions are designed for specific graph topologies, e.g., ring-shaped [13], line-shaped [15], tree-shaped [14], and other for arbitrary network [7].",
"In this paper, we address the (non-exclusive weak version of the) perpetual exploration problem, i.e., each node is visited infinitely often by a robot.",
"Robots operate in cycles that include three phases: Look, Compute, and Move (L-C-M).",
"The Look phase consists in taking a snapshot of the (local) environment of robots using the visions capabilities offered by the sensors they are equipped with.",
"The snapshot depends on the sensor capabilities with respect to environment.",
"During the Compute phase, a robot computes a destination based on the previous observation.",
"The Move phase simply consists in moving to this destination.",
"Using L-C-M cycles, several models has been proposed in the literature, capturing various degrees of synchrony between robots [17].",
"They are denoted by $\\mathcal {FSYNC}$ , $\\mathcal {SSYNC}$ , and $\\mathcal {ASYNC}$ , from the stronger to the weaker.",
"In $\\mathcal {FSYNC}$ (fully synchronous), all robots execute the L-C-M cycle synchronously and atomically.",
"In $\\mathcal {SSYNC}$ (semi- synchronous), robots are asynchronously activated to perform cycles, yet at each activation, a robot executes one cycle atomically.",
"In $\\mathcal {ASYNC}$ (asynchronous), robots execute L-C-M in a fully independent manner.",
"We assume robots having weak capabilities: they are uniform —meaning that all robots follow the same algorithm—, they are anonymous —meaning that no robot can distinguish any two other robots—, they are disoriented —they have no coherent labeling of direction—, and they have no global knowledge related to the size of the environment.",
"Furthermore, the robots have no (direct) means of communicating with each other.",
"However, each of them is endowed with persistent memory and is able to detect whether it stands alone at its current location.",
"All the aforementioned contributions assume a static environment, i.e., the graph topology explored by the robots does not evolve in function of the time.",
"In this paper, we consider dynamic environments that may change over time, for instance, a transportation network, a building in which doors are closed and open over time, or streets that are closed over time due to work in process or traffic jam in a town.",
"More precisely, we consider dynamic graphs in which edges may appear and disappear unpredictably without any stability, recurrence, nor periodicity assumption.",
"However, to ensure that the problem is not trivially unsolvable, we made the assumption that each node is infinitely often reachable from any other one through a temporal path (a.k.a. journey [6]).",
"The dynamic graphs satisfying this topological property are known as connected-over-time (dynamic) graphs [6].",
"Related work.",
"Recent work [16], [19], [20], [18], [10] deal with the terminating exploration of dynamic graphs.",
"This line of work restricts the dynamicity of the graph with various assumptions.",
"In [16] and [19], the authors focus on periodically varying graphs, i.e., the presence of each edge of the graph is periodic.",
"In [20], [18], [10], the authors assume that the graph is connected at each time instant and that there exists a stability of this connectivity in any interval of time of length $T$ (such assumption is known as $T$ -interval-connectivity [22]).",
"In [20] and [10] (resp.",
"[18]), the authors restrict their study to the case where the underlying graph (i.e., the static graph that includes all edges that are present at least once in the lifetime of the graph) forms a ring (resp.",
"a cactus) of arbitrary size.",
"In [10], the authors examine the impact of various factors (e.g., at least one node is not anonymous, knowledge of the exact number of nodes, knowledge of an upper bound on the number of nodes, sharing of a common orientation, etc.)",
"on the solvability of the terminating exploration.",
"In particular, they show that the degree of synchrony among the robot has a major impact.",
"Indeed, they prove that, independently of other assumptions, exploration is impossible in $\\mathcal {SSYNC}$ model (without extra synchronization assumptions).",
"The proof of this result relies on the possibility offered to the adversary to wake up each robot independently and to remove the edge that the robot wants to traverse at this time.",
"Note that, by its simplicity, this impossibility result is applicable to any variante of the exploration problem.",
"It is also independent of dynamicity assumptions.",
"The first attempt to solve exploration in the most general dynamicity scenario (i.e., connected-over-time assumption) has been proposed in [4].",
"The authors provide a protocol that deterministically solves the perpetual exploration problem.",
"This protocol operates in any connected-over-time ring with three synchronous robots (accordingly to the aforementioned impossibility result in [10]).",
"Further, the proposed protocol has the nice extra property of being self-stabilizing, meaning that regardless their arbitrary initial configuration, the robots eventually behave according to their specification, i.e., eventually, they explore the whole network infinitely often.",
"Note that the necessity of the assumption on the number of robots is left as an open question by this work.",
"Our contribution.",
"The main contribution of this paper is to close this question.",
"Indeed, we analyze the computability of the perpetual exploration problem in connected-over-time (dynamic) rings, i.e., we study the deterministic solvability of the problem with respect to the number of robots.",
"According to the impossibility result in [10], we restrict this study to the $\\mathcal {FSYNC}$ model.",
"As we do not consider self-stabilization (contrarily to [4]), we assume that no pair of robots have a common initial location.",
"Moreover, to ensure that the problem is not trivially solved in the initial configuration, we consider that, $k$ , the number of robots, is strictly smaller than $n$ , the number of nodes of the dynamic graph.",
"In this context, we establish the necessary and sufficient number of robots to solve the perpetual exploration for any size of connected-over-time rings (see TABLE REF for a summary).",
"Note that a connected-over-time chain can be seen as a connected-over-time ring with a missing edge.",
"So, our results are also valid on connected-over-time chains.",
"Table: Overview of the resultsIn more details, we first provide an algorithm that perpetually explores, using a team of $k\\ge 3$ robots, any connected-over-time ring of $n > k$ nodes.",
"Then, we give two non-trivial impossibility results.",
"We first show that two robots are not sufficient to perpetually explore a connected-over-time ring with a number of nodes strictly greater than three.",
"Next, we show that a single robot cannot perpetually explore a connected-over-time ring with a number of nodes strictly greater than two.",
"Finally, we close the problem by providing an algorithm for each remaining cases (one robot in a 2-node connected-over-time ring and two robots in a 3-node connected-over-time ring).",
"Outline of the paper.",
"In Section , we present formally the model considered in the remainder of the paper.",
"Section presents the algorithm to explore connected-over-time rings of size $n>k$ nodes with $k\\ge 3$ robots.",
"The impossibility result and the algorithm for two robots are both presented in Section .",
"The ones assuming a single robot are given in Section .",
"We conclude in Section ."
],
[
"Model",
"In this section, we present our formal model.",
"This model is borrowed from the one of [4] that proposes an extension of the classical model of robot networks in static graphs introduced in [21] to the context of dynamic graphs."
],
[
"Dynamic graphs",
"In this paper, we consider the model of evolving graphs introduced in [23].",
"We hence consider the time as discretized and mapped to $\\mathbb {N}$ .",
"An evolving graph $\\mathcal {G}$ is an ordered sequence $\\lbrace G_{0}, G_{1}, \\ldots \\rbrace $ of subgraphs of a given static graph $G=(V,E)$ such that, for any $i\\ge 0$ , we have $G_{i} = (V, E_{i})$ .",
"We say that the edges of $E_{i}$ are present in $\\mathcal {G}$ at time $i$ .",
"The underlying graph of $\\mathcal {G}$ , denoted $U_\\mathcal {G}$ , is the static graph gathering all edges that are present at least once in $\\mathcal {G}$ (i.e., $U_\\mathcal {G}=(V,E_\\mathcal {G})$ with $E_\\mathcal {G}=\\bigcup _{i=0}^{\\infty }E_i)$ ).",
"An eventual missing edge is an edge of $E_\\mathcal {G}$ such that there exists a time after which this edge is never present in $\\mathcal {G}$ .",
"A recurrent edge is an edge of $E_\\mathcal {G}$ that is not eventually missing.",
"The eventual underlying graph of $\\mathcal {G}$ , denoted $U_\\mathcal {G}^\\omega $ , is the static graph gathering all recurrent edges of $\\mathcal {G}$ (i.e., $U_\\mathcal {G}^\\omega =(V,E_\\mathcal {G}^\\omega )$ where $E_\\mathcal {G}^\\omega $ is the set of recurrent edges of $\\mathcal {G}$ ).",
"In this paper, we chose to make minimal assumptions on the dynamicity of our graph since we restrict ourselves on connected-over-time evolving graphs.",
"The only constraint we impose on evolving graphs of this class is that their eventual underlying graph is connected [12] (this is equivalent with the assumption that each node is infinitely often reachable from another one through a journey).",
"In the following, we consider only connected-over-time evolving graphs whose underlying graph is an anonymous and unoriented ring of arbitrary size.",
"Although the ring is unoriented, to simplify the presentation, we, as external observers, distinguish between the clockwise and the counter-clockwise (global) direction in the ring.",
"We introduce here some definitions that are used for proofs only.",
"From an evolving graph $\\mathcal {G}=\\lbrace (V, E_0), (V, E_1),(V, E_2),$ $\\ldots \\rbrace $ , we define the evolving graph $\\mathcal {G} \\backslash \\lbrace (e_{1}, \\tau _{1}), \\ldots (e_{k}, \\tau _{k})\\rbrace $ (with for any $i \\in \\lbrace 1, \\ldots , k\\rbrace $ , $e_{i} \\in E$ and $\\tau _{i} \\subseteq \\mathbb {N}$ ) as the evolving graph $\\lbrace (V, E_0^{\\prime }), (V, E_1^{\\prime }), (V, E_2^{\\prime }),\\ldots \\rbrace $ such that: $\\forall t\\in \\mathbb {N},\\forall e\\in E_\\mathcal {G},e\\in E_t^{\\prime } \\Leftrightarrow e\\in E_t\\wedge (\\forall i\\in \\lbrace 1,\\ldots ,k\\rbrace , e\\ne e_i \\vee t\\notin \\tau _i)$ .",
"A node $u$ satisfies the property $OneEdge(u, t, t^{\\prime })$ if and only if an adjacent edge of $u$ is continuously missing from time $t$ to time $t^{\\prime }$ while the other adjacent edge of $u$ is continuously present from time $t$ to time $t^{\\prime }$ .",
"We define the distance between two nodes $u$ and $v$ (denoted $d(u,v)$ ) by the length of a shortest path between $u$ and $v$ in the underlying graph."
],
[
"Robots",
"We consider systems of autonomous mobile entities called robots moving in a discrete and dynamic environment modeled by an evolving graph $\\mathcal {G}=\\lbrace (V,E_0),(V,E_1)\\ldots \\rbrace $ , $V$ being a set of nodes representing the set of locations where robots may be, $E_i$ being the set of bidirectional edges representing connections through which robots may move from a location to another one at time $i$ .",
"Robots are uniform (they execute the same algorithm), anonymous (they are indistinguishable from each other), and have a persistent memory (they can store local variables).",
"The state of a robot at time $t$ corresponds to the value of its variables at time $t$ .",
"Robots are unable to directly communicate with each other by any means.",
"Robots are endowed with local weak multiplicity detection meaning that they are able to detect if they are alone on their current node or not, but they cannot know the exact number of co-located robots.",
"When a robot is alone on its current node, we say that it is isolated.",
"A tower $T$ is a couple $(S, \\theta )$ , where $S$ is a set of robots ($|S| > 1$ ) and $\\theta =[t_{s}, t_{e}]$ is an interval of $\\mathbb {N}$ , such that all the robots of $S$ are located at a same node at each instant of time $t$ in $\\theta $ and $S$ or $\\theta $ is maximal for this property.",
"We say that the robots of $S$ form the tower at time $t_{s}$ and that they are involved in the tower between time $t_{s}$ and $t_{e}$ .",
"Robots have no a priori knowledge about the ring they explore (size, diameter, dynamicity...).",
"Finally, each robot has its own stable chirality (i.e., each robot is able to locally label the two ports of its current node with left and right consistently over the ring and time but two different robots may not agree on this labeling).",
"We assume that each robot has a variable $dir$ that stores a direction (either left or right).",
"Initially, this variable is set to $left$ .",
"At any time, we say that a robot points to left (resp.",
"right) if its $dir$ variable is equal to this (local) direction.",
"Through misuse of language, we say that a robot points to an edge when this edge is connected to the current node of the robot by the port labeled with its current direction.",
"We say that a robot considers the clockwise (resp.",
"counter-clockwise) direction if the (local) direction pointed to by this robot corresponds to the (global) direction seen by an external observer."
],
[
"Execution",
"A configuration $\\gamma $ of the system captures the position (i.e., the node where the robot is currently located) and the state of each robot at a given time.",
"Given an evolving graph $\\mathcal {G}=\\lbrace G_{0}, G_{1}, \\ldots \\rbrace $ , an algorithm $\\mathcal {A}$ , and an initial configuration $\\gamma _0$ , the execution $\\mathcal {E}$ of $\\mathcal {A}$ on $\\mathcal {G}$ starting from $\\gamma _0$ is the infinite sequence $(G_0,\\gamma _0),(G_1,\\gamma _1),(G_2,\\gamma _2),\\ldots $ where, for any $i\\ge 0$ , the configuration $\\gamma _{i+1}$ is the result of the execution of a synchronous round by all robots from $(G_i,\\gamma _i)$ as explained below.",
"The round that transitions the system from $(G_i,\\gamma _i)$ to $(G_{i+1},\\gamma _{i+1})$ is composed of three atomic and synchronous phases: Look, Compute, Move.",
"During the Look phase, each robot gathers information about its environment in $G_i$ .",
"More precisely, each robot updates the value of the following local predicates: $(i)$ $ExistsEdge(dir)$ returns true if there is an adjacent edge at the current location of the robot on its direction $dir$ , false otherwise; $(ii)$ $ExistsOtherRobotsOnCurrentNode()$ returns true if there is strictly more than one robot on the current node of the robot, false otherwise.",
"We define the local environment of a robot at a given time as the combination of the values of $ExistsEdge(dir)$ , $ExistsEdge(\\overline{dir})$ (where $\\overline{dir}$ is the opposite direction to $dir$ ), and $ExistsOtherRobotsOnCurrentNode()$ of this robot at this time.",
"The view of the robot at this time gathers its state and its local environment at this time.",
"During the Compute phase, each robot executes the algorithm $\\mathcal {A}$ that may modify its variable $dir$ depending on its current state and on the values of the predicates updated during the Look phase.",
"Finally, the Move phase consists of moving each robot through one edge in the direction it points to if there exists an edge in that direction, otherwise (i.e., the edge is missing at that time) the robot remains at its current node."
],
[
"Specification",
"We define a well-initiated execution as an execution $(G_0,\\gamma _0),(G_1,\\gamma _1),(G_2,\\gamma _2),\\ldots $ such that $\\gamma _0$ contains strictly less robots than the number of nodes of $\\mathcal {G}$ and is towerless (i.e., there is no tower in this configuration).",
"Given a class of evolving graphs $\\mathcal {C}$ , an algorithm $\\mathcal {A}$ satisfies the perpetual exploration specification on $\\mathcal {C}$ if and only if, in every well-initiated execution of $\\mathcal {A}$ on every evolving graph $\\mathcal {G}\\in \\mathcal {C}$ , every node of $\\mathcal {G}$ is infinitely often visited by at least one robot (i.e., a robot is infinitely often located at every node of $\\mathcal {G}$ ).",
"Note that this specification does not require that every robot visits infinitely often every node of $\\mathcal {G}$ ."
],
[
"With Three or More Robots",
"This section is dedicated to the more general result: the perpetual exploration exploration of connected-over-time rings of size greater than $k$ with a team of $k \\ge 3$ robots."
],
[
"Presentation of the Algorithm",
"We first describe intuitively the key ideas of our algorithm.",
"Remind that an algorithm controls the move of the robots through their variable direction.",
"Hence, designing an algorithm consists in choosing when we want a robot to keep its direction and when we want it to change its direction (in other words, turn back).",
"The first idea of our algorithm is to require that a robot keeps its direction when it is not involved in a tower (Rule 1).",
"Using this idea, some towers are necessarily formed when there exists an eventual missing edge.",
"Our algorithm reacts as follows to the formation of towers.",
"If at a time $t$ a robot does not move and forms a tower at time $t + 1$ , then the algorithm keeps the direction of the robot (Rule 2).",
"In the contrary case (that is, at time $t$ , the robot moves and forms a tower at time t+1) it changes the direction of the robot (Rule 3).",
"Let us now explain how the algorithm (Rules 1, 2, and 3) enables the perpetual exploration of any connected-over-time ring.",
"First, note that Rule 1 alone is sufficient to perpetually explore connected-over-time rings without eventual missing edge provided that the robots never meet.",
"The main property induced by Rules 2 and 3 is that any tower is broken in a finite time and that at least one robot of the tower considers each possible direction.",
"This property implies (combined with Rule 1) that $(i)$ the algorithm is able to perpetually explore any connected-over-time ring without eventual missing edge (even if robots meet); and that $(ii)$ , when the ring contains an eventual missing edge, one robot is eventually located at each extremity of the eventual missing edge and considers afterwards the direction of the eventual missing edge.",
"Let us consider this last case.",
"We call sentinels the two robots located at extremities of the eventual missing edge.",
"The other robots are called explorers.",
"By Rule 3, an explorer that arrives on a node where a sentinel is located changes its direction.",
"Intuitively, that means that the sentinel signal to the explorer that it has reached one extremity of the eventual missing edge and that it has consequently to turn back to continue the exploration.",
"Note that, by Rule 2, the sentinel keeps its direction (and hence its role).",
"Once an explorer leaves an extremity of the eventual missing edge, we know, thanks to Rule 1 and the main property induced by Rules 2 and 3, that a robot reaches in a finite time the other extremity of the eventual missing edge and that (after the second sentinel/explorer meeting) all the nodes have been visited by a robot in the meantime.",
"As we can repeat this scheme infinitely often, our algorithm is able to perpetually explore any connected-over-time ring with an eventual missing edge, that ends the informal presentation of our algorithm.",
"Refer to Algorithm REF for the formal statement of our algorithm called $\\mathbb {PEF}\\_3+$ (standing for $\\mathbb {P}$ erpetual $\\mathbb {E}$ xploration in $\\mathbb {F}$ SYNC with 3 or more robots).",
"In addition of its $dir$ variable, each robot maintains a boolean variable $HasMovedPreviousStep$ indicating if the robot has moved during its last Look-Compute-Move cycle.",
"This variable is used to implement Rules 2 and 3."
],
[
"Proof of Correctness",
"In this section, we prove the correctness of $\\mathbb {PEF}\\_3+$ with $k\\ge 3$ robots.",
"In the following, we consider a connected-over-time ring $\\mathcal {G}$ of size at least $k+1$ .",
"Let $\\varepsilon = (G_0,\\gamma _0),(G_1,\\gamma _1),\\ldots $ be any execution of $\\mathbb {PEF}\\_3+$ on $\\mathcal {G}$ .",
"Lemma 3.1 If there exists an eventual missing edge in $\\mathcal {G}$ , then at least one tower is formed in $\\varepsilon $ .",
"By contradiction, assume that $e$ is an eventual missing edge of $\\mathcal {G}$ (such that $e$ is not present in $\\mathcal {G}$ after time $t$ ) and that no tower is formed in $\\varepsilon $ .",
"Executing $\\mathbb {PEF}\\_3+$ , a robot changes the global direction it considers only when it forms a tower with another robot.",
"As, by assumption, no tower is formed in $\\varepsilon $ , each robot is always considering the same global direction.",
"All the edges of $\\mathcal {G}$ , except $e$ , are infinitely often present in $\\mathcal {G}$ .",
"Hence, any robot reaches one of the extremity of $e$ in finite time after $t$ .",
"As the robots consider a direction at each instant time and that there are at least 3 robots, at least 2 robots consider the same global direction at each instant time.",
"Hence, at least two robots reach the same extremity of $e$ .",
"A tower is formed, leading to a contradiction.",
"Lemma 3.2 If $\\varepsilon $ does not contain a tower, then every node is infinitely often visited by a robot in $\\varepsilon $ .",
"Assume that there is no tower formed in $\\varepsilon $ .",
"By Lemma REF , if there is an eventual missing edge in $\\mathcal {G}$ , then there is at least one tower formed.",
"In consequence, all the edges of $\\mathcal {G}$ are infinitely often present in $\\mathcal {G}$ .",
"Executing $\\mathbb {PEF}\\_3+$ , a robot changes the global direction it considers only when it forms a tower with another robot.",
"Hence, none of the robots change the global direction it considers in $\\varepsilon $ .",
"Since all the edges are infinitely often present, each robot moves infinitely often in the same global direction, that implies the result.",
"Lemma 3.3 If a tower $T$ of 2 robots is formed in $\\varepsilon $ , then these two robots consider two opposite global directions while $T$ exists.",
"Assume that 2 robots form a tower at a time $t$ in $\\varepsilon $ .",
"Let us consider the 2 following cases: Case 1: The two robots consider the same global direction during the Move phase of time $t - 1$ .",
"In this case, one robot (denoted $r$ ) does not move during the Move phase of time $t$ , while the other (denoted $r^{\\prime }$ ) moves and joins the first one on its current node.",
"During the Compute phase of time $t$ , $r$ still considers the same global direction, while $r^{\\prime }$ changes the global direction it considers by construction of $\\mathbb {PEF}\\_3+$ .",
"Then, the two robots consider two different global directions after the Compute phase of time $t$ .",
"Case 2: The two robots consider two opposite global directions during the Move phase of time $t - 1$ .",
"In this case, the two robots move at time $t - 1$ .",
"During the Compute phase of time $t$ , the two robots change the global direction they consider by construction of $\\mathbb {PEF}\\_3+$ .",
"Hence they consider two different global directions after the Compute phase of time $t$ .",
"A robot executing $\\mathbb {PEF}\\_3+$ changes its global direction only if it has moved during the previous step.",
"So, the robots of the tower do not change the global direction they consider as long as they are involved in the tower.",
"As the two robots consider two different global directions after the Compute phase of time $t$ , we obtain the lemma.",
"Lemma 3.4 No tower of $\\varepsilon $ involves 3 robots or more.",
"We prove this lemma by recurrence.",
"As there is no tower in $\\gamma _{0}$ by assumption, it remains to prove that, if $\\gamma _{t}$ contains no tower with 3 or more robots, so is $\\gamma _{t + 1}$ .",
"Let us study the following cases: Case 1: $\\gamma _{t}$ contains no tower.",
"The robots can cross at most one edge at each step.",
"Each node has at most 2 adjacent edges in $G_{t}$ , hence the maximum number of robots involved in a tower of $\\gamma _{t + 1}$ is 3.",
"If a tower involving 3 robots is formed in $\\gamma _{t + 1}$ , one robot $r$ has not moved during the Move phase of time $t$ , while the two other robots (located on the two adjacent nodes of its location) have moved to its position.",
"That implies that the two adjacent edges of the node where $r$ is located are present in $G_{t}$ .",
"As any robot considers a global direction at each instant time, $r$ necessarily moves in step $t$ , that is contradictory.",
"Therefore, only towers of 2 robots can be formed in $\\gamma _{t + 1}$ .",
"Case 2: $\\gamma _{t}$ contains towers of at most 2 robots.",
"Let $T$ be a tower involving 2 robots in $\\gamma _{t}$ and $u$ be the node where $T$ is located in $\\gamma _{t}$ .",
"By Lemma REF , the 2 robots of $T$ consider two opposite global directions in $\\gamma _{t}$ .",
"Consider the 3 following sub-cases: $(i)$ If there is no adjacent edge to $u$ in $G_{t}$ , then no other robot can increase the number of the robots involved in the tower.",
"$(ii)$ If there is only one adjacent edge to $u$ in $G_{t}$ , then only one robot may traverse this edge to increase the number of robots involved in $T$ .",
"Indeed, if there are multiple robots on an adjacent node to $u$ , then these robots are involved in a tower $T^{\\prime }$ of 2 robots (by assumption on $\\gamma _{t}$ ) and they are considering two opposite global directions in $\\gamma _{t}$ .",
"However, as an adjacent edge to $u$ is present in $G_{t}$ and as the robots of $T$ are considering two opposite global directions, then one robot of $T$ leaves $T$ at time $t$ .",
"In other words, even if a robot of $T^{\\prime }$ moves on $u$ , one robot of $T$ leaves $u$ .",
"Then, there is at most 2 robots on $u$ in $\\gamma _{t + 1}$ .",
"$(iii)$ If there are two adjacent edges to $u$ in $\\gamma _{t}$ , then, using similar arguments as above, we can prove that only one robot crosses each of the adjacent edges of $u$ .",
"Moreover, the robots of $T$ move in opposite global directions and leave $u$ , implying that at most 2 robots are present on $u$ in $\\gamma _{t + 1}$ .",
"Lemma 3.5 If $\\mathcal {G}$ has no eventual missing edge and $\\varepsilon $ contains towers then every node is infinitely often visited by a robot in $\\varepsilon $ .",
"Assume that $\\mathcal {G}$ has no eventual missing edge and $\\varepsilon $ contains towers.",
"We want to prove the following property.",
"If during the Look phase of time $t$ , a robot $r$ is located on a node $u$ considering the global direction $gd$ , then there exists a time $t^{\\prime } \\ge t$ such that, during the Look phase of time $t^{\\prime }$ , a robot is located on the node $v$ adjacent to $u$ in the global direction $gd$ and considers the global direction $gd$ .",
"Let $t\" \\ge t$ be the smallest time after time $t$ where the adjacent edge of $u$ in the global direction $gd$ is present in $\\mathcal {G}$ .",
"As all the edges of $\\mathcal {G}$ are infinitely often present, $t\"$ exists.",
"$(i)$ If $r$ crosses the adjacent edge of $u$ in the global direction $gd$ during the Move phase of time $t\"$ , then the property is verified.",
"$(ii)$ If $r$ does not cross the adjacent edge of $u$ in the global direction $gd$ , this implies that $r$ changes the global direction it considers during the Look phase of time $t$ .",
"While executing $\\mathbb {PEF}\\_3+$ , a robot changes its global direction when it forms a tower with another robot.",
"Therefore, at time $t$ , $r$ forms a tower with a robot $r^{\\prime }$ .",
"By Lemmas REF and REF , two robots involved in a tower consider two opposite global directions.",
"Hence, after the Compute phase of time $t$ , $r^{\\prime }$ considers the global direction $gd$ .",
"A robot executing $\\mathbb {PEF}\\_3+$ does not change the global direction it considers until it moves.",
"So, $r^{\\prime }$ considers the global direction $gd$ during the Move phase of time $t\"$ .",
"Hence, during the Look phase of time $t\" + 1$ , $r^{\\prime }$ is on node $v$ and considers the global direction $gd$ .",
"By applying recurrently this property to any robot, we prove that all the nodes are infinitely often visited.",
"Lemma 3.6 If $\\mathcal {G}$ has an eventual missing edge $e$ (such that $e$ is missing forever after time $t$ ) and, during the Look phase of a time $t^{\\prime } \\ge t$ , a robot considers a global direction $gd$ and is located on a node at a distance $d \\ne 0$ in $U_\\mathcal {G}^\\omega $ from the extremity of $e$ in the global direction $gd$ , then it exists a time $t\" \\ge t^{\\prime }$ such that, during the Look phase of time $t\"$ , a robot is on a node at distance $d - 1$ in $U_\\mathcal {G}^\\omega $ from the extremity of $e$ in the global direction $gd$ and considers the global direction $gd$ .",
"Assume that $\\mathcal {G}$ has an eventual missing edge $e$ (such that $e$ is missing forever after time $t$ ) and that, during the Look phase of time $t^{\\prime } \\ge t$ , a robot $r$ considers a global direction $gd$ and is located on a node $u$ at distance $d \\ne 0$ in $U_\\mathcal {G}^\\omega $ from the extremity of $e$ in the global direction $gd$ .",
"Let $v$ be the adjacent node of $u$ in the global direction $gd$ .",
"Let $t\" \\ge t^{\\prime }$ be the smallest time after time $t^{\\prime }$ where the adjacent edge of $u$ in the global direction $gd$ is present in $\\mathcal {G}$ .",
"As all the edges of $\\mathcal {G}$ except $e$ are infinitely often present and as $u$ is at a distance $d \\ne 0$ in $U_\\mathcal {G}^\\omega $ from the extremity of $e$ in the global direction $gd$ , then the adjacent edge of $u$ in the global direction $gd$ is infinitely often present in $\\mathcal {G}$ .",
"Hence, $t\"$ exists.",
"$(i)$ If $r$ crosses the adjacent edge of $u$ in the global direction $gd$ during the Move phase of time $t\"$ , then the property is verified.",
"$(ii)$ If $r$ does not cross the adjacent edge of $u$ in the global direction $gd$ , this implies that $r$ changes the global direction it considers during the Look phase of time $t$ .",
"While executing $\\mathbb {PEF}\\_3+$ a robot changes the global direction it considers when it forms a tower with another robot.",
"Therefore, at time $t$ , $r$ forms a tower with a robot $r^{\\prime }$ .",
"By Lemmas REF and REF , two robots involved in a tower consider two opposite global directions.",
"Hence, after the Compute phase of time $t$ , $r^{\\prime }$ considers the global direction $gd$ .",
"A robot executing $\\mathbb {PEF}\\_3+$ does not change the global direction it considers until it moves.",
"Therefore, $r^{\\prime }$ considers the global direction $gd$ during the Move phase of time $t\"$ .",
"Hence, during the Look phase of time $t\" + 1$ , $r^{\\prime }$ is on node $v$ and considers the global direction $gd$ .",
"Lemma 3.7 If $\\mathcal {G}$ has an eventual missing edge $e$ , then eventually one robot is forever located on each extremity of $e$ pointing to $e$ .",
"Assume that $\\mathcal {G}$ has an eventual missing edge $e$ such that $e$ is missing forever after time $t$ .",
"First, we want to prove that a robot reaches one of the extremities of $e$ in a finite time after $t$ and points to $e$ at this time.",
"If it is not the case at time $t$ , then there exists at this time a robot considering a global direction $gd$ and located on a node $u$ at distance $d \\ne 0$ in $U_\\mathcal {G}^\\omega $ from the extremity of $e$ in the global direction $gd$ .",
"By applying $d$ times Lemma REF , we prove that, during the Look phase of a time $t^{\\prime }\\ge t$ , a robot (denote it $r$ ) reaches the extremity of $e$ in the global direction $gd$ from $u$ (denote it $v$ and let $v^{\\prime }$ be the other extremity of $e$ ), and that this robot considers the global direction $gd$ .",
"Let us consider the following cases: Case 1: $r$ is isolated on $v$ at time $t^{\\prime }$ .",
"In this case, by construction of $\\mathbb {PEF}\\_3+$ , $r$ does not change, during the Compute phase of time $t^{\\prime }$ , the global direction that it considers during the Move phase of time $t^{\\prime } - 1$ .",
"Moreover, a robot can change the global direction it considers only if it moves during the previous step.",
"All the edges of $\\mathcal {G}$ except $e$ are infinitely often present.",
"As, at time $t^{\\prime }$ , $r$ points to $e$ , it cannot move.",
"Therefore, from time $t^{\\prime }$ , $r$ does not move and does not change the global direction it considers.",
"Then, $r$ remains located on $v$ forever after $t^{\\prime }$ considering $gd$ .",
"Case 2: $r$ is not isolated on $v$ at time $t^{\\prime }$ .",
"By Lemmas REF , $r$ forms a tower with only one another robot $r^{\\prime }$ .",
"By Lemmas REF and REF , two robots that form a tower consider two opposite global directions.",
"Hence, either $r$ or $r^{\\prime }$ considers the global direction $gd$ while the other one consider the global direction $\\overline{gd}$ .",
"As all the edges of $\\mathcal {G}$ except $e$ are infinitely often present, then in finite time either $r$ or $r^{\\prime }$ leaves $v$ .",
"We can now apply the same arguments than in Case 1 to the robot that stays on $v$ to prove that this robot remains located on $v$ forever after $t^{\\prime }$ considering $gd$ .",
"In both cases, a robot remains forever on $v$ considering $gd$ after $t^{\\prime }$ .",
"Assume without loss of generality that it is $r$ .",
"Let us consider the two following cases: Case A: It exists $r^{\\prime }\\ne r$ considering $\\overline{gd}$ at time $t^{\\prime }$ .",
"We can apply recurrently Lemma REF , and the arguments above to prove that a robot is eventually forever located on $v^{\\prime }$ considering $\\overline{gd}$ .",
"Case B: All robots $r^{\\prime }\\ne r$ considers $gd$ at time $t$ .",
"We can apply recurrently Lemma REF to prove that, in finite time, a robot forms a tower with $r$ on $v$ .",
"Then, by construction of $\\mathbb {PEF}\\_3+$ , this robot consider $\\overline{gd}$ after the Compute phase of this time (and hence during the Look phase of the next time).",
"We then come back to Case A.",
"In both cases, the lemma holds.",
"Lemma 3.8 If $\\mathcal {G}$ has an eventual missing edge and $\\varepsilon $ contains towers, then every node is infinitely often visited.",
"Assume that $\\mathcal {G}$ has an eventual missing edge $e$ that is missing forever after time $t$ .",
"By Lemma REF , there exists a time $t^{\\prime } \\ge t$ after which two robots $r_{1}$ and $r_{2}$ are respectively located on the two extremities of $e$ and pointing to $e$ .",
"As there are at least 3 robots, let $r$ be a robot (located on a node $u$ considering a global direction $gd$ ) such that $r \\ne r_{1}$ and $r \\ne r_{2}$ .",
"Let $v$ be the extremity of $e$ in the direction $gd$ of $u$ and $v^{\\prime }$ be the other extremity of $e$ .",
"Applying recurrently Lemma REF , we prove that, in finite time, all the nodes between node $u$ and $v$ in the global direction $gd$ are visited and that a robot reaches $v$ .",
"When this robot reaches $v$ , it changes its direction (hence considers $\\overline{gd}$ ) by construction of $\\mathbb {PEF}\\_3+$ since it moves during the previous step and forms a tower.",
"We can then repeat this reasoning (with $v$ and $v^{\\prime }$ alternatively in the role of $u$ and with $v^{\\prime }$ and $v$ alternatively in the role of $v$ ) and prove that all nodes are infinitely often visited.",
"Lemmas REF , REF , and REF directly imply the following result: Theorem 3.1 $\\mathbb {PEF}\\_3+$ is a perpetual exploration algorithm for the class of connected-over-time rings of arbitrary size strictly greater than the number of robots using an arbitrary number (greater than or equal to 3) of fully synchronous robots."
],
[
"With Two Robots",
"In this section, we study the perpetual exploration of rings of any size with two robots.",
"We first prove that two robots are not able to perpetually explore connected-over-time rings of size strictly greater than three (refer to Theorem REF ).",
"Then, we provide $\\mathbb {PEF}\\_2$ (see Theorem REF ), an algorithm using two robots that solves the perpetual exploration on the remaining case, i.e., connected-over-time rings of size three."
],
[
"Connected-over-Time Rings of Size 4 or More",
"The proof of our impossibility result presented in Theorem REF makes use of a generic framework proposed in [5].",
"Note that, even if this generic framework is designed for another model (namely, the classical message passing model), it is straightforward to borrow it for our current model.",
"Indeed, its proof only relies on the determinism of algorithms and indistinguishability of dynamic graphs, these arguments being directly translatable in our model.",
"We present briefly this framework here.",
"The interested reader is referred to [5] for more details.",
"This framework is based on a theorem that ensures that, if we take a sequence of evolving graphs with ever-growing common prefixes (that hence converges to the evolving graph that shares all these common prefixes), then the sequence of corresponding executions of any deterministic algorithm also converges.",
"Moreover, we are able to describe the execution to which it converges as the execution of this algorithm on the evolving graph to which the sequence converges.",
"This result is useful since it allows us to construct counter-example in the context of impossibility results.",
"Indeed, it is sufficient to construct an evolving graphs sequence (with ever-growing common prefixes) and to prove that their corresponding execution violates the specification of the problem for ever-growing time to exhibit an execution that never satisfies the specification of the problem.",
"In order to build the evolving graphs sequence suitable for the proof of our impossibility result, we need the following technical lemma.",
"Lemma 4.1 Let $\\mathcal {A}$ be a perpetual exploration algorithm in connected-over-time ring of size 4 or more using 2 robots.",
"Any execution of $\\mathcal {A}$ satisfies: For any time $t$ and any robot state $s$ , if, at time $t$ , the robots have not explored the whole ring, have not formed a tower, and each robot has only visited at most two adjacent nodes, then there exists $t^{\\prime } \\ge t$ such that a robot located on a node $u$ , on state $s$ at time $t$ , and satisfying $OneEdge(u, t, t^{\\prime })$ leaves $u$ at time $t^{\\prime }$ .",
"Consider an algorithm $\\mathcal {A}$ that deterministically solves the perpetual exploration problem for connected-over-time rings of size 4 or more using two robots.",
"Let $\\mathcal {G} = \\lbrace G_{0}=(V,E_0), G_{1}=(V,E_1), \\ldots \\rbrace $ be a connected-over-time ring (of size 4 or more).",
"Let $\\varepsilon $ be an execution of $\\mathcal {A}$ by two robots $r_{1}$ and $r_{2}$ on $\\mathcal {G}$ .",
"By contradiction, assume that there exists a time $t$ and a state $s$ such that $(i)$ the exploration of the whole ring has not been done yet; $(ii)$ from time 0 to time $t$ none of the robots have formed a tower; $(iii)$ at time $t$ each robot has only visited at most two adjacent nodes of $\\mathcal {G}$ ; and $(iv)$ at time $t$ one of the robot (without lost of generality, $r_{1}$ ) is in a state $s$ such that, for any $ t^{\\prime } \\ge t$ , if $r_{1}$ is on a node $u$ of $\\mathcal {G}$ satisfying $OneEdge(u, t, t^{\\prime })$ , then it does not leave $u$ at time $t^{\\prime }$ .",
"Let $\\mathcal {R}$ be the set of nodes visited by $r_{1}$ from time 0 to time $t$ .",
"Note that, at time $t$ , as each robot has only visited at most two adjacent nodes, then $1 \\le ǀ\\mathcal {R} ǀ\\le 2$ .",
"Let $i$ (resp.",
"$f$ ) be the node in $\\mathcal {G}$ where $r_{1}$ is located at time 0 (resp.",
"$t$ ).",
"If $ǀ\\mathcal {R} ǀ= 2$ , let $a$ be the node of $\\mathcal {R}$ such that $a \\ne i$ , otherwise (i.e., $ǀ\\mathcal {R} ǀ= 1$ ) let $a = i$ .",
"By assumption, either $f = i$ or $f$ is an adjacent node of $i$ and in this later case $a = f$ .",
"We construct a connected-over-time ring $\\mathcal {G}^{\\prime } = \\lbrace G_{0}^{\\prime }, G_{1}^{\\prime }, \\ldots \\rbrace $ (with $G^{\\prime }_i=(V^{\\prime },E^{\\prime }_i)$ for any $i\\in \\mathbb {N}$ ) such that the underlying graph of $\\mathcal {G}^{\\prime }$ contains 8 nodes in the following way.",
"Let $i_{1}^{\\prime }$ be an arbitrary node of $\\mathcal {G}^{\\prime }$ .",
"Let us construct nodes $i_{2}^{\\prime }$ , $a_{1}^{\\prime }$ , $a_{2}^{\\prime }$ , $f_{1}^{\\prime }$ , and $f_{2}^{\\prime }$ of $\\mathcal {G}^{\\prime }$ in function of $i_{1}^{\\prime }$ and of nodes $i$ , $a$ , and $f$ of $\\mathcal {G}$ as explained by Figure REF .",
"Note that this construction ensures that $f_{1}^{\\prime }$ and $f_{2}^{\\prime }$ are adjacent in $\\mathcal {G}^{\\prime }$ in any case.",
"We denote by $r(k)$ (resp.",
"$l(k)$ ) the adjacent edge in the clockwise (resp.",
"counter clockwise) direction of a node $k$ .",
"For any $j \\in \\lbrace 0,\\ldots , t-1\\rbrace $ , let $E^{\\prime }_j$ be the set $E_{\\mathcal {G}^{\\prime }}$ with the following set of additional constraintsNote that the construction of $i_{1}^{\\prime }$ , $i_{2}^{\\prime }$ , $a_{1}^{\\prime }$ , $a_{2}^{\\prime }$ , $f_{1}^{\\prime }$ , and $f_{2}^{\\prime }$ ensures us that there is no contradiction between these constraints in all cases.",
": $\\left\\lbrace \\begin{array}{ll}r(i_{1}^{\\prime }) \\in E_{j}^{\\prime } \\text{ and } l(i_{2}^{\\prime }) \\in E_{j}^{\\prime } & \\text{iff } r(i) \\in E_{j}\\\\l(i_{1}^{\\prime }) \\in E_{j}^{\\prime } \\text{ and } r(i_{2}^{\\prime }) \\in E_{j}^{\\prime } & \\text{iff } l(i) \\in E_{j}\\\\r(a_{1}^{\\prime }) \\in E_{j}^{\\prime } \\text{ and } l(a_{2}^{\\prime }) \\in E_{j}^{\\prime } & \\text{iff } r(a) \\in E_{j}\\\\l(a_{1}^{\\prime }) \\in E_{j}^{\\prime } \\text{ and } r(a_{2}^{\\prime }) \\in E_{j}^{\\prime } & \\text{iff } l(a) \\in E_{j}\\\\\\end{array}\\right.$ For any $j \\ge t$ , let $E^{\\prime }_j$ be the set $E_{\\mathcal {G}^{\\prime }}\\setminus \\lbrace (f_{1}^{\\prime },f_{2}^{\\prime })\\rbrace $ .",
"Now, we consider the execution $\\varepsilon ^{\\prime }$ of $\\mathcal {A}$ on $\\mathcal {G}^{\\prime }$ starting from the configuration where $r_{1}$ (resp.",
"$r_{2}$ ) is on node $i_{1}^{\\prime }$ (resp.",
"on node $i_{2}^{\\prime }$ ) such that the two robots have opposite chirality and that $r_{1}$ have the same chirality as in $\\varepsilon $ .",
"The execution $\\varepsilon ^{\\prime }$ satisfies the following set of claims.",
"Figure: Construction of 𝒢 i+1 \\mathcal {G}_{i + 1}, 𝒢 i+2 \\mathcal {G}_{i + 2},𝒢 i+3 \\mathcal {G}_{i + 3}, and 𝒢 i+4 \\mathcal {G}_{i + 4} in proof of Theorem .Claim 1: Until time $t$ , $r_{1}$ and $r_{2}$ execute the same actions in a symmetrical way in $\\varepsilon ^{\\prime }$ .",
"Consider that, during the Look phase of time $j$ , the two robots have the same view in $\\varepsilon ^{\\prime }$ .",
"The two robots have not the same chirality and $\\mathcal {A}$ is deterministic, then, during the Move phase of time $j$ , they are executing the same action in a symmetrical way (either not move or move in opposite directions).",
"This implies that, at time $j + 1$ , $r_{1}$ and $r_{2}$ have again the same state.",
"There are only two robots executing $\\mathcal {A}$ on $\\mathcal {G}^{\\prime }$ .",
"Hence, if a tower is formed, it is composed of $r_{1}$ and $r_{2}$ .",
"If from time 0 to time $t$ , the robots are executing the same actions in a symmetrical way, then, by construction of $\\mathcal {G}^{\\prime }$ and by the way we initially placed $r_{1}$ and $r_{2}$ on $\\varepsilon ^{\\prime }$ , the two robots see the same local environment at each instant time in $\\lbrace 0,\\ldots , t\\rbrace $ .",
"At time 0, by construction of $\\mathcal {G}^{\\prime }$ and by the way we placed $r_{1}$ and $r_{2}$ on $\\varepsilon ^{\\prime }$ , the two robots have the same view.",
"By recurrence and using the arguments of the two first paragraphs, we conclude that, from time 0 to time $t$ , $r_{1}$ and $r_{2}$ execute the same actions in a symmetrical way in $\\varepsilon ^{\\prime }$ .",
"Claim 2: Until time ${t}$ , $r_{1}$ and $r_{2}$ never form a tower in $\\varepsilon ^{\\prime }$ .",
"By construction of $\\varepsilon ^{\\prime }$ , the two robots are initially at an odd distance.",
"By Claim 1, at a time $0 <j + 1 < t$ , the two robots are either at the same distance, at a distance increased of 2, or at a distance decreased of 2 with respect to their distance at time $j$ .",
"Moreover, since $\\mathcal {G}^{\\prime }$ possesses an even number of edges, this implies that, until time $t$ , the robots are always at an odd distance from each other.",
"Claim 3: Until time ${t}$ , $r_{1}$ executes in $\\varepsilon ^{\\prime }$ the same sequence of actions than in $\\varepsilon $ .",
"Consider that, during the Look phase of time $j$ , $r_{1}$ has the same view in $\\varepsilon $ and in $\\varepsilon ^{\\prime }$ .",
"As $\\mathcal {A}$ is deterministic, then, during the Move phase of time $j$ , $r_{1}$ executes the same action (either not move, or move in the same direction) in $\\varepsilon $ and in $\\varepsilon ^{\\prime }$ .",
"This implies that, during the Look phase of time $j + 1$ , $r_{1}$ possesses the same state in $\\varepsilon $ and in $\\varepsilon ^{\\prime }$ .",
"By assumption, until time $t$ , there is no tower in $\\varepsilon $ .",
"By Claim 2, there is no tower in $\\varepsilon ^{\\prime }$ until time $t$ .",
"Hence, in the case where $r_{1}$ executes the same actions in $\\varepsilon $ and in $\\varepsilon ^{\\prime }$ from time 0 to time $t$ , $r_{1}$ sees the same local environment in $\\varepsilon $ and in $\\varepsilon ^{\\prime }$ until time $t$ (by construction of $\\mathcal {G}^{\\prime }$ and the initial location of $r_{1}$ in $\\varepsilon ^{\\prime }$ ).",
"At time 0, $r_{1}$ has the same view in $\\varepsilon $ and in $\\varepsilon ^{\\prime }$ (by construction of $\\mathcal {G}^{\\prime }$ and the initial location of $r_{1}$ in $\\varepsilon ^{\\prime }$ ).",
"By recurrence and using the arguments of the two first paragraphs, we conclude that, from time 0 to time $t$ , $r_{1}$ executes the same actions in $\\varepsilon $ and in $\\varepsilon ^{\\prime }$ .",
"Claim 4: At time ${t}$ , $r_{1}$ and $r_{2}$ are on two adjacent nodes in $\\varepsilon ^{\\prime }$ and are both in state $s$ .",
"By Claims 1 and 3 and by construction of $\\mathcal {G^{\\prime }}$ , we know that at time $t$ , $r_{1}$ is on node $f_{1}^{\\prime }$ while $r_{2}$ is on node $f_{2}^{\\prime }$ .",
"These nodes are adjacent by construction of $\\mathcal {G^{\\prime }}$ .",
"By Claim 1, as $r_{1}$ and $r_{2}$ have opposite chirality, they have the same state at time $t$ in $\\varepsilon ^{\\prime }$ .",
"By Claim 3, $r_{1}$ is in the same state at time $t$ in $\\varepsilon $ and in $\\varepsilon ^{\\prime }$ .",
"Since $r_1$ is in state $s$ at time $t$ in $\\varepsilon $ by assumption, we have the claim.",
"By construction of $\\mathcal {G}^{\\prime }$ , $f_{1}^{\\prime }$ (resp.",
"$f_{2}^{\\prime }$ ) satisfies the property $OneEdge(f_{1}^{\\prime }, t, +\\infty )$ (resp.",
"$OneEdge(f_{2}^{\\prime },$ $ t, +\\infty )$ ).",
"Then, by assumption, $r_{1}$ (resp.",
"$r_{2}$ ) does not leave node $f_{1}^{\\prime }$ (resp.",
"$f_{2}^{\\prime }$ ) after time $t$ .",
"As $\\mathcal {G}^{\\prime }$ counts 8 nodes, we obtain a contradiction with the fact that $\\mathcal {A}$ is a deterministic algorithm solving the perpetual exploration problem for connected-over-time rings using two robots.",
"Theorem 4.1 There exists no deterministic algorithm satisfying the perpetual exploration specification on the class of connected-over-time rings of size 4 or more with two fully synchronous robots.",
"By contradiction, assume that there exists a deterministic algorithm $\\mathcal {A}$ satisfying the perpetual exploration specification on any connected-over-time ring of size 4 or more with two robots $r_{1}$ and $r_{2}$ .",
"Consider the connected-over-time graph $\\mathcal {G}$ whose underlying graph $U_{\\mathcal {G}}$ is a ring of size strictly greater than 3 such that all the edges of $U_{\\mathcal {G}}$ are present at each time.",
"Consider three nodes $u$ , $v$ and $w$ of $\\mathcal {G}$ , such that node $v$ is the adjacent node of $u$ in the clockwise direction, and $w$ is the adjacent node of $v$ in the clockwise direction.",
"We denote respectively $e_{ur}$ and $e_{ul}$ the clockwise and counter clockwise adjacent edges of $u$ , $e_{vr}$ and $e_{vl}$ the clockwise and counter clockwise adjacent edges of $v$ , and $e_{wr}$ and $e_{wl}$ the clockwise and counter clockwise adjacent edges of $w$ .",
"Note that $e_{ur} = e_{vl}$ and $e_{vr} = e_{wl}$ .",
"Let $\\varepsilon $ be the execution of $\\mathcal {A}$ on $\\mathcal {G}$ starting from the configuration where $r_{1}$ (resp.",
"$r_{2}$ ) is located on node $u$ (resp.",
"$v$ ).",
"We construct a sequence of connected-over-time graphs ($\\mathcal {G}_{n}$ )$_{n \\in \\mathbb {N}}$ such that $\\mathcal {G}_{0} = \\mathcal {G}$ and for any $i \\ge 0$ , $\\mathcal {G}_{i}$ is defined as follows (denote by $\\varepsilon _i$ the execution of $\\mathcal {A}$ on $\\mathcal {G}_{i}$ starting from the same configuration as $\\varepsilon $ ).",
"We define inductively $\\mathcal {G}_{i + 1}$ , $\\mathcal {G}_{i + 2}$ , $\\mathcal {G}_{i + 3}$ , and $\\mathcal {G}_{i + 4}$ using Items 1-8 above (see also Figure REF ) under the assumption that: $(i)$ $\\mathcal {G}_{i}$ exists for a given $i \\in \\mathbb {N}$ multiple of 4; $(ii)$ $\\mathcal {G}_{i}$ is a connected-over-time ring; $(iii)$ there exists a time $t_{i}$ such that each robot has only visited at most two adjacent nodes among $\\lbrace u, v, w\\rbrace $ in $\\varepsilon _i$ ; $(iv)$ before time $t_{i}$ , the two robots never form a tower in $\\varepsilon _i$ ; and $(v)$ at time $t_{i}$ , $r_{1}$ (resp.",
"$r_{2}$ ) is located on node $u$ (resp.",
"$v$ ).",
"Due to assumptions $(ii)$ to $(v)$ , Lemma REF implies that there exists a time $t_{i}^{\\prime } \\ge t_{i}$ such that $r_{2}$ leaves $v$ at time $t_{i}^{\\prime }$ if $r_{2}$ is located on node $v$ at time $t_{i}$ and $v$ satisfies $OneEdge(v, t_{i}, t_{i}^{\\prime })$ .",
"We then define $\\mathcal {G}_{i + 1}$ such that $U_{\\mathcal {G}_{i + 1}} = U_{\\mathcal {G}_{i}}$ and $\\mathcal {G}_{i + 1} = \\mathcal {G}_{i}\\backslash \\lbrace (e_{ul}, \\lbrace t_{i}, \\ldots , t_{i}^{\\prime }\\rbrace ),$ $(e_{vl}, \\lbrace t_{i}, \\ldots , t_{i}^{\\prime }\\rbrace )\\rbrace $ .",
"Note that $\\mathcal {G}_{i}$ and $\\mathcal {G}_{i + 1}$ are indistinguishable for robots before time $t_{i}$ .",
"This implies that, at time $t_{i}$ , $r_{1}$ (resp.",
"$r_{2}$ ) is on node $u$ (resp.",
"$v$ ) in $\\varepsilon _{i + 1}$ .",
"By construction of $t_{i}^{\\prime }$ , $r_{2}$ leaves $v$ at time $t_{i}^{\\prime }$ in $\\varepsilon _{i + 1}$ .",
"Since, at time $t_{i}^{\\prime }$ , among the adjacent edges of $v$ , only $e_{vr}$ is present in $\\mathcal {G}_{i + 1}$ , $r_{2}$ crosses this edge at this time in $\\varepsilon _{i + 1}$ .",
"Hence, at time $t_{i}^{\\prime } + 1$ , $r_{2}$ is on node $w$ in $\\varepsilon _{i + 1}$ .",
"Note that none of the adjacent edges of $r_{1}$ are present between time $t_{i}$ and time $t_{i}^{\\prime }$ in $\\mathcal {G}_{i}$ .",
"That implies that, at time $t_{i}^{\\prime } + 1$ , $r_{1}$ is still on node $u$ in $\\varepsilon _{i + 1}$ .",
"Moreover, this construction ensures us that assumptions $(iii)$ and $(iv)$ are satisfied in $\\varepsilon _{i + 1}$ until time $t_{i}^{\\prime } + 1$ .",
"Finally, $\\mathcal {G}_{i + 1}$ is a connected-over-time ring (since it is indistinguishable from $\\mathcal {G}$ after $t_{i}^{\\prime } + 1$ ) and hence satisfies assumption $(ii)$ .",
"Let $t_{i + 1} = t_{i}^{\\prime } + 1$ .",
"Using similar arguments as in Item 1, we prove that there exists a time $t_{i + 1}^{\\prime }$ such that $r_{1}$ leaves $u$ at time $t_{i + 1}^{\\prime }$ if $r_{1}$ is on node $u$ at time $t_{i+1}$ and $u$ satisfies $OneEdge(u, t_{i + 1}, t_{i + 1}^{\\prime })$ .",
"We define $\\mathcal {G}_{i + 2}$ such that $U_{\\mathcal {G}_{i + 2}} = U_{\\mathcal {G}_{i + 1}}$ and $\\mathcal {G}_{i + 2} = \\mathcal {G}_{i + 1}\\backslash \\lbrace (e_{ul}, \\lbrace t_{i + 1}, \\ldots , t_{i + 1}^{\\prime }\\rbrace ),$ $(e_{wl}, \\lbrace t_{i + 1}, \\ldots , t_{i + 1}^{\\prime }\\rbrace ),$ $(e_{wr}, \\lbrace t_{i + 1}, \\ldots , t_{i + 1}^{\\prime }\\rbrace )\\rbrace $ .",
"That implies that, at time $t_{i + 1}^{\\prime } + 1$ , $r_{1}$ (resp.",
"$r_{2}$ ) is on node $v$ (resp.",
"$w$ ) in $\\varepsilon _{i + 2}$ and that assumptions $(ii)$ , $(iii)$ , and $(iv)$ are satisfied in $\\varepsilon _{i + 2}$ until time $t_{i+1}^{\\prime } + 1$ .",
"Let $t_{i + 2} = t_{i + 1}^{\\prime } + 1$ .",
"Using similar arguments as in Item 1, we prove that there exists a time $t_{i + 2}^{\\prime }$ such that $r_{1}$ leaves $v$ at time $t_{i + 2}^{\\prime }$ if $r_{1}$ is on node $v$ at time $t_{i+2}$ and $v$ satisfies $OneEdge(v, t_{i + 2}, t_{i + 2}^{\\prime })$ .",
"We define $\\mathcal {G}_{i + 3}$ such that $U_{\\mathcal {G}_{i + 3}} = U_{\\mathcal {G}_{i + 2}}$ and such that $\\mathcal {G}_{i + 3} = \\mathcal {G}_{i + 2}\\backslash \\lbrace (e_{wl}, \\lbrace t_{i + 2}, \\ldots , t_{i + 2}^{\\prime }\\rbrace ),$ $(e_{wr}, \\lbrace t_{i + 2}, \\ldots , t_{i + 2}^{\\prime }\\rbrace )\\rbrace $ .",
"That implies that, at time $t_{i + 2}^{\\prime } + 1$ , $r_{1}$ (resp.",
"$r_{2}$ ) is on node $u$ (resp.",
"$w$ ) in $\\varepsilon _{i + 3}$ and that assumptions $(ii)$ , $(iii)$ , and $(iv)$ are satisfied in $\\varepsilon _{i + 3}$ until time $t_{i+2}^{\\prime } + 1$ .",
"Let $t_{i + 3} = t_{i + 2}^{\\prime } + 1$ .",
"Using similar arguments as in Item 1, we prove that there exists a time $t_{i + 3}^{\\prime }$ such that $r_{2}$ leaves $w$ at time $t_{i + 3}^{\\prime }$ if $r_{2}$ is on node $w$ at time $t_{i+3}$ and $w$ satisfies $OneEdge(w, t_{i + 3}, t_{i + 3}^{\\prime })$ .",
"We define $\\mathcal {G}_{i + 4}$ such that $U_{\\mathcal {G}_{i + 4}} = U_{\\mathcal {G}_{i + 3}}$ and such that $\\mathcal {G}_{i + 4} = \\mathcal {G}_{i + 3}\\backslash \\lbrace (e_{ul}, \\lbrace t_{i + 3}, \\ldots , t_{i + 3}^{\\prime }\\rbrace ),$ $(e_{ur}, \\lbrace t_{i + 3}, \\ldots , t_{i + 3}^{\\prime }\\rbrace ),$ $(e_{wr}, \\lbrace t_{i + 3}, \\ldots , t_{i + 3}^{\\prime }\\rbrace )\\rbrace $ .",
"That implies that, at time $t_{i + 3}^{\\prime } + 1$ , $r_{1}$ (resp.",
"$r_{2}$ ) is on node $u$ (resp.",
"$v$ ) in $\\varepsilon _{i + 4}$ and that assumptions $(ii)$ , $(iii)$ , and $(iv)$ are satisfied in $\\varepsilon _{i + 4}$ until time $t_{i+3}^{\\prime } + 1$ .",
"Let $t_{i + 4} = t_{i + 3}^{\\prime } + 1$ .",
"Note that $\\mathcal {G}_{0}$ trivially satisfies assumptions $(i)$ to $(v)$ for $t_0=0$ (since $\\varepsilon _0=\\varepsilon $ by construction).",
"Also, given a $\\mathcal {G}_{i}$ with $i \\in \\mathbb {N}$ multiple of 4, $\\mathcal {G}_{i+4}$ exists and we proved that it satisfies assumptions $(ii)$ to $(v)$ .",
"In other words, ($\\mathcal {G}_{n}$ )$_{n \\in \\mathbb {N}}$ is well-defined.",
"We define the evolving graph $\\mathcal {G}_{\\omega }$ such that $U_{\\mathcal {G}_{\\omega }} = U_{\\mathcal {G}_{0}}$ and $\\begin{array}{r@{}c@{}l}\\mathcal {G}_{\\omega } = \\mathcal {G}_{0} \\backslash \\lbrace & (e_{ul}, & \\lbrace t_{4i}, \\ldots , t_{4i}^{\\prime }\\rbrace \\cup \\lbrace t_{4i + 1}, \\ldots t_{4i + 1}^{\\prime }\\rbrace \\cup \\lbrace t_{4i + 3}, \\ldots , t_{4i + 3}^{\\prime }\\rbrace ),\\\\& (e_{vl}, & \\lbrace t_{4i}, \\ldots , t_{4i}^{\\prime }\\rbrace \\cup \\lbrace t_{4i + 3}, \\ldots , t_{4i + 3}^{\\prime }\\rbrace ),\\\\& (e_{wl}, & \\lbrace t_{4i + 1}, \\ldots , t_{4i + 1}^{\\prime }\\rbrace \\cup \\lbrace t_{4i + 2}, \\ldots , t_{4i + 2}^{\\prime }\\rbrace ),\\\\& (e_{wr}, & \\lbrace t_{4i + 1},\\ldots , t_{4i + 1}^{\\prime }\\rbrace \\cup \\lbrace t_{4i + 2},\\ldots , t_{4i + 2}^{\\prime }\\rbrace \\cup \\lbrace t_{4i + 3},\\ldots , t_{4i + 3}\\rbrace )ǀi \\in \\mathbb {N}\\rbrace \\\\\\end{array}$ Note that, for any edge of $\\mathcal {G}_{\\omega }$ , the intervals of times where this edge is absent (if any) are finite and disjoint.",
"This edge is so infinitely often present in $\\mathcal {G}_{\\omega }$ .",
"Therefore, $\\mathcal {G}_{\\omega }$ is a connected-over-time ring.",
"For any $i\\in \\mathbb {N}$ , $\\mathcal {G}_{i}$ and $\\mathcal {G}_{\\omega }$ have a common prefix until time $t_{i}^{\\prime }$ .",
"As the sequence $(t_{n}$ )$_{n \\in \\mathbb {N}}$ is increasing by construction, this implies that the sequence ($\\mathcal {G}_{n}$ )$_{n \\in \\mathbb {N}}$ converges to $\\mathcal {G}_{\\omega }$ .",
"Applying the theorem of [5], we obtain that, until time $t_{i}^{\\prime }$ , the execution of $\\mathcal {A}$ on $\\mathcal {G}_{\\omega }$ is identical to the one on $\\mathcal {G}_{i}$ .",
"This implies that, executing $\\mathcal {A}$ on $\\mathcal {G}_{\\omega }$ (of size strictly greater than 3), $r_{1}$ and $r_{2}$ only visit the nodes $u$ , $v$ , and $w$ .",
"This is contradictory with the fact that $\\mathcal {A}$ satisfies the perpetual exploration specification on connected over time rings of size strictly greater than 3 using two robots."
],
[
"Connected-over-time Rings of Size 3",
"In this section, we present $\\mathbb {PEF}\\_2$ , a deterministic algorithm solving the perpetual exploration on connected-over-time rings of size 3 with two robots.",
"This algorithm works as follows.",
"Each robot disposes only of its $dir$ variable.",
"If at a time $t$ , a robot is isolated on a node with only one adjacent edge, then it points to this edge.",
"Otherwise (i.e., none of the adjacent edge is present, both adjacent edges are present, or the other robot is present on the same node), the robot keeps its current direction.",
"Theorem 4.2 $\\mathbb {PEF}\\_2$ is a perpetual exploration algorithm for the class of connected-over-time rings of 3 nodes using 2 fully synchronous robots.",
"Consider any execution of $\\mathbb {PEF}\\_2$ on any connected-over-time ring of size 3 with 2 robots.",
"By the connected-over-time assumption, each node has at least one adjacent edge infinitely often present.",
"This implies that any tower is broken in finite time (as robots meet only when they consider opposite directions and move as soon as it is possible).",
"Two cases are now possible.",
"Case 1: There exists infinitely often a tower in the execution.",
"Note that, if a tower is formed at a time $t$ , then the three nodes have been visited between time $t - 1$ and time $t$ .",
"Then, the three nodes are infinitely often visited by a robot in this case.",
"Case 2: There exists a time $t$ after which the robots are always isolated.",
"By contradiction, assume that there exists a time $t^{\\prime }$ such that a node $u$ is never visited after $t^{\\prime }$ .",
"As the ring has 3 nodes, that implies that, after time $max\\lbrace t,t^{\\prime }\\rbrace $ , either the robots are always switching their position or they stay on their respective nodes.",
"In the first case, during the Look phase of each time greater than $max\\lbrace t,t^{\\prime }\\rbrace $ , the respective variables $dir$ of the two robots contain the direction leading to $u$ (since it previously move in this direction).",
"As at least one of the adjacent edges of $u$ is infinitely often present, a robot crosses it in a finite time, that is contradictory with the fact that $u$ is not visited after $t^{\\prime }$ .",
"The second case implies that both adjacent edges to the location of both robots are always absent after time $t$ (since a robot moves as soon as it is possible), that is contradictory with the connected-over-time assumption.",
"In both cases, $\\mathbb {PEF}\\_2$ satisfies the perpetual exploration specification."
],
[
"With One Robot",
"This section leads a similar study than the one of Section but in the case of the perpetual exploration of rings of any size with a single robot.",
"Again, we first prove a negative result since Theorem REF states that a single robot is not able to perpetually explore connected-over-time rings of size strictly greater than 2.",
"We then provide $\\mathbb {PEF}\\_1$ (see Theorem REF ), an algorithm using a single robot that solves the perpetual exploration on connected-over-time rings of size 2."
],
[
"Connected-over-time Rings of Size 3 and More",
"Similarly to the previous section, the proof of our impossibility result presented in Theorem REF is based on the construction of an adequate sequence of evolving graphs and the application of the generic framework proposed in [5].",
"In order to build the evolving graphs sequence suitable for the proof of our impossibility result, we need the following technical lemma.",
"Lemma 5.1 Let $\\mathcal {A}$ be a perpetual exploration algorithm in connected-over-time ring of size 3 or more using one robot.",
"Any execution of $\\mathcal {A}$ satisfies: For any time $t$ and any robot state $s$ , there exists a time $t^{\\prime } \\ge t$ such that a robot located on a node $u$ , on state $s$ at time $t$ , and satisfying $OneEdge(u, t, t^{\\prime })$ leaves $u$ at time $t^{\\prime }$ .",
"Consider an algorithm $\\mathcal {A}$ that solves deterministically the perpetual exploration problem for connected-over-time rings of size 3 or more using a single robot.",
"Let $\\mathcal {G} = \\lbrace G_{0}=(V,E_0), G_{1}=(V,E_1), \\ldots \\rbrace $ be a connected-over-time ring (of size 3 or more).",
"Let $\\varepsilon $ be an execution of $\\mathcal {A}$ on $\\mathcal {G}$ by a robot $r$ .",
"By contradiction, assume that it exists a time $t$ and a state $s$ such that, for any $t^{\\prime } \\ge t$ , a robot $r$ located on a node $u$ of $\\mathcal {G}$ and in state $s$ at time $t$ with $u$ satisfying $OneEdge(u, t, t^{\\prime })$ does not leave $u$ at time $t^{\\prime }$ .",
"Let $e$ be an arbitrary adjacent edge to $u$ .",
"Let us define the connected-over-time ring $\\mathcal {G}^{\\prime } = \\lbrace G_{0}^{\\prime }=(V,E^{\\prime }_0), G_{1}^{\\prime }=(V,E^{\\prime }_1), \\ldots \\rbrace $ such that: $\\left\\lbrace \\begin{array}{ll}E_{i}^{\\prime } = E_{i} & \\text{if } i < t\\\\E_{i}^{\\prime } = E_{\\mathcal {G}}\\setminus \\lbrace e\\rbrace & \\text{if } i \\ge t\\\\\\end{array}\\right.$ Let $\\varepsilon ^{\\prime }$ be the execution of $\\mathcal {A}$ on $\\mathcal {G}^{\\prime }$ starting from the same configuration than $\\varepsilon $ .",
"As $\\mathcal {A}$ is a deterministic algorithm, $r$ is in the state $s$ and is located on node $u$ at time $t$ in $\\varepsilon ^{\\prime }$ by construction of $\\mathcal {G}^{\\prime }$ .",
"Note that the node $u$ satisfies the property $OneEdge(u, t, +\\infty )$ in $\\mathcal {G}^{\\prime }$ .",
"Then, by assumption, $r$ does not leave $u$ in $\\varepsilon ^{\\prime }$ after time $t$ .",
"This implies that, after time $t$ , only $u$ is visited in $\\varepsilon ^{\\prime }$ .",
"As $\\mathcal {G}^{\\prime }$ counts 3 or more nodes, we obtain a contradiction with the fact that $\\mathcal {A}$ is a deterministic algorithm solving the perpetual exploration problem for connected-over-time rings using a single robot.",
"Theorem 5.1 There exists no deterministic algorithm satisfying the perpetual exploration specification on the class of connected-over-time rings of size 3 or more with a single fully synchronous robot.",
"By contradiction, assume that there exists a deterministic algorithm $\\mathcal {A}$ satisfying the perpetual exploration specification on any connected-over-time ring of size 3 or more with a single robot $r$ .",
"Consider the connected-over-time graph $\\mathcal {G}$ whose underlying graph $U_{\\mathcal {G}}$ is a ring of size strictly greater than 2 such that all the edges of $U_{\\mathcal {G}}$ are present at each time.",
"Consider any node $u$ of $\\mathcal {G}$ and denote respectively by $e_{ur}$ and $e_{ul}$ the clockwise and counter clockwise adjacent edges of $u$ .",
"Let $\\varepsilon $ be the execution of $\\mathcal {A}$ on $\\mathcal {G}$ starting from the configuration where $r$ is located on node $u$ .",
"We construct a sequence of connected-over-time graphs ($\\mathcal {G}_{n}$ )$_{n \\in \\mathbb {N}}$ such that $\\mathcal {G}_{0} = \\mathcal {G}$ and for any $i \\ge 0$ , $\\mathcal {G}_{i}$ is defined as follows (denote by $\\varepsilon _i$ the execution of $\\mathcal {A}$ on $\\mathcal {G}_{i}$ starting from the same configuration as $\\varepsilon $ ).",
"We define inductively $\\mathcal {G}_{i + 1}$ and $\\mathcal {G}_{i + 2}$ using Items 1-4 above (see also Figure REF ) under the assumption that: $(i)$ $\\mathcal {G}_{i}$ exists for a given $i \\in \\mathbb {N}$ even; $(ii)$ $\\mathcal {G}_{i}$ is a connected-over-time ring; and $(iii)$ there exists a time $t_{i}$ such that $r$ is located on node $u$ at time $t$ in $\\varepsilon _i$ .",
"Due to assumptions $(ii)$ and $(iii)$ , Lemma REF implies that there exists a time $t_{i}^{\\prime } \\ge t_{i}$ such that $r$ leaves $u$ at time $t_{i}^{\\prime }$ if it is located on node $u$ at time $t_{i}$ and $u$ satisfies $OneEdge(u, t_{i}, t_{i}^{\\prime })$ .",
"We then define $\\mathcal {G}_{i + 1}$ such that $U_{\\mathcal {G}_{i + 1}} = U_{\\mathcal {G}_{i}}$ and $\\mathcal {G}_{i + 1} = \\mathcal {G}_{i}\\backslash \\lbrace (e_{ur}, \\lbrace t_{i}, \\ldots , t_{i}^{\\prime }\\rbrace )\\rbrace $ .",
"Note that $\\mathcal {G}_{i}$ and $\\mathcal {G}_{i + 1}$ are indistinguishable for $r$ before time $t_{i}$ .",
"This implies that, at time $t_{i}$ , $r$ is located on node $u$ in $\\varepsilon _{i + 1}$ .",
"By construction of $t_{i}^{\\prime }$ , $r$ leaves $u$ at time $t_{i}^{\\prime }$ in $\\varepsilon _{i + 1}$ .",
"Since, at time $t_{i}^{\\prime }$ , among the adjacent edges of $u$ , only $e_{ul}$ is present in $\\mathcal {G}_{i + 1}$ , $r$ crosses this edge at this time in $\\varepsilon _{i + 1}$ .",
"Then, at time $t_{i}^{\\prime } + 1$ , $r$ is located on node $v$ (the node adjacent to $u$ in the counter clockwise direction) in $\\varepsilon _{i + 1}$ .",
"Finally, $\\mathcal {G}_{i + 1}$ is a connected-over-time ring (since it is indistinguishable from $\\mathcal {G}$ after $t_{i}^{\\prime } + 1$ ) and hence satisfies assumption $(ii)$ .",
"Denote respectively by $e_{vr}$ and $e_{vl}$ the clockwise and counter clockwise adjacent edges of $v$ .",
"We have $e_{ul} = e_{vr}$ .",
"Let $t_{i + 1} = t_{i}^{\\prime } + 1$ .",
"Using similar arguments as in Item 1, we prove that there exists a time $t_{i + 1}^{\\prime }$ such that $r$ leaves $v$ at time $t_{i + 1}^{\\prime }$ if $r$ is located on node $v$ at time $t_{i+1}$ and $v$ satisfies $OneEdge(v, t_{i + 1}, t_{i + 1}^{\\prime })$ .",
"We define $\\mathcal {G}_{i + 2}$ such that $U_{\\mathcal {G}_{i + 2}} = U_{\\mathcal {G}_{i + 1}}$ and $\\mathcal {G}_{i + 2} = \\mathcal {G}_{i + 1}\\backslash \\lbrace (e_{vl}, \\lbrace t_{i + 1}, \\ldots t_{i + 1}^{\\prime }\\rbrace )\\rbrace $ .",
"That implies that, at time $t_{i + 1}^{\\prime } + 1$ , $r$ is on node $u$ in $\\varepsilon _{i + 2}$ and that assumptions $(ii)$ and $(iii)$ are satisfied in $\\varepsilon _{i + 2}$ until time $t_{i+1}^{\\prime } + 1$ .",
"Let $t_{i + 2} = t_{i + 1}^{\\prime } + 1$ .",
"Note that $\\mathcal {G}_{0}$ trivially satisfies assumptions $(i)$ to $(iii)$ for $t_0=0$ (since $\\varepsilon _0=\\varepsilon $ by construction).",
"Also, given a $\\mathcal {G}_{i}$ with $i \\in \\mathbb {N}$ even, $\\mathcal {G}_{i+2}$ exists and we proved that it satisfies assumptions $(ii)$ and $(iii)$ .",
"In other words, ($\\mathcal {G}_{n}$ )$_{n \\in \\mathbb {N}}$ is well-defined.",
"We define the evolving graph $\\mathcal {G}_{\\omega }$ such that $U_{\\mathcal {G}_{\\omega }} = U_{\\mathcal {G}_{0}}$ and $\\begin{array}{r@{}l}\\mathcal {G}_{\\omega } = \\mathcal {G}_{0} \\backslash \\lbrace & (e_{ur}, \\lbrace t_{2i}, \\ldots t_{2i}^{\\prime }\\rbrace ),(e_{vl}, \\lbrace t_{2i+1}, \\ldots , t_{2i+1}^{\\prime }\\rbrace ) ǀi \\in \\mathbb {N}\\rbrace \\\\\\end{array}$ Note that, for any edge of $\\mathcal {G}_{\\omega }$ , the intervals of times where this edge is absent (if any) are finite and disjoint.",
"This edge is so infinitely often present in $\\mathcal {G}_{\\omega }$ .",
"Therefore, $\\mathcal {G}_{\\omega }$ is a connected-over-time ring.",
"For any $i\\in \\mathbb {N}$ , $\\mathcal {G}_{i}$ and $\\mathcal {G}_{\\omega }$ have a common prefix until time $t_{i}^{\\prime }$ .",
"As the sequence $(t_{n}$ )$_{n \\in \\mathbb {N}}$ is increasing by construction, this implies that the sequence ($\\mathcal {G}_{n}$ )$_{n \\in \\mathbb {N}}$ converges to $\\mathcal {G}_{\\omega }$ .",
"Applying the theorem of [5], we obtain that, until time $t_{i}^{\\prime }$ , the execution of $\\mathcal {A}$ on $\\mathcal {G}_{\\omega }$ is identical to the one on $\\mathcal {G}_{i}$ .",
"This implies that, executing $\\mathcal {A}$ on $\\mathcal {G}_{\\omega }$ (of size strictly greater than 2), $r$ only visits the nodes $u$ and $v$ .",
"This is contradictory with the fact that $\\mathcal {A}$ satisfies the perpetual exploration specification on connected over time rings of size strictly greater than 2 using one robot."
],
[
"Connected-over-time Rings of Size 2",
"In this section, we present $\\mathbb {PEF}\\_1$ , a deterministic algorithm solving the perpetual exploration on connected-over-time rings of size 2 with a single robot.",
"Note that a ring of size 2 can be defined in two different ways.",
"If we consider that the graph must remain simple, such a ring is reduced to a 2-node chain (i.e., only one bidirectional edge links the two nodes).",
"Otherwise (i.e., the graph may be not simple), the two nodes are linked by two bidirectional edges.",
"In both cases, the following algorithm, $\\mathbb {PEF}\\_1$ , trivially works as follow: As soon as at least one adjacent edge to the current node of the robot is present, its variable $dir$ points arbitrarily to one of these edges.",
"Theorem 5.2 $\\mathbb {PEF}\\_1$ is a perpetual exploration algorithm for the class of connected-over-time rings of 2 nodes using a single fully synchronous robot."
],
[
"Conclusion",
"We analyzed the computability of the perpetual exploration problem on highly dynamic rings.",
"We proved that three (resp., two) robots with very few capacities are necessary to solve the perpetual exploration problem on connected-over-time rings that include strictly more than three (resp., two) nodes.",
"For the completeness of our work, we provided three algorithms: One for a single robot evolving in a 2-node ring, one for two robots exploring three nodes, and one for three or more robots moving among at least four nodes.",
"These three algorithms allow to show that the necessary number of robots is also sufficient to solve the problem."
]
] | 1612.05767 |
[
[
"A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion\n Problem"
],
[
"Abstract We study a multigrid method for solving large linear systems of equations with tensor product structure.",
"Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients.",
"When the variance in the problem is not too large, the solution can be well approximated by a low-rank object.",
"In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort.",
"The method is proved convergent with an analytic error bound.",
"Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large."
],
[
"Introduction",
"Stochastic partial differential equations (SPDEs) arise from physical applications where the parameters of the problem are subject to uncertainty.",
"Discretization of SPDEs gives rise to large linear systems of equations which are computationally expensive to solve.",
"These systems are in general sparse and structured.",
"In particular, the coefficient matrix can often be expressed as a sum of tensor products of smaller matrices [6], [13], [14].",
"For such systems it is natural to use an iterative solver where the coefficient matrix is never explicitly formed and matrix-vector products are computed efficiently.",
"One way to further reduce costs is to construct low-rank approximations to the desired solution.",
"The iterates are truncated so that the solution method handles only low-rank objects in each iteration.",
"This idea has been used to reduce the costs of iterative solution algorithms based on Krylov subspaces.",
"For example, a low-rank conjugate gradient method was given in [9], and low-rank generalized minimal residual methods have been studied in [2], [10].",
"In this study, we propose a low-rank multigrid method for solving the Galerkin systems.",
"We consider a steady-state diffusion equation with random diffusion coefficient as model problem, and we use the stochastic finite element method (SFEM, see [1], [7]) for the discretization of the problem.",
"The resulting Galerkin system has tensor product structure and moreover, quantities used in the computation, such as the solution sought, can be expressed in matrix format.",
"It has been shown that such systems admit low-rank approximate solutions [3], [9].",
"In our proposed multigrid solver, the matrix iterates are truncated to have low rank in each iteration.",
"We derive an analytic bound for the error of the solution and show the convergence of the algorithm.",
"We demonstrate using benchmark problems that the low-rank multigrid solver is often more efficient than a solver that does not use truncation, and that it is especially advantageous in reducing computing time for large-scale problems.",
"An outline of the paper is as follows.",
"In sec:model we state the problem and briefly review the stochastic finite element method and the multigrid solver for the stochastic Galerkin system from which the new technique is derived.",
"In sec:low-rank we discuss the idea of low-rank approximation and introduce the multigrid solver with low-rank truncation.",
"A convergence analysis of the low-rank multigrid solver is also given in this section.",
"The results of numerical experiments are shown in sec:numerical to test the performance of the algorithm, and some conclusions are drawn in the last section."
],
[
"Model problem",
"Consider the stochastic steady-state diffusion equation with homogeneous Dirichlet boundary conditions $ {\\left\\lbrace \\begin{array}{ll}-\\nabla \\cdot (c(x,\\omega )\\nabla u(x,\\omega )) = f(x) & \\text{in } D\\times \\Omega ,\\\\u(x,\\omega ) = 0 & \\text{on } \\partial D\\times \\Omega .\\\\\\end{array}\\right.",
"}$ Here $D$ is a spatial domain and $\\Omega $ is a sample space with $\\sigma $ -algebra ${F}$ and probability measure $P$ .",
"The diffusion coefficient $c(x,\\omega ): D\\times \\Omega \\rightarrow \\mathbb {R}$ is a random field.",
"We consider the case where the source term $f$ is deterministic.",
"The stochastic Galerkin formulation of eq:stochdiff uses a weak formulation: find $u(x,\\omega )\\in \\mathbb {V}=H_0^1(D)\\otimes L^2(\\Omega )$ satisfying $ \\int _\\Omega \\int _D c(x,\\omega )\\nabla u(x,\\omega )\\cdot \\nabla v(x,\\omega )\\text{d}x\\text{d}P = \\int _\\Omega \\int _D f(x)v(x,\\omega )\\text{d}x\\text{d}P$ for all $v(x,\\omega )\\in \\mathbb {V}$ .",
"The problem is well posed if $c(x,\\omega )$ is bounded and strictly positive, i.e., $0<c_1\\le c(x,\\omega )\\le c_2<\\infty ,\\,\\text{a.e. }",
"\\forall x\\in D,$ so that the Lax-Milgram lemma establishes existence and uniqueness of the weak solution.",
"We will assume that the stochastic coefficient $c(x,\\omega )$ is represented as a truncated Karhunen-Lo$\\grave{\\text{e}}$ ve (KL) expansion [11], [12], in terms of a finite collection of uncorrelated random variables $\\lbrace \\xi _l\\rbrace _{l=1}^m$ : $ c(x,\\omega ) \\approx c_0(x) + \\sum _{l=1}^m \\sqrt{\\lambda _l}c_l(x)\\xi _l(\\omega )$ where $c_0(x)$ is the mean function, $(\\lambda _l,c_l(x))$ is the $l$ th eigenpair of the covariance function $r(x,y)$ , and the eigenvalues $\\lbrace \\lambda _l\\rbrace $ are assumed to be in non-increasing order.",
"In sec:numerical we will further assume these random variables are independent and identically distributed.",
"Let $\\rho (\\xi )$ be the joint density function and $\\Gamma $ be the joint image of $\\lbrace \\xi _l\\rbrace _{l=1}^m$ .",
"The weak form of eq:stochdiff is then given as follows: find $u(x,\\xi )\\in \\mathbb {W}=H_0^1(D)\\otimes L^2(\\Gamma )$ s.t.",
"$ \\int _\\Gamma \\rho (\\xi )\\int _D c(x,\\xi )\\nabla u(x,\\xi )\\cdot \\nabla v(x,\\xi )\\text{d}x\\text{d}\\xi = \\int _\\Gamma \\rho (\\xi )\\int _D f(x)v(x,\\xi )\\text{d}x\\text{d}\\xi $ for all $v(x,\\xi )\\in \\mathbb {W}$ ."
],
[
"Stochastic finite element method",
"We briefly review the stochastic finite element method as described in [1], [7].",
"This method approximates the weak solution of eq:stochdiff in a finite-dimensional subspace $\\mathbb {W}^{hp} = S^h\\otimes T^p=\\text{span}\\lbrace \\phi (x)\\psi (\\xi ) \\mid \\phi (x)\\in S^h,\\psi (\\xi )\\in T^p\\rbrace ,$ where $S^h$ and $T^p$ are finite-dimensional subspaces of $H_0^1(D)$ and $L^2(\\Gamma )$ .",
"We will use quadrilateral elements and piecewise bilinear basis functions $\\lbrace \\phi (x)\\rbrace $ for the discretization of the physical space $H_0^1(D)$ , and generalized polynomial chaos [17] for the stochastic basis functions $\\lbrace \\psi (\\xi )\\rbrace $ .",
"The latter are $m$ -dimensional orthogonal polynomials whose total degree doesn't exceed $p$ .",
"The orthogonality relation means $\\int _\\Gamma \\psi _r(\\xi )\\psi _s(\\xi )\\rho (\\xi )\\text{d}\\xi =\\delta _{rs}\\int _\\Gamma \\psi _r^2(\\xi )\\rho (\\xi )\\text{d}\\xi .$ For instance, Legendre polynomials are used if the random variables have uniform distribution with zero mean and unit variance.",
"The number of degrees of freedom in $T^p$ is $N_{\\xi }=\\frac{(m+p)!}{m!p!",
"}.$ Given the subspace, now one can write the SFEM solution as a linear combination of the basis functions, $ u_{hp}(x,\\xi )=\\sum _{j=1}^{N_x}\\sum _{s=1}^{N_{\\xi }} u_{js}\\phi _j(x)\\psi _s(\\xi ),$ where $N_x$ is the dimension of the subspace $S^h$ .",
"Substituting eq:kl,eq:sfemsoln into eq:weak2, and taking the test function as any basis function $\\phi _i(x)\\psi _r(\\xi )$ results in the Galerkin system: find $\\mathbf {u}\\in \\mathbb {R}^{N_xN_{\\xi }}$ , s.t.",
"$ A\\mathbf {u}=\\mathbf {f}.$ The coefficient matrix $A$ can be represented in tensor product notation [14], $A=G_0\\otimes K_0+\\sum _{l=1}^m G_l\\otimes K_l,$ where $\\lbrace K_l\\rbrace _{l=0}^m$ are the stiffness matrices and $\\lbrace G_l\\rbrace _{l=0}^m$ correspond to the stochastic part, with entries $\\begin{aligned}G_0(r,s)&=\\int _\\Gamma \\psi _r(\\xi )\\psi _s(\\xi )\\rho (\\xi )\\text{d}\\xi ,\\,K_0(i,j)= \\int _D c_0(x)\\nabla \\phi _i(x)\\nabla \\phi _j(x)\\text{d}x,\\\\G_l(r,s)&=\\int _\\Gamma \\xi _l\\psi _r(\\xi )\\psi _s(\\xi )\\rho (\\xi )\\text{d}\\xi ,\\,K_l(i,j)= \\int _D \\sqrt{\\lambda _l}c_l(x)\\nabla \\phi _i(x)\\nabla \\phi _j(x)\\text{d}x,\\end{aligned}$ $l=1,\\dots ,m;\\,r,s=1,\\ldots ,N_\\xi ;\\,i,j=1,\\ldots ,N_x$ .",
"The right-hand side can be written as a tensor product of two vectors: $\\mathbf {f}=g_0\\otimes f_0,$ where $\\begin{aligned}g_0(r)&=\\int _\\Gamma \\psi _r(\\xi )\\rho (\\xi )\\text{d}\\xi ,\\,\\, r=1,\\ldots ,N_\\xi ,\\\\f_0(i)&=\\int _D f(x)\\phi _i(x)\\text{d}x,\\,\\, i=1,\\ldots ,N_x.\\end{aligned}$ Note that in the Galerkin system eq:galerkin, the matrix $A$ is symmetric and positive definite.",
"It is also blockwise sparse (see fig:sparsity) due to the orthogonality of $\\lbrace \\psi _r(\\xi )\\rbrace $ .",
"The size of the linear system is in general very large ($N_xN_\\xi \\times N_xN_\\xi $ ).",
"For such a system it is suitable to use an iterative solver.",
"Multigrid methods are among the most effective iterative solvers for the solution of discretized elliptic PDEs, capable of achieving convergence rates that are independent of the mesh size, with computational work growing only linearly with the problem size [8], [15].",
"Figure: Block structure of AA.",
"m=4,p=1,2,3m=4,p=1,2,3 from left to right.",
"Block size is N x ×N x N_x\\times N_x."
],
[
"Multigrid",
"In this subsection we discuss a geometric multigrid solver proposed in [4] for the solution of the stochastic Galerkin system eq:galerkin.",
"For this method, the mesh size $h$ varies for different grid levels, while the polynomial degree $p$ is held constant, i.e., the fine grid space and coarse grid space are defined as $\\mathbb {W}^{hp}=S^h\\otimes T^p,\\quad \\mathbb {W}^{2h,p}=S^{2h}\\otimes T^p,$ respectively.",
"Then the prolongation and restriction operators are of the form $ {P}=I\\otimes P,\\quad {R}=I\\otimes P^T,$ where $P$ is the same prolongation matrix as in the deterministic case.",
"On the coarse grid we only need to construct matrices $\\lbrace K^{2h}_l\\rbrace _{l=0}^m$ , and ${A}^{2h}=G_0\\otimes {K}^{2h}_0+\\sum _{l=1}^m G_l\\otimes {K}^{2h}_l.$ The matrices $\\lbrace G_l\\rbrace _{l=0}^m$ are the same for all grid levels.",
"alg:mg describes the complete multigrid method.",
"In each iteration, we apply one multigrid cycle (Vcycle) for the residual equation $A\\mathbf {c}^{(i)}=\\mathbf {r}^{(i)}=\\mathbf {f}-A\\mathbf {u}^{(i)}$ and update the solution $\\mathbf {u}^{(i)}$ and residual $\\mathbf {r}^{(i)}$ .",
"The Vcycle function is called recursively.",
"On the coarsest grid level ($h=h_0$ ) we form matrix $A$ and solve the linear system directly.",
"The system is of order $O(N_\\xi )$ since $A\\in \\mathbb {R}^{N_xN_\\xi \\times N_xN_\\xi }$ where $N_x$ is a very small number on the coarsest grid.",
"The smoothing function (Smooth) is based on a matrix splitting $A=Q-Z$ and stationary iteration $ \\mathbf {u}_{s+1} = \\mathbf {u}_s+Q^{-1}(\\mathbf {f}-A\\mathbf {u}_s),$ which we assume is convergent, i.e., the spectral radius $\\rho (I-Q^{-1}A)<1.$ The algorithm is run until the specified relative tolerance $tol$ or maximum number of iterations $maxit$ is reached.",
"It is shown in [4] that for $f\\in L^2(D)$ , the convergence rate of this algorithm is independent of the mesh size $h$ , the number of random variables $m$ , and the polynomial degree $p$ .",
"Multigrid for stochastic Galerkin systems textsc : Initinitialization Funcfunctionend VcycleVcycle SmoothSmooth : $i=0$ , $\\mathbf {r}^{(0)}=\\mathbf {f}$ , $r_0=\\Vert \\mathbf {f}\\Vert _2$ $r>tol*r_0$ $\\&$ $i\\le maxit$ $\\mathbf {c}^{(i)}=$ $A,\\mathbf {0},\\mathbf {r}^{(i)}$ $\\mathbf {u}^{(i+1)}=\\mathbf {u}^{(i)}+\\mathbf {c}^{(i)}$ $\\mathbf {r}^{(i+1)} = \\mathbf {f}-A\\mathbf {u}^{(i+1)}$ $r=\\Vert \\mathbf {r}^{(i+1)}\\Vert _2$ , $i=i+1$ $\\mathbf {u}^h=$ $A^h,\\mathbf {u}^h_0,\\mathbf {f}^h$ $h==h_0$ solve $A^h\\mathbf {u}^{h}=\\mathbf {f}^h$ directly $\\mathbf {u}^h=$ $A^h,\\mathbf {u}^h_0,\\mathbf {f}^h$ $\\mathbf {r}^h=\\mathbf {f}^h-A^h\\mathbf {u}^h$ $\\mathbf {r}^{2h}={R}\\mathbf {r}^h$ $\\mathbf {c}^{2h}=$ $A^{2h},\\mathbf {0},\\mathbf {r}^{2h}$ $\\mathbf {u}^{h}=\\mathbf {u}^{h}+{P}{\\mathbf {c}^{2h}}$ $\\mathbf {u}^h=$ $A^h,\\mathbf {u}^h,\\mathbf {f}^h$ $\\mathbf {u}=$ $A,\\mathbf {u},\\mathbf {f}$ $\\nu $ steps $\\mathbf {u} = \\mathbf {u}+Q^{-1}(\\mathbf {f}-A\\mathbf {u})$"
],
[
"Low-rank approximation",
"In this section we consider a technique designed to reduce computational effort, in terms of both time and memory use, using low-rank methods.",
"We begin with the observation that the solution vector of the Galerkin system eq:galerkin $\\mathbf {u}=[u_{11},u_{21},\\ldots ,u_{N_x1},\\ldots ,u_{1N_\\xi },u_{2N_\\xi },\\ldots ,u_{N_xN_\\xi }]^T \\in \\mathbb {R}^{N_xN_\\xi }$ can be restructured as a matrix $U = \\text{mat}(\\mathbf {u})=\\begin{pmatrix}u_{11} & u_{12} & \\cdots & u_{1N_\\xi } \\\\u_{21} & u_{22} & \\cdots & u_{2N_\\xi } \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\u_{N_x1} & u_{N_x2} & \\cdots & u_{N_xN_\\xi }\\end{pmatrix}\\in \\mathbb {R}^{N_x\\times N_\\xi }.$ Then (REF ) is equivalent to a system in matrix format, $ \\mathcal {A}(U)=F,$ where $\\begin{aligned}&\\mathcal {A}(U)=K_0UG_0^T+\\sum _{l=1}^m K_lUG_l^T,\\\\&F=\\text{mat}(\\mathbf {f})=\\text{mat}(g_0\\otimes f_0)=f_0 g_0^T.\\end{aligned}$ It has been shown in [3], [9] that the “matricized” version of the solution $U$ can be well approximated by a low-rank matrix when $N_xN_\\xi $ is large.",
"Evidence of this can be seen in fig:decay1, which shows the singular values of the exact solution $U$ for the benchmark problem discussed in sec:numerical.",
"In particular, the singular values decay exponentially, and low-rank approximate solutions can be obtained by dropping terms from the singular value decomposition corresponding to small singular values.",
"Figure: Decay of singular values of solution matrix UU.",
"Left: exponential covariance, b=5b=5, h=2 -6 h=2^{-6}, m=8m=8, p=3p=3.",
"Right: squared exponential covariance, b=2b=2, h=2 -6 h=2^{-6}, m=3m=3, p=3p=3.",
"See the benchmark problem in sec:numerical.Now we use low-rank approximation in the multigrid solver for eq:matrix.",
"Let $U^{(i)}=\\text{mat}(\\mathbf {u}^{(i)})$ be the $i$ th iterate, expressed in matricized formatIn the sequel, we use $\\mathbf {u}^{(i)}$ and $U^{(i)}$ interchangeably to represent the equivalent vectorized or matricized quantities., and suppose $U^{(i)}$ is represented as the outer product of two rank-$k$ matrices, i.e., $U^{(i)}\\approx V^{(i)}W^{(i)T}$ , where $V^{(i)}\\in \\mathbb {R}^{N_x\\times k}$ , $W^{(i)}\\in \\mathbb {R}^{N_\\xi \\times k}$ .",
"This factored form is convenient for implementation and can be readily used in basic matrix operations.",
"For instance, the sum of two matrices gives $V_1^{(i)}W_1^{(i)T}+V_2^{(i)}W_2^{(i)T}=[V_1^{(i)},V_2^{(i)}][W_1^{(i)},W_2^{(i)}]^T.$ Similarly, $\\mathcal {A}(V^{(i)}W^{(i)T})$ can also be written as an outer product of two matrices: $ \\begin{aligned}\\mathcal {A}(V^{(i)}W^{(i)T}) &=(K_0V^{(i)})(G_0W^{(i)})^T+\\sum _{l=1}^m (K_lV^{(i)})(G_lW^{(i)})^T\\\\=& \\,\\,[K_0V^{(i)},K_1V^{(i)},\\ldots ,K_mV^{(i)}][G_0W^{(i)},G_1W^{(i)},\\ldots ,G_mW^{(i)}]^T.\\\\\\end{aligned}$ If $V^{(i)},W^{(i)}$ are used to represent iterates in the multigrid solver and $k\\ll \\text{min}(N_x,N_\\xi )$ , then both memory and computational (matrix-vector products) costs can be reduced, from $O(N_xN_\\xi )$ to $O((N_x+N_\\xi )k)$ .",
"Note, however, that the ranks of the iterates may grow due to matrix additions.",
"For example, in eq:opta the rank may increase from $k$ to $(m+1)k$ in the worst case.",
"A way to prevent this from happening, and also to keep costs low, is to truncate the iterates and force their ranks to remain low."
],
[
"Low-rank truncation",
"Our truncation strategy is derived using an idea from [9].",
"Assume ${\\tilde{X}}=\\tilde{V}\\tilde{W}^T$ , $\\tilde{V}\\in \\mathbb {R}^{N_x\\times \\tilde{k}}$ , $\\tilde{W}\\in \\mathbb {R}^{N_\\xi \\times \\tilde{k}}$ , and ${X}=\\mathcal {T}(\\tilde{X})$ is truncated to rank $k$ with $X=VW^T$ , $V\\in \\mathbb {R}^{N_x\\times k}$ , $W\\in \\mathbb {R}^{N_\\xi \\times k}$ and $k<\\tilde{k}$ .",
"First, compute the QR factorization for both $\\tilde{V}$ and $\\tilde{W}$ , $\\tilde{V}=Q_{\\tilde{V}}R_{\\tilde{V}}, \\quad \\tilde{W}=Q_{\\tilde{W}}R_{\\tilde{W}},\\quad \\text{ so } {\\tilde{X}}=Q_{\\tilde{V}}R_{\\tilde{V}}R_{\\tilde{W}}^TQ_{\\tilde{W}}^T.$ The matrices $R_{\\tilde{V}}$ and $R_{\\tilde{W}}$ are of size ${\\tilde{k}}\\times {\\tilde{k}}$ .",
"Next, compute a singular value decomposition (SVD) of the small matrix $R_{\\tilde{V}}R_{\\tilde{W}}^T$ : $R_{\\tilde{V}}R_{\\tilde{W}}^T=\\hat{V} \\text{diag}(\\sigma _1,\\ldots ,\\sigma _{\\tilde{k}})\\hat{W}^T$ where $\\sigma _1,\\ldots ,\\sigma _{\\tilde{k}}$ are the singular values in descending order.",
"We can truncate to a rank-$k$ matrix where $k$ is specified using either a relative criterion for singular values, $\\sqrt{\\sigma _{k+1}^2+\\cdots +\\sigma _{{\\tilde{k}}}^2}\\le \\epsilon _{\\text{rel}}\\sqrt{\\sigma _{1}^2+\\cdots +\\sigma _{{\\tilde{k}}}^2}$ or an absolute one, $ k =\\text{max}\\lbrace k\\mid \\sigma _k\\ge \\epsilon _{\\text{abs}}\\rbrace .$ Then the truncated matrices can be written in MATLAB notation as $V=Q_{\\tilde{V}}\\hat{V}(:,1:k),\\quad W=Q_{\\tilde{W}}\\hat{W}(:,1:k)\\text{diag}(\\sigma _1,\\ldots ,\\sigma _{k}).$ Note that the low-rank matrices ${X}$ obtained from eq:reltol and eq:abstol satisfy $ \\Vert X-\\tilde{X} \\Vert _F \\le \\epsilon _\\text{rel}\\Vert {\\tilde{X}}\\Vert _F$ and $ \\Vert X-\\tilde{X} \\Vert _F \\le \\epsilon _\\text{abs}\\sqrt{\\tilde{k}-k},$ respectively.",
"The right-hand side of eq:abstolequiv is bounded by $\\sqrt{N_\\xi }\\epsilon _\\text{abs}$ since in general $N_\\xi <N_x$ .",
"The total cost of this computation is $O((N_x+N_\\xi +{\\tilde{k}}){\\tilde{k}}^2)$ .",
"In the case where $\\tilde{k}$ becomes larger than $N_\\xi $ , we compute instead a direct SVD for $\\tilde{X}$ , which requires a matrix-matrix product to compute $\\tilde{X}$ and an SVD, with smaller total cost $O(N_x N_\\xi \\tilde{k} + N_xN_\\xi ^2)$ ."
],
[
"Low-rank multigrid",
"The multigrid solver with low-rank truncation is given in alg:low-rank.",
"It uses truncation operators $\\mathcal {T}_\\text{rel}$ and $\\mathcal {T}_\\text{abs}$ , which are defined using a relative and an absolute criterion, respectively.",
"In each iteration, one multigrid cycle (Vcycle) is applied to the residual equation.",
"Since the overall magnitudes of the singular values of the correction matrix $C^{(i)}$ decrease as $U^{(i)}$ converges to the exact solution (see fig:decay2 for example), it is suitable to use a relative truncation tolerance $\\epsilon _\\text{rel}$ inside the Vcycle function.",
"In the smoothing function (Smooth), the iterate is truncated after each smoothing step using a relative criterion $ \\Vert \\mathcal {T}_\\text{rel$_1$} (U) - U \\Vert _F \\le \\epsilon _\\text{rel} \\Vert F^h-\\mathcal {A}^h(U^h_0) \\Vert _F$ where $A^h$ , $U^h_0$ , and $F^h$ are arguments of the Vcycle function, and $F^h-\\mathcal {A}^h(U^h_0)$ is the residual at the beginning of each V-cycle.",
"In Line REF , the residual is truncated via a more stringent relative criterion $ \\Vert \\mathcal {T}_\\text{rel$_2$} (R^h) - R^h \\Vert _F \\le \\epsilon _\\text{rel}h \\Vert F^h-\\mathcal {A}^h(U^h_0) \\Vert _F$ where $h$ is the mesh size.",
"In the main while loop, an absolute truncation criterion eq:abstol with tolerance $\\epsilon _\\text{abs}$ is used and all the singular values of $U^{(i)}$ below $\\epsilon _\\text{abs}$ are dropped.",
"The algorithm is terminated either when the largest singular value of the residual matrix $R^{(i)}$ is smaller than $\\epsilon _\\text{abs}$ or when the multigrid solution reaches the specified accuracy.",
"Multigrid with low-rank truncation textsc : Initinitialization Funcfunctionend VcycleVcycle SmoothSmooth : $i=0$ , $R^{(0)}=F$ in low-rank format, $r_0=\\Vert F\\Vert _F$ $r>tol*r_0$ $\\&$ $i\\le maxit$ $C^{(i)}=$ $A,0,R^{(i)}$ $\\tilde{U}^{(i+1)}=U^{(i)}+C^{(i)},$ 6cm$U^{(i+1)}=\\mathcal {T}_{\\text{abs}}(\\tilde{U}^{(i+1)})$ $\\tilde{R}^{(i+1)} = F-\\mathcal {A}(U^{(i+1)}),$ 6cm$R^{(i+1)}=\\mathcal {T}_{\\text{abs}}(\\tilde{R}^{(i+1)})$ $r=\\Vert R^{(i+1)}\\Vert _F$ , $i=i+1$ $U^h=$ $A^h,U_0^h,F^h$ $h==h_0$ solve $\\mathcal {A}^h(U^h)=F^h$ directly $U^h=$ $A^h,U_0^h,F^h$ $\\tilde{R}^h=F^h-\\mathcal {A}^h(U^h),$ 5.5cm$R^h=\\mathcal {T}_\\text{rel$_2$}(\\tilde{R}^h)$ ${R}^{2h}={R}(R^h)$ ${C}^{2h}=$ ${A}^{2h},0,R^{2h}$ $U^{h}=U^h+{P}(C^{2h})$ $U^h=$ $A^h,U^h,F^h$ $U=$ $A,U,F$ $\\nu $ steps $\\tilde{U} = U+{S}(F-\\mathcal {A}(U))$ , 5.5cm$U=\\mathcal {T}_{\\text{rel$_1$}}(\\tilde{U})$ Figure: Singular values of the correction matrix C (i) C^{(i)} at multigrid iteration i=0,1,...,5i=0,1,\\ldots ,5 (without truncation) for the benchmark problem in sec:numerical1 (with σ=0.01\\sigma =0.01, b=5b=5, h=2 -6 h=2^{-6}, m=8m=8, p=3p=3).Note that the post-smoothing is not explicitly required in alg:mg,alg:low-rank, and we include it just for sake of completeness.",
"Also, in alg:low-rank, if the smoothing operator has the form ${S}=S_1\\otimes S_2$ , then for any matrix with a low-rank factorization $X=VW^T$ , application of the smoothing operator gives $ {S}(X)={S}(VW^T)=(S_2V)(S_1W)^T,$ so that the result is again the outer product of two matrices of the same low rank.",
"The prolongation and restriction operators eq:gridtransfer are implemented in a similar manner.",
"Thus, the smoothing and grid-transfer operators do not affect the ranks of matricized quantities in alg:low-rank."
],
[
"Convergence analysis",
"In order to show that alg:low-rank is convergent, we need to know how truncation affects the contraction of error.",
"Consider the case of a two-grid algorithm for the linear system $A\\mathbf {u}=\\mathbf {f}$ , where the coarse-grid solve is exact and no post-smoothing is done.",
"Let $\\bar{A}$ be the coefficient matrix on the coarse grid, let $\\mathbf {e}^{(i)}=\\mathbf {u}-\\mathbf {u}^{(i)}$ be the error associated with $\\mathbf {u}^{(i)}$ , and let $\\mathbf {r}^{(i)}=\\mathbf {f}-A\\mathbf {u}^{(i)}=A\\mathbf {e}^{(i)}$ be the residual.",
"It is shown in [4] that if no truncation is done, the error after a two-grid cycle becomes $\\mathbf {e}_\\text{notrunc}^{(i+1)}=(A^{-1}-{P}\\bar{A}^{-1}{R})A(I-Q^{-1}A)^\\nu \\mathbf {e}^{(i)},$ and $\\Vert \\mathbf {e}_\\text{notrunc}^{(i+1)}\\Vert _A \\le C\\eta (\\nu )\\Vert \\mathbf {e}^{(i)}\\Vert _A,$ where $\\nu $ is the number of pre-smoothing steps, $C$ is a constant, and $\\eta (\\nu )\\rightarrow 0$ as $\\nu \\rightarrow \\infty $ .",
"The proof consists of establishing the smoothing property $ \\Vert A(I-Q^{-1}A)^\\nu \\mathbf {y} \\Vert _2 \\le \\eta (\\nu )\\Vert \\mathbf {y}\\Vert _A,\\quad \\forall \\mathbf {y}\\in \\mathbb {R}^{N_xN_\\xi },$ and the approximation property $ \\Vert (A^{-1}-{P}\\bar{A}^{-1}{R})\\mathbf {y} \\Vert _A \\le C\\Vert \\mathbf {y}\\Vert _2,\\quad \\forall \\mathbf {y}\\in \\mathbb {R}^{N_xN_\\xi },$ and applying these bounds to eq:errorexact.",
"Now we derive an error bound for alg:low-rank.",
"The result is presented in two steps.",
"First, we consider the Vcycle function only; the following lemma shows the effect of the relative truncations defined in eq:reltol1,eq:reltol2.",
"Let $\\mathbf {u}^{(i+1)}=\\textsc {Vcycle}(A,\\mathbf {u}^{(i)},\\mathbf {f})$ and let $\\mathbf {e}^{(i+1)}=\\mathbf {u}-\\mathbf {u}^{(i+1)}$ be the associated error.",
"Then $\\Vert \\mathbf {e}^{(i+1)}\\Vert _A \\le {C_1}(\\nu ) \\Vert \\mathbf {e}^{(i)}\\Vert _A,$ where, for small enough $\\epsilon _\\text{rel}$ and large enough $\\nu $ , $C_1(\\nu )<1$ independent of the mesh size $h$ .",
"For $s=1,\\ldots ,\\nu $ , let $\\tilde{\\mathbf {u}}^{(i)}_s$ be the quantity computed after application of the smoothing operator at step $s$ before truncation, and let $\\mathbf {u}^{(i)}_s$ be the modification obtained from truncation by $\\mathcal {T}_{\\text{rel}_1}$ of eq:reltol1.",
"For example, $\\tilde{\\mathbf {u}}_1^{(i)} = \\mathbf {u}^{(i)}+Q^{-1}(\\mathbf {f}-A\\mathbf {u}^{(i)}),\\quad {\\mathbf {u}}^{(i)}_1=\\mathcal {T}_\\text{rel$_1$}(\\tilde{\\mathbf {u}}^{(i)}_1).$ Denote the associated error as ${\\mathbf {e}}^{(i)}_s=\\mathbf {u}-{\\mathbf {u}}^{(i)}_s$ .",
"From eq:reltol1, we have ${\\mathbf {e}}_1^{(i)} = (I-Q^{-1}A)\\mathbf {e}^{(i)} + \\delta _1^{(i)},\\quad \\text{where }\\Vert \\delta _1^{(i)}\\Vert _2\\le \\epsilon _\\text{rel} \\Vert \\mathbf {r}^{(i)}\\Vert _2.$ Similarly, after $\\nu $ smoothing steps, $\\begin{aligned}{\\mathbf {e}}_\\nu ^{(i)} &= (I-Q^{-1}A)^\\nu \\mathbf {e}^{(i)} + \\Delta _\\nu ^{(i)}\\\\&=(I-Q^{-1}A)^\\nu \\mathbf {e}^{(i)} + (I-Q^{-1}A)^{\\nu -1}\\delta _1^{(i)} + \\cdots + (I-Q^{-1}A)\\delta _{\\nu -1}^{(i)} + \\delta _\\nu ^{(i)},\\end{aligned}$ where $\\Vert \\delta _s^{(i)}\\Vert _2\\le \\epsilon _\\text{rel} \\Vert \\mathbf {r}^{(i)}\\Vert _2,\\quad s=1,\\ldots ,\\nu .$ In Line REF of alg:low-rank, the residual $\\tilde{\\mathbf {r}}^{(i)}_\\nu =A{\\mathbf {e}}_\\nu ^{(i)}$ is truncated to ${\\mathbf {r}}_\\nu ^{(i)}$ via eq:reltol2, so that $\\Vert {\\mathbf {r}}^{(i)}_\\nu - \\tilde{\\mathbf {r}}^{(i)}_\\nu \\Vert _2 \\le \\epsilon _\\text{rel} h\\Vert \\mathbf {r}^{(i)}\\Vert _2.$ Let $\\tau ^{(i)}={\\mathbf {r}}^{(i)}_\\nu - \\tilde{\\mathbf {r}}^{(i)}_\\nu $ .",
"Referring to eq:errorexact,eq:errorsmooth, we can write the error associated with $\\mathbf {u}^{(i+1)}$ as $\\begin{aligned}\\mathbf {e}^{(i+1)} & = {\\mathbf {e}}_\\nu ^{(i)} - {P}\\bar{A}^{-1}{R}{\\mathbf {r}}_\\nu ^{(i)}\\\\& = (I-{P}\\bar{A}^{-1}{R}A) {\\mathbf {e}}_\\nu ^{(i)} - {P}\\bar{A}^{-1}{R} \\tau ^{(i)}\\\\& = \\mathbf {e}^{(i+1)}_\\text{notrunc} + (A^{-1}-{P}\\bar{A}^{-1}{R})A\\Delta _\\nu ^{(i)}- {P}\\bar{A}^{-1}{R} \\tau ^{(i)}\\\\& = \\mathbf {e}^{(i+1)}_\\text{notrunc} + (A^{-1}-{P}\\bar{A}^{-1}{R})(A\\Delta _\\nu ^{(i)}+\\tau ^{(i)}) - A^{-1}\\tau ^{(i)}.\\end{aligned}$ Applying the approximation property eq:approxproperty gives $\\Vert (A^{-1}-{P}\\bar{A}^{-1}{R})(A\\Delta _\\nu ^{(i)}+\\tau ^{(i)}) \\Vert _A \\le C(\\Vert A\\Delta _\\nu ^{(i)}\\Vert _2 +\\Vert \\tau ^{(i)}\\Vert _2).$ Using the fact that for any matrix $B\\in \\mathbb {R}^{N_xN_\\xi \\times N_xN_\\xi }$ , $\\sup _{\\mathbf {y}\\ne \\mathbf {0}} \\frac{\\Vert B\\mathbf {y}\\Vert _A}{\\Vert \\mathbf {y}\\Vert _A}= \\sup _{\\mathbf {y}\\ne \\mathbf {0}} \\frac{\\Vert A^{1/2}B\\mathbf {y}\\Vert _2}{\\Vert A^{1/2}\\mathbf {y}\\Vert _2}= \\sup _{\\mathbf {z}\\ne \\mathbf {0}} \\frac{\\Vert A^{1/2}BA^{-1/2}\\mathbf {z}\\Vert _2}{\\Vert \\mathbf {z}\\Vert _2}= \\Vert A^{1/2}BA^{-1/2}\\Vert _2,$ we get $\\begin{aligned} \\Vert A(I-Q^{-1}A)^{\\nu -s}\\delta _s^{(i)}\\Vert _2 & \\le \\Vert A^{1/2}\\Vert _2\\, \\Vert (I-Q^{-1}A)^{\\nu -s}\\delta _s^{(i)}\\Vert _A\\\\& \\le \\Vert A^{1/2}\\Vert _2\\, \\Vert A^{1/2}(I-Q^{-1}A)^{\\nu -s}A^{-1/2}\\Vert _2\\, \\Vert \\delta _s^{(i)}\\Vert _A\\\\& \\le \\rho (I-Q^{-1}A)^{\\nu -s}\\Vert A^{1/2}\\Vert _2^2\\, \\Vert \\delta _s^{(i)}\\Vert _2\\\\\\end{aligned}$ where $\\rho $ is the spectral radius.",
"We have used the fact that $A^{1/2}(I-Q^{-1}A)^{\\nu -s}A^{-1/2}$ is a symmetric matrix (assuming $Q$ is symmetric).",
"Define $d_1(\\nu )=(\\rho (I-Q^{-1}A)^{\\nu -1}+\\cdots +\\rho (I-Q^{-1}A)+1) \\Vert A^{1/2}\\Vert _2^2$ .",
"Then eq:reltol1,eq:reltol2 imply that $\\begin{aligned}\\Vert A\\Delta _\\nu ^{(i)}\\Vert _2 +\\Vert \\tau ^{(i)}\\Vert _2 & \\le \\epsilon _\\text{rel}(d_1(\\nu )+h)\\Vert \\mathbf {r}^{(i)}\\Vert _2\\\\& \\le \\epsilon _\\text{rel}(d_1(\\nu )+h)\\Vert A^{1/2}\\Vert _2\\,\\Vert \\mathbf {e}^{(i)}\\Vert _A.\\end{aligned}$ On the other hand, $\\begin{aligned}\\Vert A^{-1}\\tau ^{(i)} \\Vert _A =(A^{-1}\\tau ^{(i)},\\tau ^{(i)})^{1/2} &\\le \\Vert A^{-1}\\Vert _2^{1/2}\\,\\Vert \\tau ^{(i)}\\Vert _2 \\\\& \\le \\epsilon _\\text{rel}h \\Vert A^{-1}\\Vert _2^{1/2}\\, \\Vert \\mathbf {r}^{(i)}\\Vert _2\\\\&\\le \\epsilon _\\text{rel}h\\Vert A^{-1}\\Vert _2^{1/2}\\,\\Vert A^{1/2}\\Vert _2\\,\\Vert \\mathbf {e}^{(i)}\\Vert _A.\\end{aligned}$ Combining eq:boundexact,eq:errortrunc,eq:bound1,eq:bound2,eq:bound3, we conclude that $\\Vert \\mathbf {e}^{(i+1)}\\Vert _A \\le {C_1}(\\nu ) \\Vert \\mathbf {e}^{(i)}\\Vert _A$ where ${C_1}(\\nu ) = C\\eta (\\nu ) +\\epsilon _\\text{rel} (C(d_1(\\nu )+h)+h\\Vert A^{-1}\\Vert _2^{1/2})\\Vert A^{1/2}\\Vert _2.$ Note that $\\rho (I-Q^{-1}A)<1$ , $\\Vert A\\Vert _2$ is bounded by a constant, and $\\Vert A^{-1}\\Vert _2$ is of order $O(h^{-2})$ [14].",
"Thus, for small enough $\\epsilon _\\text{rel}$ and large enough $\\nu $ , $C_1(\\nu )$ is bounded below 1 independent of $h$ .",
"Next, we adjust this argument by considering the effect of the absolute truncations in the main while loop.",
"In alg:low-rank, the Vcycle is used for the residual equation, and the updated solution $\\tilde{\\mathbf {u}}^{(i+1)}$ and residual $\\tilde{\\mathbf {r}}^{(i+1)}$ are truncated to $\\mathbf {u}^{(i+1)}$ and $\\mathbf {r}^{(i+1)}$ , respectively, using an absolute truncation criterion as in eq:abstol.",
"Thus, at the $i$ th iteration ($i>1$ ), the residual passed to the Vcycle function is in fact a perturbed residual, i.e., $\\mathbf {r}^{(i)}=\\tilde{\\mathbf {r}}^{(i)}+\\beta =A\\mathbf {e}^{(i)}+\\beta ,\\quad \\text{where } \\Vert \\beta \\Vert _2\\le \\sqrt{N_\\xi }\\epsilon _\\text{abs}.$ It follows that in the first smoothing step, $\\tilde{\\mathbf {u}}_1^{(i)} = \\mathbf {u}^{(i)}+Q^{-1}(\\mathbf {f}-A\\mathbf {u}^{(i)}+\\beta ),\\quad {\\mathbf {u}}^{(i)}_1=\\mathcal {T}_\\text{rel$_1$}(\\tilde{\\mathbf {u}}^{(i)}_1),$ and this introduces an extra term in $\\Delta _\\nu ^{(i)}$ (see eq:errorsmooth), $\\Delta _\\nu ^{(i)}= (I-Q^{-1}A)^{\\nu -1}\\delta _1^{(i)} + \\cdots + (I-Q^{-1}A)\\delta _{\\nu -1}^{(i)} + \\delta _\\nu ^{(i)} -(I-Q^{-1}A)^{\\nu -1}Q^{-1}\\beta .$ As in the derivation of eq:adelta, we have $\\Vert A(I-Q^{-1}A)^{\\nu -1}Q^{-1}\\beta \\Vert _2 \\le \\rho (I-Q^{-1}A)^{\\nu -1}\\Vert A^{1/2}\\Vert _2^2\\,\\Vert Q^{-1}\\Vert _2\\,\\Vert \\beta \\Vert _2.$ In the case of a damped Jacobi smoother (see eq:jacobismoother), $\\Vert Q^{-1}\\Vert _2$ is bounded by a constant.",
"Denote $d_2(\\nu )=\\rho (I-Q^{-1}A)^{\\nu -1}\\Vert A^{1/2}\\Vert _2^2\\,\\Vert Q^{-1}\\Vert _2$ .",
"Also note that $\\Vert \\mathbf {r}^{(i)}\\Vert _2 \\le \\Vert A^{1/2}\\Vert _2\\,\\Vert \\mathbf {e}^{(i)}\\Vert _A+\\Vert \\beta \\Vert $ .",
"Then eq:bound2,eq:bound3 are modified to $ \\begin{aligned}& \\Vert A\\Delta _\\nu ^{(i)}\\Vert _2 +\\Vert \\tau ^{(i)}\\Vert _2 \\\\& \\le \\epsilon _\\text{rel}(d_1(\\nu )+h)\\Vert \\mathbf {r}^{(i)}\\Vert _2+d_2(\\nu )\\Vert \\beta \\Vert _2\\\\& \\le \\epsilon _\\text{rel}(d_1(\\nu )+h)\\Vert A^{1/2}\\Vert _2\\,\\Vert \\mathbf {e}^{(i)}\\Vert _A + (d_2(\\nu )+ \\epsilon _\\text{rel}(d_1(\\nu )+h))\\Vert \\beta \\Vert _2,\\end{aligned}$ and $ \\Vert A^{-1}\\tau ^{(i)} \\Vert _A \\le \\epsilon _\\text{rel}h\\Vert A^{-1}\\Vert _2^{1/2}\\,\\Vert A^{1/2}\\Vert _2\\,\\Vert \\mathbf {e}^{(i)}\\Vert _A + \\epsilon _\\text{rel}h\\Vert A^{-1}\\Vert _2^{1/2}\\,\\Vert \\beta \\Vert .$ As we truncate the updated solution $\\tilde{\\mathbf {u}}^{(i+1)}$ , we have $ \\mathbf {u}^{(i+1)}=\\tilde{\\mathbf {u}}^{(i+1)} + \\gamma ,\\quad \\text{where }\\Vert \\gamma \\Vert _2\\le \\sqrt{N_\\xi }\\epsilon _\\text{abs}.$ Let $ C_2(\\nu )=Cd_2(\\nu ) +\\epsilon _\\text{rel} (C(d_1(\\nu )+h)+h\\Vert A^{-1}\\Vert _2^{1/2})+\\Vert A^{1/2}\\Vert _2.$ From eq:extra1,eq:extra2,eq:extra3,eq:extra4, we conclude with the following theorem: Let $\\mathbf {e}^{(i)}=\\mathbf {u}-\\mathbf {u}^{(i)}$ denote the error at the $i$ th iteration of alg:low-rank.",
"Then $ \\Vert \\mathbf {e}^{(i+1)}\\Vert _A \\le {C_1}(\\nu )\\Vert \\mathbf {e}^{(i)}\\Vert _A + C_2(\\nu )\\sqrt{N_\\xi }\\epsilon _\\text{abs},$ where $C_1(\\nu )<1$ for large enough $\\nu $ and small enough $\\epsilon _\\text{rel}$ , and $C_2(\\nu )$ is bounded by a constant.",
"Also, eq:boundtrunc2 implies that $\\Vert \\mathbf {e}^{(i)}\\Vert _A \\le C_1^i(\\nu ) \\Vert \\mathbf {e}^{(0)}\\Vert _A + \\frac{1-C_1^{i}(\\nu )}{1-C_1(\\nu )} C_2(\\nu )\\sqrt{N_\\xi }\\epsilon _\\text{abs},$ i.e., the $A$ -norm of the error for the low-rank multigrid solution at the $i$ th iteration is bounded by $C_1^i(\\nu ) \\Vert \\mathbf {e}^{(0)}\\Vert _A+O(\\sqrt{N_\\xi }\\epsilon _\\text{abs})$ .",
"Thus, alg:low-rank converges until the $A$ -norm of the error becomes as small as $O(\\sqrt{N_\\xi }\\epsilon _\\text{abs})$ .",
"It can be shown that the same result holds if post-smoothing is used.",
"Also, the convergence of full (recursive) multigrid with these truncation operations can be established following an inductive argument analogous to that in the deterministic case (see, e.g., [5], [8]).",
"Besides, in alg:low-rank, the truncation on $\\tilde{\\mathbf {r}}^{(i+1)}$ imposes a stopping criterion, i.e., $ \\begin{aligned}\\Vert \\tilde{\\mathbf {r}}^{(i+1)} \\Vert _2 & \\le \\Vert \\tilde{\\mathbf {r}}^{(i+1)}-\\mathbf {r}^{(i+1)}\\Vert _2 + \\Vert {\\mathbf {r}}^{(i+1)} \\Vert _2\\\\& \\le \\sqrt{N_\\xi }\\epsilon _\\text{abs} + tol*r_0.\\end{aligned}$ In sec:numerical we will vary the value of $\\epsilon _\\text{abs}$ and see how the low-rank multigrid solver works compared with alg:mg where no truncation is done.",
"It is shown in [14] that for eq:galerkin, with constant mean $c_0$ and standard deviation $\\sigma $ , $\\Vert A\\Vert _2 = \\alpha (c_0 + \\sigma C_{p+1}^\\text{max} \\sum _{l=1}^m \\sqrt{\\lambda _l} \\Vert c_l(x)\\Vert _\\infty ),$ where $C_{p+1}^\\text{max}$ is the maximal root of an orthogonal polynomial of degree $p+1$ , and $\\alpha $ is a constant independent of $h$ , $m$ , and $p$ .",
"If Legendre polynomials on the interval $[-1,1]$ are used, $C_{p+1}^\\text{max}<1$ .",
"Since both $C_1$ and $C_2$ in thm:convergence are related to $\\Vert A\\Vert _2$ , the convergence rate of alg:low-rank will depend on $m$ .",
"However, if the eigenvalues $\\lbrace \\lambda _l\\rbrace $ decay fast, this dependence is negligable.",
"If instead a relative truncation is used in the while loop so that $\\mathbf {r}^{(i+1)}=\\tilde{\\mathbf {r}}^{(i+1)}+\\beta =A\\mathbf {e}^{(i+1)}+\\beta ,\\quad \\text{where } \\Vert \\beta \\Vert _2\\le \\epsilon _\\text{rel}\\Vert \\tilde{\\mathbf {r}}^{(i+1)}\\Vert ,$ then a similar convergence result can be derived, and the algorithm stops when $\\Vert \\tilde{\\mathbf {r}}^{(i+1)} \\Vert _2 \\le \\frac{tol*r}{1-\\epsilon _\\text{rel}}.$ However, the relative truncation in general results in a larger rank for $\\mathbf {r}^{(i)}$ , and the improvement in efficiency will be less significant."
],
[
"Numerical experiments",
"Consider the benchmark problem with a two-dimensional spatial domain $D=(-1,1)^2$ and constant source term $f=1$ .",
"We look at two different forms for the covariance function $r(x,y)$ of the diffusion coefficient $c(x,\\omega ).$"
],
[
"Exponential covariance",
"The exponential covariance function takes the form $ r(x,y)=\\sigma ^2\\text{exp}\\left(-\\frac{1}{b} \\Vert x-y\\Vert _1\\right).$ This is a convenient choice because there are known analytic solutions for the eigenpair ($\\lambda _l$ ,$c_l(x)$ ) [7].",
"In the KL expansion, take $c_0(x)=1$ and $\\lbrace \\xi _l\\rbrace _{l=1}^m$ independent and uniformly distributed on $[-1,1]$ : $c(x,\\omega ) = c_0(x) + \\sqrt{3}\\sum _{l=1}^m \\sqrt{\\lambda _l}c_l(x)\\xi _l(\\omega ).$ Then $\\sqrt{3}\\xi _l$ has zero mean and unit variance, and Legendre polynomials are used as basis functions for the stochastic space.",
"The correlation length $b$ affects the decay of $\\lbrace \\lambda _l\\rbrace $ in the KL expansion.",
"The number of random variables $m$ is chosen so that $ \\left(\\sum _{l=1}^m\\lambda _l\\right)\\Big /\\left(\\sum _{l=1}^M\\lambda _l\\right)\\ge 95\\%.$ Here $M$ is a large number which we set as 1000.",
"We now examine the performance of the multigrid solver with low-rank truncation.",
"We employ a damped Jacobi smoother, with $ Q=\\frac{1}{\\omega }D,\\quad D=\\text{diag}(A)=I\\otimes \\text{diag}(K_0),$ and apply three smoothing steps ($\\nu =3$ ) in the Smooth function.",
"Set the multigrid $tol=10^{-6}$ .",
"As shown in eq:stopping, the relative residual $\\Vert F-\\mathcal {A}(U^{(i)})\\Vert _F/\\Vert F\\Vert _F$ for the solution $U^{(i)}$ produced in alg:low-rank is related to the value of the truncation tolerance $\\epsilon _\\text{abs}$ .",
"In all the experiments, we also run the multigrid solver without truncation to reach a relative residual that is closest to what we get from the low-rank multigrid solver.",
"We fix the relative truncation tolerance $\\epsilon _\\text{rel}$ as $10^{-2}$ .",
"(The truncation criteria in eq:reltol1,eq:reltol2 are needed for the analysis.",
"In practice we found the performance with the relative criterion in eq:reltolequiv to be essentially the same as the results shown in this section.)",
"The numerical results, i.e., the rank of multigrid solution, the number of iterations, and the elapsed time (in seconds) for solving the Galerkin system, are given in table:n,table:m,table:sigma.",
"In all the tables, the 3rd and 4th columns are the results of low-rank multigrid with different values of truncation tolerance $\\epsilon _\\text{abs}$ , and for comparison the last two columns show the results for the multigrid solver without truncation.",
"The Galerkin systems are generated from the Incompressible Flow and Iterative Solver Software (IFISS, [16]).",
"All computations are done in MATLAB 9.1.0 (R2016b) on a MacBook with 1.6 GHz Intel Core i5 and 4 GB SDRAM.",
"table:n shows the performance of the multigrid solver for various mesh sizes $h$ , or spatial degrees of freedom $N_x$ , with other parameters fixed.",
"The 3rd and 5th columns show that multigrid with low-rank truncation uses less time than the standard multigrid solver.",
"This is especially true when $N_x$ is large: for $h=2^{-8}$ , $N_x=261121$ , low-rank approximation reduces the computing time from 2857s to 370s.",
"The improvement is much more significant (see the 4th and 6th columns) if the problem does not require very high accuracy for the solution.",
"table:m shows the results for various degrees of freedom $N_\\xi $ in the stochastic space.",
"The multigrid solver with absolute truncation tolerance $10^{-6}$ is more efficient compared with no truncation in all cases and uses only about half the time.",
"The 4th and 6th columns indicate that the decrease in computing time by low-rank truncation is more obvious with the larger tolerance $10^{-4}$ .",
"Table: Performance of multigrid solver with ϵ abs =10 -6 \\epsilon _\\text{abs}=10^{-6}, 10 -4 10^{-4}, and no truncation for various N x =(2/h-1) 2 N_x=(2/h-1)^2.",
"Exponential covariance, σ=0.01\\sigma =0.01, b=4b=4, m=11m=11, p=3p=3, N ξ =364N_\\xi =364.Table: Performance of multigrid solver with ϵ abs =10 -6 \\epsilon _\\text{abs}=10^{-6}, 10 -4 10^{-4}, and no truncation for various N ξ =(m+p)!/(m!p!",
")N_\\xi =(m+p)!/(m!p!).",
"Exponential covariance, σ=0.01\\sigma =0.01, h=2 -6 h=2^{-6}, p=3p=3, N x =16129N_x=16129.We have observed that when the standard deviation $\\sigma $ in the covariance function (REF ) is smaller, the singular values of the solution matrix $U$ decay faster (see fig:decay1), and it is more suitable for low-rank approximation.",
"This is also shown in the numerical results.",
"In the previous cases, we fixed $\\sigma $ as 0.01.",
"In table:sigma, the advantage of low-rank multigrid is clearer for a smaller $\\sigma $ , and the solution is well approximated by a matrix of smaller rank.",
"On the other hand, as the value of $\\sigma $ increases, the singular values of the matricized solution, as well as the matricized iterates, decay more slowly and the same truncation criterion gives higher-rank objects.",
"Thus, the total time for solving the system and the time spent on truncation will also increase.",
"Another observation from the above numerical experiments is that the iteration counts are largely unaffected by truncation.",
"In alg:low-rank, similar numbers of iterations are required to reach a comparable accuracy as in the cases with no truncation.",
"Table: Performance of multigrid solver with ϵ abs =10 -6 \\epsilon _\\text{abs}=10^{-6}, 10 -4 10^{-4}, and no truncation for various σ\\sigma .",
"Time spent on truncation is given in parentheses.",
"Exponential covariance, b=4b=4, h=2 -6 h=2^{-6}, m=11m=11, p=3p=3, N x =16129N_x=16129, N ξ =364.N_\\xi =364."
],
[
"Squared exponential covariance",
"In the second example we consider covariance function $ r(x,y)=\\sigma ^2\\text{exp}\\left(-\\frac{1}{b^2} \\Vert x-y\\Vert _2^2\\right).$ The eigenpair $(\\lambda _l,c_l(x))$ is computed via a Galerkin approximation of the eigenvalue problem $\\int _D r(x,y) c_l(y)\\text{d}y=\\lambda _l c_l(x).$ Again, in the KL expansion eq:exkl, take $c_0(x)=1$ and $\\lbrace \\xi _l\\rbrace _{l=1}^m$ independent and uniformly distributed on $[-1,1]$ .",
"The eigenvalues of the squared exponential covariance eq:cov2 decay much faster than those of eq:cov, and thus fewer terms are required to satisfy eq:lambda.",
"For instance, for $b=2$ , $m=3$ will suffice.",
"table:n2 shows the performance of multigrid with low-rank truncation for various spatial degrees of freedom $N_x$ .",
"In this case, we are able to work with finer meshes since the value of $N_\\xi $ is smaller.",
"In all experiments the low-rank multigrid solver uses less time compared with no truncation.",
"Table: Performance of multigrid solver with ϵ abs =10 -6 \\epsilon _\\text{abs}=10^{-6}, 10 -4 10^{-4}, and no truncation for various N x =(2/h-1) 2 N_x=(2/h-1)^2.",
"Squared exponential covariance, σ=0.01\\sigma =0.01, b=2b=2, m=3m=3, p=3p=3, N ξ =20.N_\\xi =20."
],
[
"Conclusions",
"In this work we focused on the multigrid solver, one of the most efficient iterative solvers, for the stochastic steady-state diffusion problem.",
"We discussed how to combine the idea of low-rank approximation with multigrid to reduce computational costs.",
"We proved the convergence of the low-rank multigrid method with an analytic error bound.",
"It was shown in numerical experiments that the low-rank truncation is useful in decreasing the computing time when the variance of the random coefficient is relatively small.",
"The proposed algorithm also exhibited great advantage for problems with large number of spatial degrees of freedom."
]
] | 1612.05496 |
[
[
"Approximation and simulation of infinite-dimensional Levy processes"
],
[
"Abstract In this paper approximation methods for infinite-dimensional Levy processes, also called (time-dependent) Levy fields, are introduced.",
"For square integrable fields beyond the Gaussian case, it is no longer given that the one-dimensional distributions in the spectral representation with respect to the covariance operator are independent.",
"When simulated via a Karhunen-Loeve expansion a set of dependent but uncorrelated one-dimensional Levy processes has to be generated.",
"The dependence structure among the one-dimensional processes ensures that the resulting field exhibits the correct point-wise marginal distributions.",
"To approximate the respective (one-dimensional) Levy-measures, a numerical method, called discrete Fourier inversion, is developed.",
"For this method, $L^p$-convergence rates can be obtained and, under certain regularity assumptions, mean square and $L^p$-convergence of the approximated field is proved.",
"Further, a class of (time-dependent) Levy fields is introduced, where the point-wise marginal distributions are dependent but uncorrelated subordinated Wiener processes.",
"For this specific class one may derive point-wise marginal distributions in closed form.",
"Numerical examples, which include hyperbolic and normal-inverse Gaussian fields, demonstrate the efficiency of the approach."
],
[
"Introduction",
"Uncertainty quantification plays an increasingly important role in a wide range of problems in the Engineering Sciences and Physics.",
"Examples of sources of uncertainty are imprecise or insufficient measurements and noisy data.",
"In the underlying dynamical system this is modeled via a stochastic operator, stochastic boundary conditions and/or stochastic data.",
"As an example, to model subsurface flow more realistically the coefficients of an (essentially) elliptic equation are assumed to be stochastic.",
"A common approach in the literature is to use (spatially) correlated random fields that are built from uniform distributions or colored log-normal fields.",
"The resulting point-wise marginal distributions of the field are (shifted) normally, resp.",
"log-normally distributed.",
"Neither choice is universal enough to accommodate all possible types of porosity, especially not if fractures are incorporated (see [43]).",
"In some applications it might even be necessary that the point-wise marginal distribution of the (time-dependent) random field is a pure-jump process (see [9]).",
"Here, we denominate by point-wise marginal distributions the distributions resp.",
"processes one obtains by evaluation of the random field at a fixed spatial point.",
"On a note, these are in general the distributions that may be measured in applications.",
"In the case of a (time-dependent) Gaussian random field, the approximation and simulation via its Karhunen-Loève (KL) expansion is straightforward.",
"Almost sure and $L^p$ -convergence in terms of the decay of the eigenvalues has been shown for truncated KL-expansions in [10].",
"For infinite-dimensional Lévy processes, also called Lévy fields, the approximation may still be attempted via the KL expansions: On a separable Hilbert space $(H,(\\cdot ,\\cdot )_H)$ with orthonormal basis $(e_i,i\\in \\mathbb {N})$ , a square-integrable Lévy field $L = (L(t)\\in H,t\\ge 0)$ admits the expansion $L(t)=\\sum _{i\\in \\mathbb {N}} (L(t),e_i)_H e_i,$ The sequence $((L(\\cdot ),e_i)_H, i\\in N)$ consists of one-dimensional, real-valued Lévy processes.",
"In contrast to the case of a Gaussian field, the one-dimensional processes $((L(t),e_i)_H,t\\ge 0)$ in the spectral representation are not independent but merely uncorrelated.",
"If one were to use independent Lévy processes, the resulting field would not have the desired point-wise marginal distributions and the KL expansion would, therefore, not converge to the desired Lévy field.",
"To circumvent this issue, we approximate $L$ by truncating the series after finite number of terms and generate dependent but uncorrelated processes $((L(t),e_i)_H,t\\ge 0)$ .",
"This entails, however, the simulation of one-dimensional Lévy processes.",
"A common way to do so, is to employ the so called compound Poisson approximation (CPA) (see [3], [21], [28], [39], [41] or the references therein).",
"Mean-square convergence results for the CPA are available in some cases, but require rather strong assumptions on the underlying process.",
"In addition, the obtained convergence rates are comparably low with respect to the employed time discretization, which implies that the CPA may not be suitable to sample processes involving computationally expensive components.",
"As one of the main contributions in this paper, we develop a novel approximation method for one-dimensional Lévy processes.",
"This new approach, based on Lévy bridge laws and Fourier inversion, addresses the abovementioned problems.",
"We prove $L^p$ - and almost surely convergence of the approximation under relatively weak assumptions and derive precise error bounds.",
"We show mean-square convergence of the approximation to a given infinite-dimensional Lévy process by combining the Fourier inversion method with an appropriate truncation of the KL expansion.",
"To obtain a set of dependent but uncorrelated one-dimensional processes, we utilize multi-dimensional time-changed Brownian motions.",
"The underlying variance process is represented by a positive and increasing Lévy process, a so-called subordinator.",
"As a class of subordinated processes, we consider generalized hyperbolic (GH) Lévy processes, that are based on the generalized hyperbolic distribution and cover for example normal inverse Gaussian (NIG) and hyperbolic processes.",
"These processes are widely used in applications such as Mathematical Finance, Physics and Biology (see, for instance, [4], [9], [13], [18], [19]).",
"With its fat-tailed distribution a GH-field may also be of value in the modeling of subsurface flows (see [43]).",
"For an overview on subordinated, Hilbert space-valued Lévy processes we refer to [15], [34], where this topic is treated in a rather general setting.",
"Among other subordinated Wiener processes, the construction of an infinite-dimensional NIG process can be found in [12].",
"As a further contribution of this paper, we approximate the corresponding GH Lévy fields via truncated KL expansions with dependent but uncorrelated GH-distributed one-dimensional processes and show that the approximation converges to an infinite-dimensional GH process.",
"From a simulatory point of view this entails the generation of a certain number of one-dimensional processes with a given set of parameters.",
"Conversely, we introduce a second approach, where we derive the dependence structure of the multi-dimensional GH process to obtain admissible sets of parameters such that the one-dimensional marginal GH processes are decorrelated and follow a desired distribution.",
"Using the Fourier inversion method we are able to simulate GH fields efficiently, even if a large number of one-dimensional GH processes is necessary.",
"This article is structured as follows: Section contains preliminaries on Lévy processes taking values in Hilbert spaces and the main convergence theorem for the approximation.",
"In Section , we present a new approach for the approximation of one-dimensional Lévy processes by Lévy bridge laws and prove $L^p$ - and almost sure convergence.",
"To be able to apply the algorithm in a very general setting, we introduce an extension by using Fourier inversion techniques and show how to control the $L^p$ -error.",
"We proceed by investigating the class of GH Lévy processes and state the necessary conditions for the approximated field to have point-wise GH distributed marginals.",
"In Section , we remark on some implementational details of the algorithm and conclude with NIG- and hyperbolic fields as numerical examples."
],
[
"Preliminaries",
"Throughout this paper, we consider a time interval $=[0,T]$ , with $T>0$ , and a filtered probability space $(\\Omega ,(\\mathcal {A}_t,t\\ge 0), \\mathbb {P})$ satisfying the usual conditions.",
"Let $(H,(\\cdot ,\\cdot )_H)$ be a separable Hilbert space and $(H,\\mathcal {B}(H))$ a measurable space, where $\\mathcal {B}(H)$ denotes the Borel $\\sigma $ -algebra on $H$ .",
"A Lévy process taking values in $(H,(\\cdot ,\\cdot )_H)$ is defined as follows (see [35]): Definition 2.1 A $H$ -valued stochastic process $L=(L(t),t\\in $ is called Lévy processIn the case that $H$ is an infinite-dimensional Hilbert space, sometimes $L$ is also called Lévy field to have a clear distinction from finite-dimensional Lévy processes.",
"if $L$ has stationary and independent increments, $L(0)=0$ $\\mathbb {P}$ -almost surely and $L$ is stochastically continuous, i.e.",
"for all $\\varepsilon >0$ and $t\\in holds\\begin{equation*}\\lim \\limits _{s\\rightarrow t, s\\in \\mathbb {P}(||L(t)-L(s)||_H>\\varepsilon )=0.",
"}\\end{equation*}$ The characteristic function of a Lévy process is then given by the Lévy-Khintchine formula: $\\mathbb {E}[\\exp (i(h,L(t))_H)]=\\exp (t\\Psi _L(h)),\\quad \\text{for }h\\in H,$ where the exponent is of the form $\\Psi _L(h)=i(\\iota _H,h)_H-\\frac{1}{2}(\\Sigma _H h,h)_H+\\int _H \\exp (i(h,y)_H)-1-i(h,y)_H\\mathbf {1}_{||y||_H<1}\\nu _H(dy)$ (see [35]).",
"In Eq.",
"(REF ), $\\iota _H\\in H$ , $\\Sigma _H$ is a symmetric, non-negative and nuclear operator on $H$ and $\\nu _H:\\mathcal {B}(H)\\rightarrow [0,\\infty )$ is a non-negative, $\\sigma $ -finite measure on $\\mathcal {B}(H)$ satisfying $\\nu _H(\\lbrace 0\\rbrace )=0\\quad \\text{and}\\quad \\int _H\\min (1,||y||_H^2)\\,\\nu (dy)<\\infty .$ The triplet $(\\iota _H,\\Sigma _H,\\nu _H)$ is unique for every Lévy process $L$ and called the characteristic triplet.",
"For the special case of a one-dimensional Lévy process $\\ell =(\\ell (t),t\\in $ , the Lévy-Khintchine formula simplifies to $\\mathbb {E}[\\exp (iu\\ell (t))]=\\exp \\left(t(\\iota ui -\\frac{\\sigma ^2}{2}u^2+\\int _\\mathbb {R}\\exp (iuy)-1-iuy\\mathbf {1}_{|y|<1}d\\nu (y)) \\right),\\quad u\\in \\mathbb {R},$ where $\\iota \\in \\mathbb {R}$ , $\\sigma ^2>0$ and $\\nu $ is a ($\\sigma $ -finite) measure on $\\mathcal {B}(\\mathbb {R})$ satisfying $\\nu (\\lbrace 0\\rbrace )=0\\quad \\text{and}\\quad \\int _\\mathbb {R}\\min (1,y^2)\\nu (dy)<\\infty ,$ see for instance [8] or [40].",
"The notation $\\ell $ is introduced for finite-dimensional Lévy processes to have a clear distinction from the (possibly infinite-dimensional) Lévy process $L$ .",
"If $W$ is a $H$ -valued Lévy field with characteristic triplet $(0,\\Sigma _H,0)$ , then $W$ is called $\\Sigma _H$ -Wiener process.",
"If, further, $\\Sigma _H$ is symmetric, non-negative and nuclear (see Eq.",
"(REF )) it admits, by the Hilbert-Schmidt theorem, the spectral decomposition $\\Sigma _H \\hat{e}_i = \\hat{\\rho }_i \\hat{e}_i.$ Here, $((\\hat{\\rho }_i,\\hat{e}_i),i\\in \\mathbb {N})$ is the sequence of eigenpairs of $\\Sigma _H$ , where the eigenvalues $\\hat{\\rho }_i$ are positive with zero as their only accumulation point and the sequence $(\\hat{e}_i,i\\in \\mathbb {N})$ forms an orthonormal basis of $H$ .",
"For convenience, we assume that the sequence of eigenvalues $(\\hat{\\rho }_i,i\\in \\mathbb {N})$ is given in decaying order.",
"The $\\Sigma _H$ -Wiener process $W$ admits then a unique expansion (also called Karhunen–Loève expansion) $W(t) = \\sum _{i\\in \\mathbb {N}} \\sqrt{\\hat{\\rho }_i} \\hat{e}_i w_i(t),$ where $(w_i,i\\in \\mathbb {N})$ is a sequence of independent, one-dimensional, real-valued Brownian motions.",
"An obvious way to approximate $W$ is, therefore, given by the truncated series $W_N(t) := \\sum _{i=1}^N \\sqrt{\\hat{\\rho }_i} \\hat{e}_i w_i(t).$ It can be shown that the approximations $(W_N,N\\in \\mathbb {N})$ converge in $L^2(\\Omega ;H)$ and almost surely to the $\\Sigma _H$ -Wiener process $W$ (see for instance [11]).",
"For the approximation of general (non-continuous) processes $L$ , we aim to apply a similar approach.",
"We assume $L$ is square-integrable, as otherwise $L$ does not admit a KL expansion.",
"For series representations of cylindrical Lévy processes we refer to [1], KL expansions for white noise Lévy fields may be found in [17].",
"A $H$ -valued stochastic process $(L(t),t\\in $ is said to be square-integrable if $||L(t)||_{L^2(\\Omega ;H)}:=\\mathbb {E}(||L(t)||_H^2)<+\\infty $ for all $t\\in .Obviously, mean-square convergence can only be well-defined for processes with his property.\\begin{thm}(\\cite [Theorem 4.44]{PZ07})Let L be a square-integrable Lévy process on H. Then there exists a m\\in H and a non-negative, symmetric trace class operator Q on H such that for all h_1,h_2\\in H and s,t\\in (0,T]\\begin{itemize}\\item \\mathbb {E}((L(t),h_1)_H)=t(m,h_1)_H,\\item \\mathbb {E}((L(t)-mt,h_1)_H(L(s)-ms,h_2)_H)=\\min (t,s)(Qh_1,h_2)_H\\item \\mathbb {E}(||L(t)-mt||_H^2)=t\\: tr(Q),\\end{itemize}where tr(Q) denotes the trace of Q.",
"The operator Q is also called covariance operator of L and m is called mean.\\end{thm}Note that $ Q$ in Theorem~\\ref {thm:covQ} is not necessarily equal to the operator $ H$ from the Lévy-Khintchine formula~(\\ref {eq:LKF}).They are only equal if the measure $ H$ is zero, meaning the process $ L$ has no ``jump component^{\\prime \\prime }\\footnote {This results in L being a drifted H-valued Gaussian process.",
"}.The operator $ Q$ admits a spectral decomposition with a sequence $ ((i,ei),i$\\mathbb {N}$ )$ of orthonormal eigenpairs with non-negative eigenvalues.Thus, $ L$ has the spectral expansion\\begin{equation*}L(t)=\\sum _{i\\in \\mathbb {N}}(L(t),e_i)_He_i,\\end{equation*}where the one-dimensional Lévy processes $ ((L(t),ei)H,t$ are not independent, but merely uncorrelated (see~\\cite [Section 4.8.2]{PZ07}).For the approximation of $ L$ we employ one-dimensional Lévy processes $ (ii, i$\\mathbb {N}$ )$, so that $ ii(t)$ is equal to $ (L(t),ei)H$ in distribution for all $ t and all $i\\in \\mathbb {N}$ , and define, for $N\\in \\mathbb {N}$ , the truncated sum $L_N(t):=\\sum _{i=1}^N\\sqrt{\\rho _i} e_i\\ell _i(t).$ If the spectral basis $((\\sqrt{\\rho _i}e_i),i\\in \\mathbb {N})$ of $H$ is given, the approximation of $L$ by $L_N$ reduces to the simulation of dependent but uncorrelated one-dimensional processes $\\ell _i$ .",
"In general, the processes $\\ell _i$ have infinite activity, i.e.",
"$\\mathbb {P}$ -almost all paths of the process $(\\ell _i(t),t\\in $ have an infinite number of jumps in every compact time interval.",
"Popular examples of Lévy processes with infinite activity are normal inverse Gaussian processes or hyperbolic processes, see [18].",
"As it is not possible to simulate infinitely many jumps, we need to find a suitable approximation $\\widetilde{\\ell }_i$ of $\\ell _i$ and define $\\widetilde{L}_N(t) := \\sum _{i=1}^N\\sqrt{\\rho _i} e_i \\widetilde{\\ell }_i(t).$ In the following, we derive a condition on the approximations $\\widetilde{\\ell }_i$ that ensures convergence of $\\widetilde{L}_N$ to $L$ in $L^2(\\Omega ;H)$ uniformly on $.Throughout this paper, we construct approximations $ i$ from a skeleton of discrete realizations at fixed and equidistant points in $ .",
"To this end, we introduce, for given $n\\in \\mathbb {N}$ , a time increment $\\Delta _n:=T/2^n$ and the set $\\Xi _n := \\lbrace t_j:=j\\Delta _n, \\; j=0,\\ldots ,2^n\\rbrace $ .",
"By $\\widetilde{\\ell }_i^{(n)}$ we denote some piecewise-constant càdlàg approximation of the process $\\widetilde{\\ell }_i$ (for a construction see Section ).",
"Theorem 2.2 Let $L=(L(t),t\\in $ be a square-integrable, $H$ -valued Lévy process.",
"The covariance operator $Q$ of $L$ admits a spectral decomposition by a sequence of (orthonormal) eigenpairs $((\\rho _i,e_i),i\\in \\mathbb {N})$ .",
"Assume that, for $n\\in \\mathbb {N}$ , there exists a sequence of approximations $(\\widetilde{\\ell }_i^{(n)},i\\in \\mathbb {N})$ of the one-dimensional processes $(\\ell _i,i\\in \\mathbb {N})$ on the interval $, such that the $ L2(;$\\mathbb {R}$ )$-approximation error can be bounded by\\begin{equation}\\sup _{t\\in \\mathbb {E}(|\\ell _i(t)-\\widetilde{\\ell }_i^{(n)}(t)|^2)\\le C_\\ell \\Delta _n,}where the constant C_\\ell >0 is independent of i.If, for all i\\in \\mathbb {N}, the processes \\sqrt{\\rho }_i\\ell _i are in distribution equal to (L(\\cdot ),e_i)_H then the sequence of approximations (\\widetilde{L}_N(t), N\\in \\mathbb {N}) converges in mean-square-sense to L(t), for each t\\in , and the error is bounded by\\begin{equation*}\\sup _{t\\in \\mathbb {E}(||L(t)-\\widetilde{L}_N(t)||^2_H)^{1/2}\\le \\big (T\\sum _{i=N+1}^\\infty \\rho _i\\big )^{1/2}+\\big (C_\\ell \\Delta _n\\sum _{i=1}^N\\rho _i\\big )^{1/2}.",
"}\\end{equation*}{\\begin{xmlelement*}{proof}We may assume without loss of generality that the process L has zero mean.Using the triangle inequality, the error term \\mathbb {E}(||L(t)-\\widetilde{L}_N(t)||^2_H) can be split into\\begin{equation*}\\mathbb {E}(||L(t)-\\widetilde{L}_N(t)||^2_H)^{1/2}\\le \\mathbb {E}(||L(t)-L_N(t)||^2_H)^{1/2}+\\mathbb {E}(||L_N(t)-\\widetilde{L}_N(t)||^2_H)^{1/2}.\\end{equation*}The square-integrability of L guarantees that Q is trace class and has positive eigenvalues, i.e.",
"tr(Q)=\\sum _{i\\in \\mathbb {N}}\\rho _i<+\\infty .L(t) has covariance tQ, which yields for the first error term{\\begin{@align*}{1}{-1}\\mathbb {E}(||L(t)-L_N(t)||^2_H)&=\\mathbb {E}(||L(t)||_H^2)+\\mathbb {E}(||L_N(t)||_H^2)-2\\mathbb {E}((L(t),L_N(t))_H)\\\\&= t \\;tr(Q)+\\mathbb {E}\\Big (\\sum _{i,j=1}^N\\big ( (L(t),e_i)_He_i,(L(t),e_j)_He_j \\big )_H\\Big )\\\\&\\quad -2\\mathbb {E}\\big (\\sum _{i=1}^N(L(t),(L(t),e_i)_He_i)_H\\big )\\\\&=t\\sum _{i=1}^\\infty \\rho _i+\\sum _{i=1}^N\\mathbb {E}\\big ((L(t),e_i)_H^2\\big )-2\\sum _{i=1}^N\\mathbb {E}\\big ((L(t),e_i)_H^2\\big )\\\\&=t\\sum _{i=1}^\\infty \\rho _i-\\sum _{i=1}^N\\mathbb {E}\\big ((L(t),e_i)_H^2\\big ).\\end{@align*}}With Theorem~\\ref {thm:covQ} we obtain\\begin{equation*}\\mathbb {E}((L(t),e_i)_H^2)=t(Qe_i,e_i)_H=t\\rho _i,\\end{equation*}and hence\\begin{equation*}\\sup _{t\\in \\mathbb {E}(||L(t)-L_N(t)||^2_H)= \\sup _{t\\in t\\sum _{i=N+1}^\\infty \\rho _i = T\\sum _{i=N+1}^\\infty \\rho _i.",
"}As Q is a trace class operator, the sum on the right hand side becomes arbitrary small as N\\rightarrow \\infty .This implies that L_N converges in L^2(\\Omega ;H) uniformly on to L.}For the second error term, we derive with the assumption that \\sqrt{\\rho }_i\\ell _i \\stackrel{\\mathcal {L}}{=} (L(\\cdot ),e_i)_H for all i\\in \\mathbb {N} and Ineq.~(\\ref {eq:C_ell}){\\begin{@align*}{1}{-1}\\sup _{t\\in \\mathbb {E}(||L_N(t)-\\widetilde{L}_N(t)||^2_H)&=\\sup _{t\\in \\sum _{i,j=1}^N\\mathbb {E}\\big (\\sqrt{\\rho _i\\rho _j}(\\ell _i(t)-\\widetilde{\\ell }_i^{(n)}(t))(\\ell _j(t)-\\widetilde{\\ell }_j^{(n)}(t))(e_i,e_j)_H\\big )\\\\&=\\sum _{i=1}^N\\rho _i||e_i||_H^2\\sup _{t\\in \\mathbb {E}(|\\ell _i(t)-\\widetilde{\\ell }_i^{(n)}(t)|^2)\\le C_\\ell \\Delta _n\\sum _{i=1}^N\\rho _i,}}which proves the claim.",
"Above and for the remainder of the paper we express equality in distribution by the relation \\stackrel{\\mathcal {L}}{=}.",
"}\\end{@align*}\\begin{rem}Theorem~\\ref {thm:H_error} states that the approximation \\widetilde{L}_N converges in L^2(\\Omega ;H) to L uniformly on ,for N\\rightarrow \\infty and in the case that Ineq.~(\\ref {eq:C_ell}) holds with a constant C_\\ell in the limit \\Delta _n\\rightarrow 0.We may equilibrate both error contributions by choosing N\\in \\mathbb {N} such that\\begin{equation}T\\sum _{i=N+1}^\\infty \\rho _i \\approx C_\\ell \\Delta _n\\sum _{i=1}^N\\rho _i.\\end{equation}The sum of the eigenvalues, tr(Q), is often known a priori (for example if Q is a covariance operator of the Matérn class, see Section~\\ref {sec:num}).",
"Then, only the first N eigenvalues have to be determined until Eq.~(\\ref {eq:trunc}) is fulfilled.",
"Further, optimal values for \\Delta _n and N may be chosen for given C_\\ell and (\\rho _i,i\\in \\mathbb {N}).\\end{rem}Theorem~\\ref {thm:H_error} may be generalized in an L^p-sense (the supremum is omitted for simplicity).\\begin{cor}Let the assumptions of Theorem~\\ref {thm:H_error} be fulfilled and, for p\\ge 2, \\mathbb {E}(||L(t)||^p)<+\\infty for each t\\in ,\\sum _{i\\in \\mathbb {N}}\\rho _i^{p/2}<+\\infty and\\begin{equation*}\\mathbb {E}(|\\ell _i(t)-\\widetilde{\\ell }_i^{(n)}(t)|^p)\\le C_{p,\\ell }\\Delta _n,\\end{equation*}for some C_{p,\\ell }>0 independent of i.",
"Then, the L^p(\\Omega ;H)-error is bounded by{\\begin{@align*}{1}{-1}\\mathbb {E}(||L(t)-\\widetilde{L}_N(t)||^p_H)^{1/p}&\\le \\big (\\sum _{i>N} \\rho _i\\big )^{1/2-1/p}\\Big (\\sum _{i>N} \\rho _i^{p/2} \\mathbb {E}(|\\ell _i(t)|^p)\\Big )^{1/p}\\\\&\\quad + (\\sum _{i=1}^N \\rho _i\\big )^{1/2-1/p}\\big (C_{\\ell ,p}\\Delta _n\\sum _{i=1}^N \\rho _i^{p/2}\\Big )^{1/p}.\\end{@align*}}\\end{cor}}{\\begin{xmlelement*}{proof}The proof follows closely the one of Theorem~\\ref {thm:H_error}.",
"We split the error into\\begin{equation*}\\mathbb {E}(||L(t)-\\widetilde{L}_N(t)||^p_H)^{1/p}\\le \\mathbb {E}(||L(t)-L_N(t)||^p_H)^{1/p}+\\mathbb {E}(||L_N(t)-\\widetilde{L}_N(t)||^p_H)^{1/p}.\\end{equation*}For the first term follows{\\begin{@align*}{1}{-1}\\mathbb {E}(||L(t)-L_N(t)||_H^p)=\\mathbb {E}((||L(t)-L_N(t)||_H^2)^{p/2})=\\mathbb {E}\\Big ( \\big (\\sum _{i>N} (L(t),e_i)_H^2\\big )^{p/2}\\Big ).\\end{@align*}}Since \\mathbb {E}(||L(t)||^p)<\\infty , L_N converges to L(t) in L^p(\\Omega ;H) by the Monotone Convergence Theorem.Moreover, using (L(t),e_i)\\stackrel{\\mathcal {L}}{=}\\sqrt{\\rho _i}\\ell _i(t) and Jensen^{\\prime }s inequality we may bound the above error via{\\begin{@align*}{1}{-1}\\mathbb {E}(||L(t)-L_N(t)||_H^p)&=\\mathbb {E}\\Big ( \\big (\\sum _{i>N} \\rho _i \\ell _i(t)^2 \\big )^{p/2}\\Big )\\\\&\\le \\big (\\sum _{i>N} \\rho _i\\big )^{p/2-1}\\sum _{i>N} \\rho _i^{p/2} \\mathbb {E}(|\\ell _i(t)|^p),\\end{@align*}}where we have used that \\mathbb {E}(|(L(t),e_i)_H|^p)=\\rho _i^{p/2} \\mathbb {E}(|\\ell _i(t)|^p) and \\mathbb {E}(||L(t)||^p)<+\\infty .Compared to the case p=2 with \\mathbb {E}(\\rho _i^{p/2} |\\ell _i(t)|^p)=\\rho _i, one needs additional assumptions on the p-th moment of \\ell _i to obtain an explicit bound.In a similar fashion, the second error contribution is then bounded by{\\begin{@align*}{1}{-1}\\mathbb {E}(||L_N(t)-\\widetilde{L}_N(t)||_H^p)&=\\mathbb {E}\\Big (\\big (\\sum _{i=1}^N \\rho _i|\\ell _i(t)-\\widetilde{\\ell }_i^{(n)}(t)|^2\\big )^{p/2}\\Big )\\\\&\\le \\big (\\sum _{i=1}^N \\rho _i\\big )^{p/2-1}\\sum _{i=1}^N \\rho _i^{p/2} \\mathbb {E}(|\\ell _i(t)-\\widetilde{\\ell }_i^{(n)}(t)|^p)\\\\&\\le C_{\\ell ,p}\\Delta _n\\big (\\sum _{i=1}^N \\rho _i\\big )^{p/2-1}\\sum _{i=1}^N \\rho _i^{p/2}.\\end{@align*}}\\end{xmlelement*}}\\end{equation*}By a Borel--Cantelli-type argument almost sure convergence follows from Theorem~\\ref {thm:H_error}.\\begin{cor}Let the assumptions of Theorem~\\ref {thm:H_error} hold and the eigenvalues of Q fulfill\\sum _{i\\in \\mathbb {N}} \\rho _i (i-1)<+\\infty .If for each N\\in \\mathbb {N}, n(N)\\in \\mathbb {N} is chosen such that\\begin{equation*}\\Delta _{n(N)} \\le \\frac{T\\sum _{i>N}\\rho _i}{C_\\ell \\sum _{i=1}^N\\rho _i}, \\quad N\\in \\mathbb {N},\\end{equation*}(see Remark~\\ref {rem:trunc}) the approximated Lévy process \\widetilde{L}_N converges almost surely to L in H as N\\rightarrow \\infty , where the convergence is uniform in .\\end{cor}{\\begin{xmlelement*}{proof}By Markov^{\\prime }s inequality and Theorem~\\ref {thm:H_error}, we obtain for any \\varepsilon >0 and t\\in \\begin{equation*}\\mathbb {P}(||L(t)-\\widetilde{L}_N(t)||_H>\\varepsilon )\\le \\frac{\\mathbb {E}(||L(t)-\\widetilde{L}_N(t)||_H^2)}{\\varepsilon ^2}\\le \\frac{1}{\\varepsilon ^2} \\Big (\\big (T\\sum _{i>N}\\rho _i\\big )^{1/2}+\\big (C_\\ell \\Delta _{n(N)}\\sum _{i=1}^N\\rho _i\\big )^{1/2}\\Big )^2.\\end{equation*} With \\Delta _{n(N)} as above this yields\\begin{equation*}\\sum _{N\\in \\mathbb {N}}\\mathbb {P}(||L(t)-\\widetilde{L}_N(t)||_H>\\varepsilon )\\le \\frac{4T}{\\varepsilon ^2}\\sum _{N\\in \\mathbb {N}}\\sum _{i>N}\\rho _i=\\frac{4T}{\\varepsilon ^2}\\sum _{i>N}\\rho _i(i-1)<\\infty .\\end{equation*}The claim follows by the Borel-Cantelli Lemma and by the fact that the sum on the right hand side in the inequality is independent of t.\\end{xmlelement*}}\\end{xmlelement*}}\\end{equation}For the approximation\\begin{equation*}\\widetilde{L}_N(t)=\\sum _{i=1}^N\\sqrt{\\rho _i} e_i \\widetilde{\\ell }_i(t)\\end{equation*}of $ L$, we required that the one-dimensional Lévy processes $ (i,i=1,...,N)$ are uncorrelated but not independent.Several questions may arise regarding this truncated sum:\\begin{enumerate}\\item [1.]",
"How can we efficiently simulate suitable one-dimensional approximations \\widetilde{\\ell }_i of \\ell _i and determine the constant C_\\ell to apply Theorem~\\ref {thm:H_error}?\\item [2.]",
"Is L_N again a Lévy field for arbitrary one-dimensional processes (\\ell _i,i\\in \\mathbb {N}) and can the point-wise marginal distribution of L_N(t) for a given spectral basis ((\\sqrt{\\rho }_ie_i),i\\in \\mathbb {N}) and fixed N\\in \\mathbb {N} be determined?\\item [3.]",
"Is it possible to construct L_N in a way such that its point-wise marginal processes follow a desired distribution?\\end{enumerate}In the next chapter we address the first question and present a novel approach for the approximation of arbitrary one-dimensional Lévy processes $ i$.We derive explicit error bounds and convergence results in $ Lp(;$\\mathbb {R}$ )$, hence we are able to determine $ C$ or at least bound this constant from above.The last two questions on the distribution properties of $ LN$ are then investigated in Section~\\ref {sec:GH} for an important subclass of Lévy fields.We discuss distributional features of $ LN$ so as to use the approximation methodology developed in Section~\\ref {sec:GIG_sim} efficiently to draw samples of the field $ LN$.$ Simulation of Lévy processes by Fourier inversion The simulation of an arbitrary one-dimensional Lévy process $\\ell =(\\ell (t),t\\in $ is not straightforward, as sufficiently many discrete realizations of $\\ell $ in $ are needed and the distribution of the increment $ (t+n)-(t)$ for some small time step $ n>0$ is not explicitly known in general.A well-known and common way to simulate a Lévy process with characteristic triplet $ (,2,)$ (see Equation (\\ref {eq:LKD}))is the \\textit {compound Poisson approximation} (CPA) suggested in~\\cite {R97} and~\\cite {S03}.All jumps of the process larger than some $$\\varepsilon $ >0$ are approximated by a sum of independent compound Poisson processes and the small jumps by their expected values resp.",
"by a Brownian motion.",
"For details and convergence theorems of this method we refer to \\cite {AR01,R97,S03}.Although the CPA is applicable in a very general setting, in the sense that only the triplet $ (,2,)$ has to be known for simulation, it has several drawbacks.It is possible to show that the CPA converges under certain assumptions in distribution to a Lévy process with characteristic triplet $ (,2,)$,and even strong error rates for CPA-type approximation schemes are given, for instance in \\cite {DHP12, F11, MS07}.The derived $ Lp$-error rates are, however, rather low with respect to the time discretization, only available for $ p2$ and/or require strong assumptions on the moments of the Lévy measure $$.Furthermore, if the cumulated density function (CDF) of $$ is unknown, numerical integration with respect to $$ is necessary.Evaluating the density of $$ at sufficiently many points to obtain a good approximation might be time consuming, especially if this involves computationally expensive components (e.g.",
"Bessel functions).It is, further, a-priori not clear how to discretize the measure $$ (we refer to a discussion on this matter in~\\cite [Chapter 8]{S03}).One could choose for example equidistant or equally weighted points, but this choice might have a significant impact on the precision and the speed of the simulation, and is impossible to be assessed beforehand.The disadvantages of the CPA method motivate the development of an alternative methodology.\\\\In the following, we introduce a new sampling approach which approximates the process $$ by a refining sequence of piecewise constant càdlàg processes $ ((n),n$\\mathbb {N}$ )$.We show its asymptotic convergence in $ Lp(;$\\mathbb {R}$ )$-sense and almost surely.",
"This approximation suffers from the fact that the necessary conditional densities from which we have to sample are not available for many Lévy processes.For a given refinement parameter $ n$, we develop, therefore, an algorithm to sample an approximation $ (n)$ of $(n)$ for which the resulting error may be bounded again in $ Lp(;$\\mathbb {R}$ )$-sense.This technique is based on the assumption that the characteristic function of $$ is available in closed form, which is true for a broad class of Lévy processes.We exploit this knowledge by so-called \\emph {Fourier inversion} to draw samples of the process^{\\prime } increments over an arbitrary large time step $ n>0$.In Section~\\ref {sec:num}, we then apply the described method to simulate GH Lévy fields.$ A piecewise constant approximation of $\\ell $ Throughout this chapter, we consider a one-dimensional Lévy process $\\ell =(\\ell (t),t\\in $ with characteristic function $\\phi _\\ell :\\mathbb {R}\\rightarrow \\mathbb {C}$ .",
"For any $t\\in , we denote by $ Ft$ the CDF of $ (t)$ and by $ ft$ the corresponding density function, provided that $ ft$ exists.Note that in this case $ Ft$ and $ ft$ belong to the probability distribution with characteristic function $ ()t$.To obtain a refining scheme of approximations of $$, we introduce a sampling algorithm for $$ based on the construction of \\textit {Lévy bridges}.In our context, a Lévy bridge is the stochastic process $ ((t)|t(t1,t2))$ pinned to given realizations of the boundary values $ (t1)$ and $ (t2)$ for $ 0t1<t2T$.It has been shown, for instance in \\cite [Proposition 2.3]{HHM11}, that these bridges are Markov processes.Assuming that the density $ ft$ exists for every $ t (see also Remark REF ), the distribution of the increment $\\ell (t)-\\ell (t_1)$ conditional on $\\ell (t_2)$ is well-defined whenever $f_{t_2-t_1}(\\ell (t_2))\\in (0,+\\infty )$ .",
"Its density function is then given as $f^{t_1,t_2}_{t}(x):=\\frac{f_{t-t_1}(x)f_{t_2-t}(\\ell (t_2)-x)}{f_{t_2-t_1}(\\ell (t_2))},$ with conditional expectation $\\mathbb {E}(\\ell (t)|\\ell (t_1),\\ell (t_2))=\\frac{\\ell (t_2)-\\ell (t_1)}{t_2-t_1}(t-t_1)$ (see [23],[29]).",
"This motivates the following sampling algorithm for a piecewise constant approximation of $\\ell $ : Algorithm 3.1 Let $n\\in \\mathbb {N}$ and generate a sample of the random variable $\\mathcal {X}_{0,1}$ with density $f_T$ .",
"Set $\\mathcal {X}_{0,0}:=0$ , $i:=1$ and $\\Delta _0:=T$ .",
"[1] $i\\le n$ Define $\\Delta _i=\\frac{T}{2^i}$ .",
"$j=0,2,\\dots ,2^i$ Set $\\mathcal {X}_{i,j}=\\mathcal {X}_{i-1,j/2}$ .",
"$j=1,3,\\dots ,2^i-1$ Generate the (conditional) increment $\\mathcal {X}_{i,j}-\\mathcal {X}_{i,j-1}$ within $[\\frac{(j-1)T}{2^{(i-1)}},\\frac{jT}{2^{(i-1)}}]$ .",
"That is, sample the random variable $X:\\Omega \\rightarrow \\mathbb {R}$ with density $x\\mapsto \\frac{f_{\\Delta _i}(x)f_{\\Delta _i}(\\mathcal {X}_{i,j+1}-x)}{f_{\\Delta _{i-1}}(\\mathcal {X}_{i,j+1})}$ and set $\\mathcal {X}_{i,j}:=X+\\mathcal {X}_{i,j-1}$ $i=i+1$ Define the piecewise constant process $\\overline{\\ell }^{(n)}(t):=\\mathcal {X}_{n,2^n}\\mathbf {1}_{\\lbrace T\\rbrace }(t)+\\sum \\limits _{j=1}^{2^n}\\mathcal {X}_{n,j-1}\\mathbf {1}_{\\lbrace [(j-1)T/2^n,jT/2^n)\\rbrace }(t)$ .",
"Eventually, the sequence $(\\overline{\\ell }^{(n)},n\\in \\mathbb {N})$ of càdlàg processes admits a pointwise limit in $L^p(\\Omega ;\\mathbb {R})$ which corresponds to the process $\\ell $ : Theorem 3.2 Let $\\phi _\\ell $ be a characteristic function of an infinitely divisible probability distribution.",
"For any $t\\in , assume the probability density $ ft$ corresponding to $ ()t$ exists.Further, for $ n$\\mathbb {N}$$, let $(n)$ be the process generated by Algorithm~\\ref {algo:bridge} and $ ft$ on $ (,(At,t0),P)$.If $$\\mathbb {R}$ |x|pf1(x)dx<$ for some $ p[1,)$, then\\begin{equation*}\\lim _{n\\rightarrow \\infty }\\mathbb {E}(|\\overline{\\ell }^{(n)}(t)-\\ell (t)|^p)=0,\\end{equation*}where $$ is a Lévy process with characteristic function $$ on $ (,(At,t0),P)$.$ For any $n\\in \\mathbb {N}$ and $t\\in we have that{\\begin{@align*}{1}{-1}&\\mathbb {E}[|\\overline{\\ell }^{(n+1)}(t)-\\overline{\\ell }^{(n)}(t)|^p]\\\\=&\\mathbb {E}\\Big (\\big |\\sum \\limits _{j=1}^{2^{n+1}}\\mathcal {X}_{n+1,j-1}\\mathbf {1}_{\\lbrace [(j-1)T/2^{n+1},jT/2^{n+1})\\rbrace }(t)-\\sum \\limits _{j=1}^{2^n}\\mathcal {X}_{n,j-1}\\mathbf {1}_{\\lbrace [(j-1)T/2^n,jT/2^n)\\rbrace }(t)\\big |^p\\Big )\\\\=&\\mathbb {E}\\Big (\\big |\\sum \\limits _{j=1}^{2^n}(\\mathcal {X}_{n+1,2j-1}-\\mathcal {X}_{n+1,2j-2})\\mathbf {1}_{\\lbrace [(2j-1)T/2^{n+1},2jT/2^{n+1})\\rbrace }(t)\\big |^p\\Big )\\\\\\end{@align*}}Since the increments $ Xn+1,j+1-Xn+1,j$ are i.i.d.",
"with characteristic function $ ()T/2n+1$ by construction, this yields$$\\mathbb {E}[|\\overline{\\ell }^{(n+1)}(t)-\\overline{\\ell }^{(n)}(t)|^p]\\le C_{\\ell ,T}2^{-n-1}\\int _\\mathbb {R}|x|^pf_1(x)dx=C_{\\ell ,T,p}2^{-n-1},$$where $ C,T$ resp.",
"$ C,T,p$ are positive constants that are independent of $ n$.Hence, for any $ m,n$\\mathbb {N}$$ with $ m>n$ it follows$$\\mathbb {E}[|\\overline{\\ell }^{(m)}(t)-\\overline{\\ell }^{(n)}(t)|^p]^{1/p}\\le C_{\\ell ,T,p}^{1/p}\\sum _{i=n+1}^m2^{-i/p}=C_{\\ell ,T,p}^{1/p}\\frac{2^{-n/p}-2^{-m/p}}{2^{1/p}-1},$$meaning that $ ((n)(t),n$\\mathbb {N}$ )$ is a $ Lp(;$\\mathbb {R}$ )$-Cauchy sequence and, therefore, admits a limit.The characteristic function of $(n)(t)$ is given by $ ()t2n/TT/2nn()t$.The claim follows since the distribution with characteristic function $$ is infinitely divisible, hence the limit process $ =((t),t$ is in fact a Lévy process.$ Corollary 3.3 Under the assumptions of Theorem REF with $p=1$ , $\\overline{\\ell }^{(n)}$ converges to $\\ell $ $\\mathbb {P}$ -almost surely as $n\\rightarrow +\\infty $ , uniformly in $.$ For any $t\\in and $$\\varepsilon $ >0$, we get by Markov^{\\prime }s inequality\\begin{equation*}\\mathbb {P}(|\\overline{\\ell }^{(n)}(t)-\\ell (t)|)\\le \\frac{\\mathbb {E}(|\\overline{\\ell }^{(n)}(t)-\\ell (t)|)}{\\varepsilon }\\le \\frac{C_{\\ell ,T,p}}{\\varepsilon }\\sum _{i=n}^\\infty 2^{-i}=\\frac{C_{\\ell ,T,p}2^{-n+1}}{\\varepsilon }.\\end{equation*}The claim then follows by the Borel-Cantelli Lemma since\\begin{equation*}\\sum _{n=1}^\\infty \\mathbb {P}(|\\overline{\\ell }^{(n)}(t)-\\ell (t)|)\\le \\frac{2C_{\\ell ,T,p}}{\\varepsilon }\\sum _{n=1}^\\infty 2^{-n}<+\\infty .\\end{equation*}$ Although Algorithm REF has convenient properties in terms of convergence, it may only be applied for a small class of Lévy processes.",
"For a general Lévy process $\\ell $ , the conditional densities in Eq.",
"(REF ) will be unknown and thus simulating from this distributions is impossible.",
"A few exceptions where “bridge sampling” of Lévy processes is feasible include the inverse Gaussian ([38]) and the tempered stable process ([27]).",
"However, if we consider a fixed parameter $n$ , sampling from the bridge distributions is equivalent to the following algorithm: Algorithm 3.4 [1] For $n\\in \\mathbb {N}$ , fix $\\Delta _n,\\Xi _n$ as in Section and generate $2^n$ i.i.d random variables $ X_1,\\dots ,X_{2^n}$ with density $f_{\\Delta _n}$ .",
"Set $\\ell ^{(n)}(t)=0$ if $t\\in [0,t_1)$ , $\\ell ^{(n)}(t)=\\sum _{k=1}^jX_k$ if $t\\in [t_j,t_{j+1})$ for $j=1,\\dots ,2^n-1$ and $\\ell ^{(n)}(T)=\\sum _{j=1}^{2^n} X_j$ .",
"The equivalence is in the sense that both processes are piecewise constant, càdlàg and all intermediate points follow the same conditional Lévy bridge distributions.",
"Note that $\\ell ^{(n)}$ coincides with $\\overline{\\ell }^{(n)}$ from Algorithm REF where the initial value has been chosen as $\\mathcal {X}_{0,1}=\\ell ^{(n)}(T)=\\sum _{j=1}^{2^n}X_j$ .",
"The advantage of Algorithm REF is that $2^n$ independent samples from the same distribution have to be generated, instead of $2^n$ random variables from (different) conditional distributions.",
"As we will see in the following section, sampling from the distribution with density $f_{\\Delta _n}$ may be achieved if the characteristic function $\\phi _\\ell $ is available.",
"In addition, we are still able to use the $L^p(\\Omega ;\\mathbb {R})$ error bounds from Theorem REF for a fixed $n\\in \\mathbb {N}$ .",
"Inversion of the Characteristic Function For an one-dimensional Lévy process $\\ell $ with characteristic function $\\phi _\\ell $ , the characteristic function of any increment $\\ell (t+\\Delta _n)-\\ell (t)$ can be expressed via $\\mathbb {E}[\\exp (iu(\\ell (t+\\Delta _n)-\\ell (t)))]=\\mathbb {E}[\\exp (iu(\\ell (\\Delta _n)))]=(\\phi _\\ell (u))^{\\Delta _n}$ for any time step $\\Delta _n>0$ .",
"If $F_{\\Delta _n}$ denotes again the CDF of this increment, we obtain by Fourier inversion (see [22]) $F_{\\Delta _n}(x)=\\frac{1}{2}-\\int _\\mathbb {R}\\frac{(\\phi _\\ell (u))^{\\Delta _n}}{2\\pi iu}\\exp (-iux)du.$ Using the well-known inverse transformation method (see also [2]) to sample from the CDF, allows us to reformulate Algorithm REF : Algorithm 3.5 [1] For $n\\in \\mathbb {N}$ , fix $\\Delta _n,\\Xi _n$ and generate i.i.d.",
"$U_1,\\dots ,U_{2^n}$ , where $U_j\\sim \\mathcal {U}([0,1])$ on $(\\Omega ,\\mathcal {A},\\mathbb {P})$ .",
"Determine $X_j:=\\inf \\lbrace x\\in \\mathbb {R}|F_{\\Delta _n}(x)=U_j\\rbrace $ for $j=1,\\dots ,2^n$ .",
"Set $\\ell ^{(n)}(t)=0$ if $t\\in [0,t_1)$ , $\\ell ^{(n)}(t)=\\sum _{k=1}^j X_k$ if $t\\in [t_j,t_{j+1})$ for $j=1,\\dots ,2^n-1$ and $\\ell ^{(n)}(T)=\\sum _{j=1}^{2^n}X_j$ .",
"The evaluation of $F$ is crucial and may, in general, only be done numerically.",
"To approximate the integral in Eq.",
"(REF ), we employ the discrete Fourier inversion method introduced in [24].",
"With this method the approximation error can be controlled with relatively weak assumptions on the characteristic function.",
"Hence, the resulting algorithm is applicable for a broad class of Lévy processes.",
"An alternative algorithm to approximate the CDFs of subordinating processes based on the inversion of Laplace transforms is described in [42].",
"Although this approach seems promising in terms of computational effort, here we only consider the Fourier inversion technique.",
"The latter is also applicable to Lévy processes without bounded variation and yields uniform error bounds on the approximated CDF.",
"Assumption 3.6 The distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ is continuous with finite variance and CDF $F_{\\Delta _n}$ .",
"Furthermore, there exists a constant $R>0$ and $\\eta >1$ such that $F_{\\Delta _n}(-x)\\le R|x|^{-\\eta }$ and $1-F_{\\Delta _n}(x)\\le R|x|^{-\\eta }$ for all $x>0$ .",
"there exists a constant $B > 0$ and $\\theta >0$ such that $|(\\phi _\\ell (u))^{\\Delta _n}|\\le B|\\frac{u}{2\\pi }|^{-\\theta }$ for all $u\\in \\mathbb {R}$ .",
"In case of infinite variance, we consider bounds on the density function instead: Assumption 3.7 The distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ is continuous with density $f_{\\Delta _n}$ .",
"Furthermore, there exists a constant $R>0$ and $\\eta >1$ such that $|f_{\\Delta _n}(x)|\\le R|x|^{-\\eta }$ for all $x\\in \\mathbb {R}$ .",
"there exists a constant $B > 0$ and $\\theta >0$ such that $|(\\phi _\\ell (u))^{\\Delta _n}|\\le B|\\frac{u}{2\\pi }|^{-\\theta }$ for all $u\\in \\mathbb {R}$ .",
"Remark 3.8 In the case that $\\theta >1$ , we have that $\\int _\\mathbb {R}|(\\phi _\\ell (u))^{\\Delta _n}|du\\le 2+2B\\int _1^\\infty \\left(\\frac{u}{2\\pi }\\right)^{-\\theta }<\\infty ,$ which already implies the existence of a continuous density $f_{\\Delta _n}$ in both scenarios, see for example [40].",
"Usually, $F_{\\Delta _n}$ or $f_{\\Delta _n}$ are unknown, but only the characteristic function $(\\phi _\\ell )^{\\Delta _n}$ is given.",
"To obtain $R$ and $\\eta $ , one can choose $R=(-1)^{k}\\frac{d^{2k}}{du^{2k}}((\\phi _\\ell (u))^{\\Delta _n})\\big |_{u=0}$ and $\\eta =2k$ in Ass.",
"REF , resp.",
"$R=\\frac{1}{2\\pi }\\int _\\mathbb {R}|\\frac{d^{k}}{du^{k}}((\\phi _\\ell (u))^{\\Delta _n})|du$ and $\\eta =k$ in Ass.",
"REF , where $k$ is any non-negative integer such that the derivatives exist (see [24]).",
"For example, in the first set of assumptions, the finite variance ensures that we can use $\\eta =2$ and $R$ equal to the second moment of the distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"As an approximation of $F_{\\Delta _n}$ as in Eq.",
"(REF ) we introduce the function $\\widetilde{F}_{\\Delta _n}$ given by $\\widetilde{F}_{\\Delta _n}(x):=\\sum _{k=-M/2}^{M/2}q_k\\exp (-i2\\pi kx/J),$ for $x\\in \\mathbb {R}$ , where $q_k:={\\left\\lbrace \\begin{array}{ll}1/2&\\text{for $k=0$}\\\\\\frac{1-\\cos (2\\pi \\kappa k)}{i2\\pi k}(\\phi _\\ell (-2\\pi k/J))^{\\Delta _n}&\\text{for $0<|k|<M/2$}\\\\0&\\text{for $k=M/2$}\\\\\\end{array}\\right.",
"},$ $M$ is an even integer and $\\kappa ,J>0$ are parameters which are determined below.",
"Note that $q_k=\\overline{q_{-k}}$ , where $\\overline{z}$ denotes the complex conjugate for $z\\in \\mathbb {C}$ .",
"The Hermitean symmetry also holds for the function $k\\mapsto \\exp (-i2\\pi kx/J)$ .",
"This ensures that, for every $x\\in \\mathbb {R}$ , we have $\\widetilde{F}_{\\Delta _n}(x)&=\\frac{1}{2}+\\sum _{k=1}^{M/2-1}q_k\\exp (-i2\\pi kx/J)+\\overline{q_k\\exp (-i2\\pi kx/J)}\\\\&=\\frac{1}{2}+2\\,\\text{Re}\\Big (\\sum _{k=1}^{M/2-1}q_k\\exp (-i2\\pi kx/J)\\Big )$ and hence $\\widetilde{F}_{\\Delta _n}(x)\\in \\mathbb {R}$ for any real-valued argument $x$ .",
"The last identity should be exploited during the simulation to save computational time as here only half the summation is required.",
"Lastly, we denote by $\\zeta (z,s):=\\sum _{k=0}^\\infty (k+s)^{-z}$ for $s,z\\in \\mathbb {C}$ with $\\text{Re}(s)>0$ and $\\text{Re}(z)>1$ the Hurwitz zeta function and define as in [24] $V_1(\\kappa ,\\eta )&:=(\\kappa /2)^{-\\eta }+2\\zeta (\\eta ,1-\\frac{\\kappa }{2})+\\zeta (\\eta ,1+\\frac{\\kappa }{2})+\\zeta (\\eta ,1-\\frac{3\\kappa }{2}),\\\\V_2(\\kappa ,\\eta )&:=\\frac{2^{\\eta -1}\\kappa ^{1-\\eta }}{\\eta -1}+\\frac{\\kappa }{2}\\Big (2\\zeta (\\eta ,1-\\frac{\\kappa }{2})+\\zeta (\\eta ,1+\\frac{\\kappa }{2})+\\zeta (\\eta ,1-\\frac{3\\kappa }{2})\\Big ).$ The expressions $V_1(\\kappa ,\\eta )$ and $V_2(\\kappa ,\\eta )$ establish conditions on the choice of the (not yet determined) parameter $\\kappa $ in Theorem REF .",
"For a given domain parameter $D>0$ and accuracy $\\varepsilon >0$ the approximation $\\widetilde{F}_{\\Delta _n}$ should fulfill the error bound $|\\widetilde{F}_{\\Delta _n}(x)-F_{\\Delta _n}(x)|<\\varepsilon \\quad \\text{for $x\\in [-D/2,D/2]$}.$ Once $\\kappa $ is determined, this can be achieved by choosing a sufficiently large parameter $J$ and, based on this $J$ , a sufficiently large number of summands $M$ .",
"Admissible values for $\\kappa $ , $J$ and $M$ depend on $D$ , $\\varepsilon $ and the constants in Assumption REF resp.",
"REF .",
"Theorem 3.9 ([24]) Let $D>0$ and $\\varepsilon >0$ .",
"If Assumption REF holds, choose $\\kappa $ , $J$ and $M$ such that $0<\\kappa <\\frac{2}{3}\\quad \\text{and}\\quad \\kappa ^\\eta V_1(\\kappa ,\\eta )\\le 2^{\\eta +1},$ $J\\ge \\frac{D}{\\kappa }\\quad \\text{and}\\quad J\\ge \\Big (\\frac{3RV_1(\\kappa ,\\eta )}{2\\varepsilon }\\Big )^{1/\\eta },$ and $M\\ge 2+2J\\Big (\\frac{6B}{\\varepsilon \\pi \\theta }\\Big )^{1/\\theta }.$ If Assumption REF holds, choose $\\kappa $ , $J$ and $M$ such that $0<\\kappa <\\frac{2}{3}\\quad \\text{and}\\quad \\kappa ^{\\eta -1} V_2(\\kappa ,\\eta )\\le \\frac{2^\\eta }{\\eta -1},$ $J\\ge \\frac{D}{\\kappa }\\quad \\text{and}\\quad J\\ge \\Big (\\frac{3RV_2(\\kappa ,\\eta )}{2\\varepsilon }\\Big )^{1/(\\eta -1)},$ and $M\\ge 2+2J\\Big (\\frac{6B}{\\varepsilon \\pi \\theta }\\Big )^{1/\\theta }.$ This yields, for either case, that $|F_{\\Delta _n}(x)-\\widetilde{F}_{\\Delta _n}(x)|<\\varepsilon $ for all $x\\in [-D/2,D/2]$ and it is always possible to find a $\\kappa $ that meets the given conditions.",
"Remark 3.10 In [24], by $J\\ge \\frac{2}{\\kappa }\\left(\\frac{3R}{\\varepsilon }\\right)^{1/\\eta }\\text{resp.",
"}\\quad J\\ge \\frac{2}{\\kappa }\\left(\\frac{3R}{\\varepsilon (\\eta -1)}\\right)^{1/(\\eta -1)}$ in fact stricter conditions are imposed on $J$ .",
"The proofs of Theorems 10 and 11 in [24] still give immediately a proof for Theorem REF .",
"The advantage of the bounds in Theorem REF is that they produce a smaller approximation error in the following analysis (see also Remark REF ).",
"We refer to [24] for an optimal choice of $\\kappa $ depending on $\\eta $ .",
"Once $\\kappa $ is determined, it is favorable to choose $D$ and $\\varepsilon $ in a way such that none of the parameters has a dominant effect on the resulting number of summations $M$ .",
"This is ensured if the two lower bounds on $J$ are equal, meaning for fixed $D>0$ we set $\\varepsilon =\\frac{3}{2}RV_1(\\kappa ,\\eta )\\kappa ^\\eta D^{-\\eta }\\quad \\text{resp.",
"}\\quad \\varepsilon =\\frac{3}{2}RV_2(\\kappa ,\\eta )\\kappa ^{\\eta -1}D^{-\\eta +1}$ if the first resp.",
"second set of assumptions holds.",
"Since the approximation error $|\\widetilde{F}_{\\Delta _n}(x)-F_{\\Delta _n}(x) |$ is only bounded for $x\\in [-D/2,D/2]$ , we have to modify the third step in Algorithm REF : Algorithm 3.11 [1] For $n\\in \\mathbb {N}$ , fix $\\Delta _n,\\Xi _n$ and generate i.i.d.",
"$(U_j\\sim \\mathcal {U}([0,1]),j=1,\\ldots ,2^n)$ on $(\\Omega ,\\mathcal {A},\\mathbb {P})$ .",
"Set, for $j=1,\\dots ,2^n$ $\\widetilde{X}_j={\\left\\lbrace \\begin{array}{ll}-D/2 & \\text{if $U_j<\\min \\lbrace \\widetilde{F}_{\\Delta _n}([-D/2,D/2])\\rbrace $}\\\\D/2 & \\text{if $U_j>\\max \\lbrace \\widetilde{F}_{\\Delta _n}([-D/2,D/2])\\rbrace $}\\\\\\inf \\lbrace x\\in [-D/2,D/2]\\big |\\widetilde{F}_{\\Delta _n}(x)=U_j \\rbrace &\\text{if $U_j\\in \\widetilde{F}_{\\Delta _n}([-D/2,D/2])$}\\end{array}\\right.",
"}.$ Set $\\widetilde{\\ell }^{(n)}(t)=0$ if $t\\in [0,t_1)$ , $\\widetilde{\\ell }^{(n)}(t)=\\sum _{k=1}^j\\widetilde{X}_k$ if $t\\in [t_j,t_{j+1})$ for $j=1,\\dots ,2^n-1$ and $\\widetilde{\\ell }^{(n)}(T)=\\sum _{j=1}^{2^n} \\widetilde{X}_j$ .",
"Intuitively, if we choose $D$ large and $\\varepsilon $ small enough, the atoms in the distribution of $\\widetilde{X}_i$ at $\\pm D/2$ disappear.",
"The function $\\widetilde{F}_{\\Delta _n}$ is then sufficiently close to the CDF $F_{\\Delta _n}$ , hence the generated random variables $\\widetilde{X}_i$ will have a distribution similar to $F_{\\Delta _n}$ .",
"From here on, we define $X$ as the random variable which is generated from $U\\sim \\mathcal {U}([0,1])$ by inversion of the (exact) CDF $F_{\\Delta _n}$ and $\\widetilde{X}$ as the random variable generated from $U$ by inversion of the approximated CDF $\\widetilde{F}_{\\Delta _n}$ .",
"Theorem 3.12 Let $\\widetilde{F}_{\\Delta _n}$ be the approximation of $F_{\\Delta _n}$ which is valid for parameters $D>0$ and $\\varepsilon >0$ in the sense of Theorem REF .",
"Then $\\widetilde{X}$ converges in distribution to a random variable $X$ with CDF equal to $F_{\\Delta _n}$ as $D\\rightarrow \\infty $ and $\\varepsilon \\rightarrow 0$ .",
"First, note that $\\widetilde{F}_{\\Delta _n}$ is not necessarily monotone and might admit arbitrary values outside of $[-D/2,D/2]$ , thus cannot be regarded as a CDF.",
"Since $\\widetilde{X}$ only admits values in the desired interval, we obtain probability zero for the event that $|\\widetilde{X}|>D/2$ .",
"With this in mind we construct the CDF of $\\widetilde{X}$ and show its convergence in distribution using Portmanteau's theorem.",
"We define the function $\\widehat{F}:\\mathbb {R}\\rightarrow [0,1],\\quad x\\mapsto {\\left\\lbrace \\begin{array}{ll}0 & \\text{if $x<-D/2$}\\\\\\min (1,m_D(x))\\mathbf {1}_{\\lbrace m_D(x)>0\\rbrace } & \\text{if $x\\in [-D/2,D/2]$}\\\\1 & \\text{if $x>D/2$}\\\\\\end{array}\\right.",
"},$ where $m_D(x):=\\max _{y\\in [-D/2,x]}\\widetilde{F}_{\\Delta _n}(y)$ .",
"The continuity of $\\widetilde{F}_{\\Delta _n}$ guarantees that $m_D(x)$ is well-defined for each $x\\in [-D/2,D/2]$ .",
"Clearly, $\\widehat{F}$ is monotone increasing and $\\mathbb {P}(\\widetilde{X}\\le x)=\\widehat{F}(x)$ if $|x|>D/2$ .",
"For $|x|\\le D/2$ we have that $\\begin{split}\\mathbb {P}(\\widetilde{X}\\le x)&=\\mathbb {P}(\\inf \\lbrace |y|\\le D/2\\,|\\,\\widetilde{F}_{\\Delta _n}(y)\\ge U\\rbrace \\le x)\\\\&=\\mathbb {P}(\\max \\limits _{y\\in [-D/2,x]}\\widetilde{F}_{\\Delta _n}(y)\\ge U)\\\\&=\\min (1,m_D(x))\\mathbf {1}_{\\lbrace m_D(x)>0\\rbrace }=\\widehat{F}(x),\\end{split}$ hence $\\widehat{F}$ is the CDF of $\\widetilde{X}$ .",
"With the monotonicity of $F_{\\Delta _n}$ and $|F_{\\Delta _n}-\\widetilde{F}_{\\Delta _n}|<\\varepsilon $ on $[-D/2,D/2]$ we get $\\widehat{F}(x)&=\\min (1,m_D(x))\\mathbf {1}_{\\lbrace m_D(x)>0\\rbrace }\\le \\min (1,\\max \\limits _{y\\in [-D/2,x]}\\widetilde{F}_{\\Delta _n}(y))\\\\&\\le \\min (1,\\max \\limits _{y\\in [-D/2,x]}F_{\\Delta _n}(y)+\\varepsilon )=\\min (1,F_{\\Delta _n}(x)+\\varepsilon ),$ for $x\\in [-D/2,D/2]$ and analogously $\\widehat{F}(x)\\ge \\min (1,\\max \\limits _{y\\in [-D/2,x]}F_{\\Delta _n}(y)-\\varepsilon )\\mathbf {1}_{\\lbrace \\max \\limits _{y\\in [-D/2,x]}F_{\\Delta _n}(y)-\\varepsilon >0\\rbrace }=\\max (F_{\\Delta _n}(x)-\\varepsilon ,0),$ thus $|\\widehat{F}(x)-F_{\\Delta _n}(x)|\\le \\varepsilon .$ We choose sequences $(D_k,k\\in \\mathbb {N})$ and $(\\varepsilon _m,m\\in \\mathbb {N})$ with $\\lim _{k\\rightarrow \\infty }D_k=+\\infty $ , $\\lim _{m\\rightarrow \\infty }\\varepsilon _m=0$ and denote by $\\widehat{F}_{k,m}$ the CDF of the random variables $\\widetilde{X}_{k,m}$ corresponding to each $D_k$ and $\\varepsilon _m$ .",
"For every $x\\in \\mathbb {R}$ there is some $k^*$ such that $x\\in [-D_k/2,D_k/2]$ for all $k\\ge k^*$ , hence $\\lim \\limits _{m\\rightarrow \\infty }\\lim \\limits _{k\\rightarrow \\infty }|\\widehat{F}_{k,m}(x)-F_{\\Delta _n}(x)|\\le \\lim \\limits _{m\\rightarrow \\infty }\\varepsilon _m=0$ and the claim follows by Portmanteau's theorem.",
"Remark 3.13 Before showing the convergence of $\\widetilde{X}$ to $X$ in $L^p(\\Omega ;\\mathbb {R})$ , we have to make sure that the random variables $\\widetilde{X}$ generated by Algorithm REF are actually defined on the same probability space $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ as $X$ .",
"Since $X$ represents the increment of a Lévy process $\\ell $ on $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ with CDF $F_{\\Delta _n}$ , we may define the mapping $U:=F_{ \\Delta _n}\\circ X:\\Omega \\rightarrow [0,1]$ .",
"It is then easily verified that $U$ is a $\\mathcal {U}([0,1])$ -distributed random variable.",
"For fixed parameters $D,\\varepsilon >0$ and an approximation $\\widetilde{F}_{\\Delta _n}$ of $F_{\\Delta _n}$ we define the pseudo inverse of $\\widetilde{F}_{\\Delta _n}$ (as in Algorithm REF ) as $\\widetilde{F}_{\\Delta _n}^{-1}:[0,1]\\rightarrow \\mathbb {R},\\;\\, u\\mapsto {\\left\\lbrace \\begin{array}{ll}-D/2 & \\text{if $u<\\min \\lbrace \\widetilde{F}_{\\Delta _n}([-D/2,D/2])\\rbrace $}\\\\D/2 & \\text{if $u>\\max \\lbrace \\widetilde{F}_{\\Delta _n}([-D/2,D/2])\\rbrace $}\\\\\\inf \\lbrace x\\in [-D/2,D/2]\\big |\\widetilde{F}_{\\Delta _n}(x)=u\\rbrace &\\text{if $u\\in \\widetilde{F}_{\\Delta _n}([-D/2,D/2])$}\\end{array}\\right.",
"}.$ We note that $\\widetilde{F}_{\\Delta _n}^{-1}$ is a piecewise continuous, thus measurable, mapping which implies that $\\widetilde{X}=\\widetilde{F}_{\\Delta _n}^{-1}\\circ F_{\\Delta _n}\\circ X:\\Omega \\rightarrow [-D/2,D/2]$ is a random variable on $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ .",
"Under additional, but natural, assumptions, it is possible to show stronger convergence results of the approximation for both sets of assumptions.",
"Theorem 3.14 ($L^p(\\Omega ;\\mathbb {R})$ -convergence I) Let $F_{\\Delta _n}$ be continuously differentiable on $\\mathbb {R}$ with density $f_{\\Delta _n}$ (see Remark REF ) and $(\\phi _\\ell )^{\\Delta _n}$ be bounded as in Assumption REF with $\\eta >1$ .",
"Furthermore, assume that the approximation parameters $D$ and $\\varepsilon $ fulfill $D=C\\varepsilon ^{-d}$ for $C,d>0$ .",
"If $d<\\frac{1}{p}$ , then for all $p\\in [1,\\eta )$ $\\mathbb {E}(|\\widetilde{X}-X|^p)\\rightarrow 0 \\quad \\text{as $\\varepsilon \\rightarrow 0$}.$ Let $\\varepsilon >0$ , $D=C\\varepsilon ^{-d}$ and $p\\in [1,\\eta )$ be as in the claim.",
"We split the expectation in the following way $\\mathbb {E}(|\\widetilde{X}-X|^p)=\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })+\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|\\le D/2\\rbrace }),$ and show the convergence for each term on the right hand side.",
"Recall that $\\widetilde{X}\\in [-D/2,D/2]$ by construction.",
"We obtain for the first term $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })&\\le \\int _{D/2}^\\infty |-D/2-x|^pf_{\\Delta _n}(x)dx+\\int _{-\\infty }^{-D/2}|D/2-x|^pf_{\\Delta _n}(x)dx\\\\&=\\int _{D/2}^\\infty (D/2+x)^p(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx\\\\&=\\int _{D/2}^\\infty \\int _{0}^{D/2+x}py^{p-1}dy(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx.$ Using that $(x,y)\\in (D/2,\\infty )\\times (0,D/2+x) \\Leftrightarrow (x,y)\\in (D/2,\\infty )\\times (0,D/2)\\cup (y,\\infty )\\times (D/2,\\infty ),$ we may use Fubini's theorem to exchange the order of integration and rewrite $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })&\\le \\int _{D/2}^\\infty \\int _{0}^{D/2+x}py^{p-1}dy(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx\\\\&=\\int _0^{D/2} \\int _{D/2}^\\infty (f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx py^{p-1}dy \\\\&\\quad + \\int _{D/2}^\\infty \\int _{y}^{\\infty } (f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx py^{p-1}dy\\\\&=\\int _0^{D/2} (1-F_{\\Delta _n}(D/2)+F_{\\Delta _n}(-D/2)) py^{p-1}dy \\\\&\\quad +\\int _{D/2}^\\infty (1-F_{\\Delta _n}(y)+F_{\\Delta _n}(-y)) py^{p-1}dy .\\\\$ With the bounds on $F_{\\Delta _n}$ from Assumption REF we then have $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })&\\le 2R (D/2)^{-\\eta } \\int _0^{D/2} py^{p-1}dy+2Rp\\int _{D/2}^\\infty \\frac{y^{p-1}}{y^\\eta }dy \\\\&= 2R (D/2)^{p-\\eta }+ 2Rp\\zeta (\\eta +1-p,D/2).$ Note that the Hurwitz zeta function $\\zeta $ is well-defined (as $\\eta >p$ ) and converges to 0 as $D\\rightarrow \\infty $ .",
"For the second term, consider two realizations of the random variables $X(\\omega )$ and $\\widetilde{X}(\\omega )$ for some $\\omega \\in \\Omega $ , where $|X(\\omega )|\\le D/2$ .",
"$F_{\\Delta _n}$ is continuously differentiable by assumption, hence $F_{\\Delta _n}(X(\\omega ))-F_{\\Delta _n}(\\widetilde{X}(\\omega ))=f_{\\Delta _n}(\\xi (\\omega ))(X(\\omega )-\\widetilde{X}(\\omega )),$ with $\\xi (\\omega )$ lying in between $X(\\omega )$ and $\\widetilde{X}(\\omega )$ , meaning $|\\xi (\\omega )|\\le D/2$ and $\\mathbf {1}_{\\lbrace |X(\\omega )|\\le D/2\\rbrace }(\\omega )\\le \\mathbf {1}_{\\lbrace |\\xi (\\omega )|\\le D/2\\rbrace }(\\omega ).$ For $\\widetilde{\\varepsilon }:=C^{-1}\\varepsilon ^{d+1}>0$ we split the expectation once more into $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|\\le D/2\\rbrace })\\le &\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |\\xi |\\le D/2\\rbrace })\\\\\\le &\\underbrace{\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |\\xi |\\le D/2,f(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace })}_{:=I}+\\underbrace{\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |\\xi |\\le D/2,f(\\xi )<\\widetilde{\\varepsilon }\\rbrace })}_{:=II}.$ In case that $f_{\\Delta _n}(\\xi (\\omega ))\\ge \\widetilde{\\varepsilon }$ , we can rearrange the terms to $|\\widetilde{X}(\\omega )-X(\\omega )|^p=\\frac{|F_{\\Delta _n}(\\widetilde{X}(\\omega ))-F_{\\Delta _n}(X(\\omega ))|^p}{f_{\\Delta _n}(\\xi (\\omega ))^p}.$ If $X$ and $\\widetilde{X}$ are generated by $U\\sim \\mathcal {U}([0,1])$ and $\\widehat{F}_{\\Delta _n}$ denotes again the CDF of $\\widetilde{X}$ , this yields $|\\widetilde{X}(\\omega )-X(\\omega )|^p&=\\frac{|F_{\\Delta _n}(\\widetilde{X}(\\omega ))-\\widehat{F}_{\\Delta _n}(\\widetilde{X}(\\omega ))|^p}{f_{\\Delta _n}(\\xi (\\omega ))^p}<\\frac{\\varepsilon ^p}{f_{\\Delta _n}(\\xi (\\omega ))^p},$ where we have used that $U(\\omega )=F_{\\Delta _n}(X(\\omega ))=\\widehat{F}_{\\Delta _n}(\\widetilde{X}(\\omega ))$ and $|F_{\\Delta _n}(\\widetilde{X}(\\omega ))-\\widehat{F}_{\\Delta _n}(\\widetilde{X}(\\omega ))|<\\varepsilon $ (see Theorem REF ) in the second step.",
"This gives a bound for $I$ : $ \\begin{split}I&< \\varepsilon ^p\\,\\mathbb {E}(f_{\\Delta _n}(\\xi )^{-p}\\mathbf {1}_{\\lbrace |\\xi |\\le D/2,f_{\\Delta _n}(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace })\\\\&=\\varepsilon ^p\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace }f_{\\Delta _n}(\\xi )^{1-p}d\\xi \\le \\frac{\\varepsilon ^p}{\\widetilde{\\varepsilon }^{p-1}}\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace }d\\xi .\\end{split}$ If $f_{\\Delta _n}(\\xi (\\omega ))<\\widetilde{\\varepsilon }$ , we obtain by $|\\widetilde{X}(\\omega )-X(\\omega )|\\mathbf {1}_{\\lbrace |X(\\omega )|\\le D/2\\rbrace }\\le D$ $ \\begin{split}II&\\le D^p\\,\\mathbb {E}(\\mathbf {1}_{\\lbrace |\\xi |\\le D/2,f_{\\Delta _n}(\\xi )<\\widetilde{\\varepsilon }\\rbrace })\\\\&=D^p\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )<\\widetilde{\\varepsilon }\\rbrace }f_{\\Delta _n}(\\xi )d\\xi <D^p\\,\\widetilde{\\varepsilon }\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )<\\widetilde{\\varepsilon }\\rbrace }d\\xi \\end{split}$ and hence by Eqs.",
"(REF ) and (REF ) and $\\widetilde{\\varepsilon }=C^{-1}\\varepsilon ^{1+d}$ $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|\\le D/2\\rbrace })\\le I+II< D^p\\,\\widetilde{\\varepsilon }\\Big (\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace }d\\xi +\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )<\\widetilde{\\varepsilon }\\rbrace }d\\xi \\Big )=D^{p+1}\\widetilde{\\varepsilon }.$ With the estimate for $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|> D/2\\rbrace })$ , $D=C\\varepsilon ^{-d}$ and $\\widetilde{\\varepsilon }=C^{-1}\\varepsilon ^{1+d}$ this leads to $\\mathbb {E}(|\\widetilde{X}-X|^p)&\\le 2Rp\\zeta (\\eta +1-p,D/2)+2R(D/2)^{p-\\eta }+D^{p+1}\\widetilde{\\varepsilon }\\\\&=2Rp\\zeta (\\eta +1-p,D/2)+2R(D/2)^{p-\\eta }+C^{1/d} D^{p-1/d},$ and since $0<d<\\frac{1}{p}$ and $\\eta >p$ by assumption, $\\mathbb {E}(|\\widetilde{X}-X|^p)\\rightarrow 0 \\quad \\text{as $\\varepsilon \\rightarrow 0$}.$ Remark 3.15 The relation $\\widetilde{\\varepsilon }=C^{-1}\\varepsilon ^{1+d}$ is chosen such that the factors preceding the integrals in Eqs.",
"(REF ) and (REF ) are equilibrated.",
"As only the sum of the two integrals is known a-priori, this leads to a better error estimation compared to non-equilibrated factors.",
"Theorem 3.16 ($L^p(\\Omega ;\\mathbb {R})$ -convergence II) Let $F_{\\Delta _n}$ be continuously differentiable on $\\mathbb {R}$ with density $f_{\\Delta _n}$ and $(\\phi _\\ell )^{\\Delta _n}$ be bounded as in Assumption REF with $\\eta >2$ .",
"Furthermore, assume that the approximation parameters $D$ and $\\varepsilon $ fulfill $D=C\\varepsilon ^{-d}$ for $C,d>0$ .",
"If $d<\\frac{1}{p}$ , then for all $p\\in [1,\\eta )$ $\\mathbb {E}(|\\widetilde{X}-X|^p)\\rightarrow 0 \\quad \\text{as $\\varepsilon \\rightarrow 0$}.$ Let $\\varepsilon >0$ , $D=C\\varepsilon ^{-d}$ and $p\\in [1,\\eta -1)$ .",
"Again, we split the expectation into $\\mathbb {E}(|\\widetilde{X}-X|^p)=\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })+\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|\\le D/2\\rbrace }),$ and show convergence for the first term only, as the second term can be treated analogously to Theorem REF .",
"In the same way as in Theorem REF , we may write for the first term $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })\\le \\int _{D/2}^\\infty (D/2+x)^p(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx,$ and further, by Assumption REF , follows $\\int _{D/2}^\\infty (D/2+x)^p(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx &\\le 2R\\int _{D/2}^\\infty (D/2+x)^px^{-\\eta }dx \\\\&=2^{p+1}R\\int _0^{\\infty }\\frac{(D/2+x/2)^p}{(D/2+x)^\\eta }dx\\\\&<2^{p+1}R\\zeta (\\eta -p,C\\varepsilon ^{-d}/2),$ which tends to zero as $\\varepsilon \\rightarrow 0$ , because $\\eta >p+1$ .",
"Remark 3.17 As expected, the admissible range of values for $d$ and $\\eta $ narrows as the rate of convergence $p$ increases.",
"For example, to obtain $L^2(\\Omega ;\\mathbb {R})$ -convergence, we need $d<\\frac{1}{2}$ and $\\eta >2$ in Theorem REF and $\\eta >3$ in Theorem REF .",
"Recall Remark REF , where we have concluded that optimal relations between $D$ and $\\varepsilon $ are given by $&D=\\kappa \\left(3/2\\,RV_1(\\kappa ,\\eta )\\right)^{1/\\eta }\\varepsilon ^{-1/\\eta } \\quad &&\\text{if Assumption~\\ref {ass:1} holds and}\\\\&D=\\kappa \\left(3/2\\,RV_2(\\kappa ,\\eta )\\right)^{1/(\\eta -1)}\\varepsilon ^{-1/(\\eta -1)}\\quad &&\\text{if Assumption~\\ref {ass:2} holds}.$ For $L^p(\\Omega ;\\mathbb {R})$ -convergence, we need in both cases $D=C\\varepsilon ^{-d}$ , where $C>0$ and $d\\in (0,1/p)$ .",
"Hence, we can simply use $C=\\kappa (3/2\\,RV_1(\\kappa ,\\eta ))^{1/\\eta }$ and $d=1/\\eta <1/p$ in the first scenario and $C=\\kappa (3/2\\,RV_2(\\kappa ,\\eta ))^{1/\\eta }$ and $d=1/(\\eta -1)<1/p$ for the second set of assumptions.",
"This explains the bounds on $J$ (see also Remark REF ): In Theorem REF , we obtain the expression $C^\\eta D^{p-\\eta }$ as a term of the overall error.",
"If we had used the restrictions on $J$ as in [24], we would have used $C=2(3R)^{1/\\eta }§$ instead of the choice above and this would have resulted in an error term $C^\\eta D^{p-\\eta }$ being nearly twice as large (the argumentation works analogously for Theorem REF ).",
"Example 3.18 The conditions $\\eta >p$ in Theorem REF and $\\eta >p-1$ in Theorem REF can not be relaxed as the following examples show: First, we investigate the Student's t-distribution with 3 degrees of freedom and density function $f^{t3}(x)=\\frac{\\Gamma _G(2)}{\\sqrt{3\\pi }\\Gamma _G(3/2)}\\left(1+\\frac{x^2}{3}\\right)^{-2},$ where $x\\in \\mathbb {R}$ and $\\Gamma _G(\\cdot )$ is the Gamma function.",
"As shown in [26], this distribution is infinitely divisible and has characteristic function $\\phi _{t3}(u):=\\exp (-\\sqrt{3} |u|)(\\sqrt{3} |u|+1),$ hence we can define a Lévy process $(\\ell ^{t3}(t),t\\in $ with $\\phi _{t3}(u)$ as characteristic function.",
"For simplicity we set $\\Delta _n=1$ .",
"In this case the (symmetric) distribution of the increment $\\ell ^{t3}(t+\\Delta _n)-\\ell ^{t3}(t)$ has zero mean, finite variance, and its CDF $F_1$ can be bounded for $x>0$ by $F_1(-x)=1-F_1(x)=\\int _{-\\infty }^{-x}f^{t3}(y)dy<\\frac{\\Gamma _G(2)}{\\sqrt{3\\pi }\\Gamma _G(3/2)}\\int _{-\\infty }^{-x}\\frac{3^2}{y^4}dy=\\frac{\\sqrt{3}\\Gamma _G(2)}{\\sqrt{\\pi }\\Gamma _G(3/2)}x^{-3} =:Rx^{-3}.$ Thus, this yields $\\eta =3$ .",
"The bounds for $\\phi _{t3}$ are also straightforward: $|\\phi _{t3}(u)|\\le (2\\pi )^{-1}\\max \\limits _{\\widehat{u}>0}\\exp (-\\sqrt{3} \\widehat{u})(\\sqrt{3} \\widehat{u}^2+\\widehat{u}) |\\frac{u}{2\\pi }|^{-1} =:B |\\frac{u}{2\\pi }|^{-1},$ where the maximum in $B$ is found by differentiation giving $\\widehat{u}=(1+\\sqrt{5})/(2\\sqrt{3})$ .",
"Now, all requirements for $L^3$ -convergence except $\\eta >3$ are fulfilled.",
"But the t-distribution with 3 degrees of freedom does not admit a third moment, hence we cannot have $L^3(\\Omega ;\\mathbb {R})$ -convergence although $\\eta =3$ .",
"For the second case we consider the (standard) Cauchy process with characteristic function $(\\phi _C(u))^t=\\exp (-t|u|)$ .",
"It can be shown that the increment over time $\\Delta _n>0$ is again Cauchy-distributed with density $f^C_{\\Delta _n}(x)=\\frac{\\Delta _n}{\\pi (\\Delta _n^2+x^2)}.$ This means the CDF of the increment is continuously differentiable and the bounds as in Assumption REF are easily found by $f^C_{\\Delta _n}(x)\\le (\\Delta _n/\\pi )|x|^{-2}$ for $x\\in \\mathbb {R}$ and $|(\\phi _C(u))^{\\Delta _n}|\\le (2\\pi )^{-1}\\max \\limits _{u\\in \\mathbb {R}}u\\exp (-\\Delta _n |u|)|\\frac{u}{2\\pi }|^{-1}=(2\\pi \\Delta _n)^{-1}\\exp (-1)|\\frac{u}{2\\pi }|^{-1}$ for $u\\in \\mathbb {R}$ .",
"But clearly, $L^p(\\Omega ;\\mathbb {R})$ -convergence in the sense of Theorem REF for any $p\\ge 1$ is impossible, as the Cauchy process does not have any finite moments.",
"From $L^p$ -convergence follows almost sure convergence by a Borel–Cantelli-type argument, given $\\eta $ in Assumptions REF and REF is large enough.",
"Corollary 3.19 Under the assumptions of Theorem REF , set $\\psi _1:=\\min \\left(d\\eta ,1-d\\right)$ , let $m\\in \\mathbb {N}$ and set $\\varepsilon =\\varepsilon _m=m^{-q}$ , with $q>\\psi _1^{-1}$ .",
"If $(\\widetilde{X}_m,m\\in \\mathbb {N})$ is generated based on the sequence $(\\varepsilon _m,m\\in \\mathbb {N})$ (and the corresponding $D_m=C\\varepsilon _m^{-d}$ ), then $(\\widetilde{X}_m,m\\in \\mathbb {N})$ converges to $X$ $\\mathbb {P}$ -almost surely.",
"If the assumptions of Theorem REF with $\\eta >1$ hold, we can ensure at least $L^1(\\Omega ;\\mathbb {R})$ convergence.",
"Note that the Hurwitz zeta function $\\zeta (\\eta +1-p,D/2)=\\zeta (\\eta +1-p,C/2\\varepsilon ^{-d})$ is of order $\\mathcal {O}(\\varepsilon ^{d(\\eta +1-p)})$ as $\\varepsilon \\rightarrow 0$ .",
"With Markov's inequality, $p=1$ and the given error bounds, we get that for each $\\widehat{\\varepsilon }>0$ and $m\\in \\mathbb {N}$ $\\mathbb {P}(|\\widetilde{X}_m-X|>\\widehat{\\varepsilon }\\,)\\le \\frac{\\mathbb {E}(|\\widetilde{X}_m-X|)}{\\widehat{\\varepsilon }}\\le \\frac{\\widetilde{C} }{\\widehat{\\varepsilon }}\\left(\\varepsilon _m^{d\\eta }+\\varepsilon _m^{1-d}\\right)\\le \\frac{2\\widetilde{C}}{\\widehat{\\varepsilon }}\\varepsilon _m^{\\psi _1},$ (recall that $1>\\psi _1>0$ and $\\varepsilon _m\\le 1$ ) where the constant $\\widetilde{C}>0$ depends on $R,\\eta $ and $C$ .",
"But this means $\\sum _{m=1}^\\infty \\mathbb {P}(|\\widetilde{X}_m-X|>\\widehat{\\varepsilon })\\le \\frac{2\\widetilde{C}}{\\widehat{\\varepsilon }}\\sum _{m=1}^\\infty \\varepsilon _m^{\\psi _1}= \\frac{2\\widetilde{C}}{\\widehat{\\varepsilon }}\\sum _{m=1}^\\infty m^{-q\\psi _1}<\\infty ,$ since $q\\psi _1>1$ by construction.",
"The claim then follows by the Borel-Cantelli lemma.",
"Corollary 3.20 Under the assumptions of Theorem REF , set $\\psi _2:=\\min \\left(d(\\eta -1),1-d\\right)$ , let $m\\in \\mathbb {N}$ and set $\\varepsilon =\\varepsilon _m=m^{-q}$ , with $q>\\psi _2^{-1}$ .",
"If $(\\widetilde{X}_m,m\\in \\mathbb {N})$ is generated based on the sequence $(\\varepsilon _m,m\\in \\mathbb {N})$ (and the corresponding $D_m=C\\varepsilon _m^{-d}$ ), then $(\\widetilde{X}_m,m\\in \\mathbb {N})$ converges to $X$ $\\mathbb {P}$ -almost surely.",
"We can now combine the error estimates for any increment over time $\\Delta _n>0$ with the piecewise approximation error from Algorithm REF to bound the overall error $\\ell (t)-\\widetilde{\\ell }^{(n)}(t)$ .",
"Theorem 3.21 Let $\\ell $ be a Lévy process on $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ with characteristic function $\\phi _\\ell $ , CDF $F_{t}$ and density $f_{t}$ for any $t\\in .Assume for $ n$\\mathbb {N}$$ and fixed $ n$ there are constants $ R,,B,>0$ such that either Ass~\\ref {ass:1} or Ass.~\\ref {ass:2} holds.Let $ (n)$ be the piecewise constant approximation of $$ generated by Algorithm~\\ref {algo:approx2} and the approximation $ Fn$ of $ Fn$.There are parameters $ Dn,$\\varepsilon $ n$ for $ Fn$ such that for any $ p[1,)$ resp.",
"$ p[1,-1)$ the approximation error is bounded by$$\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}\\le C_{\\ell ,T,p,R,\\eta }\\Delta _n^{1/p},\\quad t\\in $$where the constant $ C,T,p,R,>0$ only depends on the indicated parameters.$ By Theorem REF we may regard $\\ell $ as the (pointwise) $L^p(\\Omega ;\\mathbb {R})$ -limit process of the sequence $(\\overline{\\ell }^{(n)},n\\in \\mathbb {N})$ generated by Algorithm REF .",
"For fixed $n$ , we may then identify $\\overline{\\ell }^{(n)}$ with $\\ell ^{(n)}$ from Algorithm REF to obtain $\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}\\le \\mathbb {E}(|\\ell (t)-\\overline{\\ell }^{(n)}(t)|^p)^{1/p}+ \\mathbb {E}(|\\ell ^{(n)}(t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}.$ The first term is bounded by $\\mathbb {E}(|\\ell (t)-\\overline{\\ell }^{(n)}(t)|^p)^{1/p}\\le C^{1/p}_{\\ell ,T,p}\\sum _{i=n+1}^\\infty 2^{-i/p}=\\frac{C^{1/p}_{\\ell ,T,p}2^{-n/p}}{2^{1/p}-1}.$ For the treatment of the second term, we define for $j=1,\\dots ,2^n$ the random variables $X_j:=F_{\\Delta _n}^{-1}(U_j)\\stackrel{\\mathcal {L}}{=}\\ell (\\Delta _n)$ .",
"Here, $U_1,\\dots ,U_{2^n}$ is the i.i.d sequence of $\\mathcal {U}([0,1])$ random variables on $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ from Algorithm REF .",
"The increments of the approximation $\\widetilde{\\ell }^{(n)}$ are then given by $\\widetilde{X}_j:=\\widetilde{F}^{(-1)}_{\\Delta _n}(U_j)$ which yields $|\\ell ^{(n)}(t)-\\widetilde{\\ell }^{(n)}(t)|\\le \\sum _{j=1}^{2^n}|X_j-\\widetilde{X}_j|.$ The differences $(X_j-\\widetilde{X}_j, j=1,\\dots ,2^n)$ are i.i.d.",
"by construction, hence $\\mathbb {E}\\left(|\\ell ^{(n)}(t)-\\widetilde{\\ell }^{(n)}(t)|^p\\right)^{1/p}\\le \\sum _{j=1}^{2^n}\\mathbb {E}(|X_j-\\widetilde{X}_j|^p)^{1/p}\\le 2^{n}\\mathbb {E}(|X_1-\\widetilde{X}_1|^p)^{1/p}.$ Now let $\\widetilde{F}_{\\Delta _n}$ be the approximation of $F_{\\Delta _n}$ for some $\\varepsilon \\in (0,1]$ and $D=C\\varepsilon ^{-d}$ .",
"For the first set of assumptions, we apply the error estimates of Theorem REF to obtain $\\mathbb {E}\\left(|\\ell ^{(n)}(t)-\\widetilde{\\ell }^{(n)}(t)|^p\\right)^{1/p}&\\le 2^{n}\\mathbb {E}(|X_1-\\widetilde{X}_1|^p)^{1/p}\\\\&\\le 2^{n} \\big (2Rp\\zeta (\\eta +1-p,C\\varepsilon ^{-d}/2)+2R(C\\varepsilon ^{-d}/2)^{p-1/d}+C^p\\varepsilon ^{1-dp}\\big )^{1/p}\\\\&\\le 2^{n} C_{R,\\eta ,p} \\varepsilon ^{\\psi (p)/p},\\\\$ where $\\psi (p):=\\min \\left(d(\\eta +1-p),1-dp\\right)$ and $C_{R,\\eta ,p}>0$ depends on $C$ and the indicated parameters.",
"An error of order $\\Delta _n^{1/p}$ in the last inequality is then achieved by choosing $\\varepsilon =\\varepsilon _n=2^{-(np+n)/\\psi (p)}$ and $D_n=C\\varepsilon _n^{-d}$ .",
"The proof for the second set of assumptions is carried out identically with the only difference that $\\psi (p):=\\min \\left(d(\\eta -p),1-dp\\right)$ .",
"Remark 3.22 For an efficient simulation one would choose $R$ based on $(\\phi _\\ell )^{\\Delta _n}$ as in Remark REF and then $C$ based on $R$ as suggested in Remark REF .",
"Note that in this case $R=\\mathcal {O}(\\Delta _n)$ and $C=\\kappa \\left(3/2\\,RV_1(\\kappa ,\\eta )\\right)^{1/\\eta }=\\mathcal {O}(\\Delta _n^{1/\\eta })\\quad \\text{resp.",
"}\\quad C=\\kappa \\left(3/2\\,RV_2(\\kappa ,\\eta )\\right)^{1/(\\eta -1)}=\\mathcal {O}(\\Delta _n^{1/(\\eta -1)}).$ This has to be considered in the simulation of $\\widetilde{\\ell }^{(n)}$ for different $\\Delta _n$ as we point out in the setting of Ass.",
"REF : As shown in Theorem REF , the $L^p$ -error $\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)$ is bounded by $\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}&\\le 2^{n} \\Big (2Rp\\zeta (\\eta +1-p,C\\varepsilon ^{-d}/2)+2R(C\\varepsilon ^{-d}/2)^{p-1/d}+C^p\\varepsilon ^{1-dp}\\big )^{1/p}\\\\&\\quad +\\frac{C^{1/p}_{\\ell ,T,p}2^{-n/p}}{2^{1/p}-1}.$ By substituting $\\varepsilon =C^{1/d}D^{-1/d}$ , $d=1/{\\eta }$ (see Remark REF ) and $2^n={T}/{\\Delta _n}$ one obtains $ \\begin{split}\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}&\\le \\frac{(2Rp\\zeta (\\eta +1-p,D/2)+2R(D/2)^{p-\\eta }+C^\\eta D^{p-\\eta })^{1/p}}{T^{-1}\\Delta _n}\\\\&\\quad +\\frac{C^{1/p}_{\\ell ,T,p}2^{-n/p}}{2^{1/p}-1}\\end{split}$ With $R=\\mathcal {O}(\\Delta _n)$ and $C=\\mathcal {O}(\\Delta _n^{1/\\eta })$ this implies with Ineq.",
"(REF ) $\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}=\\mathcal {O}(D^{(p-\\eta )/p}\\Delta _n^{1/p-1})+\\mathcal {O}(\\Delta _n^{1/p}).$ To equilibrate both error contributions, one may choose $D:=D_n=\\Delta _n^{p/(p-\\eta )}$ in the simulation which leads to an $L^p$ -error of order $\\mathcal {O}(\\Delta _n^{1/p})$ .",
"As mentioned in the end of Section , the one-dimensional processes $(\\widetilde{\\ell }_i, i=1,\\ldots ,N)$ in the spectral decomposition are not independent, but merely uncorrelated.",
"In the next section we introduce a class of Lévy fields for which uncorrelated processes can be obtained by subordinating a multi-dimensional Brownian motion.",
"Furthermore, for the simulation of these processes the Fourier inversion method may be employed and a bound for the constant $C_\\ell $ (see Theorem REF ) can be derived.",
"Generalized hyperbolic Lévy processes Distributions which belong to the class of generalized hyperbolic distributions may be used for a wide range of applications.",
"GH distributions have been first introduced in [4] to model mass-sizes in aeolian sand (see also [5]).",
"Since then they have been successfully applied, among others, in Finance and Biology.",
"Giving a broad class the distributions are characterized by six parameters, famous representatives are the Student's t, the normal-inverse Gaussian, the hyperbolic and the variance-gamma distribution.",
"The popularity of GH processes is explained by their flexibility in modeling various characteristics of a distribution such as asymmetries or heavy tails.",
"A further advantage in our setting is, that the characteristic function is known and, therefore, the Fourier Inversion may be applied to approximate these processes.",
"This section is devoted to investigate several properties of multi-dimensional GH processes which are then used to construct an approximation of an infinite-dimensional GH field.",
"In contrast to the Gaussian case, the sum of two independent and possibly scaled GH processes is in general not again a GH process.",
"We show a possibility to approximate GH Lévy fields via Karhunen-Loève expansions in such a way that the approximated field is itself again a GH Lévy field.",
"This is essential, so as to have convergence of the approximation to a GH Lévy field in the sense of Theorem REF .",
"Furthermore, we give, for $N\\in \\mathbb {N}$ , a representation of a $N$ -dimensional GH process as a subordinated Brownian motion and show how a multi-dimensional GH process may be constructed from uncorrelated, one-dimensional GH processes with given parameters.",
"This may be exploited by the Fourier inversion algorithm in such a way that the computational expenses to simulate the approximated GH fields are virtually independent of the truncation index $N$ .",
"Assume, for $N\\in \\mathbb {N}$ , that $\\lambda \\in \\mathbb {R}$ , $\\alpha >0$ , $\\beta \\in \\mathbb {R}^N$ , $\\delta >0$ , $\\mu \\in \\mathbb {R}^N$ and $\\Gamma $ is a symmetric, positive definite (spd) $N\\times N$ -matrix with unit determinant.",
"We denote by $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ the $N$ -dimensional generalized hyperbolic distribution with probability density function $f^{GH_N}(x;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )=\\frac{\\gamma ^\\lambda \\alpha ^{N/2-\\lambda }}{(2\\pi )^{N/2}\\delta ^\\lambda K_\\lambda (\\delta \\gamma )}\\frac{K_{\\lambda -N/2}(\\alpha g(x-\\mu ))}{g(x-\\mu )^{N/2-\\lambda }}\\exp (\\beta ^{\\prime }(x-\\mu ))$ for $x\\in \\mathbb {R}^N$ , where $g(x):=\\sqrt{\\delta ^2+x^{\\prime }\\Gamma x}, \\quad \\gamma ^2:=\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta $ and $K_\\lambda (\\cdot )$ is the modified Bessel-function of the second kind with $\\lambda $ degrees of freedom.",
"The characteristic function of $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ is given by $\\begin{split}\\phi _{GH_N}(u; \\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )&:=\\exp (iu^{\\prime }\\mu )\\Big (\\frac{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta }{\\alpha ^2-(iu+\\beta )^{\\prime }\\Gamma (iu+\\beta )}\\Big )^{\\lambda /2}\\\\&\\quad \\cdot \\frac{K_\\lambda (\\delta (\\alpha ^2-(iu+\\beta )^{\\prime }\\Gamma (iu+\\beta ))^{1/2})}{K_\\lambda (\\delta \\gamma )},\\end{split}$ where $A^{\\prime }$ denotes the transpose of a matrix or vector $A$ .",
"For simplicity, we assume that the condition $\\alpha ^2>\\beta ^{\\prime }\\Gamma \\beta $ is satisfiedIf $\\alpha ^2=\\beta ^{\\prime }\\Gamma \\beta $ and $\\lambda <0$ , the distribution is still well-defined, but one has to consider the limit $\\gamma \\rightarrow 0^+$ in the Bessel functions, see [13], [39]..",
"If $N=1$ , clearly, $\\Gamma =1$ is the only possible choice for the \"matrix parameter\" $\\Gamma $ , thus we omit it in this case and denote the one-dimensional GH distribution by $GH(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu )$ .",
"Barndorff–Nielsen obtains the GH distribution in [5] as a normal variance-mean mixture of a $N$ -dimensional normal distribution and a (one-dimensional) generalized inverse Gaussian (GIG) distribution with density function $f^{GIG}(x;a,b,p)=\\frac{(b/a)^p}{2K_p(ab)}x^{p-1}\\exp (-\\frac{1}{2}(a^2x^{-1}+b^2x)),\\quad x>0$ and parameters $a,b>0$ and $p\\in \\mathbb {R}$The notation of the GIG distribution varies throughout the literature, we use the notation from [41].. To be more precise: Let $w^N(1)$ be a $N$ -dimensional standard normally distributed random vector, $\\Gamma $ a spd $N\\times N$ -structure matrix with unit determinant and $\\ell ^{GIG}(1)$ a $GIG(a,b,p)$ random variable, which is independent of $w^N(1)$ .",
"For $\\mu ,\\beta \\in \\mathbb {R}^N$ , we set $\\delta =a$ , $\\lambda =p$ , $\\alpha =\\sqrt{b^2+\\beta ^{\\prime }\\Gamma \\beta }$ and define the random variable $\\ell ^{GH_N}(1)$ as $\\ell ^{GH_N}(1):=\\mu +\\Gamma \\beta \\ell ^{GIG}(1) + \\sqrt{\\ell ^{GIG}(1)\\Gamma }w^N(1).$ Then $\\ell ^{GH_N}(1)$ is $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ -distributed, where $\\sqrt{\\Gamma }$ denotes the Cholesky decomposition of the matrix $\\Gamma $ .",
"With this in mind, one can draw samples of a GH distribution with given parameters by sampling multivariate normal and GIG-distributed random variables, as $a=\\delta >0$ and $b=\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta }>0$ is guaranteed by the conditions on the GIG parameters (this results in Eq.",
"(REF ) being fulfilled).",
"As noted in [18], for general $\\lambda \\in \\mathbb {R}$ , we cannot assume that the increments of the GH Lévy process (resp.",
"of the subordinating process) over a time length other than one follow a GH distribution (resp.",
"GIG distribution).",
"If $N=1$ , however, the (one-dimensional) GH Lévy process $\\ell ^{GH}$ has the representation $\\ell ^{GH}(t)\\stackrel{\\mathcal {L}}{=}\\mu t+\\beta \\ell ^{GIG}(t)+w(\\ell ^{GIG}(t)),\\quad \\text{for } t\\ge 0,$ where $w$ is a one-dimensional Brownian motion and $\\ell ^{GIG}$ a GIG process independent of $w$ (see [16]).",
"This result yields the following generalization: Lemma 4.1 For $N\\in \\mathbb {N}$ , the $N$ -dimensional process $\\ell ^{GH_N}=(\\ell ^{GH_N}(t),t\\in ~ $ , which is $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ -distributed, can be represented as a subordinated $N$ -dimensional Brownian motion $w^N$ via $\\ell ^{GH_N}(t)\\stackrel{\\mathcal {L}}{=}\\mu t+\\Gamma \\beta \\ell ^{GIG}(t)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(t)),$ where $(\\ell ^{GIG}(t),t\\in $ is a GIG Lévy process independent of $w^N$ and $\\sqrt{\\Gamma }$ is the Cholesky decomposition of $\\Gamma $ .",
"Since the $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ distribution may be represented as a normal variance-mean mixture (see Eq.",
"(REF )), we have, that $\\ell ^{GH_N}(1)\\stackrel{\\mathcal {L}}{=}\\mu +\\Gamma \\beta \\ell ^{GIG}(1) + \\sqrt{\\Gamma \\ell ^{GIG}(1)}w^N(1)\\stackrel{\\mathcal {L}}{=}\\mu +\\Gamma \\beta \\ell ^{GIG}(1)+ \\sqrt{\\Gamma }w^N(\\ell ^{GIG}(1))$ where $\\ell ^{GIG}(1)\\sim GIG(\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda )$ and $w^N$ is a $N$ -dimensional Brownian motion independent of $\\ell ^{GIG}(1)$ .",
"The characteristic function of the mixed density is then given by $\\phi _{GH_N}(u;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )=\\exp (iu^{\\prime }\\mu )\\mathcal {M}_{GIG}(iu^{\\prime } \\Gamma \\beta -\\frac{1}{2}u^{\\prime }\\Gamma u;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ),$ where $\\mathcal {M}_{GIG}$ denotes the moment generating function of $\\ell ^{GIG}(1)$ (see [7]).",
"The GIG distribution is infinitely divisible, thus this GIG Lévy process $\\ell ^{GIG}=(\\ell ^{GIG}(t),t\\in $ can be defined via its characteristic function for $t\\in :$$\\mathbb {E}(\\exp (iu\\ell ^{GIG}(t)))=(\\mathcal {M}_{GIG}(iu;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ))^t.$$The infinite divisibility yields further\\begin{equation*}\\begin{split}\\mathbb {E}\\big (\\exp (iu^{\\prime }\\ell ^{GH_N}(t))\\big )&=\\mathbb {E}\\big (\\exp (iu^{\\prime }\\ell ^{GH_N}(1))\\big )^t=(\\phi _{GH}(u;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma ))^t\\\\&=\\exp (iu^{\\prime }(\\mu t))(\\mathcal {M}_{GIG}(iu^{\\prime } \\Gamma \\beta -\\frac{1}{2}u^{\\prime }\\Gamma u;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ))^t.\\end{split}\\end{equation*}The expression above is the characteristic function of another normal variance-mean mixture, namely where the subordinator $ GIG$ is a GIG process with characteristic function$$\\mathbb {E}(\\exp (iu\\ell ^{GIG}(t)))=(\\mathcal {M}_{GIG}(iu;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ))^t.$$Hence, $ GHN(t)$ can be expressed as\\begin{equation*}\\ell ^{GH_N}(t)\\stackrel{\\mathcal {L}}{=}\\mu t+\\Gamma \\beta \\ell ^{GIG}(t)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(t)).\\end{equation*}$ Remark 4.2 In the special case of $\\lambda =-\\frac{1}{2}$ one obtains the normal inverse Gaussian (NIG) distribution.",
"The mixing density is, in this case, the inverse Gaussian (IG) distribution.",
"We denote the $N$ -dimensional NIG distribution by $NIG_N(\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ .",
"This is the only subclass of GH distributions which is closed under convolutions in the sense that (see [32]) $NIG_N(\\alpha ,\\beta ,\\delta _1,\\mu _1,\\Gamma )*NIG_N(\\alpha ,\\beta ,\\delta _2,\\mu _2,\\Gamma )=NIG_N(\\alpha ,\\beta ,\\delta _1+\\delta _2,\\mu _1+\\mu _2,\\Gamma ).$ For $\\lambda \\in \\mathbb {R}$ , the sum of independent GH random variables is in general not GH-distributed.",
"This implies further, that one is in general not able to derive bridge laws of these processes in closed form, meaning we need to use the algorithms introduced in Section REF for simulation.",
"As shown in [6], the GH and the GIG distribution are infinitely-divisible, thus we can define the $N$ -dimensional GH Lévy process $\\ell ^{GH_N}=(\\ell ^{GH_N}(t),t\\in $ with characteristic function $\\mathbb {E}(\\exp (iu\\ell ^{GH_N}(t))=(\\phi _{GH_N}(u;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma ))^t.$ Remark 4.3 If $\\lambda =-\\frac{1}{2}$ , the corresponding NIG Lévy process $(\\ell ^{NIG_N}(t),t\\in $ has characteristic function $\\mathbb {E}[\\exp (iu\\ell ^{NIG_N}(t))]=(\\phi _{GH_N}(u;-\\frac{1}{2},\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma ))^t=\\phi _{GH_N}(u;-\\frac{1}{2},\\alpha ,\\beta ,t\\delta ,t\\mu ,\\Gamma ).$ This is due to the fact that the characteristic function $\\phi _{IG}(u;a,b)$ of the mixing IG distribution fulfills the identity $(\\phi _{IG}(u;a,b))^t=\\phi _{IG}(u;ta,b)$ for any $t\\in and $ a,b>0$ (see \\cite {S03}).$ We consider the finite time horizon $ [0,T]$ , for $T<+\\infty $ , the probability space $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ , and a compact domain $\\mathcal {D}\\subset \\mathbb {R}^s$ for $s\\in \\mathbb {N}$ to define a GH Lévy field as a mapping $L^{GH}:\\Omega \\times \\mathbb {R},\\quad (\\omega ,x,t)\\mapsto L^{GH}(\\omega )(x)(t),$ such that for each $x\\in the point-wise marginal process\\begin{equation*}L^{GH}(\\cdot )(x)(\\cdot ):\\Omega \\times \\mathbb {R},\\quad (\\omega ,t)\\mapsto L^{GH}(\\omega )(x)(t),\\end{equation*}is a one-dimensional GH Lévy process on $ (,(At,t0),P)$ with characteristic function\\begin{equation*}\\mathbb {E}\\big (\\exp (iuL^{GH}(x)(t))\\big )=(\\phi _{GH}(u;\\lambda (x),\\alpha (x),\\beta (x),\\delta (x),\\mu (x)))^t,\\end{equation*}where the indicated parameters are given by continuous functions, i.e.",
"$ ,,C($\\mathbb {R}$ )$ and $ ,C($\\mathbb {R}$ >0)$.",
"We assume that condition (\\ref {eq:ghcond}), i.e.",
"$ (x)2>(x)2$, is fulfilled for any $ x to ensure that $L^{GH}(x)(\\cdot )$ is a well-defined GH Lévy process.",
"This, in turn, means that $L^{GH}$ takes values in the Hilbert space $H=L^2(\\mathcal {D})$ and is square integrable as $\\mathbb {E}(||L^{GH}(t)||_H^2)\\le T\\mathbb {E}(||L^{GH}(1)||_H^2)\\le T \\max \\limits _{x\\in \\mathcal {D}}\\mathbb {E}(L^{GH}(x)(1)^2)V_{\\mathcal {D}},$ where $V_{\\mathcal {D}}$ denotes the volume of $\\mathcal {D}$ .",
"The right hand side is finite since every GH distribution has finite variance (see for example [30], [41]), the parameters of the distribution of $L^{GH}(x)(1)$ depend continuously on $x$ and $\\mathcal {D}\\subset \\mathbb {R}^s$ is compact by assumption.",
"We use the Karhunen-Loève expansion from Section to obtain an approximation of a given GH Lévy field.",
"For this purpose, we consider the truncated sum $L^{GH}_N(x)(t) :=\\sum _{i=1}^{N} \\varphi _i(x)\\ell _i^{GH}(t)\\stackrel{\\mathcal {L}}{=} \\sum _{i=1}^{N} \\varphi _i(x) \\Big (\\mu _i t+\\beta _i \\ell ^{GIG}_i(t)+w_i(\\ell ^{GIG}_i(t))\\Big ),$ where $N\\in \\mathbb {N}$ and $\\varphi _i(x)=\\sqrt{\\rho _i}e_i(x)$ is the $i$ -th component of the spectral basis evaluated at the spatial point $x$ .",
"For each $i,=1,\\dots ,N$ , the processes $\\ell _i^{GH}:=(\\ell _i^{GH}(t), t\\in $ are uncorrelated but dependent $GH(\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i)$ Lévy process.",
"From Theorem REF follows that $L^{GH}_N$ converges in $L^2(\\Omega ;H)$ to $L^{GH}$ as $N\\rightarrow \\infty $ .",
"With given $\\mu _i, \\beta _i\\in \\mathbb {R}$ , we have that $\\ell ^{GH}_i(t)\\stackrel{\\mathcal {L}}{=}\\mu _i t+\\beta _i \\ell ^{GIG}_i(t)+w_i(\\ell ^{GIG}_i(t)),$ where for each $i$ , the process $(\\ell ^{GIG}_i(t), t\\in $ is a GIG Lévy process with parameters $a_i=\\delta _i, b_i=(\\alpha _i^2-\\beta _i^2)^{1/2}>0$ and $p_i=\\lambda _i\\in \\mathbb {R}$ .",
"In addition, $(w_i(t), t\\in $ is a one-dimensional Brownian motion independent of $\\ell ^{GIG}_i$ and all Brownian motions $w_1,\\dots ,w_N$ are mutually independent of each other, but the processes $\\ell ^{GIG}_1,\\dots ,\\ell ^{GIG}_N$ may be correlated.",
"We aim for an approximation $(L^{GH}_N(x)(t),t\\in $ which is a GH process for arbitrary $\\varphi _i$ and $x\\in \\mathcal {D}$ .",
"Remark REF suggests that this cannot be achieved by the summation of independent $\\ell _i^{GH}$ , but rather by using correlated subordinators $\\ell ^{GIG}_1,\\dots ,\\ell ^{GIG}_N$ .",
"Before we determine the correlation structure of the subordinators, we establish a necessary and sufficient condition on the $\\ell _i^{GH}$ to achieve the desired distribution of the approximation.",
"Lemma 4.4 Let $N\\in \\mathbb {N}$ , $t\\in and $ (iGH,i=1...,N)$ be GH processes as defined in Eq.~(\\ref {eq:Z_i}).",
"For a vector $a = (a1,...,aN)$ with arbitrary numbers $ a1,...,aN$\\mathbb {R}$ {0}$, the process $ GH,a$ defined by\\begin{equation}\\ell ^{GH,\\bf a}(t):=\\sum _{i=1}^{N} a_i\\ell _i^{GH}(t)=\\sum _{i=1}^{N} a_i(\\mu _i+\\beta _i\\ell _i^{GIG}(t)+w_i(\\ell _i^{GIG}(t)))\\end{equation}is a one-dimensional GH process, if and only if the vector $ GHN(1):=(1GH(1),...,NGH(1))'$ is multivariate $ GHN((N),(N),(N),(N),(N),)$-distributed with parameters $ (N),(N)$, $ (N)$\\mathbb {R}$$, $ (N),(N)$\\mathbb {R}$ N$and structure matrix $$\\mathbb {R}$ NN$.$ The entries of the coefficient vector ${\\bf a}$ in $\\ell ^{GH,\\bf a}$ are later identified with the basis functions $\\varphi _i(x)$ for $x\\in to show that $ LGHN(x)()$ is a one-dimensional Lévy process and the approximation $ LGHN$ a $ H$-valued GH Lévy field.",
"{\\begin{xmlelement*}{proof} We first consider the case that\\begin{equation*}\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma ).\\end{equation*}It is sufficient to show that \\ell ^{GH,\\bf a}(1) is a GH-distributed random variable, the infinite divisibility of the GH distribution then implies that (\\ell ^{GH,\\bf a}(t),t\\in is a GH process.Since the entries of the coefficient vector a_1,\\dots ,a_N are non-zero, there exists a non-singular N\\times N matrix A, such that \\ell ^{GH,\\bf a}(1) is the first component of the vector A\\ell ^{GH_N}(1).If \\ell ^{GH_N}(1) is multi-dimensional GH-distributed, then follows from~\\cite [Theorem 1]{B81}, that A\\ell ^{GH_N}(1) is also multi-dimensional GH-distributed and that the first component of A\\ell ^{GH_N}(1), namely \\ell ^{GH,\\bf a}(1), follows a one-dimensional GH distribution (the parameters of the distribution of \\ell ^{GH,\\bf a}(1) depend on A and on \\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma and are explicitly given in \\cite {B81} and below).",
"\\\\On the other hand, assume that \\ell ^{GH,\\bf a}(1) is a GH random variable (with arbitrary coefficients), but \\ell ^{GH_N}(1) is not N-dimensional GH-distributed.This means there is no representation of \\ell ^{GH_N}(1) such that\\begin{equation*}\\ell ^{GH_N}(1)\\stackrel{\\mathcal {L}}{=}\\mu +\\Gamma \\beta \\ \\ell ^{GIG}(1)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(1))\\ \\end{equation*}with \\mu ,\\beta \\in \\mathbb {R}^N, \\Gamma \\in \\mathbb {R}^{N\\times N} spd with determinant one, a GIG random variable \\ell ^{GIG}(1) and a N-dimensional Brownian motion w^N independent of \\ell ^{GIG}(1).This implies that \\ell ^{GH,\\bf a}(1)=(A\\ell ^{GH_N}(1))_1 has no representation\\begin{equation*}\\begin{split}\\ell ^{GH,\\bf a}(1)&=(A\\mu )_1+(A\\Gamma \\beta )_1\\ell ^{GIG}(1)+(A\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(1)))_1\\\\&\\stackrel{\\mathcal {L}}{=}(A\\mu )_1+(A\\Gamma \\beta )_1\\ell ^{GIG}(1)+\\sqrt{\\ell ^{GIG}(1)A_{[1]}\\Gamma A_{[1]}^{\\prime }}w^1(1),\\end{split}\\end{equation*}where A_{[1]} denotes the first row of the matrix A and w^1(1)\\sim \\mathcal {N}(0,1).For the last equality we have used the affine linear transformation property of multi-dimensional normal distributions and that \\Gamma is positive definite.Since c_A:=A_{[1]}\\Gamma A_{[1]}^{\\prime }>0, we can divide the equation above by \\sqrt{c_A} and obtain that c_A^{-1/2}\\,\\ell ^{GH,\\bf a}(1) cannot be a GH-distributed random variable, as it cannot be expressed as a normal variance-mean mixture with a GIG-distribution.But this is a contradiction, since \\ell ^{GH,\\bf a}(1) is GH-distributed by assumption and the class of GH distributions is closed under regular affine linear transformations (see~\\cite [Theorem 1c]{B81}).\\end{xmlelement*}}\\begin{rem}The condition a_i\\ne 0 is, in fact, not necessary in Lemma~\\ref {lem:GH_lin}.",
"If, for k\\in \\lbrace 1,\\dots ,N-1\\rbrace , k coefficients a_{i_1}=\\dots =a_{i_k}=0, then the summation reduces to\\begin{equation*}\\ell ^{GH,\\bf a}(t)=\\sum _{i=1}^{N}a_i \\ell _i^{GH}(t)=\\sum _{l=1}^{N-k}a_{j_l}\\ell ^{GH}_{j_l}(t),\\end{equation*}where the indices j_l are chosen such that a_{j_l}\\ne 0 for l=1,\\dots ,N-k.",
"If P\\in \\mathbb {R}^{N\\times N} is the permutation matrix with\\begin{equation*}P\\ell ^{GH_N}(1)=P(\\ell _1^{GH}(1),\\dots ,\\ell _N^{GH}(1))^{\\prime }=( \\ell ^{GH}_{j_1}(1),\\dots , \\ell ^{GH}_{j_{N-k}}(1), \\ell ^{GH}_{i_1}(1),\\dots , \\ell ^{GH}_{i_k}(1))^{\\prime },\\end{equation*}then P\\ell ^{GH_N} is again N-dimensionally GH-distributed and by~\\cite [Theorem 1a]{B81} the vector (\\ell ^{GH}_{j_1}(1),\\dots ,\\ell ^{GH}_{j_{N-k}}(1)) admits a (N-k)-dimensional GH law.",
"Thus, we only consider the case where all coefficients are non-vanishing.\\end{rem}$ The previous proposition states that the KL approximation $L^{GH}_N(x)(t) = \\sum _{i=1}^{N} \\varphi _i(x) \\ell _i^{GH}(t),$ can only be a GH process for arbitrary $(\\varphi _i(x),i=1,\\dots ,N)$ if the $\\ell _i^{GH}$ are correlated in such a way that they form a multi-dimensional GH process.",
"This rules out the possibility of independent processes $(\\ell _i^{GH},i=1,\\ldots ,N)$ , because if $\\ell ^{GH_N}(1)$ is multi-dimensional GH-distributed, it is not possible that the marginals $\\ell _i^{GH}(1)$ are independent GH-distributed random variables (see [13]).",
"The parameters $\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i$ of each process $\\ell _i^{GH}$ should remain as unrestricted as possible, so we determine in the next step the parameters of the marginals of a $GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ distribution and show how the subordinators $(\\ell ^{GIG}_i,i=1,\\ldots ,N)$ might be correlated.",
"For this purpose, we introduce the notation $A^-~\\!\\!^{\\prime }:=(A^{-1})^{\\prime }$ if $A$ is an invertible square matrix.",
"The following result allows us to determine the marginal distributions of a $N$ -dimensional GH distribution.",
"Lemma 4.5 (Masuda [30], who refers to [14], Lemma A.1.)",
"Let $\\ell ^{GH_N}(1)=(\\ell _1^{GH}(1),\\dots ,\\ell _N^{GH}(1))^{\\prime }\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma ),$ then for each $i$ we have that $\\ell _i^{GH}(1)\\sim GH(\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i)$ , where $\\begin{split}&\\lambda _i=\\lambda ^{(N)},\\quad \\alpha _i=\\Gamma _{ii}^{-1/2}\\left[(\\alpha ^{(N)})^2-\\beta _{-i}^{\\prime }\\left(\\Gamma _{-i,22}-\\Gamma _{-i,21}\\Gamma _{ii}^{-1}\\Gamma _{-i,12}\\right)\\beta _{-i}\\right]^{1/2}\\\\&\\beta _i=\\beta ^{(N)}_i+\\Gamma _{ii}^{-1}\\Gamma _{-i,12}\\beta _{-i},\\quad \\delta _i=\\sqrt{\\Gamma _{ii}}\\delta ^{(N)}_i,\\quad \\mu _i=\\mu _i^{(N)},\\end{split}$ together with $\\begin{split}&\\beta _{-i}:=(\\beta _1^{(N)},\\dots ,\\beta _{i-1}^{(N)},\\beta _{i+1}^{(N)},\\dots ,\\beta _N^{(N)})^{\\prime },\\\\&\\Gamma _{-i,12}:=(\\Gamma _{i,1},\\dots ,\\Gamma _{i,i-1},\\Gamma _{i,i+1},\\dots ,\\Gamma _{i,N}),\\quad \\Gamma _{-i,21}:=\\Gamma _{-i,12}^{\\prime }\\end{split}$ and $\\Gamma _{-i,22}$ denotes the $(N-1)\\times (N-1)$ matrix which is obtained by removing the $i$ -th row and column of $\\Gamma $ .",
"Assume that $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ , since this is a necessary (and sufficient) condition so that the (truncated) KL expansion is a GH process.",
"Lemma REF gives immediately, that for all $i=1,\\dots ,N$ , the parameters $\\lambda _i=\\lambda ^{(N)}$ have to be identical, whereas the drift $\\mu _i$ may be chosen arbitrary for each process $\\ell _i^{GH}$ .",
"Furthermore, the expectation and covariance matrix of $\\ell ^{GH_N}(1)$ is given by $ \\mathbb {E}(\\ell ^{GH_N}(1))=\\mu ^{(N)}+\\frac{\\delta ^{(N)}K_{\\lambda ^{(N)}+1}(\\delta ^{(N)}\\gamma ^{(N)})}{\\gamma ^{(N)}K_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}\\Gamma \\beta ^{(N)}$ and $ \\begin{split}\\text{Var}(\\ell ^{GH_N}(1))&=\\frac{\\delta ^{(N)}K_{\\lambda ^{(N)}+1}(\\delta ^{(N)}\\gamma ^{(N)})}{\\gamma ^{(N)}K_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}\\Gamma +\\Big (\\frac{\\delta ^{(N)}}{\\gamma ^{(N)}}\\Big )^2(\\Gamma \\beta ^{(N)})(\\Gamma \\beta ^{(N)})^{\\prime }\\\\&\\qquad \\qquad \\cdot \\Bigg (\\frac{K_{\\lambda ^{(N)}+2}(\\delta ^{(N)}\\gamma ^{(N)})}{K_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}-\\frac{K^2_{\\lambda ^{(N)}+1}(\\delta ^{(N)}\\gamma ^{(N)})}{K^2_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}\\Bigg ),\\end{split}$ where $\\gamma ^{(N)}:=((\\alpha ^{(N)})^2-\\beta ^{(N)}~\\!\\!^{\\prime }\\Gamma \\beta ^{(N)})^{1/2}$ (see [30]).",
"Example 4.6 Consider the case that the processes $\\ell _1^{GH},\\dots ,\\ell _N^{GH}$ are generated by the same subordinating $GIG(a,b,p)$ process $\\ell ^{GIG}$ , i.e.",
"$\\ell _i^{GH}(t)=\\mu _it+\\beta _i\\ell ^{GIG}(t)+w_i(\\ell ^{GIG}(t)).$ Then $\\ell _i^{GH}(1)\\sim GH(\\lambda ,\\alpha _i,\\beta _i,\\delta ,\\mu _i)$ , where $\\lambda =p$ , $\\delta =a$ are independent of $i$ and $\\alpha _i=(b^2+\\beta _i^2)^{1/2}$ .",
"If $\\mu ^{(N)}:=(\\mu _1\\dots ,\\mu _N)^{\\prime }$ , $\\beta ^{(N)}:=(\\beta _1,\\dots ,\\beta _N)^{\\prime }$ and $\\Gamma $ is the $N\\times N$ identity matrix, then $\\begin{split}\\ell ^{GH_N}(t)=(\\ell _1^{GH}(t),\\dots ,\\ell _N^{GH}(t))^{\\prime }&\\stackrel{\\mathcal {L}}{=}\\mu t+\\beta \\ell ^{GIG}(t)+w^N(\\ell ^{GIG}(t))\\\\&=\\mu t+\\Gamma \\beta \\ell ^{GIG}(t)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(t)),\\end{split}$ where $w^N$ is a $N$ -dimensional Brownian motion independent of $\\ell ^{GIG}$ .",
"Hence, $\\ell ^{GH_N}(t)$ is a multi-dimensional $GH_N(\\lambda ,\\alpha ^{(N)},\\beta ^{(N)},\\delta ,\\mu ^{(N)}$ ,$\\Gamma )$ process with $\\alpha ^{(N)}=\\sqrt{b^2+\\beta ^{\\prime }\\beta }$ .",
"One checks using Lemma REF that the parameters of the marginals of $\\ell ^{GH_N}(1)$ and $\\ell _i^{GH}(1)$ coincide for each $i$ , and that expectation and covariance of $\\ell ^{GH_N}(1)$ are given by Eq.",
"(REF ) and Eq.",
"(REF ).",
"By Lemma REF , we have that the Karhunen-Loève expansion $L_N^{GH}(x)(t)=\\sum _{i=1}^N\\varphi _i(x)\\ell _i^{GH}(t)$ in this example is a GH process for each $x\\in \\mathcal {D}$ and arbitrary basis functions $(\\varphi _i,i=1,\\ldots ,N)$ .",
"Remark 4.7 Lemma REF dictates that the subordinators $(\\ell ^{GIG}_i,i=1,\\ldots ,N)$ cannot be independent.",
"In Example REF fully correlated subordinators were used.",
"A different way to correlate the subordinators, so that Lemma REF is fulfilled, would lead to a correlation matrix, just being multiplied with $\\Gamma $ .",
"For simplicity, in the remainder of the paper, especially for the numerical examples in Section , we use fully correlated subordinators.",
"As shown in [13] the class of $N$ -dimensional GH distributions is closed under regular linear transformations: If $N\\in \\mathbb {N}$ , $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ , $A$ is an invertible $N\\times N$ -matrix and $b\\in \\mathbb {R}^N$ , then the random vector $A\\ell ^{GH_N}(1)+b$ has distribution $GH_N(\\lambda ,||A||^{-1/N}\\alpha ,A^-~\\!\\!^{\\prime }\\beta ,||A||^{1/N}\\delta ,A\\mu +b,||A||^{-2/N}A\\Gamma A^{\\prime }),$ where $||A||$ denotes the absolute value of the determinant of $A$ .",
"With this and the assumption $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ , we are also able to determine the point-wise law of $L_N^{GH}$ for given coefficients $\\varphi _1(x),\\dots ,\\varphi _N(x)$ .",
"Lemma 4.8 Let $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ and for $x\\in let $ (i(x),i = 1,...,N)$ be a sequence of non-zero coefficients (see Remark~\\ref {rem:coefficients}).Then $ (LNGH(x)(t),t$ is a GH Lévy process with parameters depending on $ x$.$ It is again sufficient to show that $L_N^{GH}(x)(1)$ follows a GH law, the resulting parameters are given below.",
"For $x\\in , define the $ NN$ matrix $ A(x)$ via\\begin{equation} A(x)_{ij}:={\\left\\lbrace \\begin{array}{ll}\\varphi _j(x) & \\text{if $i=1$ or if $i=j$} \\\\0 & \\text{elsewhere}\\end{array}\\right.",
"}.\\end{equation}The matrix $ A(x)$ is invertible with determinant $ i=1Ni(x)0$ and inverse $ A(x)-1$ given by\\begin{equation*}A(x)^{-1}_{ij}:={\\left\\lbrace \\begin{array}{ll}-\\varphi _1(x)^{-1} & \\text{if $i=1$ and $j\\ge 2$} \\\\\\varphi _i(x)^{-1} & \\text{if $i=j$} \\\\0 & \\text{elsewhere}\\end{array}\\right.",
"}.\\end{equation*}Then $ LNGH(x)(1)=i=1Ni(x)iGH(1)$ is the first entry of the random vector $ A(x)GHN(1)$.By the affine transformation property of the GH distribution and Lemma~\\ref {lem:GH_N} it follows that $ LNGH(x)(1)$ is one-dimensional GH-distributed.Now define $ :=A(x)A(x)'$, the partition\\begin{equation*}\\widetilde{\\Gamma }=\\begin{pmatrix} \\widetilde{\\Gamma }_{11} & \\widetilde{\\Gamma }_{2,1}^{\\prime }\\\\ \\widetilde{\\Gamma }_{2,1} & \\widetilde{\\Gamma }_{2,2} \\end{pmatrix}\\end{equation*}such that $ 2,1$\\mathbb {R}$ N-1$ and $ 2,2$\\mathbb {R}$ (N-1)(N-1)$ and the vector$$\\widetilde{\\beta }:=\\left(\\beta _2^{(N)}\\varphi _2(x)^{-1}-\\beta _1^{(N)}\\varphi _1(x)^{-1},\\dots ,\\beta _N^{(N)}\\varphi _N(x)^{-1}-\\beta _1^{(N)}\\varphi _1(x)^{-1}\\right)^{\\prime }\\in \\mathbb {R}^{N-1}.$$The parameters $ L,L(x),L(x),L(x)$ and $ L(x)$ of $ LNGH(x)$ are then given by{\\begin{@align*}{1}{-1}\\lambda _L&=\\lambda ^{(N)},\\\\\\alpha _L(x)&=\\widetilde{\\Gamma }_{11}^{-1/2}\\left[(\\alpha ^{(N)})^2-\\widetilde{\\beta }^{\\prime }(\\widetilde{\\Gamma }_{2,2}-\\widetilde{\\Gamma }_{11}^{-1}\\widetilde{\\Gamma }_{2,1}\\widetilde{\\Gamma }_{2,1}^{\\prime })\\widetilde{\\beta } \\right]^{1/2},\\\\\\delta _L(x)&=\\delta ^{(N)}\\sqrt{\\widetilde{\\Gamma }_{11}}=\\delta ^{(N)}\\Big (\\sum _{i,j=1}^N\\varphi _i(x)\\varphi _j(x)\\Gamma _{ij}\\Big )^{1/2},\\\\\\beta _L(x)&=\\beta _1^{(N)}\\varphi _1(x)^{-1}+\\widetilde{\\Gamma }_{11}^{-1}\\widetilde{\\Gamma }_{2,1}^{\\prime }\\widetilde{\\beta }\\quad \\text{and}\\\\\\mu _L(x)&=[A(x)\\mu ^{(N)}]_1=\\sum _{i=1}^N\\varphi _i(x)\\mu _i^{(N)}.\\end{@align*}}$ To ensure $L^2(\\Omega ;\\mathbb {R})$ convergence as in Theorem REF of the series $\\widetilde{L}_N^{GH}(x)(t) = \\sum _{i=1}^N\\sqrt{\\rho _i}e_i(x)\\widetilde{\\ell }_i^{GH}(t),$ we need to simulate approximations of uncorrelated, one-dimensional GH processes $\\ell _i^{GH}$ with given parameters $\\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i)$ .",
"To obtain a sufficiently good approximation of the Lévy field, $N$ is coupled to the time discretization of $ and the decay of the eigenvalues of $ Q$ (see Remark~\\ref {rem:trunc}).The simulation of a large number $ N$ of independent GH processes is computationally expensive, so we focus on a different approach.Instead of generating $ N$ dependent but uncorrelated, one-dimensional processes, we generate one $ N$-dimensional process with decorrelated marginals.For this approach to work we need to impose some restrictions on the target parameters $ i, i,i$ and $ i$.$ Theorem 4.9 Let $(\\ell _i^{GH},i=1\\ldots ,N)$ be one-dimensional GH processes, where, for $i=1,\\dots ,N$ , $\\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i).$ The vector $\\ell ^{GH_N}:=(\\ell _1^{GH},\\dots ,\\ell _N^{GH})^{\\prime }$ is only a $N$ -dimensional GH process if there are constants $\\lambda \\in \\mathbb {R}$ and $c>0$ such that for any $i$ $\\lambda _i=\\lambda \\quad \\text{and}\\quad \\delta _i(\\alpha _i^2-\\beta _i^2)^{1/2}=c.$ If, in addition, the symmetric matrix $U\\in \\mathbb {R}^{N\\times N}$ defined by $U_{ij}:={\\left\\lbrace \\begin{array}{ll}\\delta _i^2 &\\text{if $i=j$}\\\\\\frac{K_{\\lambda +1}(c)^2-K_{\\lambda +2}(c)K_{\\lambda }(c)}{K_{\\lambda +1}(c)K_{\\lambda }(c)}\\frac{\\beta _i\\delta _i^2\\beta _j\\delta _j^2}{c} &\\text{if $i\\ne j$}\\end{array}\\right.",
"},$ is positive definite, it is possible to construct a $N$ -dimensional GH process $\\ell ^{GH_N,U}$ with uncorrelated marginals $\\ell ^{GH,U}_i$ and $\\ell ^{GH,U}_i(1)\\stackrel{\\mathcal {L}}{=} \\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i).$ We start with the necessary condition to obtain a multi-dimensional GH distribution.",
"Let $\\ell ^{GH_N}$ be a $N$ -dimensional GH process with $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma ).$ If the law of the marginals of $\\ell ^{GH_N}$ is denoted by $ \\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i),$ then one sees immediately from Lemma REF that $\\lambda _i=\\lambda ^{(N)}$ and $\\mu _i=\\mu _i^{(N)}$ for all $i=1,\\dots ,N$ .",
"With the equations for $\\beta _i$ and $\\delta _i$ from Lemma REF , we derive for $\\Gamma \\beta ^{(N)}$ $(\\Gamma \\beta ^{(N)})_i=\\Gamma _{ii}\\beta _i^{(N)}+\\sum _{k=1, k\\ne i}^N \\Gamma _{ik}\\beta _k^{(N)}=\\Gamma _{ii}\\beta _i^{(N)}+\\Gamma _{ii}(\\beta _i-\\beta _i^{(N)})=\\big (\\frac{\\delta _i}{\\delta ^{(N)}}\\big )^2\\beta _i,$ which leads to $\\alpha _i^2&=\\Gamma _{ii}^{-1}(\\alpha ^{(N)})^2-\\Gamma _{ii}^{-1}\\sum _{k=1,k\\ne i}^N\\beta _k^{(N)}\\sum _{l=1,l\\ne i}^N\\Gamma _{kl}\\beta _l^{(N)}+\\big (\\Gamma _{ii}^{-1}\\sum _{k=1, k\\ne i}^N \\Gamma _{ik}\\beta _k^{(N)}\\big )^2\\\\&=\\big (\\frac{\\delta ^{(N)}\\alpha ^{(N)}}{\\delta _i}\\big )^2-\\big (\\frac{\\delta ^{(N)}}{\\delta _i}\\big )^2\\sum _{k=1,k\\ne i}^N\\beta _k^{(N)}((\\Gamma \\beta ^{(N)})_k-\\Gamma _{ik}\\beta _i^{(N)})+(\\beta _i-\\beta _i^{(N)})^2\\\\&=\\big (\\frac{\\delta ^{(N)}\\alpha ^{(N)}}{\\delta _i}\\big )^2-\\sum _{k=1,k\\ne i}^N\\beta _k^{(N)}\\frac{\\delta _k^2}{\\delta _i^2}\\beta _k\\\\&\\quad +\\big (\\frac{\\delta ^{(N)}}{\\delta _i}\\big )^2\\beta _i^{(N)}((\\Gamma \\beta ^{(N)})_i-\\Gamma _{ii}\\beta _i^{(N)})+\\beta _i^2-2\\beta _i\\beta _i^{(N)}+(\\beta _i^{(N)})^2\\\\&=\\big (\\frac{\\delta ^{(N)}\\alpha ^{(N)}}{\\delta _i}\\big )^2-\\sum _{k=1}^N\\beta _k^{(N)}\\frac{\\delta _k^2}{\\delta _i^2}\\beta _k+\\beta _i^2.$ The last equation is equivalent to $ \\delta _i^2(\\alpha _i^2-\\beta _i^2)=(\\delta ^{(N)}\\alpha ^{(N)})^2-\\sum _{k=1}^N\\beta _k^{(N)}\\underbrace{\\delta _k\\beta _k}_{=(\\delta ^{(N)})^2(\\Gamma \\beta _k^{(N)})}=(\\delta ^{(N)})^2\\left((\\alpha ^{(N)})^2-\\beta ^{(N)}~\\!\\!^{\\prime }\\Gamma \\beta ^{(N)}\\right),$ and since the right hand side does not depend on $i$ , we get that $\\delta _i^2(\\alpha _i^2-\\beta _i^2)>0$ has to be independent of $i$ .",
"Now assume we have a set of parameters $((\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i),i=1,\\dots ,N)$ with $\\delta _i\\sqrt{\\alpha _i^2-\\beta _i^2}=c>0\\quad \\text{and}\\quad \\lambda _i=\\lambda \\in \\mathbb {R},$ where $c$ and $\\lambda $ are independent of the index $i$ .",
"Furthermore, let the matrix $U$ as defined in the claim be positive definite.",
"We show how parameters $\\lambda ^{(U)}, \\alpha ^{(U)}, \\beta ^{(U)}, \\delta ^{(U)}, \\mu ^{(U)}$ and $\\Gamma ^{(U)}$ of a $N$ -dimensional GH process $\\ell ^{GH_N,U}$ may be chosen, such that its marginals are uncorrelated with law $\\ell ^{GH,U}_i(1)\\sim GH(\\lambda ,\\alpha _i,\\beta _i,\\delta _i,\\mu _i)$ .",
"Clearly, we have to set $\\lambda ^{(U)}:=\\lambda $ and $\\mu ^{(U)}:=(\\mu _1,\\dots ,\\mu _N)^{\\prime }$ .",
"Eq.",
"(REF ) and Eq.",
"(REF ) yield the conditions $(\\delta ^{(U)})^2(\\Gamma ^{(U)}\\beta ^{(U)})_i=\\delta _i^2\\beta _i \\quad \\text{and}\\quad \\delta ^{(U)}\\sqrt{(\\alpha ^{(U)})^2-\\beta ^{(U)T}\\Gamma ^{(U)}\\beta ^{(U)}}=\\delta _i\\sqrt{\\alpha _i^2-\\beta _i^2}=c.$ If $(\\delta ^{(U)})^2\\Gamma ^{(U)}$ fulfills the identity $(\\delta ^{(U)})^2\\Gamma ^{(U)}=U$ , we get by Eq.",
"(REF ) for $i\\ne j$ $\\text{Cov}(\\ell ^{GH,U}_i(1),\\ell ^{GH,U}_j(1))&=\\frac{K_{\\lambda +1}(c)}{cK_{\\lambda }(c)}(\\delta ^{(U)})^2\\Gamma _{ij}^{(U)}\\\\&\\quad +\\frac{K_{\\lambda +2}(c)K_{\\lambda }(c)-K_{\\lambda +1}^2(c)}{c^2K_{\\lambda }(c)^2}((\\delta ^{(U)})^2\\Gamma ^{(U)}\\beta ^{(U)})_i((\\delta ^{(U)})^2\\Gamma ^{(U)}\\beta ^{(U)})_j\\\\&=\\frac{K_{\\lambda +1}(c)}{cK_{\\lambda }(c)}U_{ij}+\\frac{K_{\\lambda +2}(c)K_{\\lambda }(c)-K_{\\lambda +1}^2(c)}{c^2K_{\\lambda }(c)^2}\\delta _i^2\\beta _i\\delta _j^2\\beta _j=0,$ hence all marginals are uncorrelated.",
"To obtain a well-defined $N$ -dimensional GH distribution, we still have to make sure that $\\Gamma ^{(U)}$ is spd with unit determinant.",
"If we define $\\delta ^{(U)}:=(\\text{det}(U))^{1/(2N)}$ , then $\\delta ^{(U)}>0$ (since $\\text{det}(U)>0$ by assumption) and $\\Gamma ^{(U)}=(\\delta ^{(U)})^{-2}U$ is spd with $\\text{det}(\\Gamma ^{(U)})=1$ .",
"It remains to determine appropriate parameters $\\alpha ^{(U)}>0$ and $\\beta ^{(U)}\\in \\mathbb {R}^N$ .",
"For $\\beta ^{(U)}$ , we use once again Lemma REF to obtain the linear equations $\\beta _i=\\beta _i+(\\Gamma _{ii}^{(U)})^{-1}\\sum _{k=1,k\\ne i}\\Gamma _{ik}^{(U)}\\beta _k^{(U)},$ for $i=1,\\dots ,N$ .",
"The corresponding system of linear equations is given by $\\begin{pmatrix}(\\Gamma _{11}^{(U)})^{-1}\\\\&\\ddots \\\\&&(\\Gamma _{NN}^{(U)})^{-1}\\\\\\end{pmatrix}\\Gamma ^{(U)}\\beta ^{(U)}=\\left(\\begin{array}{c} \\beta _1\\\\ \\vdots \\\\ \\beta _N \\end{array}\\right),$ and has a unique solution $\\beta ^{(U)}$ for any right hand side $(\\beta _1,\\dots ,\\beta _N)^{\\prime }$ , because $\\Gamma ^{(U)}$ as constructed above is invertible with positive diagonal entries.",
"Finally, we are able to calculate $\\alpha ^{(U)}$ via Equation (REF ) as $\\alpha ^{(U)}=\\Big (\\sum _{k=1}^N\\delta _k^2\\beta _k\\beta _k^{(U)}+\\big (\\frac{c}{\\delta ^{(U)}}\\big )^2\\Big )^{1/2}=\\Big (\\beta ^{(U)}~\\!\\!^{\\prime }\\Gamma ^{(U)}\\beta ^{(U)}+\\big (\\frac{c}{\\delta ^{(U)}}\\big )^2\\Big )^{1/2}$ and obtain the desired marginal distributions.",
"Note that the KL-expansion $L^{GH}_N(x)(\\cdot )$ generated by $(\\ell ^{GH,U}_i,i=1\\ldots ,N)$ in Theorem REF is a GH process for each $x\\in by Lemma~\\ref {lem:KL_GH}, whereas this is not the case if the processes $ (iGH,i=1,...,N)$ are generated independently of each other:By Lemma~\\ref {lem:GH_lin} we have that $ LGHN(x)(1)$ is only GH distributed if the vector $ (1GH(1),...,NGH(1))'$ admits a multi-dimensional GH law.As noted in~\\cite {B81} after Theorem 1, this is impossible if the processes (and hence $ (iGH(1),i=1,...,N)$) are independent.Whenever Theorem~\\ref {thm:Z_U} is applicable, we are able to approximate a GH Lévy field by generating a $ N$-dimensional GH processes, where $ N$ is the truncation index of the KL expansion.To this end, Lemma~\\ref {lem:sub} suggests the simulation of GIG processes and then subordinating $ N$-dimensional Brownian motions.With this simulation approach the question arises on why we have taken a detour via the subordinating GIG process instead of using the characteristic function a of GH process in Equation~(\\ref {eq:gh_cf}) for a ``direct^{\\prime \\prime } simulation.This has several reasons: First, the approximation of the inversion formula (\\ref {eq:finv}) can only be applied for one-dimensional GH processes, where the costs of evaluating $ GH$ or $ GIG$ are roughly the same.In comparison, the costs of sampling a Brownian motion are negligible.Second, in the multi-dimensional case, we need that all marginals of the GH process are generated by the same or correlated subordinator(s), which leaves us no choice but to sample the underlying GIG process.In addition, the simulation of a GH field requires in some cases only one subordinating process to generate a multi-dimensional GH process with uncorrelated marginals (see Theorem~\\ref {thm:Z_U}).This approach is in general more efficient than sampling a large number of uncorrelated, one-dimensional GH processes for the KL expansion.As we demonstrate in the following section, it is a straightforward application of the Fourier inversion algorithm to approximate a GIG process $ GIG$ with given parameters, since all necessary assumptions are fulfilled and the bounding parameters $ , R, $ and $ B$ may readily be calculated.$ Numerics In this section we provide some details on the implementation of the Fourier inversion method.",
"Thereafter, we apply this methodology to approximate a GH Lévy field and conclude with some numerical examples.",
"Notes on implementation Suppose we simulate a given one-dimensional Lévy process $\\ell $ which fulfills Assumption REF resp.",
"Assumption REF , using the step size $\\Delta _n>0$ and characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"Usually the parameter $\\eta $ cannot be chosen arbitrary high (as for the GIG process), but it may be possible to choose $\\eta $ within a certain range, for instance $\\eta \\in (1,2]$ for the Cauchy process in Example REF .",
"As a rule of thumb, $\\eta $ should always be determined as large as possible, as the convergence rates in Theorems REF and REF directly depend on $\\eta $ .",
"In addition, we concluded in Remark REF that $D\\simeq \\Delta _n^{p/(p-\\eta )}$ is an appropriate choice to guarantee an $L^p$ -error of order $\\mathcal {O}(\\Delta _n^{1/p})$ .",
"This means that for a given $p$ , $D$ decreases as $\\eta $ increases.",
"Since the number of summations $M$ in Algorithm REF depends on $D$ (see Theorem REF ), an increasing parameter $\\eta $ also reduces computational time.",
"Once $\\eta $ is determined, we derive $R$ by differentiation of $(\\phi _\\ell )^{\\Delta _n}$ as in Remark REF .",
"Similarly to $\\eta $ , it is often possible to choose between several values of $\\theta >0$ , but it is difficult to give a-priori a recommendation on how $\\theta $ should be selected.",
"One rather calculates for several admissible $\\theta $ the constant $C_\\theta :=\\max _{u\\in \\mathbb {R}}|u^\\theta (\\phi _\\ell (u))^{\\Delta _n}|$ numerically and deducts $B_\\theta =(2\\pi )^{-\\theta }C_\\theta $ .",
"Each combination of $(\\theta ,B_\\theta )$ then results in a valid number of summations $M_\\theta $ in the discrete Fourier Inversion algorithm.",
"Since $\\theta $ and $B_\\theta $ are only necessary to determine $M_\\theta $ , we may simply use the smallest $M_\\theta $ for the simulation.",
"To find $\\widetilde{X}$ with $\\widetilde{F}(\\widetilde{X})=U$ in Algorithm REF , we use a globalized Newton method with backtracking line search, also known as Armijo increment control.",
"The step lengths during the line search are determined by interpolation, which is a robust technique if combined with a standard Newton method.",
"Details on the globalized Newton method with backtracking may be found, for example, in [31], an example how the algorithm is used is given in [36].",
"Although convergence of this root finding algorithm is ensured by the increment control, its efficiency depends heavily on the choice of the initial value $\\widetilde{X}_0$ .",
"Clearly, $\\widetilde{X}_0$ should depend on the sampled $U\\sim \\mathcal {U}([0,1])$ and be related to the target distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"This means we should determine $\\widetilde{X}_0$ implicitly by $F^{(0)}(\\widetilde{X}_0)=U$ , where $F^{(0)}$ is a CDF of a distribution similar to the target distribution, but which can be inverted efficiently.",
"Approximation of a GH field We consider a GH Lévy field on the (separable) Hilbert space $H=L^2(\\mathcal {D})$ with a compact spatial domain $\\mathcal {D}\\subset \\mathbb {R}^s$ .",
"The operator $Q$ on $H$ is given by a Matérn covariance operator with variance $v>0$ , correlation length $r>0$ and a positive parameter $\\chi >0$ defined by $[Qh](x):=v\\int _\\mathcal {D} k_\\chi (x,y)h(y)dy,\\quad \\text{for }h\\in H,$ where $k_\\chi $ denotes the Matérn kernel.",
"For $\\chi =\\frac{1}{2}$ , we obtain the exponential covariance function and for $\\chi \\rightarrow \\infty $ the squared exponential covariance function.",
"For general $\\chi >0$ , the Matérn kernel $k_\\chi (x,y):=\\frac{2^{1-\\chi }}{\\Gamma _G(\\chi )}\\Big (\\frac{\\sqrt{2\\chi }|x-y|}{r}\\Big )^\\chi K_\\chi \\Big (\\frac{\\sqrt{2\\chi }|x-y|}{r}\\Big )$ fulfills the limit identity $k_\\chi (x,x)=\\lim _{y\\rightarrow x}k_\\chi (x,y)=1$ , which can be easily seen by [33].",
"Here $\\Gamma _G(\\cdot )$ is the Gamma function.",
"As shown in [20], this implies $tr(Q)=\\sum _{i=1}^\\infty \\rho _i=v\\int _{\\mathcal {D}}dx,$ where $(\\rho _i,i\\in \\mathbb {N})$ are the eigenvalues of the Matérn covariance operator $Q$ .",
"In general, no analytical expressions for the eigenpairs $(\\rho _i,e_i)$ of $Q$ will be available, but the spectral basis may be approximated by numerically solving a discrete eigenvalue problem and then interpolating by Nyström's method.",
"For a general overview of common covariance functions and the approximation of their eigenbasis we refer to [37] and the references therein.",
"Now let $L_N^{GH}$ be an approximation of a GH field by a $N$ -dimensional GH process $(\\ell ^{GH_N}(t),t\\in $ with fixed parameters $\\lambda ,\\alpha ,\\delta \\in \\mathbb {R}$ , $\\beta ,\\mu \\in \\mathbb {R}^N$ and $\\Gamma \\in \\mathbb {R}^{N\\times N}$ .",
"The parameters are chosen in such a way that the multi-dimensional GH process has uncorrelated marginal processes, hence the generated KL expansions $L_N^{GH}(x)(t)=\\sum _{i=1}^N\\varphi _i(x)\\ell ^{GH}_i(t)$ are again one-dimensional GH processes for any spectral basis $(\\varphi _i,i\\in \\mathbb {N})$ and fixed $x\\in \\mathcal {D}$ .",
"This in turn means, that we may draw samples of $\\ell ^{GH_N}$ by simulating a GIG process $\\ell ^{GIG}$ with parameters $a=\\delta , b=(\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta )^{1/2}$ and $p=\\lambda $ using Fourier inversion and then subordinating a $N$ -dimensional Brownian motion (see Lemma REF ).",
"The characteristic function of a GIG Lévy process $(\\ell ^{GIG}(t),t\\in $ with (fixed) parameters $a,b>0$ and $p\\in \\mathbb {R}$ is given by $ \\phi _{GIG}(u;a,b,p):=\\mathbb {E}[\\exp (iu\\ell ^{GIG}(1))]=(1-2iub^{-2})^{-p/2}\\frac{K_p(ab\\sqrt{1-2iub^{-2}})}{K_p(ab)}.$ The GIG distribution corresponding to $(\\phi _{GIG})^{\\Delta _n}$ with $\\Delta _n=1$ is continuous with finite variance (see [41]), which implies that these properties hold for all distributions with characteristic function $(\\phi _{GIG})^{\\Delta _n}$ , for any $\\Delta _n>0$ .",
"The constants as in Assumption REF are derived in the following.",
"For $k\\in \\mathbb {N}$ , the $k$ -th moment of the GIG distribution is given as $0<\\mathbb {E}\\big ((\\ell ^{GIG}(1))^k\\big )=\\big (\\frac{a}{b}\\big )^{k}\\frac{K_{p+k}(ab)}{K_p(ab)}<\\infty .$ For any $\\eta =2k$ we are, therefore, able to calculate the bounding constant $R$ via $R=(-1)^{k}\\frac{d^{2k}}{du^{2k}}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0},$ because the derivatives of $\\phi _{GIG}$ evaluated at $u=0$ are $(\\phi _{GIG}(0;a,b,p))^{(k)}=i^{-k}\\mathbb {E}\\big ((\\ell ^{GIG}(1))^k\\big )=i^{-k}\\big (\\frac{a}{b}\\big )^{k}\\frac{K_{p+k}(ab)}{K_p(ab)}.$ The calculation of the $\\eta $ -th derivative can be implemented easily by using a version of Faà di Bruno's formula containing the Bell polynomials, for details we refer to [25].",
"The bounding constants $\\theta $ and $B$ may be determined numerically as described Section REF (e.g.",
"by using the routine fminsearch in MATLAB).",
"The derivation of the bounds implies that we can ensure $L^p$ convergence of the approximated GIG process in the sense of Theorem REF for any $p\\ge 1$ , because it is possible to define $\\eta $ as any even integer and then obtain $R$ by differentiation.",
"We observe that the target distribution with characteristic function $(\\phi _{GIG}(u;a,b,p))^{\\Delta _n}$ and $\\Delta _n>0$ is not necessarily GIG, except for the Inverse Gaussian (IG) case where $p=-1/2$ and $(\\phi _{IG}(u;a,b))^{\\Delta _n}=\\phi _{IG}(u;\\Delta _na,b)$ (see Remark REF ).",
"This special feature of the IG distribution is exploited to determine the initial values $\\widetilde{X}_0$ in the Newton iteration by moment matching: Consider an $IG(a_0,b_0)$ distribution with mean $a_0/b_0$ and variance $a_0/b_0^3$ , where the parameters $a_0,b_0>0$ are “matched” to the target distribution's mean and variance via $\\frac{a_0}{b_0}&=i\\frac{d}{du}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0},\\\\\\frac{a_0}{b_0^3}&=(-1)\\frac{d^2}{du^2}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0}-\\Big (i\\frac{d}{du}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0}\\Big )^2.$ If $F_{\\Delta _n}^{IG}$ denotes the CDF of this $IG(a_0,b_0)$ distribution, the initial value of the globalized Newton method is given implicitly by $F_{\\Delta _n}^{IG}(\\widetilde{X}_0)=U$ .",
"The inversion of $F_{\\Delta _n}^{IG}$ may be executed numerically by many software packages like MATLAB.",
"With our approach, this results in the approximation of a GIG process $\\widetilde{\\ell }^{GIG}$ at discrete times $t_j\\in \\Xi _n$ .",
"The $N$ -dimensional GH process $\\ell ^{GH_N}$ may then be approximated at $t_j$ for $j=0,\\dots ,n$ by the process $\\widetilde{\\ell }^{GH_N}$ with $\\widetilde{\\ell }^{GH_N}(t_0)=0$ and the increments $\\widetilde{\\ell }^{GH_N}(t_j)-\\widetilde{\\ell }^{GH_N}(t_{j-1})=\\mu \\Delta _n+\\Gamma \\beta (\\widetilde{\\ell }^{GIG}(t_j)-\\widetilde{\\ell }^{GIG}(t_{j-1})) + \\sqrt{(\\widetilde{\\ell }^{GIG}(t_j)-\\widetilde{\\ell }^{GIG}(t_{j-1}))\\Gamma }w^N_j(1),$ for $j=1,\\dots ,n$ , where the $w_j^N(1)$ are i.i.d.",
"$\\mathcal {N}_N(0,\\mathbf {1}_{N\\times N})$ -distributed random vectors.",
"To obtain the process $\\widetilde{\\ell }^{GH_N}$ at arbitrary times $t\\in , we interpolate the samples $ (GHN(tj),j=0...,n)$ piecewise constant as in Algorithm~\\ref {algo:approx2}.With this, we are able to generate an approximation of $ LNGH$ at any point $ (x,t)D by $\\widetilde{L}_N^{GH}(x)(t):=\\sum _{i=1}^N\\varphi _i(x) \\widetilde{\\ell }^{GH}_i(t).$ The knowledge of $tr(Q)$ enables us to determine the truncation index $N$ and the constant $C_\\ell $ as in Remark : For $N\\in \\mathbb {N}$ let $(\\widetilde{\\ell }^{GH}_i,i=1,\\ldots ,N)$ be the approximations of the processes $(\\ell ^{GH}_i,i=1,\\ldots ,N)$ , where the random vector $(\\ell ^{GH}_1(1),\\dots ,\\ell _N^{GH}(1))$ is multivariate GH-distributed by assumption.",
"Hence, for every $N\\in \\mathbb {N}$ , we obtain the parameters $a(N), b(N), \\lambda (N)$ of a corresponding GIG subordinator $\\ell ^{GIG,N}$ , which is approximated through a piecewise constant process $\\widetilde{\\ell }^{GIG,N}$ as above.",
"With Eq.",
"(REF ) we calculate the error $E_{GIG,N}^p:=\\sup _{t\\in \\mathbb {E}(|\\ell ^{GIG,N}(t)-\\widetilde{\\ell }^{GIG,N}(t)|^p).",
"}for p\\in \\lbrace 1,2\\rbrace .If \\beta \\in \\mathbb {R}^N and \\Gamma \\in \\mathbb {R}^{N\\times N} denote the GH parameters corresponding to (\\ell ^{GH}_1(1),\\dots ,\\ell _N^{GH}(1)), the L^2(\\Omega ;\\mathbb {R}) approximation error of each process \\ell ^{GH}_i is given by\\begin{equation*}\\widetilde{C}_{\\ell ,i}:=\\sup _{t\\in \\frac{\\mathbb {E}(|\\ell ^{GH}_i(t)-\\widetilde{\\ell }^{GH}_i(t)|^2)}{\\Delta _n}= \\frac{E_{GIG,N}^2(\\Gamma \\beta )_i^2+E_{GIG,N}^1\\sqrt{\\Gamma _{[i]}\\Gamma _{[i]}^{\\prime }}}{\\Delta _n},}where \\Gamma _{[i]} indicates the i-th row of \\Gamma .Starting with N=1, we compute the first N eigenvalues and the difference\\begin{equation*}T\\Big (tr(Q)-\\sum _{i=1}^N\\rho _i\\Big )-\\max _{i=1,\\dots ,N}\\widetilde{C}_{\\ell ,i}\\Delta _n \\sum _{i=1}^N\\rho _i\\end{equation*}and increase N by one in every step until this expression is close to zero.If a suitable N is found, we define C_\\ell :=\\max _{i=1,\\dots ,N}\\widetilde{C}_{\\ell ,i} and thus have equilibrated truncation and approximation errors by ensuring Eq.~(\\ref {eq:trunc}).For simplicity, we have implicitly assumed here that the processes \\ell ^{GH}_i were normalized in the sense that \\text{Var}(\\ell ^{GH}_i(t))=t.This is due to the fact that \\rho _i\\ell _i (here with \\ell _i=\\ell ^{GH}_i) in Theorem~\\ref {thm:H_error} represents the scalar product (L(t),e_i)_H with variance \\rho _it.In case we have unnormalized processes, one can simply divide \\ell ^{GH}_i by its standard deviation (see Formula~(\\ref {eq:VarZ})) and adjust the constants \\widetilde{C}_{\\ell ,i} and C_\\ell accordingly.\\subsection {Numerical results}As a test for our algorithm, we generate GH fields on the time interval [0,1] with step size \\Delta _n=2^{-6}, on the spatial domain \\mathcal {D}=[0,1].For practical aspects, one is usually interested in the L^1-error \\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|) and the L^2-error (\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^2))^{1/2}.Upper bounds for both expressions depend on \\eta and D and are given by Ineq.~(\\ref {L^p error}).To obtain reasonable errors, we refer to the discussion on the choice of D in Remark~\\ref {rem:t-conv} and set D=\\Delta _n^{1/(1-\\eta )}.This ensures that the L^1-error is of order \\mathcal {O} (\\Delta _n) and is a good trade-off between simulation time and the size of the L^2-error for most values of \\eta in the GIG example below.Choosing for example D=\\Delta _n^{2/(2-\\eta )} would reduce the L^2-error to order \\mathcal {O} (\\Delta _n), but does not have a significant effect on the L^1-error and results in a higher computational time.For the Matérn covariance operator Q we use variance v=1, correlation length r=0.1 and \\chi \\in \\lbrace \\frac{1}{2},\\frac{3}{2}\\rbrace ,where a higher value of \\chi increases the regularity of the field along the x-direction.For the fixed GH parameters we choose \\alpha =5, \\beta =\\mu =0_N, \\delta =4 and \\Gamma =\\mathbf {1}_N, the shape parameter \\lambda will vary throughout our simulation and admits the values \\lambda \\in \\lbrace -\\frac{1}{2},1\\rbrace , which results in NIG resp.",
"hyperbolic GH fields.This parameter setting ensures that the multi-dimensional GH distribution has uncorrelated marginals, hence the truncated KL expansion L_N^{GH} of L^{GH} is itself an infinite dimensional GH Lévy process.Further, for every N\\in \\mathbb {N}, the constant \\widetilde{C}_{\\ell ,i} from Section~\\ref {sec:GH_app} is independent of i=1,\\dots ,N, thus the truncation index N can easily be determined to balance out the Fourier inversion and truncation error for each combination of \\lambda and \\chi .To examine the impact of \\eta on the efficiency of the simulation, we set \\eta \\in \\lbrace 4,6,8,10\\rbrace and the constant R as suggested in Section~\\ref {sec:GH_app} for each \\eta .For fixed \\eta and R, we choose \\theta \\in \\lbrace 1,1.5\\dots ,99.5,100\\rbrace and calculate for each \\theta the constant B_\\theta as in Section~\\ref {sec:imp}.This results in up to 199 different values for the number of summations M_\\theta , which all guarantee the desired accuracy \\varepsilon , meaning we can choose the smallest M_\\theta for our simulation.The optimal value \\theta _{opt} which leads to the smallest M_\\theta depends highly on the GH parameters and may vary significantly with \\eta .For \\lambda =1, we found that \\theta _{opt} ranges from 34 to 68.5, varying with each choice of \\eta \\in \\lbrace 4,6,8,10\\rbrace .In contrast, in the second example with \\lambda =-1/2, \\theta _{opt}=11 independent of \\eta .We generate 1.000 approximations \\widetilde{L}_N^{GH} for several combinations of \\lambda , \\chi and \\eta , which allows us to check if the generated samples actually follow the desired target distributions.To this end, we conduct Kolomogorov--Smirnov tests for the subordinating GIG process as well as for the distribution of the GH field at a fixed point in time and space and report on the corresponding p-values.\\begin{figure}\\centering \\subfigure [Sample of a GH field]{\\includegraphics [scale=0.45]{hyp_m0_5.eps}}\\subfigure [Empirical dist.",
"of 1.000 samples at t=x=1]{\\includegraphics [scale=0.48]{hyp_hist_m0_5_new.eps}}\\caption {Sample and empirical distribution of an hyperbolic field with parameters \\lambda =1, \\chi =1/2, \\eta =10 and truncation after N=132 terms.",
"}\\centering \\subfigure [Sample of a GH field]{\\includegraphics [scale=0.45]{nig_m1_5.eps}}\\subfigure [Empirical dist.",
"of 1.000 samples at t=x=1]{\\includegraphics [scale=0.48]{nig_hist_m1_5_new.eps}}\\caption {Sample and empirical distribution of a NIG field with parameters \\lambda =-1/2, \\chi =3/2, \\eta =10 and truncation after N=18 terms.",
"}\\end{figure}Figures~\\ref {fig:hyp} and ~\\ref {fig:nig} show samples of approximated GH random fields:Along the time axis we see the characteristic behavior of the (pure jump) GH processes for every point x\\in \\mathcal {D}.For a fixed point in time t, the paths along the x-axis vary according to their correlation, depending on the covariance parameter \\chi .As reported in \\cite {RW06}, the eigenvalues of Q decay slower if \\chi becomes smaller, meaning we need a higher number of summations N in the KL expansion so that the error contributions are equilibrated.This effect can be seen in Tables~\\ref {tab:hyp} and~\\ref {tab:nig}, where the truncation index N changes significantly with \\chi .If the KL expansion, however, can be sampled by a N-dimensional GH process as suggested in Theorem~\\ref {thm:Z_U},the number of summations N has only a minor impact on the computational costs of the KL expansion.This is due to the fact that in this case the time consuming part, namely simulating the subordinator, has to be done only once, regardless of N.Compared to these costs, the costs of subordinating a Brownian motion of any finite dimension are negligible.The histograms in Figures~\\ref {fig:hyp} and ~\\ref {fig:nig} show the empirical distribution of the approximation \\widetilde{L}_N^{GH}(x)(t) at time t=1 and x=1.The theoretical distribution at time 1 and an arbitrary point x\\in \\mathcal {D} is again GH, where the parameters are given in Lemma~\\ref {lem:KL_GH}.Obviously, the empirical distributions fit the target GH distributions from Lemma~\\ref {lem:KL_GH}.To be more precise, we have conducted a Kolmogorov-Smirnov test for both, the subordinating GIG process and the GH field at time t=1 and for the latter at x=1.We know the law of both processes at x\\in \\mathcal {D} and are able to obtain their CDFs sufficiently precise for the tests by numerical integration.The test results for 1.000 samples of the hyperbolic resp.",
"the NIG field with covariance parameters \\chi =\\frac{1}{2} resp.",
"\\chi =\\frac{3}{2} are given in Tables~\\ref {tab:hyp} and~\\ref {tab:nig} above and do not suggest that the generated samples follow another distribution than the expected one.\\end{equation*}\\begin{table}\\end{table}\\begin{center}\\small \\begin{tabular}{l|*{3}{c|}c}\\eta & E_{GIG,N}^1 & E_{GIG,N}^1/\\Delta _n &E_{GIG,N}^2&\\mathbb {E}[||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2] \\\\ \\hline 4 & 0.0143 &0.9166& 0.2584 &0.0646\\\\6 & 0.0138 &0.8835& 0.0749 & 0.0635\\\\8 & 0.0138 &0.8824& 0.0601 &0.0634\\\\10& 0.0140 &0.8975& 0.0806 &0.0636\\\\ \\hline \\hline \\eta &N & p-value GH&abs.",
"time&rel.",
"time\\\\ \\hline 4 &130& 0.8246 & 0.1945 sec.&100.00\\% \\\\6 &133& 0.3077 & 0.1093 sec.&56.19\\% \\\\8 &133& 0.3077 & 0.0851 sec.",
"&43.78\\%\\\\10&132& 0.2873 & 0.0759 sec.",
"&39.04\\%\\\\ \\hline \\end{tabular}\\end{center}\\caption {Errors, p-values and average simulation times per field based on 1.000 simulations.Stepsize \\Delta t =2^{-6} and D=\\Delta t^{1/(1-\\eta )}.GH process: \\lambda =1, \\alpha = 5, \\beta = 0_N, \\delta = 4, \\mu = 0_N, \\Gamma = 1_{N\\times N}.Covariance parameters: \\chi =1/2, r=0.1 and v=1.The KS test for the GIG subordinator returns a p-value of 0.5498 for each \\eta \\in \\lbrace 4,6,8,10\\rbrace .",
"}\\begin{center}\\small \\begin{tabular}{l|*{3}{c|}c}\\eta & E_{GIG,N}^1 & E_{GIG,N}^1/\\Delta _n &E_{GIG,N}^2&\\mathbb {E}[||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2] \\\\ \\hline 4 & 0.0132 & 0.8443 & 0.2079 & 0.0619 \\\\6 & 0.0128 & 0.8170 & 0.0584 & 0.0608 \\\\8 & 0.0127 & 0.8155 & 0.0456 & 0.0608\\\\10& 0.0129 & 0.8252 & 0.0589 & 0.0611 \\\\ \\hline \\hline \\eta &N & p-value GH&abs.",
"time&rel.",
"time\\\\ \\hline 4 & 18 & 0.9223 & 0.1039 sec.",
"& 100.00\\% \\\\6 & 18 & 0.9223 & 0.0628 sec.",
"& 60.43\\% \\\\8 & 18 & 0.9223 & 0.0460 sec.",
"& 44.29\\% \\\\10& 18 & 0.9223 & 0.0380 sec.",
"& 38.59\\% \\\\ \\hline \\end{tabular}\\end{center}\\caption {Errors, p-values and average simulation times per field based on 1.000 simulations.Stepsize \\Delta t =2^{-6} and D=\\Delta t^{1/(1-\\eta )}.GH process: \\lambda =-1/2, \\alpha = 5, \\beta = 0_N, \\delta = 4, \\mu = 0_N, \\Gamma = 1_{N\\times N}.Covariance parameters: \\chi =3/2, r=0.1 and v=1.The KS test for the GIG subordinator returns a p-value of 0.6145 for each \\eta \\in \\lbrace 4,6,8,10\\rbrace .",
"}$ We denote by $E_{GIG,N}^1$ and $E_{GIG,N}^2$ the approximation error of the subordinator as in Eq.",
"(REF ), which we have listed in absolute terms in Tables and .",
"The first error bound is also given relative to $\\Delta _n$ to show that it is in fact of magnitude $\\mathcal {O}(\\Delta _n)$ .",
"While the $L^1(\\Omega ;\\mathbb {R})$ -error $E_{GIG,N}^1$ is relatively constant for each $\\eta $ , the $L^2(\\Omega ;\\mathbb {R})$ -error $E_{GIG,N}^2$ is rather high for $\\eta =4$ , but has an acceptable upper bound for $\\eta \\ge 6$ .",
"This is not surprising, since $D=\\Delta _n^{1/(1-\\eta )}$ only guarantees that $\\mathbb {E}(|\\ell ^{GIG}(t)-\\widetilde{\\ell }^{GIG}(t)|)=\\mathcal {O}(\\Delta _n)$ .",
"We emphasize that the (theoretic) error bounds in Tables and are very conservative as the triangle inequality and similar \"coarse\" estimates were used repeatedly in their estimation in Theorem REF and REF .",
"The truncation index $N$ is highly sensitive to $\\chi $ , but has small or no variations for fixed $\\chi $ and varying $\\eta $ .",
"Since we choose $t\\in [0,1]$ , the expression $\\mathbb {E}(||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2)$ in Tables and is an upper bound for the $L^2(\\Omega ;H)$ -error $\\sup _{t\\in [0,1]}\\mathbb {E}(||L^{GH}(t)-\\widetilde{L}^{GH}_N(t)||_H^2)$ .",
"Note that this error is small in relative terms, since by our choice of $Q$ and Eq.",
"REF we have $\\mathbb {E}(||L^{GH}(1)||^2_H)=tr(Q)=1$ .",
"The p-value of the GH distribution varies if different $N$ are chosen for the KL expansion, which is natural due to statistical fluctuations.",
"More importantly, the null hypothesis, namely that the samples follow a GH distribution with the expected parameters, is never rejected at a $5\\%$ -level.",
"As expected, the speed of the simulation heavily depends on $\\eta $ .",
"Looking at the results for $\\eta =4$ , one might argue that the Fourier inversion method is only suitable for processes where this parameter can be chosen rather high, i.e.",
"for distributions which admit a large number of finite moments.",
"To qualify this objection, we consider once more the t-distribution with three degrees of freedom and the corresponding Lévy process $\\ell ^{t3}$ from Example REF .",
"Since $\\mathbb {E}(\\ell ^{t3}(\\Delta _n))=0$ and $\\text{Var}(\\ell ^{t3}(\\Delta _n))=\\sqrt{3}\\Delta _n$ , we can choose $\\eta =2$ and hence $R=\\sqrt{3}\\Delta _n$ .",
"The characteristic function of $\\ell ^{t3}(\\Delta _n)$ is given by $(\\phi _{t3}(u))^{\\Delta _n}=\\exp (-\\sqrt{3}\\Delta _n|u|)(\\sqrt{3}|u|+1)^{\\Delta _n}$ and $B$ and $\\theta $ are estimated in the same way as for the GIG process.",
"Using again $\\Delta _n=2^{-6}$ and $D=\\Delta _n^{1/(1-\\eta )}$ , we obtain that the number of summations in the approximation is $M=12.924$ for $\\theta =\\frac{19}{2}$ .",
"The simulation time for one process $\\widetilde{\\ell }^{t3}$ with $(\\Delta _n)^{-1}=2^6$ increments in the interval $[0,1]$ is on average 0.0655 seconds, where the initial values have been approximated by matching the moments of a normal distribution (the Kolmogorov-Smirnov test for a t-distribution at $t=1$ based on 1.000 samples returns a p-value of $0.5994$ ).",
"In the GIG example, we needed $M=79.086$ terms in the summation if $\\eta =4$ is chosen and still $M=33.030$ terms for $\\eta =10$ .",
"This shows that the Fourier Inversion method is also applicable if $\\eta $ can only be chosen relatively low and that the GIG (resp.",
"GH) process is a computationally expensive example of a Lévy process."
],
[
"Simulation of Lévy processes by Fourier inversion",
"The simulation of an arbitrary one-dimensional Lévy process $\\ell =(\\ell (t),t\\in $ is not straightforward, as sufficiently many discrete realizations of $\\ell $ in $ are needed and the distribution of the increment $ (t+n)-(t)$ for some small time step $ n>0$ is not explicitly known in general.A well-known and common way to simulate a Lévy process with characteristic triplet $ (,2,)$ (see Equation (\\ref {eq:LKD}))is the \\textit {compound Poisson approximation} (CPA) suggested in~\\cite {R97} and~\\cite {S03}.All jumps of the process larger than some $$\\varepsilon $ >0$ are approximated by a sum of independent compound Poisson processes and the small jumps by their expected values resp.",
"by a Brownian motion.",
"For details and convergence theorems of this method we refer to \\cite {AR01,R97,S03}.Although the CPA is applicable in a very general setting, in the sense that only the triplet $ (,2,)$ has to be known for simulation, it has several drawbacks.It is possible to show that the CPA converges under certain assumptions in distribution to a Lévy process with characteristic triplet $ (,2,)$,and even strong error rates for CPA-type approximation schemes are given, for instance in \\cite {DHP12, F11, MS07}.The derived $ Lp$-error rates are, however, rather low with respect to the time discretization, only available for $ p2$ and/or require strong assumptions on the moments of the Lévy measure $$.Furthermore, if the cumulated density function (CDF) of $$ is unknown, numerical integration with respect to $$ is necessary.Evaluating the density of $$ at sufficiently many points to obtain a good approximation might be time consuming, especially if this involves computationally expensive components (e.g.",
"Bessel functions).It is, further, a-priori not clear how to discretize the measure $$ (we refer to a discussion on this matter in~\\cite [Chapter 8]{S03}).One could choose for example equidistant or equally weighted points, but this choice might have a significant impact on the precision and the speed of the simulation, and is impossible to be assessed beforehand.The disadvantages of the CPA method motivate the development of an alternative methodology.\\\\In the following, we introduce a new sampling approach which approximates the process $$ by a refining sequence of piecewise constant càdlàg processes $ ((n),n$\\mathbb {N}$ )$.We show its asymptotic convergence in $ Lp(;$\\mathbb {R}$ )$-sense and almost surely.",
"This approximation suffers from the fact that the necessary conditional densities from which we have to sample are not available for many Lévy processes.For a given refinement parameter $ n$, we develop, therefore, an algorithm to sample an approximation $ (n)$ of $(n)$ for which the resulting error may be bounded again in $ Lp(;$\\mathbb {R}$ )$-sense.This technique is based on the assumption that the characteristic function of $$ is available in closed form, which is true for a broad class of Lévy processes.We exploit this knowledge by so-called \\emph {Fourier inversion} to draw samples of the process^{\\prime } increments over an arbitrary large time step $ n>0$.In Section~\\ref {sec:num}, we then apply the described method to simulate GH Lévy fields.$"
],
[
"A piecewise constant approximation of $\\ell $",
"Throughout this chapter, we consider a one-dimensional Lévy process $\\ell =(\\ell (t),t\\in $ with characteristic function $\\phi _\\ell :\\mathbb {R}\\rightarrow \\mathbb {C}$ .",
"For any $t\\in , we denote by $ Ft$ the CDF of $ (t)$ and by $ ft$ the corresponding density function, provided that $ ft$ exists.Note that in this case $ Ft$ and $ ft$ belong to the probability distribution with characteristic function $ ()t$.To obtain a refining scheme of approximations of $$, we introduce a sampling algorithm for $$ based on the construction of \\textit {Lévy bridges}.In our context, a Lévy bridge is the stochastic process $ ((t)|t(t1,t2))$ pinned to given realizations of the boundary values $ (t1)$ and $ (t2)$ for $ 0t1<t2T$.It has been shown, for instance in \\cite [Proposition 2.3]{HHM11}, that these bridges are Markov processes.Assuming that the density $ ft$ exists for every $ t (see also Remark REF ), the distribution of the increment $\\ell (t)-\\ell (t_1)$ conditional on $\\ell (t_2)$ is well-defined whenever $f_{t_2-t_1}(\\ell (t_2))\\in (0,+\\infty )$ .",
"Its density function is then given as $f^{t_1,t_2}_{t}(x):=\\frac{f_{t-t_1}(x)f_{t_2-t}(\\ell (t_2)-x)}{f_{t_2-t_1}(\\ell (t_2))},$ with conditional expectation $\\mathbb {E}(\\ell (t)|\\ell (t_1),\\ell (t_2))=\\frac{\\ell (t_2)-\\ell (t_1)}{t_2-t_1}(t-t_1)$ (see [23],[29]).",
"This motivates the following sampling algorithm for a piecewise constant approximation of $\\ell $ : Algorithm 3.1 Let $n\\in \\mathbb {N}$ and generate a sample of the random variable $\\mathcal {X}_{0,1}$ with density $f_T$ .",
"Set $\\mathcal {X}_{0,0}:=0$ , $i:=1$ and $\\Delta _0:=T$ .",
"[1] $i\\le n$ Define $\\Delta _i=\\frac{T}{2^i}$ .",
"$j=0,2,\\dots ,2^i$ Set $\\mathcal {X}_{i,j}=\\mathcal {X}_{i-1,j/2}$ .",
"$j=1,3,\\dots ,2^i-1$ Generate the (conditional) increment $\\mathcal {X}_{i,j}-\\mathcal {X}_{i,j-1}$ within $[\\frac{(j-1)T}{2^{(i-1)}},\\frac{jT}{2^{(i-1)}}]$ .",
"That is, sample the random variable $X:\\Omega \\rightarrow \\mathbb {R}$ with density $x\\mapsto \\frac{f_{\\Delta _i}(x)f_{\\Delta _i}(\\mathcal {X}_{i,j+1}-x)}{f_{\\Delta _{i-1}}(\\mathcal {X}_{i,j+1})}$ and set $\\mathcal {X}_{i,j}:=X+\\mathcal {X}_{i,j-1}$ $i=i+1$ Define the piecewise constant process $\\overline{\\ell }^{(n)}(t):=\\mathcal {X}_{n,2^n}\\mathbf {1}_{\\lbrace T\\rbrace }(t)+\\sum \\limits _{j=1}^{2^n}\\mathcal {X}_{n,j-1}\\mathbf {1}_{\\lbrace [(j-1)T/2^n,jT/2^n)\\rbrace }(t)$ .",
"Eventually, the sequence $(\\overline{\\ell }^{(n)},n\\in \\mathbb {N})$ of càdlàg processes admits a pointwise limit in $L^p(\\Omega ;\\mathbb {R})$ which corresponds to the process $\\ell $ : Theorem 3.2 Let $\\phi _\\ell $ be a characteristic function of an infinitely divisible probability distribution.",
"For any $t\\in , assume the probability density $ ft$ corresponding to $ ()t$ exists.Further, for $ n$\\mathbb {N}$$, let $(n)$ be the process generated by Algorithm~\\ref {algo:bridge} and $ ft$ on $ (,(At,t0),P)$.If $$\\mathbb {R}$ |x|pf1(x)dx<$ for some $ p[1,)$, then\\begin{equation*}\\lim _{n\\rightarrow \\infty }\\mathbb {E}(|\\overline{\\ell }^{(n)}(t)-\\ell (t)|^p)=0,\\end{equation*}where $$ is a Lévy process with characteristic function $$ on $ (,(At,t0),P)$.$ For any $n\\in \\mathbb {N}$ and $t\\in we have that{\\begin{@align*}{1}{-1}&\\mathbb {E}[|\\overline{\\ell }^{(n+1)}(t)-\\overline{\\ell }^{(n)}(t)|^p]\\\\=&\\mathbb {E}\\Big (\\big |\\sum \\limits _{j=1}^{2^{n+1}}\\mathcal {X}_{n+1,j-1}\\mathbf {1}_{\\lbrace [(j-1)T/2^{n+1},jT/2^{n+1})\\rbrace }(t)-\\sum \\limits _{j=1}^{2^n}\\mathcal {X}_{n,j-1}\\mathbf {1}_{\\lbrace [(j-1)T/2^n,jT/2^n)\\rbrace }(t)\\big |^p\\Big )\\\\=&\\mathbb {E}\\Big (\\big |\\sum \\limits _{j=1}^{2^n}(\\mathcal {X}_{n+1,2j-1}-\\mathcal {X}_{n+1,2j-2})\\mathbf {1}_{\\lbrace [(2j-1)T/2^{n+1},2jT/2^{n+1})\\rbrace }(t)\\big |^p\\Big )\\\\\\end{@align*}}Since the increments $ Xn+1,j+1-Xn+1,j$ are i.i.d.",
"with characteristic function $ ()T/2n+1$ by construction, this yields$$\\mathbb {E}[|\\overline{\\ell }^{(n+1)}(t)-\\overline{\\ell }^{(n)}(t)|^p]\\le C_{\\ell ,T}2^{-n-1}\\int _\\mathbb {R}|x|^pf_1(x)dx=C_{\\ell ,T,p}2^{-n-1},$$where $ C,T$ resp.",
"$ C,T,p$ are positive constants that are independent of $ n$.Hence, for any $ m,n$\\mathbb {N}$$ with $ m>n$ it follows$$\\mathbb {E}[|\\overline{\\ell }^{(m)}(t)-\\overline{\\ell }^{(n)}(t)|^p]^{1/p}\\le C_{\\ell ,T,p}^{1/p}\\sum _{i=n+1}^m2^{-i/p}=C_{\\ell ,T,p}^{1/p}\\frac{2^{-n/p}-2^{-m/p}}{2^{1/p}-1},$$meaning that $ ((n)(t),n$\\mathbb {N}$ )$ is a $ Lp(;$\\mathbb {R}$ )$-Cauchy sequence and, therefore, admits a limit.The characteristic function of $(n)(t)$ is given by $ ()t2n/TT/2nn()t$.The claim follows since the distribution with characteristic function $$ is infinitely divisible, hence the limit process $ =((t),t$ is in fact a Lévy process.$ Corollary 3.3 Under the assumptions of Theorem REF with $p=1$ , $\\overline{\\ell }^{(n)}$ converges to $\\ell $ $\\mathbb {P}$ -almost surely as $n\\rightarrow +\\infty $ , uniformly in $.$ For any $t\\in and $$\\varepsilon $ >0$, we get by Markov^{\\prime }s inequality\\begin{equation*}\\mathbb {P}(|\\overline{\\ell }^{(n)}(t)-\\ell (t)|)\\le \\frac{\\mathbb {E}(|\\overline{\\ell }^{(n)}(t)-\\ell (t)|)}{\\varepsilon }\\le \\frac{C_{\\ell ,T,p}}{\\varepsilon }\\sum _{i=n}^\\infty 2^{-i}=\\frac{C_{\\ell ,T,p}2^{-n+1}}{\\varepsilon }.\\end{equation*}The claim then follows by the Borel-Cantelli Lemma since\\begin{equation*}\\sum _{n=1}^\\infty \\mathbb {P}(|\\overline{\\ell }^{(n)}(t)-\\ell (t)|)\\le \\frac{2C_{\\ell ,T,p}}{\\varepsilon }\\sum _{n=1}^\\infty 2^{-n}<+\\infty .\\end{equation*}$ Although Algorithm REF has convenient properties in terms of convergence, it may only be applied for a small class of Lévy processes.",
"For a general Lévy process $\\ell $ , the conditional densities in Eq.",
"(REF ) will be unknown and thus simulating from this distributions is impossible.",
"A few exceptions where “bridge sampling” of Lévy processes is feasible include the inverse Gaussian ([38]) and the tempered stable process ([27]).",
"However, if we consider a fixed parameter $n$ , sampling from the bridge distributions is equivalent to the following algorithm: Algorithm 3.4 [1] For $n\\in \\mathbb {N}$ , fix $\\Delta _n,\\Xi _n$ as in Section and generate $2^n$ i.i.d random variables $ X_1,\\dots ,X_{2^n}$ with density $f_{\\Delta _n}$ .",
"Set $\\ell ^{(n)}(t)=0$ if $t\\in [0,t_1)$ , $\\ell ^{(n)}(t)=\\sum _{k=1}^jX_k$ if $t\\in [t_j,t_{j+1})$ for $j=1,\\dots ,2^n-1$ and $\\ell ^{(n)}(T)=\\sum _{j=1}^{2^n} X_j$ .",
"The equivalence is in the sense that both processes are piecewise constant, càdlàg and all intermediate points follow the same conditional Lévy bridge distributions.",
"Note that $\\ell ^{(n)}$ coincides with $\\overline{\\ell }^{(n)}$ from Algorithm REF where the initial value has been chosen as $\\mathcal {X}_{0,1}=\\ell ^{(n)}(T)=\\sum _{j=1}^{2^n}X_j$ .",
"The advantage of Algorithm REF is that $2^n$ independent samples from the same distribution have to be generated, instead of $2^n$ random variables from (different) conditional distributions.",
"As we will see in the following section, sampling from the distribution with density $f_{\\Delta _n}$ may be achieved if the characteristic function $\\phi _\\ell $ is available.",
"In addition, we are still able to use the $L^p(\\Omega ;\\mathbb {R})$ error bounds from Theorem REF for a fixed $n\\in \\mathbb {N}$ .",
"Inversion of the Characteristic Function For an one-dimensional Lévy process $\\ell $ with characteristic function $\\phi _\\ell $ , the characteristic function of any increment $\\ell (t+\\Delta _n)-\\ell (t)$ can be expressed via $\\mathbb {E}[\\exp (iu(\\ell (t+\\Delta _n)-\\ell (t)))]=\\mathbb {E}[\\exp (iu(\\ell (\\Delta _n)))]=(\\phi _\\ell (u))^{\\Delta _n}$ for any time step $\\Delta _n>0$ .",
"If $F_{\\Delta _n}$ denotes again the CDF of this increment, we obtain by Fourier inversion (see [22]) $F_{\\Delta _n}(x)=\\frac{1}{2}-\\int _\\mathbb {R}\\frac{(\\phi _\\ell (u))^{\\Delta _n}}{2\\pi iu}\\exp (-iux)du.$ Using the well-known inverse transformation method (see also [2]) to sample from the CDF, allows us to reformulate Algorithm REF : Algorithm 3.5 [1] For $n\\in \\mathbb {N}$ , fix $\\Delta _n,\\Xi _n$ and generate i.i.d.",
"$U_1,\\dots ,U_{2^n}$ , where $U_j\\sim \\mathcal {U}([0,1])$ on $(\\Omega ,\\mathcal {A},\\mathbb {P})$ .",
"Determine $X_j:=\\inf \\lbrace x\\in \\mathbb {R}|F_{\\Delta _n}(x)=U_j\\rbrace $ for $j=1,\\dots ,2^n$ .",
"Set $\\ell ^{(n)}(t)=0$ if $t\\in [0,t_1)$ , $\\ell ^{(n)}(t)=\\sum _{k=1}^j X_k$ if $t\\in [t_j,t_{j+1})$ for $j=1,\\dots ,2^n-1$ and $\\ell ^{(n)}(T)=\\sum _{j=1}^{2^n}X_j$ .",
"The evaluation of $F$ is crucial and may, in general, only be done numerically.",
"To approximate the integral in Eq.",
"(REF ), we employ the discrete Fourier inversion method introduced in [24].",
"With this method the approximation error can be controlled with relatively weak assumptions on the characteristic function.",
"Hence, the resulting algorithm is applicable for a broad class of Lévy processes.",
"An alternative algorithm to approximate the CDFs of subordinating processes based on the inversion of Laplace transforms is described in [42].",
"Although this approach seems promising in terms of computational effort, here we only consider the Fourier inversion technique.",
"The latter is also applicable to Lévy processes without bounded variation and yields uniform error bounds on the approximated CDF.",
"Assumption 3.6 The distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ is continuous with finite variance and CDF $F_{\\Delta _n}$ .",
"Furthermore, there exists a constant $R>0$ and $\\eta >1$ such that $F_{\\Delta _n}(-x)\\le R|x|^{-\\eta }$ and $1-F_{\\Delta _n}(x)\\le R|x|^{-\\eta }$ for all $x>0$ .",
"there exists a constant $B > 0$ and $\\theta >0$ such that $|(\\phi _\\ell (u))^{\\Delta _n}|\\le B|\\frac{u}{2\\pi }|^{-\\theta }$ for all $u\\in \\mathbb {R}$ .",
"In case of infinite variance, we consider bounds on the density function instead: Assumption 3.7 The distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ is continuous with density $f_{\\Delta _n}$ .",
"Furthermore, there exists a constant $R>0$ and $\\eta >1$ such that $|f_{\\Delta _n}(x)|\\le R|x|^{-\\eta }$ for all $x\\in \\mathbb {R}$ .",
"there exists a constant $B > 0$ and $\\theta >0$ such that $|(\\phi _\\ell (u))^{\\Delta _n}|\\le B|\\frac{u}{2\\pi }|^{-\\theta }$ for all $u\\in \\mathbb {R}$ .",
"Remark 3.8 In the case that $\\theta >1$ , we have that $\\int _\\mathbb {R}|(\\phi _\\ell (u))^{\\Delta _n}|du\\le 2+2B\\int _1^\\infty \\left(\\frac{u}{2\\pi }\\right)^{-\\theta }<\\infty ,$ which already implies the existence of a continuous density $f_{\\Delta _n}$ in both scenarios, see for example [40].",
"Usually, $F_{\\Delta _n}$ or $f_{\\Delta _n}$ are unknown, but only the characteristic function $(\\phi _\\ell )^{\\Delta _n}$ is given.",
"To obtain $R$ and $\\eta $ , one can choose $R=(-1)^{k}\\frac{d^{2k}}{du^{2k}}((\\phi _\\ell (u))^{\\Delta _n})\\big |_{u=0}$ and $\\eta =2k$ in Ass.",
"REF , resp.",
"$R=\\frac{1}{2\\pi }\\int _\\mathbb {R}|\\frac{d^{k}}{du^{k}}((\\phi _\\ell (u))^{\\Delta _n})|du$ and $\\eta =k$ in Ass.",
"REF , where $k$ is any non-negative integer such that the derivatives exist (see [24]).",
"For example, in the first set of assumptions, the finite variance ensures that we can use $\\eta =2$ and $R$ equal to the second moment of the distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"As an approximation of $F_{\\Delta _n}$ as in Eq.",
"(REF ) we introduce the function $\\widetilde{F}_{\\Delta _n}$ given by $\\widetilde{F}_{\\Delta _n}(x):=\\sum _{k=-M/2}^{M/2}q_k\\exp (-i2\\pi kx/J),$ for $x\\in \\mathbb {R}$ , where $q_k:={\\left\\lbrace \\begin{array}{ll}1/2&\\text{for $k=0$}\\\\\\frac{1-\\cos (2\\pi \\kappa k)}{i2\\pi k}(\\phi _\\ell (-2\\pi k/J))^{\\Delta _n}&\\text{for $0<|k|<M/2$}\\\\0&\\text{for $k=M/2$}\\\\\\end{array}\\right.",
"},$ $M$ is an even integer and $\\kappa ,J>0$ are parameters which are determined below.",
"Note that $q_k=\\overline{q_{-k}}$ , where $\\overline{z}$ denotes the complex conjugate for $z\\in \\mathbb {C}$ .",
"The Hermitean symmetry also holds for the function $k\\mapsto \\exp (-i2\\pi kx/J)$ .",
"This ensures that, for every $x\\in \\mathbb {R}$ , we have $\\widetilde{F}_{\\Delta _n}(x)&=\\frac{1}{2}+\\sum _{k=1}^{M/2-1}q_k\\exp (-i2\\pi kx/J)+\\overline{q_k\\exp (-i2\\pi kx/J)}\\\\&=\\frac{1}{2}+2\\,\\text{Re}\\Big (\\sum _{k=1}^{M/2-1}q_k\\exp (-i2\\pi kx/J)\\Big )$ and hence $\\widetilde{F}_{\\Delta _n}(x)\\in \\mathbb {R}$ for any real-valued argument $x$ .",
"The last identity should be exploited during the simulation to save computational time as here only half the summation is required.",
"Lastly, we denote by $\\zeta (z,s):=\\sum _{k=0}^\\infty (k+s)^{-z}$ for $s,z\\in \\mathbb {C}$ with $\\text{Re}(s)>0$ and $\\text{Re}(z)>1$ the Hurwitz zeta function and define as in [24] $V_1(\\kappa ,\\eta )&:=(\\kappa /2)^{-\\eta }+2\\zeta (\\eta ,1-\\frac{\\kappa }{2})+\\zeta (\\eta ,1+\\frac{\\kappa }{2})+\\zeta (\\eta ,1-\\frac{3\\kappa }{2}),\\\\V_2(\\kappa ,\\eta )&:=\\frac{2^{\\eta -1}\\kappa ^{1-\\eta }}{\\eta -1}+\\frac{\\kappa }{2}\\Big (2\\zeta (\\eta ,1-\\frac{\\kappa }{2})+\\zeta (\\eta ,1+\\frac{\\kappa }{2})+\\zeta (\\eta ,1-\\frac{3\\kappa }{2})\\Big ).$ The expressions $V_1(\\kappa ,\\eta )$ and $V_2(\\kappa ,\\eta )$ establish conditions on the choice of the (not yet determined) parameter $\\kappa $ in Theorem REF .",
"For a given domain parameter $D>0$ and accuracy $\\varepsilon >0$ the approximation $\\widetilde{F}_{\\Delta _n}$ should fulfill the error bound $|\\widetilde{F}_{\\Delta _n}(x)-F_{\\Delta _n}(x)|<\\varepsilon \\quad \\text{for $x\\in [-D/2,D/2]$}.$ Once $\\kappa $ is determined, this can be achieved by choosing a sufficiently large parameter $J$ and, based on this $J$ , a sufficiently large number of summands $M$ .",
"Admissible values for $\\kappa $ , $J$ and $M$ depend on $D$ , $\\varepsilon $ and the constants in Assumption REF resp.",
"REF .",
"Theorem 3.9 ([24]) Let $D>0$ and $\\varepsilon >0$ .",
"If Assumption REF holds, choose $\\kappa $ , $J$ and $M$ such that $0<\\kappa <\\frac{2}{3}\\quad \\text{and}\\quad \\kappa ^\\eta V_1(\\kappa ,\\eta )\\le 2^{\\eta +1},$ $J\\ge \\frac{D}{\\kappa }\\quad \\text{and}\\quad J\\ge \\Big (\\frac{3RV_1(\\kappa ,\\eta )}{2\\varepsilon }\\Big )^{1/\\eta },$ and $M\\ge 2+2J\\Big (\\frac{6B}{\\varepsilon \\pi \\theta }\\Big )^{1/\\theta }.$ If Assumption REF holds, choose $\\kappa $ , $J$ and $M$ such that $0<\\kappa <\\frac{2}{3}\\quad \\text{and}\\quad \\kappa ^{\\eta -1} V_2(\\kappa ,\\eta )\\le \\frac{2^\\eta }{\\eta -1},$ $J\\ge \\frac{D}{\\kappa }\\quad \\text{and}\\quad J\\ge \\Big (\\frac{3RV_2(\\kappa ,\\eta )}{2\\varepsilon }\\Big )^{1/(\\eta -1)},$ and $M\\ge 2+2J\\Big (\\frac{6B}{\\varepsilon \\pi \\theta }\\Big )^{1/\\theta }.$ This yields, for either case, that $|F_{\\Delta _n}(x)-\\widetilde{F}_{\\Delta _n}(x)|<\\varepsilon $ for all $x\\in [-D/2,D/2]$ and it is always possible to find a $\\kappa $ that meets the given conditions.",
"Remark 3.10 In [24], by $J\\ge \\frac{2}{\\kappa }\\left(\\frac{3R}{\\varepsilon }\\right)^{1/\\eta }\\text{resp.",
"}\\quad J\\ge \\frac{2}{\\kappa }\\left(\\frac{3R}{\\varepsilon (\\eta -1)}\\right)^{1/(\\eta -1)}$ in fact stricter conditions are imposed on $J$ .",
"The proofs of Theorems 10 and 11 in [24] still give immediately a proof for Theorem REF .",
"The advantage of the bounds in Theorem REF is that they produce a smaller approximation error in the following analysis (see also Remark REF ).",
"We refer to [24] for an optimal choice of $\\kappa $ depending on $\\eta $ .",
"Once $\\kappa $ is determined, it is favorable to choose $D$ and $\\varepsilon $ in a way such that none of the parameters has a dominant effect on the resulting number of summations $M$ .",
"This is ensured if the two lower bounds on $J$ are equal, meaning for fixed $D>0$ we set $\\varepsilon =\\frac{3}{2}RV_1(\\kappa ,\\eta )\\kappa ^\\eta D^{-\\eta }\\quad \\text{resp.",
"}\\quad \\varepsilon =\\frac{3}{2}RV_2(\\kappa ,\\eta )\\kappa ^{\\eta -1}D^{-\\eta +1}$ if the first resp.",
"second set of assumptions holds.",
"Since the approximation error $|\\widetilde{F}_{\\Delta _n}(x)-F_{\\Delta _n}(x) |$ is only bounded for $x\\in [-D/2,D/2]$ , we have to modify the third step in Algorithm REF : Algorithm 3.11 [1] For $n\\in \\mathbb {N}$ , fix $\\Delta _n,\\Xi _n$ and generate i.i.d.",
"$(U_j\\sim \\mathcal {U}([0,1]),j=1,\\ldots ,2^n)$ on $(\\Omega ,\\mathcal {A},\\mathbb {P})$ .",
"Set, for $j=1,\\dots ,2^n$ $\\widetilde{X}_j={\\left\\lbrace \\begin{array}{ll}-D/2 & \\text{if $U_j<\\min \\lbrace \\widetilde{F}_{\\Delta _n}([-D/2,D/2])\\rbrace $}\\\\D/2 & \\text{if $U_j>\\max \\lbrace \\widetilde{F}_{\\Delta _n}([-D/2,D/2])\\rbrace $}\\\\\\inf \\lbrace x\\in [-D/2,D/2]\\big |\\widetilde{F}_{\\Delta _n}(x)=U_j \\rbrace &\\text{if $U_j\\in \\widetilde{F}_{\\Delta _n}([-D/2,D/2])$}\\end{array}\\right.",
"}.$ Set $\\widetilde{\\ell }^{(n)}(t)=0$ if $t\\in [0,t_1)$ , $\\widetilde{\\ell }^{(n)}(t)=\\sum _{k=1}^j\\widetilde{X}_k$ if $t\\in [t_j,t_{j+1})$ for $j=1,\\dots ,2^n-1$ and $\\widetilde{\\ell }^{(n)}(T)=\\sum _{j=1}^{2^n} \\widetilde{X}_j$ .",
"Intuitively, if we choose $D$ large and $\\varepsilon $ small enough, the atoms in the distribution of $\\widetilde{X}_i$ at $\\pm D/2$ disappear.",
"The function $\\widetilde{F}_{\\Delta _n}$ is then sufficiently close to the CDF $F_{\\Delta _n}$ , hence the generated random variables $\\widetilde{X}_i$ will have a distribution similar to $F_{\\Delta _n}$ .",
"From here on, we define $X$ as the random variable which is generated from $U\\sim \\mathcal {U}([0,1])$ by inversion of the (exact) CDF $F_{\\Delta _n}$ and $\\widetilde{X}$ as the random variable generated from $U$ by inversion of the approximated CDF $\\widetilde{F}_{\\Delta _n}$ .",
"Theorem 3.12 Let $\\widetilde{F}_{\\Delta _n}$ be the approximation of $F_{\\Delta _n}$ which is valid for parameters $D>0$ and $\\varepsilon >0$ in the sense of Theorem REF .",
"Then $\\widetilde{X}$ converges in distribution to a random variable $X$ with CDF equal to $F_{\\Delta _n}$ as $D\\rightarrow \\infty $ and $\\varepsilon \\rightarrow 0$ .",
"First, note that $\\widetilde{F}_{\\Delta _n}$ is not necessarily monotone and might admit arbitrary values outside of $[-D/2,D/2]$ , thus cannot be regarded as a CDF.",
"Since $\\widetilde{X}$ only admits values in the desired interval, we obtain probability zero for the event that $|\\widetilde{X}|>D/2$ .",
"With this in mind we construct the CDF of $\\widetilde{X}$ and show its convergence in distribution using Portmanteau's theorem.",
"We define the function $\\widehat{F}:\\mathbb {R}\\rightarrow [0,1],\\quad x\\mapsto {\\left\\lbrace \\begin{array}{ll}0 & \\text{if $x<-D/2$}\\\\\\min (1,m_D(x))\\mathbf {1}_{\\lbrace m_D(x)>0\\rbrace } & \\text{if $x\\in [-D/2,D/2]$}\\\\1 & \\text{if $x>D/2$}\\\\\\end{array}\\right.",
"},$ where $m_D(x):=\\max _{y\\in [-D/2,x]}\\widetilde{F}_{\\Delta _n}(y)$ .",
"The continuity of $\\widetilde{F}_{\\Delta _n}$ guarantees that $m_D(x)$ is well-defined for each $x\\in [-D/2,D/2]$ .",
"Clearly, $\\widehat{F}$ is monotone increasing and $\\mathbb {P}(\\widetilde{X}\\le x)=\\widehat{F}(x)$ if $|x|>D/2$ .",
"For $|x|\\le D/2$ we have that $\\begin{split}\\mathbb {P}(\\widetilde{X}\\le x)&=\\mathbb {P}(\\inf \\lbrace |y|\\le D/2\\,|\\,\\widetilde{F}_{\\Delta _n}(y)\\ge U\\rbrace \\le x)\\\\&=\\mathbb {P}(\\max \\limits _{y\\in [-D/2,x]}\\widetilde{F}_{\\Delta _n}(y)\\ge U)\\\\&=\\min (1,m_D(x))\\mathbf {1}_{\\lbrace m_D(x)>0\\rbrace }=\\widehat{F}(x),\\end{split}$ hence $\\widehat{F}$ is the CDF of $\\widetilde{X}$ .",
"With the monotonicity of $F_{\\Delta _n}$ and $|F_{\\Delta _n}-\\widetilde{F}_{\\Delta _n}|<\\varepsilon $ on $[-D/2,D/2]$ we get $\\widehat{F}(x)&=\\min (1,m_D(x))\\mathbf {1}_{\\lbrace m_D(x)>0\\rbrace }\\le \\min (1,\\max \\limits _{y\\in [-D/2,x]}\\widetilde{F}_{\\Delta _n}(y))\\\\&\\le \\min (1,\\max \\limits _{y\\in [-D/2,x]}F_{\\Delta _n}(y)+\\varepsilon )=\\min (1,F_{\\Delta _n}(x)+\\varepsilon ),$ for $x\\in [-D/2,D/2]$ and analogously $\\widehat{F}(x)\\ge \\min (1,\\max \\limits _{y\\in [-D/2,x]}F_{\\Delta _n}(y)-\\varepsilon )\\mathbf {1}_{\\lbrace \\max \\limits _{y\\in [-D/2,x]}F_{\\Delta _n}(y)-\\varepsilon >0\\rbrace }=\\max (F_{\\Delta _n}(x)-\\varepsilon ,0),$ thus $|\\widehat{F}(x)-F_{\\Delta _n}(x)|\\le \\varepsilon .$ We choose sequences $(D_k,k\\in \\mathbb {N})$ and $(\\varepsilon _m,m\\in \\mathbb {N})$ with $\\lim _{k\\rightarrow \\infty }D_k=+\\infty $ , $\\lim _{m\\rightarrow \\infty }\\varepsilon _m=0$ and denote by $\\widehat{F}_{k,m}$ the CDF of the random variables $\\widetilde{X}_{k,m}$ corresponding to each $D_k$ and $\\varepsilon _m$ .",
"For every $x\\in \\mathbb {R}$ there is some $k^*$ such that $x\\in [-D_k/2,D_k/2]$ for all $k\\ge k^*$ , hence $\\lim \\limits _{m\\rightarrow \\infty }\\lim \\limits _{k\\rightarrow \\infty }|\\widehat{F}_{k,m}(x)-F_{\\Delta _n}(x)|\\le \\lim \\limits _{m\\rightarrow \\infty }\\varepsilon _m=0$ and the claim follows by Portmanteau's theorem.",
"Remark 3.13 Before showing the convergence of $\\widetilde{X}$ to $X$ in $L^p(\\Omega ;\\mathbb {R})$ , we have to make sure that the random variables $\\widetilde{X}$ generated by Algorithm REF are actually defined on the same probability space $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ as $X$ .",
"Since $X$ represents the increment of a Lévy process $\\ell $ on $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ with CDF $F_{\\Delta _n}$ , we may define the mapping $U:=F_{ \\Delta _n}\\circ X:\\Omega \\rightarrow [0,1]$ .",
"It is then easily verified that $U$ is a $\\mathcal {U}([0,1])$ -distributed random variable.",
"For fixed parameters $D,\\varepsilon >0$ and an approximation $\\widetilde{F}_{\\Delta _n}$ of $F_{\\Delta _n}$ we define the pseudo inverse of $\\widetilde{F}_{\\Delta _n}$ (as in Algorithm REF ) as $\\widetilde{F}_{\\Delta _n}^{-1}:[0,1]\\rightarrow \\mathbb {R},\\;\\, u\\mapsto {\\left\\lbrace \\begin{array}{ll}-D/2 & \\text{if $u<\\min \\lbrace \\widetilde{F}_{\\Delta _n}([-D/2,D/2])\\rbrace $}\\\\D/2 & \\text{if $u>\\max \\lbrace \\widetilde{F}_{\\Delta _n}([-D/2,D/2])\\rbrace $}\\\\\\inf \\lbrace x\\in [-D/2,D/2]\\big |\\widetilde{F}_{\\Delta _n}(x)=u\\rbrace &\\text{if $u\\in \\widetilde{F}_{\\Delta _n}([-D/2,D/2])$}\\end{array}\\right.",
"}.$ We note that $\\widetilde{F}_{\\Delta _n}^{-1}$ is a piecewise continuous, thus measurable, mapping which implies that $\\widetilde{X}=\\widetilde{F}_{\\Delta _n}^{-1}\\circ F_{\\Delta _n}\\circ X:\\Omega \\rightarrow [-D/2,D/2]$ is a random variable on $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ .",
"Under additional, but natural, assumptions, it is possible to show stronger convergence results of the approximation for both sets of assumptions.",
"Theorem 3.14 ($L^p(\\Omega ;\\mathbb {R})$ -convergence I) Let $F_{\\Delta _n}$ be continuously differentiable on $\\mathbb {R}$ with density $f_{\\Delta _n}$ (see Remark REF ) and $(\\phi _\\ell )^{\\Delta _n}$ be bounded as in Assumption REF with $\\eta >1$ .",
"Furthermore, assume that the approximation parameters $D$ and $\\varepsilon $ fulfill $D=C\\varepsilon ^{-d}$ for $C,d>0$ .",
"If $d<\\frac{1}{p}$ , then for all $p\\in [1,\\eta )$ $\\mathbb {E}(|\\widetilde{X}-X|^p)\\rightarrow 0 \\quad \\text{as $\\varepsilon \\rightarrow 0$}.$ Let $\\varepsilon >0$ , $D=C\\varepsilon ^{-d}$ and $p\\in [1,\\eta )$ be as in the claim.",
"We split the expectation in the following way $\\mathbb {E}(|\\widetilde{X}-X|^p)=\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })+\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|\\le D/2\\rbrace }),$ and show the convergence for each term on the right hand side.",
"Recall that $\\widetilde{X}\\in [-D/2,D/2]$ by construction.",
"We obtain for the first term $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })&\\le \\int _{D/2}^\\infty |-D/2-x|^pf_{\\Delta _n}(x)dx+\\int _{-\\infty }^{-D/2}|D/2-x|^pf_{\\Delta _n}(x)dx\\\\&=\\int _{D/2}^\\infty (D/2+x)^p(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx\\\\&=\\int _{D/2}^\\infty \\int _{0}^{D/2+x}py^{p-1}dy(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx.$ Using that $(x,y)\\in (D/2,\\infty )\\times (0,D/2+x) \\Leftrightarrow (x,y)\\in (D/2,\\infty )\\times (0,D/2)\\cup (y,\\infty )\\times (D/2,\\infty ),$ we may use Fubini's theorem to exchange the order of integration and rewrite $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })&\\le \\int _{D/2}^\\infty \\int _{0}^{D/2+x}py^{p-1}dy(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx\\\\&=\\int _0^{D/2} \\int _{D/2}^\\infty (f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx py^{p-1}dy \\\\&\\quad + \\int _{D/2}^\\infty \\int _{y}^{\\infty } (f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx py^{p-1}dy\\\\&=\\int _0^{D/2} (1-F_{\\Delta _n}(D/2)+F_{\\Delta _n}(-D/2)) py^{p-1}dy \\\\&\\quad +\\int _{D/2}^\\infty (1-F_{\\Delta _n}(y)+F_{\\Delta _n}(-y)) py^{p-1}dy .\\\\$ With the bounds on $F_{\\Delta _n}$ from Assumption REF we then have $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })&\\le 2R (D/2)^{-\\eta } \\int _0^{D/2} py^{p-1}dy+2Rp\\int _{D/2}^\\infty \\frac{y^{p-1}}{y^\\eta }dy \\\\&= 2R (D/2)^{p-\\eta }+ 2Rp\\zeta (\\eta +1-p,D/2).$ Note that the Hurwitz zeta function $\\zeta $ is well-defined (as $\\eta >p$ ) and converges to 0 as $D\\rightarrow \\infty $ .",
"For the second term, consider two realizations of the random variables $X(\\omega )$ and $\\widetilde{X}(\\omega )$ for some $\\omega \\in \\Omega $ , where $|X(\\omega )|\\le D/2$ .",
"$F_{\\Delta _n}$ is continuously differentiable by assumption, hence $F_{\\Delta _n}(X(\\omega ))-F_{\\Delta _n}(\\widetilde{X}(\\omega ))=f_{\\Delta _n}(\\xi (\\omega ))(X(\\omega )-\\widetilde{X}(\\omega )),$ with $\\xi (\\omega )$ lying in between $X(\\omega )$ and $\\widetilde{X}(\\omega )$ , meaning $|\\xi (\\omega )|\\le D/2$ and $\\mathbf {1}_{\\lbrace |X(\\omega )|\\le D/2\\rbrace }(\\omega )\\le \\mathbf {1}_{\\lbrace |\\xi (\\omega )|\\le D/2\\rbrace }(\\omega ).$ For $\\widetilde{\\varepsilon }:=C^{-1}\\varepsilon ^{d+1}>0$ we split the expectation once more into $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|\\le D/2\\rbrace })\\le &\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |\\xi |\\le D/2\\rbrace })\\\\\\le &\\underbrace{\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |\\xi |\\le D/2,f(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace })}_{:=I}+\\underbrace{\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |\\xi |\\le D/2,f(\\xi )<\\widetilde{\\varepsilon }\\rbrace })}_{:=II}.$ In case that $f_{\\Delta _n}(\\xi (\\omega ))\\ge \\widetilde{\\varepsilon }$ , we can rearrange the terms to $|\\widetilde{X}(\\omega )-X(\\omega )|^p=\\frac{|F_{\\Delta _n}(\\widetilde{X}(\\omega ))-F_{\\Delta _n}(X(\\omega ))|^p}{f_{\\Delta _n}(\\xi (\\omega ))^p}.$ If $X$ and $\\widetilde{X}$ are generated by $U\\sim \\mathcal {U}([0,1])$ and $\\widehat{F}_{\\Delta _n}$ denotes again the CDF of $\\widetilde{X}$ , this yields $|\\widetilde{X}(\\omega )-X(\\omega )|^p&=\\frac{|F_{\\Delta _n}(\\widetilde{X}(\\omega ))-\\widehat{F}_{\\Delta _n}(\\widetilde{X}(\\omega ))|^p}{f_{\\Delta _n}(\\xi (\\omega ))^p}<\\frac{\\varepsilon ^p}{f_{\\Delta _n}(\\xi (\\omega ))^p},$ where we have used that $U(\\omega )=F_{\\Delta _n}(X(\\omega ))=\\widehat{F}_{\\Delta _n}(\\widetilde{X}(\\omega ))$ and $|F_{\\Delta _n}(\\widetilde{X}(\\omega ))-\\widehat{F}_{\\Delta _n}(\\widetilde{X}(\\omega ))|<\\varepsilon $ (see Theorem REF ) in the second step.",
"This gives a bound for $I$ : $ \\begin{split}I&< \\varepsilon ^p\\,\\mathbb {E}(f_{\\Delta _n}(\\xi )^{-p}\\mathbf {1}_{\\lbrace |\\xi |\\le D/2,f_{\\Delta _n}(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace })\\\\&=\\varepsilon ^p\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace }f_{\\Delta _n}(\\xi )^{1-p}d\\xi \\le \\frac{\\varepsilon ^p}{\\widetilde{\\varepsilon }^{p-1}}\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace }d\\xi .\\end{split}$ If $f_{\\Delta _n}(\\xi (\\omega ))<\\widetilde{\\varepsilon }$ , we obtain by $|\\widetilde{X}(\\omega )-X(\\omega )|\\mathbf {1}_{\\lbrace |X(\\omega )|\\le D/2\\rbrace }\\le D$ $ \\begin{split}II&\\le D^p\\,\\mathbb {E}(\\mathbf {1}_{\\lbrace |\\xi |\\le D/2,f_{\\Delta _n}(\\xi )<\\widetilde{\\varepsilon }\\rbrace })\\\\&=D^p\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )<\\widetilde{\\varepsilon }\\rbrace }f_{\\Delta _n}(\\xi )d\\xi <D^p\\,\\widetilde{\\varepsilon }\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )<\\widetilde{\\varepsilon }\\rbrace }d\\xi \\end{split}$ and hence by Eqs.",
"(REF ) and (REF ) and $\\widetilde{\\varepsilon }=C^{-1}\\varepsilon ^{1+d}$ $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|\\le D/2\\rbrace })\\le I+II< D^p\\,\\widetilde{\\varepsilon }\\Big (\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )\\ge \\widetilde{\\varepsilon }\\rbrace }d\\xi +\\int _{-D/2}^{D/2}\\mathbf {1}_{\\lbrace f_{\\Delta _n}(\\xi )<\\widetilde{\\varepsilon }\\rbrace }d\\xi \\Big )=D^{p+1}\\widetilde{\\varepsilon }.$ With the estimate for $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|> D/2\\rbrace })$ , $D=C\\varepsilon ^{-d}$ and $\\widetilde{\\varepsilon }=C^{-1}\\varepsilon ^{1+d}$ this leads to $\\mathbb {E}(|\\widetilde{X}-X|^p)&\\le 2Rp\\zeta (\\eta +1-p,D/2)+2R(D/2)^{p-\\eta }+D^{p+1}\\widetilde{\\varepsilon }\\\\&=2Rp\\zeta (\\eta +1-p,D/2)+2R(D/2)^{p-\\eta }+C^{1/d} D^{p-1/d},$ and since $0<d<\\frac{1}{p}$ and $\\eta >p$ by assumption, $\\mathbb {E}(|\\widetilde{X}-X|^p)\\rightarrow 0 \\quad \\text{as $\\varepsilon \\rightarrow 0$}.$ Remark 3.15 The relation $\\widetilde{\\varepsilon }=C^{-1}\\varepsilon ^{1+d}$ is chosen such that the factors preceding the integrals in Eqs.",
"(REF ) and (REF ) are equilibrated.",
"As only the sum of the two integrals is known a-priori, this leads to a better error estimation compared to non-equilibrated factors.",
"Theorem 3.16 ($L^p(\\Omega ;\\mathbb {R})$ -convergence II) Let $F_{\\Delta _n}$ be continuously differentiable on $\\mathbb {R}$ with density $f_{\\Delta _n}$ and $(\\phi _\\ell )^{\\Delta _n}$ be bounded as in Assumption REF with $\\eta >2$ .",
"Furthermore, assume that the approximation parameters $D$ and $\\varepsilon $ fulfill $D=C\\varepsilon ^{-d}$ for $C,d>0$ .",
"If $d<\\frac{1}{p}$ , then for all $p\\in [1,\\eta )$ $\\mathbb {E}(|\\widetilde{X}-X|^p)\\rightarrow 0 \\quad \\text{as $\\varepsilon \\rightarrow 0$}.$ Let $\\varepsilon >0$ , $D=C\\varepsilon ^{-d}$ and $p\\in [1,\\eta -1)$ .",
"Again, we split the expectation into $\\mathbb {E}(|\\widetilde{X}-X|^p)=\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })+\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|\\le D/2\\rbrace }),$ and show convergence for the first term only, as the second term can be treated analogously to Theorem REF .",
"In the same way as in Theorem REF , we may write for the first term $\\mathbb {E}(|\\widetilde{X}-X|^p\\mathbf {1}_{\\lbrace |X|>D/2\\rbrace })\\le \\int _{D/2}^\\infty (D/2+x)^p(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx,$ and further, by Assumption REF , follows $\\int _{D/2}^\\infty (D/2+x)^p(f_{\\Delta _n}(x)+f_{\\Delta _n}(-x))dx &\\le 2R\\int _{D/2}^\\infty (D/2+x)^px^{-\\eta }dx \\\\&=2^{p+1}R\\int _0^{\\infty }\\frac{(D/2+x/2)^p}{(D/2+x)^\\eta }dx\\\\&<2^{p+1}R\\zeta (\\eta -p,C\\varepsilon ^{-d}/2),$ which tends to zero as $\\varepsilon \\rightarrow 0$ , because $\\eta >p+1$ .",
"Remark 3.17 As expected, the admissible range of values for $d$ and $\\eta $ narrows as the rate of convergence $p$ increases.",
"For example, to obtain $L^2(\\Omega ;\\mathbb {R})$ -convergence, we need $d<\\frac{1}{2}$ and $\\eta >2$ in Theorem REF and $\\eta >3$ in Theorem REF .",
"Recall Remark REF , where we have concluded that optimal relations between $D$ and $\\varepsilon $ are given by $&D=\\kappa \\left(3/2\\,RV_1(\\kappa ,\\eta )\\right)^{1/\\eta }\\varepsilon ^{-1/\\eta } \\quad &&\\text{if Assumption~\\ref {ass:1} holds and}\\\\&D=\\kappa \\left(3/2\\,RV_2(\\kappa ,\\eta )\\right)^{1/(\\eta -1)}\\varepsilon ^{-1/(\\eta -1)}\\quad &&\\text{if Assumption~\\ref {ass:2} holds}.$ For $L^p(\\Omega ;\\mathbb {R})$ -convergence, we need in both cases $D=C\\varepsilon ^{-d}$ , where $C>0$ and $d\\in (0,1/p)$ .",
"Hence, we can simply use $C=\\kappa (3/2\\,RV_1(\\kappa ,\\eta ))^{1/\\eta }$ and $d=1/\\eta <1/p$ in the first scenario and $C=\\kappa (3/2\\,RV_2(\\kappa ,\\eta ))^{1/\\eta }$ and $d=1/(\\eta -1)<1/p$ for the second set of assumptions.",
"This explains the bounds on $J$ (see also Remark REF ): In Theorem REF , we obtain the expression $C^\\eta D^{p-\\eta }$ as a term of the overall error.",
"If we had used the restrictions on $J$ as in [24], we would have used $C=2(3R)^{1/\\eta }§$ instead of the choice above and this would have resulted in an error term $C^\\eta D^{p-\\eta }$ being nearly twice as large (the argumentation works analogously for Theorem REF ).",
"Example 3.18 The conditions $\\eta >p$ in Theorem REF and $\\eta >p-1$ in Theorem REF can not be relaxed as the following examples show: First, we investigate the Student's t-distribution with 3 degrees of freedom and density function $f^{t3}(x)=\\frac{\\Gamma _G(2)}{\\sqrt{3\\pi }\\Gamma _G(3/2)}\\left(1+\\frac{x^2}{3}\\right)^{-2},$ where $x\\in \\mathbb {R}$ and $\\Gamma _G(\\cdot )$ is the Gamma function.",
"As shown in [26], this distribution is infinitely divisible and has characteristic function $\\phi _{t3}(u):=\\exp (-\\sqrt{3} |u|)(\\sqrt{3} |u|+1),$ hence we can define a Lévy process $(\\ell ^{t3}(t),t\\in $ with $\\phi _{t3}(u)$ as characteristic function.",
"For simplicity we set $\\Delta _n=1$ .",
"In this case the (symmetric) distribution of the increment $\\ell ^{t3}(t+\\Delta _n)-\\ell ^{t3}(t)$ has zero mean, finite variance, and its CDF $F_1$ can be bounded for $x>0$ by $F_1(-x)=1-F_1(x)=\\int _{-\\infty }^{-x}f^{t3}(y)dy<\\frac{\\Gamma _G(2)}{\\sqrt{3\\pi }\\Gamma _G(3/2)}\\int _{-\\infty }^{-x}\\frac{3^2}{y^4}dy=\\frac{\\sqrt{3}\\Gamma _G(2)}{\\sqrt{\\pi }\\Gamma _G(3/2)}x^{-3} =:Rx^{-3}.$ Thus, this yields $\\eta =3$ .",
"The bounds for $\\phi _{t3}$ are also straightforward: $|\\phi _{t3}(u)|\\le (2\\pi )^{-1}\\max \\limits _{\\widehat{u}>0}\\exp (-\\sqrt{3} \\widehat{u})(\\sqrt{3} \\widehat{u}^2+\\widehat{u}) |\\frac{u}{2\\pi }|^{-1} =:B |\\frac{u}{2\\pi }|^{-1},$ where the maximum in $B$ is found by differentiation giving $\\widehat{u}=(1+\\sqrt{5})/(2\\sqrt{3})$ .",
"Now, all requirements for $L^3$ -convergence except $\\eta >3$ are fulfilled.",
"But the t-distribution with 3 degrees of freedom does not admit a third moment, hence we cannot have $L^3(\\Omega ;\\mathbb {R})$ -convergence although $\\eta =3$ .",
"For the second case we consider the (standard) Cauchy process with characteristic function $(\\phi _C(u))^t=\\exp (-t|u|)$ .",
"It can be shown that the increment over time $\\Delta _n>0$ is again Cauchy-distributed with density $f^C_{\\Delta _n}(x)=\\frac{\\Delta _n}{\\pi (\\Delta _n^2+x^2)}.$ This means the CDF of the increment is continuously differentiable and the bounds as in Assumption REF are easily found by $f^C_{\\Delta _n}(x)\\le (\\Delta _n/\\pi )|x|^{-2}$ for $x\\in \\mathbb {R}$ and $|(\\phi _C(u))^{\\Delta _n}|\\le (2\\pi )^{-1}\\max \\limits _{u\\in \\mathbb {R}}u\\exp (-\\Delta _n |u|)|\\frac{u}{2\\pi }|^{-1}=(2\\pi \\Delta _n)^{-1}\\exp (-1)|\\frac{u}{2\\pi }|^{-1}$ for $u\\in \\mathbb {R}$ .",
"But clearly, $L^p(\\Omega ;\\mathbb {R})$ -convergence in the sense of Theorem REF for any $p\\ge 1$ is impossible, as the Cauchy process does not have any finite moments.",
"From $L^p$ -convergence follows almost sure convergence by a Borel–Cantelli-type argument, given $\\eta $ in Assumptions REF and REF is large enough.",
"Corollary 3.19 Under the assumptions of Theorem REF , set $\\psi _1:=\\min \\left(d\\eta ,1-d\\right)$ , let $m\\in \\mathbb {N}$ and set $\\varepsilon =\\varepsilon _m=m^{-q}$ , with $q>\\psi _1^{-1}$ .",
"If $(\\widetilde{X}_m,m\\in \\mathbb {N})$ is generated based on the sequence $(\\varepsilon _m,m\\in \\mathbb {N})$ (and the corresponding $D_m=C\\varepsilon _m^{-d}$ ), then $(\\widetilde{X}_m,m\\in \\mathbb {N})$ converges to $X$ $\\mathbb {P}$ -almost surely.",
"If the assumptions of Theorem REF with $\\eta >1$ hold, we can ensure at least $L^1(\\Omega ;\\mathbb {R})$ convergence.",
"Note that the Hurwitz zeta function $\\zeta (\\eta +1-p,D/2)=\\zeta (\\eta +1-p,C/2\\varepsilon ^{-d})$ is of order $\\mathcal {O}(\\varepsilon ^{d(\\eta +1-p)})$ as $\\varepsilon \\rightarrow 0$ .",
"With Markov's inequality, $p=1$ and the given error bounds, we get that for each $\\widehat{\\varepsilon }>0$ and $m\\in \\mathbb {N}$ $\\mathbb {P}(|\\widetilde{X}_m-X|>\\widehat{\\varepsilon }\\,)\\le \\frac{\\mathbb {E}(|\\widetilde{X}_m-X|)}{\\widehat{\\varepsilon }}\\le \\frac{\\widetilde{C} }{\\widehat{\\varepsilon }}\\left(\\varepsilon _m^{d\\eta }+\\varepsilon _m^{1-d}\\right)\\le \\frac{2\\widetilde{C}}{\\widehat{\\varepsilon }}\\varepsilon _m^{\\psi _1},$ (recall that $1>\\psi _1>0$ and $\\varepsilon _m\\le 1$ ) where the constant $\\widetilde{C}>0$ depends on $R,\\eta $ and $C$ .",
"But this means $\\sum _{m=1}^\\infty \\mathbb {P}(|\\widetilde{X}_m-X|>\\widehat{\\varepsilon })\\le \\frac{2\\widetilde{C}}{\\widehat{\\varepsilon }}\\sum _{m=1}^\\infty \\varepsilon _m^{\\psi _1}= \\frac{2\\widetilde{C}}{\\widehat{\\varepsilon }}\\sum _{m=1}^\\infty m^{-q\\psi _1}<\\infty ,$ since $q\\psi _1>1$ by construction.",
"The claim then follows by the Borel-Cantelli lemma.",
"Corollary 3.20 Under the assumptions of Theorem REF , set $\\psi _2:=\\min \\left(d(\\eta -1),1-d\\right)$ , let $m\\in \\mathbb {N}$ and set $\\varepsilon =\\varepsilon _m=m^{-q}$ , with $q>\\psi _2^{-1}$ .",
"If $(\\widetilde{X}_m,m\\in \\mathbb {N})$ is generated based on the sequence $(\\varepsilon _m,m\\in \\mathbb {N})$ (and the corresponding $D_m=C\\varepsilon _m^{-d}$ ), then $(\\widetilde{X}_m,m\\in \\mathbb {N})$ converges to $X$ $\\mathbb {P}$ -almost surely.",
"We can now combine the error estimates for any increment over time $\\Delta _n>0$ with the piecewise approximation error from Algorithm REF to bound the overall error $\\ell (t)-\\widetilde{\\ell }^{(n)}(t)$ .",
"Theorem 3.21 Let $\\ell $ be a Lévy process on $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ with characteristic function $\\phi _\\ell $ , CDF $F_{t}$ and density $f_{t}$ for any $t\\in .Assume for $ n$\\mathbb {N}$$ and fixed $ n$ there are constants $ R,,B,>0$ such that either Ass~\\ref {ass:1} or Ass.~\\ref {ass:2} holds.Let $ (n)$ be the piecewise constant approximation of $$ generated by Algorithm~\\ref {algo:approx2} and the approximation $ Fn$ of $ Fn$.There are parameters $ Dn,$\\varepsilon $ n$ for $ Fn$ such that for any $ p[1,)$ resp.",
"$ p[1,-1)$ the approximation error is bounded by$$\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}\\le C_{\\ell ,T,p,R,\\eta }\\Delta _n^{1/p},\\quad t\\in $$where the constant $ C,T,p,R,>0$ only depends on the indicated parameters.$ By Theorem REF we may regard $\\ell $ as the (pointwise) $L^p(\\Omega ;\\mathbb {R})$ -limit process of the sequence $(\\overline{\\ell }^{(n)},n\\in \\mathbb {N})$ generated by Algorithm REF .",
"For fixed $n$ , we may then identify $\\overline{\\ell }^{(n)}$ with $\\ell ^{(n)}$ from Algorithm REF to obtain $\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}\\le \\mathbb {E}(|\\ell (t)-\\overline{\\ell }^{(n)}(t)|^p)^{1/p}+ \\mathbb {E}(|\\ell ^{(n)}(t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}.$ The first term is bounded by $\\mathbb {E}(|\\ell (t)-\\overline{\\ell }^{(n)}(t)|^p)^{1/p}\\le C^{1/p}_{\\ell ,T,p}\\sum _{i=n+1}^\\infty 2^{-i/p}=\\frac{C^{1/p}_{\\ell ,T,p}2^{-n/p}}{2^{1/p}-1}.$ For the treatment of the second term, we define for $j=1,\\dots ,2^n$ the random variables $X_j:=F_{\\Delta _n}^{-1}(U_j)\\stackrel{\\mathcal {L}}{=}\\ell (\\Delta _n)$ .",
"Here, $U_1,\\dots ,U_{2^n}$ is the i.i.d sequence of $\\mathcal {U}([0,1])$ random variables on $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ from Algorithm REF .",
"The increments of the approximation $\\widetilde{\\ell }^{(n)}$ are then given by $\\widetilde{X}_j:=\\widetilde{F}^{(-1)}_{\\Delta _n}(U_j)$ which yields $|\\ell ^{(n)}(t)-\\widetilde{\\ell }^{(n)}(t)|\\le \\sum _{j=1}^{2^n}|X_j-\\widetilde{X}_j|.$ The differences $(X_j-\\widetilde{X}_j, j=1,\\dots ,2^n)$ are i.i.d.",
"by construction, hence $\\mathbb {E}\\left(|\\ell ^{(n)}(t)-\\widetilde{\\ell }^{(n)}(t)|^p\\right)^{1/p}\\le \\sum _{j=1}^{2^n}\\mathbb {E}(|X_j-\\widetilde{X}_j|^p)^{1/p}\\le 2^{n}\\mathbb {E}(|X_1-\\widetilde{X}_1|^p)^{1/p}.$ Now let $\\widetilde{F}_{\\Delta _n}$ be the approximation of $F_{\\Delta _n}$ for some $\\varepsilon \\in (0,1]$ and $D=C\\varepsilon ^{-d}$ .",
"For the first set of assumptions, we apply the error estimates of Theorem REF to obtain $\\mathbb {E}\\left(|\\ell ^{(n)}(t)-\\widetilde{\\ell }^{(n)}(t)|^p\\right)^{1/p}&\\le 2^{n}\\mathbb {E}(|X_1-\\widetilde{X}_1|^p)^{1/p}\\\\&\\le 2^{n} \\big (2Rp\\zeta (\\eta +1-p,C\\varepsilon ^{-d}/2)+2R(C\\varepsilon ^{-d}/2)^{p-1/d}+C^p\\varepsilon ^{1-dp}\\big )^{1/p}\\\\&\\le 2^{n} C_{R,\\eta ,p} \\varepsilon ^{\\psi (p)/p},\\\\$ where $\\psi (p):=\\min \\left(d(\\eta +1-p),1-dp\\right)$ and $C_{R,\\eta ,p}>0$ depends on $C$ and the indicated parameters.",
"An error of order $\\Delta _n^{1/p}$ in the last inequality is then achieved by choosing $\\varepsilon =\\varepsilon _n=2^{-(np+n)/\\psi (p)}$ and $D_n=C\\varepsilon _n^{-d}$ .",
"The proof for the second set of assumptions is carried out identically with the only difference that $\\psi (p):=\\min \\left(d(\\eta -p),1-dp\\right)$ .",
"Remark 3.22 For an efficient simulation one would choose $R$ based on $(\\phi _\\ell )^{\\Delta _n}$ as in Remark REF and then $C$ based on $R$ as suggested in Remark REF .",
"Note that in this case $R=\\mathcal {O}(\\Delta _n)$ and $C=\\kappa \\left(3/2\\,RV_1(\\kappa ,\\eta )\\right)^{1/\\eta }=\\mathcal {O}(\\Delta _n^{1/\\eta })\\quad \\text{resp.",
"}\\quad C=\\kappa \\left(3/2\\,RV_2(\\kappa ,\\eta )\\right)^{1/(\\eta -1)}=\\mathcal {O}(\\Delta _n^{1/(\\eta -1)}).$ This has to be considered in the simulation of $\\widetilde{\\ell }^{(n)}$ for different $\\Delta _n$ as we point out in the setting of Ass.",
"REF : As shown in Theorem REF , the $L^p$ -error $\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)$ is bounded by $\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}&\\le 2^{n} \\Big (2Rp\\zeta (\\eta +1-p,C\\varepsilon ^{-d}/2)+2R(C\\varepsilon ^{-d}/2)^{p-1/d}+C^p\\varepsilon ^{1-dp}\\big )^{1/p}\\\\&\\quad +\\frac{C^{1/p}_{\\ell ,T,p}2^{-n/p}}{2^{1/p}-1}.$ By substituting $\\varepsilon =C^{1/d}D^{-1/d}$ , $d=1/{\\eta }$ (see Remark REF ) and $2^n={T}/{\\Delta _n}$ one obtains $ \\begin{split}\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}&\\le \\frac{(2Rp\\zeta (\\eta +1-p,D/2)+2R(D/2)^{p-\\eta }+C^\\eta D^{p-\\eta })^{1/p}}{T^{-1}\\Delta _n}\\\\&\\quad +\\frac{C^{1/p}_{\\ell ,T,p}2^{-n/p}}{2^{1/p}-1}\\end{split}$ With $R=\\mathcal {O}(\\Delta _n)$ and $C=\\mathcal {O}(\\Delta _n^{1/\\eta })$ this implies with Ineq.",
"(REF ) $\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^p)^{1/p}=\\mathcal {O}(D^{(p-\\eta )/p}\\Delta _n^{1/p-1})+\\mathcal {O}(\\Delta _n^{1/p}).$ To equilibrate both error contributions, one may choose $D:=D_n=\\Delta _n^{p/(p-\\eta )}$ in the simulation which leads to an $L^p$ -error of order $\\mathcal {O}(\\Delta _n^{1/p})$ .",
"As mentioned in the end of Section , the one-dimensional processes $(\\widetilde{\\ell }_i, i=1,\\ldots ,N)$ in the spectral decomposition are not independent, but merely uncorrelated.",
"In the next section we introduce a class of Lévy fields for which uncorrelated processes can be obtained by subordinating a multi-dimensional Brownian motion.",
"Furthermore, for the simulation of these processes the Fourier inversion method may be employed and a bound for the constant $C_\\ell $ (see Theorem REF ) can be derived.",
"Generalized hyperbolic Lévy processes Distributions which belong to the class of generalized hyperbolic distributions may be used for a wide range of applications.",
"GH distributions have been first introduced in [4] to model mass-sizes in aeolian sand (see also [5]).",
"Since then they have been successfully applied, among others, in Finance and Biology.",
"Giving a broad class the distributions are characterized by six parameters, famous representatives are the Student's t, the normal-inverse Gaussian, the hyperbolic and the variance-gamma distribution.",
"The popularity of GH processes is explained by their flexibility in modeling various characteristics of a distribution such as asymmetries or heavy tails.",
"A further advantage in our setting is, that the characteristic function is known and, therefore, the Fourier Inversion may be applied to approximate these processes.",
"This section is devoted to investigate several properties of multi-dimensional GH processes which are then used to construct an approximation of an infinite-dimensional GH field.",
"In contrast to the Gaussian case, the sum of two independent and possibly scaled GH processes is in general not again a GH process.",
"We show a possibility to approximate GH Lévy fields via Karhunen-Loève expansions in such a way that the approximated field is itself again a GH Lévy field.",
"This is essential, so as to have convergence of the approximation to a GH Lévy field in the sense of Theorem REF .",
"Furthermore, we give, for $N\\in \\mathbb {N}$ , a representation of a $N$ -dimensional GH process as a subordinated Brownian motion and show how a multi-dimensional GH process may be constructed from uncorrelated, one-dimensional GH processes with given parameters.",
"This may be exploited by the Fourier inversion algorithm in such a way that the computational expenses to simulate the approximated GH fields are virtually independent of the truncation index $N$ .",
"Assume, for $N\\in \\mathbb {N}$ , that $\\lambda \\in \\mathbb {R}$ , $\\alpha >0$ , $\\beta \\in \\mathbb {R}^N$ , $\\delta >0$ , $\\mu \\in \\mathbb {R}^N$ and $\\Gamma $ is a symmetric, positive definite (spd) $N\\times N$ -matrix with unit determinant.",
"We denote by $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ the $N$ -dimensional generalized hyperbolic distribution with probability density function $f^{GH_N}(x;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )=\\frac{\\gamma ^\\lambda \\alpha ^{N/2-\\lambda }}{(2\\pi )^{N/2}\\delta ^\\lambda K_\\lambda (\\delta \\gamma )}\\frac{K_{\\lambda -N/2}(\\alpha g(x-\\mu ))}{g(x-\\mu )^{N/2-\\lambda }}\\exp (\\beta ^{\\prime }(x-\\mu ))$ for $x\\in \\mathbb {R}^N$ , where $g(x):=\\sqrt{\\delta ^2+x^{\\prime }\\Gamma x}, \\quad \\gamma ^2:=\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta $ and $K_\\lambda (\\cdot )$ is the modified Bessel-function of the second kind with $\\lambda $ degrees of freedom.",
"The characteristic function of $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ is given by $\\begin{split}\\phi _{GH_N}(u; \\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )&:=\\exp (iu^{\\prime }\\mu )\\Big (\\frac{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta }{\\alpha ^2-(iu+\\beta )^{\\prime }\\Gamma (iu+\\beta )}\\Big )^{\\lambda /2}\\\\&\\quad \\cdot \\frac{K_\\lambda (\\delta (\\alpha ^2-(iu+\\beta )^{\\prime }\\Gamma (iu+\\beta ))^{1/2})}{K_\\lambda (\\delta \\gamma )},\\end{split}$ where $A^{\\prime }$ denotes the transpose of a matrix or vector $A$ .",
"For simplicity, we assume that the condition $\\alpha ^2>\\beta ^{\\prime }\\Gamma \\beta $ is satisfiedIf $\\alpha ^2=\\beta ^{\\prime }\\Gamma \\beta $ and $\\lambda <0$ , the distribution is still well-defined, but one has to consider the limit $\\gamma \\rightarrow 0^+$ in the Bessel functions, see [13], [39]..",
"If $N=1$ , clearly, $\\Gamma =1$ is the only possible choice for the \"matrix parameter\" $\\Gamma $ , thus we omit it in this case and denote the one-dimensional GH distribution by $GH(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu )$ .",
"Barndorff–Nielsen obtains the GH distribution in [5] as a normal variance-mean mixture of a $N$ -dimensional normal distribution and a (one-dimensional) generalized inverse Gaussian (GIG) distribution with density function $f^{GIG}(x;a,b,p)=\\frac{(b/a)^p}{2K_p(ab)}x^{p-1}\\exp (-\\frac{1}{2}(a^2x^{-1}+b^2x)),\\quad x>0$ and parameters $a,b>0$ and $p\\in \\mathbb {R}$The notation of the GIG distribution varies throughout the literature, we use the notation from [41].. To be more precise: Let $w^N(1)$ be a $N$ -dimensional standard normally distributed random vector, $\\Gamma $ a spd $N\\times N$ -structure matrix with unit determinant and $\\ell ^{GIG}(1)$ a $GIG(a,b,p)$ random variable, which is independent of $w^N(1)$ .",
"For $\\mu ,\\beta \\in \\mathbb {R}^N$ , we set $\\delta =a$ , $\\lambda =p$ , $\\alpha =\\sqrt{b^2+\\beta ^{\\prime }\\Gamma \\beta }$ and define the random variable $\\ell ^{GH_N}(1)$ as $\\ell ^{GH_N}(1):=\\mu +\\Gamma \\beta \\ell ^{GIG}(1) + \\sqrt{\\ell ^{GIG}(1)\\Gamma }w^N(1).$ Then $\\ell ^{GH_N}(1)$ is $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ -distributed, where $\\sqrt{\\Gamma }$ denotes the Cholesky decomposition of the matrix $\\Gamma $ .",
"With this in mind, one can draw samples of a GH distribution with given parameters by sampling multivariate normal and GIG-distributed random variables, as $a=\\delta >0$ and $b=\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta }>0$ is guaranteed by the conditions on the GIG parameters (this results in Eq.",
"(REF ) being fulfilled).",
"As noted in [18], for general $\\lambda \\in \\mathbb {R}$ , we cannot assume that the increments of the GH Lévy process (resp.",
"of the subordinating process) over a time length other than one follow a GH distribution (resp.",
"GIG distribution).",
"If $N=1$ , however, the (one-dimensional) GH Lévy process $\\ell ^{GH}$ has the representation $\\ell ^{GH}(t)\\stackrel{\\mathcal {L}}{=}\\mu t+\\beta \\ell ^{GIG}(t)+w(\\ell ^{GIG}(t)),\\quad \\text{for } t\\ge 0,$ where $w$ is a one-dimensional Brownian motion and $\\ell ^{GIG}$ a GIG process independent of $w$ (see [16]).",
"This result yields the following generalization: Lemma 4.1 For $N\\in \\mathbb {N}$ , the $N$ -dimensional process $\\ell ^{GH_N}=(\\ell ^{GH_N}(t),t\\in ~ $ , which is $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ -distributed, can be represented as a subordinated $N$ -dimensional Brownian motion $w^N$ via $\\ell ^{GH_N}(t)\\stackrel{\\mathcal {L}}{=}\\mu t+\\Gamma \\beta \\ell ^{GIG}(t)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(t)),$ where $(\\ell ^{GIG}(t),t\\in $ is a GIG Lévy process independent of $w^N$ and $\\sqrt{\\Gamma }$ is the Cholesky decomposition of $\\Gamma $ .",
"Since the $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ distribution may be represented as a normal variance-mean mixture (see Eq.",
"(REF )), we have, that $\\ell ^{GH_N}(1)\\stackrel{\\mathcal {L}}{=}\\mu +\\Gamma \\beta \\ell ^{GIG}(1) + \\sqrt{\\Gamma \\ell ^{GIG}(1)}w^N(1)\\stackrel{\\mathcal {L}}{=}\\mu +\\Gamma \\beta \\ell ^{GIG}(1)+ \\sqrt{\\Gamma }w^N(\\ell ^{GIG}(1))$ where $\\ell ^{GIG}(1)\\sim GIG(\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda )$ and $w^N$ is a $N$ -dimensional Brownian motion independent of $\\ell ^{GIG}(1)$ .",
"The characteristic function of the mixed density is then given by $\\phi _{GH_N}(u;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )=\\exp (iu^{\\prime }\\mu )\\mathcal {M}_{GIG}(iu^{\\prime } \\Gamma \\beta -\\frac{1}{2}u^{\\prime }\\Gamma u;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ),$ where $\\mathcal {M}_{GIG}$ denotes the moment generating function of $\\ell ^{GIG}(1)$ (see [7]).",
"The GIG distribution is infinitely divisible, thus this GIG Lévy process $\\ell ^{GIG}=(\\ell ^{GIG}(t),t\\in $ can be defined via its characteristic function for $t\\in :$$\\mathbb {E}(\\exp (iu\\ell ^{GIG}(t)))=(\\mathcal {M}_{GIG}(iu;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ))^t.$$The infinite divisibility yields further\\begin{equation*}\\begin{split}\\mathbb {E}\\big (\\exp (iu^{\\prime }\\ell ^{GH_N}(t))\\big )&=\\mathbb {E}\\big (\\exp (iu^{\\prime }\\ell ^{GH_N}(1))\\big )^t=(\\phi _{GH}(u;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma ))^t\\\\&=\\exp (iu^{\\prime }(\\mu t))(\\mathcal {M}_{GIG}(iu^{\\prime } \\Gamma \\beta -\\frac{1}{2}u^{\\prime }\\Gamma u;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ))^t.\\end{split}\\end{equation*}The expression above is the characteristic function of another normal variance-mean mixture, namely where the subordinator $ GIG$ is a GIG process with characteristic function$$\\mathbb {E}(\\exp (iu\\ell ^{GIG}(t)))=(\\mathcal {M}_{GIG}(iu;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ))^t.$$Hence, $ GHN(t)$ can be expressed as\\begin{equation*}\\ell ^{GH_N}(t)\\stackrel{\\mathcal {L}}{=}\\mu t+\\Gamma \\beta \\ell ^{GIG}(t)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(t)).\\end{equation*}$ Remark 4.2 In the special case of $\\lambda =-\\frac{1}{2}$ one obtains the normal inverse Gaussian (NIG) distribution.",
"The mixing density is, in this case, the inverse Gaussian (IG) distribution.",
"We denote the $N$ -dimensional NIG distribution by $NIG_N(\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ .",
"This is the only subclass of GH distributions which is closed under convolutions in the sense that (see [32]) $NIG_N(\\alpha ,\\beta ,\\delta _1,\\mu _1,\\Gamma )*NIG_N(\\alpha ,\\beta ,\\delta _2,\\mu _2,\\Gamma )=NIG_N(\\alpha ,\\beta ,\\delta _1+\\delta _2,\\mu _1+\\mu _2,\\Gamma ).$ For $\\lambda \\in \\mathbb {R}$ , the sum of independent GH random variables is in general not GH-distributed.",
"This implies further, that one is in general not able to derive bridge laws of these processes in closed form, meaning we need to use the algorithms introduced in Section REF for simulation.",
"As shown in [6], the GH and the GIG distribution are infinitely-divisible, thus we can define the $N$ -dimensional GH Lévy process $\\ell ^{GH_N}=(\\ell ^{GH_N}(t),t\\in $ with characteristic function $\\mathbb {E}(\\exp (iu\\ell ^{GH_N}(t))=(\\phi _{GH_N}(u;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma ))^t.$ Remark 4.3 If $\\lambda =-\\frac{1}{2}$ , the corresponding NIG Lévy process $(\\ell ^{NIG_N}(t),t\\in $ has characteristic function $\\mathbb {E}[\\exp (iu\\ell ^{NIG_N}(t))]=(\\phi _{GH_N}(u;-\\frac{1}{2},\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma ))^t=\\phi _{GH_N}(u;-\\frac{1}{2},\\alpha ,\\beta ,t\\delta ,t\\mu ,\\Gamma ).$ This is due to the fact that the characteristic function $\\phi _{IG}(u;a,b)$ of the mixing IG distribution fulfills the identity $(\\phi _{IG}(u;a,b))^t=\\phi _{IG}(u;ta,b)$ for any $t\\in and $ a,b>0$ (see \\cite {S03}).$ We consider the finite time horizon $ [0,T]$ , for $T<+\\infty $ , the probability space $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ , and a compact domain $\\mathcal {D}\\subset \\mathbb {R}^s$ for $s\\in \\mathbb {N}$ to define a GH Lévy field as a mapping $L^{GH}:\\Omega \\times \\mathbb {R},\\quad (\\omega ,x,t)\\mapsto L^{GH}(\\omega )(x)(t),$ such that for each $x\\in the point-wise marginal process\\begin{equation*}L^{GH}(\\cdot )(x)(\\cdot ):\\Omega \\times \\mathbb {R},\\quad (\\omega ,t)\\mapsto L^{GH}(\\omega )(x)(t),\\end{equation*}is a one-dimensional GH Lévy process on $ (,(At,t0),P)$ with characteristic function\\begin{equation*}\\mathbb {E}\\big (\\exp (iuL^{GH}(x)(t))\\big )=(\\phi _{GH}(u;\\lambda (x),\\alpha (x),\\beta (x),\\delta (x),\\mu (x)))^t,\\end{equation*}where the indicated parameters are given by continuous functions, i.e.",
"$ ,,C($\\mathbb {R}$ )$ and $ ,C($\\mathbb {R}$ >0)$.",
"We assume that condition (\\ref {eq:ghcond}), i.e.",
"$ (x)2>(x)2$, is fulfilled for any $ x to ensure that $L^{GH}(x)(\\cdot )$ is a well-defined GH Lévy process.",
"This, in turn, means that $L^{GH}$ takes values in the Hilbert space $H=L^2(\\mathcal {D})$ and is square integrable as $\\mathbb {E}(||L^{GH}(t)||_H^2)\\le T\\mathbb {E}(||L^{GH}(1)||_H^2)\\le T \\max \\limits _{x\\in \\mathcal {D}}\\mathbb {E}(L^{GH}(x)(1)^2)V_{\\mathcal {D}},$ where $V_{\\mathcal {D}}$ denotes the volume of $\\mathcal {D}$ .",
"The right hand side is finite since every GH distribution has finite variance (see for example [30], [41]), the parameters of the distribution of $L^{GH}(x)(1)$ depend continuously on $x$ and $\\mathcal {D}\\subset \\mathbb {R}^s$ is compact by assumption.",
"We use the Karhunen-Loève expansion from Section to obtain an approximation of a given GH Lévy field.",
"For this purpose, we consider the truncated sum $L^{GH}_N(x)(t) :=\\sum _{i=1}^{N} \\varphi _i(x)\\ell _i^{GH}(t)\\stackrel{\\mathcal {L}}{=} \\sum _{i=1}^{N} \\varphi _i(x) \\Big (\\mu _i t+\\beta _i \\ell ^{GIG}_i(t)+w_i(\\ell ^{GIG}_i(t))\\Big ),$ where $N\\in \\mathbb {N}$ and $\\varphi _i(x)=\\sqrt{\\rho _i}e_i(x)$ is the $i$ -th component of the spectral basis evaluated at the spatial point $x$ .",
"For each $i,=1,\\dots ,N$ , the processes $\\ell _i^{GH}:=(\\ell _i^{GH}(t), t\\in $ are uncorrelated but dependent $GH(\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i)$ Lévy process.",
"From Theorem REF follows that $L^{GH}_N$ converges in $L^2(\\Omega ;H)$ to $L^{GH}$ as $N\\rightarrow \\infty $ .",
"With given $\\mu _i, \\beta _i\\in \\mathbb {R}$ , we have that $\\ell ^{GH}_i(t)\\stackrel{\\mathcal {L}}{=}\\mu _i t+\\beta _i \\ell ^{GIG}_i(t)+w_i(\\ell ^{GIG}_i(t)),$ where for each $i$ , the process $(\\ell ^{GIG}_i(t), t\\in $ is a GIG Lévy process with parameters $a_i=\\delta _i, b_i=(\\alpha _i^2-\\beta _i^2)^{1/2}>0$ and $p_i=\\lambda _i\\in \\mathbb {R}$ .",
"In addition, $(w_i(t), t\\in $ is a one-dimensional Brownian motion independent of $\\ell ^{GIG}_i$ and all Brownian motions $w_1,\\dots ,w_N$ are mutually independent of each other, but the processes $\\ell ^{GIG}_1,\\dots ,\\ell ^{GIG}_N$ may be correlated.",
"We aim for an approximation $(L^{GH}_N(x)(t),t\\in $ which is a GH process for arbitrary $\\varphi _i$ and $x\\in \\mathcal {D}$ .",
"Remark REF suggests that this cannot be achieved by the summation of independent $\\ell _i^{GH}$ , but rather by using correlated subordinators $\\ell ^{GIG}_1,\\dots ,\\ell ^{GIG}_N$ .",
"Before we determine the correlation structure of the subordinators, we establish a necessary and sufficient condition on the $\\ell _i^{GH}$ to achieve the desired distribution of the approximation.",
"Lemma 4.4 Let $N\\in \\mathbb {N}$ , $t\\in and $ (iGH,i=1...,N)$ be GH processes as defined in Eq.~(\\ref {eq:Z_i}).",
"For a vector $a = (a1,...,aN)$ with arbitrary numbers $ a1,...,aN$\\mathbb {R}$ {0}$, the process $ GH,a$ defined by\\begin{equation}\\ell ^{GH,\\bf a}(t):=\\sum _{i=1}^{N} a_i\\ell _i^{GH}(t)=\\sum _{i=1}^{N} a_i(\\mu _i+\\beta _i\\ell _i^{GIG}(t)+w_i(\\ell _i^{GIG}(t)))\\end{equation}is a one-dimensional GH process, if and only if the vector $ GHN(1):=(1GH(1),...,NGH(1))'$ is multivariate $ GHN((N),(N),(N),(N),(N),)$-distributed with parameters $ (N),(N)$, $ (N)$\\mathbb {R}$$, $ (N),(N)$\\mathbb {R}$ N$and structure matrix $$\\mathbb {R}$ NN$.$ The entries of the coefficient vector ${\\bf a}$ in $\\ell ^{GH,\\bf a}$ are later identified with the basis functions $\\varphi _i(x)$ for $x\\in to show that $ LGHN(x)()$ is a one-dimensional Lévy process and the approximation $ LGHN$ a $ H$-valued GH Lévy field.",
"{\\begin{xmlelement*}{proof} We first consider the case that\\begin{equation*}\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma ).\\end{equation*}It is sufficient to show that \\ell ^{GH,\\bf a}(1) is a GH-distributed random variable, the infinite divisibility of the GH distribution then implies that (\\ell ^{GH,\\bf a}(t),t\\in is a GH process.Since the entries of the coefficient vector a_1,\\dots ,a_N are non-zero, there exists a non-singular N\\times N matrix A, such that \\ell ^{GH,\\bf a}(1) is the first component of the vector A\\ell ^{GH_N}(1).If \\ell ^{GH_N}(1) is multi-dimensional GH-distributed, then follows from~\\cite [Theorem 1]{B81}, that A\\ell ^{GH_N}(1) is also multi-dimensional GH-distributed and that the first component of A\\ell ^{GH_N}(1), namely \\ell ^{GH,\\bf a}(1), follows a one-dimensional GH distribution (the parameters of the distribution of \\ell ^{GH,\\bf a}(1) depend on A and on \\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma and are explicitly given in \\cite {B81} and below).",
"\\\\On the other hand, assume that \\ell ^{GH,\\bf a}(1) is a GH random variable (with arbitrary coefficients), but \\ell ^{GH_N}(1) is not N-dimensional GH-distributed.This means there is no representation of \\ell ^{GH_N}(1) such that\\begin{equation*}\\ell ^{GH_N}(1)\\stackrel{\\mathcal {L}}{=}\\mu +\\Gamma \\beta \\ \\ell ^{GIG}(1)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(1))\\ \\end{equation*}with \\mu ,\\beta \\in \\mathbb {R}^N, \\Gamma \\in \\mathbb {R}^{N\\times N} spd with determinant one, a GIG random variable \\ell ^{GIG}(1) and a N-dimensional Brownian motion w^N independent of \\ell ^{GIG}(1).This implies that \\ell ^{GH,\\bf a}(1)=(A\\ell ^{GH_N}(1))_1 has no representation\\begin{equation*}\\begin{split}\\ell ^{GH,\\bf a}(1)&=(A\\mu )_1+(A\\Gamma \\beta )_1\\ell ^{GIG}(1)+(A\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(1)))_1\\\\&\\stackrel{\\mathcal {L}}{=}(A\\mu )_1+(A\\Gamma \\beta )_1\\ell ^{GIG}(1)+\\sqrt{\\ell ^{GIG}(1)A_{[1]}\\Gamma A_{[1]}^{\\prime }}w^1(1),\\end{split}\\end{equation*}where A_{[1]} denotes the first row of the matrix A and w^1(1)\\sim \\mathcal {N}(0,1).For the last equality we have used the affine linear transformation property of multi-dimensional normal distributions and that \\Gamma is positive definite.Since c_A:=A_{[1]}\\Gamma A_{[1]}^{\\prime }>0, we can divide the equation above by \\sqrt{c_A} and obtain that c_A^{-1/2}\\,\\ell ^{GH,\\bf a}(1) cannot be a GH-distributed random variable, as it cannot be expressed as a normal variance-mean mixture with a GIG-distribution.But this is a contradiction, since \\ell ^{GH,\\bf a}(1) is GH-distributed by assumption and the class of GH distributions is closed under regular affine linear transformations (see~\\cite [Theorem 1c]{B81}).\\end{xmlelement*}}\\begin{rem}The condition a_i\\ne 0 is, in fact, not necessary in Lemma~\\ref {lem:GH_lin}.",
"If, for k\\in \\lbrace 1,\\dots ,N-1\\rbrace , k coefficients a_{i_1}=\\dots =a_{i_k}=0, then the summation reduces to\\begin{equation*}\\ell ^{GH,\\bf a}(t)=\\sum _{i=1}^{N}a_i \\ell _i^{GH}(t)=\\sum _{l=1}^{N-k}a_{j_l}\\ell ^{GH}_{j_l}(t),\\end{equation*}where the indices j_l are chosen such that a_{j_l}\\ne 0 for l=1,\\dots ,N-k.",
"If P\\in \\mathbb {R}^{N\\times N} is the permutation matrix with\\begin{equation*}P\\ell ^{GH_N}(1)=P(\\ell _1^{GH}(1),\\dots ,\\ell _N^{GH}(1))^{\\prime }=( \\ell ^{GH}_{j_1}(1),\\dots , \\ell ^{GH}_{j_{N-k}}(1), \\ell ^{GH}_{i_1}(1),\\dots , \\ell ^{GH}_{i_k}(1))^{\\prime },\\end{equation*}then P\\ell ^{GH_N} is again N-dimensionally GH-distributed and by~\\cite [Theorem 1a]{B81} the vector (\\ell ^{GH}_{j_1}(1),\\dots ,\\ell ^{GH}_{j_{N-k}}(1)) admits a (N-k)-dimensional GH law.",
"Thus, we only consider the case where all coefficients are non-vanishing.\\end{rem}$ The previous proposition states that the KL approximation $L^{GH}_N(x)(t) = \\sum _{i=1}^{N} \\varphi _i(x) \\ell _i^{GH}(t),$ can only be a GH process for arbitrary $(\\varphi _i(x),i=1,\\dots ,N)$ if the $\\ell _i^{GH}$ are correlated in such a way that they form a multi-dimensional GH process.",
"This rules out the possibility of independent processes $(\\ell _i^{GH},i=1,\\ldots ,N)$ , because if $\\ell ^{GH_N}(1)$ is multi-dimensional GH-distributed, it is not possible that the marginals $\\ell _i^{GH}(1)$ are independent GH-distributed random variables (see [13]).",
"The parameters $\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i$ of each process $\\ell _i^{GH}$ should remain as unrestricted as possible, so we determine in the next step the parameters of the marginals of a $GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ distribution and show how the subordinators $(\\ell ^{GIG}_i,i=1,\\ldots ,N)$ might be correlated.",
"For this purpose, we introduce the notation $A^-~\\!\\!^{\\prime }:=(A^{-1})^{\\prime }$ if $A$ is an invertible square matrix.",
"The following result allows us to determine the marginal distributions of a $N$ -dimensional GH distribution.",
"Lemma 4.5 (Masuda [30], who refers to [14], Lemma A.1.)",
"Let $\\ell ^{GH_N}(1)=(\\ell _1^{GH}(1),\\dots ,\\ell _N^{GH}(1))^{\\prime }\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma ),$ then for each $i$ we have that $\\ell _i^{GH}(1)\\sim GH(\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i)$ , where $\\begin{split}&\\lambda _i=\\lambda ^{(N)},\\quad \\alpha _i=\\Gamma _{ii}^{-1/2}\\left[(\\alpha ^{(N)})^2-\\beta _{-i}^{\\prime }\\left(\\Gamma _{-i,22}-\\Gamma _{-i,21}\\Gamma _{ii}^{-1}\\Gamma _{-i,12}\\right)\\beta _{-i}\\right]^{1/2}\\\\&\\beta _i=\\beta ^{(N)}_i+\\Gamma _{ii}^{-1}\\Gamma _{-i,12}\\beta _{-i},\\quad \\delta _i=\\sqrt{\\Gamma _{ii}}\\delta ^{(N)}_i,\\quad \\mu _i=\\mu _i^{(N)},\\end{split}$ together with $\\begin{split}&\\beta _{-i}:=(\\beta _1^{(N)},\\dots ,\\beta _{i-1}^{(N)},\\beta _{i+1}^{(N)},\\dots ,\\beta _N^{(N)})^{\\prime },\\\\&\\Gamma _{-i,12}:=(\\Gamma _{i,1},\\dots ,\\Gamma _{i,i-1},\\Gamma _{i,i+1},\\dots ,\\Gamma _{i,N}),\\quad \\Gamma _{-i,21}:=\\Gamma _{-i,12}^{\\prime }\\end{split}$ and $\\Gamma _{-i,22}$ denotes the $(N-1)\\times (N-1)$ matrix which is obtained by removing the $i$ -th row and column of $\\Gamma $ .",
"Assume that $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ , since this is a necessary (and sufficient) condition so that the (truncated) KL expansion is a GH process.",
"Lemma REF gives immediately, that for all $i=1,\\dots ,N$ , the parameters $\\lambda _i=\\lambda ^{(N)}$ have to be identical, whereas the drift $\\mu _i$ may be chosen arbitrary for each process $\\ell _i^{GH}$ .",
"Furthermore, the expectation and covariance matrix of $\\ell ^{GH_N}(1)$ is given by $ \\mathbb {E}(\\ell ^{GH_N}(1))=\\mu ^{(N)}+\\frac{\\delta ^{(N)}K_{\\lambda ^{(N)}+1}(\\delta ^{(N)}\\gamma ^{(N)})}{\\gamma ^{(N)}K_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}\\Gamma \\beta ^{(N)}$ and $ \\begin{split}\\text{Var}(\\ell ^{GH_N}(1))&=\\frac{\\delta ^{(N)}K_{\\lambda ^{(N)}+1}(\\delta ^{(N)}\\gamma ^{(N)})}{\\gamma ^{(N)}K_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}\\Gamma +\\Big (\\frac{\\delta ^{(N)}}{\\gamma ^{(N)}}\\Big )^2(\\Gamma \\beta ^{(N)})(\\Gamma \\beta ^{(N)})^{\\prime }\\\\&\\qquad \\qquad \\cdot \\Bigg (\\frac{K_{\\lambda ^{(N)}+2}(\\delta ^{(N)}\\gamma ^{(N)})}{K_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}-\\frac{K^2_{\\lambda ^{(N)}+1}(\\delta ^{(N)}\\gamma ^{(N)})}{K^2_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}\\Bigg ),\\end{split}$ where $\\gamma ^{(N)}:=((\\alpha ^{(N)})^2-\\beta ^{(N)}~\\!\\!^{\\prime }\\Gamma \\beta ^{(N)})^{1/2}$ (see [30]).",
"Example 4.6 Consider the case that the processes $\\ell _1^{GH},\\dots ,\\ell _N^{GH}$ are generated by the same subordinating $GIG(a,b,p)$ process $\\ell ^{GIG}$ , i.e.",
"$\\ell _i^{GH}(t)=\\mu _it+\\beta _i\\ell ^{GIG}(t)+w_i(\\ell ^{GIG}(t)).$ Then $\\ell _i^{GH}(1)\\sim GH(\\lambda ,\\alpha _i,\\beta _i,\\delta ,\\mu _i)$ , where $\\lambda =p$ , $\\delta =a$ are independent of $i$ and $\\alpha _i=(b^2+\\beta _i^2)^{1/2}$ .",
"If $\\mu ^{(N)}:=(\\mu _1\\dots ,\\mu _N)^{\\prime }$ , $\\beta ^{(N)}:=(\\beta _1,\\dots ,\\beta _N)^{\\prime }$ and $\\Gamma $ is the $N\\times N$ identity matrix, then $\\begin{split}\\ell ^{GH_N}(t)=(\\ell _1^{GH}(t),\\dots ,\\ell _N^{GH}(t))^{\\prime }&\\stackrel{\\mathcal {L}}{=}\\mu t+\\beta \\ell ^{GIG}(t)+w^N(\\ell ^{GIG}(t))\\\\&=\\mu t+\\Gamma \\beta \\ell ^{GIG}(t)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(t)),\\end{split}$ where $w^N$ is a $N$ -dimensional Brownian motion independent of $\\ell ^{GIG}$ .",
"Hence, $\\ell ^{GH_N}(t)$ is a multi-dimensional $GH_N(\\lambda ,\\alpha ^{(N)},\\beta ^{(N)},\\delta ,\\mu ^{(N)}$ ,$\\Gamma )$ process with $\\alpha ^{(N)}=\\sqrt{b^2+\\beta ^{\\prime }\\beta }$ .",
"One checks using Lemma REF that the parameters of the marginals of $\\ell ^{GH_N}(1)$ and $\\ell _i^{GH}(1)$ coincide for each $i$ , and that expectation and covariance of $\\ell ^{GH_N}(1)$ are given by Eq.",
"(REF ) and Eq.",
"(REF ).",
"By Lemma REF , we have that the Karhunen-Loève expansion $L_N^{GH}(x)(t)=\\sum _{i=1}^N\\varphi _i(x)\\ell _i^{GH}(t)$ in this example is a GH process for each $x\\in \\mathcal {D}$ and arbitrary basis functions $(\\varphi _i,i=1,\\ldots ,N)$ .",
"Remark 4.7 Lemma REF dictates that the subordinators $(\\ell ^{GIG}_i,i=1,\\ldots ,N)$ cannot be independent.",
"In Example REF fully correlated subordinators were used.",
"A different way to correlate the subordinators, so that Lemma REF is fulfilled, would lead to a correlation matrix, just being multiplied with $\\Gamma $ .",
"For simplicity, in the remainder of the paper, especially for the numerical examples in Section , we use fully correlated subordinators.",
"As shown in [13] the class of $N$ -dimensional GH distributions is closed under regular linear transformations: If $N\\in \\mathbb {N}$ , $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ , $A$ is an invertible $N\\times N$ -matrix and $b\\in \\mathbb {R}^N$ , then the random vector $A\\ell ^{GH_N}(1)+b$ has distribution $GH_N(\\lambda ,||A||^{-1/N}\\alpha ,A^-~\\!\\!^{\\prime }\\beta ,||A||^{1/N}\\delta ,A\\mu +b,||A||^{-2/N}A\\Gamma A^{\\prime }),$ where $||A||$ denotes the absolute value of the determinant of $A$ .",
"With this and the assumption $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ , we are also able to determine the point-wise law of $L_N^{GH}$ for given coefficients $\\varphi _1(x),\\dots ,\\varphi _N(x)$ .",
"Lemma 4.8 Let $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ and for $x\\in let $ (i(x),i = 1,...,N)$ be a sequence of non-zero coefficients (see Remark~\\ref {rem:coefficients}).Then $ (LNGH(x)(t),t$ is a GH Lévy process with parameters depending on $ x$.$ It is again sufficient to show that $L_N^{GH}(x)(1)$ follows a GH law, the resulting parameters are given below.",
"For $x\\in , define the $ NN$ matrix $ A(x)$ via\\begin{equation} A(x)_{ij}:={\\left\\lbrace \\begin{array}{ll}\\varphi _j(x) & \\text{if $i=1$ or if $i=j$} \\\\0 & \\text{elsewhere}\\end{array}\\right.",
"}.\\end{equation}The matrix $ A(x)$ is invertible with determinant $ i=1Ni(x)0$ and inverse $ A(x)-1$ given by\\begin{equation*}A(x)^{-1}_{ij}:={\\left\\lbrace \\begin{array}{ll}-\\varphi _1(x)^{-1} & \\text{if $i=1$ and $j\\ge 2$} \\\\\\varphi _i(x)^{-1} & \\text{if $i=j$} \\\\0 & \\text{elsewhere}\\end{array}\\right.",
"}.\\end{equation*}Then $ LNGH(x)(1)=i=1Ni(x)iGH(1)$ is the first entry of the random vector $ A(x)GHN(1)$.By the affine transformation property of the GH distribution and Lemma~\\ref {lem:GH_N} it follows that $ LNGH(x)(1)$ is one-dimensional GH-distributed.Now define $ :=A(x)A(x)'$, the partition\\begin{equation*}\\widetilde{\\Gamma }=\\begin{pmatrix} \\widetilde{\\Gamma }_{11} & \\widetilde{\\Gamma }_{2,1}^{\\prime }\\\\ \\widetilde{\\Gamma }_{2,1} & \\widetilde{\\Gamma }_{2,2} \\end{pmatrix}\\end{equation*}such that $ 2,1$\\mathbb {R}$ N-1$ and $ 2,2$\\mathbb {R}$ (N-1)(N-1)$ and the vector$$\\widetilde{\\beta }:=\\left(\\beta _2^{(N)}\\varphi _2(x)^{-1}-\\beta _1^{(N)}\\varphi _1(x)^{-1},\\dots ,\\beta _N^{(N)}\\varphi _N(x)^{-1}-\\beta _1^{(N)}\\varphi _1(x)^{-1}\\right)^{\\prime }\\in \\mathbb {R}^{N-1}.$$The parameters $ L,L(x),L(x),L(x)$ and $ L(x)$ of $ LNGH(x)$ are then given by{\\begin{@align*}{1}{-1}\\lambda _L&=\\lambda ^{(N)},\\\\\\alpha _L(x)&=\\widetilde{\\Gamma }_{11}^{-1/2}\\left[(\\alpha ^{(N)})^2-\\widetilde{\\beta }^{\\prime }(\\widetilde{\\Gamma }_{2,2}-\\widetilde{\\Gamma }_{11}^{-1}\\widetilde{\\Gamma }_{2,1}\\widetilde{\\Gamma }_{2,1}^{\\prime })\\widetilde{\\beta } \\right]^{1/2},\\\\\\delta _L(x)&=\\delta ^{(N)}\\sqrt{\\widetilde{\\Gamma }_{11}}=\\delta ^{(N)}\\Big (\\sum _{i,j=1}^N\\varphi _i(x)\\varphi _j(x)\\Gamma _{ij}\\Big )^{1/2},\\\\\\beta _L(x)&=\\beta _1^{(N)}\\varphi _1(x)^{-1}+\\widetilde{\\Gamma }_{11}^{-1}\\widetilde{\\Gamma }_{2,1}^{\\prime }\\widetilde{\\beta }\\quad \\text{and}\\\\\\mu _L(x)&=[A(x)\\mu ^{(N)}]_1=\\sum _{i=1}^N\\varphi _i(x)\\mu _i^{(N)}.\\end{@align*}}$ To ensure $L^2(\\Omega ;\\mathbb {R})$ convergence as in Theorem REF of the series $\\widetilde{L}_N^{GH}(x)(t) = \\sum _{i=1}^N\\sqrt{\\rho _i}e_i(x)\\widetilde{\\ell }_i^{GH}(t),$ we need to simulate approximations of uncorrelated, one-dimensional GH processes $\\ell _i^{GH}$ with given parameters $\\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i)$ .",
"To obtain a sufficiently good approximation of the Lévy field, $N$ is coupled to the time discretization of $ and the decay of the eigenvalues of $ Q$ (see Remark~\\ref {rem:trunc}).The simulation of a large number $ N$ of independent GH processes is computationally expensive, so we focus on a different approach.Instead of generating $ N$ dependent but uncorrelated, one-dimensional processes, we generate one $ N$-dimensional process with decorrelated marginals.For this approach to work we need to impose some restrictions on the target parameters $ i, i,i$ and $ i$.$ Theorem 4.9 Let $(\\ell _i^{GH},i=1\\ldots ,N)$ be one-dimensional GH processes, where, for $i=1,\\dots ,N$ , $\\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i).$ The vector $\\ell ^{GH_N}:=(\\ell _1^{GH},\\dots ,\\ell _N^{GH})^{\\prime }$ is only a $N$ -dimensional GH process if there are constants $\\lambda \\in \\mathbb {R}$ and $c>0$ such that for any $i$ $\\lambda _i=\\lambda \\quad \\text{and}\\quad \\delta _i(\\alpha _i^2-\\beta _i^2)^{1/2}=c.$ If, in addition, the symmetric matrix $U\\in \\mathbb {R}^{N\\times N}$ defined by $U_{ij}:={\\left\\lbrace \\begin{array}{ll}\\delta _i^2 &\\text{if $i=j$}\\\\\\frac{K_{\\lambda +1}(c)^2-K_{\\lambda +2}(c)K_{\\lambda }(c)}{K_{\\lambda +1}(c)K_{\\lambda }(c)}\\frac{\\beta _i\\delta _i^2\\beta _j\\delta _j^2}{c} &\\text{if $i\\ne j$}\\end{array}\\right.",
"},$ is positive definite, it is possible to construct a $N$ -dimensional GH process $\\ell ^{GH_N,U}$ with uncorrelated marginals $\\ell ^{GH,U}_i$ and $\\ell ^{GH,U}_i(1)\\stackrel{\\mathcal {L}}{=} \\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i).$ We start with the necessary condition to obtain a multi-dimensional GH distribution.",
"Let $\\ell ^{GH_N}$ be a $N$ -dimensional GH process with $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma ).$ If the law of the marginals of $\\ell ^{GH_N}$ is denoted by $ \\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i),$ then one sees immediately from Lemma REF that $\\lambda _i=\\lambda ^{(N)}$ and $\\mu _i=\\mu _i^{(N)}$ for all $i=1,\\dots ,N$ .",
"With the equations for $\\beta _i$ and $\\delta _i$ from Lemma REF , we derive for $\\Gamma \\beta ^{(N)}$ $(\\Gamma \\beta ^{(N)})_i=\\Gamma _{ii}\\beta _i^{(N)}+\\sum _{k=1, k\\ne i}^N \\Gamma _{ik}\\beta _k^{(N)}=\\Gamma _{ii}\\beta _i^{(N)}+\\Gamma _{ii}(\\beta _i-\\beta _i^{(N)})=\\big (\\frac{\\delta _i}{\\delta ^{(N)}}\\big )^2\\beta _i,$ which leads to $\\alpha _i^2&=\\Gamma _{ii}^{-1}(\\alpha ^{(N)})^2-\\Gamma _{ii}^{-1}\\sum _{k=1,k\\ne i}^N\\beta _k^{(N)}\\sum _{l=1,l\\ne i}^N\\Gamma _{kl}\\beta _l^{(N)}+\\big (\\Gamma _{ii}^{-1}\\sum _{k=1, k\\ne i}^N \\Gamma _{ik}\\beta _k^{(N)}\\big )^2\\\\&=\\big (\\frac{\\delta ^{(N)}\\alpha ^{(N)}}{\\delta _i}\\big )^2-\\big (\\frac{\\delta ^{(N)}}{\\delta _i}\\big )^2\\sum _{k=1,k\\ne i}^N\\beta _k^{(N)}((\\Gamma \\beta ^{(N)})_k-\\Gamma _{ik}\\beta _i^{(N)})+(\\beta _i-\\beta _i^{(N)})^2\\\\&=\\big (\\frac{\\delta ^{(N)}\\alpha ^{(N)}}{\\delta _i}\\big )^2-\\sum _{k=1,k\\ne i}^N\\beta _k^{(N)}\\frac{\\delta _k^2}{\\delta _i^2}\\beta _k\\\\&\\quad +\\big (\\frac{\\delta ^{(N)}}{\\delta _i}\\big )^2\\beta _i^{(N)}((\\Gamma \\beta ^{(N)})_i-\\Gamma _{ii}\\beta _i^{(N)})+\\beta _i^2-2\\beta _i\\beta _i^{(N)}+(\\beta _i^{(N)})^2\\\\&=\\big (\\frac{\\delta ^{(N)}\\alpha ^{(N)}}{\\delta _i}\\big )^2-\\sum _{k=1}^N\\beta _k^{(N)}\\frac{\\delta _k^2}{\\delta _i^2}\\beta _k+\\beta _i^2.$ The last equation is equivalent to $ \\delta _i^2(\\alpha _i^2-\\beta _i^2)=(\\delta ^{(N)}\\alpha ^{(N)})^2-\\sum _{k=1}^N\\beta _k^{(N)}\\underbrace{\\delta _k\\beta _k}_{=(\\delta ^{(N)})^2(\\Gamma \\beta _k^{(N)})}=(\\delta ^{(N)})^2\\left((\\alpha ^{(N)})^2-\\beta ^{(N)}~\\!\\!^{\\prime }\\Gamma \\beta ^{(N)}\\right),$ and since the right hand side does not depend on $i$ , we get that $\\delta _i^2(\\alpha _i^2-\\beta _i^2)>0$ has to be independent of $i$ .",
"Now assume we have a set of parameters $((\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i),i=1,\\dots ,N)$ with $\\delta _i\\sqrt{\\alpha _i^2-\\beta _i^2}=c>0\\quad \\text{and}\\quad \\lambda _i=\\lambda \\in \\mathbb {R},$ where $c$ and $\\lambda $ are independent of the index $i$ .",
"Furthermore, let the matrix $U$ as defined in the claim be positive definite.",
"We show how parameters $\\lambda ^{(U)}, \\alpha ^{(U)}, \\beta ^{(U)}, \\delta ^{(U)}, \\mu ^{(U)}$ and $\\Gamma ^{(U)}$ of a $N$ -dimensional GH process $\\ell ^{GH_N,U}$ may be chosen, such that its marginals are uncorrelated with law $\\ell ^{GH,U}_i(1)\\sim GH(\\lambda ,\\alpha _i,\\beta _i,\\delta _i,\\mu _i)$ .",
"Clearly, we have to set $\\lambda ^{(U)}:=\\lambda $ and $\\mu ^{(U)}:=(\\mu _1,\\dots ,\\mu _N)^{\\prime }$ .",
"Eq.",
"(REF ) and Eq.",
"(REF ) yield the conditions $(\\delta ^{(U)})^2(\\Gamma ^{(U)}\\beta ^{(U)})_i=\\delta _i^2\\beta _i \\quad \\text{and}\\quad \\delta ^{(U)}\\sqrt{(\\alpha ^{(U)})^2-\\beta ^{(U)T}\\Gamma ^{(U)}\\beta ^{(U)}}=\\delta _i\\sqrt{\\alpha _i^2-\\beta _i^2}=c.$ If $(\\delta ^{(U)})^2\\Gamma ^{(U)}$ fulfills the identity $(\\delta ^{(U)})^2\\Gamma ^{(U)}=U$ , we get by Eq.",
"(REF ) for $i\\ne j$ $\\text{Cov}(\\ell ^{GH,U}_i(1),\\ell ^{GH,U}_j(1))&=\\frac{K_{\\lambda +1}(c)}{cK_{\\lambda }(c)}(\\delta ^{(U)})^2\\Gamma _{ij}^{(U)}\\\\&\\quad +\\frac{K_{\\lambda +2}(c)K_{\\lambda }(c)-K_{\\lambda +1}^2(c)}{c^2K_{\\lambda }(c)^2}((\\delta ^{(U)})^2\\Gamma ^{(U)}\\beta ^{(U)})_i((\\delta ^{(U)})^2\\Gamma ^{(U)}\\beta ^{(U)})_j\\\\&=\\frac{K_{\\lambda +1}(c)}{cK_{\\lambda }(c)}U_{ij}+\\frac{K_{\\lambda +2}(c)K_{\\lambda }(c)-K_{\\lambda +1}^2(c)}{c^2K_{\\lambda }(c)^2}\\delta _i^2\\beta _i\\delta _j^2\\beta _j=0,$ hence all marginals are uncorrelated.",
"To obtain a well-defined $N$ -dimensional GH distribution, we still have to make sure that $\\Gamma ^{(U)}$ is spd with unit determinant.",
"If we define $\\delta ^{(U)}:=(\\text{det}(U))^{1/(2N)}$ , then $\\delta ^{(U)}>0$ (since $\\text{det}(U)>0$ by assumption) and $\\Gamma ^{(U)}=(\\delta ^{(U)})^{-2}U$ is spd with $\\text{det}(\\Gamma ^{(U)})=1$ .",
"It remains to determine appropriate parameters $\\alpha ^{(U)}>0$ and $\\beta ^{(U)}\\in \\mathbb {R}^N$ .",
"For $\\beta ^{(U)}$ , we use once again Lemma REF to obtain the linear equations $\\beta _i=\\beta _i+(\\Gamma _{ii}^{(U)})^{-1}\\sum _{k=1,k\\ne i}\\Gamma _{ik}^{(U)}\\beta _k^{(U)},$ for $i=1,\\dots ,N$ .",
"The corresponding system of linear equations is given by $\\begin{pmatrix}(\\Gamma _{11}^{(U)})^{-1}\\\\&\\ddots \\\\&&(\\Gamma _{NN}^{(U)})^{-1}\\\\\\end{pmatrix}\\Gamma ^{(U)}\\beta ^{(U)}=\\left(\\begin{array}{c} \\beta _1\\\\ \\vdots \\\\ \\beta _N \\end{array}\\right),$ and has a unique solution $\\beta ^{(U)}$ for any right hand side $(\\beta _1,\\dots ,\\beta _N)^{\\prime }$ , because $\\Gamma ^{(U)}$ as constructed above is invertible with positive diagonal entries.",
"Finally, we are able to calculate $\\alpha ^{(U)}$ via Equation (REF ) as $\\alpha ^{(U)}=\\Big (\\sum _{k=1}^N\\delta _k^2\\beta _k\\beta _k^{(U)}+\\big (\\frac{c}{\\delta ^{(U)}}\\big )^2\\Big )^{1/2}=\\Big (\\beta ^{(U)}~\\!\\!^{\\prime }\\Gamma ^{(U)}\\beta ^{(U)}+\\big (\\frac{c}{\\delta ^{(U)}}\\big )^2\\Big )^{1/2}$ and obtain the desired marginal distributions.",
"Note that the KL-expansion $L^{GH}_N(x)(\\cdot )$ generated by $(\\ell ^{GH,U}_i,i=1\\ldots ,N)$ in Theorem REF is a GH process for each $x\\in by Lemma~\\ref {lem:KL_GH}, whereas this is not the case if the processes $ (iGH,i=1,...,N)$ are generated independently of each other:By Lemma~\\ref {lem:GH_lin} we have that $ LGHN(x)(1)$ is only GH distributed if the vector $ (1GH(1),...,NGH(1))'$ admits a multi-dimensional GH law.As noted in~\\cite {B81} after Theorem 1, this is impossible if the processes (and hence $ (iGH(1),i=1,...,N)$) are independent.Whenever Theorem~\\ref {thm:Z_U} is applicable, we are able to approximate a GH Lévy field by generating a $ N$-dimensional GH processes, where $ N$ is the truncation index of the KL expansion.To this end, Lemma~\\ref {lem:sub} suggests the simulation of GIG processes and then subordinating $ N$-dimensional Brownian motions.With this simulation approach the question arises on why we have taken a detour via the subordinating GIG process instead of using the characteristic function a of GH process in Equation~(\\ref {eq:gh_cf}) for a ``direct^{\\prime \\prime } simulation.This has several reasons: First, the approximation of the inversion formula (\\ref {eq:finv}) can only be applied for one-dimensional GH processes, where the costs of evaluating $ GH$ or $ GIG$ are roughly the same.In comparison, the costs of sampling a Brownian motion are negligible.Second, in the multi-dimensional case, we need that all marginals of the GH process are generated by the same or correlated subordinator(s), which leaves us no choice but to sample the underlying GIG process.In addition, the simulation of a GH field requires in some cases only one subordinating process to generate a multi-dimensional GH process with uncorrelated marginals (see Theorem~\\ref {thm:Z_U}).This approach is in general more efficient than sampling a large number of uncorrelated, one-dimensional GH processes for the KL expansion.As we demonstrate in the following section, it is a straightforward application of the Fourier inversion algorithm to approximate a GIG process $ GIG$ with given parameters, since all necessary assumptions are fulfilled and the bounding parameters $ , R, $ and $ B$ may readily be calculated.$ Numerics In this section we provide some details on the implementation of the Fourier inversion method.",
"Thereafter, we apply this methodology to approximate a GH Lévy field and conclude with some numerical examples.",
"Notes on implementation Suppose we simulate a given one-dimensional Lévy process $\\ell $ which fulfills Assumption REF resp.",
"Assumption REF , using the step size $\\Delta _n>0$ and characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"Usually the parameter $\\eta $ cannot be chosen arbitrary high (as for the GIG process), but it may be possible to choose $\\eta $ within a certain range, for instance $\\eta \\in (1,2]$ for the Cauchy process in Example REF .",
"As a rule of thumb, $\\eta $ should always be determined as large as possible, as the convergence rates in Theorems REF and REF directly depend on $\\eta $ .",
"In addition, we concluded in Remark REF that $D\\simeq \\Delta _n^{p/(p-\\eta )}$ is an appropriate choice to guarantee an $L^p$ -error of order $\\mathcal {O}(\\Delta _n^{1/p})$ .",
"This means that for a given $p$ , $D$ decreases as $\\eta $ increases.",
"Since the number of summations $M$ in Algorithm REF depends on $D$ (see Theorem REF ), an increasing parameter $\\eta $ also reduces computational time.",
"Once $\\eta $ is determined, we derive $R$ by differentiation of $(\\phi _\\ell )^{\\Delta _n}$ as in Remark REF .",
"Similarly to $\\eta $ , it is often possible to choose between several values of $\\theta >0$ , but it is difficult to give a-priori a recommendation on how $\\theta $ should be selected.",
"One rather calculates for several admissible $\\theta $ the constant $C_\\theta :=\\max _{u\\in \\mathbb {R}}|u^\\theta (\\phi _\\ell (u))^{\\Delta _n}|$ numerically and deducts $B_\\theta =(2\\pi )^{-\\theta }C_\\theta $ .",
"Each combination of $(\\theta ,B_\\theta )$ then results in a valid number of summations $M_\\theta $ in the discrete Fourier Inversion algorithm.",
"Since $\\theta $ and $B_\\theta $ are only necessary to determine $M_\\theta $ , we may simply use the smallest $M_\\theta $ for the simulation.",
"To find $\\widetilde{X}$ with $\\widetilde{F}(\\widetilde{X})=U$ in Algorithm REF , we use a globalized Newton method with backtracking line search, also known as Armijo increment control.",
"The step lengths during the line search are determined by interpolation, which is a robust technique if combined with a standard Newton method.",
"Details on the globalized Newton method with backtracking may be found, for example, in [31], an example how the algorithm is used is given in [36].",
"Although convergence of this root finding algorithm is ensured by the increment control, its efficiency depends heavily on the choice of the initial value $\\widetilde{X}_0$ .",
"Clearly, $\\widetilde{X}_0$ should depend on the sampled $U\\sim \\mathcal {U}([0,1])$ and be related to the target distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"This means we should determine $\\widetilde{X}_0$ implicitly by $F^{(0)}(\\widetilde{X}_0)=U$ , where $F^{(0)}$ is a CDF of a distribution similar to the target distribution, but which can be inverted efficiently.",
"Approximation of a GH field We consider a GH Lévy field on the (separable) Hilbert space $H=L^2(\\mathcal {D})$ with a compact spatial domain $\\mathcal {D}\\subset \\mathbb {R}^s$ .",
"The operator $Q$ on $H$ is given by a Matérn covariance operator with variance $v>0$ , correlation length $r>0$ and a positive parameter $\\chi >0$ defined by $[Qh](x):=v\\int _\\mathcal {D} k_\\chi (x,y)h(y)dy,\\quad \\text{for }h\\in H,$ where $k_\\chi $ denotes the Matérn kernel.",
"For $\\chi =\\frac{1}{2}$ , we obtain the exponential covariance function and for $\\chi \\rightarrow \\infty $ the squared exponential covariance function.",
"For general $\\chi >0$ , the Matérn kernel $k_\\chi (x,y):=\\frac{2^{1-\\chi }}{\\Gamma _G(\\chi )}\\Big (\\frac{\\sqrt{2\\chi }|x-y|}{r}\\Big )^\\chi K_\\chi \\Big (\\frac{\\sqrt{2\\chi }|x-y|}{r}\\Big )$ fulfills the limit identity $k_\\chi (x,x)=\\lim _{y\\rightarrow x}k_\\chi (x,y)=1$ , which can be easily seen by [33].",
"Here $\\Gamma _G(\\cdot )$ is the Gamma function.",
"As shown in [20], this implies $tr(Q)=\\sum _{i=1}^\\infty \\rho _i=v\\int _{\\mathcal {D}}dx,$ where $(\\rho _i,i\\in \\mathbb {N})$ are the eigenvalues of the Matérn covariance operator $Q$ .",
"In general, no analytical expressions for the eigenpairs $(\\rho _i,e_i)$ of $Q$ will be available, but the spectral basis may be approximated by numerically solving a discrete eigenvalue problem and then interpolating by Nyström's method.",
"For a general overview of common covariance functions and the approximation of their eigenbasis we refer to [37] and the references therein.",
"Now let $L_N^{GH}$ be an approximation of a GH field by a $N$ -dimensional GH process $(\\ell ^{GH_N}(t),t\\in $ with fixed parameters $\\lambda ,\\alpha ,\\delta \\in \\mathbb {R}$ , $\\beta ,\\mu \\in \\mathbb {R}^N$ and $\\Gamma \\in \\mathbb {R}^{N\\times N}$ .",
"The parameters are chosen in such a way that the multi-dimensional GH process has uncorrelated marginal processes, hence the generated KL expansions $L_N^{GH}(x)(t)=\\sum _{i=1}^N\\varphi _i(x)\\ell ^{GH}_i(t)$ are again one-dimensional GH processes for any spectral basis $(\\varphi _i,i\\in \\mathbb {N})$ and fixed $x\\in \\mathcal {D}$ .",
"This in turn means, that we may draw samples of $\\ell ^{GH_N}$ by simulating a GIG process $\\ell ^{GIG}$ with parameters $a=\\delta , b=(\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta )^{1/2}$ and $p=\\lambda $ using Fourier inversion and then subordinating a $N$ -dimensional Brownian motion (see Lemma REF ).",
"The characteristic function of a GIG Lévy process $(\\ell ^{GIG}(t),t\\in $ with (fixed) parameters $a,b>0$ and $p\\in \\mathbb {R}$ is given by $ \\phi _{GIG}(u;a,b,p):=\\mathbb {E}[\\exp (iu\\ell ^{GIG}(1))]=(1-2iub^{-2})^{-p/2}\\frac{K_p(ab\\sqrt{1-2iub^{-2}})}{K_p(ab)}.$ The GIG distribution corresponding to $(\\phi _{GIG})^{\\Delta _n}$ with $\\Delta _n=1$ is continuous with finite variance (see [41]), which implies that these properties hold for all distributions with characteristic function $(\\phi _{GIG})^{\\Delta _n}$ , for any $\\Delta _n>0$ .",
"The constants as in Assumption REF are derived in the following.",
"For $k\\in \\mathbb {N}$ , the $k$ -th moment of the GIG distribution is given as $0<\\mathbb {E}\\big ((\\ell ^{GIG}(1))^k\\big )=\\big (\\frac{a}{b}\\big )^{k}\\frac{K_{p+k}(ab)}{K_p(ab)}<\\infty .$ For any $\\eta =2k$ we are, therefore, able to calculate the bounding constant $R$ via $R=(-1)^{k}\\frac{d^{2k}}{du^{2k}}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0},$ because the derivatives of $\\phi _{GIG}$ evaluated at $u=0$ are $(\\phi _{GIG}(0;a,b,p))^{(k)}=i^{-k}\\mathbb {E}\\big ((\\ell ^{GIG}(1))^k\\big )=i^{-k}\\big (\\frac{a}{b}\\big )^{k}\\frac{K_{p+k}(ab)}{K_p(ab)}.$ The calculation of the $\\eta $ -th derivative can be implemented easily by using a version of Faà di Bruno's formula containing the Bell polynomials, for details we refer to [25].",
"The bounding constants $\\theta $ and $B$ may be determined numerically as described Section REF (e.g.",
"by using the routine fminsearch in MATLAB).",
"The derivation of the bounds implies that we can ensure $L^p$ convergence of the approximated GIG process in the sense of Theorem REF for any $p\\ge 1$ , because it is possible to define $\\eta $ as any even integer and then obtain $R$ by differentiation.",
"We observe that the target distribution with characteristic function $(\\phi _{GIG}(u;a,b,p))^{\\Delta _n}$ and $\\Delta _n>0$ is not necessarily GIG, except for the Inverse Gaussian (IG) case where $p=-1/2$ and $(\\phi _{IG}(u;a,b))^{\\Delta _n}=\\phi _{IG}(u;\\Delta _na,b)$ (see Remark REF ).",
"This special feature of the IG distribution is exploited to determine the initial values $\\widetilde{X}_0$ in the Newton iteration by moment matching: Consider an $IG(a_0,b_0)$ distribution with mean $a_0/b_0$ and variance $a_0/b_0^3$ , where the parameters $a_0,b_0>0$ are “matched” to the target distribution's mean and variance via $\\frac{a_0}{b_0}&=i\\frac{d}{du}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0},\\\\\\frac{a_0}{b_0^3}&=(-1)\\frac{d^2}{du^2}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0}-\\Big (i\\frac{d}{du}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0}\\Big )^2.$ If $F_{\\Delta _n}^{IG}$ denotes the CDF of this $IG(a_0,b_0)$ distribution, the initial value of the globalized Newton method is given implicitly by $F_{\\Delta _n}^{IG}(\\widetilde{X}_0)=U$ .",
"The inversion of $F_{\\Delta _n}^{IG}$ may be executed numerically by many software packages like MATLAB.",
"With our approach, this results in the approximation of a GIG process $\\widetilde{\\ell }^{GIG}$ at discrete times $t_j\\in \\Xi _n$ .",
"The $N$ -dimensional GH process $\\ell ^{GH_N}$ may then be approximated at $t_j$ for $j=0,\\dots ,n$ by the process $\\widetilde{\\ell }^{GH_N}$ with $\\widetilde{\\ell }^{GH_N}(t_0)=0$ and the increments $\\widetilde{\\ell }^{GH_N}(t_j)-\\widetilde{\\ell }^{GH_N}(t_{j-1})=\\mu \\Delta _n+\\Gamma \\beta (\\widetilde{\\ell }^{GIG}(t_j)-\\widetilde{\\ell }^{GIG}(t_{j-1})) + \\sqrt{(\\widetilde{\\ell }^{GIG}(t_j)-\\widetilde{\\ell }^{GIG}(t_{j-1}))\\Gamma }w^N_j(1),$ for $j=1,\\dots ,n$ , where the $w_j^N(1)$ are i.i.d.",
"$\\mathcal {N}_N(0,\\mathbf {1}_{N\\times N})$ -distributed random vectors.",
"To obtain the process $\\widetilde{\\ell }^{GH_N}$ at arbitrary times $t\\in , we interpolate the samples $ (GHN(tj),j=0...,n)$ piecewise constant as in Algorithm~\\ref {algo:approx2}.With this, we are able to generate an approximation of $ LNGH$ at any point $ (x,t)D by $\\widetilde{L}_N^{GH}(x)(t):=\\sum _{i=1}^N\\varphi _i(x) \\widetilde{\\ell }^{GH}_i(t).$ The knowledge of $tr(Q)$ enables us to determine the truncation index $N$ and the constant $C_\\ell $ as in Remark : For $N\\in \\mathbb {N}$ let $(\\widetilde{\\ell }^{GH}_i,i=1,\\ldots ,N)$ be the approximations of the processes $(\\ell ^{GH}_i,i=1,\\ldots ,N)$ , where the random vector $(\\ell ^{GH}_1(1),\\dots ,\\ell _N^{GH}(1))$ is multivariate GH-distributed by assumption.",
"Hence, for every $N\\in \\mathbb {N}$ , we obtain the parameters $a(N), b(N), \\lambda (N)$ of a corresponding GIG subordinator $\\ell ^{GIG,N}$ , which is approximated through a piecewise constant process $\\widetilde{\\ell }^{GIG,N}$ as above.",
"With Eq.",
"(REF ) we calculate the error $E_{GIG,N}^p:=\\sup _{t\\in \\mathbb {E}(|\\ell ^{GIG,N}(t)-\\widetilde{\\ell }^{GIG,N}(t)|^p).",
"}for p\\in \\lbrace 1,2\\rbrace .If \\beta \\in \\mathbb {R}^N and \\Gamma \\in \\mathbb {R}^{N\\times N} denote the GH parameters corresponding to (\\ell ^{GH}_1(1),\\dots ,\\ell _N^{GH}(1)), the L^2(\\Omega ;\\mathbb {R}) approximation error of each process \\ell ^{GH}_i is given by\\begin{equation*}\\widetilde{C}_{\\ell ,i}:=\\sup _{t\\in \\frac{\\mathbb {E}(|\\ell ^{GH}_i(t)-\\widetilde{\\ell }^{GH}_i(t)|^2)}{\\Delta _n}= \\frac{E_{GIG,N}^2(\\Gamma \\beta )_i^2+E_{GIG,N}^1\\sqrt{\\Gamma _{[i]}\\Gamma _{[i]}^{\\prime }}}{\\Delta _n},}where \\Gamma _{[i]} indicates the i-th row of \\Gamma .Starting with N=1, we compute the first N eigenvalues and the difference\\begin{equation*}T\\Big (tr(Q)-\\sum _{i=1}^N\\rho _i\\Big )-\\max _{i=1,\\dots ,N}\\widetilde{C}_{\\ell ,i}\\Delta _n \\sum _{i=1}^N\\rho _i\\end{equation*}and increase N by one in every step until this expression is close to zero.If a suitable N is found, we define C_\\ell :=\\max _{i=1,\\dots ,N}\\widetilde{C}_{\\ell ,i} and thus have equilibrated truncation and approximation errors by ensuring Eq.~(\\ref {eq:trunc}).For simplicity, we have implicitly assumed here that the processes \\ell ^{GH}_i were normalized in the sense that \\text{Var}(\\ell ^{GH}_i(t))=t.This is due to the fact that \\rho _i\\ell _i (here with \\ell _i=\\ell ^{GH}_i) in Theorem~\\ref {thm:H_error} represents the scalar product (L(t),e_i)_H with variance \\rho _it.In case we have unnormalized processes, one can simply divide \\ell ^{GH}_i by its standard deviation (see Formula~(\\ref {eq:VarZ})) and adjust the constants \\widetilde{C}_{\\ell ,i} and C_\\ell accordingly.\\subsection {Numerical results}As a test for our algorithm, we generate GH fields on the time interval [0,1] with step size \\Delta _n=2^{-6}, on the spatial domain \\mathcal {D}=[0,1].For practical aspects, one is usually interested in the L^1-error \\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|) and the L^2-error (\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^2))^{1/2}.Upper bounds for both expressions depend on \\eta and D and are given by Ineq.~(\\ref {L^p error}).To obtain reasonable errors, we refer to the discussion on the choice of D in Remark~\\ref {rem:t-conv} and set D=\\Delta _n^{1/(1-\\eta )}.This ensures that the L^1-error is of order \\mathcal {O} (\\Delta _n) and is a good trade-off between simulation time and the size of the L^2-error for most values of \\eta in the GIG example below.Choosing for example D=\\Delta _n^{2/(2-\\eta )} would reduce the L^2-error to order \\mathcal {O} (\\Delta _n), but does not have a significant effect on the L^1-error and results in a higher computational time.For the Matérn covariance operator Q we use variance v=1, correlation length r=0.1 and \\chi \\in \\lbrace \\frac{1}{2},\\frac{3}{2}\\rbrace ,where a higher value of \\chi increases the regularity of the field along the x-direction.For the fixed GH parameters we choose \\alpha =5, \\beta =\\mu =0_N, \\delta =4 and \\Gamma =\\mathbf {1}_N, the shape parameter \\lambda will vary throughout our simulation and admits the values \\lambda \\in \\lbrace -\\frac{1}{2},1\\rbrace , which results in NIG resp.",
"hyperbolic GH fields.This parameter setting ensures that the multi-dimensional GH distribution has uncorrelated marginals, hence the truncated KL expansion L_N^{GH} of L^{GH} is itself an infinite dimensional GH Lévy process.Further, for every N\\in \\mathbb {N}, the constant \\widetilde{C}_{\\ell ,i} from Section~\\ref {sec:GH_app} is independent of i=1,\\dots ,N, thus the truncation index N can easily be determined to balance out the Fourier inversion and truncation error for each combination of \\lambda and \\chi .To examine the impact of \\eta on the efficiency of the simulation, we set \\eta \\in \\lbrace 4,6,8,10\\rbrace and the constant R as suggested in Section~\\ref {sec:GH_app} for each \\eta .For fixed \\eta and R, we choose \\theta \\in \\lbrace 1,1.5\\dots ,99.5,100\\rbrace and calculate for each \\theta the constant B_\\theta as in Section~\\ref {sec:imp}.This results in up to 199 different values for the number of summations M_\\theta , which all guarantee the desired accuracy \\varepsilon , meaning we can choose the smallest M_\\theta for our simulation.The optimal value \\theta _{opt} which leads to the smallest M_\\theta depends highly on the GH parameters and may vary significantly with \\eta .For \\lambda =1, we found that \\theta _{opt} ranges from 34 to 68.5, varying with each choice of \\eta \\in \\lbrace 4,6,8,10\\rbrace .In contrast, in the second example with \\lambda =-1/2, \\theta _{opt}=11 independent of \\eta .We generate 1.000 approximations \\widetilde{L}_N^{GH} for several combinations of \\lambda , \\chi and \\eta , which allows us to check if the generated samples actually follow the desired target distributions.To this end, we conduct Kolomogorov--Smirnov tests for the subordinating GIG process as well as for the distribution of the GH field at a fixed point in time and space and report on the corresponding p-values.\\begin{figure}\\centering \\subfigure [Sample of a GH field]{\\includegraphics [scale=0.45]{hyp_m0_5.eps}}\\subfigure [Empirical dist.",
"of 1.000 samples at t=x=1]{\\includegraphics [scale=0.48]{hyp_hist_m0_5_new.eps}}\\caption {Sample and empirical distribution of an hyperbolic field with parameters \\lambda =1, \\chi =1/2, \\eta =10 and truncation after N=132 terms.",
"}\\centering \\subfigure [Sample of a GH field]{\\includegraphics [scale=0.45]{nig_m1_5.eps}}\\subfigure [Empirical dist.",
"of 1.000 samples at t=x=1]{\\includegraphics [scale=0.48]{nig_hist_m1_5_new.eps}}\\caption {Sample and empirical distribution of a NIG field with parameters \\lambda =-1/2, \\chi =3/2, \\eta =10 and truncation after N=18 terms.",
"}\\end{figure}Figures~\\ref {fig:hyp} and ~\\ref {fig:nig} show samples of approximated GH random fields:Along the time axis we see the characteristic behavior of the (pure jump) GH processes for every point x\\in \\mathcal {D}.For a fixed point in time t, the paths along the x-axis vary according to their correlation, depending on the covariance parameter \\chi .As reported in \\cite {RW06}, the eigenvalues of Q decay slower if \\chi becomes smaller, meaning we need a higher number of summations N in the KL expansion so that the error contributions are equilibrated.This effect can be seen in Tables~\\ref {tab:hyp} and~\\ref {tab:nig}, where the truncation index N changes significantly with \\chi .If the KL expansion, however, can be sampled by a N-dimensional GH process as suggested in Theorem~\\ref {thm:Z_U},the number of summations N has only a minor impact on the computational costs of the KL expansion.This is due to the fact that in this case the time consuming part, namely simulating the subordinator, has to be done only once, regardless of N.Compared to these costs, the costs of subordinating a Brownian motion of any finite dimension are negligible.The histograms in Figures~\\ref {fig:hyp} and ~\\ref {fig:nig} show the empirical distribution of the approximation \\widetilde{L}_N^{GH}(x)(t) at time t=1 and x=1.The theoretical distribution at time 1 and an arbitrary point x\\in \\mathcal {D} is again GH, where the parameters are given in Lemma~\\ref {lem:KL_GH}.Obviously, the empirical distributions fit the target GH distributions from Lemma~\\ref {lem:KL_GH}.To be more precise, we have conducted a Kolmogorov-Smirnov test for both, the subordinating GIG process and the GH field at time t=1 and for the latter at x=1.We know the law of both processes at x\\in \\mathcal {D} and are able to obtain their CDFs sufficiently precise for the tests by numerical integration.The test results for 1.000 samples of the hyperbolic resp.",
"the NIG field with covariance parameters \\chi =\\frac{1}{2} resp.",
"\\chi =\\frac{3}{2} are given in Tables~\\ref {tab:hyp} and~\\ref {tab:nig} above and do not suggest that the generated samples follow another distribution than the expected one.\\end{equation*}\\begin{table}\\end{table}\\begin{center}\\small \\begin{tabular}{l|*{3}{c|}c}\\eta & E_{GIG,N}^1 & E_{GIG,N}^1/\\Delta _n &E_{GIG,N}^2&\\mathbb {E}[||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2] \\\\ \\hline 4 & 0.0143 &0.9166& 0.2584 &0.0646\\\\6 & 0.0138 &0.8835& 0.0749 & 0.0635\\\\8 & 0.0138 &0.8824& 0.0601 &0.0634\\\\10& 0.0140 &0.8975& 0.0806 &0.0636\\\\ \\hline \\hline \\eta &N & p-value GH&abs.",
"time&rel.",
"time\\\\ \\hline 4 &130& 0.8246 & 0.1945 sec.&100.00\\% \\\\6 &133& 0.3077 & 0.1093 sec.&56.19\\% \\\\8 &133& 0.3077 & 0.0851 sec.",
"&43.78\\%\\\\10&132& 0.2873 & 0.0759 sec.",
"&39.04\\%\\\\ \\hline \\end{tabular}\\end{center}\\caption {Errors, p-values and average simulation times per field based on 1.000 simulations.Stepsize \\Delta t =2^{-6} and D=\\Delta t^{1/(1-\\eta )}.GH process: \\lambda =1, \\alpha = 5, \\beta = 0_N, \\delta = 4, \\mu = 0_N, \\Gamma = 1_{N\\times N}.Covariance parameters: \\chi =1/2, r=0.1 and v=1.The KS test for the GIG subordinator returns a p-value of 0.5498 for each \\eta \\in \\lbrace 4,6,8,10\\rbrace .",
"}\\begin{center}\\small \\begin{tabular}{l|*{3}{c|}c}\\eta & E_{GIG,N}^1 & E_{GIG,N}^1/\\Delta _n &E_{GIG,N}^2&\\mathbb {E}[||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2] \\\\ \\hline 4 & 0.0132 & 0.8443 & 0.2079 & 0.0619 \\\\6 & 0.0128 & 0.8170 & 0.0584 & 0.0608 \\\\8 & 0.0127 & 0.8155 & 0.0456 & 0.0608\\\\10& 0.0129 & 0.8252 & 0.0589 & 0.0611 \\\\ \\hline \\hline \\eta &N & p-value GH&abs.",
"time&rel.",
"time\\\\ \\hline 4 & 18 & 0.9223 & 0.1039 sec.",
"& 100.00\\% \\\\6 & 18 & 0.9223 & 0.0628 sec.",
"& 60.43\\% \\\\8 & 18 & 0.9223 & 0.0460 sec.",
"& 44.29\\% \\\\10& 18 & 0.9223 & 0.0380 sec.",
"& 38.59\\% \\\\ \\hline \\end{tabular}\\end{center}\\caption {Errors, p-values and average simulation times per field based on 1.000 simulations.Stepsize \\Delta t =2^{-6} and D=\\Delta t^{1/(1-\\eta )}.GH process: \\lambda =-1/2, \\alpha = 5, \\beta = 0_N, \\delta = 4, \\mu = 0_N, \\Gamma = 1_{N\\times N}.Covariance parameters: \\chi =3/2, r=0.1 and v=1.The KS test for the GIG subordinator returns a p-value of 0.6145 for each \\eta \\in \\lbrace 4,6,8,10\\rbrace .",
"}$ We denote by $E_{GIG,N}^1$ and $E_{GIG,N}^2$ the approximation error of the subordinator as in Eq.",
"(REF ), which we have listed in absolute terms in Tables and .",
"The first error bound is also given relative to $\\Delta _n$ to show that it is in fact of magnitude $\\mathcal {O}(\\Delta _n)$ .",
"While the $L^1(\\Omega ;\\mathbb {R})$ -error $E_{GIG,N}^1$ is relatively constant for each $\\eta $ , the $L^2(\\Omega ;\\mathbb {R})$ -error $E_{GIG,N}^2$ is rather high for $\\eta =4$ , but has an acceptable upper bound for $\\eta \\ge 6$ .",
"This is not surprising, since $D=\\Delta _n^{1/(1-\\eta )}$ only guarantees that $\\mathbb {E}(|\\ell ^{GIG}(t)-\\widetilde{\\ell }^{GIG}(t)|)=\\mathcal {O}(\\Delta _n)$ .",
"We emphasize that the (theoretic) error bounds in Tables and are very conservative as the triangle inequality and similar \"coarse\" estimates were used repeatedly in their estimation in Theorem REF and REF .",
"The truncation index $N$ is highly sensitive to $\\chi $ , but has small or no variations for fixed $\\chi $ and varying $\\eta $ .",
"Since we choose $t\\in [0,1]$ , the expression $\\mathbb {E}(||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2)$ in Tables and is an upper bound for the $L^2(\\Omega ;H)$ -error $\\sup _{t\\in [0,1]}\\mathbb {E}(||L^{GH}(t)-\\widetilde{L}^{GH}_N(t)||_H^2)$ .",
"Note that this error is small in relative terms, since by our choice of $Q$ and Eq.",
"REF we have $\\mathbb {E}(||L^{GH}(1)||^2_H)=tr(Q)=1$ .",
"The p-value of the GH distribution varies if different $N$ are chosen for the KL expansion, which is natural due to statistical fluctuations.",
"More importantly, the null hypothesis, namely that the samples follow a GH distribution with the expected parameters, is never rejected at a $5\\%$ -level.",
"As expected, the speed of the simulation heavily depends on $\\eta $ .",
"Looking at the results for $\\eta =4$ , one might argue that the Fourier inversion method is only suitable for processes where this parameter can be chosen rather high, i.e.",
"for distributions which admit a large number of finite moments.",
"To qualify this objection, we consider once more the t-distribution with three degrees of freedom and the corresponding Lévy process $\\ell ^{t3}$ from Example REF .",
"Since $\\mathbb {E}(\\ell ^{t3}(\\Delta _n))=0$ and $\\text{Var}(\\ell ^{t3}(\\Delta _n))=\\sqrt{3}\\Delta _n$ , we can choose $\\eta =2$ and hence $R=\\sqrt{3}\\Delta _n$ .",
"The characteristic function of $\\ell ^{t3}(\\Delta _n)$ is given by $(\\phi _{t3}(u))^{\\Delta _n}=\\exp (-\\sqrt{3}\\Delta _n|u|)(\\sqrt{3}|u|+1)^{\\Delta _n}$ and $B$ and $\\theta $ are estimated in the same way as for the GIG process.",
"Using again $\\Delta _n=2^{-6}$ and $D=\\Delta _n^{1/(1-\\eta )}$ , we obtain that the number of summations in the approximation is $M=12.924$ for $\\theta =\\frac{19}{2}$ .",
"The simulation time for one process $\\widetilde{\\ell }^{t3}$ with $(\\Delta _n)^{-1}=2^6$ increments in the interval $[0,1]$ is on average 0.0655 seconds, where the initial values have been approximated by matching the moments of a normal distribution (the Kolmogorov-Smirnov test for a t-distribution at $t=1$ based on 1.000 samples returns a p-value of $0.5994$ ).",
"In the GIG example, we needed $M=79.086$ terms in the summation if $\\eta =4$ is chosen and still $M=33.030$ terms for $\\eta =10$ .",
"This shows that the Fourier Inversion method is also applicable if $\\eta $ can only be chosen relatively low and that the GIG (resp.",
"GH) process is a computationally expensive example of a Lévy process."
],
[
"Generalized hyperbolic Lévy processes",
"Distributions which belong to the class of generalized hyperbolic distributions may be used for a wide range of applications.",
"GH distributions have been first introduced in [4] to model mass-sizes in aeolian sand (see also [5]).",
"Since then they have been successfully applied, among others, in Finance and Biology.",
"Giving a broad class the distributions are characterized by six parameters, famous representatives are the Student's t, the normal-inverse Gaussian, the hyperbolic and the variance-gamma distribution.",
"The popularity of GH processes is explained by their flexibility in modeling various characteristics of a distribution such as asymmetries or heavy tails.",
"A further advantage in our setting is, that the characteristic function is known and, therefore, the Fourier Inversion may be applied to approximate these processes.",
"This section is devoted to investigate several properties of multi-dimensional GH processes which are then used to construct an approximation of an infinite-dimensional GH field.",
"In contrast to the Gaussian case, the sum of two independent and possibly scaled GH processes is in general not again a GH process.",
"We show a possibility to approximate GH Lévy fields via Karhunen-Loève expansions in such a way that the approximated field is itself again a GH Lévy field.",
"This is essential, so as to have convergence of the approximation to a GH Lévy field in the sense of Theorem REF .",
"Furthermore, we give, for $N\\in \\mathbb {N}$ , a representation of a $N$ -dimensional GH process as a subordinated Brownian motion and show how a multi-dimensional GH process may be constructed from uncorrelated, one-dimensional GH processes with given parameters.",
"This may be exploited by the Fourier inversion algorithm in such a way that the computational expenses to simulate the approximated GH fields are virtually independent of the truncation index $N$ .",
"Assume, for $N\\in \\mathbb {N}$ , that $\\lambda \\in \\mathbb {R}$ , $\\alpha >0$ , $\\beta \\in \\mathbb {R}^N$ , $\\delta >0$ , $\\mu \\in \\mathbb {R}^N$ and $\\Gamma $ is a symmetric, positive definite (spd) $N\\times N$ -matrix with unit determinant.",
"We denote by $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ the $N$ -dimensional generalized hyperbolic distribution with probability density function $f^{GH_N}(x;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )=\\frac{\\gamma ^\\lambda \\alpha ^{N/2-\\lambda }}{(2\\pi )^{N/2}\\delta ^\\lambda K_\\lambda (\\delta \\gamma )}\\frac{K_{\\lambda -N/2}(\\alpha g(x-\\mu ))}{g(x-\\mu )^{N/2-\\lambda }}\\exp (\\beta ^{\\prime }(x-\\mu ))$ for $x\\in \\mathbb {R}^N$ , where $g(x):=\\sqrt{\\delta ^2+x^{\\prime }\\Gamma x}, \\quad \\gamma ^2:=\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta $ and $K_\\lambda (\\cdot )$ is the modified Bessel-function of the second kind with $\\lambda $ degrees of freedom.",
"The characteristic function of $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ is given by $\\begin{split}\\phi _{GH_N}(u; \\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )&:=\\exp (iu^{\\prime }\\mu )\\Big (\\frac{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta }{\\alpha ^2-(iu+\\beta )^{\\prime }\\Gamma (iu+\\beta )}\\Big )^{\\lambda /2}\\\\&\\quad \\cdot \\frac{K_\\lambda (\\delta (\\alpha ^2-(iu+\\beta )^{\\prime }\\Gamma (iu+\\beta ))^{1/2})}{K_\\lambda (\\delta \\gamma )},\\end{split}$ where $A^{\\prime }$ denotes the transpose of a matrix or vector $A$ .",
"For simplicity, we assume that the condition $\\alpha ^2>\\beta ^{\\prime }\\Gamma \\beta $ is satisfiedIf $\\alpha ^2=\\beta ^{\\prime }\\Gamma \\beta $ and $\\lambda <0$ , the distribution is still well-defined, but one has to consider the limit $\\gamma \\rightarrow 0^+$ in the Bessel functions, see [13], [39]..",
"If $N=1$ , clearly, $\\Gamma =1$ is the only possible choice for the \"matrix parameter\" $\\Gamma $ , thus we omit it in this case and denote the one-dimensional GH distribution by $GH(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu )$ .",
"Barndorff–Nielsen obtains the GH distribution in [5] as a normal variance-mean mixture of a $N$ -dimensional normal distribution and a (one-dimensional) generalized inverse Gaussian (GIG) distribution with density function $f^{GIG}(x;a,b,p)=\\frac{(b/a)^p}{2K_p(ab)}x^{p-1}\\exp (-\\frac{1}{2}(a^2x^{-1}+b^2x)),\\quad x>0$ and parameters $a,b>0$ and $p\\in \\mathbb {R}$The notation of the GIG distribution varies throughout the literature, we use the notation from [41].. To be more precise: Let $w^N(1)$ be a $N$ -dimensional standard normally distributed random vector, $\\Gamma $ a spd $N\\times N$ -structure matrix with unit determinant and $\\ell ^{GIG}(1)$ a $GIG(a,b,p)$ random variable, which is independent of $w^N(1)$ .",
"For $\\mu ,\\beta \\in \\mathbb {R}^N$ , we set $\\delta =a$ , $\\lambda =p$ , $\\alpha =\\sqrt{b^2+\\beta ^{\\prime }\\Gamma \\beta }$ and define the random variable $\\ell ^{GH_N}(1)$ as $\\ell ^{GH_N}(1):=\\mu +\\Gamma \\beta \\ell ^{GIG}(1) + \\sqrt{\\ell ^{GIG}(1)\\Gamma }w^N(1).$ Then $\\ell ^{GH_N}(1)$ is $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ -distributed, where $\\sqrt{\\Gamma }$ denotes the Cholesky decomposition of the matrix $\\Gamma $ .",
"With this in mind, one can draw samples of a GH distribution with given parameters by sampling multivariate normal and GIG-distributed random variables, as $a=\\delta >0$ and $b=\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta }>0$ is guaranteed by the conditions on the GIG parameters (this results in Eq.",
"(REF ) being fulfilled).",
"As noted in [18], for general $\\lambda \\in \\mathbb {R}$ , we cannot assume that the increments of the GH Lévy process (resp.",
"of the subordinating process) over a time length other than one follow a GH distribution (resp.",
"GIG distribution).",
"If $N=1$ , however, the (one-dimensional) GH Lévy process $\\ell ^{GH}$ has the representation $\\ell ^{GH}(t)\\stackrel{\\mathcal {L}}{=}\\mu t+\\beta \\ell ^{GIG}(t)+w(\\ell ^{GIG}(t)),\\quad \\text{for } t\\ge 0,$ where $w$ is a one-dimensional Brownian motion and $\\ell ^{GIG}$ a GIG process independent of $w$ (see [16]).",
"This result yields the following generalization: Lemma 4.1 For $N\\in \\mathbb {N}$ , the $N$ -dimensional process $\\ell ^{GH_N}=(\\ell ^{GH_N}(t),t\\in ~ $ , which is $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ -distributed, can be represented as a subordinated $N$ -dimensional Brownian motion $w^N$ via $\\ell ^{GH_N}(t)\\stackrel{\\mathcal {L}}{=}\\mu t+\\Gamma \\beta \\ell ^{GIG}(t)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(t)),$ where $(\\ell ^{GIG}(t),t\\in $ is a GIG Lévy process independent of $w^N$ and $\\sqrt{\\Gamma }$ is the Cholesky decomposition of $\\Gamma $ .",
"Since the $GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ distribution may be represented as a normal variance-mean mixture (see Eq.",
"(REF )), we have, that $\\ell ^{GH_N}(1)\\stackrel{\\mathcal {L}}{=}\\mu +\\Gamma \\beta \\ell ^{GIG}(1) + \\sqrt{\\Gamma \\ell ^{GIG}(1)}w^N(1)\\stackrel{\\mathcal {L}}{=}\\mu +\\Gamma \\beta \\ell ^{GIG}(1)+ \\sqrt{\\Gamma }w^N(\\ell ^{GIG}(1))$ where $\\ell ^{GIG}(1)\\sim GIG(\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda )$ and $w^N$ is a $N$ -dimensional Brownian motion independent of $\\ell ^{GIG}(1)$ .",
"The characteristic function of the mixed density is then given by $\\phi _{GH_N}(u;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )=\\exp (iu^{\\prime }\\mu )\\mathcal {M}_{GIG}(iu^{\\prime } \\Gamma \\beta -\\frac{1}{2}u^{\\prime }\\Gamma u;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ),$ where $\\mathcal {M}_{GIG}$ denotes the moment generating function of $\\ell ^{GIG}(1)$ (see [7]).",
"The GIG distribution is infinitely divisible, thus this GIG Lévy process $\\ell ^{GIG}=(\\ell ^{GIG}(t),t\\in $ can be defined via its characteristic function for $t\\in :$$\\mathbb {E}(\\exp (iu\\ell ^{GIG}(t)))=(\\mathcal {M}_{GIG}(iu;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ))^t.$$The infinite divisibility yields further\\begin{equation*}\\begin{split}\\mathbb {E}\\big (\\exp (iu^{\\prime }\\ell ^{GH_N}(t))\\big )&=\\mathbb {E}\\big (\\exp (iu^{\\prime }\\ell ^{GH_N}(1))\\big )^t=(\\phi _{GH}(u;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma ))^t\\\\&=\\exp (iu^{\\prime }(\\mu t))(\\mathcal {M}_{GIG}(iu^{\\prime } \\Gamma \\beta -\\frac{1}{2}u^{\\prime }\\Gamma u;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ))^t.\\end{split}\\end{equation*}The expression above is the characteristic function of another normal variance-mean mixture, namely where the subordinator $ GIG$ is a GIG process with characteristic function$$\\mathbb {E}(\\exp (iu\\ell ^{GIG}(t)))=(\\mathcal {M}_{GIG}(iu;\\delta ,\\sqrt{\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta },\\lambda ))^t.$$Hence, $ GHN(t)$ can be expressed as\\begin{equation*}\\ell ^{GH_N}(t)\\stackrel{\\mathcal {L}}{=}\\mu t+\\Gamma \\beta \\ell ^{GIG}(t)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(t)).\\end{equation*}$ Remark 4.2 In the special case of $\\lambda =-\\frac{1}{2}$ one obtains the normal inverse Gaussian (NIG) distribution.",
"The mixing density is, in this case, the inverse Gaussian (IG) distribution.",
"We denote the $N$ -dimensional NIG distribution by $NIG_N(\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ .",
"This is the only subclass of GH distributions which is closed under convolutions in the sense that (see [32]) $NIG_N(\\alpha ,\\beta ,\\delta _1,\\mu _1,\\Gamma )*NIG_N(\\alpha ,\\beta ,\\delta _2,\\mu _2,\\Gamma )=NIG_N(\\alpha ,\\beta ,\\delta _1+\\delta _2,\\mu _1+\\mu _2,\\Gamma ).$ For $\\lambda \\in \\mathbb {R}$ , the sum of independent GH random variables is in general not GH-distributed.",
"This implies further, that one is in general not able to derive bridge laws of these processes in closed form, meaning we need to use the algorithms introduced in Section REF for simulation.",
"As shown in [6], the GH and the GIG distribution are infinitely-divisible, thus we can define the $N$ -dimensional GH Lévy process $\\ell ^{GH_N}=(\\ell ^{GH_N}(t),t\\in $ with characteristic function $\\mathbb {E}(\\exp (iu\\ell ^{GH_N}(t))=(\\phi _{GH_N}(u;\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma ))^t.$ Remark 4.3 If $\\lambda =-\\frac{1}{2}$ , the corresponding NIG Lévy process $(\\ell ^{NIG_N}(t),t\\in $ has characteristic function $\\mathbb {E}[\\exp (iu\\ell ^{NIG_N}(t))]=(\\phi _{GH_N}(u;-\\frac{1}{2},\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma ))^t=\\phi _{GH_N}(u;-\\frac{1}{2},\\alpha ,\\beta ,t\\delta ,t\\mu ,\\Gamma ).$ This is due to the fact that the characteristic function $\\phi _{IG}(u;a,b)$ of the mixing IG distribution fulfills the identity $(\\phi _{IG}(u;a,b))^t=\\phi _{IG}(u;ta,b)$ for any $t\\in and $ a,b>0$ (see \\cite {S03}).$ We consider the finite time horizon $ [0,T]$ , for $T<+\\infty $ , the probability space $(\\Omega ,(\\mathcal {A}_t,t\\ge 0),\\mathbb {P})$ , and a compact domain $\\mathcal {D}\\subset \\mathbb {R}^s$ for $s\\in \\mathbb {N}$ to define a GH Lévy field as a mapping $L^{GH}:\\Omega \\times \\mathbb {R},\\quad (\\omega ,x,t)\\mapsto L^{GH}(\\omega )(x)(t),$ such that for each $x\\in the point-wise marginal process\\begin{equation*}L^{GH}(\\cdot )(x)(\\cdot ):\\Omega \\times \\mathbb {R},\\quad (\\omega ,t)\\mapsto L^{GH}(\\omega )(x)(t),\\end{equation*}is a one-dimensional GH Lévy process on $ (,(At,t0),P)$ with characteristic function\\begin{equation*}\\mathbb {E}\\big (\\exp (iuL^{GH}(x)(t))\\big )=(\\phi _{GH}(u;\\lambda (x),\\alpha (x),\\beta (x),\\delta (x),\\mu (x)))^t,\\end{equation*}where the indicated parameters are given by continuous functions, i.e.",
"$ ,,C($\\mathbb {R}$ )$ and $ ,C($\\mathbb {R}$ >0)$.",
"We assume that condition (\\ref {eq:ghcond}), i.e.",
"$ (x)2>(x)2$, is fulfilled for any $ x to ensure that $L^{GH}(x)(\\cdot )$ is a well-defined GH Lévy process.",
"This, in turn, means that $L^{GH}$ takes values in the Hilbert space $H=L^2(\\mathcal {D})$ and is square integrable as $\\mathbb {E}(||L^{GH}(t)||_H^2)\\le T\\mathbb {E}(||L^{GH}(1)||_H^2)\\le T \\max \\limits _{x\\in \\mathcal {D}}\\mathbb {E}(L^{GH}(x)(1)^2)V_{\\mathcal {D}},$ where $V_{\\mathcal {D}}$ denotes the volume of $\\mathcal {D}$ .",
"The right hand side is finite since every GH distribution has finite variance (see for example [30], [41]), the parameters of the distribution of $L^{GH}(x)(1)$ depend continuously on $x$ and $\\mathcal {D}\\subset \\mathbb {R}^s$ is compact by assumption.",
"We use the Karhunen-Loève expansion from Section to obtain an approximation of a given GH Lévy field.",
"For this purpose, we consider the truncated sum $L^{GH}_N(x)(t) :=\\sum _{i=1}^{N} \\varphi _i(x)\\ell _i^{GH}(t)\\stackrel{\\mathcal {L}}{=} \\sum _{i=1}^{N} \\varphi _i(x) \\Big (\\mu _i t+\\beta _i \\ell ^{GIG}_i(t)+w_i(\\ell ^{GIG}_i(t))\\Big ),$ where $N\\in \\mathbb {N}$ and $\\varphi _i(x)=\\sqrt{\\rho _i}e_i(x)$ is the $i$ -th component of the spectral basis evaluated at the spatial point $x$ .",
"For each $i,=1,\\dots ,N$ , the processes $\\ell _i^{GH}:=(\\ell _i^{GH}(t), t\\in $ are uncorrelated but dependent $GH(\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i)$ Lévy process.",
"From Theorem REF follows that $L^{GH}_N$ converges in $L^2(\\Omega ;H)$ to $L^{GH}$ as $N\\rightarrow \\infty $ .",
"With given $\\mu _i, \\beta _i\\in \\mathbb {R}$ , we have that $\\ell ^{GH}_i(t)\\stackrel{\\mathcal {L}}{=}\\mu _i t+\\beta _i \\ell ^{GIG}_i(t)+w_i(\\ell ^{GIG}_i(t)),$ where for each $i$ , the process $(\\ell ^{GIG}_i(t), t\\in $ is a GIG Lévy process with parameters $a_i=\\delta _i, b_i=(\\alpha _i^2-\\beta _i^2)^{1/2}>0$ and $p_i=\\lambda _i\\in \\mathbb {R}$ .",
"In addition, $(w_i(t), t\\in $ is a one-dimensional Brownian motion independent of $\\ell ^{GIG}_i$ and all Brownian motions $w_1,\\dots ,w_N$ are mutually independent of each other, but the processes $\\ell ^{GIG}_1,\\dots ,\\ell ^{GIG}_N$ may be correlated.",
"We aim for an approximation $(L^{GH}_N(x)(t),t\\in $ which is a GH process for arbitrary $\\varphi _i$ and $x\\in \\mathcal {D}$ .",
"Remark REF suggests that this cannot be achieved by the summation of independent $\\ell _i^{GH}$ , but rather by using correlated subordinators $\\ell ^{GIG}_1,\\dots ,\\ell ^{GIG}_N$ .",
"Before we determine the correlation structure of the subordinators, we establish a necessary and sufficient condition on the $\\ell _i^{GH}$ to achieve the desired distribution of the approximation.",
"Lemma 4.4 Let $N\\in \\mathbb {N}$ , $t\\in and $ (iGH,i=1...,N)$ be GH processes as defined in Eq.~(\\ref {eq:Z_i}).",
"For a vector $a = (a1,...,aN)$ with arbitrary numbers $ a1,...,aN$\\mathbb {R}$ {0}$, the process $ GH,a$ defined by\\begin{equation}\\ell ^{GH,\\bf a}(t):=\\sum _{i=1}^{N} a_i\\ell _i^{GH}(t)=\\sum _{i=1}^{N} a_i(\\mu _i+\\beta _i\\ell _i^{GIG}(t)+w_i(\\ell _i^{GIG}(t)))\\end{equation}is a one-dimensional GH process, if and only if the vector $ GHN(1):=(1GH(1),...,NGH(1))'$ is multivariate $ GHN((N),(N),(N),(N),(N),)$-distributed with parameters $ (N),(N)$, $ (N)$\\mathbb {R}$$, $ (N),(N)$\\mathbb {R}$ N$and structure matrix $$\\mathbb {R}$ NN$.$ The entries of the coefficient vector ${\\bf a}$ in $\\ell ^{GH,\\bf a}$ are later identified with the basis functions $\\varphi _i(x)$ for $x\\in to show that $ LGHN(x)()$ is a one-dimensional Lévy process and the approximation $ LGHN$ a $ H$-valued GH Lévy field.",
"{\\begin{xmlelement*}{proof} We first consider the case that\\begin{equation*}\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma ).\\end{equation*}It is sufficient to show that \\ell ^{GH,\\bf a}(1) is a GH-distributed random variable, the infinite divisibility of the GH distribution then implies that (\\ell ^{GH,\\bf a}(t),t\\in is a GH process.Since the entries of the coefficient vector a_1,\\dots ,a_N are non-zero, there exists a non-singular N\\times N matrix A, such that \\ell ^{GH,\\bf a}(1) is the first component of the vector A\\ell ^{GH_N}(1).If \\ell ^{GH_N}(1) is multi-dimensional GH-distributed, then follows from~\\cite [Theorem 1]{B81}, that A\\ell ^{GH_N}(1) is also multi-dimensional GH-distributed and that the first component of A\\ell ^{GH_N}(1), namely \\ell ^{GH,\\bf a}(1), follows a one-dimensional GH distribution (the parameters of the distribution of \\ell ^{GH,\\bf a}(1) depend on A and on \\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma and are explicitly given in \\cite {B81} and below).",
"\\\\On the other hand, assume that \\ell ^{GH,\\bf a}(1) is a GH random variable (with arbitrary coefficients), but \\ell ^{GH_N}(1) is not N-dimensional GH-distributed.This means there is no representation of \\ell ^{GH_N}(1) such that\\begin{equation*}\\ell ^{GH_N}(1)\\stackrel{\\mathcal {L}}{=}\\mu +\\Gamma \\beta \\ \\ell ^{GIG}(1)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(1))\\ \\end{equation*}with \\mu ,\\beta \\in \\mathbb {R}^N, \\Gamma \\in \\mathbb {R}^{N\\times N} spd with determinant one, a GIG random variable \\ell ^{GIG}(1) and a N-dimensional Brownian motion w^N independent of \\ell ^{GIG}(1).This implies that \\ell ^{GH,\\bf a}(1)=(A\\ell ^{GH_N}(1))_1 has no representation\\begin{equation*}\\begin{split}\\ell ^{GH,\\bf a}(1)&=(A\\mu )_1+(A\\Gamma \\beta )_1\\ell ^{GIG}(1)+(A\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(1)))_1\\\\&\\stackrel{\\mathcal {L}}{=}(A\\mu )_1+(A\\Gamma \\beta )_1\\ell ^{GIG}(1)+\\sqrt{\\ell ^{GIG}(1)A_{[1]}\\Gamma A_{[1]}^{\\prime }}w^1(1),\\end{split}\\end{equation*}where A_{[1]} denotes the first row of the matrix A and w^1(1)\\sim \\mathcal {N}(0,1).For the last equality we have used the affine linear transformation property of multi-dimensional normal distributions and that \\Gamma is positive definite.Since c_A:=A_{[1]}\\Gamma A_{[1]}^{\\prime }>0, we can divide the equation above by \\sqrt{c_A} and obtain that c_A^{-1/2}\\,\\ell ^{GH,\\bf a}(1) cannot be a GH-distributed random variable, as it cannot be expressed as a normal variance-mean mixture with a GIG-distribution.But this is a contradiction, since \\ell ^{GH,\\bf a}(1) is GH-distributed by assumption and the class of GH distributions is closed under regular affine linear transformations (see~\\cite [Theorem 1c]{B81}).\\end{xmlelement*}}\\begin{rem}The condition a_i\\ne 0 is, in fact, not necessary in Lemma~\\ref {lem:GH_lin}.",
"If, for k\\in \\lbrace 1,\\dots ,N-1\\rbrace , k coefficients a_{i_1}=\\dots =a_{i_k}=0, then the summation reduces to\\begin{equation*}\\ell ^{GH,\\bf a}(t)=\\sum _{i=1}^{N}a_i \\ell _i^{GH}(t)=\\sum _{l=1}^{N-k}a_{j_l}\\ell ^{GH}_{j_l}(t),\\end{equation*}where the indices j_l are chosen such that a_{j_l}\\ne 0 for l=1,\\dots ,N-k.",
"If P\\in \\mathbb {R}^{N\\times N} is the permutation matrix with\\begin{equation*}P\\ell ^{GH_N}(1)=P(\\ell _1^{GH}(1),\\dots ,\\ell _N^{GH}(1))^{\\prime }=( \\ell ^{GH}_{j_1}(1),\\dots , \\ell ^{GH}_{j_{N-k}}(1), \\ell ^{GH}_{i_1}(1),\\dots , \\ell ^{GH}_{i_k}(1))^{\\prime },\\end{equation*}then P\\ell ^{GH_N} is again N-dimensionally GH-distributed and by~\\cite [Theorem 1a]{B81} the vector (\\ell ^{GH}_{j_1}(1),\\dots ,\\ell ^{GH}_{j_{N-k}}(1)) admits a (N-k)-dimensional GH law.",
"Thus, we only consider the case where all coefficients are non-vanishing.\\end{rem}$ The previous proposition states that the KL approximation $L^{GH}_N(x)(t) = \\sum _{i=1}^{N} \\varphi _i(x) \\ell _i^{GH}(t),$ can only be a GH process for arbitrary $(\\varphi _i(x),i=1,\\dots ,N)$ if the $\\ell _i^{GH}$ are correlated in such a way that they form a multi-dimensional GH process.",
"This rules out the possibility of independent processes $(\\ell _i^{GH},i=1,\\ldots ,N)$ , because if $\\ell ^{GH_N}(1)$ is multi-dimensional GH-distributed, it is not possible that the marginals $\\ell _i^{GH}(1)$ are independent GH-distributed random variables (see [13]).",
"The parameters $\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i$ of each process $\\ell _i^{GH}$ should remain as unrestricted as possible, so we determine in the next step the parameters of the marginals of a $GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ distribution and show how the subordinators $(\\ell ^{GIG}_i,i=1,\\ldots ,N)$ might be correlated.",
"For this purpose, we introduce the notation $A^-~\\!\\!^{\\prime }:=(A^{-1})^{\\prime }$ if $A$ is an invertible square matrix.",
"The following result allows us to determine the marginal distributions of a $N$ -dimensional GH distribution.",
"Lemma 4.5 (Masuda [30], who refers to [14], Lemma A.1.)",
"Let $\\ell ^{GH_N}(1)=(\\ell _1^{GH}(1),\\dots ,\\ell _N^{GH}(1))^{\\prime }\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma ),$ then for each $i$ we have that $\\ell _i^{GH}(1)\\sim GH(\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i)$ , where $\\begin{split}&\\lambda _i=\\lambda ^{(N)},\\quad \\alpha _i=\\Gamma _{ii}^{-1/2}\\left[(\\alpha ^{(N)})^2-\\beta _{-i}^{\\prime }\\left(\\Gamma _{-i,22}-\\Gamma _{-i,21}\\Gamma _{ii}^{-1}\\Gamma _{-i,12}\\right)\\beta _{-i}\\right]^{1/2}\\\\&\\beta _i=\\beta ^{(N)}_i+\\Gamma _{ii}^{-1}\\Gamma _{-i,12}\\beta _{-i},\\quad \\delta _i=\\sqrt{\\Gamma _{ii}}\\delta ^{(N)}_i,\\quad \\mu _i=\\mu _i^{(N)},\\end{split}$ together with $\\begin{split}&\\beta _{-i}:=(\\beta _1^{(N)},\\dots ,\\beta _{i-1}^{(N)},\\beta _{i+1}^{(N)},\\dots ,\\beta _N^{(N)})^{\\prime },\\\\&\\Gamma _{-i,12}:=(\\Gamma _{i,1},\\dots ,\\Gamma _{i,i-1},\\Gamma _{i,i+1},\\dots ,\\Gamma _{i,N}),\\quad \\Gamma _{-i,21}:=\\Gamma _{-i,12}^{\\prime }\\end{split}$ and $\\Gamma _{-i,22}$ denotes the $(N-1)\\times (N-1)$ matrix which is obtained by removing the $i$ -th row and column of $\\Gamma $ .",
"Assume that $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ , since this is a necessary (and sufficient) condition so that the (truncated) KL expansion is a GH process.",
"Lemma REF gives immediately, that for all $i=1,\\dots ,N$ , the parameters $\\lambda _i=\\lambda ^{(N)}$ have to be identical, whereas the drift $\\mu _i$ may be chosen arbitrary for each process $\\ell _i^{GH}$ .",
"Furthermore, the expectation and covariance matrix of $\\ell ^{GH_N}(1)$ is given by $ \\mathbb {E}(\\ell ^{GH_N}(1))=\\mu ^{(N)}+\\frac{\\delta ^{(N)}K_{\\lambda ^{(N)}+1}(\\delta ^{(N)}\\gamma ^{(N)})}{\\gamma ^{(N)}K_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}\\Gamma \\beta ^{(N)}$ and $ \\begin{split}\\text{Var}(\\ell ^{GH_N}(1))&=\\frac{\\delta ^{(N)}K_{\\lambda ^{(N)}+1}(\\delta ^{(N)}\\gamma ^{(N)})}{\\gamma ^{(N)}K_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}\\Gamma +\\Big (\\frac{\\delta ^{(N)}}{\\gamma ^{(N)}}\\Big )^2(\\Gamma \\beta ^{(N)})(\\Gamma \\beta ^{(N)})^{\\prime }\\\\&\\qquad \\qquad \\cdot \\Bigg (\\frac{K_{\\lambda ^{(N)}+2}(\\delta ^{(N)}\\gamma ^{(N)})}{K_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}-\\frac{K^2_{\\lambda ^{(N)}+1}(\\delta ^{(N)}\\gamma ^{(N)})}{K^2_{\\lambda ^{(N)}}(\\delta ^{(N)}\\gamma ^{(N)})}\\Bigg ),\\end{split}$ where $\\gamma ^{(N)}:=((\\alpha ^{(N)})^2-\\beta ^{(N)}~\\!\\!^{\\prime }\\Gamma \\beta ^{(N)})^{1/2}$ (see [30]).",
"Example 4.6 Consider the case that the processes $\\ell _1^{GH},\\dots ,\\ell _N^{GH}$ are generated by the same subordinating $GIG(a,b,p)$ process $\\ell ^{GIG}$ , i.e.",
"$\\ell _i^{GH}(t)=\\mu _it+\\beta _i\\ell ^{GIG}(t)+w_i(\\ell ^{GIG}(t)).$ Then $\\ell _i^{GH}(1)\\sim GH(\\lambda ,\\alpha _i,\\beta _i,\\delta ,\\mu _i)$ , where $\\lambda =p$ , $\\delta =a$ are independent of $i$ and $\\alpha _i=(b^2+\\beta _i^2)^{1/2}$ .",
"If $\\mu ^{(N)}:=(\\mu _1\\dots ,\\mu _N)^{\\prime }$ , $\\beta ^{(N)}:=(\\beta _1,\\dots ,\\beta _N)^{\\prime }$ and $\\Gamma $ is the $N\\times N$ identity matrix, then $\\begin{split}\\ell ^{GH_N}(t)=(\\ell _1^{GH}(t),\\dots ,\\ell _N^{GH}(t))^{\\prime }&\\stackrel{\\mathcal {L}}{=}\\mu t+\\beta \\ell ^{GIG}(t)+w^N(\\ell ^{GIG}(t))\\\\&=\\mu t+\\Gamma \\beta \\ell ^{GIG}(t)+\\sqrt{\\Gamma }w^N(\\ell ^{GIG}(t)),\\end{split}$ where $w^N$ is a $N$ -dimensional Brownian motion independent of $\\ell ^{GIG}$ .",
"Hence, $\\ell ^{GH_N}(t)$ is a multi-dimensional $GH_N(\\lambda ,\\alpha ^{(N)},\\beta ^{(N)},\\delta ,\\mu ^{(N)}$ ,$\\Gamma )$ process with $\\alpha ^{(N)}=\\sqrt{b^2+\\beta ^{\\prime }\\beta }$ .",
"One checks using Lemma REF that the parameters of the marginals of $\\ell ^{GH_N}(1)$ and $\\ell _i^{GH}(1)$ coincide for each $i$ , and that expectation and covariance of $\\ell ^{GH_N}(1)$ are given by Eq.",
"(REF ) and Eq.",
"(REF ).",
"By Lemma REF , we have that the Karhunen-Loève expansion $L_N^{GH}(x)(t)=\\sum _{i=1}^N\\varphi _i(x)\\ell _i^{GH}(t)$ in this example is a GH process for each $x\\in \\mathcal {D}$ and arbitrary basis functions $(\\varphi _i,i=1,\\ldots ,N)$ .",
"Remark 4.7 Lemma REF dictates that the subordinators $(\\ell ^{GIG}_i,i=1,\\ldots ,N)$ cannot be independent.",
"In Example REF fully correlated subordinators were used.",
"A different way to correlate the subordinators, so that Lemma REF is fulfilled, would lead to a correlation matrix, just being multiplied with $\\Gamma $ .",
"For simplicity, in the remainder of the paper, especially for the numerical examples in Section , we use fully correlated subordinators.",
"As shown in [13] the class of $N$ -dimensional GH distributions is closed under regular linear transformations: If $N\\in \\mathbb {N}$ , $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ,\\alpha ,\\beta ,\\delta ,\\mu ,\\Gamma )$ , $A$ is an invertible $N\\times N$ -matrix and $b\\in \\mathbb {R}^N$ , then the random vector $A\\ell ^{GH_N}(1)+b$ has distribution $GH_N(\\lambda ,||A||^{-1/N}\\alpha ,A^-~\\!\\!^{\\prime }\\beta ,||A||^{1/N}\\delta ,A\\mu +b,||A||^{-2/N}A\\Gamma A^{\\prime }),$ where $||A||$ denotes the absolute value of the determinant of $A$ .",
"With this and the assumption $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ , we are also able to determine the point-wise law of $L_N^{GH}$ for given coefficients $\\varphi _1(x),\\dots ,\\varphi _N(x)$ .",
"Lemma 4.8 Let $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma )$ and for $x\\in let $ (i(x),i = 1,...,N)$ be a sequence of non-zero coefficients (see Remark~\\ref {rem:coefficients}).Then $ (LNGH(x)(t),t$ is a GH Lévy process with parameters depending on $ x$.$ It is again sufficient to show that $L_N^{GH}(x)(1)$ follows a GH law, the resulting parameters are given below.",
"For $x\\in , define the $ NN$ matrix $ A(x)$ via\\begin{equation} A(x)_{ij}:={\\left\\lbrace \\begin{array}{ll}\\varphi _j(x) & \\text{if $i=1$ or if $i=j$} \\\\0 & \\text{elsewhere}\\end{array}\\right.",
"}.\\end{equation}The matrix $ A(x)$ is invertible with determinant $ i=1Ni(x)0$ and inverse $ A(x)-1$ given by\\begin{equation*}A(x)^{-1}_{ij}:={\\left\\lbrace \\begin{array}{ll}-\\varphi _1(x)^{-1} & \\text{if $i=1$ and $j\\ge 2$} \\\\\\varphi _i(x)^{-1} & \\text{if $i=j$} \\\\0 & \\text{elsewhere}\\end{array}\\right.",
"}.\\end{equation*}Then $ LNGH(x)(1)=i=1Ni(x)iGH(1)$ is the first entry of the random vector $ A(x)GHN(1)$.By the affine transformation property of the GH distribution and Lemma~\\ref {lem:GH_N} it follows that $ LNGH(x)(1)$ is one-dimensional GH-distributed.Now define $ :=A(x)A(x)'$, the partition\\begin{equation*}\\widetilde{\\Gamma }=\\begin{pmatrix} \\widetilde{\\Gamma }_{11} & \\widetilde{\\Gamma }_{2,1}^{\\prime }\\\\ \\widetilde{\\Gamma }_{2,1} & \\widetilde{\\Gamma }_{2,2} \\end{pmatrix}\\end{equation*}such that $ 2,1$\\mathbb {R}$ N-1$ and $ 2,2$\\mathbb {R}$ (N-1)(N-1)$ and the vector$$\\widetilde{\\beta }:=\\left(\\beta _2^{(N)}\\varphi _2(x)^{-1}-\\beta _1^{(N)}\\varphi _1(x)^{-1},\\dots ,\\beta _N^{(N)}\\varphi _N(x)^{-1}-\\beta _1^{(N)}\\varphi _1(x)^{-1}\\right)^{\\prime }\\in \\mathbb {R}^{N-1}.$$The parameters $ L,L(x),L(x),L(x)$ and $ L(x)$ of $ LNGH(x)$ are then given by{\\begin{@align*}{1}{-1}\\lambda _L&=\\lambda ^{(N)},\\\\\\alpha _L(x)&=\\widetilde{\\Gamma }_{11}^{-1/2}\\left[(\\alpha ^{(N)})^2-\\widetilde{\\beta }^{\\prime }(\\widetilde{\\Gamma }_{2,2}-\\widetilde{\\Gamma }_{11}^{-1}\\widetilde{\\Gamma }_{2,1}\\widetilde{\\Gamma }_{2,1}^{\\prime })\\widetilde{\\beta } \\right]^{1/2},\\\\\\delta _L(x)&=\\delta ^{(N)}\\sqrt{\\widetilde{\\Gamma }_{11}}=\\delta ^{(N)}\\Big (\\sum _{i,j=1}^N\\varphi _i(x)\\varphi _j(x)\\Gamma _{ij}\\Big )^{1/2},\\\\\\beta _L(x)&=\\beta _1^{(N)}\\varphi _1(x)^{-1}+\\widetilde{\\Gamma }_{11}^{-1}\\widetilde{\\Gamma }_{2,1}^{\\prime }\\widetilde{\\beta }\\quad \\text{and}\\\\\\mu _L(x)&=[A(x)\\mu ^{(N)}]_1=\\sum _{i=1}^N\\varphi _i(x)\\mu _i^{(N)}.\\end{@align*}}$ To ensure $L^2(\\Omega ;\\mathbb {R})$ convergence as in Theorem REF of the series $\\widetilde{L}_N^{GH}(x)(t) = \\sum _{i=1}^N\\sqrt{\\rho _i}e_i(x)\\widetilde{\\ell }_i^{GH}(t),$ we need to simulate approximations of uncorrelated, one-dimensional GH processes $\\ell _i^{GH}$ with given parameters $\\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i)$ .",
"To obtain a sufficiently good approximation of the Lévy field, $N$ is coupled to the time discretization of $ and the decay of the eigenvalues of $ Q$ (see Remark~\\ref {rem:trunc}).The simulation of a large number $ N$ of independent GH processes is computationally expensive, so we focus on a different approach.Instead of generating $ N$ dependent but uncorrelated, one-dimensional processes, we generate one $ N$-dimensional process with decorrelated marginals.For this approach to work we need to impose some restrictions on the target parameters $ i, i,i$ and $ i$.$ Theorem 4.9 Let $(\\ell _i^{GH},i=1\\ldots ,N)$ be one-dimensional GH processes, where, for $i=1,\\dots ,N$ , $\\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i).$ The vector $\\ell ^{GH_N}:=(\\ell _1^{GH},\\dots ,\\ell _N^{GH})^{\\prime }$ is only a $N$ -dimensional GH process if there are constants $\\lambda \\in \\mathbb {R}$ and $c>0$ such that for any $i$ $\\lambda _i=\\lambda \\quad \\text{and}\\quad \\delta _i(\\alpha _i^2-\\beta _i^2)^{1/2}=c.$ If, in addition, the symmetric matrix $U\\in \\mathbb {R}^{N\\times N}$ defined by $U_{ij}:={\\left\\lbrace \\begin{array}{ll}\\delta _i^2 &\\text{if $i=j$}\\\\\\frac{K_{\\lambda +1}(c)^2-K_{\\lambda +2}(c)K_{\\lambda }(c)}{K_{\\lambda +1}(c)K_{\\lambda }(c)}\\frac{\\beta _i\\delta _i^2\\beta _j\\delta _j^2}{c} &\\text{if $i\\ne j$}\\end{array}\\right.",
"},$ is positive definite, it is possible to construct a $N$ -dimensional GH process $\\ell ^{GH_N,U}$ with uncorrelated marginals $\\ell ^{GH,U}_i$ and $\\ell ^{GH,U}_i(1)\\stackrel{\\mathcal {L}}{=} \\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i).$ We start with the necessary condition to obtain a multi-dimensional GH distribution.",
"Let $\\ell ^{GH_N}$ be a $N$ -dimensional GH process with $\\ell ^{GH_N}(1)\\sim GH_N(\\lambda ^{(N)},\\alpha ^{(N)},\\beta ^{(N)},\\delta ^{(N)},\\mu ^{(N)},\\Gamma ).$ If the law of the marginals of $\\ell ^{GH_N}$ is denoted by $ \\ell _i^{GH}(1)\\sim GH(\\lambda _i, \\alpha _i,\\beta _i, \\delta _i,\\mu _i),$ then one sees immediately from Lemma REF that $\\lambda _i=\\lambda ^{(N)}$ and $\\mu _i=\\mu _i^{(N)}$ for all $i=1,\\dots ,N$ .",
"With the equations for $\\beta _i$ and $\\delta _i$ from Lemma REF , we derive for $\\Gamma \\beta ^{(N)}$ $(\\Gamma \\beta ^{(N)})_i=\\Gamma _{ii}\\beta _i^{(N)}+\\sum _{k=1, k\\ne i}^N \\Gamma _{ik}\\beta _k^{(N)}=\\Gamma _{ii}\\beta _i^{(N)}+\\Gamma _{ii}(\\beta _i-\\beta _i^{(N)})=\\big (\\frac{\\delta _i}{\\delta ^{(N)}}\\big )^2\\beta _i,$ which leads to $\\alpha _i^2&=\\Gamma _{ii}^{-1}(\\alpha ^{(N)})^2-\\Gamma _{ii}^{-1}\\sum _{k=1,k\\ne i}^N\\beta _k^{(N)}\\sum _{l=1,l\\ne i}^N\\Gamma _{kl}\\beta _l^{(N)}+\\big (\\Gamma _{ii}^{-1}\\sum _{k=1, k\\ne i}^N \\Gamma _{ik}\\beta _k^{(N)}\\big )^2\\\\&=\\big (\\frac{\\delta ^{(N)}\\alpha ^{(N)}}{\\delta _i}\\big )^2-\\big (\\frac{\\delta ^{(N)}}{\\delta _i}\\big )^2\\sum _{k=1,k\\ne i}^N\\beta _k^{(N)}((\\Gamma \\beta ^{(N)})_k-\\Gamma _{ik}\\beta _i^{(N)})+(\\beta _i-\\beta _i^{(N)})^2\\\\&=\\big (\\frac{\\delta ^{(N)}\\alpha ^{(N)}}{\\delta _i}\\big )^2-\\sum _{k=1,k\\ne i}^N\\beta _k^{(N)}\\frac{\\delta _k^2}{\\delta _i^2}\\beta _k\\\\&\\quad +\\big (\\frac{\\delta ^{(N)}}{\\delta _i}\\big )^2\\beta _i^{(N)}((\\Gamma \\beta ^{(N)})_i-\\Gamma _{ii}\\beta _i^{(N)})+\\beta _i^2-2\\beta _i\\beta _i^{(N)}+(\\beta _i^{(N)})^2\\\\&=\\big (\\frac{\\delta ^{(N)}\\alpha ^{(N)}}{\\delta _i}\\big )^2-\\sum _{k=1}^N\\beta _k^{(N)}\\frac{\\delta _k^2}{\\delta _i^2}\\beta _k+\\beta _i^2.$ The last equation is equivalent to $ \\delta _i^2(\\alpha _i^2-\\beta _i^2)=(\\delta ^{(N)}\\alpha ^{(N)})^2-\\sum _{k=1}^N\\beta _k^{(N)}\\underbrace{\\delta _k\\beta _k}_{=(\\delta ^{(N)})^2(\\Gamma \\beta _k^{(N)})}=(\\delta ^{(N)})^2\\left((\\alpha ^{(N)})^2-\\beta ^{(N)}~\\!\\!^{\\prime }\\Gamma \\beta ^{(N)}\\right),$ and since the right hand side does not depend on $i$ , we get that $\\delta _i^2(\\alpha _i^2-\\beta _i^2)>0$ has to be independent of $i$ .",
"Now assume we have a set of parameters $((\\lambda _i,\\alpha _i,\\beta _i,\\delta _i,\\mu _i),i=1,\\dots ,N)$ with $\\delta _i\\sqrt{\\alpha _i^2-\\beta _i^2}=c>0\\quad \\text{and}\\quad \\lambda _i=\\lambda \\in \\mathbb {R},$ where $c$ and $\\lambda $ are independent of the index $i$ .",
"Furthermore, let the matrix $U$ as defined in the claim be positive definite.",
"We show how parameters $\\lambda ^{(U)}, \\alpha ^{(U)}, \\beta ^{(U)}, \\delta ^{(U)}, \\mu ^{(U)}$ and $\\Gamma ^{(U)}$ of a $N$ -dimensional GH process $\\ell ^{GH_N,U}$ may be chosen, such that its marginals are uncorrelated with law $\\ell ^{GH,U}_i(1)\\sim GH(\\lambda ,\\alpha _i,\\beta _i,\\delta _i,\\mu _i)$ .",
"Clearly, we have to set $\\lambda ^{(U)}:=\\lambda $ and $\\mu ^{(U)}:=(\\mu _1,\\dots ,\\mu _N)^{\\prime }$ .",
"Eq.",
"(REF ) and Eq.",
"(REF ) yield the conditions $(\\delta ^{(U)})^2(\\Gamma ^{(U)}\\beta ^{(U)})_i=\\delta _i^2\\beta _i \\quad \\text{and}\\quad \\delta ^{(U)}\\sqrt{(\\alpha ^{(U)})^2-\\beta ^{(U)T}\\Gamma ^{(U)}\\beta ^{(U)}}=\\delta _i\\sqrt{\\alpha _i^2-\\beta _i^2}=c.$ If $(\\delta ^{(U)})^2\\Gamma ^{(U)}$ fulfills the identity $(\\delta ^{(U)})^2\\Gamma ^{(U)}=U$ , we get by Eq.",
"(REF ) for $i\\ne j$ $\\text{Cov}(\\ell ^{GH,U}_i(1),\\ell ^{GH,U}_j(1))&=\\frac{K_{\\lambda +1}(c)}{cK_{\\lambda }(c)}(\\delta ^{(U)})^2\\Gamma _{ij}^{(U)}\\\\&\\quad +\\frac{K_{\\lambda +2}(c)K_{\\lambda }(c)-K_{\\lambda +1}^2(c)}{c^2K_{\\lambda }(c)^2}((\\delta ^{(U)})^2\\Gamma ^{(U)}\\beta ^{(U)})_i((\\delta ^{(U)})^2\\Gamma ^{(U)}\\beta ^{(U)})_j\\\\&=\\frac{K_{\\lambda +1}(c)}{cK_{\\lambda }(c)}U_{ij}+\\frac{K_{\\lambda +2}(c)K_{\\lambda }(c)-K_{\\lambda +1}^2(c)}{c^2K_{\\lambda }(c)^2}\\delta _i^2\\beta _i\\delta _j^2\\beta _j=0,$ hence all marginals are uncorrelated.",
"To obtain a well-defined $N$ -dimensional GH distribution, we still have to make sure that $\\Gamma ^{(U)}$ is spd with unit determinant.",
"If we define $\\delta ^{(U)}:=(\\text{det}(U))^{1/(2N)}$ , then $\\delta ^{(U)}>0$ (since $\\text{det}(U)>0$ by assumption) and $\\Gamma ^{(U)}=(\\delta ^{(U)})^{-2}U$ is spd with $\\text{det}(\\Gamma ^{(U)})=1$ .",
"It remains to determine appropriate parameters $\\alpha ^{(U)}>0$ and $\\beta ^{(U)}\\in \\mathbb {R}^N$ .",
"For $\\beta ^{(U)}$ , we use once again Lemma REF to obtain the linear equations $\\beta _i=\\beta _i+(\\Gamma _{ii}^{(U)})^{-1}\\sum _{k=1,k\\ne i}\\Gamma _{ik}^{(U)}\\beta _k^{(U)},$ for $i=1,\\dots ,N$ .",
"The corresponding system of linear equations is given by $\\begin{pmatrix}(\\Gamma _{11}^{(U)})^{-1}\\\\&\\ddots \\\\&&(\\Gamma _{NN}^{(U)})^{-1}\\\\\\end{pmatrix}\\Gamma ^{(U)}\\beta ^{(U)}=\\left(\\begin{array}{c} \\beta _1\\\\ \\vdots \\\\ \\beta _N \\end{array}\\right),$ and has a unique solution $\\beta ^{(U)}$ for any right hand side $(\\beta _1,\\dots ,\\beta _N)^{\\prime }$ , because $\\Gamma ^{(U)}$ as constructed above is invertible with positive diagonal entries.",
"Finally, we are able to calculate $\\alpha ^{(U)}$ via Equation (REF ) as $\\alpha ^{(U)}=\\Big (\\sum _{k=1}^N\\delta _k^2\\beta _k\\beta _k^{(U)}+\\big (\\frac{c}{\\delta ^{(U)}}\\big )^2\\Big )^{1/2}=\\Big (\\beta ^{(U)}~\\!\\!^{\\prime }\\Gamma ^{(U)}\\beta ^{(U)}+\\big (\\frac{c}{\\delta ^{(U)}}\\big )^2\\Big )^{1/2}$ and obtain the desired marginal distributions.",
"Note that the KL-expansion $L^{GH}_N(x)(\\cdot )$ generated by $(\\ell ^{GH,U}_i,i=1\\ldots ,N)$ in Theorem REF is a GH process for each $x\\in by Lemma~\\ref {lem:KL_GH}, whereas this is not the case if the processes $ (iGH,i=1,...,N)$ are generated independently of each other:By Lemma~\\ref {lem:GH_lin} we have that $ LGHN(x)(1)$ is only GH distributed if the vector $ (1GH(1),...,NGH(1))'$ admits a multi-dimensional GH law.As noted in~\\cite {B81} after Theorem 1, this is impossible if the processes (and hence $ (iGH(1),i=1,...,N)$) are independent.Whenever Theorem~\\ref {thm:Z_U} is applicable, we are able to approximate a GH Lévy field by generating a $ N$-dimensional GH processes, where $ N$ is the truncation index of the KL expansion.To this end, Lemma~\\ref {lem:sub} suggests the simulation of GIG processes and then subordinating $ N$-dimensional Brownian motions.With this simulation approach the question arises on why we have taken a detour via the subordinating GIG process instead of using the characteristic function a of GH process in Equation~(\\ref {eq:gh_cf}) for a ``direct^{\\prime \\prime } simulation.This has several reasons: First, the approximation of the inversion formula (\\ref {eq:finv}) can only be applied for one-dimensional GH processes, where the costs of evaluating $ GH$ or $ GIG$ are roughly the same.In comparison, the costs of sampling a Brownian motion are negligible.Second, in the multi-dimensional case, we need that all marginals of the GH process are generated by the same or correlated subordinator(s), which leaves us no choice but to sample the underlying GIG process.In addition, the simulation of a GH field requires in some cases only one subordinating process to generate a multi-dimensional GH process with uncorrelated marginals (see Theorem~\\ref {thm:Z_U}).This approach is in general more efficient than sampling a large number of uncorrelated, one-dimensional GH processes for the KL expansion.As we demonstrate in the following section, it is a straightforward application of the Fourier inversion algorithm to approximate a GIG process $ GIG$ with given parameters, since all necessary assumptions are fulfilled and the bounding parameters $ , R, $ and $ B$ may readily be calculated.$ Numerics In this section we provide some details on the implementation of the Fourier inversion method.",
"Thereafter, we apply this methodology to approximate a GH Lévy field and conclude with some numerical examples.",
"Notes on implementation Suppose we simulate a given one-dimensional Lévy process $\\ell $ which fulfills Assumption REF resp.",
"Assumption REF , using the step size $\\Delta _n>0$ and characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"Usually the parameter $\\eta $ cannot be chosen arbitrary high (as for the GIG process), but it may be possible to choose $\\eta $ within a certain range, for instance $\\eta \\in (1,2]$ for the Cauchy process in Example REF .",
"As a rule of thumb, $\\eta $ should always be determined as large as possible, as the convergence rates in Theorems REF and REF directly depend on $\\eta $ .",
"In addition, we concluded in Remark REF that $D\\simeq \\Delta _n^{p/(p-\\eta )}$ is an appropriate choice to guarantee an $L^p$ -error of order $\\mathcal {O}(\\Delta _n^{1/p})$ .",
"This means that for a given $p$ , $D$ decreases as $\\eta $ increases.",
"Since the number of summations $M$ in Algorithm REF depends on $D$ (see Theorem REF ), an increasing parameter $\\eta $ also reduces computational time.",
"Once $\\eta $ is determined, we derive $R$ by differentiation of $(\\phi _\\ell )^{\\Delta _n}$ as in Remark REF .",
"Similarly to $\\eta $ , it is often possible to choose between several values of $\\theta >0$ , but it is difficult to give a-priori a recommendation on how $\\theta $ should be selected.",
"One rather calculates for several admissible $\\theta $ the constant $C_\\theta :=\\max _{u\\in \\mathbb {R}}|u^\\theta (\\phi _\\ell (u))^{\\Delta _n}|$ numerically and deducts $B_\\theta =(2\\pi )^{-\\theta }C_\\theta $ .",
"Each combination of $(\\theta ,B_\\theta )$ then results in a valid number of summations $M_\\theta $ in the discrete Fourier Inversion algorithm.",
"Since $\\theta $ and $B_\\theta $ are only necessary to determine $M_\\theta $ , we may simply use the smallest $M_\\theta $ for the simulation.",
"To find $\\widetilde{X}$ with $\\widetilde{F}(\\widetilde{X})=U$ in Algorithm REF , we use a globalized Newton method with backtracking line search, also known as Armijo increment control.",
"The step lengths during the line search are determined by interpolation, which is a robust technique if combined with a standard Newton method.",
"Details on the globalized Newton method with backtracking may be found, for example, in [31], an example how the algorithm is used is given in [36].",
"Although convergence of this root finding algorithm is ensured by the increment control, its efficiency depends heavily on the choice of the initial value $\\widetilde{X}_0$ .",
"Clearly, $\\widetilde{X}_0$ should depend on the sampled $U\\sim \\mathcal {U}([0,1])$ and be related to the target distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"This means we should determine $\\widetilde{X}_0$ implicitly by $F^{(0)}(\\widetilde{X}_0)=U$ , where $F^{(0)}$ is a CDF of a distribution similar to the target distribution, but which can be inverted efficiently.",
"Approximation of a GH field We consider a GH Lévy field on the (separable) Hilbert space $H=L^2(\\mathcal {D})$ with a compact spatial domain $\\mathcal {D}\\subset \\mathbb {R}^s$ .",
"The operator $Q$ on $H$ is given by a Matérn covariance operator with variance $v>0$ , correlation length $r>0$ and a positive parameter $\\chi >0$ defined by $[Qh](x):=v\\int _\\mathcal {D} k_\\chi (x,y)h(y)dy,\\quad \\text{for }h\\in H,$ where $k_\\chi $ denotes the Matérn kernel.",
"For $\\chi =\\frac{1}{2}$ , we obtain the exponential covariance function and for $\\chi \\rightarrow \\infty $ the squared exponential covariance function.",
"For general $\\chi >0$ , the Matérn kernel $k_\\chi (x,y):=\\frac{2^{1-\\chi }}{\\Gamma _G(\\chi )}\\Big (\\frac{\\sqrt{2\\chi }|x-y|}{r}\\Big )^\\chi K_\\chi \\Big (\\frac{\\sqrt{2\\chi }|x-y|}{r}\\Big )$ fulfills the limit identity $k_\\chi (x,x)=\\lim _{y\\rightarrow x}k_\\chi (x,y)=1$ , which can be easily seen by [33].",
"Here $\\Gamma _G(\\cdot )$ is the Gamma function.",
"As shown in [20], this implies $tr(Q)=\\sum _{i=1}^\\infty \\rho _i=v\\int _{\\mathcal {D}}dx,$ where $(\\rho _i,i\\in \\mathbb {N})$ are the eigenvalues of the Matérn covariance operator $Q$ .",
"In general, no analytical expressions for the eigenpairs $(\\rho _i,e_i)$ of $Q$ will be available, but the spectral basis may be approximated by numerically solving a discrete eigenvalue problem and then interpolating by Nyström's method.",
"For a general overview of common covariance functions and the approximation of their eigenbasis we refer to [37] and the references therein.",
"Now let $L_N^{GH}$ be an approximation of a GH field by a $N$ -dimensional GH process $(\\ell ^{GH_N}(t),t\\in $ with fixed parameters $\\lambda ,\\alpha ,\\delta \\in \\mathbb {R}$ , $\\beta ,\\mu \\in \\mathbb {R}^N$ and $\\Gamma \\in \\mathbb {R}^{N\\times N}$ .",
"The parameters are chosen in such a way that the multi-dimensional GH process has uncorrelated marginal processes, hence the generated KL expansions $L_N^{GH}(x)(t)=\\sum _{i=1}^N\\varphi _i(x)\\ell ^{GH}_i(t)$ are again one-dimensional GH processes for any spectral basis $(\\varphi _i,i\\in \\mathbb {N})$ and fixed $x\\in \\mathcal {D}$ .",
"This in turn means, that we may draw samples of $\\ell ^{GH_N}$ by simulating a GIG process $\\ell ^{GIG}$ with parameters $a=\\delta , b=(\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta )^{1/2}$ and $p=\\lambda $ using Fourier inversion and then subordinating a $N$ -dimensional Brownian motion (see Lemma REF ).",
"The characteristic function of a GIG Lévy process $(\\ell ^{GIG}(t),t\\in $ with (fixed) parameters $a,b>0$ and $p\\in \\mathbb {R}$ is given by $ \\phi _{GIG}(u;a,b,p):=\\mathbb {E}[\\exp (iu\\ell ^{GIG}(1))]=(1-2iub^{-2})^{-p/2}\\frac{K_p(ab\\sqrt{1-2iub^{-2}})}{K_p(ab)}.$ The GIG distribution corresponding to $(\\phi _{GIG})^{\\Delta _n}$ with $\\Delta _n=1$ is continuous with finite variance (see [41]), which implies that these properties hold for all distributions with characteristic function $(\\phi _{GIG})^{\\Delta _n}$ , for any $\\Delta _n>0$ .",
"The constants as in Assumption REF are derived in the following.",
"For $k\\in \\mathbb {N}$ , the $k$ -th moment of the GIG distribution is given as $0<\\mathbb {E}\\big ((\\ell ^{GIG}(1))^k\\big )=\\big (\\frac{a}{b}\\big )^{k}\\frac{K_{p+k}(ab)}{K_p(ab)}<\\infty .$ For any $\\eta =2k$ we are, therefore, able to calculate the bounding constant $R$ via $R=(-1)^{k}\\frac{d^{2k}}{du^{2k}}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0},$ because the derivatives of $\\phi _{GIG}$ evaluated at $u=0$ are $(\\phi _{GIG}(0;a,b,p))^{(k)}=i^{-k}\\mathbb {E}\\big ((\\ell ^{GIG}(1))^k\\big )=i^{-k}\\big (\\frac{a}{b}\\big )^{k}\\frac{K_{p+k}(ab)}{K_p(ab)}.$ The calculation of the $\\eta $ -th derivative can be implemented easily by using a version of Faà di Bruno's formula containing the Bell polynomials, for details we refer to [25].",
"The bounding constants $\\theta $ and $B$ may be determined numerically as described Section REF (e.g.",
"by using the routine fminsearch in MATLAB).",
"The derivation of the bounds implies that we can ensure $L^p$ convergence of the approximated GIG process in the sense of Theorem REF for any $p\\ge 1$ , because it is possible to define $\\eta $ as any even integer and then obtain $R$ by differentiation.",
"We observe that the target distribution with characteristic function $(\\phi _{GIG}(u;a,b,p))^{\\Delta _n}$ and $\\Delta _n>0$ is not necessarily GIG, except for the Inverse Gaussian (IG) case where $p=-1/2$ and $(\\phi _{IG}(u;a,b))^{\\Delta _n}=\\phi _{IG}(u;\\Delta _na,b)$ (see Remark REF ).",
"This special feature of the IG distribution is exploited to determine the initial values $\\widetilde{X}_0$ in the Newton iteration by moment matching: Consider an $IG(a_0,b_0)$ distribution with mean $a_0/b_0$ and variance $a_0/b_0^3$ , where the parameters $a_0,b_0>0$ are “matched” to the target distribution's mean and variance via $\\frac{a_0}{b_0}&=i\\frac{d}{du}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0},\\\\\\frac{a_0}{b_0^3}&=(-1)\\frac{d^2}{du^2}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0}-\\Big (i\\frac{d}{du}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0}\\Big )^2.$ If $F_{\\Delta _n}^{IG}$ denotes the CDF of this $IG(a_0,b_0)$ distribution, the initial value of the globalized Newton method is given implicitly by $F_{\\Delta _n}^{IG}(\\widetilde{X}_0)=U$ .",
"The inversion of $F_{\\Delta _n}^{IG}$ may be executed numerically by many software packages like MATLAB.",
"With our approach, this results in the approximation of a GIG process $\\widetilde{\\ell }^{GIG}$ at discrete times $t_j\\in \\Xi _n$ .",
"The $N$ -dimensional GH process $\\ell ^{GH_N}$ may then be approximated at $t_j$ for $j=0,\\dots ,n$ by the process $\\widetilde{\\ell }^{GH_N}$ with $\\widetilde{\\ell }^{GH_N}(t_0)=0$ and the increments $\\widetilde{\\ell }^{GH_N}(t_j)-\\widetilde{\\ell }^{GH_N}(t_{j-1})=\\mu \\Delta _n+\\Gamma \\beta (\\widetilde{\\ell }^{GIG}(t_j)-\\widetilde{\\ell }^{GIG}(t_{j-1})) + \\sqrt{(\\widetilde{\\ell }^{GIG}(t_j)-\\widetilde{\\ell }^{GIG}(t_{j-1}))\\Gamma }w^N_j(1),$ for $j=1,\\dots ,n$ , where the $w_j^N(1)$ are i.i.d.",
"$\\mathcal {N}_N(0,\\mathbf {1}_{N\\times N})$ -distributed random vectors.",
"To obtain the process $\\widetilde{\\ell }^{GH_N}$ at arbitrary times $t\\in , we interpolate the samples $ (GHN(tj),j=0...,n)$ piecewise constant as in Algorithm~\\ref {algo:approx2}.With this, we are able to generate an approximation of $ LNGH$ at any point $ (x,t)D by $\\widetilde{L}_N^{GH}(x)(t):=\\sum _{i=1}^N\\varphi _i(x) \\widetilde{\\ell }^{GH}_i(t).$ The knowledge of $tr(Q)$ enables us to determine the truncation index $N$ and the constant $C_\\ell $ as in Remark : For $N\\in \\mathbb {N}$ let $(\\widetilde{\\ell }^{GH}_i,i=1,\\ldots ,N)$ be the approximations of the processes $(\\ell ^{GH}_i,i=1,\\ldots ,N)$ , where the random vector $(\\ell ^{GH}_1(1),\\dots ,\\ell _N^{GH}(1))$ is multivariate GH-distributed by assumption.",
"Hence, for every $N\\in \\mathbb {N}$ , we obtain the parameters $a(N), b(N), \\lambda (N)$ of a corresponding GIG subordinator $\\ell ^{GIG,N}$ , which is approximated through a piecewise constant process $\\widetilde{\\ell }^{GIG,N}$ as above.",
"With Eq.",
"(REF ) we calculate the error $E_{GIG,N}^p:=\\sup _{t\\in \\mathbb {E}(|\\ell ^{GIG,N}(t)-\\widetilde{\\ell }^{GIG,N}(t)|^p).",
"}for p\\in \\lbrace 1,2\\rbrace .If \\beta \\in \\mathbb {R}^N and \\Gamma \\in \\mathbb {R}^{N\\times N} denote the GH parameters corresponding to (\\ell ^{GH}_1(1),\\dots ,\\ell _N^{GH}(1)), the L^2(\\Omega ;\\mathbb {R}) approximation error of each process \\ell ^{GH}_i is given by\\begin{equation*}\\widetilde{C}_{\\ell ,i}:=\\sup _{t\\in \\frac{\\mathbb {E}(|\\ell ^{GH}_i(t)-\\widetilde{\\ell }^{GH}_i(t)|^2)}{\\Delta _n}= \\frac{E_{GIG,N}^2(\\Gamma \\beta )_i^2+E_{GIG,N}^1\\sqrt{\\Gamma _{[i]}\\Gamma _{[i]}^{\\prime }}}{\\Delta _n},}where \\Gamma _{[i]} indicates the i-th row of \\Gamma .Starting with N=1, we compute the first N eigenvalues and the difference\\begin{equation*}T\\Big (tr(Q)-\\sum _{i=1}^N\\rho _i\\Big )-\\max _{i=1,\\dots ,N}\\widetilde{C}_{\\ell ,i}\\Delta _n \\sum _{i=1}^N\\rho _i\\end{equation*}and increase N by one in every step until this expression is close to zero.If a suitable N is found, we define C_\\ell :=\\max _{i=1,\\dots ,N}\\widetilde{C}_{\\ell ,i} and thus have equilibrated truncation and approximation errors by ensuring Eq.~(\\ref {eq:trunc}).For simplicity, we have implicitly assumed here that the processes \\ell ^{GH}_i were normalized in the sense that \\text{Var}(\\ell ^{GH}_i(t))=t.This is due to the fact that \\rho _i\\ell _i (here with \\ell _i=\\ell ^{GH}_i) in Theorem~\\ref {thm:H_error} represents the scalar product (L(t),e_i)_H with variance \\rho _it.In case we have unnormalized processes, one can simply divide \\ell ^{GH}_i by its standard deviation (see Formula~(\\ref {eq:VarZ})) and adjust the constants \\widetilde{C}_{\\ell ,i} and C_\\ell accordingly.\\subsection {Numerical results}As a test for our algorithm, we generate GH fields on the time interval [0,1] with step size \\Delta _n=2^{-6}, on the spatial domain \\mathcal {D}=[0,1].For practical aspects, one is usually interested in the L^1-error \\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|) and the L^2-error (\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^2))^{1/2}.Upper bounds for both expressions depend on \\eta and D and are given by Ineq.~(\\ref {L^p error}).To obtain reasonable errors, we refer to the discussion on the choice of D in Remark~\\ref {rem:t-conv} and set D=\\Delta _n^{1/(1-\\eta )}.This ensures that the L^1-error is of order \\mathcal {O} (\\Delta _n) and is a good trade-off between simulation time and the size of the L^2-error for most values of \\eta in the GIG example below.Choosing for example D=\\Delta _n^{2/(2-\\eta )} would reduce the L^2-error to order \\mathcal {O} (\\Delta _n), but does not have a significant effect on the L^1-error and results in a higher computational time.For the Matérn covariance operator Q we use variance v=1, correlation length r=0.1 and \\chi \\in \\lbrace \\frac{1}{2},\\frac{3}{2}\\rbrace ,where a higher value of \\chi increases the regularity of the field along the x-direction.For the fixed GH parameters we choose \\alpha =5, \\beta =\\mu =0_N, \\delta =4 and \\Gamma =\\mathbf {1}_N, the shape parameter \\lambda will vary throughout our simulation and admits the values \\lambda \\in \\lbrace -\\frac{1}{2},1\\rbrace , which results in NIG resp.",
"hyperbolic GH fields.This parameter setting ensures that the multi-dimensional GH distribution has uncorrelated marginals, hence the truncated KL expansion L_N^{GH} of L^{GH} is itself an infinite dimensional GH Lévy process.Further, for every N\\in \\mathbb {N}, the constant \\widetilde{C}_{\\ell ,i} from Section~\\ref {sec:GH_app} is independent of i=1,\\dots ,N, thus the truncation index N can easily be determined to balance out the Fourier inversion and truncation error for each combination of \\lambda and \\chi .To examine the impact of \\eta on the efficiency of the simulation, we set \\eta \\in \\lbrace 4,6,8,10\\rbrace and the constant R as suggested in Section~\\ref {sec:GH_app} for each \\eta .For fixed \\eta and R, we choose \\theta \\in \\lbrace 1,1.5\\dots ,99.5,100\\rbrace and calculate for each \\theta the constant B_\\theta as in Section~\\ref {sec:imp}.This results in up to 199 different values for the number of summations M_\\theta , which all guarantee the desired accuracy \\varepsilon , meaning we can choose the smallest M_\\theta for our simulation.The optimal value \\theta _{opt} which leads to the smallest M_\\theta depends highly on the GH parameters and may vary significantly with \\eta .For \\lambda =1, we found that \\theta _{opt} ranges from 34 to 68.5, varying with each choice of \\eta \\in \\lbrace 4,6,8,10\\rbrace .In contrast, in the second example with \\lambda =-1/2, \\theta _{opt}=11 independent of \\eta .We generate 1.000 approximations \\widetilde{L}_N^{GH} for several combinations of \\lambda , \\chi and \\eta , which allows us to check if the generated samples actually follow the desired target distributions.To this end, we conduct Kolomogorov--Smirnov tests for the subordinating GIG process as well as for the distribution of the GH field at a fixed point in time and space and report on the corresponding p-values.\\begin{figure}\\centering \\subfigure [Sample of a GH field]{\\includegraphics [scale=0.45]{hyp_m0_5.eps}}\\subfigure [Empirical dist.",
"of 1.000 samples at t=x=1]{\\includegraphics [scale=0.48]{hyp_hist_m0_5_new.eps}}\\caption {Sample and empirical distribution of an hyperbolic field with parameters \\lambda =1, \\chi =1/2, \\eta =10 and truncation after N=132 terms.",
"}\\centering \\subfigure [Sample of a GH field]{\\includegraphics [scale=0.45]{nig_m1_5.eps}}\\subfigure [Empirical dist.",
"of 1.000 samples at t=x=1]{\\includegraphics [scale=0.48]{nig_hist_m1_5_new.eps}}\\caption {Sample and empirical distribution of a NIG field with parameters \\lambda =-1/2, \\chi =3/2, \\eta =10 and truncation after N=18 terms.",
"}\\end{figure}Figures~\\ref {fig:hyp} and ~\\ref {fig:nig} show samples of approximated GH random fields:Along the time axis we see the characteristic behavior of the (pure jump) GH processes for every point x\\in \\mathcal {D}.For a fixed point in time t, the paths along the x-axis vary according to their correlation, depending on the covariance parameter \\chi .As reported in \\cite {RW06}, the eigenvalues of Q decay slower if \\chi becomes smaller, meaning we need a higher number of summations N in the KL expansion so that the error contributions are equilibrated.This effect can be seen in Tables~\\ref {tab:hyp} and~\\ref {tab:nig}, where the truncation index N changes significantly with \\chi .If the KL expansion, however, can be sampled by a N-dimensional GH process as suggested in Theorem~\\ref {thm:Z_U},the number of summations N has only a minor impact on the computational costs of the KL expansion.This is due to the fact that in this case the time consuming part, namely simulating the subordinator, has to be done only once, regardless of N.Compared to these costs, the costs of subordinating a Brownian motion of any finite dimension are negligible.The histograms in Figures~\\ref {fig:hyp} and ~\\ref {fig:nig} show the empirical distribution of the approximation \\widetilde{L}_N^{GH}(x)(t) at time t=1 and x=1.The theoretical distribution at time 1 and an arbitrary point x\\in \\mathcal {D} is again GH, where the parameters are given in Lemma~\\ref {lem:KL_GH}.Obviously, the empirical distributions fit the target GH distributions from Lemma~\\ref {lem:KL_GH}.To be more precise, we have conducted a Kolmogorov-Smirnov test for both, the subordinating GIG process and the GH field at time t=1 and for the latter at x=1.We know the law of both processes at x\\in \\mathcal {D} and are able to obtain their CDFs sufficiently precise for the tests by numerical integration.The test results for 1.000 samples of the hyperbolic resp.",
"the NIG field with covariance parameters \\chi =\\frac{1}{2} resp.",
"\\chi =\\frac{3}{2} are given in Tables~\\ref {tab:hyp} and~\\ref {tab:nig} above and do not suggest that the generated samples follow another distribution than the expected one.\\end{equation*}\\begin{table}\\end{table}\\begin{center}\\small \\begin{tabular}{l|*{3}{c|}c}\\eta & E_{GIG,N}^1 & E_{GIG,N}^1/\\Delta _n &E_{GIG,N}^2&\\mathbb {E}[||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2] \\\\ \\hline 4 & 0.0143 &0.9166& 0.2584 &0.0646\\\\6 & 0.0138 &0.8835& 0.0749 & 0.0635\\\\8 & 0.0138 &0.8824& 0.0601 &0.0634\\\\10& 0.0140 &0.8975& 0.0806 &0.0636\\\\ \\hline \\hline \\eta &N & p-value GH&abs.",
"time&rel.",
"time\\\\ \\hline 4 &130& 0.8246 & 0.1945 sec.&100.00\\% \\\\6 &133& 0.3077 & 0.1093 sec.&56.19\\% \\\\8 &133& 0.3077 & 0.0851 sec.",
"&43.78\\%\\\\10&132& 0.2873 & 0.0759 sec.",
"&39.04\\%\\\\ \\hline \\end{tabular}\\end{center}\\caption {Errors, p-values and average simulation times per field based on 1.000 simulations.Stepsize \\Delta t =2^{-6} and D=\\Delta t^{1/(1-\\eta )}.GH process: \\lambda =1, \\alpha = 5, \\beta = 0_N, \\delta = 4, \\mu = 0_N, \\Gamma = 1_{N\\times N}.Covariance parameters: \\chi =1/2, r=0.1 and v=1.The KS test for the GIG subordinator returns a p-value of 0.5498 for each \\eta \\in \\lbrace 4,6,8,10\\rbrace .",
"}\\begin{center}\\small \\begin{tabular}{l|*{3}{c|}c}\\eta & E_{GIG,N}^1 & E_{GIG,N}^1/\\Delta _n &E_{GIG,N}^2&\\mathbb {E}[||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2] \\\\ \\hline 4 & 0.0132 & 0.8443 & 0.2079 & 0.0619 \\\\6 & 0.0128 & 0.8170 & 0.0584 & 0.0608 \\\\8 & 0.0127 & 0.8155 & 0.0456 & 0.0608\\\\10& 0.0129 & 0.8252 & 0.0589 & 0.0611 \\\\ \\hline \\hline \\eta &N & p-value GH&abs.",
"time&rel.",
"time\\\\ \\hline 4 & 18 & 0.9223 & 0.1039 sec.",
"& 100.00\\% \\\\6 & 18 & 0.9223 & 0.0628 sec.",
"& 60.43\\% \\\\8 & 18 & 0.9223 & 0.0460 sec.",
"& 44.29\\% \\\\10& 18 & 0.9223 & 0.0380 sec.",
"& 38.59\\% \\\\ \\hline \\end{tabular}\\end{center}\\caption {Errors, p-values and average simulation times per field based on 1.000 simulations.Stepsize \\Delta t =2^{-6} and D=\\Delta t^{1/(1-\\eta )}.GH process: \\lambda =-1/2, \\alpha = 5, \\beta = 0_N, \\delta = 4, \\mu = 0_N, \\Gamma = 1_{N\\times N}.Covariance parameters: \\chi =3/2, r=0.1 and v=1.The KS test for the GIG subordinator returns a p-value of 0.6145 for each \\eta \\in \\lbrace 4,6,8,10\\rbrace .",
"}$ We denote by $E_{GIG,N}^1$ and $E_{GIG,N}^2$ the approximation error of the subordinator as in Eq.",
"(REF ), which we have listed in absolute terms in Tables and .",
"The first error bound is also given relative to $\\Delta _n$ to show that it is in fact of magnitude $\\mathcal {O}(\\Delta _n)$ .",
"While the $L^1(\\Omega ;\\mathbb {R})$ -error $E_{GIG,N}^1$ is relatively constant for each $\\eta $ , the $L^2(\\Omega ;\\mathbb {R})$ -error $E_{GIG,N}^2$ is rather high for $\\eta =4$ , but has an acceptable upper bound for $\\eta \\ge 6$ .",
"This is not surprising, since $D=\\Delta _n^{1/(1-\\eta )}$ only guarantees that $\\mathbb {E}(|\\ell ^{GIG}(t)-\\widetilde{\\ell }^{GIG}(t)|)=\\mathcal {O}(\\Delta _n)$ .",
"We emphasize that the (theoretic) error bounds in Tables and are very conservative as the triangle inequality and similar \"coarse\" estimates were used repeatedly in their estimation in Theorem REF and REF .",
"The truncation index $N$ is highly sensitive to $\\chi $ , but has small or no variations for fixed $\\chi $ and varying $\\eta $ .",
"Since we choose $t\\in [0,1]$ , the expression $\\mathbb {E}(||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2)$ in Tables and is an upper bound for the $L^2(\\Omega ;H)$ -error $\\sup _{t\\in [0,1]}\\mathbb {E}(||L^{GH}(t)-\\widetilde{L}^{GH}_N(t)||_H^2)$ .",
"Note that this error is small in relative terms, since by our choice of $Q$ and Eq.",
"REF we have $\\mathbb {E}(||L^{GH}(1)||^2_H)=tr(Q)=1$ .",
"The p-value of the GH distribution varies if different $N$ are chosen for the KL expansion, which is natural due to statistical fluctuations.",
"More importantly, the null hypothesis, namely that the samples follow a GH distribution with the expected parameters, is never rejected at a $5\\%$ -level.",
"As expected, the speed of the simulation heavily depends on $\\eta $ .",
"Looking at the results for $\\eta =4$ , one might argue that the Fourier inversion method is only suitable for processes where this parameter can be chosen rather high, i.e.",
"for distributions which admit a large number of finite moments.",
"To qualify this objection, we consider once more the t-distribution with three degrees of freedom and the corresponding Lévy process $\\ell ^{t3}$ from Example REF .",
"Since $\\mathbb {E}(\\ell ^{t3}(\\Delta _n))=0$ and $\\text{Var}(\\ell ^{t3}(\\Delta _n))=\\sqrt{3}\\Delta _n$ , we can choose $\\eta =2$ and hence $R=\\sqrt{3}\\Delta _n$ .",
"The characteristic function of $\\ell ^{t3}(\\Delta _n)$ is given by $(\\phi _{t3}(u))^{\\Delta _n}=\\exp (-\\sqrt{3}\\Delta _n|u|)(\\sqrt{3}|u|+1)^{\\Delta _n}$ and $B$ and $\\theta $ are estimated in the same way as for the GIG process.",
"Using again $\\Delta _n=2^{-6}$ and $D=\\Delta _n^{1/(1-\\eta )}$ , we obtain that the number of summations in the approximation is $M=12.924$ for $\\theta =\\frac{19}{2}$ .",
"The simulation time for one process $\\widetilde{\\ell }^{t3}$ with $(\\Delta _n)^{-1}=2^6$ increments in the interval $[0,1]$ is on average 0.0655 seconds, where the initial values have been approximated by matching the moments of a normal distribution (the Kolmogorov-Smirnov test for a t-distribution at $t=1$ based on 1.000 samples returns a p-value of $0.5994$ ).",
"In the GIG example, we needed $M=79.086$ terms in the summation if $\\eta =4$ is chosen and still $M=33.030$ terms for $\\eta =10$ .",
"This shows that the Fourier Inversion method is also applicable if $\\eta $ can only be chosen relatively low and that the GIG (resp.",
"GH) process is a computationally expensive example of a Lévy process."
],
[
"Numerics",
"In this section we provide some details on the implementation of the Fourier inversion method.",
"Thereafter, we apply this methodology to approximate a GH Lévy field and conclude with some numerical examples."
],
[
"Notes on implementation",
"Suppose we simulate a given one-dimensional Lévy process $\\ell $ which fulfills Assumption REF resp.",
"Assumption REF , using the step size $\\Delta _n>0$ and characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"Usually the parameter $\\eta $ cannot be chosen arbitrary high (as for the GIG process), but it may be possible to choose $\\eta $ within a certain range, for instance $\\eta \\in (1,2]$ for the Cauchy process in Example REF .",
"As a rule of thumb, $\\eta $ should always be determined as large as possible, as the convergence rates in Theorems REF and REF directly depend on $\\eta $ .",
"In addition, we concluded in Remark REF that $D\\simeq \\Delta _n^{p/(p-\\eta )}$ is an appropriate choice to guarantee an $L^p$ -error of order $\\mathcal {O}(\\Delta _n^{1/p})$ .",
"This means that for a given $p$ , $D$ decreases as $\\eta $ increases.",
"Since the number of summations $M$ in Algorithm REF depends on $D$ (see Theorem REF ), an increasing parameter $\\eta $ also reduces computational time.",
"Once $\\eta $ is determined, we derive $R$ by differentiation of $(\\phi _\\ell )^{\\Delta _n}$ as in Remark REF .",
"Similarly to $\\eta $ , it is often possible to choose between several values of $\\theta >0$ , but it is difficult to give a-priori a recommendation on how $\\theta $ should be selected.",
"One rather calculates for several admissible $\\theta $ the constant $C_\\theta :=\\max _{u\\in \\mathbb {R}}|u^\\theta (\\phi _\\ell (u))^{\\Delta _n}|$ numerically and deducts $B_\\theta =(2\\pi )^{-\\theta }C_\\theta $ .",
"Each combination of $(\\theta ,B_\\theta )$ then results in a valid number of summations $M_\\theta $ in the discrete Fourier Inversion algorithm.",
"Since $\\theta $ and $B_\\theta $ are only necessary to determine $M_\\theta $ , we may simply use the smallest $M_\\theta $ for the simulation.",
"To find $\\widetilde{X}$ with $\\widetilde{F}(\\widetilde{X})=U$ in Algorithm REF , we use a globalized Newton method with backtracking line search, also known as Armijo increment control.",
"The step lengths during the line search are determined by interpolation, which is a robust technique if combined with a standard Newton method.",
"Details on the globalized Newton method with backtracking may be found, for example, in [31], an example how the algorithm is used is given in [36].",
"Although convergence of this root finding algorithm is ensured by the increment control, its efficiency depends heavily on the choice of the initial value $\\widetilde{X}_0$ .",
"Clearly, $\\widetilde{X}_0$ should depend on the sampled $U\\sim \\mathcal {U}([0,1])$ and be related to the target distribution with characteristic function $(\\phi _\\ell )^{\\Delta _n}$ .",
"This means we should determine $\\widetilde{X}_0$ implicitly by $F^{(0)}(\\widetilde{X}_0)=U$ , where $F^{(0)}$ is a CDF of a distribution similar to the target distribution, but which can be inverted efficiently."
],
[
"Approximation of a GH field",
"We consider a GH Lévy field on the (separable) Hilbert space $H=L^2(\\mathcal {D})$ with a compact spatial domain $\\mathcal {D}\\subset \\mathbb {R}^s$ .",
"The operator $Q$ on $H$ is given by a Matérn covariance operator with variance $v>0$ , correlation length $r>0$ and a positive parameter $\\chi >0$ defined by $[Qh](x):=v\\int _\\mathcal {D} k_\\chi (x,y)h(y)dy,\\quad \\text{for }h\\in H,$ where $k_\\chi $ denotes the Matérn kernel.",
"For $\\chi =\\frac{1}{2}$ , we obtain the exponential covariance function and for $\\chi \\rightarrow \\infty $ the squared exponential covariance function.",
"For general $\\chi >0$ , the Matérn kernel $k_\\chi (x,y):=\\frac{2^{1-\\chi }}{\\Gamma _G(\\chi )}\\Big (\\frac{\\sqrt{2\\chi }|x-y|}{r}\\Big )^\\chi K_\\chi \\Big (\\frac{\\sqrt{2\\chi }|x-y|}{r}\\Big )$ fulfills the limit identity $k_\\chi (x,x)=\\lim _{y\\rightarrow x}k_\\chi (x,y)=1$ , which can be easily seen by [33].",
"Here $\\Gamma _G(\\cdot )$ is the Gamma function.",
"As shown in [20], this implies $tr(Q)=\\sum _{i=1}^\\infty \\rho _i=v\\int _{\\mathcal {D}}dx,$ where $(\\rho _i,i\\in \\mathbb {N})$ are the eigenvalues of the Matérn covariance operator $Q$ .",
"In general, no analytical expressions for the eigenpairs $(\\rho _i,e_i)$ of $Q$ will be available, but the spectral basis may be approximated by numerically solving a discrete eigenvalue problem and then interpolating by Nyström's method.",
"For a general overview of common covariance functions and the approximation of their eigenbasis we refer to [37] and the references therein.",
"Now let $L_N^{GH}$ be an approximation of a GH field by a $N$ -dimensional GH process $(\\ell ^{GH_N}(t),t\\in $ with fixed parameters $\\lambda ,\\alpha ,\\delta \\in \\mathbb {R}$ , $\\beta ,\\mu \\in \\mathbb {R}^N$ and $\\Gamma \\in \\mathbb {R}^{N\\times N}$ .",
"The parameters are chosen in such a way that the multi-dimensional GH process has uncorrelated marginal processes, hence the generated KL expansions $L_N^{GH}(x)(t)=\\sum _{i=1}^N\\varphi _i(x)\\ell ^{GH}_i(t)$ are again one-dimensional GH processes for any spectral basis $(\\varphi _i,i\\in \\mathbb {N})$ and fixed $x\\in \\mathcal {D}$ .",
"This in turn means, that we may draw samples of $\\ell ^{GH_N}$ by simulating a GIG process $\\ell ^{GIG}$ with parameters $a=\\delta , b=(\\alpha ^2-\\beta ^{\\prime }\\Gamma \\beta )^{1/2}$ and $p=\\lambda $ using Fourier inversion and then subordinating a $N$ -dimensional Brownian motion (see Lemma REF ).",
"The characteristic function of a GIG Lévy process $(\\ell ^{GIG}(t),t\\in $ with (fixed) parameters $a,b>0$ and $p\\in \\mathbb {R}$ is given by $ \\phi _{GIG}(u;a,b,p):=\\mathbb {E}[\\exp (iu\\ell ^{GIG}(1))]=(1-2iub^{-2})^{-p/2}\\frac{K_p(ab\\sqrt{1-2iub^{-2}})}{K_p(ab)}.$ The GIG distribution corresponding to $(\\phi _{GIG})^{\\Delta _n}$ with $\\Delta _n=1$ is continuous with finite variance (see [41]), which implies that these properties hold for all distributions with characteristic function $(\\phi _{GIG})^{\\Delta _n}$ , for any $\\Delta _n>0$ .",
"The constants as in Assumption REF are derived in the following.",
"For $k\\in \\mathbb {N}$ , the $k$ -th moment of the GIG distribution is given as $0<\\mathbb {E}\\big ((\\ell ^{GIG}(1))^k\\big )=\\big (\\frac{a}{b}\\big )^{k}\\frac{K_{p+k}(ab)}{K_p(ab)}<\\infty .$ For any $\\eta =2k$ we are, therefore, able to calculate the bounding constant $R$ via $R=(-1)^{k}\\frac{d^{2k}}{du^{2k}}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0},$ because the derivatives of $\\phi _{GIG}$ evaluated at $u=0$ are $(\\phi _{GIG}(0;a,b,p))^{(k)}=i^{-k}\\mathbb {E}\\big ((\\ell ^{GIG}(1))^k\\big )=i^{-k}\\big (\\frac{a}{b}\\big )^{k}\\frac{K_{p+k}(ab)}{K_p(ab)}.$ The calculation of the $\\eta $ -th derivative can be implemented easily by using a version of Faà di Bruno's formula containing the Bell polynomials, for details we refer to [25].",
"The bounding constants $\\theta $ and $B$ may be determined numerically as described Section REF (e.g.",
"by using the routine fminsearch in MATLAB).",
"The derivation of the bounds implies that we can ensure $L^p$ convergence of the approximated GIG process in the sense of Theorem REF for any $p\\ge 1$ , because it is possible to define $\\eta $ as any even integer and then obtain $R$ by differentiation.",
"We observe that the target distribution with characteristic function $(\\phi _{GIG}(u;a,b,p))^{\\Delta _n}$ and $\\Delta _n>0$ is not necessarily GIG, except for the Inverse Gaussian (IG) case where $p=-1/2$ and $(\\phi _{IG}(u;a,b))^{\\Delta _n}=\\phi _{IG}(u;\\Delta _na,b)$ (see Remark REF ).",
"This special feature of the IG distribution is exploited to determine the initial values $\\widetilde{X}_0$ in the Newton iteration by moment matching: Consider an $IG(a_0,b_0)$ distribution with mean $a_0/b_0$ and variance $a_0/b_0^3$ , where the parameters $a_0,b_0>0$ are “matched” to the target distribution's mean and variance via $\\frac{a_0}{b_0}&=i\\frac{d}{du}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0},\\\\\\frac{a_0}{b_0^3}&=(-1)\\frac{d^2}{du^2}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0}-\\Big (i\\frac{d}{du}((\\phi _{GIG}(u;a,b,p))^{\\Delta _n})\\big |_{u=0}\\Big )^2.$ If $F_{\\Delta _n}^{IG}$ denotes the CDF of this $IG(a_0,b_0)$ distribution, the initial value of the globalized Newton method is given implicitly by $F_{\\Delta _n}^{IG}(\\widetilde{X}_0)=U$ .",
"The inversion of $F_{\\Delta _n}^{IG}$ may be executed numerically by many software packages like MATLAB.",
"With our approach, this results in the approximation of a GIG process $\\widetilde{\\ell }^{GIG}$ at discrete times $t_j\\in \\Xi _n$ .",
"The $N$ -dimensional GH process $\\ell ^{GH_N}$ may then be approximated at $t_j$ for $j=0,\\dots ,n$ by the process $\\widetilde{\\ell }^{GH_N}$ with $\\widetilde{\\ell }^{GH_N}(t_0)=0$ and the increments $\\widetilde{\\ell }^{GH_N}(t_j)-\\widetilde{\\ell }^{GH_N}(t_{j-1})=\\mu \\Delta _n+\\Gamma \\beta (\\widetilde{\\ell }^{GIG}(t_j)-\\widetilde{\\ell }^{GIG}(t_{j-1})) + \\sqrt{(\\widetilde{\\ell }^{GIG}(t_j)-\\widetilde{\\ell }^{GIG}(t_{j-1}))\\Gamma }w^N_j(1),$ for $j=1,\\dots ,n$ , where the $w_j^N(1)$ are i.i.d.",
"$\\mathcal {N}_N(0,\\mathbf {1}_{N\\times N})$ -distributed random vectors.",
"To obtain the process $\\widetilde{\\ell }^{GH_N}$ at arbitrary times $t\\in , we interpolate the samples $ (GHN(tj),j=0...,n)$ piecewise constant as in Algorithm~\\ref {algo:approx2}.With this, we are able to generate an approximation of $ LNGH$ at any point $ (x,t)D by $\\widetilde{L}_N^{GH}(x)(t):=\\sum _{i=1}^N\\varphi _i(x) \\widetilde{\\ell }^{GH}_i(t).$ The knowledge of $tr(Q)$ enables us to determine the truncation index $N$ and the constant $C_\\ell $ as in Remark : For $N\\in \\mathbb {N}$ let $(\\widetilde{\\ell }^{GH}_i,i=1,\\ldots ,N)$ be the approximations of the processes $(\\ell ^{GH}_i,i=1,\\ldots ,N)$ , where the random vector $(\\ell ^{GH}_1(1),\\dots ,\\ell _N^{GH}(1))$ is multivariate GH-distributed by assumption.",
"Hence, for every $N\\in \\mathbb {N}$ , we obtain the parameters $a(N), b(N), \\lambda (N)$ of a corresponding GIG subordinator $\\ell ^{GIG,N}$ , which is approximated through a piecewise constant process $\\widetilde{\\ell }^{GIG,N}$ as above.",
"With Eq.",
"(REF ) we calculate the error $E_{GIG,N}^p:=\\sup _{t\\in \\mathbb {E}(|\\ell ^{GIG,N}(t)-\\widetilde{\\ell }^{GIG,N}(t)|^p).",
"}for p\\in \\lbrace 1,2\\rbrace .If \\beta \\in \\mathbb {R}^N and \\Gamma \\in \\mathbb {R}^{N\\times N} denote the GH parameters corresponding to (\\ell ^{GH}_1(1),\\dots ,\\ell _N^{GH}(1)), the L^2(\\Omega ;\\mathbb {R}) approximation error of each process \\ell ^{GH}_i is given by\\begin{equation*}\\widetilde{C}_{\\ell ,i}:=\\sup _{t\\in \\frac{\\mathbb {E}(|\\ell ^{GH}_i(t)-\\widetilde{\\ell }^{GH}_i(t)|^2)}{\\Delta _n}= \\frac{E_{GIG,N}^2(\\Gamma \\beta )_i^2+E_{GIG,N}^1\\sqrt{\\Gamma _{[i]}\\Gamma _{[i]}^{\\prime }}}{\\Delta _n},}where \\Gamma _{[i]} indicates the i-th row of \\Gamma .Starting with N=1, we compute the first N eigenvalues and the difference\\begin{equation*}T\\Big (tr(Q)-\\sum _{i=1}^N\\rho _i\\Big )-\\max _{i=1,\\dots ,N}\\widetilde{C}_{\\ell ,i}\\Delta _n \\sum _{i=1}^N\\rho _i\\end{equation*}and increase N by one in every step until this expression is close to zero.If a suitable N is found, we define C_\\ell :=\\max _{i=1,\\dots ,N}\\widetilde{C}_{\\ell ,i} and thus have equilibrated truncation and approximation errors by ensuring Eq.~(\\ref {eq:trunc}).For simplicity, we have implicitly assumed here that the processes \\ell ^{GH}_i were normalized in the sense that \\text{Var}(\\ell ^{GH}_i(t))=t.This is due to the fact that \\rho _i\\ell _i (here with \\ell _i=\\ell ^{GH}_i) in Theorem~\\ref {thm:H_error} represents the scalar product (L(t),e_i)_H with variance \\rho _it.In case we have unnormalized processes, one can simply divide \\ell ^{GH}_i by its standard deviation (see Formula~(\\ref {eq:VarZ})) and adjust the constants \\widetilde{C}_{\\ell ,i} and C_\\ell accordingly.\\subsection {Numerical results}As a test for our algorithm, we generate GH fields on the time interval [0,1] with step size \\Delta _n=2^{-6}, on the spatial domain \\mathcal {D}=[0,1].For practical aspects, one is usually interested in the L^1-error \\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|) and the L^2-error (\\mathbb {E}(|\\ell (t)-\\widetilde{\\ell }^{(n)}(t)|^2))^{1/2}.Upper bounds for both expressions depend on \\eta and D and are given by Ineq.~(\\ref {L^p error}).To obtain reasonable errors, we refer to the discussion on the choice of D in Remark~\\ref {rem:t-conv} and set D=\\Delta _n^{1/(1-\\eta )}.This ensures that the L^1-error is of order \\mathcal {O} (\\Delta _n) and is a good trade-off between simulation time and the size of the L^2-error for most values of \\eta in the GIG example below.Choosing for example D=\\Delta _n^{2/(2-\\eta )} would reduce the L^2-error to order \\mathcal {O} (\\Delta _n), but does not have a significant effect on the L^1-error and results in a higher computational time.For the Matérn covariance operator Q we use variance v=1, correlation length r=0.1 and \\chi \\in \\lbrace \\frac{1}{2},\\frac{3}{2}\\rbrace ,where a higher value of \\chi increases the regularity of the field along the x-direction.For the fixed GH parameters we choose \\alpha =5, \\beta =\\mu =0_N, \\delta =4 and \\Gamma =\\mathbf {1}_N, the shape parameter \\lambda will vary throughout our simulation and admits the values \\lambda \\in \\lbrace -\\frac{1}{2},1\\rbrace , which results in NIG resp.",
"hyperbolic GH fields.This parameter setting ensures that the multi-dimensional GH distribution has uncorrelated marginals, hence the truncated KL expansion L_N^{GH} of L^{GH} is itself an infinite dimensional GH Lévy process.Further, for every N\\in \\mathbb {N}, the constant \\widetilde{C}_{\\ell ,i} from Section~\\ref {sec:GH_app} is independent of i=1,\\dots ,N, thus the truncation index N can easily be determined to balance out the Fourier inversion and truncation error for each combination of \\lambda and \\chi .To examine the impact of \\eta on the efficiency of the simulation, we set \\eta \\in \\lbrace 4,6,8,10\\rbrace and the constant R as suggested in Section~\\ref {sec:GH_app} for each \\eta .For fixed \\eta and R, we choose \\theta \\in \\lbrace 1,1.5\\dots ,99.5,100\\rbrace and calculate for each \\theta the constant B_\\theta as in Section~\\ref {sec:imp}.This results in up to 199 different values for the number of summations M_\\theta , which all guarantee the desired accuracy \\varepsilon , meaning we can choose the smallest M_\\theta for our simulation.The optimal value \\theta _{opt} which leads to the smallest M_\\theta depends highly on the GH parameters and may vary significantly with \\eta .For \\lambda =1, we found that \\theta _{opt} ranges from 34 to 68.5, varying with each choice of \\eta \\in \\lbrace 4,6,8,10\\rbrace .In contrast, in the second example with \\lambda =-1/2, \\theta _{opt}=11 independent of \\eta .We generate 1.000 approximations \\widetilde{L}_N^{GH} for several combinations of \\lambda , \\chi and \\eta , which allows us to check if the generated samples actually follow the desired target distributions.To this end, we conduct Kolomogorov--Smirnov tests for the subordinating GIG process as well as for the distribution of the GH field at a fixed point in time and space and report on the corresponding p-values.\\begin{figure}\\centering \\subfigure [Sample of a GH field]{\\includegraphics [scale=0.45]{hyp_m0_5.eps}}\\subfigure [Empirical dist.",
"of 1.000 samples at t=x=1]{\\includegraphics [scale=0.48]{hyp_hist_m0_5_new.eps}}\\caption {Sample and empirical distribution of an hyperbolic field with parameters \\lambda =1, \\chi =1/2, \\eta =10 and truncation after N=132 terms.",
"}\\centering \\subfigure [Sample of a GH field]{\\includegraphics [scale=0.45]{nig_m1_5.eps}}\\subfigure [Empirical dist.",
"of 1.000 samples at t=x=1]{\\includegraphics [scale=0.48]{nig_hist_m1_5_new.eps}}\\caption {Sample and empirical distribution of a NIG field with parameters \\lambda =-1/2, \\chi =3/2, \\eta =10 and truncation after N=18 terms.",
"}\\end{figure}Figures~\\ref {fig:hyp} and ~\\ref {fig:nig} show samples of approximated GH random fields:Along the time axis we see the characteristic behavior of the (pure jump) GH processes for every point x\\in \\mathcal {D}.For a fixed point in time t, the paths along the x-axis vary according to their correlation, depending on the covariance parameter \\chi .As reported in \\cite {RW06}, the eigenvalues of Q decay slower if \\chi becomes smaller, meaning we need a higher number of summations N in the KL expansion so that the error contributions are equilibrated.This effect can be seen in Tables~\\ref {tab:hyp} and~\\ref {tab:nig}, where the truncation index N changes significantly with \\chi .If the KL expansion, however, can be sampled by a N-dimensional GH process as suggested in Theorem~\\ref {thm:Z_U},the number of summations N has only a minor impact on the computational costs of the KL expansion.This is due to the fact that in this case the time consuming part, namely simulating the subordinator, has to be done only once, regardless of N.Compared to these costs, the costs of subordinating a Brownian motion of any finite dimension are negligible.The histograms in Figures~\\ref {fig:hyp} and ~\\ref {fig:nig} show the empirical distribution of the approximation \\widetilde{L}_N^{GH}(x)(t) at time t=1 and x=1.The theoretical distribution at time 1 and an arbitrary point x\\in \\mathcal {D} is again GH, where the parameters are given in Lemma~\\ref {lem:KL_GH}.Obviously, the empirical distributions fit the target GH distributions from Lemma~\\ref {lem:KL_GH}.To be more precise, we have conducted a Kolmogorov-Smirnov test for both, the subordinating GIG process and the GH field at time t=1 and for the latter at x=1.We know the law of both processes at x\\in \\mathcal {D} and are able to obtain their CDFs sufficiently precise for the tests by numerical integration.The test results for 1.000 samples of the hyperbolic resp.",
"the NIG field with covariance parameters \\chi =\\frac{1}{2} resp.",
"\\chi =\\frac{3}{2} are given in Tables~\\ref {tab:hyp} and~\\ref {tab:nig} above and do not suggest that the generated samples follow another distribution than the expected one.\\end{equation*}\\begin{table}\\end{table}\\begin{center}\\small \\begin{tabular}{l|*{3}{c|}c}\\eta & E_{GIG,N}^1 & E_{GIG,N}^1/\\Delta _n &E_{GIG,N}^2&\\mathbb {E}[||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2] \\\\ \\hline 4 & 0.0143 &0.9166& 0.2584 &0.0646\\\\6 & 0.0138 &0.8835& 0.0749 & 0.0635\\\\8 & 0.0138 &0.8824& 0.0601 &0.0634\\\\10& 0.0140 &0.8975& 0.0806 &0.0636\\\\ \\hline \\hline \\eta &N & p-value GH&abs.",
"time&rel.",
"time\\\\ \\hline 4 &130& 0.8246 & 0.1945 sec.&100.00\\% \\\\6 &133& 0.3077 & 0.1093 sec.&56.19\\% \\\\8 &133& 0.3077 & 0.0851 sec.",
"&43.78\\%\\\\10&132& 0.2873 & 0.0759 sec.",
"&39.04\\%\\\\ \\hline \\end{tabular}\\end{center}\\caption {Errors, p-values and average simulation times per field based on 1.000 simulations.Stepsize \\Delta t =2^{-6} and D=\\Delta t^{1/(1-\\eta )}.GH process: \\lambda =1, \\alpha = 5, \\beta = 0_N, \\delta = 4, \\mu = 0_N, \\Gamma = 1_{N\\times N}.Covariance parameters: \\chi =1/2, r=0.1 and v=1.The KS test for the GIG subordinator returns a p-value of 0.5498 for each \\eta \\in \\lbrace 4,6,8,10\\rbrace .",
"}\\begin{center}\\small \\begin{tabular}{l|*{3}{c|}c}\\eta & E_{GIG,N}^1 & E_{GIG,N}^1/\\Delta _n &E_{GIG,N}^2&\\mathbb {E}[||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2] \\\\ \\hline 4 & 0.0132 & 0.8443 & 0.2079 & 0.0619 \\\\6 & 0.0128 & 0.8170 & 0.0584 & 0.0608 \\\\8 & 0.0127 & 0.8155 & 0.0456 & 0.0608\\\\10& 0.0129 & 0.8252 & 0.0589 & 0.0611 \\\\ \\hline \\hline \\eta &N & p-value GH&abs.",
"time&rel.",
"time\\\\ \\hline 4 & 18 & 0.9223 & 0.1039 sec.",
"& 100.00\\% \\\\6 & 18 & 0.9223 & 0.0628 sec.",
"& 60.43\\% \\\\8 & 18 & 0.9223 & 0.0460 sec.",
"& 44.29\\% \\\\10& 18 & 0.9223 & 0.0380 sec.",
"& 38.59\\% \\\\ \\hline \\end{tabular}\\end{center}\\caption {Errors, p-values and average simulation times per field based on 1.000 simulations.Stepsize \\Delta t =2^{-6} and D=\\Delta t^{1/(1-\\eta )}.GH process: \\lambda =-1/2, \\alpha = 5, \\beta = 0_N, \\delta = 4, \\mu = 0_N, \\Gamma = 1_{N\\times N}.Covariance parameters: \\chi =3/2, r=0.1 and v=1.The KS test for the GIG subordinator returns a p-value of 0.6145 for each \\eta \\in \\lbrace 4,6,8,10\\rbrace .",
"}$ We denote by $E_{GIG,N}^1$ and $E_{GIG,N}^2$ the approximation error of the subordinator as in Eq.",
"(REF ), which we have listed in absolute terms in Tables and .",
"The first error bound is also given relative to $\\Delta _n$ to show that it is in fact of magnitude $\\mathcal {O}(\\Delta _n)$ .",
"While the $L^1(\\Omega ;\\mathbb {R})$ -error $E_{GIG,N}^1$ is relatively constant for each $\\eta $ , the $L^2(\\Omega ;\\mathbb {R})$ -error $E_{GIG,N}^2$ is rather high for $\\eta =4$ , but has an acceptable upper bound for $\\eta \\ge 6$ .",
"This is not surprising, since $D=\\Delta _n^{1/(1-\\eta )}$ only guarantees that $\\mathbb {E}(|\\ell ^{GIG}(t)-\\widetilde{\\ell }^{GIG}(t)|)=\\mathcal {O}(\\Delta _n)$ .",
"We emphasize that the (theoretic) error bounds in Tables and are very conservative as the triangle inequality and similar \"coarse\" estimates were used repeatedly in their estimation in Theorem REF and REF .",
"The truncation index $N$ is highly sensitive to $\\chi $ , but has small or no variations for fixed $\\chi $ and varying $\\eta $ .",
"Since we choose $t\\in [0,1]$ , the expression $\\mathbb {E}(||L^{GH}(1)-\\widetilde{L}^{GH}_N(1)||_H^2)$ in Tables and is an upper bound for the $L^2(\\Omega ;H)$ -error $\\sup _{t\\in [0,1]}\\mathbb {E}(||L^{GH}(t)-\\widetilde{L}^{GH}_N(t)||_H^2)$ .",
"Note that this error is small in relative terms, since by our choice of $Q$ and Eq.",
"REF we have $\\mathbb {E}(||L^{GH}(1)||^2_H)=tr(Q)=1$ .",
"The p-value of the GH distribution varies if different $N$ are chosen for the KL expansion, which is natural due to statistical fluctuations.",
"More importantly, the null hypothesis, namely that the samples follow a GH distribution with the expected parameters, is never rejected at a $5\\%$ -level.",
"As expected, the speed of the simulation heavily depends on $\\eta $ .",
"Looking at the results for $\\eta =4$ , one might argue that the Fourier inversion method is only suitable for processes where this parameter can be chosen rather high, i.e.",
"for distributions which admit a large number of finite moments.",
"To qualify this objection, we consider once more the t-distribution with three degrees of freedom and the corresponding Lévy process $\\ell ^{t3}$ from Example REF .",
"Since $\\mathbb {E}(\\ell ^{t3}(\\Delta _n))=0$ and $\\text{Var}(\\ell ^{t3}(\\Delta _n))=\\sqrt{3}\\Delta _n$ , we can choose $\\eta =2$ and hence $R=\\sqrt{3}\\Delta _n$ .",
"The characteristic function of $\\ell ^{t3}(\\Delta _n)$ is given by $(\\phi _{t3}(u))^{\\Delta _n}=\\exp (-\\sqrt{3}\\Delta _n|u|)(\\sqrt{3}|u|+1)^{\\Delta _n}$ and $B$ and $\\theta $ are estimated in the same way as for the GIG process.",
"Using again $\\Delta _n=2^{-6}$ and $D=\\Delta _n^{1/(1-\\eta )}$ , we obtain that the number of summations in the approximation is $M=12.924$ for $\\theta =\\frac{19}{2}$ .",
"The simulation time for one process $\\widetilde{\\ell }^{t3}$ with $(\\Delta _n)^{-1}=2^6$ increments in the interval $[0,1]$ is on average 0.0655 seconds, where the initial values have been approximated by matching the moments of a normal distribution (the Kolmogorov-Smirnov test for a t-distribution at $t=1$ based on 1.000 samples returns a p-value of $0.5994$ ).",
"In the GIG example, we needed $M=79.086$ terms in the summation if $\\eta =4$ is chosen and still $M=33.030$ terms for $\\eta =10$ .",
"This shows that the Fourier Inversion method is also applicable if $\\eta $ can only be chosen relatively low and that the GIG (resp.",
"GH) process is a computationally expensive example of a Lévy process."
]
] | 1612.05541 |
[
[
"On the spectrum of an operator in truncated Fock space"
],
[
"Abstract We study the spectrum of an operator matrix arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus.",
"An analytic description of the essential spectrum is established.",
"Further, a criterion for the finiteness of the number of eigenvalues below the bottom of the essential spectrum is derived."
],
[
"Introduction",
"In this paper we study the essential spectrum and discrete spectrum of the tridiagonal operator matrix $H:=\\begin{pmatrix}H_{00} & H_{01} & 0\\\\[1ex]H_{01}^* & H_{11} & H_{12}\\\\[1ex]0 & H_{12}^* & H_{22}\\\\[1ex]\\end{pmatrix}$ in the so-called truncated Fock space ${\\mathcal {H}}:={\\mathcal {H}}_0\\oplus {\\mathcal {H}}_1\\oplus {\\mathcal {H}}_2$ with ${\\mathcal {H}}_0:={\\mathbb {C}}$ , ${\\mathcal {H}}_1:=L^2(\\Omega ,{\\mathbb {C}})$ and ${\\mathcal {H}}_2:=L^2_{\\rm sym}(\\Omega ^2,{\\mathbb {C}})$ .",
"Here $\\Omega $ is a $d$ -dimensional open cube $(-a,a)^d$ , $d\\in {\\mathbb {N}}$ , $a\\in (0,\\infty )$ , and $L^2_{\\rm sym}(\\Omega ^2,{\\mathbb {C}})$ stands for the subspace of $L^2(\\Omega ^2,{\\mathbb {C}})$ consisting of symmetric functions (with respect to the two variables).",
"The operator entries $H_{ij}: {\\mathcal {H}}_j \\rightarrow {\\mathcal {H}}_i$ , $|i-j|\\le 1$ , $i,j=0,1,2$ , are given by $\\nonumber &H_{00} f_0 = w_0 f_0, \\quad H_{01}f_1 = \\int \\limits _{\\Omega } v_0(s)f_1(s) \\,{\\rm {d}}s,\\\\[-1mm] &(H_{11} f_1)(x) = w_1(x)f_1(x), \\quad (H_{12}f_2)(x)=\\int \\limits _{\\Omega }v_1(x,s)f_2(x,s) \\,{\\rm {d}}s,\\\\[1mm] \\nonumber &(H_{22}f_2)(x,y) = w_2(x,y)f_2(x,y), \\\\[-1.5mm] \\nonumber $ for almost all (a.a.) $x,y\\in \\Omega $ with parameter functions satisfying certain rather weak conditions to be specified below.",
"Operator matrices of this form play a key role for the study of the energy operator of the spin-boson Hamiltonian with two bosons on the torus.",
"In fact, the latter is a $6\\times 6$ operator matrix which is unitarily equivalent to a $2\\times 2$ block diagonal operator with two copies of a particular case of $H$ on the diagonal, see e.g.",
"[11].",
"Consequently, the essential spectrum and finiteness of discrete eigenvalues of the spin-boson Hamiltonian are determined by the corresponding spectral information on the operator matrix $H$ in (REF ).",
"Independently of whether the underlying domain is a torus or the whole space ${\\mathbb {R}}^d$ , the full spin-boson Hamiltonian is an infinite operator matrix in Fock space for which rigorous results are very hard to obtain.",
"One line of attack is to consider the compression to the truncated Fock space with a finite number $N$ of bosons, and in fact most of the existing literature concentrates on the case $N\\le 2$ .",
"For the case of ${\\mathbb {R}}^d$ there are some exceptions, see e.g.",
"Hübner, Spohn [5], [6] for arbitrary finite $N$ and Zhukov, Minlos [16] for $N=3$ , where a rigorous scattering theory was developed for small coupling constants.",
"For the case when the underlying domain is a torus, the spectral properties of a slightly simpler version of $H$ were investigated by Muminov, Neidhardt and Rasulov [11], Albeverio, Lakaev and Rasulov [2], Lakaev and Rasulov [8], Rasulov [12], see also the references therein.",
"In the case when $v_1$ is a function of a single variable and all parameter functions are continuous (sometimes even real-analytic) with special properties on a closed torus of specific dimension, an analytic description of the essential spectrum was first given in [8]; a Birman-Schwinger type result was first established in [2]; the finiteness of the discrete spectrum was analysed in [11] for $d=1$ with real-analytic parameter functions.",
"In this paper we establish an analytic description of the essential spectrum, a Birman-Schwinger type result as well as a criterion guaranteeing the finiteness of discrete eigenvalues below the bottom of the essential spectrum of $H$ .",
"Compared to earlier work, we achieve these results in a more general setting with weaker conditions on the parameter functions.",
"For example, the dimension $d\\in {\\mathbb {N}}$ is arbitrary, the parameter function $v_1$ is required to be neither of one variable nor real-analytic or continuous.",
"In fact, our analysis shows that it suffices to require the boundedness of the functions $x\\mapsto \\Vert v_1(x,\\cdot )\\Vert _{L^{2+\\varepsilon }(\\Omega )}$ and $y\\mapsto \\Vert v_1(\\cdot ,y)\\Vert _{L^{2+4/\\varepsilon }(\\Omega )}$ on $\\Omega $ for some $\\varepsilon >0$ .",
"Although we consider the case $a<\\infty $ throughout the paper, our methods are of local nature and also apply to the case $a=\\infty $ where $\\Omega ={\\mathbb {R}}^d$ .",
"< Another difference to earlier work is that we employ more abstract me-thods, allowing for simpler proofs of the first two results mentioned above; in particular, we do not make use of the so-called generalized Friedrichs model in our analysis.",
"However, in spite of being self-adjoint and bounded (with compact underlying domain), the operator matrix $H$ in (REF ) is, up to our knowledge, not covered by any of the currently existing abstract results such as [3], [1], [10], [7], [9].",
"The abstract results on the essential/discrete spectrum in [3], [1], [10] do not apply since the required compactness assumptions on certain auxiliary operators are violated mainly due to the non-compactness of partial-integral operators.",
"The variational principles of [7], [9] do not give information on the finiteness/infiniteness of discrete eigenvalues either because none of the diagonal entries of $H$ has infinitely many discrete eigenvalues.",
"For the present approach, since the last diagonal entry $H_{22}$ of $H$ is a multiplication operator, it turned out to be natural to use singular sequences to describe one part of the essential spectrum and to employ a Schur complement approach to describe the second part.",
"We mention that, in a more concrete setting, the infiniteness of the discrete eigenvalues below the bottom of the essential spectrum of $H$ and corresponding eigenvalue asymptotics were also discussed in the literature, see e.g.",
"Albeverio, Lakaev and Rasulov [2]; these results were obtained using the machinery developed in Sobolev [14].",
"To achieve analogous results in our general setting seems to be very challenging and is beyond the scope of this paper.",
"The paper is organized as follows.",
"In Section 2 we formulate the hypotheses on the parameter functions, explain the reduction of the problem to a $2\\times 2$ operator matrix and describe the Schur complement of the latter.",
"In Sections 3 and 4 we establish the analytic description of the essential spectrum and a Birman-Schwinger type result, respectively.",
"In Section 5, inspired by the methods of [2], we derive the criterion for the finiteness of the discrete spectrum below the bottom of the essential spectrum of $H$ .",
"Section 6 contains some concluding remarks e.g.",
"on the limiting case $a=\\infty $ and on modifications of the assumptions under which our results continue to hold.",
"The following notations will be used in the sequel: ${\\rm {cl}}\\,(X)$ denotes the closure of a set $X\\subset {\\mathbb {R}}^d$ in ${\\mathbb {R}}^d$ (w.r.t.",
"the standard topology); for a complex-valued function $\\varphi $ , we denote by $\\varphi ^*$ the complex conjugate of $\\varphi $ ; $\\operatornamewithlimits{ran}(f)$ and $\\operatornamewithlimits{ess\\,ran}(f)$ respectively denote the range and the essential range of a (measurable) function $f$ on $\\Omega $ or $\\Omega ^2$ , respectively; a function $f$ on $\\Omega ^2$ is called symmetric if $f(x,y) = f^*(y,x)$ for (a.a. if applicable) $x,y \\in \\Omega $ ."
],
[
"The block operator matrix",
"Throughout the paper we assume that the parameter functions in (REF ) satisfy the following hypotheses.",
"(A)Assumption (A) $w_0\\in {\\mathbb {R}}, v_0\\in L^2(\\Omega , {\\mathbb {C}})$ , $w_1\\in L^\\infty (\\Omega , {\\mathbb {R}})$ , $w_2\\in C(\\Omega ^2)\\cap L^\\infty (\\Omega ^2, {\\mathbb {R}})$ with $w_2(x,y)=w_2(y,x)$ , $x,y\\in \\Omega $ .",
"For some $\\varepsilon >0$ the functions $x\\mapsto v_1(x,\\, \\cdot \\,)$ and $y\\mapsto v_1(\\, \\cdot \\,,y)$ belong to $L^\\infty (\\Omega , L^{2+\\varepsilon }(\\Omega ,{\\mathbb {C}}))$ and $L^\\infty (\\Omega , L^{2+4/\\varepsilon }(\\Omega ,{\\mathbb {C}}))$ , respectively, i.e.",
"$\\operatornamewithlimits{ess\\,sup}_{x\\in \\Omega }\\Vert v_1(x,\\, \\cdot \\,)\\Vert _{L^{2+\\varepsilon }(\\Omega )}<\\infty , \\quad \\operatornamewithlimits{ess\\,sup}_{y\\in \\Omega }\\Vert v_1(\\, \\cdot \\,,y)\\Vert _{L^{2+4/\\varepsilon }(\\Omega )}<\\infty .$ Remark 2.1 (i) Under Assumption , it is easy to see that $H:{\\mathcal {H}}\\rightarrow {\\mathcal {H}}$ is an everywhere defined bounded self-adjoint operator.",
"(ii) Since $H_{00}$ , $H_{01}$ and $H_{10}$ are finite-rank operators and the essential spectrum as well as the finiteness of (parts of) the discrete spectrum of self-adjoint operators are invariant with respect to finite-rank perturbations (see e.g.",
"[4]), we can restrict ourselves to studying the spectrum of the $2\\times 2$ operator matrix ${\\mathcal {A}}:=\\begin{pmatrix}H_{11} & H_{12}\\\\[1.5ex]H^*_{12} & \\!H_{22}\\end{pmatrix}.$ acting in the Hilbert space ${\\mathcal {H}}_1\\oplus {\\mathcal {H}}_2$ .",
"(iii) Since ${\\rm {Vol}}\\,(\\Omega )<\\infty $ , Hölder's inequality together with the second condition in (REF ) yields that $v_1(x,\\, \\cdot \\,)\\in L^2(\\Omega , {\\mathbb {C}})$ for a.a. $x\\in \\Omega $ and $\\displaystyle \\operatornamewithlimits{ess\\,sup}_{x\\in \\Omega }\\Vert v_1(x,\\, \\cdot \\,)\\Vert _{L^2(\\Omega )} \\le (2a)^{\\frac{\\varepsilon d}{4+2\\varepsilon }}\\operatornamewithlimits{ess\\,sup}_{x\\in \\Omega }\\Vert v_1(x,\\, \\cdot \\,)\\Vert _{L^{2+\\varepsilon }(\\Omega )}<\\infty .$ It is easy to check that the adjoint operator $H^*_{12}:{\\mathcal {H}}_1\\rightarrow {\\mathcal {H}}_2$ is given by $(H^*_{12}f)(x,y) = \\frac{1}{2}v_1(x,y)^*f(x)+\\frac{1}{2}v_1(y,x)^*f(y), \\quad f\\in {\\mathcal {H}}_1,$ for a.a. $x,y\\in \\Omega $ .",
"Schur complements have proven to be useful tools when dealing with $2\\times 2$ operator matrices (see e.g.",
"[15]).",
"The first Schur complement associated with the operator matrix ${\\mathcal {A}}-z$ is given by $S(z) &= H_{11}-z-H_{12}(H_{22}-z)^{-1}H^*_{12} =: \\Delta (z)+K(z)$ for $z\\notin \\sigma (H_{22})={\\rm {cl}}\\,(\\operatornamewithlimits{ran}w_2)$ where $\\Delta (z):{\\mathcal {H}}_1\\rightarrow {\\mathcal {H}}_1$ is the multiplication operator by the function $\\Delta (\\, \\cdot \\,;z)$ defined as $\\Delta (x;z) := w_1(x)-z-\\frac{1}{2}\\int _{\\Omega }\\frac{|v_1(x,y)|^2}{w_2(x,y)-z} \\,{\\rm {d}}y, \\quad x\\in \\Omega ,$ and $K(z)\\!",
": {\\mathcal {H}}_1\\rightarrow {\\mathcal {H}}_1$ is the integral operator with kernel $K(\\, \\cdot \\,,\\, \\cdot \\,;z)$ given by $K(x,y;z) := -\\frac{1}{2}\\frac{v_1(x,y) v_1(y,x)^*}{w_2(x,y)-z}, \\quad (x,y)\\in \\Omega ^2.$ For every $z\\!\\in \\!",
"{\\mathbb {R}}\\setminus {\\rm {cl}}\\,(\\operatornamewithlimits{ran}w_2)$ , the Schur complement $S(z)$ is bounded and self-adjoint in ${\\mathcal {H}}_1$ , the function $\\Delta (\\, \\cdot \\,; z)$ is real-valued and $K(x,y;z)=K(y,x;z)^*$ , $x,y\\in \\Omega $ ; thus the operators $\\Delta (z)$ and $K(z)$ are self-adjoint, too.",
"Moreover, it follows from (REF ) that $\\operatornamewithlimits{ess\\,sup}_{x\\in \\Omega }\\Delta (x;z)<\\infty $ for every $z\\in {\\mathbb {R}}\\setminus {\\rm {cl}}\\,(\\operatornamewithlimits{ran}w_2)$ .",
"Therefore, the multiplication operator $\\Delta (z):{\\mathcal {H}}_1\\rightarrow {\\mathcal {H}}_1$ is bounded for every $z\\in {\\mathbb {R}}\\setminus {\\rm {cl}}\\,(\\operatornamewithlimits{ran}w_2)$ , and hence so is $K(z):{\\mathcal {H}}_1\\rightarrow {\\mathcal {H}}_1$ .",
"In fact, we have more than just the boundedness of the integral operator $K(z)$ as a corollary of the next lemma.",
"Lemma 2.2 Let Assumption be satisfied.",
"For every $z\\in {\\mathbb {R}}\\setminus {\\rm {cl}}\\,(\\operatornamewithlimits{ran}w_2)$ , the integral operator $K(z):{\\mathcal {H}}_1\\rightarrow {\\mathcal {H}}_1$ is Hilbert-Schmidt.",
"Let $z\\in {\\mathbb {R}}\\setminus {\\rm {cl}}\\,(\\operatornamewithlimits{ran}w_2)$ be fixed.",
"By Young's inequality, we have $\\displaystyle |v_1(x,y)|^2|v_1(y,x)|^2 \\le \\frac{2}{2+\\varepsilon } |v_1(x,y)|^{2+\\varepsilon } + \\frac{\\varepsilon }{2+\\varepsilon }|v_1(y,x)|^{2+4/\\varepsilon }$ for a.a. $x,y\\in \\Omega $ .",
"Therefore, $|K(x,y;z)|^2 \\le \\frac{2|v_1(x,y)|^{2+\\varepsilon } + \\varepsilon |v_1(y,x)|^{2+4/\\varepsilon }}{4(2+\\varepsilon )\\operatornamewithlimits{dist}(z,\\operatornamewithlimits{ran}w_2)^2}$ for a.a. $x,y\\in \\Omega $ .",
"On the other hand, in view of Assumption , it is easy to see that the following estimates hold $& \\displaystyle \\Vert v_1\\Vert _{L^{2+\\varepsilon }(\\Omega ^2)}^{2+\\varepsilon } \\le (2a)^d \\operatornamewithlimits{ess\\,sup}_{x\\in \\Omega }\\Vert v_1(x,\\, \\cdot \\,)\\Vert _{L^{2+\\varepsilon }(\\Omega )}^{2+\\varepsilon }<\\infty ,\\\\& \\displaystyle \\displaystyle \\Vert v_1\\Vert _{L^{2+4/\\varepsilon }(\\Omega ^2)}^{2+4/\\varepsilon } \\le (2a)^d \\operatornamewithlimits{ess\\,sup}_{y\\in \\Omega }\\Vert v_1(y,\\, \\cdot \\,)\\Vert _{L^{2+4/\\varepsilon }(\\Omega )}^{2+4/\\varepsilon }<\\infty .$ Hence $K(\\, \\cdot \\,,\\, \\cdot \\,;z)\\in L^2(\\Omega ^2)$ and thus $K(z)$ is Hilbert-Schmidt."
],
[
"Analytic description of the essential spectrum",
"The following theorem provides an explicit formula for the essential spectrum of $H$ in terms of the functions $w_2$ and $\\Delta $ given by (REF ) and (REF ).",
"Theorem 3.1 Let Assumption be satisfied and let $m:=\\!\\inf _{(x,y)\\in \\Omega ^2} \\!",
"w_2(x,y), \\quad M:=\\!\\sup _{(x,y)\\in \\Omega ^2} \\!",
"w_2(x,y).$ Then $\\sigma _{\\rm ess}(H)=\\Sigma _1\\cup \\Sigma _2$ where $\\Sigma _1 := {\\rm {cl}}\\,(\\operatornamewithlimits{ran}w_2)=[m,M], \\quad \\Sigma _2 := {\\rm {cl}}\\,\\lbrace z\\in {\\mathbb {R}}\\setminus \\Sigma _1: \\; 0\\in \\operatornamewithlimits{ess\\,ran}\\Delta (\\, \\cdot \\,;z)\\rbrace .$ Recall that, by Remark REF , $\\sigma _{\\rm ess}(H)=\\sigma _{\\rm ess}({\\mathcal {A}})$ and thus it suffices to establish that $\\sigma _{\\rm ess}({\\mathcal {A}})=\\Sigma _1\\cup \\Sigma _2$ .",
"First we show $\\Sigma _1 \\subset \\sigma _{\\rm ess}({\\mathcal {A}})$ .",
"Since the essential spectrum is closed, we only have to prove the inclusion $\\lbrace w_2(x,y) : x,y\\in \\Omega \\rbrace \\subset \\sigma _{\\rm ess}({\\mathcal {A}}).$ To this end, let $z_0 \\in \\lbrace w_2(x,y) : x,y\\in \\Omega \\rbrace $ be arbitrary.",
"Since $w_2:\\Omega ^2\\rightarrow {\\mathbb {R}}$ is continuous on $\\Omega ^2$ by Assumption , it follows that $z_0=w_2(x_0,y_0)$ for some $(x_0, y_0) \\in \\Omega ^2$ .",
"Let $\\chi $ be the normalized characteristic function of the annulus $\\bigl \\lbrace x\\!\\in \\!",
"\\Omega : \\!\\frac{1}{2} \\le \\Vert x\\Vert \\le 1\\bigr \\rbrace $ and define the sequences $\\lbrace \\varphi _n\\rbrace _{n \\in {\\mathbb {N}}}$ , $\\lbrace \\phi _n\\rbrace _{n \\in {\\mathbb {N}}} \\subset {\\mathcal {H}}_1=L^2(\\Omega ,{\\mathbb {C}})$ by $\\varphi _n(x) := 2^{\\frac{nd}{2}} \\chi (2^n(x-x_0)), \\quad \\phi _n(y) := 2^{\\frac{nd}{2}} \\chi (2^n(y-y_0)), \\quad x,y \\in \\Omega ;$ note that $\\varphi _n=\\phi _n$ if $x_0=y_0$ .",
"It is easy to check that $\\operatorname{supp}(\\varphi _n) \\cap \\operatorname{supp}(\\varphi _m) = \\operatorname{supp}(\\phi _n) \\cap \\operatorname{supp}(\\phi _m)=\\emptyset $ for all $n,m\\in {\\mathbb {N}}$ with $n\\ne m$ and that there is $N_0\\in {\\mathbb {N}}$ such that $\\Vert \\varphi _n\\Vert _{L^2(\\Omega )}=\\Vert \\phi _n\\Vert _{L^2(\\Omega )}=1, \\quad \\operatorname{supp}(\\varphi _n) \\cap \\operatorname{supp}(\\phi _k)= \\emptyset $ for all positive integers $n,k\\ge N_0$ .",
"So both $\\lbrace \\varphi _n\\rbrace ^\\infty _{n=N_0}$ and $\\lbrace \\phi _n\\rbrace ^\\infty _{n=N_0}$ are orthonormal systems in ${\\mathcal {H}}_1$ .",
"Now consider the sequence $\\lbrace \\psi _n\\rbrace ^\\infty _{n=N_0}$ defined by $\\psi _n(x,y)={\\left\\lbrace \\begin{array}{ll}\\varphi _n(x) \\phi _n(y) = \\varphi _n(x) \\varphi _n(y) &\\mbox{if } x_0=y_0,\\\\\\frac{1}{\\sqrt{2}}\\bigl (\\varphi _n(x)\\phi _n(y)+\\varphi _n(y)\\phi _n(x)\\bigr ) &\\mbox{if } x_0\\ne y_0,\\end{array}\\right.",
"}$ for $x,y\\in \\Omega $ .",
"It is easy to see that the sequence $\\lbrace \\psi _n\\rbrace ^\\infty _{n=N_0}$ is an orthonormal system in ${\\mathcal {H}}_2=L^2_{\\rm {sym}}(\\Omega ,{\\mathbb {C}})$ .",
"Hence the sequence $\\lbrace \\widetilde{\\psi }_n\\rbrace ^\\infty _{n=N_0}$ given by $\\widetilde{\\psi }_n(x,y)= \\binom{0}{\\psi _n(x,y)}, \\quad x,y \\in \\Omega ,$ is an orthonormal system in ${\\mathcal {H}}_1\\oplus {\\mathcal {H}}_2$ .",
"Thus, if we show $\\Vert ({\\mathcal {A}}-z_0) \\widetilde{\\psi }_n\\Vert _{{\\mathcal {H}}} \\rightarrow 0$ as $n\\rightarrow \\infty $ , it follows that $\\lbrace \\widetilde{\\psi }_n\\rbrace ^\\infty _{n=N_0}$ is a singular sequence for ${\\mathcal {A}}-z_0$ and thus $z_0\\in \\sigma _{\\rm ess}({\\mathcal {A}})$ , see [13].",
"Note that $\\Vert ({\\mathcal {A}}-z_0) \\widetilde{\\psi }_n\\Vert ^2_{{\\mathcal {H}}}= \\Vert H_{12} \\psi _n\\Vert ^2_{L^2(\\Omega )}+ \\Vert (H_{22}-z_0) \\psi _n\\Vert ^2_{L^2(\\Omega ^2)}.$ By construction of the sequence $\\lbrace \\psi _n\\rbrace _{n \\in {\\mathbb {N}}}$ , it easily follows that $\\Vert (H_{22}-z_0) \\psi _n\\Vert _{L^2(\\Omega ^2)} = \\Vert (w_2-w_2(x_0,y_0))\\psi _n\\Vert _{L^2(\\Omega ^2)} \\rightarrow 0, \\quad n \\rightarrow \\infty ,$ so it is left to be shown that $\\Vert H_{12} \\psi _n\\Vert _{L^2(\\Omega )} \\rightarrow 0, \\quad n \\rightarrow \\infty .$ By Assumption , there are constants $C>0$ and $\\varepsilon >0$ such that $\\Vert v_1(x, \\, \\cdot \\,) \\Vert _{L^{2+\\varepsilon }(\\Omega )} \\le C$ for a.a. $x \\in \\Omega $ .",
"Applying Hölder's inequality with $p=2+\\varepsilon $ and $q= \\frac{p}{p-1}$ , we thus obtain $\\begin{split}\\left|\\int _{\\Omega } v_1(x,y) \\varphi _n(y) \\,{\\rm {d}}y\\right| &\\le \\Vert v_1(x, \\, \\cdot \\,)\\Vert _{L^p(\\Omega )} \\Vert \\varphi _n\\Vert _{L^q(\\Omega )}\\le C 2^{n d(\\frac{1}{2}-\\frac{1}{q})}\\end{split}$ for a.a. $x \\in \\Omega $ .",
"In the same way it follows that $\\begin{split}\\left|\\int _{\\Omega } v_1(x,y) \\phi _n(y) \\,{\\rm {d}}y\\right| \\le C 2^{n d(\\frac{1}{2}-\\frac{1}{q})}\\end{split}$ for a.a. $x \\in \\Omega $ .",
"Therefore, using $\\Vert \\varphi _n\\Vert _{L^2(\\Omega )}=\\Vert \\phi _n\\Vert _{L^2(\\Omega )}=1$ and applying the triangle inequality, we easily obtain $\\Vert H_{12} \\psi _n\\Vert _{L^2(\\Omega )} \\le C 2^{n d(\\frac{1}{2}-\\frac{1}{q})+1}, \\quad n\\ge N_0.$ This proves (REF ) because $q= 2-\\frac{\\varepsilon }{1+\\varepsilon }<2$ .",
"Now it remains to be shown that $({\\mathbb {R}}\\setminus \\Sigma _1) \\cap \\sigma _{\\rm ess}({\\mathcal {A}}) = \\Sigma _2$ .",
"To this end, let $z \\in {\\mathbb {R}}\\backslash \\Sigma _1$ be arbitrary.",
"It is not difficult to check that [15] applies and yields $z \\in \\sigma _{\\rm ess}({\\mathcal {A}}) \\quad \\Longleftrightarrow \\quad 0 \\in \\sigma _{\\rm ess}(S(z)).$ Since $K(z)$ is compact, we have $\\sigma _{\\rm ess}(S(z)) \\!=\\!",
"\\sigma _{\\rm ess}(\\Delta (z))\\!=\\!\\operatornamewithlimits{ess\\,ran}(\\Delta (\\, \\cdot \\,;z))$ .",
"Therefore, by (REF ), $z \\in \\sigma _{\\rm ess}({\\mathcal {A}}) \\quad \\Longleftrightarrow \\quad 0\\in \\operatornamewithlimits{ess\\,ran}\\Delta (\\, \\cdot \\,;z) \\quad \\Longleftrightarrow \\quad z \\in \\Sigma _2.$ Remark 3.2 While it is always the case that $\\Sigma _1\\ne \\emptyset $ , the following example shows that $\\Sigma _2=\\emptyset $ may occur.",
"Let $d\\in {\\mathbb {N}}$ be arbitrary and let $\\Omega =(-a,a)^d$ with $a=2^{(1-d)/d}$ so that ${\\rm {vol}}\\,(\\Omega )=2$ .",
"Let $w_2$ be an arbitrary function satisfying Assumption and denote its continuous extension to ${\\rm {cl}}\\,(\\Omega ^2)$ also by $w_2$ .",
"If $m,M$ are defined as in (REF ) and we choose the parameter functions $w_1$ and $v_1$ as $& v_1(x,y) = (w_2(x,y)-m)^{1/2}(M-w_2(x,y))^{1/2}, \\quad x,y\\in {\\rm {cl}}\\,(\\Omega ), \\\\& w_1(x) = m+M-\\frac{1}{2}\\int _\\Omega w_2(x,y) \\,{\\rm {d}}y, \\quad x\\in {\\rm {cl}}\\,(\\Omega ),$ then, clearly, Assumption is satisfied and $\\Delta (\\, \\cdot \\,;m) \\equiv \\Delta (\\, \\cdot \\,;M) \\equiv 0$ on ${\\rm {cl}}\\,(\\Omega )$ .",
"On the other hand, it is easy to see that the function $z\\mapsto \\Delta (x;z)$ is strictly decreasing on $(-\\infty ,m) \\cup (M,\\infty )$ for each fixed $x\\in {\\rm {cl}}\\,(\\Omega )$ .",
"Therefore, for each $z<m$ , we have $\\Delta (x;z)>\\Delta (x;m)=0$ for all $x\\in {\\rm {cl}}\\,(\\Omega )$ and for each $z>M$ , we have $\\Delta (x;z)<\\Delta (x;M)=0$ for all $x\\in {\\rm {cl}}\\,(\\Omega )$ .",
"Consequently, $\\Sigma _2=\\emptyset $ ."
],
[
"Birman-Schwinger type principle",
"For a bounded self-adjoint operator $A:{\\mathcal {H}}\\rightarrow {\\mathcal {H}}$ and a constant ${\\lambda }\\in {\\mathbb {R}}$ , we define the quantity $n({\\lambda }; A) := \\sup _{\\mathcal {L} \\subset {\\mathcal {H}}} \\lbrace \\dim \\mathcal {L} : (Au,u) >{\\lambda }\\Vert u\\Vert _{{\\mathcal {H}}}^2, u \\in \\mathcal {L}\\rbrace = \\dim \\mathcal {L}_{({\\lambda },\\infty )}(A),$ where $\\mathcal {L}_{({\\lambda },\\infty )}(A)$ is the spectral subspace of $A$ corresponding to the interval $({\\lambda },\\infty )$ .",
"Note that, if $n({\\lambda }; A)$ is finite, then it is equal to the number of the eigenvalues of $A$ larger than ${\\lambda }$ (counted with multiplicities), see e.g., [4].",
"For ${\\lambda }\\le \\min \\sigma _{\\rm ess}(A)$ , we denote by $N({\\lambda };A)$ the number of eigenvalues of $A$ that are less than ${\\lambda }$ ; observe that, for $ z < \\min \\sigma _{\\rm ess}(A)$ , $N(z; A)= n(-z; -A).$ In the sequel, we will use the so-called Weyl inequality (see e.g.",
"[4]) $n({\\lambda }_1+{\\lambda }_2; V_1+V_2) \\le n({\\lambda }_1;V_1)+n({\\lambda }_2;V_2)$ for compact self-adjoint operators $V_1$ , $V_2:{\\mathcal {H}}\\rightarrow {\\mathcal {H}}$ , and real numbers ${\\lambda }_1,{\\lambda }_2$ .",
"The following lemma plays a crucial role in the analysis of the discrete spectrum.",
"Lemma 4.1 Let Assumption be satisfied.",
"For every $z<\\min \\sigma _{\\rm ess}({\\mathcal {A}})$ , $N(z;{\\mathcal {A}}) = N(0; S(z)).$ Let $ z < \\min \\sigma _{\\rm ess}({\\mathcal {A}})$ be fixed.",
"Then $z<\\min \\sigma (H_{22})$ by Theorem REF .",
"In the Hilbert space ${\\mathcal {H}}_1\\oplus {\\mathcal {H}}_2$ , we consider the operators $W(z):={\\rm {diag}}(S(z), H_{22}-z), \\quad V(z):=\\begin{pmatrix}I & \\:-H_{12}(H_{22}-z)^{-1}\\\\[1ex]0 & I\\end{pmatrix}.$ Clearly, $V(z):{\\mathcal {H}}\\rightarrow {\\mathcal {H}}$ is bijective and $W(z)=V(z)({\\mathcal {A}}-z)V(z)^*$ due to the Frobenius-Schur factorization, see e.g.",
"[3].",
"Therefore, $N(z;{\\mathcal {A}}) &= N(0;{\\mathcal {A}}-z)\\\\&=N(0;V(z)({\\mathcal {A}}-z)V(z)^*)\\\\&=N(0;W(z)).$ On the other hand, $N(0;W(z))=N(0;S(z))+N(0;H_{22}-z)=N(0;S(z)).$ Lemma 4.2 Let Assumption be satisfied.",
"For every $z < \\min \\sigma _{\\rm ess}({\\mathcal {A}})$ , we have $\\displaystyle \\operatornamewithlimits{ess\\,inf}_{x \\in \\Omega } \\Delta (x;z) > 0.$ Suppose, to the contrary, that there exists $z^*<\\min \\sigma _{\\rm ess}({\\mathcal {A}})$ such that $\\operatornamewithlimits{ess\\,inf}_{x \\in \\Omega } \\Delta (x;z^*) \\le 0.$ Then we must have $\\operatornamewithlimits{ess\\,inf}_{x \\in \\Omega } \\Delta (x;z^*) < 0,$ for otherwise we would have $z^*\\in \\sigma _{\\rm ess}({\\mathcal {A}})$ contradicting $z^*<\\min \\sigma _{\\rm ess}({\\mathcal {A}})$ .",
"By (REF ) and Assumption , there exists a sequence $\\lbrace x_n\\rbrace _{n\\in {\\mathbb {N}}}\\subset \\Omega $ satisfying the conditions $\\Delta (x_n;z^*)<0, \\quad v_1(x_n,\\, \\cdot \\,)\\in L^{2+\\varepsilon }(\\Omega ), \\quad n\\in {\\mathbb {N}}.$ Consider the sequence of functions $z\\mapsto \\Delta (x_n;z)$ , $n\\in {\\mathbb {N}}$ , on $(-\\infty ,z^*]$ .",
"For every fixed $n\\in {\\mathbb {N}}$ , it is easy to see that $\\lim _{z\\rightarrow -\\infty }\\Delta (x_n;z)=+\\infty ,$ and $z\\mapsto \\Delta (x_n;z)$ is a continuous, strictly decreasing function on $(-\\infty , z^*]$ as $\\frac{\\partial }{\\partial z}\\Delta (x_n;z)= -1-\\frac{1}{2}\\int _{\\Omega }\\frac{|v_1(x_n,y)|^2}{(w_2(x_n,y)-z)^2} \\,{\\rm {d}}y\\le -1.$ Hence, in view of the first condition in (REF ), the mean-value theorem implies that there exists a sequence $\\lbrace z_n\\rbrace _{n\\in {\\mathbb {N}}}\\subset (-\\infty ,z^*)$ with $\\Delta (x_n;z_n)=0$ for each $n\\in {\\mathbb {N}}$ .",
"On the other hand, (REF ) implies that the sequence $\\lbrace z_n\\rbrace _{n\\in {\\mathbb {N}}}$ is bounded from below, too.",
"Therefore, by Bolzano-Weierstrass' theorem, there is a subsequence $\\lbrace z_{n_k}\\rbrace _{k\\in {\\mathbb {N}}}$ converging to some $z_0\\in (-\\infty , z^*]$ .",
"If we write, $\\Delta (x_{n_k};z_0) &= \\Delta (x_{n_k};z_0) - \\Delta (x_{n_k};z_{n_k}) \\\\[2ex]&= (z_{n_k}-z_0)\\Bigg (1+\\frac{1}{2}\\int _{\\Omega }\\frac{|v_1(x_{n_k},y)|^2}{(w_2(x_{n_k},y)-z_0)(w_2(x_{n_k},y)-z_{n_k})} \\,{\\rm {d}}y\\Bigg ),$ it follows from the second relation in (REF ) (see also Remark REF (iii)) that the integral in the bracket is finite and thus $\\Delta (x_{n_k};z_0)\\rightarrow 0$ , $k\\rightarrow \\infty $ .",
"Therefore, Theorem REF shows that $z_0\\in \\sigma _{\\rm ess}({\\mathcal {A}})$ , contradicting $z_0<\\min \\sigma _{\\rm ess}({\\mathcal {A}})$ .",
"It follows from Lemma REF that the function $x \\mapsto \\Delta (x;z)^{-1/2}, \\quad x \\in \\Omega ,$ is well-defined and bounded for every $ z < \\min \\sigma _{\\rm ess}({\\mathcal {A}})$ .",
"Let $T(z)$ be the integral operator with kernel $T(x,y;z) := - \\Delta (x;z)^{-1/2} K(x,y;z) \\Delta (y;z)^{-1/2}, \\quad (x,y)\\in \\Omega ^2,$ where $K(\\, \\cdot \\,, \\, \\cdot \\,; z)$ is defined as in (REF ).",
"Proposition 4.3 Let Assumption be satisfied and let $ z < \\min \\sigma _{\\rm ess}({\\mathcal {A}})$ be arbitrary.",
"Then $T(z) : {\\mathcal {H}}_1 \\rightarrow {\\mathcal {H}}_1$ is Hilbert-Schmidt and $N(z; {\\mathcal {A}})= n(1; T(z)).$ Let $ z < \\min \\sigma _{\\rm ess}({\\mathcal {A}})$ be fixed.",
"Lemma REF implies that $\\Delta (z)^{-1/2}:{\\mathcal {H}}_1\\rightarrow {\\mathcal {H}}_1$ is well-defined and positive operator.",
"Since $K(z):{\\mathcal {H}}_1\\rightarrow {\\mathcal {H}}_1$ is Hilbert-Schmidt by Lemma REF , and $\\Delta (z)^{-1/2}:{\\mathcal {H}}_1\\rightarrow {\\mathcal {H}}_1$ is bounded, it follows that the operator $T(z)= - \\Delta (z)^{-1/2} K(z) \\Delta (z)^{-1/2}$ is Hilbert-Schmidt, too.",
"Recalling that $S(z)= \\Delta (z)+ K(z)$ , we have $\\begin{split}\\Delta (z)^{-1/2} S(z) \\Delta (z)^{-1/2} &=I + \\Delta (z)^{-1/2} K(z) \\Delta (z)^{-1/2} = I- T(z).\\end{split}$ Therefore, $n(1, T(z)) &= N(-1; -T(z)) = N(0; I- T(z)) \\\\&= N\\bigl (0; \\Delta (z)^{-1/2} S(z) \\Delta (z)^{-1/2}\\bigr )\\\\&= N(0; S(z)).$ Applying Lemma REF , we thus obtain $N(z;{\\mathcal {A}}) = N(0; S(z)) = n(1, T(z)).",
"$"
],
[
"Criterion for the finiteness of the discrete spectrum below\nthe bottom of the essential spectrum",
"For $\\delta >0$ , denote by $B_{\\delta }(0)$ the ball of radius $\\delta $ with centre at the origin in ${\\mathbb {R}}^d$ .",
"For $s\\ge 0$ , we define functions $\\Phi _s:\\Omega ^2\\rightarrow {\\mathbb {R}}$ , $\\Phi _{s}(x,y) ={\\left\\lbrace \\begin{array}{ll}\\Vert x\\Vert ^s+\\Vert y\\Vert ^s & {\\rm {if}} \\quad x\\in B_{\\delta }(0)\\times B_{\\delta }(0), \\\\\\hspace{19.91692pt} 1 & {\\rm {otherwise}}.\\end{array}\\right.",
"}$ (B)Assumption (B) There exist constants $\\alpha \\ge 0$ , $\\beta \\in {\\mathbb {R}}$ , $C_1,C_2>0$ , $\\delta \\!\\in \\!",
"(0,a)$ and a unique point $(t_0,t_0)\\in \\Omega ^2$ such that, for a.a. $x,y\\in \\Omega $ , $&w_2(x,y)-\\min \\sigma _{\\rm ess}(H) \\ge C_1 \\Phi _{\\alpha }(x-t_0, y-t_0), \\\\&\\chi _{B_{\\delta }(t_0)}(y)\\, |v_1(x,y)|\\le C_2 \\Phi _\\beta (0,y-t_0).$ We denote by $\\alpha ^*$ and $\\beta ^*$ , respectively, the infimum and the supremum of the values of $\\alpha $ and $\\beta $ satisfying Assumption .",
"Remark 5.1 If $\\min \\sigma _{\\rm ess}(H)\\notin \\Sigma _1$ , then $\\alpha ^*=0$ and the function $\\Delta (\\, \\cdot \\,; \\min \\sigma _{\\rm ess}(H)):\\Omega \\rightarrow [0,\\infty )$ is well-defined.",
"If $\\min \\sigma _{\\rm ess}(H)\\in \\Sigma _1$ and Assumption is satisfied, then the function (REF ) is well-defined provided that $\\alpha ^*<2\\beta ^*+d$ .",
"(C)Assumption (C) There exist constants $\\gamma \\ge 0$ and $C_3>0$ such that for $\\delta $ as in Assumption , whenever $\\alpha ^* < 2\\beta ^*+d$ , then for a.a. $x\\in \\Omega $ , $\\Delta (x; \\min \\sigma _{\\rm ess}(H))\\ge C_3\\Phi _{\\gamma }(x-t_0,0).$ We denote by $\\gamma ^*$ the infimum of the values of $\\gamma $ satisfying Assumption .",
"Theorem 5.2 Let Assumptions , and be satisfied and let $\\alpha ^*+\\gamma ^* < 2\\beta ^*+d.$ Then the operator $H$ has a finite number of eigenvalues (counted with multiplicities) below $\\min \\sigma _{\\rm ess}(H)$ .",
"Throughout the proof we adopt the notation $E_{\\min }:=\\min \\sigma _{\\rm ess}({\\mathcal {A}})$ .",
"Recall that, in view of Remark REF , $\\min \\sigma _{\\rm ess}(H)=\\min \\sigma _{\\rm ess}({\\mathcal {A}})$ and $N(E_{\\min };H)<\\infty \\quad \\Longleftrightarrow \\quad N(E_{\\min };{\\mathcal {A}})<\\infty .$ By Proposition REF , $T(z)$ is Hilbert-Schmidt for all $z<\\min \\sigma _{\\rm ess}({\\mathcal {A}})$ .",
"Next, we show that $T(E_{\\min })$ is Hilbert-Schmidt as well.",
"It follows from Assumptions , and that the kernel of $T(E_{\\min })$ is square-integrable if the function $(x,y) \\,\\mapsto \\, \\Vert x\\Vert ^{\\beta ^*-{\\gamma ^*}/2} \\Vert y\\Vert ^{\\beta ^*-{\\gamma ^*}/2}\\frac{1}{\\Vert x\\Vert ^{\\alpha ^*}+\\Vert y\\Vert ^{\\alpha ^*}}$ is square-integrable over $B_{\\delta }(0) \\times B_{\\delta }(0)$ .",
"Passing to generalized polar coordinates, it is easy to see that the latter is equivalent to $\\displaystyle \\int _0^\\delta \\int _0^\\delta \\frac{1}{(r_1^{\\alpha ^*}+r_2^{\\alpha ^*})^2} \\, r_1^{2\\beta ^*-\\gamma ^*+d-1} r_2^{2\\beta ^*-\\gamma ^*+d-1} \\,{\\rm {d}}r_1 \\,{\\rm {d}}r_2 <\\infty .$ Using the elementary inequality between the arithmetic and geometric means, $\\frac{r_1^{\\alpha ^*}+r_2^{\\alpha ^*}}{2} \\ge \\sqrt{r_1^{\\alpha ^*} r_2^{\\alpha ^*}} ,$ it is not difficult to check that (REF ) holds if $\\int _0^\\delta t^{(2\\beta ^*+d)-(\\alpha ^*+\\gamma ^*)-1} \\,{\\rm {d}}t < \\infty ,$ which, in turn, holds if and only if $\\alpha ^*+\\gamma ^* < 2\\beta ^*+d$ .",
"Therefore, $T(E_{\\min })$ is Hilbert-Schmidt if (REF ) is satisfied.",
"Summing up, $T(z)$ is Hilbert-Schmidt for every $z\\le E_{\\min }$ .",
"Moreover, it is an immediate consequence of Lebesgue's dominated convergence theorem that the map $T(\\, \\cdot \\,): (-\\infty , E_{\\min }] \\rightarrow L({\\mathcal {H}}_1)$ is continuous.",
"Next, let $z\\le E_{\\min }$ be arbitrary.",
"Since $T(z)$ is compact, we obviously have $n(1/2; T(z))<\\infty .$ Using the Weyl inequality (REF ) for the compact self-adjoint operators $V_1=T(E_{\\min }), \\quad V_2=T(z)-T(E_{\\min }),$ and ${\\lambda }_1={\\lambda }_2=1/2$ , we obtain $n(1;T(z)) \\le n(1/2; T(E_{\\min })) + n(1/2; T(z)-T(E_{\\min })).$ Since $T(\\, \\cdot \\,): (-\\infty , E_{\\min }] \\rightarrow L({\\mathcal {H}}_1)$ is continuous, we thus have $\\lim _{z\\nearrow E_{\\min }} \\!\\!",
"n(1;T(z)) & \\le n(1/2; T(E_{\\min })) + \\lim _{z\\nearrow E_{\\min }} \\!\\!",
"n(1/2; T(z)-T(E_{\\min }))\\\\&= n(1/2; T(E_{\\min })).$ This together with Proposition REF yields $N(E_{\\min };{\\mathcal {A}}) = \\lim _{z\\nearrow E_{\\min }} \\!\\!",
"N(z;{\\mathcal {A}}) \\le n(1/2; T(E_{\\min }))<\\infty .",
"$ Remark 5.3 Whenever $\\min \\sigma _{\\rm ess}(H)\\notin \\Sigma _2$ , condition (REF ) is sharp in the sense that, if $\\alpha ^*+\\gamma ^* \\ge 2\\beta ^*+d$ , then the operator $H$ may have an infinite number of eigenvalues below $\\min \\sigma _{\\rm ess}(H)$ .",
"This occurs, for example, for $d=1$ , $\\Omega =[-\\pi ,\\pi ]$ , and $\\begin{array}{l}w_1(x)=1+\\sin ^2(x), \\quad v_1(x,y)=\\sqrt{\\dfrac{3}{\\pi }} \\sin (x), \\\\w_2(x,y)=\\varepsilon (x)+2\\varepsilon (x+y)+\\varepsilon (y),\\end{array}\\quad x,y\\in [-\\pi ,\\pi ],$ where $\\varepsilon (x):=1-\\cos (x)$ , $x\\in [-\\pi ,\\pi ]$ , see [11] for more details."
],
[
"Concluding remarks",
"We conclude the paper with some remarks on modifications of our assumptions and results, and on the case $a=\\infty $ .",
"6.1.",
"The uniqueness of the point $(x_0,y_0)\\in \\Omega $ in Assumption was assumed just for simplicity.",
"Theorem REF can be generalized if we assume that there exist finitely many points $(t_j,t_j)$ , $j=0,1,\\ldots ,N$ , and constants $\\alpha _j\\ge 0$ , $\\beta _j\\in {\\mathbb {R}}$ , $\\gamma _j\\ge 0$ , $j=0,\\ldots ,N$ , $C_1, C_2, C_3>0$ and $\\delta >0$ with $\\delta < \\min _{\\begin{array}{c}k\\ne l\\\\0\\le k,l\\le N\\end{array}} \\frac{1}{2} \\operatornamewithlimits{dist}((t_k,t_k),(t_l,t_l))$ such that for a.a. $x,y\\in \\Omega $ and each $j=0,1,\\ldots ,N$ , $w_2(x,y)-\\min \\sigma _{\\rm ess}(H) \\ge C_1 \\prod _{j=0}^N \\Phi _{\\alpha _j}(x-t_j, y-t_j)$ , $\\chi _{B_{\\delta }(t_j)}(y) |v_1(x,y)|\\le C_2\\Phi _{\\beta _j}(0,y-t_j)$ , $\\Delta (x; \\min \\sigma _{\\rm ess}(H))\\ge C_3\\Phi _{\\gamma _j}(x-t_j,0)$ with $\\Phi _s$ as in (REF ).",
"Defining the constants $\\alpha _j^*$ , $\\beta _j^*$ and $\\gamma _j^*$ in an analogous way and replacing condition (REF ) by $\\alpha _j^*+\\gamma _j^*<2\\beta _j^*+d, \\quad j=0,1,\\ldots ,N,$ the same analysis as above in a sufficiently small neighborhood of every point $(t_j,t_j)$ shows that the number of eigenvalues below $\\min \\sigma _{\\rm ess}(H)$ remains finite in this case.",
"Note that this is no longer true in general if the number of such points is infinite, see [11] for an example in the smooth setting.",
"6.2.",
"We mention that in the previous studies, e.g.",
"in [11], it was always assumed that the parameter function $w_2$ has a unique non-degenerate global minimum, which implies that $\\alpha ^*=2$ in our Assumption .",
"While uniqueness in [11] was assumed just for simplicity, our analysis shows that the non-degeneracy of the global minimum in [11] is not always needed to guarantee the finiteness of the discrete spectrum below $\\min \\sigma _{\\rm ess}(H)$ , at least if $\\min \\sigma _{\\rm ess}(H)\\notin \\Sigma _2$ .",
"6.3.",
"Under assumptions analogous to Assumptions , and with the same method, one immediately obtains an analogue of Theorem REF guaranteeing the finiteness of discrete eigenvalues above $\\max \\sigma _{\\rm ess}(H)$ .",
"6.4.",
"Motivated by the application to spin-boson Hamiltonians on the torus in ${\\mathbb {R}}^d$ (which was studied e.g.",
"in [11]), we focused on the case $a<\\infty $ throughout the paper.",
"However, our methods are of local nature and thus readily apply to the case of $\\Omega ={\\mathbb {R}}^d$ where $a=\\infty $ .",
"By requiring $v_1$ to have a compact support in ${\\mathbb {R}}^2$ and the conditions (REF ) of Assumption to hold on the support of $v_1$ , and assuming the rest of the hypotheses in Assumption as well as in Assumptions , for $\\Omega ={\\mathbb {R}}^d$ , we obtain the same conclusions of Theorems REF and REF ."
]
] | 1612.05459 |
[
[
"Studies of three-particle correlations and reaction-plane correlators\n from STAR"
],
[
"Abstract We present STAR measurements of various harmonics of three-particle correlations in $\\sqrt{s_{NN}}=200$ GeV Au+Au collisions at RHIC.",
"The quantity $\\langle\\cos(m\\phi_1+n\\phi_2-(m+n)\\phi_3)\\rangle$ is measured for inclusive charged particles for different harmonics $m$ and $n$ as a function of collision centrality, transverse momentum $p_T$ and relative pseudorapidity $\\Delta\\eta$.",
"These observables provide detailed information on global event properties like correlations between event planes of different harmonics and are particularly sensitive to the expansion dynamics of the matter produced in the collisions.",
"We compare our measurements to different viscous hydrodynamic models.",
"We argue that these measurements probe the three-dimensional structure of the initial state and provide unique ways to constrain the transport parameters involved in hydrodynamic modeling of heavy-ion collisions."
],
[
"Introduction",
"By now it has been established that relativistic heavy-ion collisions produce a strongly correlated Quark Gluon Plasma (sQGP).",
"Recent experimental studies have focused on studying the properties of such sQGP.",
"One of the striking properties of such a phase of matter is that it exhibits nearly perfect fluidity characterized by the smallest viscosity-to-entropy-density ratio $\\eta /s$ amongst all known fluids in the nature.",
"Over the past years, combined insights from both experiment through measurements of anisotropic flow coefficients $v_n$ and from theory through viscous hydrodynamic simulations, have made precise extraction of $\\eta /s$ possible.",
"However a very intrinsic characteristic about such transport parameter, i.e.",
"its temperature dependence is not yet fully constrained by experimental measurements.",
"A primary goal of the current measurements at RHIC is therefore to go beyond conventional measurements of flow coefficients and provide new observables that can be useful to constrain $\\eta /s (T)$ .",
"In this work we present the measurements of three particle correlations from the STAR experiment using the observable $C_{m,n,m+n} = \\langle \\langle \\cos (m\\phi _1 + n \\phi _2 - (m+n) \\phi _3)\\rangle \\rangle $ , where $m,n$ defines the harmonic coefficients and $\\phi _{1,2,3}$ , the azimuthal angles of three particles [1].",
"The inner and the outer averages are taken over all triplets and events respectively.",
"The observable $C_{m,n,m+n}$ can be approximated as correlations of flow harmonics, $v_n$ s, and corresponding event plane angles, $\\Psi _n$ s, as ${\\langle v_m v_n v_{m+n} \\cos (m \\Psi _m + n \\Psi _n - (m+n) \\Psi _{m+n}) \\rangle }$ .",
"Theoretical studies show that such an observable can probe non-linear hydrodynamic response and therefore become more sensitive to viscosity than individual flow harmonics $v_n$ [2], [3], [4], [5], [1], [6], [7], [8], [9], [10].",
"Better sensitivity to viscous effects can be very useful towards more precise extraction of different transport parameters and possibly their temperature dependence by comparison to hydrodynamic simulations [11], [12].",
"Measurements of event plane and flow harmonic correlations have been performed at LHC by ATLAS collaboration [13] and recently by ALICE collaboration[14].",
"However measurements at a single energy is not sufficient to constrain $\\eta /s~(T)$ .",
"LHC measurements are sensitive to the $\\eta /s$ at higher temperatures, meanwhile full constraint on $\\eta /s~(T)$ can only be achieved with complementary measurements of $C_{m,n,m+n}$ at RHIC [15], [16], [11], [12].",
"In fact, measurements over the entire range of energy available under the Beam Energy Scan (BES) program at RHIC will be most preferable in this context.",
"Measurements at RHIC have additional advantages.",
"Since the beam rapidity is smaller one expects stronger variation of initial geometry, fluctuations, energy density, temperature, baryon density etc.",
"over a relatively smaller window of rapidity as compared to LHC.",
"In this context, measurements of $C_{m,n,m+n}$ on the pseudorapidity separation between particles may allow us to study the breaking of longitudinal invariance, three dimension structure of the initial state [3], [17], [18], [19], [20] over relatively smaller widow of acceptance available for measurements at RHIC than LHC.",
"In addition to constraining initial state and transport parameters, the charge dependence of three particle correlation can be used to search for the signals of the chiral magnetic effect (CME) [21], [22], [23], [24].",
"In this work we will not study such charge dependence, however, we expect that the results presented here for inclusive charged particles will provide important baseline for the CME measurements."
],
[
"Experiment and analysis",
"We analyze the data on Au+Au collisions at $\\sqrt{s_{NN}}=200$ GeV collected by the STAR detector [25] during 2011 year running of RHIC.",
"For the measurements of $C_{m,n,m+n}$ we use charged particles within the pseudorapidity range of $|\\eta |\\!<\\!1$ and transverse momentum of $p_T\\!>\\!0.2$ GeV/$c$ detected by the Time Projection Chamber (TPC), the primary tracking systems of STAR situated inside a 0.5 Tesla solenoidal magnetic field [26].",
"We use algebra based on Q-vectors and in order to account for imperfections in the detector acceptance we apply track-by-track weights [27], [28].",
"We also apply momentum dependent tracking efficiency.",
"In such estimation, we correct for the track-merging artifacts by measuring the relative pseudo rapidity separation between any two tracks and correcting for missing pairs apparent at $\\Delta \\eta \\approx 0$ .",
"We estimate systematic uncertainties in our measurements by analyzing datasets of different time periods, from different years, with different tracking algorithms, with different efficiency estimates, by varying z-vertex position of the collision, and by varying track selection criteria.",
"In addition we also quantify the effects of short-range quantum and Coulomb correlations in the systematic uncertainties by studying $\\Delta \\eta $ dependence of $C_{m,n,m+n}$ .",
"Finally for data-model comparison we estimate the number of participant nucleons $N_{\\rm part}$ using a Monte-Carlo Glauber model for different centrality intervals ($0-5\\%, 5-10\\%, 10-20\\%, ..., 70-80\\%$ ) used in this analysis [29], [30].",
"For selection of such centrality bins we use the distribution of minimum bias uncorrected multiplicity of charged particles in the pseudorapidity region $|\\eta |<0.5$ measured by the TPC."
],
[
"Results and discussion",
"In this conference proceedings we present results for the correlators $C_{1,1,2}, C_{1,2,3}, C_{2,2,4}, C_{2,3,5}$ .",
"We first present differential measurements such as $\\Delta \\eta $ and $p_T$ dependence of these correlators, the goal of such study is to understand how different physical scenarios effect these observables.",
"We later on present integrated measurements i.e.",
"the centrality dependence of $C_{m,n,m+n}$ and make comparisons to viscous hydrodynamic model calculations with different assumptions of $\\eta /s (T)$ ."
],
[
"$\\Delta \\eta $ dependence",
" In Fig.",
"REF we show the $\\Delta \\eta $ dependence of $C_{1,2,3}$ and $C_{2,2,4}$ .",
"A similar measurement for $C_{1,1,2}$ was previously presented by STAR in Ref [22] The $\\Delta \\eta $ dependence for all other harmonics of $C_{m,n,m+n}$ will be presented in a future publication.. One can clearly see in Fig.",
"REF (left) that $C_{1,2,3}$ correlator shows a very strong dependence on $\\Delta \\eta _{1,3}$ (i.e.",
"between the first and third order harmonics) but a very weak dependence on $\\Delta \\eta _{1,2}$ (i.e.",
"between first and second order harmonics).",
"We omit the curve for the variation of $C_{1,2,3}$ with $\\Delta \\eta _{2,3}$ for clarity which looks very similar to the curve shown for $\\Delta \\eta _{1,2}$ .",
"A strong dependence of $C_{1,1,2}$ correlator for $\\Delta \\eta _{1,2}$ (i.e.",
"between two first order harmonics) was also observed in the previous STAR measurement in Ref [22].",
"In contrast, the similar measurement shown in Fig.REF (right) for the correlator $C_{2,2,4}$ with two possible combinations of $\\Delta \\eta $ shows a much weaker dependence compared to its absolute magnitude.",
"These observations indicate a very specific pattern for three particle correlations.",
"The relative rapidity dependence between either “first-first” or “first-third” harmonics show strong variations and even change of sign, whereas between second and any other harmonics the correlations show much weaker variation in relative rapidity.",
"Variations of $C_{m,n,m+n}$ with $\\Delta \\eta $ can come from hydrodynamic response to the three-dimensional structure of initial state [3], [17], [18], [19], [20].",
"They can also arise from artifacts such as short-range correlations, non-flow and resonance decays [21], etc., that give rise to two-particle correlations that are correlated to an event plane (determined by the third particle) and do not vanish after averaging over many events.",
"However, if such a variation persist up to large $\\Delta \\eta $ , e.g.",
"as shown in Fig.REF for $C_{1,2,3}$ vs $\\Delta \\eta _{1,3}$ , they can not be driven by short range correlations.",
"In a flow scenario, strong variation in $\\Delta \\eta $ can come from de-correlation in initial state geometry, e.g.",
"driven by a breaking of longitudinal invariance through a forward-backward rapidity dependence of harmonic planes particularly between $\\Psi _1$ and $\\Psi _3$ [3].",
"In case of $\\Psi _2$ , one do not expect strong variation with rapidity due to geometry of collisions.",
"In a non-flow scenario, in case of $C_{1,2,3}$ one possible source of $\\Delta \\eta _{1,3}$ dependence could be momentum conservation that leads to back-to-back correlations between two particles from jets that are correlated to second order event plane.",
"We discuss such scenario in the next section."
],
[
"$p_T$ dependence",
"The effect of momenta conservation is expected to be dominant at higher transverse momentum and for low multiplicity events.",
"Therefore, measurements performed in peripheral events can be a good baseline for such studies.",
"In the central events due to large number of particles, quenching of jet-like correlations etc., the effect of momentum conservation will not be dominant.",
"It is therefore essential to perform this exercise in both central (e.g.",
"$0\\!-\\!5\\%$ ) and peripheral (e.g.",
"$70\\!-\\!80\\%$ ) and contrast the trend seen in data.",
"From $\\Delta \\eta $ dependence of $C_{m,n,m+n}$ , as discussed in previous section, we find that the correlators involving first order harmonics can be sensitive to non-flow effects such as momentum conservations from jets etc.",
"In Fig.",
"REF we therefore study the variations of the correlators $C_{1,1,2}$ and $C_{1,2,3}$ with the transverse momentum $p_T$ of the particle corresponding to the first order harmonic, i.e.",
"for the first particle as denoted by $p_{T,1}$ .",
"In order to remove trivial increase of first order harmonic $v_1$ with transverse momentum and trivial dilution of correlation while going from peripheral to central events we multiply the correlator by a factor of $N_{\\rm part}^2/p_{T,1}$ .",
"The results for $70\\!-\\!80\\%$ indicates that at high $p_{T,1}$ both $C_{1,1,2}$ and $C_{1,2,3}$ becomes negative.",
"Such trend is consistent with a picture of momentum conservation and can be understood as follows.",
"If a pair of back-to-back particles gets aligned along $\\Psi _2$ , they will lead to negative values for these correlators since then we have $C_{1,1,2}\\approx \\cos (\\pi )$ and $C_{1,2,3}\\approx \\cos (\\pm 3\\pi )$ .",
"This might explain the decreasing trend for $70\\!-\\!80\\%$ events.",
"However such a scenario can not explain the trend seen $0-5\\%$ events where one finds negative signal at small $p_{T,1}$ and nearly zero or positive signal at large $p_{T,1}$ .",
"This qualitatively different trend seen in central events can not be due to non-flow correlations from back-to-back pairs.",
"Clearly the differential measurements of these correlators can provide better insights of the relative contributions of different sources of correlations that can affect $C_{m,n,m+n}$ .",
"Model calculations that include full treatment of three-dimensional initial geometry, fluctuations and different other sources of correlations can improve our understanding in this context [31].",
"Figure: Transverse momentum dependence of the three particle correlator C 1,1,2 C_{1,1,2} and C 1,2,3 C_{1,2,3}."
],
[
"Centrality dependence",
"We measure the centrality dependence of $C_{m,n,m+n}$ and compare our results with three different viscous hydrodynamic model calculations.",
"They include 1) hydrodynamic simulations by Teaney and Yan [3], [5], 2) the perturbative QCD$+$ saturation$+$ hydro based “EKRT\" model [11] and 3) hydrodynamic simulations MUSIC [32] with IP-Glasma initial conditions [33].",
"In addition we also estimate the correlations from initial state geometry using Monte Carlo Glauber model by approximating $C_{m,n,m+n}={\\langle \\varepsilon _m \\varepsilon _n \\varepsilon _{m+n} \\cos (m \\Phi _m + n \\Phi _n - (m+n) \\Phi _{m+n}) \\rangle }$ , where $\\varepsilon _n$ s and $\\Phi _n$ s are the initial eccentricities and the participant planes respectively.",
"All of these models have been previously constrained by the measurements of $v_n$ and other data on azimuthal correlations from RHIC and LHC, but they do not include longitudinal dependence in the initial state and assume boost invariance.",
"From Fig.REF it is evident that the correlator $C_{2,2,4}$ has the least variation on $\\Delta \\eta $ and will provide the best opportunity for comparison to boost-invariant hydrodynamic simulations.",
"We therefore present the centrality dependence of the correlator $C_{2,2,4}$ in Fig.REF (left).",
"In Fig.REF (right) we also compare the centrality dependence of $C_{2,3,5}$ .",
"In order to scale out the trivial dilution of correlations due to increase of number of pairs while going from peripheral to central events we have multiplied the correlators by $N_{\\rm part}^2$ .",
"Figure: Centrality dependence of three particle correlator C 1,1,2 C_{1,1,2} and C 2,2,4 C_{2,2,4} compared to different viscous hydrodynamic model calculations.The Glauber model calculations predict that purely initial state correlation of eccentricities and participant planes leads to negative values for all the correlators.",
"Both $C_{2,2,4}$ and $C_{2,3,5}$ being positive in data indicates the dominance of non-linear hydrodynamic response of the medium to initial state geometry.",
"This observation is consistent to the measurement at LHC by the ATLAS collaboration in Ref [13].",
"We however, find that although the qualitative trends predicted by different viscous hydrodynamic simulations are similar to data, some quantitative differences exist.",
"Particularly for $C_{2,2,4}$ one can see that the current precision of the data can very well differentiate between constant and temperature dependent viscosity used in the EKRT simulations.",
"Such comparisons would be key to constrain $\\eta /s (T)$ .",
"Apart from analysis at $\\sqrt{s_{NN}}=200$ GeV, our future studies will be focused on measurements for other lower energies under RHIC Beam Energy Scan program which will provide better constraints of $\\eta /s (T)$ ."
],
[
"Summary",
"In summary, we have presented the first measurements of three-particle correlations $C_{m,n,m+n}= \\langle \\langle \\cos (m\\phi _1 + n \\phi _2 - (m+n) \\phi _3)\\rangle \\rangle $ in $\\sqrt{s_{NN}}=200$ GeV Au+Au collisions at RHIC.",
"In comparison to conventional flow harmonic measurements these correlators can provide additional information such as de-correlation of event planes driven by three dimensional structure of the initial state and non-linear hydrodynamic response of the medium.",
"When compared to viscous hydrodynamic models these measurements with the precision presented here have the potential to constrain transport parameters and their temperature dependence.",
"This work was supported under Department of Energy Contract No.",
"DE-SC0012704.",
"We thank Li Yan, Risto Paatelainen, Harri Niemi and Gabriel Denicol for providing their model predictions and helpful discussion."
]
] | 1612.05593 |
[
[
"Optimal Differentially Private Mechanisms for Randomised Response"
],
[
"Abstract We examine a generalised Randomised Response (RR) technique in the context of differential privacy and examine the optimality of such mechanisms.",
"Strict and relaxed differential privacy are considered for binary outputs.",
"By examining the error of a statistical estimator, we present closed solutions for the optimal mechanism(s) in both cases.",
"The optimal mechanism is also given for the specific case of the original RR technique as introduced by Warner in 1965."
],
[
"Background",
"Stanley L. Warner first proposed the Randomised Response (RR) technique as a means to eliminate bias in surveying in 1965 [31].",
"Respondents would be handed a spinner by the surveyor to decide which of two questions the respondent would answer, for example, Have you ever cheated on your spouse/partner?",
"Have you always been faithful to your spouse/partner?",
"Respondents would spin the spinner in private and answer the given question truthfully with a `yes' or `no'.",
"Respondents would be afforded plausible deniability as the surveyor would not know the question to which the answer refers.",
"This would encourage respondents to engage with the survey and answer the question truthfully.",
"The spinner can be replaced by any appropriate randomisation device, such as coin flips, dice or drawing from a pack of cards.",
"A rich body of literature now exists on RR.",
"The inefficiencies of Warner's original RR model have been examined by a number of authors and many new RR models have been proposed.",
"These include the unrelated question model [13], the forced response model [2], Moor's procedure [26] and two-stage RR models [25], [24].",
"More comprehensive lists of RR models can be found in [21], [1].",
"RR is actively used in surveying when asking questions of a sensitive nature.",
"Examples include surveys on doping and drug use in elite athletes [27], cognitive-enhancing drug use among university students [5], faking on a CV [6], corruption [11], sexual behaviour [3], and child molestation [8].",
"Researchers remain divided on the effectiveness of RR.",
"While some works have shown RR to be an improvement on different survey techniques, including direct questioning (where no randomisation is involved), [29], [12], [22], [20], [28], others remain sceptical on its advantage [32], [33], [23].",
"Public trust in RR has also been shown to be lacking [4].",
"Separately, differential privacy has emerged as a model of interest in privacy-preserving data publishing since being presented in 2006 [7].",
"Differential privacy gives a quantitative mathematical definition to measure the level of privacy achieved in a given data release.",
"This definition determines the amount of manipulation that needs to be applied to the data to achieve the desired level of privacy.",
"Under differential privacy, privacy is quantified by how statistically indistinguishable the privacy-preserved outputs from two similar datasets are.",
"When applied to randomised response, where the output from a single individual is binary, differential privacy requires the output from any two individuals to be statistically indistinguishable, to a specified degree."
],
[
"Our Results",
"In this paper we examine a generalisation of Warner's original RR technique, and establish conditions under which such a model satisfies differential privacy.",
"By calculating the estimator of minimal variance, we determine the optimal differentially private RR mechanism.",
"We examine strict $\\epsilon $ -differential privacy and relaxed ($\\epsilon $ ,$\\delta $ )-differential privacy.",
"Complete solutions for the optimal mechanisms are presented for both cases.",
"The optimal mechanism is also given for Warner's RR model satisfying ($\\epsilon $ , $\\delta $ )-differential privacy."
],
[
"Related Work",
"The application of differential privacy to randomised response has been limited to date.",
"[30] examined using randomised response to differentially privately collect data, although their analysis only considered strict $\\epsilon $ -differential privacy and a comparison of its efficiency with respect to the Laplace mechanism, a mechanism popular in the differential privacy literature.",
"Randomised response has been used in conjunction with differential privacy in a more general context in the form of local privacy, also known as input perturbation.",
"For example, extreme mechanisms for local differential privacy have been studied in [18], [16], while differential privacy was applied to social network data in the form of graphs with randomised response in [19].",
"Outside randomised response and local privacy, optimal mechanisms in differential privacy have received some attention, including work on strict differential privacy [10] and relaxed differential privacy [9]."
],
[
"Structure of Paper",
"We begin in Section with an introduction to the Randomised Response (RR) technique, and derive the statistical estimator and associated bias and error; we also present Warner's original RR model.",
"We introduce differential privacy in Section and present a number of preliminary results for later use in Section .",
"The main results are given in Sections , and , relating to strict differential privacy, relaxed differential privacy and Warner's model respectively.",
"Concluding remarks are given in Section ."
],
[
"Introduction",
"We are looking to determine the proportion $\\pi $ of people in the population possessing a particular sensitive attribute, where possession of the attribute is binary.",
"We conduct a survey on $n$ individuals of the population by uniform random sampling with replacement.",
"A single respondent's answer $X_i\\in \\lbrace 0,1\\rbrace $ is a randomised version of their truthful answer $x_i\\in \\lbrace 0,1\\rbrace $ , in order to protect their privacy.",
"The randomised response will therefore not definitively reveal a respondent's truthful answer.",
"By convention, a value of 1 denotes possession of the sensitive attribute, while 0 denotes that the respondent does not possess the attribute.",
"We denote by $N$ the number of randomised responses that return 1, hence $N=\\sum _{i \\in [n]} X_i$ where $[n]=[1,n] \\cap \\mathbb {Z}$ .",
"We are therefore looking to estimate $\\pi $ from $\\frac{N}{n}$ ."
],
[
"Generalised RR Model",
"In keeping with standard notation, $(\\Omega , \\mathcal {F}, \\mathbb {P})$ denotes a probability space.",
"$X_i: \\Omega \\rightarrow \\lbrace 0,1\\rbrace $ is then a random variable for each $i \\in [n]$ , dependent on the truthful value $x_i$ .",
"We define the randomised response mechanism by $\\mathbb {P}(X_i=k \\mid x_i = j)=p_{jk},$ which leads us to defining the design matrix of the mechanism as follows.",
"Definition 1 (Design Matrix) A randomised response mechanism as defined in (REF ) is uniquely determined by its design matrix, $P=\\left(\\begin{array}{cc} p_{00} & p_{01} \\\\ p_{10} & p_{11}\\end{array}\\right).$ For the probability mass functions of each $X_i$ to sum to 1, we require $p_{00} + p_{01}=1$ and $p_{10} + p_{11} = 1$ .",
"The design matrix therefore simplifies to $P = \\left(\\begin{array}{cc} p_{00} & 1-p_{00} \\\\ 1-p_{11} & p_{11}\\end{array}\\right),$ where $p_{00},p_{11} \\in [0,1]$ .",
"As $\\pi $ is the true proportion of individuals in the population possessing the sensitive attribute, we can calculate the probability mass function of each $X_i$ : $\\begin{aligned}\\mathbb {P}(X_i = 0) &= (1-\\pi ) p_{00} + \\pi (1-p_{11})\\\\&= p_{00} - \\pi (p_{00} + p_{11} - 1),\\end{aligned}\\\\[1em]\\begin{aligned}\\mathbb {P}(X_i = 1) &= \\pi p_{11} + (1-\\pi )(1-p_{00})\\\\&= 1-p_{00} + \\pi (p_{00} + p_{11} - 1).\\end{aligned}$ Remark: Direct questioning corresponds to the case where $p_{00}=p_{11}=1$ ."
],
[
"Estimator, Bias and Error",
"Having presented the RR mechanism previously, we now need to establish an estimator of $\\pi $ from the parameters of the mechanism, $p_{00}$ and $p_{11}$ , and from the distribution of randomised responses, namely $\\frac{N}{n}$ .",
"We first establish a maximum likelihood estimator (MLE) for the mechanism and then examine its bias and error.",
"Theorem 1 Let $p_{00} + p_{11} \\ne 1$ .",
"Then the MLE for $\\pi $ of the randomised response mechanism given by (REF ) is $\\hat{\\Pi }(p_{00},p_{11}) = \\frac{p_{00}-1}{p_{00} + p_{11} - 1} + \\frac{N}{(p_{00} + p_{11} - 1)n}.$ Let us first index the sample so that $X_i=1$ for each $i\\le N$ , and $X_i=0$ for each $i>N$ .",
"Then the likelihood $L$ of the sample is $L = \\mathbb {P}(X_i=1)^N \\mathbb {P}(X_i=0)^{n-N}.$ The log-likelihood is $\\log (L) = N \\log \\mathbb {P}(X_i=1) + (n-N) \\log \\mathbb {P}(X_i=0),$ whose derivatives are $\\frac{\\partial \\log (L)}{\\partial \\pi } &= \\frac{N}{\\mathbb {P}(X_i=1)}\\frac{\\partial \\mathbb {P}(X_i=1)}{\\partial \\pi }+\\frac{n-N}{\\mathbb {P}(X_i=0)}\\frac{\\partial \\mathbb {P}(X_i=0)}{\\partial \\pi },\\\\\\frac{\\partial ^2 \\log (L)}{\\partial \\pi ^2} &= -\\frac{N}{\\mathbb {P}(X_i=1)^2}\\left(\\frac{\\partial \\mathbb {P}(X_i=1)}{\\partial \\pi }\\right)^2-\\frac{n-N}{\\mathbb {P}(X_i=0)^2}\\left(\\frac{\\partial \\mathbb {P}(X_i=0)}{\\partial \\pi }\\right)^2.$ We note that $\\frac{\\partial ^2 \\log (L)}{\\partial \\pi ^2}<0$ , hence the maximum of $\\log (L)$ occurs when $\\frac{\\partial \\log (L)}{\\partial \\pi }=0$ .",
"Solving for $\\pi $ completes the proof.",
"We note the following standard identity in probability and statistics, $\\operatorname{Var}(Y) = \\mathbb {E}[Y^2] - \\mathbb {E}[Y]^2,$ for any random variable $Y$ .",
"We now calculate the bias and error of $\\hat{\\Pi }$ .",
"We use the variance of the estimator to characterise error in line with conventional practice.",
"Similarly by convention, we characterise the bias of an estimator as its expected deviation from the quantity it is estimating (i. e. $\\mathbb {E}[\\hat{\\Pi }-\\pi ]$ ).",
"We remind the reader of the dependence of $\\operatorname{Var}(\\hat{\\pi })$ on $\\pi $ by writing $\\operatorname{Var}(\\hat{\\Pi }| \\pi )$ .",
"Corollary 1 The MLE $\\hat{\\Pi }$ constructed in Theorem REF is unbiased and has error $\\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})| \\pi ) = \\frac{\\frac{1}{4} - \\left(p_{00}-\\frac{1}{2}-\\pi (p_{00} + p_{11} - 1)\\right)^2}{(p_{00}+p_{11}-1)^2 n}.$ Since the survey we are conducting is by uniform random sampling with replacement, $N$ is a sum of independent and identically distributed random variables.",
"Therefore, $\\mathbb {E}[N] = n \\mathbb {E}[X_i]$ and $\\operatorname{Var}(N) = n \\operatorname{Var}(X_i)$ .",
"Since $X_i \\in \\lbrace 0,1\\rbrace $ , it can be shown that $\\mathbb {E}[X_i] = \\mathbb {E}[X_i^2] = \\mathbb {P}(X_i=1) = 1-p_{00} + \\pi (p_{00} + p_{11}-1)$ .",
"Hence, $\\mathbb {E}[\\hat{\\Pi }] &= \\frac{p_{00}-1}{p_{00} + p_{11} - 1} + \\frac{\\mathbb {E}[N]}{(p_{00} + p_{11} - 1)n}\\\\&= \\frac{p_{00}-1}{p_{00} + p_{11} - 1} + \\frac{\\mathbb {E}[X_i]}{p_{00} + p_{11} - 1}\\\\&= \\pi ,$ and so $\\hat{\\Pi }$ is unbiased as claimed.",
"Secondly, $\\operatorname{Var}(\\hat{\\Pi }| \\pi ) &= \\frac{\\operatorname{Var}(N)}{(p_{00}+p_{11}-1)^2 n^2}\\\\&= \\frac{\\operatorname{Var}(X_i)}{(p_{00}+p_{11}-1)^2 n}\\\\&= \\frac{\\mathbb {E}[X_i^2]-\\mathbb {E}[X_i]^2}{(p_{00}+p_{11}-1)^2 n}\\\\&= \\frac{\\mathbb {P}(X_i=1)\\mathbb {P}(X_i=0)}{(p_{00}+p_{11}-1)^2 n},$ which can be simplified to (REF ).",
"When conducting a survey on a population, it is often useful and necessary to estimate the margin of error of the estimate on a sample.",
"For a confidence level $c \\in [0,1]$ , the margin of error of a sample is given by $\\omega \\ge 0$ , where $\\mathbb {P}(|\\hat{\\Pi }-\\pi | \\le \\omega ) \\ge c.$ In practical applications, a 95% confidence interval is typically used [17].",
"In the absence of any additional information on the distribution of $\\hat{\\Pi }$ , Chebyshev's inequality can be used to derive a general, but conservative, margin of error, assuming $\\hat{\\Pi }$ has finite variance.",
"In such a scenario, the margin of error of a sample is given to be $4.5\\sigma $ , where the standard deviation $\\sigma $ is given by $\\sqrt{\\operatorname{Var}(\\hat{\\Pi }|\\pi )}$ , since $\\mathbb {P}\\left(|\\hat{\\Pi }-\\pi | \\le 4.5\\sqrt{\\operatorname{Var}(\\hat{\\Pi }|\\pi )}\\right) \\ge 0.95.$ In many practical situations, the central limit theorem is invoked to determine heuristically a margin of error.",
"For a random variable $G$ that is normally distributed with mean $\\mu $ and variance $\\sigma ^2$ , we have $\\mathbb {P}(|G-\\mu | \\le 1.96 \\sigma ) \\ge 0.95,$ hence $1.96 \\sigma $ is typically taken as the margin of error in such scenarios [17].",
"However, this non-rigorous approach only gives a loose representation of the margin of error, given that the guarantee of the central limit theorem only applies in the limit as the sample size $n$ approaches infinity.",
"Due to this variability in defining the margin of error of a sample, we only focus on determining the error of the estimator, $\\operatorname{Var}(\\hat{\\Pi }|\\pi )$ , in this paper.",
"This error can be used to calculate the margin of error for a particular application, as outlined above."
],
[
"Warner's RR model",
"Warner's model [31] is a specific case of the generalised model introduced in Section REF .",
"Warner proposed that surveyors would present respondents with a spinner which they would spin in private to decide which one of two questions to answer.",
"The spinner would point to a question (e. g. “Have you ever cheated on your spouse/partner?”) with probability $p_w$ , and to the complement of that question (e. g. “Have you always been faithful to your spouse/partner?”) with probability $1-p_w$ .",
"Respondents would then be asked to answer the chosen question truthfully, but without revealing which question they were answering.",
"As before, $x_i$ denotes the truthful response of respondent $i$ , while $X_i$ denotes the randomised response, as determined by the process outlined above.",
"Warner's model corresponds to the case where $p_{00} = p_{11} = p_w$ .",
"We denote by $P_w$ the design matrix of Warner's model, which is given by $P_w = \\left(\\begin{array}{cc} p_w & 1-p_w \\\\ 1-p_w & p_w\\end{array}\\right),$ while the probability mass function of each $X_i$ is defined as $\\mathbb {P}(X_i = 0) &= p_w - \\pi (2p_w - 1),\\\\\\mathbb {P}(X_i = 1) &= 1-p_w + \\pi (2p_w - 1).$ Using the same unbiased MLE in (REF ), we denote by $\\hat{\\Pi }_w$ the estimator for Warner's model and, by (REF ), find its error to be $\\operatorname{Var}(\\hat{\\Pi }_w(p_w)| \\pi ) = \\frac{\\frac{1}{4} - \\left(p_w-\\frac{1}{2}-\\pi (2p_w - 1)\\right)^2}{(2p_w-1)^2 n}.$"
],
[
"Differential Privacy",
"Differential privacy was first proposed by Dwork in 2006 [7] as a way to measure the level of privacy achieved when publishing data.",
"Using the same notation as in [14], we denote by $D^m$ the space of all $m$ -row datasets (let $D$ be the space of each row) and by $\\mathbf {d}\\in D^m$ a dataset in this space.",
"We then denote by $X_\\mathbf {d}: \\Omega \\rightarrow D^n$ a randomised version of $\\mathbf {d}$ .",
"If $D$ is assumed to be discrete, the mechanism $X_\\mathbf {d}$ is said to satisfy ($\\epsilon $ ,$\\delta $ )-differential privacy if $\\mathbb {P}(X_\\mathbf {d}\\in A) \\le e^\\epsilon \\mathbb {P}(X_{\\mathbf {d}^\\prime } \\in A) + \\delta ,$ for each $\\mathbf {d}, \\mathbf {d}^\\prime \\in D^m$ that differ in exactly one row (i. e. there exists exactly one $j \\in [m]$ such that $d_j \\ne d_j^\\prime $ ) and for each subset $A \\subset D^m$ .",
"This set-up simplifies in the case of randomised response introduced in Section .",
"Firstly, the datasets contain only one row ($m=1$ ), and the row-space is $\\lbrace 0,1\\rbrace $ .",
"We are therefore only required to show that (REF ) holds for $\\mathbf {d}\\ne \\mathbf {d}^\\prime \\in \\lbrace 0,1\\rbrace $ and for $A=\\lbrace 0\\rbrace , \\lbrace 1\\rbrace $ .",
"Formally, ($\\epsilon $ ,$\\delta $ )-differential privacy is satisfied if $\\mathbb {P}(X_i = j) \\le e^\\epsilon \\mathbb {P}(X_k = j) + \\delta ,$ for any $i, k \\in [n]$ and $j \\in \\lbrace 0,1\\rbrace $ .",
"For the RR mechanism given by (REF ) to satisfy ($\\epsilon $ , $\\delta $ )-differential privacy, we require the following to hold: $p_{11} &\\le e^\\epsilon (1-p_{00}) + \\delta ,\\\\p_{00} &\\le e^\\epsilon (1-p_{11}) + \\delta ,\\\\1-p_{00} &\\le e^\\epsilon p_{11} + \\delta ,\\nonumber \\\\1-p_{11} &\\le e^\\epsilon p_{00} + \\delta .\\nonumber $ We can now define the set of pairs $(p_{00}, p_{11})$ that correspond to a RR mechanism which satisfies ($\\epsilon $ , $\\delta $ )-differential privacy.",
"Definition 2 (Region of Feasibility) A RR mechanism, given by (REF ), satisfies ($\\epsilon $ , $\\delta $ )-differential privacy if $(p_{00},p_{11})\\in \\mathcal {R}$ , where $\\mathcal {R} \\subset \\mathbb {R}^2$ is defined as $\\mathcal {R}=\\left\\lbrace (p_{00},p_{11})\\in \\mathbb {R}^2:\\begin{array}{l} p_{00},p_{11} \\in [0,1],\\\\p_{00}\\le e^\\epsilon (1-p_{11})+\\delta ,\\\\p_{11}\\le e^\\epsilon (1-p_{00})+\\delta ,\\\\1-p_{11} \\le e^\\epsilon p_{00}+\\delta ,\\\\1-p_{00} \\le e^\\epsilon p_{11}+\\delta .\\end{array}\\right\\rbrace .$ We consider the case where $p_{00}+p_{11} > 1$ .",
"Note that the estimator error, and hence the optimal mechanism, is undefined when $p_{00}+p_{11} = 1$ .",
"If $p_{00} + p_{11} < 1$ , we permute all responses such that $X^\\prime _i = 1-X_i$ .",
"This corresponds to the columns of the design matrix being swapped, giving $p^\\prime _{00}=1-p_{00}$ and $p^\\prime _{11}=1-p_{11}$ , hence $p^\\prime _{00} + p^\\prime _{11}=2-p_{00}-p_{11}>1$ .",
"We can therefore assume $p_{00} + p_{11} > 1$ without loss of generality.",
"When $p_{00} + p_{11} > 1$ , we note that (i) $1-p_{11} < p_{00} \\le e^\\epsilon (1-p_{11})+\\delta < e^\\epsilon p_{00}+\\delta $ and (ii) $1-p_{00} < p_{11} \\le e^\\epsilon (1-p_{00})+\\delta < e^\\epsilon p_{11}+\\delta $ .",
"Hence, the region of feasibility simplifies to $\\mathcal {R}^\\prime $ as follows: $\\mathcal {R}^\\prime &= \\left\\lbrace (p_{00},p_{11})\\in \\mathcal {R} : p_{00} + p_{11} > 1 \\right\\rbrace \\\\&= \\left\\lbrace (p_{00},p_{11})\\in \\mathbb {R}:\\begin{array}{l} p_{00}, p_{11} \\le 1, \\\\p_{00} + p_{11} > 1 ,\\\\p_{00}\\le e^\\epsilon (1-p_{11})+\\delta ,\\\\p_{11}\\le e^\\epsilon (1-p_{00})+\\delta .\\end{array}\\right\\rbrace .$ Furthermore, we denote by $\\mathcal {R}^{\\prime \\prime }$ the boundary of $\\mathcal {R}^\\prime $ which satisfies at least one of inequalities (): $\\mathcal {R}^{\\prime \\prime }= \\mathcal {R}^\\prime \\setminus \\left\\lbrace (p_{00}, p_{11})\\in \\mathbb {R}: \\begin{array}{l} p_{00} < e^\\epsilon (1-p_{11})+\\delta ,\\\\p_{11} < e^\\epsilon (1-p_{00})+\\delta .\\end{array}\\right\\rbrace .$ The set $\\mathcal {R}^{\\prime \\prime }$ therefore consists of the union of two line segments in the unit square, where (REF ) and () are tight.",
"We are therefore looking to find the RR mechanism which minimises estimator error, while still being ($\\epsilon $ , $\\delta $ )-differentially private.",
"Hence, we seek to find $\\operatornamewithlimits{arg\\,min}_{(p_{00},p_{11})\\in \\mathcal {R}^\\prime } \\operatorname{Var}\\left(\\left.\\hat{\\Pi }(p_{00},p_{11})\\right| \\pi \\right).$"
],
[
"Preliminary Results",
"We begin by presenting two results which will be of use later in the paper.",
"The first result concerns the non-negativity of a non-linear function on the unit square.",
"Lemma 1 Let $f:\\mathbb {R}\\times \\mathbb {R}\\rightarrow \\mathbb {R}$ be defined by $f(x, y)=2xy-x-y+1.$ Then, $f(x,y)\\ge 0$ for all $x, y \\in [0,1]$ .",
"Furthermore, $\\operatornamewithlimits{arg\\,min}_{x, y \\in [0,1]} f(x,y) = \\lbrace (0, 1), (1, 0)\\rbrace .$ Let's first consider $\\min _{x\\in [0,1]} f(x,y)$ : $\\min _{x \\in [0,1]} f(x,y) &= \\min _{x \\in [0,1]} (2xy-x)-y+1\\nonumber \\\\&= \\min _{x \\in [0,1]} \\left((2y-1)x\\right)-y+1\\nonumber \\\\&={\\left\\lbrace \\begin{array}{ll} y & \\text{if } y \\le \\frac{1}{2},\\\\1-y & \\text{if } y > \\frac{1}{2}.\\end{array}\\right.",
"}$ It follows that $\\min _{y \\in [0,1]} \\left(\\min _{x \\in [0,1]} f(x,y)\\right) = 0.$ By symmetry of $f$ , it also follows that $\\min _{x \\in [0,1]} \\left(\\min _{y \\in [0,1]} f(x,y)\\right) = 0,$ hence $f(x,y) \\ge 0$ for all $x, y \\in [0,1]$ .",
"We note that $f(1,0)=f(0,1)=0$ , and by (REF ) we see that these values uniquely minimise $f(x,y)$ for all $x,y \\in [0,1]$ .",
"In the second result of this section we prove that an optimal mechanism exists on $\\mathcal {R}^{\\prime \\prime }$ (i. e. on the boundary of $\\mathcal {R}^\\prime $ where at least one of inequalities () is tight), and additionally that when $\\pi \\in (0,1)$ , optimal mechanisms only occur on $\\mathcal {R}^{\\prime \\prime }$ .",
"Lemma 2 Let $p_{00} + p_{11} > 1$ .",
"Then there exists $(p_{00}^*, p_{11}^*) \\in \\operatornamewithlimits{arg\\,min}_{\\mathcal {R}^\\prime } \\operatorname{Var}(\\hat{\\Pi }|\\pi )$ such that $(p_{00}^*, p_{11}^*) \\in \\mathcal {R}^{\\prime \\prime }$ .",
"Furthermore, when $0 < \\pi < 1$ , $\\operatornamewithlimits{arg\\,min}_{\\mathcal {R}^\\prime } \\operatorname{Var}(\\hat{\\Pi }|\\pi ) \\subseteq \\mathcal {R}^{\\prime \\prime }$ .",
"Let's consider $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{00}}$ and $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{11}}$ .",
"Firstly, after some rearranging/manipulation, $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{11}} = -\\frac{2p_{00}(1-p_{00})(1-\\pi )+\\pi (2p_{00}p_{11}-p_{00}-p_{11}+1)}{(p_{00}+p_{11}-1)^3n}.$ By Lemma REF , we know that $2p_{00}p_{11}-p_{00}-p_{11}+1 \\ge 0$ , and since $p_{00}+p_{11} - 1 > 0$ by hypothesis, we conclude that $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{11}} \\le 0$ .",
"We further note that $2p_{00}p_{11}-p_{00}-p_{11}+1 > 0$ by Lemma REF , since the assumption that $p_{00}+p_{11} > 1$ means $p_{00},p_{11} > 0$ .",
"Hence $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{11}} = 0$ only when $\\pi = 0$ and $p_{00} = 1$ .",
"Equivalently, $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{11}} < 0 \\text{ when } \\pi > 0 \\text{ or } p_{00} < 1.$ Secondly, after some rearranging/manipulation, $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{00}} =-\\frac{(2p_{00}p_{11}-p_{00}-p_{11}+1)(1-\\pi )+2p_{11}\\pi (1-p_{11})}{(p_{00}+p_{11}-1)^3 n}.$ Since, by assumption, we have $2p_{00}p_{11}-p_{00}-p_{11}+1 \\ge 0$ and since $p_{11} \\in [0,1]$ , we see that $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{00}} \\le 0$ .",
"Similar to the reasoning above, since $2p_{00}p_{11}-p_{00}-p_{11}+1 > 0$ and $p_{11} > 0$ , $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{00}} = 0$ only when $\\pi = 1$ and $p_{11} = 1$ .",
"Equivalently, $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{00}} < 0 \\text{ when } \\pi < 1 \\text{ or } p_{11} < 1.$ Since $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{00}} \\le 0$ and $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{11}} \\le 0$ , there exists a mechanism on the boundary of $\\mathcal {R}^\\prime $ which minimises the estimator error, i. e. $\\partial \\mathcal {R}^\\prime \\cap \\left(\\operatornamewithlimits{arg\\,min}_{(p_{00},p_{11})\\in \\mathcal {R}^\\prime } \\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})| \\pi )\\right) \\ne \\emptyset .$ However, if $0 < \\pi < 1$ , we see from (REF ) and (REF ) that $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{00}} < 0$ and $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{11}} < 0$ .",
"Hence, $\\operatornamewithlimits{arg\\,min}_{(p_{00},p_{11})\\in \\mathcal {R}^\\prime } \\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})| \\pi ) \\subseteq \\partial \\mathcal {R}^\\prime ,$ i. e. the optimal mechanisms only occur on the boundary of $\\mathcal {R}^\\prime $ .",
"Finally, suppose $(p_{00},p_{11}) \\in \\partial \\mathcal {R}^\\prime $ , but neither of the inequalities in () are tight.",
"Then there exist $\\Delta _0,\\Delta _1 \\ge 0$ , $\\Delta _0+\\Delta _1>0$ where $(p_{00}+\\Delta _0,p_{11}+\\Delta _1)\\in \\partial \\mathcal {R}^\\prime $ , but because $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{00}} \\le 0$ and $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }| \\pi )}{\\partial p_{11}} \\le 0$ , then $\\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})| \\pi ) \\ge \\operatorname{Var}(\\hat{\\Pi }(p_{00}+\\Delta _0,p_{11}+\\Delta _1)| \\pi )$ .",
"Hence minimal error is achieved when at least one of the inequalities () is tight, i. e. $&\\operatornamewithlimits{arg\\,min}_{(p_{00},p_{11})\\in \\mathcal {R}^\\prime } \\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})| \\pi ) \\subseteq \\mathcal {R}^{\\prime \\prime }.$ For the remainder of this paper, we assume $\\pi \\in (0,1)$ .",
"Note that the results on optimal mechanisms still hold for $\\pi \\in [0,1]$ , however these optima may not be unique."
],
[
"Optimal Mechanism for $\\epsilon $ -Differential Privacy",
"We have already established that the parameters for the optimal ($\\epsilon $ , $\\delta $ )-differentially private mechanism lie on $\\mathcal {R}^{\\prime \\prime }$ .",
"We now examine the case of $\\epsilon $ -differential privacy, where $\\delta =0$ , with the additional assumption that $\\epsilon > 0$ .",
"Theorem 2 Let $\\pi \\in (0,1)$ , $p_{00} + p_{11} > 1$ and $\\epsilon > 0$ .",
"The $\\epsilon $ -differentially private RR mechanism which minimises estimator error is given by the design matrix $P_\\epsilon = \\left(\\begin{array}{cc} \\frac{e^\\epsilon }{e^\\epsilon +1} & \\frac{1}{e^\\epsilon +1} \\\\ \\frac{1}{e^\\epsilon +1} & \\frac{e^\\epsilon }{e^\\epsilon +1} \\end{array}\\right).$ By Lemma REF , we know that the parameters $(p_{00}, p_{11})$ of the optimal mechanism exist on the boundary of $\\mathcal {R}^\\prime $ , with at least one of the inequalities () tight.",
"We now separately consider the cases where (REF ) and () are tight.",
"By hypothesis, $\\delta =0$ and $\\epsilon \\ne 0$ .",
"(REF ) tight: $p_{11}=e^\\epsilon (1-p_{00})$ , constrained by $p_{11}\\ge 0$ and $p_{00}\\le e^\\epsilon (1-p_{11})$ .",
"By () and since $p_{00} = 1-e^{-\\epsilon }p_{11}$ , we have $e^\\epsilon p_{11} &\\le e^\\epsilon - p_{00}\\\\&=e^\\epsilon -(1-e^{-\\epsilon }p_{11})\\\\&=e^\\epsilon -1+e^{-\\epsilon }p_{11},$ which we rewrite as $p_{11}(e^\\epsilon - e^{-\\epsilon })\\le e^\\epsilon -1,$ and noting that $e^{2\\epsilon }-1=(e^\\epsilon -1)(e^\\epsilon +1)$ , we see that $p_{11} &\\le \\frac{e^\\epsilon -1}{e^{-\\epsilon }(e^{2\\epsilon }-1)}\\\\&= \\frac{e^\\epsilon }{e^\\epsilon +1}.$ We are therefore considering $\\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})| \\pi )$ on the line $p_{00} = 1-e^{-\\epsilon }p_{11}$ for $0 \\le p_{11} \\le \\frac{e^\\epsilon }{e^\\epsilon +1}$ .",
"We parametrise this line as follows, where $0 < t \\le 1$ , $p_{00} = r(t)$ and $p_{11} = s(t)$ (we require $t>0$ since $p_{00}+p_{11} > 1$ ): $\\begin{aligned}r(t) &= (1-t)+\\frac{e^\\epsilon }{1+e^\\epsilon }t=1-e^{-\\epsilon }s(t),\\\\s(t) &= \\frac{e^\\epsilon }{1+e^\\epsilon }t.\\end{aligned}$ For simplicity, we let $\\hat{\\Pi }(r(t), s(t))=\\hat{\\Pi }_1(t)$ .",
"After some manipulation, we see that $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }_1(t)| \\pi )}{\\partial t} = -\\frac{(1+e^\\epsilon )(1+\\pi (e^\\epsilon -1))}{(e^\\epsilon -1)^2t^2n},$ and noting that $e^\\epsilon > 1$ , we see that $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }_1(t)| \\pi )}{\\partial t}<0$ .",
"Hence, $\\operatornamewithlimits{arg\\,min}_{t \\in (0,1]} \\operatorname{Var}(\\hat{\\Pi }_1(t)| \\pi ) = \\lbrace 1\\rbrace .$ () tight: By symmetry of the equations (), we simply let $p_{00}=s(t)$ and $p_{11}=r(t)$ .",
"By examining (REF ) and (REF ), we see that $\\operatorname{Var}(\\hat{\\Pi }(p_{00}, p_{11})| 1-\\pi ) = \\operatorname{Var}(\\hat{\\Pi }(p_{11}, p_{00})| \\pi ),$ and by letting $\\hat{\\Pi }(s(t), r(t))=\\hat{\\Pi }_2(t)$ , we get $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }_2(t)| \\pi )}{\\partial t} = -\\frac{(1+e^\\epsilon )(1+(1-\\pi )(e^\\epsilon -1)}{(e^\\epsilon -1)^2t^2n}.$ Again it follows that $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }_2(t)| \\pi )}{\\partial t}<0$ , and so $\\operatornamewithlimits{arg\\,min}_{t \\in (0,1]} \\operatorname{Var}(\\hat{\\Pi }_2(t)| \\pi ) = \\lbrace 1\\rbrace .$ By (REF ), (REF ) and (REF ), we can now conclude that $\\operatornamewithlimits{arg\\,min}_{(p_{00},p_{11})\\in \\mathcal {R}^\\prime } \\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})| \\pi )= \\left\\lbrace \\left(\\frac{e^\\epsilon }{e^\\epsilon +1}, \\frac{e^\\epsilon }{e^\\epsilon +1}\\right)\\right\\rbrace ,$ and so the result follows.",
"Remark: When $\\epsilon =0$ , all rows of the design matrix must be identical, i. e. $p_{00}=1-p_{11}$ and $p_{11}=1-p_{00}$ .",
"This gives $p_{00}+p_{11} = 1$ , leading to an unbounded estimator error (REF ).",
"In practical terms, 0-differential privacy enforces the same output distribution for every respondent, hence nothing meaningful can be learned."
],
[
"Optimal Mechanism for ($\\epsilon $ , {{formula:55512451-cf2e-41af-a36e-9d913f0152a0}} )-Differential Privacy",
"Let's now consider the optimal mechanism for ($\\epsilon $ , $\\delta $ )-differential privacy.",
"We parametrise $\\mathcal {R}^{\\prime \\prime }$ as follows.",
"If we let $\\begin{aligned}r_\\delta (t) &= \\left(1+e^{-\\epsilon }\\delta \\right)(1-t)+\\frac{e^\\epsilon +\\delta }{e^\\epsilon +1}t,\\\\&=1-e^{-\\epsilon }(s_\\delta (t)-\\delta ),\\\\s_\\delta (t) &= \\frac{e^\\epsilon +\\delta }{e^\\epsilon +1} t,\\end{aligned}$ for $t\\in [0,1]$ , then the boundary where (REF ) holds is parametrised by $p_{00} = r_\\delta (t)$ and $p_{11} = s_\\delta (t)$ ; by symmetry, the boundary where () holds is parametrised by $p_{00} = s_\\delta (t)$ and $p_{11} = r_\\delta (t)$ .",
"We note that $t=1$ denotes an extreme point of $\\mathcal {R}^\\prime $ (and $\\mathcal {R}^{\\prime \\prime }$ ), the point at which both inequalities () are tight.",
"Here $p_{00} = p_{11} = r_\\delta (1) = s_\\delta (1)= \\frac{e^\\epsilon +\\delta }{e^\\epsilon +1}$ ."
],
[
"Preliminary Lemmas",
"Before proceeding to the main result of this section, we first present a collection of lemmas for later use.",
"The first result states that the minimal variance of $\\hat{\\Pi }$ on $\\mathcal {R}^{\\prime \\prime }$ will occur at one of its extreme points (i. e. at one of the endpoints of the two line segments which comprise $\\mathcal {R}^{\\prime \\prime }$ ).",
"Lemma 3 Let $r_\\delta $ and $s_\\delta $ be given by (REF ), let $\\delta > 0$ and let $a \\le b \\in [0,1]$ .",
"Then, $\\operatornamewithlimits{arg\\,min}_{t \\in [a,b]} \\operatorname{Var}(\\hat{\\Pi }(r_\\delta (t), s_\\delta (t))| \\pi ) \\subseteq \\lbrace a,b\\rbrace .$ For simplicity, we denote $\\hat{\\Pi }(r_\\delta (t), s_\\delta (t))$ by $\\hat{\\Pi }_{1,\\delta }(t)$ .",
"By some manipulation, it can be shown that the numerator of $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }_{1,\\delta }(t)| \\pi )}{\\partial t}$ is linear in $t$ , hence it has at most one root at $t = \\frac{\\delta (1+e^\\epsilon )(2e^\\epsilon +2\\delta -1-\\pi (e^\\epsilon +2\\delta -1))}{(e^\\epsilon +\\delta )(e^\\epsilon +2\\delta -1)(1+(e^\\epsilon -1)\\pi )}.$ By substitution, we find that $\\frac{\\partial ^2 \\operatorname{Var}(\\hat{\\Pi }_{1,\\delta }(t)| \\pi )}{\\partial t^2} = -\\frac{(e^\\epsilon +\\delta )^2(e^\\epsilon +2\\delta -1)^4(1+(e^\\epsilon -1)\\pi )^4}{8e^{2\\epsilon }\\delta ^3(e^\\epsilon +\\delta -1)^3(1+e^\\epsilon )^2n},$ when $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }_{1,\\delta }(t)| \\pi )}{\\partial t}=0$ .",
"By inspection, and since $\\delta > 0$ , we see that $\\frac{\\partial ^2 \\operatorname{Var}(\\hat{\\Pi }_{1,\\delta }(t)| \\pi )}{\\partial t^2} < 0$ when $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }_{1,\\delta }(t)| \\pi )}{\\partial t}=0$ , and so this point is the maximum of $\\operatorname{Var}(\\hat{\\Pi }_{1,\\delta }(t)| \\pi )$ .",
"Hence, the minimum of $\\operatorname{Var}(\\hat{\\Pi }_{1,\\delta }(t)| \\pi )$ cannot occur at a mid-point of an interval.",
"The result follows.",
"We next show that the error of $\\hat{\\Pi }$ along the boundary constrained by (REF ) is uniformly greater than along the boundary constrained by () when $\\pi \\le \\frac{1}{2}$ .",
"Lemma 4 Let $r_\\delta $ and $s_\\delta $ be given by (REF ) and let $\\delta > 0$ .",
"Then, when $\\pi \\le \\frac{1}{2}$ , $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (t), s_\\delta (t))| \\pi ) \\le \\operatorname{Var}(\\hat{\\Pi }(s_\\delta (t), r_\\delta (t))| \\pi ),$ for $t \\in [0,1]$ .",
"Conversely, if $\\pi \\ge \\frac{1}{2}$ , then $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (t), s_\\delta (t))| \\pi ) \\ge \\operatorname{Var}(\\hat{\\Pi }(s_\\delta (t), r_\\delta (t))| \\pi ),$ for $t \\in [0,1]$ .",
"After manipulation of the terms, we can show that $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (t), s_\\delta (t))| \\pi ) - \\operatorname{Var}(\\hat{\\Pi }(s_\\delta (t), r_\\delta (t))| \\pi ) =-\\frac{(e^\\epsilon +1)(e^\\epsilon +\\delta )(1-2\\pi )(1-t)}{(e^\\epsilon (e^\\epsilon -1)t+\\delta (1-t+e^\\epsilon (1+t)))n}.$ We see that $1-2\\pi \\ge 0$ when $\\pi \\le \\frac{1}{2}$ , and $1-2\\pi \\le 0$ when $\\pi \\ge \\frac{1}{2}$ , and, since $t\\in [0,1]$ and $\\delta > 0$ , the result follows.",
"Finally, we present $t_0(\\epsilon , \\delta )$ as the $t$ -value which gives the endpoints of the line segments of $\\mathcal {R}^{\\prime \\prime }$ at the boundary of the unit square.",
"Lemma 5 Define $t_0:\\mathbb {R}\\times \\mathbb {R}\\rightarrow [0,1]$ by $t_0(\\epsilon ,\\delta ) = \\frac{\\delta (e^\\epsilon +1)}{e^\\epsilon +\\delta },$ then, $(r_\\delta (t_0(\\epsilon ,\\delta )), s_\\delta (t_0(\\epsilon , \\delta ))) \\in \\partial \\mathcal {R}^\\prime .$ By explicit calculation, $r_\\delta (t_0(\\epsilon ,\\delta )) &= 1,\\\\s_\\delta (t_0(\\epsilon ,\\delta )) &= \\delta .$ By definition, it follows that $(1, \\delta ) \\in \\mathcal {R}^\\prime \\cup \\partial \\mathcal {R}^\\prime $ , and since $p_{00} \\le 1$ is a boundary of $\\lbrace (p_{00},p_{11}) \\in \\mathcal {R}^\\prime \\rbrace $ , it follows that $(1, \\delta ) \\in \\partial \\mathcal {R}^\\prime $ .",
"Remark: When $\\delta = 0$ , $\\left(r_\\delta (t_0(\\epsilon ,\\delta )), s_\\delta (t_0(\\epsilon , \\delta ))\\right) \\notin \\mathcal {R}^\\prime $ , since we require $r_\\delta + s_\\delta > 1$ .",
"Remark: By linearity, it follows that $(r_\\delta (t), s_\\delta (t)) \\in \\mathcal {R}^\\prime $ for all $t_0(\\epsilon ,\\delta ) < t \\le 1$ , and that $(r_\\delta (t), s_\\delta (t)) \\notin \\mathcal {R}^\\prime $ when $t < t_0(\\epsilon ,\\delta )$ ."
],
[
"Main Result",
"We now present the main results of this paper, which establish the optimal ($\\epsilon $ , $\\delta $ )-differentially private RR mechanism(s).",
"The following results assume $\\delta > 0$ ; the optimal mechanism when $\\delta = 0$ was presented in Theorem REF .",
"Note that we continue to assume $\\pi \\in (0,1)$ to ensure uniqueness of the optima.",
"The following theorem establishes the optimal RR mechanism(s) when $\\pi \\le \\frac{1}{2}$ .",
"Theorem 3 Let $\\delta > 0$ and $0 < \\pi \\le \\frac{1}{2}$ , and define $g:\\mathbb {R}\\times \\mathbb {R}\\rightarrow \\mathbb {R}$ by $g(\\epsilon ,\\delta ) = \\frac{\\delta (e^\\epsilon +\\delta )}{(e^\\epsilon +2\\delta -1)^2}.$ Then, for $r_\\delta $ and $s_\\delta $ given by (REF ), $\\operatornamewithlimits{arg\\,min}_{(p_{00},p_{11}) \\in \\mathcal {R}^\\prime } \\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})| \\pi )={\\left\\lbrace \\begin{array}{ll} \\lbrace (r_\\delta (t_0),s_\\delta (t_0))\\rbrace , & \\text{if } g(\\epsilon ,\\delta ) > \\pi ,\\\\\\lbrace (r_\\delta (1),s_\\delta (1))\\rbrace , & \\text{if } g(\\epsilon ,\\delta ) < \\pi ,\\\\\\lbrace (r_\\delta (t_0),s_\\delta (t_0)), (r_\\delta (1),s_\\delta (1))\\rbrace , & \\text{if } g(\\epsilon ,\\delta ) = \\pi .\\end{array}\\right.",
"}$ where $t_0 = t_0(\\epsilon ,\\delta )$ .",
"By Lemmas REF , REF and REF , we know that when $0 < \\pi \\le \\frac{1}{2}$ and $\\delta > 0$ , $\\operatornamewithlimits{arg\\,min}_{(p_{00}, p_{11})\\in \\mathcal {R}^\\prime } \\operatorname{Var}(\\hat{\\Pi }(p_{00}, p_{11}) | \\pi ) \\subseteq \\lbrace (r_\\delta (t_0), s_\\delta (t_0)), (r_\\delta (1), s_\\delta (1))\\rbrace .$ We are therefore considering two candidate points, which can be shown to resolve to $r_\\delta (t_0) &= 1, & s_\\delta (t_0) &= \\delta ,\\\\r_\\delta (1) &= \\frac{e^\\epsilon + \\delta }{e^\\epsilon +1}, & s_\\delta (1) &= \\frac{e^\\epsilon +\\delta }{e^\\epsilon +1}.$ We are therefore seeking to determine the sign of $\\operatorname{Var}(\\hat{\\Pi }(1, \\delta ) | \\pi )-\\operatorname{Var}\\left(\\left.\\hat{\\Pi }\\left(\\frac{e^\\epsilon + \\delta }{e^\\epsilon +1},\\frac{e^\\epsilon + \\delta }{e^\\epsilon +1}\\right) \\right| \\pi \\right).$ After some manipulation, we can show that (REF ) simplifies to $\\frac{(1-\\delta )(\\pi (e^\\epsilon +2\\delta -1)-\\delta (e^\\epsilon +\\delta ))}{\\delta (e^\\epsilon +2\\delta -1)^2n},$ and we note that its denominator is strictly positive since $\\delta > 0$ .",
"Note additionally that (REF ) simplifies to zero when $\\delta = 1$ , which is trivial since $r_1(t_0)=s_1(t_0)=r_1(1)=s_1(1)=1$ .",
"The sign of (REF ) is therefore determined by the sign of $\\pi (e^\\epsilon +2\\delta -1)-\\delta (e^\\epsilon +\\delta )$ , which gives $g(\\epsilon , \\delta )$ when solved for $\\pi $ .",
"Hence, $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (t_0),s_\\delta (t_0))| \\pi ) < \\operatorname{Var}(\\hat{\\Pi }(r_\\delta (1),s_\\delta (1))| \\pi )$ when $g(\\epsilon ,\\delta ) > \\pi $ .",
"The other results follow similarly.",
"Remark: When $g(\\epsilon ,\\delta ) \\le \\pi $ , the optimal mechanism corresponds with that established for $\\epsilon $ -differential privacy on RR (with an added dependence for $\\delta $ ) and also with the optimal mechanism established in Theorem 10 of [15] for mechanisms on categorical data.",
"However, when $g(\\epsilon ,\\delta ) > \\pi $ , the optimal mechanism is one which we have not encountered previously.",
"The next corollary establishes the optimal mechanism(s) when $\\pi \\ge \\frac{1}{2}$ , and follows from Theorem REF by the symmetry of $\\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})| \\pi )$ in $p_{00}$ and $p_{11}$ .",
"Corollary 2 Let $\\delta > 0$ and $\\frac{1}{2} \\le \\pi < 1$ .",
"Then, for $r_\\delta $ and $s_\\delta $ given by (REF ) and $g$ given by (REF ), $\\operatornamewithlimits{arg\\,min}_{(p_{00},p_{11}) \\in \\mathcal {R}^\\prime } \\operatorname{Var}(\\hat{\\Pi }(p_{00},p_{11})|\\pi )= {\\left\\lbrace \\begin{array}{ll} \\lbrace (s_\\delta (t_0),r_\\delta (t_0))\\rbrace , & \\text{if } g(\\epsilon ,\\delta ) > 1-\\pi ,\\\\\\lbrace (s_\\delta (1),r_\\delta (1))\\rbrace , & \\text{if } g(\\epsilon ,\\delta ) < 1-\\pi ,\\\\\\lbrace (s_\\delta (t_0),r_\\delta (t_0)), (s_\\delta (1),r_\\delta (1))\\rbrace , & \\text{if } g(\\epsilon ,\\delta ) = 1-\\pi ,\\end{array}\\right.",
"}$ where $t_0 = t_0(\\epsilon ,\\delta )$ .",
"The result follows from Theorem REF since $\\operatorname{Var}(\\hat{\\Pi }(p_{00}, p_{11})|\\pi ) &= \\operatorname{Var}(\\hat{\\Pi }(p_{11}, p_{00})| 1-\\pi ).$ Example REF and Figure REF illustrate the conclusion of Theorem REF .",
"Example 1 Consider Theorem REF and Corollary REF for various values of $\\epsilon $ , $\\delta $ and $\\pi $ .",
"For simplicity, in each of these examples we set $n=1$ .",
"$\\epsilon =\\frac{1}{2}$ , $\\delta =\\frac{1}{10}$ , $\\pi =\\frac{1}{4}$ : In this case, we have $g(\\epsilon ,\\delta ) = 0.243 < \\pi $ .",
"Hence, the design matrix of the optimal mechanism is denoted by $\\left(\\begin{array}{cc} \\frac{e^\\epsilon +\\delta }{e^\\epsilon +1} & \\frac{1-\\delta }{e^\\epsilon +1}\\\\\\frac{1-\\delta }{e^\\epsilon +1} & \\frac{e^\\epsilon +\\delta }{e^\\epsilon +1}\\end{array}\\right).$ This can be verified by noting that $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (1), s_\\delta (1))| \\pi ) = 2.372$ and $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (t_0),s_\\delta (t_0))| \\pi ) = 2.438$ .",
"$\\epsilon =1$ , $\\delta =\\frac{2}{5}$ , $\\pi =\\frac{1}{10}$ : In this case, $g(\\epsilon ,\\delta )=0.197 > \\pi $ .",
"Hence, the design matrix of the optimal mechanism is denoted by $\\left(\\begin{array}{cc} 1 & 0 \\\\1-\\delta & \\delta \\end{array}\\right).$ Again, this can be verified by noting that $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (1), s_\\delta (1))| \\pi ) = 0.385$ and $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (t_0),s_\\delta (t_0))| \\pi ) = 0.24$ .",
"$\\epsilon =\\frac{1}{2}$ , $\\delta =\\frac{1}{3}$ , $\\pi =\\frac{9}{10}$ : Since $\\pi \\ge \\frac{1}{2}$ , we use Corollary REF for this example.",
"We note that $g(\\epsilon ,\\delta ) = 0.382 > 1-\\pi $ .",
"Hence, the design matrix of the optimal mechanism is denoted by $\\left(\\begin{array}{cc} \\delta & 1-\\delta \\\\0 & 1\\end{array}\\right).$ We see that $\\operatorname{Var}(\\hat{\\Pi }(s_\\delta (1), r_\\delta (1))| \\pi ) = 0.854$ and $\\operatorname{Var}(\\hat{\\Pi }(s_\\delta (t_0), r_\\delta (t_0))| \\pi ) = 0.143$ .",
"Note also that $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (0), s_\\delta (0))| \\pi ) = 1.911$ , corresponding with the conclusion of Lemma REF $\\epsilon =\\ln (2), \\delta =\\frac{1}{4}, \\pi =\\frac{1}{4}$ : In this case, we have $g(\\epsilon , \\delta ) = \\frac{1}{4} = \\pi $ , hence there are two optimal mechanisms, $\\left(\\begin{array}{cc} \\frac{e^\\epsilon +\\delta }{e^\\epsilon +1} & \\frac{1-\\delta }{e^\\epsilon +1}\\\\\\frac{1-\\delta }{e^\\epsilon +1} & \\frac{e^\\epsilon +\\delta }{e^\\epsilon +1}\\end{array}\\right), \\left(\\begin{array}{cc} 1 & 0 \\\\1-\\delta & \\delta \\end{array}\\right).$ This can be verified by noting that $\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (1), s_\\delta (1))| \\pi )=\\operatorname{Var}(\\hat{\\Pi }(r_\\delta (t_0), s_\\delta (t_0))| \\pi )=\\frac{15}{16}$ .",
"Figure: A contour plot of various level sets of g(ϵ,δ)g(\\epsilon ,\\delta ).",
"Given π\\pi , ϵ\\epsilon and δ\\delta , these level sets can be used to determine the optimal (ϵ\\epsilon , δ\\delta )-differentially private RR mechanism."
],
[
"Optimal Warner Mechanism for ($\\epsilon $ , {{formula:aa543ea0-f18d-4ded-9668-c3c8902a6e17}} )-Differential Privacy",
"In the final result of this paper, we examine the optimal mechanism for Warner's RR mechanism.",
"We recall that Warner's mechanism imposed the additional constraint that $p_{00} = p_{11} = p_w$ , so the design matrix becomes $\\left(\\begin{array}{cc} p_w & 1-p_w \\\\ 1-p_w & p_w \\end{array}\\right).$ The error of such a mechanism is only a function of $p_w$ and the population proportion $\\pi $ , as shown in (REF ).",
"As before, we require $2p_w > 1$ .",
"Our region of feasibility is therefore $\\mathcal {R}_w = \\left(\\frac{1}{2}, \\frac{e^\\epsilon +\\delta }{e^\\epsilon +1}\\right].$ Theorem 4 Consider Warner's RR mechanism as presented in Section REF .",
"Then, $\\operatornamewithlimits{arg\\,min}_{p_w \\in \\mathcal {R}_w} \\operatorname{Var}(\\hat{\\Pi }_w(p_w)|\\pi ) = \\left\\lbrace \\frac{e^\\epsilon +\\delta }{e^\\epsilon +1}\\right\\rbrace .$ By (REF ), we note that $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }_w(p_w)|\\pi )}{\\partial p_w} = \\frac{1}{(1-2p_w)^3 n},$ hence $\\frac{\\partial \\operatorname{Var}(\\hat{\\Pi }_w(p_w)|\\pi )}{\\partial p_w}<0$ when $p_w > \\frac{1}{2}$ .",
"Therefore, $\\operatornamewithlimits{arg\\,min}_{p_w \\in \\mathcal {R}_w} \\operatorname{Var}(\\hat{\\Pi }_w(p_w)|\\pi ) = \\max (\\mathcal {R}_w),$ and the result follows."
],
[
"Conclusions",
"We have presented the optimal differentially private RR mechanisms with respect to a maximum likelihood estimator, where both strict and relaxed differential privacy were considered.",
"For a given desired level of privacy, as determined by $\\epsilon $ and $\\delta $ , we presented a method to quickly determine the optimal mechanism.",
"This will allow for the optimal implementation of differential privacy in any randomised response survey."
],
[
"Acknowledgement",
"The first named author was supported by the Science Foundation Ireland grant SFI/11/PI/1177."
]
] | 1612.05568 |
[
[
"A Distributed Algorithm for Throughput Optimal Routing in Overlay\n Networks"
],
[
"Abstract We address the problem of optimal routing in overlay networks.",
"An overlay network is constructed by adding new overlay nodes on top of a legacy network.",
"The overlay nodes are capable of implementing any dynamic routing policy, however, the legacy underlay has a fixed, single path routing scheme and uses a simple work-conserving forwarding policy.",
"Moreover, the underlay routes are pre-determined and unknown to the overlay network.",
"The overlay network can increase the achievable throughput of the underlay by using multiple routes, which consist of direct routes and indirect routes through other overlay nodes.",
"We develop a throughput optimal dynamic routing algorithm for such overlay networks called the Optimal Overlay Routing Policy (OORP).",
"OORP is derived using the classical dual subgradient descent method, and it can be implemented in a distributed manner.",
"We show that the underlay queue-lengths can be used as a substitute for the dual variables.",
"We also propose various schemes to gather the information about the underlay that is required by OORP and compare their performance via extensive simulations."
],
[
"Introduction",
"Optimal routing algorithmsA routing algorithm is throughput optimal if it can stabilize any traffic that can be stabilized by some routing algorithm.",
"have received a significant amount of attention in the literature for the past two decades (e.g.",
"[1], [2], [10], [11]), however, they have had limited success in terms of implementations.",
"One of the main reasons behind the lack of traction is that these policies require additional functionalities that are not supported by the legacy devices.",
"For example, most of these algorithms need the network to be composed of homogeneous nodes that possess the ability to implement a dynamic routing policy.",
"In contrast, many legacy networks use a single path routing scheme with a work-conserving forwarding policy such as FIFO, and hence can support only a fraction of the achievable throughput.",
"Thus, an implementation of a throughput optimal scheme usually requires a complete overhaul of the network.",
"An overlay architecture for a gradual move towards optimal routing was proposed in [6].",
"This architecture integrates overlay nodes capable of dynamic routing into an underlay network of legacy devices (see Figure REF for an example).",
"In this paper, we develop a throughput optimal dynamic routing algorithm for such overlay networks.",
"Overlay networks have been used to improve the performance and capabilities of computer networks for a long time.",
"The Internet itself started as a data network built on top of the telephone network.",
"An overlay architecture to improve the robustness of the Iinternet was proposed in [3], where alternate overlay paths are used to overcome path loss in the underlay network.",
"Placement for the overlay node to improve path diversity was studied in [4].",
"Architectures for designing overlay networks that improve different quality of service metrics have been proposed in [18], [19].",
"Currently overlay is being used extensively for applications such as content delivery, multicast, etc.",
"Figure: Overlay network architecture.",
"If the overlay node A has traffic for node D, it can either route it directly using the tunnel from A to D or relay it through other overlay node B or C.In [6], the authors study the problem of placing the minimum number of overlay nodes into an existing underlay in order to maximize network throughput.",
"In particular, the authors show that with just a few overlay nodes, maximum network throughput can be achieved.",
"However, [6] also shows that the backpressure routing algorithm of [1], which is known to be optimal in a wide range of scenarios, leads to a loss in throughput when used in an overlay network.",
"Then the authors of [6] proposes a heuristic for optimal routing called the Overlay Backpressure Policy (OBP).",
"An optimal backpressure like routing algorithm for a special case, where the underlay paths do not overlap with each other, was given in [7].",
"This paper also proposes a threshold based heuristic for general overlay networks.",
"The schemes presented in [6] and [7] are very similar and were conjectured to be throughput optimal.",
"In this paper, we provide a counterexample to show that OBP is in fact not throughput optimal and develop a new optimal routing policy.",
"To derive the optimal policy, we notice that the suboptimality of backpressure arises from its failure to accurately account for congestion in the underlay paths.",
"Traditional backpressure doesn't keep track of the packets in the underlay which can lead the overlay nodes to send too many packets into the underlay creating instability of underlay queues.",
"We will first develop a centralized solution which achieves optimality by limiting the traffic injected into the underlay so that the underlay queues are always bounded.",
"Then we use the intuition gained from this policy to develop a distributed solution which uses the queue backlog information in the underlay to compute the amount of flow transmitted into each underlay path.",
"This policy implicitly favors underlay paths that are less congested and preserves stability of all the queues.",
"This paper is organized as follows.",
"We describe our model in the next section.",
"In section III, we provide a counterexample to the OBP routing policy.",
"Then in section IV, we provide a centralized stochastic policy that is throughput optimal for overlay networks.",
"In section V, we develop a distributed policy based on the dual subgradient descent method that requires the underlay queue-lengths.",
"In section VI, we propose three approaches to estimating the queue-lengths if they are not available to the overlay nodes.",
"Finally, we verify the performance of our algorithm with extensive simulations."
],
[
"Model",
"We model the network as a graph $(N, E)$ where $N$ is the set of nodes and $E$ is the set of directed links.",
"The links are capacitated and the capacity of a link $(i,j)\\in E$ is given by $c_{ij}$ .",
"The nodes can be of two types: underlay or overlay.",
"We represent the set of all underlay nodes by $U$ and the set of all overlay nodes by $\\mathcal {O} = N \\backslash U$ .",
"The network supports a set of commodities, $K$ , where each commodity $k\\in K$ is defined by a source-destination pair.",
"For the ease of exposition, we will formulate the problem with all the sources and destinations being overlay nodes.",
"In Section REF , we discuss how the same solution can be applied when this is not the case.",
"The time is slotted and indexed by $t$ .",
"We remove the time index for notational simplicity if removing it doesn't create ambiguity."
],
[
"Overlay",
"The overlay network consists of the controllable nodes $\\mathcal {O}$ which are capable of implementing a dynamic routing algorithm.",
"The links between two overlay nodes can either be a direct edge or a path through the underlay referred to as a tunnel.",
"A tunnel $l$ is a sequence of nodes $l_1, l_2, \\dots , l_{|l|}$ where $|l|$ is the length of the tunnel.",
"We represent the set of all the tunnels in the network by $L$ .",
"Since a tunnel connects two overlay nodes, $l_1$ and $l_{|l|}$ are overlay nodes, and $l_{|2|}, ..., l_{|l|-1}$ are underlay nodes.",
"When a packet is sent into a tunnel $l$ , node $l_1$ encapsulates it into a packet destined to node $l_{|l|}$ and forwards it onto the underlay node $l_2$ .",
"The route taken by the tunnel is dictated by the path from $l_2$ to $l_{|l|}$ which is assigned by the underlay.",
"When the packet reaches $l_{|l|}$ , it is decapsulated and enqueued at the node.",
"An example of the different type of links in an overlay network is given in Figure REF .",
"This overlay network consists of one direct link (1,4) and three tunnels (1,3,4), (2,3,4) and (2,3,5).",
"Figure: A physical network and its corresponding overlay network.Each overlay node $i$ maintains a queue for each commodity $k$ and the backlog is represented by $Q^k_i$ .",
"The number of external commodity $k$ packets that arrive at node $i$ represented by $A^k_i$ .",
"Let $F^k_{l}$ represents the amount of packets injected into the tunnel $l$ , and $\\bar{F}^k_{l}$ represents the number of packets that exit tunnel $l$ .",
"The quantities are different because a packet sent into the tunnel might not exit the tunnel for several time-steps.",
"Let $F^k_{ij}$ represent the number of commodity $k$ packets that are transmitted on an overlay to overlay link $(i,j)$ .",
"Figure REF illustrates the meaning of these variables on a simple network.",
"The backlog of commodity $k$ packets at overlay node $i$ evolves as follows: $Q^k_i(t+1) &= \\left[ Q^k_i(t) - \\sum _{j\\in \\mathcal {O}} F^k_{ij}(t)- \\sum _{l\\in L: l_1 = i} F^k_{l}(t) + \\right.",
"\\nonumber \\\\&\\qquad \\qquad \\left.",
"\\sum _{j\\in \\mathcal {O}} F^k_{ji}(t) + \\sum _{l\\in L:l_{|l|}=i} \\bar{F}^k_{l}(t) + A^k_i (t) \\right]^+ \\nonumber $ Here, $\\lbrace l\\in L: l_1 = i\\rbrace $ are all the tunnels that start at node $i$ , $\\lbrace l\\in L:l_{|l|}=i\\rbrace $ are the tunnels that end at node $i$ , and $[.",
"]^+ = \\max (.,0)$ .",
"Packets are removed at the destination node, hence the backlog of a commodity at its destination is zero.",
"We assume that all the traffic arrivals $A^k_i$ are i.i.d.",
"with a mean of $\\lambda ^k_i$ .",
"We also assume that the arrival rate vector $\\lambda $ is in the interior of the throughput region of the overlay network $\\Lambda $ [6].",
"We will be designing a dynamic routing policy that controls $F^k_{l}$ and $F^k_{ij}$ at each time-step so that both the overlay and the underlay queues stabilize."
],
[
"Underlay",
"The underlay network consists of the uncontrollable nodes $U$ .",
"These nodes have a static routing policy which assigns a fixed path between each pair of nodes in the underlay.",
"The paths are assumed to be acyclic and unique, which ensures that all the tunnels are acyclic an that they take a fixed route through the underlay.",
"An underlay node maintains a queue per outgoing link.",
"The backlog on the queue associated with the link $(a,b)$ is represented by $Q_{ab}$ .",
"The queues have infinite buffer space hence packets are not dropped.",
"When a packet arrives at an underlay node, the node looks up the link assigned to it based on its destination and enqueues it on the corresponding link.",
"Since several tunnels of the overlay network can pass through the same underlay link the underlay queues accumulates packets from several different tunnels and commodities.",
"An example of an underlay queue that is shared by several tunnels is presented in Figure REF .",
"Packets from both the tunnels (1,3,4) and (2,3,4) are queued on the link (3,4).",
"The underlay employs a work-conserving forwarding scheme that is “universally stable” as defined in [5].",
"This assumption ensures that if the number of packets injected into the underlay at each timeslot satisfies the capacity constraints of the tunnels, then the underlay queues are deterministically bounded.",
"Specifically, under a universally stable forwarding policy, an underlay queue corresponding to link $(a,b)$ is always deterministically bounded if $\\sum _{l\\in L:(a,b)\\in l} \\sum _k F^k_l(t) <c_{ab} \\forall t. $ Here $\\lbrace l\\in L:(a,b)\\in l\\rbrace $ is the set of tunnels that pass through the link $(a,b)$ .",
"We refer to such constraints as the tunnel capacity constraints.",
"Several work-conserving policies that are universally stable are given in [5]."
],
[
"Background",
"The problem of optimal routing in an overlay network was first studied in [6], where it was shown that backpressure routing, which is known to be throughput optimal in a range of scenarios, is not optimal for overlay networks, and proposed a heuristic called the Overlay Backpressure Policy (OBP).",
"The OBP heuristic was conjectured to be throughput optimal.",
"For each tunnel $l$ and commodity $k$ OBP keeps track of the packets in flight $H^k_l$ , which is the number of packets that have been transmitted into the tunnel by node $l_1$ but haven't reached node $l_{|l|}$ .",
"The weight for each commodity over the tunnel $W^k_{l}(t)$ is computed as follows $W^k_{l}(t) = Q^k_{l_1}(t) - H_{l}^k (t)- Q_{l_{|l|}}^k(t).$ A link $(i,j)$ that connects two overlay nodes can be thought of as a tunnel $l=(i,j)$ with no underlay node, hence the weight is computed as $W^k_{l}(t) = Q^k_{l_1}(t) - Q_{l_{|l|}}^k(t).$ Then, the commodity with the highest weight sends its packets into the tunnel provided that the weight is positive.",
"A precise description of the OBP is given in Algorithm .",
"[h!]",
"Overlay Backpressure Policy (OPB): For each tunnel $l$ at each time-step $t$ : Compute the commodity $k^*$ that maximizes the weight $W^k_{l}(t)$ , $k^* \\in \\arg \\max _k W_{l}^k(t).$ Ties are broken arbitrarily.",
"Transmit $\\mu $ packets into the tunnel where $\\mu = \\left\\lbrace \\begin{array}{l} c_{l_1l_2} \\text{ if }W_{l}^{k^*}(t) > 0\\\\ 0, \\text{ otherwise,} \\end{array} \\right.$ where $c_{l_1l_2}$ is the capacity of the first link of tunnel $l$ .",
"This policy makes sense intuitively because it encourages utilizing the tunnels that have less packets in them.",
"When a tunnel is congested, the number of packets in flight is high, which encourages the overlay nodes to use alternate routes and send packets into the tunnel only when the backlog in the overlay is extremely high.",
"This behavior is common to backpressure-based optimal routing algorithms.",
"Moreover, OBP reduces to backpressure routing when all the nodes are overlay nodes.",
"We present the following counterexample to show that the OBP is not throughput optimal.",
"Consider a network topology given in Figure REF where all the links are unit capacity.",
"There are three commodities with source $s_i$ and destination $d_i$ , $i=1,2,3$ .",
"The source and the destination are overlay nodes, whereas the nodes 1, 2 and 3 (in gray) are underlay nodes.",
"The underlay nodes use the FIFO queuing disciplineFrom [8] we know that FIFO is throughput optimal for a ring which is the underlay topology in this example.. Each commodity in this network has two tunnels to the destination, e.g.",
"$(s_1, 1, 2, d_1)$ and $(s_1, 3, 1, 2, d_1)$ .",
"Note that the shorter tunnels do not share any links between them.",
"So, if the shorter tunnel is chosen by each commodity, this network can support an arrival rate vector of $[1,1,1]$ .",
"Figure: Counterexample for throughput optimality of the Overlay Backpressure Policy of .Let us consider Poisson arrivals with the rate vector of $[0.8, 0.8, 0.8]$ , which is clearly inside the stability region.",
"To support this rate OBP has to send most of its traffic through the shorter tunnels.",
"However, as we show below, congestion can lead traffic to use longer tunnels, which leads to instability.",
"A simulation result showing this instability is given in Figure REF .",
"This instability is caused by overlapping tunnels where congestion in one tunnel forces commodities to use the longer tunnels which in turn leads to more congestion.",
"Consider the situation where the number of packets in flight is large for tunnel $(s_1,1,2,d_1)$ .",
"So, commodity 1 traffic is routed through the tunnel $(s_1,3,1,2,d_1)$ .",
"This means that the link (3,1) is being used by commodity 1 packets, which creates congestion for commodity 3 over tunnel $(s_3,3,1,d_3)$ forcing commodity 3 traffic onto tunnel $(s_3,2,3,1,d_3)$ .",
"This problem continues for the tunnels used by commodity 2, which in turn create congestion for the tunnels of commodity 1 forcing its traffic onto tunnel $(s_1,3,1,2,d_1)$ further exacerbating the situation.",
"This cyclical nature of increased congestion makes all the commodities unstable."
],
[
"Centralized solution",
"We begin by providing a centralized optimal routing policy for overlay networks.",
"In this section we assume that the underlay topology is known, and a centralized controller can make the routing decisions at the overlay.",
"The key to obtaining a throughput optimal policy is to realize that the underlay cannot make dynamic decisions, hence, the overlay necessarily has to take into account the capacities of the underlay links while making scheduling decisions.",
"Our algorithm works by choosing the scheduling decision which minimizes the $T$ -slot drift of the quadratic Lyapunov function of the overlay queues [15].",
"This is similar in spirit to the backpressure routing algorithm, which implements a schedule that minimizes the Lyapunov drift at every slot.",
"In our set-up, multislot drift needs to be considered since packets that are sent into the tunnel take several timeslots to come out of the tunnel.",
"In addition to minimizing the drift, we also have to make sure that the underlay queues are bounded.",
"Because we assume that the underlay forwarding scheme is universally stable, we are able to guarantee that underlay queues are bounded once the tunnel capacity constraints (REF ) are met.",
"Thus the algorithm seeks a scheduling decision that minimizes the drift subject to the tunnel capacity constraints.",
"To simplify the notation, in this section, a link between two overlay nodes will be viewed as a tunnel which does not comprise of any underlay nodes.",
"We divide the time into $T$ -slot duration frames and consider minimizing the $T$ -slot drift.",
"At the beginning of each frame, we solve the optimization problem (REF ) in a centralized fashion.",
"The solution to (REF ) minimizes the drift of a quadratic Lyapunov function, while simultaneously satisfying the tunnel capacity constraints.",
"The solution gives us ${F^k_l}^*$ , which is the number of packets commodity $k$ must send into tunnel $l$ in order to minimize the drift.",
"A complete description of the policy is given in Algorithm .",
"Centralized Policy At the beginning of each frame solve the following optimization problem: ${F^k_l}^* = \\operatornamewithlimits{arg\\,max}_{F^k_l} &\\sum _{k,l}F^k_{l} [Q^k_{l_1}(t) - Q^k_{l_{|l|}}(t)] \\\\s.t.",
"&\\sum _{l\\in L:(a,b)\\in l} \\sum _k F^k_{l} \\le c_{ab}, \\forall (a,b) \\in E \\\\& F^k_l \\ge 0 $ Send ${F_l^k}^*$ packets of commodity $k$ into tunnel $l$ each time slot in the frame.",
"If ${F_l^k}^*$ is not an integer, approximate it by sending $p$ packets every $q$ slots so that $\\frac{p}{q} \\approx {F_l^k}^*$ .",
"In the special case when a tunnel $l$ does not share any link with other tunnels, we see that ${F_l^k}^\\star $ can be computed independently of the other tunnels.",
"The commodity $k^*$ is the one with the highest differential backlog $Q^k_{l_1}(t) - Q^k_{l_{|l|}}(t)$ and $F_l^{k^*}$ is chosen to be the capacity of the smallest link in the tunnel.",
"Thus our algorithm resembles the backpressure routing except for the fact that the packets can face large delays while passing the underlay.",
"However, when there is a shared link, all the tunnels that share the link are required to exchange their backlog information.",
"Next we show that the Algorithm stabilizes the network queues if the arrival rate vector $\\lambda $ lies in the interor of the stability region.",
"We use $T$ -slot Lyapunov drift analysis to prove that these queues are strongly stable [15].",
"Theorem 1 For any arrival rate vector $\\lambda $ in the stability region $\\Lambda $ and a large enough frame length $T$ , the policy given in Algorithm stabilizes all the queues in the network.",
"The proof of the theorem is given in the Appendix."
],
[
"Fluid Formulation and Distributed Solution",
"The centralized policy in the previous section requires the knowledge of the underlay topology which might not be known to the overlay.",
"Moreover, having a centralized controller is often impractical.",
"We now consider the fluid model of the network and develop a decentralized policy.",
"Fluid models have been successfully utilized to establish the stability of queueing networks (e.g.",
"[17], [16]).",
"Let $f_{ij}^k$ be the flow assigned to commodity $k$ on the link $(i,j)\\in E$ , and $f_l^k$ be the flow assigned to commodity $k$ on the tunnel $l\\in L$ .",
"Let $f$ denote the vector containing all the flow variables.",
"The arrival rate of commodity $k$ at overlay node $i$ is represented by $\\lambda ^k_i$ , and we assume that the vector of arrival rates $\\lambda $ is in the interior of the stability region.",
"For simplicity, we will assume $\\lambda $ to be a constant, however, if it is time-variying, we note that the technical results hold as long as the arrival rate is bounded at each time-step and the expected value $E[\\lambda (t)]$ exists.",
"The problem of stabilizing the network queues can be formulated as a linear program that finds a feasible flow allocation on all the links and tunnels, $\\max & \\text{ 0 } \\nonumber \\\\s.t.",
"&\\sum _{l:(i,j)\\in l} \\sum _k f^k_{l} \\le c_{ij}, \\forall (i,j): i\\in U, j\\in N \\\\&\\sum _{l:(i,j)\\in l} \\sum _k f^k_{l} \\le c_{ij}, \\forall (i,j): i\\in \\mathcal {O}, j\\in U \\\\& \\sum _j f_{ij}^k + \\sum _{l:l_1=i} f^k_l - \\sum _{j} f^k_{ji} - \\nonumber \\\\&\\qquad \\qquad \\qquad \\sum _{l:l_{|l|} = i} f_l^k - \\lambda ^k_i \\ge 0, \\forall i\\in \\mathcal {O},k \\\\&\\sum _k f^k_{ij} \\le c_{ij}, \\forall i,j \\in \\mathcal {O} \\\\&f^k_{ij}, f^k_l \\ge 0, $ Here, the inequalities (REF ) are the tunnel capacity constraints which are the fluid version of (REF ).",
"Each one of these constraints correspond to an uncontrollable link, i.e.",
"a link between two underlay nodes or a link that goes from underlay to an overlay node.",
"Inequalities () are the link capacity constraints corresponding to the first link in the tunnel, i.e.",
"the links that go from an overlay node to an underlay node.",
"This link is responsible for controlling the rate received by the underlay links.",
"Inequalities () are the flow conservation constraints on the overlay network.",
"Note that the flow conservation constraints are not required for the underlay because for each tunnel $l$ , there is a single route and the flows coming into the underlay are feasible because of (REF ).",
"That is, for a tunnel $l$ , when $f$ is a feasible solution, $f^k_l = f^k_{l_1,l_2} = f^k_{l_2,l_3} = ... = f^k_{l_{|l|-1}, l_{|l|}}.$ Constraints () are the capacity constraints for the overlay links."
],
[
"Dual problem",
"We now formulate the dual problem so that it can be solved with the subgradient descent method [9], [14].",
"Let $q_{ij}$ and $q_i^k$ denote the dual variables for the tunnel constraints (REF ) and the flow conservation constraints () respectively, and let $q$ represent the vector containing all the dual variables.",
"The Lagrangian function is given by, $L(f, q ) &= \\sum _{(i,j):i\\in U}q_{ij} \\left( c_{ij} - \\sum _{l: (i,j)\\in l} \\sum _k f^k_{l} \\right) + \\sum _{i\\in \\mathcal {O}, k} q^k_i \\nonumber \\\\&\\left(\\sum _j f_{ij}^k + \\sum _{l:l_1=i} f^k_l - \\sum _{j} f^k_{ji} - \\sum _{l:l_{|l|} = i} f_l^k - \\lambda ^k_i \\right) \\nonumber \\\\&= \\sum _l \\sum _k f^k_l \\left(q^k_{l_1}-\\sum _{(i,j)\\in l:i \\in U}q_{ij}-q^k_{l_{|l|}}\\right) + \\nonumber \\\\& \\sum _{(i,j)} \\sum _k f^k_{ij} ( q^k_{i} - q^k_{j}) + \\sum _{(i,j):i\\in U} q_{ij}c_{ij} - \\sum _{i\\in \\mathcal {O}, k} q^k_i \\lambda ^k_i ,$ where the second equality is obtained by rearranging the terms so that the flow variables are factored out instead of the dual variables.",
"Let $X$ be a set such that any $f \\in X$ satisfies the constraints (), () and ().",
"Note that these constraints can be enforced locally by an overlay node using only locally available information.",
"This property will be essential in designing the decentralized algorithm.",
"The dual objective function corresponding to the problem (REF ) is $D(q) = \\max _{f \\in X} L(f,q).$ The dual problem is given by, $\\min _q &\\quad D(q) \\\\\\text{s.t.}",
"&\\quad q \\ge 0.",
"\\nonumber $ Since the primal problem (REF ) is a linear program, the duality gap is zero (Slater's condition [14]).",
"Hence, solution of the dual (REF ) yields a feasible flow allocation."
],
[
"Distributed solution",
"The subgradient method works by initializing the dual variables with a value $q(0)\\ge 0$ , and then iterating on them until it converges to optimal $q^\\star $ .",
"Each iteration involves computing a subgradient $g$ of $D$ at the current value of the dual variables, then updating the dual variables as follows: $ q(t+1) = \\left[q(t) - \\alpha (t) g(t) \\right]^+.",
"$ Here $\\alpha (t)$ is positive scalar step-size.",
"The dual variables are known to converge to the optimal $q^\\star $ if the step-sizes $\\alpha (t)$ are chosen appropriately.",
"However, if $\\alpha (t)\\equiv \\alpha $ , then the iterates (REF ) converge to a bounded neighbourhood of $q^\\star $ [14].",
"Let ${f^k_l}^*$ and ${f^k_{ij}}^*$ be the values of flow variables which maximize the Lagrangian $L(f,q)$ over $f \\in X$ for a fixed $q$ , i.e.",
"$D(q) = L(f^*,q)$ .",
"From [14] we know that a subgradient of $D(q)$ is given by a vector $g$ with entries as, $& g_{ij} = c_{ij} - \\sum _{l: (i,j) \\in l} \\sum _k {f^k_{l}}^*, \\text{ and } \\\\&g^k_{i} =\\sum _j {f_{ij}^k}^* + \\sum _{l:l_1=i} {f^k_l}^* - \\sum _{j} {f^k_{ji}}^* - \\sum _{l:l_{|l|} = i} {f_l^k}^* - \\lambda ^k_i.",
"$ Now we can use the recursive equation (REF ) to update the dual variables.",
"The only necessary step that we haven't covered so far is the computation of ${f^k_l}^*$ and ${f^k_{ij}}^*$ .",
"A careful observation of equation (REF ) and the set $X$ shows that this is a simple optimization problem that can be solved in a decentralized fashion.",
"The objective is a weighted sum of the flow variables, and the constraints that form $X$ are the link capacity constraints.",
"At a high level, for each overlay link, the solution chooses the maximum value of the flow variable that corresponds to the commodity with the highest positive weight.",
"A complete algorithm to compute the optimal flow variables and update the dual variables is given in Algorithm REF .",
"[h!]",
"Optimal Overlay Routing Policy (OORP) At each time-step $t$ , overlay node $i$ does the following: Optimal flow variables computation (used to obtain the subgradients): An overlay to overlay link $(i,j)$ computes the flow variables ${f^k_{ij}}^*$ : Let $k^{opt} \\in \\operatornamewithlimits{arg\\,max}_k q^k_i - q^k_j,$ ties are broken arbitrarily.",
"The weight of commodity $k^{opt}$ in this link is $W^{opt}_{ij} = q^{k^{opt}}_i - q^{k^{opt}}_j$ .",
"For $k=k^{opt}$ , ${f^{k}_{ij}}^* = \\left\\lbrace \\begin{array}{l} c_{ij} \\text{ if } W^{opt}_{ij} > 0 \\\\ 0, \\text{ otherwise} \\end{array} \\right.$ For all $k \\ne k^{opt}$ , ${f^k_{ij}}^* = 0$ .",
"Each overlay to underlay link $(i,j)$ computes the flow variable $f^k_l$ for all $l:(l_1,l_2) = (i,j)$ : Let $(l^{opt},k^{opt}) \\in \\operatornamewithlimits{arg\\,max}_{l:(l_1,l_2) = (i,j),k} q^k_{l_1}-\\sum _{(a,b)\\in l:a \\in U}q_{ab}-q^k_{l_{|l|}}.",
"$ Ties are broken arbitrarily.",
"Let the weight of commodity $k^{opt}$ in the tunnel be $W^{opt}_l = q^{k^{opt}}_{{l^{opt}}_1}-\\sum _{(a,b)\\in l^*:a \\in U}q_{ab}-q^{k^{opt}}_{{l^{opt}}_{|l^{opt}|}}.$ For $(l,k) = (l^{opt},k^{opt})$ , ${f^k_l}^* = \\left\\lbrace \\begin{array}{l} c_{ij} \\text{ if } W^{opt}_l> 0 \\\\ 0, \\text{ otherwise} \\end{array} \\right.$ For all $(l,k):l \\ne l^{opt}$ or $k \\ne k^{opt}$ , ${f^k_l}^* = 0$ .",
"Data transmission: Transmit ${f_{ij}^k}^*$ amount of commodity $k$ traffic into each overlay to overlay link $(i,j)$ and ${f_{l}^k}^*$ amount of commodity $k$ traffic into each tunnel $l$ .",
"Dual variables update: Performed by an overlay node i: $q^k_i(t+1) &= \\left[q(t) - \\alpha (t) \\left(\\sum _j {f_{ij}^k}^* + \\sum _{l:l_1=i} {f^k_l}^* \\right.",
"\\right.",
"\\nonumber \\\\& \\qquad \\left.",
"\\left.",
"- \\sum _{j:(j,i)\\in E} {f^k_{ji}}^* - \\sum _{l:l_{|l|} = i} {f_l^k}^* - \\lambda ^k_i \\right) \\right]^+ $ Performed by an underlay node i: $q_{ij}(t+1) &= \\left[q(t) - \\alpha (t) \\left( c_{ij} - \\sum _{l: (i,j) \\in l} \\sum _k {f^k_{l}}^* \\right) \\right]^+ $"
],
[
"Queue-lengths as dual variables",
"The subgradient descent algorithm presented in the Algorithm REF requires the network to explicitly keep track of the dual variables.",
"In order to implement the algorithm in a decentralized fashion, each underlay node $i$ needs to maintain a dual variable $q_{ij}$ for each link $(i,j)$ , and each overlay node $i$ needs to maintain a dual variable $q^k_i$ for each commodity $k$ .",
"This is a reasonable assumption for the overlay nodes, but not justified for the uncontrollable underlay.",
"To get around similar problems of not having a dual variable, works such as [12], [11], etc.",
"have proposed approximating them with the corresponding queue lengths.",
"The argument behind this procedure is that the subgradients are proportional to the change in queue-lengths, so that the queue-lengths will move in the same direction as the dual variables.",
"Next, we give an example in which this proportionality does not hold.",
"In spite of this issue, we show that the queue-lengths can provide a good approximation for the dual variables.",
"We first observe that the dual variable update equations (REF ) and (REF ) are the same as the queue update equations when the flows sent into the tunnels $f_l^k$ are feasible for the underlay, i.e.",
"when no queues buildup in the underlay.",
"But when the flows do not satisfy the tunnel capacity constraints, the underlay queues build up, and the flows get reduced from their initial value as they pass through the bottleneck links.",
"This decrease in the flow size is not captured in these dual variable update equations (REF ), (REF ).",
"Consider the simple network shown in Figure REF .",
"There is one commodity, $k=1$ , with source node 1 and destination node 4, and a single tunnel $l=(1,2,3,4)$ .",
"Suppose that at a certain iteration, $q_1^1 > q_4^1$ , hence ${f_l^k}^* = 3$ .",
"This flow into the tunnel gets bottlenecked at link $(2,3)$ so node 3 only receives a flow of 1.",
"In this situation, equation (REF ) predicts that the queue-length for $q_{34}$ would increase because a flow of size 3 was sent into the tunnel and the capacity of the link is 2, however this queue can only decrease or stay unchanged at 0.",
"Figure: Link (3,4)(3,4) never builds a queue as the flow gets bottlenecked by (2,3).To capture this reduction of the flow sizes in the tunnel, we model the queuing in the network as follows: $\\hat{q}^k_i(t+1) &= \\left[\\hat{q}(t) - \\alpha (t) \\left(\\sum _j {f_{ij}^k}^* + \\sum _{l:l_1=i} {f^k_l}^* \\right.",
"\\right.",
"\\nonumber \\\\& \\qquad \\left.",
"\\left.",
"- \\sum _{j:(j,i)\\in E} {f^k_{ji}}^* - \\sum _{l:l_{|l|} = i} \\epsilon ^k_l(i) {f_l^k}^* - \\lambda ^k_i \\right) \\right]^+ \\\\\\hat{q}_{ij}(t+1) &= \\left[\\hat{q}(t) - \\alpha (t) \\left( c_{ij} - \\sum _{l: (i,j) \\in l} \\sum _k \\epsilon ^k_l(i,j) {f^k_{l}}^* \\right) \\right]^+ $ where $\\epsilon ^k_l(i), \\epsilon ^k_l(i,j) \\in [0,1]$ represent the reduction suffered by the corresponding flows before arriving at node $i$ .",
"These quantities are implicitly determined by the network at each time-step depending on the scheduling policy in the underlay.",
"In the example presented above, for any work conserving scheme, $\\epsilon _l^k(3,4) = 1/3$ .",
"We will show that for any value of $\\epsilon $ in the set $[0,1]$ the queue-lengths will converge to the optimal dual variables.",
"Let $g$ be the true subgradient of $D$ at $q$ , and $\\hat{g}$ be the approximate subgradient after the reduction, then we can represent the queuing equation as $\\hat{q}(t+1) = \\left[\\hat{q}(t) - \\alpha (t) \\hat{g}(t) \\right]^+,$ and $\\hat{g} \\ge g$ .",
"Before we prove the convergence, we state the following preliminary lemma.",
"Lemma 1 The vector $q^*=0$ is an optimal solution to the dual problem (REF ).",
"Since the objective of the primal problem is 0, a feasible solution to the primal is given by any feasible flow allocation $f_{ij}^k$ .",
"Since $q=0$ is a feasible dual solution, and any feasible $f_{ij}^k$ together with $q=0$ satisfy the complementary slackness condition (Theorem 4.5 in [13]), the proof follows.",
"This shows that the optimal solution corresponds to queue lengths equal to zero which makes, sense intuitively because any feasible flow allocation in the fluid domain doesn't require queuing.",
"Let $G$ be a constant such that it bounds the Euclidean norm of the subgradients of the dual function $D(q)$ for all possible values of $q$ , i.e.",
"$G > \\Vert g\\Vert $ .",
"From equations (REF )-(), we can see that the subgradients are bounded because the flow variables are bounded by link capacities and arrival rates are bounded by assumption.",
"So G is finite.",
"For simplicity we fix $\\alpha (t) = 1$ and present the following convergence result.",
"Theorem 2 Let us approximate the dual variables $q$ with the queue-lengths $\\hat{q}$ that evolve according to equations (REF )-().",
"Using the dual subgradient descent algorithm with $\\alpha (t)=1$ , the queue lengths converge to the set $S= \\left\\lbrace \\hat{q}: D(\\hat{q}) \\le \\frac{1}{2}G^2 \\right\\rbrace $ .",
"We will show that $||\\hat{q}(t+1) - q^*||^2 < ||\\hat{q}(t) - q^*||^2$ when $q(t)$ is outside the set S. Because $q^*=0$ from Lemma REF , it suffices to show that $||\\hat{q}(t+1)||^2 < ||\\hat{q}(t)||^2$ .",
"We have, $||\\hat{q}(t+1)||^2 &= \\left\\Vert \\hat{q}(t) - \\hat{g} \\right\\Vert ^2$ Since $\\hat{g} \\ge g$ , $||\\hat{q}(t+1)||^2 &\\le \\left\\Vert \\hat{q}(t) - g \\right\\Vert ^2 \\\\&= \\Vert \\hat{q}(t)\\Vert ^2 - 2 \\hat{q}(t)^T g + \\left\\Vert g\\right\\Vert ^2$ Our algorithm chooses $g$ to be a subgradient of $D(.",
")$ at $\\hat{q}(t)$ .",
"So, $ D(x) \\ge D(\\hat{q}(t)) + \\left(x - \\hat{q}(t) \\right)^Tg, \\forall x \\in \\mathbb {R}^{m},$ wehre $m$ is the dimension of $\\hat{q}$ .",
"Taking $x = 0$ , $D(\\hat{q}(t)) \\le \\hat{q}(t)^T g (f(t)^*)$ So, $||\\hat{q}(t+1)||^2 \\le ||\\hat{q}(t)||^2 - 2 D(\\hat{q}(t)) + G^2.$ Hence when, the $\\hat{q}$ is far away from the optimal, specifically when $D(\\hat{q}(t)) > \\frac{1}{2}G^2$ , it moves towards the optimum in the next time-step.",
"Hence, we will use queue-lengths instead of the dual variables in the implementation of OORP.",
"This will allow us to use the policy presented in Algorithm REF without having to perform the dual variables update."
],
[
"Underlay sources and destinations",
"The problem formulation given in beginning of Section V assumes that all the flows go from one overlay node to another.",
"However, this assumption can be easily removed.",
"Any flow that originates in the underlay be routed over a single path using the underlay routing scheme.",
"Hence these flows can be represented simply as a reduction in the link capacities in the constraints (5) and (6) for the links traversed by this flow.",
"Our algorithm stays unchanged because it is agnostic to the change in the link capacities at the underlay.",
"OORP is also optimal when the underlay is a destination because destination nodes do not perform any routing."
],
[
"Rate control",
"It is well known that subgradient descent is a general method to solve convex optimization problems.",
"The dual gradient descent algorithm has been used derive distributed solutions to network utility maximization problems (e.g.",
"[9], [11]).",
"In the overlay network setting we can get a distributed solution to the utility maximization problem of the form: $\\max \\sum _{k\\in K, i\\in \\mathcal {O}} U^k_i(\\lambda _i^k)$ where $U^k_i(.",
")$ is concave and strictly increasing.",
"In this setting, we can assume that there is an infinite backlog at the sources, and the rates $\\lambda _i^k$ are chosen to maximize the total network utility.",
"We can use the same derivation technique as in section REF to obtain a distributed algorithm.",
"The algorithm is very similar to OORP with an added rate controller at each source.",
"The rate control algorithm so obtained is standard, and the joint rate control and routing algorithm can be written as follows: [ht] Rate control algorithm for the utility maximization problem At each time-step $t$ : Source node $i\\in \\mathcal {O}$ for commodity $k$ chooses the rate ${\\lambda _i^k}^*$ as follows: ${\\lambda _i^k}^* = \\operatornamewithlimits{arg\\,max}_{0\\le \\lambda _i^k \\le \\mathcal {M}^i_k} {U^k_i(\\lambda _i^k)}-q^k_i \\lambda ^k_i.$ Here, $\\mathcal {M}^i_k$ is a finite upper bound on the rate that the source $i$ can receive.",
"All the overlay nodes use OORP for routing."
],
[
"Unknown Underlay Queues",
"In the previous section we showed that the dual subgradient descent algorithm can be used to compute a feasible rate for each commodity on each link.",
"We also showed that the queue lengths can be used to approximate the subgradient.",
"However, typically legacy devices may not be able to send queue-lengths to the sources.",
"In this section, we will present two approaches to estimate the required queue-length information.",
"The first approach will estimate it using the delay experienced by the packets.",
"The second approach will involve sending probe packets at a certain time intervals that collect the queue-length information in the tunnels."
],
[
"Delay based approaches",
"From equation (REF ) it can be seen that in order to compute the subgradients we only need the total backlog in the tunnel, i.e.",
"we don't need the length of individual queues.",
"A natural approach to estimate the total backlog in a tunnel is by using the time it takes for a packet to traverse it.",
"To implement this method, each tunnel $l$ maintains a delay variable $D_l$ .",
"When a packet is sent into a tunnel, the sending node stamps the packet with the current time.",
"When the packet exits the tunnel, the difference between the current time and the time-stamp on the packet is used to update $D_l$ .",
"When computing the optimal flow variables, in equation (REF ) of OORP we substitute the sum of the underlay queues-lengths, $\\sum _{(a,b)\\in l:a \\in U}q_{ab}$ , with $D_l$ .",
"For this approach, we assume that the underlay uses a FIFO queuing model which is a common forwarding scheme.",
"Hence, this method requires no modification to the underlay.",
"A similar approach has been used by TCP Vegas to solve a network utility maximization problem [12].",
"Although this approach is simple and does not require cooperation from the underlay, the queue-length estimates obtained by this method can be arbitrarily bad.",
"Consider a FIFO queue that is empty at time zero.",
"As shown in Figure REF (a), it has an incoming rate of 2 and outgoing capacity of 1.",
"We want to estimate the queue-length at time $t$ by using packet delays.",
"To see the problem with this approach, let us consider a situation when 2 packets arrive at the queue at every time-slot for the first $\\tau $ time-slots, and no arrivals happen after that.",
"In this situation, the actual queue length grows at the rate of 1 for the first $\\tau $ time-slots, and then it decreases at the rate of 1 packet per time-slot until the queue is empty.",
"On the other hand, the delay increases at the rate of $\\frac{1}{2}$ , and the last packet (that arrives at the $\\tau $ th time-slot) sees a delay of 100 because there are 99 packets in the queue at that time.",
"So at time $2\\tau $ when the queue is emptied, the packet received will have suffered a delay of $\\tau $ time-slots giving a queue-length estimate of $\\tau $ , whereas the actual queue-length at that time is zero.",
"Furthermore, the estimate stays bad until another arrival happens.",
"This problem is illustrated in Figure REF (b).",
"These arbitrarily bad estimates lead to sub-optimality of OORP which we will observe in the simulations.",
"A simple way to improve the estimate is to send empty probe packets when real packets are not available for some time period $\\mathcal {T}$ .",
"A similar approach has been shown to achieve throughput optimality in a special scenario in [20].",
"This approach quickly identifies when a queue becomes empty in the absence of new data packets, and the control algorithm can react accordingly.",
"Although this approach corrects the estimate within P time-slots, it can still suffers from the arbitrarily bad estimation errors.",
"As shown in Figure REF (c), at time $2\\tau $ the estimate is $\\tau $ whereas the actual queue-length is zero.",
"Thus, we propose the following approach using explicit probes.",
"Figure: The actual queue-length of a single FIFO queue and its estimate calculated using delay.",
"The arrivals happen at the rate of 2 packets per time-slot for the first τ\\tau time-slots, and there are no arrivals after that.",
"Service rate is fixed at 1 packet per time-slot."
],
[
"Priority probe approach",
"In this approach, we assume that the underlay nodes are capable of stamping the current queue-lengths into a special type of packets called the probe packets.",
"We also assume that these packets are given higher priority compared to the data packets, and they do not consume link capacity because they are very small in size.",
"These packets are generated by each tunnel at a fixed time-intervals $\\mathcal {T}$ .",
"When a probe packet exits a tunnel, the sum of the queue-lengths it has collected can be used to compute the optimal flow variables in Algorithm REF .",
"We can see that this approach results in a much more accurate estimation of the backlog compared to the delay based approach.",
"However, the value of $\\mathcal {T}$ can have a significant impact on the performance on the algorithm.",
"We will study its impact in the next section via simulations."
],
[
"Simulation Results",
"We present several simulation results to evaluate the performance of the optimal overlay routing policy (OORP) given in Algorithm REF .",
"First, we will ascertain that this algorithm is in fact optimal for the network in which the OBP policy of [6] was suboptimal.",
"Then we evaluate the effect that different methods of estimating the queue-lengths have on the performance of the algorithm.",
"Next we will study the performance of our algorithm when there is uncontrolled background traffic in the underlay.",
"Finally, we will simulate the rate control algorithm to show that it achieves the maximum throughput."
],
[
"OORP on the counterexample to OBP",
"We reconsider the network from Section III, shown in Figure REF .",
"The network has three commodities and it can support a maximum arrival rate vector of $\\lambda _{\\max }^{simA} = [1,1,1]$ .",
"We simulate the network under three different policies: the backpressure policy (BP), the overlay backpressure policy (OBP) and our policy, OORP.",
"The simulations are conducted at different loads $\\rho = 0.5, 0.55, ..., 1$ .",
"For each policy the arrivals are Poisson distributed with rates $\\lambda = \\rho \\lambda _{\\max }$ .",
"The result of the simulations is given in Figure REF .",
"Figure: Performance of different routing algorithms on the overlay network shown in Figure .The BP algorithm executed on the overlay network does not account for the underlay nodes and simply views a tunnel $l$ as a link between two overlay nodes with capacity $c_{l_1 l_2}$ .",
"In the plot we can see that this algorithm becomes unstable around the load of $0.56$ .",
"This is as expected because for each commodity the backpressure policy uses both tunnels equally since they have equal weights.",
"The end node of both the tunnels is a destination, which has zero backlog, and BP does not account for the backlog in the underlay.",
"So the weigh for each tunnel of commodity $k$ is equal to $Q_{s_k}^k$ .",
"When the algorithm uses the longer tunnel, the network becomes unstable for relatively low load.",
"The plot also shows that OBP is suboptimal and OORP achieves maximum throughput.",
"We discussed the suboptimality of OBP in Section III.",
"The main reason was that the OBP policy could not avoid using the longer tunnel which gave raise to a cycle of increased congestion.",
"But in OORP, when the underlay queues-lengths are positive, the shorter tunnels have a higher weight than the longer tunnels.",
"For example, for commodity 1, the weight of the shorter tunnel $(s_1,1,2,d_1)$ is $Q^1_{s_1} - Q_{12}-Q_{2d_1}$ and the weight of the longer tunnel $(s_1,3,1,2,d_1)$ is $Q^1_{s_1}-Q_{31} - Q_{12}-Q_{2d_1}$ .",
"So when the underlay queues are large, the longer tunnel needs a lot more backlog at the source than the shorter tunnel for its weight to be positive.",
"This causes OORP to avoid using the longer tunnels when the network is congested."
],
[
"Estimated Tunnel Backlog",
"We consider the network given in Figure REF to observe the effect of estimating the backlog in the tunnels.",
"In this network, all the links are bidirectional, composed of two unidirectional links.",
"The links between an overlay and an underlay node have capacity 2 in each direction.",
"All other links have unit capacity in both directions.",
"We will simulate the network with two commodities.",
"The first commodity is defined by the source-destination pair (1,3) and the second is defined by (2,4).",
"For these commodities the network supports a max-flow vector of $\\lambda _{\\max }^{simB} = [2,2]$ .",
"The simulation is performed at various load levels and the arrivals are Poisson distributed.",
"The underlay uses the shortest path routing hence creating a large number of available tunnels.",
"Node 1 can send packets to node 3 directly via node 7 or 10 using the tunnels (1,7,5,6,3) and (1,10,7,5,6,3) respectively.",
"However, these tunnels overlap, hence there is no benefit in using both of them.",
"To achieve the throughput of two, node 1 must send its traffic through node 2 and have it forward it to node 3.",
"Similarly node 2 must send some of its traffic through node 1 in order to achieve high throughput.",
"Observing the organization of the tunnels in the network, we can see that using the wrong tunnel might cause the network to lose throughput.",
"In addition, the tunnels form cycles in the overlay topology.",
"These features make this topology challenging for a routing algorithm to achieve the optimal throughput.",
"Figure: Physical and overlay network topology for simulations in Sections B and C.The result of the simulations under different load levels is given in Figure REF .",
"We can see that the delay approach, which uses packet delay as an estimate of the tunnel backlog, does not provide optimal throughput.",
"Although probing the delay in the network with control packets improves the performance, it is still suboptimal.",
"This happens because when the backlog is large the estimation error of this approach can also be large as described in Section REF .",
"We can also see that the probing approach achieves optimal throughput, and its performance is close to that of using the actual queue-lengths.",
"The estimates obtained by this approach are much more accurate than those from the delay approach because they are not affected by the amount of backlog in the network.",
"When $\\mathcal {T}$ is increased, the stale estimate is used for a longer time period, so the performance of the algorithm degrades.",
"Figure: Performance of OORP under different measures of tunnel backlog."
],
[
"Background traffic",
"So far we have assumed that all the traffic in the network belongs to the overlay network.",
"However, in real networks the underlay can be routing other traffic not generated by the overlay nodes.",
"Next, we will study the performance of our algorithm under such traffic.",
"We expect the OORP to be throughput optimal under stable background traffic in the underlay because such traffic can be thought of as a reduction in the link capacities in inequalities (REF ) as described in Section REF .",
"We again consider the network from Figure REF with two commodities (1,3) and (2,4).",
"We inject two flows of background traffic: first going from node 7 to 6 along the path (7,5,6) with the arrival rate of 0.5, and second going from 8 to 14 along the path (8, 9, 12, 14) with the arrival rate of 0.2.",
"The arrivals happen according to the Poisson process.",
"Note that the first background flow blocks commodity 1's tunnel (1,7,4,5) and the second flow blocks commodity 2's tunnel (2,12,14,13,4); both the tunnels are essential for achieving the max flow vector $\\lambda ^{simB}_{\\max }$ .",
"This reduces the maximum supportable arrival rates for the two commodities to $\\lambda ^{simC}_{\\max }=[1.5, 1.8]$ .",
"Figure REF shows the result of the simulation.",
"We can see that all the approaches except for the delay based approaches achieve the maximum throughput.",
"Figure: Performance of OORP in a network with background traffic."
],
[
"Rate control",
"To observe our rate controller at work, we consider the network from Section REF with a minor modification as shown in Figure REF .",
"We add a new overlay node 15 and a new commodity (15, 14) to the network.",
"Node 15 connects to node 5 with a directed link (15, 4) which has a unit capacity.",
"We constrain the third commodity to use the tunnel provided by the shortest path (5, 6, 9, 12, 14).",
"For all three sources, the utility function is chosen to be $20\\log (\\lambda )$ and $M_i^k=20$ .",
"Note that the addition of the third commodity makes the simulation more challenging because the rate that maximizes total throughput is not the same as the rates that maximizes utility.",
"We assume that the backlog information of each tunnel is available to the overlay nodes instantaneously.",
"Figure: Topology for the rate control experiment.",
"The dotted lines show the tunnel assigned to the third commodity and the background traffic.Constrained by the link capacities and the background traffic, the maximum throughputs for the commodities 1,2, and 3 are 1.5, 1.8, and 1 respectively.",
"However, the third commodity interferes with both commodities 1 and 2, hence the throughput of [1.5, 1.8, 1] is not achievable.",
"From the plot in Figure REF we can see that the throughput vector converges to [1, 1.3, 0.5] which maximizes the total utility.",
"We can see that this throughput vector has a smaller sum than the sum of the maximum throughput supported in Section REF .",
"That is, the network could have supported higher throughput by giving zero throughput to the third commodity, but that would have decreased the utility of the network.",
"Figure: Throughput achieved by the rate control algorithm with OORP converges to the rate that maximizes utility."
],
[
"Conclusion",
"We showed that the existing algorithms for routing traffic in an overlay network are suboptimal, and developed a throughput optimal policy called the Optimal Overlay Routing Policy (OORP).",
"This policy is distributed and can also be used with a rate controller to maximize network utility.",
"Our algorithm requires the knowledge of congestion at the underlay, which might not be available to the overlay nodes.",
"Hence we proposed different approaches to estimating underlay congestion.",
"Simulations results show that OORP outperforms OBP and that estimating congestion using probing mechanism is effective.",
"Future research will include obtaining better estimates for the congestion in the tunnels with minimum support from the underlay nodes and reducing the delay experienced by the packets in the network.",
"[Proof of Theorem REF ]"
],
[
"Stationary policy $\\pi $ ",
"In order to prove the stability of the centralized policy, we need a stationary policy that stabilizes the network.",
"For any arrival rates ${\\bf \\lambda }$ such that ${\\bf \\lambda } + {\\bf \\epsilon } \\in \\Lambda $ , $\\epsilon >0$ , we know that there exist a feasible flow allocation vectors $(f^k_{l})_{l\\in L, k \\in K}$ such that for any overlay node $n$ , $ \\sum _{l\\in L: l_1 = n} f^k_{l} - \\sum _{l\\in L: l_{|l|} = n} f^k_{l} = \\lambda ^k_n + \\epsilon .$ This vector can be obtained by solving the multi-commodity flow problem.",
"We assume that these flow variables can be closely approximated by rational numbers.",
"So there exists integers $p^k_{l}$ and $q$ such that $f^k_{l} = p^k_{l}/q$ .",
"The time-slot are divided into $T$ slot long frames.",
"The policy $\\pi $ simply sends $p^k_{l}$ amount of commodity $k$ packets every $q$ time-slots in each frame.",
"Because the underlay is using a universally stable forwarding scheme and the burstiness constraints are satisfied, the underlay queues are deterministically bounded by a constant $B$ [5].",
"Also note that all the capacity constraints are satisfied every $q$ time-slots.",
"Hence, $\\pi $ stabilizes $\\lambda $ .",
"Let $F^k_{l}(t+\\tau ,\\pi )$ represent the number of packets sent into tunnel $l$ by node $l_1$ at time $t+\\tau $ under policy $\\pi $ .",
"Let $\\bar{F}^k_{l}(t+\\tau ,\\pi )$ represent the number of packets that are received by node $l_{|l|}$ at time $t$ from tunnel $l$ under policy $\\pi $ .",
"Note that $F^k_{l}(t+\\tau ,\\pi ) = \\bar{F}^k_{l}(t+\\tau ,\\pi )$ only if the tunnel $l$ is a direct link between two overlay nodes.",
"If a tunnel passes through the underlay, it can take a bounded amount of time for the packets to exit the tunnel.",
"Now, we prove the following lemma that will be used in proving the theorem.",
"Lemma 2 For the proposed randomized policy $\\pi $ $&\\mathbb {E}\\left[ \\sum _{l\\in L: l_1 = n} \\sum _{\\tau =0}^{T-1}F^k_{l}(t+\\tau ,\\pi ) - \\right.",
"\\nonumber \\\\& \\left.\\left.",
"\\sum _{{l\\in L: l_{|l|} = n} }\\sum _{\\tau =0}^{T-1} \\bar{F}^k_{l}(t+\\tau ,\\pi ) \\right|{\\bf Q}(t)\\right] \\ge T(\\lambda _n^k + \\epsilon ) - B, \\forall n. \\nonumber $ $&\\mathbb {E}\\left[ \\sum _{l\\in L: l_1 = n} \\sum _{\\tau =0}^{T-1}F^k_{l}(t+\\tau ,\\pi ) - \\right.",
"\\nonumber \\\\&\\qquad \\qquad \\left.",
"\\left.",
"\\sum _{{l\\in L: l_{|l|} = n} }\\sum _{\\tau =0}^{T-1} \\bar{F}^k_{l}(t+\\tau ,\\pi ) \\right|{\\bf Q}(t)\\right] \\nonumber \\\\&=\\mathbb {E}\\left[ \\sum _{l\\in L: l_1 = n} \\sum _{\\tau =0}^{T-1}F^k_{l}(t+\\tau ,\\pi ) - \\right.",
"\\nonumber \\\\& \\qquad \\qquad \\left.",
"\\sum _{{l\\in L: l_{|l|} = n} }\\sum _{\\tau =0}^{T-1} \\bar{F}^k_{l}(t+\\tau ,\\pi ) \\right] \\\\&\\ge \\mathbb {E}\\left[ \\sum _{l\\in L: l_1 = n} \\sum _{\\tau =0}^{T-1}F^k_{l}(t+\\tau ,\\pi ) - \\right.",
"\\nonumber \\\\& \\qquad \\qquad \\left.",
"\\sum _{{l\\in L: l_{|l|} = n} }\\sum _{\\tau =0}^{T-1} F^k_{l}(t+\\tau ,\\pi ) - B \\right] \\\\&= T\\sum _{l\\in L: l_1 = n} \\frac{p^k_{l}}{q} - T\\sum _{l\\in L: l_{|l|} = n} \\frac{p^k_{l}}{q} - B\\\\&= T\\sum _{l\\in L: l_1 = n} f^k_{l} - T\\sum _{l\\in L: l_{|l|} = n} f^k_{l} - B\\\\&= T(\\lambda _n^k + \\epsilon ) - B \\nonumber $ Here $B$ is a finite constant representing the maximum amount of backlog in the underlay network.",
"We use this constant to obtain inequality (REF ) and equation (REF ) to obtain the last equality."
],
[
"Analysis of TBP",
"We know that the underlay queues are stable because the traffic injected into the tunnels satisfy the burstiness constraints and the underlay employs a universally stable forwarding policy [5].",
"Next we prove the stability of overlay queues.",
"The queue evolution of an overlay node $n$ can be written as: $Q^k_n(t+1) &= \\left[ Q^k_n(t) - \\sum _{l: l_1 = n} F^k_{l}(t) + \\sum _{l:l_{|l|}=n} \\bar{F}^k_{l}(t) + A^k_n (t)\\right]^+ \\\\&\\le \\left[ Q^k_n(t) - \\sum _{l: l_1 = n} F^k_{l}(t) \\right]^+ + \\sum _{l:l_{|l|}=n} \\bar{F}^k_{l}(t) + A^k_n (t)$ Here $F^k_{l}(t)$ represents the amount of packets injected into the tunnel $l$ at time $t$ , and $\\bar{F}^k_{l}(t)$ represents the number of packets that exit tunnel $l$ at time $t$ .",
"Then the queue length after $T$ slots can be bounded as follows $Q^k_n(t+T) \\le &\\left[ Q^k_n(t) - \\sum _{l: l_1 = n} \\sum _{\\tau =0}^{T-1}F^k_{l}(t+\\tau ) \\right]^+ +\\nonumber \\\\& \\sum _{l:l_{|l|}=n} \\sum _{\\tau =0}^{T-1} \\bar{F}^k_{l}(t+\\tau ) + \\sum _{\\tau =0}^{T-1}A^k_n (t+\\tau ) \\nonumber \\\\\\le &\\left[ Q^k_n(t) - \\sum _{l: l_1 = n} \\sum _{\\tau =0}^{T-1}F^k_{l}(t+\\tau ) \\right]^+ +\\nonumber \\\\& \\sum _{l:l_{|l|}=n} \\sum _{\\tau =0}^{T-1} F^k_{l}(t+\\tau ) + \\sum _{\\tau =0}^{T-1}A^k_n (t+\\tau ) + B \\nonumber $ The first inequality comes from considering all the arrivals and departures in a T-slot interval in a single slot.",
"We get the second inequality by bounding the backlog in the underlay nodes by a constant $B$ .",
"Now to prove the theorem, consider the quadratic Lyapunov function $L({\\bf Q}(t)) = \\sum _{k,n} \\left( Q^k_n(t) \\right)^2.$ The T-slot drift is given by: $\\Delta _T = &\\mathbb {E}\\left[L({\\bf Q}(t+T)) - L({\\bf Q}(t)) | {\\bf Q}(t) \\right] \\nonumber \\\\\\le &T^2K + \\sum _{k,n} Q^k_n(t) \\left(T \\lambda ^k_n + B\\right) + \\sum _{k,n} Q^k_n \\nonumber \\\\&\\left.",
"\\mathbb {E} \\left[ \\sum _{\\tau =0}^{T-1} \\sum _{l:l_1=n} F^k_{l}(t+\\tau ) - \\sum _{\\tau =0}^{T-1} \\sum _{l:l_{|l|}=n} F^k_{l}(t+\\tau ) \\right|{\\bf Q}(t)\\right] \\\\= &T^2K + \\sum _{k,n} Q^k_n(t) \\left(T \\lambda ^k_n + B\\right) \\nonumber \\\\&- \\mathbb {E}\\left[\\left.",
"\\sum _{\\tau =0}^{T-1} \\sum _{k,l} F^k_{l}(t+\\tau ) \\left( Q^k_{l_1}(t) - Q^k_{l_{|l|}}(t) \\right) \\right| {\\bf Q}(t) \\right] $ The TBP policy minimizes the right hand side of inequality (REF ) at every time-slot.",
"Hence it also minimizes the right hand side of inequality ().",
"So we can bound the drift by the rate variables chosen by the stationary policy.",
"$\\Delta _T \\le &T^2K + \\sum _{k,n} Q^k_n(t) \\left(T \\lambda ^k_n + B\\right) + \\sum _{k,n} Q^k_n \\nonumber \\\\& \\mathbb {E} \\left[ \\sum _{\\tau =0}^{T-1} \\sum _{l:l_1=n} F^k_{l}(t+\\tau , \\pi ) \\right.\\nonumber \\\\&\\left.\\left.- \\sum _{\\tau =0}^{T-1} \\sum _{l:l_{|l|}=n} F^k_{l}(t+\\tau , \\pi ) \\right|{\\bf Q}(t)\\right] \\\\\\le & T^2K + \\sum _{k,n} Q^k_n(t) \\left(T \\lambda ^k_n + B \\right) - \\sum _{k,n} Q^k_n (T\\lambda _n^k \\nonumber \\\\&+ T\\epsilon -B) \\\\\\Delta ^{TBP}_T \\le &T^2K - \\sum _{k,n}Q^k_n(t) (T\\epsilon -2B)$ We use Lemma REF to obtain (REF ).",
"The drift is negative when $T>2B/\\epsilon $ and the queues are large.",
"From [15] we know that the overlay queues are strongly stable."
]
] | 1612.05537 |
[
[
"Optical properties of azobenzene-functionalized self-assembled\n monolayers: Intermolecular coupling and many-body interactions"
],
[
"Abstract In a joint theoretical and experimental work the optical properties of azobenzene-functionalized self-assembled monolayers (SAMs) are studied at different molecular packing densities.",
"Our results, based on density-functional and many-body perturbation theory, as well as on differential reflectance (DR) spectroscopy, shed light on the microscopic mechanisms ruling photo-absorption in these systems.",
"While the optical excitations are intrinsically excitonic in nature, regardless of the molecular concentration, in densely-packed SAMs intermolecular coupling and local-field effects are responsible for a sizable weakening of the exciton binding strength.",
"Through a detailed analysis of the character of the electron-hole pairs, we show that distinct excitations involved in the photo-isomerization at low molecular concentrations are dramatically broadened by intermolecular interactions.",
"Spectral shifts in the calculated DR spectra are in good agreement with the experimental results.",
"Our findings represent an important step forward to rationalize the excited-state properties of these complex materials."
],
[
"Introduction",
"Controlled architectures of photo-responsive chromophores can be obtained in an elegant way by self-assembly of functionalized molecules in ordered monolayers.",
"In this way, photo-switching materials can be potentially exploited in view of realistic applications.",
"[1], [2], [3], [4], [5], [6] Azobenzene-functionalized self-assembled monolayers (SAMs) of alkyl chains have been successfully synthesized in the last two decades.",
"[7], [8], [9], [10], [11], [12] Unfortunately, such systems come with a substantial drawback: The photo-isomerization rate is drastically suppressed due to steric hindrance or excitonic coupling.",
"[13], [14] A number of experimental strategies have been suggested to overcome this problem, such as replacing the aliphatic linker by an aromatic one [15] or even bulkier groups, [16], [17], [18] or diluting the azobenzene moieties in densely-packed alkanethiolate SAMs.",
"[14], [19], [20], [21], [22] From a theoretical viewpoint the issue of hindered photo-isomerization in densely-packed azobenzene-functionalized SAMs has been addressed by a few works, mainly focused on the excitonic coupling between the chromophores, [23], [24] as well as on the effects of a metal substrate.",
"[25] While these investigations have significantly contributed to describe the excited-state properties of azobenzene derivatives, a very recent publication elucidates the role of defects in the photo-isomerization of such SAMs.",
"[26] Still, a full understanding of the fundamental physical mechanisms ruling optical absorption at increasing molecular concentration is still missing.",
"Identifying the nature of the excitations is an essential step in view of defining new strategies that enable to restore the photo-switching efficiency exhibited by azobenzene in solution.",
"For this purpose, we provide here an in-depth analysis of the basic physical mechanisms governing photo-absorption in well-ordered azobenzene-functionalized SAMs at increasing packing density of the chromophores.",
"To do so, we combine ab initio many-body theory with differential reflectance spectroscopy.",
"We investigate the effects of increasing molecular density on the photo-absorption properties of these systems, specifically focusing on the role of intermolecular coupling and local-field effects.",
"The analysis of the spectra is supported by a detailed characterization of the optical excitations, in view of explaining how many-body effects rule the excitation process.",
"Good agreement between theory and experiment corroborates our conclusions."
],
[
"Self-assembled monolayers of azobenzene-functionalized alkanethiols",
"In this work, we focus on azobenzene-functionalized alkanethiolate SAMs, which form well-ordered, densely-packed structures with the trans-configuration representing the ground state.",
"Intermolecular interactions in such systems result in a predominantly upright orientation of the alkyl chains as well as of the chromophore (see Fig.",
"REF a).",
"On average, the plane of the azobenzene moiety is tilted by 73$^{\\circ }$ with respect to the surface.",
"[13] The lateral SAM structure has been investigated by atomic-force microscopy (AFM) and scanning tunneling microscopy (STM).",
"[27], [28] Densely-packed azobenzene-alkanethiolate SAMs form a nearly rectangular two-dimensional structure with lattice constants $a$ =6.05 Å and $b$ =7.80 Å including two molecules in the unit cell.",
"Without sacrifying the overall structure of the SAM, the density of azobenzene moieties can be reduced by mixing functionalized and unfunctionalized alkanethiolates.",
"As a consequence of the weaker interaction between the chromophores photo-isomerization is enabled.",
"Additionally, the azobenzene moieties tend to orient more parallel to the surface.",
"For a dilution up to 20$\\%$ of a densely-packed azobenzene SAM the average tilt angle of the aromatic plane is about 45$^{\\circ }$ .",
"[21] In our first-principles calculations, azobenzene-functionalized SAMs are modeled by a two-dimensional crystal structure, where the alkyl chains and the gold surface are not considered.",
"This choice is motivated by the fact that the alkyl chains do not contribute significantly to the optical absorption at photon energies below 6 eV, and that the azobenzene moieties are expected to be decoupled from the gold substrate by the alkyl chains.",
"To mimic the chemical environment of the bond to the alkyl chain, we add a methoxy (OCH3) group terminating the azobenzene molecules (see Fig.",
"REF b).",
"We model a densely-packed SAM (p-SAM) by considering an orthorhombic unit cell For the reciprocal-space representation of an orthorhombic unit cell and the identification of the corresponding high-symmetry points, we refer to the Bilbao Crystallographic Server.",
"[65] of lattice parameters $a$ and $b$ , incorporating two molecules oriented parallel to each other and tilted by 30$^{\\circ }$ with respect to the surface normal corresponding to the $ab$ plane (Fig.",
"REF c).",
"In this configuration the chromophores are separated by $\\sim $ 2 Å and $\\sim $ 3.8 Å in the direction parallel and perpendicular to the plane of the phenyl rings, respectively.",
"We include $\\sim $ 14 Å of vacuum in the normal direction, to avoid unphysical interactions between the replicas.",
"To model a SAM with reduced packing density, we consider a system including only one molecule per identical unit cell.",
"In the following, we refer to the structure shown in Fig.",
"REF d as diluted SAM (d-SAM).",
"For comparison with the isolated molecule, we study also a single azobenzene chromophore in an orthorhombic supercell, incorporating $\\sim $ 6 Å of vacuum in each direction.",
"The electronic and optical properties of the investigated systems depend essentially on intermolecular distances.",
"Only minor effects are expected to come from the orientation of the molecules in the unit cell and with respect to each other.",
"For this reason, we can rely on the geometries adopted in our first-principles calculations, although rather simplified compared to the experimental sample."
],
[
"Theoretical background ",
"Ground-state properties are obtained in the framework of density-functional theory (DFT), adopting the linearized augmented planewave (LAPW) plus local orbital method as implemented in the exciting code.",
"[30] Optical absorption spectra are calculated from many-body perturbation theory, following a two-step procedure: Quasi-particle (QP) energies are evaluated from $G_0W_0$ [31], [32] and optical excitations are computed with the Bethe-Salpeter equation (BSE), an effective equation of motion for the electron-hole ($e$ -$h$ ) two-particle Green's function.",
"[33], [34] This methodology allows for describing optical excitations in molecular materials from the gas-phase to the condensed matter.",
"[35], [36], [37], [38], [39], [40], [41], [42] In matrix form, the BSE is expressed as $\\sum _{v^{\\prime }c^{\\prime }\\mathbf {k^{\\prime }}} \\hat{H}^{BSE}_{vc\\mathbf {k},v^{\\prime }c^{\\prime }\\mathbf {k^{\\prime }}} A^{\\lambda }_{v^{\\prime }c^{\\prime }\\mathbf {k^{\\prime }}} = E^{\\lambda } A^{\\lambda }_{vc\\mathbf {k}} ,$ where the indexes $v$ and $c$ indicate valence and conduction states, respectively.",
"In a spin-unpolarized system the effective two-particle Hamiltonian [43], [44] reads: $\\hat{H}^{BSE} = \\hat{H}^{diag} + 2 \\gamma _x \\hat{H}^x + \\gamma _c \\hat{H}^{dir}.$ $\\hat{H}^{diag}$ is the diagonal term, which accounts for single-particle transition energies.",
"Considering only this term in Eq.",
"REF corresponds to the independent-particle approximation (IPA).",
"The term $\\hat{H}^x$ includes the short-range exchange Coulomb interaction $\\bar{v}$ , which accounts for local-field effects (LFE): $\\hat{H}^x = \\int d^3\\mathbf {r} \\int d^3\\mathbf {r}^{\\prime } \\phi _{v\\mathbf {k}} (\\mathbf {r}) \\phi ^*_{c\\mathbf {k}} (\\mathbf {r}) \\bar{v}(\\mathbf {r},\\mathbf {r}^{\\prime }) \\phi ^*_{v^{\\prime }\\mathbf {k}^{\\prime }} (\\mathbf {r}^{\\prime }) \\phi _{c^{\\prime }\\mathbf {k}^{\\prime }} (\\mathbf {r}^{\\prime }),$ where $\\phi $ are QP wave-functions.",
"$\\hat{H}^{dir}$ contains the statically screened Coulomb potential $W$ , which describes the attractive $e$ -$h$ interaction: $\\hat{H}^{dir} \\!",
"\\!",
"= \\!",
"- \\!",
"\\!",
"\\int \\!",
"\\!",
"\\!",
"d^3\\mathbf {r} \\!",
"\\!",
"\\int \\!",
"\\!",
"d^3\\mathbf {r}^{\\prime } \\phi _{v\\mathbf {k}} (\\mathbf {r}) \\phi ^*_{c\\mathbf {k}} (\\mathbf {r}^{\\prime }) W(\\mathbf {r},\\mathbf {r}^{\\prime }) \\phi ^*_{v^{\\prime }\\mathbf {k}^{\\prime }} (\\mathbf {r}) \\phi _{c^{\\prime }\\mathbf {k}^{\\prime }} (\\mathbf {r}^{\\prime }).$ The screened Coulomb potential $W$ is calculated from the inverse of the macroscopic dielectric tensor, evaluated within the random-phase approximation.",
"Details on the implementation within the LAPW formalism can be found in Refs. pusc-ambr02prb,sagm-ambr09pccp.",
"The coefficients $\\gamma _x$ and $\\gamma _c$ in Eq.",
"(REF ) enable turning on and off the exchange and the direct term, $\\hat{H}^x$ and $\\hat{H}^{dir}$ (Eq.",
"REF and REF ), respectively.",
"The solutions of the full Hamiltonian in Eq.",
"REF ($\\gamma _x$ = $\\gamma _c$ = 1) correspond to singlet excitations.",
"When the $e$ -$h$ exchange interaction is neglected ($\\gamma _x$ = 0 and $\\gamma _c$ = 1) triplet solutions are obtained.",
"Further details on the BSE Hamiltonian and on its spin structure can be found in Refs. rohl-loui00prb,pusc-ambr02prb.",
"For bound excitons below the QP gap ($E_g$ ) obtained from $G_0W_0$ binding energies ($E_b$ ) are computed as the difference between $E_g$ and the excitation energies $E^{\\lambda }$ .",
"The eigenvectors $A^{\\lambda }$ of Eq.",
"REF provide information about the character and composition of the excitations.",
"In particular, they are used to define the weight of each transition between valence and conduction states at a given $\\mathbf {k}$ -point: $w^{\\lambda }_{v\\mathbf {k}} = \\sum _c |A^{\\lambda }_{vc\\mathbf {k}}|^2 , \\, \\, \\, w^{\\lambda }_{c\\mathbf {k}} = \\sum _v |A^{\\lambda }_{vc\\mathbf {k}}|^2,$ where the sums are performed over the range of occupied and unoccupied states included in the solution of the BSE (Eq.",
"REF ).",
"The eigenvectors $A^{\\lambda }$ enter the expression of the oscillator strength, given by the square modulus of $\\mathbf {t}^{\\lambda }= \\sum _{vc\\mathbf {k}} A^{\\lambda }_{vc\\mathbf {k}} \\dfrac{\\langle v\\mathbf {k}|\\widehat{\\mathbf {p}}|c\\mathbf {k}\\rangle }{\\varepsilon _{c\\mathbf {k}} - \\varepsilon _{v\\mathbf {k}}} ,$ where $\\hat{\\mathbf {p}}$ is the momentum operator, and $\\varepsilon _{v\\mathbf {k}}$ and $\\varepsilon _{c\\mathbf {k}}$ are the QP energies of the involved valence and conduction states, respectively.",
"The optical absorption spectra are represented by the imaginary part of the macroscopic dielectric function ($\\epsilon _M$ ) $\\mathrm {Im}\\epsilon _M = \\dfrac{8\\pi ^2}{\\Omega } \\sum _{\\lambda } |\\mathbf {t}^{\\lambda }|^2 \\delta (\\omega - E^{\\lambda }) ,$ where $\\Omega $ is the unit cell volume and $\\omega $ is the energy of the incoming photon."
],
[
"Computational details",
"All calculations are performed with the all-electron full-potential code exciting, [30] implementing the LAPW method.",
"The ground state of the investigated systems is computed by means of DFT, adopting the Perdew-Wang local-density approximation for the exchange-correlation functional.",
"[46] For the sampling of the Brilloiun zone (BZ), we employ a 6$\\times $ 4$\\times $ 1 (3$\\times $ 2$\\times $ 1) $\\mathbf {k}$ -point mesh for the p-SAM (d-SAM) in both ground-state and many-body perturbation theory calculations.",
"Concerning the basis functions, a planewave cutoff $G_{max}$ =5 bohr is adopted for the calculation of the single molecule.",
"For the SAMs, this value is 4.625 bohr.",
"Muffin-tin spheres with radii of 0.8 bohr are considered for hydrogen, 1.1 bohr for nitrogen, and 1.2 bohr for carbon and oxygen.",
"Each structure is optimized by minimizing atomic forces within a threshold of 0.025 eV/Å, with the lattice parameters of the unit cell kept fixed.",
"The corresponding geometries are displayed in Figs.",
"REF b-d.",
"In our $G_0W_0$ implementation, [47] the dynamically screened Coulomb potential $W_0$ is computed within the random-phase approximation.",
"200 empty Kohn-Sham (KS) states are included in the calculation.",
"The so-obtained correction of 3.42 eV, 3.32 eV, and 1.8 eV to the band gap of the isolated molecule, the d-SAM, and the p-SAM, respectively, is then applied to the KS electronic structure through a scissors operator.",
"BSE calculations are performed within the Tamm-Dancoff approximation.",
"To calculate the screened Coulomb interaction 500 unoccupied states for the molecule and 400 conduction bands for the SAMs are considered.",
"Approximately 1000 (500) $|\\mathbf {G}+\\mathbf {q}|$ vectors are included for the d-SAM (p-SAM) and more than 5000 for the single molecule.",
"In the solution of the BSE Hamiltonian (Eq.",
"REF ) for the single molecule we include 16 occupied and 22 unoccupied states, corresponding to about 8 and 5 eV below and above the Fermi energy, respectively.",
"In the case of the d-SAMs we consider 11 valence and 15 conduction bands, corresponding to 5 and 4.5 eV below and above the Fermi energy, respectively.",
"For the p-SAM, 10 occupied and 11 unoccupied bands are taken into account, corresponding to 3 and 4 eV below and above the Fermi energy, respectively.",
"For a quantitative comparison with the experimental data, we calculate optical coefficients, specifically absorbance and reflectance, by adopting the 4$\\times $ 4 matrix formulation of Maxwell's equations, [48], [49] as implemented in the LayerOptics code.",
"[50]"
],
[
"Electronic properties",
"We start our analysis by inspecting the KS electronic properties of the isolated azobenzene and of the SAMs.",
"In Fig.",
"REF we show the energy levels of the single molecule, along with the real-space representation of a set of molecular orbitals closest to the Fermi energy.",
"In the proximity of the KS gap, we find a number of states with pronounced localization on the azo group.",
"These findings are in agreement with previous studies on trans-azobenzene in the gas phase.",
"[51], [52], [53] In particular, the highest occupied molecular orbital, HOMO (H), is distributed almost exclusively on the N=N bond, thus being a non-bonding ($n$ ) orbital.",
"On the other hand, the H-1 and the lowest unoccupied molecular orbital, LUMO (L) exhibit $\\pi $ character with the electron density spread over the phenyl rings.",
"Further away from the KS gap, the states are localized only on either carbon ring: this is the case of L+1 and L+2, as well as of H-2 and H-4.",
"These pairs of states are energetically very close to each other.",
"In between, H-3 is spread over the whole molecule, including the OCH3 group.",
"In the conduction region, additional delocalized levels, such as L+3 and L+4, are found at higher energies, more than 3 eV above the gap.",
"The electronic properties of the single molecule are reflected also in the SAMs.",
"By inspecting the band structure of the d-SAM (Fig.",
"REF b), which includes only one azobenzene in the unit cell, we notice that the KS gap has basically the same size as in the isolated molecule.",
"The valence-band maximum (VBM) and the conduction-band minimum (CBM) maintain their localized character also in the SAMs.",
"In the mean-field DFT picture, their energy difference is almost unaffected by the packing arrangement.",
"Also the very low dispersion of the bands near the fundamental gap confirms the localized molecular character of the electronic states in the d-SAM.",
"In the valence region, $\\sim $ 2.5 eV below the Fermi energy ($E_F$ ), a set of almost flat bands appears.",
"Again, they are similar to the KS states of the isolated molecule: VBM-2 is rather localized on the OCH3 group, whereas VBM-3 and VBM-4 are the counterparts of H-2 and H-4, respectively.",
"The lowest-energy state in this manifold of bands has delocalized $\\pi $ character.",
"Also in the conduction region, we notice strong similarities with the electronic properties of the isolated azobenzene.",
"About 1.5 eV above the lowest unoccupied band, we find three flat bands, with CBM+1 and CBM+2 being the counterparts of L+1 and L+2, respectively.",
"The highest-energy band of this group has instead extended $\\pi ^*$ character, like the L+3 state in Fig.",
"REF a.",
"Below -4 eV and above 3 eV, the KS states exhibit a delocalized intermolecular character, as confirmed by the more pronounced dispersion of the corresponding bands.",
"In Fig.",
"REF c the band structure of the p-SAM is shown.",
"Since this system has two molecules per unit cell (Fig.",
"REF c), bands have double multiplicity.",
"This is especially evident at the top of the valence region, where the two uppermost nearly dispersionless bands, VBM and VBM-1, have almost the same energy.",
"They are localized on the azo group and hence exhibit $n$ character like the HOMO in the isolated molecule.",
"At lower energy in the valence region we find another pair of bands with more pronounced dispersion, namely VBM-2 and VBM-3.",
"The latter are $\\pi $ states, corresponding to H-1 in the single molecule (Fig.",
"REF a).",
"Even deeper in energy, about 2 eV below $E_F$ , a manifold of bands appears, including states which are the counterparts of H-2, H-3, and H-4 of the isolated molecule.",
"In the conduction region, the two lowest-energy bands (CBM and CBM+1) are almost flat and degenerate along the $\\Gamma $ -X direction, which is almost parallel to the projection of the long molecular axis onto the two-dimensional plane of the SAM (see Fig REF c).",
"These states are delocalized over the whole molecule, similar to the LUMO of the isolated compound (Fig.",
"REF a).",
"A gap of about 1.5 eV separates these two lowest unoccupied states from higher-energy ones, which exhibit much more pronounced dispersion.",
"In this region the KS states show enhanced intermolecular coupling, which results in an increased wave-function delocalization.",
"Figure: (Color online) Optical spectra of azobenzene molecule and SAMs, expressed by Imϵ M \\mathrm {\\epsilon _M}.",
"The G 0 W 0 G_0W_0 gap is marked by a vertical dashed line.",
"a) Singlet excitations, computed from the full BSE.",
"An average over the three Cartesian components is shown for the molecule, while the in-plane (∥\\parallel ) and out-of-plane (⊥\\perp ) components are plotted for the SAMs.",
"b) BSE spectra, averaged over all Cartesian components, computed without LFE (triplet excitations, solid line) and within the independent-particle approximation (IPA, shaded area).",
"A Lorentzian broadening of 0.1 eV is applied to all the spectra."
],
[
"Optical properties",
"In Fig.",
"REF a, the calculated optical spectra of the azobenzene molecule and of the SAMs are shown.",
"The energies of the main excitations are summarized in Table REF .",
"We start our analysis from the spectrum of an isolated molecule (Fig.",
"REF a, top panel), which is expressed by Im$\\epsilon _M$ averaged over the three Cartesian components.",
"The lowest-energy excitation, $S_1$ , is forbidden by symmetry.",
"It has $n$ -$\\pi ^*$ character, corresponding to an almost pure transition between the HOMO and the LUMO, as reported in Table REF .",
"At higher energy, at about 3.9 eV, a strong peak dominates the spectrum.",
"This excitation, labeled $S_2$ , is responsible for the photo-isomerization of azobenzene, switching from the trans to the cis conformation.",
"[54] $S_2$ is a bound exciton, with $E_b \\sim $ 1.5 eV.",
"It exhibits $\\pi $ -$\\pi ^*$ character, being dominated by the H-1$\\rightarrow $ L transition.",
"At 3.5 eV $S_2$ ' appears as a weak shoulder of $S_2$ , given by two almost degenerate excitations from H-2 and H-3 to the LUMO.",
"The localization of these occupied states on a single phenyl ring (Fig REF a) explains the low oscillator strength of $S_2$ '.",
"The last bound exciton appearing in the spectrum is $S_3$ , which shows a remarkably mixed character with contributions from a number of occupied $\\pi $ states to the $\\pi ^*$ orbitals L+1 and L+2 (see Table REF ).",
"The excitation energies of the molecule considered here are lower by a few hundred meV compared to those reported in the literature for bare trans-azobenzene.",
"[55], [56], [57], [52], [53] This is expected, since the presence of the oxygen-based group terminating the molecule (see Fig.",
"REF a) is known to cause a red-shift of the $\\pi $ -$\\pi ^*$ transitions.",
"[23] On the other hand, the excitation energy of $S_1$ is underestimated by more than 1.5 eV compared to results from literature (see, e.g., Ref.",
"crec-roit06jpca and references therein).",
"This can be ascribed to the starting-point dependence of the $G_0W_0$ step, which becomes particularly severe in case of localized KS states, such as the HOMO of azobenzene.",
"We expect that a hybrid functional as starting point for the $G_0W_0$ calculation [58] or self-consistent $GW$ [59] would improve the result.",
"However, this problem does not affect the nature of the excitations, and therefore it is irrelevant for the essence of the present work.",
"Table: Composition, in terms of single-particle transitions, of the main excitations of the isolated azobenzene molecule.",
"The weight of each transition is given in brackets, including only contributions larger than 10%\\%.",
"In BSE-triplet and IPA the contributions of all active excitations embraced by the peaks are listed.For the analysis of the absorption spectra of the SAMs, we plot the parallel ($\\parallel $ ) and perpendicular ($\\perp $ ) components of the imaginary part of the macroscopic dielectric function: Im$\\epsilon _M^{\\parallel }$ represents an average of the $ab$ plane of the unit cell depicted in Fig.",
"REF c-d, while Im$\\epsilon _M^{\\perp }$ corresponds to the component along the $c$ axis.",
"The overall pronounced differences reflect the orientation of the molecules in the SAM.",
"The spectrum of the d-SAM exhibits strong similarities with the one of the single molecule, starting from the position of the QP gap.",
"In the lowest energy region, we find the n-$\\pi ^*$ excitation $S_1$ , which is still symmetry-forbidden.",
"At higher energy ($\\sim $ 4.5 eV), the strong $\\pi $ -$\\pi ^*$ peak $S_2$ appears.",
"Its excitation energy is blue-shifted by about 0.6 eV compared to the isolated molecule, as a result of two counteracting effects.",
"On the one hand, the QP gap, represented by the IPA onset (Table REF and Fig.",
"REF b) decreases by 0.13 eV in the d-SAM, due to intermolecular coupling.",
"On the other hand, the exciton binding energy is reduced by almost 0.4 eV, owing to the enhanced screening and wave-function overlap resulting from the packing of the chromophores in the unit cell.",
"A similar effect is noticed also for $S_3$ , which is unbound, i.e., appears above the QP gap.",
"Compared to the spectrum of the isolated molecule, the excitation energy of $S_3$ undergoes a blue-shift of more than 0.3 eV (Table REF ).",
"For a quantitative analysis of the character of these excitations, we display in Fig.",
"REF the corresponding weights (Eq.",
"REF ) on top of the KS band structure.",
"Overall, the excitations of the d-SAM show analogous character to those of the isolated molecule.",
"$S_1$ is an almost pure transition between the VBM and the CBM, which are the counterparts of the HOMO and the LUMO, respectively.",
"In the same way, $S_2$ is clearly dominated by the transition from the VBM-1 to the CBM, and $S_2$ ' by VBM-3 to CBM.",
"Also in the d-SAM, $S_3$ exhibits a remarkably mixed character, with the most significant contributions coming from VBM-1 and VBM-3 in the valence region and from the four lowest conduction bands.",
"The polarization of $S_2$ and $S_3$ along the long molecular axis is further emphasized by the predominance of Im$\\epsilon _M^{\\perp }$ over Im$\\epsilon _M^{\\parallel }$ (Fig.",
"REF a).",
"Figure: (Color online) Weights of singlet (a) and triplet (b) excitations of the d-SAM plotted on top of the Kohn-Sham band structure.",
"The size of the red and blue circles is representative of their magnitude.The optical spectrum of the p-SAM is completely different from the previous ones.",
"With increasing packing density of the chromophores and thus enhanced intermolecular interactions, the optical gap, represented by the IPA absorption onset, is significantly reduced to about 3.5 eV (Table REF ).",
"In addition, two specific features are immediately visible in the spectrum displayed on the bottom panel of Fig.",
"REF a.",
"First, the lowest-energy excitation $S_1$ is optically active, although weak.",
"It is still found below the QP gap, with $E_b \\approx $ 1 eV.",
"Second, the sharp peak $S_2$ , associated with a bound excitonic state in the spectra of the molecule and of the d-SAM, now appears above the absorption onset of the material in a broad absorption band along with other intense excitations.",
"This behavior is mainly due to LFE, counteracting the red-shift induced by the band-gap narrowing, as an effect of intermolecular electronic interactions.",
"[60], [61], [62] Moreover, contrary to the case of the d-SAM where the out-of-plane component of Im$\\epsilon _M$ dominates the spectrum in the considered frequency range, here Im$\\epsilon _M^{\\parallel }$ and Im$\\epsilon _M^{\\perp }$ show different spectral weights in specific energy regions.",
"This is obviously related to the character of the excitations involved.",
"To gain deeper insight, we again consider the excitation weights plotted on top of the KS band structure (Fig.",
"REF ).",
"As discussed in Sec.",
"REF , the presence of two molecules in the unit cell gives rise to two bands at the top of the valence region (VBM and VBM-1) and at the bottom of the conduction region (CBM and CBM+1) having the same character as the HOMO and the LUMO of the isolated molecule.",
"The first exciton $S_1$ arises from a transition between these occupied and unoccupied states, and is now dipole-allowed, due to symmetry breaking.",
"The weights are homogeneously distributed within the BZ in both valence and conduction regions revealing the intramolecular character of this excitation.",
"The $n$ -$\\pi ^*$ nature of $S_1$ explains the predominance of Im$\\epsilon _M^{\\parallel }$ at the absorption onset (Fig REF a).",
"Around 3 eV, where a manifold of excitations with similar character as $S_1$ takes place, the in-plane component of the macroscopic dielectric function dominates over the out-of-plane one.",
"Excitations with $\\pi $ -$\\pi ^*$ character start appearing $\\sim $ 1 eV above the QP gap, at about 4 eV, where also the relative intensity of Im$\\epsilon _M^{\\perp }$ overcomes the one of Im$\\epsilon _M^{\\parallel }$ .",
"In the spectrum of the p-SAM, we identify $S_2$ as the most intense excitation among a manifold of solutions of the BSE exhibiting rather mixed character.",
"From Fig.",
"REF a we notice that the largest weights come from transitions between VBM-5 and VBM-6 to CBM and CBM+1, and non-negligible contributions arise also from transitions from the VBM to higher conduction bands.",
"The weights are not uniformly distributed throughout the BZ, pointing to an intermolecular character of the excitation.",
"Excitations with the same mixed character but larger contributions from transitions between VBM-2/VBM-3 to CBM/CBM+1 appear in the same energy window (3.9 – 4.5 eV).",
"However, they display lower oscillator strength than $S_2$ .",
"Also $S_2$ ' shows a very mixed character, and is dominated by contributions similar to those of $S_2$ .",
"Again, we notice an inhomogeneous distribution of the weights in the BZ, indicating an intermolecular character of the $e$ -$h$ wave-function in real space.",
"Finally, we identify $S_3$ , in analogy with the molecule and the d-SAM, as an excitation from VBM-2 and VBM-3 to higher-energy conduction bands, namely CBM+2 and CBM+3.",
"Figure: (Color online) Weights of singlet (a) and triplet (b) excitations of the p-SAM plotted on top of the Kohn-Sham band structure.",
"The size of the red and blue circles is representative of their magnitude.The striking differences in the optical absorption properties of the p-SAM compared to its diluted counterpart and the isolated molecule are mainly to be ascribed to intermolecular interactions that act in different ways.",
"The array of closely-packed molecules gives rise to enhanced screening effects, related to more delocalized $e$ -$h$ wave-functions, which exhibit a reduced binding strength.",
"From the spectra in Fig.",
"REF a and the data reported in Table REF , this trend is very clear.",
"Considering for example $S_2$ , we notice that its binding energy decreases systematically with increasing molecular concentration, up to the p-SAM, where this excitation is found more than 0.5 eV above the QP gap.",
"Remarkably, a qualitatively similar behavior is exhibited by the core-level excitations from the nitrogen $K$ -edge of azobenzene-functionalized SAMs.",
"[63] In that case, the binding energy of the first bound exciton is reduced by about 2 eV going from the isolated molecule to the p-SAM.",
"Another important effect of intermolecular interactions is the redistribution of the spectral weight to higher energies, due to dipole coupling.",
"This mechanism has been rationalized also for azobenzene dimers from a quantum-chemistry perspective.",
"[23], [24] In the present work we demonstrate that such spectral blue-shift is directly associated to LFE, expressed by the short-range $e$ -$h$ exchange interaction (Eq.",
"REF ).",
"In order to quantitatively determine the role of LFE in the spectra of azobenzene-functionalized SAMs, we now compare Fig.",
"REF a and Fig.",
"REF b.",
"In the latter, the dielectric function is calculated by neglecting LFE, i.e., by setting $\\gamma _{x}$ =0 in Eq.",
"REF , thereby turning off the exchange interaction.",
"The solutions of the resulting Hamiltonian are the triplet excitations.",
"Being spin-forbidden, they cannot be probed by an optical-absorption experiment.",
"However, the comparison between singlet and triplet spectra helps us explaining the role of local fields and excitonic effects in the optical excitations of these systems.",
"All spectra in Fig.",
"REF b show the same features, namely the lowest-energy dark excitation $T_1$ , the intense peaks $T_2$ and $T_3$ , and the weaker $T_2$ ' in between.",
"The corresponding excitation energies are obviously red-shifted compared to their singlet counterparts (Fig.",
"REF a), due to the missing $e$ -$h$ repulsive term.",
"All the triplet excitations exhibit similar relative intensity regardless of the packing density, and likewise their nature remains the same from the isolated monomer to the p-SAM.",
"Specifically, we inspect the character of the $\\pi $ -$\\pi ^*$ transition $T_2$ and compare it with its singlet counterpart $S_2$ .",
"When intermolecular interactions are negligible or missing, as in the case of the single molecule, this excitation stems almost completely from H-1$\\rightarrow $ L, no matter whether LFE are included or not (Table REF ).",
"In fact, already in the IPA spectrum of the molecule shown in Fig.",
"REF b, a sharp peak appears, given by the $\\pi $ -$\\pi ^*$ excitation $T_2$ .",
"Also for the d-SAM we find an intense resonance at the onset of the IPA spectrum.",
"By comparing the character of $S_2$ and $T_2$ in this system (Fig.",
"REF ), the growing influence of intermolecular interactions is evident.",
"While $T_2$ is given by a pure VBM-1$\\rightarrow $ CBM transition, in $S_2$ we find minor contributions also from deeper (higher) valence (conduction) bands.",
"Also $S_3$ and $T_3$ exhibit a similar behavior.",
"While the mixed character of this excitation is enhanced upon inclusion of the Coulomb interaction already in the isolated molecule (Table REF ), in the d-SAM we notice that many more interband transitions are involved in the composition of $S_3$ compared to $T_3$ .",
"In case of the p-SAM, intermolecular interactions captured by LFE significantly affect the character of the bright excitations $S_2$ , $S_2$ ', and $S_3$ .",
"In the IPA spectrum, that is shown for comparison, the absorption onset is rather featureless.",
"In the BSE spectrum computed without accounting for LFE (triplet excitations) excitonic peaks are prominent, and resemble in energy and nature those characterizing the spectra of the molecule and of the d-SAM.",
"$T_2$ is an almost pure transition from VBM-2 and VBM-3 to CBM and CBM+1.",
"Likewise, $T_2$ ' shows a predominant contribution from lower-energy valence bands to CBM/CBM+1.",
"$T_3$ retains its mixed character, already discussed for the molecule and for the d-SAM.",
"Concerning the excitation energies, we notice a similar behavior as in the singlet spectra in Fig.",
"REF a: $T_2$ blue-shifts by more than 0.3 eV from the molecule to the d-SAM.",
"It results from narrowing of the QP gap ($\\sim $ 0.1 eV), due to the increased molecule-molecule electronic interactions, and the enhanced exciton delocalization, which reduces the binding energy by about 0.5 eV.",
"When going from the diluted to the packed SAM, the excitations undergo a red-shift by approximately 0.6 eV.",
"In this case, the reduction of the QP gap is so large ($\\sim $ 2 eV) that it overcomes the pronounced decrease of exciton binding energies ($\\sim $ 1.4 eV, see Table REF ).",
"It is worth noting that these spectral shifts are unrelated to LFE, which are not present in the calculation of triplet excitations.",
"Similar effects, due to the coupling of the electronic wave-functions and to the consequent delocalization of the resulting $e$ -$h$ pairs as an effect of the increased intermolecular interactions, have been discussed already in the context of organic crystals.",
"[39], [41], [60] When the effects of local fields are accounted for, as in the singlet spectra, they additionally contribute to an overall blue-shift of the oscillator strength and, most importantly, to its redistribution to higher energies.",
"At this point, we briefly discuss our results in the context of the existing theoretical literature.",
"[23], [24] In the present work, we investigate optical absorption properties of azobenzene-functionalized SAMs from a solid-state physics perspective.",
"We treat the ordered SAMs as periodic systems, such that the electronic wave-functions are allowed to spread over the infinitely extended structure.",
"In this way, the interactions between the chromophores are quantitatively taken into account in the ground state as well as in the excited state.",
"Moreover, our approach for calculating optical excitations based on the solution of the BSE enables us to consider explicitly the effects of the repulsive exchange and of the attractive direct $e$ -$h$ Coulomb interaction, which counteract each other.",
"From the results of this analysis we assert that the blue-shift experienced by the main absorption peaks in the spectrum of the p-SAM is mainly due to intermolecular coupling.",
"LFE enhance the mixing of single-particle transitions contributing to the excitations, thereby turning a sharp resonance into a broad absorption band.",
"The $e$ -$h$ pairs become delocalized, assuming intermolecular character.",
"In this regard, our findings are in line with the rationale expressed by previous theoretical works, [23], [24] where SAMs have been modeled in a quantum-chemical framework, accounting for dipole-dipole interactions in molecular dimers and/or clusters."
],
[
"Differential reflectance spectra",
"To experimentally investigate the effect of increasing density of chromophores on the optical properties of azobenzene-functionalized alkanethiolate SAMs (Fig.",
"REF a), we use the azo compound 11-(4-(phenyldiazenyl)phenoxy)-undecane-1-thiol (referred to as Az11) synthesized in the group of Rafal Klajn (Weizmann Institute of Science, Revohot, Israel), as well as dodecane-1-thiol (C12, Sigma-Aldrich).",
"A SAM is prepared by immersing a gold substrate into a methanolic solution of the thiols for 20 hours, as detailed in Ref. mold+15lang.",
"Depending on the relative concentrations of Az11 and C12 in solution, SAMs with different chromphore densities are obtained.",
"Since the mixing of the molecules in the SAM is mainly statistical, the local environment of the azobenzene moieties is not homogeneous, in contrast to our first-principles calculations.",
"The optical properties of the SAM are determined by differential reflectance (DR) spectroscopy, measuring the reflectance of the gold before and after the SAM preparation.",
"All measurements are performed at ambient conditions, with an angle of incidence of 45$^{\\circ }$ , with both $s$ - and $p$ -polarized light.",
"Additional experimental details are reported in Ref. mold+15lang.",
"For a direct comparison between theoretical and experimental data, we calculate DR spectra of azobenzene-functionalized SAMs on gold by employing the LayerOptics code.",
"[50] The contribution of the SAMs, assumed to form a 2 nm thick layer, is given by the full dielectric tensors computed from BSE.",
"The effects of the gold substrate are incorporated in a dielectric constant $\\epsilon _{sub} = \\epsilon _1 + i\\epsilon _2$ = -0.36 + $i$ 6.475.",
"This value is extracted from the average of the complex refractive index of gold, in the frequency region between 3 and 4.5 eV, [64] corresponding to the $S_2$ excitation band of the azobenzene-functionalized SAM.",
"The choice of the dielectric permittivity of the metal substrate is crucial to capture the essential features of the experimental DR spectra.",
"The angle of incidence is set to 45$^{\\circ }$ as in the experiment.",
"Interference effects at the layer boundaries, as well as the influence of the polarization of the incoming light are taken into account.",
"In the case of the isolated molecule, absorption measurements are performed in solution.",
"The resulting absorbance curve is shown in Fig.",
"REF d, while its theoretical counterpart is displayed in Fig.",
"REF a.",
"Both spectra exhibit the same structure, with an intense peak at lower energy ($S_2$ ), followed by a weaker one at higher energy ($S_3$ ).",
"The first resonance $S_2$ is blue-shifted by approximately 0.3 eV in the computed absorbance, compared to the experimental result.",
"This difference can be ascribed to the solvent effects.",
"The relative intensity of the peaks $S_2$ and $S_3$ is well reproduced by theory, while the absolute energy position of the latter peak is underestimated by about 0.3 eV.",
"This behavior can be attributed to the inclusion of the QP correction to the gap, associated to the self-energy of the LUMO, as a scissors operator, that rigidly up-shifts all conduction states by the same amount of energy.",
"Considering that $S_3$ mostly stems from transitions to states above the LUMO (see Table REF ), the application of a scissors operator can be responsible for the underestimation of its excitation energy.",
"Also the $G_0W_0$ starting-point, as discussed in Sec.",
"REF , can play a role in this regard.",
"The DR spectra of the packed and diluted SAMs are displayed in Fig.",
"REF b, c (theory) and Fig.",
"REF e, f (experiment).",
"In the first-principles results, the peaks are slightly shifted compared to the maxima in Im$\\epsilon _M$ (Fig.",
"REF a).",
"Specifically, in the case of the d-SAM the first intense peak in Fig.",
"REF b is 0.25 eV higher in energy with respect to the $S_2$ resonance identified in the macroscopic dielectric function in Fig.",
"REF a.",
"This difference is not unexpected, since the DR is not only obtained from the imaginary part of $\\epsilon _M$ , but also depends on its real part.",
"In agreement with the predominantly upright orientation of the chromophores, the $s$ -polarized component is weak in the entire spectral window explored in Fig.",
"REF b, with a single broad peak at the low-energy edge of the $S_3$ band.",
"For the p-SAM, the energy region between 4.5 and 5.5 eV is characterized by two peaks associated with $S_2$ ' ($p$ -polarized component only) and $S_3$ (both $s$ - and $p$ -polarized components) in Im$\\epsilon _M$ (Fig.",
"REF a).",
"At lower energy, around 4 eV, we identify the spectral features associated with the $S_2$ resonance.",
"Overall, the $s$ -polarized component is enhanced in comparison with its counterpart in the d-SAM.",
"Between 3 and 4 eV we notice a weak hump that can be related to a corresponding local maximum in the in-plane component of Im$\\epsilon _M$ in Fig.",
"REF a.",
"The experimental DR spectra of the diluted and densely-packed SAMs (Fig.",
"REF e, f) are quite similar.",
"They both exhibit a broad hump in the region of the $S_2$ band of the isolated azobenzene, between approximately 3.2 and 4.5 eV.",
"In the case of the d-SAM the maximum in the $p$ -polarized component is shifted to higher energies by 0.4 eV compared to the single molecule, while for the p-SAM this shift increases to 0.65 eV.",
"It should be noted, however, that the peak maximum in the DR does not simply correspond to a blue-shifted $S_2$ resonance, but it rather originates from the multitude of excitations appearing in that energy region, as discussed in Sec.",
"REF .",
"The discrepancy between the measured spectrum of an Az11-SAM diluted with C12 chains and its theoretical counterpart (d-SAM) can be rationalized from the structural difference between the experimental sample and the system modeled from first principles.",
"While in the latter the orientation of the chromophores is assumed to be the same as in the p-SAM, the diluted SAM sample comprises a mixture of different local molecular environments.",
"The probability of finding two chromophores on directly neighboring sites is still very high for a dilution to 50$\\%$ .",
"Additionally, the chromophores tend to orient themselves more parallel to the surface.",
"A detailed characterization of the diluted SAM samples is provided in Ref. mold+15lang.",
"Since the interaction with nearest-neighbors contributes dominantly to the spectral shifts, the experimental DR spectra of diluted and densely-packed SAMs have a similar shape.",
"Only the energetic position of the peak maximum changes with the chromophore density.",
"The trends shown by our theoretical and experimental results agree qualitatively, confirming the validity of our analysis and supporting the interpretation of our results.",
"As demonstrated by the analysis of our BSE spectra, local-fields enhance the effects of the Coulomb screening, giving rise to a significant blue-shift of the oscillator strength.",
"As such, they play a crucial role in determining the spectral features at large packing density of the chromophores, being responsible for the pronounced mixing of single-particle transitions, which gives rise to the multitude of excitations in the spectral range of interest.",
"Fine details in the spectra cannot be captured due to the differences between the first-principles description and the experimental setup discussed above.",
"These differences, however, do not affect the essence of our results."
],
[
"Summary and Conclusions",
"In summary, we have presented a joint theoretical and experimental study on the optical properties of azobenzene-functionalized SAMs.",
"Our results indicate that at low molecular concentrations the spectra are dominated by an intense $\\pi $ -$\\pi ^*$ resonance in the near-UV region, which is known to be involved in the trans-cis photo-isomerization of the azobenzene moiety.",
"At increasing packing density, this resonance is quenched, and the entire spectra undergo a significant redistribution of oscillator strengths to higher energies.",
"From a thorough analysis based on many-body perturbation theory, we are able to ascribe this behavior to a competition between band-gap narrowing, which tends to red-shift the spectra, and exciton delocalization, which blue-shifts the excitation energies, acting in the same direction as local-field effects.",
"The latter, in particular, are significantly enhanced by the strong coupling between the chromophores and are responsible for a remarkable mixing of the interband transitions contributing to the main excitations.",
"While clearly excitonic in nature, excitations become delocalized and the resulting absorption band is significantly broadened.",
"By lowering the concentration of molecules, the role of intermolecular coupling decreases accordingly, with the optical features of the isolated molecules being restored.",
"With this analysis we have revealed the fundamental physical mechanisms ruling the photo-absorption properties of azobenzene-functionalized SAMs.",
"The behavior of these systems upon light absorption is intrinsically dominated by many-body interactions.",
"This knowledge represents an important step forward in view of understanding and rationalizing the excited-state properties of these complex systems.",
"Relevant input and output files of first-principles calculations can be found in the NoMaD Repository, with the corresponding DOI: http://dx.doi.org/10.17172/NOMAD/2016.12.07-1."
],
[
"Acknowledgement",
"This work was funded by the German Research Foundation (DFG), through the Collaborative Research Center SFB 658.",
"C.C.",
"acknowledges financial support from the Berliner Chancengleichheitsprogramm (BCP) and from IRIS Adlershof.",
"T.M., C.G., and M.W.",
"thank Rafal Klajn and coworkers (Weizmann Institute of Science, Israel) for providing the azobenzene compound."
]
] | 1612.05399 |
[
[
"Higher derivative corrections to incoherent metallic transport in\n holography"
],
[
"Abstract Transport in strongly-disordered, metallic systems is governed by diffusive processes.",
"Based on quantum mechanics, it has been conjectured that these diffusivities obey a lower bound $D/v^2\\gtrsim \\hbar/k_B T$, the saturation of which provides a mechanism for the T-linear resistivity of bad metals.",
"This bound features a characteristic velocity $v$, which was later argued to be the butterfly velocity $v_B$, based on holographic models of transport.",
"This establishes a link between incoherent metallic transport, quantum chaos and Planckian timescales.",
"Here we study higher derivative corrections to an effective holographic action of homogeneous disorder.",
"The higher derivative terms involve only the charge and translation symmetry breaking sector.",
"We show that they have a strong impact on the bound on charge diffusion $D_c/v_B^2\\gtrsim \\hbar/k_B T$, by potentially making the coefficient of its right-hand side arbitrarily small.",
"On the other hand, the bound on energy diffusion is not affected."
],
[
"Introduction",
"It has long been argued that strongly-coupled quantum matter without quasiparticles has the shortest equilibration timescale allowed by quantum mechanics, $\\tau _P\\sim \\hbar /k_B T$ [1], [2], [3].",
"This is believed to underpin many of the unusual transport properties of bad metals, like the $T$ -linearity of their resistivity [4], [5], the violation of the Mott-Ioffe-Regel (MIR) bound [6] or thermal diffusion [7].",
"If quasiparticles are short-lived, the dynamics is governed by the collective excitations of the strongly-coupled quantum fluid, which are simply the conserved quantities of the system (assuming no symmetry is spontaneously broken).",
"From the point of view of transport at late times, there are two distinct regimes, depending on the strength of momentum relaxation.",
"When momentum relaxes slowly, thermoelectric transport is dominated by a single purely imaginary pole in the complex frequency plane, lying parametrically closer to the real axis than other `UV' poles.",
"The dynamics is effectively truncated to keeping track only of this Drude-like pole, and the DC and AC electric conductivities take a simple form at low frequencies: $\\sigma (\\omega )=\\frac{\\chi _{JP}^2}{\\chi _{PP}(\\Gamma -i\\omega )}+O(\\Gamma ^0,\\omega ^0)\\,,\\qquad \\sigma _{DC}=\\frac{\\chi _{JP}^2}{\\chi _{PP}}\\,.$ The $\\chi $ 's are static susceptibilities and similar expressions hold for the other thermoelectric conductivities.",
"$\\Gamma $ is the momentum relaxation rate, and can be computed using the memory matrix formalism [8], [9], [10], [11] or gauge/gravity duality techniques [9], [12], [13], [14], [15] by considering the operator breaking translation symmetry in the state.",
"By assumption, $\\Gamma \\ll k_B T$ to avoid mixing with other, UV poles at scales $\\sim k_B T$ .",
"DC conductivities in this regime are typically high and do not violate the MIR bound.",
"However, the optical conductivity of bad metals displays broad Drude peaks, with a width $\\Gamma \\sim 1/\\tau _P\\sim T$ , [4].",
"This is the incoherent limit where momentum relaxes quickly and does not govern the late time transport properties.",
"The collective excitations are simply diffusion of charge and energy [5], as can be checked in explicit holographic models of incoherent transport [14].",
"In this case, DC conductivities are expected to be small, as there is no low-lying pole (compared to the temperature scale): this suggests an avenue towards violating the MIR bound, at least in principle.",
"Hartnoll conjectured [5] that the diffusivities obeyed a lower bound in this regime: $\\frac{D_{e,c}}{v^2}\\gtrsim \\frac{\\hbar }{k_B T}$ Here $v$ stands for some characteristic velocity of the system, which in a weakly-coupled metal would be the Fermi velocity.",
"By making use of Einstein relations $D_c=\\sigma /\\chi $ (neglecting thermoelectric effects), a linear in $T$ resistivity follows when the bound is saturated, provided the charge static susceptibility carries no temperature dependence.",
"Two questions come to mind when considering (REF ): What is $v$ at strong coupling?",
"Can the validity of this bound be tested in explicit models of incoherent transport?",
"Motivated by Gauge/Gravity duality computations, Blake proposed to replace $v$ in (REF ) by the “butterfly velocity\" $v_B$ [16], [17].",
"Indeed, the butterfly velocity appears in certain out-of-time-order four-point correlation functions and is a measure of how fast quantum information scrambles.",
"This provides a natural velocity at strong coupling, in contrast to the Fermi velocity which strictly speaking can only be defined in the presence of long-lived quasiparticles.",
"The butterfly velocity can be computed holographically in terms of horizon data by considering shockwave geometries [18], [16], [19], which encode the propagation of energy after a particle falls in the black hole horizon.",
"The butterfly velocity is closely linked to the Lyapunov time $\\tau _L$ , which also obeys a lower bound featuring the Planckian timescale, $\\tau _L\\geqslant \\hbar /2\\pi k_B T$ [20].",
"This bound is saturated by quantum field theories with Einstein holographic duals.",
"Thus, relating quantum chaos to incoherent metallic transport via Planckian timescales is an appealing proposal.",
"Another hint comes from recent progress in computing holographic DC thermoelectric conductivities.",
"It has been shown that these are given by formulæ evaluated on the black hole horizon under very general assumptions [21], [22], [23], [24], [25], [26].",
"As the metric and matter field expansion close to the horizon are independent from details of the UV asymptotics, these formulæ are in this sense universal.",
"By way of the Einstein relations, the diffusivities are therefore connected to physics at the black hole horizon, as is the butterfly velocity.",
"[16], [17] showed that the bound (REF ) held at low temperatures for particle-hole symmetric states which violate hyperscaling, both for exactly translation invariant black holes [27], [28], [29] as well as in the incoherent limit [23], [30].",
"In these specific examples, the precise coefficient on the right-hand side of (REF ) is given in terms of the set of critical exponents, but is not expected to be universal.",
"Of course, these holographic examples do not directly apply to bad metals, which are at finite density and not particle-hole symmetric.",
"They do provide evidence that some version of the bound of [5] is at work when transport is diffusion-dominated.",
"It is also important to note that no general proof of the bound (REF ) exists, as static susceptibilities depend in general on the full bulk solution and not just the horizon.",
"Said otherwise, the diffusivities are not given by horizon formulæ (though see the recent preprint where such a case is studied [31]).",
"More evidence for the bound (REF ) on energy diffusion was provided for finite density, AdS$_2$ horizons in [31], [32].",
"In this work, our goal is to study the sensitivity of the combined proposal of [5], [16], [17] to higher derivative terms in the effective holographic action.",
"As the Einstein-Hilbert action is really only a leading two-derivative term in what should be thought of as a low energy effective action, it is natural to include higher-derivative terms.",
"In passing, it also allows us to study the bound for a different class of finite density AdS$_2$ horizons than those of [31], [32].",
"Holographic bounds and higher derivative corrections have a rich common history [33], [34], [35], [36], [37], [38].",
"Whenever a bound of the kind (REF ) is formulated, the coefficient on the right-hand side of the inequality should really be understood as an $O(1)$ number: $\\frac{D_{c}}{v_B^2}\\ge \\mathcal {A}\\frac{\\hbar }{k_B T},\\qquad \\frac{D_{e}}{v_B^2}\\ge \\mathcal {B}\\frac{\\hbar }{k_B T},\\qquad \\mathcal {A},\\mathcal {B}\\sim O(1)$ The name of the game is now to find out how higher-derivative terms affect $\\mathcal {A}$ and $\\mathcal {B}$ , taking into account that: the higher-derivative couplings need to be small in some sense for the effective field theory approach to be well-defined; their allowed values are constrained by requiring the dual field theory to be causal.",
"For instance, the KSS bound [33] is lowered at most to ${\\eta \\over s}\\ge {16\\over 25\\pi }~ {\\hbar \\over k_B}$ upon including a Gauss-Bonnet term [34], so that some version of the original bound is still believed to hold.",
"On the other hand, while [35] proved a lower bound on the electric conductivity in Einstein-Maxwell theory, in [37], [38] it was shown how certain higher-derivative terms may lower this bound all the way to zero.",
"That is to say, these couplings are sufficiently unconstrained by the stability analysis to allow in principle the coefficient on the right hand side of the bound to vanish.",
"The specific holographic models we will use to study the bound (REF ) are given below in REF and REF .",
"They include quartic derivative terms between the Maxwell field strength and the translation-symmetry breaking scalar sector.",
"The first contains the higher-derivative coupling $\\frac{\\mathcal {J}}{4}\\,Tr[\\mathcal {X}\\,F^2]$ while the second contains $\\mathcal {K}\\,Tr[\\mathcal {X}]\\frac{F^2}{4}$ where $\\mathcal {X}$ involves the massless scalars and is defined in (REF ).",
"Our main results is that while the bound on the diffusion of energy remains impervious to these terms, they strongly affect the diffusion of charge in the incoherent limit.",
"For our two models, we find that $\\frac{D_c\\,T}{v_B^2}\\ge \\left(1-\\frac{3}{2}\\mathcal {J}\\right)\\,\\frac{1}{\\pi }\\,\\frac{\\hbar }{k_B}\\,,\\qquad \\frac{D_c\\,T}{v_B^2}\\ge \\left(1+6\\,\\mathcal {K}\\right)\\,f(\\mathcal {K})\\,\\frac{1}{\\pi }\\,\\frac{\\hbar }{k_B}$ with $f(\\mathcal {K})$ some function defined from (REF ).",
"Our analysis of stability and causality constraints restricts the couplings to $0\\le \\mathcal {J}\\le 2/3\\,,\\qquad -1/6\\le \\mathcal {K}\\le 1/6\\,.$ Unlike higher-derivative corrections to the KSS bound, they seem to allow for an arbitrary violation of the bound (REF ), namely the right hand side may be tuned as small as desired.",
"We pause here to note that it was already pointed out in [17] that the number $\\mathcal {B}$ on the right hand side of the energy diffusion bound could be arbitrarily small, provided the dynamical exponent $z$ is also small.",
"Inhomogeneous setups, both holographic or generalizations of SYK, also lead to violations of the bound featuring the butterfly velocity, [39], [40].",
"There, it is shown that the inequality sign in (REF ) is actually reversed.",
"Using higher derivative (gravitational) theories in order to investigate holographic phenomena is not without pitfalls.",
"The actions we use in this paper do not lead to higher order than second derivatives in the classical equations of motion, and so do not contain ghosts.",
"In the context of effective field theories, the higher-derivative couplings (including ours) should be considered as suppressed by appropriate powers of the Planck length or the effective string scale, and so do not typically give rise to causality violations in the absence of ghosts.",
"However, as pointed out above, we are also interested in situations where these corrections might be $\\mathcal {O}(1)$ .",
"This happens for instance in classical (large $N$ ), weakly-coupled string theory: the curvature corrections to the Einstein-Hilbert action are set by the string coupling $\\alpha ^{\\prime }$ , and become important at energies much lower than the Planck scale.",
"Then [41] showed that theories with such higher-derivative gravitational terms would necessarily violate causality, unless an infinite number of spin $\\ge 2$ particles were added at these energies.",
"Their calculation amounts to showing that the higher derivative corrections can induce time advances in high energy scattering experiments in shockwave backgrounds, which in turn can lead to close timelike curves.",
"We do not believe such causality violations can be triggered by the higher derivative terms we consider, since they do not involve higher derivatives of the metrics, which can be seen in [41] to ultimately be the source of the time advances.",
"To summarize this discussion: Rigorously speaking we cannot fully trust truncated derivative corrections in string theory.",
"Experience from many exact results in $\\alpha ^{\\prime }$ in string theory suggests that if the truncations do not violate basic principles of the theory (unitarity, the proper Cauchy problem etc), they are expected to give qualitatively trustworthy results.",
"We do not consider terms due to string loop corrections that may violate the large-N expansions at finite string coupling.",
"We are therefore confident that the physics we analyze is characteristic of healthy higher-derivative corrections in string theory, and that our results give a glimpse into the finite coupling constant regime of the associated dual theories.",
"In the remainder of the paper, we present our results in more detail.",
"Section is devoted to our holographic models, their black hole solutions and constraints coming from stability.",
"In section , we present the expressions for their DC thermoelectric conductivities.",
"In section we compute the charge and energy diffusion constants in the incoherent regime, and show how the charge diffusivity bound is affected by the higher derivative couplings.",
"Some technical details are relegated to a number of appendices."
],
[
"The holographic models",
"Our starting point is the Einstein-Hilbert action in 4 bulk dimensions with negative cosmological constant $\\Lambda $ (and $1/16\\,\\pi \\, G_N=1$ ): $\\mathcal {S}_g\\,=\\,\\int \\,d^4x\\,\\sqrt{-g}\\,\\left(\\,R\\,-\\,2\\,\\Lambda \\,\\right)$ To accommodate finite density states, we add a U(1) vector field $A_\\mu $ with associated field strength defined as $F_{\\mu \\nu }=\\partial _{[\\mu }A_{\\mu ]}$ .",
"We will break translation invariance by introducing two massless Stückelberg fields with a bulk profile $\\phi ^I=k\\,\\delta ^I_i\\, x^i$ [42].",
"We construct the mixed tensor: ${\\mathcal {X}^\\mu }_\\nu \\equiv \\frac{1}{2}\\sum _{I=x,y}\\partial ^\\mu \\phi ^I \\partial _\\nu \\phi ^I={1\\over 2}\\sum _{I=x,y}g^{\\mu \\rho }\\partial _{\\rho }\\phi ^I\\partial _\\nu \\phi ^I\\,\\,.$ and consider the generic action, coupling the electromagnetic and translation-symmetry breaking sectors: $\\mathcal {S}&=&\\mathcal {S}_g\\,+\\,\\mathcal {S}_a\\\\\\mathcal {S}_a&=&-\\,\\int \\,d^4x\\,\\sqrt{-g}\\,\\mathcal {Z}\\left(Tr[\\mathcal {X}^m],\\,Tr[\\mathcal {X}^n\\,F^2]\\,\\right)\\,.$ where $Tr[\\mathcal {X}^m]\\,\\equiv \\, \\mathcal {X}^\\mu _{\\,\\,\\,\\,\\nu _1\\dots }\\mathcal {X}^{\\nu _{m-1}}_{\\hspace{19.91684pt}\\mu }\\;\\;\\;,\\;\\;\\;Tr[\\mathcal {X}^n\\,F^2]\\,\\equiv \\,[\\mathcal {X}^n]^\\mu _{\\,\\,\\,\\,\\nu }~F^{\\nu }_{\\hspace{5.69046pt}\\nu ^{\\prime }}~F^{\\nu ^{\\prime }}_{\\hspace{5.69046pt}\\mu }$ and the indices run over non-negative integers $m,n\\,=\\,0,\\,1,\\,2\\dots $ .",
"For convenience we also define $Tr[\\mathcal {X}]\\equiv X$ .",
"We focus on the two following classes of models: Model 1: $\\mathcal {S}_a\\,=\\,-\\,\\,\\int \\,d^4x\\,\\sqrt{-g}\\,\\left(\\,X\\,+\\,\\frac{1}{4}\\,F^2\\,+\\,\\frac{\\mathcal {J}}{4}\\,Tr[\\mathcal {X}\\,F^2]\\,\\right)$ This model was introduced and analyzed recently in [38].",
"Model 2: $\\mathcal {S}_a\\,=\\,-\\,\\int \\,d^4x\\,\\sqrt{-g}\\,\\,\\mathcal {W}\\left(X,F^2/4\\right)$ This is a rather general class of models.",
"Within this class we will mostly focus on a special benchmark case: $\\textbf {Model\\,2_U}\\,:\\qquad \\mathcal {W}(X,F^2/4)=X+U(X)\\,\\frac{F^2}{4}$ Moreover, in some cases we will specialize further and define: $\\textbf {Model\\,2_\\mathcal {K}}\\,:\\qquad U(X)\\,=\\,1\\,+\\,\\mathcal {K}\\,X$ also studied in [38].",
"Furthermore we consider an isotropic ansatz for the bulk metric and other fields: $ds^2=-D(r)\\,dt^2+B(r)\\,dr^2+C(r)\\, dx^idx_i, \\ \\ A_\\mu =A_t (r)\\,dt, \\ \\ \\phi ^I=k\\,\\delta ^I_i\\, x^i,$ where $i=x,y$ denotes the two spatial directions.",
"The aim of this paper is to study the effects of the the higher derivative terms (REF ), (REF ) on the transport properties of the dual CFT at finite temperature T and charge density $\\rho $ .",
"If we set $\\mathcal {J}=0$ or $U(X)=1$ , then we recover the “linear axion model” of [42]."
],
[
"Model 1: the $\\mathcal {J}$ coupling",
"The $\\mathcal {J}$ coupling does not affect the solution to the background equations given our Ansatz (REF ).",
"This follows from how indices are contracted in $Tr(\\mathcal {X}^n F^2)$ and it holds for all $n>1$ .",
"The background is then identical to the one found in [43], [42]: $&ds^2\\,=\\,-\\,D(r)\\,dt^2\\,+\\,\\frac{dr^2}{D(r)}\\,+\\,r^2\\,dx^i\\,dx_i\\,,\\nonumber \\\\&D(r)\\,=\\,r^2\\,\\left[\\,1\\,-\\,\\frac{r_h^3}{r^3}\\,-\\,\\left(\\frac{k^2}{2\\,r^2}\\,+\\,\\frac{\\mu ^2\\,r_h}{4\\,r^3}\\right)\\,\\left(1\\,-\\,\\frac{r_h}{r}\\right)\\,\\right],\\nonumber \\\\&A\\,\\equiv A_tdt=\\,\\left(\\mu -\\frac{\\rho }{r}\\right)\\,dt$ where we fix $\\Lambda =-3$ , and $r_h$ is the location of the event horizon.",
"Regularity of the gauge field at the horizon implies that we have $\\rho =\\mu \\,r_h$ , and the temperature of the background can be identified with the surface gravity at the horizon: $T\\,=\\,\\frac{D^{\\prime }(r_h)}{4\\,\\pi }\\,=\\,\\frac{3 \\,r_h}{4 \\,\\pi }\\,-\\frac{k^2}{8 \\,\\pi \\,r_h}-\\frac{\\mu ^2}{16\\, \\pi \\,r_h}$ These are the background data we will use later in computing the conductivities."
],
[
"Model 2: $\\mathcal {W}(X,F^2/4)$ action",
"This class of models represents a generalization of what was already presented and studied in [38], [44], [45], [37], [46].",
"To simplify notation, we define $Y={1\\over 4}F^2 \\;\\;\\;,\\;\\;\\;\\mathcal {W}_{Y}(Y,X)\\equiv \\frac{\\partial \\mathcal {W}(Y,X)}{\\partial Y}\\;\\;\\;,\\;\\;\\;\\mathcal {W}_{X}(Y,X)\\equiv \\frac{\\partial \\mathcal {W}(Y,X)}{\\partial X}$ The solution for the background metric takes the form: $&ds^2\\,=\\,-\\,D(r)\\,dt^2\\,+\\,\\frac{1}{D(r)}\\,dr^2\\,+\\,r^2\\,dx^i\\,dx_i\\,,\\nonumber \\\\&D(r)\\,=\\,\\frac{1}{2\\,r}\\int _{r_h}^r d\\tilde{r} \\left[6\\,\\tilde{r} ^2-\\mathcal {W}(\\bar{Y},\\bar{X})\\,\\tilde{r} ^2-\\frac{\\rho ^2}{\\tilde{r} ^2\\,\\mathcal {W}_{Y}(\\bar{Y},\\bar{X})}\\right]\\,,$ where $r=r_h$ is again the position of the event horizon.",
"The time component of the Maxwell equations for the gauge field $A=A_t(r)dt$ yields: $\\rho =r^2\\,\\mathcal {W}_{Y} \\left(\\bar{Y},\\bar{X}\\right)\\,A_t^{\\prime }$ where the constant $\\rho $ represents the charge density of our system.",
"The background values $\\bar{X},\\bar{Y}$ for the $X,Y$ scalar invariants turn out to be: $\\bar{X}\\,=\\,\\frac{k^2}{r^2}\\,,\\qquad \\bar{Y}\\,=\\,\\frac{1}{2}\\,A_t^{\\prime }(r)^2$ The temperature of the solution is given as always by: $T\\,=\\,\\frac{D^{\\prime }(r_h)}{4\\,\\pi }\\,=\\,\\frac{1}{4\\,\\pi }\\left[3\\,r-\\frac{\\mathcal {W}(\\bar{Y},\\bar{X})\\,r}{2}-\\frac{\\rho ^2}{2\\,r^3\\,\\mathcal {W}_{Y}(\\bar{Y},\\bar{X})}\\right]_{r=r_h}$ More details about the specific models $2_U$ and $2_\\mathcal {K}$ are presented in appendix .",
"In particular, when $\\mathcal {K}<0$ , some care must be exercised to derive the background solution.",
"However, physical quantities expressed in terms of field theory data $(T,\\mu )$ can safely be analytically continued to from $\\mathcal {K}>0$ to $\\mathcal {K}<0$ ."
],
[
"Stability",
"The higher-derivative couplings $\\mathcal {J}, \\mathcal {K}$ were constrained in [38] by imposing positivity of the DC electric conductivity and studying the stability of the $a_x$ linear perturbation at zero density: $0\\,\\le \\,\\mathcal {J}\\,\\le 2/3\\,,\\qquad \\qquad -1/6\\,\\le \\,\\mathcal {K}\\,\\le \\,1/6\\,.$ Here it is worth emphasizing that only the lower bound on $\\mathcal {K}$ comes from considering the stability of the linear fluctuations at non-zero frequency – a significantly harder problem than in the DC limit, where closed form expressions for all DC conductivity can be obtained and their inspection yields the other constraints.",
"We have extended the analysis in [38] by looking at both at background and linearized probes.",
"The null energy condition (NEC) and the local thermodynamic stability (positivity of the specific heat and charge susceptibility) can be studied directly from the background solution.",
"We find that the static susceptibilities are positive for all values of the higher-derivative couplings and do not constrain them at all.",
"On the other hand, the NEC requires $\\mathcal {J}\\le 2/3$ and $\\mathcal {K}\\ge -1/6$ .",
"Further details are given in appendix .",
"Here we simply comment on the NEC.",
"It implies in general $\\bar{X}\\,\\mathcal {W}_X(\\bar{Y},\\bar{X})\\,-\\,2\\,\\bar{Y}\\,\\mathcal {W}_Y(\\bar{Y},\\bar{X})\\,\\ge \\,0\\,.$ where $\\bar{X},\\bar{Y}$ are the background values for $X,Y$ .",
"This constraint coincides with the absence of ghosts and matches with previous studies [44], [37], [46], [38].",
"In particular it leads to a positive effective graviton mass squared $m_g^2\\ge 0$ .",
"Extending to linear fluctuations, we could perform two checks: the stability of the parity-odd fluctuations at zero wavevector and zero density; and the analysis of the scaling dimensions of the IR operators in the AdS$_2\\times R^2$ zero temperature spacetime, both in the transverse and longitudinal sector and at non-zero wavevector $q$ .",
"If these dimensions become complex for certain values of the couplings and a certain range of wavevectors, we have found an instability.",
"At non-zero density, the linear fluctuation equations are coupled and we could not rewrite them as decoupled Schrödinger equations.",
"One way to confirm our stability analysis would be to inspect the spectrum of quasi-normal modes and check they are all in the lower half of the complex frequency plane.",
"This analysis would be quite involved and beyond the scope of this paper.",
"So we content ourselves with the necessary conditions (REF ).",
"The analysis of the scaling dimensions of the IR operators is simplest when the linear equations around AdS$_2\\times R^2$ can be decoupled in terms of gauge-invariant master variables.",
"These decoupled equations can be integrated, imposing ingoing boundary conditions.",
"And the scaling dimensions can be read off from the asymptotics of the resulting solutions.",
"This program can only be carried out in very special, highly symmetric cases, like the AdS-Reissner-Nordstrom black hole.",
"It does not seem possible in our setup, as the equations do not decouple.",
"We can however work out the scaling dimensions by plugging in a power law Ansatz for the perturbations.",
"Details of the derivation are provided in appendix .",
"We do not write here the final expressions for the scaling dimensions, which are very messy.",
"For the model 1, we could check analytically that the scaling dimensions are always real in the range $0\\le \\mathcal {J}\\le 2/3$ .",
"The model $2_{\\mathcal {K}}$ is harder to analyze in full generality.",
"We can show that the transverse scaling dimensions are real in the range $-1/6\\le {\\mathcal {K}}\\le 1/6$ .",
"In the longitudinal sector, we can only do this when picking random values for ${\\mathcal {K}}$ in the same range but we cannot prove it in general.",
"All in all, we take it that the arguments above make a very good case for stability of both models given the condition (REF )."
],
[
"DC thermoelectric conductivities",
"Thermoelectric transport in the dual CFT can be described by the generalized Ohm's law: $\\left( \\begin{array}{c}J^x \\\\Q^x \\\\\\end{array}\\right)=\\left(\\begin{array}{cc}\\sigma & \\alpha \\,T \\\\\\bar{\\alpha }\\, T & \\bar{\\kappa } \\,T \\\\\\end{array}\\right)\\left( \\begin{array}{c}E_x \\\\-\\nabla _x T/T \\\\\\end{array}\\right)$ where the matrix of thermoelectric conductivities parametrizes the linear response to electric fields and temperature gradients.",
"In the absence of parity violation, the conductivity matrix is symmetric $\\alpha =\\bar{\\alpha }$ .",
"DC conductivities can be computed holographically in terms of data on the black hole horizon using the techniques described in [23], [24].",
"We simply quote the final results in the main text and relegate the details of the computation to appendix ."
],
[
"Model 1",
"The DC conductivities for the model described in REF read as follows: $&\\sigma \\,=\\,1\\,-\\,\\frac{\\mathcal {J}\\, k^2}{4 \\,r_h^2}+\\,\\frac{\\mu ^2 \\left(1-\\frac{\\mathcal {J} \\,k^2}{4 \\,r_h^2}\\right)^2}{k^2 \\left(1+\\frac{\\mathcal {J} \\,\\mu ^2}{4\\,r_h^2}\\right)}\\,,\\\\&\\bar{\\kappa }\\,=\\,\\frac{4\\, \\pi \\, s\\, T}{k^2 \\left(1+\\frac{\\mathcal {J} \\,\\mu ^2}{4\\,r_h^2}\\right)}\\,,\\\\&\\alpha \\,=\\,\\bar{\\alpha }\\,=\\,\\frac{4\\,\\pi \\,\\mu \\,r_h\\,\\left(1-\\frac{\\mathcal {J} \\,k^2}{4 \\,r_h^2}\\right)}{k^2 \\left(1+\\frac{\\mathcal {J} \\,\\mu ^2}{4\\,r_h^2}\\right)}\\,,\\\\&\\kappa \\,=\\,\\frac{16\\, \\pi ^2\\, r_h^2\\, T}{k^2+\\mu ^2}\\,.$ where the entropy density $s=4\\,\\pi \\, r_h^2$ .",
"Here, $\\bar{\\kappa }$ is the thermal conductivity at zero electric field, while $\\kappa $ is the thermal conductivity at zero current.",
"They are related through $\\kappa =\\bar{\\kappa }-\\,\\bar{\\alpha }\\,\\alpha \\, T/\\sigma $ ."
],
[
"Model 2",
"The DC conductivities for the model described in REF read: $&\\sigma =\\left[\\mathcal {W}_{Y}\\left(\\bar{Y},\\bar{X}\\right)+\\frac{4\\,\\pi \\,\\rho ^2}{k^2\\,s\\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)}\\right]_{r=r_h}, \\\\&\\alpha =\\bar{\\alpha }=\\frac{4\\,\\pi \\,\\rho }{k^2\\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)}\\Big |_{r=r_h},\\\\&\\bar{\\kappa }=\\frac{4 \\,\\pi \\, s\\,T}{ k^2\\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)}\\Big |_{r=r_h} \\\\&\\kappa =\\frac{4\\,\\pi \\, s\\,T}{k^2\\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)+\\frac{4\\,\\pi \\,\\rho ^2}{s\\,\\mathcal {W}_{Y}\\left(\\bar{Y},\\bar{X}\\right)}}\\Big |_{r=r_h}.$ We can additionally define the Lorentz ratios: $&\\bar{L}\\,\\equiv \\,\\frac{\\bar{\\kappa }\\,T}{\\sigma }\\,=\\,\\frac{4\\,\\pi \\, s^2\\,T^2}{k^2\\,s\\,\\mathcal {W}_Y\\,\\mathcal {W}_X\\,+\\,4\\,\\pi \\, \\rho ^2}\\,,\\\\&L\\,\\equiv \\,\\frac{\\kappa \\,T}{\\sigma }\\,=\\,\\frac{4\\,\\pi \\, k^2\\, s^3\\, \\mathcal {W}_Y\\,\\mathcal {W}_X\\,T^2}{\\left[k^2\\,s\\,\\mathcal {W}_Y\\,\\mathcal {W}_X\\,+\\,4\\,\\pi \\, \\rho ^2\\right]^2}\\,.$ The values of the DC transport coefficients (REF )-() for the particular models $2_U$ and $2_\\mathcal {K}$ are presented in appendix ."
],
[
"About the Kelvin formula ",
"Recently, the relation $\\frac{\\alpha }{\\sigma }\\Big |_{T=0}\\,\\equiv \\,\\lim _{T\\rightarrow 0}\\frac{\\partial s}{\\partial \\rho }\\Big |_T$ has been highlighted as a feature of any AdS$_2\\times R^2$ horizon [31], [32].",
"[32] argued further that this was fixed by the symmetries of AdS$_2$ .",
"Indeed, we observe that (REF ) is verified in all the models we considered.",
"We give more details about this check in appendix ."
],
[
"Impact of higher derivative couplings on the diffusivity bounds",
"The incoherent limit, i.e.",
"the limit of strong momentum dissipation, is defined by: $T,\\mu \\ll k$ while keeping the dimensionless ratio $T/\\mu $ finite.",
"This is the regime where transport is governed by diffusive processes [5] rather than by slow momentum relaxation, as was checked in the linear axion model [14].",
"In this limit, both the off-diagonal conductivities decay faster with $k$ than the diagonal ones (which are actually non-zero).",
"Effectively, the charge and heat flows decouple [15] in spite of the fact that this is not a zero density limit.",
"The same is true for the matrix of static susceptibilities.",
"Consequently, in the incoherent regime, the charge and energy diffusivities can be independently defined as: $T,\\mu \\ll k:\\qquad \\qquad D_c\\,=\\,\\frac{\\sigma }{\\chi }\\,,\\qquad D_e\\,=\\,\\frac{\\kappa }{c_v}$ where $\\chi ={\\partial \\rho \\over \\partial \\mu }\\Big |_{T}\\;\\;\\;,\\;\\;\\;c_v=T \\,{\\partial s\\over \\partial T}\\Big |_{\\mu }\\,.$ $\\chi $ is the charge susceptibility at constant temperature and $c_v$ the specific heat of the system at constant chemical potential (which in this limit is the same as at constant charge density).",
"The butterfly velocity of the system, describing the spreading of quantum information in the dual QFT, has been already computed in [16] for a generic background of the form (REF ) and it turns out to be: $v_B^2\\,=\\,\\frac{2\\,\\pi \\,T}{C^{\\prime }(r_h)}$ Because we have chosen a radial gauge so that $C(r)=r^2$ we obtain the general expression: $v_B^2\\,=\\,\\frac{\\pi \\,T}{r_h}$ The linear axion model was defined in the beginning of section REF and corresponds to setting the higher derivative couplings $\\mathcal {J}$ and $\\mathcal {K}$ to zero.",
"In this model and in the incoherent limit defined in (REF ), both the charge and the energy diffusivities, appropriately normalized, are bounded from below as shown in [16] (in passing generalizing the analysis there to finite density), $\\frac{D_c\\,T}{v_B^2}\\ge \\frac{1}{\\pi }\\;\\;\\;,\\;\\;\\;\\frac{D_e\\,T}{v_B^2}\\ge \\frac{1}{2\\,\\pi }\\;.$ Our aim is to investigate if this inequality is still valid once higher derivative corrections are taken in consideration."
],
[
"Model 1: the $\\mathcal {J}$ coupling ",
"Since the background is not affected by $\\mathcal {J}$ , it is straightforward to perform the same computations at finite $\\mathcal {J}$ .",
"The susceptibility is given by: $\\chi \\,=\\,r_h\\;,$ while the conductivity in the incoherent limit is: $\\sigma _{DC}^{(inc)}\\,=\\,\\left(1\\,-\\,\\frac{3}{2}\\,\\mathcal {J}\\right)$ We note that in the incoherent limit, the radius of the horizon becomes proportional to the momentum dissipation strength $k$ .",
"In particular, for models 1, $2_U$ and $2_{{\\mathcal {K}}}$ considered in this paper, we have $r_h\\,=\\, k/\\sqrt{6}\\;.",
"$ The equation above implies that in such models the butterfly velocity in the incoherent limit becomes: $\\left(v_B^2\\right)^{(inc)}\\,=\\,\\frac{\\sqrt{6}\\,\\pi \\,T}{k}$ In addition, the heat capacity and the thermal conductivity in the combined incoherent limit are given by: $\\kappa ^{(inc)}\\,=\\,\\frac{8\\,\\pi ^2}{3}\\,T,\\qquad c_v^{(inc)}\\,=\\,\\frac{8}{3}\\,\\sqrt{\\frac{2}{3}}\\,\\pi ^2\\,T\\,k\\,.$ Using (REF )-(REF ) we obtain the following equalities in this limit $\\frac{D_c\\,T}{v_B^2}\\Big |_{inc}= \\frac{1}{\\pi }\\,\\left(1\\,-\\,\\frac{3}{2}\\,\\mathcal {J}\\right)\\;\\;\\;,\\;\\;\\;\\frac{D_e\\,T}{v_B^2}\\Big |_{inc}=\\frac{1}{2\\,\\pi }\\;.$ The charge diffusivity is modified to leading order in the incoherent limit and the dimensionless ratio $\\frac{D_c\\,T}{v_B^2}$ vanishes for $\\mathcal {J}=2/3$ .",
"At that same value of $\\mathcal {J}$ the incoherent DC conductivity $\\sigma _{DC}^{(inc)}$ vanishes.",
"We believe this to be a generic feature of all effective actions where momentum relaxing terms couple directly to the Maxwell term.",
"We obtain the same results considering higher order deformations of the type: $\\sim \\,\\,Tr\\left(\\mathcal {X}^n\\,F^2\\right)\\,,\\qquad \\text{with}\\quad n>1\\,.$ for all of which the background would still remain unchanged."
],
[
"Model $2_U$ ",
"We will now investigate the $2_U$ class of models defined as $\\mathcal {W}(X,Y)\\,=\\,X\\,+\\,U(X)\\,Y\\;.$ In this case, the static susceptibility in the incoherent limit is: $\\chi \\,=\\,\\left(\\,\\int _{r_h}^{\\infty }\\,\\frac{1}{y^2\\,U(k^2/y^2)}\\,dy\\,\\right)^{-1}$ The precise derivation of this formula is shown in appendix .",
"The susceptibility above is finite, because $U(0)=1$ (in order to have the correctly normalized Maxwell term near the boundary) but it is manifestly not given in terms of horizon data and depends on the full bulk geometry [21], [16].",
"The DC conductivity in the incoherent limit can be extracted from the generic formulæ of the previous section and it reads: $\\sigma _{DC}^{(inc)}=\\,U(k^2/r_h^2)$ Combining the previous results we conclude that in the combined incoherent limit the diffusivity asymptotes to $\\frac{D_c\\,T}{v_B^2}\\Big |_{inc}\\,=\\,\\lim _{\\begin{array}{c}\\\\T\\,\\rightarrow \\, 0 \\\\ \\mu \\,\\rightarrow \\, 0\\end{array}}\\,\\frac{U(k^2/r_h^2)\\,r_h}{\\pi }\\,\\int ^{\\infty }_{r_h}\\,\\frac{1}{y^2\\,U(k^2/y^2)}$ The key point is that the dimensionless ratio $\\frac{D_c\\,T}{v_B^2}$ becomes zero every time $U(X)$ vanishes in the incoherent limit.",
"This is the same point were $\\sigma _{DC}^{(inc)}$ vanishes.",
"We find this correlation robust and present in all the models we considered.",
"Figure: Physical quantities in the incoherent limit for the specific model 2 𝒦 2_\\mathcal {K}: U(X)=1+𝒦XU(X)=1+\\mathcal {K}X.",
"Left: Susceptibility in units of k as a function of 𝒦{\\mathcal {K}}.",
"The dashed red line has been added manually because Mathematica was not able to plot the function in its whole domain.",
"Right : Incoherent conductivity as a function of 𝒦{\\mathcal {K}}.On the other hand we can show that the $U(X)$ coupling does not affect energy diffusion and we still have: $\\frac{D_c\\,T}{v_B^2}\\Big |_{inc}=\\frac{1}{2\\,\\pi }$ This is due to the fact that the value of the heat capacity and the thermal conductivity in the combined incoherent limit are not modified by the U coupling and they still take the form indicated in (REF )."
],
[
"Model $2_\\mathcal {K}$ : the {{formula:cb04f7f8-4317-44e6-8051-4728ee340741}} coupling ",
"To illustrate the previous paragraph, we choose the function: $U(X)\\,=\\,1\\,+\\mathcal {K}\\,X$ In the allowed range of parameters, the DC conductivity in the incoherent limit is given by: $\\sigma _{DC}^{(inc)}=1+6\\,\\mathcal {K} \\quad \\text{with } -1/6\\le \\,\\mathcal {K}\\,\\le 1/6$ and the charge susceptibility by: $&\\chi ^{(inc)}={\\left\\lbrace \\begin{array}{ll}\\frac{2 \\,k\\, \\sqrt{\\mathcal {K}}}{\\pi -2 \\,ArcTan\\left(\\frac{1}{ \\sqrt{6\\,\\mathcal {K}}}\\right)} & \\quad \\text{if }\\,\\,\\,\\, 0<\\mathcal {K}\\le 1/6\\\\[0.2cm]1/\\sqrt{6} & \\quad \\text{if }\\,\\,\\mathcal {K}\\,=\\,0\\\\[0.2cm]\\frac{k\\, \\sqrt{|\\mathcal {K}|}}{\\log \\left(\\frac{\\sqrt{6\\,|\\mathcal {K}|}+1}{\\sqrt{1-6 \\,|\\mathcal {K}|}}\\right)} & \\quad \\text{if } \\,\\,-1/6\\le \\mathcal {K}<0\\\\\\end{array}\\right.",
"}$ These two quantities are shown in fig.REF .",
"The incoherent heat capacity and the thermal conductivity are not affected by the $\\mathcal {K}$ coupling and they take the form (REF ).",
"Using the definition of the butterfly velocity given previously, we compute the diffusivities and obtain the dimensionless ratios: $&\\frac{D_c\\,T}{v_B{}^2}\\Big |_{inc}={\\left\\lbrace \\begin{array}{ll}\\frac{(6 \\,\\mathcal {K}+1) \\left(\\pi \\,-\\,2\\, ArcTan\\left(\\frac{1}{\\sqrt{6 \\,\\mathcal {K}} }\\right)\\right)}{2 \\,\\pi \\, \\sqrt{6\\,\\mathcal {K}}}\\,+\\,\\dots & \\quad \\text{if }\\,\\,\\,\\, 0<\\mathcal {K}\\le 1/6\\\\[0.2cm]\\frac{1}{\\pi }& \\quad \\text{if }\\,\\,\\mathcal {K}\\,=\\,0\\\\[0.2cm]\\frac{(1-6 \\,|\\mathcal {K}|) \\log \\left(\\frac{ \\sqrt{6\\,|\\mathcal {K}|}+1}{\\sqrt{1-6 \\,|\\mathcal {K}|}}\\right)}{\\pi \\, \\sqrt{6\\,|\\mathcal {K}|}}\\,+\\,\\dots & \\quad \\text{if } \\,\\,-1/6\\le \\mathcal {K}<0\\\\\\end{array}\\right.",
"}\\\\[0.2cm]&\\frac{D_e\\,T}{v_B{}^2}\\Big |_{inc}=\\frac{1}{2\\,\\pi }$ Figure: Incoherent limit of D c T v B 2 \\frac{D_c\\,T}{v_B^2} in function of the 𝒦\\mathcal {K} parameter for the choice U(X)=1+𝒦XU(X)=1+\\mathcal {K}X.",
"The dashed line is the bound in the case 𝒦=0\\mathcal {K}=0.The behaviour of $D_c\\,T/{v_B}^2$ in function of $\\mathcal {K}$ is shown in fig.REF .",
"It vanishes when $\\mathcal {K}=-1/6$ , at the boundary of the stability region.",
"There, the DC conductivity vanishes linearly while the charge susceptibility does so only logarithmically.",
"The version of the charge diffusivity bound proposed in [16], [17] can be violated in this model as well.",
"In contrast, the higher-derivative term does not affect the ratio $D_e\\,T/{v_B}^2$ ."
],
[
"Arbitrary Stückelberg potential $V(X)$ ",
"As we have seen above, the bound on the diffusion of energy is not affected by the higher-derivative couplings we have turned on.",
"A natural extension is to introduce an arbitrary potential $V(X)$ for the Stückelberg fields, rather than the linear version $V(X)=X$ we have been using throughout the draft.",
"However, as we show below, this has no effect on the diffusion of energy in the incoherent limit.",
"For simplicity, we consider $W(X,F^2)=V(X)$ in (REF ), that is we consider zero density states.",
"The temperature of the model is defined by: $T\\,=\\,\\frac{3 \\,r_h}{4 \\,\\pi }-\\frac{r_h\\, V\\left(\\frac{k^2}{r_h^2}\\right)}{8 \\,\\pi }$ The radius of the horizon in the incoherent limit is still proportional to the momentum dissipation strength k via the relation: $r_h^{(inc)}\\,=\\,\\frac{k}{\\sqrt{V^{-1}(6)}}$ which is in agreement with (REF ) if we set $V(X)=X$ .",
"In addition the thermal conductivity and the heat capacity are defined by: $\\kappa \\,=\\,\\frac{4\\,\\pi \\,s\\,T}{k^2\\,V^{\\prime }\\left(\\frac{k^2}{r_h^2}\\right)}\\,,\\qquad c_v\\,=\\,8\\,\\pi \\,r_h\\,T\\,\\left(\\frac{dT}{d r_h}\\right)^{-1}=\\frac{64 \\,\\pi ^2 \\,r_h^3\\, T}{2\\, k^2\\, V^{\\prime }\\left(\\frac{k^2}{r_h^2}\\right)-r_h^2 \\left(V\\left(\\frac{k^2}{r_h^2}\\right)-6\\right)}$ In the incoherent limit (REF ) we discover that their ratio reads as: $\\frac{\\kappa }{c_v}\\Big |_{inc}\\,=\\,\\frac{\\sqrt{V^{-1}(6)}}{2\\,k}$ Once we combine the previous result with the definition of the butterfly velocity we obtain: $\\frac{D_e\\,T}{v_B^2}\\Big |_{inc}\\,=\\,\\frac{\\kappa }{c_v}\\,\\frac{T}{v_B^2}\\Big |_{inc}\\,=\\,\\frac{1}{2\\,\\pi }$ which is the same expression found for the linear choice $V(X)=X$ .",
"Therefore we conclude that the $V(X)$ generalization has no impact on the energy diffusion and the following inequality $\\frac{D_e}{v^2}\\gtrsim \\frac{\\hbar }{k_B T}$ still holds.",
"We observe that this originates from two successive cancellations, such that in the end the general potential $V(X)$ does not affect the bound (REF ).",
"Firstly, some factors of $V^{\\prime }(X)$ drop out when computing the energy diffusivity in the incoherent limit.",
"Secondly, the remaining factor $V^{-1}(6)$ in (REF ) is compensated by an analogous term in the expression for the butterfly velocity, leading finally to (REF )."
],
[
"Discussion",
"In this paper, we studied higher derivative couplings $g_i$ between the charge and translation symmetry breaking sectors in toy-models of holographic thermoelectric transport.",
"Focusing on the limit of fast momentum relaxation, we pointed out that these terms have a very strong impact on a recently proposed bound on charge diffusion [16], [17] (elaborating on a previous proposal [5]): $\\frac{D_c}{v_B^2}\\gtrsim \\frac{\\hbar }{k_B T}$ where $v_B$ is the butterfly velocity.",
"While the proposal in [5] essentially came from general considerations as well as experimental data on so-called bad metals, its refinement in [16], [17] was justified using holographic computations.",
"As such, it is rather natural to test it further by including higher derivative terms in the effective holographic action.",
"For simplicity, we restricted our investigation to models with quartic couplings only (see section ).",
"We paid particular attention to the stability and the consistency of the models restricting the allowed values for the couplings: $g_i^{min}\\,\\le \\,g_i\\,\\le \\,g_i^{max}$ where the edge values depend on the specific features of the model (see section REF ).",
"In more detail for all those cases we found a relation of the type: $\\frac{D_c}{v_B^2}\\,=\\,\\mathcal {A}\\left(g_i\\right)\\,\\frac{\\hbar }{k_B T}$ where $\\mathcal {A}$ is an order one number which only depends on the higher derivative couplings.",
"It vanishes for particular finite values of the higher derivative couplings $g_i^{*}$ .",
"Of course one should keep in mind that these higher derivative couplings should be suppressed by powers of the string coupling, so it is unclear how realistic the values leading to $\\mathcal {A}(g_i^{*})=0$ are.",
"We note that since no higher derivative gravitational term is involved, the couplings we consider may be $\\mathcal {O}(1)$ without violating causality along the lines of [41].",
"It is very intriguing that the values $g_i^{*}$ lie at the edge of the range allowed by the stability analysis.",
"This is also true for the hyperscaling violating metrics examined in [16], [17].",
"It would be worthwhile to understand this better.We are grateful to Mike Blake for discussions on this point.",
"Let us pause to compare with the analogous violation of the KSS bound $\\eta /s\\ge \\hbar /4\\pi k_B$ by higher derivative terms, like Gauss-Bonnet [47].",
"Including these terms modify the order one number on the right hand side and indeed can lower it, but causality prevents its vanishing.",
"So the notion that there should be a lower bound on the ratio of shear viscosity to entropy density in strongly-coupled quantum field theories still survives.",
"Our case is crucially different since the violation can be arbitrary down to zero value, at least up to the validity of our stability analysis.",
"Admittedly, we have not fully carried it out as the lack of decoupling of the fluctuation equations render it untractable analytically.",
"A more elaborate numerical analysis is needed and beyond the scope of this work.",
"We hope to return to it in the future.",
"Two more features of our analysis, specifically due to the incoherent limit $T,\\mu \\ll k$ are very noteworthy.",
"First, thermal and electrical transport always decouple.",
"This was already noted in [15]: there, two decoupled, gauge-invariant bulk variables were found to be dual to two decoupled currents (with, in the language of (REF ), zero off-diagonal static susceptibility), which in the incoherent limit asymptoted to the charge and heat currents respectively.",
"The same physical mechanism is at work here, upon turning on higher-derivative terms: while we have not been able to find decoupled bulk variables, the off diagonal elements of the conductivity and susceptibility matrices decay faster than the diagonal one.",
"It would be very interesting if this was a general feature of thermoelectric incoherent transport, beyond these specific holographic examples.",
"We also found a strong correlation between the vanishing of the dimensionless parameter $\\mathcal {A}$ , which controls charge diffusion, and the vanishing of the corresponding DC electric conductivity in the incoherent limit.",
"In all the models we considered the charge susceptibility remains finite in the incoherent limit implying the relation: $\\mathcal {A}\\left(g_i^{*}\\right)\\,=\\,0\\qquad \\Longleftrightarrow \\qquad \\sigma _{DC}^{(inc)}\\left(g_i^{*}\\right)\\,=\\,0$ In other words, the charge diffusion bound is badly violated every time the corresponding incoherent electric DC conductivity vanishes.",
"One way out would be if the bound shown in [35] in four-dimensional Einstein-Maxwell theories could be generalized to our setup.",
"But as we have argued, and unless a more refined stability analysis narrows the allowed range for the couplings, this does not seem to be the case.",
"A relevant question here is to what extend such a bound is independent from those on the conductivity [37], [38].",
"This depends on the behaviour of the static susceptibility and butterfly velocity in the incoherent limit.",
"In our model 1, the ratio $v_B^2\\chi /T$ is $T$ and $r_h$ independent.",
"This is a special feature of this model, whereby the background is not affected by the higher-derivative coupling.",
"It affects only transport and so indeed the vanishing of the diffusivity bound follows from the vanishing of the dc conductivity.",
"The behaviour of the charge diffusivity is less trivial in the model with the $\\mathcal {K}$ coupling, as seen from eqns (REF )-(REF ).",
"Background thermodynamics are affected by the higher-derivative coupling.",
"From (REF ), we see that $\\chi \\rightarrow 0$ when $\\mathcal {K}\\rightarrow -1/6$ .",
"This means that in this limit, no electric current propagates (the dc conductivity is zero), and introducing a small chemical potential does not create a charge density in linear response (the susceptibility is zero).",
"However, the dc conductivity vanishes faster than the susceptibility, so the charge diffusivity also vanishes.",
"In the two models we consider, it thus appears that there is a close relation between the vanishing of the dc conductivity and the violation of the diffusivity bound.",
"It would be interesting to prove that the static susceptibility can never vanish fast enough to spoil this.",
"To get a better handle on how higher-derivative couplings affect the bound on charge diffusion, it would be interesting to consider other models, such as non-linear electrodynamics [46], including non-linear DBI setups [48].",
"The higher derivative couplings we have considered do not affect the energy diffusion bound in the incoherent limit, including when an arbitrary potential $V(X)$ for the Stückelberg fields is included.",
"A natural future direction would be to consider higher derivative couplings between the gravity and Stückelberg sector, responsible for momentum relaxation.",
"A careful analysis of causality along the lines of [41] will be required in this case.",
"More recently, it was shown in [40] that inhomogeneities could lead to a sign reversal of the bound.",
"Understanding better the validity of the diffusion bounds featuring the butterfly velocity and the interplay with translation symmetry breaking is clearly an important issue."
],
[
"Acknowledgments",
"We would like to thanks Mike Blake and Sean Hartnoll for interesting and useful comments on the manuscript.",
"MB would like to thank R. Nepomechie and the University of Miami for the warm hospitality during the completion of this work.",
"This work was supported in part by the Advanced ERC grant SM-grav, No 669288.",
"BG is partially supported by the Marie Curie International Outgoing Fellowship nr 624054 within the 7th European Community Framework Programme FP7/2007-2013.",
"The work of BG was partially performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293.",
"WJL is financially supported by the Fundamental Research Funds for the Central Universities No.",
"DUT 16 RC(3)097, NSFC Grants No.",
"11275208 as well as 11375026."
],
[
"Equations of motion",
"For the sake of completeness we show the equations of motion for the models considered in this short appendix."
],
[
"Model 1 ",
"The $\\mathcal {J}$ coupling is not affecting the background equations of motion.",
"Therefore the latter coincide exactly with the EOMs for the linear Stückelbergs model presented in [42].",
"We omit them."
],
[
"Model 2 ",
"In order to be coincise we define $Y\\equiv F^2/4$ .",
"The equations of motion for the model 2 defined in REF generically read: $&\\partial _\\mu \\left[\\sqrt{-g}\\,\\mathcal {W}_{Y}(Y,X)\\,F^{\\mu \\nu }\\right]=0 \\\\&\\partial _\\mu \\left[\\sqrt{-g}\\,\\mathcal {W}_{X}(Y,X)g^{\\mu \\nu }\\partial _\\nu \\phi ^I\\right]=0 \\\\&R_{\\mu \\nu }-\\left[3+\\frac{1}{2}R-\\frac{1}{2}\\,\\mathcal {W}(Y,X)\\right]g_{\\mu \\nu }\\\\ \\nonumber &=\\frac{1}{2}\\,\\mathcal {W}_{Y}(Y,X)\\,F_{\\mu \\sigma }{F_\\nu }^\\sigma +\\frac{1}{2}\\,\\mathcal {W}_{X}(Y,X)\\partial _\\mu \\phi ^I\\partial _\\nu \\phi ^I .$ Taking the ansatz (REF ), we obtain the equations of motion for $A_t$ , $B$ , $C$ and $D$ as follows: $&\\left[\\mathcal {W}_{Y}\\, \\frac{C}{\\sqrt{BD}}\\, {A_t^{\\prime }}^2\\right]^{\\prime }=0\\\\&\\left(6\\,-\\,\\mathcal {W}\\right)\\,B\\,D+\\frac{C^{\\prime 2}\\,D}{2\\,C^2}+\\frac{B^{\\prime }\\,C^{\\prime }\\,D}{B\\,C}-\\frac{2\\,C^{\\prime \\prime }\\,D}{C}-\\frac{1}{2}\\,\\mathcal {W}_{Y}\\,{A_t^{\\prime }}^2=0,\\\\&\\left(\\mathcal {W}-6\\right)\\,B\\,D+\\frac{C^{\\prime }\\,D^{\\prime }}{C}+ \\frac{1}{2}\\,\\frac{C^{\\prime 2}\\,D}{C^2}+\\frac{1}{2}\\mathcal {W}_{Y}\\,A_t^{\\prime 2}=0,\\\\&\\left(\\mathcal {W}-6-\\frac{k^2}{C}\\,\\mathcal {W}_{X}\\right)\\,B\\,D+\\frac{1}{2}\\left(\\frac{C^{\\prime }\\,D^{\\prime }}{C}-\\frac{B^{\\prime }\\,C^{\\prime }\\,D}{B\\,C}\\right)-\\frac{1}{2}\\frac{C^{\\prime 2}\\,D}{C^2}+\\frac{C^{\\prime \\prime }\\,D}{C}+D^{\\prime \\prime }\\\\ \\nonumber &-\\frac{1}{2}\\left(\\frac{B^{\\prime }\\,D^{\\prime }}{B}+\\frac{D^{\\prime 2}}{D}\\right)-\\frac{1}{2}\\mathcal {W}_{Y}\\,A_t^{\\prime 2}=0$"
],
[
"Derivation of the thermoelectric conductivities ",
"To compute the DC conductivities, we consider the following time-dependent perturbations around the background $&\\delta A_x=(\\zeta A_t(r)-E) t+a_x(r),\\nonumber \\\\&\\delta g_{tx}=-\\zeta D(r) t+r^2h_{tx}(r),\\nonumber \\\\&\\delta g_{rx}=r^2h_{rx}(r),\\nonumber \\\\&\\delta \\phi ^I=\\psi ^x(r).$"
],
[
"Model 1 ",
"The equations of motion are given by $&\\left[\\left(1-\\frac{\\mathcal {J}\\,k^2}{4\\,r^2}\\right)\\left(D \\,a_x^{\\prime }+\\rho \\, h_{tx}\\right)\\right]^{\\prime }=0\\\\ &h_{rx}-\\left(1-\\frac{\\mathcal {J}\\,k^2}{4\\,r^2}\\right)\\frac{\\rho \\,(\\zeta A_t-E)}{k^2 \\,\\left(1+\\frac{\\mathcal {J}\\,\\rho ^2}{4\\,r^4}\\right)\\,D\\,r^2}+\\frac{r^2\\,\\zeta }{k^2 \\left(1+\\frac{\\mathcal {J}\\,\\rho ^2}{4\\,r^4}\\right)D}\\left(\\frac{D}{r^2}\\right)^{\\prime }-\\frac{{\\psi ^x}^{\\prime }}{k}=0\\\\ &\\left[r^2 \\,\\left(1+\\frac{\\mathcal {J}\\,\\rho ^2}{4\\,r^4}\\right) D ({\\psi ^x}^{\\prime }-k\\,h_{rx})\\right]^{\\prime }-k\\, \\left(1+\\frac{\\mathcal {J}\\,\\rho ^2}{4\\,r^4}\\right)\\zeta =0\\\\ &h_{tx}^{\\prime \\prime }+\\frac{4}{r}\\,h_{tx}^{\\prime }+\\frac{k^2\\left(1+\\frac{\\mathcal {J}\\,\\rho ^2}{4\\,r^4}\\right)}{D\\,r^2}h_{tx}+\\frac{\\rho }{r^{4}}\\left(1-\\frac{\\mathcal {J}\\,k^2}{4\\,r^2}\\right)a_{x}^{\\prime }=0.$ We will adopt the strategy of [24] to express the currents $J^x$ and $Q^x$ in terms of horizon quantities.",
"From the Maxwell equation (REF ), we define a conserved current along the radial direction in the bulk $J\\equiv -\\left[\\left(1-\\frac{\\mathcal {J}\\,k^2}{4\\,r^2}\\right)\\left(D\\, a_x^{\\prime }+\\rho \\, h_{tx}\\right)\\right],$ which one can check that it equals the U(1) current in the boundary theory $<J^x>\\equiv \\frac{\\delta S}{\\delta A_x}\\Big |_{r\\rightarrow \\infty }=-\\lim _{r\\rightarrow \\infty }\\sqrt{-g}\\left[F^{rx}-\\mathcal {J}\\left(XF\\right)^{[rx]}\\right]$ with the ansatz on fluctuations.",
"Then we are going to construct a conserved current in the bulk which corresponds to the heat current on boundary $Q^x\\equiv T^{t x}-\\mu \\,J^x$ .",
"Finally, we find that the following quantity $Q=D^2\\left(\\frac{r^2h_{tx}}{D}\\right)^{\\prime }-A_{t} J,$ is constant along the radial direction, namely $\\partial _r Q=0$ .",
"And one can further prove that the first term is related to the time-independent part of the stress tensor $T_0^{tx}$ and the second term equals $\\mu J^x$ as $r\\rightarrow \\infty $ .",
"Then $Q$ corresponds to the heat current $Q^x$ in the boundary theory.",
"The regular boundary conditions at the horizon can be chosen as follows $&&a_x\\approx -\\frac{E}{4\\pi T}\\,ln(r-r_h)+...\\\\&&h_{tx}\\approx D \\,h_{rx}|_{r=r_h}-\\frac{\\zeta \\, D}{4\\,\\pi \\,T\\,r_h^2}\\,ln(r-r_h)+...$ Then the electric and thermal currents can be expressed in terms of horizon quantities $&&J=\\left[E\\left(1-\\frac{\\mathcal {J}k^2}{4\\,r^2}\\right)\\left(1+\\left(1-\\frac{\\mathcal {J}k^2}{4\\,r^2}\\right)\\frac{\\rho ^2}{k^2\\left(1+\\frac{\\mathcal {J}\\rho ^2}{4\\,r^4}\\right) r^2}\\right)+\\zeta \\left(1-\\frac{\\mathcal {J}k^2}{4\\,r^2}\\right)\\frac{\\rho \\,D^{\\prime }(r)}{k^2 \\left(1+\\frac{\\mathcal {J}\\rho ^2}{4\\,r^4}\\right)}\\right]_{r=r_h},\\nonumber \\\\&&Q=\\left[E \\left(1-\\frac{\\mathcal {J}k^2}{4\\,r^2}\\right) \\frac{\\rho \\, D^{\\prime }(r)}{k^2\\left(1+\\frac{\\mathcal {J}\\rho ^2}{4\\,r^4}\\right)}+\\zeta \\,\\frac{r^2\\, D^{\\prime }(r)^2}{k^2 \\left(1+\\frac{\\mathcal {J}\\rho ^2}{4\\,r^4}\\right)}\\right]_{r=r_h}.$ From these expressions the conductivities (REF ) follow directly."
],
[
"Model 2 ",
"The equations of motion are given by $&\\left[\\mathcal {W}_{Y}\\left(\\bar{Y},\\bar{X}\\right)D \\,a_x^{\\prime }+\\rho \\, h_{tx}\\right]^{\\prime }=0\\\\ &h_{rx}-\\frac{\\rho \\,(\\zeta A_t-E)}{k^2 \\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)D\\,r^2}+\\frac{r^2\\,\\zeta }{k^2 \\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right) D}\\left(\\frac{D}{r^2}\\right)^{\\prime }-\\frac{{\\psi ^x}^{\\prime }}{k}=0\\\\ &\\left[r^2 \\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right) D ({\\psi ^x}^{\\prime }-k\\,h_{rx})\\right]^{\\prime }-k\\, \\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)\\zeta =0\\\\ &h_{tx}^{\\prime \\prime }+\\frac{4}{r}h_{tx}^{\\prime }-\\frac{k^2 \\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)}{Dr^2}h_{tx}+\\frac{\\rho }{r^{4}}a_x^{\\prime }=0.$ From the Maxwell equation (REF ), we define a conserved current along the radial direction in the bulk $J\\equiv -[\\mathcal {W}_{Y}\\left(\\bar{Y},\\bar{X}\\right)D \\,a_x^{\\prime }+\\rho \\, h_{tx}],$ which one can check that it equals the U(1) current in the boundary theory $<J^x>\\equiv \\frac{\\delta S}{\\delta A_x}\\Big |_{r\\rightarrow \\infty }=-\\lim _{r\\rightarrow \\infty }\\sqrt{-g}\\,\\mathcal {W}_{Y}(\\bar{Y},\\bar{X})\\,F^{r x}$ with the ansatz on fluctuations.",
"Then we are going to construct a conserved current in the bulk which corresponds to the heat current on boundary $Q^x\\equiv T^{t x}-\\mu J^x$ .",
"Finally, we find that the following quantity $Q=D^2\\left(\\frac{r^2h_{tx}}{D}\\right)^{\\prime }-A_{t} J,$ is constant along the radial direction, namely $\\partial _r Q=0$ .",
"And one can further prove that the first term is related to the time-independent part of the stress tensor $T_0^{tx}$ and the second term equals $\\mu J^x$ as $r\\rightarrow \\infty $ .",
"Then $Q$ corresponds to the heat current $Q^x$ in the boundary theory.",
"The regular boundary conditions at the horizon can be chosen as follows $&&a_x\\approx -\\frac{E}{4\\pi T}\\,ln(r-r_h)+...\\\\&&h_{tx}\\approx D \\,h_{rx}|_{r=r_h}-\\frac{\\zeta \\, D}{4\\,\\pi \\,T\\,r_h^2}\\,ln(r-r_h)+...$ Then the electric and thermal currents can be expressed in terms of horizon quantities $&&J=\\left[E\\left(\\mathcal {W}_{Y}\\left(\\bar{Y},\\bar{X}\\right)+\\frac{\\rho ^2}{k^2\\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)r^2}\\right)+\\zeta \\frac{\\rho \\,D^{\\prime }(r)}{k^2 \\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)}\\right]_{r=r_h}, \\\\&&Q=\\left[E \\,\\frac{\\rho \\,D^{\\prime }(r)}{k^2 \\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)}+\\zeta \\,\\frac{r^2 \\,D^{\\prime }(r)^2}{k^2 \\,\\mathcal {W}_{X}\\left(\\bar{Y},\\bar{X}\\right)}\\right]_{r=r_h}.$ Just taking the appropriate derivatives of the previous current we derive the conductivity matrix shown in (REF )."
],
[
"Background and thermoelectric conductivities for the specific models $2_U$ and {{formula:d41079d8-8ab1-439b-b48b-162630308be8}}",
"In this appendix we will provide detailed formulae that give the background and conductivities of the special 2 models."
],
[
"Model $2_U$ ",
"For this particular choice the background solution takes the form: $&A_t(r)=\\rho \\int _{r_h}^{r}\\, \\frac{1}{y^2\\,U(k^2/y^2)}\\,dy\\,, \\\\&D(r)=\\frac{1}{2\\,r}\\int _{r_h}^{r}\\, \\left[6\\,y^2-k^2-\\frac{\\rho ^2}{2\\,y^2\\,U(k^2/y^2)}\\right]\\,dy\\,.$ and the Hawking temperature reads: $T=\\frac{1}{4\\pi }\\left[3r_h-\\frac{1}{2}\\frac{k^2}{r_h}-\\frac{\\rho ^2}{4\\,r_h^3U(k^2/r_h^2)}\\right]$ The thermoelectric DC data are given by: $&\\sigma =U\\left(X_h\\right)+\\frac{4\\,\\pi \\,\\rho ^2}{k^2s\\left(1-\\frac{8\\,\\pi ^2\\,\\rho ^2\\,U^{\\prime }(X_h)}{s^2\\,U(X_h)^2 }\\right)}, \\\\&\\alpha =\\bar{\\alpha }=\\frac{4\\,\\pi \\,\\rho }{k^2\\left(1-\\frac{8\\,\\pi ^2\\,\\rho ^2\\,U^{\\prime }(X_h)}{s^2\\,U(X_h)^2 }\\right)}, \\\\&\\bar{\\kappa }=\\frac{4\\,\\pi \\, s\\,T}{ k^2\\left(1-\\frac{8\\,\\pi ^2\\,\\rho ^2\\,U^{\\prime }(X_h)}{s^2\\,U(X_h)^2 }\\right)} \\\\&\\kappa =\\frac{4\\,\\pi \\, s\\,T}{k^2\\,\\left(1-\\frac{8\\,\\pi ^2\\,\\rho ^2\\,U^{\\prime }(X_h)}{s^2\\,U(X_h)^2 }\\right)+\\frac{4\\,\\pi \\,\\rho ^2}{s\\,U(X_h)}}.$ where for convenience we defined $X_h=k^2/r_h^2$ and s is the entropy density $s=4\\,\\pi \\,r_h^2$ .",
"The chemical potential for the system can be defined as usual by: $\\mu \\,=\\,A_t(\\infty )\\,-\\,A_t(r_h)\\,=\\rho \\,\\int _{r_h}^{\\infty }\\, \\frac{1}{y^2\\,U(k^2/y^2)}\\,dy$ i.e.",
"the leading value of the gauge field at the boundary once the regularity condition $A_t(r_h)=0$ is provided.",
"Since in the incoherent limit the radius of the horizon $r_h$ is just a function of the momentum dissipation strength $k$ , it is straightforward to compute the susceptibility in that limit as: $\\chi ^{(inc)}\\,=\\,\\frac{\\partial \\rho }{\\partial \\mu }\\,=\\,\\left(\\frac{\\partial \\mu }{\\partial \\rho }\\right)^{-1}\\,=\\,\\left(\\int _{r_h}^{\\infty }\\, \\frac{1}{y^2\\,U(k^2/y^2)}\\,dy\\right)^{-1}$ which is the result presented in the main text."
],
[
"Model $2_\\mathcal {K}$ ",
"In this subsection we give more details about the solution for the $2_\\mathcal {K}$ model.",
"Assuming the $U(X)$ function to be of the form: $U(X)\\,=\\,1\\,+\\,\\mathcal {K}\\,X$ the background solution for the gauge field is: $A_t(r)\\,=\\,\\frac{\\rho \\left(ArcTan\\left(\\frac{r}{k\\, \\sqrt{\\,\\mathcal {K}}}\\right)\\,-\\,ArcTan\\left(\\frac{r_h}{k \\,\\sqrt{\\mathcal {K}}}\\right)\\right)}{k\\, \\sqrt{\\mathcal {K}}}$ while the temperature and the electric DC conductivity are: $&T\\,=\\,-\\frac{\\rho ^2}{16 \\,\\pi \\, r_h\\,\\left(k^2 \\,\\mathcal {K}+r_h^2\\right)}-\\frac{k ^2}{8 \\,\\pi \\,r_h}+\\frac{3 \\,r_h}{4\\, \\pi }\\,\\\\&\\sigma _{DC}\\,=\\,1+\\frac{\\mathcal {K}\\, k^2}{r_h^2}+\\frac{\\rho ^2}{k^2 \\,r_h^2\\, \\left(1-\\frac{\\mathcal {K}\\, \\rho ^2}{2 \\,\\left(\\mathcal {K}\\, k^2+r_h^2\\right)^2}\\right)}\\,$ The other thermoelectric conductivities for the choice $U(X)=1+\\mathcal {K}X$ can be directly extracted from the results above and for brevity we omit them.",
"Note that for $\\mathcal {K}<0$ the solution for the gauge field in the $r$ coordinate becomes: $A_t(r)\\,=\\,\\frac{\\rho \\left(ArcTanh\\left(\\frac{r_h}{k\\, \\sqrt{\\,|\\mathcal {K}|}}\\right)\\,-\\,ArcTanh\\left(\\frac{r}{k \\,\\sqrt{|\\mathcal {K}|}}\\right)\\right)}{k\\, \\sqrt{|\\mathcal {K}|}}$ and it is clearly problematic.",
"Indeed from the previous expression we see that: $\\frac{r}{k \\,\\sqrt{|\\mathcal {K}|}}\\,<\\,1\\,.$ which cannot be the case since the boundary is located at $r=\\infty $ .",
"In order to have a well defined solution we have to redefine the radial coordinate as follows: $z\\,=\\,\\sqrt{r^2\\,-\\,|\\mathcal {K}|\\,k^2}$ Of course all the physical quantities turn out to be independent of the radial coordinate choice and they are continuous with respect to the coupling $\\mathcal {K}$ .",
"Note that $\\mathcal {K}\\ge -1/6$ for consistency.",
"In more detail, in this new radial coordinate we have that the functions appearing in the metric become: $C(z)\\,=\\,z^2\\,+\\,|\\mathcal {K}|\\,k^2\\;\\;\\;,\\;\\;\\;B(z)\\,=\\,\\frac{z^2}{D(z) \\left(k^2 \\,|\\mathcal {K}|+z^2\\right)}$ The solution for gauge field is: $A_t(z)\\,=\\,\\frac{\\rho \\, \\log \\left(\\frac{z \\left(k \\sqrt{|\\mathcal {K}|}+\\sqrt{k^2 \\,|\\mathcal {K}|+z_h^2}\\right)}{z_h\\left(k\\,\\sqrt{|\\mathcal {K}|}+\\sqrt{k^2 \\,|\\mathcal {K}|+z^2}\\right)}\\right)}{k\\, \\sqrt{|\\mathcal {K}|}}$ We can check that using this new radial coordinate there is no issue for any value in the range $-1/6\\le \\mathcal {K}<0$ .",
"The formula for the temperature gets modified into: $T\\,=\\,\\frac{1}{4\\,\\pi }\\,\\frac{g_{tt}^{\\prime }(z_h)}{\\sqrt{g_{tt}(z_h)\\,g_{rr}(z_h)}}\\,=\\,\\frac{1}{4\\,\\pi }\\frac{D^{\\prime }(z_h)}{\\sqrt{B(z_h)\\,D(z_h)}}$ which gives: $T\\,=\\,-\\frac{2 \\,k^2 (1-6 \\,|\\mathcal {K}|) \\,z_h^2+\\rho ^2-12 \\,z_h^4}{16 \\,\\pi \\,z_h^2 \\sqrt{k^2 \\,|\\mathcal {K}|+z_h^2}}$ In the incoherent limit we now have: $&z_h^{(inc)}\\,=\\,\\frac{\\sqrt{1-6\\,|\\mathcal {K}|}\\,k}{\\sqrt{6}}\\,,\\\\&\\sigma _{DC}^{(inc)}\\,=\\,\\frac{\\left(z_h^{(inc)}\\right)^2}{\\left(z_h^{(inc)}\\right)^2\\,+\\,|\\mathcal {K}|\\,k^2}\\,=\\,1\\,-\\,6\\,|\\mathcal {K}|\\,,\\\\&\\chi ^{(inc)}\\,=\\,\\frac{k \\sqrt{|\\mathcal {K}|}}{\\log \\left(\\frac{k\\, \\sqrt{|\\mathcal {K}|}+\\sqrt{k^2\\, |\\mathcal {K}|+\\,\\left(z_h^{(inc)}\\right)^2}}{z_h^{(inc)}}\\right)}\\,=\\,\\frac{k\\, \\sqrt{|\\mathcal {K}|}}{\\log \\left(\\frac{\\sqrt{6\\,|\\mathcal {K}|}+1}{\\sqrt{1-6 \\,|\\mathcal {K}|}}\\right)}\\,.$ Considering also that the expression for the butterfly velocity in this new $z$ coordinate reads as follows: $v_B^2\\,=\\,\\frac{\\pi \\,T}{\\sqrt{z_h^2\\,+\\,|\\mathcal {K}|\\,k^2}}$ we arrive at the final expression for the charge diffusion appearing in the main text: $\\frac{D_c\\,T}{v_B^2}\\Big |_{inc}\\,=\\,\\frac{(1-6 \\,|\\mathcal {K}|) \\log \\left(\\frac{ \\sqrt{6\\,|\\mathcal {K}|}+1}{\\sqrt{1-6 \\,|\\mathcal {K}|}}\\right)}{\\pi \\, \\sqrt{6\\,|\\mathcal {K}|}}$ which is valid for $\\mathcal {K}<0$ and it joins continuosly with the expression for positive $\\mathcal {K}$ as expected."
],
[
"Kelvin formula",
"In this appendix we prove explicitely that the Kelvin formula: $\\frac{\\alpha }{\\sigma }\\Big |_{T=0}\\,\\equiv \\,\\lim _{T\\rightarrow 0}\\frac{\\partial s}{\\partial \\rho }\\Big |_T$ holds for all the models we considered."
],
[
"$\\mathcal {J}$ model ",
"For this model the extremal horizon is located at: $r_0\\,=\\,\\frac{\\sqrt{2\\,k^2\\,+\\,\\mu ^2}}{2\\,\\sqrt{3}}$ The Seebeck coefficient at zero temperature is: $\\frac{\\alpha }{\\sigma }\\Big |_{T=0}\\,=\\,\\frac{2\\,\\pi \\,\\mu \\,\\sqrt{2\\,k^2\\,+\\,\\mu ^2}}{\\sqrt{3}\\,(k^2\\,+\\,\\mu ^2)}$ Using the chain rule: $\\frac{\\partial s}{\\partial \\rho }\\,=\\,\\frac{\\partial s}{\\partial \\mu }\\,\\left(\\frac{\\partial \\rho }{\\partial \\mu }\\right)^{-1}$ and noticing that $\\rho =\\mu \\,r_h$ and the finite temperature horizon is located at: $r_h\\,=\\,\\frac{1}{6} \\,\\left(\\sqrt{6 \\,k^2+3\\, \\mu ^2+16\\, \\pi ^2 \\,T^2}\\,+\\,4 \\,\\pi \\,T\\right)$ it is straightforward to show that (REF ) holds."
],
[
"$\\mathcal {W}$ model ",
"In this model the Seebeck coefficient is generically given by: $\\frac{\\alpha }{\\sigma }\\,=\\,\\frac{4 \\,\\pi \\, \\rho \\, r_h^2}{k^2\\,r_h^2\\,\\mathcal {W}_Y\\,\\mathcal {W}_X\\,+\\,\\rho ^2}\\,\\underbrace{=}_{Maxwell\\,eq.",
"}\\,\\frac{4 \\,\\pi \\, r_h^2\\, A_t^{\\prime }(r_h)}{\\rho \\,A_t^{\\prime }(r_h)+k^2 \\,\\mathcal {W}_X}$ where its zero temperature value is just obtained replacing $r_h$ with the position of the extremal horizon $r_0$ .",
"In order to compute the thermal derivative is convenient to use: $\\frac{\\partial s}{\\partial \\rho }\\Big |_T\\,=\\,\\frac{\\partial s}{\\partial r_h}\\,\\frac{\\partial r_h}{\\partial \\rho }\\Big |_T$ where the last term can be derived using the equation of state as follows: $dT\\,=\\,\\frac{\\partial T}{\\partial r_h}\\,d r_h\\,+\\,\\frac{\\partial T}{\\partial \\rho }\\,d \\rho \\,=\\,0$ Using the Maxwell equation the thermal derivative at fixed temperature becomes: $\\frac{\\partial s}{\\partial \\rho }\\Big |_T\\,=\\,\\frac{8 \\,\\pi \\, r_h^2 \\,A_t^{\\prime }(r_h)}{\\rho \\,A_t^{\\prime }(r_h)\\,+\\,2\\, k^2 \\,\\mathcal {W}_X\\,-\\,2\\, \\Lambda \\, r_h^2\\,-\\,r_h^2\\, \\mathcal {W}}$ Imposing the zero temperature limit: $\\Lambda \\,=\\,\\frac{1}{2} \\left(-\\mathcal {W}_Y\\, A_t^{\\prime }(r_0)^2\\,-\\,\\mathcal {W}\\right)$ we obtain: $\\lim _{T \\rightarrow 0}\\,\\frac{\\partial s}{\\partial \\rho }\\Big |_{T}\\,=\\,\\frac{4 \\,\\pi \\, r_0^2\\, A_t^{\\prime }(r_0)}{\\rho \\,A_t^{\\prime }(r_0)+k^2 \\,\\mathcal {W}_X}$ which coincides with (REF ) at zero temperature.",
"In conclusion, also in the generic $\\mathcal {W}$ model, the Kelvin formula holds."
],
[
"Null Energy Condition",
"In this short appendix we summarize and give more details about the consistency analysis performed."
],
[
"Model 1",
"The consistency of this model has already been analyzed in [38] and constrains the coupling $\\mathcal {J}$ to satisfy: $0\\,\\le \\,\\mathcal {J}\\,\\le 2/3$ We refer the reader to [38] for details."
],
[
"Model 2 ",
"Generically the NEC is given by $T_{\\mu \\nu }k^\\mu k^\\nu \\ge 0,$ where $k^\\mu $ is a null vector $k^\\mu k_\\mu =0$ .",
"Recall that the stress tensor is $T_{\\mu \\nu }=-g_{\\mu \\nu }\\mathcal {W}(Y,X)+\\mathcal {W}_Y(Y,X)\\,F_{\\mu \\rho }F_{\\nu \\sigma }g^{\\rho \\sigma }+\\mathcal {W}_X(Y,X)\\partial _\\mu \\phi ^I\\partial _\\nu \\phi ^I.$ We then have $\\mathcal {W}_Y(Y,X) F_{\\mu \\rho }F_{\\nu \\sigma }g^{\\rho \\sigma }k^\\mu k^\\nu +\\mathcal {W}_X(Y,X)\\partial _\\mu \\phi ^I\\partial _\\nu \\phi ^Ik^\\mu k^\\nu \\ge 0$ We can construct a complete basis for the null vectors space, which is given by $&&k_{(1)}^\\mu =(D(r)^{-1/2},D(r)^{1/2},0,0)\\nonumber \\\\&&k_{(2)}^\\mu =(D(r)^{-1/2},0,1/r^2,0)\\nonumber \\\\&&k_{(3)}^\\mu =(D(r)^{-1/2},0,0,1/r^2)$ All in all, we derive the following constraint $\\bar{X}\\,\\mathcal {W}_X(\\bar{Y},\\bar{X})-2\\,\\bar{Y}\\,\\mathcal {W}_Y(\\bar{Y},\\bar{X})\\ge 0$ which is presented in the main text.",
"Let us focus now on the benchmark model $\\mathcal {W}(X,Y)=X+U(X)Y$ with $U(X)=1+ \\mathcal {K}X$ .",
"It has already been proven in [38] that the coupling has to satisfy $-1/6\\,\\le \\,\\mathcal {K}\\,\\le \\,1/6$ We checked the behaviour of various other quantities such as the heat capacity and the charge susceptibility in order to analyze the stability of the background solutions.",
"As a result, we have not found stricter constraints than the ones already mentioned.",
"The full analysis confirms the consistency range already obtained in [38]."
],
[
"Scaling dimensions of IR operators",
"In this appendix we analyze the conformal dimension of the IR operators in the zero temperature limit.",
"More concretely we study the transverse and longitudinal sectors of the linearized fluctuations around the AdS$_2\\times R^2$ geometry.",
"The complete transverse and longitudinal sectors are defined by the following sets of (not independent) fluctuations: $&\\text{transverse: } \\Big \\lbrace h_{ty},\\,h_{xy},\\,h_{uy},\\,A_{y},\\,\\delta \\phi ^y\\Big \\rbrace \\\\&\\text{longitudinal: } \\Big \\lbrace h_{tt},\\,h_{xx},\\,h_{yy},\\,h_{tu},\\,h_{uu},\\,A_t,\\,A_u,\\,\\delta \\phi ^x\\Big \\rbrace $ where the momentum $q$ is taken for simplicity along the $x$ direction.",
"The correct way of proceeding would be to define gauge invariant independent variables but for simplicity we decide to work in gauge variant variables; in this way not all the fluctuations are independent and most of the equations read as constraints.",
"The AdS$_2\\times R^2$ solution is defined by the following: $ds^2\\,=\\,-\\,\\frac{1}{u^2}\\,dt^2\\,+\\,\\frac{L_0^2}{u^2}\\,du^2\\,+\\,dx^2\\,+\\,dy^2\\,,\\qquad A_t(u)\\,=\\,\\frac{Q}{u}\\,.$ The equations of motion fix the AdS$_2$ length $L_0$ and the IR charge $Q$ in terms of the cosmological constant $\\Lambda $ and the momentum dissipation rate $k$ .",
"In order to find the conformal dimensions of the IR operators we will perform a scaling ansatz for all the fields of the type: $\\Psi _i(t,u,x)\\,=\\,\\alpha _i\\,u^{\\Delta _i}\\,e^{i\\,(q\\,x\\,-\\,\\,\\omega \\,t)}$ where q and $\\omega $ are the momentum and the frequency of the fluctuations.",
"The power $\\Delta _i$ is related to the conformal dimension of the IR operator dual to the bulk field $\\Psi _i$ and $\\alpha _i$ is just a normalization constant.",
"We will then solve the algebraic equations around the AdS$_2\\times R^2$ background and extract the powers $\\Delta _i$ .",
"In order for the background to be stable the conformal dimensions of the IR operators, and more practically the solutions for $\\Delta _i$ , have to be real; this requirement could possibly constrain the possible values of the higher derivatives couplings.",
"For simplicity we focus just on the $\\mathcal {J}$ and the $2_\\mathcal {K}$ models and we omit most of the lengthy computations."
],
[
"$\\mathcal {J}$ model ",
" For the $\\mathcal {J}$ model defined in sec.",
"REF the AdS$_2\\times R^2$ solution is defined by: $L_0\\,=\\,\\frac{2\\,-\\,Q^2}{k^2},\\,\\qquad \\Lambda \\,=\\,\\frac{(Q^2-4)\\,k^2}{4\\,(Q^2-2)}\\,.$ In the following we will normalize the AdS$_2$ length to 1 by fixing: $Q\\,=\\,\\sqrt{2-k^2}$ This choice will force $k^2<2$ .",
"In the transverse sector we adopt the radial gauge $h_{uy}=0$ and in order to find a solution we take the scaling ansatz: $A_y=\\bar{a}_y\\,u^{\\Delta _T\\,+\\,3}\\,,\\,\\,\\,h_{ty}=\\bar{h}_{ty}\\,u^{\\Delta _t}\\,,\\,\\,\\,h_{xy}=\\omega \\,\\bar{h}_{xy}\\,u^{\\Delta _T\\,+\\,2}\\,,\\,\\,\\,\\delta \\phi ^y=\\omega \\,\\bar{\\phi }^y\\,u^{\\Delta _T\\,+\\,4}\\,.$ We can then solve for all the normalization constants (note that one of them is not physical and can be set to the identity) and consequently determine the power $\\Delta _T$ which will fix the conformal dimensions of the IR operators in the transverse sector.",
"All in all we are left with the following equations for $\\Delta _T$ (removing some modes which can be checked are pure gauge): $&4\\,q^2\\,+\\left(-k^2+\\Delta _T \\,(\\Delta _T+7)+12\\right) \\left(\\left(k^2-2\\right) \\mathcal {J}-4\\right)\\,=\\,0\\,,\\\\[0.2cm]&q^2 \\left(-2 \\,k ^2 ((3 \\,\\Delta _T (\\Delta _T+5)+20) J+4)+16 (\\Delta _T+2) (\\Delta _T+3)+k^6 \\left(-\\mathcal {J}^2\\right)+2 \\,k^4\\, \\mathcal {J} (\\mathcal {J}+3)\\right)+\\\\&+2(\\Delta _T+1) (\\Delta _T+2) (\\Delta _T+3) (\\Delta _T+4) \\left(k^2 \\,\\mathcal {J}-4\\right)+4 \\,q^4 \\left(k^2\\, \\mathcal {J}-2\\right)\\,=\\,0\\,.$ We can solve these equations with Mathematica and check that in the range: $0\\,\\le \\,\\mathcal {J}\\,\\le \\,\\frac{2}{3}$ all the roots are real.",
"We now proceed with the longitudinal sector.",
"The independent fields are taken to be: $A_t=\\bar{a}_t\\,u^{\\Delta _L\\,+\\,2}\\,,\\,\\,\\,h_{tt}=\\bar{h}_{tt}\\,u^{\\Delta _L}\\,,\\,\\,\\,h_{xx}=\\bar{h}_{xx}\\,u^{\\Delta _L\\,+\\,2}\\,,\\,\\,\\,h_{yy}=\\bar{h}_{yy}\\,u^{\\Delta _L\\,+\\,2}\\,,\\,\\,\\,\\delta \\phi ^y\\,=\\,\\frac{\\bar{\\phi }^y}{k}\\,u^{\\Delta _L\\,+\\,2}\\,.$ Following a similar procedure we obtain the equations: $&k^4 \\left(-4\\, q^2 ((\\Delta _L+1) (\\Delta _L+2) \\mathcal {J}+1)+\\Delta _L (\\Delta _L+1) (\\Delta _L+2) (\\Delta _L+3) \\mathcal {J}+2 \\,\\mathcal {J}\\, q^4\\right)+\\\\&+2 \\left(-2 (\\Delta _L+1)(\\Delta _L+2) q^2+\\Delta _L (\\Delta _L+1) (\\Delta _L+2) (\\Delta _L+3)+q^4\\right) \\\\ &\\Big ((\\Delta _L+1) (\\Delta _L+2) (\\mathcal {J}+2)-2 \\,q^2\\big )-k^2\\big (q^4 (3 (\\Delta _L+1) (\\Delta _L+2) \\mathcal {J}+8)\\\\&-(\\Delta _L+1) (\\Delta _L+2) q^2 ((3 \\Delta _L (\\Delta _L+3)+10) \\mathcal {J}+12)+\\Delta _L (\\Delta _L+1)(\\Delta _L+2) (\\Delta _L+3) \\\\&((\\Delta _L (\\Delta _L+3)+4) \\mathcal {J}+4)-\\mathcal {J}\\, q^6\\Big )+k^6 \\,\\mathcal {J}\\, q^2\\,=\\,0\\,.$ Again it is possible to prove that once we restrict: $0\\,\\le \\,\\mathcal {J}\\,\\le \\,\\frac{2}{3}$ all the roots are real."
],
[
"$\\mathcal {K}$ model ",
" We can run the same arguments as before for the $2_{\\mathcal {K}}$ model.",
"The AdS$_2\\times R^2$ solution is now defined by: $L_0\\,=\\,\\frac{4\\,-\\,2\\,Q^2\\,-\\,\\mathcal {K}\\,Q^2\\,k^2}{2\\,k^2},\\,\\qquad \\Lambda \\,=\\,\\frac{(Q^2\\,-\\,4)\\,k^2}{2\\,(4\\,-\\,2\\,Q^2\\,-\\,\\mathcal {K}\\,Q^2\\,k^2)}\\,.$ We can again normalize the AdS$_2$ length to 1 by fixing: $Q\\,=\\,\\sqrt{\\frac{2\\,(2\\,-\\,k^2)}{2\\,+\\,\\mathcal {K}\\,k^2}}$ This choice will force $\\frac{2\\,-\\,k^2}{2\\,+\\,\\mathcal {K}\\,k^2}>0$ .",
"For the transverse sector we are left with the following equations: $&k^2 \\left(\\mathcal {K} \\left(\\Delta _T (\\Delta _T+7)-q^2+14\\right)-2\\right)+2 \\left(\\Delta _T (\\Delta _T+7)-q^2+12\\right)-2 \\,k^4\\, \\mathcal {K}\\,=\\,0\\,,\\\\[0.2cm]& 2 \\left(\\Delta _T^4+10 \\Delta _T^3+35 \\Delta _T^2-2 (\\Delta _T+2) (\\Delta _T+3) q^2+50 \\Delta _T+q^4+24\\right)+\\\\&+k^2 \\left(\\mathcal {K}\\left(\\Delta _T (\\Delta _T+3)-q^2+2\\right) \\left(\\Delta _T (\\Delta _T+7)-q^2+12\\right)+2 q^2\\right)+2 \\,k^4\\, \\mathcal {K}\\, q^2\\,=\\,0\\,.$ and again if: $-1/6\\,\\le \\,\\mathcal {K}\\,\\le \\,1/6$ all the roots are real.",
"In the longitudinal sector it is convenient to perform the following redefinition: $\\Delta _L\\,=\\,\\frac{-3+\\sqrt{\\tilde{\\Delta }_L}}{2}$ In that way the equation for $\\tilde{\\Delta }_L$ becomes of the cubic form: $a\\,\\tilde{\\Delta }_L^3\\,+\\,b\\,\\tilde{\\Delta }_L^2\\,+\\,c\\,\\tilde{\\Delta }_L\\,+\\,d\\,=\\,0$ where: $a\\,=&\\,\\left(k^2 \\,\\mathcal {K}+1\\right)^2 \\left(k^2\\, \\mathcal {K}+2\\right)^2 \\left(\\left(k ^2-1\\right) \\mathcal {K}+1\\right)\\,,\\\\[0.2cm]b\\,=&\\,(-\\,k^2 \\mathcal {K}-1) (k^2\\,\\mathcal {K}+2)(8 \\,k^2+\\mathcal {K} (8 \\,k^8 \\,\\mathcal {K}^2+k^6\\, \\mathcal {K}(\\mathcal {K} (8 q^2-5)+24)\\\\&+k^4 (-4 (\\mathcal {K}-10) \\mathcal {K}\\, q^2-3 (\\mathcal {K}-4) \\mathcal {K}+24)\\\\&+k^2(-20 (\\mathcal {K}-3) q^2-25 \\,\\mathcal {K}+39)-24 q^2-22)+24 q^2+22)\\,,\\\\[0.2cm]c\\,=&\\,(k^2 \\mathcal {K}+1) (16 k^{10} \\mathcal {K}^4 (2 q^2+5)+k ^8 \\mathcal {K}^3 (32 \\mathcal {K} q^4+16 (\\mathcal {K}+14)q^2-141 \\mathcal {K}+400)\\\\&+k^6 \\mathcal {K}^2 (\\mathcal {K} (-16 (\\mathcal {K}-14) q^4-8 (\\mathcal {K}-14) q^2+61 \\mathcal {K}-526)+544q^2+720)\\\\&+k ^4 \\mathcal {K} (145 \\mathcal {K}^2-112 (\\mathcal {K}-5) \\mathcal {K} q^4+8 ((47-7 \\mathcal {K}) \\mathcal {K}+68) q^2-553\\mathcal {K}+560)\\\\&+4 k^2 (16 (9-4 \\mathcal {K}) \\mathcal {K} q^4+8 ((17-6 \\mathcal {K}) \\mathcal {K}+6) q^2+\\mathcal {K} (2 \\mathcal {K}-23))\\\\&-4\\mathcal {K} (48 q^4+56 q^2+19)+192 q^4+4 (40 k^2+56 q^2+19))\\,,\\\\[0.2cm]d\\,=&\\,(k^2 \\mathcal {K}+1) (k^6 (-\\mathcal {K}^4) (4 q^2+9) (8 k ^4-15 k ^2+16 (k^2-1) q^4+4 (9 k ^2-8) q^2+7)\\\\&-k ^4 \\mathcal {K}^3 (4 q^2+9) (40 k ^4-58 k ^2+16 (6k^2-5) q^4+8 (8 k ^4+4 k ^2-5) q^2+19)\\\\&+k ^2 \\mathcal {K}^2 (-9 (72 k ^4-67 k^2+8)+64 (8-13 k ^2) q^6-16 (64 k ^4+67 k ^2-48) q^4\\\\&-4 (64 k ^6+328 k ^4-5 k^2-32) q^2)+4 \\mathcal {K}(-126 k ^4+45 k ^2+64 (1-3 k ^2) q^6\\\\&-16 (20 k ^4+9 k ^2-3)q^4-4 (32 k ^6+50 k ^4+25 k ^2-11) q^2+9)\\\\&-4 (4 k ^2+4 q^2+1) (8 q^2 (2 k ^2+2q^2+1)+9))\\,.$ In principle we have to show that all the roots of such a cubic equation in terms of the new variable $\\tilde{\\Delta }_L$ are real and positive.",
"In order to do so it is better to recast the equation in the form: $t^3\\,+\\,p\\,t\\,+\\,q\\,=\\,0\\,.$ using the change of variable: $\\tilde{\\Delta }_L\\,\\longrightarrow \\,t\\,-\\,\\frac{b}{3\\,a}$ and define the discriminant: $\\mathcal {D}\\,=\\,\\frac{q^2}{4}\\,+\\,\\frac{p^3}{27}$ If the discriminat is negative: $\\mathcal {D}\\,<\\,0$ the cubic equation has 3 real roots.",
"Additionally if: $a\\,b\\,<\\,0\\,.$ all the roots are positive.",
"Because of the complexity of the expressions we have not been able to prove analytically the previous statements.",
"Nevertheless we have performed several numerical checks and plots in order to assess their validity.",
"We have found that in the range: $-1/6\\,\\le \\,\\mathcal {K}\\,\\le \\,1/6$ no imaginary root appears."
],
[
"Final outcome ",
"The analysis of the conformal dimensions of the IR operators in the transverse and longitudinal sectors does not constraint further the range of validity of our higher derivative theories."
]
] | 1612.05500 |
[
[
"Homogenization and boundary layers in domains of finite type"
],
[
"Abstract This paper is concerned with the homogenization of Dirichlet problem of elliptic systems in a bounded, smooth domain of finite type.",
"Both the coefficients of the elliptic operator and the Dirichlet boundary data are assumed to be periodic and rapidly oscillating.",
"We prove the theorem of homogenization and obtain an algebraic rate of convergence that depends explicitly on dimension and the type of the domain."
],
[
"Introduction",
"In this paper, we consider the oscillating Dirichlet problem for uniformly elliptic systems in general domains, $\\left\\lbrace \\begin{aligned}\\mathcal {L}_\\varepsilon u_\\varepsilon (x) &= 0 &\\quad & \\text{in } \\Omega , \\\\u_\\varepsilon (x) &= f(x,x/\\varepsilon ) &\\quad & \\text{on } \\partial \\Omega ,\\end{aligned}\\right.$ where $\\mathcal {L}_\\varepsilon = -\\text{div} (A(x/\\varepsilon ) \\nabla ) = - \\frac{\\partial }{\\partial x_i} \\bigg \\lbrace a^{\\alpha \\beta }_{ij} \\Big ( \\frac{x}{\\varepsilon }\\Big ) \\frac{\\partial }{\\partial x_j}\\bigg \\rbrace $ is a second-order elliptic operator in divergence form with rapidly oscillating periodic coefficients (Einstein's convention for summation will be used throughout).",
"Here $\\varepsilon >0$ is a small parameter and $\\Omega \\subset \\mathbb {R}^d$ is a smooth, bounded domain without the assumption of strict convexity and $d\\ge 2$ .",
"We assume that the coefficients $A = A(y) = (a_{ij}^{\\alpha \\beta })$ , with $1\\le i,j\\le d$ and $1\\le \\alpha ,\\beta \\le m$ , satisfies the ellipticity condition, $\\lambda |\\xi |^2 \\le a_{ij}^{\\alpha \\beta } \\xi _i^\\alpha \\xi _j^\\beta \\le \\lambda ^{-1}|\\xi |^2, \\qquad \\text{for any } \\xi = (\\xi _i^\\alpha ) \\in \\mathbb {R}^{m\\times d},$ where $\\lambda \\in (0,1)$ is a fixed constant.",
"Both $A(y)$ and the Dirichlet boundary data $f(x,y)$ are assumed to be 1-periodic in $y$ , i.e., $A(y+z) = A(y) \\quad \\text{and} \\quad f(x,y+z) = f(x,y) \\quad \\forall x\\in \\partial \\Omega , y\\in \\mathbb {R}^d, z\\in \\mathbb {Z}^d.$ Since we will not try to compute the minimal regularity required for $A$ and $f$ , we simply assume that $A \\in C^\\infty (\\mathbb {R}^d) \\quad \\text{and} \\quad f\\in C^\\infty (\\partial \\Omega \\times \\mathbb {R}^d).$ The asymptotic analysis of problem (REF ), which is closely related to the higher order convergence rates of homogenization problems with non-oscillating boundary data (see Theorem REF ), was raised in [9], for instance, and remained open for decades until recently; see [6], [15], [16], [3], [4], [5], [2], [7], [21], [13], [18] and references therein.",
"The pioneering work was due to Gérard-Varet and Masmoudi in [16].",
"Under the extra assumption that $\\Omega $ is strictly convex, they proved that as $\\varepsilon \\rightarrow 0$ , the unique solution of (REF ) $u_\\varepsilon $ converges strongly in $L^2(\\Omega )$ to some function $u_0$ , which is a solution of $\\left\\lbrace \\begin{aligned}\\mathcal {L}_0 u_0(x) &= 0 &\\quad & \\text{in } \\Omega , \\\\u_0(x) &= \\bar{f}(x) &\\quad & \\text{on } \\partial \\Omega ,\\end{aligned}\\right.$ where the operator $\\mathcal {L}_0$ is given by $\\mathcal {L}_0 = -\\text{div}(\\widehat{A}\\nabla )$ , with $\\widehat{A}$ being the usual homogenized matrix of $A$ , and $\\bar{f}$ is the homogenized Dirichlet boundary data that depends non-trivially on $f,A$ and also $\\Omega $ (see REF ).",
"Moreover, they showed that for each $\\delta >0$ , $\\Vert u_\\varepsilon - u_0\\Vert _{L^2(\\Omega )} \\le C \\varepsilon ^{\\frac{d-1}{3d+5} - \\delta },$ where $C$ depends on $\\delta ,d,m,A,f$ and $\\Omega $ .",
"Most recently, under the same conditions, remarkable improvement was made in [7] for Dirichlet problems, where the authors obtained nearly optimal convergence rates for $d\\ge 4$ and improved suboptimal convergence rates for $d=2,3$ .",
"Then soon in [21], with new ingredients from $A_p$ weighted estimates, Shen and the author of this paper obtained the nearly optimal convergence rates for Neumann problems with first-order oscillating Neumann boundary data for all dimensions, as well as the lower dimensional cases ($d=2,3$ ) of Dirichlet problems.",
"Precisely, for Dirichlet problems, it was proved in [7] and [21] that $\\Vert u_\\varepsilon - u_0\\Vert _{L^2(\\Omega )} \\le \\left\\lbrace \\begin{aligned}C\\varepsilon ^{\\frac{1}{4} - \\delta } \\qquad & \\text{ for }& d=2, \\\\C \\varepsilon ^{\\frac{1}{2} - \\delta } \\qquad & \\text{ for } & d\\ge 3.\\end{aligned}\\right.$ and the homogenized data $\\bar{f}\\in W^{1,q}\\cap L^\\infty (\\partial \\Omega )$ for any $q<d-1$ .",
"The convergence rates in (REF ) are optimal up to an arbitrarily small loss on the exponent, for the optimal convergence rates, i.e., $O(\\varepsilon ^{1/4})$ for $d=2$ and $O(\\varepsilon ^{1/2})$ for $d\\ge 3$ , were shown in [5] for operators with constant coefficients.",
"The regularity for $\\bar{f}$ is also sharp in a certain sense as shown in [21].",
"We emphasize that the previous results and their proofs rely essentially on the geometry of the boundary while domains with a different geometry will have completely different behaviors, for example, the difference between strictly convex domains [16], [5] and polygon domains [15], [4].",
"Also in a very recent paper [2], it was shown constructively that the rate of convergence for (REF ) can be arbitrarily slow if the domain is merely non-strictly convex.",
"In fact, the strict convexity of the domain plays an essential role in the arguments of [16], [7], [21], for which we will give a brief interpretation in two aspects as follows.",
"On one hand, near a given point on $\\partial \\Omega $ , the local behavior of the solution depends whether or not the direction of the normal vector to $\\partial \\Omega $ is non-resonant (with the lattice $\\mathbb {Z}^d$ ).",
"If it is non-resonant, the quantitative analysis depends further on the so called Diophantine condition of the normal vector.",
"For this reason, the regularity of the Diophantine function on the boundary, which is definitely guaranteed by the strict convexity of a domain, does have a crucial impact on the rates of convergence.",
"On the other hand, when approximating $u_\\varepsilon $ , a solution of an oscillating Dirichlet problem, near a given point on $\\partial \\Omega $ by a solution of a half-space problem with the corresponding tangent plane being the boundary, the convexity of $\\Omega $ ensures that the domain lies on one side of the tangent plane and hence $u_\\varepsilon $ could be well approximated near the given point.",
"This fact obviously fails if $\\Omega $ is a non-convex domain.",
"The main purpose of this paper is to remove the assumption of strict convexity and obtain an algebraic rate of convergence that depends explicitly on certain quantitative property of the boundary.",
"We should point out that this is not the first paper dealing with oscillating boundary value problems in general domains.",
"In [13], the authors established the qualitative homogenization for Dirichlet problems of (scalar) fully nonlinear uniformly elliptic operators in general domains, whose irrational normals have a small Hausdorff dimension.",
"In [12] and [14], the same type of equations with oscillating Neumann boundary data was studied in general domains without flatness.",
"However, as far as we know, this should be the first paper that studies the quantitative homogenization of periodic oscillating boundary value problem in general non-convex smooth domains.",
"Though the assumption of strict convexity on the domains will be removed, as we have mentioned before, we do require some restrictive condition to rule out any boundary with nontrivial resonant portion, for example, a domain with flat portion with rational normal on the boundary.",
"In this paper, we introduce a mild condition on $\\partial \\Omega $ to characterize the non-flatness of $\\partial \\Omega $ , namely, the hypersurfaces of finite type.",
"The notion of finite type has been well studied in many references mainly in Fourier analysis and could be defined geometrically in terms of the condition of finite order of contact on the principle curvatures.",
"For our convenience in this paper, we prefer to introduce an equivalent analytical definition.",
"We first give a definition of functions of finite type; also see [22].",
"Definition 1.1 A smooth function $g$ is of type $k$ ($k \\ge 2$ ) in some connected open set $U$ , if there exist some multi-index $\\alpha $ with $1<|\\alpha | \\le k$ and $\\delta > 0$ such that $|\\partial ^\\alpha g| \\ge \\delta $ in $U$ .",
"Let $S$ be a smooth hypersurface in $\\mathbb {R}^d$ .",
"For any $x_0\\in S$ , we can translate and rotate the hypersurface so that $x_0$ is moved to the origin and the tangent plane becomes $x_d = 0$ .",
"Meanwhile, near $x_0$ , $S$ is transformed to a graph of some function $x_d = \\phi = \\phi _{x_0}:\\mathbb {R}^{d-1} \\rightarrow \\mathbb {R}$ , which satisfies $\\phi (0) = 0$ and $\\nabla \\phi (0) = 0$ .",
"We call $x_d = \\phi (x^{\\prime }) = \\phi _{x_0}(x^{\\prime })$ the local graph of $S$ at $x_0$ .",
"Definition 1.2 A smooth hypersurface $S\\subset \\mathbb {R}^d$ is called type $k$ , if for each $x_0 \\in S$ , there exist constants $r>0$ and $\\delta >0$ such that the corresponding local graph $x_d = \\phi _{x_0}(x^{\\prime })$ is of type $k$ in $B(0,r) \\subset \\mathbb {R}^{d-1}$ with a lower bound $\\delta $ .",
"For a smooth domain $\\Omega $ , we also say $\\Omega $ is of type $k$ if its boundary $\\partial \\Omega $ is of type $k$ .If $S$ is a closed smooth hypersurface (i.e., the boundary of a bounded smooth domain), the constants $r$ and $\\delta $ can be chosen uniformly with respect to $x_0$ .",
"The condition of finite type on a hypersurface sometimes is referred as finite order of contact with the tangent plane.",
"Now, we state the main theorem as follows.",
"Theorem 1.3 Assume that $\\Omega $ is a bounded, smooth domain of type $k$ , and that (REF ), (REF ) and (REF ) hold.",
"Then problem (REF ) admits homogenization, i.e., the solution $u_\\varepsilon $ of (REF ) converges to $u_0$ in $L^2(\\Omega )$ , where $u_0$ is the unique solution of the corresponding homogenized problem (REF ) with boundary data $\\bar{f}$ .",
"Moreover, there exist some $q^*>0$ and $\\alpha ^* > 0$ depending explicitly on $k$ and $d$ such that $\\bar{f} \\in W^{1,q}\\cap L^\\infty (\\partial \\Omega ), \\qquad \\text{for any } q<q^*,$ and $\\Vert u_\\varepsilon - u_0\\Vert _{L^2(\\Omega )} \\le C \\varepsilon ^{\\frac{1}{2}\\alpha ^* - \\delta },$ where $\\delta >0$ is an arbitrarily small number and $C$ depends on $\\delta ,k, d,m,A,f$ and $\\Omega $ .",
"The explicit formula for $q^*$ is given by $q^* = \\frac{d-1}{2\\gamma -1},$ with $\\gamma = (d-1)(k-1)$ .",
"While the explicit formula for $\\alpha ^*$ , which comes from optimizing several error terms, is given by $\\alpha ^* = \\max _{s\\in [1/2,1]} \\bigg [ s \\wedge 4(s - \\frac{1}{2}) \\wedge (d-1)(s - \\frac{1}{2})\\wedge \\frac{(1-s)(d-1)}{\\gamma -1} \\wedge \\frac{s(d-1)}{1+\\gamma }\\bigg ],$ where $\\gamma $ is the same and $a\\wedge b := \\min \\lbrace a, b\\rbrace $ .",
"Observe that given specific $d$ and $k$ , $\\alpha ^*$ can be computed by solving a linear programming problem.",
"We remark that except for lower dimensions and $k=2$ (see Remark REF ), the formula for $\\alpha ^*$ seems suboptimal in general, in view of the exponent obtained for non-oscillating operators ($\\alpha ^* = 1/k$ , regardless of dimensions; see Theorem REF for details).",
"Nevertheless, finding a better or optimal rate for general $k$ and $d$ should also be an interesting problem.",
"The first reduction of the proof of Theorem REF is due to a new idea for quantifying the non-flatness of the boundary.",
"Recall that one of the key points in oscillating boundary value problems is the so called Diophantine condition; see Definition REF .",
"It has been shown in [16] that if $\\Omega $ is a bounded, smooth and strictly convex domain, then the reciprocal of Diophantine function $\\varkappa \\circ n(x) = \\varkappa (n(x))$ is in $L^{d-1,\\infty }(\\partial \\Omega ,d\\sigma )$ , where $\\sigma $ is the surface measure of $\\partial \\Omega $ and $n(x)$ is the unit outer normal to $\\partial \\Omega $ at $x$ .",
"This condition played a key role in [16], [7], [21] for oscillating boundary value problems in strictly convex domains.",
"Similarly in our setting, to prove Theorem REF , we would like to show that if $S$ is a closed (i.e., compact and boundaryless) smooth hypersurface of finite type, then $(\\varkappa \\circ n)^{-1} \\in L^{p,\\infty }(S,d\\sigma )$ for some $p>0$ depending explicitly on the type (actually, $p=1/(k-1)$ if $k$ is the type).",
"Surprisingly, we are able to show that $S$ is finite type if and only if there exists some $p>0$ such that $(\\varkappa \\circ On)^{-1} \\in L^{p,\\infty }(S,d\\sigma )$ uniformly for any orthogonal matrix $O$ .",
"This equivalence is a consequence of the famous van de Corput's lemma and a criteria established in Proposition REF which involves the so called sublevel set estimate.",
"The orthogonal matrix involved here is a necessary requirement for rotation invariance of the finite type assumption.",
"As another easy corollary of the criteria, we also show that $S$ is strictly convex if and only if $(\\varkappa \\circ On)^{-1} \\in L^{d-1,\\infty }(S,d\\sigma )$ uniformly for all orthogonal matrix $O$ .",
"In addition, the relationships between finite type condition, oscillatory integrals, the property of Diophantine function and homogenization are summarized in Figure REF .",
"As a consequence, Theorem REF is reduced to Theorem 1.4 Assume that $\\Omega $ is a bounded, smooth domain such that $(\\varkappa \\circ n)^{-1} \\in L^{p,\\infty }(\\partial \\Omega ,d\\sigma )$ , and let $A$ and $f$ be the same as Theorem REF .",
"Then (REF ) admits homogenization, and (REF ) and (REF ) hold with $q^*$ and $\\alpha ^*$ given by (REF ) and (REF ), respectively, where $\\gamma = (d-1)/p$ .",
"Notice that Theorem REF is an even more general result which recovers the special case of strictly convex domains.",
"Actually, if $\\Omega $ is bounded, smooth and strictly convex, then $(\\varkappa \\circ n)^{-1} \\in L^{d-1,\\infty }(\\partial \\Omega ,d\\sigma )$ and hence $\\gamma = 1$ in (REF ) and (REF ).",
"Thus, one sees that $q^* = d-1$ and $\\alpha ^* = 1\\wedge \\frac{d-1}{2}$ , which exactly coincides with the exponents of (REF ) for all dimensions $d\\ge 2$ .",
"For $p<d-1$ , it is unknown whether $q^*$ or $\\alpha ^*$ are optimal.",
"We now present a sketch of the proof of Theorem REF .",
"Our approach follows the line of [7] and [21] with some technical modifications.",
"The starting point is the Poisson integral formula for $u_\\varepsilon $ and an expansion for Poisson kernel established in [8], $\\begin{aligned}u_\\varepsilon (x) & = \\int _{\\partial \\Omega } P_{\\Omega ,\\varepsilon }(x,y) f\\Big (y,\\frac{y}{\\varepsilon }\\Big ) d\\sigma (y)\\\\& = \\int _{\\partial \\Omega } P_{\\Omega }(x,y) \\omega _{\\varepsilon }(y) f\\Big (y,\\frac{y}{\\varepsilon }\\Big ) d\\sigma (y) + \\text{ small error}\\end{aligned}$ where $P_{\\Omega ,\\varepsilon }$ and $ P_{\\Omega }$ are the Poisson kernels associated with $\\mathcal {L}_\\varepsilon $ and $\\mathcal {L}_0$ , respectively, in $\\Omega $ , and $\\omega _\\varepsilon $ is an oscillating function correcting the Poisson kernel.",
"The integral in the second line of (REF ) will be denoted by $\\tilde{u}_\\varepsilon $ and it is sufficient to consider the $L^2$ error of $\\tilde{u}_\\varepsilon - u_0$ .",
"In view of (REF ), the only obscure factor in $\\omega _\\varepsilon $ is $\\nabla \\Phi ^*_\\varepsilon $ , the adjoint Dirichlet correctors introduced in [17].",
"We then show that in a neighborhood of some given point on $\\partial \\Omega $ , $\\nabla \\Phi ^*_\\varepsilon $ can be approximated by a function in the form of $\\nabla \\Phi ^*_\\varepsilon (x) \\sim I + \\nabla \\chi ^*(x/\\varepsilon ) + \\nabla \\bar{v}^*_\\varepsilon (x)$ where $I$ is the identity matrix and $\\chi ^*$ is the usual adjoint corrector.",
"In the case of convex domains, the third term on the right-hand side of (REF ) is a solution of a half-space problem depending on the given point, as $\\Omega $ is on one side of the tangent plane.",
"However, for non-convex domains in our setting, $\\Omega $ may lie on both sides of the tangent plane.",
"To handle this situation, we first obtain the approximation (REF ) in a sub-domain of $\\Omega $ which inscribes $\\partial \\Omega $ at the given point and lies on one side of the tangent plane.",
"Then we extend (REF ) to the other side by a standard extension argument with $C^2$ regularity preserved.",
"It turns out that the convexity actually plays no role in the approximation (REF ).",
"We can then proceed as in [21] by introducing a Carderón-Zygmund-type decomposition of the boundary adapted to the function $F = \\varkappa ^{1/\\gamma }$ , where $\\gamma $ is given as in Theorem REF , and then for each small surface cube in the decomposition, approximating the integral on $\\partial \\Omega $ by the integral on a carefully selected tangent plane.",
"The sizes of surface cubes in the decomposition sit in $(\\tau ,\\sqrt{\\tau })$ , where $\\tau $ is a small parameter related to $\\varepsilon $ .",
"Unlike the case of strictly convex domains where $\\tau = \\varepsilon ^{1-}$ is fixed, for general domains considered in Theorem REF , we have to set $\\tau = \\varepsilon ^s$ for some $s\\in (1/2,1)$ to be determined and eventually identify the best $s$ by optimizing several error terms.",
"Actually, with the decomposition proposed, we split the $L^2$ error of $\\tilde{u}_\\varepsilon - u_0$ into two parts.",
"The trivial part is the $L^2$ estimate of $\\tilde{u}_\\varepsilon - u_0$ over a small boundary layer $\\Gamma _\\varepsilon $ whose volume is $O(\\varepsilon ^s)$ .",
"The interior part is much involving and splits further into 5 errors ($I_k, k=1,2,3,4,5$ ), which are contained in the following: $\\begin{aligned}\\text{Total error of }u_\\varepsilon - u_0 = & \\text{ Error } E_1 \\text{ coming from (\\ref {eq_ue_exp})} \\\\&+ \\text{Error } E_2 \\text{ coming from trivial estimate in } \\Gamma _\\varepsilon \\\\&+ \\text{Error } I_1 \\text{ coming from (\\ref {eq_dPhi*})} \\\\&+ \\text{Error } I_2 \\text{ coming from projection onto a tangent plane} \\\\&+ \\text{Error } I_3 \\text{ coming from quantitative ergodic theorem} \\\\&+ \\text{Error } I_4 \\text{ coming from regularity of homogenized boundary data} \\\\&+ \\text{Error } I_5 \\text{ coming from changing variables back to } \\partial \\Omega .\\end{aligned}$ The right-hand side of the last expression includes all the error terms needed to bound the total $L^2$ error of $u_\\varepsilon - u_0$ .",
"We point out that the error $E_1$ is $O(\\varepsilon ^{1-})$ ; the error $E_2$ contributes the exponent $s$ in (REF ); $I_1$ and $I_2$ contribute to exponent $4(s - \\frac{1}{2}) \\wedge (d-1)(s - \\frac{1}{2})$ ; $I_3$ contributes to exponent $\\frac{(1-s)(d-1)}{\\gamma -1}$ ; $I_4$ contributes to exponent $\\frac{s(d-1)}{1+\\gamma }$ and $I_5$ is smaller and ignored.",
"Note that the parameter $\\gamma $ , which quantifies the non-flatness of the boundary, is only involved in $I_3$ and $I_4$ , due to the presence of the Diophantine function in the arguments.",
"We should also mention that the quantitative ergodic theorem for the estimate of $I_3$ was introduced in [7] and the (optimal) regularity of the homogenized data involved in the estimate of $I_4$ was first proved in [21] via an $A_p$ weighted estimate.",
"The outline of the paper is described as follows.",
"The preliminaries are given in Section 2, including standard notations used throughout.",
"Section 3 is devoted to the quantification of the non-flatness of hypersurfaces of finite type.",
"In Section 4, we establish the approximation (REF ).",
"In Section 5, we introduce a partition of unity on $\\partial \\Omega $ with the assumption $\\varkappa (\\cdot )^{-1} \\in L^{p,\\infty }(\\partial \\Omega )$ for some $p>0$ .",
"Finally, Theorem REF and hence Theorem REF are proved in Section 6, as well as a theorem concerning the higher order convergence rate for non-oscillating boundary value problems.",
"Acknowledgment.",
"The author would like to thank professor Zhongwei Shen for helpful discussions and suggestions."
],
[
"Preliminaries",
"Throughout we will use the following notations.",
"The indices $i,j,k,\\ell $ usually denote integers ranging between 1 and $d$ , whereas the small Greek letters $\\alpha ,\\beta ,\\gamma $ usually denote integers ranging between 1 and $m$ .",
"The vector $e_i \\in \\mathbb {R}^d$ stands for $i$ -th vector of canonical basis of $\\mathbb {R}^d$ and $e^\\alpha \\in \\mathbb {R}^m$ stands for $\\alpha $ -th vector in canonical basis of $\\mathbb {R}^m$ .",
"For $x = (x_1,x_2,\\cdots ,x_d) \\in \\mathbb {R}^d$ , we write $x = (x^{\\prime },x_d)$ where $x^{\\prime }\\in \\mathbb {R}^{d-1}$ .",
"We use $\\mathbb {S}^{d-1}$ and $d$ to denote the $d-1$ dimensional unit sphere and $d$ dimensional torus, respectively.",
"For the coefficient matrix $A$ , we write $A_\\varepsilon (x) = A(x/\\varepsilon )$ for simplicity.",
"For real numbers $a,b$ , we let $a\\wedge b = \\min \\lbrace a,b\\rbrace $ and $a\\vee b = \\max \\lbrace a,b\\rbrace $ .",
"As usual, $C$ and $c$ are positive constants that may vary from line to line and they depend at most on $d,m,A$ and $\\Omega $ as well as other parameters, but never on $\\varepsilon $ or the Diophantine constant $\\varkappa $ .",
"The dependence on $A$ should be interpreted as both the ellipticity constant $\\lambda $ and $\\Vert A\\Vert _{C^k(d)}$ for some $k=k(d)>1$ .",
"In Section 6, we use $\\delta $ to denote an arbitrarily small exponent which may also differ in each occurrence.",
"Assume that $A$ satisfies (REF ) and (REF ).",
"For each $1\\le j\\le d, 1\\le \\beta \\le m$ , let $\\chi = (\\chi _j^\\beta ) = (\\chi _j^{1\\beta },\\chi _j^{2\\beta },\\cdots ,\\chi _j^{m\\beta })$ denote the correctors for $\\mathcal {L}_\\varepsilon $ , which are 1-periodic functions satisfying the cell problem $\\left\\lbrace \\begin{aligned}\\mathcal {L}_1 (\\chi ^{\\beta }_{j} + P_j^\\beta )(x) &= 0 \\qquad \\text{ in } d, \\\\\\int _{d} \\chi _j^\\beta & = 0,\\end{aligned}\\right.$ where $P_j^\\beta (x) = x_je^\\beta $ .",
"Note that $\\nabla P_j^\\beta = e_j e^\\beta $ .",
"We introduce the matrix of Dirichlet boundary correctors $\\Phi _\\varepsilon = \\Phi _{\\varepsilon ,j}^\\beta = (\\Phi _{\\varepsilon ,j}^{1\\beta },\\Phi _{\\varepsilon ,j}^{2\\beta },\\dots ,\\Phi _{\\varepsilon ,j}^{m\\beta })$ associated with $\\mathcal {L}_\\varepsilon $ in a bounded domain $\\Omega $ .",
"Indeed, for each $1\\le j\\le d, 1\\le \\beta \\le m$ , $\\Phi _{\\varepsilon ,j}^{\\beta }$ is the solution of $\\left\\lbrace \\begin{aligned}\\mathcal {L}_\\varepsilon \\Phi ^{\\beta }_{\\varepsilon ,j}(x) &= 0 \\qquad & \\text{ in }& \\Omega , \\\\\\Phi ^{\\beta }_{\\varepsilon ,j}(x) &= P_j^\\beta (x) \\qquad & \\text{ on } &\\partial \\Omega .\\end{aligned}\\right.$ The homogenized operator is given by $\\mathcal {L}_0 = -\\text{div}(\\widehat{A}\\nabla )$ , where the homogenized matrix $\\widehat{A} = (\\widehat{a}_{ij}^{\\alpha \\beta })$ is defined by $\\widehat{A} = \\int _{d} A(I+\\nabla \\chi ) \\quad \\text{or in component form} \\quad \\widehat{a}_{ij}^{\\alpha \\beta } = \\int _{d} \\bigg \\lbrace a_{ij}^{\\alpha \\beta } + a_{ik}^{\\alpha \\gamma } \\frac{\\partial }{\\partial y_k} (\\chi _j^{\\gamma \\beta }) \\bigg \\rbrace .$ We also introduce the adjoint operator $\\mathcal {L}^*_\\varepsilon = - \\text{div}(A^*_\\varepsilon \\nabla )$ , where $A^* = (a_{ij}^{*\\alpha \\beta })$ and $a_{ij}^{*\\alpha \\beta } = a_{ji}^{\\beta \\alpha }$ .",
"Note that $A^*$ also satisfies (REF ), (REF ) and (REF ).",
"Let $\\chi ^*$ and $\\Phi _{\\varepsilon }^*$ be the adjoint correctors and the adjoint Dirichlet boundary correctors, respectively, associated with $\\mathcal {L}^*$ .",
"Let $\\Omega $ be a bounded $C^{2,\\sigma }$ domain and $\\sigma \\in (0,1)$ .",
"The matrix of Poisson kernel $P_{\\Omega ,\\varepsilon }: \\Omega \\times \\partial \\Omega \\mapsto \\mathbb {R}^{m\\times m}$ , associated with $\\mathcal {L}_\\varepsilon $ in $\\Omega $ , is defined by $P_{\\Omega ,\\varepsilon }^{\\alpha \\beta }(x,y) = -n(y)\\cdot a^{\\gamma \\beta }(y/\\varepsilon ) \\nabla _y G_{\\Omega ,\\varepsilon }^{\\alpha \\gamma }(x,y),$ where $n(y)$ is the unit outer normal and $G_{\\Omega ,\\varepsilon }$ is the matrix of Green's function associated with $\\mathcal {L}_\\varepsilon $ in $\\Omega $ .",
"The following uniform estimates in [8] will be useful, $|P_{\\Omega ,\\varepsilon }(x,y)| \\le \\frac{C}{|x-y|^{d-1}},$ and $|P_{\\Omega ,\\varepsilon }(x,y)| \\le \\frac{C\\text{dist}(x,\\partial \\Omega )}{|x-y|^{d}}.$ Let $P_{\\Omega }$ be the Poisson kernel associated with the homogenized operator $\\mathcal {L}_0$ in $\\Omega $ .",
"Clearly, $P_{\\Omega }$ possesses the same estimates (REF ) and (REF ).",
"Recall that the two-scale expansion of the Poisson kernel of $\\mathcal {L}_\\varepsilon $ in $\\Omega $ was established in [17], $P_{\\Omega ,\\varepsilon }^{\\alpha \\beta }(x,y) = P_{\\Omega }^{\\alpha \\gamma }(x,y) \\omega _\\varepsilon ^{\\gamma \\beta }(y) + R_\\varepsilon ^{\\alpha \\beta }(x,y) \\qquad \\text{for } x\\in \\Omega , y\\in \\partial \\Omega ,$ where $R_\\varepsilon $ is the remainder term satisfying $|R_\\varepsilon (x,y)| \\le \\frac{C \\varepsilon \\ln (2+\\varepsilon ^{-1} |x-y|)}{|x-y|^d}.$ The highly oscillating factor $\\omega _\\varepsilon (y)$ in (REF ) is given by $\\omega _\\varepsilon ^{\\gamma \\beta }(y) = h^{\\gamma \\nu }(y) \\cdot n_k(y) n_{\\ell }(y) \\frac{\\partial }{\\partial y_\\ell } \\Phi _{\\varepsilon ,k}^{*\\rho \\nu }(y) \\cdot a_{ij}^{\\rho \\beta }(y/\\varepsilon ) n_i(y)n_j(y),$ and $h(y)$ is the inverse matrix of $\\widehat{a}_{ij}(y) n_i(y) n_j(y)$ .",
"Let $u_\\varepsilon $ be the solution of (REF ).",
"By Poisson integral formula, we have $u_\\varepsilon (x) = \\int _{\\partial \\Omega } P_{\\Omega ,\\varepsilon }(x,y) f(y,y/\\varepsilon ) d\\sigma (y).$ Note that (REF ) implies the Agmon-type maximum principle $\\Vert u_\\varepsilon \\Vert _{L^\\infty (\\Omega )} \\le C\\Vert f\\Vert _{L^\\infty (\\partial \\Omega \\times d)}$ , which we will often refer to.",
"Define $\\tilde{u}_\\varepsilon (x) = \\int _{\\partial \\Omega } P_{\\Omega }(x,y) \\omega _\\varepsilon (y) f(y,y/\\varepsilon ) d\\sigma (y).$ Lemma 2.1 Let $\\Omega $ be a bounded $C^{2,\\sigma }$ domain and let (REF ), (REF ) and (REF ) hold.",
"Then $\\Vert u_\\varepsilon - \\tilde{u}_\\varepsilon \\Vert _{L^q} \\le C\\varepsilon ^{1/q}(1+|\\ln \\varepsilon |) \\Vert f\\Vert _{L^\\infty (\\partial \\Omega \\times d)}.$ for any $1\\le q<\\infty $ .",
"This follows readily from (REF ) and a similar proof can be found in [21].",
"Thanks to Lemma REF , the estimate for $\\Vert u_\\varepsilon - u_0\\Vert _{L^2(\\Omega )}$ is reduced to $\\Vert \\tilde{u}_\\varepsilon - u_0\\Vert _{L^2(\\Omega )}$ ."
],
[
"Quantification of non-flatness",
"In this section, we develop a new idea involving the Diophantine condition to quantify the non-flatness of hypersurfaces of a domain which could be used in homogenization of oscillating boundary value problems.",
"It turns out that this idea has a close connection with the sublevel set estimate and van de Corput's lemma in the theory of oscillatory integrals.",
"To begin with, we recall the Diophantine condition for a unit vector $n \\in \\mathbb {S}^{d-1}$ .",
"Definition 3.1 Given $n\\in \\mathbb {S}^{d-1}$ .",
"We say $n$ satisfies the Diophantine condition with some fixed $\\mu >0$ , if there exists some constant $C > 0$ such thatThe matrix $I-n\\otimes n$ is the orthogonal projection onto $n^\\perp $ .",
"$|(I - n\\otimes n)\\xi | \\ge C |\\xi |^{-\\mu } \\qquad \\text{for all } \\xi \\in \\mathbb {Z}^d\\setminus \\lbrace 0\\rbrace .$ We call $\\varkappa = \\varkappa (n)$ the Diophantine constant if it is the largest constant such that (REF ) holds.",
"As a function of $n$ , $\\varkappa (n)$ will also be called the Diophantine function on $\\mathbb {S}^{d-1}$ .",
"One can slightly generalize the Diophantine function from $\\mathbb {S}^{d-1}$ to any closed smooth hypesurface $S$ by considering $ \\varkappa \\circ n(x)$ for all $x\\in S$ , where $n(x)$ is the outer normal to $S$ at $x$ .",
"As we have mentioned in Introduction, the fact $(\\varkappa \\circ n)^{-1} \\in L^{d-1,\\infty }(\\partial \\Omega )$ played a crucial role in oscillating boundary value problems in strictly convex domains considered in [16], [7], [21].",
"For general smooth compact hypersurfaces, one might also ask if there is still some possible $p$ less than $d-1$ such that $(\\varkappa \\circ n)^{-1} \\in L^{p,\\infty }(S,d\\sigma )$ and what condition would exactly guarantee this.",
"In this section, we will give positive answers to these two questions.",
"As stated in Introduction, for any $x_0\\in S$ , we can translate and rotate the hypersurface so that $S$ is given by its local graph $x_d = \\phi (x^{\\prime }) = \\phi _{x_0}(x^{\\prime })$ near $x_0$ .",
"Proposition 3.2 Let $S$ be a closed smooth hypersurface in $\\mathbb {R}^d$ .",
"The following statements are equivalent: (i) there is some $\\mu >0$ so that $(\\varkappa \\circ On(\\cdot ))^{-1} \\in L^{p,\\infty }(S,d\\sigma )$ uniformly for any orthogonal matrix $O$ ; (ii) the function $h_{\\omega }$ defined below satisfies $h_\\omega (x) := \\frac{1}{\\sqrt{1-[\\omega \\cdot n(x)]^2}} \\in L^{p,\\infty }(S,d\\sigma ),$ uniformly for any $\\omega \\in \\mathbb {S}^{d-1}$ ; (iii) there exist some $r_0>0$ and $C_0>0$ such that for all $x_0 \\in S$ , its local graph satisfies $\\sigma \\lbrace x^{\\prime }\\in B(0,r_0) \\subset \\mathbb {R}^{d-1}: |\\nabla \\phi _{x_0}(x^{\\prime })| \\le t \\rbrace \\le C_0t^p.$ for all $0<t<1$ .",
"First, we show that (i) is equivalent to (ii).",
"We assume the statement (i) holds.",
"For a fixed $\\omega \\in \\mathbb {S}^{d-1}$ , we can find an orthogonal matrix $O$ such that $\\omega $ is the first column of $O^t$ , where $O^t$ is the transpose of $O$ .",
"This implies that $\\omega \\cdot n = e_1\\cdot On$ .",
"Now a key observation is $\\sqrt{1-[\\omega \\cdot n]^2} = |(I-n\\otimes n) \\omega |, \\quad \\forall \\omega , n\\in \\mathbb {S}^{d-1}.$ It follows $\\sqrt{1-[\\omega \\cdot n]^2} = \\sqrt{1-[e_1\\cdot On]^2} = |(I-On\\otimes On) e_1| \\ge \\varkappa \\circ On |e_1|^{-\\mu }.$ As a result $\\sigma \\lbrace x\\in S: \\sqrt{1-[\\omega \\cdot n(x)]^2} < t\\rbrace \\le \\sigma \\lbrace x\\in S: \\varkappa \\circ On(x) < t \\rbrace \\le Ct^p,$ where in the second inequality we used the condition $(\\varkappa \\circ On)^{-1}\\in L^{p,\\infty }(S,d\\sigma )$ .",
"Clearly, this implies that $h_\\omega $ is in $L^{p,\\infty }(S,d\\sigma )$ , uniformly.",
"Now we assume that the statement (ii) holds.",
"Let $0<t<1$ .",
"Observe that $\\lbrace x\\in \\partial \\Omega : (\\varkappa \\circ O n(x))^{-1} > t^{-1}\\rbrace \\subset S_{t} = \\bigcup _{\\xi \\in \\mathbb {Z}^d \\setminus \\lbrace 0 \\rbrace } \\lbrace x\\in \\partial \\Omega : |(I-On\\otimes On)\\xi | < t |\\xi |^{-\\mu } \\rbrace .$ Using (REF ) and (REF ), we have $&\\sigma \\lbrace x\\in \\partial \\Omega : |(I-On\\otimes On)\\xi | < t |\\xi |^{-\\mu } \\rbrace \\\\&\\qquad = \\sigma \\lbrace x\\in \\partial \\Omega : |(I-On\\otimes On)\\omega | < t |\\xi |^{-1-\\mu }, \\omega = |\\xi |^{-1}\\xi \\rbrace \\\\&\\qquad = \\sigma \\lbrace x\\in \\partial \\Omega : \\sqrt{1-[O^t\\omega \\cdot n(x)]^2} < t |\\xi |^{-1-\\mu }, \\omega = |\\xi |^{-1}\\xi \\rbrace \\\\& \\qquad \\le Ct^{p} |\\xi |^{-p(1+\\mu )}.$ Now we choose $\\mu $ sufficiently large so that $p(1+\\mu )> d$ .",
"Then it follows $\\sigma \\lbrace x\\in \\partial \\Omega : (\\varkappa \\circ n(x))^{-1} > t^{-1}\\rbrace \\le \\sigma (S_t) \\le C t^p,$ for any $0<t<1$ .",
"This finishes the proof of equivalence between (i) and (ii).",
"Next we show that (ii) is equivalent to (iii).",
"Assume that (ii) is true.",
"Let $x_0$ be a point on $S$ and $x_d = \\phi (x^{\\prime }) = \\phi _{x_0}(x^{\\prime })$ be the local graph of $S \\cap B(x_0,r_0)$ , where $r_0$ is given in Definition REF .",
"Recall that by definition, $\\phi (0) = 0$ and $\\nabla \\phi (0) = 0$ .",
"Note that in local coordinates $n(x^{\\prime }) = \\frac{(\\nabla \\phi (x^{\\prime }),-1)}{\\sqrt{1+|\\nabla \\phi (x^{\\prime })|^2}}.$ Put $w = e_d$ .",
"Then, observe that $1 - (n(x^{\\prime })\\cdot \\omega )^2 = \\frac{|\\nabla \\phi (x^{\\prime })|^2 }{1+|\\nabla \\phi (x^{\\prime })|^2}.$ It follows from (ii) that $\\sigma \\lbrace x^{\\prime }\\in B(0,r_0): |\\nabla \\phi (x^{\\prime })| < t \\rbrace \\le \\sigma \\lbrace x^{\\prime }\\in B(0,r_0): \\sqrt{1-(n(x^{\\prime })\\cdot \\omega )^2} < t \\rbrace \\le Ct^p.\\\\$ On the contrary, we now assume that the statement (ii) is false, which means that for any large $M>0$ , there exist $w$ and $t$ (depending on $M$ ) such that $\\sigma \\lbrace x\\in S: \\sqrt{1-[\\omega \\cdot n(x)]^2} < t \\rbrace \\ge Mt^p.$ Fix such $w$ and $t$ .",
"Let $r_0$ be given by Definition REF .",
"Then it is not hard to see that there exists some point $x_0 \\in S$ such that $\\sqrt{1-[\\omega \\cdot n(x_0)]^2} < t,$ and $\\sigma \\lbrace x\\in S\\cap B(x_0,r_0): \\sqrt{1-[\\omega \\cdot n(x)]^2} < t \\rbrace \\ge C^{-1}Mt^p,$ where $C$ depends only on $S$ .",
"Without loss of generality, we may assume $\\omega \\cdot n(x) \\ge 0$ for $x$ in the set involved above.",
"Now we have a simple observation: for any $u,v\\in \\mathbb {S}^{d-1}$ and $u\\cdot v \\ge 0$ , $\\sqrt{1-(u\\cdot v)^2} \\le |u-v| \\le 2\\sqrt{1-(u\\cdot v)^2}.$ As a result, $\\begin{aligned}\\sqrt{1-(n(x_0)\\cdot n(x))^2} & \\le |u(x_0)-n(x)| \\\\& \\le |u(x_0)-\\omega | + |\\omega -n(x)| \\\\& \\le 2\\sqrt{1-(n(x_0)\\cdot \\omega )^2} + 2\\sqrt{1-(\\omega \\cdot n(x))^2} \\\\& < 4t,\\end{aligned}$ for all $x\\in \\lbrace y\\in S\\cap B(x_0,r_0): \\sqrt{1-[\\omega \\cdot n(y)]^2} < t \\rbrace $ , where we also used (REF ) in the last inequality.",
"It follows that $\\begin{aligned}&\\sigma \\lbrace x\\in S\\cap B(x_0,r_0): \\sqrt{1-[n(x_0)\\cdot n(x)]^2} < 4t \\rbrace \\\\&\\qquad \\qquad \\ge \\sigma \\lbrace x\\in S\\cap B(x_0,r_0): \\sqrt{1-[\\omega \\cdot n(x)]^2} < t \\rbrace \\ge C^{-1}M t^p.\\end{aligned}$ Again, we apply the local graph $x_d = \\phi (x^{\\prime }) = \\phi _{x_0}(x^{\\prime })$ at $x_0$ and use (REF ) and (REF ) to obtain $\\begin{aligned}&\\sigma \\lbrace x\\in S\\cap B(x_0,r_0): \\sqrt{1-[n(x_0)\\cdot n(x)]^2} < 4t \\rbrace \\\\& \\qquad \\le C \\sigma \\lbrace x^{\\prime }\\in B(x_0,r_0): \\sqrt{1-[e_d \\cdot n(x^{\\prime })]^2} < 4t \\rbrace \\\\& \\qquad \\le C \\sigma \\lbrace x^{\\prime }\\in B(x_0,r_0): |\\nabla \\phi (x^{\\prime })| < Ct \\rbrace .\\end{aligned}$ This, together with (REF ), leads to $\\sigma \\lbrace x^{\\prime }\\in B(x_0,r_0): |\\nabla \\phi (x^{\\prime })| < Ct \\rbrace \\ge C^{-1}Mt^p.$ Since $M$ can be arbitrarily large, the last inequality contradicts to (iii).",
"This completes the proof.",
"A straightforward application of Proposition REF is to verify that every strictly convex domain satisfies the statements in Proposition REF with $p = d-1$ .",
"Actually, they are even sufficient and necessary condition of each other.",
"Proposition 3.3 A closed smooth hypersurface in $\\mathbb {R}^d$ is strictly convex if and only if there is some $\\mu >0$ so that $(\\varkappa \\circ On(\\cdot ))^{-1} \\in L^{d-1,\\infty }(S,d\\sigma )$ uniformly for any orthogonal matrix $O$ .",
"First we assume that $S$ is strictly convex.",
"It is sufficient to verify (iii) of Proposition REF .",
"For any fixed $x_0\\in S$ , let $\\phi _{x_0}$ be the local graph of $S$ at $x_0$ .",
"Since $S$ is strictly convex, the Hessian matrix $\\nabla ^2 \\phi _{x_0}(0)$ is positive definite and its eigenvalues are bounded below uniformly in $x_0$ .",
"It follows from the mean value theorem that $|\\nabla \\phi _{x_0}(x^{\\prime })| = |\\nabla ^2 \\phi _{x_0}(\\xi ) x^{\\prime } | \\ge c|x^{\\prime }|,$ if $|x|<r_0$ for some sufficiently small $r_0$ depending only on $S$ .",
"Hence $\\sigma \\lbrace x^{\\prime }\\in B(0,r_0): |\\nabla \\phi _{x_0}(x^{\\prime })| < t \\rbrace \\le \\sigma \\lbrace x^{\\prime }\\in B(0,r_0): |x^{\\prime }| < Ct \\rbrace \\le Ct^{d-1}.$ This proves the statement (iii) with $p = d-1$ .",
"On the contrary, suppose that $S$ is not strictly convex, then there must be some point $x_0\\in S$ such that the Hessian matrix $\\nabla ^2 \\phi _{x_0}(0)$ is degenerate.",
"Without loss of generality, we can rotate the coordinates such that $\\nabla ^2 \\phi _{x_0}(0)$ is diagonal, i.e., $\\nabla ^2 \\phi _{x_0}(0) =\\begin{bmatrix}\\lambda _1 & & &\\\\& \\ddots & &\\\\& & \\lambda _{d-2} &\\\\& & & 0\\end{bmatrix}.$ Now writing $x^{\\prime } = (x^{\\prime \\prime },x_{d-1}), x^{\\prime \\prime }\\in \\mathbb {R}^{d-2}$ , we see that $|\\nabla ^2 \\phi _{x_0}(0)x^{\\prime }| \\le C|x^{\\prime \\prime }|$ .",
"Thus, in view of $\\nabla \\phi _{x_0}(x^{\\prime }) = \\nabla ^2 \\phi _{x_0}(0) x^{\\prime } + O(|x^{\\prime }|^2)$ , we have $\\begin{aligned}\\lbrace x^{\\prime }\\in B(0,r_0): |\\nabla \\phi _{x_0}(x^{\\prime })| < t \\rbrace &\\supset \\lbrace x^{\\prime }\\in B(0,r_0): |\\nabla ^2\\phi _{x_0}(0) x^{\\prime }| + C|x^{\\prime }|^2 < t\\rbrace \\\\& \\supset \\lbrace x^{\\prime }\\in B(0,r_0): C|x^{\\prime \\prime }| + C|x^{\\prime }|^2 < t\\rbrace \\\\& \\supset \\lbrace x^{\\prime }\\in B(0,r_0): C|x^{\\prime \\prime }| < t, C|x_{d-1}|^2 < t \\rbrace \\end{aligned}$ for sufficiently small $t$ .",
"It follows that, $\\sigma \\lbrace x^{\\prime }\\in B(0,r_0): |\\nabla \\phi _{x_0}(x^{\\prime })| < t \\rbrace \\ge C^{-1} t^{d-2+1/2}$ for some constant $C$ and for sufficiently small $t$ .",
"This contradicts to (iii) of Proposition REF with $p = d-1$ and hence the proof is complete.",
"Remark 3.4 The statement (i) is stronger than $(\\varkappa \\circ n)^{-1} \\in L^{p,\\infty }(S,d\\sigma )$ since it requires rotation invariance for the hypersurfaces, which actually rules out any hypersurfaces containing a non-trivial portion of an affine hypersurface.",
"This is even clear if we notice that the statement (iii), a type of sublevel set estimate, alleges that the hypersurface must bend at a certain degree at each point.",
"However, it is possible to have a flat portion on $S$ so that $(\\varkappa \\circ n)^{-1}$ is in $L^{p,\\infty }(S,d\\sigma )$ .",
"In fact, if $D$ is the union of all the flat portions of $S$ whose normals satisfy the Diophantine condition, then it suffices to consider the remaining portion $S\\setminus D$ .",
"Fortunately, with slight modification in (iii), Proposition REF still holds with $S$ replaced by $S\\setminus D$ ."
],
[
"Hypersurfaces of finite type",
"In what follows, we will consider the hypersurfaces of finite type.",
"Note that Definition REF is given in a pure analytical way.",
"To see that this condition is relatively mild, we state an equivalent geometrical definition: a compact hypersurface $S$ is of finite type if at least one of the principle curvatures of $S$ does not vanish to infinite order, uniformly at each point.",
"In view of this, we remark that, except for $d-1$ dimensional linear submanifolds (i.e., a portion of an affine hypersurface), some typical cases are allowed for hypersurfaces of finite type, including disconnected hypersurfaces (related to multiply connected domains), saddle points or saddle surfaces and lower dimensional linear submanifolds (such as the side surface of a 3 dimensional cylinder).",
"Also, we mention that any compact real-analytic hypersurface not lying in any affine hypersurface must be of finite type; see [22].",
"The next theorem, in connection with the well-known van de Corput's lemma and sublevel set estimate, indicates the importance of the notion of finite type for our application.",
"Theorem 3.5 Let $\\phi $ be a smooth function of type $k$ in some sphere $B \\subset \\mathbb {R}^d$ , then (i) (van de Corput) for any $\\psi \\in C_0^\\infty (B)$ , $\\bigg | \\int _{\\mathbb {R}^d} e^{i\\lambda \\phi (x)} \\psi (x) dx \\bigg | \\le C \\lambda ^{-1/k} \\Vert \\nabla \\psi \\Vert _{L^1(B)},$ where the constant depends only on $d,k,\\delta $ and $\\Vert \\phi \\Vert _{C^{k+1}(B)}$ .",
"(ii) (Sublevel set estimate) for any $t>0$ , $\\sigma \\lbrace x\\in (1/2)B: |\\phi (x)| \\le t \\rbrace \\le C t^{1/k},$ where the constant depends only on $d,k,\\delta $ and $\\Vert \\phi \\Vert _{C^{k+1}(B)}$ .",
"The proof of part (i) can be found in, e.g., [22].",
"The proof of part (ii) follows from part (i) via a simple trick (see [11]): by writing $u(\\phi (x)) = \\int _{\\mathbb {R}} e^{i\\lambda \\phi (x)} \\hat{u}(\\lambda ) d\\lambda $ , decay estimate (REF ) translate directly to the estimate on $\\int u(\\phi (x)) f(x) dx$ .",
"In particular, choose non-negative function $f\\in C_0^\\infty (B)$ so that $f = 1$ on $(1/2)B$ and $u(t) = \\chi _{[-1,1]}(t/\\alpha )$ , where $\\chi _{[-1,1]}$ is the characteristic function of $[-1,1]$ .",
"Then one sees that estimate (REF ) implies (REF ).",
"The following surprising results indicates the equivalence between $(\\varkappa \\circ On(\\cdot ))^{-1} \\in L^{p,\\infty }(S,d\\sigma )$ and the finite type condition.",
"Proposition 3.6 A closed smooth hypersurface $S\\subset \\mathbb {R}^d$ is of finite type if and only if the statements of Proposition REF hold with some $p>0$ .",
"More precisely, we have: (i) If $S$ is of type $k$ , then the statements of Proposition REF hold with $p = 1/(k-1)$ ; (ii) Conversely, if Proposition REF holds with some $p>(d-1)/k$ , then $S$ is of type $k$ ; (iii) In particular, if $d=2$ , then $S$ is of type $k$ if and only if Proposition REF holds with $p = 1/(k-1)$ .",
"(i) Let $S$ be of type $k$ .",
"By the definition, for any $x_0\\in S$ , the local graph $x_d = \\phi _{x_0}(x^{\\prime })$ is a function of type $k$ in $B(0,r)$ for some $r>0$ with lower bound $\\delta >0$ .",
"Since $S$ is compact and smooth, the parameters $r$ and $\\delta $ involved above can be chosen uniformly in $x_0$ .",
"Let $r_0$ and $\\delta _0$ be the universal parameters for $S$ .",
"It follows that there is some $1\\le j\\le d-1$ such that $\\frac{\\partial }{\\partial x_j} \\phi _{x_0}$ is type $k-1$ .",
"By the sublevel set estimate (REF ), there exists some constant $C$ independent of $t$ and $x_0$ such that $\\sigma \\Big \\lbrace x^{\\prime }\\in B(0,r_0): \\Big |\\frac{\\partial }{\\partial x_j}\\phi _{x_0} (x^{\\prime })\\Big | \\le t \\Big \\rbrace \\le Ct^{1/(k-1)}.$ Since $\\lbrace x^{\\prime }\\in B(0,r_0): |\\nabla \\phi _{x_0}(x^{\\prime })| \\le t \\rbrace \\subset \\lbrace x^{\\prime }\\in B(0,r_0): |\\frac{\\partial }{\\partial x_j} \\phi _{x_0}(x^{\\prime })| \\le t \\rbrace $ , one concludes that $\\sigma \\lbrace x^{\\prime }\\in B(0,r_0): |\\nabla \\phi _{x_0}(x^{\\prime })| \\le t \\rbrace \\le C t^{1/(k-1)}.$ This proves (iii) of Proposition REF and therefore implies all the statements in Proposition REF are true due to the equivalence.",
"(ii) If $S$ is not type $k$ , then by Definition REF and the compactness of $S$ , there must be some point $x_0\\in S$ such that $\\partial ^\\alpha \\phi _{x_0}(0) = 0$ for all $|\\alpha | \\le k$ .",
"It follows that $|\\nabla \\phi _{x_0}(x^{\\prime })| = O(|x^{\\prime }|^k)$ and thereby $\\sigma \\lbrace x^{\\prime }\\in B(0,r_0): |\\nabla \\phi _{x_0}(x^{\\prime })| \\le t \\rbrace \\ge \\sigma \\lbrace x^{\\prime }\\in B(0,r_0): C|x^{\\prime }|^k \\le t \\rbrace = Ct^{(d-1)/k}.$ Obviously, this contradicts to the assumption that Proposition REF holds with some $p > (d-1)/k$ .",
"(iii) Finally, if $d=2$ , combining (i) and (ii), we obtain (iii).",
"We remark that for $d\\ge 3$ , there is a gap between the exponents of (i) and (ii) in Proposition REF , which arises naturally since, in the worst case, the finite type condition is only satisfied in a certain direction along the tangent plane.",
"So to fill this gap by using the sublevel set estimate, a condition of (strong) finite type applied to all directions and an extra assumption of convexity of certain type on the hypersurfaces may be required; see, e.g., [10], [19] Next, we will apply the van de Corput's estimate (REF ) to prove a homogenization theorem for operators with constant coefficients in a domain of finite type.",
"We establish the convergence rate which is optimal in the sense of (REF ) and the purpose of doing so is to provide a comparison with later result dealing with oscillating coefficients.",
"Precisely, we consider the following Dirichlet problem with constant coefficients $\\left\\lbrace \\begin{aligned}-\\nabla \\cdot (A_0\\nabla u_\\varepsilon )(x) &= 0 \\quad & \\text{ in }& \\Omega , \\\\u_\\varepsilon (x) &= f(x,x/\\varepsilon ) \\quad & \\text{ on } &\\partial \\Omega .\\end{aligned}\\right.$ where constant matrix $A_0$ satisfies (REF ) and $f(x,y)$ satisfies (REF ) and (REF ).",
"Then we have Theorem 3.7 Let $\\Omega $ be a bounded, smooth domain of type $k$ .",
"Then the solutions of system (REF ) converges strongly in $L^2(\\Omega )$ , as $\\varepsilon \\rightarrow 0$ , to some function $u_0$ , which is the solution of $\\left\\lbrace \\begin{aligned}-\\nabla \\cdot (A_0\\nabla u_0)(x) &= 0 \\quad & \\text{ in }& \\Omega , \\\\u_0(x) &= \\bar{f}(x) \\quad & \\text{ on } &\\partial \\Omega .\\end{aligned}\\right.$ where $\\bar{f}(x) = \\int _{d} f(x,y)dy$ .",
"Moreover, there exists $C$ independent of $\\varepsilon $ such that $\\Vert u_\\varepsilon - u_0\\Vert _{L^2(\\Omega )} \\le C \\varepsilon ^{1/(2k)}.$ Write $f_\\varepsilon (y) = f(y,y/\\varepsilon )$ .",
"Let $P(x,y)$ be the Poisson kernel of operator $ - \\nabla \\cdot A_0\\nabla $ in $\\Omega $ .",
"By (REF ), (REF ) and the Poisson integral formula, one has $u_\\varepsilon (x) - u_0(x) = \\int _{\\partial \\Omega } P(x,y) ( f_\\varepsilon (y) - \\bar{f}(y) ) d\\sigma (y).$ Now we can localize the integral by applying a partition of unity on $\\partial \\Omega $ .",
"Precisely, we can construct finite smooth functions $\\lbrace \\eta _i : 1\\le i\\le N\\rbrace $ such that $1 = \\sum \\eta _i$ on $\\partial \\Omega $ and $\\text{supp}(\\eta _i) \\subset B(y_i,r_0), y_i\\in \\partial \\Omega $ , where $r_0$ is chosen suitably small.",
"Moreover, $|\\nabla \\eta _i| \\le C$ .",
"Therefore, $u_\\varepsilon (x) - u_0(x) = \\sum _{i=1}^{N}\\int _{\\partial \\Omega } P(x,y) ( f_\\varepsilon (y) - \\bar{f}(y) ) \\eta _{i}(y) d\\sigma (y).$ Now we fix some $\\eta = \\eta _i$ with $y_0 = y_i$ and $\\text{supp}(\\eta ) \\subset B(y_0,r_0)$ , and consider the integral in (REF ) with $\\eta $ involved.",
"By translation and rotation we can transform the surface integral to the usual one in $\\mathbb {R}^{d-1}$ .",
"Precisely, we assume that $z = O^t(y-y_0)$ moves $y_0 \\in \\partial \\Omega $ to origin and transforms the tangent plane at $y_0$ to $z_d = 0$ , where $O$ is an orthogonal matrix.",
"As a result, $\\partial \\Omega \\cap B(y_0,r_0)$ is transformed to the local graph $z_d = \\phi (z^{\\prime }) = \\phi _{y_0}(z^{\\prime })$ which satisfies $\\phi (0) = 0$ and $\\nabla \\phi (0) = 0$ .",
"Thus it is sufficient to estimate $\\begin{aligned}&\\int _{\\partial \\Omega \\cap B(x_0,r_0)} P(x,y)( f_\\varepsilon (y) - \\bar{f}(y) ) \\eta (y) dy \\\\& \\quad = \\int _{\\lbrace |z^{\\prime }|<r_0, z_d = \\phi (z^{\\prime })\\rbrace } P(x,Oz+y_0)( f_\\varepsilon - \\bar{f} )(Oz+y_0) \\eta (Oz+y_0) d\\sigma (z) \\\\& \\quad = \\int _{B(0,r_0)} P(x,Oz+y_0) ( f_\\varepsilon - \\bar{f})(Oz+y_0) \\eta (Oz+y_0) \\sqrt{1+|\\nabla \\phi (z^{\\prime })|^2} dz^{\\prime },\\end{aligned}$ where $z = (z^{\\prime },\\phi (z^{\\prime }))$ .",
"Next we expand $f_\\varepsilon (y) - \\bar{f}(y)$ in Fourier series, i.e., $f_\\varepsilon (y) - \\bar{f}(y) = \\sum f_m(y) e^{i\\varepsilon ^{-1} m\\cdot y}$ , where the sum is taken over all $m\\in \\mathbb {Z}^d,m\\ne 0$ .",
"The last integral now is reduced to the estimate of $e^{i\\varepsilon ^{-1}m\\cdot y_0} \\int _{B(0,r_0)} P(x,Oz+y_0)f_m(Oz+y_0) e^{i\\varepsilon ^{-1}O^tm \\cdot z} \\eta (Oz+y_0) \\sqrt{1+|\\nabla \\phi (z^{\\prime })|^2} dz^{\\prime }.$ To simplify the expression, let $n = |Q^t m|^{-1} O^tm = (n^{\\prime },n_d)$ and $\\varphi (z^{\\prime }) = n\\cdot z = n^{\\prime }\\cdot z^{\\prime } + n_d \\phi (z^{\\prime })$ .",
"Also, let $\\lambda = \\varepsilon ^{-1} |m|^{-1}$ and $g_{m}(z) = P(x,Oz+y_0)f_m(Oz+y_0) \\eta (Oz+y_0) \\sqrt{1+|\\nabla \\phi (z^{\\prime })|^2}.$ Then, (REF ) becomes $e^{i\\varepsilon ^{-1}m\\cdot y_0} \\int _{B(0,r_0)} g_m(z^{\\prime }) e^{i\\lambda \\varphi (z^{\\prime })} dz^{\\prime }.$ In view of the form of $\\varphi (z^{\\prime })$ , the estimate of oscillatory integral (REF ) can be divided into two cases.",
"Case 1: $|n_d| < \\delta _0$ for some sufficiently small $\\delta _0$ , say, $\\delta _0 < (1/10) \\min \\lbrace 1, \\Vert \\nabla \\phi \\Vert _{L^\\infty (B(0,r_0))}^{-1} \\rbrace $ .",
"In this case, it is easy to see that $|\\nabla \\varphi (z^{\\prime })| \\ge 1/2$ on $B(0,1/2)$ .",
"By a standard estimate of oscillatory integral, we have $\\bigg | \\int _{B(0,r_0)} g_m(z^{\\prime }) e^{i\\lambda \\varphi (z^{\\prime })} dz^{\\prime } \\bigg | \\le C \\lambda ^{-1} \\int _{B(0,r_0)} |\\nabla g_m|.$ Note that $|\\nabla g_m| \\le C \\Vert f_m\\Vert _{C^1} \\sum _{k=0,1} |\\nabla ^k P(x,Oz+y_0)| \\le C \\Vert f_m\\Vert _{C^1} |x-(Oz+y_0)|^{d}$ .",
"It follow that $\\int _{B(0,r_0)} |\\nabla g_m| \\le C \\Vert f_m\\Vert _{C^1(\\partial \\Omega )} \\text{dist}(x,\\partial \\Omega )^{-1}.$ Hence $\\bigg | \\int _{B(0,r_0)} g_m(z^{\\prime }) e^{i\\lambda \\varphi (z^{\\prime })} dz^{\\prime } \\bigg | \\le C \\varepsilon |m| \\Vert f_m\\Vert _{C^1(\\partial \\Omega )} \\text{dist}(x,\\partial \\Omega )^{-1}.$ Case 2: $n_d \\ge \\delta _0$ , where $\\delta _0$ is the same as case 1.",
"Now one takes advantage of the assumption that $\\partial \\Omega $ is of type $k$ and use Theorem REF (i) to obtain $\\bigg | \\int _{B(0,r_0)} g_m(z^{\\prime }) e^{i\\lambda \\varphi (z^{\\prime })} dz^{\\prime } \\bigg | & \\le C\\lambda ^{-1/k} \\int _{B(0,r_0)} |\\nabla g_m| \\\\& \\le C \\varepsilon ^{1/k} |m|^{1/k} \\Vert f_m\\Vert _{C^1(\\partial \\Omega )} \\text{dist}(x,\\partial \\Omega )^{-1}.$ Combining Case 1 and Case 2, and summing $m$ over all $m\\in \\mathbb {Z}^d\\setminus \\lbrace 0\\rbrace $ , one obtains the estimate for (REF ), $\\bigg | \\int _{\\partial \\Omega \\cap B(x_0,r_0)} P(x,y)( f_\\varepsilon (y) - \\bar{f}(y) ) \\eta (y) dy \\bigg | \\le C \\frac{\\varepsilon ^{1/k}}{\\text{dist}(x,\\partial \\Omega )} \\sum _{m\\in \\mathbb {Z}^d\\setminus \\lbrace 0\\rbrace } |m|\\Vert f_m\\Vert _{C^1(\\partial \\Omega )}.$ Using the smoothness of $f$ , one can easily verifies that $\\sum _{m\\in \\mathbb {Z}^d\\setminus \\lbrace 0\\rbrace } |m|\\Vert f_m\\Vert _{C^1(\\partial \\Omega )} & \\le \\beta \\sum _{m\\in \\mathbb {Z}^d\\setminus \\lbrace 0\\rbrace } |m|^{-d-1} + \\beta ^{-1} \\sum _{m\\in \\mathbb {Z}^d\\setminus \\lbrace 0\\rbrace } |m|^{d+3} \\Vert f_m\\Vert ^2_{C^1(\\partial \\Omega )} \\\\& \\le C\\sup _{x\\in \\partial \\Omega } \\big (\\Vert f(x,\\cdot )\\Vert _{H^{(d+3)/2}(d)} + \\Vert \\nabla _x f(x,\\cdot )\\Vert _{H^{(d+3)/2}(d)} \\big ),$ where $\\beta $ is selected to minimize the second inequality.",
"As a consequence, we obtain $|u_\\varepsilon (x) - u_0(x)| \\le \\frac{C \\varepsilon ^{1/k}}{\\text{dist}(x,\\partial \\Omega )}, \\qquad \\forall x\\in \\Omega $ On the other hand, the Agmon-type maximal principle implies $|u_\\varepsilon (x) - u_0(x)| \\le C$ for all $x\\in \\Omega $ .",
"Hence, $\\int _{\\Omega } |u_\\varepsilon - u_0|^2 & = \\int _{\\lbrace \\text{dist}(x,\\partial \\Omega ) > \\varepsilon ^{1/k} \\rbrace } |u_\\varepsilon - u_0|^2 + \\int _{\\lbrace \\text{dist}(x,\\partial \\Omega ) \\le \\varepsilon ^{1/k} \\rbrace } |u_\\varepsilon - u_0|^2 \\\\& \\le C\\varepsilon ^{2/k} \\int _{\\lbrace \\text{dist}(x,\\partial \\Omega ) > \\varepsilon ^{1/k} \\rbrace } \\text{dist}(x,\\partial \\Omega )^{-2} dx + C|\\lbrace \\text{dist}(x,\\partial \\Omega ) \\le \\varepsilon ^{1/k} \\rbrace | \\\\& \\le C\\varepsilon ^{1/k}.$ This ends the proof.",
"We end this section by a diagram illustrating the relationships between finite type condition, oscillatory integrals, the property of Diophantine function and homogenization; see Figure REF .",
"Note that the arrows represent implications.",
"Besides Theorem REF , all the implications listed in the diagram have been proved or interpreted in this section.",
"It is of independent interest to observe the close relationship between the Diophantine function and oscillatory integrals while further developments regarding this would be interesting as well.",
"Figure: Relationships between properties of hypersurfaces"
],
[
"Auxiliary problems in half-space",
"For $n\\in \\mathbb {S}^{d-1}$ and $a\\in \\mathbb {R}$ , let $\\mathbb {H}^d_n(a)$ denote the half-space $\\lbrace x\\in \\mathbb {R}^d: x\\cdot n < -a\\rbrace $ with $n$ being the unit outer normal to its boundary $\\partial \\mathbb {H}^d_n(a) = \\lbrace x\\in \\mathbb {R}^d: x\\cdot n = -a \\rbrace $ .",
"Consider the Dirichlet problem $\\left\\lbrace \\begin{aligned}- \\text{div} (A\\nabla u(x)) &= 0 \\qquad & \\text{ in }& \\mathbb {H}^d_n(a), \\\\u(x) &= f(x) \\qquad & \\text{ on } &\\partial \\mathbb {H}^d_n(a),\\end{aligned}\\right.$ where $A$ satisfies (REF ), (REF ) and (REF ), and $f$ is smooth and 1-periodic.",
"Instead of solving (REF ) directly, we try to find a solution of (REF ) with a particular form, i.e., $u(x) = V^a(x-(x\\cdot n)n, -x\\cdot n)$ , where $V^a = V^a(\\theta ,t)$ is a function of $(\\theta ,t) \\in d\\times [a,\\infty )$ .",
"To identify the system satisfied for $V^a$ , let $M$ be a $d\\times d$ orthogonal matrix whose last column is $-n$ .",
"Let $N$ denote the $d\\times (d-1)$ matrix of the first $d-1$ columns of $M$ .",
"Since $MM^T = I$ , we see that $NN^T + n\\otimes n = I$ .",
"It follows from (REF ) and the previous settings that $V^a$ must be a solution of $\\left\\lbrace \\begin{aligned}- \\Bigg ( \\begin{aligned}N^T \\nabla _\\theta \\\\ \\partial _t \\ \\ \\;\\end{aligned}\\Bigg ) \\cdot B\\Bigg ( \\begin{aligned}N^T \\nabla _\\theta \\\\ \\partial _t \\ \\ \\;\\end{aligned}\\Bigg ) V&= 0 \\qquad & \\text{ in }& d\\times (a,\\infty ), \\\\V &= F \\qquad & \\text{ on } & d\\times \\lbrace a\\rbrace ,\\end{aligned}\\right.$ where $B(\\theta ,t) = M^TA(\\theta -tn) M$ and $F(\\theta ) = f(\\theta )$ .",
"Observe that if $V^a$ is a solution of (REF ) with $a\\in \\mathbb {R}$ , then $ V^a(\\theta ,t) = V^0(\\theta -an,t-a) $ , which reduces the problem to the particular case $a = 0$ .",
"Now we collect some important results concerning the lifted system (REF ) in the following theorem.",
"Theorem 4.1 Let $n\\in \\mathbb {S}^{d-1},a = 0$ and $F\\in C^\\infty (d)$ .",
"Then (i) The system (REF ) has a smooth solution $V$ such that for all $k,s\\ge 0$ , $\\int _0^\\infty \\Vert N^T \\nabla _\\theta \\partial _t^k V\\Vert _{H^s(d)}^2 + \\Vert \\partial _t^{k+1} V\\Vert _{H^s(d)}^2 dt \\le C,$ where $C$ depends only on $d,m,k,s,A$ and $F$ .",
"(ii) If $n$ satisfies the Diophantine condition with constant $\\varkappa >0$ and $V$ is the solution of (REF ) given in (i), then there exists a constant $V_\\infty $ such that for all $\\alpha \\in \\mathbb {N}^d,k\\ge 0$ and $s\\ge 0$ , $|N^T\\nabla _\\theta \\partial _\\theta ^\\alpha \\partial _t^k V|+ |\\partial _\\theta ^\\alpha \\partial _t^{k+1} V| + \\varkappa |\\partial _\\theta ^\\alpha \\partial _t^{k}(V - V_\\infty )| \\le \\frac{C}{(1+\\varkappa t)^s},$ where $C$ depends only on $d,m,k,\\alpha ,s,A$ and $F$ .",
"(iii) Let $n$ satisfy the Diophantine condition with constant $\\varkappa >0$ and $\\tilde{n}$ be any other unit vector in $\\mathbb {S}^{d-1}$ .",
"Let $V$ and $\\widetilde{V}$ be the solutions of (REF ) corresponding to $n$ and $\\tilde{n}$ , respectively.",
"Define $W = V - \\widetilde{V}$ .",
"Then for any $0<\\sigma <1$ , $\\int _0^1 \\int _{d} |\\widetilde{N}^T \\nabla _\\theta W|^2 + |\\partial _t W|^2 d\\theta dt \\le C \\bigg ( \\frac{|n-\\tilde{n}|^4}{\\varkappa ^{4+\\sigma }} + \\frac{|n-\\tilde{n}|^2}{\\varkappa ^{2+\\sigma }} \\bigg ).$ where $(\\widetilde{N},-\\tilde{n})$ is an orthogonal matrix and $C$ depends only on $d,m,\\sigma ,A$ and $F$ .",
"The proofs of (i) and (ii) are more or less well-known and can be found in [15], [7], [18].",
"Statement (iii) was established in [21] recently for Neumann problems by applying a weighted estimate.",
"The proof for Dirichlet problems is similar without any real difficulty.",
"For problems with non-convex domains, We will also need the extension of $V$ from $d\\times [a,\\infty )$ to the whole space $d\\times \\mathbb {R}$ .",
"We state the result as follows.",
"Proposition 4.2 If $V \\in C^k(d\\times [a,\\infty ))$ , then it has an extension $\\bar{V} \\in C^k(d\\times \\mathbb {R})$ such that $\\bar{V}(\\theta ,t) = V(\\theta ,t) \\qquad \\text{if }\\ t\\ge a,$ and $\\Vert \\bar{V}\\Vert _{C^k(d\\times [a-r,a+r])} \\le C \\Vert V\\Vert _{C^k(d\\times [a,a+(k+1)r])},$ where $C$ depends only on $k$ and $d$ .",
"This may be proved by a standard construction; see [1], for example.",
"Without loss of generality, we may assume $a = 0$ .",
"Then define $\\bar{V}(\\theta ,t) = \\left\\lbrace \\begin{aligned}&V(\\theta ,t) \\qquad & \\text{ if }& t\\ge 0, \\\\& \\sum _{j=1}^{k+1} \\lambda _j V(\\theta ,-jt) \\qquad & \\text{ if } & t<0,\\end{aligned}\\right.$ where $\\lambda _1,\\lambda _2,\\cdots ,\\lambda _{k+1}$ are the unique solution of the system of $k+1$ linear equations $\\sum _{j=1}^{k+1} (-j)^i \\lambda _j=1,\\qquad i = 0,1,\\cdots , k.$ Then one may verify that $\\bar{V} \\in C^k(d\\times \\mathbb {R})$ and (REF ) is satisfied.",
"As mentioned before, one solution of (REF ) can be given by $u(x) = V^a(x-(x\\cdot n)n, -x\\cdot n) = V(x-(x\\cdot n+a)n, -x\\cdot n -a),$ where $V$ is the solution of (REF ) with $a = 0$ , given by Theorem REF .",
"Due to Proposition REF , we can extend $u$ from $\\mathbb {H}^d_n(a)$ to $\\mathbb {R}^d$ .",
"Actually, one can define $\\begin{aligned}\\bar{u}(x) & = \\bar{V}(x-(x\\cdot n+a)n, -x\\cdot n -a) \\\\& =\\left\\lbrace \\begin{aligned}& V(x-(x\\cdot n+a)n, -x\\cdot n -a) \\qquad & \\text{ if }& x\\cdot n \\le -a, \\\\& \\sum _{j=1}^{k+1} \\lambda _j V(x-(x\\cdot n+a)n, j(x\\cdot n +a)) \\qquad & \\text{ if } & x\\cdot n > -a,\\end{aligned}\\right.\\end{aligned}$ where $\\lambda _j$ is given by (REF ).",
"Then $\\bar{u}$ is a $C^k$ extension of $u$ in $\\mathbb {R}^d$ .",
"Moreover, there exist constants $C,c >1$ depending at most on $k,d$ and $m$ such that for any $x_0 \\in \\partial \\mathbb {H}^d_n(a)$ and $r>0$ , $\\Vert \\bar{u}\\Vert _{C^k(B(x_0,r))} \\le C \\Vert u\\Vert _{C^k(B(x_0,cr)\\cap \\mathbb {H}_n^d(a))}.$"
],
[
"Local two-scale expansion",
"Throughout this subsection we assume that $\\Omega $ is a bounded smooth domain in $\\mathbb {R}^d,d\\ge 2$ , and that $A$ satisfies (REF ), (REF ) and (REF ).",
"Consider the Dirichlet problem $\\left\\lbrace \\begin{aligned}\\mathcal {L}_\\varepsilon u_\\varepsilon (x) &= 0 \\qquad & \\text{ in }& \\Omega , \\\\u_\\varepsilon (x) &= f_\\varepsilon (x) = \\varepsilon f(x/\\varepsilon ) \\qquad & \\text{ on } &\\partial \\Omega ,\\end{aligned}\\right.$ where $f(y)$ is 1-periodic and smooth.",
"Fix $x_0\\in \\partial \\Omega $ .",
"The main goal of this subsection is to find an approximation of $u_\\varepsilon $ in a neighborhood of $x_0$ .",
"To this end, we solve the Dirichlet problem in a half-space $\\left\\lbrace \\begin{aligned}\\mathcal {L}_\\varepsilon v_\\varepsilon (x) &= 0 \\qquad & \\text{ in }& \\mathbb {H}^d_{n_0}(a), \\\\v_\\varepsilon (x) &= f_\\varepsilon (x) \\qquad & \\text{ on } &\\partial \\mathbb {H}^d_{n_0}(a),\\end{aligned}\\right.$ where $a = -x_0\\cdot n_0$ and $\\partial \\mathbb {H}^d_{n_0}(a)$ is the tangent plane of $\\partial \\Omega $ at $x_0$ .",
"Note that $v_\\varepsilon $ has a form of $v_\\varepsilon (x) = \\varepsilon v_1(x/\\varepsilon )$ , and $v_1$ is the solution of $\\left\\lbrace \\begin{aligned}\\mathcal {L}_1 v_1(x) &= 0 \\qquad & \\text{ in }& \\mathbb {H}^d_{n_0}(a/\\varepsilon ), \\\\v_1(x) &= f(x) \\qquad & \\text{ on } &\\partial \\mathbb {H}^d_{n_0}(a/\\varepsilon ),\\end{aligned}\\right.$ The existence of the solution of (REF ) or (REF ) as well as its estimates have been established via the half-space problem in Theorem REF (i) and formula (REF ).",
"Note that $\\Omega \\setminus \\mathbb {H}^d_{n_0}(a)$ is non-empty for non-convex domain $\\Omega $ .",
"To approximate $u_\\varepsilon $ in $\\Omega $ , we could extend $v_1$ and hence $v_\\varepsilon $ , in the form of (REF ) with $k = 2$ , to the whole space $\\mathbb {R}^d$ .",
"Let $\\bar{v}_\\varepsilon $ denote the extended function of $v_\\varepsilon $ .",
"It follows from (REF ), (REF ) and (REF ) that $\\bar{v}_\\varepsilon = v_\\varepsilon \\text{ in } \\mathbb {H}^d_{n_0}(a) \\quad \\text{and} \\quad \\sup _{x\\in B_r}|\\nabla ^k \\bar{v}_\\varepsilon (x)| \\le C\\sup _{B_{cr}\\cap \\mathbb {H}_{n_0}^d(a)} |\\nabla ^k v_\\varepsilon (x)|, \\quad k = 0,1,2.$ for any $r>0$ , where $B_r$ and $B_{cr}$ are centered at $x_0$ .",
"Define $w_\\varepsilon (x) = u_\\varepsilon (x) - \\bar{v}_\\varepsilon (x)$ .",
"Observe that by the definition of $\\bar{v}_\\varepsilon $ , $w_\\varepsilon $ is a solution of $\\mathcal {L}_\\varepsilon w_\\varepsilon (x) = 0$ only in $\\Omega \\cap \\mathbb {H}^d_{n_0}(a)$ .",
"Now we prove the following.",
"Theorem 4.3 Let $w_\\varepsilon $ be constructed as above.",
"Let $\\varepsilon \\le r\\le \\sqrt{\\varepsilon }$ .",
"Then for any $\\sigma \\in (0,1)$ , $\\Vert \\nabla w_\\varepsilon \\Vert _{L^\\infty (B(x_0,r) \\cap \\Omega )} \\le C\\sqrt{\\varepsilon } + C\\frac{r^{2+\\sigma }}{\\varepsilon ^{1+\\sigma }},$ where $C$ depends on $d,m,\\mu ,\\sigma ,\\Omega ,A$ and $f$ .",
"To prove the theorem, we require the following lemmas.",
"Lemma 4.4 Let $u_\\varepsilon $ be a solution of (REF ), then one has for any $k\\ge 0$ , $\\Vert \\nabla ^k u_\\varepsilon \\Vert _{L^\\infty (\\Omega )} \\le C\\varepsilon ^{1-k},$ where $C$ is independent of $\\varepsilon $ .",
"For $k=0$ , we use the Agmon-type maximal principle to obtain $\\Vert u_\\varepsilon \\Vert _{L^\\infty (\\Omega )} \\le C\\Vert f_\\varepsilon \\Vert _{L^\\infty (\\partial \\Omega )} \\le C\\varepsilon .$ For $k > 0$ , we apply a blow-up argument.",
"Set $u_\\varepsilon (x) = \\varepsilon u_1(x/\\varepsilon )$ .",
"Then $u_1$ is a solution of $\\left\\lbrace \\begin{aligned}\\mathcal {L}_1 u_1(x) &= 0 \\qquad & \\text{ in }& \\Omega ^{\\varepsilon }, \\\\u_1(x) &= f(x) \\qquad & \\text{ on } &\\partial \\Omega ^{\\varepsilon },\\end{aligned}\\right.$ where $\\Omega ^\\varepsilon = \\lbrace x: \\varepsilon x \\in \\Omega \\rbrace $ .",
"Note that the $C^k$ character of $\\Omega ^{\\varepsilon }$ is controlled by that of $\\Omega $ .",
"It follows from the local Schauder's estimate that for any $x\\in \\overline{\\Omega ^{\\varepsilon }}$ , $\\Vert \\nabla ^k u_1\\Vert _{L^\\infty (B(x,1)\\cap \\Omega ^{\\varepsilon })} \\le C\\Vert u_1\\Vert _{L^\\infty (B(x,2)\\cap \\Omega ^{\\varepsilon })} + \\Vert f\\Vert _{C^{k,\\alpha }(B(x,2)\\cap \\Omega ^{\\varepsilon })}.$ Since $f$ is 1-periodic, then $\\Vert f\\Vert _{C^{k,\\alpha }(B(x,2)\\cap \\Omega ^{\\varepsilon })} \\le C\\Vert f\\Vert _{C^{k,\\alpha }(d)}$ .",
"And by (REF ), $\\Vert u_1\\Vert _{L^\\infty (\\Omega ^{\\varepsilon })} \\le C$ .",
"It follows that $\\Vert \\nabla ^k u_1\\Vert _{L^\\infty (\\Omega ^{\\varepsilon })} \\le C,$ for any $k> 0$ , where $C$ depends also on $k$ .",
"Changing variables back to $u_\\varepsilon $ , we obtain the desired estimates (REF ).",
"Lemma 4.5 Let $\\bar{v}_\\varepsilon $ be constructed as above, then one has for $k=0,1,2$ , $\\Vert \\nabla ^k \\bar{v}_\\varepsilon \\Vert _{L^\\infty (\\mathbb {R}^d)} \\le C\\varepsilon ^{1-k},$ where $C$ is independent of $\\varepsilon $ .",
"In view of (REF ), it suffices to consider the estimates for $v_\\varepsilon $ .",
"Let $v_\\varepsilon (x) = \\varepsilon v_1(x/\\varepsilon )$ .",
"Then $v_1$ is the solution of (REF ), which can also be given by the Poisson integral formula $v_1(x) = \\int _{\\partial \\mathbb {H}_{n_0}^d(a/\\varepsilon )} P_{\\mathbb {H}}(x,y) f(y) d\\sigma (y),$ where $P_{\\mathbb {H}}$ is the Poisson kernel of $\\mathcal {L}_1$ in the half-space $ \\mathbb {H}_{n_0}^d(a/\\varepsilon )$ .",
"A similar estimate as (REF ) in half-spaces was established in [16], i.e., $P_{\\mathbb {H}}(x,y) \\le \\frac{C \\text{dist}(x,\\partial \\mathbb {H}_{n_0}^d(a/\\varepsilon ))}{|x-y|^d}, \\qquad \\text{for all } x\\in \\mathbb {H}_{n_0}^d(a/\\varepsilon ).$ Then it follows from (REF ) that $\\Vert v_1\\Vert _{L^\\infty ( \\mathbb {H}_{n_0}^d(a/\\varepsilon ))} \\le C\\Vert f\\Vert _{L^\\infty (\\partial \\mathbb {H}_{n_0}^d(a/\\varepsilon ))}$ (Agmon-type maximal principle).",
"Thus, $\\Vert v_\\varepsilon \\Vert _{L^\\infty (\\mathbb {H}_{n_0}^d(a))} \\le C \\varepsilon $ as desired for $k=0$ .",
"The estimates for $k>0$ follow similarly as Lemma REF by the local Schauder's estimates.",
"The proof follows a line of [7], with modifications made to adjust to our setting of non-convex domains.",
"Step 1: Set up and conventions.",
"First of all, since $\\Omega $ is smooth, then for each point $x_0$ on $\\partial \\Omega $ , there is another domain $\\widetilde{\\Omega }$ satisfying the following: $\\widetilde{\\Omega }\\subset \\Omega \\cap \\mathbb {H}^d_{n_0}(a)$ ; $\\widetilde{\\Omega }$ shares the same tangent hyperplane $\\partial \\mathbb {H}^d_{n_0}(a)$ of $\\Omega $ at point $x_0$ ; $\\widetilde{\\Omega }$ is a $C^{2,\\alpha }$ domain whose $C^{2,\\alpha }$ character is controlled by that of $\\Omega $ .",
"The existence of such domains is obvious since smooth domains always satisfy the uniform interior spheres condition.",
"Let $y\\in \\partial \\Omega $ and $|y-x_0|\\le r_0$ for some $r_0$ depending only on $\\Omega $ .",
"We will use the following conventions: let $\\hat{y}$ denote the projection of $y$ on $\\partial \\mathbb {H}^d_{n_0}(a)$ such that $y - \\hat{y}$ is a multiple of $n_0$ ; let $\\tilde{y}$ denote the first point on $\\partial \\widetilde{\\Omega }$ such that $y - \\tilde{y}$ is a multiple of $n_0$ .",
"Since both $\\Omega $ and $\\widetilde{\\Omega }$ are at least $C^2$ near $x_0$ , it is easy to see that for all $y$ satisfying $|y-x_0| \\le r_0$ , $|y - \\tilde{y}| + |y - \\hat{y}| + |\\tilde{y} - \\hat{y}| \\le C |y - x_0|^2.$ This also implies $|y-x_0| \\approx |\\tilde{y} - x_0| \\approx |\\hat{y} - x_0|$ if $r_0$ is sufficiently small.",
"On the other hand, let $n(y)$ and $\\tilde{n}(\\tilde{y})$ denote the unit outer normal of $\\partial \\Omega $ and $\\partial \\widetilde{\\Omega }$ , respectively.",
"Then $|n(y) - n_0| + |\\tilde{n}(\\tilde{y}) - n_0| + |\\tilde{n}(\\tilde{y}) - n(y)| \\le C|y-x_0|.$ Step 2: We prove the estimate (REF ) in $B(x_0,r)\\cap \\widetilde{\\Omega }$ which is a subset of $B(x_0,r)\\cap \\Omega $ , i.e., $\\Vert \\nabla w_\\varepsilon \\Vert _{L^\\infty (B_r \\cap \\widetilde{\\Omega })} \\le C\\sqrt{\\varepsilon } + C\\frac{r^{2+\\sigma }}{\\varepsilon ^{1+\\sigma }},$ where $B_r = B(x_0,r)$ and $\\varepsilon \\le r\\le \\sqrt{\\varepsilon }$ .",
"The idea for the proof of (REF ) is similar to the case of convex domains since, by the definition of $w_\\varepsilon $ , $w_\\varepsilon $ is a solution of $\\mathcal {L}_\\varepsilon w_\\varepsilon = 0 \\quad \\text{subject to certain Dirichlet boundary condition on } \\partial \\widetilde{\\Omega }.$ Indeed, it follows from the uniform Lipschitz estimate in $C^{1,\\alpha }$ domains that $\\begin{aligned}\\Vert \\nabla w_\\varepsilon \\Vert _{L^\\infty (B_r\\cap \\widetilde{\\Omega })} & \\le Cr^{-1} \\Vert w_\\varepsilon \\Vert _{L^\\infty (B_{2r}\\cap \\widetilde{\\Omega })} \\\\& \\qquad + C \\Vert \\nabla _{\\tan } w_\\varepsilon \\Vert _{L^\\infty (B_{2r}\\cap \\partial \\widetilde{\\Omega })} + C r^\\sigma \\Vert \\nabla _{\\tan } w_\\varepsilon \\Vert _{C^\\sigma (B_{2r}\\cap \\partial \\widetilde{\\Omega })}.\\end{aligned}$ Note that $\\nabla _{\\tan } $ can be written as $ (I - \\tilde{n}\\otimes \\tilde{n}) \\nabla $ (which can be viewed as the projection of $\\nabla $ onto the tangent planes $\\tilde{n}^\\perp $ ), where $\\tilde{n}$ is the unit outer normal of $\\partial \\widetilde{\\Omega }$ .",
"We now deal with the estimate of $\\nabla _{\\tan } w_\\varepsilon $ on $B_{2r}\\cap \\widetilde{\\Omega }$ .",
"Recall that $w_\\varepsilon = u_\\varepsilon - v_\\varepsilon $ in $\\widetilde{\\Omega }$ since $\\bar{v}_\\varepsilon = v_\\varepsilon $ in $\\widetilde{\\Omega }$ .",
"Using the fact $u_\\varepsilon = \\varepsilon f(x/\\varepsilon )$ on $\\partial \\Omega $ , we know $(I - n\\otimes n)\\nabla (u_\\varepsilon - \\varepsilon f(x/\\varepsilon ))(y) = 0$ on $\\partial \\Omega $ .",
"It follows that $\\begin{aligned}& |(I - \\tilde{n}\\otimes \\tilde{n}) \\nabla (u_\\varepsilon - f_\\varepsilon )(\\tilde{y})| \\\\&\\qquad = |(I - \\tilde{n}\\otimes \\tilde{n}) \\nabla (u_\\varepsilon - f_\\varepsilon )(\\tilde{y}) - (I - n\\otimes n) \\nabla (u_\\varepsilon - f_\\varepsilon )(y)| \\\\& \\qquad \\le |\\tilde{n}\\otimes \\tilde{n} - n\\otimes n| \\Vert \\nabla (u_\\varepsilon -f_\\varepsilon )\\Vert _{L^\\infty (B_{2r}\\cap \\Omega )} + |\\tilde{y} - y| \\Vert \\nabla ^2(u_\\varepsilon -f_\\varepsilon )\\Vert _{L^\\infty (B_{2r}\\cap \\Omega )} \\\\& \\qquad \\le C|y-x_0| + C \\frac{|y-x_0|^2}{\\varepsilon },\\end{aligned}$ where we have used the mean value theorem in the first inequality, and used (REF ), (REF ) and (REF ) in the second one.",
"Similarly, taking advantage of the fact $v_\\varepsilon = f_\\varepsilon $ on the hyperplane $\\partial \\mathbb {H}^d_{n_0}(a)$ , we have $(I - n_0\\otimes n_0)\\nabla (v_\\varepsilon - f_\\varepsilon )(\\hat{y}) = 0$ .",
"By the same argument as (REF ), we obtain $|(I - \\tilde{n}\\otimes \\tilde{n}) \\nabla (v_\\varepsilon - f_\\varepsilon )(\\tilde{y})| \\le C|y-x_0| + C \\frac{|y-x_0|^2}{\\varepsilon }.$ Combining (REF ) and (REF ), we have $\\Vert \\nabla _{\\tan } w_\\varepsilon \\Vert _{L^\\infty (B_{2r}\\cap \\partial \\widetilde{\\Omega })} = \\Vert (I-\\tilde{n}\\otimes \\tilde{n})\\nabla (u_\\varepsilon - v_\\varepsilon )\\Vert _{L^\\infty (B_{2r}\\cap \\partial \\widetilde{\\Omega })} \\le Cr + C\\frac{r^2}{\\varepsilon } \\le C\\frac{r^2}{\\varepsilon },$ where the last inequality holds for $r\\ge \\varepsilon $ .",
"A similar argument also shows that $\\Vert \\nabla _{\\tan }^2 w_\\varepsilon \\Vert _{L^\\infty (B_{2r}\\cap \\partial \\widetilde{\\Omega })} \\le C \\varepsilon ^{-2} r^2$ , which, by interpolation, implies $\\Vert \\nabla _{\\tan } w_\\varepsilon \\Vert _{C^{\\sigma }(B_{2r}\\cap \\partial \\widetilde{\\Omega })} \\le C\\varepsilon ^{-1-\\sigma } r^{2}$ for any $0<\\sigma <1$ .",
"As a result, to see (REF ), it is left to estimate $\\Vert w_\\varepsilon \\Vert _{L^\\infty (B_{2r} \\cap \\widetilde{\\Omega })}$ .",
"Step 3: To estimate $w_\\varepsilon (x)$ in $B_{2r}\\cap \\widetilde{\\Omega }$ , we first claim that $|w_\\varepsilon (\\tilde{y})| \\le C |y-x_0|^2 \\qquad \\text{for all } \\tilde{y} \\in \\partial \\widetilde{\\Omega }\\cap B(x_0,r_0).$ Actually, write agian $w_\\varepsilon = (u_\\varepsilon - f_\\varepsilon ) - (v_\\varepsilon - f_\\varepsilon )$ .",
"Using the cancellation $u_\\varepsilon - f_\\varepsilon = 0$ on $\\partial \\Omega $ and mean value theorem, we have $|u_\\varepsilon (\\tilde{y}) - f_\\varepsilon (\\tilde{y})| & = |u_\\varepsilon (\\tilde{y}) - f_\\varepsilon (\\tilde{y}) - (u_\\varepsilon (y) - f_\\varepsilon (y))| \\\\& \\le C|\\tilde{y} - y| \\Vert \\nabla (u_\\varepsilon -f_\\varepsilon )\\Vert _{L^\\infty (B_{2r}\\cap \\Omega )} \\\\& \\le C|y-x_0|^2,$ where in the last inequality we have used (REF ) and (REF ).",
"The estimate for $|v_\\varepsilon (\\tilde{y}) - f_\\varepsilon (\\tilde{y})|$ is the same, which proves (REF ).",
"Then we take advantage of the Poisson integral formula and split it into two parts, $\\begin{aligned}w_\\varepsilon (x) & = \\int _{\\partial \\widetilde{\\Omega }} P_{\\widetilde{\\Omega },\\varepsilon }(x,\\tilde{y}) w_\\varepsilon (\\tilde{y}) d\\sigma (\\tilde{y}) \\\\& = \\int _{\\partial \\widetilde{\\Omega }\\cap \\lbrace |\\tilde{y} - x_0|\\le c\\sqrt{\\varepsilon }\\rbrace } P_{\\widetilde{\\Omega },\\varepsilon }(x,\\tilde{y}) w_\\varepsilon (\\tilde{y}) d\\sigma (\\tilde{y}) + \\int _{\\partial \\widetilde{\\Omega }\\cap \\lbrace |\\tilde{y} - x_0| > c\\sqrt{\\varepsilon }\\rbrace } P_{\\widetilde{\\Omega },\\varepsilon }(x,\\tilde{y}) w_\\varepsilon (\\tilde{y}) d\\sigma (\\tilde{y})\\end{aligned}$ where $P_{\\widetilde{\\Omega },\\varepsilon }$ is the Poisson kernel of $\\mathcal {L}_\\varepsilon $ in $\\widetilde{\\Omega }$ and satisfies the same estimate as (REF ) with $\\Omega $ replaced by $\\widetilde{\\Omega }$ .",
"To estimate the first term on the right-hand side of (REF ), we apply (REF ) and (REF ), $\\begin{aligned}&\\Bigg | \\int _{\\partial \\widetilde{\\Omega }\\cap \\lbrace |\\tilde{y} - x_0|\\le c\\sqrt{\\varepsilon }\\rbrace } P_{\\widetilde{\\Omega },\\varepsilon }(x,\\tilde{y}) w_\\varepsilon (\\tilde{y}) d\\sigma (\\tilde{y}) \\Bigg | \\\\& \\le C \\int _{\\partial \\widetilde{\\Omega }\\cap \\lbrace |\\tilde{y} - x_0|\\le c\\sqrt{\\varepsilon }\\rbrace } \\text{dist}(x,\\partial \\widetilde{\\Omega })\\frac{|y - x_0|^2}{|x-\\tilde{y}|^d} d\\sigma (\\tilde{y}) \\\\& \\le C \\int _{\\partial \\widetilde{\\Omega }\\cap \\lbrace |\\tilde{y} - x_0|\\le c\\sqrt{\\varepsilon }\\rbrace } \\text{dist}(x,\\partial \\widetilde{\\Omega })\\frac{|x - x_0|^2}{|x-\\tilde{y}|^d} d\\sigma (\\tilde{y}) + C \\int _{\\partial \\widetilde{\\Omega }\\cap \\lbrace |\\tilde{y} - x_0|\\le c\\sqrt{\\varepsilon }\\rbrace } \\frac{\\text{dist}(x,\\partial \\widetilde{\\Omega })}{|x-\\tilde{y}|^{d-2}} d\\sigma (\\tilde{y}) \\\\& \\le C |x-x_0|^2 + C\\text{dist}(x,\\partial \\widetilde{\\Omega }) \\sqrt{\\varepsilon } \\\\& \\le Cr^2 + r\\sqrt{\\varepsilon },\\end{aligned}$ where we have used the observation $|y-x_0|^2 \\le C|\\tilde{y} - x_0|^2 \\le C|\\tilde{y} - x|^2 + C|x-x_0|^2$ .",
"To bound the second term on the right-hand side of (REF ), we note that (REF ) and (REF ) give $\\Vert w_\\varepsilon \\Vert _{L^\\infty (\\widetilde{\\Omega })} \\le C\\varepsilon $ .",
"Then $\\begin{aligned}\\Bigg | \\int _{\\partial \\widetilde{\\Omega }\\cap \\lbrace |\\tilde{y} - x_0| > c\\sqrt{\\varepsilon }\\rbrace } P_{\\widetilde{\\Omega },\\varepsilon }(x,\\tilde{y}) w_\\varepsilon (\\tilde{y}) d\\sigma (\\tilde{y}) \\Bigg | & \\le C\\varepsilon \\int _{\\partial \\widetilde{\\Omega }\\cap \\lbrace |\\tilde{y} - x_0| > c\\sqrt{\\varepsilon }\\rbrace } \\frac{\\text{dist}(x,\\partial \\widetilde{\\Omega })}{|\\tilde{y} - x|^d} d\\sigma (\\tilde{y}) \\\\& \\le C\\varepsilon \\text{dist}(x,\\partial \\widetilde{\\Omega }) (\\sqrt{\\varepsilon })^{-1} \\le Cr\\sqrt{\\varepsilon }.\\end{aligned}$ It follows $|w_\\varepsilon (x)| \\le Cr^2 + Cr\\sqrt{\\varepsilon }, \\qquad \\text{for all } x\\in B(0,2r)\\cap \\widetilde{\\Omega }.$ This, together with (REF ) and the estimates for $\\nabla _{\\tan } w_\\varepsilon $ in Step 2, proves (REF ).",
"Step 4: Finally, to extend estimate (REF ) to $B_r\\cap \\Omega $ , it suffices to note that $\\partial \\widetilde{\\Omega }$ and $\\partial \\Omega $ are very close near $x_0$ and the $C^2$ regularity of $\\bar{v}_\\varepsilon $ are preserved, thanks to REF .",
"Actually, for any point $y^* \\in B_r \\cap \\Omega \\setminus \\widetilde{\\Omega }$ , there exist $y\\in B_{cr}\\cap \\partial \\Omega $ and corresponding $\\tilde{y} \\in B_{cr} \\cap \\partial \\widetilde{\\Omega }$ such that $y^*$ is on the segment connecting $y$ and $\\tilde{y}$ .",
"Then $|\\nabla w_\\varepsilon (y^*) - \\nabla w_\\varepsilon (\\tilde{y})| \\le C\\Vert \\nabla ^2 w_\\varepsilon \\Vert _{L^\\infty (B_{2r})} |y^* - \\tilde{y}| \\le C\\frac{|y - \\tilde{y}|}{\\varepsilon } \\le C\\frac{r^2}{\\varepsilon },$ where we have used (REF ) in the last inequality.",
"Finally, applying (REF ) for $\\nabla w_\\varepsilon (\\tilde{y})$ , we obtain $|\\nabla w_\\varepsilon (y^*) \\le C\\sqrt{\\varepsilon } + C\\frac{r^{2+\\sigma }}{\\varepsilon ^{1+\\sigma }}, \\qquad \\text{for all } y^* \\in B_r \\cap \\Omega \\setminus \\widetilde{\\Omega }.$ Combing this with (REF ), we obtain (REF ) as desired.",
"In view of (REF ), to study the oscillating behavior of $\\omega _\\varepsilon $ , the difficulty is to understand the behavior of $ \\nabla \\Phi _\\varepsilon ^*$ near the boundary.",
"This can be done by applying Theorem REF to $u_{\\varepsilon ,j}^{*\\beta } = \\Phi ^{*\\beta }_{\\varepsilon ,j}(x) - P_j^\\beta (x) - \\varepsilon \\chi _j^{*\\beta }(x/\\varepsilon )$ for each $1\\le j\\le d, 1\\le \\beta \\le m$ .",
"Clearly, by definitions of $\\Phi $ and $\\chi $ , $u_{\\varepsilon ,j}^\\beta $ satisfies $\\left\\lbrace \\begin{aligned}\\mathcal {L}_\\varepsilon ^* u_{\\varepsilon ,j}^{*\\beta }(x) &= 0 \\qquad & \\text{ in }& \\Omega , \\\\u_{\\varepsilon ,j}^{*\\beta }(x) &= -\\varepsilon \\chi _j^{*\\beta }(x/\\varepsilon ) \\qquad & \\text{ on } &\\partial \\Omega .\\end{aligned}\\right.$ For each fixed $x_0\\in \\partial \\Omega $ , the system (REF ) associated with the adjoint operator $\\mathcal {L}_\\varepsilon ^*$ and $f_\\varepsilon = -\\varepsilon \\chi _j^{*\\beta }(x/\\varepsilon )$ has a solution $v_{\\varepsilon ,j}^{*\\beta }$ of form $v_{\\varepsilon ,j}^{*\\beta }(x) = \\varepsilon V_j^{*\\beta } \\bigg ( \\frac{x-(x\\cdot n_0 + a)n_0}{\\varepsilon }, - \\frac{x\\cdot n_0 + a}{\\varepsilon } \\bigg ), \\qquad \\text{for } x\\cdot n_0 \\le -a,$ where $a = -x_0\\cdot n_0$ and $V_j^{*\\beta } = V_j^{*\\beta }(\\theta ,t)$ is a solution of $\\left\\lbrace \\begin{aligned}- \\Bigg ( \\begin{aligned}N^T \\nabla _\\theta \\\\ \\partial _t \\ \\ \\;\\end{aligned}\\Bigg ) \\cdot B^*\\Bigg ( \\begin{aligned}N^T \\nabla _\\theta \\\\ \\partial _t \\ \\ \\;\\end{aligned}\\Bigg ) V_j^{*\\beta }&= 0 \\qquad & \\text{ in }& d\\times (0,\\infty ), \\\\V_j^{*\\beta } &= -\\chi _j^{*\\beta } \\qquad & \\text{ on } & d\\times \\lbrace 0\\rbrace ,\\end{aligned}\\right.$ given by Theorem REF .",
"Note that $V_j^{*\\beta }$ also depends on $n_0$ .",
"Now let $\\bar{v}_{\\varepsilon ,j}^\\beta $ be the extension of $v_{\\varepsilon ,j}^{*\\beta }$ given by (REF ) with $k=2$ and a change of variables.",
"Precisely, $\\bar{v}_{\\varepsilon ,j}^{*\\beta }(x) =\\left\\lbrace \\begin{aligned}& \\varepsilon V_j^{*\\beta } \\bigg ( \\frac{x-(x\\cdot n_0 + a)n_0}{\\varepsilon }, - \\frac{x\\cdot n_0 + a}{\\varepsilon } \\bigg ) \\qquad & \\text{ if }& x\\cdot n_0 \\le -a, \\\\& \\sum _{j=1}^3 \\varepsilon \\lambda _j V_j^{*\\beta } \\bigg ( \\frac{x-(x\\cdot n_0 + a)n_0}{\\varepsilon }, \\frac{j(x\\cdot n_0 + a)}{\\varepsilon } \\bigg ) \\qquad & \\text{ if } & x\\cdot n_0 > -a.\\end{aligned}\\right.$ Then, one may deduce from Theorem REF that Theorem 4.6 Let $\\varepsilon \\le r\\le \\sqrt{\\varepsilon }$ and $\\sigma \\in (0,1)$ .",
"Then for any $x\\in B(x_0,r) \\cap \\Omega $ , $\\bigg | \\nabla \\bigg ( \\Phi ^{*\\beta }_{\\varepsilon ,j}(x) - P_j^\\beta (x) - \\varepsilon \\chi _j^{*\\beta }(x/\\varepsilon ) - \\bar{v}_{\\varepsilon ,j}^{*\\beta }(x) \\bigg )\\bigg |\\le C\\sqrt{\\varepsilon } + C\\frac{r^{2+\\sigma }}{\\varepsilon ^{1+\\sigma }},$ where $C$ depends on $d,m,\\mu ,\\sigma ,\\Omega ,A$ and $f$ ."
],
[
"A partition of unity",
"For simplicity of notation, throughout this section we will write $\\varkappa \\circ n(x)$ as $\\varkappa (x)$ if no ambiguity.",
"For a large class of smooth domains, such as domains of finite type addressed in this paper, it is reasonable to assume that $\\varkappa (\\cdot )^{-1} \\in L^{p,\\infty }(\\partial \\Omega ),$ for some fixed $p\\in (0,d-1]$ , where $d\\ge 2$ .",
"Define $\\gamma = \\frac{d-1}{p}.$ Obviously, $\\gamma \\ge 1$ .",
"In the next lemma, we will construct a Calderón-Zygmund-type decomposition adapted to the function $\\varkappa (x)$ .",
"Essentially this $L^\\infty $ -based decomposition is a modified version of [7] or a special case of $L^q$ -based decomposition in [21].",
"However, in our application for general domains, the $L^\\infty $ based decomposition is more flexible and convenient.",
"We mention that the partition of unity, provided by the next lemma, will play a crucial role in the analysis of oscillating Dirichlet problems.",
"As in [21], we first describe such construction in flat spaces.",
"Lemma 5.1 Let $F$ be a bounded non-negative function on some cube $Q_0\\subset \\mathbb {R}^{d-1}$ .",
"Let $\\tau >0$ be a small parameter.",
"Then there exists a finite sequence of dyadic cubes (obtained by bisecting $Q_0$ ) $\\mathcal {P}= \\mathcal {P}_\\tau = \\lbrace Q_j:j=1,2,\\cdots \\rbrace $ such that (i) The interiors of these cubes are disjoint.",
"(ii) $Q_0 = \\cup _j Q_j$ .",
"(iii) For each $Q_j$ , $\\Vert F\\Vert _{L^\\infty (6Q_j)} \\le \\frac{\\tau }{\\ell (Q_j)},$ and $\\Vert F\\Vert _{L^\\infty (6Q^+_j)} > \\frac{\\tau }{\\ell (Q^+_j)},$ where $Q_j^+$ is the parent of $Q_j$ .",
"(iv) If $\\text{dist}(Q_j,Q_k) = 0$ , then $\\frac{1}{2} \\ell (Q_j) \\le |Q_k| \\le 2 \\ell (Q_j).$ (v) There exists an absolute constant $C>0$ such that $\\#\\lbrace Q_j: \\ell (Q_j) \\ge \\lambda \\tau \\rbrace \\le C(\\lambda \\tau )^{-(d-1)} \\sigma (\\lbrace x\\in Q_0: F(x) \\le \\lambda ^{-1} \\rbrace ).$ Let $U$ be the set of all dyadic cubes in $Q_0$ such that $(\\ref {est_Fle})$ holds.",
"We say $Q$ is a maximal element of $U$ if $Q$ is not properly contained in any other cube in $U$ .",
"Let $\\mathcal {P}$ denote the set of all maximal elements of $U$ .",
"Clearly, by definition, the interiors of cubes in $\\mathcal {P}$ are disjoint and (REF ), (REF ) are satisfied for each $Q_j$ in $\\mathcal {P}$ .",
"To see (ii), it suffices to note that $F$ is bounded and hence (REF ) must be satisfied for sufficiently small cubes.",
"And (iv) follows by a similar argument as [20] which we will omit here.",
"Finally, to see (v), let $t = 2^{-k} \\ell (Q_0)$ be fixed.",
"We consider the number of cubes $Q_j\\in \\mathcal {P}$ with $\\ell (Q_j) = t$ .",
"It follows from (REF ) that $\\#\\lbrace Q_j: \\ell (Q_j) = t \\rbrace & = t^{1-d} \\sigma \\Big (\\bigcup _{\\ell (Q_j) = t} Q_j\\Big ) \\\\& \\le t^{1-d} \\sigma ( \\lbrace x\\in \\partial \\Omega : F(x) \\le t^{-1}\\tau \\rbrace ).$ Set $\\lambda = t \\tau ^{-1}$ , we obtain $\\#\\lbrace Q_j: \\ell (Q_j) = \\lambda \\tau \\rbrace \\le (\\lambda \\tau )^{-(d-1)} \\sigma ( \\lbrace x\\in \\partial \\Omega : F(x) \\le \\lambda ^{-1} \\rbrace ).$ Finally, replacing $\\lambda $ with $2^{k} \\lambda $ in the last inequality and summing up all $k\\ge 0$ , we obtain the desired estimate (REF ).",
"Let $F\\in L^\\infty (\\partial \\Omega )$ .",
"Following the lines of [21], we can perform a partition of unity on $\\partial \\Omega $ based on Lemma REF .",
"For readers' convenience, we will provide the outline of the construction.",
"Fix $x_0\\in \\partial \\Omega $ .",
"Let $r_0$ be sufficiently small so that $B(x_0,r_0)\\cap \\partial \\Omega $ is given by the local graph in a coordinate system.",
"Let $\\partial \\mathbb {H}_{n_0}^d(a)$ denote the tangent plane for $\\partial \\Omega $ at $x_0$ , where $n_0 = n(x_0)$ and $a = - x_0\\cdot n_0$ .",
"For $x\\in B(x_0,r_0)\\cap \\partial \\Omega $ , let $P(x) = x - ((x-x_0)\\cdot n_0) n_0$ denote its projection on $\\partial \\mathbb {H}_{n_0}^d(a)$ .",
"Note that $P$ is one-to-one from $B(x_0,r_0)\\cap \\partial \\Omega $ to its image in $\\partial \\mathbb {H}_{n_0}^d(a)$ and keeps length and measure comparable.",
"To construct a partition of unity on $B(x_0,r_0)\\cap \\partial \\Omega $ for $F$ , we use the inverse map $P^{-1}$ to lift a partition on the tangent plane, given in Lemma REF , to $\\partial \\Omega $ .",
"Precisely, for a fixed cube $Q_0$ in $\\partial \\mathbb {H}_{n_0}^d(a)$ such that $B(x_0,2r_0)\\cap \\partial \\Omega \\subset P^{-1}(Q_0) \\subset B(x_0,4r_0\\sqrt{d})\\cap \\partial \\Omega $ , we apply Lemma REF to $Q_0$ for function $F\\circ P^{-1}$ .",
"This generates a finite sequence of dyadic cubes $\\lbrace Q_j\\rbrace $ satisfying the properties in Lemma REF .",
"Let $x_j$ be the center of $Q_j$ and $r_j$ be the side length.",
"Let $\\widetilde{Q}_j = P^{-1}(Q_j)$ .",
"Then $\\widetilde{Q}_0 = P^{-1}(Q_0) = \\bigcup _j \\widetilde{Q}_j$ gives a decomposition of $\\widetilde{Q}_0$ .",
"Also, let $\\tilde{x}_j = P^{-1}(x_j)$ and $t\\widetilde{Q}_j = P^{-1}(tQ_j)$ .",
"Now for each $\\widetilde{Q}_j$ , we choose $\\eta _j \\in C_0^\\infty (\\mathbb {R}^d)$ such that $0\\le \\eta _j \\le 1, \\eta _j = 1$ on $\\widetilde{Q}_j, \\eta _j = 0$ on $\\partial \\Omega \\setminus 2\\widetilde{Q}_j$ , and $|\\nabla ^k \\eta _j| \\le Cr_j^{-k}$ .",
"Note that by Lemma REF (iv), $1\\le \\sum _j \\eta _j \\le C_0$ on $\\widetilde{Q}_0$ , where $C_0$ is a constant depending only on $d$ and $\\Omega $ .",
"Finally, we set $\\varphi _j(x) = \\frac{\\eta _j(x)}{\\sum _k \\eta _k(x)}.$ Clearly, $\\sum _j \\varphi _j = 1$ on $\\widetilde{Q}_0$ , $0\\le \\varphi _j \\le 1, \\varphi _j \\ge C_0^{-1}$ on $\\widetilde{Q}_j, \\varphi _j = 0$ on $\\partial \\Omega \\setminus 2\\widetilde{Q}_j$ , and $|\\nabla ^k \\varphi _j| \\le Cr_j^{-k}$ .",
"Further more, the properties (iii) and (v) in Lemma REF are preserved, i.e., $\\Vert F\\Vert _{L^\\infty (6\\widetilde{Q}_j)} \\le \\frac{\\tau }{r_j}, \\qquad \\Vert F\\Vert _{L^\\infty (18\\widetilde{Q}_j)} > \\frac{\\tau }{r_j},$ and $\\#\\lbrace \\widetilde{Q}_j: r_j \\ge \\lambda \\tau \\rbrace \\le C(\\lambda \\tau )^{-(d-1)} \\sigma (\\lbrace x\\in \\widetilde{Q}_0: F(x) \\le \\lambda ^{-1} \\rbrace ).$ For our application in homogenization, we will apply the above decomposition to $F = \\varkappa ^{1/\\gamma }$ , where $\\gamma $ is defined in (REF ).",
"Let $\\lbrace \\widetilde{Q}_j: j=1,2,\\cdots \\rbrace $ be the generated cubes on $\\partial \\Omega $ and other notations are also kept as before.",
"By (REF ), for each $\\widetilde{Q}_j$ , there exists $z_j \\in 18\\widetilde{Q}_j$ such that $\\varkappa (z_j) > \\Big ( \\frac{\\tau }{r_j} \\Big )^\\gamma .$ Lemma 5.2 There exists some $C>0$ such that for each $j$ , $\\tau \\le r_j \\le C \\sqrt{\\tau }.$ Note that $\\varkappa \\le 1$ .",
"This implies that $r_j \\ge \\tau $ , due to (REF ).",
"To see the other direction, we first claim that for any $0<q<p$ , $x\\in \\partial \\Omega $ and $0<r<\\text{diam}(\\Omega )$ , $\\bigg ( _{B(x,r)\\cap \\partial \\Omega } \\varkappa ^{-q} \\bigg )^{1/q} \\le \\frac{C}{r^\\gamma },$ where $C$ depends only on $d,q,\\gamma $ and $\\Omega $ .",
"This claim is an extension of [21], whose proof is almost the same.",
"We omit the details here.",
"Now by (REF ) and (REF ), we have $1 \\le \\bigg ( _{6\\widetilde{Q}_j} \\varkappa ^{-q} \\bigg )^{1/q} \\Vert \\varkappa \\Vert _{L^\\infty (6\\widetilde{Q}_j)} \\le C r_j^{-\\gamma } \\Big ( \\frac{\\tau }{r_j} \\Big )^{\\gamma },$ which implies $r_j \\le C\\sqrt{\\tau }$ .",
"The following lemma is the same as [21] which will be useful to us.",
"We give a simpler proof here based on (REF ).",
"Lemma 5.3 Let $0<\\alpha <d-1$ , then $\\sum _{j} r_j^{d-1+\\alpha } \\le C\\tau ^\\alpha ,$ where $C$ depends only on $\\alpha , \\gamma , d$ and $\\Omega $ .",
"It follows from (REF ) that $\\sum _j r_j^{d-1+\\alpha } & \\le \\sum _k \\sum _{2^{k-1}\\tau \\le r_j < 2^k \\tau } (2^k \\tau )^{d-1+\\alpha } \\\\& \\le C \\int _0^\\infty (\\lambda \\tau )^{\\alpha } \\lambda ^{-1} \\sigma \\lbrace x\\in \\widetilde{Q}_0: \\varkappa ^{-1/\\gamma }(x) > \\lambda \\rbrace d\\lambda \\\\& = C \\tau ^{\\alpha } \\int _{\\partial \\Omega } \\varkappa ^{-\\alpha /\\gamma } d\\sigma \\\\& \\le C \\tau ^{\\alpha },$ for any $\\alpha <d-1$ ."
],
[
"Proof of main theorem",
"We first prove Theorem REF and then Theorem REF follows readily from Proposition REF .",
"The line of argument is similar to [21], [7].",
"Due to Lemma REF , it is sufficient to estimate $\\Vert \\tilde{u}_\\varepsilon - u_0\\Vert _{L^2(\\Omega )}$ , where $\\tilde{u}_\\varepsilon $ and $u_0$ are defined by $\\tilde{u}_\\varepsilon ^\\alpha (x) = \\int _{\\partial \\Omega } P_{\\Omega }^{\\alpha \\gamma }(x,y) \\omega _\\varepsilon ^{\\gamma \\beta }(y) f^\\beta (y,y/\\varepsilon )d\\sigma (y)$ and $u_0^\\alpha (x) = \\int _{\\partial \\Omega } P_{\\Omega }^{\\alpha \\gamma }(x,y) \\bar{f}^\\gamma (y) d\\sigma (y).$ Now we need to find an explicit expression for the homogenized data $\\bar{f}$ .",
"Roughly speaking, the homogenized data $\\bar{f}$ in (REF ) should be the weak limit of $\\omega _\\varepsilon (y) f (y/\\varepsilon )$ as $\\varepsilon \\rightarrow 0$ .",
"By (REF ) and (REF ), for $y\\in B(x_0,r)\\cap \\partial \\Omega $ , one has $\\begin{aligned}&\\omega _\\varepsilon ^{\\gamma \\beta }(y) f^\\beta (y/\\varepsilon )\\\\& \\quad = h^{\\gamma \\nu }(y) \\cdot n_{\\ell }(y) \\frac{\\partial }{\\partial y_\\ell } [ P_k^{\\rho \\nu }(y) + \\varepsilon \\chi _k^{*\\rho \\nu }(y/\\varepsilon ) + \\bar{v}^{*\\rho \\nu ,x_0}_{\\varepsilon ,k}(y) ] n_k(y) \\cdot a_{ij}^{\\rho \\beta }(y/\\varepsilon ) n_i(y)n_j(y) f^\\beta (y,y/\\varepsilon ) \\\\& \\qquad + \\text{Error terms}.\\end{aligned}$ Note that $\\bar{v}_\\varepsilon ^{*,x_0}(y)$ is given in (REF ) which depends also on $x_0$ .",
"For a fixed $y \\in \\partial \\Omega $ , in view of the quantitative ergodic theorem [7], we know that $\\omega _\\varepsilon (y) f (y/\\varepsilon )$ converges to its average on the tangent plane $\\mathbb {H}_{n}^d(a)$ at $y$ , where $n = n(y)$ .",
"The only unclear term in (REF ) is $n\\cdot \\nabla \\bar{v}_{\\varepsilon ,k}^{*\\nu ,x_0}$ .",
"Actually, in view of (REF ), for $z \\in \\mathbb {H}_{n}^d(a)$ , one has $n\\cdot \\nabla \\bar{v}_{\\varepsilon ,k}^{*\\nu ,x_0}(z) = n\\cdot (1-n\\otimes n, -n) \\Bigg ( \\begin{aligned}\\nabla _\\theta \\\\ \\partial _t \\;\\end{aligned}\\Bigg ) V_k^{*\\nu ,x_0}\\Big ( \\frac{z}{\\varepsilon },0 \\Big ) = -\\partial _t V_k^{*\\nu ,x_0}\\Big ( \\frac{z}{\\varepsilon },0 \\Big ).$ Note that $V_k^{*,x_0}(\\theta ,t)$ is 1-periodic in $\\theta $ .",
"As a consequence, without justification, we can define the homogenized boundary data as follows: $\\begin{aligned}&\\bar{f}^\\gamma (y) \\\\& = h^{\\gamma \\nu }(y) \\int _{d} [\\delta ^{\\rho \\nu } + n(y) \\cdot \\nabla \\chi ^{*\\rho \\nu }(\\theta )\\cdot n(y) - \\partial _t V^{*\\rho \\nu ,y}(\\theta ,0) \\cdot n(y)] n_i(y)n_j(y)a_{ij}^{\\rho \\beta }(\\theta ) f^\\beta (y,\\theta ) d\\theta \\end{aligned}$ Remark 6.1 If the coefficient matrix $A = (a_{ij}^{\\alpha \\beta })$ is constant (or divergence free), then $\\chi ^* = 0$ and hence $V^* = 0$ in (REF ).",
"Also in this case, one has $\\widehat{A} = A$ .",
"By the definition of $h$ , this implies that $h^{\\gamma \\nu } \\delta ^{\\rho \\nu } n_i n_j a_{ij}^{\\rho \\beta } = \\delta ^{\\gamma \\beta }$ .",
"As a result, (REF ) is reduced to $\\bar{f}(y) = \\int _{d} f(y,\\theta ) d\\theta .$ This exactly coincides with the homogenized boundary data defined in Theorem REF for Dirichlet problems with constant coefficients.",
"Proposition 6.2 Let $x,y\\in \\partial \\Omega $ and $|x-y|<r_0$ .",
"Suppose that $n(x),n(y)$ satisfies the Diophantine condition with constant $\\varkappa (x)$ and $\\varkappa (y)$ respectively.",
"Let $\\bar{f}$ be defined by (REF ).",
"Then (i) For any $\\sigma \\in (0,1)$ , $|\\bar{f}(x) - \\bar{f}(y)| \\le C\\bigg ( \\frac{|x-y|^2}{\\varkappa ^{2+\\sigma }} + \\frac{|x-y|}{\\varkappa ^{1+\\sigma }}\\bigg ) \\sup _{z\\in d} \\Vert f(\\cdot ,z)\\Vert _{C^1(\\partial \\Omega )}.$ where $\\varkappa = \\varkappa (x) \\vee \\varkappa (y)$ and $C$ depends only on $d,m,\\sigma ,\\Omega $ and $A$ .",
"(ii) For any $0<q<q^* = (d-1)/(2\\gamma -1)$ , one has $\\bar{f} \\in W^{1,q}\\cap L^\\infty (\\partial \\Omega ).$ Part (i) of the last proposition is taken from [21], which actually holds for Dirichlet problem as well and follows from Theorem REF (iii).",
"The proof of part (ii) is similar as [21] with an obvious modification.",
"Note that the convexity is not necessary in the proofs.",
"The rest of the proof is devoted to estimating $\\Vert u_\\varepsilon - u_0\\Vert _{L^2(\\Omega )}$ .",
"To begin with, we perform a partition of unity on $\\partial \\Omega $ and restrict ourself on $B(x_0,r_0) \\cap \\partial \\Omega $ for some $x_0$ and $r_0>0$ sufficiently small.",
"So without any loss of generality, we may assume $\\text{supp}(f(\\cdot ,y)) \\subset B(x_0,r_0)$ for any $y\\in d$ .",
"Then we construct another partition of unity on $B(x_0,r_0) \\cap \\partial \\Omega $ adapted to $F = \\varkappa ^{1/\\gamma }$ , by the method described in the last section, with $\\tau = \\varepsilon ^{s}, $ for some constant $s\\in [1/2,1]$ , which will be properly selected by optimizing several errors.",
"Thus, there exist a finite sequence of $\\lbrace \\varphi _j \\rbrace $ of $C_0^\\infty $ positive functions in $\\mathbb {R}^d$ and a finite of sequence of surface cubes $\\lbrace \\widetilde{Q}_j\\rbrace $ on $\\partial \\Omega $ , such that $\\sum _j \\varphi _j = 1$ on $B(x_0,2r_0)\\cap \\partial \\Omega $ .",
"Note that $\\varphi _j$ is supported in $2\\widetilde{Q}_j$ and $|\\nabla ^k \\varphi _j| \\le Cr_j^{-k}$ , where $r_j$ is the side length of $\\widetilde{Q}_j$ as before.",
"Note that $\\tilde{x}_j$ is the center of $\\widetilde{Q}_j$ .",
"Let $\\Gamma _\\varepsilon $ denote a boundary layer $\\Gamma _\\varepsilon = \\Omega \\cap \\bigg ( \\bigcup _{j} B(\\tilde{x}_j, Cr_j) \\bigg )$ and $D_\\varepsilon = \\Omega \\setminus \\Gamma _\\varepsilon $ .",
"By Lemma REF , $|\\Gamma _\\varepsilon | \\le \\sum _j |B(\\tilde{x}_j,Cr_j)| \\le C \\sum _j r_j^d \\le C\\tau = C\\varepsilon ^{s}.$ Thus for any $q>0$ , $\\int _{\\Gamma _\\varepsilon } |u_\\varepsilon - u_0|^q \\le C\\varepsilon ^{s},$ where we have used the boundedness of $u_\\varepsilon $ and $u_0$ .",
"To deal with the $L^q$ norm of $u_\\varepsilon - u_0$ on $D_\\varepsilon $ , we introduce a function (see [21]) $\\Theta _t(x) = \\sum _j \\frac{r_j^{d-1+t}}{|x - \\tilde{x}_j|^{d-1}},$ where $0\\le t < d-1$ .",
"Lemma 6.3 Let $\\Theta _t(x)$ be defined by (REF ).",
"Then if $q>0$ and $0\\le qt < d-1$ , $\\int _{D_\\varepsilon } (\\Theta _t(x))^q dx \\le C \\tau ^{qt}.$ The original lemma in [21] was proved for $q\\ge 1$ .",
"It follows trivially from Hölder's inequality that the lemma holds also for $0<q< 1$ , which will also be useful for us.",
"This lemma will play a key role and be used repeatedly in the following context.",
"As in [7], [21], we split $\\tilde{u}_\\varepsilon - u_0$ into five parts $\\begin{aligned}\\tilde{u}_\\varepsilon (x) - u_0(x) & = \\int _{\\partial \\Omega } P_{\\Omega }(x,y) \\omega _\\varepsilon (y) f(y,y/\\varepsilon )d\\sigma (y) - \\int _{\\partial \\Omega } P_{\\Omega }(x,y) \\bar{f}(y) d\\sigma (y)\\\\& = I_1 + I_2 + I_3 + I_4 + I_5,\\end{aligned}$ where $I_k, 1\\le k \\le 5,$ will be defined below and handled separately.",
"We point out in advance that estimates for $I_3$ and $I_4$ essentially distinguish from the case of strictly convex domains and need more careful calculations.",
"Let $\\delta >0$ be an arbitrarily small exponent that might differ in each occurrence.",
"Estimate of $I_1$ : Let $\\begin{aligned}I_1 =& \\int _{\\partial \\Omega } P_{\\Omega }^{\\alpha \\gamma }(x,y) \\omega _\\varepsilon ^{\\gamma \\beta }(y) f^\\beta (y,y/\\varepsilon )d\\sigma (y) \\\\&\\qquad - \\sum _j \\int _{\\partial \\Omega } \\varphi _j(y)P_{\\Omega }^{\\alpha \\gamma }(x,y) \\tilde{\\omega }_{\\varepsilon }^{\\gamma \\beta ,z_j}(y) f^\\beta (y,y/\\varepsilon )d\\sigma (y),\\end{aligned}$ where $\\tilde{\\omega }_{\\varepsilon }^{\\gamma \\beta ,z_j}(y) = h^{\\gamma \\nu }(y) n_{\\ell }(y) \\frac{\\partial }{\\partial y_\\ell } \\Big [P_k^{\\rho \\nu }(y) + \\varepsilon \\chi _k^{*\\rho \\nu }(y/\\varepsilon ) + \\bar{v}^{*\\rho \\nu ,z_j}_{\\varepsilon ,k}(y)\\Big ] n_k(y) a_{im}^{\\rho \\beta }(y/\\varepsilon ) n_i(y)n_m(y),$ and $z_j$ 's are specially selected as in (REF ).",
"Note that $I_1$ comes from the error terms in (REF ), which by (REF ) is bounded by $C \\sum _j \\int _{\\partial \\Omega } \\varphi _j(y) |P_{\\Omega }(x,y)| \\bigg ( \\sqrt{\\varepsilon } + \\frac{r_j^{2+\\sigma }}{\\varepsilon ^{1+\\sigma }}\\wedge 1 \\bigg ) d\\sigma (y) = R_1 + R_2,$ for any $\\sigma \\in (0,1)$ .",
"Observe that $R_1 \\le C\\sqrt{\\varepsilon } \\int _{\\partial \\Omega } |P_{\\Omega }(x,y)| \\le C\\sqrt{\\varepsilon }.$ For $R_2$ , using $|P_{\\Omega }(x,y)| \\le C|x-y|^{1-d}$ and $|x-y| \\approx |x - \\tilde{x}_j|$ for $x\\in D_\\varepsilon , y \\in B(\\tilde{x}_j,Cr_j)$ , we have $R_2 = C \\sum _j \\int _{\\partial \\Omega } \\varphi _j(y) |P_{\\Omega }(x,y)| \\bigg ( \\frac{r_j^{2+\\sigma }}{\\varepsilon ^{1+\\sigma }}\\wedge 1 \\bigg ) d\\sigma (y)\\le C \\varepsilon ^{-1-\\sigma } \\sum _j \\frac{r_j^{2+\\sigma +d-1}}{|x - \\tilde{x}_j|^{d-1}}$ Now we estimate $R_2$ by Lemma REF in two separate cases.",
"If $2(2+\\sigma )<d-1$ , then we apply Lemma REF directly with $q = 2$ and obtain $\\int _{D_\\varepsilon } |R_2(x)|^2 dx \\le C \\varepsilon ^{-2(1+\\sigma )} \\tau ^{2(2+\\sigma )} \\le C\\varepsilon ^{4(s - \\frac{1}{2}) -\\delta },$ where we have used $\\tau = \\varepsilon ^s$ and chosen $\\sigma $ sufficiently small.",
"Otherwise, we choose suitable $q<2$ such that $q(2+\\sigma ) = d-1-\\sigma <d-1$ and then apply Lemma REF $\\int _{D_\\varepsilon } |R_2(x)|^q dx \\le C \\varepsilon ^{-q(1+\\sigma )} \\tau ^{q(2+\\sigma )} \\le C\\varepsilon ^{(s - \\frac{1}{2})(d-1)-\\delta },$ where again, $\\sigma $ is chosen sufficiently small.",
"Clearly, (REF ) also implies $|R_2| \\le C$ .",
"Thus, a simple interpolation leads to $\\int _{D_\\varepsilon } |R_2(x)|^2 dx \\le C\\varepsilon ^{(s - \\frac{1}{2} )(d-1)-\\delta }.$ Combining (REF ), (REF ) and (REF ), we obtain $\\int _{D_\\varepsilon } |I_1(x)|^2 dx \\le C \\varepsilon ^{1 \\wedge 4(s - \\frac{1}{2}) \\wedge (d-1)(s - \\frac{1}{2}) -\\delta }.$ Estimate of $I_2$ : Set $\\begin{aligned}I_2 &= \\sum _j \\int _{\\partial \\Omega } \\varphi _j(y)P_{\\Omega }^{\\alpha \\gamma }(x,y) \\tilde{\\omega }_{\\varepsilon }^{\\gamma \\beta ,z_j}(y) f^\\beta (y,y/\\varepsilon )d\\sigma (y)\\\\& \\qquad - \\sum _j \\int _{\\partial \\mathbb {H}^d_j} \\varphi _j(P_j^{-1}(y))P_{\\Omega }^{\\alpha \\gamma }(x,P_j^{-1}(y)) \\tilde{\\omega }_{\\varepsilon }^{\\gamma \\beta ,z_j}(y) f^\\beta (z_j,y/\\varepsilon )d\\sigma (y)\\end{aligned}$ where $\\partial \\mathbb {H}^d_j$ denotes the tangent plane for $\\partial \\Omega $ at $z_j$ and $P^{-1}_j$ is the inverse of the projection map from $B(z_j,Cr_j) \\cap \\partial \\Omega $ to $\\partial \\mathbb {H}^d_j$ .",
"We clarify that in (REF ), $n(y)$ is the outer normal of $y \\in \\partial \\Omega $ .",
"But in the second term of (REF ), $y$ needs to belong to $\\partial \\mathbb {H}^d_j$ and hence we need to update $n(y) = n(z_j)$ for all $y\\in \\partial \\mathbb {H}^d_j$ .",
"This modification leads to some harmless errors bounded by $Cr_j \\le Cr^2_j/\\varepsilon $ .",
"Then, for the same reason as the term $T_2$ in [7] or $I_2$ in [21], we are able to bound $I_2$ by $|I_2| \\le C\\varepsilon ^{-1} \\sum _j \\frac{r_j^{2+d-1}}{|x - \\tilde{x}_j|^{d-1}}.$ Similar as (REF ), we estimate this in two cases and obtain $\\int _{D_\\varepsilon } |I_2(x)|^2 dx \\le C\\varepsilon ^{4(s - \\frac{1}{2} ) \\wedge (d-1)(s - \\frac{1}{2}) -\\delta }.$ Estimate of $I_3$ : Set $\\begin{aligned}I_3 & = \\sum _j \\int _{\\partial \\mathbb {H}^d_j} \\varphi _j(P_j^{-1}(y))P_{\\Omega }^{\\alpha \\gamma }(x,P_j^{-1}(y)) \\tilde{\\omega }_{\\varepsilon }^{\\gamma \\beta ,z_j}(y) f^\\beta (z_j,y/\\varepsilon )d\\sigma (y) \\\\& \\qquad - \\sum _j \\int _{\\partial \\mathbb {H}^d_j} \\varphi _j(P_j^{-1}(y))P_{\\Omega }^{\\alpha \\gamma }(x,P_j^{-1}(y)) \\bar{f}^\\gamma (z_j) d\\sigma (y),\\end{aligned}$ where $\\bar{f}$ is defined in (REF ).",
"To estimate $I_3$ , we apply the quantitative ergodic theorem in [7].",
"As we have mention in the estimate of $I_2$ , the outer normal in the definition of $\\tilde{\\omega }_{\\varepsilon }^{\\gamma \\beta ,z_j}(y)$ is constant on $\\partial \\mathbb {H}^d_j$ with Diophantine constant $\\varkappa (z_j)$ , and therefore $\\tilde{\\omega }_{\\varepsilon }^{\\gamma \\beta ,z_j}(y)$ is nothing but a slice of some 1-periodic function in $\\mathbb {R}^d$ (see (REF )).",
"Note that by (REF ), $\\varkappa (z_j) > (\\tau /r_j)^{\\gamma }$ .",
"Then it follows from [7] that for any $N>0$ , $\\begin{aligned}|I_3| & \\le C\\sum _j \\Big ( \\frac{\\varepsilon r_j^\\gamma }{\\tau ^\\gamma } \\Big )^{N} \\int _{2\\widetilde{Q}_j} |\\nabla ^N(\\varphi _j(y) P_{\\Omega }(x,y))| d\\sigma (y) \\\\& \\le C\\sum _j \\Big ( \\frac{\\varepsilon r_j^\\gamma }{\\tau ^\\gamma } \\Big )^{N} \\sum _{k=0}^{N} \\frac{r_j^{d-1-N+k}}{|x - \\tilde{x}_j|^{d-1+k}} \\\\& \\le C \\varepsilon ^N \\tau ^{-\\gamma N} \\sum _j \\frac{r_j^{N(\\gamma - 1) +d-1}}{|x - \\tilde{x}_j|^{d-1}},\\end{aligned}$ where we have used $|\\nabla ^k \\varphi _j| \\le Cr_j^{-k}$ , $|\\nabla ^k P_{\\Omega }(x,y)| \\le C|x-y|^{1-d-k}$ and $r_j \\le C|x-\\tilde{x}_j| \\approx C|x-y|$ for all $x\\in D_\\varepsilon $ and $y\\in 2\\widetilde{Q}_j$ .",
"Now we choose $q\\le 2$ and $N\\ge 1$ properly so that $qN(\\gamma - 1) = d-1-\\delta <d-1$ and apply Lemma REF $\\int _{D_\\varepsilon } |I_3|^q \\le C\\varepsilon ^{qN} \\tau ^{-q\\gamma N} \\tau ^{qN(\\gamma - 1)} = C \\varepsilon ^{(1-s)(d-1)/(\\gamma -1) -\\delta }.$ This implies, as before, $\\int _{D_\\varepsilon } |I_3|^2 \\le C \\varepsilon ^{(1-s)(d-1)/(\\gamma -1)-\\delta }.$ Estimate of $I_4$: Set $\\begin{aligned}I_4 & = \\sum _j \\int _{\\partial \\mathbb {H}^d_j} \\varphi _j(P_j^{-1}(y))P_{\\Omega }^{\\alpha \\gamma }(x,P_j^{-1}(y)) \\bar{f}^\\gamma (z_j) d\\sigma (y) \\\\& \\qquad - \\sum _j \\int _{\\partial \\mathbb {H}^d_j} \\varphi _j(P_j^{-1}(y))P_{\\Omega }^{\\alpha \\gamma }(x,P_j^{-1}(y)) \\bar{f}^\\gamma (P_j^{-1}(y)) d\\sigma (y).\\end{aligned}$ The estimate for $I_4$ essentially relies on the regularity of homogenized data $\\bar{f}$ .",
"Indeed, by Proposition REF $|\\bar{f}(z_j) - \\bar{f}(P_j^{-1}(y))| & \\le C\\Bigg ( \\frac{r_j^2}{\\varkappa (z_j)^{2+\\sigma }} + \\frac{r_j}{\\varkappa (z_j)^{1+\\sigma }}\\Bigg ) \\\\& \\le C\\Bigg ( \\frac{r_j^{2+\\gamma (2+\\sigma )}}{\\tau ^{\\sigma (2+\\sigma )}} + \\frac{r_j^{1+\\gamma (1+\\sigma )}}{\\tau ^{\\sigma (1+\\sigma )}}\\Bigg ),$ where we also used $|z_j - P_j^{-1}(y)|\\le Cr_j$ .",
"This leads to a bound for $I_4$ $|I_4| \\le C \\tau ^{-\\gamma (2+\\sigma )} \\sum _j \\frac{r_j^{2+\\gamma (2+\\sigma ) + d-1}}{|x - x_j|^{d-1}} + C \\tau ^{-\\gamma (1+\\sigma )} \\sum _j \\frac{r_j^{1+\\gamma (1+\\sigma ) + d-1}}{|x - x_j|^{d-1}},$ of which we denote the terms on the right-hand side by $J_1$ and $J_2$ in proper order.",
"Using Lemma REF and a familiar argument as before, we are able to show $\\begin{aligned}\\int _{D_\\varepsilon } |I_4|^2 &\\le C \\int _{D_\\varepsilon } |J_1|^2 + C\\int _{D_\\varepsilon } |J_2|^2 \\\\& \\le C \\varepsilon ^{4s \\wedge s(d-1)/(1+\\gamma )-\\delta } + C\\varepsilon ^{2s \\wedge s(d-1)/(1+\\gamma )-\\delta } \\\\& \\le C\\varepsilon ^{2s \\wedge s(d-1)/(1+\\gamma )-\\delta }.\\end{aligned}$ Estimate of $I_5$: Finally, let $\\begin{aligned}I_5 & = \\sum _j \\int _{\\partial \\mathbb {H}^d_j} \\varphi _j(P_j^{-1}(y))P_{\\Omega }^{\\alpha \\gamma }(x,P_j^{-1}(y)) \\bar{f}^\\gamma (P_j^{-1}(y)) d\\sigma (y) \\\\&\\qquad - \\int _{\\partial \\Omega } P_{\\Omega }(x,y) \\bar{f}(y) d\\sigma (y).\\end{aligned}$ A change of variables gives $|I_5| \\le C \\sum _j \\frac{r_j^{1+d-1}}{|x - \\tilde{x}_j|^{d-1}}.$ Then by Lemma REF and a familiar argument, we obtain $\\int _{D_\\varepsilon } |I_5|^2 \\le C \\varepsilon ^{2s \\wedge s(d-1)-\\delta }.$ Now it suffices to choose $s \\in [1/2,1]$ properly to maximize the exponents for the bounds of $I_k$ 's, as well as (REF ).",
"To simplify, we note that the bound of $I_5$ is controlled by $I_4$ and the exponent $2s$ in the bound of $I_4$ can be ignored since $2s\\ge 1$ .",
"As a result, it is sufficient to maximize $\\alpha ^* = \\max _{s\\in [1/2,1]} \\bigg [ s \\wedge 4(s - \\frac{1}{2}) \\wedge (d-1)(s - \\frac{1}{2})\\wedge \\frac{(1-s)(d-1)}{\\gamma -1} \\wedge \\frac{s(d-1)}{1+\\gamma }\\bigg ].$ It is easy to see that $\\alpha ^*$ is well-defined and $0 <\\alpha ^* \\le 1$ , if $\\gamma >1$ .",
"If $\\gamma = 1$ , i.e., $p = d-1$ , we should replace (REF ) by $\\alpha ^* = \\max _{s\\in [1/2,1]} \\bigg [ s \\wedge 4(s - \\frac{1}{2}) \\wedge (d-1)(s - \\frac{1}{2} ) \\wedge \\frac{s(d-1)}{2}\\bigg ],$ since the term involving $\\gamma -1$ is positive infinity as long as $s \\ne 1$ .",
"Note that (REF ) is an increasing function of $s$ and thus the maximum is attained as $s$ approaching 1.",
"Thus in this case, $\\alpha ^* = 1 \\wedge \\frac{d-1}{2} = 1\\wedge \\frac{p}{2}.$ By Proposition REF , this is exactly the case of strictly convex domains and (REF ) conincides with (REF ) as expected.",
"Therefore, we have shown that $\\int _{\\Omega } |\\tilde{u}_\\varepsilon - u_0|^2 \\le C\\varepsilon ^{\\alpha ^* - \\delta },$ for arbitrarily small $\\delta >0$ , where $\\alpha ^*$ is given by (REF ) if $p<d-1$ and by (REF ) if $p = d-1$ .",
"This, together with Lemma REF and Proposition REF (ii), ends the proof of Theorem REF .",
"Remark 6.4 Note that the exponent $\\alpha ^*$ for $\\gamma > 1$ in (REF ) can be computed precisely by solving a linear programming problem.",
"However, for lower dimensional cases ($d\\le 5$ ), we can determine $\\alpha ^*$ easily.",
"And for higher dimensional cases , it is not hard to find a lower bound for $\\alpha ^*$ .",
"These will be treated separately in the following.",
"Case 1: $d=2,3$ .",
"Note that in this situation, the second term and the last term in the brackets of (REF ) can be ignored since $4\\ge d-1$ and $d-1\\le 1+\\gamma $ .",
"For either $d=2$ or $d=3$ , the maximum is attained by setting $(d-1)(s - \\frac{1}{2})= \\frac{(1-s)(d-1)}{\\gamma -1},$ which gives $s = (1+\\gamma )/(2\\gamma )$ .",
"Substituting this back we obtain that $\\alpha ^* = \\frac{d-1}{2\\gamma } = \\frac{p}{2}.$ Case 2: $d = 4,5$ .",
"In this situation, only the second term in the brackets of (REF ) can be ignored.",
"And we need to consider two subcases.",
"If $d-1\\le \\gamma +1$ , i.e., $\\gamma \\ge d-2$ , then $\\alpha ^* = \\max _{s\\in [1/2,1]} \\bigg [ (d-1)(s - \\frac{1}{2})\\wedge \\frac{(1-s)(d-1)}{\\gamma -1} \\wedge \\frac{s(d-1)}{1+\\gamma }\\bigg ] = \\frac{p}{2},$ since all the three terms are equal when $s = (1+\\gamma )/(2\\gamma )$ .",
"Now if $1\\le \\gamma < d-2$ , $\\alpha ^* = \\max _{s\\in [1/2,1]} \\bigg [s \\wedge (d-1)(s - \\frac{1}{2})\\wedge \\frac{(1-s)(d-1)}{\\gamma -1}\\bigg ].$ In this subcase, it is not hard to verify the intersections of the graph for three corresponding linear functions and obtain the maximum point by solving $s = \\frac{(1-s)(d-1)}{\\gamma -1}.$ This gives $\\alpha ^* = s = \\frac{d-1}{\\gamma + d-2} = \\frac{(d-1)p}{d-1+(d-2)p}.$ Combining the two sub-cases together, we have for both $d = 4$ and 5, $\\alpha ^* = \\frac{p}{2} \\wedge \\frac{(d-1)p}{d-1+(d-2)p}.$ Case 3: $d>5$ .",
"In view of Case 1 and Case 2, it is natural to pick $s = (1+\\gamma )/(2\\gamma )$ , which actually optimizes the last three terms of (REF ).",
"Then we have $\\alpha ^* \\ge \\frac{1+\\gamma }{2\\gamma } \\wedge \\frac{4}{2\\gamma } \\wedge \\frac{d-1}{2\\gamma } = \\frac{p+d-1}{2(d-1)} \\wedge \\frac{2p}{d-1} \\wedge \\frac{p}{2}.$ If $\\Omega $ is a smooth compact domain of type $k$ , then Proposition REF claims that $\\varkappa (\\cdot )^{-1} \\in L^{p,\\infty }(\\partial \\Omega ,d\\sigma )$ with $p = 1/(k-1)$ .",
"Then Theorem REF follows readily from Theorem REF .",
"Remark 6.5 For the domains of finite type, the rate of convergence in Theorem REF is not nearly close to the result of Theorem REF , except for $k=2$ and lower dimensions.",
"Actually, if $2\\le d\\le 5$ and $\\partial \\Omega $ is of type $k$ , by Proposition REF , $\\varkappa (n(\\cdot ))^{-1} \\in L^{p,\\infty }$ with $p = 1/(k-1)$ .",
"Then applying Theorem REF and Remark REF , we have $\\Vert u_\\varepsilon - u_0\\Vert _{L^2(\\Omega )} \\le C\\varepsilon ^{\\frac{1}{4(k-1)} - \\delta }.$ The only case coincides with Theorem REF is $k =2$ .",
"We end this paper by stating an application of Theorem REF concerning the higher order convergence rate for non-oscillating Dirichlet boundary value problem.",
"Theorem 6.6 Let $A$ and $\\Omega $ be the same as Theorem REF .",
"Assume that $u_\\varepsilon $ is the solution of $\\left\\lbrace \\begin{aligned}\\mathcal {L}_\\varepsilon u_\\varepsilon (x) &= 0 &\\quad & \\text{in } \\Omega , \\\\u_\\varepsilon (x) &= g(x) &\\quad & \\text{on } \\partial \\Omega ,\\end{aligned}\\right.$ where $g$ is smooth.",
"Let $u_0$ be the solution of the homogenized problem of (REF ).",
"Then there exist a unique function $v^{bl}$ independent of $\\varepsilon $ such that $\\Vert u_\\varepsilon (x) - u_0(x) - \\varepsilon \\chi (x/\\varepsilon ) \\nabla u_0(x) - \\varepsilon v^{\\text{bl}}(x)\\Vert _{L^2(\\Omega )} \\le C\\varepsilon ^{1+\\alpha ^*-\\delta },$ for any $\\delta >0$ , where $\\alpha ^*$ is given by (REF ).",
"Furthermore, the function $v^{\\text{bl}}$ is the solution of a non-oscillating Dirichlet problem $\\left\\lbrace \\begin{aligned}\\mathcal {L}_0 v^{\\text{bl}}(x) &= 0 &\\quad & \\text{in } \\Omega , \\\\v^{\\text{bl}}(x) &= g_*(x) &\\quad & \\text{on } \\partial \\Omega ,\\end{aligned}\\right.$ which is the homogenized problem of $\\left\\lbrace \\begin{aligned}\\mathcal {L}_\\varepsilon u_{1,\\varepsilon }^{\\text{bl}}(x) &= 0 &\\quad & \\text{in } \\Omega , \\\\u_{1,\\varepsilon }^{\\text{bl}}(x) &= -\\chi \\Big (\\frac{x}{\\varepsilon }\\Big ) \\nabla u_0(x) &\\quad & \\text{on } \\partial \\Omega .\\end{aligned}\\right.$ Note that (REF ) is a special case of (REF ).",
"The proof of Theorem REF is the same as [16].",
"From the improved $L^2$ convergence rate (REF ), one can also have an improved $H^1$ convergence rate $O(\\varepsilon ^{1+\\alpha ^*-\\delta })$ in any relatively compact subset of $\\Omega $ .",
"The details will be omitted here; see [16] for reference."
]
] | 1612.05383 |
[
[
"Recent progress on intrinsic charm"
],
[
"Abstract Over the past $\\sim\\!\\!",
"10$ years, the topic of the nucleon's nonperturbative or $\\textit{intrinsic}$ charm (IC) content has enjoyed something of a renaissance, largely motivated by theoretical developments involving quark modelers and PDF fitters.",
"In this talk I will briefly describe the importance of intrinsic charm to various issues in high-energy phenomenology, and survey recent progress in constraining its overall normalization and contribution to the momentum sum rule of the nucleon.",
"I end with the conclusion that progress on the side of calculation has now placed the onus on experiment to unambiguously resolve the proton's intrinsic charm component."
],
[
"Introduction",
"The nucleon's nonperturbative (i.e., intrinsic) component has been a largely unresolved issue for the past several decades following the idea's first incarnation in the seminal paper of Brodsky, Hoyer, Peterson, and Sakai (BHPS) [1], which I briefly review in Sec.",
"REF below.",
"Despite this long history, the problem of incontrovertibly establishing the existence of intrinsic charm (IC) empirically and determining an overall numerical magnitude for its contribution to the proton wave function has been something of a Gordian knot.",
"At the time of its proposal, the BHPS framework exploited recent developments in light-front field theory [2], [3] to formulate a simple picture based upon a Fock-state expansion of the nucleon wave function to include 5-quark states $|uudc\\bar{c}\\rangle $ involving charm not generated through the usual pQCD (or extrinsic) mechanism(s).",
"Despite considerable variation, models of intrinsic charm (IC) unavoidably involve some expression of this fundamental idea, and in this talk I survey recent progress developing calculations of this sort, as well as numerical work to constrain the range of possibilities for the size of IC that has proceeded apace."
],
[
"Modeling the proton's intrinsic charm",
"Various theoretical approaches have proliferated in the past several decades, involving a number of assumed mechanisms for generating the intrinsic component of the charm PDF.",
"Here I highlight several of these in increasing order of complexity."
],
[
"Scalar frameworks",
"As the original scalar framework formulated in the infinite momentum frame (IMF), the above-mentioned BHPS description treats the transition probability for a proton with mass $M$ to go through a transition $p \\rightarrow uudc\\bar{c}$ (or, indirectly, to an internal 5-quark state containing any heavy quark pair) in terms of an old-fashioned perturbation theory energy denominator expressible through the masses $m_i$ and momentum fractions $x_i$ of the constituents of the 5-quark state, $P \\Big ( p \\rightarrow uudc\\bar{c} \\Big )\\ \\sim \\ \\left[ M^2 - \\sum _{i=1}^5 \\frac{m_{\\perp i}^2}{x_i}\\right]^{-2}.$ In the expression above $x_i$ and $m_{\\perp i}$ are, respectively, the light-front fraction and transverse mass of the $i^{th}$ quark, the latter explicitly given by $m_{\\perp i}^2 \\equiv k_{\\perp i}^2 + m_i^2$ , and the indices 4 and 5 are taken to apply to the heavy quark pair ($c$ and $\\bar{c}$ ).",
"A special merit of this scheme is its simplicity: if one assumes the energy denominator of Eq.",
"(REF ) to be controlled by the charm quark mass (i.e., $m^2_c = m^2_{\\bar{c}} \\gg m^2_{u/d},\\ M^2$ ), the expression for the 5-quark probability can be integrated to obtain a compact form for the the $x$ dependence of the IC PDF: $P(x)\\ =\\ \\frac{N x^2}{2}\\left[ \\frac{(1-x)}{3}\\left( 1 + 10x + x^2 \\right)\\ +\\ 2 x\\, (1+x) \\ln (x)\\right],$ in which I have taken $x_5 \\rightarrow x$ , and the overall normalization $N$ is connected to the total intrinsic charm probability in the proton, subject to the constraint $\\int dx\\, c(x) = \\int dx\\, \\bar{c}(x)$ , which ensures the correct zero charm valence structure.",
"Building on the approach above, in a model-based analysis [4] of the nucleon's heavy-quark content and subsequent QCD global fit [5], Pumplin and (for the global analysis) collaborators considered a series of models for the Fock space wave function on the light-front for a proton to make a transition to a four quark plus one antiquark system, with the heavy $q\\bar{q}$ pair composed of either charm or bottom quarks.",
"This ansatz ultimately envisioned a simplified case wherein a spinless point particle of mass $m_0$ interacts with coupling strength $g$ to $N$ scalar particles having masses $m_1, m_2, \\ldots , m_N$ .",
"Pumplin then found the unintegrated light-front probability density to have the form [4] dP = g2(162)N-1(N-2)!",
"j=1N dxj ( 1-j=1N xj) s0ds (s-s0)N-2(s-m02)2 |F(s)|2, where the invariant mass is $s_0 = \\sum _{j=1}^N m_j^2/x_j$ , and a vertex function $F(s)$ must be stipulated to control behavior in the ultraviolet.",
"In particular, if the transverse momenta and the factors of $1/x_j$ appearing in Eq.",
"(REF ) are ignored and the charm mass is taken to be much larger than all other mass scales, one obtains the distribution prescribed by the BHPS model [1] after also assuming a pointlike vertex factor $F(s) = 1$ .",
"While these schemes are convenient and produce testable predictions for IC, a model with more physical interactions accounting for the structure of the charm spectrum is also desirable, and I sketch this idea in Sec.",
"REF now."
],
[
"Meson-baryon models",
"As higher-mass extensions of the pion-cloud picture of nucleon structure, meson-baryon models (MBMs) are a natural framework for studying the proton's interactions with external electromagnetic probes, relying on the advantageous properties of time-ordered perturbative theory (TOPT) and the convolution approach to compute corrections to the nucleon's hadronic tensor, $W^{\\mu \\nu }$ .",
"Figure: Diagrams relevant for the dominant contribution to the charm structure function inthe MBM of Ref. .",
"(Left) The TOPT diagram for the contribution of the dissociatedΛ c D * \\Lambda _c D^* state to the hadronic tensor of the proton.",
"(Right) An analogous diagram leadingto the charm quark distribution within a spin-1/21/2 baryon (here, the Λ c \\Lambda _c), c B (z)c_B(z),assuming a quark-diquark picture for the baryon's constituent substructure; such processes are calculatedin detail in Ref.",
".Physically, a MBM describes the nucleon's intrinsic charm content in a two-step ansatz formulated in terms of hadronic degrees of freedom as well as at quark level.",
"Moreover, unlike the BHPS formalism with $m_c = m_{\\bar{c}}$ in Sec.",
"REF , MBMs can readily produce experimentally testable asymmetries between the $c$ and $\\bar{c}$ distributions in the nucleon.",
"The basic goal of these models is the probability for the nucleon to spontaneously fluctuate into states involving an intermediate meson $M$ and baryon $B$ , according to $|N\\rangle \\ =\\ \\sqrt{Z_2}\\, \\left| N \\right.\\rangle _0\\ +\\ \\sum _{M,B} \\int \\!",
"dy\\, d^2{k}_\\perp \\,\\phi _{MB}(y,k^2_\\perp )\\,\\big | M(y,{k}_\\perp ); B(1-y,-{k}_\\perp ) \\big \\rangle ,$ in which $\\left| N \\right.\\rangle _0$ represents the undressed three-quark nucleon state, and $Z_2$ is an associated renormalization constant.",
"The quantity $\\phi _{MB}(y,k^2_\\perp )$ is an amplitude for the process whereby the nucleon reconfigures into an intermediate meson $M$ carrying a fraction $y$ of the proton's longitudinal momentum and transverse momentum ${k}_\\perp $ , and a baryon $B$ with longitudinal momentum fraction $1-y \\equiv \\bar{y}$ and transverse momentum $-{k}_\\perp $ .",
"The invariant mass squared $s_{MB}$ of this intermediate state appearing in the derivations below can then be expressed in the IMF by $s_{MB}(y,k^2_\\perp )\\ =\\ \\frac{k^2_\\perp + m_M^2}{y} + \\frac{k^2_\\perp + M_B^2}{1-y}\\ \\equiv \\ s\\, ,$ where the internal meson and baryon masses are respectively given by $m_M$ and $M_B$ .",
"Ultimately, the dominant mechanism determining the IC distributions in the MBM of Ref.",
"[6] originated with the reconfiguration of the proton into intermediate states consisting of spin-1 charmed mesons $D^* = \\bar{D}^{*0}$ or $D^{*-}$ and corresponding charm-containing baryons.",
"The associated probabilistic splitting function for this mode is related to the amplitude of Eq.",
"(REF ) by $f_{MB}(y) = \\int _0^\\infty d^2{k}_\\perp \\,|\\phi _{MB}(y,k^2_\\perp )|^2$ , and, due to the higher $N$ -$D^*$ -$\\Lambda _c$ spin interaction, arises from a linear combination of vector ($G_v$ ), tensor ($G_t$ ), and vector-tensor interference ($G_{vt}$ ) pieces.",
"Viz., fD* B(y) = TB 1162 dk2 y (1-y) |F(s)|2 (s - M2)2 [ g2 Gv(y,k2) + g f M Gvt(y,k2) + f2M2 Gt(y,k2) ] , where Gv(y,k) = - 6 M MB + 4(P k) (p k)mD2 + 2 P p , Gvt(y,k) = 4(M + MB)(P p - M MB) - 2mD2 [ MB (P k)2 - (M + MB)(P k)(p k) + M (p k)2 ] , Gt(y,k) = -(P p)2 + (M + MB)2 P p - M MB (M2 + MB2 + M MB) + 12mD2 [ (P p - M MB) [(P-p) k]2 - 2 (MB2 Pk - M2 pk) [(P-p) k] + 2 (P k) (p k) (2P p - MB2 - M2) ] , and, as depicted in the left panel of Fig.",
"REF , $p$ represents the 4-momentum of the interacting baryon (e.g., $\\Lambda ^+_c$ ), $T_B$ is an isospin factor, and the products $P \\cdot p$ , $P \\cdot k$ , $p \\cdot k$ can all be evaluated in terms of explicit TOPT expressions defined in the IMF as in the Appendices of Ref. [6].",
"The hadronic interaction strengths $g$ and $f$ of Eq.",
"(REF ) are fixed using SU$(4)$ quark model symmetry constraints á la standard Lippmann-Schwinger analyses [7].",
"A standard feature of this approach is the necessity of regulating the inevitable divergences that appear at large $k^2_\\perp $ — a fact which follows from the essential status of MBMs as loop corrections or dressings to the photon-nucleon vertex.",
"Typically, regularization is carried out phenomenologically, and implemented through a specific parametric choice for the relativistic vertex factor $F(s)$ appearing in Eq.",
"(REF ), for example.",
"The analysis of Ref.",
"[6] made use of a Gaussian with the form $F(s) = \\exp [-(s-M^2)/\\Lambda ^2]\\ ,$ in which $\\Lambda $ is a cutoff parameter fixed across the various meson-baryon modes involved in the model and determined by fitting $pp \\rightarrow \\Lambda _c + X$ hadroproduction data from ISR [8]; this yielded $\\Lambda = (3.0 \\pm 0.2)$ GeV, leading to the predicted model bands plotted in Fig.",
"REF below.",
"Using this general formalism, one can compute other intermediate spin-isospin combinations for the possible charm-containing meson-baryon states of Eq.",
"(REF ), and use an analogous framework embodied by the right panel of Fig.",
"REF to determine normalized distributions of (anti-)charm quarks within these hadronic states — e.g., the $c$ distribution within the $\\Lambda _c$ meson, $c_{\\Lambda }(z)$ , as a function of a quark momentum fraction $z$ .",
"Having assembled these various ingredients, the MBM specifies the nonperturbative IC distribution at the partonic threshold $Q^2 = m^2_c$ as an incoherent sum over various meson-baryon states: c(x) = M,B [ x1 dyy fMB(y) cM(xy) + x1 dyy fBM(y) cB(xy) ] , c(x) = B,M [ x1 dyy fBM(y) cB(xy) + x1 dyy fMB(y) cM(xy) ] .",
"Figure: (Left) Intrinsic contributions to the proton's charm PDF at the startingscale Q 0 2 =m c 2 Q^2_0 = m^2_c of QCD evolution under different assumed prescriptions forthe infrared behavior of the quark-hadron vertex that determines the distributionsc ¯ M (z)\\bar{c}_M(z) and c B (z)c_B(z).",
"(Right) The charm sector part of the proton structurefunction F 2 cc ¯ =4 x / 9c(x,Q 2 )+c ¯(x,Q 2 )F^{c\\bar{c}}_2 = \\big ( 4x \\big / 9 \\big ) \\left[ c(x,Q^2) + \\bar{c}(x,Q^2) \\right]at an evolved scale of Q 2 =60Q^2 = 60 GeV 2 ^2.",
"The two red points belong to the highest EMC bin〈Q 2 〉=60\\langle Q^2 \\rangle = 60 GeV 2 ^2 ; while these points overhang the predictions of pQCD,they are at the lower periphery of the range predicted by the MBM informed by hadroproductiondata.The expressions in Eqs.",
"(REF ) and (REF ) are the culmination of the MBM of Ref.",
"[6] and depend crucially upon assumed schemes to counter numerical poles in TOPT energy denominators that occur at infrared momenta in the quark-level amplitudes used to compute the distributions $\\bar{c}_M(z)$ and $c_B(z)$ .",
"As described at length in Ref.",
"[6], three main prescriptions were employed (a “confining” scheme to cancel the offending TOPT denominators, an “effective mass” approach in which the charm quark is taken to be sufficiently heavy as to avoid numerical poles, and a simple delta-function assumption for the quark distributions); after fixing UV regulators to hadroproduction data, starting-scale IC distributions such as those shown for $c(x)$ in the left panel of Fig.",
"REF were then obtained.",
"For the sake of the data comparisons shown in the right panel of Fig.",
"REF , these distributions were evolved according to pQCD to the empirical scale of the European Muon Collaboration (EMC) [9], which measured the charm sector contribution to the $F_2(x,Q^2)$ structure function of the proton via $\\mu $ -Fe DIS."
],
[
"A QCD global analyses of the charm PDF",
"The model-based treatment of IC in Sec.",
"REF served the purpose of enlightening possible physical mechanisms for generating nonperturbative charm in a manner that led to direct constraints from experimental information.",
"However, the ambiguity in the resulting magnitude of IC in the proton as suggested by the panels of Fig.",
"REF demanded a more systematic numerical approach.",
"The most comprehensive such method involves performing a QCD global analysis of the world's data explicitly including IC as a hypothesis.",
"This was recently undertaken in Ref.",
"[10] using the $\\mathcal {O}(\\alpha _s)$ formalism of Hoffmann and Moore [11] for the charm structure function and, for the DGLAP starting-scale IC itself, the parametric shapes computed in the analysis of Ref. [6].",
"In fact, in what follows the “confining” prescription corresponding to the central solid red curve in the LHS of Fig.",
"REF was used by default, and the commonly-used total momentum fraction was taken as a proxy for the overall normalization of IC in the proton: $\\langle x \\rangle _{\\rm IC}\\ \\equiv \\ \\int _0^1 dx\\, x \\left[ c(x) + \\bar{c}(x) \\right]\\ .$ QCD global analyses of this quantity (as well as of an analogous fraction for intrinsic bottom [12]) had been carried out by a number of groups over the years as exemplified by a recent CT14 calculation [13] that assessed several IC scenarios (IC$=0$ , as well as BHPS and a low $x$ -dominated “sealike” shape); not unlike the earlier work of Ref.",
"[5], this determination found considerable levels of IC could be tolerated by a global fit — in particular, CT14 discovered that IC as large as $\\langle x \\rangle _{\\rm IC} \\sim 2-3\\%$ could be accommodated within their framework, depending upon the specific IC scenario assumed.",
"Figure: Aside from the EMC data which were not included in the global fit shownhere, the full set of the JR14 analysis was first used to study the constraintsimposed by the world's data on the total IC fraction 〈x〉 IC \\langle x \\rangle _{\\rm IC}.",
"Asnoted in TABLE I of Ref.",
"and tabulated in the legend here, thisinvolved 4296 data points from 26 independent sets of measurements.",
"Moreover,the JR14 fitting technology incorporates a means of parametrizing theeffects of dynamical higher twist, target mass, and nuclear corrections, enabling substantiallyless restrictive kinematical cuts: Q 2 ≥1Q^2 \\ge 1 GeV 2 ^2 and W 2 ≥3.5W^2 \\ge 3.5 GeV 2 ^2.As a result, results were additionally constrained by important inputs from fixed-targetSLAC data on proton and deuteron targets (brown circles and blue triangles,respectively), which drove the very rapid growth in the χ 2 \\chi ^2 of the globalfit relative to χ 0 2 \\chi ^2_0 — the corresponding value at〈x〉 IC =0\\langle x \\rangle _{\\rm IC} = 0.Unlike previous analyses, however, our recent effort in Ref.",
"[10] relied upon an updated formalism [14] which accounted for various sub-leading $1/Q^2$ corrections (e.g., higher-twist effects and target mass corrections) and nuclear effects for DIS observables.",
"This enhancement enabled a description of information at much lower $W^2$ and $Q^2$ than typically allowed in most global fits, and we therefore leveraged this to perform our QCD global analysis of IC with less restrictive kinematical cuts on the included data sets ($Q^2 \\ge 1$ GeV$^2$ and $W^2 \\ge 3.5$ GeV$^2$ ).",
"In Fig.",
"REF , I summarize the contributions of the data sets included in the global analysis of Ref.",
"[10] to the total growth in the $\\chi ^2$ resulting from the fit.",
"From the parabolas of Fig.",
"REF , it is evident that some data sets (e.g., the H1 $F_2$ measurements corresponding to the blue-pentagon curve at bottom) do little to constrain $\\langle x \\rangle _{\\rm IC}$ ; rather, the bulk of the constraint to the proton's total IC (given by solid black disks) is driven by SLAC fixed-target information — on the deuteron (blue triangles) and proton (brown circles).",
"The pronounced sensitivity of these data sets to IC can be interpreted as arising indirectly from the tightened constraints provided by the SLAC data to the light-quark sector of the global fit, which in turn limits the flexibility of the fitting framework in tolerating larger magnitudes for $\\langle x \\rangle _{\\rm IC}$ .",
"This point is further driven home by the left panel of Fig.",
"REF , which separates the `SLAC' contributions to the $\\chi ^2$ profile from the `rest' of the data set summarized in Fig.",
"REF .",
"In the end, this global fit places quite stringent constraints on the magnitude of IC: $\\langle x \\rangle _{\\rm IC} < 0.1\\%$ at the $5\\sigma $ level.",
"Figure: Plots of the χ 2 \\chi ^2 growth profiles in the QCD globalanalysis of Ref.",
"for runs in which EMC data werenot included (left), and for global fits constrained by EMC(right).An additional motivation for the analysis of Ref.",
"[10], however, was the need to evaluate the suggestive measurements of EMC [9].",
"As has been pointed out numerous times in the literature, the highest $Q^2$ bins of the EMC data seem to provide some hint of the existence of IC, but before Ref.",
"[10] this information had not been treated in the context of a global analysis.",
"Ultimately, incorporating the EMC points into the data set summarized in Fig.",
"REF led to a slight preference for nonzero IC, with the EMC set alone favoring $\\langle x \\rangle _{\\rm IC} = 0.3-0.4\\%$ as indicated by the minimum in the green double-dotted curve on the RHS of Fig.",
"REF .",
"The modest preference for nonzero IC is diluted by the presence of other mitigating data sets, however, and the full global fit results in $\\langle x \\rangle _{\\rm IC} = 0.13 \\pm 0.04\\%$ — a significantly reduced magnitude relative to the results of earlier analyses like Ref. [13].",
"We also found the EMC points to be poorly fit (with $\\chi ^2/{\\rm datum} = 4.3$ ) and in some apparent tension with lower $Q^2$ points from HERA [15].",
"These issues necessitate further evaluation of the EMC set, and demand additional experimental measurement which might clarify the magnitude of IC without residual tension or ambiguity."
],
[
"Recent developments in IC phenomenology",
"Following the completion of the work in Ref.",
"[10], a number of intriguing developments in the phenomenology of IC have emerged, of which I highlight a couple of recent examples.",
"Partly motivated by the work of Refs.",
"[10], [16], Ball et al.",
"of the NNPDF Collaboration announced an independent global analysis [17] of the charm PDF in which the possibility of intrinsic charm was explicitly allowed via separate “Perturbative” and “Fitted” distributions which were then constrained through the neural networks-based methodology of NNPDF.",
"Moreover, like Refs.",
"[10], [16] this work also incorporated the provocative EMC data, confronting them with a global fitting technology that does not presuppose a particular model-derived shape for the IC distribution; the relaxation of this aspect of typical QCD global fits affords the NNPDF framework greater flexibility in describing the world's data, but can also result in distributions that are difficult to reconcile with the predictions of model-building.",
"This fact is evident in the especially hard shape for the “Fitted” charm distribution obtained in Fig.",
"3 of Ref.",
"[17], which corresponded to $\\langle x \\rangle _{\\rm IC} = 0.7 \\pm 0.3\\%$ , and demands further study.",
"Aside from direct measurements of the charm structure function, more oblique means of accessing the nucleon's nonperturbative charm content can be constructed.",
"An archetypal example of this sort of approach can be found in the precise observation of prompt neutrino production [18] in dedicated cosmic-ray experiments like IceCube [19].",
"The analysis in this latter reference demonstrates that IC in accord with the upper reaches of the model predictions of Ref.",
"[6] (which allows larger IC normalizations and preceded the more systematic global analysis of Ref.",
"[10]) might in principle be observable in the IceCube prompt neutrino spectrum; this suggest that such measurements could serve the role of an additional avenue to either observing or constraining the proton's nonperturbative charm."
],
[
"Conclusion and outlook",
"I have described the results of several closely related lines of investigation into the intrinsic charm problem.",
"Given that direct experimental information is relatively limited, much of this work has been constrained in its ability to recommend a robust value for, e.g., the total IC momentum fraction $\\langle x \\rangle _{\\rm IC}$ .",
"The present situation therefore seems to be one in which the reach of modeling and analyses of data is approaching a point of diminishing return, and additional empirical inputs would be invaluable.",
"For this purpose modern, direct measurements of $F^{c\\bar{c}}_2(x,Q^2)$ — especially at large $x$ and low/intermediate $Q^2$ along the lines of the original EMC data — could better constrain both models and global analyses and settle some of the questions related to the interpretation of the EMC set itself as raised in Refs.",
"[10], [16].",
"Measurements of this type might ideally be carried out at a future electron-ion collider [20] and be complemented by work at the proposed AFTER@CERN fixed-target $pp$ experiment [21], which would putatively operate at $\\sqrt{s} = 115$ GeV; this might similarly be the case for a suggested measurement of the forward production of $Z$ bosons in coincidence with charm-containing jets at LHCb [22], possibly exposing the behavior of the charm PDF in the critical valence region."
],
[
"Acknowledgements",
"I thank Wally Melnitchouk, Tim Londergan, and Pedro Jimenez-Delgado for discussions and collaboration on various portions of this work.",
"For additional useful conversations, I thank Stan Brodsky, Ranjan Laha, Gerald Miller, and Jean-Phillipe Lansberg.",
"This work was supported by the U.S. Department of Energy Office of Science, Office of Basic Energy Sciences program under Award Number DE-FG02-97ER-41014."
]
] | 1612.05686 |
[
[
"Testing the axion-conversion hypothesis of 3.5 keV emission with\n polarization"
],
[
"Abstract The recently measured 3.5 keV line in a number of galaxy clusters, the Andromeda galaxy (M31) and the Milky Way (MW) center can be well accounted for by a scenario in which dark matter decays to axion-like particles (ALPs) and subsequently convert to 3.5 keV photons in magnetic fields of galaxy clusters or galaxies.",
"We propose to test this hypothesis by performing X-ray polarization measurements.",
"Since ALPs can only couple to photons with polarization orientation parallel to magnetic field, we can confirm or reject this model by measuring the polarization of 3.5 keV line and comparing it to the orientation of magnetic field.",
"We discuss luminosity and polarization measurements for both galaxy cluster and spiral galaxy, and provide a general relation between polarization and galaxy inclination angle.",
"This effect is marginally detectable with X-ray polarimetry detectors currently under development, such as the enhanced X-ray Timing and Polarization (eXTP), the Imaging X-ray Polarimetry Explorer (IXPE) and the X-ray Imaging Polarimetry Explorer (XIPE).",
"The sensitivity can be further improved in the future with detectors of larger effective area or better energy resolutions."
],
[
"Testing the axion-conversion hypothesis of 3.5 keV emission with polarization Yan Gong Key Laboratory of Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Beijing 100012, China Xuelei Chen Key Laboratory of Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Beijing 100012, China University of Chinese Academy of Sciences, Beijing 100049, China Center of High Energy Physics, Peking University, Beijing 100871, China Hua Feng Department of Engineering Physics and Center for Astrophysics, Tsinghua University, Beijing 100084, China The recently measured 3.5 keV line in a number of galaxy clusters, the Andromeda galaxy (M31) and the Milky Way (MW) center can be well accounted for by a scenario in which dark matter decays to axion-like particles (ALPs) and subsequently convert to 3.5 keV photons in magnetic fields of galaxy clusters or galaxies.",
"We propose to test this hypothesis by performing X-ray polarization measurements.",
"Since ALPs can only couple to photons with polarization orientation parallel to magnetic field, we can confirm or reject this model by measuring the polarization of 3.5 keV line and comparing it to the orientation of magnetic field.",
"We discuss luminosity and polarization measurements for both galaxy cluster and spiral galaxy, and provide a general relation between polarization and galaxy inclination angle.",
"This effect is marginally detectable with X-ray polarimetry detectors currently under development, such as the enhanced X-ray Timing and Polarization (eXTP), the Imaging X-ray Polarimetry Explore (IXPE) and the X-ray Imaging Polarimetry Explorer (XIPE).",
"The sensitivity can be further improved in the future with detectors of larger effective area or better energy resolutions.",
"95.35.+d, 98.80.Cq, 98.56.Ne, 95.85.Nv Introduction.",
"A weak emission line at 3.5 keV is detected recently by XMM-Newton and $Chandra$ observations in a number of galaxy clusters, as well as in the M31, and possibly the MW center [1], [2], [3], though some detections are still debated [4], [5], [6].",
"The origin of this line is still unclear.",
"It might be an unidentified atomic line or lines emitted from the hot ionized intracluster medium [1], [7], [8].",
"Another interesting possibility is that it originates from the decay or annihilation of dark matter (such as axion-like particle (ALP), axinos and sterile neutrinos) to photons, (e.g.",
"[9], [10], [11], [12], [13]).",
"However, observations are somewhat puzzling and not entirely consistent with the dark matter direct-production-of-photons interpretation [1], [14].",
"The 3.5 keV emission from the Perseus cluster is much stronger than other clusters, but the dark matter decay model predicts that the signal should be proportional to dark matter quantity in a cluster.",
"Also, most of 3.5 keV emission comes from the cool cores of clusters, which have much smaller scales than dark matter halos.",
"These seem to suggest that the line is associated with an unknown astrophysical process in the cool cores of galaxy clusters or galaxies.",
"Recent works (e.g.",
"[14], [4]) proposed that this line is produced by dark matter decay but via an indirect process, ${\\rm DM}\\rightarrow a\\rightarrow \\gamma $ , i.e.",
"dark matter particles first decay to ALPs which have very small mass, and then subsequently the ALPs are converted to photons by interacting with local magnetic field.",
"For example, such decay may happen if the dark matter is a sterile neutrino [14], [16].",
"This model naturally explains why the line is stronger in clusters with cool cores: the conversion probability of ALPs to photons is proportional to $\\mathbf {B}_{\\perp }^2$ , where $\\mathbf {B_\\perp }$ is the magnetic field perpendicular to the relativistically moving ALP path.",
"In this scenario, the generation of the 3.5 keV X-ray photon is enhanced in regions with strong magnetic field, such as the central region of galaxy clusters with cool cores, or the central region and spiral arms of spiral galaxies such as M31 and MW [2], [3], [14].",
"If the 3.5 keV emission line is generated by this process, how can we confirm it?",
"Here we propose a test experiment by using the X-ray polarization.",
"Only photons with polarization in the direction of magnetic field can couple to ALPs.",
"If the 3.5 keV signal indeed originates from the ALP-photon conversion process, these photons should be polarized and the orientation of polarization would parallel the magnetic field in the region where they are produced.",
"For observations of nearby galaxies (e.g.",
"M31) and clusters (e.g.",
"Perseus, Coma and Centaurus), we can identify the 3.5 keV line emission regions, then compare the polarization of the 3.5 keV photons with the magnetic field in that region to check if they are aligned.",
"Other polarization mechanisms are relatively weak at this wavelength and cannot affect the axion-conversion signal.",
"For distant galaxies and clusters, the individual emission regions may not be resolvable, but the total strength, orientation and average degree of polarization from the entire cluster or galaxy can still be used.",
"Model.",
"In the presence of background magnetic field, the ALP field $a$ is coupled to electromagnetic field via the effective Lagrangian $\\mathcal {L}_{a\\gamma \\gamma } = g_{a\\gamma \\gamma }~ a ~{\\bf E}\\cdot {\\bf B} \\equiv \\frac{a}{M}{\\bf E}\\cdot {\\bf B} \\, ,$ where $g_{a\\gamma \\gamma }$ is coupling factor, and the suppression mass scale $M \\gtrsim 10^{10}{\\rm GeV}$ [15].",
"In the presence of a static magnetic field, this interaction term allows the conversion of the $a$ particle to a photon with its polarization parallel to $\\mathbf {B}$ .",
"For a single domain with homogenous magnetic field, the conversion probability for ALP with energy $\\omega $ is given in the linear approximation by [18], [17], $P_{a\\rightarrow \\gamma } = {\\rm sin}^2(2\\theta )\\,{\\rm sin}^2(\\frac{\\Delta }{{\\rm cos}\\, 2\\theta }).$ Here tan$\\,2\\theta =\\frac{2B_{\\perp }\\omega }{M m^2_{\\rm eff}}$ , and $\\Delta =\\frac{m^2_{\\rm eff}L}{4\\omega }$ , where $B_{\\perp }$ is the magnetic field perpendicular to the direction of motion of ALPs, $L$ is the size of the domain, and $m^2_{\\rm eff}=m^2_a-\\omega ^2_{\\rm pl}$ where $\\omega _{\\rm pl}=\\sqrt{4\\pi \\alpha n_e/m_e}$ and $n_e$ is the electron number density.",
"We also have $\\Delta /\\cos 2\\theta =\\Delta _{\\rm osc} L/2$ , where $\\Delta _{\\rm osc}^2=(\\Delta _{\\gamma }-\\Delta _a)^2+4\\Delta _{a\\gamma }^2$ , with $\\Delta _{\\gamma }=-\\omega _{\\rm pl}^2/2\\omega $ , $\\Delta _{a}=-m_a^2/2\\omega $ , and $\\Delta _{a\\gamma }=B_{\\perp }/2M$ .",
"When $2|\\Delta _{a\\gamma }|\\ll |\\Delta _{\\gamma }-\\Delta _a|$ , we find $\\rm cos\\,2\\theta \\sim 1$ , and Eq.",
"(REF ) is then reduced to the form used in the study of neutrino oscillation.",
"Significant production rate is only achieved for the nearly massless ALP case [14], i.e.",
"$m_a \\ll \\omega _{\\rm pl}$ .",
"Thus, for $m_a \\approx 0$ , $|\\tan 2\\theta | \\sim B_{\\perp }\\omega /n_eM$ and $\\Delta \\sim n_eL/\\omega $ [4], [14].",
"For typical galaxy or galaxy cluster environments, $n_e \\sim 10^{-2}-10^{-3}\\ {\\rm cm}^{-3}$ and $B \\sim 10^0-10^1{\\mu \\rm Gs}$ , we find $|\\theta | \\ll 1$ and $\\Delta \\ll 1$ where the small angle approximation applies, and then we have $P_{a\\rightarrow \\gamma } \\simeq \\frac{1}{4}\\left(\\frac{B_{\\perp }L}{M}\\right)^2.$ The magnetic field distribution is found empirically to follow electron number density, which takes the form $B(r) = B_0 \\left[\\frac{n_e(r)}{n_0}\\right]^p,$ where we assume $p=0.5$ and 1, and $n_0=n_{c0}$ and $n_{g0}$ for galaxy clusters and galaxies, respectively [14], [20], [21].",
"Here $B_0\\sim \\mathcal {O}(1\\,{{\\mu \\rm Gs}})$ for non-cool core galaxy cluster, and $\\mathcal {O}(10\\,{{\\mu \\rm Gs}})$ for cool core cluster and spiral galaxy [14], [22], [23].",
"We model electron density with the $\\beta $ -model for clusters, $n_e(r) = n_{c0} [1+({r^2}/{r^2_c})]^{-3\\beta /2}$ , where typical values are $n_{c0}\\sim 10^{-2}\\ \\rm cm^{-3}$ , $r_c\\sim 300$ kpc and $\\beta \\sim 1$ .",
"The exponential disk model [19] can be used for spiral galaxies, $n_e(r, z^{\\prime }) = n_{g0} \\exp (-r/r_0) \\exp (-|z^{\\prime }|/z^{\\prime }_0)$ , where $r$ is the radial distance from the galaxy center in galactic plane, and $z^{\\prime }$ is the vertical height above galactic plane.",
"Typical values are $n_{g0}\\sim 10^{-2}$ $\\rm cm^{-3}$ , $r_0\\sim 15$ kpc, and $z^{\\prime }_0\\sim 1$ kpc for thick disks, and $r_0\\sim 4$ kpc, and $z^{\\prime }_0\\sim 0.05$ kpc for thin disks [19].",
"However, we find that the results are quite similar for both cases.",
"The photon emission rate from dark matter decay to ALPs followed by ALP-photon conversion is given by $\\dot{N}_{\\gamma } =N_{\\gamma }/\\tau _{d\\rightarrow a}$ where $N_{\\gamma }$ is the number of photons, and $\\tau _{d\\rightarrow a}$ is the lifetime of dark matter particles decaying to ALPs, which we assume to be $\\tau _{d\\rightarrow a}\\sim \\Lambda ^2/m_d^3$ .",
"For $m_d\\simeq 7.1$ keV and the dark matter-ALP coupling $\\Lambda =10^{17}$ GeV, it is about $10^{27}$ s [14].",
"We assume that the dark matter density distribution follows the Navarro-Frenk-White (NFW) profile [24], and the dark matter particle density is denoted as $n_d(r) = \\rho _d(r)/m_d$ , where $m_d$ is dark matter particle mass.",
"For galaxy cluster, $N_{\\gamma } = \\int P_{a\\rightarrow \\gamma }\\, n_d(r)\\ d^3 r$ , with typical cluster size $R \\sim $ 1 Mpc.",
"For spiral galaxy, we model $N_{\\gamma } = D \\iint d\\phi \\, dr\\, r \\, n_d(r) P_{a\\rightarrow \\gamma }$ , where $D$ is the thickness of the layer parallel to the galactic plane.",
"In the small-angle approximation, the photon production rate $\\dot{N}_{\\gamma } \\sim (B_{\\perp }L\\,m_d/M\\Lambda )^2$ .",
"In the above we have simply integrated the converted photon flux.",
"A rigorous treatment would require consideration of the photon-ALP mixing problem, with photoelectric absorption by the intervening gas.",
"This can be computed by solving a differential equation for the density matrix, as shown in Ref.",
"[4].",
"For the usual spiral galaxy and galaxy cluster, we find that $P_{\\gamma \\rightarrow a}(r)$ are at most $\\sim 10^{-3}$ and $\\sim 10^{-2}$ , respectively.",
"The photoelectric absorption is at the level of $1\\% \\sim 10\\%$ level depending on the inclination angle of the line of sight with respect to the disk.",
"Given the current uncertainties on the model parameters, here we neglect such effects.",
"Next, we estimate the polarization of 3.5 keV photons from galaxy clusters and spiral galaxies.",
"Since cluster magnetic field is irregular [26], the total polarization of ALP-converted photons which depend on the magnetic field and properties of member galaxies should vary with different clusters.",
"For typical large clusters with nearly spherical symmetry, we expect the total polarization should be close to zero as the differently polarized photons from different parts of the cluster cancel out.",
"However, in resolved observations, we should be able to compare the orientation of magnetic field inside the cluster with the polarization of the photons.",
"If the polarization are aligned with the magnetic field, it would be a strong evidence in support of the ALP-photon conversion scenario, and vice versa, this scenario can be falsified if it is found otherwise.",
"Figure: The magnetic field model for spiral galaxy.",
"The coordinate yy is along the line of sight, and xx and zz axes are along the long and short axis of the projected galactic plane, respectively.",
"The z ' z^{\\prime }-axis points to the actual galactic pole, which is inclined with angle α\\alpha , and φ\\phi denotes the polar coordinate on the galactic plane.",
"The dotted lines in both panels denote the middle planes of the two layers above and below the central layer which includes the galactic disk.In spiral galaxies, in addition to the small scale stochastic magnetic field, there are large scale regular magnetic fields [22], [23].",
"We consider a two component model of magnetic field: the field within the galactic disk which aligns with the spiral arms, and the X-shaped halo field above and below the disk [22], [23] (Fig.",
"REF ).",
"The upper panel shows the galactic plane and magnetic field in the disk $B_{\\phi }$ .",
"For simplicity and as a good approximation in our calculation, we assume $B_{\\phi }$ is oriented in circular direction, but have different sense of rotation (clockwise or counter clockwise) between different spiral arms [22], [23].",
"Note that the different sense of rotation of $B_{\\phi }$ does not affect our calculation since the polarization is the same for either case.",
"The lower panel shows the magnetic field $B_X$ above and below galactic disk.",
"Only the $B_X$ close to the disk can affect the ALP-photon conversion, since the field strength $B$ and the coherent length $L$ declines quickly as the galactic latitude increases [4], [14].",
"We assume the two outer layers has the same thickness $D$ as the central layer which includes the galactic plane, and both $B_{\\phi }$ and $B_X$ follow the $n_e$ as Eq.",
"(REF ), and $B_{\\phi }(r) = B(r, z^{\\prime }=0)$ , $B_X = B(r, z^{\\prime }=\\pm z^{\\prime }_0)$ .",
"We can then compute the projected components $B_{\\phi }$ and $B_X$ .",
"Here we also assume $B_X^{z^{\\prime }}/B_X^r=f_X^{z^{\\prime }r}$ , where $B_X^{z^{\\prime }}$ and $B_X^r$ are the $B_X$ components along $z^{\\prime }$ and $r$ directions, respectively.",
"For simplicity $f_X^{z^{\\prime }r}$ is assumed to be a constant at different $r$ [22], [23].",
"The conversion rate can then be estimated with Eq.",
"(REF ) by substituting $P_{\\phi }^{xz} &=& P_{a\\rightarrow \\gamma }[ B_{\\phi }^{xz}, n_e(r,z^{\\prime }=0), L_{\\rm eff} ] \\\\P_X^{xz} &=& 2\\, P_{a\\rightarrow \\gamma }[ B_X^{xz}, n_e(r,z^{\\prime }=z^{\\prime }_0), L_{\\rm eff} ].$ The superscript `$xz$ ' denotes the $x-z$ plane perpendicular to the observer.",
"The effective coherent length is given by $L_{\\rm eff}=L_d\\, {\\rm sin}\\, \\alpha $ where $L_d$ is the coherent length of magnetic field in the galactic plane.",
"If $L_{\\rm eff}<D_{\\rm disk}$ we set $L_{\\rm eff}=D_{\\rm disk}$ , where $D_{\\rm disk}$ is the thickness of galaxy disk.",
"The photons of different polarizations are produced at different places so these are incoherent light.",
"The net polarization is given by $f_{\\rm Pol} = \\frac{N_{\\rm gal}^x-N_{\\rm gal}^z}{N_{\\rm gal}^{xz}},$ where $N_{\\rm gal}^x=N_{\\phi }^x+N_X^x$ and $N_{\\rm gal}^z=N_{\\phi }^z+N_X^z$ , and the total number is $N_{\\rm gal}^{xz}=N_{\\phi }^{xz}+N_X^{xz}$ .",
"The degrees of polarization for the central and two outer layers are $ f_{\\rm Pol}^{\\phi } = (N_{\\phi }^x-N_{\\phi }^z)/N_{\\phi }^{xz}$ and $f_{\\rm Pol}^X = (N_X^x-N_X^z)/N_X^{xz}$ , respectively.",
"The orientation of polarization is along the $x$ direction (long axis) when $f_{\\rm Pol}>0$ , and along the $z$ direction (short axis) when $f_{\\rm Pol}<0$ .",
"Figure: The emission rate N γ ˙\\dot{N_{\\gamma }} from ALP-photon conversion as a function of magnetic field B 0 B_0.",
"The red and blue solid lines show the estimations with magnetic coherent length L=10L=10 kpc.",
"The shaded light red and blue regions are derived by taking L=1∼100L=1 \\sim 100 kpc for clusters and L=1∼30L=1 \\sim 30 kpc for galaxies with inclination angles α=90 ∘ \\alpha =90^{\\circ }.",
"The blue dash-dotted, long dashed, short dashed and dotted lines are for spiral galaxy with α=60 ∘ \\alpha = 60^{\\circ }, 45 ∘ 45^{\\circ }, 30 ∘ 30^{\\circ } and 0 ∘ 0^{\\circ }, respectively.",
"We assume the coupling factors M=10 13 M=10^{13} GeV and Λ=10 16 \\Lambda =10^{16} GeV here.",
"The measurements of 3.5 keV emission for clusters and galaxies are also shown as hatched boxes , , .Figure: Degree of polarization as a function of galactic inclination angle α\\alpha .",
"The f Pol f_{\\rm Pol}, f Pol φ f_{\\rm Pol}^{\\phi } and f Pol X f_{\\rm Pol}^X are shown in solid, dashed and dotted lines, respectively.",
"The upper and lower bounds of light blue shaded region is obtained with f X z ' r =0.1f_X^{z^{\\prime }r}=0.1 and 10, and f X z ' r =0.5f_X^{z^{\\prime }r}=0.5 and 2 for deep blue region.Results.",
"In Figure REF , the predicted emission rate $\\dot{N}_{\\gamma }$ is shown as a function of the strength of central magnetic field $B_0$ for galaxy clusters or spiral galaxies.",
"On the same plot we also show the measured 3.5 keV emission rates (derived from fluxes [1], [2], [3] and distances) with uncertainties as the hatched regions.",
"The flux of cluster 3.5 keV emission are obtained from measurements of the Coma+Centaurus+Ophiuchus clusters and also the stacking result of 69 clusters (marked as “all others\") [1], in the figure we plot the per cluster value obtained by dividing the stacking result by the number of stacked samples.",
"The flux of the M31 [2] and MW [3] are $\\sim 5$ and $(29\\pm 5)$ $\\times $ $10^{-6}\\ \\rm phts\\ cm^{-2} s^{-1}$ respectively.",
"The strongest 3.5 keV line emission is found in the Perseus cluster, with $52.0_{-21.3}^{+37.0} \\times 10^{-6}$ phts cm$^{-2}$ s$^{-1}$ [1], though Ref.",
"[2] found a lower value, and the Hitomi satellite did not detect it [6].",
"These differences in the measured values may be due to the different resolution of the observations.",
"For example, the Hitomi has lower angular resolution than the Chandra and the XMM-Newton, so the central AGN and diffuse cluster emissions are mixed.",
"The $Chandra$ data indicates a strong dip in the AGN spectrum around 3.5 keV [32], so such mixing may give a mild dip at 3.5 keV, which is indeed seen in the Hitomi data.",
"Thus, the Perseus 3.5 keV signal could still be real and we assume so below.",
"We find that the prediction is generally in good agreements with the observations.",
"The 3.5 keV emission of Perseus is stronger than other clusters as expected, since it is a cool core cluster and has strong central magnetic field as high as $B_0\\sim 25\\ \\mu $ G [27].",
"The data for the MW center is lower than the prediction, and also several orders of magnitude lower than M31.",
"This may be due partly to the higher magnetic field strength in M31, and partly to the fact that not the whole galaxy but only the central region is measured for the MW.",
"The emission rate from spiral galaxy is much lower than that from galaxy cluster with the same $B_0$ , which may explain why there are more detections of 3.5 keV emission in clusters than galaxies [1], [2], [3].",
"In Fig.",
"REF , we plot $f_{\\rm Pol}$ , $f_{\\rm Pol}^{\\phi }$ and $f_{\\rm Pol}^X$ as a function of galaxy inclination angle $\\alpha $ in blue solid, dashed and dotted lines respectively.",
"These results are obtained by taking the ratio $f_X^{z^{\\prime }r}=B_X^{z^{\\prime }}/B_X^r=1$ for the X-shaped magnetic field $B_X$ .",
"We also show the results with different $f_X^{z^{\\prime }r}$ in shaded regions.",
"There is no polarization for $\\alpha =0^{\\circ }$ (face-on galaxy), since the magnetic field in our model is axisymmetric in this case and the polarizations from different parts cancelled out.",
"At $\\alpha =90^{\\circ } $ (edge-on galaxy), $f_{\\rm Pol}^{\\phi }$ is always equal to 1, which means photons of ALP-photon conversion from galactic plane are totally polarized and the polarization is parallel to the galactic plane.",
"On the other hand, $f_{\\rm Pol}^X$ can vary from $-1$ to 1 at $\\alpha =90^{\\circ }$ depending on the value of $f_X^{z^{\\prime }r}$ .",
"For $f_X^{z^{\\prime }r}=1$ , we find $f_{\\rm Pol}^X\\simeq -0.27$ at $\\alpha =90^{\\circ }$ .",
"The total degree of polarization $f_{\\rm Pol}$ is between $f_{\\rm Pol}^X$ and $f_{\\rm Pol}^{\\phi }$ .",
"Our results are obtained from integration along the line of sight, which is not very sensitive to the precise shape of the magnetic field.",
"The variations of the result with respect to the inclination angle and strength shown here also give some sense of how the results would vary if the shape of the magnetic field varies.",
"Note that $f_{\\rm Pol}$ is greater than 0 as long as $B_{\\phi }$ is larger than $B_X$ , which should be the case for most spiral galaxies.",
"Hence, we predict that the net polarization of the ALP-converted photons is along the long axis of the projected galactic disk for most spiral galaxies.",
"This can be used to test whether the 3.5 keV line originates from this mechanism.",
"Of course, if we can measure the inclination angle of a spiral galaxy, we could also compare the measured $f_{\\rm Pol}$ to our prediction.",
"Here we are using a simple magnetic field model, the predication could be refined if the actually measured value of its magnetic field is available.",
"For example, $\\alpha =77.5^{\\circ }$ for M31, and using the above formula we obtain $f_{\\rm Pol}\\simeq 0.43$ .",
"With the actually measured M31 magnetic field [28], we find $f_{\\rm Pol} \\sim 0.5$ .",
"Discussions.",
"Sensitive X-ray polarimetry based on the photoelectric effect has become available in recent years [29].",
"Several space missions dedicated to or capable of X-ray polarimetry have been proposed and are undergoing phase-A studies, including the eXTP and IXPE/XIPE [30].",
"The energy resolution (FWHM) of the polarization measurement is about $\\Delta E=0.8$ keV [31].",
"Assuming the continuum is unpolarized and the line fully polarized, the observed degree of polarization would be $ f_{\\rm pol}^{\\rm eff} \\approx I_{\\rm line}/ S_{\\rm cont} \\Delta E$ , where $S_{\\rm cont}$ is the continuum spectral flux density, $I_{\\rm line}$ is the line flux, and $\\Delta E$ is the energy resolution.",
"For the Perseus cluster, $f_{\\rm pol}^{\\rm eff} \\sim 0.2\\% - 0.6\\%$ taken into account of uncertainty.",
"For the 0.6% polarized case, the signal can be detected with an exposure time of $1.7 \\times 10^6$ seconds for eXTP or IXPE/XIPE at a confidence level of 99% neglecting the instrumental and cosmic X-ray backgrounds.",
"For the other sources, the MW center is most promising case, which needs an exposure time comparable to that of Perseus, while for M31 or the off-center region of Perseus cluster it could be one order of magnitude longer.",
"Thus, we see that the polarized signals can possibly be detected with X-ray polarimeters currently under development.",
"In the future, X-ray polarimeters equipped on large area telescopes or with better energy resolution or lower systematics will enable us to measure the 3.5 keV line from a large sample of clusters, and test the axion conversion hypothesis.",
"Acknowledgements.",
"We thank Prof. Shuang-Nan Zhang for helpful discussions.",
"YG acknowledges the support of Bairen program from the Chinese Academy of Sciences (CAS).",
"XLC acknowledges the support of the MoST 863 program grant 2012AA121701, the CAS Frontier Science Key Project No.",
"QYZDJ-SSW-SLH017, the CAS Strategic Priority Research Program XDB09020301, and the NSFC through grant No.",
"11373030 and 11633004.",
"HF acknowledges funding support from the NSFC under grant No.",
"11633003, and the Tsinghua University Initiative Scientific Research Program."
]
] | 1612.05697 |
[
[
"Hyperbolic Geometry and Moduli of Real Curves of Genus Three"
],
[
"Abstract The moduli space of smooth real plane quartic curves consists of six connected components.",
"We prove that each of these components admits a real hyperbolic structure.",
"These connected components correspond to the six real forms of a certain hyperbolic lattice over the Gaussian integers.",
"We will study this Gaussian lattice in detail.",
"For the connected component that corresponds to maximal real quartic curves we obtain a more explicit description.",
"We construct a Coxeter diagram that encodes the geometry of this component."
],
[
"Introduction",
"Recently there has been a great deal of progress in the construction of period maps from moduli spaces to ball quotients.",
"This allows for a new approach to the study of questions of reality for these moduli spaces.",
"The main example of this in the literature is the work of Allcock, Carlson and Toledo on the moduli space of cubic surfaces.",
"In [2] they construct a period map from this moduli space to a ball quotient of dimension four.",
"The question of reality for this period map is studied in [3].",
"One of the five connected components of this real moduli space, the one where all 27 lines on the smooth real cubic surface surface are real, was previously studied by Yoshida [28] using the period map of [2].",
"The moduli space of real hyperelliptic curves of genus three has been studied by Chu [7] using the period map of Deligne and Mostow [8].",
"In this article we will focus mostly on smooth nonhyperelliptic curves of genus three.",
"The canonical map of such a curve is an embedding onto a smooth plane quartic.",
"For the moduli space of smooth plane quartic curves there is a period map due to Kondo [13].",
"It maps the moduli space to a ball quotient of dimension six.",
"We will study the question of reality for this period map.",
"The classification of smooth real plane quartic curves is classical.",
"The set of real points of such a curve consists of up to four ovals in the real projective plane.",
"There are six possible configuration of the ovals.",
"Each of them determines a connected component in the space of smooth real plane quartic curves.",
"This is the projective space $P_{4,3}(\\mathbb {R})=\\mathbb {P}\\operatorname{Sym}^4(\\mathbb {R}^3)$ of dimension 14 without the discriminant locus $\\Delta (\\mathbb {R})$ , that represents singular quartics.",
"Since the group $PGL_3(\\mathbb {R})$ is connected, the moduli space $ \\mathcal {Q}^\\mathbb {R} = PGL_3(\\mathbb {R}) \\backslash \\left( P_{4,3}(\\mathbb {R})-\\Delta (\\mathbb {R}) \\right) $ also consists of six components which we denote by $\\mathcal {Q}_j^\\mathbb {R}$ with $j=1,\\ldots ,6$ .",
"The correspondence between these components and the topological types of the set of real points of the curves is shown in Figure REF .",
"Table: The topological types of representative curves C(ℝ)C(\\mathbb {R}) for the six components of 𝒬 i ℝ ⊂𝒬 ℝ \\mathcal {Q}_i^\\mathbb {R} \\subset \\mathcal {Q}^\\mathbb {R} for i=1,...,6i=1,\\ldots ,6.In this article we will prove that each of the components $\\mathcal {Q}_j^\\mathbb {R}$ is isomorphic to a divisor complement in an arithmetic real ball quotient.",
"In order to formulate this more precisely we introduce some notation on Gaussian lattices.",
"Let $\\mathcal {G}=\\mathbb {Z}[i]$ be the Gaussian integers and let $\\Lambda _{1,6}$ be the Gaussian lattice $\\mathcal {G}^7$ equipped with the Hermitian form $h(\\cdot ,\\cdot )$ defined by the matrix $ H = \\begin{pmatrix} -2&1+i\\\\1-i&-2 \\end{pmatrix}^{\\oplus 3}\\oplus (2).",
"$ We denote the group of unitary transformations of this lattice by $\\Gamma =U(\\Lambda )$ .",
"The lattice $\\Lambda _{1,6}$ has hyperbolic signature $(1,6)$ and determines a complex ball of dimension six by the expression $\\mathbb {B}_6 = \\mathbb {P} \\lbrace z\\in \\Lambda _{1,6} \\otimes _\\mathcal {G} \\mathbb {C} \\ ; \\ h(z,z)>0 \\rbrace .$ A root is an element $r\\in \\Lambda _{1,6}$ such that $h(r,r)=-2$ and for every root $r$ we define its root mirror to be the hypersurface $H_r = \\lbrace z\\in \\mathbb {B}_6 \\ ; \\ h(r,z)=0 \\rbrace $ .",
"We denote by $\\mathbb {B}_6^\\circ $ the complement in $\\mathbb {B}_6$ of all root mirrors.",
"Our main result is the following theorem.",
"Theorem 1.1 There are six projective classes of antiunitary involutions $[\\chi _j]$ with $j=1,\\dots ,6$ of the lattice $\\Lambda _{1,6}$ up to conjugation by $P\\Gamma $ .",
"Each of them determines a real ball $\\mathbb {B}_6^{\\chi _j}\\subset \\mathbb {B}_6$ and there are isomorphisms of real analytic orbifolds $\\mathcal {Q}_j^\\mathbb {R} \\longrightarrow P\\Gamma ^{\\chi _j} \\backslash \\left( \\mathbb {B}_6^{\\chi _j} \\right)^\\circ .$ The group $P\\Gamma ^{\\chi _j}$ is the stabilizer of the real ball $\\mathbb {B}_6^{\\chi _j}$ in $P\\Gamma $ .",
"It is an arithmetic subgroup of $PO(\\Lambda _{1,6}^{\\chi _j})$ for each $j=1,\\ldots ,6$ .",
"In fact we obtain more information on the lattices $\\Lambda _{1,6}^{\\chi _j}$ and the groups $P\\Gamma ^{\\chi _j}$ for $j=1,\\ldots ,6$ .",
"They are finite index subgroups of hyperbolic Coxeter groups of finite covolume and we determine the Coxeter diagrams for these latter groups using Vinberg's algorithm.",
"For the group $P\\Gamma ^{\\chi _1}$ that corresponds to the component $\\mathcal {Q}_1^\\mathbb {R}$ of maximal quartic curves we obtain a very explicit description: it is the semidirect product of a hyperbolic Coxeter group of finite covolume by its group of diagram automorphisms.",
"The fundamental domain of this Coxeter group is a convex hyperbolic polytope $C_6$ whose Coxeter diagram is shown in Figure REF .",
"Its group of diagram automorphisms is the symmetric group $S_4$ .",
"The locus of fixed points in $C_6$ of this group is a hyperbolic line segment.",
"It corresponds to a pencil of smooth real quartic curves that was previously studied by W.L.",
"Edge [9].",
"It consist of four ovals with an $S_4$ -symmetry and we determine this family explicitly.",
"The walls of the polyhedron $C_6$ represent either singular quartics or hyperelliptic curves.",
"The Coxeter diagram $C_5$ of the wall representing hyperelliptic curves is shown in Figure REF on the right.",
"It is the Coxeter diagram that corresponds to the connected component of the moduli space of real binary octics where all eight points are real.",
"This component is described by Chu in [7].",
"We complement this work by explicitly computing the Coxeter diagram of $C_5$ .",
"The automorphism group of this diagram is isomorphic to $D_8$ and there is a unique fixed point in $C_5$ .",
"It correspond to the isomorphism class of the binary octic where the zeroes are image of the eighth roots of unity under a Cayley transform $z \\mapsto i(1-z)/(1+z)$ .",
"Acknowledgements The results of this article are contained in the PhD thesis of the second author.",
"This research was supported by NWO free competition grant number 613.000.909.",
"The authors would like to thank Professor Allcock and Professor Kharlamov for useful comments.",
"Figure: The Coxeter diagram of the hyperbolic Coxeter polytope C 6 C_6 (left) and the wall that corresponds to C 5 C_5 (right).",
"The nodes represent the walls and a double edge connecting two nodes means their walls meet at an angle of π/4\\pi /4, a thick edge means they are parallel and no edge means they are orthogonal."
],
[
"Lattices",
"A lattice is a pair $(L,(\\cdot ,\\cdot ))$ with $L$ a free $\\mathbb {Z}$ -module of finite rank $r$ and $(\\cdot ,\\cdot )$ a nondegenerate, symmetric bilinear form on $L$ taking values in $\\mathbb {Z}$ .",
"This bilinear form extends naturally to a bilinear form $(\\cdot ,\\cdot )_\\mathbb {Q}$ on the rational vector space $L\\otimes _\\mathbb {Z} \\mathbb {Q}$ and its signature $(r_+,r_-)$ is called the signature of $L$ .",
"The dual of $L$ is the group $L^\\vee =\\operatorname{Hom}(L,\\mathbb {Z})$ and the lattice $L$ is naturally embedded in $L^\\vee $ by the assignment $x\\mapsto (x,\\cdot )$ .",
"The group $L^\\vee $ is naturally embedded in the vector space $L\\otimes _\\mathbb {Z} \\mathbb {Q}$ by the identification $ L^\\vee = \\lbrace x\\in L\\otimes _\\mathbb {Z} \\mathbb {Q} \\ ; \\ (x,y)_\\mathbb {Q} \\in \\mathbb {Z} \\text{ for all } y\\in L \\rbrace .$ Note that the induced bilinear form on $L^\\vee $ need not be integer valued, but by abuse of language we still call $L^\\vee $ a lattice.",
"An isomorphism between lattices $L_1$ and $L_2$ is a group isomorphism $\\phi :L_1\\rightarrow L_2$ that preserves the bilinear forms of $L_1$ and $L_2$ .",
"If $\\lbrace e_1,\\ldots ,e_r\\rbrace \\subset L$ is a basis for $L$ then the matrix $B=\\begin{pmatrix}(e_1,e_1)&\\cdots & (e_1,e_r)\\\\\\vdots &\\ddots & \\vdots \\\\(e_r,e_1)&\\cdots & (e_r,e_r)\\end{pmatrix}$ is called the Gram matrix.",
"Its determinant $d(L)$ is an invariant called the discriminant of the lattice.",
"A lattice is called unimodular if $d(L)=\\pm 1$ or equivalently if $L^\\vee =L$ .",
"A lattice $L$ is called even if $(x,x)\\in 2\\mathbb {Z}$ for all $x\\in L$ , otherwise it is called odd.",
"We denote the automorphism group of a lattice $L$ by $O(L)$ .",
"An important class of automorphisms of a lattice $L$ of signature $(r_+,r_-)$ with $r_+ \\le 1$ is given by its reflections.",
"For $r \\in L$ primitive (that is $q\\cdot r \\in L$ for $q\\in \\mathbb {Q}$ only if $q\\in \\mathbb {Z}$ ) and of negative norm $(r,r)$ we define the reflection $s_r$ in $r$ by the formula $s_r(x) = x-2\\frac{(r,x)}{(r,r)}r.$ This reflection is an automorphism of the lattice $L$ if and only if $2(r,x)\\in (r,r)\\mathbb {Z}$ for all $x\\in L$ .",
"In that case we call the negative norm vector $r$ a root in $L$ .",
"Since conjugation by an element of $O(L)$ of a reflection is again a reflection, the reflections in roots generate a normal subgroup $W(L)\\triangleleft O(L)$ .",
"Let $L$ be an even lattice.",
"The quotient $A_L=L^\\vee / L$ is called the discriminant group of $L$ .",
"It is a finite abelian group of order $d(L)$ .",
"We denote the minimal number of generators of $A_L$ by $l(A_L)$ .",
"If $A_L\\cong (\\mathbb {Z}/2\\mathbb {Z})^a$ for some $a\\in \\mathbb {N}$ then $L$ is called 2-elementary.",
"Proposition 2.1 (Nikulin [17], Thm.",
"3.6.2) An indefinite, even 2-elementary lattice with $r_+>0$ and $r_->0$ is determined up to isomorphism by the invariants $(r_+,r_-,a,\\delta )$ .",
"The invariant $\\delta $ is defined by $ \\delta ={\\left\\lbrace \\begin{array}{ll}0 & \\text{ if } (x,x)_\\mathbb {Q} \\in \\mathbb {Z} \\text{ for all } x\\in L^\\vee \\\\1 & \\text{ else }\\end{array}\\right.",
"}$ The discriminant quadratic form $q_L$ on $A_L$ takes values in $\\mathbb {Q}/2\\mathbb {Z}$ and is defined by the expression $q_L(x+L) &\\equiv (x,x)_\\mathbb {Q} \\bmod {2\\mathbb {Z}} \\quad \\text{for } x\\in L^\\vee .",
"\\\\$ The group of automorphisms of $A_L$ that preserve the discriminant quadratic form $q_L$ is denoted by $O(A_L)$ and there is a natural homomorphism: $O(L)\\rightarrow O(A_L)$ .",
"If $\\phi _L \\in O(L)$ then we denote by $q(\\phi _L)\\in O(A_L)$ the induced automorphism of $A_L$ .",
"Theorem 2.2 (Nikulin, [17], Thm.",
"3.6.3) Let $L$ be an even, indefinite 2-elementary lattice.",
"Then the natural homomorphism $O(L) \\rightarrow O(A_L)$ is surjective.",
"Proposition 2.3 (Nikulin, [17], Prop.",
"1.6.1) Let $L$ be an even unimodular lattice and $M$ a primitive sublattice of $L$ with orthogonal complement $M^\\perp =N$ .",
"There is a natural isomorphism $\\gamma :A_M\\rightarrow A_N$ for which $q_N\\circ \\gamma = -q_M$ .",
"Let $\\phi _M \\in O(M)$ and $\\phi _N\\in O(N)$ .",
"The automorphism $(\\phi _M , \\phi _N)$ of $M\\oplus N$ extends to $L$ if and only if $q(\\phi _N) \\circ \\gamma =\\gamma \\circ q(\\phi _M)$ .",
"Theorem 2.4 (Nikulin, [17], Thm.",
"1.14.4) Let $M$ be an even lattice of signature $(s_+,s_-)$ and let $L$ be an even unimodular lattice of signature $(r_+,r_-)$ .",
"There is a unique primitive embedding of $M$ into $L$ provided the following hold: $s_+<r_+$ $s_-<r_-$ $l(A_M) \\le \\operatorname{rank}(L) - \\operatorname{rank}(M)-2$ We denote by $L(n)$ the lattice $L$ where the bilinear form is scaled by a factor $n\\in \\mathbb {Z}$ and we write $U$ for the even unimodular hyperbolic lattice of rank 2 with Gram matrix $\\left( {\\begin{matrix}0&1\\\\1&0\\end{matrix}} \\right)$ .",
"Furthermore we denote by $A_i,D_j,E_k$ with $i,j\\in \\mathbb {N}$ , $j\\ge 4$ and $k=6,7,8$ the lattices associated to the negative definite Cartan matrices of this type.",
"For example $A_2 = \\begin{pmatrix}-2&1\\\\1&-2\\end{pmatrix} \\ , \\ A_1\\oplus A_1(2) = \\begin{pmatrix}-2&0\\\\0&-4\\end{pmatrix} \\ , \\ D_4 = \\begin{pmatrix}-2&1&0&0\\\\1&-2&1&1\\\\0&1&-2&0\\\\0&1&0&-2\\end{pmatrix}.$ Determining if two lattices are isomorphic can be challenging.",
"In the following lemma we describe some isomorphic lattices that we will encounter frequently when studying Gaussian lattices.",
"Lemma 2.5 There are isomorphisms of hyperbolic lattices $(4) \\oplus A_1 &\\cong (2) \\oplus A_1(2) \\\\U(2) \\oplus A_1 &\\cong (2) \\oplus A_1^2 \\\\(2) \\oplus A_1(2) \\oplus D_4(2) &\\cong (2) \\oplus A_1^2 \\oplus A_1(2)^3$ For the first isomorphism we explicitly determine a base change: $ \\begin{pmatrix}1&-1\\\\-1&2\\end{pmatrix}^t \\begin{pmatrix}4&0\\\\0&-2\\end{pmatrix} \\begin{pmatrix}1&-1\\\\-1&2\\end{pmatrix} =\\begin{pmatrix}2&0\\\\0&-4\\end{pmatrix}.$ For the second isomorphism we calculate the invariants $(r_+,r_-,a,\\delta )$ of Proposition REF .",
"They are easily seen to be $(1,2,3,1)$ for both lattices so that the lattices are isomorphic.",
"The third isomorphism is the least obvious.",
"We also determine an explicit base change: $ B^t \\left( (2) \\oplus A_1^2 \\oplus A_1(2)^3 \\right)B = (2) \\oplus A_1(2) \\oplus D_4(2) $ where $B$ is the unimodular matrix: $ B = \\begin{pmatrix}3&2&1&0&1&1\\\\-1&0&-1&1&-1&-1\\\\-1&0&0&-1&0&0\\\\-1&-1&0&0&0&-1\\\\-1&-1&0&0&-1&0\\\\-1&-1&-1&0&0&0\\end{pmatrix}.$"
],
[
"Hyperbolic reflection groups",
"Most of the results of this section can be found in [26].",
"Let $L$ be a hyperbolic lattice of hyperbolic signature $(1,n)$ .",
"We can associate to $L$ the space $V=L\\otimes _\\mathbb {Z} \\mathbb {R}$ with isometry group $O(V)\\cong O(1,n)$ .",
"A model for real hyperbolic $n$ -space $\\mathbb {H}_n$ is given by one of the sheets of the two sheeted hyperboloid $\\lbrace x\\in V \\ ; \\ (x,x)=1 \\rbrace $ in $V$ .",
"Its isometry group is the subgroup $O(V)^+<O(V)$ of index two of isometries that preserves this sheet.",
"Another model for $\\mathbb {H}_n$ which we will use most of the time is the ball defined by $ \\mathbb {B}_n = \\mathbb {P}\\lbrace x\\in L\\otimes _\\mathbb {Z} \\mathbb {R} \\ ; \\ (x,x)>0 \\rbrace $ whose isometry group is naturally identified with the group $O(\\mathbb {B})\\cong PO(1,n)$ .",
"The group $O(L)^+=O(L)\\cap O(V)^+$ is a discrete subgroup of $O(V)^+$ and it has finite covolume by a theorem of Siegel [22].",
"Let $W(L)<O(L)^+$ be the normal subgroup generated by the reflections in roots of negative norm of $L$ .",
"We can write the group $O(L)^+$ as $O(L)^+ = W(L) \\rtimes S(C)$ where $C\\subset \\mathbb {B}_n$ is a fundamental chamber of $W(L)$ and $S(C)$ is the subgroup of $O(L)^+$ that maps $C$ to itself.",
"The lattice $L$ is called reflective if $W(L)$ has finite index in $O(L)^+$ .",
"In this case $C$ is a hyperbolic polytope of finite volume which we assume from now on.",
"We say that $\\lbrace r_i\\rbrace _{i\\in I}$ with $I=\\lbrace 1,\\dots ,k\\rbrace $ is a set of simple roots for $C$ if all pairwise inner products are nonnegative and $C$ is the polyhedron bounded by the mirrors $H_{r_i}$ so that $C = \\lbrace x \\in L\\otimes _{\\mathbb {Z}} \\mathbb {R} \\ ; \\ (x,x)>0 \\ ,\\ (r_i,x) \\ge 0 \\text{ for } i=1,\\ldots ,k \\rbrace / \\mathbb {R}_+.$ The root mirrors meet at dihedral angles $\\pi /m_{ij}$ with $m_{ij} = 2,3,4,6$ or they are disjoint in $\\mathbb {B}_n$ .",
"In this last case we say that two root mirrors $H_{r_i}$ and $H_{r_j}$ are parallel if they meet at infinity so that $m_{ij}=\\infty $ , or ultraparallel if they do not meet even at infinity.",
"The matrix $G$ with entries $(r_i,r_j)_{i,j\\in I}$ is called the Gram matrix of $C$ and in case two mirrors are not ultraparallel the $m_{ij}$ can be calculated from $G$ by the relation $ (r_i,r_j)^2 = (r_i,r_i)(r_j,r_j)\\cos ^2 \\left( \\frac{\\pi }{m_{ij}} \\right).",
"$ The polytope $C$ is described most conveniently by its Coxeter diagram $D_I$ .",
"This is a graph with $k$ nodes labeled by simple roots $\\lbrace r_i\\rbrace _{i\\in I}$ .",
"Nodes $i$ and $j$ are connected by $4\\cos ^2(\\pi /m_{ij})$ edges in case $m_{ij} < \\infty $ .",
"If $m_{ij}=\\infty $ we connect the vertices by a thick edge.",
"In addition we connect two nodes by a dashed edge if their corresponding mirrors are ultraparallel.",
"In the examples that come from Gaussian lattices we will only encounter roots of norm $-2,-4$ and $-8$ so we also subdivide the corresponding nodes into $0,2$ and 4 parts respectively.",
"These conventions are illustrated in Figure REF .",
"Figure: Conventions for Coxeter graphsA Coxeter subdiagram $D_J\\subset D_I$ with $J\\subset I$ is called elliptic if the corresponding Gram matrix is negative definite of rank $\\vert J \\vert $ and parabolic if it is negative semidefinite of rank $\\vert J\\vert -\\#$ components of $D_J$ .",
"An elliptic subdiagram is a disjoint union of finite Coxeter diagrams and a parabolic subdiagram the disjoint union of affine Coxeter diagrams.",
"The elliptic subdiagrams of $D$ of rank $r$ correspond to the $(n-r)$ -faces of the polyhedron $C\\in \\mathbb {B}_n$ .",
"A parabolic subdiagram of rank $n-1$ corresponds to a cusp of $C$ .",
"By the type of a face or cusp of $C$ we mean the type of the corresponding Coxeter subdiagram."
],
[
"Vinberg's algorithm",
"Suppose we are given a hyperbolic lattice $L$ of signature $(1,n)$ .",
"Vinberg [26] describes an algorithm to determine a set of simple roots of $W(L)$ .",
"If the algorithm terminates these simple roots determine a hyperbolic polyhedron $C\\subset \\mathbb {B}_n$ of finite volume which is a fundamental chamber for the reflection subgroup $W(L)$ .",
"We start by choosing a controlling vector $p\\in L$ such that $(p,p)>0$ .",
"This implies that $[p]\\in \\mathbb {B}_n$ .",
"The idea is to determine a sequence of roots $r_1,r_2,\\ldots $ so that the hyperbolic distance of $p$ to the mirrors $H_{r_i}$ is increasing.",
"Since the hyperbolic distance $d(p,H_{r_i})$ is given by $\\sinh ^2 d(p,H_{r_i}) = \\frac{ -(r_i,p)^2}{(r_i,r_i)\\cdot (p,p)}$ the height $h(r_i)$ of a root defined by $h(r_i)=-2(r_i,p)^2/(r_i,r_i)$ is a measure for this distance.",
"First we determine the roots of height 0.",
"They form a finite root system $R$ and we choose a set of simple roots $r_1,\\ldots ,r_i$ to be our first batch of roots.",
"For the inductive step in the algorithm we consider all roots of height $h$ and assume that all roots of smaller height have been enumerated.",
"A root of height $h$ is accepted if and only if it has nonnegative inner product with all previous roots of the sequence.",
"The algorithm terminates if the accute angled polyhedron spanned by the mirrors $H_{r_1},\\ldots $ has finite volume.",
"This can be checked using the following criterion also due to Vinberg.",
"Proposition 3.1 A Coxeter polyhedron $C\\subset \\mathbb {B}_n$ has finite volume if and only if every elliptic subdiagram of rank $n-1$ can be extended in exactly two ways to an elliptic subdiagram of rank $n$ or to a parabolic subdiagram of rank $n-1$ .",
"Furthermore there should be at least 1 elliptic subdiagram of rank $n-1$ .",
"Since an elliptic subdiagram of rank $n-1$ corresponds to an edge of the polyhedron $C$ the geometrical content of this criterion is that every edge connects either two actual vertices, two cusps or a vertex and a cusp.",
"The following example is due to Vinberg, see [25] §4.",
"Example 3.2 Consider the hyperbolic lattice $\\mathbb {Z}_{1,n}(2)=(2)\\oplus A_1^n$ with its standard orthogonal basis $\\lbrace e_0,\\ldots ,e_n\\rbrace $ where $2\\le n \\le 9$ .",
"The possible root norms are $-2$ and $-4$ .",
"We take as controlling vector $p=e_0$ with $(p,p)=2$ .",
"The height 0 root system is of type $B_n$ and a basis of simple roots is given by $r_1=e_1-e_2 \\ , \\ \\ldots , r_n = e_{n-1}-e_n \\ , \\ r_n= e_n.$ The next root accepted by Vinberg's algorithm is the root $r_{n+1}=e_0-e_1-e_2$ of height 2 for $n=2$ and the root $r_{n+1}=e_0-e_1-e_2-e_3$ of height 1 for $3\\le n \\le 9$ .",
"This root indeed satisfies $(r_{n+1},r_i)>0$ for $i=1,\\ldots ,n$ .",
"The resulting Coxeter polyhedron is a simplex and has finite volume so the algorithm terminates.",
"In all the cases there is a single cusp of type $\\widetilde{A}_1$ for $n=2$ and of type $\\widetilde{B}_{n-1}$ for $n=3,\\ldots ,8$ , except when $n=9$ in which case there are 2 cusps of type $\\widetilde{B}_{8}$ and $\\widetilde{E}_8$ .",
"The Coxeter diagrams are shown in Figure REF .",
"Figure: The Coxeter diagrams of the groups O(ℤ 1,n (2)) + O(\\mathbb {Z}_{1,n}(2))^+ for n=2,...,9n=2,\\ldots ,9."
],
[
"Gaussian lattices",
"This section is in a sense the technical heart of this article.",
"We study Gaussian lattices of hyperbolic signature and show how these give rise to arithmetic complex ball quotients.",
"Antiunitary involutions of the Gaussian lattice then correspond to real forms of these ball quotients.",
"The main examples are the two Gaussian lattices $\\Lambda _{1,5}$ and $\\Lambda _{1,6}$ whose ball quotients correspond to the moduli spaces of smooth binary octics and smooth quartic curves.",
"An excellent reference on the topic of Gaussian lattices is [1].",
"It also contains many examples of lattices over the Eisenstein and Hurwitz integers.",
"A Gaussian lattice is a pair $(\\Lambda ,\\rho )$ with $\\Lambda $ a lattice and $\\rho \\in O(\\Lambda )$ an automorphism of order four such that the powers $\\rho ,\\rho ^2$ and $\\rho ^3$ act without nonzero fixed points.",
"Such a lattice $\\Lambda $ can be considered as a module over the ring of Gaussian integers $\\mathcal {G}=\\mathbb {Z}[i]$ by assigning $(a+ib)x=ax+b\\rho (x)$ for all $x\\in \\Lambda $ and $a,b\\in \\mathbb {Z}$ .",
"The expression $h(x,y)=(x,y) + i(\\rho (x),y)$ defines a $\\mathcal {G}$ -valued nondegenerate Hermitian form on $\\Lambda $ which is linear in its second argument and antilinear in its first argument.",
"Conversely suppose that $\\Lambda $ is a free $\\mathcal {G}$ -module of finite rank equipped with a $\\mathcal {G}$ -valued Hermitian form $h(\\cdot ,\\cdot )$ .",
"We define a symmetric bilinear form on the underlying $\\mathbb {Z}$ -lattice of $\\Lambda $ by taking the real part of the Hermitian form: $(x,y)=\\operatorname{Re}h(x,y)$ .",
"Multiplication by $i$ defines an automorphism $\\rho $ of order 4 so the pair $(\\Lambda ,\\rho )$ is a Gaussian lattice.",
"It is easily checked that these two constructions are inverse to each other.",
"Another way of defining a Gaussian lattice is by prescribing a Hermitian Gaussian matrix.",
"Such a matrix $H$ satisfies $\\overline{H}^t=H$ and defines a Hermitian form on $\\mathcal {G}^n$ by the formula $h(x,y)=\\bar{x}^tHy$ .",
"The dual of a Gaussian lattice $\\Lambda $ is the lattice $\\Lambda ^\\vee = \\operatorname{Hom}(\\Lambda ,\\mathcal {G})$ .",
"It is naturally embedded in the vector space $\\Lambda \\otimes _\\mathcal {G} \\mathbb {Q}(i)$ by the identification $ \\Lambda ^\\vee = \\left\\lbrace x \\in \\Lambda \\otimes _\\mathcal {G} \\mathbb {Q}(i) \\ ; \\ h(x,y)\\in \\mathcal {G} \\ \\text{for all} \\ y\\in \\Lambda \\right\\rbrace .$ From now on we only consider nondegenerate Gaussian lattices that satisfy the condition $h(x,y)\\in (1+i)\\mathcal {G}$ for al $x,y\\in \\Lambda $ .",
"This is equivalent to $\\Lambda \\subset (1+i)\\Lambda ^\\vee $ and implies that the underlying $\\mathbb {Z}$ -lattice of $\\Lambda $ is even.",
"Lemma 4.1 The group $U(\\Lambda )$ of unitary transformations of a Gaussian lattice $\\Lambda $ is equal to the group $ \\Gamma = \\lbrace \\gamma \\in O(\\Lambda ) \\ ; \\ \\gamma \\circ \\rho = \\rho \\circ \\gamma \\rbrace $ of orthogonal transformations of the underlying $\\mathbb {Z}$ -lattice of $\\Lambda $ that commute with $\\rho $ .",
"If $\\gamma \\in U(\\Lambda )$ then by definition $h(\\gamma x,\\gamma y) = h(x,y)$ for all $x,y\\in \\Lambda $ .",
"Using the definition of the Hermitian form $h$ this is equivalent to $(\\gamma x,\\gamma y) + i (\\rho \\gamma x, \\gamma y)&= (x,y) + i(\\rho x,y).\\\\$ By considering the real part of this equality we see that $\\gamma \\in O(\\Lambda )$ .",
"Combining this with the equality of the imaginary parts of the equation we obtain $(\\rho \\gamma x,\\gamma y)=(\\gamma \\rho x,\\gamma y)$ for all $x,y \\in \\Lambda $ .",
"This is equivalent to: $\\rho \\circ \\gamma =\\gamma \\circ \\rho $ .",
"This proves the inclusion $U(\\Lambda )\\subset \\Gamma $ .",
"For the other inclusion we can reverse the argument.",
"A root $r \\in \\Lambda $ is a primitive element of norm $-2$ .",
"For every root $r$ we define a complex reflection $t_{r}$ of order 4 (a tetraflection) by $t_{r}(x) = x - (1-i)\\frac{h(r,x)}{h(r,r)} r ,$ which is an element of $U(\\Lambda )$ because $\\Lambda \\subset (1+i)\\Lambda ^\\vee $ .",
"It is a unitary transformation of $\\Lambda $ that maps $r \\mapsto ir$ and fixes pointwise the mirror $H_r=\\lbrace x\\in \\Lambda \\ ; \\ h(r,x)=0 \\rbrace $ .",
"The tetraflection $t_{r}$ and the mirror $H_r$ only depend on the orbit of $r$ under the group of units $\\mathcal {G}^\\ast =\\lbrace 1,i,-1,-i\\rbrace $ of $\\mathcal {G}$ .",
"We call such an orbit a projective root and denote it by $[r]$ .",
"If the group generated by tretraflections in the roots has finite index in $\\Gamma = U(\\Lambda )$ we say that the lattice $\\Lambda $ is tetraflective.",
"Example 4.2 (The Gaussian lattice $\\Lambda _2$ ) The lattice $D_4$ is given by $ D_4 = \\left\\lbrace x \\in \\mathbb {Z}^4 \\ ; \\ \\sum x_i \\equiv 0 \\bmod {2} \\right\\rbrace $ with the symmetric bilinear form induced by the standard form of $\\mathbb {Z}^4$ scaled by a factor $-1$ so that $(x,y)=-\\sum x_iy_i$ .",
"We choose a basis for this lattice given by the roots $\\lbrace \\beta _i \\rbrace $ with the Gram matrix $B$ shown below.",
"$\\begin{array}{lcr}\\beta _1=e_3-e_1 & {3}{*}{\\ \\ B=\\begin{pmatrix}-2&0&1&-1\\\\0&-2&1&1\\\\1&1&-2&0\\\\-1&1&0&-2\\end{pmatrix}} & {3}{*}{\\ \\ \\rho =\\begin{pmatrix}0&-1&0&0\\\\1&0&0&0\\\\0&0&0&-1\\\\0&0&1&0\\end{pmatrix}}\\\\\\beta _2=-e_1-e_3 \\\\\\beta _3=e_1-e_2 \\\\\\beta _4=e_3-e_4\\end{array}$ The matrix $\\rho $ defines an automorphism of order 4 without fixed points which turns the lattice $D_4$ into a Gaussian lattice which we will call $\\Lambda _2$ .",
"A basis for $\\Lambda _2$ is given by the roots $\\lbrace \\beta _1,\\beta _3\\rbrace $ and the Gram matrix $H$ with respect to this basis is given by $ H = \\begin{pmatrix}-2&1+i\\\\1-i&-2\\end{pmatrix}.",
"$ A small calculation shows that there are 6 projective roots which are the $\\mathcal {G}^\\ast $ -orbits of the roots $ \\left\\lbrace \\begin{pmatrix}1\\\\0\\end{pmatrix} , \\begin{pmatrix}0\\\\1\\end{pmatrix} , \\begin{pmatrix}1\\\\1\\end{pmatrix} , \\begin{pmatrix}1\\\\-i\\end{pmatrix}, \\begin{pmatrix}1\\\\1-i\\end{pmatrix} , \\begin{pmatrix}1+i\\\\1\\end{pmatrix} \\right\\rbrace $ The group generated by the tetraflections in these roots is the complex reflection group $G_8$ of order 96 in the Shephard-Todd classification [21].",
"A basis for the dual lattice $\\Lambda _2^\\vee $ is given by $\\left\\lbrace \\frac{1}{1+i}\\beta _1,\\frac{1}{1+i}\\beta _3 \\right\\rbrace $ so that $(1+i)\\Lambda _2^\\vee = \\Lambda _2$ .",
"Example 4.3 Consider the Gaussian lattice $\\Lambda _{1,1}$ with basis $\\lbrace e_1,e_2\\rbrace $ and Hermitian form defined by the matrix: $H=\\begin{pmatrix} 0&1+i\\\\ 1-i&0 \\end{pmatrix}.$ It is easy to verify that $(1+i)\\Lambda _{1,1}^\\vee =\\Lambda _{1,1}$ .",
"A basis $\\lbrace \\beta _1,\\ldots ,\\beta _4\\rbrace $ for the underlying $\\mathbb {Z}$ -lattice and its Gram matrix $B$ are shown below.",
"$\\begin{array}{lr}\\beta _1 = e_2 & {3}{*}{B=\\begin{pmatrix}0&1&0&0\\\\1&0&0&0\\\\0&0&0&2\\\\0&0&2&0\\end{pmatrix}} \\\\\\beta _2 = ie_1 \\\\\\beta _3 = e_1-ie_1 \\\\\\beta _4 = e_2-ie_2\\end{array}$ We conclude that the underlying $\\mathbb {Z}$ -lattice is isomorphic to $U\\oplus U(2)$ .",
"Using these two examples of Gaussian lattices we can construct many more by forming direct sums.",
"We are especially interested in the Gaussian lattices of hyperbolic signature since these occur in the study of certain moduli problems.",
"For example the Gaussian lattice $ \\Lambda _{1,1} \\oplus \\Lambda _2 \\oplus \\Lambda _2 $ plays an important role in the study of the moduli space $\\mathcal {M}_{0,8}$ of 8 points on the projective line.",
"Yoshida and Matsumoto [15] prove that the unitary group of this lattice is generated by 7 tetraflections so that it is in particular tetraflective.",
"This also follows from the work of Deligne and Mostow [8]."
],
[
"Antiunitary involutions of Gaussian lattices",
"Let $\\Lambda $ be a Gaussian lattice of rank $n$ and signature $(n_+,n_-)$ .",
"An antiunitary involution of $\\Lambda $ is an involution $\\chi $ of the underlying $\\mathbb {Z}$ -lattice that satisfies $ h\\left( \\chi (x),\\chi (y) \\right) = \\overline{h\\left( x,y \\right)}.",
"$ Equivalently it is an involution that anticommutes with $\\rho $ so that: $\\chi \\circ \\rho = - \\rho \\circ \\chi $ .",
"An antiunitary involution $\\chi $ naturally extends to the $\\mathbb {Q}(i)$ -vectorspace $\\Lambda _{\\mathbb {Q}}=\\Lambda \\otimes _\\mathcal {G} \\mathbb {Q}(i)$ which can be regarded as a $\\mathbb {Q}$ -vectorspace of dimension $2n$ and signature $(2n_+,2n_-)$ .",
"The fixed point subspace $\\Lambda _\\mathbb {Q}^\\chi $ is a $\\mathbb {Q}$ -vectorspace of dimension $n$ and signature $(n_+,n_-)$ .",
"Consider the fixed point lattice $\\Lambda ^\\chi = \\Lambda \\cap \\Lambda _\\mathbb {Q}^\\chi $ .",
"The Hermitian form restricted to $\\Lambda ^\\chi $ takes on real values in $(1+i)\\mathcal {G}$ and therefore has in in fact even values.",
"This implies that $\\Lambda ^\\chi (\\frac{1}{2})$ in an integral lattice.",
"Proposition 4.4 Let $\\Lambda $ be the Gaussian lattice defined by a Hermitian matrix $H$ , so that in particular $\\Lambda \\cong \\mathcal {G}^n$ .",
"Every antiunitary involution of $\\Lambda $ is of the form $\\chi = M \\circ \\operatorname{conj}$ where $\\operatorname{conj}$ is standard complex conjugation on $\\Lambda \\cong \\mathcal {G}^n$ .",
"The matrix $M$ has coefficients in $\\mathcal {G}$ and satisfies $\\overline{M}M=I$ and: $\\overline{M}^tHM=\\overline{H}$ .",
"Suppose that $\\chi $ is a antiunitary involution of $\\Lambda $ .",
"Since every antiunitary involution on the vector space $\\Lambda \\otimes _\\mathcal {G} \\mathbb {Q}(i)$ is conjugate to standard complex conjugation there is a matrix $N$ such that $N\\circ \\chi \\circ N^{-1} =\\operatorname{conj}$ .",
"We can rewrite this as $\\chi = M \\circ \\operatorname{conj}$ where $M=N^{-1}\\overline{N}$ and $M$ has coefficients in $\\mathcal {G}$ .",
"It is clear that $\\overline{M}M=I$ .",
"Finally we can rewrite the equality $h(\\chi (x),\\chi (y)) = \\overline{h(x,y)}$ as $ x^t\\overline{M}^tHM\\overline{y} = x^t\\overline{H}y.",
"$ This holds for all $x,y\\in \\Lambda $ so that the last equality of the proposition follows.",
"Let $\\chi $ be an antiunitary involution of the lattice $\\Lambda $ and let $[\\chi ]$ be its projective equivalence class.",
"The elements of $[\\chi ]$ are the involutions $i^k\\chi $ with $k=0,1,2,3$ .",
"By conjugation with the scalar $i$ we see that the two involutions $\\lbrace \\chi ,-\\chi \\rbrace $ and also $\\lbrace i\\chi ,-i\\chi \\rbrace $ are conjugate in $\\Gamma $ .",
"The antiunitary involutions $\\chi $ and $i\\chi $ need not be $\\Gamma $ -conjugate, so in particular their fixed point lattices need not be isomorphic.",
"This can already be seen in the simplest case of antiunitary involutions on $\\mathcal {G}$ .",
"The fixed points lattice of the antiunitary involution $\\operatorname{conj}$ and $i\\circ \\operatorname{conj}$ are $\\mathbb {Z}$ and $(1+i)\\mathbb {Z}$ respectively.",
"We now present some computational lemma's on antiunitary involutions of Gaussian lattices of small rank.",
"These will be very useful later on and will be referenced to throughout this text.",
"Lemma 4.5 Let $\\psi _1,\\psi _2,\\psi _2^\\prime $ and $\\psi _4$ be the transformations obtained by composing the following matrices with complex conjugation: $ M_1 = (1) \\ , \\ M_2=\\begin{pmatrix}i&0\\\\0&1\\end{pmatrix} \\ , \\ M_2^\\prime =\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix} \\ , \\ M_4=\\begin{pmatrix}0&0&i&0\\\\0&0&0&1\\\\i&0&0&0\\\\0&1&0&0\\end{pmatrix}.",
"$ They define antiunitary involutions $\\chi $ on certain Gaussian lattices $\\Lambda $ shown in Table REF .",
"The fixed point lattices $\\Lambda ^\\chi $ are also computed along with a matrix $B^\\chi $ such that the columns of this matrix form a $\\mathbb {Z}$ -basis for the fixed point lattice $\\Lambda ^\\chi $ .",
"Table: Some antiunitary transformations of Gaussian lattices of small rank.Using the conditions on $M_i$ from Proposition REF it is a straightforward calculation to prove that the $\\psi _i$ are antiunitary involutions.",
"Furthermore we need to check that the columns of $B^{\\psi _i}$ form a basis for the fixed point lattice $\\Lambda ^{\\psi _i}$ and that $\\Lambda ^{\\psi _i}=\\overline{B^{\\psi _i}}^t\\Lambda B^{\\psi _i}$ .",
"For example the fixed point lattice $(\\Lambda _2^2)^{\\psi _4}$ is given by the subset $ \\lbrace (z_1,z_2,i\\bar{z}_1,\\bar{z}_2) \\ ; \\ z_1,z_2\\in \\mathcal {G} \\rbrace \\subset \\Lambda _2^2 $ and it is not difficult to check that the columns of $B^{\\psi _4}$ indeed form a $\\mathbb {Z}$ -basis.",
"The verification for the other lattices proceeds similarly.",
"Lemma 4.6 The antiunitary involution $\\psi _2$ is conjugate in $U(\\Lambda _2)$ to $i\\psi _2$ , likewise $\\psi _2^\\prime $ is conjugate in $U(\\Lambda _{1,1})$ to $i\\psi _2^\\prime $ , and likewise $\\psi _4$ is conjugate in $U(\\Lambda _2^2)$ to $i\\psi _4$ .",
"The matrix $N=\\left( {\\begin{matrix}0&i\\\\1&0\\end{matrix}} \\right)$ satisfies $\\overline{N}^t\\Lambda _2 N =\\Lambda _2$ and $\\overline{N}^t\\Lambda _{1,1} N =\\Lambda _{1,1}$ so it is contained in $U(\\Lambda _2)$ and $U(\\Lambda _{1,1})$ .",
"It also satisfies $N\\psi _2\\overline{N}^{-1}=i\\psi _2$ and $N\\psi _2^\\prime \\overline{N}^{-1}=i\\psi _2^\\prime $ .",
"Similarly conjugation by the matrix $\\left( {\\begin{matrix}0&N\\\\N&0\\end{matrix}} \\right) \\in U(\\Lambda _2^2)$ maps $\\psi _4$ to $i\\psi _4$ ."
],
[
"Ball quotients from hyperbolic lattices",
"Let $\\Lambda $ be a Gaussian lattice of hyperbolic signature $(1,n)$ with $n\\ge 2$ such that $\\Lambda \\subset (1+i)\\Lambda ^\\vee $ .",
"We can associate to $\\Lambda $ a complex ball: $ \\mathbb {B} = \\mathbb {P}\\lbrace x \\in \\Lambda \\otimes _{\\mathcal {G}} \\mathbb {C} \\ ; \\ h(x,x) > 0 \\rbrace .",
"$ The group $P\\Gamma =PU(\\Lambda )$ acts properly discontinuously on $\\mathbb {B}$ .",
"The ball quotient $P\\Gamma \\backslash \\mathbb {B}$ is a quasi-projective variety of finite hyperbolic volume by the theorem of Baily-Borel [4].",
"Recall that a root $r \\in \\Lambda $ is an element of norm $-2$ .",
"We denote by $\\mathcal {H}\\subset \\mathbb {B}$ the union of all the root mirrors $H_r$ and write $\\mathbb {B}^\\circ = \\mathbb {B} - \\mathcal {H}$ .",
"In all the examples we consider later on the space $P\\Gamma \\backslash \\mathbb {B}^\\circ $ is a moduli space for certain smooth objects.",
"The image of $\\mathcal {H}$ in this space is called the discriminant and parametrizes certain singular objects.",
"The following lemma describes how two mirrors in $\\mathcal {H}$ can intersect.",
"Lemma 4.7 Let $r_1,r_2$ be two roots in $\\Lambda $ such that $H_{r_1} \\cap H_{r_2} \\ne \\emptyset $ .",
"The projective classes $[ r_1]$ and $[r_2]$ are either identical, orthogonal or they span a Gaussian lattice of type $\\Lambda _2$ .",
"Since the images of $H_{r_1}$ and $H_{r_2}$ meet in $\\mathbb {B}$ there is a vector $x\\in \\Lambda $ with $h(x,x)>0$ orthogonal to both $r_1$ and $r_2$ .",
"This implies that $r_1$ and $r_2$ span a negative definite space so that the Hermitian matrix $ \\begin{pmatrix}-2&h(r_1,r_2)\\\\ h(r_2,r_1)&-2 \\end{pmatrix} $ is negative definite.",
"This is equivalent to $\\vert h(r_1,r_2)\\vert ^2 < 4$ and since $h(r_1,r_2)\\in (1+i)\\mathcal {G}$ we see that either $h(r_1,r_2)=0$ or $h(r_1,r_2)=\\pm 1 \\pm i$ .",
"In the second case we can assume that $h(r_1,r_2)=1+i$ by multiplying $r_1$ and $r_2$ by suitable units in $\\mathcal {G}^\\ast $ .",
"Let $\\mathbb {B}^\\chi $ be the fixed point set in $\\mathbb {B}$ of the real form $[\\chi ]$ .",
"Since the fixed point lattice $\\Lambda ^\\chi $ is of hyperbolic signature this is a real ball given by $ \\mathbb {B}^\\chi = \\mathbb {P}\\lbrace x \\in \\Lambda ^\\chi \\otimes _\\mathbb {Z} \\mathbb {R} \\ ; \\ h(x,x)>0 \\rbrace .",
"$ Note that the lattice $\\Lambda ^{i\\chi }$ defines the same real ball.",
"The isomorphism type of the unordered pair $(\\Lambda ^\\chi ,\\Lambda ^{i\\chi })$ is an invariant of the $P\\Gamma $ -conjugacy class of $[\\chi ]$ as shown by the following lemma.",
"This invariant will prove very useful to distinguish between classes up to $P\\Gamma $ -conjugacy.",
"Lemma 4.8 If the projective classes $[\\chi ]$ and $[\\chi ^\\prime ]$ of two antiunitary involutions $\\chi $ and $\\chi ^\\prime $ of $\\Lambda $ are conjugate in $P\\Gamma $ then the isomorphism classes of the unordered pairs of lattices $( \\Lambda ^\\chi , \\Lambda ^{i\\chi } )$ and $( \\Lambda ^{\\chi ^\\prime } , \\Lambda ^{i\\chi ^\\prime } )$ are equal.",
"Suppose $[\\chi ]$ and $[\\chi ^\\prime ]$ are conjugate in $P\\Gamma $ .",
"Then there is a $g\\in \\Gamma $ such that $[g\\chi g^{-1}]=[\\chi ^\\prime ]$ .",
"This implies that $g\\chi g^{-1} = \\lambda \\chi ^\\prime $ for some unit $\\lambda \\in \\mathcal {G}^\\ast $ so that the antiunitary involutions $\\chi $ and $\\lambda \\chi ^\\prime $ are conjugate in $\\Gamma $ .",
"From this we deduce that $\\Lambda ^\\chi \\cong \\Lambda ^{\\lambda \\chi ^\\prime }$ .",
"Since $g\\in \\Gamma $ commutes with multiplication by $i$ the involutions $i\\chi $ and $i\\lambda \\chi ^\\prime $ are also conjugate in $\\Gamma $ and we get $\\Lambda ^{i\\chi } \\cong \\Lambda ^{i\\lambda \\chi ^\\prime }$ .",
"Proposition 4.9 Let $P\\Gamma ^\\chi $ be the stabilizer of $\\mathbb {B}^\\chi $ in $P\\Gamma $ .",
"Then we have $ P\\Gamma ^\\chi = \\lbrace [g] \\in P\\Gamma \\ ; \\ [g] \\circ [\\chi ] = [\\chi ] \\circ [g] \\rbrace $ The following statements are equivalent: $&[g]\\in P\\Gamma ^\\chi ,&\\\\&[gx] \\in \\mathbb {B}^\\chi &\\text{ for all } [x]\\in \\mathbb {B}^\\chi ,\\\\&[\\chi (gx)] = [g(\\chi x)] &\\text{ for all } [x]\\in \\mathbb {B}^\\chi ,\\\\&[\\chi (gz)] = [g(\\chi z)] &\\text{ for all } [z]=[x+iy] \\ , \\ [x],[y] \\in \\mathbb {B}^\\chi , \\\\&[g \\circ \\chi ] = [\\chi \\circ g].&$ From Proposition REF we see that for every element $[g]\\in P\\Gamma ^\\chi $ precisely one of the following holds: There is a $g\\in [g]$ such that: $g\\chi g^{-1}=\\chi $ so that: $g\\Lambda ^\\chi = \\Lambda ^\\chi $ .",
"There is a $g\\in [g]$ such that $g\\chi g^{-1}=i\\chi $ so that: $g\\Lambda ^\\chi = \\Lambda ^{i\\chi }$ .",
"We use Chu's convention from [7] and say that $[g]\\in P\\Gamma ^\\chi $ is of type $I$ or of type $II$ respectively.",
"The elements of type $I$ form a subgroup of $P\\Gamma ^\\chi $ which we denote by $P\\Gamma ^\\chi _I$ .",
"If there exists an element of type $II$ then this subgroup is of index 2, otherwise every element of $P\\Gamma ^\\chi $ is of type $I$ .",
"Every element $[g]\\in P\\Gamma ^\\chi $ of type $I$ determines a unique element in $PO(\\Lambda ^\\chi )$ so there is a natural embedding $P\\Gamma ^\\chi _I \\hookrightarrow PO(\\Lambda ^\\chi )$ .",
"In general not every element $[g]\\in PO(\\Lambda ^\\chi )$ extends to the group $P\\Gamma $ .",
"Let $B^\\chi $ be a matrix whose columns represent a basis for the lattice $\\Lambda ^\\chi $ in $\\Lambda $ .",
"Then we have $P\\Gamma ^\\chi _I = \\lbrace [M]\\in PO(\\Lambda ^\\chi ) \\ ; \\ B^\\chi M(B^\\chi )^{-1} \\in \\operatorname{Mat}_{n+1}(\\mathcal {G}) \\rbrace $ so that $P\\Gamma ^\\chi _I$ is the subgroup of $PO(\\Lambda ^\\chi )$ consisting of all elements that extend to unitary transformations of the Gaussian lattice $\\Lambda $ .",
"Theorem 4.10 The groups $P\\Gamma ^\\chi $ and $PO(\\Lambda ^\\chi )$ are commensurable.",
"We have seen that the intersection of the two groups is given by $ P\\Gamma ^\\chi \\cap PO(\\Lambda ^\\chi ) = P\\Gamma ^\\chi _I$ and has at most index 2 in $P\\Gamma ^\\chi $ .",
"We now prove that this intersection is a congruence subgroup of $PO(\\Lambda ^\\chi )$ so that in particular it has finite index.",
"Recall that the adjoint matrix $B^{\\operatorname{adj}}$ has coefficients in $\\mathcal {G}$ and satisfies $(\\det B)B^{-1}=B^{\\operatorname{adj}}$ .",
"If we write $M=1+X$ then by Equation REF we have $[M]\\in P\\Gamma ^\\chi _I$ if and only if $\\det B$ divides $BXB^{\\operatorname{adj}}$ .",
"This is certainly the case if $\\det B$ divides $X$ so if $M \\equiv 1 \\bmod {(\\det B)}$ .",
"This implies that $P\\Gamma ^\\chi _I$ contains the principal congruence subgroup $ \\lbrace [M]\\in PO(\\Lambda ^\\chi ) \\ ; \\ M \\equiv 1 \\bmod {(\\det B)} \\rbrace .",
"$ In the examples we encounter the lattice $\\Lambda ^\\chi $ is reflective so that the reflections generate a finite index subgroup in $PO(\\Lambda ^\\chi )$ .",
"By the results of Section the group $PO(\\Lambda ^\\chi )$ is of the form $W(C)\\rtimes S(C)$ where $C\\subset \\mathbb {B}^\\chi $ is a Coxeter polytope of finite volume, $W(C)$ its reflection group and $S(C)$ a group of automorphisms of $C$ .",
"The polytope $C$ can be determined by Vinberg's algorithm.",
"We will see that in many cases the reflection subgroup of the group $P\\Gamma ^\\chi _I$ is also of finite index.",
"This can be determined by applying Vinberg's algorithm with the condition that in every step we only accept roots $r$ such that the reflection $s_r \\in PO(\\Lambda ^\\chi )$ satisfies Equation REF .",
"This is equivalent to the condition $\\frac{2r}{h(r,r)} \\in \\Lambda ^\\vee .$ We finish this section by describing how a root mirror $H_r \\in \\mathcal {H}$ can meet the real ball $\\mathbb {B}^\\chi $ .",
"This intersection can be of codimension one or two as shown by the following lemma.",
"Lemma 4.11 Suppose $r \\in \\Lambda $ is a root such that $\\mathbb {B}^\\chi \\cap H_r \\ne \\emptyset $ .",
"Then $\\mathbb {B}^\\chi \\cap H_r$ is equal to $\\mathbb {B}^\\chi \\cap L$ with $L^\\perp $ a lattice in $\\Lambda ^\\chi $ of type $A_1,A_1(2),A_1\\oplus A_1(2)$ or $A_1(2)^2$ .",
"If $x\\in H_r \\cap \\mathbb {B}^\\chi $ then $x$ is fixed by both $s_r$ and $s_{\\chi r}$ so the intersection $H_r \\cap H_{\\chi r}$ is nonempty and we are in the situation of Lemma REF .",
"Suppose that $\\chi [r] =[r]$ .",
"If $\\chi r = \\pm r$ then either $r$ or $ir$ is a root of $\\Lambda ^\\chi $ .",
"Both have norm $-2$ so they span a root system of type $A_1$ .",
"If $\\chi r = \\pm i r$ then one of $(1\\pm i)r$ is a root of $\\Lambda ^\\chi $ .",
"Both have norm $-4$ so they span a root system of type $A_1(2)$ .",
"If $\\chi [r] \\ne [r]$ then the roots $r$ and $\\chi r$ span a rank two Gaussian lattice that is either $(-2)\\oplus (-2)$ or $\\Lambda _2$ according to Lemma REF .",
"The involution $\\chi $ acts on these lattices as the antiunitary involution $\\psi _2^\\prime $ .",
"The fixed point lattice for $(-2)\\oplus (-2)$ is $A_1(2)^2$ as follows from a straightforward computation.",
"For $\\Lambda _2$ we get the fixed point lattice $A_1\\oplus A_1(2)$ as follows from Lemma REF ."
],
[
"The Gaussian lattice $\\Lambda _{1,2}$",
"The lattice $\\Lambda _{1,2}=\\Lambda _2\\oplus (2)$ of signature $(1,2)$ is related to the moduli space $\\mathcal {M}(321^3)$ of eight-tuples of points on $\\mathbb {P}^1$ such that there are unique points of multiplicity 3 and 2 and three distinct points of multiplicity 1.",
"We study the antiunitary involutions of this lattice in some detail.",
"Using Table REF we can immediately write down two antiunitary involutions of $\\Lambda _{1,2}$ , namely $\\psi _2\\oplus \\psi _1$ and $\\psi _2^\\prime \\oplus \\psi _1$ .",
"We will prove that their projective classes are distinct modulo conjugation in $P\\Gamma =PU(\\Lambda _{1,2})$ .",
"There is however a another antiunitary involution of $\\Lambda _{1,2}$ given by $\\psi _3 =M_3\\circ \\operatorname{conj}$ where $M_3$ is the complicated matrix $ M_3=\\begin{pmatrix}-2+i&2-2i&-2-2i\\\\2&-1&2i\\\\1+3i&-2-2i&-3+2i\\end{pmatrix}.$ This antiunitary involution takes on a much simpler form if we change to a different basis for $\\Lambda _{1,2}$ as shown by the following lemma.",
"Lemma 4.12 The Gaussian lattices $\\Lambda _2\\oplus (2)$ and $(-2)\\oplus \\Lambda _{1,1}$ are isomorphic.",
"The antiunitary involution $\\psi _3$ of $\\Lambda _2\\oplus (2)$ maps to the antiunitary involution $\\psi _1\\oplus \\psi _2$ of $(-2)\\oplus \\Lambda _{1,1}$ under this isomorphism.",
"The underlying $\\mathbb {Z}$ -lattices of the Gaussian lattices $\\Lambda _2\\oplus (2)$ and $(-2)\\oplus \\Lambda _{1,1}$ are $D_4\\oplus (2)$ and $U\\oplus U(2)\\oplus A_1$ .",
"Both are even 2-elementary lattices and the invariants $(r_+,r_-,l,\\delta )$ of Theorem REF are easily seen to be $(1,2,3,1)$ for both lattices hence they are isomorphic.",
"An explicit base change is given by $\\overline{B}^t(\\Lambda _2\\oplus (2))B=(-2)\\oplus \\Lambda _{1,1}$ for the unimodular matrix $ B = \\begin{pmatrix}1+i&i&0\\\\1-i&0&1\\\\1&1&i\\end{pmatrix} .",
"$ The final statement follows from the equality $B(\\psi _1\\oplus \\psi _2)\\overline{B}^{-1}=\\psi _3$ .",
"Proposition 4.13 The projective classes of the three antiunitary involutions $\\chi $ given by $\\psi _2\\oplus (2), \\ \\psi _2^\\prime \\oplus (2)$ and $\\psi _3$ of $\\Lambda _{1,2}$ are distinct modulo conjugation in $P\\Gamma $ .",
"The groups $P\\Gamma ^\\chi $ of these involutions are hyperbolic Coxeter groups and their Coxeter diagrams are shown in Table REF .",
"We will use Lemma REF to show that the projective classes of the three antiunitary involutions are not $P\\Gamma $ -conjugate.",
"For this we need to calculate the fixed point lattices of $\\chi $ and $i\\chi $ for all three antiunitary involutions.",
"These can be read off from Table REF for the antiunitary involutions $\\psi _2\\oplus \\psi _1$ and $\\psi _2^\\prime \\oplus \\psi _1$ .",
"For $\\psi _3$ we use Lemma REF combined with Table REF .",
"We also use Lemma REF to simplify the lattices.",
"For example one has $\\Lambda _{1,2}^{i({\\psi _2\\oplus \\psi _1})} &\\cong (4) \\oplus A_1^2 \\\\&\\cong (2) \\oplus A_1 \\oplus A_1(2)$ where the first isomorphism follows from Table REF and Lemma REF and the second follows from Lemma REF .",
"The results are listed in Table REF .",
"The lattices $(2) \\oplus A_1\\oplus A_1(2)$ and $U(2) \\oplus A_1(2)$ in this table are not isomorphic.",
"Indeed, if we scale them by a factor $\\frac{1}{2}$ then one is even while the other is not.",
"This proves that the $P\\Gamma $ -conjugation classes of $\\psi _2\\oplus \\psi _1$ and $\\psi _3$ are distinct.",
"We can distinguish the fixed point lattices of $\\psi _2^\\prime \\oplus \\psi _1$ from the previous two by calculating their discriminants.",
"Table: The three classes of antiunitary involutions of the lattice Λ 1,2 \\Lambda _{1,2}.The moduli space $\\mathcal {M}(321^3)$ has three connected components so the three projective classes actually form a complete set of representatives for $P\\Gamma $ -conjugation classes of antiunitary involutions in $\\Lambda _{1,2}$ .",
"For more information we refer to [19] Section 3.5.",
"The lattice $\\Lambda _{1,6} = \\Lambda _2^3\\oplus (2)$ is related to the moduli space of plane quartic curves.",
"In this section we collect some useful properties of this lattice that will be used in later sections.",
"We start by introducing a very convenient basis.",
"Lemma 4.14 There is a basis $\\lbrace e_1,\\ldots ,e_7\\rbrace $ for $\\Lambda _{1,6}$ so that the basis vectors are enumerated by the vertices of the Coxeter diagram of type $E_7$ as in Figure REF .",
"By this we mean that the basis satisfies $h(e_i,e_j) = {\\left\\lbrace \\begin{array}{ll}-2 & \\text{if } i=j \\\\1+\\operatorname{sign}(j-i)i & \\text{if } i,j \\text{ connected} \\\\0 & \\text{else}.\\end{array}\\right.",
"}$ An example of such a basis is given by the column vectors of the matrix $ B_{E_7} =\\begin{pmatrix}1&-1-i&0&0&0&0&0\\\\0&-1&0&0&0&0&0\\\\0&-1-i&1&-1-i&0&0&0\\\\0&-1&0&-1&0&0&1\\\\0&0&0&-1-i&1&0&0\\\\0&0&0&-1&0&1&0\\\\0&1&0&1&0&0&0\\end{pmatrix}.$ The tetraflections $t_{e_i}\\in U(\\Lambda _{1,6})$ with $i=1,\\ldots ,7$ satisfy the commutation and braid relations of the Artin group $A(E_7)$ of type $E_7$ so that they induce a representation $A(E_7)\\rightarrow U(\\Lambda _{1,6})$ by tetraflections.",
"In fact this homomorphism extends to an epimorphism $A(\\widetilde{E}_7) \\rightarrow U(\\Lambda _{1,6})$ as follows from [10], [14].",
"Hence the lattice $\\Lambda _{1,6}$ is tetraflective.",
"Proposition 4.15 Let $V$ be the orthogonal vectorspace over $\\mathbb {F}_2$ defined by $ V = \\Lambda _{1,6}/(1+i)\\Lambda _{1,6} \\cong (\\mathbb {F}_2)^7 $ with the invariant quadratic form $q(x) \\equiv \\frac{1}{2}h(x,x) \\bmod {2}$ .",
"Reduction modulo $(1+i)$ induces a surjective homomorphism $ U(\\Lambda _{1,6}) \\rightarrow O(V,q) \\cong W(E_7)^+.",
"$ where we denote by $W(E_7)^+$ the Weyl group of type $E_7$ divided modulo its center $\\lbrace \\pm 1\\rbrace $ .",
"This group is generated by the images of the tetraflections $t_{e_i}$ with $i=1,\\ldots ,7$ .",
"The tetraflections $t_{e_i}$ with $i=1,\\dots ,7$ act as reflections on the vectorspace $V$ since their squares act as the identity.",
"This defines a representation of the Weyl group $W(E_7)$ on $V$ .",
"The matrices of these tetraflections modulo $(1+i)$ are identical to the matrices of the simple generating reflections of $W(E_7)$ modulo 2.",
"These act naturally on the $\\mathbb {F}_2$ -vectorspace $V^\\prime = Q/2Q$ where $Q$ is the root lattice of type $E_7$ .",
"This space is equipped with the invariant quadratic form defined by $q^\\prime (x) \\equiv \\frac{1}{2}(x,x) \\bmod {2}$ where $(\\cdot ,\\cdot )$ is the natural bilinear form on $Q$ defined by the Gram matrix of type $E_7$ .",
"We conclude that the representation spaces $(V,q)$ and $(V^\\prime ,q^\\prime )$ for $W(E_7)$ are isomorphic.",
"The proposition now follows from Exercise 3 in §4 of Ch VI of [5] where it is shown that $ 1 \\rightarrow \\lbrace \\pm 1 \\rbrace \\rightarrow W(E_7) \\rightarrow O(V,q) \\rightarrow 1$ is an exact sequence.",
"Let $U(\\Lambda _{1,6})^a$ be the set of antiunitary transformations of $\\Lambda _{1,6}$ .",
"Reduction modulo $(1+i)$ also induces a map $ U(\\Lambda _{1,6})^a \\rightarrow O(V,q)\\cong W(E_7)^+$ since complex conjugation induces the identity map on $V$ .",
"The projective class of an antiunitary involution $[\\chi ]$ maps to an involution $u$ of $W(E_7)^+$ under this map.",
"Its image does not depend on the choice of representative for the class $[\\chi ]$ since multiplication by $i$ acts as the identity on $V$ .",
"This implies that the conjugation class of the involution $u$ in $W(E_7)^+$ is an invariant of the $P\\Gamma $ -conjugation class of $[\\chi ]$ .",
"The conjugation classes of involutions of $W(E_7)$ are determined by Wall[27].",
"This can also be derived from more general results by Richardson [18].",
"We review these results in the Appendix of this article.",
"There are ten conjugation classes that come in five pairs $\\lbrace u,-u\\rbrace $ .",
"Since both $u$ and $-u$ map to the same involution $\\overline{u} \\in W(E_7)^+$ each pair determines a unique conjugation class in $W(E_7)^+$ .",
"We will use this to prove the following theorem.",
"Theorem 4.16 If we define (in the notation of Table REF and Lemma REF ) antiunitary involutions $\\chi _i$ of $\\Lambda _2^3\\oplus (2)$ for $i=1,\\ldots ,6$ by $\\chi _1 &=\\psi _2^3 \\oplus \\psi _1 &\\chi _2 &=\\psi _2^2 \\oplus \\psi _2^\\prime \\oplus \\psi _1 &\\chi _3 &=\\psi _2 \\oplus (\\psi _2^\\prime )^2 \\oplus \\psi _1 \\\\\\chi _4 &= (\\psi _2^\\prime )^3 \\oplus \\psi _1 &\\chi _5 &= \\psi _4 \\oplus \\psi _2 \\oplus \\psi _1 &\\chi _6 &= \\psi _4 \\oplus \\psi _3$ then their projective classes are distinct modulo conjugation by $P\\Gamma $ .",
"According to Lemma REF and the previous example it is clear that the $\\chi _i$ are antiunitary involutions of the lattice $\\Lambda _2^3\\oplus (2)$ .",
"By reducing the $\\chi _i$ modulo $(1+i)$ they map to involutions $\\overline{u}_i$ in $W(E_7)^+$ .",
"To distinguish them we calculate the dimensions of the fixed point spaces in $V$ and compare them to those of the involutions in $W(E_7)^+$ .",
"From this we conclude that $\\overline{u}_1,\\overline{u}_2$ and $\\overline{u}_4$ are of type $(1,E_7),(A_1,D_6)$ and $(A_1^3,A_1^4)$ respectively.",
"It is clear that $\\overline{u}_5=\\overline{u}_6$ .",
"We used the computer algebra package SAGE to determine that both are of type $(D_4,A_1^{3\\prime })$ and that $\\overline{u}_3$ is of type $(A_1^2,D_4A_1)$ .",
"All of this is summarized in Table REF .",
"Table: The six projective classes of antiunitary involutions of Λ 1,6 \\Lambda _{1,6} and the type of the involution they induce in W(E 7 ) + W(E_7)^+ by reducing modulo 1+i1+i.This method is insufficient to distinghuish the classes of $\\chi _5$ and $\\chi _6$ .",
"For this we determine the fixed point lattice $\\Lambda _{1,6}^{\\chi _i}$ and $\\Lambda _{1,6}^{i\\chi _i}$ for $i=5,6$ and use Lemma REF .",
"The lattices $\\Lambda _{1,6}^{\\chi _5}$ and $\\Lambda _{1,6}^{\\chi _6}$ are both isomorphic to $(2) \\oplus A_1^2 \\oplus D_4(2)$ .",
"The lattice $\\Lambda _{1,6}^{i\\chi _5}$ is isomorphic to $(2) \\oplus A_1 \\oplus A_1(2) \\oplus D_4(2) \\cong (2) \\oplus A_1^3 \\oplus A_1(2)^3 $ where we used Lemma REF .",
"The fixed point lattice $\\Lambda _{1,6}^{i\\chi _6}$ is isomorphic to $U(2) \\oplus A_1(2) \\oplus D_4(2)$ .",
"After scaling by a factor $\\frac{1}{2}$ we see that $\\Lambda _{1,6}^{i\\chi _5}$ is odd while the $\\Lambda _{1,6}^{i\\chi _6}$ is even so that they are not isomorphic.",
"Consequently the $P\\Gamma $ -conjugacy classes of the $[\\chi _5]$ and $[\\chi _6]$ are distinct.",
"Table: The fixed point lattices for χ j \\chi _j and iχ j i\\chi _j for j=1,...,6j=1,\\ldots ,6 and their discriminants.Remark 4.17 The question remains whether the list of antiunitary involutions from Theorem REF is complete.",
"This is in fact the case as we will see in Proposition REF .",
"It is a consequence of the fact that the moduli space of smooth real quartics consists of six connected components.",
"Theorem 4.18 The hyperbolic lattices $\\Lambda _{1,6}^\\chi $ for $\\chi =\\chi _j,i\\chi _j$ where $j=1,\\ldots ,6$ from Table REF are all reflective and the hyperbolic Coxeter diagrams for the groups $PO(\\Lambda _{1,6}^\\chi )$ are shown in Figure REF .",
"Figure: The Coxeter diagram of the groups PO(Λ 1,6 χ )PO(\\Lambda _{1,6}^{\\chi }) for χ=χ j ,iχ j \\chi =\\chi _j,i\\chi _j with j=1,...,6j=1,\\ldots ,6.We observe from Table REF that there are seven distinct hyperbolic lattices.",
"To prove that they are reflective we apply Vinberg's algorithm.",
"We demonstrate this for the hyperbolic lattice $(2)\\oplus A_1^2\\oplus D_4(2)$ corresponding to the antiunitary involutions $\\chi _5$ and $\\chi _6$ .",
"Let $\\lbrace e_0,e_1,e_2\\rbrace $ be an orthonormal basis for $(2)\\oplus A_1^2$ .",
"Recall that the root lattice $D_4(2)$ is given by $ D_4(2) = \\lbrace (u_1,u_2,u_3,u_4)\\in \\mathbb {Z}^4 \\ ; \\ \\sum _{i=0}^4 u_i \\equiv 0 \\pmod {-2}\\rbrace .",
"$ It contains roots of norm $-4$ and $-8$ and both form a root system of type $D_4$ .",
"Together these roots form a root system of type $F_4$ .",
"If we choose the controlling vector $e_0$ the height 0 root system is of type $B_2F_4$ spanned by the roots $\\lbrace r_1,\\ldots ,r_6\\rbrace $ from Table REF .",
"This table also shows how the algorithm proceeds.",
"The resulting Coxeter diagram is shown in Figure REF .",
"The Coxeter diagrams for the other six hyperbolic lattices can be computed similarly and are also shown in this figure.",
"Table: Vinberg's algorithm for the hyperbolic lattice (2)⊕A 1 2 ⊕D 4 (2)(2)\\oplus A_1^2\\oplus D_4(2).",
"This lattice corresponds to the two antiunitary involutions χ 5 \\chi _5 and χ 6 \\chi _6."
],
[
"Kondo's period map",
"In this section we review Kondo's construction of a period map for complex plane quartic curves [13].",
"Let $C$ be a smooth quartic curve in $\\mathbb {P}^2$ defined by a homogeneous polynomial $f(x,y,z)$ of degree four.",
"We briefly recall some terminology from Mumford's geometric invariant theory of quartic curves [16].",
"A complex quartic curve is called stable if it has at worst ordinary nodes and cusps as singularities and semistable if it has at worst tacnodes as singularities or is a smooth conic of multiplicity two.",
"We define the surface $X$ to be the fourfold cyclic cover of $\\mathbb {P}^2$ ramified along $C$ so that $X = \\lbrace w^4 = f(x,y,z) \\rbrace \\subset \\mathbb {P}^3.$ The surface $X$ is a $K3$ -surface of degree four with an action of the group of covering transformations of the cover $\\pi :X\\rightarrow \\mathbb {P}^2$ .",
"This group is cyclic of order four and a generator is given by the transformation $ \\rho _X \\cdot [w:x:y:z]=[i w : x: y: z].$ The involution $\\tau _X=\\rho _X^2$ also acts on $X$ and the quotient surface $Y=X/\\tau _X$ is a double cover of $\\mathbb {P}^2$ ramified over the quartic $C$ .",
"It is a del Pezzo surface of degree two.",
"The situation is summarized by the following commutative diagram.",
"[row sep = small] X dl1 dd Y dr2 P2 The cohomology group $H^2(X,\\mathbb {Z})$ is even, unimodular of signature $(3,19)$ and so is isomorphic to the $K3$ lattice $L=E_8^2\\oplus U^3$ .",
"A choice of isomorphism $\\phi : H^2(X,\\mathbb {Z}) \\rightarrow L$ is called a marking of $X$ .",
"We fix a marking and let $\\rho $ and $\\tau $ denote the automorphisms of $L$ induced by $\\rho _X$ and $\\tau _X$ .",
"Kondo [13] proves that the eigenlattices of $\\tau $ for the eigenvalues $+1$ and $-1$ are isomorphic to $L_+ \\cong A_1^7 \\oplus (2) \\quad , \\quad L_- \\cong D_4^3 \\oplus (2)^2 .$ Remark 5.1 The expression for $L_-$ in Equation REF is different from the lattice $U(2)^2 \\oplus D_8 \\oplus A_1^2$ given by Kondo.",
"Since the lattice $L_-$ is even and 2-elementary its isomorphism type is determined by the invariants $(r_+,r_-,a,\\delta )$ from Theorem REF .",
"These invariants are $(2,12,8,1)$ for both lattices so that the lattices are isomorphic.",
"For the lattice $U\\oplus U(2) \\oplus D_4^2 \\oplus A_1^2$ the invariants also take these values so that it is isomorphic to the previous two lattices.",
"For applications later on it is convenient to have a more explicit desciption of the involution $\\tau $ .",
"This is provided by the following lemma.",
"Lemma 5.2 Let $L=U^3\\oplus E_8^2$ be the $K3$ lattice.",
"The involution $\\tau $ is conjugate in $O(L)$ to the involution given by $-I_2 \\oplus \\begin{pmatrix}0&I_2\\\\I_2&0\\end{pmatrix} \\oplus u \\oplus u$ where $u\\in O(E_8)$ is an involution of type $D_4A_1$ .",
"Since the involution $u$ is of type $D_4A_1$ , its negative $-u$ is of type $A_1^3$ .",
"This implies that the eigenlattice for the eigenvalue 1 of $u$ in $E_8$ is isomorphic to $A_1^3$ .",
"The $\\pm 1$ eigenlattices in $L$ of the involution in Equation REF are then given by $\\begin{aligned}U(2) \\oplus A_1^6 &\\cong (2) \\oplus A_1^7 \\\\U \\oplus U(2) \\oplus D_4^2 \\oplus A_1^2 &\\cong D_4^3 \\oplus (2)^2\\end{aligned}$ These eigenlattices are isomorphic to those of $\\tau $ in Equation REF .",
"The lattice $(2)\\oplus A_1^7$ has a unique embedding into the $K3$ lattice $L$ up to automorphisms in $O(L)$ by Theorem REF .",
"This implies that the involution of Equation REF is conjugate to $\\tau $ in $O(L)$ .",
"The map $\\pi _1$ induces a primitive embedding of lattices $\\pi _1^\\ast :\\operatorname{Pic}Y \\rightarrow \\operatorname{Pic}X$ and the image is precisely the lattice $\\phi ^{-1}(L_+)$ .",
"It is the Picard group of the del Pezzo surface $Y$ scaled by a factor two which comes from the fact that the map $\\pi _1$ is of degree two.",
"The powers $\\rho ,\\rho ^2$ and $\\rho ^3$ act on the lattice $L_-$ without fixed points.",
"This action turns $L_-$ into a Gaussian lattice of signature $(1,6)$ isomorphic to the Gaussian lattice $\\Lambda _{1,6}=\\Lambda _2^3\\oplus (2)$ .",
"From now on we identify $L_-$ considered as a Gaussian lattice with $\\Lambda _{1,6}$ and write $L_-$ for the underlying $\\mathbb {Z}$ -lattice.",
"If $\\gamma \\in \\pi _1^\\ast \\operatorname{Pic}(Y)$ then $(\\omega ,\\gamma )=0$ for all $\\omega \\in H^{2,0}(X,\\mathbb {C})$ so that the complex ball: $ \\mathbb {B} = \\mathbb {P}\\lbrace x \\in \\Lambda _{1,6} \\otimes _\\mathcal {G} \\mathbb {C} \\ ; \\ h(x,x)>0 \\rbrace $ is a period domain for smooth plane quartic curves.",
"Let $\\Gamma =U(\\Lambda _{1,6})$ be the unitary group of the Gaussian lattice $\\Lambda _{1,6}$ .",
"Equivalently it is the group of orthogonal transformations of the lattice $L_-$ that commute with $\\rho $ .",
"The period map $\\operatorname{Per}: \\mathcal {Q} \\rightarrow P\\Gamma \\backslash \\mathbb {B}$ is injective by the Torelli theorem for $K3$ surfaces but not surjective.",
"Its image misses certain divisors in $\\mathbb {B}$ which we now describe.",
"An element $r \\in \\Lambda _{1,6}$ is called a root if $h(r,r)=-2$ and for every root we define the mirror $H_r = \\lbrace z\\in \\mathbb {B} \\ ; \\ h(r,z)=0 \\rbrace $ .",
"We denote by $\\mathcal {H}\\subset \\mathbb {B}$ the union of all the root mirrors $H_r$ and write $\\mathbb {B}^\\circ = \\mathbb {B} \\setminus \\mathcal {H}$ .",
"Theorem 5.3 (Kondo) The period map defines an isomorphism of holomorphic orbifolds $ \\operatorname{Per}: \\mathcal {Q} \\rightarrow P \\Gamma \\backslash \\mathbb {B}^\\circ .$ The proof consists of constructing an inverse map of the period map.",
"We give a brief sketch of the main arguments used in [13].",
"Let $z\\in \\mathbb {B}^\\circ $ .",
"There is a $K3$ surface $X$ together with a marking $\\phi :H^2(X,\\mathbb {Z})\\rightarrow L$ such that the period point of $X$ is $z$ .",
"This $K3$ surface $X$ has an automorphism $\\rho _X$ of order four such that its action on $H^2(X,\\mathbb {Z})$ corresponds to the action of $\\rho $ on $L$ .",
"The quotient surface $Y=X/\\left< \\tau _X \\right>$ with $\\tau _X = \\rho _X^2$ is a del Pezzo surface of degree two.",
"Its anticanonical map: $|K_Y|:Y\\rightarrow \\mathbb {P}^2$ is a double cover of $\\mathbb {P}^2$ ramified over a smooth plane quartic curve $C$ .",
"The inverse period map associates to the $P\\Gamma $ -orbit of $z\\in \\mathbb {B}^\\circ $ the isomorphism class of this quartic curve $C$ .",
"Furthermore Kondo proves in [13] Lemma 3.3 that there are two $\\Gamma $ -orbits of roots in $\\Lambda _{1,6}$ .",
"This determines a decomposition $\\mathcal {H}=\\mathcal {H}_n \\cup \\mathcal {H}_h$ where: $\\begin{aligned}\\mathcal {H}_n &= \\left\\lbrace H_r \\in \\mathcal {H} \\ ; \\ H_r\\cap \\Lambda _{1,6} \\cong \\Lambda _2^2\\oplus \\left( {\\begin{matrix}-2&0\\\\0&2\\end{matrix}}\\right) \\right\\rbrace \\\\\\mathcal {H}_h &= \\left\\lbrace H_r \\in \\mathcal {H} \\ ; \\ H_r\\cap \\Lambda _{1,6} \\cong \\Lambda _2^2 \\oplus \\Lambda _{1,1} \\right\\rbrace .\\end{aligned}$ A smooth point of a mirror $H_r \\in \\mathcal {H}_n$ corresponds to a plane quartic curve with a node and a smooth point of a mirror $H_r \\in \\mathcal {H}_h$ corresponds to a smooth hyperelliptic curve of genus three."
],
[
"The lattices $L_+$ and {{formula:f3d75126-77d2-41d6-8bba-fa4905637900}}",
"The main result of this section is Lemma REF which states that an antiunitary involution of the Gaussian lattice $\\Lambda _{1,6}$ can be lifted to an involution of the $K3$ lattice such that its fixed point lattice is of hyperbolic signature.",
"This will be an important ingredient in the proof of one of our main results: the real analogue of Kondo's period map for real quartic curves in Section REF .",
"We start with a detailed analysis of the lattices $L_+$ and $L_-$ .",
"The lattice $L_+ \\cong (2) \\oplus A_1^7$ has an orthogonal basis $\\lbrace e_0,\\ldots ,e_7\\rbrace $ that satisfies $(e_0,e_0)=2$ and $(e_i,e_i)=-2$ for $i=1,\\ldots ,7$ .",
"According to Kondo the automorphism $\\rho $ acts on $L_+$ by fixing the element $k=-3e_0+e_1+\\ldots +e_7$ and acting as $-1$ on its orthogonal complement $k^\\perp $ in $L_+$ .",
"This special element $k$ satisfies $(k,k)=4$ and represents the canonical class of the del Pezzo surface $Y$ .",
"The orthogonal complement $k^\\perp $ is isomorphic to the root lattice $E_7(2)$ .",
"By the results of Section there is an isomorphism of groups: $O(L_+)\\cong O(L_+)^+ \\times \\mathbb {Z}/2\\mathbb {Z}$ where the second factor is generated by $-1\\in O(L_+)$ .",
"The group $O(L_+)^+$ is a hyperbolic Coxeter group as we have seen in Example REF and its Coxeter diagram shown is Figure REF .",
"Figure: The Coxeter diagram of the group O(L + ) + O(L_+)^+.From this diagram we see that the reflections in the long negative simple roots of $L_+$ form a subgroup $W(E_7)<O(L_+)^+$ of type $E_7$ .",
"It is precisely the stabilizer of the element $k \\in L_+$ .",
"Recall from Section that the discriminant group of a lattice $L$ is defined by $A_L=L^\\vee /L$ .",
"Since the dual lattice $L_+^\\vee $ can be naturally identified with the lattice $\\frac{1}{2}L_+$ we have: $A_{L_+} = \\frac{1}{2}L_+/L_+ \\cong (\\mathbb {Z}/2\\mathbb {Z})^8.$ Proposition 5.4 The natural map $O(L_+) \\rightarrow O(A_{L_+})$ maps the subgroup $W(E_7)<O(L_+)^+$ isomorphically onto $O(A_{L_+})$ .",
"The bilinear form on $L_+\\cong (2)\\oplus A_1^7$ is even valued so that a reflection $s_r$ in a short root $r$ of norm $\\pm 2$ satisfies: $ s_r(x) = x\\pm (r,x)r \\equiv x \\bmod {L_+} $ for $x\\in \\frac{1}{2}L_+$ .",
"This implies that these reflections are contained in the kernel of the map $O(L_+)\\rightarrow O(A_{L_+})$ .",
"As a consequence the image of this map is generated by the subgroup $W(E_7)<O(L_+)^+$ of reflections in negative simple long roots.",
"According to Kondo [13] Lemma 2.2 the group $O(A_{L_+})$ is isomorphic to $W(E_7)^+ \\times \\mathbb {Z}/2\\mathbb {Z} \\cong W(E_7)$ .",
"Since the natural map $O(L_+)\\rightarrow O(A_{L_+})$ is surjective, the proposition follows by Theorem REF .",
"The $K3$ lattice $L$ is an even unimodular lattice and the primitive sublattices $L_+$ and $L_-$ satisfy: $L_-^\\perp = L_+$ .",
"According to Proposition REF there is a natural isomorphism $O(A_{L_-}) \\cong O(A_{L_+})$ which allows us to identify these groups.",
"In particular we have $O(A_{L_-})\\cong W(E_7)^+ \\times \\mathbb {Z}/2\\mathbb {Z}$ .",
"We prefer to consider $L_-$ as the Gaussian lattice $\\Lambda _{1,6}$ so that $\\begin{aligned}A_{\\Lambda _{1,6}}&=\\Lambda _{1,6}^\\vee / \\Lambda _{1,6} \\\\& \\cong \\left( \\frac{1}{1+i}\\mathcal {G}/\\mathcal {G} \\right)^6 \\times \\frac{1}{2}\\mathcal {G}/\\mathcal {G}\\end{aligned}$ because $\\Lambda _{1,6} = \\Lambda _{1,5} \\oplus \\mathcal {G}(-2)$ with $\\Lambda _{1,5}=\\Lambda _2^2 \\oplus \\Lambda _{1,1}$ and $\\Lambda _{1,5} = (1+i)\\Lambda _{1,5}^\\vee $ .",
"Remark 5.5 Note that there are isomorphism of additive groups $\\frac{1}{1+i}\\mathcal {G}/\\mathcal {G} \\cong \\mathbb {Z}/2\\mathbb {Z}$ and $\\frac{1}{2}\\mathcal {G}/\\mathcal {G} \\cong \\mathbb {Z}/2\\mathbb {Z} \\times \\mathbb {Z}/2\\mathbb {Z}$ .",
"The generators of this last group are $\\frac{1}{2}$ and $\\frac{i}{2}$ and they are exchanged by multiplication by $i$ .",
"Proposition 5.6 The composition of homomorphisms: $ U(\\Lambda _{1,6}) \\rightarrow O(A_{\\Lambda _{1,6}}) \\cong W(E_7)^+ \\times \\mathbb {Z}/2\\mathbb {Z}$ is given by reduction modulo $1+i$ on the first factor and the second factor is generated by the image of the central element $\\rho \\in U(\\Lambda _{1,6})$ .",
"Let $A_{\\Lambda _{1,6}}^\\prime $ be the subset of $A_{\\Lambda _{1,6}}$ where the discriminant quadratic form takes values in $\\mathbb {Z}/2\\mathbb {Z}$ .",
"The Gaussian lattice $\\Lambda _{1,6}$ satisfies $\\Lambda _{1,6}\\subset (1+i)\\Lambda _{1,6}^\\vee $ so that the following equalities hold: $A_{\\Lambda _{1,6}}^\\prime &= \\lbrace x\\in \\Lambda _{1,6}^\\vee /\\Lambda _{1,6} \\ ; \\ h(x,x)\\in \\mathbb {Z} \\rbrace \\\\&= \\frac{1}{1+i}{\\Lambda _{1,6}} / {\\Lambda _{1,6}}.$ By writing: $h(\\frac{1}{1+i}x,\\frac{1}{1+i}x)=\\frac{1}{2}h(x,x)$ for $x\\in \\Lambda _{1,6}$ we see that the $\\mathbb {F}_2$ -vectorspace $A_{\\Lambda _{1,6}}^\\prime $ with its induced quadratic form $q_{\\Lambda _{1,6}}$ is isomorphic to the quadratic space $(V,q)$ from Proposition REF .",
"According to this proposition there is an isomorphism $O(A_{\\Lambda _{1,6}}^\\prime ) \\cong W(E_7)^+$ and the composition of natural maps: $U(\\Lambda _{1,6})\\rightarrow O(A_{L_-})\\rightarrow O(A_{L_-}^\\prime ) \\cong W(E_7)^+$ corresponds to mapping an element $g\\in U(\\Lambda _{1,6})$ to its reduction $\\overline{g}$ modulo $(1+i)$ .",
"The automorphism $\\rho \\in U(\\Lambda _{1,6})$ corresponds to multiplication by $i$ and by definition commutes with every element in $U(\\Lambda _{1,6})$ .",
"It maps to the identity in $O(A_{\\Lambda _{1,6}}^\\prime )$ but acts as a nontrivial involution in $O(A_{\\Lambda _{1,6}})$ by Remark REF .",
"This implies that $O(A_{\\Lambda _{1,6}})$ is isomorphic to the direct product of $O(A_{\\Lambda _{1,6}}^\\prime )$ with the subgroup $\\mathbb {Z}/2\\mathbb {Z} \\triangleleft O(A_{\\Lambda _{1,6}})$ generated by $\\rho $ .",
"Lemma 5.7 Let $\\chi _- \\in U(\\Lambda _{1,6})^a$ be an antiunitary involution of $\\Lambda _{1,6}$ .",
"There is a unique $\\chi \\in O(L)$ that restricts to $\\chi _-$ on $L_-$ so that the fixed point lattice $L^\\chi $ is of hyperbolic signature.",
"Since complex conjugation on $\\Lambda _{1,6}$ induces the identity on $O(A_{\\Lambda _{1,6}})$ the statement of Proposition REF is also true for the composition of homomorphisms: $ U(\\Lambda _{1,6})^a \\rightarrow O(A_{\\Lambda _{1,6}})\\cong W(E_7)^+\\times \\mathbb {Z}/2\\mathbb {Z}.$ Consider the image of the antiunitary involution $\\chi _- \\in U(\\Lambda _{1,6})^a$ under this composition.",
"This image is of the form $(\\bar{u},\\pm 1)$ where the involution $\\bar{u}\\in W(E_7)^+$ is obtained by reducing $\\chi _- \\in U(\\Lambda _{1,6})^a$ modulo $(1+i)$ .",
"Observe that if the antiunitary involution $\\chi _-$ maps to $(\\bar{u},1)$ then $i\\chi $ maps to $(\\bar{u},-1)$ .",
"The involution $ \\chi _+=(\\pm u,-1) \\in W(E_7) \\times \\mathbb {Z}/2\\mathbb {Z} < O(L_+) $ maps to $(\\bar{u},\\pm 1) \\in O(A_{L_+})$ by Proposition REF .",
"Since $\\chi _+$ maps $k \\mapsto -k$ and $(k,k)=4$ the lattice of fixed points $L_+^{\\chi _+}$ is negative definite.",
"By Proposition REF there is a unique involution $\\chi \\in O(L)$ that restricts to $\\chi _- \\in U(\\Lambda _{1,6})$ and $\\chi _+ \\in O(L_+)$ respectively.",
"Since $\\Lambda _{1,6}^{\\chi _-}$ is of hyperbolic signature and $L_+^{\\chi _+}$ is negative definite the fixed point lattice $L^\\chi $ is of hyperbolic signature.",
"Proposition 5.8 Consider the 12 antiunitary involutions $\\chi _j$ and $i\\chi _j$ for $j=1,\\ldots ,6$ from Theorem REF .",
"For each of them the corresponding involution $\\chi _+ \\in O(L_+)$ is of the form $(u,-1)\\in W(E_7)\\times \\mathbb {Z}/2\\mathbb {Z}$ .",
"The conjugation classes of the involutions $u\\in W(E_7)$ are shown in Table REF .",
"Table: The conjugation classes in W(E 7 )W(E_7) of the 12 antiunitary involutions χ j ,iχ j ∈U(Λ 1,6 ) a \\chi _j,i\\chi _j \\in U(\\Lambda _{1,6})^a for j=1,...,6j=1,\\ldots ,6.This follows from Table REF and the proof of Lemma REF ."
],
[
"Periods of real quartic curves",
"Let $C=\\lbrace f(x,y,z) = 0 \\rbrace \\subset \\mathbb {P}^2$ be a smooth real plane quartic curve.",
"This means that $C$ is invariant under complex conjugation of $\\mathbb {P}^2(\\mathbb {C})$ or equivalently that the polynomial $f$ has real coefficients.",
"The $K3$ surface $X$ that corresponds to $C$ is also defined by an equation with real coefficients.",
"Complex conjugation on $\\mathbb {P}^3(\\mathbb {C})$ induces an antiholomorphic involution $\\chi _X$ on $X$ .",
"Definition 5.9 A $K3$ surface $X$ is called real if it is equipped with an antiholomorphic involution $\\chi _X$ .",
"We will also call such an involution a real form of $X$ .",
"The real points of $X$ , which we denote by $X(\\mathbb {R})$ , are the fixed points of the real form.",
"Theorem 5.10 Let $\\chi $ be an involution on the $K3$ lattice $L$ .",
"There exists a marked $K3$ surface $(X,\\phi )$ such that $\\chi _X=\\phi ^{-1}\\circ \\chi \\circ \\phi $ induces a real form on $X$ if and only if the lattice of fixed points $L^\\chi $ has hyperbolic signature.",
"See [23] Chapter VIII Theorem $2.3$ .",
"Suppose $(X,\\chi _X)$ is a real $K3$ surface.",
"By choosing a marking we obtain an involution $\\chi $ of the $K3$ lattice $L$ .",
"By Theorem REF the fixed point lattice of this involution $L^\\chi $ is of hyperbolic signature.",
"Since the $K3$ lattice is an even unimodular lattice, the lattice $L^\\chi $ is even and 2-elementary.",
"According to Proposition REF the isomorphism type of $L^\\chi $ is determined by three invariants $(r,a,\\delta )$ where $r=r_++r_-=1+r_-$ .",
"It is clear that these invariants do not depend on the marking of $X$ .",
"The following theorem originally due to Kharlamov [11] shows that they determine the topological type of the real point set $X(\\mathbb {R})$ .",
"We will write $S_g$ for a real orientable surface of genus $g$ and $kS$ for the disjoint union of $k$ copies of a real surface $S$ .",
"Theorem 5.11 (Nikulin [17] Thm.",
"3.10.6) Let $(X,\\chi _X)$ be a real $K3$ surface.",
"Then: $X(\\mathbb {R}) = {\\left\\lbrace \\begin{array}{ll}\\emptyset & \\text{if } (r,a,\\delta ) = (10,10,0) \\\\ 2S_1 & \\text{if } (r,a,\\delta ) = (10,8,0) \\\\S_g \\sqcup kS_0 & \\text{otherwise} \\end{array}\\right.",
"}$ where $g=\\frac{1}{2}(22-r-a)$ and $k=\\frac{1}{2}(r-a)$ .",
"Remark 5.12 There are two antiholomorphic involutions on the $K3$ surface $X = \\lbrace w^4 = f(x,y,z)\\rbrace $ .",
"Since we chose the sign of $f(x,y,z)$ to be positive on the interior of the curve $C(\\mathbb {R})$ the antiholomorphic involution $\\chi _X$ is determined without ambiguity.",
"By fixing a marking $\\phi :H^2(X,\\mathbb {Z}) \\rightarrow L$ of the $K3$ surface $X$ we associate to $\\chi _X$ the involution: $ \\chi =\\phi \\circ \\chi _X^\\ast \\circ \\phi ^{-1}$ of the $K3$ lattice $L$ .",
"Since the involution $\\chi $ commutes with $\\tau $ it preserves the $\\pm 1$ -eigenlattices of the involution $\\tau $ .",
"We denote by $\\chi _-$ (resp.",
"$\\chi _+$ ) the induced involution on $L_-$ (resp.",
"$L_+$ ).",
"It is clear that $\\chi $ and $\\rho $ satisfy the relation: $ \\rho \\circ \\chi = \\tau \\circ \\chi \\circ \\rho $ so that on the eigenlattice $L_-$ where $\\tau $ acts as $-1$ they anticommute and on $L_+$ they commute.",
"This implies that $\\chi _-$ is an antiunitary involution of the Gaussian lattice $\\Lambda _{1,6}$ .",
"By the results of Section REF on Kondo's period map we can associate to a smooth real plane quartic curve $C$ a period point $[x]\\in \\mathbb {B}^\\circ $ and the real form $[\\chi _-]$ of $\\mathbb {B}$ we just defined fixes $[x]$ .",
"The following lemma shows that the $P\\Gamma $ -conjugation class of $[\\chi _-]$ does not change if we vary $C$ in its connected component of $\\mathcal {Q}^\\mathbb {R}$ .",
"Lemma 5.13 If two smooth real plane quartic curves $C$ and $C^\\prime $ are real isomorphic then the projective classes $[\\chi _-]$ and $[\\chi _-^\\prime ]$ of their corresponding antiunitary involutions in $\\Lambda _{1,6}$ are conjugate in $P\\Gamma $ .",
"Since $C$ and $C^\\prime $ are real plane curves a real isomorphism $C\\rightarrow C^\\prime $ is induced from an element in $PGL(3,\\mathbb {R})$ .",
"We can lift this element to $PGL(4,\\mathbb {R})$ so that it induces an isomorphism $\\alpha _C:X\\rightarrow X^\\prime $ that commutes with the covering transformations $\\rho _X$ and $\\rho _{X^\\prime }$ of $X$ and $X^\\prime $ .",
"Since the real forms $\\chi _X$ and $\\chi _X^\\prime $ of $X$ and $X^\\prime $ are both induced by complex conjugation on $\\mathbb {P}^3$ they satisfy $\\chi _X^\\prime = \\alpha _C \\circ \\chi _X \\circ \\alpha _C^{-1}$ .",
"By fixing markings of the $K3$ surfaces $X$ and $X^\\prime $ we obtain induced orthogonal transformations $\\chi ,\\chi ^\\prime $ and $\\alpha $ of the $K3$ -lattice $L$ such that $\\chi ^\\prime = \\alpha \\circ \\chi \\circ \\alpha ^{-1}$ .",
"Since $\\alpha $ commutes with $\\rho $ the restriction $\\alpha _-$ of $\\alpha $ to $L_-$ is contained in $\\Gamma $ .",
"This proves the lemma.",
"Let $\\mathbb {B}^{\\chi _-}$ be the fixed point set in $\\mathbb {B}$ of the real form $[\\chi _-]$ .",
"The fixed point lattice $\\Lambda _{1,6}^{\\chi _-}$ has hyperbolic signature $(1,6)$ so that $\\mathbb {B}^{\\chi _-}$ is the real hyperbolic ball $ \\mathbb {B}^{\\chi _-} = \\mathbb {P}\\lbrace x\\in \\Lambda _{1,6}^{\\chi _-} \\otimes _\\mathbb {Z} \\mathbb {R} \\ ; \\ h(x,x) >0 \\rbrace .$ As before we denote by $P\\Gamma ^{\\chi _-}$ the stabilizer of $\\mathbb {B}^{\\chi _-}$ in the ball $\\mathbb {B}$ .",
"Since the period point of a smooth real quartic curve $C$ is fixed by $[\\chi _-]$ it lands in the real ball quotient: $P\\Gamma ^{\\chi _-}\\backslash (\\mathbb {B}^{\\chi _-})^\\circ $ .",
"This gives rise to a real period map.",
"More precisely we have the following real analogue of Theorem REF .",
"Theorem 5.14 The real period map $\\operatorname{Per}^\\mathbb {R}$ that maps a smooth real plane quartic to its period point in $P\\Gamma \\backslash \\mathbb {B}^\\circ $ defines an isomorphism of real analytic orbifolds: $ \\operatorname{Per}^\\mathbb {R}: \\mathcal {Q}^\\mathbb {R} \\rightarrow \\coprod _{[\\chi _-]} P \\Gamma ^{\\chi _-} \\big \\backslash \\left( \\mathbb {B}^{\\chi _-} \\right)^\\circ $ where $[\\chi _-]$ varies over the $P\\Gamma $ -conjugacy classes of projective classes of antiunitary involutions of $\\Lambda _{1,6}$ .",
"We construct an inverse to the real period map.",
"Let $z \\in \\mathbb {B}^\\circ $ be such that $\\chi _-(z)=z$ for a certain antiunitary involution of $\\Lambda _{1,6}$ .",
"From the proof of Theorem REF we see that there is a marked $K3$ surface $X$ that corresponds to $z$ .",
"According to Lemma REF the involution $\\chi _-$ lifts to an involution $\\chi \\in O(L)$ such that for its restriction $\\chi _+$ to $L_+$ the fixed point lattice $L_+^{\\chi _+}$ is negative definite.",
"Since $\\Lambda _{1,6}^{\\chi _-}$ is of hyperbolic signature the lattice $L^\\chi $ is also of hyperbolic signature.",
"According to Theorem REF this implies that the marked $K3$ surface $X$ is real.",
"Its real form $\\chi _X$ commutes with $\\tau _X$ so that it induces a real form on $\\chi _Y$ on the del Pezzo surface $Y=X/\\left< \\tau _X \\right>$ .",
"The anticanonical system $|-K_Y|:Y\\rightarrow \\mathbb {P}^2$ is the double cover of $\\mathbb {P}^2$ ramified over a smooth real plane quartic curve $C$ .",
"The inverse of the real period map associates to the $P\\Gamma ^{\\chi _-}$ orbit of $z\\in (\\mathbb {B}^{\\chi _-})^\\circ $ the real isomorphism class of the real quartic curve $C$ ."
],
[
"The six components of $\\mathcal {Q}^\\mathbb {R}$",
"In this section we complete our description of the real period map $\\operatorname{Per}^\\mathbb {R}$ by connecting the six connected components of the moduli space $\\mathcal {Q}^\\mathbb {R}$ of smooth real plane quartic curves to the six projective classes of antiunitary involutions of the Gaussian lattice $\\Lambda _{1,6}$ from Theorem REF .",
"We first prove that these six antiunitary involutions are in fact all of them.",
"Proposition 5.15 There are six projective classes of antiunitary involutions of the Gaussian lattice $\\Lambda _{1,6}$ up to conjugation by $P\\Gamma $ .",
"Since $\\mathcal {Q}^\\mathbb {R}$ consists of six connected components and the real period map $\\operatorname{Per}^\\mathbb {R}$ is surjective the number of projective classes is at most six.",
"In Theorem REF we found six projective classes of antiunitary involutions up to conjugation by $P\\Gamma $ so these six are all of them.",
"The following corollary follows from the proof of Theorem REF .",
"Corollary 5.16 Suppose $z\\in \\mathbb {B}^\\circ $ is a real period point so that it is fixed by an antiunitary involution $\\chi \\in U(\\Lambda _{1,6})^a$ .",
"By the real period map we associate to $z \\in \\mathbb {B}^\\circ $ a real del Pezzo surface $Y$ of degree two together with a marking $ H^2(Y,\\mathbb {Z})\\rightarrow L_+\\left(\\tfrac{1}{2}\\right) $ such that the induced involution of the real form of $Y$ on $L_+(\\frac{1}{2})$ is given by $\\chi _+$ .",
"We review some results of [27] on real del Pezzo surfaces of degree two.",
"Other references on this subject are Kollár [12] and Russo [20].",
"A real del Pezzo surface $Y$ of degree two is the double cover of the projective plane $\\mathbb {P}^2$ ramified over a smooth real plane quartic curve $C\\subset \\mathbb {P}^2$ so that: $ Y = \\lbrace w^2 = f(x,y,z) \\rbrace .",
"$ We choose the sign of $f$ so that $f>0$ on the orientable interior part of $C(\\mathbb {R})$ .",
"By using the deck transformation $\\rho _Y$ of the cover we see that $\\begin{aligned}\\chi _Y^+: \\left[ w:x:y:z \\right] & \\mapsto [\\bar{w}:\\bar{x}:\\bar{y}:\\bar{z}] \\\\\\chi _Y^-: [w:x:y:z] & \\mapsto [-\\bar{w}:\\bar{x}:\\bar{y}:\\bar{z}] \\\\\\end{aligned}$ are the two real forms of $Y$ .",
"These real forms satisfy $\\chi _Y^-=\\rho _Y \\circ \\chi _Y^+$ and we denote the real point sets of $\\chi _Y^+$ and $\\chi _Y^-$ by $Y^+(\\mathbb {R})$ and $Y^-(\\mathbb {R})$ respectively.",
"Note that $Y^+(\\mathbb {R})$ is an orientable surface while $Y^-(\\mathbb {R})$ is nonorientable.",
"Suppose that $H^2(Y,\\mathbb {Z})\\rightarrow L_+(\\frac{1}{2})$ is a marking of $Y$ .",
"The deck transformation $\\rho _Y$ induces the involution: $ \\rho = (-1,1) \\in W(E_7)\\times \\mathbb {Z}/2\\mathbb {Z}.",
"$ in $O(L_+(\\frac{1}{2}))$ .",
"This implies that the two real forms $\\chi _Y^\\pm $ form a pair $ (\\chi _Y^+,\\chi _Y^-) \\longleftrightarrow (\\pm u , -1) \\in W(E_7)\\times \\mathbb {Z}/2\\mathbb {Z}.",
"$ In [27] Wall determines the correspondence between the conjugation classes of the $u\\in W(E_7)$ and the topological type of $Y(\\mathbb {R})$ .",
"The results are shown in Table REF .",
"We use the notation $kX$ for the disjoint union and $\\# kX$ for the connected sum of $k$ copies of a real surface $X$ .",
"From this table we see that except for the classes of $D_4$ and $A_1^{3\\prime }$ the conjugation class of $u\\in W(E_7)$ determines the topological type of the real plane quartic curve $C(\\mathbb {R})$ .",
"Table: The real topological types of real del Pezzo surfaces of degree two and their corresponding involutions in the Weyl group W(E 7 )W(E_7).Theorem 5.17 The correspondence between the six projective classes of antiunitary involutions of the lattice $\\Lambda _{1,6}$ up to conjugation by $P\\Gamma $ and the real components of $\\mathcal {Q}^\\mathbb {R}$ is given by $ \\mathcal {Q}_j^\\mathbb {R} \\longleftrightarrow \\chi _j \\qquad j=1,\\ldots ,6.",
"$ The index $j$ on the left is given by Table REF and the index $j$ on the right by Table REF .",
"For $j=1,2,3,4$ the statement follows by comparing Table REF and Table REF .",
"Unfortunately this does not work for the projective classes antiunitary involutions $\\chi _5$ and $\\chi _6$ since both correspond to the involutions $D_4\\in W(E_7)$ .",
"To distinguish these two we will prove that the antiunitary involution $i\\chi _6$ extends to an involution of the $K3$ lattice whose real $K3$ surface $X$ has no real points.",
"This implies that the projective class of $\\chi _6$ corresponding to the component $\\mathcal {Q}^\\mathbb {R}_6$ of smooth real quartic curves with no real points.",
"For this let $L=U^3 \\oplus E_8^2$ be the $K3$ lattice and consider the involution: $\\chi = -I_2 \\oplus \\begin{pmatrix}0&I_2\\\\I_2&0\\end{pmatrix} \\oplus \\begin{pmatrix}0&I_8\\\\I_8&0\\end{pmatrix} \\in O(L).$ It is clear from the expression for $\\chi $ that the fixed point lattice $L^\\chi $ is isomorphic to $U(2)\\oplus E_8(2)$ .",
"The invariants $(r,a,\\delta )$ of this lattice are given by $(10,10,0)$ so that $X(\\mathbb {R}) = \\emptyset $ according to Theorem REF .",
"Using the explicit embedding of $L_+$ and $L_-$ into the $K3$ lattice $L$ from Lemma REF it is easily seen that: $L_-^{\\chi _-} \\cong U(2)\\oplus D(4) \\oplus A_1(2) \\quad , \\quad L_+^{\\chi _+} \\cong A_1(2)^3.",
"$ By consulting Table REF we now deduce that $\\chi $ is conjugate to $i\\chi _6$ in $P\\Gamma $ ."
],
[
"The geometry of maximal quartics",
"We now study the component $\\mathcal {Q}_1^\\mathbb {R} \\cong P\\Gamma ^{\\chi _1} \\backslash \\mathbb {B}_6^{\\chi _1}$ that corresponds to $M$ -quartics in more detail.",
"An $M$ -quartic is a smooth real plane quartic curve $C$ such that its set of real points $C(\\mathbb {R})$ consists of four ovals.",
"Much of the geometry of such quartics is encoded by a hyperbolic polytope $C_6 \\subset \\mathbb {B}_6^{\\chi _1}$ .",
"Theorem 5.18 The group $P\\Gamma ^{\\chi _1}$ is isomorphic to the semidirect product $ W(C_6) \\rtimes \\operatorname{Aut}(C_6) $ where $C_6\\subset \\mathbb {B}_6^{\\chi _1}$ is the hyperbolic Coxeter polytope whose Coxeter diagram is shown in Figure REF .",
"Its automorphism group $\\operatorname{Aut}(C_6)$ is isomorphic to the symmetric group $S_4$ .",
"Recall that an element $ [g]\\in P\\Gamma ^\\chi $ is of type $II$ if and only if there is a $g\\in [g]$ such that $g\\Lambda _{1,6}^\\chi = \\Lambda _{1,6}^{i\\chi }$ .",
"We see from Table REF that the lattice $\\Lambda _{1,6}^{\\chi }$ is not isomorphic to the lattice $\\Lambda _{1,6}^{i\\chi }$ so that the group $P\\Gamma ^\\chi $ does not contain elements of type $II$ .",
"Therefore the group $P\\Gamma ^{\\chi _1}$ consists of all element of $PO(\\Lambda ^{\\chi _1}_{6,1})$ that are induced from $U(\\Lambda _{1,6})$ .",
"The lattice $\\Lambda _{1,6}^{\\chi _1}$ is isomorphic to $(2) \\oplus A_1^6$ .",
"A basis $\\lbrace e_0,\\ldots ,e_6\\rbrace $ in $\\Lambda _{1,6}$ is given by the columns of the matrix: $B_1 = \\begin{pmatrix}0&1+i&0&0&0&0&0\\\\0&1&1&0&0&0&0\\\\0&0&0&1+i&0&0&0\\\\0&0&0&1&1&0&0\\\\0&0&0&0&0&1+i&0\\\\0&0&0&0&0&1&1\\\\1&0&0&0&0&0&0\\end{pmatrix}.$ It is a reflective lattice and the group $PO(\\Lambda _{1,6}^{\\chi _1})$ is a Coxeter group whose Coxeter diagram can be found in Figure REF .",
"A reflection $s_{r}\\in PO(\\Lambda _{1,6}^{\\chi _1})$ is induced from $U(\\Lambda _{1,6})$ if and only if the root $r$ satisfies Equation REF .",
"Note that a vector $r=(z_1,\\ldots ,z_7)\\in \\Lambda _{1,6}\\otimes _\\mathcal {G} \\mathbb {Q}$ is contained in $\\Lambda _{1,6}^\\vee $ if and only if $z_i \\in \\frac{1}{1+i}\\mathcal {G}$ for $i=1,\\ldots ,6$ and $z_7 \\in \\frac{1}{2}\\mathcal {G}$ so that we can rewrite this equation as: $\\frac{2(1+i)z_i}{h(r,r)}\\in \\mathcal {G} \\quad \\text{for } i=1,\\ldots ,6 \\quad , \\quad \\frac{4z_7}{h(r,r)}\\in \\mathcal {G}.$ These equations are automatically satisfied if $h(r,r)=-2$ and if $h(r,r)=-4$ they are equivalent to: $(1+i)$ divides $z_i$ for $i=1,\\ldots ,6$ .",
"This can be checked from the matrix $B_1$ .",
"Now we run Vinberg's algorithm with this condition and the result is the hyperbolic Coxeter polytope $C_6$ shown in Figure REF .",
"The vertices $r_1,r_3,r_5$ and $r_{13}$ of norm $-4$ roots form a tetrahedron.",
"Every symmetry of this tetrahedron extends to the whole Coxeter diagram.",
"Consequently the symmetry group of the Coxeter diagram is the symmetry group of a tetrahedron which is isomorphic to $S_4$ .",
"Consider the two elements $s,t\\in PO(\\Lambda _{1,6}^{\\chi _1})$ defined by $\\begin{aligned}s&=s_{e_4-e_6}\\cdot s_{e_3-e_5} \\cdot s_{e_1-e_3} \\cdot s_{e_2-e_4} \\\\t&=s_{e_0-e_1-e_3-e_4} \\cdot s_{e_0-e_1-e_5-e_6}.\\end{aligned}$ The element $s$ has order three and corresponds to the rotation of the tetrahedron that fixes $r_{13}$ and cyclically permutes $(r_1r_5r_3)$ .",
"The element $t$ has order two and corresponds to the reflection of the tetrahedron that interchanges $r_1$ and $r_{13}$ and fixes $r_3$ and $r_5$ .",
"Together these transformations generate $S_4$ .",
"We can check that both are contained in $P\\Gamma ^{\\chi _1}$ by using Equation REF .",
"Figure: The Coxeter diagram of the reflection part of the group PΓ 1 P\\Gamma ^1We see from the Coxeter diagram of the polytope $C_6$ that there are three orbits of roots under the automorphism group $\\operatorname{Aut}(C_6)\\cong S_4$ .",
"The orbit of a root $r$ corresponding to a grey node of norm $-2$ satisfies $r^\\perp \\cong \\Lambda _2^2 \\oplus \\Lambda _{1,1}$ .",
"According to Equation REF the mirror of such a root is of hyperelliptic type.",
"This means that the smooth points of such a mirror correspond to a smooth hyperelliptic genus three curves.",
"The Coxeter diagram of the wall that corresponds to the hyperelliptic root $r_{11}$ is the subdiagram consisting of the nodes belonging to the roots $ \\lbrace r_1,r_2,r_3,r_5,r_6,r_7,r_9,r_{13}\\rbrace .",
"$ It is isomorphic to the Coxeter diagram on the right hand side of Figure REF .",
"This is also the case for the other two hyperelliptic roots so they correspond to the maximal real component of real hyperelliptic genus three curves.",
"The other two orbits of roots satisfy $r^\\perp \\cong \\Lambda _2^2 \\oplus \\left( {\\begin{matrix}-2&0\\\\0&2\\end{matrix}} \\right)$ so that their mirrors are of nodal type.",
"For a white root of norm $-2$ the orthogonal complement $r^\\perp $ in the lattice $\\Lambda _{1,6}^{\\chi _1}$ is isomorphic to $(2)\\oplus A_1^5$ .",
"The smooth points of such a mirror correspond to quartic curves with a nodal singularity such that the tangents at the node are real.",
"Locally such a node is described by the equation $x^2-y^2=0$ .",
"This happens when two ovals touch each other.",
"Since there are four ovals this can happen in $\\binom{4}{2}=6$ ways; hence there are six mirrors of this type.",
"For a nodal root of norm $-4$ the orthogonal complement is given by $r^\\perp \\cong (2)\\oplus A_1^4 \\oplus A_1(2)$ in $\\Lambda _{1,6}^{\\chi _1}$ .",
"The smooth points of such a mirror correspond to quartic curves with a nodal singularity such the tangents at the node are complex conjugate.",
"Locally this is described by $x^2+y^2=0$ .",
"It happens when an oval shrinks to a point which can occur for each of the four ovals, and so there are four mirrors of this type.",
"A point $[x] \\in C_6$ that is invariant under the action of $\\operatorname{Aut}(C_6)\\cong S_4$ corresponds to an $M$ -quartic whose automorphism group is isomorphic to $S_4$ .",
"These points are described by the following lemma.",
"Lemma 5.19 A point $[x]\\in C_6$ with $x=(x_0,\\ldots ,x_6) \\in \\Lambda _{1,6}^{\\chi _1}\\otimes _{\\mathbb {Z}}\\mathbb {R}$ is invariant under $\\operatorname{Aut}(C_6)\\cong S_4$ if and only if it lies on the hyperbolic line segment $ L= \\lbrace (-2b-a,b,a,b,a,b,a) \\ ; \\ a,b,\\in \\mathbb {R}, \\ b \\le a \\le 0 \\rbrace /\\mathbb {R}_+ \\subset C_6.",
"$ The line segment $L$ has fixed distances $d_1,d_2$ and $d_3$ to mirrors of type c] (1) at (0,0) ;,c,fill=LightGray] (1) at (0,0) ; and c,vert] (1) at (0,0) ; respectively, and these distances satisfy $ [\\sinh ^2 d_1:\\sinh ^2 d_2:\\sinh ^2 d_3]=[a^2:b^2:(a-b)^2/2].$ The group $\\operatorname{Aut}(C_6)$ is generated by the two elements $s$ and $t$ from Equation REF .",
"A small computation shows that a point $x \\in \\Lambda _{1,6}^{\\chi _1} \\otimes _\\mathbb {Z} \\mathbb {R}$ is invariant under these two generators if and only if it is of the form $ x = (-2b-a,b,a,b,a,b,a).$ The second statement of the Lemma follows from the formula for hyperbolic distance (Equation REF ) and the equalities $(x,r_i) = {\\left\\lbrace \\begin{array}{ll} -a & i=2,4,6,7,8,9 \\\\-b & i=10,11,12 \\\\a-b & i=1,3,5,13 .\\end{array}\\right.",
"}$ The line segment $L$ connects the vertex $L_0=(-2,1,0,1,0,1,0)\\in C_6$ of type $A_1^6$ to the point $L_1=(-3,1,1,1,1,1,1)$ .",
"A consequence of the real period map of Theorem REF is that there is a unique one-parameter family of smooth plane quartics with automorphism group $S_4$ that corresponds to the line segment $L\\subset C_6$ .",
"This pencil was previously studied by W.L Edge [9].",
"It is described by the following proposition.",
"Proposition 5.20 The one-parameter family of quartic curves $C_t$ by: $ C_t = \\prod (\\pm x \\pm y +z) + t(x^4+y^4+z^4) \\quad , \\quad 0\\le t\\le 1.",
"$ corresponds to the line segment $L$ under the real period map.",
"This family is invariant under permutations of the coordinates $(x,y,z)$ and the transformations: $(x,y,z)\\mapsto (\\pm x,\\pm y,z)$ .",
"Together these generate a group $S_3\\rtimes V_4 \\cong S_4$ .",
"The curve $C_0$ is a degenerate quartic that consists of four lines and has six real nodes corresponding to the intersection points of the lines.",
"For $0<t<1$ the curve $C_t$ is an $M$ -quartic.",
"The quartic $C_1$ has no real points except for four isolated nodes.",
"Remark 5.21 A plane quartic curve with a single node has 22 bitangents, because the 6 bitangents through the node all have multiplicity 2.",
"If a real plane quartic has 4 smooth ovals in the real projective plane, then this curve has 24 real bitangents intersecting the quartic in 2 real points on 2 distinct ovals (indeed, one has 4 such bitangents for each pair of ovals).",
"Each nonconvex oval gives a real bitangent intersecting that oval in 2 real points.",
"Each convex oval gives a real bitangent intersecting the quartic in 2 complex conjugate points.",
"The conclusion is that a real plane quartic with 4 smooth ovals in the real projective plane has 28 bitangents, and therefore this plane quartic curve can have no complex singular points.",
"In turn this implies that the moduli space of maximal real quartics is a contractible orbifold, and is in fact an open convex polytope modulo an action of $S_4$ .",
"The same phenomenon holds for the moduli space of maximal real octics, which is again a contractible orbifold.",
"However nonmaximal real octics with a smooth real locus might have complex singular points, and a similar phenomenon is to be expected for nonmaximal real quartic curves.",
"Figure: The one-parameter family of quartic curves C t C_tRemark 5.22 It would be interesting to also describe the Weyl chambers of the other five components of the moduli space of smooth real plane quartic curves.",
"A similar question can be asked for the other components of the moduli space of smooth real binary octics.",
"For the component that corresponds to binary octics with six points real and one pair of complex conjugate points we managed to compute by hand the Coxeter diagram of this chamber.",
"The result was already much more complicated then the diagram of the polytope $C_5$ of of Figure REF .",
"This leads us to believe that the Coxeter diagrams of the remaining five components of the moduli space of smooth real plane quartics will be even more complicated.",
"Computing them would require implementing our version of Vinberg's algorithm in a computer.",
"We expect that this will produce complicated Coxeter diagrams that do not provide much insight."
],
[
"Appendix: Involutions in Coxeter groups",
"In this Appendix we will determine the conjugation classes of involutions in the Weyl group of type $E_7$ .",
"Weyl groups can be realized as finite Coxeter groups.",
"The classification of conjugacy classes of involutions in a Coxeter group was done by Richardson [18] and Springer [24].",
"Before this the classification of conjugacy classes of elements of finite Coxeter groups was obtained by Carter [6].",
"We will give a brief overview of these results.",
"Definition 5.23 A Coxeter system is a pair $(W,S)$ with $W$ a group presented by a finite set of generators $S=\\lbrace s_1,\\ldots ,s_r\\rbrace $ subject to relations $(s_is_j)^{m_{ij}}=1 \\quad \\text{with} \\quad 1\\le i,j \\le r$ where $m_{ii}=1$ and $m_{ij}=m_{ji}$ are integers $\\ge 2$ .",
"We also allow $m_{ij}=\\infty $ in which case there is no relation between $s_i$ and $s_j$ .",
"These relations are encoded by the Coxeter graph of $(W,S)$ .",
"This is a graph with $r$ nodes labeled by the generators.",
"Nodes $i$ and $j$ are not connected if $m_{ij}=2$ and are connected by an edige if $m_{ij}\\ge 3$ with mark $m_{ij}$ if $m_{ij} \\ge 4$ .",
"For a Coxeter system $(W,S)$ we define an action of the group $W$ on the real vector space $V$ with basis $\\lbrace e_s\\rbrace _{s\\in S}$ .",
"First we define a symmetric bilinear form $B$ on $V$ by the expression $ B(e_i,e_j) = -2 \\cos \\left(\\frac{\\pi }{m_{ij}}\\right).",
"$ Then for each $s_i\\in S$ the reflection: $s_i(x)=x-B(e_i,x)e_i$ preserves this form $B$ .",
"In this way we obtain a homomorphism $W\\rightarrow GL(V)$ called the geometric realization of $W$ .",
"For each subset $I \\subseteq S$ we can form the standard parabolic subgroup $W_I<W$ generated by the elements $\\lbrace s_i ; i \\in I \\rbrace $ acting on the subspace $V_I$ generated by $\\lbrace e_i\\rbrace _{i\\in I}$ .",
"We say that $W_I$ (or also $I$ ) satisfies the $(-1)$ -condition if there is a $w_I\\in W_I$ such that $w_I \\cdot x = -x$ for all $x\\in V_I$ .",
"The element $w_I$ necessarily equals the longest element of $(W_I,S_I)$ .",
"This implies in particular that $W_I$ is finite.",
"Let $I,J\\subseteq S$ , we say that $I$ and $J$ are $W$ -equivalent if there is a $w\\in W$ that maps $\\lbrace e_i \\rbrace _{i \\in I}$ to $\\lbrace e_j\\rbrace _{j \\in J}$ .",
"Now we can formulate the main theorem of [18].",
"Theorem 5.24 (Richardson) Let $(W,S)$ be a Coxeter system and let $\\mathcal {J}$ be the set of subsets of $S$ that satisfy the $(-1)$ -condition.",
"Then: If $c \\in W$ is an involution, then $c$ is conjugate in $W$ to $w_I$ for some $I \\in \\mathcal {J}$ .",
"Let $I,J \\in \\mathcal {J}$ .",
"The involutions $w_I$ and $w_J$ are conjugate in $W$ if and only if $I$ and $J$ are $W$ -equivalent.",
"This theorem reduces the problem of finding all conjugacy classes of involutions in $W$ to finding all $W$ -equivalent subsets in $S$ satisfying the $(-1)$ -condition.",
"First we determine which subsets $I\\subseteq S$ satisfy the $(-1)$ -condition, then we present an algorithm that determines when two subsets $I,J\\subseteq S$ are $W$ -equivalent.",
"If $(W_I,S_I)$ is irreducible and satisfies the $(-1)$ -condition then it is of one of the following types $A_1,B_n,D_{2n},E_7,E_8,F_4,G_2,H_3,H_4,I_2(2p)$ with $n,p\\in \\mathbb {N}$ and $p\\ge 4$ .",
"If $(W_I,S_I)$ is reducible and satisfies the $(-1)$ -condition then $W_I$ is the direct product of irreducible, finite standard parabolic subgroups $(W_i,S_i)$ from (REF ).",
"The Coxeter diagrams of the $(W_i,S_i)$ occur as disjoint subdiagrams of the types in the list of the diagram of $(W,S)$ .",
"The element $w_I$ is the product of the $w_{I_i}$ which act as $-1$ on the $V_{I_i}$ .",
"Now let $K \\subseteq S$ be of finite type and let $w_K$ be the longest element of $(W_K,S_K)$ .",
"The element $\\tau _K=-w_K$ defines a diagram involution of the Coxeter diagram of $(W_K,S_K)$ which is nontrivial if and only if $w_K \\ne -1$ .",
"If $I,J\\subseteq K$ are such that $\\tau _K I = J$ then $I$ and $J$ are $W$ -equivalent.",
"To see this, observe that $w_Kw_I \\cdot I = w_K \\cdot (-I) = \\tau _K I=J$ .",
"Now we define the notion of elementary equivalence.",
"Definition 5.25 We say that two subsets $I,J\\subseteq S$ are elementary equivalent, denoted by $I \\vdash J$ , if $\\tau _K I = J$ with $K=I\\cup \\lbrace \\alpha \\rbrace = J\\cup \\lbrace \\beta \\rbrace $ for some $\\alpha ,\\beta \\in S$ .",
"It is proved in [18] that $I$ and $J$ are $W$ -equivalent if and only if they are related by a chain of elementary equivalences: $I = I_1 \\vdash I_2 \\vdash \\ldots \\vdash I_n = J$ .",
"This provides a practical algorithm to determine all the conjugation classes of involutions in a given Coxeter group $(W,S)$ using its Coxeter diagram: Make a list of all the subdiagrams of the Coxeter diagram of $(W,S)$ that satisfy the $(-1)$ -condition.",
"These are exactly the disjoint unions of diagrams in the list (REF ).",
"Every involution in $W$ is conjugate to $w_K$ with $K$ a subdiagram in this list.",
"Find out which subdiagrams of a given type are $W$ -equivalent by using chains of elementary equivalences.",
"Example 5.26 ($E_7$ ) We use the procedure described above to determine all conjugation classes of involutions in the Weyl group of type $E_7$ .",
"This result will be used many times later on.",
"Since $W_7$ contains the element $-1$ the conjugation classes of involutions come in pairs $\\lbrace u,-u\\rbrace $ .",
"We label the vertices of the Coxeter diagram as in Figure REF Figure: The labelling of the nodes of the E 7 E_7 diagramIt turns out that all involutions of a given type are equivalent with the exception of type $A_1^3$ .",
"In that case there are two nonequivalent involutions as seen in Figure REF .",
"The types of involutions that occur are: $\\lbrace 1,E_7\\rbrace \\ ,\\ \\lbrace A_1,D_6\\rbrace \\ ,\\ \\lbrace A_1^2,D_4A_1\\rbrace \\ ,\\ \\lbrace A_1^3,A_1^4\\rbrace \\ ,\\ \\lbrace D_4,A_1^{3\\prime }\\rbrace .$ Figure: The involutions A 1 3 A_1^3 (left) and A 1 3' A_1^{3\\prime } (right).For example, consider the two subdiagrams of type $A_1$ with vertices $\\lbrace 1\\rbrace $ and $\\lbrace 2\\rbrace $ .",
"The diagram automorphism $\\tau _{\\lbrace 1,2\\rbrace }$ which is of type $A_2$ exchanges the vertices $\\lbrace 1\\rbrace $ and $\\lbrace 2\\rbrace $ , so they are elementary equivalent.",
"One shows in a similar way that all diagrams of type $A_1$ are equivalent."
]
] | 1612.05785 |
[
[
"Polynomial-time classical simulation of quantum ferromagnets"
],
[
"Abstract We consider a family of quantum spin systems which includes as special cases the ferromagnetic XY model and ferromagnetic Ising model on any graph, with or without a transverse magnetic field.",
"We prove that the partition function of any model in this family can be efficiently approximated to a given relative error E using a classical randomized algorithm with runtime polynomial in 1/E, system size, and inverse temperature.",
"As a consequence we obtain a polynomial time algorithm which approximates the free energy or ground energy to a given additive error.",
"We first show how to approximate the partition function by the perfect matching sum of a finite graph with positive edge weights.",
"Although the perfect matching sum is not known to be efficiently approximable in general, the graphs obtained by our method have a special structure which facilitates efficient approximation via a randomized algorithm due to Jerrum and Sinclair."
],
[
"Proof of Theorem ",
"In this Appendix we prove Theorem REF , following Section 5 of Ref.",
"[16] closely with a few small modifications.",
"We first introduce some additional notation.",
"Throughout this Section $\\Gamma =(V,E,w)$ is a graph with $|V|=2N$ vertices and positive edge weights $w(e)>0$ for all $e\\in E$ .",
"Recall from the main text that we write $M_k (\\Gamma )$ for the set of all matchings of $\\Gamma $ containing exactly $k$ edges.",
"We also define the set of all matchings $M_{*}(\\Gamma )=\\bigcup _{k=0,1,\\ldots ,N} M_k(\\Gamma )$ .",
"Define a positive weight function on matchings $W(\\Gamma ,M)=\\prod _{e\\in M} w(e)\\qquad \\qquad M\\in M_{*}(\\Gamma )$ and weighted sums $Z_k(\\Gamma )=\\sum _{M\\in M_k(\\Gamma )}W(\\Gamma ,M) \\qquad \\qquad Z(\\Gamma )=\\sum _{M\\in M_*(\\Gamma )} W(\\Gamma ,M).$ Here $\\mathrm {PerfMatch}(\\Gamma )=Z_N(\\Gamma )$ and $\\mathrm {NearPerfMatch}(\\Gamma )=Z_{N-1}(\\Gamma )$ .",
"In the following we say that $X$ approximates $Y$ within ratio $R$ iff $YR^{-1}\\le X\\le YR.$ We use the following theorem which is Corollary 4.3 of Ref. [16].",
"While Ref.",
"[16] does not explicitly state the runtime bound, it is implicit in the proof.",
"Theorem 2 (Approximate sampler [16]) There exists a classical probabilistic algorithm $\\mathcal {A}(\\Gamma ,\\epsilon )$ which takes as input a graph $\\Gamma =(V,E,w)$ and a precision parameter $\\epsilon >0$ , and outputs a matching $M\\in M_{*}(\\Gamma )$ according to a probability distribution $P$ .",
"Moreover, for each $M\\in M_{*}(\\Gamma )$ the probability $P(M)$ approximates $\\frac{W(\\Gamma ,M)}{Z(\\Gamma )}$ within ratio $1+\\epsilon $ .",
"The runtime of the algorithm is $O\\left(|E|^3|V|w_{max}^4 \\log (w_{max}/w_{min})+|E|^2w_{max}^4 \\log (\\epsilon ^{-1})\\right).$ The following “log-concavity” theorem was proven by Heilmann and Lieb in Ref.",
"[24] (a different proof for graphs with uniform edge weights is given in Ref.",
"[16]).",
"Theorem 3 (Theorem 7.1 of Ref.",
"[24]) $Z_k(\\Gamma )^2\\ge Z_{k-1}(\\Gamma )Z_{k+1}(\\Gamma ).$ As a direct consequence of log-concavity we obtain: Corollary 1 $\\frac{1}{\\sum _{e\\in E}{w(e)}}=\\frac{Z_{0}(\\Gamma )}{Z_1(\\Gamma )}\\le \\frac{Z_{1}(\\Gamma )}{Z_{2}(\\Gamma )}\\le \\ldots \\le \\frac{Z_{N-1}(\\Gamma )}{Z_N(\\Gamma )}.$ We are now going to show that, given $\\Gamma $ and an integer $1\\le k\\le N$ , we can modify the edge weights such that that the probability of matchings containing $k$ or $k+1$ edges is non-negligible.",
"We shall write $\\Gamma (\\alpha )$ for the graph obtained from $\\Gamma $ by multiplying all edge weights by $\\alpha $ , i.e., $\\Gamma (\\alpha )=(V,E,w^{\\prime })$ with $w^{\\prime }(e)=w(e)\\alpha $ .",
"Define $P_k(\\alpha )=Z_k(\\Gamma (\\alpha ))/Z(\\Gamma (\\alpha )).$ Lemma 2 Let an integer $1\\le k\\le N$ and $\\Gamma =(V,E,w)$ be given.",
"Suppose $\\frac{Z_{N-1}(\\Gamma )}{Z_N(\\Gamma )}\\le q(N)$ where $q$ is a polynomial.",
"Furthermore, suppose that $\\alpha _k>0$ approximates $\\frac{Z_{k-1}(\\Gamma )}{Z_{k}(\\Gamma )}$ within ratio $1+\\frac{\\epsilon }{2N}$ for some $\\epsilon \\in (0,1)$ .",
"Then $P_k(\\alpha _k)=\\Omega (N^{-1}) \\qquad \\text{and} \\qquad P_{k+1}(\\alpha _k)=\\Omega \\left(N^{-1}|E|^{-1}q(N)^{-1}w_{max}^{-1}\\right).$ In the following for ease of notation we write $\\alpha =\\alpha _k$ .",
"First let $i\\ge k$ and note that $\\frac{P_{k}(\\alpha )}{P_{i}(\\alpha )}=\\frac{Z_k(\\Gamma )}{Z_{i}(\\Gamma )}\\alpha ^{k-i}=\\alpha ^{k-i}\\prod _{j=k}^{i-1}\\frac{Z_j(\\Gamma )}{Z_{j+1}(\\Gamma )}\\ge \\alpha ^{k-i}\\left(\\frac{Z_{k-1}(\\Gamma )}{Z_{k}(\\Gamma )}\\right)^{i-k}\\ge \\left(1+\\frac{\\epsilon }{2N}\\right)^{-N}\\ge \\frac{1}{2}.$ where in the third inequality we used Corollary REF and in the last inequality we used the fact that $(1+\\epsilon /2N)^N\\le 1+\\epsilon \\le 2$ .",
"A symmetric argument establishes the same bound in the case $i<k$ , i.e., $P_{k}(\\alpha ) \\ge P_{i}(\\alpha )/2$ for all $i$ .",
"Since $\\sum _{i=0}^{N}P_i(\\alpha )=1$ this implies $P_{k}(\\alpha )\\ge (2N+2)^{-1}$ which establishes the first claim in Eq.",
"(REF ).",
"We also have $P_{k+1}(\\alpha )=\\alpha \\frac{Z_{k+1}(\\Gamma )}{Z_k(\\Gamma )} P_{k}(\\alpha )\\ge \\left(1+\\frac{\\epsilon }{2N}\\right)^{-1}\\frac{Z_{k-1}(\\Gamma )}{Z_{k}(\\Gamma )} \\frac{Z_{k+1}(\\Gamma )}{Z_k(\\Gamma )} P_{k}(\\alpha )\\ge \\frac{1}{2q(N)\\sum _{e\\in E} w(e)}P_k(\\alpha ).$ where in the last inequality we used Corollary REF twice, along with the fact that $\\frac{Z_{N-1}(\\Gamma )}{Z_N(\\Gamma )}\\le q(N)$ .",
"Substituting $\\sum _{e\\in E}w(e)\\le |E|w_{max}$ and $P_k(\\alpha )=\\Omega (N^{-1})$ gives the second claim in Eq.",
"(REF ).",
"In the remainder of this Section we describe how the following algorithm, denoted $\\mathcal {B}$ , can be used to provide the randomized approximation scheme claimed in Theorem REF .",
"The algorithm takes as input a graph $\\Gamma $ , a positive integer $T$ , a polynomial $q$ and a precision parameter $0<\\delta <1$ .",
"[H] $\\mathcal {B}(\\Gamma ,T,q,\\delta )$ [1] $\\alpha _1 \\leftarrow \\left(\\sum _{e\\in E} w(e)\\right)^{-1}$ Set $\\alpha _1=\\frac{Z_{0}(\\Gamma )}{Z_{1}(\\Gamma )}$ $\\Pi \\leftarrow \\sum _{e\\in E} w(e)$ .",
"Set $\\Pi =Z_1(\\Gamma )$ $k=1$ to $N-1$ $\\alpha _k>2q(N)$ or $\\alpha _k< \\left(2\\sum _{e\\in E} w(e)\\right)^{-1}$ 0 Make $T$ calls to $\\mathcal {A}(\\Gamma (\\alpha _k),\\delta )$ , resulting in outputs $Y=\\lbrace y_1,\\ldots ,y_T\\rbrace \\in M_{*}(\\Gamma )$ .",
"$p_k \\leftarrow T^{-1} |Y\\cap M_{k}(\\Gamma )|$ $p_{k+1} \\leftarrow T^{-1} |Y\\cap M_{k+1}(\\Gamma )|$ $p_k=0$ or $p_{k+1}=0$ 0 $\\alpha _{k+1} \\leftarrow \\alpha _k p_k/p_{k+1}$ $\\alpha _{k+1}$ is our estimate of $\\frac{Z_{k}(\\Gamma )}{Z_{k+1}(\\Gamma )}$ $\\Pi \\leftarrow \\Pi /\\alpha _{k+1}$ $\\Pi $ is our estimate of $Z_{k+1}(\\Gamma )$ $\\Pi $ Theorem 4 Let $q$ be a polynomial and let $\\epsilon >0$ be given.",
"Let $\\Gamma =(V,E,w)$ satisfy $\\frac{Z_{N-1}(\\Gamma )}{Z_N(\\Gamma )}\\le q(N)$ .",
"One can choose $T=\\tilde{\\Theta }\\left(\\epsilon ^{-2} N^4 |E|^2 w_{max}^2q(N)^2\\right) \\qquad \\text{and}\\qquad \\delta =\\Theta (\\epsilon N^{-1})$ such that, with probability at least $3/4$ , the output of algorithm $\\mathcal {B}(\\Gamma ,T,q,\\delta )$ approximates $Z_N(\\Gamma )$ within ratio $1+\\epsilon $ .",
"For each $1\\le k\\le N$ let $\\mathcal {E}_k$ denote the event that the algorithm $\\mathcal {B}(\\Gamma ,T,q,\\delta )$ assigns a value to variable $\\alpha _k$ before terminating and that this value approximates $\\frac{Z_{k-1}(\\Gamma )}{Z_{k}(\\Gamma )}$ within ratio $1+\\frac{\\epsilon }{2N}$ .",
"We shall prove inductively that $\\mathrm {Pr}\\left[\\mathcal {E}_k\\right]\\ge \\left(1-\\frac{1}{4N^2}\\right)^k \\qquad \\qquad 1\\le k\\le N.$ The theorem then follows directly from Eq.",
"(REF ).",
"Let $X$ denote the output of the algorithm.",
"If events $\\mathcal {E}_1,\\mathcal {E}_2,\\ldots \\mathcal {E}_N$ all occur then $X=(\\alpha _1\\alpha _2\\ldots \\alpha _N)^{-1}$ , and $\\mathrm {Pr}\\left[X \\text{ approximates $Z_N(\\Gamma )$ within ratio $(1+\\epsilon /2N)^N$}\\right]\\ge \\left(1-\\frac{1}{4N^2}\\right)^{\\sum _{k=1}^{N} k}.$ Noting that $(1+\\frac{\\epsilon }{2N})^N\\le 1+\\epsilon $ and that $\\left(1-\\frac{1}{4N^2}\\right)^{\\sum _{k=1}^{N} k}\\ge \\left(1-\\frac{1}{4N^2}\\right)^{N^2}\\ge 3/4$ then completes the proof.",
"It remains to establish Eq.",
"(REF ).",
"It holds trivially for $k=1$ since $\\alpha _1=\\left(\\sum _{e\\in E}w(e)\\right)^{-1}=Z_0(\\Gamma )/Z_1(\\Gamma )$ .",
"For the inductive step let us suppose that Eq.",
"(REF ) holds for $k$ .",
"Then $\\mathrm {Pr}[\\mathcal {E}_{k+1}]\\ge \\mathrm {Pr}[\\mathcal {E}_{k+1} | \\mathcal {E}_{k}]\\mathrm {Pr}[\\mathcal {E}_{k}]\\ge \\left(1-\\frac{1}{4N^2}\\right)^{k}\\mathrm {Pr}[\\mathcal {E}_{k+1} | \\mathcal {E}_{k}].$ To complete the proof it suffices to show that the conditional probability above satisfies $\\mathrm {Pr}[\\mathcal {E}_{k+1} | \\mathcal {E}_{k}]\\ge \\left(1-\\frac{1}{4N^2}\\right).$ So now suppose that event $\\mathcal {E}_{k}$ has occured.",
"Let us examine what happens during the $k$ th iteration of the for loop in the algorithm.",
"By our inductive hypothesis, when the algorithm reaches the $k$ th iteration of line 3 we have $\\left(1+\\frac{\\epsilon }{2N}\\right)^{-1} \\frac{Z_{k-1}(\\Gamma )}{Z_{k}(\\Gamma )}\\le \\alpha _{k} \\le \\left(1+\\frac{\\epsilon }{2N}\\right) \\frac{Z_{k-1}(\\Gamma )}{Z_{k}(\\Gamma )}$ and since $ \\left(\\sum _{e\\in E} w(e)\\right)^{-1} \\le \\frac{Z_{k-1}(\\Gamma )}{Z_{k}(\\Gamma )}\\le q(N)$ (by Corollary REF ) we see that the algorithm continues past line 4 without terminating.",
"Let us now consider the values $p_k$ and $p_{k+1}$ which are subsequently assigned in lines 7 and 8.",
"Both of these quantities are averages of i.i.d $0/1$ -valued random variables: $p_{k}=\\frac{1}{T}\\sum _{i=1}^{T} \\mathbb {I}_k (y_i) \\qquad p_{k+1}=\\frac{1}{T}\\sum _{i=1}^{T} \\mathbb {I}_{k+1} (y_i) \\qquad \\mathbb {I}_j(M)={\\left\\lbrace \\begin{array}{ll}1, & M\\in M_j(\\Gamma )\\\\0, &\\text{otherwise.}\\end{array}\\right.",
"}$ Here each $y_i$ is drawn from the output of $\\mathcal {A}(\\Gamma (\\alpha _k),\\delta )$ .",
"Applying Theorem REF we see that $\\mathbb {E}[p_j] \\text{ approximates } P_j(\\alpha _k) \\text{ within ratio } (1+\\delta ) \\qquad j=k,k+1.$ Applying Hoeffding's inequality we get $\\mathrm {Pr}\\left[|p_j-\\mathbb {E}[p_j]|\\ge \\mathbb {E}[p_j] \\gamma \\right]\\le 2e^{-2T(\\mathbb {E}[p_j])^2 \\gamma ^2}\\le 2e^{-T(P_j(\\alpha _k))^2 \\gamma ^2/2} \\qquad j=k,k+1.$ where in the last inequality we used the fact that Eq.",
"(REF ) implies $\\mathbb {E}[p_j]\\ge P_j(\\alpha )/2$ .",
"Using Eq.",
"(REF ) and applying Lemma REF we see that $P_k(\\alpha _k)$ and $P_{k+1}(\\alpha _k)$ are bounded as in Eq.",
"(REF ).",
"Thus by choosing $T=\\Theta \\left(\\frac{\\log (N)}{(P_{k+1}(\\alpha _k))^2\\gamma ^2}\\right)=\\tilde{\\Theta }\\left(\\gamma ^{-2} N^2 |E|^2 w_{max}^2q(N)^2\\right),$ we can ensure that the right-hand side of Eq.",
"(REF ) is at most $\\frac{1}{8N^2}$ and therefore, with probability at least $1-\\frac{1}{4N^2}$ we have: $\\frac{p_k}{p_{k+1}} \\text{ approximates } \\frac{\\mathbb {E}[p_k]}{\\mathbb {E}[p_{k+1}]} \\text{ within ratio } \\frac{1+\\gamma }{1-\\gamma }.$ To complete the proof we now show that, for suitably chosen $\\gamma ,\\delta $ , Eqs.",
"(REF ,REF ) together imply that event $\\mathcal {E}_{k+1}$ occurs.",
"Since Eq.",
"(REF ) was shown to hold with probability at least $1-\\frac{1}{4N^2}$ this proves Eq.",
"(REF ).",
"Putting together Eqs.",
"(REF ,REF ) we get that $\\frac{p_k}{p_{k+1}} \\text{ approximates } \\frac{P_k(\\alpha _k)}{P_{k+1}(\\alpha _k)} \\text{ within ratio } R(\\gamma ,\\delta )=\\left(\\frac{1+\\gamma }{1-\\gamma }\\right)\\left(1+\\delta \\right)^2.$ Choose $\\gamma =\\epsilon /(c_1N)$ and $\\delta =\\epsilon /(c_2N)$ for absolute constants $c_1,c_2>0$ such that $R(\\gamma ,\\delta )\\le 1+\\frac{\\epsilon }{2N}$ .",
"With this choice, and noting that $\\alpha _k\\frac{P_k(\\alpha _k)}{P_{k+1}(\\alpha _k)}=Z_{k}(\\Gamma )/Z_{k+1}(\\Gamma )$ , we see that Eq.",
"(REF ) implies that $\\alpha _{k+1}=\\alpha _k p_k/p_{k+1}$ approximates $Z_{k}(\\Gamma )/Z_{k+1}(\\Gamma )$ within ratio $1+\\frac{\\epsilon }{2N}$ .",
"In other words event $\\mathcal {E}_{k+1}$ occurs.",
"This completes the proof.",
"Finally we now complete the proof of Theorem REF .",
"[Proof of Theorem REF ] Theorem REF states that the output $X$ of the algorithm $\\mathcal {B}(\\Gamma ,T,q,\\delta )$ satisfies $(1+\\epsilon )^{-1}Z_N(\\Gamma )\\le X\\le (1+\\epsilon )Z_N(\\Gamma )$ with probability at least $3/4$ .",
"Using the fact that $(1+\\epsilon )^{-1}\\ge (1-\\epsilon )$ we see that the algorithm provides a randomized approximation scheme for $Z_N(\\Gamma )=\\mathrm {PerfMatch}(\\Gamma )$ (provided that $T,\\delta $ are chosen as specified in Theorem REF ).",
"Now let us upper bound the runtime of the algorithm.",
"Each time the subroutine $\\mathcal {A}$ is called its graph argument has edge weights $w(e)\\cdot \\alpha _k$ for some $k$ , which is always upper bounded by $w_{max} (2q(N))$ due to the condition in Line 4 of the algorithm.",
"Using Eq.",
"(REF ) with $w_{max}\\rightarrow 2w_{max}q(N)$ the runtime of each such call is upper bounded by $\\tilde{O}\\left(N|E|^3w_{max}^4 q(N)^4\\right).$ Multiplying this by the maximum total number $NT$ of calls to $\\mathcal {A}$ and substituting $|V|=2N$ we obtain the claimed runtime bound from theorem REF ."
],
[
"Proof of Lemma ",
"In this Appendix we prove Lemma REF .",
"We begin by stating bounds of the form Eqs.",
"(, ).",
"Proposition 1 For $0<t<1$ we have $g(t)&=e^{-t/2(Y\\otimes Y-X\\otimes X)+E(t)} \\qquad \\;\\;\\Vert E(t)\\Vert \\le t^2 \\\\h(t)&=e^{-t/2(-Y\\otimes Y-X\\otimes X)+F(t)} \\qquad \\Vert F(t)\\Vert \\le t^2.$ We defer the (straightforward) proof of Proposition REF .",
"We shall also use the following bound.",
"Lemma 3 Let $H_1,H_2,\\ldots , H_L$ be Hermitian operators with $\\Vert H_i\\Vert \\le \\delta $ for all $i$ , where $0\\le \\delta L\\le 1/2$ .",
"Define $C=e^{H_L/2}e^{H_{L-1}/2}\\ldots e^{H_1/2}.$ Then there exists a Hermitian operator $\\Delta $ such that $CC^{\\dagger }=e^{H_1+H_2+\\ldots +H_L+\\Delta } \\qquad \\text{and} \\quad \\Vert \\Delta \\Vert \\le 2\\pi (\\delta L)^3.$ Consider a time dependent Hamiltonian $H(t)$ with $t\\in [-L,L]$ defined as follows: $H(t)=\\left\\lbrace \\begin{array}{rcl}H_a/2 &\\mbox{if}& a-1\\le |t|<a \\quad \\mbox{for some $a=1,\\ldots ,L$} \\\\0 && \\mbox{otherwise.}",
"\\\\\\end{array}\\right.$ Let $U(t)$ be the solution of a differential equation $\\frac{dU(t)}{dt}=H(t) U(t), \\quad -L\\le t\\le L.$ We choose initial conditions $U(-L)=I$ .",
"Note that $U(L)=CC^{\\dagger }$ .",
"The Magnus expansion gives $CC^{\\dagger }=U(L)=\\exp {[\\Omega ]}, \\quad \\Omega =\\sum _{k=1}^\\infty \\Omega _k,$ where $\\Omega _1=\\int _{-L}^L dt \\, H(t)=H_1+H_2+\\ldots + H_L,$ and $\\Omega _2=\\frac{1}{2} \\int _{-L}^L dt \\int _{-L}^t ds\\, [H(t),H(s)].$ The norm of the higher order terms can be bounded as $\\Vert \\Omega _k\\Vert \\le \\pi \\left( \\int _{-L}^L \\Vert H(t) \\Vert dt \\right)^k \\le \\pi (\\delta L)^k.$ see page 29 of Ref. [25].",
"Here in the last inequality we used the bound $\\Vert H(t)\\Vert \\le \\delta /2$ .",
"Let us choose $\\Delta =\\Omega -\\Omega _1.$ Since $\\Omega _1$ and $\\Omega $ are hermitian, we infer that $\\Delta $ is hermitian.",
"A direct inspection shows that $\\Omega _2=0$ .",
"Therefore Eq.",
"(REF ) gives $\\Vert \\Delta \\Vert =\\Vert \\Omega -\\Omega _1-\\Omega _2\\Vert \\le \\sum _{k=3}^\\infty \\Vert \\Omega _k\\Vert \\le \\pi \\sum _{k=3}^\\infty (\\delta L)^k \\le \\pi (\\delta L)^3 \\sum _{k=0}^\\infty 2^{-k} = 2\\pi (\\delta L)^3.$ We now use Eqs.",
"(REF ,REF ,) and Lemma REF to prove Lemma REF .",
"[Proof of Lemma REF ] It will be convenient to rewrite the Hamiltonian Eq.",
"(REF ) using coefficients $p_{ij}=(b_{ij}-c_{ij})/2$ and $q_{ij}=(b_{ij}+c_{ij})/2$ , i.e., $H=\\sum _{1\\le i<j\\le n} p_{ij}(-X_iX_j-Y_i Y_j)+\\sum _{1\\le i<j\\le n}q_{ij}(-X_iX_j+Y_i Y_j)+\\sum _{i=1}^{n} d_i(I+Z_i).$ Using the fact that $|c_{ij}|<b_{ij}\\le 1$ we see that $p_{ij},q_{ij}\\in [0,1]$ .",
"Let $0<\\epsilon <1$ and $\\beta >0$ be given.",
"Let $r>2\\beta $ be a positive integer which we will fix later.",
"Define a rescaled Hamiltonian and rescaled coefficients $H^{\\prime }=\\frac{\\beta }{r} H \\qquad p^{\\prime }_{ij}=\\frac{\\beta }{r}p_{ij} \\qquad q^{\\prime }_{ij}=\\frac{\\beta }{r}q_{ij} \\qquad d^{\\prime }_{i}= \\frac{\\beta }{r}d_{i}.$ The rescaled coefficients satisfy $0\\le p^{\\prime }_{ij},q^{\\prime }_{ij},|d^{\\prime }_i|\\le \\frac{\\beta }{r}< \\frac{1}{2}.$ Consider a product $C&=\\prod _{1\\le i\\le n} f_{i}(e^{-d^{\\prime }_i}) \\prod _{1\\le i<j\\le n} g_{ij}(q^{\\prime }_{ij})\\prod _{1\\le i<j\\le n} h_{ij}(p^{\\prime }_{ij})\\\\&=\\prod _{1\\le i\\le n} e^{-d^{\\prime }_i (I+Z_i)/2} \\prod _{1\\le i<j\\le n} e^{-q^{\\prime }_{ij}/2(Y_iY_j-X_iX_j)+E_{ij}} \\prod _{1\\le i<j\\le n}e^{-p^{\\prime }_{ij}/2(-Y_iY_j-X_iX_j)+F_{ij}}$ where in the second line we used Eqs.",
"(REF ,REF ,).",
"Here the Hermitian operators $E_{ij},F_{ij}$ satisfy $\\Vert E_{ij}\\Vert \\le (q^{\\prime }_{ij})^2\\le \\frac{\\beta ^2}{r^2} \\qquad \\Vert F_{ij}\\Vert \\le (p^{\\prime }_{ij})^2\\le \\frac{\\beta ^2}{r^2}.$ The bounds Eq.",
"(REF ) and the fact that $e^{-d^{\\prime }_i}\\le e^{1/2}<2$ ensure that Eq.",
"(REF ) is a product of $n^2$ gates from the set $\\mathcal {G}$ defined in Eq.",
"(REF ).",
"Furthermore, Eq.",
"() has the form Eq.",
"(REF ) with $L=n^2$ , and $\\Vert H_i\\Vert \\le \\max _{jk} \\bigg \\lbrace |2d^{\\prime }_j|, \\; 2q^{\\prime }_{jk}+2\\Vert E_{jk}\\Vert , \\; 2p^{\\prime }_{jk}+2\\Vert F_{jk}\\Vert \\bigg \\rbrace \\le \\left(2\\beta /r+2\\beta ^2/r^2\\right)\\le \\frac{3\\beta }{r}$ where we used Eq.",
"(REF ).",
"Applying Lemma REF and using Eqs.",
"(REF ,REF ) gives $CC^{\\dagger }=\\exp \\left[-H^{\\prime }+\\sum _{1\\le i<j\\le n} (2E_{ij}+2F_{ij})+\\Delta \\right] \\qquad \\Vert \\Delta \\Vert \\le 2\\pi \\left(\\frac{3\\beta n^2}{r}\\right)^3$ as long as our choice of $r$ satisfies $6 \\beta n^2 r^{-1}\\le 1$ (which will be the case, see below).",
"Using Eq.",
"(REF ) and the triangle inequality gives $CC^{\\dagger }=e^{-H^{\\prime }+D} \\qquad \\Vert D\\Vert \\le \\frac{2n^2\\beta ^2}{r^2}+2\\pi \\left(\\frac{3\\beta n^2}{r}\\right)^3,$ and $(CC^{\\dagger })^r=e^{-rH^{\\prime }+rD}\\equiv e^{-\\beta H+Q} \\qquad \\Vert Q\\Vert \\le \\frac{2n^2\\beta ^2}{r}+\\frac{2\\pi 3^3 \\beta ^3 n^6}{r^2}.$ Since $C$ is a product of $n^2$ gates from $\\mathcal {G}$ the left hand side is a product of $J=2n^2r$ such gates.",
"At the end of the proof we will choose $r$ to ensure that $\\Vert Q\\Vert \\le \\epsilon /4$ .",
"Next consider a partial product of the form given on the left hand side of Eq.",
"(REF ).",
"Since $CC^{\\dagger }$ is a product of $2n^2$ gates and $G_JG_{J-1}\\ldots G_1=(CC^{\\dagger })^r$ , we may write $G_jG_{j-1}\\ldots G_i= L_{ij}(CC^{\\dagger })^{K}R_{ij}$ where $K\\ge 0$ and $R_{ij}$ and $L_{ij}$ are each products of at most $2n^2-1$ gates $\\lbrace G_t\\rbrace $ , and furthermore $R_{1j}=L_{iJ}=I$ .",
"Since each gate is of the form $G_t=e^{H_t/2}$ where $H_i$ satisfies Eq.",
"(REF ), we have $\\Vert G_t\\Vert \\le e^{\\frac{3\\beta }{2r}}$ , and thus $\\Vert R_{ij}\\Vert ,\\Vert L_{ij}\\Vert \\le e^{3\\frac{\\beta }{2r} (2n^2-1)}\\le e^{1/2}\\le 2$ where we used Eq.",
"(REF ).",
"Moreover, since the left-hand side of Eq.",
"(REF ) contains $j-i+1$ gates in total and $R_{ij},L_{ij}$ contain at most $2n^2-1$ gates each, we have $0\\le (j-i+1)-2n^2K\\le 2(2n^2-1),$ and therefore $\\left|K-\\frac{(j-i+1)}{2n^2}\\right|\\le 2.$ Combining Eqs.",
"(REF ,REF ) we obtain $G_jG_{j-1}\\ldots G_i=L_{ij}e^{-KH^{\\prime }+KD}R_{ij},$ where $\\Vert KD\\Vert \\le \\Vert rD\\Vert =\\Vert Q\\Vert $ and, using $H^{\\prime }=\\beta /r H$ and Eq.",
"(REF ), $\\Vert KH^{\\prime }- \\frac{(j-i+1)\\beta }{2n^2r}H\\Vert \\le \\frac{2\\beta }{r}\\Vert H\\Vert \\le 4n^2\\beta /r.$ Combining Eqs.",
"(REF ,REF ,REF ) and using the fact that $J=2n^2r$ gives Eq.",
"(REF ) with $\\Vert W_{ij}\\Vert \\le \\frac{4n^2\\beta }{r}+\\Vert Q\\Vert \\le \\frac{4n^2\\beta }{r}+\\frac{2n^2\\beta ^2}{r}+\\frac{2\\pi 3^3 \\beta ^3 n^6}{r^2},$ where in the last inequality we used Eq.",
"(REF ).",
"Now choose $r=O(n^3\\lceil \\beta \\rceil ^2\\epsilon ^{-1})$ to make the right hand side at most $\\epsilon /4$ and such that Eq.",
"(REF ) is also satisfied.",
"This gives Eqs.",
"(REF ,REF ) with $\\Vert Q\\Vert \\le \\Vert W_{ij}\\Vert \\le \\epsilon /4$ .",
"Noting that $J=2n^2r=O(n^5(\\beta ^2+1)\\epsilon ^{-1})$ completes the proof.",
"Finally, we prove Proposition REF .",
"[Proof of Proposition REF ] We have the equality $g(t)=\\exp {\\left[\\frac{1}{2}R(t)(X\\otimes X-Y\\otimes Y)+\\frac{t}{4} R(t)\\left(Z\\otimes I +I\\otimes Z\\right)\\right]}$ for all $t>0$ , where $R(t)=\\frac{1}{\\sqrt{1+t^2/4}}\\cosh ^{-1}(1+t^2/2)=\\frac{1}{\\sqrt{1+t^2/4}}\\log \\left(1+t\\sqrt{1+t^2/4}+t^2/2\\right).$ Using a second order Taylor expansion about $t=0$ one can confirm that $|R(t)-t|\\le t^3/6 \\qquad 0<t< 1.$ Indeed, $R(t)$ has a continuous second derivative on $[0,1]$ , and is thrice differentiable on the open interval $(0,1)$ .",
"Applying Taylor's theorem we obtain $R(t)=t+0\\cdot t^2+\\mathrm {Error} \\qquad \\qquad \\left|\\mathrm {Error}\\right|\\le \\frac{t^3}{3!",
"}\\max _{(0,1)}\\left|\\frac{d^{3}R}{dt^3}\\right| \\qquad \\qquad 0<t<1.$ Here $\\frac{d^{3}R}{dt^3}=\\frac{44t^2-64}{(t^2+4)^3}+\\frac{72t-12t^3}{(t^2+4)^{7/2}}\\cosh ^{-1}(1+t^2/2) \\qquad t>0.$ The first term is negative and has magnitude at most 1 on the interval $(0,1)$ whereas the second term is nonnegative and has magnitude at most $\\frac{72}{4^{7/2}}\\cosh ^{-1}(3/2)=0.54...$ on $(0,1)$ .",
"Therefore $\\max _{(0,1)}\\left|\\frac{d^{3}R}{dt^3}\\right|\\le 1$ and plugging into Eq.",
"(REF ) gives Eq.",
"(REF ).",
"From Eq.",
"(REF ) and $\\Vert X\\otimes X-Y\\otimes Y\\Vert =\\Vert \\left(Z\\otimes I +I\\otimes Z\\right)\\Vert =2$ we obtain $\\left\\Vert \\frac{1}{2}(R(t)-t)(X\\otimes X-Y\\otimes Y)+\\frac{t}{4}R(t)\\left(Z\\otimes I +I\\otimes Z\\right)\\right\\Vert \\le \\frac{t^3}{6}+\\frac{t}{2}\\left(t+\\frac{t^3}{6}\\right)\\le t^2 \\qquad \\qquad 0<t<1.$ Using this bound in Eq.",
"(REF ) we arrive at Eq.",
"(REF ).",
"Finally, from Eq.",
"(REF ) we see that $h(t)=(I\\otimes X)g(t)(I\\otimes X)$ and therefore Eq.",
"(REF ) implies Eq.",
"() where $F(t)=(I\\otimes X)E(t)(I\\otimes X)$ ."
]
] | 1612.05602 |
[
[
"Effect of Nonlinear Energy Transport on Neoclassical Tearing Mode\n Stability in Tokamak Plasmas"
],
[
"Abstract An investigation is made into the effect of the reduction in anomalous perpendicular electron heat transport inside the separatrix of a magnetic island chain associated with a neoclassical tearing mode in a tokamak plasma, due to the flattening of the electron temperature profile in this region, on the overall stability of the mode.",
"The onset of the neoclassical tearing mode is governed by the ratio of the divergences of the parallel and perpendicular electron heat fluxes in the vicinity of the island chain.",
"By increasing the degree of transport reduction, the onset of the mode, as the divergence ratio is gradually increased, can be made more and more abrupt.",
"Eventually, when the degree of transport reduction passes a certain critical value, the onset of the neoclassical tearing mode becomes discontinuous.",
"In other words, when some critical value of the divergence ratio is reached, there is a sudden bifurcation to a branch of neoclassical tearing mode solutions.",
"Moreover, once this bifurcation has been triggered, the divergence ratio must reduced by a substantial factor to trigger the inverse bifurcation."
],
[
"Introduction",
"Neoclassical tearing modes are large-scale magnetohydrodynamical instabilities that cause the axisymmetric, toroidally-nested, magnetic flux-surfaces of a tokamak plasma to reconnect to form helical magnetic island structures on low mode-number rational magnetic flux surfaces.",
"[1] Island formation leads to a degradation of plasma energy confinement.",
"[2] Indeed, the confinement degradation associated with neoclassical tearing modes constitutes a major impediment to the development of effective operating scenarios in present-day and future tokamak experiments.",
"[3] Neoclassical tearing modes are driven by the flattening of the temperature and density profiles within the magnetic separatrix of the associated island chain, leading to the suppression of the neoclassical bootstrap current in this region, which has a destabilizing effect on the mode.",
"[4] The degree of flattening of a given profile (i.e., either the density, electron temperature, or ion temperature profile) within the island separatrix depends on the ratio of the associated perpendicular (to the magnetic field) and parallel transport coefficients.",
"[5] The dominant contribution to the perpendicular transport in tokamak plasmas comes from small-scale drift-wave turbulence, driven by plasma density and temperature gradients.",
"[1] The fact that the density and temperature profiles are flattened within the magnetic separatrix of a magnetic island chain implies a substantial reduction in the associated perpendicular transport coefficients in this region.",
"Such a reduction has been observed in gyrokinetic simulations,[6], [7], [8], [9] as well as in experiments.",
"[10], [11], [12], [13], [14] A strong reduction in perpendicular transport within the magnetic separatrix calls into question the conventional analytic theory of neoclassical tearing modes in which the perpendicular transport coefficients are assumed to spatially uniform in the island region.",
"[5] The aim of this paper is to investigate the effect of the reduction in perpendicular transport inside the separatrix of a neoclassical magnetic island chain, due to profile flattening in this region, on the overall stability of the mode.",
"For the sake of simplicity, we shall only consider the influence of the flattening of the electron temperature profile on mode stability.",
"However, the analysis contained in this paper could be generalized, in a fairly straightforward manner, to take into account the influence of the flattening of the ion temperature and density profiles.",
"Consider a large aspect-ratio, low-$\\beta $ , circular cross-section, tokamak plasma equilibrium.",
"Let us adopt a right-handed cylindrical coordinate system ($r$ , $\\theta $ , $z$ ) whose symmetry axis ($r=0$ ) coincides with the magnetic axis of the plasma.",
"The system is assumed to be periodic in the $z$ -direction with period $2\\pi \\,R_0$ , where $R_0$ is the simulated major plasma radius.",
"It is helpful to define the simulated toroidal angle $\\varphi =z/R_0$ .",
"The coordinate $r$ serves as a label for the unperturbed (by the tearing mode) magnetic flux-surfaces.",
"Let the equilibrium toroidal magnetic field, $B_z$ , and the equilibrium toroidal plasma current both run in the $+z$ direction.",
"Suppose that a neoclassical tearing mode generates a helical magnetic island chain, with $m_\\theta $ poloidal periods, and $n_\\varphi $ toroidal periods, that is embedded in the aforementioned plasma.",
"The island chain is assumed to be radially localized in the vicinity of its associated rational surface, minor radius $r_s$ , which is defined as the unperturbed magnetic flux-surface at which $q(r_s)=m_\\theta /n_\\varphi $ .",
"Here, $q(r)$ is the safety-factor profile (which is assumed to be a monotonically increasing function of $r$ ).",
"Let the full radial width of the island chain's magnetic separatrix be $W$ .",
"In the following, it is assumed that $\\epsilon _s \\equiv r_s/R_0\\ll 1$ and $W/r_s\\ll 1$ .",
"It is convenient to employ a frame of reference that co-rotates with the magnetic island chain.",
"All fields are assumed to depend (spatially) only on the radial coordinate $r$ and the helical angle $\\zeta =m_\\theta \\,\\theta -n_\\varphi \\,\\varphi $ .",
"Let $k_\\theta =m_\\theta /r_s$ , $q_s=m_\\theta /n_\\varphi $ , and $s_s=d\\ln q/d\\ln r|_{r_s}$ .",
"The magnetic shear length at the rational surface is defined $L_s= R_0\\,q_s/s_s$ .",
"Moreover, the unperturbed (by the magnetic island) electron temperature gradient scale-length at the rational surface takes the form $L_T = -1/(d\\ln T_0/dr)_{r_s}$ , where $T_0(r)$ is the unperturbed electron temperature profile.",
"In the following, it is assumed that $L_T>0$ , as is generally the case in conventional tokamak plasmas.",
"[1] The helical magnetic flux is defined $\\chi (x,\\zeta ) = -\\frac{B_z}{R_0}\\int _0^x \\left(\\frac{1}{q}-\\frac{1}{q_s}\\right) (r_s+x)\\,dx + \\delta \\chi (x,\\zeta ),$ where $x=r-r_s$ , and the magnetic field perturbation associated with the tearing mode is written $\\delta {\\bf B}=\\nabla \\times (\\delta \\chi \\,{\\bf e}_z)$ .",
"It is easily demonstrated that ${\\bf B}\\cdot \\nabla \\chi =0$ , where ${\\bf B}$ is the total magnetic field.",
"[15] Hence, $\\chi $ is a magnetic flux-surface label.",
"It is helpful to introduce the normalized helical magnetic flux, $\\psi = (L_s/B_z\\,w^{\\,2})\\,\\chi $ , where $w=W/4$ .",
"The normalized flux in the vicinity of the rational surface is assumed to take the form [15] $\\psi (X,\\zeta ) = \\frac{1}{2}\\,X^{\\,2} + \\cos \\zeta ,$ where $X=x/w$ .",
"As is well-known, the contours of $\\psi $ map out a symmetric (with respect to $X=0$ ), constant-$\\psi $ ,[16] magnetic island chain whose O-points lie at $\\zeta =\\pi $ , $X=0$ , and $\\psi =-1$ , and whose X-points lie at $\\zeta =0$ , $2\\pi $ , $X=0$ , and $\\psi =+1$ .",
"The chain's magnetic separatrix corresponds to $\\psi =+1$ , the region inside the separatrix to $-1\\le \\psi <1$ , and the region outside the separatrix to $\\psi >1$ .",
"The full radial width of the separatrix (in $X$ ) is 4.",
"Finally, the electron temperature profile in the vicinity of the rational surface is written $T(X,\\zeta )=T_s\\left[1-\\left(\\frac{w}{L_T}\\right)\\delta T(X,\\zeta )\\right],$ where $T_s=T_0(r_s)$ , and $\\left.\\delta T(X,\\zeta )\\right|_{\\lim |X|\\rightarrow \\infty } = X.$ Note that $\\delta T(X,\\zeta )$ is an odd function of $X$ .",
"In the following, it is assumed that $w/L_T\\ll 1$ ."
],
[
"Electron Energy Conservation Equation",
"The steady-state electron temperature profile in the vicinity of the island chain is governed by the following well-known electron energy conservation equation: [17], [5] $\\left(\\frac{W}{W_c}\\right)^{4}\\left[\\left[\\delta T,\\psi \\right],\\psi \\right] + \\frac{\\partial ^{\\,2} \\delta T}{\\partial X^{\\,2}} =0,$ where $[A,B] \\equiv \\frac{\\partial A}{\\partial X}\\,\\frac{\\partial B}{\\partial \\zeta }-\\frac{\\partial A}{\\partial \\zeta }\\,\\frac{\\partial B}{\\partial X},$ and $W_c =4\\left(\\frac{\\kappa _\\perp }{\\kappa _\\parallel }\\right)^{1/4}\\left(\\frac{L_s}{k_\\theta }\\right)^{1/2}.$ Here, $\\kappa _\\perp $ and $\\kappa _\\parallel $ are the perpendicular (to the magnetic field) and parallel electron thermal conductivities, respectively.",
"The first term on the right-hand side of Eq.",
"(REF ) represents the divergence of the parallel (to the magnetic field) electron heat flux, whereas the second term represents the divergence of the perpendicular electron heat flux.",
"[In fact, because $[[\\delta T,\\psi ],\\psi ]$ and $\\partial ^{\\,2} \\delta T/\\partial X^{\\,2}$ are both ${\\cal O}(1)$ in our normalization scheme, the ratio of the divergences of the parallel and perpendicular heat fluxes is effectively measured by $(W/W_c)^4$ .]",
"The quantity $W_c$ is the critical island width above which the former term dominates the latter, causing the temperature profile to flatten within the island separatrix.",
"[5] In writing Eq.",
"(REF ), we have neglected any localized sources or sinks of heat in the island region.",
"We have also assumed that $\\kappa _\\perp $ and $\\kappa _\\parallel $ are spatially uniform in the vicinity of the rational surface.",
"The latter assumption is relaxed in Sect."
],
[
"Narrow-Island Limit",
"Consider the so-called narrow-island limit in which $W\\ll W_c$ .",
"[5] Let $Y = \\left(\\frac{w}{w_c}\\right)X.$ Equation (REF ) transforms to give $\\frac{\\partial ^{\\,2} \\delta T}{\\partial Y^{\\,2}}+Y^{\\,2}\\,\\frac{\\partial ^{\\,2} \\delta T}{\\partial \\zeta ^{\\,2}}&=-\\left(\\frac{W}{W_c}\\right)^{\\,2}\\left(\\sin \\zeta \\,\\frac{\\partial \\,\\delta T}{\\partial \\zeta } + 2\\,Y\\,\\sin \\zeta \\,\\frac{\\partial ^{\\,2} \\delta T}{\\partial Y\\,\\partial \\zeta }+Y\\,\\cos \\zeta \\,\\frac{\\partial \\, \\delta T}{\\partial Y}\\right)\\nonumber \\\\[0.5ex]&\\phantom{=}-\\left(\\frac{W}{W_c}\\right)^{\\,4} \\sin ^2\\zeta \\,\\frac{\\partial ^{\\,2} \\delta T}{\\partial Y^{\\,2}}.$ We can write $\\delta T(Y,\\zeta ) = \\left(\\frac{W_c}{W}\\right)Y + \\left(\\frac{W}{W_c}\\right) T_1(Y,\\zeta ) +{\\cal O}\\left(\\frac{W}{W_c}\\right)^{3},$ where $\\frac{\\partial ^{\\,2} T_1}{\\partial Y^{\\,2}}+Y^{\\,2}\\,\\frac{\\partial ^{\\,2} T_1}{\\partial \\zeta ^{\\,2}}=-Y\\,\\cos \\zeta ,$ subject to the boundary conditions $T_1(0,\\zeta )=0$ , and $T_1\\rightarrow 0$ as $|Y|\\rightarrow \\infty $ .",
"Note that the solution (REF ) automatically satisfies the boundary condition (REF ).",
"It follows that $T_1 (Y,\\zeta ) = \\frac{\\sqrt{2}}{4}\\,f\\left(\\sqrt{2}\\,Y\\right)\\,\\cos \\zeta ,$ where $f(p)$ is the well-behaved solution of $\\frac{d^{\\,2}f}{dp^{\\,2}}-\\frac{1}{4}\\,p^{\\,2}\\,f = -p$ that satisfies $f(0)=0$ , and $f\\rightarrow 0$ as $|p|\\rightarrow \\infty $ .",
"Note that $f(-p)=-f(p)$ .",
"Hence, in the narrow-island limit,[5] $\\delta T(X,\\zeta ) = X + \\left(\\frac{W}{W_c}\\right)\\frac{\\sqrt{2}}{4}\\,f\\left(\\sqrt{2}\\,\\frac{W}{W_c} X\\right)\\,\\cos \\zeta +{\\cal O}\\left(\\frac{W}{W_c}\\right)^{3}.$"
],
[
"Wide-Island Limit",
"Consider the so-called wide-island limit in which $W\\gg W_c$ .",
"[5] We can write $\\delta T (X,\\zeta )=\\bar{T}(\\psi ) + \\left(\\frac{W_c}{W}\\right)^{\\,4}\\tilde{T}(\\psi ,\\zeta ),$ where $\\bar{T}$ and $\\tilde{T}$ are both ${\\cal O}(1)$ , and $\\langle \\tilde{T}\\rangle = 0.$ Here, $\\langle \\cdots \\rangle $ is the so-called flux-surface average operator.",
"[15] This operator is defined as follows: $\\langle A\\rangle = \\int _{\\zeta _0}^{2\\pi -\\zeta _0}\\frac{A_+(\\psi ,\\zeta )}{\\sqrt{2\\,(\\psi -\\cos \\zeta )}}\\,\\frac{d\\zeta }{2\\pi }$ for $-1\\le \\psi \\le 1$ , and $\\langle A\\rangle = \\int _0^{2\\pi }\\,\\frac{A(s,\\psi ,\\zeta )}{\\sqrt{2\\,(\\psi -\\cos \\zeta )}}\\,\\frac{d\\zeta }{2\\pi }$ for $\\psi >1$ , where $s={\\rm sgn}(X)$ , $\\zeta _0=\\cos ^{-1}({\\psi })$ , and $A_+(\\psi ,\\zeta ) = \\frac{1}{2}\\left[A(+1,\\psi ,\\zeta ) +A(-1,\\psi ,\\zeta )\\right].$ Note that $\\langle [A,\\psi ]\\rangle \\equiv 0$ for all $A$ .",
"Equations (REF ) and (REF ) can be combined to give $[[\\tilde{T},\\psi ],\\psi ] + \\left(\\frac{W_c}{W}\\right)^4 \\frac{\\partial ^{\\,2} \\tilde{T}}{\\partial X^{\\,2}} + \\frac{\\partial ^{\\,2}\\bar{T}}{\\partial X^{\\,2}}=0.$ The flux-surface average of the previous equation yields $\\left\\langle \\frac{\\partial ^{\\,2}\\bar{T}}{\\partial X^{\\,2}}\\right\\rangle ={\\cal O}\\left(\\frac{W_c}{W}\\right)^4,$ which implies that $\\frac{d}{d\\psi }\\left(\\langle X^{\\,2}\\rangle \\,\\frac{d\\bar{T}}{d\\psi }\\right) \\simeq 0.$ The previous equation can be integrated to give $\\bar{T}(\\psi )=\\left\\lbrace \\begin{array}{ccc}0&\\mbox{\\hspace{28.45274pt}} & -1\\le \\psi \\le 1\\\\[0.5ex]s\\int _1^\\psi \\frac{d\\psi ^{\\prime }}{\\langle X^{\\,2}\\rangle (\\psi ^{\\prime })}&&\\psi >1\\end{array}\\right.,$ which satisfies the boundary condition (REF ).",
"Hence, in the wide-island limit,[5] $\\delta T(X,\\zeta ) =\\bar{T}(\\psi ) + {\\cal O}\\left(\\frac{W_c}{W}\\right)^{\\,4}.$"
],
[
"Modified Rutherford Equation",
"The temporal evolution of the island width is governed by the so-called modified Rutherford equation, which takes the form [15], [5], [4] $G_1\\,\\tau _R\\,\\frac{d}{dt}\\!\\left(\\frac{W}{r_s}\\right) = {\\Delta }^{\\prime }\\,r_s + G_2\\,\\alpha _b\\,\\frac{L_s}{L_T}\\,\\frac{r_s}{W},$ where $G_1 &= 2\\int _{-1}^\\infty \\frac{\\langle \\cos \\zeta \\rangle ^{\\,2}}{\\langle 1\\rangle }\\,d\\psi ,\\\\[0.5ex]G_2 &= 16\\int _{-1}^\\infty \\left\\langle \\frac{\\partial T}{\\partial X}\\right\\rangle \\frac{\\langle \\cos \\zeta \\rangle }{\\langle 1\\rangle }\\,d\\psi .$ Here, $\\tau _R=\\mu _0\\,r_s^{\\,2}/\\eta (r_s)$ is the resistive evolution timescale at the rational surface, and $\\eta (r)$ is the unperturbed plasma resistivity profile.",
"Moreover, ${\\Delta }^{\\prime }<0$ is the standard linear tearing stability index.",
"[16] Finally, $\\alpha _b = f_s\\,(q_s/\\epsilon _s)\\,\\beta $ , where $f_s=1.46\\,\\epsilon _s^{\\,1/2}$ is the fraction of trapped electrons, $\\beta =\\mu _0\\,n_s\\,T_s/B_z^{\\,2}$ , and $n_s$ is the unperturbed electron number density at the rational surface.",
"The second term on the right-hand side of Eq.",
"(REF ) parameterizes the destabilizing influence of the perturbed bootstrap current.",
"[5], [4] Note that, in this paper, for the sake of simplicity, we have employed the so-called lowest-order asymptotic matching scheme described in Ref. rf1.",
"This accounts for the absence of higher-order island saturation terms in Eq.",
"(REF )."
],
[
"Introduction",
"In conventional tokamak plasmas, the dominant contribution to the perpendicular electron thermal conductivity, $\\kappa _\\perp $ , comes from small-scale drift-wave turbulence driven by electron temperature gradients.",
"[1] The fact that the electron temperature gradient is flattened within the magnetic separatrix of a sufficiently wide magnetic island chain implies a substantial reduction in the perpendicular electron thermal conductivity in this region.",
"There is clear experimental evidence that this is indeed the case.",
"[11], [12], [14] In particular, Ref.",
"ex5 reports a reduction in $\\kappa _\\perp $ at the O-point of the magnetic island chain associated with a typical neoclassical tearing mode by 1 to 2 orders of magnitude.",
"Obviously, such a strong reduction in $\\kappa _\\perp $ within the magnetic separatrix calls into question the conventional analytic model of neoclassical tearing modes, described in Sect.",
", in which $\\kappa _\\perp $ is assumed to spatially uniform in the vicinity of the rational surface."
],
[
"Nonuniform Perpendicular Electron Conductivity Model",
"As a first attempt to model the reduction in $\\kappa _\\perp $ due to temperature flattening within the magnetic separatrix of a neoclassical island chain, let us write $\\kappa _\\perp = \\left\\lbrace \\begin{array}{ccc}\\kappa _{\\perp \\,1}&\\mbox{\\hspace{28.45274pt}}& -1\\le \\psi \\le 1\\\\[0.5ex]\\kappa _{\\perp \\,0} &&\\psi > 1\\end{array}\\right.,$ where $\\kappa _{\\perp \\,1}$ and $\\kappa _{\\perp \\,0}$ are spatial constants, with $\\kappa _{\\perp \\,1}\\le \\kappa _{\\perp \\,0}$ .",
"Since the mean temperature gradient outside the separatrix of a neoclassical magnetic island chain is similar in magnitude to the equilibrium temperature gradient [see Eq.",
"()], it is reasonable to assume that $\\kappa _{\\perp \\,0}$ is equal to the local (to the rational surface) perpendicular electron thermal conductivity in the absence of an island chain.",
"Let $W_{c\\,0} &=4\\left(\\frac{\\kappa _{\\perp \\,0}}{\\kappa _\\parallel }\\right)^{1/4}\\left(\\frac{L_s}{k_\\theta }\\right)^{1/2},\\\\[0.5ex]W_{c\\,1} &=4\\left(\\frac{\\kappa _{\\perp \\,1}}{\\kappa _\\parallel }\\right)^{1/4}\\left(\\frac{L_s}{k_\\theta }\\right)^{1/2},$ be the critical island widths outside and inside the separatrix, respectively.",
"Likewise, let $\\xi _0 &= \\left(\\frac{W}{W_{c\\,0}}\\right)^4,\\\\[0.5ex]\\xi _1 &= \\left(\\frac{W}{W_{c\\,1}}\\right)^4,$ measure the ratios of the divergences of the parallel and perpendicular electron heat fluxes outside and inside the separatrix, respectively.",
"Finally, let the parameter $\\lambda = \\frac{\\kappa _{\\perp \\,1}}{\\kappa _{\\perp 0}} =\\frac{\\xi _0}{\\xi _1}$ measure the relative reduction of perpendicular electron heat transport within the island separatrix.",
"Let us adopt the following simple model: $\\lambda = {\\rm e}^{-\\xi _1}+ \\left(1-{\\rm e}^{-\\xi _1}\\right)\\delta ,$ where $0<\\delta \\le 1$ .",
"According to this model, the degree of perpendicular transport reduction within the separatrix is controlled by the parameter $\\xi _1$ , which measures ratio of the divergences of the parallel and perpendicular electron heat fluxes inside the separatrix.",
"(See Sects.",
"REF and REF .)",
"If $\\xi _1$ is much less than unity then there is no temperature flattening within the separatrix, which implies that $\\lambda =1$ (i.e., there is no reduction in transport).",
"On the other hand, if $\\xi _1$ is much greater than unity then the temperature profile is completely flattened inside the separatrix, and the transport is reduced by some factor $\\delta $ (say).",
"The previous formula is designed to interpolate smoothly between these two extremes as $\\xi _1$ varies.",
"Equations (REF ) and (REF ) can be combined to give $\\xi _0=\\lambda \\,\\ln \\left(\\frac{1-\\delta }{\\lambda -\\delta }\\right).$ If follows that $\\delta \\le \\lambda \\le 1$ , with $\\xi _0=0$ when $\\lambda =1$ , and $\\xi _0\\rightarrow \\infty $ as $\\lambda \\rightarrow \\delta $ .",
"It is easily demonstrated that the function $\\xi _0(\\lambda )$ has a point of inflection when $\\delta =\\delta _{\\rm crit} = 1/(1+{\\rm e}^{\\,2})= 0.1192$ .",
"This point corresponds to $\\xi _0=4\\,\\delta _{\\rm crit}= 0.4768$ and $\\lambda =2\\,\\delta _{\\rm crit}=0.2384$ .",
"Figure REF shows the perpendicular electron transport reduction parameter, $\\lambda $ , plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, $\\xi _0$ , for various values of the maximum transport reduction parameter, $\\delta $ .",
"It can be seen that if $\\delta >\\delta _{\\rm crit}$ then the $\\xi _0$ –$\\lambda $ curves are such that $d\\xi _0/d\\lambda <0$ for $\\delta \\le \\lambda \\le 1$ .",
"This implies that $\\lambda $ decreases smoothly and continuously as $\\xi _0$ increases, and vice versa.",
"We shall refer to these solutions as continuous solutions of Eq.",
"(REF ).",
"On the other hand, if $\\delta <\\delta _{\\rm crit}$ then the $\\xi _0$ –$\\lambda $ curves are such that $d\\xi _0/d\\lambda >0$ for some intermediate range of $\\lambda $ values lying between $\\delta $ and 1.",
"As illustrated in Fig.",
"REF , the fact that if $\\delta <\\delta _{\\rm crit}$ then $d\\xi _0/d\\lambda >0$ for intermediate values of $\\lambda $ implies that there are two separate branches of solutions to Eq.",
"(REF )—the first characterized by $d\\xi _0/d\\lambda <0$ and relatively large $\\lambda $ , and the second characterized by $d\\xi _0/d\\lambda < 0$ and relatively small $\\lambda $ .",
"We shall refer to the former solution branch as the large-temperature-gradient branch [because it is characterized by a relatively large value of $\\lambda $ , which, from Eq.",
"(REF ), implies a relatively small value of $\\xi _1$ , which, from Eq.",
"(REF ), implies a relatively large electron temperature gradient inside the separatrix], and the latter as the small-temperature-gradient branch [because it is characterized by a relatively small value of $\\lambda $ , which, from Eq.",
"(REF ), implies a relatively large value of $\\xi _1$ , which, from Eq.",
"(REF ), implies a relatively small electron temperature gradient inside the separatrix].",
"The two solution branches are separated by a dynamically inaccessible branch characterized by $d\\xi _0/d\\lambda >0$ .",
"We shall refer to this branch of solutions as the inaccessible branch.",
"Referring to Fig.",
"REF , as $\\xi _0$ increases from zero, we start off on the large-temperature-gradient solution branch, and $\\lambda $ decreases smoothly.",
"However, when a critical value of $\\xi _0$ is reached (at which $d\\xi _0/d\\lambda =0$ ) there is a bifurcation to the small-temperature-gradient solution branch.",
"We shall refer to this bifurcation as the temperature-gradient-flattening bifurcation, because it is characterized by a sudden decrease in the transport ratio parameter, $\\lambda $ , which implies a sudden decrease in the electron temperature gradient within the island separatrix.",
"Once on the small-temperature-gradient solution branch, the control parameter $\\xi _0$ must be reduced significantly in order to trigger a bifurcation back to the large-temperature-gradient solution branch.",
"We shall refer to this bifurcation as the temperature-gradient-restoring bifuration, because it is characterized by a sudden increase in the transport ratio parameter, $\\lambda $ , which implies a sudden increase in the electron temperature gradient within the island separatrix Figure REF shows the critical values of the control parameter $\\xi _0$ below and above which a temperature-gradient-flattening and a temperature-gradient-restoring bifurcation, respectively, are triggered, plotted as a function of $\\delta /\\delta _{\\rm crit}$ .",
"Figure REF shows the extents of the various solution branches (i.e., the continuous, large-temperature-gradient, small-temperature-gradient, and inaccessible branches) plotted in $\\xi _0$ –$\\xi _1$ space.",
"It is clear that the large-temperature-gradient solution branch is characterized by $\\xi _0\\ll 1$ and $\\xi _1\\stackrel{_{\\normalsize <}}{_{\\normalsize \\sim }}1$ .",
"In other words, the region outside the island separatrix lies in the narrow-island limit, $W\\ll W_{c\\,0}$ , whereas that inside the separatrix lies in the narrow/intermediate island limit, $W\\stackrel{_{\\normalsize <}}{_{\\normalsize \\sim }}W_{c\\,1}$ .",
"[See Eqs.",
"(REF ) and ().]",
"This implies weak to moderate flattening of the temperature gradient within the separatrix.",
"On the other hand, the small-temperarture-gradient solution branch is characterized by $\\xi _0\\ll 1$ and $\\xi _1\\gg 1$ .",
"In other words, the region outside the island separatrix lies in the narrow-island limit, $W\\ll W_{c\\,0}$ , whereas that inside the separatrix lies in the wide-island limit, $W\\gg W_{c\\,1}$ .",
"This implies strong flattening of the temperature gradient within the separatrix.",
"Figure REF suggests that bifurcated solutions of Eq.",
"(REF ) occur because it is possible for the regions inside and outside the island separatrix to lie in opposite asymptotic limits (the two possible limits being the wide-island and the narrow-island limits).",
"Obviously, this is not possible in the conventional model in which $\\kappa _\\perp $ is taken to be spatially uniform in the island region.",
"Finally, according to our simple model, the critical value of the maximum transport reduction parameter, $\\delta $ , below which bifurcated solutions of the electron energy transport equation occur is $0.1192$ .",
"As we have seen, there is experimental evidence for a transport reduction within the separatrix of a neoclassical island chain by between 1 and 2 orders of magnitude.",
"[14] According to our model, such a reduction would be large enough to generate bifurcated solutions."
],
[
"Composite Island Temperature Profile Model",
"Let $\\delta T_{\\rm narrow}(X,\\zeta ,\\xi ) = X + \\xi ^{1/4}\\,\\frac{\\sqrt{2}}{4}\\,f\\left(\\sqrt{2}\\,\\xi ^{1/4}\\, X\\right)\\,\\cos \\zeta $ be the island temperature profile in the narrow-island limit.",
"[See Eq.",
"(REF ).]",
"Here, $\\xi =(W/W_c)^{\\,4}$ .",
"Likewise, let $\\delta T_{\\rm wide}(X,\\zeta ) = \\left\\lbrace \\begin{array}{ccc}0&\\mbox{\\hspace{28.45274pt}}& -1\\le \\psi \\le 1\\\\[0.5ex]s\\int _0^\\psi \\frac{d\\psi ^{\\prime }}{\\langle X^{\\,2}\\rangle (\\psi ^{\\prime })}&&\\psi >1\\end{array}\\right..$ be the island temperature profile in the wide-island limit.",
"[See Eqs.",
"(REF ) and (REF ).]",
"Let us write $\\delta T(X,\\zeta ) = \\left\\lbrace \\begin{array}{ccc}\\delta T_{\\rm inside}(X,\\zeta )&\\mbox{\\hspace{28.45274pt}}&-1\\le \\psi \\le 1\\\\[0.5ex]\\delta T_{\\rm outside}(X,\\zeta )&&\\psi >1\\end{array}\\right.,$ where [cf., Eq.",
"(REF )] $\\delta T_{\\rm inside}(X,\\zeta )= {\\rm e}^{-\\xi _1}\\,\\delta T_{\\rm narrow}(X,\\zeta ,\\xi _1)+(1-{\\rm e}^{-\\xi _1})\\,\\delta T_{\\rm wide}(X,\\zeta )$ and $\\delta T_{\\rm outside}(X,\\zeta )= {\\rm e}^{-\\xi _0}\\,\\delta T_{\\rm narrow}(X,\\zeta ,\\xi _0)+(1-{\\rm e}^{-\\xi _0})\\,\\delta T_{\\rm wide}(X,\\zeta ).$ It follows that $\\frac{\\partial \\,\\delta T_{\\rm inside}}{\\partial X}&\\simeq {\\rm e}^{-\\xi _1}\\left[1+ \\frac{f^{\\prime }(0)}{2}\\,\\xi _1^{\\,1/2}\\,\\cos \\zeta + {\\cal O}(\\xi _1)\\right],\\\\[0.5ex]\\frac{\\partial \\,\\delta T_{\\rm outside}}{\\partial X}&\\simeq {\\rm e}^{-\\xi _0}\\left[1+ \\frac{f^{\\prime }(0)}{2}\\,\\xi _0^{\\,1/2}\\,\\cos \\zeta +{\\cal O}(\\xi _0)\\right]+(1-{\\rm e}^{-\\xi _0})\\,\\frac{X}{\\langle X^{\\,2}\\rangle }.$ Here, $f^{\\prime }(0)=1.1981$ , as determined from the numerical solution of Eq.",
"(REF )."
],
[
"Evaluation of Integrals",
"According to Eqs.",
"(REF ), (), (REF ), and (), $G_1 &= 2\\,(I_2+ I_3),\\\\[0.5ex]G_2 &= 16\\,I_1\\,({\\rm e}^{-\\xi _0}-{\\rm e}^{-\\xi _1})+ 8\\,f^{\\prime }(0)\\,I_2\\,\\xi _1^{\\,1/2}\\,{\\rm e}^{-\\xi _1}+ 8\\,f^{\\prime }(0)\\,I_3\\,\\xi _0^{\\,1/2}\\,{\\rm e}^{-\\xi _0}\\nonumber \\\\[0.5ex]&\\phantom{=}+16\\,I_4\\,(1-{\\rm e}^{-\\xi _0}),$ where $I_1 &= -\\int _{-1}^1\\langle \\cos \\zeta \\rangle \\,d\\psi ,\\\\[0.5ex]I_2 &= \\int _{-1}^{1}\\frac{\\langle \\cos \\zeta \\rangle ^{\\,2}}{\\langle 1\\rangle }\\,d\\psi ,\\\\[0.5ex]I_3&= \\int _{1}^\\infty \\frac{\\langle \\cos \\zeta \\rangle ^{\\,2}}{\\langle 1\\rangle }\\,d\\psi ,\\\\[0.5ex]I_4&=\\int _1^{\\infty } \\frac{\\langle \\cos \\zeta \\rangle }{\\langle X^{\\,2}\\rangle \\,\\langle 1\\rangle }\\,d\\psi .$ Here, use has been made of the easily proved result $\\int _{-1}^\\infty \\langle \\cos \\zeta \\rangle \\,d\\psi = 0.$ Let $\\psi =2\\,k^{\\,2}-1$ .",
"It follows that $d\\psi =4\\,k\\,dk$ .",
"In the region $0\\le k \\le 1$ , we can write $\\zeta &= 2\\,\\cos ^{-1}(k\\,\\sin \\vartheta ),\\\\[0.5ex]X &= 2\\,k\\,\\cos \\vartheta ,\\\\[0.5ex]\\cos \\zeta &= 1-2\\,(1-k^{\\,2}\\,\\sin ^2\\vartheta ),\\\\[0.5ex]\\langle A\\rangle &=\\int _{-\\pi /2}^{\\pi /2}\\frac{A(k,\\vartheta )}{\\sqrt{1-k^{\\,2}\\,\\sin ^2\\vartheta }}\\,\\frac{d\\vartheta }{2\\pi }.$ On the other hand, in the region $k>1$ , we can write $\\zeta &=\\pi -2\\,\\vartheta ,\\\\[0.5ex]X &= 2\\sqrt{k^{\\,2}-\\sin ^2\\vartheta },\\\\[0.5ex]\\cos \\zeta &= 2\\,k^{\\,2}-1-2\\,(k^{\\,2}-\\sin ^2\\vartheta ),\\\\[0.5ex]\\langle A\\rangle &=\\int _{-\\pi /2}^{\\pi /2}\\frac{A(k,\\vartheta )}{\\sqrt{k^{\\,2}-\\sin ^2\\vartheta }}\\,\\frac{d\\vartheta }{2\\pi }.$ Here, it is assumed that $A$ is an even function of $X$ .",
"Let ${\\cal A}(k)&= 2\\,k\\,\\langle 1\\rangle ,\\\\[0.5ex]{\\cal B}(k)&=2\\,k\\,\\langle \\cos \\zeta \\rangle ,\\\\[0.5ex]{\\cal C}(k) &=\\frac{\\langle X^{\\,2}\\rangle }{2\\,k}.$ It follows from Eqs.",
"(REF )–(REF ) that in the region $0\\le k \\le 1$ , ${\\cal A}(k) &= \\frac{2}{\\pi }\\,k\\,K(k),\\\\[0.5ex]{\\cal B}(k) &=\\frac{2}{\\pi }\\,k\\left[K(k)-2\\,E(k)\\right].$ On the other hand, in the region $k>1$ , ${\\cal A}(k) &= \\frac{2}{\\pi }\\,K(1/k),\\\\[0.5ex]{\\cal B}(k) &=\\frac{2}{\\pi }\\left[(2\\,k^{\\,2}-1)\\,K(1/k) -2\\,k^{\\,2}\\,E(1/k)\\right],\\\\[0.5ex]{\\cal C}(k) &= \\frac{2}{\\pi }\\,E(1/k).$ Here, $K(k) &=\\int _0^{\\pi /2} (1-k^{\\,2}\\,\\sin ^2\\vartheta )^{-1/2}\\,d\\vartheta ,\\\\[0.5ex]E(k) &= \\int _0^{\\pi /2} (1-k^{\\,2}\\,\\sin ^2\\vartheta )^{1/2}\\,d\\vartheta $ are complete elliptic integrals.",
"[19] Hence, according to Eqs.",
"(REF )–() and (REF )–(), $I_1 &=-2 \\int _0^1 {\\cal B}\\,dk=0.4244,\\\\[0.5ex]I_2 &= 2\\int _0^1\\frac{{\\cal B}^{\\,2}}{\\cal A}\\,dk=0.3527,\\\\[0.5ex]I_3&= 2\\int _1^\\infty \\frac{{\\cal B}^{\\,2}}{\\cal A}\\,dk=0.0587,\\\\[0.5ex]I_4 &= 2\\int _1^\\infty \\frac{{\\cal B}}{{\\cal A}\\,{\\cal C}}\\,dk=0.3838.$ Thus, Eqs.",
"(REF ) and () yield $G_1 &=0.8227, \\\\[0.5ex]G_2 &= 6.791\\,({\\rm e}^{-\\xi _0}-{\\rm e}^{-\\xi _1})+3.380\\,\\xi _1^{\\,1/2}\\,{\\rm e}^{-\\xi _1}+0.562\\,\\xi _0^{\\,1/2}\\,{\\rm e}^{-\\xi _0}+6.140\\,(1-{\\rm e}^{-\\xi _0}),$ respectively."
],
[
"Destabilizing Effect of Perturbed Bootstrap Current",
"The dimensionless parameter $G_2$ , appearing in the modified Rutherford equation, (REF ), measures the destabilizing influence of the perturbed bootstrap current.",
"Figure REF shows $G_2$ plotted as a function of the so-called neoclassical tearing mode control parameter, $\\xi _0 = \\left(\\frac{W}{W_{c\\,0}}\\right)^4 = \\left(\\frac{W}{4}\\right)^4\\left(\\frac{\\kappa _\\parallel }{\\kappa _{\\perp \\,0}}\\right)\\left(\\frac{k_\\theta }{L_s}\\right)^{2},$ which measures the ratio of the divergences of the parallel to the perpendicular electron heat fluxes outside the island separatrix.",
"[See Eqs.",
"(REF ) and (REF ).]",
"The curves shown in this figure are obtained from Eqs.",
"(REF ), (REF ), and (REF ).",
"Note that $\\kappa _\\parallel $ and $\\kappa _{\\perp \\,0}$ are the local (to the rational surface) parallel and perpendicular electron thermal conductivities, respectively, in the absence of an island chain.",
"It can be seen, from Fig.",
"REF , that if the maximum transport reduction parameter, $\\delta $ , is relatively close to unity (implying a relatively weak reduction in the perpendicular electron thermal conductivity inside the island separatrix when the electron temperature profile is completely flattened in this region) then the bootstrap destabilization parameter, $G_2$ , increases monotonically with increasing $\\xi _0$ , taking the value $3.492\\,\\xi _0^{\\,1/2}$ when $\\xi _0\\ll 1$ , and approaching the value $6.140$ asymptotically as $\\xi _0\\rightarrow \\infty $ .",
"[5] [These two limits follow from Eq.",
"(REF ), given that $\\xi _1\\simeq \\xi _0$ when $\\lambda \\simeq 1$ .]",
"According to Fig.",
"REF , as $\\delta $ decreases significantly below unity (implying an increasingly strong reduction in the perpendicular electron thermal conductivity inside the island separatrix when the electron temperature profile is completely flattened in this region) it remains the case that $G_2=3.492\\,\\xi _0^{\\,1/2}$ when $\\xi _0\\ll 1$ , and $G_2\\rightarrow 6.140$ as $\\xi _0\\rightarrow \\infty $ .",
"However, at intermediate values of $\\xi _0$ [i.e., $\\xi _0 \\sim {\\cal O}(1)$ ], the rate of increase of $G_2$ with $\\xi _0$ becomes increasingly steep.",
"This result suggests that a substantial reduction in the perpendicular electron thermal conductivity inside the island separatrix, when the electron temperature profile is completely flattened in this region, causes the bootstrap destabilization term in the modified Rutherford equation, (REF ), to “switch on” much more rapidly as the neoclassical tearing mode control parameter, $\\xi _0$ , is increased, compared to the standard case in which there is no reduction in the conductivity.",
"Finally, it is apparent from Fig.",
"REF that if $\\delta $ falls below the critical value $\\delta _{\\rm crit}=0.1192$ then the bootstrap destabilization parameter, $G_2$ , becomes a multi-valued function of $\\xi _0$ at intermediate values of $\\xi _0$ .",
"As illustrated in Fig.",
"REF , this behavior is due to the existence of separate branches of solutions of the electron energy conservation equation.",
"(See Sect.",
"REF .)",
"The large-temperature-gradient branch is characterized by relatively weak flattening of the electron temperature profile within the island separatrix, and a consequent relatively small value (i.e., significantly smaller than the asymptotic limit $6.140$ ) of the bootstrap destabilization parameter, $G_2$ .",
"On the other hand, the small-temperature-gradient branch is characterized by almost complete flattening of the electron temperature profile within the island separatrix.",
"Consequently, the bootstrap destabilization parameter, $G_2$ , takes a value close to the asymptotic limit $6.140$ on this solution branch.",
"The large-temperature-gradient and small-temperature-gradient solution branches are separated by a dynamically inaccessible branch of solutions.",
"Referring to Fig.",
"REF , as the neoclassical tearing mode control parameter, $\\xi _0$ , increases from a value much less than unity, we start off on the large-temperature-gradient solution branch, and the bootstrap destabilization parameter, $G_2$ , increases smoothly and monotonically from a small value.",
"However, when a critical value of $\\xi _0$ is reached, there is a gradient-flattening-bifurcation to the small-temperature-gradient solution branch.",
"This bifurcation is accompanied by a sudden increase in $G_2$ to a value close to its asymptotic limit $6.140$ .",
"Once on the small-temperature-gradient solution branch, $\\xi _0$ must be decreased by a significant amount before a gradient-restoring-bifurcation to the large-temperature-gradient solution branch is triggered.",
"Moreover, the gradient-restoring-bifurcation is accompanied by a very large reduction in $G_2$ ."
],
[
"Long Mean-Free-Path Effects",
"The parallel electron thermal conductivity takes the form [17] $\\kappa _\\parallel \\sim n_e\\,v_e\\,\\lambda _e$ in a collisional plasma, where $n_e$ is the electron number density, $v_e$ is the electron themal velocity, and $\\lambda _e$ is the electron mean-free-path.",
"However, in a conventional tokamak plasma the mean-free-path $\\lambda _e$ typically exceeds the parallel (to the magnetic field) wavelength $\\lambda _\\parallel $ of low-mode-number helical perturbations.",
"Under these circumstances, the simple-minded application of Eq.",
"(REF ) yields unphysically large parallel heat fluxes.",
"The parallel conductivity in the physically-relevant long-mean-free-path limit ($\\lambda _e\\ll \\lambda _\\parallel $ ) can be crudely estimated as [5], [20] $\\kappa _\\parallel \\sim n_e\\,v_e\\,\\lambda _\\parallel ,$ which is equivalent to replacing parallel conduction by parallel convection in the electron energy conservation equation, (REF ).",
"For a magnetic island of full radial width $W$ , the typical value of $\\lambda _\\parallel $ is $n_\\varphi \\,s_s\\,w/R_0$ .",
"Hence, in the long-mean-free-path limit, the expression for the neoclassical tearing mode control parameter (REF ) is replaced by $\\xi _0 = \\left(\\frac{W}{4}\\right)^5\\left(\\frac{\\kappa _\\parallel ^{\\prime }}{\\kappa _{\\perp \\,0}}\\right)\\left(\\frac{k_\\theta }{L_s}\\right)^{2},$ where $\\kappa _\\parallel ^{\\prime } = n_\\varphi \\,n_e\\,v_e\\,s_s/R_0$ , and $n_e$ and $v_e$ are evaluated at the rational surface."
],
[
"Summary and Discussion",
"In this paper, we have investigated the effect of the reduction in anomalous perpendicular electron heat transport inside the separatrix of a magnetic island chain associated with a neoclassical tearing mode in a tokamak plasma, due to the flattening of the electron temperature profile in this region, on the overall stability of the mode.",
"Our model (which is described in Sect. )",
"is fairly crude, in that the perpendicular electron thermal conductivity, $\\kappa _\\perp $ , is simply assumed to take different spatially-uniform values in the regions inside and outside the separatrix.",
"Moreover, when the temperature profile is completely flattened within the island separatrix, $\\kappa _\\perp $ in this region is assumed to be reduced by some factor $\\delta $ , where $0<\\delta \\le 1$ .",
"The degree of temperature flattening inside the separatrix is ultimately controlled by a dimensionless parameter $\\xi _0$ that measures the ratio of the divergences of the parallel and perpendicular electron heat fluxes in the vicinity of the island chain.",
"Expressions for $\\xi _0$ in the short-mean-free-path and the more physically-relevant long-mean-free-path limits are given in Eqs.",
"(REF ) and (REF ), respectively.",
"Finally, the destabilizing influence of the perturbed bootstrap current is parameterized in terms of a dimensionless quantity $G_2>0$ that appears in the modified Rutherford equation.",
"[See Eqs.",
"(REF ) and ().]",
"A large value of $G_2$ implies substantial destabilization, and vice versa.",
"In the standard case $\\delta =1$ (in which there is no reduction in the perpendicular electron thermal conductivity inside the island separatrix when the electron temperature profile is completely flattened in this region), the bootstrap destabilization parameter $G_2$ increases smoothly and monotonically as the control parameter $\\xi _0$ increases, from a value much less than unity when $\\xi _0\\ll 1$ , to the asymptotic limit $6.140$ when $\\xi _0\\gg 1$ .",
"[5] (See Section REF .)",
"As $\\delta $ decreases significantly below unity (implying an increasingly strong reduction in the perpendicular electron thermal conductivity inside the island separatrix when the electron temperature profile is completely flattened in this region), the small-$\\xi _0$ and large-$\\xi _0$ behaviors of the bootstrap destabilization parameter remain unchanged.",
"However, at intermediate values of the control parameter $\\xi _0$ [i.e., $\\xi _0\\sim {\\cal O}(1)$ ], the rate of increase of $G_2$ with $\\xi _0$ becomes increasingly steep.",
"(See Fig.",
"REF .)",
"In other words, a substantial reduction in the perpendicular electron thermal conductivity inside the island separatrix, when the electron temperature profile is completely flattened in this region, causes the bootstrap destabilization parameter $G_2$ to “switch on\" much more rapidly as the control parameter $\\xi _0$ is increased, compared to the standard case in which $\\delta =1$ .",
"(See Section REF .)",
"Finally, if $\\delta $ falls below the critical value $0.1192$ then the bootstrap destabilization parameter, $G_2$ , becomes a multi-valued function of the control parameter $\\xi _0$ , at intermediate values of $\\xi _0$ .",
"This behavior is due to the existence of separate branches of solutions of the electron energy conservation equation.",
"(See Sect.",
"REF .)",
"The large-temperature-gradient branch is characterized by relatively weak flattening of the electron temperature profile within the island separatrix, and a consequent relatively small value (i.e., significantly smaller than the asymptotic limit $6.140$ ) of the bootstrap destabilization parameter, $G_2$ .",
"On the other hand, the small-temperature-gradient branch is characterized by almost complete flattening of the electron temperature profile within the island separatrix.",
"Consequently, the bootstrap destabilization parameter, $G_2$ , takes a value close to the asymptotic limit $6.140$ on this solution branch.",
"The large-temperature-gradient and small-temperature-gradient solution branches are separated by a dynamically inaccessible branch of solutions.",
"As the control parameter, $\\xi _0$ , increases from a value much less than unity, the system starts off on the large-temperature-gradient solution branch, and the bootstrap destabilization parameter, $G_2$ , increases smoothly and monotonically from a small value.",
"However, when a critical value of $\\xi _0$ is reached, there is a gradient-flattening-bifurcation to the small-temperature-gradient solution branch.",
"(See Fig.",
"REF .)",
"This bifurcation is accompanied by a sudden increase in $G_2$ to a value close to its asymptotic limit $6.140$ .",
"Once on the small-temperature-gradient solution branch, $\\xi _0$ must be decreased by a significant amount before a gradient-restoring-bifurcation to the large-temperature-gradient solution branch is triggered.",
"Moreover, the gradient-restoring-bifurcation is accompanied by a very large reduction in $G_2$ .",
"(See Section REF .)",
"The behavior described in the preceding paragraph points to the disturbing possibility that a neoclassical tearing mode in a tokamak plasma could become essentially self-sustaining.",
"In other words, once the mode is triggered, and the electron temperature profile is flattened within the island separatrix, the consequent substantial reduction in the perpendicular thermal conductivity in this region reinforces the temperature flattening, making it very difficult to remove the mode from the plasma."
],
[
"Acknowledgements",
"This research was funded by the U.S. Department of Energy under contract DE-FG02-04ER-54742."
]
] | 1612.05798 |
[
[
"iDroop: A dynamic droop controller to decouple power grid's steady-state\n and dynamic performance"
],
[
"Abstract This paper presents a novel Dynam-i-c Droop (iDroop) control mechanism to perform primary frequency control with gird-connected inverters that improves the network dynamic performance.",
"The work is motivated by the dynamic degradation experienced by the power grid due to the increase in asynchronous inverted-based generation.",
"We show that the widely suggested virtual inertia solution suffers from unbounded noise amplification (infinite $\\mathcal H_2$ norm) when measurement noise is considered.",
"This suggests that virtual inertia could potentially further degrade the grid performance once broadly deployed.",
"This motivates the proposed solution in this paper that overcomes the limitations of virtual inertia controllers while sharing the same advantages of traditional droop control.",
"In particular, our iDroop controllers are decentralized, rebalance supply and demand, and provide power sharing.",
"Furthermore, our solution can improve the dynamic performance without affecting the steady state solution.",
"Our algorithm can be incrementally deployed and can be guaranteed to be stable using a decentralized sufficient stability condition on the parameter values.",
"We illustrate several features of our solution using numerical simulations."
],
[
"Introduction",
"Droop control has a long history in power system frequency control [1].",
"It is perhaps one of the simplest – and yet very effective – decentralized mechanisms that achieve synchronization and supply-demand balance in a network with several generating resources running in parallel [2].",
"While its implementation may vary depending on the specific device, its basic operational principle remains unchanged: whenever frequency is above (below) the nominal value, decrease (increase) power proportionally to the frequency deviation.",
"Thus it is usually referred as the primary layer of the frequency control architecture [3].",
"Not surprisingly, its many benefits have made droop control one of the features of power system engineering that have successfully survived decades of technological advances.",
"Unfortunately, the very principles that this mechanism relies on are becoming less and less valid due to several reasons.",
"Firstly, droop control relies on the fact that demand is frequency dependent, yet with the increase of power electronic based loads the aggregate load is becoming less sensitive to frequency [4].",
"Secondly, while traditional generation always provides some level of droop control, renewable generation is insensitive to frequency fluctuations.",
"The remaining conventional generators are forced to handle the whole burden of regulating the frequency.",
"Thirdly, newer types of generators have little or no inertia at all when compared with traditional ones, which is slowly introducing a dynamic degradation that concerns many utilities [5].",
"Recently, a large body of literature has been developed on the design of distributed control mechanisms with the objective to synthetically generate frequency responsive generation and demand that can overcome the diminishing participation of these elements in primary frequency control [6], [7], [8], [9].",
"In particular, the proposed strategies seek to introduce a more frequency responsive devices, either on the load side [6], [10] or on the generation side [9], that not only implements droop characteristics for primary frequency control, but also guarantee higher level operational constraints such as restoring frequency to nominal value [11], preserving inter area flow constraints, respecting thermal limits, and providing efficiency[10].",
"While these solutions directly address the loss of frequency responsiveness in the grid, they do not explicitly address dynamic degradation.",
"Interestingly, droop control can in principle also provide dynamic performance improvement by properly selecting the droop coefficients [12].",
"For example, in an under-damped power grid, decreasing the droop coefficient can reduce the frequency Nadir (maximum frequency excursion) in the same way increasing friction can reduce the overshoot of an under-damped spring-mass system.",
"However this is not a practical solution since it also requires the generator to take a larger share of the supply-demand imbalance.",
"As a result, the efficiency of the steady-state resource allocation becomes intrinsically coupled with the possible dynamic performance improvement.",
"It is the purpose of this work to eliminate this coupling.",
"This paper proposes a novel dynam-i-c droop (iDroop) control algorithm that is able to maintain all the desired features of traditional droop control while providing enough design flexibility to improve the dynamic performance.",
"More precisely, we present a control scheme –whose input is frequency and output is power generation– that preserves the same steady-state characteristics as traditional droop control, maintains grid stability, and provides supply-demand balance.",
"Our iDroop controller can be implemented by inverters providing power from renewable sources or by intelligent loads.",
"Finally, we numerical demonstrate that one can use the additional flexibility to improve the $\\mathcal {H}_2$ -norm of the system.",
"Paper Organization: Section describes the power network model as well as several inverter operational modes used to interface with the grid.",
"Section uses dynamic and steady state performance metrics to motivate the need for a novel droop control solution.",
"Section introduces the proposed iDroop control and shows how our solution is able to preserve the same steady state as droop control while providing enough flexibility to improve dynamic performance.",
"Moreover, we provide a decentralized sufficient condition on the parameter values that guarantee the stability of our controllers.",
"We numerically illustrate the functionalities of iDroop in Section and conclude in Section ."
],
[
"Network Model",
"We consider a power system composed by $n$ buses denoted by either $i$ or $j$ , i.e.",
"$i,j\\in N:=\\lbrace 1,\\dots , n\\rbrace $ .",
"We use $ij$ to denote the transmission line that connects bus $i$ and $j$ , and use $E$ to denote the set of lines, i.e.",
"$ij\\in E$ .",
"Thus the topology of the power system is described by the graph $G=(N,E)$ .",
"The admittance of the line $ij$ is given by $y_{ij}=g_{ij}-\\mathbf {i}b_{ij}$ , where $g_{ij}$ and $b_{ij}$ denote the conductance and susceptance of line $ij$ , respectively.",
"The state of the network is described by the complex voltages $(V_i)_{i\\in N}=(v_ie^{j\\theta _i})_{i\\in N}$ where $v_i$ and $\\theta _i$ represent the voltage magnitude and phase of bus $i$ , respectively."
],
[
"Generators and Loads",
"We model the dynamics of each conventional generator using the standard swing equations [13], [14].",
"We denote the frequency of each generator by $\\omega _i$ which evolves according to i = i, Mii = pin +qri - (Di+1Rig)i -Pie, where $M_i$ denotes the aggregate generator inertia, $D_i$ is the aggregate damping and frequency dependent load coefficient, and $R_i^g$ is the droop coefficient.",
"$p_i^{in}$ denotes the net constant power injection at bus $i$ , $q^r_i$ is the controllable input power injected by grid-connected inverters, and $P_i^e$ denotes the net electric power drawn by the grid."
],
[
"Inverters",
"In this paper we seek to develop a new control scheme that can be implemented by inverters to improve the dynamic performance of the power grid.",
"Since the power electronics of the inverters are significantly faster than the electromechanical dynamics of the generators, we assume that inverters can statically update its power, i.e.",
"$q^r_i = u_i,$ where $u_i$ is the command input.",
"We assume that (REF ) represents the aggregate power of all the inverters connected at bus $i$ .",
"There are different operational modes in which inverters can be interfaced with the power grid [15], [16], [17], [18].",
"Here we briefly review the most common ones: Constant Power: This is the default operational mode in today's grid and amounts to setting $u_i = q_i^{r,0},$ where $q_i^{r,0}$ is a constant parameter representing power generation set point.",
"Droop Control: This mode aims to change the power injection of the inverter to provide additional droop capabilities by setting $u_i = q_i^{r,0} - \\frac{1}{R^r_i}\\omega _i,$ where $R^r_i$ is the droop coefficient.",
"Virtual Inertia: This operational mode has been recently proposed [19], [20] as an alternative method to compensate the loss of inertia and is given by $u_i = q_i^{r,0} -\\frac{1}{R^r_i}\\omega _i - M^v_i\\dot{\\omega }_i,$ where $M^v_i$ represents the virtual inertia."
],
[
"Power Flow Model",
"We consider a Kron-reduced network model [21], where constant impedance loads are implicitly included in the line impedances of the reduced network.",
"Thus every remaining bus represents a grid generator.",
"We further use the DC network model which has been widely adopted for purpose of designing frequency controllers for a long time [22], [23].",
"Therefore, the total electric power drained by the network at bus $i$ is $P_i^{e} = -\\sum _{j\\in \\mathcal {N}_i} b_{ij}(\\theta _i-\\theta _j),$ where the set $\\mathcal {N}_i$ denotes the set of neighboring buses adjacent to bus $i$ ."
],
[
"Network Dynamics",
"Combining (REF )-(REF ) we arrive to the following compact description of the system dynamics = M = pin + qr - (Rg-1+D)- LB where $M:=\\operatorname{diag}(M_i,\\,i\\in N)$ , $D:=\\operatorname{diag}(D_i,\\,i\\in N)$ , ${R^g}^{-1}=\\operatorname{diag}(\\frac{1}{R_i^g},\\,i\\in N)$ , $\\omega :=(\\omega _i,\\,i\\in N)$ , $p^{in}=(p_i^{in},\\,i\\in N)$ , $L_B$ is the $b_{ij}$ -weighted Laplacian matrix $(L_B)_{ij} = {\\left\\lbrace \\begin{array}{ll}-b_{ij}, & \\text{if $i\\ne j$, $ij\\in E$}, \\\\\\sum _{k\\in \\mathcal {N}_i} b_{ik},& \\text{if $i=j$,}\\\\0,&\\text{otherwise,}\\end{array}\\right.",
"}$ and $q^r:=(q^r_i,\\,\\,i\\in N)$ is given by $q^r_i = {\\left\\lbrace \\begin{array}{ll}q_i^{r,0}, & \\text{if $i\\in CP$,}\\\\q_i^{r,0} -\\frac{1}{R_i^r}\\omega _i,& \\text{if $i\\in DC$,}\\\\q_i^{r,0} -\\frac{1}{R_i^r}\\omega _i - M^v_i\\dot{\\omega }_i,& \\text{if $i\\in VI$.}\\\\\\end{array}\\right.",
"}$ The sets $CP$ , $DC$ and $VI$ are the subsets of buses that have inverters operating in constant power, droop control and virtual inertia modes, respectively.",
"In the absence of higher layer controllers, such as automatic generation control [24], the system can synchronize with a nontrivial frequency deviation from the nominal ($\\omega _i^*\\ne 0$ ).We assume here w.l.o.g.",
"that the nominal frequency is 0 Thus we refer to the vector $(\\theta ^*(t):=\\theta ^*+\\omega ^*t,\\omega ^*,{q^r}^*)$ as a steady state solution of the system (REF ), with (REF ) equal zero and $\\frac{\\text{d}}{\\text{d}t} \\theta ^*(t)=\\omega ^*$ .",
"Furthermore, using (REF ) it is easy to see that $L_B\\theta ^*(t)$ is constant, which implies that $\\omega ^*=\\mathbf {1}_n \\omega ^*_0$ , with $\\mathbf {1}_n\\in {R}^n$ being the vector of all ones, and the scalar $\\omega _0^*$ given by $\\omega _0^* = \\frac{\\sum _{i=1}^n p_i^{in}+{q_i^{r,0}}}{\\sum _{i=1}^n (D_i + {R_i^g}^{-1})+\\sum _{i\\in DC\\cup VI} {R_i^r}^{-1} }.$ In summary, the steady state solution of (REF ) and (REF ) is given by $(\\theta ^*(t),\\omega ^*,{q^r}^*)=(\\theta ^*+\\mathbf {1}_n \\omega _0^*t, q^{r*} )$ , where $q_i^{r*}=q_i^{r,0}$ , if $i\\in CP$ , or $q_i^{r*}=q_i^{r,0} - {R_i^r}^{-1}\\omega _0^*$ , if $i\\in DC\\cup VI$ .",
"Thus the smallest invariant set that includes all possible steady states is given by $\\mathcal {E}\\!",
":=\\!\\lbrace \\!",
"(\\theta \\!+\\!\\lambda \\mathbf {1}_n,\\omega ,q^r)\\!\\in \\!R^{3n}\\!\\!",
":\\!\\text{$\\lambda \\in R$},(\\ref {eq:system-b})=0,\\omega _i\\!=\\!\\omega _0^*\\rbrace .$"
],
[
"Steady State and Dynamic Performance ",
"In this section we introduce two metrics, one for steady state and another for dynamic performance, and illustrate how the existing solutions for inverter control either cannot produce any dynamic performance improvement, or the performance improvement comes at the cost of steady state deviation from the desired operational point."
],
[
"Steady State Performance",
"We now define the steady state performance metric used in this paper.",
"Let $\\delta q_i^g:=-\\frac{1}{R_i^g}\\omega _i$ and $\\delta q_i^r:=q_i^r-q_i^{r,0}$ be the power deviation of generators and inverters, respectively.",
"After a disturbance these quantities need to be modified in order to compensate a supply-demand imbalance, i.e.",
"$\\sum _{i=1}^n \\delta q_i^g + \\sum _{i\\in DC\\cup VI}\\delta q_i^r = \\Delta P,$ where $\\Delta P$ denotes the power imbalance.",
"We assign to both conventional and inverter-based generators a cost $c_i^g(q_i^g):=\\frac{\\alpha ^g_i}{2}(q_i^g)^2$ and $c_i^r(q_i^r):=\\frac{\\alpha ^r_i}{2}(q_i^r)^2$ respectively.",
"Thus given a set deviations satisfying (REF ) the total system cost of mitigating an imbalance of $\\Delta P$ is given by SS-Cost: i=1n gi2(qig)2+ iDCVIri2(qir)2 One of the main attractive features of droop control is its ability to share the supply-demand mismatch among different resources.",
"Recent works have shown that the steady state allocation of droop control can be represented as the solution of an optimization problem (see e.g. [6]).",
"In particular, if we define let $\\delta d_i=:D_i\\omega _i$ denote the frequency dependent demand deviation, then it can be shown that the system (REF ) solves: qgi,qri,diminimize i=1n Rgi(qig )22 + (di)22Di + iDCVIRri(qri )22 subject to i=1n piin +qir,0+ qig -di+iDCVIqi =0 The proof of this claim is already standard in the community (see e.g.",
"[6], [25], [7], [26], [27]), we refer the reader to [6] for a proof of a similar statement.",
"As a result of the steady state characteristic of droop control, it is possible to optimally minimize the steady state cost as it is summarized in the next theorem.",
"Theorem 1 (Droop Control Optimality) Let $(\\theta ^*(t),\\omega ^*,{q^r}^*)$ be the steady state solution of (REF ) where ${q_i^r}^*$ is given by (REF ).",
"If $R_i^r=\\alpha _i^r$ and $R_i^g=\\alpha _i^g$ , then ($\\delta {q_i^g}^* := -\\frac{1}{R_i^g}\\omega _0^*$ , $\\delta {q_i^r}^* := {q_i^r}^*-q_i^{r,0}= -\\frac{1}{R_i^g}\\omega _0^*$ ) is the unique allocation that minimizes the steady state cost (REF ) subject to (REF ), where $\\Delta P:=\\sum _{i=1}^n p_i^{in}+q_i^{r,0} - D_i\\omega _0^*$ .",
"We start by characterizing the optimal solution of minimizing (REF ) subject to (REF ).",
"Since the problem has a strictly convex objective with linear constraints then there is a unique solution that is characterized by the Karush-Kuhn-Tucker (KKT) conditions for optimality[28].",
"Thus, $\\hat{\\delta q^g}^*$ and $\\hat{\\delta q^r}^*$ is an optimal solution if and only if there exists a scalar $\\lambda ^*$ such that $\\alpha _i^g\\hat{\\delta q_i^g}^* = \\lambda ^*\\,\\forall i,\\;\\;\\; \\alpha _i^r\\hat{\\delta q_i^r}^* = \\lambda ^*\\,i\\in DC\\cup VI,$ and $\\sum _{i=1}^n\\hat{\\delta q_i^g}^*+\\hat{\\delta q_i^r}^*=\\Delta P$ .",
"To finalize the proof we just need to show that when $R_i^r=\\alpha _i^r$ and $R_i^g=\\alpha _i^g$ , then the steady state deviations of droop control (${\\delta {q_i^g}^*} := -\\frac{1}{R_i^g}\\omega _0^*$ and $\\delta {q_i^r}^*:= {q_i^r}^*-q_i^{r,0}$ ) satisfy the KKT conditions.",
"By (REF ),qir*:=qir* - qir,0 = {ll -1Rir0*, if $i\\in DC \\cup VI$ , 0, otherwise.",
".",
"Therefore, since by definition ${\\delta q^g}^*= -\\frac{1}{R_i^g}\\omega _0^*$ , then we have that when $R_i^r=\\alpha _i^r$ and $R_i^g=\\alpha _i^g$ , $({\\delta q^g}^*,{\\delta q^r}^*)$ satisfies (REF ) for $\\lambda ^*=\\omega _0^*$ .",
"Finally, feasibility follows by definition of $\\Delta P$ since i=1n qig*+qir*= -(i=1nRig-1+iDCVI Rir-1)0* =i=1n piin+qir*-Di0*=:P where the second steps follows from (REF ).",
"Thus (${\\delta q_i^g}^*,{\\delta q_i^r}^*$ ) is feasible and satisfies the KKT conditions.",
"Therefore, (${\\delta q_i^g}^*,{\\delta q_i^r}^*$ ) is an optimal allocation.",
"Since the optimal solution is unique, we must have $({\\delta q_i^g}^*,{\\delta q_i^r}^*)=(\\hat{\\delta q_i^g}^*,\\hat{\\delta q_i^r}^*)$ .",
"Remark 1 Theorem REF illustrates the versatility of droop control and how it can be used to optimally accommodate supply-demand imbalances.",
"However, it also highlights the need to tune parameters that have a direct effect on the network dynamics in order to achieve optimal steady state performance.",
"Remark 2 The use of quadratic costs is standard in the power system literature [29].",
"However, since this paper focuses mostly on local analysis, the analysis can be extended to include nonlinear costs by substituting $\\alpha _i^{z}$ with $\\frac{\\partial }{\\partial x}c_i^{z}(x)|_{x=\\delta {q_i^{z}}^*}$ where $\\delta {q_i^{z}}^*$ denotes the equilibrium value of $q_i^{z}$ ($z$ refers to either $g$ or $r$ )."
],
[
"Dynamic Performance",
"We now focus our attention on the dynamic performance metric.",
"In general, there are several metrics that can be considered [30], and the change of droop control coefficients can have direct effect in all of them.",
"Here we focus $\\mathcal {H}_2$ norm of the system when the output is the frequency vector $\\omega $ and the system is being driven by stochastic white noise.",
"The main motivation of this particular choice of metric is the need to characterize the effect of the increasing generation volatility as well as the effect of the measurement noise.",
"We argue that while generator-based droop control can be usually implemented without major measurement noise, implementing droop control on inverters that are distributed all over the transmission and distribution network needs to account for these errors.",
"Thus the use of $\\mathcal {H}_2$ -norm seems natural as it already accounts for stochastic inputs.",
"We assume that the net power injection of each bus is given by $P_i^{in} + k_i^1w_i^1(t)$ , where $w_1(t)=(w_i^1(t),i\\in N)$ is a vector of uncorrelated stochastic white noise with unit variance ($E[w_1^{T}(\\tau )w_1(t)]=\\delta (t-\\tau )I_n$ ) that represents demand fluctuations.",
"Similarly, we assume that for the purposes of implementing the droop control on the inverter the measured frequency is given by $\\omega _i(t) + k_i^2w_i^2(t)$ where again $w_2(t)=(w_i^2(t),i\\in N)$ is such that $E[w_2^{T}(\\tau )w_2(t)]=\\delta (t-\\tau )I_n$ .",
"Finally, since to estimate $\\dot{\\omega }_i(t)$ one needs to obtain first $\\omega _i$ , we assume that the measurement value of the frequency derivative is $\\dot{\\omega }_i + k_i^3w_i^3$ , where $w_3=(w_i^3:=\\frac{\\text{d}}{\\text{d}t}w_i^2,\\,i\\in N)$ .We use here the notation $\\frac{\\text{d}}{\\text{d}t}w_i^2$ to represent the frequency weighted noise process with weight function given by $\\text{w}_i(\\mathbf {i}2\\pi f)=\\mathbf {i}2\\pi f$ , see Remark REF for more details.",
"Without loss of generality we make the following change of variables $\\delta \\theta (t) = \\theta (t)-\\omega _0^*t,\\text{ and } \\delta \\omega (t) = \\omega (t) - \\omega _0^*.$ Thus, defining the system output to be $y(t)=\\delta \\omega (t)$ we can combine (REF ) together with (REF ) to get the following MIMO system I 0 0M = 0 I - L -D + 0 0 0 K1 -Rr-1K2 -MvK3 w1 w2 w3 y = 0 I where $M^v=\\operatorname{diag}(M^v_i)$ , $K_x = \\operatorname{diag}(k_i^x)$ and $w_x=(w_i^x)$ , with $x\\in \\lbrace 1,\\,2,\\,3\\rbrace $ , $\\hat{M}=M+M^v$ , $\\hat{D}=D+\\hat{R}^{-1}$ and $\\hat{R}^{-1}={R^{g}}^{-1}+{R^{r}}^{-1}$ .",
"Notice that for simplicity the system (REF ) implicitly assumes that all the inverters in the system are operated using the virtual inertia operation mode.",
"However, it is possible to model using (REF ) the droop controlled mode by setting $M^v_i=k_i^3=0$ .",
"Moreover, if one wants to model the constant power mode one just needs to additionally set $\\frac{1}{R_i^r}=0$ .",
"To simplify the discussion we will only consider two cases: 1) All the inverters implement droop control ($DC=N$ ); 2) All the inverters implement the virtual inertia control ($VI=N$ ).",
"It will also be useful to write (REF ) in standard form = A + B w1 w2 w3 , y = C , where A= 0 I - M-1L -M-1D , C= 0 I , and B = 0 0 0 M-1K1 -M-1R-1K2 -M-1MvK3 .",
"Let $H$ denote the LTI system (REF ).",
"Then the square of the $\\mathcal {H}_2$ norm of (REF ) can be formally defined using Dyn-Cost: ||H||H22 = t E [yT(t)y(t)] where $y(t)$ is the output of (REF ) when the input $w(t)=(w^1(t),w^2(t),w^3(t))$ is composed by a white noise process with unit covariance (i.e., $E[w_k(\\tau )w_k^T(t)] = \\delta (t -\\tau )I$ for $k\\in \\lbrace 1,2\\rbrace $ ) and its derivative ($w_3=\\frac{d}{dt}w_2$ ).",
"The computation of the $\\mathcal {H}_2$ norm has been widely studied in modern control theory.",
"In particular, in the case when $w_2$ and $w_3$ are not correlated processes ($w_3\\ne \\frac{d}{dt}w_2$ ), one very useful procedure to compute $||H||_{\\mathcal {H}_2}$ (see [31]) is based on using $||H||_{\\mathcal {H}_2}^2=\\operatorname{tr}(B^TXB)$ where $X$ is the observability Grammian, i.e.",
"$X$ solves the Lyapunov equation $A^TX+XA=-C^TC.$ In the context of power systems the use of this methodology has been first used in [32], where the authors seek to compute the power losses incurred by the network in the process of resynchronizing generators after a disturbance.",
"Since then, several works have used similar metrics to evaluate effect of controllers on the power system performance, see e.g.",
"[33], [34].",
"Remark 3 It is important to notice that in our case the system is driven also by the derivative of the noise process $w_2$ .",
"Thus, (REF ) can only be applied when $K_3=0$ .",
"When $K_3\\ne 0$ , then (REF ) corresponds to a frequency weighted $\\mathcal {H}_2$ -norm.",
"More precisely, then noise process $w_3$ is a frequency weighted process with weight function given by $W(s)=sI$ , i.e.",
"$\\hat{w}_3(s) = s\\hat{w}_2(s)$ where $\\hat{w}_k(s)$ is the Laplace Transform of $w_k(t)$ .",
"The next theorem shows that droop controlled inverters can indeed affect the performance by changing droop parameters, and that inverters implementing virtual inertia can drastically degrade the performance when measurement noise is considered.",
"Theorem 2 ($\\mathcal {H}_2$ -norm Computation) Assume homogeneous parameter values, i.e.",
"$M_i=m$ , $M^v_i=m^v$ , $D_i=d$ , $R_i^g= r^g$ , $R_i^r=r^r$ , $k_i^1=k_1$ , $k_i^2=k_2$ and $k_i^3=k_3$ .",
"Let $\\hat{r}^{-1} = {r^g}^{-1}+{r^r}^{-1},\\;\\hat{d}= d+\\hat{r}^{-1},$ and let $H_{DC}$ and $H_{VI}$ denote the MIMO system (REF ) when all inverters implement droop control and virtual inertia, respectively.",
"Then the squared $\\mathcal {H}_2$ norm of $H_{DC}$ and $H_{VI}$ is given by ||HDC||H22 =n((k1)2+(k2rr-1)2 )2m(d+rg-1+rr-1) , and ||HVI||H22 =+, respectively.",
"The proof of this theorem is analogous to [32].",
"We study the two cases (REF ) and (REF ) separately.",
"Computing $||H_{DC}||_{\\mathcal {H}_2}^2$ : Notice that in this case $M^v=0$ .",
"In order to compute $||H_{DC}||_{\\mathcal {H}_2}^2$ we first make the same change of variable used in [32], $\\delta \\theta = U\\theta ^{\\prime } \\text{ and } \\delta \\omega = U\\omega ^{\\prime },$ where $U$ is the orthonormal transformation that diagonalizes $L$ , i.e.",
"$U^TL_BU= \\Gamma $ where $\\Gamma =\\operatorname{diag}\\lbrace \\lambda _1=0,\\dots ,\\lambda _n\\rbrace $ and $U$ is assumed w.l.o.g.",
"to be of the form $U=[\\frac{1}{\\sqrt{n}}\\mathbf {1}_n \\,|\\,U_{n-1}]$ .",
"Thus, if we further transform $y=Uy^{\\prime }$ , $w_1=Uw_1^{\\prime }$ and $w_2=Uw_2^{\\prime }$ , we can decouple (REF ) into $n$ subsystems given by ${H_{DC,i}}$ : cl i' i' = 0 1 -im -dm ' ' + 0 0 k1m -k2mrr w1i' w2i' yi' = 0 1 i' i' , where a simple computation using (REF ) and (REF ) shows that $||H_{DC,i}||_{\\mathcal {H}_2}^2=\\frac{\\left((k_1)^2+(k_2{r^r}^{-1})^2 \\right)}{2m(d+{r^g}^{-1}+{r^r}^{-1}) }.$ Therefore, since $||H_{DC}||_{\\mathcal {H}_2}^2 = \\sum _{i=1}^n||H_{DC,i}||_{\\mathcal {H}_2}^2$ we obtain (REF ).",
"Computing $||H_{VI}||_{\\mathcal {H}_2}^2$ : To show that the norm $||H_{VI}||_{\\mathcal {H}_2}^2=+\\infty $ we will show that the transfer function of $H_{VI}$ has nonzero feedthrough ($\\sigma _{\\max }(H(\\mathbf {i}\\infty ))\\ge \\varepsilon >0$ ).",
"To compute the transfer function we first notice that $w_3=\\dot{w}_2$ which implies that we can model in the Laplace domain $\\hat{w}_3(s) = s\\hat{w}_2(s)$ .",
"Similar to the DC case, we can use a change of variable to decouple the system into $n$ different modes given by i' i' = Ai ' ' + Bi w1i' w2i' w3i' , yi' = Ci i' i' with Ai=0 1 -im -dm , Bi=0 0 0 k1m -k2mrr-k3mvm Ci=0 1 where $\\hat{m} =m+m^v$ .",
"We drop the subscript $i$ and define $\\hat{w}^{\\prime }(s) = [\\hat{w}^{\\prime }_1(s)\\,\\hat{w}^{\\prime }_2(s)\\,\\hat{w}^{\\prime }_3(s)]^T$ .",
"Thus we can compute the transfer function $H(s)$ using y(s)=H(s)w'(s)= C(sI-A)-1B w'(s) =1(s)Cs +dm 1 -m sB w'(s) =1(s)-m sB w'(s) =s(s) k1m -k2mrr-k3mvm w'(s) =s(s) k1m -k2mrr-k3mvm w1'(s) w2'(s) w3'(s) =ss2+dms + m k1m -(sk3mvm+ k2mrr) w1'(s) w2'(s) where $\\Delta (s)=s^2+\\frac{\\hat{d}}{\\hat{m}}s + \\frac{\\lambda }{\\hat{m}}$ and in the last step we used the relationship $\\hat{w}_3^{\\prime }(s) = s\\hat{w}_2^{\\prime }(s)$ .",
"It follows that when $f \\rightarrow +\\infty $ , $H(\\mathbf {i}2\\pi f)\\rightarrow \\begin{bmatrix}0& -\\frac{k_3m^v}{\\hat{m}}\\end{bmatrix},$ which implies that $||H_{VI}||_{\\mathcal {H}_2}^2=+\\infty $ .",
"Theorem REF provides an interesting insight on how the different controllers described in Section affect the system performance and illustrates the effect of measurement errors on it.",
"To understand the performance changes, we provide the $\\mathcal {H}_2$ norm of the swing equations ($H_{SW}$ ) without any additional control ($r^r=m^v=k_2=k_3=0$ ) $||H_{SW}||_{\\mathcal {H}_2}^2 = \\frac{n}{2m(d+{r^g}^{-1})}(k_1)^2.$ Thus it is easy to see that adding droop control introduces a larger damping $(d+{r^g}^{-1}+{r^r}^{-1})>d+{r^g}^{-1}$ .",
"However, there is an intrinsic tradeoff since if $r^r$ is small enough, then the noise amplification takes over and increases the norm.",
"But perhaps the most interesting result from this theorem is (REF ), which implies that the massive use of virtual inertia in the network can amplify the stochasticity of the system and introduce huge volatility in the frequency fluctuations."
],
[
"The Need for a Better Solution",
"The analysis provided in sections REF and REF shows that none of the existing solutions can simultaneously achieve efficient steady state operation while improving dynamic performance.",
"On the one hand, while Theorem REF shows indeed that droop control can be used to optimally allocate resources by carefully setting the parameters $R^r_i=\\alpha _i^r$ and $R^g_i=\\alpha _i^g$ , this tie between control parameters and economic efficiency makes it impossible to further improve the dynamic performance without incurring on an additional steady state cost (REF ).",
"Therefore, if one wants to operate the system in an efficient steady state, then $R^r_i$ cannot be used to improve the dynamic performance.",
"On the other hand, while the use of virtual inertia has been suggested to be a viable solution to improve the dynamic performance without losing steady state efficiency, Theorem REF shows that this solution cannot be widely adopted since it will induce a large noise amplification that can hinder the secure operation of the power grid.",
"As a result, all the existing solutions for inverter control cannot provide a dynamic performance improvement without sacrificing either steady-state or dynamic performance."
],
[
"Dynam-i-c Droop Control (iDroop)",
"We now introduce our iDroop control.",
"The main underlying idea on the design of the proposed controller is to leverage the flexibility that virtual inertia controllers provide while controlling the noise amplification using a filtering stage.",
"Let $ID\\subset N$ be the set of network buses that implement iDroop.",
"Then we propose to control the power injected to the grid by the inverter using iDroop: rl qi = qir,0 + xi xi = i(-1Rrii - xi )-ii A few comments are in order.",
"Firstly, the parameter $\\nu _i$ place a role similar to $M^v_i$ in the virtual inertia controller.",
"However, since this term is integrated in (), it is not longer interpreted as a virtual inertia.",
"Secondly, the parameters $\\delta _i$ and $\\nu _i$ can be independently tuned to reduce the noise introduced by the frequency ($k_i^2w^2_i$ ) or the frequency derivative ($k_i^3w^3_i=k_i^3\\dot{w}^2_i$ ).",
"This can be easily seen when these noise processes are introduced in () giving $\\dot{x}_i = \\delta _i\\left(-\\frac{1}{R^r_i}\\omega _i - x_i \\right)-\\nu _i\\dot{\\omega }_i - \\frac{\\delta _ik_i^2}{R_i^r}w_i^2 - \\nu _ik_i^3\\dot{w}_i^2.$ Finally, while the stability (in the absence of noise) of (REF ) with (REF ) is trivially guaranteed, the additional integration stage requires a more detailed analysis.",
"We do not provide here an explicit formula for $||H_{iDroop}||_{\\mathcal {H}_2}$ in this paper and leave it for future research.",
"Instead we will numerically illustrate in the next section the effect of $\\nu _i$ and $\\delta _i$ on this norm and how the performance of iDroop compares with the one of droop controlled inverters.",
"In the rest of this section we show that indeed our controllers are able preserve the steady state behavior for arbitrary parameter values $\\delta _i$ and $\\nu _i$ , and characterize a sufficient condition for asymptotic stability."
],
[
"Steady State Optimality",
"We now show that our iDroop controllers provide the same steady state properties as traditional droop control.",
"We do this by showing that iDroop achieves the minimum SS-Cost (REF ).",
"Theorem 3 (iDroop Optimality) Let $(\\theta ^*(t),\\omega ^*\\!\\!,{q^r}^*)$ be the steady state solution of (REF ) where $q_i^r$ is given by ().",
"Then the steady state solution of (REF ) and () is given by $ \\theta (t)^* \\!\\!= \\theta ^* + \\mathbf {1}_n\\omega _0^*t,\\,\\, \\omega _i^* = \\omega _0^*\\,\\,\\forall i \\text{ and }q_i^*-q_i^{r,0}= x_i^*=-\\frac{\\omega _0^*}{R_i^r}{}.$ Moreover, if $R_i^r=\\alpha _i^r$ and $R_i^g=\\alpha _i^g$ , then $\\delta {q_i^g}^* := -\\frac{1}{R_i^g}\\omega _0^*$ , $\\delta {q_i^r}^* := {q_i^r}^*-q_i^{r,0}$ is the unique allocation that minimizes the steady state cost (REF ) subject to (REF ), where $\\Delta P=\\sum _{i=1}^n p_i^{in}+q_i^{r,0} - D_i\\omega _0^*$ .",
"The proof of this theorem relies on Theorem REF .",
"We will show that iDroop has a steady state behavior that is identical to the standard droop control when $DC\\cup VI=ID$ .",
"Thus, we can use then use Theorem REF to show that indeed iDroop preserves the optimality characteristics of the traditional droop control.",
"We now characterize the steady state behavior of (REF ) and ().",
"Similarly to (REF ) and (REF ), we can use (REF ) to show that $\\dot{\\omega }=0$ only if $\\theta (t)=\\theta ^*(t) := \\theta ^*+\\mathbf {1}_n\\omega _0^*t$ .",
"Using () with $\\dot{x}=0$ and $\\dot{\\omega }=0$ , we get that $R_i^rx_i^* = -\\omega _i^*=-\\omega _0^*$ .",
"Therefore, the steady state deviation of the inverter $i$ is given by $\\delta {q_i^r}^*=q_i^*-q_i^{r,0} = x_i^*=-\\frac{1}{R_i^r}\\omega _0^*.$ Since the steady state deviation of conventional generators is by definition ${\\delta q_i^g}^*=-\\frac{1}{R_i^g}\\omega _0^*$ , then we have obtained the same allocation described in Theorem REF .",
"It follows then that when $\\alpha _i^r=R_i^r$ and $\\alpha _i^g=R_i^g$ , the steady state solution of () is an allocation that minimizes (REF ) subject to (REF )."
],
[
"Stability Analysis",
"We now show that the controllers described in () can preserve the stability of the network.",
"We use the same change of variable used in (REF ) together with $ \\delta x(t) = x(t) - x^*$ .",
"Therefore (REF ) and () become = M = - (D + Rg-1) -LB + x x= -K(Rr-1 + x) -K where $K_\\nu =\\operatorname{diag}(\\nu _i,i\\in N)$ .",
"To simplify notation we assume here that every bus iDroop.",
"However, the results can be generalized for any combination of iDroop with the controllers in (REF ).",
"It is easy to see that now the steady state solutions of Theorem REF become $( \\delta \\theta ^*, \\delta \\omega ^*, \\delta x^*)=(\\mathbf {1}_n\\alpha ,0,0)$ for any $\\alpha \\in R$ .",
"Thus the set of equilibria of (REF ) is given by $\\hat{\\mathcal {E}}=\\lbrace ( \\delta \\theta , \\delta \\omega , \\delta x): \\delta \\omega = \\delta x=0,\\, \\delta \\theta =\\mathbf {1}_n\\alpha ,\\,\\alpha \\in R\\rbrace .$ Theorem 4 (Asymptotic Convergence) Whenever the following condition holds, $\\frac{\\nu _i}{\\delta _i(\\nu _i\\!+\\!",
"{R_i^r}^{-1})}>0\\;\\text{ and }\\;(D_i\\!+\\!",
"{R_i^g}^{-1})+\\frac{\\nu _i{R_i^r}^{-1}}{\\nu _i+{R_i^r}^{-1}}>0,$ the iDroop control () converges asymptotically to the set of equilibria $\\hat{\\mathcal {E}}$ described in (REF ).",
"We will show that under the conditions of the theorem, the following Lyapunov function decreases along trajectories: $\\begin{aligned}V( \\delta \\theta , \\delta \\omega , \\delta x)=& \\frac{1}{2} \\delta \\theta ^T\\!L_B \\delta \\theta +\\frac{1}{2} \\delta \\omega ^TM \\delta \\omega \\\\&+ \\frac{1}{2}( \\delta x+K_\\nu \\delta \\omega )^TT( \\delta x+K_\\nu \\delta \\omega ).\\end{aligned}$ where $T\\in {R}^{n\\times n}$ is a positive definite diagonal matrix to be defined later ($T\\succ 0$ and $T=\\operatorname{diag}(v)$ , with $v\\in {R}^n$ ).",
"The Lyapunov function $V$ is inspired on [35] where a similar derivative term is used to damp oscillations.",
"The change of $V$ along the trajectories of (REF ) is given by V = TLB + TM + (x+K)TT(x+K) = TLB+ T(-(D + Rg-1)- LB+x) +(x+K)TTK(-Rr-1-x) = -T(D + Rg-1 + KTKRr-1) -xTTKx +T(I-KTK-TKRr-1)x where (REF ) follows from (REF ), and (REF ) from rearranging the terms.",
"We now choose $T$ so that the cross term in (REF ) is zero.",
"Since $T$ is assumed to be a diagonal matrix, then we get 0=I-KTK-TKRr-1 0=I-TK(K+ Rr-1) T = K-1 (K+ Rr-1)-1 Therefore, using (REF ), it follows that V= -T(D + Rg-1 + KRr-1(K+ Rr-1)-1 ) -xTKK-1(K+ Rr-1)-1 x Thus, it follows from (REF ) that $\\dot{V}\\le 0$ and we can now apply LaSalle's Invariance Principle[36] to show that $(\\delta \\theta (t),\\delta \\omega (t),\\delta x(t))$ converges to the largest invariant set $M\\subset \\lbrace \\dot{V}\\equiv 0\\rbrace $ .",
"Using (REF ) we can show that $\\dot{V}\\equiv 0$ implies that $\\delta \\omega (t)\\equiv 0$ and $\\delta x(t)\\equiv 0$ , which implies in turn (through (REF )) that $\\dot{\\delta \\theta }\\equiv 0$ .",
"Therefore, it must follow that $\\delta \\theta (t)\\equiv \\delta \\theta ^*$ .",
"Finally, using (REF ) we get $0\\equiv L_B\\delta \\theta (t)\\equiv L_B\\delta \\theta ^*$ which, since the graph is connected, implies that $\\theta ^*=\\mathbf {1}_n\\alpha $ for some $\\alpha \\in R$ .",
"Therefore, every trajectory converges to the set of equilibria $\\hat{\\mathcal {E}}$ given in (REF ), i.e.",
"$M\\subset \\hat{\\mathcal {E}}$ .",
"Figure: IEEE 39 Bus System (New England)"
],
[
"Numerical Illustrations",
"In this section we numerically illustrate some of the features of our iDroop control using the IEEE 39 bus system shown in Fig.",
"REF .",
"The network parameters as well as the stationary starting point were obtained from the Power System Toolbox (PST)[37] dataset.",
"Before building the dynamic model, we perform the Kron reduction for every load bus.",
"Thus the final system has 10 buses that correspond to each generator in the nework.",
"The inertia parameters of each generators are also obtained from the PST dataset.",
"Throughout the time domain simulations we assume that the aggregate generator damping and load frequency sensitivity parameter is $D_i=0.1$ .",
"We also set the droop coefficient of each generator to $R_i^g=15$ .",
"On each bus we add an additional inverter-based generator and vary their operational mode using either one of the three modes described in (REF ) (CP=constant power, DC=droop control and VI=virtual inertia), or the iDroop mode.",
"The droop coefficient is set to $R_i^r=15$ in all the cases.",
"For illustration purposes we select parameters so that the VI mode and iDroop have a similar frequency transient behavior.",
"In particular, we choose $M_i^v=0.15$ , $\\delta _i=6$ , $\\nu _i=0.9$ .",
"Figure: Frequency deviations after perturbationWe initialize the system in steady state and at time $t=1$ we introduce a step change $\\Delta P_{30}=-0.5$ p.u.",
"in the power injection at bus 30 (where generator 10 is located).",
"Fig.",
"REF shows the evolution of the frequency deviations for the four cases: (a) CP, (b) DC, (c) VI and (d) iDroop.",
"It can be seen that both VI (Fig.",
"REF -c) and iDroop (Fig.",
"REF -d) can reduce the Nadir (minimum frequency achieved), while achieving the the same steady state as the DC mode (Fig.",
"REF -b).",
"Figure: Power deviation of inverterHowever, the way this is achieved is in the case of VI and iDroop is completely different.",
"This can be seen in Fig.",
"REF where we show the power deviation experienced by each individual inverter.",
"In particular, the step change in the power injection introduces a discontinuity in $\\dot{\\omega }_{30}$ .",
"This is shown in Fig.",
"REF -c where the inverter at bus 30 has a step change in power.",
"iDroop on the other hand is able to perform a similar task using less peak power and with a more desirable behavior.",
"This drastic change on the power experienced by the VI also illustrates why it is expected that this solution will produce noise amplification.",
"Figure: ℋ 2 \\mathcal {H}_2-norm comparison between iDroop and Droop ControlFigure: Frequency fluctuations when δ i =6\\delta _i=6 and ν i =0.01\\nu _i=0.01Finally, since the VI mode has unbounded $\\mathcal {H}_2$ -norm, we show in Fig.",
"REF the $\\mathcal {H}_2$ -norm of the system when the inverters are using only the DC mode and iDroop on a high measurement noise regime ($k_i^1=0.1$ , $k_i^2=k_i^3=5$ ).",
"We can see that DC does not vary with $\\delta $ and $\\nu $ (as expected) while iDroop does.",
"However, more importantly, iDroop is able to achieve better performance than DC despite using frequency derivative measurements.",
"This is also illustrated in Figure REF where we simulate the frequency fluctuations for the values of $\\delta _i=6$ and $\\nu _i=0.01$ ."
],
[
"Concluding Remarks",
"This paper studies the intrinsic trade off between steady state and dynamic performance of inverter-based droop control and several of its variants.",
"We show that while the standard droop control can improve the dynamic performance, it can only achieve it by losing steady state efficiency.",
"Moreover, our analysis also shows that the popular alternative of adding virtual inertia is very sensitive to measurement noise and can increase the frequency variance unboundedly.",
"To solve these issues we propose a new control scheme (iDroop) that is able to tune dynamic performance without altering steady state efficiency.",
"We characterize the set of parameter values that can guarantee the stability of the steady state solution and illustrate its behavior numerically.",
"In particular, we show that iDroop is able to reduce the Nadir and variance when compared with Constant Power (CP) and Droop Control (DC) modes."
]
] | 1612.05804 |
[
[
"Determination of Baum-Bott residues of higher codimensional foliations"
],
[
"Abstract Let $\\mathscr{F}$ be a singular holomorphic foliation, of codimension $k$, on a complex compact manifold such that its singular set has codimension $\\geq k+1$.",
"In this work we determinate Baum-Bott residues for $\\mathscr{F}$ with respect to homogeneous symmetric polynomials of degree $k+1$.",
"We drop the Baum-Bott's generic hypothesis and we show that the residues can be expressed in terms of the Grothendieck residue of an one-dimensional foliation on a $(k+1)$-dimensional disc transversal to a $(k+1)$-codimensional component of the singular set of $\\mathscr{F}$.",
"Also, we show that Cenkl's algorithm for non-expected dimensional singularities holds dropping the Cenkl's regularity assumption."
],
[
"Introduction",
"In [2] P. Baum and R. Bott developed a general residue theory for singular holomorphic foliations on complex manifolds.",
"More precisely, they proved the following result: Theorem 1.1 (Baum-Bott) Let $ {F}$ be a holomorphic foliation of codimension $k$ on a complex manifold $M$ and $\\varphi $ be a homogeneous symmetric polynomials of degree $d$ satisfying $ k < d \\le n$ .",
"Let $Z$ be a compact connected component of the singular set $\\mathrm {Sing}( {F})$ .",
"Then, there exists a homology class $\\mathrm {Res}_{\\varphi }( {F}, Z) \\in \\text{\\rm H}_{2(n - d)}(Z; \\mathbb {C})$ such that: $\\mathrm {Res}_{\\varphi }( {F}, Z) $ depends only on $\\varphi $ and on the local behavior of the leaves of $ {F}$ near $Z$ , Suppose that $M$ is compact and denote by $\\mathrm {Res}({\\varphi }, {F}, Z):=\\alpha _{\\ast }\\mathrm {Res}_{\\varphi }( {F}, Z)$ , where $\\alpha _{\\ast }$ is the composition of the maps $\\displaystyle \\text{\\rm H}_{2(n - d )}(Z; \\mathbb {C}) \\stackrel{i^{\\ast }}{\\longrightarrow }\\text{\\rm H}_{2(n - d )}(M; \\mathbb {C}) $ and $\\displaystyle \\text{\\rm H}_{2(n - d )}(M; \\mathbb {C}) \\stackrel{P}{\\longrightarrow }\\text{\\rm H}^{2d}(M; \\mathbb {C}) $ with $i^{\\ast }$ is the induced map of inclusion $i : Z \\longrightarrow M$ and $P$ is the Poincaré duality.",
"Then $\\displaystyle \\varphi (\\mathcal {N}_{ {F}}) = \\sum _{Z} \\mathrm {Res}({\\varphi }, {F}, Z).$ The computation and determination of the residues is difficult in general.",
"If the foliation $ {F}$ has dimension one with isolated singularities , Baum and Bott in [1] show that residues can be expressed in terms of a Grothendieck residue, i.e, for each $p\\in \\mathrm {Sing}( {F})$ we have $\\mathrm {Res}_{\\varphi }( {F}, Z)= \\mbox{Res}_{p}\\Big [\\varphi (JX)\\frac{dz_{1}\\wedge \\cdots \\wedge dz_{n}}{X_{1}\\cdots X_{n}}\\Big ] ,$ where $X$ is a germ of holomorphic vector field at $p$ tangent to $ {F}$ and $JX$ is the jacobian of $X$ .",
"The subset of $\\mathrm {Sing}( {F})$ composed by analytic subsets of codimension $k+1$ will be denoted by $\\mathrm {Sing}_{k+1}( {F})$ and it is called the singular set of $ {F}$ with expected codimension .",
"Baum and Bott in [2] exibes the residues for generic componentes of Sing$_{k + 1}( {F})$ .",
"Let us recall this result: An irreducible component $Z$ of $\\mathrm {Sing}_{k + 1}( {F})$ comes endowed with a filtration.",
"For given point $p \\in Z$ choose holomorphic vector fields $v_{1},\\dots , v_{s}$ defined on an open neighborhood $U_{p}$ of $p \\in M$ and such that for all $x \\in U_{p}$ , the germs at $x$ of the holomorphic vector fields $v_{1},\\dots ,v_{s}$ are in $ {F}_{x}$ and span $ {F}_{x}$ as a $\\mathcal {O}_{x}$ -module.",
"Define a subspace $V_{p}( {F}) \\subset T_{p}M$ by letting $V_{p}( {F})$ be the subspace of $T_{p}M$ spanned by $v_1(p),\\dots v_s(p)$ .",
"We have $ Z^{(i)} = \\lbrace p \\in Z ; \\ \\ \\mathrm {dim}(V_{p}( {F})) \\le n - k - i \\rbrace \\ \\ \\mbox{for} \\ \\ i = 1,\\dots ,n - k. $ Then, $ Z \\supseteq Z^{(1)} \\supseteq Z^{(2)} \\supseteq \\dots \\supseteq Z^{(n - k)}$ is a filtration of $Z$ .",
"Now, consider a symmetric homogeneous polynomial $\\varphi $ of degree $k+1$ .",
"Let $Z \\subset \\mathrm {Sing}_{k+1}( {F})$ be an irreducible component.",
"Take a generic point $p \\in Z$ such that $p$ is a point where $Z$ is smooth and disjoint from the other singular components.",
"Now, consider $B_{p}$ a ball centered at $p $ , of dimension $k+1$ sufficiently small and transversal to $Z$ in $p$ .",
"In [2] Baum and Bott proved under the following generic assumption $ \\mbox{cod}(Z) = k + 1 \\ \\ \\ \\mbox{and} \\ \\ \\ \\mbox{cod}(Z^{(2)} ) < k + 1 $ that we have $ \\mathrm {Res}( {F}, \\varphi ; Z) = \\mathrm {Res}_{\\varphi }( {F}|_{B_p}; p)[Z], $ where $ \\mathrm {Res}_{\\varphi }( {F}|_{B_p}; p)$ represents the Grothendieck residue at $p$ of the one dimensional foliation $ {F}|_{B_p}$ on $B_p$ and $[Z]$ denotes the integration current associated to $Z$ .",
"In [5] and [8] the authors determine the residue $\\mathrm {Res}( {F}, c_1^{k+1} ; Z)$ , but even in this case they do not show that we can calculate these residues in terms of the Grothendieck residue of a foliation on a transversal disc.",
"In [15] Vishik proved the same result under the Baum-Bott's generic hypotheses but supposing that the foliation has locally free tangent sheaf.",
"In [3] F. Bracci and T. Suwa study the behavior of the Baum-Bott residues under smooth deformations, providing an effective way of computing residues.",
"In this work we drop the Baum-Bott's generic hypotheses and we prove the following : Theorem 1.2 Let $ {F}$ be a singular holomorphic foliation of codimension $k$ on a compact complex manifold $M$ such that $\\mathrm {cod} (\\mathrm {Sing}( {F}) )\\ge k + 1$ .",
"Then, $ \\mathrm {Res}( {F}, \\varphi ; Z) = \\mathrm {Res}_{\\varphi }( {F}|_{B_p}; p)[Z], $ where $\\mathrm {Res}_{\\varphi }( {F}|_{B_p}; p)$ represents the Grothendieck residue at $p$ of the one dimensional foliation $ {F}|_{B_p}$ on a $(k+1)$ -dimensional transversal ball $B_p$ .",
"Finally, in the last section we apply Cenkl's algorithm for non-expected dimensional singularities [7].",
"Moreover, we drop Cenkl's regularity hypothesis and we conclude that it is possible to calculate the residues for foliations whenever $\\mathrm {cod} (\\mathrm {Sing}( {F})) \\ge k+s$ , with $s\\ge 1$ ."
],
[
"Acknowledgments",
"We are grateful to Jean-Paul Brasselet, Tatsuo Suwa and Marcio G. Soares for interesting conversations.",
"This work was partially supported by CNPq, CAPES, FAPEMIG and FAPESP-2015/20841-5.",
"We are grateful to Institut de Mathématiques de Luminy- Marseille and Imecc–Unicamp for hospitality.",
"Finally, we would like to thank the referee by the suggestions, comments and improvements to the exposition."
],
[
"Holomorphic foliations",
"Denote by $ \\Theta _M$ the tangent sheaf of $M$ .",
"A foliation $ {F}$ of codimension $k$ on an $n$ -dimensional complex manifold $M$ is given by a exact sequence of coherent sheaves $0 \\longrightarrow T {F}\\longrightarrow \\Theta _M \\rightarrow N_{ {F}} \\longrightarrow 0,$ such that $ [T {F},T {F}] \\subset T {F}$ and the normal sheaf $N_{ {F}}$ of $ {F}$ is a torsion free sheaf of rank $k\\le n -1$ .",
"The sheaf $T {F}$ is called the tangent sheaf of $ {F}$ .",
"The singular set of ${F}$ is defined by $\\mathrm {Sing}( {F}):=\\mathrm {Sing}(N_{ {F}})$ .",
"The dimension of $ {F}$ is $\\mathrm {dim}( {F})=n-k$ .",
"Also, a foliation $ {F}$ , of codimension $k$ , can be induced by a exact sequence $0 \\longrightarrow N_{ {F}}^{\\vee } \\longrightarrow \\Omega _M^1\\rightarrow \\mbox{${\\mathcal {Q}}$}_{ {F}} \\longrightarrow 0,$ where $\\mbox{${\\mathcal {Q}}$}_{ {F}}$ is a torsion free sheaf of rank $n-k$ .",
"Moreover, the singular set of ${F}$ is $\\mathrm {Sing}(\\mbox{${\\mathcal {Q}}$}_{ {F}})$ .",
"Now, by taking the wedge product of the map $ N_{ {F}}^{\\vee } \\longrightarrow \\Omega _M^1 $ we get a morphism $\\bigwedge ^k N_{ {F}}^{\\vee } \\longrightarrow \\Omega _M^k$ and twisting by $(\\bigwedge ^k N_{ {F}}^{\\vee })^{\\vee }=\\det (N_{ {F}})$ we obtain a morphism $\\omega : \\mathcal {O}_M \\longrightarrow \\Omega _M^k\\otimes \\det (N_{ {F}}).$ Therefore, a foliation is induced by a twisted holomorphic $k$ -form $\\omega \\in \\text{\\rm H}^0(X,\\Omega _M^k\\otimes \\det (N_{ {F}}) )$ which is locally decomposable outside the singular set of $ {F}$ .",
"That is, by the classical Frobenius Theorem for each point $p\\in X\\setminus \\mathrm {Sing}( {F})$ there exists a neighbourhood $U$ and holomorphics 1-forms $\\omega _1, \\dots ,\\omega _k \\in \\text{\\rm H}^0(U, \\Omega _U^1)$ such that $\\omega |_{U}=\\omega _1 \\wedge \\cdots \\wedge \\omega _k$ and $d \\omega _i \\wedge \\omega _1 \\wedge \\cdots \\wedge \\omega _k=0$ for all $i=1,\\dots ,k$ ."
],
[
"Proof of the Theorem",
"Given a multi-index $\\alpha = (\\alpha _{1},\\dots ,\\alpha _{k})$ with $\\alpha _{j} \\ge 0 $ for $j = 1,\\dots ,k$ , consider the homogeneous symmetric polynomial of degree $k + 1$ , $\\varphi = c_{1}^{\\alpha _{1}}c_{2}^{\\alpha _{2}}\\cdots c_{k}^{\\alpha _{k}} $ such that $1\\alpha _{1} + 2\\alpha _{2} + \\cdots + k\\alpha _{k} = k + 1 $ .",
"Let us consider the twisted $k$ -form $ \\omega \\in \\text{\\rm H}^{0}(M, \\Omega ^{k}_{M} \\otimes \\det (N_{ {F}}))$ induced by $ {F}$ .",
"Denote by Sing$_{ k+1}( {F})$ the union of the irreducible components of $\\mathrm {Sing}( {F})$ of pure codimension $k + 1$ .",
"Consider an open subset $U\\subset M\\setminus \\mathrm {Sing}( {F}) $ .",
"Thus, the form $\\omega |_U$ is decomposable and integrable.",
"That is, $\\omega |_U$ is given by a product of $k$ 1-forms $\\omega _{1} \\wedge \\cdots \\wedge \\omega _{k}$ .",
"Then, it is possible to find a matrix of $(1,0)$ -forms $(\\theta _{ls}^*)$ such that $ \\partial \\omega _{l} = \\sum _{s = 1}^{k} \\theta _{ls}^* \\wedge \\omega _{s}, \\ \\ \\overline{\\partial } \\omega _{l} =0, \\ \\ \\forall \\ \\ l = 1,\\dots ,k.$ We have that $\\omega _{1} , \\ldots , \\omega _{k}$ is a local frame for $N_{ {F}}^*|_{U}$ and the identity above induces on $U$ the Bott partial connection $\\nabla : C^{\\infty }( N_{ {F}}^*|_{U} ) \\rightarrow C^{\\infty }( (T {F}^*\\oplus \\overline{TM}) \\otimes N_{ {F}}^* |_{U} )$ defined by $\\nabla _v(\\omega _{l})= i_{v} ( \\partial \\omega _{l}), \\ \\ \\ \\ \\nabla _u(\\omega _{l})= i_{u} ( \\overline{\\partial } \\omega _{l})=0,$ where $v \\in C^{\\infty }( T {F}|_{U} )$ and $u\\in C^{\\infty }( \\overline{TM} |_{U} )$ which can be extended to a connection $D^*:C^{\\infty }( N_{ {F}}^*|_{U} ) \\rightarrow C^{\\infty }( (TM^*\\oplus \\overline{TM}) \\otimes N_{ {F}}^* |_{U} )$ in the following way $D^*_v(\\omega _{l})=\\sum _{s=1}^{k} i_v(\\pi (\\theta _{ls}^*)) \\omega _{s} , \\ \\ \\ \\ D^*_u(\\omega _{l})= i_{u} ( \\overline{\\partial } \\omega _{l})=0$ where $v \\in C^{\\infty }( TM|_{U} )$ and $u\\in C^{\\infty }( \\overline{TM} |_{U} )$ and $\\pi :TM^*|_{U}\\rightarrow N_{ {F}}^*|_{U} $ is the natural projection.",
"Let $\\theta ^{*} $ be the matrix of the connection $D^*$ , then $\\theta :=[-\\theta ^{*}]^t$ is the matrix of the induced connection $D$ with respect to the frame $\\lbrace \\omega _{1} , \\dots , \\omega _{k}\\rbrace $ .",
"Let $K$ be the curvature of the connection $D$ of $N_{ {F}}$ on $M\\setminus \\mathrm {Sing}( {F})$ .",
"It follows from Bott's vanishing Theorem [13] that $\\varphi (K) =0 $ .",
"Let $V$ be a small neighborhood of $\\mathrm {Sing}_{k + 1}( {F})$ .",
"We regularize $\\theta $ and $K$ on $V$ , i.e.",
"we choose a matrix of smooth forms $\\widehat{\\theta }$ and $\\widehat{K}$ coinciding with $\\theta $ and $K$ outside of $V$ , respectively.",
"By hypothesis $\\mathrm {dim}(\\mathrm {Sing}( {F}))\\le n- k-1$ we conclude by a dimensional reason that, for $\\deg (\\varphi ) = k+1$ , only the components of dimension $n-k -1$ of $\\mathrm {Sing}( {F})$ play a role.",
"In fact, since $\\mathrm {Res}_{\\varphi }( {F}, Z) \\in \\text{\\rm H}_{2(n-k-1)}(Z, \\mathbb {C})$ , components of dimension smaller than $n-k -1$ contribute nothing.",
"This means that $\\varphi (\\widehat{K})$ localizes on Sing$_{k + 1}( {F})$ .",
"Then, $\\varphi (\\widehat{K})$ has compact support on $V$ , where $V$ is a small neighborhood of $\\mathrm {Sing}_{k + 1}( {F})$ .",
"That is, $\\mathrm {Supp}(\\varphi (\\widehat{K})) \\subset \\overline{V} .$ Then $\\varphi (\\widehat{K})= \\sum _{Z_i} \\widehat{\\lambda _i}(\\varphi ) [Z_i],$ where $Z_i$ is an irreducible component of Sing$_{k+1}( {F})$ and $\\widehat{\\lambda _i}(\\varphi ) \\in \\mathbb {C}$ .",
"On the other hand, we have that $\\varphi (N_{ {F}})=\\sum _{Z_i} \\mathrm {Res}({\\varphi }, {F}, Z_i) = \\sum _{Z_i} \\lambda _i(\\varphi ) [Z_i].$ We will show that $\\lambda _i(\\varphi )=\\widehat{\\lambda _i}(\\varphi )$ , for all $i$ .",
"In particular, this implies that $\\varphi (\\widehat{K})=\\varphi (N_{ {F}})$ .",
"Thereafter, we will determinate the numbers $\\widehat{\\lambda _i}(\\varphi )$ .",
"Consider the unique complete polarization of the polynomial $\\varphi $ , denoted by $\\widetilde{\\varphi }$ .",
"That is, $\\widetilde{\\varphi }$ is a symmetric $k$ -linear function such that $\\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\widetilde{\\varphi }(\\widehat{K},\\dots ,\\widehat{K}) = \\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\varphi (\\widehat{K}).",
"$ Take a generic point $p \\in Z_i$ , that is, $p$ is a point where $Z_i$ is smooth and disjoint from the other components.",
"Let us consider $L \\subset M$ a $(k+1)$ -ball intersecting transversally Sing$_{k+1}( {F})$ at a single point $p \\in Z_i$ and non intersecting other component.",
"Define $BB( {F},\\varphi ; Z_i) := \\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{L} \\varphi ( \\widehat{K}).$ Then $\\widehat{ \\lambda _i}(\\varphi ) = BB( {F},\\varphi ; Z_i)$ .",
"In fact $BB( {F},\\varphi ; Z_i)=\\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{L} \\varphi (\\widehat{K})= [L] \\cap [\\varphi ( \\widehat{K})]= \\widehat{ \\lambda _i}(\\varphi ) [L] \\cap [Z_i]=\\widehat{ \\lambda _i}(\\varphi )$ since $[L] \\cap [Z_i]=1$ and $[L] \\cap [Z_i]=0$ for all $i\\ne j$ .",
"For each $j = 1,\\dots ,k$ , define the polynomial $ \\varphi _{j}(\\widehat{\\theta }, \\widehat{K}) := \\widetilde{\\varphi }( \\widehat{\\theta },\\underbrace{-2 \\widehat{\\theta } \\wedge \\widehat{\\theta } ,\\dots ,-2\\widehat{\\theta }\\wedge \\widehat{\\theta }}_{j - 1},\\underbrace{ \\widehat{K},\\dots ,\\widehat{K}}_{k - j}).",
"$ Now, we consider the $(2k + 1)$ - form $\\displaystyle \\varphi _{\\alpha }(\\widehat{\\theta }, \\widehat{K}) = \\sum _{j = 0}^{k - 1}(-1)^{j} \\frac{(k - 1)!",
"}{2^{j}(k - j - 1)!",
"(k + j)! }",
"\\varphi _{j + 1}(\\widehat{\\theta }, \\widehat{K}) .$ It follows from [15] that on $X\\setminus Sing_{k+1}( {F})$ we have $ d(\\varphi _{\\alpha }(\\widehat{\\theta }, \\widehat{K}) ) = \\varphi (\\widehat{K}).",
"$ Consider $i: B \\rightarrow M$ an embedding transversal to $Z_i$ on $p$ as above, i.e, $i(B)=L$ .",
"We have then an one-dimensional foliation $ {F}|_{L}=i^* {F}$ on $B$ singular only on $i^{-1}(p)=0$ .",
"We have that $\\widehat{\\lambda _i}(\\varphi )= BB( {F},\\varphi ; Z_i)= \\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{L} \\varphi ( \\widehat{K})= \\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{B} \\varphi (i^* \\widehat{K}).$ Now, by Stokes's theorem we obtain $\\widehat{\\lambda _i}(\\varphi ) = \\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{B} \\varphi (i^* \\widehat{K})= \\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{B} d(\\varphi _{\\alpha }(i^*\\widehat{\\theta },i^* \\widehat{K}) )=\\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{\\partial B} \\varphi _{\\alpha }(i^*\\widehat{\\theta },i^* \\widehat{K}).$ Firstly, it follows from [15] that $\\widehat{\\lambda _i}(\\varphi )=\\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{\\partial B} \\varphi _{\\alpha }(i^*\\widehat{\\theta },i^* \\widehat{K})= \\mbox{ Res}_{\\varphi }(i^* {F}; 0)$ Now, we will adopt the Baum and Bott construction [2].",
"Denote by $\\mathcal {A}_M$ the sheaf of germs of real-analytic functions on $M$ .",
"Consider on $M$ a locally free resolution of $N_{ {F}}$ $0\\rightarrow \\mathcal {E}_r \\rightarrow \\mathcal {E}_{r-1} \\rightarrow \\cdots \\rightarrow \\mathcal {E}_{0} \\rightarrow N_{ {F}}\\otimes \\mathcal {A}_M\\rightarrow 0.$ Let $D_{q},D_{q-1}, \\dots , D_0$ be connections for $\\mathcal {E}_q , \\mathcal {E}_{q-1} , \\dots , \\mathcal {E}_{0}$ , respectively.",
"Set the curvature of $D_{i}$ by $K_i=K(D_{i})$ .",
"By using Baum-Bott notation [2] we have that $\\varphi (K_{q}| K_{q-1}| \\cdots | K_0)= \\varphi ( N_{ {F}}).$ Consider on $V$ a locally free resolution of the tangent sheaf of $ {F}$ : $0\\rightarrow \\mathcal {E}_q \\rightarrow \\mathcal {E}_{q-1} \\rightarrow \\cdots \\rightarrow \\mathcal {E}_{1} \\rightarrow T {F}\\otimes \\mathcal {A}_V\\rightarrow 0.$ Combining this sequence with the sequence $0\\rightarrow T {F}\\otimes \\mathcal {A}_V \\rightarrow TV \\rightarrow N_{ {F}}\\otimes \\mathcal {A}_V\\rightarrow 0.$ we get $ 0\\rightarrow \\mathcal {E}_q \\rightarrow \\mathcal {E}_{q-1} \\rightarrow \\cdots \\rightarrow \\mathcal {E}_{1} \\rightarrow TV \\rightarrow N_{ {F}}\\otimes \\mathcal {A}_V\\rightarrow 0.$ Pulling back the sequence (REF ) by $i:B\\rightarrow V$ we obtain an exact sequence on $B$ : $0\\rightarrow i^*\\mathcal {E}_q \\rightarrow i^* \\mathcal {E}_{q-1} \\rightarrow \\cdots \\rightarrow i^*\\mathcal {E}_{1} \\rightarrow i^*(T {F}\\otimes \\mathcal {A}_V)\\rightarrow 0.$ Since $B$ is a small ball we have the splitting $i^*TV=TB\\oplus N_{B|V}$ , where $N_{B|V}$ denotes its normal bundle.",
"We consider the projection $\\xi :i^*TV \\rightarrow TB$ and we map $i^*TV$ to $N_{i^* {F}}$ via $i^*TV \\stackrel{\\xi }{\\rightarrow } TB \\rightarrow N_{i^* {F}}$ which give us an exact sequence $0\\rightarrow i^*(T {F}\\otimes \\mathcal {A}_V) \\rightarrow i^*TV \\rightarrow N_{i^* {F}} \\otimes \\mathcal {A}_B\\rightarrow 0$ Now, combining the exact sequences (REF ) and (REF ) we obtain an exact sequence $0\\rightarrow i^*\\mathcal {E}_q \\rightarrow i^* \\mathcal {E}_{q-1} \\rightarrow \\cdots \\rightarrow i^*\\mathcal {E}_{1} \\rightarrow i^*TV \\rightarrow N_{i^* {F}}\\otimes \\mathcal {A}_B\\rightarrow 0.$ Let $D_{q},D_{q-1}, \\dots , D_0$ be connections for $\\mathcal {E}_q , \\mathcal {E}_{q-1} , \\dots , \\mathcal {E}_{1} , TV$ , respectively.",
"Observe that $i^*\\varphi (K_{q}| K_{q-1}| \\cdots | K_0)=\\varphi (i^*K_{q}| i^*K_{q-1}| \\cdots | i^*K_0)= \\varphi ( N_{i^* {F}}).$ Finally, it follows from [2] $\\mbox{ Res}_{\\varphi }(i^* {F}; 0)= \\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{ B} \\varphi (i^*K_{q}| i^*K_{q-1}| \\cdots | i^*K_0)=\\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{ B} i^*\\varphi (K_{q}| K_{q-1}| \\cdots | K_0)$ and [2] that $\\mbox{ Res}_{\\varphi }(i^* {F}; 0)= \\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{ B} i^*\\varphi (K_{q}| K_{q-1}| \\cdots | K_0) = \\lambda _i(\\varphi ).$ Thus, we conclude from (REF ) and (REF ) that $\\lambda _i(\\varphi )=\\widehat{\\lambda _i}(\\varphi ),$ for all $i$ .",
"This implies that $\\varphi (\\widehat{K})=\\varphi (N_{ {F}})$ .",
"Now, we will determinate the numbers $\\widehat{\\lambda _i}(\\varphi )$ .",
"Let $X=\\sum _{r=1}^{k+1} X_i\\partial / \\partial z_i$ be a vector field inducing $i^* {F}$ on $B$ and $J(X)$ denotes the Jacobian of $X$ .",
"Let $\\omega $ be the 1-form on $B\\setminus \\lbrace 0\\rbrace $ such that $i_X(\\omega )=1$ .",
"It follows from [15] that $\\widehat{\\lambda _i}(\\varphi ) = \\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{\\partial B} \\varphi _{\\alpha }(i^*\\widehat{\\theta },i^* \\widehat{K})=\\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{\\partial B} \\omega \\wedge (\\overline{\\partial }\\,\\omega )^{k}\\varphi (-J(X)).$ Thus, $\\widehat{\\lambda _i}(\\varphi )=\\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{\\partial B} (-1)^{k+1} \\omega \\wedge (\\overline{\\partial }\\,\\omega )^{k}\\varphi (J(X)),$ By using Martinelli's formula [9] we have $\\widehat{\\lambda _i}(\\varphi ) =\\left(\\frac{1}{2 \\pi i}\\right)^{k+1} \\int _{\\partial B} (-1)^{k+1} \\omega \\wedge (\\overline{\\partial }\\,\\omega )^{k}\\varphi (J(X))=\\mbox{Res}_{0}\\Big [\\varphi (JX)\\frac{dz_{1}\\wedge \\cdots \\wedge dz_{k+1}}{X_{1}\\cdots X_{k+1}}\\Big ].$ Therefore, $\\widehat{\\lambda _i}(\\varphi ) = \\mbox{ Res}_{\\varphi }(i^* {F}; 0) = \\mbox{Res}_{\\varphi }( {F}|_{L}; p),$ where $\\mathrm {Res}_{\\varphi }( {F}|_{L}; p)$ represents the Grothendieck residue at $p$ of the one dimensional foliation $ {F}|_{L}$ on a $(k+1)$ -dimensional transversal ball $L$ ."
],
[
"Examples ",
"In the next examples, with a slight abuse of notation, we write $ \\mathrm {Res}( {F}, \\varphi ; Z_{i}) = \\lambda _i(\\varphi ).",
"$ Example 4.1 Let $ {F}$ be the logarithmic foliation on $\\mathbb {P}^{3}$ induced, locally in $(\\mathbb {C}^{3}, (x,y,z))$ by the polynomial 1-form $ \\omega = yzdx + xzdy + xy dz.",
"$ In this chart, the singular set of $\\omega $ is the union of the lines $ Z_{1} = \\lbrace x = y = 0 \\rbrace ; \\ \\ Z_{2} = \\lbrace x = z = 0 \\rbrace \\ \\ \\mbox{and} \\ \\ Z_{3} = \\lbrace y = z = 0 \\rbrace $ .",
"We have $ \\mathrm {Res}( {F},c_{1}^{2}; Z_{i}) = \\mathrm {Res}_{c_{1}^{2}} (\\mbox{${G}$}; p_{i})$ , where $\\mbox{${G}$}$ is a foliation on $D_{i}$ with $D_{i}$ a 2-disc cutting transversally $Z_{i}$ .",
"Consider $D_{1}=\\lbrace ||(x,y) ||\\le 1, \\ \\ z = 1 \\rbrace $ then, we have $ \\omega |_{D_{1}} =: \\omega _{1} = y dx + x dy \\ \\ \\ \\ \\mbox{with} \\ \\ \\mbox{dual} \\ \\ \\mbox{vector} \\ \\ \\mbox{field} \\ \\ \\ \\ X_{1} = x \\frac{\\partial }{\\partial x} - y \\frac{\\partial }{\\partial y}.$ Then, $D_{1} \\cap Z_{1} = \\lbrace p_{1} = (0,0, 1) \\rbrace .",
"$ Now, a straightforward calculation shows that $J X_{1} = \\begin{pmatrix} 1 & 0 \\\\0 & -1 \\end{pmatrix}.$ Thus, $ \\mathrm {Res}_{c_{1}^{2}}(\\mbox{${G}$}; p_{1}) = \\frac{c_{1}^{2}(J X_{1}(p_{1}))}{\\det (J X_{1}(p_{1}))} = 0.",
"$ The same holds for $Z_{2}$ and $Z_{3}$ .",
"The foliation $ {F}$ is induced, in homogeneous coordinates $[X, Y, Z, T]$ , by the form $ \\widetilde{\\omega } = YZT dX + XZT dY + XYT dZ -3XYZ dT.",
"$ The singular set of $ {F}$ is the union of the lines $Z_{1}$ , $Z_{2}$ , $Z_{3}$ , and $ Z_{4} = \\lbrace T = X = 0 \\rbrace , \\ \\ Z_{5} = \\lbrace T = Y = 0 \\rbrace \\ \\ \\mbox{and} \\ \\ Z_{6} = \\lbrace T = X = 0 \\rbrace .",
"$ For $ Z_{4} = \\lbrace X = T = 0 \\rbrace $ we can consider the local chart $U_{y} = \\lbrace Y = 1 \\rbrace $ .",
"Then, we have, $ \\omega _{y} := \\widetilde{\\omega }|_{U_{y}} = z t dx + x t dz - 3xz d t. $ Take a 2-disc transversal $D_{2} = \\lbrace ||(x,t)||\\le 1,\\ z = 1 \\rbrace $ .",
"$ \\omega _{2} := \\omega _{y}|_{D_{2}} = t dx - 3x dt \\ \\ \\ \\ \\mbox{with} \\ \\ \\mbox{dual} \\ \\ \\mbox{vector} \\ \\ \\mbox{field} \\ \\ \\ \\ \\displaystyle X_{2} = -3x \\frac{\\partial }{ \\partial x} - t \\frac{\\partial }{\\partial t}.$ Thus, $ Z_{4} \\cap D_{2} = \\lbrace (0,1,0) =: p_{4} \\rbrace $ and $J X_{2}(p_{4}) = \\begin{pmatrix} -3 & 0 \\\\0 & -1 \\end{pmatrix}.$ Therefore, $ \\mathrm {Res}\\displaystyle _{c_{1}^{2}}(\\mbox{${G}$}; p_{4}) = \\frac{c_{1}^{2}(JX_{2})(p_{4})}{\\det (JX_{2})(p_{4})} = \\frac{16}{3}$ .",
"An analogous calculation shows that $\\mathrm {Res}\\displaystyle _{c_{1}^{2}}(\\mbox{${G}$}; p_{5}) =\\mathrm {Res}\\displaystyle _{c_{1}^{2}}(\\mbox{${G}$}; p_{6}) = \\frac{16}{3}.$ Now, we will verify the formula $\\displaystyle c_{1}^{2}(N_{ {F}}) = \\sum _{i = 1}^{6}\\mathrm {Res}( {F}, c_{1}^{2} ; Z_{i})[Z_{i}].$ On the one hand, Since $\\det (N_{ {F}})=\\mathcal {O}_{\\mathbb {P}^3}(4)$ , then $c_{1}^{2}(N_{ {F}}) = c_{1}^{2}(\\det (N_{ {F}}))=16h^2,$ where $h$ represents the hyperplane class.",
"On the other hand, by the above calculations and since $[Z_{i}]=h^2$ , for all $i$ , we have $\\sum _{i = 1}^{6} \\mathrm {Res}( {F}, c_{1}^{2} ; Z_{i})[Z_{i}]= 0[Z_{1}] + 0[Z_{2}] + 0[Z_{3}] + \\frac{16}{3}[Z_{4}] + \\frac{16}{3}[Z_{5}] + \\frac{16}{3}[Z_{6}]=16h^2.",
"$ The following example is due to D. Cerveau and A. Lins Neto, see [6].",
"It originates from the so-called exceptional component of the space of codimension one holomorphic foliations of degree 2 of $\\mathbb {P}^{n}$ .",
"We can simplify the computation as done by M. Soares in [12].",
"Example 4.2 Consider $ {F}$ be a holomorphic foliation of codimension one on $\\mathbb {P}^{3}$ , given locally by the 1-form $ \\omega = z(2y^{2} - 3x )dx + z(3z - xy)dy - (xy^{2} -2x^{2} + yz )dz.",
"$ The singular set of this foliation has one connect component, denoted by $Z$ , with 3 irreducible components, given by: 1) the twisted cubic $ \\Gamma : \\ \\ \\ y \\longmapsto (2/3 y^{2}, y, 2/9 y^{3})$ , 2) the quadric $Q: \\ \\ \\ y \\longmapsto (y^{2}/2, y, 0) $ , 3) the line $L: \\ \\ \\ y \\longmapsto (0, y, 0).",
"$ We consider a transversal 2-disc $ D \\subset \\lbrace y = 1 \\rbrace $ and we take the restriction of $ {F}$ on the afine open $\\lbrace y = 1 \\rbrace $ .",
"We have an one-dimensional holomorphic foliation, denoted by $\\mbox{${G}$}$ , given by the 1-form on $H$ $ \\widetilde{\\omega } = (2z - 3xz)dx + (2x^{2} -x - z )dz $ with dual vector field $ X = (2x^{2} -x - z )\\frac{\\partial }{\\partial x} + (- 2z + 3xz)\\frac{\\partial }{\\partial z}.",
"$ The singular set of $\\mbox{${G}$}$ is given by $ \\mathrm { Sing}( X) = \\Big \\lbrace p_{1} = (2/3, 1, 2/9) ; p_{2} = (1/2, 1, 0) ; p_{3} = (0, 1, 0) \\Big \\rbrace .",
"$ We know how to calculate the Grothendieck residue of the foliation $\\mbox{${G}$}$ : $\\mathrm {Res}\\displaystyle _{c_{1}^{2}}(\\mbox{${G}$}; p_{1}) = \\frac{c_{1}^{2}(JX(p_{1}))}{\\det (JX(p_{1}))} = \\frac{25}{6},$ $\\mathrm {Res}\\displaystyle _{c_{1}^{2}}(\\mbox{${G}$}; p_{2}) = \\frac{c_{1}^{2}(JX(p_{2}))}{\\det (JX(p_{2}))} = - \\frac{1}{2},$ $\\mathrm {Res}\\displaystyle _{c_{1}^{2}}(\\mbox{${G}$}; p_{3}) = \\frac{c_{1}^{2}(JX(p_{3}))}{\\det (JX(p_{3}))} = \\frac{9}{2}.$ Now, we will verify the formula $\\displaystyle c_{1}^{2}(N_{ {F}}) = \\mathrm {Res}( {F}, c_{1}^{2} ; \\Gamma )[\\Gamma ]+ \\mathrm {Res}( {F}, c_{1}^{2} ; Q)[Q] + \\mathrm {Res}( {F}, c_{1}^{2} ; L)[L]$ On the one hand, Since $\\det (N_{ {F}})=\\mathcal {O}_{\\mathbb {P}^3}(4)$ , then $c_{1}^{2}(N_{ {F}}) = c_{1}^{2}(\\det (N_{ {F}}))=16h^2,$ where $h$ represents the hyperplane class.",
"On the other hand, by the above calculations and using that $[\\Gamma ]=3h^2$ , $[Q]=2h^2$ and $[L]=h$ we have $\\sum _{i = 1}^{3} \\mathrm {Res}( {F}, c_{1}^{2} ; Z_{i})[Z_{i}].= \\frac{25}{6}[\\Gamma ] - \\frac{ 1}{2}[Q] + \\frac{9}{2}[L]= \\frac{25}{6}[3h^2] - \\frac{1}{2}[2h^2] + \\frac{9}{2}[L]=16h^2.",
"$ Example 4.3 Let $f:M \\dashrightarrow N$ be a dominant meromorphic map such that $\\mathrm {dim}(N)= k+1$ and $\\mbox{${G}$}$ is an one-dimensional foliation on $N$ with isolated singular set $\\mathrm {Sing}(\\mbox{${G}$})$ .",
"Suppose that $f:M \\dashrightarrow N$ is a submersion outside its indeterminacy locus $Ind(f)$ .",
"Then, the induced foliation $ {F}=f^*\\mbox{${G}$}$ on $M$ has codimension $k$ and $\\mathrm {Sing}( {F})=f^{-1} (\\mathrm {Sing}(\\mbox{${G}$}))\\cup Ind(f)$ .",
"If $Ind(f)$ has codimension $\\ge k+1$ , we conclude that $\\mathrm {cod} (\\mathrm {Sing}( {F}))\\ge k+1$ .",
"If $q\\in f^{-1}(p)\\subset M$ is a regular point of the map $ f:M \\dashrightarrow N$ , then $ \\mathrm {Res}(f^*\\mbox{${G}$}, \\varphi ; f^{-1}(p)) = \\mathrm {Res}_{\\varphi }(\\mbox{${G}$}; p)[f^{-1}(p)], $ where $ \\mathrm {Res}_{\\varphi }(\\mbox{${G}$}; p)$ represents the Grothendieck residue at $p\\in \\mathrm {Sing}(\\mbox{${G}$}).$ In fact, there exist open sets $U\\subset M $ and $V \\subset N$ , with $q\\in f^{-1}(p)\\subset U$ and $p\\in V$ , such that $U\\simeq f^{-1}(p) \\times V $ .",
"Now, if we take a $(k+1)$ -ball $B$ in $V$ then by theorem REF we have $ \\mathrm {Res}(f^*\\mbox{${G}$}, \\varphi ; f^{-1}(p)) = \\mathrm {Res}_{\\varphi }(\\mbox{${G}$}|_{B}; p)[f^{-1}(p)]=\\mathrm {Res}_{\\varphi }(\\mbox{${G}$}|; p)[f^{-1}(p)].",
"$ For instance, if $f:\\mathbb {P}^n \\dashrightarrow ( \\mathbb {P}^{k+1}, \\mbox{${G}$}) $ is a rational linear projection and $\\mbox{${G}$}$ is an one-dimensional foliation with isolated singularities.",
"Since $Ind(f)= \\mathbb {P}^{k+1}$ , then $ \\mathrm {cod} (\\mathrm {Sing}(f*\\mbox{${G}$}))=k+1$ .",
"Therefore $\\mathrm {Res}(f^*\\mbox{${G}$}, \\varphi ; f^{-1}(p)) = \\mathrm {Res}(f^*\\mbox{${G}$}, \\varphi ; \\mathbb {P}^{k+1})= \\mathrm {Res}_{\\varphi }(\\mbox{${G}$}; p)[ \\mathbb {P}^{k+1}].$"
],
[
"Cenkl algorithm for singularities with non-expected dimension ",
"In [7] Cenkl provided an algorithm to determinate residues for non-expected dimensional singularities, under a certain regularity condition on the singular set of the foliation.",
"We observe that this condition is not necessary.",
"In fact, Cenkl 's conditions are the following: Suppose that the singular set $S:=\\mathrm {Sing}( {F})$ of $ {F}$ has pure codimension $k+s$ , with $s\\ge 1$ , and $\\mathrm {cod}(S) \\ge 4$ .",
"there exists a closed subset $W \\subset M$ such that $S\\subset W$ with the property $\\text{\\rm H}^j(W, \\mathbb {Z}) \\simeq \\text{\\rm H}^j(W\\setminus S , \\mathbb {Z}), \\ \\ j=1,2.$ Denote by $M^{\\prime }=M\\setminus S$ , Cenkl show that under the above condition the line bundle $\\wedge ^k( N_{ {F}}|_{M^{\\prime }}^{\\vee } )$ on $M^{\\prime }$ can be extended a line bundle on $M$ .",
"We observe that there always exists a line bundle $\\det (N_{ {F}})^{\\vee }= [\\wedge ^k (N_{ {F}})^\\vee ]^{\\vee \\vee }$ on $M$ which extends $\\wedge ^k( N_{ {F}}|_{M^{\\prime }}^{\\vee } )$ , since $N_{ {F}}$ is a torsion free sheaf and $S=\\mathrm {Sing}(N_{ {F}})$ .",
"See, for example [11].",
"Now, consider the vector bundle $E_{ {F}}= \\det (N_{ {F}})^{\\vee }\\oplus \\det (N_{ {F}})^{\\vee }.$ Observe that $E_{ {F}}|_{M^{\\prime }}= \\wedge ^k( N_{ {F}}|_{M^{\\prime }}^{\\vee } ) \\oplus \\wedge ^k( N_{ {F}}|_{M^{\\prime }}^{\\vee } ) $ .",
"Thus, we conclude that Lemma 1 in [7] holds in general: Lemma 5.1 Consider the projective bundle $\\pi : \\mathbb {P}(E_{ {F}}) \\rightarrow M$ .",
"Then there exist a holomorphic foliation $ {F}_{\\pi }$ on $ \\mathbb {P}(E_{ {F}}) $ with singular set $\\mathrm {Sing}( {F}_{\\pi })=\\pi ^{-1}(S)$ such that $\\mathrm {dim}( {F}_{\\pi })=\\mathrm {dim}( {F})$ and $\\mathrm {dim}(\\mathrm {Sing}( {F}_\\pi ))=\\mathrm {dim}(S)+1.$ We succeeded in replacing the compact manifold M with a foliation $ {F}$ and the singular set $S$ such that $\\mathrm {dim}( {F})- \\mathrm {dim}(\\mathrm {Sing}( {F}))=n-s$ by another compact manifold $\\mathbb {P}(E_{ {F}})$ and a foliation $ {F}_{\\pi }$ with singular set $\\mathrm {dim}( {F}_\\pi )- \\mathrm {dim}(\\mathrm {Sing}( {F}_\\pi ))=n-s-1$ .",
"If this procedure is repeated $(n-s-1)$ -times we end up with a compact complex analytic manifold with a holomorphic foliation whose singular set is a subvariety of complex dimension one less than the leaf dimension of the foliation.",
"That is, we have a tower of foliated manifolds $\\nonumber {(P_{n-s-1}, {F}^{n-s-1}) [r]^{\\pi _{n-s-1}} & (P_{n-s-2}, {F}^{n-s-2}) [r] & \\cdots [r]^{\\pi _{2}} & (P_{1}, {F}^{1}) [r]^{\\pi _1:=\\pi } & (M, {F})}$ where $(P_{i}, {F}^{i}) $ is such that $P_{i}= \\mathbb {P}(E_{ {F}^{i-1}})$ and $(P_{1}, {F}^{1})=( \\mathbb {P}(E_{ {F}}), {F}_{\\pi }) $ .",
"Thus, by Lemma REF we conclude that on $ P_{n-s-1}$ we have a foliations $ {F}^{n-s-1}$ such that $\\mathrm {Sing}( {F}^{n-s-1})=(\\pi _{n-s-1} \\circ \\cdots \\circ \\pi _2\\circ \\pi _1)^{-1}(S) $ and $\\mathrm {dim}(\\mathrm {Sing}( {F}^{n-s-1}))=\\mathrm {dim}( {F}^{n-s-1})-1.$ That is, $\\mathrm {cod}(\\mathrm {Sing}( {F}^{n-s-1}))= \\mathrm {cod}( {F}^{n-s-1})+1$ .",
"On the one hand, we can apply the Theorem REF to determinate the residues of $ {F}^{n-s-1}$ .",
"On the other hand, Cenkl show that we can calculate the residue $\\mathrm {Res}_{\\varphi }( {F}^1, Z_1)$ in terms of the residue $\\mathrm {Res}_{\\varphi }( {F}, Z)$ for symmetric polynomial $\\varphi $ of degree $k+1$ .",
"Let us recall the Cenkl's construction: Let $\\sigma _1,\\dots ,\\sigma _{\\ell }$ be the elementary symmetric functions in the $n$ variables $x_1,\\dots ,x_n$ and let $\\rho _1,\\dots , \\rho _{\\ell }$ be the elementary symmetric functions in the $n+1$ variables $x_1,\\dots ,x_n,y$ .",
"It follows from [7] that for any polynomial $\\phi $ , of degree $\\ell $ , can be associated a polynomial $\\psi $ of degree $\\ell +1$ such that $\\psi (\\rho _1,\\dots , \\rho _{\\ell } ) = \\phi (\\sigma _1,\\dots ,\\sigma _{\\ell }) y + \\phi ^0(\\sigma _1,\\dots ,\\sigma _{\\ell }) + \\sum _{j\\ge 2} \\phi ^j(\\sigma _1,\\dots ,\\sigma _{\\ell }) \\cdot y^j,$ where $\\phi ^0$ has degree $\\ell +1$ and $ \\phi ^j$ has degree $\\ell -j+1$ .",
"Let $T_{P/M}$ be the tangent bundle associated the one-dimensional foliation induced by the $\\mathbb {P}^1$ -fibration $(P, {F}_{\\pi }) \\rightarrow (M, {F})$ .",
"Therefore, it follows from Lemma REF , Cenkl's construction [7] and Theorem REF the following : Theorem 5.2 Suppose that $\\mathrm {cod}(\\mathrm {Sing}( {F}))\\ge \\mathrm {cod}( {F})+2$ .",
"If $\\varphi $ is a homogeneous symmetric polynomials of degree $\\mathrm {cod}( {F})+1$ , then $\\mathrm {Res}_{\\psi }( {F}^1|_{B_p}; p)[Z_1] = \\pi ^*\\mathrm {Res}_{\\varphi }( {F}, Z) \\cap c_1(T_{P/M}) + \\pi ^*(\\phi ^0(N_{ {F}})) + \\sum _{j\\ge 2} \\pi ^*(\\phi ^j(N_{ {F}}))\\cap c_1(T_{P/M})^j,$ where $\\mathrm {Res}_{\\psi }( {F}^1|_{B_p}; p)$ represents the Grothendieck residue at $p$ of the one dimensional foliation $ {F}^1|_{B_p}$ on a $(k+1)$ -dimensional transversal ball $B_p$ .",
"We believe that this algorithm can be adapted to the context of residues for flags of foliations [4]."
]
] | 1612.05787 |
[
[
"Learning to predict where to look in interactive environments using deep\n recurrent q-learning"
],
[
"Abstract Bottom-Up (BU) saliency models do not perform well in complex interactive environments where humans are actively engaged in tasks (e.g., sandwich making and playing the video games).",
"In this paper, we leverage Reinforcement Learning (RL) to highlight task-relevant locations of input frames.",
"We propose a soft attention mechanism combined with the Deep Q-Network (DQN) model to teach an RL agent how to play a game and where to look by focusing on the most pertinent parts of its visual input.",
"Our evaluations on several Atari 2600 games show that the soft attention based model could predict fixation locations significantly better than bottom-up models such as Itti-Kochs saliency and Graph-Based Visual Saliency (GBVS) models."
],
[
"Introduction",
"Human visual attention is attracted either by salient stimuli reflected from the environment or task demands where an observer is attracted to objects of interest [7].",
"Researchers have noticed that human visual system does not perceive and process the whole visual information provided at once.",
"Instead, humans selectively pay attention to different parts of the visual input to gather relevant information sequentially and try to combine information from each step over time to build an abstract representation of the entire input [7], [25].",
"Works done in neuroscience and cognitive science literature have revealed selection of regions of interest is highly dependent on two modes of visual attention: bottom-up and top-down.",
"The bottom-up mode attends to low-level features of potential importance and attention is represented in the form of a saliency map [15], [11].",
"The top-down mode deals with task demands and goals which strongly influence scene parts to which humans should fixate [35], [13].",
"Attention models inspired by the human perception have shown good results in a variety of applications of computer vision such as object recognition and detection [23], [37], video compression [14], [9] and virtual reality [27].",
"Moreover, with the increased interest in deep learning paradigm in recent years, visual attention mechanisms incorporated in these methods have also shown astonishing results in a wide range of applications including, image captioning [19], [40], machine translation [2], speech recognition and object recognition [8].",
"Reinforcement learning [34] is one of the most powerful frameworks in solving sequential decision making problems, where decision making is based on a sequence of observations of the task’s environment.",
"Until recently, applying reinforcement learning to some real world applications where the input data is high dimensional (e.g., vision and speech) was a major challenge.",
"Recent advances in deep learning has resulted in powerful tools for automatic feature extraction from raw data, e.g.",
"raw pixels of an image.",
"Research has shown that deep neural networks can be combined with reinforcement learning in order to learn useful representations.",
"For example, Deep Q-Network (DQN) algorithm [20], [22], which is a combination of Q-learning with a deep neural network, has achieved good performances on several games in the Atari 2600 domain and in some games it can gain even higher scores than the human player.",
"This combination, deep neural networks and reinforcement learning framework, has also been shown to achieve promising results in the computer vision domain [1], [28], specifically in visual attention based models.",
"The main challenge in sequential attention models is leaning where to look.",
"Reinforcement learning techniques such as policy gradients are good choices to address this challenge [21].",
"Overall, there are two types of attention models, the soft attention models and the hard attention models.",
"The soft attention models are end-to-end approaches and differentiable deterministic mechanisms that can be learned by gradient based methods.",
"However, the hard attention models are stochastic processes and not differentiable.",
"Thus, they can be trained by using the REINFORCE algorithm [39] in the reinforcement learning framework.",
"In this paper, a soft attention mechanism is integrated into the deep Q-network (as shown in Figure REF ) to enable a deep Q-learning agent to learn to play Atari 2600 games by focusing on the most pertinent parts of its visual input.",
"In fact, the model tries to learn the control actions and the attention locations simultaneously.",
"To test our model, we compare predicted fixation locations with explicit attention judgements of people with a specific scenario as explained in section REF .",
"The results give better fixation prediction accuracy compared to two popular bottom-up (BU) saliency models: Itti-Koch’s saliency model [15] and Graph-Based Visual Saliency (GBVS) model [11].",
"We also demonstrate that the proposed model can learn how to play Atari 2600 games (i.e.",
"leaning the control actions of the game) and where to look (i.e.",
"learning the fixation locations) on video game frames effectively."
],
[
"Related Work",
"Experiments studying eye movements in natural behaviour have proved that the task plays a strong role in learning where and when human fixate, and that the eyes are guided to the points that are sometimes non-salient [35], [13], [8].",
"For instance, some works on daily activities such as driving [18], tea making [17] and making a sandwich [13] have demonstrated that almost all fixations fall on task-relevant objects and very few irrelevant regions are fixated.",
"Recognizing the influence of reward signals [26] on eye movements and in directing eye gaze, some researchers (e.g., [32]) have shown that reinforcement learning is useful for modelling eye movement.",
"They suggested an RL-based method to learn visio-motor behaviours by considering uncertainty costs when an eye movement action is made in a sidewalk navigation task of a virtual urban environment [33], where the goal is maximizing discounted sum of future rewards (or at least cost) over time.",
"Combining deep learning and reinforcement learning has achieved impressive results in learning to play various video games [20], [22], [36], [10], [30] as well as in different problems of computer vision field.",
"For instance, combining deep Convolutional Neural Networks (CNNs) with multi-layered Recurrent Neural Networks (RNNs) in particular, with Long Short-Term Memory (LSTM) components that use reinforcement learning, have led to end-to-end systems for deciding where to look.",
"Such techniques could perform well in classification and recognition tasks, e.g.",
"recurrent neural models of attention for fine-grained categorization on the Stanford Dogs data set [28], cluttered digit classification on the MNIST date set and playing a toy visual control problem [21], and the recognition of house number sequences from Google Street View images [1].",
"Both soft attention and hard attention mechanisms have gained remarkable performance over different challenging problems.",
"Recent research done by [40] has applied both mechanisms to generate image captions.",
"The attention mechanism in this work aims to generate each word of the caption by focusing on the relevant parts of the image.",
"Another approach using an attention mechanism is called spatial transformer networks [16].",
"Spatial transformer networks have used affine transformations as a soft attention mechanism to focusing on the relevant part of an image.",
"More recently, [29] used a recurrent soft attention based model for action recognition in videos and showed that their model performs better than some baselines (without the attention mechanism).",
"In this paper, inspired by the work of [29], we propose a soft attention mechanism incorporated in a Deep Q-Network [20], [22] to learn where to look and which actions to select.",
"[38] have proposed a new neural network architecture to play Atari 2600 games, which is an extension of the Double Deep Q-Network (Double DQN) in [36].",
"Applying their method to several Atari 2600 games, they obtained state-of-the-art results.",
"They also visualized saliency maps, indicating where the agent looks at, while taking an action.",
"To this end, they generated the saliencies by computing the Jacobians of the state-value and action-value functions, inspired by the way introduced in [31].",
"In contrast, here, we use multi-layered LSTMs and the soft attention mechanism to predict fixation locations and compare our model’s accuracy with the bottom-up saliency models.",
"Figure: () Architecture of our model (a deep Q-network with possibility of visual attention).",
"The CNN part of the model takes as input a sequence of states of the game and extracts feature maps.",
"It then computes vertical feature slices, C t C_t with dimension DD.",
"() The attention model.",
"at each time step tt, the attention model uses C t C_t along with the previous internal state of the LSTM part, h t-1 h_{t-1} to produce an expected value, c t c_t of vertical feature slices, C t,i C_{t,i} regarding α t,i \\alpha _{t,i}, the importance of each part of input frame."
],
[
"Model Architecture: ConvNet + Attention Model + LSTM",
"The idea behind the proposed model is similar to the work done in [29], but with differences in terms of the way the model is trained (i.e.",
"reinforcement learning) and the application domain.",
"Here we are interested in learning to play Atari 2600 games by selecting the appropriate control actions provided for each game while paying attention to the most important locations.",
"Our model is an aggregation of CNN, soft attention mechanism and RNN (i.e.",
"LSTM layers), as shown in Figure REF .",
"In order to extract convolutional features, we used the first three layers of the Deep Q-Network (DQN) [22] as the CNN part of the model, three stacks of convolutions plus rectifier nonlinearity layers with filters $32\\ 8\\times 8$ , 64 $4\\times 4$ and $64\\ 3\\times 3$ , and strides $4, 2$ and 1, respectively.",
"At each time-step $t$ , a visual frame is fed into the system and the last convolutional layer of the CNN part outputs $D$ feature maps $K \\times K$ (for example, here 64 feature maps $7\\times 7$ ).",
"Then, feature maps are converted to $K^2$ vectors in which each vector has $D$ dimension as follows: $C_t = [C_{t,1},C_{t,2}, \\ldots ,C_{t,K^2}], \\qquad C_{t,i}\\in \\mathbb {R}^D.$ Next, they are fed to the attention model to compute the probabilities corresponding to the importance of each part of the input frame.",
"In other words, the input frame is divided into $K^2$ regions and the attention mechanism tries to attend to the most relevant region.",
"We utilize the soft attention mechanism of [40] (See [2], [6] for a detailed discussion).",
"Figure REF shows the structure of the attention mechanism.",
"The attention model takes $K^2$ vectors, $C_{t,1},C_{t,2}, \\ldots ,C_{t,K^2}$ and a hidden state $h_{t-1}$ .",
"It then produces a vector $c_t$ which is a linear weighted combination of the values of $C_{t,i}$ .",
"The $h_{t-1}$ is the internal state of the LSTM at the previous time step.",
"Each vector $C_{t,i}$ is a representation of different regions of the input frame.",
"More formally, the attention module tries to select the vectors using a linear combination of $h_{t-1}$ and $C_{t,i}$ , i.e: $&c_t=E[C_t]=\\sum _{i=1}^{k \\times k} \\alpha _{t,i} C_{t,i}, \\\\&\\alpha _{t,i}= \\frac{\\exp ({f_{att}(C_{t,i},h_{t-1}}))}{\\sum _{j=1}^{k \\times k} \\exp ({f_{att}(C_{t,i},h_{t-1}}))} \\qquad i \\in 1,2,\\ldots ,{k \\times k}, \\\\&f_{att}(C_{t,i},h_{t-1})=\\tanh ({W_{hatt}}^T h_{t-1}+{W_{catt}}^T C_{t,i}),$ At each time step $t$ , attention module computes $f_{att}$ , a $\\tanh $ layer which is a composition of the values of $C_{t,i}$ and $h_{t-1}$ .",
"Then uses it to calculate $\\alpha _{t,i}$ , a softmax over ${k \\times k}$ regions.",
"They can be considered as the amount of the importance of the corresponding vector $C_{t,i}$ among of ${k \\times k}$ vectors in the input image.",
"Once $\\alpha _{t,i}$ values are ready, the attention model calculates $c_t$ , a weighted sum of all vectors $C_{t,i}$ based on given $\\alpha _{t,i}$ s. Thus, in this way, the RL agent can learn to emphasize on the interesting part of the input frame based on the given state.",
"Note that the soft attention model is fully differentiable which allows training the systems in an end-to-end manner.",
"The RNN part of the network, which is a stack of two LSTM layers with the LSTM sizes 64, uses the previous hidden state $h_{t-1}$ and the output of the attention module $c_t$ to calculate the next hidden state $h_{t}$ .",
"The $h_{t}$ is used as input of the output layer, which is a fully-connected linear layer with the output neurons for each legal action of the game.",
"It is also utilized as input of the attention module in order to calculate the value $c_{t+1}$ at the next time-step."
],
[
"Model Training",
"The goal of the RL agent is to learn an optimal policy $\\pi $ , the probability of selecting action $a$ in state $s$ by focusing on the most relevant part of the given state, such that by following the underlying policy the sum of the discounted rewards over time is maximized.",
"Since the proposed model is an end-to-end learning system, all of the network parameters (i.e.",
"three parts of the model: ConvNet, the attention network and the LSTMs), $\\theta _t$ can be learned by trying to minimize the following loss function of mean-squared error in Q values: $&L(\\theta )=E[(r+\\gamma max_{a^\\prime }Q(s^\\prime ,a^\\prime ;\\theta _{t-1})-Q(s,a;\\theta _t))^2],$ where $r+\\gamma max_{a^\\prime }Q(s^\\prime ,a^\\prime ;\\theta _{t-1})$ is the target value, $r$ is a scalar reward which the agent receives after taking action $a$ in state $s$ .",
"Parameter $0 \\leqslant \\gamma \\leqslant 1$ is called the discount factor.",
"For optimizing the above loss function, we utilize the stochastic gradient descent method.",
"Thus, in the Q-learning algorithm, the parameters are updated as follows: $&\\theta _i= \\theta _{i-1}+ \\alpha {(y_i-Q(s,a;\\theta _i))} \\frac{\\partial Q(s,a;\\theta _i)}{\\partial \\theta _i},$ where it is implicit that $y_i=r+\\gamma max_{a^\\prime }Q(s^\\prime ,a^\\prime ;\\theta _{t-1}$ ) is the target value for iteration $i$ , and $\\alpha $ is a learning rate.",
"Figure: Visualization of attended locations by the proposed model over sample frames of the five Atari 2600 games.",
"The white regions of frames indicate where the model is interested to look at.",
"Higher brightness means more attention."
],
[
"Evaluation metrics and Hyperparameters",
"To compare the power of the proposed model for predicting fixation locations against the other models, we used two metrics: Normalized Scanpath Saliency (NSS) [24] and Area Under Curve (AUC) [5].",
"NSS is used for evaluating the salience map values using fixated locations.",
"It is a response value at the human eye position $(x_h,y_h)$ of the saliency map ($sm$ ) suggested by a model, where saliency scores of the predicted map density have zero mean and unit standard deviation.",
"NSS can be formulated as follows: $&NSS= \\frac{sm(x_h,y_h)-\\mu _{sm}}{\\sigma _{sm}},$ where ${\\sigma ^2}_{sm}$ and $\\mu _{sm}$ are the variance and the mean of the saliency map ($sm$ ).",
"NSS equal 1 means that the subject’s eye position fall in a region where the predicted map density is one standard deviation above average, while NSS $=$ 0 suggests the model would not perform better than random.",
"NSS $<$ 0 indicates attention to non-salient locations predicted by the model.",
"AUC is the area under the Receiver Operating Characteristics (ROC) curve [5].",
"It is one the most commonly used ways to evaluate the performance of a binary classifier.",
"The ROC curve is created based on the true positive rate (TPR) against the false positive rate (FPR) for different cut-off points of a parameter (here, the salience value).",
"Each pair (TPR, FPR) of the ROC curve is computed with a particular threshold.",
"To use this metric in this work, human fixations on the image are considered as ground truth and the saliency map can be viewed as a binary classifier to classify each pixel of the image to fixated and non-fixated samples.",
"By drawing the ROC curve and calculating the area under the ROC curve, we can evaluate how well the saliency map predicts actual human eye fixations.",
"The perfect predication has a score of 1.0, while a score of 0.5 indicates that the model is no better chance.",
"In all of our experiments, we trained the system with a 2-layer LSTM with 64 hidden/cell units and a value of 4 for the number of sequence steps in backpropagation through time.",
"In order to ensure stable learning, LSTM gradients were clipped to a value of 10 [12].",
"At each step of training the network, the initial LSTM hidden and cell states were set to zero.",
"Exploration strategy of the agent was $\\epsilon $ -greedy policy with $\\epsilon $ decreasing linearly from 1 to 0.1 over the first million steps.",
"The discount factor was set to $\\gamma = 0.99$ .",
"The networks were trained for 2 million steps and the size of the replay memory was 500, 000.",
"All network weights were updated by the RMSProp optimizer [41] with mini batches of size 32, a momentum of 0.95, and a learning rate of $\\alpha = 0.00025$ .",
"Training for all the games was done with the same network architecture and hyperparameters.",
"Figure: Each row shows a sample frame of an Atari 2600 game and corresponding saliency maps overlaid on the frame from different models.",
"The red circles indicate the human fixation (i.e., the clicked positions by the subjects).",
"The green and blue circles indicate the location of the maximum point in each map for GBVS and Itti-Koch algorithms, respectively.",
"The white parts indicate attended regions by the model.",
"Higher brightness means more attention.Figure: NSS and ROC scores of predicted fixation locations via the different models."
],
[
"The Arcade Learning Environment",
"We have used Atari 2600 games in the Arcade Learning Environment (ALE) [3].",
"The ALE provides an environment that emulates the Atari 2600 games.",
"It presents a very challenging environment for the reinforcement learning and other approaches that have a high dimensional visual input ($210 \\times 160$ RGB video at 60Hz), which might also be partially observable.",
"It presents a wide range of interesting games that are useful to be a standard testbed for evaluating the novel algorithms.",
"In our experiments, we used 5 Atari games: Pong, Phoenix, Enduro, Breakout and Seaquest.",
"Our goal is training a RL agent to learn a certain optimal policy to play each of the games and attention to task-relevant parts of the input frame.",
"In fact, learning to choose the optimal action based on the region selected from among provided candidates (vertical feature slices resulting from the last convolution layer of the CNN part of the model, see $C_t$ in section ).",
"Figure REF shows some sample frames of the utilized games as well as the learned attended regions."
],
[
"Psychophysics and Collecting Data",
"To test the accuracy of the model, we recruited three subjects (two males and one female).",
"They were not experts in playing Atari 2600 games.",
"So, at first we provided them five games (Pong, Phoenix, Enduro, Breakout and Seaquest) of the set of Atari 2600 games.",
"We asked subjects to play the games to learn how to play and familiarize themselves with the environment.",
"Second, we showed the recorded videos played by the RL agent to subjects (the frame rate: 5fps; higher rate was a little fast for the subjects to concentrate).",
"Subjects were asked to click at frame locations where they thought should be looked at while actually playing the games.",
"Then, we collected the clicked positions and corresponding frames as the ground truths (surrogates for actual eye positions of the subjects) in our analyses.",
"For each game, we considered one episode, which is a trajectory from initial to a terminal state."
],
[
"Bottom-up saliency models",
"In order to evaluate the suggested model, we compared the performance of our model against two popular bottom-up saliency (BU) models: Itti-Koch’s saliency model [15] and Graph-Based Visual Saliency (GBVS) introduced by [11].",
"We used the freely available implementations of the saliency models provided by HarelJ.",
"Harel, A saliency implementation in MATLAB: http://www.klab.caltech.edu/~harel/share/gbvs.php.",
"Itti-Koch’s bottom up model uses twelve features extracted from the image, including color contrast (red/green and blue/yellow), temporal luminance flicker, luminance contrast, four orientations (0, 45, 90, 135), and four oriented motion energies (up, down, left, right).",
"Next, it computes conspicuity maps of individual features through center surround architecture inspired by biological receptive fields.",
"It then linearly combines saliency from different features to generate a unique master saliency map representing the conspicuity of each location in the visual input.",
"The GBVS algorithm attempts to create feature maps in the way done by Itti-Koch’s method, but normalizes them using a graph based approach to reach a better combination of conspicuous maps.",
"Table: Comparison of performance of the proposed model against bottom-up models on several frames of the five Atari 2600 games"
],
[
"Results",
"We ran the two bottom-up saliency models (i.e.",
"Itti-Koch saliency model and the GBVS algorithm) over five Atari 2600 games and compared their NSS and AUC scores to scores achieved by our suggested model.",
"Table REF reports a comparison of the proposed model with those bottom-up models.",
"The results on Table REF demonstrate that the soft attention model significantly performs better than the Itti-Koch and the GBVS methods using both evaluation metrics over all five games.",
"We observe that NSS and ROC scores for some games like Breakout, Enduro and Pong are better than others.",
"Further, movements of objects (especially, enemies) in the games Phoenix and Seaquest are high compared to the other games which makes learning an appropriate control policy difficult.",
"As a result to better predict fixation locations, the algorithm needs more than 2 million time steps.",
"For fair comparison, similar to the other games, we stopped training the networks of these games after 2 million steps.",
"Figure REF shows several sample frames of five Atari games and their corresponding saliency maps overlaid on the frames from the investigated models.",
"The distance between interesting places of the frame for the subject (shown in red circles in the figure) and predicted fixation locations (i.e.",
"the location of the maximum point in the saliency map) of the models (shown in blue and green circles) illustrates how well a model’s predictions match subjects’ click positions.",
"We see that the soft attention model (RL-based model) is able to suggest significantly better attention predictions compared to the GBVS and Itti-Koch methods.",
"Figure REF shows the NSS and ROC scores of the model across all frames of five games.",
"The plot illustrates a big performance difference between bottom-up saliency models, and the proposed model.",
"For instance, the soft attention model obtains mean NSS and ROC scores of 0.74 and 0.70 while bottom-up saliency models do not perform better than random.",
"From these results, we learn that bottom-up saliency models performs poorly in interactive tasks where humans are actively engaged in tasks (e.g.",
"playing video games).",
"The interested readers are referred to [4], where they propose several top-down attention models and exhaustively compare them with bottom-up saliency models."
],
[
"Conclusion",
"In this paper, we utilized a combination of Deep Q-network algorithm and recurrent soft attention mechanism to decide where to look as well as learn action selection in complex interactive environments.",
"We applied the proposed model to predict fixation locations, while a reinforcement learning agent is playing Atari 2600 games.",
"To show fixation prediction accuracy of our model, we set up a scenario to collect click locations (as measure of explicit attention) from a number of subjects while they are watching the videos of played games.",
"We ran the two most commonly used bottom-up saliency models over five recorded video game video clips which have been played by a RL agent.",
"We compared the model to those bottom up models.",
"Our experiments show that saliency maps suggested by our model obtain significantly better AUC and NSS scores.",
"Despite the remarkable results, we are already feeding the entire frame into the system, which is computationally expensive.",
"Using only focused parts of the given frame as input of the system (i.e., foveated representation) would be an interesting future research direction."
]
] | 1612.05753 |
[
[
"Efficient sparse polynomial factoring using the Funnel heap"
],
[
"Abstract This work is a comprehensive extension of Abu-Salem et al.",
"(2015) that investigates the prowess of the Funnel Heap for implementing sums of products in the polytope method for factoring polynomials, when the polynomials are in sparse distributed representation.",
"We exploit that the work and cache complexity of an Insert operation using Funnel Heap can be refined to de- pend on the rank of the inserted monomial product, where rank corresponds to its lifetime in Funnel Heap.",
"By optimising on the pattern by which insertions and extractions occur during the Hensel lifting phase of the polytope method, we are able to obtain an adaptive Funnel Heap that minimises all of the work, cache, and space complexity of this phase.",
"Additionally, we conduct a detailed empirical study confirming the superiority of Funnel Heap over the generic Binary Heap once swaps to external memory begin to take place.",
"We demonstrate that Funnel Heap is a more efficient merger than the cache oblivious k-merger, which fails to achieve its optimal (and amortised) cache complexity when used for performing sums of products.",
"This provides an empirical proof of concept that the overlapping approach for perform- ing sums of products using one global Funnel Heap is more suited than the serialised approach, even when the latter uses the best merging structures available."
],
[
"Introduction",
"Hensel lifting techniques are at the basis of several polynomial factoring algorithms that are fast in practice.",
"The classical algorithms are designed for generic bivariate polynomials over finite fields without reference to sparsity (e.g.",
"[6], [13]).",
"The polytope method of [1] is intended to factor sparse polynomials more efficiently, by exploiting the structure of their Newton polygon.",
"It promises to be significantly fast when the polygon has a few decompositions, and can help factor families of polynomials which possess the same Newton polytope.",
"While the pre-processing stages of the polytope method benefit from the sparsity of the input in reference to its Newton polygon, the Hensel lifting phase that pursues the boundary factorisations does not do so.",
"Our chain of work in [2], [3] reveals that the inner workings of Hensel lifting remain oblivious to the sparsity of the input as well as fluctuations in the sparsity of intermediary output, so long as one is designing the Hensel lifting phase using the dense model for polynomial representation.",
"In contrast, the sparse distributed representation considers the problem size to be a function of the number of non-zero terms of the polynomails treated, which captures the fluctuation in sparsity throughout the factorisation process.",
"In [2], we revised the analysis of the Hensel lifting phase when polynomials are in sparse distributed representation.",
"We derived that the asymptotic performance in work, space, and cache complexity is critically affected not only by the degree of the input polynomial, but also by the following factors: (i) the sparsity of each polynomial multiplication, and (ii) the sparsity of the resulting polynomial products to be merged into a final summand.",
"We further showed that even with advanced additive (merging) data structures like the cache aware tournament tree or the cache oblivious $k$ -merger, the asymptotic performance of the serialised version in all three metrics is still poor.",
"This was a result of the straightforward implementation which performed polynomial products first off, to be followed by sums of those products, a process that we dubbed serialised.",
"We remedied this by re-engineering the Hensel lifting phase such that sums of polynomial products are computed simultaneously using a MAX priority queue.",
"This generalises the approach of [14], [16], [17], [18] for a single polynomial multiplication.",
"We derived orders of magnitude reduction in work, space, and cache complexity even against a serialised version that employs many possible enhancements, and succeeded in evading expression swell.",
"Hereafter, we label the serialised and the priority queue versions of Hensel lifting as SER-HL and PQ-HL respectively.",
"More specifically and with regard s to the latter algorithm, we will denote by PQ-HL$^{\\mathcal {B}}$ the version that uses Binary Heap as a priority queue, and by PQ-HL$^{\\mathcal {F}}$ the version that uses Funnel Heap instead.",
"Our experiments in [2], [3] demonstrate that the polytope method is now able to adapt significantly more efficiently to sparse input when its Newton polygon consists of a few edges, something not to have been observed when employing SER-HL.",
"In [3], we shifted to enhancing the overlapping algorithm PQ-HL$^{\\mathcal {F}}$ .",
"The motivation lies in the fact that Binary Heap is not scalable, which, on a serial machine, is interpreted to say that its performance will deteriorate once data no longer fits in in-core memory, thus restricting the number of non-zero terms that input and intermediary output polynomials are permitted to possess.",
"By performing priority queue operations using optimal cache complexity and in a cache oblivious fashion, Funnel Heap beats Binary Heap at large scale.",
"The fact that Funnel Heap assumes no knowledge of the underlying parameters such as memory level, memory level size, or word length, makes it ideal for applications where polynomial arithmetic is susceptible to fluctuations in sparsity.",
"However, all of those features can also be observed when adopting an alternate cache oblivious priority queue (see for example, [9], [4]).",
"As such, we pursued Funnel Heap for further attributes that can improve on its asymptotic performance, as well as exploit it at small scale, specifically for Hensel lifting.",
"In [3], we addressed the chaining optimisation, and how Funnel Heap can be tailored to implement it in a highly efficient manner.",
"We exploited that Funnel Heap is able to identify equal order monomials “for free” as part of its inner workings whilst it re-organises itself over sufficiently many updates during one of its special operations known as the “SWEEP”.",
"By this we were able to eliminate entirely the requirement for searching from the chaining process.",
"We designed a batched mode for chaining that gets overlapped with Funnel Heap's mechanism for emptying its in-core components.",
"In addition to also managing expression swell and irregularity in sparsity, batched chaining is sensitive to the number of distinct monomials residing in Funnel Heap, as opposed to the number of replicas chained.",
"This allows the overhead due to batched chaining to decrease with increasing replicas.",
"For sufficiently large input size with respect to the cache-line length, and also sufficiently sparse input and intermediary polynomials, batched chaining that is “search free” leads to an implementation of Hensel lifting that exhibits optimal cache complexity in the number of replicas found in Funnel Heap, and one that achieves an order of magnitude reduction in space, as well as a reduction in the logarithmic factor in work and cache complexity, when comparing against PQ-HL$^{\\mathcal {B}}$ of [2].",
"We label as FH-HL the enhancement of Hensel lifting using Funnel Heap and batched chaining.",
"This paper extends all of the above work in garnering the prowess of Funnel Heap.",
"To this end, we incorporate analytical as well as experimental algorithmics techniques as follows: In Section , we provide proofs of results introduced in [3] pertaining to properties of Funnel Heap, several of which are of independent worth extending beyond Hensel lifting.",
"For example, we provide complete proofs for the following: We establish where the replicas will reside immediately after each insertion into Funnel Heap.",
"We determine the number of times one is expected to call SWEEP on each link of Funnel Heap throughout a given sequence of insertions.",
"Given an upper bound on the maximum constituency of Funnel Heap at any one point in time across a sequence of operations, we compute the total number of links required by Funnel Heap.",
"We establish that the cache complexity by which one performs batched chaining within FH-HL is optimal.",
"In Section , we exploit that the work and cache complexity of an Insert operation using Funnel Heap can be refined to depend on the rank of the inserted monomial product, where rank corresponds to its lifetime in Funnel Heap.",
"By optimising on the pattern by which insertions and extractions occur during the Hensel lifting phase of the polytope method, we are able to obtain an adaptive Funnel Heap that minimises all of the work, cache, and space complexity of this phase.",
"This, in turn, maximises the chances of having all polynomial arithmetic performed in the innermost levels of the memory hierarchy, and observes nearly optimal spatial locality.",
"We show that the asymptotic costs of such preprocessing can be embedded in the overall costs to perform Hensel lifting with batched chaining (FH-HL), independently of the amount of minimisation taking place.",
"We call the resulting algorithm FH-RANK.",
"In Section , we develop the experimental algorithmics component to our work addressing various facets: We conduct a detailed empirical study confirming the scalability of Funnel Heap over the generic Binary Heap.",
"By simulating out of core behaviour, Funnel Heap is superior once swaps to external memory begin to take place, despite that it performs considerably more work than Binary Heap.",
"This supports the notion that Funnel Heap should be employed even when performing a single polynomial multiplication or division once data grows out of core.",
"We support the theoretical analysis of the cache and space complexity in [3] using accounts of cache misses and memory consumption of FH-HL.",
"This can be seen as an extension of [3], as the performance measures presented there capture only the real execution time.",
"We benchmark FH-RANK against several other variants of Hensel lifting, which include PQ-HL$^{\\mathcal {B}}$ , PQ-HL$^{\\mathcal {B}}$ with the chaining method akin to [18], PQ-HL$^{\\mathcal {F}}$ , and FH-HL.",
"Our empirical account of time, space and cache complexity of FH-RANK confirm the predicted asymptotic analysis in all three metrics.",
"We demonstrate that Funnel Heap is a more efficient merger than the cache oblivious $k$ -merger, which fails to achieve its optimal (and amortised) cache complexity when used for performing sums of products.",
"We attribute this to the fact that the polynomial streams to be merged during Hensel lifting cannot be guaranteed to be of equal size (as a result of fluctuating sparsity).",
"This provides an empirical proof of concept that the overlapping approach for performing sums of products using one global Funnel Heap is more suited than the serialised approach, even when the latter uses the best merging structures available.",
"We now begin with the following section on background literature and results."
],
[
"Background",
"In the remainder of this paper, we will consider that in-core memory is of size $M$ .",
"It is organised using cache lines (disk blocks), respectively, each consisting of $B$ consecutive words.",
"All words in a single line are transferred together between in-core and out-of-core memory in one round (I/O operation) referred to as a cache miss (disk block transfer)."
],
[
"Funnel Heap:",
"Funnel Heap implements Insert and Extract-Max operations in a cache oblivious fashion.",
"For $N$ elements, Funnel Heap can perform these operations using amortised (and optimal) $O(\\frac{1}{B} \\log _{M/B} \\frac{N}{B})$ cache misses [8].",
"At the innermost level, Funnel heap is first constructed using simple binary mergers.",
"Each binary merger processes two input sorted streams and produces their final merge.",
"The heads of the input streams and the tail of the output stream reside in buffers of a limited size.",
"A binary merger is invoked using a FILL function when merge steps are repetitively performed until its output buffer is full or both its input streams are exhausted.",
"One can construct binary merge trees by letting the output buffer of one merger be an input buffer of another merger.",
"Now let $k = 2^i$ for $i \\in \\mathbb {Z}^{+}$ .",
"A $k$ -merger is a binary merge tree with exactly $k$ input streams.",
"The size of the output buffer is $k^3$ , and the sizes of the remaining buffers are defined recursively in a Van Emde Boas fashion (See [7], [8], [12]).",
"Funnel Heap consists of a sequence $\\lbrace K_i\\rbrace $ of $k$ -mergers, where $k$ increases doubly exponentially across the sequence.",
"The $K_i$ 's are linked together in a list, with the help of extra binary mergers and buffers at each juncture of the list.",
"In Fig.",
"REF , the circles are binary mergers, rectangles are buffers, and triangles are $k$ -mergers.",
"Link $i$ in the linked list consists of a binary merger $v_i$ , two buffers $A_i$ and $B_i$ , and a merger $K_i$ with $k_i$ input buffers labeled as $S_{i,1}, \\ldots , S_{i,k_i}$ .",
"Link $i$ has an associated counter $c_i$ for which $1 \\le c_i \\le k_i+1$ .",
"Initially, $c_i = 1$ .",
"It will be an invariant that $S_{i,c_i},\\ldots ,S_{i,k_{i}}$ are empty.",
"The first structure in Funnel Heap is a buffer $S_{0,1}$ of extremely small size $s_1$ , dedicated for insertion.",
"This buffer occupies in-core memory at all times.",
"Funnel Heap is now laid out in memory in the order $S_{0,1}$ , link 1, link 2, etc.",
"Within link $i$ the layout order is $c_i$ , $A_i$ , $v_i$ , $B_i$ , $K_i$ , $S_{i,1}$ , $\\ldots $ , $S_{i,k_{i}}$ .",
"Figure: Funnel HeapThe linked list of buffers and mergers constitute one binary tree $T$ with root $v_1$ and with sorted sequences of elements on the edges.",
"This tree is heap-ordered: when traversing any path towards the root, elements will be passed in increasing order.",
"If buffer $A_1$ is non-empty, the maximum element will reside in $A_1$ or in $S_{0,1}$ .",
"The smaller mergers in Funnel Heap are meant to occupy primary memory, and can process sufficiently many insertions and extractions in-core before an expensive operation is encountered.",
"In contrast, the larger mergers tend to be out of core, and contain elements that are least likely to be accessed in the near future.",
"To perform an Extract-Max, we call FILL on $v_1$ if buffer $A_1$ is empty.",
"We return the largest element residing in both $S_{0,1}$ and $A_1$ .",
"To insert into Funnel Heap, an element has to be inserted into $S_{0,1}$ .",
"If $S_{0,1}$ is full, a SWEEP function is called.",
"Its purpose is to free the insertion buffer $S_{0,1}$ together with all the heavily occupied links in Funnel Heap which are closer to in-core memory.",
"During a SWEEP, all elements residing in those dense links are extracted then merged into one single stream.",
"This stream is then copied sufficiently downwards in Funnel Heap, towards the first link which has at least one empty input buffer.",
"As a result of SWEEP, the dense links are now free and Funnel Heap operations are resumed within in-core memory.",
"The SWEEP kernel is considerably expensive, yet, sufficiently many insertions and all the extractions can be accounted for between any two SWEEPs."
],
[
"The polytope method:",
"Let $\\mathbb {F}$ denote a finite field of characteristic $p$ , and consider a polynomial $f \\in \\mathbb {F}[x,y]$ with total degree $n$ .",
"We wish to obtain a polynomial factorisation of $f$ into two factors $g$ and $h$ such that $f = gh$ and $g$ , $h \\in {\\mathbb {F}}[x,y]$ .",
"Let $Newt(f)$ denote the Newton polygon $\\mathbb {R}^2$ of $f$ defined as the convex hull of the support vector of $f$ .",
"One identifies suitable subsets $\\lbrace \\Delta _i\\rbrace $ of edges belonging to $Newt(f)$ , such that all lattice points can be accounted for by a proper translation of this set of edges.",
"One then specialises terms of $f$ along each edge $\\delta _j^{(i)} \\in \\Delta _i$ .",
"Those specialisations are derived from the nonzero terms of $f$ whose exponents make up integral points on each $\\delta _j^{(i)}$ , and we label them as $f_{0}^{\\delta _{j}}$ .",
"These can be transformed into Laurent polynomials in one variable.",
"For at least one $\\Delta _i$ , the associated edge polynomials $f_{0}^{\\delta _{j}}$ ought to be squarefree, for all $\\delta _j \\in \\Delta _i$ .",
"One then begins lifting using the boundary factorisations given by $f_0^{\\delta _{j}} =g_{0}^{\\delta _{j}}h_{0}^{\\delta _{j}}$ , for all $\\delta _j \\in \\Delta _i$ .",
"For each boundary factorisation, we determine the associated $\\lbrace g_k\\rbrace $ 's and $\\lbrace h_k\\rbrace $ 's that satisfy the Hensel lifting equation $g_{0}^{\\delta _j} h_{k}^{\\delta _j} + h_{0}^{\\delta _j}g_{k}^{\\delta _j} = f_{k}^{\\delta _j} - \\sum _{j = 1}^{k-1}g_{j}^{\\delta _j} h_{k-j}^{\\delta _j} $ for $k = 1,\\ldots , \\min (\\deg (g_0), \\deg (h_0))$ ."
],
[
"Sums of products using a priority queue",
"In [2] we revised the analysis associated with the bottleneck in computation arising in Eq.",
"(REF ), using the sparse distributed representation.",
"In this model of representation, a polynomial is exclusively represented as the sum of its non-zero terms, sorted upon some decreasing monomial ordering.",
"Eq.",
"(REF ) can be modeled using the input and output requirements shown in Alg.",
"1: Local-Iterative [1]An integer $k$ designating one iterative step in the Hensel lifting process.",
"Two sets of univariate polynomials over $\\mathbb {F}$ , $\\lbrace g_i\\rbrace _{i = 1}^{k-1}$ , $\\lbrace h_i\\rbrace _{i = 1}^{k-1}$ , in sparse distributed monomial order representation.",
"The polynomial $S_{k} = \\sum _{i = 1}^{k-1} g_{i} \\cdot h_{j}$ , where $j = k-i$ .",
"$i = 1$ to $k-1$ Compute $p_{i} \\leftarrow g_{i} \\cdot h_{j}$ .",
"Compute $S_k = \\sum _{i = 1}^{k-1} p_i$ .",
"We distinguish between the serialised approach (SER-HL) and the overlapping approach (PQ-HL) for performing the required arithmetic.",
"In the serialised version, one performs all polynomial multiplications first, and then merges all the resulting polynomial products.",
"In the overlapping approach, one handles all arithmetic simultaneously using a single Max priority queue.",
"In [2], we analysed the work, space, and cache complexity, when polynomials are in sparse distributed representation.",
"We derived that the performance of the serialised version in all three metrics is critically affected not only by the degree of the input polynomial, but also by the following factors: (i) the sparsity of each polynomial multiplication, and (ii) the sparsity of the resulting polynomial products to be merged into a final summand.",
"We further showed that this remains the case even with advanced additive (merging) data structures like the cache aware tournament tree or the cache oblivious $k$ -merger, for performing the sums of resulting polynomial products, and that the serialised approach is not able to fully exploit the cache efficiency of these structures.",
"In the overlapping approach, the priority queue is initialised using the highest order monomial products generated from each product $g_i \\cdot h_j$ .",
"Then, terms of $S_k$ are produced in decreasing order of degree, via successive invocations of Extract-Max upon the priority queue.",
"In [3], we pursued Funnel Heap as an alternative to the generic Binary Heap for implementing the overlapping approach.",
"Beyond its cache oblivious nature and optimal cache complexity, we showed that Funnel Heap allows for a mechanism of chaining that significantly improves its overall performance.",
"Chaining replicas outside the priority queue following insertions is a well known technique (e.g.",
"see [16], [17], [18]) for the case of single polynomial multiplication using binary heap).",
"It helps reduce several parameters tied to performance, such as the total number of extractions required to perform a single polynomial multiplication and the size of the priority queue.",
"In turn, the latter results in reducing the number of monomial comparisons as well as the cache complexity required to perform each priority queue operation.",
"In the straight-forward implementation, one has to search for a replica immediately after an insertion and then chain the newly inserted element to the end of a linked list tied to that replica in the priority queue.",
"When using Binary Heap, chaining hinders performance critically.",
"Each insertion into the linked list denoting the chain incurs a random miss, whereas a single search query may require traversing the entire heap.",
"It follows that the work and cache complexity of a single insertion amounts to that of traversal of $N$ elements for a heap of size $N$ .",
"When employed in the priority queue that is implementing sums of products arising in Hensel lifting, chaining becomes daunting as the size of the queue and the amount of replication change irregularly from one iteration to the other.",
"In [3] we showed how to exploit the expensive SWEEP kernel of Funnel Heap in order to develop a cache friendly batched chaining mechanism (BATCHED-CHAIN) that gets intertwined with the SWEEP's internal operations.",
"The crux behind our approach lies in delaying chaining and performing it in batches, somehow at the “right time”.",
"In the interim, a prescribed amount of replication is tolerated, whose effect is shown to be insignificant at scale.",
"Here, we restrict chaining to only two specific phases in Funnel Heap's operations.",
"If one is inserting a monomial product into the (sorted) insertion buffer $S_{0,1}$ , a replica that resides in $S_{0,1}$ is immediately identified and chaining can take place.",
"One does not attempt to find a replica outside of $S_{0,1}$ .",
"If such a replica exists, chaining will be deferred until $S_{0,1}$ is full.",
"That is when SWEEP is invoked upon some link $i$ as well as one of its input buffers $S_{i,c_i}$ .",
"In the duration of SWEEP, one is forming the stream $\\sigma $ which contains the merged output of all elements in the buffers leading from $A_i$ to $S_{i,c_i}$ together with all elements in links $1, \\ldots , i-1$ .",
"During the merge, the replicas residing in those specified regions of Funnel Heap will be aligned consecutively and thus identified.",
"One can then chain them all and at once outside of Funnel Heap.",
"BATCHED-CHAIN eliminates entirely the need for searching for replicas, and lesser links would be allocated to Funnel Heap, which reduces garbage collection.",
"BATCHED-CHAIN is further sensitive to the number of distinct monomials in Funnel Heap, and not the number of replicas chained.",
"This can be understood to mean that the overhead due to chaining decreases with increasing replicas, which is intuitively appealing, since chaining is likely to be disabled once the number of replicas is lower than an acceptable threshold.",
"When incorporating Funnel Heap and BATCHED-CHAIN into the priority queue algorithm for sums of products, Alg.",
"FH-HL was shown to be significantly fast.",
"The timings reported in [3] correspond to overall run-time, with the following percentages of improvement recorded, attained with increasing input size: about 90%-98% (FH-HL to Magma 2.18-7), about 90%- 99% (FH-HL to SER-HL), about 10%-60% (FH-HL to PQ-HL$^{\\mathcal {B}}$ ).",
"The dramatic reduction in run-time over SER-HL is largely attributed to substantial expression swell, and that over PQ-HL$^{\\mathcal {B}}$ is attributed to BATCHED-CHAIN.",
"In this section we revisit several claims made in [3] and provide their complete proofs.",
"Those results pertain to the behaviour of Funnel Heap in general and not necessarily only in relation to Hensel lifting, and thus are of independent worth.",
"Unless otherwise stated, all lemmas and corollaries in this specific section are stated in [3].",
"We begin by the following invariant which identifies where the replicas will reside immediately after each Insert into Funnel HEap: Lemma 3.1 Let $\\ell $ denote the index of the last link in Funnel Heap.",
"Using BATCHED-CHAIN, and immediately following each insertion, there will be no replication within the constituency of any buffer $\\lbrace \\lbrace S_{i,j}\\rbrace _{j = 1}^{k_{i}}\\rbrace _{i = 0}^{\\ell }$ .",
"As a result, a given element in some buffer $S_{i,j}$ may only be replicated at most once in each of the preceding buffers $\\lbrace S_{i,j^{\\prime }}\\rbrace _{j^{\\prime } = 1}^{j-1}$ in its own link or in each of the buffers $\\lbrace \\lbrace S_{i^{\\prime },j}\\rbrace _{j = 1}^{k_{i^{\\prime }}}\\rbrace _{i^{\\prime } = i+1}^{\\ell }$ in the larger links.",
"Consider the case when one is inserting immediately into the insertion buffer $S_{0,1}$ .",
"Alg.",
"BATCHED-CHAIN ensures that chaining is happening immediately, and so there will be no replicas in this particular buffer.",
"Now consider a random $S_{i,j}$ for $i > 0$ .",
"We know that one can only write elements to $S_{i,j}$ upon a call onto $SWEEP(i)$ .",
"This call produces the stream $\\sigma $ which merges the content of all links $1,\\ldots ,i-1$ together with the content of the path $p$ leading from $A_1$ down to $S_{i,j}$ .",
"Since BATCHED-CHAIN employs chaining during the formation of $\\sigma $ , buffer $S_{i,j}$ will not contain any replicas.",
"Now, by the first claim above, each buffer $S_{i,j}$ in Funnel Heap contains distinct elements.",
"When $i = 0$ , it is straightforward to see that since $S_{0,1}$ has no buffers which precede it, each of its elements is replicated at most once in each of the following buffers.",
"Now take $i > 0$ .",
"We know that once SWEEP is called onto $S_{i,j}$ , each buffer $\\lbrace S_{i,j^{\\prime }}\\rbrace _{j^{\\prime } > j}$ in the $i$ 'th link must be empty.",
"Also, as we form $\\sigma $ – the end of which is written to $S_{i,j}$ – we exclude the elements residing in each buffer that is also in the same link as $S_{i,j}$ but which precede it in that link.",
"It follows that the only possible replicas of each element in $S_{i,j}$ will be in each of the buffers $\\lbrace S_{i,j^{\\prime }}\\rbrace _{j^{\\prime } = 1}^{j-1}$ preceding it in its own link, as well as each of the buffers $\\lbrace \\lbrace S_{i^{\\prime },j}\\rbrace _{j = 1}^{k_i}\\rbrace _{i^{\\prime } = i}^{\\ell }$ in the larger links.",
"The following result captures the number of times one is expected to call SWEEP on each link of Funnel Hap throughout a given sequence of insertions and extractions: Lemma 3.2 Let $\\ell $ denote the index of the last link in Funnel Heap and let $T_j$ denote the total number of times SWEEP$(j)$ is called, across a given sequence of insertions and extractions.",
"Then $T_j = c_{\\ell }\\cdot \\overset{\\ell -1}{\\underset{i = j}{\\prod }} k_i$ We proceed by backward induction on $j$ .",
"Take $j = \\ell $ .",
"Link $\\ell $ has $k_{\\ell }$ input buffers.",
"Since this is the last link, not all of its input buffers $S_{i,j}$ may be written onto using SWEEP.",
"In fact, exaclty $c_{\\ell }$ of them will be so.",
"We thus have $T_{\\ell } = c_{\\ell }$ .",
"We now show that $T_j = c_{\\ell } \\cdot \\overset{\\ell -1}{\\underset{i = j}{\\prod }} k_i$ assuming the property holds for $T_{j+1}$ .",
"Observe that before any SWEEP on link $j+1$ has occurred, there should have preceded it exactly $k_{j}$ SWEEPs, in order to fill each of the input buffers in link $j$ .",
"Also, by the inductive hypothesis, the total number of SWEEPs on link $j+1$ is given by $T_{j+1} = c_{\\ell }\\overset{\\ell -1}{\\underset{i = j+1}{\\prod }} k_i$ .",
"Combining, we get that there are $\\begin{array}{rcl}T_j & = & k_j \\cdot c_{\\ell } \\cdot \\overset{\\ell }{\\underset{i = j+1}{\\prod }} k_i \\\\& = & c_{\\ell } \\cdot \\overset{\\ell }{\\underset{i = j}{\\prod }} k_i\\\\\\end{array}$ SWEEPs on link $j$ .",
"Given an upper bound on the maximum constituency of Funnel Heap at any one point in time across a sequence of operations, we now determine the total number of links the heap requires: Lemma 3.3 Let $\\ell $ denote the index of the last link in Funnel Heap.",
"Then $\\ell = \\theta (\\left|T \\right|\\log \\log \\left|T \\right|),$ where $\\left|T \\right|$ designates the maximum number of elements residing in Funnel Heap at any point in time.",
"From [8] we invoke the following proven results which we require for our proof: The space usage $s_i$ of each input buffer in link $i$ satisfies $s_i = \\theta (k_i^3)$ , where $k_i$ is the number of input buffers in link $i$ .",
"The space usage of link $i$ is $\\theta (k_is_i)$ , i.e.",
"it is dominated by the space usage of all of its $k_i$ input buffers.",
"$k_i = \\theta (k_{i-1}^{4/3})$ Since link $\\ell $ is the last link required by Funnel Heap to host all elements of its elements, those elements will consume at least one path leading to the first input buffer of link $\\ell $ , and at most all $k_{\\ell }$ such possible paths.",
"By (2) above, the space usage of each such path is dominated by the size of the input buffer itself and we thus have $\\left|T \\right|= O(k_{\\ell }s_{\\ell })$ and $\\left|T \\right|= \\Omega (s_{\\ell })$ .",
"By $T = O(k_{\\ell }s_{\\ell })$ we have: $\\begin{array}{rcl}T & = & O(k_{\\ell }s_{\\ell })\\\\& = & O(k_{\\ell }^{4}) \\quad \\mbox{ by (1) above}\\\\& = & O\\left(\\left(k_{1}^{(4/3)^{\\ell -1}}\\right)^{4}\\right)\\end{array}$ where the last equality follows by (3) above and by unrolling the recursive relation down to the base case.",
"Using $k_1 = 2$ and composing the logarithm function on the two bases 2 and 4/3 respectively, we get $\\ell = O(\\log \\log T)$ .",
"Taking $T = \\Omega (s_{\\ell })$ one can proceed analogously as above and obtain $\\ell = \\Omega (\\log \\log T)$ .",
"This concludes the proof.",
"As in [16], [17], [18], reasoning in the sparse distributed representation produces worst-case versus best case polynomial multiplication, depending on the structure of the output.",
"In the worst case, a given multiplication $g_i\\cdot h_j$ is sparse as it yields a product with $\\theta (\\#g_i \\cdot \\#h_j)$ non-zero terms, an incidence of a memory bound computation.",
"At best, the multiplication is dense as it yields a product with $\\theta (\\#g_i + \\#h_j)$ terms.",
"When the product has significantly fewer terms due to cancelation of terms, the operation is said to suffer from expression swell.",
"We now establish that the cache complexity by which one performs BATCHED-CHAIN within FH-HL is optimal.",
"For this, we require a few notations from [3] that will be helpful in the forthcoming sections as well.",
"Let $\\bar{g} = \\max \\lbrace \\#g_i\\rbrace _{i = 1}^{k}$ and $\\bar{h} = \\max \\lbrace \\#h_j\\rbrace _{j = 1}^{k}$ , which denote the maximum number of non-zero monomials comprising each $g_j$ and $h_j$ respectively.",
"Let $\\tau $ denote the fraction of reduction in the size of the heap during chaining, such that the largest size the priority queue attains during the $k$ 'th lifting step is $\\theta (k \\bar{g}/\\tau )$ .",
"Let $\\tau ^{\\prime }$ denote the fraction of replication in the total number of monomial products such that the total number of replicas chained during the $k$ 'th Hensel lifting step is $\\theta (k\\bar{g}\\bar{h}/\\tau ^{\\prime })$ .",
"The two parameters $\\tau $ and $\\tau ^{\\prime }$ reflect, in an asymptotic sense, the changes in the size of the queue as a function of the amount of replicas.",
"Particularly, the bounds on $\\tau $ and $\\tau ^{\\prime }$ are as follows.",
"When no replicas are encountered at all during any one lifting step, we have that $\\tau = 1$ and $\\tau ^{\\prime } = \\theta (k \\bar{g}\\bar{h})$ .",
"In contrast, when each polynomial in the pair $(g_i,h_j)$ is totally dense and all resulting products in one lifting step are of the same degree, the heap will contain only one element, leading to $\\tau = \\theta (k \\bar{g})$ and $\\tau ^{\\prime } = \\theta (1)$ .",
"We now have the following: Corollary 3.4 Assume the sparse distributed representation for polynomials.",
"Assume further that $B = O \\left( \\frac{\\tau \\bar{h} \\log (k \\bar{g}/\\tau )}{\\log \\log (k\\bar{g}/\\tau )}\\right)$ .",
"In the worst case analysis when each polynomial multiplication $g_ih_j$ is sparse, the cache complexity by which one performs BATCHED-CHAIN within FH-HL is optimal.",
"Following the analysis in Prop.",
"3.6 of [3], the cache complexity of FH-HL is split into two major parts.",
"The first part accounts for all the insertions into Funnel Heap using $O(k\\bar{g}\\bar{h} \\frac{1}{B} \\log _{M/B} \\frac{k\\bar{g}}{\\tau })$ cache misses.",
"The second part accounts for the cost to perform BATCHED-CHAIN using $O(\\frac{k \\bar{g}\\bar{h}}{\\tau ^{\\prime }\\, B} + \\frac{k \\bar{g}}{\\tau } \\log \\log \\frac{k \\bar{g}}{\\tau })$ cache misses.",
"When $B = O \\left( \\frac{\\tau \\bar{h} \\log (k \\bar{g}/\\tau )}{\\log \\log (k\\bar{g}/\\tau )}\\right)$ , we get that the second summand in the cache complexity incurred by BATCHED-CHAIN is dominated by the cost to perform all the insertions into Funnel Heap, or that the cost for BATCHED-CHAIN is dominated by $O(\\frac{k \\bar{g}\\bar{h}}{\\tau ^{\\prime }\\, B})$ , where $\\theta (\\frac{k \\bar{g}\\bar{h}}{\\tau ^{\\prime }})$ denotes the total number of replicas chained.",
"It follows that the cache complexity of BATCHED-CHAIN corresponds to that of traversal, and hence is optimal.",
"In the following, we provide a detailed proof that FH-HL, and thanks to BATCHED-CHAIN, outperforms PQ-HL$^{\\mathcal {F}}$ (and thus by transitivity, also PQ-HL$^{\\mathcal {B}}$ ).",
"In other words, performing sums of products using Funnel Heap with BATCHED-CHAIN is provably more efficient in work, space, and cache complexity than if we were to resort to a standalone Funnel Heap implementation.",
"Corollary 3.5 Assume the sparse distributed representation for polynomials, and assume further the conditions in Cor.",
"REF .",
"In the worst case analysis when each polynomial multiplication $g_ih_j$ is sparse, FH-HL achieves an order of magnitude reduction in space, as well as a reduction in the logarithmic factor in work and cache complexity, over PQ-HL$^{\\mathcal {F}}$ .",
"From [2], Alg.",
"PQ-HL$^{\\mathcal {F}}$ requires the following costs: Table: NO_CAPTIONFrom [3], Alg.",
"FH-HL requires the following costs: Table: NO_CAPTIONReductions in space borne by FH-HL are obvious by comparing $\\theta \\left(\\frac{k \\bar{g}}{\\tau }\\right)$ and $\\theta \\left(k \\bar{g}\\right)$ respectively, and noting that $\\tau \\ge 1$ .",
"The logarithmic factor reductions in work are obvious by comparing $\\theta \\left(k \\bar{g}\\bar{h} \\log \\frac{k \\bar{g}}{\\tau }\\right)$ and $\\theta \\left(k \\bar{g}\\bar{h} \\log k \\bar{g}\\right)$ respectively.",
"Similarly, for reductions in cache complexity, we require $\\left(k \\bar{g} \\bar{h} \\frac{1}{B} \\log _{M/B} \\frac{k\\bar{g}}{\\tau }\\right) + \\frac{k\\bar{g}\\bar{h}}{\\tau ^{\\prime }B}= O \\left(k \\bar{g} \\bar{h}\\frac{1}{B} \\log _{M/B} k\\bar{g}\\right)$ or $\\left(\\frac{1}{B} \\log _{M/B} \\frac{k\\bar{g}}{\\tau }\\right) + \\frac{1}{\\tau ^{\\prime }B}= O \\left(\\frac{1}{B} \\log _{M/B} k\\bar{g}\\right)$ Recall the bounds established earlier for $\\tau $ and $\\tau ^{\\prime }$ .",
"When $\\tau = 1$ , we have $\\tau ^{\\prime } = \\theta \\left(k \\bar{g}\\bar{h}\\right)$ , for which we have: $\\left(\\frac{1}{B} \\log _{M/B} \\frac{k\\bar{g}}{\\tau }\\right) + \\frac{1}{\\tau ^{\\prime }B} = \\theta \\left( \\frac{1}{B} \\log k \\bar{g} + \\frac{1}{k \\bar{g}\\bar{h} \\, B}\\right) = \\theta \\left( \\frac{1}{B} \\log k \\bar{g}\\right)$ and (REF ) holds.",
"As $\\tau $ increases, $\\tau ^{\\prime }$ satisfies $\\tau ^{\\prime } = \\Omega (1)$ and so $\\left(\\frac{1}{B} \\log _{M/B} \\frac{k\\bar{g}}{\\tau }\\right) = O\\left(\\frac{1}{B} \\log _{M/B} k\\bar{g} \\right)$ and $\\frac{1}{\\tau ^{\\prime } \\, B} = O \\left( \\frac{1}{B}\\right) = O\\left(\\frac{1}{B} \\log _{M/B} k\\bar{g} \\right)$ for which (REF ) holds again.",
"This concludes the proof."
],
[
"Adaptive Funnel Heap and the Sequence of Insertions/Extractions",
"The canonical SWEEP function described in Sec.",
"works by identifying the smallest link in Funnel Heap that is completely empty, which necessitates that one keeps pushing the content of the heap downwards in the direction of larger and larger links, which are also more likely to be out-of-core.",
"An enhanced version of SWEEP exploits the smaller links in Funnel Heap that are sufficiently sparse, instead of always sweeping onto totally empty, yet significantly larger buffers.",
"The refined SWEEP operation identifies the first link $i$ whose total number of elements residing in its input buffers $\\lbrace S_{i,j}\\rbrace $ is less than half of its total size.",
"The input buffer with minimal occupancy in that link, say $S_{i,j_1}$ , is then recycled and its content moved onto another input buffer, say $S_{i,j_2}$ , with second largest occupancy.",
"SWEEP is now called with $S_{i,j_1}$ as the destination buffer.",
"That smaller buffers are effectively used instead of the larger, out-of-core buffers causes Funnel Heap to adapt to various modes of usage.",
"The analysis in [8] shows that the amortised cost for the $r$ 'th insertion is now $O(\\frac{1}{B} \\log _{M/B} \\frac{N_r}{B})$ , where $N_r$ denotes some notion of the lifetime of the $r$ 'th inserted element in Funnel Heap.",
"Particularly, if the $r$ 'th inserted element is removed by an Extract-Max prior to the $t$ 'th inserted element, then $N_r = t-r$ .",
"In this section, we exploit the idea that the cache complexity of an Insert operation can be refined to depend on the rank of the inserted monomial product, by optimising on the pattern by which insertions and extractions occur during the Hensel lifting phase, with the notion of lifetime in hindsight.",
"We achieve this by efficiently delaying all the insertions that come from polynomial pairs that “can wait”, as indicated by their total order and their rank in relation to the maximal element residing in Funnel Heap.",
"The refined process, which we label as FH-RANK, proceeds as follows.",
"We first accumulate the set of all distinct monomial orders $\\alpha $ appearing in the sum of products $S_k$ .",
"When the input polynomials are univariate, the monomial order can be understood to denote the total degree of a given monomial product.",
"For each $\\alpha $ , let $\\psi (\\alpha )$ denote the set of indices $\\lbrace i \\left|\\right.",
"1 \\le i \\le k \\wedge o(g_ih_{j=k-i}) = \\alpha \\rbrace $ , which maps each monomial order $\\alpha $ to the polynomial operands that resulted in a product of this particular order.",
"Let $\\mathcal {O} = \\lbrace (\\alpha ,\\psi (\\alpha ))\\rbrace $ , where $\\mathcal {O}$ is sorted on $\\alpha $ in strictly decreasing order.",
"We manipulate the sequence of insertions into Funnel Heap based on information derived from $\\mathcal {O}$ as follows.",
"Let $\\alpha _{\\max } = \\max \\lbrace \\alpha \\rbrace $ .",
"Initially, we insert monomial products generated from pairs pointed to by $\\psi (\\alpha _{\\max })$ only.",
"No other polynomial pair of total order $\\alpha ^{\\prime } < \\alpha _{\\max }$ may be involved, until at least one monomial product from $\\psi (\\alpha _{\\max })$ has been inserted into Funnel Heap, whose order is less than or equal to $\\alpha ^{\\prime }$ .",
"This point in time is identified by knowledge of the next maximal order to be encountered before an upcoming Extract-Max is called.",
"The function NEXT-MAX-ORDER introduced below answers this particular query, by calling EXTRACT-MAX on funnel heap whilst refraining from actually extracting the maximal element and only reporting on its order.",
"Alg.",
"2 summarises the details of this adaptive technique, which we label as FH-RANK.",
"We further demonstrate its impact asymptotically speaking, particularly with regards to the costs associated with preprocessing the sorted list $\\mathcal {O}$ , as well as invoking NEXT-MAX-ORDER on top of the existing calls to Extract-Max.",
"Alg.",
"FH-RANK [1] An integer $k$ designating one iterative step in the Hensel lifting process.",
"Two sets of univariate polynomials over $\\mathbb {F}$ , $\\lbrace g_i\\rbrace _{i = 1}^{k-1}$ , $\\lbrace h_i\\rbrace _{i = 1}^{k-1}$ , in sparse and sorted monomial order representation.",
"Also, two arrays $Ord_g$ and $Ord_h$ , such that $Ord_g(i)$ and $Ord_h(i)$ designate respectively the maximal order of the polynomials $g_i$ and $h_i$ under the assumed monomial ordering, for $i = 1,\\ldots ,k$ .",
"The polynomial $S_{k} = \\sum _{i = 1}^{k-1} g_{i} \\cdot h_{j}$ , where $j = k-i$ .",
"For each product pair $(g_i,h_j)$ , calculate $o(i,j)$ , the total order of their product under the assumed monomial ordering, by a forward scan of the array $Ord_g$ and a backward scan of $Ord_h$ .",
"Collect the set $\\lbrace \\alpha \\rbrace $ of distinct total orders.",
"Set $\\mathcal {O} = \\lbrace \\left(\\alpha , \\psi (\\alpha )\\right)\\rbrace $ , where $\\psi (\\alpha )= \\lbrace i \\left|\\right.",
"1 \\le i \\le k \\wedge o(g_ih_{j=k-i}) = \\alpha \\rbrace $ .",
"If $k \\in \\theta (n)$ , sort $\\mathcal {O}$ on the $\\lbrace \\alpha \\rbrace $ 's using Counting Sort.",
"Else, use a cache efficient comparison based algorithm.",
"Consider $O_{ind} = (\\alpha _{ind}, \\psi (\\alpha _{ind}))$ , where $ind \\leftarrow 1$ .",
"$i \\in \\psi (\\alpha _{ind})$ , and $j = k-i$ Call BATCHED-CHAIN$(X_{1}^{(i)}Y_{1}^{(j)}, {\\bf g}^{(i)}, {\\bf h}^{(j)})$ to insert those monomial products into Funnel Heap while chaining.",
"Set $t \\leftarrow 0$ .",
"Let $(XY, {\\bf g}, {\\bf h})$ denote the the maximal element in Funnel Heap, $\\beta $ denote the rank of $XY$ under the assumed monomial ordering.",
"Set $t \\leftarrow t+1$ , $a_{t} \\leftarrow 0$ , $R_t \\leftarrow XY$ .",
"Call Extract-Max on Funnel Heap to return the maximal element $(XY, {\\bf g}, {\\bf h})$ .",
"Return all monomial products of order $\\beta $ chained outside of Funnel Heap.",
"the maximal element in Funnel Heap has rank equal to $\\beta $ Repeat Steps 10-11 above each element $(X_{u}^{(i)}Y_{w}^{(j)}, {\\bf g}^{(i)}, {\\bf h}^{(j)})$ returned in Steps 10 and 11 Perform the coefficient arithmetic required to accumulate in $a_t$ by reading the coefficients of terms pointed to by ${\\bf g}^{(i)}$ and ${\\bf h}^{(j)}$ , then set $S_k \\leftarrow S_k + a_tR_t$ .",
"myalg Alg.",
"FH-RANK (continued) myalg If $w < \\# h_{j}$ , insert into Funnel Heap the horizontal successor by calling BATCHED-CHAIN$(X_{u}^{(i)}Y_{w+1}^{(j)})$ .",
"If $w = 1$ and $u < \\# g_{i}$ , insert into Funnel Heap the vertical successor by calling BATCHED-CHAIN$(X_{u+1}^{(i)}Y_{w}^{(j)})$ .",
"$\\beta ^{\\prime } \\leftarrow NEXT-MAX-ORDER$ .",
"$\\alpha _{ind+1} \\ge \\beta ^{\\prime }$ $i \\in \\psi (\\alpha _{ind+1})$ , and $j = k-i$ Call BATCHED-CHAIN$(X_{1}^{(i)}Y_{1}^{(j)}, {\\bf g}^{(i)}, {\\bf h}^{(j)})$ to insert those monomial products into Funnel Heap while chaining.",
"$ind \\leftarrow ind+1$ .",
"no monomials can be inserted into Funnel Heap.",
"Return $S_k$ .",
"Lem.",
"REF , Cor.",
"REF and Cor.REF we argue that with FH-RANK, the space required to handle sums of products using the priority queue approach is minimised, and so are the work and cache complexity required to perform insertions: Lemma 4.1 In Alg.",
"FH-RANK, the lifetime of each inserted element in the priority queue is minimised.",
"We establish the proof by showing that no monomial product enters the priority queue prior to the time when it is necessary for it to be there.",
"Put differently, a monomial product is inserted into Funnel Heap at a point in time when its insertion can no longer be deferred.",
"To show the claim, assume that Funnel Heap contains a monomial product $(X_{u}^{(i)}Y_{w}^{(j)}, {\\bf g}^{(i)}, {\\bf h}^{(j)})$ whose insertion could have been safely deferred.",
"Then one of those two cases must hold: The polynomial pair $(g_i,h_j)$ should have not been engaged in the insertion process.",
"But that is impossible since the pair $(g_i,h_j)$ must have been identified by the latest call to NEXT-MAX-ORDER in Step 20, which ensures that a polynomial pair is chosen only when its highest order monomial product is larger than the maximum residing element in Funnel Heap.",
"The pair $(g_i,h_j)$ is already engaged in the insertion process but this particular monomial product $X_{u}^{(i)}Y_{w}^{(j)}$ can wait.",
"This is also impossible since the sequence of insertions and extractions ensures that a given monomial product is inserted only after one of its horizontal or vertical predecessors ($X_{u-1}^{(i)}Y_{w}^{(j)}$ or $X_{u}^{(i)}Y_{w-1}^{(j)}$ ) have been extracted.",
"This means that ($X_{u}^{(i)}Y_{w}^{(j)}$ ) can potentially be the next maximum, and so must be inserted into Funnel Heap.",
"Corollary 4.2 In Alg.",
"FH-RANK, the cache complexity of Insert is minimised.",
"Consider the $r$ 'th insertion in the sequence prescribed by Alg.",
"FH-RANK.",
"Let $t$ be the index of the first insertion that takes place immediately following the extraction of the $r$ 'th element.",
"The refined SWEEP operation attains the amortised cache complexity of the $r$ 'th insertion to be $O(\\frac{1}{B} \\log _{M/B} \\frac{N_r}{B})$ , where $N_r = t-r$ .",
"By Prop.",
"REF , the lifetime in Funnel Heap of the $r$ 'th element is minimised, and hence, so is $t-r$ .",
"This concludes the proof.",
"Moreover, we have: Corollary 4.3 In Alg.",
"FH-RANK, the size of the priority queue is minimised.",
"Put differently, the likelihood that it can operate in as innermost as possible levels of the memory hierarchy is maximised.",
"The size of the priority queue is minimised as an immediate consequence of Lemma REF , since no element enters the queue prior to the time when it has to be there.",
"As an immediate consequence of Cor.",
"REF , the work required to perform each monomial product using insertions and extractions is also minimised.",
"Finally, we establish that spatial locality associated with BATCHED-CHAIN in Alg.",
"FH-RANK is nearly optimal.",
"By observing optimal spatial locality, the length of each stride in the address space is at most 1.",
"Our definition of nearly optimal relaxes this requirement: it suffices to have that the length of each stride in the address space is minimised.",
"Corollary 4.4 BATCHED-CHAIN invoked by Alg.",
"FH-RANK exhibits nearly optimal spatial locality.",
"From [3], the mechanism for chaining in batches introduced in Sec.",
"is achieved as follows.",
"We store all monomials of a given order $\\alpha $ and that have to be excluded from the queue in a dynamic array $D[\\alpha ]$ .",
"The set $\\lbrace D[\\alpha ]\\rbrace _{\\alpha }$ over all monomial orders $\\alpha $ encountered during the latest SWEEP represents pointers to the heads of chains.",
"These pointers are aligned consecutively in a static array $D$ in increasing monomial order, where the size of $D$ grows like the bound on $\\deg (S_k)$ .",
"We will label the memory accesses to the dynamic array pointed to by $D[\\alpha ]$ as horizontal accesses.",
"In contrast, we will label the memory accesses to the static array $D$ , as we hop from one pointer $D[\\alpha ]$ to another, as vertical accesses.",
"Observe that replicas of each given monomial order $\\alpha $ are chained consecutively into the single chain pointed to by $D[\\alpha ]$ , which maintains sequential spatial locality corresponding to strides of length equal to 1 in the horizontal direction.",
"Because of BATCHED-CHAIN, all pointers $\\lbrace D[\\alpha ]\\rbrace _{\\alpha }$ are accessed in increasing monomial order.",
"Additionally, because of FH-RANK, no element being chained could have been delayed entry into Funnel Heap.",
"It follows that jumps in the vertical direction are minimised.",
"We devote the remainder of this section to showing that the pre-processing costs associated with Alg.",
"FH-RANK can be embedded in the costs to perform all monomial insertions, independently of the amount of minimisation taking place.",
"This is taken up in Lem.",
"REF , Lem.",
"REF and Cor.",
"REF .",
"Lemma 4.5 The cost to perform Next-Max-Order is $\\theta (1)$ if buffer $A_1$ of Funnel Heap is non-empty.",
"Else, the cost to perform Next-Max-Order accounts for the cost to Call one ensuing Extract-Max.",
"If buffer $A_1$ is non-empty, Next-Max-Order returns the maximum over all elements residing in the insertion buffer $S_{0,1}$ and $A_1$ .",
"Else, Next-Max-Order will trigger a FILL on buffer $A_1$ (see Sec.",
").",
"Merely revealing the maximum element, however, does not alter the physical constituency of $A_1$ .",
"Hence, this buffer can be queried again with respect to the maximum residing in it, when the first Extract-Max to be encountered after the call to Next-Max-Order is issued.",
"This can also be done without the need for FILL This completes the proof.",
"Lemma 4.6 Consider an invocation of Alg.",
"FH-RANK during the $k$ 'th lifting step.",
"If $k \\in \\theta (n)$ , then sorting polynomial pairs on their total order in Step 2 of FH-RANK requires $\\theta (k)$ work and $\\theta ((M+k)/B)$ cache misses.",
"Else, we have $k \\in O(n)$ but $k \\notin \\Omega (n)$ , for which this step requires $O(k \\log k)$ work and $O(\\frac{k}{B} \\log _{M/B} k)$ cache misses.",
"Collecting the total order of products in $\\lbrace (g_i\\cdot h_j)\\rbrace _{k =1,\\ldots ,n}$ using a forward and backward scan of the arrays $Ord_g$ and $Ord_h$ respectively requires only cache miss operations, whose total cost amounts to $\\theta (k/B)$ .",
"We now address the costs for sorting.",
"Case 1: Using Counting Sort, the number of records to be sorted is equal to $k-1$ .",
"Since each record represents the total degree of a monomial product, it is then less than or equal to $n-k$ , and so, the work of counting sort is $\\theta (k + (n - k)) = \\theta (n)$ .",
"Using a cache efficient variant of counting sort attains a cache complexity nearly linear in the number of records as well, and is equal to $\\theta (\\frac{n+M}{B})$ cache misses (see [15]).",
"When $k = \\theta (n)$ , the space, work, and cache complexity of Counting Sort simplify to those as stated in the Lemma above.",
"Case 2: Here, we know that $k \\notin \\theta (n)$ .",
"But $k \\le n$ , so we must have $k \\in O(n)$ but $k \\notin \\Omega (n)$ .",
"Here, counting sort is no longer linear in the number of records being sorted.",
"Using any of the comparison-based, cache efficient sorting algorithms requires $O(k)$ space, optimal $O(k \\log k)$ work, and optimal $O(\\frac{k}{B}\\log _{M/B} k)$ cache misses (See [12], for example).",
"In the following Corollary, we conclude that the cost for sorting and pre-fetching the maximal order can be embedded in the asymptotic costs for performing all monomial products comprising $S_k$ , independently of the amount of minimisation exerted onto Funnel Heap.",
"Corollary 4.7 Assume the sparse distributed representation for polynomials.",
"Let $\\left|T \\right|_{\\min }$ denote the size of Funnel Heap following the minimisation incurred by Alg.",
"FH-RANK in the $k$ 'th lifting step, $\\bar{g} = \\max \\lbrace \\#g_i\\rbrace _{i = 1}^{k}$ and $\\bar{h} = \\max \\lbrace \\#h_j\\rbrace _{j = 1}^{k}$ .",
"Assume further that the cache size $M$ satisfies $M \\in O(n)$ , and, if $k \\in O(n)$ but $k \\notin \\Omega (n)$ , that $\\bar{g}\\bar{h} = \\Omega (\\log k)$ .",
"In the worst case analysis when each polynomial multiplication $g_ih_j$ is sparse, the cost to sort all polynomial products $\\lbrace (g_i\\cdot h_j)\\rbrace $ on their total order and to issue all calls to NEXT-MAX-ORDER can be embedded in the cost for performing all insertions into Funnel Heap.",
"The gist of the proof lies in deriving bounds on the costs for the pre-processing phase that do not depend on $\\left|T_{\\min } \\right|$ .",
"By Lem.",
"REF , each call to NEXT-MAX-ORDER accounts for the cost of the ensuing Extract-Max.",
"Performing all $\\theta \\left(k \\bar{g}\\bar{h}\\right)$ monomial extractions and insertions into Funnel Heap requires $O \\left(k \\bar{g}\\bar{h} \\log \\left|T \\right|_{\\min } \\right)$ work and amortised $O \\left(k \\bar{g}\\bar{h} \\frac{1}{B} \\log _{M/B} \\left|T \\right|_{\\min } \\right)$ cache misses.",
"From Lem.",
"REF , if $k \\in \\theta (n)$ , we know that cache friendly counting sort requires $\\theta (k)$ work and $\\theta ((M+k)/B)$ cache misses, and for these costs to be embedded in their respective counterparts, we require $k \\in O\\left(k \\bar{g}\\bar{h} \\log \\left|T \\right|_{\\min } \\right)$ and $\\frac{(M+k)}{B} \\in O\\left(k \\bar{g}\\bar{h} \\frac{1}{B} \\log _{M/B}\\left|T \\right|_{\\min } \\right).$ The requirement in (REF ) trivially holds.",
"For (REF ), the assumption that $M \\in O(n)$ when $k \\in \\theta (n)$ leads to $(M+k)/B = O(k/B) \\in O\\left(k \\bar{g}\\bar{h} \\frac{1}{B} \\log _{M/B}\\left|T \\right|_{\\min } \\right)$ .",
"When $k \\in O(n)$ but $k \\notin \\Omega (n)$ , the work of sorting ought to satisfy $k \\log k \\in O \\left(k \\bar{g}\\bar{h} \\log \\left|T \\right|_{\\min } \\right),$ or $\\log k \\in O \\left(\\bar{g}\\bar{h} \\log \\left|T \\right|_{\\min } \\right),$ which is attained if each monomial product $g_ih_j$ , known to be sparse and having $O(\\bar{g}\\bar{h})$ non-zero terms, also has at least $\\Omega (\\log k)$ non-zero terms.",
"This is also a very realistic assumption since $k$ is sufficiently small in this branch of the proof.",
"Using this same requirement, we also get that the cache complexity required by sorting is embedded by that to perform all monomial insertions and extractions, as we can see from: $\\frac{k}{B}\\log _{M/B} k \\in O \\left(k \\bar{g}\\bar{h} \\frac{1}{B} \\log _{M/B} \\left|T \\right|_{\\min } \\right).$ Remark 4.8 The condition $M \\in O(n)$ entails that the input polynomial has sufficiently high degree with respect to the size of in-core memory.",
"This is a very reasonable requirement at scale bearing contemporary cache sizes.",
"To require that $\\log k \\in O(\\bar{g}\\bar{h})$ when $k \\in O(n)$ but $k \\notin \\Omega (n)$ means that the sparsest polynomial product encountered in the $k$ 'th lifting step has $\\Omega (\\log k)$ non-zero terms.",
"But any such product has degree at most $n-k$ , which means that for significantly small iteration indices $k$ , the polynomial products arising tend to be of significantly large degrees.",
"The lower bound requirement on the sparsity of polynomial products indicated by $\\bar{g}\\bar{h} \\in \\Omega (\\log k)$ is thus very permissive."
],
[
"Experimental Results",
"We implement all algorithms in C++ and compile our code using g++ version 4.4.6 20120305 with optimization level -O3.",
"We run the experiments on an Intel(R) Xeon(R) CPU E5645 with 43GB of RAM, 12MB in L3 cache, and 256KB in L2 cache.",
"To record cache misses, we use the STXXL library in Sec.",
"REF and the profiler tool perf in Sec.",
"REF and REF ."
],
[
"Funnel Heap Benchmarks: Performance at Scale",
"In this section we present a preamble where we benchmark Funnel Heap against Binary Heap for performing a generic sequence of priority queue operations outside the scope of Hensel lifting.",
"The only available benchmarking appears in the unpublished work of [19]The authors of the current manuscript were unable to locate any standardised implementations of funnel heap available for public use..",
"The specific goal of this section is to reproduce the conclusions derived in [19] on our own machines and to reveal the cut-off line when Funnel Heap is able to beat Binary Heap.",
"A careful examination is required before one is able to witness the performance predicted by the asymptotic analysis at large scale.",
"This is because Funnel Heap performs more computations than Binary Heap, making it expensive to use at small scale, when the cost to perform memory accesses does not dominate performance.",
"In line with [19], we simulate out of core behaviour by constraining the RAM of our machine to 16MB through the use of STXXL vectors (see STXXL version 1.3.1 and [11]).",
"In this case, we force both heaps to store part of their structure on disk despite that the input suites are not too large.",
"We generate a list of random integers and perform the following sequence of insertions and extractions onto the queues: $N$ elements are pushed into the heap $N/2$ elements are popped off the heap $N/2$ elements are pushed into the heap $N$ elements are popped off the heap In Table REF below, we present an account of the overall runtime as well as cache misses incurred by each of the two heaps.",
"The term “Max Capacity” denotes the largest number of elements each heap occupies at any one point in time.",
"In the first three rows, Funnel Heap loses out to Binary Heap in terms of overall runtime.",
"For this particular range, the input is too small, and Funnel Heap's performance is computation bound, as it attempts to maintain its structure by calling FILL and SWEEP.",
"Once Binary Heap grows out-of-core, however, its runtime becomes memory-bound and performance deteriorates significantly.",
"This point in time is obtained when Binary Heap contains about $N = 4\\times 10^6$ elements of size four bytes each, the expected cutoff line representing the customised size of RAM.",
"Beyond that point, both heaps start swapping to disk and the cache complexity begins to dominate the computation cost.",
"Funnel Heap now beats Binary Heap by orders of magnitude, as predicted by the asymptotic analysis.",
"Table: Generic Priority Queue OperationsThe results in this section should be construed in the following sense.",
"The input suite to our polynomial factorisation experiments below does not attain a level of growth sufficient to solicit out-of core behaviour.",
"As such, any significant improvements in performance shown hereafter can only be attributed to the optimisations incurred onto Funnel Heap, particularly, batched chaining and the optimised sequence of insertions, but not to Funnel Heap alone.",
"On the other hand, we do expect Funnel Heap to outperform Binary Heap considerably, without any of those mentioned optimisations.",
"This will remain applicable even when tackling a single polynomial multiplication or division – and not just sums of products – once the input data grows sufficiently large."
],
[
"Performance of FH-RANK",
"This section is dedicated to the performance of FH-RANK.",
"We start off with a summary of relevant results from [3], where we observe all of the following.",
"Both overlapping implementations PQ-HL$^{\\mathcal {B}}$ and FH-HL are always significantly faster than SER-HL, despite that the Newton polygons treated there are extremely sparse.",
"Even when the polygon has a few edges, the polytope method remains susceptible to fluctuations in the sparsity of the intermediary polynomials, which is attributed to expression swell.",
"Both PQ-HL$^{\\mathcal {B}}$ and FH-HL are also always faster than Magma 2.18-7, whose built-in function for factoring bivariate polynomials relies on the standard algorithms in [5], [20].",
"Starting with bivariate polynomials of total degree equal to $10,000$ , and despite that the input polynomials are significantly sparse, Magma runs out of memory.",
"Alg.",
"FH-HL is always fastest, with a dramatic reduction in run-time over SER-HL which is largely attributed to substantial expression swell, and also over PQ-HL$^{\\mathcal {B}}$ , which is largely attributed to BATCHED-CHAIN.",
"Hereafter we address the performance of FH-RANK, by benchmarking it against all of PQ-HL$^{\\mathcal {F}}$ , PQ-HL$^{\\mathcal {B}}$ , PQ-HL$^{\\mathcal {B}}$ with chainining (PQ-HL$^{\\mathcal {B}}$ -CHAIN), and FH-HL.",
"We use the sparse distribued representation for encoding all polynomials.",
"Our input suite consists of random bivariate polynomials over $\\mathbb {F}_3$ that turn out to factor into two irreducibles, and that are extremely sparse.",
"In several instances, we specifically generate random polynomials whose Newton polygon is the sparsest possible, consisting of the triangle $(0,n)$ , $(n,0$ ), and $(0,0)$ .",
"For sorting the polynomial pairs on their total degree, we note that the input degrees we treat here fall within a certain range for which the standard GCC quicksort implementation is known to be highly competitive over cache efficient alternatives.",
"This is established thoroughly by the algorithmic engineering study of [10] (see for example the experiments in Sec.",
"5).",
"The results for each input polynomial tested are reported using a pair of tables.",
"In all of the captions, the parameter $n$ denotes the total degree of the input polynomial, and $t$ denotes the total number of its non-zero terms.",
"Our requirement for sparse input is to have $t \\ll n^2$ .",
"The parameter $F$ corresponds to the total number of boundary factorisations attempted before the two irreducible factors are produced.",
"The first table for each input polynomial provides an account of run-time in seconds as well as cache misses.",
"The second and third ensuing tables show the number of SWEEPs called upon each link in the Funnel Heap used in Alg.",
"PQ-HL$^{\\mathcal {F}}$ and Alg.",
"FH-RANK respectively.",
"In those tables we also indicate the size of each link.",
"FH-RANK against all other variants: Both FH-RANK and FH-HL are faster than all of PQ-HL$^{\\mathcal {F}}$ , PQ-HL$^{\\mathcal {B}}$ , and PQ-HL$^{\\mathcal {B}}$ -CHAIN, and they incur considerably less cache misses thanks to BATCHED-CHAIN.",
"In turn, FH-RANK improves over FH-HL at a larger scale thanks to the optimised sequence of insertions.",
"This is demonstrated by an order of magnitude reduction in time as well as cache misses over FH-HL.",
"For all of the input polynomials tested, the second and third tables show that FH-RANK brings about an order of magnitude reducion in space, as demonstrated in the reduction of the number of links required, as well as the corresponding size of each link.",
"Of significance also is the notable reduction in the number of sweeps to each link.",
"The improvement in the amount of space required as well as the number of SWEEPs incurred explain the significant reduction in the run-time and cache misses associated with FH-RANK.",
"In contrast, we note that FH-HL consumed the same amount of peak memory as PQ-HL$^{\\mathcal {F}}$ : as explained in [3], FH-HL requires the same amount of space as FH, except that an asymptotically large amount of space shifts from being “working space” to “auxiliary space”, which improves on runtime and rate of cache misses of FH-HL.",
"The effect of chaining on Binary Heap: In several instances, PQ-HL$^{\\mathcal {B}}$ -CHAIN incurs the same order of cache misses as PQ-HL$^{\\mathcal {B}}$ , and on a few occasions it is actually slower.",
"This demonstrates that chaining is not consistently efficient when employing Binary Heap.",
"In [3], we elaborate on the reasons behind this behaviour.",
"For example, each insertion into the linked list denoting the chain incurs a random miss.",
"Also, a single search for a replica to be chained may very well require traversing the entire heap of size $N$ , bringing the work and cache complexity for performing a single search to be that of traversal of $N$ elements.",
"Summing up, neither temporal locality nor spatial locality are observed in PQ-HL$^{\\mathcal {B}}$ -CHAIN.",
"Funnel Heap without any of the optimisations: The results reported for PQ-HL$^{\\mathcal {F}}$ are also not promising for the range of input degrees we had treated, specifically as taken against PQ-HL$^{\\mathcal {B}}$ and PQ-HL$^{\\mathcal {B}}$ -CHAIN.",
"Funnel Heap inherently performs more work, and incurs more cache misses for smaller levels of the memory hierarchy.",
"The input polynomials treated here do not solicit access to disk.",
"As a result, the extra work performed by Funnel Heap is not compensated for.",
"Yet, it is evident from the benchmarks reported in Section REF , that PQ-HL$^{\\mathcal {F}}$ is set to outperform PQ-HL$^{\\mathcal {B}}$ and PQ-HL$^{\\mathcal {B}}$ -CHAIN once the input is large enough to solicit disk swaps.",
"At smaller scale, all the benefits observed are attributed solely to the techniques in BATCHED-CHAIN and optimising the sequence of insertions (FH-HL and FH-RANK respectively)."
],
[
"Funnel Heap versus the $k$ -merger",
"In this section we gather further insight into the bottleneck associated with the irregularity in computations as a result of the varying density of the intermediary output arising during Hensel lifting.",
"In [2], we observed that if space is not a concern and one is willing to store all polynomial products during each Hensel lifting step before their final merge into $S_k$ , the only cache oblivious competitor at scale for the global priority queue approach would be the following scenario.",
"One would perform each polynomial multplication separately using a dedicated (local) priority queue, followed by merging of all the polynomial products using the static and cache oblivious $k$ -merger of [12].",
"Local priority queues tackling one polynomial multiplication at a time are more likely to reside in cache when the multiplication is sparse.",
"As such, a data structure suited for internal memory such as a MAX-heap should be used, since it performs less work than any external memory implementation.",
"The goal of this section, however, is to demonstrate that even in this scenario, the $k$ -merger still fails to achieve its optimal (and amortised) work and cache complexity, since the streams to be merged are of varying density.",
"This is in contrast to its typical mode of usage in merge-based sorting algorithms where the streams tend to be of equal size.",
"This section provides an empirical proof of concept that overlapping arithmetic using a global funnel heap is not only cache-oblivious, but is further guaranteed to attain optimal performance, bearing the irregularity in computation.",
"An interesting conclusion we draw is that, despite that the $k$ -merger incurs optimal work for merging a given set of elements, it is beaten by funnel heap, albeit at a higher work complexity, once the $k$ -merger fails to amortise its cache complexity in the presence of irregular input.",
"To this end, we demonstrate by merging streams of random integers, and we simulate three scenarios depending on the density of streams.",
"In Table REF , we process $k^2$ streams each containing $k$ elements, in Table REF we process $k$ streams with $k$ elements each, and in Table REF , we process $k$ streams with $k^2$ elements each.",
"In addition to cache misses and total execution time, we record the total number of integer comparisons required by merging sub-routines, including the merging that is required by the SWEEP function in Funnel Heap.",
"On integer comparisons: In all three tables, Funnel Heap does more integer comparisons than the $k$ -merger, something we attribute to the cost of its SWEEP function.",
"Despite that it incurs the least number of comparisons, the $k$ -merger lags behind in cache miss rate and then finally overall execution time.",
"On cache misses: In all three tables, Funnel Heap terminates using significantly lower cache misses than the $k$ -merger.",
"We note, however, that the rate of improvement of Funnel Heap in Table REF is less than what is observed for Tables REF and REF , which is to be expected, since the $k$ -merger is now able to produce $k^3$ elements at the end of its invocation.",
"This is in line with the $k$ -merger's cache complexity analysis, by which we know that the amortised cache complexity is met so long as the output buffer produces $k^3$ elements.",
"On overall runtime: In all three categories, Funnel Heap terminates faster than the single, fixed size $k$ -merger.",
"In Tables REF and REF , Funnel Heap attains a significantly lower cache miss rate that justifies its fast execution time.",
"In Table REF , and despite that the $k$ -merger catches up by performing its best cache misse rate as opposed to Tables REF and REF , it still lags behind Funnel Heap in total execution time.",
"Table: k 2 k^2 streams of kk elements eachTable: kk streams with kk elements eachTable: kk streams with k 2 k^2 elements each"
],
[
"Conclusion",
"In this paper we presented a comprehensive design and analysis that extends the work in [2] and [3].",
"Alg.",
"FH-RANK exploits all the features of Funnel Heap for implementing sums of products arising in Hensel lifting of the polytope method, when polynomials are in sparse distributed representation.",
"Those features involve a batched mechanism for chaining replicas as well as optimising on the sequence of insertions and extractions in order to minimise the size of the priority queue as well as the work and cache complexity.",
"The competitive asymptotics are validated by empirical results, which, in addition to asserting the high efficieny of FH-RANK whether or not data fits in in-core memory, help us derive two other main conclusions.",
"Firstly, we confirm that at a large scale, all polynomial arithmetic employing a priority queue will benefit substantially from using Funnel Heap over Binary Heap, even without the proposed mechanisms for chaining and/or optimising the sequence of insertions/extractions.",
"Secondly, Funnel Heap is confirmed to be superior in practice as a merger when tested against the provably optimal $k$ -merger structure, despite having a higher work complexity.",
"This is attributed to its ability to adapt to merging input streams of fluctuating density, which in turn, makes Funnel Heap ideal for performing polynomial arithmetic in the sparse distributed representation, where such fluctuation affects overall performance.",
"This supports our argument that one should resort to the overlapping approach using a single priority queue, as opposed to handling each of the the local multiplications separately using a local priority queue, to be followed by additive merging of all polynomial streams.",
"This conclusion remains valid whether or not expression swell is taking place."
],
[
"Acknowledgments",
"We thank the Lebanese National Council for Scientific Research and the University Research Board – American University of Beirut, for supporting this work."
]
] | 1612.05403 |
[
[
"Cosmic microwave background limits on accreting primordial black holes"
],
[
"Abstract Interest in the idea that primordial black holes (PBHs) might comprise some or all of the dark matter has recently been rekindled following LIGO's first direct detection of a binary-black-hole merger.",
"Here we revisit the effect of accreting PBHs on the cosmic microwave background (CMB) frequency spectrum and angular temperature/polarization power spectra.",
"We compute the accretion rate and luminosity of PBHs, accounting for their suppression by Compton drag and Compton cooling by CMB photons.",
"We estimate the gas temperature near the Schwarzschild radius, and hence the free-free luminosity, accounting for the cooling resulting from collisional ionization when the background gas is mostly neutral.",
"We account approximately for the velocities of PBHs with respect to the background gas.",
"We provide a simple analytic estimate of the efficiency of energy deposition in the plasma.",
"We find that the spectral distortions generated by accreting PBHs are too small to be detected by FIRAS, as well as by future experiments now being considered.",
"We analyze Planck CMB temperature and polarization data and find, under our most conservative hypotheses, and at the order-of-magnitude level, that they rule out PBHs with masses >~ 10^2 M_sun as the dominant component of dark matter."
],
[
"Introduction",
"The idea of primordial black holes (PBHs) was first put forward by Zel'dovich and Novikov in the sixties [1].",
"Developing it further, Hawking argued that early-Universe fluctuations could lead to the formation of PBHs with masses down to the Planck mass [2].",
"Chapline was the first to suggest that PBHs could make the dark matter (DM) [3].",
"Though this class of DM candidate has taken a back seat to the notion that DM is a new elementary particle [4], [5], [6], [7], [8], the idea of PBH dark matter was recently rekindled [9], [10], following the first detection of two merging $\\sim 30\\, M_\\odot $ black holes by LIGO [11].",
"Given the increasingly constraining null searches for particle DM, PBHs and their observational consequences are worth reconsidering [12], [13].",
"The abundance of PBHs is constrained by a variety of observations in several mass ranges (for a comprehensive review see Refs.",
"[12], [13]).",
"To cite only a few constraints, null microlensing searches exclude compact objects with masses $\\lesssim 10 \\, M_\\odot $ [14], [15], and wide-binary surveys exclude those with masses $\\gtrsim 10^2 \\,M_\\odot $ [16], [17].",
"For PBHs more massive than $\\sim 1\\, M_{\\odot }$ , strong constraints were derived by Ricotti, Ostriker, and Mack [18] (hereafter ROM) from the cosmic microwave background (CMB) frequency spectrum and temperature and polarization anisotropies.",
"The basic idea behind these limits is that PBHs accrete primordial gas in the early Universe and then convert a fraction of the accreted mass to radiation.",
"The resulting injection of energy into the primordial plasma then affects its thermal and ionization histories [19], and thus leads to distortions to the frequency spectrum of the CMB and to its temperature/polarization power spectra.",
"ROM estimate that CMB anisotropy measurements by WMAP [20] and limits on CMB spectral distortions by FIRAS [21] exclude PBHs with masses $M \\gtrsim 1\\, M_{\\odot }$ and $M \\gtrsim 0.1 \\, M_{\\odot }$ , respectively, as the dominant component of dark matter.",
"Using ROM's results, Ref.",
"[22] strengthened these constraints with Planck data.",
"Here we re-examine in detail CMB limits to the PBH abundance, building on and expanding the work of ROM.",
"It is notoriously difficult to estimate from first principles and self-consistently the accretion rate onto a central object and the corresponding radiative efficiency (see, e.g., the discussion in Chapter 14 of Ref. [23]).",
"In this work, we strive to estimate the minimum physically-plausible PBH luminosity in order to set the most conservative constraints to the PBH abundance.",
"The bounds we derive are significantly weaker than those of ROM: using Planck temperature and polarization data [24], we find that only PBHs with masses $M \\gtrsim 10^2\\, M_{\\odot }$ can be conservatively excluded as the dominant component of the dark matter.",
"Moroever, we find that CMB spectral-distortion measurements, both current and upcoming, do not place any constraints on PBHs.",
"The single largest difference between our work and ROM's lies in the adopted radiative efficiency $\\epsilon \\equiv L/\\dot{M}c^2$ to convert the mass accretion rate $\\dot{M}$ to luminosity $L$ .",
"For masses $M \\lesssim 10^4 \\, M_{\\odot }$ , both ROM and this work conservatively assume a quasi-spherical accretion flow.",
"Shapiro [25] provided a first-principles estimate of the radiative efficiency for this problem.",
"This shows that $\\epsilon \\propto \\dot{m} \\equiv \\dot{M} c^2/L_{\\rm Edd}$ , where $L_{\\rm Edd}$ is the Eddington luminosity.",
"While ROM assume a fixed $\\epsilon /\\dot{m} =0.011$ for $\\dot{m} \\le 1$ , we generalize Shapiro's calculation, in particular accounting for Compton cooling by ambient CMB photons, and explicitly compute $\\epsilon /\\dot{m}$ as a function of PBH mass and redshift.",
"We find that $\\epsilon /\\dot{m}$ never exceeds $\\sim 10^{-3}$ (corresponding to Shapiro's result for accretion from an HII region), and can be as low as $\\sim 10^{-5}$ after recombination (corresponding to Shapiro's result for accretion from an HI region), or even lower at high redshifts and for large PBH masses for which Compton cooling is important.",
"A few other differences moreover contribute to lowering the mass accretion rate with respect to that derived by ROM, as detailed in the remainder of this article.",
"The rest of this paper is organized as follows.",
"The core of our calculation is laid out in Section : there we compute the accretion rate and luminosity of an accreting black hole in the early Universe.",
"We discuss the local feedback of this radiation in Section .",
"In Section we estimate the efficiency with which the energy injected by PBHs is deposited into the plasma.",
"We then estimate the effect of PBHs on CMB observables and derive the resulting constraints in Section .",
"Finally, we conclude in Section .",
"To keep the calculation tractable analytically we must make several approximations and assumptions.",
"In order to not disrupt the flow of the calculation, we defer the verification of these assumptions to Appendix .",
"In Appendix we compare our analytic approximation for the efficiency of energy deposition in the plasma to existing studies.",
"The first aspect to consider is the geometry of the accretion.",
"If the characteristic angular momentum of the accreted gas (at the Bondi radius) is smaller than the angular momentum at the innermost stable circular orbit, the accretion is mostly spherical.",
"Otherwise, an accretion disk forms.",
"Disk accretion is typically much more efficient than spherical accretion at converting accreted mass into radiation.",
"Indeed, while in the latter case the dominant source of luminosity is bremsstrahlung radiation from the hot ionized plasma near the event horizon, in the former case the large viscous heating required to dissipate angular momentum leads to radiating a significant fraction of the rest-mass energy [23].",
"It is difficult to estimate the angular momentum of the gas accreting onto PBHs, as it requires knowledge of the PBH-baryon relative velocity on scales of the order of the Bondi radius, which is much smaller than any currently observed cosmological scale.",
"A correct estimate of this relative velocity would moreover require accounting for the (non-linear) clustering of PBHs.",
"Following our philosophy to derive the most conservative and physically-motivated accretion rate and luminosity, we shall therefore adopt a spherical accretion model, expanding on the classic work of Shapiro [25].",
"We note that this is also the underlying assumption made in ROM for PBH masses $M\\lesssim 10^3 -10^4\\, M_{\\odot }$ (see their Section 3.3).",
"Another difficulty is that of local feedback.",
"The radiation emanating from the accreting PBH may indeed ionize and/or heat the accreting gas, which would in turn affect the radiative output.",
"We show in Appendix REF that thermal feedback is negligible for all masses and redshifts considered (consistent with ROM's Section 4.2.1 results).",
"We will also see in Section REF that the Strömgren radius is always significantly smaller than the Bondi radius (consistent with our luminosity being significantly lower than that of ROM, who find that the photoionized region is marginally smaller than the Bondi radius).",
"Hence one can assume that in the outermost region of the accretion flow, the ionization fraction is approximately equal to the background value.",
"Close enough to the black hole, the gas eventually becomes fully ionized, either through photoionizations by the outgoing radiation field, or collisional ionizations, or both.",
"We will see in Section REF that neither ionization process clearly prevails.",
"To circumvent a complex self-consistent calculation of the luminosity and ionization profile, we shall consider the two limiting cases where one of the two ionization processes is dominant, and quote our results for both.",
"In the first case, we shall completely neglect any radiative feedback, and assume that the ionization fraction $x_e$ is equal to the background value $\\overline{x}_e$ , until the temperature of the gas reaches $ \\sim 10^4$ K, at which point the gas gets collisionally ionized.",
"In our second limiting case, we assume that the radiation from the PBH photoionizes the gas up to a radius beyond that at which $T \\sim 10^4$ K (so collisional ionizations are not relevant), yet inside the Bondi radius.",
"In all figures we refer to the former case by collisional ionization and the latter by photoionization.",
"The correct result (within our overall model) lies somewhere between these two limiting cases.",
"The difference between the final results in the two limits illustrates the relatively large theoretical uncertainty associated with this calculation.",
"In what follows we split the calculation of the hydrodynamical and thermal state of the gas accreting onto a BH into three regions.",
"First, in Section REF , we study the outermost region where we assume a constant free-electron fraction $x_e$ equal to the background value $\\overline{x}_e$ .",
"We solve for the steady-state fluid and heat equations, as well as the accretion rate, accounting for Compton drag (as in Ref. [26]).",
"We also include, for the first time in this context, for Compton cooling by CMB photons.",
"Secondly, in Section REF we consider the (re)ionization of hydrogen in the collisional ionization case, if the background gas is already partially neutral.",
"We assume that hydrogen gets collisionally ionized once the gas reaches a characteristic temperature $T_{\\rm ion} \\sim 10^4$ K, and that this ionization proceeds roughly at constant temperature.",
"Thirdly, in Section REF we study the innermost region where the gas is fully ionized and adiabatically compressed.",
"We account for the change of the adiabatic index once electrons become relativistic.",
"The final outcome of this calculation is the gas temperature near the event horizon, which, alongside the accretion rate, determines the luminosity of the accreting BH, as we shall see in Section REF .",
"Figure REF illustrates the temperature profile in the various regions considered.",
"We conclude this Section by considering the effect of PBH velocities.",
"Figure: Schematic temperature profile for the gas accreting onto a BH.",
"If Compton cooling is efficient (γ≫1\\gamma \\gg 1), the gas temperature remains close to the CMB temperature down to r∼γ -2/3 r B r \\sim \\gamma ^{-2/3} r_{\\rm B}, where r B r_{\\rm B} is the Bondi radius.",
"The temperature then increases adiabatically as T∝ρ 2/3 ∝1/rT \\propto \\rho ^{2/3} \\propto 1/r.",
"If photoionizations can be neglected, and if the background gas is partially neutral, the gas gets collisionally ionized at nearly constant temperature once it reaches T ion ≈1.5×10 4 T_{\\rm ion} \\approx 1.5 \\times 10^4 K. Once the gas is fully ionized, the temperature resumes increasing adiabatically as T∝1/rT \\propto 1/r until electrons become relativistic, at which point the change in the adiabatic index implies T∝ρ 4/9 ∝r -2/3 T \\propto \\rho ^{4/9} \\propto r^{-2/3}.",
"If the luminosity of the accreting PBH is large enough, the gas is photoionized instead of collisionally ionized.",
"In that case the gas temperature reaches larger values near the black hole horizon.We consider spherical accretion of a pure hydrogenAccounting for helium is conceptually straightforward but would add unneeded complications for the order-of-magnitude calculation presented here.",
"gas onto an isolated point mass $M$ , bathed in the quasi-uniform CMB radiation field (we check the validity of the isolated-PBH assumption in Appendix REF ).",
"In general, one should solve for the time-dependent fluid, heat and ionization equations, all of which are coupled.",
"For simplicity we shall assume a constant ionization fraction $x_e = \\overline{x}_e$ in the outermost region, equal to the background value.",
"As long as the characteristic accretion timescale is much shorter than the Hubble timescale, one can make the steady-state approximation.",
"Ref.",
"[26] showed that this is the case for $M \\lesssim 3 \\times 10^4\\, M_{\\odot }$ , so we shall limit ourselves to this mass range.",
"In this outermost region, far from the BH horizon, a Newtonian treatment is very accurate.",
"We denote by $v \\equiv v_r < 0$ the peculiar radial velocity (i.e.",
"the velocity with respect to the Hubble flow) of the accreted gas.",
"The steady-state mass and momentum equations for the fluid are $4 \\pi r^2 \\rho |v| &=& \\dot{M} = \\textrm {const},\\\\v \\frac{d v}{d r} &=& - \\frac{G M}{r^2} - \\frac{1}{\\rho } \\frac{d P}{dr} - \\frac{4}{3} \\frac{\\overline{x}_e \\sigma _{\\rm T} \\rho _{\\rm cmb}}{m_p c} v, $ where the pressure $P$ is $P = \\frac{\\rho }{m_p}(1 + \\overline{x}_e) T,$ and the last term in the momentum equation is the drag force due to Compton scattering of the ambient nearly homogeneous CMB photons with energy density $\\rho _{\\rm cmb}$ [26], $\\sigma _{\\rm T}$ being the Thomson cross section.",
"Note that we have neglected the self-gravity of the accreted gas, which is valid for $M \\lesssim 3 \\times 10^5\\, M_{\\odot }$ [26].",
"Consistent with our steady-state approximation, we also neglected the Hubble drag term $H v$ , which is of the same order as the neglected partial time derivative $\\partial v/\\partial t$ .",
"The fluid equation must be complemented by the heat equation.",
"For simplicity we shall only consider Compton cooling by CMB photons [27] as a heat sink in this region.",
"The steady-state heat equation is then $v \\rho ^{2/3} \\frac{d}{dr}\\left( \\frac{T}{\\rho ^{2/3}}\\right) = \\frac{8 \\overline{x}_e \\sigma _{\\rm T}\\rho _{\\rm cmb}}{3 m_e c (1 + \\overline{x}_e)}(T_{\\rm cmb} - T), $ where $T_{\\rm cmb}$ is the temperature of CMB photons.",
"Since we only consider PBH masses for which the accretion timescale is shorter than the Hubble timescale, whenever Compton cooling becomes relevant to the accretion problem, it is even more important for the background temperature evolution, and enforces $T_{\\infty } = T_{\\rm cmb}$ .",
"If Compton drag and cooling were negligible, one would recover the the classic Bondi accretion problem [28], the characteristic velocity, length and timescales of which are $v_{\\rm B} \\equiv \\sqrt{P_{\\infty }/\\rho _{\\infty }},\\ \\ \\ r_{\\rm B} \\equiv \\frac{G M}{v_B^2}, \\ \\ \\ t_{\\rm B} \\equiv \\frac{G M}{v_B^3}, $ where $P_{\\infty }$ and $\\rho _{\\infty }$ are the gas pressure and density far from the point mass ($\\rho _{\\infty } = \\overline{\\rho }_b$ , the mean baryon density).",
"It is best to rewrite the problem in terms of dimensionless variables $x \\equiv r/r_{\\rm B}$ , $u \\equiv v/v_{\\rm B}$ , $\\hat{\\rho } \\equiv \\rho /\\rho _{\\infty }$ , $\\hat{T} \\equiv T/T_{\\infty }$ .",
"We also define the dimensionless constants $\\lambda &\\equiv & \\frac{\\dot{M}}{4 \\pi \\rho _{\\infty } r_{\\rm B}^2 v_{\\rm B}}, \\\\\\beta &\\equiv & \\frac{4}{3} \\frac{\\overline{x}_e \\sigma _{\\rm T}\\rho _{\\rm cmb} }{m_p c} t_{\\rm B}, \\\\\\gamma &\\equiv & \\frac{8 \\overline{x}_e \\sigma _{\\rm T}\\rho _{\\rm cmb} }{3 m_e c (1 + \\overline{x}_e)} t_{\\rm B}= \\frac{2 m_p}{m_e (1 + \\overline{x}_e)} \\beta \\gg \\beta .",
"$ We show in Fig.",
"REF the dimensionless Compton drag and cooling rates $\\beta $ and $\\gamma $ , as a function of redshift and PBH mass.",
"In terms of these variables the problem to solve is $\\hat{\\rho } x^2 |u| &=& \\lambda , \\\\u \\frac{du}{dx} &=& - \\frac{1}{x^2} - \\frac{1}{\\hat{\\rho }} \\frac{d}{dx}(\\hat{\\rho } \\hat{T}) - \\beta u, \\\\u \\hat{\\rho }^{2/3} \\frac{d}{dx}\\left( \\frac{\\hat{T}}{\\hat{\\rho }^{2/3}}\\right) &=& \\gamma (1 - \\hat{T}), $ with asymptotic conditions $\\hat{\\rho } \\rightarrow 1$ and $\\hat{T} \\rightarrow 1$ at $x \\rightarrow \\infty $ .",
"Before moving on, let us note that the PBH mass does not grow significantly in a Hubble time [29].",
"Indeed, $\\frac{\\dot{M}}{H M} = \\frac{4 \\pi \\lambda \\overline{\\rho }_b (G M)^2}{H M v_{\\rm B}^3} = \\frac{4 \\pi \\lambda G \\overline{\\rho }_b t_B}{H}= \\frac{3}{2} \\lambda \\frac{\\overline{\\rho }_b}{\\overline{\\rho }_{\\rm tot}} H t_{\\rm B} ,~~~$ where in the last equality we used Friedman's equation for the Hubble rate $H$ .",
"Therefore, provided the steady-state approximation is valid (i.e.",
"$t_{\\rm B} \\ll H^{-1}$ ), we see that $\\dot{M}\\ll H M$ .",
"Figure: Characteristic dimensionless Compton drag rate β\\beta [Eq.",
"(), upper panel] and Compton cooling rate γ\\gamma [Eq.",
"(), lower panel], as a function of redshift, and for PBH masses M=1,10 2 M = 1, 10^2 and 10 4 M ⊙ 10^4\\, M_{\\odot }, from bottom to top.",
"Both are evaluated for a standard recombination and thermal history, with the substitution v B →v eff v_{\\rm B} \\rightarrow v_{\\rm eff} as described in Section ."
],
[
"Solution for $\\beta \\ll \\gamma \\ll 1$",
"When both Compton drag and cooling are negligible, we recover the classic Bondi problem [28] for an adiabatic gas, with $\\hat{T} = \\hat{\\rho }^{2/3}$ .",
"In this case the momentum equation can be rewritten as a conservation equation $\\frac{1}{2} u^2 - \\frac{1}{x} + \\frac{5}{2} (\\hat{\\rho }^{2/3} - 1) = 0.",
"$ Using Eq.",
"(REF ) and multiplying by $2 x$ , we get $x u^2 + \\frac{5 \\lambda ^{2/3}}{(x u^2)^{1/3}} = 5 x + 2.$ The left-hand-side reaches a minimum at $x u^2 = (5/3)^{3/4} \\lambda ^{1/2}$ , with value $4(5/3)^{3/4} \\lambda ^{1/2}$ .",
"For a solution to exist for all $x$ , this has to be less than 2, the minimum of the right-hand-side, implyingThe difference of our maximum value of $\\lambda $ and the usually quoted value of 1/4 comes from our normalization of velocities with $v_{\\rm B}$ rather than the adiabatic sound speed at infinity, which is $(5/3)^{1/2} v_{\\rm B}$ .",
"$\\lambda \\le \\lambda _{\\rm ad} \\equiv \\frac{1}{4} (3/5)^{3/2}$ .",
"Though all solutions with sub-critical $\\lambda \\le \\lambda _{\\rm ad}$ are a priori acceptableThis is not the case for the Bondi problem with adiabatic index $< 5/3$ , for which sub-critical solutions have a velocity that tends to zero near the origin, which is unphysical., we shall assume, like Bondi, that the physically realized solution is that of maximum accretion, i.e.",
"that $\\lambda = \\lambda _{\\rm ad} \\equiv \\frac{1}{4} \\left(\\frac{3}{5}\\right)^{3/2} \\approx 0.12.$ Combining Eqs.",
"(REF ) and (REF ) one can show that the asymptotic behaviors of fluid variables for $x \\ll 1$ are $u(x) &\\approx & - \\frac{1}{\\sqrt{2}} x^{-1/2},\\\\\\hat{\\rho }(x) &\\approx & \\left(\\frac{3}{10}\\right)^{3/2} x^{-3/2},\\\\\\hat{T}(x) &\\approx & \\frac{3}{10} x^{-1}.",
"$"
],
[
"Solution for $\\beta \\ll 1$ and {{formula:70388956-f9ba-4d99-b5e8-d313742a0048}}",
"If $\\gamma \\gg 1$ Compton cooling efficiently maintains $\\hat{T} \\approx 1$ down to $x \\sim \\gamma ^{-2/3} \\ll 1$ .",
"At that point pressure forces are negligible relative to gravity, and the temperature is no longer relevant to the other fluid variables.",
"We may therefore first solve the isothermal Bondi problem for the fluid variables, and deduce the temperature profile from them.",
"For the isothermal Bondi problem with $\\hat{T} = 1$ , the conserved quantity is now $\\frac{1}{2} u^2 - \\frac{1}{x} + \\ln (\\hat{\\rho }) = 0.$ Here again one can show that there exists a maximum value of $\\lambda $ for which the problem has a solution.",
"For sub-critical $\\lambda $ , however, the velocity tends to zero towards the origin and the density diverges unphysically as $\\textrm {e}^{1/x}$ .",
"Therefore the physically valid solution is that with the critical accretion rate $\\lambda = \\lambda _{\\rm iso} \\equiv \\frac{1}{4} \\textrm {e}^{3/2} \\approx 1.12.$ It is sensible that the accretion rate is larger in the isothermal case than in the adiabatic case.",
"Indeed, the temperature is larger in the adiabatic case, providing a larger pressure support counterbalancing gravity.",
"For $x \\ll 1$ the velocity reaches the free-fall solution $u(x) \\approx - \\sqrt{2/x}$ and the density is then $\\hat{\\rho }(x) \\propto x^{-3/2}$ .",
"Inserting these asymptotic forms into the heat equation, we get $\\frac{\\sqrt{2}}{x^{3/2}} \\frac{d}{dx}(x \\hat{T}) = \\gamma (\\hat{T}-1).$ One can write an explicit integral solution to this equation.",
"In particular we find the asymptotic limit for $x \\ll \\gamma ^{-2/3}$ , $\\hat{T}(x) \\approx \\left(\\frac{4}{3}\\right)^{1/3} \\frac{\\Gamma (2/3) }{\\gamma ^{2/3}x} \\approx \\frac{1.5}{\\gamma ^{2/3} x}, $ where $\\Gamma $ is Euler's Gamma function."
],
[
"Solution for $\\beta \\ll 1$ and arbitrary {{formula:e55e9f52-a42a-43a6-9ac7-81d475a6b414}}",
"For arbitrary values of $\\gamma $ (while $\\beta \\ll 1$ ) the momentum equation can no longer be rewritten as a conservation equation and one must solve explicitly the coupled fluid and heat equations, and determine the accretion “eigenvalue\" $\\lambda $ numerically.",
"We re-write the system ()-() in the form $\\left(u - \\frac{5}{3} \\frac{\\hat{T}}{u}\\right) \\frac{du}{dx} &=& \\frac{10}{3} \\frac{\\hat{T}}{x} - \\frac{1}{x^2} - \\gamma \\frac{1-\\hat{T}}{u}, \\\\\\frac{d\\hat{T}}{dx} &=& - \\frac{2}{3} \\frac{\\hat{T}}{u} \\frac{d u}{dx} - \\frac{4}{3} \\frac{\\hat{T}}{x} + \\gamma \\frac{1-\\hat{T}}{u}.$ The boundary conditions at large radii are $u(x) = -\\lambda /x^2, \\hat{T}(x) = 1$ .",
"We see that the system is singular at the point $x_*$ where the velocity reaches the local adiabatic sound speed, $u_* =-\\sqrt{ 5 \\hat{T}_*/3}, $ unless this condition is met simultaneously with $\\frac{10}{3} \\frac{\\hat{T}_*}{x_*} - \\frac{1}{x_*^2} - \\gamma \\frac{1-\\hat{T}_*}{u_*} = 0, $ so that the right-hand-side of Eq.",
"(REF ) vanishes, leading to a finite derivative.",
"There is a single value $\\lambda _*$ for which these two conditions are satisfied simultaneously: larger $\\lambda $ lead to a singularity while for lower values $du/dx$ changes sign before the singularity, and the velocity unphysically tends to zero at the origin.",
"We find $\\lambda _*$ by bisection: starting with $\\lambda _{\\rm min} \\equiv 0 < \\lambda _* < \\lambda _{\\rm max} \\equiv 2$ , we set $\\lambda = (\\lambda _{\\rm min} + \\lambda _{\\max })/2$ and integrate the system numerically from $x = 100$ towards $x=0$ , until either the singularity or $du/dx = 0$ is reached.",
"In the former case, we set $\\lambda _{\\max } = \\lambda $ at the next step, and in the latter case, we set $\\lambda _{\\min } = \\lambda $ , so that $\\lambda _{\\min } < \\lambda _* < \\lambda _{\\max }$ at each step.",
"We do so until the fractional difference between $\\lambda _{\\max }$ and $\\lambda _{\\min }$ is less than a small error tolerance, typically $10^{-6}$ .",
"We show the resulting function $\\lambda (\\gamma )$ in Fig.",
"REF .",
"We find that the following analytic expression is a good fit to the numerical results: $\\lambda (\\gamma ; \\beta \\ll 1) \\approx \\lambda _{\\rm ad} + (\\lambda _{\\rm iso} - \\lambda _{\\rm ad}) \\left(\\frac{\\gamma ^2}{88 + \\gamma ^2}\\right)^{0.22}.$ Figure: Dimensionless accretion rate λ\\lambda as a function of the dimensionless Compton cooling rate γ\\gamma .",
"Black circles are our numerical results and the purple line is our analytic fit, Eq.",
"().While it is relatively simple to obtain a very precise value of $\\lambda $ numerically, obtaining a precise asymptotic limit of $\\hat{T}(x)$ at $x \\rightarrow 0$ proved to be more challenging.",
"Keeping in mind that this calculation is an order-of-magnitude estimate, we simply assume the following expression, interpolating between the adiabatic case (REF ) and the quasi-isothermal case (REF ): $\\tau \\equiv \\underset{x \\rightarrow 0}{\\lim }(x \\hat{T}) \\approx \\frac{1.5}{5 + \\gamma ^{2/3}}.",
"$ Inserting $T \\approx \\tau /x$ , $u \\approx - \\omega /x^{1/2}$ , $\\rho \\propto x^{-3/2}$ in the momentum equation (), we find $\\omega = \\sqrt{2 - 5 \\tau }$ .",
"In summary, the asymptotic values of the temperature, velocity and density fields are $\\hat{T}(x) &\\approx & \\frac{\\tau }{x}, \\\\u(x) &\\approx & -\\sqrt{\\frac{2 - 5 \\tau }{x}}, \\\\\\hat{\\rho }(x) &\\approx & \\frac{\\lambda }{\\sqrt{2 - 5 \\tau }} x^{-3/2}.$"
],
[
"Solution for $1 \\lesssim \\beta \\ll \\gamma $",
"When Compton drag is significant ($\\beta \\gtrsim 1$ ), there is no longer any conserved quantity, even in the quasi-isothermal case.",
"We can simply determine the asymptotic value of $\\lambda $ for $\\beta \\gg 1$ by considering the momentum equation at $x \\ll 1$ , where the pressure force is negligible with respect to gravity.",
"In this regime we find $u \\approx - 1/(\\beta x^2)$ , implying that $\\lambda \\rightarrow \\beta ^{-1}$ for large $\\beta $ .",
"Physically, the drag force balances the gravitational force, i.e.",
"the velocity reaches the terminal velocity.",
"Once $x \\lesssim \\beta ^{-2/3} \\gg \\gamma ^{-2/3}$ , the advection term $u (du/dx)$ becomes dominant over the drag term $-\\beta u$ and the velocity reaches the free-fall solution $u \\approx -\\sqrt{2/x}$ .",
"Since this occurs at a radius much larger than $\\gamma ^{-2/3}$ , the asymptotic behavior or $\\hat{T}$ , is still given by Eqs.",
"(REF ) and (REF ).",
"The effect of Compton drag is therefore only to change the accretion rate.",
"Ref.",
"[26] find the following analytic approximation for $\\lambda (\\beta )$ , valid for all values of $\\beta $ (but for $\\gamma \\gg 1$ only, as they consider isothermal accretion): $\\lambda (\\gamma \\gg 1; \\beta ) \\approx \\exp \\left[ \\frac{9/2}{3 + \\beta ^{3/4}}\\right] \\frac{1}{(\\sqrt{1 + \\beta }+1)^2}.$ For general $\\gamma $ and $\\beta $ we may use the following approximation for the dimensionless accretion rate: $\\lambda (\\gamma , \\beta ) = \\frac{\\lambda (\\gamma ; \\beta \\ll 1) \\lambda (\\gamma \\gg 1; \\beta )}{\\lambda _{\\rm iso}}.",
"$ This approximation is well justified since $\\beta \\ll \\gamma $ .",
"As a consequence, either $\\beta \\ll 1$ or $\\gamma \\gg 1$ .",
"The dimensionless accretion rate $\\lambda $ is the first main result of this Section.",
"We show its evolution as a function of redshift for several PBH masses in Fig.",
"REF .",
"While ROM do account for Compton drag following the analysis of Ref.",
"[26], they implicitly assume that $\\gamma \\gg 1$ at all times.",
"In other words, they do not account for the factor of $\\sim 10$ decrease of $\\lambda $ at low redshift when Compton cooling becomes negligible and the accretion becomes mostly adiabatic.",
"Figure REF also shows the evolution of the accretion rate normalized to the Eddington rate, $\\dot{m} \\equiv \\dot{M} c^2/L_{\\rm Edd}$ .",
"Figure: Characteristic dimensionless accretion rate λ\\lambda (upper panel) and accretion rate normalized to the Eddington value m ˙≡M ˙c 2 /L Edd \\dot{m} \\equiv \\dot{M} c^2/L_{\\rm Edd} (lower panel) as a function of redshift, for PBH masses 1,10 2 1, 10^2 and 10 4 M ⊙ 10^4\\, M_{\\odot }.",
"These quantities are evaluated with substitution v B →v eff v_{\\rm B} \\rightarrow v_{\\rm eff} as described in Section ."
],
[
"Collisional ionization region",
"If the emerging radiation field is too weak to photoionize the gas, it eventually gets collisionally ionized as it is compressed and heated up.",
"We assume that this proceeds roughly at constant temperature $T \\approx T_{\\rm ion}\\approx 1.5 \\times 10^4$ .",
"Indeed, if ionization proceeds through collisional ionizations balanced by radiative recombinations, the equilibrium ionization fraction only depends on temperature, with a sharp transition at $ T \\approx 1.5 \\times 10^4$ K (for instance, using Eq.",
"(2) or Ref.",
"[30], we get $x_e = (0.01, 0.5, 0.99)$ at $T = (1.1, 1.5, 2.5) \\times 10^4$ K, respectively).",
"Getting back to dimensionful variables, we found in the previous section that at small radii, $T(r) \\approx \\tau T_{\\infty } \\frac{r_{\\rm B}}{r}, $ where $\\tau $ is a dimensionless constant at most equal to 3/10, and smaller when Compton cooling is important.",
"The effect of the ionization region is only relevant once the global free-electron fraction $\\overline{x}_e$ falls significantly below unity, i.e.",
"for $T_{\\infty } \\lesssim 3000$ K $\\ll T_{\\rm ion}$ .",
"Therefore we expect the ionization region to be reached deep inside the Bondi radius, where the asymptotic behavior (REF ) is accurate.",
"The ionization region therefore starts at radius $r_{\\rm ion}^{\\rm start} \\approx \\tau \\frac{T_{\\infty }}{T_{\\rm ion}} r_{\\rm B}, $ where the density is, from Eq.",
"() $\\rho _{\\rm ion}^{\\rm start} \\approx \\frac{\\lambda }{\\sqrt{2 - 5 \\tau }} \\rho _{\\infty }\\left(\\frac{T_{\\rm ion}}{\\tau T_{\\infty }}\\right)^{3/2}.",
"$ We assume that ionization proceeds through collisions of neutral hydrogen atoms with free electrons.",
"This process only redistributes the internal energy of the gas, i.e.",
"does not generate any net heat.",
"However it does lead to a temperature decrease as the number of free particles increase and some of their energy is used to ionize the gas.",
"If the temperature is to remain constant through the ionization region, this effect must be compensated by the temperature increase due to the adiabatic compression of the gas.",
"In equations, we write the first law of thermodynamics: $\\Delta \\left(\\frac{3}{2} (1+ x_e) T - (1 - x_e) E_{\\rm I}\\right) = - (1 + x_e) T \\rho \\Delta (1/\\rho ),~~~~~$ where the second term in the internal energy on the left-hand-side accounts for the binding energy $E_{\\rm I} = 13.6$ eV of neutral hydrogen atoms.",
"Assuming the temperature remains constant throughout the ionization region we arrive at the simple relation between changes in density and ionization fraction: $\\Delta \\ln \\rho = \\left(\\frac{3}{2} + \\frac{E_{\\rm I}}{T_{\\rm ion}}\\right) \\Delta \\ln (1 + x_e).$ Therefore the ratio of the density at the end of the ionization region to that at its beginning is $\\frac{\\rho _{\\rm ion}^{\\rm end}}{\\rho _{\\rm ion}^{\\rm start}} = \\left(\\frac{2}{1 + \\overline{x}_e}\\right)^{\\frac{3}{2} + \\frac{E_{\\rm I}}{T_{\\rm ion}}} \\approx \\left(\\frac{2}{1 + \\overline{x}_e}\\right)^{12}, $ where we took $T_{\\rm ion} \\approx 1.5 \\times 10^4$ K. Assuming that $\\rho \\propto r^{-3/2}$ throughout the region, we get $\\frac{r_{\\rm ion}^{\\rm end}}{r_{\\rm ion}^{\\rm start}} \\approx \\left(\\frac{1 + \\overline{x}_e}{2}\\right)^{8} .$ We see that the ionization region may extend by a factor of $\\sim 300$ in radius if $\\overline{x}_e \\ll 1$ .",
"This is consistent with Shapiro's results [25], who finds an ionization region extending over a factor $\\sim 10^3$ in radius for accretion from a neutral gas.",
"Note that we have neglected the heat loss due to collisional excitations followed by radiative decays, as they cannot be simply included in our basic treatment.",
"We have also neglected Compton cooling by CMB photons, which may become relevant again once the ionization fraction increases.",
"Accounting for these cooling mechanisms would imply a larger density contrast $\\rho ^{\\rm end}/\\rho ^{\\rm start}$ hence a more extended ionization region, and an overall larger suppression of the temperature near the PBH horizon.",
"If the radiation field from the accreting PBH is intense enough, it may photoionize the gas beyond $r_{\\rm ion}$ , in which case there is no collisional ionization region.",
"To group both cases we define $\\rho _{\\rm ion}^{\\rm end} =\\rho _{\\rm ion}^{\\rm start}$ in that case, so that in general $\\frac{\\rho _{\\rm ion}^{\\rm end}}{\\rho _{\\rm ion}^{\\rm start}} = \\chi \\equiv {\\left\\lbrace \\begin{array}{ll}\\left(\\frac{2}{1 + \\overline{x}_e}\\right)^8 &\\textrm {(collisional ionization)}, \\\\1 &\\textrm {(photoionization)}.",
"\\end{array}\\right.",
"}$"
],
[
"Innermost adiabatic region",
"Once the gas is fully ionized, it resumes adiabatic compression (we justify in Appendix REF that free-free cooling can neglected for the mass range considered).",
"The thermal energy density of the ionized plasma is $u = \\frac{3}{2} n_e \\left(1 + f(T/m_e c^2)\\right) T,$ where the dimensionless function $f$ accounts for the fact that electrons are potentially relativistic, and has asymptotic limits $f(X \\ll 1) = 1$ and $f(X \\gg 1) = 2$ .",
"The pressure remains unchanged $P = 2n_e T$ , and the first law of thermodynamics can then be written $\\frac{3}{2} \\left[1 + f(X) + X f^{\\prime }(X)\\right] \\frac{dT}{T} = 2 \\frac{d \\rho }{\\rho }, \\ \\ \\ \\ X \\equiv \\frac{T}{m_e c^2}.$ This can be integrated to give $\\frac{\\rho _2}{\\rho _1} = \\frac{3}{4} \\int _{X_1}^{X_2} \\left[1 + f(X) + X f^{\\prime }(X)\\right] \\frac{dX}{X}.",
"$ We have computed the function $f$ explicitly and find that it is well approximated by the simple functional form $f(X) \\approx 1 + \\frac{X}{X+0.73}.",
"$ With this simple analytic form Eq.",
"(REF ) can be integrated analytically to obtain $\\rho _2(T_2)$ .",
"We invert this relation numerically and obtain the following approximation, valid for $T_1 \\ll m_e c^2$ and arbitrary $T_2$ : $\\frac{T_2}{m_e c^2} &\\approx & \\mathcal {F}\\left(\\frac{T_1}{m_e c^2} \\left(\\frac{\\rho _2}{\\rho _1}\\right)^{2/3}\\right),\\\\\\mathcal {F}(Y) &\\equiv & Y \\left(1 + \\frac{Y}{0.27}\\right)^{-1/3}.",
"$ This recovers the expected asymptotic behaviors $T \\propto \\rho ^{2/3}$ for $T \\lesssim m_e c^2$ and $T \\propto \\rho ^{4/9}$ for $T \\gtrsim m_e c^2$ and moreover gives an accurate result for arbitrary temperatures.",
"We may now finally compute the gas temperature near the Schwarzschild radius $r_{\\rm S}$ .",
"The velocity there nears the speed of light, $|v| \\approx c$ , so the density is $\\rho _{\\rm S} &=& \\frac{\\lambda }{(c/v_{\\rm B}) (r_{\\rm S}/r_{\\rm B})^2} \\rho _{\\infty }= \\frac{\\lambda }{4 (v_{\\rm B}/c)^3} \\rho _{\\infty } \\nonumber \\\\&=& \\frac{\\lambda }{4} \\left(\\frac{m_p c^2}{(1+ \\overline{x}_e) T_{\\infty }}\\right)^{3/2} \\rho _{\\infty }.$ At the end of the ionization region, the temperature is $T_{\\rm ion}$ and, from Eqs.",
"(REF ) and (REF ), the density is $\\rho _{\\rm ion}^{\\rm end} = \\chi \\frac{\\lambda }{\\sqrt{2 - 5 \\tau }} \\left(\\frac{T_{\\rm ion}}{\\tau T_{\\infty }}\\right)^{3/2} \\rho _{\\infty },$ Using Eq.",
"(REF ) with $T_1 = T_{\\rm ion}$ and $\\rho _1 = \\rho _{\\rm ion}^{\\rm end}$ , we finally obtain the temperature $T_{\\rm S}$ at the Schwarzschild radius: $T_{\\rm S} = m_e c^2 \\mathcal {F}(Y_{\\rm S}),$ where $\\mathcal {F}$ is given by Eq.",
"(REF ) and $Y_{\\rm S} &\\equiv & \\frac{T_{\\rm ion}}{m_e c^2} \\left(\\frac{\\rho _{\\rm S}}{\\rho _{\\rm ion}^{\\rm end}}\\right)^{2/3}\\nonumber \\\\&=& \\chi ^{-2/3} \\left(\\frac{2}{1 + \\overline{x}_e}\\right) \\frac{\\tau }{4} \\left(1 - \\frac{5}{2} \\tau \\right)^{1/3}\\frac{m_p}{m_e}.",
"$ It is interesting to compare this result to those of Shapiro [25], who did not consider Compton cooling (i.e.",
"$\\tau = 3/10$ ), assumed that photoionizations are negligible (in which case we have $\\chi ^{-2/3} \\approx [(1+ \\overline{x}_e)/2]^8$ ), and only studied the cases $\\overline{x}_e = 1$ or 0.",
"In the former case, we find $Y_{\\rm S} \\approx \\frac{3}{40} 4^{-1/3} \\frac{m_p}{m_e} \\approx 10^2 \\gg 1$ and as a result electrons are relativistic at the Schwarzschild radius, with temperature $T_{\\rm S} &\\approx & m_e c^2 0.27^{1/3} Y_{\\rm S}^{2/3} \\approx 0.08 (m_p c^2)^{2/3} (m_e c^2)^{1/3} \\nonumber \\\\&\\approx & 0.7 \\times 10^{11} \\textrm {K},$ in excellent agreement with Shapiro's result (see also [23]).",
"In the case of a neutral background, taking $T_{\\rm ion} = 1.5 \\times 10^4$ K, $Y_{\\rm S}$ is a factor $\\sim 2^{-7}$ smaller, i.e.",
"$Y_{\\rm S} \\approx 0.7$ , so electrons are marginally relativistic at the horizon, with $T_{\\rm S} \\approx 0.4~ m_e c^2 \\approx 2.5 \\times 10^9$ K. This is a factor of $\\sim 2$ higher than Shapiro's result, consistent with our neglect of collisional excitations in the ionization region.",
"Equations (REF ), (REF ), (REF ) and (REF ) constitute the second main result of this Section.",
"They give the gas temperature near the BH horizon, accounting for Compton cooling and an arbitrary background ionization fraction, in the two limiting cases of collisional ionization or photoionization.",
"We show the temperature $T_{\\rm S}$ as a function of redshift and PBH mass in Fig.",
"REF .",
"At high redshift, the temperature is suppressed by the strong Compton cooling.",
"In the collisional ionization case, once the Universe becomes neutral, some thermal energy is used in ionizing the gas, lowering $T_{\\rm S}$ by a factor up to $\\sim 300$ , corresponding to the radial extent of the ionization region.",
"Figure: Characteristic temperature of the accreting gas near the Schwarzschild radius, evaluated with the substitution v B →v eff v_{\\rm B} \\rightarrow v_{\\rm eff} as described in Section ."
],
[
"Luminosity of an accreting black hole",
"The luminosity of the accreting BH arises mostly from Bremsstrahlung (free-free) radiation near the Schwarzschild radius.",
"We show in Appendix REF that free-bound radiation is negligible with respect to free-free radiation.",
"The frequency-integrated emissivity (in ergs/s/cm$^3$ ) of a fully-ionized thermal electron-proton plasma can be written in the general form (see e.g. Ref.",
"[31]) $j^{\\rm ff} = n_e^2 ~\\alpha c \\sigma _{\\rm T} T ~ \\mathcal {J}(T/m_e c^2), $ where $\\alpha $ is the fine-structure constant and $\\mathcal {J}(X)$ is a dimensionless function.",
"Ref.",
"[32] provide a simple fitting formula for the $e-p$ free-free emissivity, accurate to a few percent, and Ref.",
"[33] provide a sub-percent accuracy code for the $e-e$ free-free emissivity.",
"We fit the sum of the two within a few percent by the following analytic approximation, generalizing that of Ref.",
"[32]: $\\mathcal {J}(X) \\approx {\\left\\lbrace \\begin{array}{ll}\\frac{4}{\\pi } \\sqrt{2/\\pi } X^{-1/2} \\left(1 + 5.5 X^{1.25} \\right), & X < 1,\\\\[6pt]\\frac{27}{2 \\pi } \\left[\\ln (2 X \\textrm {e}^{- \\gamma _{\\rm E}} + 0.08) + \\frac{4}{3}\\right], & X > 1,~~~~~\\end{array}\\right.",
"}$ where $\\gamma _{\\rm E} \\approx 0.577$ is Euler's Gamma constant.",
"Assuming the plasma is optically thin (which we show explicitly in Appendix REF ), the luminosity is then obtained by integrating the emissivity over volume, $L = \\int 4 \\pi r^2 dr j$ .",
"Let us note that this purely Newtonian expression does not properly account for relativistic effects which become relevant near the horizon [25]; they results in order-unity corrections which are below our theoretical uncertainty.",
"Near the Schwarzschild radius the gas is in free-fall, $|v| \\approx c \\sqrt{r_{\\rm S}/r}$ , and the electron density results from the mass-conservation equation: $n_e = \\frac{\\dot{M}}{4 \\pi m_p r^2 |v|} = \\frac{\\dot{M}}{4 \\pi m_p r_{\\rm S}^2 c} (r/r_{\\rm S})^{-3/2}.",
"$ The radial dependence of the temperature near the horizon depends on $T/m_e c^2$ .",
"For the range of temperature considered we find that $ 0.8 \\lesssim - d \\ln (T \\mathcal {J})/d \\ln r \\lesssim 1.1$ .",
"Approximating $T \\mathcal {J}(T) \\propto r^{-1}$ , we therefore get $L \\approx \\alpha \\frac{T_{\\rm S} }{m_p c^2}\\mathcal {J}(T_{\\rm S}) \\frac{\\dot{M} c^2}{L_{\\rm Edd}} \\dot{M} c^2, $ where we recall that the Eddington luminosity is $L_{\\rm Edd} \\equiv \\frac{4 \\pi G M m_p c}{\\sigma _{\\rm T}}.",
"$ With this we see that the radiative efficiency $\\epsilon \\equiv L/\\dot{M} c^2$ is proportional to $\\dot{m} \\equiv \\dot{M}c^2/L_{\\rm Edd}$ , with $\\frac{\\epsilon }{\\dot{m}} \\approx \\alpha \\frac{T_{\\rm S} }{m_p c^2}\\mathcal {J}(T_{\\rm S}).$ The highest temperature, hence efficiency, is achieved when Compton cooling is negligible and the background is fully ionized, in which case we find $T_{\\rm S} \\approx 10^{11}$ K and $\\epsilon /\\dot{m} \\approx 0.0015$ .",
"This is nearly one order of magnitude below the value $\\epsilon /\\dot{m} = 0.011$ assumed in ROM, and is further suppressed at most times, as we show in Fig.",
"REF .",
"Figure: Radiative efficiency ϵ≡L/M ˙c 2 \\epsilon \\equiv L/\\dot{M} c^2 divided by the dimensionless accretion rate m ˙≡M ˙c 2 /L Edd \\dot{m} \\equiv \\dot{M} c^2/L_{\\rm Edd}, evaluated with the substitution v B →v eff v_{\\rm B} \\rightarrow v_{\\rm eff} as described in Section ."
],
[
"Accounting for BH velocities",
"All of the calculations so far assume perfectly spherically-symmetric accretion.",
"In practice, the accreting PBHs are moving with respect to the ambient gas with some velocity $v$ .",
"It is not at all clear what the best way is to account for the black hole peculiar velocity without performing a full time-dependent hydrodynamical simulation.",
"Bondi and Hoyle [34] studied analytically accretion on a point mass moving highly supersonically, and found $\\dot{M} \\approx 2.5 \\pi (G M)^2 \\rho _{\\infty }/v^3$ .",
"Inspired by this result, Bondi [28] suggested substituting the sound speed at infinity $c_s$ by $\\sqrt{c_s^2 + v^2}$ in the accretion rate, which, he argued, ought to give the correct order of magnitude for the result.",
"Though this provides a prescription for the accretion rate, it is not clear how to self-consistently account for relative velocities in the estimate of the gas temperature.",
"For definiteness, and lacking a better theory, we shall approximate the effect of relative velocities by substituting $v_{\\rm B}^2 \\rightarrow v_{\\rm B}^2 + v^2$ throughout the calculation.",
"This is equivalent to substituting $T_{\\infty } \\rightarrow T_{\\infty } + m_p v^2/(1 + \\overline{x}_e)$ .",
"The same route was followed in ROM.",
"The relative velocity $v$ is comprised of two pieces: a Gaussian linear contribution on large scales, $v_{\\rm L}$ , whose power spectrum and variance can be extracted from linear Boltzmann codes, and a small-scale contribution due to non-linear clustering of PBHs, $v_{\\rm NL}$ .",
"We shall not consider the latter here, but point out that it would further suppress the effect of PBHs on the CMB.",
"If PBHs make up the dark matter, the linear velocity $v_{\\rm L}$ is nothing but the relative velocity of baryons and dark matter.",
"After kinematic decoupling at $z \\approx 10^3$ , dark matter and baryons fall in the same gravitational potentials on scales larger than the baryon Jeans scale and hence $v_{\\rm L} \\propto 1/a$ , independent of scale [35].",
"Before then, however, the relative velocity has a more complex time and scale-dependence since baryons undergo acoustic oscillations while the dark-matter overdensities grow.",
"Ref.",
"[36] explicitly compute $\\langle v_{\\rm L}^2 \\rangle $ as a function of time and find that it is mostly constant for $z \\gtrsim 10^3$ (see their Fig. 1).",
"Since, as we shall see, most of the effect of accreting PBHs on the CMB takes place after decoupling, we need not have a very precise estimate of $v_{\\rm L}$ before then, and assume the following simple redshift dependence: $\\langle v_{\\rm L}^2 \\rangle ^{1/2} \\approx \\min \\left[1, z/10^3\\right] \\times 30 ~\\textrm {km/s}.",
"$ Let us point out that the relative velocity adopted in ROM is quite different from what we use here (see their Fig.",
"2); in particular they under-estimate it for $z \\gtrsim 200$ , leading to an over-estimate of the accretion rate.",
"As we saw in Section REF , the BH luminosity is quadratic in the accretion rate, and therefore, in the standard Bondi case, proportional to $(v_{\\rm B}^2 + v_{\\rm L}^2)^{-3}$ .",
"The total energy injected in the plasma is obtained by averaging the BH luminosity over the Gaussian distribution of relative velocities.",
"We defineThis is equivalent to the quantity $\\langle v_{\\rm eff} \\rangle _A$ in ROM.",
"$v_{\\rm eff} \\equiv \\langle ( v_{\\rm B}^2 + v_{\\rm L}^2)^{-3} \\rangle ^{-1/6}$ .",
"It has the following approximate limits: $v_{\\rm eff} \\approx {\\left\\lbrace \\begin{array}{ll} \\sqrt{v_{\\rm B} \\langle v_{\\rm L}^2 \\rangle ^{1/2}}, \\ \\ &v_{\\rm B} \\ll \\langle v_{\\rm L}^2 \\rangle ^{1/2}\\\\v_{\\rm B}, \\ \\ &v_{\\rm B} \\gg \\langle v_{\\rm L}^2 \\rangle ^{1/2}\\end{array}\\right.}",
"$ We show $v_{\\rm B}$ , $\\langle v_{\\rm L}^2 \\rangle ^{1/2}$ and $v_{\\rm eff}$ in Fig.",
"REF .",
"Figures REF , REF , REF and REF where all obtained by setting $v_{\\rm B} \\rightarrow v_{\\rm eff}$ , in order to illustrate the characteristic accretion rate and radiative efficiency.",
"The final result of this Section is the luminosity of accreting PBHs, averaged over the distribution of relative velocities, which we show in Fig.",
"REF .",
"We emphasize that to obtain $\\langle L \\rangle $ , we have replaced $v_{\\rm B} \\rightarrow \\sqrt{v_{\\rm B}^2 + v_{\\rm L}^2}$ throughought the calculation, and then averaged the luminosity over the three-dimensional Gaussian distribution of ${v}_{\\rm L}$ .",
"Figure: Characteristic velocities in the problem at hand: the isothermal sound speed v B v_{\\rm B} (dotted), rms BH-baryon relative velocity 〈v L 2 〉 1/2 \\langle v_{\\rm L}^2\\rangle ^{1/2} (dashed) and effective velocity v eff v_{\\rm eff} defined in Eq.",
"() (solid), used in Figures , , and to illustrate characteristic values of intermediate quantities.Figure: Luminosity of accreting PBHs as a function of redshift, averaged over the Gaussian distribution of large-scale relative velocities."
],
[
"Local radiation feedback",
"Before estimating the effect of the PBH radiation on the global thermal and ionization history, let us first examine whether it can affect the local accretion flow itself."
],
[
"Local thermal feedback",
"Throughout the calculation we have neglected local Compton heating by the radiation produced by the accreting PBH.",
"Here we discuss the validity of this assumption.",
"The rate of energy injection per electron by Compton scattering with the PBH radiation is $\\int dE \\frac{1}{4 \\pi r^2} \\frac{1}{E} \\frac{d L}{d E} \\langle \\sigma \\Delta E \\rangle \\approx 0.1~ \\frac{\\sigma _{\\rm T} L}{4 \\pi r^2},$ where we used the approximation (REF ) for $\\langle \\sigma \\Delta E\\rangle $ .",
"Hence the rate of Compton heating by the PBH radiation is $\\dot{T}_{\\textrm {Compt}, L} \\approx \\frac{2}{3} \\frac{x_e}{1 + x_e} 0.1~ \\frac{\\sigma _{\\rm T} L}{4 \\pi r^2}.$ We need to compare this rate to the largest of the Compton cooling rate by CMB photons and the rate of adiabatic heating: $\\dot{T}_{\\rm Compt, cmb} &\\equiv & \\frac{8}{3} \\frac{x_e}{1 + x_e} \\sigma _{\\rm T} \\frac{\\rho _{\\rm cmb} T_{\\rm cmb}}{m_e c},\\\\\\dot{T}_{\\rm ad} &\\approx & T \\frac{|v|}{r}.$ If $\\gamma \\gg 1$ the latter two rates are approximately equal at $r_* \\approx \\gamma ^{-2/3} r_{\\rm B}$ , adiabatic heating being dominant for $r \\lesssim r_*$ and Compton cooling by CMB photons for $r \\gtrsim r_*$ (see Section REF ).",
"For $r < r_*$ , $T \\propto 1/r$ and $|v| \\propto 1/r^{1/2}$ so $\\dot{T}_{\\textrm {Compt}, L}/\\dot{T}_{\\rm ad} \\propto r^{1/2}$ .",
"For $r > r_*$ , $\\dot{T}_{\\textrm {Compt}, L}/\\dot{T}_{\\rm Compt, cmb} \\propto r^{-2}$ .",
"Therefore the impact of thermal feedback is maximized at $r \\approx r_*$ .",
"If $\\gamma \\ll 1$ , then we only need to compare the Compton heating rate to adiabatic cooling, at the Bondi radius where this ratio is maximized.",
"We see that for arbitrary $\\gamma $ the relevant radius at which to compare Compton heating to adiabatic cooling is $r \\approx r_{\\rm B}/(1+ \\gamma ^{2/3})$ , where $T \\approx T_{\\rm cmb}$ in both cases.",
"After some algebra we arrive at $\\max \\left[\\frac{\\dot{T}_{\\textrm {Compt}, L}}{\\dot{T}}\\right] \\approx 0.07 \\frac{x_e}{1 + x_e} \\frac{L}{L_{\\rm Edd}} \\frac{v_{\\rm B}}{c} \\frac{m_p c^2}{T_{\\rm cmb}} \\sqrt{1 + \\gamma ^{2/3}}.$ We show this ratio in Fig.",
"REF , where we see that it is always less than unity for $M \\le 10^4 M_{\\odot }$ .",
"We can therefore safely neglect local thermal feedback for the mass range we consider.",
"Figure: Estimated maximum fractional importance of local thermal feedback from Compton heating by the PBH radiation."
],
[
"Local ionization feedback",
"Througout the paper we have computed all relevant quantities in both the “collisional ionization\" and the “photoionization\" limits.",
"In the former case, we assumed that the radiation field from the accreting BH does not affect the ionization state of the gas in the immediate vicinity of the BH, so the gas gets eventually gets collisionally ionized, which reduces its temperature near the horizon.",
"In the latter case, we assumed that the neighboring gas is fully photoionized.",
"We now show that neither case is accurate and that within the adopted model, the level of feedback is expected to be somewhat intermediate between the two.",
"To do so, we estimate the extent of the photoionized region (the Strömgren sphere) around an accreting PBH, in the absence of collisional ionizations.",
"Following the standard derivation (see e.g.",
"Ref.",
"[37]), $\\int _0^{\\infty } 4 \\pi r^2 dr n_e n_p \\alpha _{\\rm B}(T) = \\int _{\\nu _0}^{\\infty } d \\nu \\frac{L_{\\nu }}{h \\nu },$ where $\\nu _0 = 13.6 \\, \\textrm {eV}/h$ is the ionization threshold and $\\alpha _{\\rm B}$ is the case-B recombination coefficient.",
"This equation states that the total rate of recombinations is equal to the emission rate of ionizing photons.",
"Note that it does not depend on the exact shape of the photoionization cross section (and in particular also accounts for ionizations by inelastic Compton scattering at high energies).",
"Now we assume that the gas is fully ionized up to a radius $R$ , after which it quickly becomes neutral.",
"We also approximate $\\alpha _{\\rm B}(T) \\propto T^{-q}$ .",
"Finally, in the free-fall limit, $n_e \\propto 1/r^{3/2}$ and $T \\propto 1/r$ , implying $\\alpha _{\\rm B}(T) = \\alpha _{\\rm B, \\rm ion} (r/r_{\\rm ion})^q,$ where $\\alpha _{\\rm B, \\rm ion} \\equiv \\alpha _{\\rm B}(T_{\\rm ion})$ and $r_{\\rm ion}$ is the radius at which $T = T_{\\rm ion} \\equiv 1.5 \\times 10^4$ K. Using Eq.",
"(REF ) we arrive at $\\int _0^{\\infty } 4 \\pi r^2 dr n_e n_p \\alpha _{\\rm B}(T) = \\frac{\\dot{M}^2 \\, \\alpha _{\\rm B, \\rm ion}}{4 \\pi (m_p c)^2 r_{\\rm S}} \\frac{(R/r_{\\rm ion})^q}{q}.$ To compute the number of ionizing photons, we assume an approximately flat spectrum $L_{\\nu } \\approx L/\\nu _{\\max }$ for $\\nu \\le \\nu _{\\max } \\equiv T_{\\rm S}/h$ , so that $\\int _{\\nu _0}^{\\infty } d \\nu \\frac{L_{\\nu }}{h \\nu } \\approx \\frac{L}{T_{\\rm S}} \\ln (T_{\\rm S}/h \\nu _0).$ Using Eq.",
"(REF ) for $L$ , we arrive at the following expression for the radius $R$ , that does not explicitly depend on the luminosity or the accretion rate, but does depend weakly on $T_{\\rm S}$ , the temperature at the horizon: $\\frac{R}{r_{\\rm ion}} \\approx \\left[\\frac{\\alpha c \\sigma _{\\rm T}}{\\alpha _{\\rm B, \\rm ion}}\\mathcal {J}(T_{\\rm S}) \\ln (T_{\\rm S}/h\\nu _0) \\right]^{1/q}.$ From Ref.",
"[38] we get $\\alpha _{\\rm B, \\rm ion} \\approx 1.8 \\times 10^{-13}$ cm$^{-3}$ s$^{-1}$ , with a local power law $q \\approx 0.86$ , hence $\\frac{R}{r_{\\rm ion}} \\approx 2 \\times 10^{-4} \\left[\\mathcal {J}(T_{\\rm S}) \\ln (T_{\\rm S}/h\\nu _0) \\right]^{1.16}.$ Let us now consider the two limiting regimes once the background ionization fraction drops significantly below unity.",
"Assuming the gas is photoionized by the radiation field rather than collisionally ionized, we found $T_{\\rm S} \\approx 10^{11}$ K at low redshift, implying $\\frac{R}{r_{\\rm ion}} \\approx 0.1 \\ \\ \\ (T_{\\rm S} = 10^{11} \\, \\rm K).$ Since this is less than unity, this implies that assuming ionizations proceed exclusively through photoionizations is not self-consistent, as the radiation from the BH cannot photoionize the gas all the way to $r_{\\rm ion}$ .",
"Let us notice that this implies a fortiori that the photoionized region does not extend to the outermost region where $T \\approx T_{\\rm cmb}$ , and that we are hence justified in assuming $x_e = \\overline{x}_e$ there.",
"If we instead take the “collisional ionization\" limit, for which $T_{\\rm S} \\approx 3 \\times 10^9$ K, at low redshift, we get $\\frac{R}{r_{\\rm ion}} \\approx 0.02 \\ \\ \\ (T_{\\rm S} = 3 \\times 10^{9} \\, \\rm K).$ This radius is larger than the innermost edge of the collisional ionization region, which we found to be $\\sim 0.003 \\, r_{\\rm ion}$ .",
"This implies that it is also not self-consistent to assume that the gas is exclusively collisionally ionized, as the photoionization region from the resulting radiation field would extend inside the collisional ionization region.",
"We therefore conclude that neither approximation is self-consistent, and that the actual luminosity (within our assumed spherical accretion model) is intermediate between these two limiting cases.",
"We now move on to compute the global effects of the the PBH luminosity on the background gas."
],
[
"Total energy deposition rate",
"Assuming PBHs make a fraction $f_{\\rm pbh}$ of the dark matter, the volumetric rate of energy injection (in ergs/cm$^3$ /s) by accreting PBHs is $\\dot{\\rho }_{\\rm inj} = f_{\\rm pbh} \\frac{\\rho _{\\rm dm}}{M} \\langle L\\rangle .$ This energy is injected in the form of a nearly flat photon spectrum (i.e.",
"the free-free luminosity per frequency interval $dL/d \\nu $ is approximately constant), up to maximum energy $E_{\\max } \\approx T_{\\rm S}$ , typically $\\sim 0.2$ MeV for $z \\lesssim 10^3$ and up to $\\sim 6$ MeV at higher redshifts.",
"What is relevant for cosmological observables is the volumetric rate of energy deposited in the plasma (in the form of heat or ionizations), which we denote by $\\dot{\\rho }_{\\rm dep}$ .",
"The two rates are not necessarily equal, unless energy is deposited on-the-spot.",
"At the characteristic energies considered, the dominant photon cooling process is inelastic Compton scattering off electrons, whether bound or free [39], [40].",
"In principle, in order to obtain the energy deposition rate one should solve for the time evolution of the photon distribution, as well as that of the secondary high-energy electrons resulting from Compton scattering.",
"To simplify matters we shall assume that the latter deposit their energy on-the-spot, so we only need to follow the photon distribution $\\mathcal {N}_E$ (in photons/cm$^3$ /erg).",
"The differential scattering cross section for Compton scattering is [40] $\\frac{d \\sigma (E)}{d E^{\\prime }} &=& \\frac{3}{8} \\sigma _{\\rm T} \\frac{m_e c^2}{E^2}\\nonumber \\\\&\\times & \\left[\\frac{E^{\\prime }}{E} + \\frac{E}{E^{\\prime }} - 1 + \\left(1 + \\frac{m_e c^2}{E} - \\frac{m_e c^2}{E^{\\prime }}\\right)^2 \\right], ~~~~$ where $E$ is the initial energy of the photon and $E^{\\prime }$ is its final energy, restricted to the range $E^{\\prime }_{\\min }(E) \\equiv \\frac{E}{1 + 2 E/m_e c^2} \\le E^{\\prime } \\le E. $ In principle the photon distribution $\\mathcal {N}_E$ should be obtained by solving an integro-differential Boltzmann equation.",
"To simplify, we approximate the Boltzmann equation by the continuity equationOne can also think of this equation as a Fokker-Planck equation without a diffusion term.",
"$a^{-2}\\frac{d}{d t}(a^2 \\mathcal {N}_E) \\approx \\frac{1}{E} \\frac{d \\dot{\\rho }_{\\rm inj}}{dE} +\\frac{\\partial }{\\partial E}\\left( \\dot{\\mathcal {E}}(E) \\mathcal {N}_E \\right), $ where $d/dt \\equiv \\partial /\\partial t - H E \\partial /\\partial E$ is the derivative along the photon geodesics and $\\dot{\\mathcal {E}}(E) \\equiv \\overline{n}_{\\rm H} c \\langle \\sigma \\Delta E \\rangle $ is the rate of energy loss due to Compton scattering, where $\\langle \\sigma \\Delta E \\rangle \\equiv \\int _{E^{\\prime }_{\\min }(E)}^E d E^{\\prime } \\frac{d \\sigma (E)}{dE^{\\prime }}(E - E^{\\prime }).$ The $a^2$ factors in Eq.",
"(REF ) ensure that $\\mathcal {N}_E \\propto a^{-2}$ in the absence of the source and collision terms, and the form of the differential operator for Compton scattering explicitly conserves the number of photons.",
"We show the ratio $\\langle \\sigma \\Delta E \\rangle / \\sigma _{\\rm T} E$ in Fig.",
"REF .",
"We see that for the range of energies considered, within a factor of 2 at most, $\\langle \\sigma \\Delta E \\rangle \\approx 0.1 ~ \\sigma _{\\rm T} E. $ The factor of 0.1 can be understod as follows.",
"For $E \\gtrsim m_e c^2$ , photons lose most of their energy in each scattering event, but the Compton cross-section is suppressed with respect to the Thomson limit.",
"For $E \\lesssim m_e c^2$ , the Compton cross section tends to the Thomson limit, but photons only lose a small fraction of their energy in each scattering event.",
"Within our set of approximations, the differential energy deposition rate is $\\frac{d \\dot{\\rho }_{\\rm dep}}{d E} \\approx \\dot{\\mathcal {E}}(E) \\mathcal {N}_E \\approx 0.1 ~\\overline{n}_{\\rm H} c \\sigma _{\\rm T} E \\mathcal {N}_E.$ From Eq.",
"(REF ) we find that this quantity satisfies the following equation $a^{-6}\\frac{d}{dt}\\left(a^6 \\frac{d \\dot{\\rho }_{\\rm dep}}{dE} \\right) &\\approx & 0.1~ \\overline{n}_{\\rm H} c \\sigma _{\\rm T} \\nonumber \\\\&\\times &\\left[ \\frac{d \\dot{\\rho }_{\\rm inj}}{dE} + E \\frac{\\partial }{\\partial E}\\left(\\frac{d \\dot{\\rho }_{\\rm dep}}{dE} \\right)\\right].~~~$ Integrating over energies (and recalling that $d/dt = \\partial /\\partial t - H E \\partial /\\partial E$ ), we arrive at the following very simple differential equation for the total energy deposition rate: $a^{-7} \\frac{d}{dt}(a^7 \\dot{\\rho }_{\\rm dep}) \\approx 0.1~ \\overline{n}_{\\rm H} c \\sigma _{\\rm T} (\\dot{\\rho }_{\\rm inj} - \\dot{\\rho }_{\\rm dep}).",
"$ We compare and contrast our results to existing analytic calculations in Appendix .",
"Physically, Eq.",
"(REF ) implies that $\\dot{\\rho }_{\\rm dep} \\approx \\dot{\\rho }_{\\rm inj}$ (i.e.",
"that the energy is deposited “on the spot\") as long as the Compton cooling timescale $(0.1 c \\sigma _{\\rm T} \\overline{n}_{\\rm H})^{-1}$ is much shorter than the characteristic timescale over which $\\dot{\\rho }_{\\rm inj}$ changes.",
"Once this is no longer the case, the deposited energy rapidly decays as $1/a^7$ .",
"The Compton cooling timescale becomes longer than the Hubble timescale at $z \\approx 200$ .",
"However $\\dot{\\rho }_{\\rm inj}$ can change on a timescale significantly shorter than a Hubble time, in particular around recombination (see Fig.",
"REF ), so $\\dot{\\rho }_{\\rm dep}$ may deviate from $\\dot{\\rho }_{\\rm inj}$ even earlier on.",
"We show the ratio $\\dot{\\rho }_{\\rm dep}/\\dot{\\rho }_{\\rm inj}$ as a function of redshift for a $10^2 \\, M_{\\odot }$ PBH in Fig.",
"REF .",
"Note that this is conceptually equivalent to the dimensionless efficiency $f(z)$ usually computed in the context of dark-matter annihilation (see e.g. Ref. [40]).",
"We see that this ratio goes to unity at $z \\gtrsim 10^3$ , and is suppressed for $z \\lesssim 300$ , as expected.",
"Interestingly, in the collisional ionization case, this ratio can actually be larger than unity around $z \\sim 10^3$ .",
"This is due to the sharp decrease of the PBH average luminosity at recombination for $M \\lesssim 10^2\\, M_{\\odot }$ (see Fig.",
"REF ), hence of the instantaneous injected energy, and the non-negligible time-delay between injection and deposition already present at that redshift.",
"Figure: Ratio of the cross-section-averaged energy loss per Compton scattering event to σ T E\\sigma _{\\rm T} E, as a function of photon energy.Figure: Ratio of the energy deposition rate to the instantaneous energy injection rate (equivalent of the dimensionless efficiency f(z)f(z) usually computed in the context of dark-matter annihilation), as a function of redshift.",
"We only show the case M=10 2 M ⊙ M = 10^2\\, M_{\\odot } as other cases are very similar."
],
[
"Effect on the thermal and ionization histories",
"To conclude this Section, we must describe how exactly the energy is deposited in the plasma.",
"We follow the simple prescription of Ref.",
"[39], assuming that for a neutral gas the deposited energy is equally split among heating, ionizations and excitations, and rescale these fractions for arbitrary ionization fractions.",
"We only consider the effect on hydrogen recombination for simplicity.",
"Specifically, we take the following prescriptions for the additional rates of change of gas temperature, direct ionizations and excitations: $\\Delta \\dot{T}_{\\rm gas} &=& \\frac{2}{3 n_{\\rm tot}} \\frac{1 + 2 x_e}{3} \\dot{\\rho }_{\\rm dep}, \\\\\\Delta \\dot{x}_e^{\\rm direct} &=& \\frac{1- x_e}{3} \\frac{\\dot{\\rho }_{\\rm dep}}{E_{\\rm I} n_{\\rm H}},\\\\\\Delta \\dot{x}_2 &=& \\frac{1- x_e}{3} \\frac{\\dot{\\rho }_{\\rm dep}}{E_{\\rm 2} n_{\\rm H}},$ where $n_{\\rm tot}$ is the total number density of free particles, $x_2$ is the fraction of excited hydrogen and $E_2 \\equiv 10.2$ eV is the first excitation energy (we assume that all excitations are to the first excited state for simplicity).",
"Note that in our previous notation $x_e \\equiv \\overline{x}_e$ is the background ionization fraction and similarly $T_{\\rm gas} \\equiv T_{\\infty }$ .",
"We implement these modifications in the recombination code hyrec [41], [42].",
"We self-consistently account for the heating of the gas into the PBH luminosity, i.e.",
"account for the global feedback of PBHs.",
"We show the resulting changes in the ionization history in Fig.",
"REF .",
"Comparing with Fig.",
"3 of ROM, we see that we obtain a significantly smaller effect on the ionization history.",
"Figure: Upper panel: global free electron fraction x e (z)x_e(z) in the standard scenario (lower black curve), and accounting for PBHs with parameters (M pbh /M ⊙ ,f pbh )=(10 2 ,1),(10 3 ,10 -2 ),(10 4 ,10 -4 )(M_{\\rm pbh}/M_{\\odot }, f_{\\rm pbh}) = (10^2, 1), (10^3, 10^{-2}), (10^4, 10^{-4}), in that order from bottom to top at low redshift.",
"Lower panel: change in the ionization history due to accreting PBHs for the same parameters.",
"We only show the collisional ionization case here."
],
[
"Effect of global heating",
"Energy deposited in the photon-baryon plasma at redshift $z \\lesssim 2\\times 10^6$ does not get fully thermalized, and results in distortions to the CMB spectrum.",
"Depending on when the energy is deposited, the distortion generated is either a chemical potential ($\\mu $ -type) or a Compton-$y$ distortion.",
"Their amplitudes are approximately given by (see e.g.",
"[43]) $\\mu &\\approx & 1.4 \\int _{5\\times 10^4}^{2 \\times 10^6} d \\ln (1 + z) \\frac{\\dot{\\rho }_{\\rm dep}^{\\rm heat}}{H \\rho _{\\rm cmb}},\\\\y &\\approx & \\frac{1}{4} \\int _{200}^{5 \\times 10^4} d \\ln (1 + z) \\frac{\\dot{\\rho }_{\\rm dep}^{\\rm heat}}{H \\rho _{\\rm cmb}}.",
"$ The relevant ratio is therefore that of the volumetric rate of heat deposition per Hubble time to the CMB photon energy density: $\\frac{\\dot{\\rho }_{\\rm dep}^{\\rm heat}}{H \\rho _{\\rm cmb}} &\\approx & \\frac{1 + 2 x_e}{3} \\frac{\\dot{\\rho }_{\\rm dep}}{\\dot{\\rho }_{\\rm inj}} f_{\\rm pbh} \\frac{\\rho _{\\rm dm}}{\\rho _{\\rm cmb}} \\frac{\\langle L \\rangle }{H M}\\nonumber \\\\&\\approx & 4\\times 10^{-4}~ \\frac{1 + 2 x_e}{3} \\frac{\\dot{\\rho }_{\\rm dep}}{\\dot{\\rho }_{\\rm inj}} f_{\\rm pbh} \\frac{\\langle L\\rangle }{L_{\\rm Edd}} \\frac{(z_{\\rm eq}/z)^3}{\\sqrt{1 + z_{\\rm eq}/z}},~~~~~$ where $z_{\\rm eq} \\approx 3400$ is the redshift of matter-radiation equality.",
"In the $\\mu $ -era $z \\gtrsim 5 \\times 10^4$ , we have $x_e \\rightarrow 1$ (neglecting Helium), $\\dot{\\rho }_{\\rm dep} = \\dot{\\rho }_{\\rm inj}$ , and $z \\gg z_{\\rm eq}$ , and we arrive at $\\mu \\le 6 \\times 10^{-8} f_{\\rm pbh} ~ \\underset{z \\ge 5 \\times 10^4}{\\max }\\left(\\frac{\\langle L\\rangle }{L_{\\rm Edd}}\\right).$ This is always significantly below the sensitivity of FIRAS [21], and would be within the reach of proposed spectral distortion experiments such as PIXIE [44] only if PBHs radiated near the Eddington luminosity.",
"In practice, $L \\ll L_{\\rm Edd}$ at all times (see Fig.",
"REF ), hence we conclude that accreting PBHs are not and will never be detectable through $\\mu $ -type spectral distortions.",
"The $y$ -parameter integral (REF ) is dominated by the lower redshift cutoff $z \\approx 200$ corresponding to the thermal decoupling of gas and CMB photons.",
"Since the luminosity is a slowly varying function near $z \\approx 200$ and $\\dot{\\rho }_{\\rm dep} \\approx \\dot{\\rho }_{\\rm inj}$ , we find $y \\approx 0.02 ~ f_{\\rm pbh} \\frac{\\langle L\\rangle }{L_{\\rm Edd}}\\Big {|}_{z \\approx 200}.$ From Fig.",
"REF , we see that for the mass range considered $M \\le 10^4\\, M_{\\odot }$ this is always below the sensitivity of FIRAS [21].",
"For $M = 10^4\\, M_{\\odot }$ , the $y$ -parameter may be as large as $y \\sim 2 \\times 10^{-7} f_{\\rm pbh}$ .",
"This is within the projected sensitivity of PIXIE for $f_{\\rm pbh} = 1$ , but is one order of magnitude below the expected foreground $y$ -parameter from the low-redshift intra-cluster medium [45].",
"UPDATE To conclude, we find that the global heating of the plasma due to accreting PBHs does not leave any observable imprint on CMB spectral distortions, neither for current instruments, nor for proposed ones."
],
[
"Distortion from local Compton cooling",
"There is another source of energy injection in the CMB, which occurs in the immediate vicinity of the PBH: when Compton cooling is efficient, the volumetric rate of energy transfer from the gas to the CMB is $\\frac{d\\dot{E}}{4 \\pi r^2 dr} = n_{\\rm H} \\frac{4 \\overline{x}_e \\sigma _{\\rm T} \\rho _{\\rm cmb} }{m_e c (1 + \\overline{x}_e)}(T - T_{\\rm cmb}) \\nonumber \\\\\\approx -\\frac{3}{2} n_{\\rm H} \\rho ^{2/3} v \\frac{d}{dr}(T_{\\rm cmb}/\\rho ^{2/3}),$ where the second equality is obtained by setting $T \\approx T_{\\rm cmb}$ in the left-hand-side of Eq.",
"(REF ), which holds as long as Compton cooling is efficient.",
"Therefore the rate of energy injection per PBH is $\\dot{E} &=& T_{\\rm cmb} \\int _{r_{\\min }}^{\\infty } dr 4 \\pi r^2 v~ n_{\\rm H} \\frac{d (\\ln \\rho )}{dr}\\nonumber \\\\&=& \\frac{\\dot{M}}{m_p} T_{\\rm cmb} \\log (\\rho (r_{\\rm min})/\\rho _{\\infty }),$ where $r_{\\rm min} \\sim \\gamma ^{-2/3} r_{\\rm B}$ is the radius at which Compton cooling becomes inefficient, $\\gamma $ being the dimensionless Compton cooling parameter defined in Eq. ().",
"With $\\rho (r_{\\rm min}) \\approx \\rho _{\\infty } (r_{\\rm min}/r_{\\rm B})^{-3/2}$ , we arrive at $\\dot{E} \\sim \\frac{\\dot{M}}{m_p} T_{\\rm cmb} \\log (\\gamma ).$ We therefore get a characteristic distortion amplitude $\\frac{\\dot{\\rho }_{\\rm inj}}{H \\rho _{\\rm cmb}} &\\sim & f_{\\rm pbh} \\frac{\\dot{M}}{H M} \\frac{T_{\\rm cmb} \\rho _{\\rm dm}}{\\rho _{\\rm cmb} m_p} \\log (\\gamma ) \\nonumber \\\\&\\sim & f_{\\rm pbh} \\frac{\\dot{M}}{H M} \\frac{n_{\\rm H}}{n_{\\rm cmb}},$ where we have used $\\rho _{\\rm dm} \\sim \\rho _b$ and $n_{\\rm cmb} \\sim \\rho _{\\rm cmb}/T_{\\rm cmb}$ is the number density of CMB photons.",
"We see that this is proportional to the baryon-to-photon ratio $n_{\\rm H}/n_{\\rm cmb} \\sim 10^{-10}$ , and moreover multiplied by $H^{-1}\\dot{M}/M$ which, as we discussed near Eq.",
"(REF ), is always less than unity for the mass range we consider.",
"Therefore local Compton cooling by CMB photons does not lead to any observable spectral distortion."
],
[
"Effect on CMB anisotropy power spectra",
"The change in the ionization history shown in Fig.",
"REF affects the visibility function for CMB anisotropies, and as a consequence the angular power spectra of temperature and polarization fluctuations.",
"We have incorporated the modified hyrec into the Boltzmann code class [46].",
"We show in Fig.",
"REF the changes in CMB power spectra for the same parameters used in Fig.",
"REF .",
"The effect is qualitatively similar to an increase in the reionization optical depth: fluctuations are damped on small angular scales due to scattering of photons out of the line of sight, and the polarization is enhanced on relatively large angular scales.",
"The latter are smaller than the scales affected by reionization, as the effect of PBHs is at larger redshifts.",
"For small PBH masses, the suppression on small scales is accompanied by oscillations, resulting from the change of the redshift of last scattering.",
"Indeed, as can be seen in Fig.",
"REF , low-mass PBHs affect the recombination history near the last-scattering surface $z \\sim 10^3$ more than high-mass PBHs, whose effect is mostly on the freeze-out free-electron fraction.",
"Figure: Fractional change in the CMB temperature (upper panel) and EE-mode polarization (lower panel) power spectra resulting from accreting PBHs.",
"The parameters are (M pbh /M ⊙ ,f pbh )=(10 2 ,1),(10 3 ,10 -2 ),(10 4 ,10 -4 )(M_{\\rm pbh}/M_{\\odot }, f_{\\rm pbh}) = (10^2, 1), (10^3, 10^{-2}), (10^4, 10^{-4}), in that order with increasing overall amplitude.",
"We only show the collisional ionization case here."
],
[
"Analysis of Planck data",
"To analyze the CMB anisotropy data from Planck, one should in principle run a Monte Carlo Markov Chain (MCMC), accounting for foreground nuisance parameters (see e.g. [24]).",
"However, this approach is too computationally taxing if we are to set an upper bound on the abundance of PBHs as a function of PBH mass, as it would require running a MCMC simulation for every mass considered.",
"Instead, we performed a simplified yet accurate data analysis as follows.",
"We use the Plik$\\_$ lite best-fit $\\hat{C}_{\\ell }$ and covariance matrix ${\\Sigma }$ for the high-$\\ell $ binned CMB-only $TT$ , $TE$ and $EE$ power spectra provided by the Planck collaborationAvailable at http://pla.esac.esa.int/pla/ [47].",
"These spectra and their covariance matrix are obtained by marginalizing over foreground nuisance parameters.",
"Since they are only provided for multipoles $\\ell \\ge 30$ , we moreover assume a prior on the optical depth to reionization $\\tau _{\\rm reio} = \\tau _0 \\pm \\sigma _{\\tau } \\equiv 0.0596 \\pm 0.0089$ as obtained by the latest Planck data analysis [48].",
"This prior on $\\tau _{\\rm reio}$ accounts approximately for the large-scale temperature and polarization data (see Refs.",
"[49], [50] for an analysis similar in spirit).",
"Given the relatively large effect of accreting PBHs on low-$\\ell $ polarization (see Fig.",
"REF ), a full data analysis might change the constraints by order-unity factors; however this is below our theoretical uncertainty.",
"For a given set of cosmological parameters $\\vec{\\theta } = (H_0, \\Omega _b h^2, \\Omega _c h^2, A_s, n_s, \\tau _{\\rm reio}, f_{\\rm pbh})$ the $\\chi ^2$ is then $\\chi ^2(\\vec{\\theta }) &=& \\frac{1}{2} \\left(C_{\\ell }^X(\\vec{\\theta }) - \\hat{C}^X_{\\ell }\\right) ({\\Sigma ^{-1}})^{X X^{\\prime }}_{\\ell \\ell ^{\\prime }}\\left(C^{X^{\\prime }}_{\\ell ^{\\prime }}(\\vec{\\theta }) - \\hat{C}^{X^{\\prime }}_{\\ell ^{\\prime }}\\right)\\nonumber \\\\&+& \\frac{1}{2} \\frac{(\\tau _{\\rm reio} - \\tau _0)^2}{\\sigma _{\\tau }^2},$ where we sum over repeated indices, $X \\in (TT, TE, EE)$ , and the $C_{\\ell }^{X}(\\vec{\\theta })$ are the theoretical power spectra obtained with our modified hyrec and class.",
"Taylor-expanding about the best-fit standard cosmological parameters $\\vec{\\theta }_0$ given in Ref.",
"[48] (with $f_{\\rm pbh, 0} = 0$ ), we rewrite this as $\\chi ^2(\\vec{\\theta }) &\\approx & \\chi ^2(\\vec{\\theta }_0) + \\Delta \\theta _i \\frac{\\partial C_{\\ell }^X}{\\partial \\theta _i} \\Big {|}_{\\theta _0} ({\\Sigma ^{-1}})^{X X^{\\prime }}_{\\ell \\ell ^{\\prime }}\\left(C_{\\ell ^{\\prime }}^{X^{\\prime }}(\\vec{\\theta }_0) - \\hat{C}^{X^{\\prime }}_{\\ell ^{\\prime }}\\right)\\nonumber \\\\&+& \\frac{1}{2} \\Delta \\theta _i F_{ij} \\Delta \\theta _j ,$ where $\\Delta \\theta _i \\equiv \\theta _i -\\theta _{0, i}$ and $F_{ij} \\approx \\frac{\\partial C^X_{\\ell }}{\\partial \\theta _i} ({\\Sigma ^{-1}})^{X X^{\\prime }}_{\\ell \\ell ^{\\prime }}\\frac{\\partial C^{X^{\\prime }}_{\\ell ^{\\prime }}}{\\partial \\theta _j} + \\frac{\\delta _{i, i_{\\tau }} \\delta _{j, i_\\tau }}{\\sigma _{\\tau }^2}$ is the Fisher-information or curvature matrix [51], for which we have neglected the smaller term linear in $(C^X_{\\ell }(\\vec{\\theta }_0) - \\hat{C}^X_{\\ell })$ .",
"Maximizing this quadratic approximation of the $\\chi ^2$ allows us to find the best-fit cosmological parameters $\\hat{\\vec{\\theta }}$ , with their covariance given by $(F^{-1})_{ij}$ .",
"We have checked that without PBHs this simple analysis recovers very accurately the best-fit standard 6 cosmological parameters obtained in Ref.",
"[47], with biases of at most $0.17\\,\\sigma $ .",
"The variances we derive match those of Ref.",
"[24] for $H_0, \\Omega _b h^2, \\Omega _c h^2$ and $n_s$ and those of Ref.",
"[48] for $A_s$ and $\\tau _{\\rm reio}$ , as expected since we are using the same high-$\\ell $ covariance as in the former reference, and the prior on $\\tau _{\\rm reio}$ (strongly degenerate with $A_s$ ) from the latter.",
"We apply this analysis to derive the best-fit and 1-$\\sigma $ error on $f_{\\rm pbh}$ , as a function of $M_{\\rm pbh}$ .",
"We explicitly checked that for the limits we obtain, the change in the anisotropy power spectra is indeed linear in $f_{\\rm pbh}$ (ROM find an effect that goes as $f_{\\rm pbh}^{1/2}$ because they obtain a much larger effect on the freeze-out free-electron abundance than we do).",
"Though we consider a limit on the normalization of a Dirac-function mass distribution, this analysis can be generalized to any extended mass function [52], by replacing $f_{\\rm pbh}L(M)/M \\rightarrow \\int dM \\frac{d f_{\\rm pbh}}{d M}L(M)/M$ , where $df_{\\rm pbh}/dM$ is the differential DM-PBH fraction.",
"For all PBH masses we consider, $M \\le 10^4 \\,M_{\\odot }$ (as the steady-state approximation breaks down beyond that mass), the best-fit $\\hat{f}_{\\rm pbh}$ is always less than a fraction of standard deviationThe astute reader may wonder why even given several probed PBH masses, some best-fit $\\hat{f}_{\\rm pbh}$ do not deviate by more than one standard deviation from 0; the reason is that the effect of PBHs of different masses on the CMB is very similar, hence the best-fit values are expected to be correlated.",
"$\\sigma _{f_{\\rm pbh}}$ .",
"We show $\\sigma _{f_{\\rm pbh}}$ in Fig.",
"REF , as a simple proxy for the upper limit on this parameterStrictly speaking, given the prior $f_{\\rm pbh} \\ge 0$ , defining the 68%-confidence interval is a bit more subtle; given the large uncertainties of the calculation, we shall not delve into such technical details here.. We see that in the collisional ionization limit, CMB anisotropy measurements by Planck exclude PBHs with masses $M\\gtrsim 10^2\\, M_{\\odot }$ as the dominant component of the dark matter.",
"In the photoionization limit, this threshold is lowered to $\\sim 10\\, M_{\\odot }$ .",
"In either case, our bound is significantly weaker than that of ROM.",
"The up to two orders of magnitude difference in the constraint between the two limiting cases illustrates the level of uncertainty in the calculation.",
"Nevertheless, we believe our most conservative bound is robust and difficult to evade, at least at the order-of-magnitude level.",
"Figure: Approximate CMB-anisotropy constraints on the fraction of dark matter made of PBHs derived in this work (thick black curves).",
"The “collisional ionization\" case assumes that the radiation from the PBH does not ionize the local gas, which eventually gets collisionally ionized.",
"The “photoionization\" case assumes that the local gas is ionized due to the PBH radiation, up to a radius larger than the collisional ionization region, yet smaller than the Bondi radius.",
"The former case is the most conservative, as collisional ionization leads to a smaller temperature near the black hole horizon, hence a smaller luminosity, and weaker bounds.",
"The correct result lies somewhere between these two limiting cases.",
"For comparison, we also show the CMB bound previously derived by ROM (thin dashed curve), as well as various dynamical constraints: micro-lensing constraints from the EROS (purple curve) and MACHO (blue curve) collaborations (but see Ref.",
"for caveats), limits from Galactic wide binaries , and ultra-faint dwarf galaxies (in all cases we show the most conservative limits provided in the referenced papers)."
],
[
"Discussion and conclusions",
"In this work we have revisited and revised existing CMB limits to the abundance of primordial black holes.",
"We showed that CMB-anisotropy measurements by the Planck satellite exclude PBHs as the dominant component of dark matter for masses $\\gtrsim \\, 10^2\\, M_{\\odot }$ .",
"The physical mechanism involved is that PBHs would radiate a fraction of the rest-mass energy they accrete, heating up and partially reionizing the Universe.",
"Such an increase in the free-electron abundance would change the CMB temperature and polarization power spectra.",
"Planck measurements do not allow for large deviations from the standard recombination history [24], which leads to tight bounds for large and luminous PBHs.",
"The constraints we derive are significantly weaker than the previous result of Ricotti et al.",
"(ROM) [18], so it is instructive to briefly summarize the differences in our respective calculations.",
"First and foremost, we compute the radiative efficiency $\\epsilon \\equiv L/\\dot{M} c^2$ from first principles, generalizing Shapiro's classic calculation for spherical accretion around a black hole [25].",
"We account for Compton drag and cooling as well as ionization cooling once the background gas is neutral.",
"At fixed accretion rate, the efficiency we derive is at least a factor of ten and up to three orders of magnitude lower than what is assumed in ROM for spherically-accreting PBHs.",
"The second largest difference is in the accretion rate itself.",
"ROM compute the accretion rate for an isothermal equation of state, assuming that Compton cooling by CMB photons is always very efficient.",
"In fact, for sufficiently low redshift and low PBH masses Compton cooling is negligle and the gas is adiabatically heated.",
"In this case the higher gas temperature, and hence pressure, imply an accretion rate that is lower by a factor of $\\sim 10$ .",
"Since the PBH luminosity is quadratic in the accretion rate, this translates to a factor of $\\sim 100$ reduction in the effect of PBHs on CMB observables.",
"A third difference is the relative velocity between PBHs and baryons, which ROM significantly underestimates around $z\\sim 10^3$ , leading to an over-estimate of the accretion rate.",
"There are considerable theoretical uncertainties in the calculation of the accretion rate and luminosity of PBHs, as we have illustrated by considering two limiting cases for the radiative feedback on the local ionization fraction, leading to largely different results.",
"Let us recall the most critical uncertainties here.",
"First, we have only considered spherical accretion.",
"Extrapolating the measured primordial power spectrum to the very small scale corresponding to the Bondi radius, ROM estimated the angular momentum of the accreted gas; they argued that the accretion is indeed spherical for PBHs less massive than $\\sim 10^3- 10^4\\, M_{\\odot }$ .",
"However, there is no direct measurements of the ultra-small-scale power spectrum, and all bets are open for a Universe containing PBHs.",
"If small-scale fluctuations are larger (for instance due to non-linear clustering of PBHs), an accretion disk could form, with a significantly enhanced luminosity with respect to spherical accretion.",
"On the other hand, non-spherical accretion could conceivably also lead to complex three-dimensional flows near the black hole giving rise to a turbulent pressure that lowers the accretion rate and radiative output.",
"Secondly, we have accounted for the motion of PBHs with an approximate and very uncertain rescaling of the accretion rate.",
"Given that dark-matter-baryon relative velocities are typically supersonic, we expect shocks and a much more complex accretion flow in general.",
"Thirdly, we have assumed a steady-state flow, but have not established whether such a flow is stable, even for a static black hole.",
"Last but not least, if PBHs only make a fraction of the dark matter, an assumption must be made about the rest of it, the simplest one being that it is made of weakly interacting massive particles (WIMPs).",
"If so, these WIMPs ought to be accreted by PBHs, whose mass may grow significantly after matter-radiation equality [55], and as a consequence increase the accretion rate of baryons [26], [18].",
"For the sake of simplicity, and given the added uncertainty associated with the accretion of collisionless particles, we have not accounted for this possibility in this work.",
"Given these major qualitative uncertainties, we have made several simplifications leading to additional factors of a few inaccuracies: for instance, our calculation is purely Newtonian, and our analytic treatments of the ionization region and of energy deposition into the plasma are only approximate.",
"We have also only explicitly analyzed Planck's temperature and polarization data for multipoles $\\ell \\ge 30$ , approximating the effect of large-scale measurements by a simple prior on the optical depth to reionization.",
"In a nutshell, the reader should keep in mind that this is a complex problem and that many simplifying assumptions underly our results, which we expect to be accurate at the order-of-magnitude level only.",
"To conclude, we find that, up to the theoretical uncertainties aforementioned, CMB anisotropies conservatively rule out PBHs more massive than $\\sim 10^2\\, M_{\\odot }$ as the dominant form of dark matter.",
"This bound could be tighter by up to one order of magnitude if the local gas is predominantly photoionized rather than collisionally ionized.",
"Given the recent renewed interest in the $\\sim 10-100 \\, M_{\\odot }$ window, it would be very interesting to generalize our accretion model to self-consistently account for ionization feedback, a task beyond the scope of this article, and to be pursued in future work.",
"In the mean time, there are a number of other interesting astrophysical probes in that mass range.",
"These include future measurements of the stochastic gravitational-wave background [56], [57], [58], [59], [60] and of the mass spectrum [61], redshift distribution [62], and orbital eccentricies [63] for future binary-black-hole mergers; lensing of fast radio bursts by PBHs [64]; pulsar timing [65], [66]; radio/x-ray sources [67] or the cosmic infrared background [68]; the dynamics of compact stellar systems [54]; strong-lensing systems [69]; and perhaps clustering of GW events [70], [71], [72], [73].",
"The conclusions of our work suggest that it will be important to pursue vigorously these alternative avenues."
],
[
"Acknowledgments",
"We are grateful to Graeme Addison, Julián Muñoz and Vivian Poulin for helpful conversations about the Planck likelihood, Fisher analysis, and energy deposition, respectively.",
"We also thank Massimo Ricotti for insightful feedback on this work.",
"We thank Simeon Bird, William Dawson, Alvise Raccanelli and Pasquale Serpico for useful comments on this manuscript.",
"We also acknowledge conversations with Jens Chluba, Ilias Cholis, Sébastien Clesse, Juan García-Bellido, Ely Kovetz, Julian Krolik, Katie Mack, and Hong-Ming Zhu.",
"This work was supported by the Simons Foundation, NSF grant PHY-1214000, and NASA ATP grant NNX15AB18G."
],
[
"Isolated PBH assumption",
"Our calculation assumes gas accreting on an isolated BH.",
"This approximation is valid as long as the Bondi radius is significantly smaller than the characteristic proper separation $\\overline{r}$ between PBHs.",
"Numerically, we get $r_{\\rm B} &=& \\frac{G M}{v_B^2} \\approx 6 \\times 10^{14} \\textrm {cm} \\frac{M}{M_{\\odot }} \\frac{10^3}{1+z}\\\\\\overline{r} &=& \\left(\\frac{3 M}{4 \\pi f_{\\rm pbh} \\overline{\\rho }_{\\rm dm}}\\right)^{1/3} \\nonumber \\\\&\\approx & 6 \\times 10^{17} \\textrm {cm} \\left(\\frac{M}{f_{\\rm pbh} M_{\\odot }}\\right)^{1/3} \\frac{10^3}{1+z},$ where we estimated the Bondi radius for a PBH at rest and assuming $T_{\\rm gas} = T_{\\rm cmb}$ and $x_e \\ll 1$ , valid for $200 \\lesssim z \\lesssim 1100$ .",
"We therefore find that the isolated PBH approximation holds for $M \\lesssim 3 \\times 10^4 f_{\\rm pbh}^{-1/2} M_{\\odot }.$ Given that our conservative bound is $f_{\\rm pbh} \\lesssim (100\\, M_{\\odot }/M)^2$ , PBHs can indeed be considered as isolated for all masses considered.",
"Note, however, that this estimate only holds for quasi-uniformly distributed PBHs.",
"In practice PBHs may cluster significantly if they make up a significant fraction of the dark matter, due to Poisson fluctuations in their initial clustering [74].",
"We do not attempt to account for this effect here."
],
[
"Free-Free cooling",
"Free-free cooling is efficient when the associated timescale $t_{\\rm ff} \\sim n_e T/j_{\\rm ff}$ is much shorter than the local accretion timescale $t_{\\rm acc} \\sim r/|v|$ .",
"The ratio of the two timescales is $\\frac{t_{\\rm acc}}{t_{\\rm ff}} \\sim \\frac{r/|v|}{n_e T/j_{\\rm ff}}\\sim \\frac{\\alpha c \\sigma _{\\rm T} n_e r}{|v|}\\mathcal {J} \\sim \\frac{\\alpha c \\sigma _{\\rm T} \\dot{M}}{4 \\pi m_p r v^2} \\mathcal {J},$ where we have used $n_e = \\rho /m_p = \\dot{M}/(4 \\pi m_p r^2 |v|)$ .",
"In the innermost region, the gas is in near free-fall, so that $r v^2 \\sim GM$ .",
"Recalling that the Eddington luminosity is given by Eq.",
"(REF ), we may rewrite this as $\\frac{t_{\\rm acc}}{t_{\\rm ff}} \\sim \\dot{m} \\alpha \\mathcal {J}.$ Therefore, as long as $\\dot{m} \\lesssim $ a few, we may safely neglect free-free cooling.",
"This is indeed the case for the mass range $M \\lesssim 10^4 M_{\\odot }$ that we consider (see Fig.",
"REF )."
],
[
"Free-bound radiation",
"At low frequencies, near the ionization threshold of hydrogen, radiative recombinations also contribute to the radiation of the plasma [25].",
"We consider only recombinations to the ground state of hydrogen, for which the cross-section is well known and has the approximate dependence near threshold $\\sigma _{\\rm pi}(\\nu ) \\approx \\sigma _0 \\left(\\frac{\\nu _{\\rm I}}{\\nu }\\right)^3,$ with $\\sigma _0 \\approx 6 \\times 10^{-18}$ cm$^2$ and $\\nu _{\\rm I} \\equiv E_{\\rm I}/h$ is the threshold photoionization frequency.",
"Assuming Saha equilibrium, detailed balance considerations allow us to compute the corresponding free-bound emissivity (see e.g. Ref.",
"[37]): $j_{\\nu }^{\\rm fb} = n_e^2 (3 \\pi m_e T)^{-3/2} \\frac{8 \\pi h^4 \\nu ^3}{c^2} \\textrm {e}^{- h (\\nu - \\nu _{\\rm I})/T}\\sigma _{\\rm pi}(\\nu ).$ Therefore the free-bound emissivity near threshold is nearly independent of frequencyThis result differs from Shapiro's assumed free-bound spectrum..",
"The ratio of free-bound to free-free emissivities is $\\frac{j^{\\rm fb}}{j^{\\rm ff}} \\sim \\left(\\frac{h \\nu _{\\rm I}}{\\sqrt{m_e c^2 T}}\\right)^3 \\frac{\\sigma _0}{\\alpha \\sigma _{\\rm T}} \\ll 1.$ Even though $\\sigma _0 \\gg \\alpha \\sigma _{\\rm T}$ , this ratio is largely suppressed due to the first factor."
],
[
"Optical thickness",
"In our estimate of the luminosity we have assumed that the plasma is optically thin.",
"Here we show that the plasma is indeed optically thin to both Compton scattering and free-free absorption.",
"The Compton optical depth is dominated by the densest regions near the horizon.",
"Since the Compton cross section is lower than Thomson for relativistic photons, the Compton optical depth is less than $\\tau _{\\rm Com} \\lesssim r_{\\rm S} n_e \\sigma _{\\rm T} \\sim \\dot{m},$ where we used Eqs.",
"(REF ) and (REF ).",
"Therefore, as long as $\\dot{m} \\lesssim 1$ the plasma is optically thin to Compton scattering [18].",
"The free-free absorption coefficient $\\alpha _{\\nu }^{\\rm ff}$ (with dimensions of inverse length) is [75] $\\alpha _{\\nu }^{\\rm ff} = \\frac{j_{\\nu }^{\\rm ff}}{B_{\\nu }(T)},$ where $j_{\\nu }^{\\rm ff} \\approx \\frac{j^{\\rm ff}}{4 \\pi } h/T$ is the emissivity and $B_{\\nu }(T)$ is the Planck function.",
"Since $j_{\\nu }^{\\rm ff} \\propto n_e^2$ the total optical depth is dominated by the region near the horizon.",
"The optical depth is then $\\tau ^{\\rm ff} \\sim r_{\\rm S} \\alpha _{\\nu }^{\\rm ff} \\sim r_{\\rm S} \\frac{j c^2 }{h \\nu _{\\max }^4},$ where we approximated $j_{\\nu } \\sim j/\\nu _{\\max }$ and $B_{\\nu } \\sim h \\nu _{\\max }^3/c^2$ , where $\\nu _{\\max } \\sim T_{\\rm S}/h$ .",
"Using Eq.",
"(REF ), we get $\\tau _{\\rm ff} \\sim \\alpha ~\\tau _{\\rm Th} \\left(\\frac{h c}{T_{\\rm S}} n_e^{1/3}\\right)^3.",
"$ The last term is the degeneracy factor: if it is greater than unity one ought to account for electron degeneracy pressure.",
"It is easy to check that this term is always much smaller than unity for all cases considered."
],
[
"Energy deposition rate: comparison with the existing literature",
"Several papers attempt an analytic estimate of the energy deposition rate as we do in Section , as opposed to a fully numerical treatment as in Ref. [40].",
"Here we compare and contrast our results to the existing literature.",
"Reference [76] gives the following integral expression for the photon density per energy interval [their equation (2.12), rewritten in our notation]: $\\mathcal {N}_E(t) &=& \\int ^t dt^{\\prime } \\frac{1}{E^{\\prime }} \\frac{d \\dot{\\rho }_{\\rm inj}}{dE^{\\prime }} \\left(\\frac{a^{\\prime }}{a}\\right)^3 \\nonumber \\\\&&\\times \\exp \\left[- \\int _{t^{\\prime }}^t dt^{\\prime \\prime } n_A^{\\prime \\prime } c \\sigma _{\\rm tot}(E^{\\prime \\prime })\\right],$ where $E^{\\prime } \\equiv E a/a^{\\prime }$ , $E^{\\prime \\prime } \\equiv E a/a^{\\prime \\prime }$ , $n_A$ is the number density of absorbers and $\\sigma _{\\rm tot}(E)$ is total cross section for all the interactions suffered by the DM-sourced photon and that result in the production of free electrons.",
"This integral equation is equivalent to the following partial differential equation: $a^{-3}\\frac{d}{dt}(a^3 \\mathcal {N}_E) = \\frac{1}{E} \\frac{d \\dot{\\rho }_{\\rm inj}}{dE} - n_A c \\sigma _{\\rm tot}(E) \\mathcal {N}_E.",
"$ This equation differs from our Eq.",
"(REF ) in two ways.",
"First, in the absence of photon sources or sinks, Eq.",
"(REF ) does not recover the correct scaling $\\mathcal {N}_E \\propto a^{-2}$ .",
"Second, the second term on the right-hand side implies that photons are destroyed in the ionization process.",
"While this is the case for direct photoionization events $\\gamma + H \\rightarrow p + e$ , it is not the case for ionizations following Compton scattering events $\\gamma + H \\rightarrow p + e + \\gamma ^{\\prime }$ , for which part of the energy of the incoming photon is used for ionizing the atom, but the photon is not destroyed.",
"From Eq.",
"(REF ), Ref.",
"[77] deduces the energy deposition rate (correcting a mistake in Refs.",
"[78], [79]).",
"Assuming $\\sigma _{\\rm tot}(E) \\approx \\sigma _{\\rm T}$ (valid for $E \\lesssim m_e c^2$ ), the resulting energy deposition rate is (Eq.",
"(4.21) of Ref.",
"[79], corrected in Appendix B of Ref.",
"[77]) $&&\\dot{\\rho }_{\\rm dep} = \\int ^t dt^{\\prime } \\textrm {e}^{- \\kappa (t, t^{\\prime })} \\left(\\frac{a^{\\prime }}{a}\\right)^8 \\overline{n}_{\\rm H}^{\\prime }c \\sigma _{\\rm T} \\dot{\\rho }_{\\rm inj}^{\\prime } \\ \\ (\\textrm {Poulin et al.})",
"\\nonumber ,\\\\&&\\textrm {with} \\ \\ \\kappa (t, t^{\\prime }) \\equiv \\int _{t^{\\prime }}^t d t^{\\prime \\prime } \\overline{n}_{\\rm H}^{\\prime \\prime } c \\sigma _{\\rm T}.$ Rewriting this as differential equation would lead to $a^{-8} \\frac{d (a^8 \\dot{\\rho }_{\\rm dep})}{dt} = \\overline{n}_{\\rm H} c \\sigma _{\\rm T} \\left[\\dot{\\rho }_{\\rm inj} - \\dot{\\rho }_{\\rm dep} \\right].$ This differs from our Eq.",
"(REF ) in two ways.",
"First, the incorrect scaling $\\dot{\\rho }_{\\rm dep} \\propto a^{-8}$ instead of $a^{-7}$ once energy deposition becomes inefficient is a direct consequence of the incorrect scaling in Eq.",
"(2.12) of Ref. [76].",
"Secondly, our right-hand-side is smaller by an (approximate) factor $0.1$ .",
"This translates the fact that, even for $E \\lesssim m_e c^2$ , only a small fraction of the energy of Compton-scattered photons is lost to ionizations, as opposed to the totality of it as implicitly assumed in Eq.",
"(REF )."
]
] | 1612.05644 |
[
[
"Saturn rings: fractal structure and random field model"
],
[
"Abstract This study is motivated by the observation, based on photographs from the Cassini mission, that Saturn's rings have a fractal structure in radial direction.",
"Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of that fractal structure is possible?",
"(2) What kinematics model of such fractal rings can be formulated?",
"Both challenges are based on taking Saturn's rings' spatial structure as being statistically stationarity in time and statistically isotropic in space, but statistically non-stationary in space.",
"An answer to the first challenge is given through the calculus in non-integer dimensional spaces and basic mechanics arguments (Tarasov (2006) \\textit{Celest.",
"Mech.",
"Dyn.",
"Astron.}",
"\\textbf{94}).",
"The second issue is approached in Section~3 by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle.",
"Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square $F^{2}$ of the radial cross-section $F$ of the rings, where $F$ is a fat fractal."
],
[
"Introduction",
"A recent study of the photographs of Saturn's rings taken during the Cassini mission has demonstrated their fractal structure [6].",
"This leads us to ask these questions: Q1: What mechanics argument in support of that fractal structure is possible?",
"Q2: What kinematics model of such fractal rings can be formulated?",
"These issues are approached from the standpoint of Saturn's rings' spatial structure having (i) statistical stationarity in time and (ii) statistical isotropy in space, but (iii) statistical non-stationarity in space.",
"The reason for (i) is an extremely slow decay of rings relative to the time scale of orbiting around Saturn.",
"The reason for (ii) is the obviously circular, albeit disordered and fractal, pattern of rings in the radial coordinate.",
"The reason for (iii) is the lack of invariance with respect to arbitrary shifts in Cartesian space which, on the contrary and for example, holds true for a basic model of turbulent velocity fields.",
"Hence, the model we develop is one of rotational fields of all the particles, each travelling on its circular orbit whose radius is dictated by basic orbital mechanics.",
"The Q1 issue is approached in Section 2 from the standpoint of calculus in non-integer dimensional space, based on an approach going back to [10], [11].",
"We compare total energies of two rings — one of non-fractal and another of fractal structure, both carrying the same mass — and infer that the fractal ring is more likely.",
"We also compare their angular momenta.",
"The Q2 issue is approached in Section 3 in the following way.",
"Assume that the angular velocity vector of a rotating particle is a single realisation of a random field.",
"Mathematically, the above field is a random section of a special vector bundle.",
"Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions $\\lbrace \\,B_{k}(r,s)\\colon k\\ge 0\\,\\rbrace $ with $\\sum _{k=0}^{\\infty }\\operatorname{tr}(B_{k}(r,r))<\\infty ,$ where the real-valued parameters $r$ and $s$ run over the radial cross-section $F$ of Saturn's rings.",
"To reflect the observed fractal nature of Saturn's rings, [2] and independently [9] supposed that the set $F$ is a fat fractal subset of the set $\\mathbb {R}$ of real numbers.",
"The set $F$ itself is not a fractal, because its Hausdorff dimension is equal to 1.",
"However, the topological boundary $\\partial F$ of the set $F$ , that is, the set of points $x_{0}$ such that an arbitrarily small interval $(x_{0}-\\varepsilon ,x_{0}+\\varepsilon )$ intersects with both $F$ and its complement, $\\mathbb {R}\\setminus F$ , is a fractal.",
"The Hausdorff dimension of $\\partial F$ is not an integer number."
],
[
"Basic considerations",
"We begin with the standard gravitational parameter, $\\mu =GM_{\\mathrm {Saturn}}$ ; its value for Saturn ($\\mu =37,931,187$ $km^{3}/s^{2}$ ) is known but will not be needed in the derivations that follow.",
"For any particle of mass $m$ located within the ring, we take $m\\ll M_{\\mathrm {Saturn}}$ with dimensions also much smaller than the distance to the center of Saturn.",
"Each particle is regarded as a rigid body, with its orbit about the spherically symmetric Saturn being circular.",
"We are using the cylindrical coordinate system $\\left( r,\\theta ,z\\right) $ , such that the $z$ -axis is aligned with the normal to the plane of rings, Fig.",
"REF .",
"The particle's orbital frame of reference with the origin $O$ at its center of mass is made of three axes: $a_{1}$ in the radial direction, $a_{2}$ tangent to the orbit in the direction of motion, and $a_{3}$ normal to the orbit plane.",
"All the particles orbit around Saturn in the same plane.",
"The attitude of any given particle is described by the vector of body axes $\\left\\lbrace \\mathbf {x}\\right\\rbrace ^{T}=\\lbrace x_{1},x_{2},x_{3}\\rbrace ^{T}$ , which are related to the vector $\\left\\lbrace \\mathbf {a}\\right\\rbrace $ in the orbital frame of reference of the particle by $\\left\\lbrace \\mathbf {x}\\right\\rbrace =\\left[ \\mathbf {l}\\right] \\left\\lbrace \\mathbf {a}\\right\\rbrace .$ Here $\\left[ \\mathbf {l}\\right] $ is the matrix of direction cosines $l_{i}$ , $i=1,2,3$ .",
"Figure: The planar ring of particles, adapted from , showing the Saturnian (Cartesian and cylindrical)coordinate systems as well as the orbital frame of reference (a 1 ,a 2 ,a 3 )\\mathbf {(}a_{1},a_{2},a_{3}) and the body axes (x 1 ,x 2 ,x 3 )\\mathbf {(}x_{1},x_{2},x_{3}) of atypical particle.Henceforth, we consider two rings: Euclidean (i.e.",
"non-fractal) and a fractal one; both rings are planar, Fig.",
"REF .",
"Hereinafter the subscript $_{\\mathbb {E}}$ denotes any Euclidean object.",
"Next, we must consider the mass of a Euclidean ring (body $B_{\\mathbb {E}}$ ) versus a fractal ring (body $B_{\\alpha }$ ).",
"From a discrete system point of view, the ring is made of $I$ particles $\\lbrace i=1,...,I\\rbrace $ , each with a respective mass $m_{i}$ , moment of inertia $\\mathbf {j}_{i}$ , and positions $\\mathbf {x}_{i}$ .",
"The mass of a Euclidean ring $B_{\\mathbb {E}}$ , with radius $r\\in [R_{D},R]$ and thickness $h$ in $z$ -direction, is now taken in a continuum sense $\\begin{array}{c}M_{\\mathbb {E}}=\\sum _{i=1}^{I}m_{i}\\rightarrow \\int _{B_{\\mathbb {E}}}\\rho dB_{\\mathbb {E}}=h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }\\rho _{\\mathbb {E}}hdS_{2} \\\\=2\\pi h\\rho _{\\mathbb {E}}\\int _{R_{D}}^{R}rdr=\\rho _{\\mathbb {E}}h\\pi \\left(R^{2}-R_{D}^{2}\\right) .\\end{array}$ In the above we have assumed the mass to be homogeneously distributed throughout the ring with a mass density $\\rho _{\\mathbb {E}}$ .",
"To get quantitative results, one may take: $R=140\\times 10^{6}m$ as the outer radius of Saturn's F ring, $R_{D}=74.5\\times 10^{6}m$ as the radius of the (inner) D ring, and the rings' thickness $h=100m$ ."
],
[
"Mass densities",
"All the rings constituting the fractal ring $B_{\\alpha }$ are embedded in $\\mathbb {R}^{3}$ , also with radius $r\\in [R_{D},R]$ and thickness $h$ in $z$ -direction.",
"The parameter $\\alpha $ ($<1$ ) denotes the fractal dimension in the radial direction, i.e.",
"on any ray(any because the ring is axially symmetric about $z$ ).",
"Thus, the (planar) fractal dimension, such as seen and measured on photographs, is $D=\\alpha +1<2$ , consistent with the fact that Saturn's rings are partially plane-filling if interpreted as a planar body.",
"In order to do any analysis involving $B_{\\alpha }$ , in the vein of [10], [11], we employ the integration in non-integer dimensional space.",
"That is, we take the infinitesimal element $dB_{\\alpha }$ of $B_{\\alpha }$ according to [5]: $dB_{\\alpha }=h\\text{ }dS_{\\alpha },\\text{ \\ \\ }dS_{\\alpha }=\\alpha \\left(\\frac{r}{R}\\right) ^{\\alpha -1}dS,\\text{ \\ \\ }dS=rdrd\\theta .$ Now, the mass of a fractal ring is $\\begin{array}{c}M_{\\alpha }=\\sum _{i=1}^{I}m_{i}\\rightarrow \\int _{B}\\rho _{\\alpha }dB_{\\alpha }=h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }\\rho _{\\alpha }dS_{\\alpha } \\\\=\\displaystyle 2\\pi h\\rho _{\\alpha }\\int _{R_{D}}^{R}\\alpha \\left( \\frac{r}{R}\\right) ^{\\alpha -1}rdr=2\\pi h\\rho _{\\alpha }\\frac{\\alpha }{\\alpha +1}\\left(R^{2}-\\frac{R_{D}^{\\alpha +1}}{R^{\\alpha -1}}\\right) ,\\end{array}$ which involves an effective mass density $\\rho _{\\alpha }$ of a fractal ring.",
"Note that the above correctly reduces to (1) for $\\alpha \\rightarrow 1$ .",
"Since the rings in both interpretations must have the same mass, requiring $M_{\\alpha }=M_{\\mathbb {E}}$ for any $\\alpha $ , gives $\\rho _{\\alpha }=\\frac{\\alpha +1}{2\\alpha }\\rho _{\\mathbb {E}},$ which is a decreasing function of $\\alpha $ (i.e.",
"we must have $\\rho _{\\alpha }>\\rho _{\\mathbb {E}}$ for $\\alpha <1$ ) and which correctly gives $\\lim _{\\alpha \\rightarrow 1}\\rho _{\\alpha =1}=\\rho _{\\mathbb {E}}$ for $\\alpha =1$ , i.e.",
"when the fractal ring becomes non-fractal.",
"Thus, a fractal ring has a higher effective mass density than the homogeneous Euclidean ring of the same overall dimensions."
],
[
"Moments of inertia",
"The moment of inertia of the Euclidean ring ($r\\in [0,R]$ and thickness $h$ in $z$ -direction), assuming $\\rho _{\\mathbb {E}}=\\mathrm {const}$ , is $I_{\\mathbb {E}}=\\frac{1}{2}\\pi h\\rho _{\\mathbb {E}}\\left(R^{4}-R_{D}^{4}\\right) =\\frac{1}{2}M\\left( R^{2}+R_{D}^{2}\\right) ,$ while the moment of inertia of a fractal ring is $\\begin{array}{c}I_{\\alpha }=h\\int _{B}\\rho _{\\alpha }r^{2}dB_{E}=h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }r^{2}\\rho _{\\alpha }hdS_{\\alpha } \\\\\\displaystyle 2\\pi h\\rho _{\\alpha }\\int _{R_{D}}^{R}r^{2}\\alpha \\left( \\frac{r}{R}\\right) ^{\\alpha -1}rdr=2\\pi h\\rho _{\\alpha }\\frac{\\alpha }{\\alpha +3}\\left( R^{4}-\\frac{R_{D}^{\\alpha +3}}{R^{\\alpha -1}}\\right) .\\end{array}$ Now, take the limit $\\alpha \\rightarrow 1$ : $\\lim _{\\alpha \\rightarrow 1}I_{\\alpha }=\\frac{1}{2}\\pi h\\rho _{\\mathbb {E}}\\left( R^{4}-R_{D}^{4}\\right) =I_{\\mathbb {E}},$ as expected.",
"Note that $I_{\\alpha }$ is an increasing function of $\\alpha $ (i.e.",
"we must have $I_{\\alpha }<I_{\\mathbb {E}}$ for $\\alpha <1$ ) and which correctly gives $\\lim _{\\alpha \\rightarrow 1}I_{\\alpha }=I_{\\mathbb {E}}$ for $\\alpha =1$ .",
"We also observe from (6) that a fractal ring has a lower moment of inertia than the homogeneous Euclidean ring with the same overall dimensions."
],
[
"Energies",
"Since for an object of mass $m$ on a circular orbit the total energy is $E=-\\mu /2r$ , the total energy (sum of kinetic and potential) of the Euclidean ring is $\\begin{array}{c}\\displaystyle E_{\\mathbb {E}}=-\\sum _{i=1}^{I}\\frac{\\mu m_{i}}{2r_{i}}\\rightarrow -\\int _{B}\\frac{\\mu \\rho _{E}}{2r}\\text{ }dB \\\\=\\displaystyle -\\frac{1}{2}h\\mu \\rho _{\\mathbb {E}}\\int _{0}^{R}\\int _{0}^{2\\pi }r^{-1}rdrd\\theta =-\\pi h\\mu \\rho _{\\mathbb {E}}\\left( R-R_{D}\\right) .\\end{array}$ On the other hand, the total energy of the fractal ring $B_{\\alpha }$ is [again with $dS_{\\alpha }=\\alpha \\left( \\frac{r}{R}\\right) ^{\\alpha -1}rdrd\\theta $ ] $\\begin{array}{c}E_{\\alpha }=\\displaystyle -\\sum _{i=1}^{I}\\frac{\\mu m_{i}}{2r_{i}}\\rightarrow -\\int _{B}\\frac{\\mu \\rho _{\\alpha }}{2r}\\text{ }dB=-\\int _{R_{D}}^{R}\\frac{1}{2}h\\mu \\rho _{\\alpha }\\frac{\\alpha +1}{2\\alpha }r^{-1}dS_{\\alpha } \\\\=\\displaystyle -\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }\\frac{1}{2}h\\mu \\rho _{\\alpha }\\alpha \\frac{\\alpha +1}{2\\alpha }r^{-1}\\left( \\frac{r}{R}\\right) ^{\\alpha -1}rdrd\\theta =-\\pi h\\mu \\rho _{\\alpha }\\frac{\\alpha +1}{2\\alpha }\\left(R-R_{D}\\right) .\\end{array}$ Now, take the limit $\\alpha \\rightarrow 1$ : $\\lim _{\\alpha \\rightarrow 1}E_{\\alpha }=\\frac{1}{2}\\pi h\\rho _{\\mathbb {E}}\\left( R-R_{D}\\right) =E_{\\mathbb {E}},$ as expected.",
"Comparing $E_{\\alpha }$ with $E_{\\mathbb {E}}$ , gives $E_{\\alpha }=\\frac{\\alpha +1}{2\\alpha }E_{\\mathbb {E}},$ which is a decreasing function of $\\alpha $ .",
"Thus, given the minus sign in (8) and (9), the fractal ring has a lower total energy than the homogeneous Euclidean ring with the same overall dimensions and the same mass.",
"In other words, with reference to question Q1 in the Introduction, the ring having a fractal structure is more likely than that with a non-fractal one.",
"The foregoing argument extends the approach of [13], who showed that a Euclidean ring has a lower energy than a Euclidean spherical shell, which in turn is lower than that of a Euclidean ball.",
"Putting all the inequalities together, we have $E_{\\alpha }\\le E_{\\mathbb {E}}\\le E_{\\mathrm {shell}}\\le E_{\\mathrm {ball}}.$"
],
[
"Angular Momenta",
"For any particle of velocity $v$ on a circular orbit of radius $r$ around a planet: $\\mu =rv^{2}=r^{3}\\Omega ^{2}=4\\pi ^{2}r^{3}/T^{2},$ where $\\Omega $ is the angular velocity and $T$ is the period.",
"This implies: $v=\\sqrt{\\mu /r}\\text{\\ \\ \\ and \\ \\ }\\Omega =\\sqrt{\\mu /r^{3}}.$ For the Euclidean ring ($r\\in [0,R]$ and thickness $h$ in $z$ -direction), the angular momentum is $\\begin{array}{c}\\displaystyle H_{\\mathbb {E}}=\\sum _{i=1}^{I}m_{i}r_{i}v_{i}\\rightarrow h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }\\rho _{\\mathbb {E}}rv\\text{ }rdrd\\theta \\\\=\\displaystyle h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }\\rho _{\\mathbb {E}}r\\sqrt{\\mu /r}\\text{ }rdrd\\theta =2\\pi h\\rho _{\\mathbb {E}}\\sqrt{\\mu }\\frac{2}{5}\\left(R^{5/2}-R_{D}^{5/2}\\right) ,\\end{array}$ while for the fractal ring $B_{\\alpha }$ , the angular momentum is $\\begin{array}{c}H_{\\alpha }=\\sum _{i=1}^{I}m_{i}r_{i}v_{i}\\rightarrow h\\int _{0}^{R}\\int _{R_{D}}^{2\\pi }\\rho (r)rv\\text{ }dS_{\\alpha }=\\displaystyle 2\\pi h\\rho _{\\alpha }\\int _{R_{D}}^{R}r\\sqrt{\\frac{\\mu }{r}}\\alpha \\left(\\frac{r}{R}\\right) ^{\\alpha -1}\\text{ }rdr \\\\=\\displaystyle 2\\pi h\\rho _{\\alpha }\\sqrt{\\mu }\\frac{\\alpha }{\\alpha +3/2}R^{1-\\alpha }\\left( R^{3/2+\\alpha }-R_{D}^{3/2+\\alpha }\\right) .\\end{array}$ This correctly reduces to $H_{\\mathbb {E}}$ above for $\\alpha \\rightarrow 1$ .",
"Comparing $H_{\\alpha }$ with $H_{\\mathbb {E}}$ , shows that $H_{\\alpha }$ is an increasing function of $\\alpha $ and this correctly gives $\\lim _{\\alpha \\rightarrow 1}H_{\\alpha =1}=H_{\\mathbb {E}}$ , i.e.",
"the fractal ring has a lower angular momentum than the homogeneous Euclidean ring with the same overall dimensions.",
"At this point, we note that in inelastic collisions the momentum is conserved (just as in elastic collisions), but the kinetic energy is not as it is partially converted to other forms of energy.",
"If this argument is applied to the rings, one may argue that $H_{\\alpha }=H_{\\mathbb {E}}$ should hold for any $\\alpha $ , which can be satisfied by accounting for the angular momentum of particles due to rotation about their own axes .",
"Thus, instead of (13), writing $j_{i}$ for the moment of inertia of the particle $i$ , we have the contribution of the angular momentum of that rotation in terms of the Euler angle $\\phi $ about the $a_{3}$ axis: $H_{\\mathbb {E}}=\\sum _{i=1}^{I}m_{i}r_{i}v_{i}+\\sum _{i\\in I}j_{i}\\omega _{zi}\\rightarrow h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }\\rho _{\\mathbb {E}}rv\\text{}rdrd\\theta +h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }j\\phi \\text{ }rdrd\\theta .$ The first integral can be calculated as before, while in the second one we could assume $j=const$ although this would still leave the microrotation $\\omega _{z}$ as an unknown function of $r$ .",
"Turning to the fractal ring we also have two terms $H_{\\alpha }=\\sum _{i=1}^{I}m_{i}r_{i}v_{i}+\\sum _{i\\in I}j_{i}\\omega _{zi}\\rightarrow h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }\\rho _{E}rv\\text{ }dS_{\\alpha }+h\\int _{R_{D}}^{R}\\int _{0}^{2\\pi }j\\phi \\text{ }dS_{\\alpha },$ showing that the statistics $\\omega _{z}\\left( r\\right) $ needs to be determined.",
"At this point we turn to the question Q2."
],
[
"A stochastic model of kinematics",
"First, we consider the particles in Saturn's rings at a time moment 0.",
"Introduce a spherical coordinate system $(r,\\varphi ,\\theta )$ with origin $O $ in the centre of Saturn such that the plane of Saturn's rings corresponds to the polar angle's value $\\theta =\\pi /2$ .",
"Let $\\overline{\\mathbf {\\omega }}(r,\\varphi )\\in \\mathbb {R}^{3}$ be the angular velocity vector of a rotating particle located at $(r,\\varphi )$ .",
"We assume that $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ is a single realisation of a random field.",
"To explain the exact meaning of this construction, we proceed as follows.",
"Let $(x,y,z)$ be a Cartesian coordinate system with origin in the centre of Saturn such that the plane of Saturn's rings corresponds to the $xy$ -plane, Fig.",
"REF .",
"Let $O(2)$ be the group of real orthogonal $2\\times 2$ matrices, and let $SO(2)$ be its subgroup consisting of matrices with determinant equal to 1.",
"Put $G=O(2)\\times SO(2)$ , $K=O(2)$ .",
"The homogeneous space $C=G/K=SO(2)$ can be identified with a circle, the trajectory of a particle inside rings.",
"Consider the real orthogonal representation $U$ of the group $O(2)$ in $\\mathbb {R}^3$ defined by $ g=\\begin{pmatrix}g_{11} & g_{12} \\\\g_{21} & g_{22}\\end{pmatrix}\\mapsto U(g)=\\begin{pmatrix}g_{11} & g_{12} & 0 \\\\g_{21} & g_{22} & 0 \\\\0 & 0 & \\det g\\end{pmatrix}.$ Introduce an equivalence relation in the Cartesian product $G\\times \\mathbb {R}^3$ : two elements $(g_1,\\mathbf {x}_1)$ and $(g_2,\\mathbf {x}_2)$ are equivalent if and only if there exists an element $g\\in O(2)$ such that $(g_2,\\mathbf {x}_2)=(g_1g,U(g^{-1})\\mathbf {x}_1)$ .",
"The projection map maps an element $(g,\\mathbf {x})\\in G\\times \\mathbb {R}^3$ to its equivalence class and defines the quotient topology on the set $E_U$ of equivalence classes.",
"Another projection map, $\\pi \\colon E_U\\rightarrow C,\\qquad \\pi (g,\\mathbf {x})=gK,$ determines a vector bundle $\\xi =(E_U,\\pi ,C)$ .",
"The topological space $R=\\mathbb {R}^2\\setminus \\lbrace \\mathbf {0}\\rbrace $ is the union of circles $C_r$ of radiuses $r>0$ .",
"Every circle determines the vector bundle $\\xi _r=(E_{Ur},\\pi _r,C_r)$ .",
"Consider the vector bundle $\\eta =(E,\\pi ,R) $ , where $E$ is the union of all $E_{Ur}$ , and the restriction of the projection map $\\pi $ to $E_{Ur}$ is equal to $\\pi _r$ .",
"The random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ is a random section of the above bundle, that is, $\\overline{\\mathbf {\\omega }}(r,\\varphi )\\in \\pi ^{-1}(r,\\varphi )=\\mathbb {R}^3$ .",
"In what follow we assume that the random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ is second-order, i.e., $\\mathsf {E}[\\Vert \\overline{\\mathbf {\\omega }}(r,\\varphi )\\Vert ^2]<\\infty $ for all $(r,\\varphi )\\in R$ .",
"There are at least three different (but most probably equivalent) approaches to the construction of random sections of vector bundles, the first by [4], the second by [7], [8], and the third by [3].",
"In what follows, we will use the second named approach.",
"It is based on the following fact: the vector bundle $\\eta =(E,\\pi ,R)$ is homogeneous or equivariant.",
"In other words, the action of the group $O(2)$ on the bundle base $R$ induces the action of $O(2)$ on the total space $E$ by $(g_0,\\mathbf {x})\\mapsto (gg_0,\\mathbf {x})$ .",
"This action identifies the spaces $\\pi ^{-1}(r_0,\\varphi )$ for all $\\varphi \\in [0,2\\pi )$ , while the action of the multiplicative group $\\mathbb {R}^+$ on R, $\\lambda (r,\\varphi )=(\\lambda r,\\varphi )$ , $\\lambda >0$ , identifies the spaces $\\pi ^{-1}(r,\\varphi _0)$ for all $r>0$ .",
"We suppose that the random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ is mean-square continuous, i.e., $\\lim _{\\Vert \\mathbf {x}-\\mathbf {x}_0\\Vert \\rightarrow 0}\\mathsf {E}[\\Vert \\overline{\\mathbf {\\omega }}(\\mathbf {x})-\\overline{\\mathbf {\\omega }}(\\mathbf {x}_0))\\Vert ^2]=0$ for all $\\mathbf {x}_0\\in R$ .",
"Let $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle =\\mathsf {E}[\\overline{\\mathbf {\\omega }}(\\mathbf {x})]$ be the one-point correlation vector of the random field $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ .",
"On the one hand, under rotation and/or reflection $g\\in O(2)$ the point $\\mathbf {x}$ becomes the point $g\\mathbf {x}$ .",
"Evidently, the axial vector $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ transforms according to the representation (REF ) and becomes $U(g)\\overline{\\mathbf {\\omega }}(g\\mathbf {x})$ .",
"The one-point correlation vector of the so transformed random field remains the same, i.e., $\\langle \\overline{\\mathbf {\\omega }}(g\\mathbf {x})\\rangle =U(g)\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle .$ On the other hand, the one-point correlation vector of the random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ should be independent upon an arbitrary choice of the $x$ - and $y$ -axes of the Cartesian coordinate systems, i.e., it should not depend on $\\varphi $ .",
"Then we have $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle =U(g)\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle $ for all $g\\in O(2)$ , i.e., $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle $ belongs to a subspace of $\\mathbb {R}^{3}$ where a trivial component of $U$ acts.",
"Then we obtain $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x})\\rangle =\\mathbf {0}$ , because $U$ does not contain trivial components.",
"Similarly, let $\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x}),\\overline{\\mathbf {\\omega }}(\\mathbf {y})\\rangle =\\mathsf {E}[\\overline{\\mathbf {\\omega }}(\\mathbf {x}) \\otimes \\overline{\\mathbf {\\omega }}(\\mathbf {y})]$ be the two-point correlation tensor of the random field $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ .",
"Under the action of $O(2)$ we should have $\\langle \\overline{\\mathbf {\\omega }}(g\\mathbf {x}),\\overline{\\mathbf {\\omega }}(g\\mathbf {y})\\rangle =(U\\otimes U)(g)\\langle \\overline{\\mathbf {\\omega }}(\\mathbf {x}),\\overline{\\mathbf {\\omega }}(\\mathbf {y})\\rangle .$ In other words, the random field $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ is wide-sense isotropic with respect to the group $O(2)$ and its representation $U$ .",
"Consider the restriction of the field $\\overline{\\mathbf {\\omega }}(\\mathbf {x})$ to a circle $C_r$ , $r>0$ .",
"The spectral expansion of the field $\\lbrace \\,\\overline{\\mathbf {\\omega }}(r,\\varphi )\\colon \\varphi \\in C_r\\,\\rbrace $ can be calculated using [7] or [8].",
"The representation $U$ is the direct sum of the two irreducible representations $\\lambda _-(g)=\\det g$ and $\\lambda _1(g)=g$ .",
"The vector bundle $\\eta $ is the direct sum of the vector bundles $\\eta _-$ and $\\eta _1$ , where the bundle $\\eta _-$ (resp.",
"$\\eta _1$ ) is generated by the representation $\\lambda _-$ (resp.",
"$\\lambda _1$ ).",
"Let $\\mu _0$ be the trivial representation of the group $SO(2)$ , and let $\\mu _k$ be the representation $\\mu _k(\\varphi )=\\begin{pmatrix}\\cos (k\\varphi ) & \\sin (k\\varphi ) \\\\-\\sin (k\\varphi ) & \\cos (k\\varphi )\\end{pmatrix}.$ The representations $\\lambda _-\\otimes \\mu _k$ , $k\\ge 0$ are all irreducible orthogonal representations of the group $G=O(2)\\times SO(2)$ that contain $\\lambda _-$ after restriction to $O(2)$ .",
"The representations $\\lambda _1\\otimes \\mu _k$ , $k\\ge 0$ are all irreducible orthogonal representations of the group $G=O(2)\\times SO(2)$ that contain $\\lambda _1$ after restriction to $O(2)$ .",
"The matrix entries of $\\mu _0$ and of the second column of $\\mu _k$ form an orthogonal basis in the Hilbert space $L^2(SO(2),\\mathrm {d}\\varphi )$ .",
"Their multiples $e_k(\\varphi )={\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\sqrt{2\\pi }}, & \\mbox{if } k=0, \\\\\\frac{1}{\\sqrt{\\pi }}\\cos (k\\varphi ), & \\mbox{if } k\\le -1 \\\\\\frac{1}{\\sqrt{\\pi }}\\sin (k\\varphi ), & \\mbox{if } k\\ge 1\\end{array}\\right.",
"}$ form an orthonormal basis of the above space.",
"Then we have $ \\overline{\\mathbf {\\omega }}(r,\\varphi )=\\sum _{k=-\\infty }^{\\infty }e_k(\\varphi )\\mathbf {Z}^k(r),$ where $\\lbrace \\,\\mathbf {Z}^k(r)\\colon k\\in \\mathbb {Z}\\,\\rbrace $ is a sequence of centred stochastic processes with $\\begin{aligned}\\mathsf {E}[\\mathbf {Z}^k(r)\\otimes \\mathbf {Z}^l(r)]&=\\delta _{kl}B^{(k)}(r),\\\\\\sum _{k\\in \\mathbb {Z}}\\operatorname{tr}(B^{(k)}(r))&<\\infty .",
"\\end{aligned}$ It follows that $\\mathbf {Z}^k(r)=\\int _{0}^{2\\pi }\\overline{\\mathbf {\\omega }}(r,\\varphi )e_k(\\varphi )\\,\\mathrm {d}\\varphi .$ Then we have $ \\mathsf {E}[\\mathbf {Z}^k(r)\\otimes \\mathbf {Z}^l(s)]=\\int _{0}^{2\\pi }\\int _{0}^{2\\pi } \\mathsf {E}[\\overline{\\mathbf {\\omega }}(r,\\varphi _1)\\otimes \\overline{\\mathbf {\\omega }}(s,\\varphi _2)] e_k(\\varphi _1)\\,\\mathrm {d}\\varphi _1e_l(\\varphi _2)\\,\\mathrm {d}\\varphi _2.$ The field is isotropic and mean-square continuous, therefore $\\mathsf {E}[\\overline{\\mathbf {\\omega }}(r,\\varphi _1)\\otimes \\overline{\\mathbf {\\omega }}(s,\\varphi _2)] =B(r,s,\\cos (\\varphi _1-\\varphi _2))$ is a continuous function.",
"Note that $e_k(\\varphi )$ are spherical harmonics of degree $|k|$ .",
"Denote by $\\mathbf {x\\cdot y}$ the standard inner product in the space $\\mathbb {R}^d$ , and by $\\mathrm {d}\\omega (\\mathbf {y})$ the Lebesgue measure on the unit sphere $S^{d-1}=\\lbrace \\,\\mathbf {x}\\in \\mathbb {R}^d\\colon \\Vert \\mathbf {x}\\Vert =1\\,\\rbrace $ .",
"Then $\\int _{S^{d-1}}\\,\\mathrm {d}\\omega (\\mathbf {x})=\\omega _d=\\frac{2\\pi ^{d/2}}{\\Gamma (d/2)},$ where $\\Gamma $ is the Gamma function.",
"Now we use the Funk–Hecke theorem, see [1].",
"For any continuous function $f$ on the interval $[-1,1]$ and for any spherical harmonic $S_k(\\mathbf {y})$ of degree $k$ we have $\\int _{S^{d-1}}f(\\mathbf {x\\cdot y})S_k(\\mathbf {x})\\,\\mathrm {d}\\omega (\\mathbf {x})=\\lambda _kS_k(\\mathbf {y}),$ where $\\lambda _k=\\omega _{d-1}\\int _{-1}^{1}f(u)\\frac{C^{(n-2)/2}_k(u)}{C^{(n-2)/2}_k(1)} (1-u^2)^{(n-3)/2}\\,\\mathrm {d}u,$ $d\\ge 3$ , and $C^{(n-2)/2}_k(u)$ are Gegenbauer polynomials.",
"To see how this theorem looks like when $d=2$ , we perform a limit transition as $n\\downarrow 2$ .",
"By [1], $\\lim _{\\lambda \\rightarrow 0}\\frac{C^{\\lambda }_k(u)}{C^{\\lambda }_k(1)}=T_k(u),$ where $T_k(u)$ are Chebyshev polynomials of the first kind.",
"We have $\\omega _1=2$ , $\\mathbf {x\\cdot y}$ becomes $\\cos (\\varphi _1-\\varphi _2)$ , and $\\mathrm {d}\\omega (\\mathbf {x})$ becomes $\\mathrm {d}\\varphi _1$ .",
"We obtain $\\int _{0}^{2\\pi }B(r,s,\\cos (\\varphi _1-\\varphi _2))e_k(\\varphi _1)\\,\\mathrm {d}\\varphi _1 =B^{(k)}(r,s)e_k(\\varphi _2),$ where $B^{(k)}(r,s)=2\\int _{-1}^{1}B(r,s,u)T_{|k|}(u)(1-u^2)^{-1/2}\\,\\mathrm {d}u,$ Equation (REF ) becomes $\\mathsf {E}[\\mathbf {Z}^k(r)\\otimes \\mathbf {Z}^l(s)]=\\int _{0}^{2\\pi }B^{(k)}(r,s) e_k(\\varphi _2)e_l(\\varphi _2)\\,\\mathrm {d}\\varphi _2=\\delta _{kl}B^{(k)}(r,s).$ In particular, if $k\\ne l$ , then the processes $\\mathbf {Z}^k(r)$ and $\\mathbf {Z}^l(r)$ are uncorrelated.",
"Calculate the two-point correlation tensor of the random field $\\overline{\\mathbf {\\omega }}(r,\\varphi )$ .",
"We have $ \\begin{aligned}\\mathsf {E}[\\overline{\\mathbf {\\omega }}(r,\\varphi _1)\\otimes \\overline{\\mathbf {\\omega }}(s,\\varphi _2)]&=\\sum _{k=-\\infty }^{\\infty }e_k(\\varphi _1)e_k(\\varphi _2)B^{(k)}(r,s)\\\\&=\\frac{1}{2\\pi }B^{(0)}(r,s)+\\frac{1}{\\pi }\\sum _{k=1}^{\\infty }\\cos (k(\\varphi _1-\\varphi _2))B^{(k)}(r,s).",
"\\end{aligned}$ Now we add a time coordinate, $t$ , to our considerations.",
"A particle located at $(r,\\varphi )$ at time moment $t$ , was located at $(r,\\varphi -\\sqrt{GM}t/r^{3/2})$ at time moment 0.",
"It follows that $\\overline{\\mathbf {\\omega }}(t,r,\\varphi )=\\overline{\\mathbf {\\omega }}\\left(r,\\varphi -\\frac{\\sqrt{GM}t}{r^{3/2}}\\right),$ where $G$ is Newton's gravitational constant and $M$ is the mass of Saturn.",
"Equation (REF ) gives $ \\overline{\\mathbf {\\omega }}(t,r,\\varphi )=\\sum _{k=-\\infty }^{\\infty }e_k\\left(\\varphi -\\frac{\\sqrt{GM}t}{r^{3/2}}\\right)\\mathbf {Z}^k(r),$ while Equation (REF ) gives $\\begin{aligned}\\mathsf {E}[\\overline{\\mathbf {\\omega }}(t_1,r,\\varphi _1)\\otimes \\overline{\\mathbf {\\omega }}(t_2,s,\\varphi _2)] &=\\frac{1}{2\\pi }B^{(0)}(r,s)\\\\&\\quad +\\frac{1}{\\pi }\\sum _{k=1}^{\\infty }\\cos \\left(k\\left(\\varphi _1-\\varphi _2-\\frac{\\sqrt{GM}(t_1-t_2)}{r^{3/2}}\\right)\\right)B^{(k)}(r,s).",
"\\end{aligned}$ Conversely, let $\\lbrace \\,B^{(k)}(r,s)\\colon k\\ge 0\\,\\rbrace $ be a sequence of continuous positive-definite matrix-valued functions with $ \\sum _{k=0}^{\\infty }\\operatorname{tr}(B^{(k)}(r,r))<\\infty ,$ and let $\\lbrace \\,\\mathbf {Z}_k(r)\\colon k\\in \\mathbb {Z}\\,\\rbrace $ be a sequence of uncorrelated centred stochastic processes with $\\mathsf {E}[\\mathbf {Z}^k(r)\\otimes \\mathbf {Z}^l(s)]=\\delta _{kl}B^{(|k|)}(r,s).$ The random field (REF ) may describe rotating particles inside Saturn's rings, if all the functions $B^{(k)}(r,s)$ are equal to 0 outside the rectangle $[R_0,R_1]^2$ , where $R_0$ (resp.",
"$R_1$ ) is the inner (resp.",
"outer) radius of Saturn's rings.",
"To make our model more realistic, we assume that all the functions $B^{(k)}(r,s)$ are equal to 0 outside the Cartesian square $F^{2}$ , where $F $ is a fat fractal subset of the interval $[R_{0},R_{1}]$ , see [12].",
"[9] calls these sets dusts of positive measure.",
"Such a set has a positive Lebesgue measure, its Hausdorff dimension is equal to 1, but the Hausdorff dimension of its boundary is not an integer number.",
"A classical example of a fat fractal is a fat Cantor set.",
"In contrast to the ordinary Cantor set, where we delete the middle one-third of each interval at each step, this time we delete the middle $3^{-n}$ th part of each interval at the $n$ th step.",
"To construct an example, consider an arbitrary sequence of continuous positive-definite matrix-valued functions $\\lbrace \\,B^{(k)}(r,s)\\colon k\\ge 0\\,\\rbrace $ satisfying (REF ) of the following form: $B^{(k)}(r,s)=\\sum _{i\\in I_k}\\mathbf {f}_{ik}(r)\\mathbf {f}^{\\top }_{ik}(s),$ where $\\mathbf {f}_{ik}(r)\\colon [R_0,R_1]\\rightarrow \\mathbb {R}^3$ are continuous functions, satisfying the following condition: for each $r\\in [R_0,R_1]$ the set $I_{kr}=\\lbrace \\,i\\in I_k\\colon f_i(r)\\ne 0\\,\\rbrace $ is as most countable and the series $\\sum _{i\\in I_{kr}}\\Vert \\mathbf {f}_i(r)\\Vert ^2$ converges.",
"The so defined function is obviously positive-definite.",
"Put $\\tilde{B}^{(k)}(r,s)=\\sum _{i\\in I_k}\\tilde{\\mathbf {f}}_{ik}(r)\\tilde{\\mathbf {f}}^{\\top }_{ik}(s),\\qquad r,s\\in F.$ The functions $\\tilde{B}^{(k)}(r,s)$ are the restrictions of positive-definite functions $B^{(k)}(r,s)$ to $F^2$ and are positive-definite themselves.",
"Consider the centred stochastic process $\\lbrace \\,\\tilde{\\mathbf {Z}}^k(r)\\colon r\\in F\\,\\rbrace $ with $\\mathsf {E}[\\tilde{\\mathbf {Z}}^k(r)\\otimes \\tilde{\\mathbf {Z}}^l(s)]=\\delta _{kl}\\tilde{B}^{(|k|)}(r,s),\\qquad r,s\\in F.$ Condition (REF ) guarantees the mean-square convergence of the series $\\overline{\\mathbf {\\omega }}(t,r,\\varphi )=\\sum _{k=-\\infty }^{\\infty }e_k\\left(\\varphi -\\frac{\\sqrt{GM}t}{r^{3/2}}\\right)\\tilde{\\mathbf {Z}}^k(r)$ for all $t\\ge 0$ , $r\\in F$ , and $\\varphi \\in [0,2\\pi ]$ ."
],
[
"Closure",
"This paper reports an investigation of the fractal character of Saturnian rings.",
"First, working with the calculus in a non-integer dimensional space, by energy arguments, we infer that the fractally structured ring is more likely than a non-fractal one.",
"Next, we develop a kinematics model in which angular velocities of particles form a random field."
]
] | 1612.05499 |
[
[
"Quantifying the Heat Dissipation from a Molecular Motor's Transport\n Properties in Nonequilibrium Steady States"
],
[
"Abstract Theoretical analysis, which maps single molecule time trajectories of a molecular motor onto unicyclic Markov processes, allows us to evaluate the heat dissipated from the motor and to elucidate its dependence on the mean velocity and diffusivity.",
"Unlike passive Brownian particles in equilibrium, the velocity and diffusion constant of molecular motors are closely inter-related to each other.",
"In particular, our study makes it clear that the increase of diffusivity with the heat production is a natural outcome of active particles, which is reminiscent of the recent experimental premise that the diffusion of an exothermic enzyme is enhanced by the heat released from its own catalytic turnover.",
"Compared with freely diffusing exothermic enzymes, kinesin-1 whose dynamics is confined on one-dimensional tracks is highly efficient in transforming conformational fluctuations into a locally directed motion, thus displaying a significantly higher enhancement in diffusivity with its turnover rate.",
"Putting molecular motors and freely diffusing enzymes on an equal footing, our study offers thermodynamic basis to understand the heat enhanced self-diffusion of exothermic enzymes."
],
[
"Aknowledgement",
"We thank Bae-Yeun Ha, Hyunggyu Park, Dave Thirumalai, and Anatoly Kolomeisky for helpful comments and illuminating discussions.",
"We acknowledge the Center for Advanced Computation in KIAS for providing computing resources.",
"Figure: Analysis of experimental data, digitized from Ref.",
", using (NN=4)-state cyclic model.The solid lines are the fits to the dataA.",
"VV vs [ATP] at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).B.",
"VV vs ff at [ATP] = 5 μ\\mu M.C.",
"VV vs ff at [ATP] = 2 mM.D.",
"Stall force as a function of [ATP], measured by `Position clamp' (red square) or `Fixed trap' (blue circle) methods.E.",
"DD vs [ATP] at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).DD was estimated from r=2D/Vd 0 r=2D/Vd_0.",
"F. DD vs ff at [ATP] = 2 mM.Table: Parameters determined from the fit using (N=4)-state model.",
"The unit of {u n }\\lbrace u_n\\rbrace and {w n }\\lbrace w_n\\rbrace is s -1 s^{-1} except for u 1 o u_1^o ([u 1 o ]=μM -1 s -1 [u_1^o]={\\mu M}^{-1} s^{-1}).Supplementary Information"
],
[
"1. Derivation of the third degree polynomial dependence of $D$ on {{formula:bfe6300b-f497-4028-bece-5a1753aeadd7}} .",
"Here, we show a polynomial dependence of $D$ on $V$ using a few specific examples of the $N$ -state periodic reaction model [19] whose reaction scheme is demonstrated in Fig.",
"REF .",
"When $N$ =1, the master equation to solve is: $\\dot{\\pi }_\\mu (t)=u_1\\pi _{\\mu -1}(t)+w_1\\pi _{\\mu +1}(t)-(u_1+w_1)\\pi _\\mu (t),$ where $\\pi _\\mu (t)$ is the probability of motor being in the $\\mu $ -th reaction cycle at time $t$ .",
"Using generating function $F(z,t)=\\sum _{\\mu =-\\infty }^{\\infty }z^\\mu \\pi _\\mu (t)$ with $\\pi _\\mu (0)=\\delta _{\\mu ,0}$ [49], the master equation is written in terms of $F(z,t)$ as $\\partial _tF(z,t)&=\\left(u_1z+\\frac{w_1}{z}-(u_1+w_1)\\right)F(z,t)\\nonumber \\\\F(z,t)&=e^{\\left(u_1z+\\frac{w_1}{z}-(u_1+w_1)\\right)t}.$ Now, it is straightforward to obtain the mean velocity ($V$ ) and diffusion constant ($D$ ) using $\\partial _z\\log {F(z,t)}|_{z=1}=\\langle \\mu (t)\\rangle $ and $\\partial ^2_z\\log {F(z,t)}|_{z=1}=\\langle \\mu ^2(t)\\rangle -\\langle \\mu (t)\\rangle ^2-\\langle \\mu (t)\\rangle $ , where $\\mu (t)$ is the number of steps taken by the molecular motor until time $t$ .",
"$V\\equiv \\lim _{t\\rightarrow \\infty }\\frac{d_0\\langle \\mu (t)\\rangle }{t}= d_0(u_1-w_1)$ and $D\\equiv \\lim _{t\\rightarrow \\infty }\\frac{d^2_0(\\langle \\mu ^2(t)\\rangle -\\langle \\mu (t)\\rangle ^2)}{2t} = \\frac{d_0^2(u_1+w_1)}{2}.$ Provided that only $u_1$ changes (for example by increasing ATP concentration) while $w_1$ remains constant.",
"elimination of $u_1$ from $V(u_1)$ and $D(u_1)$ relates $D$ to $V$ as $ D(V) = D_0+\\frac{d_0}{2}V$ where $D_0\\equiv d_0^2w_1,$ showing that for ($N$ =1)-state kinetic model, $D$ is linear in $V$ ."
],
[
"($N$ =2)-state kinetic model",
"For the ($N$ =2)-kinetic model [20], $V = d_0\\frac{u_1 u_2 - w_1 w_2}{u_1 + u_2 + w_1 + w_2}$ and $D=\\frac{d_0^2}{2}\\left[\\frac{u_1u_2}{w_1w_2}+1-2\\left(\\frac{u_1u_2}{w_1w_2}-1\\right)^2\\frac{w_1w_2}{\\sigma ^2}\\right]\\frac{w_1w_2}{\\sigma }$ where $\\sigma =u_1+u_2+w_1+w_2$ .",
"Then, $D=D(V)$ is obtained by eliminating $u_1$ between Eq.REF and Eq.REF : $D(v)/d_0^2&=\\left(\\frac{u_2}{\\kappa +1}\\right)+\\frac{1}{2}\\left(\\frac{\\kappa -1}{\\kappa +1}\\right)u_2v-\\left(\\frac{u_2}{\\kappa +1}\\right)\\frac{u_2^2}{w_1w_2}v^2+\\left(\\frac{u_2}{\\kappa +1}\\right)\\frac{u_2^2}{w_1w_2}v^3$ where $\\kappa \\equiv \\frac{u_2(u_2+w_1+w_2)}{w_1w_2}$ , and $v\\equiv V/V_{max}$ ($0\\le v\\le 1$ ) with $V_{max}=d_0u_2$ .",
"Eq.REF confirms that $D$ is a third order polynomial in $V$ .",
"Incidentally, the ($N$ =2)-kinetic model is reduced to the Michaelis-Menten equation by setting $u_1=u_1^o[S]$ and $w_2=0$ .",
"$V= d_0\\frac{u_1 u_2}{u_1 + u_2 + w_1} = \\frac{V_{max}[S]}{ K_M + [S] }$ where $K_M = (u_2 + w_1)/u_1^o$ , and $D(v)=D_{max}v\\left[1-2\\phi v+2\\phi v^2\\right]$ with $\\phi \\equiv \\frac{k_{cat}}{u_1^oK_M}$ .",
"Note that $D\\le D_{max}=d_0V_{max}/2$ and that for $D(v)$ to be positive for all the range of $v$ , the parameter $\\phi $ should be in a rather narrow range of $0\\le \\phi \\le 2$ ."
],
[
"General case: $N$ -state kinetic model",
"The above two examples of one-dimensional hopping model was extended to the $N$ -state kinetic model by Derrida [19].",
"He obtained exact expressions for the mean velocity ($V^{(D)}$ ) and diffusion constant ($D^{(D)}$ ), where the superscript $(D)$ refers to Derrida's, in terms of the rate constants $\\lbrace u_n\\rbrace $ and $\\lbrace w_n\\rbrace $ .",
"Derrida's expression for $V^{(D)}$ and $D^{(D)}$ are related to $V$ and $D$ as $V\\equiv (d_0/N)V^{(D)}$ and $D\\equiv (d_0^2/N^2)D^{(D)}$ .",
"$V^{(D)} = \\frac{N}{\\sum _{n=1}^{N} r_n}\\left(1- \\frac{\\prod _{n=1}^{N} w_n}{\\prod _{n=1}^{N} u_n }\\right)$ and $D^{(D)}&=\\frac{1}{\\left(\\sum _{n=1}^Nr_n\\right)^2}\\left[V^{(D)}\\sum _{n=1}^Nq_n\\sum _{i=1}^Nir_{n+i}+N\\sum _{n=1}^Nu_nq_nr_n\\right]-V^{(D)}\\frac{N+2}{2}$ where $r_n = \\frac{1}{u_n} \\left[1 + \\sum _{i=1}^{N-1} \\prod _{j=1}^{i} \\frac{w_{n+j-1}}{u_{n+j}}\\right]$ , and $q_n=\\frac{1}{u_n}\\left[1+\\sum _{i=1}^{N-1}\\prod _{j=1}^i\\frac{w_{n-j}}{u_{n-j}}\\right]$ with periodic boundary conditions $u_{n+N} = u_n$ , and $w_{n+N} = w_n$ .",
"$D^{(D)} = D^{(D)}(V^{(D)})$ is obtained by eliminating $u_1$ between Eq.REF and Eq.REF .",
"For that, we first express various terms in Eq.REF in terms of $u_1$ : $ \\sum _{n=1}^Nr_n&=\\frac{A}{u_1}+B,\\nonumber \\\\\\sum _{n=1}^Nq_n\\sum _{i=1}^Nir_{n+i}&=\\frac{\\alpha }{u_1^2}+\\frac{\\beta }{u_1}+\\gamma ,\\nonumber \\\\\\sum _{n=1}^Nu_nq_nr_n&=\\frac{\\xi }{u_1^2}+\\frac{\\eta }{u_1}+\\zeta .$ where A, $B\\left(=\\sum _{n=2}^N\\frac{1}{u_n}\\left[1+\\sum _{i=1}^{N-n}\\prod _{j=1}^i\\frac{w_{n+j-1}}{u_{n+j}}\\right]\\right)$ , $\\alpha $ , $\\beta $ , $\\gamma $ , $\\xi $ , $\\eta $ , and $\\zeta $ are all positive constants independent of $u_1$ .",
"Next, $\\sum _{n=1}^Nr_n$ in Eq.REF substituted to Eq.REF gives $ V^{(D)}=N\\frac{1-C/u_1}{A/u_1+B},$ where $C\\left(= \\prod _{n=1}^Nw_n/\\prod _{n=2}^Nu_n \\right)$ , and $u_1$ is expressed in terms of $V^{(D)}$ $\\frac{1}{u_1}=\\frac{1-B\\frac{V^{(D)}}{N}}{C+A\\frac{V^{(D)}}{N}}.$ Finally, with Eqs.REF and REF , we show that $D^{(D)}$ (Eq.REF ) can be expressed as a third degree polynomial in $V^{(D)}$ $D^{(D)}&=V^{(D)}\\frac{\\left[\\alpha (1-B\\frac{V^{(D)}}{N})^2+\\beta (1-B\\frac{V^{(D)}}{N})(A\\frac{V^{(D)}}{N}+C)+\\gamma (A\\frac{V^{(D)}}{N}+C)^2\\right]}{(A+BC)^2}\\nonumber \\\\&\\hspace{14.22636pt}+N\\frac{\\xi (1-B\\frac{V^{(D)}}{N})^2+\\eta (1-B\\frac{V^{(D)}}{N})(A\\frac{V^{(D)}}{N}+C)+\\zeta (A\\frac{V^{(D)}}{N}+C)^2}{(A+BC)^2}-V^{(D)}\\frac{N+2}{2}\\nonumber \\\\&=z_0+z_1V^{(D)}+z_2(V^{(D)})^2+z_3(V^{(D)})^3\\nonumber \\\\D&=\\alpha _0+\\alpha _1V+\\alpha _2V^2+\\alpha _3V^3.$ with $\\alpha _i=(d_0/N)^{2-i}z_i$ , and $z_0&=\\frac{(\\xi +\\eta C+\\zeta C^2)}{(A+BC)^2}\\nonumber \\\\z_1&=\\left[\\frac{(\\alpha +\\beta C+\\gamma C^2)+(-2\\xi B+\\eta (A-BC)+2\\zeta AC)}{(A+BC)^2}-\\frac{N+2}{2}\\right]\\nonumber \\\\z_2&=\\frac{(-2\\alpha B+\\beta (A-BC) +2\\gamma AC)+(\\xi B^2-\\eta AB +\\zeta A^2)}{(A+BC)^2}\\nonumber \\\\z_3&=\\frac{(\\alpha B^2-\\beta AB +\\gamma A^2)}{(A+BC)^2}.",
"\\nonumber $"
],
[
"Alternative derivation of $D(V)$",
"In addition to Derrida's result [19], the sign of $\\alpha _i$ can be determined by deriving the relation between $V^{ (D) }$ and $D^{ (D) }$ using the result in ref.",
"[50].",
"From Eq.",
"(23) in ref.",
"[50], $ \\begin{aligned}V^{ (D) } = - i \\frac{c^{\\prime }_0}{c_1}\\end{aligned}$ where $c^{\\prime }_0 = i N ( \\prod _{n=1}^{N} u_n - \\prod _{n=1}^{N} w_n )$ and $c_1 = c_1( \\lbrace u_n\\rbrace , \\lbrace w_n\\rbrace )$ .",
"By combining two expressions of $V^{ (D) }$ , Eq.REF and Eq.REF , we get $c_1& = \\prod _{n=1}^{N} u_n\\times \\sum _{m=1}^{N-1} \\left[ \\frac{1}{u_m} \\left( 1 + \\sum _{i=1}^{N-1} \\prod _{j=1}^{i} \\frac{w_{m+j}}{u_{m+j}}\\right) \\right] \\nonumber \\\\& = u_1 \\prod _{n=2}^{N} u_n \\left[\\frac{1}{u_1} \\left( 1 + \\sum _{i=1}^{N-1} \\prod _{j=1}^{i} \\frac{w_{1+j}}{u_{1+j}} \\right)+ \\sum _{m=2}^{N-1} \\frac{1}{u_m} \\left( 1 + \\sum _{i=1}^{N-1} \\prod _{j=1}^{i} \\frac{w_{m+j}}{u_{m+j}} \\right)\\right]\\nonumber \\\\& = \\mathcal {A} u_1 + \\mathcal {B}$ where $\\mathcal {A}$ and $\\mathcal {B}$ are constants depending on ($u_2,\\ldots ,u_N$ ) and ($w_1,w_2,\\ldots w_N$ ).",
"Eq.",
"(REF ) substituted to Eq.",
"(REF ) gives $\\begin{aligned}V^{ (D) } = N \\frac{u_1 \\prod _{n=2}^{N} u_n - \\prod _{n=1}^{N} w_n}{\\mathcal {A}u_1 + \\mathcal {B}}\\end{aligned}$ and hence $u_1$ can be written as $\\begin{aligned}u_1 = \\frac{\\mathcal {B} V^{ (D) } + N \\prod _{n=1}^N w_n}{N \\prod _{n=2}^{N} u_n - \\mathcal {A}V^{ (D) }}.\\end{aligned}$ From Eq.REF and Eq.REF , $\\mathcal {A} = B \\prod _{n=2}^N u_n$ and $\\mathcal {B} = A \\prod _{n=2}^N u_n$ where $A$ and $B$ are the same constants used in Eq.REF .",
"Now using Eq.",
"(REF ), Eq.",
"(REF ), and general expression of $D^{ (D) }$ from Eq.",
"(24) of ref.",
"[50], we have $\\begin{aligned}D^{ (D) } & = \\frac{c_0^{\\prime \\prime } - 2 c_2 (V^{ (D) })^2 }{2 c_1 } \\\\&=\\frac{c_0^{\\prime \\prime } - 2 c_2 (V^{ (D) })^2 }{2 (\\frac{B}{\\prod _{n=2}^N u_n} u_1 + \\frac{A}{\\prod _{n=2}^N u_n}) } \\\\& = \\frac{( c_0^{\\prime \\prime } - 2 c_2 (V^{ (D) })^2 ) (\\prod _{n=2}^{N} u_n (N- B V^{(D)}) )}{2N( A + C B)}\\end{aligned}$ where $c_0^{\\prime \\prime } = N^2 ( u_1 \\prod _{n=2}^{N} u_n + \\prod _{n=1}^{N} w_n ) = N^2 (\\prod _{n=2}^N u_n) (u_1 + C)$ , and $c_2 = \\beta _1 u_1 + \\beta _2$ where $\\beta _1$ and $\\beta _2$ are positive constants depending on ($u_2, u_3, \\cdots , u_N$ ) and ($w_1, w_2, \\cdots , w_N$ ) (see Eq.",
"(53, 54) of ref.",
"[50]).",
"Since $u_1 (N - B V^{ (D) }) = NC + AV^{ (D) }$ (Eq.REF ), we get $D^{ (D) } & = \\frac{1}{2} \\frac{( c_0^{^{\\prime \\prime }} - 2 c_2 (V^{ (D) })^2 ) ( (N- B V^{(D)}) )\\prod _{n=2}^{N} u_n}{N A + N C B} \\nonumber \\\\& = \\frac{1}{2} \\frac{( N^2 (\\prod _{n=2}^N u_2) (u_1 + C) - 2 (\\beta _1 u_1 + \\beta _2) (V^{ (D) })^2 ) ( (N- B V^{(D)}) )\\prod _{n=2}^{N} u_n}{N A + N C B} \\nonumber \\\\& = \\frac{\\prod _{n=2}^N u_n}{2} \\frac{ \\left( N^2 (\\prod _{n=2}^N u_2) ( NC + A V^{(D)} + C(N-B V^{ (D) } ) ) - 2 ( \\beta _1 (N C + A V^{(D)}) + \\beta _2 ( N- B V^{(D)} ) ) (V^{ (D) })^2 \\right)}{N A + N C B} \\nonumber \\\\&= z_0 + z_1 V^{ (D) } + z_2 (V^{ (D) })^2 + z_3 (V^{ (D) })^3$ where $z_0&=\\frac{N^2 (\\prod _{n=2}^N u_n )^2 C}{ A + BC }>0\\nonumber \\\\z_1&=\\frac{N (\\prod _{n=2}^N u_n )^2 (A- BC)}{2 (A + BC)}\\nonumber \\\\z_2&=- \\frac{(\\prod _{n=2}^N u_n )( \\beta _1 C + \\beta _2 )}{A + BC}<0\\nonumber \\\\z_3&= \\frac{(\\prod _{n=2}^N u_n )( - \\beta _1 A + \\beta _2 B)}{N(A + BC)}.\\nonumber $ It is obvious that $z_0 >0$ and $z_2 < 0$ since $A$ , $B$ , $C$ , $\\beta _1$ , and $\\beta _2$ are all positive constants."
],
[
"2. The 1D hopping model with a finite processivity",
"Because of a probability of being dissociated from microtubules, kinesin motors display a finite processivity.",
"However, since the mean velocity and diffusion constant are calculated from the trajectories that remain on the track, the expressions of $V$ and $D$ in terms of the rate constants are unchanged.",
"To make this point mathematically more explicit, we consider the master equation assuming a constant dissociation rate $k_d$ from each chemical state.",
"$\\frac{d P_{\\mu ,n} (t)}{d t}&=u_{n-1} P_{\\mu , n-1}(t)+w_{n}P_{\\mu , n+1}(t)\\nonumber \\\\&-(u_n+w_{n-1}+k_d)P_{\\mu , n}(t)$ where $P_{\\mu , n}(t)$ is the probability of being in the $n$ -th chemical state at the $\\mu $ -th reaction cycle.",
"The probability of the motor remaining on the track (survival probability of motor) is $ S (t) \\equiv \\sum _{\\mu =\\infty }^{\\infty } \\sum _{n=1}^{N} P_{\\mu , n}(t) = e^{-k_d t}.$ The expectation value of an observable, which can be used to calculate $\\langle x(t)\\rangle $ or $\\langle x^2(t)\\rangle $ , is expressed as $\\langle A(t)\\rangle = \\sum _{\\mu =-\\infty }^{\\infty } \\sum _{n=1}^{N} \\Phi _{\\mu ,n}(t) A(\\mu (t))$ with a probability density function renormalized with respect to the survival probability $ \\Phi _{\\mu , n}(t) \\equiv \\frac{P_{\\mu ,n}(t)}{S(t)}=P_{\\mu ,n}(t)e^{k_dt}.$ Incidentally, $\\Phi _{\\mu , n}(t)$ satisfies the following master equation.",
"$\\frac{d\\Phi _{\\mu ,n} (t)}{dt}&=u_{n-1}\\Phi _{\\mu , n-1}(t)+w_{n}\\Phi _{\\mu , n+1}(t) \\\\&\\hspace{28.45274pt}-(u_n+w_{n-1})\\Phi _{\\mu , n}(t),$ which is identical to Eq.REF , but now the probability of interest is explicitly confined to the ensemble of trajectories remaining on the track.",
"For an arbitrary value of $k_d$ and for any $N$ , the expressions of $V$ , $D$ , and Eq.REF remain unchanged except that the range of ensemble is specific to the motor trajectories remaining on the track.",
"Furthermore, the expression of $\\dot{Q}$ , which depends only on $V$ and rate constants, remains identical in the presence of detachment (finite $k_d>0$ ).",
"Therefore, our formalisms remain valid for motors with a finite processivity."
],
[
"3. Mapping the master equation for $N$ -state kinetic model onto Langevin and Fokker-Planck equations",
"The master equation (Eq.REF ) can be mapped onto a Langevin equation for position $x(t)$ as $\\dot{x}(t)=V+\\sqrt{2D}\\eta (t)$ where for (N=1)-state model $V=d_0(u_1-w_1)$ and $D=d_0^2/2\\times (u_1+w_1)$ as in Eqs.REF and REF , and $P[\\eta (t)]\\propto \\exp {\\left(-\\frac{1}{2}\\int _0^td\\tau \\eta ^2(\\tau )\\right)}$ .",
"Then, with the transition probability (propagator), $P(x_{t+\\epsilon }|x_t)&=\\left(\\frac{1}{4\\pi D\\epsilon }\\right)^{1/2}e^{-\\frac{\\lbrace x_{t+\\epsilon }-x_t-\\epsilon V\\rbrace ^2}{4D\\epsilon }},$ where $x_{t}\\equiv x(t)$ , and starting from an initial condition, $P[x(0)]=\\delta [x(0)-x_0]$ , it is straightforward to obtain the position of motor at time $t$ : $P[x(t)]&=\\int dx_0\\int dx_{\\epsilon }\\cdots \\int dx_{t-\\epsilon }P(x_t|x_{t-\\epsilon })\\cdots P(x_{2\\epsilon }|x_{\\epsilon })P(x_{\\epsilon }|x_0)P(x_0) \\nonumber \\\\&=\\left(\\frac{1}{4\\pi Dt}\\right)^{1/2}\\exp {\\left(-\\frac{(x(t)-x_0-Vt)^2}{4Dt}\\right)}\\nonumber \\\\&=\\left(\\frac{1}{2\\pi d_0^2(u_1+w_1)t}\\right)^{1/2} \\exp {\\left(-\\frac{[x(t)-x_0-d_0(u_1-w_1)t]^2}{2d_0^2(u_1+w_1)t}\\right)},$ where we plugged $V$ and $D$ from Eqs.REF , REF for ($N$ =1)-state kinetic model in the last line.",
"Unlike the normal Langevin equation, where the noise strength determined by FDT is associated with an ambient temperature ($\\sim \\sqrt{T}$ ), the noise strength in Eq.REF is solely determined by the forward and backward rate constants, which fundamentally differs from the Brownian motion of a thermally equilibrated colloidal particle in a heat bath.",
"Next, Fokker-Planck equation follows from Eq.REF , $\\partial _t P(x,t) &= D \\partial _x^2 P(x,t) - V \\partial _x P(x,t)\\nonumber \\\\&=-\\partial _x j(x,t)$ with the probability current being defined as $j (x,t) = - D \\partial _x P(x,t) + V P(x,t).$ Then, mean local velocity $v(x,t)\\equiv j(x,t)/P(x,t)$ is defined $v(x,t)= V - D \\partial _x \\log { P(x,t) }.$ In order to relate this definition of the mean local velocity to heat dissipated from the molecular motor moving along microtubules in a NESS, we consider $\\gamma _{\\text{eff}}$ , an effective friction coefficient, and introduce a nonequilibrium potential $\\phi (x)\\equiv -\\log {P^{ss}(x)}$ [51], [52].",
"By integrating the both side of Eq.REF in a NESS with respect to the displacement corresponding to a single step, we obtain $\\int _x^{x+d_0} \\gamma _{\\text{eff}} v^{ss}(x)dx &= \\gamma _{\\text{eff}} V d_0 +\\gamma _{\\text{eff}}D(\\phi (x+d_0)-\\phi (x)).$ Following the literature on NESS thermodynamics [52], [53], [51], we endowed each term of Eq.REF with its physical meaning.",
"(i) housekeeping heat: $Q_{hk}=\\int _x^{x+d_0} \\gamma _{\\text{eff}} v(x)dx,$ (ii) total heat: $Q=\\gamma _{\\text{eff}} V d_0$ and (iii) excess heat: $Q_{ex}=-\\gamma _{\\text{eff}}D(\\phi (x+d_0)-\\phi (x)).$ Eqs.REF , REF , and REF satisfy $Q_{hk}=Q-Q_{ex}.$ and in fact $Q_{ex}=0$ because of the periodic boundary condition implicit to our problem of molecular motor, which leads to $\\phi (x+d_0)=\\phi (x)$ .",
"Hence, $Q_{hk}=Q=\\gamma _{\\text{eff}}Vd_0.$ Although we introduced the effective friction coefficient $\\gamma _{\\text{eff}}$ in Eq.REF to define the heats produced at nonequilibrium, Eq.REF finally allows us to associate $\\gamma _{\\text{eff}}$ with other physically well-defined quantities.",
"$\\gamma _{\\text{eff}}=\\frac{Q_{hk}}{Vd_0} = \\frac{k_B T}{d^2_0(j_+-j_-)} \\log { \\left( \\frac{j_+}{j_-} \\right) }.$ Here, note that we for the first time introduced the temperature $T$ , which was discussed neither in the master equation (Eq.REF ) nor in the Langevin equation (Eq.REF ).",
"Of special note is that $\\gamma _{\\text{eff}}$ does not remain constant, but depends on the steady state flux $j^{ss}=j=j_+-j_-$ .",
"Similar to the effective diffusion constant of bacterium, $D_{\\text{eff}}$ , discussed in the main text, $\\gamma _{\\text{eff}}$ is defined operationally.",
"At equilibrium, when the detailed balance (DB) is established ($j_+=j_-=j_0$ ), $\\gamma _{\\text{eff}}^{\\text{DB}}$ approaches to: $\\gamma _{\\text{eff}}^{\\text{DB}} &=\\lim _{j_+\\rightarrow j_-}\\frac{k_B T}{d^2_0(j_+-j_-)} \\log { \\left( \\frac{j_+}{j_-} \\right) }\\rightarrow \\frac{k_B T}{d^2_0j_{0}}.$ In fact, $D_0=k_BT/\\gamma _{\\text{eff}}^{\\text{DB}}=d_0^2j_0$ satisfies the FDT for passive particle at thermal equilibrium, i.e., $k_BT=D_0\\gamma ^{\\text{DB}}_{\\text{eff}}$ .",
"For ($N$ =1)-state model, $D_0=d_0^2w_1$ , which is identical to Eq.REF ."
],
[
"4. Nonequilibrium steady state thermodynamics.",
"To drive a system out of equilibrium, one has to supply a proper form of energy into the system.",
"Molecular motors move in one direction because transduction of chemical free energy into conformational change is processed.",
"Relaxation from a nonequilibrium state is accompanied with heat and entropy production.",
"In the presence of external nonconservative force (chemical or mechanical force), the system reaches the nonequilibrium steady state.",
"If one considers a Markov dynamics for microscopic state $i$ , described by the master equation $\\partial _t p_i(t)=-\\sum _j (W_{ij}p_i(t)-W_{ji}p_j(t))$ , the system relaxes to nonequilibrium steady state at long time, establishing time-independent steady state probability $\\lbrace p_i^{ss}\\rbrace $ for each state satisfying the zero flux condition $\\sum _j(W_{ij}p^{ss}_i-W_{ji}p^{ss}_j)=0$ .",
"A removal of the nonconservative force is led to further relaxation to the equilibrium ensemble, in which the detailed balance (DB) is (locally) established in every pair of the states such that $p_i^{eq}W_{ij}=p_j^{eq}W_{ji}$ for all $i$ and $j$ .",
"An important feature of the equilibrium, which differentiates itself from NESS, is the condition of DB.",
"Over the decade, there have been a number of endeavors to better characterize the system out-of-equilibrium [51].",
"One of them is to define the heat and entropy production in the context of Master equation.",
"The heat and entropy productions in reference to either steady state or equilibrium are defined to better characterize the process of interest.",
"The aim is to associate the time dependent probability for state ($\\lbrace p_i(t)\\rbrace $ ) and transition rates between the states $\\lbrace W_{ij}\\rbrace $ with newly defined macroscopic thermodynamic quantities at nonequilibrium [54].",
"Here, we review NESS thermodynamics formalism developed by Ge and Qian [54].",
"For nonequilibrium relaxation processes one can consider three relaxation processes: (i) relaxation process of a system far-from-equilibrium (FFE) to a nonequilibrium steady state (NESS); (ii) relaxation process of a system far-from-equilibrium (FFE) to an equilibrium (EQ).",
"(iii) relaxation process of a system in NESS to an equilibrium (EQ).",
"To describe these relaxation processes using the probabilities for state, we introduce a phenomenological definition of an internal energy of state $i$ at a steady state by $u^{ss}_i=-k_BT\\log {p_i^{ss}}$ , and at equilibrium by $u^{eq}_i=-k_BT\\log {p_i^{eq}}$ .",
"Then the following thermodynamic quantities are defined either in reference to NESS or equilibrium.",
"First, the thermodynamic potentials are defined in reference to the NESS: the total energy $U(t)=\\sum _i^Np_i(t)u^{ss}_i$ ; the total free energy $F(t)=U(t)-TS(t)=k_BT\\sum _i^Np_i(t)\\log {(p_i(t)/p_i^{ss})}$ .",
"Second, the thermodynamic potentials are defined in reference to the equilibrium: the total energy $U^{eq}(t)=\\sum _i^Np_i(t)u^{eq}_i$ ; the total free energy $F^{eq}(t)=U^{eq}(t)-TS(t)=k_BT\\sum _i^Np_i(t)\\log {(p_i(t)/p_i^{eq})}$ .",
"In both cases, Gibbs entropy, $S(t)=-k_B\\sum _i^Np_i(t)\\log {p_i(t)}$ , is defined as usual.",
"Next, the above definitions of generalized thermodynamic potentials, one can define the heat and entropy productions associated with the relaxation processes (i), (ii), (iii).",
"The diagram in Fig.REF depicts the relaxation processes mentioned here.",
"Figure: A diagram illustrating the balances between various thermodynamic quantities discussed in the text.",
"The curvy arrows denote the heat and entropy production from relaxation processes.$-dF(t)/dt$ is the rate of entropy production in the relaxation from FFE to NESS, $\\frac{dF(t)}{dt}\\equiv -\\dot{f}_d&=-T\\sum _{i>j}\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\right]\\log {\\left[\\frac{p_i(t)p_j^{ss}}{p_j(t)p_i^{ss}}\\right]}.$ and $-dU(t)/dt$ is the rate of heat production in the relaxation from FFE to NESS.",
"$\\frac{dU(t)}{dt}\\equiv -\\dot{Q}_{ex}&=-\\sum _{i>j}(W_{ij}p_i(t)-W_{ji}p_j(t))(u^{ss}_i-u^{ss}_j)\\nonumber \\\\&=T\\sum _{i>j}(W_{ij}p_i-W_{ji}p_j)\\log {\\left(\\frac{p_i^{ss}}{p_j^{ss}}\\right)}.$ Similarly, $-dF^{eq}(t)/dt$ is the rate of entropy production during the relaxation to equilibrium, $\\frac{dF^{eq}(t)}{dt}\\equiv -T\\dot{e}_p&=-T\\sum _{i>j}\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\right]\\log {\\left[\\frac{p_i(t)W_{ij}}{p_j(t)W_{ji}}\\right]}$ and $-dU^{eq}(t)/dt$ is the rate of heat production.",
"$\\frac{dU^{eq}(t)}{dt}\\equiv -\\dot{h}_d&=-\\sum _{i>j}(W_{ij}p_i(t)-W_{ji}p_j(t))(u^{eq}_i-u^{eq}_j)\\nonumber \\\\&=T\\sum _{i>j}(W_{ij}p_i-W_{ji}p_j)\\log {\\left(\\frac{W_{ij}}{W_{ji}}\\right)}$ where the condition of DB ($p_i^{eq}/p_j^{eq}=W_{ji}/W_{ij}$ ) was used to derive the last line.",
"Furthermore, the heat production involved with the relaxation from NESS to equilibrium ($\\dot{Q}_{hk}$ ), namely housekeeping heat which is introduced in NESS stochastic thermodynamics from the realization that maintaining NESS requires some energy, is defined by either using $\\dot{Q}_{hk}=(-dF^{eq}(t)/dt)-(-dF(t)/dt)=T\\dot{e}_p-\\dot{f}_d$ or $\\dot{Q}_{hk}=(-dU^{eq}(t)/dt)-(-dU(t)/dt)=\\dot{h}_d-\\dot{Q}_{ex}$ .",
"Explicit calculations using the representation of thermodynamic potential in terms of master equation lead to $\\dot{Q}_{hk}=T\\sum _{i>j}\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\right]\\log {\\left[\\frac{p^{ss}_iW_{ij}}{p_j^{ss}W_{ji}}\\right]}.$ Lastly, from the definition of Gibbs entropy ($S(t)=-k_B\\sum _ip_i(t)\\log {p_i(t)}$ ), or from the thermodynamic relationships $TdS/dt=dF/dt-dU/dt=dF^{eq}/dt-dU^{eq}/dt$ , it is straightforward to show that $T\\frac{dS}{dt}&=\\frac{dF}{dt}-\\frac{dU}{dt}=\\frac{dF^{eq}}{dt}-\\frac{dU^{eq}}{dt}\\nonumber \\\\&=-T\\sum _{i>j}\\left[W_{ij}p_i(t)-W_{ji}p_j(t)\\right]\\log {\\left(\\frac{p_i(t)}{p_j(t)}\\right)}\\nonumber \\\\&=\\dot{h}_d-T\\dot{e}_p.$ Now, with the various heat and entropy production defined from generalized potentials $F(t)$ , $U(t)$ , and $F^{eq}(t)$ , $U^{eq}(t)$ ($\\dot{f}_d$ , $\\dot{Q}_{ex}$ , $T\\dot{e}_p$ , $\\dot{h}_d$ , and $\\dot{Q}_{hk}$ ) we acquire two important balance laws in nonequilibrium thermodynamics: $T\\dot{e}_p&=\\dot{f}_d+\\dot{Q}_{hk}\\nonumber \\\\\\dot{h}_d&=\\dot{Q}_{hk}+\\dot{Q}_{ex}$ Thus, (i) the total entropy production of a system, $T\\dot{e}_p(=-dF^{eq}/dt)$ , is contributed by the free energy dissipation due to the relaxation to NESS, $\\dot{f}_d(=-dF/dt)$ , and the housekeeping heat, $\\dot{Q}_{hk}(=dF/dt-dF^{eq}/dt)$ , that is required to maintain the NESS.",
"(ii) The total heat production $\\dot{h}_d(=-dU^{eq}/dt)$ of a system is decomposed into $\\dot{Q}_{hk}(=dU/dt-dU^{eq}/dt)$ and the excess heat $\\dot{Q}_{ex}(=-dU/dt)$ .",
"The diagram in Fig.REF recapitulates the various heat and entropy production terms and their balance.",
"When the system is already in NESS, then neither the production of entropy nor excess heat is anticipated ($\\dot{f}_d=0$ , $\\dot{Q}_{ex}=0$ ), and hence it follows that the amount of heat, entropy, and housekeeping heat required to sustain NESS are identical ($T\\dot{e}_p=\\dot{Q}_{hk}=\\dot{h}_d$ ).",
"Figure: Relaxation dynamics of various nonequilibrium thermodynamic quantities from far-from-equilibrium states calculated using (N=2)-state system.The parameters used for the plots are:[ATP] = 1 mM, ff = 1 pN;u 1 0 =1.8s -1 μM -1 u_1^0 = 1.8 ~ s^{-1} \\mu M^{-1}, u 2 =108s -1 u_2 = 108 ~ s^{-1}, w 1 =6.0s -1 w_1 = 6.0 ~ s^{-1}, and w 2 =16s -1 w_2 = 16 ~ s^{-1} at zero load;θ 1 + =0.135,θ 2 + =0.035,θ 1 - =0.080\\theta ^+_1 = 0.135, \\theta ^+_2 = 0.035, \\theta ^-_1 = 0.080, and θ 2 - =0.75\\theta ^-_2 = 0.75.Plots were made using three different initial conditions: A. p 1 (0)=1,p 2 (0)=0p_1(0) = 1, p_2(0) = 0; B. p 1 (0)=0.5,p 2 (0)=0.5p_1(0) = 0.5, p_2(0) = 0.5; and C. p 1 (0)=0,p 2 (0)=1p_1(0)=0, p_2(0) =1.In order to gain a better insight into the energy balance of molecular motor that operates in nonequilibrium steady state, we consider the dynamics of molecular motor systems by means of a cyclic Markov model and relate the essential parameters of the model with NESS thermodynamics.",
"The thermodynamic quantities associated with nonequilibrium process ($\\dot{f}_d$ , $\\dot{Q}_{hk}$ , $T\\dot{e}_p$ , $\\dot{h}_d$ , $\\dot{Q}_{ex}$ ) can be evaluated explicitly using $(N=2)$ -state Markov model; the time evolution of each state is given by $p_1(t)=p_1^{ss}+(p_1(0)-p_1^{ss})e^{-\\sigma t}$ and $p_2(t)=p_2^{ss}-(p_1(0)-p_1^{ss})e^{-\\sigma t}$ with $p_1^{ss}=(u_2+w_1)/\\sigma $ , $p_2^{ss}=(u_1+w_2)/\\sigma $ , and $\\sigma =u_1+u_2+w_1+w_2$ .",
"Using the conditions satisfied in 2-state model ($p_{1+2}(t)=p_1(t)$ ) $W_{12}=u_1$ , $W_{21}=w_1$ , $W_{23}=u_2$ , $W_{32}=w_2$ ; otherwise $W_{ij}=0$ , we obtain $\\frac{\\dot{Q}_{hk}(t)}{T}&=[w_1p_2(t)-u_1p_1(t)]\\log {\\left[\\frac{(u_2+w_1)w_1}{(u_1+w_2)u_1}\\right]}+[w_2p_1(t)-u_2p_2(t)]\\log {\\left[\\frac{(u_1+w_2)w_2}{(u_2+w_1)u_2}\\right]}\\nonumber \\\\&= \\frac{u_1u_2-w_1w_2}{u_1+u_2+w_1+w_2}\\log {\\left[\\frac{u_1u_2}{w_1w_2}\\right]}-\\lambda (p_1(0)-p_1^{ss})e^{-\\sigma t}\\nonumber \\\\&\\xrightarrow{}(j_+-j_-)\\log {\\left(\\frac{j_+}{j_-}\\right)}\\ge 0$ where $\\lambda =\\left\\lbrace (u_1+w_1)\\log {\\left[\\frac{(u_2+w_1)w_1}{(u_1+w_2)u_1}\\right]}-(u_2+w_2)\\log {\\left[\\frac{(u_1+w_2)w_2}{(u_2+w_1)u_2}\\right]}\\right\\rbrace $ .",
"$\\dot{e}_p(t)&=[w_1p_2(t)-u_1p_1(t)]\\log {\\left[\\frac{p_2(t)w_1}{p_1(t)u_1}\\right]}+[w_2p_1(t)-u_2p_2(t)]\\log {\\left[\\frac{p_1(t)w_2}{p_2(t)u_2}\\right]}\\nonumber \\\\&\\xrightarrow{} \\frac{u_1u_2-w_1w_2}{u_1+u_2+w_1+w_2}\\log {\\left[\\frac{u_1u_2}{w_1w_2}\\right]}=(j_+-j_-)\\log {\\left(\\frac{j_+}{j_-}\\right)}\\ge 0$ $\\frac{\\dot{h}_d(t)}{T}&=[w_1p_2(t)-u_1p_1(t)]\\log {\\left[\\frac{w_1}{u_1}\\right]}+[w_2p_1(t)-u_2p_2(t)]\\log {\\left[\\frac{w_2}{u_2}\\right]}\\nonumber \\\\&\\xrightarrow{} \\frac{u_1u_2-w_1w_2}{u_1+u_2+w_1+w_2}\\log {\\left[\\frac{u_1u_2}{w_1w_2}\\right]}=(j_+-j_-)\\log {\\left(\\frac{j_+}{j_-}\\right)}\\ge 0$ $\\frac{\\dot{f}_d(t)}{T}&=[w_1p_2(t)-u_1p_1(t)]\\log {\\left[\\frac{p_2(t)p_1^{ss}}{p_1(t)p_2^{ss}}\\right]}+[w_2p_1(t)-u_2p_2(t)]\\log {\\left[\\frac{p_1(t)p_2^{ss}}{p_2(t)p_1^{ss}}\\right]}\\nonumber \\\\&=\\sigma (p_2(t)p_1^{ss}-p_1(t)p_2^{ss})\\log {\\left[\\frac{p_2(t)p_1^{ss}}{p_1(t)p_2^{ss}}\\right]}\\ge 0\\nonumber \\\\&\\xrightarrow{} 0$ The relaxation time to a steady state (NESS) from an arbitrary state in a far-from-equilibrium is $\\sim \\sigma ^{-1}=(u_1+u_2+w_1+w_2)^{-1}$ , and it is noteworthy that the entropy production inside the system ($T\\dot{e}_p$ ), the total heat production that will be discharged to the surrounding ($\\dot{h}_d$ ), and the housekeeping heat ($\\dot{Q}_{hk}$ ) are all identical at the steady state as $T\\dot{e}_p=\\dot{h}_d=\\dot{Q}_{hk}\\rightarrow (j_+-j_-)\\log {(j_+/j_-)}\\ge 0$ , and $\\dot{f}_d=0$ .",
"Here, $\\dot{Q}_{ex}$ , the residual of heat (excess heat, $\\dot{Q}_{ex}=\\dot{h}_d-\\dot{Q}_{hk}$ ) for the nonequilibrium process, is zero at the steady state.",
"Although obtained for 2-state model, the above expression, especially the total heat production (or housekeeping heat) at the steady state, $\\dot{h}_d=T\\dot{e}_p=\\dot{Q}_{hk}=T(j_+-j_-)\\log {j_+/j_-}$ can easily be generalized for the $N$ -state model."
],
[
"5. Relationship between motor diffusivity and heat dissipation.",
"For $(N=2)$ -state model one can obtain an explicit expression that relates $D$ with $\\dot{Q}$ (for the case of $f=0$ ) as follows.",
"From the expressions of $V$ (Eq.REF ), $D$ (Eq.REF ), and $\\dot{Q}$ , $\\dot{Q}=\\frac{V}{d_0}k_BT\\log {\\frac{u_1u_2}{w_1w_2}}$ Substitution of $u_1=u_1(V)$ from Eq.REF into Eq.REF gives an expression of $\\dot{Q}$ as a function of $V$ : $\\frac{\\dot{Q}}{k_{cat}k_BT}&=v\\log {\\left[\\frac{1+\\kappa v}{1-v}\\right]}\\nonumber \\\\&=\\sum _{n=1}^{\\infty }\\frac{1}{n}\\left(1+(-1)^{n-1}\\kappa ^n\\right)v^{n+1}$ where $v=V/V_{max}$ ($0\\le v\\le 1$ ) and $\\kappa \\equiv \\frac{k_{cat}(k_{cat}+w_1+w_2)}{w_1w_2}$ .",
"$\\dot{Q}$ diverge as $v\\rightarrow 1$ ; but for small $v\\ll 1$ , $\\dot{Q}/(k_{cat} k_B T) \\sim (\\kappa +1)v^2$ , thus $v\\sim \\dot{Q}^{1/2}$ .",
"As long as $\\dot{Q}$ is small, one should expect from Eq.REF that $D$ increases with $\\dot{Q}$ as $D=D_0+\\gamma _1\\dot{q}^{1/2}+\\gamma _2\\dot{q}+\\gamma _3\\dot{q}^{3/2}$ where $\\dot{q}\\equiv \\frac{\\dot{Q}}{k_BTk_{cat}}$ , $D_0=\\frac{d_0^2k_{cat}}{\\kappa +1}$ , $\\gamma _1=d_0^2k_{cat}\\frac{\\kappa -1}{2 (\\kappa +1)^{3/2}}$ , $\\gamma _2=-d_0^2k_{cat}\\frac{k_{cat}^2}{w_1w_2}$ , and $\\gamma _3=\\frac{d_0^2k_{cat}}{(\\kappa +1)^{5/2}}\\frac{k_{cat}^2}{w_1w_2}$ .",
"For arbitrary number of states $N$ , by using Eq.REF and Eq.REF , $\\dot{Q}$ can be written as $\\dot{Q}/k_BT & = \\frac{V}{d_0} \\log {\\frac{ \\prod _{i=1}^N u_i }{ \\prod _{i=1}^N w_i } } \\nonumber \\\\& = \\frac{V^{(D)}}{N} \\log {\\left( 1 + V^{(D)} \\frac{ \\sum _{n=1}^N r_n }{ N }\\frac{ \\prod _{i=1}^N u_i }{ \\prod _{i=1}^N w_i }\\right)} \\nonumber \\\\& = \\frac{V^{(D)}}{N} \\log {\\left( 1 + \\frac{V^{(D)}}{N} \\left( A + B u_1 \\right) \\frac{ 1 }{C }\\right)} \\nonumber \\\\&=\\frac{V^{ (D) }}{N} \\log { \\left( 1 + \\frac{ V^{(D)} }{N} f(V^{ (D) }/N ) \\right) }$ where we used $V^{ (D) } = V N / d_0 $ , $f(V^{ (D) }/N ) =\\frac{A}{C} + B \\left( 1 - B \\frac{V^{ (D) } }{N} \\right)^{-1} \\left( 1 + \\frac{A}{C} \\frac{ V^{ (D) }}{N} \\right)$ , and Eqs.REF , REF , REF .",
"The definitions of $A, B$ , and $C$ are identical to those in Eq.REF .",
"Here $V^{ (D) } / N$ corresponds to ATP hydrolysis rate.",
"For $V^{ (D) } \\rightarrow 0$ , $\\dot{Q} \\rightarrow 0$ is expected.",
"Also for small $V^{ (D) }$ , $\\dot{Q} \\sim ( \\frac{A}{C} + B ) \\left(\\frac{V^{(D)}}{N} \\right)^2$ .",
"Thus, $V \\sim \\dot{Q}^{1/2}$ .",
"Since $D\\sim V+\\mathcal {O}(V^2)$ for small $V$ , it follows that $D$ can be written as a function of $\\dot{q}$ as in the same form as Eq.REF ."
],
[
"6. Rate constants, enhancement of diffusion, and conversion efficiency determined from the (N=2)-state kinetic model.",
"The values in the Table 1 were compiled based on the followings."
],
[
"Catalase",
"In Ref.",
"[5] $(\\Delta D / D_0)=0.28$ at $ V = 1.7 \\times 10^4$ $s^{-1}$ ; however, $V= 1.7 \\times 10^4$ $s^{-1}$ is not the maximum catalytic rate.",
"Because $\\Delta D/D_0$ is approximately linear in $V$ , the enhancement of diffusion at the maximal turnover rate $V_{\\text{max}}=u_2= 5.8 \\times 10^4$ is estimated as $(\\Delta D / D_0)_{\\text{obs}} = 0.28 \\times \\frac{5.8 \\times 10^4}{1.7 \\times 10^4} = 0.96 \\approx 1$ ."
],
[
"Alkaline phosphatase",
"Similar to catalase, $(\\Delta D / D_0)_{\\text{obs}} = 0.77 \\times \\frac{1.4 \\times 10^4}{5.5 \\times 10^3} = 2.5 \\approx 3$ ."
],
[
"Estimate of $(\\Delta D/D_0)_{\\text{max}}$",
"Freely diffusing enzymes effectively perform no work on the surrounding environment; thus $-\\Delta \\mu _{\\text{eff}}=Q$ with $W=0$ , which leads to $e^{Q/k_B T}= u_1 u_2/w_1 w_2$ .",
"By assuming that the substrate concentration $[S] \\sim K_M=(u_2 +w_1) / u_1^o$ , we get $\\begin{aligned}e^{Q/k_B T}&= \\frac{ u_1 u_2 }{w_1 w_2}\\ \\sim \\frac{ u_1^o K_M u_2 }{w_1 w_2}= \\frac{ (u_2 + w_1) u_2 }{w_1 w_2} \\\\&\\ge \\frac{ u_2^2 }{w_1 w_2}= \\frac{ k_{cat}^2 }{w_1 w_2}.\\end{aligned}$ This relation allows us to estimate the upper bound of $(\\Delta D/D_0)_{\\text{max}}$ as follows when $u_2 \\gg w_1, w_2$ is satisfied.",
"$\\left(\\frac{\\Delta D}{D_0}\\right)_{\\text{max}}\\approx \\frac{ k_{cat}^2 }{2 w_1 w_2} \\le \\frac{1}{2}e^{Q/k_BT}.$ Alternatively, $u_1$ and $w_2$ of enzymes can be estimated by assuming (i) that the reaction is diffusion limited, $u_1^o = 10^8 s^{-1} M^{-1}$ , and (ii) that the substrate concentration $[S]$ is similar to Michaelis-Menten constant $K_M$ ($[S]\\sim K_M$ ).",
"The two conditions $u_1 = u_1^o [S] \\sim K_M \\times 10^8$ ($s^{-1}$ ) and $K_M(=(u_2+w_1)/u_1^o)$ , and $Q$ (heat measured by the calorimeter in ref.",
"[5]), $u_2$ , $K_M$ which are available in ref.",
"[5], provide all the rate constants including $w_1=u_1^oK_M-u_2$ and $w_2=\\frac{u_1u_2}{w_1e^{Q/k_BT}}$ , allowing us to calculate $\\left(\\frac{\\Delta D}{D_0}\\right)_{\\text{max}}=\\frac{u_2^2+(w_1+w_2)u_2-w_1w_2}{2w_1w_2}$ .",
"Figure: Analysis of experimental data, extracted from Ref.",
", but using (NN=2)-state model.The solid lines are the fits to the dataA.",
"VV vs ATP at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).B.",
"VV vs load at [ATP] = 5 μ\\mu M.C.",
"VV vs load at [ATP] = 2 mM.D.",
"Stall force as a function of [ATP], measured by `Position clamp' (red square) or `Fixed trap' (blue circle) methods.E.",
"DD vs ATP at f=f=1.05 pN (red square), 3.59 pN (blue circle), and 5.63 pN (black triangle).DD was estimated from r=2D/Vd 0 r=2D/Vd_0.",
"F. DD vs load at [ATP] = 2 mM.G-I.",
"Motor diffusivity (DD) as a function of mean velocity (VV) for kinesin-1.",
"(VV,DD) measured at varying [ATP] (= 0 – 2 mM) and a fixed (G) f=f=1.05 pN, (H) 3.59 pN, and (I) 5.63 pN .The black dashed lines in G and H are the fits using Eq..The solid lines in magenta in G-I are plotted using the (NN=2)-kinetic model."
]
] | 1612.05747 |
[
[
"Dark Matter Relics and the Expansion Rate in Scalar-Tensor Theories"
],
[
"Abstract We study the impact of a modified expansion rate on the dark matter relic abundance in a class of scalar-tensor theories.",
"The scalar-tensor theories we consider are motivated from string theory constructions, which have conformal as well as disformally coupled matter to the scalar.",
"We investigate the effects of such a conformal coupling to the dark matter relic abundance for a wide range of initial conditions, masses and cross-sections.",
"We find that exploiting all possible initial conditions, the annihilation cross-section required to satisfy the dark matter content can differ from the thermal average cross-section in the standard case.",
"We also study the expansion rate in the disformal case and find that physically relevant solutions require a nontrivial relation between the conformal and disformal functions.",
"We study the effects of the disformal coupling in an explicit example where the disformal function is quadratic."
],
[
"Introduction",
"The most recent cosmological observations support the so called $\\Lambda CDM$ model of cosmology.",
"These observations provide overwhelming evidence for the existence of non-baryonic (cold) dark matter (DM), constituting about $27\\%$ of the Universe's energy density budget, while another $\\sim 68\\%$ is believed to be in the form of dark energy, which can simply be a cosmological constant, $\\Lambda $ .",
"The other $\\sim 5\\%$ being formed by baryonic matter, described by the standard model (SM) of particles.",
"A popular framework to understand the origin of DM is the thermal relic scenario.",
"In this scenario, at very early times when the universe was at a very high temperature, thermal equilibrium was obtained and the number density of DM particles $\\chi $ was roughly equal to the number density of photons.",
"During equilibrium the dark matter number density decayed exponentially as $n_\\chi ^{eq} \\sim e^{-m_\\chi /T}$ for a non-relativistic DM candidate, where $m_\\chi $ is the mass of the DM particle $\\chi $ .",
"As the universe cooled down as it expanded, DM interactions became less frequent and eventually, the DM interaction rate dropped below the expansion rate ($\\Gamma _\\chi < H$ ).",
"At this point the density number froze-out and the universe was left with a “relic\" of DM particles.",
"Therefore, the dependence of the number density at the time of freeze-out is crucial to determine the DM relic abundance.",
"The longer the DM particles remain in equilibrium, the lower its density will be at freeze-out and vice-versa.",
"In the standard $\\Lambda CDM$ scenario, particle freeze-out happens during the radiation era and DM species with weak scale interaction cross-section freeze-out with an abundance that matches the present observed value.",
"The weakness of the interactions is reflected in the predicted thermally-averaged annihilation cross section, $\\langle \\sigma v \\rangle $ , which is around $3.0\\times 10^{-26}cm^3s^{-1}$ .",
"Despite such a small value, the Fermi-LAT and Planck experiments have been exploring upper bounds on $\\langle \\sigma v \\rangle $ (see [1], [2]).",
"From observations, it appears that the annihilation cross-section can be smaller than the thermal average value for lower dark matter masses ($\\le 100$ GeV), whereas an annihilation cross-section larger than the thermal average value can still be allowed for larger DM mass.",
"If, from future measurements, $\\langle \\sigma v \\rangle \\ne 3.0\\times 10^{-26}cm^3s^{-1}$ is established, what can we say about the origin of the dark matter?",
"Can it still be the thermal dark matter or do we need non-thermal origin of dark matter?",
"In the case of non-thermal origin, the DM can arise from the decay of a heavy particle, e.g., moduli, and can satisfy the DM content with any value of $\\langle \\sigma v \\rangle $ .",
"The primary motivation of this paper is to find out whether the DM content can still have a thermal origin with larger or smaller $\\langle \\sigma v \\rangle $ by utilising non-standard cosmology.",
"The phenomenological $\\Lambda CDM$ model complemented with the inflationary paradigm to provide the seeds of large scale structure, is very successful in describing our current universe.",
"However, the physics describing the universe's evolution from the end of inflation (reheating) to just before big-bang nucleosynthesis (BBN) ($t\\lesssim 200$ s) remains relatively unconstrained.",
"During this period the universe may have gone through a “non-standard\" period of expansion, and still be compatible with BBN.",
"If such modification happened during DM decoupling, the DM freeze-out may be modified with measurable consequences for the relic DM abundances.",
"Departures from the standard cosmology between reheating and BBN will mainly be a consequence of a modified expansion rate ($\\tilde{H}$ ), due to a modification of General Relativity (GR).",
"Such modifications are well motivated by attempts to embed the $\\Lambda CDM$ and inflation models into a fundamental theory of gravity and particle physics, such as theories with extra-dimensions, supergravity and string theory.",
"Indeed, our main motivation in this paper is to develop further tools that may allow us to connect such fundamental theories with observations.",
"In this paper our approach will be mostly phenomenological, but we have in mind a scenario that can be derived in the context of a fundamental theory of gravity, such as string theory.",
"Furthermore, we will be concerned with modifications to the standard picture due to the presence and interactions with scalar fields only.",
"Modifications to the relic abundances were first discussed by Catena et al.",
"[3] in the context of conformally coupled scalar-tensor theories (ST), such as generalisations of the Brans-Dicke theory.",
"Further studies on conformally coupled ST models have been performed in the last years in [4], [5], [6], [7] (see also [8], [9], [10], [11], [12], [13], [14]) with the most recent work of [15] where it is shown that the boundary conditions used in [3] cannot have $\\langle \\sigma v \\rangle \\ne 3.0\\times 10^{-26}cm^3s^{-1}$ .",
"Indeed ST theories where dark matter and dark energy are correlated constitute an attractive way to address the dark matter and dark energy problems via an attraction mechanism towards standard general relativity (GR) [3].",
"In the context of ST theories, the most general physically consistent relation between two metrics in the presence of a scalar field, is given byIn the more general case, $C$ and $D$ can be functions of $X=\\frac{1}{2} (\\partial \\phi )^2$ as well.",
"However, we will not consider this case in the present paper.",
"[16]: $\\tilde{g}_{\\mu \\nu } = C(\\phi ) g_{\\mu \\nu } + D(\\phi ) \\partial _\\mu \\phi \\partial _\\nu \\phi \\,.$ The first term in (REF ) is the well-known conformal transformation which characterises the Brans-Dicke class of scalar-tensor theories explored in [3], [4], [5], [6], [7], [15].",
"However, in reference [15], it is shown that there is no change in the thermal cross-section arising from the conformally modified metric compared to the standard cosmology after satisfying all the constraints based on the boundary conditions chosen in [3], [4].",
"The second term is the disformal contribution, which is generic in extensions of general relativity.",
"In particular, it arises naturally in D-brane models, as discussed in [17] in a model of coupled dark matter and dark energy.",
"In this paper we revisit the expansion rate modification and impact on the DM relic abundances for the conformal case, providing new interesting results with new boundary conditions to show that $\\langle \\sigma v \\rangle \\ne 3.0\\times 10^{-26}cm^3s^{-1}$ , while satisfying the DM content.",
"We further discuss the general modifications to the expansion rate and Boltzmann equation, due to the disformal coupling and present an explicit non-trivial example for the case in which the conformal term in (REF ) is a monomial.",
"The paper is organised as follows.",
"We start in section introducing the scalar-tensor theory conformally and disformally coupled to matter.",
"Then, we examine the formulation of this theory in the Einstein and Jordan frames, comment on their physical interpretation and derive the equations that describe the cosmological evolution of the Universe.",
"Subsequently, after discussing the expansion rate modifications caused by the presence of the conformal and disformally coupled scalar field in section , we investigate in detail its impact on the dark matter relic abundance by exploring a concrete pure conformal example as well as an example where we also turn on the disformal contribution.",
"Finally, in section we conclude."
],
[
"The scalar-tensor theory set-up",
"We are interested in scalar-tensor theories coupled to matter both conformally and disformally [16].",
"Our motivation comes from theories with extra dimensions and in particular string theory compactifications, where several additional scalar fields appear, from closed and open string theory sectors of the theory [17].",
"Our approach in this paper nonetheless, will be phenomenological and therefore our equations will be simplified.",
"However, we present the more general set-up, which can accommodate a realisation from concrete string theory compactifications in appendix and .",
"The action we want to consider is given by: $S_{EH}=\\frac{1}{2\\kappa ^2} \\!\\!\\int {\\!d^4x\\sqrt{-g}\\,R}- \\!\\!\\int {\\!d^4x\\sqrt{- g} \\left[\\frac{1}{2} (\\partial \\phi )^2+V(\\phi )\\right]}- \\!\\!\\int {\\!d^4x\\sqrt{-\\tilde{g}} \\,{\\cal L}_{M}(\\tilde{g}_{\\mu \\nu }) } \\,.$ Here the disformally coupled metric is given by $\\tilde{g}_{\\mu \\nu } = C(\\phi ) g_{\\mu \\nu } + D(\\phi ) \\partial _\\mu \\phi \\partial _\\nu \\phi \\,,$ and the inverse by: $\\tilde{g}^{\\mu \\nu } = \\frac{1}{C}\\left[ g^{\\mu \\nu } - \\frac{D\\,\\partial ^\\mu \\phi \\partial ^\\nu \\phi }{C+D(\\partial \\phi )^2}\\right]\\,.$ Moreover, $\\kappa ^2=M_{P}^{-2}=8\\pi G$ , but keep in mind that that $G$ is not in general equal to Newton's constant as measured by e.g.",
"local experiments.",
"Further, $C(\\phi ), D(\\phi )$ are functions of $\\phi $ , which can be identified as a conformal and disformal couplings of the scalar to the metric, respectively (note that the conformal coupling is dimensionless, whereas the disformal one has units of $mass^{-4}$ ).",
"The action in (REF ) is written in the Einstein frameIn string theory, the Einstein frame refers to the frame in which the dilaton and graviton degrees of freedom are decoupled, while the string (or Jordan) frame is that in which they are not.",
"Further, the dilaton field as well as all other moduli (scalar) fields not relevant for the cosmological discussion are stabilised, massive, and are therefore decoupled from the low energy effective theory.",
"In the literature of scalar-tensor theories however, the Einstein and Jordan frames are identified with respect to the (usually single) scalar field to which gravity is coupled, but such scalar has no particular physical nor geometrical interpretation., which is identified in the literature of scalar-tensor theories (including conformal and disformal couplings) with the frame respect to which the scalar field, gravity is coupled.",
"We follow this use and refer to “Jordan\" or “disformal frame\" to identify the frame in which dark matter is coupled only to the metric $\\tilde{g}_{\\mu \\nu }$ , rather than to the metric $g_{\\mu \\nu } $ and a scalar field $\\phi $ .",
"The equations of motion obtained from (REF ) are: $R_{\\mu \\nu } -\\frac{1}{2}g_{\\mu \\nu } R = \\kappa ^2\\left(T^\\phi _{\\mu \\nu } + T_{\\mu \\nu }\\right)\\,,$ where, in the frame relative to $g_{\\mu \\nu }$ , the energy-momentum tensors are defined as $T^\\phi _{\\mu \\nu } = -\\frac{2}{\\sqrt{-g}} \\frac{\\delta S_{\\phi }}{\\delta g^{\\mu \\nu }} \\,,\\qquad \\quad T_{\\mu \\nu }= -\\frac{2}{\\sqrt{-g}} \\frac{\\delta \\left(-\\sqrt{-\\tilde{g}} \\,{\\cal L}_{M}\\right) }{\\delta g^{\\mu \\nu }}\\,,$ and we model the energy-momentum tensor for matter and both dark components as perfect fluids, that is: $T_{\\mu \\nu }^i = P_i g_{\\mu \\nu } +(\\rho _i + P_i) u_\\mu u_\\nu $ where $\\rho _i$ , $P_i$ are the energy density and pressure for each fluid $i$ with equation of state $P_i/\\rho _i = \\omega _i$ .",
"For the scalar field, the energy-momentum tensor takes the form: $T_{\\mu \\nu }^{\\phi } = - g_{\\mu \\nu } \\left[ \\frac{1}{2} (\\partial \\phi )^2+ V \\right]+ \\partial _\\mu \\phi \\, \\partial _\\nu \\phi \\,,$ and one can define the energy density and pressure of the scalar field as: $\\rho _\\phi =- \\frac{1}{2}(\\partial \\phi )^2 + V \\,, \\qquad P_\\phi = - \\frac{1}{2}(\\partial \\phi )^2 - V \\,.$ Finally the equation of motion for the scalar field dark energy becomes: $&&\\hspace{-28.45274pt}- \\nabla _\\mu \\nabla ^\\mu \\phi \\!",
"+ V^{\\prime }- \\frac{T^{\\mu \\nu }}{2}\\!\\left[\\frac{C^{\\prime }}{C} g_{\\mu \\nu } +\\frac{D^{\\prime }}{C}\\partial _\\mu \\phi \\partial _\\nu \\phi \\right]+\\nabla _\\mu \\left[\\frac{D}{C}T^{\\mu \\nu } \\partial _\\nu \\phi \\right] =0 \\,.$ Due to the nontrivial coupling, the individual conservation equations for the two fluids are modified.",
"However, the conservation equation for the full system is preserved, and given in the usual way by $\\nabla _\\mu \\left(T^{\\mu \\nu }_{\\phi } + T^{\\mu \\nu }\\right) =0\\,.$ Thus using (REF ) and the equation of motion for the scalar field we can write $\\nabla _\\mu T^{\\mu \\nu }_\\phi = Q \\,\\partial ^\\nu \\phi = - \\nabla _\\mu T^{\\mu \\nu }\\,,$ where $Q\\equiv \\nabla _\\mu \\left[\\frac{D}{C} \\,T^{\\mu \\lambda } \\,\\partial _\\lambda \\phi \\right] - \\frac{T^{\\mu \\nu } }{2} \\left[\\frac{C^{\\prime }}{C} g_{\\mu \\nu } +\\frac{D^{\\prime }}{C} \\,\\partial _\\mu \\phi \\,\\partial _\\nu \\phi \\right]\\,.$ In the Jordan, or disformal frame, as defined above, matter is conserved, $\\tilde{\\nabla }_\\mu \\tilde{T}^{\\mu \\nu } =0 \\,,$ where $\\tilde{\\nabla }_\\mu $ is the covariant derivative computed with respect to the disformal metric (REF ) with the Christoffel symbols given by $\\tilde{\\Gamma }^\\mu _{\\alpha \\beta } =\\Gamma ^\\mu _{\\alpha \\beta } + \\frac{C^{\\prime }}{C}\\, \\delta ^\\mu _{(\\alpha }\\partial _{\\beta )}\\phi - \\gamma ^2 \\, \\frac{C^{\\prime }}{2C} \\,\\partial ^\\mu \\phi \\, g_{\\alpha \\beta }+\\frac{D}{C}\\,\\gamma ^2\\,\\partial ^\\mu \\phi \\left[\\!\\nabla _\\alpha \\nabla _\\beta \\phi + \\!\\left(\\frac{D^{\\prime }}{2D} -\\!\\frac{C^{\\prime }}{C}\\right) \\!\\partial _\\alpha \\phi \\partial _\\beta \\phi \\right]\\,, \\nonumber \\\\$ and we have introduced the “Lorentz factor\" $\\gamma $ defined as $\\gamma = \\frac{1}{\\sqrt{1+\\frac{D}{C} (\\partial \\phi )^2}}\\,.$ In this frame, the energy-momentum tensor is defined as $\\tilde{T}^{\\mu \\nu } = \\frac{2}{\\sqrt{-\\tilde{g}}} \\frac{\\delta S_{M}}{\\delta \\tilde{g}_{\\mu \\nu } } \\,$ and the disformal energy-momentum tensor can be written as: $\\tilde{T}^{\\mu \\nu } = (\\tilde{\\rho }+\\tilde{P}) \\tilde{u}^{\\mu } \\tilde{u}^{\\mu } + \\tilde{P}\\, \\tilde{g}^{\\mu \\nu }\\,,$ where $\\tilde{u}^{\\mu } = C^{-1/2} \\gamma \\,u^{\\mu } $ .",
"Using (REF ), we obtain a relation between the energy momentum tensor in both frames as: $\\tilde{T}^{\\mu \\nu } = C^{-3} \\gamma \\, T^{\\mu \\nu }\\,.$ Further using (REF ) we arrive at a relation among the energy densities and pressures in both frames, given by $\\tilde{\\rho }= C^{-2} \\gamma ^{-1} \\rho \\,, \\qquad \\tilde{P} = C^{-2}\\gamma \\, P,$ and therefore the equations of state in both frames are related by $\\tilde{\\omega }= \\omega \\, \\gamma ^2$ .",
"Note that in the pure conformal case, $D=0$ , $\\gamma =1$ and therefore $\\tilde{\\omega }= \\omega $ ."
],
[
"Cosmological equations",
"Consider an homogeneous and isotropic FRW metric $g_{\\mu \\nu }$ , $ds^2 = - dt^2 + a(t)^2 dx_i dx^i \\,,$ where $a(t)$ is the scale factor.",
"In this background, the Einstein and Klein-Gordon equations become, respectively $&& H^2 =\\frac{\\kappa ^2}{3} \\left[\\rho _\\phi +\\rho \\right]\\,, \\\\&& \\dot{H} + H^2 = -\\frac{\\kappa ^2}{6}\\left[ \\rho _\\phi + 3P_\\phi +\\rho +3 P \\right]\\,,\\\\&& \\ddot{\\phi }+3H\\dot{\\phi }+ V_{,\\phi }+Q_0 =0 \\,.$ where, $H= \\frac{\\dot{a}}{a}$ , dots are derivatives with respect to $t$ and we have denoted $V_{,\\phi } \\equiv \\frac{dV}{d\\phi }$ .",
"Also the Lorentz factor becomes $\\gamma = (1-D \\,\\dot{\\phi }^2/C)^{-1/2}.$ The continuity equations for the scalar field and matter are given by $&&\\dot{\\rho }_\\phi + 3H(\\rho _{\\phi }+P_{\\phi }) = -Q_0\\dot{\\phi }\\,, \\\\&&\\dot{\\rho }+ 3H(\\rho +P) = Q_0\\,\\dot{\\phi }\\,.$ where $Q_0$ is given by $Q_0 = \\rho \\left[ \\frac{D}{C} \\,\\ddot{\\phi }+ \\frac{D}{C} \\,\\dot{\\phi }\\left(\\!3H + \\frac{\\dot{\\rho }}{\\rho } \\right) \\!+ \\!\\left(\\!\\frac{D_{,\\phi }}{2C}-\\frac{D}{C}\\frac{C_{,\\phi }}{C}\\!\\right) \\dot{\\phi }^2 +\\frac{C_{,\\phi }}{2\\,C} (1-3\\,\\omega )\\right].",
"\\nonumber \\\\$ Using () we can rewrite this in a more compact and useful form as $Q_0 = \\rho \\left( \\frac{\\dot{\\gamma }}{\\dot{\\phi }\\, \\gamma } + \\frac{C_{,\\phi }}{2C} (1-3\\,\\omega \\,\\gamma ^2) -3H\\omega \\,\\frac{(\\gamma ^{2}-1) }{\\dot{\\phi }}\\right)\\,.$ Plugging this into the (non-)conservation equation for dark matter (), gives: $\\dot{\\rho }+ 3H (\\rho + P\\,\\gamma ^{2}) = \\rho \\left[\\frac{\\dot{\\gamma }}{\\gamma } + \\frac{C_{,\\phi } }{2C} \\,\\dot{\\phi }\\, (1-3\\,\\omega \\gamma ^2)\\right]\\,.$ Using the relations for the physical proper time and the scale factors in the two frames, given by $\\tilde{a} = C^{1/2} a \\,, \\qquad \\quad d\\tilde{\\tau }= C^{1/2} \\gamma ^{-1} d\\tau \\,,$ we can define the disformal-frame Hubble parameter $\\tilde{H} \\equiv \\frac{d \\ln {\\tilde{a}}}{d\\tilde{\\tau }}$ , as $\\tilde{H} = \\frac{\\gamma }{C^{1/2}}\\left[ H + \\frac{C_{,\\phi }}{2C}\\dot{\\phi }\\right]\\,,$ so that (REF ) takes the standard form in terms of $\\tilde{H}$ : $\\frac{d{\\tilde{\\rho }}}{d\\tilde{\\tau }} + 3 \\tilde{H}( \\tilde{\\rho }+\\tilde{P}) =0 \\,.$ Equations (REF ) and (REF ) give the background evolution equations for the modified expansion rate and matter's density evolution."
],
[
"Master equations",
"In order to solve the cosmological equations, it is convenient to replace time derivatives with derivatives with respect to the number of e-folds $N$ , defined as $ N = \\ln a/a_0$ and define $\\lambda = \\frac{V}{ \\rho }(=\\frac{\\tilde{V}}{\\tilde{\\rho }})$ .",
"With these definitions, we can rewrite the Friedmann equation (REF ) and $Q_0$ as: $H^2 &=& \\frac{\\kappa ^2 \\rho }{3} \\frac{(1+\\lambda )}{\\left(1- \\frac{\\kappa ^2\\phi ^{\\prime 2}}{6 }\\right)} \\,, \\\\\\!\\!\\frac{Q_0}{\\rho }& =& \\frac{\\gamma ^2 H^2}{2}\\Bigg [ \\frac{2D}{C} \\phi ^{\\prime \\prime } -\\frac{2D}{C} \\phi ^{\\prime } \\!\\left(3\\,\\omega + \\frac{\\kappa ^2\\phi ^{\\prime 2}}{2} + \\frac{3(1+\\omega ) B}{2(1+\\lambda )} \\right)+ \\left(\\frac{D}{C}\\right)_{\\!\\!,\\phi } \\!\\!\\phi ^{\\prime 2} + \\frac{C_{\\!,\\phi }}{H^2 C}(\\gamma ^{-2}-3\\omega ) \\Bigg ], \\nonumber \\\\$ where here we denote $^{\\prime }= d/dN$ .",
"Note also that (REF ) implies that $\\kappa \\,\\phi ^{\\prime } \\le \\pm \\sqrt{6}$ .",
"Using these equations and further defining a dimensionless scalar field $\\varphi = \\kappa \\,\\phi $ , we can rewrite () and () as: $&& H^{\\prime } = - H \\left[ \\frac{3B}{2(1+\\lambda )} (1+ \\omega ) + \\frac{\\varphi ^{\\prime 2}}{2}\\right], \\\\ \\nonumber \\\\&& \\varphi ^{\\prime \\prime } \\left[1\\!+\\!",
"\\frac{3H^2\\gamma ^2 B}{\\kappa ^2(1+\\lambda )} \\frac{D}{C} \\right]+ 3\\,\\varphi ^{\\prime } \\left[1- \\omega \\frac{3H^2\\gamma ^2 B}{\\kappa ^2(1+\\lambda )} \\frac{D}{C} \\right] +\\frac{H^{\\prime }}{H} \\varphi ^{\\prime } \\left[1+ \\frac{3H^2\\gamma ^2 B}{\\kappa ^2(1+\\lambda )} \\frac{D}{C} \\right]\\nonumber \\\\&& \\hspace{42.67912pt}+ \\frac{3B}{1+\\lambda } \\alpha (\\varphi )(1-3 \\,\\omega \\gamma ^2)+ \\frac{3B\\lambda }{(1+\\lambda )} \\frac{V_{,\\varphi }}{V} + \\frac{3 H^2 \\gamma ^2 B}{\\kappa ^2(1+\\lambda )} \\frac{D}{C}\\left[ (\\delta (\\varphi ) - \\alpha (\\varphi )) \\,\\varphi ^{\\prime 2} \\right] =0\\,,\\nonumber \\\\$ where we defined: $&&B \\equiv 1-\\frac{\\varphi ^{\\prime 2}}{6}, \\\\&& \\gamma ^{-2} = 1- \\frac{H^2}{\\kappa ^2}\\frac{D}{C} \\varphi ^{\\prime 2}\\,, \\\\&& \\alpha (\\varphi ) = \\frac{d \\ln C^{1/2}}{d\\varphi }, \\\\&& \\delta (\\varphi ) = \\frac{d \\ln D^{1/2}}{d\\varphi } \\,.$ One can solve the system of coupled equations above for $H$ and $\\varphi $ as functions of $N$ .",
"However, in some cases it is simpler to use (REF ) into () and solve the following disformal master equation: $\\frac{2(1+\\lambda )}{3 B} \\,\\varphi ^{\\prime \\prime } &+& \\left(2\\lambda +1 -\\omega \\right)\\varphi ^{\\prime } + 2 \\lambda \\, \\frac{d\\ln V}{d\\varphi } +2 (1 -3\\,\\omega \\,\\gamma ^{2})\\,\\alpha (\\varphi )\\nonumber \\\\&+ & \\, \\frac{2\\gamma ^2(1+\\lambda )}{3B}\\frac{D \\rho }{C} \\left(\\varphi ^{\\prime \\prime } -3\\,\\varphi ^{\\prime }\\left[ \\omega + \\frac{\\varphi ^{\\prime 2}}{6} + \\frac{(1+\\omega ) B}{2(1+\\lambda )}\\right]+ \\frac{C}{2D}\\left(\\frac{D}{C}\\right)_{\\!,\\varphi } \\varphi ^{\\prime 2} \\right) \\!=0 \\,, \\nonumber \\\\$ with $\\gamma $ given by: $\\gamma ^{-2} = 1- \\frac{(1+\\lambda )}{3B}\\frac{D \\rho }{C} \\varphi ^{\\prime 2}\\,.$ From (REF ) we see that the conformal case is recovered for $D=0$ , when the second line vanishes.",
"Moreover, the disformal piece appears always together with derivatives of the scalar field, as expected and also nontrivially coupled to the energy density.",
"This complicates considerably the analysis of the disformal case, as we will see below."
],
[
"Modified expansion rate",
"The effect of the expansion rate during the early time evolution due to the presence of a scalar field can be extracted from the Hubble parameter evolution in the disformal frame defined as: $\\tilde{H} = d (\\log \\tilde{a} )/d\\tilde{\\tau },$ which can be written using (REF ) as: $\\tilde{H} = \\frac{H\\gamma }{C^{1/2}} \\left(1+ \\alpha (\\varphi ) \\varphi ^{\\prime } \\right)\\,,$ where remember that $\\gamma $ depends on $H$ (or $\\rho $ ) as seen from (REF ), while in the pure conformal case $D=0$ and $\\gamma = 1$ .",
"Note that in principle, the factor $(1+ \\alpha (\\varphi ) \\varphi ^{\\prime } )$ can be positive or negative, indicating an expansion or contraction modified rate.",
"We stick to positive definite values for this factor and therefore only modified expansion rates, though in principle, one could have a brief contraction period during the early universe evolution, before the onset of BBNSee [18] for a review on scenarios with a possible contraction phase in the early universe..",
"Moreover, notice that while $\\tilde{H}$ can grow during the cosmological evolution, the null energy condition (NEC) is not violated.",
"This is because the Einstein frame expansion rate $H$ is dictated by the energy density $\\rho $ and pressure $p$ , which obey the NEC and therefore $\\dot{H}<0$ during the whole evolution, as it should (see for example [19]).",
"We further want to relate the modified expansion rate to the expected expansion rate in general relativity (GR), that is: $H_{GR}^2 = \\frac{\\kappa _{GR}^2}{3} \\, \\tilde{\\rho }\\,.$ We can do this be using the Friedmann equation (REF ) and the relation between the energy densities (REF ) to write $\\gamma ^{-1} H^2 = \\frac{\\kappa ^2}{\\kappa _{GR}^2} \\frac{ C^2\\,(1+\\lambda )}{B} \\, H_{GR}^2 \\,.$ Using the definition of $\\gamma $ (see (REF )) into this equation, one finds a cubic equation for $H^2$ in terms of all the other parameters.",
"The real positive solution to that equation can then be replaced into (REF ) to find the modified expansion rate $\\tilde{H}$ , which will thus be a complicated function of $H_{GR}$ as we now see.",
"The cubic equation for $H$ takes the form: $d_1 H^6-H^4 + d_2^2=0\\,,$ where $d_1=\\frac{D}{C}\\frac{\\varphi ^{\\prime 2}}{\\kappa ^2} \\,, \\hspace{28.45274pt} d_2=\\frac{\\kappa ^2}{\\kappa _{GR}^2}\\frac{C^2 (1+\\lambda )H_{GR}^2}{B}\\,.$ The solutions to (REF ) can be written as $H^2=\\frac{1}{3 d_1}\\left(1+\\left(\\frac{2}{\\Delta }\\right)^{1/3}+\\left(\\frac{\\Delta }{2}\\right)^{1/3}\\right)$ with $\\Delta =2-27d_1^2d_2^2+d_1d_2\\sqrt{27(27d_1^2d_2^2-4)}$ .",
"The other two solutions can be obtained by replacing $ \\left(\\frac{2}{\\Delta }\\right)^{1/3} \\rightarrow e^{2\\pi i/3} \\left(\\frac{2}{\\Delta }\\right)^{1/3}\\qquad {\\rm and } \\qquad \\left(\\frac{2}{\\Delta }\\right)^{1/3} \\rightarrow e^{4\\pi i/3} \\left(\\frac{2}{\\Delta }\\right)^{1/3}\\,.$ We are interested in real positive solutions for $H^2$ .",
"One possibility to get this is to have the imaginary part of $(\\Delta /2)^{1/3}$ vanish by requiring that $\\Delta >0$ , which is impossible.",
"Therefore, the way to obtain real solutions for $H$ is to have the imaginary parts of $(\\Delta /2)^{1/3}$ and $(\\Delta /2)^{-1/3}$ cancel each other, leaving a real positive solution.",
"For this, we need that $27d_1^2d_2^2 \\le 4$ , which implies the following relation between the conformal and disformal functions: $\\frac{3\\sqrt{3} \\,D C\\, \\varphi ^{\\prime 2} (1+\\lambda )}{B} \\frac{H^2_{GR}}{\\kappa _{GR}^2}\\le 2\\,.$ Under this condition, we can rewrite $\\Delta $ as: $\\Delta = 2-27d_1^2d_2^2+ i d_1d_2\\sqrt{27(4- 27d_1^2d_2^2)}\\,, $ which allows us to define a complex number $Z\\equiv \\Delta /2$ and it is easy to check that $\\bar{Z} = 2/\\Delta $ and thus $|Z|^2 =1$ .",
"Denoting further $Z_i$ with $i=1,2,3$ denoting the three solutions to $H$ as explained above, the solutions for $H$ , (REF ) takes the simple form: $H^2_i=\\frac{1}{3 d_1}\\left[1+Z_i^{1/3} + \\bar{Z}_i^{1/3}\\right] \\,,$ and remember that we are interested only in the real positive solution.",
"We can now plug in (REF ), as well as the expression for $\\gamma $ in terms of $H$ into the Jordan frame expansion rate (REF ), can be written as: $\\tilde{H}^2 = \\frac{\\kappa ^2}{\\kappa ^2_{GR}} \\frac{ \\gamma ^3\\, C(1+\\lambda ) (1+\\alpha (\\varphi )\\varphi ^{\\prime })^2 }{B} \\,\\,H_{GR}^2\\,,$ where there is a non-trivial dependence of $H_{GR}$ encoded in $\\gamma _i = \\frac{1}{3d_1d_2}\\left[1+Z_i^{1/3} + \\bar{Z}_i^{1/3}\\right] \\,.$ In the conformal case, $D=0$ , $\\gamma =1$ and therefore (REF ) is simply $\\tilde{H}^2 = \\frac{\\kappa ^2}{\\kappa _{GR}^2}\\frac{C (1+\\lambda ) (1+\\alpha (\\varphi ) \\varphi ^{\\prime })^2}{B}\\, H^2_{GR} \\,.$ From this relation we define a speed-up parameter $\\xi $ , which will be useful below to measure the departures from the $GR$ expansion rate result: $\\xi \\equiv \\frac{\\tilde{H}}{H_{GR}}\\,.$"
],
[
"Modifications of the dark matter relic abundances",
"In this section we discuss the modifications to the DM relic abundance's predictions due to modifications of the expansion rate before the onset of nucleosynthesis caused by the presence of a scalar field conformal and disformally coupled to matter.",
"We start by revisiting the conformal case, discussed originally in [3]Modifications to the Boltzmann equation due to a conformal coupling in (non-critical) string theory have been discussed in [8].. We first solve (numerically) the master equation for the scalar field (REF ) in order to compute the modified expansion rate $\\tilde{H}$ and compare it with the standard expansion rate, $H_{GR}$ .",
"We then use this to compute the modifications to the dark matter relic abundances by solving the Boltzmann equation using the modified expansion rate.",
"We start revisiting by the conformal case by exploring a wide range of initial conditions, masses and cross-sections.",
"We then look at an explicit disformal example.",
"Before solving the master equation (REF ), we would like to write it in terms of Jordan frame quantities $\\tilde{\\omega }= \\omega \\gamma ^2$ , $\\tilde{\\rho }= C^{-2}\\gamma ^{-1} \\rho $ .",
"Moreover, the number of e-folds $N$ can be expressed in terms of Jordan frame quantities as follows.",
"In this frame, the entropy is conserved and is given by $\\tilde{S}=\\tilde{a}\\, \\tilde{s}$ , where $\\tilde{s}= \\frac{2\\pi }{45} g_s(\\tilde{T}) \\tilde{T}^3$ .",
"So, the conservation of entropy and (REF ) show that $N$ is a function of temperature and the scalar field as: $N\\equiv \\ln \\frac{a}{a_0}=\\ln \\left[\\frac{\\tilde{T}_0}{\\tilde{T}}\\left(\\frac{g_s(\\tilde{T}_0)}{g_s(\\tilde{T})}\\right)^{1/3}\\right]+\\ln \\left[\\frac{C_0}{C}\\right]^{1/2}.$ Therefore, we can introduce the parameter, $\\tilde{N}$ , defined as $\\tilde{N} \\equiv \\ln \\left[\\frac{\\tilde{T}_0}{\\tilde{T}}\\left(\\frac{g_s(\\tilde{T}_0)}{g_s(\\tilde{T})}\\right)^{1/3}\\right].$ and transform to derivatives w.r.t.",
"$\\tilde{N}$ (assuming well behaved functions): $\\varphi ^{\\prime } = \\frac{1}{\\left(1-\\alpha (\\varphi ) \\frac{d\\varphi }{d\\tilde{N}}\\right)} \\frac{d\\varphi }{d\\tilde{N}} \\,,\\qquad \\varphi ^{\\prime \\prime } = \\frac{1}{\\left(1-\\alpha (\\varphi ) \\frac{d\\varphi }{d\\tilde{N}}\\right)^3} \\left( \\frac{d^2\\varphi }{d\\tilde{N}^2} + \\frac{d\\alpha }{d\\varphi } \\left(\\frac{d\\varphi }{d\\tilde{N}}\\right)^3\\right)\\,.",
"\\\\ \\nonumber $ In a slight abuse of notation and to keep expressions neat, in what follows we denote derivatives w.r.t.",
"$\\tilde{N}$ with a prime $^{\\prime }$ ."
],
[
"Conformal case",
"We start with the pure conformal case.",
"That is, we take $D(\\phi ) =0$ in (REF ) and therefore $\\gamma =1$ (and $\\tilde{\\omega }= \\omega $ ).",
"Moreover, during the radiation and matter dominated eras, of interest for us, the potential energy of the scalar field is subdominant and therefore, we take $\\lambda \\sim 0$ .",
"Therefore the master equation (REF ) simplifies to: $\\frac{2}{3 (1-\\varphi ^{\\prime 2}/6)} \\,\\varphi ^{\\prime \\prime } &+& \\left(1 -\\tilde{\\omega }\\right)\\varphi ^{\\prime } + 2 (1 -3\\,\\tilde{\\omega })\\,\\alpha (\\varphi ) =0,$ which in terms of derivatives wrt $\\tilde{N}$ takes the form: $\\frac{1 }{3B\\left[1-\\alpha (\\varphi ) \\varphi ^{\\prime } \\right]^3} \\left(\\varphi ^{\\prime \\prime } + \\frac{d\\alpha }{d\\varphi } \\left(\\varphi ^{\\prime }\\right)^3\\right)+ \\frac{(1-\\tilde{\\omega }) }{\\left[1-\\alpha (\\varphi )\\varphi ^{\\prime }\\right]} \\,\\varphi ^{\\prime }+(1-3\\,\\tilde{\\omega }) \\, \\alpha (\\varphi )=0 \\,, \\nonumber \\\\$ where $B= 1-\\frac{\\left(\\varphi ^{\\prime }\\right)^2 }{6\\left(1-\\alpha (\\varphi ) \\varphi ^{\\prime }\\right)^2} $ .",
"Using the relation between $\\tilde{H}$ and $H_{GR}$ defined in (REF ), we can write the speed-up parameter as $\\xi = \\frac{\\tilde{H}}{H_{GR}} = \\frac{C^{1/2}(\\varphi )}{C^{1/2}(\\varphi _0)} \\frac{1}{\\left(1-\\alpha (\\varphi ) \\varphi ^{\\prime }\\right)\\sqrt{B}} \\frac{1}{\\sqrt{1+ \\alpha ^2(\\varphi _0)}}$ where we have used the relation between the bare gravitational constant and that measured by local experiments for conformally coupled theories [20]: $\\kappa _{GR}^2 = \\kappa ^2 C(\\varphi _0) [1+\\alpha ^2(\\varphi _0)]\\,,$ where $\\varphi _0$ is the value of the scalar field at present time."
],
[
"Expansion rate modification",
"The scalar equation in the conformal case (REF ), as function of $N$ (for $\\lambda =0$ ) contains a term which can be interpreted as an effective potential, dictated by $V_{eff} = \\ln C^{1/2}$ .",
"For a strictly radiation dominated era, $\\tilde{\\omega }=1/3$ , the effective potential term vanishes and we are left with an equation that can be solved analytically [21], giving $\\varphi ^{\\prime } \\propto e^{-N}$ .",
"That is, any initial velocity will rapidly go to zero (remember that from the Friedmann equation (REF ), $\\varphi ^{\\prime }$ is constrained to be $\\varphi ^{\\prime } \\lesssim \\pm \\sqrt{6}$ ).",
"Therefore we explore the effects of having a non-zero initial velocity in our analysis below (see also Appendix for further examples).",
"Since the scalar field is expressed in Planck units, we focus on order one or smaller field variations $\\Delta \\varphi $ .",
"One can check, using the analytic solution to (REF ) deep in the radiation era, that for initial velocities $\\varphi ^{\\prime }_0 \\ll \\pm \\sqrt{6}$ , the total field displacement is of order $\\Delta \\varphi \\sim \\varphi ^{\\prime }_0$ [21].",
"However, given that we don't know much about the theory before BBN, we explore different initial values for $(\\varphi _0, \\varphi ^{\\prime }_0)$ and study their consequences.",
"In particular we explore initial values $\\varphi _0$ and $\\varphi ^{\\prime }_0 \\in (-1.0, 1.0)$ .",
"We now concentrate on an explicit conformal factor.",
"We use the same conformal factor as that studied in [3], which is given by: $C(\\varphi ) = (1+ b \\,e^{-\\beta \\, \\varphi })^2$ with the values $b=0.1$ , $\\beta =8$ , which have been shown to satisfy the constraints imposed by tests of gravity, for the parameters $\\alpha , \\beta $ , $\\xi $ .",
"As we will see, the requirement of reaching the GR expansion rate value by the time of the onset of BBN, drives these parameters to very small values, which are thus consistent with the constraints from gravity for their values today.",
"As we discussed above, in the equation of motion for $\\varphi $ , with $\\tilde{\\omega }\\ne 1/3 $ , the conformal factor acts as an effective potential on which the scalar field moves, damped by the Hubble friction (see ()).",
"Since any initial velocity $\\varphi ^{\\prime }$ goes rapidly to zero deep in the radiation era, in the subsequent evolution of $\\varphi $ , the term $(\\varphi ^{\\prime } )^2$ in the master equation will be negligible.",
"In this regime, the equation is that of a particle moving in an effective potential with a damping term.",
"Therefore, one can understand the evolution of $\\varphi $ from the form of the effective potential ($V_{eff}=\\ln C^{1/2}$ ) and the initial conditions chosen.",
"For the conformal function we are considering (REF ), one sees that there is a set of initial conditions that will give rise to an interesting behaviour in $\\varphi $ , and therefore an interesting modified expansion rate in the Jordan frame $\\tilde{H}$ (REF ), as we now explain.",
"In general, both the initial position and velocity of the scalar field can take any value, positive and negative.",
"In the runaway effective potential dictated by $\\ln C^{1/2}$ for the conformal factor (REF ) we consider here, we have the following possibilities.",
"i) The scalar field starting somewhere up in the runaway effective potential with zero initial velocity.",
"In this case the scalar field will roll-down the potential, eventually stopping due to Hubble friction, at some constant value of $\\varphi $ , which depends on the its initial value.",
"So long as $(1+\\alpha (\\varphi ) \\,\\varphi ^{\\prime })$ stays positive (see (REF )), $C$ will evolve rapidly towards 1.",
"More generally, the initial velocity can be different from zero.",
"If the initial velocity is positive, the behaviour will be similar to the previous case.",
"The field will roll-down the effective potential towards its final terminal value.",
"ii) A more interesting possibility arises when one allows for negative initial velocities.",
"In this case, the field will start rolling-up the effective potential towards smaller values of the field, eventually turning back down and moving towards its terminal value.",
"It is easy to see that an interesting effect happens when the field starts at an initial positive value.",
"Given a sufficient initial negative velocity the field will move towards negative values until its velocity becomes zero and then positive again, as it rolls back down the effective potential.",
"This change in sing for the scalar evolution will produce a pick in the conformal function that will give rise to a non-trivial modification of the Jordan's frame expansion rate $\\tilde{H}$ , as we are looking for.",
"As mentioned before, we are interested in (sub-)Planckian initial values $(\\varphi _0, \\varphi ^{\\prime }_0)$ , such that $\\tilde{H}>0$ .",
"With these requirements, one can see that given an initial negative velocity, there is a suitable initial value of the scalar field such that the behaviour just described holds and the expansion rate $\\tilde{H}$ has an interesting evolution before the onset of BBN.",
"At late times the conformal function goes to one and the GR expansion is recovered.",
"We show this behaviour explicitly in Figures REF and REF where we plot the numerical solution for the evolution of $\\varphi $ and $C(\\varphi )$ as functions of the temperature.",
"In these plots we find $\\varphi =\\varphi (\\tilde{T})$ by first solving (REF ) numerically with initial conditions $ (\\varphi _0,\\varphi ^{\\prime }_0) = (0.2, -0.99)$ and then use (REF ) to express $\\varphi (\\tilde{N})$ as function of $\\tilde{T}$ In appendix we show further examples of the thermal evolution of the scalar field.. As we can see, the conformal factor starts growing towards a maximum value as $\\varphi $ moves to negative values, to rapidly drop down towards its GR value at $C\\rightarrow 1$ as $\\varphi $ moves down the effective potential towards positive values.",
"This non-trivial effect will give rise to the possibility of re-annihilation, as we discuss below.",
"Figure: Typical evolution of the scalar field as temperature decreases.",
"The initial values are (ϕ,dϕ/dN ˜)=(0.2,-0.994)(\\varphi ,d\\,\\varphi /d\\,\\tilde{N}) = (0.2,-0.994).Figure: Behaviour of the conformal factor, C(ϕ)C(\\varphi ) as a function of the temperature for the same initial values as in Fig.",
".Based on the discussion above, we have solved the master equation (REF ), to find the the scalar field as a function of $\\tilde{N}$ for various initial conditions, where we see the interesting behaviour explained above.",
"The resulting modified expansion rate and its comparison with the standard case is shown in Figure REF for the same initial conditions as in Figures REF and REF .",
"In our numerical exploration, we choose initial conditions for which the notch in the expansion rate (see Fig.",
"REF ) occurs closer to the BBN timeIn appendix we show more examples of modified expansion rate using different initial conditions..",
"This has interesting consequences for the dark matter annihilation, as we discuss below.",
"Figure: Comparing the Hubble expansion rate H ˜\\tilde{H} in the Jordan Frame with the standard Hubble expansion rate H GR H_{GR}.The presence of the scalar field enhances and decreases the expansion rate during the radiation dominated era.",
"This plot corresponds toinitial conditions given by (ϕ 0 ,ϕ 0 ' )=(0.2,-0.994)(\\varphi _0,\\varphi ^{\\prime }_0) = (0.2,-0.994).When solving the master equation (REF ), we have taken into account an important effect that occurs during the radiation dominated era.",
"Deep in this epoch, the equation of state is given by $\\tilde{\\omega }=1/3$ .",
"When a particle species in the cosmic soup becomes non-relativistic, $\\tilde{\\omega }$ differs slightly from $1/3$ .",
"When the temperature of the universe drops below the rest mass of each of the particle types, there are non-zero contributions to $1-3\\tilde{\\omega }$ .",
"This activates the effective potential, which can be seen in the last term of (REF ), and displaces, or “kicks“ the field along $V_{eff}$ .",
"To examine this effect in more detail, we start by writing $1-3\\,\\tilde{\\omega }$ during the early stages of the universe as in [3] and [15] $1-3\\,\\tilde{\\omega }= \\frac{\\tilde{\\rho }- 3\\, \\tilde{p}}{\\tilde{\\rho }} =\\sum _{A} \\frac{\\tilde{\\rho }_A - 3\\tilde{p}_A}{\\tilde{\\rho }} +\\frac{\\tilde{\\rho }_m}{\\tilde{\\rho }}\\,,$ where the sum runs over all particles in thermal equilibrium during the radiation dominated era and $\\tilde{\\rho }_m$ is the contribution from the non-relativistic decoupled and pressureless matter.",
"The summation over all the particle is responsible for the kicking effect discussed above.",
"Then, a kick function is defined as $\\Sigma (\\tilde{T}) \\equiv \\sum _{A} \\frac{\\tilde{\\rho }_A - 3\\tilde{p}_A}{\\tilde{\\rho }}\\,,$ where the energy density $\\tilde{\\rho }_A$ and pressure $\\tilde{p}_A$ of each type $A$ of particle are given by $\\tilde{\\rho }_A(\\tilde{T}) =\\frac{g_A}{2\\pi ^2}\\int ^\\infty _{m_A}\\frac{\\left(E^2-m_A^2\\right)^{1/2}}{\\exp (E/\\tilde{T})\\pm 1}E^2dE$ $\\tilde{p}_A(\\tilde{T}) =\\frac{g_A}{6\\pi ^2}\\int ^\\infty _{m_A}\\frac{\\left(E^2-m_A^2\\right)^{3/2}}{\\exp (E/\\tilde{T})\\pm 1}E^2dE$ with $g_A$ being the number of internal degrees of freedom of species of type $A$ and the plus (minus) sign in the integral corresponds to fermions (bosons).",
"To compute (REF ), we use the Standard Model particle spectrum.",
"In particular, we take into account the top quark, the Higgs boson, Z boson, W bosons, bottom quark, tau lepton, charm quark, charged pions, neutral pion, moun lepton and the electronWe present more details of the kick function in appendix .. As we show in Figure , $\\Sigma $ is mostly zero, except when the kicks happen.",
"During the radiation dominated era $\\tilde{\\rho }\\simeq \\pi ^2 g_{eff}(\\tilde{T}) \\tilde{T}^4/30$ , where $\\tilde{T}$ is the Jordan frame temperature and $g_{eff}$ is the total number of relativistic degrees of freedom.",
"Also, during this stage $\\tilde{\\rho }_m$ is negligible and as we have shown $\\Sigma $ is slightly different than zero.",
"Therefore, we can compute the equation of state from (REF ) as $\\tilde{\\omega }=(1-\\Sigma (\\tilde{T}))/3$ .",
"In Figure REF we show the evolution of $\\tilde{\\omega }$ between 10 TeV and 10 eV.",
"This figure shows four troughs, which are the “kicks” mentioned above.",
"Each kick corresponds to the transition of one or more particles to the non-relativistic regime.",
"For example, the trough at around 0.5 MeV is due to the electron, while the one at around 100 GeV is due to the heavy particles ($t$ , $H$ , $Z$ and $W$ ).",
"Towards the end of the radiation era, approaching the transition to the matter dominated era, eq.",
"(REF ) takes the approximate form: $1-3\\,\\tilde{\\omega }\\simeq \\frac{\\tilde{\\rho }_{m}}{\\tilde{\\rho }_m + \\tilde{\\rho }_r} \\simeq \\frac{1}{1+ \\tilde{T}/\\tilde{T}_{eq}} \\,,$ where $\\tilde{T}_{eq} \\sim {\\cal O}(10^{-9})$ GeV is the temperature at matter-radiation equality, that is, $\\tilde{\\rho }_m(\\tilde{T}_{eq}) = \\tilde{\\rho }_r(\\tilde{T}_{eq})$ .",
"We can now combine (REF ) and (REF ) to compute the thermal evolution of (REF ) in the radiation dominated and matter dominated eras and use it in the master equation.",
"Figure: Evolution of ω ˜\\tilde{\\omega } in () as function of temperature during the radiation dominated era."
],
[
"Parameter Constraints",
"In scalar-tensor theories of gravity, there are some constraints on the parameters that need to be taken into account.",
"Deviations from GR can be parametrised in terms of the post-Newtonian parameters $\\gamma _{PN}$ and $\\beta _{PN}$ , which are given in terms of $\\alpha (\\varphi _0)$ defined in () and its derivative $\\alpha ^{\\prime }_0= d\\alpha / d\\varphi |_{\\varphi _0}$ as [22], [23]: $\\gamma _{PN} -1 = -\\frac{2\\alpha _0^2}{1+\\alpha _0^2} \\,, \\qquad \\beta _{PN} -1 = \\frac{1}{2}\\frac{\\alpha ^{\\prime }_0\\alpha _0^2}{(1+\\alpha _0^2)^2} \\,,$ Solar system tests of gravity, including the perihelion shift of Mercury, Lunar Laser Ranging experiments, and the measurements of the Shapiro time delay by the Cassini spacecraft [24], [25], [26] indicate that $\\alpha _0$ should be very small, with values $\\alpha _0^2 \\lesssim 10^{-5}$ , while binary pulsar observations impose that $\\alpha ^{\\prime }_0\\gtrsim -4.5$ .",
"The last constraint applies to the the speed-up factor $\\xi $ , which has to be of order 1 before the onset of BBN.",
"In our examples we have $\\alpha _0^2 \\simeq 2\\times 10^{-5}$ , $\\alpha ^{\\prime }_0 > 0$ and $\\xi \\approx 1.05$ ."
],
[
"Impact on relic abundances ",
"We are now ready to discuss the impact of the modified expansion rates on the relic abundance of dark matter species.",
"For a dark matter species $\\chi $ with mass $m_\\chi $ and annihilation cross-section $\\langle \\sigma v \\rangle $ , where $v$ is the relative velocity, the dark matter number density $n_\\chi $ evolves according to the Boltzmann equation $\\frac{d n_\\chi }{dt} = -3 \\tilde{H} n_\\chi - \\langle \\sigma v \\rangle \\left( n_\\chi ^2 - (n_\\chi ^{eq})^2 \\right)\\,,$ where, as we have discussed above, the relevant expansion rate is the Jordan frame one, which can give interesting effects due to the presence of the scalar field.",
"Further $n_\\chi ^{eq}$ is the equilibrium number density.",
"We can rewrite this equation in terms of $x=m_\\chi /\\tilde{T}$ $\\frac{d Y}{dx} = - \\frac{\\tilde{s} \\langle \\sigma v \\rangle }{x \\tilde{H}} \\left( Y^2 - Y_{eq}^2 \\right) \\,.$ where $Y = \\frac{n_\\chi }{\\tilde{s}}$ , $\\tilde{s}= \\frac{2\\pi }{45} g_s(\\tilde{T}) \\tilde{T}^3$ .",
"Numerical solutions to the Boltzmann equation (REF ) with the modified expansion rate $\\tilde{H}$ were found for dark matter particles with masses ranging from 5 GeV to 1000 GeV.",
"For instance, we show solutions in figures REF and REF for two different masses.",
"As we can see from (REF ), the annihilation cross-section influences the evolution of the abundance $Y$ .",
"The current value of $Y$ determines the present dark matter content of the universe.",
"This can be seen clearly by recalling the current value of the energy density parameter $\\Omega _0=\\frac{\\rho _0}{\\rho _{c,0}}=\\frac{m\\,Y_0\\,s_0}{\\rho _{c,0}}$ , where $\\rho _{c,0}$ and $s_0$ are the well-known current values of the critical energy density and the entropy density of the universe, respectively.",
"So, for each single mass, the thermally-averaged annihilation cross section, $\\langle \\sigma v \\rangle $ , was chosen such as the current DM content of the universe is 27 %, so $\\Omega _0=0.27$ .",
"In Figure REF we show the annihilation cross-section, $\\langle \\sigma v \\rangle _{Conformal}$ , found for all masses and compare it to the annihilation cross sections for the standard cosmology model, $\\langle \\sigma v \\rangle _{Standard}$ .",
"As it is shown, for large masses $\\langle \\sigma v \\rangle _{Conformal}$ is larger than $\\langle \\sigma v \\rangle _{Standard}$ , up to a factor of four.",
"As the mass decreases $\\langle \\sigma v \\rangle _{Conformal}$ decreases up to the point where is smaller than $\\langle \\sigma v \\rangle _{Standard}$ .",
"Then, for masses smaller than 100 GeV the figure shows that $\\langle \\sigma v \\rangle _{Conformal} \\approx \\langle \\sigma v\\rangle _{Standard}$ .",
"Thus, we have found that the annihilation cross-sections can be larger or smaller than the thermal average cross-section.",
"Just to give an example of larger and smaller cross-section, the following table compares the numerical values of $\\langle \\sigma v \\rangle _{Conformal}$ and $\\langle \\sigma v \\rangle _{Standard}$ for two dark matter masses, 1000 GeV and 130 GeV.",
"Table: NO_CAPTIONFigures REF and REF show the evolution of the abundance $\\tilde{Y}(x)$ for DM particles with masses 130 GeV and 1000 GeV, respectively.",
"These figures also include the abundance $Y_{GR}(x)$ calculated in the standard cosmology model and the equilibrium abundance $Y_{Eq}(x)$ .",
"Figure: Evolution of the abundance as temperature changes for a DM particle of mass 130 GeV.The temperature evolution of the abundance for a 130 GeV mass is not noticeable affected by the presence of the scalar field $\\phi $ .",
"In this case, $\\tilde{Y}$ and $Y_{GR}$ are almost indistinguishable from one another.",
"On the other hand, the scalar field $\\phi $ has a prominent effect on the temperature evolution of the abundance for a 1000 GeV DM particle.",
"First of all, the freeze-out happens earlier than expected due to the enhancement of the expansion rate, $\\tilde{H}$ .",
"Then, an unusual effect appears.",
"As the temperature decreases, $\\tilde{H}$ becomes smaller than the interaction rateThe interaction rate is defined as $\\tilde{\\Gamma }\\equiv \\langle \\sigma v \\rangle _{Conformal}\\,\\tilde{s}\\,\\tilde{Y}$ .",
"$\\tilde{\\Gamma }$ and a short period of annihilation starts again called “re-annihilation”.",
"The re-annihilation process reduces the abundance of dark matter until a second and final freeze-out happens.",
"After this final freeze-out the abundance remains constant.",
"Figure: Abundance for a mass of 1000 GeV.Figure: Annihilation cross section as function of mass.",
"The presence of the scalar field enhances the 〈σv〉\\langle \\sigma v \\rangle for large masses,and diminishes 〈σv〉\\langle \\sigma v \\rangle for masses around 130 GeV, while small mass the effect is almost negligible.Figure: Expansion rate (as in figure ) and interaction rate as function of temperature.",
"The interaction rate, Γ ˜\\tilde{\\Gamma }, is given by 〈σv〉 Conformal s ˜Y ˜\\langle \\sigma v \\rangle _{Conformal}\\,\\tilde{s}\\,\\tilde{Y}.",
"We use Y ˜\\tilde{Y} from figures and and the values of 〈σv〉 Conformal \\langle \\sigma v \\rangle _{Conformal} presented previously for 130 GeV and 1000 GeV masses.The re-annihilation phase can be described better by discussing the relation between the expansion rate $\\tilde{H}$ and the interaction rate $\\tilde{\\Gamma }$ .",
"The first freeze-out happens when $\\tilde{\\Gamma }$ becomes smaller than $\\tilde{H}$ which can be seen in figure REF to happen around a temperature of 50 GeV for a 1000 GeV particle.",
"Then, near to 7 GeV $\\tilde{H}$ drops below $\\tilde{\\Gamma }$ , and so the re-annihilation process starts and goes on until the second freeze-out occurs.",
"Around 2 GeV $\\tilde{H}$ becomes much larger than $\\tilde{\\Gamma }$ and so the abundance becomes almost constant.",
"Our analysis shows that, as found in [3], re-annihilation occurs for this particular choice of conformal factor.",
"However, we found that when fully integrating the master equation, the re-annihilation occurs only for very large masses of the dark matter particles (in [3] it was found for $m=50$ GeV).",
"On the other hand, in [15], no re-annihilation was foundAlthough [15] used a different conformal factor to [3], we expect the re-annihilation effect to be present also in that case., which was probably due to the initial conditions used and the values of the DM masses explored."
],
[
"Disformal case",
"We now discuss briefly the effect of the disformal factor in the metric (REF ) to the expansion rate of the universe, $\\tilde{H}$ , and compare it to the conformal modification to $\\tilde{H}$We leave a detailed exploration for a future publication..",
"Hence, we explore $D(\\phi )\\ne 0$ for the same conformal factor studied before, that is, $C(\\varphi ) = (1+ b \\,e^{-\\beta \\, \\varphi })^2$ for $b=0.1$ , $\\beta =8$ .",
"To investigate these modifications, we first need to look at the the scalar field evolution with temperature.",
"In the pure conformal case studied above, we found the thermal evolution of the scalar field by solving the master equation (REF ) numerically, which is (REF ) for $D(\\phi )=0$ .",
"However, to study the effects of the disformal factor on the scalar field, it is more convenient to solve the system of two coupled equations (REF ) and ().",
"Using these equations we find solutions for the dimensionless scalar field $\\varphi $ , and for the expansion rate in the Einstein frame $H$ .",
"Notice that solving the system of coupled equations or solving the master equation to find the thermal evolution of the scalar field are equivalent methods (as we have explicitly checked), because (REF ) it is nothing but a combination (REF ) and ().",
"However, while in the pure conformal case the master equation can be made independent of $H$ (or $\\rho $ ), this is not the case for the more general disformal case, as we can see in eq.",
"(REF ).",
"In the same way as for the conformal case, we are interested mainly in the radiation and matter eras and therefore we can neglect the potential energy of the scalar field.",
"Thus, we consider $V\\sim 0$ and $\\lambda =0$ .",
"Also, while solving the coupled equations we have to express $\\omega $ in the Jordan frame by using $\\tilde{\\omega }= \\omega \\gamma ^2$ and transform all derivatives w.r.t.",
"$N$ to derivatives w.r.t.",
"$\\tilde{N}$ by using (REF ).",
"With this information, we solve the system of coupled equations numerically to find the dimensionless scalar field $\\varphi $ and the Hubble parameter $H$ , as functions of the number of e-folds $\\tilde{N}$ (and the temperature).",
"We choose the same initial conditions for the scalar field and its derivative as in the conformal case and to obtain the initial condition for $H$ , we use (REF ).",
"Once we have the solutions for $\\varphi $ and $H$ as functions of temperature, we can go back to (REF ) and (REF ) to obtain the expansion rate for the disformal model.",
"As an example, in Figure REF we show the effects of a disformal factor given by $D(\\varphi ) = D_0\\,\\varphi ^2$ with $D_0 =-4.9\\times 10^{-14}$ .",
"In this plot, we illustrate the effect of the disformal contribution on the expansion rate ($\\tilde{H}_{Disformal}$ ) and compare it to the modified expansion rate for the conformal case ($\\tilde{H}_{Conformal}$ ) and the standard case ($H_{GR}$ ).",
"We use the same initial conditions as in Figures REF and REF for the scalar field and its derivative.",
"Also, it is important to mention that for the case shown the parameter constraints described in section REF are satisfied.",
"In particular we find $\\alpha _0^2 \\simeq 2\\times 10^{-5}$ , $\\alpha ^{\\prime }_0 > 0$ and $\\xi \\approx 1.02$ .",
"Figure: Comparing the modified expansion rate of the universe in the disformal and conformal scenarios for the same initial conditions as in Fig.",
".From our example, with $C$ and $D$ as indicated above, we can clearly see the differences from the disformally modified expansion rate $\\tilde{H}_{Disformal}$ compared to the conformally modified and standard case, $H_{GR}$ .",
"The evolution of $\\tilde{H}_{Disformal}$ is similar to that of $\\tilde{H}_{Conformal}$ , having an (two in our example) enhancement and a decrement compared to the standard expansion rate $H_{GR}$ .",
"Moreover, the main differences with respect to the conformal modification are the position of the notch and its shape.",
"The notch is moved to higher temperatures and it becomes a little bit sharper.",
"These differences between the expansion rates can be understood from ().",
"First, in this equation we see that the factor $\\frac{3H^2\\gamma ^2BD}{\\kappa ^2(1+\\lambda )C}$ in the coefficients of $\\varphi ^{\\prime \\prime }$ , $\\varphi ^{\\prime }$ and $\\varphi ^{\\prime 2}$ vanishes when $D=0$ .",
"For the disformal example shown in Figure REF , this factor is a very small correction to the equation, which is reflected in the slight shape modification of $\\tilde{H}_{Disformal}$ compared to $\\tilde{H}_{Conformal}$ .",
"Second, the term proportional to $\\delta (\\varphi )$ plays a more important role, being responsible for the shifting of the notch.",
"As was discussed previously in section REF , a modification to the expansion rate of the universe prior to BBN has tremendous consequences on the abundance, $\\tilde{Y}$ , of dark matter particles.",
"Also, this modification implies that the thermally-averaged annihilation cross section, $\\langle \\sigma v \\rangle $ , for dark matter particles differs significantly from the one predicted by the standard cosmological model, which is approximately $3.0 \\times 10^{-26}\\rm {cm^3/s}$ (see Figure REF ).",
"We have seen than the enhancement of $\\tilde{H}$ allows bigger values of $\\langle \\sigma v \\rangle $ for particles with masses within certain range, and also, the lower value of $\\tilde{H}$ implies smaller $\\langle \\sigma v \\rangle $ for particles with masses within a small interval.",
"Thus, the location and shape of the notch determines for which masses the annihilation cross section is smaller.",
"In the disformal scenario, the notch has been moved to higher temperatures, which allows particles with higher masses to have smaller and larger annihilation cross sections for the observed DM content.",
"We leave the detailed study of the modification of $\\tilde{H}$ for more general conformal and disformal factors for a separate forthcoming publication."
],
[
"Conclusions",
"Scalar-tensor theories of gravity are a useful method to explore departures of the expansion rate of the universe from the standard cosmological model in the early universe.",
"The expansion rate of the universe had a strong influence on the evolution of the dark matter abundance during the early stages of the universe's evolution, specially prior to BBN.",
"Modifications to the expansion rate during that time would be reflected in the calculation of the dark matter relic abundance and so can be used as a probe to the predictions of scalar-tensor theories.",
"In this paper, we explored the role played by the scalar field in the modification of the expansion rate of the universe on scalar-tensor theories coupled both conformally and disformally to matter.",
"For the conformal case, we explored a conformal factor of the type $C=(1+b\\,e^{-\\beta \\,\\varphi })^2$ .",
"We made no approximations and solved numerically the master equation for the scalar (REF ) for a suitable range of initial conditions.",
"Using this result we then computed the expansion rate modification $\\tilde{H}$ under the presence of the scalar field during the radiation dominated era prior to BBN.",
"When comparing the expansion rate, $\\tilde{H}$ , to the standard expansion rate, $H_{GR}$ , we found that the speed-up factor, $\\xi =\\frac{\\tilde{H}}{H_{GR}}$ , increases up to 200 and then become of order 1 prior to BBN (see Fig.",
"REF ).",
"Previously, in reference [15], it was shown that there is no change in the expansion rate arising from the modification due to $C$ compared to the standard cosmology, after satisfying all the constraints based on the boundary conditions chosen in [3], [4].",
"This enhancement on the expansion rate has important consequences on the evolution of the abundance of dark matter particles.",
"So, we also investigated the effect on the abundance of dark matter particles.",
"We observed that for dark matter particles of large mass ($m\\sim 10^3$ GeV in our example, compared to the $m\\sim 50$ GeV reported in [3] for which we found no re-annihilation), the particles undergo a second annihilation process and then freeze-out once and for all in (see Fig.",
"REF ).",
"Moreover, we found that for large masses the annihilation cross-section has to be up to four times larger than that of standard cosmology models in order to satisfy the dark matter content of the universe of 27 %.",
"On the other hand, for small masses this re-annihilation process is not present, but we found that for masses around 130 GeV, the annihilation cross-section can be smaller than the annihilation cross-section for the standard cosmological model (see Fig.",
"REF ).",
"We also started to investigate the effects on the early evolution of the expansion rate of a disformal factor in the metric (REF ).",
"We noticed that in order to have a consistent solution, i.e.",
"a real positive $\\tilde{H}$ , the conformal and disformal factors need to satisfy a very specific relation, (REF ).",
"We studied the effect of a disformal factor by turning on a small disformal contribution to the conformal case, given by $D(\\varphi )=-4.9\\times 10^{-14}\\varphi ^2$ .",
"To find the modified expansion rate $\\tilde{H}$ during the radiation dominated era prior to BBN, we solved numerically the disformal system of coupled equations () and (REF ).",
"We found that when the disformal function is turned on, $\\tilde{H}$ has a very similar profile as for pure conformal case with an enhancement and a notch compare to the standard expansion rate.",
"However, the position of the notch changes and there may be a second enhancement of $\\tilde{H}$ compared to $H_{GR}$ , depending on the initial conditions (see also Appendix ).",
"The disformal factor, is moving the notch to higher temperatures which means that we can have larger and smaller annihilation cross-section for any mass of the DM candidate for the observed DM content.",
"We have shown the analysis of the disformal case in a concrete example.",
"We plan to study the disformal effects in more detail in order to asses their general features in a forthcoming publication.",
"Moreover, as we mentioned at the beginning of the paper, we are aiming at models that can be embedded in a more fundamental theory of gravity such as string theory.",
"We plan to analyse these cases in a future publication.",
"We would like to thank, David Cerdeño and Nicolao Fornengo for discussions and Gianmassimo Tasinato for discussions and comments on the manuscript.",
"BD and EJ acknowledge support from DOE Grant DE-FG02-13ER42020."
],
[
"Numerical Implementation",
"In this section we describe in detail our numerical method to study the master equation and expansion rate evolution and present additional examples for different initial conditions.",
"As we explained in section REF the result shown in Figure REF for $\\tilde{\\omega }(\\tilde{T})$ was obtained using the so-called “kick” function, defined in (REF ).",
"After using $\\tilde{\\rho }\\simeq \\pi ^2 g_{eff}(\\tilde{T}) \\tilde{T}^4/30$ , (REF ) and (REF ) $\\Sigma $ becomes $\\Sigma (\\tilde{T}) = \\sum _{A} \\frac{15}{\\pi ^4} \\frac{g_A}{g_{eff}(\\tilde{T})} \\,y_A^2 \\int _{y_A}^{\\infty } dx \\frac{\\sqrt{x^2-y^2_A}}{e^x\\pm 1}\\,,$ where $g_{eff}(\\tilde{T})$ is the total number of relativistic degrees of freedomTo calculate $g_{eff}$ we follow the numerical procedure described in Appendix A of [27]., $g_A$ is the number of internal degrees of freedom of particles of type $A$ , $y_A=m_A/\\tilde{T}$ and the plus (minus) sign in the integral is for fermions (bosons).",
"The particle spectrum used to compute $\\Sigma $ is shown in Table REF and in Figure we plot the resulting kick function during the radiation dominated era.",
"Then, we compute the equation of state from (REF ) as $\\tilde{\\omega }=(1-\\Sigma (\\tilde{T}))/3$ , which is shown in Figure REF .",
"Table: Spectrum of particles used to calculate the kick function ().",
"For each particle, we show its mass and number of internal degrees of freedom, g A g_A.Figure: Thermal evolution of the kick function during the radiation dominated era.",
"Outside the interval of temperatures shown, Σ\\Sigma vanishes."
],
[
"Conformal case solutions",
"Using the numerical solution for $\\tilde{\\omega }$ , we plugged it into the master equation, and solve it numerically for the interesting initial conditions as described in section REF for the conformal case.",
"To look for suitable solutions, we fixed the initial value of the velocity and then look for an initial value of the scalar such that $(1+\\alpha (\\varphi )\\varphi ^{\\prime })$ stayed positive, thus giving a positive modified expansion rate.",
"As we explained in that section, we were aiming at solutions where the scalar field passed from positive to negative to positive values again.",
"In Figure REF , we show the thermal evolution of the scalar field in the pure conformal scenario for different initial velocities $\\varphi ^{\\prime }$ .",
"We let the initial value of $\\varphi $ , fixed at $\\varphi _i=0.2$ and solve the master equation (REF ) for the different initial velocities and initial temperature to be 1000 GeV.",
"This plot shows the behaviour of the scalar field as described in section REF .",
"In Figure REF , we show the resulting modified expansion rates for the different initial conditions considered in Fig.",
"REF ."
],
[
"Disformal case solutions",
"As discussed in the main text, when both conformal and disformal functions are non-zero, we solve the system of coupled equations () and (REF ) numerically with Mathematica.",
"We explored different boundary conditions for the scalar field and its first derivative.",
"With the solutions of these equations we used (REF ) to find the modified expansion rate.",
"In Figure REF we show the modified expansion rates for a disformal factor given by $D=D_0\\varphi ^2$ with $D_0=-4.9\\times 10^{-14}$ and the conformal factor being the same as before, that is $C= (1+b e^{-\\beta \\varphi })^2$ with $b=0.1$ , $\\beta = 8$ .",
"We have added one additional initial condition with respect to the conformal case, $\\varphi ^{\\prime }_{initial}=-1.$ It is interesting to notice that for this initial condition, the pure conformal case does not give a solution satisfying the necessary constraints explained in the main text.",
"In this sense, we see that the disformal contribution is important in order to find solutions otherwise excluded.",
"Figure: Modified expansion rate as function of temperature in the disformal scenario for various initial conditions.We use ϕ i =0.2\\varphi _{i}= 0.2 and every curve shown corresponds to a different value of ϕ i ' \\varphi ^{\\prime }_i"
],
[
"General Disformal Set-Up",
"The general scalar-tensor action coupled to matter, which can include a realisation in string theory compactifications is given by: $S=S_{EH} + S_\\phi + S_{m}\\,,$ where: $&& S_{EH}=\\frac{1}{2\\kappa ^2} \\!\\!\\int {\\!d^4x\\sqrt{-g}\\,R}, \\\\&& S_\\phi = - \\!\\!\\int {\\!d^4x\\sqrt{- g} \\left[\\frac{b}{2} (\\partial \\phi )^2+ M^4 C_1 ^2(\\phi )\\sqrt{1+\\frac{D_1(\\phi )}{C_1(\\phi )} (\\partial \\phi )^2}+V(\\phi )\\right]} \\,,\\\\&& S_{m} = - \\!\\!\\int {\\!d^4x\\sqrt{-\\tilde{g}} \\,{\\cal L}_{DM}(\\tilde{g}_{\\mu \\nu }) } \\,,$ and the disformally coupled metric is given by $\\tilde{g}_{\\mu \\nu } = C_2(\\phi ) g_{\\mu \\nu } + D_2(\\phi ) \\partial _\\mu \\phi \\partial _\\nu \\phi \\,.$ $b$ is a constant equal to 1 or 0, depending on the model one wants to consider; $C_i(\\phi ), D_i(\\phi )$ are functions of $\\phi $ , which can be identified as conformal and disformal couplings of the scalar to the metric, respectively.",
"Finally, we have introduced the mass scale $M$ to keep units right (remember that the conformal coupling is dimensionless, whereas the disformal has units of $Mass^{-4}$ .)",
"The connection of the general action (REF ) to the different models in the literature can be obtained as follows: the case $C_1 = D_2$ , $D_1=D_2$ , $b=0$ arises when considering a D-brane moving along an extra dimension.",
"This case was studied in [17] as a model of a coupled dark matter dark energy sector scenario, where scaling solutions arise naturally.",
"Note that in this case, the kinetic term for the scalar field, identified with dark energy for example, is automatically non-canonical and dictated by the DBI action (see [17]).",
"On the other hand, phenomenological models considering a disformal coupling between matter and a scalar field, usually consider a canonical kinetic term, and therefore, in that case, $C_1=D_1=0$ and $b=1$ ($C_1$ can be taken to be non-zero and will be part of the scalar potential).",
"Furthermore, the widely studied case of a conformal coupling is obtained for $b=1, C_1=D_1=D_2=0$ or, as in the case of a D-brane for exampleFor the system corresponding to a D-brane moving in a typically warped compactification in string theory, the functions $C(\\phi )$ and $D(\\phi )$ are identified with powers of the so-called warp factor, usually denoted as $h(\\phi )$ .",
"In this approach, the longitudinal and transverse fluctuations of the D-brane are identified with the dark matter and dark energy fluids respectively [17]., simply considering small velocities with $b=0, C_1=C_2$ and $D_1=D_2$ , and normalising canonically the scalar field (see Appendix ).",
"Finally, let us clarify further our nomenclature on frames.",
"The action in (REF ) is written in the Einstein frame, which in string theory, is usually related to the frame in which the dilaton and the graviton degrees of freedom are decoupled.",
"From this point of view, the dilaton field as well as all other moduli fields not relevant for the cosmological discussion are considered as stabilised, massive, and are therefore decoupled from the low energy effective theory.",
"In the literature of scalar-tensor theories however (including conformal and disformal couplings), the Einstein and Jordan frames are identified with respect to the (usually single) scalar field to which gravity is coupled.",
"In this paper, we follow this and call “Jordan\" or “disformal frame\" the frame in which dark matter is coupled only to the metric $\\tilde{g}_{\\mu \\nu }$ , rather than to the metric $g_{\\mu \\nu } $ and a scalar field $\\phi $ .",
"The equations of motion obtained from (REF ) are (REF ): $R_{\\mu \\nu } -\\frac{1}{2}g_{\\mu \\nu } R = \\kappa ^2\\left(T^\\phi _{\\mu \\nu } + T^{DM}_{\\mu \\nu }\\right)\\,,$ where in the frame relative to $g_{\\mu \\nu }$ the energy momentum tensors are defined in (REF ) and (REF ).",
"The energy-momentum tensor for the scalar field in the general case is modified from (REF ) to: $T_{\\mu \\nu }^{\\phi } = - g_{\\mu \\nu } \\left[M^4C_1^2 \\gamma _1^{-1} + \\frac{b}{2} (\\partial \\phi )^2+ V \\right]+ \\left(M^4 C_1D_1 \\,\\gamma _1+ b\\right) \\partial _\\mu \\phi \\, \\partial _\\nu \\phi $ where now the energy density and pressure are given by: $\\rho _\\phi =- \\frac{b}{2}(\\partial \\phi )^2 + M^4C_1^2 \\gamma _1 + V \\,, \\qquad P_\\phi = - \\frac{b}{2}(\\partial \\phi )^2 - M^4C_1^2\\gamma _1^{-1} - V \\,,$ and the “Lorentz factor\" $\\gamma _1$ introduced above is defined by $\\gamma _1 \\equiv \\left(1+ \\frac{D_1}{C_1}\\, (\\partial \\phi )^2\\right)^{-1/2}\\,.$ We can rewrite (REF ) in a more succinct way, by defining ${\\mathcal {V}} \\equiv V + C_1^2 M^4$ $\\rho _\\phi =- \\left[\\frac{b}{2}+ \\frac{M^4C_1 D_1\\gamma _1}{\\gamma +1}\\right] (\\partial \\phi )^2 + {\\mathcal {V}} \\,, \\qquad P_\\phi = - \\left[\\frac{b}{2}+ \\frac{M^4C_1 D_1\\gamma _1^{-1}}{\\gamma +1}\\right] (\\partial \\phi )^2 - {\\mathcal {V}} \\,.$ The equation of motion for the scalar field becomes (compare with (REF )) $&&\\hspace{-28.45274pt}- \\nabla _\\mu \\!\\left[(M^4 D_1C_1\\gamma _1\\, +b)\\,\\partial ^\\mu \\phi \\right] \\!+ \\!\\frac{\\gamma _1^{-1}M^4 C_1^2}{2} \\!\\left[\\!\\frac{D_1^{\\prime }}{D_1} +3\\frac{C_1^{\\prime }}{C_1} \\right] \\!+ \\!\\frac{\\gamma _1 \\,M^4 C_1^2}{2} \\!\\left[\\!\\frac{C_1^{\\prime }}{C_1} -\\frac{D_1^{\\prime }}{D_1} \\right] \\!+\\!",
"V^{\\prime }\\nonumber \\\\&& \\hspace{85.35826pt}- \\frac{T^{\\mu \\nu }}{2}\\!\\left[\\frac{C_2^{\\prime }}{C_2} g_{\\mu \\nu } +\\frac{D_2^{\\prime }}{C_2}\\partial _\\mu \\phi \\partial _\\nu \\phi \\right]+\\nabla _\\mu \\left[\\frac{D_2}{C_2}T^{\\mu \\nu } \\partial _\\nu \\phi \\right] =0\\,.",
"\\nonumber \\\\$ Finally, the energy-momentum conservation equation gives rise to (REF ), where $Q$ now is given in terms of $C_2, D_2$ : $Q\\equiv \\nabla _\\mu \\left[\\frac{D_2}{C_2} \\,T^{\\mu \\lambda } \\,\\partial _\\lambda \\phi \\right] - \\frac{T^{\\mu \\nu } }{2} \\left[\\frac{C_2^{\\prime }}{C_2} g_{\\mu \\nu } +\\frac{D_2^{\\prime }}{C_2} \\,\\partial _\\mu \\phi \\,\\partial _\\nu \\phi \\right]\\,.$"
],
[
"General cosmological equations",
"The equations of motion for the general system in an FRW background become: $&& H^2 =\\frac{\\kappa ^2}{3} \\left[\\rho _\\phi +\\rho \\right]\\,, \\\\&& \\dot{H} + H^2 = -\\frac{\\kappa ^2}{6}\\left[ \\rho _\\phi + 3P_\\phi +\\rho +3 P \\right]\\,,\\\\&& \\ddot{\\phi }\\left[1+ \\frac{b}{M^4C_1D_1\\gamma _1^3}\\right]+3H\\dot{\\phi }\\,\\gamma _1^{-2}\\left[\\frac{b}{M^4C_1D_1\\gamma _1}+ 1\\right] \\nonumber \\\\&&\\hspace{14.22636pt}+ \\frac{C_1}{2D_1}\\left(\\gamma _1^{-2}\\left[\\frac{5 C_1^{\\prime }}{C_1} - \\frac{D_1^{\\prime }}{D_1}\\right]+ \\frac{D_1^{\\prime }}{D_1}- \\frac{C_1^{\\prime }}{C_1} -4\\gamma ^{-3}_1 \\frac{C^{\\prime }}{C} \\right)+ \\frac{1}{M^4C_1D_1\\gamma _1^{3}}\\, ({\\mathcal {V}}^{\\prime }+Q_0) =0 \\,, \\nonumber \\\\$ where, $H= \\frac{\\dot{a}}{a}$ , dots are derivatives with respect to $t$ , $^{\\prime }=d/d\\phi $ and $\\gamma _1= (1-D_1 \\,\\dot{\\phi }^2/C_1)^{-1/2}.$ We also have the continuity equations for the scalar field and matter given by $&&\\dot{\\rho }_\\phi + 3H(\\rho _{\\phi }+P_{\\phi }) = -Q_0\\dot{\\phi }\\,, \\\\&&\\dot{\\rho }+ 3H(\\rho +P) = Q_0\\,\\dot{\\phi }\\,.$ where $Q_0$ is given by $Q_0 = \\rho \\left[ \\frac{D_2}{C_2} \\,\\ddot{\\phi }+ \\frac{D_2}{C_2} \\,\\dot{\\phi }\\left(\\!3H + \\frac{\\dot{\\rho }}{\\rho } \\right) \\!+ \\!\\left(\\!\\frac{D_2^{\\prime }}{2C_2}-\\frac{D_2}{C_2}\\frac{C_2^{\\prime }}{C_2}\\!\\right) \\dot{\\phi }^2 +\\frac{C_2^{\\prime }}{2\\,C_2} (1-3\\,\\omega )\\right].",
"\\nonumber \\\\$ Using () we can rewrite this in a more compact and useful form as $Q_0 = \\rho \\left( \\frac{\\dot{\\gamma }_2}{\\dot{\\phi }\\, \\gamma _2} + \\frac{C_2^{\\prime }}{2C_2} (1-3\\,\\omega \\,\\gamma _2^2) -3H\\omega \\,\\frac{(\\gamma _2 -1) }{\\dot{\\phi }}\\right) \\,,$ where $\\gamma _2= (1-D_2 \\,\\dot{\\phi }^2/C_2)^{-1/2}.$ Plugging this into the (non-)conservation equation for dark matter (), gives: $\\dot{\\rho }+ 3H (\\rho + P\\,\\gamma _2^{2}) = \\rho \\left[\\frac{\\dot{\\gamma }_2}{\\gamma _2} + \\frac{C_2^{\\prime } }{2C_2} \\,\\dot{\\phi }\\, (1-3\\,\\omega \\gamma _2^2)\\right]\\,.$ The energy densities and pressures in the Einstein and Jordan frames are now related similarly to (REF ), replacing $\\gamma \\rightarrow \\gamma _2$ : $\\tilde{\\rho }= C_2^{-2} \\gamma _2^{-1} \\rho \\,, \\qquad \\tilde{P} = C_2^{-2}\\gamma _2\\, P,$ and therefore the equation of states in both frames are related by $\\tilde{\\omega }= \\omega \\gamma _2^2$ .",
"Similarly the physical proper time and the scale factors in the two frames are related via $\\gamma _2$ : $\\tilde{a} = C_2^{1/2} a \\,, \\qquad \\quad d\\tilde{\\tau }= C_2^{1/2} \\gamma _2^{-1} d\\tau \\,.$ Defining the disformal frame Hubble parameter $\\tilde{H} \\equiv \\frac{d \\ln {\\tilde{a}}}{d\\tilde{\\tau }}$ , gives: $\\tilde{H} = \\frac{\\gamma _2}{C_2^{1/2}}\\left[ H + \\frac{C_2^{\\prime }}{2C_2}\\dot{\\phi }\\right]\\,.$ To solve the equations of motion one now can proceed as in section REF to write the equations in terms of derivatives w.r.t.",
"the number of e-folds $N$ and consider different cases by choosing appropriately the parameters $b,C_i,D_i$ .",
"We leave the analysis of these for a future publication."
],
[
"The conformal case in D-brane scenarios",
"In this section we show how to recover the pure conformal case from the D-brane picture, that is, $b=0$ , $C_1=C_2$ , $D_1=D_2$ .",
"We start by expanding the square root in the scalar part of the action (REF ).",
"Doing this we get $S_\\phi & =& -\\int {d^4x \\sqrt{-g} \\left[ M^4C_1^2 \\left( 1+ \\frac{D_1}{2C_1} (\\partial \\phi )^2 + \\dots \\right) + V(\\phi )\\right]}\\nonumber \\\\&=& -\\int {d^4x \\sqrt{-g} \\left[ \\frac{M^4C_1 D_1}{2} (\\partial \\phi )^2 + M^4C_1^2(\\phi )+ V(\\phi ) + \\dots \\right]}\\nonumber \\\\&=& -\\int {d^4x \\sqrt{-g} \\left[ \\frac{M^4C_1 D_1}{2} (\\partial \\phi )^2 + {\\mathcal {V}}(\\phi ) + \\dots \\right]} \\,,$ On the other hand, the matter Lagrangean takes the form $S_{DM} &=& -\\int {d^4x \\sqrt{-\\tilde{g}} \\,{\\cal L}_{DM}(\\tilde{g}_{\\mu \\nu })}\\nonumber \\\\&=& -\\int {d^4x \\sqrt{-g} \\, C_1^2(\\phi ) \\left(1+ \\frac{D_1}{2C_1} (\\partial \\phi )^2 + \\dots \\right){\\cal L}_{DM}( \\tilde{g}_{\\mu \\nu })} \\nonumber \\\\ &=& -\\int {d^4x \\sqrt{- g} \\, C_1^2 (\\phi ) {\\cal L}_{DM}( \\tilde{g}_{\\mu \\nu }) + \\dots } =-\\int {d^4x \\sqrt{- \\tilde{g}} \\, {\\cal L}_{DM}( \\tilde{g}_{\\mu \\nu }) + \\dots } \\nonumber \\\\$ where now $\\tilde{g}_{\\mu \\nu } = C_1(\\phi ) g_{\\mu \\nu }$ (and we have used that $\\det \\tilde{g}_{\\mu \\nu } = C_1^4 (1+ D_1/C_1 (\\partial \\phi )^2)$ ).",
"Finally, to compare the D-brane case with the pure conformal case, we need to canonically normalise $\\phi $ .",
"Calling $\\varphi $ the canonically normalised field, this is obtained from $\\phi $ as $\\varphi = \\int M^2\\sqrt{D_1C_1} \\, d\\phi \\,.$ It is clear that when $D_1=1/(M^4 C_1)$ , $\\varphi =\\phi $ and therefore the action for the scalar field (REF ) is already in the required form.",
"We can now take the limit $\\gamma \\rightarrow 1$ into the equations of motion (REF )-() and make the identification $D_1=1/(M^4C_1)$ to recover the conformal case equations of motion.",
"Note that in this limit $Q_0 \\rightarrow \\rho C_1^{\\prime }/2C_1$ , and is independent of $D_1$ (see (REF ) with $C_1=C_2, D_1=D_2$ )."
]
] | 1612.05553 |
[
[
"Scheme variations of the QCD coupling and tau decays"
],
[
"Abstract The QCD coupling, $\\alpha_s$, is not a physical observable since it depends on conventions related to the renormalization procedure.",
"Here we discuss a redefinition of the coupling where changes of scheme are parametrised by a single parameter $C$.",
"The new coupling is denoted $\\hat \\alpha_s$ and its running is scheme independent.",
"Moreover, scheme variations become completely analogous to renormalization scale variations.",
"We discuss how the coupling $\\hat \\alpha_s$ can be used in order to optimize predictions for the inclusive hadronic decays of the tau lepton.",
"Preliminary investigations of the $C$-scheme in the presence of higher-order terms of the perturbative series are discussed here for the first time."
],
[
"Introduction",
"The perturbative expansion in the strong coupling $\\alpha _s$ is the main approach to predictions in quantum chromodynamics (QCD) at sufficiently high energies.",
"However, the expansion parameter, $\\alpha _s$ , is not a physical observable of the theory.",
"Its definition carries a dependence on conventions related to the renormalization procedure, such as the renormalization scale and renormalization scheme.",
"Physical observables should, of course, be independent of any such conventions.",
"This requirement leads, in the case of the renormalization scale, to well defined Renormalization Group Equations (RGE) that must be satisfied by physical quantities.",
"The situation regarding the renormalization scheme is more complicated and perturbative computations are, most often, performed in conventional schemes such as ${\\overline{\\rm MS}}$ [1].",
"In this work we discuss a new definition of the QCD coupling, that we denote $\\hat{\\alpha }_s$ , recently introduced in Ref.",
"[2], and its applications to the QCD description of inclusive hadronic $\\tau $ decays.",
"The running of this new coupling is renormalization scheme independent, i.e.",
"in its $\\beta $ function only scheme independent coefficients intervene.",
"The scheme dependence of $\\hat{\\alpha }_s$ is parametrised by a single continuous parameter $C$ .",
"The evolution of $\\hat{\\alpha }_s$ with respect to this new parameter is governed by the same $\\beta $ function that governs the scale evolution.",
"We refer to the coupling $\\hat{\\alpha }_s$ as the $C$ -scheme coupling.",
"An important aspect is the fact that perturbative expansions in $\\alpha _s$ are divergent series that are assumed to be asymptotic expansions to a “true” value, which is unknown [3].F.",
"Dyson formulated the first form of this reasoning in 1952, in the context of Quantum Electrodynamics [4].",
"In this spirit, different schemes correspond to different asymptotic expansions to the same scheme invariant physical quantity, and should be interpreted as such.",
"One can then use the parameter $C$ to interpolate between perturbative series with larger or smaller coupling values, and exploit this dependence in order to optimize the predictions for observables of the theory.",
"The idea of exploiting the scheme dependence in order to optimize the series differs from the approach of other celebrated methods used for the optimisation of perturbative predictions.",
"In methods such as Brodsky-Lepage-Mackenzie (BLM) [5] or the Principle of Maximum Conformality [6], [7] the idea is to obtain a scheme independent result through a well defined algorithm for setting the renormalization scale, regardless of the intermediate scheme used for the perturbative calculation (which most often is ${\\overline{\\rm MS}}$ ).",
"The “effective charge\" method [8], on the other hand, involves a process dependent definition of the coupling.",
"In the procedure described here, one defines a process independent class of schemes, parametrised by the parameter $C$ .",
"The optimal value of $C$ must be set independently for each process considered.",
"We begin with the scale running of the QCD coupling that we write as $-\\,Q\\,\\frac{{\\rm d}a_Q}{{\\rm d}Q} \\,\\equiv \\, \\beta (a_Q) \\,=\\,\\beta _1\\,a_Q^2 + \\beta _2\\,a_Q^3 + \\beta _3\\,a_Q^4 + \\cdots $ We will work with $a_Q \\equiv \\alpha _s(Q)/\\pi $ , with $Q$ being a physically relevant scale.",
"Since the recent five-loop computation of Ref.",
"[9], the first five coefficients of the QCD $\\beta $ -function are known analytically.",
"The coefficients $\\beta _1$ and $\\beta _2$ are scheme independent.",
"Let us consider a scheme transformation to a new coupling $a^{\\prime }$ , which, perturbatively, takes the general form $a^{\\prime } \\,\\equiv \\, a + c_1\\,a^2 + c_2\\,a^3 + c_3\\,a^4 + \\cdots \\,$ The QCD scale $\\Lambda $ is also different in the two schemes and obeys the relation $\\Lambda ^{\\prime } \\,=\\, \\Lambda \\,{\\rm e}^{c_1/\\beta _1}.$ The shift in $\\Lambda ^{\\prime }$ depends only on a single constant [10], governed by $c_1$ of Eq.",
"(REF ).",
"This fact motivates the definition of the new coupling $\\hat{a}_Q$ , which is scheme invariant except for shifts in $\\Lambda $ parametrised by a parameter $C$ as $\\frac{1}{\\hat{a}_Q} + \\frac{\\beta _2}{\\beta _1} \\ln \\hat{a}_Q \\,&\\equiv &\\,\\beta _1 \\Big ( \\ln \\frac{Q}{\\Lambda } + \\frac{C}{2} \\Big ) \\nonumber \\\\&& \\hspace{-51.21495pt} \\,=\\, \\frac{1}{a_Q} + \\frac{\\beta _1}{2}\\,C +\\frac{\\beta _2}{\\beta _1}\\ln a_Q - \\beta _1 \\!\\int \\limits _0^{a_Q}\\,\\frac{{\\rm d}a}{\\tilde{\\beta }(a)},$ where $\\frac{1}{\\tilde{\\beta }(a)} \\,\\equiv \\, \\frac{1}{\\beta (a)} - \\frac{1}{\\beta _1 a^2}+ \\frac{\\beta _2}{\\beta _1^2 a}$ is free of singularities in the limit $a\\rightarrow 0$ and we have used the scale invariant form of $\\Lambda $ .",
"The coupling $\\hat{a}_Q$ is a function of the parameter $C$ but we do not make this dependence explicit to keep the notation simple.",
"The definition of Eq.",
"(REF ) should be interpreted in perturbation theory in an iterative sense, which allows one to deduce the corresponding coefficients $c_i$ of Eq.",
"(REF ) (their explicit expressions are given in [2] using the ${\\overline{\\rm MS}}$ as the input scheme).",
"One should remark that a combination similar to (REF ), but without the logarithmic term on the left-hand side, was already discussed in Refs.",
"[11], [12].",
"However, without this term, an unwelcome logarithm of $a_Q$ remains in the perturbative relation between the couplings $\\hat{a}_Q$ and $a_Q$ .",
"This non-analytic term is avoided by the construction of Eq.",
"(REF ).",
"From the definition of the new coupling $\\hat{a}_Q$ we can derive its $\\beta $ function that reads $-\\,Q\\,\\frac{{\\rm d}\\hat{a}_Q}{{\\rm d}Q} \\,\\equiv \\, \\hat{\\beta }(\\hat{a}_Q) \\,=\\,\\frac{\\beta _1 \\hat{a}_Q^2}{\\left(1 - \\mbox{$\\frac{\\beta _2}{\\beta _1}$}\\, \\hat{a}_Q\\right)} .$ The function $\\hat{\\beta }$ takes a simple form and is scheme independent since only the coefficients $\\beta _1$ and $\\beta _2$ intervene.",
"The evolution with the parameter $C$ obeys an analogous equation $-2 \\frac{{\\rm d}\\hat{a}_Q}{{\\rm d}C} =\\,\\frac{\\beta _1 \\hat{a}_Q^2}{\\left(1 - \\mbox{$\\frac{\\beta _2}{\\beta _1}$}\\, \\hat{a}_Q\\right)}.$ Therefore, there is a complete analogy between the coupling evolution with respect to the scale and with respect to the scheme parameter $C$ .",
"The dependence of $\\hat{a}_Q$ on $C$ is displayed in Fig.",
"REF using the ${\\overline{\\rm MS}}$ as input scheme and setting the scale to the $\\tau $ mass, $M_\\tau $ .",
"The new coupling becomes smaller for larger values of $C$ and perturbativity breaks down for values below roughly $C=-2$ .",
"Therefore, we restrict our analysis to $C\\ge -2$ .",
"Figure: The coupling a ^(M τ )\\hat{a}(M_\\tau ) according to Eq.",
"() as a functionof CC, and for the MS ¯{\\overline{\\rm MS}} input value α s (M τ )=0.316(10)\\alpha _s(M_\\tau )=0.316(10).",
"Theyellow band corresponds to the α s \\alpha _s uncertainty."
],
[
"Application to $\\tau $ decays",
"As a phenomenological application of the $C$ -scheme coupling, we focus here on the perturbative expansion of the total $\\tau $ hadronic width.",
"The chief observable is the ratio $R_\\tau $ of the total hadronic branching fraction to the electron branching fraction.",
"It is conventionally decomposed as $R_\\tau \\,=\\, 3\\, S_{\\rm EW} (|V_{ud}|^2 + |V_{us}|^2)\\, ( 1 + \\delta ^{(0)}+ \\cdots ),$ where $S_{\\rm EW}$ is an electroweak correction and $V_{ud}$ , as well as $V_{us}$ , CKM matrix elements.",
"Perturbative QCD is encoded in $\\delta ^{(0)}$ (see Refs.",
"[13], [14] for details) and the ellipsis indicate further small sub-leading corrections.",
"The calculation of $\\delta ^{(0)}$ is performed from a contour integral of the so called Adler function in the complex energy plane, exploiting analyticity properties, which allows one to avoid the low energy region where perturbative QCD is not valid.",
"In doing so, one must adopt a procedure in order to deal with the renormalization scale.",
"The scale logarithms can be summed either before or after performing the contour integration.",
"The first choice, where the integrals are performed over the running QCD coupling, is called Contour Improved Perturbation Theory (CIPT), while the second, where the coupling is evaluated at a fixed scale and the integrals are performed over the logarithms, is called Fixed Order Perturbation Theory (FOPT).",
"Analytic results for the coefficients of the Adler function are available up to five loops, or $\\alpha _s^4$ [15].",
"Here we consider an estimate for the yet unknown fifth order coefficient of the Adler function, namely $c_{51}=283$ [14].",
"In FOPT, the perturbative series of $\\delta ^{(0)}(a_Q)$ in terms of the ${\\overline{\\rm MS}}$ coupling $a_Q$ is given by [15], [14] $\\delta _{\\rm FO}^{(0)}(a_Q) =a_Q + 5.202a_Q^2 + 26.37a_Q^3 + 127.1a_Q^4 +\\cdots $ In the $C$ -scheme coupling $\\hat{a}_Q$ , the expansion for $\\delta _{\\rm FO}^{(0)}$ is $&\\delta _{\\rm FO}^{(0)}(\\hat{a}_Q) = \\hat{a}_Q + (5.202 + 2.25 C)\\,\\hat{a}_Q^2 \\nonumber \\\\& \\hspace{5.69046pt} + (27.68 + 27.41 C + 5.063 C^2)\\,\\hat{a}_Q^3 \\nonumber \\\\& \\hspace{5.69046pt} + (148.4 + 235.5 C + 101.5 C^2 + 11.39 C^3)\\,\\hat{a}_Q^4\\nonumber \\\\& \\hspace{5.69046pt} + \\cdots $ Figure: δ FO (0) (a ^ Q )\\delta _{\\rm FO}^{(0)}(\\hat{a}_Q) of Eq.",
"() as a function ofCC.",
"The yellow band arises from either removing or doubling the fifth-orderterm.",
"In the red dots, the 𝒪(a ^ 5 ){\\cal O}(\\hat{a}^5) vanishes, and 𝒪(a ^ 4 ){\\cal O}(\\hat{a}^4) is taken asthe uncertainty.For further explanation, see the text.In Fig.",
"REF , we display $\\delta _{\\rm FO}^{(0)}(\\hat{a}_Q)$ as a function of $C$ .",
"Assuming $c_{5,1}=283$ , the yellow band corresponds to removing or doubling the ${\\cal O}(\\hat{a}^5)$ term.",
"A plateau is found for $C\\approx -1$ .",
"Taking $c_{5,1}=566$ and then doubling the ${\\cal O}(\\hat{a}^5)$ results in the blue curve that does not show this stability.",
"Hence, this scenario is disfavoured.",
"In the red dots, which lie at $C=-0.882$ and $C=-1.629$ , the ${\\cal O}(\\hat{a}^5)$ correction vanishes, and the ${\\cal O}(\\hat{a}^4)$ term is taken as the uncertainty, in the spirit of asymptotic series.",
"The point to the right has a substantially smaller error, and yields $\\delta _{\\rm FO}^{(0)}(\\hat{a}_{M_\\tau },C=-0.882) \\,=\\,0.2047 \\pm 0.0034 \\pm 0.0133 \\,.$ The second error covers the uncertainty of $\\alpha _s(M_\\tau )$ .",
"In this case, the direct ${\\overline{\\rm MS}}$ prediction of Eq.",
"(REF ) is $\\delta _{\\rm FO}^{(0)}(a_{M_\\tau }) \\,=\\, 0.1991 \\pm 0.0061 \\pm 0.0119\\,\\,\\,\\,\\, ({\\overline{\\rm MS}}) \\,.$ This value is somewhat lower, but within $1\\,\\sigma $ of the higher-order uncertainty.",
"In CIPT, contour integrals over the running coupling have to be computed, and hence the result cannot be given in analytical form.",
"The general behaviour is very similar to FOPT, with the exception that now also for $c_{5,1}=566$ a zero of the ${\\cal O}(\\hat{a}^5)$ term is found.",
"Employing the value of $C$ which leads to the smaller uncertainty one finds $\\delta _{\\rm CI}^{(0)}(\\hat{a}_{M_\\tau },C=-1.246) \\,=\\,0.1840 \\pm 0.0062 \\pm 0.0084 \\,.$ As has been discussed many times in the past (see e.g.",
"[14]) the CIPT prediction lies substantially below the FOPT results.",
"On the other hand, the parametric $\\alpha _s$ uncertainty in CIPT turns out to be smaller."
],
[
"Higher-order terms",
"The behaviour of the series at higher orders is not known exactly.",
"However, realistic models of the Adler function can be constructed in the Borel plane, in which the singularities of the function, namely its renormalon content, is partially known [3].",
"In Ref.",
"[14] (see also Ref.",
"[16]), models of the Adler function were constructed using the leading renormalons, that largely dominate the higher-order behaviour of the perturbative series.",
"The model is matched to the exactly known coefficients in order to fully reproduce QCD for terms up to $a_Q^4$ .",
"This allows for a complete reconstruction of the series, to arbitrarily high orders in the coupling, and, moreover, one is able to obtain the “true” value of the asymptotic series by means of the Borel sum.",
"In fact, the series is not strictly Borel summable because infra-red renormalons obstruct integration on the positive real axis.",
"The “true” value has, therefore, an inherent ambiguity that stems from the prescription adopted to circumvent the singularities along the contour of integration.",
"This ambiguity is related to non-perturbative physics [3], [14].",
"Here we perform a preliminary investigation of the behaviour of $\\delta ^{(0)}$ at higher orders using the $C$ -scheme coupling.",
"The Adler function coefficients for terms higher than $a_Q^5$ are obtained in the ${\\overline{\\rm MS}}$ scheme from the central model of Ref. [14].",
"The series can then be translated to the $C$ -scheme by means of the perturbative relation between the couplings $a_Q$ and $\\hat{a}_Q$ [2].",
"Fig.",
"REF shows four different series that should approach the same Borel summed result, showed as a horizontal band.",
"The four series use as input the coefficients exactly known in QCD with the addition of the estimate $c_{5,1}=283$ .",
"One observes that the optimised version of $\\delta _{\\rm FO}^{(0)}$ (filled circles) approaches the Borel sum of the series faster than the ${\\overline{\\rm MS}}$ result (empty circles).",
"Of course, because the optimised series has a larger coupling (see Fig.",
"1) asymptoticity sets in earlier and the divergent character is clearly visible already around the 10th order.",
"The FOPT result with $C=0.7$ shows that smaller couplings do not necessarily lead to a better approximation at lower orders, requiring many more terms to give a good approximation to the Borel summed result.",
"Finally, the optimal CIPT series does not give a good approximation to the Borel summed result (this is also the case in the ${\\overline{\\rm MS}}$ [14]).",
"Unfortunately, the use of the $C$ -scheme coupling does not make the CIPT prediction closer to FOPT.",
"The $C$ -scheme FOPT, on the other hand, is in excellent agreement with the central Borel model which suggests that FOPT should be the favoured expansion.",
"Figure: Four series for δ (0) \\delta ^{(0)} with higher-order coefficients from the central model of Ref. .",
"In all cases α s (M τ )=0.316\\alpha _s(M_\\tau )=0.316 which corresponds to the central value of the present world average .",
"The optimised FOPT (filled circles) and CIPT (filled squares) series can be compared with the FOPT MS ¯{\\overline{\\rm MS}} results (empty circles) and FOPT for C=0.7C=0.7 (triangles).",
"The shaded band gives the Borel summed result, the “true\" value of the series, with its associated ambiguity ."
],
[
"Acknowledgements",
" It is a pleasure to thank the organisers of this very fruitful meeting.",
"DB is supported by the São Paulo Research Foundation (FAPESP) grant 2015/20689-9, and by CNPq grant 305431/2015-3.",
"The work of MJ and RM has been supported in part by MINECO Grant number CICYT-FEDER-FPA2014-55613-P, by the Severo Ochoa excellence program of MINECO, Grant SO-2012-0234, and Secretaria d'Universitats i Recerca del Departament d'Economia i Coneixement de la Generalitat de Catalunya under Grant 2014 SGR 1450."
]
] | 1612.05558 |
[
[
"First estimation of the fission dynamics of the spectator created in\n heavy-ion collisions"
],
[
"Abstract In peripheral high-energy heavy-ion collisions only parts of colliding nuclei interact leading to the production of a fireball.",
"The remnants of such nuclei are called spectators.",
"We estimated the excitation energy as a function of impact parameter.",
"Their excitation energy is of the order of 100 MeV.",
"The dynamical evolution of hot nuclei is described by solving a set of Langevin equations in four-dimensional collective coordinate space.",
"The range of nuclear masses and excitation energies suits very well to ability of our model.",
"Thus for the first time we investigate dynamically the fission and evaporation channels in de-excitation of the spectators produced in heavy-ions collision."
],
[
"Introduction",
"The physics of high-energy heavy-ion collisions develops very effectively since many years.",
"At CERN e.g.",
"$^{208}$ Pb+$^{208}$ Pb collisions were considered at SPS and LHC.",
"Two lead nuclei flying with velocities comparable with light velocity produce the fireball in the place of collision, but depending on the centrality of the collision the Pb parts which are not participating in the collision (called ”spectators”) follow their initial path and deexcites later on.",
"There are many experimental and theoretical [1], [2] investigations of the processes connected to the fireball – a piece of quark-gluon plasma.",
"The density of the matter in the contact area is so high that the quark-gluon plasma is created and the high-energy particles such as pions, kaons, and others emerge in the hadronisation process.",
"This part of the reaction requires dedicated models and methods that are beyond the scope of the present discussion.",
"The interesting issue is the behavior of the spectators produced as the remnants of the collision [3].",
"There are not so many experimental observables published up to now, which can help with constraining the physics of spectator evolution.",
"The fast collision of two spherical ions provides exotic, for nuclear physics, shapes of the remnants which are strongly dependent on the impact parameter.",
"The deformation energy of the produced system turns into excitation energy.",
"Thus we treat the spectator as an excited nucleus, which can deexcite by shape changes or emission of light particles and/or $\\gamma $ -rays.",
"This allows to apply the stochastic approach based on the solving transport equations in collective coordinate space [4], [5], [6], [7]."
],
[
"Theoretical method",
"The stochastic approach assumes that the evolution of the nucleus in collective coordinate space is similar to the Brownian motion of particles which interact stochastically involving a large number of internal degrees of freedom, which constitutes the surrounding \"heat bath\".",
"The motion of the nucleus is slowed down by a friction, considered here as the hydrodynamical friction force.",
"The transport coefficients are derived from the random force averaged over a time larger than the collision time scale between collective and internal variables.",
"The Gaussian white noise is taken for the random part, producing fluctuations of the physical observables such as the mass/charge distribution of the fission fragments or evaporation residua.",
"The coupled Langevin equations describing the fission reaction have the form: $\\frac{d q_{i}}{d t}&=&\\mu _{ij}p_{j}, \\\\\\frac{d p_{i}}{d t}&=&- \\frac{1}{2}p_{j}p_{k}\\frac{\\partial \\mu _{jk}}{\\partial q_{i}}- \\left( \\frac{\\partial F}{\\partial q_i} \\right)_T- \\gamma _{ij}\\mu _{jk}p_{k}+\\theta _{ij}\\xi _{j}\\left(t\\right),\\nonumber $ where ${\\bf q}$ are the collective coordinates, ${\\bf p}$ are the conjugate momenta, $F({\\bf q},K)=V({\\bf q},K) - a({\\bf q}) T^2$ is the Helmholtz free energy, $V({\\bf q})$ is the potential energy, $m_{ij}({\\bf q})$ ($\\Vert \\mu _{ij}\\Vert =\\Vert m_{ij}\\Vert ^{-1}$ ) is the tensor of inertia, and $\\gamma _{ij}({\\bf q})$ is the friction tensor.",
"The $\\xi _j\\left(t\\right)$ is a random variable satisfying the relations $<\\xi _{i}>&=&0, \\nonumber \\\\<\\xi _{i}(t_{1})\\xi _{j}(t_{2})>&=&2\\delta _{ij}\\delta (t_{1}-t_{2}).$ Thus, a Markovian approximation is assumed to be valid.",
"The strength of the random force $\\theta _{ij}$ is given by the Einstein relation $\\sum \\theta _{ik}\\theta _{kj} = T\\gamma _{ij}$ .",
"The temperature of the \"heat bath\" $T$ is determined by the Fermi-gas model formula $T=(E_{\\rm {int}}/a)^{1/2}$ , where $E_{\\rm {int}}$ is the internal excitation energy of the nucleus and $a({\\bf q})$ is the level-density parameter.",
"The energy conservation law is used in the form $E^{*}=E_{\\rm {int}}+E_{\\rm {coll}}+V+ E_{\\rm {evap}}(t)$ at each time step during a random walk along the Langevin trajectory in the collective coordinate space.",
"The total excitation energy of the nucleus is $E^{*}$ and the collective kinetic energy is taken as: $E_{\\rm {coll}}= 0.5 \\sum \\mu _{ij} p_i p_j$ .",
"The energy carried away by the evaporated particles by the time $t$ is marked as $E_{\\rm {evap}}(t)$ .",
"The collective coordinate space is constructed with three deformation parameters: elongation of the nucleus $(q_1)$ , its neck $(q_2)$ and octupole shape $(q_3)$ based on the ”funny hills” parametrization proposed initially by Brack [8] and the forth degree of freedom is the projection of the angular momentum vector on the fission axis.",
"This four collective coordinates allow to describe rich ensemble of shapes and aditinally it allows to orient the nucleus in the laboratory space, because $K$ -is the projection of the spin on the fission axis.",
"The 4D Langevin method describes the behavior of the hot nucleus in the reference frame connected with the center-of-mass of each nucleus, thus in the case of the fast moving systems in the laboratory frame, additional transformations are necessary as the velocities of nuclei are in relativistic regime.",
"The initial condition to start evolution of the nucleus are usually the mass and charge of the beam and target nuclei, their excitation energy and maximal angular momentum, which can be produced in the reaction.",
"Our idea of investigating the spectator with 4D Langevin method is very recent.",
"There are no direct measurements that could identify any fission/evaporation reactions occurring in the remnant of heavy-ions collisions, but new developments in the detection techniques [9] could make possible the investigations of new effects accompanying the main reaction.",
"The estimation of the shape of the spectator just after the ion collision is the gateway to introduce the fission dynamics.",
"We propose three possible scenarios of the non-central collisions.",
"Let us assume that the beam and target nuclei have the spherical forms in their rest frame.",
"When they fly in opposite directions, collide and they graze each other.",
"The central part produces the quark-gluon plasma and all kinds of high-energy particle phenomena.",
"Remnants could have extremely exotic forms and sometimes it is assumed that they are a collection of free nucleons.",
"Here we discuss the following scenarios of initial shape production: ”sphere-plane” means the sphere cut by the plane at various impact parameters; ”sphere-sphere” gives the shape of the spectator which is sphere but with missing spherical cap; ”sphere-cylinder” is the globe shot by the bullet.",
"These three scenarios gives shapes, very strange for nuclear physics.",
"These exotic deformations are impossible to describe with any standard nuclear shape parameterizations.",
"Thus we make an assumption that the time to change nuclear deformation from initial (complicated) to spherical, is small compared to the time-scale of the rest of the dynamics.",
"This rough assumption will be investigated in the future in detail.",
"The difference in the surface energy for spherical and deformed nucleus give the preliminary excitation energy of the spectator.",
"The potential energy of the nucleus is obtained with the Lublin-Strasbourg Drop formula [10], [11]: $E_{lsd} (Z,N;q)&=&E_{vol}+E_{surf}+E_{curv}+E_{_{Coul}}\\\\&=&b_{vol} (1-\\kappa _{vol} I^2) \\,A+b_{surf} (1-\\kappa _{surf} I^2)\\, A^{2/3}B_{surf} (q)\\nonumber \\\\&+&b_{curv} (1-\\kappa _{curv} I^2)\\, A^{1/3}B_{curv} (q)+\\frac{3}{5}e^2 \\frac{Z^2}{r^{ch}_{0}A^{1/3}} B_{Coul} (q),\\quad $ where $ I =(N-Z)/A$ .",
"The geometric factors: $B_{surf}$ , $B_{curv}$ and $B_{Coul}$ are defined as: $B_{surf}=\\frac{S(def)}{S(0)},\\quad B_{curv}=\\frac{C(def)}{C(0)}\\quad \\mathrm { and }\\quad B_{Coul}=\\frac{E_{Coul}(def)}{E_{Coul}(0)}$ that are the ratios between the surface($S$ )/curvature($C$ )/Coulomb energy ($E_{Coul}$ ) of the deformed and spherical nucleus.",
"The details are presented e.g.",
"in [10].",
"Thus for each of the scenarios, the surface and the volume of the spectator were calculated as functions of the impact parameter.",
"The volume of the spectator and its mass were obtained using the geometrical properties.",
"Assuming $N/Z_{Pb}=N/Z_{spectator}$ also the charge was assessed.",
"The deformation together with the mass and charge gives the surface energy estimation."
],
[
"Results",
"The surface energy of the remnant just after the collision reduced by the energy of the spherical nucleus gives the deformation energy which is presented in Fig.",
"REF .",
"The deformation energy depends on the initial condition.",
"The highest deformation energies (almost 500 MeV) are predicted for the case of ”sphere-sphere” grazing as the shapes are more curved than in other cases.",
"The smallest energies are obtained for the sphere cut by the plane as the forms are less curved and the surface of the deformed nucleus is closer to a spherical shape.",
"Figure: (Color on line) The deformation energy for three scenarios: ”sphere-sphere”, ”sphere-plane” and ”sphere-cylinder”.",
"The impact distance is normalized to the radius of the spherical lead nucleus.The impact parameter is good observable to discuss the central-peripheral collision, as it is presented in Fig.",
"REF , but for fission dynamics calculation the mass and charge of the spectators are necessary.",
"Thus in Fig.",
"REF the deformation energy is shown as a function of the mass of the remnant.",
"The deformation energy has the main contribution to the excitation energy of the spectator.",
"The black crosses mark the nuclei taken in further calculations.",
"Figure: (Color on line) The deformation energy versus the spectator mass estimated from its volume.",
"The crosses represent nuclei taken for dynamical evaluation.The 4D Langevin method is dedicated to describe the fission dynamics of the hot nuclei at excitation energies $E^{\\star }$ =50–250 MeV and the driving forces are derived macroscopically.",
"The excitation energy was taken as the deformation energy from the scenario: ”sphere-cylinder” as the energy range fits exactly limitation of our model.",
"From phenomenological point of view this model is also the most realistic.",
"Figure: (Color on line) The average values of the mass (a), total kinetic energy (c), temperature (e) and fission time (f) resulting from initial systems withM=64, 100, 142, 168 and 194.",
"The widths of the mass (b) and TKE distribution (d) are also displayed.",
"The dashed line in panel (a) shows the mean mass of the fission fragments assuming symmetric scission of the nucleus.Five nuclei are chosen arbitrarily for the fission dynamics studies: $^{64}$ Mn, $^{100}$ Y, $^{142}$ Ba, $^{168}$ Dy and $^{194}$ Os.",
"The maximal angular momentum was taken as 2 $\\hbar $ just to avoid possible technical problems.",
"For each nucleus around 300 000 trajectories were calculated and the results are presented in Fig.",
"REF .",
"The panel (a) compares the mean mass coming from an assumption of the symmetric division of the fissioning nucleus (red dashed line) and centroid of the mass distribution whose width is shown in panel (b).",
"The medium mass system demonstrates a slight deviation from the static assumption as during the path to fission various particles (protons.",
"neutrons, deuteron, $\\alpha $ -particles) are emitted.",
"Looking at panel (c) for the Total Kinetic Energy (TKE) of the fission fragments, it is visible that almost linear decrease of the excitation energy (Fig.",
"REF ) provides the exponential increase of the average TKE as the system is more and more heavy.",
"The effects of inertia and/or the excitation energy is also visible in panel (f) for the increased fission time."
],
[
"Summary",
"The preliminary results discussed here show a possibility of investigating the fission of spectators produced in ultrarelativistic heavy-ions collisions.",
"The three scenarios of initial shape estimation for the remnants of the Pb-Pb reaction have been considered to estimate the excitation energies available for various impact parameters.",
"Several observables have been extracted from the state-of-art Langevin calculations for fission dynamics.",
"The average values and widths of mass, charge and total kinetic energy distribution of the fission fragments as well as mean values of the temperature and fission times have been shown for the first time.",
"Acknowledgements The work was partially supported by the Polish National Science Centre under Contract No.",
"2013/08/M/ST2/00257 (LEA COPIGAL) (Project No.",
"18) and IN2P3-COPIN (Project No.",
"12-145, 09-146), and by the Russian Foundation for Basic Research (Project No.",
"13-02-00168)."
]
] | 1612.05397 |
[
[
"A method for obtaining nonspreading solutions of the Schr\\\"odinger\n equation"
],
[
"Abstract In this paper, we present a simple analytical method for obtaining a nonspreading solution of the time-dependent Schr\\\"odinger equation, which is given by the Airy function.",
"The solution is derived by imposing a restriction on the phase factor of the ansatz that is taken to solve the differential equation.",
"Considering at first the free particle, we show that nonspreading solutions can also be obtained for a time-dependent linear potential.",
"The method is shown to work in both one and two dimensions, and can be easily extended if required.",
"The applicability of the method is discussed in relation to the nonlinear case."
],
[
"Introduction",
"Berry and Balazs showed that the free-particle Schrödinger equation admits non-trivial solutions whose amplitude is given by the Airy differential equation [1].",
"Specifically, they showed that $-\\frac{\\hbar ^2}{2m}\\psi _{xx} = i\\hbar \\psi _{t} \\,,$ has a unique solution given by $\\psi (x,t) = \\mathrm {Ai}\\left[\\frac{B}{\\hbar ^{2/3}}\\left(x-\\frac{B^3t^2}{4m^2}\\right)\\right]e^{\\left(iB^3t/2m\\hbar \\right)\\left[x-\\left(B^3t^2/6m^2\\right)\\right]}\\,,$ where $\\displaystyle \\mathrm {Ai}(z)$ is the Airy function and $\\displaystyle B$ is a positive constant.",
"Unlike plane wave solution, the Airy solution given above is not separable in the variables $\\displaystyle \\lbrace x,\\, t\\rbrace $ .",
"The essential feature of these solutions is that the time-evolution of the wavepacket is non-dispersive.",
"The solution given by Eq.",
"(REF ) was proved to be the only unique solution of Eq.",
"(REF ) having this property.",
"This result was originally explained in terms of the behaviour of the corresponding families of the semiclassical orbits in phase space.",
"Subsequently, much work has been done to interpret this surprising result, with a proper quantum mechanical derivation being given in [2].",
"Recently, the nonspreading solution has evoked considerable interest in the community.",
"It has been shown that, under the paraxial approximation the wave equation allows a nonspreading solution [3].",
"It has been verified experimentally by generating optical beams with wave packets that are accelerated [4].",
"This leads further possibility of using these solutions in various optical systems.",
"In this paper, we show that it is possible to obtain the result from techniques based on ordinary calculus.",
"It is shown that the in the ansatz $\\displaystyle A(x,t)e^{i\\phi (x,t)}$ that is used to obtain wave-like solutions of Eq.",
"(REF ), a particular choice of the phase $\\displaystyle \\phi $ will lead to a reduction of the Schrödinger equation to the Airy differential equation.",
"In this paper, we first describe a method for obtaining the nonspreading solution for the Schrödinger equation for a free particle in one dimension.",
"We also show that we can easily extend the solution for linear and time-dependent potentials.",
"In section , we show that the method can be extended to higher dimension by taking the two dimensional case as an example."
],
[
"The method for obtaining solution",
"We are looking for non-trivial solutions of $\\psi _{xx} + \\mathrm {i}\\kappa \\psi _t - \\bar{V}(x,t)\\psi = 0\\,,$ where $\\displaystyle \\kappa = 2m/\\hbar $ and $\\displaystyle \\bar{V} = (\\kappa /\\hbar )V$ .",
"We start with the free particle case first with $\\displaystyle \\bar{V}(x,t)=0$ .",
"After demonstrating the method, we extend the solution with a potential term."
],
[
"The one-dimensional Schrödinger equation",
"In this section we solve for the non-disperssive solution in one dimension.",
"We start with $\\displaystyle V=0$ in Eq.",
"(REF ).",
"First, we propose a trial solution of the form, $\\psi = A(x,t)\\,e^{i\\phi (x,t)}\\,,$ where $\\displaystyle A(x,t)$ and $\\displaystyle \\phi (x,t)$ are assumed to be real functions.",
"Separating real and imaginary parts after substituting Eq.",
"(REF ) in Eq.",
"(REF ), we obtain: $A_{xx} - A(\\phi _x)^2 -\\kappa A \\phi _t & = & 0\\,, \\nonumber \\\\A\\phi _{xx} + 2 A_x\\phi _x + \\kappa A_t &= &0\\,.$ We start with the ansatz that the phase term $\\displaystyle \\phi (x,t)$ must satisfy the Laplace equation, $\\frac{\\partial ^2\\phi }{\\partial x^2} = 0\\,, $ With this we get, $\\phi = \\phi _1 \\cdot x + \\phi _0\\,,$ where $\\displaystyle \\lbrace \\phi _1,\\phi _0\\rbrace $ are functions of time only.",
"Substituting $\\displaystyle \\phi $ in Eq.",
"(REF ), we get $A_{xx} - Ay = 0\\,,$ where, $y = \\phi _1^2+\\kappa \\left(\\dot{\\phi }_1 x+\\dot{\\phi }_0\\right)\\,.$ In the above, $\\displaystyle \\dot{\\phi }=\\frac{d\\phi }{dt}$ .",
"We further assume that $\\displaystyle \\partial y/\\partial x$ is constant implying $\\displaystyle \\dot{\\phi }_1 = P$ (a constant).",
"Then we have reduced Eq.",
"(REF ) to an Airy differential equation with the solution $A = \\mathrm {Ai}\\left[\\frac{y}{(\\kappa P)^{2/3}}\\right]\\,.$ The condition for $\\displaystyle \\phi _0$ is given from Eq.",
"(REF ) by $\\kappa \\ddot{\\phi }_0 + 4P^2t = 0\\,,$ resulting in, $\\phi _0 = -\\frac{2}{3\\kappa }P^2t^3 + Qt\\,.$ Here $\\displaystyle P$ and $\\displaystyle Q$ are constants.",
"It can be shown that one can get Berry's solution with $\\displaystyle Q=0$ and $\\displaystyle P=B^3/2m\\hbar $ where $\\displaystyle B$ is an arbitrary positive constant.",
"At this point, we would like to physically justify the ansatz used in obtaining the solution.",
"It is closely related to the symmetry properties of the Schrödinger equation.",
"The Schrödinger equation admits a Galilean transformation which preserves the form of the differential equation [5].",
"The transformed wave-function is then just the original wave-function multiplied by a phase which is linear in the spatial coordinates.",
"Thus, we must have $\\displaystyle \\partial ^2 \\phi / \\partial x^2 = 0$ ."
],
[
"With linear potential",
"In this section, we generalise to the case with potential.",
"Starting with the same trial solution to obtain, $A_{xx} - A(\\phi _x)^2 -\\kappa A \\phi _t - \\bar{V}A = 0\\,, \\nonumber \\\\A\\phi _{xx} + 2 A_x \\phi _x + \\kappa A_t = 0\\,.$ Since the second equation is the same as given in Eq.",
"(REF ), the form of $\\displaystyle \\phi $ is modified by an additional term due to the potential.",
"The equation for the amplitude $\\displaystyle A$ also takes the same form in terms of the variable $\\displaystyle y = \\phi _{1}^2+\\kappa \\left(\\dot{\\phi }_1 x + \\dot{\\phi }_0\\right) + \\bar{V}$ , as before, $\\displaystyle \\partial y/\\partial x$ is assumed constant.",
"Like before, we take $\\displaystyle \\mathrm {d}\\phi _{1}/\\mathrm {d}t = P$ with the addition that $\\displaystyle \\partial \\bar{V}/\\partial x = \\bar{V_1}$ , both of them being constants.",
"With these conditions, we have $\\frac{\\partial y}{\\partial x} = \\kappa \\frac{\\mathrm {d}\\phi _{1}}{\\mathrm {d}t} + \\frac{\\partial \\bar{V}}{\\partial x} = \\kappa P + \\bar{V_1}\\,,$ from which we finally get $\\phi _1 = Pt~,\\qquad \\bar{V} = \\bar{V_1}x + \\bar{V_0}\\,.$ Here, we have freedom to choose $\\displaystyle \\bar{V_0}$ to be a function of time.",
"In order to get $\\displaystyle \\phi _0$ we solve an equation similar to Eq.",
"(REF ) to obtain $\\phi _0 = -\\frac{1}{\\kappa ^2}\\left[\\int ^t\\bar{V_0}(t^{\\prime })\\mathrm {d}t^{\\prime } + \\frac{2}{3}\\kappa P^2t^3 -\\frac{1}{3}P\\bar{V_1}t^3\\right] + Qt~,$ where $\\displaystyle Q$ is a constant.",
"It can be readily seen this is a validation of the results of Berry and Balazs for more general potentials of the form $\\displaystyle F(t)x$ ."
],
[
"Solution in two dimensions",
"In this section, we consider the Schrödinger equation in two spatial dimensions.",
"For the potential-free case we wish to solve, $\\psi _{xx} + \\psi _{yy} + i\\kappa \\psi _t = 0\\,,$ with the trial solution $\\displaystyle \\psi = A(x,y,t)e^{i\\psi (x,y,t)}.$ Proceeding similarly as before, the real and imaginary parts are separated as $A_{xx} + A_{yy} - A(\\phi _x)^2 - A(\\phi _y)^2 - \\kappa A \\phi _t &=& 0\\,,\\nonumber \\\\A\\phi _{xx} + A\\phi _{yy} + 2A_x\\phi _x + 2A_y\\phi _y + \\kappa A_t & = &0\\,.$ Thus, the required conditions on $\\displaystyle \\phi $ are $\\displaystyle \\phi _{xx}=\\phi _{yy}=0$ , which gives us $\\phi = \\phi _0 + \\phi _1 x + \\phi _2 y\\,.$ Here all $\\displaystyle \\phi _i$ $\\displaystyle (i=0,1,2)$ are functions of time only.",
"Substituting back into Eq.",
"(REF ) we get $A_{xx} + A_{yy} - Az = 0\\,,$ where, $z = \\phi _{1}^2 + \\phi _{2}^2 + \\kappa \\left(\\dot{\\phi }_1 x + \\dot{\\phi }_2 y +\\dot{\\phi }_0 \\right)\\,.$ Thus, the required solution for the amplitude is $A = \\mathrm {Ai}\\left[\\frac{z}{\\kappa ^{2/3}(P_{1}^2+P_{2}^2)^{1/3}}\\right]~,$ where $\\displaystyle \\dot{\\phi }_1 =P_{1}$ and $\\displaystyle \\dot{\\phi }_2=P_{2}$ are both constants.",
"With these conditions, we get, $ \\phi _{0} = -\\left(\\frac{2}{3\\kappa }\\right)\\left(P_{1}^2+P_{2}^2\\right)t^3-Qt\\,,$ where $\\displaystyle Q$ is a constant as before.",
"A similar calculation holds for the case with potential.",
"This solution is essentially non-separable in the variables $\\displaystyle \\lbrace x,\\,y,\\, t\\rbrace $ Figure: The plot of the amplitude in the two dimensional as function of x\\displaystyle x and y\\displaystyle y.",
"Here, we take P 1 =10.1\\displaystyle P_1 = 10.1, P 2 =2.3\\displaystyle P_2 =2.3 , Q=0\\displaystyle Q = 0, κ=1\\displaystyle \\kappa = 1 and t=0\\displaystyle t = 0."
],
[
"A nonlinear example",
"In this section, we apply this method in solving a particular example of the nonlinear type - the Gross-Pitaevskii equation [7], which is given (in atomic units $\\displaystyle m=\\hbar =1$ ) by $i\\psi _t = -\\psi _{xx} + V\\psi + |\\psi |^2\\psi ~.$ Here, $\\displaystyle V=V(x,t)$ is the potential as given in section-REF .",
"With the substitution of the same trial solution as before, we find that the resulting equations obtained by separating the real and imaginary parts are $A_{xx} - A(\\phi _x)^2 - A \\phi _t -V A -A^3 &= &0\\, \\nonumber \\\\A \\phi _{xx} + 2 A_x \\phi _x + A_t = 0\\,.$ The second equation is exactly the same as before given by Eq.",
"(REF ), while we get an additional nonlinear term $\\displaystyle A^3$ is the first.",
"With the condition that $\\displaystyle \\partial ^2\\phi /\\partial x^2$ be zero, we get, after simplification, the following differential equation $A_{zz} = Az + \\mu A^3\\,,$ with $\\displaystyle z$ defined similarly to the variable $\\displaystyle y$ in section 2 (multiplied by an approriate constant) and $\\displaystyle \\mu $ is a constant.",
"This equation is a variant of the Type-II Painlevé transcendent [8], which are known to have solutions in terms of integral equations involving Airy functions [9].",
"Thus we see a specific example of the utility of the present method in obtaining special solutions of nonlinear Schrödinger equations."
],
[
"Discussion",
"Here, we have showed that it is possible to solve for the nondispersive solution given by Berry and Balazs in a systematic way.",
"It is possible to obtain the solution from a purely analytical calculation, without any direct quantum-mechanical arguments.",
"The method we have presented, which is due to the fact that the second derivative of the phase in the spatial coordinate can be set to zero, leads to an analogy with the modification of the wavefunction under a Galilean coordinate transformation.",
"This requirement is shown to work in a broad sense, through the derivation of the Airy wavepacket in two dimensions and the reduction of the nonlinear Gross-Pitaevskii equation to an ordinary diffenetial equation related to the second Painlevé transcendent.",
"The latter has analytical solutions and numerical approximations which have a striking resemblance with the Airy functions [10]."
],
[
"Acknowledgement",
"KRN would like to thank Dr Bhavtosh Bansal and Dr Vivek Vyas for fruitful discussions.",
"KRN would wish to acknowledge the Visiting Associateship programme of Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune.",
"A part of this work was carried out during the visit to IUCAA under this programme."
]
] | 1612.05380 |
[
[
"Origin of doping-induced suppression and reemergence of magnetism in\n LaFeAsO$_{1-x}$H$_x$"
],
[
"Abstract We investigate the evolution of magnetic properties as a function of hydrogen doping in iron based superconductor LaFeAsO$_{1-x}$H$_x$ using the dynamical mean-field theory combined with the density-functional theory.",
"We find that two independent consequences of the doping, the increase of the electron occupation and the structural modification, have the opposite effects on the strength of electron correlation and magnetism, resulting in the minimum of the calculated magnetic moment around the intermediate doping level as a function of $x$.",
"Our result provides a natural explanation for the puzzling recent experimental discovery of the two separated antiferromagnetic phases at low and high doping limits.",
"Furthermore, the increase of orbital occupation and correlation strength with the doping results in reduced orbital polarization of $d_{xz/yz}$ orbitals and the enhanced role of $d_{xy}$ orbital in the magnetism at high doping levels, and their possible implications to the superconductivity are discussed in line with the essential role of the magnetism."
],
[
"Introduction",
"Iron-based high temperature ($T$ ) superconductors including the seminal material LaFeAsO$_{1-x}$ F$_x$[1] share a common feature that the impurity doping results in the suppression of magnetic and/or structural orders in undoped parent compounds and subsequent emergence of superconductivity.",
"The underlying mechanism is still not well understood, however, as the impurity doping has several independent effects on the system such as the change of the electron occupancy, structural modification, and occurrence of the disorder, etc.",
"One of the popular explanations has been based on the itinerant picture of the antiferromagnetic (AFM) ordering in undoped samples.",
"As the doping changes the number of carriers and the position of the Fermi level, the Fermi surface (FS) nesting condition for the spin density wave (SDW) formation becomes poorer.",
"[2], [3], [4], [5] However, the non-negligible role of the electron correlation in these materials has been pointed out,[6], [7], [8] and the validity of the FS nesting picture alone, which assumes the rigid band against the carrier doping, often turns out to be doubtful in explaining the emergence and suppression of the AFM order.",
"On the other hand, the structural modification effect is known to alter significantly the magnetic property with the strong magnetostructural coupling in this system.",
"[2], [9], [10] Even the isovalent impurity doping or the hydrostatic pressure alone, which introduce no extra carrier, can give rise to the similar phase diagram with the case of the carrier doping.",
"[11], [12], [13], [14], [15] There is a general consensus about the importance of the magnetism in understanding the superconductivity of iron-based materials,[16], [17] as the magnetism is omnipresent in this class of materials at least in the form of the short-range spin fluctuation.",
"[18], [19], [20] However, the basic nature of the magnetism has been controversial among the SDW of itinerant electrons,[2], [3], [4], [5], [21] the Heisenberg type interactions of localized spins,[22], [23], [24], [25] as well as their intermediate picture.",
"[26], [27], [28], [29], [30], [31], [32], [33], [34], [35] Nevertheless, the spin fluctuation has been widely accepted as the most probable candidate for the pairing glue for the superconductivity.",
"[36], [37], [38], [39], [40], [41], [42] In the meanwhile, there are also growing arguments and evidences for the presence of the orbital order and fluctuations in these materials.",
"[43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54] Recent experiments on FeSe suggest that the orbital degree of freedom drives the electronic nematicity and spontaneous symmetry breaking instead of the spin degree of freedom, [55], [56] drawing attention for the alternative mechanism of the pairing mediated by the orbital fluctuation.",
"[57], [58] Because the spin and orbital degrees of freedom are coupled each other, [43], [59] however, there can be an inherent ambiguity in determining which order forms first and drives the other.",
"Recently, a series of experiments has revived the attention to the seminal material of the family by the hydrogen (H) doping, namely, LaFeAsO$_{1-x}$ H$_x$ .",
"[60], [61], [62] Overcoming the solubility limit of the conventional fluorine dopant, hydrogen can increase the doping concentration up to $x=0.6$ .",
"Surprisingly, another superconducting and AFM phases adjacent to each other are found at high doping levels, analogously to their conventional counterparts at the low doping level (see Fig. 1(a)).",
"Posing fundamental questions on the nature of magnetism and superconductivity, this finding is expected to give a clue for still unresolved issues mentioned earlier.",
"Although some theoretical attempts have been made to explain the appearance of the second AFM phase mostly focusing on the FS nesting property in the itinerant electron picture,[60], [61], [63] first-principles approach simultaneously incorporating the itinerant and localized aspects of the system is desirable when we consider the 'moderately correlated' nature of these materials [6], [7], [8], [64], [65] and some unsatisfactory conclusions from the itinerant picture such as the prediction of an incorrect magnetic ordering vector.",
"[61] In this paper, we investigate the magnetic and electronic properties of LaFeAsO$_{1-x}$ H$_x$ as a function of $x$ using the combined method of density-functional theory plus dynamical mean-field theory (DFT+DMFT), which captures the material-specific electronic correlation.",
"[66], [67] Considering changes of both electron occupancy and lattice structure caused by the hydrogen doping which turn out to have the opposite effects on the electron correlation and magnetism, we find that both static magnetic moment and local magnetic susceptibility initially decrease to the minimum at around $x=0.3$ and then increase again up to $x=0.6$ , in agreement with the experimental phase diagram of the two separate AFM phases centered at $x=0$ and 0.5.",
"More electron occupation at $d_{xz/yz}$ orbitals with the doping enhances the importance of the $d_{xy}$ orbital in the static magnetic moment and also in spin dynamics, while reducing the orbital polarization.",
"Our results emphasize the importance of the electron correlation and structural modification in understanding the doping induced evolution of the electronic structure, and also the magnetism as an indispensable ingredient for the emergence of the superconductivity in these materials."
],
[
"Calculation Method",
"We use the modern implementation of DFT+DMFT method within all electron embedded DMFT approach,[67] where in addition to correlated Fe atoms the itinerant states of other species are included in the Dyson self-consistent equation.",
"The strong correlations on the Fe ion are treated by DMFT, adding self-energy $\\Sigma (\\omega )$ on a quasi atomic orbital in real space, to ensure stationarity of the DFT+DMFT approach.",
"The self-energy $\\Sigma (\\omega )$ contains all Feynman diagrams local to the Fe ion.",
"No downfolding or other approximations were used, and the calculations are all-electron as implemented in Ref.",
"67, which is based on Wien2k.",
"[68] We used the GGA exchange-correlation functional,[69] and the quantum impurity model was solved by the continuous time quantum Monte Carlo impurity solver [70] using $U=5.0$ eV and $J=0.72$ eV.",
"Brillouin zone integration is done on the 12$\\times $ 12$\\times $ 6 k-point mesh for the AFM unitcell of LaFeAsO containing 4 Fe atoms.",
"All calculations are done for $T=150$ K. We consider both paramagnetic (PM) and AFM states at this temperature.",
"The AFM state is considered to represent the actual stable phase observed in experiments, while the PM state is also calculated to understand the driving force with which the AFM state is stabilized from a 'bare' state."
],
[
"Electronic correlation and magnetic strengths as functions of doping",
"Schematic phase diagram of LaFeAsO$_{1-x}$ H$_x$ in the $x-T$ space is depicted in Fig. 1(a).",
"The first AFM phase with a stripe-type order is rapidly suppressed and disappears around $x=0.05$ with the emergence of the first superconducting phase followed by the adjacent second superconducting phase.",
"Further doping initiates the second AFM phase, of the same ordering pattern with the first AFM phase, but with a slightly different atomic displacement.",
"[62] We perform the DFT+DMFT calculations to check if this suppression and reappearance of the AFM phase can be reproduced.",
"To take the electron doping effect into account, we adopt the virtual crystal approximation.",
"[60] In addition, the structural change due to the H doping is incorporated by interpolating both the lattice constants and internal atomic coordinates of the available experimental values at $x=0$ [71] and $x=0.51$ [62] in the PM states with the tetragonal lattice symmetry.",
"This should be a reasonable approximation considering the almost linear As height dependence on $x$ observed experimentally,[60] avoiding the difficulty in the structural optimization of alloy structures within DFT which would require rather complicated statistical treatment, besides concerns about the general reliability of DFT in predicting the accurate lattice structure of iron-based superconducting materials.",
"Using this doping scheme, the static magnetic moment and the local spin susceptibility, $\\chi _{local}(\\omega =0)=\\int _0^\\beta \\!",
"\\langle S(\\tau )S(0) \\rangle \\mathrm {d}\\tau $ , are calculated as a function of $x$ for the stripe-type AFM phase and shown in Fig. 1(b).",
"The magnetic moment at $x=0$ is estimated to be 0.66 $\\mu _B$ with an agreement with the measured value 0.63 $\\mu _B$ ,[72] and decreases to the minimum value of 0.12 $\\mu _B$ at $x=0.4$ , and then exhibits a rapid increase to reach 0.68 $\\mu _B$ at $x=0.6$ .",
"The local susceptibility shows a similar behavior with a minimum at $x=0.3$ , implying that the overall magnetic strength is suppressed and then re-enhanced with the doping.",
"Therefore, we verify that the DFT+DMFT method captures the essential underlying physics of two separate AFM phases in this material and produces a consistent behavior of the local magnetic strength, while the complete suppression of magnetic phase and emergence of the superconducting phase in the intermediate $x$ as shown in Fig.",
"1(a) could not be properly described within the current calculation scheme.",
"For comparison, we perform the DFT calculation on the relative stability of the AFM phase and the magnetic moment as functions of $x$ .",
"For $x=0$ , the AFM phase is found to be 180 meV/Fe more stable than the nonmagnetic phase with the magnetic moment of 2.15 $\\mu _B$ .",
"Upon increasing $x$ , the stability of AFM phase and the magnetic moment show no discernible change, suggesting that the normal DFT calculation cannot properly describe the observed evolution of the magnetism.",
"Then we investigate the underlying mechanism of the doping-induced change of the electronic structure by considering the electron addition and structural modification separately.",
"Because the nominal number of valence electrons in a Fe atom for the undoped material is six, which corresponds to an electron doped system from the half-filled orbitals, further electron doping should result in the monotonic decrease of the correlation strength.",
"[73] On the other hand, the H doping increases the distance between Fe and surrounding As atoms as determined experimentally,[71], [62] which would lead to the localization of Fe $d$ orbitals.",
"[74] To confirm this speculation, we estimate the mass enhancement $1/Z = 1-\\frac{\\partial \\Sigma (\\omega )}{\\partial \\omega }\\vert _{\\omega =0}$ for the two effects separately.",
"First, we calculate $1/Z$ of the $d_{xy}$ in the PM phase as a function of $x$ considering only the electron addition effect by fixing the lattice structure to that of $x=0$ as in Fig. 1(c).",
"Indeed, $1/Z$ monotonically increases with the electron addition.",
"On the contrary, when only the structural effect is included without extra electron, $1/Z$ monotonically decreases with increasing $x$ .",
"When these two competing effects are combined, $1/Z$ increases overall with the doping, which means that the localization by the structural modification becomes more dominant at the highly doped system.",
"This competing effects are also reflected on the magnetic strength of the AFM phase as shown in Fig. 1(d).",
"When only electron addition effect is considered, the local susceptibility is found to monotonically decrease with increasing $x$ , while it increases monotonically when only the structural modification is taken into account (with the number of extra electrons fixed to 0.6), in agreement with the behavior of the mass enhancement factors in Fig. 1(c).",
"Therefore, we can conclude that the initial suppression and the later re-enhancement of the magnetism with the H doping originates from the two competing effects : the electron addition and increasing Fe-As distance which suppresses and enhances the local correlation and hence the local magnetism, respectively.",
"Our analysis naturally draws attention to the important role of the electron correlation and also the structural effect in understanding the doping induced phase diagram of this material.",
"Suppression of the magnetism and the existence of the quantum criticality in phosphorus-doped Ba122 systems BaFe$_2$ As$_{2-x}$ P$_x$ [75], [76], [77] are another set of examples which demonstrate the dramatic effect of the pure structure modification, where decreased Fe-anion distance was pointed out to cause the suppression of the magnetism.",
"[76]"
],
[
"Fermi surfaces",
"We also investigate the evolution of the FS with the doping which is generally considered to be relevant to the existence of the AFM phase.",
"[61], [63] The FSs for three different doping levels, $x=$ 0, 0.3, and 0.5, are calculated with both the DFT+DMFT and DFT methods and displayed in Fig. 2.",
"Starting with a relatively good nesting between the hole and electrons surfaces at $x=0$ for the DFT case, the doping degrades the nesting with shrinking the hole surfaces at the $\\Gamma $ point and enlarging the electron surfaces at the $M$ point (see Fig.",
"2(g)-(i)), as the electron doping raises the Fermi level.",
"Our result shows a good agreement with the previous DFT calculation using the experimentally determined lattice structure,[60] confirming that our assumption of the linear dependence of the lattice constants and internal atomic coordinates is reasonable.",
"The DFT+DMFT results shown in Figs.",
"2(a)-(f) are qualitatively similar, but the $d_{xy}$ hole FS expands compared with the DFT results, because of more correlated nature of the $d_{xy}$ orbital than $d_{xz/yz}$ orbitals as pointed out in the DFT+DMFT study of LiFeAs.",
"[78] The decrease of the overall spectral weight with the doping and relatively larger incoherence of the $d_{xy}$ surface reflect the larger correlation at high doping levels and for the $d_{xy}$ orbital (Figs. 2(a)-(c)).",
"Nevertheless, both levels of the theory indicate the monotonic degradation of the FS nesting with the doping, as already indicated by previous calculations,[60], [63] manifesting that the FS nesting alone cannot explain the appearance of the second AFM phase.",
"Again, we conclude that the electron correlation and many-body effects should be incorporated to understand the doping-induced evolution of the magnetism."
],
[
"Spin resolved spectral function",
"To understand the doping-induced suppression and the reemergence of the magnetism in detail, we investigate the spin-resolved spectral function of the $d_{xy}$ orbital in the AFM phase as a function of $x$ as shown in Fig. 3.",
"At $x=0$ , the majority and minority spin states exhibit a large exchange splitting reflecting the overall magnetic moment of 0.66 $\\mu _B$ , with a distinct pseudo-gap feature (a dip in spectral function) at the Fermi energy induced by the coupling between the electron and hole bands at the Fermi energy.",
"[31] With increasing doping level up to $x=0.3$ (see Fig.",
"3(a)), spectral weights moves from the peak just above the Fermi level to one below the Fermi energy in the minority spin channel, suggesting the doped electrons fills the minority spin states.",
"The electron filling in the minority spin states with the doping naturally leads to the gradual reduction of the exchange splitting and the magnetic moment, along with the size of the pseudo-gap.",
"On the other hand, further doping over $x=0.3$ enhances the exchange splitting as shown in Fig.",
"3(b), which seems to almost retrace the evolution of the spectral function from $x=0$ to $x=0.3$ in Fig. 3(a).",
"However, there are several noticeable differences as well.",
"First, the pseudo-gap position is constantly shifting deeper in the valence states with its size and the magnetic moment increasing with increasing $x$ , which indicates the rise of the Fermi energy as a result of the electron doping.",
"More importantly, doping over $x=0.3$ develops a shoulder growing with $x$ near -1 eV in the majority spin channel as indicated by a arrow in Fig.",
"3(b), contributing to build up the magnetic moment against the electron filling on the minority spin states with the doping.",
"The spectral weight piled up in this position results from many-body effects and hence is incoherent, rather than from the shift of coherent quasi-particle states.",
"The inset of Fig.",
"3(b) depicts the imaginary part of the $d_{xy}$ component of the electron self energy (Im$\\Sigma $ ) along with the spectral function for $x=0.6$ .",
"One can identify a strong peak of the Im$\\Sigma $ around -0.7 eV, close to the $J$ value 0.72 eV adopted in this study, and the shoulder structure of the spectral function at a nearby position, suggesting that the shoulder structure originates from the incoherent excitations related to the self energy.",
"Similar energy scales between this incoherent excitation and $J$ is also consistent with the suggestion that iron-based superconductors are Hund's metals where $J$ plays more important role than $U$ .",
"[7]"
],
[
"Orbital polarization",
"As mentioned earlier, the orbital order is of great interest for these materials regarding the electronic nematicity and also superconductivity itself.",
"Here we compare the orbital polarization, i.e., the imbalance between $d_{xz/yz}$ orbitals, in the AFM state for low and high doping cases.",
"For $x=0$ as displayed in Fig.",
"4(a), $d_{xz}$ and $d_{yz}$ spectral functions show noticeable difference, where the spin polarization is larger for $d_{yz}$ as well as $d_{xy}$ orbitals while $d_{xz}$ spin splitting is smaller.",
"On the other hand, for the high doping case of $x=0.5$ in Fig.",
"4(b), $d_{xz/yz}$ components of the spectral function becomes much more similar with each other and now the $d_{xy}$ orbital has the most significant spin polarization.",
"Indeed, our estimated orbital polarization $(n_{xz}-n_{yz})/(n_{xz}+n_{yz})$ decreases from 3.9 $\\%$ at $x=0$ to 1.5 $\\%$ at $x=0.5$ while the magnetic moments for the two cases are comparable.",
"Increasing $x$ enhances the crystal field splitting pushing up the $d_{xy}$ level above $d_{xz/yz}$ level, so that the doped electrons fill $d_{xz/yz}$ orbitals first rather than the $d_{xy}$ orbital, reducing the imbalance between $d_{xz/yz}$ orbitals as well as between their spin components.",
"Meanwhile, besides the less electron filling, the elevated As height in the high doping case further enhances the electron correlation for the $d_{xy}$ orbital via the `kinetic frustration' [79] compared with the $d_{xz/yz}$ orbitals.",
"So $d_{xy}$ becomes the most significant component for the local spin fluctuations in the PM phase and for static magnetic moments in the AFM phase.",
"Therefore, at high doping levels, strong local magnetism appears mainly from $d_{xy}$ orbital and the orbital polarization from $d_{xz/yz}$ is largely suppressed."
],
[
"Spin excitation spectrum",
"So far, we have considered the doping induced evolution of the magnetically ordered state, and we will conclude our discussion by investigating the dynamic spin fluctuations in the PM state, which is more relevant to the superconductivity, as a function of the doping.",
"Although $T=150$ K at which the calculation is done is close to the AFM transition temperature, the spin susceptibility in the PM state is expected to be a smooth varying function of $T$ (except at the AFM ordering wave vector for which the susceptibility diverges at the AFM transition temperature), so we expect qualitatively similar results for other temperatures.",
"We evaluate the dynamical spin structure factor $S({\\bf q},\\omega )=\\frac{\\chi ^{\\prime \\prime }({\\bf q},\\omega )}{1-e^{\\hbar \\omega /k_BT}}$ using the DFT+DMFT method as displayed in Fig.",
"5, where both the one-particle Green's function and the local two-particle vertex function are determined ab-initio.",
"[80] For $x=0$ , the spin excitation spectrum has strong peaks near the zero energy around the wave vector ${\\bf q}=(\\pi ,0)$ , which corresponds to the magnetic ordering vector of the AFM phase, and disperses over the path shown in the spectrum, reaching a maximum energy at the zone boundary ${\\bf q}=(\\pi ,\\pi )$ , all consistent with previous results.",
"[80], [81], [41] As the doping level $x$ increases, the overall spin excitation spectral weights tend to shift to lower energies as the spin wave dispersion decreases with the increasing correlation strength.",
"The excitation near ${\\bf q}=(\\pi /2,\\pi /2)$ noticeably goes down towards the zero energy with the doping, and a new possible static magnetic order for this wave vector is suggested for $x=0.5$ .",
"However, the intensity of excitations has always the maximum at the conventional AFM ordering vector ${\\bf q}=(\\pi ,0)$ for all the doping cases, consistent with the experimentally found second AFM phase for the high doping levels.",
"[62] For $S({\\bf q},\\omega )$ at ${\\bf q}=(\\pi +\\delta ,0)$ which is slightly off the magnetic ordering vector, as shown in Fig.",
"5(d), the peak height near the zero energy is reduced for $x=0.3$ compared with that for $x=0$ indicating the suppressed tendency towards the static magnetic order, and then it becomes pronounced again for the higher doping level $x=0.5$ suggesting the re-enhanced magnetism, which shows a qualitative agreement with the initial decrease and re-enhancement of the calculated magnetic moment in the AFM phase shown in Fig.",
"1 and again also with the motivating experiments.",
"[61], [62] When decomposed by orbitals (see Figs.",
"5(e)-(g)), the $d_{xz/yz}$ components show a large anisotropy with the $d_{yz}$ component peak being dominant at $x=0$ .",
"As the doping level increases, the $d_{xz/yz}$ anisotropy keeps decreasing while the $d_{xy}$ component becomes most prominent.",
"The decreasing $d_{xz/yz}$ anisotropy and the enhancement of the $d_{xy}$ component with the increasing doping is consistent with the features observed in our result for the AFM phase."
],
[
"Discussion",
"Our results remind us of the indispensable role of the electron correlation in the iron-based superconducting materials, as well as the impact of the structural change.",
"The doping-induced evolution of the electronic and magnetic properties cannot be understood by simply adopting the rigid shift of the Fermi level or even the self-consistent addition of carriers without taking the structural effect into account.",
"In addition, a natural view on the doping-induced evolution of spin and orbital orders can be obtained.",
"The `ferro-orbital order', which is coupled to the AFM spin order in the undoped materials, is the lowest-energy configuration for the nominally half-filled orbitals to maximize the kinetic energy gain.",
"[43] Both the orbital and spin orders of $d_{xz}$ or $d_{yz}$ are suppressed when the doping supplies more electrons to these orbitals away from the half-filling.",
"The $d_{xy}$ orbital, for which the spin order can form but the orbital order is no longer relevant, becomes the dominant channel for the electron hopping to reduce the kinetic energy as discussed above.",
"As a result, spin order/fluctuation is present near the both superconducting domes found in LaFeAsO$_{1-x}$ H$_x$ while orbital order/fluctuation is expected to be strong only near the first superconducting phase in the lower doping level.",
"Our results consequently suggests that the spin fluctuation is more closely related to the superconductivity, at least for the second superconducting phase in this alloy, while the orbital fluctuation, which is significant only at low doping levels, might not be a prerequisite for the superconductivity in general.",
"The enhanced role of the $d_{xy}$ orbital in magnetism is expected to naturally lead to its dominant role also in the superconductivity with a larger FS hole pocket of this orbital as shown in Fig. 2.",
"Although the enhanced electron correlation and consequent stronger spin fluctuation in the higher doping level might contribute to the strong superconductivity, too strong correlation would be harmful to the superconductivity, out of several reasons,[79] due to the lowered magnon energy scale (Fig.",
"5) which is directly coupled to the size of the superconducting gap.",
"Further theoretical study which directly attacks the superconductivity as a function of the doping level will be desirable."
],
[
"Conclusion",
"In summary, by adopting the DFT+DMFT method, where the local dynamic correlation effect is taken exactly, we successfully reproduce the hydrogen-doping-induced suppression and revitalization of the magnetism in LaFeAsO$_{1-x}$ H$_x$ which has been recently established experimentally.",
"Taking the structural modification by the doping into account along with the carrier addition is found to be essential, as the two factors induce independent and opposite effects on the electron correlation strength and the magnetism in this alloy.",
"Doping reduces the orbital imbalance between $d_{xz/yz}$ orbitals as well as their magnetic activity, while the $d_{xy}$ orbital becomes the dominant electron hopping channel with increased electron correlation and the magnetic strength for high doping levels.",
"Indispensable role of the electron correlation and detailed atomic structure is identified in understanding the electronic and magnetic properties, and the magnetism possibly as more fundamental ingredient in realizing the superconductivity is suggested over the orbital degrees of freedom.",
"This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2016R1C1B1014715 and 2015R1D1A1A01059621)."
]
] | 1612.05520 |
[
[
"The relation between globular cluster systems and supermassive black\n holes in spiral galaxies. The case study of NGC 4258"
],
[
"Abstract We aim to explore the relationship between globular cluster total number, $N_{\\rm GC}$, and central black hole mass, $M_\\bullet$, in spiral galaxies, and compare it with that recently reported for ellipticals.",
"We present results for the Sbc galaxy NGC 4258, from Canada France Hawaii Telescope data.",
"Thanks to water masers with Keplerian rotation in a circumnuclear disk, NGC 4258 has the most precisely measured extragalactic distance and supermassive black hole mass to date.",
"The globular cluster (GC) candidate selection is based on the ($u^*\\ -\\ i^\\prime$) vs. ($i^\\prime\\ -\\ K_s$) diagram, which is a superb tool to distinguish GCs from foreground stars, background galaxies, and young stellar clusters, and hence can provide the best number counts of GCs from photometry alone, virtually free of contamination, even if the galaxy is not completely edge-on.",
"The mean optical and optical-near infrared colors of the clusters are consistent with those of the Milky Way and M 31, after extinction is taken into account.",
"We directly identify 39 GC candidates; after completeness correction, GC luminosity function extrapolation and correction for spatial coverage, we calculate a total $N_{\\rm GC} = 144\\pm31^{+38}_{-36}$ (random and systematic uncertainties, respectively).",
"We have thus increased to 6 the sample of spiral galaxies with measurements of both $M_\\bullet$ and $N_{\\rm GC}$.",
"NGC 4258 has a specific frequency $S_{\\rm N} = 0.4\\pm0.1$ (random uncertainty), and is consistent within 2$\\sigma$ with the $N_{\\rm GC}$ vs. $M_\\bullet$ correlation followed by elliptical galaxies.",
"The Milky Way continues to be the only spiral that deviates significantly from the relation."
],
[
"[ht] -Band Shape Parameters of Globular Cluster Candidates Table: NO_CAPTION Table: NO_CAPTION Note—Table REF is published in its entirety in the machine-readable format.",
"A portion is shown here for guidance regarding its form and content."
]
] | 1612.05655 |
[
[
"Tuning the hopping parameter in the Oktay-Kronfeld action for charm and\n bottom quarks on a MILC HISQ ensemble"
],
[
"Abstract The first step in the calculation of semi-leptonic form factors in the decay of heavy mesons is the tuning of the hopping parameter $\\kappa$ for the charm and bottom quark masses.",
"Results for the Oktay-Kronfeld (OK) action are presented for one $N_f=2+1+1$ HISQ ensemble generated by the MILC collaboration at $a\\approx 0.12\\,\\mathrm{fm}$ and $M_\\pi\\approx 310$ MeV.",
"Estimates of hyperfine splitting of heavy-light and heavy-heavy mesons are presented and the inconsistency parameter is evaluated."
],
[
"Introduction",
"There are two independent methods to extract $V_{cb}$ with $B$ -meson decays.",
"One is the heavy quark expansion method based on QCD sum rules with inclusive $B$ -meson decays: $\\bar{B}\\rightarrow X_{c}\\ell \\bar{\\nu }$ , and the other is the lattice QCD method to calculate the semileptonic form factors in the analysis of the exclusive $B$ -meson decays: $\\bar{B} \\rightarrow D^{(\\ast )} \\ell \\bar{\\nu }$ .",
"There exists about $3\\sigma $ tension in $|V_{cb}|$ between the inclusive and exclusive decay channels [1], [2].",
"The future experiment at KEK (Belle 2) will increase statistics for $B$ -meson decays dramatically (by a factor of 50).",
"It is time to improve the lattice results of semileptonic form factors for the exclusive $B$ -meson decays.",
"Since the dominant error in lattice QCD results for $|V_{cb}|$ comes from the heavy quark discretization, we simulate the Oktay-Kronfeld (OK) action [3], a highly improved version of the Fermilab formulation.",
"If we use the OK action instead of the clover action (the original action of the Fermilab formulation [4]), then the power counting estimate suggests that the discretization error due to charm quarks can be reduced from 1.0% (clover) down to 0.2% (OK) for the semileptonic form factor for the $\\bar{B} \\rightarrow D^* \\ell \\bar{\\nu }$ decay at zero recoil.",
"The OK action is improved to $\\mathcal {O}(\\lambda ^3)$ in HQET power counting, and $\\mathcal {O}(v^6)$ in NRQCD power counting, while the clover action is improved to $\\mathcal {O}(\\lambda ^2)$ in HQET and to $\\mathcal {O}(v^4)$ in NRQCD.",
"One drawback is that the OK action takes significantly more computing resources (by a factor of $\\approx 50$ ) to calculate its propagator.",
"We measured heavy-light (HL) and heavy-heavy (HH) meson spectra to probe the improvement by the OK action, and the inconsistency parameter and hyperfine splitting showed clear improvement [5].",
"In this paper, we tune hopping parameters using the physical $B_s$ and $D_s$ meson spectrum on the coarse MILC HISQ ensemble at $a \\approx 0.12 \\,\\mathrm {fm}$ ."
],
[
"Simulation Details",
"We use the coarse ($a\\approx 0.12$ fm) ensemble of the MILC HISQ lattices [6].",
"The lattice geometry is $24^3\\times 64$ .",
"The tadpole improvement coefficient is $u_0=0.86372$ from the plaquette Wilson loop.",
"The sea quark masses are $am_\\ell =0.0102$ for light quarks, $am_s=0.0509$ for the strange quark, and $am_c=0.635$ for the charm quark.",
"For the HL mesons, $B_s^{(\\ast )}$ and $D_s^{(\\ast )}$ , we use the HISQ action for the strange quark, and the OK action for charm and bottom quarks.",
"Heavy quark propagators are generated using an optimized BiCGStab inverter [7].",
"In the OK action, the tadpole improved bare quark mass $m_0$ is related to the hopping parameter $\\kappa $ as follows, $am_0 = \\frac{1}{2u_0\\kappa } -(1+3\\zeta r_s + 18c_4)$ where $c_4$ is the tree-level matching coefficient of a dimension-7 operator in the OK action [3].",
"Here, we set $\\zeta =1$ for isotropic lattices and $r_s=1$ as is standard for the Wilson clover action.",
"To tune the hopping parameter $\\kappa $ to the physical values, we simulate four $\\kappa $ values each for the bottom and the charm quarks as shown in Table REF .",
"The parameters of covariant Gaussian smearing used at both the source and sink of heavy quark propagators to reduce the excited state contamination [8] are given in Table REF .",
"For HISQ valence quarks, we use point source and sink.",
"Table: Parameters used for generating the valence quark propagators.",
"table:valenceHISQm s v m^v_s is set to the physical strange quark mass, and ϵ\\epsilon is the coefficient of the Naik term in the HISQ action .table:valenceOK κ\\kappa values for the bottom and charmquarks.",
"The covariant Gaussian smearing parameters σ\\sigma and N GS N_{GS} are defined in Ref.",
".We calculate HL and HH meson correlators on 500 configurations using 6 sources for bottom quarks and 3 sources for charm quarks.",
"We use jackknife resampling to estimate the statistical error.",
"We fix the time separation between sources to $\\Delta t = 6$ .",
"We choose the initial source time slice randomly for each configuration.",
"We use 11 different momentum projections for the two-point meson correlation functions.",
"To increase the statistics, we use the time reflection symmetry of the two-point correlation functions."
],
[
"Fits to the Meson Correlators and the Dispersion Relation",
"The numerical data for the two-point meson correlators is fit using $f^\\text{HL}(t; \\mathbf {p}) &= Ae^{-Et} \\big ( 1-(-1)^t re^{-\\Delta Et} \\big ) + Ae^{-E(T-t)} \\big (1-(-1)^t re^{-\\Delta E(T-t)}\\big )\\\\f^\\text{HH}(t; \\mathbf {p}) &= Ae^{-Et}+Ae^{-E(T-t)}.$ The HL meson correlator, $f^\\text{HL}$ , has 4 fit parameters: the ground state energy and amplitude ($E$ , $A$ ), an amplitude ratio ($r=A^p/A$ ), and energy difference ($\\Delta E=E^p-E$ ), where the superscript $p$ stands for the opposite parity partner state that is present in staggered fermion correlation functions.",
"$f^\\text{HH}$ is the function used to fit the HH mesons.",
"The range $12 \\le t \\le 19$ is used to fit the HL mesons and $12 \\le t \\le 16$ for the HH mesons.",
"The ground state energy $E(\\mathbf {p})$ is then fit using the following dispersion relation: $E(\\mathbf {p})=M_1 + \\frac{\\mathbf {p}^2}{2M_2} - \\frac{(\\mathbf {p}^2)^2}{8M_4^3} - \\frac{a^3W_4}{6}\\sum _{i=1}^3 p_i^4 \\,,$ to obtain $M_1$ the rest mass, $M_2$ the kinetic mass, $M_4$ the quartic mass, and $W_4$ the Lorentz symmetry breaking term.",
"In both fits we use the full covariance matrix with trivial priors."
],
[
"Kappa Tuning",
"We determine the hopping parameters $\\kappa _b$ and $\\kappa _c$ such that the kinetic masses are equal to the physical $B_s$ and $D_s$ masses, respectively.",
"We tune the kinetic mass $M_2$ rather than the rest mass $M_1$ .",
"The form factors and decay constants which we are interested in are independent of the rest mass $M_1$ in the Fermilab interpretation of improved Wilson fermions.",
"We use the HQET inspired fitting function for kinetic HL meson masses, $aM_2(\\kappa )=d_0+am_2(\\kappa )+\\frac{d_1}{am_2(\\kappa )}+\\frac{d_2}{(am_2 (\\kappa ))^2} \\,,$ where $M_2$ is the kinetic mass of the HL meson, and $m_2$ is the kinetic mass of the heavy quark.",
"We determine $d_0$ , $d_1$ and $d_2$ using the correlated least $\\chi ^2$ fitting.",
"Here, $m_2(\\kappa )$ is related to the bare mass $m_0(\\kappa )$ at the tree level as follows, $\\frac{1}{ am_2} &= \\frac{2 \\zeta ^2}{ am_0 (2+ am_0)} + \\frac{ r_s\\zeta }{1+ am_0 }.$ Figure REF shows the interpolation of $m_2$ to the physical values for the bottom and charm quarks.",
"Results for $\\kappa _b$ and $\\kappa _c$ are summarized in Table REF .",
"In Table REF , we present the $\\kappa $ -tuning results using physical values of the pseudoscalar meson mass ($M_X$ ), vector meson mass ($M_{X^\\ast }$ ), and the spin-averaged mass $(M_X+3M_{X^\\ast })/4$ for $X=B_s$ or $D_s$ .",
"We find that all the results for $\\kappa $ determined from different spin states are consistent within statistical uncertainty.",
"We also perform another fit using a simpler fitting function: $aM_2=d_0+am_2+d_1/(am_2)$ , and take the difference in $\\kappa $ as the systematic error due to ambiguity in the fitting function.",
"Figure: Plot of the pseudoscalar meson mass D s D_s (B s B_s) versusthe kinetic quark mass m 2 c m_2^c (m 2 b m_2^b) defined inEq. ().",
"The physical κ c \\kappa _c andκ b \\kappa _b (shown as red triangles and given in Table) are determined by tuning the D s D_s and B s B_s pseudoscalar massesto their experimental values.Table: Results of tuning the κ\\kappa for the bottom (κ b \\kappa _b)and charm (κ c \\kappa _c) quarks.",
"For converting the experimental MMto aMaM, we use a=0.12520(22) fm a=0.12520(22)\\,\\mathrm {fm} .",
"Inκ b,c \\kappa _{b,c}, the first error is statistical, the second erroris propagation of experimental error in M X M^X, and the third erroris systematic to account for the uncertainty in the fit ansatz."
],
[
"Inconsistency Parameter",
"The inconsistency parameter $I$ [10], [11] is used to see $O(\\mathbf {p}^4)$ improvement in the OK action.",
"Let us use $Q$ for heavy quarks and $q$ for light quarks, and define $\\delta M \\equiv M_{2}-M_{1}$ as the difference between the kinetic and rest masses.",
"Then the inconsistency parameter $I$ is $I\\equiv \\frac{2\\delta M_{\\bar{Q} q} -(\\delta M_{\\bar{Q} Q}+\\delta M_{\\bar{q} q})}{2M_{2\\bar{Q} q}}= \\frac{2\\delta B_{\\bar{Q} q} -(\\delta B_{\\bar{Q} Q}+\\delta B_{\\bar{q} q})}{2M_{2\\bar{Q} q}}$ Here the binding energies $B_{1,2}$ are $M_{1\\bar{Q} q}= m_{1\\bar{Q} } + m_{1 q} + B_{1\\bar{Q} q }, \\qquad M_{2\\bar{Q} q}= m_{2\\bar{Q} } + m_{2 q} + B_{2\\bar{Q} q }$ for HL mesons.",
"Here the quark masses $m_{1,2}$ are defined by the quark dispersion relation, which is similar to Eq.",
"(REF ).",
"We neglect the light quarkonium contribution $\\delta M_{\\bar{q} q}$ (and $\\delta B_{\\bar{q} q}$ ).",
"In Fig.",
"REF we present results for $I$ for pseudoscalar mesons.",
"Near the $B_s$ region, $I$ is consistent with the continuum limit, $I=0$ , within the error bars, which indicates a dramatic improvement from that of the Fermilab action: $I\\approx -0.6$ [5].",
"Figure: (Left) Inconsistency parameter II versus the pseudoscalarmass aM 2Q ¯q a M_{2\\bar{Q}q} for charm (green squares) and bottom (bluecircles) quarks.",
"(Right) Dispersion relation for the HL bottommeson with κ=0.040\\kappa =0.040.",
"The rest mass, M 1 =2.112(2)M_1=2.112(2), is givenby the blue circle at 𝐩=0\\mathbf {p}=0.",
"The kinetic mass,M 2 =3.408(176)M_2=3.408(176), is extracted from the fit and shown by the redtriangle translated to E=2.112E= 2.112.",
"Note that the determinationof the error in M 2 M_2 is much larger than in M 1 M_1."
],
[
"Hyperfine Splittings",
"We define the hyperfine splittings of HL and HH pseudoscalar mesons, $\\Delta _1$ and $\\Delta _2$ as $\\Delta _1=M_1^\\ast -M_1,\\qquad \\Delta _2=M_2^\\ast -M_2 \\,,$ and plot $\\Delta _2$ versus $\\Delta _1$ in Fig.",
"REF .",
"As illustrated in Fig.",
"REF , $M_2$ has much larger errors than $M_1$ since it is extracted from the slope versus momentum.",
"Consequently, $\\Delta _2$ has larger errors than $\\Delta _1$ .",
"Figure: Hyperfine splitting for HH mesonsThe HQET expansion for $\\Delta _1$ in the HL meson system is given in Ref.",
"[12]: $\\Delta _1 = M_1^{\\ast } - M_1 &= \\frac{4\\lambda _2}{2m_B} -\\frac{4\\rho _2}{4m_E^2}+ \\frac{8T_2}{2m_2 2m_B} + \\frac{4(T_4-T_2)}{4m_B^2} +\\mathcal {O}\\left(\\frac{1}{m^3}\\right),$ where $\\lambda _2$ , $\\rho _2$ , $T_2$ , $T_4$ are HQET matrix elements defined in Ref. [12].",
"For the OK action, the matching conditions are $m_2=m_B=m_E$ [3].",
"Thus, $\\Delta _1$ defined in Eq.",
"(REF ) is in terms of the kinetic quark mass, which was used to tune the $\\kappa $ to the physical value.",
"To analyze $\\Delta _1$ , we recast Eq.",
"REF as $a\\Delta _{1} = h_0 + \\frac{h_1}{am_2} + \\frac{h_2}{(am_2)^2}+ \\frac{h_3}{(am_2)^3} \\,,$ where $h_1 = 2 a^2 \\lambda _2$ and $h_2 = a^3 (-\\rho _2 + T_2 +T_4)$ .",
"Because we have only 4 data points, we set $h_3=0$ in the fits.",
"Correlated fits, shown in Fig.",
"REF , give $h_0 = 0$ within statistical uncertainty, consistent with the theoretical prediction.",
"Our results, with $h_0$ set to zero in the fits are summarized in Table REF .",
"The corresponding $h_i$ from fits to $\\Delta _2$ were very poorly determined.",
"We are performing simulations at other values of the lattice spacing and quark mass in order to perform the continuum-chiral extrapolation and compare with the experimental value.",
"Table: Hyperfine splittings, λ 2 \\lambda _2 and A≡-ρ 2 +T 2 +T 4 A \\equiv -\\rho _2+T_2 +T_4 for the D s D_s and B s B_s mesons at the physical valuesof κ c \\kappa _c and κ b \\kappa _b.",
"Δ exp \\Delta _\\text{exp} is theexperimental value ."
],
[
"Summary and Plan",
"We tuned the bottom and charm quark masses using physical values for $B^{(*)}_s$ and $D^{(*)}_s$ mesons.",
"Estimates from fits to the pseudoscalar, vector, and spin-averaged mesons masses are consistent within their statistical uncertainty (see Table REF ).",
"We used estimates from the pseudoscalar mesons for the analysis of the hyperfine splittings.",
"These values of $\\kappa _b$ and $\\kappa _c$ are now being used to measure the semileptonic form factors for the exclusive decays $\\bar{B}\\rightarrow D^{(*)} \\ell \\bar{\\nu }$ .",
"Figure: Plots of Δ 1 (D s )\\Delta _1(D_s) (left panel) andΔ 1 (B s )\\Delta _1(B_s) (right panel) versus the kinetic masses, m 2 c m_2^cand m 2 b m_2^b, of the quarks.",
"Results at the physical quark masses,tuned using the pseudoscalar meson masses, are shown by the red trianglesand given in Table .We thank Jon A. Bailey for helpful comments and suggestions.",
"The research of W. Lee is supported by the Creative Research Initiatives Program (No.",
"20160004939) of the NRF grant funded by the Korean government (MEST).",
"W. Lee would like to acknowledge the support from the KISTI supercomputing center through the strategic support program for the supercomputing application research (No. KSC-2014-G2-002).",
"Computations were carried out in part on the DAVID GPU clusters at Seoul National University.",
"The research of T. Bhattacharya, R. Gupta and Y-C. Jang is supported by the U.S. Department of Energy, Office of Science of High Energy Physics under contract number DE-KA-1401020, the LANL LDRD program and Institutional Computing."
]
] | 1612.05707 |
[
[
"Nonlocal Schrodinger Equations for Integro-Differential Operators with\n Measurable Kernels and Asymptotic Potentials"
],
[
"Abstract In this paper, we investigate the existence of nonnegative solutions for the problem $$ -\\mathcal{L}_{K}u+V(x)u=f(u) $$ in $\\mathbb R^n$, where $-\\mathcal{L}_{K}$ is a integro-differential operator with measurable kernel $K$ and $V$ is a continuous potential.",
"Under apropriate hypothesis, we prove, using variational methods, that the above equation has solution."
],
[
"Introduction",
"In this article we consider the class of integro-differential Schrödinger equations $\\begin{array}{lcl}- \\mathcal {L}_{K} u +V(x)u = f(u), &\\mbox{ in }& \\mathbb {R}^n,\\end{array}\\qquad \\mathrm {(P)}$ where $- \\mathcal {L}_{K}$ is a integro-differential operator given by $- \\mathcal {L}_{K}u(x)= 2 \\int _{\\mathbb {R}^{n}}(u(x)-u(y))K(x-y)dy$ and $K$ satisfy general properties.",
"This study leads both to nonlocal and to nonlinear difficulties.",
"For example, we can not benefit from the $s$ -harmonic extension of or commutator properties (see ).",
"The study of nonlocal operators is important because they intervene in a quantity of applications and models.",
"For example, we mention their use in phase transition models (see , ), image reconstruction problems (see ), obstacle problem, optimization, finance, phase transitions.",
"Integro-differential equations arise naturally in the study of stochastic processes with jumps, and more precisely of Lévy processes.",
"This paper was motivated by , where the authors study the existence of positive solutions for the problem $\\left\\lbrace \\begin{array}{lcl}- \\Delta u +V(x)u = f(u), &\\mbox{ in }& \\mathbb {R}^n, \\\\ u \\in D^{1,2}(\\mathbb {R}^{n})\\end{array}\\right.$ where $V$ and $f$ are continuous functions with $V$ being a nonnegative function and $f$ having a subcritical or critical growth.",
"Our purpose is to study an analogously problem, considering the operator $-\\mathcal {L}_{k}$ instead of the Laplacian operator.",
"Several papers have studied the problem $(P)$ when $K (x) = \\frac{C_{n,s}}{2}|x|^{n + 2s}$ , where $C_{n,s}=\\left(\\int _{\\mathbb {R}^{n}}\\frac{1-\\cos (\\xi _{1})}{|\\xi |^{n+2s}}d \\xi \\right)^{-1},$ that is, when the operator $-\\mathcal {L}_{k}$ is the fractional Laplacian operator (see ).",
"Next, we will mention some of these papers.",
"In , the author has proved the existence of positive solutions from $(P)$ when $V$ is a constant small enough.",
"Also, in , the problem was studied when $f$ is asymptotically linear and $V$ is constant.",
"In , the authors study the problem $(P)$ when $V \\in C^{n}(\\mathbb {R}^{n},\\mathbb {R})$ , $V$ is positive and $\\lim \\limits _{n \\rightarrow \\infty }V(|x|) \\in (0, \\infty ].$ In , the authors has Studied $(P)$ when $ V $ and $ f $ are asymptotically periodic.",
"When $V=1$ , Felmer et al.",
"has studied the existence, regularity and qualitative properties of ground states solutions for problem $(P)$ (see ).",
"In , Teng and He have shown the existence of solution for $(P)$ when $f(x,u)=P(x)|u|^{p-2}u+Q(x)|u|^{2^{\\ast }_{s}-2}u,$ where $2 < p < 2^{\\ast }_{s}$ and the potential functions $P(x)$ and $Q(x)$ satisfy certain hypothesis.",
"In , the authors have shown the existence of solution for $(P)$ when $V \\in C^{n}(\\mathbb {R}^{n}, \\mathbb {R})$ and there exists $r_{0} > 0$ such that, for any $M > 0$ , $\\mbox{meas}(\\left\\lbrace x \\in B_{r_{0}}(y); V (x)\\le M\\right\\rbrace )\\rightarrow 0\\mbox{ as }|y| \\rightarrow \\infty .$ In the problem $(P)$ was studied when $V \\in C^1(\\mathbb {R}^{n},\\mathbb {R})$ , $\\liminf _{|x|\\rightarrow \\infty }V(x)\\ge V_{\\infty }$ where $V_{\\infty }$ is constant, and $f \\in C^{1}(\\mathbb {R}^{n}, \\mathbb {R})$ .",
"By method of the Nehari manifold , Sechi has shown that the problem $(P)$ has a solution if $V\\le V_{\\infty }$ , but $V$ is not identically equal to $V_{\\infty }$ , where $V_{\\infty }$ is a constant.",
"Also in , Secchi have obtained the existence of ground state solutions of $(P)$ for general $s \\in (0,1)$ when $V(x)\\rightarrow \\infty $ as $|x| \\rightarrow \\infty $ .",
"In , the authors obtain the existence of a sequence of radial and non radial solutions for the problem $(P)$ when $V$ and $f$ are radial functions.",
"Some other interesting studies by variational methods of the problem $(P)$ can be found in , , , , , , , , , , , , , and .",
"Many of them use strong tools that we can not use here in our problem, as the $s$ -harmonic extension and commutator properties.",
"Here, we will admit that the potential $V$ is continuous and satisfies, $(V_{1}-)$ $\\inf _{x \\in \\mathbb {R}^{n}}V(x)>0;$ $(V_{2}-)$ $V(x)\\le V_{\\infty }$ for same constant $V_{\\infty }>0$ and for all $x \\in B_{1}(0)$ .",
"Note that, $(V_{1})$ implies that $(V_{3}-)$ There are $R>0$ and $\\Lambda >0$ such that $V(x)\\ge \\Lambda $ for all $|x|\\ge R$ .",
"Also, we will assume that $f\\in C(\\mathbb {R},\\mathbb {R})$ is a function satisfying: $(f_{1}-)$ $|f(s)|\\le c_{0}|s|^{p-1}$ , for some constant $C>0$ and $p \\in (2,2^{\\ast }_{s})$ ; $(f_{2}-)$ There is $\\theta >2$ such that $\\theta F(s)\\le sf(s)$ for all $s>0$ ; $(f_{3}-)$ $f(t)>0$ for all $t>0$ and $f(t)=0$ for all $t<0$ .",
"The kernel $K:\\mathbb {R}^{n}\\rightarrow \\left(0, \\infty \\right)$ is a measurable function such that $(K_{1}-)$ $K(x)=K(-x)$ for all $x \\in \\mathbb {R}^{n}$ ; $(K_{2}-)$ There is $\\lambda >0$ and $s \\in (0,1)$ such that $\\lambda \\le K(x)|x|^{n+2s}$ almost everywere in $\\mathbb {R}^{n}$ ; $(K_{3}-)$ $\\gamma K \\in L^{1}(\\mathbb {R}^{n})$ , where $\\gamma (x)=\\min \\left\\lbrace |x|^{2},1\\right\\rbrace $ .",
"Note that, when $K(x)=\\frac{C_{n,s}}{2}|x|^{-(n+2s)}$ we have that $-\\mathcal {L}_{k}$ is a fractional laplacian, $(-\\Delta )^{s}$ .",
"Our paper is organized as follows.",
"In section 2, we will present some properties of the space in which we will study the problem $(P)$ .",
"In section 3, we will define an auxiliary problem and we will show that the functional energy associated with the auxiliary problem satisfies the condition of Palais-Smale.",
"By difficulty nonlocal of the operator $-\\mathcal {L}_{K}$ , we will can not use the same technique used in .",
"Therefore, we will present an another technique to show this result.",
"In section 4, we will prove that a general estimative for weak solution of $-\\mathcal {L}_{K}u+b(x)u=g(x,u),$ where $b\\ge 0$ , $|g(x,t)|\\le h(x)|t|$ and $h\\in L^{q}(\\mathbb {R}^{n})$ with $q>\\frac{n}{2s}$ .",
"We will show that there is $M=M(q,||h||_{L^{q}})$ such that the solution $u$ satisfies $||u||_{\\infty } \\le M||u||_{2^{\\ast }_{s}}.$ In , using the $s$ -harmonic extension of , the authors has shown the same estimate when $ -\\mathcal {L}_{K}$ is the fractional Laplacian operator.",
"In our case, we can not use the $s$ -harmonic extension, because we do not have an analogously extension for integro-differential operators.",
"In section 5, we show our main result in this paper, the Theorem REF ."
],
[
"Preliminaries",
"Let $s \\in (0,1)$ , we denote by $H^{s}(\\mathbb {R}^{n})$ the fractional sobolev space.",
"It is defined as $H^{s}(\\mathbb {R}^{n}):=\\left\\lbrace u \\in L^{2}(\\mathbb {R}^{n});\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\frac{(u(x)-u(y))^{2}}{|x-y|^{n+2s}}dxdy<\\infty \\right\\rbrace .$ The space $H^{s}(\\mathbb {R}^{n})$ is a Hilbert space with the norm $||u||_{H^{s}}=\\left(\\int _{\\mathbb {R}^{n}}|u|^{2}dx+\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\frac{(u(x)-u(y))^{2}}{|x-y|^{n+2s}}dxdy\\right)^{\\frac{1}{2}}$ We define $X$ as the linear space of Lebesgue measurable functions from $\\mathbb {R}^{n}$ to $\\mathbb {R}$ such that any function $u$ in $X$ belongs to $L^{2}(\\mathbb {R}^{n})$ and the function $(x,y)\\longmapsto (u(x)-u(y))\\sqrt{K(x-y)}$ is in $L^{2}(\\mathbb {R}^{n}\\times \\mathbb {R}^{n})$ .",
"The function $||u||_{X}:= \\left(\\int _{\\mathbb {R}^{n}}u^{2}dx+\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}(u(x)-u(y))^{2}K(x-y)dxdy\\right)^{\\frac{1}{2}}$ defines a norm in $X$ and $(X, ||\\cdot ||_{X})$ is a Hilbert space.",
"By $(K_{2})$ , the space $X$ is continuously embedded in $H^{s}(\\mathbb {R}^{n})$ .",
"Therefore, $X$ is continuously embedded in $L^{p}(\\mathbb {R}^{n})$ for $p \\in \\left[2,2^{\\ast }_{s}\\right]$ , where $2^{\\ast }_{s}=\\frac{2n}{n-2s}$ .",
"If $\\Omega \\subset \\mathbb {R}^{n}$ , we define $X_{0}(\\Omega )=\\left\\lbrace u \\in X; u=0 \\mbox{ in } \\mathbb {R}^{n}\\setminus \\Omega .\\right\\rbrace .$ The space $X_{0}(\\Omega )$ is a Hilbert Space with the norm $u \\longmapsto ||u||_{X_{0}(\\Omega )}:=\\left(\\int _{\\Omega }u^{2}dx+\\int _{Q}\\left(u(x)-u(y)\\right)^{2}K(x-y)dxdy\\right)^{\\frac{1}{2}},$ where $Q=(\\mathbb {R}^{n}\\times \\mathbb {R}^{n}) \\setminus (\\Omega ^{c} \\times \\Omega ^{c})$ (see Lemma 7 in ).",
"It is continuously embedded in $H_{0}^{s}(\\mathbb {R}^{n})$ .",
"For definition and properties of $H_{0}^{s}(\\mathbb {R}^{n})$ we indicate .",
"In the problem $(P)$ we will consider the space $E$ defined as $E=\\left\\lbrace u \\in X; \\int _{\\mathbb {R}^{n}}V(x)u^{2}dx< \\infty \\right\\rbrace $ The space $E$ is a Hilbert space with the norm $u \\longmapsto ||u||:=\\left(\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)^{2}K(x-y)dxdy+\\int _{\\mathbb {R}^{n}}V(x)u^{2}dx\\right)^{\\frac{1}{2}}.$ If $u,v \\in C_{0}^{\\infty }(\\mathbb {R}^{n})$ then $(-\\mathcal {L}_{k}u,v)_{L^{2}}=[u,v].$ where $[u,v]=\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)\\left(v(x)-v(y)\\right)K(x-y)dxdy.$ Therefore, we say that $u\\in E$ is a solution for the problem $(P)$ if $[u,v]+\\int _{\\mathbb {R}^{n}}V(x)uvdx = \\int _{\\mathbb {R}^{n}}f(u)vdx$ for all $v \\in E$ , that is $\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)\\left(v(x)-v(y)\\right)K(x-y)dxdy+\\int _{\\mathbb {R}^{n}}V(x)uvdx = \\int _{\\mathbb {R}^{n}}f(u)vdx.$ Let $A, B \\subset \\mathbb {R}^{n}$ and $u,v \\in X$ .",
"We will denote $[u,v]_{A\\times B}=\\int _{A}\\int _{B}(u(x)-u(y))(v(x)-v(y))K(x-y)dxdy$ and we will denote $[u,v]_{\\mathbb {R}^{n}\\times \\mathbb {R}^{n}}$ by $[u,v]$ .",
"The Euler-Lagrange functional associated with $(P)$ is given by $I(u)=\\frac{1}{2}||u||^{2}-\\int _{\\mathbb {R}^{n}}F(u)dx,$ where $F(t)=\\int _{0}^{t}f(s)ds.$ From hypothesis about $f$ , the functional is $C^{1}(E,\\mathbb {R})$ and $I^{\\prime }(u)v=[u,v]+\\int _{\\mathbb {R}^{n}}V(x)uvdx-\\int _{\\mathbb {R}^{n}}f(u)vdx.$ We will denote by $B$ the unitary ball of $\\mathbb {R}^{n}$ .",
"Define $I_{0}:X_{0}(B)\\longrightarrow \\mathbb {R}$ by $I_{0}(u)=:\\int _{\\mathbb {R}^{n}} \\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)^{2}K(x-y)dxdy+\\int _{\\mathbb {R}^{n}}V_{\\infty }u^{2}dx-\\int _{\\mathbb {R}^{n}}F(u)dx,$ where $V_{\\infty }$ is the constant of $(V_{2})$ .",
"The functional $I_{0}$ has the mountain pass geometry.",
"We will denote by $d$ the mountain pass level associated with $I_{0}$ , that is $d= \\inf _{\\gamma \\in \\Gamma } \\max _{t \\in [0,1]}I_{0}(\\gamma (t)),$ where $\\Gamma =\\left\\lbrace \\gamma \\in C([0,1],X_{0}(\\Omega )); \\gamma (0)=0\\mbox{ and }\\gamma (1)=e\\right\\rbrace ,$ with $e$ fixed and verifying $I_{0}(e)<0$ .",
"Note that $d$ depends only on $V_{\\infty }$ , $\\theta $ and $f$ ."
],
[
"An Auxiliary Problem",
"According to , we will modified the problem defining an auxiliary problem.",
"But, as the operator $-\\mathcal {L}_{K}$ is nonlocal, we can not use the same ideas of to prove that the functional associated the auxiliary problem satisfies the Palais-Smale condition.",
"It is necessary that we use an another technics.",
"For $k=\\frac{2 \\theta }{\\theta -2}$ we consider $\\begin{array}{lll}\\tilde{f}(x,t)&=& \\left\\lbrace \\begin{array}{lll}f(t) & if & kf(t) \\le V(x)t \\\\\\frac{V(x)}{k}t& if & kf(t)>V(x)t\\end{array}\\right.\\end{array}$ and $\\begin{array}{lll}g(x,t)&=& \\left\\lbrace \\begin{array}{lll}f(t) & if & |x|\\le R \\\\\\tilde{f}(x,t)& if & |x|>R.\\end{array}\\right.\\end{array}$ And we define the auxiliary problem $\\left\\lbrace \\begin{array}{lllll}-\\mathcal {L}_{K}u+V(x)u&=&g(x,u)& in &\\mathbb {R}^{n} \\\\u \\in E&&&&\\end{array}\\right.$ We have that, for all $t \\in \\mathbb {R}$ and $x \\in \\mathbb {R}^{n}$ $\\tilde{f}(x,t)\\le f(t)$ ; $g(x,t) \\le \\frac{V(x)}{k}t$ ,se $|x|\\ge R$ ; $G(x,t)=F(t)$ se $|x|\\le R$ $G(x,t)\\le \\frac{V(x)}{2k}t^{2}$ se $|x|>R$ ; where $G(x,t)=\\int _{0}^{t}g(x,s)ds.$ The Euler-Lagrange functional associated with the auxiliary problem is given by $J(u)=\\frac{1}{2}||u||^{2} - \\int _{\\mathbb {R}^{n}}G(x,u)dx.$ The functional $J \\in C^{1}(X,\\mathbb {R})$ and $\\begin{array}{ll}J^{\\prime }(u)v&=\\int _{\\mathbb {R}^{n}}\\int _{\\mathbb {R}^{n}}\\left(u(x)-u(y)\\right)\\left(v(x)-v(y)\\right)K(x-y)dxdy \\\\&+\\int _{\\mathbb {R}^{n}}V(x)uvdx - \\int _{\\mathbb {R}^{n}}g(x,u)vdx.\\end{array}$ The functional $J$ has the mountain pass geometry.",
"Then, there is a sequence $\\left\\lbrace u_{n}\\right\\rbrace _{n \\in \\mathbb {N}}$ such that $J^{\\prime }(u_{n})\\rightarrow 0\\mbox{ and }J(u_{n})\\rightarrow c,$ where $c>0$ is the mountain pass level associated with $J$ , that is $c= \\inf _{\\gamma \\in \\Gamma } \\max _{t \\in [0,1]}J(\\gamma (t))$ where $\\Gamma =\\left\\lbrace \\gamma \\in C([0,1],E); \\gamma (0)=0\\mbox{ and }, \\gamma (1)=e\\right\\rbrace .$ and $e$ is the function fixed in $\\ref {eq1}$ .",
"By definition $c \\le d$ uniformly in $R>0$ .",
"Lemma 3.1 The sequence $\\left\\lbrace u_{n}\\right\\rbrace _{n \\in \\mathbb {N}}$ is bounded.",
"By $(f_{2})$ , $(3)$ and $(4)$ $\\begin{array}{ll}&J(u)-\\frac{1}{\\theta }J^{\\prime }(u)u\\\\& = \\left(\\frac{\\theta -2}{4\\theta }\\right)||u||^{2}+ \\frac{1}{2k}||u||^{2} + \\int _{\\mathbb {R}^{n}}\\frac{1}{\\theta }g(x,u)u-G(x,u)dx \\\\&\\ge \\left(\\frac{\\theta -2}{4\\theta }\\right)||u||^{2}+ \\frac{1}{2k}||u||^{2} + \\int _{|x|>R}\\frac{1}{\\theta }g(x,u)u - \\frac{1}{2k}\\int _{|x|>R}V(x)u^{2}dx \\\\& \\ge \\left(\\frac{1}{2k}\\right)||u||^{2}.\\end{array}$ Thereby $|J(u)|+|J^{\\prime }(u)u|\\ge \\left(\\frac{\\theta -2}{4\\theta }\\right)||u||^{2}$ for all $u \\in E$ .",
"This last inequality ensures that the sequence is bounded.",
"Let $r>R$ and $A=\\left\\lbrace x \\in \\mathbb {R}^{n}; r<||x||<2r\\right\\rbrace $ .",
"Consider $\\eta :\\mathbb {R}^{n}\\rightarrow \\mathbb {R}$ a function such that $\\eta =1$ in $B_{2r}^{c}(0)$ , $\\eta =0$ in $B_{r}(0)$ , $0\\le \\eta \\le 1$ and $|\\nabla \\eta |<\\frac{2}{r}$ .",
"Note that $(B_{r} \\times B_{r})^{c} = (B_{r}^{c}\\times \\mathbb {R}^{n}) \\cup ( B_{r} \\times B_{r}^{c}),$ where $B_{r}=B_{r}(0)$ and $B_{2r}=B_{2r}(0)$ .",
"We will decompose $B_{r}^{c}\\times \\mathbb {R}^{n} = (A \\times \\mathbb {R}^{n}) \\cup (B_{2r}^{c}\\times B_{r}) \\cup (B_{2r}^{c}\\times A) \\cup (B_{2r}^{c}\\times B_{2r}^{c})$ and $B_{r} \\times B_{r}^{c} = (B_{r} \\times A) \\cup (B_{r} \\times B_{2r}^{c})$ Lemma 3.2 We have that $\\begin{array}{ll}&\\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))(\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y))K(x-y)dxdy\\\\&+\\int _{B_{2r}^{c}}\\int _{B_{r}}(u_{n}(x)-u_{n}(y))(\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y))K(x-y)dxdy \\\\& \\ge -\\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy\\end{array}$ $\\begin{array}{ll}&\\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))(\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y))K(x-y)dxdy \\\\& +\\int _{B_{2r}^{c}}\\int _{B_{r}}(u_{n}(x)-u_{n}(y))(\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y))K(x-y)dxdy \\\\& =\\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(x)(u_{n}(x)-u_{n}(y))K(x-y)dxdy\\\\&- \\int _{B_{2r}^{c}}\\int _{B_{r}}u_{n}(y)(u_{n}(x)-u_{n}(y))K(x-y)dxdy \\\\&= \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(x)(u_{n}(x)-u_{n}(y))K(x-y)dxdy\\\\&- \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)(u_{n}(x)-u_{n}(y))K(x-y)dydx \\\\& = \\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy \\\\&+ \\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(y)+u_{n}(x))(u_{n}(x)-u_{n}(y))K(x-y)dxdy\\\\& = \\int _{B_{r}}\\int _{B_{2r}^{c}}(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy \\\\&+ \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(x)^{2}-u_{n}(y)^{2}K(x-y)dxdy \\\\& \\ge -\\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy\\end{array}$ Lemma 3.3 Let $\\epsilon >0$ .",
"There is $r_{0}>0$ such that if $r>r_{0}$ then $\\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy<\\epsilon ,$ for all $n \\in \\mathbb {N}$ .",
"For each $y\\in B_{r}(0)$ $B_{r}(y) \\subset B_{2r}(0).$ Then $\\begin{array}{ll}&\\int _{B_{2r}(0)^{c}}K(x-y)dx \\\\&\\le \\int _{B_{r}(y)^{c}}K(x-y)dx\\\\& = \\int _{B_{r}(0)^{c}}K(z)dz.",
"\\\\\\end{array}$ By Lemma REF , there is $L>0$ such that $||u_{n}||_{L^{2}}^{2}<L$ for all $n \\in \\mathbb {N}$ .",
"By $(K_{3})$ , there is $r_{0}>0$ such that $\\int _{B_{r}(0)^{c}}K(z)dz<\\frac{\\epsilon }{L},$ for all $r>r_{0}$ .",
"Then by REF , for all $n \\in \\mathbb {N}$ and $r>r_{0}$ $\\begin{array}{ll}&\\int _{B_{r}(0)}\\int _{B_{2r}(0)^{c}}u_{n}(y)^{2}K(x-y)dxdy\\\\& =\\int _{B_{r}(0)}u_{n}(y)^{2}\\int _{B_{2r}(0)^{c}}K(x-y)dxdy\\\\& \\le \\int _{B_{r}(0)}u_{n}(y)^{2}\\int _{B_{r}(0)^{c}}K(z)dzdy \\\\& = \\int _{B_{r}(0)^{c}}K(z)dz\\int _{B_{r}(0)}u_{n}(y)^{2}dy\\\\& \\le \\epsilon \\end{array}$ Lemma 3.4 There are constants $K_{1}>0$ and $K_{2}>0$ such that $\\begin{array}{ll}&\\int _{A}\\int _{\\mathbb {R}^{n}}|u_{n}(y)||(u_{n}(x)-u_{n}(y))||(\\eta (x)-\\eta (y))|K(x-y)dxdy \\\\&\\le \\frac{2K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ Note that $\\begin{array}{ll}\\int _{\\mathbb {R}^{n}}|\\eta (x)-\\eta (y)|^{2}K(x-y)dx& = \\int _{\\mathbb {R}^{n}}|\\eta (z+y)-\\eta (y)|^{2}K(z)dz \\\\& =\\int _{B_{1}(0)}|\\eta (z+y)-\\eta (y)|^{2}K(z)dz\\\\&+\\int _{B_{1}^{c}(0)}|\\eta (z+y)-\\eta (y)|^{2}K(z)dz \\\\& \\le \\frac{4}{r^{2}}\\int _{B_{1}(0)}|z|^{2}K(z)dz +4\\int _{B_{1}(0)^{c}}K(z)dz \\\\& \\le \\frac{4}{r^{2}}P_{1}+4P_{2},\\end{array}$ where $P_{1}=\\int _{B_{1}}|z|^{2}K(z)dz$ and $P_{2}= \\int _{B_{1}^{c}}K(z)dz.$ Let $K_{1}=2\\sqrt{P_{1}}$ and $K_{2}=2\\sqrt{P_{2}}$ .",
"Then, by Holder inequality $\\begin{array}{ll}&\\int _{A}\\int _{\\mathbb {R}^{n}}|u_{n}(y)||(u_{n}(x)-u_{n}(y))||(\\eta (x)-\\eta (y))|K(x-y)dxdy \\\\& \\le (\\frac{2\\sqrt{P_{1}}}{r}+2\\sqrt{P_{2}})\\int _{A}|u_{n}(y)|\\left(\\int _{\\mathbb {R}^{n}}|(u_{n}(x)-u_{n}(y))|^{2}K(x-y)dx\\right)^{\\frac{1}{2}}dy\\\\& \\le (\\frac{K_{1}}{r}+K_{2})||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ Lemma 3.5 For the same constants $K_{1}>0$ and $K_{2}>0$ of the Lemma REF $\\begin{array}{ll}&\\int _{B_{r}}\\int _{A}|u_{n}(x)-u_{n}(y)||\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y)|K(x-y)dxdy \\\\&\\le \\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ Indeed, by property $(K_{1})$ of $K$ $\\begin{array}{ll}&\\int _{B_{r}}\\int _{A}|u_{n}(x)-u_{n}(y)||\\eta (x)u_{n}(x)-\\eta (y)u_{n}(y)|K(x-y)dxdy\\\\& = \\int _{B_{r}}\\int _{A}|u_{n}(x)||u_{n}(x)-u_{n}(y)||\\eta (x)|K(x-y)dxdy \\\\& = \\int _{A}\\int _{B_{r}}|u_{n}(x)||u_{n}(x)-u_{n}(y)||n(x)-n(y)|K(x-y)dydx\\\\& =\\int _{A}\\int _{B_{r}}|u_{n}(y)||u_{n}(y)-u_{n}(x)||n(y)-n(x)|K(y-x)dxdy\\end{array}$ $Using $ K1)$ and Lemma \\ref {lem34}, we prove this lemma.$ Lemma 3.6 We have that $\\begin{array}{ll}&-\\int _{B_{2r}^{c}}\\int _{A}u_{n}(y)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy \\\\&\\le \\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ $\\begin{array}{ll}&-\\int _{B_{2r}^{c}}\\int _{A}u_{n}(y)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy\\\\&=\\int _{B_{2r}^{c}}\\int _{A}(u_{n}(x)-u_{n}(y))^{2}(\\eta (x)-\\eta (y))K(x-y)dxdy \\\\&-\\int _{B_{2r}^{c}}\\int _{A}u_{n}(x)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy \\\\& = \\int _{B_{2r}^{c}}\\int _{A}(u_{n}(x)-u_{n}(y))^{2}(n(x)-1)K(x-y)dxdy \\\\&-\\int _{B_{2r}^{c}}\\int _{A}u_{n}(x)(u_{n}(x)-u_{n}(y))(n(x)-n(y))K(x-y)dxdy \\\\& \\le -\\int _{B_{2r}^{c}}\\int _{A}u_{n}(x)(u_{n}(x)-u_{n}(y))(n(x)-n(y))K(x-y)dxdy \\\\& = -\\int _{B_{2r}^{c}}\\int _{A}u_{n}(x)(u_{n}(y)-u_{n}(x))(n(y)-n(x))K(x-y)dxdy \\\\& \\le \\int _{B_{2r}^{c}}\\int _{A}|u_{n}(x)||u_{n}(y)-u_{n}(x)||n(y)-n(x)|K(x-y)dxdy \\\\& =\\int _{A}\\int _{B_{2r}^{c}}|u_{n}(x)||u_{n}(y)-u_{n}(x)||n(y)-n(x)|K(y-x)dydx \\\\& \\le \\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}.\\end{array}$ In the last inequality, we have used the Lemma REF and $(K_{1})$ .",
"We will prove that the functional $J$ satisfies the Palais-Smale condition.",
"The next proposition ensures the existence of solution in the level $c$ for the auxiliary problem (see REF ).",
"We emphasize that by a nonlocal difficulty, we can not repeat the same arguments used in to show that the functional energy associated at the auxiliary problem satisfies the Palais-Smale condition, therefore we use another technique to show this.",
"Proposition 3.7 Suppose that $f$ and $V$ satisfy $(V_{1})$ , $(f_{1})-(f_{3})$ .",
"Then the functional $J$ satisfies the Palais-Smale condition.",
"By Lemma REF the Palais-Smale sequence $\\left\\lbrace u_{n}\\right\\rbrace _{n \\in \\mathbb {N}}$ is bounded.",
"We can suppose that $\\left\\lbrace u_{n}\\right\\rbrace _{n \\in \\mathbb {N}}$ converges weakly to $u$ .",
"By $K$ properties we have that $\\eta u_{n} \\in X$ and $||\\eta u_{n}|| \\le ||u_{n}||$ (see Lemma $5.1$ in ).",
"Then, the sequence $\\left\\lbrace \\eta u_{n}\\right\\rbrace $ is bounded in $X$ .",
"Therefore $I^{^{\\prime }}(u_{n})(\\eta u_{n})=o_{n}(1)$ .",
"That is $\\begin{array}{ll}&[u_{n},\\eta u_{n}]+\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx = \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ o_{n}(1).\\end{array}$ But, note that $[u_{n},\\eta u_{n}] = [u_{n},\\eta u_{n}]_{C(B_{r}\\times B_{r})}$ , because $\\eta =0$ in $B_{r}$ .",
"By REF , REF and REF we have: $\\begin{array}{ll}&[u_{n},\\eta u_{n}]_{A \\times \\mathbb {R}^{n}}+[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times A}+[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times B_{2r}^{c}}\\\\&+[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times B_{r}}+[u_{n},\\eta u_{n}]_{B_{r}\\times B_{2r}^{c}}+[u_{n},\\eta u_{n}]_{B_{r}\\times A}\\\\&+\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx = \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ o_{n}(1)\\end{array}$ By Lemma REF , $\\begin{array}{ll}&[u_{n},\\eta u_{n}]_{A \\times \\mathbb {R}^{n}}+[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times A}+\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\&\\le \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy-[u_{n},\\eta u_{n}]_{B_{r}\\times A}+o_{n}(1)\\end{array}$ Above, we have used that $[u_{n},\\eta u_{n}]_{B_{2r}^{c}\\times B_{2r}^{c}} =[u_{n}, u_{n}]_{B_{2r}^{c}\\times B_{2r}^{c}} \\ge 0$ .",
"Because $\\eta = 1$ in $B_{2r}^{c}$ .",
"If $C$ and $D$ are subsets of $\\mathbb {R}^{n}$ and $u \\in E$ , then $\\begin{array}{ll}[u, \\eta u]_{C \\times D}&=\\int _{C}\\int _{D}(u(x)-u(y))(\\eta u(x)-\\eta u(y))K(x-y)dxdy \\\\&=\\int _{C}\\int _{D}\\eta (x)(u(x)-u(y))^{2}K(x-y)dxdy\\\\& + \\int _{C}\\int _{D}u(y)(u(x)-u(y))(\\eta (x)-\\eta (y))K(x-y)dxdy.\\end{array}$ Thereby, $\\begin{array}{ll}&\\int _{A}\\int _{\\mathbb {R}^{n}}\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy+\\\\&+\\int _{B_{2r}^{c}}\\int _{A}\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy +\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\& \\le \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy-[u_{n},\\eta u_{n}]_{B_{r}\\times A} \\\\& -\\int _{A}\\int _{\\mathbb {R}^{n}}u_{n}(y)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy \\\\& -\\int _{B_{2r}^{c}}\\int _{A}u(y)(u_{n}(x)-u_{n}(y))(\\eta (x)-\\eta (y))K(x-y)dxdy+o_{n}(1).\\end{array}$ From Lemmas REF , REF and REF , we obtain constants $K_{1},K_{2}>0$ such that $\\begin{array}{ll}&\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\&\\le \\int _{A}\\int _{\\mathbb {R}^{n}}\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy\\\\&+\\int _{B_{2r}^{c}}\\int _{A}\\eta (x)(u_{n}(x)-u_{n}(y))^{2}K(x-y)dxdy +\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\& \\le \\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx+ \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy \\\\&+\\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}+o_{n}(1).\\end{array}$ By $(2)$ , $(f_{3})$ and $r>R$ we have $\\int _{\\mathbb {R}^{n}}g(x,u_{n})\\eta u_{n}dx \\le \\frac{1}{k}\\int _{\\mathbb {R}^{n}}\\eta V(x)u_{n}^{2}dx.$ Thereby, $\\begin{array}{ll}&\\left(1-\\frac{1}{k}\\right)\\int _{\\mathbb {R}^{n}}V(x)u_{n}^{2}\\eta dx \\\\& \\le \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x-y)dxdy \\\\&+\\frac{K_{1}}{r}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}} + K_{2}||u_{n}||_{L^{2}(A)}[u_{n},u_{n}]^{\\frac{1}{2}}+o_{n}(1)\\end{array}$ By Lemma REF , there is $C_{1}>0$ such that $||u_{n}||\\le C_{1}$ .",
"Then, for some constant $C>0$ $\\begin{array}{ll}&\\left(1-\\frac{1}{k}\\right)\\int _{|x|>2r}V(x)u_{n}^{2}dx \\\\& \\le \\int _{B_{r}}\\int _{B_{2r}^{c}}u_{n}(y)^{2}K(x,y)dxdy +C(\\frac{1}{r}+1)||u_{n}||_{L^{2}(A)}+o_{n}(1)\\end{array}$ Let $\\epsilon >0$ .",
"By Lemma REF , we can take $r$ , large enough, such that $\\begin{array}{ll}\\int _{|x|>2r}V(x)u_{n}^{2}dx \\le \\frac{\\epsilon (k-1)}{3k}+C(\\frac{1}{r}+1)||u_{n}||_{L^{2}(A)} +o_{n}(1)\\end{array}$ for all $n \\in \\mathbb {N}$ .",
"Also, we can assume that $||u||_{L^{2}(A)}< \\frac{\\epsilon (k-1)}{6 C(\\frac{1}{r}+1)k}$ By property $(2)$ of $g$ $g(x,u_{n})u_{n} \\le \\frac{V(x)}{k}u_{n}^{2}.$ for all $x$ , with $|x|>2r>R$ .",
"Therefore, by REF $\\int _{|x|>2r}g(x,u_{n})u_{n}dx \\le \\frac{\\epsilon }{3}+C\\left(\\frac{1}{r}+1\\right)\\frac{k}{k-1}||u_{n}||_{L^{2}(A)} +o_{n}(1).$ By REF , REF and compact embedding of the fractional Sobolev spaces, we can take $n_{1}\\in \\mathbb {N}$ such that if $n>n_{1}$ then $\\int _{|x|>2r}g(x,u_{n})u_{n}dx \\le \\frac{5\\epsilon }{6}.$ Note that, we can suppose that $r>0$ satisfies $\\int _{|x|>2r}g(x,u)udx \\le \\frac{\\epsilon }{12}.$ By compact embedding of fractional sobolev spaces we have that, there is, $n_{0} \\in \\mathbb {N}$ with $n_{0}>n_{1}$ and if $n>n_{0}$ then $\\left|\\int _{|x|\\le 2r}g(x,u_{n})u_{n}dx - \\int _{|x|\\le 2r}g(x,u)udx\\right|< \\frac{\\epsilon }{12}.$ Thereby, for $n>n_{0}$ we have $\\left|\\int _{\\mathbb {R}^{n}}g(x,u_{n})u_{n}dx - \\int _{\\mathbb {R}^{n}}g(x,u)udx\\right|< \\epsilon $ that is $\\lim \\limits _{n \\rightarrow \\infty }\\int _{\\mathbb {R}^{n}}g(x,u_{n})u_{n}dx = \\int _{\\mathbb {R}^{n}}g(x,u)udx$ From $I^{\\prime }(u_{n})u_{n}=o_{n}(1)$ , we conclude that $||u_{n}|| \\rightarrow ||u||$ and therefore $\\left\\lbrace u_{n}\\right\\rbrace $ converges to $u$ in $X$ .",
"Corollary 3.8 Suppose $(V_{1})$ , $(f_{1})-(f_{3})$ .",
"Then, there is $u \\in X$ such that $J(u)=c$ and $J^{\\prime }(u)=0$ .",
"Moreover, $u\\ge 0$ almost everywere in $\\mathbb {R}^{n}$ .",
"By REF and Proposition REF , there is $u \\in X$ such that $J(u)=c$ and $J^{\\prime }(u)=0$ .",
"Let $A=\\left\\lbrace x \\in \\mathbb {R}^{n};|x|>R;\\right\\rbrace \\cap \\left\\lbrace x \\in \\mathbb {R}^{n}; u(x)<0;\\right\\rbrace $ .",
"If $x \\in A$ , then $g(x,u(x))=\\frac{V(x)}{k}u(x)$ and in if $x \\in A^{c}$ , then $g(x,u)\\ge 0$ .",
"We have $0\\ge [u,u^{-}]+\\int _{A^{c}}V(x)uu^{-} = \\left(\\frac{1}{k}-1\\right)\\int _{A}V(x)uu^{-} + \\int _{A^{c}}g(x,u)u^{-}dx \\ge 0$ where $u^{-}(x)=\\max \\left\\lbrace -u(x),0\\right\\rbrace $ .",
"Then $[u,u^{-}]=0$ This implies that $u^{-}=0$ (see proof of Lemma 4.1 in ).",
"As a consequence of inequalities REF and REF we have the following proposition.",
"Proposition 3.9 If $V$ and $f$ satisfies $(V_{1}),(V_{2}), (f_{1})-(f_{3})$ , then the solution $u$ of the auxiliary problem satisfies $||u||^{2} \\le 2kd$ uniformly in $R>0$ .",
"$L^{\\infty }$ estimative for solution of auxiliary problem In this section, we will prove a Brezis-Kato estimative.",
"We will prove that, admitting some hypothesis, there is $M>0$ such that the solution of the problem $\\left\\lbrace \\begin{array}{rlll}-\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\mathbb {R}^{n} \\\\\\end{array}\\right.$ satisfies $||u||_{L^{\\infty }(\\mathbb {R}^{n})}\\le M$ and $M$ does not depend on $||u||$ (see Proposition REF ).",
"We emphasize that, in the best of our knowlegends, this result is being presented for the first time in the literature.",
"In , the authors have shown this result when the operator $\\mathcal {L}_{K}$ is the fractional Laplacian operator, that is, when $K(x)=\\frac{C_{n,s}}{2}|x|^{-n-2s}$ .",
"But, in our case, we can not use the same technique used in , because we do not have a version of the $s$ -harmonic extension for more general integro-differential operators.",
"Therefore, we present an another technique.",
"Remark 4.1 Let $\\beta >1$ .",
"Define the real functions $m(x):= (\\lambda -1)(x^{\\beta }+x^{-\\beta })-\\lambda (x^{\\beta -1}+x^{1-\\beta })+2$ and $p(x):= (\\lambda -1)(x^{\\beta }+x^{-\\beta })+\\lambda (x^{\\beta -1}+x^{1-\\beta })-2.$ where $\\lambda :=\\frac{\\beta ^{2}}{2\\beta -1}$ .",
"Then $m(x)\\ge 0$ and $p(x)\\ge 0$ for all $x> 0$ .",
"We will omit the proof of the claim that appears in the Remark REF .",
"Let $\\beta >1$ and define $f(x)=x|x|^{2(\\beta -1)}$ and $g(x)=x|x|^{\\beta -1}.$ The functions $f$ and $g$ are continuous and differentiable for all $x \\in \\mathbb {R}$ .",
"Consider $x,y\\in \\mathbb {R}$ with $x \\ne y$ .",
"By Mean Value Theorem, there are $\\theta _{1}(x,y)$ , $\\theta _{2}(x,y) \\in \\mathbb {R}$ such that $f^{^{\\prime }}(\\theta _{1}(x,y))=\\frac{f(x)-f(y)}{x-y}$ and $g^{^{\\prime }}(\\theta _{2}(x,y))=\\frac{g(x)-g(y)}{x-y},$ that is $(2\\beta -1)|\\theta _{1}(x,y)|^{2(\\beta -1)}=\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}$ and $\\beta |\\theta _{2}(x,y)|^{(\\beta -1)}=\\frac{x|x|^{(\\beta -1)}-y|y|^{(\\beta -1)}}{x-y}.$ Implying that $|\\theta _{1}(x,y)|=\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}\\right)^{\\frac{1}{2(\\beta -1)}}$ and $|\\theta _{2}(x,y)|=\\left(\\frac{1}{\\beta }\\frac{x|x|^{(\\beta -1)}-y|y|^{(\\beta -1)}}{x-y}\\right)^{\\frac{1}{(\\beta -1)}}.$ We consider $\\theta _{1}(x,x)=\\theta _{2}(x,x)=0$ for all $x \\in \\mathbb {R}$ .",
"Remark 4.2 Note that $|\\theta _{1}(x,y)|=|\\theta _{1}(y,x)|$ and $|\\theta _{2}(x,y)| = |\\theta _{2}(y,x)|$ for all $x,y \\in \\mathbb {R}$ .",
"Lemma 4.3 With the same notation, if $x \\ne 0$ then $|\\theta _{1}(x,0)|\\ge |\\theta _{2}(x,0)|.$ By equalities REF and REF , we have $|\\theta _{1}(x,0)|=\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}}{x}\\right)^{\\frac{1}{2(\\beta -1)}} = \\left(\\frac{1}{2\\beta -1}\\right)^{\\frac{1}{2(\\beta -1)}}|x|$ and $|\\theta _{2}(x,0)|=\\left(\\frac{1}{\\beta }\\frac{x|x|^{\\beta -1}}{x}\\right)^{\\frac{1}{(\\beta -1)}} = \\left(\\frac{1}{\\beta }\\right)^{\\frac{1}{(\\beta -1)}}|x|.$ Thereby, $|\\theta _{1}(x,0)|\\ge |\\theta _{2}(x,0)|.$ Lemma 4.4 If $x,y \\in \\mathbb {R}$ , then $|\\theta _{1}(x,y)|\\ge |\\theta _{2}(x,y)|.$ If $x = 0$ or $y = 0$ then the inequality was proved by Lemma REF and Remark REF .",
"The case $x=y$ is trivial.",
"We can suppose that $x \\ne y$ , $x \\ne 0$ and $y \\ne 0$ .",
"By equalities REF and REF we have that $|\\theta _{1}(x,y)|\\ge |\\theta _{2}(x,y)|$ if, and only if $\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}\\right)^{\\frac{1}{2(\\beta -1)}} \\ge \\left(\\frac{1}{\\beta }\\frac{x|x|^{\\beta -1}-y|y|^{\\beta -1}}{x-y}\\right)^{\\frac{1}{(\\beta -1)}}.$ This last inequality is true if, and only if $\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y} \\ge \\frac{1}{\\beta ^{2}}\\left(\\frac{x|x|^{\\beta -1}-y|y|^{\\beta -1}}{x-y}\\right)^{2}.$ This last inequality occurs if, and only if $\\lambda (x-y)(x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}) \\ge \\left(x|x|^{\\beta -1}-y|y|^{\\beta -1}\\right)^{2},$ that is $\\lambda (|x|^{2\\beta }-xy|y|^{2(\\beta -1)}-yx|x|^{2(\\beta -1)}+|y|^{2\\beta })\\ge |x|^{2\\beta }-2xy|x|^{\\beta -1}|y|^{\\beta -1}+|y|^{2\\beta }.$ But, we are supposing that $x\\ne 0$ and $y\\ne 0$ , then the last inequality is equivalent to $\\begin{array}{ll}&\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] \\\\&\\ge \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - 2\\frac{x}{|x|}\\frac{y}{|y|} + \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\end{array}$ We will prove that the inequality REF is true.",
"If $x\\cdot y>0$ , then $\\begin{array}{ll}&\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2\\frac{x}{|x|}\\frac{y}{|y|} - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& =\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(\\frac{|y|}{|x|}\\right)^{\\beta -1}-\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2 - \\left(\\frac{|y|}{|x|}\\right)^{\\beta } \\\\& = (\\lambda -1)\\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta }+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta }\\right]-\\lambda \\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta +1}\\right]+2 \\\\& = m(\\frac{|x|}{|y|}).\\end{array}$ If $x \\cdot y<0$ , then $\\begin{array}{ll}& \\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2\\frac{x}{|x|}\\frac{y}{|y|} - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& =\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + \\left(\\frac{|y|}{|x|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right]- \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - 2 - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& = (\\lambda -1)\\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta }+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta }\\right]+\\lambda \\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta +1}\\right]-2\\\\& = p(\\frac{|x|}{|y|}).\\end{array}$ By Remark REF , we have that $m(\\frac{|x|}{|y|})\\ge 0$ and $p(\\frac{|x|}{|y|}) \\ge 0$ .",
"This proves the inequality REF and the Lemma REF .",
"Our main result of this section will establishes an important estimate involving the $L^{\\infty }(\\mathbb {R}^{n})$ norm of the solution $u$ of the auxiliary problem.",
"It states that: Proposition 4.5 Leq $h \\in L^{q}(\\mathbb {R}^{n})$ with $q>\\frac{n}{2s}$ , and $v \\in E \\subset X$ be a weak solution of $\\left\\lbrace \\begin{array}{rlll}-\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\mathbb {R}^{n} \\\\\\end{array}\\right.$ where $g$ is a continuous functions satisfying $|g(x,s)|\\le h(x)|s|$ for $s\\ge 0$ , $b$ is a positive function in $\\mathbb {R}^{n}$ and $E$ is definded as in REF .",
"Then, there is a constant $M=M(q,||h||_{L^{q}})$ such that $||v||_{\\infty }\\le M||v||_{2^{\\ast }_{s}}.$ Let $\\beta >1$ .",
"For any $n \\in \\mathbb {N}$ define $A_{n}=\\left\\lbrace x \\in \\mathbb {R}^{n}; |v(x)|^{\\beta -1}\\le n\\right\\rbrace $ and $B_{n}:=\\mathbb {R}^{n}\\setminus A_{n}.$ Consider $\\begin{array}{ll}f_{n}(t):=&\\left\\lbrace \\begin{array}{lll}t|t|^{2(\\beta -1)}&se&|t|^{\\beta -1}\\le n\\\\n^{2}t&se&|t|^{\\beta -1}>n\\end{array}\\right.\\end{array}$ and $\\begin{array}{ll}g_{n}(t):=&\\left\\lbrace \\begin{array}{lll}t|t|^{(\\beta -1)}&se&|t|^{\\beta -1}\\le n\\\\nt&se&|t|^{\\beta -1}>n\\end{array}\\right.\\end{array}$ Note that $f_{n}$ and $g_{n}$ are continuous functions and they are differentiable with the exception of $n^{\\frac{1}{\\beta -1}}$ and $-n^{\\frac{1}{\\beta -1}}$ and its derivatives are limited.",
"Then $f_{n}$ and $g_{n}$ are lipschtz.",
"Therefore, setting $v_{n}:=f_{n}\\circ v$ and $w_{n}:=g_{n}\\circ v$ we have that $v_{n},w_{n}\\in E$ .",
"Note that $\\begin{array}{ll}\\left[v,v_{n}\\right]&= \\int _{A_{n}}\\int _{A_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy \\\\ & +\\int _{B_{n}}\\int _{B_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy\\\\ &+2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ By equality REF , if $x,y \\in A_{n}$ then $v_{n}(x)-v_{n}(y)=f_{n}^{\\prime }(\\theta _{1}(x,y))(v(x)-v(y)),$ where we are denoting $\\theta _{1}(v(x),v(y))$ by $\\theta _{1}(x, y)$ .",
"Thereby, $\\begin{array}{ll}\\left[v,v_{n}\\right] &= \\int _{A_{n}}\\int _{A_{n}}(2\\beta -1)|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ Analogously, by REF $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &= \\int _{A_{n}}\\int _{A_{n}}\\beta ^{2}|\\theta _{2}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}},\\end{array}$ where we are denoting $\\theta _{2}(v(x),v(y))$ by $\\theta _{2}(x, y)$ .",
"By Lemma REF , $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &\\le \\int _{A_{n}}\\int _{A_{n}}\\beta ^{2}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ This implies that $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\left[v,v_{n}\\right]- \\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\&\\le (\\beta -1)^{2}\\int _{A_{n}}\\int _{A_{n}}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\& +2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}-2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ The equation REF implies that $\\begin{array}{ll}&\\left[v,v_{n}\\right]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx - 2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}\\\\ &\\ge (2\\beta -1)\\int _{A_{n}}\\int _{A_{n}}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy,\\end{array}$ because $b(x)vv_{n}=b(x)w_{n}^{2}\\ge 0$ .",
"Replacing REF in the inequality REF we obtain $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\left[v,v_{n}\\right]- \\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\ &\\le \\frac{(\\beta -1)^{2}}{2\\beta -1}\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}},\\end{array}$ that is $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx\\\\ &\\le (\\frac{(\\beta -1)^{2}}{2\\beta -1}+1)\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\\\&= \\frac{\\beta ^{2}}{2\\beta -1}\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}bvv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\\\&\\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ In short, $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &+ \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx \\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ But, if $n \\in \\mathbb {N}$ and $C=2+\\frac{2(\\beta -1)^{2}}{2\\beta -1},$ then a simple calculation shows that the function $r(s,t)=2(ns-t|t|^{\\beta -1})^{2}-C(s-t)(n^{2}s-t|t|^{2(\\beta -1)}),$ satisfies $r(s,t)\\le 0$ for all $|s|>n^{\\frac{1}{\\beta -1}}$ and $|t|\\le n^{\\frac{1}{\\beta -1}}$ .",
"Then, taking $s=v(x)$ and $t=v(y)$ for $x\\in B_{n}$ and $y \\in A_{n}$ and replacing in REF we obtain $2(w_{n}(x)-w_{n}(y))^{2}-C(v(x)-v(y))(v_{n}(x)-v_{n}(y))\\le 0.$ Thereby $+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\le 0.$ By inequality REF , $\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx.$ Let $S>0$ be the best constant verifying $||u||_{2^{\\ast }_{s}}^{2} \\le S||u||_{X}^{2},$ for all $u \\in X$ , that is $S=\\sup _{u \\in X}\\frac{||u||_{2^{\\ast }_{s}}^{2}}{||u||_{X}^{2}}.$ By inequality REF , $\\begin{array}{ll}\\left(\\int _{A_{n}}|w_{n}|^{2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}}&\\le \\left(\\int _{\\mathbb {R}^{n}}|w_{n}|^{2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}}\\\\&\\le S||w_{n}||^{2} \\\\& \\le S\\beta \\int _{\\mathbb {R}^{n}}g(x,v(x))v_{n}dx \\\\& \\le S\\beta \\int _{\\mathbb {R}^{n}}h(x)w_{n}^{2}dx \\\\& \\le S\\beta ||h||_{q}||w_{n}||_{\\frac{2q}{q-1}}^{2}.\\end{array}$ But, we have that $|w_{n}(x)|\\le |v(x)|^{\\beta }$ for all $x \\in B_{n}$ and $|w_{n}(x)|=|v(x)|^{\\beta }$ for all $x \\in A_{n}$ .",
"Thereby, $\\begin{array}{ll}\\left(\\int _{A_{n}}|v|^{\\beta 2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}} \\le S\\beta ||h||_{q}\\left[\\int _{\\mathbb {R}^{n}}|v|^{\\frac{2q\\beta }{q-1}}dx\\right]^{\\frac{q-1}{q}}.\\end{array}$ By Monotone Convergence Theorem, $||v||_{2^{\\ast }_{s}\\beta } \\le (\\beta S||h||_{q})^{\\frac{1}{2\\beta }}||v||_{2\\beta q_{1}},$ where $q_{1}=\\frac{q}{q-1}$ .",
"Define $\\eta :=\\frac{2^{\\ast }_{s}}{2q_{1}}.$ and note that $\\eta >1$ .",
"When $\\beta =\\eta $ we have that $2\\beta q_{1} = 2^{\\ast }_{s}$ .",
"Then, by REF $||v||_{2^{\\ast }_{s}\\eta } \\le (\\eta S||h||_{q})^{\\frac{1}{2\\eta }}||v||_{2^{\\ast }_{s}}.$ Taking $\\beta =\\eta ^{2}$ in REF we obtain $||v||_{2^{\\ast }_{s}\\eta ^{2}} \\le \\eta ^{\\frac{1}{\\eta ^{2}}} (S||h||_{q})^{\\frac{1}{2\\eta ^{2}}}||v||_{2^{\\ast }_{s}\\eta }.$ By inequalities REF and REF we have $||v||_{2^{\\ast }_{s}\\eta ^{2}} \\le \\eta ^{\\frac{1}{(\\eta ^{2}}+\\frac{1}{2\\eta })} (S||h||_{q})^{(\\frac{1}{2\\eta ^{2}}+\\frac{1}{2\\eta })}||v||_{2^{\\ast }_{s}}.$ Inductively, we can prove that $||v||_{2^{\\ast }_{s}\\eta ^{m}} \\le \\eta ^{(\\frac{1}{2\\eta }+\\frac{1}{\\eta ^{2}}+...+\\frac{m}{2\\eta ^{m}})} (S||h||_{q})^{(\\frac{1}{2\\eta }+\\frac{1}{2\\eta ^{2}}+...+\\frac{1}{2\\eta ^{m}})}||v||_{2^{\\ast }_{s}}$ for all $m \\in \\mathbb {N}$ .",
"But, $\\sum _{i=1}^{\\infty }\\frac{m}{2\\eta ^{m}}=\\frac{1}{2(\\eta -1)^{2}}$ and $\\sum _{i=1}^{\\infty }\\frac{1}{2\\eta ^{m}}=\\frac{1}{2(\\eta -1)}.$ Thereby, for all $m>0$ $||v||_{2^{\\ast }_{s}\\eta ^{m}} \\le \\eta ^{\\frac{1}{2(\\eta -1)^{2}}} (S||h||_{q})^{\\frac{1}{2(\\eta -1)}}||v||_{2^{\\ast }_{s}}$ Recalling that $||v||_{\\infty }=\\lim \\limits _{n \\rightarrow \\infty }||v||_{p}$ and that $\\eta >1$ we have that $||v||_{\\infty }\\le M||v||_{2^{\\ast }_{s}}$ for $M=\\eta ^{\\frac{1}{2(\\eta -1)^{2}}} (S||h||_{q})^{\\frac{1}{2(\\eta -1)}}$ and $\\eta = \\frac{n(q-1)}{q(n-2s)}$ We conclude the proof of Proposition REF noting that $M$ depends only on $q$ , $||h||_{q}$ .",
"Solution for Problem (P) In this section, we prove the main result, the Theorem REF .",
"By Corollary REF , there is $u \\in X$ such that $J(u)=c$ and $J^{\\prime }(u)=0$ .",
"We have the following estimate for $||u||_{\\infty }$ .",
"Lemma 5.1 The solution $u$ of the auxiliary problem satisfies $||u||_{\\infty } \\le M(2Skd)^{\\frac{1}{2}}$ Consider the functions $\\begin{array}{lll}H(x,t)&=&\\left\\lbrace \\begin{array}{lllll}f(t)&if&|x|<R&or&f(t)\\le \\frac{V(x)}{k}t\\\\0&if&|x|\\ge R&and&f(t)> \\frac{V(x)}{k}t\\end{array}\\right.\\end{array}$ and $\\begin{array}{lll}b(x)&=&\\left\\lbrace \\begin{array}{lllll}V(x)&if&|x|<R&or&f(u)\\le \\frac{V(x)}{k}u\\\\\\left(1-\\frac{1}{k}\\right)V(x)&if&|x|\\ge R&and&f(u)> \\frac{V(x)}{k}u.\\end{array}\\right.\\end{array}$ Note that the function $u$ is solution of $\\left\\lbrace \\begin{array}{ll}&-\\mathcal {L}_{k}u+b(x)u=H(x,u)\\\\&u \\in E.\\end{array}\\right.$ By $(f_{1})$ , $|H(x,t)|\\le c_{0}|t|^{p-1}$ for $p \\in (2,2^{\\ast }_{s})$ .",
"Thereby, $|H(x,u)|\\le h(x)|u|,$ where $h(x)=C_{0}|u|^{p-2}$ .",
"Note that $h \\in L^{\\frac{2^{\\ast }_{s}}{p-2}}$ with $||h||_{L^{q}(\\mathbb {R}^{n})}\\le C(2ksd)^{\\frac{p-2}{2^{\\ast }_{s}}}.$ The number $p$ satisfies $p<2^{\\ast }_{s} = 2+\\frac{2s}{n}2^{\\ast }_{s}.$ Then $q =\\frac{2^{\\ast }_{s}}{p-2}>\\frac{n}{2s}.$ By Proposition REF and sobolev embedding $||u||_{\\infty }\\le M||u||_{2^{\\ast }_{s}} \\le MS^{\\frac{1}{2}}||u||,$ where $M=M(q,||h||_{q})$ .",
"By Proposition REF , we have $||u||_{\\infty } \\le M(2kSd)^{\\frac{1}{2}}.$ Theorem 5.2 Suppose that $V$ satisfies $(V_{1})$ -$(V_{2})$ and that $f$ satisfies $(f_{1})$ -$(f_{3})$ .",
"There is $\\Lambda ^{\\ast }=\\Lambda ^{\\ast }(V_{\\infty },\\theta ,p,c_{0},s)>0$ such that if $\\Lambda >\\Lambda ^{\\ast }$ in $(V_{3})$ , then the problem $(P)$ has a nonnegative nontrivial solution.",
"Indeed, let $|x|\\ge R$ .",
"If $u(x)=0$ then by denfition $f(u(x))=g(x,u(x))$ .",
"If $u(x)>0$ then $\\begin{array}{ll}\\frac{f(u(x))}{u(x)}&\\le c_{0}|u|^{p-2} \\\\&\\le c_{0}||u||_{\\infty }^{p-2} \\\\& = \\frac{ c_{0}||u||_{\\infty }^{p-2}}{\\Lambda }\\Lambda \\\\& \\le \\frac{k^{\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\frac{p-2}{2}}}{\\Lambda }\\frac{V(x)}{k}\\end{array}$ Define $\\Lambda ^{\\ast }=k^{\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\frac{p-2}{2}}.$ If $\\Lambda >\\Lambda ^{\\ast }$ then $\\frac{f(u(x))}{u(x)} \\le \\frac{V(x)}{k}.$ By definition of $g$ we have $g(x,u(x))=f(u(x))$ .",
"Then $g(x,u(x))=f(u(x))$ for all $x \\in \\mathbb {R}^{n}$ .",
"Therefore, $u$ is a solution nonnegative and nontrivial of problem $(P)$ .",
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],
[
"$L^{\\infty }$ estimative for solution of auxiliary problem",
"In this section, we will prove a Brezis-Kato estimative.",
"We will prove that, admitting some hypothesis, there is $M>0$ such that the solution of the problem $\\left\\lbrace \\begin{array}{rlll}-\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\mathbb {R}^{n} \\\\\\end{array}\\right.$ satisfies $||u||_{L^{\\infty }(\\mathbb {R}^{n})}\\le M$ and $M$ does not depend on $||u||$ (see Proposition REF ).",
"We emphasize that, in the best of our knowlegends, this result is being presented for the first time in the literature.",
"In , the authors have shown this result when the operator $\\mathcal {L}_{K}$ is the fractional Laplacian operator, that is, when $K(x)=\\frac{C_{n,s}}{2}|x|^{-n-2s}$ .",
"But, in our case, we can not use the same technique used in , because we do not have a version of the $s$ -harmonic extension for more general integro-differential operators.",
"Therefore, we present an another technique.",
"Remark 4.1 Let $\\beta >1$ .",
"Define the real functions $m(x):= (\\lambda -1)(x^{\\beta }+x^{-\\beta })-\\lambda (x^{\\beta -1}+x^{1-\\beta })+2$ and $p(x):= (\\lambda -1)(x^{\\beta }+x^{-\\beta })+\\lambda (x^{\\beta -1}+x^{1-\\beta })-2.$ where $\\lambda :=\\frac{\\beta ^{2}}{2\\beta -1}$ .",
"Then $m(x)\\ge 0$ and $p(x)\\ge 0$ for all $x> 0$ .",
"We will omit the proof of the claim that appears in the Remark REF .",
"Let $\\beta >1$ and define $f(x)=x|x|^{2(\\beta -1)}$ and $g(x)=x|x|^{\\beta -1}.$ The functions $f$ and $g$ are continuous and differentiable for all $x \\in \\mathbb {R}$ .",
"Consider $x,y\\in \\mathbb {R}$ with $x \\ne y$ .",
"By Mean Value Theorem, there are $\\theta _{1}(x,y)$ , $\\theta _{2}(x,y) \\in \\mathbb {R}$ such that $f^{^{\\prime }}(\\theta _{1}(x,y))=\\frac{f(x)-f(y)}{x-y}$ and $g^{^{\\prime }}(\\theta _{2}(x,y))=\\frac{g(x)-g(y)}{x-y},$ that is $(2\\beta -1)|\\theta _{1}(x,y)|^{2(\\beta -1)}=\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}$ and $\\beta |\\theta _{2}(x,y)|^{(\\beta -1)}=\\frac{x|x|^{(\\beta -1)}-y|y|^{(\\beta -1)}}{x-y}.$ Implying that $|\\theta _{1}(x,y)|=\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}\\right)^{\\frac{1}{2(\\beta -1)}}$ and $|\\theta _{2}(x,y)|=\\left(\\frac{1}{\\beta }\\frac{x|x|^{(\\beta -1)}-y|y|^{(\\beta -1)}}{x-y}\\right)^{\\frac{1}{(\\beta -1)}}.$ We consider $\\theta _{1}(x,x)=\\theta _{2}(x,x)=0$ for all $x \\in \\mathbb {R}$ .",
"Remark 4.2 Note that $|\\theta _{1}(x,y)|=|\\theta _{1}(y,x)|$ and $|\\theta _{2}(x,y)| = |\\theta _{2}(y,x)|$ for all $x,y \\in \\mathbb {R}$ .",
"Lemma 4.3 With the same notation, if $x \\ne 0$ then $|\\theta _{1}(x,0)|\\ge |\\theta _{2}(x,0)|.$ By equalities REF and REF , we have $|\\theta _{1}(x,0)|=\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}}{x}\\right)^{\\frac{1}{2(\\beta -1)}} = \\left(\\frac{1}{2\\beta -1}\\right)^{\\frac{1}{2(\\beta -1)}}|x|$ and $|\\theta _{2}(x,0)|=\\left(\\frac{1}{\\beta }\\frac{x|x|^{\\beta -1}}{x}\\right)^{\\frac{1}{(\\beta -1)}} = \\left(\\frac{1}{\\beta }\\right)^{\\frac{1}{(\\beta -1)}}|x|.$ Thereby, $|\\theta _{1}(x,0)|\\ge |\\theta _{2}(x,0)|.$ Lemma 4.4 If $x,y \\in \\mathbb {R}$ , then $|\\theta _{1}(x,y)|\\ge |\\theta _{2}(x,y)|.$ If $x = 0$ or $y = 0$ then the inequality was proved by Lemma REF and Remark REF .",
"The case $x=y$ is trivial.",
"We can suppose that $x \\ne y$ , $x \\ne 0$ and $y \\ne 0$ .",
"By equalities REF and REF we have that $|\\theta _{1}(x,y)|\\ge |\\theta _{2}(x,y)|$ if, and only if $\\left(\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y}\\right)^{\\frac{1}{2(\\beta -1)}} \\ge \\left(\\frac{1}{\\beta }\\frac{x|x|^{\\beta -1}-y|y|^{\\beta -1}}{x-y}\\right)^{\\frac{1}{(\\beta -1)}}.$ This last inequality is true if, and only if $\\frac{1}{2\\beta -1}\\frac{x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}}{x-y} \\ge \\frac{1}{\\beta ^{2}}\\left(\\frac{x|x|^{\\beta -1}-y|y|^{\\beta -1}}{x-y}\\right)^{2}.$ This last inequality occurs if, and only if $\\lambda (x-y)(x|x|^{2(\\beta -1)}-y|y|^{2(\\beta -1)}) \\ge \\left(x|x|^{\\beta -1}-y|y|^{\\beta -1}\\right)^{2},$ that is $\\lambda (|x|^{2\\beta }-xy|y|^{2(\\beta -1)}-yx|x|^{2(\\beta -1)}+|y|^{2\\beta })\\ge |x|^{2\\beta }-2xy|x|^{\\beta -1}|y|^{\\beta -1}+|y|^{2\\beta }.$ But, we are supposing that $x\\ne 0$ and $y\\ne 0$ , then the last inequality is equivalent to $\\begin{array}{ll}&\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] \\\\&\\ge \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - 2\\frac{x}{|x|}\\frac{y}{|y|} + \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\end{array}$ We will prove that the inequality REF is true.",
"If $x\\cdot y>0$ , then $\\begin{array}{ll}&\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2\\frac{x}{|x|}\\frac{y}{|y|} - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& =\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(\\frac{|y|}{|x|}\\right)^{\\beta -1}-\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2 - \\left(\\frac{|y|}{|x|}\\right)^{\\beta } \\\\& = (\\lambda -1)\\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta }+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta }\\right]-\\lambda \\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta +1}\\right]+2 \\\\& = m(\\frac{|x|}{|y|}).\\end{array}$ If $x \\cdot y<0$ , then $\\begin{array}{ll}& \\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - \\left(xy\\frac{|y|^{\\beta -2}}{|x|^{\\beta }}\\right)-\\left(\\frac{xy|x|^{\\beta -2}}{|y|^{\\beta }}\\right)+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right] - \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + 2\\frac{x}{|x|}\\frac{y}{|y|} - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& =\\lambda \\left[ \\left(\\frac{|x|}{|y|}\\right)^{\\beta } + \\left(\\frac{|y|}{|x|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\right]- \\left(\\frac{|x|}{|y|}\\right)^{\\beta } - 2 - \\left(\\frac{|y|}{|x|}\\right)^{\\beta }\\\\& = (\\lambda -1)\\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta }+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta }\\right]+\\lambda \\left[\\left(\\frac{|x|}{|y|}\\right)^{\\beta -1}+\\left(\\frac{|x|}{|y|}\\right)^{-\\beta +1}\\right]-2\\\\& = p(\\frac{|x|}{|y|}).\\end{array}$ By Remark REF , we have that $m(\\frac{|x|}{|y|})\\ge 0$ and $p(\\frac{|x|}{|y|}) \\ge 0$ .",
"This proves the inequality REF and the Lemma REF .",
"Our main result of this section will establishes an important estimate involving the $L^{\\infty }(\\mathbb {R}^{n})$ norm of the solution $u$ of the auxiliary problem.",
"It states that: Proposition 4.5 Leq $h \\in L^{q}(\\mathbb {R}^{n})$ with $q>\\frac{n}{2s}$ , and $v \\in E \\subset X$ be a weak solution of $\\left\\lbrace \\begin{array}{rlll}-\\mathcal {L}_{k}v+b(x)v&=g(x,v)&in&\\mathbb {R}^{n} \\\\\\end{array}\\right.$ where $g$ is a continuous functions satisfying $|g(x,s)|\\le h(x)|s|$ for $s\\ge 0$ , $b$ is a positive function in $\\mathbb {R}^{n}$ and $E$ is definded as in REF .",
"Then, there is a constant $M=M(q,||h||_{L^{q}})$ such that $||v||_{\\infty }\\le M||v||_{2^{\\ast }_{s}}.$ Let $\\beta >1$ .",
"For any $n \\in \\mathbb {N}$ define $A_{n}=\\left\\lbrace x \\in \\mathbb {R}^{n}; |v(x)|^{\\beta -1}\\le n\\right\\rbrace $ and $B_{n}:=\\mathbb {R}^{n}\\setminus A_{n}.$ Consider $\\begin{array}{ll}f_{n}(t):=&\\left\\lbrace \\begin{array}{lll}t|t|^{2(\\beta -1)}&se&|t|^{\\beta -1}\\le n\\\\n^{2}t&se&|t|^{\\beta -1}>n\\end{array}\\right.\\end{array}$ and $\\begin{array}{ll}g_{n}(t):=&\\left\\lbrace \\begin{array}{lll}t|t|^{(\\beta -1)}&se&|t|^{\\beta -1}\\le n\\\\nt&se&|t|^{\\beta -1}>n\\end{array}\\right.\\end{array}$ Note that $f_{n}$ and $g_{n}$ are continuous functions and they are differentiable with the exception of $n^{\\frac{1}{\\beta -1}}$ and $-n^{\\frac{1}{\\beta -1}}$ and its derivatives are limited.",
"Then $f_{n}$ and $g_{n}$ are lipschtz.",
"Therefore, setting $v_{n}:=f_{n}\\circ v$ and $w_{n}:=g_{n}\\circ v$ we have that $v_{n},w_{n}\\in E$ .",
"Note that $\\begin{array}{ll}\\left[v,v_{n}\\right]&= \\int _{A_{n}}\\int _{A_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy \\\\ & +\\int _{B_{n}}\\int _{B_{n}}(v_{n}(x)-v_{n}(y))(v(x)-v(y)K(x-y)dxdy\\\\ &+2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ By equality REF , if $x,y \\in A_{n}$ then $v_{n}(x)-v_{n}(y)=f_{n}^{\\prime }(\\theta _{1}(x,y))(v(x)-v(y)),$ where we are denoting $\\theta _{1}(v(x),v(y))$ by $\\theta _{1}(x, y)$ .",
"Thereby, $\\begin{array}{ll}\\left[v,v_{n}\\right] &= \\int _{A_{n}}\\int _{A_{n}}(2\\beta -1)|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ Analogously, by REF $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &= \\int _{A_{n}}\\int _{A_{n}}\\beta ^{2}|\\theta _{2}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy +2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}},\\end{array}$ where we are denoting $\\theta _{2}(v(x),v(y))$ by $\\theta _{2}(x, y)$ .",
"By Lemma REF , $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &\\le \\int _{A_{n}}\\int _{A_{n}}\\beta ^{2}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\&+n^{2}\\int _{B_{n}}\\int _{B_{n}}(v(x)-v(y))^{2}K(x-y)dxdy+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ This implies that $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\left[v,v_{n}\\right]- \\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\&\\le (\\beta -1)^{2}\\int _{A_{n}}\\int _{A_{n}}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy\\\\& +2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}-2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ The equation REF implies that $\\begin{array}{ll}&\\left[v,v_{n}\\right]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx - 2\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}\\\\ &\\ge (2\\beta -1)\\int _{A_{n}}\\int _{A_{n}}|\\theta _{1}(x,y)|^{2(\\beta -1)}(v(x)-v(y))^{2}K(x-y)dxdy,\\end{array}$ because $b(x)vv_{n}=b(x)w_{n}^{2}\\ge 0$ .",
"Replacing REF in the inequality REF we obtain $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx-\\left[v,v_{n}\\right]- \\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\\\ &\\le \\frac{(\\beta -1)^{2}}{2\\beta -1}\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}},\\end{array}$ that is $\\begin{array}{ll}&\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx\\\\ &\\le (\\frac{(\\beta -1)^{2}}{2\\beta -1}+1)\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}b(x)vv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\\\&= \\frac{\\beta ^{2}}{2\\beta -1}\\left([v,v_{n}]+\\int _{\\mathbb {R}^{n}}bvv_{n}dx\\right)\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\\\&\\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx\\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ In short, $\\begin{array}{ll}\\left[w_{n},w_{n}\\right] &+ \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx \\\\&+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}}.\\end{array}$ But, if $n \\in \\mathbb {N}$ and $C=2+\\frac{2(\\beta -1)^{2}}{2\\beta -1},$ then a simple calculation shows that the function $r(s,t)=2(ns-t|t|^{\\beta -1})^{2}-C(s-t)(n^{2}s-t|t|^{2(\\beta -1)}),$ satisfies $r(s,t)\\le 0$ for all $|s|>n^{\\frac{1}{\\beta -1}}$ and $|t|\\le n^{\\frac{1}{\\beta -1}}$ .",
"Then, taking $s=v(x)$ and $t=v(y)$ for $x\\in B_{n}$ and $y \\in A_{n}$ and replacing in REF we obtain $2(w_{n}(x)-w_{n}(y))^{2}-C(v(x)-v(y))(v_{n}(x)-v_{n}(y))\\le 0.$ Thereby $+2\\left[w_{n},w_{n}\\right]_{A_{n}\\times B_{n}}+(-2-\\frac{2(\\beta -1)^{2}}{2\\beta -1})\\left[v,v_{n}\\right]_{A_{n}\\times B_{n}} \\le 0.$ By inequality REF , $\\left[w_{n},w_{n}\\right] + \\int _{\\mathbb {R}^{n}}b(x)w_{n}^{2}dx \\le \\beta \\int _{\\mathbb {R}^{n}}g(x,v)v_{n}dx.$ Let $S>0$ be the best constant verifying $||u||_{2^{\\ast }_{s}}^{2} \\le S||u||_{X}^{2},$ for all $u \\in X$ , that is $S=\\sup _{u \\in X}\\frac{||u||_{2^{\\ast }_{s}}^{2}}{||u||_{X}^{2}}.$ By inequality REF , $\\begin{array}{ll}\\left(\\int _{A_{n}}|w_{n}|^{2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}}&\\le \\left(\\int _{\\mathbb {R}^{n}}|w_{n}|^{2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}}\\\\&\\le S||w_{n}||^{2} \\\\& \\le S\\beta \\int _{\\mathbb {R}^{n}}g(x,v(x))v_{n}dx \\\\& \\le S\\beta \\int _{\\mathbb {R}^{n}}h(x)w_{n}^{2}dx \\\\& \\le S\\beta ||h||_{q}||w_{n}||_{\\frac{2q}{q-1}}^{2}.\\end{array}$ But, we have that $|w_{n}(x)|\\le |v(x)|^{\\beta }$ for all $x \\in B_{n}$ and $|w_{n}(x)|=|v(x)|^{\\beta }$ for all $x \\in A_{n}$ .",
"Thereby, $\\begin{array}{ll}\\left(\\int _{A_{n}}|v|^{\\beta 2^{\\ast }_{s}}dx\\right)^{\\frac{2}{2^{\\ast }_{s}}} \\le S\\beta ||h||_{q}\\left[\\int _{\\mathbb {R}^{n}}|v|^{\\frac{2q\\beta }{q-1}}dx\\right]^{\\frac{q-1}{q}}.\\end{array}$ By Monotone Convergence Theorem, $||v||_{2^{\\ast }_{s}\\beta } \\le (\\beta S||h||_{q})^{\\frac{1}{2\\beta }}||v||_{2\\beta q_{1}},$ where $q_{1}=\\frac{q}{q-1}$ .",
"Define $\\eta :=\\frac{2^{\\ast }_{s}}{2q_{1}}.$ and note that $\\eta >1$ .",
"When $\\beta =\\eta $ we have that $2\\beta q_{1} = 2^{\\ast }_{s}$ .",
"Then, by REF $||v||_{2^{\\ast }_{s}\\eta } \\le (\\eta S||h||_{q})^{\\frac{1}{2\\eta }}||v||_{2^{\\ast }_{s}}.$ Taking $\\beta =\\eta ^{2}$ in REF we obtain $||v||_{2^{\\ast }_{s}\\eta ^{2}} \\le \\eta ^{\\frac{1}{\\eta ^{2}}} (S||h||_{q})^{\\frac{1}{2\\eta ^{2}}}||v||_{2^{\\ast }_{s}\\eta }.$ By inequalities REF and REF we have $||v||_{2^{\\ast }_{s}\\eta ^{2}} \\le \\eta ^{\\frac{1}{(\\eta ^{2}}+\\frac{1}{2\\eta })} (S||h||_{q})^{(\\frac{1}{2\\eta ^{2}}+\\frac{1}{2\\eta })}||v||_{2^{\\ast }_{s}}.$ Inductively, we can prove that $||v||_{2^{\\ast }_{s}\\eta ^{m}} \\le \\eta ^{(\\frac{1}{2\\eta }+\\frac{1}{\\eta ^{2}}+...+\\frac{m}{2\\eta ^{m}})} (S||h||_{q})^{(\\frac{1}{2\\eta }+\\frac{1}{2\\eta ^{2}}+...+\\frac{1}{2\\eta ^{m}})}||v||_{2^{\\ast }_{s}}$ for all $m \\in \\mathbb {N}$ .",
"But, $\\sum _{i=1}^{\\infty }\\frac{m}{2\\eta ^{m}}=\\frac{1}{2(\\eta -1)^{2}}$ and $\\sum _{i=1}^{\\infty }\\frac{1}{2\\eta ^{m}}=\\frac{1}{2(\\eta -1)}.$ Thereby, for all $m>0$ $||v||_{2^{\\ast }_{s}\\eta ^{m}} \\le \\eta ^{\\frac{1}{2(\\eta -1)^{2}}} (S||h||_{q})^{\\frac{1}{2(\\eta -1)}}||v||_{2^{\\ast }_{s}}$ Recalling that $||v||_{\\infty }=\\lim \\limits _{n \\rightarrow \\infty }||v||_{p}$ and that $\\eta >1$ we have that $||v||_{\\infty }\\le M||v||_{2^{\\ast }_{s}}$ for $M=\\eta ^{\\frac{1}{2(\\eta -1)^{2}}} (S||h||_{q})^{\\frac{1}{2(\\eta -1)}}$ and $\\eta = \\frac{n(q-1)}{q(n-2s)}$ We conclude the proof of Proposition REF noting that $M$ depends only on $q$ , $||h||_{q}$ ."
],
[
"Solution for Problem (P)",
"In this section, we prove the main result, the Theorem REF .",
"By Corollary REF , there is $u \\in X$ such that $J(u)=c$ and $J^{\\prime }(u)=0$ .",
"We have the following estimate for $||u||_{\\infty }$ .",
"Lemma 5.1 The solution $u$ of the auxiliary problem satisfies $||u||_{\\infty } \\le M(2Skd)^{\\frac{1}{2}}$ Consider the functions $\\begin{array}{lll}H(x,t)&=&\\left\\lbrace \\begin{array}{lllll}f(t)&if&|x|<R&or&f(t)\\le \\frac{V(x)}{k}t\\\\0&if&|x|\\ge R&and&f(t)> \\frac{V(x)}{k}t\\end{array}\\right.\\end{array}$ and $\\begin{array}{lll}b(x)&=&\\left\\lbrace \\begin{array}{lllll}V(x)&if&|x|<R&or&f(u)\\le \\frac{V(x)}{k}u\\\\\\left(1-\\frac{1}{k}\\right)V(x)&if&|x|\\ge R&and&f(u)> \\frac{V(x)}{k}u.\\end{array}\\right.\\end{array}$ Note that the function $u$ is solution of $\\left\\lbrace \\begin{array}{ll}&-\\mathcal {L}_{k}u+b(x)u=H(x,u)\\\\&u \\in E.\\end{array}\\right.$ By $(f_{1})$ , $|H(x,t)|\\le c_{0}|t|^{p-1}$ for $p \\in (2,2^{\\ast }_{s})$ .",
"Thereby, $|H(x,u)|\\le h(x)|u|,$ where $h(x)=C_{0}|u|^{p-2}$ .",
"Note that $h \\in L^{\\frac{2^{\\ast }_{s}}{p-2}}$ with $||h||_{L^{q}(\\mathbb {R}^{n})}\\le C(2ksd)^{\\frac{p-2}{2^{\\ast }_{s}}}.$ The number $p$ satisfies $p<2^{\\ast }_{s} = 2+\\frac{2s}{n}2^{\\ast }_{s}.$ Then $q =\\frac{2^{\\ast }_{s}}{p-2}>\\frac{n}{2s}.$ By Proposition REF and sobolev embedding $||u||_{\\infty }\\le M||u||_{2^{\\ast }_{s}} \\le MS^{\\frac{1}{2}}||u||,$ where $M=M(q,||h||_{q})$ .",
"By Proposition REF , we have $||u||_{\\infty } \\le M(2kSd)^{\\frac{1}{2}}.$ Theorem 5.2 Suppose that $V$ satisfies $(V_{1})$ -$(V_{2})$ and that $f$ satisfies $(f_{1})$ -$(f_{3})$ .",
"There is $\\Lambda ^{\\ast }=\\Lambda ^{\\ast }(V_{\\infty },\\theta ,p,c_{0},s)>0$ such that if $\\Lambda >\\Lambda ^{\\ast }$ in $(V_{3})$ , then the problem $(P)$ has a nonnegative nontrivial solution.",
"Indeed, let $|x|\\ge R$ .",
"If $u(x)=0$ then by denfition $f(u(x))=g(x,u(x))$ .",
"If $u(x)>0$ then $\\begin{array}{ll}\\frac{f(u(x))}{u(x)}&\\le c_{0}|u|^{p-2} \\\\&\\le c_{0}||u||_{\\infty }^{p-2} \\\\& = \\frac{ c_{0}||u||_{\\infty }^{p-2}}{\\Lambda }\\Lambda \\\\& \\le \\frac{k^{\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\frac{p-2}{2}}}{\\Lambda }\\frac{V(x)}{k}\\end{array}$ Define $\\Lambda ^{\\ast }=k^{\\frac{p}{2}}c_{0}M^{p-2}(2Sd)^{\\frac{p-2}{2}}.$ If $\\Lambda >\\Lambda ^{\\ast }$ then $\\frac{f(u(x))}{u(x)} \\le \\frac{V(x)}{k}.$ By definition of $g$ we have $g(x,u(x))=f(u(x))$ .",
"Then $g(x,u(x))=f(u(x))$ for all $x \\in \\mathbb {R}^{n}$ .",
"Therefore, $u$ is a solution nonnegative and nontrivial of problem $(P)$ .",
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]
] | 1612.05696 |
[
[
"Geometric properties of the Markov and Lagrange spectra"
],
[
"Abstract We prove several results on (fractal) geometric properties of the classical Markov and Lagrange spectra.",
"In particular, we prove that the Hausdorff dimensions of intersections of both spectra with half-lines always coincide, and may assume any real value in the interval $[0, 1]$."
],
[
"Introduction and statement of the results",
"Let ${\\alpha }$ be an irrational number.",
"According to Dirichlet's theorem, the inequality $|{\\alpha }-\\frac{p}{q}|<\\frac{1}{q^2}$ has infinitely many rational solutions $\\frac{p}{q}$ .",
"Hurwitz improved this result by proving that $|{\\alpha }-\\frac{p}{q}|<\\frac{1}{\\sqrt{5} q^2}$ also has infinitely many rational solutions $\\frac{p}{q}$ for any irrational ${\\alpha }$ , and that $\\sqrt{5}$ is the largest constant that works for any irrational ${\\alpha }$ .",
"However, for particular values of ${\\alpha }$ we can improve this constant.",
"More precisely, if we define $k({\\alpha }):=\\sup \\lbrace k>0\\mid |{\\alpha }-\\frac{p}{q}|<\\frac{1}{kq^2}$ has infinitely many rational solutions $\\frac{p}{q}\\rbrace =\\limsup _{q\\rightarrow +\\infty }\\, ((q|q {\\alpha }-p|)^{-1})$ , we have $k({\\alpha })\\ge \\sqrt{5}$ , $\\forall {\\alpha }\\in {\\mathbb {R}}\\setminus Q$ and $k\\left(\\frac{1+\\sqrt{5}}{2} \\right)=\\sqrt{5}$ .",
"Definition 1: The Lagrange spectrum is the set $L=\\lbrace k({\\alpha }) \\mid {\\alpha }\\in {\\mathbb {R}}\\setminus Q$ , $k({\\alpha })<+\\infty \\rbrace $ .",
"Hurwitz-Markov theorem determines the first element of $L$ , which is $\\sqrt{5}$ .",
"This set $L$ encodes many diophantine properties of real numbers.",
"The study of the geometric structure of $L$ is a classical subject, which began with Markov, proving in 1879 ([Ma]) that $L\\cap (-\\infty , 3)=\\large \\lbrace k_1=\\sqrt{5}< k_2=2 \\sqrt{2}<k_3= \\frac{\\sqrt{221}}{5}<\\cdots \\large \\rbrace $ where $k_n$ is a sequence (of irrational numbers whose squares are rational) converging to 3 - more precisely, the elements $k_n$ of $L\\cap (-\\infty , 3)$ are the numbers the form $\\sqrt{9-\\frac{4}{z^2}}$ , where $z$ is a positive integer such that there are other positive integers $x, y$ with $x\\le y\\le z$ and $x^2+y^2+z^2=3xyz$ .",
"This means that the “beginning” of the set $L$ is discrete.",
"This is not true for the whole set $L$ .",
"Indeed, M. Hall proved in 1947 ([H]) a result on sums of continued fractions with coefficients bounded by 4: if $C_4=\\lbrace {\\alpha }=[0; a_1, a_2,\\ldots ]\\in [0,1]| a_n\\le 4, \\forall n\\ge 1\\rbrace $ then $C_4+C_4=\\lbrace x+y|x, y\\in C_4\\rbrace =[\\sqrt{2}-1, 4(\\sqrt{2}-1)]$ .",
"This implies that $L$ contains the whole half line $[6,+\\infty )$ .",
"In 1975, G. Freiman ([F]) determined the biggest half line $[c, +\\infty )$ that is contained in $L$ : he proved that $c=\\frac{2221564096 + 283748\\sqrt{462}}{491993569} \\cong 4,52782956616\\dots \\,.$ This half-line is known as Hall's ray.",
"These last two results are based on the study of arithmetic sums of regular Cantor sets, whose relationship with the Lagrange spectrum will be explained below.",
"Since the best rational approximations of an irrational number are its convergents (from its continued fraction representation), it is not surprising that $k({\\alpha })$ is related to the continued fraction of ${\\alpha }$ .",
"In fact, if the continued fraction of ${\\alpha }$ is ${\\alpha }= [a_0;a_1,a_2,\\dots ] = a_0 +\\cfrac{1}{a_1 + \\cfrac{1}{a_2 + {\\atop \\ddots }}}$ then we have the following formula: $k({\\alpha })=\\limsup _{n\\rightarrow \\infty }({\\alpha }_n+{\\beta }_n),$ where $\\; {\\alpha }_n=[a_n;a_{n+1},a_{n+2},\\dots ]\\;\\text{ and}\\;{\\beta }_n=[0; a_{n-1}, a_{n-2},\\dots ,a_1]$ .",
"This follows from the equality $\\left|{\\alpha }-\\frac{p_n}{q_n}\\right| =\\frac{1}{q_n({\\alpha }_{n+1}q_n+q_{n-1})}=\\frac{1}{({\\alpha }_{n+1}+{\\beta }_{n+1}) q_n^2}\\: ,\\quad \\forall n\\in N.$ This formula for $k({\\alpha })$ implies the following alternative definition of the Lagrange spectrum $L$ , due to Perron ([P]): let $\\Sigma =({N^*})^{{\\mathbb {Z}}}$ be the set of all bi-infinite sequences of positive integers.",
"If $\\underline{\\theta }=(a_n)_{n\\in {\\mathbb {Z}}}\\in \\Sigma $ , let ${\\alpha }_n=[a_n;a_{n+1},a_{n+2},\\dots ]$ and ${\\beta }_n=[0;a_{n-1},a_{n-2},\\dots ], \\forall n \\in {\\mathbb {Z}}$ .",
"We define $f(\\underline{\\theta })={\\alpha }_0+{\\beta }_0=[a_0; a_1,a_2,\\dots ]+[0; a_{-1}, a_{-2},\\dots ].$ Then, if $\\sigma \\colon \\Sigma \\rightarrow \\Sigma $ is the shift map defined by $\\sigma ((a_n)_{n\\in {\\mathbb {Z}}})=(a_{n+1})_{n\\in {\\mathbb {Z}}}$ , then the Lagrange spectrum is equal to $L=\\lbrace \\limsup _{n\\rightarrow \\infty } f(\\sigma ^n \\underline{\\theta }), \\underline{\\theta }\\in \\Sigma \\rbrace $ .",
"In this context we can also define the Markov spectrum.",
"Definition 2: The Markov spectrum is the set $M=\\lbrace \\sup _{n\\in {\\mathbb {Z}}} f(\\sigma ^n\\underline{\\theta }), \\underline{\\theta }\\in \\Sigma \\rbrace $ .",
"It also has an arithmetical interpretation (see [P]), namely $M=\\lbrace (\\inf _{(x,y)\\in {{\\mathbb {Z}}}^2\\setminus (0,0)} |f(x,y)|)^{-1},\\quad f(x,y)=a x^2 + bxy+cy^2, \\quad b^2-4ac=1\\rbrace .$ It is well-known (see [CF]) that $M$ and $L$ are closed sets of the real line and $L\\subset M$ .",
"In particular, $M$ also contains the Hall's ray $[c,+\\infty )$ .",
"Freiman also proved in [F] that this is the biggest half-line contained in $M$ .",
"In this paper, we study the geometrical behaviour of $L$ and $M$ between 3 and $c$ .",
"Consider the function $d:{\\mathbb {R}}\\rightarrow [0,1]$ defined by $d(t)=HD(L\\cap (-\\infty ,t))$ , where $HD$ denotes Hausdorff dimension (see [Fa] for the definitions and basic properties of the notions of dimension used in this paper).",
"We will prove the following results about the Markov and Lagrange spectra: Theorem 1: Given $t\\in {\\mathbb {R}}$ we have $d(t)=HD(L\\cap (-\\infty ,t))=HD(M\\cap (-\\infty ,t))=\\overline{\\dim }(L\\cap (-\\infty ,t))=\\overline{\\dim }(M\\cap (-\\infty ,t))$ , where $\\overline{\\dim }$ denotes upper box dimension.",
"Moreover, $d(t)$ is a continuous non-decreasing surjective function from ${\\mathbb {R}}$ to $[0,1]$ , and we have: i) $d(t)=\\min \\lbrace 1,2 D(t)\\rbrace $ , where $D(t):=HD(k^{-1}(-\\infty ,t)) = HD(k^{-1}(-\\infty ,t])$ is a continuous function from ${\\mathbb {R}}$ to $[0,1)$ .",
"ii) $\\max \\lbrace t\\in {\\mathbb {R}}\\mid d(t)=0\\rbrace =3$ .",
"iii) There is $\\delta >0$ such that $d(\\sqrt{12}-\\delta )=1$ .",
"This theorem solves affirmatively Problem 3 of [B].",
"It also gives some answers to Problem 5 of the same paper: the continuous function $d(t)=HD(L\\cap (-\\infty ,t))$ , which coincides (for $t>0$ ) with $\\sigma (1/t)$ , in the notation of [B], is a Cantor stair function: it is constant in the connected components of the complement of $L \\cap (-\\infty , t_1]$ , where $t_1:=\\min \\lbrace t \\in {\\mathbb {R}}\\mid d(t)=1\\rbrace \\le \\sqrt{12}-\\delta <\\sqrt{12}$ ; notice that $L \\cap (-\\infty , t_1]$ is a compact set with zero Lebesgue measure, and so with empty interior.",
"On the other hand, we have the following Corollary: $d(t)$ is not a Hölder continuous function.",
"Proof: Suppose by contradiction that $d(t)$ is Hölder continuous with exponent $\\alpha >0$ .",
"By the previous theorem, there is ${\\varepsilon }>0$ such that $0<d(3+{\\varepsilon })<\\alpha $ .",
"Since $d(t)$ is constant in connected components of the open set $\\mathbb {R} \\setminus L$ , the function $d$ maps the set $L \\cap (-\\infty ,3+{\\varepsilon }]$ , whose Hausdorff dimension is $d(3+{\\varepsilon })<\\alpha $ , to the nontrivial interval $[0,d(3+{\\varepsilon })]$ .",
"This is a contradiction, since the image of any set of Hausdorff dimension smaller than $\\alpha $ by a Hölder continuous function with exponent ${\\alpha }$ has zero Lebesgue measure (and indeed Hausdorff dimension smaller than one).",
"$\\Box $ Remark: The proof of Theorem 1 doesn't give any estimate on the modulus of continuity of $d(t)$ .",
"However it is possible to give such an estimate by modifying the proof.",
"See the discussion at the end of section 6.",
"The proof of Theorem 1 is based on the idea of approximating parts of the spectra from inside and from outside by sums of regular Cantor sets.",
"Theorem 1 uses techniques developed in a joint work with J.C. Yoccoz about sums of Cantor sets that implies that the sum of two non essentially affine regular Cantor sets have Hausdorff dimension equal to the minimum between one and the sum of their Hausdorff dimensions.",
"This result will be discussed in the next section.",
"The other results are Theorems 2 and 3 below.",
"Bugeaud defines in [B], for $c>0$ , the sets $\\text{Exact}(c)=\\lbrace \\alpha \\in {\\mathbb {R}}| \\left|\\alpha -\\frac{p}{q}\\right|<\\frac{c}{q^2}\\text{ for infinitely many }(p, q)\\in {\\mathbb {Z}}\\times {\\mathbb {N}}^*\\text{ but, for every } {\\varepsilon }>0,$ $\\left|\\alpha -\\frac{p}{q}\\right|<\\frac{c-{\\varepsilon }}{q^2}\\text{ has only a finite number of solutions }(p, q)\\in {\\mathbb {Z}}\\times {\\mathbb {N}}^*\\rbrace \\text{ and}$ $\\text{Exact}^{\\prime }(c)=\\lbrace \\alpha \\in {\\mathbb {R}}|\\text{ For every } {\\varepsilon }>0, \\left|\\alpha -\\frac{p}{q}\\right|<\\frac{c+{\\varepsilon }}{q^2}\\text{ for infinitely many }(p, q)\\in {\\mathbb {Z}}\\times {\\mathbb {N}}^*\\text{ but}$ $\\left|\\alpha -\\frac{p}{q}\\right|<\\frac{c}{q^2}\\text{ has only a finite number of solutions }(p, q)\\in {\\mathbb {Z}}\\times {\\mathbb {N}}^*\\rbrace .$ Clearly Exact$(c) \\,\\cup \\,$ Exact$^{\\prime }(c)=k^{-1}(c^{-1})$ .",
"Theorem 2: $\\lim _{c\\rightarrow 0}HD(\\text{Exact}(c))=\\lim _{c\\rightarrow 0}HD(\\text{Exact}^{\\prime }(c))=1$ .",
"Consequently, $\\lim _{t\\rightarrow \\infty }HD(k^{-1}(t))=1$ and $\\lim _{t\\rightarrow \\infty }HD(k^{-1}(-\\infty ,t))=\\lim _{t\\rightarrow \\infty }D(t)=1$ .",
"This solves affirmatively Problem 4 of [B].",
"We also prove a result on the topological structure of the Lagrange spectrum $L$ .",
"Theorem 3: $L^{\\prime }$ is a perfect set, i.e., $L^{\\prime \\prime }=L^{\\prime }$ .",
"The proof of this theorem uses the fact that an element of the Lagrange spectrum associated to an infinite sequence $\\underline{\\theta }$ is accumulated by infinitely many sums of the type ${\\alpha }_n+{\\beta }_n$ , which is not necessarily true for elements of the Markov spectrum.",
"The question of whether $M^{\\prime \\prime }=M^{\\prime }$ is still open.",
"There are still some important questions left on the structure of the Markov and Lagrange spectra.",
"For instance: 1) Consider the function $d_{loc}: L^{\\prime }\\rightarrow {\\mathbb {R}}$ given by $d_{loc}(t)=\\lim _{{\\varepsilon }\\rightarrow 0} HD(L\\cap (t-{\\varepsilon }, t+{\\varepsilon }))$ .",
"Is $d_{loc}$ a non-decreasing function ?",
"2) Describe the geometric structure of the difference set $M\\setminus L$ .",
"3) Let, as before, $t_1=\\min \\lbrace t \\in {\\mathbb {R}}\\mid d(t)=1\\rbrace <\\sqrt{12}$ .",
"Is it true that $t_1=\\inf $ int $L$ ?",
"This would imply that int$(L \\cap (-\\infty , \\sqrt{12}])\\ne \\emptyset $ , which in its turn implies that int$(C_2+C_2)\\ne \\emptyset $ , where $C_2=\\lbrace {\\alpha }=[0; a_1, a_2,\\ldots ]\\in [0,1]| a_n\\le 2, \\forall n\\ge 1\\rbrace $ .",
"We should mention that, in relation to question 2), there are some progresses in recent preprints by the author and C. Matheus: in [MM3] and [MM1], the authors describe the intersections of $M\\setminus L$ with the maximal intervals not intersecting $L$ containing the first examples $\\gamma =3.11812017815993...$ and $\\alpha _{\\infty }=3.293044265...$ by Freiman of elements in $M\\setminus L$ .",
"These intersections have positive Hausdorff dimensions (which coincide with the Hausdorff dimensions of certain regular Cantor sets we describe in these works).",
"In [MM4], the authors exhibit a regular Cantor set diffeomorphic to $C_2$ (and so with Hausdorff dimensions larger than $0.53128$ ) near $3.7096998597502$ contained in $M\\setminus L$ .",
"And, in [MM3], they prove that the Hausdorff dimension of $M\\setminus L$ is smaller than $0.986927$ , and indicate, using heuristic estimates, that this Hausdorff dimension is smaller than $0.888$ .",
"In relation to question 4), the question whether int$(C_2+C_2)\\ne \\emptyset $ was posed in page 71 of [CF].",
"I would like to thank Yann Bugeaud, Aline Gomes Cerqueira, Carlos Matheus, Túlio Carvalho and Yuri Lima for helpful comments and suggestions which substantially improved this work."
],
[
"A dimension formula for arithmetic sums of regular Cantor sets",
"We say that $K \\subset {\\mathbb {R}}$ is a regular Cantor set of class $C^k$, $k\\ge 1$ , if: i) there are disjoint compact intervals $I_1,I_2,\\dots ,I_r$ such that $K\\subset I_1 \\cup \\cdots \\cup I_r$ and the boundary of each $I_j$ is contained in $K$ ; ii) there is a $C^k$ expanding map $\\psi $ defined in a neighbourhood of $I_1\\cup I_2\\cup \\cdots \\cup I_r$ such that $\\psi (I_j)$ is the convex hull of a finite union of some intervals $I_s$ satisfying: ii.1) for each $j$ , $1\\le j\\le r$ , and $n$ sufficiently big, $\\psi ^n(K\\cap I_j)=K$ ; ii.2) $K=\\bigcap \\limits _{n\\in \\mathbb {N}} \\psi ^{-n}(I_1\\cup I_2\\cup \\cdots \\cup I_r)$ .",
"We say that $\\lbrace I_1,I_2,\\dots ,I_r\\rbrace $ is a Markov partition for $K$ and that $K$ is defined by $\\psi $ .",
"Let $K$ be regular Cantor sets of class $C^2$ defined by the expansive function $\\psi $ .",
"It is a general fact, due originally to Poincaré, that, given a periodic point $p$ of period $r$ of $\\psi $ , there is a $C^2$ diffeomorphism $h$ of the support interval $I$ of $K$ such that $\\tilde{\\psi }=h^{-1}\\circ \\psi ^r\\circ h$ is affine in $h^{-1}(J)$ , where $J$ is the connected component of the domain of $\\psi ^r$ which contains $p$ .",
"We say that $K$ is non essentially affine if ${\\tilde{\\psi }}^{^{\\prime \\prime }}(x)\\ne 0$ for some $x\\in h^{-1}(K)$ .",
"In [Mo], we use the Scale Recurrence Lemma of [MY] in order to prove the following Theorem.",
"If $K$ and $K^{\\prime }$ are regular Cantor sets of class $C^2$ and $K$ is non essentially affine, then $HD(K+K^{\\prime })=\\min \\lbrace HD(K)+HD(K^{\\prime }), 1\\rbrace $.",
"This result will be a central tool in the proof of Theorem 1."
],
[
"Regular Cantor sets defined by the Gauss map",
"The Gauss map is the map $g\\colon (0,1]\\rightarrow [0,1]$ given by $g(x)=\\lbrace \\frac{1}{x}\\rbrace =\\frac{1}{x}-\\lfloor \\frac{1}{x}\\rfloor , \\,\\forall \\, x\\in (0,1].$ It acts as a shift on continued fractions: if $a_n\\in {\\mathbb {N}}^*, \\forall n\\ge 1$ then $g([0; a_1, a_2, a_3, \\dots ]=[0; a_2, a_3, a_4,\\dots ]$ .",
"Regular Cantor sets defined by the Gauss map (or iterates of it) restricted to some finite union of intervals are closely related to continued fractions with bounded partial quotients.",
"We will often consider such regular Cantor sets associated to complete shifts.",
"A complete shift is associated to finite sets of finite sequences of positive integers in the following way: given a finite set $B=\\lbrace {\\beta }_1,{\\beta }_2,\\dots ,{\\beta }_m\\rbrace $ , $m\\ge 2$ , where ${\\beta }_j\\in ({\\mathbb {N}}^*)^{r_j}$ , $r_j\\in {\\mathbb {N}}^*$ , $1\\le j\\le m$ and ${\\beta }_i$ does not begin by ${\\beta }_j$ for $i \\ne j$ , the complete shift associated to $B$ is the set $\\Sigma (B)\\subset ({\\mathbb {N}}^*)^{{\\mathbb {N}}}$ of the infinite sequences obtained by concatenations of elements of $B$ : $\\Sigma (B)=\\lbrace ({\\alpha }_0,{\\alpha }_1,{\\alpha }_2,\\dots ) \\, \\mid \\, {\\alpha }_j\\in B,\\,\\, \\forall \\, j\\in {\\mathbb {N}}\\rbrace .$ Here (and in the rest of the paper), we use the following notation for concatenations of finite sequences: if ${\\alpha }_j=(a_j^{(1)},a_j^{(2)},\\dots ,{\\alpha }_j^{(m_j)})$ then $({\\alpha }_0,{\\alpha }_1,{\\alpha }_2,\\dots )$ means the sequence $(a_0^{(1)},a_0^{(2)},\\dots ,{\\alpha }_0^{(m_0)},a_1^{(1)},a_1^{(2)},\\dots ,{\\alpha }_1^{(m_1)},a_2^{(1)},a_2^{(2)},\\dots ,{\\alpha }_2^{(m_2)},\\dots )$ .",
"In some cases, when there is no ambiguity, we will write ${\\alpha }_0 {\\alpha }_1 {\\alpha }_2\\cdots $ and also ${\\alpha }_0^N$ to represent the concatenation of $N$ copies of ${\\alpha }_0$ .",
"In some cases some of the ${\\alpha }_j$ are finite sequences and some cases are single numbers, which are viewed as one-element sequences.",
"Associated to $\\Sigma (B)$ is the Cantor set $K(B)\\subset [0,1]$ of the real numbers whose continued fractions are of the form $[0;{\\gamma }_1,{\\gamma }_2,{\\gamma }_3,\\dots ]$ , where ${\\gamma }_j\\in B$ , $\\forall \\, j\\ge 1$ .",
"This is a regular Cantor set.",
"Indeed, if $a_j$ and $b_j$ are respectively the smallest and largest elements of $K(B)$ whose continued fractions begin by $[0; {\\beta }_j]$ , for $1\\le j\\le m$ , and $I_j=[a_j,b_j]$ , then $K(B)$ is the regular Cantor set defined by the map $\\psi $ with domain $\\bigcup \\limits _{j=1}^m I_j$ given by $\\psi |_{I_j}=g^{r_j}$ , $1\\le j\\le m$ .",
"We have the following Proposition 1.",
"The Cantor sets $K(B)$ defined by the Gauss map associated to complete shifts are non essentially affine.",
"Proof: Let $B=\\lbrace {\\beta }_1,{\\beta }_2,\\dots ,{\\beta }_m\\rbrace $ , ${\\beta }_j=(b_1^{(j)},b_2^{(j)},\\dots ,b_{r_j}^{(j)})\\in ({\\mathbb {N}}^*)^{r_j}$ , $1\\le j\\le m$ .",
"For each $j\\le m$ , let $x_j=[0; {\\beta }_j,{\\beta }_j,{\\beta }_j,\\dots ]\\in I_j$ be the fixed point of $\\psi |_{I_j}=g^{r_j}$ .",
"Notice that, since ${\\beta }_i$ does not begin by ${\\beta }_j$ for $i \\ne j$ , the $x_j, 1\\le j\\le m$ are all distinct.",
"Moreover, according to the classical theory of continued fractions, if $p_k^{(j)}/q_k^{(j)}:=[0;b_1^{(j)},b_2^{(j)},\\dots ,b_k^{(j)}]$ , for $1\\le j\\le m$ , $1\\le k\\le r_j$ , we have $I_j\\subset \\lbrace [0;{\\beta }_j,{\\alpha }],\\,\\, {\\alpha }\\ge 1\\rbrace $ and $\\psi |_{I_j}(x)$ is given by $\\psi |_{I_j}(x)=\\frac{q_{r_j}^{(j)}x-p_{r_j}^{(j)}}{-q_{r_j-1}^{(j)}x+p_{r_j-1}^{(j)}}$ (see the appendix); so $x_j$ is the positive root of $q_{r_j-1}^{(j)}x^2+(q_{r_j}^{(j)}-p_{r_j-1}^{(j)})x-p_{r_j}^{(j)}$ (since $x_j$ is the fixed point of $\\psi |_{I_j}$ ).",
"For each $j\\le m$ , since $\\psi |_{I_j}$ is a Möbius function with a hyperbolic fixed point $x_j$ , there is a Möbius function ${\\alpha }_j(x)=\\frac{a_jx+b_j}{c_jx+d_j}$ with ${\\alpha }_j(x_j)=x_j$ , ${\\alpha }_j^{\\prime }(x_j)=1$ such that ${\\alpha }_j\\circ (\\psi |_{I_j})\\circ {\\alpha }_j^{-1}$ is an affine map.",
"If we show that the Möbius functions ${\\alpha }_1\\circ (\\psi |_{I_2})\\circ {\\alpha }_1^{-1}$ is not affine then we are done, since the second derivative of a non-affine Möbius function never vanishes.",
"Suppose by contradiction that ${\\alpha }_1\\circ (\\psi |_{I_2})\\circ {\\alpha }_1^{-1}$ is affine.",
"Since ${\\alpha }_1\\circ (\\psi |_{I_1})\\circ {\\alpha }_1^{-1}$ is also affine these two functions have a common fixed point at $\\infty $ , so ${\\alpha }_1^{-1}(\\infty )=-d_1/c_1$ is a common fixed point of $\\psi |_{I_2}$ and $\\psi |_{I_1}$ , which implies that ${\\alpha }_1^{-1}(\\infty )$ is a common root of $q_{r_1-1}^{(1)}x^2+(q_{r_1}^{(1)}-p_{r_1-1}^{(1)})x-p_{r_1}^{(1)}$ and $q_{r_2-1}^{(2)}x^2+(q_{r_2}^{(2)}-p_{r_2-1}^{(2)})x-p_{r_2}^{(2)}$ .",
"Since these polynomials of ${\\mathbb {Q}}[x]$ are irreducible (indeed their roots $x_1$ and $x_2$ are irrational because their continued fractions expansions are infinite), they must be associates in ${\\mathbb {Q}}[x]$ , and so their remaining roots $x_1$ and $x_2$ must coincide, which is a contradiction.",
"$\\Box $ Corollary 1.",
"$HD(K(B)+K(B^{\\prime }))=\\min \\lbrace 1,HD(K(B))+HD(K(B^{\\prime }))\\rbrace $ , for every sets $B$ , $B^{\\prime }$ of finite sequences of positive integers.",
"Definition 2: If ${\\beta }=(b_1,b_2,\\dots ,b_{n-1},b_n)$ , then ${\\beta }^t:=(b_n,b_{n-1},\\dots ,b_2,b_1)$ .",
"Given a set of finite sequences $B$ , we define $B^t:=\\lbrace {\\beta }^t, {\\beta }\\in B\\rbrace $ .",
"Proposition 2.",
"$HD(K(B))=HD(K(B^t))$ , for any finite set $B$ of finite sequences.",
"Proof: This follows from $q_n({\\beta })=q_n({\\beta }^t)$ , $\\forall {\\beta }$ (see the appendix of [CF] on properties of continuants), and from the fact that, if $\\psi |_{I_j(x)}=\\frac{q_{n}^{(j)}x-p_{n}^{(j)}}{-q_{n-1}^{(j)}x+p_{n-1}^{(j)}}$ , then $\\psi ^{\\prime }|_{I_j(x)}=\\frac{-(p_{n}^{(j)}q_{n-1}^{(j)}-p_{n-1}^{(j)}q_{n}^{(j)})}{(-q_{n-1}^{(j)}x+p_{n-1}^{(j)})^2}=\\frac{(-1)^n}{(-q_{n-1}^{(j)}x+p_{n-1}^{(j)})^2}$ satisfies $(q_{n}^{(j)})^2\\le |\\psi ^{\\prime }|_{I_j(x)}|\\le 4(q_{n}^{(j)})^2$ , since $\\frac{1}{2q_{n}^{(j)}}\\le \\frac{1}{q_{n}^{(j)}+q_{n-1}^{(j)}} \\le |q_{n-1}^{(j)}x-p_{n-1}^{(j)}|\\le \\frac{1}{q_{n}^{(j)}}.$ $\\Box $ Corollary 2.",
"$HD(K(B)+K(B^t))=\\min \\lbrace 1,2 \\cdot HD(K(B))\\rbrace $ , for every set $B$ of finite sequences of positive integers."
],
[
"Fractal dimensions of the spectra",
"We recall that the Lagrange spectrum is given by $L=\\lbrace \\ell (\\underline{{\\theta }}),\\,\\, \\underline{{\\theta }}\\in \\Sigma \\rbrace $ , where $\\Sigma =({\\mathbb {N}}^*)^{{\\mathbb {Z}}}$ and, for $\\underline{{\\theta }}=(a_n)_{n\\in {\\mathbb {Z}}}\\in \\Sigma $ , $\\ell (\\underline{{\\theta }}):=\\limsup _{n\\rightarrow +\\infty }({\\alpha }_n+{\\beta }_n)$ , where ${\\alpha }_n$ and ${\\beta }_n$ are defined as the continued fractions ${\\alpha }_n:=[a_n; a_{n+1},a_{n+2},\\dots ]$ and ${\\beta }_n:=[0;a_{n-1},a_{n-2},\\dots ]$ , while the Markov spectrum is given by $M=\\lbrace m(\\underline{{\\theta }}),\\underline{{\\theta }}\\in \\Sigma \\rbrace $ , where $m(\\underline{{\\theta }})=\\sup \\lbrace {\\alpha }_n+{\\beta }_n,\\,\\, n\\in {\\mathbb {Z}}\\rbrace $ .",
"Given a finite sequence ${\\alpha }=(a_1,a_2,\\dots ,a_n)\\in ({\\mathbb {N}}^*)^n$ , we define its size by $s({\\alpha }):=|I({\\alpha })|$ , where $I({\\alpha })$ is the interval $\\lbrace x\\in [0,1] \\mid x=[0; a_1,a_2,\\dots ,a_n,{\\alpha }_{n+1}]$ , ${\\alpha }_{n+1}\\ge 1\\rbrace $ .",
"If we take $p_0=0$ , $q_0=1$ , $p_1=1$ , $q_1=a_1$ and, for $k\\ge 0$ , $p_{k+2}=a_{k+2}p_{k+1}+p_k$ and $q_{k+2}=a_{k+2} q_{k+1}+q_k$ , then $I({\\alpha })$ is the interval with extremities $[0;a_1,a_2,\\dots ,a_n]=p_n/q_n$ and $[0;a_1,a_2,\\dots ,a_{n-1},a_n+1]=\\frac{p_n+p_{n-1}}{q_n+q_{n-1}}$ and so $s({\\alpha })=\\left| \\frac{p_n}{q_n}-\\frac{p_n+p_{n-1}}{q_n+q_{n-1}} \\right| = \\frac{1}{q_n(q_n+q_{n-1})},$ since $p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}$ .",
"We define $r({\\alpha })=\\lfloor \\log s({\\alpha })^{-1} \\rfloor $ which controls the order of magnitude of the size of $I({\\alpha })$ .",
"We also define, for $r \\in {\\mathbb {N}}, P_r= \\lbrace {\\alpha }=(a_1,a_2,\\dots ,a_n) \\mid r({\\alpha }) \\ge r$ and $r((a_1,a_2,\\dots ,a_{n-1}))<r\\rbrace $ .",
"Write $\\Sigma =\\Sigma ^- \\times \\Sigma ^+$ , where $\\Sigma ^-=({\\mathbb {N}}^*)^{{\\mathbb {Z}}_-}$ and $\\Sigma ^+=({\\mathbb {N}}^*)^{{\\mathbb {N}}}$ , and let $\\sigma \\colon \\Sigma \\rightarrow \\Sigma $ be the shift given by $\\sigma ((a_n)_{n\\in {\\mathbb {Z}}})=(a_{n+1})_{n\\in {\\mathbb {Z}}}$ .",
"We will work with a one-parameter family of subshifts of $\\Sigma $ given by $\\Sigma _t=\\lbrace \\underline{{\\theta }}\\in \\Sigma \\mid m(\\underline{{\\theta }})\\le t\\rbrace $ for $t\\in {\\mathbb {R}}$ (in fact we will take $t\\ge 3$ ).",
"Note that $\\Sigma _t$ is invariant by transposition and by $\\sigma $ .",
"Note that if $\\underline{{\\theta }}=(a_n)_{n\\in {\\mathbb {Z}}}\\in \\Sigma $ then $\\alpha _n+\\beta _n>\\alpha _n\\ge a_n$ for every $n$ , and so $m(\\underline{{\\theta }})>\\sup \\lbrace a_n,n\\in {\\mathbb {Z}}\\rbrace $ .",
"So, if $m(\\underline{{\\theta }})\\le t$ we have $a_n\\le \\lfloor t \\rfloor $ , $\\forall \\,\\, n\\in {\\mathbb {N}}$ .",
"Given $t\\in [3,+\\infty )$ and $r\\in {\\mathbb {N}}$ , let $T:=\\lfloor t \\rfloor $ and $C(t,r)$ be the set $\\lbrace {\\alpha }=(a_1,a_2,\\dots ,a_n) \\in P_r \\mid K_t\\cap I({\\alpha })\\ne \\emptyset \\rbrace $ .",
"Here $K_t:=\\lbrace [0;{\\gamma }]| {\\gamma }\\in \\pi _+(\\Sigma _t)\\rbrace $ , where $\\pi _+\\colon \\Sigma \\rightarrow \\Sigma ^+$ is the projection associated to the decomposition $\\Sigma =\\Sigma ^-\\times \\Sigma ^+$ .",
"Since $\\Sigma _t$ is invariant by transposition and by $\\sigma $ , $K_t$ is invariant by the Gauss map $g$ and $M\\cap (-\\infty , t) \\subset ({\\mathbb {N}}^* \\cap [1, T])+K_t+K_t$ .",
"We define $N(t,r):=|C(t,r)|$ , where $|\\cdot |$ denotes cardinality.",
"Notice that if $r\\le s$ then $N(t,r)\\le N(t,s)$ and, if $t\\le \\tilde{t}$ , then $N(t,r)\\le N(\\tilde{t},r)$ .",
"For any finite sequences ${\\alpha },{\\beta }$ and any positive integers $k_1, k_2\\le T$ we have $r({\\alpha }{\\beta }k_1 k_2)\\ge r({\\alpha })+r({\\beta })$ (see the appendix), so if $C(t,r)=\\lbrace {\\alpha }_1,{\\alpha }_2,\\dots ,{\\alpha }_u\\rbrace $ and $C(t,s)=\\lbrace {\\beta }_1,{\\beta }_2,\\dots ,{\\beta }_v\\rbrace $ , we may cover $K_t$ by the $T^2uv=T^2N(t,r)N(t,s)$ intervals $I({\\alpha }_i{\\beta }_j k_1 k_2)$ , $1\\le i \\le u$ , $1\\le j\\le v$ , $1\\le k_1, k_2\\le T$ , which satisfy $r({\\alpha }_i{\\beta }_j k_1 k_2)\\ge r+s$ , $\\forall \\,\\, i,j,k$ .",
"Replacing, if necessary, some of these intervals by larger intervals $I({\\gamma })$ in $P_{r+s}$ , we conclude that $N(t,r+s)\\le T^2N(t,r)N(t,s)$ and so $\\log (T^2N(t,r+s))\\le \\log (T^2N(t,r))+\\log (T^2N(t,s)),\\quad \\forall \\,\\, r,s.$ This implies that $\\lim _{m\\rightarrow \\infty }\\frac{1}{m} \\log (T^2N(t,m))=\\inf _{m\\in {\\mathbb {N}}^*}\\frac{1}{m} \\log (T^2N(t,m))=\\lim _{m\\rightarrow \\infty }\\frac{1}{m}\\log (N(t,m))$ exists.",
"We will call this limit $D(t)$ (which coincides with the (upper) box dimension of $K_t$ , as follows easily from its definition).",
"Notice that $D(t)$ is a non-decreasing function.",
"We will see in the proof of Theorem 1 that $D(t)$ is continuous and that $HD(k^{-1}(-\\infty ,t))=D(t)$ .",
"Lemma 1.",
"$D(t)$ is right-continuous: given $t_0\\in [3,+\\infty )$ and $\\eta >0$ there is $\\delta >0$ such that for $t_0\\le t\\le t+\\delta $ we have $D(t_0)\\le D(t) \\le D(t+\\delta )<D(t_0)+\\eta $ .",
"Proof: If for every $t > t_0$ , $r$ large, $\\frac{\\log N(t,r)}{r} \\ge D(t_0)+\\eta $ we would have $D(t_0) \\ge D(t_0) + \\eta $ , contradiction (indeed $C(t,r) \\subset C(s,r)$ for $t \\le s$ , and, by compacity, $C(t_0,r) ={\\bigcap }_{t>t_0}\\,C(t,r)$ ).",
"$\\Box $ Lemma 2.",
"Given $t\\in (3,+\\infty )$ and $\\eta \\in (0,1)$ there is $\\delta >0$ and a Cantor set $K(B)$ defined by the Gauss map associated to a complete shift $\\Sigma (B)\\subset \\Sigma $ such that $\\Sigma (B)\\subset \\Sigma _{t-\\delta }$ and $HD(K(B))>(1-\\eta )D(t)$ .",
"Since the proof of this Lemma is somewhat technical, we will postpone it to Section 6.",
"Lemma 3.",
"Given a complete shift $\\Sigma (X)\\subset \\Sigma $ (where X is a finite set of finite sequences of positive integers), we have $HD(\\ell (\\Sigma (X)))=HD(m(\\Sigma (X)))=\\overline{\\dim }(\\ell (\\Sigma (X)))=\\overline{\\dim }(m(\\Sigma (X)))=\\min \\lbrace 2 \\cdot HD(K(X)), 1 \\rbrace .$ Proof: Let $T$ be the largest element of a sequence in $X$ .",
"First of all we clearly have $\\ell (\\Sigma (X)) \\subset m(\\Sigma (X))\\subset \\bigcup \\limits _{1\\le a\\le T\\atop 1\\le i, j\\le R} (a+g^i(K(X))+g^j(K(X))),\\;\\;\\text{where $R$ is the length of}$ the largest word of $X$ , so $HD(\\ell (\\Sigma (X)) \\le HD(m(\\Sigma (X)) \\le \\min \\lbrace 2 \\cdot HD(K(X)), 1 \\rbrace $ .",
"Let ${\\varepsilon }>0$ be given.",
"We will show that there are regular Cantor sets $K, K^{\\prime }$ defined by iterates of the Gauss map with $HD(K),HD(K^{\\prime })>HD(K(X))-{\\varepsilon }$ such that $K+K^{\\prime } \\subset \\ell (\\Sigma (X)) \\subset m(\\Sigma (X))$ .",
"Since, by the dimension formula stated in section 2, $HD(K+K^{\\prime })=\\min \\lbrace HD(K)+HD(K^{\\prime }), 1\\rbrace >\\min \\lbrace 2 \\cdot HD(K(X)), 1 \\rbrace -2{\\varepsilon }$ and ${\\varepsilon }>0$ is arbitrary, the result will follow.",
"Given a positive integer $n$ , let $X^n=\\lbrace ({\\gamma }_1,{\\gamma }_2,\\dots ,{\\gamma }_n)|{\\gamma }_j \\in X, \\forall j \\le n \\rbrace $ .",
"We have $\\Sigma (X^n)=\\Sigma (X)$ and $K(X^n)=K(X)$ .",
"Replacing $X$ by $X^n$ for some $n$ large, we may assume without loss of generality that for any $A \\subset X$ (resp.",
"$A^t \\subset X^t$ ) with $|A|\\le 2$ (resp.",
"$|A^t|\\le 2$ ), we have $HD(K(X\\setminus A))>HD(K(X))-{\\varepsilon }$ (resp.",
"$HD(K(X^t\\setminus A^t))>HD(K(X^t))-{\\varepsilon }=HD(K(X))-{\\varepsilon }$ ).",
"Order $X$ and $X^t$ in the following way: given ${\\gamma }, \\tilde{{\\gamma }}\\in X \\, (\\text{resp.",
"}{\\gamma }, \\tilde{{\\gamma }}\\in X^t)$ , we say that ${\\gamma }< \\tilde{{\\gamma }}$ if and only if $[0;{\\gamma }]<[0;\\tilde{{\\gamma }}]$ .",
"Suppose that the maximum of $m(\\Sigma (X))$ is attained at $\\tilde{\\underline{\\theta }}= (\\dots ,\\tilde{{\\gamma }}_{-1},\\tilde{{\\gamma }}_0,\\tilde{{\\gamma }}_1,\\dots ),\\tilde{{\\gamma }}_i \\in X, \\forall i \\in {\\mathbb {Z}}$ , in a position belonging to the sequence $\\tilde{{\\gamma }}_0$ .",
"Let $X^* = X\\backslash \\lbrace \\min X, \\max X\\rbrace $ , $(X^t)^* = X^t\\backslash \\lbrace \\min X^t, \\max X^t\\rbrace $ .",
"Essentially, $K(X^*)$ and $K((X^t)^*)$ will be the required Cantor sets, but first we have to control the positions where the $\\limsup $ is attained (the idea is somewhat similar to the proof that Hall's theorem ([H]) on sums of continued fractions with coefficients bounded by 4 implies that the Lagrange spectrum contains $[6,+\\infty )$ ) and which words can appear in the beginning of the elements.",
"For each positive integer $m$ , let $C^m$ be the set of sequences $(\\dots , {\\gamma }_{-m-2}, {\\gamma }_{-m-1}, \\tilde{{\\gamma }}_{-m}, \\tilde{{\\gamma }}_{-m+1}, \\dots , \\tilde{{\\gamma }}_{-1}, \\tilde{{\\gamma }}_0, \\tilde{{\\gamma }}_1, \\dots , \\tilde{{\\gamma }}_{m-1}, \\tilde{{\\gamma }}_m, {\\gamma }_{m+1}, {\\gamma }_{m+2}, \\dots )$ where ${\\gamma }_k \\in X^*$ for $k \\ge m+1$ , ${\\gamma }_k^t \\in (X^t)^*$ for $k \\le -m-1$ .",
"Then, for $m$ large enough, there is $\\eta >0$ such that for each $\\underline{\\theta }\\in C^m$ , $\\text{sup}({\\alpha }_n+{\\beta }_n) =m(\\underline{\\theta })$ is attained only for values of $n$ corresponding to the piece $\\tau =(\\tilde{{\\gamma }}_{-m}, \\tilde{{\\gamma }}_{-m+1}, \\dots , \\tilde{{\\gamma }}_{-1}, \\tilde{{\\gamma }}_0, \\tilde{{\\gamma }}_1, \\dots , \\tilde{{\\gamma }}_{m-1}, \\tilde{{\\gamma }}_m)$ of $\\underline{\\theta }$ and, if $n$ does not correspond to the piece $\\tau $ , then ${\\alpha }_n+{\\beta }_n<m(\\underline{\\theta })-\\eta $ .",
"Indeed, if it is not the case, we may assume without loss of generality that there is a sequence $(m_k)$ tending to $+\\infty $ and, for each $k$ , $\\underline{\\theta }^{(k)} \\in C^{m_k}$ and $n_k$ corresponding to a piece ${\\gamma }_{r(k)}$ , with $r(k)>m_k$ such that ${\\alpha }_{n_k}(\\underline{\\theta }^{(k)})+{\\beta }_{n_k}(\\underline{\\theta }^{(k)})>m(\\underline{\\theta }^{(k)})-1/k$ .",
"Since $\\underline{\\theta }^{(k)}$ converges to $\\tilde{\\underline{\\theta }}$ , $m(\\underline{\\theta }^{(k)})$ converges to $m(\\tilde{\\underline{\\theta }})$ and, by compacity, if $N_k$ denotes the size of the sequence $\\tilde{{\\gamma }}_0, \\tilde{{\\gamma }}_1, \\dots , \\tilde{{\\gamma }}_{m_k-1}, \\tilde{{\\gamma }}_{m_k}, {\\gamma }_{m_k+1}, {\\gamma }_{m_k+2}, \\dots , {\\gamma }_{r(k)-1}$ , $(\\sigma ^{N_k}(\\underline{\\theta }^{(k)}))$ has a subsequence which converges to some $\\hat{\\underline{\\theta }}= (\\dots ,\\hat{{\\gamma }}_{-1},\\hat{{\\gamma }}_0,\\hat{{\\gamma }}_1,\\dots ) \\in \\Sigma (X)$ , with $\\hat{{\\gamma }}_i \\in X^*, \\forall i \\ge 0$ , such that $\\text{sup}({\\alpha }_n+{\\beta }_n)=m(\\hat{\\underline{\\theta }})=m(\\tilde{\\underline{\\theta }})$ is attained for some $n$ corresponding to the piece $\\hat{{\\gamma }}_0$ .",
"This is a contradiction, since $m(\\tilde{\\underline{\\theta }})$ is the maximum of $m(\\Sigma (X))$ and, changing $\\hat{{\\gamma }}_1$ by $\\min X$ or $\\max X$ , we strictly increase the value of $m(\\hat{\\underline{\\theta }})$ .",
"Notice that the same argument shows that for any $\\underline{\\theta }\\in C^m$ and $\\underline{\\theta }^* \\in \\Sigma (X^*)$ , we have $m(\\underline{\\theta }^*)<m(\\underline{\\theta })-\\eta $ (for $m$ large enough).",
"Now, fixing $m$ with the above properties and ${\\gamma }^{(0)} \\in X$ such that $({\\gamma }^{(0)})^t \\in (X^t)^*$ , we may associate to each $x=[0;{\\gamma }_1(x),{\\gamma }_2(x),{\\gamma }_3(x),\\dots ] \\in K(X^*)$ an element $\\underline{\\Theta }(x) \\in C^m$ given by $\\underline{\\Theta }(x)=(\\dots , {\\gamma }^{(0)}, {\\gamma }^{(0)}, \\tilde{{\\gamma }}_{-m}, \\tilde{{\\gamma }}_{-m+1}, \\dots , \\tilde{{\\gamma }}_{-1}, \\tilde{{\\gamma }}_0, \\tilde{{\\gamma }}_1, \\dots , \\tilde{{\\gamma }}_{m-1}, \\tilde{{\\gamma }}_m, {\\gamma }_1(x), {\\gamma }_2(x), \\dots )=$ $=(\\dots , {\\gamma }^{(0)}, {\\gamma }^{(0)}, \\tau , {\\gamma }_1(x), {\\gamma }_2(x), \\dots ).$ For each position $n$ corresponding to the piece $\\tau $ of $\\underline{\\Theta }(x)$ , we write $g_n(x)={\\alpha }_n(\\underline{\\Theta }(x))+{\\beta }_n(\\underline{\\Theta }(x))$ ; in fact ${\\beta }_n(\\underline{\\Theta }(x))$ does not depend on $x$ , so, for distinct values of $n$ , the functions $g_n$ are distinct rational maps of $x$ .",
"This implies that, except for finitely many values of $x$ , the values of $g_n(x)$ for these values of $n$ are all distinct.",
"Let $x^{\\#}=[0;{\\gamma }_1^{\\#},{\\gamma }_2^{\\#},{\\gamma }_3^{\\#},\\dots ]$ be one of these values.",
"Since $\\text{sup}({\\alpha }_n+{\\beta }_n) =m(\\underline{\\Theta }(x^{\\#}))$ is attained for values of $n$ corresponding to the piece $\\tau $ of $\\underline{\\Theta }(x^{\\#})$ , let $n_0$ be the position in $\\tau $ for which $m(\\underline{\\Theta }(x^{\\#}))={\\alpha }_{n_0}(\\underline{\\Theta }(x^{\\#}))+{\\beta }_{n_0}(\\underline{\\Theta }(x^{\\#}))$ .",
"For $N$ large enough, taking $\\tau ^{\\#}=(({\\gamma }^{(0)})^N, \\tau , {\\gamma }_1^{\\#},{\\gamma }_2^{\\#},\\dots , {\\gamma }_N^{\\#})$ , the following holds: if $\\underline{\\theta }=(\\dots , {\\gamma }_{-2}, {\\gamma }_{-1}, \\tau ^{\\#}, {\\gamma }_{1}, {\\gamma }_{2}, \\dots )$ , with ${\\gamma }_k \\in X^*, ({\\gamma }_{-k})^t \\in (X^t)^*, \\forall k \\ge 1$ , writing $\\tau ^{\\#}=({\\overline{a}}_{-N_1},\\dots ,{\\overline{a}}_{-2}, {\\overline{a}}_{-1}, {\\overline{a}}_0, {\\overline{a}}_1, {\\overline{a}}_2, \\dots , {\\overline{a}}_{N_2})$ , where ${\\overline{a}}_0$ is in the position $n_0$ of $\\tau $ , we have $m(\\underline{\\theta })=[{\\overline{a}}_0; {\\overline{a}}_1, {\\overline{a}}_2, \\dots , {\\overline{a}}_{N_2},{\\gamma }_1, {\\gamma }_2, {\\gamma }_3, \\dots ]+[0; {\\overline{a}}_{-1}, {\\overline{a}}_{-2}, \\dots , {\\overline{a}}_{-N_1}, {{\\gamma }_{-1}}^t, {{\\gamma }_{-2}}^t, {{\\gamma }_{-3}}^t, \\dots ]$ .",
"It follows that, defining $K:=\\lbrace [{\\overline{a}}_0; {\\overline{a}}_1, {\\overline{a}}_2, \\dots , {\\overline{a}}_{N_2},{\\gamma }_1, {\\gamma }_2, {\\gamma }_3, \\dots ]| {\\gamma }_j \\in X^*, \\forall j \\ge 1\\rbrace \\text{ and}$ $K^{\\prime }:=\\lbrace [0; {\\overline{a}}_{-1}, {\\overline{a}}_{-2}, \\dots , {\\overline{a}}_{-N_1}, {{\\gamma }^{\\prime }_1}^t, {{\\gamma }^{\\prime }_2}^t, {{\\gamma }^{\\prime }_3}^t, \\dots ]| {{\\gamma }^{\\prime }_j}^t \\in (X^t)^*, \\forall j \\ge 1\\rbrace ,$ we have $K+K^{\\prime } \\subset \\ell (\\Sigma (X))$ .",
"In order to show this, given $x=[{\\overline{a}}_0; {\\overline{a}}_1, {\\overline{a}}_2, \\dots , {\\overline{a}}_{N_2},{\\gamma }_1, {\\gamma }_2, {\\gamma }_3, \\dots ] \\in K$ and $y=[0; {\\overline{a}}_{-1}, {\\overline{a}}_{-2}, \\dots , {\\overline{a}}_{-N_1}, {{\\gamma }^{\\prime }_1}^t, {{\\gamma }^{\\prime }_2}^t, {{\\gamma }^{\\prime }_3}^t, \\dots ]\\in K^{\\prime }$ , and defining, for each positive integer $m$ , $\\tau ^{(m)}=({\\gamma }^{\\prime }_m,{\\gamma }^{\\prime }_{m-1},\\dots ,{\\gamma }^{\\prime }_1, \\tau ^{\\#}, {\\gamma }_1,{\\gamma }_2,\\ldots ,{\\gamma }_m)$ , we have, for $\\underline{\\Theta }^*(x,y)= (\\dots , {\\gamma }^{(0)}, {\\gamma }^{(0)}, \\tau ^{(1)}, \\tau ^{(2)}, \\tau ^{(3)}, \\dots ),\\text{ and }\\hat{\\underline{\\Theta }}(x,y)=(\\dots , {\\gamma }^{\\prime }_3, {\\gamma }^{\\prime }_2, {\\gamma }^{\\prime }_1, \\tau ^{\\#}, {\\gamma }_1, {\\gamma }_2, {\\gamma }_3, \\dots ),$ $\\ell (\\underline{\\Theta }^*(x,y))=m(\\hat{\\underline{\\Theta }}(x,y))=x+y.$ Indeed, there is a sequence of positions $(s_k)$ with $s_k$ corresponding to the piece $\\tau ^{(k)}$ of $\\underline{\\Theta }^*(x,y)$ such that $\\sigma ^{s_k}(\\underline{\\Theta }^*(x,y))$ converges to $\\sigma ^{n_0}(\\hat{\\underline{\\Theta }}(x,y))$ , so ${\\alpha }_{s_k}(\\underline{\\Theta }^*(x,y))+{\\beta }_{s_k}(\\underline{\\Theta }^*(x,y))$ converges to ${\\alpha }_{n_0}(\\hat{\\underline{\\Theta }}(x,y))+{\\beta }_{n_0}(\\hat{\\underline{\\Theta }}(x,y))=m(\\hat{\\underline{\\Theta }}(x,y))=x+y$ , and, in particular, $\\ell (\\underline{\\Theta }^*(x,y)) \\ge m(\\hat{\\underline{\\Theta }}(x,y))=x+y$ .",
"On the other hand, there are increasing sequences $(m_k)$ and $(r_k)$ such that the position $m_k$ corresponds to the piece $\\tau ^{(r_k)}$ in $\\underline{\\Theta }^*(x,y)$ and ${\\alpha }_{m_k}(\\underline{\\Theta }^*(x,y))+{\\beta }_{m_k}(\\underline{\\Theta }^*(x,y))$ converges to $\\ell (\\underline{\\Theta }^*(x,y))$ .",
"Now, if $|m_k-s_{r_k}|$ has a bounded subsequence, then there is $b \\in {\\mathbb {Z}}$ such that $\\sigma ^{m_k}(\\underline{\\Theta }^*(x,y))$ has a subsequence converging to $\\sigma ^b(\\hat{\\underline{\\Theta }}(x,y))$ , so $\\ell (\\underline{\\Theta }^*(x,y)) =\\lim ({\\alpha }_{m_k}(\\underline{\\Theta }^*(x,y))+{\\beta }_{m_k}(\\underline{\\Theta }^*(x,y))) \\le m(\\hat{\\underline{\\Theta }}(x,y))=x+y$ .",
"On the other hand, if $|m_k-s_{r_k}|$ is unbounded, there is $c \\in {\\mathbb {Z}}$ and a subsequence of $\\sigma ^{m_k}(\\underline{\\Theta }^*(x,y))$ which converges to $\\sigma ^c(\\underline{\\theta }^*)$ , where $\\underline{\\theta }^*$ is an element of $\\Sigma (X^*)$ , but in this case we would have $\\ell (\\underline{\\Theta }^*(x,y)) \\le m(\\underline{\\theta }^*)<m(\\hat{\\underline{\\Theta }}(x,y))-\\eta $ , which is a contradiction.",
"Finally, notice that $K$ and $K^{\\prime }$ are diffeomorphic respectively to $K(X^*)$ and $K((X^t)^*)$ , so $HD(K)=HD(K(X^*))>HD(K(X))-{\\varepsilon }$ and $HD(K^{\\prime })=HD(K((X^t)^*))>$$HD(K(X^t))-{\\varepsilon }=HD(K(X))-{\\varepsilon }$ .",
"$\\Box $"
],
[
"Proofs of the main results",
"Proof of Theorem 1: Applying Lemma 3 to the complete shift $\\Sigma (B)$ obtained in Lemma 2, we get that, for any $\\eta >0$ , there is $\\delta >0$ such that $\\min \\lbrace 2(1-\\eta )D(t),1\\rbrace \\le \\min \\lbrace 2HD(K(B)),1\\rbrace \\le HD(L\\cap (-\\infty ,t-\\delta ]) \\le HD(L \\cap (-\\infty ,t)) \\le HD(M \\cap (-\\infty ,t)) \\le \\overline{\\dim }(M\\cap (-\\infty ,t))\\le \\min \\lbrace 2 \\cdot HD(K_t),1\\rbrace \\le \\min \\lbrace 2 \\cdot D(t),1\\rbrace $ (since $\\ell (\\Sigma (B))\\subset L\\cap (-\\infty ,t-\\delta ]$ , $L\\cap (-\\infty , t) \\subset M\\cap (-\\infty , t) \\subset ({\\mathbb {N}}^* \\cap [1, T])+K_t+K_t$ and $D(t)$ is the upper box dimension of $K_t$ ), and so, if $d(t):=HD(L\\cap (-\\infty ,t))$ , we have $d(t)= HD(M\\cap (-\\infty ,t))=\\overline{\\dim }(L\\cap (-\\infty ,t))=\\overline{\\dim }(M\\cap (-\\infty ,t))=\\min \\lbrace 2 \\cdot D(t),1\\rbrace $ .",
"In order to conclude the proof of the first assertion of i), it is enough to show that $HD(k^{-1}(-\\infty ,t))=D(t)$ .",
"In the notation of Lemma 3, let $x\\in K, y\\in K^{\\prime }$ .",
"For each $z=[0; {\\alpha }_1, {\\alpha }_2, \\dots ]\\in K(X^*)$ , define $\\underline{{\\lambda }}(z)={\\underline{{\\lambda }}}_{x,y}(z)=({\\alpha }_{1!",
"},\\tau ^{(1)},{\\alpha }_{2!",
"},\\tau ^{(2)},{\\alpha }_3,{\\alpha }_4,{\\alpha }_5,{\\alpha }_{3!",
"},\\tau ^{(3)},{\\alpha }_7,\\dots ,{\\alpha }_{4!",
"},\\tau ^{(4)},\\\\{\\alpha }_{25},{\\alpha }_{26},\\dots ,{\\alpha }_{5!",
"},\\tau ^{(5)},\\;\\dots \\;,{\\alpha }_{r!",
"},\\tau ^{(r)},{\\alpha }_{r!+1},\\dots ),$ and $h(z)=[0;\\underline{{\\lambda }}(z)]$ .",
"We have, as before, $k(h(z))=x+y$ .",
"On the other hand, given any $\\rho >0$ , we have $|z-z^{\\prime }|=O(|h(z)-h(z^{\\prime })|^{1-\\rho })$ for $|z-z^{\\prime }|$ small, so $HD(k^{-1}(x+y)) \\ge HD(K(B^*))>HD(K(B))-{\\varepsilon }$ .",
"As before, we get $HD(k^{-1}(-\\infty ,t)) \\ge HD(k^{-1}(-\\infty ,t-\\delta ])$ $\\ge HD(k^{-1}(x+y)) > HD(K(B))-{\\varepsilon }> (1-\\eta )D(t)-{\\varepsilon }.$ Since $\\eta $ and ${\\varepsilon }$ are arbitrary, $HD(k^{-1}(-\\infty ,t)) \\ge D(t)$ .",
"For the reverse inequality, let $w \\in k^{-1}(-\\infty ,t)$ .",
"We have $\\limsup _{n\\rightarrow \\infty }({\\alpha }_n(w)+{\\beta }_n(w))=k(w)<t$ , so there is $n_0 \\in {\\mathbb {N}}$ such that $n \\ge n_0 \\Rightarrow {\\alpha }_n(w)+{\\beta }_n(w)<t$ .",
"This implies that $k^{-1}(-\\infty ,t) \\subset {\\bigcup }_{n \\in {\\mathbb {N}}}(g^{-n}(K_t))$ , where $g$ is the Gauss map, so $HD(k^{-1}(-\\infty ,t)) \\le D(t)$ .",
"Thus we have $HD(k^{-1}(-\\infty ,t))=D(t)$ .",
"Recall that, by Lemma 1, $D(t)$ is a right-continuous function.",
"Thus we have $D(t)=HD(k^{-1}(-\\infty ,t)) &\\le & HD(k^{-1}(-\\infty ,t]) \\le \\lim \\limits _{s\\rightarrow t+} HD(k^{-1}(-\\infty ,s)) \\\\&=& \\lim \\limits _{s\\rightarrow t+} D(s) = D(t).$ Then $HD(k^{-1}(-\\infty ,t])=HD(k^{-1}(-\\infty ,t))=D(t)$ , and we conclude that $d(t) =\\min \\lbrace 2HD(k^{-1}(-\\infty ,t)),1\\rbrace = \\min \\lbrace 2HD(k^{-1}(-\\infty ,t]),1\\rbrace $ .",
"Finally, $D(t)$ is left-continous (and so is continuous), since, by Lemma 2, given $t\\in [3,+\\infty )$ and $\\eta \\in (0,1)$ , there is $\\delta >0$ such that $D(t-\\delta )\\ge HD(K(B))>(1-\\eta )D(t)$ , so $\\lim \\limits _{s\\rightarrow t-} D(s) = D(t)$ .",
"In order to conclude, notice that, for each positive integer $m$ , $\\Sigma (\\lbrace 21^{2m}2,21^{2m+2}2\\rbrace ) \\subset \\Sigma _{3+2^{-m}}$ (notice that $[2;1, 1, 1,\\ldots ]+[0;2, 1, 1, 1,\\ldots ]=3$ ), so $D(3+{\\varepsilon })>0$ for every ${\\varepsilon }>0$ and, since $\\Sigma _{\\sqrt{12}}=\\lbrace 1,2\\rbrace ^{{\\mathbb {Z}}}$ , $D(\\sqrt{12})=HD(K_{\\sqrt{12}})=HD(K(\\lbrace 1,2\\rbrace ))=HD(C_2)=0,53128\\ldots >1/2$ , so, since $D(t)$ is continuous, there is $\\delta >0$ such that $D(\\sqrt{12}-\\delta )>1/2$ , and thus we have $d(\\sqrt{12}-\\delta )=\\min \\lbrace 2 \\cdot D(\\sqrt{12}-\\delta ),1\\rbrace =1$ .",
"$\\Box $ Remark: It follows from the above proof and from the general estimates of fractal dimensions of regular Cantor sets of Chapter 4 of [PT] that there is a constant $C>0$ such that, for each positive integer $m$ , $D(3+2^{-m})\\ge HD(K(\\lbrace 21^{2m}2,21^{2m+2}2\\rbrace ))>C/m$ .",
"This gives another proof of the fact that the functions $D(t)$ and $d(t)$ are not Hölder continuous.",
"Proof of Theorem 2: Given $m \\ge 2$ , let $C_m=\\lbrace {\\alpha }=[0;a_1,a_2,a_3,\\ldots ] \\in [0,1]| a_k \\le m, \\forall k \\ge 1\\rbrace $ .",
"M. Hall proved in [H] that $C_4+C_4=\\lbrace {\\alpha }+{\\beta }| {\\alpha },{\\beta }\\in C_4\\rbrace =[\\sqrt{2}-1,4(\\sqrt{2}-1)]$ .",
"On the other hand, we have $\\lim _{m\\rightarrow \\infty }HD(C_m)=1$ .",
"In fact, Jarník proved in [J] that $HD(C_m)>1-\\frac{1}{m \\cdot \\log 2}, \\forall m>8.$ Let now $t \\ge 7$ be given.",
"Let $m=\\lfloor t \\rfloor -3$ .",
"There are an integer $n \\in \\lbrace m+2,m+3\\rbrace $ and ${\\alpha }=[0;a_1,a_2,a_3,\\ldots ], {\\beta }=[0;b_1,b_2,b_3,\\ldots ] \\in C_4$ such that $t=n+{\\alpha }+{\\beta }$ .",
"For each $r\\ge 1$ , let $\\tilde{\\tau }^{(r)}$ and $\\hat{\\tau }^{(r)}$ be respectively the sequences $(m+1,b_{2r-1},b_{2r-2},\\ldots ,b_2,b_1,n,a_1,a_2,\\ldots ,a_{2r-2},a_{2r-1},m+1)$ and $(m+1,b_{2r},b_{2r-1},\\ldots ,b_2,b_1,n,a_1,a_2,\\ldots ,a_{2r-1},a_{2r},m+1)$ .",
"Consider now the maps $\\tilde{h}, \\hat{h}: C_m \\rightarrow [0,1]$ given by $\\tilde{h}(z) = \\tilde{h}([0;c_1,c_2,c_3,\\ldots ] )=[0;c_{1!",
"},\\tilde{\\tau }^{(1)},c_{2!",
"},\\tilde{\\tau }^{(2)}, c_3,c_4,c_5,c_{3!",
"},\\tilde{\\tau }^{(3)}, \\\\c_7, c_8,\\dots ,c_{4!",
"},\\tilde{\\tau }^{(4)}, c_{25}, \\dots ,c_{5!",
"}, \\tilde{\\tau }^{(5)},\\: \\dots \\:, c_{r!",
"},\\tilde{\\tau }^{(r)},c_{r!+1},\\dots ].$ $\\hat{h}(z) = \\hat{h}([0;c_1,c_2,c_3,\\ldots ] )=[0;c_{1!",
"},\\hat{\\tau }^{(1)},c_{2!",
"},\\hat{\\tau }^{(2)}, c_3,c_4,c_5,c_{3!",
"},\\hat{\\tau }^{(3)}, \\\\c_7, c_8,\\dots ,c_{4!",
"},\\hat{\\tau }^{(4)}, c_{25}, \\dots ,c_{5!",
"}, \\hat{\\tau }^{(5)},\\: \\dots \\:, c_{r!",
"},\\hat{\\tau }^{(r)},c_{r!+1},\\dots ].$ It is easy to see that $k(\\tilde{h}(z))=t$ for every $z \\in C_m$ .",
"Moreover, since $[x_0;x_1,x_2,x_3,x_4,\\dots ]$ is increasing in $x_0, x_2, x_4,\\ldots $ and decreasing in $x_1, x_3, x_5, \\ldots $ , $\\tilde{h}(z)\\in \\text{Exact}(t^{-1})$ and $\\hat{h}(z)\\in \\text{Exact}^{\\prime }(t^{-1})$ .",
"On the other hand, given any $\\rho >0$ , we have $|z-z^{\\prime }|=O(|\\tilde{h}(z)-\\tilde{h}(z^{\\prime })|^{1-\\rho })$ and $|z-z^{\\prime }|=O(|\\hat{h}(z)-\\hat{h}(z^{\\prime })|^{1-\\rho })$ for $|z-z^{\\prime }|$ small, so $HD(\\text{Exact}(t^{-1})) \\ge HD(C_m)$ and $HD(\\text{Exact}^{\\prime }(t^{-1})) \\ge HD(C_m)$ .",
"Since $\\lim _{m\\rightarrow \\infty }HD(C_m)=1$ , we are done.",
"$\\Box $ Proof of Theorem 3: Let $x \\in L^{\\prime }$ .",
"Consider a sequence $x_n$ converging to $x$ , $x_n \\in L$ , $x_n \\ne x$ .",
"Choose $\\underline{{\\theta }}^{(n)} \\in \\Sigma $ such that $x_n = \\ell (\\underline{{\\theta }}^{(n)})$ .",
"Let $\\underline{{\\theta }}^{(n)}=(b_j^{(n)})_{j\\in {\\mathbb {Z}}}$ and assume $b_j^{(n)} \\le 4$ , $\\forall \\, j$ , $\\forall \\, n$ (which is possible since me may assume that the $x_n$ are not in Hall's ray).",
"We have $x_n ={\\limsup }_{j\\rightarrow \\infty }({\\alpha }_j^{(n)} + {\\beta }_j^{(n)})$ .",
"Given $\\delta >0$ , $\\exists \\, n_0 \\in {\\mathbb {N}}$ large such that $n \\ge n_0 \\Rightarrow |\\ell (\\underline{{\\theta }}^{(n)})-x| < \\delta $ and there are infinitely many $j \\in {\\mathbb {N}}$ such that $|{\\alpha }_j^{(n)} + {\\beta }_j^{(n)}-x| < \\delta $ .",
"Let $N =\\lceil \\delta ^{-1} \\rceil $ .",
"Given such a pair $(j,n)$ consider the finite sequence with $2N+1$ terms $(b_{j-N}^{(n)},b_{j-N+1}^{(n)},\\dots , b_j^{(n)},\\dots , b_{j+N}^{(n)}) =: S(j,n)$ .",
"There is a sequence $S$ such that for infinitely many values of $n$ , $S$ appears infinitely many times as $S(j,n), \\, j \\in {\\mathbb {N}}$ , i.e., there are $j_1(n) < j_2(n) <\\dots $ with $\\lim _{i\\rightarrow \\infty } (j_{i+1}(n)-j_i(n))=\\infty $ and $S(j_i(n),n) = S$ , $\\forall \\,i \\ge 1$ , for all $n$ in some infinite set $A \\subset {\\mathbb {N}}$ .",
"Consider the sequences ${\\beta }(i,n)$ for $i\\ge 1$ , $n \\in A$ given by ${\\beta }(i,n) = (b_{j_i(n)+N+1}^{(n)}, b_{j_i(n)+N+2}^{(n)},\\dots ,b_{j_{i+1}(n)+N}^{(n)}).$ There are $(i_1,n_1)$ and $(i_2,n_2)$ for which there is no sequence ${\\gamma }$ such that ${\\beta }(i_1,n_1)$ and ${\\beta }(i_2,n_2)$ are concatenations of copies of ${\\gamma }$ , otherwise $x_n$ would be constant for $n \\in A$ .",
"This implies that, taking $B = \\lbrace {\\beta }(i_1,n_1) {\\beta }(i_2,n_2), {\\beta }(i_2,n_2) {\\beta }(i_1,n_1)\\rbrace $ , $K(B)$ is a regular Cantor set, so, as in Lemma 3, $\\ell (K(B))$ contains a regular Cantor set $\\hat{K}$ with $d(x,\\hat{K}) \\le 2\\delta $ .",
"$\\Box $"
],
[
"Proof of Lemma 2",
"Let $\\tau =\\eta /40$ .",
"Since $t>3$ , we have $D(t)>0$ , and so we may choose $r_0\\in {\\mathbb {N}}$ large such that, for $r\\ge r_0$ , $|\\frac{\\log N(t,r)}{r}-D(t)|<\\frac{\\tau }{2} D(t)$ .",
"Let $B_0:=C(t,r_0)$ and $N_0:=N(t,r_0)=|B_0|$ .",
"Let $k=8N_0^2\\lceil 2/\\tau \\rceil $ .",
"Take $\\tilde{B}=\\lbrace {\\beta }={\\beta }_1{\\beta }_2\\cdots {\\beta }_k \\mid {\\beta }_j\\in B_0,1\\le j\\le k $ and $K_t\\cap I({\\beta })\\ne \\emptyset \\rbrace $ .",
"Given ${\\beta }={\\beta }_1{\\beta }_2\\cdots {\\beta }_k\\in \\tilde{B}$ (with ${\\beta }_i\\in B_0$ , $1\\le i\\le k$ ), we say that $j$ , $1\\le j\\le k$ , is a right-good position of ${\\beta }$ if there are elements ${\\beta }^{(s)}={\\beta }_1{\\beta }_2\\cdots {\\beta }_{j-1} {\\beta }_j^{(s)} {\\beta }_{j+1}^{(s)}\\cdots {\\beta }_k^{(s)}$ , $s=1,2$ , of $\\tilde{B}$ such that we have the following inequality of continued fractions: $[0; {\\beta }_j^{(1)}]<[0;{\\beta }_j]<[0;{\\beta }_j^{(2)}]$ .",
"We say that $j$ is a left-good position if there are elements ${\\beta }^{(s)}={\\beta }_1^{(s)}{\\beta }_2^{(s)}\\cdots {\\beta }_{j-1}^{(s)}{\\beta }_j^{(s)}{\\beta }_{j+1}{\\beta }_{j+2}\\cdots {\\beta }_k$ , $s=3,4$ , of $\\tilde{B}$ such that $[0;({\\beta }_j^{(3)})^t]<[0;{\\beta }_j^t]<[0;({\\beta }_j^{(4)})^t]$ .",
"Finally, we say that $j$ is a good position if it is both right-good and left-good.",
"We will show that most positions of most words of $\\tilde{B}$ are good.",
"Let us first estimate $|\\tilde{B}|$ .",
"It follows from Lemma A2 of the appendix that, for ${\\beta }\\in \\tilde{B}$ , $s({\\beta })<(2e^{-r_0})^k<e^{-k(r_0-1)}$ .",
"Moreover, since $N(t,k(r_0-1))\\ge \\frac{1}{T^2} e^{k(r_0-1)D(t)}$ , $\\lbrace I({\\beta }); {\\beta }\\in \\tilde{B}\\rbrace $ is a covering of $K_t$ by intervals of size smaller than $e^{-k(r_0-1)}$ and the function $h:\\tilde{B} \\rightarrow C(t,k(r_0-1))$ defined by $h({\\beta })=h(({\\beta }_1{\\beta }_2\\dots {\\beta }_k))=({\\beta }_1{\\beta }_2\\dots {\\beta }_j)$ , where $j = \\min \\lbrace i\\; ;i\\le k\\; \\text{ and }\\;r(({\\beta }_1{\\beta }_2\\dots {\\beta }_i))\\ge k(r_0-1)\\;\\rbrace $ is onto, we have: $|\\tilde{B}|\\ge \\; \\frac{1}{T^2} e^{k(r_0-1)D(t)}&>& 2\\,e^{k(r_0-2)D(t)},\\quad \\text{since $k$ is large} \\nonumber \\\\&\\ge & 2\\,e^{(1-\\tau /2)r_0 kD(t)},\\quad \\text{since $r_0$ is large} \\nonumber \\\\&>& 2\\,e^{(1-\\tau )(1+\\tau /2)r_0kD(t)}\\nonumber \\\\&>& 2\\,N_0^{(1-\\tau )k},\\quad \\text{since}\\: N(t,r_0)<e^{\\left(1+\\frac{\\tau }{2}\\right)D(t) r_0} .\\nonumber \\\\\\nonumber $ Now, let us estimate the number of words ${\\beta }\\in \\tilde{B}$ such that at least $k/20$ positions of ${\\beta }$ are not right good: we have at most $2^k$ choices for the set of the $m\\ge k/20$ positions which are not right-good.",
"Once we choose this set of positions, if $j$ is such a position and ${\\beta }_1,{\\beta }_2,\\dots ,{\\beta }_{j-1}\\in B_0$ are already chosen, there are at most two (the largest and the smallest) choices for ${\\beta }_j\\in B_0$ such that for some ${\\beta }={\\beta }_1{\\beta }_2\\cdots {\\beta }_{j-1}{\\beta }_j{\\beta }_{j+1}\\cdots {\\beta }_k\\in \\tilde{B}$ the position $j$ is not right good.",
"If $j$ is any other position, we have of course at most $N_0=|B_0|$ possible choices for ${\\beta }_j$ , so we have at most $2^m\\cdot N_0^{k-m}\\le 2^{k/20}N_0^{19k/20}$ words in $\\tilde{B}$ with this chosen set of $m$ positions which are not right-good.",
"Therefore, the number of words ${\\beta }\\in \\tilde{B}$ for which the number of positions which are not right-good is at least $k/20$ is bounded by $2^k\\cdot 2^{k/20}\\cdot N_0^{19k/20}=2^{21k/20}\\cdot N_0^{19k/20}$ .",
"Analogously, the number of words ${\\beta }\\in \\tilde{B}$ for which there are at least $k/20$ positions which are not left-good is also bounded by $2^{21k/20}\\cdot N_0^{19k/20}$ .",
"This implies that for at least $|\\tilde{B}|-2\\cdot 2^{21k/20}\\cdot N_0^{19k/20} > 2N_0^{(1-\\tau )k}-2^{1+21k/20}\\cdot N_0^{19k/20} > N_0^{(1-\\tau )k}$ words of $\\tilde{B}$ , the number of good positions is at least $9k/10$ .",
"Let us call such an element of $\\tilde{B}$ an excellent word.",
"If ${\\beta }={\\beta }_1{\\beta }_2\\cdots {\\beta }_k\\in \\tilde{B}$ (with ${\\beta }_j\\in B_0$ , $1\\le j\\le k$ ) is an excellent word, we may find $\\lceil 2k/5\\rceil $ positions $i_1,i_2,\\dots ,i_{\\lceil 2k/5\\rceil }\\le k$ with $i_{s+1}\\ge i_s+2$ , $\\forall \\, s< \\lceil 2k/5\\rceil $ , such that the positions $i_1,i_1+1$ , $i_2, i_{2}+1,\\dots ,i_{\\lceil 2k/5\\rceil }$ , $i_{\\lceil 2k/5\\rceil }+1$ are good.",
"Since $k=8N_0^2\\lceil 2/\\tau \\rceil $ , we may take, for $1\\le s\\le 3N_0^2$ , $j_s:=i_{s\\lceil 2/\\tau \\rceil }$ (notice that $3N_0^2\\lceil 2/\\tau \\rceil <\\frac{16}{5}N_0^2\\lceil 2/\\tau \\rceil =2k/5$ ), so we have $j_{s+1}-j_s\\ge 2\\lceil 2/\\tau \\rceil $ , $\\forall \\, s<3N_0^2$ , and the positions $j_s,j_s+1$ are good for $1\\le s\\le 3N_0^2$ .",
"Now, the number of possible choices of $(j_1,j_2,\\dots ,j_{3N_0^2})$ is bounded by $\\binom{k}{3N_0^2}<2^k$ and, given $(j_1,j_2,\\dots ,j_{3N_0^2})$ the number of choices of $({\\beta }_{j_1},{\\beta }_{j_1+1},\\dots ,{\\beta }_{j_{3N_0^2}},{\\beta }_{j_{3N_0^2}+1})$ is bounded by $N_0^{6N_0^2}$ .",
"So, we may choose $\\hat{\\j }_1,\\hat{\\j }_2,\\dots ,\\hat{\\j }_{3N_0^2}$ with $\\hat{\\j }_{s+1}-\\hat{\\j }_s\\ge 2\\lceil 2/\\tau \\rceil $ , $\\forall \\, s<3N_0^2$ , and words $\\hat{{\\beta }}_{\\hat{\\j }_1},\\hat{{\\beta }}_{\\hat{\\j }_1+1}, \\hat{{\\beta }}_{\\hat{\\j }_2},\\hat{{\\beta }}_{\\hat{\\j }_2+1},\\dots ,\\hat{{\\beta }}_{\\hat{\\j }_{3N_0^2}},\\hat{{\\beta }}_{\\hat{\\j }_{3N_0^2}+1} \\in B_0$ such that the set $X:=\\lbrace {\\beta }={\\beta }_1{\\beta }_2\\cdots {\\beta }_k\\in \\tilde{B}$ excellent$\\,|\\,\\, \\hat{\\j }_s,\\hat{\\j }_s+1$ are good positions and ${\\beta }_{\\hat{\\j }_s}=\\hat{{\\beta }}_{\\hat{\\j }_s}, {\\beta }_{\\hat{\\j }_s+1}=\\hat{{\\beta }}_{\\hat{\\j }_s+1}, \\forall \\, s\\le 3N_0^2\\rbrace $ has at least $\\frac{N_0^{(1-\\tau )k}}{2^k \\cdot N_0^{6N_0^2}} > N_0^{(1-2\\tau )k}$ elements, as $N_0$ and $k$ are large.",
"Since $N_0=|B_0|$ , there are $N_0^2$ possible choices for the pairs $(\\hat{{\\beta }}_{\\hat{\\j }_s},\\hat{{\\beta }}_{\\hat{\\j }_s+1})$ .",
"We will consider, for $1\\le s<t\\le 3N_0^2$ , the projections $\\pi _{s,t}\\colon X\\rightarrow B_0^{\\hat{\\j }_t-\\hat{\\j }_s}$ given by $\\pi _{s,t}({\\beta }_1{\\beta }_2\\cdots {\\beta }_k)=({\\beta }_{\\hat{\\j }_s+1},{\\beta }_{\\hat{\\j }_s+2},\\dots ,{\\beta }_{\\hat{\\j }_t})$ .",
"We will show that the images of many of these projections are large.",
"For each pair $(s,t)$ with $1\\le s<t\\le 3N_0^2$ such that $|\\pi _{s,t}(X)| < N_0^{(1-10\\tau )(\\hat{\\j }_t-\\hat{\\j }_s)}$ , we will exclude from $\\lbrace 1,2,\\dots ,3N_0^2\\rbrace $ the indices $s,s+1,\\dots ,t-1$ .",
"Let us estimate the total number of indices excluded: the set of excluded indices is the union of the intervals $[s,t)$ (intersected with ${\\mathbb {Z}}$ ) over the pairs $(s,t)$ as above.",
"Now we use the elementary fact that, given a finite family of intervals, there is a subfamily of disjoint intervals whose sum of lenghts is at least half of the measure of the union of the intervals of the original family.",
"We apply this fact to the above intervals $[s,t)$ .",
"Suppose that the total number of indices excluded is at least $2N_0^2$ .",
"By the above fact, we may find a disjoint collection of intervals $[s,t)$ as above whose sum of lenghts is at least $N_0^2$ .",
"Let us call $\\cal P$ the set of these pairs $(s,t)$ .",
"Since $\\hat{\\j }_t-\\hat{\\j }_s\\ge 2(t-s)\\lceil 2/\\tau \\rceil , \\forall t>s$ , the sum of $(\\hat{\\j }_t-\\hat{\\j }_s)$ for $(s,t) \\in \\cal P$ is at least $2N_0^2\\lceil 2/\\tau \\rceil $ .",
"Since for each pair $(s,t) \\in \\cal P$ we have $|\\pi _{s,t}(X)|<N_0^{(1-10\\tau )(\\hat{\\j }_t-\\hat{\\j }_s)}$ , we get $N_0^{(1-2\\tau )k} <|X| &<& N_0^{(1-10\\tau )\\sum \\limits _{(s,t)\\in \\cal P}(\\hat{\\j }_t-\\hat{\\j }_s)} \\cdot N_0^{\\#\\lbrace i; i\\: \\notin [\\hat{\\j }_s,\\hat{\\j }_t), \\forall (s,t) \\in \\cal P\\rbrace } \\nonumber \\\\&<&N_0^{(1-10\\tau )\\cdot 2N_0^2\\lceil 2/\\tau \\rceil } \\cdot N_0^{k-2N_0^2\\lceil 2/\\tau \\rceil }, \\nonumber \\nonumber $ since we have at most $N_0$ choices for ${\\beta }_i$ for each index $i$ which does not belong to the union of the intervals $[\\hat{\\j }_s,\\hat{\\j }_t)$ associated to these pairs $(s,t)$ .",
"However, this is a contradiction, since this inequality is equivalent to $N_0^{20\\tau N_0^2\\lceil 2/\\tau \\rceil }<N_0^{2\\tau k}$ , which cannot hold, because $2\\tau k=16\\tau N_0^2\\lceil 2/\\tau \\rceil <20\\tau N_0^2\\lceil 2/\\tau \\rceil $ .",
"So, the total number of excluded indices is smaller than $2N_0^2$ .",
"Now, there are at least $N_0^2+1$ indices which are not excluded.",
"We will have two non-excluded indices $s<t$ such that $\\hat{{\\beta }}_{\\hat{\\j }_s}=\\hat{{\\beta }}_{\\hat{\\j }_t}$ and $\\hat{{\\beta }}_{\\hat{\\j }_s+1}=\\hat{{\\beta }}_{\\hat{\\j }_t+1}$ .",
"We claim that, for $B:=\\pi _{s,t}(X)$ , the shift $\\Sigma (B)$ satisfies the conclusions of the statement.",
"Indeed, since $s$ and $t$ are not excluded, we have $|B|\\ge N_0^{(1-10\\tau )(\\hat{\\j }_t-\\hat{\\j }_s)}$ .",
"Moreover, by Proposition A1 of the appendix, for every ${\\alpha }\\in B$ we have $|I({\\alpha })|=s({\\alpha })>(2(T+1)^2 e^{r_0})^{-(\\hat{\\j }_t-\\hat{\\j }_s)} > e^{-(\\hat{\\j }_t-\\hat{\\j }_s)(r_0+\\lceil \\log (2(T+1)^2) \\rceil )}.$ So, the Hausdorff dimension of $K(B)$ is at least $\\frac{(1-10\\tau )\\log N_0}{r_0+\\lceil \\log (2(T+1)^2) \\rceil } > \\frac{(1-10\\tau )r_0}{r_0+\\lceil \\log (2(T+1)^2) \\rceil }\\cdot (1-\\frac{\\tau }{2})D(t) > \\left(1-12\\tau \\right)D(t)> (1-\\eta )D(t).$ On the other hand, if $\\tilde{k}:=\\hat{\\j }_t-\\hat{\\j }_s$ , ${\\gamma }_1:=\\hat{{\\beta }}_{\\hat{\\j }_s+1}=\\hat{{\\beta }}_{\\hat{\\j }_t+1}$ and ${\\gamma }_2:=\\hat{{\\beta }}_{\\hat{\\j }_t}=\\hat{{\\beta }}_{\\hat{\\j }_s}$ , all elements of $B$ are of the form ${\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2$ , where ${\\gamma }_1,{\\beta }_2,{\\beta }_3,\\dots ,{\\beta }_{\\tilde{k}-1}$ , ${\\gamma }_2\\in B_0$ and there are ${\\gamma }_1^{\\prime },{\\gamma }_1^{\\prime \\prime }$ , ${\\gamma }_2^{\\prime },{\\gamma }_2^{\\prime \\prime }\\in B_0$ with $[0;{\\gamma }_2^{\\prime }]<[0;{\\gamma }_2]< [0;{\\gamma }_2^{\\prime \\prime }]$ and $[0;({\\gamma }_1^{\\prime })^t]<[0;{\\gamma }_1^t]<[0;({\\gamma }_1^{\\prime \\prime })^t]$ such that $I({\\gamma }_1^{\\prime }{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2{\\gamma }_1)\\cap K_t\\ne \\emptyset ,\\quad I({\\gamma }_1^{\\prime \\prime }{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2{\\gamma }_1)\\cap K_t\\ne \\emptyset , \\nonumber \\\\I({\\gamma }_2{\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2^{\\prime })\\cap K_t\\ne \\emptyset ,\\quad I({\\gamma }_2{\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2^{\\prime \\prime })\\cap K_t\\ne \\emptyset .\\nonumber $ We will show that this implies the existence of $\\delta >0$ such that $\\Sigma (B)\\subset \\Sigma _{t-\\delta }$ .",
"Let ${\\gamma }_1^t=(c_1,c_2,\\dots ,c_{m_1})$ , with $c_j\\in {\\mathbb {N}}^*$ , $\\forall \\, j\\le m_1$ , and ${\\gamma }_2=(d_1,d_2,\\dots ,d_{m_2})$ with $d_j\\in {\\mathbb {N}}^*$ , $\\forall \\, j\\le m_2$ .",
"Let ${\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2\\in B$ where ${\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}=a_1a_2\\cdots a_{\\tilde{m}}$ with $a_j\\in {\\mathbb {N}}^*$ , $\\forall \\, j\\le \\tilde{m}$ .",
"We want to estimate three kinds of sums of continued fractions.",
"The first one are sums of continued fractions beginning by $[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2,{\\gamma }_1,\\dots ]+[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\dots ].$ Let us assume, without loss of generality, that $q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2) \\le q_{m_1+j-1}(a_{j-1},\\ldots , a_1, {\\gamma }_1^t)$ (the other case, when the reverse inequality $q_{m_1+j-1}(a_{j-1},\\ldots , a_1, {\\gamma }_1^t) \\le q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2)$ holds, is symmetric).",
"Assume also that $[a_j; a_{j+1},\\ldots ,a_{\\tilde{m}},{\\gamma }_2]<[a_j; a_{j+1},\\ldots ,a_{\\tilde{m}},{\\gamma }_2^{\\prime }]$ (otherwise we change ${\\gamma }_2^{\\prime }$ by ${\\gamma }_2^{\\prime \\prime }$ ).",
"This allows us to exhibit $\\delta > 0$ such that, for any $\\underline{\\theta }^{(i)} \\in \\lbrace 1, 2, \\dots , T\\rbrace ^{{\\mathbb {N}}},\\; 1 \\le i \\le 4$ , ${[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2,\\underline{\\theta }^{(1)}]+[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(2)}] <}\\\\& <[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2^{\\prime },\\underline{\\theta }^{(3)}]+[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(4)}]-\\delta .$ Indeed, by the Lemma A1 of the appendix, $[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2^{\\prime },\\underline{\\theta }^{(3)}]-[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2,\\underline{\\theta }^{(1)}]>\\frac{1}{(T+1)(T+2)q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2)^2}$ $\\text{ and }\\,\\mid [0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(4)}]-[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(2)}]\\mid < \\\\\\frac{1}{q_{m_1+m_2+j-1}(a_{j-1},\\ldots , a_1, {\\gamma }_1^t, {\\gamma }_2^t)^2}<\\frac{1}{(F_{m_2+1}q_{m_1+j-1}(a_{j-1}, \\ldots , a_1, {\\gamma }_1^t))^2} \\le \\\\\\frac{1}{(F_{m_2+1}q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2))^2}\\le \\frac{1}{2(T+1)(T+2)q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2)^2}\\\\(\\text{here we use that}\\; m_2 \\text{ is large};\\\\ (F_n)\\text{ denotes Fibonacci's sequence, given by } F_0=0, F_1=1, F_{n+2}=F_{n+1}+F_n, \\forall n \\ge 0).\\\\$ So, the inequality holds with $\\delta :=\\frac{1}{(T+1)^{2(m_1+m_2+\\tilde{m})}}<\\frac{1}{2(T+1)(T+2)q_{m_2+\\tilde{m} -j}(a_{j+1},\\ldots , a_{\\tilde{m}}, {\\gamma }_2)^2}.$ On the other hand, $I({\\gamma }_2{\\gamma }_1{\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2^{\\prime })\\cap K_t\\ne \\emptyset $ , so there are $\\underline{\\theta }^{(3)}$ and $\\underline{\\theta }^{(4)}$ such that $(\\underline{\\theta }^{(4)})^t {\\gamma }_2{\\gamma }_1 {\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2^{\\prime } \\underline{\\theta }^{(3)} \\in \\Sigma _t$ , and thus $[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2^{\\prime },\\underline{\\theta }^{(3)}]+ [0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(4)}]\\le t$ , which implies that, for any $\\underline{\\theta }^{(i)} \\in \\lbrace 1, 2, \\dots , T\\rbrace ^{{\\mathbb {N}}}, i=1,2$ , $[a_j;a_{j+1},\\dots ,a_{\\tilde{m}},{\\gamma }_2,\\underline{\\theta }^{(1)}]+[0;a_{j-1},\\dots ,a_1,{\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(2)}] < t-\\delta $ .",
"The other two kinds of sums of continued fractions we want to estimate are sums of continued fractions beginning by $[d_j; d_{j+1},\\ldots ,d_{m_2},{\\gamma }_1,\\ldots ]+[0; d_{j-1},\\ldots , d_1, a_{\\tilde{m}},\\ldots , a_1, {\\gamma }_1^t,\\ldots ]$ and, symmetrically, sums of continued fractions beginning by $[0; c_{j+1},\\ldots , c_{m_1}, {\\gamma }_2^t,\\ldots ]+[c_j; c_{j-1},\\ldots , c_1, a_1,\\ldots , a_{\\tilde{m}}, {\\gamma }_2,\\ldots ]$ .",
"We have: $q_{m_2-j+m_1}(d_{j+1},\\ldots ,d_{m_2}, {\\gamma }_1) \\le q_{j-1+\\tilde{m} +m_1}(d_{j-1},\\ldots , d_1, a_{\\tilde{m}},\\ldots , a_1, {\\gamma }_1^t)$ (indeed, $\\tilde{m}/(m_1+m_2)$ is large when $\\eta $ and $\\tau $ are small, depending on $T$ ).",
"Assume that $[d_j; d_{j+1},\\ldots ,d_{m_2}, {\\gamma }_1] < [d_j; d_{j+1},\\ldots ,d_{m_2}, {\\gamma }_1^{\\prime }]$ (otherwise we change ${\\gamma }_1^{\\prime }$ by ${\\gamma }_1^{\\prime \\prime }$ ).",
"Since $I({\\gamma }_2{\\gamma }_1 {\\beta }_2{\\beta }_3\\cdots {\\beta }_{\\tilde{k}-1}{\\gamma }_2{\\gamma }_1^{\\prime })\\cap K_t\\ne \\emptyset $ , estimates analogous to the previous ones imply that, for any $\\underline{\\theta }^{(i)} \\in \\lbrace 1, 2, \\dots , T\\rbrace ^{{\\mathbb {N}}},\\: i=1,2$ , we have $[d_j; d_{j+1},\\ldots ,d_{m_2}, {\\gamma }_1,\\underline{\\theta }^{(1)}]+$$[0; d_{j-1},\\ldots , d_1, a_{\\tilde{m}},\\ldots , a_1, {\\gamma }_1^t,{\\gamma }_2^t,\\underline{\\theta }^{(2)}] < t-\\delta $ .",
"This implies that the complete shift $\\Sigma (B)$ satisfies the conditions of the statement, which concludes the proof of the Lemma.",
"$\\Box $ As we said before, this proof doesn't give any estimate on the modulus of continuity of $d(t)$ .",
"Indeed, in the beginning of the proof of Lemma 2, we used the fact that $u(r)=\\log (T^2N(t,r))$ is subadditive in order to guarantee the existence of $r_0\\in {\\mathbb {N}}$ large such that, for $r\\ge r_0$ , $|\\frac{\\log N(t,r)}{r}-D(t)|<\\frac{\\tau }{2} D(t)$ (recall that $D(t)=\\lim _{m\\rightarrow \\infty }\\frac{1}{m} \\log (T^2N(t,m))=\\lim _{m\\rightarrow \\infty }\\frac{1}{m}\\log (N(t,m))$ ).",
"However, this gives no estimate on $r_0$ .",
"Consider, for instance, the function $v(n)$ given by $v(n)=2n$ for $n\\le M_0$ and $v(n)=n+M_0$ for $n>M_0$ , where $M_0$ is a large positive integer.",
"It is subadditive, increasing, $\\lim _{n\\rightarrow \\infty } v(n)/n=1$ but $v(M_0)/M_0=2$ , and $M_0$ can be taken arbitrarily large.",
"However it is possible to adapt the proof in order to give an estimate on the modulus of continuity of $d(t)$ , using an idea of [FMM].",
"Given ${\\varepsilon }>0$ (which we may assume to be smaller than $\\frac{1}{7}<\\frac{1}{10\\log 2}$ ), we want to obtain $\\delta \\in (0,1)$ as an explicit function of ${\\varepsilon }$ such that $D(t-\\delta )>D(t)-{\\varepsilon }$ .",
"Of course there is no loss of generality in assuming $D(t)\\ge {\\varepsilon }$ .",
"We may also assume that $T=\\lfloor t \\rfloor < 4+{\\varepsilon }^{-1}/\\log 2$ (and thus $t<T+1<3{\\varepsilon }^{-1}$ ) since, by the proof of Theorem 2, if $\\lfloor t\\rfloor \\ge 4+{\\varepsilon }^{-1}/\\log 2\\ge 14$ , for $m=\\lfloor t\\rfloor -4\\ge \\text{max}\\lbrace 9,{\\varepsilon }^{-1}/\\log 2\\rbrace $ , we have $D(t-1)\\ge HD(C_m)>1-\\frac{1}{m\\log 2}>1-{\\varepsilon }$ (and so $D(t-1)>D(t)-{\\varepsilon }$ ).",
"Under these hypothesis, we will apply the conclusions of Lemma 2 for $\\eta ={\\varepsilon }$ .",
"In its proof, in this case, it is enough to assume $r_0\\ge 1/\\tau ^2$ and that, for $k=8N(t,r_0)^2\\lceil 2/\\tau \\rceil $ , $\\frac{\\log N(t,r_0)}{r_0}<(1+\\tau /2)\\frac{\\log N(t,k(r_0-1))}{k(r_0-1)}$ (indeed, assuming the above bounds for $t$ , it is not difficult to check that, except for this inequality relating $N(t,r_0)$ and $N(t,k(r_0-1))$ , the claims in other parts of the proof of the Lemma that use the assumptions that $r_0$ and $k$ are large are satisfied provided $r_0\\ge 1/\\tau ^2$ ).",
"We define a sequence $(c_n)_{n\\ge 0}$ recursively by $c_0=\\lceil \\frac{1}{\\tau ^2}\\rceil $ and, for every $n\\ge 0$ , $c_{n+1}=8N(t,c_n)^2\\lceil \\frac{2}{\\tau }\\rceil (c_n-1)$ .",
"We claim that, for some integer $s_0<(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })$ , we will have $\\frac{\\log N(t,c_{s_0})}{c_{s_0}}<(1+\\frac{\\tau }{2})\\frac{\\log N(t,c_{s_0+1})}{c_{s_0+1}}=(1+\\frac{\\tau }{2})\\frac{\\log N(t,k(c_{s_0}-1))}{k(c_{s_0}-1)}$ , with $k=8N(t,c_{s_0})^2\\lceil \\frac{2}{\\tau }\\rceil $ .",
"Indeed, if it is not the case, then $\\frac{\\log N(t,c_{n+1})}{c_{n+1}}\\le (1+\\frac{\\tau }{2})^{-1}\\frac{\\log N(t,c_n)}{c_n}$ for $0\\le n<(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })$ , and so, for $M=\\lceil (1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })\\rceil $ , we would have $\\frac{\\log N(t,c_M)}{c_M}\\le (1+\\frac{\\tau }{2})^{-M}\\cdot \\frac{\\log N(t,c_0)}{c_0}<({\\varepsilon }/4)\\cdot \\frac{\\log N(t,c_0)}{c_0}$ , since $(1+\\frac{\\tau }{2})^{-(1+\\frac{2}{\\tau })}<e^{-1}$ .",
"On the other hand, it follows by Lemma A3 that, for every $m\\ge c_0$ , $N(t,m)<(T+1)^2e^m<e^{2m}$ (recall that $c_0=\\lceil \\frac{1}{\\tau ^2}\\rceil =\\lceil \\frac{1600}{{\\varepsilon }^2}\\rceil $ ), and so $\\frac{\\log N(t,c_M)}{c_M}\\le ({\\varepsilon }/4)\\cdot \\frac{\\log N(t,c_0)}{c_0}<{\\varepsilon }/2$ .",
"This leads to a contradiction since, for every positive integer $m$ , ${\\varepsilon }\\le D(t)\\le \\frac{\\log (T^2N(t,m))}{m}$ and, in particular, $\\frac{\\log N(t,c_M)}{c_M}>{\\varepsilon }-\\frac{2\\log T}{c_M}\\ge {\\varepsilon }/2$ , since $c_M\\ge c_0\\ge \\frac{1600}{{\\varepsilon }^2}$ and $T<3/{\\varepsilon }$ .",
"Now let $r_0=c_{s_0}$ .",
"By the previous discussion, the proof of Lemma 2 works for this $r_0$ (and $k=8N(t,r_0)^2\\lceil 2/\\tau \\rceil $ ), so, for $\\delta =\\frac{1}{(T+1)^{2(m_1+m_2+\\tilde{m})}}\\ge \\frac{1}{(T+1)^{2k\\cdot \\text{max}\\lbrace |\\beta |, \\beta \\in C(t,r_0)\\rbrace }}\\ge \\frac{1}{(T+1)^{2k\\cdot r_0/\\log 2}},$ we have $D(t-\\delta )>(1-{\\varepsilon })D(t)>D(t)-{\\varepsilon }$ .",
"We will now give an explicit positive lower bound for $\\delta $ in terms of ${\\varepsilon }$ .",
"In order to do this we define recursively, for each integer $n\\ge 0$ and $x\\in {\\mathbb {R}}$ , the functions ${\\cal T}(x,n)$ and ${\\cal T}(n)$ by ${\\cal T}(x,0)=x$ , ${\\cal T}(x,n+1)=e^{{\\cal T}(x,n)}$ and ${\\cal T}(n)={\\cal T}(1,n)$ .",
"We have, for every $n\\ge 0$ , $c_{n+1}=8N(t,c_n)^2\\lceil \\frac{2}{\\tau }\\rceil (c_n-1)\\le 8e^{4c_n}\\cdot \\frac{3}{\\tau }\\cdot c_n<e^{e^{c_n}},$ since, for every $n\\ge 0$ , $c_n\\ge c_0\\ge \\frac{1}{\\tau ^2}=\\frac{1600}{{\\varepsilon }^2}$ and $N(t,c_n)\\le e^{2c_n}$ , therefore $r_0=c_{s_0}<{\\cal T}(c_0,2s_0)={\\cal T}(\\lceil \\frac{1}{\\tau ^2}\\rceil ,2s_0)$ and $2\\log (T+1)\\cdot k\\cdot \\frac{r_0}{\\log 2}=16\\log (T+1)\\cdot N(t,r_0)^2\\lceil \\frac{2}{\\tau }\\rceil \\cdot \\frac{r_0}{\\log 2}\\le 16\\log (3/{\\varepsilon })\\cdot e^{4r_0}\\cdot \\frac{3}{\\tau }\\cdot \\frac{r_0}{\\log 2}<e^{e^{r_0}},$ so $\\delta \\ge \\frac{1}{(T+1)^{2k\\cdot r_0/\\log 2}}=e^{-2\\log (T+1)\\cdot k\\cdot r_0/\\log 2}>e^{-e^{e^{r_0}}}>\\frac{1}{{\\cal T}(c_0,2s_0+3)}.$ Finally, since $2^k\\ge k^2$ for every $k\\ge 4$ , it follows by induction that, for every $n\\ge 4$ , ${\\cal T}(n)\\ge (n+1)^6$ for every $n\\ge 0$ .",
"Indeed, ${\\cal T}(4)>2^{16}>5^6$ and, for $n\\ge 4$ , ${\\cal T}(n+1)>2^{{\\cal T}(n)}\\ge {\\cal T}(n)^2\\ge (n+1)^{12}>(n+2)^6$ .",
"This implies that ${\\cal T}(\\lfloor 1/{\\varepsilon }\\rfloor )\\ge (1/{\\varepsilon })^6>1601/{\\varepsilon }^2>\\lceil \\frac{1600}{{\\varepsilon }^2}\\rceil =\\lceil \\frac{1}{\\tau ^2}\\rceil =c_0$ (recall that $0<{\\varepsilon }<1/7$ ), so ${\\cal T}(c_0,2s_0+3)<{\\cal T}({\\cal T}(\\lfloor 1/{\\varepsilon }\\rfloor ),2s_0+3)={\\cal T}(\\lfloor 1/{\\varepsilon }\\rfloor +2s_0+3)$ , and, since $s_0<(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })$ , we have $\\lfloor 1/{\\varepsilon }\\rfloor +2s_0+3<3+\\lfloor 1/{\\varepsilon }\\rfloor +2(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })\\le 3+1/{\\varepsilon }+2(1+\\frac{2}{\\tau })\\log (4/{\\varepsilon })=$ $=3+1/{\\varepsilon }+2(1+\\frac{80}{{\\varepsilon }})\\log (4/{\\varepsilon })<\\frac{161}{{\\varepsilon }}\\log (4/{\\varepsilon }),$ and therefore $\\delta >\\frac{1}{{\\cal T}(c_0,2s_0+3)}>\\frac{1}{{\\cal T}(\\lfloor 1/{\\varepsilon }\\rfloor +2s_0+3)}\\ge \\frac{1}{{\\cal T}(\\lfloor \\frac{161}{{\\varepsilon }}\\log (4/{\\varepsilon })\\rfloor )}.$"
],
[
"Appendix: Basic facts and estimates on continued fractions",
"We will prove here some elementary facts on continued fractions used in the previous sections.",
"We refer to [CF] for the facts used but not proved here.",
"Let $x = [a_0;a_1,a_2,a_3,\\dots ]$ be a real number, and $(\\frac{p_n}{q_n})_{n \\in \\mathbb {N}}, \\frac{p_n}{q_n}= [a_0;a_1,a_2,\\dots ,a_n]$ its sequence of convergents.",
"We have, for every $n\\in {\\mathbb {N}}$ , $p_{n+1}q_n-p_nq_{n+1}=(-1)^n$ , $x=[a_0; a_1, a_2, \\dots , a_n, {\\alpha }_{n+1}]=\\frac{\\alpha _{n+1}p_n+p_{n-1}}{\\alpha _{n+1}q_n+q_{n-1}},$ and so $\\alpha _{n+1}=\\frac{p_{n-1}-q_{n-1}x}{q_{n}x-p_{n}}$ and $x-\\frac{p_n}{q_n}= \\frac{(-1)^n}{(\\alpha _{n+1}+\\beta _{n+1})q_n^2},$ where $\\beta _{n+1}= \\frac{q_{n-1}}{q_n}= [0;a_n,a_{n-1},a_{n-2},\\dots ,a_1].$ In particular, $ \\left| x-\\frac{p_n}{q_n} \\right|=\\frac{1}{(\\alpha _{n+1}+\\beta _{n+1})q_n^2}.",
"$ Recall that, given a finite sequence ${\\alpha }=(a_1,a_2,\\dots ,a_n)\\in ({\\mathbb {N}}^*)^n$ , we define its size by $s({\\alpha }):=|I({\\alpha })|$ , where $I({\\alpha })$ is the interval $\\lbrace x\\in [0,1] \\mid x=[0; a_1,a_2,\\dots ,a_n,{\\alpha }_{n+1}], {\\alpha }_{n+1}\\ge 1\\rbrace $ , whose endpoints are $p_n/q_n$ and $\\frac{p_n+p_{n-1}}{q_n+q_{n-1}}$ , $r({\\alpha })=\\lfloor \\log s({\\alpha })^{-1} \\rfloor $ and, for $r \\in {\\mathbb {N}}, P_r= \\lbrace {\\alpha }=(a_1,a_2,\\dots ,a_n) \\mid r({\\alpha }) \\ge r$ and $r((a_1,a_2,\\dots ,a_{n-1}))<r\\rbrace $ .",
"Since $\\alpha _{n+1}=\\frac{p_{n-1}-q_{n-1}x}{q_{n}x-p_{n}}$ , the $n$ -th iterate of the Gauss map restricted to the interval $I({\\alpha })$ is given by ${g^n}|_{{I({\\alpha })}}(x)=g^n(\\alpha _1^{-1})=\\alpha _{n+1}^{-1}=\\frac{q_{n}x-p_{n}}{p_{n-1}-q_{n-1}x}.$ Lemma A1.",
"If $a_0, a_1, a_2, \\ldots $ , $b_n, b_{n+1}, \\ldots $ are positive integers with $a_{n-1}, a_n, a_{n+1}, b_n, b_{n+1} \\le T$ and $a_n\\ne b_n$ , then $|[a_0;a_1, a_2,\\ldots ,a_n, a_{n+1}, a_{n+2}, \\ldots ]-[a_0;a_1, a_2,\\ldots ,b_n, b_{n+1}, b_{n+2}, \\ldots ]|>\\frac{1}{(T+1)(T+2)q_n^2},$ where $q_n=q_n(a_1, a_2,\\ldots , a_n)$ .",
"Proof: Let $x=[a_0;a_1, a_2,\\ldots ,a_n, a_{n+1}, a_{n+2}, \\ldots ]$ and $y=[a_0;a_1, a_2,\\ldots ,a_{n-1}, b_n, b_{n+1}, b_{n+2}, \\ldots ]$ .",
"We have $x=\\frac{\\alpha _{n}(x)p_{n-1}+p_{n-2}}{\\alpha _{n}(x)q_{n-1}+q_{n-2}}, y=\\frac{\\alpha _{n}(y)p_{n-1}+p_{n-2}}{\\alpha _{n}(y)q_{n-1}+q_{n-2}},$ and so $|x-y|=\\left|\\frac{\\alpha _{n}(x)p_{n-1}+p_{n-2}}{\\alpha _{n}(x)q_{n-1}+q_{n-2}}-\\frac{\\alpha _{n}(y)p_{n-1}+p_{n-2}}{\\alpha _{n}(y)q_{n-1}+q_{n-2}}\\right|=\\left|\\frac{(\\alpha _{n}(x)-\\alpha _{n}(y))(p_{n-1}q_{n-2}-p_{n-2}q_{n-1})}{(\\alpha _{n}(x)q_{n-1}+q_{n-2})(\\alpha _{n}(y)q_{n-1}+q_{n-2})}\\right|$ $=\\left|\\frac{(\\alpha _{n}(x)-\\alpha _{n}(y))(-1)^{n-2}}{(\\alpha _{n}(x)q_{n-1}+q_{n-2})(\\alpha _{n}(y)q_{n-1}+q_{n-2})}\\right|=\\frac{|\\alpha _{n}(x)-\\alpha _{n}(y)|}{(\\alpha _{n}(x)q_{n-1}+q_{n-2})(\\alpha _{n}(y)q_{n-1}+q_{n-2})}.$ We have $\\lfloor \\alpha _n(x) \\rfloor =a_n$ , $\\lfloor \\alpha _n(y) \\rfloor =b_n$ and $a_n\\ne b_n$ , so $|\\alpha _{n}(x)-\\alpha _{n}(y)|>[1;T+1]-[0;1,T+1]=\\frac{1}{T+1}+\\frac{1}{T+2}>\\frac{2}{T+2}$ .",
"Moreover, $\\alpha _{n}(x)q_{n-1}+q_{n-2}<(1+a_n)q_{n-1}+q_{n-2}=q_n+q_{n-1}<2q_n$ and $\\alpha _{n}(y)q_{n-1}+q_{n-2}<\\alpha _{n}(y)(q_{n-1}+q_{n-2})\\le \\alpha _{n}(y)q_n<(T+1)q_n$ , and so $|x-y|>\\frac{2}{T+2} \\cdot \\frac{1}{2q_n\\cdot (T+1)q_n}=\\frac{1}{(T+1)(T+2)q_n^2}.\\Box $ Lemma A2.",
"If $\\alpha =a_1a_2\\cdots a_m$ and $\\beta =b_1b_2\\cdots b_n$ are finite words, then $\\dfrac{1}{2}s(\\alpha )s(\\beta )<s(\\alpha \\beta )<2s(\\alpha )s(\\beta ).$ Proof: By Euler's property of continuants (cf.",
"Appendix 2 of [CF]), we have $q_{m+n}(\\alpha \\beta )=q_m(\\alpha )q_n(\\beta )+q_{m-1}(a_1a_2\\cdots a_{m-1})q_{n-1}(b_2b_3\\cdots b_n)$ and then $q_m(\\alpha )q_n(\\beta )<q_{m+n}(\\alpha \\beta )<2q_m(\\alpha )q_n(\\beta ).$ From the left inequality, we have $s(\\alpha \\beta )&=&\\dfrac{1}{q_{m+n}(\\alpha \\beta )\\left[q_{m+n}(\\alpha \\beta )+q_{m+n-1}(\\alpha \\beta )\\right]}\\\\&<&\\dfrac{1}{q_m(\\alpha )q_n(\\beta )\\left[q_m(\\alpha )q_n(\\beta )+q_m(\\alpha )q_{n-1}(\\beta )\\right]}\\\\&<&\\dfrac{2}{q_m(\\alpha )q_n(\\beta )\\left[q_m(\\alpha )+q_{m-1}(\\alpha )\\right]\\left[q_n(\\beta )+q_{n-1}(\\beta )\\right]}\\\\ &=&2\\cdot \\dfrac{1}{q_m(\\alpha )\\left[q_m(\\alpha )+q_{m-1}(\\alpha )\\right]}\\cdot \\dfrac{1}{q_n(\\beta )\\left[q_n(\\beta )+q_{n-1}(\\beta )\\right]}\\\\&=&2s(\\alpha )s(\\beta ),$ where in the second inequality we used that $2q_m(\\alpha )q_n(\\beta )+2q_m(\\alpha )q_{n-1}(\\beta )&>&q_m(\\alpha )q_n(\\beta )+q_m(\\alpha )q_{n-1}(\\beta )+\\\\&&q_{m-1}(\\alpha )q_n(\\beta )+q_{m-1}(\\alpha )q_{n-1}(\\beta )\\\\\\iff \\hspace{14.22636pt} q_m(\\alpha )q_n(\\beta )+q_m(\\alpha )q_{n-1}(\\beta )&>&q_{m-1}(\\alpha )q_n(\\beta )+q_{m-1}(\\alpha )q_{n-1}(\\beta ),$ which is obviously true.",
"For the other inequality, proceed analogously: $s(\\alpha \\beta )&=&\\dfrac{1}{q_{m+n}(\\alpha \\beta )\\left[q_{m+n}(\\alpha \\beta )+q_{m+n-1}(\\alpha \\beta )\\right]}\\\\&>&\\dfrac{1}{2}\\cdot \\dfrac{1}{q_m(\\alpha )q_n(\\beta )\\left[q_{m+n}(\\alpha \\beta )+q_{m+n-1}(\\alpha \\beta )\\right]}\\\\&>&\\dfrac{1}{2}\\cdot \\dfrac{1}{q_m(\\alpha )q_n(\\beta )\\left[q_m(\\alpha )+q_{m-1}(\\alpha )\\right]\\left[q_n(\\beta )+q_{n-1}(\\beta )\\right]}\\,,$ as $q_{m+n}(\\alpha \\beta )+q_{m+n-1}(\\alpha \\beta )&<&\\left[q_m(\\alpha )+q_{m-1}(\\alpha )\\right]\\left[q_n(\\beta )+q_{n-1}(\\beta )\\right]\\\\\\iff \\hspace{5.69046pt} q_{m-1}(\\alpha ){\\tilde{q}}_{n-1}(\\beta )+q_{m-1}(\\alpha ){\\tilde{q}}_{n-2}(\\beta )&<&q_{m-1}(\\alpha )q_n(\\beta )+q_{m-1}(\\alpha )q_{n-1}(\\beta )\\\\\\iff \\hspace{78.24507pt}{\\tilde{q}}_{n-1}(\\beta )+{\\tilde{q}}_{n-2}(\\beta )&<&q_n(\\beta )+q_{n-1}(\\beta ),$ where ${\\tilde{q}}_{n-1}(\\beta )=q_{n-1}(b_2b_3\\cdots b_n)$ and ${\\tilde{q}}_{n-2}(\\beta )=q_{n-2}(b_2b_3\\cdots b_{n-1})$ , and the last inequality is true, since we have ${\\tilde{q}}_{n-1}(\\beta )<q_n(\\beta )$ and ${\\tilde{q}}_{n-2}(\\beta )<q_{n-1}(\\beta )$ .",
"This concludes the proof.",
"$\\Box $ Lemma A3.",
"If ${\\alpha }=(a_1,a_2,\\dots ,a_n)\\in ({\\mathbb {N}}^*)^n$ belongs to $P_r$ and $1\\le a_j\\le T$ for $1\\le j\\le n$ , then $s({\\alpha })>((T+1)^2e^r)^{-1}$ .",
"Proof: We have $r(a_1,a_2,\\dots ,a_{n-1})<r$ , so $s(a_1,a_2,\\dots ,a_{n-1})=(q_{n-1}(q_{n-1}+q_{n-2}))^{-1}>e^{-r}$ , and thus $s({\\alpha })^{-1}=q_n(q_n+q_{n-1})\\le (Tq_{n-1}+q_{n-2})((T+1)q_{n-1}+q_{n-2})<(T+1)q_{n-1}\\cdot (T+1)(q_{n-1}+q_{n-2})=(T+1)^2q_{n-1}(q_{n-1}+q_{n-2})<(T+1)^2e^r$ , so $s({\\alpha })>((T+1)^2e^r)^{-1}$ .",
"$\\Box $ Proposition A1.",
"Let $r, k$ be positive integers and $\\alpha _i, 1\\le i \\le k$ finite sequences which belong to $P_r$ and whose elements are bounded by $T$ .",
"Then, if ${\\alpha }={\\alpha }_1{\\alpha }_2\\cdots {\\alpha }_k$ , we have $s(\\alpha )>(2(T+1)^2 e^r)^{-k}$ .",
"Proof: For $1\\le i\\le k$ we have, by lemma A3, $s({\\alpha }_i)>((T+1)^2e^r)^{-1}$ .",
"So, using the lemma A2, we get $s({\\alpha })=s({\\alpha }_1{\\alpha }_2\\cdots {\\alpha }_k)>\\left(\\frac{1}{2}\\right)^{k-1}s({\\alpha }_1)s({\\alpha }_2)\\cdots s({\\alpha }_k)>$ $>\\left(\\frac{1}{2}\\right)^{k-1}(((T+1)^2e^r)^{-1})^k>(2(T+1)^2 e^r)^{-k}.\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\Box $ References [B] Bugeaud, Y.",
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]
] | 1612.05782 |
[
[
"The geometric lattice of embedded subsets"
],
[
"Abstract This work proposes an alternative approach to the so-called lattice of embedded subsets, which is included in the product of the subset and partition lattices of a finite set, and whose elements are pairs consisting of a subset and a partition where the former is a block of the latter.",
"The lattice structure proposed in a recent contribution relies on ad-hoc definitions of both the join operator and the bottom element, while also including join-irreducible elements distinct from atoms.",
"Conversely, here embedded subsets obtain through a closure operator defined over the product of the subset and partition lattices, where elements are generic pairs of a subset and a partition.",
"Those such pairs that coincide with their closure are precisely embedded subsets, and since the Steinitz exchange axiom is also satisfied, what results is a geometric (hence atomic) lattice given by a simple matroid (or combinatorial geometry) included in the product of the subset and partition lattices (as the partition lattice itself is the polygon matroid defined on the edges of a complete graph).",
"By focusing on its M\\\"obius function, this geometric lattice of embedded subsets of a n-set is shown to be isomorphic to the lattice of partitions of a n+1-set."
],
[
"Introduction",
"For a finite set $N=\\lbrace 1,\\ldots ,n\\rbrace $ , the framework developed in the sequel obtains from both the lattice $(2^N,\\cap ,\\cup )$ of subsets of $N$ and the lattice $(\\mathcal {P}^N,\\wedge ,\\vee )$ of partitions of $N$ .",
"Generic subsets and partitions are denoted, respectively, by $A,B\\in 2^N$ and $P,Q\\in \\mathcal {P}^N$ .",
"Recall that a partition $P=\\lbrace A_1,\\ldots ,A_{|P|}\\rbrace $ is a family of (non-empty) pair-wise disjoint subsets $A_1,\\ldots ,A_{|P|}\\in 2^N$ , called blocks, whose union is $N$ .",
"As usual [2], [18], partitions are (partially) ordered by coarsening $\\geqslant $ , meaning that any $P,Q\\in \\mathcal {P}^N$ satisfy $P\\geqslant Q$ if every block $B\\in Q$ is included in a block $A\\in P$ , i.e.",
"$A\\supseteq B$ .",
"Hence the bottom partition is $P_{\\bot }=\\lbrace \\lbrace 1\\rbrace ,\\ldots ,\\lbrace n\\rbrace \\rbrace $ , while the top one is $P^{\\top }=\\lbrace N\\rbrace $ .",
"Denote by $X^N=2^N\\times \\mathcal {P}^N$ the product of the subset and partition lattices, whose elements $(A,P),(B,Q)\\in X^N$ are pairs consisting of a subset $A,B\\in 2^N$ and a partition $P,Q\\in \\mathcal {P}^N$ .",
"Lattice $(X^N,\\sqcap ,\\sqcup )$ is partially ordered by $\\sqsupseteq $ and naturally obtains through product of the meet, join and order relations for the subset and partition lattices.",
"That is to say, $(A,P)\\sqcap (B,Q)=(A\\cap B,P\\wedge Q)$ as well as $(A,P)\\sqcup (B,Q)=(A\\cup B,P\\vee Q)$ , while $(A,P)\\sqsupseteq (B,Q)\\Leftrightarrow A\\supseteq B,P\\geqslant Q$ , where $P\\wedge Q$ is the coarsest partition finer than both $P,Q$ and $P\\vee Q$ is the finest partition coarser than both $P,Q$ .",
"Thus the bottom element is $(\\emptyset ,P_{\\bot })$ while the top one is $(N,P^{\\top })$ .",
"The existing lattice $(\\mathfrak {C}(N)_{\\bot },\\sqcap ,\\sqcup _*)$ of embedded subsets has elements given by those pairs $(A,P)\\in X^N$ such that $A\\in P$ , together with “an artificial bottom element $\\perp $ , which is introduced for mathematical convenience and that may be considered as $(\\emptyset ,P_{\\bot })$” [6].",
"It is not hard to see that if $A,B\\in 2^N$ and $P,Q\\in \\mathcal {P}^N$ satisfy $A\\in P$ as well as $B\\in Q$ , then non-emptiness $A\\cap B\\ne \\emptyset $ entails $(A\\cap B,P\\wedge Q)\\in \\mathfrak {C}(N)_{\\bot }$ , since $A\\cap B$ is a block of $P\\wedge Q$ .",
"However, in general $A\\cup B$ is not a block of $P\\vee Q$ , i.e.",
"$(A\\cup B,P\\vee Q)\\notin \\mathfrak {C}(N)_{\\bot }$ .",
"For this reason, the existing lattice of embedded subsets $\\mathfrak {C}(N)_{\\bot }$ has the same meet $\\sqcap $ and order $\\sqsupseteq $ applying to product lattice $X^N$ , but the join $\\sqcup _*$ is different from $\\sqcup $ as detailed below.",
"The resulting structure [6] has $n$ atoms of the form $(i,P_{\\bot }),i\\in N$ together with additional $\\binom{n}{2}(n-2)$ join-irreducible elements [2], and thus the lattice is non-atomic (hence non-geometric) precisely because many elements cannot be decomposed as a join of atoms.",
"In fact, the only embedded subsets $(A,P)\\in \\mathfrak {C}(N)_{\\bot }$ admitting such a decomposition [6] are those where partition $P$ is the modular element [2], [15] of the partition lattice whose unique non-singleton block, if any, is precisely $A$ [2] [17].",
"This is denoted by $P=P^A_{\\bot }$ , with $P^A_{\\bot }=\\lbrace A,\\lbrace i_1\\rbrace ,\\ldots ,\\lbrace i_{n-|A|}\\rbrace \\rbrace $ and $\\lbrace i_1,\\ldots ,i_{n-|A|}\\rbrace =N\\backslash A=A^c$ .",
"Since $P^i_{\\bot }=P_{\\bot }$ for all $i\\in N$ and $P^N_{\\bot }=P^{\\top }$ , there are $2^n-n$ such modular partitions $P^A_{\\bot },\\emptyset \\ne A\\in 2^N$ .",
"They play a key role in the approach proposed here, which relies on a closure operator $cl:X^N\\rightarrow X^N$ over poset $(X^N,\\sqsupseteq )$ , meaning [2] that for all $(A,P),(B,Q)\\in X^N$ the following hold: (i) $cl(A,P)\\sqsupseteq (A,P)$ , (ii) $(A,P)\\sqsupseteq (B,Q)\\Rightarrow cl(A,P)\\sqsupseteq cl(B,Q)$ , (iii) $cl(cl(A,P))=cl(A,P)$ .",
"This closure is defined by $cl(\\emptyset ,P)=(\\emptyset ,P)$ for all $P\\in \\mathcal {P}^N$ , while for $\\emptyset \\ne A$ $cl(A,P)=(\\bar{A},P\\vee P^A_{\\bot })$ such that $A\\subseteq \\bar{A}\\in P\\vee P^A_{\\bot }$ .",
"In words, $\\bar{A}$ is the block of $P\\vee P^A_{\\bot }$ satisfying $\\bar{A}\\supseteq A$ , and $P^A_{\\bot }$ is the modular partition where the only non-singleton block, if any, is $A$ .",
"The remainder of this work is concerned with poset $(\\mathcal {E}^N,\\sqsupseteq )$ , with $\\mathcal {E}^N=\\lbrace (A,P):cl(A,P)=(A,P)\\rbrace $ consisting of those pairs that coincide with their closure.",
"As $\\perp $ corresponds to $(\\emptyset ,P_{\\bot })$ (see above), $\\mathfrak {C}(N)_{\\bot }\\subset \\mathcal {E}^N\\subset X^N$ , and in particular $\\mathcal {E}^N\\backslash \\mathfrak {C}(N)_{\\bot }=\\lbrace (\\emptyset ,P):P\\ne P_{\\bot }\\rbrace $ .",
"From the enumerative perspective, $|\\mathfrak {C}(N)_{\\bot }|=1+\\sum _{1\\le k\\le n}k\\mathcal {S}_{n,k}$ , where $\\mathcal {S}_{n,k}$ is the Stirling number of the second kind, i.e.",
"the number of partitions of a $n$ -set into $k$ blocks, while $|\\mathcal {E}^N|=\\sum _{1\\le k\\le n}(k+1)\\mathcal {S}_{n,k}=\\mathcal {B}_{n+1}$ , where $\\mathcal {B}_{n+1}$ is the Bell number, i.e.",
"the number of partitions of a $n+1$ -set [2], [7], [12].",
"The join obtained through closure $cl(\\cdot )$ coincides, over $\\mathfrak {C}(N)_{\\bot }$ , with the existing one $\\sqcup _*$ (defined in [6]), i.e.",
"for $(A,P),(B,Q)\\in \\mathfrak {C}(N)_{\\bot }$ , $cl((A,P)\\sqcup (B,Q))=(A,P)\\sqcup _*(B,Q).$ However, the novel enlarged lattice $\\mathcal {E}^N$ of embedded subsets is substantially different from the existing one $\\mathfrak {C}(N)_{\\bot }$ , as all its $n+\\binom{n}{2}=\\binom{n+1}{2}$ join-irreducible elements (apart from $(\\emptyset ,P_{\\bot })$ ) are atoms.",
"Hence $\\mathcal {E}^N$ is a geometric lattice with the same atoms as product lattice $X^N$ , and in fact the former reproduces within the latter a situation similar to the inclusion of partition lattice $(\\mathcal {P}^N,\\wedge ,\\vee )$ within Boolean lattice $(2^{N_2},\\cap ,\\cup )$ , where $N_2=\\lbrace \\lbrace i,j\\rbrace :1\\le i<j\\le n\\rbrace $ is the $\\binom{n}{2}$ -set whose elements are all $A\\in 2^N$ such that $|A|=2$ .",
"Since $N_2$ is the edge set of the complete graph $K_N=(N,N_2)$ on vertex set $N$ , partition lattice $(\\mathcal {P}^N,\\wedge ,\\vee )$ obtains through the closure $cl:2^{N_2}\\rightarrow 2^{N_2}$ such that for every edge set $E\\in 2^{N_2}$ graph $G^{\\prime }=(N,cl(E))$ obtains from graph $G=(N,E)$ by adding all edges within each component, i.e.",
"each component of $G^{\\prime }$ is a clique (or maximal complete subgraph) [2].",
"Thus partition lattice $\\mathcal {P}^N$ corresponds to the family $\\lbrace E:E=cl(E)\\rbrace \\subset 2^{N_2}$ of edge sets that coincide with their closure, with the resulting structure known as the polygon matroid defined on the edges of the complete graph on vertex set $N$ [2].",
"In the sequel, closure $cl:X^N\\rightarrow X^N$ identifying the novel lattice $\\mathcal {E}^N$ of embedded subsets is shown to provide a simple matroid (or combinatorial geometry) [2].",
"The following Section 2 details how the closure defined by expression (1) above satisfies the Steinitz exchange axiom, thereby showing that the resulting lattice of embedded subsets is geometric.",
"Section 3 identifies the modular elements of this lattice for a $n$ -set, while also highlighting the isomorphism with the lattice of partitions of a $n+1$ -set.",
"Section 4 is devoted to Möbius inversion, determining the Möbius function of the geometric lattice of embedded subsets of a $n$ -set, which equals the Möbius function of the lattice of partitions of a $n+1$ -set.",
"Since functions taking real values on embedded subsets were firstly considered in cooperative game theory (as “games in partition function form”), Section 5 focuses on the (free) vector space of such functions and on the associated Möbius algebra, with emphasis on the vector subspaces identified by those lattice functions whose Möbius inversion lives only on modular elements.",
"Section 6 concludes the paper with some final remarks."
],
[
"Partitions, atoms and closure",
"The rank $r:\\mathcal {P}^N\\rightarrow \\mathbb {Z}_+$ of partitions is $r(P)=n-|P|$ , and atoms are immediately above the bottom element, with rank 1 [2].",
"Hence they consist of $n-1$ blocks, out of which $n-2$ are singletons while the remaining one is a pair.",
"As already outlined, this means that $\\mathcal {P}^N$ has essentially the same $\\binom{n}{2}$ atoms as $2^{N_2}$ .",
"For $1\\le i<j\\le n$ , denote by $[ij]$ the atom of $\\mathcal {P}^N$ whose non-singleton block is pair $\\lbrace i,j\\rbrace $ .",
"A fundamental difference between Boolean lattice $(2^{N_2},\\cap ,\\cup )$ and partition lattice $(\\mathcal {P}^N,\\wedge ,\\vee )$ is in terms of join-decompositions [2].",
"In fact, every $\\lbrace \\lbrace i,j\\rbrace _1,\\ldots ,\\lbrace i,j\\rbrace _{|E|}\\rbrace =E\\in 2^{N_2}$ admits a unique decomposition as a join of atoms, namely $E=\\lbrace i,j\\rbrace _1\\cup \\cdots \\cup \\lbrace i,j\\rbrace _{|E|}$ , which is irredundant.",
"Conversely, a generic partition $P\\in \\mathcal {P}^N$ admits several such decompositions $P=[ij]_1\\vee \\cdots \\vee [ij]_k$ (with $k\\ge r(P)$ ), most of which redundant, while the unique maximal one (in terms of inclusion) clearly is the join of all atoms $[ij]$ such that $[ij]\\leqslant P$ .",
"In particular, $r(P)$ is the minimum number of atoms in a join-decomposition of $P$ , thus any $P=[ij]_1\\vee \\cdots \\vee [ij]_{r(P)}$ is irredundant.",
"The classical examples of semimodular lattices [18] come from sets endowed with a closure operator satisfying the Steinitz exchange axiom.",
"Specifically, set $N_2$ endowed with the closure operator $cl:2^{N_2}\\rightarrow 2^{N_2}$ introduced in Section 1 yields the semimodular lattice $\\mathcal {P}^N$ of partitions.",
"Subsets $E\\in 2^{N_2}$ such that $cl(E)=E$ are said to be closed, and the family of closed subsets forms a complete lattice under inclusion with meet given by intersection and join given by the closure of the union.",
"That is, if $E,E^{\\prime }$ are closed, then $E\\cap E^{\\prime }$ is closed as well, while $cl(E\\cup E^{\\prime })$ is the smallest closed subset containing both $E$ and $E^{\\prime }$ .",
"Furthermore, for all $E\\in 2^{N_2}$ and $\\lbrace i,j\\rbrace ,\\lbrace i,j\\rbrace ^{\\prime }\\in N_2$ , the closure also satisfies the following Steinitz exchange axiom: $\\text{if }\\lbrace i,j\\rbrace \\notin cl(E)\\text{ and }\\lbrace i,j\\rbrace \\in cl(E\\cup \\lbrace i,j\\rbrace ^{\\prime })\\text{ then }\\lbrace i,j\\rbrace ^{\\prime }\\in cl(E\\cup \\lbrace i,j\\rbrace )\\text{.",
"}$ Hence the family of closed subsets is a matroid (included in $2^{N_2}$ ) which, in particular, is said to be simple (or a combinatorial geometry) in that $cl(\\emptyset )=\\emptyset $ and $cl(\\lbrace i,j\\rbrace )=\\lbrace i,j\\rbrace $ for all $\\lbrace i,j\\rbrace \\in N_2$ [2].",
"As already outlined, a simple matroid also obtains as the novel enlarged lattice $\\mathcal {E}^N$ of embedded subsets, whose elements are closed pairs, i.e.",
"pairs $(A,P)\\in X^N$ that coincide with their closure $cl(A,P)$ as defined by expression (1) above.",
"To see this, firstly consider the covering relation between closed pairs, denoted by $\\sqsupset ^*$ , i.e.",
"for $(A,P),(B,Q)\\in \\mathcal {E}^N$ , it holds $(A,P)\\sqsupset ^*(B,Q)$ whenever $\\lbrace (B^{\\prime },Q^{\\prime }):(A,P)\\sqsupseteq (B^{\\prime },Q^{\\prime })\\sqsupseteq (B,Q),(B^{\\prime },Q^{\\prime })\\in \\mathcal {E}^N\\rbrace =\\lbrace (A,P),(B,Q)\\rbrace $ (see [2]).",
"The general situation is when either $A\\ne \\emptyset \\ne B$ or else $A=\\emptyset =B$ , in which case $(A,P)\\sqsupset ^*(B,Q)$ attains whenever $P\\gtrdot Q$ , where $\\gtrdot $ denotes the covering relation between partitions, i.e.",
"$P$ obtains by merging two blocks of $Q$ into a unique block.",
"In addition, it can also be $A\\ne \\emptyset =B$ , and in this case $(A,P)\\sqsupset ^*(B,Q)$ attains for $P=Q$ .",
"This means that $\\mathcal {E}^N$ has $n+\\binom{n}{2}=\\binom{n+1}{2}$ atoms, in that the bottom pair $(\\emptyset ,P_{\\bot })$ is covered by $n$ embedded subsets $(i,P_{\\bot })$ for $i\\in N$ , and by further $\\binom{n}{2}$ embedded subsets $(\\emptyset ,[ij])$ for $\\lbrace i,j\\rbrace \\in N_2$ .",
"Let $\\mathcal {A}_1=\\lbrace (i,P_{\\bot }):i\\in N\\rbrace $ and $\\mathcal {A}_2=\\lbrace (\\emptyset ,[ij]):\\lbrace i,j\\rbrace \\in N_2\\rbrace $ .",
"Hence the set of atoms of $\\mathcal {E}^N$ is $\\mathcal {A}=\\mathcal {A}_1\\cup \\mathcal {A}_2$ .",
"The closure operator defined by expression (1) above can now be regarded as a mapping $cl:2^{\\mathcal {A}}\\rightarrow 2^{\\mathcal {A}}$ , with closed subsets of $\\mathcal {A}$ being the elements of lattice $\\mathcal {E}^N$ .",
"In this view, consider a non-closed subset $S\\in \\mathcal {A}$ , with $S=S_1\\cup S_2,S_1\\in 2^{\\mathcal {A}_1},S_2\\in 2^{\\mathcal {A}_2}$ .",
"Firstly focus on the simple case where $S_1=\\emptyset $ , thus $S=S_2=\\lbrace (\\emptyset ,[ij]_1),\\ldots ,(\\emptyset ,[ij]_{|S_2|})\\rbrace $ .",
"Here the closure $cl(S)$ of $S$ obtains by adding all atoms $(\\emptyset ,[ij])\\in \\mathcal {A}_2$ such that $[ij]$ is finer than $[ij]_1\\vee \\cdots \\vee [ij]_{|S_2|}$ , i.e.",
"$cl(S)=S\\cup \\lbrace (\\emptyset ,[ij]):[ij]\\leqslant [ij]_1\\vee \\cdots \\vee [ij]_{|S_2|}\\rbrace $ .",
"Hence for $S_1=\\emptyset $ the closure $cl:2^{\\mathcal {A}}\\rightarrow 2^{\\mathcal {A}}$ applying to embedded subsets reproduces the closure $cl:2^{N_2}\\rightarrow 2^{N_2}$ applying to partitions described above.",
"For the general case $\\lbrace (i_1,P_{\\bot }),\\ldots ,(i_{|S_1|},P_{\\bot })\\rbrace =S_1\\ne \\emptyset \\ne S_2=\\lbrace (\\emptyset ,[ij]_1),\\ldots ,(\\emptyset ,[ij]_{|S_2|})\\rbrace $ , let $\\lbrace i_1,\\ldots ,i_{|S_1|}\\rbrace =A$ and $[ij]_1\\vee \\cdots \\vee [ij]_{|S_2|}=P$ , with $(A,P)\\in X^N$ .",
"In view of the closure applying to partitions, like in the previous case, if $[ij]\\leqslant P$ , then $(\\emptyset ,[ij])\\in cl(S)$ .",
"At this point, the closure applying to embedded subsets defined in Section 1 works as follows.",
"If $A\\in P$ (entailing $P=P\\vee P^A_{\\bot }$ or, equivalently, $P\\geqslant P^A_{\\bot }$ , of course), then the only atoms to be added are those just described, i.e.",
"$cl(S)=S\\cup \\lbrace (\\emptyset ,[ij]):[ij]\\leqslant P\\rbrace $ , and $(A,P)\\in \\mathcal {E}^N$ .",
"Otherwise, for $k\\le |P|$ , let $B_1,\\cdots ,B_k\\in P$ be all the blocks of $P$ whose intersection with $A$ is non-empty.",
"That is, $\\lbrace B:B\\in P,B\\cap A\\ne \\emptyset \\rbrace =\\lbrace B_1,\\ldots ,B_k\\rbrace $ .",
"Then, $\\bar{A}=B_1\\cup \\cdots \\cup B_k$ is a block of partition $P\\vee P^A_{\\bot }$ , with proper inclusion $\\bar{A}\\supset A$ , and therefore $(\\bar{A},P\\vee P^A_{\\bot })\\in \\mathcal {E}^N$ is the minimal (in terms of $\\sqsupseteq $ ) embedded subset $(B,Q)$ satisfying $(B,Q)\\sqsupseteq (A,P)$ .",
"On the other hand, the smallest closed subset of $\\mathcal {A}$ containing $S$ is $cl(S)=\\lbrace (i,P_{\\bot }):i\\in \\bar{A}\\rbrace \\cup \\lbrace (\\emptyset ,[ij]):[ij]\\leqslant P\\vee P^A_{\\bot }\\rbrace $ .",
"Proposition 1 Closure $cl:2^{\\mathcal {A}}\\rightarrow 2^{\\mathcal {A}}$ satisfies the Steinitz exchange axiom.",
"Proof: For $(A,P)\\in \\mathcal {E}^N$ , let atoms $(i,P_{\\bot }),(i^{\\prime },P_{\\bot })\\in \\mathcal {A}_1,(\\emptyset ,[ij]),(\\emptyset ,[ij]^{\\prime })\\in \\mathcal {A}_2$ satisfy $(A,P)\\lnot \\sqsupseteq (i,P_{\\bot }),(i^{\\prime },P_{\\bot }),(\\emptyset ,[ij]),(\\emptyset ,[ij]^{\\prime })$ , i.e.",
"$i,i^{\\prime }\\notin A$ and $[ij],[ij]^{\\prime }\\lnot \\leqslant P$ .",
"In terms of subsets of atoms, let $S_{AP}=\\lbrace (\\tilde{i},P_{\\bot }):\\tilde{i}\\in A\\rbrace \\cup \\lbrace (\\emptyset ,\\tilde{[ij]}):\\tilde{[ij]}\\leqslant P\\rbrace $ , hence $S_{AP}=cl(S_{AP})$ and $(i,P_{\\bot }),(i^{\\prime },P_{\\bot }),(\\emptyset ,[ij]),(\\emptyset ,[ij]^{\\prime })\\notin S_{AP}$ .",
"Two cases $(i^{\\prime },P_{\\bot }),(\\emptyset ,[ij]^{\\prime })\\in cl(S_{AP}\\cup (i,P_{\\bot }))$ and $(i^{\\prime },P_{\\bot }),(\\emptyset ,[ij]^{\\prime })\\in cl(S_{AP}\\cup (\\emptyset ,[ij]))$ must be distinguished.",
"In the first one, there is a block $A^{\\prime }\\in P,A^{\\prime }\\ne A$ such that $i\\in A^{\\prime }$ , and both $i^{\\prime }\\in A^{\\prime }$ and $|A\\cap \\lbrace i,j\\rbrace ^{\\prime }|=1=|A^{\\prime }\\cap \\lbrace i,j\\rbrace ^{\\prime }|$ hold.",
"Therefore, $cl(S_{AP}\\cup (i^{\\prime },P_{\\bot }))=cl(S_{AP}\\cup (i,P_{\\bot }))=cl(S_{AP}\\cup (\\emptyset ,[ij]^{\\prime }))$ , entailing that the exchange axiom is satisfied.",
"Analogously, in the second case there is a block $A^{\\prime }\\in P,A^{\\prime }\\ne A$ such that $|A\\cap \\lbrace i,j\\rbrace |=1=|A^{\\prime }\\cap \\lbrace i,j\\rbrace |$ , and both $i^{\\prime }\\in A^{\\prime }$ and $|A\\cap \\lbrace i,j\\rbrace ^{\\prime }|=1=|A^{\\prime }\\cap \\lbrace i,j\\rbrace ^{\\prime }|$ hold.",
"Hence again the exchange axiom is satisfied as $cl(S_{AP}\\cup (i^{\\prime },P_{\\bot }))=cl(S_{AP}\\cup (\\emptyset ,[ij]))=cl(S_{AP}\\cup (\\emptyset ,[ij]^{\\prime }))$ .",
"Hence, $\\mathcal {E}^N$ is a matroid and, in particular, a simple one, in that all $\\binom{n+1}{2}+1$ subsets of atoms given by the empty set and the $\\binom{n+1}{2}$ singletons are closed.",
"Proposition 2 Poset $(\\mathcal {E}^N,\\sqsupseteq )$ is a geometric lattice where the meet and join of any $(A,P),(B,Q)\\in \\mathcal {E}^N$ are $(A,P)\\sqcap (B,Q)$ and $cl((A,P)\\sqcup (B,Q))$ , respectively.",
"Proof: Since $|\\mathcal {E}^N|<\\infty $ , it only has to be verified [2] that for all $(A,P),(B,Q)\\in \\mathcal {E}^N$ the following hold: - if $(A,P)\\sqsupset ^*(B,Q)$ , then there is an atom $(i^{\\prime },P_{\\bot })\\in \\mathcal {A}_1,(B,Q)\\lnot \\sqsupseteq (i^{\\prime },P_{\\bot })$ and/or $(\\emptyset ,[ij])\\in \\mathcal {A}_2,(B,Q)\\lnot \\sqsupseteq (\\emptyset ,[ij])$ such that $(A,B)=cl((B,Q)\\sqcup (i^{\\prime },P_{\\bot }))$ and/or $(A,B)=cl((B,Q)\\sqcup (\\emptyset ,[ij]))$ , - if $(A,B)=cl((B,Q)\\sqcup (i^{\\prime },P_{\\bot }))$ and/or $(A,B)=cl((B,Q)\\sqcup (\\emptyset ,[ij]))$ for some atom $(i^{\\prime },P_{\\bot })\\in \\mathcal {A}_1,(B,Q)\\lnot \\sqsupseteq (i^{\\prime },P_{\\bot })$ and/or $(\\emptyset ,[ij])\\in \\mathcal {A}_2,(B,Q)\\lnot \\sqsupseteq (\\emptyset ,[ij])$ , then $(A,P)\\sqsupset ^*(B,Q)$ .",
"Both conditions are trivially satisfied if $(B,Q)$ is the bottom element, hence let $(B,Q)\\ne (\\emptyset ,P_{\\bot })$ .",
"Then $(A,P)\\sqsupset ^*(B,Q)$ attains for $P\\gtrdot Q,A\\supseteq B$ .",
"If $A=B$ , then $(A,P)=cl((B,Q)\\sqcup (\\emptyset ,[ij]))$ for all atoms $(\\emptyset ,[ij])\\in \\mathcal {A}_2,(B,Q)\\lnot \\sqsupseteq (\\emptyset ,[ij])$ such that $\\lbrace i,j\\rbrace \\cap B=\\emptyset $ (where of course $Q\\lnot \\geqslant [ij]$ ), while proper inclusion $A\\supset B$ entails that for some block $B^{\\prime }\\in Q,B^{\\prime }\\ne B$ the covering embedded subset $(A,P)$ obtains as $(A,P)=cl((B,Q)\\sqcup (\\emptyset ,[ij]))=cl((B,Q)\\sqcup (i^{\\prime },P_{\\bot }))$ for all atoms $(i^{\\prime },P_{\\bot })\\in \\mathcal {A}_1,(B,Q)\\lnot \\sqsupseteq (i^{\\prime },P_{\\bot }),(\\emptyset ,[ij])\\in \\mathcal {A}_2,(B,Q)\\lnot \\sqsupseteq (\\emptyset ,[ij])$ such that $i^{\\prime }\\in B^{\\prime }$ and $|B\\cap \\lbrace i,j\\rbrace |=1=|B^{\\prime }\\cap \\lbrace i,j\\rbrace |$ .",
"Accordingly, $A=B\\cup B^{\\prime }$ .",
"Turning to the second condition, for all $(i^{\\prime },P_{\\bot })\\in \\mathcal {A}_1,(B,Q)\\lnot \\sqsupseteq (i^{\\prime },P_{\\bot })$ , $(\\emptyset ,[ij])\\in \\mathcal {A}_2,(B,Q)\\lnot \\sqsupseteq (\\emptyset ,[ij])$ it is easily checked that both $cl((B,Q)\\sqcup (i^{\\prime },P_{\\bot }))$ and $cl((B,Q)\\sqcup (\\emptyset ,[ij]))$ cover $(B,Q)$ .",
"In fact, the general case is where there is a block $B^{\\prime }\\in Q,B^{\\prime }\\ne B$ such that $i^{\\prime }\\in B^{\\prime }$ and $|B^{\\prime }\\cap \\lbrace i,j\\rbrace |=1=|B\\cap \\lbrace i,j\\rbrace |$ , entailing $cl((B,Q)\\sqcup (i^{\\prime },P_{\\bot }))=cl((B,Q)\\sqcup (\\emptyset ,[ij]))=(B\\cup B^{\\prime },Q\\vee P^{B\\cup B^{\\prime }}_{\\bot })$ , with $Q\\vee P^{B\\cup B^{\\prime }}_{\\bot }$ obtained by merging blocks $B$ and $B^{\\prime }$ in $Q$ , hence $Q\\vee P^{B\\cup B^{\\prime }}_{\\bot }\\gtrdot Q$ .",
"Yet, like for the first condition, if $B\\cap \\lbrace i,j\\rbrace =\\emptyset $ , then there are blocks $B^{\\prime },B^{\\prime \\prime }\\in Q$ , with $B^{\\prime }\\ne B\\ne B^{\\prime \\prime }$ , such that $|B^{\\prime }\\cap \\lbrace i,j\\rbrace |=1=|B^{\\prime \\prime }\\cap \\lbrace i,j\\rbrace |$ .",
"Accordingly, $cl((B,Q)\\sqcup (\\emptyset ,[ij]))=(B,Q^{\\prime })$ , where $Q^{\\prime }$ obtains by merging blocks $B^{\\prime },B^{\\prime \\prime }$ in $Q$ , i.e.",
"$Q^{\\prime }\\gtrdot Q$ .",
"Having recognized the lattice structure characterizing $\\mathcal {E}^N$ , it may now be readily seen that the Jordan-Dedekind chain condition [2] is satisfied, and the rank function $r:\\mathcal {E}^N\\rightarrow \\mathbb {Z}_+$ is $r(A,P)=r(P)+\\min \\lbrace |A|,1\\rbrace $ .",
"In fact, for any $(A,P),(B,Q)\\in \\mathcal {E}^N$ such that $(A,P)\\sqsupset (B,Q)$ (that is, $(A,P)\\sqsupseteq (B,Q)$ , $(A,P)\\ne (B,Q)$ ), all maximal chains $\\lbrace (B^{\\prime },Q^{\\prime })_0,(B^{\\prime },Q^{\\prime })_1,\\ldots ,(B^{\\prime },Q^{\\prime })_k\\rbrace $ from $(B,Q)=(B^{\\prime },Q^{\\prime })_0$ to $(A,P)=(B^{\\prime },Q^{\\prime })_k$ (i.e.",
"$(B^{\\prime },Q^{\\prime })_{k^{\\prime }+1}\\sqsupset ^*(B^{\\prime },Q^{\\prime })_{k^{\\prime }}$ for $0\\le k^{\\prime }<k$ ) have same length $k$ , in that $(B^{\\prime },Q^{\\prime })_{k^{\\prime }+1}=cl((B^{\\prime },Q^{\\prime })_{k^{\\prime }}\\sqcup (i,P_{\\bot }))$ and/or $(B^{\\prime },Q^{\\prime })_{k^{\\prime }+1}=cl((B^{\\prime },Q^{\\prime })_{k^{\\prime }}\\sqcup (\\emptyset ,[ij]))$ with atoms $(i,P_{\\bot }),(\\emptyset ,[ij])$ such that $(B^{\\prime },Q^{\\prime })_{k^{\\prime }+1}\\sqsupseteq (i,P_{\\bot }),(\\emptyset ,[ij])$ while $(B^{\\prime },Q^{\\prime })_{k^{\\prime }}\\lnot \\sqsupseteq (i,P_{\\bot }),(\\emptyset ,[ij])$ ."
],
[
"Modular elements and isomorphism",
"By definition (of modular elements of geometric lattices [2]), modular embedded subsets are those $(A,P)\\in \\mathcal {E}^N$ such that for all $(B,Q)\\in \\mathcal {E}^N$ it holds $r(A,P)+r(B,Q)=r(cl((A,P)\\sqcup (B,Q)))+r((A,P)\\sqcap (B,Q))$ .",
"For $Q\\in \\mathcal {P}^N$ and $\\emptyset \\ne A\\in 2^N$ , let $Q^A=\\lbrace B\\cap A:B\\in Q,B\\cap A\\ne \\emptyset \\rbrace $ denote the partition of $A$ induced by $Q$ .",
"Proposition 3 An embedded subset $(A,P)\\in \\mathcal {E}^N$ is modular if and only if $P$ is a modular partition and either $A=\\emptyset $ , or else $A$ is the non-singleton block of $P$ when $P\\ne P_{\\bot }$ .",
"Proof: For the ‘if’ part, the bottom element $(\\emptyset ,P_{\\bot })$ is evidently modular.",
"Accordingly, firstly consider embedded subset $(\\emptyset ,P^A_{\\bot })$ , with $|A|>1$ .",
"The rank is $r(\\emptyset ,P^A_{\\bot })=n-(n-|A|+1)=|A|-1$ .",
"On the other hand, any $(B,Q)$ has rank $n-|Q|+1$ (since the case $B=\\emptyset $ is trivial precisely because $P^A_{\\bot }$ is a modular partition).",
"Thus $r(\\emptyset ,P^A_{\\bot })+r(B,Q)=n-|Q|+|A|$ .",
"Now consider that the meet has rank $r((\\emptyset ,P^A_{\\bot })\\sqcap (B,Q))=n-(|Q^A|+n-|A|)=|A|-|Q^A|$ , while the (closure of the) join yields $r(cl((\\emptyset ,P^A_{\\bot })\\sqcup (B,Q)))=n-(|Q|-|Q^A|+1)+1=n-|Q|+|Q^A|$ .",
"Therefore, $|A|-|Q^A|+n-|Q|+|Q^A|=n-|Q|+|A|$ as desired.",
"In the general case $(A,P^A_{\\bot })$ , with $r(A,P^A_{\\bot })=r(\\emptyset ,P^A_{\\bot })+1=|A|$ , the sum is $r(A,P^A_{\\bot })+r(B,Q)=n-|Q|+1+|A|$ .",
"The rank $r((A,P^A_{\\bot })\\sqcap (B,Q))$ of the meet equals $|A|-|Q^A|$ if $A\\cap B=\\emptyset $ and $|A|-|Q^A|+1$ if $A\\cap B\\ne \\emptyset $ .",
"Correspondingly, the rank $r(cl((A,P^A_{\\bot })\\sqcup (B,Q)))$ of the join equals $n-(|Q|-|Q^A|+1-1)+1=n-|Q|+|Q^A|+1$ if $A\\cap B=\\emptyset $ and $n-(|Q|-|Q^A|+1)+1=n-|Q|+|Q^A|$ if $A\\cap B\\ne \\emptyset $ .",
"Thus the sought sum is $|A|-|Q^A|+n-|Q|+|Q^A|+1=n-|Q|+1+|A|$ in the former case and $|A|-|Q^A|+1+n-|Q|+|Q^A|=n-|Q|+1+|A|$ in the latter, i.e.",
"the same.",
"Turning to the ‘only if’ part, it is not difficult to check that, just like the rank of partitions, the rank of embedded subsets is submodular, that is to say $r(A,P)+r(B,Q)\\ge r(cl((A,P)\\sqcup (B,Q)))+r((A,P)\\sqcap (B,Q))$ , with strict inequality whenever $(A,P),(B,Q)$ is not a modular pair [2] of lattice elements.",
"Let $N_+=\\lbrace 1,\\ldots ,n,n+1\\rbrace $ , i.e.",
"$N_+=N\\cup \\lbrace n+1\\rbrace $ and denote by $\\mathcal {P}^{N_+}$ the lattice of partitions of $N_+$ , with generic element denoted by $P^+\\in \\mathcal {P}^{N_+}$ .",
"From Section 1, $|\\mathcal {E}^N|=\\mathcal {B}_{n+1}=|\\mathcal {P}^{N_+}|$ .",
"In fact, there is a lattice isomorphism $f:\\mathcal {E}^N\\rightarrow \\mathcal {P}^{N_+}$ , meaning that for all $(A,P),(B,Q)\\in \\mathcal {E}^N$ the following holds: $f((A,P)\\sqcap (B,Q))&=&f(A,P)\\wedge f(B,Q)\\text{,}\\\\f(cl((A,P)\\sqcup (B,Q)))&=&f(A,P)\\vee f(B,Q)\\text{.",
"}$ To see this, consider that any partition $\\lbrace A_1,\\ldots ,A_{|P|}\\rbrace =P\\in \\mathcal {P}^N$ has associated $|P|+1$ partitions $P^+_1,\\ldots ,P^+_{|P|},P^+_{|P|+1}\\in \\mathcal {P}^{N_+}$ , all inducing the same partition $P$ of $N$ .",
"The first $|P|$ ones $P^+_1,\\ldots ,P^+_{|P|}$ obtain each by placing additional element $n+1$ into block $A_k\\in P,1\\le k\\le |P|$ , while the last one $P^+_{|P|+1}$ obtains by placing $n+1$ into a singleton block $\\lbrace n+1\\rbrace \\in P^+_{|P|+1}$ .",
"Of course, the former are pair-wise incomparable, i.e.",
"$P^+_k\\lnot \\geqslant P^+_{k^{\\prime }}\\lnot \\geqslant P^+_k$ for $1\\le k<k^{\\prime }\\le |P|$ , while the latter is strictly finer than all the others: $P^+_k>P^+_{|P|+1},1\\le k\\le |P|$ .",
"The same applies to embedded subsets $(A_1,P),\\ldots ,(A_{|P|},P),(\\emptyset ,P)\\in \\mathcal {E}^N$ , in that $(A_k,P)\\lnot \\sqsupseteq (A_{k^{\\prime }},P)\\lnot \\sqsupseteq (A_k,P)$ for $1\\le k<k^{\\prime }\\le |P|$ as well as $(A_k,P)\\sqsupset (\\emptyset ,P)$ , $1\\le k\\le |P|$ .",
"In other terms, letting $P=\\lbrace A,B_1,\\ldots ,B_{|P|-1}\\rbrace \\in \\mathcal {P}^N$ for all $(A,P)\\in \\mathcal {E}^N$ such that $A\\ne \\emptyset $ , isomorphism $f:\\mathcal {E}^N\\rightarrow \\mathcal {P}^{N_+}$ is defined by $f(A,P)=P^+\\text{ such that }P^+=\\lbrace A\\cup \\lbrace n+1\\rbrace ,B_1,\\ldots ,B_{|P|-1}\\rbrace \\text{.",
"}$ On the other hand, for every $P=\\lbrace B_1,\\ldots ,B_{|P|}\\rbrace \\in \\mathcal {P}^N$ and $(\\emptyset ,P)\\in \\mathcal {E}^N$ , $f(\\emptyset ,P)=P^+\\text{ such that }P^+=\\lbrace \\lbrace n+1\\rbrace ,B_1,\\ldots ,B_{|P|}\\rbrace \\text{.",
"}$ Evidently, in this way the $2^{n+1}-(n+1)$ modular elements of $\\mathcal {P}^{N_+}$ bijectively correspond to the $2^n-n+2^n-1=2^{n+1}-(n+1)$ modular elements of $\\mathcal {E}^N$ ."
],
[
"Möbius inversion",
"Möbius inversion applies to locally finite posets [13].",
"In particular, for a finite lattice $(L,\\wedge ,\\vee )$ ordered by $\\geqslant $ , with generic elements $x,y,z\\in L$ and bottom element $x_{\\bot }$ , any lattice function $f: L\\rightarrow \\mathbb {R}$ has Möbius inversion (from below) $\\mu ^f:L\\rightarrow \\mathbb {R}$ given by $\\mu ^f(x)=\\sum _{x_{\\bot }\\leqslant y\\leqslant x}\\mu _L(y,x)f(y)$ , where $\\mu _L$ is the Möbius function of $L$ , defined recursively on ordered pairs $(y,x)\\in L\\times L$ by $\\mu _L(y,x)=1$ if $y=x$ as well as $\\mu _L(y,x)=-\\sum _{y\\leqslant z<x}\\mu _L(y,z)$ if $y<x$ (i.e.",
"$y\\leqslant x$ and $y\\ne x$ ), while $\\mu _L(y,x)=0$ if $y\\lnot \\leqslant x$ .",
"Accordingly, $f$ and $\\mu ^f$ are related by $f(x)=\\sum _{y\\leqslant x}\\mu ^f(y)$ for all $x\\in L$ .",
"The Möbius function of the subset lattice is $\\mu _{2^N}(B,A)=(-1)^{|A\\backslash B|}$ , with $B\\subseteq A$ .",
"Concerning the Möbius function of $\\mathcal {P}^N$ , given any two partitions $P,Q\\in \\mathcal {P}^N$ , if $Q<P=\\lbrace A_1,\\ldots ,A_{|P|}\\rbrace $ , then for every $A\\in P$ there are $B_1,\\ldots ,B_{k_A}\\in Q$ such that $A=B_1\\cup \\cdots \\cup B_{k_A}$ , with $k_A>1$ for at least one $A\\in P$ .",
"Segment (or interval) $[Q,P]=\\lbrace P^{\\prime }:Q\\leqslant P^{\\prime }\\leqslant P\\rbrace $ is isomorphic to product $\\times _{A\\in P}\\mathcal {P}^{k_A}$ , where $\\mathcal {P}^k$ denotes the lattice of partitions of a $k$ -set.",
"Let $c^{QP}_k=|\\lbrace A:A\\in P,k_A=k\\rbrace |,1\\le k\\le n$ be the number of blocks of $P$ obtained as the union of $k$ blocks of $Q$ , where $c^{QP}=(c^{QP}_1,\\ldots ,c^{QP}_n)\\in \\mathbb {Z}_+^n$ is the class (or type) of segment $[Q,P]$ .",
"Then [13], $\\mu _{\\mathcal {P}^N}(Q,P)=(-1)^{-n+\\sum _{1\\le k\\le n}c^{QP}_k}\\prod _{1<k<n}(k!)^{c^{QP}_{k+1}}\\text{.",
"}$ This result obtains since the Möbius function is multiplicative [2] [9] [3] [17], which also entails here that for any two pairs $(A,P),(B,Q)\\in X^N$ the Möbius function $\\mu _{X^N}$ of $X^N$ is given by $\\mu _{X^N}((B,Q),(A,P))=\\mu _{2^N}(B,A)\\mu _{\\mathcal {P}^N}(Q,P)$ .",
"Another fundamental result in the present setting concerns the link between the Möbius function and a closure operator [2] [9] [8] [13] [17].",
"For closed pairs $(A,P),(B,Q)\\in \\mathcal {E}^N$ , and denoting generic pairs by $(A^{\\prime },P^{\\prime })\\in X^N$ , the Möbius function $\\mu _{\\mathcal {E}^N}$ of $\\mathcal {E}^N$ is $\\mu _{\\mathcal {E}^N}((B,Q),(A,P))=\\sum _{\\underset{cl(A^{\\prime },P^{\\prime })=(A,P)}{(A^{\\prime },P^{\\prime })\\in X^N}}\\mu _{X^N}((B,Q),(A^{\\prime },P^{\\prime }))\\text{.",
"}$ The Möbius function $\\mu _{\\mathcal {E}^N}$ of $\\mathcal {E}^N$ is now determined through recursion following [13].",
"Of course, this leads to re-obtain the isomorphism observed in Section 3.",
"Firstly some further isomorphisms concerning segments $[(B,Q),(A,P)]\\subseteq \\mathcal {E}^N$ have to be highlighted.",
"Extending the notation introduced above, for $0\\le m\\le n$ denote by $\\mathcal {P}^m$ the lattice of partitions of a $m$ -set.",
"Also let $\\mathcal {E}^m$ denote the lattice of embedded subsets of a $m$ -set, hence $\\mathcal {E}^0=\\lbrace (\\emptyset ,\\emptyset )\\rbrace $ consists of only one element.",
"Similarly, recall that the recursions for the Stirling numbers of the second kind and for the Bell numbers [2] rely on the convention that the number of partitions of the empty set is 1, i.e.",
"$\\mathcal {P}^0=\\lbrace \\emptyset \\rbrace $ .",
"Finally, set $\\mu _{\\mathcal {P}^m}=\\mu _{\\mathcal {P}^m}(\\bot ,\\top )$ and $\\mu _{\\mathcal {E}^m}=\\mu _{\\mathcal {E}^m}(\\bot ,\\top )$ , where $\\bot $ and $\\top $ are the bottom and top elements of the corresponding lattice, with $\\mu _{\\mathcal {P}^m}=(-1)^{m-1}(m-1)!$ from [13], and $\\mu _{\\mathcal {P}^0}=1=\\mu _{\\mathcal {E}^0}$ (see [17]; also, $-1!=-1$ [7]).",
"Now consider any non-empty segment $[(B,Q),(A,P)]\\subseteq \\mathcal {E}^N$ , that is to say $(A,P)\\sqsupseteq (B,Q)$ .",
"Firstly, if $B=A$ , then $[(B,Q),(A,P)]\\cong [Q^{A^c},P^{A^c}]$ , where $\\cong $ denotes isomorphism and $[Q^{A^c},P^{A^c}]\\subseteq \\mathcal {P}^{|A^c|}$ , hence $\\mu _{\\mathcal {E}^N}((B,Q),(A,P))$ is given by expression (2), i.e.",
"$\\mu _{\\mathcal {E}^N}((B,Q),(A,P))&=&\\mu _{\\mathcal {P}^{|A^c|}}(Q^{A^c},P^{A^c})\\\\&=&(-1)^{-|A^c|+\\sum _{1\\le k\\le |A^c|}c^{Q^{A^c}P^{A^c}}_k}\\prod _{1<k<|A^c|}(k!",
")^{c^{Q^{A^c}P^{A^c}}_{k+1}}\\text{,}$ with $c^{Q^{A^c}P^{A^c}}\\in \\mathbb {Z}_+^{|A^c|}$ being the class of segment $[Q^{A^c},P^{A^c}]$ .",
"Secondly, consider the case where $\\emptyset \\ne B\\subset A$ .",
"Letting $B=B^A_1$ , partition $Q$ has form $Q=\\lbrace B_1^A,\\ldots ,B_{|Q^A|}^A\\rbrace \\cup Q^{A^c}$ , where $\\lbrace B_1^A,\\ldots ,B_{|Q^A|}^A\\rbrace =Q^A$ .",
"It is readily recognized that for any atom $(i,P_{\\bot })\\in \\mathcal {A}_1$ of $\\mathcal {E}^N$ (see Section 2), segment $[(i,P_{\\bot }),(N,P^{\\top })]\\subset \\mathcal {E}^N$ is isomorphic to $\\mathcal {P}^N$ (see also [6]).",
"Analogously, denoting by $\\lbrace A\\rbrace $ the top element of $\\mathcal {P}^{|A|}$ , segment $[(B,Q^A),(A,\\lbrace A\\rbrace )]\\subset \\mathcal {E}^{|Q^A|}$ is isomorphic to $\\mathcal {P}^{|A|}$ .",
"Therefore, $\\mu _{\\mathcal {E}^N}((B,Q),(A,P))&=&\\mu _{\\mathcal {P}^{|Q^A|}}\\mu _{\\mathcal {P}^{|A^c|}}(Q^{A^c},P^{A^c})\\\\&=&(-1)^{|Q^A|-1}(|Q^A|-1)!\\mu _{\\mathcal {P}^{|A^c|}}(Q^{A^c},P^{A^c})\\text{.",
"}$ The third and most relevant case is when $\\emptyset =B\\subset A$ .",
"Like in the previous case, partition $Q$ has form $Q^A\\cup Q^{A^c}$ , and thus $[(\\emptyset ,Q^A),(A,\\lbrace A\\rbrace )]\\cong \\mathcal {E}^{|Q^A|}$ .",
"Accordingly, $[(\\emptyset ,Q),(A,P)]\\cong \\mathcal {E}^{|Q^A|}\\times [Q^{A^c},P^{A^c}]$ , entailing $\\mu _{\\mathcal {E}^N}((B,Q),(A,P))=\\mu _{\\mathcal {E}^{|Q^A|}}\\mu _{\\mathcal {P}^{|A^c|}}(Q^{A^c},P^{A^c})\\text{.",
"}$ Hence the Möbius function $\\mu _{\\mathcal {E}^N}:\\mathcal {E}^N\\times \\mathcal {E}^N\\rightarrow \\mathbb {Z}$ is fully determined once $\\mu _{\\mathcal {E}^N}((\\emptyset ,P_{\\bot }),(N,P^{\\top }))=\\mu _{\\mathcal {E}^n}$ is known.",
"To this end, focus on the dual atoms of $\\mathcal {E}^N$ , i.e.",
"closed pairs covered by the top element $(N,P^{\\top })$ .",
"There are $2^n-2$ such dual atoms with form $(A,\\lbrace A,A^c\\rbrace )$ for $\\emptyset \\ne A\\ne N$ , plus a further one with form $(\\emptyset ,P^{\\top })$ , where $2^n-1$ is precisely the number of dual atoms of $\\mathcal {P}^{n+1}$ .",
"Also recall that from Weisner's theorem [3] [13] [17], for any $(A,P),(B,Q)\\in \\mathcal {E}^N$ with $(N,P^{\\top })\\sqsupset (A,B)$ $\\sum _{\\underset{(A^{\\prime },P^{\\prime })\\sqcap (A,P)=(B,Q)}{(A^{\\prime },P^{\\prime })\\in \\mathcal {E}^N}}\\mu _{\\mathcal {E}^N}((A^{\\prime },P^{\\prime }),(N,P^{\\top }))=0\\text{.",
"}$ Proposition 4 The Möbius function $\\mu _{\\mathcal {E}^n}$ of the lattice of embedded subsets of a $n$ -set is $\\mu _{\\mathcal {E}^n}=\\mu _{\\mathcal {P}^{n+1}}=(-1)^nn!$ .",
"Proof: In expression (4), let $(B,Q)=(\\emptyset ,P_{\\bot })$ and choose $(A,P)$ to be a dual atom of the form $(N\\backslash i,\\lbrace N\\backslash i,i\\rbrace )$ for $i\\in N$ .",
"Then, $(A^{\\prime },P^{\\prime })\\sqcap (A,P)=(\\emptyset ,P_{\\bot })$ entails that $(A^{\\prime },P^{\\prime })\\in \\mathcal {A}$ is an atom and, in particular, either $(A^{\\prime },P^{\\prime })\\in \\mathcal {A}_2$ has form $(\\emptyset ,[ij])$ with $j\\in N\\backslash i$ , or else $(A^{\\prime },P^{\\prime })\\in \\mathcal {A}_1$ has form $(i,P_{\\bot })$ .",
"Accordingly, $\\mu _{\\mathcal {E}^n}=-\\sum _{\\underset{(A^{\\prime },P^{\\prime })\\sqcap (A,P)=(\\emptyset ,P_{\\bot })}{(A^{\\prime },P^{\\prime })\\in \\mathcal {A}}}\\mu _{\\mathcal {E}^N}((A^{\\prime },P^{\\prime }),(N,P^{\\top }))\\text{,}$ where the sum ranges over the $n$ atoms just described.",
"From the above isomporphisms, $[(i,P_{\\bot }),(N,P^{\\top })]\\cong \\mathcal {P}^n$ as well as $[(\\emptyset ,[ij]),(N,P^{\\top })]\\cong \\mathcal {E}^{n-1}$ .",
"Thus $\\mu _{\\mathcal {E}^n}=-(\\mu _{\\mathcal {P}^n}+(n-1)\\mu _{\\mathcal {E}^{n-1}})$ .",
"Since $\\mu _{\\mathcal {E}^0}=\\mu _{\\mathcal {P}^0}=1$ and $\\mu _{\\mathcal {E}^1}=\\mu _{\\mathcal {P}^2}=-1$ , the desired conclusion $\\mu _{\\mathcal {E}^n}=-(\\mu _{\\mathcal {P}^n}+(n-1)\\mu _{\\mathcal {P}^n})=-n\\mu _{\\mathcal {P}^n}=\\mu _{\\mathcal {P}^{n+1}}$ follows.",
"It may be noted that this route aims at following tightly [13].",
"However, if $(A,P)$ is (straightforwardly) chosen to be the dual atom $(\\emptyset ,P^{\\top })$ (rather than $(N\\backslash i,\\lbrace N\\backslash i,i\\rbrace )$ ), then the above argument immediately yields $\\mu _{\\mathcal {E}^n}=-n\\mu _{\\mathcal {P}^n}$ , as the $n$ involved atoms are all the elements of $\\mathcal {A}_1$ , i.e.",
"$(j,P_{\\bot }),j\\in N$ .",
"Proposition 4 confirms that the lattice $\\mathcal {E}^n$ of embedded subsets of a $n$ -set is isomorphic to the lattice $\\mathcal {P}^{n+1}$ of partitions of a $n+1$ -set.",
"In particular, the characteristic polynomial [2], [17], [20] is $\\chi _{\\mathcal {E}^n}(x)=\\chi _{\\mathcal {P}^{n+1}}(x)=(x-1)(x-2)\\cdots (x-n)\\text{,}$ with the Whitney numbers of the first kind given by the Stirling numbers of the first kind [1], [2], i.e.",
"$\\sum _{\\underset{r(A,P)=k}{(A,P)\\in \\mathcal {E}^N}}\\mu _{\\mathcal {E}^N}((\\emptyset ,P_{\\bot }),(A,P))=s_{n+1,n+1-k}\\text{ for }0\\le k\\le n\\text{,}$ and the Whitney numbers of $\\mathcal {E}^n$ of the second kind given by the Stirling numbers of the second kind, i.e.",
"$|\\lbrace (A,P):(A,P)\\in \\mathcal {E}^N,r(A,P)=k\\rbrace |=S_{n+1,n+1-k}\\text{ for }0\\le k\\le n\\text{.",
"}$ Furthermore, $\\mathcal {E}^n$ is a supersolvable lattice, since in view of Proposition 3 it admits $\\frac{(n+1)!",
"}{2}$ maximal chains from the bottom element to the top one consisting of modular elements [16] [17].",
"It seems also worth pointing out (again, see Section 1) that while in the non-atomic lattice $\\mathfrak {C}(N)_{\\bot }$ of embedded subsets only the $2^n-1$ modular elements $(A,P^A_{\\bot }),A\\ne \\emptyset $ admit a decomposition as a join of the $n$ available atoms $(i,P_{\\bot })$ (see [6]), in the geometric lattice $\\mathcal {E}^N$ all elements admit a decomposition as a join of the $\\binom{n+1}{2}$ atoms.",
"Accordingly [3], $\\mu _{\\mathcal {C}(N)_{\\bot }}((\\emptyset ,P_{\\bot }),(A,P))=\\left\\lbrace \\begin{array}{c} (-1)^{|A|}\\text{ if }P=P^A_{\\bot }\\\\0\\text{ otherwise} \\end{array}\\right.$ while $\\mu _{\\mathcal {E}^N}((\\emptyset ,P_{\\bot }),(A,P))=(-1)^{|A|}|A|!\\prod _{B\\in P^{A^c}}(-1)^{|B|-1}(|B|-1)!$ and $\\mu _{\\mathcal {E}^N}((\\emptyset ,P_{\\bot }),(A,P))$ equals the difference between the number of subsets $\\lbrace (A^{\\prime },P^{\\prime })_1,\\ldots ,(A^{\\prime },P^{\\prime })_{|S|}\\rbrace =S\\subseteq \\mathcal {A}$ of atoms of $\\mathcal {E}^N$ such that $|S|$ is even as well as $cl((A^{\\prime },P^{\\prime })_1\\sqcup \\cdots \\sqcup (A^{\\prime },P^{\\prime })_{|S|})=(A,P)$ and the number of subsets $\\lbrace (A^{\\prime },P^{\\prime })_1,\\ldots ,(A^{\\prime },P^{\\prime })_{|S^{\\prime }|}\\rbrace =S^{\\prime }\\subseteq \\mathcal {A}$ of atoms such that $|S^{\\prime }|$ is odd as well as $cl((A^{\\prime },P^{\\prime })_1\\sqcup \\cdots \\sqcup (A^{\\prime },P^{\\prime })_{|S^{\\prime }|})=(A,P)$ [9].",
"Finally, it seems worth noticing that $(-1)^{r(N,P^{\\top })}\\mu _{\\mathcal {E}^N}((\\emptyset ,P_{\\bot }),(N,P^{\\top }))=(-1)^n(-1)^nn!=(-1)^{2n}n!>0$ for all $n\\ge 0$ , which is consistent with [3]."
],
[
"Möbius algebra, vector subspaces and games",
"The purpose of this section is to highlight that all cooperative games are in fact elements of subspaces of the free vector space $V(\\mathcal {E}^N)$ over $\\mathbb {R}$ generated by $\\mathcal {E}^N$ [1], [2].",
"That is, the elements of $V(\\mathcal {E}^N)\\subset \\mathbb {R}^{\\mathcal {B}_{n+1}}$ are real-valued lattice functions $g:\\mathcal {E}^N\\rightarrow \\mathbb {R}$ .",
"Recall that, for $N$ regarded as a player set, cooperative game theory deals with three types of lattice functions: coalitional games [14] or set functions $v:2^N\\rightarrow \\mathbb {R}_+,v(\\emptyset )=0$ , global games [4] or partition functions $h:\\mathcal {P}^N\\rightarrow \\mathbb {R}_+,h(P_{\\bot })=0$ , games in partition function form [6], [10], [19], hereafter referred to as PFF games for short, or functions $f:\\mathfrak {C}(N)_{\\bot }\\rightarrow \\mathbb {R}_+,f(\\emptyset ,P_{\\bot })=0$ .",
"A further fourth type of cooperative games may be defined, which incorporates all previous ones $(a)$ , $(b)$ and $(c)$ , namely extended PFF games, or those elements $g\\in V(\\mathcal {E}^N)$ of the free vector space under concern such that $g:\\mathcal {E}^N\\rightarrow \\mathbb {R}_+,g(\\emptyset ,P_{\\bot })=0$ .",
"Together with the Möbius function, the other element of the incidence algebra (of locally finite posets [2] [13]) playing a fundamental role in cooperative game theory is the zeta function $\\zeta $ , as it provides the so-called ‘unanimity games’ [14].",
"In the incidence algebra of $\\mathcal {E}^N$ over $\\mathbb {R}$ , the zeta function $\\zeta _{\\mathcal {E}^N}:\\mathcal {E}^N\\times \\mathcal {E}^N\\rightarrow \\lbrace 0,1\\rbrace $ is defined on ordered pairs of embedded subsets by $\\zeta _{\\mathcal {E}^N}((B,Q),(A,P))=\\left\\lbrace \\begin{array}{c} 1\\text{ if }(A,P)\\sqsupseteq (B,Q)\\text{,}\\\\0\\text{ otherwise.}",
"\\end{array}\\right.$ In these terms, unanimity (coalitional) games $u_B,\\emptyset \\ne B\\in 2^N$ are defined by $u_B(A)=\\zeta _{2^N}(B,A)$ for all subsets (or coalitions) $A\\in 2^N$ , where $\\zeta _{2^N}$ is the zeta function in the incidence algebra of $2^N$ .",
"They are linearly independendent elements of the vector space of coalitional games and thus form a basis.",
"This fact is essential since the very beginning of cooperative game theory.",
"In the present setting, for $(B,Q)\\in \\mathcal {E}^N$ , let $\\zeta ^{B,Q}(A,P)=\\zeta _{\\mathcal {E}^N}((B,Q),(A,P))$ for all $(A,P)\\in \\mathcal {E}^N$ .",
"Then, $\\lbrace \\zeta ^{B,Q}:(B,Q)\\in \\mathcal {E}^N\\rbrace $ is a basis of $V(\\mathcal {E}^N)$ .",
"Also denote by $V(\\mathcal {E}^N)_{\\bot }\\subset \\mathbb {R}^{\\mathcal {B}_{n+1}-1}_+$ the subspace of extended PFF games.",
"Furthermore, let the Möbius inversion of $g$ be $\\mu ^g:\\mathcal {E}^N\\rightarrow \\mathbb {R}$ , i.e.",
"$\\mu ^g(A,P)&=&\\sum _{\\underset{(B,Q)\\in \\mathcal {E}^N}{(B,Q)\\sqsubseteq (A,P)}}\\mu _{\\mathcal {E}^N}((B,Q),(A,P))g(B,Q)=\\\\&=&g(A,P)-\\sum _{\\underset{(B,Q)\\in \\mathcal {E}^N}{(B,Q)\\sqsubset (A,P)}}\\mu ^g(B,Q)\\text{,}$ with $\\mu ^g(\\emptyset ,P_{\\bot })=g(\\emptyset ,P_{\\bot })$ ($=0$ if $g\\in V(\\mathcal {E}^N)_{\\bot }$ ), and where $\\mu ^{\\zeta ^{B,Q}}(A,P)=\\left\\lbrace \\begin{array}{c} 1\\text{ if }(A,P)=(B,Q)\\\\0\\text{ otherwise} \\end{array}\\right.$ for every element $\\zeta ^{B,Q}$ of the chosen basis, thereby providing the canonical basis (or Kronecker delta) elements (see [2], while crucially noticing that the alternative definition $\\zeta ^{B,Q}(A,P)=\\zeta _{\\mathcal {E}^N}((A,P),(B,Q))$ applies there.)",
"Specifically, let $\\epsilon _{B,Q}\\in V(\\mathcal {E}^N)$ be the $\\mathcal {B}_{n+1}$ -vector with entries indexed by elements $(A,P)\\in \\mathcal {E}^N$ and such that its unique non-zero entry is the $(B,Q)$ one, which equals 1.",
"Then the Möbius algebra of $\\mathcal {E}^N$ over $\\mathbb {R}$ (denoted by Möb$(\\mathcal {E}^N)$ , see [1] [2]) is defined by the system of orthogonal idempotents $\\lbrace \\epsilon _{B,Q}:(B,Q)\\in \\mathcal {E}^N\\rbrace $ and linear extension to all of $V(\\mathcal {E}^N)$ through multiplication $\\epsilon _{B,Q}\\cdot \\epsilon _{A,P}=\\left\\lbrace \\begin{array}{c} 1\\text{ if }(A,P)=(B,Q)\\text{,}\\\\0\\text{ otherwise.}",
"\\end{array}\\right.$ In terms of the above basis $\\lbrace \\zeta ^{B,Q}:(B,Q)\\in \\mathcal {E}^N\\rbrace $ of $V(\\mathcal {E}^N)$ , this multiplication yields $\\zeta ^{B,Q}\\cdot \\zeta ^{A,P}=\\zeta ^{cl((A,P)\\sqcup (B,Q))}$ (see [2], crucially noticing again that since the alternative definition of the basis detailed above applies there, then $\\zeta ^{B,Q}\\cdot \\zeta ^{A,P}=\\zeta ^{(A,P)\\sqcap (B,Q)}$ ).",
"It is well known that Möbius inversion $\\mu ^g$ quantifies precisely the coefficients identifying functions $g\\in V(\\mathcal {E}^N)$ as linear combinations of the basis elements, that is $g(A,P)=\\sum _{(B,Q)\\in \\mathcal {E}^N}\\mu ^g(B,Q)\\zeta ^{B,Q}(A,P)\\text{ for all }(A,P)\\in \\mathcal {E}^N\\text{.",
"}$ If $Y\\subset \\mathcal {E}^N$ and $\\mu ^g(A,P)=0$ for all $(A,P)\\in \\mathcal {E}^N\\backslash Y$ , then Möbius inversion $\\mu ^g$ may be said to live only $Y$ .",
"In these terms, the following observations are immediate: (traditional) PFF games are elements of the subspace $\\hat{V}(\\mathcal {E}^N)\\subset \\mathbb {R}^{\\mathcal {B}_{n+1}-\\mathcal {B}_n}$ of $V(\\mathcal {E}^N)$ consisting of those extended PFF games $g\\in V(\\mathcal {E}^N)$ whose Möbius inversion lives only on embedded subsets $(A,P)$ such that $A\\ne \\emptyset $ , i.e.",
"$A=\\emptyset \\Rightarrow \\mu ^g(A,P)=0$ , entailing $g(\\emptyset ,P)=0$ for all $P\\in \\mathcal {P}^N$ ; global games are elements of the subspace $\\tilde{V}(\\mathcal {E}^N)\\subset \\mathbb {R}^{\\mathcal {B}_n-1}$ consisting of extended PFF games $g\\in V(\\mathcal {E}^N)$ whose Möbius inversion lives only on embedded subsets $(A,P)$ such that $A=\\emptyset $ and $P\\ne P_{\\bot }$ , entailing $g(A,P)=g(\\emptyset ,P)$ for all $A\\in P$ (thereby somehow strengthening the idea of a global level of satisfaction, common to all players, see [4]); coalitional games are elements of the subspace $\\bar{V}(\\mathcal {E}^N)\\subset \\mathbb {R}^{2^n-1}$ consisting of those $g\\in V(\\mathcal {E}^N)$ whose Möbius inversion lives only on the $2^n-1$ modular elements $(A,P^A_{\\bot })$ of $\\mathcal {E}^N$ defined in Section 3 such that $A\\ne \\emptyset $ , yielding $g(\\emptyset ,P)=0$ for all $P\\in \\mathcal {P}^N$ as well as $g(A,P)=g(A,P^A_{\\bot })$ for all $P\\in \\mathcal {P}^N$ such that $A\\in P$ .",
"Another subspace of $V(\\mathcal {E}^N)$ appearing in cooperative game theory is given by the so-called “additively separable” global games (or partition functions, see [4], [5]), namely those $h:\\mathcal {P}^N\\rightarrow \\mathbb {R}_+$ such that $h(P)=h_v(P):=\\sum _{A\\in P}v(A)$ for some coalitional game (or set function) $v:2^N\\rightarrow \\mathbb {R}_+$ .",
"The Möbius inversion of these partition functions lives only on the $2^n-n$ modular elements $P^A_{\\bot }$ of the partition lattice, see Section 1.",
"This is detailed hereafter (see also [4], [5] and [11]).",
"Proposition 5 If $h=h_v$ and $\\sum _{i\\in N}v(\\lbrace i\\rbrace )=h_v(P_{\\bot })>0$ , then a continuuum of set functions $w:2^N\\rightarrow \\mathbb {R}_+,w\\ne v$ satisfies $h=h_w$ .",
"Proof: By direct substitution, the Möbius inversion $\\mu ^{h_v}:\\mathcal {P}^N\\rightarrow \\mathbb {R}$ of partition function $h_v$ satisfies $\\mu ^{h_v}(P)=\\sum _{A\\in P}\\sum _{B\\subseteq A}v(B)\\sum _{Q\\leqslant P:B\\in Q}\\mu _{\\mathcal {P}^N}(Q,P)\\text{ for all }P\\in \\mathcal {P}^N\\text{.",
"}$ If $P\\ne P^A_{\\bot }$ , then the recursion for Möbius function $\\mu _{\\mathcal {P}^N}:\\mathcal {P}^N\\times \\mathcal {P}^N\\rightarrow \\mathbb {R}$ yields $\\sum _{Q\\leqslant P:A\\in Q}\\mu _{\\mathcal {P}^N}(Q,P)=\\sum _{P^A_{\\bot }\\leqslant Q\\leqslant P}\\mu _{\\mathcal {P}^N}(Q,P)=0\\text{,}$ and the same for proper subsets $B\\subset A$ .",
"Möbius inversion $\\mu ^{h_v}$ thus lives only on modular partitions, where it obtains recursively by $\\mu ^{h_v}(P_{\\bot })=\\sum _{i\\in N}v(\\lbrace i\\rbrace )$ and $\\mu ^{h_v}(P_{\\bot }^A)=\\mu ^v(A)$ for $1<|A|\\le n$ , with $P^N_{\\bot }=P^{\\top }$ .",
"Hence any $w\\ne v$ satisfying $\\sum _{i\\in N}v(\\lbrace i\\rbrace )=\\sum _{i\\in N}w(\\lbrace i\\rbrace )$ and $\\mu ^v(A)=\\mu ^w(A)$ for all $A\\in 2^N$ such that $|A|>1$ also additively separates $h$ , i.e.",
"$h_v=h_w$ .",
"If $\\sum _{i\\in N}v(\\lbrace i\\rbrace )>0$ , then all $w:2^N\\rightarrow \\mathbb {R}_+$ such that $h_v=h_w$ may be chosen from within a $n-1$ -dimensional simplex $\\Delta \\subset \\mathbb {R}^n_+$ .",
"However, if $\\sum _{i\\in N}v(\\lbrace i\\rbrace )=0$ , then there is no $w:2^N\\rightarrow \\mathbb {R}_+$ such that $h_v=h_w,v\\ne w$ .",
"Nevertheless, this technical distinction plays no role as soon as attention is placed on generic partition and set functions $h:\\mathcal {P}^N\\rightarrow \\mathbb {R},w:2^N\\rightarrow \\mathbb {R}$ , i.e.",
"not required to take only non-negative real values.",
"In any case, in view of observation $(ii)$ above, it is evident that additively separable partition functions may be regarded as elements of the subspace of $V(\\mathcal {E}^N)$ consisting of those extended PFF games $g$ with Möbius inversion $\\mu ^g$ such that $\\mu ^g(B,Q)\\ne 0$ only if $B=\\emptyset $ and $Q$ is a modular partition.",
"Developing from additively separable partition functions, now consider those $g\\in V(\\mathcal {E}^N)$ with Möbius inversion $\\mu ^g$ living only on the $2^{n+1}-(n+1)$ modular elements of $\\mathcal {E}^N$ (see Section 3).",
"Proposition 6 If $g\\in V(\\mathcal {E}^N)$ has Möbius inversion $\\mu ^g$ living only on modular elements of $\\mathcal {E}^N$ , then there are set functions $v,w:2^N\\rightarrow \\mathbb {R},v(\\emptyset )=0$ such that $g(A,P)=v(A)+\\sum _{B\\in P}w(B)\\text{ for all }(A,P)\\in \\mathcal {E}^N\\text{.",
"}$ Proof: Under the above conditions, and given expression (5) above, $g(A,P)=\\mu ^g(\\emptyset ,P_{\\bot })+\\sum _{B\\in P}\\sum _{\\emptyset \\ne B^{\\prime }\\subseteq B}\\mu ^g(\\emptyset ,P^{B^{\\prime }}_{\\bot })+\\sum _{\\emptyset \\ne A^{\\prime }\\subseteq A}\\mu ^g(A^{\\prime },P^{A^{\\prime }}_{\\bot })$ for all $(A,P)\\in \\mathcal {E}^N$ .",
"In view of Proposition 5, $\\mu ^g(\\emptyset ,P_{\\bot })+\\sum _{B\\in P}\\sum _{\\emptyset \\ne B^{\\prime }\\subseteq B}\\mu ^g(\\emptyset ,P^{B^{\\prime }}_{\\bot })=\\sum _{B\\in P}w(B)$ for any set function $w$ with Möbius inversion $\\mu ^w$ such that $\\mu ^g(\\emptyset ,P_{\\bot })=\\sum _{i\\in N}w(\\lbrace i\\rbrace )\\text{ and }\\mu ^g(\\emptyset ,P^{B^{\\prime }}_{\\bot })=\\mu ^w(B^{\\prime })\\text{ for }B^{\\prime }\\in 2^N,|B^{\\prime }|>1\\text{.",
"}$ On the other hand, $\\sum _{\\emptyset \\ne A^{\\prime }\\subseteq A}\\mu ^g(A^{\\prime },P^{A^{\\prime }}_{\\bot })=v(A)=\\sum _{A^{\\prime }\\subseteq A}\\mu ^v(A^{\\prime })$ for any set function $v$ with Möbius inversion $\\mu ^v$ satisfying $\\mu ^g(A^{\\prime },P^{A^{\\prime }}_{\\bot })=\\mu ^v(A^{\\prime })$ for $A^{\\prime }\\in 2^N$ , $|A^{\\prime }|>0$ and $\\mu ^v(\\emptyset )=v(\\emptyset )=0$ .",
"Like for additively separable partition functions, if the Möbius inversion of extended PFF games lives only on the modular elements of the lattice, then all the values taken on embedded subsets can be recovered from only two set functions, out of which one is unique while the other can be chosen from within a continuum (in view of Proposition 5)."
],
[
"Concluding remarks",
"This paper essentially proposes to look at the so-called lattice of embedded subsets from a wider perspective, such that the novel resulting structure is a combinatorial geometry obtained through a closure operator that satisfies the Steinitz exchange axiom.",
"With respect to the lattice previously proposed in [6], the geometric one presented here additionally includes all elements given by a (non-bottom) partition paired with the empty set, and this evidently yields a most natural lattice embedding, i.e.",
"of $\\mathcal {P}^N$ into $\\mathcal {E}^N$ .",
"However, from this perspective what seems mostly important is that the geometric lattice of embedded subsets of a $n$ set is isomorphic to the lattice of partitions of a $n+1$ -set.",
"In general, this enables to re-obtain a variety of results applying to partitions, ranging from the characteristic polynomial to supersolvability.",
"Since embedded subsets, or embedded coalitions, were firstly used as modeling tools in cooperative game theory, a meaningful direction for investigation focuses on the free vector space of functions taking real values on lattice elements.",
"In this view, a novel type of games has been defined, namely extended PFF games, which englobe all existing cooperative games as elements of suitable vector subspaces.",
"But perhaps most importantly, the geometric lattice of embedded subsets yields that all existing cooperative games are real-valued functions defined on atomic lattices, and this may have fundamental implications for the solution concept (i.e.",
"how to share the fruits of cooperation) associated with these games.",
"This shall be addressed in future work."
]
] | 1612.05814 |
[
[
"Five-loop quark mass and field anomalous dimensions for a general gauge\n group"
],
[
"Abstract We present analytical five-loop results for the quark mass and quark field anomalous dimensions, for a general gauge group and in the MSbar scheme.",
"We confirm the values known for the gauge group SU(3) from an independent calculation, and find full agreement with results available from large-Nf studies."
],
[
"Introduction",
"High-precision determinations of Standard Model (SM) parameters are crucially important for precise predictions for the observables measured in collider physics experiments, such as currently performed at the LHC, or possibly at a future linear collider.",
"A thorough comparison of these predictions with experimental results allows to scrutinize the details of the hugely successful SM, and might shed light on possible physics beyond the SM.",
"In order to pin down the theory's fundamental parameters, such as coupling constants and masses, with sufficient precision, the knowledge of higher order perturbative corrections is required.",
"This encompasses the evaluation of multi-loop Feynman diagrams and Feynman integrals, for which significant progress has been made in recent years, mainly with respect to the formulation of advanced algorithms that allow to treat the complexity level met in those higher-loop integrals.",
"We focus here on the strong interactions, which are embedded into the SM via Quantum Chromodynamics (QCD).",
"The relevant parameters are then the (strong) gauge coupling and the quark masses, both of which run with the energy scale according to the renormalization group (RG) equations.",
"In order to evolve e.g.",
"the low-energy value of the coupling constant (measured with high precision from tau lepton decay) to high energies, the anomalous dimension of the gauge coupling (the so-called Beta function) is needed, as coefficient in the corresponding RG equation.",
"Likewise, a high-order evaluation of the quark mass anomalous dimension gives access to precise values for e.g.",
"charm and bottom quark masses, which are measured at low energies (typically a few GeV) but whose uncertainty at the high-energy scale of the Higgs mass $m_H=125$ GeV is important in Higgs decay rates into such quark pairs.",
"In particular, to match the precision of the known five-loop inclusive decay width of $H\\rightarrow q\\bar{q}$ [1], one should evolve the parameters (which are $\\alpha _s(\\mu )$ and the running quark mass $m_q(\\mu )$ with $\\mu $ being the renormalization scale) from low energies to $\\mu =m_H$ at the same perturbative order, for full consistency and to avoid large logarithms $\\ln (\\mu ^2/m_H^2)$ .",
"Given this clear phenomenological motivation, we will present new results for the quark mass anomalous dimension here, applying a number of the above-mentioned algorithmic advances.",
"While this renormalization constant has been studied previously up to five loops in perturbation theory [2], [3], [4], [5], [6], [7], we generalize it from the gauge group SU(3) to a semi-simple Lie group.",
"At the same time, we provide a truly independent check on the available SU(3) result, since we utilize largely independent methods, as described below.",
"We will also give results for a related quantity needed to renormalize the quark sector at five loops, namely the quark field anomalous dimension in Feynman gauge, again generalizing known SU(3) results to a semi-simple Lie group.",
"The paper is organized as follows.",
"We start by explaining our computational setup in section .",
"Using some notation defined in section , we then present and discuss results in sections and , before concluding in section .",
"For convenience, an appendix reproduces large-$N_{\\mathrm {f}}$ results from the literature, which we use for consistency checks."
],
[
"Setup",
"Let us start by explaining our computational setup, which closely follows the one employed and tested in [8].",
"We base our highly automated setup on the diagram generator qgraf [9], [10] and various in-house FORM [11], [12], [13] codes.",
"After generating the required fermionic 2-point functions, we apply projectors and perform the group algebra with color [14].",
"To extract the ultraviolet (UV) divergences, we then use the freedom to change the low-momentum behavior and make all propagators massive, which regulates the infrared at the cost of introducing a new counterterm for the unphysical regulator mass.",
"Expanding to sufficient depth in the external momentum [15], [16], [17] and keeping all potentially UV divergent structures results in nullifying the external momentum.",
"The coefficients of this expansion can then be mapped onto a family of fully massive vacuum integrals, which are labelled by 15 indices (corresponding to maximally 12 propagators plus 3 scalar products) at five loops [18].",
"As a next and fairly time-consuming step, we reduce those integrals to a small set of master integrals, powered by our own codes crusher [19] and TIDE [18], which are based on integration-by-parts (IBP) identities [20] and use a Laporta-type algorithm [21] for a systematic integral reduction.",
"At five loops, we end up with a set of 110 master integrals.",
"These have been evaluated in an $\\varepsilon $ -expansion around $d=4-2\\varepsilon $ dimensions in previous works [18], [22], using an approach based on IBP reductions and difference equations [21] that has been realized in C++ and uses Fermat [23] to perform the polynomial algebra that arises in solving systems of linear equations with large rational coefficients.",
"The resulting high-precision numerical results for the coefficients of the $\\varepsilon $ -expansions finally allow us to utilize the integer-relation finding algorithm PSLQ [24] to discover the analytic content of some of these numbers, and to find relations between others.",
"As a consequence, we are able to provide all our results given below in analytic form.",
"Figure: 5-loop master integrals with 12 lines that contribute to eqs. ()-().",
"Each line denotes a massive propagator 1/(k 2 +m 2 )1/(k^2+m^2), and each dot stands for an extra power of the corresponding propagator.As has already been mentioned elsewhere [8], our high-precision evaluation of all 5-loop master integrals has not yet produced results for the 12-line families, see figure REF .",
"These do not contain divergences in four dimensions, and could therefore be avoided in evaluations of anomalous dimensions.",
"It turns out, however, that with our integral reduction criteria and lexicographic ordering prescription, we do get contributions from these integral classes since they are multiplied by prefactors with spurious poles as $d\\rightarrow 4$ .",
"To fix the values of the 12-line master integrals that we need, we first note that in all our results, only three independent linear combinations appear: $\\ell _0&=& 689 \\,I_{30527.1.1, 5} - 3934 \\,I_{30527.1.2, 5} + 5464 \\,I_{30527.1.3, 5} + 1152 \\,I_{30527.1.4, 5} - 5228 \\,I_{30527.6.3, 5} \\;,\\\\\\ell _1&=& 689 \\,I_{30527.1.1, 6} - 3934 \\,I_{30527.1.2, 6} + 5464 \\,I_{30527.1.3, 6} + 1152 \\,I_{30527.1.4, 6} - 5228 \\,I_{30527.6.3, 6} \\nonumber \\\\&&+ 1968 \\,I_{30527.1.1, 5} + 3890 \\,I_{30527.1.2, 5} + 3844 \\,I_{30527.1.3, 5} - 6912 \\,I_{30527.1.4, 5} - 70 \\,I_{30527.6.3, 5} \\;,\\\\\\ell _2&=& 11 \\,I_{31740.1.1, 5} - 72 \\,I_{31740.1.3, 5} \\;.$ Here, each integral $I_{\\#,n}$ corresponds to the $\\varepsilon ^n$ -coefficient of the respective fully massive master integral of figure REF , divided by the fifth power of the 1-loop tadpole $J$ for normalization reasons$J=\\int {\\rm d}^dk/(k^2+m^2)\\sim \\Gamma (1-d/2)$ has a simple pole as $d\\rightarrow 4$ ; hence, finite 5-loop terms correspond to $\\varepsilon ^5$ ..",
"While it is conceivable that there exists a suitable basis transformation that eliminates these linear combinations altogether from the final results, we have not yet performed a systematic search of such transformations in our integral reduction tables, but opted for other criteria to fix the numerical values of the three linear combinations, with high precision, as we will explain now.",
"As a crude order-of-magnitude estimate, we have evaluated the set of 12-line integrals via Feynman parametric representations (see e.g.",
"[25]) and subsequent (primary) sector decomposition, using the strategy explained in [26], [27] and as implemented in FIESTA [28] as well as own code (see [29]).",
"Due to the large prefactors and cancellations in eqs.",
"(REF )-(), a 6-digit evaluation results in the estimates $\\ell _0\\approx -7.47(1)\\;,\\quad \\ell _1\\approx -50.6(1)\\;,\\quad \\ell _2\\approx -0.673(1)\\;,$ with 3-digit accuracy.",
"Turning now to high-precision evaluations, we first recall that the higher-order $\\varepsilon $ -poles of renormalization constants are completely determined by lower-order coefficients.",
"Checking these, we obtain one consistency condition that allows to fix $\\ell _0$ .",
"Second, we observe the occurrence of some rank-12 group invariants in another renormalization constant that we have evaluated using the same setup, namely for the ghost-gluon vertex [30].",
"Using eq.",
"(REF ), to determine their coefficients, we find that they vanish at least to this low accuracy.",
"Taking this zero for granted, we turn the argument around and require e.g.",
"the structure $d^{\\,444}_{FAA}\\,N_{\\mathrm {f}}$ (where the group invariant is in the notation of [14]) to be absent from the final result; this gives us another condition, fixing $\\ell _2$ .",
"Third, for fixing the remaining linear combination, we choose to compare our results in the SU(3) limit to the previously known 5-loop results.",
"To be concrete, out of the many possible coefficients we choose the $n_f$ term of $\\gamma _m$ as given in [7], giving us one constraint which fixes $\\ell _1$ .",
"Along these lines, we obtain numerical values for all three linear combinations with 260 digits, the first 50 of which read $\\ell _0&=& -7.4750787021276651819913288152084850401974826928834\\dots \\;,\\\\\\ell _1&=& -50.563714841071996428539372592222326105092965639946\\dots \\;,\\\\\\ell _2&=& -0.67332086607447050046759024439428336720209195028580\\dots \\;,$ and which can be seen to be consistent with our low-precision estimates of eq.",
"(REF ) that had been obtained by direct integration."
],
[
"Notation",
"Let us fix our notation here: we work with a semi-simple Lie algebra with hermitian generators $T^a$ , whose real and antisymmetric structure constants $f^{abc}$ are fixed by the commutation relation $[T^a,T^b]=i f^{abc}T^c$ .",
"As usual, the quadratic Casimir operators of the fundamental (adjoint) representation (of dimensions $N_{\\mathrm {F}}$ and $N_{\\mathrm {A}}$ , respectively) are defined as $T^aT^a=C_{\\mathrm {F}}1\\!\\!1$ ($f^{acd}f^{bcd}=C_{\\mathrm {A}}\\delta ^{ab}$ ).",
"Furthermore, traces are normalized as ${\\rm Tr}(T^aT^b)=T_{\\mathrm {F}}\\delta ^{ab}$ , we denote the number of quark flavors with $N_{\\mathrm {f}}$ , and find it convenient to define the following normalized combinations: $c_f=\\frac{C_{\\mathrm {F}}}{C_{\\mathrm {A}}} \\quad ,\\quad n_f=\\frac{N_{\\mathrm {f}}\\,T_{\\mathrm {F}}}{C_{\\mathrm {A}}} \\;.$ In our multi-loop diagrams, we will encounter traces of more than two group generators, giving rise to higher-order group invariants.",
"It is useful to define traces over combinations of symmetric tensors [14], of which we need the following (writing $[F^a]_{bc}=-if^{abc}$ for the generators of the adjoint representation): $d_1=\\frac{[{\\rm sTr}(T^aT^bT^cT^d)]^2}{N_{\\mathrm {A}}T_{\\mathrm {F}}^2C_{\\mathrm {A}}^2} \\;,\\;d_2=\\frac{{\\rm sTr}(T^aT^bT^cT^d)\\,{\\rm sTr}(F^aF^bF^cF^d)}{N_{\\mathrm {A}}T_{\\mathrm {F}}C_{\\mathrm {A}}^3} \\;,\\;d_3=\\frac{[{\\rm sTr}(F^aF^bF^cF^d)]^2}{N_{\\mathrm {A}}C_{\\mathrm {A}}^4} \\;.$ Here, ${\\rm sTr}$ stands for a fully symmetrized trace (such that ${\\rm sTr}(ABC)=\\frac{1}{2}{\\rm Tr}(ABC+ACB)$ etc.).",
"While dealing with the quark sector, it might seem more natural to normalize these traces with respect to the dimension of the fundamental representation; to this end, we note the relation $N_{\\mathrm {A}}T_{\\mathrm {F}}=N_{\\mathrm {F}}C_{\\mathrm {F}}$ which holds in general [14], [31].",
"For the gauge group SU($N$ ) (where $T_{\\mathrm {F}}=\\frac{1}{2}$ and $C_{\\mathrm {A}}=N$ ), the normalized group invariants introduced above read [14] $n_f=\\frac{N_{\\mathrm {f}}}{2N}\\;,\\quad c_f=\\frac{N^2-1}{2N^2}\\;,\\quad d_1=\\frac{N^4-6N^2+18}{24N^4}\\;,\\quad d_2=\\frac{N^2+6}{24N^2}\\;,\\quad d_3=\\frac{N^2+36}{24N^2}\\;.$ The corresponding SU(3) values, relevant for physical QCD, hence read $\\mbox{SU}(3):\\quad n_f=\\frac{N_{\\mathrm {f}}}{6}\\;,\\quad c_f=\\frac{4}{9}\\;,\\quad d_1=\\frac{5}{216}\\;,\\quad d_2=\\frac{5}{72}\\;,\\quad d_3=\\frac{5}{24}\\;.$"
],
[
"Quark mass renormalization",
"The renormalization constant for the quark mass $m_{\\rm bare}=Z_m\\,m_{\\rm ren}$ , or equivalently its anomalous dimension $\\gamma _m = -\\partial _{\\ln \\mu ^2}\\ln Z_m$ , has been known at two [2] and three loops [3], [4] for a long time.",
"At four loops, $\\gamma _m$ is known for SU(N) and QED [6] as well as for a general Lie group [5].",
"Presently, at five loops only the SU(3) value is publicly available [7].",
"We present our corresponding result for a general Lie group below.",
"The structure of the quark mass anomalous dimension is $\\partial _{\\ln \\mu ^2}\\ln m_{\\rm q}(\\mu ) &\\equiv & \\gamma _m(a) \\;=\\;-c_f\\,a\\,\\Big \\lbrace 3 +\\gamma _{m1}\\,a +\\gamma _{m2}\\,a^2 +\\gamma _{m3}\\,a^3 +\\gamma _{m4}\\,a^4 +\\dots \\Big \\rbrace \\;,\\\\a&\\equiv &\\frac{C_{\\mathrm {A}}\\,g^2(\\mu )}{16\\pi ^2}\\;,$ with $g(\\mu )$ being the renormalized QCD gauge coupling constant that depends on the renormalization scale $\\mu $ (we prefer to use the expansion parameter $a$ which is nothing but a rescaled version of the renormalized strong coupling constant $\\alpha _s=\\frac{g^2(\\mu )}{4\\pi }$ ).",
"We work in $d=4-2\\varepsilon $ dimensions and employ the ${\\overline{\\mbox{\\rm {MS}}}}$ scheme.",
"The coefficients $\\gamma _{mn}$ are polynomials in $n_f$ and can be written in terms of our normalized group factors.",
"Up to four loops, they read [5] $3^1\\,\\gamma _{m1} &=& n_f\\Big [-10\\Big ]+\\Big [(9c_f+97)/2\\Big ] \\;,\\\\3^3\\,\\gamma _{m2} &=& n_f^2\\Big [\\!-\\!140\\Big ] +n_f\\Big [54(24\\zeta _3\\!-\\!23)c_f-4(139\\!+\\!324\\zeta _3)\\Big ]+\\Big [(6966c_f^2\\!-\\!3483c_f\\!+\\!11413)/4\\Big ],\\\\3^4\\,\\gamma _{m3} &=& n_f^3\\Big [-8(83-144\\zeta _3)\\Big ] +n_f^2\\Big [48(19-270\\zeta _3+162\\zeta _4)c_f+2(671+6480\\zeta _3-3888\\zeta _4)\\Big ]\\nonumber \\\\&&+n_f\\Big [-216(35-207\\zeta _3+180\\zeta _5)c_f^2-3(8819-9936\\zeta _3+7128\\zeta _4-2160\\zeta _5)c_f\\nonumber \\\\&&-(65459/2+72468\\zeta _3-21384\\zeta _4-32400\\zeta _5)+2592(2-15\\zeta _3)d_1\\Big ]\\nonumber \\\\&&+\\tfrac{9}{8}\\Big [-9(1261+2688\\zeta _3)c_f^3+6(15349+3792\\zeta _3)c_f^2-2(34045+5472\\zeta _3-15840\\zeta _5)c_f\\nonumber \\\\&&+(70055+11344\\zeta _3-31680\\zeta _5)-1152(2-15\\zeta _3)d_2\\Big ] \\;,$ where we have denoted values of the Riemann Zeta function as $\\zeta _s=\\zeta (s)=\\sum _{n>0}n^{-s}$ .",
"At five loops, from app.",
", we have LO and NLO large-$N_{\\mathrm {f}}$ terms to all orders, coinciding with the leading terms above, and predicting the first two terms of the 5-loop contributions as $6^5\\,\\gamma _{m4} &=& \\gamma _{m44}\\,\\Big [4n_f\\Big ]^4+\\gamma _{m43}\\,\\Big [4n_f\\Big ]^3+\\gamma _{m42}\\,\\Big [4n_f\\Big ]^2+\\gamma _{m41}\\,\\Big [4n_f\\Big ]+\\gamma _{m40}\\;,\\\\\\gamma _{m44} &=& -6(65+80\\zeta _3-144\\zeta _4)\\;,\\\\\\gamma _{m43} &=& 3(4483\\!+\\!4752\\zeta _3\\!-\\!12960\\zeta _4\\!+\\!6912\\zeta _5)c_f+(18667/2\\!+\\!32208\\zeta _3\\!+\\!29376\\zeta _4\\!-\\!55296\\zeta _5)\\;.$ We have evaluated the remaining coefficients, mapping all diagrams onto fully massive vacuum tadpoles; IBP-reducing them to master integrals; using high-precision numerical evaluations thereof plus some additional consistency conditions to fix linear combinations of 12-line master integrals as explained in section ; and finally employing PSLQ at 200 digits for discovery, and at 250 digits for confirmation, we confirm the two large-$N_{\\mathrm {f}}$ expressions in eqs.",
"() and (), and obtain the three missing coefficients of eq.",
"(REF ) in analytic form, containing only Zeta valuesTo make the group structure more visible, we resort to a vector notation here and below, where a dot between two curly brackets denotes a scalar product as e.g.",
"in $\\lbrace c_f,1\\rbrace .\\lbrace a,b\\rbrace =c_fa+b$ .",
": $\\gamma _{m42} &=& \\Big \\lbrace c_f^2,c_f,d_1,1\\Big \\rbrace .\\Big \\lbrace 9 (45253-230496 \\zeta _{3}+48384 \\zeta _{3}^2+70416 \\zeta _{4}+144000 \\zeta _{5}-86400 \\zeta _{6}),\\nonumber \\\\&&375373+323784 \\zeta _{3}-1130112 \\zeta _{3}^2+905904 \\zeta _{4}-672192 \\zeta _{5}+129600 \\zeta _{6},\\nonumber \\\\&&-864 (431 - 1371 \\zeta _3 + 432 \\zeta _4 + 420 \\zeta _5),\\nonumber \\\\&&4 (13709+394749 \\zeta _{3}+173664 \\zeta _{3}^2-379242 \\zeta _{4}-119232 \\zeta _{5}+162000 \\zeta _{6})\\Big \\rbrace \\;,\\\\\\gamma _{m41} &=& \\Big \\lbrace c_f^3,c_f^2,c_fd_1,c_f,d_1,d_2,1\\Big \\rbrace .\\Big \\lbrace -54 (48797-247968 \\zeta _{3}+24192 \\zeta _{4}+444000 \\zeta _{5}-241920 \\zeta _{7}),\\nonumber \\\\&&-18 (406861+216156 \\zeta _{3}-190080 \\zeta _{3}^2+254880 \\zeta _{4}-606960 \\zeta _{5}-475200 \\zeta _{6}+362880 \\zeta _{7}),\\nonumber \\\\&&-62208 (11+154 \\zeta _{3}-370 \\zeta _{5}),\\nonumber \\\\&&753557+15593904 \\zeta _{3}-3535488 \\zeta _{3}^2-6271344 \\zeta _{4}-17596224 \\zeta _{5}+1425600 \\zeta _{6}+1088640 \\zeta _{7},\\nonumber \\\\&&1728 (3173-6270 \\zeta _{3}+1584 \\zeta _{3}^2+2970 \\zeta _{4}-13380 \\zeta _{5}),\\nonumber \\\\&&1728 (380 - 5595 \\zeta _3 - 1584 \\zeta _3^2 - 162 \\zeta _4 + 1320 \\zeta _5),\\\\&&-2 (4994047+11517108 \\zeta _{3}-57024 \\zeta _{3}^2-5931900 \\zeta _{4}-15037272 \\zeta _{5}+4989600 \\zeta _{6}+3810240 \\zeta _{7})\\Big \\rbrace \\;,\\nonumber \\\\\\gamma _{m40} &=& \\Big \\lbrace c_f^4,c_f^3,c_f^2,c_fd_2,c_f,d_2,d_3,1\\Big \\rbrace .\\Big \\lbrace 972 (50995+6784 \\zeta _{3}+16640 \\zeta _{5}),\\nonumber \\\\&&-54 (2565029+1880640 \\zeta _{3}-266112 \\zeta _{4}-1420800 \\zeta _{5}),\\nonumber \\\\&&108 (2625197+1740528 \\zeta _{3}-125136 \\zeta _{4}-2379360 \\zeta _{5}-665280 \\zeta _{7}),\\nonumber \\\\&&373248 (141+80 \\zeta _{3}-530 \\zeta _{5}),\\nonumber \\\\&&-8 (25256617+16408008 \\zeta _{3}+627264 \\zeta _{3}^2-812592 \\zeta _{4}-40411440 \\zeta _{5}+3920400 \\zeta _{6}-5987520 \\zeta _{7}),\\nonumber \\\\&&-6912 (9598+453 \\zeta _{3}+4356 \\zeta _{3}^2+1485 \\zeta _{4}-26100 \\zeta _{5}-1386 \\zeta _{7}),\\nonumber \\\\&&5184 (537 + 2494 \\zeta _3 + 5808 \\zeta _3^2 + 396 \\zeta _4 - 7820 \\zeta _5 - 1848 \\zeta _7),\\\\&&4 (22663417+\\!10054464 \\zeta _{3}+\\!1254528 \\zeta _{3}^2-\\!1695276 \\zeta _{4}-\\!41734440 \\zeta _{5}+\\!7840800 \\zeta _{6}+\\!5987520 \\zeta _{7})\\Big \\rbrace \\;.\\nonumber $ Let us now discuss some checks on this new result.",
"The authors of [7] have published the full 5-loop result for the case of SU(3).",
"To compare, we recall the definition of our expansion parameter in eq.",
"(REF ) and put all group invariants to their SU(3) values as given in eq.",
"(REF ); the $\\gamma _{m1..4}$ as listed above then coincide with the expressions given in [7].",
"Furthermore, the same group has very recently generalized their work to a general Lie group as well [32].",
"We have cross-checked their preliminary result with our $\\gamma _{m4}$ as given above, and found full agreement.",
"Since both five-loop results have been obtained with completely different methods (with the exception of also relying on qgraf for diagram generation, in [32] the 5-loop renormalization constants are mapped onto massless 4-loop two-point functions [33], [34], [35], and integral reduction is done via $1/d$ expansions [36], [37]), this agreement constitutes an extremely strong check."
],
[
"Quark field renormalization",
"For completeness, let us also present our new five-loop result for the quark field anomalous dimension $\\gamma _2 = -\\partial _{\\ln \\mu ^2}\\ln Z_2$ , where the renormalization constant $Z_2$ relates bare and renormalized quark fields as $\\psi _{\\rm bare}=\\sqrt{Z_2}\\psi _{\\rm ren}$ .",
"As opposed to the quark mass, this quantity is not physical and hence gauge dependent.",
"Lower-loop results can be found for SU(N) and covariant gauge in [38], and for a general Lie group with $\\xi ^0,\\xi ^1$ terms (which corresponds to a NLO expansion around Feynman gauge) in [39].",
"At five loops, $\\gamma _2$ is known for SU(3) in Feynman gauge [7], and we will below once more present the generalization to a general Lie group.",
"Up to four loops, we obtain ($\\xi $ being the covariant gauge parameter, such that the values $\\xi =0/1$ correspond to Feynman/Landau gauge) $\\gamma _2&=&-c_f\\,a\\,\\Big \\lbrace (1-\\xi )+\\gamma _{21}\\,a+\\gamma _{22}\\,a^2+\\gamma _{23}\\,a^3+\\gamma _{24}\\,a^4+\\dots \\Big \\rbrace \\;,\\\\2^2\\,\\gamma _{21} &=& n_f\\Big [-8\\Big ]+\\Big [-6c_f+34-10\\xi + \\xi ^2\\Big ]\\;,\\\\2^53^2\\,\\gamma _{22} &=& n_f^2\\Big [640\\Big ]+ n_f\\Big [8 (108 c_f- 1301 + 153 \\xi )\\Big ] \\\\&& + \\Big [432 c_f^2 -\\!",
"72 (143 \\!-\\!48 \\zeta _3)c_f+\\!2 (10559 -\\!1080 \\zeta _3) -\\!9 \\xi (371 +\\!48 \\zeta _3) +\\!27 \\xi ^2 (23 +\\!4 \\zeta _3) -\\!90 \\xi ^3 \\Big ]\\;,\\nonumber \\\\2^43^5\\,\\gamma _{23} &=&n_f^3 \\Big [ 13440 \\Big ]+ n_f^2 \\Big [ 6912 (19 - 18 \\zeta _3)c_f+ 16 (6835 + 9072 \\zeta _3) + 64 \\xi (269 - 324 \\zeta _3) \\Big ]\\nonumber \\\\&&+ n_f\\Big [5184 (19 - 48 \\zeta _3) c_f^2+ \\big (-108 (2407 - 1584 \\zeta _3 - 1296 \\zeta _4 - 5760 \\zeta _5)\\nonumber \\\\&&+ 324 \\xi (767 - 528 \\zeta _3 - 144 \\zeta _4) \\big )c_f+ 497664 d_1-(1365691 + 154224 \\zeta _3 + 97200 \\zeta _4 + 311040 \\zeta _5)\\nonumber \\\\&&+ \\xi (48865 + 152928 \\zeta _3 + 29160 \\zeta _4)- 54 \\xi ^2 (109 + 84 \\zeta _3 - 18 \\zeta _4)\\Big ]\\nonumber \\\\&&+\\Big [- 486 (1027 + 3200 \\zeta _3 - 5120 \\zeta _5) c_f^3+ 324 (5131 + 10176 \\zeta _3 - 17280 \\zeta _5) c_f^2\\nonumber \\\\&&+ \\big (-108 (23777 + 7704 \\zeta _3 + 2376 \\zeta _4 - 28440 \\zeta _5) - 1944 \\xi (6 - 7 \\zeta _3 + 10 \\zeta _5)\\big ) c_f\\nonumber \\\\&&+ 486 \\big (16 (-33 + 95 \\zeta _3 - 85 \\zeta _5) - 8 \\xi (1 + 48 \\zeta _3 - 70 \\zeta _5)- 8 \\xi ^2 (7\\zeta _3+5\\zeta _5)+ 20 \\xi ^3 (2\\zeta _3-\\zeta _5)\\nonumber \\\\&&- \\xi ^4 (7\\zeta _3-5\\zeta _5)\\big ) d_2+ (10059589/4 - 241218 \\zeta _3 + 168156 \\zeta _4 - 604260 \\zeta _5)\\nonumber \\\\&&- \\xi (2127929/8 + 164106 \\zeta _3 - 21141 \\zeta _4 - 107730 \\zeta _5)+ 27 \\xi ^2 (13883 + 9108 \\zeta _3 - 1548 \\zeta _4\\nonumber \\\\&&- 1920 \\zeta _5)/8- 81 \\xi ^3 (263 + 65 \\zeta _3 - 9 \\zeta _4 + 20 \\zeta _5)/2+ 81 \\xi ^4 (57 + \\zeta _3 + 10 \\zeta _5)/4\\Big ] \\;.$ We have presented the full gauge parameter dependence above.",
"Note that this fills a gap in the literature and constitutes new information at four loops: in [39] only the terms linear in $\\xi $ have been evaluated for a general Lie group, while with full gauge dependence only the SU(N) result is available [38].",
"However, due to the degeneracies $2d_1=6c_f^2-5c_f+13/12$ and $2d_2=7/12-c_f$ in the SU(N) limit, one cannot uniquely extract the Lie group structure from the latter reference.",
"Needless to say that our result for $\\gamma _{23}$ given in eq.",
"() reproduces the $\\xi ^0$ and $\\xi ^1$ terms given in [39], and in the SU(N) limit reduces to the respective expressions of [38], for all powers of $\\xi $ .",
"Expanding the all-order large-$N_{\\mathrm {f}}$ Landau gauge result of eq.",
"(REF ) in the coupling $a_f$ allows to confirm eqs.",
"(REF )-() to NLO in $n_f$ , and to predict the first two terms of $\\gamma _{24}$ in that gauge as $24^3\\,\\gamma _{24}|_{\\xi =1}&=&\\frac{83-144\\zeta _3}{72}\\,\\Big [16n_f\\Big ]^4+\\gamma _{243}^{\\,\\xi =1}\\,\\Big [16n_f\\Big ]^3+\\dots \\;,\\\\\\gamma _{243}^{\\,\\xi =1} &=& \\Big \\lbrace c_f, 1\\Big \\rbrace .\\Big \\lbrace -659/18 + 312 \\zeta _3 - 216 \\zeta _4, -1783/36 - 248 \\zeta _3 + 216 \\zeta _4 \\Big \\rbrace \\;.$ At five loops, along the same steps as explained in section , we have obtained the new Feynman gauge resultThe restriction to $\\xi =0$ is for practical reasons only.",
"To evaluate the $\\xi $ -dependent coefficients, one would need to enlarge the integral reduction tables as produced by crusher and TIDE to integrals with higher propagator powers (or dots), roughly one more dot per power of the gauge parameter.",
"Since the present calculation is at the limit of what the computing resources available to us are able to handle, we defer this to future work.",
"$24^3\\,\\gamma _{24} &=&\\frac{83-144\\zeta _3}{72}\\,\\Big [16n_f\\Big ]^4+\\gamma _{243}\\,\\Big [16n_f\\Big ]^3+\\gamma _{242}\\,\\Big [16n_f\\Big ]^2+\\gamma _{241}\\,\\Big [16n_f\\Big ]+\\gamma _{240}+{\\cal O}(\\xi )\\;,$ where the coefficients again contain only Zeta values up to weight 7, $\\gamma _{243} &=& \\Big \\lbrace c_f, 1\\Big \\rbrace .\\Big \\lbrace -659/18 + 312 \\zeta _3 - 216 \\zeta _4, -3443/48 - 255 \\zeta _3 + 252 \\zeta _4\\Big \\rbrace \\;,\\\\\\gamma _{242} &=& \\Big \\lbrace c_f^2, c_f, d_1, 1\\Big \\rbrace .\\Big \\lbrace -2 (2497 - 1200 \\zeta _3 + 3456 \\zeta _4 - 8640 \\zeta _5), \\nonumber \\\\&&477433/12 - 45636 \\zeta _3 + 4608 \\zeta _3^2 + 11448 \\zeta _4 - 65088 \\zeta _5 + 28800 \\zeta _6,-384 (115 - 33 \\zeta _3 - 90 \\zeta _5),\\nonumber \\\\&&3015955/72 + 69509 \\zeta _3 - 2304 \\zeta _3^2 - 12861 \\zeta _4 + 16662 \\zeta _5 - 14400 \\zeta _6 - 11907 \\zeta _7 \\Big \\rbrace \\;,\\\\\\gamma _{241} &=& \\Big \\lbrace c_f^3, c_f^2, c_fd_1, c_f, d_1, d_2, 1\\Big \\rbrace .\\Big \\lbrace 24 (29209 + 89984 \\zeta _3 + 12288 \\zeta _3^2 - 28800 \\zeta _4 - 187520 \\zeta _5 +76800 \\zeta _6), \\nonumber \\\\&& -4 (296177 + 517020 \\zeta _3 + 26784 \\zeta _3^2 - 469908 \\zeta _4 -4104720 \\zeta _5 + 1069200 \\zeta _6 + 3011904 \\zeta _7), \\nonumber \\\\&& -2304 (748 + 4536 \\zeta _3 -1368 \\zeta _3^2 - 6780 \\zeta _5 + 3255 \\zeta _7), \\nonumber \\\\&& 8 (115334 - 37764 \\zeta _3 -123012 \\zeta _3^2 - 49923 \\zeta _4 - 1124556 \\zeta _5 + 133650 \\zeta _6 +1519308 \\zeta _7), \\nonumber \\\\&& 192 (16732 + 39912 \\zeta _3 - 10944 \\zeta _3^2 - 72960 \\zeta _5 +36771 \\zeta _7), \\nonumber \\\\&&96 (6158 - 13952 \\zeta _3 - 372 \\zeta _3^2 + 2880 \\zeta _4 - 39475 \\zeta _5 - 3900 \\zeta _6 + 45696 \\zeta _7),\\\\&& -34919359/9 - 753797 \\zeta _3 + 548148 \\zeta _3^2 - 135063 \\zeta _4 + 1759474 \\zeta _5 +265350 \\zeta _6 - 2647806 \\zeta _7\\Big \\rbrace \\;,\\nonumber \\\\\\gamma _{240} &=& \\Big \\lbrace c_f^4, c_f^3, c_f^2, c_fd_2, c_f, d_2, d_3, 1\\Big \\rbrace .\\Big \\lbrace 1728 (4977 + 128000 \\zeta _3 + 19968 \\zeta _3^2 + 180800 \\zeta _5 -381024 \\zeta _7), \\nonumber \\\\&& -96 (835739 + 8494144 \\zeta _3 + 1182336 \\zeta _3^2 - 316800 \\zeta _4 +3983360 \\zeta _5 + 844800 \\zeta _6 - 17852688 \\zeta _7), \\nonumber \\\\&& 192 (825361 + 5472068 \\zeta _3 +651816 \\zeta _3^2 - 335808 \\zeta _4 - 1140420 \\zeta _5 + 950400 \\zeta _6 -8056377 \\zeta _7), \\nonumber \\\\&& 4608 (10 + 53226 \\zeta _3 - 15264 \\zeta _3^2 + 2145 \\zeta _5 -45885 \\zeta _7), \\; -16 (84040774/9 \\nonumber \\\\&& + 33396648 \\zeta _3 + 2804616 \\zeta _3^2 - 838782 \\zeta _4 - 18160944 \\zeta _5 +6252300 \\zeta _6 - 41015331 \\zeta _7), \\nonumber \\\\&& -384 (43066 + 628802 \\zeta _3 - 160998 \\zeta _3^2 + 36540 \\zeta _4 - 201125 \\zeta _5 -53475 \\zeta _6 - 403263 \\zeta _7), \\nonumber \\\\&&-72 (20566 - 218812 \\zeta _3 - 79080 \\zeta _3^2 - 13212 \\zeta _4 + 760220 \\zeta _5 + 20100 \\zeta _6 - 660667 \\zeta _7),\\; 804023630/9 \\nonumber \\\\&& + 101490400 \\zeta _3 + 3143352 \\zeta _3^2 + 7356024 \\zeta _4 -86186276 \\zeta _5 + 18372900 \\zeta _6 - 115799439 \\zeta _7 \\Big \\rbrace \\;.$ As a speculation, comparing the Landau gauge prediction eq.",
"(REF ) with the Feynman gauge result eq.",
"(REF ), the full result for $\\gamma _{243}$ could be simply adding $\\delta \\gamma _{243}=\\xi (3197/144+7\\zeta _3-36\\zeta _4)$ ; more generally however, consistency only requires that $\\delta \\gamma _{243}=c_f\\,u_1(\\xi )+u_2(\\xi )$ with $u_1(0)=0=u_1(1)$ as well as $u_2(0)=0$ and $u_2(1)=3197/144+7\\zeta _3-36\\zeta _4$ , the simplest choice being the one speculated above.",
"As a check, replacing the group invariants with the values of eq.",
"(REF ) in our Feynman gauge result for $\\gamma _{24}$ , we find perfect agreement with the known SU(3) result of [7] (see also eq.",
"(46) of [40])."
],
[
"Conclusions",
"We have provided new results for two fundamental renormalization coefficients, at five loops and for a semi-simple Lie group.",
"In particular, our expression for the gauge-invariant quark mass anomalous dimension $\\gamma _m$ coincides in various limits (large $N_{\\mathrm {f}}$ as well as SU(3)) with previously known results, and coincides exactly with recent results of another group [32].",
"We have also provided the Feynman gauge result for the quark field anomalous dimension $\\gamma _2$ , which again could be checked against known expressions in the abovementioned limits.",
"From these two quantities, one can reconstruct the two renormalization constants $Z_m$ and $Z_2$ of the quark sector, an electronic version of which is available by downloading the source of this article from http://arXiv.org/abs/1612.05512.",
"As had already been observed in [5], looking at e.g.",
"the 4-loop result for the quark mass anomalous dimension eq.",
"(REF ), all Zeta terms (and also the higher group invariants $d_n$ ) vanish at $\\lbrace c_f=1,n_f=\\frac{1}{2},d_1=d_2\\rbrace $ , which corresponds to ${\\cal N}=1$ supersymmetry.",
"The same had happened for the 4-loop Beta function (generalizing the last condition to $d_1=d_2=d_3$ ).",
"For these parameters values, from section we have $\\gamma _m &=& -a\\Big \\lbrace 3+\\!16\\,a+\\!\\tfrac{310}{3}\\,a^2+\\!\\tfrac{2228}{3}\\,a^3+\\!\\Big (\\tfrac{671075}{108} -\\!194\\,d_1+\\!\\big (\\tfrac{1483}{2} -\\!2028\\,d_1\\big ) \\zeta _3-\\!20\\big (55 +\\!354\\,d_1\\big ) \\zeta _5\\Big ) a^4\\Big \\rbrace \\;,\\nonumber $ where we observe that the cancellation pattern does not hold through five loops – although the structure becomes much simpler, due to cancellation of all terms containing $\\lbrace \\zeta _3^2,\\zeta _4,\\zeta _6,\\zeta _7\\rbrace $ .",
"To conclude the renormalization program at five loops, one needs to determine three more renormalization constants, which can be chosen to be those of the gluon and ghost fields, $Z_3$ and $Z_3^{c}$ , respectively, and of the ghost-gluon vertex $Z_1^{ccg}$ .",
"From these, due to gauge invariance, one can then construct the renormalization constant for the gauge coupling (aka the Beta function, whose five-loop coefficient is so far known for SU(3) only [41]) as well as those for the remaining vertices.",
"While the same methods that we have used here are sufficient to calculate those missing coefficients (which indeed already led to the NLO terms at large $N_{\\mathrm {f}}$ [8]), due to the complexity of the determination of $Z_3$ we leave the evaluation of the full coefficients for future work.",
"We are indebted to K. Chetyrkin for sending us their new five-loop results for $\\gamma _m$ prior to publication [32], to enable the important check discussed at the end of section .",
"The work of T.L.",
"has been supported in part by DFG grants GRK 881 and SCHR 993/2.",
"A.M. is supported by a European Union COFUND/Durham Junior Research Fellowship under EU grant agreement number 267209.",
"P.M. was supported in part by the EU Network HIGGSTOOLS PITN-GA-2012-316704.",
"Y.S.",
"acknowledges support from FONDECYT project 1151281 and UBB project GI-152609/VC.",
"All diagrams were drawn with Axodraw [42], [43]."
],
[
"Summary of large-$N_{\\mathrm {f}}$ results",
"Some coefficients of QCD anomalous dimensions are known to all loop orders, from a large $N_{\\mathrm {f}}$ expansion.",
"Taking $n_f$ and $c_f$ as above and defining $a_f\\equiv \\frac{N_{\\mathrm {f}}T_{\\mathrm {F}}g^2(\\mu )}{12\\pi ^2}\\;=\\;\\frac{4\\,n_f\\,a}{3}\\quad ,\\qquad \\eta (\\varepsilon )\\equiv \\frac{(2\\varepsilon -3)\\Gamma (4-2\\varepsilon )}{16\\Gamma ^2(2-\\varepsilon )\\Gamma (3-\\varepsilon )\\Gamma (\\varepsilon )}\\;,$ the all-order leading-$N_{\\mathrm {f}}$ [44] and next-to-leading-$N_{\\mathrm {f}}$ [45], [46] expressions that we need here read $\\gamma _m&=&\\frac{4c_f}{n_f}\\Big \\lbrace \\eta (a_f)+\\frac{\\eta _3(a_f)}{8\\,n_f}+{\\cal O}\\Big (\\frac{1}{n_f^2}\\Big )\\Big \\rbrace \\;,\\\\\\gamma _2|_{\\xi =1}&=&-\\frac{2a_fc_f}{n_f}\\Big \\lbrace \\eta (a_f)+\\frac{1}{n_f}\\frac{\\eta _4(a_f)}{4a_f}+{\\cal O}\\Big (\\frac{1}{n_f^2}\\Big )\\Big \\rbrace \\;,$ where the fact that the asymptotic expansions have been performed in Landau gauge only does not affect the physical and gauge invariant quark mass anomalous dimension $\\gamma _m$ .",
"To define the coefficient functions $\\eta _3$ and $\\eta _4$ , it is convenient to define the linear combinations $\\Psi (\\varepsilon ) &=& \\psi _0(1 - 2 \\varepsilon ) + \\psi _0(1 + \\varepsilon ) -\\psi _0(1 - \\varepsilon ) - \\psi _0(1) \\;,\\\\\\Phi (\\varepsilon ) &=& \\psi _1(1 - 2 \\varepsilon ) - \\psi _1(1 + \\varepsilon ) -\\psi _1(1 - \\varepsilon ) + \\psi _1(1) \\;,\\\\\\Theta (\\varepsilon ) &=& \\psi _1(1 - \\varepsilon ) - \\psi _1(1) \\;,$ where $\\psi _n(x)=\\partial _x^{n+1}\\ln \\Gamma (x)$ is the PolyGamma function.",
"Then [45], [46] $\\eta _3(\\varepsilon ) &\\equiv &\\Big (\\!-\\!\\frac{11}{4} +\\!",
"\\sum _{n>0}\\frac{f_n\\varepsilon ^n}{n}\\Big ) 8 \\varepsilon \\partial _\\varepsilon \\eta (\\varepsilon )-\\frac{16 \\eta ^2(\\varepsilon )}{(3 \\!-\\!",
"2 \\varepsilon )(1 \\!-\\!",
"\\varepsilon )}\\Big \\lbrace 3 (2 - \\varepsilon )^2 (1 - \\varepsilon )^2 \\Theta (\\varepsilon )-(5 + 5 \\varepsilon - 11 \\varepsilon ^2 + 4 \\varepsilon ^3) ,\\nonumber \\\\&&(88 \\!-\\!",
"372 \\varepsilon \\!+\\!",
"551 \\varepsilon ^2 \\!-\\!",
"380 \\varepsilon ^3 \\!+\\!",
"160 \\varepsilon ^4 \\!-\\!",
"64 \\varepsilon ^5 \\!+\\!",
"16 \\varepsilon ^6)\\!-\\!",
"4 \\varepsilon (3 \\!-\\!",
"2 \\varepsilon ) (1 \\!-\\!",
"2 \\varepsilon ) (2 \\!-\\!",
"\\varepsilon ) (1 \\!-\\!",
"\\varepsilon )^2 \\big (\\Psi ^2(\\varepsilon ) \\!+\\!",
"\\Phi (\\varepsilon )\\big )\\nonumber \\\\&&+ 2 (1 - \\varepsilon ) (24 - 144 \\varepsilon + 249 \\varepsilon ^2 - 146 \\varepsilon ^3 + 12 \\varepsilon ^4 + 8 \\varepsilon ^5) \\Psi (\\varepsilon )\\Big \\rbrace .\\Big \\lbrace \\tfrac{2}{2 - \\varepsilon }\\,c_f, \\tfrac{1}{4\\varepsilon (3 - 2 \\varepsilon )(1 - 2 \\varepsilon )}\\Big \\rbrace \\;,\\\\\\eta _4(\\varepsilon ) &=& \\frac{\\varepsilon \\eta _3(\\varepsilon )}{2}+\\Big (\\!-\\!\\frac{11}{4} +\\!",
"\\sum _{n>0}\\frac{f_n\\varepsilon ^n}{n}\\Big ) 4\\varepsilon \\eta (\\varepsilon )+\\frac{2\\eta ^2(\\varepsilon )}{3-2\\varepsilon } \\Big \\lbrace -8(1-4\\varepsilon +2\\varepsilon ^2),\\tfrac{(2-5\\varepsilon +2\\varepsilon ^2)^2}{1-\\varepsilon }\\Big \\rbrace .",
"\\Big \\lbrace c_f,1\\Big \\rbrace \\;,\\\\{\\rm with}&&\\sum _{j>0}f_j\\,\\varepsilon ^j \\;\\equiv \\; -\\eta (\\varepsilon )\\Big \\lbrace 4(1+\\varepsilon )(1-2\\varepsilon )c_f+\\tfrac{4\\varepsilon ^4-14\\varepsilon ^3+32\\varepsilon ^2-43\\varepsilon +20}{(1-\\varepsilon )(3-2\\varepsilon )}\\Big \\rbrace \\;.$ The coefficients $a_f$ , $\\eta (\\varepsilon )$ and $f_j$ are the same as we had defined in [8]."
]
] | 1612.05512 |
[
[
"How long does a quantum particle or wave stay in given region of space?"
],
[
"Abstract The delay time associated with a scattering process is one of the most important dynamical aspects in quantum mechanics.",
"A common measure of this is the Wigner delay time based on the group velocity description of a wave-packet, which my easily indicate super-luminal or even negative times of interaction that are unacceptable.",
"Many other measures such as dwell times have been proposed, but also suffer from serious deficiencies, particularly for evanescent waves.",
"One important way of realising a timescale that is causally connected to the spatial region of interest has been to utilize the dynamical evolution of extra degrees of freedom called quantum clocks, such as the spin of an electron in an applied magnetic field or coherent decay or growth of light in an absorptive or amplifying medium placed within the region of interest.",
"Here we provide a review of the several approaches developed to answer the basic question - how much time does a quantum particle (or wave) spend in a specified region of space?",
"While a unique answer still evades us, important progress has been made in understanding the timescales and obtaining positive definite times of interaction by noting that all such clocks are affected by spurious scattering concomitant with the very clock potentials, however, weak they be and by eliminating the spurious scattering."
],
[
"Introduction",
"One of the first things a student of Physics learns is to calculate the time at which an event, such as the location of a particle at a given position, occurs.",
"A related question would be to ask how long does a particle stay in a given region of space during the course of its motion.",
"Classically it is possible to simultaneously specify the position and the momentum of a particle, and these questions about the time instants or the time intervals for given events have unambiguous answers.",
"Even as we proceed to statistically understand systems with very large number of particles through probability distributions, where the classical motion of each particle is described only stochastically, concepts such as the first passage times [1] remain meaningful.",
"This well entrenched concept, however, becomes difficult to define for quantum mechanical systems, and indeed, for any form of waves.",
"Figure: Left panel: Schematic picture of a quantum particle that is initially localized in a fixed region and arrives at a detector array.",
"The first passage time refers to the first point in time when the particle is registered at a detector.",
"Right panel: The particle may take any virtual path toget to the detector, including ones that cross the boundary many times.",
"The final amplitude will be a sum over the amplitudes for any such path.Imagine an experiment where a quantum-mechanical particle is released from some fixed region inside a box.",
"On one side of the box there is a screen with detectors which click as soon as the particle \"arrives\" at the screen.",
"One expects that the time of arrival of the particle is a stochastic variable and it is interesting to ask for it's probability distribution.",
"This is similar to asking for the distribution of the time of absorption of a Brownian particle at some point.",
"However, the quantum problem turns out to be very subtle and there is as yet no clear answer to the question.",
"The point is that the particle that is initially localized and released must subsequently cross a given boundary in space when it is detected for the very first time.",
"Yet this very calculation includes amplitudes from paths that may very well extend all over space even beyond the given boundary.",
"Very recently the problem of first time arrival of a quantum mechanical particle has been considered satisfactorily utilizing a path integral approach that with a restricted path decomposition and appears to succeed in obtaining positive definite quantum first passage times for motion on a one dimensional lattice [2].",
"The principal difficulty arises from the fact that waves are infinitely deformable objects and many aspects of motion arise through interference effects.",
"Hence it becomes impossible to define well defined start and finish lines for wave packets.",
"In the quantum mechanical perspective, where all dynamical observables have a corresponding operator, there has been great difficulty in defining a “good” Hermitian operator that conforms to classical notions of time or a time interval [3].",
"For classical waves such as electromagnetic waves also, related difficulties exist.",
"For example, we would like to classically demand that a time of stay in a given region of space should be (i) Real, (ii) Positive definite, (iii) tend to classically calculable times in the limit of large energies.",
"Inspite of serious efforts over many years [4], no definition for such a quantum mechanical time has been universally accepted.",
"In fact, when we say that a wave moves, it becomes imperative to clearly define what is the quantity related to the motion of a wave that is being talked about.",
"Regardless of these difficulties, it must be emphasized that the times of traverse or dwell associated with a wave are calculable quantities that can be useful for understanding processes and process rates in a given system.",
"In this tutorial, we will discuss the several approaches, the related difficulties and some recent possibilities that have arisen in this context.",
"First, we will go through some of the basic definitions and concepts regarding wave motion and the times that can be related to the motion of a wave.",
"It turns out that there is a large variety of times arising from different definitions related to various different quantities associated with a wave.",
"Surprisingly, these definitions can result in apparent superluminal times or even negative delay times for the transit of a wave through dispersive media or potentials.",
"Subsequently, we will describe certain clocking mechanisms associated with physical processes such as the precession of a spin in a magnetic field [5] that have been developed to calculate theoretically the traversal or dwell times of a wave [4].",
"These clocking mechanisms have to be used with care as they can contribute to scattering and change the very problem being discussed [6].",
"These ideas lead to the possibility of using quantum dephasing or stochastic absorption as a clock."
],
[
"1. Time-scales based on the group velocity ",
"Here we will try to explore these questions in the general context of waves to cover both electromagnetism and quantum mechanics.",
"Both the time independent Schrodinger equation for the wavefunction of a quantum particle and the Maxwell equations in frequency domain for the amplitude of a time harmonic electromagnetic wave (of only one polarization) reduce to the Helmholtz wave equation: $\\nabla ^2 \\psi + k^2 \\psi = 0.$ For the electromagnetic wave, k is the wave vector given by $k^2 = n^2 \\omega ^2 /c^2$ , $\\omega $ is the angular frequency of the wave and $n$ is the refractive index of the medium with $c$ being the speed of light in vacuum.",
"For the quantum particle in comparison, $k$ is given by $k^2 = 2m(E-V_0)/\\hbar $ , where$E$ and $m$ are the energy and the mass of the particle and$V_0$ is the potential.",
"The principle difference between the the two systems is in free space where the refractive index is unity.",
"Consequently the wave-vector for an electromagnetic wave is linearly proportional to the frequency.",
"In the case of the quantum particle, even for $V_0=0$ , the wave-vector disperses with the quadratic root of the energy.",
"This has a non-trivial manifestation in the undistorted propagation of electromagnetic pulses in free space while a quantum mechanical wavepacket spreads out and disperses even in free space.",
"Thus, it is realized that the fundamental properties of propagation of the waves are governed by the potentials or the refractive index of the medium through the dispersion relation between the wave-vector and the frequency or energy.",
"Consider the scalar wave $\\psi (\\vec{r},t) = a(\\vec{r}) \\exp [i\\phi (\\vec{r}) - i\\omega t]$ where $\\phi (\\vec{r})$ is some scalar function.",
"For this wave, $\\phi (\\vec{r}) = \\mathrm {constant}$ denotes the constant phase surfaces.",
"To trace the motion of these surfaces, let us look at the condition at two points $(\\vec{r},t)$ , and $(\\vec{r} + \\delta \\vec{r}, t+\\delta t)$ .",
"The phase front is the same if, and only if, $\\phi (\\vec{r} + \\delta \\vec{r}) -\\omega ( t+\\delta t) \\simeq \\phi (\\vec{r}) + \\delta \\vec{r} \\cdot \\nabla \\phi (\\vec{r}),t) - \\omega t - \\omega \\delta t = \\phi (\\vec{r}) - \\omega t,$ where we have included the first order term only in the infinitesimal $ \\delta \\vec{r}$ .",
"FRom the above, we obtain $ v_p = \\left| \\frac{\\delta \\vec{r}}{\\delta t} \\right| = \\frac{\\omega }{| \\nabla \\phi |} $ as the phase velocity for a wave with an arbitrary wave front.",
"The ratio , where $\\phi (\\vec{r})$ is the phase of the wave [7], is known as the phase velocity of the wave and represents the rate at which the equiphase surfaces of the wave move through the medium.",
"Note that the phase velocity can be just about any number (positive or negative) depending on the phase structure (gradient) of the wave.",
"In one dimension or when there is transverse invariance along two dimensions and plane waves result, the phase velocity reduces to the familiar relation.",
"This gives rise to the conventional notion that the phase velocity for a wave is $c/n$ , and is hence mistaken for the speed of a wave in a medium.",
"As pointed out, the phase velocity can easily be superluminal for materials with refractive index $n < 1$ .",
"Further, in negative refractive index materials [8] and in waveguides that support backward wave propagation, this phase velocity is obviously negative.",
"Thus, the phase velocity does not really specify anything concrete about the rate of the motion of the wave or the time spent by a wave in a given region of space.",
"In a material medium, the refractive index (polarization) of a medium will, in general, be frequency dependent and a complex quantity.",
"This is a simple consequence of the medium having certain natural frequencies at which it will resonantly polarize corresponding to atomic or molecular levels of the constituents of the medium.",
"Due to the different refractive index at different frequencies, the constituent time harmonic waves present in a wave packet essentially travel at their own phase velocities resulting in the interference pattern changing completely in time and leading to a dispersion of the wave packet.",
"This is easily seen by writing down the field amplitude of the wave at different times.",
"If $E(\\vec{r},0)$ be the field amplitude at time$t =0$ , and $E(\\vec{k} = \\int _{-\\infty }^\\infty E(\\vec{r},0) e^{-i\\vec{k}\\cdot \\vec{r}} ~d^3r$ is is the spatial Fourier transform of the field amplitude, the field amplitude at any other time, $t$ , is given by $E(\\vec{r}, t) = \\left(\\frac{1}{2\\pi }\\right)^3 \\int _{-\\infty }^\\infty E(\\vec{k}) e^{i(\\vec{k}\\cdot \\vec{r}-\\omega t)} ~d^3k .$ If the dispersion was linear, this would just correspond to the same function shifted to a new position.",
"For an arbitrary dispersion $\\omega (\\vec{k})$ , it becomes difficult to say anything in general.",
"If we assume, however, that most of the amplitude is concentrated in a small frequency band about a central (carrier) frequency $\\omega _0$ , then one can carry out a Taylor’s series expansion $ \\omega (\\vec{k}) = \\omega (\\vec{k}_0) + (\\vec{k} -\\vec{k}_0) \\cdot \\nabla _k \\omega (\\vec{k}_0) + \\cdots , $ where the subscript $k$ indicates that the derivative is with respect to the wave-vector and retain only the linear term.",
"Substituting this in the expression for the field amplitude, one obtains, $E(\\vec{r}, t) = \\left(\\frac{1}{2\\pi }\\right)^3 e^{i\\varphi } \\int _{-\\infty }^\\infty E(\\vec{k}) e^{i\\vec{k}\\cdot [\\vec{r} - \\nabla _k\\omega (\\vec{k}_0) - \\omega t]} ~d^3k ,$ which is essentially the same waveform that is shifted by an amount $\\nabla _k \\omega (\\vec{k}_0)t$ in space, apart from a trivial extra phase of $\\varphi = \\omega _0 t + k_0 \\cdot \\nabla _k \\omega (\\vec{k}_0)t$ .",
"This brings up another rate at which the envelope of the wave propagates in the medium and defines the so-called group velocity $v_g = \\nabla _k \\omega (\\vec{k}_0)$ .",
"This tracks the rate at which fiducial (well recognisable) points on the waveform, such as the peak of the wavepacket, move and it is assumed that waveform is largely undistorted.",
"This is rarely satisfied for a quantum mechanical wave even in free space.",
"For an electromagnetic wave, however, this is satisfied for quasi-monochromatic fields and when absorption in the medium is minimal.",
"Again, it should be noted that the spread in the wavevectors is also equivalently small and the superposition should consist of waves moving along a common direction.",
"Noting that the group delay time accumulated upon traversing a distance $L$ in the medium is $ \\frac{L}{\\nabla _k \\omega (\\vec{k}_0)} = \\left.",
"\\frac{\\partial (kL)}{\\partial \\omega }\\right|_{\\omega _0} ,$ one can equivalently define a group delay time for scattering problems where the kinematic phase is replaced by the phase change upon scattering ($\\phi $ ).",
"In one dimension, this would be the phase shift upon reflection or transmittance.",
"This yields the Wigner’s group delay time $\\tau _w = \\left.",
"\\frac{\\partial \\phi }{\\partial \\omega } \\right|_{\\omega _0}.$ Note the extrapolation made here from a free propagation problem into the time delay obtained in a scattering event.",
"This time essentially measures the time difference between the entry and emergence of fiducial points into and out of the scattering volume.",
"Figure: The change in refractive index due to dispersion is plotted with a magnified scale.",
"The dispersion of the real part of the refractive index for a single Lorentz resonance (red dotted curve) and for a highly dispersive medium that can be produced, for example by electromagnetically induced transparency or by a Fano resonance.",
"Such dispersions enable very large changes of the refractive index, either normal or anomalous as marked in the figure.",
"The group velocity can easily become very small or very large and even negative in such dispersive regimes.Using the dispersion for light, $k^2 = n(\\omega )^2 \\omega ^2/c^2$ in a homogenous isotropic medium with a dispersive refractive index $n(\\omega )$ , the group velocity is also often conveniently written in the following manner $v_g = \\frac{c}{n(\\omega _0) + \\omega _0 (dn/ d\\omega )_{\\omega _0}}.$ The group velocity from this definition immediately shows that the group velocity can be very small when the dispersion is normal and large, i.e., $(dn/d\\omega ) \\gg 0$ (see Fig.",
"REF ).",
"This is the origin of ultra-slow light which has been demonstrated using the large dispersion possible in atomic gas vapours and Bose-Einstein condensates [9].",
"On the other, if the dispersion were anomalous and large , $(dn/d\\omega ) < 0~\\mathrm {and}~ |dn/d\\omega | \\gg 0$ (see Fig.",
"REF ), then we have that the group velocity can become greater than c, or even negative in such situations.",
"Indeed there have been several experiments on the apparent motion of light in a medium at a rate faster than light in vacuum [REF].",
"Negative group velocities and negative Wigner delay times are even more problematic – that would imply that the pulse exited region of space before it even entered, in a sense violating causality.",
"Landauer [4] had severely criticised the Wigner delay time to give information about the dwell times on grounds of causality and emphasized that there was no causal connection between the peaks (or fiducial points) of the incoming and the outgoing waveforms.",
"This difficulty is accentuated when the scattering potential strongly deforms the wave-packet.",
"The large dispersion responsible for the deformation is also usually accompanied by severe dissipation or gain that causes large spectral modifications with corresponding distortions of the temporal pulse shape.",
"Hence the analysis of the motion of a spectrally broad or distorted pulse in terms of the conventional group velocity is severely limited, as the pulse can lose its very identity after propagating to a large distance in the dispersive medium.",
"These arguments apply and hold true even for definitions of the times based on the center-of-mass of the wavepacket or in an alternative approaches where the motion of the forward edge of the wavepacket is followed.",
"These issues raise questions about the group velocity, or the Wigner delay time that is based on the group velocity, to actually provide an answer to the question that we seek about the time that the wave spends in a given region of space.",
"It has been emphasized by most authors that the superluminal or negative Wigner delay times do not violate causality or the Special theory of relativity.",
"It simply turns out that the functions that we considered to describe the fields have been smooth analytic functions.",
"This is a consequence of the fields being the solutions of the Helmhotz differential equation.",
"Analytic functions are problematic because they have an infinite support – the function extends all over the space, although it may be very small.",
"A good example of such functions is the Gaussian function.",
"In principle, one may always take the analytic function at a given point and carry out a Taylor's series expansion of the function using the values of the function and its derivatives at the given point, to obtain its value at any point in space, however, far off from that point.",
"Since the function is analytic, all the derivatives at the given point exist.",
"Thus, knowledge of the analytic signal even if localized in the manner of a Gaussian function, already exists at all the other points in principle, even if the value of the signal were to be infinitesimally small.",
"Thus, there is no extra information being conveyed to the other points with the wave motion as the information was already present.",
"Thus, there is no violation of the special theory of relativity or of causality.",
"It has been emphasized that in principle only meromorphic functions (with discontinuities in the function or its derivatives) can be used to encode information.",
"Any such discontinuity will generate very high frequency components in the power spectrum of the signal.",
"These high frequency components will always propagate at the speed of light in vacuum (c) as the refractive index as This is a consequence of finite inertia for the charge carriers in all material media."
],
[
"The Dwell times based on current fluxes or energy transfer",
"A second approach to this problem has been to define the time in terms of probability of finding the particle within the spatial volume of interest.",
"Smith [10] defined the dwell time for a mono-energetic quantum particle in the region $[0, L]$ (in one dimension) as $\\tau _d = \\frac{1}{J} \\int _0^L \\vert \\psi (x) \\vert ^2 ~dx$ where $\\psi (x)$ is the wavefunction and $ J = \\mathrm {Re} (\\hbar /im) (\\psi ^*\\;\\nabla \\psi )$ is the current flux associated with the incoming particle.",
"Note that the Smith dwell time, is independent of the scattering channel (reflection or transmission) and is hence, an unconditional time.",
"In case of a time-varying pulsed waveform, this dwell time can be generalized by integrating over all times as $\\tau _d = \\int _{-\\infty }^{\\infty } dt ~ \\int _0^L \\vert \\psi (x) \\vert ^2 ~dx.$ In the case of electromagnetic waves, a similar approach can be adopted, but using the energy of the fields.",
"Thus the dwell time in a region of volume $(V)$ could be defined as $ \\tau _d = \\frac{\\int _V U d^3r}{ \\int _A \\vec{S} \\cdot d\\vec{a}} , $ where $U$ is the energy density associated with the electromagnetic wave, $\\vec{S}$ is the Poynting vector denoting the power flow per unit area of the incoming wave and A is the surface area of volume through which the incoming wave is incident.",
"While there is no problem with the Poynting vector defined as $\\vec{S} = \\vec{E}\\times \\vec{H}$ in terms of the electric and magnetic fields, there are severe difficulties in defining an energy density solely associated with the wave in a dispersive and dissipative (or amplifying) medium.",
"The difficulties of separating the energy associated with the wave and the polarization in the medium are well known in this scenario where the two fields continuously exchange energy.",
"In fact, it is well known that the energy density defined by first order Taylor expansions of the dispersion [11] can easily be negative for severe dispersion of the material parameters.",
"In some sense, the above quantity and the Smith Dwell time are equivalent as both the quantities involve quadratic expressions of the underlying fields.",
"The issue of dispersion and polarization in material media complicate the issues further in the context of electromagnetic waves.",
"Another fruitful approach to define arrival times that is based on the centroid of power flow of the electromagnetic wave is noteworthy [12].",
"An arrival time at a point can be defined as a time average (first moment of time) over the component of the Poynting vector normal to a surface $\\langle t \\rangle _r = \\frac{\\hat{u} \\cdot \\int _{-\\infty }^\\infty t \\vec{S}(\\vec{r},t) dt}{ \\hat{u} \\cdot \\int _{-\\infty }^\\infty \\vec{S}(\\vec{r},t) dt},$ where $\\hat{u}$ denotes the unit normal to the given surface, which could very well be that of a detector.",
"The time for traverse between two points $\\vec{r}_i$ and $\\vec{r}_f$ can now be thought of as the difference between the arrival times at the two points as $\\Delta t = \\langle t\\rangle _{\\vec{r}_f} - \\langle t\\rangle _{\\vec{r}_i}$ .",
"A basic theorem was proven to show that the propagation delay could be decomposed in terms of a net group delay and a reshaping delay.",
"The net group delay consists of a spectrally wieghted average group delay at the final point $\\vec{r}_f$ given by $\\Delta t_G = = \\frac{\\hat{u} \\cdot \\int _{-\\infty }^\\infty t \\vec{S}(\\vec{r}_f,\\omega ) [(\\partial \\mathrm {Re}(k) / \\partial \\omega ) \\cdot \\Delta r] d\\omega }{ \\hat{u} \\cdot \\int _{-\\infty }^\\infty \\vec{S}(\\vec{r}_f,\\omega ) d\\omega },$ The reshaping delay, as the very name suggests, arises from the deformation due to spectral modulation by the medium and can be calculated in terms of the spectral fields at the initial point as $\\Delta t_R = \\wp \\lbrace \\exp [-\\mathrm {Im}(\\vec{k}) \\cdot \\Delta \\vec{r}] \\vec{E}(\\vec{r}_i, \\omega )\\rbrace \\wp \\lbrace \\vec{E}(\\vec{r}_i, \\omega )\\rbrace ,$ where the operator $ \\wp \\lbrace \\vec{E}(\\vec{r}, \\omega )\\rbrace =\\frac{\\hat{u} \\cdot \\int _{-\\infty }^\\infty \\mathrm {Re} [-i \\frac{\\partial \\vec{E}(\\vec{r}, \\omega )}{\\partial \\omega }\\; \\times \\; \\vec{H}^*(\\vec{r},\\omega )] d\\omega }{ \\hat{u} \\cdot \\int _{-\\infty }^\\infty \\vec{S}(\\vec{r},\\omega ) d\\omega }, $ This approach which is applicable to an arbitrary waveform or dispersive medium showed that both the components in general were always significant.",
"The most important aspect of this proposal is that it does not involve any perturbative expansion of the wave number around the carrier frequency.",
"A few salient points may be noted in this context: For a narrowband pulse, the reshaping delay tends to zero and the total delay time is dominated by the group delay time.",
"We note that even the Wigner delay time would describe the situation quite well in this case.",
"For a broadband pulse with only propagating components, the net group delay can become negative, but the corresponding reshaping delay causes the overall delay time to remain luminal as the pulse experiences a strong reshaping during propagation.",
"This makes this proposal a very strong candidate to represent the traversal time that is causal and non-negative.",
"It has been shown that the definition is equivalent to another definition based on the rate of energy absorbed by a detector $ \\langle t \\rangle _r = \\frac{ \\int _{-\\infty }^\\infty t \\frac{d A(\\vec{r},t)}{dt} dt}{ \\int _{-\\infty }^\\infty \\frac{d A(\\vec{r},t)}{dt} dt}, $ where$(dA/dt)$ is the rate of absorption of energy per unit volume inside the detector placed at $\\vec{r}$ , and is given by $ \\frac{dA}{dt} = \\int \\int \\omega \\left[ \\varepsilon _0 \\mathrm {Im}(\\varepsilon ) E^*(\\omega ^{\\prime }) E(\\omega ) + \\mu _0\\mathrm {Im}(\\mu )H^*(\\omega ) H(\\omega ) \\right]d\\omega ^{\\prime }~d\\omega .",
"$ This is integrated spatially over the detector volume to obtain the total rate of absorption within the detector.",
"Obviously the spatial extent of the detector is assumed to be small compared to the length scales of propagation or spatial pulse widths.",
"In a medium such as a plasma, however, it has been shown [13] that a positivity of the traversal time in not obtained, particularly for broad-band pulses, which have large amounts of evanescent frequency components when the carrier frequency is smaller than the plasma frequency.",
"This is related to the Hartmann effect [14], whereby the time for tunelling at far-sub-barrier energies becomes almost constant and negates the possibility of using this definition in principle for all situations.",
"Nevertheless, it should be noted that it is still operationally one of the most useful definitions of the traversal times and has been validated in experiments involving both temporally dispersive and angularly dispersive situations."
],
[
"Quantum clocks ",
"Due to issues with the various timescales which fail to conform to our intuitive understanding of the traversal timescales, attempts were made to develop“quantum clocks” that measure the time the particle spends in a given volume.",
"In analogy with a classical clock, the clock should tick only when the particle is within the region of interest.",
"This is accomplished by coupling other degrees of freedom to other fields localized within the region of interest.",
"Dynamical evolution of those degrees of freedom occur only when the particle is present in the region of interest.",
"Thus, the expectation values of quantities associated with those degrees of freedom will translate to expectation values of the times spent in the region of space.",
"While clocks have been proposed in principle for a long time [15], three powerful methods that can have direct experimental implementation have been proposed and are popular.",
"We will discuss the main ideas behind these proposals."
],
[
"Büttiker-Landauer oscillating barrier times",
"Büttiker and Landauer proposed [16] that the time of traverse of a charged particle through a potential barrier could be timed by super-imposing a time-harmonic electromagnetic field on top of the potential only within the region of interest.",
"Suppose is the region in which we seek the time of sojourn, the potential would be $V_0(\\vec{r}) + V_1(\\vec{r}) \\cos (\\omega t)$ where $V_0$ is the original static potential and the perturbing oscillating field has a magnitude ($V_1$ ) that is constant within the region of interest and is zero outside (see Fig.",
"REF for a schematic depiction).",
"Interaction of the charged particle with the electromagnetic field would cause cause the absorption or emission of quanta of radiation.",
"Thus, if the oscillation frequency was very small, the energy broadening of the transmitted or reflected spectrum would not be visible.",
"The particle effectively sees a static potential and the transmission and reflected fluxes will adiabatically vary in time with the potential barrier height that changes harmonically.",
"Thus, the times of interaction or traversal are very small compared to the time period of the oscillation.",
"On the other hand, at high frequencies of oscillation, the particle would see the potential undergoing many oscillations during the time of its stay in the region and it would exchange quanta with the field.",
"Then the energy spectrum of the reflected or transmitted particle would have energy sidebands separated from the incident energy by the energy of the quanta ($\\pm \\hbar \\omega $ ).",
"In this limit, the period of the electromagnetic field is clearly much smaller than the traversal time ($\\tau _s$ ).",
"There would be a crossover between the two regimes of low or high frequencies and the period of the electromagnetic field during the crossover regime ($\\omega \\tau _s \\simeq 1$ ) would a good measure of the time of traversal, regardless of whether the quantum wavefunction has a propagating nature or evanescent nature.",
"Figure: Schematic picture depicted the tunnelling of a particle through a potential barrier with a time dependent strength.",
"The particle exchanges quanta of energy (photons) with the radiation field and the transmission develops energy sidebands.",
"The particles with higher energy would tunnel through more efficiently than the particles with lower energy.",
"The period of the potential oscillations when the energy sidebands develop gives a timescale for the traversal of the particle through the barrierIn the one-dimensional case of a particle tunneling through a rectangular potential barrier, Buttiker and Landauer [16] obtain a tunneling time of $\\tau _{BL} = \\left[ \\frac{m}{2(V_0-E)} \\right]^{1/2} d,$ where $E$ is the energy of the tunneling particle and $d$ is the width of the barrier.",
"Using the WKB approximation at energies well below the barrier height, this was further generalized to spatially varying potential barrier in one-dimension as $\\tau _{BL} = \\int _{x_1}^{x_2} \\left[ \\frac{m}{2(V_0-E)} \\right]^{1/2} dx = \\int _{x_1}^{x_2} \\frac{m}{\\hbar \\kappa (x)} dx,$ where $\\hbar \\kappa (x) = \\sqrt{2m(V_0(x) - E)}$ behaves as the instantaneous momentum of the tunneling particle.",
"For more general potential shapes and energies, the calculations of the traversal times by this approach become very difficult.",
"This approach, however, clearly sets out a timescale for the problem, particularly for the limiting case of low energy tunneling, which any other valid approach would need to reproduce.",
"An approach where the flow of the particle through the barrier region was constructed in terms of two counter-propagating streams within the WKB approximation also obtained the traversal times consistent with the one obtained here [17]."
],
[
"Büttiker's spin clock (Larmor precession and spin flip)",
"A second clock related to using the Larmor precession of a quantum particle with an associated magnetic moment (or spin), such as an electron or a neutron, in an applied magnetic field (B).",
"The rate of the precession of the spin $\\omega _L = g\\mu _B B/\\hbar $ , ($\\mu _B$ is the Bohr magneton) is constant in a spatially constant magnetic field and could, thus, act as a possible clocking mechanism.",
"Consider a magnetic spin that is initially polarized along the x-axis and moving along the y-axis through a region with potential, $V(y)$ , wherein a uniform magnetic field along the z-axis is applied (see Fig.",
"REF for a schematic depiction).",
"If the particle takes a time ($\\tau _y$ ) to traverse through the region of the magnetic field, the spin components of the transmitted flux would be $\\langle S_y\\rangle = -(\\hbar /2) \\omega _L\\tau _y$ to the lowest order in the magnetic field.",
"Thus, measurement of the spin precession could yield the traversal time as $\\tau _y = \\frac{2}{g\\mu _B} \\lim _{B\\rightarrow 0} \\frac{\\partial \\langle S_y\\rangle }{\\partial B}$ Büttiker [5] recognized that the spin has a tendency to align along the magnetic field direction that he called spin rotation in addition to the Larmor precession in the place perpendicular to the magnetic field (spin precession).",
"Thus, there could be two time scales associated with the extents of the spin rotation ($\\tau _z$ ) and the spin precession ($\\tau _y$ ).",
"The Hamiltonian for the case of a rectangular barrier is $H = \\left\\lbrace \\begin{array}{l} (p^2/2m + V(y)) I - (\\hbar \\omega _L/2) \\sigma _z ~~~\\forall ~~~ 0 < y < L, \\\\ p^2/2m I~~~~\\mathrm {elsewhere} \\end{array} \\right.$ where $V(y)$ is the spatially constant potential, $I$ is the identity matrix and $\\sigma _z$ is the z-component of the Pauli spin matrix and $H$ acts on the spinor $ \\psi = \\left( \\begin{array}{c} \\psi _+(y) \\\\ \\psi _-(y) \\end{array} \\right)$ , where $\\psi _\\pm $ are are the Zeeman components that represent the anti-parallel and parallel spin amplitudes.",
"The input spinor for a particle polarized along the x-axis can be written as equal superpositions of these components outside the region of the magnetic field due to which the expectation value of the Sz component would be zero.",
"In the presence of the magnetic field , however, the kinetic energy for these components differ by the Zeeman energy of $\\pm \\hbar \\omega _L/2$ , and this gives a different value to the wave-vectors of the two components in the potential region.",
"This is particularly severe in the case of tunneling at energies $(E)$ below the barrier when the exponential decay for the wavefunctions becomes very different: $\\kappa _\\pm = \\left\\lbrace \\frac{2m}{\\hbar ^2} (V_0 -E) \\mp \\frac{m\\omega _L}{\\hbar } \\right\\rbrace ^{1/2} \\simeq \\kappa \\mp \\frac{m\\omega _L}{2\\hbar \\kappa },$ where $\\kappa ^2 = 2m(V_0-E)/\\hbar ^2$ .",
"Thus, one spin component has a greater probability to tunnel across than the other and the transmitted flux becomes spin polarized.",
"The transmittance amplitudes for the two spin components can be approximately calculated in the case of energies far below the barrier height as $T_\\pm \\simeq T \\exp (\\pm \\omega _L\\tau _z)$ , where $\\tau _z = ml/\\hbar \\kappa $ .",
"The expectation values of the z-spin component of the transmitted flux is easily obtained as $ \\langle S_z \\rangle = \\frac{\\hbar }{2} \\frac{T_+ - T_-}{T_++T_-} = \\frac{\\hbar }{2} \\tanh (\\omega _L\\tau _z) \\simeq \\frac{\\hbar }{2} \\omega _L\\tau _z $ where the last approximation is made in the limit of small magnetic fields.",
"Thus, the z-component of the spin scales linearly with the magnetic field and presents yet another clocking mechanism and the spin rotation time can be defined as $\\tau _z = \\frac{2}{g\\mu _B} \\lim _{B \\rightarrow 0} \\frac{\\partial \\langle S_z\\rangle }{\\partial B}$ It turns out that the spin precession ($S_y$ ) dominates for energies far above the barrier and the spin rotation ($S_z$ ) dominates for energies far below the barrier.For a rectangular barrier, the spin precession time for energies far above the barrier tend to the Wigner delay times and far below the barrier, the spin rotation times tend to the Büttiker-Landauer times for the oscillating barrier.",
"Büttiker proposed that a net traversal time could be defined as the Pythogorean sum of the two spin times as $\\tau ^2 = \\tau _y^2 + \\tau _z^2$ , which reflects the vectorial nature of the change of the spin.",
"This prescription does not, however, have any fundamental basis."
],
[
"Absorption and amplification as a clock",
"A third interesting manner to clock the time of sojourn in a given region of space would be absorption or amplification, which would be at a rate proportional to the amplitude of the wave.",
"Hence, a measurement of the wave amplitude as it enters and exits the region of interest can yield the time that it spends inside.",
"Since the growth is exponential, a logarithmic derivative of the transmittance / reflectance with respect to the imaginary potential is needed.",
"For light, both absorption and stimulated emission are coherent processes that leave the phase of a coherent mode unchanged, due to the Bosonic nature and this definition in terms of absorption or amplification is a natural definition.",
"Such a process would not be strictly applicable to Fermionic particles like electrons or neutrons.",
"However, the effects of absorption or amplification would eventually need to be discussed in the limit of infinitesimally small levels of absorption/amplification that tends to zero, so that the original problem remains unchanged.",
"Hence, we may assume that the formal procedure works for Fermionic particles as well.",
"For a scalar wave, coherent absorption / amplification may be implemented by adding an imaginary potential ($iV_I$ ) only in the region of interest (instead of the magnetic field as in Fig.",
"REF ) , which will eventually be made zero ($V_I \\rightarrow 0$ ).",
"The Schrodinger wave equation for the wave function of a particle in the presence of the imaginary potential becomes $i\\hbar \\frac{\\partial \\psi }{\\partial t} = -\\frac{\\hbar ^2}{2m} \\nabla ^2 \\psi + [ V_0(\\vec{r}) + i V_I] \\psi .$ Then the traversal times for the transmission and reflection can be defined as $\\tau ^{(T)} = \\frac{\\hbar }{2} \\lim _{V_I \\rightarrow 0} \\frac{\\partial \\ln \\vert T \\vert ^2 }{\\partial V_I} , ~~\\mathrm {and}~~\\tau ^{(R)} = \\frac{\\hbar }{2} \\lim _{V_I \\rightarrow 0} \\frac{\\partial \\ln \\vert R\\vert ^2 }{\\partial V_I}$ respectively, where T and R are the complex transmission and reflection coefficients.",
"The idea was originally suggested by by Pippard to Buttiker [18], who found that while it reproduced the spin precession times, it did not recover the spin rotation time for waves that are evanescent in the region of interest (sub-barrier tunneling).",
"It reproduces the Wigner delay times in the limit of large energy above the barrier height.",
"This identity can be understood in terms of the analytic properties of the complex reflection and transmission coefficients.",
"Let us consider the transmission coefficient in the complex energy plane $(E_r, E_i)$ .",
"Its logarithm $\\ln (T) - \\ln \\vert T \\vert + i \\mathrm {Arg}(T)$ would be an analytic function within the Riemann sheet and would satisfy the Cauchy Riemann conditions: $ \\frac{\\partial \\ln \\vert T \\vert }{\\partial E_r} = - \\frac{\\partial \\mathrm {Arg}(T) }{\\partial E_i }, ~~\\mathrm {and} ~ ~\\frac{\\partial \\ln \\vert T \\vert }{\\partial E_i} = \\frac{\\partial \\mathrm {Arg}(T) }{\\partial E_r }.",
"$ Thus, for energies far above the barrier, the left hand side of the second relation is directly proportional to the traversal times obtained by the imaginary potential ($E_i = V_I$ ), while the right hand side relates directly to the kinetic energy of the particle as potential energy is comparatively small ($E_r \\gg V_0$ ).",
"This is why the traversal times tend to the Wigner delay time for large energies.",
"A similar argument will hold for the reflection delay times as well.",
"Note, however, that it is assumed that the potentials are varied everywhere in space in these arguments.",
"Usually for the a quantum clock, the potential should only be varied locally within the region of interest (see Fig.",
"REF ).",
"Figure: Schematic diagram depicting a possible pathway in the Feynman path integral sense.",
"(a) shows the portion affected by a global variation of the potential, and (b) shows the portion (solid line) affected by a local variation of the potential.",
"The dotted portion of the path is not affected and should not be counted towards the traversal time.The issues with the traversal times obtained by the imaginary potential clocks refuse to remain positive for all potentials and further do not tend to the Büttiker-Landauer times for energies far below the barrier.",
"For the case of tunneling across a rectangular barrier, the ratio $\\tau ^{(T,R)} / \\tau _{BL} \\rightarrow 0$ in the low energy limit.",
"The Büttiker-Landauer times are very compelling in that limit and we would really want the traversal to remain positive definite."
],
[
" Sojourn times: correcting the clocks",
"It turns out that most of the problems of defining a positive definite traversal time for the transmission arise from spurious scattering concomitant with the very clocking mechanism / potential and a procedure was duly outlined to separate out this extra scattering and correct the traversal times [6].",
"We will call these corrected traversal times as sojourn times as they literally relate to the times of journey through the region of interest.",
"We will primarily deal with the imaginary potential clock here, but will point out the exact analogies with the Larmor clock.",
"Figure: Schematic diagram partial transmission and reflection coefficients associated with the potentials (regions 1 and 3) on either side of a constant potential region (2).",
"These partial transmission(t jk t_{jk} and reflection (r jk r_{jk}) coefficients may be used to construct the net transmission and reflection coefficients using equations ( ).First of all, it is important to appreciate that the presence of the clock potential not only invokes a response in the relevant extra degree of freedom, but also modifies the scattering due to a change of potential.",
"One would expect this extra scattering to go to zero in the limit of zero clock potential.",
"Our procedure, however, involves taking a derivative with respect to the clock potential, which can have contributions that would not vanish as the clock potential is made zero.",
"Let us consider the transmission of a wave with energy above the barrier (case of propagating waves) through the constant potential region encased between two arbitrary potentials on either side as shown in Fig.",
"REF .",
"The space may be divided into three regions and the partial coefficients of transmission and reflection for a wave incident from region (j) unto region (k) are $t_{jk}$ and $r_{jk}$ as shown schematically in Fig.",
"REF .",
"The transmission and reflection coefficients can be written as a sum of the partial waves through the region [7] $T&=& t_{12} t_{23} e^{ik^{\\prime }L} + t_{12} r_{23} r_{21} t_{23} e^{3ik^{\\prime }L} + t_{12} r_{23} r_{21} r_{23} r_{21}t_{23} e^{5ik^{\\prime }L} + \\cdots \\nonumber \\\\R&=& r_{12} + t_{12} r_{23} t_{21} e^{2ik^{\\prime }L} + t_{12} r_{23} r_{21} r_{23} t_{21} e^{4ik^{\\prime }L} +\\cdots $ the coefficients $t_{jk}$ and $r_{jk}$ also depend on the clock potential and a derivative with respect to the total transmission or reflection coefficient would leave behind some terms arising from this dependence, which do not vanish as the clock potential is made to go to zero.",
"This is the origin of the spurious scattering.",
"An analysis of the structure of the partial wave superposition quickly reveals that the growth or attenuation related to the imaginary potential would only involve the paired combination of ($V_I L$ ) where $L$ is the lenght of the spatial region of interest.",
"Afterall the traversal time should directly relate to spatial region of interest.",
"The spurious scattering on the other hand would only involve unpaired $V_I$ .",
"A formal procedure to isolate the effects of this spurious scattering can now be given.",
"We will treat$\\xi = V_I L $ and $V_I$ as independent variables, keep $\\xi $ formally constant and let $V_IL \\rightarrow 0$ in the expression for the transmission coefficient.",
"The sojourn time for transmission for propagating waves is now obtained as $\\tau _s^{(T)} = \\frac{\\hbar L}{2} \\lim _{\\xi \\rightarrow 0} \\frac{\\partial \\ln \\vert T(\\xi , V_I=0) \\vert ^2}{\\partial \\xi } .$ For the case of wave tunneling (energy lesser than the barrier height $E < V_0$ ), the wave vector is principally imaginary in the barrier region.",
"The real part of the potential or refractive index essentially affects the rate of exponential decay / growth of the evanescent wave.",
"On the other hand, the imaginary potential or imaginary part of the refractive index causes a phase shift with distance of this evanescent wave [13].",
"Consider the complex wave-vector for propagating waves ($E > V_0$ ), $ k = \\sqrt{ \\frac{2m}{\\hbar ^2} [E - (V_0+iV_I)]} ~~\\simeq k_r - i \\frac{mV_I}{k_r \\hbar ^2} ~~ \\forall ~~ V_I \\ll E, $ where $\\hbar k_r = \\sqrt{2m(E-V_0)}$ .",
"When we have evanescent waves ($E < V_0$ ) on the other hand, we can write $ k = \\sqrt{ \\frac{2m}{\\hbar ^2} [E - (V_0+iV_I)]} ~~\\simeq i \\kappa _r - \\frac{mV_I}{\\kappa _r \\hbar ^2} ~~ \\forall ~~ V_I \\ll V_0, $ where $\\hbar \\kappa = \\sqrt{2m(V_0-E)}$ .We hence realise the principle effect of the clock with respect to the spatial region is in the phase of the wave and not the amplitude for the evanescent wave.",
"This is completely analogous to the spin rotation being predominant over the spin precession in the case of the Larmor clock.",
"Mathematically, we have a square root singularity for the wave-vector in the complex energy plane and we are unable to analytically continue the behaviour for propagating waves to the case of evanescent waves across the branch-cut in either case of the imaginary potential clock or the Larmor clock.",
"Hence, we define for the case of sub-barrier tunneling or evanescent wave, the sojourn time as $\\tau ^{(T)}_s = \\frac{\\hbar L}{2} ~ \\lim _{\\xi \\rightarrow 0}~\\frac{\\partial }{\\partial \\xi } \\left[ \\ln \\left(\\frac{T(\\xi , V_I=0)}{T^*(\\xi , V_I=0)} \\right) \\right] ,$ where $T*$ is the complex conjugate of $T$ and the ratio $T/T*$ essentially yields twice the phase of the transmission coefficient.",
"For the case of the region with constant potential of height $V_0$ enclosed between the two arbitrary potentials and energy above the barrier, the sojourn time reduces to a positive definite quantity: $ \\tau ^{(T)}_s = \\frac{1 - \\vert r_{21} r_23 \\vert ^2 }{1 + \\vert r_{21} r_{23} \\vert ^2 - 2\\mathrm {Re}(r_{21} r_{23} e^{2i k_r L})} \\tau _{BL}.",
"$ A similar positive definite expression is obtained in the case of evanesent waves (tunneling below the barrier) as $ \\tau ^{(T)}_s = \\frac{1 - \\vert r_{21} r_23 \\vert ^2 e^{-4\\kappa L} }{1 + \\vert r_{21} r_{23} \\vert ^2e^{-4\\kappa L} - 2\\mathrm {Re}(r_{21} r_{23} e^{-2\\kappa L})} \\tau _{BL}.",
"$ The sojourn time tends directly to the Büttiker-Landauer times for an opaque barrier (energies far below the barrier or long barrier lengths), which is very important.",
"In the high energy limit, it tends to the classical Wigner delay times.",
"Thus it has the correct limits.",
"Further, it was shown in Ref.",
"[6] that the sojourn times defined in this manner for two non-overlapping regions is additive and hence, the times for any region of space with any arbitrary applied potential can be concluded to be positive definite.",
"The case of reflection is a little bit more complex.",
"Applying the above definitions for the reflection coefficient in place of the transmission coefficient does not yield a positive definite sojourn time.",
"From the expansion in terms of the partial waves, one notes an essential difference between the reflection and the transmission.",
"All the partial waves for the transmitted wave sample the region of interest and pick the paired combination $\\xi = V_IL$ in the amplitude or the phase.",
"In the case of reflection, there is one partial wave $r_{12}$ corresponding to the prompt reflection from the edge of the region of interest that never enters the region of interest.",
"Yet, this partial wave interferes with the others to produce the net reflection amplitude.",
"In the spirit of our earlier arguments that the sojourn time should be causally related to the region of interest, it would be necessary to eliminate the weightage of this partial wave that should not be affected by the clocking potential.",
"Hence, we explicitly subtract this prompt reflection amplitude out of the total reflection amplitude $R^{\\prime } = R - r_{12}$ and define the sojourn times for reflection using $R^{\\prime }$ in place of $T$ as before for both cases of the propagating waves as well as the evanescent waves.",
"A simple and general result is obtained as $\\tau ^{(R)}_s = \\tau ^{(T)}_s + \\tau _{BL}$ which is consequently always greater than the sojourn time for transmission and positive definite.",
"An experimental measurement of this procedure is possible by interfering destructively the reflection from a modified potential whose reflection is $r_{12}$ with the reflectance from the given potential.",
"For example, one can use the same optical system that is index matched to the continuum form beyond the point where the absorption or gain is applied.",
"Alternatively, one may use the recently developed metamaterial perfect absorbers [19] for light, where there is the possibility of matching the impedance and preventing any reflection.",
"It should be noted that such a corection procedure can be analogously applied to the spin precession and spin rotation times for the Larmor clock as well, where the paired variable $\\xi = BL$ would be taken to zero after explicitly putting the magnetic field $B=0$ while keeping $\\xi $ formally constant.",
"We then obtain identical results to the imaginary potential clock.",
"The procedure outlined above was shown equivalent to stochastic absorption [20], whereby the absorption does not cause any scattering but contributes only to loss of the wave flux.",
"This can be viewed as the case when there is inelastic scattering out of the given mode of the mesoscopic and the coherence of the mode is not affected.",
"Only the coherent part of the wave is measured and the scattering into other modes manifests as a loss for this mode.",
"Another example could be that the scattering leads to decoherence of the wave.",
"In any interferometric measurement only the coherent part of the wave is measured and the decohered part would appear as a loss.",
"Thus, a finite rate of decoherence itself could be utilized as a clocking mechanism ."
],
[
"The problem of the quantum first passage time",
"Consider a free quantum particle that is released from a localized region at some instant of time and thereafter subjected to instantaneous projective measurements to detect its arrival at a particular region of space.",
"The measurements are made at regular time intervals , and the system is allowed to evolve until the time a detection occurs.",
"The main question addressed is: what is the probability that the particle is detected for the first time after time $t$ i.e.",
"at the $n=(t/\\tau )^{th}$ measurement [21].",
"Conversely, one can ask for the probability of particle not being detected (i.e., surviving) upto a given time.",
"It is proposed by a general perturbative approach for understanding the dynamics which maps the evolution operator, which consists of successive unitary transformations followed by projections, to one described by a non-Hermitian Hamiltonian.",
"For some examples of a particle moving on one- and two-dimensional lattices with one or more detection sites, use this approach to find exact expressions for the survival probability and find excellent agreement with direct numerical results.",
"For the one- and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle.",
"It is shown that an interesting and nontrivial connection between the dynamics of the particle in their model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites [22], [23].",
"If continuous projective measurements are done then famous Zeno comes in to effect and particle does not evolve.",
"Thus quantum first passage time indeed a nontrivial model.",
"It is very subtle and as mentioned before in quantum mechanics there is no dynamical operator for time travel between two points.",
"We may conclude this brief disussion on this topic by saying that the quantum first passage time is still remains a mystery"
],
[
"Conclusions and Outlook ",
"We have outlined here various attempts to answer a fundamental question, namely, “what is the time that a quantum particle or wave spends in a specified region of space ?” We do not have a self-adjoint operator for the arrival time in quantum mechanics, and the arrival time is not an observable.",
"Yet it is intuitive and important to ask about timescales of any physical problem and the time of stay or sojourn appears to be a calculable quantity and a useful one to compare timescales.",
"The sojourn time can provide for a meaningful alternative view-point within quantum mechanics.",
"Knowledge of the sojourn times obviously cannot provide answers to questions that cannot be answered within the paradigm of quantum mechanics.",
"Here we have exploited the fact that light as well as quantum particles are mathematically described by the same Helmholtz equation and talk about both systems in the same breath.",
"Starting with the description of wavepackets by the group velocity and its extension by Wigner, we have indicated a variety of manners in which this question has been approached.",
"The Smith dwell time is a positive definite time, but it is an unconditional time that does not depend on the output scattering channel.",
"The path integral approach [21] or an approach based on the WKB method [17] are closely related to this, but can define conditional traversal times.",
"Beyond these, we discuss the proposals to consider the dynamical evolution of an extra degree of freedom attached to the traversing particle due to the local interaction with a potential applied only in the region of interest.",
"One has to meaningfully identify the extent of evolution of the clocking mechanism, which is an observable, with the time spent in that region.",
"We have discussed three examples: (i) exchange of quanta via interaction with a radiation field; (ii) the spin precession; (iii) rotation in a magnetic field or the growth / attenuation due to an imaginary potential (amplifying or absorbing medium).",
"A rather subtle problem that arises with these clocks is that they give rise to an extra spurious scattering that interferes with the very process and effectively changes the potential in the region of interest.",
"This extra scatterings is carefully identified and is explicitly eliminated by a formal mathematical procedure.",
"Further, it is shown that the effect of the clocking potential manifests in different quantities for propagating and evanescent waves: In the case of the imaginary potentials, it manifests in the amplitude for propagating waves and in the phase for evanescent waves (tunneling below the barrier); and in the case of the Larmor spin clock, it mainfests in the spin precession for propagating waves and in the spin rotation for evanescent waves.",
"As a final caveat, it is shown that partial waves that reflect from the surface of the region should also be eliminated from reckoning as they do not spend anytime within the region of interest and this can, indeed, be done by interferometric measurements.",
"Including these considerations yields a sojourn time that is Real and positive definite; Additive for non-overlapping regions of space; Related causally to the region of interest; Calculable and directly related to a measurable quantity.",
"The main issue in defining traversal or sojourn times for a quantum system has been due to interference between partial waves (the alternative paths in quantum mechanics), which defies naive realism.",
"With these advances in understanding, however, the sojourn time becomes a calculable quantity that is practically useful for estimating other quantities and understanding physical phenomena, for example, the dephasing rates in a quantum system."
],
[
"Acknowledgement",
"Both the authors acknowledge illuminating and educative discussions with Prof. N. Kumar who introduced them to these topics as well as the constant encouragement they have received from him.",
"They would like to dedicate this tutorial to him.",
"SAR acknowledges funding from the DST India through a Swarna Jayanti Fellowship.",
"AMJ thanks DST, India for financial support (through J. C. Bose National Fellowship)"
]
] | 1612.05709 |
[
[
"A continuum of accretion burst behavior in young stars observed by K2"
],
[
"Abstract We present 29 likely members of the young $\\rho$ Oph or Upper Sco regions of recent star formation that exhibit \"accretion burst\" type light curves in $K2$ time series photometry.",
"The bursters were identified by visual examination of their ~80 day light curves, though all satisfy the $M < -0.25$ flux asymmetry criterion for burst behavior defined by Cody et al.",
"(2014).",
"The burst sources represent $\\approx$9% of cluster members with strong infrared excess indicative of circumstellar material.",
"Higher amplitude burster behavior is correlated with larger inner disk infrared excesses, as inferred from $WISE$ $W1-W2$ color.",
"The burst sources are also outliers in their large H$\\alpha$ emission equivalent widths.",
"No distinction between bursters and non-bursters is seen in stellar properties such as multiplicity or spectral type.",
"The frequency of bursters is similar between the younger, more compact $\\rho$ Oph region, and the older, more dispersed Upper Sco region.",
"The bursts exhibit a range of shapes, amplitudes (~10-700%), durations (~1-10 days), repeat time scales (~3-80 days), and duty cycles (~10-100%).",
"Our results provide important input to models of magnetospheric accretion, in particular by elucidating the properties of accretion-related variability in the low state between major longer duration events such as EX Lup and FU Ori type accretion outbursts.",
"We demonstrate the broad continuum of accretion burst behavior in young stars -- extending the phenomenon to lower amplitudes and shorter timescales than traditionally considered in the theory of pre-main sequence accretion history."
],
[
"Introduction",
"Variable mass flux has long been recognized as an important element of protostellar and pre-main sequence accretion.",
"Accretion rates are believed to be higher ($\\sim 10^{-5} M_\\odot $ yr$^{-1}$ ) during the first 10$^5$ years of protostellar evolution, with frequent outbursts of up to $10^{-4} M_\\odot $ yr$^{-1}$ [70].",
"The bursts are predicted as a consequence of unstable pile-up of gas in the inner disk, which then intermittently releases a cascade of material onto the star due to viscous-thermal disk instabilities [18], [168], [45].",
"The burst frequency and perhaps amplitude decline over time [71], [161].",
"While most of the stellar mass is thought to accumulate in the early phases of protostellar evolution, accretion at rates of $10^{-10}$ –$10^{-6} M_\\odot $ yr$^{-1}$ persists through the T Tauri phase (ages up to a few Myr), with less frequent bursts [69].",
"The currently accepted picture of T Tauri star accretion involves magnetic funnel flows channeling gas from the inner disk onto the central star.",
"Where this material impacts the surface, shocks arise and thermal hot spots form.",
"There may be one spot near each magnetic pole, or multiple spot complexes that are distributed about the stellar surface.",
"The number and geometry of the funnel flows is thought to depend on the accretion rate [138].",
"Empirically, time series monitoring of young stellar objects (YSOs; ages $<$ 1–10 Myr) has an extensive history.",
"It has been known since or before e.g.",
"[83] that T Tauri stars display flux variations at a wide variety of timescales and magnitudes.",
"The outbursting FU Ori stars and their lower amplitude, repeating cousins the EX Lup stars, are at the extreme end of the variabiity spectrum – and rare.",
"More common photometric variability is characterized by smaller amplitude and shorter timescale fluctuations.",
"The studies of [75], [60], [145], and [55] have illustrated and quantified much of the “typical\" young star photometric phenomena occurring from sub-hour to multi-decade time scales.",
"One cause of the routine brightness variations is sporadic infall of material from the surrounding disk.",
"Even in their predominant low-state accretion phases, young stars are understood as variable accretors.",
"The photometric variability is complemented by highly variable emission line profiles and veiling [34], [40] including in stars with relatively low accretion rates such as TW Hya and V2129 Oph [2], [3].",
"Nevertheless, the variability time scales and accretion rate changes remain poorly quantified for typical young accreting star/disk systems.",
"There may be a continuum of “burst\" behavior with a range of amplitudes and time scales that have not yet been appropriately sampled or appreciated in existing ground-based data sets.",
"While most of the stellar mass has been assumed to accumulate in the episodic and dramatic fashion of the rare large FU Ori and Ex Lup type events, the role and implications of discrete lower amplitude accretion events [155] and continuously stochastic accretion behavior [156] is not well understood in the context of stellar mass accumulation and inner disk evolution.",
"In probing accretion variability, space-based photometric campaigns have several advantages over ground-based work, including near-continuous sampling (versus interruptions for daytime, weather, etc), higher measurement precision, and fainter signal detection limits.",
"A detailed analysis of optical and infrared variability among disk-bearing stars in the $\\sim $ 3 Myr NGC 2264 was conducted by [38] based on a forty-day optical time series from the CoRoT space telescope at 10-minute cadence, along with 30 days of Spitzer Space Telescope monitoring at 100-minute cadence.",
"These data enabled an unprecedented view of YSO brightness changes on a variety of timescales.",
"Among the detected variability groups was a new class of “stochastic accretion burst” light curves – dominated by brightening events of duration 0.1-1 days and amplitude 5-50% the quiescent flux value [155].",
"It was speculated that these events were caused by the unsteady infall of material onto the stellar surface, as predicted by [143].",
"They may thus represent “normal\" discrete accretion variations and bursts, in contrast to the FU Ori and EX Lup outbursts described above.",
"The NASA $K2$ mission Campaign 2 observations included the young $\\rho $ Ophiuchus molecular cloud region at $<$ 1-2 Myr, and the adjacent Upper Scorpius OB association, which is debated from analysis of HR diagrams to be either $\\sim $ 3-5 Myr based on the low mass stellar population [122], [76] or $\\sim $ 11 Myr based on the solar and super-solar mass population [115]; the latter age is beginning to be favored by results on eclipsing binaries [85], [47] and asteroseismology [136].",
"By sampling stars with ages comparable to and extending to much older than NGC 2264 (in $\\rho $ Oph and Upper Sco, respectively), the $K2$ time series data can be used to compare accretion burst behavior as a function of age and therefore presumably disk properties which are expected to evolve with time.",
"We report here on a continuum of accretion burst behavior among members of $\\rho $ Oph and Upper Sco.",
"We observe discrete brightening events that range in amplitude from 0.1 to 2.5 magnitudes and in timescale from $<$ 1 day to $>$ 1 week.",
"We demonstrate that the burst phenomenon is seen only in those stars with evidence for strongly accreting disks, distinct from the typical disks in the region with weaker infrared excess and H$\\alpha $ emission.",
"Section 2 contains description of the $K2$ observations and pixel file processing, and of follow-up high dispersion spectroscopy.",
"Section 3 presents the light curve analysis and identification of burst type variables, Section 4 a discussion of the corresponding accretion and disk properties, and Section 5 the spatial and time domain characteristics of bursting sources.",
"We discuss the implications of these observations in Section 6 and summarize the results in Section 7.",
"The $K2$ mission [80] observed nearly 2000 stars in the young $\\rho $ Ophiuchus and Upper Scorpius regions during Campaign 2.",
"We have mainly considered objects submitted under programs GO2020, GO2047, GO2052, GO2056, GO2063, and GO2085 of the Campaign 2 solicitation, which comprise both secure cluster members and less secure candidates.",
"We later noted aperiodically variable stars among a number of other programs targeting cool dwarfs and therefore added objects from programs GO2104, GO2051, GO2069 GO2029, GO2106, GO2089, GO2092, GO2049, GO2045, GO2107, GO2075, and GO2114 if they also had proper motions consistent with Upper Sco.",
"[37], cull some of the less confident young star candidates using WISE photometry to eliminate giant stars and other contaminants.",
"This vetting results in a reduced set of 1443 $\\rho $ Oph and Upper Sco candidate members.",
"For each star in this set, we downloaded the target pixel file (TPF) from the Mikulski Archive for Space Telescopes (MAST).",
"Each target is stored under its Ecliptic Plane Input Catalog (EPIC) identification number, as listed in Table 1.",
"Data for each object includes 3811 $\\sim $ 10$\\times $ 12 pixel stamp images, obtained between 2014 23 August and 10 November.",
"Since the loss of a second reaction wheel during the Kepler mission in May of 2013, telescope pointing for $K2$ has suffered reduced stability and requires corrective thruster firings approximately every six hours.",
"We find a corresponding target centroid drift at a rate of $\\sim $ 0.1, or $\\sim $ 0.02 pixels, per hour.",
"While this movement is relatively small, associated detector sensitivity variations at the few percent level per pixel compromise the otherwise exquisite photometry, introducings jumps in measured flux on the same six-hour timescales.",
"It is helpful to track the $x$ -$y$ position drift for each star over time, as this can be used for aperture placement and later detrending of the light curves.",
"TPF headers provide a rough world coordinate system solution which is the same for all images, but these are not precise enough to center the target.",
"We therefore cut out a 5$\\times $ 5 pixel region around the specified target position, and used this to calculate a flux-weighted centroid.",
"We carried out photomety with moving apertures, the centers of each specified by the measured centroid locations.",
"This approach helps to minimize the effect of detector drift on the photometry.",
"Circular apertures were used with radii ranging from 1.0 to 4.0 pixels, in intervals of 0.5 pixels.",
"We found that photometric noise levels after detrending for position jump effects were generally minimized with the 2-pixel aperture, though for a few objects we selected the 1.5 or 3-pixel apertures.",
"These sizes have the additional advantage of being small enough so as to avoid flux contamination from other stars lying $\\sim $ 12 away.",
"To clean the data, we discarded the first 93 light curve points, for which the pixel positions were particularly errant compared to the rest of the time series.",
"We also removed points with detector anomaly flags.",
"Finally, we pruned points lying more than five standard deviations off the median light curve trend.",
"This was accomplished by median smoothing on $\\sim $ 2-day timescales, removing outliers, and then adding the median trend back in.",
"In general, our raw moving aperture photometry consists of lower levels of pointing-related systematic jitter than for the fixed aperture case.",
"For all of the stars discussed in this work, the amplitude of intrinsic variability dwarfs these systematics, and no further corrections are needed.",
"However, this is not the case for the less variable cluster members which form a control population for comparison of variability demographics.",
"In these light curves, a prominent sawtooth pattern appears on the $\\sim $ 6-hour timescales corresponding to thruster firings, an effect that was mitigated with the detrending procedure described in [1], as described in [37].",
"In one exceptional case (EPIC 203954898/2MASS J16263682-2415518), the light of a highly variable star was contaminated by a close neighbor 8 away.",
"Single pixel photometry shows that this object undergoes high amplitude bursts, while the neighboring star is relatively constant and slightly fainter.",
"Restricting the aperture to encompass only EPIC 203954898 results in degraded precision.",
"We therefore used a 3-pixel aperture encircling both stars, and then removed the average flux of the companion by scaling the data to match the amplitude of the bursts in the single pixel light curves (the lower precision here affects only the detailed light curve morphology and not its overall amplitude).",
"The result is a light curve with maximum burst amplitude of nearly eight times the quiescent flux level."
],
[
"New spectra and compiled spectroscopic data",
"We collected both archival and new spectroscopic data at high dispersion with the Keck/HIRES spectrograph [160].",
"The new observations were obtained on one of: 2016 May 17 and 20, 2015 June 1 and 2, or 2013 June 4, UT, and covered the spectral range $\\sim $ 4800 Å to 9200 Å at resolution $R\\approx 36,000$ .",
"The images were processed and the spectra were extracted and calibrated using the $makee$ software written by Tom Barlow.",
"H$\\alpha $ emission line strengths and spectral type estimates are tabulated in Table ; other emission lines that were observed in these high dispersion data are discussed in the Appendix notes on individual stars.",
"Table 1 as well as the notes section includes literature information in addition to our spectroscopic findings."
],
[
"Speckle imaging",
"We obtained high-resolution speckle imaging for six of our young stars to assess multiplicity properties.",
"Our observations used the Differential Speckle Survey Instrument [77] on the Gemini-South telescope in 2016 June.",
"Speckle observations were simultaneously made in two medium band filters with central wavelengths and bandpass FWHM values of ($\\lambda _c$ , $\\delta \\lambda $ ) = (692,47) and (883,54) nm.",
"Each star was observed for approximately 10 minutes during which time we obtained 3-5 image sets consisting of 1000, 60 ms simultaneous frames.",
"These observations were made during clear weather at airmass 1.0 to 1.3, when the native seeing was 0.4-0.6 arcsec.",
"Details of speckle observations using the Gemini telescope and our data reduction procedures can be found in [78] and [79].",
"We conducted visual examination of all 1443 young star light curves in order to identify stars in a “bursting\" state.",
"Such objects were selected by identifying behavior consistent with that presented in [38] and [155].",
"To confirm our visually identified bursters, we also computed quantitative variability metrics, specifically the $M$ and $Q$ statistics defined by us in the papers above.",
"$M$ describes the degree of symmetry of the light curve about its mean value.",
"It is calculated by determining the ratio of the mean of magnitude data in the top and bottom deciles to the median of all light curve points.",
"$M$ achieves negative values when there is a significant number of points brighter than the median, but not so many faint points.",
"Unlike what was done in [38], we calculated the $M$ statistic from flux, rather than magnitude values.",
"The main effect of this change is to lower $M$ values (by $\\sim $ 10% on average), since the magnitude to flux conversion makes bright peaks more pronounced.",
"We argue that flux units are a more natural choice here, as flux correlates with luminosity and accretion rate.",
"Bursting light curves are highly asymmetric with frequent flux increases over the mean, resulting in $M$ values from approximately -0.3 to -1.3.",
"While the boundary is somewhat subjective, all objects selected by eye meet the previously defined $M<-0.25$ criterion for burster status.",
"The $Q$ statistic defined in [38] describes tendency towards or away from periodicity over the time series.",
"It is a measurement of how much the standard deviation shrinks when the light curve is phased to its dominant periodicity and the associated pattern is repeated and subtracted out from the raw time series.",
"Strictly periodic behavior (i.e., complete removal of the phase pattern) returns $Q=0$ while light curves with no repeating behavior have $Q=1$ .",
"In a few cases for which the light curves are entirely aperiodic, this removal process actually increases the underlying standard deviation; this is why the computed $Q$ value is occasionally greater than 1.0.",
"In previous work, we have denoted light curves with moderate $Q$ values of 0.15–0.60 as “quasi-periodic.” We emphasize here that this range is somewhat subjective and was based on a by-eye analysis of CoRoT data on the NGC 2264 cluster.",
"As explained in Cody et al.",
"2016, the present K2 dataset contains some objects with $Q>0.6$ that nevertheless display repeating components upon visual examination.",
"The $M$ and $Q$ values for the selected bursters are provided in Table 2.",
"They are also plotted in Fig.",
"REF , alongside the values for other disk-bearing young stars in the $K2$ field, as selected in [37].",
"The $K2$ burster sample behavior ranges from periodic to quasi-periodic to aperiodic, with light curves most tending towards aperiodicity.",
"Objects falling in the bursting section of the diagram but not highlighted as such tend to be long-timescale variables for which the trend removal failed.",
"We favor our by-eye classification over the $M$ and $Q$ statistics here and thus do not consider these bursters.",
"Several other objects with highly negative $M$ values are dominated by periodic modulation and only display zero or one bursting event.",
"We also leave these out of the sample, in light of classification ambiguity.",
"Overall, 18 objects with $M<-0.25$ (i.e., a value that would qualify them as busters) were removed from the sample.",
"Most of these may be seen in Figure REF as the black points lying above the $M=-0.25$ line (apart from five that are hidden under orange burster points); the majority are only marginally above it.",
"In total, we have selected 29 stars as bursters in $\\rho $ Oph and Upper Sco; division into the two regions was based on a $1.2\\times 1.2$ square surrounding the position RA=246.79, Dec=-24.60 to define the extent of $\\rho $ Oph [37].",
"We list the basic properties of the bursters in Table 1 and show their light curves in Figure REF .",
"The burster class exhibits several subsets of behavior, with some light curves displaying a nearly continuous series of events, and others exhibiting more discrete brightening events.",
"We discuss the timescales of bursting in Section 6. llcccccc 8 0pt Young stars exhibiting bursting behavior in $K2$ Campaign 2 EPIC id 2MASS id Other ids SpT EW H$\\alpha $ H$\\alpha $ 10% Refs Region (Å) (km s$^{-1}$ ) 203382255 J16144265-2619421 M4-M5.5 -77 154 1 USco 203725791 J16012902-2509069 USco CTIO 7 M2/M3.5 -170, -129 437 1, 2 USco 203786695 J16245974-2456008 WSB 18 M3.5 -8.4, -140 - 3 $\\rho $ Oph 203789507 J15570490-2455227 - - - USco 203794605 J16302339-2454161 WSB 67 M3.5-M5 -69 485 1 $\\rho $ Oph 203822485 J16272297-2448071 WSB 49, MHO 2111, DROXO 57 M4.25 -37 - 4 $\\rho $ Oph 203856109 J16095198-2440197 M5-M5.5 -15 155 1 USco 203899786 J16252434-2429442 V852 Oph, SR 22, DoAr 19, WSB 23 M4.5/M3 -31, -170 - 4, 5, 6 $\\rho $ Oph 203905576 J16261886-2428196 VSSG 1, Elias 20, YLW 31, ISO-Oph 24, K7–mid-M/M0 -70 416 1,7 $\\rho $ Oph IRAS 16233-2421, MHO 2103 203905625 J16284527-2428190 V853 Oph, SR 13, DoAr 40, WSB 62, M3.75 -30, -48, -46 - 4 $\\rho $ Oph ISO-Oph 199, HBC 266 203913804 J16275558-2426179 V2059 Oph, DoAr 37, SR 10, ISO-Oph 187, M2 -43, -56, -108 - 4,8 $\\rho $ Oph WSB 57, YLW 56, HBC 265, SVS 1771 203928175 J16282333-2422405 SR 20W K5 -35 - 3 $\\rho $ Oph 203935537 J16255615-2420481 V2058 Oph, DoAr 20, Elias 13, K4.5 -220, -87, -67 - 4 $\\rho $ Oph SR 4, WSB 25, YLW 25, IRAS 16229-2413, MHA 365-12, ISO-Oph 6 203954898 J16263682-2415518 ISO-Oph 51 M0 -10 - 9 $\\rho $ Oph 204130613 J16145026-2332397 BV Sco M4.5 -108 - 10 USco 204226548 J15582981-2310077 USco CTIO 33, USco 42 M3 -158, -250 - 11,12 USco 204233955 J16072955-2308221 M3 -150 - 10 USco 204342099 J16153456-2242421 VV Sco, IRAS 16126-2235, PDS 82a M1/K9-M0 -20, -31 337 13, 1 USco 204347422 J16195140-2241266 - - - USco 204360807 J16215741-2238180 M6 -140 341 1 USco 204397408 J16081081-2229428 M5.75/M5 -22, -49, -31 - 10, 14, 15 USco 204440603 J16142312-2219338 M5.75 -95 - 10 USco 204830786 J16075796-2040087 IRAS 16050-2032 M1/G6-K5 -357, -165 684 16, 1 USco 204906020 J16070211-2019387 KSA 68 M5 -8, -30 - 11, 17 USco 204908189 J16111330-2019029 M1/M3 -160 324 1,18 USco 205008727 J16193570-1950426 K7-M3 -55 322 1 USco 205061092 J16145178-1935402 M5-M6 -70 173 1 USco 205088645 J16111237-1927374 M5, M6 -50, -50 - 12, 19 USco 205156547 J16121242-1907191 M5-M6 -15 127 1 USco Stars in the $K2$ Campaign 2 burster sample, in order of EPIC id.",
"EPIC 203786695/2MASS J16245974-2456008 has a companion at 1.1 separation, and the two H$\\alpha $ values belong to the distinct components of the system.",
"The H$\\alpha $ 10% widths in column 6 are derived from data presented in this paper.",
"We have highlighted in bold the other new values derived as part of this work.",
"References: 1) this work, 2) [137], 3) [27], 4) [165], 5) [121], 6) [100], 7) [6], 8) [8], 9) [52], 10) [96], 11) [43], 12) [122], 13) [124], 14) [154], 15) [44], 16) [88], 17) [123], 18) [99], 19) [101].",
"Figure: Light curves of selected bursters over the 80-day duration of K2K2 Campaign 2, in approximate order of amplitude.Figure: Cont.Figure: Cont.Figure: Cont.Figure: Cont.Figure: Cont."
],
[
"Disk and accretion properties of the burst sample",
"A hypothesis for the short-lived, often repeatable, brightening events in our 29 $K2$ light curves is that they are caused by episodic accretion from the circumstellar disk onto the young star.",
"A specific requirement for the accretion-driven burst hypothesis is that the objects exhibit both infrared excess indicative of circumstellar dust, serving as the reservoir for the accretion, and either ultraviolet excess or line emission from hot gas in the nearby circumstellar environment.",
"The dust criterion is satisfied by our burster sample, as all stars have been already selected as infrared excess sources in [37].",
"As shown in that work, there are 344 disk bearing stars in the entire $K2$ Campaign 2 sample, of which 299 are bright enough to obtain light curves.",
"This number includes all 29 burster stars identified here.",
"Therefore, the fraction of bursters among the total disk-bearing sample is at least 8$\\pm $ 2%.",
"The error comes from consideration of the Poisson uncertainties.",
"These values can also be considered separately for the $\\rho $ Oph (128 disked stars; 92 with light curves) and Upper Sco (216 disked stars; 207 with light curves) samples.",
"There are 10 bursters in $\\rho $ Oph and 19 bursters in Upper Sco; both numbers lead to roughly the same value of 8–10%, with a $\\sim $ 2% error on this fraction."
],
[
"Circumstellar Dust",
"Infrared color-magnitude diagrams can also shed light on what photometric aspects, if any, separate burster stars from other disk bearing sources.",
"We show the color-color diagrams $J-K$ versus $K-W3$ and $J-K$ versus $K-W4$ in Figure REF and the spectral energy distributions in Appendix Figure REF , to illustrate the strength of emission at farther, cooler locations in the disk.",
"The vast majority of the bursters have excesses in the three longest wavelength WISE bands $W2$ , $W3$ , and $W4$ ($4.6 \\mu $ m, $12 \\mu $ m, and $22 \\mu $ m, respectively).",
"The individual spectral energy distributions in the Appendix (Fig.",
"REF ) provide a finer look at the circumstellar flux patterns.",
"This finding is in stark contrast to the overall disk sample, in which only 50% of objects (or a total of 172) have excesses in all three bands.",
"Thus, the presence of a full, minimally evolved disk appears to be preferred for bursting behavior.",
"Figure: Near and mid-infrared color-color diagram of stars in the Upper Sco/ρ\\rho Oph regions (grey), with disk-bearing stars in black and bursters highlighted in red.",
"Burster point sizes are scaled by light curve amplitude.It has traditionally been thought that young stars with prominent fading events are surrounded by nearly edge-on disks; the dust clumps routinely obscure the central star [25], [24], causing decrements in the light curve.",
"Conversely, one might imagine that for disk systems farther from edge-on orientation, we would have a more direct view of the accretion columns and shocks near the stellar poles.",
"This could allow relatively unfettered observation of accretion bursts.",
"Looking at the selection of 172 full disks identified in [37], 90% are variable, but only 17% are bursting.",
"If this is a reflection of geometric selection, then one might hypothesize that burster disks are viewed at angles ranging from face-on to $\\sim $ 34.",
"However, this assumes that face-on is the best angle at which to view bursting; as suggested by the ALMA data described below, this may not be the case.",
"The idea that bursting behavior may be a function of viewing angle can be further explored by considering resolved disk imaging.",
"Five of the bursters discussed in this paper have been observed with ALMA at 0.88 mm [31], [15].",
"EPIC 204830786 (2MASS J16075796-2040087) has a broad CO $J=3-2$ line detection, with velocities from -17 to 17 km s$^{-1}$ ; the profile suggests that the disk is not face-on.",
"EPIC 204342099 (2MASS J16153456-2242421) is the only other disk with a CO detection, albeit a weak one.",
"The broad velocity distribution again suggests that this system is not oriented face-on.",
"The other three disk-bearing sources in our sample (EPIC 204906020/2MASS J16070211-2019387, EPIC 204226548/2MASS J15582981-2310077, EPIC 204908189/2MASS J16111330-2019029) have no CO detection.",
"The latter two do show continuum, so CO may be highly depleted or the disks in these cases could be physically small.",
"The continuum fluxes can also be used to infer disk properties.",
"[14] infers from modeling SEDs and size constraints that EPIC 204830786 (2MASS J16075796-2040087) has inclination 42$^{+12}_{-9}$ , while EPIC 204342099 (2MASS J16153456-2242421) is inclined at 43$^{+15}_{-16}$ .",
"Dust masses were inferred by [15] and range from $<$ 0.5 $M_\\oplus $ (EPIC 204906020/2MASS J16070211-2019387; non-detection) to 9.3 $M_\\oplus $ in the case of EPIC 204830786/2MASS J16075796-2040087.",
"All five bursters in the ALMA sample are noted as having been classified as full disks (as opposed to evolved or transitional) by [99].",
"These intermediate inclination values are consistent with the idea proposed above that we are not looking through disk material, which would be expected to produce “dipping\" rather than “bursting” light curves.",
"But they are inconclusive regarding whether we have a direct view of material accreting onto the central star."
],
[
"Gas and Accretion",
"In addition to infrared disk indicators, we obtained spectroscopic data from both the literature and our own high-resolution follow-up spectroscopy (§2.2).",
"The H$\\alpha $ emission equivalent width (EW) and 10% width values (Table 1) are indicative of significant accretion.",
"As a control sample, we gathered H$\\alpha $ EWs for other disk-bearing (but not necessarily bursting) stars in Upper Sco from [137], [43], and [122].",
"We cross-matched them against our Upper Sco/$\\rho $ Oph K2 $WISE$ excess star list and eliminated any objects not in common.",
"We then compared the H$\\alpha $ values of this general Upper Sco sample with those of the bursters in Figure REF .",
"Since some of the bursters have multiple H$\\alpha $ measurements (see Table 1), we have taken the average of all available values.",
"We find that the bursters occupy a large range of H$\\alpha $ EW values, from -10Å (presumably a low-state value) to several hundred angstroms in emission.",
"The non-burster stars, on the other hand, display weak H$\\alpha $ , with EWs primarily from 0 to -15Å and a tail out to -50Å with one value around -120Å.",
"Figure REF illustrates the emission line profiles for the 12 out of the 29 bursters for which we have high resolution spectra in H$\\alpha $ , Ca2 8542Å and He1 5876Å.",
"[163] advocated the designation of accreting stars based on H$\\alpha $ line widths at 10% of the maximum line strength that are larger than 270 km s$^{-1}$ .",
"Although all 12 sources have velocities larger than 100 km s$^{-1}$ (see Table 1), only slightly more than 1/2 meet the [163] requirement.",
"There is a tendency for the narrower velocity stars (EPIC 203856109/2MASS J16095198-2440197, EPIC 205156547/2MASS 16121242-1907191, EPIC 203382255/2MASS J16144265-2619421, EPIC 205061092/2MASS J16145178-1935402, EPIC 205008727/2MASS J16193570-1950426) to have lower duty cycles in their burst patterns (Figure REF ) (Where available, H$\\alpha $ velocity and duty cycle are correlated at a significance level of 1.2$\\times 10^{-3}$ ).",
"It may be that our spectra were taken at non-burst epochs.",
"This is speculative at best, however.",
"The weak H$\\alpha $ sources all exhibit only narrow component emission in their Ca2 profiles, as does EPIC 204342099 (2MASS J16153456-2242421) which has a broad H$\\alpha $ profile.",
"Generally, the broad H$\\alpha $ sources exhibit both broad component and narrow component Ca2.",
"[13] review the classic literature on Ca2 profile morphology in young stars, and discuss magnetospheric models of it.",
"Our profiles do not seem to exhibit the asymmetries predicted by these models (e..g blueshifted peaks, redshifted depressions), however.",
"Notably, our broad-lined Ca2 sources also exhibit evidence for forbidden line emission, e.g.",
"[O1] 6300Å.",
"The morphology of He1 emission lines in young stars was studied by [19] who also designated narrow line, broad line, and narrow$+$ broad profile categories.",
"Our stars are dominated by their narrow component emission, with widths ranging between 40-70 km s$^{-1}$ , but some may also have weak broad components that would require line decomposition to characterize.",
"Most of the He1 profiles appear to have slight asymmetries, however, in the sense of broader redshifted emission than blueshifted emission with the line peaks at zero-velocity.",
"Beyond emission line morphology, accretion rates are estimated for a handful of our burster stars by [111].",
"Values range from 10$^{-9.9}$ to 10$^{-6.7}$ $M_\\odot $ yr$^{-1}$ – a large range.",
"It should be noted that neither the H$\\alpha $ EWs presented above nor the accretion rates referenced here were necessarily measured during a time when the stars were undergoing bursting events.",
"Thus it is plausible that accretion is preferentially high in these sources, but only at certain times.",
"Under the hypothesis that burst events in light curves are due to higher than average mass flow, we can convert the flux to a quantitative increase in the mass accretion rate.",
"This requires several assumptions.",
"First, we assign to all stars in our sample a low-level baseline accretion rate, $\\dot{M}_{\\rm low}$ , which then increases to a larger rate $\\dot{M}_{\\rm high}$ during bursts.",
"We assume that this increase in mass flow can be equated to the ratio of the accretion flux $F_{\\rm acc}$ in and out of the burst state through the accretion luminosity, $L_{\\rm acc}$ : $\\frac{\\dot{M}_{\\rm high}}{\\dot{M}_{\\rm low}} = \\frac{L_{\\rm acc,high}}{L_{\\rm acc,low}}\\sim \\frac{F_{\\rm acc,high}}{F_{\\rm acc,low}}.$ The accretion luminosity is related to the stellar mass $M_*$ and radius $R_*$ by $L_{\\rm acc} = 1.25(GM_*\\dot{M}/R_*)$ where the pre-factor is that appropriate for an assumed magnetospheric accretion scenario.",
"The measured flux density $F$ (i.e., in the $Kepler$ band) contains contributions from both accretion and the underlying stellar luminosity, $L_*$ .",
"We label the measured flux ratio “$r$ ”: $r \\equiv \\frac{F_{\\rm high}}{F_{\\rm low}} = \\frac{F_*+F_{\\rm acc,high}}{F_*+F_{\\rm acc,low}}$ $F_{\\rm low}$ and $F_{\\rm high}$ are approximately the minimum and maximum flux values respectively attained in a given light curve.",
"To determine $F_{\\rm acc,high}/F_{\\rm acc,low}$ and hence $\\dot{M}_{\\rm high}/\\dot{M}_{\\rm low}$ , we must estimate the ratio of stellar to accretion luminosity.",
"This quantity has been studied by e.g.",
"[110] and we adopt the fit based on their Figure 2.",
"For each star in the burster sample, we estimate the stellar luminosity by considering the $J$ -band magnitudes of similar spectral type non-accreting young stars in the $K2$ Campaign 2 set, and applying bolometric and extinction corrections as outlined in [111].",
"Combining this with Equations 1 and 2, it can be shown that $\\frac{\\dot{M}_{\\rm high}}{\\dot{M}_{\\rm low}}\\sim \\frac{L_*}{L_{\\rm acc}}(r-1)+r.$ Given our estimates of $L_*/L_{\\rm acc}$ and $r$ as measured from the $K2$ photometry, we have calculated the increase in mass accretion rate during bursts for each of our sources.",
"Figure REF illustrates the resulting values as a function of spectral type, in cases where this is known to a subclass or better.",
"We find that they range from $\\sim $ 10 to over 400 for the most prominent burst events.",
"Figure: We show the cumulative distributions of Hα\\alpha equivalent width for disk-bearing stars in our K2 young star set with available spectroscopy; this set includes 28 bursters and 46 non-bursters.",
"Where more than one Hα\\alpha measurement is available, we adopt the mean value.",
"The sample was binned in 10Å wide sets to produce the distribution.",
"While the non-bursting stars tend to predominate between 0 and -15Å (i.e., emission), the burster Hα\\alpha values show a much wider dispersion, reaching much values up to -250Å.",
"Thus the burst phenomenon seems to favor stars with high accretion rates.",
"We note that there is a single disk-bearing star with quasi-periodic light curve that has a reported Hα\\alpha equivalent width around -120Å.",
"Such a value is highly unusual for a star with a non-bursting light curve.Figure: Line profiles in Hα\\alpha , Ca2 8542 Å, and He1 5876 Å measured by Keck/HIRES for 12 of the 29 bursters, labeled by EPIC identifier.",
"The Hα\\alpha panels include a horizontal line at 10% of the peak (un-normalized) flux in addition to the horizontal line indicating the continuum level.",
"Note the change in velocity scale for the helium panels.",
"Broad width and structured velocity profiles indicate accretion and wind phenomena.Figure: Estimated increase in accretion rate during bursts, as a function of stellar spectral type."
],
[
"Spatial distribution",
"The sample we have analyzed here is likely a mixture of ages from young ($<1-2$ Myr) but still optically visible stars associated with the young $\\rho $ Oph cloud, to somewhat older (5-10 Myr) stars in the off-cloud Upper Sco region which still retain accreting circumstellar disks.",
"Bursters as identified here make up $\\sim $ 9% of the disk-bearing young star sample in the $K2$ /C2 dataset.",
"We initially hypothesized that they would preferentially appear in the compact, young $\\rho $ Oph cluster, as opposed to the more dispersed and older Upper Sco region.",
"But the spatial distribution shown in Figure REF surprisingly reveals equal proportions of bursters in the two regions.",
"Furthermore, there is no correlation with global extinction measures.",
"This suggests that either the young population extends from $\\rho $ Oph out into the surrounding areas, or that the burster phenomenon is less dependent on age than on disk properties such as mass.",
"[52] found evidence for an intermediate age population ($\\sim $ 3 Myr) of YSOs outside of the main $\\rho $ Oph cloud core, but within the main L 1688 cloud.",
"Furthermore, [165] found a negligible age difference between sets of Upper Sco and L 1688 association members; both regions were estimated to be $\\sim $ 3 Myr old.",
"Figure: Top: Spatial distribution of young stars (gray), including variables (black) and specifically bursters (red), overlaid on the K2 field of view.",
"The concentration of stars near RA=246.8, Dec=-24.6 is the ρ\\rho Oph cluster.",
"Bottom: Young stars with disks (cyan) and bursters (red) overlaid on the extinction map of the ρ\\rho Oph region."
],
[
"Spectral types",
"To assess the mass distribution of stars displaying bursts, we have gathered spectral types from the literature.",
"Those found for the bursters are displayed in Table 1.",
"We have also selected a control sample of non-bursting stars from [99].",
"That work presents spectral types for hundreds of USco members; we cull the list to include only non-bursting, inner disk-bearing stars observed in K2 Campaign 2.",
"The resulting set of 31 objects has spectral types ranging from B8 to M8, as shown in Figure REF .",
"We compare this against the distribution of spectral types for the bursters identified here, finding a significant difference only for the B–F spectral types.",
"There are early type stars in the sample but none of their variability is of the bursting type.",
"K2 light curves for these objects show mainly low-level quasi-periodic modulation.",
"However, the spectral type distributions of bursters and non-bursters is very similar for the K–M range.",
"Statistically, the two sets are indistinguishable here.",
"Thus we conclude that the preponderance of late-M spectral types among bursters is consistent with the increased fraction of young stars with disks at low masses.",
"Figure: Spectral type distributions for bursting and non-bursting young disk-bearing stars observed in K2 Campaign 2.",
"Two distributions are very different for the B–F range, but indistinguishable for K and M spectral types."
],
[
"Multiplicity",
"The presence of close stellar companions to the young stars in our sample may influence their variability properties.",
"Indeed it has been hypothesized that, for high-amplitude outbursting stars such as FUors, events could be triggered by the presence of a perturbing companion [129], [94].",
"However, support for this idea is mixed [61].",
"It is nevertheless worthwhile to check which, if any, of our burster sample may be in binary systems.",
"We have vetted over half of the sample for multiplicity by examining the available high-resolution imaging as well as obtaining new speckle observations."
],
[
"Literature assessment",
"We mined the literature for adaptive optics and speckle imaging of these targets, and the details are tabulated as part of the individual object commentary in the Appendix.",
"Only three objects have evidence for a close companion published in the literature: EPIC 204906020 (2MASS J16070211-2019387), EPIC 203786695 (2MASS J16245974-2456008), EPIC 203905625 (2MASS J16284527-2428190).",
"The first is a close binary [89].",
"Likewise, the second has a companion at $\\sim $ 15 AU (100.4 mas) [84].",
"The last of these has an even closer companion at $\\sim $ 2 AU (13 mas) [153].",
"Interestingly, all three systems also have wider ($>$ 50 AU) companions.",
"None of their light curves shows any hint of a periodicity.",
"One additional object, EPIC 204342099 (2MASS J16153456-2242421), may have a 274 AU separation companion [63], but it is not clear whether the two stars are bound.",
"EPIC 204830786 (2MASS J16075796-2040087) is associated with another star 21.5 away ($>$ 3000 AU separation) but has not been surveyed for closer companions.",
"Other than these five objects, 9 other bursters in our sample have been surveyed for multiplicity, at a variety of sensitivities and separations: EPIC 204226548 (2MASS J15582981-2310077), EPIC 203899786 (2MASS J16252434-2429442), EPIC 203935537 (2MASS J16255615-2420481), EPIC 203905576 (2MASS J16261886-2428196), EPIC 203954898 (2MASS J16263682-2415518), EPIC 203822485 (2MASS J16272297-2448071), EPIC 203913804 (2MASS J16275558-2426179), EPIC 203928175 (2MASS J16282333-2422405), EPIC 203794605 (2MASS J16302339-2454161).",
"No companions were found in these cases.",
"However, the separations probed are very non-uniform and range from 10 mas (1.5 AU) in some cases to 1–30 in others ($>$ 150 AU); details are provided in the Appendix individual objects section."
],
[
"DSSI targets",
"In addition to data compiled from the literature, we also have speckle imaging observations of six bursters using DSSI (§2.3).",
"For EPIC 204342099 (2MASS J16153456-2242421), we recover the companion reported at 1.9 by [63]; however, we measure the separation to be 1.50 ($\\sim $ 218 AU), and a magnitude difference of $\\Delta $ m=3.38 at the 880 nm band.",
"This star previously had direct imaging and aperture masking by [87] that ruled out further objects down to 240 mas (35 AU).",
"EPIC 204830786 (2MASS J16075796-2040087) has a previously noted possible companion at thousands of AU separation, but the DSSI observations otherwise support the hypothesis that this is a single star.",
"EPIC 204440603 (2MASS J16142312-2219338) has no previous multiplicity information, but the 880 nm image suggests a possible companion at a separation of 0.1.",
"However, the lack of a similar detection at 692 nm and the faintness of this star makes speckle reconstruction challenging; thus the existence of such a companion remains indeterminate.",
"EPIC 204360807 (2MASS J16215741-2238180) has no multiplicity information in literature, but we find it to be a 0.48 separation binary (70 AU), with $\\Delta $ m=0.74 at 880 nm.",
"EPIC 203935537 (2MASS J16255615-2420481) has no reported evidence for multiplicity, and we do not detect companions down to 0.1 separation (15 AU) at 4–5 magnitudes of contrast in the 692 and 880 nm bands.",
"Finally, EPIC 203928175 (2MASS J16282333-2422405) was reported by [33] to host no companions down to 20 mas (3 AU) at 1-3 magnitude contrast; our observations support the lack of binarity, with no detections outward of 0.1 at $\\Delta m\\sim $ 4.4 magnitudes."
],
[
"Time domain behavior",
"In order to appreciate the diversity of the bursting behavior among our sample of objects, we must go beyond just the $M$ and $Q$ metrics discussed above.",
"We quantified the peak-to peak amplitudes of the bursters by first cleaning and normalizing the light curves and then computing the maximum-minus-minimum values.",
"There are several ways to quantify light curve timescales.",
"First, we measure the burst duty cycle, which is the fraction of time each object spends in a bursting state.",
"This is by nature somewhat subjective, as bursts display a range of amplitudes and shapes.",
"We identified bursting portions of each light curve by first fitting and removing a low-order median trend to the light curve.",
"This flattens the “continuum” level from which bursts arrive.",
"We then measure the typical point-to-point scatter by shifting each point by one, subtracting from the original light curve, and dividing the standard deviation of the result by $\\sqrt{2}$ .",
"Using this measure of scatter, we have found that burst behavior, as detected by eye, includes points that lie about 15 times the scatter above the minimum of the continuum-flattened light curve.",
"We thereby selected bursting and non-bursting sections for each time series.",
"This method only failed for three objects (EPIC 204397408/2MASS J16081081-2229428, EPIC 205156547/2MASS J16121242-1907191, and EPIC 203856109/2MASS J16095198-2440197) that displayed intermittent quasi-periodic behavior that was picked up as bursts.",
"We manually removed these light curve portions for the statistical analysis.",
"In Figure REF we display the peak-to-peak amplitudes versus duty cycle of each burster.",
"The duty cycles exhibit a large range of values, from almost 100% down to $\\sim $ 10%.",
"Typically the light curves with the highest amplitudes have higher duty cycles of 60% and above, with the exception of outlier EPIC 203954898/2MASS J16263682-2415518.",
"This object may represent a distinct form of bursting behavior.",
"We also quantify the burst timescales by applying a method similar to the one described in [38] (§6.5 of that paper).",
"In brief, this involves identifying peaks that rise above a particular amplitude level compared to the surrounding light curve.",
"Once peaks are found, the median timescale separating them is computed.",
"This procedure is repeated for a variety of amplitudes, from the noise level up to the maximum light curve extent.",
"The result is a plot of timescale versus amplitude (e.g., Fig.",
"32 in [38]).",
"Finally, we take as a “representative” timescale the value corresponding to an amplitude that is 40% of the maximum peak-to-peak value (we note that this is different from the value of 70% adopted in [38] and appears more appropriate for the burster light curves examined here).",
"This computation only fails for the light curve of EPIC 203382255 (2MASS J16144265-2619421), which displays only one burst event; here the timescale is indeterminate.",
"In Figure REF we display the peak-to-peak amplitudes versus estimated timescale for each burster.",
"Again, there is a large range of values, but no clear correlation with amplitude.",
"The burst duration is another way to quantify the observed events.",
"This is a challenging measurement, as there is a superposition of bursts with varying widths and heights.",
"We simplify as above by only considering peaks that rise to a level of at least 40% of the maximum peak-to-peak value.",
"For each peak, we identify the surrounding points that are more than 15 times the point to point scatter above the minimum light curve value (as was done for the burst duty cycle calculation).",
"We then adopt the median burst duration of all such peaks in each light curve.",
"The result is plotted against peak-to-peak amplitude in Figure REF .",
"We also compare the durations with the repeat timescales in Figure REF .",
"Here we find that burst duration is correlated with repeat timescale.",
"This is somewhat expected since, by definition, a duration is typically larger than the average timescale between bursts.",
"However, we observe a distinct lack of short bursts (duration $\\le $ 2 days) with long repeat timescales.",
"This may be rooted in the physical mechanism of the bursts.",
"We have also generated periodograms to identify any repeating components in the light curves.",
"Most bursters do not exhibit periodicity but instead adhere to stochastic behavior, with any quasi-periodicity quantified via the $Q$ statistic (see §3).",
"Those that do show significant periodicities (as indicated by $Q\\le 0.61$ and/or a strong, isolated periodogram peak) are EPIC 203794605/2MASS J16302339-2454161 ($P=4.5$ d), EPIC 203899786/2MASS 16252434-2429442 ($P=6.0$ d), EPIC 203928175/2MASS J16282333-2422405 ($P=4.4$ d), EPIC 203954898/2MASS J16263682-2415518 ($P=20.8$ d), and EPIC 204347422/2MASS J16195140-2241266 ($P=6.9$ d).",
"The measured periods are similar to the burst repeat timescales inferred above.",
"In addition, EPIC 203856109 (2MASS J16095198-2440197), EPIC 204233955 (2MASS J16072955-2308221), EPIC 204397408 (2MASS J16081081-2229428), and EPIC 205156547 (2MASS J16121242-1907191) display short-timescale ($P$ less than a few days) periodic behavior outside of their bursting states.",
"These periods are more typical of the K2 $\\rho $ Oph and Upper Sco sample as a whole [127] and likely measure stellar rotation, whereas those of the bursts are longer by factors at least several.",
"Figure: Maximum burst amplitude (in units of normalized flux) versus duty cycle, i.e., fraction of time spent bursting.Figure: Burst repeat timescales versus peak-to-peak light curve amplitude, in normalized flux.Figure: Burst durations versus peak-to-peak light curve amplitude, in normalized flux.Figure: Burst repeat timescale versus durations.",
"The dashed line indicates where these two quantities are equal.",
"We have left out EPIC 203382255 (2MASS J16144265-2619421) since it only has one burst and the repeat timescale is thus indeterminate.Few of the timescale metrics show any relation to [circum]stellar properties, but one potential correlation stands out in peak-to-peak amplitude versus the infrared $W1-W2$ color (Figure REF ), which is indicative of a dusty inner disk.",
"These two quantities are correlated at a significance level of 4$\\times 10^{-5}$ (Pearson $r$ coefficient of 0.69).",
"This is also borne out in Figure REF , which suggests that the dustiest objects have the highest light curve amplitudes.",
"Figure: Peak-to-peak light curve amplitude for bursters is shown against the W1-W2W1-W2 color.",
"All included objects have W1-W2W1-W2 values indicative of inner disk dust, but the highest amplitude bursters tend to have larger infrared excesses, with one prominent exception (EPIC 205008727/2MASS J16193570-1950426; W1-W2∼1.3W1-W2\\sim 1.3).llcccccccc 8 0pt Light curve metrics for bursting young stars in $K2$ Campaign 2 EPIC id 2MASS id Amplitude Q M Timescale Duration Duty cycle Period (norm.",
"flux) [d] [d] [d] 203382255 J16144265-2619421 1.43 1.00 -1.14 $>$ 77.74 7.95 0.08 - 203725791 J16012902-2509069 0.68 0.87 -0.29 5.18 1.90 0.66 - 203786695 J16245974-2456008 0.25 1.00 -0.59 5.52 4.54 0.43 - 203789507 J15570490-2455227 0.36 0.97 -0.41 9.12 4.45 0.06 - 203794605 J16302339-2454161 0.87 0.55 -0.25 4.64 3.67 0.93 4.46 203822485 J16272297-2448071 0.62 0.84 -0.29 9.36 7.66 0.90 - 203856109 J16095198-2440197 0.35 1.00 -1.00 3.11 5.25 0.15 - 203899786 J16252434-2429442 1.13 0.61 -0.83 6.42 3.98 0.96 5.95 203905576 J16261886-2428196 3.85 1.00 -0.66 7.48 5.97 0.88 - 203905625 J16284527-2428190 0.32 1.0 -0.31 7.80 5.17 0.72 - 203913804 J16275558-2426179 0.41 1.0 -0.37 5.47 3.56 0.70 - 203928175 J16282333-2422405 3.19 0.54 -0.66 4.68 2.53 0.73 4.39 203935537 J16255615-2420481 0.24 1.00 -0.31 4.92 3.09 0.69 - 203954898 J16263682-2415518 6.82 0.61 -1.35 31.17 6.84 0.43 20.83 204130613 J16145026-2332397 1.87 0.85 -0.35 3.43 0.90 0.62 - 204226548 J15582981-2310077 0.47 1.00 -0.53 8.54 3.44 0.73 - 204233955 J16072955-2308221 1.99 0.85 -0.82 3.42 2.08 0.98 - 204342099 J16153456-2242421 0.83 0.91 -0.72 35.53 11.26 0.88 - 204347422 J16195143-2241332 1.38 0.75 -1.11 7.23 1.65 0.10 6.94 204360807 J16215741-2238180 0.55 0.87 -0.49 5.21 2.53 0.65 - 204397408 J16081081-2229428 0.15 0.59 -0.68 5.01 5.30 0.22 1.65 204440603 J16142312-2219338 0.43 1.00 -0.93 4.55 3.15 0.28 - 204830786 J16075796-2040087 1.91 1.00 -0.67 14.70 4.31 0.91 - 204906020 J16070211-2019387 0.34 0.93 -0.47 3.90 0.91 0.63 - 204908189 J16111330-2019029 1.28 0.76 -0.59 7.99 7.97 0.90 19.23 205008727 J16193570-1950426 0.91 1.00 -0.68 10.25 7.93 0.76 - 205061092 J16145178-1935402 0.35 1.00 -0.51 4.85 1.94 0.35 - 205088645 J16111237-1927374 0.10 1.00 -0.53 11.30 3.18 0.04 - 205156547 J16121242-1907191 0.16 1.00 -1.01 4.38 3.62 0.23 - We tabulate basic statistical properties of the burster light curves.",
"$Q$ and $M$ are discussed in §3 as well as Cody et al.",
"(2014).",
"Amplitudes represent peak-to-peak measurements."
],
[
"Discussion and Summary",
"$K2$ data from Campaign 2 have probed the optical burst properties of young stars on time scales ranging from hours up to several months, with approximately 8–10% ($\\pm $ 2%) of strong disk sources exhibiting burst behavior.",
"This is roughly in agreement with the 13$^{+3}_{-2}$ % fraction found for NGC 2264 [38], [155].",
"It is possible that an even larger fraction of young stars undergo bursting, but are not detected as such if the amplitude is low (i.e., $<100$ % peak to peak).",
"Burst behavior could in some cases be masked by other types of variability.",
"Burster stars host inner circumstellar disks, as evidenced by WISE excesses and SEDs (Figure REF ).",
"Furthermore, there is a positive correlation between the $W1-W2$ color and light curve peak-to-peak amplitude.",
"Stronger inner disk excesses appear to be associated with bigger bursts.",
"Most of these objects have exceptionally strong H$\\alpha $ emission, as well as other Balmer lines, He1, and Ca2 in emission, as is typical for strongly accreting stars.",
"Viewing geometry may play a role in setting the observability of the bursting phenomenon, but thus far we only have disk inclination constraints from ALMA on two sources in the sample, both of which are tilted at $\\sim $ 42$\\pm $ 15.",
"Members of the bursting sample typically exhibit multiple discrete brightening events, some lasting up to or just over one week.",
"Time domain properties of the bursters are diverse, with some stars exhibiting a nearly continuous series of bursts and others displaying one or two isolated episodes superimposed on otherwise lower amplitude or quasi-periodic behavior.",
"The majority of these light curves show flux variations of less than a factor of two and are similar to the objects in the $\\sim $ 3 Myr NGC 2264 highlighted by [155].",
"However, seven of 29 bursters display discrete brightenings of more than 100% on day to week timescales, unlike most heretofore classified young stellar variables that we are aware of.",
"Of particular interest is the subset of these for which the bursts repeat quasi-periodically: EPIC 204347422 (2MASS J16195140-2241266), EPIC 203928175 (2MASS J16282333-2422405), and EPIC 203954898 (2MASS J16263682-2415518).",
"It is unclear as to what physical phenomenon sets the periods of 6.9 d, 4.4 d, and 20.8 d (respectively) in these cases; it appears to be relatively independent of stellar mass, as indicated by spectral type.",
"There also is a lack of evidence for binarity in most of the bursters, although in some cases limits from imaging are relatively shallow.",
"We thus speculate that burst events are not triggered by any companion, but rather by a repetitive interaction between the stellar magnetic field and inner disk.",
"From the theoretical perspective, magnetically channeled accretion is not predicted as steady in numerical simulations.",
"An inner disk is truncated at a balance point between the inward pressure from accretion and the outward pressure of the magnetosphere.",
"This produces variable and possibly cyclic mass flow due to instabilities and pulsational behavior.",
"Such variations in the mass loading of accretion columns are modeled under different physical scenarios by, e.g., [98], [59], [140], [141], [142], and [45], [46].",
"The disk corotation radius ($r_c$ ) and the magnetospheric radius ($r_m$ ) are critical in determining the regime of accretion under which a star-disk system falls.",
"When $r_m<r_c$ , gas at the inner disk edge rotates faster than the star and its magnetosphere, causing it to flow along magnetic field lines onto the central star.",
"As long as $r_m>0.7r_c$ , “stable” accretion occurs [20] and gas follows two funnel streams onto the star [144], [141].",
"For smaller values of $r_m$ , the Rayleigh-Taylor instability sets in and accretion becomes chaotic, with many tongues of matter extending from the inner disk to the stellar surface [138], [91].",
"Numerous hot spots are present on the stellar surface, and the associated light curves display irregular bursting.",
"This regime may be responsible for the large subset of bursters that we observe with high duty cycles.",
"When $r_m$ is up to a factor of two larger than $r_c$ , on the other hand, gas in the inner disk rotates slower than the magnetosphere and is accelerated azimuthally.",
"Models [46] predict a “trapped disk\" regime in which relatively continuous accretion bursts occur on short timescales.",
"When $r_m>>r_c$ , material tends to be flung out azimuthally in what is known as the “strong propeller\" regime [139].",
"The light curves of several of our objects (EPIC 203954898/2MASS JJ16263682-2415518 and EPIC 204347422/2MASS J16195140-2241266) strongly resemble the simulated mass flow variations predicted by [167] and [92] for propeller behavior.",
"In this scenario, the matter accretes episodically in three phases: First, material accumulates at the inner disk boundary, unable to flow inward.",
"The magnetosphere is compressed toward the stellar surface.",
"Next, compression reaches the point at which the magnetosphere can no longer withstand the gravitational forces on the accumulated material; gas accretes rapidly onto the star in a funnel flow.",
"Finally, the compression pressure is relieved, accretion ceases, and the magnetosphere is able to expand outward again.",
"Following this sequence, the cycle repeats.",
"The time between bursts is longer than in other regimes, and the accretion rate can follow a flare-like pattern (rapid rise; slower decline) as a function of time.",
"Strong outflows may also be present.",
"EPIC 203382255 (2MASS 16144265-2619421) shows outflow signatures and also has the longest repeat timescale, making it an additional propeller regime candidate.",
"In contrast, a few other objects with outflow-related spectral lines do not display the expected alternating burst and quiescence pattern.",
"Further investigation of these targets is necessary to estimate magnetic and corotation radii and compare with the theoretical expectations.",
"The ratio $r_m/r_c$ depends on mass accretion rate, magnetic field strength, and stellar spin period– parameters that we are unable to determine independently for this sample of burster stars.",
"In the meantime, the burst timescales measured for our sample provide perhaps the best indication of the physical mechanisms at work.",
"Duty cycles range from 10% and all the way up to nearly 100%– suggesting that we are seeing different modes of accretion from continuous to episodic and less frequent.",
"The possible correlation between burst duration and repeat timescale (Figure REF ) may imply a relationship between mass loading timescale and accretion rate.",
"A major open question stemming from this work is what the origin of the day to (multi-)week repeat timescales that we observe is, and how it relates to young outbursting stars with much longer duty cycles (e.g., EXors and FUors).",
"Prominent examples from the literature include V899 Mon, which has had repeated bursts separated by $\\sim $ 1 year [112], and V1647 Ori, which repeats at $\\sim $ 2 years [11].",
"On even longer timescales, other EXor type stars tend to outburst once or twice per decade.",
"Unlike the $K2$ objects, the longer among these time scales are thought to be related to the viscous time scale in the disk.",
"Here, the material drains inward and must undergo replenishment before the next instability-driven episode can occur.",
"It has been speculated that the frequency and amplitude of outbursts may be set by the accretion rate, with younger and higher amplitude outbursting objects accreting more rapidly [12].",
"We would then expect accretion rates for our own sources to be somewhat lower.",
"This does indeed seem to be the case, as the median accretion rate where available for bursters is 10$^{-8.1}$ $M_\\odot $ yr$^{-1}$ (for exact values, see the notes on individual objects in the Appendix).",
"We have found (Section 4.2) that this increases one to two orders of magnitude during the most extreme $K2$ light curve peaks.",
"Further spectroscopic measurements during times of definitive bursting may confirm these estimates.",
"In summary, the $K2$ mission is providing an unprecedented view of the time domain properties of young stars, and showing that the optical photometric manifestations of accretion phenomena take on a wide variety of timescales and amplitudes.",
"Follow-up observations, including spectroscopy, should be carried out to investigate changes in spectral emission and inner disk structure in the highest amplitude objects presented here.",
"The work of A.M.C was supported by a NASA/NPP fellowship.",
"We thank the referee for useful feedback that improved this paper.",
"We acknowledge Luisa Rebull for calling our attention to one of the burst type objects that we had overlooked in our initial examination.",
"Thanks also to Nic Scott for assistance with DSSI observations on the Gemini South telescope.",
"This paper includes data collected by the $K2$ mission.",
"Funding for the $K2$ mission is provided by the NASA Science Mission directorate.",
"The spectroscopic data were obtained at the W.M.",
"Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.",
"The Observatory was made possible by the generous financial support of the W.M.",
"Keck Foundation.",
"These results are also based on observations obtained as part of the program GS-2016A-Q-64 at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil).",
"This paper has utilized the SIMBAD and Vizier services, through which data from USNO-B, APASS, 2MASS, and WISE were collected."
],
[
"Notes on individual objects",
"Not all of our sample stars appear in the literature, but for those that have been studied previously, we highlight the key results here.",
"We also incorporate information from our new speckle and spectroscopic observations."
],
[
"EPIC 203382255 / 2MASS 16144265-2619421",
"This source is cataloged by [95] as an astrometric and photometric member, but is not otherwise studied in the literature.",
"Our spectra suggest a spectral type of M4-M5.5 depending on the spectral range (earlier at bluer wavelengths).",
"Lithium is present with $W_{Li} = 0.30 A$ along with the H$\\alpha $ emission reported in Table , He1, weak NaD, and Ca2 are seen in emission.",
"Broad [O1] is also present.",
"The system is a clear accretion/outflow source."
],
[
"EPIC 203725791 / 2MASS J16012902-2509069",
"This star was first proposed as an Upper Sco member by [9], based on $RIZ$ photometry.",
"It was only recently confirmed by [137], based on significant lithium absorption.",
"No other data has been reported for this star.",
"Our HIRES spectrum indicates a spectral type of M2 with lithium absorption strength $W_{Li}=0.3$ Å.",
"Strong H$\\alpha $ emission is seen, as indicated in Table , along with lorentz-broadened He1.",
"NaD, weak narrow Fe2 as well as Mg1 emission, two-component Ca2, and O1 emission.",
"Very weak [O1] is also exhibited."
],
[
"EPIC 203786695 / 2MASS J16245974-2456008",
"This star, WSB 18, was first noted as part of Wilking et al.",
"(1987)'s H$\\alpha $ emission survey of the the $\\rho $ Ophiuchi complex.",
"Lithium absorption was detected by [52], confirming youth.",
"It is a visual binary with a separation of 1.1 (138 AU) and a 0.49 flux ratio [131].",
"Therefore, both components contribute to our $K2$ light curve.",
"According to [27], the primary has a spectral type of M2, whereas the secondary is M2.5.",
"Both show H$\\alpha $ in emission, but the primary does not appear to have a disk [104], whereas the secondary does.",
"Further, [84] found that the primary itself is a double star with 0.1 separation.",
"No detections of these sources at $>70$ $\\mu $ m were made with Herschel.",
"In the submillimeter, [5] put an upper limit of 0.003 $M_\\odot $ on the dust mass surrounding this source, although it is unclear which components were included."
],
[
"EPIC 203794605 / 2MASS J16302339-2454161",
"According to [166], this star displayed an H$\\alpha $ emission intensity of 3 on a scale of 1 to 5 (where 1 is weak and 5 is strong).",
"Simon et al.",
"(1995) performed a direct imaging search for companions in the 0.005-10 separation range, but did not find anything.",
"Neither were any companions detected in the Catalog of High Angular Resolution Measurements [134], its successor CHARM2 [135], or the [125] speckle imaging survey of $\\rho $ Oph.",
"For the latter, no companions were detected down to 0.04 (0.14) times the stellar brightness at 0.5 (0.15) separation.",
"This star was included in the Spitzer c2d legacy survey.",
"Our HIRES spectrum is veiled, but consistent with a spectral type of M3.5-M5.",
"Lithium is present with strength $W_{Li} = 0.32 Å$ .",
"H$\\alpha $ emission as reported in Table is strong with several components and a blueward-displaced central absorption.",
"Additional emission includes He1 with lorentzian wings, broad NaD, many multi-component Fe2 lines, strong Ca2 with multiple component and O1 8446.",
"Among the outflow lines, only multi-component [O1] is seen."
],
[
"EPIC 203822485 / 2MASS J16272297-2448071",
"This star, WSB 49, is in the $\\rho $ Ophiuchus cluster.",
"It was first detected as part of Wilking et al.",
"(1987)'s H$\\alpha $ emission line survey.",
"It is also an IRAS source [164].",
"It was first discovered as an x-ray source with the ROSAT High Resolution Imager [64], and also noted XMM Newton observations by [118].",
"[165] confirmed its youth via detection of lithium absorption.",
"[52] estimate a mass of 0.17 $M_\\odot $ .",
"The star was surveyed for multiplicity, but no companions were detected down to $>0.15$ separation at flux ratios of 0.06.",
"An infrared excess is detected with Spitzer [53].",
"[5] observed it in the submillimeter, classifying the disk as class II, with $<0.005$ $M_\\odot $ of material.",
"In addition, the disk is detected at 70 $\\mu $ m with Herschel/PACS [126].",
"It was listed as a long timescale near-infrared variable by [113], while [66] detected $\\sim $ 20% variations in the mid-infrared with Spitzer."
],
[
"EPIC 203856109 / 2MASS J16095198-2440197",
"There is no previous literature on this source.",
"Our spectrum indicates a spectral type of M5-M5.5 with lithium at $W_{Li} = 0.61 Å$ .",
"Moderate H$\\alpha $ emission, as indicated in Table 1, is present and has a multi-component profile.",
"Weak, also multi-component, He1 is also present, but no other emission lines."
],
[
"EPIC 203899786 / 2MASS J16252434-2429442",
"This star in the $\\rho $ Oph cluster is also known as V852 Oph and SR 22.",
"Its emission line spectrum was reported as early as the mid-20th century [157], with prominent hydrogen, Ca2, and Fe2 noted.",
"[74] and [73] included it in their emission line star catalogs.",
"It is a ROSAT x-ray source [32].",
"Irregular variability was also reported early on, by [148] and [54].",
"[4] label the object as class III based on its 2.2–10 $\\mu $ m slope, and despite its previous classification as a classical T Tauri star by [39].",
"It was relabeled as class II by [62], based on their 1.1–2.4 $\\mu $ m survey.",
"[5] confirmed the disk with submillimeter observations, detecting 0.002 $M_\\odot $ of material.",
"Similarly, [107] report 2.05 $M_{\\rm Jup}$ of material, based on SCUBA-2 850 $\\mu $ m observations.",
"[126] detected the disk at 70 $\\mu $ m with Herschel, and reported that there is a gap in the spectral energy distribution that suggests a hole.",
"The source is evidently a single star.",
"[125] ruled out any companions down to a flux ratio of 0.07 at a separation of 0.15, and [62] did not detect any radial velocity variations indicative of a spectroscopic binary.",
"[33] reported no companions down to 10 mas and contrasts of several magnitudes."
],
[
"EPIC 203905576 / 2MASS J16261886-2428196",
"This star– better known as VSSG 1– boasts a disk that was first detected by IRAS [36].",
"The mid-infrared slope, $\\alpha $ , is -0.4, making it a class II disk according to [4] (although Wilking et al.",
"1989 earlier classified it as class I).",
"[105] used SPEX to measure an $n_{2-25}$ spectral index of -1.26, confirming the class II categorization.",
"They also reported a small 10 $\\mu $ m silicate feature.",
"[6] have observed this source's disk with the Submillimeter Array at 0.87 mm and estimated a dust mass of 0.029 $M_\\odot $ , along with an accretion rate of $10^{-7}$ $M_\\odot $ yr$^{-1}$ .",
"H$_2$ O was detected in the Spitzer IRS spectrum obtained by [120], along with HCN, C$_2$ H$_2$ and CO$_2$ .",
"Salyk et al.",
"(2011) confirm these detections and estimate a disk mass of 0.029 $M_\\odot $ .",
"They infer a disk inclination of 53.",
"Submillimeter observations with ATCA [133] resulted in an estimated disk outer radius of 100–300 AU and a much lower dust mass of $4.5\\times 10^{-5}$ –$1.9\\times 10^{-4}$ $M_\\odot $ .",
"[7] looked for mid-infrared variability in this source and concluded that it is not an EXor candidate.",
"[111] report a fairly high accretion rate of 10$^{-7.19}$ $M_\\odot $ yr$^{-1}$ , and a Pa$\\beta $ equivalent width of 8.9Å in emission.",
"According to the work of [125], no companions are visible down to a flux ratio of 0.04 at a separation of 0.15.",
"Our spectrum exhibits strong H$\\alpha $ as indicated in Table with a double-peaked broad profile, as well as strong and broad Ca2 triplet as well as O1 8446 emission.",
"There is very little in the way of absorption, presumably due to heavy veiling, and spectral types from K7 to mid-M are plausible.",
"For the same reason, aggravated by low signal-to-noise in this region of the spectrum, there is no lithium measurement."
],
[
"EPIC 203905625 / 2MASS J16284527-2428190",
"EPIC 203905625, also known as V853 Oph and SR 13, is a late-type star in $\\rho $ Ophiuchus with reported spectral types from M2 to M4 [165].",
"Accretion signatures include H$\\alpha $ as well as calcium in emission.",
"[147] first reported strong veiling in the star's spectrum.",
"[23] found that the H$\\alpha $ emission is variable (30–48Å).",
"[111] estimated an accretion rate of $10^{-8.31}$ $M_\\odot $ yr$^{-1}$ , from near-infrared spectra.",
"They detected Pa$\\beta $ in emission, at an equivalent width of -1.7Å.",
"X-rays from this object were first detected with Einstein [108].",
"A disk around this star was also observed, with IRAS [36].",
"[4] used 1.3mm observations to detect the disk; they classify it as class II, based on a 2.2–10 $\\mu $ m slope of -0.8.",
"[120] detect H$_2$ O, OH, HCN, and C$_2$ H$_2$ in a Spitzer IRS spectrum of this target.",
"[126] report Herschel detections at 70 through 500 $\\mu $ m; they classify the system as transitional, based on 12–24 $\\mu $ m data.",
"The star is a multiple system, with a companion first detected at 0.4 separation via speckle imaging [58].",
"[10] and [125] confirmed the 0.4 and a 0.238 flux ratio.",
"[153] conducted an IR imaging survey that revealed a closer companion at separation 13 mas.",
"[104] found that both the primary and its 0.4 companion have class II disks.",
"[5] measured a total disk mass of 0.01 $M_\\odot $ using submillimeter observations, while [107] measured a similar 7.81 $M_J$ from SCUBA-2 850 $\\mu $ m observations.",
"Variability in this object was initially reported by [147].",
"[149] found optical variations from 12.6 to 14.7.",
"[75] observed V magnitude fluctuations from 12.83 to 13.52, while [60] reported $V$ magnitudes between 12.61 and 13.87.",
"Likewise, [152] conducted photometric monitoring, reporting a $V$ -band amplitude of 0.91 magnitudes.",
"The light curves are too sparsely sampled to identify any morphological features, although they are classified as “irregular.\"",
"The broadband K2 light curve exhibits 0.2 mag events, suggesting that amplitudes are higher at bluer wavelengths.",
"This is confirmed by Herbst et al.",
"(1994)'s estimate of d$U$ /d$V$ : 2.39."
],
[
"EPIC 203913804 / 2MASS J16275558-2426179",
"This target is also known as SR 10 and V2059 Oph.",
"It originally appeared in Herbig & Rao's (1972) catalog of emission line stars.",
"Both [148], and [90] list it among their variable star compilations; [149] labeled the variations as “irregular.\"",
"H$\\alpha $ emission at 40Å was reported early on by [146].",
"[8] also observed strong H$\\alpha $ , as well as He1 and Fe2 in emission.",
"[166] noted this star in their H$\\alpha $ emission survey, and [165] again measured H$\\alpha $ in emission as well as lithium in absorption.",
"[111] estimated the accretion rate to be 10$^{-7.95}$ $M_\\odot $ yr$^{-1}$ ; they detected Pa$\\beta $ emission at an equivalent width of -5.6Å.",
"[109] report a slightly lower accretion rate of 10$^{-8.3}$ $M_\\odot $ yr$^{-1}$ .",
"[153] conducted an imaging survey for binary companions at project separations from 0.005 to 10 down to a $K$ magnitude of 11.1.",
"Likewise, [125] searched for companions at separations of 0.15 and 0.50 but did not identify any down to flux ratios of 0.04 and 0.02, respectively.",
"[134], [135], [16], and [33] did not detect companions either, down to 10–20 mas separation at several magnitudes contrast.",
"Thus this object appears to be a single star.",
"EPIC 203913804/2MASS J16275558-2426179 is a ROSAT x-ray source [32].",
"The star is also an IRAS source [164].",
"[4] identified a class II circumstellar disk, and [67] later confirmed with Spitzer data.",
"[5] studied the system in the submillimeter and deduced an upper limit on the disk mass of 0.005 $M_\\odot $ , based on the 1.3 mm flux.",
"Similarly, [107] found an upper limit of 5.0284 $M_{\\rm Jup}$ .",
"[126] detected the disk at 70 $\\mu $ m with Herschel."
],
[
"EPIC 203928175 / 2MASS J16282333-2422405",
"EPIC 203928175 is a star in the $\\rho $ Ophiuchus region that is also known as SR 20 W [157] and ROXC J162823.4.",
"[165] report a spectral type of K5, along with variable H$\\alpha $ emission and an equivalent width of 35Å.",
"It was surveyed for binarity by [125], but no companions were detected down to 0.08 times the stellar flux at a separation of 0.15, or 0.04 times the stellar flux at 0.50.",
"[33] also did not find any companions down to 20 mas separation, at contrasts of 1–3 magnitudes.",
"With DSSI, we do not make any detections outward of 0.1 at $\\Delta m\\sim $ 4.4 magnitudes (flux contrast $\\sim $ 0.02) in the 692 or 880 nm bands.",
"The object is encircled by a class II disk, as reported by [53].",
"The disk was subsequently detected at 70, 160, 250, 350, and 500 $\\mu $ m with Herschel by [126], who tentatively classified it as transitional."
],
[
"EPIC 203935537 / 2MASS J16255615-2420481",
"This star is also known as SR 4 and V2058 Oph.",
"It has a long history of photometric and spectroscopic study, dating back to Struve & Rudkjøbing's (1949) publication of emission line stars.",
"[74], [166], and [73] listed it in their catalogs of emission line stars.",
"It has had a range of H$\\alpha $ equivalent widths measured from 84Å to 220Å as well as a low vsin$i$ of $\\sim $ 9 km s$^{-1}$ [22].",
"[159] acquired blue spectra of this target, which revealed H($\\beta $ , $\\gamma $ , $\\delta $ ) and Ca in emission, as well as significant veiling at 4450Å.",
"[51] measured the veiling factor, $r$ , to be $\\sim $ 1.5.",
"[165] detected lithium absorption and H$\\alpha $ emission in this source.",
"[130] classified the H$\\alpha $ emission line profile as type IIR, in which there are two peaks of similar height.",
"The accretion rate is estimated by [111] to be a fairly high 10$^{-6.74}$ $M_\\odot $ yr$^{-1}$ , and the same authors detected Pa$\\beta $ in emission at an equivalent width of -19.0Å.",
"As suggested by [117], the star may be the driver for a nearby Herbig Haro flow (HH 312) in the region.",
"The star is a ROSAT x-ray source.",
"It is also an IRAS point source [81], [162], [36].",
"[21] also observed the class II disk with ISO.",
"[5] detected the disk at 850 $\\mu $ m with SCUBA, and they estimated a mass of 0.004 $M_\\odot $ based on SED fitting.",
"[107] used SCUBA-2 to measure a larger disk mass of 9.4 $M_{\\rm Jup}$ ($\\sim $ 0.009 $M_\\odot $ ).",
"Spitzer/IRS observations revealed a 10 $\\mu $ m silicate feature, with a typical equivalent width of 2.29 $\\mu $ m (Furlan et al.",
"2009).",
"Interferometric data and modeling led to an inferred inner disk radius of 0.112 AU [51].",
"[119] estimated a very similar inner ring radius of 0.118 AU from near-infrared interferometry.",
"[6] observed the disk with the Submillimeter Array and found a centrally peaked morphology.",
"Their modeling predicts an inner disk radius of 0.07 AU and an inclination of 39; they infer an accretion rate of 10$^{-6.8}$ $M_\\odot $ yr$^{-1}$ , consistent with previous values.",
"Millimeter and submillimeter ATCA observations by [133] led to an outer disk radius of 100–300 AU and a dust mass of $\\sim $ 2$\\times $ 10$^{-5}$ $M_\\odot $ .",
"The disk is also detected with Herschel at 160 and 250 $\\mu $ m [126].",
"The object is a known variable, as originally reported by [148], [149] and [90].",
"It was followed up by [75], who found variations in the $V$ band of 12.73–12.93 during over 4000 days of monitoring.",
"The amplitude was larger at blue wavelengths, with a typical d$U$ /d$V$ of 2.4 magnitudes.",
"[152] reported a $V$ -band amplitude of 0.41 magnitudes over both short (hour–day) and long (years) timescales, with no detectable periodicity.",
"[60] monitored the star for over 7 years in the optical, finding a similar $V$ -magnitude range of 12.60–13.09.",
"EPIC 203935537/2MASS J16255615-2420481 is, to the best of our knowledge, a single star.",
"Ghez et al.",
"(1993)'s speckle imaging campaign did not reveal any companions down to 0.1 (0.2), at a flux ratio of 17 (18).",
"Neither did Simon et al.",
"(1995)'s imaging survey, which was sensitive to separations of 0.005–1.",
"Our own DSSI observations did not show any companions down to 0.1 separation at 4–5 magnitudes of contrast (flux ratio $\\sim $ 0.02) in the 692 and 880 nm bands.",
"[125] searched for companions in high resolution imaging but did not find any down to a flux ratio of 0.05 at a separation of 0.15.",
"Neither [106] nor [65] detected any radial velocity variations indicative of spectroscopic binary status.",
"Further high resolution imaging [134], [135] failed to reveal companions."
],
[
"EPIC 203954898 / 2MASS J16263682-2415518",
"This object was the subject of a multiplicity survey by [125], but no companions were detected down to 0.05 (0.12) times the stellar brightness at 0.5 (0.15) separation.",
"Likewise, [49] did not detect any companion, and reported that two faint stars observed at 6.1 and 6.3 separation [50] are likely background objects.",
"The star is a known X-ray emitter [64], [82], [57] and has a significant infrared excess, as indicated by ISO observations [21], Spitzer data [53], [35] and the AllWISE catalog [41].",
"The spectral index, $\\alpha $ , is 0.08, indicating a flat disk [53].",
"A spectral type range of K7-M1 was reported by [52].",
"Our own spectrum from the Palomar 200-inch telescope Double Spectrograph suggests K8.",
"We adopt M0.",
"[48] report an effective temperature of 3700$\\pm $ 56 K, consistent with this spectral type.",
"They also find a $v$ sin$i$ of 27$\\pm $ 4.7 km s$^{-1}$ .",
"The mass has been estimated to be 0.18 $M_\\odot $ [111], which is very low considering the temperature and youth of the object.",
"We suspect that several groups have confused the source with a neighboring star, 2MASS J16263713-2415599, which is $\\sim $ 9 away.",
"For example, [165] list the object name ROXRA22, a ROSAT X-ray source at RA=16:26:36.9, Dec=-24:15:53 [64].",
"However, the [165] coordinates match the companion star at RA=16:26:37.1, Dec=-24:15:59.9, and the listed spectral type is later, at M5.",
"The companion paper by [52] provides the same effective temperature, luminosity, and mass estimate, but an earlier spectral type range and lower extinction ($A_V$ =4.9).",
"Because of these discrepancies, we derive our own spectral data, apart from the $v$ sin$i$ measurement.",
"With the clear disk signatures, we can be confident that it is a young member of $\\rho $ Ophiuchus.",
"[111] list an accretion rate of $<10^{-9.71}$ $M_\\odot $ yr$^{-1}$ , based on near-infrared spectroscopy.",
"EPIC 203954898/2MASS J16263682-2415518 was monitored in the mid-infrared with Spitzer as part of the Young Stellar Object Variability project [128].",
"[66] obtained 81 datapoints spread over 34 days at 3.6 $\\mu $ m. The mean magnitude in this band was 8.15, in line with previous brightness estimates [35], [53], and the standard deviation was 0.09 magnitudes.",
"The star showed variability at the $\\sim $ 0.05-magnitude level for the first 15 days of monitoring, followed by a $\\sim $ 0.2-magnitude increase followed by a similar decrease over the remaining 20 days of observation.",
"WISE data also display a $\\sim $ 0.3-magnitude drop in the W1 band from February to August 2010."
],
[
"EPIC 204130613 / 2MASS J16145026-2332397",
"This star, BV Sco, was listed as an irregular variable by [149].",
"It has been erroneously classified as an RR Lyrae star in SIMBAD; this type of variability is inconsistent with the stochasticity seen in our $K2$ light curve.",
"Like EPIC 204233955/2MASS J16072955-2308221, it was labeled by [97] as a photometric non-member of Upper Scorpius (based on $ZYJHK$ data), before it was re-classified as a strongly accreting member [96].",
"It showed both H$\\alpha $ and He1 in emission (-108Å and -3.0Å equivalent widths, respectively).",
"Furthermore, they detected the calcium triplet lines and forbidden O1 emission, suggesting outflows."
],
[
"EPIC 204226548 / 2MASS J15582981-2310077",
"Also known as USco CTIO 33, [9] first identified this star as a candidate Upper Sco member.",
"[122] confirmed its youth via measurement of lithium absorption; they also detected strong H$\\alpha $ emission.",
"With a spectral type of M3, it is estimated to be 0.36 $M_\\odot $ by [86].",
"These authors also searched for spectroscopic and wide (1–30) binary companions, but did not find any.",
"This star was detected as a ROSAT x-ray source (RX J155829.5-231026) by [151].",
"[29] were the first to detect an infrared excess, at 8 and 16 $\\mu $ m. Cieza et al.",
"(2008) labeled it a transition disk source, with a mass of $<1.5\\times 10^{-3}$ $M_\\odot $ worth of material based on submillimeter data.",
"No millimeter flux was detected by [103].",
"[31] report a submillimeter detection with ALMA, but the disk is unresolved.",
"They constrain the dust mass to be $0.58\\pm 0.13$ $M_\\oplus $ .",
"[43] measure significant H$\\alpha $ emission from a broad, flat-topped peak; their estimated accretion rate is $10^{-9.91}$ $M_\\odot $ yr$^{-1}$ .",
"Similar values were derived by [42].",
"Molecular gas is detected with Herschel/PACS (both C$_2$ H$_2$ and HCN) by [114].",
"[102] estimate less than 0.9 $M_{Jup}$ worth of gas mass."
],
[
"EPIC 204233955 / 2MASS J16072955-2308221",
"EPIC 204233955 is a spectral type M3 low-mass star in the Upper Scorpius association [96].",
"It was initially classified as a photometric non-member by [97], but [96] found it to be an accreting source with strong emission lines, including H$\\alpha $ , He1, and O1 forbidden lines.",
"H$\\alpha $ and He1 equivalent widths are -150Å and -3.0Å, respectively.",
"Mid-infrared data from the AllWISE survey [41] reveal a significant infrared excess in all bands, indicative of a disk.",
"Other than the work of [97], [96], this star has not been studied in detail."
],
[
"EPIC 204342099 / 2MASS J16153456-2242421",
"EPIC 204342099, otherwise known as VV Sco, is a T Tauri star in the Upper Scorpius association.",
"This object is also a known X-ray emitter from ROSAT observations [68], XMM Newton observations [93].",
"[124] obtained low and medium resolution spectra of this star as part of an X-ray selected sample of candidate Upper Scorpius members.",
"They reported a spectral type of M1 and confirmed its youth via lithium absorption.",
"EPIC 204342099 is a known disk-bearing source, originally discovered with IRAS [81].",
"It was studied in detail with Spitzer/IRS by [56], who list it under the id 16126-2235.",
"They find a very strong 10 $\\mu $ m silicate feature.",
"This star was first presented as a visual binary by [63]; the separation is 1.9.",
"[86] measured a spectral type of M3.5 for the companion.",
"It is not clear whether this object is a co-moving Upper Scorpius member or a serendipitous field object; if the former, then the separation is 274 AU.",
"With DSSI speckle imaging, we measure the separation to be smaller at 1.50, with a magnitude difference of $\\Delta $ m=3.38 at the 880 nm band.",
"[87] also searched for closer companions with direct imaging and aperture masking, but did not find any within 240 mas, at a magnitude difference of 2.8.",
"Sparse variability data is available for EPIC 204342099/2MASS J16153456-2242421.",
"The star was first noted as variable by [116].",
"[17] monitored it for optical rotation signatures but did not detect any periodicities in the light curve.",
"Our HIRES spectrum suggests a spectral type of K9-M0 with lithium absorption present at strength $W_{Li}=0.45$ Å.",
"The H$\\alpha $ emission is consistent with accretion (Table ) and the profile exhibits a blueward asymmetry along with a redshifted absorption notch against the emission.",
"Very weak and narrow profiles in He1, Fe2, and Ca2 are also present in our data, along with very weak and narrow [O1]."
],
[
"EPIC 204360807 / 2MASS J16215741-2238180",
"There is no previous literature on this source.",
"As with other objects under consideration here, there is significant veiling present with the spectral type changing from M2 in the bluer orders of our HIRES to possibly as late as M6 by about 8800 Å. Lithium has strength $W_{Li} = 0.27 A$ .",
"H$\\alpha $ emission as reported in Table 1 is very strong and there is He1, broad NaD, and many Fe2 lines.",
"The Ca2 triplet has multiple components and O1 8446 is present.",
"Of the outflow lines only weak [O1] is seen.",
"With our DSSI speckle observations, we identify a companion at 0.48 separation with $\\Delta $ m=0.74 at 880 nm."
],
[
"EPIC 204397408 / 2MASS J16081081-2229428",
"This object was first identified as a candidate USco member based on proper motion by [97] with [26] assigning it a membership probability of 99.9% based on the USNO-B proper motion.",
"[154] confirmed youth via variability and spectroscopy, assigning a spectral type of M5.",
"They noted the star as an active accretor, based on a strong H$\\alpha $ emission line.",
"Likewise, [96] also spectroscopically confirmed this object as a USco member.",
"[44] measured a radial velocity of about -11 km/s, slightly lower than the cluster mean, and a rotation velocity of $v\\sin i=16.52\\pm 4.05$ .",
"[132] analyzed WISE photometry for EPIC 204397408/2MASS J16081081-2229428, concluding that it harbors a class II disk.",
"[99] also labeled it as a full disk."
],
[
"EPIC 204440603 / 2MASS J16142312-2219338",
"This very low mass star was classified by [97] as a photometric and proper motion member of Upper Scorpius based on UKIDSS data.",
"Following up with the Anglo-Australian Telecope AAOmega spectrograph, [96] obtained intermediate-resolution spectra of EPIC 204440603/2MASS J16142312-2219338 from 5750 to 8800Å, deriving a spectral type of M5.75 and H$\\alpha $ equivalent width of -94.5Å.",
"Their measured Na I and K I gravity-sensitive equivalent widths as well as detection of Li I absorption cements the classification of this object as a young low-mass star."
],
[
"EPIC 204830786 / 2MASS J16075796-2040087",
"This Upper Scorpius member was first identified as a strong H$\\alpha $ emission-line star by [158].",
"[88] obtained low-resolution spectra, which confirmed H$\\alpha $ emission at an equivalent width of -357Å and Ca2 triplet emission as well.",
"Detection of further emission lines (N2, S2, Fe2, Ni2, O1, and the Paschen series) suggested accretion-driven jets.",
"These authors also associated EPIC 204830786/2MASS J16075796-2040087 with a wide-separation companion some 21.5 (3120 AU) away.",
"It has a significant infrared excess, as shown with IRAS [28] and later with Spitzer and WISE by [99].",
"Our HIRES spectrum suggests a spectral type of late G to early K but the spectrum is clearly heavily veiled; the lithium strength is $W_{Li}=0.20$ Å.",
"Strong H$\\alpha $ emission is seen, as indicated in Table , with a blue-side absorption notch in the profile.",
"Strong and broad He1, NaD, Fe2, Ca2, O1 8446, and perhaps other emission is present.",
"Strong multi-component forbidden emission lines of [O1] and [S2] are seen, along with single-component [N2] and many [Fe2] lines.",
"Our DSSI speckle observations do not identify any companions outward of 0.1 from this star, at a magnitude difference of $\\Delta m\\sim 4$ in the 692 and 880 nm bands."
],
[
"EPIC 204906020 / 2MASS J16070211-2019387",
"[123] detected EPIC 204906020 as a youthful member of the Upper Sco association via spectroscopic measurement of lithium absorption.",
"This M5 star also has H$\\alpha $ emission, although it was weak enough in some observations to lead to a weak-lined T Tauri star classification.",
"[86] reported the object to be an M5/M5.5 wide binary with a 1.63 separation.",
"[88] found that the primary is itself a binary, with 55 mas projected separation (8 AU at the distance of Upper Sco).",
"[29] reported infrared excesses at 8 and 16 $\\mu $ m, indicating a disk.",
"[30] also detected an excess at 24 $\\mu $ m with Spitzer/MIPS.",
"[103] did not detect any cool dust around this system at millimeter wavelengths, at a 3-$\\sigma $ upper limit of 3.7$\\times 10^{-3}$ $M_{\\rm Jup}$ .",
"However, [102] used Herschel/PACS to detect 6.6$\\times 10^{-6}$ $M_\\odot $ worth of dust.",
"[31] detected but did not resolve the disk with ALMA.",
"EPIC 204906020/2MASS J16142312-2219338 is also known to have a circumstellar disk, as first reported by [132] and confirmed by [99].",
"The SED slope is -1.3, making it a class II disk [132].",
"Our speckle observations with DSSI rule out any companions beyond 0.1 at 4.6 magnitudes contrast in the 692 and 880 nm bands."
],
[
"EPIC 204908189 / 2MASS J16111330-2019029",
"This source appears in the literature only in the [99] $WISE$ sample and the [15] ALMA study.",
"Our HIRES spectrum suggests a spectral type of M1 with lithium absorption present at strength $W_{Li}=0.15$ Å.",
"Strong H$\\alpha $ emission is seen, as indicated in Table , along with He1, weak but broad NaD, weak and narrow Fe2, weak and narrow Ca2, but moderately broad O1 8446 emission.",
"Weak and narrow [O1] is also present."
],
[
"EPIC 205008727 / 2MASS J16193570-1950426",
"There is no previous literature on this source.",
"The HIRES spectrum is heavily veiled but appears to be a late K to M3 type, with lithium present at strength $W_{Li} = 0.55 A$ .",
"H$\\alpha $ emission as reported in Table is strong and has multiple components.",
"Additional emission includes He1, NaD, Fe2, Ca2 with the same profile shape as the H$\\alpha $ and O1 8446 is present.",
"Outflow lines of [O1], [N2], and [S2] are also present, along with [Fe2]."
],
[
"EPIC 205061092 / 2MASS J16145178-1935402",
"There is no previous literature on this source.",
"Our HIRES data indicate a spectral type of M5-M6 with lithium at $W_{Li} = 0.59 A$ .",
"Strong H$\\alpha $ emission, as indicated in Table 1, is present along with He1, but no other emission lines."
],
[
"EPIC 205088645 / 2MASS J16111237-1927374",
"[122] first identified this star as an M5 member of the USco association, based on lithium absorption and broad H$\\alpha $ emission (-50Å).",
"[26] assigned it a membership probability of 99.9% based on the USNO-B proper motion.",
"[101] found a slightly later spectral type of M6 based on low-resolution spectra, and similarly broad H$\\alpha $ emission.",
"The object displays an infrared excess confirmed by WISE to come from a full disk [99]."
],
[
"EPIC 205156547 / 2MASS J16121242-1907191",
"There is no previous literature on this source.",
"Our spectrum indicates a spectral type of M5-M6 with lithium at $W_{Li} = 0.56 A$ .",
"Moderate H$\\alpha $ emission, as indicated in Table 1, is apparent with with an asymmetric extension on the blue side of the profile.",
"There is also He1 but no other emission lines."
]
] | 1612.05599 |
[
[
"A Complete Characterization of Pretty Good State Transfer on Paths"
],
[
"Abstract We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian.",
"If $n$ is the length of the path, and the vertices are indexed by the positive integers from 1 to $n$, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices $a$ and $b$ are that (a) $a + b = n + 1$, (b) $n + 1$ has at most one odd non-trivial divisor, and (c) if $n = 2^t r - 1$, for $r$ odd and $r \\neq 1$, then $a$ is a multiple of $2^{t - 1}$."
],
[
"Introduction and Preliminaries",
"Many quantum algorithms may be modelled as a quantum process occurring on a graph.",
"In [1], Childs shows that any quantum computation can be encoded as a quantum walk in some graph and thus quantum walks can be regarded as a quantum computation primitive.",
"The protocol for quantum communication through unmeasured and unmodulated spin chains was presented by Bose [2], and led to the interpretation of quantum channels implemented by spin chains as wires for transmission of states.",
"We model such a spin chain of $n$ interacting qubits by the graph of a path of $n$ vertices, denoted $P_n$ , with the vertices labelled from 1 to $n$ corresponding to qubits and the edges $\\lbrace i, i + 1\\rbrace $ , $1 \\le i < n$ corresponding to their interactions.",
"These interactions are defined by a time-independent Hamiltonian; that is, a symmetric matrix that acts on the Hilbert space of dimension $2^n$ .",
"We are concerned here with the $XY$ -Hamiltonian, whose action on the 1-excitation subspaces is equivalent to the action of the 01-symmetric adjacency matrix of $P_n$ on $n$ , and whose process evolves according to the transition matrix $U(t) := \\exp (itA)$ .",
"To be more specific, we consider the Hamiltonian $ H = \\sum _{a = 1}^{n - 1} \\sigma ^x_{a \\ a + 1}\\sigma ^y_{a \\ a + 1}, $ where $\\sigma ^x_{a \\ a + 1}$ and $\\sigma ^y_{a \\ a + 1}$ are the operators that apply the Pauli matrices $\\sigma ^x$ and $\\sigma ^y$ at the qubits located at vertices $a$ and $a + 1$ , and act as the identity at all other qubits.",
"The sum is over all pairs of vertices that are edges of the underlying graph.",
"Consequently, if $|u\\rangle $ stands for the system state in which the qubit at vertex $u$ is at $|1\\rangle $ and all others are at $|0\\rangle $ , then $ H |a\\rangle = {\\left\\lbrace \\begin{array}{ll}| 2 \\rangle , & a = 1 \\\\| n - 1 \\rangle , & a = n \\\\| a - 1 \\rangle + | a + 1 \\rangle , & a \\notin \\lbrace 1, n\\rbrace .\\end{array}\\right.}",
"$ Because of this, the action of $H$ on the set $\\lbrace |a\\rangle : a \\in V(G)\\rbrace $ is equivalent to the action of the adjacency matrix $A(P_n)$ on the canonical basis of $n$ .",
"In other words, $H$ can be block diagonalized, and one of its blocks is equal to $A(P_n)$ .",
"The subspace spanned by $\\lbrace |a\\rangle : a \\in V(G)\\rbrace $ is the 1-excitation subspace of the Hilbert space, and the quantum dynamics restricted to this subspace corresponds to the scenario where one qubit, say the one at $a$ , is initialized at $|1\\rangle $ and all others at $|0\\rangle $ .",
"Due to Schrödinger's equation $ |\\langle a |\\exp (t H) | b \\rangle |^2 $ indicates the probability that the state $|1\\rangle $ is measured at $b$ after time $t$ .",
"We are concerned solely with the 1-excitation subspace.",
"Let $_u$ denote the vector of the canonical basis of ${n}$ that is 1 at the entry corresponding to vertex $u$ at the ordering of the rows and columns of $A(G)$ .",
"From the remarks above, if the system is initialized with state $|1\\rangle $ at vertex $a$ and all others at $|0\\rangle $ , it follows that $ |_a^T \\exp (t A) _b|^2 $ indicates the probability that the state $|1\\rangle $ is measured at $b$ after time $t$ .",
"One of the major goals of quantum communication on spin chains is to transfer a state with high fidelity.",
"At the maximum fidelity of 1, we say we have achieved perfect state transfer.",
"Analogously, we say we have perfect state transfer between vertices $a$ and $b$ if there exists a time $\\tau $ such that $|| _a^T U(\\tau ) _b|| = 1$ .",
"The concept of perfect state transfer was first introduced by Christandl et al.",
"[3], who also showed perfect state transfer is only possible on spin chains of two or three qubits.",
"However, even without perfect state transfer, the fidelity may be quite high, and the notion of pretty good state transfer was isolated by Godsil [4] as a relaxation of perfect state transfer.",
"Formally, we say there is pretty good state transfer if, for any $\\epsilon > 0$ , there exists a time $\\tau $ such that the fidelity is $1 - \\epsilon $ .",
"Analogously, we say we have pretty good state transfer between vertices $a$ and $b$ if, for any $\\epsilon > 0$ , there exists a time $\\tau $ such that $|| U(\\tau )_{a,b} || > 1 - \\epsilon $ , or equivalently, if for any $\\epsilon > 0$ , there exists a $\\lambda \\in , $ || = 1$, such that $ || U() a - b || < $; this condition is abbreviated to $ U() a b$ for convenience.",
"Godsil et al.~\\cite {GKSS12} demonstrated the following result.$ [5] Pretty good state transfer occurs between the end vertices of $P_n$ if and only if $n = p - 1, 2p - 1$ , where $p$ is a prime, or $n = 2^m - 1$ .",
"Moreover, when pretty good state transfer occurs between the end vertices of $P_n$ , then it occurs between vertices $a$ and $n + 1 - a$ for all $a \\ne (n + 1) /2$ .",
"Banchi et al.",
"[6] showed that pretty good state transfer occurs between the $j$ th and $(n - j +1)$ -th vertices of spin chains with $XYZ$ -Hamiltonian, whose action on the 1-exication subspaces is equivalent to the action of the Laplacian adjacency matrix on $n$ , if the number of vertices is a power of 2, and that the condition is necessary for $j = 1$ , but possibly not for other values of $j$ .",
"Moreover, they present the open question of pretty good state transfer between inner vertices with $XY$ -Hamiltonian.",
"Coutinho, Guo, and van Bommel [7] investigated this question and determined the following infinite family of paths that admit pretty good state transfer between inner vertices but not between the two end vertices.",
"[7] Given any odd prime $p$ and positive integer $t$ , there is pretty good state transfer in $P_{2^t p -1}$ between vertices $a$ and $2^t p -a$ , whenever $a$ is a multiple of $2^{t-1}$ .",
"In this paper, we present necessary and sufficient conditions for pretty good state transfer between any two vertices of a path, demonstrating that the above result is the only family of paths that admit pretty good state transfer between inner vertices but not between the two end vertices.",
"In the next section, we present definitions and preliminary results that will be required to prove this characterization."
],
[
"Preliminaries",
"If $M$ is a symmetric matrix with $d$ distinct eigenvalues $\\theta _1 > \\theta _2 > \\cdots > \\theta _d$ , then the spectral decomposition of $M$ is $ M = \\sum _{j = 1}^d \\theta _j E_j, $ where $E_r$ denotes the idempotent projection onto the eigenspace corresponding to $\\theta _r$ .",
"If $a \\in V(G)$ , then the eigenvalue support of $a$ is the following subset of the eigenvalues: $ \\Theta _a = \\lbrace \\theta _j : E_j _a \\ne 0\\rbrace .",
"$ We say that vertices $a$ and $b$ are strongly cospectral if $E_r _a = \\pm E_r _b$ for all $r$ .",
"The following result is given by Banchi et al. [6].",
"[6] If pretty good state transfer occurs between $a$ and $b$ , then they are strongly cospectral vertices.",
"The spectrum of the adjacency matrix of $P_n$ (see [8] for example) is $ \\theta _j = 2 \\cos \\frac{\\pi j}{n + 1}, \\quad 1 \\le j \\le n, $ and the eigenvector corresponding to $\\theta _j$ is given by $ (\\beta _1, \\beta _2, \\ldots , \\beta _k), \\mbox{ where } \\beta _k = \\sin \\frac{k \\pi j}{n + 1}.",
"$ As stated by Coutinho, Guo, and van Bommel, the following lemma immediately follows.",
"[7] Vertices $a$ and $b$ of $P_n$ are strongly cospectral if and only if $a + b = n + 1$ .",
"Moreover, we observe that when $a + b = n + 1$ , $E_r _a = E_r _b$ when $r$ is odd, and $E_r _a = - E_r _b$ when $r$ is even.",
"We will derive our next result, which gives a sufficient condition to show pretty good state transfer does not occur between a given pair of vertices of $P_n$ , from Kronecker's Theorem, stated below.",
"[Kronecker, see [9]] Let $\\theta _0, \\ldots , \\theta _d$ and $\\zeta _0, \\ldots , \\zeta _d$ be arbitrary real numbers.",
"For an arbitrarily small $\\epsilon $ , the system of inequalities $ | \\theta _r y - \\zeta _r | < \\epsilon \\pmod {2 \\pi }, \\quad (r = 0, \\ldots , d), $ admits a solution for $y$ if and only if, for integers $l_0, \\ldots , l_d$ , if $ \\ell _0 \\theta _0 + \\cdots + \\ell _d \\theta _d = 0, $ then $ \\ell _0 \\zeta _0 + \\cdots + \\ell _d \\zeta _d \\equiv 0 \\pmod {2 \\pi }.",
"$ Let $a$ and $b$ be vertices of $P_n$ such that $a + b = n + 1$ .",
"If there is a set of integers $\\lbrace \\ell _j : \\theta _j \\in \\Theta _a,\\ j \\mbox{ odd}\\rbrace $ such that $ \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ odd}\\end{array}} \\ell _j \\theta _j = 0 \\quad \\mbox{and} \\quad \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ odd}\\end{array}} \\ell _j \\mbox{ is odd} $ and there is a set of integers $\\lbrace \\ell _j : \\theta _j \\in \\Theta _a,\\ j \\mbox{ even}\\rbrace $ such that $ \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ even}\\end{array}} \\ell _j \\theta _j = 0 \\quad \\mbox{and} \\quad \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ even}\\end{array}} \\ell _j \\mbox{ is odd} $ then pretty good state transfer does not occur between vertices $a$ and $b$ .",
"Proof: Suppose pretty good state transfer occurs between vertices $a$ and $b$ .",
"We see that the condition $U(\\tau ) _a \\approx \\lambda _b$ is equivalent to $ e^{i \\theta _r \\tau } E_r _a \\approx \\lambda E_r _b$ for all $r$ .",
"Writing $\\lambda = e^{i \\delta }$ , this condition is equivalent to $\\theta _r \\tau \\approx \\delta + q_r \\pi $ for all $r$ such that $\\theta _r \\in \\Theta _a$ , where $q_r$ is even when $r$ is odd and $q_r$ is odd when $r$ is even.",
"So, we wish to solve the system of inequalities $ |\\theta _r \\tau - (\\delta + \\sigma _r \\pi ) | < \\epsilon \\pmod {2 \\pi }, \\quad (r : \\theta _r \\in \\Theta _a), $ where $\\sigma _r = 0$ when $r$ is odd and $\\sigma _r = 1$ when $r$ is even.",
"Hence, by Kronecker's Theorem, if $ \\sum _{\\theta _r \\in \\Theta _a} \\ell _r \\theta _r = 0, $ then $ \\sum _{\\theta _r \\in \\Theta _a} \\ell _r (\\delta + \\sigma _r \\pi ) \\equiv 0 \\pmod {2 \\pi }.",
"$ Now, since there is a set of integers $\\lbrace \\ell _j : \\theta _j \\in \\Theta _a,\\ j \\mbox{ odd}\\rbrace $ such that $ \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ odd}\\end{array}} \\ell _j \\theta _j = 0 \\quad \\mbox{and} \\quad \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ odd}\\end{array}} \\ell _j \\mbox{ is odd} $ then we must have $ \\delta \\equiv 0 \\pmod {2 \\pi }.",
"$ However, since there is a set of integers $\\lbrace \\ell _j : \\theta _j \\in \\Theta _a,\\ j \\mbox{ even}\\rbrace $ such that $ \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ even}\\end{array}} \\ell _j \\theta _j = 0 \\quad \\mbox{and} \\quad \\sum _{\\begin{array}{c}\\theta _j \\in \\Theta _a \\\\ j \\mbox{ even}\\end{array}} \\ell _j \\mbox{ is odd} $ then we must also have $ \\delta \\equiv \\pi \\pmod {2 \\pi }, $ which is the contradiction completing the proof ."
],
[
"Necessary and Sufficient Conditions",
"We first state the following results about sums of cosines which we will use in the proof of the main theorem.",
"Their proofs are included for completeness.",
"The first result uses the fact that $2 \\cos x = e^{i x} + e^{-i x}$ , and then we sum the resulting geometric series.",
"Let $q$ be an odd integer.",
"Then $ 2 \\sum _{k = 1}^{\\frac{q - 1}{2}} (-1)^k \\cos \\left( \\frac{k \\pi }{q} \\right) + 1 = 0.",
"$ Proof: $& 2 \\sum _{k = 1}^{\\frac{q - 1}{2}} (-1)^k \\cos \\left( \\frac{k \\pi }{q} \\right) + 1 \\\\&= \\sum _{k = 1}^{\\frac{q - 1}{2}} (-1)^k (e^{\\frac{i k \\pi }{q}} + e^{\\frac{- i k \\pi }{q}}) + 1 \\\\&= \\sum _{k = 1}^{\\frac{q - 1}{2}} (-e^{\\frac{i \\pi }{q}})^k + \\sum _{k = 1}^{\\frac{q - 1}{2}} (-e^{-\\frac{i \\pi }{q}})^k + 1 \\\\&= \\frac{-e^{\\frac{i \\pi }{q}} + (-e^{\\frac{i \\pi }{q}})^{\\frac{q + 1}{2}}}{1 + e^{\\frac{i \\pi }{q}}} + \\frac{(-e^{\\frac{i \\pi }{q}})^{-1} + (-e^{\\frac{i \\pi }{q}})^{-\\frac{q + 1}{2}}}{1 + e^{-\\frac{i \\pi }{q}}} + 1 \\\\&= \\frac{-e^{\\frac{i \\pi }{q}} + (-e^{\\frac{i \\pi }{q}})^{\\frac{q + 1}{2}}}{1 + e^{\\frac{i \\pi }{q}}} - \\frac{1+ (-e^{\\frac{i \\pi }{q}})^{-\\frac{q - 1}{2}}}{1 + e^{\\frac{i \\pi }{q}}} + \\frac{1 + e^{\\frac{i \\pi }{q}}}{1 + e^{\\frac{i \\pi }{q}}} \\\\&= \\frac{(-e^{\\frac{i \\pi }{q}})^{\\frac{q + 1}{2}} - (-e^{\\frac{i \\pi }{q}})^{-\\frac{q - 1}{2}}}{1 + e^{\\frac{i \\pi }{q}}} \\\\&= 0 \\square $ The second result makes use of the following sum-product identity: $ \\cos \\alpha + \\cos \\beta = 2 \\cos \\frac{\\alpha + \\beta }{2} \\cos \\frac{\\alpha - \\beta }{2} .",
"$ The final step is an application of the previous lemma.",
"Let $n = km$ , where $m$ is an odd integer, and $0 \\le a < k$ be an integer.",
"Then $ \\sum _{j = 0}^{m - 1} (-1)^j \\cos \\left( \\frac{(a + jk) \\pi }{n} \\right) = 0.",
"$ Proof: $& \\sum _{j = 0}^{m - 1} (-1)^j \\cos \\left( \\frac{(a + jk) \\pi }{n} \\right) \\\\&= (-1)^{\\frac{m - 1}{2}} \\left[ \\cos \\left( \\frac{\\left(a + \\frac{m - 1}{2} k \\right) \\pi }{n} \\right) \\right.",
"\\\\ &\\quad \\left.+ \\sum _{j = 1}^{\\frac{m - 1}{2}} (-1)^j \\left(\\cos \\left( \\frac{\\left(a + (\\frac{m - 1}{2} - j) k \\right) \\pi }{n} \\right) + \\cos \\left( \\frac{\\left(a + (\\frac{m - 1}{2} + j) k \\right) \\pi }{n} \\right) \\right) \\right] \\\\&= (-1)^{\\frac{m - 1}{2}} \\left[ \\cos \\left( \\frac{\\left(a + \\frac{m - 1}{2} k \\right) \\pi }{n} \\right) \\right.",
"\\\\ &\\quad \\left.+ \\sum _{j = 1}^{\\frac{m - 1}{2}} (-1)^j 2 \\cos \\left( \\frac{\\left(a + \\frac{m - 1}{2} k \\right) \\pi }{n} \\right) \\cos \\left( \\frac{j k \\pi }{n} \\right) \\right] \\\\&= (-1)^{\\frac{m - 1}{2}} \\cos \\left( \\frac{\\left(a + \\frac{m - 1}{2} k \\right) \\pi }{n} \\right) \\left( 1 + 2 \\sum _{j = 1}^{\\frac{m - 1}{2}} (-1)^j \\cos \\left( \\frac{j \\pi }{m} \\right) \\right) \\\\&= 0 \\square $ We will now state and prove the main theorem.",
"There is pretty good state transfer on $P_n$ between vertices $a$ and $b$ if and only if $a + b = n + 1$ and either: $n = 2^t - 1$ , where $t$ is a positive integer; or, $n = 2^t p - 1$ , where $t$ is a nonnegative integer and $p$ is an odd prime, and $a$ is a multiple of $2^{t - 1}$ .",
"Proof: The sufficiency of the conditions is given by Theorem and Theorem .",
"It remains to show that the conditions are necessary.",
"The necessity of the first condition follows from Lemma and Lemma .",
"Henceforth, we need only consider the possibility of pretty good state transfer between vertices $a$ and $n + 1 - a$ .",
"Suppose first that there is pretty good state transfer on $P_n$ between vertices $a$ and $n + 1 - a$ when $n = 2^t r - 1$ , where $t$ is a positive integer and $r$ is an odd composite number.",
"Let $p$ be a prime factor of $r$ .",
"If $p \\mid \\frac{2^t r}{\\gcd (a, 2^t r)}$ , then if $\\theta _k \\notin \\Theta _a$ , then $k$ is a multiple of $p$ .",
"But then, for $c \\in \\lbrace 1, 2\\rbrace $ , we have $ \\sum _{i = 0}^{r/p - 1} (-1)^i \\theta _{c + i 2^t p} = \\sum _{i = 0}^{r/p - 1} (-1)^i \\cos \\left( \\frac{(c + i 2^t p) \\pi }{n + 1} \\right) = 0 $ by Lemma .",
"Hence, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , a contradiction.",
"Hence, we can now assume that $r \\mid a$ .",
"Since it follows from condition (a) that $a \\ne 2^{t-1} r$ , we have that $t \\ge 2$ and $4 \\mid \\frac{2^t r}{\\gcd (a, 2^t r)}$ , so if $\\theta _k \\notin \\Theta _a$ , then $k$ is a multiple of 4.",
"But then, for $c \\in \\lbrace 1, 2\\rbrace $ , we have $ \\sum _{i = 0}^{r - 1} (-1)^i \\theta _{c + i 2^t} = \\sum _{i = 0}^{r - 1} (-1)^i \\cos \\left( \\frac{(c + i 2^t) \\pi }{n + 1} \\right) = 0 $ by Lemma .",
"Hence, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , a contradiction.",
"Now suppose that there is pretty good state transfer on $P_n$ between vertices $a$ and $n + 1 - a$ when $n = r - 1$ , where $r$ is an odd composite number.",
"Let $p$ be a prime factor of $r$ .",
"If $p \\mid \\frac{2^t r}{\\gcd (a, 2^t r)}$ , then if $\\theta _k \\notin \\Theta _a$ , then $k$ is a multiple of $p$ .",
"But then, for $c \\in \\lbrace 1, 2\\rbrace $ , we have $&\\sum _{i = 0}^{\\frac{1}{2} (r/p - 1)} \\theta _{c + i 2 p} + \\sum _{i = 0}^{\\frac{1}{2} (r/p - 1) - 1} \\theta _{n + 1 - (c + p + i 2 p)} \\\\&= \\sum _{i = 0}^{r/p - 1} (-1)^i \\theta _{c + i p} = \\sum _{i = 0}^{r/p - 1} (-1)^i \\cos \\left( \\frac{(c + i p) \\pi }{n + 1} \\right) = 0$ by Lemma .",
"Hence, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , a contradiction.",
"Hence, we have shown the necessity of the conditions on $n$ .",
"It remains to show the necessity of the conditions on $a$ when $n = 2^t p - 1$ , where $t$ is a positive integer and $p$ is an odd prime.",
"Suppose $a$ is not a multiple of $2^{t - 1}$ .",
"Then again we have that if $\\theta _k \\notin \\Theta _a$ , then $k$ is a multiple of 4.",
"So as above, it follows from Lemma that $P_n$ does not have pretty good state transfer between $a$ and $n + 1 - a$ , which is the contradiction completing the proof .",
"Acknowledgements The author thanks Chris Godsil for his guidance and support and Gabriel Coutinho for his introduction to this problem.",
"References"
]
] | 1612.05603 |
[
[
"Occulting Light Concentrators in Liquid Scintillator Neutrino Detectors"
],
[
"Abstract The experimental efforts characterizing the era of precision neutrino physics revolve around collecting high-statistics neutrino samples and attaining an excellent energy and position resolution.",
"Next generation liquid-based neutrino detectors, such as JUNO, HyperKamiokande, etc, share the use of a large target mass, and the need of pushing light collection to the edge for maximal calorimetric information.",
"Achieving high light collection implies considerable costs, especially when considering detector masses of several kt.",
"A traditional strategy to maximize the effective photo-coverage with the minimum number of PMTs relies on Light Concentrators (LC), such as Winston Cones.",
"In this paper, the authors introduce a novel concept called Occulting Light Concentrators (OLC), whereby a traditional LC gets tailored to a conventional PMT, by taking into account its single-photoelectron collection efficiency profile and thus occulting the worst performing portion of the photocathode.",
"Thus, the OLC shape optimization takes into account not only the optical interface of the PMT, but also the maximization of the PMT detection performances.",
"The light collection uniformity across the detector is another advantage of the OLC system.",
"By considering the case of JUNO, we will show OLC capabilities in terms of light collection and energy resolution."
],
[
"Introduction",
"The traditional goal of non-imaging light concentrators (LCs) has always been to maximize the collected light (see for instance [1] and [2]).",
"This is particularly true for detectors with a limited ($\\sim $ 30%) geometrical coverage, such as SNO, CTF, Borexino, etc.",
"Moreover, in a liquid based detector the amount of collected light depends on the vertex position, due to light absorption and scattering.",
"By limiting the field of view of the PMT, LCs reduce the amount of collected light for events at large radii, thus helping to increase the energy reconstruction uniformity.",
"In detectors as JUNO and RENO-50, planning very high geometrical coverages (around 70%), LC remains a valid tool also against another non-homogeneity effect.",
"Actually, in large PMTs the photoelectron (p.e.)",
"Detection Efficiency (DE) is uneven throughout their surface - the PMT edge being the worst-performing region.",
"The dispersive effects induced by such a non-homogeneous DE affect the total number of collected p.e., worsening significantly the detector-level energy resolution when compared to the intrinsic stochastic component.",
"Standard LCs can become Occulting Light Concentrator (OLC) with the aim of reducing PMT-related dispersive effects while maximizing light collection and making the detector response more uniform."
],
[
"OLCs in a gigantic (15 kt) liquid scintillator detector",
"An ideal two-dimensional Compound Tangential Concentrator (CTC) is designed in order to transmit to the photocathode all the light incident at the entrance aperture with an angle $<\\theta _i$ and to avoid the transmission of all the light with incident angle $>\\theta _i$ [1].",
"Thus, $\\theta _i$ is a CTC construction parameter and defines the maximum accepted incident photon angle.",
"In 3D, where the CTC is obtained by rotating the 2D profile around the symmetry axis, the perfect performances in light transmission are slightly degraded for angles $<\\theta _i$ .",
"We call $\\theta ^*$ the incident angle where this degradation starts and $\\theta _{FV}$ the maximal photon incident angle at the PMT level (see Figure REF ).",
"We consider two extreme configurations: OLC$_1$ , obtained by requiring $\\theta _i = \\theta _{FV}$ and OLC$_2$ , by requiring $\\theta ^* = \\theta _{FV}$ .",
"We simulate a detector Fiducial Volume (FV) as a finite spherical light source of 16 m radius and the PMT as a semi-sphere of 25.4 cm of radius, 4 m away from the edge of the FV.",
"Figure: Left: Geometry of the detector Fiducial Volume and of the PMT.",
"θ FV \\theta _{FV} corresponds to the maximal photon incident angle at the PMT level.",
"Center: 2D profile of OLC 1 _{1} (orange) and OLC 2 _2 (green), as obtained by requiring the maximum radius of the LC not to exceed the PMT radius via the Tangent Ray Method.Right: Acceptance of the optical module obtained with a 3D OLC 1 _{1} (orange) and OLC 2 _2 (green).",
"In our case, θ FV \\theta _{FV} is about 55 ∘ ^{\\circ }.Figure: Left: Collection Efficiency Profiles as a function of ρ PMT \\rho _{PMT}.",
"Center: Light Yield for different CE profiles.",
"Right: Detector Resolution for different CE profiles.Last panel of Figure REF shows the acceptances of the two 3D OLC designs: only OLC$_2$ has been configured in order to maintain the maximal acceptance up to $\\theta _{FV}$ .",
"We consider 12 possible CE profiles as a function of the distance from the PMT axis ($\\rho _{PMT}$ ), as shown in Figure REF (left).",
"A non-flat CE profile acts on the detector resolution by reducing the number of detected photons (increasing the stochastic resolution), as clearly shown by blue dots in central panel of Figure REF .",
"Moreover, it introduces a dependence on the event vertex position, resulting in an additional non-stochastic resolution term, as show in the right panel of Figure REF (cf.",
"empty and filled dots).",
"OLC focuses the light meant to hit the external crown of the photocathode (large $\\rho $ , close to the PMT equator) towards the central region of the photocathode, effectively mitigating the dispersive behaviour of the CE profiles.",
"This is shown again in the central and right panel of Figure REF for OLC$_1$ (orange) and OLC$_2$ (green).",
"Plots in Figure REF are obtained by applying a radial non-uniformity correction; we assume either perfect knowledge of CE profiles (filled dots) or flat CE (empty dots).",
"For CE profiles up to number 7, OLCs clearly improve the energy resolution.",
"Results are comparable with those obtained by full knowledge of CE profile.",
"Thanks to its tallest shape, OLC$_1$ not only collects more light in the case of a flat CE, but it also manages to have the best light collection across all the CE models.",
"However, in terms of energy resolution, OLC$_2$ has better performance than OLC$_1$ also for extreme CE profiles, due to its peculiar acceptance curve."
],
[
"OLCs in JUNO",
"The largest (20 kt) liquid scintillator detector currently under construction, JUNO [3], will be equipped with about 18 k 20” PMTs, with the aim to reach about 75% of geometrical coverage.",
"The main goal of JUNO is to determine the neutrino Mass Hierarchy by measuring the energy spectrum of $\\overline{\\nu }_e$ coming from nuclear reactors at 52 km far from the detector.",
"To achieve this goal, an unprecedented 3% energy resolution at 1 MeV is required.",
"The JUNO collaboration is presently considering to implement OLCs in the detector design, in order to: (1) occult the PMT edge, where the CE decreases; (2) improve the uniformity in the light collection across the detector; (3) recover good light in case the total number of PMTs has to be reduced.",
"The CE profile has been measured on a sub-sample of JUNO PMTs.",
"The analytical behavior as a function of $\\rho _{PMT}$ is reported in Figure REF (left), showing an almost constant CE up to $\\rho _{PMT}\\sim $ 24 cm.",
"The OLC profile has thus been tailored on JUNO PMT, via the CTC method, in order to occult the photocathode area at $\\rho _{PMT}>$ 24 cm, for R$_{FV}=$ 17.2 m, R$_{buffer}=$ 19.5 m and by asking $\\theta _i = \\theta _{FV}$ .",
"Since the complete CTC shape does not fit JUNO geometrical constraints, it must be cut in order to avoid overlapping with near-by OLCs, as shown in Figure REF (middle).",
"The reference clearance between two PMTs is 25 mm.",
"Due to mechanical constraints, larger clearances are begin considered.",
"To increase the clearance also means to reduce the total number of PMTs.",
"As shown in Figure REF (right), OLCs help to recover light when increasing the clearance.",
"Moreover, the light collection becomes more uniform across the detector volume.",
"Thanks to their occulting action and to the uniformity in the light detection, also the energy resolution improves.",
"In particular, when requiring a larger clearance, OLC allow to recover the energy resolution level of the standard configuration (no-OLC and 25 mm clearance), thus avoiding any degradation in the energy reconstruction."
]
] | 1612.05444 |
[
[
"Semiclassical Formulation of Gottesman-Knill and Universal Quantum\n Computation"
],
[
"Abstract We give a path integral formulation of the time evolution of qudits of odd dimension.",
"This allows us to consider semiclassical evolution of discrete systems in terms of an expansion of the propagator in powers of $\\hbar$.",
"The largest power of $\\hbar$ required to describe the evolution is a traditional measure of classicality.",
"We show that the action of the Clifford operators on stabilizer states can be fully described by a single contribution of a path integral truncated at order $\\hbar^0$ and so are \"classical,\" just like propagation of Gaussians under harmonic Hamiltonians in the continuous case.",
"Such operations have no dependence on phase or quantum interference.",
"Conversely, we show that supplementing the Clifford group with gates necessary for universal quantum computation results in a propagator consisting of a finite number of semiclassical path integral contributions truncated at order $\\hbar^1$ , a number that nevertheless scales exponentially with the number of qudits.",
"The same sum in continuous systems has an infinite number of terms at order $\\hbar^1$."
],
[
"Introduction",
"The study of contextuality in quantum information has led to progress in our understanding of the Wigner function for discrete systems.",
"Using Wootters' original derivation of discrete Wigner functions [1], Eisert [2], Gross [3], and Emerson [4] have pushed forward a new perspective on the quantum analysis of states and operators in finite Hilbert spaces by considering their quasiprobability representation on discrete phase space.",
"Most notably, the positivity of such representations has been shown to be equivalent to non-contextuality, a notion of classicality [4], [5].",
"Quantum gates and states that exhibit these features are the stabilizer states and Clifford operations used in quantum error correction and stabilizer codes.",
"The non-contextuality of stabilizer states and Clifford operations explains why they are amenable to efficient classical simulation [6], [7].",
"This progress raises the question of how these discrete techniques are connected to prior established methods for simulating quantum mechanics in phase space.",
"A particularly relevant method is trajectory-based semiclassical propagation, which has been widely used in the continuous context.",
"Perhaps, when applied to the discrete case, semiclassical propagators can lend their physical intuition to outstanding problems in quantum information.",
"Conversely, concepts from quantum information may serve to illuminate the comparatively older field of continuous semiclassics.",
"Quantum information attempts to classify the “quantumness” of a system by the presence or absence of various quantum resources.",
"Semiclassical analysis proceeds by successive approximation using $\\hbar $ as a small parameter, where the power of $\\hbar $ required is a measure of “quantumness”.",
"Can these two views of quantum vs. classical be related?",
"In the current paper, we build a bridge from the continuous semiclassical world to the discrete world and examine the classical-quantum characteristics of discrete quantum gates found in circuit models and their stabilizer formalism.",
"Stabilizer states are eigenvalue one eigenvectors of a commuting set of operators making up a group which does not contain $-\\mathbb {I}$ .",
"The set of stabilizer states is preserved by elements of the Clifford group, which is the normalizer of the Pauli group, and can be simulated very efficiently.",
"More precisely, by the Gottesman-Knill Theorem, for $n$ qubits, a quantum circuit of a Clifford gate can be simulated using $\\mathcal {O}(n)$ operations on a classical computer.",
"Measurements require $\\mathcal {O}(n^2)$ operations with $\\mathcal {O}(n^2)$ bits of storage [6], [7].",
"The reason that stabilizer evolution by Clifford gates can be efficiently simulated classically has been explained in various ways.",
"For instance, as already mentioned, stabilizer states have been shown to be non-contextual in qudit [4] and rebit [8] systems.",
"The obstacle to proving this for qubits is that qubit systems possess state-independent contextuality [9].",
"Of course, we know how to simulate qubit stabilizer states and Clifford operations efficiently by the Gottesmann-Knill theorem [7].",
"For recent progress relating non-contextuality to classical simulatability for qubits we refer the reader to [10].",
"It has also been shown for dimensions greater than two that a state of a discrete system is a stabilizer state if and only if its appropriately defined discrete Wigner function is non-negative [3].",
"Therefore, when acted on by positive-definite operators, it can be considered as a proper positive-definite (classical) distribution.",
"Here, we instead relate the concept of efficient classical simulation to the power of $\\hbar $ that a path integral treatment must be expanded to in order to describe the quantum evolution of interest.",
"It is well known that Gaussian propagation in continuous systems under harmonic Hamiltonians can be described with a single contribution from the path integral truncated at order $\\hbar ^0$ [11].",
"We show that the corresponding case in discrete systems exists.",
"In the discrete case, stabilizer states take the place of Gaussians and harmonic Hamiltonians that additionally preserve the discrete phase space take the place of the general continuous harmonic Hamiltonians.",
"In the discrete case we will only consider $d$ -dimensional systems for odd $d$ since their center representation (or Weyl formalism) is far simpler.",
"As a consequence, we will show that operations with Clifford gates on stabilizer states can be treated by a path integral independent of the magnitude of $\\hbar $ and are thus fundamentally classical.",
"Such operations have no dependence on phase or quantum interference.",
"This can be viewed as a restatement of the Gottesman-Knill theorem in terms of powers of $\\hbar $ .",
"We also consider more general propagation for discrete quantum systems.",
"Quantum propagation in continuous systems can be treated by a sum consisting of an infinite number of contributions from the path integral truncated at order $\\hbar ^1$ .",
"In discrete systems, we show that the corresponding sum consists of a finite number of terms, albeit one that scales exponentially with the number of qudits.",
"This work also answers a question posed by the recent work of Penney et al.",
"that explored a “sum-over-paths” expression for Clifford circuits in terms of symplectomorphisms on phase-space and associated generating actions.",
"Penney et al.",
"raised the question of how to relate the dynamics of the Wigner representation of (stabilizer) states to the dynamics which are the solutions of the discrete Euler-Lagrange equations for an associated functional [12].",
"By relying on the well-established center-chord (or Wigner-Weyl-Moyal) formalism in continuous [13] and discrete systems [14], we show how the dynamics of Wigner representations are governed by such solutions related to a “center generating” function and that these solutions are harmonic and classical in nature.",
"We begin by giving an overview of the center-chord representation in continuous systems in Section .",
"Then, Section introduces the expansion of the path integral in powers of $\\hbar $ .",
"This leads us to show what “classical” simulability of states in the continuous case corresponds to and to what higher order of $\\hbar $ an expansion is necessary to treat any quantum operator.",
"Section then introduces the discrete variable case and defines its corresponding conjugate position and momentum operators.",
"The path integral in discrete systems is then introduced in Section and, in Section , we define the Clifford group and stabilizer states.",
"We prove that stabilizer state propagation within the Clifford group is captured fully up to order $\\hbar ^0$ and so is efficiently simulable classically.",
"Section shows that extending the Clifford group to a universal gate set necessitates an expansion of the semiclassical propagator to a finite sum at order $\\hbar ^1$ .",
"Finally, we close the paper with some discussion and directions for future work in Section ."
],
[
"Center-Chord Representation in Continuous Systems",
"We define position operators $\\hat{q}$ , $\\hat{q} \\left|q^{\\prime }\\right\\rangle = q^{\\prime } \\left|q^{\\prime }\\right\\rangle $ , and momentum operators $\\hat{p}$ as their Fourier transform, $\\hat{p} = \\hat{\\mathcal {F}}^\\dagger \\hat{q} \\hat{\\mathcal {F}}$ , where $\\hat{F} = h^{\\frac{n}{2}} \\int ^\\infty _{-\\infty } \\mbox{d} p \\int ^\\infty _{-\\infty } \\mbox{d} q \\exp \\left(\\frac{2 \\pi i}{\\hbar } p \\cdot q \\right) \\left|p\\right\\rangle \\left\\langle q\\right|.$ Since $[\\hat{q}, \\hat{p}] = i \\hbar $ , these operators produce a particularly simple Lie algebra and are the generators of a Lie group.",
"In this Lie group we can define the “boost” operator: $\\hat{Z}^{\\delta p} \\left|q^{\\prime }\\right\\rangle = e^{\\frac{i}{\\hbar } \\hat{q} \\delta p} \\left|q^{\\prime }\\right\\rangle = e^{\\frac{i}{\\hbar } q^{\\prime } \\delta p} \\left|q^{\\prime }\\right\\rangle ,$ and the “shift” operator: $\\hat{X}^{\\delta q} \\left|q^{\\prime }\\right\\rangle = e^{-\\frac{i}{\\hbar } \\hat{p} \\delta q} \\left|q^{\\prime }\\right\\rangle = \\left|q^{\\prime } + \\delta q\\right\\rangle .$ Using the canonical commutation relation and $e^{\\hat{A} + \\hat{B}} = e^{\\hat{A}} e^{\\hat{B}} e^{-\\frac{1}{2}[\\hat{A}, \\hat{B}]}$ if $[\\hat{A}, \\hat{B}]$ is a constant, it follows that $\\hat{Z} \\hat{X} = e^{\\frac{i}{\\hbar }} \\hat{X} \\hat{Z}.$ This is known as the Weyl relation and shows that the product of a shift and a boost (a generalized translation) in phase space is only unique up to a phase governed by $\\hbar $ .",
"We proceed to introduce the chord representation of operators and states [13].",
"The generalized phase space translation operator (often called the Weyl operator) is defined as a product of the shift and boost: $\\hat{T}(\\xi _p, \\xi _q) = e^{-\\frac{i}{2 \\hbar } \\xi _p \\cdot \\xi _q} \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q},$ where $\\xi \\equiv (\\xi _p, \\xi _q) \\in \\mathbb {R}^{2n}$ define the chord phase space.",
"$\\hat{T}(\\xi _p, \\xi _q)$ is a translation by the chord $\\xi $ in phase space.",
"This can be seen by examining its effect on position and momentum states: $\\hat{T}(\\xi _p, \\xi _q) \\left|q\\right\\rangle = e^{\\frac{i}{\\hbar } \\left(q + \\frac{\\xi _q}{2}\\right) \\cdot \\xi _p} \\left|q+\\xi _q\\right\\rangle ,$ and $\\hat{T}(\\xi _p, \\xi _q) \\left|p\\right\\rangle = e^{-\\frac{i}{\\hbar } \\left(p + \\frac{\\xi _p}{2}\\right) \\cdot \\xi _q} \\left|p+\\xi _p\\right\\rangle ,$ which are shown in Fig.",
"REF .",
"Changing the order of shifts $\\hat{X}$ and boosts $\\hat{Z}$ changes the phase of the translation in phase space by $\\xi $ , as given by the Weyl relation above (Eq.",
"REF ).",
"An operator $\\hat{A}$ can be expressed as a linear combination of these translations: $\\hat{A} = \\int ^\\infty _{-\\infty } \\mbox{d} \\xi _p \\int ^\\infty _{-\\infty } \\mbox{d} \\xi _q \\, A_\\xi (\\xi _p, \\xi _q) \\hat{T}(\\xi _p, \\xi _q),$ where the weights are: $A_\\xi (\\xi _p, \\xi _q) = \\operatorname{Tr}\\left( \\hat{T}(\\xi _p, \\xi _q)^\\dagger \\hat{A} \\right).$ These weights give the chord representation of $\\hat{A}$ .",
"Figure: Translation of a) a position state and b) a momentum state along the chord (ξ p ,ξ q )(\\xi _p, \\xi _q) in phase space.The Weyl function, or center representation, is dual to the chord representation.",
"It is defined in terms of reflections instead of translations.",
"We can define the reflection operator $\\hat{R}$ as the symplectic Fourier transform of the translation operator: $&&\\hat{R}(x_p, x_q) = \\left(2 \\pi \\hbar \\right)^{-n} \\int ^\\infty _{-\\infty } \\mbox{d} \\xi e^{\\frac{i}{\\hbar } {\\xi }^T \\mathcal {J} {x} } \\hat{T}(\\xi )$ where $x \\equiv \\left(x_p, x_q\\right) \\in \\mathbb {R}^{2n}$ are a continuous set of Weyl phase space points or centers and $\\mathcal {J}$ is the symplectic matrix $\\mathcal {J} = \\left( \\begin{array}{cc} 0 & -\\mathbb {I}_{n}\\\\ \\mathbb {I}_{n} & 0 \\end{array}\\right),$ for $\\mathbb {I}_n$ the $n$ -dimensional identity.",
"The association of this operator with reflection can be seen by examining its effect on position and momentum states: $\\hat{R}(x_p, x_q) \\left|q\\right\\rangle = e^{\\frac{i}{\\hbar } 2 (x_q - q) \\cdot x_p} \\left|2x_q-q\\right\\rangle ,$ and $\\hat{R}(x_p, x_q) \\left|p\\right\\rangle = e^{-\\frac{i}{\\hbar } 2 (x_p - p) \\cdot x_q} \\left|2x_p-p\\right\\rangle ,$ which are sketched in Fig.",
"REF .",
"It is thus evident that $\\hat{R}(x_p, x_q)$ reflects the phase space around $x$ .",
"Note that while we refer to reflections in the symplectic sense here, and in the rest of the paper; Eqs.",
"REF and REF show that they are in fact “an inversion around $x$ ” in every two-plane of conjugate ${x_p}_i$ and ${x_q}_i$ .",
"However, we will keep to the established nomenclature [13].",
"Figure: Reflection of a) a position state and b) a momentum state across the center (x p ,x q )(x_p, x_q) in phase space.An operator $\\hat{A}$ can now be expressed as a linear combination of reflections: $\\hat{A} = \\left(2 \\pi \\hbar \\right)^{-n} \\int ^\\infty _{-\\infty } \\mbox{d} x_p \\int ^\\infty _{-\\infty } \\mbox{d} x_q \\, A_x(x_p, x_q) \\hat{R}( x_p, x_q ),$ where $A_x(x_p, x_q) = \\operatorname{Tr}\\left( \\hat{R}(x_p, x_q)^\\dagger \\hat{A} \\right),$ and is called the center representation of $\\hat{A}$ .",
"This representation is of particular interest to us because we can rewrite the components $A_x$ for unitary transformations $\\hat{A}$ as: $A_x(x_p, x_q) = e^{\\frac{i}{\\hbar } S(x_p, x_q)},$ where $S(x_p, x_q)$ is equivalent to the action the transformation $A_x$ produces in Weyl phase space in terms of reflections around centers $x$ [13].",
"Thus, $S$ is also called the “center generating” function.",
"For a pure state $\\left|\\Psi \\right\\rangle $ , the Wigner function given by Eq.",
"REF simplifies to: $&& {\\Psi }_x(x_p, x_q) = \\left(2 \\pi \\hbar \\right)^{-n}\\\\&& \\int ^\\infty _{-\\infty } \\mbox{d} \\xi _q \\, \\Psi \\left( x_q + \\frac{\\xi _q }{2} \\right) {\\Psi ^*} \\left( x_q - \\frac{\\xi _q}{2} \\right) e^{-\\frac{i}{\\hbar } \\xi _q \\cdot x_p}.",
"\\nonumber $ The center representation for quantum states immediately yields the well-known Wigner function for continuous systems.",
"The chord representation is the symplectic Fourier transform of the Wigner function.",
"The center and chord representations are dual to each other, and are the Wigner and Wigner characteristic functions res[ectively.",
"Identifying the Wigner functions with the center representation, and the center representation as dual to the chord representation motivates the development of both center (Wigner) and chord representations for discrete systems in Section ."
],
[
"Path Integral Propagation in continuous Systems",
"Propagation from one quantum state to another can be expressed in terms of the path integral formalism of the quantum propagator.",
"For one degree of freedom, with an initial position $q$ and final position $q^{\\prime }$ , evolving under the Hamiltonian $H$ for time $t$ , the propagator is $\\left\\langle q|e^{-i H t/\\hbar }|q^{\\prime }\\right\\rangle = \\int \\mathcal {D}[q_t] \\, \\exp \\left(\\frac{i}{\\hbar } G[q_t]\\right)$ where $G[q_t]$ is the action of the trajectory $q_t$ , which starts at $q$ and ends at $q^{\\prime }$ a time $t$ later [15], [16].",
"Eq.",
"REF can be reexpressed as a variational expansion around the set of classical trajectories (a set of measure zero) that start at $q$ and end at $q^{\\prime }$ a time $t$ later.",
"This is an expansion in powers of $\\hbar $ : $&& \\left\\langle q^{\\prime }|e^{-\\frac{i}{\\hbar }Ht}|q\\right\\rangle =\\\\&& \\sum _j^\\text{cl.",
"paths} \\int \\mathcal {D}[q_{tj}] \\, e^{\\frac{i}{\\hbar } \\left( G[q_{tj}] + \\delta G[q_{tj}] + \\frac{1}{2} \\delta ^2 G[q_{tj}] + \\ldots \\right) }, \\nonumber $ where $\\delta G[q_{tj}]$ denotes a functional variation of the paths $q_{tj}$ and for classical paths $\\delta G[q_{tj}] = 0$ (For further details we refer the reader to Section 10.3 of [17]).",
"Terminating Eq.",
"REF to first order in $\\hbar $ produces the position state representation of the van Vleck-Morette-Gutzwiller propagator [18], [19], [20]: $&& \\left\\langle q^{\\prime }|e^{-\\frac{i}{\\hbar }Ht}|q\\right\\rangle = \\\\&& \\sum _j\\left( \\frac{- \\frac{\\partial ^2 G_{jt}(q,q^{\\prime })}{\\partial q \\partial q^{\\prime }}}{2 \\pi i \\hbar } \\right)^{1/2} e^{i \\frac{G_{jt}(q,q^{\\prime })}{\\hbar }} + \\mathcal {O}(\\hbar ^2).\\nonumber $ where the sum is over all classical paths that satisfy the boundary conditions.",
"In the center representation, for $n$ degrees of freedom, the semiclassical propagator $U_t(x_p, x_q)$ becomes [13]: $&&U_t(x_p, x_q) = \\\\&& \\sum _j \\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} e^{\\frac{i}{\\hbar } S_{tj}(x_p, x_q)} + \\mathcal {O}(\\hbar ^2),\\nonumber $ where $S_{tj}(x_p, x_q)$ is the center generating function (or action) for the center $x = (x_p, x_q)\\equiv \\frac{1}{2}\\left[(p, q)+(p^{\\prime }, q^{\\prime })\\right]$ .",
"In general this is an underdetermined system of equations and there are an infinite number of classical trajectories that satisfy these conditions.",
"The accuracy of adding them up as part of this semiclassical approximation is determined by how separated these trajectories are with respect to $\\hbar $ —the saddle-point condition for convergence of the method of steepest descents.",
"However, some Hamiltonians exhibit a single saddle point contribution and are thus exact at order $\\hbar ^1$ See Ehrenfest's Theorem, e.g.",
"in [42]: Lemma 1 There is only one classical trajectory $(p, q)\\underset{t}{\\rightarrow } (p^{\\prime }, q^{\\prime })$ that satisfies the boundary conditions $(x_p, x_q) = \\frac{1}{2}\\left[(p, q)+(p^{\\prime }, q^{\\prime })\\right]$ and $t$ under Hamiltonians that are harmonic in $p$ and $q$ .",
"Proof For a quadratic Hamiltonian, the diagonalized equations of motion for $n$ -dimensional $(p^{\\prime }, q^{\\prime })$ are of the form: $p^{\\prime }_i &=& \\alpha (t)_i p_i + \\beta (t)_i q_i + \\gamma (t)_i,\\\\q^{\\prime }_i &=& \\delta (t)_i p_i + \\epsilon (t)_i q_i + \\eta (t)_i,$ for $i \\in \\lbrace 1, 2, \\ldots , n\\rbrace $ .",
"Since $t$ is known and $(p, q)$ can be written in terms of $(p^{\\prime }, q^{\\prime })$ by using $(x_p, x_q)$ , this brings the total number of linear equations to $2n$ with $2n$ unknowns and so there exists one unique solution.$\\Box $ Since the equations of motion for a harmonic Hamiltonian are linear, we can write their solutions as: $\\left( \\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array} \\right) = \\mathcal {M}_t \\left[ \\left( \\begin{array}{c}p\\\\ q \\end{array}\\right) + \\frac{1}{2} \\alpha _t \\right] + \\frac{1}{2} \\alpha _t,$ where $\\mathcal {\\alpha }_t$ is an $n$ -vector and $\\mathcal {M}_t$ is an $n\\times n$ symplectic matrix, both with entries in $\\mathbb {R}$ .",
"In this case, the center generating function $S_t(x_p, x_q)$ is also quadratic, in particular $S_t(x_p, x_q) = \\alpha _t^T \\mathcal {J} \\left(\\begin{array}{c}x_p\\\\ x_q\\end{array}\\right) + (x_p, x_q) \\mathcal {B}_t \\left(\\begin{array}{c}x_p\\\\ x_q\\end{array}\\right),$ where $\\mathcal {B}_t$ is a real symmetric $n\\times n$ matrix that is related to $\\mathcal {M}_t$ by the Cayley parameterization of $\\mathcal {M}_t$ [22]: $\\mathcal {J} \\mathcal {B}_t = \\left( 1 + \\mathcal {M}_t \\right)^{-1} \\left( 1 - \\mathcal {M}_t \\right) = \\left( 1 - \\mathcal {M}_t \\right) \\left( 1 + \\mathcal {M}_t \\right)^{-1}.$ Since one classical trajectory contribution is sufficient in this case, if the overall phase of the propagated state is not important, then the expansion w.r.t.",
"$\\hbar $ in Eq.",
"REF can be truncated at order $\\hbar ^0$ .",
"Dropping terms that are higher order than $\\hbar ^0$ and ignoring phase is equivalent to propagating the classical density $\\rho (x)$ corresponding to the $(p, q) $ -manifold, under the harmonic Hamiltonian and determining its overlap with the $(p^{\\prime }, q^{\\prime })$ -manifold after time $t$ .",
"Such a treatment under a harmonic Hamiltonian results in just the absolute value of the prefactor of Eq.",
"REF : $\\left| \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right|^{\\frac{1}{2}}$ .",
"Indeed, this was van Vleck's discovery before quantum mechanics was formalized [18].",
"The relative phases of different classical contributions are no longer a concern and the higher order terms only weigh such contributions appropriately.",
"Here we are interested in propagating between Gaussian states in the center representation.",
"In continuous systems, a Gaussian state in $n$ dimensions can be defined as: $\\Psi _\\beta (q) = \\left[\\pi ^{-n} \\det \\left( \\operatorname{\\text{Re}}{\\Sigma }_\\beta \\right) \\right]^{\\frac{1}{4}} \\exp \\left( \\varphi \\right),$ where $\\varphi = \\frac{i}{\\hbar } p_\\beta \\cdot \\left( q - {q}_\\beta \\right) - \\frac{1}{2} \\left( q - {q}_\\beta \\right)^T {\\Sigma }_\\beta \\left( q - q_\\beta \\right).$ $q_\\beta \\in \\mathbb {R}^n$ is the central position, $p_\\beta \\in \\mathbb {R}^n$ is the central momentum, and $\\Sigma _\\beta $ is a symmetric $n\\times n$ matrix where $\\operatorname{\\text{Re}}\\Sigma _\\beta $ is proportional to the spread of the Gaussian and $\\operatorname{\\text{Im}}\\Sigma _\\beta $ captures $p$ -$q$ correlation.",
"This state describes momentum states ($\\delta (p - p_\\beta )$ in momentum representation) when $\\Sigma _\\beta \\rightarrow 0$ and position states $\\delta (q- q_\\beta )$ when $\\Sigma _\\beta \\rightarrow \\infty $ .",
"Rotations between these two cases corresponds to $\\operatorname{\\text{Re}}\\Sigma _\\beta = 0$ and $\\operatorname{\\text{Im}}\\Sigma _\\beta \\ne 0$ .",
"Gaussians remain Gaussians under evolution by a harmonic Hamiltonian, even if it is time-dependent.",
"This can be shown by simply making the ansatz that the state remains a Gaussian and then solving for its time-dependent $\\Sigma _\\beta $ , $p_\\beta $ , $q_\\beta $ and phase from the time-dependent Schrödinger equation [17], or just by applying the analytically known Feynman path integral, which is equivalent to the van Vleck path integral, to a Gaussian [23].",
"Moreover, with the propagator in the center representation known to only have have one saddle-point contribution for a harmonic Hamiltonian, it is fairly straight-forward to show that this is also true for its coherent state representation (that is, taking a Gaussian to another Gaussian).",
"Applying the propagator to an initial and final Gaussian in the center representation $&& \\left[ \\left|\\Psi _\\beta \\right\\rangle \\left\\langle \\Psi _\\beta \\right| U_t \\left|\\Psi _\\alpha \\right\\rangle \\left\\langle \\Psi _\\alpha \\right| \\right]_x (x) = \\\\&& \\left( \\pi \\hbar \\right)^{-3n} \\int ^\\infty _{-\\infty } \\mbox{d} x_1 \\int ^\\infty _{-\\infty } \\mbox{d} x_2 \\, U_t(x_1 + x_2- x) \\nonumber \\\\&& \\qquad \\qquad \\qquad \\times {\\Psi _\\beta }_x(x_2) {\\Psi _\\alpha }_x (x_1) \\nonumber \\\\&& \\qquad \\qquad \\qquad \\times e^{2\\frac{i}{\\hbar }(x_1^T \\mathcal {J} {x_2} + x_2^T \\mathcal {J} {x} + x^T \\mathcal {J} {x_1})}, \\nonumber $ we see that since $U_t$ is a Gaussian from Eq.",
"REF ($S_{tj}$ is quadratic for harmonic Hamiltonians) and since the Wigner representations of the Gaussians, ${\\Psi _\\alpha }_x$ and ${\\Psi _\\beta }_x$ , are also known to be Gaussians, the full integral in the above equation is a Gaussian integral and thus evaluates to produce a Gaussian with a prefactor.",
"This is equivalent to evaluating the integral by the method of steepest descents which finds the saddle points to be the points that satisfy $\\frac{\\partial \\phi }{\\partial x} = 0$ where $\\phi $ is the phase of the integrand's argument.",
"Since this argument is quadratic, its first derivative is linear and so again there is only one unique saddle point.",
"Indeed, such an evaluation produces the coherent state representation of the vVMG propagator [24].",
"Just as we found with the center representation, the absolute value of its prefactor corresponds to the order $\\hbar ^0$ term.",
"As a consequence of this single contribution at order $\\hbar ^0$ , it follows that the Wigner function of a state, $\\Psi _x(x)$ , evolves under the operator $\\hat{V}$ with an underlying harmonic Hamiltonian by $\\Psi _x(\\mathcal {M}_{\\hat{V}}(x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , where $\\mathcal {M}_{\\hat{V}}$ is the symplectic matric and $\\alpha _{\\hat{V}}$ is the translation vector associated with $\\hat{V}$ 's action [14].",
"Before proceeding to the discrete case, we note that the center representation that we have defined allows for a particularly simple way to express how far the path integral treatment must be expanded in $\\hbar $ in order to describe any unitary propagation (not necessarily harmonic) in continuous quantum mechanics.",
"Reflections and translations can also be described by truncating Eq.",
"REF at order $\\hbar ^1$ (or $\\hbar ^0$ if overall phase isn't important) since they correspond to evolution under a harmonic Hamiltonian.",
"In particular, translations are displacements along a chord $\\xi $ and so have Hamiltonians $H \\propto \\xi _q \\cdot p - \\xi _p \\cdot q$ .",
"Reflections are symplectic rotations around a center $x$ and so have Hamiltonians $H \\propto \\frac{\\pi }{4}\\left[ (p- x_p)^2 + (q- x_q)^2 \\right]$ .",
"From Eq.",
"REF we see that any operator can be expressed as an infinite Riemann sum of reflections.",
"Therefore, since reflections are fully described by a truncation at order $\\hbar ^1$ , it follows that an infinite Riemann sum of path integral solutions truncated at order $\\hbar ^1$ can describe any unitary evolution.",
"The same statement can be made by considering the chord representation in terms of translations.",
"Hence, quantum propagation in continuous systems can be fully treated by an infinite sum of contributions from a path integral approach truncated at order $\\hbar ^1$ .",
"As an aside, in general this infinite sum isn't convergent and so it is often more useful to consider reformulations that involve a sum with a finite number of contributions.",
"One way to do this is to apply the method of steepest descents directly on the operator of interest and use the area between saddle points as the metric to determine the order of $\\hbar $ necessary, instead of dealing with an infinity of reflections (or translations).",
"This results in the semiclassical propagator already presented, but associated with the full Hamiltonian instead of a sum of reflection Hamiltonians.",
"In summary, we have explained why propagation between Gaussian states under Hamiltonians that are harmonic is simulable classically (i.e.",
"up to order $\\hbar ^0$ ) in continuous systems.",
"We will see that the same situation holds in discrete systems for stabilizer states, with the additional restriction that the propagation takes the phase space points, which are now discrete, to themselves.",
"Discrete Center-Chord Representation We now proceed to the discrete case and introduce the center-chord formalism for these systems.",
"It will be useful for us to define a pair of conjugate degrees of freedom $p$ and $q$ for discrete systems.",
"Unfortunately, this isn't as straight-forward as in the continuous case, since the usual canonical commutation relations cannot hold in a finite-dimensional Hilbert space where the operators are bounded (since $\\operatorname{Tr}[\\hat{p}, \\hat{q}] = 0$ ).",
"We begin in one degree of freedom.",
"We label the computational basis for our system by $n \\in {0, 1, \\ldots , d-1}$ , for $d$ odd and we assume that $d$ is odd for the rest of this paper.",
"We identify the discrete position basis with the computational basis and define the “boost” operator as diagonal in this basis: $\\hat{Z}^{\\delta p} \\left|n\\right\\rangle \\equiv \\omega ^{n \\delta p} \\left|n\\right\\rangle ,$ where $\\omega $ will be defined below.",
"We define the normalized discrete Fourier transform operator to be equivalent to the Hadamard gate: $\\hat{F} = \\frac{1}{\\sqrt{d}}\\sum _{m,n \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{m n} \\left|m\\right\\rangle \\left\\langle n\\right|.$ This allows us to define the Fourier transform of $\\hat{Z}$ : $\\hat{X} \\equiv \\hat{F}^\\dagger \\hat{Z} \\hat{F}$ Again, as before, we call $\\hat{X}$ the “shift” operator since $\\hat{X}^{\\delta q} \\left|n\\right\\rangle \\equiv \\left|n\\oplus \\delta q\\right\\rangle ,$ where $\\oplus $ denotes mod-$d$ integer addition.",
"It follows that the Weyl relation holds again: $\\hat{Z} \\hat{X} = \\omega \\hat{X} \\hat{Z}.$ The group generated by $\\hat{Z}$ and $\\hat{X}$ has a $d$ -dimensional irreducible representation only if $\\omega ^d=1$ for odd $d$ .",
"Equivalently, there are only reflections relating any two phase space points on the Weyl phase space “grid” if $d$ is odd [25].",
"We take $\\omega \\equiv \\omega (d) = e^{2 \\pi i/d}$ [26].",
"This was introduced by Weyl [27].",
"Note that this means that $\\hbar = \\frac{d}{2 \\pi }$ or $h = d$ .",
"This means that the classical regime is most closely reached when the dimensionality of the system is reduced ($d \\rightarrow 0$ ) and thus the most “classical” system we can consider here is a qutrit (since we keep $d$ odd and greater than one).",
"This is the opposite limit considered by many other approaches where the classical regime is reached when $d \\rightarrow \\infty $ .",
"One way of interpreting the classical limit in this paper is by considering $h$ to be equal to the inverse of the density of states in phase space (i.e.",
"in a Wigner unit cell).",
"As $\\hbar \\rightarrow 0$ phase space area decreases as $\\hbar ^2$ but the number of states only decreases as $\\hbar $ leading to an overall density increase of $\\hbar $ .",
"This agrees with the notion that states should become point particles of fixed mass in the classical limit.",
"By analogy with continuous finite translation operators, we reexpress the shift $\\hat{X}$ and boost $\\hat{Z}$ operators in terms of conjugate $\\hat{ p}$ and $\\hat{ q}$ operators: $ \\hat{Z} \\left|n\\right\\rangle = e^{\\frac{2 \\pi i }{d}\\hat{q}} \\left|n\\right\\rangle = e^{\\frac{2 \\pi i}{d} n} \\left|n\\right\\rangle , $ and $ \\hat{X} \\left|n\\right\\rangle = e^{-\\frac{2 \\pi i }{d} \\hat{p}} \\left|n\\right\\rangle = \\left|n\\oplus 1\\right\\rangle .",
"$ Hence, in the diagonal “position” representation for $\\hat{Z}$ : $\\hat{Z} = \\left( \\begin{array}{cccccc} 1 & 0 & 0 & \\cdots & 0\\\\0 & e^{\\frac{2 \\pi i}{d}} & 0 & \\cdots & 0\\\\0 & 0 & e^{\\frac{4 \\pi i}{d}} & \\cdots & 0\\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & 0 & e^{\\frac{2 (d-1) \\pi i}{d}} \\end{array} \\right),$ and $\\hat{X} = \\left( \\begin{array}{cccccc} 0 & 0 & \\cdots & 0 & 1\\\\1 & 0 & \\cdots & 0 & 0\\\\0 & 1 & 0 & \\cdots & 0\\\\\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\0 & \\cdots & 0 & 1 & 0 \\end{array} \\right).$ Thus, $\\hat{q} = \\frac{d}{2 \\pi i} \\log \\hat{Z} = \\sum _{n \\in \\mathbb {Z}/d\\mathbb {Z}} n \\left|n\\right\\rangle \\left\\langle n\\right|,$ and $\\hat{ p} = \\hat{F}^\\dagger \\hat{ q} \\hat{F}.$ Therefore, we can interpret the operators $\\hat{ p}$ and $\\hat{ q}$ as a conjugate pair similar to conjugate momenta and position in the continuous case.",
"However, they differ from the latter in that they only obey the weaker group commutation relation $e^{i j \\hat{ q}/\\hbar }e^{i k \\hat{ p}/\\hbar } e^{-i j \\hat{ q}/\\hbar }e^{-i k \\hat{ p}/\\hbar } = e^{-i jk /\\hbar } \\hat{\\mathbb {I}}.$ This corresponds to the usual canonical commutation relation for $p$ and $q$ 's algebra at the origin of the Lie group ($j = k = 0$ ); expanding both sides of Eq.",
"REF to first order in $p$ and $q$ yields the usual canonical relation.",
"We proceed to introduce the Weyl representation of operators and states in discrete Hilbert spaces with odd dimension $d$ and $n$ degrees of freedom [1], [28], [29].",
"The generalized phase space translation operator (the Weyl operator) is defined as a product of the shift and boost with a phase appropriate to the $d$ -dimensional space: $\\hat{T}(\\xi _p, \\xi _q) = e^{-i \\frac{\\pi }{d} \\xi _p \\cdot \\xi _q} \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q},$ where $\\xi \\equiv (\\xi _p, \\xi _q) \\in (\\mathbb {Z} / d \\mathbb {Z})^{2n}$ and form a discrete “web” or “grid” of chords.",
"They are a discrete subset of the continous chords we considered in the infinite-dimensional context in Section and their finite number is an important consequence of the discretization of the continuous Weyl formalism.",
"Again, an operator $\\hat{A}$ can be expressed as a linear combination of translations: $\\hat{A} = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} A_\\xi (\\xi _p, \\xi _q) \\hat{T}(\\xi _p, \\xi _q),$ where the weights are the chord representation of the function $\\hat{A}$ : ${A}_\\xi (\\xi _p, \\xi _q) = d^{-n} \\operatorname{Tr}\\left( \\hat{T}(\\xi _p, \\xi _q)^\\dagger \\hat{A} \\right).$ When applied to a state $\\hat{\\rho }$ , this is also called the “characteristic function” of $\\hat{\\rho }$ [30].",
"As before, the center representation, based on reflections instead of translations, requires an appropriately defined reflection operator.",
"We can define the discrete reflection operator $\\hat{R}$ as the symplectic Fourier transform of the discrete translation operator we just introduced: $\\hat{R}(x_p, x_q) = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{\\frac{2 \\pi i}{d} (\\xi _p, \\xi _q) \\mathcal {J} (x_p, x_q)^T} \\hat{T}(\\xi _p, \\xi _q).$ With this in hand, we can now express a finite-dimensional operator $\\hat{A}$ as a superposition of reflections: $\\hat{A} = d^{-n} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} {A}_x(x_p, x_q) \\hat{R}( x_p, x_q ),$ where ${A}_x(x_p, x_q) = d^{-n} \\operatorname{Tr}\\left( \\hat{R}(x_p, x_q)^\\dagger \\hat{A} \\right).$ $x \\equiv (x_p, x_q) \\in (\\mathbb {Z} / d \\mathbb {Z})^{2n}$ are centers or Weyl phase space points and, like their $(\\xi _p, \\xi _q)$ brethren, form a discrete subgrid of the continuous Weyl phase space points considered in Section .",
"Again, the center representation is of particular interest to us because for unitary gates $\\hat{A}$ we can rewrite the components ${A}_x(x_p, x_q)$ as: ${A}_x(x_p, x_q) = \\exp \\left[\\frac{i}{\\hbar } S(x_p, x_q)\\right]$ where $S(x_p, x_q)$ is the argument to the exponential, and is equivalent to the action of the operator in center representation (the center generating function).",
"Aside from Eq.",
"REF , the center representation of a state $\\hat{\\rho }$ can also be directly defined as the symplectic Fourier transform of its chord representation, $\\rho _\\xi $ [3]: $\\rho _x(x_p, x_q) = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{\\frac{2 \\pi i}{d} (\\xi _p, \\xi _q) \\mathcal {J} (x_p, x_q)^T} \\rho _\\xi (\\xi _p,\\xi _q).$ We note again that for a pure state $\\left|\\Psi \\right\\rangle $ , the Wigner function from Eqs.",
"REF and REF simplifies to: ${\\Psi }_x(x_p, x_q) &=& d^{-n} \\sum _{\\begin{array}{c}\\xi _q \\in \\\\(\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{-\\frac{2 \\pi i}{d} \\xi _q \\cdot x_p} \\Psi \\left( x_q + \\frac{(d+1) \\xi _q }{2} \\right) {\\Psi ^*} \\left( x_q - \\frac{(d+1) \\xi _q}{2} \\right).$ It may be clear from this short presentation that the chord and center representations are dual to each other.",
"A thorough review of this subject can be found in [13].",
"Path Integral Propagation in Discrete Systems Rivas and Almeida [14] found that the continuous infinite-dimensional vVMG propagator can be extended to finite Hilbert space by simply projecting it onto its finite phase space tori.",
"This produces: $&& {U}_t(x) = \\\\&& \\left< \\sum _j \\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} e^{\\frac{i}{\\hbar } S_{tj}(x)} e^{i \\theta _k} \\right>_k + \\mathcal {O}(\\hbar ^2),\\nonumber $ where an additional average must be taken over the $k$ center points that are equivalent because of the periodic boundary conditions.",
"Maintaining periodicity requires that they accrue a phase $\\theta _k$ See Eq.",
"$5.18$ in [14] for a definition of the phase..",
"The derivative $\\frac{\\partial ^2 S_{tj}}{\\partial x^2}$ is performed over the continuous function $S_{tj}$ defined after Eq.",
"REF , but only evaluated at the discrete Weyl phase space points $x \\equiv (x_p, x_q)\\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{2n}$ .",
"The prefactor can also be reexpressed: $\\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} = \\left\\lbrace 2^d \\det \\left[ 1 + \\mathcal {M}_{tj} \\right] \\right\\rbrace ^{-\\frac{1}{2}},$ which is perhaps more pleasing in the discrete case as it does not involve a continuous derivative.",
"As in the continuous case, for a harmonic Hamiltonian $H(p, q)$ , the center generating function $S(x_p, x_q)$ is equal to $\\alpha ^T \\mathcal {J} x + x^T \\mathcal {B} x$ where Eq.",
"REF and Eq.",
"REF hold.",
"Moreover, if the Hamiltonian takes Weyl phase space points to themselves, then by the same equations it follows that $\\mathcal {M}$ and $\\alpha $ must have integer entries.",
"This implies that for $m, n \\in \\mathbb {Z}^{ n}$ , $&& \\mathcal {M} \\left(\\begin{array}{c}p+ m d + \\alpha _p/2\\\\q + n d + \\alpha _q/2\\end{array}\\right) + \\left( \\begin{array}{c} \\alpha _p/2\\\\ \\alpha _q/2 \\end{array} \\right) \\nonumber \\\\&=& \\mathcal {M} \\left(\\begin{array}{c}p + \\alpha _p/2 \\\\ q + \\alpha _q/2 \\end{array}\\right) + \\left( \\begin{array}{c} \\alpha _p/2\\\\ \\alpha _q/2 \\end{array} \\right) + d \\mathcal {M}\\left(\\begin{array}{c}m\\\\ n\\end{array}\\right) \\\\&=& \\left(\\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array}\\right) \\mod {d. \\nonumber }Therefore, phase space points (p, q) that lie on Weyl phase space points go to the equivalent Weyl phase space points (p^{\\prime }, q^{\\prime }).$ Moreover, again if $m, n \\in \\mathbb {Z}^n$ , $&& S(x_p + m d, x_q + n d) \\nonumber \\\\&=& \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right)^T A \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right) \\nonumber \\\\&& + b \\cdot \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right)\\\\&=& \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right)^T A \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right) + b \\cdot \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right) \\nonumber \\\\&& + d \\left[ 2 \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right)^T A \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right) \\right.\\nonumber \\\\&& \\qquad \\left.+ d \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right)^T A \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right) + b \\cdot \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right)\\right] \\nonumber \\\\&=& S(x_p, x_q) \\mod {d}, \\nonumber $ for some symmetric $A \\in \\mathbb {Z}^{n\\times n}$ and $b \\in \\mathbb {Z}^{n}$ .",
"Therefore, these equivalent trajectories also have equivalent actions (since the action is multiplied by $\\frac{2 \\pi i}{d}$ and exponentiated).",
"Hence, there is only one term to the sum in Eq.",
"REF .",
"Moreover, [25] showed that the sum over the phases $\\theta _k$ produces only a global phase that can be factored out.",
"Therefore, if we can neglect the overall phase, ${U}_t(x) = \\left| 2^d \\det \\left[ 1 + \\mathcal {M} \\right] \\right|^{\\frac{1}{2}},$ where the classical trajectories whose centers are $(x_p, x_q)$ satisfy the periodic boundary conditions.",
"As in the continuous case, we point out that this means that translations and reflections are fully captured by a path integral treatment that is truncated at order $\\hbar ^1$ (or order $\\hbar ^0$ if their overall phase isn't important) because their Hamiltonians are harmonic, but in the discrete case there is an additional requirement that they are evaluated at chords/centers that take Weyl phase space points to themselves.",
"Just as in the continuous case, the single contribution at order $\\hbar ^0$ implies that the propagator of the Wigner function of states, $\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(x)$ under gates $\\hat{V}$ with underlying harmonic Hamiltonians is captured by $\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ for $\\mathcal {M}_{\\hat{V}}$ and $\\alpha _{\\hat{V}}$ associated with $\\hat{V}$ .",
"Stabilizer Group Here we will show that the Hamiltonians corresponding to Clifford gates are harmonic and take Weyl phase space points to themselves.",
"Thus they can be captured by only the single contribution of Eq.",
"REF at lowest order in $\\hbar $ .",
"This then implies that stabilizer states can also be propagated to each other by Clifford gates with only a single contribution to the sum in Eq.",
"REF .",
"The Clifford gate set of interest can be defined by three generators: a single qudit Hadamard gate $\\hat{F}$ and phase shift gate $\\hat{P}$ , as well as the two qudit controlled-not gate $\\hat{C}$ .",
"We examine each of these in turn.",
"Hadamard Gate The Hadamard gate was defined in Eq.",
"REF and is a rotation by $\\frac{\\pi }{2}$ in phase space counter-clockwise.",
"Hence, for one qudit, it can be written as the map in Eq.",
"REF where $\\mathcal {M}_{\\hat{F}} = \\left( \\begin{array}{cc} 0 & 1\\\\ -1 & 0 \\end{array} \\right),$ and $\\alpha _{\\hat{F}} = (0,0)$ .",
"We have set $t=1$ and drop it from the subscripts from now on.",
"Since $\\alpha $ is vanishing and $\\mathcal {M}$ has integer entries, this is a cat map and such maps have been shown to correspond to Hamiltonians [32] $&&H(p,q) =\\\\&&f(\\operatorname{Tr}\\mathcal {M} ) \\left[ \\mathcal {M}_{12} p^2 - \\mathcal {M}_{21} q^2 + \\left(\\mathcal {M}_{11} - \\mathcal {M}_{22} \\right) pq \\right],\\nonumber $ where $f(x) = \\frac{\\sinh ^{-1}(\\frac{1}{2}\\sqrt{x^2-4})}{\\sqrt{x^2-4}}.$ For the Hadamard $\\mathcal {M}_{\\hat{F}}$ this corresponds to $H_{\\hat{F}} = \\frac{\\pi }{4} (p^2 + q^2)$ , a harmonic oscillator.",
"The center generating function $S(x_p, x_q)$ is thus $(x_p, x_q ) \\mathcal {B} (x_p, x_q)^T$ and solving Eq.",
"REF finds for the one-qudit Hadamard, $\\mathcal {B}_{\\hat{F}} = \\left(\\begin{array}{cc}1 & 0\\\\ 0 & 1 \\end{array} \\right).$ Thus, $S_{\\hat{F}}(x_p, x_q) = x_p^2 + x_q^2$ .",
"Indeed, applying Eq.",
"REF to Eq.",
"REF reveals that the Hadamard's center function (up to a phase) is: ${F}_x(x_p,x_q) = e^{\\frac{2 \\pi i}{d}(x_p^2+x_q^2)}.$ Eq.",
"REF shows how to map Weyl phase space to Weyl phase space under the Hadamard transformation.",
"Furthermore this map is pointwise, which implies the quadratic form of the center generating function obtained in Eq.",
"REF .",
"Phase Shift Gate The phase shift gate can be generalized to odd $d$ -dimensions [33] by setting it to: $\\hat{P} = \\sum _{j \\in \\mathbb {Z}/d \\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{2}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ Examining its effect on stabilizer states, it is clear that it is a $q$ -shear in phase space from an origin displaced by $\\frac{d-1}{2}\\equiv -\\frac{1}{2}$ to the right.",
"This can be expressed as the map in Eq.",
"REF with $\\mathcal {M}_{\\hat{P}} = \\left( \\begin{array}{cc} 1 & 1\\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{P}} = \\left( -\\frac{1}{2}, 0 \\right)$ .",
"This corresponds to $\\mathcal {B}_{\\hat{P}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & \\frac{1}{2} \\end{array} \\right).$ Solving Eq.",
"REF with this $\\mathcal {B}_{\\hat{P}}$ and $\\alpha _{\\hat{P}}$ reveals that $S_{\\hat{P}}(x_p, x_q) = -\\frac{1}{2} x_q + \\frac{1}{2} x_q^2$ .",
"Again, this agrees with the argument of the center representation of the phase-shift gate obtained by applying Eq.",
"REF to Eq.",
"REF : ${P}_x(x_p,x_q) = e^{\\frac{2 \\pi i}{d} \\frac{1}{2} (-x_q + x_q^2)}.$ Discretization the equations of motion for harmonic evolution for unit timesteps leads to: $\\left(\\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array}\\right) = \\left(\\begin{array}{c}p\\\\ q\\end{array}\\right) + \\mathcal {J} \\left(\\frac{\\partial H}{\\partial p}, \\frac{\\partial H}{\\partial q}\\right)^T,$ where the last derivative is on the continuous function $H$ , but only evaluated on the discrete Weyl phase space points.",
"It follows that $H_{\\hat{P}} = -\\frac{d+1}{2} q^2 + \\frac{d+1}{2} q.$ We have obtained the Hamiltonian for the phase-shift gate by a different procedure than that used for the Hadamard where we appealed to the result given in Eq.",
"REF for quantum cat maps.",
"However, as the phase-shift gate is a quantum cat map as well, we could have obtained Eq.",
"REF in this manner.",
"Similarly, the approach we used to find the phase-shift Hamiltonian by discretizing time in Eq.",
"REF would work for the Hadamard gate but it is a bit more involved since the latter contains both $p$ - and $q$ -evolution.",
"Nevertheless, this produces Eq.",
"REF as well.",
"We presented both techniques for illustrative purposes.",
"Controlled-Not Gate Lastly, the controlled-not gate can be generalized to $d$ -dimensions [33] by $\\hat{C} = \\sum _{j,k \\in \\mathbb {Z}/d \\mathbb {Z}} \\left|j,k \\oplus j\\right\\rangle \\left\\langle j,k\\right|.$ It is clear that this translates the $q$ -state of the second qudit by the $q$ -state of the first qudit.",
"As a result, as is evident by examining the gate's action on stabilizer states, the first qudit experiences an “equal and opposite reaction” force that kicks its momentum by the $q$ -state of the second qudit.",
"This is the phase space picture of the well-known fact that a CNOT examined in the $\\hat{X}$ basis has the control and target reversed with respect to the $\\hat{Z}$ basis.",
"This can also seen by looking at its effect in the momentum ($\\hat{X}$ ) basis: ${\\hat{F}}^\\dagger \\hat{C} \\hat{F} = \\sum _{j,k \\in \\mathbb {Z}/d \\mathbb {Z}} \\left|j \\ominus k,k\\right\\rangle \\left\\langle j,k\\right|.$ As a result, this gate is described by the map: $\\mathcal {M}_{\\hat{C}} = \\left( \\begin{array}{ccccc} 1 & -1 & 0 & 0\\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1 & 0\\\\ 0 & 0 & 1 & 1\\end{array}\\right),$ and $\\alpha _{\\hat{C}} = (0,0,0,0)$ .",
"This corresponds to $\\mathcal {B}_{\\hat{C}} = \\left( \\begin{array}{ccccc} 0 & 0 & 0 & 0\\\\ 0 & 0 & -\\frac{1}{2} & 0\\\\ 0 & -\\frac{1}{2} & 0 & 0\\\\ 0 & 0 & 0 & 0\\end{array}\\right).$ Hence its center generating function $S_{\\hat{C}}(x_p, x_q) = -x_{p_2} x_{q_1}$ .",
"Again, this corresponds with the argument of the center representation of the controlled-not gate, which can be found to be: ${C}_x(x_p, x_q) = e^{-\\frac{2 \\pi i}{d} x_{q_1} x_{p_2}}.$ Therefore, this gate can be seen to be a bilinear $p$ -$q$ coupling between two qudits and corresponds to the Hamiltonian $H_{\\hat{C}} = p_1 q_2,$ as can be found from Eq.",
"REF again.",
"As a result, it is now clear that all the Clifford group gates have Hamiltonians that are harmonic and that take Weyl phase space points to themselves.",
"Therefore, their propagation can be fully described by a truncation of the semiclassical propagator Eq.",
"REF to order $\\hbar ^0$ as in Eq.",
"REF and they are manifestly classical in this sense.",
"To summarize the results of this section, using Eq.",
"REF , the Hadamard, phase shift, and CNOT gates can be written as: $\\hat{F} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d \\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ -(x_p^2 + x_q^2) - d \\left(x_p \\xi _p - x_q \\xi _q \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q},$ $\\hat{P} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d \\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{2} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q},$ and $&&\\hat{C} = \\\\&& d^{-4} \\sum _{\\begin{array}{c}x_p, x_q, \\\\ \\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z}/d \\mathbb {Z})^{ 2}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ x_{q_1} x_{p_2} - d \\left(x_p \\cdot \\xi _q - x_q \\cdot \\xi _p \\right) \\right] } \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q}, \\nonumber $ (up to a phase).",
"This form emphasizes their quadratic nature.",
"As for the continuous case, there exists a particularly simple way in discrete systems to see to what order in $\\hbar $ the path integral must be kept to handle unitary propagation beyond the Clifford group.",
"We describe this in the next section.",
"We note that the center generating actions $S(x_p, x_q)$ found here are related to the $G(q^{\\prime }, q)$ found by Penney et al.",
"[12], which are in terms of initial and final positions, by symmetrized Legendre transform [13]: $G(q^{\\prime }, q, t) = F\\left(\\frac{q^{\\prime } + q}{2}, p(q^{\\prime } - q)\\right),$ where the canonical generating function $F\\left(\\frac{q^{\\prime } + q}{2}, p\\right) = S\\left( x_p = p, x_q = \\frac{q^{\\prime } + q}{2} \\right) + p \\cdot \\left( q^{\\prime } - q \\right),$ for $p (q^{\\prime } - q)$ given implicitly by $\\frac{\\partial F}{\\partial p} = 0$ .",
"Applying this to the actions we found reveals that $G_{\\hat{F}}(q^{\\prime }, q, t) = q^{\\prime } q,$ $G_{\\hat{P}}(q^{\\prime }, q, t) = \\frac{d+1}{2} (q^2 - q),$ and $G_{\\hat{C}}((q^{\\prime }_1, q^{\\prime }_2), (q_1, q_2), t) = 0,$ which is in agreement with [12].",
"Classicality of Stabilizer States In this subsection we show that stabilizer states evolve to stabilizer states under Clifford gates, and that it is possible to describe this evolution classically.",
"For odd $d \\ge 3$ the positivity of the Wigner representation implies that evolution of stabilizer states is non-contextual, and so here we are investigating in detail what this means in our semiclassical picture.",
"To begin, it is instructive to see the form stabilizer states take in the discrete position representation and in the center representation.",
"Gross proved that [3]: Theorem 1 Let $d$ be odd and $\\Psi \\in L^2((\\mathbb {Z}/d\\mathbb {Z})^n)$ be a state vector.",
"The Wigner function of $\\Psi $ is non-negative if and only if $\\Psi $ is a stabilizer state.",
"Gross also proved [3] Corollary 1 Given that $\\Psi (q) \\ne 0$ $\\forall \\, q$ , a vector $\\Psi $ is a stabilizer state if and only if it is of the form $\\Psi _{\\theta _\\beta ,\\eta _\\beta }(q) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( q^T \\theta _\\beta q + \\eta _\\beta \\cdot q \\right) \\right].$ where $\\theta _\\beta \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{n\\times n}$ and $q, \\eta _\\beta \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^n$ .",
"Applying Eq.",
"REF to Eq.",
"REF , the Wigner function of such maximally supported stabilizer states can be found to be: $&& {\\Psi _{\\theta _\\beta ,\\eta _\\beta }}_x(x_p, x_q) \\propto \\\\&& d^{-n} \\sum _{\\begin{array}{c}\\xi _q \\in \\\\\\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{ n}\\end{array}} \\exp \\left[\\frac{2 \\pi i}{d} \\xi _q \\cdot \\left( \\eta _\\beta - x_p + 2 \\theta _\\beta x_q \\right)\\right].\\nonumber $ Therefore, one finds that the Wigner function is the discrete Fourier sum equal to $\\delta _{\\eta _\\beta - x_p + 2\\theta _\\beta x_q}$ .",
"For $\\theta _\\beta = 0$ the state is a momentum state at $x_p$ .",
"Finite $\\theta _\\beta $ rotates that momentum state in phase space in “steps” such that it always lies along the discrete Weyl phase space points $(x_p, x_q) \\in (\\mathbb {Z}/d\\mathbb {Z})^{2n}$ .",
"This Gaussian expression only captures stabilizer states that are maximally supported in $q$ -space.",
"One may wonder what the stabilizer states that aren't maximally supported in $q$ -space look like in Weyl phase space.",
"Of course, it is possible that some may be maximally supported in $p$ -space and so can be captured by the following corollary: Corollary 2 If $\\Psi (p) \\ne 0$ for all $p$ 's then there exists a $\\theta _{\\beta p} \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{n\\times n}$ and an $\\eta _{\\beta p} \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^n$ such that $\\Psi _{\\theta _{\\beta p},\\eta _{\\beta p}}(p) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( p^T \\theta _{\\beta p} p + \\eta _{\\beta p} \\cdot p \\right) \\right].$ Proof This can be shown following the same methods employed by Gross [3] but in the discrete $p$ -basis.",
"Unfortunately, it is easy to show that Corollary REF and REF do not provide an expression for all stabilizer states (except for the odd prime $d$ case, as we shall see shortly) as there exist stabilizer states for odd non-prime $d$ that are not maximally supported in $p$ - or $q$ -space or any finite rotation between those two.",
"To find an expression that encompasses all stabilizer states, we must turn to the Wigner function of stabilizer states.",
"An equivalent definition of stabilizer states on $n$ qudits is given by states $\\hat{V} \\underbrace{\\left|0\\right\\rangle \\otimes \\cdots \\otimes \\left|0\\right\\rangle }_{n}$ where $\\hat{V}$ is a quantum circuit consisting of Clifford gates.",
"We know that the Clifford circuits are generated by the $\\hat{P}$ , $\\hat{F}$ and $\\hat{C}$ gates, and that the Wigner functions $\\Psi _x(x)$ of stabilizer states propagate under $\\hat{V}$ as $\\Psi _x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , it follows that the Wigner function of stabilizer states is: $\\delta _{\\Phi _0 \\cdot \\mathcal {M}_{\\hat{V}} \\cdot x, r_0},$ where $\\Phi _0 = \\left(\\begin{array}{cc} 0 & 0\\\\ 0 & \\mathbb {I}_n\\end{array} \\right)$ and $r_0 = (0, 0)$ .",
"We have therefore proved the next theorem: Theorem 2 The Wigner function $\\Psi _x(x)$ of a stabilizer state for any odd $d$ and $n$ qudits is $\\delta _{\\Phi \\cdot x, r}$ for $2n \\times 2n$ matrix $\\Phi $ and $2n$ vector $r$ .",
"As an aside, Theorem REF allows us to develop an all-encompassing Gaussian expression for stabilizer states for the restricted case that $d$ is odd prime.",
"In this case, the following Corollary shows that a “mixed” representation is always possible: where each degree of freedom is expressed in either the $p$ - or $q$ -basis: Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state for $n$ qudits, then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .",
"Proof We begin with a one-qudit case.",
"We examine the equation specified by $\\Phi \\cdot x = r$ : $\\alpha q_1 + \\beta p_1 = \\gamma $ for $\\alpha $ , $\\beta $ , and $\\gamma \\in \\mathbb {Z}/d\\mathbb {Z}$ .",
"If $\\alpha = 0$ then $\\Psi (q_1)$ is maximally supported and if $\\beta = 0$ then $\\Psi (p_1)$ is maximally supported since the equation specifies a line on $\\mathbb {Z}/d \\mathbb {Z}$ in $q_1$ and $p_1$ respectively.",
"If $\\alpha \\ne 0$ and $\\beta \\ne 0$ then the equation can be rewritten as $q_1 + (\\beta /\\alpha ) p_1 = \\gamma /\\alpha ,$ and it follows that $q_1$ can take any values on $\\mathbb {Z}/d \\mathbb {Z}$ and so $\\Psi (q_1)$ is maximally supported.",
"However, it is also possible to reexpress the equation as: $p_1 + (\\alpha /\\beta ) q_1 = \\gamma /\\beta .$ It follows that $p_1$ can also take any values on $\\mathbb {Z}/d \\mathbb {Z}$ —$\\Psi (p_1)$ is also maximally supported.",
"Therefore, one can always choose either a $p_1$ - or $q_1$ -basis such that the state is maximally supported and so is representable by a Gaussian function.",
"We now consider adding another qudit such that the state becomes ${\\Psi }_x(p_1,p_2,q_1,q_2)$ .",
"There are now two equations specified by $\\Phi \\cdot x = r$ and it follows that it is always possible to combine the two equations such that $p_1$ and $q_1$ are only in one equation and written in terms of each other (and generally the second degree of freedom): $\\alpha q_1 + \\beta p_1 + \\gamma q_2 + \\delta p_2 = \\epsilon ,$ for $\\alpha $ , $\\beta $ , $\\gamma $ , $\\delta $ and $\\epsilon \\in \\mathbb {Z}/d\\mathbb {Z}$ .",
"It will turn out that the $\\gamma q_2 + \\delta p_2$ term is irrelevant.",
"We can rewrite the above equation as: $q_1 + (\\beta /\\alpha ) p_1 + (\\gamma /\\alpha ) q_2 + (\\delta /\\alpha ) p_2 = \\epsilon /\\alpha ,$ if $\\alpha \\ne 0$ .",
"Since there is no other equation specifying $p_1$ , this is an equation for a line on $\\mathbb {Z}/ d \\mathbb {Z}$ and so $\\Psi $ is is maximally supported on $q_1$ .",
"Otherwise, rewriting the above equation as: $p_1 + (\\alpha /\\beta ) q_1 +(\\gamma /\\beta ) q_2 + (\\delta /\\beta ) p_2 = \\epsilon /\\beta ,$ if $\\beta \\ne 0$ shows that $\\Psi $ is maximally supported on $p_1$ .",
"If $\\alpha = \\beta = 0$ then both $p_1$ and $q_1$ are undetermined and so either representation produces a maximally supported state.",
"The same procedure can be performed to find if $q_2$ or $p_2$ produce a maximally supported state.",
"As can be seen, we are really just repeating the same procedure as we did when there was only one qudit because the other degrees of freedom have no impact on this determination.",
"Expressing $\\Psi $ in the basis that is maximally supported in every degree of freedom means that it is therefore a Gaussian.",
"Therefore, it follows that every degree of freedom (corresponding to a qudit) is maximally supported in either the $p$ - or $q$ - basis and so Eq.",
"REF always describes stabilizer states for odd prime $d$ .$\\Box $ The form of Eq.",
"REF is more general than Eq.",
"REF because it does not depend on the support of the state.",
"As we saw in the proof, this representation is generally not unique; for every qudit $i$ that is not a position or momentum state, $x_i$ can be either $p_i$ or $q_i$ .",
"However, if it is a position state then $x_i = p_i$ and if it is a momentum state then $x_i = q_i$ ; position and momentum states must be expressed in their conjugate representation in order to be captured by a Gaussian of the form in Eq.",
"REF instead of Kronecker deltas.",
"The reason this mixed representation doesn't hold for non-prime odd $d$ is that the coefficients above can be (multiples of) prime factors of $d$ and so no longer produce “lines” in $p_i$ or $q_i$ that cover all of $\\mathbb {Z}/d \\mathbb {Z}$ .",
"An alternative proof of this corollary that explores this case further is presented in the Appendix.",
"An example of the different classes of stabilizer states that are possible for odd prime $d$ , in terms of their support, is shown in Fig.",
"REF .",
"There it can be seen that a stabilizer state is either maximally supported in $p_i$ or $q_i$ , and is a Kronecker delta function in the other degree of freedom, or it is maximally supported in both.",
"Figure: The two classes of stabilizer states possible for odd prime d=7d=7 in terms of support: a) maximally supported in pp or qq and b) maximally supported in pp and qq.",
"The central grids denote the Wigner function ΨΨ x (p,q)\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(p,q) of a stabilizer state Ψ\\Psi with d=7d=7.",
"The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\Psi (q)|^2).On the other hand, for odd non-prime $d$ , we see in Fig.",
"REF that another class is possible: stabilizer states that are maximally supported in neither $p_i$ or $q_i$ .",
"In fact, rotating the basis in any of the discrete angles afforded by the grid still does not produce a basis that is maximally supported (as discussed in the Appendix).",
"Notice also, that Fig.",
"REF b shows that it is no longer true that a state that is maximally supported in only $q_i$ or $p_i$ is automatically a Kronecker delta when expressed in terms of the other.",
"Figure: The four classes of stabilizer states possible for odd non-prime d=15d=15 in terms of support: a) & b) maximally supported in pp or qq, c) maximally supported in pp and qq, and d) not maximally supported in pp or qq.",
"The central grids denote the Wigner function ΨΨ x (p,q)\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(p,q) for a stabilizer state Ψ\\Psi with d=15d=15.",
"The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\Psi (q)|^2).In summary, stabilizer states have Wigner function $\\delta _{\\Phi \\cdot x, r}$ and, for odd prime $d$ , are Gaussians in mixed representation that lie on the Weyl phase space points $(x_p, x_q)$ .",
"Heuristically, they correspond to Gaussians in the continuous case that spread along their major axes infinitely.",
"The only reason that they aren't always expressible as Gaussians in the mixed representation is that they sometimes “skip” over some of the discrete grid points due to the particular angle they lie along phase space for odd non-prime $d$ .",
"Wigner functions $\\Psi _x(x)$ of stabilizer states propagate under $\\hat{V}$ as $\\Psi _x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , and this preserves the form of the state.",
"In other words, Clifford gates take stabilizer states to other stabilizer states, as expected, just like in the continuous case Gaussians go to other Gaussians under harmonic evolution.",
"It is also clear that stabilizer state propagation under Clifford gates can be expressed by a path integral at order $\\hbar ^0$ .",
"Discrete Phase Space Representation of Universal Quantum Computing A similar statement to the one we made in Section —that any operator can be expressed as an infinite sum of path integral contribution truncated at order $\\hbar ^1$ —can be made in discrete systems.",
"However, there is an important difference in the number of terms making up the sum.",
"To see this we can follow reasoning that is similar to that employed in the continuous case.",
"Namely, from Eq.",
"REF we see that any discrete operator can also be expressed as a linear combination of reflections, but unlike the continuous case, this sum has a finite number of terms.",
"Since reflections can be expressed fully by the discrete path integral truncated at order $\\hbar ^1$ , as discussed previously, it follows that any unitary operator in discrete systems can be expressed as a finite sum of contributions from path integrals truncated at order $\\hbar ^1$ .",
"Again, the same statement can be made by considering the chord representation in terms of translations.",
"Hence, quantum propagation in discrete systems can be fully treated by a finite sum of contributions from a path integral approach truncated at order $\\hbar ^1$ .",
"To gather some understanding of this statement, we can consider what is necessary to add to our path integral formulation when we complete the Clifford gates with the T-gate, which produces a universal gate set.",
"The T-gate is generalized to odd $d$ -dimensions by $\\hat{T} = \\sum _{j \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{4}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ This gate can no longer be characterized by an $\\mathcal {M}$ with integer entries.",
"In particular, $\\mathcal {M}_{\\hat{T}} = \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{T}} = \\left( -\\frac{1}{4}, 0 \\right)$ .",
"This corresponds to $\\mathcal {B}_{\\hat{T}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & -\\frac{1}{4}\\end{array} \\right).$ Thus, the center function $T_x(x_p,x_q) = e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)},$ corresponding to the phase shift Hamiltonian applied for only half the unit of time.",
"The operator can thus be written: $\\hat{T} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d\\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{4} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q}.$ Though this operator is quadratic, it no longer takes the Weyl center points to themselves.",
"This means that the $\\hbar ^0$ limit of Eq.",
"REF is now insufficient to capture all the dynamics because the overlap with any $\\left|q\\right\\rangle $ will now involve a linear superposition of partially overlapping propagated manifolds.",
"It must therefore be described by a path integral formulation that is complete to order $\\hbar ^1$ .",
"In particular, $\\hat{T} = d^{-1} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})\\end{array}} e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)} \\hat{R}( x_p, x_q ),$ where $\\hat{R}$ should be substituted by its path integral.",
"Note that this does not imply efficient classical simulation of quantum computation but quite the opposite.",
"Indeed, for $n$ qudits, there are $d^{2n}$ terms in the sum above.",
"While every Weyl phase space point has only a single associated path when acted on by Clifford gates, this is no longer true in any calculation of evolution under the T-gate.",
"Eq.",
"REF expresses the T-gate as a sum over phase space operators (the reflections) evaluated on all the phase space points.",
"Thus, it can be interpreted as associating an exponentially large number of paths to every phase space point, instead of the single paths found for Clifford gates.",
"Therefore, any simulation of the T-gate naively necessitates adding up an exponential large sum over paths and so is comparably inefficient.",
"Conclusion The treatment presented here formalizes the relationship between stabilizer states in the discrete case and Gaussians in the continuous case, which has often been pointed out [3].",
"Namely, only Gaussians that lie along Weyl phase space points directly correspond to Gaussians in the continuous world in terms of preserving their form under a harmonic Hamiltonian, an evolution that is fully describable by truncating the path integral at order $\\hbar ^0$ .",
"Furthermore, we showed that the Clifford group gates, generated by the Hadamard, phase shift and controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order.",
"We found that this was because their Hamiltonians are harmonic and take Weyl phase space points to themselves.",
"This proves the Gottesman-Knill theorem.",
"The T-gate, needed to complete a universal set with the Hadamard, was shown not to satisfy these properties, and so requires a path integral treatment that is complete up to $\\hbar ^1$ .",
"The latter treatment includes a sum of terms for which the number of terms scales exponentially with the number of qudits.",
"We note that our observations pertaining to classical propagation in continuous systems have long been very well known.",
"In the continuous case, the Wigner function of a quantum state is non-negative if and only if the state is a Gaussian [34] and it has also long been known that quantum propagation from one Gaussian state to another only requires propagation up to order $\\hbar ^0$ [11].",
"Indeed, it has been shown that this is a continuous version of “stabilizer state propagation” in finite systems [35], and is therefore, in principle, useful for quantum error correction and cluster state quantum computation [36], [37].",
"It is also well known in the discrete case that quadratic Hamiltonians can act classically and be represented by symplectic transformations in the study of quantum cat maps [38], [14], [25] and linear transformations between propagated Wigner functions [39].",
"Interestingly though, this latter work appears to have predated the discovery that stabilizer states have positive-definite Wigner functions [3] and therefore, as far as we know, has not been directly related to stabilizer states and the $\\hbar ^0$ limit of their path integral formulation, which is a relatively recent topic of particular interest to the quantum information community and those familiar with Gottesman-Knill.",
"Otherwise, this claim has been pointed out in terms of concepts related to positivity and related concepts in past work [40], [2].",
"We also note that our exploration of continous systems is not meant to explore the highly related topic of continuous-variable quantum information.",
"Many topics therein apply to our discussion here, such as the continuous stabilizer state propagation we mentioned above.",
"However, our intention in introducing the continuous infinite-dimensional case was not to address these topics but to instead relate the established continuous semiclassical formalism to the discrete case, and thereby bridge the notions of phase space and dynamics between the two worlds.",
"There is an interesting observation to be made of the weights of the reflections that make up the complete path integral formulation of a unitary operator.",
"Namely, as is clear in Eqs.",
"REF and REF , the coefficients consist of the exponentiated center generating function multiplied by $\\frac{i}{\\hbar }$ .",
"This is very similar to the form of the vVMG path integral in Eqs.",
"REF and REF .",
"However, in Eqs.",
"REF and REF , reflections serve as the prefactors measuring the reflection spectral overlap of a propagated state with its evolute and the center generating actions provide the quantal phase.",
"Thus, this formulation can be interpreted as an alternative path integral formulation of the vVMG, one consisting of reflections as the underlying classical trajectory only, instead of the more tailored trajectories that result from applying the method of steepest descents directly on an operator.",
"The fact that any unitary operator in the discrete case can be expressed as a sum consisting of a finite number of order $\\hbar ^1$ path integral contributions, has the added interesting implication that uniformization—higher order $\\hbar $ corrections to the “primitive” semiclassical forms such as Eq.",
"REF —isn't really necessary in discrete systems.",
"Uniformization is characterized by the proper treatment of coalescing saddle points and has long been a subject of interest in continuous systems where “anharmonicity” bedevils computationally efficient implementation.",
"It seems that this problem isn't an issue in the discrete case since a fully complete sum with a finite number of terms, naively numbering $d^2$ for one qudit, exists.",
"As a last point, there is perhaps an alternative way to interpret the results presented here, one in terms of “resources”.",
"Much like “magic” (or contextuality) and quantum discord can be framed as a resource necessary to perform quantum operations that have more power than classical ones, it is possible to frame the order in $\\hbar $ that is necessary in the underlying path integral describing an operation as a resource necessary for quantumness.",
"In this vein, it can be said that Clifford gate operations on stabilizer states are operations that only require $\\hbar ^0$ resources while supplemental gates that push the operator space into universal quantum computing require $\\hbar ^1$ resources.",
"The dividing line between these two regimes, the classical and quantum world, is discrete, unambiguous and well-defined.",
"Acknowledgments The authors thank Prof. Alfredo Ozorio de Almeida for very fruitful discussions about the center-chord representation in discrete systems and Byron Drury for his help proof-reading and bringing [12] to our attention.",
"This work was supported by AFOSR award no.",
"FA9550-12-1-0046.",
"Appendix Gross proved that for odd prime $d$ [41]: Lemma 2 Let $\\Psi $ be a state vector with positive Wigner function for odd prime $d$ .",
"If $\\Psi $ is supported on two points, then it has maximal support.",
"With this lemma in mind, we can offer an alternative proof of Corollary REF : Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .",
"Proof We will show that for odd prime $d$ , every degree of freedom can only be fully supported or a Kronecker delta, for all other degrees of freedom fixed; WLOG we will consider a two-dimensional stabilizer state $\\Psi (q_1,q_2)$ and show that if $\\exists \\, q^{\\prime }_1$ such that $\\Psi (q^{\\prime }_1, q_2) \\ne 0$ $\\forall q_2$ then $\\Psi (q_1,q_2)\\ne 0$ $\\forall \\, q_1, q_2$ and vice-versa (if $\\exists \\, q^{\\prime }_1$ s.t.",
"$\\Psi (q^{\\prime }_1, q_2)$ is a delta function then $\\Psi (q_1,q_2)$ is a delta function in $q_2$ $\\forall q_1$ ).",
"Therefore, if a degree of freedom is maximally supported in one degree of freedom for all others fixed, then it is maximally supported for all values of the other degrees of freedom.",
"On the other hand, if it is a delta function in one degree of freedom for all others fixed, then it is a delta function for all values of the other degrees of freedom.",
"Assume that for $q_1, q_2 \\in \\lbrace 0, \\ldots , d-1\\rbrace $ , $\\exists \\, q^{\\prime }_1, q^{\\prime \\prime }_1$ such that $\\Psi (q^{\\prime }_1,q_2) = 0$ for some $q_2$ and $\\Psi (q^{\\prime \\prime }_1, q_2) \\ne 0$ $\\forall q_2$ .",
"We proceed to prove by contradiction.",
"Hence $\\Psi (q^{\\prime \\prime }_1, q_2) \\equiv \\Psi _{q^{\\prime \\prime }_1} \\propto \\left[ \\frac{2 \\pi i}{d} \\left( \\theta _{q^{\\prime \\prime }_1} q^2_2 + \\eta _{q^{\\prime \\prime }_1} q_2 \\right) \\right]$ by Corollary REF and $\\Psi (q^{\\prime }_1,q_2) \\equiv \\Psi _{q^{\\prime }_1}(q_2) \\propto \\delta _{q_2, q(q^{\\prime }_1)}$ for some $q(q^{\\prime }_1)\\in \\mathbb {Z}/d\\mathbb {Z}$ by [3].",
"We can rotate in $p_2$ -$q_2$ space to form a new basis $q^*_2$ in $d$ discrete angles (since $\\theta _{q^{\\prime \\prime }_1} \\in \\mathbb {Z}/d\\mathbb {Z}$ ) such that $\\Psi _{q^{\\prime }_1}(q^*_2) \\ne 0$ (since a delta function is not maximally supported only at the one angle perpendicular to it).",
"Since there exists $(d-1)$ other values of $q_1$ other than $q^{\\prime }_1$ , it follows that there exists at least one such angle such that $\\Psi _{q_1}(q^*_2) \\ne 0$ $\\forall q_1, q^*_2 \\in \\lbrace 0,\\ldots ,d-1\\rbrace $ .",
"We define $q^*_2$ as the basis that is rotated by this angle with respect to $q_2$ .",
"By Corollary REF , this means that $\\Psi ^*(q_1,q^*_2) \\propto \\exp ( \\theta ^{\\prime }_{11} q^2_1 + \\theta ^{\\prime }_{22} {q^*_2}^2 + 2 \\theta ^{\\prime }_{12} q_1 q^*_2 + \\eta ^{\\prime }_1 q_1 + \\eta ^{\\prime }_2 q_2),$ where by $\\Psi ^*$ we mean $\\Psi $ expressed in the new basis $q^*_2$ in its second degree of freedom.",
"Hence, $\\Psi ^*_{q_1}(q^*_2) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( \\theta ^{\\prime }_{q_1} {q^*_2}^2 + \\eta ^{\\prime }_{q_1} q^*_2 \\right) \\right] \\exp \\left[ \\frac{2 \\pi i}{d} \\theta _{12} q_1 q^*_2 \\right].$ Acting on this last equation to rotate back to $q_2$ , we must produce $\\Psi _{q^{\\prime \\prime }_1}(q_2) \\propto \\delta _{q_2,q(q^{\\prime \\prime }_1)}$ .",
"But then Eq.",
"REF implies that $\\Psi _{q^{\\prime }_1}(q^*_2)$ must also be proportional to $\\delta _{q_2,q(q^{\\prime }_1)}$ .",
"This is a contradiction.",
"Therefore, if a degree of freedom is maximally supported for all others fixed, then it is maximally supported for all values of the other degrees of freedom and vice-versa.",
"In the latter case, a position state in the $i$ th degree of freedom can be represented as a Gaussian by using the $p$ -basis where it becomes a plane wave ($\\theta _i = 0$ ).",
"In other words, one can always choose $x_i$ to be $p_i$ or $q_i$ such that Eq.",
"REF holds for odd prime $d$ .",
"$\\Box $ Finally, the reason this result does not hold for odd non-prime $d$ is that for discrete Wigner space there are $d + 1$ unique angles minus all the prime factors of $d$ .",
"For non-prime $d$ , there is more than one such prime factor and so there are cases when one cannot “rotate” away all non-maximally supported states."
],
[
"Discrete Center-Chord Representation",
"We now proceed to the discrete case and introduce the center-chord formalism for these systems.",
"It will be useful for us to define a pair of conjugate degrees of freedom $p$ and $q$ for discrete systems.",
"Unfortunately, this isn't as straight-forward as in the continuous case, since the usual canonical commutation relations cannot hold in a finite-dimensional Hilbert space where the operators are bounded (since $\\operatorname{Tr}[\\hat{p}, \\hat{q}] = 0$ ).",
"We begin in one degree of freedom.",
"We label the computational basis for our system by $n \\in {0, 1, \\ldots , d-1}$ , for $d$ odd and we assume that $d$ is odd for the rest of this paper.",
"We identify the discrete position basis with the computational basis and define the “boost” operator as diagonal in this basis: $\\hat{Z}^{\\delta p} \\left|n\\right\\rangle \\equiv \\omega ^{n \\delta p} \\left|n\\right\\rangle ,$ where $\\omega $ will be defined below.",
"We define the normalized discrete Fourier transform operator to be equivalent to the Hadamard gate: $\\hat{F} = \\frac{1}{\\sqrt{d}}\\sum _{m,n \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{m n} \\left|m\\right\\rangle \\left\\langle n\\right|.$ This allows us to define the Fourier transform of $\\hat{Z}$ : $\\hat{X} \\equiv \\hat{F}^\\dagger \\hat{Z} \\hat{F}$ Again, as before, we call $\\hat{X}$ the “shift” operator since $\\hat{X}^{\\delta q} \\left|n\\right\\rangle \\equiv \\left|n\\oplus \\delta q\\right\\rangle ,$ where $\\oplus $ denotes mod-$d$ integer addition.",
"It follows that the Weyl relation holds again: $\\hat{Z} \\hat{X} = \\omega \\hat{X} \\hat{Z}.$ The group generated by $\\hat{Z}$ and $\\hat{X}$ has a $d$ -dimensional irreducible representation only if $\\omega ^d=1$ for odd $d$ .",
"Equivalently, there are only reflections relating any two phase space points on the Weyl phase space “grid” if $d$ is odd [25].",
"We take $\\omega \\equiv \\omega (d) = e^{2 \\pi i/d}$ [26].",
"This was introduced by Weyl [27].",
"Note that this means that $\\hbar = \\frac{d}{2 \\pi }$ or $h = d$ .",
"This means that the classical regime is most closely reached when the dimensionality of the system is reduced ($d \\rightarrow 0$ ) and thus the most “classical” system we can consider here is a qutrit (since we keep $d$ odd and greater than one).",
"This is the opposite limit considered by many other approaches where the classical regime is reached when $d \\rightarrow \\infty $ .",
"One way of interpreting the classical limit in this paper is by considering $h$ to be equal to the inverse of the density of states in phase space (i.e.",
"in a Wigner unit cell).",
"As $\\hbar \\rightarrow 0$ phase space area decreases as $\\hbar ^2$ but the number of states only decreases as $\\hbar $ leading to an overall density increase of $\\hbar $ .",
"This agrees with the notion that states should become point particles of fixed mass in the classical limit.",
"By analogy with continuous finite translation operators, we reexpress the shift $\\hat{X}$ and boost $\\hat{Z}$ operators in terms of conjugate $\\hat{ p}$ and $\\hat{ q}$ operators: $ \\hat{Z} \\left|n\\right\\rangle = e^{\\frac{2 \\pi i }{d}\\hat{q}} \\left|n\\right\\rangle = e^{\\frac{2 \\pi i}{d} n} \\left|n\\right\\rangle , $ and $ \\hat{X} \\left|n\\right\\rangle = e^{-\\frac{2 \\pi i }{d} \\hat{p}} \\left|n\\right\\rangle = \\left|n\\oplus 1\\right\\rangle .",
"$ Hence, in the diagonal “position” representation for $\\hat{Z}$ : $\\hat{Z} = \\left( \\begin{array}{cccccc} 1 & 0 & 0 & \\cdots & 0\\\\0 & e^{\\frac{2 \\pi i}{d}} & 0 & \\cdots & 0\\\\0 & 0 & e^{\\frac{4 \\pi i}{d}} & \\cdots & 0\\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & 0 & e^{\\frac{2 (d-1) \\pi i}{d}} \\end{array} \\right),$ and $\\hat{X} = \\left( \\begin{array}{cccccc} 0 & 0 & \\cdots & 0 & 1\\\\1 & 0 & \\cdots & 0 & 0\\\\0 & 1 & 0 & \\cdots & 0\\\\\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\0 & \\cdots & 0 & 1 & 0 \\end{array} \\right).$ Thus, $\\hat{q} = \\frac{d}{2 \\pi i} \\log \\hat{Z} = \\sum _{n \\in \\mathbb {Z}/d\\mathbb {Z}} n \\left|n\\right\\rangle \\left\\langle n\\right|,$ and $\\hat{ p} = \\hat{F}^\\dagger \\hat{ q} \\hat{F}.$ Therefore, we can interpret the operators $\\hat{ p}$ and $\\hat{ q}$ as a conjugate pair similar to conjugate momenta and position in the continuous case.",
"However, they differ from the latter in that they only obey the weaker group commutation relation $e^{i j \\hat{ q}/\\hbar }e^{i k \\hat{ p}/\\hbar } e^{-i j \\hat{ q}/\\hbar }e^{-i k \\hat{ p}/\\hbar } = e^{-i jk /\\hbar } \\hat{\\mathbb {I}}.$ This corresponds to the usual canonical commutation relation for $p$ and $q$ 's algebra at the origin of the Lie group ($j = k = 0$ ); expanding both sides of Eq.",
"REF to first order in $p$ and $q$ yields the usual canonical relation.",
"We proceed to introduce the Weyl representation of operators and states in discrete Hilbert spaces with odd dimension $d$ and $n$ degrees of freedom [1], [28], [29].",
"The generalized phase space translation operator (the Weyl operator) is defined as a product of the shift and boost with a phase appropriate to the $d$ -dimensional space: $\\hat{T}(\\xi _p, \\xi _q) = e^{-i \\frac{\\pi }{d} \\xi _p \\cdot \\xi _q} \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q},$ where $\\xi \\equiv (\\xi _p, \\xi _q) \\in (\\mathbb {Z} / d \\mathbb {Z})^{2n}$ and form a discrete “web” or “grid” of chords.",
"They are a discrete subset of the continous chords we considered in the infinite-dimensional context in Section and their finite number is an important consequence of the discretization of the continuous Weyl formalism.",
"Again, an operator $\\hat{A}$ can be expressed as a linear combination of translations: $\\hat{A} = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} A_\\xi (\\xi _p, \\xi _q) \\hat{T}(\\xi _p, \\xi _q),$ where the weights are the chord representation of the function $\\hat{A}$ : ${A}_\\xi (\\xi _p, \\xi _q) = d^{-n} \\operatorname{Tr}\\left( \\hat{T}(\\xi _p, \\xi _q)^\\dagger \\hat{A} \\right).$ When applied to a state $\\hat{\\rho }$ , this is also called the “characteristic function” of $\\hat{\\rho }$ [30].",
"As before, the center representation, based on reflections instead of translations, requires an appropriately defined reflection operator.",
"We can define the discrete reflection operator $\\hat{R}$ as the symplectic Fourier transform of the discrete translation operator we just introduced: $\\hat{R}(x_p, x_q) = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{\\frac{2 \\pi i}{d} (\\xi _p, \\xi _q) \\mathcal {J} (x_p, x_q)^T} \\hat{T}(\\xi _p, \\xi _q).$ With this in hand, we can now express a finite-dimensional operator $\\hat{A}$ as a superposition of reflections: $\\hat{A} = d^{-n} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} {A}_x(x_p, x_q) \\hat{R}( x_p, x_q ),$ where ${A}_x(x_p, x_q) = d^{-n} \\operatorname{Tr}\\left( \\hat{R}(x_p, x_q)^\\dagger \\hat{A} \\right).$ $x \\equiv (x_p, x_q) \\in (\\mathbb {Z} / d \\mathbb {Z})^{2n}$ are centers or Weyl phase space points and, like their $(\\xi _p, \\xi _q)$ brethren, form a discrete subgrid of the continuous Weyl phase space points considered in Section .",
"Again, the center representation is of particular interest to us because for unitary gates $\\hat{A}$ we can rewrite the components ${A}_x(x_p, x_q)$ as: ${A}_x(x_p, x_q) = \\exp \\left[\\frac{i}{\\hbar } S(x_p, x_q)\\right]$ where $S(x_p, x_q)$ is the argument to the exponential, and is equivalent to the action of the operator in center representation (the center generating function).",
"Aside from Eq.",
"REF , the center representation of a state $\\hat{\\rho }$ can also be directly defined as the symplectic Fourier transform of its chord representation, $\\rho _\\xi $ [3]: $\\rho _x(x_p, x_q) = d^{-n} \\sum _{\\begin{array}{c}\\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{\\frac{2 \\pi i}{d} (\\xi _p, \\xi _q) \\mathcal {J} (x_p, x_q)^T} \\rho _\\xi (\\xi _p,\\xi _q).$ We note again that for a pure state $\\left|\\Psi \\right\\rangle $ , the Wigner function from Eqs.",
"REF and REF simplifies to: ${\\Psi }_x(x_p, x_q) &=& d^{-n} \\sum _{\\begin{array}{c}\\xi _q \\in \\\\(\\mathbb {Z} / d \\mathbb {Z})^{ n}\\end{array}} e^{-\\frac{2 \\pi i}{d} \\xi _q \\cdot x_p} \\Psi \\left( x_q + \\frac{(d+1) \\xi _q }{2} \\right) {\\Psi ^*} \\left( x_q - \\frac{(d+1) \\xi _q}{2} \\right).$ It may be clear from this short presentation that the chord and center representations are dual to each other.",
"A thorough review of this subject can be found in [13]."
],
[
"Path Integral Propagation in Discrete Systems",
"Rivas and Almeida [14] found that the continuous infinite-dimensional vVMG propagator can be extended to finite Hilbert space by simply projecting it onto its finite phase space tori.",
"This produces: $&& {U}_t(x) = \\\\&& \\left< \\sum _j \\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} e^{\\frac{i}{\\hbar } S_{tj}(x)} e^{i \\theta _k} \\right>_k + \\mathcal {O}(\\hbar ^2),\\nonumber $ where an additional average must be taken over the $k$ center points that are equivalent because of the periodic boundary conditions.",
"Maintaining periodicity requires that they accrue a phase $\\theta _k$ See Eq.",
"$5.18$ in [14] for a definition of the phase..",
"The derivative $\\frac{\\partial ^2 S_{tj}}{\\partial x^2}$ is performed over the continuous function $S_{tj}$ defined after Eq.",
"REF , but only evaluated at the discrete Weyl phase space points $x \\equiv (x_p, x_q)\\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{2n}$ .",
"The prefactor can also be reexpressed: $\\left\\lbrace \\det \\left[ 1 + \\frac{1}{2} \\mathcal {J} \\frac{\\partial ^2 S_{tj}}{\\partial x^2} \\right] \\right\\rbrace ^{\\frac{1}{2}} = \\left\\lbrace 2^d \\det \\left[ 1 + \\mathcal {M}_{tj} \\right] \\right\\rbrace ^{-\\frac{1}{2}},$ which is perhaps more pleasing in the discrete case as it does not involve a continuous derivative.",
"As in the continuous case, for a harmonic Hamiltonian $H(p, q)$ , the center generating function $S(x_p, x_q)$ is equal to $\\alpha ^T \\mathcal {J} x + x^T \\mathcal {B} x$ where Eq.",
"REF and Eq.",
"REF hold.",
"Moreover, if the Hamiltonian takes Weyl phase space points to themselves, then by the same equations it follows that $\\mathcal {M}$ and $\\alpha $ must have integer entries.",
"This implies that for $m, n \\in \\mathbb {Z}^{ n}$ , $&& \\mathcal {M} \\left(\\begin{array}{c}p+ m d + \\alpha _p/2\\\\q + n d + \\alpha _q/2\\end{array}\\right) + \\left( \\begin{array}{c} \\alpha _p/2\\\\ \\alpha _q/2 \\end{array} \\right) \\nonumber \\\\&=& \\mathcal {M} \\left(\\begin{array}{c}p + \\alpha _p/2 \\\\ q + \\alpha _q/2 \\end{array}\\right) + \\left( \\begin{array}{c} \\alpha _p/2\\\\ \\alpha _q/2 \\end{array} \\right) + d \\mathcal {M}\\left(\\begin{array}{c}m\\\\ n\\end{array}\\right) \\\\&=& \\left(\\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array}\\right) \\mod {d. \\nonumber }Therefore, phase space points (p, q) that lie on Weyl phase space points go to the equivalent Weyl phase space points (p^{\\prime }, q^{\\prime }).$ Moreover, again if $m, n \\in \\mathbb {Z}^n$ , $&& S(x_p + m d, x_q + n d) \\nonumber \\\\&=& \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right)^T A \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right) \\nonumber \\\\&& + b \\cdot \\left( \\begin{array}{c}x_p+ m d\\\\ x_q+ n d\\end{array} \\right)\\\\&=& \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right)^T A \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right) + b \\cdot \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right) \\nonumber \\\\&& + d \\left[ 2 \\left( \\begin{array}{c}x_p\\\\ x_q\\end{array} \\right)^T A \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right) \\right.\\nonumber \\\\&& \\qquad \\left.+ d \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right)^T A \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right) + b \\cdot \\left( \\begin{array}{c}m\\\\ n\\end{array} \\right)\\right] \\nonumber \\\\&=& S(x_p, x_q) \\mod {d}, \\nonumber $ for some symmetric $A \\in \\mathbb {Z}^{n\\times n}$ and $b \\in \\mathbb {Z}^{n}$ .",
"Therefore, these equivalent trajectories also have equivalent actions (since the action is multiplied by $\\frac{2 \\pi i}{d}$ and exponentiated).",
"Hence, there is only one term to the sum in Eq.",
"REF .",
"Moreover, [25] showed that the sum over the phases $\\theta _k$ produces only a global phase that can be factored out.",
"Therefore, if we can neglect the overall phase, ${U}_t(x) = \\left| 2^d \\det \\left[ 1 + \\mathcal {M} \\right] \\right|^{\\frac{1}{2}},$ where the classical trajectories whose centers are $(x_p, x_q)$ satisfy the periodic boundary conditions.",
"As in the continuous case, we point out that this means that translations and reflections are fully captured by a path integral treatment that is truncated at order $\\hbar ^1$ (or order $\\hbar ^0$ if their overall phase isn't important) because their Hamiltonians are harmonic, but in the discrete case there is an additional requirement that they are evaluated at chords/centers that take Weyl phase space points to themselves.",
"Just as in the continuous case, the single contribution at order $\\hbar ^0$ implies that the propagator of the Wigner function of states, $\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(x)$ under gates $\\hat{V}$ with underlying harmonic Hamiltonians is captured by $\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ for $\\mathcal {M}_{\\hat{V}}$ and $\\alpha _{\\hat{V}}$ associated with $\\hat{V}$ ."
],
[
"Stabilizer Group",
"Here we will show that the Hamiltonians corresponding to Clifford gates are harmonic and take Weyl phase space points to themselves.",
"Thus they can be captured by only the single contribution of Eq.",
"REF at lowest order in $\\hbar $ .",
"This then implies that stabilizer states can also be propagated to each other by Clifford gates with only a single contribution to the sum in Eq.",
"REF .",
"The Clifford gate set of interest can be defined by three generators: a single qudit Hadamard gate $\\hat{F}$ and phase shift gate $\\hat{P}$ , as well as the two qudit controlled-not gate $\\hat{C}$ .",
"We examine each of these in turn."
],
[
"Hadamard Gate",
"The Hadamard gate was defined in Eq.",
"REF and is a rotation by $\\frac{\\pi }{2}$ in phase space counter-clockwise.",
"Hence, for one qudit, it can be written as the map in Eq.",
"REF where $\\mathcal {M}_{\\hat{F}} = \\left( \\begin{array}{cc} 0 & 1\\\\ -1 & 0 \\end{array} \\right),$ and $\\alpha _{\\hat{F}} = (0,0)$ .",
"We have set $t=1$ and drop it from the subscripts from now on.",
"Since $\\alpha $ is vanishing and $\\mathcal {M}$ has integer entries, this is a cat map and such maps have been shown to correspond to Hamiltonians [32] $&&H(p,q) =\\\\&&f(\\operatorname{Tr}\\mathcal {M} ) \\left[ \\mathcal {M}_{12} p^2 - \\mathcal {M}_{21} q^2 + \\left(\\mathcal {M}_{11} - \\mathcal {M}_{22} \\right) pq \\right],\\nonumber $ where $f(x) = \\frac{\\sinh ^{-1}(\\frac{1}{2}\\sqrt{x^2-4})}{\\sqrt{x^2-4}}.$ For the Hadamard $\\mathcal {M}_{\\hat{F}}$ this corresponds to $H_{\\hat{F}} = \\frac{\\pi }{4} (p^2 + q^2)$ , a harmonic oscillator.",
"The center generating function $S(x_p, x_q)$ is thus $(x_p, x_q ) \\mathcal {B} (x_p, x_q)^T$ and solving Eq.",
"REF finds for the one-qudit Hadamard, $\\mathcal {B}_{\\hat{F}} = \\left(\\begin{array}{cc}1 & 0\\\\ 0 & 1 \\end{array} \\right).$ Thus, $S_{\\hat{F}}(x_p, x_q) = x_p^2 + x_q^2$ .",
"Indeed, applying Eq.",
"REF to Eq.",
"REF reveals that the Hadamard's center function (up to a phase) is: ${F}_x(x_p,x_q) = e^{\\frac{2 \\pi i}{d}(x_p^2+x_q^2)}.$ Eq.",
"REF shows how to map Weyl phase space to Weyl phase space under the Hadamard transformation.",
"Furthermore this map is pointwise, which implies the quadratic form of the center generating function obtained in Eq.",
"REF ."
],
[
"Phase Shift Gate",
"The phase shift gate can be generalized to odd $d$ -dimensions [33] by setting it to: $\\hat{P} = \\sum _{j \\in \\mathbb {Z}/d \\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{2}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ Examining its effect on stabilizer states, it is clear that it is a $q$ -shear in phase space from an origin displaced by $\\frac{d-1}{2}\\equiv -\\frac{1}{2}$ to the right.",
"This can be expressed as the map in Eq.",
"REF with $\\mathcal {M}_{\\hat{P}} = \\left( \\begin{array}{cc} 1 & 1\\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{P}} = \\left( -\\frac{1}{2}, 0 \\right)$ .",
"This corresponds to $\\mathcal {B}_{\\hat{P}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & \\frac{1}{2} \\end{array} \\right).$ Solving Eq.",
"REF with this $\\mathcal {B}_{\\hat{P}}$ and $\\alpha _{\\hat{P}}$ reveals that $S_{\\hat{P}}(x_p, x_q) = -\\frac{1}{2} x_q + \\frac{1}{2} x_q^2$ .",
"Again, this agrees with the argument of the center representation of the phase-shift gate obtained by applying Eq.",
"REF to Eq.",
"REF : ${P}_x(x_p,x_q) = e^{\\frac{2 \\pi i}{d} \\frac{1}{2} (-x_q + x_q^2)}.$ Discretization the equations of motion for harmonic evolution for unit timesteps leads to: $\\left(\\begin{array}{c}p^{\\prime }\\\\ q^{\\prime }\\end{array}\\right) = \\left(\\begin{array}{c}p\\\\ q\\end{array}\\right) + \\mathcal {J} \\left(\\frac{\\partial H}{\\partial p}, \\frac{\\partial H}{\\partial q}\\right)^T,$ where the last derivative is on the continuous function $H$ , but only evaluated on the discrete Weyl phase space points.",
"It follows that $H_{\\hat{P}} = -\\frac{d+1}{2} q^2 + \\frac{d+1}{2} q.$ We have obtained the Hamiltonian for the phase-shift gate by a different procedure than that used for the Hadamard where we appealed to the result given in Eq.",
"REF for quantum cat maps.",
"However, as the phase-shift gate is a quantum cat map as well, we could have obtained Eq.",
"REF in this manner.",
"Similarly, the approach we used to find the phase-shift Hamiltonian by discretizing time in Eq.",
"REF would work for the Hadamard gate but it is a bit more involved since the latter contains both $p$ - and $q$ -evolution.",
"Nevertheless, this produces Eq.",
"REF as well.",
"We presented both techniques for illustrative purposes."
],
[
"Controlled-Not Gate",
"Lastly, the controlled-not gate can be generalized to $d$ -dimensions [33] by $\\hat{C} = \\sum _{j,k \\in \\mathbb {Z}/d \\mathbb {Z}} \\left|j,k \\oplus j\\right\\rangle \\left\\langle j,k\\right|.$ It is clear that this translates the $q$ -state of the second qudit by the $q$ -state of the first qudit.",
"As a result, as is evident by examining the gate's action on stabilizer states, the first qudit experiences an “equal and opposite reaction” force that kicks its momentum by the $q$ -state of the second qudit.",
"This is the phase space picture of the well-known fact that a CNOT examined in the $\\hat{X}$ basis has the control and target reversed with respect to the $\\hat{Z}$ basis.",
"This can also seen by looking at its effect in the momentum ($\\hat{X}$ ) basis: ${\\hat{F}}^\\dagger \\hat{C} \\hat{F} = \\sum _{j,k \\in \\mathbb {Z}/d \\mathbb {Z}} \\left|j \\ominus k,k\\right\\rangle \\left\\langle j,k\\right|.$ As a result, this gate is described by the map: $\\mathcal {M}_{\\hat{C}} = \\left( \\begin{array}{ccccc} 1 & -1 & 0 & 0\\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1 & 0\\\\ 0 & 0 & 1 & 1\\end{array}\\right),$ and $\\alpha _{\\hat{C}} = (0,0,0,0)$ .",
"This corresponds to $\\mathcal {B}_{\\hat{C}} = \\left( \\begin{array}{ccccc} 0 & 0 & 0 & 0\\\\ 0 & 0 & -\\frac{1}{2} & 0\\\\ 0 & -\\frac{1}{2} & 0 & 0\\\\ 0 & 0 & 0 & 0\\end{array}\\right).$ Hence its center generating function $S_{\\hat{C}}(x_p, x_q) = -x_{p_2} x_{q_1}$ .",
"Again, this corresponds with the argument of the center representation of the controlled-not gate, which can be found to be: ${C}_x(x_p, x_q) = e^{-\\frac{2 \\pi i}{d} x_{q_1} x_{p_2}}.$ Therefore, this gate can be seen to be a bilinear $p$ -$q$ coupling between two qudits and corresponds to the Hamiltonian $H_{\\hat{C}} = p_1 q_2,$ as can be found from Eq.",
"REF again.",
"As a result, it is now clear that all the Clifford group gates have Hamiltonians that are harmonic and that take Weyl phase space points to themselves.",
"Therefore, their propagation can be fully described by a truncation of the semiclassical propagator Eq.",
"REF to order $\\hbar ^0$ as in Eq.",
"REF and they are manifestly classical in this sense.",
"To summarize the results of this section, using Eq.",
"REF , the Hadamard, phase shift, and CNOT gates can be written as: $\\hat{F} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d \\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ -(x_p^2 + x_q^2) - d \\left(x_p \\xi _p - x_q \\xi _q \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q},$ $\\hat{P} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d \\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{2} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q},$ and $&&\\hat{C} = \\\\&& d^{-4} \\sum _{\\begin{array}{c}x_p, x_q, \\\\ \\xi _p, \\xi _q \\in \\\\ (\\mathbb {Z}/d \\mathbb {Z})^{ 2}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ x_{q_1} x_{p_2} - d \\left(x_p \\cdot \\xi _q - x_q \\cdot \\xi _p \\right) \\right] } \\hat{Z}^{ \\xi _p} \\hat{X}^{ \\xi _q}, \\nonumber $ (up to a phase).",
"This form emphasizes their quadratic nature.",
"As for the continuous case, there exists a particularly simple way in discrete systems to see to what order in $\\hbar $ the path integral must be kept to handle unitary propagation beyond the Clifford group.",
"We describe this in the next section.",
"We note that the center generating actions $S(x_p, x_q)$ found here are related to the $G(q^{\\prime }, q)$ found by Penney et al.",
"[12], which are in terms of initial and final positions, by symmetrized Legendre transform [13]: $G(q^{\\prime }, q, t) = F\\left(\\frac{q^{\\prime } + q}{2}, p(q^{\\prime } - q)\\right),$ where the canonical generating function $F\\left(\\frac{q^{\\prime } + q}{2}, p\\right) = S\\left( x_p = p, x_q = \\frac{q^{\\prime } + q}{2} \\right) + p \\cdot \\left( q^{\\prime } - q \\right),$ for $p (q^{\\prime } - q)$ given implicitly by $\\frac{\\partial F}{\\partial p} = 0$ .",
"Applying this to the actions we found reveals that $G_{\\hat{F}}(q^{\\prime }, q, t) = q^{\\prime } q,$ $G_{\\hat{P}}(q^{\\prime }, q, t) = \\frac{d+1}{2} (q^2 - q),$ and $G_{\\hat{C}}((q^{\\prime }_1, q^{\\prime }_2), (q_1, q_2), t) = 0,$ which is in agreement with [12]."
],
[
"Classicality of Stabilizer States",
"In this subsection we show that stabilizer states evolve to stabilizer states under Clifford gates, and that it is possible to describe this evolution classically.",
"For odd $d \\ge 3$ the positivity of the Wigner representation implies that evolution of stabilizer states is non-contextual, and so here we are investigating in detail what this means in our semiclassical picture.",
"To begin, it is instructive to see the form stabilizer states take in the discrete position representation and in the center representation.",
"Gross proved that [3]: Theorem 1 Let $d$ be odd and $\\Psi \\in L^2((\\mathbb {Z}/d\\mathbb {Z})^n)$ be a state vector.",
"The Wigner function of $\\Psi $ is non-negative if and only if $\\Psi $ is a stabilizer state.",
"Gross also proved [3] Corollary 1 Given that $\\Psi (q) \\ne 0$ $\\forall \\, q$ , a vector $\\Psi $ is a stabilizer state if and only if it is of the form $\\Psi _{\\theta _\\beta ,\\eta _\\beta }(q) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( q^T \\theta _\\beta q + \\eta _\\beta \\cdot q \\right) \\right].$ where $\\theta _\\beta \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{n\\times n}$ and $q, \\eta _\\beta \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^n$ .",
"Applying Eq.",
"REF to Eq.",
"REF , the Wigner function of such maximally supported stabilizer states can be found to be: $&& {\\Psi _{\\theta _\\beta ,\\eta _\\beta }}_x(x_p, x_q) \\propto \\\\&& d^{-n} \\sum _{\\begin{array}{c}\\xi _q \\in \\\\\\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{ n}\\end{array}} \\exp \\left[\\frac{2 \\pi i}{d} \\xi _q \\cdot \\left( \\eta _\\beta - x_p + 2 \\theta _\\beta x_q \\right)\\right].\\nonumber $ Therefore, one finds that the Wigner function is the discrete Fourier sum equal to $\\delta _{\\eta _\\beta - x_p + 2\\theta _\\beta x_q}$ .",
"For $\\theta _\\beta = 0$ the state is a momentum state at $x_p$ .",
"Finite $\\theta _\\beta $ rotates that momentum state in phase space in “steps” such that it always lies along the discrete Weyl phase space points $(x_p, x_q) \\in (\\mathbb {Z}/d\\mathbb {Z})^{2n}$ .",
"This Gaussian expression only captures stabilizer states that are maximally supported in $q$ -space.",
"One may wonder what the stabilizer states that aren't maximally supported in $q$ -space look like in Weyl phase space.",
"Of course, it is possible that some may be maximally supported in $p$ -space and so can be captured by the following corollary: Corollary 2 If $\\Psi (p) \\ne 0$ for all $p$ 's then there exists a $\\theta _{\\beta p} \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^{n\\times n}$ and an $\\eta _{\\beta p} \\in \\left(\\mathbb {Z}/d\\mathbb {Z}\\right)^n$ such that $\\Psi _{\\theta _{\\beta p},\\eta _{\\beta p}}(p) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( p^T \\theta _{\\beta p} p + \\eta _{\\beta p} \\cdot p \\right) \\right].$ Proof This can be shown following the same methods employed by Gross [3] but in the discrete $p$ -basis.",
"Unfortunately, it is easy to show that Corollary REF and REF do not provide an expression for all stabilizer states (except for the odd prime $d$ case, as we shall see shortly) as there exist stabilizer states for odd non-prime $d$ that are not maximally supported in $p$ - or $q$ -space or any finite rotation between those two.",
"To find an expression that encompasses all stabilizer states, we must turn to the Wigner function of stabilizer states.",
"An equivalent definition of stabilizer states on $n$ qudits is given by states $\\hat{V} \\underbrace{\\left|0\\right\\rangle \\otimes \\cdots \\otimes \\left|0\\right\\rangle }_{n}$ where $\\hat{V}$ is a quantum circuit consisting of Clifford gates.",
"We know that the Clifford circuits are generated by the $\\hat{P}$ , $\\hat{F}$ and $\\hat{C}$ gates, and that the Wigner functions $\\Psi _x(x)$ of stabilizer states propagate under $\\hat{V}$ as $\\Psi _x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , it follows that the Wigner function of stabilizer states is: $\\delta _{\\Phi _0 \\cdot \\mathcal {M}_{\\hat{V}} \\cdot x, r_0},$ where $\\Phi _0 = \\left(\\begin{array}{cc} 0 & 0\\\\ 0 & \\mathbb {I}_n\\end{array} \\right)$ and $r_0 = (0, 0)$ .",
"We have therefore proved the next theorem: Theorem 2 The Wigner function $\\Psi _x(x)$ of a stabilizer state for any odd $d$ and $n$ qudits is $\\delta _{\\Phi \\cdot x, r}$ for $2n \\times 2n$ matrix $\\Phi $ and $2n$ vector $r$ .",
"As an aside, Theorem REF allows us to develop an all-encompassing Gaussian expression for stabilizer states for the restricted case that $d$ is odd prime.",
"In this case, the following Corollary shows that a “mixed” representation is always possible: where each degree of freedom is expressed in either the $p$ - or $q$ -basis: Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state for $n$ qudits, then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .",
"Proof We begin with a one-qudit case.",
"We examine the equation specified by $\\Phi \\cdot x = r$ : $\\alpha q_1 + \\beta p_1 = \\gamma $ for $\\alpha $ , $\\beta $ , and $\\gamma \\in \\mathbb {Z}/d\\mathbb {Z}$ .",
"If $\\alpha = 0$ then $\\Psi (q_1)$ is maximally supported and if $\\beta = 0$ then $\\Psi (p_1)$ is maximally supported since the equation specifies a line on $\\mathbb {Z}/d \\mathbb {Z}$ in $q_1$ and $p_1$ respectively.",
"If $\\alpha \\ne 0$ and $\\beta \\ne 0$ then the equation can be rewritten as $q_1 + (\\beta /\\alpha ) p_1 = \\gamma /\\alpha ,$ and it follows that $q_1$ can take any values on $\\mathbb {Z}/d \\mathbb {Z}$ and so $\\Psi (q_1)$ is maximally supported.",
"However, it is also possible to reexpress the equation as: $p_1 + (\\alpha /\\beta ) q_1 = \\gamma /\\beta .$ It follows that $p_1$ can also take any values on $\\mathbb {Z}/d \\mathbb {Z}$ —$\\Psi (p_1)$ is also maximally supported.",
"Therefore, one can always choose either a $p_1$ - or $q_1$ -basis such that the state is maximally supported and so is representable by a Gaussian function.",
"We now consider adding another qudit such that the state becomes ${\\Psi }_x(p_1,p_2,q_1,q_2)$ .",
"There are now two equations specified by $\\Phi \\cdot x = r$ and it follows that it is always possible to combine the two equations such that $p_1$ and $q_1$ are only in one equation and written in terms of each other (and generally the second degree of freedom): $\\alpha q_1 + \\beta p_1 + \\gamma q_2 + \\delta p_2 = \\epsilon ,$ for $\\alpha $ , $\\beta $ , $\\gamma $ , $\\delta $ and $\\epsilon \\in \\mathbb {Z}/d\\mathbb {Z}$ .",
"It will turn out that the $\\gamma q_2 + \\delta p_2$ term is irrelevant.",
"We can rewrite the above equation as: $q_1 + (\\beta /\\alpha ) p_1 + (\\gamma /\\alpha ) q_2 + (\\delta /\\alpha ) p_2 = \\epsilon /\\alpha ,$ if $\\alpha \\ne 0$ .",
"Since there is no other equation specifying $p_1$ , this is an equation for a line on $\\mathbb {Z}/ d \\mathbb {Z}$ and so $\\Psi $ is is maximally supported on $q_1$ .",
"Otherwise, rewriting the above equation as: $p_1 + (\\alpha /\\beta ) q_1 +(\\gamma /\\beta ) q_2 + (\\delta /\\beta ) p_2 = \\epsilon /\\beta ,$ if $\\beta \\ne 0$ shows that $\\Psi $ is maximally supported on $p_1$ .",
"If $\\alpha = \\beta = 0$ then both $p_1$ and $q_1$ are undetermined and so either representation produces a maximally supported state.",
"The same procedure can be performed to find if $q_2$ or $p_2$ produce a maximally supported state.",
"As can be seen, we are really just repeating the same procedure as we did when there was only one qudit because the other degrees of freedom have no impact on this determination.",
"Expressing $\\Psi $ in the basis that is maximally supported in every degree of freedom means that it is therefore a Gaussian.",
"Therefore, it follows that every degree of freedom (corresponding to a qudit) is maximally supported in either the $p$ - or $q$ - basis and so Eq.",
"REF always describes stabilizer states for odd prime $d$ .$\\Box $ The form of Eq.",
"REF is more general than Eq.",
"REF because it does not depend on the support of the state.",
"As we saw in the proof, this representation is generally not unique; for every qudit $i$ that is not a position or momentum state, $x_i$ can be either $p_i$ or $q_i$ .",
"However, if it is a position state then $x_i = p_i$ and if it is a momentum state then $x_i = q_i$ ; position and momentum states must be expressed in their conjugate representation in order to be captured by a Gaussian of the form in Eq.",
"REF instead of Kronecker deltas.",
"The reason this mixed representation doesn't hold for non-prime odd $d$ is that the coefficients above can be (multiples of) prime factors of $d$ and so no longer produce “lines” in $p_i$ or $q_i$ that cover all of $\\mathbb {Z}/d \\mathbb {Z}$ .",
"An alternative proof of this corollary that explores this case further is presented in the Appendix.",
"An example of the different classes of stabilizer states that are possible for odd prime $d$ , in terms of their support, is shown in Fig.",
"REF .",
"There it can be seen that a stabilizer state is either maximally supported in $p_i$ or $q_i$ , and is a Kronecker delta function in the other degree of freedom, or it is maximally supported in both.",
"Figure: The two classes of stabilizer states possible for odd prime d=7d=7 in terms of support: a) maximally supported in pp or qq and b) maximally supported in pp and qq.",
"The central grids denote the Wigner function ΨΨ x (p,q)\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(p,q) of a stabilizer state Ψ\\Psi with d=7d=7.",
"The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\Psi (q)|^2).On the other hand, for odd non-prime $d$ , we see in Fig.",
"REF that another class is possible: stabilizer states that are maximally supported in neither $p_i$ or $q_i$ .",
"In fact, rotating the basis in any of the discrete angles afforded by the grid still does not produce a basis that is maximally supported (as discussed in the Appendix).",
"Notice also, that Fig.",
"REF b shows that it is no longer true that a state that is maximally supported in only $q_i$ or $p_i$ is automatically a Kronecker delta when expressed in terms of the other.",
"Figure: The four classes of stabilizer states possible for odd non-prime d=15d=15 in terms of support: a) & b) maximally supported in pp or qq, c) maximally supported in pp and qq, and d) not maximally supported in pp or qq.",
"The central grids denote the Wigner function ΨΨ x (p,q)\\left|\\Psi \\right\\rangle \\left\\langle \\Psi \\right|_x(p,q) for a stabilizer state Ψ\\Psi with d=15d=15.",
"The projection of this state onto pp-space is shown in the upper right (|Ψ(p)| 2 |\\Psi (p)|^2) and the projection onto qq-space is shown in the upper left (|Ψ(q)| 2 |\\Psi (q)|^2).In summary, stabilizer states have Wigner function $\\delta _{\\Phi \\cdot x, r}$ and, for odd prime $d$ , are Gaussians in mixed representation that lie on the Weyl phase space points $(x_p, x_q)$ .",
"Heuristically, they correspond to Gaussians in the continuous case that spread along their major axes infinitely.",
"The only reason that they aren't always expressible as Gaussians in the mixed representation is that they sometimes “skip” over some of the discrete grid points due to the particular angle they lie along phase space for odd non-prime $d$ .",
"Wigner functions $\\Psi _x(x)$ of stabilizer states propagate under $\\hat{V}$ as $\\Psi _x(\\mathcal {M}_{\\hat{V}} (x + \\alpha _{\\hat{V}}/2) + \\alpha _{\\hat{V}}/2)$ , and this preserves the form of the state.",
"In other words, Clifford gates take stabilizer states to other stabilizer states, as expected, just like in the continuous case Gaussians go to other Gaussians under harmonic evolution.",
"It is also clear that stabilizer state propagation under Clifford gates can be expressed by a path integral at order $\\hbar ^0$ .",
"Discrete Phase Space Representation of Universal Quantum Computing A similar statement to the one we made in Section —that any operator can be expressed as an infinite sum of path integral contribution truncated at order $\\hbar ^1$ —can be made in discrete systems.",
"However, there is an important difference in the number of terms making up the sum.",
"To see this we can follow reasoning that is similar to that employed in the continuous case.",
"Namely, from Eq.",
"REF we see that any discrete operator can also be expressed as a linear combination of reflections, but unlike the continuous case, this sum has a finite number of terms.",
"Since reflections can be expressed fully by the discrete path integral truncated at order $\\hbar ^1$ , as discussed previously, it follows that any unitary operator in discrete systems can be expressed as a finite sum of contributions from path integrals truncated at order $\\hbar ^1$ .",
"Again, the same statement can be made by considering the chord representation in terms of translations.",
"Hence, quantum propagation in discrete systems can be fully treated by a finite sum of contributions from a path integral approach truncated at order $\\hbar ^1$ .",
"To gather some understanding of this statement, we can consider what is necessary to add to our path integral formulation when we complete the Clifford gates with the T-gate, which produces a universal gate set.",
"The T-gate is generalized to odd $d$ -dimensions by $\\hat{T} = \\sum _{j \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{4}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ This gate can no longer be characterized by an $\\mathcal {M}$ with integer entries.",
"In particular, $\\mathcal {M}_{\\hat{T}} = \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{T}} = \\left( -\\frac{1}{4}, 0 \\right)$ .",
"This corresponds to $\\mathcal {B}_{\\hat{T}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & -\\frac{1}{4}\\end{array} \\right).$ Thus, the center function $T_x(x_p,x_q) = e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)},$ corresponding to the phase shift Hamiltonian applied for only half the unit of time.",
"The operator can thus be written: $\\hat{T} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d\\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{4} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q}.$ Though this operator is quadratic, it no longer takes the Weyl center points to themselves.",
"This means that the $\\hbar ^0$ limit of Eq.",
"REF is now insufficient to capture all the dynamics because the overlap with any $\\left|q\\right\\rangle $ will now involve a linear superposition of partially overlapping propagated manifolds.",
"It must therefore be described by a path integral formulation that is complete to order $\\hbar ^1$ .",
"In particular, $\\hat{T} = d^{-1} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})\\end{array}} e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)} \\hat{R}( x_p, x_q ),$ where $\\hat{R}$ should be substituted by its path integral.",
"Note that this does not imply efficient classical simulation of quantum computation but quite the opposite.",
"Indeed, for $n$ qudits, there are $d^{2n}$ terms in the sum above.",
"While every Weyl phase space point has only a single associated path when acted on by Clifford gates, this is no longer true in any calculation of evolution under the T-gate.",
"Eq.",
"REF expresses the T-gate as a sum over phase space operators (the reflections) evaluated on all the phase space points.",
"Thus, it can be interpreted as associating an exponentially large number of paths to every phase space point, instead of the single paths found for Clifford gates.",
"Therefore, any simulation of the T-gate naively necessitates adding up an exponential large sum over paths and so is comparably inefficient.",
"Conclusion The treatment presented here formalizes the relationship between stabilizer states in the discrete case and Gaussians in the continuous case, which has often been pointed out [3].",
"Namely, only Gaussians that lie along Weyl phase space points directly correspond to Gaussians in the continuous world in terms of preserving their form under a harmonic Hamiltonian, an evolution that is fully describable by truncating the path integral at order $\\hbar ^0$ .",
"Furthermore, we showed that the Clifford group gates, generated by the Hadamard, phase shift and controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order.",
"We found that this was because their Hamiltonians are harmonic and take Weyl phase space points to themselves.",
"This proves the Gottesman-Knill theorem.",
"The T-gate, needed to complete a universal set with the Hadamard, was shown not to satisfy these properties, and so requires a path integral treatment that is complete up to $\\hbar ^1$ .",
"The latter treatment includes a sum of terms for which the number of terms scales exponentially with the number of qudits.",
"We note that our observations pertaining to classical propagation in continuous systems have long been very well known.",
"In the continuous case, the Wigner function of a quantum state is non-negative if and only if the state is a Gaussian [34] and it has also long been known that quantum propagation from one Gaussian state to another only requires propagation up to order $\\hbar ^0$ [11].",
"Indeed, it has been shown that this is a continuous version of “stabilizer state propagation” in finite systems [35], and is therefore, in principle, useful for quantum error correction and cluster state quantum computation [36], [37].",
"It is also well known in the discrete case that quadratic Hamiltonians can act classically and be represented by symplectic transformations in the study of quantum cat maps [38], [14], [25] and linear transformations between propagated Wigner functions [39].",
"Interestingly though, this latter work appears to have predated the discovery that stabilizer states have positive-definite Wigner functions [3] and therefore, as far as we know, has not been directly related to stabilizer states and the $\\hbar ^0$ limit of their path integral formulation, which is a relatively recent topic of particular interest to the quantum information community and those familiar with Gottesman-Knill.",
"Otherwise, this claim has been pointed out in terms of concepts related to positivity and related concepts in past work [40], [2].",
"We also note that our exploration of continous systems is not meant to explore the highly related topic of continuous-variable quantum information.",
"Many topics therein apply to our discussion here, such as the continuous stabilizer state propagation we mentioned above.",
"However, our intention in introducing the continuous infinite-dimensional case was not to address these topics but to instead relate the established continuous semiclassical formalism to the discrete case, and thereby bridge the notions of phase space and dynamics between the two worlds.",
"There is an interesting observation to be made of the weights of the reflections that make up the complete path integral formulation of a unitary operator.",
"Namely, as is clear in Eqs.",
"REF and REF , the coefficients consist of the exponentiated center generating function multiplied by $\\frac{i}{\\hbar }$ .",
"This is very similar to the form of the vVMG path integral in Eqs.",
"REF and REF .",
"However, in Eqs.",
"REF and REF , reflections serve as the prefactors measuring the reflection spectral overlap of a propagated state with its evolute and the center generating actions provide the quantal phase.",
"Thus, this formulation can be interpreted as an alternative path integral formulation of the vVMG, one consisting of reflections as the underlying classical trajectory only, instead of the more tailored trajectories that result from applying the method of steepest descents directly on an operator.",
"The fact that any unitary operator in the discrete case can be expressed as a sum consisting of a finite number of order $\\hbar ^1$ path integral contributions, has the added interesting implication that uniformization—higher order $\\hbar $ corrections to the “primitive” semiclassical forms such as Eq.",
"REF —isn't really necessary in discrete systems.",
"Uniformization is characterized by the proper treatment of coalescing saddle points and has long been a subject of interest in continuous systems where “anharmonicity” bedevils computationally efficient implementation.",
"It seems that this problem isn't an issue in the discrete case since a fully complete sum with a finite number of terms, naively numbering $d^2$ for one qudit, exists.",
"As a last point, there is perhaps an alternative way to interpret the results presented here, one in terms of “resources”.",
"Much like “magic” (or contextuality) and quantum discord can be framed as a resource necessary to perform quantum operations that have more power than classical ones, it is possible to frame the order in $\\hbar $ that is necessary in the underlying path integral describing an operation as a resource necessary for quantumness.",
"In this vein, it can be said that Clifford gate operations on stabilizer states are operations that only require $\\hbar ^0$ resources while supplemental gates that push the operator space into universal quantum computing require $\\hbar ^1$ resources.",
"The dividing line between these two regimes, the classical and quantum world, is discrete, unambiguous and well-defined.",
"Acknowledgments The authors thank Prof. Alfredo Ozorio de Almeida for very fruitful discussions about the center-chord representation in discrete systems and Byron Drury for his help proof-reading and bringing [12] to our attention.",
"This work was supported by AFOSR award no.",
"FA9550-12-1-0046.",
"Appendix Gross proved that for odd prime $d$ [41]: Lemma 2 Let $\\Psi $ be a state vector with positive Wigner function for odd prime $d$ .",
"If $\\Psi $ is supported on two points, then it has maximal support.",
"With this lemma in mind, we can offer an alternative proof of Corollary REF : Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .",
"Proof We will show that for odd prime $d$ , every degree of freedom can only be fully supported or a Kronecker delta, for all other degrees of freedom fixed; WLOG we will consider a two-dimensional stabilizer state $\\Psi (q_1,q_2)$ and show that if $\\exists \\, q^{\\prime }_1$ such that $\\Psi (q^{\\prime }_1, q_2) \\ne 0$ $\\forall q_2$ then $\\Psi (q_1,q_2)\\ne 0$ $\\forall \\, q_1, q_2$ and vice-versa (if $\\exists \\, q^{\\prime }_1$ s.t.",
"$\\Psi (q^{\\prime }_1, q_2)$ is a delta function then $\\Psi (q_1,q_2)$ is a delta function in $q_2$ $\\forall q_1$ ).",
"Therefore, if a degree of freedom is maximally supported in one degree of freedom for all others fixed, then it is maximally supported for all values of the other degrees of freedom.",
"On the other hand, if it is a delta function in one degree of freedom for all others fixed, then it is a delta function for all values of the other degrees of freedom.",
"Assume that for $q_1, q_2 \\in \\lbrace 0, \\ldots , d-1\\rbrace $ , $\\exists \\, q^{\\prime }_1, q^{\\prime \\prime }_1$ such that $\\Psi (q^{\\prime }_1,q_2) = 0$ for some $q_2$ and $\\Psi (q^{\\prime \\prime }_1, q_2) \\ne 0$ $\\forall q_2$ .",
"We proceed to prove by contradiction.",
"Hence $\\Psi (q^{\\prime \\prime }_1, q_2) \\equiv \\Psi _{q^{\\prime \\prime }_1} \\propto \\left[ \\frac{2 \\pi i}{d} \\left( \\theta _{q^{\\prime \\prime }_1} q^2_2 + \\eta _{q^{\\prime \\prime }_1} q_2 \\right) \\right]$ by Corollary REF and $\\Psi (q^{\\prime }_1,q_2) \\equiv \\Psi _{q^{\\prime }_1}(q_2) \\propto \\delta _{q_2, q(q^{\\prime }_1)}$ for some $q(q^{\\prime }_1)\\in \\mathbb {Z}/d\\mathbb {Z}$ by [3].",
"We can rotate in $p_2$ -$q_2$ space to form a new basis $q^*_2$ in $d$ discrete angles (since $\\theta _{q^{\\prime \\prime }_1} \\in \\mathbb {Z}/d\\mathbb {Z}$ ) such that $\\Psi _{q^{\\prime }_1}(q^*_2) \\ne 0$ (since a delta function is not maximally supported only at the one angle perpendicular to it).",
"Since there exists $(d-1)$ other values of $q_1$ other than $q^{\\prime }_1$ , it follows that there exists at least one such angle such that $\\Psi _{q_1}(q^*_2) \\ne 0$ $\\forall q_1, q^*_2 \\in \\lbrace 0,\\ldots ,d-1\\rbrace $ .",
"We define $q^*_2$ as the basis that is rotated by this angle with respect to $q_2$ .",
"By Corollary REF , this means that $\\Psi ^*(q_1,q^*_2) \\propto \\exp ( \\theta ^{\\prime }_{11} q^2_1 + \\theta ^{\\prime }_{22} {q^*_2}^2 + 2 \\theta ^{\\prime }_{12} q_1 q^*_2 + \\eta ^{\\prime }_1 q_1 + \\eta ^{\\prime }_2 q_2),$ where by $\\Psi ^*$ we mean $\\Psi $ expressed in the new basis $q^*_2$ in its second degree of freedom.",
"Hence, $\\Psi ^*_{q_1}(q^*_2) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( \\theta ^{\\prime }_{q_1} {q^*_2}^2 + \\eta ^{\\prime }_{q_1} q^*_2 \\right) \\right] \\exp \\left[ \\frac{2 \\pi i}{d} \\theta _{12} q_1 q^*_2 \\right].$ Acting on this last equation to rotate back to $q_2$ , we must produce $\\Psi _{q^{\\prime \\prime }_1}(q_2) \\propto \\delta _{q_2,q(q^{\\prime \\prime }_1)}$ .",
"But then Eq.",
"REF implies that $\\Psi _{q^{\\prime }_1}(q^*_2)$ must also be proportional to $\\delta _{q_2,q(q^{\\prime }_1)}$ .",
"This is a contradiction.",
"Therefore, if a degree of freedom is maximally supported for all others fixed, then it is maximally supported for all values of the other degrees of freedom and vice-versa.",
"In the latter case, a position state in the $i$ th degree of freedom can be represented as a Gaussian by using the $p$ -basis where it becomes a plane wave ($\\theta _i = 0$ ).",
"In other words, one can always choose $x_i$ to be $p_i$ or $q_i$ such that Eq.",
"REF holds for odd prime $d$ .",
"$\\Box $ Finally, the reason this result does not hold for odd non-prime $d$ is that for discrete Wigner space there are $d + 1$ unique angles minus all the prime factors of $d$ .",
"For non-prime $d$ , there is more than one such prime factor and so there are cases when one cannot “rotate” away all non-maximally supported states."
],
[
"Discrete Phase Space Representation of Universal Quantum Computing",
"A similar statement to the one we made in Section —that any operator can be expressed as an infinite sum of path integral contribution truncated at order $\\hbar ^1$ —can be made in discrete systems.",
"However, there is an important difference in the number of terms making up the sum.",
"To see this we can follow reasoning that is similar to that employed in the continuous case.",
"Namely, from Eq.",
"REF we see that any discrete operator can also be expressed as a linear combination of reflections, but unlike the continuous case, this sum has a finite number of terms.",
"Since reflections can be expressed fully by the discrete path integral truncated at order $\\hbar ^1$ , as discussed previously, it follows that any unitary operator in discrete systems can be expressed as a finite sum of contributions from path integrals truncated at order $\\hbar ^1$ .",
"Again, the same statement can be made by considering the chord representation in terms of translations.",
"Hence, quantum propagation in discrete systems can be fully treated by a finite sum of contributions from a path integral approach truncated at order $\\hbar ^1$ .",
"To gather some understanding of this statement, we can consider what is necessary to add to our path integral formulation when we complete the Clifford gates with the T-gate, which produces a universal gate set.",
"The T-gate is generalized to odd $d$ -dimensions by $\\hat{T} = \\sum _{j \\in \\mathbb {Z}/d\\mathbb {Z}} \\omega ^{\\frac{(j-1)j}{4}} \\left|j\\right\\rangle \\left\\langle j\\right|.$ This gate can no longer be characterized by an $\\mathcal {M}$ with integer entries.",
"In particular, $\\mathcal {M}_{\\hat{T}} = \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & 1 \\end{array} \\right),$ and $\\alpha _{\\hat{T}} = \\left( -\\frac{1}{4}, 0 \\right)$ .",
"This corresponds to $\\mathcal {B}_{\\hat{T}} = \\left( \\begin{array}{cc} 0 & 0\\\\ 0 & -\\frac{1}{4}\\end{array} \\right).$ Thus, the center function $T_x(x_p,x_q) = e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)},$ corresponding to the phase shift Hamiltonian applied for only half the unit of time.",
"The operator can thus be written: $\\hat{T} = d^{-2} \\sum _{\\begin{array}{c}x_p,x_q, \\\\ \\xi _p,\\xi _q \\in \\\\ \\mathbb {Z}/d\\mathbb {Z}\\end{array}} e^{ -\\frac{2 \\pi i}{d} \\left[ \\frac{1}{4} (x_q - x_q^2) - d \\left(x_p \\xi _q - x_q \\xi _p \\right) \\right] } \\hat{Z}^{\\xi _p} \\hat{X}^{\\xi _q}.$ Though this operator is quadratic, it no longer takes the Weyl center points to themselves.",
"This means that the $\\hbar ^0$ limit of Eq.",
"REF is now insufficient to capture all the dynamics because the overlap with any $\\left|q\\right\\rangle $ will now involve a linear superposition of partially overlapping propagated manifolds.",
"It must therefore be described by a path integral formulation that is complete to order $\\hbar ^1$ .",
"In particular, $\\hat{T} = d^{-1} \\sum _{\\begin{array}{c}x_p, x_q \\in \\\\ (\\mathbb {Z} / d \\mathbb {Z})\\end{array}} e^{-\\frac{2 \\pi i}{d} \\frac{1}{4} (x_q-x_q^2)} \\hat{R}( x_p, x_q ),$ where $\\hat{R}$ should be substituted by its path integral.",
"Note that this does not imply efficient classical simulation of quantum computation but quite the opposite.",
"Indeed, for $n$ qudits, there are $d^{2n}$ terms in the sum above.",
"While every Weyl phase space point has only a single associated path when acted on by Clifford gates, this is no longer true in any calculation of evolution under the T-gate.",
"Eq.",
"REF expresses the T-gate as a sum over phase space operators (the reflections) evaluated on all the phase space points.",
"Thus, it can be interpreted as associating an exponentially large number of paths to every phase space point, instead of the single paths found for Clifford gates.",
"Therefore, any simulation of the T-gate naively necessitates adding up an exponential large sum over paths and so is comparably inefficient."
],
[
"Conclusion",
"The treatment presented here formalizes the relationship between stabilizer states in the discrete case and Gaussians in the continuous case, which has often been pointed out [3].",
"Namely, only Gaussians that lie along Weyl phase space points directly correspond to Gaussians in the continuous world in terms of preserving their form under a harmonic Hamiltonian, an evolution that is fully describable by truncating the path integral at order $\\hbar ^0$ .",
"Furthermore, we showed that the Clifford group gates, generated by the Hadamard, phase shift and controlled-not gates, can be fully described by a truncation of their semiclassical propagator at lowest order.",
"We found that this was because their Hamiltonians are harmonic and take Weyl phase space points to themselves.",
"This proves the Gottesman-Knill theorem.",
"The T-gate, needed to complete a universal set with the Hadamard, was shown not to satisfy these properties, and so requires a path integral treatment that is complete up to $\\hbar ^1$ .",
"The latter treatment includes a sum of terms for which the number of terms scales exponentially with the number of qudits.",
"We note that our observations pertaining to classical propagation in continuous systems have long been very well known.",
"In the continuous case, the Wigner function of a quantum state is non-negative if and only if the state is a Gaussian [34] and it has also long been known that quantum propagation from one Gaussian state to another only requires propagation up to order $\\hbar ^0$ [11].",
"Indeed, it has been shown that this is a continuous version of “stabilizer state propagation” in finite systems [35], and is therefore, in principle, useful for quantum error correction and cluster state quantum computation [36], [37].",
"It is also well known in the discrete case that quadratic Hamiltonians can act classically and be represented by symplectic transformations in the study of quantum cat maps [38], [14], [25] and linear transformations between propagated Wigner functions [39].",
"Interestingly though, this latter work appears to have predated the discovery that stabilizer states have positive-definite Wigner functions [3] and therefore, as far as we know, has not been directly related to stabilizer states and the $\\hbar ^0$ limit of their path integral formulation, which is a relatively recent topic of particular interest to the quantum information community and those familiar with Gottesman-Knill.",
"Otherwise, this claim has been pointed out in terms of concepts related to positivity and related concepts in past work [40], [2].",
"We also note that our exploration of continous systems is not meant to explore the highly related topic of continuous-variable quantum information.",
"Many topics therein apply to our discussion here, such as the continuous stabilizer state propagation we mentioned above.",
"However, our intention in introducing the continuous infinite-dimensional case was not to address these topics but to instead relate the established continuous semiclassical formalism to the discrete case, and thereby bridge the notions of phase space and dynamics between the two worlds.",
"There is an interesting observation to be made of the weights of the reflections that make up the complete path integral formulation of a unitary operator.",
"Namely, as is clear in Eqs.",
"REF and REF , the coefficients consist of the exponentiated center generating function multiplied by $\\frac{i}{\\hbar }$ .",
"This is very similar to the form of the vVMG path integral in Eqs.",
"REF and REF .",
"However, in Eqs.",
"REF and REF , reflections serve as the prefactors measuring the reflection spectral overlap of a propagated state with its evolute and the center generating actions provide the quantal phase.",
"Thus, this formulation can be interpreted as an alternative path integral formulation of the vVMG, one consisting of reflections as the underlying classical trajectory only, instead of the more tailored trajectories that result from applying the method of steepest descents directly on an operator.",
"The fact that any unitary operator in the discrete case can be expressed as a sum consisting of a finite number of order $\\hbar ^1$ path integral contributions, has the added interesting implication that uniformization—higher order $\\hbar $ corrections to the “primitive” semiclassical forms such as Eq.",
"REF —isn't really necessary in discrete systems.",
"Uniformization is characterized by the proper treatment of coalescing saddle points and has long been a subject of interest in continuous systems where “anharmonicity” bedevils computationally efficient implementation.",
"It seems that this problem isn't an issue in the discrete case since a fully complete sum with a finite number of terms, naively numbering $d^2$ for one qudit, exists.",
"As a last point, there is perhaps an alternative way to interpret the results presented here, one in terms of “resources”.",
"Much like “magic” (or contextuality) and quantum discord can be framed as a resource necessary to perform quantum operations that have more power than classical ones, it is possible to frame the order in $\\hbar $ that is necessary in the underlying path integral describing an operation as a resource necessary for quantumness.",
"In this vein, it can be said that Clifford gate operations on stabilizer states are operations that only require $\\hbar ^0$ resources while supplemental gates that push the operator space into universal quantum computing require $\\hbar ^1$ resources.",
"The dividing line between these two regimes, the classical and quantum world, is discrete, unambiguous and well-defined."
],
[
"Acknowledgments",
"The authors thank Prof. Alfredo Ozorio de Almeida for very fruitful discussions about the center-chord representation in discrete systems and Byron Drury for his help proof-reading and bringing [12] to our attention.",
"This work was supported by AFOSR award no.",
"FA9550-12-1-0046."
],
[
"Appendix",
"Gross proved that for odd prime $d$ [41]: Lemma 2 Let $\\Psi $ be a state vector with positive Wigner function for odd prime $d$ .",
"If $\\Psi $ is supported on two points, then it has maximal support.",
"With this lemma in mind, we can offer an alternative proof of Corollary REF : Corollary 3 For odd prime $d$ , if $\\Psi $ is a stabilizer state then there always exists a mixed representation in position and momentum such that: $\\Psi _{\\theta _{\\beta x},\\eta _{\\beta x}}(x) = \\frac{1}{\\sqrt{d}} \\exp \\left[\\frac{2 \\pi i}{d} \\left( x^T \\theta _{\\beta x} x + \\eta _{\\beta x} \\cdot x \\right) \\right],$ where $x_i$ can be either $p_i$ or $q_i$ .",
"Proof We will show that for odd prime $d$ , every degree of freedom can only be fully supported or a Kronecker delta, for all other degrees of freedom fixed; WLOG we will consider a two-dimensional stabilizer state $\\Psi (q_1,q_2)$ and show that if $\\exists \\, q^{\\prime }_1$ such that $\\Psi (q^{\\prime }_1, q_2) \\ne 0$ $\\forall q_2$ then $\\Psi (q_1,q_2)\\ne 0$ $\\forall \\, q_1, q_2$ and vice-versa (if $\\exists \\, q^{\\prime }_1$ s.t.",
"$\\Psi (q^{\\prime }_1, q_2)$ is a delta function then $\\Psi (q_1,q_2)$ is a delta function in $q_2$ $\\forall q_1$ ).",
"Therefore, if a degree of freedom is maximally supported in one degree of freedom for all others fixed, then it is maximally supported for all values of the other degrees of freedom.",
"On the other hand, if it is a delta function in one degree of freedom for all others fixed, then it is a delta function for all values of the other degrees of freedom.",
"Assume that for $q_1, q_2 \\in \\lbrace 0, \\ldots , d-1\\rbrace $ , $\\exists \\, q^{\\prime }_1, q^{\\prime \\prime }_1$ such that $\\Psi (q^{\\prime }_1,q_2) = 0$ for some $q_2$ and $\\Psi (q^{\\prime \\prime }_1, q_2) \\ne 0$ $\\forall q_2$ .",
"We proceed to prove by contradiction.",
"Hence $\\Psi (q^{\\prime \\prime }_1, q_2) \\equiv \\Psi _{q^{\\prime \\prime }_1} \\propto \\left[ \\frac{2 \\pi i}{d} \\left( \\theta _{q^{\\prime \\prime }_1} q^2_2 + \\eta _{q^{\\prime \\prime }_1} q_2 \\right) \\right]$ by Corollary REF and $\\Psi (q^{\\prime }_1,q_2) \\equiv \\Psi _{q^{\\prime }_1}(q_2) \\propto \\delta _{q_2, q(q^{\\prime }_1)}$ for some $q(q^{\\prime }_1)\\in \\mathbb {Z}/d\\mathbb {Z}$ by [3].",
"We can rotate in $p_2$ -$q_2$ space to form a new basis $q^*_2$ in $d$ discrete angles (since $\\theta _{q^{\\prime \\prime }_1} \\in \\mathbb {Z}/d\\mathbb {Z}$ ) such that $\\Psi _{q^{\\prime }_1}(q^*_2) \\ne 0$ (since a delta function is not maximally supported only at the one angle perpendicular to it).",
"Since there exists $(d-1)$ other values of $q_1$ other than $q^{\\prime }_1$ , it follows that there exists at least one such angle such that $\\Psi _{q_1}(q^*_2) \\ne 0$ $\\forall q_1, q^*_2 \\in \\lbrace 0,\\ldots ,d-1\\rbrace $ .",
"We define $q^*_2$ as the basis that is rotated by this angle with respect to $q_2$ .",
"By Corollary REF , this means that $\\Psi ^*(q_1,q^*_2) \\propto \\exp ( \\theta ^{\\prime }_{11} q^2_1 + \\theta ^{\\prime }_{22} {q^*_2}^2 + 2 \\theta ^{\\prime }_{12} q_1 q^*_2 + \\eta ^{\\prime }_1 q_1 + \\eta ^{\\prime }_2 q_2),$ where by $\\Psi ^*$ we mean $\\Psi $ expressed in the new basis $q^*_2$ in its second degree of freedom.",
"Hence, $\\Psi ^*_{q_1}(q^*_2) \\propto \\exp \\left[\\frac{2 \\pi i}{d} \\left( \\theta ^{\\prime }_{q_1} {q^*_2}^2 + \\eta ^{\\prime }_{q_1} q^*_2 \\right) \\right] \\exp \\left[ \\frac{2 \\pi i}{d} \\theta _{12} q_1 q^*_2 \\right].$ Acting on this last equation to rotate back to $q_2$ , we must produce $\\Psi _{q^{\\prime \\prime }_1}(q_2) \\propto \\delta _{q_2,q(q^{\\prime \\prime }_1)}$ .",
"But then Eq.",
"REF implies that $\\Psi _{q^{\\prime }_1}(q^*_2)$ must also be proportional to $\\delta _{q_2,q(q^{\\prime }_1)}$ .",
"This is a contradiction.",
"Therefore, if a degree of freedom is maximally supported for all others fixed, then it is maximally supported for all values of the other degrees of freedom and vice-versa.",
"In the latter case, a position state in the $i$ th degree of freedom can be represented as a Gaussian by using the $p$ -basis where it becomes a plane wave ($\\theta _i = 0$ ).",
"In other words, one can always choose $x_i$ to be $p_i$ or $q_i$ such that Eq.",
"REF holds for odd prime $d$ .",
"$\\Box $ Finally, the reason this result does not hold for odd non-prime $d$ is that for discrete Wigner space there are $d + 1$ unique angles minus all the prime factors of $d$ .",
"For non-prime $d$ , there is more than one such prime factor and so there are cases when one cannot “rotate” away all non-maximally supported states."
]
] | 1612.05649 |
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